[["KEK-TH-2094\n\n[**Unbroken $E_7\\times E_7$ nongeometric heterotic strings,\\\nstable degenerations and\\\nenhanced gauge groups in F-theory duals**]{}\n\n1.2cm\n\nYusuke Kimura$^1$ 0.4cm [*$^1$KEK Theory Center, Institute of Particle and Nuclear Studies, KEK,\\\n1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan*]{} 0.4cm E-mail: kimurayu@post.kek.jp\n\n1.5cm\n\nIntroduction\n============\n\nF-theory/heterotic duality [@Vaf; @MV1; @MV2; @Sen; @FMW] states that F-theory [@Vaf; @MV1; @MV2] compactification on an elliptic K3 fibered Calabi\u2013Yau $(n+1)$-fold describes a theory physically equivalent to heterotic compactification[^1] on an elliptic Calabi\u2013Yau $n$-fold. Non-perturbative aspects of heterotic theory can be studied by utilizing this duality. F-theory/heterotic duality is strictly formulated when the stable degeneration limit[^2] [@FMW; @AM] is taken on the F-theory side in which K3 fibers split into pairs of half K3 surfaces.\n\nRecently, eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_8\\mathfrak{e}_7$ algebra were constructed by Malmendier and Morrison [@MM] by utilizing the F-theory/heterotic duality. The Narain space [@Narain] $$D_{2,18} / O(\\Lambda^{2,18})$$ gives the moduli space of eight-dimensional heterotic strings, and the double cover of this space, $$D_{2,18} / O^+ (\\Lambda^{2,18}),$$ is equivalent to the moduli space of F-theory on elliptic K3 surfaces with a section. This is the statement of F-theory/heterotic duality. Malmendier and Morrison considered F-theory compactifications on elliptic K3 surfaces with $H\\oplus E_8\\oplus E_7$ lattice polarization, namely elliptically fibered K3 surfaces with a type $II^*$ fiber and a type $III^*$ fiber with a global section, and they constructed the moduli of heterotic strings with unbroken $\\mathfrak{e}_8\\mathfrak{e}_7$ algebra as the heterotic duals of them on the 2-torus. The moduli space of the non-geometric heterotic strings with unbroken $\\mathfrak{e}_8\\mathfrak{e}_7$ algebra constructed in [@MM] is given by $$D_{2,3} / O^+(L^{2,3}).$$ Here $L^{2,3}$ denotes the orthogonal complement of $H\\oplus E_8\\oplus E_7$ inside the K3 lattice $\\Lambda_{K3}$, and the non-geometric heterotic strings constructed in [@MM] possess $O^+(L^{2,3})$-symmetry. Here $O^+(L^{2,3})$[^3] mixes the complex structure moduli, the K\u00e4hler moduli and the moduli of Wilson line values. Therefore, the resulting heterotic strings do not have a geometric interpretation[^4]; for this reason, the resulting heterotic strings are called [*non-geometric*]{} heterotic strings. A single Wilson line expectation value is non-zero for the non-geometric heterotic strings with unbroken $\\mathfrak{e}_8\\mathfrak{e}_7$ gauge algebra as constructed in [@MM]. The mathematical results of Kumar [@Kumar] and Clingher and Doran [@ClingherDoran2011; @ClingherDoran2012], which gave the Weierstrass equations of elliptic K3 surfaces with a global section with $E_8 E_7$ singularity, the coefficients of which are expressed as Siegel modular forms of even weight, were used in their construction.\n\nClingher, Malmendier, and Shaska [@CMSE7] extended the construction of non-geometric heterotic strings by Malmendier and Morrison to non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra. F-theory compactifications on elliptic K3 surfaces with $H\\oplus E_7\\oplus E_7$ lattice polarization, namely K3 surfaces with a global section with two type $III^*$ fibers, were considered, and eight-dimensional non-geometric heterotic strings on $T^2$ were obtained as the heterotic duals in their construction. The moduli of the resulting heterotic strings is parametrized by the space $$D_{2,4} / O^+(L^{2,4}).$$ Here $D_{2,4}$ is the symmetric space of $O(2,4)$, namely, $D_{2,4}$ is defined as $O(2)\\times O(4)\\backslash O(2,4)$. The symmetric space $D_{2,4}$ is also referred to as the [*bounded symmetric domain of type $IV$*]{}. Here $L^{2,4}$ denotes the orthogonal complement of $H\\oplus E_7\\oplus E_7$ in the K3 lattice $\\Lambda_{K3}$. The complex structure moduli, K\u00e4hler moduli, and the moduli of Wilson line expectation values are mixed under the symmetry $O^+(L^{2,4})$, thus the heterotic strings constructed in [@CMSE7] also do not have a geometric interpretation. Two Wilson line expectation values are non-trivial in non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra. See also [@GJ1412; @LMV1508; @FGLMM1603; @MS1609; @GLMM1611; @FM1708; @FGLMM1712; @Kimura1810; @Plauschinn2018] for recent progress on non-geometric heterotic strings.\n\nIn this note, we analyze theories that correspond to the points in the moduli of eight-dimensional non-geometric heterotic strings on the 2-torus $T^2$ constructed in [@CMSE7], at which the ranks of the non-Abelian gauge groups are enhanced to 18 on the F-theory side. These are the maximal enhancements of the non-Abelian gauge groups on the F-theory side. We mainly consider $E_8\\times E_8$ heterotic strings, rather than $SO(32)$ heterotic strings. (However, we do consider some applications to $SO(32)$ heterotic strings.) As only up to an $E_8\\times E_8 \\times U(1)^4$ gauge group can arise in eight-dimensional $E_8 \\times E_8$ heterotic strings compactified on the 2-torus [@Kimura1810], a consideration of the ranks of the non-Abelian gauge groups reveals that the gauge groups of the dual heterotic theories of these F-theory models do not allow for a perturbative interpretation. These heterotic strings include the non-perturbative effects of 5-branes. We find that these theories can be described as the deformations of the heterotic strings from the stable degeneration limit, in which the F-theory/heterotic duality strictly holds, and that these deformations result from the coincident 7-branes on the F-theory side. In the heterotic language, the effect of coincident 7-branes corresponds to the presence of 5-branes.\n\nWhen the non-Abelian gauge groups on F-theory on an elliptic K3 surface are enhanced to rank 18, K3 surfaces become [*extremal*]{} K3 surfaces. A K3 surface is called attractive, when it has the Picard number $\\rho=20$, which is the highest value for a complex K3 surface. When an attractive K3 surface has an elliptic fibration with a section with the singularity type of rank 18, the elliptic fibration is referred to as extremal. Owing to the classification result in [@SZ], the complex structures of extremal K3 surfaces on which non-Abelian gauge groups on F-theory compactifications are enhanced to rank 18 in the moduli can be determined, and this enables us to deduce the Weierstrass equations of extremal K3 surfaces. By analyzing the deduced Weierstrass equations, we study F-theory compactifications and the non-geometric heterotic duals at these special points in the moduli.\n\nWe also discuss applications to $SO(32)$ heterotic strings in this study. We deduce the Weierstrass equations of elliptic K3 surfaces appearing as the compactification spaces of the F-theory duals of some $SO(32)$ heterotic strings, which are obtained as the transformations of $\\mathfrak{e}_7\\mathfrak{e}_7$ non-geometric heterotic strings.\n\nIn addition, we consider fibering elliptic K3 surfaces that belong to the F-theory side of the moduli of eight-dimensional $\\mathfrak{e}_7\\mathfrak{e}_7$ non-geometric heterotic strings, over $\\P^1$, to build elliptically fibered Calabi\u2013Yau 3-folds with a global section. We study F-theory compactifications on the resulting elliptic Calabi\u2013Yau 3-folds[^5]. We find that highly enhanced gauge groups arise in these compactifications. It is mainly local F-theory model buildings that have been discussed in recent studies [@DWmodel; @BHV; @BHV2; @DWGUT]. However, the global aspects of the geometry need to be considered to discuss the issues of gravity. We investigate F-theory on elliptically fibered Calabi\u2013Yau 3-folds from the global perspective in this study.\n\nA similar organization can be found in [@Kimura1810].\n\nThis note is structured as follows. In Section \\[sec2\\], we briefly review F-theory compactifications, and we also review attractive K3 surfaces and extremal K3 surfaces that are technically necessary to analyze special points in the moduli of eight-dimensional non-geometric heterotic strings and F-theory duals. We also review the construction of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra in [@CMSE7].\n\nIn Section \\[sec3\\], we discuss the special points in the eight-dimensional non-geometric heterotic moduli with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ at which the ranks of the non-Abelian gauge symmetries on the F-theory side are enhanced to 18. The gauge groups in the heterotic strings which correspond to these points do not allow for the perturbative interpretations. We demonstrate that these theories can be seen as deformations of the stable degenerations as a result of the coincident 7-branes on the F-theory side. We also discuss applications to $SO(32)$ heterotic strings. We derive the Weierstrass equations of K3 elliptic fibrations appearing as the compactification spaces of the F-theory duals of some $SO(32)$ heterotic strings. We determine the gauge groups that arise on F-theory compactifications, including the global structures of the gauge groups.\n\nWe build elliptically fibered Calabi\u2013Yau 3-folds in Section \\[sec4\\] by fibering examples of elliptic K3 surfaces, which belong to the F-theory side of the moduli of eight-dimensional unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ non-geometric heterotic strings, over $\\P^1$. We analyze F-theory compactifications on the resulting elliptic Calabi\u2013Yau 3-folds. First, we consider the higher-dimensional analog of the construction of genus-one fibered K3 surfaces without a global section[^6] to build genus-one fibered Calabi\u2013Yau 3-folds without a section. This construction ensures that the resulting 3-folds in fact satisfy the Calabi\u2013Yau condition. Similar constructions of genus-one fibered Calabi\u2013Yau 4-folds without a section using double covers can be found in [@KCY4]. Taking the Jacobian fibration[^7] of the resulting genus-one fibered Calabi\u2013Yau 3-folds yields elliptically fibered Calabi\u2013Yau 3-folds with a global section. K3 fibers of these elliptic Calabi\u2013Yau 3-folds belong to the F-theory side of the moduli of eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra. Therefore, the obtained elliptic Calabi\u2013Yau 3-folds can be seen as the fibering of such K3 surfaces over the base curve $\\P^1$. We deduce the gauge groups on F-theory compactifications on the elliptic Calabi\u2013Yau 3-folds, and we find that some specific models do not have a $U(1)$ gauge field. We determine the Mordell\u2013Weil groups of some models, and we obtain the global structures of the gauge groups of these models. We also deduce candidate matter spectra on F-theory on the constructed elliptically fibered Calabi\u2013Yau 3-folds that satisfy the six-dimensional anomaly cancellation condition. We determine these candidate matter spectra directly from the global defining equations of the elliptically fibered Calabi\u2013Yau 3-folds. We state our concluding remarks in Section \\[sec5\\].\n\nReview of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra, F-theory, and extremal K3 surfaces {#sec2}\n==================================================================================================================================\n\nReview of F-theory compactifications {#ssec2.1}\n------------------------------------\n\nWe briefly review F-theory compactifications on elliptic K3 surfaces. A similar review can be found in [@Kimura1810]. F-theory is compactified on spaces that admit a genus-one fibration. The complex structure of the genus-one fiber is identified with the axio-dilaton in F-theory compactification. This formulation allows the axio-dilaton to have $SL(2,\\Z)$ monodromy. Genus-one fibrations do not necessarily admit a global section; there are situations in which they have a global section, and those in which they do not. F-theory compactifications on elliptic fibrations with a global section have been investigated in recent studies, for example, in [@MorrisonPark; @MPW; @BGK; @BMPWsection; @CKP; @BGK1306; @CGKP; @CKP1307; @CKPS; @AL; @EKY1410; @LSW; @CKPT; @CGKPS; @MP2; @BMW2017; @CL2017; @BMW1706; @EKY1712; @KimuraMizoguchi; @EK1802; @Kimura1802; @LRW2018; @EK1808; @MizTani2018; @CMPV1811; @TT2019]. Although Calabi\u2013Yau genus-one fibration lacking a global section cannot be expressed in the Weierstrass form, when the Jacobian fibration of it exists, the Jacobian fibration yields an elliptic fibration with a global section. Calabi\u2013Yau genus-one fibration $Y$ and the Jacobian fibration $J(Y)$ have the identical types of the singular fibers, and they have the same discriminant loci.\n\nGenus-one fibers degenerate over the codimension 1 locus in the base space, and this locus is referred to as the discriminant locus. Such degenerate fibers are called the singular fibers. When genus-one fiber degenerates, it becomes either $\\P^1$ with a single singularity, or a sum of smooth $\\P^1$\u2019s meeting in specific ways. The types of the singular fibers of genus-one fibered surfaces were classified by Kodaira [@Kod1; @Kod2]. Methods to determine the singular fibers of elliptic surfaces can be found in [@Ner; @Tate].\n\nIn F-theory compactifications on genus-one fibrations, the non-Abelian gauge groups that form on the 7-branes correspond to the singular fibers of genus-one fibrations [@MV2; @BIKMSV]. The correspondences of the singular fibers and the singularity types of the compactification spaces are shown in Table \\[tablemonodromy in 2.1\\] below. The corresponding monodromies and j-invariants of the singular fibers are also presented in the table.\n\n  --------------------------------------------------------------------------------------\n   Fiber type   J-invariant            Monodromy  Order of Monodromy   Singularity Type\n  ------------ ------------- ------------------- -------------------- ------------------\n    $I^*_0$       regular      $-\\begin{pmatrix}          2                 $D_4$\n                                        1 & 0 \\\\                      \n                                        0 & 1 \\\\                      \n                                  \\end{pmatrix}$                      \n\n     $I_m$       $\\infty$       $\\begin{pmatrix}       infinite           $A_{m-1}$\n                                        1 & m \\\\                      \n                                        0 & 1 \\\\                      \n                                  \\end{pmatrix}$                      \n\n    $I^*_m$      $\\infty$      $-\\begin{pmatrix}       infinite           $D_{m+4}$\n                                        1 & m \\\\                      \n                                        0 & 1 \\\\                      \n                                  \\end{pmatrix}$                      \n\n      $II$           0          $\\begin{pmatrix}          6                 none.\n                                        1 & 1 \\\\                      \n                                       -1 & 0 \\\\                      \n                                  \\end{pmatrix}$                      \n\n     $II^*$          0          $\\begin{pmatrix}          6                 $E_8$\n                                       0 & -1 \\\\                      \n                                        1 & 1 \\\\                      \n                                  \\end{pmatrix}$                      \n\n     $III$         1728         $\\begin{pmatrix}          4                 $A_1$\n                                        0 & 1 \\\\                      \n                                       -1 & 0 \\\\                      \n                                  \\end{pmatrix}$                      \n\n    $III^*$        1728         $\\begin{pmatrix}          4                 $E_7$\n                                       0 & -1 \\\\                      \n                                        1 & 0 \\\\                      \n                                  \\end{pmatrix}$                      \n\n      $IV$           0          $\\begin{pmatrix}          3                 $A_2$\n                                        0 & 1 \\\\                      \n                                      -1 & -1 \\\\                      \n                                  \\end{pmatrix}$                      \n\n     $IV^*$          0          $\\begin{pmatrix}          3                 $E_6$\n                                      -1 & -1 \\\\                      \n                                        1 & 0 \\\\                      \n                                  \\end{pmatrix}$                      \n  --------------------------------------------------------------------------------------\n\n  : \\[tablemonodromy in 2.1\\]Monodromies, j-invariants and the corresponding types of the singularities of singular fibers. \u201cRegular\u201d for j-invariant of $I_0^*$ fiber means that j-invariant can take any finite value in $\\C$ for $I_0^*$ fiber.\n\nThe types of singular fibers of elliptic surfaces can be determined from the vanishing orders of the coefficients of the Weierstrass equations. The correspondences of the fiber types and the vanishing orders of the Weierstrass coefficients are shown in Table \\[tableWeierstrasscoefficients in 2.1\\].\n\n       Fiber Type        Ord($f$)   Ord($g$)   Ord($\\Delta$)\n  --------------------- ---------- ---------- ---------------\n         $I_0 $          $\\ge 0$    $\\ge 0$          0\n    $I_n $ ($n\\ge 1$)       0          0            $n$\n          $II $          $\\ge 1$       1             2\n         $III $             1       $\\ge 2$          3\n          $IV $          $\\ge 2$       2             4\n         $I_0^*$         $\\ge 2$       3             6\n                            2       $\\ge 3$   \n   $I^*_m$ ($m \\ge 1$)      2          3           $m+6$\n         $IV^*$          $\\ge 3$       4             8\n         $III^*$            3       $\\ge 5$          9\n         $II^*$          $\\ge 4$       5            10\n\n  : \\[tableWeierstrasscoefficients in 2.1\\]List of the types of the singular fibers, and the corresponding vanishing orders of the coefficients, $f,g$, of the Weierstrass equation $y^2=x^3+f\\, x+g$, and the orders of the discriminant, $\\Delta$.\n\nWhen an elliptic fibration has a global section, the set of sections form a group, known as the Mordell\u2013Weil group. The rank of the Mordell\u2013Weil group gives the number of the $U(1)$ gauge fields in F-theory compactification on the elliptic fibration [@MV2].\n\nThe second integral cohomology group $H^2(S, \\Z)$ of K3 surface $S$ includes the information of the geometry of the K3 surface. This group has the lattice structure, and it is called the K3 lattice, $\\Lambda_{K3}$. The K3 lattice is unimodular, even lattice of signature (3,19), and it is isometric to the direct sum of two $E_8$\u2019s and three hyperbolic planes [@Mil] $$\\Lambda_{K3} \\cong E_8^2 \\oplus H^3.$$ The group of divisors (modulo algebraic equivalence) constitutes a sublattice inside the K3 lattice, called the N\u00e9ron-Severi lattice $NS(S)$. When a K3 surface has an elliptic fibration with a global section, an elliptic fiber and a global section generate the hyperbolic plane $H$ inside the N\u00e9ron-Severi lattice. K3 surface $S$ admitting an elliptic fibration with a section is equivalent to the condition that the N\u00e9ron-Severi lattice $NS(S)$ contains the hyperbolic plane $H$ [@Kondoauto]. When an elliptic K3 surface has the singular fibers, the N\u00e9ron-Severi lattice $NS(S)$ contains the $ADE$ lattices that correspond to the types of the singular fibers. For example, that a K3 surface $S$ is $H\\oplus E_7\\oplus E_7$-lattice polarized means that the N\u00e9ron-Severi lattice $NS(S)$ includes the lattice $ H\\oplus E_7\\oplus E_7$, and this is equivalent to the condition that the K3 surface $S$ is elliptically fibered with a section, the singular fibers of which include two type $III^*$ fibers (or worse). K3 surfaces with $H\\oplus E_7\\oplus E_7$ lattice polarization are parametrized by the bounded symmetric domain of type $IV$, $D_{2,4}$, modded out by the symmetry of the orthogonal complement of the lattice $ H\\oplus E_7\\oplus E_7$ inside the K3 lattice $\\Lambda_{K3}$: $$\\label{symmetry 2,4 in 2.1}\nD_{2,4} / O^+(L^{2,4}).$$ $L^{2,4}$ denotes the orthogonal complement of $ H\\oplus E_7\\oplus E_7$ inside the K3 lattice $\\Lambda_{K3}$. Because the K3 lattice $\\Lambda_{K3}$ is isometric to $E_8^2\\oplus H^3$, $L^{2,4}$ can also be defined as the orthogonal complement of $E_7\\oplus E_7$ in the lattice $E_8^2\\oplus H^2$.\n\nBy utilizing the F-theory/heterotic duality, eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$, the moduli space of which is equivalent to (\\[symmetry 2,4 in 2.1\\]) were constructed in [@CMSE7]. In this note, we study the points in the moduli of such non-geometric heterotic strings at which the non-Abelian gauge symmetries are enhanced to rank 18 on the F-theory side.\n\nConstruction of $\\mathfrak{e}_7\\mathfrak{e}_7$ non-geometric heterotic strings by Clingher, Malmendier, and Shaska {#ssec2.2}\n------------------------------------------------------------------------------------------------------------------\n\nWe briefly review the construction of eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ by Clingher, Malmendier and Shaska [@CMSE7].\n\nAs stated previously, the moduli of elliptic K3 surfaces with a global section with two $E_7$ singularities, namely the K3 surfaces with $H\\oplus E_7\\oplus E_7$ lattice polarization, are parameterized by the following space: $$D_{2,4} / O^+(L^{2,4}).$$ The bounded symmetric domain of type $IV$, $D_{2,4}$, is known to be isomorphic to ${\\bf H}_2$ [@Matsumoto]: $$\\label{H2 isom in 2.2}\n{\\bf H_2} \\cong D_{2,4}.$$ ${\\bf H_2}$ is defined as $${\\bf H_2}:=\\Big\\{ \\begin{pmatrix}\nz_1 & z_2 \\\\\nz_3 & z_4 \\\\\n\\end{pmatrix} \\in M_2(\\C) \\hspace{2mm} | \\hspace{2mm} 4 {\\rm Im} \\, z_1 {\\rm Im} \\, z_4 > |z_2-\\overline{z_3}|^2 \\hspace{2mm} {\\rm and} \\hspace{2mm} {\\rm Im} \\, z_4 > 0 \\Big\\}.$$ As mentioned in [@CMSE7], ${\\bf H_2}$ is a generalization of the Siegel upper-half space $\\H_2$ in the following sense: $$\\H_2 = \\Big\\{ \\omega \\in {\\bf H_2} \\hspace{2mm} | \\hspace{2mm} \\omega^t=\\omega \\Big\\}.$$ The modular group $\\Gamma$ acting on ${\\bf H_2}$ is defined as $$\\Gamma  = \\Big\\{ G\\in GL_4 (\\Z[i]) \\hspace{2mm} | \\hspace{2mm} G^\\dg \\,  \\begin{pmatrix}\n0 & {\\bf 1}_2 \\\\\n-{\\bf 1}_2 & 0 \\\\\n\\end{pmatrix} \\, G = \\begin{pmatrix}\n0 & {\\bf 1}_2 \\\\\n-{\\bf 1}_2 & 0 \\\\\n\\end{pmatrix} \\Big\\}.$$ ${\\bf 1}_2$ denotes the 2 $\\times$ 2 identity matrix. $\\begin{pmatrix}\nA & B \\\\\nC & D \\\\\n\\end{pmatrix}$ in the modular group $\\Gamma$ acts on $\\omega \\in {\\bf H}_2$ as $$\\begin{pmatrix}\nA & B \\\\\nC & D \\\\\n\\end{pmatrix}\\cdot \\omega = (A\\, \\omega+B) (C\\, \\omega+D)^{-1}.$$ There is an involution, ${\\cal T}$, that acts on ${\\bf H}_2$ as $${\\cal T}\\cdot \\omega =\\omega^t.$$ The group $\\Gamma_{\\cal T}$ is defined to be the semi-direct product of the modular group $\\Gamma$ and $<{\\cal T}>$: $$\\Gamma_{\\cal T} := \\Gamma \\, \\rtimes <{\\cal T}>.$$ There is an isomorphism $\\Gamma_{\\cal T} \\cong O^+(L^{2,4})$, and this induces the isomorphism (\\[H2 isom in 2.2\\]) [@Matsumoto].\n\nUnder the isomorphism $\\Gamma_{\\cal T} \\cong O^+(L^{2,4})$, the ring of $O^+(L^{2,4})$-modular forms corresponds to the ring of $\\Gamma_{\\cal T}$-modular forms of even characteristic [@Vinberg2010], generated by the five modular forms $J_k$ of weights $2k$, $k=2, \\cdots, 6$ [@CMSE7]. See [@CMSE7] for definitions of the modular forms $J_k$. In a special situation, the modular forms $J_2, J_3, J_5, J_6$ restrict to Igusa\u2019s generators [@Igusa], $\\psi_4, \\psi_6, \\chi_{10}, \\chi_{12}$ (and $J_4$ vanishes in this situation) [@CMSE7].\n\nThe periods of $H\\oplus E_7\\oplus E_7$ lattice polarized K3 surfaces $S$ in $H^2(S, \\Z)$ determine points in ${\\bf H}_2$. The Weierstrass coefficients of such elliptically fibered K3 surfaces were given in terms of $\\Gamma_{\\cal T}$-modular forms of even characteristic [@CMSE7].\n\nThe Weierstrass equation of a K3 surface with $H\\oplus E_7\\oplus E_7$ lattice polarization is given by [@CMSE7]: $$\\label{Weierstrass with param in 2.2}\ny^2 = x^3+ (e\\, t^4+c\\, t^3+a\\, t^2) x + t^7+g\\, t^6+(d\\, e+f)\\, t^5+c\\, d\\, t^4 + b\\, t^3.$$ Up to some scale factors, the coefficients are given in terms of the modular forms $J_2, J_3, J_4, J_5, J_6$ [@CMSE7]: $$\\begin{aligned}\n\\label{coeffs in 2.2}\nc= -J_5(\\omega), & d=-\\frac{1}{3}J_4(\\omega), & e = -3J_2(\\omega) \\\\ \\nonumber\nf= J_6(\\omega), & &  g=-2J_3(\\omega) \\\\ \\nonumber \na=-3 d^2=-\\frac{1}{3}J_4(\\omega)^2, & & b=-2d^3=\\frac{2}{27}J_4(\\omega)^3.\\end{aligned}$$ The elliptically fibered K3 surface determines a point in $D_{2,4}$, and this also determines a point in ${\\bf H}_2$ under the isomorphism (\\[H2 isom in 2.2\\]), which we denote by $\\omega$.\n\nNow, consider a manifold $M$ and a line bundle $\\Lambda$ on $M$, and choose sections $a,b,c,d,e,f, g$ of the line bundles $\\Lambda^{\\otimes 16}$, $\\Lambda^{\\otimes 24}$, $\\Lambda^{\\otimes 10}$, $\\Lambda^{\\otimes 8}$, $\\Lambda^{\\otimes 4}$, $\\Lambda^{\\otimes 12}$ and $\\Lambda^{\\otimes 6}$, respectively. When the sections $a,b,c,d,e,f,g$ are identified as (\\[coeffs in 2.2\\]), because $\\Gamma_{\\cal T}$ is isomorphic to $O^+(L^{2,4})$, the compactification on $M$ (which is the 2-torus when we consider 8D heterotic strings) gives a heterotic string theory with $O^+(L^{2,4})$-symmetry. The moduli space of eight-dimensional heterotic strings on the 2-torus $T^2$ decomposes into the product of the complex structure moduli, the Wilson line expectation values and K\u00e4hler moduli, in a suitable limit [@NSWheterotic]. The complex structure moduli, the Wilson line expectation values and K\u00e4hler moduli are mixed under the $O^+(L^{2,4})$-symmetry. This represents the construction of non-geometric heterotic strings with $\\mathfrak{e}_7\\mathfrak{e}_7$ gauge algebra in [@CMSE7].\n\nThe locus in the moduli in which the singularity ranks of elliptic K3 surfaces are enhanced satisfies 5-brane solutions on the heterotic side. The generic 5-brane solutions of non-geometric heterotic strings with $\\mathfrak{e}_7\\mathfrak{e}_7$ gauge algebra are discussed in [@CMSE7].\n\nElliptic K3 surfaces with the lattice polarization $H\\oplus E_7 \\oplus E_7$ were described in [@ClingherDoran2011] as the minimal resolution of the quartic hypersurfaces in $\\P^3$ given by the following equations: $$\\label{hypersurface in 2.2}\nY^2ZW-4X^3Z+3\\alpha XZW^2+\\beta ZW^3+\\gamma XZ^2W- \\frac{1}{2} (\\zeta W^4+\\delta Z^2W^2)+\\varepsilon XW^3=0,$$ where $[X:Y:Z:W]$ are homogeneous coordinates on $\\P^3$. $\\alpha, \\beta, \\gamma, \\delta, \\varepsilon, \\zeta$ are parameters, and $(\\gamma, \\delta)\\ne (0,0)$, and $(\\varepsilon, \\zeta)\\ne (0,0)$.\n\nMaking the following substitutions $$\\begin{aligned}\nX & = & tx \\\\ \\nonumber\nY & = & y \\\\ \\nonumber\nW & = & 4t^3 \\\\ \\nonumber \nZ & = & 4t^4\\end{aligned}$$ yields the Weierstrass equation with two type $III^*$ fibers as follows [@CMSE7] : $$\\label{first fibration in 2.2}\ny^2 = x^3 + 4t^3\\, (\\gamma t^2 -3\\alpha t+\\varepsilon)\\, x-8t^5\\, (\\delta t^2+2\\beta t+\\zeta).$$ Type $III^*$ fibers are at $t=0$ and at $t=\\infty$.\n\nK3 surface with the lattice polarization $H\\oplus E_7 \\oplus E_7$ given by (\\[hypersurface in 2.2\\]) always admits another fibration with a type $II^*$ fiber and a type $I^*_2$ fiber, as shown in [@CMSE7], and the Weierstrass equation of this fibration is: $$\\label{second fibration in 2.2}\n\\begin{split}\ny^2 = & x^3 - \\frac{1}{3} \\big[9\\alpha t^4+3(\\gamma\\zeta +\\delta\\varepsilon)\\, t^3+ (\\gamma\\varepsilon)^2\\, t^2 \\big] x\\\\\n & + \\frac{1}{27} \\big[27t^7 - 54\\beta t^6 + 27 (\\alpha\\gamma\\varepsilon+\\delta\\zeta)\\, t^5+9\\gamma\\varepsilon (\\delta\\varepsilon +\\gamma\\zeta)\\, t^4+2 (\\gamma\\varepsilon)^3\\, t^3 \\big].\n \\end{split}$$ The Weierstrass equation (\\[second fibration in 2.2\\]) was used in [@CMSE7] to construct eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$. (Compare the equation (\\[second fibration in 2.2\\]) with the equation (\\[Weierstrass with param in 2.2\\]).) Although the presence of two $E_7$ singularities in the Weierstrass equation is explicit in the equation (\\[first fibration in 2.2\\]), as stated in [@CMSE7], the Weierstrass equation (\\[first fibration in 2.2\\]) does not necessarily extend over the entire parameter space. For this reason, the Weierstrass equation (\\[second fibration in 2.2\\]) was instead used to construct non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ in [@CMSE7].\n\nThe K3 surface with the lattice polarization $H\\oplus E_7 \\oplus E_7$ (\\[hypersurface in 2.2\\]) also admits another elliptic fibration further, the singular fibers of which include a type $I_8^*$ fiber (or worse) [@CMSE7]. This alternate fibration relates to $SO(32)$ heterotic string. The Weierstrass equation of this fibration is obtained by making the following substitutions into the equation (\\[hypersurface in 2.2\\]) [@CMSE7]: $$\\begin{aligned}\nX & = & tx^3 \\\\ \\nonumber\nY & = & \\sqrt{2}x^2 y \\\\ \\nonumber\nW & = & 2x^3 \\\\ \\nonumber \nZ & = & 2x^2 (-\\varepsilon t+\\zeta).\\end{aligned}$$ The Weierstrass equation is [@CMSE7] : $$y^2 = x^3 + A x^2 + B x,$$ where $$\\begin{aligned}\nA & = & t^3-3\\alpha t-2\\beta \\\\ \\nonumber\nB & = & (\\gamma t-\\delta) (\\varepsilon t-\\zeta).\\end{aligned}$$ The discriminant is given by $$\\Delta = B^2 \\, (A^2 -4B).$$\n\nExtremal K3 surfaces {#ssec2.3}\n--------------------\n\nBy the Shioda\u2013Tate formula [@Shiodamodular; @Tate1; @Tate2], the following equality holds for an elliptic surface $S$ with a global section: $${\\rm rk} ADE + {\\rm rk} MW +2 = \\rho(S).$$ We have used rk $ADE$ to denote the rank of the singularity type of an elliptic surface $S$. The Picard number $\\rho(S)$ ranges from 2 to 20 for an elliptic K3 surface with a section. Thus, the rank of the singularity of an elliptic K3 surface $S$ with a section is bounded by: $${\\rm rk} ADE = \\rho(S)-2-{\\rm rk} MW \\le 18-{\\rm rk} MW.$$ Therefore, the rank of the singularity of an elliptic K3 surface $S$ with a section can be 18 at the highest, and this value is achieved precisely when the Picard number attains the highest value 20, and the Mordell\u2013Weil rank is 0. Physically, this means that the rank of the gauge group on F-theory compactification on an elliptic K3 surface is at most 18, and when the non-Abelian gauge group has the rank 18, it does not have a $U(1)$ gauge field.\n\nK3 surfaces with Picard number 20 are called [*attractive*]{} K3 surfaces[^8]. The complex structure moduli of the attractive K3 surfaces is known to be parametrized by three integers. The transcendental lattice $T(S)$ of a K3 surface $S$ is the orthogonal complement of the N\u00e9ron\u2013Severi lattice $NS(S)$ inside the K3 lattice $\\Lambda_{K3}$, and the transcendental lattices $T(S)$ of attractive K3 surfaces are positive-definite, even 2 $\\times$ 2 lattices. The complex structure of an attractive K3 surface is determined by the transcendental lattice [@PS-S; @SI]. The intersection form of the transcendental lattice of an attractive K3 surface can be transformed into the following form under the $GL_2(\\Z)$ action: $$\\label{general intersection form in 2.3}\n\\begin{pmatrix}\n2a & b \\\\\nb & 2c \\\\\n\\end{pmatrix}.$$ Here $a,b,c$ are integers, $a,b,c\\in\\Z$, and satisfy the relations: $$a \\ge c \\ge b  \\ge 0.$$ Thus, the triplet of integers, $a,b,c$, parameterizes the complex structure moduli of the attractive K3 surfaces. We denote an attractive K3 surface, whose transcendental lattice has the intersection form (\\[general intersection form in 2.3\\]) as $S_{[2a \\hspace{1mm} b \\hspace{1mm} 2c]}$ in this note.\n\nAn elliptic attractive K3 surfaces with a section is said to be [*extremal*]{} when it has Mordell\u2013Weil rank 0. This condition is equivalent to an elliptic K3 surface with a section having singularity rank 18. Thus, the non-Abelian gauge group forming in F-theory compactification on an elliptic K3 surface has rank 18 precisely when the K3 surface is extremal. In Section \\[sec3\\], we study the points in the moduli of eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra at which the non-Abelian gauge groups are enhanced to rank 18 on the F-theory side. Elliptic K3 surfaces on the F-theory side become extremal at these points.\n\nElliptically fibered K3 surfaces generally admit distinct elliptic fibrations[^9], and distinct elliptic fibrations have different singularity types and different Mordell\u2013Weil groups. Physically, this means that the gauge groups and $U(1)$ gauge fields that arise in F-theory compactification on an elliptic K3 surface with the fixed complex structure vary, because there still remains freedom to choose a fibration structure among the distinct choices of elliptic fibrations of that elliptic K3 surface[^10].\n\nThe attractive K3 surface whose transcendental lattice has the intersection form $$\\begin{pmatrix}\n2 & 0 \\\\\n0 & 2 \\\\\n\\end{pmatrix}$$ is particularly relevant to the contents of this study. The elliptic fibrations of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ were classified in [@Nish] and there are 13 types. We list these 13 types of elliptic fibration of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ in Appendix \\[secA\\].\n\nSpecial points in the moduli of eight-dimensional non-geometric heterotic strings and F-theory duals with enhanced gauge groups {#sec3}\n===============================================================================================================================\n\nSummary {#ssec3.1}\n-------\n\nThere are finitely many points in the moduli of eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra, at which the non-Abelian gauge groups on the F-theory side are enhanced to rank 18.\n\nIn Section \\[ssec3.2\\], we show that the heterotic strings at these special points in the moduli can be described as deformations of the stable degenerations, as a result of the coincident 7-branes on the F-theory side. This effect can be seen as the insertion of 5-branes in the heterotic language. We also discuss applications to $SO(32)$ heterotic strings.\n\nAs stated in Section \\[ssec2.3\\], K3 surfaces become extremal on the F-theory side at these points in the moduli. The complex structures of the extremal K3 surfaces were classified in [@SZ], and using this result, the complex structures of the extremal K3 surfaces at these points in the moduli can be determined. This enables us to determine the Weierstrass equations of the extremal K3 surfaces that appear as compactification spaces on the F-theory side in the moduli. By using this approach, we study the physics of the theories at the enhanced special points in the moduli.\n\nIn eight-dimensional $E_8\\times E_8$ heterotic strings on the 2-torus $T^2$, only the gauge groups up to $E_8\\times E_8\\times U(1)^4$ can arise in the perturbative description [@Kimura1810]. This implies that the heterotic dual of F-theory on an extremal elliptic K3 surface with the non-Abelian gauge group of rank 18 does not allow for the perturbative interpretation of the gauge group. This can reflect some non-perturbative aspects of the non-geometric heterotic strings.\n\nF-theory on extremal K3 surfaces and non-geometric heterotic duals in the moduli {#ssec3.2}\n--------------------------------------------------------------------------------\n\nWe discuss the points in the moduli of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra, at which the non-Abelian gauge symmetries on the F-theory side are enhanced to rank 18. K3 surfaces as compactification spaces on the F-theory side become extremal at these points. There are finitely many such points in the moduli, and the complex structures and the singularity types of the extremal K3 surfaces that appear in the moduli can be determined from Table 2 of [@SZ]. Among these, those of the singularity types, which include $E_8 E_7$, are studied in [@Kimura1810]. We do not discuss these extremal K3 surfaces in this note. Instead, we discuss the extremal K3 surfaces that belong to the moduli, the singularity types of which include $E_7^2$ [^11].\n\nThe singularity types of the extremal K3 surfaces in the moduli of K3 surfaces with $H\\oplus E_7 \\oplus E_7$ lattice polarization, which do not include $E_8$, are as follows [@SZ]: $$E_7^2 A_3 A_1, \\hspace{2mm} E_7^2D_4, E_7^2 A_4, \\hspace{2mm} E_7^2 A_2^2.$$ We study F-theory on the extremal K3 surfaces possessing the first two singularity types in this note.\n\nBecause the perturbative eight-dimensional heterotic strings on $T^2$ can have up to $E_8\\times E_8\\times U(1)^4$ gauge group, the heterotic duals of F-theory on these extremal K3 surfaces do not allow for the perturbative interpretation of these gauge groups [@Kimura1810]. As we demonstrate in Sections \\[sssec3.2.1\\] and \\[sssec3.2.2\\], these theories can be seen as deformations of the stable degenerations as a result of the coincident 7-branes on the F-theory side. These theories satisfy multiple 5-brane solutions on the heterotic side.\n\n### Extremal K3 surface with $E_7^2 A_3 A_1$ singularity {#sssec3.2.1}\n\nThe complex structure of the extremal K3 surface with $E_7^2 A_3 A_1$ singularity is uniquely determined, and its transcendental lattice has the following intersection form [@SZ]: $$\\begin{pmatrix}\n4 & 0 \\\\\n0 & 2 \\\\\n\\end{pmatrix}.$$ Therefore, the attractive K3 surface $S_{[4 \\hspace{1mm} 0 \\hspace{1mm} 2]}$[^12] admits an extremal fibration with the singularity type $E_7^2 A_3 A_1$, and F-theory on this extremal fibration has non-geometric heterotic dual with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$. The Weierstrass form of this extremal fibration can be found in [@KRES] as $$\\label{extremal fibration 402 in 3.2.1}\ny^2=x^3-\\frac{9}{16}(t^2+s^2+\\frac{10}{3}ts)\\, t^3s^3\\, x+\\frac{9}{4}t^5s^5\\, (\\frac{1}{4}t^2+\\frac{1}{4}s^2+\\frac{7}{18}ts),$$ the singular fibers of which consist of two type $III^*$ fibers, a type $I_4$ fiber, and a type $I_2$ fiber. The above Weierstrass equation was obtained in [@KRES] as the quadratic base change of an extremal rational elliptic surface. Geometrically, the quadratic base change of a rational elliptic surface is to glue a pair of identical rational elliptic surfaces. Extremal rational elliptic surfaces are the rational elliptic surfaces with a global section, the singularity types of which have rank 8. The types of singular fibers of the extremal rational elliptic surfaces were classified in [@MP]. The fiber types of the extremal rational elliptic surfaces are listed in Appendix \\[secB\\].\n\nThe complex structures of extremal rational elliptic surfaces are uniquely specified by the fiber types, except those with two fibers of type $I^*_0$ (see [@MP]). The complex structures of extremal rational elliptic surfaces with two fibers of type $I_0^*$ depend on the $j$-invariants of the fibers. The $j$-invariant $j$ of an extremal rational elliptic surface with two type $I_0^*$ fibers is constant over the base, and the fixed $j$ specifies the complex structure [@MP]. In this study, we denote, for example, the extremal rational elliptic surface with a type $III^*$ fiber and a type $III$ fiber as $X_{[III, \\hspace{1mm} III^*]}$. We simply use $n$ to denote a singular fiber of type $I_n$ and $m^*$ to represent a fiber of type $I^*_m$. The extremal rational elliptic surface with a type $III^*$ fiber, a type $I_2$ fiber and a type $I_1$ fiber is denote as $X_{[III^*, \\hspace{1mm} 2, 1]}$. Because the complex structure of an extremal rational elliptic surface with two type $I_0^*$ fibers depends on the $j$-invariant of the elliptic fibers, we use $X_{[0^*, \\hspace{1mm} 0^*]}(j)$ to denote this extremal rational elliptic surface.\n\nNow we demonstrate that F-theory on an extremal K3 surface (\\[extremal fibration 402 in 3.2.1\\]) can be seen as a deformation of stable degeneration, owing to an effect of coincident 7-branes. As deduced in [@KRES], the K3 extremal fibration (\\[extremal fibration 402 in 3.2.1\\]) is obtained as the quadratic base change of the extremal rational elliptic surface $X_{[III^*, \\hspace{1mm} 2, 1]}$ in which two type $I_2$ fibers and two type $I_1$ fibers collide. Whereas the quadratic base change of a rational elliptic surface generally yields an elliptic K3 surface, with twice as many singular fibers as the original rational elliptic surface, at the special limits at which singular fibers collide, the singularity type of the resulting K3 surface is enhanced. As discussed in [@KRES], two identical extremal rational elliptic surfaces $X_{[III^*, \\hspace{1mm} 2, 1]}$ are glued together to yield an elliptic K3 surface, which we denote as $S_1$, the singular fibers of which consist of two type $III^*$ fibers, two type $I_2$ fibers, and two type $I_1$ fibers. In the special limit at which 7-branes over which type $I_2$ fiber lies coincide with those over which type $I_2$ fiber lies, and 7-brane over which type $I_1$ fiber lies coincides with 7-brane over which type $I_1$ fiber lies, two type $I_2$ fibers are enhanced to type $I_4$ fiber, and two type $I_1$ fibers are enhanced to type $I_2$ fiber. Because a K3 surface with two type $III^*$ fibers, a type $I_4$ fiber, and a type $I_2$ fiber has the singularity type $E_7^2 A_3 A_1$, the K3 surface $S_1$ deforms and it becomes an extremal K3 surface (\\[extremal fibration 402 in 3.2.1\\]) in this limit. In short, F-theory on the extremal K3 surface (\\[extremal fibration 402 in 3.2.1\\]) can be seen as deformation of the stable degeneration because of the coincident 7-branes.\n\nAs the singularity rank of a rational elliptic surface is up to 8, the non-Abelian gauge group that arises on F-theory on a generic K3 surface obtained as the reverse of the stable degeneration has rank up to 16. Here, by generic we mean a situation in which singular fibers of rational elliptic surfaces do not collide when they are glued together to yield an elliptic K3 surface. When the large radius limit is taken, the heterotic dual of this compactification admits a geometric interpretation. In special situations in which singular fibers collide, 7-branes become coincident and the singularity ranks of the resulting K3 surfaces enhance to become greater than 16. The gauge groups of the heterotic duals of F-theory compactifications on these K3 surfaces do not allow for geometric interpretation.\n\nThe Mordell\u2013Weil group of the K3 extremal fibration (\\[extremal fibration 402 in 3.2.1\\]) is isomorphic to $\\Z_2$ [@SZ; @BLe]. Thus, the gauge group that arises in F-theory compactification on this extremal K3 surface is [@KRES] $$E_7 \\times E_7 \\times SU(4) \\times SU(2) / \\Z_2.$$\n\nComparing the Weierstrass equation (\\[extremal fibration 402 in 3.2.1\\]) with equation (\\[first fibration in 2.2\\]), we find that the following substitutions: $$\\begin{aligned}\n\\label{substitution 402 in 3.2.1}\n\\alpha & = & \\frac{5}{32} \\\\ \\nonumber\n\\beta & = & -\\frac{7}{128} \\\\ \\nonumber\n\\gamma = \\varepsilon & = & -\\frac{9}{64} \\\\ \\nonumber\n\\delta = \\zeta & = & -\\frac{9}{128}   \\end{aligned}$$ into equation (\\[first fibration in 2.2\\]) yield the Weierstrass equation (\\[extremal fibration 402 in 3.2.1\\]). Plugging the substitutions (\\[substitution 402 in 3.2.1\\]) into the equation of the alternate fibration (\\[second fibration in 2.2\\]), we obtain the following equation: $$\\label{alternate 402 in 3.2.1}\ny^2=x^3-\\frac{1}{3}\\, (10t^4+8t^3+t^2)\\, x+t^7+\\frac{56}{27}t^6+\\frac{26}{9}t^5+\\frac{8}{9}t^4+\\frac{2}{27}t^3,$$ with the discriminant $$\\label{disc alternate 402 in 3.2.1}\n\\Delta \\sim t^{11} (t+2)^2 (27t+4).$$ From equations (\\[alternate 402 in 3.2.1\\]) and (\\[disc alternate 402 in 3.2.1\\]), we find that the fibration (\\[alternate 402 in 3.2.1\\]) has a type $II^*$ fiber at $t=\\infty$, a type $I_5^*$ fiber at $t=0$, a type $I_2$ fiber at $t=-2$, and a type $I_1$ fiber at $t=-4/27$. Thus, the fibration (\\[alternate 402 in 3.2.1\\]) has singularity type $E_8 D_9 A_1$, and because the singularity type has rank 18, we deduce that this fibration is also extremal. Therefore, we find that the attractive K3 surface $S_{[4 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ also admits an extremal fibration (\\[alternate 402 in 3.2.1\\]) with singularity type $E_8 D_9 A_1$. This agrees with the results in [@SZ; @BLe].\n\n### Extremal K3 surface with $E_7^2 D_4$ singularity {#sssec3.2.2}\n\nThe complex structure of the extremal K3 surface with the singularity type $E_7^2 D_4$ is uniquely determined, and the intersection form of the transcendental lattice is [@SZ]: $$\\begin{pmatrix}\n2 & 0 \\\\\n0 & 2 \\\\\n\\end{pmatrix}.$$ The Weierstrass equation of this extremal fibration of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ is given as follows: $$\\label{extremal 202 in 3.2.2}\ny^2=x^3+ 4t^3(t-s)^2 \\, s^3 \\, x,$$ with the discriminant $$\\Delta \\sim t^9s^9 \\, (t-s)^6.$$ $[t:s]$ in the equation (\\[extremal 202 in 3.2.2\\]) denotes the homogeneous coordinate of the base $\\P^1$. Two type $III^*$ are at $[t:s]=[0:1]$ and $[1:0]$, and a type $I^*_0$ fiber is at $[t:s]=[1:1]$.\n\nAs shown in [@KRES], the extremal K3 fibration (\\[extremal 202 in 3.2.2\\]) can be seen as deformation of the stable degeneration. Gluing two identical extremal rational elliptic surfaces $X_{[III, \\hspace{1mm} III^*]}$ yields an elliptic K3 surface, $S_2$, the singular fibers of which have two type $III^*$ fibers and two type $III$ fibers. This is technically given by a generic quadratic base change of the extremal rational elliptic surface $X_{[III, \\hspace{1mm} III^*]}$, and this is the reverse of the stable degeneration. In a special limit at which two type $III$ fibers collide, the elliptic K3 surface $S_2$ deforms to yield the extremal fibration (\\[extremal 202 in 3.2.2\\]) of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ [@KRES]. 7-branes over which type $III$ fiber lies coincide with those over which type $III$ fiber lies in this limit, at which colliding two type $III$ fibers are enhanced to a type $I_0^*$ fiber. Therefore, F-theory on the extremal K3 surface (\\[extremal 202 in 3.2.2\\]) can be seen as deformation of the stable degeneration as the consequence of the coincident 7-branes.\n\nThe Mordell\u2013Weil group of the extremal elliptic fibration (\\[extremal 202 in 3.2.2\\]) is isomorphic to $\\Z_2$ [@Nish; @SZ]. Therefore, the gauge group on F-theory compactification on the extremal fibration (\\[extremal 202 in 3.2.2\\]) is [@KRES] $$E_7 \\times E_7 \\times SO(8) / \\Z_2.$$\n\nComparing the equation (\\[extremal 202 in 3.2.2\\]) with the equation (\\[first fibration in 2.2\\]), we find that the following substitutions $$\\begin{aligned}\n\\label{substitution 202 in 3.2.2}\n\\alpha & = & \\frac{2}{3} \\\\ \\nonumber\n\\beta = \\delta = \\zeta & = & 0 \\\\ \\nonumber\n\\gamma = \\varepsilon & = & 1  \\end{aligned}$$ into (\\[first fibration in 2.2\\]) yield the Weierstrass equation (\\[extremal 202 in 3.2.2\\]).\n\nBy plugging the substitutions (\\[substitution 202 in 3.2.2\\]) into the equation (\\[second fibration in 2.2\\]), we obtain the following Weierstrass equation: $$\\label{alternate 202 in 3.2.2}\ny^2=x^3-\\frac{1}{3}t^2(6t^2+1)x+\\frac{1}{27}t^3(27t^4+18t^2+2),$$ with the discriminant $$\\label{disc alternate 202 in 3.2.2}\n\\Delta \\sim t^{12} (27t^2+4).$$ We can confirm from the equations (\\[alternate 202 in 3.2.2\\]) and (\\[disc alternate 202 in 3.2.2\\]) that this alternate fibration in fact has a type $II^*$ fiber at $t=\\infty$, a type $I_6^*$ fiber at $t=0$, and two type $I_1$ fibers at the roots of $27t^2+4=0$. Thus, the singularity type of the alternate fibration is $E_8 D_{10}$, and we find that this fibration is also extremal. This gives the Weierstrass equation of the fibration no. 2 in Table \\[tablefibrations202 in A\\] of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ in Appendix \\[secA\\].\n\nDouble cover of $\\P^1\\times \\P^1$ ramified along a bidegree (4,4) curve, given by the following equation: $$\\label{genusone K3 in 3.2.2}\n\\tau^2=(t-\\alpha_1)^3(t-\\alpha_2)\\, x^4+(t-\\alpha_3)^3 (t-\\alpha_2)$$ yields a genus-one fibered K3 surface lacking a global section, but admitting a bisection, and this K3 surface was considered in [@K2] in the context of F-theory compactifications on genus-one fibrations without a global section. $x$ denotes the inhomogeneous coordinate of the first $\\P^1$, and $t$ denotes the inhomogeneous coordinate of the second $\\P^1$, in the product $\\P^1\\times \\P^1$, respectively. $\\alpha_1$, $\\alpha_2$, $\\alpha_3$ are distinct points in $\\P^1$. $\\alpha$\u2019s are superfluous parameters, and these can be mapped to: $$\\alpha_1=0, \\hspace{2mm} \\alpha_2=1, \\hspace{2mm} \\alpha_3=\\infty$$ under some appropriate automorphism of the base $\\P^1$. The K3 genus-one fibration (\\[genusone K3 in 3.2.2\\]) has two type $III^*$ fibers at $t=\\alpha_1, \\alpha_3$, and a type $I_0^*$ fiber at $t=\\alpha_2$ [@K2].\n\nThe Jacobian fibration of the K3 genus-one fibration (\\[genusone K3 in 3.2.2\\]) gives the extremal K3 elliptic fibration (\\[extremal 202 in 3.2.2\\]), as demonstrated in [@K2]. Utilizing this fact, in Section \\[sec4\\] we build an elliptically fibered Calabi\u2013Yau 3-fold, by fibering the K3 genus-one fibration (\\[genusone K3 in 3.2.2\\]) over the base $\\P^1$, then taking the Jacobian fibration of it. It turns out that the resulting Calabi\u2013Yau 3-fold is a fibration of the extremal K3 surface (\\[extremal 202 in 3.2.2\\]) over the base $\\P^1$. We also build a family of elliptic Calabi\u2013Yau 3-folds which includes this Calabi\u2013Yau 3-fold. These constructions will be discussed in detail in Section \\[sec4\\].\n\nApplications to $SO(32)$ heterotic strings {#ssec3.3}\n------------------------------------------\n\nAs reviewed in Section \\[ssec2.2\\], the K3 surface (\\[hypersurface in 2.2\\]) with $H\\oplus E_7 \\oplus E_7$ lattice polarization admits another elliptic fibration, the singular fibers of which include a type $I_8^*$ fiber, given by the Weierstrass equation: $$\\label{third fibration in 3.3}\ny^2=x^3+ (t^3-3\\alpha t-2\\beta)\\, x^2+(\\gamma t-\\delta)(\\varepsilon t-\\zeta)\\, x$$ with the discriminant $$\\Delta \\sim (\\gamma t-\\delta)^2(\\varepsilon t-\\zeta)^2 \\big[(t^3-3\\alpha t-2\\beta)^2-4(\\gamma t-\\delta)(\\varepsilon t-\\zeta)\\big].$$ Therefore, there is a birational map that transforms the elliptic fibration with two type $III^*$ fibers (\\[first fibration in 2.2\\]) into an alternate fibration (\\[third fibration in 3.3\\]) with a type $I_8^*$ fiber (or worse). Using this birational map, we send the extremal K3 elliptic fibrations with two $E_7$ singularities studied in Section \\[ssec3.2\\] to another fibration with a type $I_8^*$ fiber (or worse). This relates to $SO(32)$ heterotic strings.\n\nAs we saw previously in Section \\[sssec3.2.2\\], the Weierstrass equation of the extremal fibration of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ with singularity type $E_7^2 D_4$ is given by (\\[extremal 202 in 3.2.2\\]), with $$\\begin{aligned}\n\\label{substitution 202 in 3.3}\n\\alpha & = & \\frac{2}{3} \\\\ \\nonumber\n\\beta = \\delta = \\zeta & = & 0 \\\\ \\nonumber\n\\gamma = \\varepsilon & = & 1.  \\end{aligned}$$ By plugging these values (\\[substitution 202 in 3.3\\]) into the alternate fibration (\\[third fibration in 3.3\\]), we obtain the Weierstrass equation as $$\\label{transf 202 in 3.3}\ny^2=x^3+t(t^2-2)\\, x^2+t^2\\, x,$$ with the discriminant $$\\begin{aligned}\n\\Delta & \\sim & t^4 \\, \\big( t^2(t^2-2)^2-4t^2 \\big) \\\\ \\nonumber\n& = & t^8\\, (t^2-4).\\end{aligned}$$ In the homogeneous form, the discriminant is $$\\label{disc tranf 202 in 3.3}\n\\Delta \\sim t^8 s^{14}\\, (t^2-4s^2).$$ From equations (\\[transf 202 in 3.3\\]) and (\\[disc tranf 202 in 3.3\\]), we deduce that the alternate fibration (\\[transf 202 in 3.3\\]) has a type $I_8^*$ fiber at $[t:s]=[1:0]$, a type $I_2^*$ fiber at $[t:s]=[0:1]$, and two type $I_1$ fibers at $[t:s]=[2:1], [-2:1]$. Thus, the alternate fibration (\\[transf 202 in 3.3\\]) has singularity type $D_{12}D_6$, and we find that this fibration is also extremal. We conclude that the alternate fibration (\\[transf 202 in 3.3\\]) yields fibration no.\u00a08 in Table \\[tablefibrations202 in A\\] in Appendix \\[secA\\] of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$.\n\nThe Mordell\u2013Weil group of the fibration (\\[transf 202 in 3.3\\]) is isomorphic to $\\Z_2$ (see [@Nish; @SZ]); therefore, the gauge group on F-theory compactification on the fibration (\\[transf 202 in 3.3\\]) is $$SO(24)\\times SO(12)/ \\Z_2.$$\n\nThe attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ has another extremal fibration with the singularity type $D_8^2 A_1^2$ [@Nish]. (This is fibration no.9 in Table \\[tablefibrations202 in A\\] in Appendix \\[secA\\].) As deduced in [@KRES], the Weierstrass equation of this extremal fibration is given as follows: $$\\label{D8 202 in 3.3}\ny^2=x^3 -3t^2s^2\\, (t^4+s^4-t^2s^2)\\, x + (t^2+s^2)\\, t^3s^3\\, (2t^4-5t^2s^2+2s^4),$$ with the discriminant $$\\Delta \\sim t^{10}s^{10}\\, (t-s)^2\\, (t+s)^2.$$ Type $I^*_4$ fibers are at $[t:s]=[1:0], [0:1]$, and type $I_2$ fibers are at $[t:s]=[1:1], [1:-1]$.\n\nAs shown in [@KRES], extremal fibration (\\[D8 202 in 3.3\\]) can be seen as deformation of the stable degeneration in which two extremal rational elliptic surfaces $X_{[4^*, \\hspace{1mm} 1,1]}$ are glued together. Gluing of two extremal rational elliptic surfaces $X_{[4^*, \\hspace{1mm} 1,1]}$ yields an elliptic K3 surface, which we denote by $S_3$, the singular fibers of which are two type $I_4^*$ fibers and four type $I_1$ fibers. In a limit at which two pairs of type $I_1$ fibers collide, type $I_1$ fibers collide and they are enhanced to a type $I_2$ fiber. In this limit, K3 surface $S_3$ deforms to yield the attractive K3 surface with the extremal fibration (\\[D8 202 in 3.3\\]) [@KRES]. Therefore, extremal fibration (\\[D8 202 in 3.3\\]) can be seen as deformation of the stable degeneration, as a result of coincident 7-branes over which type $I_1$ fibers lie.\n\nThe Mordell\u2013Weil group of the fibration (\\[D8 202 in 3.3\\]) is isomorphic to $\\Z_2\\times\\Z_2$ [@Nish; @SZ]; therefore, the gauge group on F-theory compactification on the fibration (\\[D8 202 in 3.3\\]) is [@KRES] $$SO(16)\\times SO(16)\\times SU(2)^2/ \\Z_2\\times \\Z_2.$$\n\nWe saw previously in Section \\[sssec3.2.1\\] that the attractive K3 surface $S_{[4 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ admits an extremal fibration with the singularity type $E_7^2 A_3 A_1$, and the Weierstrass equation of this fibration is given by (\\[extremal fibration 402 in 3.2.1\\]), with $$\\begin{aligned}\n\\label{substitution 402 in 3.3}\n\\alpha & = & \\frac{5}{32} \\\\ \\nonumber\n\\beta & = & -\\frac{7}{128} \\\\ \\nonumber\n\\gamma = \\varepsilon & = & -\\frac{9}{64} \\\\ \\nonumber\n\\delta = \\zeta & = & -\\frac{9}{128}.   \\end{aligned}$$ By plugging these into the equation (\\[third fibration in 3.3\\]), we obtain an alternate fibration given by: $$\\label{transf 402 in 3.3}\ny^2=x^3+(t^3-\\frac{15}{32}t+\\frac{7}{64})\\, x^2+(\\frac{9}{64})^2(t-\\frac{1}{2})^2 \\, x,$$ with the discriminant $$\\label{disc transf 402 in 3.3}\n\\Delta \\sim (t-\\frac{1}{2})^7 (4t+1)^2 (t+1).$$ From the equations (\\[transf 402 in 3.3\\]) and (\\[disc transf 402 in 3.3\\]), we find that the alternate fibration has a type $I_8^*$ fiber at $t=\\infty$, a type $I_1^*$ fiber at $t=\\frac{1}{2}$, a type $I_2$ fiber at $t=-\\frac{1}{4}$, and a type $I_1$ fiber at $t=-1$. Thus, the alternate fibration (\\[transf 402 in 3.3\\]) has the singularity type $D_{12} D_5 A_1$, and this is also extremal. This result agrees with the elliptic fibrations with a section of the attractive K3 surface $S_{[4 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ obtained in [@BLe].\n\nThe Mordell\u2013Weil group of the alternate fibration (\\[transf 402 in 3.3\\]) is isomorphic to $\\Z_2$ [@SZ; @BLe]. Therefore, the gauge group on F-theory compactification on the fibration (\\[transf 402 in 3.3\\]) is $$SO(24) \\times SO(10) \\times SU(2) / \\Z_2.$$\n\nJacobian Calabi\u2013Yau 3-folds and F-theory compactifications {#sec4}\n==========================================================\n\nIn this section, we fiber elliptic K3 surfaces over $\\P^1$ to yield elliptically fibered Calabi\u2013Yau 3-folds[^13] with a global section, and we study six-dimensional F-theory compactifications with $N=1$ supersymmetry on the constructed Calabi\u2013Yau 3-folds. K3 fibers in this construction include an elliptic K3 surface that belongs to the F-theory side of the moduli of eight-dimensional non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra, discussed in Section \\[sssec3.2.2\\].\n\nTo be clear, we first consider genus-one fibered K3 surfaces lacking a global section, built as double covers of $\\P^1\\times \\P^1$ ramified along a (4,4) curve. The built genus-one fibered K3 surfaces do not have a global section, but they have a bisection [@K2]. We consider higher-dimensional analogs of these K3 surfaces to yield genus-one fibered Calabi\u2013Yau 3-folds, built as double covers of $\\P^1\\times\\P^1\\times\\P^1$ ramified along a tridegree (4,4,4) surface. As we show in Section \\[ssec4.1\\], the constructed Calabi\u2013Yau 3-folds are genus-one fibered, but they lack a global section. These Calabi\u2013Yau 3-folds still have a bisection [@MTsection]. The pullback of $\\cal{O}$(1) class in $\\P^1$ yields a bisection [@Kdisc].\n\nTaking the Jacobian fibrations of these Calabi\u2013Yau genus-one fibrations yields elliptically fibered Calabi\u2013Yau 3-folds with a global section. The resulting elliptic Calabi\u2013Yau 3-folds are also K3 fibered, and when we tune the coefficients of the defining equation, we obtain Calabi\u2013Yau 3-folds, K3 fibers of which are the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ that belongs to the F-theory side of the moduli of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra.\n\nWe deduce the non-Abelian gauge groups on F-theory compactifications on the Jacobian Calabi\u2013Yau 3-folds. We also perform a consistency check of the obtained gauge groups, by considering the symmetry that the elliptic fibers possess. Highly enhanced gauge groups arise when we choose specific coefficients of the defining equations of the Jacobian Calabi\u2013Yau 3-folds. We determine the Mordell\u2013Weil groups of some specific Calabi\u2013Yau 3-folds, and we deduce the global structures of the gauge groups for F-theory on these spaces. We obtain some models without a $U(1)$ gauge field. Furthermore, we deduce viable candidate matter spectra on F-theory on the constructed elliptically fibered Calabi\u2013Yau 3-folds that satisfy the six-dimensional anomaly cancellation condition.\n\nCalabi\u2013Yau 3-folds as double covers, Jacobian fibrations, and the discriminant locus {#ssec4.1}\n------------------------------------------------------------------------------------\n\nDouble covers of the product of projective lines, $\\P^1\\times \\P^1\\times \\P^1$, ramified over a $(4,4,4)$ surface have the trivial canonical bundles, $K=0$; therefore, they describe Calabi\u2013Yau 3-folds. Fiber of projection onto $\\P^1\\times\\P^1$ is a double cover of $\\P^1$ branched over four points, namely, it is an elliptic curve. Thus, projection onto $\\P^1\\times\\P^1$ gives a genus-one fibration. Fiber of projection onto $\\P^1$ is a double cover of $\\P^1\\times\\P^1$ ramified along a $(4,4)$ curve, which yields a genus-one fibered K3 surface. Therefore, projection onto $\\P^1$ yields a K3 fibration. These K3 surfaces do not have a section, but they have a bisection [@K2].\n\nIn this note, we consider in particular the double covers of $\\P^1\\times\\P^1\\times\\P^1$ given by the following equations: $$\\label{doublecover 3-fold in 4.1}\n\\tau^2= f_1(t) g_1(u) \\, x^4+ f_2(t) g_2(u),$$ where $x$ is the inhomogeneous coordinate on the first $\\P^1$, and $t$ and $u$ are the inhomogeneous coordinates on the second and the third $\\P^1$ in the product $\\P^1\\times\\P^1\\times\\P^1$. Here $f_1, f_2$ are polynomials in the variable $t$ of degree four, and $g_1, g_2$ are polynomials of degree four in the variable $u$. By splitting the polynomials $f_1, f_2, g_1, g_2$ into linear factors, equation (\\[doublecover 3-fold in 4.1\\]) can be rewritten as follows: $$\\label{doublecover rewritten in 4.1}\n\\tau^2= \\Pi_{i=1}^4 (t-\\alpha_i)\\cdot \\Pi_{j=1}^4 (u-\\beta_j) \\cdot x^4+ \\Pi_{k=5}^8 (t-\\alpha_k) \\cdot \\Pi_{l=5}^8 (u-\\beta_l).$$ K3 fiber of this genus-one fibered Calabi\u2013Yau 3-fold is given by $$\\label{genus-one K3 fiber in 4.1}\n\\tau^2= \\Pi_{i=1}^4 (t-\\alpha_i)\\cdot x^4+ \\Pi_{k=5}^8 (t-\\alpha_k).$$ As shown in [@K2], K3 fiber (\\[genus-one K3 fiber in 4.1\\]) is genus-one fibered, but it does not have a global section. K3 fiber (\\[genus-one K3 fiber in 4.1\\]) has a bisection [@K2].\n\nUsing an argument similar to that in [@KCY4], we can show that the Calabi\u2013Yau 3-fold (\\[doublecover rewritten in 4.1\\]) indeed lacks a rational section. Suppose it admits a rational section. Then, it restricts to a K3 fiber, and this gives a global section to the K3 fiber, leading to a contradiction. By an argument similar to those in [@K2; @KCY4; @Kdisc], the genus-one fibered Calabi\u2013Yau 3-fold (\\[doublecover rewritten in 4.1\\]) has a bisection[^14].\n\nWe can consider a special situation in which $$\\label{coeff in 4.1}\n\\alpha_1=\\alpha_2=\\alpha_3, \\hspace{5mm} \\alpha_4=\\alpha_8, \\hspace{5mm} \\alpha_5=\\alpha_6=\\alpha_7.$$ This yields a genus-one fibered Calabi\u2013Yau 3-fold $$\\tau^2= (t-\\alpha_1)^3(t-\\alpha_4)\\cdot \\Pi_{j=1}^4 (u-\\beta_j) \\cdot x^4+ (t-\\alpha_5)^3(t-\\alpha_4) \\cdot \\Pi_{l=5}^8 (u-\\beta_l),$$ and K3 fiber given by $$\\label{K3 fiber special in 4.1}\n\\tau^2= (t-\\alpha_1)^3(t-\\alpha_4)\\, x^4+ (t-\\alpha_5)^3(t-\\alpha_4).$$ This is the genus-one fibered K3 surface (\\[genusone K3 in 3.2.2\\]) lacking a section, which we discussed in Section \\[sssec3.2.2\\][^15].\n\nThe Jacobian fibrations of the genus-one fibered Calabi\u2013Yau 3-folds (\\[doublecover rewritten in 4.1\\]) yield elliptically fibered Calabi\u2013Yau 3-folds with a section. The Jacobian fibrations are given by [@Muk] $$\\label{Jacobian 3-fold in 4.1}\n\\tau^2=\\frac{1}{4}x^3-\\Pi_{i=1}^8 (t-\\alpha_i)\\cdot \\Pi_{j=1}^8 (u-\\beta_j)\\cdot x.$$ Projection onto $\\P^1\\times\\P^1$ gives an elliptic fibration, and projection onto $\\P^1$ yields a K3 fibration. K3 fiber of the projection onto $\\P^1$ is given by $$\\label{Jacobian K3 fiber in 4.1}\n\\tau^2=\\frac{1}{4}x^3-\\Pi_{i=1}^8 (t-\\alpha_i)\\cdot x.$$\n\nWhen parameters $\\alpha$ are tuned as in (\\[coeff in 4.1\\]), the Weierstrass equation of the Jacobian fibration becomes $$\\label{enhanced Jacobian 3-fold in 4.1}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot \\Pi_{j=1}^8 (u-\\beta_j)\\, x,$$ and K3 fiber (\\[Jacobian K3 fiber in 4.1\\]) becomes $$\\label{enhanced Jacobian K3 fiber in 4.1}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\, x.$$ This is the Jacobian fibration of the K3 fiber (\\[K3 fiber special in 4.1\\]), and this is the extremal fibration (\\[extremal 202 in 3.2.2\\]) of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$[^16], which belongs to the F-theory side of the moduli of the non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra.\n\nThe obtained Jacobian Calabi\u2013Yau 3-folds (\\[Jacobian 3-fold in 4.1\\]) yield fibrations of K3 surfaces (\\[Jacobian K3 fiber in 4.1\\]) over $\\P^1$, and this family includes fibrations of the extremal K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$, which we discussed in Section \\[sssec3.2.2\\], over $\\P^1$.\n\nThe discriminant of the Calabi\u2013Yau Jacobian fibration (\\[Jacobian 3-fold in 4.1\\]) is given by the following equation: $$\\label{disc in 4.1}\n\\Delta \\sim \\Pi_{i=1}^8 (t-\\alpha_i)^3\\cdot \\Pi_{j=1}^8 (u-\\beta_j)^3.$$ The discriminant locus of the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) is given by the vanishing of the discriminant (\\[disc in 4.1\\]). Genus-one fibered Calabi\u2013Yau 3-fold (\\[doublecover rewritten in 4.1\\]) and the Jacobian fibration (\\[Jacobian 3-fold in 4.1\\]) have the identical discriminant loci, and the identical configurations of the singular fibers.\n\nFrom the equation (\\[disc in 4.1\\]), we find that the discriminant components of the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) are given as follows: $$\\begin{aligned}\nA_i & = & \\{t=\\alpha_i\\}  \\hspace{5mm} (i=1, \\cdots, 8) \\\\ \\nonumber\nB_j & = & \\{u=\\beta_j\\}  \\hspace{5mm} (j=1, \\cdots, 8).\\end{aligned}$$ In F-theory on the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]), 7-branes are wrapped on these components. Components $A_i$ are isomorphic to $\\{{\\rm pt}\\} \\times \\P^1$, and components $B_j$ are isomorphic to $\\P^1\\times \\{{\\rm pt}\\}$. Therefore, these are isomorphic to $\\P^1$. The types of the singular fibers and the non-Abelian gauge groups on the 7-branes will be discusses in Section \\[ssec4.2\\].\n\nNon-Abelian gauge groups {#ssec4.2}\n------------------------\n\nWe determine the non-Abelian gauge groups on F-theory on the Jacobian Calabi\u2013Yau 3-folds constructed in Section \\[ssec4.1\\]. We also check the consistency of the obtained gauge groups.\n\n### Singular fibers of the Jacobian Calabi\u2013Yau 3-folds, and non-Abelian gauge groups {#sssec4.2.1}\n\nFrom the Weierstrass equation (\\[Jacobian 3-fold in 4.1\\]) of the Jacobian Calabi\u2013Yau 3-fold and the discriminant (\\[disc in 4.1\\]), we find that for a generic situation in which the coefficients $\\alpha$\u2019s and $\\beta$\u2019s are mutually distinct, the types of the singular fibers on the components $A_i$, $i=1, \\cdots, 8$, and $B_j$, $j=1, \\cdots, 8$, are $III$. In this case, the non-Abelian gauge group that arises on F-theory compactification on the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) is $$SU(2)^{16}.$$\n\nWhen two of the coefficients, $\\alpha_i$ and $\\alpha_k$, become coincident, a pair of type $III$ fibers on the components $A_i$ and $A_k$ collides, and it is enhanced to a type $I^*_0$ fiber. Because the polynomial $$x^3 - \\Pi_{j=1}^8 (u-\\beta_j)\\cdot x$$ splits into the linear factor and the quadratic factor as: $$x\\, \\big(x^2 - \\Pi_{j=1}^8 (u-\\beta_j) \\big),$$ type $I^*_0$ fiber is semi-split [@BIKMSV]. The non-Abelian gauge group that arises on the 7-branes wrapped on the component $A_i$ is thus enhanced to $SO(7)$ in this situation [^17] .\n\nWhen three of the coefficients, $\\alpha_i$, $\\alpha_k$ and $\\alpha_l$, become coincident, type $III$ fibers on the components $A_i$, $A_k$, $A_l$ collide, and they are enhanced to a type $III^*$ fiber. Further enhancement breaks the Calabi\u2013Yau condition, as stated in [@K2; @KCY4]. An argument similar to that stated previously equally applies to $\\beta$\u2019s and the components $B_j$. The results are presented in Table \\[tablelistgaugegroup in 4.2.1\\] below.\n\n       Component       Fiber type   non-Abel. Gauge Group\n  ------------------- ------------ -----------------------\n   $A_{1,\\cdots, 8}$     $III$             $SU(2)$\n   $A_{1,\\cdots, 8}$    $I^*_0$            $SO(7)$\n   $A_{1,\\cdots, 8}$    $III^*$             $E_7$\n   $B_{1,\\cdots, 8}$     $III$             $SU(2)$\n   $B_{1,\\cdots, 8}$    $I^*_0$            $SO(7)$\n   $B_{1,\\cdots, 8}$    $III^*$             $E_7$\n\n  : \\[tablelistgaugegroup in 4.2.1\\]Fiber types and the gauge groups on the discriminant components.\n\nAs discussed in Section \\[ssec4.1\\], K3 fiber becomes most enhanced when parameters $\\alpha$ are tuned as: $$\\alpha_1=\\alpha_2=\\alpha_3, \\hspace{5mm} \\alpha_4=\\alpha_8, \\hspace{5mm} \\alpha_5=\\alpha_6=\\alpha_7.$$ In this case, the non-Abelian gauge group that arises on F-theory compactification on the Jacobian Calabi\u2013Yau 3-fold (\\[enhanced Jacobian 3-fold in 4.1\\]) is $$E_7^2 \\times SO(7) \\times SU(2)^8.$$ K3 fiber becomes the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ with the singularity type $E_7^2 D_4$ (\\[enhanced Jacobian K3 fiber in 4.1\\]) in this situation, and this attractive K3 surface was discussed in Section \\[sssec3.2.2\\]. The singularity type of the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) is most enhanced, when the following equalities hold among coefficients $\\beta$\u2019s further: $$\\beta_1=\\beta_2=\\beta_3, \\hspace{5mm} \\beta_4=\\beta_8, \\hspace{5mm} \\beta_5=\\beta_6=\\beta_7.$$ In this case, the Weierstrass equation of the Jacobian Calabi\u2013Yau 3-fold becomes $$\\label{jacobian 3-fold most enhanced in 4.2.1}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^3(u-\\beta_5)^3(u-\\beta_4)^2 \\, x.$$ The types of the singular fibers over the components, $A_1$, $A_5$, $B_1$, $B_5$, are enhanced to type $III^*$, and the types of the singular fibers over the components $A_4$ and $B_4$ are enhanced to type $I_0^*$. In this situation, the non-Abelian gauge group on the F-theory compactification of the Jacobian Calabi\u2013Yau 3-fold (\\[jacobian 3-fold most enhanced in 4.2.1\\]) is enhanced to: $$E_7^4 \\times SO(7)^2.$$\n\n### Consistency check of the gauge groups {#sssec4.2.2}\n\nBy an argument similar to those given in [@K2; @KCY4], smooth genus-one fibers of the Calabi\u2013Yau double covers (\\[doublecover 3-fold in 4.1\\]) are invariant under the following transformation: $$x  \\rightarrow  e^{\\frac{2 \\pi i}{4}}\\cdot x.$$ We find from this that genus-one fibers of the Calabi\u2013Yau double covers (\\[doublecover 3-fold in 4.1\\]) possess complex multiplication of order 4; therefore, the generic genus-one fiber of the Calabi\u2013Yau double cover (\\[doublecover 3-fold in 4.1\\]) has j-invariant 1728. This requires the singular fibers to have j-invariant 1728 [@K2; @KCY4]. Because the types of the singular fibers of Calabi\u2013Yau genus-one fibration and those of the Jacobian fibration are identical, this means that the singular fibers of the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) also have j-invariant 1728.\n\nAccording to Table \\[tablemonodromy in 2.1\\] in Section \\[ssec2.1\\], the types of the singular fibers with j-invariant 1728 are: $III$, $III^*$, and $I^*_0$. (j-invariant of type $I^*_0$ fiber can take the value 1728.) Thus, the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) can have the singular fibers, only of types $III$, $I^*_0$ and $III^*$. This agrees with the results obtained in Section \\[sssec4.2.1\\]. The monodromies of orders 2 and 4 characterize the non-Abelian gauge groups that form on F-theory compactifications of the Jacobian Calabi\u2013Yau 3-folds (\\[Jacobian 3-fold in 4.1\\]).\n\nMordell\u2013Weil groups of some models, and models without $U(1)$ gauge field {#ssec4.3}\n-------------------------------------------------------------------------\n\nWe determine the Mordell\u2013Weil groups of F-theory models on the Jacobian Calabi\u2013Yau 3-folds (\\[enhanced Jacobian 3-fold in 4.1\\]). We deduce that they do not have a $U(1)$ gauge field.\n\nWe saw previously that the Weierstrass equation of the Jacobian Calabi\u2013Yau 3-fold becomes (\\[enhanced Jacobian 3-fold in 4.1\\]): $$\\label{enhanced Jacobian in 4.3}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot \\Pi_{j=1}^8 (u-\\beta_j)\\, x,$$ when the K3 fibers are most enhanced, namely when K3 fibers become the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ with the singularity type $E_7^2 D_4$. The Mordell\u2013Weil group of this extremal K3 surface is known [@Nish; @SZ], and it is isomorphic to $\\Z_2$. (See fibration no.4 in Table \\[tablefibrations202 in A\\] in Appendix \\[secA\\].) Using an argument similar to those given in [@KCY4; @Kdisc], we consider the specialization to the K3 fiber to deduce that the Mordell\u2013Weil group of the Jacobian Calabi\u2013Yau 3-fold (\\[enhanced Jacobian in 4.3\\]) is isomorphic to that of the K3 fiber. Therefore, we find that the Mordell\u2013Weil group of the Jacobian Calabi\u2013Yau 3-fold $J(Y)$ (\\[enhanced Jacobian in 4.3\\]) is isomorphic to $\\Z_2$: $$MW (J(Y)) \\cong \\Z_2.$$ Thus, the global structure [@AspinwallGross; @AMrational; @MMTW] of the gauge group forming on the 7-branes is given as follows: $$E_7^2 \\times SO(7) \\times SU(2)^8 / \\Z_2.$$ The F-theory on the Jacobian Calabi\u2013Yau 3-folds (\\[enhanced Jacobian in 4.3\\]) does not have a $U(1)$ gauge field.\n\nMatter spectra {#ssec4.4}\n--------------\n\nIn this section, we deduce the candidate matter spectra on six-dimensional F-theory on the Jacobian Calabi\u2013Yau 3-folds constructed in Section \\[ssec4.1\\]. As we stated previously, 7-branes are wrapped on the discriminant components given by $$\\begin{aligned}\nt & = & \\alpha_i \\hspace{5mm} (i=1, \\ldots, 8) \\\\ \\nonumber\nu & = & \\beta_j \\hspace{5mm} (j=1, \\ldots, 8).\\end{aligned}$$ 7-branes wrapping the components are isomorphic to $\\P^1$. 7-branes intersect at the points $(t,u)=(\\alpha_i, \\beta_j)$, $i,j=1, \\ldots, 8$, in the base surface, and matter arises at these points.\n\nThe base surface $B$ of the Jacobian Calabi\u2013Yau 3-folds constructed in Section \\[ssec4.1\\] is isomorphic to $\\P^1\\times \\P^1$, $B\\cong \\P^1\\times\\P^1$, thus the number of tensor multiplets that arise in F-theory compactification on the Jacobian Calabi\u2013Yau 3-folds is [@MV2] $$\\begin{aligned}\nT & = & h^{1,1}(B=\\P^1\\times\\P^1)-1 = 2-1 \\\\ \\nonumber\n& = & 1. \\end{aligned}$$ The six-dimensional anomaly cancellation condition [@GSW6d; @Sagnotti; @Erler; @Sch6d] then requires that $$\\label{anomaly cancellation in 4.4}\nH-V=273-29=244.$$\n\nFor simplicity, we only consider the case in which K3 fibers are most enhanced, namely K3 fibers become the extremal K3 surface with the singularity type $E_7^2 D_4$. This corresponds to the case where the coefficients $\\alpha$ satisfy the relations (\\[coeff in 4.1\\]) $$\\label{coeff in 4.3}\n\\alpha_1=\\alpha_2=\\alpha_3, \\hspace{5mm} \\alpha_4=\\alpha_8, \\hspace{5mm} \\alpha_5=\\alpha_6=\\alpha_7.$$ As we saw in Section \\[ssec4.3\\], for this situation the Mordell\u2013Weil rank of the Jacobian Calabi\u2013Yau 3-folds is 0, and the Mordell\u2013Weil group is isomorphic to $\\Z_2$.\n\nFirst, we consider the case where a pair of $\\beta$ coincide: $$\\beta_1=\\beta_2.$$ Then the equation of the Jacobian Calabi\u2013Yau 3-fold (\\[Jacobian 3-fold in 4.1\\]) becomes $$\\label{1st 3-fold in 4.4}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^2\\cdot \\Pi_{j=3}^8 (u-\\beta_j)\\cdot x$$ and the discriminant is given as follows $$\\Delta \\sim (t-\\alpha_1)^9(t-\\alpha_5)^9(t-\\alpha_4)^6\\cdot (u-\\beta_1)^6\\cdot \\Pi_{j=3}^8 (u-\\beta_j)^3.$$ The non-Abelian gauge group forming in F-theory compactification is $$E_7^2 \\times SO(7)^2 \\times SU(2)^6.$$ Therefore, we have $$V=133\\times 2+21\\times 2+6\\times 3=326.$$ The anomaly cancellation condition (\\[anomaly cancellation in 4.4\\]) requires that $$H=V+244=326+244=570.$$\n\nMatter arises at the 21 intersections $$(\\alpha_i, \\beta_j) \\hspace{1cm} (i=1,4,5, \\hspace{2mm} j=1,3,4,5,6,7,8).$$ Parameters $\\alpha_1, \\alpha_4, \\alpha_5$ can be sent to fixed values, e.g., 0, 1, $\\infty$, under an automorphism of the first $\\P^1$ in the base $\\P^1\\times\\P^1$. Therefore, these are superfluous and are not actual parameters of the complex structure deformation. Among parameters of the complex structure deformation $\\beta_1, \\beta_3, \\beta_4, \\beta_5, \\beta_6, \\beta_7, \\beta_8$, three of these can be fixed to specific values under an automorphism of the second $\\P^1$ in the base $\\P^1\\times\\P^1$. Thus, the number of the effective parameters of the complex structure deformation is four. The number of the neutral hypermultiplets arising from the complex structure deformations is therefore given by $$H^0=1+4=5.$$ Here, $H^0$ is used to denote the number of the neutral hypermultiplets. It follows that the sum of the dimensions of the representations of matter arising at the 21 intersections $(\\alpha_i, \\beta_j)$, $i=1,4, 5$, $j=1,3,4,5,6,7,8$, must be $$570-5=565$$ to cancel the anomaly.\n\n$E_7$ angularity and $D_4$ singularity collide at two intersections $(t,u)=(\\alpha_1, \\beta_1), (\\alpha_5, \\beta_1)$, and $D_4$ singularities collide at the intersection $(t,u)=(\\alpha_4, \\beta_1)$. $E_7$ singularity and $A_1$ singularity collide at the 12 intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=3, \\ldots, 8$. $D_4$ singularity and $A_1$ singularity collide at the six intersections $(t,u)=(\\alpha_4, \\beta_j)$, $j=3, \\ldots, 8$. When the types of colliding singularities are fixed, from a symmetry argument, the representations of matter arising at the intersections at which the fixed types of singularities collide should be identical.\n\nIf we assume that ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the two intersections where $E_7$ angularity and $D_4$ singularity collide, ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the intersection where $D_4$ singularities collide, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$ arises at the 12 intersections where $E_7$ singularity and $A_1$ singularity collide and ${\\bf 7}\\oplus {\\bf 1}$ arises at the six intersections where $D_4$ singularity and $A_1$ singularity collide, then the net dimension of the matter representations arising at the intersections of the 7-branes is $$(56+7+1)\\times 2+(21+7+1)+(28+2)\\times 12+(7+1)\\times 6=565.$$ Therefore, these matter representations yield a candidate of consistent matter spectrum on F-theory compactification on the Calabi\u2013Yau 3-fold (\\[1st 3-fold in 4.4\\]) that satisfies the anomaly cancellation condition. Here $\\frac{1}{2}{\\bf 56}$ denotes the $\\frac{1}{2}$-hypermultiplet of ${\\bf 56}$ of $E_7$[^18]. Having $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 1}$ instead of $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$ at the 12 intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=3, \\ldots, 8$, and ${\\bf 7}\\oplus {\\bf 2}\\oplus {\\bf 1}$ instead of ${\\bf 7}\\oplus {\\bf 1}$ at the six intersections $(t,u)=(\\alpha_4, \\beta_j)$, $j=3, \\ldots, 8$ also yields another viable candidate of matter spectrum. Additionally, having matter representation ${\\bf 56}\\oplus {\\bf 8}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, or matter representation ${\\bf 21}\\oplus {\\bf 8}$, instead of ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where two $D_4$ singularities collide, also yield other viable candidate matter spectra. Furthermore, having matter representation ${\\bf 56}\\oplus {\\bf 8}\\oplus {\\bf 1}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, matter representation ${\\bf 21}$, instead of ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersection where two $D_4$ singularities collide, and matter representation ${\\bf 7}\\oplus {\\bf 2}$, instead of ${\\bf 7}\\oplus {\\bf 1}$, at the intersections where $D_4$ singularity and $A_1$ singularity collide also yields another viable candidate matter spectrum.\n\nBecause the Mordell\u2013Weil group of the Calabi\u2013Yau elliptic fibration (\\[enhanced Jacobian 3-fold in 4.1\\]) is isomorphic to $\\Z_2$, the global structure of the gauge group forming in F-theory compactification on the Calabi\u2013Yau 3-fold (\\[1st 3-fold in 4.4\\]) is $$E_7^2 \\times SO(7)^2 \\times SU(2)^6 /\\Z_2.$$\n\nNext, we discuss the case where three pairs of the parameters $\\beta$ coincide: $$\\begin{aligned}\n\\beta_1=\\beta_2 \\\\ \\nonumber\n\\beta_3=\\beta_4 \\\\ \\nonumber\n\\beta_5=\\beta_6.\\end{aligned}$$ The equation of the Calabi\u2013Yau 3-fold (\\[enhanced Jacobian 3-fold in 4.1\\]) becomes $$\\label{2nd 3-fold in 4.4}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^2(u-\\beta_3)^2(u-\\beta_5)^2\\cdot \\Pi_{j=7}^8 (u-\\beta_j)\\cdot x$$ and the discriminant is given as follows $$\\Delta \\sim (t-\\alpha_1)^9(t-\\alpha_5)^9(t-\\alpha_4)^6\\cdot (u-\\beta_1)^6(u-\\beta_3)^6(u-\\beta_5)^6\\cdot \\Pi_{j=7}^8 (u-\\beta_j)^3.$$ Non-Abelian gauge group forming in F-theory compactification is: $$E_7^2 \\times SO(7)^4 \\times SU(2)^2.$$ Thus, $$V=133\\times 2+21\\times 4+3\\times 2=356.$$ Anomaly cancellation condition (\\[anomaly cancellation in 4.4\\]) requires that $$H=356+244=600.$$ Among the parameters of the complex structure deformation, $\\beta_1, \\beta_3, \\beta_5, \\beta_7, \\beta_8$, three can be sent to fixed values under an automorphism of $\\P^1$. Therefore, the actual number of the parameters of the complex structure deformation is two. From this, we obtain the number of the neutral hypermultiplets arising from the complex structure deformations as $$H^0=1+2=3.$$ 7-branes intersect in fifteen points at $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,4,5$, $j=1,3,5,7,8$, and matter arises at these intersections. The net representation dimensions of matter arising from these intersections must be $$600-3=597$$ owing to the anomaly cancellation condition.\n\n$E_7$ angularity and $D_4$ singularity collide at six intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=1,3,5$, and $D_4$ singularities collide at the three intersections $(t,u)=(\\alpha_4, \\beta_j)$, $j=1,3,5$. $E_7$ singularity and $A_1$ singularity collide at the four intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=7,8$. $D_4$ singularity and $A_1$ singularity collide at the two intersections $(t,u)=(\\alpha_4, \\beta_j)$, $j=7,8$.\n\nWe assume that ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the six intersections where $E_7$ angularity and $D_4$ singularity collide, ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the three intersection where $D_4$ singularities collide, $\\frac{1}{2}{\\bf 56}$ arises at the four intersections where $E_7$ singularity and $A_1$ singularity collide and ${\\bf 7}$ arises at the two intersections where $D_4$ singularity and $A_1$ singularity collide, then the net dimension of the matter representations arising at the intersections of the 7-branes is $$(56+7+1)\\times 6+(21+7+1)\\times 3+28\\times 4+7\\times 2=597.$$ Thus, this yields a consistent matter candidate on F-theory on the Jacobian Calabi\u2013Yau 3-fold (\\[2nd 3-fold in 4.4\\]). Having ${\\bf 56}\\oplus {\\bf 7}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the six intersections where $E_7$ angularity and $D_4$ singularity collide, and ${\\bf 7}\\oplus {\\bf 2}\\oplus {\\bf 1}$, instead of ${\\bf 7}$, at the two intersections where $D_4$ singularity and $A_1$ singularity collide, also yields another viable candidate matter spectrum. Furthermore, having ${\\bf 56}\\oplus {\\bf 7}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the six intersections where $E_7$ angularity and $D_4$ singularity collide, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 1}$, instead of $\\frac{1}{2}{\\bf 56}$, at the four intersections where $E_7$ singularity and $A_1$ singularity collide and ${\\bf 7}\\oplus {\\bf 1}$, instead of ${\\bf 7}$, at the two intersections where $D_4$ singularity and $A_1$ singularity collide, yields another viable candidate matter spectrum. In addition to these, having ${\\bf 56}\\oplus {\\bf 8}\\oplus {\\bf 1}\\oplus {\\bf 1}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the six intersections where $E_7$ angularity and $D_4$ singularity collide, ${\\bf 21}$, instead of ${\\bf 21}\\oplus {\\bf 7} \\oplus {\\bf 1}$, at the three intersections where two $D_4$ singularities collide, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$, instead of $\\frac{1}{2}{\\bf 56}$, at the four intersections where $E_7$ singularity and $A_1$ singularity collide and ${\\bf 7}\\oplus {\\bf 2}$, instead of ${\\bf 7}$, at the two intersections where $D_4$ singularity and $A_1$ singularity collide, yields another viable candidate matter spectrum.\n\nThe global structure of the gauge group forming in F-theory compactification on the Calabi\u2013Yau 3-fold (\\[2nd 3-fold in 4.4\\]) is $$E_7^2 \\times SO(7)^4 \\times SU(2)^2 /\\Z_2.$$\n\nWe then consider the case a triplet of parameters $\\beta$ coincide: $$\\beta_1=\\beta_2=\\beta_3.$$ In this situation, the equation of the Calabi\u2013Yau 3-fold (\\[enhanced Jacobian 3-fold in 4.1\\]) becomes $$\\label{3rd 3-fold in 4.4}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^3\\cdot \\Pi_{j=4}^8 (u-\\beta_j)\\cdot x$$ and the discriminant is given as follows $$\\Delta \\sim (t-\\alpha_1)^9(t-\\alpha_5)^9(t-\\alpha_4)^6\\cdot (u-\\beta_1)^9\\cdot \\Pi_{j=4}^8 (u-\\beta_j)^3.$$ Non-Abelian gauge group forming in F-theory compactification is: $$E_7^3 \\times SO(7) \\times SU(2)^5.$$ Therefore, we have $$133\\times 3+21+3\\times 5=435.$$ The anomaly cancellation condition requires that $$H=435+244=679.$$ Among the parameters of the complex structure deformation $\\beta_1, \\beta_4, \\beta_5, \\beta_6, \\beta_7, \\beta_8$, three can be sent to fixed values under an automorphism of $\\P^1$. Therefore, the number of the effective parameters of the complex structure deformation is three. From this, we obtain the number of the neutral hypermultiplets arising from the complex structure deformations as $$H^0=1+3=4.$$ 7-branes intersect in eighteen points at $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,4,5$, $j=1,4,5,6,7,8$. The net representation dimensions of matter arising from these intersections must be $$679-4=675$$ owing to the anomaly cancellation condition.\n\n$E_7$ angularities collide at the two intersections $(t,u)=(\\alpha_i, \\beta_1)$, $i=1,5$, and $D_4$ singularity and $E_7$ singularity collide at the intersection $(t,u)=(\\alpha_4, \\beta_1)$. $E_7$ singularity and $A_1$ singularity collide at the ten intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=4, \\cdots, 8$. $D_4$ singularity and $A_1$ singularity collide at the five intersections $(t,u)=(\\alpha_4, \\beta_j)$, $j=4, \\cdots ,8$.\n\nWe assume that ${\\bf 133}$ arises at the two intersections where two $E_7$ angularities collide, ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the intersection where $D_4$ and $E_7$ singularities collide, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$ arises at the ten intersections where $E_7$ singularity and $A_1$ singularity collide and ${\\bf 7}\\oplus {\\bf 2}$ arises at the five intersections where $D_4$ singularity and $A_1$ singularity collide, then the net dimension of the matter representations arising at the intersections of the 7-branes is $$133\\times 2+(56+7+1)+(28+2)\\times 10+(7+2)\\times 5=675.$$ Thus, the anomaly cancels and this yields a consistent matter candidate on F-theory on the Jacobian Calabi\u2013Yau 3-fold (\\[3rd 3-fold in 4.4\\]). Having matter representation ${\\bf 56}\\oplus {\\bf 8}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersection where $E_7$ angularity and $D_4$ singularity collide also yields another viable candidate matter spectrum.\n\nThe global structure of the gauge group forming in F-theory compactification on the Calabi\u2013Yau 3-fold (\\[3rd 3-fold in 4.4\\]) is $$E_7^3 \\times SO(7) \\times SU(2)^5 /\\Z_2.$$\n\nWe now consider the case a triplet and a pair of parameters $\\beta$ coincide: $$\\begin{aligned}\n\\beta_1=\\beta_2=\\beta_3 \\\\ \\nonumber\n\\beta_4=\\beta_8.\\end{aligned}$$ In this situation, the equation of the Jacobian Calabi\u2013Yau 3-fold (\\[enhanced Jacobian 3-fold in 4.1\\]) becomes: $$\\label{4th 3-fold in 4.4}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^3(u-\\beta_4)^2\\cdot \\Pi_{j=5}^7 (u-\\beta_j)\\cdot x.$$ The discriminant is given as follows $$\\Delta \\sim (t-\\alpha_1)^9(t-\\alpha_5)^9(t-\\alpha_4)^6\\cdot (u-\\beta_1)^9(u-\\beta_4)^6\\cdot \\Pi_{j=5}^7 (u-\\beta_j)^3.$$ Non-Abelian gauge group forming in F-theory compactification is: $$E_7^3 \\times SO(7)^2 \\times SU(2)^3.$$ Therefore, we have $$133\\times 3+21\\times 2+3\\times 3=450.$$ The anomaly cancellation condition requires that $$H=450+244=694.$$ Among the parameters of the complex structure deformation $\\beta_1, \\beta_4, \\beta_5, \\beta_6, \\beta_7$, three can be sent to fixed values under an automorphism of $\\P^1$. Therefore, the number of the effective parameters of the complex structure deformation is two. From this, we obtain the number of the neutral hypermultiplets arising from the complex structure deformations as $$H^0=1+2=3.$$ 7-branes intersect in fifteen points at $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,4,5$, $j=1,4,5,6,7$. The net representation dimensions of matter arising from these intersections must be $$694-3=691$$ owing to the anomaly cancellation condition.\n\n$E_7$ angularities collide at the two intersections $(t,u)=(\\alpha_i, \\beta_1)$, $i=1,5$, and $D_4$ singularity and $E_7$ singularity collide at the three intersection points $(t,u)=(\\alpha_4, \\beta_1), (\\alpha_i, \\beta_4)$, $i=1,5$. $D_4$ singularities collide at the intersection point $(t,u)=(\\alpha_4, \\beta_4)$. $E_7$ singularity and $A_1$ singularity collide at the six intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=5,6,7$. $D_4$ singularity and $A_1$ singularity collide at the three intersections $(t,u)=(\\alpha_4, \\beta_j)$, $j=5,6,7$.\n\nWe assume that ${\\bf 133}$ arises at the two intersections where two $E_7$ angularities collide, ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the three intersection points where $D_4$ and $E_7$ singularities collide, ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the intersection point where two $D_4$ singularities collide, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$ arises at the six intersections where $E_7$ singularity and $A_1$ singularity collide and ${\\bf 7}\\oplus {\\bf 1}$ arises at the three intersections where $D_4$ singularity and $A_1$ singularity collide, then the net dimension of the matter representations arising at the intersections of the 7-branes is $$133\\times 2+(56+7+1)\\times 3+(21+7+1)+(28+2)\\times 6+(7+1)\\times 3=691.$$ Thus, the anomaly cancels and this yields a consistent matter candidate on F-theory on the Jacobian Calabi\u2013Yau 3-fold (\\[4th 3-fold in 4.4\\]). Having matter representation ${\\bf 56}\\oplus {\\bf 7}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, and matter representation ${\\bf 7}\\oplus {\\bf 2}$, instead of ${\\bf 7}\\oplus {\\bf 1}$, at the intersections where $D_4$ and $A_1$ singularities collide also yields another viable candidate matter spectrum. Furthermore, having matter representation ${\\bf 133}\\oplus {\\bf 1}$, instead of ${\\bf 133}$, at the intersections where two $E_7$ angularities collide, matter representation ${\\bf 56}\\oplus {\\bf 8}\\oplus {\\bf 1}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, matter representation ${\\bf 21}$, instead of ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersection where two $D_4$ singularities collide, and matter representation ${\\bf 7}\\oplus {\\bf 2}$, instead of ${\\bf 7}\\oplus {\\bf 1}$, at the intersections where $D_4$ and $A_1$ singularities collide yields another viable candidate matter spectrum.\n\nThe global structure of the gauge group forming in F-theory compactification on the Calabi\u2013Yau 3-fold (\\[4th 3-fold in 4.4\\]) is $$E_7^3 \\times SO(7)^2 \\times SU(2)^3 /\\Z_2.$$\n\nWe consider the case a triplet and two pairs of parameters $\\beta$ coincide: $$\\begin{aligned}\n\\beta_1=\\beta_2=\\beta_3 \\\\ \\nonumber\n\\beta_4=\\beta_8 \\\\ \\nonumber\n\\beta_5=\\beta_6.\\end{aligned}$$ In this situation, the equation of the Jacobian Calabi\u2013Yau 3-fold (\\[enhanced Jacobian 3-fold in 4.1\\]) becomes: $$\\label{5th 3-fold in 4.4}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^3(u-\\beta_4)^2(u-\\beta_5)^2(u-\\beta_7)\\cdot x.$$ The discriminant is given as follows $$\\Delta \\sim (t-\\alpha_1)^9(t-\\alpha_5)^9(t-\\alpha_4)^6\\cdot (u-\\beta_1)^9(u-\\beta_4)^6(u-\\beta_5)^6(u-\\beta_7)^3.$$ Non-Abelian gauge group forming in F-theory compactification is: $$E_7^3 \\times SO(7)^3 \\times SU(2).$$ Therefore, we have $$133\\times 3+21\\times 3+3=465.$$ The anomaly cancellation condition requires that $$H=465+244=709.$$ Among the parameters of the complex structure deformation $\\beta_1, \\beta_4, \\beta_5, \\beta_7$, three can be sent to fixed values under an automorphism of $\\P^1$. Therefore, the number of the effective parameters of the complex structure deformation is one. From this, we obtain the number of the neutral hypermultiplets as $$H^0=1+1=2.$$ 7-branes intersect in twelve points at $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,4,5$, $j=1,4,5,7$. The net representation dimensions of matter arising from these intersections must be $$709-2=707$$ owing to the anomaly cancellation condition.\n\n$E_7$ angularities collide at the two intersections $(t,u)=(\\alpha_i, \\beta_1)$, $i=1,5$, and $E_7$ singularity and $D_4$ singularity collide at the five intersection points $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5, j=4,5$, $(\\alpha_4, \\beta_1)$. $D_4$ singularities collide at the two intersection points $(t,u)=(\\alpha_4, \\beta_j)$, $j=4,5$. $E_7$ singularity and $A_1$ singularity collide at the two intersections $(t,u)=(\\alpha_i, \\beta_7)$, $i=1,5$. $D_4$ singularity and $A_1$ singularity collide at the intersection $(t,u)=(\\alpha_4, \\beta_7)$.\n\nWe assume that ${\\bf 133}$ arises at the two intersections where two $E_7$ angularities collide, ${\\bf 56}\\oplus {\\bf 7}$ arises at the five intersection points where $E_7$ and $D_4$ singularities collide, ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the two intersection points where two $D_4$ singularities collide, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$ arises at the two intersections where $E_7$ singularity and $A_1$ singularity collide, and ${\\bf 7}\\oplus {\\bf 1}$ arises at the intersection where $D_4$ singularity and $A_1$ singularity collide, then the net dimension of the matter representations arising at the intersections of the 7-branes is $$133\\times 2+(56+7)\\times 5+(21+7+1)\\times 2+(28+2)\\times 2+(7+1)=707.$$ Thus, the anomaly cancels and this yields a consistent matter candidate on F-theory on the Jacobian Calabi\u2013Yau 3-fold (\\[5th 3-fold in 4.4\\]). Having matter representation ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, instead of ${\\bf 56}\\oplus {\\bf 7}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, matter representation $\\frac{1}{2}{\\bf 56}$, instead of $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$, at the intersections where $E_7$ and $A_1$ singularities collide, and matter representation ${\\bf 7}$, instead of ${\\bf 7}\\oplus {\\bf 1}$, at the intersection where $D_4$ and $A_1$ singularities collide also yields another viable candidate matter spectrum. Furthermore, having matter representation ${\\bf 133}\\oplus {\\bf 1}$, instead of ${\\bf 133}$, at the intersections where two $E_7$ angularities collide, and matter representation $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 1}$, instead of $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 2}$, at the intersections where $E_7$ and $A_1$ singularities collide yields another viable candidate matter spectrum. Additionally, having matter representation ${\\bf 56}\\oplus {\\bf 8}\\oplus {\\bf 1}\\oplus {\\bf 1}$, instead of ${\\bf 56}\\oplus {\\bf 7}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, matter representation ${\\bf 21}$, instead of ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where two $D_4$ singularities collide, and matter representation ${\\bf 7}\\oplus {\\bf 2}$, instead of ${\\bf 7}\\oplus {\\bf 1}$, at the intersection where $D_4$ and $A_1$ singularities collide also yields another viable candidate matter spectrum.\n\nThe global structure of the gauge group forming in F-theory compactification on the Calabi\u2013Yau 3-fold (\\[5th 3-fold in 4.4\\]) is $$E_7^3 \\times SO(7)^3 \\times SU(2) /\\Z_2.$$\n\nWe finally discuss the case two triplets and a pair of parameters $\\beta$ coincide: $$\\begin{aligned}\n\\beta_1=\\beta_2=\\beta_3 \\\\ \\nonumber\n\\beta_4=\\beta_8 \\\\ \\nonumber\n\\beta_5=\\beta_6=\\beta_7.\\end{aligned}$$ In this situation, the equation of the Jacobian Calabi\u2013Yau 3-fold (\\[enhanced Jacobian 3-fold in 4.1\\]) becomes: $$\\label{6th 3-fold in 4.4}\n\\tau^2=\\frac{1}{4}x^3-(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^3(u-\\beta_5)^3(u-\\beta_4)^2\\cdot x.$$ The discriminant is given as follows $$\\Delta \\sim (t-\\alpha_1)^9(t-\\alpha_5)^9(t-\\alpha_4)^6\\cdot (u-\\beta_1)^9(u-\\beta_5)^9(u-\\beta_4)^6.$$ Non-Abelian gauge group forming in F-theory compactification is: $$E_7^4 \\times SO(7)^2.$$ Thus, we have $$133\\times 4+21\\times 2=574.$$ The anomaly cancellation condition requires that $$H=574+244=818.$$ The parameters of the complex structure deformation, $\\beta_1, \\beta_4, \\beta_5$, can be sent to fixed values under an automorphism of $\\P^1$. Therefore, the number of the effective parameters of the complex structure deformation is zero, and the complex structure is fixed for this situation. From this, we obtain the number of the neutral hypermultiplets as $$H^0=1+0=1.$$ 7-branes intersect in nine points at $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,4,5$, $j=1,4,5$. The net representation dimensions of matter arising from these intersections must be $$818-1=817$$ owing to the anomaly cancellation condition.\n\n$E_7$ angularities collide at the four intersections $(t,u)=(\\alpha_i, \\beta_j)$, $i=1,5$, $j=1,5$, and $E_7$ singularity and $D_4$ singularity collide at the four intersection points $(t,u)=(\\alpha_4, \\beta_j)$, $j=1,5$, $(\\alpha_i, \\beta_4)$, $i=1,5$. $D_4$ singularities collide at the intersection point $(t,u)=(\\alpha_4, \\beta_4)$.\n\nWe assume that ${\\bf 133}$ arises at the four intersections where two $E_7$ angularities collide, ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the four intersection points where $E_7$ and $D_4$ singularities collide, and ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$ arises at the intersection point where two $D_4$ singularities collide, then the net dimension of the matter representations arising at the intersections of the 7-branes is $$133\\times 4+(56+7+1)\\times 4+(21+7+1)=817.$$ Thus, the anomaly cancels and this yields a consistent matter candidate on F-theory on the Jacobian Calabi\u2013Yau 3-fold (\\[6th 3-fold in 4.4\\]). Having matter representation ${\\bf 56}\\oplus {\\bf 8}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ angularity and $D_4$ singularity collide, or matter representation ${\\bf 21}\\oplus {\\bf 8}$, instead of ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersection where two $D_4$ singularities collide, also yield other viable candidate matter spectra. Additionally, having matter representation ${\\bf 133}\\oplus {\\bf 1}$, instead of ${\\bf 133}$, at the intersections where two $E_7$ angularities collide, and matter representation ${\\bf 56}\\oplus {\\bf 7}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ and $D_4$ singularities collide, also yields another viable candidate matter spectrum. Furthermore, having matter representation ${\\bf 133}\\oplus {\\bf 1}$, instead of ${\\bf 133}$, at the intersections where two $E_7$ angularities collide, matter representation ${\\bf 56}\\oplus {\\bf 8}\\oplus {\\bf 1}$, instead of ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersections where $E_7$ and $D_4$ singularities collide, and matter representation ${\\bf 21}$, instead of ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, at the intersection where two $D_4$ singularities collide, yields another viable candidate matter spectrum.\n\nThe global structure of the gauge group forming in F-theory compactification on the Calabi\u2013Yau 3-fold (\\[6th 3-fold in 4.4\\]) is $$E_7^4 \\times SO(7)^2 /\\Z_2.$$\n\nWe have deduced candidate matter spectra on F-theory on the constructed elliptically fibered Calabi\u2013Yau 3-folds. We have observed that it is natural to expect that ${\\bf 133}$ (or ${\\bf 133}\\oplus {\\bf 1}$) arises at the collision of two $E_7$ singularities [^19], to cancel the anomaly[^20]. Under this assumption, we observed that matter representation arising at the collision of $E_7$ and $A_1$ singularities should include the $\\frac{1}{2}$-hypermultiplet $\\frac{1}{2}{\\bf 56}$ of $E_7$. If matter representation at this intersection includes ${\\bf 56}$, the net dimension of matter representations exceeds the number required by the anomaly cancellation condition by a large amount. There appear a few possibilities for matter representation at the collision of $E_7$ and $D_4$ singularities, such as matter representations ${\\bf 56}\\oplus {\\bf 7}\\oplus {\\bf 1}$ or ${\\bf 56}\\oplus {\\bf 8}$, and whether to include ${\\bf 1}$. We observed a few possibilities for matter representation at the collision of two $D_4$ singularities, such as ${\\bf 21}\\oplus {\\bf 7}\\oplus {\\bf 1}$, ${\\bf 21}\\oplus {\\bf 8}$, or ${\\bf 21}$. There are also a few possibilities for matter at the collision of $E_7$ and $A_1$ singularities, such as $\\frac{1}{2}{\\bf 56}$, $\\frac{1}{2}{\\bf 56}\\oplus {\\bf 1}$, or $\\frac{1}{2}{\\bf 56} \\oplus {\\bf 2}$, and similarly for the collision of $D_4$ and $A_1$ singularities.\n\nConclusions {#sec5}\n===========\n\nIn this study, we have investigated the points in the eight-dimensional moduli of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra, at which the ranks of the non-Abelian gauge groups on the F-theory side are enhanced to 18. The gauge groups at these points do not allow for the perturbative interpretation on the heterotic side. We demonstrated in this study that these theories can be seen as deformations of the stable degenerations owing to an effect of coincident 7-branes. This effect corresponds to the insertion of 5-branes from the heterotic viewpoint. We also discussed the application to $SO(32)$ heterotic strings.\n\nK3 surfaces on the F-theory side of the moduli become extremal, when the non-Abelian gauge groups are enhanced to rank 18. We studied the Weierstrass equations of the extremal K3 surfaces that appear on the F-theory side of the eight-dimensional moduli of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$. The points in the moduli at which the ranks of the non-Abelian gauge groups are enhanced to 17 on the F-theory side also do not allow for the perturbative interpretations of the gauge groups on the heterotic side. It can be interesting to study these points in the moduli, and this can be a direction of future study.\n\nWe have also built elliptically fibered Calabi\u2013Yau 3-folds, by fibering an elliptic K3 surface, which belongs to the F-theory side of the eight-dimensional moduli of non-geometric heterotic strings with unbroken $\\mathfrak{e}_7\\mathfrak{e}_7$ algebra, over $\\P^1$. We analyzed six-dimensional F-theory compactifications on the built elliptic Calabi\u2013Yau 3-folds. When we tune the parameters for the defining equations of these elliptic Calabi\u2013Yau 3-folds, highly enhanced gauge groups form on the 7-branes. Eight-dimensional F-theory compactified on the extremal K3 fibers $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ of these specific tuned Calabi\u2013Yau spaces has non-geometric heterotic duals. Determining whether this duality extends to six-dimensional theories, namely whether F-theory compactifications on the total Calabi\u2013Yau 3-folds have dual non-geometric heterotic strings, is a likely direction of future study.\n\nWe have also deduced viable candidate matter spectra on F-theory on the constructed elliptically fibered Calabi\u2013Yau 3-folds, for the case when K3 fibers are most enhanced (\\[coeff in 4.1\\]). There are certain ambiguities such as whether matter arising at intersections of 7-branes includes ${\\bf 1}$, or ${\\bf 2}$, or does not include these. There are also ambiguities of whether ${\\bf 7}\\oplus {\\bf 1}$ or ${\\bf 8}$ arises where $D_4$ singularities and $D_4$ and $A_1$ singularities collide. There is also an ambiguity of whether matter arising at the collision of $D_4$ singularities includes ${\\bf 7}\\oplus {\\bf 1}$, or ${\\bf 8}$, or does not include these. Except for these ambiguities, the possibility of candidate matter spectra appears unique. We have observed that either ${\\bf 133}$ or ${\\bf 133}\\oplus {\\bf 1}$ arises where two $E_7$ singularities collide. We have also observed that matter arising where $E_7$ and $A_1$ singularities collide should include the $\\frac{1}{2}$-hypermultiplet, $\\frac{1}{2}{\\bf 56}$ of $E_7$, to cancel the anomaly. Confirming the actual matter spectra by analyzing the resolution of the singularities of the Jacobian Calabi\u2013Yau 3-folds is also a likely direction of future study.\n\nAcknowledgments {#acknowledgments .unnumbered}\n===============\n\nWe would like to thank Shun\u2019ya Mizoguchi for discussions. This work is partially supported by Grant-in-Aid for Scientific Research [\\#]{}16K05337 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.\n\nElliptic fibrations of attractive K3 $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ {#secA}\n============================================================================\n\n[@Nish] classified the types of the elliptic fibrations of the attractive K3 surface with the discriminant four, $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$, and computed the Mordell\u2013Weil groups of the fibrations. We present in Table \\[tablefibrations202 in A\\] the types of the elliptic fibrations and the Mordell\u2013Weil groups of the attractive K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ determined in [@Nish].\n\n  --------------------------------------------------------------------------------------------\n                          $                          type of singularity        MW group\n                  \\begin{array}{c}                                        \n           \\mbox{Elliptic fibrations of}\\\\                                \n   \\mbox{$S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$}                         \n                     \\end{array}                                          \n                          $                                               \n  ------------------------------------------------- --------------------- --------------------\n                        No.1                            $E^2_8 A^2_1$              0\n\n                        No.2                            $E_8 D_{10}$               0\n\n                        No.3                           $D_{16} A^2_1$            $\\Z_2$\n\n                        No.4                             $E^2_7 D_4$             $\\Z_2$\n\n                        No.5                          $E_7 D_{10} A_1$           $\\Z_2$\n\n                        No.6                            $A_{17} A_1$             $\\Z_3$\n\n                        No.7                              $D_{18}$                 0\n\n                        No.8                            $D_{12} D_6$             $\\Z_2$\n\n                        No.9                            $D^2_8 A^2_1$      $\\Z_2 \\oplus \\Z_2$\n\n                        No.10                           $A_{15} A_3$             $\\Z_4$\n\n                        No.11                           $E_6 A_{11}$        $\\Z\\oplus \\Z_3$\n\n                        No.12                              $D^3_6$          $\\Z_2\\oplus\\Z_2$\n\n                        No.13                              $A^2_9$               $\\Z_5$\n  --------------------------------------------------------------------------------------------\n\n  : \\[tablefibrations202 in A\\]List of the singularity types of the elliptic fibrations of K3 surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$, and the Mordell\u2013Weil groups of the fibrations.\n\nTypes of the singular fibers of extremal rational elliptic surfaces {#secB}\n===================================================================\n\nThe types of the singular fibers of the extremal rational elliptic surfaces [@MP] are presented in Table \\[tablelisttypes extremalRES in B\\]. The complex structures of the extremal rational elliptic surfaces are uniquely specified by the types of the singular fibers, except rational elliptic surfaces with two type $I^*_0$ fibers, $X_{[0^*, \\hspace{1mm} 0^*]}(j)$ [@MP].\n\n  ----------------------------------------------------------------------------------\n                  $                              $                       $\n           \\begin{array}{c}              \\begin{array}{c}        \\begin{array}{c}\n      \\mbox{Extremal rational}\\\\         \\mbox{Type of}\\\\        \\mbox{Type of}\\\\\n       \\mbox{elliptic surface}        \\mbox{singular fiber}     \\mbox{singularity} \n             \\end{array}                    \\end{array}             \\end{array}\n                  $                              $                       $\n  ---------------------------------- ------------------------- ---------------------\n    $X_{[II, \\hspace{1mm} II^*]}$          $II$, $II^*$                $E_8$\n\n   $X_{[III, \\hspace{1mm} III^*]}$        $III$, $III^*$             $E_7A_1$\n\n    $X_{[IV, \\hspace{1mm} IV^*]}$          $IV$, $IV^*$              $E_6A_2$\n\n   $X_{[0^*, \\hspace{1mm} 0^*]}(j)$      $I^*_0$, $I^*_0$             $D_4^2$\n\n   $X_{[II^*, \\hspace{1mm} 1, 1]}$      $II^*$ $I_1$ $I_1$             $E_8$\n\n   $X_{[III^*, \\hspace{1mm} 2, 1]}$     $III^*$ $I_2$ $I_1$          $E_7A_1$\n\n   $X_{[IV^*, \\hspace{1mm} 3, 1]}$      $IV^*$ $I_3$ $I_1$           $E_6A_2$\n\n    $X_{[4^*, \\hspace{1mm} 1, 1]}$      $I_4^*$ $I_1$ $I_1$            $D_8$\n\n    $X_{[2^*, \\hspace{1mm} 2, 2]}$      $I^*_2$ $I_2$ $I_2$         $D_6A_1^2$\n\n    $X_{[1^*, \\hspace{1mm} 4, 1]}$      $I_1^*$ $I_4$ $I_1$          $D_5A_3$\n\n          $X_{[9, 1, 1, 1]}$          $I_9$ $I_1$ $I_1$ $I_1$          $A_8$\n\n          $X_{[8, 2, 1, 1]}$          $I_8$ $I_2$ $I_1$ $I_1$        $A_7A_1$\n\n          $X_{[6, 3, 2, 1]}$          $I_6$ $I_3$ $I_2$ $I_1$       $A_5A_2A_1$\n\n          $X_{[5, 5, 1, 1]}$          $I_5$ $I_5$ $I_1$ $I_1$         $A_4^2$\n\n          $X_{[4, 4, 2, 2]}$          $I_4$ $I_4$ $I_2$ $I_2$      $A_3^2A_1^2$\n\n          $X_{[3, 3, 3, 3]}$          $I_3$ $I_3$ $I_3$ $I_3$         $A_2^4$\n  ----------------------------------------------------------------------------------\n\n  : \\[tablelisttypes extremalRES in B\\]List of the types of the singular fibers of extremal rational elliptic surfaces.\n\n[99]{} C.\u00a0Vafa, \u201cEvidence for F-theory\u201d, [*Nucl. 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F.\u00a0Apruzzi, J.\u00a0J.\u00a0Heckman, D.\u00a0R.\u00a0Morrison and L.\u00a0Tizzano, \u201c4D Gauge Theories with Conformal Matter\u201d, [*JHEP*]{} [**09**]{} (2018) 088 \\[arXiv:1803.00582 \\[hep-th\\]\\].\n\n[^1]: Recent progress of heterotic strings can be found, for example, in [@BHW0504; @BHW0507; @KM2009; @ACGLP1307; @AGS1402; @OS1402; @AT1405; @OHS1509; @COO1605; @AOMSS1806; @COOS1810].\n\n[^2]: Stable degenerations in F-theory/heterotic duality have been studied recently, for example, in [@AHK; @BKW; @BKL; @CGKPS; @MizTan; @KRES].\n\n[^3]: The authors of [@MMS] discussed connections of K3 surfaces with lattice polarizations, non-geometric heterotic strings, and $O^+(\\Lambda^{2,2})$-modular forms.\n\n[^4]: The authors of [@HMW] discussed non-geometric type II theories.\n\n[^5]: Recent discussions of F-theory compactifications on elliptic Calabi\u2013Yau 3-folds can be found, e.g., in [@GM; @KT0906; @KMT0911; @KMT1008; @MTmatter; @GM2; @BG1112; @MT1201; @MT1204; @T1205; @JT1605; @MPT1610; @MMP1711; @HT1805; @LLW1810; @Kimura1810]. The authors of [@BM; @AGW1612; @GW1804] discussed F-theory on Calabi\u2013Yau 3-folds with terminal singularities.\n\n[^6]: Recent studies of F-theory compactifications on genus-one fibered spaces lacking a global section can be found in, for example, [@BM; @MTsection; @AGGK; @KMOPR; @GGK; @MPTW; @MPTW2; @BGKintfiber; @CDKPP; @LMTW; @K; @K2; @KCY4; @CGP; @Kdisc; @Kimura1801; @AGGO1801; @Kimura1806; @TasilectWeigand; @CLLO; @TasilectCL; @HT].\n\n[^7]: [@Cas] discussed the Jacobians of elliptic curves.\n\n[^8]: We refer to complex K3 surfaces with the highest Picard number 20 as attractive K3 surfaces, following the convention for the term used in [@M].\n\n[^9]: Genus-one fibered K3 surfaces in general admit both genus-one fibrations without a section, as well as elliptic fibrations with a section. However, as shown in [@Keum], the attractive K3 surface with discriminant four, $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$, only admits elliptic fibrations with a global section. The authors of [@KimuraMizoguchi] discussed F-theory compactification on the surface $S_{[2 \\hspace{1mm} 0 \\hspace{1mm} 2]}$, in relation to the appearances of the $U(1)$ factor.\n\n[^10]: This point is discussed in [@BKW].\n\n[^11]: The singularity types of extremal K3 surfaces can also be enhanced to $E_8D_{7}$, as discussed in [@CMSE7].\n\n[^12]: The elliptic fibrations and the Weierstrass equations of the attractive K3 surface $S_{[4 \\hspace{1mm} 0 \\hspace{1mm} 2]}$ were obtained in [@BLe].\n\n[^13]: In [@Nak; @DG; @G] the elliptic fibrations of 3-folds were discussed.\n\n[^14]: Thus, a discrete $\\Z_2$ symmetry [@MTsection] arises in six-dimensional F-theory compactifications on the genus-one fibered Calabi\u2013Yau 3-folds (\\[doublecover rewritten in 4.1\\]).\n\n[^15]: As we stated previously in Section \\[sssec3.2.2\\], $\\alpha_1, \\alpha_4, \\alpha_5$ in equation (\\[K3 fiber special in 4.1\\]) are superfluous parameters, and they can be mapped to $0, 1, \\infty$ under certain automorphism of the base $\\P^1$.\n\n[^16]: Using a coordinate transformation, equation (\\[enhanced Jacobian K3 fiber in 4.1\\]) can be replaced with $$\\tau^2=x^3+4(t-\\alpha_1)^3(t-\\alpha_4)^2(t-\\alpha_5)^3\\, x.$$ As stated previously, we can send $\\alpha_1, \\alpha_4, \\alpha_5$ to $0,1,\\infty$, and this yields (\\[extremal 202 in 3.2.2\\]).\n\n[^17]: In a special situation in which there are four pairs of identical $\\beta$\u2019s, e.g. $\\beta_1=\\beta_5$, $\\beta_2=\\beta_6$, $\\beta_3=\\beta_7$, $\\beta_4=\\beta_8$, the polynomial splits into three linear factors: $$x\\, \\big(x-\\Pi_{j=1}^4 (u-\\beta_j) \\big) \\, \\big(x+\\Pi_{j=1}^4 (u-\\beta_j). \\big)$$ In this special situation, type $I_0^*$ fiber over the component $A_i$ becomes split, and the gauge group that forms on the 7-branes wrapped on the component $A_i$ is enhanced to $SO(8)$.\n\n[^18]: The appearance of $\\frac{1}{2}$-hypermultiplets of ${\\bf 56}$ of $E_7$ in F-theory compactification was discussed in [@MV1; @MV2; @BIKMSV]. The base of elliptically fibered Jacobian Calabi\u2013Yau 3-folds (\\[1st 3-fold in 4.4\\]), $\\P^1\\times\\P^1$, is isomorphic to Hirzebruch surface $\\mathbb{F}_0$. Weierstrass coefficient $f$ of the Weierstrass equation $y^2=x^3+f\\, x+g$ of Jacobian Calabi\u2013Yau 3-fold (\\[1st 3-fold in 4.4\\]) is given by $$f=(t-\\alpha_1)^3(t-\\alpha_5)^3(t-\\alpha_4)^2\\cdot (u-\\beta_1)^2 \\cdot \\Pi^8_{j=3} (u-\\beta_j).$$ Candidate $\\frac{1}{2}$-hypermultiplets $\\frac{1}{2}{\\bf 56}$ localized at the twelve intersections have an interpretation as localized at the intersections of $E_7$ loci $t=\\alpha_{1,5}$ and the zeroes of $\\Pi^8_{j=3} (u-\\beta_j)$ [@BIKMSV].\n\n[^19]: The authors of [@AHMT1803] discussed the collision of two $E_7$ singularities in the context of four-dimensional conformal matter in F-theory.\n\n[^20]: A similar observation was made for the collision of two $E_6$ singularities for elliptically fibered Calabi\u2013Yau 3-folds of \u201cFermat-types\u201d in [@Kimura1810], where it was argued that ${\\bf 78}\\oplus {\\bf 1}$ is expected to arise at this intersection.\n"], ["---\nabstract: 'When dealing with zeta-function regularized functional determinants of matrix valued differential operators, an additional term, overlooked until now and due to the multiplicative anomaly, may arise. The presence and physical relevance of this term is discussed in the case of a charged bosonic field at finite charge density and other possible applications are mentioned.'\naddress: |\n    Theoretical Physics Group, Imperial College,\\\n    Prince Consort Road, London SW7 2BZ, United Kingdom\nauthor:\n- 'Antonio Filippi [^1]'\ntitle: Multiplicative anomaly and finite charge density\n---\n\nINTRODUCTION\n============\n\nIn field theory we often have to deal with functional determinants of differential operators. These, as formal products of infinite eigenvalues, are divergent objects (UV divergence) and a regularization scheme is therefore necessary. One of the most successful and powerful ones is the zeta-function regularization method [@raysin; @dowcri; @haw; @bcvz; @libro]. It permits us to give a meaning to the ill defined quantity $\\ln\\det A $, where $A$ is a second order elliptic differential operator, through the zeta function $\\zeta(s|A)=\\mbox{Tr}\\, A^{-s}$, which is well defined for a sufficiently large real part of $s$ and can be analytically continued to a function meromorphic in all the plane and analytic at $s=0$. As such its derivative to respect to $s$ at zero is well defined and the logarithm of the zeta-function regularized functional determinant will then be defined by $$\\begin{aligned}\n\\ln\\det\\frac{A}{M^2} =-\\zeta'(0|A)-\\zeta(0|A)\\ln M^2 ,\\end{aligned}$$ where $M^2$ is a renormalization scale mass.\n\nSometimes, however, the differential operator takes a matrix form in the field space, as is the case with the two real components $\\phi_i$ of a complex scalar field. In this case we end up evaluating a quantity of the form $\\ln \\det (A B)$, with $A$ and $B$ two commuting pseudo-differential operators. The fact is that it is not always true that the equality $\\ln \\det\n(AB)=\\ln \\det(A)+\\ln \\det (B)$ holds. On the contrary, an additional term $a(A,B)$, called the multiplicative anomaly [@wod; @konvis; @kas], may be present on the right hand-side and eventually have physical relevance [@evz; @efvz]. In this work I will introduce this quantity, compute it and analyse its physical relevance in the case of a charged scalar field at finite temperature and charge density, as well as present other possible physical systems in which it could play a role.\n\nThis work has been developed in collaboration with E. Elizalde, in Barcelona (Spain), and L. Vanzo and S. Zerbini, in Trento (Italy). My thank goes also to R. Rivers and T. Evans for stimulating discussions.\n\nTHE BOSE GAS AT FINITE CHARGE DENSITY\n=====================================\n\nThe relativistic complex scalar field at finite temperature in the presence of a net charge density has given rise to a certain interest during recent years [@kap; @habwel; @bbd; @bd; @kirtom; @chemical].\n\nIn our recent paper [@efvz] it is shown that, in a coherent regularized approach, the multiplicative anomaly, overlooked until now, could play a role in this system. I will try here to outline the results avoiding the mathematical machinery. For clarity, I will mainly restrict myself to four space-time dimensions although the system has been studied in generic $D$ dimensions.\n\nThe relevant quantity for my proposes is the grand canonical partition function, which, for this system, is [@kap; @habwel; @chemical] $$\\begin{aligned}\nZ_{\\beta}(\\mu)=\\mbox{Tr}\\,   e^{-\\beta (H-\\mu Q)} =\n\\int_{\\phi(\\tau)=\\phi(\\tau+\\beta)}[d \\phi_i]\ne^{-\\frac{1}{2}\\int_0^\\beta d\\tau \\int d^3x \\phi_iA_{ij}\\phi_j}\n\\:,\\label{dddd}\\end{aligned}$$ where $H$ is the Hamiltonian of the system, $Q$ the charge and $\\beta$ the inverse of the temperature. $A_{ij}$ is the elliptic, non-self-adjoint, matrix valued, differential operator\n\n(\n\n[cc]{} [-[\\_\\^[2]{}]{}-[\\^[2]{}]{}+m\\^[2]{}-e\\^2\\^[2]{}]{} & [-2ie]{}\\\n[2ie]{} & [-[\\_\\^[2]{}]{}-[\\^[2]{}]{}+m\\^[2]{}-e\\^2\\^[2]{}]{}\n\n)\n\n.\n\nIn this case computing the partition function requires taking both an algebraic determinant and a functional one. The standard procedure consists in taking the algebraic one first [@kap; @bbd; @bd; @kirtom]. Stimulated by a recent criticism [@dowker], we showed the validity of this procedure [@rdowker].\n\nNow, we have two possible factorizations for this algebraic determinant: Z\\_()=- =-=-\\[o2\\], where: K\\_=-[\\^[2]{}]{}+m\\^[2]{}+(i[\\_]{}ie)\\^2 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 L\\_=-[\\_\\^[2]{}]{}+(e)\\^2 .\n\nI will avoid here the standard steps that lead to the computation of the logarithm of the partition function, as the reader will find them in greater detail in ref. [@efvz].\n\nAssuming, as in the precedent literature, the validity of the identity $\\ln \\det(AB)=\\ln \\det(A)+\\ln \\det (B)$ and disregarding the anomaly, we obtain, for the $K_{\\pm}$ factorization, $$\\begin{aligned}\n\\!\\!\\!\\!\\!\\ln Z_{\\be}(K_+,K_-)\\!\\!\\!\\!&=&\\!\\!\\!\\!\\frac{\\be V }{32\\pi^2}\\aq\nm^4 (\\ln {\\mbox{$\\frac{m^2}{M^2}$}}-3/2) \\cq \\nn \\\\\n&-&\\!\\!\\!\\! V\\!\\! \\int\\!\\frac{d^{3}k}{(2\\pi)^{3}}\\!\n\\ln(1\\!-\\!e^{-\\be(\\sqrt{k^{2}+m^{2}}-e\\mu)})\\! -\\! V\\!\\! \\int\\!\\frac{d^{3}k}{(2\\pi)^{3}}\\! \\ln(1\\!-\\!e^{-\\be(\\sqrt{k^{2}+m^{2}}+e\\mu)}) ,\n\\label{441}\n\\enq\nwhere the expected contributions for vacuum, particles and\nantiparticles are manifest. {}From now on I will\nrepresent the thermal contributions as $S(\\be,\\mu)$.\n\nSimilar manipulations can be done for the other factorization\n$L_{\\pm}$. In this case though,  the chemical potential does not\nappear in the sum over Matsubara frequencies, but remains with the\nmomentum integral and therefore the term linear in $\\beta$ will be chemical\npotential dependent:\n\\beq\n\\ln Z_{\\be}(L_+,L_-)&=&\\frac{\\be V }{32\\pi^2}\\aq\nm^4 (\\ln {\\mbox{$\\frac{m^2}{M^2}$}}-3/2) \\cq+ \\frac{\\be V }{8\\pi^2} \\at\n\\frac{e^4\\mu^4}{3}-e^2\\mu^2 m^2 \\ct + S(\\be,\\mu)\\: .\n\\label{44}\n\\enq\nIn this system the importance of the multiplicative anomaly is\ntherefore manifest. Despite having $ \\ln (K_{-}K_{+}) =\\ln (L_{-}L_{+})$, \nthese two options give two different results for a zeta-function regularized partition function {\\em if the multiplicative\nanomaly is disregarded}.\n\n\\section{THE MULTIPLICATIVE ANOMALY}\n\nThe multiplicative anomaly \\cite{wod,konvis,kas} is defined as\n\\begin{eqnarray} \na_D(A,B)=\\ln \\det (AB)-\\ln \\det(A)-\\ln \\det (B) \\end{aligned}$$ where the determinants of the two elliptic operators are defined by means of the zeta-function method. I recall that $D$ are the space-time dimensions. In principle, it could be computed directly as difference of the involved quantities. In reality, actual calculations are very complicated even for simpler operators. We can fortunately resort to Wodzicki\u2019s results for a remarkably neat recipe.\n\nFor any classical pseudo-differential operator $A$ there exists a complete symbol $A(x,k)=e^{-ikx}Ae^{ikx}$. This admits an asymptotic expansion for $|k| \\to \\infty$, $$\\begin{aligned}\nA(x,k)\\sim\\sum_{j=0} A_{a-j}(x,k) \\:,\\label{sy1}\\end{aligned}$$ where the coefficients (their number is infinite) fulfil the homogeneity property $ A_{a-j}(x,tk)= t^{a-j}A_{a-j}(x,k)$, for $t>0$. The number $a$ is called the order of $A$. Now, Wodzicki [@wod] proved that for two invertible, self-adjoint, elliptic, commuting, pseudodifferential operators on a smooth compact manifold without boundaries $M_D$: $$\\begin{aligned}\na(A,B)=\\frac{\\mbox{res}\\left[\n(\\ln(A^bB^{-a}))^2 \\right]}{2ab(a+b)}=a(B,A)\n\\:,\\label{wod3}\\end{aligned}$$ where $a >0$ and $ b> 0$ are the orders of $A$ and $B$, respectively. Here the quantity $\\mbox{res}(A)$ is the Wodzicki non-commutative residue. It can be computed easily using the homogeneous component $A_{-D}(x,k)$ of order $-D$ of the complete symbol, $$\\begin{aligned}\n\\mbox{res}(A)=\\int_{M_D}\\frac{dx}{(2\\pi)^{D}}\\int_{|k|=1}A_{-D}(x,k)dk\n\\:.\\label{wod2}\\end{aligned}$$\n\nAll this can be applied to our operators. As an example: A(x,k)\\_[K\\_]{}=k\\^2+m\\^2-e\\^2\\^2+ i2ek\\_ -k\\^2+m\\^2-e\\^2\\^2- i2ek\\_\\^2 .\\[903\\] Simply expanding (\\[903\\]) and performing the above integration (\\[wod2\\]) we obtain the non-commutative residue. Remembering (\\[wod3\\]) and that the order of our operators is 2, we have the related multiplicative anomaly as a\\_4(K\\_+,K\\_-)= e\\^2\\^2( m\\^2-) .\\[ex1\\] The same can be done for $L_{\\pm}$, obtaining another expression for $a_4(L_+,L_-)$.\n\nFinally, including this two results in (\\[441\\]) and (\\[44\\]) respectively, we obtain Z\\_(K\\_+,K\\_-)=Z\\_(L\\_+,L\\_-)&=&m\\^4 (-3/2) +S(,)\\\n&-& e\\^2\\^2( m\\^2-) ,\\[441bis\\] and the logarithm of the partition function turns to be the same for the two different approaches. Although consistent now, our result is remarkably different from the one in the literature where the multiplicative anomaly was disregarded. The physical relevance of this additional term will be discussed in the next section.\n\nMore generally, this term can be easily computed for any space-time dimension $D$ and turns out to be always vanishing for odd $D$ [@evz; @efvz]. It has also been computed for the self-interacting field [@bbd; @bd], but there many difficulties arise when dealing with the regularized determinants of the complicated operators involved [@efvz].\n\nPHYSICAL RELEVANCE\n==================\n\nTo investigate the physical relevance of the multiplicative anomaly the crucial quantity is the effective potential in presence of external sources, which can be expressed as a function of the charge density $\\rho=\\frac{1}{\\be V}\\frac{\\partial \\ln\nZ_{\\be}(\\mu,J_i)}{\\partial \\mu}=\\frac{<Q>}{V}$ and the mean field $x^2=\\Phi^2 $ as F(,,x)=-Z\\_() + +(m\\^2+e\\^2\\^2)x\\^2 , \\[v1\\] $$\\begin{aligned}\n\\rho=\\frac{1}{\\be V}\\frac{\\partial \\ln\nZ_{\\be}(\\mu)}{\\partial \\mu}+ e^2\\mu x^2 \\:,\\label{v2}\\end{aligned}$$ where the later is an implicit expression for the chemical potential as a function of $\\rho$.\n\nThe physical states correspond to the minima of the effective potential, located in $\\frac{\\partial F}{\\partial x}= x\n(m^2-e^2\\mu^2)=0$. We find therefore: 1) an unbroken phase, $x=0$, $e\\mu <m$; 2) a symmetry breaking solution, $x\\neq 0$, $e\\mu=\\pm m$, giving the relativistic Bose-Einstein condensation. For our system, explicitly, the unbroken and broken phase are respectively \\_= min F &=&[E]{}\\_V- S(,)++ e\\^2\\^2( m\\^2-) ,\\\n&=&-- e( m\\^2-) , \\[sbp111\\] where ${\\cal E}_V$ is the vacuum contribution, and \\_&=&[E]{}\\_V- S(,e=m)+ + ,\\\n&=&-|\\_[e=m]{}- +e m x\\^2 .\\[sbp1\\] It is possible to see, under a detailed generic $D$ analysis, that these expressions, as they are, give some inconsistencies in the broken phase [@efvz]. We have to remember, though, that we worked until now with regularized but \u201cunrenormalized\u201d charge density. Since it appears in the partition function multiplied by $\\mu$, any ambiguity in it will correspond to an uncertainty in the free energy density of the kind $\\mu K$. This $K$ has to be fixed following physical requirements. A very reasonable one is that the symmetry is unbroken at $T=0$, $\\rho=0$. For $D=4$, $K$ will be $K=-\\frac{e m^3}{24\\pi^2}$. For $D=4$ only, this choice also removes the multiplicative anomaly contribution to the charge density, so that the anomaly does not alter the broken phase in any aspect and we get $$\\begin{aligned}\n{\\cal F}_\\beta &=&{\\cal E}_V-\\frac{1}{\\beta V}\nS(\\beta,m)+\\frac{m}{e}  \\rho^R, \\\\ e x^2&=&\n\\left. \\frac{1}{m}\\left( \\rho^R+\\frac{1}{\\beta V}\\frac{\\partial S(\\beta,\\mu)}{\\partial \\mu}\\right|_{e\\mu=m} \\right)\n\\:.\\label{sbp2}\\end{aligned}$$ Also the critical temperature ($x=0$, $e\\mu=m$) remains unchanged with this renormalization. Is different the unbroken phase, where the anomalous term remains: $$\\begin{aligned}\n {\\cal F}_\\beta &=&{\\cal E}_V-\\frac{1}{\\beta V} S(\\beta,\\mu)+\\mu \\rho^R -\\frac{\\mu e m^3}{12 \\pi^2}+\\frac{1}{8\n\\pi^2}\\left( e^2\\mu^2m^2-\\frac{1}{3}e^4\\mu^4 \\right), \\\\\n\\rho^R&=&-\\frac{1}{\\beta V}\\frac{\\partial S(\\beta,\\mu)}{\\partial\n\\mu} -\\frac{1}{8 \\pi^2}\\left( e^2\\mu m^2-\\frac{2}{3}e^4\\mu^3 \\right)\n+\\frac{em^3}{24\\pi^2}\\:. \\label{sspp}\\end{aligned}$$ We should now observe that the anomalous contribution to the free energy is non leading at ultra relativistic temperatures $T > m$, since the thermal terms go as $T^4$. On the other hand, it does not even contribute to the low temperature limit, corresponding to the broken phase, so that it could give relevant corrections only in a intermediate range $T\\simeq m $. Notice, finally, that the anomalous term is vanishing as $e \\to 0$, and the correct expression of the free energy density for the uncharged boson gas is recovered.\n\nGENERAL CONSIDERATIONS\n======================\n\nThis analysed is just one of the many possible physical systems were the multiplicative anomaly could play a role [@nuovozerbini]. The first question is therefore if this additional terms will always have physical relevance [@dowker; @rdowker]. This will in general depend on the system. As an example, for the above case the anomaly is vanishing for any odd dimension. In other cases [@evz] the anomaly could be simply non physical, as it could be reabsorbed in the renormalization procedure. Work is currently in progress on other systems, including fermionic ones. It is not difficult to see that for a single free fermionic field the anomaly is always vanishing, too. On the other side, it could play a role for neutrino mixing, in relation with recent results regarding inequivalent representations of the vacuum [@mixing] and, in general, any time when there is a possible mixing or rotation in the functional space of the fields [@rdowker]. This needs further investigation due to the deep connection between the multiplicative anomaly and the functional measure, which goes to the roots of the definition of the functional integral itself.\n\nThe other relevant question is, of course, if the anomaly is regularization dependent [@evans]. Zeta function regularization is just one example of a wider class of regularizations called: \u201cgeneralized proper-time regularizations\u201d [@sch; @bal], for which we showed the anomaly to be present [@bcvz; @efvz]. This topic is also under further investigations and created a vivid debate lately. Here too, the answers are probably to be found in a proper and consistent definition of the ill-defined functional determinant itself, where this regularization approach has, up to now, proved to be rigorous and coherent [@revans].\n\n[*Note added in proof:*]{} After my talk, a work [@mcktom] by McKenzie-Smith and Toms appeared. There, the relevance of considering the multiplicative anomaly within a functional integral approach is recognized although they do not agree on its physical relevance for the relativistic charged bosonic field.\n\n[99]{}\n\nD.B.\u00a0Ray and I.M.\u00a0Singer, Advances in Math. [**7**]{}, 145 (1971).\n\nJ.S.\u00a0Dowker and R.\u00a0Critchley, Phys.\u00a0Rev. [**D 13**]{}, 3224 (1976).\n\nS.W.\u00a0Hawking, Commun.\u00a0Math.\u00a0Phys. [**55**]{}, 133 (1977).\n\nA.A.\u00a0Bytsenko, G.\u00a0Cognola, L.\u00a0Vanzo and S.\u00a0Zerbini, Phys.\u00a0Rep. 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B.\n\nM.\u00a0Blasone, G.\u00a0Vitiello, Ann. Phys. [**244**]{}, 283, (1995); M.\u00a0Blasone, P.A.\u00a0Henning, G.\u00a0Vitiello, [*Green\u2019s function approach to fermion mixing*]{}, hep-th/9803157 (1998).\n\nT.S. Evans, [*Regularization schemes and the multiplicative anomaly*]{}, hep\u2013th/9803184 (1998).\n\nJ.\u00a0Schwinger, Phys.\u00a0Rev. [**82**]{}, 664 (1951).\n\nR.D.\u00a0Ball, Phys.\u00a0Rep. [**182**]{}, 1 (1989).\n\nE.\u00a0Elizalde, A.\u00a0Filippi, L.\u00a0Vanzo and S.\u00a0Zerbini, [*Is the multiplicative anomaly dependent on the regularization\u00a0?* ]{}, hep\u2013th/9804071 (1998).\n\nJ.J.\u00a0McKenzie-Smith and D.J.\u00a0Toms, [*There is no new physics in the multiplicative anomaly*]{}, hep-th/9805184 (1998).\n\n[^1]: Talk given at the Workshop on QCD at Finite Baryon Density, April 27-30, 1998, Bielefeld, Germany. The author wishes to acknowledge financial support from the European Commission under TMR contract N. ERBFMBICT972020 and, previously, from the Foundation Blanceflor Boncompagni-Ludovisi, n\u00e9e Bildt.\n"], ["---\nabstract: 'Motivated by recent research of Nikitin [*et al*]{} (J.Phys.D [**49**]{},055301(2009)), we examine the effects of interatomic interactions on adatom surface diffusion. By using a mean-field approach in the random walk problem, we derive a nonlinear diffusion equation and analyze its solutions. The results of our analysis are in good agreement with direct numerical simulations of the corresponding discrete model. It is shown that by analyzing a time dependence of adatom concentration profiles one can estimate the type and strength of interatomic interactions.'\nauthor:\n- 'Yuri B. Gaididei'\n- 'Vadim M. Loktev'\n- 'Anton G. Naumovets'\n- 'Anatoly G. Zagorodny'\ntitle: Adatom interaction effects in surface diffusion\n---\n\nIntroduction\n============\n\nDiffusion is ubiquitous in Nature. It determines the behavior and controls the efficiency of many biological and technological processes. Examples include the wetting, conductivity of biological membranes, catalysis, growth of crystals, sintering, soldering [*etc*]{}. Surface diffusion is particularly important in nano-technological processes which are aimed to obtain objects of submicron sizes where the surface properties are of the same importance as the bulk ones. Macroscopic description of diffusion is based on Fick\u2019s law, which postulates proportionality between the particle flux and the concentration gradient. Establishing a link between macroscopic laws of diffusion and microscopic non-equilibrium density matrix approach is one of the most challenging and important problems of non-equilibrium statistical mechanics (for reviews see, e.g. [@gomer; @naumovets_rev; @gouyet; @alanissila; @naumovets2005; @antezak]). Surface diffusion is essentially many-particle process. Even at very low coverage when the interaction between adatoms is negligible, the random walk of an isolated adsorbed particle is a collective motion due its interaction with substrate atoms [@alanissila; @naumovets2005]. The walker moves in a potential landscape which is changed by the walker [@lam]. The walk on a deformable medium when the walker leaves behind a trail and due to slow relaxation the trail affects the next walker, is also a collective process [@gaididei; @huang]. At finite coverage particles at surfaces, in addition to the interaction with a substrate, experience lateral interactions of different origin: the attractive van der Waals, oscillating electronic exchange and multipole-multipole electrostatic interactions. The dipole-dipole interaction which is due to a polar (mainly dipolar) character of adsorption bonds is long-ranged (as $r^{-3}$) and generally repulsive since all dipole moments are essentially parallel (see review papers, e.g. [@alanissila; @naumovets2005]).\n\nBowker and King [@bowker] used Monte-Carlo simulations in order to clarify the effect of lateral interactions of adatoms on the shape of evolving concentration profiles in surface diffusion. They showed that the intersection point of the diffusion profiles with the initial stepwise profile lies above $\\theta_{max}/2$ in the case of lateral repulsion and below $\\theta_{max}/2$ in the case of attraction ($\\theta_{max}$ is the maximum concentration in the initial step).\n\nIn a quite recent paper [@nikitin] an approach based on error function expansion was proposed to fit experimental concentration profiles. This algorithm provides a high-accuracy fitting and allows extracting the concentration dependence of diffusivity from experimental data. The goal of our paper is to model and examine the effects of interatomic interactions on adatom surface diffusion. Starting with nonlinear random walk equations where the interatomic interactions are considered in the mean-field approach, we derive a nonlinear diffusion equation and analyze its solutions. The results of our analysis are in good agreement with direct numerical simulations of the corresponding discrete model. It is shown that by analyzing a time dependence of adatom concentration profiles one can estimate the type and strength of interatomic interactions. The paper is organized as follows. In Sec.1 we present the model. In Sec.2 we study both analytically and numerically interaction effects for the case of low adatom coverage. Sec.3 is devoted to analytical treatment of nonlinear diffusion in the case of non-monotonic concentration dependence of the diffusion coefficient. We also compare our results with the results of full scale numerical simulations and results of experimental observation. Sec.4 presents some concluding remarks.\n\nModel and equations of motion\n=============================\n\nThe transport of particles on the surface is described by the set of random walk equations $$\\begin{aligned}\n\\label{random_walk_eqn}\n\\frac{d}{dt}\\,\\theta_{\\vec{n}}(t)=\\sum_{\\vec{\\rho}}\\,\\Big[W_{\\vec{n}+\\vec{\\rho}\\rightarrow\\vec{n}}\\,\n\\theta_{\\vec{n}+\\vec{\\rho}}(t)\\,\n\\Big(1-\\theta_{\\vec{n}}(t)\\Big)-\nW_{\\vec{n}\\rightarrow\\vec{n}+\\vec{\\rho}}\\,\\theta_{\\vec{n}}(t)\\,\\Big(1-\\theta_{\\vec{n}+\\vec{\\rho}}(t)\\Big)\\Big]\\end{aligned}$$ where $\\theta_{\\vec{n}}$ is the probability for a particle occupying the $\\vec{n}$-th binding site on the surface (in the literature on surface science this quantity has a meaning of coverage), $W_{\\vec{n}\\rightarrow\\vec{n}+\\vec{\\rho}}$ gives the rate of the jumps from the binding site $\\vec{n}$ to a neighboring site $\\vec{n}+\\vec{\\rho}$ (the vector $\\vec{\\rho}$ connects nearest neighbors) . The terms $\\Big(1-\\theta_{\\vec{n}}(t)\\Big)$ in Eqs. (\\[random\\_walk\\_eqn\\]) take into account the fact that there may be only one adatom at a given site or, in other words, a so-called kinematic interaction. The probability with which the particle jumps from site $\\vec{n}$ to a nearest neighbor $\\vec{n}+\\vec{\\rho}$ satisfies the detailed balance condition, $$\\begin{aligned}\n\\label{det-rate}\nW_{\\vec{n}\\rightarrow\\vec{n}+\\vec{\\rho}}\\,e^{-\\beta E_{\\vec{n}}}=W_{\\vec{n}+\\vec{\\rho}\\rightarrow\\vec{n}}\\,e^{-\\beta E_{\\vec{n}+\\vec{\\rho}}},\\end{aligned}$$ where $E_{\\vec{n}}$ is the binding energy of the particle situated at site $\\vec{n}$ , $\\beta=1/k_B T, ~k_B$ is the Boltzmann constant and $T$ is the temperature of the system. For the transition rates we choose $$\\begin{aligned}\n\\label{rate}W_{\\vec{n}\\rightarrow\\vec{n}+\\vec{\\rho}}=w_{\\vec{\\rho}}\\,e^{\\beta E_{\\vec{n}}}\\end{aligned}$$ which corresponds to setting the activation energy for a jump to the initial binding energy. Here $ w_{\\vec{\\rho}}=\\nu_{\\vec{\\rho}}\\,e^{-\\beta E_b} \\,\\,(w_{\\vec{\\rho}}=w_{-\\vec{\\rho}})$ is the jump rate of an isolated particle with standard notations: $\\nu_{\\vec{\\rho}}$ is a frequency factor and $E_b-E_{\\vec{n}}$ is the height of the random walk barrier. Inserting Eq. (\\[rate\\]) into Eqs. (\\[random\\_walk\\_eqn\\]) we obtain that the random walk of particles on the surface is described by a set of equations $$\\begin{aligned}\n\\label{random_walk_eqn_mod}\\frac{d}{dt}\\,\\theta_{\\vec{n}}(t)=\n \\sum_{\\vec{\\rho}}\\,w_{\\vec{\\rho}}\\,\\Big[\\Big(1-\\theta_{\\vec{n}}(t)\\Big)\n\\,\\theta_{\\vec{n}+\\vec{\\rho}}(t)\\,e^{\\beta\\,E_{\\vec{n}+\\vec{\\rho}}}\n-\\,\\theta_{\\vec{n}}(t)\\,\\Big(1-\\theta_{\\vec{n}+\\vec{\\rho}}(t)\\Big)\\,e^{\\beta\\,E_{\\vec{n}}}\n\\Big]\\;.\\end{aligned}$$\n\nIn the case when the characteristic size of the particle distribution inhomogeneity is much larger than the lattice spacing one can replace $\\theta_{\\vec{n}}$ and $E_{\\vec{n}}$ by the functions $\\theta(\\vec{r})$ and $E(\\vec{r})$ of the continuous variable $\\vec{r}$ and, by expanding the functions $\\theta(\\vec{r}+\\vec{\\rho})$ and $E(\\vec{r}+\\vec{\\rho})$ into a Taylor series, obtain from Eqs. (\\[random\\_walk\\_eqn\\_mod\\]) that in the continuum approximation the transport of particles on the surface is described by the equation of the form $$\\begin{aligned}\n\\label{nonl_diff_eq-1}\\partial_t \\theta=w\\,\\nabla\\Big\\{\\Big(\\nabla\\,\\theta+\\beta\\,\\theta\\,(1-\\theta)\\,\\nabla E\\Big)\\,e^{\\beta E}\\Big\\}\\end{aligned}$$ where the notation $w=\\frac{1}{2}\\,\\sum\\limits_{\\vec{\\rho}}\\,\\vec{\\rho}^2\\,w_{\\vec{\\rho}}$ is used.\n\nWe will study the particle kinetics in the mean field approach when the binding energy $E$ is assumed to be a functional of the particle density $\\theta(\\vec{r},t)$: $E(\\vec{r})= {\\cal E}(\\theta)$. In this case Eq. (\\[nonl\\_diff\\_eq-1\\]) takes a form of nonlinear diffusion equation $$\\begin{aligned}\n\\label{nonl_diff_eq}\\partial_t \\theta=\\,\\nabla\\Big\\{D(\\theta)\\,\\nabla \\theta\\Big\\}\\;,\\end{aligned}$$ where $$\\begin{aligned}\n\\label{nonl_diff_coeff}D(\\theta)=w\\,\\Big(1+\\beta\\,\\theta\\,(1-\\theta)\\,\\frac{\\delta {\\cal E}}{\\delta \\theta}\\Big)\\,e^{\\beta {\\cal E}}\\end{aligned}$$ is a nonlinear (collective) diffusion coefficient.\n\nSurface diffusion at low coverage\n=================================\n\nIn what follows we restrict ourselves to studying particle distributions spatially homogeneous along the $y$ coordinate: $\\theta(\\vec{r},t)\\equiv \\theta(x,t)$. We assume that initially the particles are step-like distributed $$\\begin{aligned}\n\\label{initial}\\theta(x,0)=\\theta_{max}\\, H(-x)\\end{aligned}$$ where $H(x)$ is the Heaviside step function. By introducing a centered particle density $\\xi(x,t)=\\,\\Big(\\theta(x,t)-0.5\\,\\theta_{max}\\Big)/\\theta_{max}$, we see that the initial distribution $\\xi(x,0)$ is an odd function of the spatial variable $x$. It is obvious that in the no-interaction case (${\\cal E}=const$), when the diffusion equation (\\[nonl\\_diff\\_eq\\]) is linear, the antisymmetric character of the function $\\xi(x,t)$ is preserved for all $t>0$. This means that in the case of noninteracting particles the concentration profile for each time moment $t$ passes through the point $\\Big(0,\\frac{\\theta_{max}}{2}\\Big).$ However, interacting diffusing particles exhibit quite a different behavior. In 1969, Vedula and one of the authors detected for the first time that concentration profiles formed in the process of surface diffusion of thorium on tungsten intersected the initial step-like profile at a point lying well above $\\theta_{max}$ (see [@vedula; @naumovets_rev]). Since then, similar behavior has been found for many electropositive adsorbates whose adatoms are known to interact repulsively. A recent example obtained in the case of surface diffusion of Li on the Dy-Mo\u00a0(112) surface was discussed in [@nikitin]. It is worth noticing that the above mentioned behavior was observed even for rather low coverage: $\\theta_{max}\\,< 0.3$ (see Fig. 2 in [@nikitin]). Therefore to explain such a behavior one may assume that the binding energy $E(\\vec{r})$ is linearly dependent on the particle concentration: $$\\begin{aligned}\n\\label{mean_field}\nE(\\vec{r})=E_0+\\int d\\vec{r}'\\,V(\\vec{r}-\\vec{r}')\\,\\theta(\\vec{r}')\\approx E_0+\\theta(\\vec{r})\\,\\int d\\vec{r}'\\,V(\\vec{r}')\\end{aligned}$$ where $E_0$ is a site energy and $V(\\vec{r}-\\vec{r}')$ is an interaction parameter which includes all types of lateral interactions. In this case the nonlinear diffusion coefficient (\\[nonl\\_diff\\_coeff\\]) takes a form $$\\begin{aligned}\n\\label{nonl_diff_coeff_mod}\nD(\\theta)=D^*\\,\\Big(1+\\alpha \\,\\theta\\,(1-\\theta)\\Big)\\,e^{\\alpha\\theta}\\end{aligned}$$ where $$D^*=w\\,e^{\\beta\\,E_0}\\equiv\\nu\\,e^{-\\beta\\,(E_b-E_0)}$$ is the diffusion coefficient for an isolated particle (or a so-called tracer diffusion coefficient) and the dimensionless parameter $$\\alpha=\\beta\\,V_0,~~~V_0=\\int d\\vec{r}'\\,V(\\vec{r}')$$ characterizes the strength of the lateral interaction.\n\nThe aim of this section is to develop an approach which allows estimating the effects of interparticle interactions in the surface diffusion. It is seen from Eqs. (\\[nonl\\_diff\\_eq\\]) that the spatio-temporal behavior of the centered particle density $\\xi(x,t)$ is governed by the equation $$\\begin{aligned}\n\\label{nonl_diff_eq_2}\\partial_{\\tau} \\xi=\\,\\partial_x^2\\Big(\\xi + P(\\xi)\\Big)\\;, \\end{aligned}$$ where $\\tau=D_0\\,t$ is a rescaled time and the quantity $$\\begin{aligned}\n\\label{P}P(\\xi)=\\frac{1}{D^{*}}\\,\\int\\limits_0^\\xi\\,d\\xi'\\,D(\\xi')-\\xi\\end{aligned}$$ describes the nonlinear properties of the diffusion and vanishes when $\\alpha\\rightarrow 0$. Taking into account that Eq. (\\[nonl\\_diff\\_eq\\_2\\]) with the initial condition given by Eq. (\\[initial\\]) is invariant under gauge transformations $\\tau\\rightarrow \\lambda^2 \\,\\tau,x\\rightarrow \\lambda \\,x,\\theta\\rightarrow \\theta$ ($\\lambda$ is an arbitrary number) one can look for a solution of Eq. (\\[nonl\\_diff\\_eq\\_2\\]) in terms of the Boltzmann variable $z=\\frac{x}{2\\sqrt{\\tau}}$ : $\\xi(x,\\tau)=\\zeta(z)$ where the function $\\zeta(z)$ satisfies the equation $$\\begin{aligned}\n\\label{nonl_diff_eq_3}\\frac{d^2}{d z^2}\\Big(\\zeta+P(\\zeta)\\Big)+2 z \\frac{d \\zeta}{d z}=0,\\end{aligned}$$ with the boundary conditions $$\\begin{aligned}\n\\label{nonl_diff_eq_4}\n\\zeta(z)\\rightarrow\\mp\\frac{1}{2},~~~z\\rightarrow\\pm{\\infty}\\;.\\end{aligned}$$ Eqs. (\\[nonl\\_diff\\_eq\\_3\\]), (\\[nonl\\_diff\\_eq\\_4\\]) can be rewritten in the form of the following integral equation $$\\begin{aligned}\n\\label{nonl_int_eq}\\zeta(z)= \\frac{\\sqrt{\\pi}}{4}\\,\\int\\limits_{0}^{\\infty}dz\\,w_+(z)-\\frac{1}{2}\\Big(1- \\frac{\\sqrt{\\pi}}{2}\\,\\int\\limits_{0}^{\\infty}dz\\,w_-(z)\\Big)\\,\\mathrm{erf}(z)\n-\\nonumber\\\\\n\\frac{\\sqrt{\\pi}}{2}\\,\n\\int\\limits_{0}^{z}dz_1\\,e^{z_1^2}\\,\\Big(\\mathrm{erf}(z)-\\mathrm{erf}(z_1)\\Big)\\,\\frac{d^2}{d z_1^2}\\,P\\Big(\\zeta(z_1)\\Big),\n\\nonumber\\\\\nw_{\\pm}(z)=e^{z^2}\\,\\Big(1-\\mathrm{erf}(z)\\Big)\\,\\frac{d^2}{d z^2}\\,\\Big[P\\Big(\\zeta(z)\\Big)\\pm P\\Big(\\zeta(-z)\\Big)\\Big]\\;.\\end{aligned}$$ where $\\mathrm{erf}(z)$ is the error function [@abr]. It is seen from Eq. (\\[nonl\\_int\\_eq\\]) that the concentration profiles $\\xi(x,t)$ for different time moments intersect at the point $\\Big(0,\\zeta(0)\\Big)$ with $$\\begin{aligned}\n\\label{intersect}\\zeta(0)=\n\\frac{\\sqrt{\\pi}}{4}\\,\\int\\limits_{0}^{\\infty}dz\\,w_+(z)\\;.\\end{aligned}$$ In the weak interaction/low coverage limit when $\\alpha\\,\\theta_{max}<\\,1$ one can replace the function $\\zeta(z)$ in the right-hand-side of Eqs. , (\\[intersect\\]) by its expression obtained in the linear case:$~\\zeta_0(z)=\\frac{1}{2}\\,\\mathrm{erf}(z)$ and obtain approximately that under the step-like initial condition (\\[initial\\]) the concentration profiles $\\theta(x,\\tau)$ for different time moments intersect at the point which corresponds to the concentration $$\\begin{aligned}\n\\label{intersect_point}\\theta_0=\n\\Big(\\frac{1}{2}+\\zeta(0)\\Big)\\,\\theta_{max}\\;,\\nonumber\\\\\n\\zeta(0)=\\frac{\\pi-2}{4 \\,\\pi}\\,\\alpha\\,\\theta_{max}\\approx 0.1\\,\\alpha\\,\\theta_{max}\\;.\\end{aligned}$$ Thus the concentration value $\\theta_0$ at which the concentration profiles intersect changes in the presence of lateral interatomic interactions: $\\theta_0>\\theta_{max}/2~~(\\theta_0<\\theta_{max}/2)$ when the interaction is repulsive (attractive). We applied Eq.(\\[intersect\\_point\\]) to analyze the results obtained in [@nikitin] for the diffusion of Li on the Dy-Mo (112) surface at low coverage ($\\theta_{max}\\approx 0.33$) for which $\\theta_0\\approx 0.19$ and found out that $\\alpha\\,\\theta_{max}\\approx 0.97$. It is seen that strictly speaking it is not fully legitimate to use our simple analytical perturbation approach (which is valid for $\\alpha\\,\\theta_{max} \\ll 1$) to analyze the results of experiments [@nikitin] but a qualitative agreement takes place. To validate our analytical results we carried out numerical simulations of Eqs. (\\[random\\_walk\\_eqn\\_mod\\]), (\\[rate\\]) which in the 1D-case for a system with $N$ binding sites have the form $$\\begin{aligned}\n\\label{random_walk_eqn_1D}\n\\frac{d}{d\\tau}\\,\\theta_1=(1-\\theta_{1})\\,e^{\\beta\\,{\\cal E}_{2}}\\,\\theta_{2}-\\Big(1-\\theta_{2}\\Big)\\,\ne^{\\beta\\,{\\cal E}_{1}}\\,\\theta_{1},\\nonumber\\\\\n\\frac{d}{d\\tau}\\,\\theta_n=(1-\\theta_{n})\\,\\Big(e^{\\beta\\,{\\cal E}_{n+1}}\\,\\theta_{n+1}+\ne^{\\beta\\,{\\cal E}_{n-1}}\\,\\theta_{n-1}\\Big)-\\Big(2-\\theta_{n+1}-\\theta_{n-1}\\Big)\\,\ne^{\\beta\\,{\\cal E}_{n}}\\,\\theta_{n},\n~~(n=2,... N-1),\\nonumber\\\\\n\\frac{d}{d\\tau}\\,\\theta_N=(1-\\theta_{N})\\,e^{\\beta\\,{\\cal E}_{N-1}}\\,\\theta_{N-1}-\\Big(1-\\theta_{N-1}\\Big)\\,\ne^{\\beta\\,{\\cal E}_{N}}\\,\\theta_{N}\\end{aligned}$$ where $\\beta \\,{\\cal E}_{n}=\\alpha\\,\\theta_n$. Thus, in our model the total number of particles is a conserved quantity. As an initial state we used a step-like distribution $$\\begin{aligned}\n\\theta_n=\\theta_{max},~~~~\\mathrm{for}~~1\\leq n\\leq \\frac{N}{4}\\;,\\nonumber\\\\\n\\theta_n=0,~~~~\\mathrm{otherwise\\;.}\\end{aligned}$$We found out that for $\\theta_{max}=0.33$ the concentration profiles intersect at the point $(0,0.19)$ ( as it was observed in the experiment [@nikitin]) for $\\alpha\\approx (3\\div 3.5)$ ( see Fig. \\[fig:coverage\\_low\\]) or $\\alpha\\,\\theta_{max}\\approx (1\\div 1.5)$ which is a good agreement with our analytics. Thus basing on our approach one can conclude that Li adatoms on the Dy-Mo (112) surface mostly repel each other and the intensity of the repulsion is $V_0\\approx (3\\div 3.5) \\,k_B\\,T$.\n\nDiffusion of Li adatoms on Dy/Mo(112) was investigated experimentally at $T=600 K$ [@nikitin], so the estimated repulsion energy $V_0$ amounts to $0.16-0.18 $ eV. Let us assess this value in terms of the dipole-dipole interaction. The energy of the repulsive interaction between two dipoles having moments $p$ and situated on the surface at a distance $r$ is $$\\begin{aligned}\n\\label{dipo-int}U_{dd}=2\\,\\frac{p^2}{r^3}\\approx\\frac{1.25\\, p^2 \\,[Debyes]}{r^3\\,[Angstroms]}\\,[eV]\\end{aligned}$$ The dipole moment can be determined from the work function change $\\Delta\\varphi$ using the Helmholtz formula for the double electric layer: $$\\begin{aligned}\n\\label{dipo}\\mid\\Delta\\varphi\\mid= 4\\,\\pi\\,n\\,p\\,e\\;,   \\end{aligned}$$ where $n$ is the surface concentration of adatoms and e is the electronic charge. For Li on the Mo(112) surface, the $p$ value at low coverage was found to be $1.4$ Debyes [@braun]. Then, using Eq. (\\[dipo-int\\]) we can find that for two such dipoles the interaction energy $U_{dd}=0.16 $eV can be attained at a distance $r\\approx 2.5 \\r{A}$, which is close to the distance between the nearest adsorption sites $(2.73\\r{A})$ within the atomic troughs on Mo(112). This estimation shows that the intensity of the lateral interaction deduced from the diffusion data in the way presented above seems physically reasonable. Recall, however, that $V_0$ determines a resultant effect experienced by a jumping particle from all its counterparts which, in the case of heterodiffusion, are non-uniformly distributed over the surface and provide an additional driving force (supplementary to the coverage gradient) that favors a faster diffusion of repulsing particles.\n\nThe adatom interaction effects fade away for the late stage of the evolution when the particle density becomes small and the diffusion process transfers into a linear regime ( see Fig. \\[fig:coverage\\_low\\] for $t=60000$).\n\nMean-square deviation\n=====================\n\nThe adatom interactions manifest themselves also in the integral characteristics of kinetics of adatom diffusion. It is well known that in the linear regime the variance $$\\begin{aligned}\n\\label{mean_sq_def}\n\\langle x^2\\rangle=\n\\frac{\\int\\limits_{-\\infty}^{\\infty}\\,dx\\,x^2\\,\\theta(x,\\tau)}\n{\\int\\limits_{-\\infty}^{\\infty}\\,dx\\,\\,\\theta(x,\\tau)}\\end{aligned}$$ behaves (in one-dimensional case) as $\\langle x^2\\rangle=2\\tau$. Therefore it is naturally to introduce a variance rate $$\\begin{aligned}\n\\label{mean_sq_rate}\n\\Delta(\\tau)=\\Big(2-\\frac{d}{d \\tau}\\langle x^2\\rangle\\Big)^2\\end{aligned}$$ whose time dependence provides a useful information about nonlinear effects in the diffusion process.\n\n![Concentration profile obtained from nonlinear random walk equations (\\[random\\_walk\\_eqn\\_1D\\]) for $\\alpha=3$ and initial distribution given by a thin dashed curve. The left panel shows the early stage of evolution: $w t=500$ (solid curve), $w t=2000$ (dashed curve), the right panel shows the late stage of evolution: $~w t=40000$ (dashed),$~w t=60000$ (solid). The line which corresponds to $\\theta=\\theta_{max}/2$ is shown as a horizontal dotted line in the left panel and the line which corresponds to the final state of the evolution $\\theta=\\theta_{max}/4$ is shown as a horizontal dotted line in the right panel.[]{data-label=\"fig:coverage_low\"}](coverage_low_row.eps){width=\"50.00000%\"}\n\nFor this quantity we obtain from Eq. (\\[nonl\\_diff\\_eq\\_4\\]) that $$\\begin{aligned}\n\\label{mean_sq_eq}\\Delta(\\tau)=\\alpha^2\\,\\Big(\\int\\limits_{-\\infty}^{\\infty}\\,dx\\,\n\\,\\theta^2(x,\\tau)\\Big{/}\\int\\limits_{-\\infty}^{\\infty}\\,dx\\,\\,\\theta(x,\\tau)\\Big)^2\\end{aligned}$$ Assuming that initially the particles are concentrated in a finite domain in a $\\Pi$-like form: $$\\begin{aligned}\n\\label{initial1}\\theta(x,0)=\\theta_{max}\\,\\Big(H(x+l)-H(x-l)\\Big)\\end{aligned}$$ where $2 l$ is the size of the initial domain, for small nonlinearities $\\alpha$ and low coverage $\\theta_{max}\\ll 1$ we obtain approximately that $$\\begin{aligned}\n\\label{mean_sq_eq_ap}\\Delta(\\tau)=\\alpha^2\\,\\Big(\\int\\limits_{-\\infty}^{\\infty}\\,dx\\,\n\\,\\theta_{lin}^2(x,\\tau)\\Big{/}\\int\\limits_{-\\infty}^{\\infty}\\,dx\\,\\,\\theta_{lin}(x,\\tau)\\Big)^2\\end{aligned}$$ where $$\\begin{aligned}\n\\label{theta0}\\theta_{lin}(x,\\tau)=\\frac{\\theta_{max}}{2}\\,\\Big[\\mathrm{erf} \\Big(\\frac{l-x}{2\\,\\sqrt{\\tau}}\\Big)-\\mathrm{erf} \\Big(-\\frac{l+x}{2\\,\\sqrt{\\tau}}\\Big)\\Big]\\end{aligned}$$ is the solution of the linear diffusion equation with the initial condition (\\[initial1\\]). In the limit of small $l$ we obtain that $$\\begin{aligned}\n\\label{mean_sq_t}\\Delta(\\tau)\\,\n\\approx\\frac{\\alpha^2\\,\\theta_{max}^2}{2\\,\\pi\\,\\tau}\\,l^2\\;.\\end{aligned}$$\n\nWe checked our analytical considerations by carrying out numerical simulations of Eqs. (\\[random\\_walk\\_eqn\\_1D\\]) with the initial concentration profile given by Eq. (\\[initial1\\])(see Fig. \\[fig:profile\\_variance\\]) for different values of the nonlinearity parameter $\\alpha$. The results of these simulations are presented in Fig. \\[fig:variance\\]. The figure shows that the numerically evaluated temporal behavior of the rate function $\\Delta$ is in a good agreement with our analytical expression given by Eq. (\\[mean\\_sq\\_t\\]). Moreover the slopes of the curves as it is prescribed by the analytics relate as $0.51:0.91:2.0:3.6\\approx \\alpha_1^2:\\alpha_2^2:\\alpha_3^2:\\alpha_4^2=0.15^2:0.2^2:0.3^2:0.4^2\\;.$ Thus, by measuring the temporal behavior of concentration profiles, it is possible to estimate the strength of interatomic interactions.\n\nConcentration profiles with plateau\n===================================\n\nIn general, the diffusion coefficient is a non-monotonic function of atomic concentration (see e g [@naumovets2005]). There is a number of physical reasons which can cause a non-monotonic coverage dependence of the diffusion coefficient. In thermodynamic terms, the diffusion flux is proportional to the gradient of chemical potential of adsorbed particles $\\mu$ which can be written as [@adamson],[@loburets] $$\\begin{aligned}\n\\label{mu}\\mu=\\mu_0-q(\\theta)+\\frac{1}{\\beta}\\,\\ln\\Big(\\frac{\\theta}{1-\\theta}\\Big)\\end{aligned}$$ The first term in this equation is the standard chemical potential of the adsorbate, $q(\\theta)$ is the differential heat of adsorption and the third term stems from the entropy of mixing of adatoms with the vacant adsorption sites on the substrate. (Note that this simplified expression relates only to the first monolayer and does not take into account the possibility of formation of the second and next monolayers). The diffusion coefficient can be represented as a product $$\\begin{aligned}\n\\label{Dif}D(\\theta)=D_j\\,\\beta\\,\\frac{\\partial \\mu}{\\partial\\,\\ln\\theta}\\;,\\end{aligned}$$ where $D_j$ is a so-called kinetic factor (or jump diffusion coefficient) [@gomer],[@naumovets2005],[@loburets] and the derivative in the brackets is named the thermodynamic factor. In a simplest case when the cross-correlations between the velocities of diffusing particles are absent, $D_j$ coincides with the tracer diffusion coefficient $D*$ given by Eq. (10). Inserting Eq. (\\[mu\\]) into Eq. (\\[Dif\\]), we get $$\\begin{aligned}\n\\label{Diff}D(\\theta)=D_j\\,\\Big(-\\beta\\,\\theta\\,\\frac{\\partial q}{\\partial \\theta}+\\frac{1}{1-\\theta}\\Big)\\;.\\end{aligned}$$ It is seen from Eq. (\\[Diff\\]) that any effect which entails a sharp decrease in the heat of adsorption as a function of coverage will result in a maximum of the diffusion coefficient in this coverage range [@note]. For instance, such a situation occurs when all energetically profitable sites at the surface are occupied and adatoms start to fill less favorable sites. Actually, Bowker and King [@bowker] found in their Monte Carlo simulations that a well-pronounced maximum in the $D(\\theta)$ dependence observed by Butz and Wagner [@butz] can be explained by existence of two types of lateral interactions: repulsive one between the nearest neighbors and attractive between next-nearest neighbors. A similar effect is typical for volume diffusion of interstitial atoms in disordered binary alloys having a BCC structure with two nonequivalent interstitial positions [@smirnovaa]. In the framework of local equilibrium statistical operator approach [@zubarev] it was shown that the physical reason for a non-monotonic concentration dependence coefficient is a combined action of lateral interaction and adatom density fluctuations [@tarasenko]. A sharp drop in the heat of adsorption is also observed in the transition from filling the first, strongly bound (chemisorbed) monolayer to filling the second, weakly bound (e.g., physisorbed) monolayer. In such a case, the spreading of the first monolayer proceeds through diffusion in the mobile uppermost (second or next) monolayer (the so called \u201cunrolling carpet\u201d mechanism) [@gomer]. This example shows that a change in the heat of adsorption can be accompanied not only by variation of the diffusion parameters (the activation energy and prefactor $D_0$ in the Arrhenius equation), but also by a change in the atomistic diffusion mechanism itself.\n\nIt is worth noting also that the non-monotonic concentration dependence may be phenomenologically connected with a step-like dependence of the heat of adsorption on the coverage ( see a review paper [@naumovets89]) . Fig. \\[fig:diffusion\\_coeff\\] shows the diffusion coefficient calculated from Eq. (\\[nonl\\_diff\\_coeff\\]) by assuming that the on-site adatom energy ${\\cal E}(\\theta)$ (which in most cases is proportional to the heat adsorption $q(\\theta)$) has the form $$\\begin{aligned}\n\\label{on_site_step}\n{\\cal E}(\\theta)=\\alpha\\,\\Big(1+\\tanh\\kappa\\,(\\theta-\\theta_{thr})\\Big)\\end{aligned}$$ where the parameter $\\theta_{thr}$ gives the threshold value of the coverage and the parameter $\\kappa$ characterizes the sharpness of the transition to the new state [@naum].\n\nThe aim of this section is to consider the diffusion process with a step-like concentration dependent on-site energy given by Eq. (\\[on\\_site\\_step\\]) and clarify what kind of new information one can derive by comparing theoretically obtained concentration profiles with experimentally obtained ones.\n\nIt is very hard and may be hopeless to solve the equation (\\[nonl\\_diff\\_eq\\]) with the diffusion coefficient given by Eqs. (\\[nonl\\_diff\\_coeff\\]) and (\\[on\\_site\\_step\\]). However, the problem can be solved and some insight into the kinetics can be achieved in the limiting case of very sharp energy concentration dependence: $~\\kappa\\rightarrow \\infty\\;.$ In this case the diffusion coefficient (\\[nonl\\_diff\\_coeff\\]) and (\\[on\\_site\\_step\\]) takes the form $$\\begin{aligned}\n\\label{nonl_diff_coeff_step}\n D(\\theta)=D^{*}\\,\\Big(1+a\\,\\delta(\\theta-\\theta_{thr})+b\\,H(\\theta-\\theta_{thr})\\Big)\\end{aligned}$$ where $$\\begin{aligned}\n\\label{cs}a=\\alpha\\,e^{\\alpha}\\theta_{thr}\\,(1-\\theta_{thr}),\n ~~b=e^{2\\,\\alpha}-1.\\end{aligned}$$ The nonlinear diffusion equation (\\[nonl\\_diff\\_eq\\]) with the diffusion coefficient (\\[nonl\\_diff\\_coeff\\_step\\]) and the initial condition (\\[initial\\]) has a self-similar solution $\\theta(x,\\tau)\\equiv\\Theta(z),~(z=x/2\\sqrt{\\tau})$ which can be presented in the form (see Appendix for a detailed derivation) $$\\theta(x,\\tau)=\\left\\{ \\begin{array}{rl}\\theta_{thr}\\,\\frac{\\mathrm{erfc}\\Big(\\frac{x}{2\\,\\sqrt{\\tau}}\\Big)}\n {\\mathrm{erfc}\\big(z_1\\big)}\\,, &\\mbox{when}~~x\\geq 2\\,z_1\\,\\sqrt{\\tau}\\\\\n\\theta_{thr}\\,\\;,&\\mbox{when}~~-\\,2\\,z_2\\,\\sqrt{\\tau}\\,e^{\\alpha} \\leq x \\leq 2\\,z_1\\,\\sqrt{\\tau}\\\\\n\\theta_{max}-\\Big(\\theta_{max}-\\theta_{thr}\\Big)\n\\,\\frac{\\mathrm{erfc}\\Big(-\\frac{x}{2\\,\\sqrt{\\tau}}\\,e^{-\\alpha}\\Big)}{\\mathrm{erfc}\\big(z_2\\big)}\\;, &\\mbox{when}~~x\\leq -2\\,z_2\\,e^{\\alpha}\\,\\sqrt{\\tau}\n\\end{array}\\right.$$ Here the parameters $z_1$ and $z_2$ are determined by the equations $$\\begin{aligned}\n\\label{z12s}z_2=\\frac{\\sqrt{\\pi}}{2}\\,\\alpha\\,\\,\n(1-\\theta_{thr})\n \\,e^{z_1^2}\\,\\mathrm{erfc}(z_1)-\\,z_1\\,e^{-\\alpha}\\;,\\nonumber\\\\\ne^{\\alpha}\\,\\Big(\\theta_{max}-\\theta_{thr}\\Big)\\,e^{z_2^2}\\,\\mathrm{erfc}(z_2)=\n\\theta_{thr} \\,e^{z_1^2}\\,\\mathrm{erfc}(z_1)\\end{aligned}$$ which are obtained from Eqs. (\\[z12\\]) taking into account the definition (\\[cs\\]). Thus, the concentration profile in the case of non-monotonic diffusion coefficient is characterized by existence of a plateau where the concentration of adatoms does not depend on the spatial variable $x$. The length of the plateau $\\ell_p=2\\sqrt{\\tau} (z_1+z_2\\,e^{\\alpha})$ increases with time. Such a behavior is shown in Fig. \\[fig:front\\_anal\\]. The rate with which the length $\\ell+p$ of the plateau increases is determined by the nonlinear parameter $\\alpha$ and the threshold coverage $\\theta_{thr}$. We carried out also numerical simulations of Eqs. (\\[random\\_walk\\_eqn\\_1D\\]) with the the step-like on-site energy ${\\cal E}_n=\\alpha\\,\\Big[1+\\tanh\\Big(\\kappa\\,(\\theta_n-\\theta_{thr})\\Big)\\Big]$ and as it is seen from Fig. \\[fig:front\\_numer\\_ext\\] our simple model (\\[nonl\\_diff\\_coeff\\_step\\]) is in reasonable agreement with numerics. Note that the plateau in the concentration dependence develops only on intermediate stage of the evolution. For large enough times the height of the concentration profile becomes small, effects of interatomic interactions are negligible and the profile evolutes in accordance with the linear diffusion equation (see Fig. \\[fig:front\\_numer\\_ext\\]).\n\nIt is worth noting that the diffusion of Dy adatoms absorbed by Mo (1 1 2) for the initial coverage $\\theta(x,0)\\approx 0.7 H(-x)$ shows a very well pronounced plateau in the concentration profile dependence both on the spatial coordinate for different time moments and on the Boltzmann variable ( see Figs. 7 and 8 in [@nikitin]). It means that as it is prescribed by our analytical considerations the length of the plateau increases as $t^{1/2}$. This suggests that our simple analytical model may be a useful tool in analyzing an experimentally observed concentration behavior.\n\nConclusions and discussion\n==========================\n\nIn this paper, we have investigated the role of interactions between adatoms in surface diffusion. The problem was considered analytically in the mean-field approach. By analyzing discrete nonlinear random walk equations and corresponding nonlinear diffusion equations with an initial condition in a form of step-like concentration profile, we have found that the interactions between adatoms influence significantly the concentration profile development on early and intermediate stage of the process. In the case of low coverage the interaction between adatoms makes the concentration profile asymmetric: it is shifted to the side of high concentration in the case of repulsive interactions and to the side of low concentration for attractive interactions. By calculating the magnitude of the shift one can estimate the intensity of lateral interactions between adatoms. At the late stage of kinetics the role of interatomic interactions becomes negligible. By studying the nonlinear random walk process which is characterized by a sharp maximum in the concentration dependence of the diffusivity, we have found that a well-pronounced plateau develops in the concentration profile. The length of the plateau increases in time as $t^{1/2}$ . The height of the plateau $\\theta_{thr}$ corresponds to the maximum of the diffusion coefficient which in the frame of our approach corresponds to a sharp decrease in the heat of adsorption as a function of coverage. The rate with which the length of the plateau increases with time is determined by an amount at which the adsorption heat drops at the threshold coverage $\\theta_{thr}$ . All above mentioned results can be verified experimentally.\n\nAcknowledgements {#acknowledgements .unnumbered}\n================\n\nThe authors acknowledge support from a Goal-oriented program of the National Academy of Sciences of Ukraine.\n\n![Concentration profiles for initial pulse-like distribution (dashed line) and for $\\tau=200$ (solid line). The nonlinearity parameter $\\alpha=0.2$.[]{data-label=\"fig:profile_variance\"}](profile_variance.eps){width=\"40.00000%\"}\n\n![Numerically obtained variance rate $\\Delta$ given Eq. (\\[mean\\_sq\\_rate\\]) as a function of the inverse time $1/t$ for three different values of the nonlinearity parameter $\\alpha$: $\\alpha=0.15$ (dotted line), $\\alpha=0.2$ (dashed line), $~\\alpha=0.3$ (solid line).[]{data-label=\"fig:variance\"}](variance_vs_inv_time.eps){width=\"40.00000%\"}\n\n![Diffusion coefficient for a step-like on-site energy. The parameters used are $~a=1,~\\ell=0.05,~\\theta_0=0.5$ []{data-label=\"fig:diffusion_coeff\"}](diffusion_coeff.eps){width=\"40.00000%\"}\n\n![Analytically obtained concentration profile for the diffusion coefficient in the $\\delta$-function limit (\\[on\\_site\\_step\\]) with $\\alpha=1,~\\theta_0=0.5,~\\Theta=1$ for three different time moments : $~\\tau=0.005$ (dotted line), $~\\tau=1$ (dashed line), $~\\tau=5$ (solid line).[]{data-label=\"fig:front_anal\"}](profile_anal.eps){width=\"40.00000%\"}\n\n![(color on-line) Numerically obtained concentration profiles in the case of the step function energy dependence given by Eq. (\\[on\\_site\\_step\\]) with $\\theta_0=0.5,~~\\alpha=1,~~\\ell=0.005$ for different time moments: $t=0$ (dotted gray line), $t=6000$ (dotted line), $t=12000$ (dashed line), $t=20000$ (solid line),$~t=40000$ (dashed, red, thick line). []{data-label=\"fig:front_numer_ext\"}](diffusion_front_numer_tanh_ext.eps){width=\"40.00000%\"}\n\n {#appendA}\n\nThe nonlinear diffusion equation (\\[nonl\\_diff\\_eq\\]) with the diffusion coefficient (\\[nonl\\_diff\\_coeff\\_step\\]) and the initial condition (\\[initial\\]) has a self-similar solution $\\theta(x,\\tau)\\equiv\\Theta(z),~(z=x/2\\sqrt{\\tau})$ which satisfies the equation $$\\begin{aligned}\n\\label{nonl_diff_eq_b}-2\\,z\\,\\frac{d\\Theta}{d z}=\\frac{d}{d z}\\Big(D(\\Theta)\\,\\frac{d\\Theta}{d z}\\Big)\\;.\\end{aligned}$$ The boundary conditions for Eq. (\\[nonl\\_diff\\_eq\\_b\\]) are $$\\begin{aligned}\n\\label{bound_cond}\\Theta(z)\\rightarrow\\Theta,~~~\\text{for}~~z\\rightarrow-\\infty\\;,\\nonumber\\\\\n\\Theta(z)\\rightarrow 0,~~~\\text{for}~~z\\rightarrow\\infty\\;.\\end{aligned}$$\n\nFrom Eqs. (\\[nonl\\_diff\\_eq\\_b\\]) and (\\[nonl\\_diff\\_coeff\\_step\\]) we see that the function $$\\begin{aligned}\n\\label{y}y(\\Theta)=\\int\\limits_0^\\Theta\\,d\\Theta'\\,z(\\Theta')\\;,\\end{aligned}$$ where $z(\\Theta)$ is an inverse function with respect to $\\Theta(z)$, satisfies the equation $$\\begin{aligned}\n\\label{eq_y}-2\\,\\frac{d^2 y}{d \\,\\Theta^2}=D(\\Theta)\\frac{1}{y} \\end{aligned}$$ or equivalently, two equations $$\\begin{aligned}\n\\label{eqs_y}-2\\,\\frac{d^2 y}{d\\,\\Theta^2}=\\frac{1}{y},~~~~\\text{for}~~\\Theta < \\theta_{thr}\\;,\\nonumber\\\\\n-2\\,\\frac{d^2 y}{d\\,\\Theta^2}=\\frac{1+b}{y},~~~~\\text{for}~~\\Theta > \\theta_{thr}\\;\n \\end{aligned}$$ augmented by the jump condition $$\\begin{aligned}\n\\label{jump} \\frac{d y}{d\\Theta}\\Big|_{\\Theta=\\theta_{thr}+0}-\\frac{d y}{d \\Theta}\\Big|_{\\Theta=\\theta_{thr}-0}=\n-\\frac{a}{2\\,y(\\theta_c)}\\;,\\end{aligned}$$ and the continuity condition $$\\begin{aligned}\n\\label{cont} y(\\theta_{thr}+0)=y(\\theta_{thr}-0)=\ny(\\theta_c)\\;.\\end{aligned}$$ By integrating Eqs. (\\[eqs\\_y\\]) once, we get $$\\begin{aligned}\n\\label{eqs_y_i} \\frac{d y}{d \\Theta}=\\sqrt{2 z_1+\\ln\\frac{y(\\theta_{thr})}{y}},~~~~\\text{for}\\,\\,\\,\\Theta < \\theta_{thr}\\;,\\nonumber\\\\\n-\\frac{1}{\\sqrt{1+b}}\\frac{d y}{d \\Theta}=\\sqrt{2\\,z_2+\\ln \\frac{y(\\theta_{thr})}{y}},~~~~\\text{for}\\,\\,\\,\\Theta > \\theta_{thr}\\;,\\end{aligned}$$ where the constants $y(\\theta_{thr})$, $z_1$ and $z_2$ satisfy the equations $$\\begin{aligned}\n\\label{z12}\\theta_{thr}=\\frac{\\sqrt{\\pi}}{2}\\,\\frac{a}{z_1+\\sqrt{1+b}\\,z_2} \\,e^{z_1^2}\\,\\mathrm{erfc}(z_1)\\;,\\nonumber\\\\\n\\sqrt{1+b}\\,\\Big(\\theta_{max}-\\theta_{thr}\\Big)=\\frac{\\sqrt{\\pi}}{2}\\,\\frac{a}{z_1+\\sqrt{1+b}\\,z_2} \\,e^{z_2^2}\\,\\mathrm{erfc}(z_2),\\nonumber\\\\\n y(\\theta_{thr})=\\frac{a}{z_1+\\sqrt{1+b}\\,z_2}\\end{aligned}$$ which were obtained from the jump condition (\\[jump\\]) and the continuity condition (\\[cont\\]). Taking into account the definition (\\[y\\]), we obtain eventually from Eqs. (\\[eqs\\_y\\]) that the concentration profile is determined by the following expressions $$\\begin{aligned}\n\\label{theta_sol} \\theta(z)=\\theta_{thr}\\,\\,\\frac{\\mathrm{erfc}(z)}{\\mathrm{erfc}(z_1)} ~~~~~\\text{when}~~z\\geq z_1\\;,\\nonumber\\\\\n\\theta(z)=\\theta_{thr}~~~~~\\text{when}~~-\\sqrt{1+b}\\,\\,z_2 \\leq z \\leq z_1\\;,\\nonumber\\\\\n\\theta(z)=\\theta_{max}-\\Big(\\theta_{max}-\\theta_{thr}\\Big)\\,\\,\\frac{\\mathrm{erfc}\n\\Big(-\\frac{z}{\\sqrt{1+b}}\\Big)}{\\mathrm{erfc}(z_2)} ~~~~~\\text{when}~~z\\leq -z_2\\,\\sqrt{1+b}\\;.\\end{aligned}$$\n\n[14]{} Gomer R 1990 Diffusion of adsorbates on metal surfaces Rep.Prog.Phys. [**53**]{} 917-1002 Naumovets A G, Vedula A S 1985 Surface Diffusion of Adsorbates, Surf. Sci. Rep. [**4**]{} 365-434 Gouyet J F, Plapp M, Dietrich W, and Maas P 2003 Adv. Phys. [**52**]{} 523 Ala-Nissilla T, Ferrando R, Ying S C 2002 Collective and single particle diffusion on surfaces Advances in Physics [**51**]{} 949-1078 Naumovets A G 2005 Collective surface diffusion:An experimentalist\u2019s point of view Physica A [**357**]{} 189-215 Antezak G, Ehrlich G 2010 [*Surface diffusion:metals, metal atoms, and clusters*]{} (Cambridge: Cambridge Univ. Press) Freimuth R D and Lam L 1992 [*Modeling complex phenomena*]{} (New York:Springer) Gaididei Yu B 1993 Ion-conformational interaction and charge transport through channels of biological membranes J. Biological Physics [**19**]{} 19-38 Sheng-You Huang, Xian-Wu Zou, Wen-Bing Zhang, and Zhun-Zhi Jin 2002 Random walks on a (2+1)-dimensional deformable medium Phys. Rev.Lett. [**88**]{} 056102-056105 Bowker M, King D A 1978 Adsorbate diffusion on single crystal surfaces II Extension to next nearest neighbour interactions, Surface Sci. [**72**]{} 208-212 Nikitin A G, Spichak S V, Vedula Yu S and Naumovets A G 2009 Symmetries and modelling functions for diffusion processes J. Phys. D:Appl.Phys. [**42**]{} 055301-055313 Vedula Yu S, Naumovets A G 1969 A study of surface diffusion of thorium adatoms on a tungsten sigle crystal, in: Surface Diffusion and Spreading (Poverkhostnaya Diffuziya i Rastekanie), ed. by Ya. Geguzin, Nauka, Moscow, 149-160 (in Russian) Braun O M and Medvedev V K 1989 Interaction between particles adsorbed on metal surfaces Sov. Phys. Uspekhi [**32**]{} 328 Adamson N W and Mermin N D 1976 [*Physical Chemistry of Surfaces*]{}, 3rd ed. (Wiley, N.Y.) Ch. XIV Loburets A T, Naumovets A and Vedula Yu S 1997 Surface Diffusion and Phase Transitions in Atomic Overlayers, in [*Surface Diffusion. Atomistic and Collective Processes*]{}, ed. by Tringides M C (Plenum, N.Y.) 509-528 It is worth noting that Eq. (\\[Dif\\]) is valid for $\\theta\\,<\\,1$ and the divergence $D(\\theta)$ for $\\theta\\rightarrow 1$ is a model effect. It does not appear when the possibility of filling the second and next monolayers is taken into consideration. K[\u00fc]{}ntz M, Lavall[\u00e9]{}e P 2003 Anomalous spreading of a density front from an infinite continuous source in a concentration-dependent lattice gas automaton diffusion model J. Phys. D:Appl.Phys. [**36**]{} 1135-1142 K[\u00fc]{}ntz M, Lavall[\u00e9]{}e P 2004 Anomalous diffusion is the rule in concentration-dependent diffusion processes J. Phys. D:Appl.Phys. [**37**]{} L5-L8 M.\u00a0Abramowitz and I.\u00a0Stegun, 1972 [*Handbook of Mathematical Functions*]{} (Dover Publications, Inc., New York) Butz R and Wagner H 1977 Diffusion of oxygen on tungsten $\\Big(1 1 0\\Big)$ Surf.Sci. [**63**]{} 448-459 Smirnov A A 1982 [*Theory of Diffusion in Iterstitial Alloys*]{} (Kiev: Naukova Dumka) Zubarev D N 1971 [*Non-Equilibrium Statistical Thermodynamics*]{}(Moscow: Nauka) Chumak A A and Tarasenko A A 1980 Diffusion and density fluctuations of atoms absorbed on solid surfaces, Surface Sci. [**91**]{} 694-706 Naumovets A G 1989 Phase transitions in two dimensions, Contemporary Physics, [**30**]{} 187-201 For a number of metal-on-metal systems a sharp D maximum is observed at submonolayer coverages corresponding to an initial stage of commensurate-incommensurate (C-I) phase transition \\[Adamson A W 1976 Physical Chemistry of Surfaces, 3rd ed. (Wiley, N.Y.) Ch. XIV, Masuda T, Barnes T J, Hu P and King D A 1992 Surface Sci. [**276**]{} 122\\]. The transition starts with a local breaking of the commensurability, namely with formation of incommensurate walls which separate commensurate domains \\[ Lyuksyutov I F, Naumovets A G and Pokrovsky V L 1992 Two-Dimensional Crystals (Academic Press, Boston) \\]. The domain walls can be considered as misfit dislocations in the commensurate phase and described as topological solitons \\[Loburets A T, Naumovets A G and Vedula Yu S 1997. Surface diffusion and phase transitions in atomic overlayers. In: Surface Diffusion. Atomistic and Collective Processes, ed. by Tringides M C (Plenum, N.Y.) p. 509\\]. Such objects were predicted and treated theoretically, and also detected experimentally by low-energy electron diffraction and scanning tunneling microscopy \\[ Andryushechkin B V, Eltsov K N and Shevlyuga V M 2001 Surface Sci. [**472**]{} 80\\]. The solitons were shown to possess a high mobility and thus play a role of effective mass carriers in surface diffusion process. It is interesting to note that the highest diffusion rate is observed at the initial stages of C-I transition when the number of solitons is small. As their number grows, the diffusion rate decreases, so the coverage dependence of the diffusion coefficient is non-monotonic. This behaviour resembles the non-monotonic dependence of material strength on the concentration of dislocations: the strength increases when the dislocations are so numerous that they are pinning each other.\n"], ["---\nauthor:\n- |\n    Cunxi Yu$^1$, Daniel Holcomb$^2$\\\n    Cornell University $^1$\\\n    University of Massachusetts Amherst$^2$\\\n    cunxi.yu@cornell.edu\nbibliography:\n- 'security.bib'\n- 'verification\\_ycunxi.bib'\n- 'synthesis.bib'\n- 'dan.bib'\ntitle: 'Algorithmic Obfuscation over GF([$2^m$]{})'\n---\n\n\n"], ["---\nabstract: 'Many automatically analyzable scientific questions are well-posed and offer a variety of information about the expected outcome *a priori*. Although often being neglected, this prior knowledge can be systematically exploited to make automated analysis operations sensitive to a desired phenomenon or to evaluate extracted content with respect to this prior knowledge. For instance, the performance of processing operators can be greatly enhanced by a more focused detection strategy and the direct information about the ambiguity inherent in the extracted data. We present a new concept for the estimation and propagation of uncertainty involved in image analysis operators. This allows using simple processing operators that are suitable for analyzing large-scale spatiotemporal (3D+t) microscopy images without compromising the result quality. On the foundation of fuzzy set theory, we transform available prior knowledge into a mathematical representation and extensively use it enhance the result quality of various processing operators. All presented concepts are illustrated on a typical bioimage analysis pipeline comprised of seed point detection, segmentation, multiview fusion and tracking. Furthermore, the functionality of the proposed approach is validated on a comprehensive simulated 3D+t benchmark data set that mimics embryonic development and on large-scale light-sheet microscopy data of a zebrafish embryo. The general concept introduced in this contribution represents a new approach to efficiently exploit prior knowledge to improve the result quality of image analysis pipelines. Especially, the automated analysis of terabyte-scale microscopy data will benefit from sophisticated and efficient algorithms that enable a quantitative and fast readout. The generality of the concept, however, makes it also applicable to practically any other field with processing strategies that are arranged as linear pipelines.'\nauthor:\n- 'Johannes Stegmaier$^\\ast$ Ralf\u00a0Mikut[^1]'\ntitle: 'Fuzzy-based Propagation of Prior Knowledge to Improve Large-Scale Image Analysis Pipelines'\n---\n\nBackground {#sec:UncertaintyFramework}\n==========\n\nAvailable prior knowledge is often not sufficiently considered by automatic processing pipelines and a great amount potentially useful extra information remains unused. Particularly in the domain of image processing and image analysis, the visual analysis of acquired image data offers a large repository of usable *a priori* information that can often easily be verbalized by experts of the respective application field. Examples of the successful incorporation of prior knowledge are, [*e.g.*]{}, the approaches described in [@Al-Kofahi10; @Bourgine10] that make use of information about the expected object number as well as their associated physical size in order to adjust and improve seed point detection algorithms. Analogously, properties like size, shape, geometry, intensity distributions and the like can be used to improve the algorithmic performance of image segmentation algorithms [@Fernandez10; @Lou11; @Khan14a]. Such prior knowledge is often embedded into the algorithm via shape penalization terms that are appended to the energy functional of graph-cut [@Lou11; @Vu08] or a level-set segmentation [@Leventon00] or by generalized Hough transforms that can detect arbitrary shapes [@Ballard81]. Object properties like size, shape and movement dynamics can also be used to formulate efficient correction heuristics for object tracking algorithms [@Bao06; @Schiegg13; @Amat14].\n\nA great but often underestimated potential for algorithmic improvements lies in the estimation of uncertainties of the automatically produced results and should ideally be considered by subsequent processing steps [@Santo04]. On the pixel level, this uncertainty can be used to assess the information quality of a single pixel due to sensor imperfections or temperature dependence [@Santo04; @Maji11]. Furthermore, the localization uncertainty of geometric features such as corners, centroids, edges and lines in images has been assessed in [@Pal01; @Chen09; @Chen10; @Anchini07]. An approach to evaluate the quality of image registration algorithms was presented in [@Kybic10]. Besides many applications from the field of quality quantification, a part of research focuses on uncertainty quantification in areas such as face recognition and other biometric technologies [@Callejas06; @Betta11; @Betta12], the tracking of shapes in ultrasound images [@Zhou05] or to evaluate the impact of noisy measurements on the validity of diagnosis results [@Mencattini10]. An uncertainty formulation based on fuzzy set theory has been employed to perform pixel- or object-based classification tasks [@Boskovitz02; @Tizhoosh05; @Radojevic15]. A further possibility to exploit the uncertainty information is to optimize parameter values of a respective operator in a feedback fashion such that the outcome minimizes a previously defined optimization criterion as demonstrated in [@Khan13smps; @Khan13Krusebook]. Another example is the improvement of a graph-based watershed implementation, where uncertainties are used to assess the influence of individual edges on the final segmentation outcome [@Straehle12].\n\nHitherto, however, a uniform approach to systematically transform, embed and use the available prior knowledge to improve both existing and new algorithms is missing. Although the sequential arrangement of processing operators is a broadly used concept in image analysis, results propagated through the pipelines are mostly not assessed by the individual pipeline components with respect to their result quality. Thus, errors made in early processing steps tend to accumulate and may negatively affect the final result quality. Additionally, many existing methods for processing tasks such as seed point detection, segmentation and tracking are often not directly applicable to large-scale 3D+t data sets due to enormous memory or computation time demands.\n\nThroughout the present contribution, uncertainty is considered as the imperfect knowledge about the validity of a piece of extracted information produced by the respective image analysis operators and is used to derive efficient improvement heuristics to enhance image analysis pipelines [@Bouchon00; @Stegmaier12a]. We present and apply a new concept for the estimation and propagation of uncertainty involved in image analysis operators that improves the result quality awareness of each processing operator and can be used to trigger adapted processing strategies. On the foundation of our previous work [@Stegmaier12a], we use fuzzy set theory to transform available prior knowledge into a mathematical representation and extensively use it to enhance the performance of image analysis operators by data filtering, uncertainty propagation and explicit exploitation of information uncertainty for result improvements. In particular, we extend an exemplary image analysis pipeline comprised of seed point detection, segmentation, multiview fusion and tracking with uncertainty handling. After introducing the general concept, we demonstrate how simple processing operators can be extended with uncertainty handling to improve large-scale analyses of 3D+t microscopy images. All methods are quantitatively validated on a comprehensive simulated validation benchmark data set that mimics embryonic development and is inspired by epiboly movements of zebrafish embryos. Moreover, we show qualitative results obtained with the presented framework on large-scale light-sheet microscopy data of developing zebrafish embryos.\n\nMethods\n=======\n\nUncertainty Propagation in Image Analysis Pipelines\n---------------------------------------------------\n\n### The Image Analysis Pipeline Concept {#sec:UncertaintyPropagation:PipelineConcept}\n\nMost image analysis pipelines make use of multiple processing operators that are arranged as a linear processing pipeline and perform specialized tasks, such as improving or transforming the image signal, or to extract information from the images (\\[fig:OperatorPipelineReduced\\]). The $N_{\\text{op}}$ sequentially connected operators receive either a raw or processed version of the input image (denoted by $\\mathbf{I}_i$), extracted features (denoted by $\\mathcal{X}_i$) or both from their preceding processing operator with $i \\in \\{1, ..., N_\\text{op} \\}$ being the ID of the operator.\n\n![General image analysis pipeline comprised of $N_{\\text{op}}$ sequentially arranged processing operators. Each operator directly depends on the quality of the input images ($\\mathbf{I_\\ast}$) or features ($\\mathcal{X}_\\ast$) provided by its predecessor (adapted from [@Stegmaier12a]).[]{data-label=\"fig:OperatorPipelineReduced\"}](Figure1.png){width=\"1.0\\hsize\"}\n\nThe output set $\\mathcal{X}_i$ of processing operator $i$ is an ${(N_i \\times N_{\\text{f},i})}$ matrix with $N_i$ data tuples and $N_{\\text{f},i}$ features. For processing operators without any feature output, $\\mathcal{X}_i$ is an empty matrix and only the processed image is passed to the next operator.\n\n### Identification of Suitable Prior Knowledge {#sec:PriorKnowledgeIdentification}\n\nPrior knowledge can be obtained from literature, expert knowledge, experimental evidence or knowledge databases. Visual analysis of acquired data often allows experts to easily identify recurring patterns, intensity properties or the appearance of objects contained in the images that can be described in natural language. An exemplary overview of such prior information derived from microscopy images is summarized in \\[tab:PriorKnowledgeTable\\].\n\n[m[0.12]{}m[0.55]{}m[0.26]{}]{} **Source** & **Description** & **Example**\\\nImage Acquisition & Acquisition-specific prior knowledge such as illumination conditions, detection path, image resolution, physical spacing of voxels, high-quality image regions, point spread function (PSF) or the detection path. & Image quality decreases from XY to YZ.\\\nIntensity & Signal-dependent information like the intensity range, time-variant characteristics of objects ([*e.g.*]{}, photobleaching in fluorescence microscopy), signal-to-noise ratio and global statistical properties of the image intensity values. & Valid objects are brighter than XY.\\\nLocalization & Positional information of the objects or object properties in absolute image coordinates. Furthermore, localization of extracted properties or objects relative to each other can be used to define neighborhood relations. & Object type XY only appears close to location YZ.\\\nSpatial Extent & Object properties such as size, volume, principal components, convex hull extents or bounding volumes. & Object type X is larger than Y but smaller than Z.\\\nGeometry & Geometrical properties like dimensionality, symmetry, shape, proportions and relative localization of features within an object. & Object type XY has a line-like shape with a central symmetry axis.\\\nMorphology & Combination of intensity-based and geometrical properties, [*e.g.*]{}, to link information about patterning, texture, structure and color to geometrical properties such as shape and symmetry. & Object XY is spherical, bright and has a textured surface.\\\nObject Interaction & Characterization of between-object properties like clustering, adhesion, repulsion, division or regional density changes. & Object behavior XY rather appears in dense regions.\\\nSpatio-Temporal Coherence & Dynamically changing quantities such as object growth, movement direction, speed, object appearance and disappearance. & Object moves maximally XY pixels between two subsequent frames.\\\n\n\\[tab:PriorKnowledgeTable\\]\n\nThe prior information are listed in bottom-up order, [*i.e.*]{}, from the acquisition stage over the content of a single image through to the spatio-temporal comparison between time series of images. Naturally, the listing is not exhaustive and the suitable features have to be carefully selected to match the underlying image material and analysis problem. In the following sections, the presented natural language expressions will be used to transform the prior knowledge of different sources to a consistent mathematical representation using the concept of fuzzy sets.\n\n### Quantifying Prior Knowledge using Fuzzy Set Membership Functions {#sec:FuzzySetMembershipFunctions}\n\nTo transform the prior knowledge presented in \\[tab:PriorKnowledgeTable\\] into a mathematical representation, we make use of fuzzy sets that have been introduced by Zadeh in 1965 [@Zadeh65]. Analogous to the characteristic function of a classical set, a fuzzy set $\\mathcal{A}$ can be defined by its associated membership function (MBF) $\\mu_{\\mathcal{A}}:\\mathcal{X}\\mapsto [0,1]$ that maps each element of a universe of discourse $\\mathcal{X}$ to a value in the range $[0,1]$ [@Zadeh65]. This assigned value in turn directly reflects the fuzzy set membership degree (FSMD) of the respective element to the fuzzy set $\\mathcal{A}$. The special cases $\\mu_{\\mathcal{A}}(x)=1$ and $\\mu_{\\mathcal{A}}(x)=0$ indicate that $x$ is fully included or not part of the fuzzy set $\\mathcal{A}$, respectively [@Bede13]. The most common membership functions used in practice are trapezoidal membership functions, which can be parameterized to model singletons, triangular and rectangular MBFs. A trapezoidal membership function can be formulated as $$\\begin{gathered}\n    \\mu_\\mathcal{A}(x,\\pmb{\\theta}) = \\max\\left( \\min\\left( \\frac{x-a}{b-a}, 1, \\frac{d-x}{d-c} \\right), 0\\right),\n    \\label{eq:TrapezoidalMBF}\\end{gathered}$$ with the parameter vector $\\pmb{\\theta}=\\left(a,b,c,d\\right)^\\top$ that is used to control the start and end points of the respective transition regions. Here, we make use of a standard partition, [*i.e.*]{}, maximally two neighboring fuzzy sets overlap and have non-zero membership values for a certain value of $x$ and the respective membership degrees for any input value of $x$ sum up to $1$ (\\[fig:TrapezoidalMembershipFunctionComparison\\]).\n\n![Different possibilities to partition the input space of a feature $x$ using trapezoidal membership functions. In (A), each of the linguistic terms has a separate fuzzy set and (B) shows a reduced version with only two fuzzy sets that correspond to the desired class and its complement. In (B), different possibilities to summarize the correct objects arise. Besides restricting the class to the correct set as done in (B), the correct fuzzy set could be extended by the potentially useful classes (*Small* and *Large*). However, the appropriate formulation has to be chosen application dependent.[]{data-label=\"fig:TrapezoidalMembershipFunctionComparison\"}](Figure2.png){width=\"1.0\\hsize\"}\n\nConsidering, for instance, the detection of a specific kind of object it might make sense to use the size feature of the object as an indicator of its appropriateness. As described in [@Stegmaier12], the linguistic terms could in this case for example be determined by five possible outcomes, where the extracted feature ...\n\n1.  ... perfectly matches the expected value (*Correct*).\n\n2.  ... is smaller than expected but might be useful (*Small*).\n\n3.  ... is larger than expected but might contain useful information (*Large*).\n\n4.  ... is too small and not useful (*Too Small*, [*e.g.*]{}, noise or artifacts).\n\n5.  ... is too large and not useful (*Too Large*, [*e.g.*]{}, segments in background regions).\n\nAvailable prior knowledge can be used to determine the parameterization of the associated fuzzy sets and an exemplary standard partition is shown in \\[fig:TrapezoidalMembershipFunctionComparison\\]A. If only one outcome of the operators is of importance ([*e.g.*]{}, Case 1 in the above-mentioned example), it is possible to use only one linguistic term and to aggregate all other cases by its complement (\\[fig:TrapezoidalMembershipFunctionComparison\\]B). We use $\\mu_{\\mathcal{A}_{ifl}}: \\mathbb{R} \\rightarrow [0,1]$ to denote the fuzzy set membership function for image analysis operator $i$, feature $f \\in \\lbrace 1,...,N_{\\text{f},i} \\rbrace$ and linguistic term $l \\in \\lbrace 1,...,N_\\text{l} \\rbrace$. Thus, the $n$-th data tuple produced by operator $i$ obtains the FSMD value $\\mu_{\\mathcal{A}_{ifl}}(x_i[n,f])$ to the fuzzy set $\\mathcal{A}_{ifl}$ for each feature $f$ and each linguistic term $l$.\n\n### Combination of Fuzzy Set Membership Functions {#sec:MembershipFunctionCombination}\n\nFuzzy set membership degree values of multiple features that characterize a linguistic term ([*e.g.*]{}, if an object of interest is bright and elongated at the same time) can be combined using a fuzzy pendant to a logical conjunction [@Bouchon00]. A conjunction of $N_{\\text{f},i}$ fuzzy set membership functions for linguistic term $l$ can be defined using the minimum operator: $$\\begin{gathered}\n    \\mu_{\\text{lc},\\mathcal{A}_{il}}(\\mathbf{x}_{i}[n]) = \\min_{f=1,...N_{\\text{f},i}} \\left( \\mu_{A_{ifl}}(x_{i}[n,f]) \\right).\n    \\label{eq:MembershipFunctionCombinationMin}\\end{gathered}$$ Features that should not contribute to the combined fuzzy set membership function can be disabled by setting the corresponding MBFs to the constant value $1$ (identity element of the conjunction) and the complement of the combined linguistic term is simply given by $1-\\mu_{\\text{lc},\\mathcal{A}_{il}}(\\mathbf{x}_{i}[n])$ as illustrated in \\[fig:TrapezoidalMembershipFunctionComparison\\]B. Compared to the multiplication-based conjunction used in [@Stegmaier12a], the minimum-based formulation in \\[eq:MembershipFunctionCombinationMin\\] is more informative, as a non-zero value directly represents the lowest FSMD value of the considered fuzzy sets.\n\n### Uncertainty Propagation in Image Analysis Pipelines {#sec:UncertaintyPropagation}\n\nWe use the FSMD values associated with each data tuple to perform a feed-forward propagation of the reliability of extracted data to downstream operators. For each data vector that is produced by operator $i$, we calculate the degree of membership to the respective fuzzy sets and append it to the feature output $\\mathcal{X}_i$. If only a classification into correct vs.\u00a0incorrect objects needs to be performed (\\[fig:TrapezoidalMembershipFunctionComparison\\]B) or if a linguistic term is described by a combination of different fuzzy sets (\\[eq:MembershipFunctionCombinationMin\\]), a single FSMD value is appended per data tuple. The FSMD values can be used to perform object filtering, extended information propagation and to resolve ambiguities.\n\n#### Uncertainty-based Object Rejection: {#sec:ObjectRejection}\n\nThe first application of the uncertainty framework is to filter the extracted output information $\\mathcal{X} _i$ produced by an operator $i$ using thresholds $\\alpha_{il} \\in [0,1]$. According to the FSMD values $\\mu_{\\text{lc},\\mathcal{A}_{il}}(\\mathbf{x}_i[n])$ calculated for each data tuple $\\mathbf{x}_i[n] \\in \\mathcal{X} _i$, $\\mathbf{x}_i[n]$ is only passed to the next pipeline component if $\\mu_{\\text{lc},\\mathcal{A}_{il}}(\\mathbf{x}_i[n]) \\geq \\alpha_{il}$ for the membership to a desired set. The reduced set which serves as input for operator $i+1$ is denoted by $\\tilde{\\mathcal{X}}_{i}$. To keep all extracted information, the threshold is set to $\\alpha_{il}=0$. Contrary, for $\\alpha_{il}=1$, no uncertain information is passed to operator $i+1$. Based on application specific criteria or object properties, this FSMD-based object rejection allows to easily filter out false positive detections as demonstrated in \\[sec:SeedDetection\\] and \\[sec:Segmentation\\], respectively.\n\n#### Extended Information Propagation to Compensate Operator Flaws: {#sec:InformationPropagation}\n\nSecond, we allow operators to fall back on information of penultimate processing steps if predecessors do not deliver good results. For instance, if an operator $i$ fails to sufficiently extract information from its provided input data ([*e.g.*]{}, missing, merged or misshapen objects), it can inform downstream operators about these flawed results. Using a second threshold $\\beta_{il} \\in \\left[\\alpha_{il}, 1\\right]$ for each operator, the FSMD level below which the information of the previous steps should be additionally propagated can be controlled. More formally this means that instead of only forwarding the $\\alpha_{il}$-filtered set $\\tilde{\\mathcal{X}}_{i} \\subseteq \\mathcal{X}_{i}$ produced by operator $i$ to operator $i+1$ a set $\\Omega_i = \\tilde{\\mathcal{X}}_{i} \\cup \\tilde{\\Omega}_{i-1}$ with $i \\geq 2$ and $\\Omega_1 = \\tilde{\\mathcal{X}}_1$ is passed through the pipeline.\n\n$\\tilde{\\Omega}_{i-1}$ represents the subset of elements in $\\Omega_{i-1}$ which were not successfully transferred into useful information by operator $i$, [*i.e.*]{}, elements $\\mathbf{x}_{i-1}[n] \\in {\\Omega}_{i-1}$ that generated output $\\mathbf{x}_{i}[n] \\in \\tilde{\\mathcal{X}} _{i}$ with $\\alpha_{il} \\leq \\mu_{\\text{lc},\\mathcal{A}_{il}}(\\mathbf{x}_{i}[n]) < \\beta_{il}$. Such elements characterize information of operator $i-1$ that might be useful in later steps to correct flawed results of operator $i$. If $\\beta_{il}=1$ all information of $i-1$ that produced an uncertain outcome is propagated to the successor $i+1$. If $\\beta_{il} = \\alpha_{il}$ only the information $\\tilde{\\mathcal{X}_i}$ produced by operator $i$ is propagated. In the current version of the framework, the respective processing operators are responsible for calculating $\\Omega_i$ and to appropriately calculate the respective FSMD values.\n\nThis approach was successfully used to resolve tracking conflicts that originated from under-segmentation errors as described in \\[sec:Tracking\\].\n\n#### Resolve Ambiguities using Propagated Uncertainty: {#sec:ResolvingAmbiguities}\n\nIn addition to filtering and propagating the information of the operators within the pipeline, the provided uncertainty information can explicitly be used by the processing operators to improve their results. Depending on the degree of uncertainty of provided information, parameters or even whole processing methods can be adapted if needed. Of course, the adaptions required by a particular algorithm cannot be generalized. However, we showcase two potential applications in the next sections, namely the fusion of redundant seed points (\\[sec:SeedDetection\\]) and the correction of under-segmentation errors (\\[sec:Segmentation\\]).\n\nThe general scheme for the proposed uncertainty propagation framework is summarized in \\[fig:OperatorPipeline\\] and is applied to an exemplary image analysis pipeline in the next sections.\n\n![image](Figure3.png){width=\"1.0\\hsize\"}\n\nExtending and Enhancing Algorithms with Uncertainty Treatment\n-------------------------------------------------------------\n\nBased on the general concept presented in the previous section, we applied it to an exemplary image analysis pipeline comprised of seed point detection, segmentation, multiview fusion and object tracking. For each operator, FSMD values were estimated based on prior knowledge and used for algorithmic improvements where possible. We use a simulated benchmark data set that mimics 3D microscopy images containing fluorescently labeled nuclei of an artificial embryo.\n\nThe advantages of using simulated image data for validation is the possibility to have a single comprehensive data set for the validation of all pipeline components. Instead of testing each component separately on different benchmarks, this allows to uncover specific bottlenecks or error sources in the processing pipelines. Furthermore, different acquisition deficiencies such as different point-spread-functions, decreasing signal-to-noise ratios or multiview acquisition deficiencies can be simulated. The immediate availability of a reliable ground truth enables a quantitative validation without the bias observed for manually annotated benchmark data that suffers from intra- and inter-expert variability. As the simulated benchmark is close to the target application of the pipeline, namely quantitatively analyzing terabyte-scale 3D+t fluorescence microscopy images, the developed concepts and algorithms can easily be put into practice, [*e.g.*]{}, for false positive reduction of a segmentation algorithm or for segmentation-based multiview fusion [@Kobitski15; @Stegmaier16Diss].\n\nDetails on the benchmark generation can be found in \\[sec:chap4:Benchmark\\] and [@Stegmaier16a]. Abbreviations for the different algorithms are given in round brackets and a quantitative comparison of the result quality is provided in \\[sec:Experiments\\].\n\n### Seed Point Detection {#sec:SeedDetection}\n\nIn [@Stegmaier14], a blob detection method based on the Laplacian-of-Gaussian (LoG) maximum intensity projection was used to localize fluorescently labeled cellular nuclei in 3D microscopy images. In brief, a 3D input image is filtered with differently scaled LoG filters with standard deviations $\\sigma$ matching the expected object radius $r$ using the relation $\\sigma= r / \\sqrt{2}$. Subsequently, the 3D maximum projection of these LoG-filtered images is formed and local extrema are extracted from this projection image (LoGSM). Although, the proposed method worked well in many scenarios, it frequently missed objects that did not exhibit a strict local maximum due to an intensity plateau ([*e.g.*]{}, elongated objects, overexposure or discretization artifacts). To get rid of this behavior, we used the $\\leq$-operator instead of the $<$-operator to additionally detect non-strict local extrema (LoGNSM). However, this increased the amount of false positive detections in background regions and along elongated objects.\n\nDetections in background regions were removed using an intensity threshold ($t_{\\text{wmi}}$) applied on the mean intensity of a small window surrounding the potential detection. The remaining seed points were mostly located properly on the detected objects and remaining false positive detections largely originated from objects that were detected multiple times. To combine redundant objects to a single one, a fusion approach based on hierarchical clustering was used (LoGNSM+F). The hierarchical cluster tree was computed using Ward\u2019s minimum variance method to compute distances between clusters, [*i.e.*]{}, the within-cluster variance was minimized to obtain equally sized clusters [@WardJr63]. The final clustering was obtained from the complete cluster tree using a distance-based cutoff $t_{\\text{dbc}}$ that was set to the smallest expected object radius $r_{\\text{min}}$, to fuse close redundant detections and to prevent fusion of neighboring objects. A single detection per object was obtained by averaging the feature vectors of all detected seeds in a cluster.\n\nAs the seed detection stage usually represents one of the first analysis steps, no preceding uncertainty information was considered. To inform down-stream operators about the expected result quality, the uncertainty of the detected seed points was estimated using the window mean intensity, the maximum seed intensity and the z-position features of the extracted objects (LoGNSM+F+U). Besides discarding obvious false positive detections in background regions (intensity-based thresholds), the fuzzy sets for the z-position were adjusted such that seed detections in low contrast regions (farther away from the detection objective) had lower membership degrees to the class of correct objects than objects in the high contrast regions (closer to the detection objective). The final fuzzy set membership degree of an object to the fuzzy set of correct objects was determined using the minimum of the obtained membership degrees and was appended as a new feature to the output matrix of the seed detection algorithm. To filter false positives the forward threshold slightly above zero to $\\alpha_{11}=0.0001$ (forward threshold for Operator 1 and Linguistic Term 1). A color-coded visualization of the detected seed points and the fuzzy sets for the different features are shown in \\[fig:UncertaintyVisualizationAndFuzzySets\\].\n\n![Maximum intensity projection of a 3D benchmark image along the Z, Y and X axis superimposed with detected seed points (A, B, C). Seed points are colored according to their FSMD to the class of a correct detection ranging from red over blue to green for low, medium and high membership degree, respectively. The fuzzy sets used for the individual features are depicted in (D) and the `min`-operator was used as a fuzzy conjunction to obtain the final membership degree. The uncertainty gradient along the z-axis was introduced due to the signal attenuation at locations farther away from the detection objective and was used in later steps to resolve multiview fusion ambiguities.[]{data-label=\"fig:UncertaintyVisualizationAndFuzzySets\"}](Figure4.png){width=\"\\hsize\"}\n\n### Segmentation {#sec:Segmentation}\n\nAfter the seed detection stage, a segmentation operator is used to extract the regions and regional properties of all detected objects from the simulated 3D image stacks, [*e.g.*]{}, using the algorithms presented in [@Al-Kofahi10; @Stegmaier14; @Faure16]. For demonstration purposes, we further improved an algorithm based on adaptive thresholding using Otsu\u2019s method [@Otsu79] and a watershed-based splitting of merged objects [@Beare06] (OTSUWW) as described in [@Stegmaier14]. Therefore, propagated information from the seed detection stage and estimated FSMD values of extracted segments are used to improve the algorithmic efficiency and the segmentation quality (OTSUWW+U).\n\nBased on the extracted statistical quantities of the benchmark images (\\[tab:Segmentation:GroundTruthStatistics\\]), we derive the parameter vector $\\pmb{\\theta}=(a,b,c,d)^\\top$ of the trapezoidal fuzzy set membership function for each considered feature using the minimum and maximum values as $a,d$ parameters, respectively. The remaining parameters $b,c$ were set to the 5%-quantile and the 95%-quantile. This parameterization ensured that all values smaller or larger than the maximum values obtained a membership degree of zero and that 90% of the data range has a membership value of one assigned. Of course, this parameterization is application and data dependent and can be customized, [*e.g.*]{}, to adjust the behavior for extrema at the lower and upper spectrum of the value range. For simplicity, the focus was put on volume and size information of the objects. In the absence of ground truth data, the transition regions for the fuzzy sets can be identified by a manual analysis of objects that deviate from the expectation at the lower and the upper feature value range, [*e.g.*]{}, using software tools such as Fiji, ICY or Vaa3D [@Schindelin12; @Chaumont12; @Peng14].\n\nMinimum, maximum and quantile values were used to formulate fuzzy sets for each of the features. Individual fuzzy sets were combined using the minimum operator, to obtain a single membership degree value to the fuzzy set of being a valid object. \\[tab:Segmentation:GroundTruthStatistics\\]\n\nAs depicted in \\[fig:Segmentation:GroundTruthStatistics\\], the shapes of the fuzzy set membership functions derived from the statistical quantities resemble the respective distribution observed in the feature histograms. We used the $\\min$-operator to combine the individual fuzzy set membership degrees to a single value, [*i.e.*]{}, the combined FSMD value directly corresponded to the membership degree of the feature that deviated the most from the specified expected range. Of course, the size criteria discussed here should only be considered as an exemplary illustration. There are various other features that can potentially be used to assess and improve segmentation results, [*e.g.*]{}, integrated intensity, edge information, local entropy, local signal-to-noise ratios (SNR), principal components, weighted centroids and many more. Furthermore, if colocalized channels are investigated, complementary information can be used to formulate more complex decision rules. In the case of a fluorescently labeled nuclei and membranes that are imaged in different channels [@Fernandez10; @Khan14a; @Stegmaier16], rules like *\u201ceach cell has exactly one nucleus\u201d* can be formalized in the same way using the fuzzy set membership functions for a quantification of the available prior knowledge.\n\n![Feature histograms (top) and fuzzy set membership functions (bottom) for volume ($\\pmb{\\theta}_\\text{vol}=(449,617,1405,2016)^\\top$), width ($\\pmb{\\theta}_\\text{w}=(13, 15, 24, 31)^\\top$), height ($\\pmb{\\theta}_\\text{h}=(13, 15, 24, 34)^\\top$) and depth ($\\pmb{\\theta}_\\text{w}=(3, 5, 8, 11)^\\top$).[]{data-label=\"fig:Segmentation:GroundTruthStatistics\"}](Figure5.png){width=\"\\hsize\"}\n\nThe identified FSMD values of segmented objects, were then used to detect under-segmentation errors produced by Otsu\u2019s method that needed be to further split to match the expected object size. This was realized in the investigated OTSUWW+U implementation using the seed points of the LoGNSM+F+U method (\\[sec:SeedDetection\\]). In contrast to the previous approach (OTSUWW, [@Beare06; @Stegmaier14]), the watershed-based splitting technique was only applied to objects that were known to be larger than expected. Using a parallelization strategy similar to the one discussed in [@Stegmaier14], all segments with combined FSMD values below $\\beta_{21}$ (backward threshold for Operator $2$ and Linguistic Term $1$) that corresponded to objects larger than expected (Case 3, \\[sec:FuzzySetMembershipFunctions\\]) were distributed among the available CPU cores and a seeded watershed approach was used for object splitting in each of the cropped regions of the image [@Beare06]. This approach was much faster than directly applying the watershed algorithm on the entire image (OTSUWW), due to the uncertainty-guided, locally applied processing of erroneous objects. After splitting merged objects, the connected components of the image needed to be identified once more and the uncertainty values were re-evaluated as well, to provide the updated information to the subsequent processing operators.\n\nTo further improve the results with respect to false positive detections observed for higher noise levels, segments with combined FSMD values below $\\alpha_{21}=0.1$ (forward threshold for Operator $2$ and Linguistic Term $1$), [*i.e.*]{}, objects that were smaller than the expected object size (Case 2 and Case 4, \\[sec:FuzzySetMembershipFunctions\\]) were removed from the label images. To facilitate the implementation of the uncertainty-guided segmentation, it was realized using just a single fuzzy set for the correct class of objects (Case 1, \\[sec:FuzzySetMembershipFunctions\\]) to identify objects that needed further consideration. To determine if objects with low FSMD values were smaller or larger than the expectation, a comparison to the boundaries of the expected valid range was performed (trapezoidal fuzzy set parameters $b, c$). Threshold values were identified using the interactive graphical user interface presented in [@Stegmaier16].\n\n### Tracking {#sec:Tracking}\n\nThe final step of the investigated image analysis pipeline was the tracking of all detected objects, [*i.e.*]{}, to identify the correct correspondences of detected objects in subsequent time points of the acquired images. For illustration purposes, we used a straightforward nearest-neighbor tracking approach implemented in the open-source MATLAB toolbox Gait-CAD [@Stegmaier12]. Each object present in a frame was associated with the spatially closest object in the subsequent frame. This procedure was applied to every frame of the data set in order to get a complete linkage of all objects.\n\nIn addition to tracking the results obtained from the segmentation methods introduced in the previous section, we present an alternative approach that combines a flawed segmentation with provided seeds (OTSU+NN+U). Inspired by conservation tracking methods [@Schiegg13], we leave the flawed segmentation results produced by the respective algorithms unchanged and additionally provide detected seed points to the tracking algorithm instead of actually using the seed points for splitting in the image domain [@Stegmaier12a]. As shown in \\[tab:Segmentation:DetectionPerformance\\], the segmentation quality achieved by OTSU is in principle not reasonably usable without the watershed-based object splitting. Nevertheless, the tracking algorithm could be extended to decide which of the information was reliable and suitable for tracking and could optionally fall back to the provided seed points if the segmentation quality was insufficient. Therefore, we used the FSMD values provided by the segmentation stage. Using an empirically determined threshold of $\\beta_{31} = 0.9$ (backward threshold for Operator $3$ and Linguistic Term $1$), all objects with an aggregated FSMD value lower than the threshold were not tracked with the actual segment, but with the seed points the respective segment contained. The forward threshold parameter was set to $\\alpha_{31} = 0.0$ in order to report all tracking results.\n\nResults and Discussion {#sec:Experiments}\n======================\n\nThe functionality of the proposed approaches was validated on simulated 3D benchmark images. The data sets contained images with different numbers of objects (`SBDE1`), different noise levels (`SBDE2`) and a set of $50$ sequential time points with $1000$ moving and interacting objects (`SBDE3`). The `SBDE3` data set additionally included a multiview simulation, [*i.e.*]{}, at each time point, two simultaneous images from opposite direction were generated. An overview of the generated benchmark data sets is provided in \\[tab:Appendix:BenchmarkDatasetsEmbryo\\] and a brief description on the data set generation is provided in \\[sec:chap4:Benchmark\\] as well as \\[fig:chap4:Benchmark:Illustration\\] and \\[fig:chap4:Benchmark:TimeSeries\\].\n\nSeed Point Detection Validation\n-------------------------------\n\nTo validate the proposed improvements of the LoG-based seed detection algorithm, the `SBDE1` and `SBDE2` benchmark data sets were used (\\[tab:Appendix:BenchmarkDatasetsEmbryo\\]) with the parameters listed in \\[tab:chap4:SeedDetection:ParameterizationTable\\] and the performance measures described in \\[sec:PerformanceAssessment\\]. The obtained values are summarized in \\[tab:SeedDetection:DetectionPerformance\\], whereas each entry of the table corresponds to the arithmetic mean value of the independently obtained results on the ten benchmark images of `SBDE1`.\n\nQuantitative performance assessment of the LoG-based seed detection methods. The criteria are true positives (TP), false positives (FP), false negatives (FN), recall, precision, F-Score, the distance to the reference (Dist., smaller values are better) as well as the achieved time performance measures in seconds (smaller values are better) and voxels per second (larger values are better). All values represent the arithmetic mean of the individually processed benchmark images. \\[tab:SeedDetection:DetectionPerformance\\]\n\nThe quantitative analysis confirmed that the proposed extensions of LoGSM could improve the algorithmic performance by up to $9.2\\%$ with respect to the F-Score. LoGSM had few false positive detections but on the other hand missed many objects due to the strict maximum detection (recall of $0.77$ and precision of $1.0$). The recall could be improved by $18.2\\%$ to a value of $0.91$ by additionally allowing non-strict maxima (LoGNSM). However, this adaption concurrently raised the number of false positives and thus lowered the precision by $16.0\\%$ to $0.84$, as objects with maximum plateaus were detected multiple times. These multi-detection errors could be successfully removed using the proposed fusion technique, which was reflected in an F-Score value of $0.95$ for LoGNSM+F(+U), [*i.e.*]{}, compared to the LoGSM method, the F-Score was increased by $9.2\\%$. Regarding the processing times, the additional effort for a redundant detection was almost negligible, as the non-strict maximum detection simply detected more seed points during the same iteration over the image. The seed point fusion was performed directly in the feature space and was therefore also insignificant compared to the preceding processing steps. For the feature set described here, using the uncertainty-based object rejection (LoGNSM+F+U) only slightly improved the results compared to directly fusing and filtering the data using the hard intensity threshold but increased the processing time by $35\\%$. LoGNSM+F yielded almost identical results and required only $6\\%$ more processing time compared to LoGSM. Nevertheless, all objects were equipped with an uncertainty value that was propagated through the pipeline and proved to be beneficial to filter, fuse and correct the extracted data in subsequent steps. In addition, it should be noted that the processing time required for the image analysis easily exceeds the fuzzy set calculations as soon as the images get larger.\n\nFurthermore, we tested the performance of the seed detection under different image noise conditions using the `SBDE2` data set, which contained images with different settings for the additive Gaussian noise standard deviation ($\\sigma_{\\text{agn}} \\in [0.0005, 0.01]$). Seeds from these images were extracted using the LoGSM, LoGNSM, LoGNSM+F and LoGNSM+F+U algorithms and the intensity thresholds were determined for each of the noise levels individually using the semi-automatic graphical user interface described in [@Stegmaier16AT]. For higher noise levels, the number of detections in background regions heavily increased and it became ambiguous to determine true positive detections in low-contrast regions. The manual threshold was therefore adjusted such that the false positive detections were minimized and only unambiguous seeds were considered. This continuous threshold adaption is also the reason for constant (\\[fig:SeedDetection:NoiseLevelInfluence\\]A, C) or increasing precision (\\[fig:SeedDetection:NoiseLevelInfluence\\]B), as it was easier to identify false positives rather than false negative detections in noisy image regions. Objects were robustly detected down to a signal-to-noise ratio of $5$ (\\[fig:SeedDetection:NoiseLevelInfluence\\]), which was close to the visual limit of detection [@Murphy12] and emphasized the uncertainty-based improvements.\n\n![Assessment of the seed detection performance for the different noise levels of the `SBDE2` data set. The performance measures recall, precision and F-Score are plotted versus the additive Gaussian noise level parameter $\\sigma_{\\text{agn}}$ for LoGSM (A), LoGNSM (B) and LoGNSM+F(+U) (C). As LoGNSM+F and LoGNSM+F+U produced identical results with respect to recall, precision and F-Score, the plots are combined to a single panel (\\[tab:SeedDetection:DetectionPerformance\\]). The influence of the noise level on the signal-to-noise ratio of the images is plotted in (D).[]{data-label=\"fig:SeedDetection:NoiseLevelInfluence\"}](Figure6.png){width=\"\\hsize\"}\n\nSegmentation Validation\n-----------------------\n\nThe segmentation performance was validated using the `SBDE1` and the `SBDE2` data sets (\\[tab:Appendix:BenchmarkDatasetsEmbryo\\]). In addition to the algorithms described in \\[sec:Segmentation\\] (OTSU, OTSUWW, OTSUWW+U), the segmentation quality obtained by the TWANG method (see [@Stegmaier14] for details) was added and quantitatively compared to a TWANG version (TWANG+U) that relied on the improved seed detection operator introduced in the previous section (LoGNSM+F+U). The respective parameterization and a brief description of each algorithm is provided in \\[tab:chap4:Segmentation:ParameterizationTable\\] and the validation measures are summarized in \\[sec:PerformanceAssessment\\] and [@Coelho09]. In \\[tab:Segmentation:DetectionPerformance\\], the quantitative segmentation quality results obtained on the `SBDE1` data set are summarized. The Rand index (RI) value was almost identical for all algorithms and OTSU yielded the highest value. The enhanced adaptive threshold-based techniques yielded an 11.0% better Jaccard index (JI) value than TWANG+U and the best normalized sum of distances (NSD) value was obtained by OTSUWW+U. Considering the results for RI, JI and NSD, the global threshold-based techniques (OTSU\\*) produced slightly more accurate results (0.3%, 5.3% and 12.0%, respectively) for the objects they were still able to resolve compared to the best results of the TWANG-based methods. Both TWANG-based methods on the other hand produced the minimal amount of topological errors with respect to split and merged objects compared to all OTSU-based methods. The number of added objects was minimal for OTSUWW+U. TWANG+U produced slightly more added objects than TWANG but efficiently detected far more objects. Thus, the F-Score values achieved by TWANG+U were further increased by $8.4$% compared to TWANG, [*i.e.*]{}, TWANG+U produced the best results with the fewest topological errors (F-Score $0.9$). The low amount of split and merged nuclei for TWANG originate from the single-cell extraction strategy, rather than using a global threshold as performed in the OTSU-based methods. The relatively large amount of added objects detected by the TWANG segmentation were mostly no real false positive detections, but segments where most of the extracted region intersected with the image background instead of the actual object and are thus considered as false positives.\n\nThe criteria used to compare the algorithms are the Rand index (RI), the Jaccard index (JI), the normalized sum of distances (NSD) and the Hausdorff metric (HM) as described in [@Coelho09]. Additionally, the topological errors were assessed by counting split, merged, added and missing objects. Precision, recall and F-Score are based on the topological errors by considering split and added nuclei as false positives and merged and missing objects as false negatives, respectively. The achieved time performance was measured in seconds (smaller values are better) and voxels per second (larger values are better). All values represent the arithmetic mean of the individually processed benchmark images. \\[tab:Segmentation:DetectionPerformance\\]\n\nRegarding the processing times OTSU was the fastest approach, but due to the poor quality without the uncertainty-based extension it was not really an option for a reliable analysis of the image data. TWANG and TWANG+U were $1.8$ times slower than the plain OTSU method but $2.4$ and $3.8$ times faster than OTSUWW and OTSUWW+U, respectively. At the same time, TWANG+U was the most precise approach (F-Score $0.9$). Furthermore, OTSUWW+U was $1.6$ times faster than OTSUWW due to the focused object splitting and additionally produced better results due to the improved seed detection and noise reduction.\n\nThese results confirmed that the uncertainty information could be efficiently exploited to guide computationally demanding processing operators to specific locations and thus, to speed up processing operations while the result quality was preserved or even improved.\n\n![image](Figure8.png){width=\"\\hsize\"}\n\nExemplary FSMDs of the final segmentation results of different algorithms are depicted in \\[fig:Segmentation:Comparison\\] and provide a convenient visualization to instantly assess the segmentation quality and to identify potential problems of the methods even by non-experts. All algorithms suffered from the light attenuation in the axial direction. Especially, the techniques that relied on a single global intensity threshold had problems to identify the objects located in these low-contrast regions. Besides missing many objects, OTSU additionally merged many of the high intensity objects to a single large blob. Especially, in the z-direction many mergers occurred due to the lower sampling in this direction. However, these merged regions could be successfully split to a large extent using the proposed seed-based splitting techniques (OTSUWW, OTSUWW+U). As TWANG directly operated on the provided seeds, it was still able to extract most of the objects in these regions and yielded even higher recall values using the LoGNSM+F+U seed points. However, due to the low-contrast, the segmentation quality of the extracted segments in these regions was reduced. In \\[tab:SeedDetection:DetectionPerformance\\], this is reflected by the increased number of added objects for TWANG and TWANG+U, which were mostly no real false positives as described above.\n\nTo investigate the impact of the signal-to-noise ratio of the images to the segmentation quality, the benchmark data set `SBDE2` was processed using all five algorithms. The segmentation quality of all adaptive thresholding-based methods was heavily affected by the noise level of the images yielding poor precision and recall values even for the lowest noise levels (\\[fig:Segmentation:NoiseLevelInfluence\\]A, B). This was caused by the global threshold that on the one hand merged a lot of objects and on the other hand detected a high amount of false positive segments. The uncertainty-based method OTSUWW+U successfully preserved the increased recall of OTSUWW and at the same time substantially increased the precision to an almost perfect level for noise parameters of $\\sigma_\\text{agn} < 0.003$ (\\[fig:Segmentation:NoiseLevelInfluence\\]C). The increasing number of small segments observed for OTSU and OTSUWW could be efficiently filtered using the uncertainty-based object rejection. As TWANG heavily depended on the quality of the provided seeds, the observed curves in \\[fig:Segmentation:NoiseLevelInfluence\\]C, D show a high correlation to the seed detection performance (\\[fig:SeedDetection:NoiseLevelInfluence\\]A, C) and render it as a suitable method even for higher noise levels. The improved seed detection of LoGNSM+F+U also directly affected the quality of the TWANG+U segmentation. Note that the seed points were adjusted for each of the noise levels, [*i.e.*]{}, subtle variations in the precision and recall (\\[fig:Segmentation:NoiseLevelInfluence\\]D, E) values are caused by a subjective manual threshold adaption.\n\n![Performance evaluation of the segmentation methods OTSU (A), OTSUWW (B), OTSUWW+U (C), TWANG (D) and TWANG+U (E) on images of the `SBDE2` data set with different signal-to-noise ratios. The methods based on adaptive thresholding (OTSU, OTSUWW) suffered from high noise levels and produced a successively increased amount of false positive detections, which could be efficiently suppressed using the uncertainty framework-based extension (OTSUWW+U). The result quality of both TWANG versions directly correlated with the quality of the provided seed points, [*i.e.*]{}, TWANG+U benefited from the improved detection rate of LoGNSM+F+U.[]{data-label=\"fig:Segmentation:NoiseLevelInfluence\"}](Figure7.png){width=\"\\hsize\"}\n\nTracking Validation {#sec:TrackingValidation}\n-------------------\n\nThe tracking validation was performed on the `SBDE3` data set, which consisted of $50$ frames with two simultaneously acquired rotation images for each frame, yielding a total number of $100$ frames that needed to be processed. Segmentation was performed using OTSUWW, OTSUWW+U, TWANG and TWANG+U separately for all time points and view angles. To obtain a single set of objects for each time point, segmentation results of different view angles were fused using a segment-based fusion approach described in \\[sec:MultiviewFusion\\] (Additional file 1). The centroids of all detected objects were then used to perform the nearest neighbor tracking (NN). In addition, the method described in \\[sec:Tracking\\] (OTSU+F+NN+U) was applied to the test data set and the Otsu-based threshold was applied to both rotation images independently and the resulting binary images were then fused by simply using the maximum pixel value of the two images. FSMD values were then estimated on the connected components of the fused image using the same fuzzy sets as for the segmentation step. The obtained tracking results are summarized in \\[tab:Tracking:PerformanceTable\\] and \\[fig:Tracking:Results\\].\n\nQuantitative performance assessment of a nearest neighbor tracking algorithm (NN) applied on different segmentation results. Two algorithms without uncertainty-based improvements (OTSUWW, TWANG) were compared to enhanced pipelines that explicitly incorporated prior knowledge-based uncertainty treatment (OTSUWW+U, TWANG+U). Furthermore, an OTSU-based segmentation with additional seed points from LoGNSM+F+U (OTSU+F+NN+U) was used with an adapted tracking algorithm. Note that the segmentation produced by OTSU+F+NN+U was not usable for other purposes than tracking due to many merged regions (indicated by ($^\\ast$)). The validation measures correspond to true positives (TP), false positives (FP), false negatives (FN), redundant edges (Red.), missing edges (Miss.) and merged objects (Merg.). Furthermore, recall, precision, F-Score and the TRA measure were calculated as described in \\[sec:PerformanceAssessment\\] and [@Maska14]. Processing times are average values for applying segmentation and tracking on a single image and were measured in seconds (lower values are better) and voxels per second (higher values are better). \\[tab:Tracking:PerformanceTable\\]\n\nBoth pipelines without uncertainty treatment reached the lowest tracking accuracy with respect to the tracking (TRA, see \\[sec:PerformanceAssessment\\]) quality measure due to an increased number of missing objects (recall of $0.89$ for OTSUWW+NN, and $0.88$ for TWANG+NN) and a high number of false positive detections (precision of $0.82$ for OTSUWW+NN). Of course, these missing objects directly correlated with the number of missing edges and explain the $222.4\\%$ (OTSUWW+NN) and $139.9\\%$ (TWANG+NN) higher amount of missing edges compared to the best scoring algorithm in this category (OTSU+F+NN+U). Furthermore, OTSUWW+NN suffered from many merged regions that also contributed to the $8.2\\%$ lower recall value compared to OTSU+F+NN+U. In contrast to this, all uncertainty-enhanced methods provided comparable results, with the best results achieved by OTSU+F+NN+U and TWANG+U+NN. Although the amount of false negatives of TWANG+U+NN was almost halved compared to TWANG+NN, the detection rate was still the largest problem of this pipeline resulting in a $21.2\\%$ higher amount of missing edges compared to OTSU+F+NN+U. On the other hand, OTSU+F+NN+U still suffered from the same under-segmentation tendency as observed for OTSUWW+NN and OTSUWW+U+NN and consequently missing edges were its main problem. With respect to false positive detections both TWANG-based methods provided the best results, with precision values of $1.0$ due to a high quality seed detection that benefitted from the simultaneous multiview acquisition. This reflects an improvement of the precision obtained by the TWANG-based methods of $22.0\\%$ compared to OTSUWW+NN, which did not have an uncertainty-based object exclusion and thus an increased amount of false positive detections in background regions. The reason for the slightly higher number of redundant edges observed for the two TWANG-based methods is not yet fully clear. Most likely, the seed detection already provided a redundant seed to the segmentation method, which produced two nearby segments that were in turn counted as a redundant segment by the tracking evaluation. However, even for the worst algorithm in this category (TWANG+NN) redundant edges were only observed for $1.4\\%$ of the tracked objects and thus play a minor role compared to the other tracking errors. As shown in \\[fig:Tracking:Results\\], the number of false negatives and merged objects directly correlated with the density of the objects, [*i.e.*]{}, the closer the objects are to each other, the more under-segmentation errors occurred. However, this was not the case for TWANG-based methods due to the explicit prior knowledge about the object size that is incorporated to the algorithm. The OTSU+F+NN+U method produced the best tracking result and was the second fastest method (TRA value of $0.94$ and average processing time of $18.7s$) closely followed by TWANG+U+NN (TRA value of $0.92$ and average processing time of $21.6s$). Compared to OTSUWW+NN, TWANG+U+NN and OTSU+F+NN+U provide superior quality in all categories and in particular an increase of the TRA measure by $ 13.6\\%$ and $16.0\\%$, and a decrease of processing times by $58.0\\%$ and $63.7\\%$, respectively. Thus, the two latter methods represent the best quality vs.\u00a0speed trade-off and are suitable for large-scale analyses. Although OTSU+F+NN+U provided excellent results in this comparison, it should be noted that the extracted segmentation masks were largely merged and an object splitting approach as performed for OTSUWW would be required if object properties need to be known. An additional object splitting approach, however, would eradicate the performance benefit of the method and, [*e.g.*]{}, the TWANG+U+NN pipeline should be favored in this case.\n\n![Quantitative performance assessment of a nearest neighbor tracking algorithm (NN) applied on different segmentation results obtained on the `SBDE3` data set. Two algorithms without uncertainty-based improvements as described in [@Stegmaier14] (OTSUWW, TWANG) were compared to enhanced pipelines that explicitly incorporate prior knowledge-based uncertainty treatment (OTSUWW+U, TWANG+U, OTSU+F+NN+U). (\\*) indicates that the respective algorithm did not produce a usable segmentation image, as the correction was solely performed at the tracking step.[]{data-label=\"fig:Tracking:Results\"}](Figure9.png){width=\"\\hsize\"}\n\nTo fall back on seed point information has no benefit for segmentation methods like TWANG, where the algorithmic design already only extracts a single segment per seed point and literally no merged objects exist. However, an interesting extension to consider in upcoming work might be a combination of the LoGNSM+F+U and the OTSU-based segmentation for seed detection and to feed these seeds to the TWANG algorithm, to reach both a further reduced amount of missed objects and a reduced amount of merged objects. Moreover, the temporal coherence was not yet considered in the investigated framework, [*i.e.*]{}, additionally allowing a nearest neighbor matching over multiple frames could potentially also help to reduce the number of missing and redundant detections.\n\nApplication to Light-Sheet Microscopy Images of Zebrafish Embryos {#sec:ApplicationExample}\n-----------------------------------------------------------------\n\nThe presented framework was successfully used for the automated analysis of large-scale 3D+t microscopy images of developing zebrafish embryos [@Stegmaier14; @Kobitski15; @Stegmaier16a; @Stegmaier16AT]. In particular, we used the LoGNSM+U method, [*i.e.*]{}, seed points were detected using a non-strict local maximum detection with a subsequent fusion of redundant detections and a false positive suppression based on the axial location of the seeds as well as their fluorescence intensity information (\\[sec:SeedDetection\\]). These seeds were then provided to the TWANG algorithm as described in [@Stegmaier14] and the segments of different views were combined using a segment-based fusion approach (\\[sec:MultiviewFusion\\] in Additional file 1 and [@Kobitski15]). Finally, a nearest-neighbor tracking was applied to the detected objects to obtain the movement trajectories (TWANG+U+NN). As there were about $8$ million dynamically interacting objects in total for the investigated time period of about 3-10 hours post fertilization (hpf), manual labeling and validation were impossible. Qualitative results of the seed detection, the segmentation and the tracking stage are depicted in \\[fig:ApplicationExample\\].\n\n![image](Figure10.png){width=\"\\hsize\"}\n\nConclusion\n==========\n\nIn this contribution, we presented a general concept for the mathematical formulation of prior knowledge and showed how image analysis pipelines can be equipped with the formalized prior knowledge to make more elaborate decisions. The framework includes the propagation of estimated result uncertainties, to be able to inform downstream pipeline operators about the validity of their input data and to potentially improve their results. Besides these general concepts, we demonstrated how an exemplary pipeline consisting of seed point detection, segmentation, multiview fusion and tracking could be systematically extended by the proposed uncertainty considerations, in order to filter, repair and fuse produced data. The performance of all proposed improvements was quantitatively assessed on a new and comprehensive validation benchmark inspired by light-sheet microscopy recordings of live specimen. The extensions proofed their superior performance compared to the plain pipelines and only had a low impact on the processing times due to the lightweight adaptions of the propagated feature matrices. Thus, the proposed framework represents a powerful approach to improve the quality and efficiency of image analysis pipelines. Several components of the presented framework were successfully used to analyze large-scale 3D+t light-sheet microscopy images of developing zebrafish embryos as described in \\[sec:ApplicationExample\\] and [@Stegmaier12; @Kobitski15].\n\nFor simplicity, we used mostly focused on simple processing methods to illustrate the general concepts. However, extending more complex seed detection, segmentation or tracking algorithms with the presented concepts of filtering, splitting and fusion should work analogously if the uncertainty-based corrections are considered as a post-processing strategy of each processing operator. Especially the tracking step offers a lot of potential to be further improved. The uncertainty framework could be exploited to classify movement events, [*e.g.*]{}, to detect object divisions or to reconstruct missing objects using the temporal coherence of the objects. However, this was beyond the scope of this paper and will be addressed in upcoming work. In addition to the algorithmic improvements, we showed how the respective fuzzy sets can be parameterized based on available prior knowledge, such as feature histograms or knowledge about acquisition deficiencies. The respective shape of the fuzzy sets has to be determined based on the desired outcome. For instance, the false positive suppression at the seed point detection stage could be performed with a fixed threshold instead of an explicit usage of fuzzy sets for the intensity-based features. However, to model the increasing uncertainty (decreasing FSMD value) in regions farther away from the detection objective using a trapezoidal shape was more appropriate. Further work has to be put on the automatic determination of the involved fuzzy sets, [*e.g.*]{}, using a semi-automatic approach for a manual classification of a representative subset of data.\n\nAcknowledgements {#acknowledgements .unnumbered}\n================\n\nThe work was funded by the German Research Foundation DFG (JS, Grant No MI 1315/4, associated with SPP 1736 Algorithms for Big Data) and to the Helmholtz Association in the program BioInterfaces in Technology and Medicine (RM).\n\nValidation Benchmark {#sec:chap4:Benchmark}\n====================\n\nTo provide a thorough validation of the entire pipeline comprised of seed point detection, segmentation, multiview fusion and tracking, we used our recently presented approach for generating comprehensive validation benchmarks [@Stegmaier16a] with an adapted object movement simulation. In brief, object locations, object movements and object interactions were simulated over multiple time points to obtain movement behaviors that resembled biological specimens. At each simulated position, a video snippet containing a simulated fluorescent cellular nucleus [@Svoboda12] was added to an artificial 3D image, to form the data basis for each time point. To simulate acquisition deficiencies, the simulated images were disrupted by additive Gaussian noise, Poisson shot noise, a point-spread-function simulation [@Preibisch14], light attenuation and multiview acquisition simulation as detailed in [@Stegmaier16a]. All acquisition deficiencies can be added to the simulated raw images using an XPIWIT pipeline [@Bartschat16] and the pipeline can be downloaded from <https://bitbucket.org/jstegmaier/embryomicsbenchmark/downloads/AcquisitionSimulationPipeline.zip>.\n\nFor the sake of simplicity, we simulated objects that were moving on a spherical surface. Instead of using object displacements of a real embryo as described in [@Stegmaier16a], the simulated objects only moved due to density changes and the resulting repulsive and adhesive forces acting between neighboring objects. Furthermore, the simulation was constrained to a spherical surface to prevent arbitrary movement in the simulation space. A schematic illustration of the simulated specimen and an overview of the involved simulation steps of the benchmark is shown in \\[fig:chap4:Benchmark:Illustration\\]. The repulsive ($\\Delta\\mathbf{x}^{\\text{rep}}$) and adhesive forces ($\\Delta\\mathbf{x}^{\\text{adh}}$) as well as the parameterization was taken from [@Macklin12] and the boundary constraint was defined as: $$\\begin{gathered}\n    \\Delta\\mathbf{x}^{\\text{bdr}}(\\mathbf{x}, \\mathbf{c}, r_i, r_o, a) = \n    \\begin{cases} \n        \\frac{\\mathbf{x}-\\mathbf{c}}{\\Vert\\mathbf{x}-\\mathbf{c}\\Vert} \\cdot \\left( 1-\\frac{1}{e^{-a(\\Vert\\mathbf{x}-\\mathbf{c}\\Vert-r_i)}} \\right), & \\Vert\\mathbf{x}-\\mathbf{c}\\Vert < r_i \\\\ \n        -\\frac{\\mathbf{x}-\\mathbf{c}}{\\Vert\\mathbf{x}-\\mathbf{c}\\Vert} \\cdot \\left(1-\\frac{1}{e^{a(\\Vert\\mathbf{x}-\\mathbf{c}\\Vert-r_o)}} \\right), & \\Vert\\mathbf{x}-\\mathbf{c}\\Vert > r_o \\\\ \n        \\mathbf{0}, & \\text{else}.\n     \\end{cases} \n\\label{eq:chap4:Benchmark:BoundaryForce}\\end{gathered}$$ In \\[eq:chap4:Benchmark:BoundaryForce\\], $\\mathbf{x}$ is the centroid of the considered object, $\\mathbf{c}$ is the center of the bounding volumes, $r_i$ and $r_o$ are the radii of the inner and the outer sphere, respectively, and finally $a$ controls the shape of the sigmoidal boundary potential function. $\\Delta\\mathbf{x}^{\\text{bdr}}$ only contributed to the displacement of a simulated object if the object was already out of the boundary. This additional displacement component prevented objects from entering the inner bounding sphere and from escaping the outer bounding sphere of the simulated embryo (\\[fig:chap4:Benchmark:Illustration\\]A). Analogous to the formulation used in [@Stegmaier16a], the total displacement vector of a single object at a given time point can be summarized to:\n\n![image](SupplementaryFigure1.png){width=\"\\hsize\"}\n\n$$\\begin{aligned}\n    \\Delta\\mathbf{x}_i^{\\text{tot}} = &w_{\\text{bdr}} \\cdot \\Delta\\mathbf{x}^{\\text{bdr}}(\\mathbf{x}_i) + \\nonumber \\\\\n     & \\sum_{j\\in\\{1,...,N\\}}^{i\\neq j}{\\left[w_{\\text{rep}} \\cdot \\Delta\\mathbf{x}^{\\text{rep}}(\\Vert\\mathbf{x}_i-\\mathbf{x}_j\\Vert) + w_{\\text{adh}} \\cdot \\Delta\\mathbf{x}^{\\text{adh}}(\\Vert\\mathbf{x}_i-\\mathbf{x}_j\\Vert)\\right]}.\n    \\label{eq:chap4:Benchmark:TotalForce}\\end{aligned}$$\n\nThe weights of the adhesive and repulsive displacement components were set to the default values mentioned in [@Macklin12]: $w_{\\text{adh}}=0.52$, $w_{\\text{rep}}=1.0$. Moreover, the weight for the boundary constraint $w_{\\text{bdr}}=3.0$ was manually adjusted, such that the interacting objects remained within the spherical boundaries. Note that these parameters were empirically determined and that the presented model does not necessarily represent an accurate physical simulation of the interacting objects. However, the determined parameters produced movement behaviors that were similar to the epiboly movements observed during early zebrafish development due to increased object densities and boundary constraints that caused a directed movement on a sphere surface. Exemplary benchmark images are shown in \\[fig:chap4:Benchmark:TimeSeries\\] for different time points and different additive Gaussian noise levels. Due to the availability of a complete ground truth, a detailed quantitative analysis of all involved pipeline steps could be performed with a single benchmark using the performance measures described in the following section. All validation datasets ($\\approx$70GB) used in this publication are available upon request.\n\n![image](SupplementaryFigure2.png){width=\"\\textwidth\"}\n\nPerformance Assessment {#sec:PerformanceAssessment}\n======================\n\nSeed Detection\n--------------\n\nThe seed detection quality was evaluated using the benchmark datasets `SBDE1` and `SBDE2` (\\[tab:Appendix:BenchmarkDatasetsEmbryo\\]). The intersections of the detected seeds with the labeled ground truth image were calculated. True positives (TP) were counted as ground truth objects that contained at least one seed point. Seed points that were detected in background regions or redundant detections of ground truth objects were considered as false positives (FP). Ground truth objects that did not contain a seed point were counted as false negatives (FN). Using TP, FP and FN, recall, precision and the F-Score (harmonic mean of precision and recall) were calculated. For all true positives, the average distance to the centroids of the respective ground truth objects was additionally calculated.\n\nSegmentation {#segmentation}\n------------\n\nThe segmentation quality was assessed using the benchmark datasets `SBDE1` and `SBDE2` (\\[tab:Appendix:BenchmarkDatasetsEmbryo\\]). As the provided ground truth of the benchmark contained the complete label images of each frame, a detailed quantitative assessment of the automatic segmentation quality could be performed. The set of segmentation validation measures proposed by Coelho [*et al.*]{}\u00a0was used, namely the Rand index (RI), the Jaccard index (JI), the normalized sum of distances (NSD) and the Hausdorff metric (HM). A detailed description of the measures can be found in [@Coelho09]. Topological errors produced by the automatic segmentation were separated into added, missing, split or merged objects. Besides the error counts, this topological information was used to define the number of false positives as the sum of split and added cells and analogously the false negatives as the sum of merged and missing cells. These values were then used to calculate recall, precision and F-Score.\n\nTracking {#tracking}\n--------\n\nTo assess the tracking quality, the `SBDE3` dataset was used (\\[tab:Appendix:BenchmarkDatasetsEmbryo\\]). The comparison of the investigated algorithms was performed using the `TRA` measure as described by Ma[\u0161]{}ka [*et al.*]{}[@Maska14]. This measure was calculated by considering the tracking result as an acyclic oriented graph and by comparing this graph to the respective ground truth graph. The inverted, weighted and normalized number of required changes to transform the automatically generated graph to the ground truth graph yielded the normalized `TRA` measure (higher values are better with 1 being ideal). As the centroids of all objects, the complete temporal association and the object ancestry was known for the simulated `SBDE3` dataset, the required ground truth graph could directly be generated using this data. To obtain a more detailed view on the errors made by the respective tracking algorithms, the number of false positive detections, false negative detections, incorrect edges, missing edges, redundant edges and merged objects were counted. Detailed descriptions of the validation measures are provided in [@Maska14].\n\nEvaluation Platform {#sec:Appendix:EvaluationPlatform}\n-------------------\n\nAll measurements with respect to processing times were performed on a desktop PC equipped with an Intel Core i7-2600 CPU @ 3.4GHz and 32GB of memory installed using the Windows 7 (x64) operating system.\n\nMultiview Fusion {#sec:MultiviewFusion}\n================\n\nA frequently used approach to partly overcome quality deficiencies in 3D microscopy that are caused by attenuation and scattering of light along the axial direction is the acquisition of multiple views from different perspectives [@Krzic12; @Chhetri15]. Such complementary image stacks of a specimen can be obtained either by using multiple oppositely arranged detection paths [@Tomer12; @Chhetri15] or by a rotation of the probe using a single camera [@Kobitski15]. Both strategies for the acquisition of multiview images require a subsequent fusion step of the information present in the individual views into a single consistent representation. The benchmark dataset `SBDE3` used for the tracking validation contains such a simulated multiview acquisition, [*i.e.*]{}, at each time point, two opposite images were simulated. To fuse to complementary views, we use a segmentation-based fusion approach that combines the results of separately applied segmentation on the different views to a single segmentation image. As the image transformation of $180^{\\circ}$ was already known from the benchmark generation step, the registration process was skipped. Corresponding segments were identified using a histogram-based approach, [*i.e.*]{}, a 2D label histogram was filled by iterating over all voxels of both 3D segmentation images and by successively increasing the 2D histogram bins indicated by the respective image label pairs. Segments that were only present in one or the other image could be found by searching for empty columns and rows of the histogram, respectively, and by copying these segments without any further consideration into the new result image. In the next step, the assignments of the remaining segments in both images were identified by searching for the respective label pair with the maximum overlap. In the case of segmentation fusion, it was not desired to perform a weighted average of the segments but rather to use the better segment for the new image. This was accomplished by simply selecting the segment with a higher FSMD value to the desired class of objects based on the FSMD values of the segments and the seed points that contained the information about the validity of the segment and the axial localization of the extracted objects, respectively. 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A compactness criterion for sets of automorphisms is established, generalizing the theorem by Burns and Spatzier that the full automorphism group, endowed with the compact-open topology, is a locally compact group.'\nauthor:\n- 'Nicolas Radu[^1]'\ndate: 'July 15, 2016'\ntitle: |\n    A topological characterization of the Moufang\\\n    property for compact polygons\n---\n\nIntroduction\n============\n\nCompact buildings form a natural generalization of compact projective planes. They were introduced by Burns and Spatzier in\u00a0[@Burns], and used in their proof of the Rank Rigidity Theorem for Riemannian manifolds of non-positive sectional curvature, see [@Burns2]. A prominent example of a compact building is provided by the spherical building associated with a semisimple algebraic group over a non-discrete locally compact field. Since the origin, the problem of characterizing those buildings associated with semisimple algebraic groups among all compact buildings has attracted much attention; in some sense, this is the topic of the encyclopaedic book\u00a0[@Salzmann], mainly devoted to the study of *connected* compact projective planes. The relevance of that problem is also well illustrated by the work of Burns and Spatzier: a key result from [@Burns] that is used in the proof of Rank Rigidity in [@Burns2] asserts that a thick irreducible *connected* compact spherical building of rank\u00a0$\\geq 2$ is the spherical building associated with a simple Lie group if and only if its automorphism group is strongly transitive (see Subsection\u00a02.2 for the definition of strong transitivity). This result has been improved by Grundh\u00f6fer, Knarr and Kramer who showed in [@GKK] and [@GKK2] that the same classification is true for connected buildings admitting a chamber transitive automorphism group. The starting point of the present article is the question whether such a statement could hold beyond the connected case. A precise formulation can be stated as\u00a0follows.\n\nLet $\\Delta$ be an infinite thick irreducible compact spherical building of rank at least\u00a0$2$. Then $\\Delta$ is the spherical building associated with a semisimple algebraic group over a non-discrete locally compact field if and only if $\\Delta$ is strongly transitive.\n\nHere again, one could even replace strong transitivity by chamber transitivity in this conjecture. Historically, a similar problem had been suggested by J.\u00a0Tits in the 1970\u2019s for finite buildings. More precisely, Tits conjectured that a finite thick irreducible building of rank\u00a0$\\geq 2$ should be the building associated with a finite simple group of Lie type if and only if the building has a strongly transitive automorphism group, see [@Tits]\\*[Conjecture\u00a011.5.1]{}. Tits\u2019 hope was that this could be helpful to the classification of the finite simple groups, which was an ongoing project at the time. Ironically, the latter classification was achieved first, and then used by Buekenhout and Van Maldeghem in their proof that Tits\u2019 conjecture is indeed accurate, see [@Buekenhout].\n\nAs mentioned above, the conjecture is certainly true for connected buildings by the work of Burns and Spatzier; we may therefore assume that $\\Delta$ is totally disconnected, since any disconnected compact building is so. Moreover, Grundh\u00f6fer, Kramer, Van Maldeghem and Weiss have proved that, under the same hypotheses as the conjecture, the building $\\Delta$ is the spherical building associated with a semisimple algebraic group over a non-discrete locally compact field if and only if $\\Delta$ is Moufang: this follows from\u00a0[@Grundhofer]\\*[Theorem\u00a01.1]{}. As is well known, Tits proved that every thick irreducible spherical building of rank\u00a0$\\geq 3$ is Moufang (see [@Tits]\\*[Addenda]{}), so that the conjecture above can be reduced to the following one, which appears as Question\u00a01.4 in [@Grundhofer].\n\nLet $\\Delta$ be an infinite thick irreducible compact totally disconnected spherical building of rank\u00a0$2$. If $\\Delta$ is strongly transitive, then $\\Delta$ is Moufang.\n\nIn the rest of this paper, we shall therefore focus on compact spherical buildings of rank\u00a0$2$, also called **compact polygons**. More precisely, a compact polygon of diameter $m$ is called a **compact $m$-gon**. It is irreducible if and only if $m \\geq 3$.\n\nOur first main result may be viewed as a first step towards the conjecture above.\n\n\\[theorem:Characterization\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $G$ be a closed subgroup of the group $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ of topological automorphisms which is strongly transitive on $\\Delta$. The following assertions are equivalent.\n\n(i) $\\Delta$ is Moufang.\n\n(ii) For each panel $\\pi$ of $\\Delta$, the closure in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ of the group of projectivities $\\Pi(\\pi)$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\n(iii) For each panel $\\pi$ of $\\Delta$, the closure of the natural image of $\\operatorname{\\mathrm{Stab}}_{G}(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\nThe notion of a convergence group is recalled in Subsection\u00a0\\[subsection:convergence\\] while the group of projectivities $\\Pi(\\pi)$ is defined in Subsection\u00a0\\[subsection:projectivities\\]. Conditions (i) and (ii) are both independent of the group $G$. In fact, Condition (ii) in Theorem\u00a0\\[theorem:Characterization\\] implies in particular that the closure in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ of the group of projectivities $\\Pi(\\pi)$ is a locally compact group (see Lemma\u00a0\\[lemma:lc\\] below). In the case of *connected* compact polygons, the latter property in turn is known to be equivalent to the Moufang condition, without assuming any extra homogeneity assumption on $\\Delta$ a priori (see [@Lowen]\\*[Theorem\u00a05.1]{} or [@Salzmann]\\*[Theorem\u00a066.1]{}). If one believes that the connected case is representative of the general case, one should therefore not expect Condition\u00a0(ii) from Theorem\u00a0\\[theorem:Characterization\\] to be satisfied by all infinite compact totally disconnected polygons.\n\nWe point out the following corollary of Theorem\u00a0\\[theorem:Characterization\\].\n\n\\[corollary:affine\\] Let $X$ be a locally finite thick irreducible affine building of rank\u00a0$3$. If $X$ is strongly transitive, then the building at infinity $X_\\infty$ is Moufang. In particular $X$ is the Bruhat\u2013Tits building associated with a semisimple algebraic group over a non-Archimedean local field.\n\nThe special case of Corollary\u00a0\\[corollary:affine\\] where $X$ is of type $\\tilde{A}_2$ was obtained by Van Maldeghem and Van Steen in\u00a0[@VM]\\*[Main Theorem]{}, while the general case has been obtained by Caprace and Monod in\u00a0[@CapraceMonod]\\*[Corollary\u00a0E]{} using CAT($0$) geometry. Our approach to Theorem\u00a0\\[theorem:Characterization\\] is different, but was partly inspired by theirs: instead of the CAT($0$) Levi decomposition used in their work, we use a purely algebraic counterpart of that result, due to Baumgartner and Willis\u00a0[@Baumgartner], and valid in the realm of totally disconnected locally compact groups.\n\nThe basic result that makes the structure theory of locally compact groups available in the study of compact buildings is the theorem, due to Burns\u2013Spatzier and valid for all thick irreducible compact spherical building $\\Delta$ of rank\u00a0$\\geq 2$, asserting that the topological automorphism group $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is a locally compact group, see [@Burns]\\*[Theorem\u00a02.1]{}. Equivalently, there exists an identity neighbourhood in the group $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ endowed with the compact-open topology that is compact. Our second main result is a compactness criterion for more general subsets of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ in the rank\u00a0$2$ case. Notice that we do not make any transitivity assumption on the automorphism group.\n\n\\[theorem:Criterion\\] Let $\\Delta$ be a thick compact $m$-gon with $m \\geq 3$ and $C, C'$ be opposite chambers of $\\Delta$. Denote by $D_0$ and $D_1$ the two chambers adjacent to (but different from) $C$ in the apartment containing $C$ and $C'$ and by $v_0$ (resp. $v_1$) the common vertex of $C$ and $D_0$ (resp. $D_1$). Let also $E_0$ (resp. $E_1$) be a chamber having vertex $v_0$ (resp. $v_1$) but different from $C$ and $D_0$ (resp. $D_1$). Let finally $U$ and $U'$ be closed subsets of $\\operatorname{\\mathrm{Cham}}\\Delta$ such that every chamber of $U$ is opposite every chamber of $U'$. Then for all $\\varepsilon > 0$, the (possibly empty) set $$J_\\varepsilon(C,C',U,U',E_0,E_1) := \\left\\{ \\varphi \\in \\operatorname{\\mathrm{Auttop}}(\\Delta) {\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\;}\\begin{array}{l}\n\\varphi(C) \\in U, \\ \\varphi(C') \\in U', \\\\\n\\rho(\\varphi(C), \\varphi(B_{v_i}(D_i, \\varepsilon))) \\geq \\varepsilon  \\ \\ \\forall i \\in \\{0,1\\},\\\\\n\\rho(\\varphi(D_i), \\varphi(B_{v_i}(C, \\varepsilon))) \\geq \\varepsilon  \\ \\ \\forall i \\in \\{0,1\\},\\\\\n\\rho(\\varphi(C), \\varphi(E_i)) \\geq \\varepsilon  \\ \\ \\forall i \\in \\{0,1\\},\\\\\n\\rho(\\varphi(D_i), \\varphi(E_i)) \\geq \\varepsilon  \\ \\ \\forall i \\in \\{0,1\\}\\\\\n\\end{array}\\\n\\right\\}$$ is compact, where $\\rho$ is some fixed metric associated to the topology on $\\operatorname{\\mathrm{Cham}}\\Delta$ and $B_{v}(C, r)$ denotes the open ball in $\\operatorname{\\mathrm{Cham}}(v)$ centered at $C$ and of radius $r$ (with respect to the metric $\\rho$).\n\n![Illustration of Theorem\u00a0\\[theorem:Criterion\\].[]{data-label=\"picture:Criterion\"}](figure1.eps)\n\nRemark that the conclusion of Theorem\u00a0\\[theorem:Criterion\\] is also valid for irreducible compact spherical buildings of higher rank, but the proof that we are aware of in that case is very indirect: one deduces it from the fact that those buildings are Moufang, hence are buildings at infinity of Bruhat-Tits buildings, via the works of Tits\u00a0[@Tits]\\*[Addenda]{} and Grundh\u00f6fer\u2013Kramer\u2013Van Maldeghem\u2013Weiss\u00a0[@Grundhofer]\\*[Theorem\u00a01.1]{}.\n\nA particular case of this criterion is the following result, which can be seen as a topological version of a well-known theorem [@Tits]\\*[Theorem\u00a04.1.1]{} of Tits. Indeed, the latter basically states that $L_0(C,C')$ (as defined below) is reduced to the identity in every thick (not necessarily topological) spherical building.\n\n\\[corollary:Epsilon\\] Let $\\Delta$ be a thick compact $m$-gon with $m \\geq 3$ and $C, C'$ be opposite chambers of $\\Delta$. There exists $\\varepsilon > 0$ such that the set $$L_\\varepsilon(C,C') := \\left\\{ \\varphi \\in \\operatorname{\\mathrm{Auttop}}(\\Delta) \\mid \\rho(X, \\varphi (X)) \\leq \\varepsilon \\ \\ \\forall X \\in E_1(C) \\cup \\{C'\\}\n \\right\\}$$ is compact, where $\\rho$ is some fixed metric associated to the topology on $\\operatorname{\\mathrm{Cham}}\\Delta$ and $E_1(C)$ denotes the set of chambers adjacent to $C$.\n\nWe remark that the local compactness theorem of Burns and Spatzier in the rank\u00a0$2$ case can now be recovered as an immediate consequence of Corollary\u00a0\\[corollary:Epsilon\\]. Actually, our proof of Theorem\u00a0\\[theorem:Criterion\\] was inspired by their work; note however that our approach is uniform in the sense that it does not require to distinguish the cases where the diameter of $\\Delta$ is even or odd. In the particular case of compact projective planes (i.e. for $m = 3$), this result was also proved by Grundh\u00f6fer in a very short and elegant way, see\u00a0[@Grundhoferbis]\\*[Theorem\u00a01]{}.\n\n\\[corollary:lc\\] Let $\\Delta$ be a thick compact $m$-gon with $m \\geq 3$. Then $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is locally compact.\n\nOne could ask if the definition of the set $J_\\varepsilon(C,C',U,U',E_0,E_1)$ whose compactness is ensured by Theorem\u00a0\\[theorem:Criterion\\] is optimal: is there any natural bigger subset of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ whose compactness could be proved as well for all compact polygons? In that direction, we suggest the following question; we feel that the answer is likely to be positive.\n\nUnder the same hypotheses and with the same notation as in Theorem\u00a0\\[theorem:Criterion\\], is the following set compact? $$\\tilde{J}_\\varepsilon(C,C',U,U',E_0,E_1) := \\left\\{ \\varphi \\in \\operatorname{\\mathrm{Auttop}}(\\Delta) {\\;\\ifnum\\currentgrouptype=16 \\middle\\fi|\\;}\\begin{array}{l}\n\\varphi(C) \\in U, \\ \\varphi(C') \\in U', \\\\\n\\rho(\\varphi(C), \\varphi(E_i)) \\geq \\varepsilon  \\ \\ \\forall i \\in \\{0,1\\},\\\\\n\\rho(\\varphi(D_i), \\varphi(E_i)) \\geq \\varepsilon  \\ \\ \\forall i \\in \\{0,1\\}\\\\\n\\end{array}\\\n \\right\\}.$$\n\nActually, this question is a natural generalization of the well-known conjecture that in any compact projective plane, the stabilizer of four points, none three of which are collinear, is compact (see\u00a0[@Salzmann]\\*[Section\u00a044]{}). It should also be emphasized that this question is strongly related to the conjecture mentioned at the beginning of this introduction. Indeed, in order to prove the conjecture, it only remains to show that Condition (ii) or (iii) in Theorem\u00a0\\[theorem:Characterization\\] is satisfied by any strongly transitive compact polygon. A positive answer to the above question would ensure that a slightly weaker condition than (iii) holds for all compact polygons; in view of Theorem\u00a0\\[theorem:Characterization\\], this would be an important step toward the proof of the conjecture.\n\nThe proof of Theorem\u00a0\\[theorem:Criterion\\] is given in Section\u00a0\\[section:criterion\\], while Theorem\u00a0\\[theorem:Characterization\\] is proved in Section\u00a0\\[section:characterization\\] (see Corollary\u00a0\\[corollary:MoufangCG\\], Proposition\u00a0\\[proposition:proj-CG\\] and Theorem\u00a0\\[theorem:Moufang\\]). Some of the preparatory results of Section\u00a0\\[section:criterion\\] are also used in Section\u00a0\\[section:characterization\\] to prove the implication (iii) $\\Rightarrow$ (i) of Theorem\u00a0\\[theorem:Characterization\\].\n\nAcknowledgement {#acknowledgement .unnumbered}\n---------------\n\nThe results of this paper are the outcomes of my Master\u2019s thesis, which was supervised by Pierre-Emmanuel Caprace. I\u00a0am very grateful to him for suggesting the topic and for his support and guidance throughout the preparation of this work. I\u00a0also thank Linus Kramer for several comments on the history of compact polygons.\n\nPreliminaries\n=============\n\nIn this section, we give the definition and first properties of compact spherical buildings and recall what the strong transitivity and the Moufang property exactly are.\n\nCompact spherical buildings\n---------------------------\n\nLet $\\Delta$ be a spherical building of rank\u00a0$k$ viewed as a simplicial complex, and denote by $\\Delta_r$ the set of simplices of dimension\u00a0$r-1$ in $\\Delta$ (for $r \\in \\{1, \\ldots, k\\}$). We fix an ordering of the $k$ types of vertices in $\\Delta$. Then $\\Delta$ is called a **compact spherical building** if the set $\\operatorname{\\mathrm{Vert}}\\Delta := \\Delta_1$ carries a compact topology such that the set $\\operatorname{\\mathrm{Cham}}\\Delta := \\Delta_k$ is closed in $(\\operatorname{\\mathrm{Vert}}\\Delta)^k$ (where a chamber of $\\Delta$ is viewed as a $k$-tuple of vertices, ordered according to their type). For each $r \\in \\{1, \\ldots, k\\}$, $\\Delta_r$ is given the induced topology from the product topology on $(\\operatorname{\\mathrm{Vert}}\\Delta)^r$. We also say that $\\Delta$ is connected or totally disconnected when $\\operatorname{\\mathrm{Cham}}\\Delta$ has the said property. When $\\Delta$ is thick and has no factor of rank\u00a0$1$, one can actually show that the topology on $\\operatorname{\\mathrm{Vert}}\\Delta$ is metrizable (see\u00a0[@Grundhofer]\\*[Proposition\u00a06.14]{}). We will therefore always suppose that the topology comes from a metric. The **topological automorphism group** $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ of $\\Delta$ is the group of all automorphisms of $\\Delta$ whose restriction to each $\\Delta_r$ are homeomorphisms. The group $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is endowed with the topology induced from the compact-open topology of $C(\\operatorname{\\mathrm{Cham}}\\Delta,\\operatorname{\\mathrm{Cham}}\\Delta)$ (where $C(X,Y)$ denotes the set of continuous maps from $X$ to $Y$). Since $\\operatorname{\\mathrm{Cham}}\\Delta$ is compact and metrizable, it also actually is the topology of uniform convergence.\n\nWe can now give the first properties of compact buildings. The following results all come from article\u00a0[@Burns] of Burns and Spatzier and are almost direct consequences of the definition of compact buildings, except Lemma\u00a0\\[lemma:non-isolated\\] whose proof is more technical.\n\n\\[proposition:opposite\\] Let $\\Delta$ be a compact spherical building of rank\u00a0$k$. The set $$\\{(R,R') \\in \\Delta_r^2 \\mid R \\text{ is opposite } R'\\}$$ is open in $\\Delta_r^2$ for each $1 \\leq r \\leq k$.\n\nSee [@Burns]\\*[Proposition\u00a01.9]{}.\n\n\\[proposition:projections\\] Let $C$ be a chamber of a compact spherical building $\\Delta$ and $R$ be a face of a chamber opposite $C$. If $R_n \\to R$ and $C_n \\to C$, then $\\operatorname{\\mathrm{proj}}_{R_n}C_n \\to \\operatorname{\\mathrm{proj}}_R C$.\n\nSee [@Burns]\\*[Proposition\u00a01.10]{}.\n\n\\[corollary:homeo\\] Let $R$ and $R'$ be two opposite residues of a compact spherical building $\\Delta$. The projection $\\operatorname{\\mathrm{proj}}_R \\mid_{R'} : \\operatorname{\\mathrm{Cham}}R' \\to \\operatorname{\\mathrm{Cham}}R$ is a homeomorphism.\n\nThis follows from Proposition\u00a0\\[proposition:projections\\].\n\nIn a polygon (i.e. a spherical building of rank\u00a0$2$), we will say that a gallery $(C_1, \\ldots, C_n)$ **stammers** if $C_i = C_{i-1}$ for some $i \\in \\{2, \\ldots, n\\}$. The two following results are true in any usual (non-topological) polygon. Recall that an $m$-gon is a polygon of diameter $m$.\n\n\\[lemma:minimal\\] Let $\\Delta$ be an $m$-gon. A gallery $(C_1, \\ldots, C_n)$ in $\\Delta$ with $n \\leq m+1$ is minimal if and only if it does not stammer. In particular, there is no non-stammering closed gallery in $\\Delta$ having less than $2m$ chambers.\n\nThis is a direct consequence of\u00a0[@Burns]\\*[Lemma\u00a00.4]{}.\n\nLet $\\Delta$ be a polygon and $x, y$ be distinct non-opposite vertices of\u00a0$\\Delta$. There is a unique minimal gallery with initial vertex $x$ and final vertex $y$, which we denote by $[x,y]$.\n\nThe existence of two minimal galleries from $x$ to $y$ would contradict Lemma\u00a0\\[lemma:minimal\\].\n\nWhen $\\Delta$ is a polygon, we denote by $\\operatorname{\\mathrm{D}}: \\operatorname{\\mathrm{Vert}}\\Delta \\times \\operatorname{\\mathrm{Vert}}\\Delta \\to \\operatorname{\\mathbf{N}}$ the graph distance in the $1$-dimensional simplicial complex $\\Delta$.\n\n\\[lemma:gallery\\] Let $\\Delta$ be a compact polygon and $x, y$ be distinct non-opposite vertices of $\\Delta$. If $x_n \\to x$, $y_n \\to y$ and $\\operatorname{\\mathrm{D}}(x_n, y_n) = \\operatorname{\\mathrm{D}}(x, y)$ for all $n \\in \\operatorname{\\mathbf{N}}$, then $[x_n, y_n] \\to [x, y]$.\n\nSee\u00a0[@Burns]\\*[Lemma\u00a01.12]{}.\n\nWe say that two vertices $v$ and $v'$ in an $m$-gon are **almost opposite** if $\\operatorname{\\mathrm{D}}(v,v') = m-1$ (note that $\\operatorname{\\mathrm{D}}(v,v') = m$ if and only if $v$ and $v'$ are opposite). We then have a result similar to Proposition\u00a0\\[proposition:opposite\\] for almost opposite vertices.\n\n\\[proposition:almost-opposite\\] Let $\\Delta$ be a compact polygon. The set $$\\{(v,v') \\in (\\operatorname{\\mathrm{Vert}}\\Delta)^2 \\mid v \\text{ is almost opposite } v'\\}$$ is open in $(\\operatorname{\\mathrm{Vert}}\\Delta)^2$.\n\nSee\u00a0[@Burns]\\*[Lemma\u00a01.13]{}.\n\nThe following result will finally be helpful in the proofs of Theorems\u00a0\\[theorem:Characterization\\] and\u00a0\\[theorem:Criterion\\].\n\n\\[lemma:non-isolated\\] Let $\\Delta$ be an infinite thick compact $m$-gon with $m \\geq 3$. For each $v \\in \\operatorname{\\mathrm{Vert}}\\Delta$, no chamber of $\\operatorname{\\mathrm{Cham}}(v)$ is isolated in $\\operatorname{\\mathrm{Cham}}(v)$. In particular, $\\operatorname{\\mathrm{Cham}}(v)$ is infinite.\n\nSee\u00a0[@Burns]\\*[Lemma\u00a01.14]{}.\n\nStrong transitivity and the Moufang property {#subsection:transitivity}\n--------------------------------------------\n\nLet $\\Delta$ be a spherical building and $\\operatorname{\\mathrm{Aut}}(\\Delta)$ be the automorphism group of $\\Delta$. A subgroup $G$ of $\\operatorname{\\mathrm{Aut}}(\\Delta)$ is said to be **strongly transitive** if $G$ is transitive on the set of all pairs $(A,C)$ where $A$ is an apartment of $\\Delta$ and $C$ is a chamber of $A$. If $\\Delta$ is a usual (non-topological) building, we say that $\\Delta$ is **strongly transitive** when $\\operatorname{\\mathrm{Aut}}(\\Delta)$ is strongly transitive. If $\\Delta$ is a compact building, this terminology is rather used when $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is strongly transitive.\n\nThe following fact is almost immediate but worth mentioning.\n\n\\[lemma:strongtransitivity\\] Let $\\Delta$ be a spherical building and $\\pi, \\pi'$ be opposite panels of $\\Delta$. Let also $G$ be a strongly transitive subgroup of $\\operatorname{\\mathrm{Aut}}(\\Delta)$. Then the stabilizer $\\operatorname{\\mathrm{Stab}}_G(\\pi, \\pi')$ of $\\pi$ and $\\pi'$ in $G$ acts $2$-transitively on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\nLet $C_1 \\neq D_1$ and $C_2 \\neq D_2$ be chambers in $\\operatorname{\\mathrm{Cham}}(\\pi)$. We want to find $g \\in \\operatorname{\\mathrm{Stab}}_G(\\pi, \\pi')$ such that $g(C_1) = C_2$ and $g(D_1) = D_2$. Let $A_1$ (resp. $A_2$) be the apartment of $\\Delta$ containing $C_1$ (resp. $C_2$), $D_1$ (resp. $D_2$) and $\\pi'$. By strong transitivity of $G$, there is $g \\in G$ such that $g(A_1) = A_2$ and $g(C_1) = C_2$. Clearly, we also have $g(\\pi) = \\pi$, $g(\\pi') = \\pi'$ and $g(D_1) = D_2$.\n\nThe Moufang property is another transitivity hypothesis on the automorphism group of a building. We say that a thick irreducible spherical building $\\Delta$ is **Moufang** (or satisfies the **Moufang property**) if for each root $\\alpha$ of $\\Delta$, the **root group** $$U_\\alpha := \\{g \\in \\operatorname{\\mathrm{Aut}}(\\Delta) \\mid g \\text{ fixes every chamber having a panel in } \\alpha \\setminus \\partial\\alpha\\}$$ acts transitively on the set of apartments of $\\Delta$ containing $\\alpha$. A natural wish would be to replace $\\operatorname{\\mathrm{Aut}}(\\Delta)$ with $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ in the case where $\\Delta$ is a compact building, but this is actually not necessary. Indeed, one can show that every element of a root group is automatically continuous, hence contained in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ (see\u00a0[@Grundhofer]\\*[Corollary\u00a06.17]{}).\n\nA compactness criterion for subsets of Auttop(Delta) {#section:criterion}\n====================================================\n\nThis section is dedicated to the proof of Theorem\u00a0\\[theorem:Criterion\\] which gives, in a compact polygon $\\Delta$, a sufficient condition on subsets of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ for being compact.\n\nRelative compactness in Auttop(Delta)\n-------------------------------------\n\nTo prove Theorem\u00a0\\[theorem:Criterion\\], we need some characterization of relative compactness in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$. The tool that will help us is the Arzela-Ascoli theorem. For the reader\u2019s convenience, we recall its statement below. First recall that when $X$ is a topological space and $Y$ is a metric space, a subset $\\mathcal{F}$ of $C(X,Y)$ is said to be **equicontinuous at $x_0 \\in X$** if, for all $\\varepsilon > 0$, there exists a neighbourhood $U$ of $x_0$ such that $$f(U) \\subseteq B(f(x_0), \\varepsilon) \\quad\\text{for all } f \\in \\mathcal{F},$$ where $B(x, r)$ denotes the open ball of radius $r$ centered at $x$. The set $\\mathcal{F}$ is then called **equicontinuous on $X$** if it is equicontinuous at each point of $X$.\n\nLet $X$ be a topological space and $Y$ be a metric space. Let $\\mathcal{F}$ be a subset of $C(X,Y)$. If $\\mathcal{F}$ is equicontinuous on $X$ and if $\\{f(x) \\mid f \\in \\mathcal{F}\\}$ is relatively compact in $Y$ for all $x \\in X$, then $\\mathcal{F}$ is relatively compact in $C(X,Y)$ equipped with the compact-open topology. The converse holds if $X$ is locally compact.\n\nSee\u00a0[@Munkres]\\*[Theorem\u00a047.1]{}.\n\nWe can now already give some characterization of relative compactness in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$.\n\n\\[lemma:equicontinuous\\] Let $\\Delta$ be a compact spherical building. A set $S \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$ is relatively compact in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ if and only if $S$ and $S^{-1}$ are equicontinuous on $\\operatorname{\\mathrm{Cham}}\\Delta$.\n\nFirst assume that $S$ is relatively compact in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$. The set $S$ is thus also relatively compact in $C(\\operatorname{\\mathrm{Cham}}\\Delta, \\operatorname{\\mathrm{Cham}}\\Delta)$, hence $S$ is equicontinuous on $\\operatorname{\\mathrm{Cham}}\\Delta$ by Arzela-Ascoli. Since $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is a topological group, the inverse function of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is a homeomorphism and $S^{-1}$ is in turn relatively compact and equicontinuous on $\\operatorname{\\mathrm{Cham}}\\Delta$.\n\nConversely, assume that $S$ and $S^{-1}$ are equicontinuous. We want to show that $S$ is relatively compact in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$. We already know by Arzela-Ascoli that $S$ is relatively compact in $C(\\operatorname{\\mathrm{Cham}}\\Delta, \\operatorname{\\mathrm{Cham}}\\Delta)$. A way to conclude is therefore to prove that the closure of $S$ in $C(\\operatorname{\\mathrm{Cham}}\\Delta, \\operatorname{\\mathrm{Cham}}\\Delta)$ is contained in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$. We thus consider a sequence $\\varphi_n \\to \\varphi$ with $\\{\\varphi_n\\} \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$ and $\\varphi \\in C(\\operatorname{\\mathrm{Cham}}\\Delta, \\operatorname{\\mathrm{Cham}}\\Delta)$ and prove that $\\varphi \\in \\operatorname{\\mathrm{Auttop}}(\\Delta)$. As a direct consequence of the fact that the adjacency relation is closed in a compact building, $\\varphi$ is a (continuous) building morphism. To complete the proof, it suffices to find another such morphism $\\psi$ so that $\\varphi \\psi = \\psi \\varphi = \\operatorname{\\mathrm{id}}_\\Delta$. Since $S^{-1}$ is equicontinuous and thus relatively compact in $C(\\operatorname{\\mathrm{Cham}}\\Delta, \\operatorname{\\mathrm{Cham}}\\Delta)$, it follows that some subsequence of $(\\varphi_n^{-1})$ converges to an element of $C(\\operatorname{\\mathrm{Cham}}\\Delta, \\operatorname{\\mathrm{Cham}}\\Delta)$. Taking this limit for $\\psi$, we obtain $\\varphi \\psi = \\psi \\varphi = \\operatorname{\\mathrm{id}}_\\Delta$.\n\nTo prove that a given subset of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ is relatively compact, we will generally proceed by contradiction. In this context, the next proposition will be helpful. We introduce a convenient definition to state it.\n\nLet $\\Delta$ be a compact polygon. Two convergent sequences $a_n \\to a$ and $b_n \\to b$ in $\\operatorname{\\mathrm{Vert}}\\Delta$ are said to be **collapsed by** a sequence $\\{\\varphi_n\\} \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$ if $a \\neq b$ and if $\\varphi_n a_n \\to x$ and $\\varphi_n b_n \\to x$ for some $x \\in \\operatorname{\\mathrm{Vert}}\\Delta$.\n\n\\[proposition:closer\\] Let $\\Delta$ be a compact polygon and $S \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$. If $S$ is not relatively compact in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$, then there exist two sequences $a_n \\to a$ and $b_n \\to b$ in $\\operatorname{\\mathrm{Vert}}\\Delta$ which are collapsed by some sequence $(\\varphi_n)$ in $S$ or in $S^{-1}$.\n\nLet $S$ be a subset of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ which is not relatively compact. By Lemma\u00a0\\[lemma:equicontinuous\\], $S$ or $S^{-1}$ is not equicontinuous on $\\operatorname{\\mathrm{Cham}}\\Delta$ and therefore also not equicontinuous on $\\operatorname{\\mathrm{Vert}}\\Delta$. Suppose $S$ is not equicontinuous. Then there exist some $x \\in \\operatorname{\\mathrm{Vert}}\\Delta$ and $\\varepsilon > 0$ such that for every neighbourhood $U$ of $x$, there exists $\\psi \\in S$ satisfying $\\psi(U) \\not \\subseteq B(\\psi x, \\varepsilon)$. In other words, for every $n \\in \\operatorname{\\mathbf{N}}^*$ there exist $\\psi_n \\in S$ and $y_n \\in B\\left(x, \\frac{1}{n}\\right)$ such that $\\psi_n y_n \\not \\in B(\\psi_n x, \\varepsilon)$. Clearly $y_n \\to x$ and we can assume by passing to a subsequence that $\\psi_n x \\to a$ and $\\psi_n y_n \\to b$ for some $a \\neq b$. Hence, the sequences $a_n = \\psi_n x \\to a$ and $b_n = \\psi_n y_n \\to b$ are collapsed by $\\{\\varphi_n = \\psi_n^{-1}\\} \\subseteq S^{-1}$. The case where $S^{-1}$ is not equicontinuous is identical but gives two sequences which are collapsed by a sequence in $(S^{-1})^{-1} = S$.\n\nSequences collapsed by a sequence of automorphisms\n--------------------------------------------------\n\nIn this section, we consider a compact polygon $\\Delta$. Proposition\u00a0\\[proposition:closer\\] states that one can deduce from non-relative compactness of a subset $S$ of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ that there exist two sequences $a_n \\to a$ and $b_n \\to b$ of vertices of $\\Delta$ collapsed by a sequence in $S$ or in $S^{-1}$. The goal of this subsection is to show that we can actually obtain additional constraints on these two sequences. The following result of Burns and Spatzier already goes in this direction by asserting that we can assume $\\operatorname{\\mathrm{D}}(a_n, b_n) = \\operatorname{\\mathrm{D}}(a, b) = 2$.\n\n\\[proposition:centered\\] Let $\\Delta$ be a thick compact polygon. If there exist two sequences $a_n \\to a$ and $b_n \\to b$ in $\\operatorname{\\mathrm{Vert}}\\Delta$ collapsed by a sequence $\\{\\varphi_n\\} \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$, then there exist two sequences $a'_n \\to a'$ and $b'_n \\to b'$ collapsed by $(\\varphi_n)$ and such that $\\operatorname{\\mathrm{D}}(a'_n, b'_n) = \\operatorname{\\mathrm{D}}(a', b') = 2$ for all $n \\in \\operatorname{\\mathbf{N}}$.\n\nSee [@Burns]\\*[Discussion between Assertions 2.3 and 2.4]{}.\n\nNow suppose that we are given a converging sequence of apartments $A_n \\to A$, i.e. $2m$ converging sequences $v_i^{(n)} \\to v_i$ ($i \\in \\{0, \\ldots, 2m-1\\}$) where $v_0^{(n)}, \\ldots, v_{2m-1}^{(n)}$ (resp. $v_0, \\ldots, v_{2m-1}$) are the vertices of an apartment $A_n$ (resp. $A$). In this context, we can generally even assume that the middle vertex $c_n$ of $a_n$ and $b_n$ (which are such that $\\operatorname{\\mathrm{D}}(a_n, b_n) = 2$) is a vertex of $A_n$. For this reason we introduce the following definition.\n\nLet $\\Delta$ be a compact polygon. Two convergent sequences $a_n \\to a$ and $b_n \\to b$ in $\\operatorname{\\mathrm{Vert}}\\Delta$ are said to be **centered at** $c_n \\to c$ if $\\operatorname{\\mathrm{D}}(a_n, c_n) = \\operatorname{\\mathrm{D}}(c_n, b_n) = 1$ for all $n \\in \\operatorname{\\mathbf{N}}$. This also implies that $\\operatorname{\\mathrm{D}}(a, c) = \\operatorname{\\mathrm{D}}(c, b) = 1$.\n\nBefore proving the announced result, we show the next lemma, which provides a way to go from a sequence of vertices to a sequence of opposite vertices.\n\n\\[lemma:opposite\\] Let $\\Delta$ be a compact polygon and let $a_n \\to a$ and $b_n \\to b$ be two sequences in $\\operatorname{\\mathrm{Vert}}\\Delta$ centered at $c_n \\to c$ and collapsed by a sequence $\\{\\varphi_n\\} \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$. Let also $c'_n \\to c'$ be a sequence in $\\operatorname{\\mathrm{Vert}}\\Delta$ such that $c'_n$ (resp. $c'$) is opposite $c_n$ (resp. $c$) for all $n \\in \\operatorname{\\mathbf{N}}$. Suppose that $\\varphi_n c_n \\to \\tilde{c}$ and $\\varphi_n c'_n \\to \\tilde{c}'$ for some opposite vertices $\\tilde{c}$ and $\\tilde{c}'$. Denote by $C_n$ (resp.\u00a0$D_n$) the chamber whose vertices are $c_n$ and $a_n$ (resp. $b_n$) and by $a'_n$ (resp. $b'_n$) the vertex of $\\operatorname{\\mathrm{proj}}_{c'_n}(C_n)$ (resp. $\\operatorname{\\mathrm{proj}}_{c'_n}(D_n)$) different from $c'_n$. Then the sequences $(a'_n)$ and $(b'_n)$ converge, are centered at $(c'_n)$ and are collapsed by $(\\varphi_n)$.\n\n![Illustration of Lemma\u00a0\\[lemma:opposite\\].[]{data-label=\"picture:opposite\"}](figure2.eps)\n\nBy Proposition\u00a0\\[proposition:projections\\], the sequence $(a'_n)$ converges to $a'$, the vertex of $\\operatorname{\\mathrm{proj}}_{c'}(C)$ different from $c'$ where $C$ is the chamber having vertices $c$ and $a$. The sequence $(b'_n)$ converges to $b'$ defined in the same way and $a' \\neq b'$ since $a \\neq b$. The fact that these two sequences are collapsed by $(\\varphi_n)$ directly comes from the observation that the two sequences $\\left(\\varphi_n \\operatorname{\\mathrm{proj}}_{c'_n}(C_n)\\right) = \\left(\\operatorname{\\mathrm{proj}}_{\\varphi_n c'_n}(\\varphi_n C_n)\\right)$ and $\\left(\\varphi_n \\operatorname{\\mathrm{proj}}_{c'_n}(D_n)\\right) = \\left(\\operatorname{\\mathrm{proj}}_{\\varphi_n c'_n}(\\varphi_n D_n)\\right)$ have the same limit by Proposition\u00a0\\[proposition:projections\\].\n\nWe draw attention to the fact that it is essential to have $\\tilde{c}$ and $\\tilde{c}'$ opposite each other to apply the previous lemma.\n\n\\[proposition:centerok\\] Let $\\Delta$ be a thick compact polygon and $A_n \\to A$ be a converging sequence of apartments of $\\Delta$. If there exist two sequences $a_n \\to a$ and $b_n \\to b$ in $\\operatorname{\\mathrm{Vert}}\\Delta$ collapsed by a sequence $\\{\\varphi_n\\} \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$ and if $\\varphi_n A_n \\to \\tilde{A}$ for some apartment\u00a0$\\tilde{A}$ of $\\Delta$, then there exist two sequences $a'_n \\to a'$ and $b'_n \\to b'$ centered at $c'_n \\to c'$ and collapsed by some subsequence of $(\\varphi_n)$, with $c'_n \\in A_n$ for all $n \\in \\operatorname{\\mathbf{N}}$.\n\nBy Proposition\u00a0\\[proposition:centered\\], we can assume that the two sequences $a_n \\to a$ and $b_n \\to b$ are centered at some sequence of vertices $c_n \\to c$. Let $d$ be a vertex of $A$ opposite $c$ (to find such a vertex, one can take a gallery from $c$ to any vertex of $A$ and extend this one with chambers of $A$ until an opposite vertex is reached) and let $d_n$ be the vertex of $A_n$ (for $n \\in \\operatorname{\\mathbf{N}}$) such that $d_n \\to d$. Denote also by $\\tilde{d}$ the limit of $(\\varphi_n d_n)$. By Proposition\u00a0\\[proposition:opposite\\], $c_n$ is opposite $d_n$ for sufficiently large $n$. We can also assume by passing to a subsequence that $(\\varphi_n c_n)$ converges to some $\\tilde{c}$. If $\\tilde{c}$ is opposite $\\tilde{d}$, then we can immediately complete the proof using Lemma\u00a0\\[lemma:opposite\\].\n\n![Illustration of Proposition\u00a0\\[proposition:centerok\\].[]{data-label=\"picture:centerok\"}](figure3.eps)\n\nIf, on the contrary, $\\tilde{c}$ is not opposite $\\tilde{d}$, then we cannot proceed in this way. In this case, pick a sequence $e_n \\to e$ with $e_n$ (resp. $e$) adjacent to $d_n$ (resp. $d$) but not contained in $A_n$ (resp. $A$). It is easy to show that such a sequence exists, using for example Proposition\u00a0\\[proposition:projections\\]. Since $e$ is almost opposite $c$, $e_n$ is almost opposite $c_n$ for sufficiently large $n$ (Proposition\u00a0\\[proposition:almost-opposite\\]). For these $n$, there is a gallery from $d_n$ to $c_n$ of length $m = \\operatorname{\\mathrm{diam}}\\Delta$ passing through $e_n$. Let $d_n = e_0^{(n)}, e_1^{(n)} = e_n, \\ldots, e_{m-1}^{(n)}, e_m^{(n)} = c_n$ be the vertices of this gallery, as illustrated in Figure\u00a0\\[picture:centerok\\]. After passing to subsequences, we can suppose that $e_i^{(n)} \\to e_i$ and $\\varphi_n e_i^{(n)} \\to \\tilde{e}_i$ for each $i \\in \\{0, \\ldots, m\\}$. In particular, $\\tilde{e}_0 = \\tilde{d}$ and $\\tilde{e}_m = \\tilde{c}$. Since $\\tilde{c}$ is not opposite $\\tilde{d}$, the gallery of vertices $\\tilde{d} = \\tilde{e}_0, \\tilde{e}_1, \\ldots, \\tilde{e}_m = \\tilde{c}$ must stammer. This means that there exists $k \\in \\{0, \\ldots, m-2\\}$ such that $\\tilde{e}_k = \\tilde{e}_{k+2}$ (if there is more than one such $k$, then we choose the smallest one). In other words, since $e_k \\neq e_{k+2}$, the two sequences $e_k^{(n)} \\to e_k$ and $e_{k+2}^{(n)} \\to e_{k+2}$ are centered at $e_{k+1}^{(n)} \\to e_{k+1}$ and collapsed by $(\\varphi_n)$. Now take $c'$ a vertex of $A$ such that $\\operatorname{\\mathrm{D}}(c',d) = m-k-1$ and $c'_n$ the vertex of $A_n$ (for $n \\in \\operatorname{\\mathbf{N}}$) such that $c'_n \\to c'$. Thanks to this choice, $e_{k+1}$ is opposite $c'$ and $\\tilde{e}_{k+1}$ is opposite the limit $\\tilde{c}'$ of $(\\varphi_n c'_n)$ (in the particular case where $\\tilde{e}_1 \\in \\tilde{A}$, we choose $c'$ so that the minimal gallery from $\\tilde{d}$ to $\\tilde{c}'$ does not contain $\\tilde{e}_1$). We can therefore apply Lemma\u00a0\\[lemma:opposite\\] to the sequences $e_k^{(n)} \\to e_k$ and $e_{k+2}^{(n)} \\to e_{k+2}$ centered at $e_{k+1}^{(n)} \\to e_{k+1}$ and the opposite sequence $c'_n \\to c'$.\n\nThe following result eventually shows that we can even suppose that one of the two sequences which are collapsed is contained in the sequence of apartments $A_n \\to A$. The proof only works when the diameter $m$ of $\\Delta$ is at least $3$, i.e. when $\\Delta$ is irreducible.\n\n\\[proposition:centerokbetter\\] Let $\\Delta$ be a thick compact $m$-gon with $m \\geq 3$ and $A_n \\to A$ be a converging sequence of apartments of $\\Delta$. If there exist two sequences $a_n \\to a$ and $b_n \\to b$ in $\\operatorname{\\mathrm{Vert}}\\Delta$ collapsed by a sequence $\\{\\varphi_n\\} \\subseteq \\operatorname{\\mathrm{Auttop}}(\\Delta)$ and if $\\varphi_n A_n \\to \\tilde{A}$ for some apartment $\\tilde{A}$ of $\\Delta$, then there exist two sequences $a'_n \\to a'$ and $b'_n \\to b'$ centered at $c'_n \\to c'$ and collapsed by some subsequence of $(\\varphi_n)$, with $c'_n \\in A_n$ and $a'_n \\in A_n$ for all $n \\in \\operatorname{\\mathbf{N}}$.\n\nBy Proposition\u00a0\\[proposition:centerok\\], we can assume that the sequences $a_n \\to a$ and $b_n \\to b$ are centered at $v_n \\to v$ with $v_n \\in A_n$ (and thus $v \\in A$). Let $\\tilde{x}$ be the common limit of $(\\varphi_n a_n)$ and $(\\varphi_n b_n)$.\n\n![Illustration of Proposition\u00a0\\[proposition:centerokbetter\\].[]{data-label=\"picture:centerokbetter\"}](figure4.eps)\n\nIf $\\tilde{x}$ is a vertex of $\\tilde{A}$, then consider $s_n \\to s$ the sequence of vertices with $s_n \\in A_n$ and $s \\in A$ such that $\\varphi_n s_n \\to \\tilde{x}$. Since $a \\neq b$, we can assume without loss of generality that $a \\neq s$. The sequences $a_n \\to a$ and $s_n \\to s$ are then collapsed by $(\\varphi_n)$, which ends the proof.\n\nSuppose now that $\\tilde{x} \\not \\in \\tilde{A}$. Let $v'_n$ (resp. $v'$) be the vertex of $A_n$ (resp. $A$) opposite $v_n$ (resp. $v$), $\\tilde{v}'$ be the limit of $(\\varphi_n v'_n)$ and $w_n \\to w$ be a sequence of vertices where $w_n$ (resp.\u00a0$w$) is a vertex of $A_n$ (resp. $A$) and is adjacent to $v'_n$ (resp. $v'$). Consider also a sequence $u_n \\to u$ with $u_n$ (resp. $u$) adjacent to $w_n$ (resp. $w$) but outside $A_n$ (resp. $A$). Passing to a subsequence if necessary, we can assume that $\\varphi_n u_n \\to \\tilde{u}$ for some vertex $\\tilde{u}$ (adjacent to the limit of $(\\varphi_n w_n)$). Once again, if $\\tilde{u} \\in \\tilde{A}$ then the proof is completed. We therefore assume that $\\tilde{u} \\not \\in \\tilde{A}$. For sufficiently large $n$, $a_n$ is almost opposite $u_n$ and $b_n$ is almost opposite $v'_n$ (see Proposition\u00a0\\[proposition:almost-opposite\\]). We can thus draw $[a_n, u_n]$ and $[b_n, v'_n]$ as in Figure\u00a0\\[picture:centerokbetter\\]. Thanks to Lemma\u00a0\\[lemma:gallery\\], $\\varphi_n[a_n, u_n] \\to [\\tilde{x}, \\tilde{u}]$ and $\\varphi_n[b_n, v'_n] \\to [\\tilde{x}, \\tilde{v}']$. The fact that $\\tilde{u} \\not \\in \\tilde{A}$ ensures that the two galleries $[\\tilde{x}, \\tilde{u}]$ and $[\\tilde{x}, \\tilde{v}']$ have no shared chambers. Let $c_n$ be the vertex of $[b_n, v'_n]$ adjacent to $b_n$, $d_n$ be the vertex of $[u_n, a_n]$ adjacent to $u_n$, $e_n$ be the vertex of $[d_n, a_n]$ adjacent to $d_n$, and $C_n$ be the chamber whose vertices are $d_n$ and $e_n$. The vertices $e_n$ and $c_n$ are opposite for large\u00a0$n$. Now, let $C'_n$ be the projection of $C_n$ on $c_n$ and $f_n$ be the vertex of $C'_n$ different from $c_n$. Since $d_n$ and $c_n$ are almost opposite as well as the limits of $(\\varphi_n d_n)$ and $(\\varphi_n c_n)$, Lemma\u00a0\\[lemma:gallery\\] gives $\\varphi_n f_n \\to \\tilde{x}$ and the two sequences $(b_n)$ and $(f_n)$ centered at $(c_n)$ are collapsed by $(\\varphi_n)$.\n\nLet finally $c'$ be a vertex of $A$ at distance $2$ from $v'$ such that $c'$ is opposite the limit of $(c_n)$ and let $c'_n$ be the vertex of $A_n$ (for $n \\in \\operatorname{\\mathbf{N}}$) such that $c'_n \\to c'$. As the limits of $(\\varphi_n c_n)$ and $(\\varphi_n c'_n)$ are opposite, we can apply Lemma\u00a0\\[lemma:opposite\\] to get two sequences of vertices centered at $c'_n \\to c'$, collapsed by $(\\varphi_n)$ and so that one of them is contained in $A_n \\to A$.\n\nProof of the compactness criterion\n----------------------------------\n\nThe following lemma is almost evident but will play an important role in the next results. As in the statement of Theorem\u00a0\\[theorem:Criterion\\], we write $B_v(C,r)$ for the open ball in $\\operatorname{\\mathrm{Cham}}(v)$ centered at $C$ and of radius $r$.\n\n\\[lemma:technical\\] Let $\\Delta$ be a compact polygon and $v^{(n)} \\to v$ and $v'^{(n)} \\to v'$ be two sequences in $\\operatorname{\\mathrm{Vert}}\\Delta$ such that $v^{(n)}$ (resp. $v$) is opposite $v'^{(n)}$ (resp. $v'$) for all $n \\in \\operatorname{\\mathbf{N}}$. Let also $X^{(n)} \\to X$ be a sequence in $\\operatorname{\\mathrm{Cham}}\\Delta$ such that $X^{(n)}$ (resp. $X$) has vertex $v^{(n)}$ (resp.\u00a0$v$) for all $n \\in \\operatorname{\\mathbf{N}}$. For each $\\eta > 0$, there exists $\\eta' > 0$ such that $$B_{v'^{(n)}}(\\operatorname{\\mathrm{proj}}_{v'^{(n)}}(X^{(n)}), \\eta') \\subseteq \\operatorname{\\mathrm{proj}}_{v'^{(n)}}(B_{v^{(n)}}(X^{(n)}, \\eta)) \\ \\text{ for all $n \\in \\operatorname{\\mathbf{N}}$.}$$\n\nFix $\\eta > 0$ and suppose by contradiction that $\\eta'$ does not exist. This means that we can find, after passage to a subsequence, a sequence $(Y^{(n)})$ with $Y^{(n)} \\in B_{v'^{(n)}}(\\operatorname{\\mathrm{proj}}_{v'^{(n)}}(X^{(n)}), \\frac{1}{n})$ but $Y^{(n)} \\not \\in \\operatorname{\\mathrm{proj}}_{v'^{(n)}}(B_{v^{(n)}}(X^{(n)}, \\eta))$ for all $n \\in \\operatorname{\\mathbf{N}}^*$. By Proposition\u00a0\\[proposition:projections\\], $Y^{(n)} \\to \\operatorname{\\mathrm{proj}}_{v'}(X)$ and thus $\\operatorname{\\mathrm{proj}}_{v^{(n)}}(Y^{(n)}) \\to \\operatorname{\\mathrm{proj}}_{v}(\\operatorname{\\mathrm{proj}}_{v'}(X)) = X$. This contradicts the fact that $\\operatorname{\\mathrm{proj}}_{v^{(n)}}(Y^{(n)}) \\not \\in B_{v^{(n)}}(X^{(n)}, \\eta)$ for all $n \\in \\operatorname{\\mathbf{N}}^*$.\n\nThe next definition will turn out to be convenient.\n\nLet $X^{(n)} \\to X$ and $Y^{(n)} \\to Y$ be two sequences in $\\operatorname{\\mathrm{Cham}}\\Delta$ with $X^{(n)}$ (resp.\u00a0$X$) adjacent to $Y^{(n)}$ (resp. $Y$) in vertex $v^{(n)}$ (resp. $v$) for all $n \\in \\operatorname{\\mathbf{N}}$. Let $(\\varphi_n)$ be a sequence in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ such that $\\varphi_n Y^{(n)} \\to \\tilde{Y}$ for some $\\tilde{Y} \\in \\operatorname{\\mathrm{Cham}}\\Delta$. For $\\eta > 0$, we say that $(X, Y)$ is **$\\eta$-correct** (the sequences $X_n \\to X$, $Y_n \\to Y$ and $(\\varphi_n)$ being clear from the context) if for each sequence $(T^{(n)})$ with $T^{(n)} \\in B_{v^{(n)}}(X^{(n)}, \\eta)$ the sequence $(\\varphi_n T^{(n)})$ does not accumulate at $\\tilde{Y}$. We say that $(X,Y)$ is **correct** if it is $\\eta$-correct for some $\\eta > 0$.\n\nThe following lemma is the key result for the proof of Theorem\u00a0\\[theorem:Criterion\\]. Indeed, it makes it possible to find some kind of uniform convergence from just a simple convergence. Figure\u00a0\\[picture:key\\] should make the statement easier to digest.\n\n\\[lemma:key\\] Let $\\Delta$ be a compact polygon and $A^{(n)} \\to A$ be a converging sequence of apartments of $\\Delta$. Let $v_0^{(n)}, \\ldots, v_{2m-1}^{(n)}$ (resp. $v_0, \\ldots, v_{2m-1}$) be the vertices of $A^{(n)}$ (resp.\u00a0$A$) such that $v_i^{(n)} \\to v_i$ for $i \\in \\{0, \\ldots, 2m-1\\}$ and let $C_i^{(n)}$ (resp. $C_i$) be the chamber having vertices $v_i^{(n)}$ (resp. $v_i$) and $v_{i+1}^{(n)}$ (resp. $v_{i+1}$) for $i \\in \\{0, \\ldots, 2m-1\\}$, where $v_{2m}^{(n)} := v_0^{(n)}$ and $v_{2m} := v_0$. Let also $b^{(n)} \\to b \\not \\in A$ be a sequence in $\\operatorname{\\mathrm{Vert}}\\Delta$ such that $b^{(n)}$ is adjacent to $v_0^{(n)}$ for all $n \\in \\operatorname{\\mathbf{N}}$. Fix $\\eta > 0$. Then there exists $\\eta' > 0$ such that if $(\\varphi_n)$ is a sequence in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ satisfying\n\n(a) $\\varphi_n v_i^{(n)} \\to \\tilde{v}_i$ for each $i \\in \\{0, \\ldots, 2m-1\\}$ and the set of vertices $\\tilde{v}_0, \\ldots, \\tilde{v}_{2m-1}$ form an apartment $\\tilde{A}$ (whose chambers are denoted by $\\tilde{C}_0, \\ldots, \\tilde{C}_{2m-1}$ with $\\varphi_n C_i^{(n)} \\to \\tilde{C}_i$),\n\n(b) $(C_{m+1}, C_m)$ is $\\eta$-correct (with respect to $(\\varphi_n)$),\n\n(c) $\\varphi_n b^{(n)} \\to \\tilde{v}_1$;\n\nthen for each sequence $(R^{(n)})$ with $R^{(n)} \\in B_{v_{m-1}^{(n)}}(C_{m-2}^{(n)}, \\eta')$ for all $n \\in \\operatorname{\\mathbf{N}}$, $\\varphi_n R^{(n)} \\to \\tilde{C}_{m-2}$.\n\n![Illustration of Lemma\u00a0\\[lemma:key\\].[]{data-label=\"picture:key\"}](figure5.eps)\n\nSince $b \\not\\in A$, we can assume that $b^{(n)} \\not \\in A^{(n)}$ for all $n \\in \\operatorname{\\mathbf{N}}$. We consider the balls $B_{v_{m+1}^{(n)}}(C_{m+1}^{(n)}, \\eta)$ and apply Lemma\u00a0\\[lemma:technical\\] twice: first with the sequences $v_{m+1}^{(n)} \\to v_{m+1}$ and $b^{(n)} \\to b$ and then with the sequences $b^{(n)} \\to b$ and $v_{m-1}^{(n)} \\to v_{m-1}$. This gives $\\eta' > 0$ such that $$B_{v_{m-1}^{(n)}}(C_{m-2}^{(n)}, \\eta') \\subseteq \\operatorname{\\mathrm{proj}}_{v_{m-1}^{(n)}}(\\operatorname{\\mathrm{proj}}_{b^{(n)}}(B_{v_{m+1}^{(n)}}(C_{m+1}^{(n)}, \\eta))) \\ \\text{ for all $n \\in \\operatorname{\\mathbf{N}}$.} \\quad (*)$$ We now prove that $\\eta'$ satisfies the statement.\n\nConsider a sequence $(R^{(n)})$ with $R^{(n)} \\in B_{v_{m-1}^{(n)}}(C_{m-2}^{(n)}, \\eta')$ and suppose for a contradiction that there exists a sequence $(\\varphi_n)$ satisfying the conditions of the statement but such that $(\\varphi_n R^{(n)})$ does not converge to $\\tilde{C}_{m-2}$. Passing to a subsequence, we can assume that $\\varphi_n R^{(n)} \\to \\tilde{R} \\neq \\tilde{C}_{m-2}$ and $R^{(n)} \\to R$. Now take $S^{(n)} = \\operatorname{\\mathrm{proj}}_{b^{(n)}}(R^{(n)})$ and $S = \\operatorname{\\mathrm{proj}}_b(R)$. By Proposition\u00a0\\[proposition:projections\\], $S^{(n)} \\to S$. Finally, define $T^{(n)} = \\operatorname{\\mathrm{proj}}_{v_{m+1}^{(n)}}(S^{(n)})$ and $T = \\operatorname{\\mathrm{proj}}_{v_{m+1}}(S)$. Once again, $T^{(n)} \\to T$. By $(*)$, $T^{(n)} \\in B_{v_{m+1}^{(n)}}(C_{m+1}^{(n)}, \\eta)$ for all $n \\in \\operatorname{\\mathbf{N}}$. We now show that the convergence of $(\\varphi_n R^{(n)})$ to $\\tilde{R} \\neq \\tilde{C}_{m-2}$ necessarily implies $\\varphi_n T^{(n)} \\to \\tilde{C}_m$, which is impossible since $(C_{m+1},C_m)$ is $\\eta$-correct.\n\nDenote by $r^{(n)}$ the vertex of $R^{(n)}$ different from $v_{m-1}^{(n)}$ and by $\\tilde{r}$ the vertex of $\\tilde{R}$ different from $\\tilde{v}_{m-1}$. Since $b^{(n)}$ and $r^{(n)}$ are almost opposite (for large enough $n$) as well as $\\tilde{v}_1$ and\u00a0$\\tilde{r}$, $\\varphi_n[b^{(n)}, r^{(n)}] \\to [\\tilde{v}_1, \\tilde{r}]$ (Lemma\u00a0\\[lemma:gallery\\]) and in particular $\\varphi_n S^{(n)} \\to \\tilde{C}_1$. As $v_1$ is opposite $v_{m+1}$, we therefore get by Proposition\u00a0\\[proposition:projections\\]: $$\\varphi_n T^{(n)} = \\varphi_n \\left( \\operatorname{\\mathrm{proj}}_{v_{m+1}^{(n)}}(S^{(n)}) \\right) = \\operatorname{\\mathrm{proj}}_{\\varphi_nv_{m+1}^{(n)}}(\\varphi_n S^{(n)}) \\to \\operatorname{\\mathrm{proj}}_{\\tilde{v}_{m+1}}(\\tilde{C}_1) = \\tilde{C}_m. \\qedhere$$\n\nWe are now able to prove Theorem\u00a0\\[theorem:Criterion\\].\n\nIf $\\Delta$ is finite, then $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ and $J_\\varepsilon := J_\\varepsilon(C,C',U,U',E_0,E_1)$ are finite and the latter is thus compact. We will therefore assume that $\\Delta$ is infinite. By Lemma\u00a0\\[lemma:non-isolated\\], this implies that $\\operatorname{\\mathrm{Cham}}(v)$ is infinite for each vertex $v$ of $\\Delta$.\n\nSince $J_\\varepsilon$ is closed in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$, it is compact if and only if it is relatively compact in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$. Suppose for a contradiction that it is not relatively compact. By Proposition\u00a0\\[proposition:closer\\], there exist two sequences $a^{(n)} \\to a$ and $b^{(n)} \\to b$ collapsed by a sequence $(\\varphi_n)$ in $J_\\varepsilon$ or in $J_\\varepsilon^{-1}$. Observe both cases:\n\n(1) If $\\{\\varphi_n\\} \\subseteq J_\\varepsilon$, then we define $C^{(n)} = C$, $C'^{(n)} = C'$, $D_i^{(n)} = D_i$ and $E_i^{(n)} = E_i$ for all $n \\in \\operatorname{\\mathbf{N}}$ and $i \\in \\{0,1\\}$. Let also $A^{(n)} = A$ be the apartment containing $C$ and $C'$ (for all $n \\in \\operatorname{\\mathbf{N}}$). Passing to a subsequence, we can assume that $\\varphi_n C^{(n)} \\to \\tilde{C}$ and $\\varphi_n C'^{(n)} \\to \\tilde{C}'$. By definition of $J_\\varepsilon$, $\\tilde{C}$ and $\\tilde{C}'$ are opposite, which means that $\\varphi_n A^{(n)} \\to \\tilde{A}$, the apartment containing $\\tilde{C}$ and $\\tilde{C}'$. The definition of $J_\\varepsilon$ is also such that $(C, D_i)$ and $(D_i, C)$ are $\\varepsilon$-correct for $i \\in \\{0, 1\\}$. Finally, we can assume that $\\varphi_n E_i^{(n)} \\to \\tilde{E}_i$ and $\\tilde{E}_i \\not \\in \\tilde{A}$ for $i \\in \\{0,1\\}$.\n\n(2) If $\\{\\varphi_n\\} \\subseteq J_\\varepsilon^{-1}$, then we define $F^{(n)} = \\varphi_n^{-1} C$, $F'^{(n)} = \\varphi_n^{-1} C'$, $G_i^{(n)} = \\varphi_n^{-1} D_i$ and $H_i^{(n)} = \\varphi_n^{-1} E_i$ for all $n \\in \\operatorname{\\mathbf{N}}$ and $i \\in \\{0,1\\}$. Passing to a subsequence, we can assume that $F^{(n)} \\to F$, $F'^{(n)} \\to F'$, $G_i^{(n)} \\to G_i$ and $H_i^{(n)} \\to H_i$. Note that $F$ is opposite $F'$ since $\\{\\varphi_n^{-1}\\} \\subseteq J_\\varepsilon$, which allows us to denote by $A^{(n)}$ (resp. $A$) the apartment containing $F^{(n)}$ (resp. $F$) and $F'^{(n)}$ (resp. $F'$) for all $n \\in \\operatorname{\\mathbf{N}}$. By definition of $J_\\varepsilon$, $H_i \\not \\in A$. Moreover, $\\varphi_n F^{(n)} = C \\to C =: \\tilde{F}$, $\\varphi_n F'^{(n)} = C' \\to C'=: \\tilde{F'}$ and $\\varphi_n H_i^{(n)} = E_i \\to E_i =: \\tilde{H}_i$. Hence, $\\varphi_n A^{(n)} \\to \\tilde{A}$, the apartment containing $\\tilde{F}$ and $\\tilde{F}'$. It is easy to see that $(F, G_i)$ and $(G_i, F)$ are $\\varepsilon$-correct (with respect to $(\\varphi_n)$) and clearly $\\tilde{H}_i \\not \\in \\tilde{A}$ for $i \\in \\{0,1\\}$.\n\nThe situations in (1) and (2) are exactly the same, replacing letters $F$, $G$ and $H$ in (2) by letters $C$, $D$ and $E$ respectively. For this reason, we from now on assume that we are in case (1) and only use the objects $C^{(n)} \\to C$, $C'^{(n)} \\to C'$, $D_i^{(n)} \\to D_i$ and $E_i^{(n)} \\to E_i$ satisfying the conditions stated above. Let $v_0, \\ldots, v_{2m-1}$ be the vertices of $A$ in the natural order so that $E_0$ has vertex $v_0$ and $E_1$ has vertex $v_1$. Once again, $v_0^{(n)}, \\ldots, v_{2m-1}^{(n)}$ denotes the corresponding vertices in $A^{(n)}$ and $\\tilde{v}_0, \\ldots, \\tilde{v}_{2m-1}$ the corresponding vertices in $\\tilde{A}$.\n\n\\[claim:correct\\] $(X,Y)$ is correct for any two adjacent chambers $X$ and $Y$ in $A$.\n\n![Illustration of Claim\u00a0\\[claim:correct\\].[]{data-label=\"picture:nonconvergence\"}](figure6.eps)\n\nIn the proof of this claim, we say that $v_i$ is correct if $(X,Y)$ and $(Y,X)$ are correct where $X$ and $Y$ are the two chambers of $A$ having vertex $v_i$. We already know that $v_0$ and $v_1$ are correct, and we want to prove that every vertex of $A$ is correct.\n\nWe actually show that if $v_i$ is correct, then $v_{-i}$ and $v_{2-i}$ are also correct (the indices being considered modulo $2m$). To do so, draw the minimal gallery from $E_0$ (resp. $E_0^{(n)}$) to $v_m$ (resp. $v_m^{(n)}$) and denote by $w_1, \\ldots, w_{m-1}$ (resp. $w_1^{(n)}, \\ldots, w_{m-1}^{(n)}$) its vertices (see Figure\u00a0\\[picture:nonconvergence\\]). By Lemma\u00a0\\[lemma:gallery\\], $w_i^{(n)} \\to w_i$ for each $i \\in \\{1,\\ldots, m-1\\}$. Now assume that $v_i$ is correct for some $i \\in \\{1, \\ldots, m-1\\}$. To show that $v_{-i}$ is also correct, consider $X$ a chamber of $A$ having vertex $v_{-i}$ and $Y$ the other chamber of this apartment having vertex $v_{-i}$. Project these two chambers on $w_{m-i}$ by defining $X' = \\operatorname{\\mathrm{proj}}_{w_{m-i}}(X)$ and $Y' = \\operatorname{\\mathrm{proj}}_{w_{m-i}}(Y)$, and then on $v_i$ by defining $X'' = \\operatorname{\\mathrm{proj}}_{v_i}(X')$ and $Y'' = \\operatorname{\\mathrm{proj}}_{v_i}(Y')$. The chambers $X''$ and $Y''$ obviously are the chambers of $A$ having vertex $v_i$. Define $X^{(n)}$, $Y^{(n)}$, $X'^{(n)}$, $Y'^{(n)}$, $X''^{(n)}$ and $Y''^{(n)}$ in the natural way. Since $v_i$ is correct, there exists $\\eta > 0$ such that $(X'',Y'')$ is $\\eta$-correct. We can assume that $w_{m-i}^{(n)}$ is opposite $v_i^{(n)}$ and $v_{-i}^{(n)}$ for all $n \\in \\operatorname{\\mathbf{N}}$ since $E_0 \\not \\in A$. This allows us to go from $v_i$ to $v_{-i}$ via $w_{m-i}$ using Lemma\u00a0\\[lemma:technical\\] twice. This gives $\\eta' > 0$ such that $$B_{v_{-i}^{(n)}}(X^{(n)}, \\eta') \\subseteq \\operatorname{\\mathrm{proj}}_{v_{-i}^{(n)}}(\\operatorname{\\mathrm{proj}}_{w_{m-i}^{(n)}}(B_{v_i^{(n)}}(X''^{(n)}, \\eta))) \\ \\text{ for all $n \\in \\operatorname{\\mathbf{N}}$.} \\quad (**)$$ We now show that $(X,Y)$ is $\\eta'$-correct. Suppose for a contradiction that there exists $(T^{(n)})$ with $T^{(n)} \\in B_{v_{-i}^{(n)}}(X^{(n)}, \\eta')$ such that $(\\varphi_n T^{(n)})$ accumulates at $\\tilde{Y}$, the limit of $(\\varphi_{n} Y^{(n)})$. Considering $T'^{(n)} = \\operatorname{\\mathrm{proj}}_{w_{m-i}^{(n)}}(T^{(n)})$ and $T''^{(n)} = \\operatorname{\\mathrm{proj}}_{v_i^{(n)}}(T'^{(n)})$, Proposition\u00a0\\[proposition:projections\\] directly shows that $(\\varphi_n T''^{(n)})$ accumulates at $\\tilde{Y}''$, the limit of $(\\varphi_{n} Y''^{(n)})$. This is a contradiction since $(X'',Y'')$ is $\\eta$-correct and $T''^{(n)} \\in  B_{v_i^{(n)}}(X''^{(n)}, \\eta)$ for all $n \\in \\operatorname{\\mathbf{N}}$ by $(**)$. Note that we assumed $i \\in \\{1,\\ldots,m-1\\}$ but the reasoning is obviously the same for $i \\in \\{m+1, \\ldots,2m-1\\}$.\n\nThe same construction with $E_1$ instead of $E_0$ enables us to go from $v_i$ to $v_{2-i}$. We directly conclude that all vertices are correct since $v_0$ and $v_1$ are correct and $v_{i+2} = v_{2-(-i)}$ is correct as soon as $v_i$ is. The claim stands proven.\n\nFrom now on, we denote by $C_i$ the chamber having vertices $v_i$ and $v_{i+1}$ (indices being taken modulo $2m$).\n\n\\[claim:b-c\\] After possible passage to a subsequence, there exists for each $i \\in \\{0, \\ldots, 2m-1\\}$ a sequence $b_i^{(n)} \\to b_i \\not \\in A$ with $b_i^{(n)}$ adjacent to $v_i^{(n)}$ for all $n \\in \\operatorname{\\mathbf{N}}$ and $\\varphi_n b_i^{(n)} \\to \\tilde{v}_{i+1}$. Similarly, there exists a sequence $c_i^{(n)} \\to c_i \\not \\in A$ with $c_i^{(n)}$ adjacent to $v_i^{(n)}$ and $\\varphi_n c_i^{(n)} \\to \\tilde{v}_{i-1}$.\n\n![Illustration of Claim\u00a0\\[claim:b-c\\].[]{data-label=\"picture:A1\"}](figure7.eps)\n\nRecall that there are two sequences $a^{(n)} \\to a$ and $b^{(n)} \\to b$ collapsed by $(\\varphi_n)$. All the hypotheses of Proposition\u00a0\\[proposition:centerokbetter\\] being met, we can assume that $a^{(n)} \\to a$ and $b^{(n)} \\to b$ are centered at some $v_j^{(n)} \\to v_j$ and that $a^{(n)} \\in A^{(n)}$ for all $n \\in \\operatorname{\\mathbf{N}}$. Without loss of generality, we may assume that $a^{(n)} = v_{j+1}^{(n)}$. Clearly $b \\neq v_{j+1}$, and $b \\neq v_{j-1}$ since $(C_{j-1},C_j)$ is correct. The sequence $b_j^{(n)} \\to b_j$ defined by $b_j^{(n)} = b^{(n)}$ satisfies the statement of the claim for $i=j$.\n\nWe now prove that one can construct $b_{j-1}^{(n)} \\to b_{j-1}$ from $b_j^{(n)} \\to b_j$. Since $(C_{j+m+1},C_{j+m})$ is correct, there is by Lemma\u00a0\\[lemma:key\\] a sequence $d^{(n)} \\to d \\not \\in A$ with $d^{(n)}$ adjacent to $v_{j+m-1}^{(n)}$ such that $\\varphi_n d^{(n)} \\to \\tilde{v}_{j+m-2}$. After projection on $v_{j-1}^{(n)} \\to v_{j-1}$ (see Lemma\u00a0\\[lemma:opposite\\]), there is a sequence $b_{j-1}^{(n)} \\to b_{j-1} \\not \\in A$ with $b_{j-1}^{(n)}$ adjacent to $v_{j-1}^{(n)}$ such that $\\varphi_n b_{j-1}^{(n)} \\to \\tilde{v}_j$. By repeating this process, we get the sequences $b_i^{(n)} \\to b_i$ satisfying the statement for each $i \\in \\{0, \\ldots, 2m-1\\}$. Using Lemma\u00a0\\[lemma:opposite\\], there obviously also is for each vertex $v_i$ a sequence $c_i^{(n)} \\to c_i \\not \\in A$ with $c_i^{(n)}$ adjacent to $v_i^{(n)}$ such that $\\varphi_n c_i^{(n)} \\to \\tilde{v}_{i-1}$.\n\nLet us now focus on $\\operatorname{\\mathrm{Cham}}(v_0)$. In the rest of this proof, we associate to each $X \\in \\operatorname{\\mathrm{Cham}}(v_0)$ a sequence $X^{(n)} \\to X$ with $X^{(n)} \\in \\operatorname{\\mathrm{Cham}}(v_0^{(n)})$. Such a sequence exists, take for instance $X^{(n)} = \\operatorname{\\mathrm{proj}}_{v_0^{(n)}}(\\operatorname{\\mathrm{proj}}_{v_m}(X))$ ($v_m$ being opposite $v_0^{(n)}$ for sufficiently large $n$).\n\n\\[claim:chamv0\\] For each $X \\in \\operatorname{\\mathrm{Cham}}(v_0)$, there exists $\\eta_X > 0$ such that if $\\varphi_{k(n)} X^{(k(n))} \\to \\tilde{X}$ for some subsequence $(\\varphi_{k(n)})$ of $(\\varphi_n)$ and some chamber $\\tilde{X}$, then $\\varphi_{k(n)} R^{(k(n))} \\to \\tilde{X}$ for each sequence $(R^{(n)})$ with $R^{(n)} \\in B_{v_0^{(n)}}(X^{(n)}, \\eta_X)$ for all $n \\in \\operatorname{\\mathbf{N}}$.\n\nLet $X \\in \\operatorname{\\mathrm{Cham}}(v_0)$ and first assume that $X \\neq C, D_0$. Consider the sequence of apartments $A_1^{(n)} \\to A_1$ where $A_1^{(n)}$ is the apartment containing $C^{(n)}$, $X^{(n)}$ and $v_m^{(n)}$, and the sequence $A_2^{(n)} \\to A_2$ where $A_2^{(n)}$ is the apartment containing $D_0^{(n)}$, $X^{(n)}$ and $v_m^{(n)}$ ($X^{(n)}$ is different from $C^{(n)}$ and $D_0^{(n)}$ for sufficiently large $n$). The idea is to apply Lemma\u00a0\\[lemma:key\\] to these two sequences of apartments simultaneously. We do it for $A_1^{(n)} \\to A_1$. Denote by $w_{m+1}^{(n)}, \\ldots, w_{2m-1}^{(n)}$ the new vertices of $A_1^{(n)}$ as in Figure\u00a0\\[picture:A2\\]. The sequence $c_1^{(n)} \\to c_1 \\not \\in A$ is such that $\\varphi_n c_1^{(n)} \\to \\tilde{v}_0$ and, after projecting on $w_{m+1}^{(n)} \\to w_{m+1}$ (see Lemma\u00a0\\[lemma:opposite\\]), there also is a sequence $d^{(n)} \\to d \\not \\in A_1$ with $d^{(n)}$ adjacent to $w_{m+1}^{(n)}$. Moreover, by Claim\u00a0\\[claim:correct\\] there exists $\\eta_1 > 0$ such that $(C_2,C_1)$ is $\\eta_1$-correct. We can therefore apply Lemma\u00a0\\[lemma:key\\] which provides $\\eta'_1 > 0$ satisfying the properties stated in this same lemma. In the same way with $A_2^{(n)} \\to A_2$, there is $\\eta'_2 > 0$ with similar properties.\n\nWe now prove that $\\eta_X = \\min(\\eta'_1, \\eta'_2)$ is adequate. Suppose that $\\varphi_{k(n)}X^{(k(n))} \\to \\tilde{X}$ for some subsequence $(\\varphi_{k(n)})$ of $(\\varphi_n)$ and some chamber $\\tilde{X}$. If $\\tilde{X} \\neq C$, then the sequence of apartments $\\varphi_{k(n)} A_1^{(k(n))}$ converges to the apartment $\\tilde{A}_1$ containing $\\tilde{C}$, $\\tilde{X}$ and $\\tilde{v}_m$. Also, we have in this case $\\varphi_{k(n)} d^{(k(n))} \\to \\tilde{w}_{m+2}$ (the limit of $(\\varphi_{k(n)} w_{m+2}^{(k(n))})$) because $\\varphi_n c_1^{(n)} \\to \\tilde{v}_0$. This means that $(\\varphi_{k(n)})$ satisfies the three conditions (a), (b) and (c) of Lemma\u00a0\\[lemma:key\\] for $\\eta_1$ and thus $\\varphi_{k(n)} R^{(k(n))} \\to \\tilde{X}$ as soon as $(R^{(n)})$ is a sequence with $R^{(n)} \\in B_{v_0^{(n)}}(X^{(n)}, \\eta'_1)$ for all $n \\in \\operatorname{\\mathbf{N}}$. On the other hand, if $\\tilde{X} = C$ then $\\tilde{X} \\neq D_0$ and we can do the same trick with $A_2^{(n)} \\to A_2$ and $\\eta'_2$. In either case, we have shown that $\\eta_X = \\min(\\eta'_1, \\eta'_2)$ is adequate.\n\n![Illustration of Claim\u00a0\\[claim:chamv0\\].[]{data-label=\"picture:A2\"}](figure8.eps)\n\nNow if $X = C$, the convergence $\\varphi_{k(n)} X^{(k(n))} \\to \\tilde{D}_0$ cannot happen because $(C,D_0)$ is correct. We can therefore in this case do the same reasoning with only $A_2^{(n)} \\to A_2$ and take $\\eta_X = \\eta'_2$. Similarly, for $X = D_0$ we only consider $A_1^{(n)} \\to A_1$ and take $\\eta_X = \\eta'_1$.\n\nTo complete the proof, we show that the situation described in Claim\u00a0\\[claim:chamv0\\] is actually impossible. Indeed, by compactness of $\\operatorname{\\mathrm{Cham}}(v_0)$ we get $$\\operatorname{\\mathrm{Cham}}(v_0) = \\bigcup_{j=1}^r B_{v_0}(X_j, \\eta_{X_j})$$ for some $X_1, \\ldots, X_r \\in \\operatorname{\\mathrm{Cham}}(v_0)$. Passing to a subsequence, we can assume that $\\varphi_n X_j^{(n)} \\to \\tilde{X}_j$ for each $j \\in \\{1, \\ldots, r\\}$. Now consider some $S \\in \\operatorname{\\mathrm{Cham}}(\\tilde{v}_0) \\setminus \\{\\tilde{X}_1, \\ldots, \\tilde{X}_r\\}$ (which is non-empty since each panel is infinite). By a construction similar to the construction of $X^{(n)}$ from $X$, we can build a sequence $S^{(n)} \\to S$ with $S^{(n)} \\in \\operatorname{\\mathrm{Cham}}(\\varphi_n v_0^{(n)})$ for all $n \\in \\operatorname{\\mathbf{N}}$. Let $R^{(n)} = \\varphi_n^{-1}S^{(n)}$; we can assume that $R^{(n)} \\to R \\in \\operatorname{\\mathrm{Cham}}(v_0)$. There exists $j \\in \\{1, \\ldots, r\\}$ such that $R \\in B_{v_0}(X_j, \\eta_{X_j})$ and this means that $R^{(n)} \\in B_{v_0^{(n)}}(X_j^{(n)}, \\eta_{X_j})$ for sufficiently large\u00a0$n$. Hence, $(\\varphi_n R^{(n)})$ should converge to $\\tilde{X}_j$, which is impossible as $\\varphi_n R^{(n)} = S^{(n)} \\to S \\neq \\tilde{X}_j$.\n\nDefine $D_0$, $D_1$, $v_0$ and $v_1$ as in Theorem\u00a0\\[theorem:Criterion\\] and take $\\varepsilon > 0$ sufficiently small so that\n\n(A) $\\overline{B(C, \\varepsilon)} \\times \\overline{B(C', \\varepsilon)} \\subseteq \\{(X,X') \\in (\\operatorname{\\mathrm{Cham}}\\Delta)^2 \\mid X \\text{ is opposite } X'\\}$,\n\n(B) $\\rho(C,D_i) \\geq 4\\varepsilon$ for $i \\in \\{0, 1\\}$,\n\n(C) there exists $E_i \\in \\operatorname{\\mathrm{Cham}}(v_i)$ such that $\\rho(C, E_i) \\geq 3 \\varepsilon$ and $\\rho(D_i, E_i) \\geq 3 \\varepsilon$ for $i \\in \\{0,1\\}$.\n\nIt is obvious that there exists $\\varepsilon > 0$ satisfying (B) and (C), and Proposition\u00a0\\[proposition:opposite\\] tells us that we can choose it so that it also satisfies (A). It is then immediate to see that $L_\\varepsilon(C,C')$ is a closed subset of $J_\\varepsilon(C,C',\\overline{B(C, \\varepsilon)},\\overline{B(C', \\varepsilon)},E_0,E_1)$, the latter being compact by Theorem\u00a0\\[theorem:Criterion\\].\n\nConsider two opposite chambers $C$ and $C'$ of $\\Delta$. The set $L_\\varepsilon(C,C')$ is a neighbourhood of the identity in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ and is compact for some $\\varepsilon > 0$ by Corollary\u00a0\\[corollary:Epsilon\\].\n\nTopological characterization of the Moufang property {#section:characterization}\n====================================================\n\nIn this last section, our ultimate goal is to prove Theorem\u00a0\\[theorem:Characterization\\]. The implications (i) $\\Rightarrow$ (ii) and (ii) $\\Rightarrow$ (iii) are explained in the first two subsections while the most difficult part (iii) $\\Rightarrow$ (i) is the subject of the rest of the section.\n\nConvergence groups {#subsection:convergence}\n------------------\n\nIf $M$ is a topological space and $G$ is a topological group acting continuously on $M$, then the action of $G$ on $M$ is **proper** if for every compact set $K \\subseteq M$, the set $\\{g \\in G \\mid g(K) \\cap K \\neq \\varnothing\\}$ is compact. When $M$ is compact, we also say that the action of $G$ on $M$ is **$n$-proper** if the componentwise action of $G$ on the space $$M^{(n)} = \\{(x_1, \\ldots, x_n\\} \\in M^n \\mid x_i \\neq x_j \\ \\ \\forall i \\neq j\\}$$ is proper, where $M^{(n)}$ is equipped with the induced topology from the product topology on $M^{n}$. We finally say that $G$ acts as a **convergence group** on $M$ if $G$ acts $3$-properly on $M$.\n\nWe will be particularly interested in subgroups of the homeomorphism group of a compact metric space $M$ that act as a convergence group on $M$.\n\n\\[lemma:lc\\] Let $(M, d)$ be a compact metric space and $G$ be a subgroup of $\\operatorname{\\mathrm{Homeo}}(M)$ (equipped with the compact-open topology) acting as a convergence group on $M$. Then $G$ is closed in $\\operatorname{\\mathrm{Homeo}}(M)$ and $G$ is locally compact.\n\nIf $M$ has less than $3$ elements, then $\\operatorname{\\mathrm{Homeo}}(M)$ is finite and hence $G$ is locally compact and closed in $\\operatorname{\\mathrm{Homeo}}(M)$. Now suppose that $|M| \\geq 3$.\n\nWe first show that $G$ is closed in $\\operatorname{\\mathrm{Homeo}}(M)$. Let $g_n \\to g$ be a sequence with $g_n \\in G$ for all $n \\in \\operatorname{\\mathbf{N}}$ and $g \\in \\operatorname{\\mathrm{Homeo}}(M)$. We want to prove that $g \\in G$. Take $x_1, x_2, x_3$ three distinct elements of $M$ and consider $\\varepsilon > 0$ such that $d(g(x_i),g(x_j)) > 2\\varepsilon$ for each $i \\neq j$. Define $$K = \\{(x_1,x_2,x_3)\\} \\cup (\\overline{B(g(x_1), \\varepsilon)} \\times \\overline{B(g(x_2), \\varepsilon)} \\times \\overline{B(g(x_3), \\varepsilon)}) \\subseteq M^{(3)}.$$ This is clearly a compact subset of $M^{(3)}$, and $g_n \\in \\{h \\in G \\mid h(K) \\cap K \\neq \\varnothing\\}$ for sufficiently large $n$. This set being compact by hypothesis, it is closed in $G$ and $g \\in G$.\n\nWe now prove that $G$ is locally compact. Once again, take $x_1, x_2, x_3$ three distinct elements of $M$. Choose $\\varepsilon > 0$ such that $d(x_i, x_j) > 3\\varepsilon$ for each $i \\neq j$ and define $$K = \\{(y_1, y_2, y_3) \\in M^3 \\mid d(y_i, y_j) \\geq \\varepsilon \\ \\ \\forall i \\neq j\\} \\subseteq M^{(3)}.$$ This is a compact subset of $M^{(3)}$ and thus $V := \\{h \\in G \\mid h(K) \\cap K \\neq \\varnothing\\}$ is compact. But $V$ contains the ball $$B(\\operatorname{\\mathrm{id}}, \\varepsilon) := \\{h \\in G \\mid d(h(x), x) < \\varepsilon \\ \\ \\forall x \\in M\\}.$$ Indeed, if $h \\in B(\\operatorname{\\mathrm{id}}, \\varepsilon)$, then $(h(x_1),h(x_2),h(x_3)) \\in h(K) \\cap K$. Therefore, $V$ is a compact neighbourhood of the identity of $G$.\n\nThe notion of convergence group appears in the theory of trees.\n\n\\[lemma:tree\\] Let $T$ be a locally finite tree. The automorphism group $\\operatorname{\\mathrm{Aut}}(T)$ of $T$ with the topology of pointwise convergence acts as a convergence group on the space of ends $T_\\infty$ of $T$.\n\nThe action of $\\operatorname{\\mathrm{Aut}}(T)$ on the set of vertices of $T$ (endowed with the discrete topology) is proper, since $T$ is locally finite. The result then follows from the fact that one can canonically associate a vertex of $T$ to each element of $T_\\infty^{(3)}$.\n\nThe reason why we are interested in convergence groups is the existence of the following reciprocal result, due to Carette and Dreesen.\n\n\\[theorem:Carette\\] Let $G$ be a $\\sigma$-compact, locally compact, non-compact group acting continuously and transitively as a convergence group on an infinite compact totally disconnected space $M$. Then $G$ acts continuously, properly and faithfully on some locally finite tree $T$ and the spaces $M$ and $T_\\infty$ are equivariantly homeomorphic.\n\nThis is a particular case of [@Carette]\\*[Theorem\u00a0D]{}.\n\nGroup of projectivities {#subsection:projectivities}\n-----------------------\n\nIf $\\pi$ and $\\pi'$ are two opposite panels of a compact spherical building $\\Delta$, then the projection $\\operatorname{\\mathrm{proj}}_\\pi\\mid_{\\pi'} : \\operatorname{\\mathrm{Cham}}(\\pi') \\to \\operatorname{\\mathrm{Cham}}(\\pi)$ is a homeomorphism (see Corollary\u00a0\\[corollary:homeo\\]). A map of the form $$[\\pi_0, \\pi_1, \\ldots, \\pi_n] := \\operatorname{\\mathrm{proj}}_{\\pi_n} \\circ \\operatorname{\\mathrm{proj}}_{\\pi_{n-1}} \\circ \\ldots \\circ \\operatorname{\\mathrm{proj}}_{\\pi_1} \\mid_{\\pi_0}\\ : \\operatorname{\\mathrm{Cham}}(\\pi_0) \\to \\operatorname{\\mathrm{Cham}}(\\pi_n)$$ where $\\pi_0, \\pi_1, \\ldots, \\pi_n$ is a sequence of panels with $\\pi_i$ opposite $\\pi_{i-1}$ for each $i \\in \\{1, \\ldots, n\\}$ is called a **projectivity** from $\\pi_0$ to $\\pi_n$. The **group of projectivities** $\\Pi(\\pi)$ associated to a panel $\\pi$ is the group of all projectivities from $\\pi$ to $\\pi$ (seen as a subgroup of $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$, so that two projectivities acting in the same way on $\\operatorname{\\mathrm{Cham}}(\\pi)$ are considered equal).\n\nNow let $X$ be a locally finite thick affine building of rank at least\u00a0$3$ and of irreducible type. It is well known (see [@Grundhofer]\\*[Proposition\u00a06.4]{}) that the spherical building $X_\\infty$ at infinity of $X$ can be given a structure of totally disconnected compact building. Also, one can construct for each panel $\\pi$ of $X_\\infty$ a locally finite thick tree $T(\\pi)$, often called *panel-tree*, whose space of ends $T_\\infty(\\pi)$ (we should actually write $T(\\pi)_\\infty$) is in bijective correspondence with $\\operatorname{\\mathrm{Cham}}(\\pi)$ (see [@Ronan]\\*[Lemma\u00a010.4]{} or [@Weiss]\\*[Proposition\u00a011.22]{}). This bijection $\\psi : T_\\infty(\\pi) \\to \\operatorname{\\mathrm{Cham}}(\\pi)$ is a homeomorphism if we equip $T_\\infty(\\pi)$ with the usual topology. There also is a natural action of the group of projectivities $\\Pi(\\pi)$ on $T(\\pi)$ and $\\psi$ is equivariant under $\\Pi(\\pi)$. These remarks are summarized in the following proposition.\n\n\\[proposition:equivariant\\] Let $X$ be a locally finite thick irreducible affine building of rank at least\u00a0$3$. For each panel $\\pi$ of $X_\\infty$, there exist a locally finite thick tree $T(\\pi)$ such that the group of projectivities $\\Pi(\\pi)$ acts on $T(\\pi)$ and a homeomorphism $\\psi : T_\\infty(\\pi) \\to \\operatorname{\\mathrm{Cham}}(\\pi)$ equivariant under $\\Pi(\\pi)$.\n\nSee the results mentioned in the previous discussion.\n\nWe can now prove the implication (i) $\\Rightarrow$ (ii) in Theorem\u00a0\\[theorem:Characterization\\].\n\n\\[proposition:affine\\] Let $X$ be a locally finite thick irreducible affine building of rank at least\u00a0$3$. For each panel $\\pi$ of $X_\\infty$, the closure of the group of projectivities $\\Pi(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\nIn view of Proposition\u00a0\\[proposition:equivariant\\], given a panel $\\pi$ of $X_\\infty$ there is a tree $T(\\pi)$ such that $\\Pi(\\pi)$ acts on $T(\\pi)$ and the spaces $T_\\infty(\\pi)$ and $\\operatorname{\\mathrm{Cham}}(\\pi)$ are equivariantly homeomorphic. We can therefore see $\\Pi(\\pi)$ as a subgroup of $\\operatorname{\\mathrm{Aut}}(T(\\pi))$, and the closure $\\overline{\\Pi}(\\pi)$ of $\\Pi(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(T_\\infty(\\pi))$ remains in $\\operatorname{\\mathrm{Aut}}(T(\\pi))$ since $\\operatorname{\\mathrm{Aut}}(T(\\pi))$ is closed in $\\operatorname{\\mathrm{Homeo}}(T_\\infty(\\pi))$. Lemma\u00a0\\[lemma:tree\\] then states that $\\operatorname{\\mathrm{Aut}}(T(\\pi))$ acts as a convergence group on $T_\\infty(\\pi)$. Since the restriction to a closed subgroup of an action as a convergence group is itself an action as a convergence group, this is also the case of $\\overline{\\Pi}(\\pi)$. Recalling that $T_\\infty(\\pi)$ and $\\operatorname{\\mathrm{Cham}}(\\pi)$ are equivariantly homeomorphic, we see that $\\overline{\\Pi}(\\pi)$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\n\\[corollary:MoufangCG\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$. If $\\Delta$ is Moufang, then for each panel $\\pi$ of $\\Delta$ the closure in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ of the group of projectivities $\\Pi(\\pi)$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\nBy\u00a0[@Grundhofer]\\*[Theorem\u00a01.1]{}, $\\Delta$ is the building at infinity of some locally finite thick irreducible affine building $X$ of rank\u00a0$3$, i.e. $\\Delta \\cong X_\\infty$. The result then follows from Proposition\u00a0\\[proposition:affine\\].\n\nTo prove the implication (ii) $\\Rightarrow$ (iii) in Theorem\u00a0\\[theorem:Characterization\\], we need some basic observations about groups acting on a locally finite thick tree $T$ so that the induced action on $T_\\infty$ is $2$-transitive. We use a result of Tits to prove the following lemma, but the reader should be aware that it can also be shown without using such a deep result.\n\n\\[lemma:hyperbolic\\] Let $T$ be a locally finite thick tree and let $G \\leq \\operatorname{\\mathrm{Aut}}(T)$. If $G$ acts $2$-transitively on $T_\\infty$, then there exists a hyperbolic element in $G$, i.e. some $g \\in G$ that stabilizes a bi-infinite geodesic $\\ell$ of $T$ and acts non-trivially on $\\ell$ by translation.\n\nAssume for a contradiction that $G$ does not contain any hyperbolic element. Then by\u00a0[@Titsarbres]\\*[Proposition\u00a03.4]{}, $G$ is contained in the stabilizer of a vertex, an edge or an end of $T$. However, none of these three situations is possible since $G$ acts $2$-transitively on $T_\\infty$.\n\n\\[lemma:ray\\] Let $T$ be a locally finite thick tree and let $G \\leq \\operatorname{\\mathrm{Aut}}(T)$. Consider an infinite ray $r = (v_i)_{i \\geq 1}$ in $T$ and let $v_0$ and $v'_0$ be two vertices of $T$ adjacent to $v_1$ and different from $v_2$. If $G$ acts $2$-transitively on $T_\\infty$, then there exists $g \\in G$ such that $g(r) = r$ and $g(v_0) = v'_0$.\n\nConsider a bi-infinite geodesic $\\ell$ of $T$ containing $r$ and $v_0$. By Lemma\u00a0\\[lemma:hyperbolic\\], there exists a hyperbolic element in $G$, and by $2$-transitivity of $G$ there even is an element $t \\in G$ stabilizing $\\ell$ and acting on it non-trivially by translation. Replacing $t$ by $t^{-1}$ if necessary, we can assume that $t(v_0) = v_m$ for some $m > 0$. Now let $\\ell'$ be a bi-infinite geodesic containing $r$ and $v'_0$. By $2$-transitivity of $G$, we can conjugate $t$ by an adequate element of $G$ to find $t' \\in G$ stabilizing $\\ell'$ and such that $t'(v'_0) = v_m$. Then $t'^{-1} t \\in G$ fixes $r$ and sends $v_0$ on $v'_0$.\n\nThe implication (ii) $\\Rightarrow$ (iii) in Theorem\u00a0\\[theorem:Characterization\\] can now be shown.\n\n\\[proposition:proj-CG\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $\\pi$ be a panel of $\\Delta$. Suppose that the closure of the group of projectivities $\\Pi(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$. Then for all $G \\leq \\operatorname{\\mathrm{Auttop}}(\\Delta)$, the closure of the natural image of $\\operatorname{\\mathrm{Stab}}_G(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\nBy\u00a0[@Knarr]\\*[Lemma\u00a01.2]{}, the action of $\\Pi(\\pi)$ on $\\operatorname{\\mathrm{Cham}}(\\pi)$ is $2$-transitive. Now suppose that the closure $\\overline{\\Pi}(\\pi)$ of $\\Pi(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(\\pi)$.\n\n\\[claim:equivariant\\] The group $\\overline{\\Pi}(\\pi)$ acts on a locally finite thick tree $T(\\pi)$ such that $T_\\infty(\\pi)$ and $\\operatorname{\\mathrm{Cham}}(\\pi)$ are equivariantly homeomorphic.\n\nWe want to apply Theorem\u00a0\\[theorem:Carette\\] and thus check its hypotheses. The group $\\overline{\\Pi}(\\pi)$ is locally compact (Lemma\u00a0\\[lemma:lc\\]), $\\sigma$-compact (it is second-countable by [@Arens]\\*[Theorem\u00a05]{} and a locally compact second-countable space is always $\\sigma$-compact), $\\operatorname{\\mathrm{Cham}}(\\pi)$ is infinite (Lemma\u00a0\\[lemma:non-isolated\\]), compact and totally disconnected, and the action of $\\overline{\\Pi}(\\pi)$ on $\\operatorname{\\mathrm{Cham}}(\\pi)$ is continuous and transitive (even $2$-transitive). It only remains to show that $\\overline{\\Pi}(\\pi)$ is non-compact. By contradiction, suppose that $\\Pi(\\pi)$ is relatively compact in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$. It is therefore also relatively compact in $C(\\operatorname{\\mathrm{Cham}}(\\pi), \\operatorname{\\mathrm{Cham}}(\\pi))$ and, by the Arzela-Ascoli theorem, $\\Pi(\\pi)$ is equicontinuous on $\\operatorname{\\mathrm{Cham}}(\\pi)$. This is a contradiction with the fact that the action of $\\Pi(\\pi)$ on $\\operatorname{\\mathrm{Cham}}(\\pi)$ is $2$-transitive. By Theorem\u00a0\\[theorem:Carette\\], we therefore have a locally finite tree $T(\\pi)$ as wanted. This tree may be not thick, but we can actually assume it is. Indeed, the vertices of degree\u00a0$1$ can clearly be deleted, as well as the vertices of degree\u00a0$2$ by joining their two neighbours. Note that there is no infinite ray in $T(\\pi)$ only containing vertices of degree\u00a0$2$ (which would prevent the deletion of these vertices) since this would imply the existence of an isolated chamber in $\\operatorname{\\mathrm{Cham}}(\\pi)$, which is impossible in view of Lemma\u00a0\\[lemma:non-isolated\\].\n\nThe next claim shows that the structure of $T(\\pi)$ is encoded in the group structure of $\\overline{\\Pi}(\\pi)$. For any vertex $v$ of $T(\\pi)$, we write $S_v$ for the stabilizer of $v$ in $\\overline{\\Pi}(\\pi)$.\n\n\\[claim:encoded\\] The groups $S_v$ are pairwise distinct and are exactly the maximal compact subgroups $K$ of $\\overline{\\Pi}(\\pi)$ such that $[K : (K \\cap H)] \\geq 3$ for any other maximal compact subgroup $H$ of $\\overline{\\Pi}(\\pi)$. Moreover, two distinct vertices $v$ and $v'$ of $T(\\pi)$ are adjacent if and only if $[S_v : (S_v \\cap S_{v'})] \\leq [S_v : (S_v \\cap S_w)]$ for any vertex $w$ different from $v$.\n\nIn view of Lemma\u00a0\\[lemma:ray\\], the groups $S_v$ are pairwise distinct. Moreover, they are maximal compact subgroups of $\\overline{\\Pi}(\\pi)$ and the only other maximal compact subgroups are the stabilizers of those edges $e$ for which there is an element of $\\overline{\\Pi}(\\pi)$ inverting $e$ (see\u00a0[@HarmonicAnalysis]\\*[Theorem\u00a05.2]{}). If $S_e$ denotes the stabilizer in $\\overline{\\Pi}(\\pi)$ of such an edge $e = [v_1,v_2]$, then $[S_e : (S_e \\cap S_{v_1})] = 2$. On the other hand, if $v$ is a vertex of $T(\\pi)$ then by Lemma\u00a0\\[lemma:ray\\] and since $T(\\pi)$ is thick we have $[S_v : (S_v \\cap S_{v'})] \\geq 3$ for $v'$ a vertex different from $v$ and $[S_v : (S_v \\cap S_e)] \\geq 3$ for $e$ an edge of $T(\\pi)$. The first part of the claim is therefore proven.\n\nNow fix a vertex $v$ of $T$. By Lemma\u00a0\\[lemma:ray\\], for any other vertex $w$ the equality $$[S_v : (S_v \\cap S_w)] = \\deg(v) \\cdot \\prod_{i=1}^{n-1}(\\deg(v_i)-1)$$ holds, where $v, v_1, \\ldots, v_{n-1}, w$ is the path in $T(\\pi)$ joining $v$ and $w$ (this formula can be obtained by induction on $n$). The second part of the claim follows from this observation.\n\n\\[claim:actiontree\\] The stabilizer $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$ of $\\pi$ in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ acts on $T(\\pi)$ and the homeomorphism between $T_\\infty(\\pi)$ and $\\operatorname{\\mathrm{Cham}}(\\pi)$ is equivariant under $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$.\n\nFirst remark that $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$ normalizes $\\Pi(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$. Indeed, if $[\\pi, \\pi_1, \\ldots, \\pi_{n-1}, \\pi]$ is a projectivity and $g \\in \\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$, one can check that $$g \\circ [\\pi, \\pi_1, \\ldots, \\pi_{n-1}, \\pi] \\circ g^{-1} = [\\pi, g(\\pi_1), \\ldots, g(\\pi_{n-1}), \\pi].$$ In view of Claim\u00a0\\[claim:encoded\\], the action by conjugation of $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$ on $\\Pi(\\pi)$ induces an action of $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$ on $T(\\pi)$. It remains to show that the homeomorphism $\\psi : T_\\infty(\\pi) \\to \\operatorname{\\mathrm{Cham}}(\\pi)$ given by Claim\u00a0\\[claim:equivariant\\] is equivariant under $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$. Consider $g \\in \\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$, $C \\in \\operatorname{\\mathrm{Cham}}(\\pi)$ and $(v_i)_{i \\geq 1}$ a ray in $T(\\pi)$ representing the end corresponding to $C$. We want to show that the ray $(g(v_i))_{i \\geq 1}$ represents the end corresponding to $g(C)$. To do so, remark that $$\\bigcup_{k=1}^\\infty \\bigcap_{i=k}^\\infty S_{v_i}$$ fixes $C$ but does not fix any chamber different from $C$ (this follows from Lemma\u00a0\\[lemma:ray\\]). Now we also have $$g\\left(\\bigcup_{k=1}^\\infty \\bigcap_{i=k}^\\infty S_{v_i}\\right)g^{-1} = \\bigcup_{k=1}^\\infty \\bigcap_{i=k}^\\infty S_{g(v_i)} =: R$$ by definition of the action of $\\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(\\pi)$ on $T(\\pi)$. But $R$ fixes $g(C)$ and also fixes the chamber corresponding to the end represented by the ray $(g(v_i))_{i \\geq 1}$. Since $R$ can only fix one chamber, the claim stands proven.\n\nIn view of Claim\u00a0\\[claim:actiontree\\] and since $\\operatorname{\\mathrm{Aut}}(T(\\pi))$ is closed in $\\operatorname{\\mathrm{Homeo}}(T_\\infty(\\pi))$, for any $G \\leq \\operatorname{\\mathrm{Auttop}}(\\Delta)$ the closure of the image of $\\operatorname{\\mathrm{Stab}}_G(\\pi)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(\\pi))$ can be seen as a (closed) subgroup of $\\operatorname{\\mathrm{Aut}}(T(\\pi))$. The conclusion thus follows from Lemma\u00a0\\[lemma:tree\\], as the restriction to a closed subgroup of an action as a convergence group is itself an action as a convergence group.\n\nConventions for the rest of the section {#subsection:conventions}\n---------------------------------------\n\nThe rest of this section is dedicated to the proof of the implication (iii) $\\Rightarrow$ (i) in Theorem\u00a0\\[theorem:Characterization\\]. Unless otherwise stated, we will from now on study an infinite thick compact totally disconnected $m$-gon $\\Delta$ with $m \\geq 3$. We will also assume that we are in presence of a strongly transitive closed subgroup $G$ of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ such that $\\Delta$ satisfies the *property $\\operatorname{\\mathcal{CG}}_G$*:\n\nLet $\\Delta$ be a compact polygon and let $G$ be a subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$. We will say that $\\Delta$ has the **property $\\operatorname{\\mathcal{CG}}_G$** if for each vertex (i.e. panel) $v$ of $\\Delta$, the closure of the image of $\\operatorname{\\mathrm{Stab}}_{G}(v)$ in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(v))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(v)$.\n\nOur ultimate goal is to prove that, under these conditions, $\\Delta$ has the Moufang property. We can already directly use Theorem\u00a0\\[theorem:Carette\\] to construct a tree associated to each vertex of $\\Delta$.\n\n\\[proposition:tree\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $G$ be a strongly transitive subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ satisfying property $\\operatorname{\\mathcal{CG}}_G$. Then for each vertex $v$ of $\\Delta$ there exists a locally finite thick tree $T(v)$ such that $\\operatorname{\\mathrm{Stab}}_{G}(v)$ acts on $T(v)$ and the spaces $T_\\infty(v)$ and $\\operatorname{\\mathrm{Cham}}(v)$ are equivariantly homeomorphic.\n\nThe property $\\operatorname{\\mathcal{CG}}_G$ tells us that the group $\\overline{i(\\operatorname{\\mathrm{Stab}}_G(v))}$ where $i$ is the natural map from $\\operatorname{\\mathrm{Stab}}_{G}(v)$ to $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(v))$ acts as a convergence group on $\\operatorname{\\mathrm{Cham}}(v)$. All the hypotheses of Theorem\u00a0\\[theorem:Carette\\] are then satisfied. Indeed, $\\overline{i(\\operatorname{\\mathrm{Stab}}_G(v))}$ is locally compact (Lemma\u00a0\\[lemma:lc\\]) and $\\sigma$-compact (it is second-countable by [@Arens]\\*[Theorem 5]{} and a locally compact second-countable space is always $\\sigma$-compact), $\\operatorname{\\mathrm{Cham}}(v)$ is infinite (Lemma\u00a0\\[lemma:non-isolated\\]), compact and totally disconnected, and the action of $\\overline{i(\\operatorname{\\mathrm{Stab}}_G(v))}$ on $\\operatorname{\\mathrm{Cham}}(v)$ is continuous and transitive (even $2$-transitive, see Lemma\u00a0\\[lemma:strongtransitivity\\]). It only remains to show that $\\overline{i(\\operatorname{\\mathrm{Stab}}_G(v))}$ is non-compact. Suppose for a contradiction that $i(\\operatorname{\\mathrm{Stab}}_G(v))$ is relatively compact in $\\operatorname{\\mathrm{Homeo}}(\\operatorname{\\mathrm{Cham}}(v))$. It is therefore also relatively compact in $C(\\operatorname{\\mathrm{Cham}}(v), \\operatorname{\\mathrm{Cham}}(v))$ and, by the Arzela-Ascoli theorem, $i(\\operatorname{\\mathrm{Stab}}_G(v))$ is equicontinuous on $\\operatorname{\\mathrm{Cham}}(v)$. This is a contradiction with the fact that the action of $i(\\operatorname{\\mathrm{Stab}}_G(v))$ on $\\operatorname{\\mathrm{Cham}}(v)$ is $2$-transitive.\n\nHence, there is a locally finite tree $T(v)$ such that $\\overline{i(\\operatorname{\\mathrm{Stab}}_G(v))}$ acts on $T(v)$ and so that $T_\\infty(v)$ and $\\operatorname{\\mathrm{Cham}}(v)$ are equivariantly homeomorphic. The group $\\operatorname{\\mathrm{Stab}}_G(v)$ obviously acts on $T(v)$ too, and the homeomorphism remains equivariant under $\\operatorname{\\mathrm{Stab}}_G(v)$. We can also assume that $T(v)$ is thick since the vertices of degree $\\leq 2$ can be deleted (for those of degree\u00a0$2$, the two neighbours have to be joined by an edge). Note that there is no infinite ray only containing vertices of degree $2$ since it would contradict Lemma\u00a0\\[lemma:non-isolated\\].\n\nAutomorphisms fixing an apartment\n---------------------------------\n\nIn this subsection, we focus on the fixator $H = \\operatorname{\\mathrm{Fix}}_G(A)$ of an apartment $A$ of $\\Delta$ (where $\\Delta$ and $G$ are as described in Subsection\u00a0\\[subsection:conventions\\]). By Proposition\u00a0\\[proposition:tree\\], for each vertex $v$ of $A$ there exists a tree $T(v)$ whose space of ends corresponds to $\\operatorname{\\mathrm{Cham}}(v)$. The group $H$ acts simultaneously on all these trees. Moreover, for each vertex $v$ of $A$, $H$ fixes the two chambers of $A$ having vertex $v$, which means that it fixes the two corresponding ends of $T(v)$. In this tree, $H$ thus stabilizes the bi-infinite geodesic between these two ends, which we denote by $\\ell_v$. We therefore have for each vertex $v$ of $A$ an induced action of $H$ on $\\ell_v$.\n\nOur goal is now to show that for every vertex $v$ of $A$, there exists some element of $H$ fixing $\\ell_{v}$ and $\\ell_{v'}$ pointwise where $v'$ is the vertex of $A$ opposite $v$ but acting non-trivially on every other $\\ell_{w}$. We will actively use this result in the last subsection to prove that $\\Delta$ is Moufang.\n\nAs a first step in this direction, we show that for each vertex $v$ of $A$, there is an element of $H$ that acts non-trivially on $\\ell_v$.\n\n\\[lemma:notfix\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $G$ be a strongly transitive subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ satisfying property $\\operatorname{\\mathcal{CG}}_G$. Let $A$ be an apartment of $\\Delta$, $v$ be a vertex of $A$ and $H = \\operatorname{\\mathrm{Fix}}_G(A)$. There is an element of $H$ that acts non-trivially on $\\ell_v$.\n\nLet $v'$ be the vertex of $A$ opposite $v$ and let $L = \\operatorname{\\mathrm{Stab}}_G(v,v')$. Since $G$ is strongly transitive, the natural image $\\tilde{L}$ of $L$ in $\\operatorname{\\mathrm{Aut}}(T(v))$ acts $2$-transitively on $T_\\infty(v)$ (see Lemma\u00a0\\[lemma:strongtransitivity\\]). By Lemma\u00a0\\[lemma:hyperbolic\\], there is a hyperbolic element in $\\tilde{L}$ and hence an element that stabilizes $\\ell_v$ but does not fix it pointwise. It corresponds to an element of $H$ that does not fix $\\ell_v$ pointwise.\n\nTo go further in the discussion, we denote by $v_0, \\ldots, v_{2m-1}$ the vertices of $A$ in the natural order and distinguish three types of possible actions of an element $t \\in H$ around a vertex $v_i$:\n\n(a) If $t$ fixes $\\ell_{v_i}$ pointwise, we will say that $t$ acts **trivially around** $v_i$.\n\n(b) If $t$ translates $\\ell_{v_i}$ toward the end representing the chamber of $A$ having vertex $v_{i+1}$, we will say that $t$ acts **positively around** $v_i$.\n\n(c) If $t$ translates $\\ell_{v_i}$ toward the end representing the chamber of $A$ having vertex $v_{i-1}$, we will say that $t$ acts **negatively around** $v_i$.\n\nAn example of such an action of an element $t \\in H$ on $A$ is shown in Figure\u00a0\\[picture:action\\]. An arrow in the anticlockwise direction means that the action is positive while an arrow in the clockwise direction means that it is negative. In this figure, the action is positive around $v_0, v_1$ and $v_2$ and negative around $v_3, v_4$ and $v_5$.\n\nThe three types of action can easily be spotted by observing how $t^n$ acts on $\\Delta$ when $n$ goes to infinity. Indeed, denoting by $C_{+}$ the chamber having vertices $v_i$ and $v_{i+1}$, by $C_{-}$ the chamber having vertices $v_i$ and $v_{i-1}$, and by $C$ any chamber having vertex $v_i$ different from $C_+$ and $C_-$, we get the following characterizations.\n\n(a) $t$ acts trivially around $v_i$ $\\Leftrightarrow$ $t^n(C)$ does not accumulate at $C_+$ nor at $C_-$.\n\n(b) $t$ acts positively around $v_i$ $\\Leftrightarrow$ $t^n(C) \\to C_+$.\n\n(c) $t$ acts negatively around $v_i$ $\\Leftrightarrow$ $t^n(C) \\to C_-$.\n\n![Representation of an action.[]{data-label=\"picture:action\"}](figure9.eps)\n\nThese characterizations already allow us to prove the following lemma giving a relationship between the action of an element $t$ around $v_i$ and its action around $v_{i+m}$, the opposite vertex in $A$. The action represented in Figure\u00a0\\[picture:action\\] is coherent with this information.\n\n\\[lemma:oppositeaction\\] The action of an element $t \\in H$ around a vertex $v_i$ of $A$ is opposite to its action around $v_{i+m}$: $t$ either acts trivially around the two vertices, or it acts positively around one and negatively around the other.\n\nLet $C \\not \\in A$ be a chamber having vertex $v_i$. Then $\\operatorname{\\mathrm{proj}}_{v_{i+m}}(C)$ is a chamber having vertex $v_{i+m}$ and $t^n(\\operatorname{\\mathrm{proj}}_{v_{i+m}}(C)) = \\operatorname{\\mathrm{proj}}_{v_{i+m}}(t^n(C))$. By Proposition\u00a0\\[proposition:projections\\], the action of $t$ around $v_i$ is opposite to its action around $v_{i+m}$.\n\nWe can now prove that $H$ always contains some particular elements.\n\n\\[proposition:particular\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $G$ be a strongly transitive subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ satisfying property $\\operatorname{\\mathcal{CG}}_G$. Let $A$ be an apartment of $\\Delta$, $v_0, \\ldots, v_{2m-1}$ be the vertices of $A$ and $H = \\operatorname{\\mathrm{Fix}}_G(A)$. For each $i \\in \\{0, \\ldots, 2m-1\\}$, there is an element of $H$ that acts positively around all vertices of $A$ closer to $v_i$ than to $v_{i+1}$ and negatively around all vertices of $A$ closer to $v_{i+1}$ than to $v_i$ (indices being taken modulo $2m$).\n\nWe divide the proof into three claims.\n\n\\[claim:somewhere\\] There exists an element of $H$ that acts positively around $v_s$ and negatively around $v_{s+1}$ for some $s \\in \\{0, \\ldots, 2m-1\\}$.\n\nIn view of Lemma\u00a0\\[lemma:notfix\\], there is an element of $H$ acting positively around at least one vertex of $A$. We consider $t \\in H$ an element acting positively around a maximal number of consecutive vertices of $A$, and denote by $v_r, v_{r+1}, \\ldots, v_s$ these vertices. Note that $t$ does not act positively around all vertices of $A$ in view of Lemma\u00a0\\[lemma:oppositeaction\\]. Hence, $t$ acts negatively or trivially around $v_{s+1}$. Suppose that $t$ acts trivially around $v_{s+1}$. By Lemma\u00a0\\[lemma:notfix\\], there is $t' \\in H$ acting non-trivially around $v_{s+1}$. Replacing $t'$ by $t'^{-1}$ if necessary, we can assume that $t'$ acts positively around $v_{s+1}$. Then for sufficiently large $n \\in \\operatorname{\\mathbf{N}}$, $t^n t' \\in H$ acts positively around $v_r, \\ldots, v_s, v_{s+1}$ which contradicts the maximality of $t$. This means that $t$ acts negatively around $v_{s+1}$, and the claim stands proven.\n\n\\[claim:small\\] For each $i \\in \\{0, \\ldots, 2m-1\\}$, there exists an element of $H$ that acts positively around $v_i$ and negatively around $v_{i+1}$.\n\nBy Claim\u00a0\\[claim:somewhere\\], there is $t \\in H$ acting positively around $v_s$ and negatively around $v_{s+1}$ for some $s \\in \\{0, \\ldots, 2m-1\\}$. Now take $i \\in \\{0, \\ldots, 2m-1\\}$. Since $G$ is strongly transitive, there is an element $g \\in H$ sending the chamber with vertices $v_s$ and $v_{s+1}$ to the chamber having vertices $v_i$ and $v_{i+1}$. It can be directly checked that $gtg^{-1} \\in H$ acts positively around $v_i$ and negatively around $v_{i+1}$.\n\n\\[claim:particular\\] Let $s \\in \\{0, \\ldots, 2m-1\\}$ and let $t \\in H$ be an element acting positively around $v_s$ and negatively around $v_{s+1}$. Then $t$ acts positively around all vertices of $A$ closer to $v_s$ than to $v_{s+1}$ and negatively around all vertices of $A$ closer to $v_{s+1}$ than to $v_s$.\n\n![Illustration of Claim\u00a0\\[claim:particular\\].[]{data-label=\"picture:particular\"}](figure10.eps)\n\nBy Lemma\u00a0\\[lemma:oppositeaction\\], $t$ acts negatively around $v_{s+m}$ and positively around $v_{s+m+1}$ (see Figure\u00a0\\[picture:particular\\]). We now want to show that $t$ acts negatively around each vertex between $v_{s+1}$ and $v_{s+m}$ and positively around all other vertices. We therefore consider some vertex $v_i$ with $s+1 < i < s+m$ and show that the action around $v_i$ is negative (and it will follow that the action around $v_{i+m}$ is positive). Denoting by $C_j$ the chamber having vertices $v_j$ and $v_{j+1}$ for $j \\in \\{0, \\ldots, 2m-1\\}$, it suffices to find some chamber $C \\not \\in A$ having vertex $v_i$ and such that $t^n(C) \\to C_{i-1}$. To get one, consider a chamber $D \\not \\in A$ having vertex $v_{s+m+1}$ and a chamber $D' \\not \\in A$ having vertex $v_s$. Let also $a$ (resp. $a'$) be the vertex of $D$ (resp. $D'$) different from $v_{s+m+1}$ (resp. $v_s$). The vertices $a$ and $a'$ are almost opposite while $t^n(a) \\to v_{s+m+2}$ and $t^n(a') \\to v_{s+1}$ with $v_{s+m+2}$ and $v_{s+1}$ almost opposite. Hence, $t^n([a, a']) \\to [v_{s+m+2}, v_{s+1}]$ (Lemma\u00a0\\[lemma:gallery\\]). Now consider $b$ the first vertex of $[a,a']$ opposite $v_i$ and $C'$ the chamber just after $b$ in this new gallery from $v_{s+m+1}$ to $v_s$ (see Figure\u00a0\\[picture:particular\\]). Clearly $t^n(b) \\to v_{i+m}$ and $t^n(C') \\to C_{i+m}$. Define finally $C = \\operatorname{\\mathrm{proj}}_{v_i}(C')$ to get $t^n(C) \\to C_{i-1}$ as wanted (see Proposition\u00a0\\[proposition:projections\\]). Note that $C \\neq C_{i-1}$ since the vertex of $C'$ different from $b$ is opposite $v_{i-1}$.\n\nThe proposition follows from Claim\u00a0\\[claim:small\\] and Claim\u00a0\\[claim:particular\\].\n\nThe desired result can finally be shown.\n\n\\[proposition:particular2\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $G$ be a strongly transitive subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ satisfying property $\\operatorname{\\mathcal{CG}}_G$. Let $A$ be an apartment of $\\Delta$, $v_0, \\ldots, v_{2m-1}$ be the vertices of $A$ and $H = \\operatorname{\\mathrm{Fix}}_G(A)$. For each $i \\in \\{0, \\ldots, m-1\\}$, there is an element of $H$ that acts trivially around $v_i$ and $v_{i+m}$ but non-trivially around all other vertices of $A$.\n\nFix $i \\in  \\{0,\\ldots,m-1\\}$. By Proposition\u00a0\\[proposition:particular\\], there exists $t_1 \\in H$ acting positively around all vertices of $A$ closer to $v_{i-1}$ than to $v_i$ and negatively around all vertices of $A$ closer to $v_i$ than to $v_{i-1}$. There also exists $t_2 \\in H$ acting positively around all vertices of $A$ closer to $v_i$ than to $v_{i+1}$ and negatively around all vertices of $A$ closer to $v_{i+1}$ than to $v_i$. Denote by $T_1 \\in \\operatorname{\\mathbf{N}}^*$ (resp. $T_2 \\in \\operatorname{\\mathbf{N}}^*$) the length of the translation of $\\ell_{v_i}$ induced by $t_1$ (resp. $t_2$). Now consider the element $t = t_1^{T_2} t_2^{T_1} \\in H$. Clearly, $t$ acts trivially around $v_i$ (and hence around $v_{i+m}$) and non-trivially around all other vertices of $A$.\n\nProof of the characterization of the Moufang property\n-----------------------------------------------------\n\nThe proof of implication (iii) $\\Rightarrow$ (i) in Theorem\u00a0\\[theorem:Characterization\\] will use the theory of contraction groups. If $G$ is a topological group and $t \\in G$, the **contraction group** associated to $t$ is the subgroup of $G$ defined by $$\\operatorname{\\mathrm{Con}}_G(t) = \\{g \\in G \\mid t^n g t^{-n} \\to e\\}$$ where $e$ is the identity element of $G$. The **parabolic subgroup** associated to $t$ is the subgroup of $G$ defined by $$\\operatorname{\\mathrm{Par}}_G(t) = \\{g \\in G \\mid \\{t^n g t^{-n}\\}_{n \\in \\operatorname{\\mathbf{N}}} \\text{ is relatively compact in } G\\}.$$\n\nThe following lemma is immediate.\n\n\\[lemma:inclusion\\] Let $f : G \\to H$ be a continuous homomorphism of topological groups and let $t \\in G$. We have the following inclusions. $$f(\\operatorname{\\mathrm{Con}}_G(t)) \\subseteq \\operatorname{\\mathrm{Con}}_H(f(t)) \\ \\ \\text{ and } \\ \\ f(\\operatorname{\\mathrm{Par}}_G(t)) \\subseteq \\operatorname{\\mathrm{Par}}_H(f(t))$$\n\nUsing the continuity of $f$, we directly get $f(\\operatorname{\\mathrm{Con}}_G(t)) \\subseteq \\operatorname{\\mathrm{Con}}_H(f(t))$. Similarly, the inclusion $f(\\operatorname{\\mathrm{Par}}_G(t)) \\subseteq \\operatorname{\\mathrm{Par}}_H(f(t))$ follows from the fact that a continuous image of a compact set is compact.\n\nIn the case where $G$ is the automorphism group of a tree, we can compute the contraction groups and the parabolic subgroups associated to some particular elements.\n\n\\[lemma:treegroups\\] Let $T$ be a locally finite tree and let $t$ be an element of $\\operatorname{\\mathrm{Aut}}(T)$ stabilizing a bi-infinite geodesic $(a,b)$ in $T$ (where $a, b \\in T_\\infty$).\n\n(i) If $t$ fixes $(a, b)$ pointwise, then $\\operatorname{\\mathrm{Con}}_{\\operatorname{\\mathrm{Aut}}(T)}(t) = \\{e\\}$.\n\n(ii) If $t$ translates $(a, b)$ toward $b$, then $\\operatorname{\\mathrm{Par}}_{\\operatorname{\\mathrm{Aut}}(T)}(t) = \\operatorname{\\mathrm{Fix}}_{\\operatorname{\\mathrm{Aut}}(T)}(a)$.\n\nPoint (i) directly comes from the fact that $\\{t^n\\}_{n \\in \\operatorname{\\mathbf{N}}}$ is relatively compact in $\\operatorname{\\mathrm{Aut}}(T)$ when $t$ fixes $(a,b)$ pointwise. The proof of (ii) is given in\u00a0[@Caprace]\\*[Lemma\u00a02.3]{}.\n\nFinally, the following theorem is due to Baumgartner and Willis.\n\n\\[theorem:Baumgartner\\] Let $G$ be a totally disconnected locally compact group and let $t \\in G$. Then $$\\operatorname{\\mathrm{Par}}_G(t) = \\operatorname{\\mathrm{Con}}_G(t) \\cdot (\\operatorname{\\mathrm{Par}}_G(t) \\cap \\operatorname{\\mathrm{Par}}_G(t^{-1})).$$\n\nSee\u00a0[@Baumgartner]\\*[Corollary\u00a03.17]{}.\n\nWe are now able to prove the implication (iii) $\\Rightarrow$ (i) in Theorem\u00a0\\[theorem:Characterization\\].\n\n\\[theorem:Moufang\\] Let $\\Delta$ be an infinite thick compact totally disconnected $m$-gon with $m \\geq 3$ and let $G$ be a closed strongly transitive subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ satisfying property $\\operatorname{\\mathcal{CG}}_G$. Then $\\Delta$ is Moufang.\n\nFix some root $\\alpha$ of $\\Delta$ and denote by $v_0, v_1, \\ldots, v_m$ its vertices in the natural order. We want to prove that the root group $$U_\\alpha = \\{g \\in \\operatorname{\\mathrm{Aut}}(\\Delta) \\mid g \\text{ fixes every chamber having a panel in } \\alpha \\setminus \\partial\\alpha\\}$$ acts transitively on the set of apartments containing $\\alpha$. This is equivalent to saying that $U_\\alpha$ acts transitively on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$ where $C$ is the chamber having vertices $v_0$ and $v_1$. First remark that $V_0 := \\operatorname{\\mathrm{Fix}}_G(\\alpha)$ acts transitively on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$ since $G$ is strongly transitive (see\u00a0Lemma\u00a0\\[lemma:strongtransitivity\\]). We will therefore proceed by induction, by defining successively $$\\begin{aligned}\nV_1 &= \\operatorname{\\mathrm{Ker}}(V_0 \\curvearrowright \\operatorname{\\mathrm{Cham}}(v_1)), \\\\\nV_2 &= \\operatorname{\\mathrm{Ker}}(V_1 \\curvearrowright \\operatorname{\\mathrm{Cham}}(v_2)), \\\\\n\\vdots \\hspace{0.1cm} & \\hspace{2cm} \\vdots \\\\\nV_{m-1} &= \\operatorname{\\mathrm{Ker}}(V_{m-2} \\curvearrowright \\operatorname{\\mathrm{Cham}}(v_{m-1})).\\end{aligned}$$ In this way, $V_1$ is the fixator in $G$ of $\\alpha$ and every chamber having vertex $v_1$, $V_2$ is the fixator in $G$ of $\\alpha$ and every chamber having vertex $v_1$ or $v_2$, and so on until $V_{m-1}$ which is exactly $U_\\alpha \\cap G$. The action of $V_0$ on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$ is transitive and we would like to prove that the action of $V_{m-1}$ on this set remains transitive. It thus suffices to show that if the action of $V_i$ is transitive, then the action of $V_{i+1}$ is transitive too.\n\nSuppose that the action of $V_i$ on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$ is transitive for some $i \\in \\{0, \\ldots, m-2\\}$. Consider $A$ any apartment containing $\\alpha$ and $H = \\operatorname{\\mathrm{Fix}}_G(A)$. By Proposition\u00a0\\[proposition:particular2\\], there exists $t \\in H$ fixing $\\ell_{v_{i+1}}$ pointwise but not $\\ell_{v_j}$ for $j \\in \\{0, \\ldots, i\\}$. Let $a$ be the end of $T(v_0)$ corresponding to the chamber $C$ and $b$ be the end of $T(v_0)$ corresponding to the other chamber of $A$ having vertex $v_0$. Replacing $t$ by $t^{-1}$ if necessary, we can assume that $t$ acts negatively around $v_0$, i.e. that it translates $\\ell_{v_0} = (a,b)$ toward $b$. Now consider $W_i$ the semidirect product in $G$ of $V_i$ and $\\langle t \\rangle$ (this is well defined as $H$ normalizes $V_i$). To prove that the action of $V_{i+1}$ on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$ remains transitive, we show that $\\operatorname{\\mathrm{Con}}_{W_i}(t)$ is included in $V_{i+1}$ and that its action on the set remains transitive.\n\n\\[claim:ParW\\] The equality $\\operatorname{\\mathrm{Par}}_{W_i}(t) = W_i$ holds.\n\nWe prove the claim by contradiction. Suppose that there is some element $g \\in W_i$ such that $\\{t^n g t^{-n}\\}_{n \\geq 0}$ is not relatively compact in $W_i$. Since $W_i$ is closed in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$, it is not relatively compact in $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ either. By Proposition\u00a0\\[proposition:closer\\] with $S = \\{t^n g t^{-n}\\}_{n \\geq 0}$, there exist two sequences in $\\operatorname{\\mathrm{Vert}}\\Delta$ which are collapsed by a subsequence of $(t^ngt^{-n})$ or $(t^ng^{-1}t^{-n})$. Replacing $g$ by $g^{-1}$ if necessary, we can assume that it is a subsequence of $(t^ngt^{-n})$. We would then like to use Proposition\u00a0\\[proposition:centerok\\] with the sequence of apartments $A \\to A$. Let $v_{m+1}, \\ldots, v_{2m-1}$ be the vertices of $A$ outside $\\alpha$ (in the natural order) and $D$ the chamber having vertices $v_0$ and $v_{2m-1}$. Passing to a subsequence, we can assume that $t^ngt^{-n} D \\to \\tilde{D}$ for some chamber $\\tilde{D}$. If $\\tilde{D} \\neq C$, then $t^ngt^{-n} A \\to \\tilde{A}$ where $\\tilde{A}$ is the unique apartment containing $\\alpha$ and $\\tilde{D}$ and Proposition\u00a0\\[proposition:centerok\\] gives two sequences centered at $v_k \\to v_k$ for some $k \\in \\{0, \\ldots, 2m-1\\}$ and collapsed by a subsequence of $(t^ngt^{-n})$. By Lemma\u00a0\\[lemma:opposite\\], we can assume that $k \\in \\{0, \\ldots, m-1\\}$. On the other hand, if $\\tilde{D} = C$, then $v_1 \\to v_1$ and $v_{2m-1} \\to v_{2m-1}$ are sequences centered at $v_0 \\to v_0$ and collapsed by $(t^ngt^{-n})$.\n\nIn either case, there are two sequences centered at $v_k \\to v_k$ for some $k \\in \\{0, \\ldots, m-1\\}$ and collapsed by a subsequence of $(t^ngt^{-n})$. We introduce the notation (for $j \\in \\{0, \\ldots, m\\}$) $$\\varphi_j : W_i \\to \\operatorname{\\mathrm{Aut}}(T(v_j))$$ for the natural map from $W_i \\subseteq \\operatorname{\\mathrm{Stab}}_{\\operatorname{\\mathrm{Auttop}}(\\Delta)}(v_j)$ to $\\operatorname{\\mathrm{Aut}}(T(v_j))$. For each $j \\in \\{1, \\ldots, m-~1\\}$, since $t^ngt^{-n}$ translates $\\ell_{v_j}$ exactly as $g$ does for all $n \\in \\operatorname{\\mathbf{N}}$, the set $\\{\\varphi_j(t^ngt^{-n})\\}_{n \\geq 0}$ is relatively compact in $\\operatorname{\\mathrm{Aut}}(T(v_j))$. It is also relatively compact in $\\operatorname{\\mathrm{Homeo}}(T_\\infty(v_j))$ and hence equicontinuous on $T_\\infty(v_j)$ by the Arzela-Ascoli theorem. It is therefore impossible to have two sequences centered at $v_j \\to v_j$ and collapsed by a subsequence of $(t^ngt^{-n})$ for $j \\in \\{1, \\ldots, m-1\\}$. But it is not possible either to have them centered at $v_0 \\to v_0$. Indeed, the existence of two such sequences would imply that $\\varphi_0(g) \\not \\in \\operatorname{\\mathrm{Par}}_{\\operatorname{\\mathrm{Aut}}(T(v_0))}(\\varphi_0(t)) = \\operatorname{\\mathrm{Fix}}_{\\operatorname{\\mathrm{Aut}}(T(v_0))}(a)$ (see Lemma\u00a0\\[lemma:treegroups\\] (ii)) which is impossible since $g \\in W_i$ fixes $a$. The claim is therefore proven.\n\n\\[claim:transitiveCon\\] $\\operatorname{\\mathrm{Con}}_{W_i}(t)$ acts transitively on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$.\n\nApplying Theorem\u00a0\\[theorem:Baumgartner\\] to $t \\in W_i$ (note that $W_i$ is locally compact as a closed subgroup of $\\operatorname{\\mathrm{Auttop}}(\\Delta)$ which is locally compact (Corollary\u00a0\\[corollary:lc\\]) and totally disconnected since it acts continuously and faithfully on the totally disconnected space $\\operatorname{\\mathrm{Cham}}\\Delta$), we get $$\\operatorname{\\mathrm{Par}}_{W_i}(t) = \\operatorname{\\mathrm{Con}}_{W_i}(t) \\cdot (\\operatorname{\\mathrm{Par}}_{W_i}(t) \\cap \\operatorname{\\mathrm{Par}}_{W_i}(t^{-1})).$$ Now by Lemma\u00a0\\[lemma:inclusion\\], $$\\varphi_0(\\operatorname{\\mathrm{Par}}_{W_i}(t)) \\subseteq \\varphi_0(\\operatorname{\\mathrm{Con}}_{W_i}(t)) \\cdot (\\operatorname{\\mathrm{Par}}_{\\operatorname{\\mathrm{Aut}}(T(v_0))}(\\varphi_0(t)) \\cap \\operatorname{\\mathrm{Par}}_{\\operatorname{\\mathrm{Aut}}(T(v_0))}(\\varphi_0(t^{-1}))).$$ Claim\u00a0\\[claim:ParW\\] together with Lemma\u00a0\\[lemma:treegroups\\] (ii) finally gives $$\\varphi_0(W_i) \\subseteq \\varphi_0(\\operatorname{\\mathrm{Con}}_{W_i}(t)) \\cdot \\operatorname{\\mathrm{Fix}}_{\\operatorname{\\mathrm{Aut}}(T(v_0))}(a, b).$$ But $V_i$ acts transitively on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$ and so does $W_i$, which means that $\\varphi_0(W_i)(b) = T_\\infty(v_0) \\setminus \\{a\\}$. Hence, $$T_\\infty(v_0) \\setminus \\{a\\} = \\varphi_0(W_i)(b) \\subseteq \\varphi_0(\\operatorname{\\mathrm{Con}}_{W_i}(t))(b)$$ and $\\operatorname{\\mathrm{Con}}_{W_i}(t)$ also acts transitively on $\\operatorname{\\mathrm{Cham}}(v_0) \\setminus \\{C\\}$.\n\n\\[claim:inclusionCon\\] The inclusion $\\operatorname{\\mathrm{Con}}_{W_i}(t) \\subseteq V_{i+1}$ holds.\n\nIt is not even clear for the moment that $\\operatorname{\\mathrm{Con}}_{W_i}(t)$ is included in $V_i$. If $i = 0$, then $t \\in H \\subseteq V_0$ and $W_0 = V_0$ so this is true. If $i > 0$, then $t \\not \\in V_i$ (that is actually why we introduced the semidirect product $W_i$) but we can prove that $\\operatorname{\\mathrm{Con}}_{W_i}(t) \\subseteq V_i$. Indeed, suppose by contradiction that there is an element of $\\operatorname{\\mathrm{Con}}_{W_i}(t)$ that is not in $V_i$, i.e. of the form $xt^k$ with $x \\in V_i$ and $k \\in \\operatorname{\\mathbf{Z}}_0$. This means that $t^n x t^{k-n} \\to e$ in $W_i$ when $n$ goes to infinity. But $t$ does not fix $\\ell_{v_i}$ pointwise while $x$ does as it acts trivially on $\\operatorname{\\mathrm{Cham}}(v_i)$. Hence, $t^n x t^{k-n}$ does not fix $\\ell_{v_i}$ pointwise for each $n \\in \\operatorname{\\mathbf{N}}$ and $t^n x t^{k-n} \\not\\to e$. This proves $\\operatorname{\\mathrm{Con}}_{W_i}(t) \\subseteq V_i$. Finally, since $$\\varphi_{i+1}(\\operatorname{\\mathrm{Con}}_{W_i}(t)) \\subseteq \\operatorname{\\mathrm{Con}}_{\\operatorname{\\mathrm{Aut}}(T(v_{i+1}))}(\\varphi_{i+1}(t)) = \\{e\\}$$ by Lemma\u00a0\\[lemma:treegroups\\] (i), we get $\\operatorname{\\mathrm{Con}}_{W_i}(t) \\subseteq V_{i+1}$.\n\nAs explained above, the conclusion follows from Claim\u00a0\\[claim:transitiveCon\\] and Claim\u00a0\\[claim:inclusionCon\\].\n\nWe finally give the proof of Corollary\u00a0\\[corollary:affine\\].\n\nLet $X$ be a locally finite thick affine building of rank\u00a0$3$ and of irreducible type which is strongly transitive. It is clear that $\\operatorname{\\mathrm{Aut}}(X)$ is a closed subgroup of $\\operatorname{\\mathrm{Auttop}}(X_\\infty)$ and, by Proposition\u00a0\\[proposition:affine\\] and Proposition\u00a0\\[proposition:proj-CG\\], $X_\\infty$ has the property $\\operatorname{\\mathcal{CG}}_{\\operatorname{\\mathrm{Aut}}(X)}$. Moreover, $\\operatorname{\\mathrm{Aut}}(X)$ is strongly transitive on $X$ and hence on $X_\\infty$. By Theorem\u00a0\\[theorem:Moufang\\], $X_\\infty$ is Moufang. In other words, $X$ is a Bruhat-Tits building. Those buildings have been classified by Bruhat and Tits; this classification is the subject of an important part of the book\u00a0[@Weiss].\n\n[^1]: F.R.S.-FNRS Research Fellow.\n"], ["---\nabstract: |\n    To exploit synergies between the [*Herschel*]{} Space Observatory and next generation radio facilities, we have extended the semi-empirical extragalactic radio continuum simulation of Wilman et al. to the mid- and far-infrared. Here we describe the assignment of infrared spectral energy distributions (SEDs) to the star-forming galaxies and active galactic nuclei, using [*Spitzer*]{} 24, 70 and 160\u00a0and [*SCUBA*]{} 850\u00a0survey results as the main constraints.\n\n    Star-forming galaxies dominate the source counts, and a model in which their far-infrared\u2013radio correlation and infrared SED assignment procedure are invariant with redshift underpredicts the observed 24 and 70\u00a0source counts. The 70\u00a0deficit can be eliminated if the star-forming galaxies undergo stronger luminosity evolution than originally assumed for the radio simulation, a requirement which may be partially ascribed to known non-linearity in the far-infrared\u2013radio correlation at low luminosity if it evolves with redshift. At 24, the shortfall is reduced if the star-forming galaxies develop SEDs with cooler dust and correspondingly stronger Polycyclic Aromatic Hydrocarbon (PAH) emission features with increasing redshift at a given far-infrared luminosity, but this trend may reverse at $z>1$ in order not to overproduce the sub-mm source counts. The resulting model compares favourably with recent [*BLAST*]{} results and we have extended the simulation database to aid the interpretation of [*Herschel*]{} surveys. Such comparisons may also facilitate further model refinement and revised predictions for the [*Square Kilometer Array*]{} and its precursors.\nauthor:\n- \ntitle: 'An infrared\u2013radio simulation of the extragalactic sky: from the Square Kilometer Array to Herschel'\n---\n\n\\#1[to 0pt[\\#1]{}]{} \\#1 \\#1\n\nto size\n\n\\#1\n\nto size\n\n\u00df =cmr10 scaled2 =cmbx10 scaled2 =cmti10 scaled2 PS. 2\n\n=6truein\n\ngalaxies:evolution \u2013 galaxies:active \u2013 galaxies:starburst \u2013 cosmology:observations \u2013 infrared:galaxies \u2013 radio continuum:galaxies\n\nINTRODUCTION\n============\n\nIn Wilman et al.\u00a0(2008) (hereafter W08) we presented a semi-empirical simulation of the extragalactic radio continuum sky primarily intended to aid the design of scientific programmes for the next generation of radio telescope facilities, culminating in the [*Square Kilometer Array*]{} (SKA). The simulation covers a sky area of $20 \\times 20$\u00a0deg$^{2}$ and contains radio-loud and radio-quiet active galactic nuclei (AGN) and star-forming galaxies out to redshift $z=20$ within a framework for their large-scale clustering. The full source catalogue \u2013 containing 320 million sources above 10\u00a0nJy at five frequencies ranging from 151\u00a0MHz to 18\u00a0GHz \u2013 can be accessed at the SKADS Simulated Skies ($S^{3}$) database[^1] under $S^{3}$\u2013SEX (semi-empirical extragalactic simulation).\n\nThere are by necessity numerous uncertainties and limitations in the $S^{3}$\u2013SEX simulation, including but not limited to issues such as the form of the high-redshift evolution of the AGN and galaxies, the lack of star-forming/AGN hybrid galaxies, and the abundance of highly-obscured Compton-thick AGN. In so far as possible, flexibility was built into the simulations to allow post-processing to improve their accuracy as observations in the years ahead lead to improved constraints. Major advances in these areas are expected from the far-infrared surveys to be performed by the [*Herschel*]{} Space Observatory (Pilbratt 2008). To facilitate such comparisons we have post-processed the $S^{3}$\u2013SEX simulation to cover these wavelengths. In this paper, we describe the recipes for assigning infrared spectral energy distributions (SEDs) to the radio sources, using existing mid- and far-infrared results from [*Spitzer*]{} and sub-mm survey data from [*SCUBA*]{} as the primary constraints. We then present our predictions for [*Herschel*]{} surveys, bolstered by a comparison with results from the Balloon-borne Large Aperture Submillimetre Telescope ([*BLAST*]{}) which offer a foretaste of [*Herschel*]{}\u2019s capabilities. In keeping with the philosophy of the S$^{3}$ project to maximise the degree of interaction between the user and the database, the infrared fluxes are also provided on the $S^{3}$ webpage from which users can generate simulation products for comparison with observations. The radio\u2013infrared connection is of immense empirical and theoretical interest for extragalactic surveys, for the identification and follow-up of [*Herschel*]{} sources, and for probing the physics of the far-infrared\u2013radio correlation and its possible evolution. The simulation can nevertheless also serve as a standalone [*Herschel*]{} simulation, for comparison with others such as the phenomenologically-inspired backward evolution models of Valiante et al.\u00a0(2009) and Pearson & Khan\u00a0(2009), and the model of Lacey et al.\u00a0(2009) which is based on a semi-analytical galaxy formation model.\n\nTHE ASSIGNMENT OF INFRARED SEDs TO THE RADIO SOURCE POPULATIONS\n===============================================================\n\nThe input radio source catalogue was obtained from the $S^{3}$-SEX online database and negative evolution at high redshift was imposed using the \u2018default post-processing options\u2019 described in W08. Each radio source was assigned an infrared SED from various template libraries appropriate for star-forming galaxies and AGN, and the output flux density of each infrared model component for each galaxy \\[in log(F$_{\\rm{\\nu}}$(Jy)\\] was computed at observed wavelengths of 24, 70, 100, 160, 250, 350, 450, 500, 850, and 1200 . The wavelengths of 24, 70 and 160 \u00a0are those of the [*Spitzer*]{} MIPS instrument, for which an abundance of extragalactic survey data is available to guide the construction of our model and to compare against its output (see the review of Soifer et al.\u00a02008). [*Herschel*]{} will conduct surveys at 250, 350 and 500 \u00a0with the SPIRE instrument and at 70, 100 and 160\u00a0with PACS. Longer-wavelength constraints are available from submillimetre surveys by SCUBA at 450 and 850, and from the MAMBO instrument on the 30-metre IRAM telescope at 1200. The catalogued output flux densities are monochromatic values, with the exception of the [*Spitzer*]{} 24 and 70\u00a0MIPS bands for which the modelled spectra were convolved with the MIPS bandpass transmission curves, following the [*Spitzer*]{} Synthetic Photometry Recipe[^2]. This is due to the spectral complexity in the rest-frame 10\u00a0region, particularly for the star formation component due to the presence of Polycyclic Aromatic Hydrocarbon (PAH) features.\n\nThe format of the output from the infrared extension, as it appears in the $S^{3}$-SEX online database, is described in Appendix A.\n\nStar-forming galaxies\n---------------------\n\nThe population of star-forming galaxies in W08 comprises two populations, normal (or quiescent) galaxies and starbursts, identified respectively with the low and high-luminosity Schechter function components of the local 1.4\u00a0GHz luminosity function of Yun, Reddy & Condon (2001). The entire population was evolved with pure luminosity evolution (1+z)$^{3.1}$ out to z=1.5 (defined in an Einstein-de Sitter cosmology but adapted to the flat-$\\Lambda$ cosmology used for the simulation). The default post-processing option consists of negative space density evolution of the form $(1+z)^{-7.9}$ above $z=4.8$. We stress that the terms \u2018normal galaxy\u2019 and \u2018starburst\u2019 are merely convenient labels for these two components of the local luminosity function in our phenomenological model. Physically speaking, the terms may not necessarily offer an accurate description of the nature of the star formation in these populations, especially beyond the local Universe.\n\nThe first step in assigning an infrared SED is to use the far-infrared\u2013radio correlation for star-forming galaxies (eqn.\u00a05 of Yun, Reddy & Condon\u00a02001) in order to derive the rest-frame far-infrared luminosity, L(FIR), given the rest-frame 1.4\u00a0GHz luminosity, L$_{\\rm{1.4 GHz}}$. The relation is characterised by the \u201cq\u201d parameter:\n\n$$q = \\rm{ log [L(FIR,W)}/3.75 \\times 10^{12} Hz] - log [\\rm{L_{\\rm{1.4 GHz}} (W Hz^{-1})}],$$\n\nfor which we assume a gaussian distribution with $\\mu = 2.34$ and $\\sigma = 0.26$ (Yun, Reddy & Condon). There is evidence that this relation holds out to redshift $z > 1$ (e.g. Appleton et al.\u00a02004, Garn et al.\u00a02009b), although such conclusions are sensitive to the assumed infrared SEDs and associated \u201ck-corrections\u201d. Indeed, using recent [*BLAST*]{} stacking measurements, Ivison et al.\u00a0(2009) reported evidence for a tentative decline in $q_{\\rm{IR}}$ (defined as the logarithmic ratio of the the bolometric 8\u20131000\u00a0infrared to monochromatic 1.4\u00a0GHz radio fluxes), of the form $q_{\\rm{IR}} \\propto (1+z)^{-0.15 \\pm 0.03}$ (see also Kovacs et al.\u00a02006).\n\nThe far-infrared luminosity, L(FIR), is traditionally defined through a linear combination of IRAS 60 and 100\u00a0flux densities\u00a0(see Sanders & Mirabel\u00a01996), which for an SED consisting of a superposition of modified blackbodies yields the 42.5\u2013122.5\u00a0luminosity to sub-percent accuracy (Helou et al.\u00a01988). Given L(FIR), galaxies were assigned a model SED from the library of star-forming galaxy templates assembled by Rieke et al.\u00a0(2009). The latter consists of 14 SEDs uniformly spaced in total infrared luminosity (8\u20131000), L(TIR), from log L(TIR)() = 9.75 to 13.00. The templates were taken from the online version of Table 4 in Rieke et al.\u00a0(2009) but they terminate on the short wavelength side at 5.26\u00a0for those with log L(TIR) () $\\leq 11$ and at 4.02\u00a0at higher luminosities. In order to predict 24\u00a0fluxes for galaxies at redshift $z \\geq 3$, we need to extend these templates to shorter wavelengths. For this purpose we employed the spectra of the individual local luminous (LIRGs) and ultraluminous infrared galaxies (ULIRGs) in Table 3 of Rieke et al.\u00a0(2009), which extend down to 0.4. From these, we computed mean LIRG and ULIRG spectra and matched them onto the original Rieke templates below the cut-off wavelengths of 4.02 and 5.26. The ULIRG spectrum was used for the templates with log L(TIR)() $>$ 12, and the LIRG spectrum for the remaining templates. The resulting templates should be considered as schematic below $\\sim 3$, as we did not attempt to model the stellar population and rest-frame optical extinction in a self-consistent fashion.\n\nThe template SEDs were integrated over the appropriate wavelengths ranges to yield the mapping between L(TIR) to L(FIR) shown in Fig.\u00a0\\[fig:LTIRtoFIR\\_Rieke\\]; the adopted relation between the two quantities is mildly non-linear: log L(FIR) = 1.1log L(TIR) \u2013 1.42. Based on the IRAS Bright Galaxy Sample (Soifer et al.\u00a02007), Marcillac et al.\u00a0(2006) assumed a linear relation: L(FIR) = 1.91 L(TIR). In Fig.\u00a0\\[fig:RiekesameFIR\\] we show the Rieke templates scaled to a common L(FIR).\n\nAt a given L(FIR), the local galaxy population exhibits a distribution in 60\u2013100\u00a0colour or, equivalently, dust temperature, as characterised by Chapin et al.\u00a0(2009). We use this distribution (Chapin et al.\u00a02009, equations 8 and 9) to assign each star-forming galaxy a 60\u2013100\u00a0colour, $C$, defined as the ratio of the rest-frame monochromatic 60 and 100\u00a0flux densities: $C = \\rm{log} L_{\\rm{60}}/L_{\\rm{100}}$. The template nearest in colour is then selected from the Rieke library using the empirical $C$\u2013L(TIR) conversion for these templates shown in Fig.\u00a0\\[fig:RiekeCOLOR\\], for which a functional fit is:\n\n$$C = 0.33 \\rm{tanh}[1.1(\\rm{log L(TIR)} - 11.0)] - 0.23.$$\n\nHaving selected the template shape from the Rieke library by this process, the final step is to normalise it to actual value of L(FIR) originally specified by the far-infrared\u2013radio relation. This is done using the non-linear L(FIR)\u2013L(TIR) relation given in Fig.\u00a0\\[fig:LTIRtoFIR\\_Rieke\\]. Chaplin et al.\u00a0(2009) presented evidence for evolution in the colour-luminosity relation, but we initially assume that the local relation applies at all redshifts.\n\n![The symbols show the far-infrared luminosity (42.5\u2013122.5\u00a0), L(FIR)(), of the star-forming galaxy templates of Rieke et al.\u00a0(2009) as a function of total infrared luminosity, L(TIR). To assign SED templates to the simulated galaxies, we assume the non-linear relation given by the solid line: log L(FIR) = 1.1log L(TIR) \u2013 1.42. The dashed line shows the linear relation L(FIR) = 1.91 L(TIR) assumed by Marcillac et al.\u00a0(2006).[]{data-label=\"fig:LTIRtoFIR_Rieke\"}](RIEKE_FIRvsTIR.ps){width=\"47.00000%\"}\n\n![The star-forming galaxy template SEDs scaled to a common far-infrared luminosity using the L(TIR)\u2013L(FIR) scaling in Fig.\u00a0\\[fig:LTIRtoFIR\\_Rieke\\]. With increasing L(TIR), the SEDs peak at shorter wavelengths and exhibit weaker spectral features in the $\\lambda <20$\u00a0region.[]{data-label=\"fig:RiekesameFIR\"}](RIEKEtemplates_NORMsameFIR_tempex.ps){width=\"47.00000%\"}\n\n![Colour of the Rieke templates \\[$C = \\rm{log} L_{\\rm{60}}/L_{\\rm{100}}$\\] as a function of log L(TIR)(), with the fit given by equation (2) of section 2.1.[]{data-label=\"fig:RiekeCOLOR\"}](RIEKE_COLORvsTIR.ps){width=\"47.00000%\"}\n\nRadio-quiet AGN\n---------------\n\nIn W08 a population of radio-quiet AGN was incorporated using the Ueda et al.\u00a0(2003) hard X-ray luminosity function and a relation between 2\u201310\u00a0and 1.4\u00a0GHz luminosities. To assign infrared SEDs the first task is to split the population into subgroups of unobscured AGN, and Compton-thin and Compton-thick obscured AGN. This was not done in W08 but is now necessary because the infrared SEDs are sensitive to this classification. We start from the findings of Hasinger\u00a0(2008), who used a compilation of hard X-ray surveys to parameterise the fraction of obscured Compton-thin AGN (as a proportion of a total population comprising unobscured AGN and obscured Compton-thin AGN, but excluding Compton-thick obscured AGN) as a function of $L_{\\rm{X}}$ (2\u201310\u00a0luminosity; ):\n\n$$f_{\\rm{2}} = [0.27 + \\beta(log L_{\\rm{X}} - 43.75)]g(z),$$\n\nwhere $\\beta = -0.281$ and $g(z)$ is evolution of the form (1+z)$^{0.62}$ out to $z=2$ and flat thereafter. The number of Compton-thick obscured AGN is not as well constrained, and in W08 we simply boosted the space density of the Ueda et al.\u00a0(2003) luminosity function by 50\u00a0per cent in a notional allowance for them, but left the issue open for subsequent refinement in post-processing. Ueda et al. showed that a reasonable match to the X-ray background results if the number of Compton-thick AGN simply equals the number of Compton-thin obscured AGN. We thus assume that the abundance of Compton-thick AGN population is a factor $f_{\\rm{Cthick}}$ times that of the combined population of unobscured and Compton-thin obscured AGN, where $f_{\\rm{Cthick}}$ = min\\[0.5,$f_{\\rm{2}}$\\], with the upper bound of 0.50 hard-wired into the W08 simulation. After the removal of any excess Compton-thick AGN, the remaining sources in the W08 catalogue are probabilistically identified as unobscured (type I), Compton-thin or Compton-thick obscured (type II) AGN. Finally, 10\u00a0per cent of all sources are flagged for removal because W08 included radio-loud AGN with a separate radio luminosity function but this population is also implicitly present in the Ueda et al. luminosity function. The fractions of retained radio-quiet AGN which are unobscured, Compton-thin and Compton-thick obscured are shown as functions of $L_{\\rm{X}}$ and redshift in Fig.\u00a0\\[fig:RQAGNfrac\\].\n\n![image](RQAGN_hist1_2deg.ps){width=\"35.00000%\"} ![image](RQAGN_hist2_2deg.ps){width=\"35.00000%\"} ![image](RQAGN_hist3_2deg.ps){width=\"35.00000%\"} ![image](RQAGN_hist4_2deg.ps){width=\"35.00000%\"}\n\n### AGN emission\n\nFor the retained sources, $L_{\\rm{X}}$ is inferred from the radio luminosity using equation (2) of W08. Sources with $L_{\\rm{X}}$ above and below a threshold of $3 \\times 10^{44}$\u00a0are classified as quasars and Seyferts, respectively. Unobscured (type I) quasars are assigned at random one of the three templates for optically-selected QSOs compiled by Polletta et al.\u00a0(2007) (termed QSO1, BQSO1 and TQSO1, which differ in their relative optical/infrared flux ratios)[^3]. The unobscured Seyferts are assigned SEDs from the online library of models computed by Nenkova et al.\u00a0(2008) using the [*CLUMPY*]{} code for dust radiative transfer in a clumpy AGN torus. To narrow down the parameter space of available models, we use the results of Thompson et al.\u00a0(2009) which are based on analysis of Seyfert 1 AGN mid-infrared spectra. They prefer a range of [*CLUMPY*]{} parameter space defined by the parameters $Y=30$, $q=1$, $\\sigma = 45$\u00a0deg, $m=2$, $i=30$\u00a0deg, $\\tau_{\\rm{V}}=30-60$ and $N_{\\rm{0}} \\leq 6$ (see Nenkova et al.\u00a02008 for the meaning of these parameters). 18 CLUMPY models satisfy these constraints and were assigned at random to our type I Seyferts, with the SED consisting of the reprocessed torus emission and direct AGN broken power-law emission (defined as a power-law $F_{\\rm{\\lambda}} \\propto \\lambda^{-1.5}$ over 0.1\u20131\u00a0with a Rayleigh-Jeans slope at longer wavelengths).\n\nFor the type II Compton-thin AGN, we assigned one of the type 2 QSO templates given by Polletta et al.\u00a0(2006) and Polletta et al.\u00a0(2007) (termed QSO2 and Torus, respectively) for the quasars, and one of the aforementioned CLUMPY models (but without the direct AGN emission component) to the Seyferts.\n\nFor the Compton-thick AGN, we assigned half the population SEDs using the same prescription as for the type II Compton-thin sources; for the other half of the population, we reflect the growing body of evidence which suggests that a large proportion of Compton-thick AGN are obscured by an extended distribution of cold dust in the host galaxy and not by a nuclear torus (see e.g. Polletta et al., 2006, 2008; Martinez-Sansigre et al. 2006, Lacy et al. 2007). Accordingly we assign these sources an intrinsic type I or Compton-thin type II SED, extinguished by a dust screen with $A_{\\rm{V}} = 4-  27$\u00a0mag\u00a0(as found by Polletta et al.\u00a02008 for obscured quasars). An extinction curve extending into the infrared is taken from McClure\u00a0(2009). These AGN infrared SED components are normalised using the correlation between 12 micron and hard X-ray emission found by Gandhi et al.\u00a0(2009), which appears to hold for obscured and unobscured AGN (including Compton-thick sources) and to extend into the quasar regime:\n\n$$\\rm{ log L_{\\rm{MIR}}} = (-4.37 \\pm 3.08) + (1.106 \\pm 0.071)\\rm{log} L_{\\rm{X}},$$\n\nwhere L$_{\\rm{MIR}} = \\lambda L_{\\rm{\\lambda}}$ at 12.3\u00a0in . Gaussian scatter of $\\sigma = 0.36$ is assumed on the relation.\n\n### Star-formation\n\nThere is abundant evidence for ongoing star formation in the host galaxies of radio-quiet AGN and this is likely to dominate the far-infrared SED. The consensus emerging from targeted observations of local Seyferts (Buchanan et al.\u00a02006; Thompson et al.\u00a02009) and studies of more distant samples and cosmological surveys (e.g. Silverman et al.\u00a02009; Hern\u00e1n-Caballero et al.\u00a02009; Serjeant & Hatziminaoglou\u00a02009) is that the star formation is generally higher in obscured AGN compared with their unobscured counterparts, and increases sub-linearly with AGN luminosity. We embody these findings with a star formation rate (SFR) $\\propto \\sqrt{L_{\\rm{X}}} (1+z)^{1.6}$, but assume different normalizations for the type I and II AGN. This luminosity and redshift dependence is similar to that measured for quasars by Serjeant & Hatziminaoglou\u00a0(2009) and the luminosity dependence matches that of local Seyfert 1s (Thompson et al.\u00a02009).\n\nFor type I AGN, we normalise the relation using the measurements of type I SDSS AGN at z=0.15 by Kim et al.\u00a0(2006), which have a median SFR of 0.5. The median log L(\\[OIII\\]$\\lambda5007$) of this sample, as inferred from Fig.\u00a04(b) of Kim et al., is 41.4 () which equates to log L$_{\\rm{X}}=42.6$ using the log L$_{\\rm{X}}$\u2013log L(\\[OIII\\]$\\lambda5007$) relation of eqn.\u00a0(1) of Silverman et al.\u00a0(2009). Thus for type I AGN:\n\n$${\\rm SFR} = 0.63 \\sqrt{L_{\\rm{X}} /10^{43}} (1+z)^{1.6} \\Msunpyr.$$\n\nFor the type II AGN (Compton-thin and Compton-thick), we base our results on the $z=0.1$ SDSS type II AGN sample of Kauffmann et al.\u00a0(2003), focussing on the sub-sample of strong-lined sources with log L(\\[OIII\\])$>$ 40.5 indicated by the red dots in Fig.\u00a010 of Silverman et al.\u00a0(2009). Referring once more to the log L(\\[OIII\\]) distributions in Fig.\u00a04(b) of Kim et al.\u00a0(2006), we infer for this particular luminosity-limited sample a median log L(\\[OIII\\])=41.1, which translates into log L$_{\\rm{X}}=42.25$ and hence boosts the pre-factor of our assumed SFR(L$_{\\rm{X}}$,z) relation from 0.63 to 2. A scatter of 1 dex is assumed on the SFR(L$_{\\rm{X}}$,z) relation and the star-formation rates are assumed to decline as $(1+z)^{-7.9}$ at $z > 4.8$ following the assumed evolution of the global star formation rate density. SEDs for the star formation component are assigned from the Rieke et al.\u00a0(2009) library, using the relation given by Yun, Reddy & Condon\u00a0(2001) (their eqn.\u00a013) to relate the star formation rate and 1.4\u00a0GHz radio luminosity; the far-infrared\u2013radio relation (eqn.\u00a01) was then used to assign L(FIR) and hence an appropriate template from the Rieke library using Fig.\u00a0\\[fig:LTIRtoFIR\\_Rieke\\].\n\nRadio-loud AGN\n--------------\n\nRadio-loud AGN were modelled by W08 using the Willott et al.\u00a0(2001) 151\u00a0MHz luminosity function comprising a weakly-evolving low luminosity component and a rapidly-evolving high luminosity component, identified with FRI and FRII radio sources, respectively. We consider them separately for the purposes of infrared SED assignment.\n\n### FRIs\n\nThere is scant evidence for the presence of intrinsically luminous optical-UV AGN in the radio population we associate with FRI sources (e.g. Vardoulaki et al.\u00a02008). We therefore model their infrared SEDs with a template representing host galaxy starlight only, which we produced using the GALSYNTH online interface to the GRASIL code developed by Silva et al.\u00a0(1998). The models are based on their \u2018Elliptical model\u2019, which gives the evolving SED (taking account of dust reprocessing) of a stellar population formed in a burst of 1\u00a0Gyr duration (see description of this particular model at http://adlibitum.oat.ts.astro.it/silva/grasil/modlib/modlib.html). We generated models with ages separated by 0.1\u00a0Gyr for 0.1\u20131.5\u00a0Gyr, and by 1\u00a0Gyr for ages 2\u201313\u00a0Gyr. We assume formation redshifts distributed uniformly in the range z=10\u201320, and select the template nearest in age at the redshift of the galaxy under consideration. To normalise the SEDs, we draw upon the bivariate 1400\u00a0MHz\u2013K-band luminosity function derived by Mauch & Sadler\u00a0(2007) for a local sample: given a radio power, the absolute K-band magnitude is drawn from a Gaussian distribution of specified mean and variance and this is used to normalise the SED. For simplicity, we assume the same (local) relation between K-band luminosity and radio power at all redshifts. An artifact of this choice of model is that the mid-far infrared regions of the SED template increase sharply for ages below the assumed duration of the starburst (1\u00a0Gyr). This causes discontinuities in the mid- and far-infrared fluxes at $z \\simeq 4$ where the galaxy age first drops below this figure.\n\n### FRIIs\n\nThe FRIIs are first split into populations of quasars and radio galaxies for which the nuclear emission is seen directly and through obscuration, respectively. Willott et al.\u00a0(2000) found that for log L(151\u00a0MHz)\\[W/Hz/sr\\]$>26.5$ the quasar fraction is 0.4, independent of redshift and luminosity, and around 0.1 at lower radio luminosity. Taking into account the contamination by FRIs at lower luminosity, we simply assume that 40 per cent of our FRIIs are seen as quasars, i.e. those seen at viewing angle of $\\leq 53$\u00a0degrees to the jet axis. To assign AGN and starburst infrared SEDs, we recall that quasars are, on average, more intrinsically luminous AGN than the radio galaxies (see e.g. Simpson 2003). To work with a parameter that is closely tied to the intrinsic strength of the nuclear emission, we convert from L(151\u00a0MHz) \\[W/Hz/sr\\] to L(\\[OIII\\]$\\lambda5007$) \\[W\\] using a relation from Grimes, Rawlings & Willott\u00a0(2004) (GRW04):\n\n$$\\rm{log L([OIII]\\lambda5007}) = 7.53 + 1.045\\rm{log L(151~MHz).}$$\n\nEqn.\u00a0(6) applies to the population as a whole, but following GRW04 we assume that the quasars and radio galaxies lie 0.3 dex systemtically above and below it, respectively, with an additional 0.5 dex of scatter on each distribution. To assign an AGN component to the infrared SED, we draw upon the findings of Haas et al.\u00a0(2008) who presented [*Spitzer*]{} 3\u201324\u00a0photometry of 3CR steep-spectrum quasars and radio galaxies at redshift $1<z<2.5$. The quasar mid-infrared SEDs are very uniform, roughly constant in $\\nu F_{\\rm{\\nu}}$. There is a near proportionality between $\\nu F_{\\rm{\\nu}}$(8\u00a0, rest) and $\\nu F_{\\rm{\\nu}}$(178\u00a0MHz, rest) with 0.5 dex scatter. The radio galaxies are dominated by host galaxy starlight below 3\u00a0(rest) and at longer wavelengths by a quasar nucleus reddened by up to A$_{\\rm{V}}=50$\u00a0mag. The radio galaxies are a factor 3\u201310 less luminous in this spectral range than the quasars, which they attribute entirely to extinction. For a sample of 3CR sources at $0.4 \\leq z <1.2$, Cleary et al.\u00a0(2007) find that the quasars are 4 times as luminous as the radio galaxies at 15, half of which they attribute to extinction, and half to synchrotron contamination in the quasars. However, their sample includes a significant number of steep-spectrum compact and super-luminal quasars. They also assumed a spherically symmetric dust model, which is likely to be inadequate.\n\nWe assume that the quasars and radio galaxies follow the same intrinsic relation between $\\nu L_{\\rm{\\nu}}$(8\u00a0\u00a0rest) \\[W\\] and L(\\[OIII\\]$\\lambda5007$) \\[W\\], derived from the quasar points in Fig.\u00a01 of Haas et al.\u00a0(2008) using eqn.\u00a0(6) to convert from L(151\u00a0MHz) to L(\\[OIII\\]$\\lambda5007$). We assume:\n\n$$\\rm{log \\nu L_{\\rm{\\nu}} (8~\\micron~rest)} = 3.3 + \n0.9569\\rm{log L([OIII]\\lambda5007}).$$\n\nNo intrinsic scatter is assumed on this, as the scatter in eqn.\u00a0(6) can reproduce the scatter in Fig.\u00a01 of Haas et al.\u00a0(2008). Using this normalisation, we assign the quasars randomly to one of the three type I quasar templates from Polletta et al.\u00a0(2007). The radio galaxies are also assigned one of these template SEDs in an analogous manner, and then extinction of $A_{\\rm{V}}$=0\u201340\u00a0mag is applied. An elliptical host galaxy model from the GRASIL library is assigned as for the FRIs but in this case normalised using the K\u2013z relation of Willott et al.\u00a0(2003) with an additional offset of $\\Delta K = -0.36(\\rm{log L(151 MHz)[W/Hz/sr]} - 26.55)$\u00a0mag (following Willott et al.\u00a02003, Fig.\u00a03).\n\nIn principle, the evolving elliptical template from the GRASIL library should self-consistently reproduce the far-infrared dust emission. To compare with this, we included in addition a far-infrared dust component to explicitly match the starburst properties to the sub-millimetre observations of radio galaxies and quasars by Archibald et al.\u00a0(2001), Willott et al.\u00a0(2002) and Reuland et al.\u00a0(2004). From Willott et al.\u00a0(2002), the radio quasars appear to be a factor of several brighter than the radio galaxies at 850. In Grimes, Willott & Rawlings\u00a0(2005), this was enshrined as a universal relationship between L(\\[OIII\\]$\\lambda5007$) \\[W\\] and L(850) \\[W/Hz/sr\\]:\n\n$$\\rm{log L(850\\micron)} = -2.5753 + 0.682\\rm{log L([OIII]\\lambda5007}),$$\n\nwith additional scatter drawn from a gaussian of $\\sigma = 0.44$. With this normalisation, we adopt a modified blackbody with $\\beta = 1.95$, $T=41$, as found by Priddey & McMahon\u00a0(2001) for high-redshift quasars. Archibald et al.\u00a0(2001) found L(850\u00a0(rest)) to be a strongly increasing function of redshift, but for our adopted SED parameters, the redshift dependence is a more modest $(1+z)^{1.5}$ (see Reuland et al., Fig.\u00a04); this dependence is added to eqn.\u00a0(8) and normalised at $z=2.5$. There is evidence for a decline in L(850\u00a0(rest)) at $z>4$, so in common with other parts of the simulation we assume a decline of the form $(1+z)^{-7.9}$ at $z>4.8$. We note, however, that Rawlings et al.\u00a0(2004) found any explicit redshift dependence to be small or absent. Although Willott et al.\u00a0(2002) found tentative evidence for an anti-correlation between radio-source size and sub-mm luminosity for quasars, no such trend was found for the radio galaxy sample of Reuland et al.\u00a0(2004) and we have chosen not to incorporate one.\n\nFinally, we remark that our treatment of the infrared SED is confined to reprocessed dust emission due to star formation and AGN heating. As a result, we have not endeavoured to incorporate an infrared synchrotron emission component in the beamed radio-loud AGN. If required, such a component could be added as an extrapolation of the radio SED given in the $S^{3}$ database, following the beaming treatment described by W08. It would only have a significant effect on the predicted source counts at the brightest flux densities.\n\nCOMPARISON WITH EXISTING OBSERVATIONS\n=====================================\n\nThe prescriptions outlined in section 2 constitute our \u2018baseline model\u2019. Here we compare its source counts and other derived quantities against observations. In Fig.\u00a0\\[fig:BASELINEdsc\\] we show the normalised differential source counts at 24, 70 and 160\u00a0for comparison with the most recent [*Spitzer*]{} MIPS survey results. To generate these plots we used a 4\u00a0deg$^{2}$ area of the input radio simulation, coupled with a shallow subset of the full 400\u00a0deg$^{2}$ radio simulation area to generate the bright flux ends (above 1.6\u00a0at 24, 30\u00a0at 70\u00a0and 100\u00a0at 160). The simulated counts fall decisively short of the observations below 1\u00a0and 10\u00a0at 24 and 70, respectively. There is a modest excess of simulated sources above $\\sim 100$\u00a0at 70 and 160, and above $\\sim 10$\u00a0at 850\u00a0(Fig.\u00a0\\[fig:BASELINE850csc\\]) where we compare with results from the [*SCUBA*]{} Half Degree Extragalactic Survey (SHADES; Coppin et al.\u00a02006).\n\nSince the source counts are dominated by star-forming galaxies, we show in Fig.\u00a0\\[fig:BASELINEqparams\\] the quantity $q_{\\rm{IR}} = log (F_{\\rm{IR}}/F_{\\rm{20 cm}})$ in the 24 and 70\u00a0bands for the simulated normal galaxy population. $F_{\\rm{IR}}$ and $F_{\\rm{20 cm}}$ are the simulated infrared and radio fluxes which would be observed in these bands, i.e. without the application of any K-corrections. The simulated sample is dominated by galaxies substantially fainter than those in current observational samples, but our $q_{\\rm{24}}$ and $q_{\\rm{70}}$ values compare reasonably well with those measured locally (Appleton et al.\u00a02004, Beswick et al.\u00a02008 and Garn et al.\u00a02009a,b). For a better comparison with the observations, we applied cuts in flux density to yield the sub-samples also shown in Fig.\u00a0\\[fig:BASELINEqparams\\]: at 24, we selected galaxies with $F_{\\rm{24}} > 0.3$\u00a0and $F_{\\rm{20 cm}} > 90$\u00a0to mimic the selection of Appleton et al.\u00a0(2004), but we note that these authors only included sources with spectroscopic redshifts and appear not to discriminate between star-forming galaxies and AGN, biases which we cannot easily accommodate. At 70, we apply cuts of $F_{\\rm{70}} > 6$\u00a0and $F_{\\rm{20 cm}} > 30$\u00a0to match the selection of Seymour et al.\u00a0(2009), who use photometric and spectroscopic redshifts and apply star-forming galaxy/AGN classification. The resulting sample shows only a very modest decline in $q_{\\rm{70}}$ out to $z=2$, consistent with the findings of Seymour et al.\n\n![image](24micDSC_RIEKEtemps_AUG1_2010FINAL.ps){width=\"47.00000%\"} ![image](70micDSC_RIEKEtemp_AUG1_2010FINAL.ps){width=\"47.00000%\"} ![image](160micDSC_RIEKEtemp_AUG1_2010FINAL.ps){width=\"47.00000%\"}\n\n![The thin continuous line shows the simulated integral source counts at 850\u00a0for the baseline model using a 4\u00a0deg$^{2}$ region of the input radio simulation; the contributions of normal and starburst galaxies are shown by the dashed and dot-dashed lines, respectively. The asterisks and thick lines show the measured counts and uncertainty region for the SCUBA Half-Degree Extragalactic Survey (Coppin et al. 2006).[]{data-label=\"fig:BASELINE850csc\"}](850micronCUMSC_AUG1_2010FINAL.ps){width=\"47.00000%\"}\n\n![The quantity $q_{\\rm{IR}} = log (F_{\\rm{IR}}/F_{\\rm{20 cm}})$ is used to characterise the far-infrared:radio correlation for star-forming galaxies and is plotted here for the entire population of simulated normal galaxies (black dots) at 24 and 70, using the baseline model of section 2. The red crosses and diamonds are the sources remaining after the application of infrared and radio flux cuts to mimic the selection of Appleton et al.\u00a0(2004) and Seymour et al.\u00a0(2009), respectively. []{data-label=\"fig:BASELINEqparams\"}](q24_Rieketemplates_NG_AUG1_plusOBS_tempex.ps \"fig:\"){width=\"47.00000%\"} ![The quantity $q_{\\rm{IR}} = log (F_{\\rm{IR}}/F_{\\rm{20 cm}})$ is used to characterise the far-infrared:radio correlation for star-forming galaxies and is plotted here for the entire population of simulated normal galaxies (black dots) at 24 and 70, using the baseline model of section 2. The red crosses and diamonds are the sources remaining after the application of infrared and radio flux cuts to mimic the selection of Appleton et al.\u00a0(2004) and Seymour et al.\u00a0(2009), respectively. []{data-label=\"fig:BASELINEqparams\"}](q70_Rieketemplates_NG_AUG1_plusOBS.ps \"fig:\"){width=\"47.00000%\"}\n\nREVISIONS TO THE BASELINE MODEL\n===============================\n\nWe now adapt the baseline model to provide a better match to the existing infrared source count data. We focus entirely on the star-forming galaxies and perform the modifications in two steps, motivated by recent observational findings. Firstly, we use post-processing to modify the cosmological luminosity evolution of the star-forming galaxies to remove the shortfall in the 70\u00a0source counts. Secondly, we allow the SED template assignment procedure to evolve towards cooler templates at higher redshift in an attempt to rectify the deficit in the 24\u00a0counts.\n\nModifications to the luminosity evolution of the star-forming galaxies\n----------------------------------------------------------------------\n\nIn W08 we assumed that the radio luminosity function of the star-forming galaxy population undergoes pure luminosity evolution (PLE) of the form $(1+z)^{p}$ with p=3.1 out to z=1.5, applied in an [*Einstein \u2013 de Sitter cosmology*]{}, but adapted to the flat $\\Lambda$ cosmology of the simulation. Since then, new observational constraints on the evolutionary form have been published. At 70, Huynh et al.\u00a0(2007) constrained the evolution out to $z \\sim 1$ and found a degeneracy between luminosity and density evolution with $p=2.78$ in the PLE case; using several [*Spitzer*]{} surveys, Magnelli et al.\u00a0(2009) found redshift evolution consistent with PLE with $p=3.6 \\pm 0.4$ out to $z \\sim 1.3$. Both these studies derived the evolution for a flat $\\Lambda$ cosmology, which for a given functional form of PLE, leads to a higher abundance of galaxies than present in the W08 simulation. The discrepancy is especially pronounced at and below the break in the luminosity function. We can, however, correct for it by boosting the luminosity of the star-forming galaxies in the W08 catalogue by an amount $\\Delta log L(L,z)$. The boost required to bring the W08 luminosity function into agreement with PLE of $p=3.1$ to $z=1.5$ in a $\\Lambda$ universe is shown in Fig.\u00a0\\[fig:lumboost\\]. \\[We note, however, that W08 assumed that the local luminosity function is flat below 20.6 W/Hz, in which regime no luminosity boost can compensate for the deficit in space density; for continuity, the boost is accordingly held constant below this evolving break luminosity\\]. After the application of this luminosity boost, the 24 and 70\u00a0source counts are as shown in Fig.\u00a0\\[fig:LUMBOOSTdsc\\]. The 70 \u00a0counts below 10\u00a0are now in satisfactory agreement with the observational data but a deficit remains at 24, which we address in the next subsection.\n\n![The boost in luminosity, $\\Delta log L(L,z)$ (dex), which must be applied to the star-forming galaxies in the W08 catalogue to transform to pure luminosity evolution of the form $(1+z)^{3.1}$ to $z=1.5$ in a $\\Lambda$-Universe. The lines show redshifts z=0.08 (solid), 0.45 (short dashed), 0.83 (dotted), 1.2 (dot-dashed) and 1.5 (long dashed).[]{data-label=\"fig:lumboost\"}](LFboost_evol4_corrected.ps){width=\"47.00000%\"}\n\n![image](24micDSC_RIEKEtemps_OCT100_2010FINAL.ps){width=\"47.00000%\"} ![image](70micDSC_RIEKEtemp_OCT100_2010FINAL.ps){width=\"47.00000%\"} ![image](160micDSC_RIEKEtemp_OCT100_2010FINAL.ps){width=\"47.00000%\"}\n\nEvolution towards cooler dust templates and the final model\n-----------------------------------------------------------\n\nThere is growing evidence that the one-to-one correspondence between infrared luminosity and template SED established in the local Universe breaks down at earlier cosmic epochs. To cite several examples, Symeonidis et al.\u00a0(2009) showed that luminous and ultraluminous galaxies out to redshift $z \\sim 1$ have much cooler dust distributions (i.e. SEDs peaking at longer wavelengths) than their local counterparts in the IRAS Bright Galaxy Sample (see also Symeonidis et al.\u00a02008); and in the mid-infrared, Farrah et al.\u00a0(2008) showed that ULIRGs at $z \\sim 1.7$ have spectral features characteristic of starbursts with luminosities $10^{10-11}$, rather than local ULIRGs. Such trends may reflect more spatially extended star-forming regions at higher redshifts with lower dust optical depths. Based on a study of sub-mm galaxies at $z \\sim 2.5$, Chapin et al.\u00a0(2009) argued for luminosity evolution in the colour-luminosity relation of the form $(1+z)^{3}$, in the sense that the SED becomes cooler at higher redshift (see also Pope et al.\u00a02006).\n\nBuilding on these findings, we now introduce phenomenological modifications to the SED template assignment procedure with the aim of reducing the remaining deficit in the 24\u00a0source counts (Fig.\u00a0\\[fig:LUMBOOSTdsc\\]). With reference to the Rieke templates in Fig.\u00a0\\[fig:RiekesameFIR\\], we see that for a given L(FIR), evolution towards cooler dust templates with increasing redshift can significantly boost the observed 24\u00a0flux relative to that at 70. Accordingly, the L(FIR) is scaled down by a factor $f(z)$ for the purposes of selecting the far-infrared colour, $C$, from the $C$\u2013L(FIR) distribution (see section 2.1). The chosen template is, however, still normalised to the original value of L(FIR). After some experimentation, we chose a function which evolves as $f(z) = (1 + z)^{2.5}$ out to $z=1$. Beyond $z \\sim 1$, the evolution towards cooler templates must stop or go into reverse in order not to overproduce the 850\u00a0counts. We chose a decline of the form $(1+z)^{-1.5}$ from $z=1-2$, flat thereafter. Even with this in place, the starbursts still overproduce the bright end of the 850\u00a0source counts above 10\u00a0if we retain the \u2018default post-processing option\u2019, i.e. a $(1+z)^{-7.9}$ decline in space density at $z>4.8$. As shown in Fig.\u00a0\\[fig:TEMPEVdsc850\\], one possible way to avoid this is to remove the entire starburst population (i.e. the high-luminosity component) at $z>1.5$ and we adopt this solution for our final model.\n\nThe removal of the high-redshift starburst component of the luminosity function implies that the evolving normal galaxy population in the simulation is sufficient to account for the star formation at these epochs. Although we used the terms \u2018normal galaxy\u2019 and \u2018starburst\u2019 merely as labels to refer to the two Schechter function components of the local luminosity function, this trend may find some physical justification in the work of Obreschkow & Rawlings\u00a0(2009). The latter authors showed that, with increasing redshift, galaxy disks are smaller, leading to higher gas densities and pressures and hence higher molecular gas fractions and star formation rates. In the local Universe, star formation rates in excess of $\\sim 10$\u00a0can only occur in dense gas disks created by zero angular momentum gas in major mergers characteristic of starburst galaxies, e.g. the merger-triggered local ULIRGs.\n\nEven with the starburst cut-off in place, the modest (factor $\\sim 2$) excesses of simulated sources at the bright flux ends of all three [*Spitzer*]{} bands persist. In Fig.\u00a0\\[fig:fluxredshift\\], we show the flux\u2013redshift planes for the normal galaxy and starburst populations for 70 and 160\u00a0fluxes exceeding 100; much of the excess normal galaxy population occurs at redshifts $z<0.5$, particularly at 70. It may arise from a combination of effects, such as a departure from strictly power-law pure luminosity evolution or from an inaccurate treatment of the scatter on the far-infrared relation in eqn.\u00a0(1) (i.e. from galaxies scattered into the high-L(FIR) gaussian wing of the \u201cq\u201d parameter distribution). In support of the first possibility, Huynh et al.\u00a0(2007) found no evidence for significant evolution at infrared luminosities below $10^{11}$\u00a0for redshifts $z<0.4$.\n\nA modest (factor $\\sim 2$) deficit also persists in the 24\u00a0counts around 0.3, which suggests that the required template evolution may be more extreme than assumed, possibly requiring a dependence on luminosity in addition to redshift. It is also possible that the Rieke templates underestimate the PAH strength in the higher redshift galaxies.\n\nFinally, we note that gravitational lensing biases (e.g. Negrello et al.\u00a02007) are not accounted for in these simulations, but can be critical when the source counts are very steep.\n\n![Upper panel: 850\u00a0source counts for the final model of section 4.2 as determined from a 4\u00a0deg$^{2}$\u00a0simulation area; the observed SHADES counts are the same as in Fig.\u00a0\\[fig:BASELINE850csc\\]; the dashed line shows the contribution of the normal galaxies and the dot-dashed line represents the starbursts, assuming a sharp cut-off in the latter population at $z>1.5$; the dotted line shows the contribution of the starbursts under the assumption of the \u2018default post-processing\u2019 cut-off in the space density, i.e. $(1+z)^{-7.9}$ at $z>4.8$. Lower panel: 850\u00a0redshift distributions for flux limits of 2.4 and 5, assuming the $z>1.5$ cut-off in the starburst population.[]{data-label=\"fig:TEMPEVdsc850\"}](850micronCUMSC_OCT107_2x2_test1pt5plus.ps \"fig:\"){width=\"47.00000%\"} ![Upper panel: 850\u00a0source counts for the final model of section 4.2 as determined from a 4\u00a0deg$^{2}$\u00a0simulation area; the observed SHADES counts are the same as in Fig.\u00a0\\[fig:BASELINE850csc\\]; the dashed line shows the contribution of the normal galaxies and the dot-dashed line represents the starbursts, assuming a sharp cut-off in the latter population at $z>1.5$; the dotted line shows the contribution of the starbursts under the assumption of the \u2018default post-processing\u2019 cut-off in the space density, i.e. $(1+z)^{-7.9}$ at $z>4.8$. Lower panel: 850\u00a0redshift distributions for flux limits of 2.4 and 5, assuming the $z>1.5$ cut-off in the starburst population.[]{data-label=\"fig:TEMPEVdsc850\"}](850micron_zhist_OCT107_2pt4mJy_and5mJy_zed1pt5CUTOFF_2x2deg_newbin.ps \"fig:\"){width=\"47.00000%\"}\n\n![image](24micDSC_RIEKEtemps_OCT107_2010FINAL.ps){width=\"47.00000%\"} ![image](70micDSC_RIEKEtemps_OCT107_2010FINAL.ps){width=\"47.00000%\"} ![image](160micDSC_RIEKEtemps_OCT107_2010FINAL.ps){width=\"47.00000%\"}\n\n![Flux-redshift distributions for simulated normal galaxies and starbursts contributing to the excess differential source counts above 100\u00a0in Fig.\u00a0\\[fig:TEMPEVdsc\\] at 70 and 160, demonstrating that much of this excess arises from a local population at $z<0.5$; determined using a shallow subset of the full $20 \\times 20$\u00a0deg$^{2}$ input radio simulation.[]{data-label=\"fig:fluxredshift\"}](flux_redshift20x20.ps){width=\"47.00000%\"}\n\nComparison with observed redshift distributions\n-----------------------------------------------\n\nHaving achieved reasonable agreement with the [*Spitzer*]{} MIPS source counts, particularly at the faint end of the observed flux range, we now compare the redshift distributions. In Fig.\u00a0\\[fig:24micZDIST\\], we show them for 24\u00a0sources above flux limits of 0.08, 0.15 and 0.3\u00a0mJy. The observational data were taken from Le Floc\u2019h et al.\u00a0(2009) and are based on photometric redshift analysis of 30,000 sources in a $\\sim 1.68$\u00a0deg$^{2}$ region of the COSMOS field. The simulated distributions were derived from a 0.25\u00a0deg$^{2}$ region and are in good agreement with the observations. We note, however, that Le Floc\u2019h et al.\u00a0excluded galaxies brighter than $I_{\\rm{AB}}=20$\u00a0mag to minimise cosmic variance fluctuations at low redshift. As they also remarked, the small cosmic volume covered by the observations below $z \\sim 0.4$ prevents any useful comparison in this regime.\n\nAt 70\u00a0we compare with results from the COSMOS and GOODS-N fields (Fig.\u00a0\\[fig:70micZDIST\\]) using a 4\u00a0deg$^{2}$ simulation area. The COSMOS results are based on photometric redshift analysis by Hickey et al.\u00a0(in prep.) for a shallow field covering 1.43\u00a0deg$^{2}$ and a deep sub-region of 0.165\u00a0deg$^{2}$. The deep field is complete to 10, but the shallow field is only 60\u00a0per cent complete at this flux level; in calculating dn/dz for the shallow field we compensated for this by weighting each source by $1/c(f)$, where $c(f)$ is the completeness level at its flux, as read off from Fig.\u00a03 of Frayer et al.\u00a0(2009). The shallow field dn/dz agrees better with the simulation at $z<0.4$, whilst at $z>0.75$ the match to the deeper field is more favourable. Redshift distributions for 70\u00a0COSMOS sources have also been compiled by Kartaltepe et al.\u00a0(2009).\n\nAt a deeper flux level of 2, we compare with the raw redshift distribution derived by Huynh et al.\u00a0(2007) for the 0.05\u00a0deg$^{2}$ GOODS-N field, using a 0.25\u00a0deg$^{2}$ simulation area; there is again a deficit of observed sources but the GOODS-N source catalogue is only $\\sim 50$\u00a0per cent complete at the flux limit (see Frayer et al.\u00a02006a, Table 1), and the raw redshift distribution of Huynh et al.\u00a0(2007; their Fig.\u00a01) appears not to correct for this incompleteness. The effects of cosmic variance for the small field observed may also be important.\n\nAt 850, the redshift distributions for flux limits of a few \u00a0(Fig.\u00a0\\[fig:TEMPEVdsc850\\]) have median redshifts at $z = 2-3$ and high-redshift tails extending beyond $z=4$. Thus, even though the final model has eliminated the starburst (high-L) population at $z>1.5$, it is nevertheless able to reproduce the population of sub-mm (SCUBA) galaxies which reside on average at $z \\simeq 2.3$, as deduced from spectroscopic (Chapman et al.\u00a02005) and radio-mm-FIR photometric measurements (Aretxaga et al.\u00a02007). There is also observational evidence that the distribution extends to redshift $z=4$ and beyond (Coppin et al.\u00a02009). This reinforces our earlier statement that the terms \u2018normal galaxy\u2019 and \u2018starburst\u2019, which we use to label the two Schechter function components of the $z=0$ star-forming galaxy luminosity function, should not be invested with much physical significance beyond the local Universe.\n\n![image](24micron_zhist_OCT106_flim0.ps){width=\"47.00000%\"} ![image](24micron_zhist_OCT106_flim1.ps){width=\"47.00000%\"} ![image](24micron_zhist_OCT106_flim2.ps){width=\"47.00000%\"}\n\n![Comparison of the simulated redshift distribution for the final model of section 4.2 (solid lines) with observations at 70. Upper panel: results from the COSMOS field from Hickey et al.\u00a0(in prep) for S(70)$>10$\u00a0(dashed and dotted histograms are for the deep and shallow fields, respectively \u2013 see section 4.3 for details); lower panel: the observed data (histogram) are the raw redshift distribution of Huynh et al.\u00a0(2007, Fig.\u00a01) for the GOODS-N field above 2, for which the 70\u00a0source catalogue is, however, only about 50 per cent complete.[]{data-label=\"fig:70micZDIST\"}](70micron_zhist_OCT107_COSMOS_both.ps \"fig:\"){width=\"47.00000%\"} ![Comparison of the simulated redshift distribution for the final model of section 4.2 (solid lines) with observations at 70. Upper panel: results from the COSMOS field from Hickey et al.\u00a0(in prep) for S(70)$>10$\u00a0(dashed and dotted histograms are for the deep and shallow fields, respectively \u2013 see section 4.3 for details); lower panel: the observed data (histogram) are the raw redshift distribution of Huynh et al.\u00a0(2007, Fig.\u00a01) for the GOODS-N field above 2, for which the 70\u00a0source catalogue is, however, only about 50 per cent complete.[]{data-label=\"fig:70micZDIST\"}](70micron_zhist_OCT107_Huynh.ps \"fig:\"){width=\"47.00000%\"}\n\nPossible evolution in the far-infrared\u2013radio correlation?\n---------------------------------------------------------\n\n[*Our default assumption is that the luminosity boost introduced in section 4.1 applies equally to the radio and infrared luminosities and that it thus reflects a modification to the cosmological evolution of the star-forming galaxy population as a whole.*]{} Indeed, as we show in Fig.\u00a0\\[fig:RadioSCcheck\\], the 1.4\u00a0GHz source counts for the star-forming galaxies come into better agreement with the observations after the application of the luminosity boost. The relevant observations are taken from Seymour et al.\u00a0(2008) and provide a break-down of the 1.4\u00a0GHz sources counts into contributions from AGN and star-forming galaxies, with the latter dominating below 0.1\u00a0mJy. With reference to the radio source counts in Fig.\u00a04 of W08, the increase in the star-forming galaxy contribution brings the total radio source count at 1.4\u00a0GHz into better agreement with the observations at this flux level. In light of this, the luminosity boost should be applied when using the simulated radio catalogues, as described in Appendix A.\n\n![Differential 1.4\u00a0GHz source counts for the star-forming galaxies in the simulation with (dashed line) and without (continuous line) the application of the luminosity boost introduced in section 4.1. The observational data points are the star-forming galaxy source counts measured by Seymour et al.\u00a0(2008).[]{data-label=\"fig:RadioSCcheck\"}](RadioSCcheck2x2deg.ps){width=\"47.00000%\"}\n\nNevertheless, we cannot exclude the possibility that some contribution to $\\Delta log L(L,z)$ may arise from redshift evolution and non-linearity in the far-infrared:radio correlation. Observational evidence for redshift evolution is at present limited, as discussed in section 3, but strong upward evolution in the relation is expected at $z>3$ due to increased inverse Compton scattering off the cosmic microwave background (e.g. Murphy\u00a02009). In the local Universe, there is evidence that the relation may become non-linear at low luminosities (e.g. Condon\u00a01992; Yun, Reddy & Condon\u00a01992; Best et al.\u00a02005). Best et al.\u00a0(2005) found that below log L(1.4\u00a0GHz) = 22.5\u00a0W\u00a0Hz$^{-1}$ the measured local radio luminosity function falls below that derived in the far-infrared. Such non-linearity could partially account for the luminosity boosting, but it would need to evolve strongly with redshift in order to match that required in Fig.\u00a0\\[fig:lumboost\\].\n\nAt higher redshift, major contributions in this area are expected from [*Herschel*]{} surveys, as suggested by initial results from [*BLAST*]{} (Ivison et al.\u00a02009). To compare with the latter we show in Fig.\u00a0\\[fig:qIvison\\] the simulated quantity $q_{\\rm{IR}}$, defined analogously to eqn.\u00a0(1) but with the far-infrared luminosity replaced by the rest-frame 8\u20131000\u00a0luminosity, L(TIR), (eqn.\u00a04 of Ivison et al.). The simulated values are in good accord with the value of $q_{\\rm{IR}}=2.41 \\pm 0.2$ for the 250-selected sample of Ivison et al.\u00a0(2009), with no evidence of redshift dependence. Ivison et al.\u00a0also determined $q_{\\rm{IR}}$ for a sample derived by stacking at the positions of 24\u00a0sources and found tentative evidence for a decline of the form $(1+z)^{-0.15 \\pm 0.03}$, but we do not attempt to replicate the selection effects associated with the definition of this particular sample. The lack of redshift dependence in the simulated $q_{\\rm{IR}}$ primarily reflects the redshift-invariance we assumed for the far-infrared\u2013radio relation in eqn.\u00a0(1), in conjunction with the only mildly non-linear dependence of L(TIR) on L(FIR) in Fig.\u00a0\\[fig:LTIRtoFIR\\_Rieke\\].\n\nFor completeness, we also show in Fig.\u00a0\\[fig:qIvison\\] the quantities $q_{\\rm{24}}$ and $q_{\\rm{70}}$ as functions of redshift, for comparison with those of the baseline model shown in Fig.\u00a0\\[fig:BASELINEqparams\\]. Reflecting the imposed template evolution, $q_{\\rm{24}}$ increases more strongly with redshift out to $z=1$ than in the baseline model, whilst $q_{\\rm{70}}$ remains effectively constant out to $z=1$.\n\n![Upper panel: the bolometric (8\u20131000) $q_{\\rm{IR}}$ for the simulated star-forming galaxies. The dashed lines represent the $\\pm 3 \\sigma$ range measured by Ivison et al.\u00a0(2009) for a 250\u00a0sample. The continuous line shows the tentative evolution measured by Ivison et al.\u00a0from stacking measurements. Middle and lower panels: $q_{\\rm{24}}$ and $q_{\\rm{70}}$ for the final model, with symbols defined as in Fig.\u00a0\\[fig:BASELINEqparams\\]. []{data-label=\"fig:qIvison\"}](qIR_Ivison.ps \"fig:\"){width=\"47.00000%\"} ![Upper panel: the bolometric (8\u20131000) $q_{\\rm{IR}}$ for the simulated star-forming galaxies. The dashed lines represent the $\\pm 3 \\sigma$ range measured by Ivison et al.\u00a0(2009) for a 250\u00a0sample. The continuous line shows the tentative evolution measured by Ivison et al.\u00a0from stacking measurements. Middle and lower panels: $q_{\\rm{24}}$ and $q_{\\rm{70}}$ for the final model, with symbols defined as in Fig.\u00a0\\[fig:BASELINEqparams\\]. []{data-label=\"fig:qIvison\"}](q24_Rieketemplates_NG_OCT107_plusOBS_tempex.ps \"fig:\"){width=\"47.00000%\"} ![Upper panel: the bolometric (8\u20131000) $q_{\\rm{IR}}$ for the simulated star-forming galaxies. The dashed lines represent the $\\pm 3 \\sigma$ range measured by Ivison et al.\u00a0(2009) for a 250\u00a0sample. The continuous line shows the tentative evolution measured by Ivison et al.\u00a0from stacking measurements. Middle and lower panels: $q_{\\rm{24}}$ and $q_{\\rm{70}}$ for the final model, with symbols defined as in Fig.\u00a0\\[fig:BASELINEqparams\\]. []{data-label=\"fig:qIvison\"}](q70_Rieketemplates_NG_OCT107_plusOBS_tempex.ps \"fig:\"){width=\"47.00000%\"}\n\nPREDICTIONS FOR HERSCHEL SURVEYS\n================================\n\n![Upper panel: predicted [*Herschel*]{} integral source counts at 100\u00a0(black solid line), 250\u00a0(red dotted line), 350\u00a0(green dashed) and 500\u00a0(blue dot-dashed line); lower panel: in differential format for comparison with the [*BLAST*]{} counts from Table S2 of Devlin et al.\u00a0(2009), which are shown by the points with error bars (asterisks, 250; squares, 350; triangles, 500).[]{data-label=\"fig:Herschel_SC\"}](Herschel_CUMSC_OCT107_2x2_colour_zed1pt5SBcutoff.ps \"fig:\"){width=\"47.00000%\"} ![Upper panel: predicted [*Herschel*]{} integral source counts at 100\u00a0(black solid line), 250\u00a0(red dotted line), 350\u00a0(green dashed) and 500\u00a0(blue dot-dashed line); lower panel: in differential format for comparison with the [*BLAST*]{} counts from Table S2 of Devlin et al.\u00a0(2009), which are shown by the points with error bars (asterisks, 250; squares, 350; triangles, 500).[]{data-label=\"fig:Herschel_SC\"}](HerschelBLAST_DSC_OCT107_BLAST_DEVLIN_2x2_colour_zed1pt5SBcutoff.ps \"fig:\"){width=\"47.00000%\"}\n\nWe now present some predictions for forthcoming [*Herschel*]{} surveys and a comparison with results from the [*BLAST*]{} mission which flew a prototype of the SPIRE instrument on a telescope half [*Herschel*]{}\u2019s size. In Fig.\u00a0\\[fig:Herschel\\_SC\\], we show predicted integral source counts for surveys at 100, 250, 350 and 500\u00a0which correspond closely with those predicted by the backward-evolution model of Valiante et al.\u00a0(2009). Also shown in this figure are normalised differential source counts at 250, 350 and 500, which compare reasonably well with the [*BLAST*]{} counts as determined from the \u2018P(D)\u2019 analysis of Devlin et al.\u00a0(2009) (see also Patanchon et al.\u00a02009). At 250 and 350, the model marginally exceeds the [*BLAST*]{} measurements in the highest flux bins, as also hinted at in the 160\u00a0comparison above $\\sim 100$. In Fig.\u00a0\\[fig:BLAST250zdist\\] we compare with the \u2018complete\u2019 redshift distribution derived by Dunlop et al.\u00a0(2009) for the 150\u00a0arcmin$^{2}$\u00a0GOODS-S field at 250; the observed sample consists of 20 sources with redshifts down to a nominal flux limit of 35, although small number statistics and cosmic variance limit the scope of the comparison.\n\n![A comparison of the redshift distribution of [*BLAST*]{} 250\u00a0sources brighter than 35\u00a0in GOODS-S (Dunlop et al.\u00a02009) (histogram) with the simulated distributions for flux limits of 25, 30, 35, 40 and 50\u00a0(solid lines, top to bottom), determined from a 0.25\u00a0deg$^{2}$ simulation area.[]{data-label=\"fig:BLAST250zdist\"}](250micron_zhist_OCT107_BLAST_N_zed1pt5SBcutoff.ps){width=\"47.00000%\"}\n\nAlthough the $S^{3}$ online interface allows the user to query the database to generate redshift distributions for arbitrary radio and infrared flux cuts and survey areas, we follow Lacey et al.\u00a0(2009) and show in Fig.\u00a016 dn/dz plots for a selection of [*Herschel*]{} Key Programme[^4] surveys: (i) the ultra-deep 0.012\u00a0deg$^{2}$\u00a0\u2018pencil beam\u2019 GOODS-[*Herschel*]{} field[^5]; (ii) Levels 1 and 5 of the [*Herschel*]{} Multi-tiered Extragalactic Survey (HERMES [^6]); (iii) the wide ($\\sim 600$\u00a0deg$^{2}$) but shallow ATLAS survey (Eales et al.\u00a02009). Table\u00a01 shows their area coverages, multi-band sensitivities, predicted numbers of galaxies and the fractions at $z>1$ and $z>2$. Our dn/dz predictions were derived from a $0.25$\u00a0deg$^{2}$ simulation area and scaled to the required survey areas, except for the ATLAS and HERMES Level 5 predictions for which we used a 4\u00a0deg$^{2}$ area . The predicted redshift distributions correspond quite well with those of Lacey et al.\u00a0(2009) (L09), despite the differences in our simulation methodologies. For GOODS-[*Herschel*]{}, our predicted redshift distributions exhibit broader peaks than those of L09, extending beyond $z=1$; similarly, for HERMES Level 1 and 5 at 250\u00a0our distribution peaks just above $z=1$ and that of L09 just below. At 100 and 160\u00a0in HERMES Level 5, we predict broad redshift distribution with a tail extending beyond $z=1$ whereas L09 predict distributions strongly peaked at $z=0.2$. For the ATLAS survey, the distributions of L09 exhibit marginally higher peaks than ours at $z \\leq 0.2$; the L09 distributions cut-off sharply above $z \\sim 1.5$, as do ours, due to the step function cut-off in the starburst population above $z=1.5$.\n\n  ---------------------- ------------- ------ ------- ----------------- ------------------- --------------------------\n  Survey                 Area          Band   Depth   Total$^{\\star}$   f(z$>$1)$\\dagger$   f(z$>$2)$\\dagger\\dagger$\n                         (deg$^{2}$)   ()     ()                                            \n  GOODS-[*Herschel*]{}   0.012         100    0.6     320               0.61                0.14\n  Ultra-deep                           160    0.9     460               0.69                0.25\n  Hermes Level 1         0.11          250    4.2     1850              0.69                0.24\n                                       350    5.7     1080              0.75                0.29\n                                       500    4.9     680               0.82                0.37\n  Hermes Level 5         27            100    27      11440             0.24                0.0018\n                                       160    39      13420             0.41                0.033\n                                       250    14      85700             0.63                0.18\n                                       350    19      27000             0.69                0.24\n                                       500    16      10470             0.79                0.34\n  ATLAS                  600           100    67      59700             0.15                1.6E-04\n                                       160    94      57170             0.30                8.3E-03\n                                       250    46      165100            0.51                0.10\n                                       350    62      28530             0.51                0.12\n                                       500    53      5720              0.63                0.19\n  ---------------------- ------------- ------ ------- ----------------- ------------------- --------------------------\n\n\\\n$\\star$ Total number of simulated galaxies in survey area at $z=1-4$\\\n$\\dagger$ Fraction at $z>1$\\\n$\\dagger\\dagger$ Fraction at $z>2$\\\n\n![Predicted redshift distributions for a selection of [*Herschel*]{} Key Programme surveys (listed in Table 1) at 100\u00a0(black solid lines), 160\u00a0(purple dot-dot-dashed lines), 250\u00a0(red dotted lines), 350\u00a0(green dashed lines) and 500\u00a0(blue dot-dashed lines)](ZDIST_GOODSULTRADEEP_colour_zed1pt5SBcutoff.ps \"fig:\"){width=\"7.8cm\"} ![Predicted redshift distributions for a selection of [*Herschel*]{} Key Programme surveys (listed in Table 1) at 100\u00a0(black solid lines), 160\u00a0(purple dot-dot-dashed lines), 250\u00a0(red dotted lines), 350\u00a0(green dashed lines) and 500\u00a0(blue dot-dashed lines)](ZDIST_HERMES_LEVEL1_colour_zed1pt5SBcutoff.ps \"fig:\"){width=\"7.8cm\"} ![Predicted redshift distributions for a selection of [*Herschel*]{} Key Programme surveys (listed in Table 1) at 100\u00a0(black solid lines), 160\u00a0(purple dot-dot-dashed lines), 250\u00a0(red dotted lines), 350\u00a0(green dashed lines) and 500\u00a0(blue dot-dashed lines)](ZDIST_HERMES_LEVEL5_colour_2x2_10nJycat_zed1pt5SBcutoff.ps \"fig:\"){width=\"7.8cm\"} ![Predicted redshift distributions for a selection of [*Herschel*]{} Key Programme surveys (listed in Table 1) at 100\u00a0(black solid lines), 160\u00a0(purple dot-dot-dashed lines), 250\u00a0(red dotted lines), 350\u00a0(green dashed lines) and 500\u00a0(blue dot-dashed lines)](ZDIST_ATLAS_2x2_colour_10nJycatNG_zed1pt5SBcutoff.ps \"fig:\"){width=\"7.8cm\"}\n\n\\[fig:Herschelzdist\\]\n\nCONCLUSIONS\n===========\n\nWe have post-processed the S$^{3}$-SEX semi-empirical simulation of the extragalactic radio continuum sky (W08) to make predictions for [*Herschel*]{} surveys at far-infrared wavelengths. Existing observations in the mid-infrared with [*Spitzer*]{} at 24, 70 and 160\u00a0and at 850\u00a0with [*SCUBA*]{}, together impose strong constraints on the assignment of infrared SEDs to the star-forming galaxies as a function of redshift. Our principal findings, incorporated into the final model, are as follows:\n\n\\(i) In order to match the 70\u00a0counts, the star-forming galaxies are required to undergo stronger luminosity evolution than assumed by W08 for the radio simulation. It may be that W08 simply used an inaccurate evolutionary prescription based on the information available at that time; alternatively, it could be that there is genuine differential evolution between the far-infrared and radio populations, as a result of an evolving non-linearity in the far-infrared:radio correlation which is already apparent locally at low luminosity . Results from [*Herschel*]{} and [*SKA*]{} precursors will resolve this issue.\n\n\\(ii) From the local Universe to redshift $z=1$, star-forming galaxies are required to develop progressively cooler SED templates at fixed L(FIR) once the latter has been set by the far-infrared\u2013radio correlation; the rest-frame 60\u2013100\u00a0colour is assumed to evolve as $\\rm{log} L_{\\rm{60}}/L_{\\rm{100}} \\sim (1+z)^{-2.5}$; beyond $z=1$, this evolution must go into reverse in order not to overproduce the 850\u00a0source counts.\n\nOur chosen model compares favourably with recent far-infrared survey results from [*BLAST*]{}. This inspires confidence in using it to make predictions for [*Herschel*]{} Key Programme surveys. Our predicted source counts and redshift distributions correspond closely with those of Valiante et al.\u00a0(2009) and Lacey et al.\u00a0(2009), respectively, despite the clear differences in the methodologies of our models. This suggests that the combination of [*Spitzer*]{} mid-infrared and 850\u00a0data impose strong constraints on the allowable model parameter space. Data products from our simulation are available on the S$^{3}$ website[^7] and we expect them to serve as a valuable resource for the interpretation of [*Herschel*]{} surveys.\n\nACKNOWLEDGMENTS {#acknowledgments .unnumbered}\n===============\n\nRJW is supported by the Square Kilometre Array Design Study. 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Dwelly T., Moss D., McHardy I., Zoghbi A., Rieke G., Page M., Hopkins A., Loaring N., 2008, MNRAS, 386, 1695 Seymour N., Huynh M., Dwelly T., Symeonidis M., Hopkins A., McHardy I.M., Page M., Rieke G., 2009, MNRAS, 398, 1573 Silva L., Granato G.L., Bressan A., Danese L., 1998, ApJ, 509, 103 Silverman J.D., et al., 2009, ApJ, 696, 396 Simpson C.J., 2003, NewAR, 47, 211 Soifer B.T., Sanders D.B., Madore B.F., Neugebauer G., Danielson G.E., Elias J.H., Lonsdale C.J., Rice W.L., 1987, ApJ, 320, 238 Soifer B.T., Helou G., Werner M., 2008, ARA&A, 2008, 46, 201 Symeonidis M., Willner S.P., Rigopoulou D., Huang J.-S., Fazio G.G., Jarvis M.J., 2008, MNRAS, 385, 1015 Symeonidis M., Page M., Seymour N., Dwelly T., Coppin K., McHardy I., Rieke G.H., Huynh M., 2009, arXiv:0905.0854 Thomspson G.D., Levenson N.A., Uddin S.A., Sirocky M.M., 2009, ApJ, 697, 182 Valiante E., Lutz D., Sturm E., Genzel R., Chapin E.L., 2009, ApJ, 701, 1814 Vardoulaki E., Rawlings S., Simpson C., Bonfield D., Ivison R.J., Eduardo I., 2008, MNRAS, 387, 505 Wilman R.J., et al., 2008, MNRAS, 388, 1335 (W08) Willott C.J., Rawlings S., Blundell K.M., Lacy M., 2000, MNRAS, 316, 449 Willott C.J., Rawlings S., Blundell K.M., Lacy M., Eales S.A., 2001, MNRAS, 322, 536 Willott C.J., Rawlings S., Jarvis M.J., Blundell K.M., 2003, MNRAS, 339, 173 Ueda Y., Akiyama M., Ohta K., Miyaji T., 2003, ApJ, 598, 886 Yun M.S., Reddy N.A., Condon J.J., 2001, ApJ, 554, 803\n\nDescription of database format\n==============================\n\nThe simulated infrared fluxes are available from our online interactive database at http://s-cubed.physics.ox.ac.uk/s3sex. They are provided as supplementary columns to the [*Galaxies Table*]{}, the first 16 columns of which contain the existing output from the W08 radio simulation. Each row of this table refers to an individual galaxy; multi-component radio sources (i.e. FRI and FRII sources) are denoted by their core position and integrated radio fluxes. A description of the supplementary columns is given in Table A1.\n\nIn addition to the mid- and far-infrared flux densities (24\u20131200), we also provide 2.2\u00a0K-band magnitudes. For the radio-loud AGN, the K-band magnitude is used to normalize the SED; for the radio-quiet AGN and star-forming galaxies, the K-band magnitude should be considered as schematic as no attempt was made to model accurately the stellar population and dust extinction at rest-frame optical wavelengths, even though the SED templates extend into this regime.\n\nThe [*RQ-AGN classification flag*]{} applies only to the RQ-AGN and specifies whether the source is classified as unobscured, Compton-thin or Compton-thick obscured, or whether it is part of the \u2018excess\u2019 population which is filtered out (as described in section 2.2).\n\nThe [*Space-density filter flag*]{} indicates which sources have been filtered out due to the imposed cut-off in the space density at high-redshift. The [*infrared*]{} fluxes of all filtered-out sources are set to -999.\n\n$\\Delta$log(L,z) is the luminosity boost which has been applied to the star-forming galaxies to generate the infrared fluxes, over and above the level set by the far-infrared:radio correlation. Note that the tabulated [*radio*]{} fluxes for the star-forming galaxies are the original values from W08, even though the boost may also be applicable to them, as discussed in sections 4.1 and 4.4.\n\n[|ll|]{}Column & Attribute description\\\n\\\n17 & log$_{\\rm{10}}$(24\u00a0flux density) \\[Jy\\]\\\n18 & log$_{\\rm{10}}$(70\u00a0flux density) \\[Jy\\]\\\n19 & log$_{\\rm{10}}$(100\u00a0flux density) \\[Jy\\]\\\n20 & log$_{\\rm{10}}$(160\u00a0flux density) \\[Jy\\]\\\n21 & log$_{\\rm{10}}$(250\u00a0flux density) \\[Jy\\]\\\n22 & log$_{\\rm{10}}$(350\u00a0flux density) \\[Jy\\]\\\n23 & log$_{\\rm{10}}$(450\u00a0flux density) \\[Jy\\]\\\n24 & log$_{\\rm{10}}$(500\u00a0flux density) \\[Jy\\]\\\n25 & log$_{\\rm{10}}$(850\u00a0flux density) \\[Jy\\]\\\n26 & log$_{\\rm{10}}$(1200\u00a0flux density) \\[Jy\\]\\\n27 & 2.2\u00a0K-band magnitude\\\n28 & RQ-AGN classification flag $^{\\star}$:\\\n& 1=TYPE 1 (unobscured),\\\n& 2=TYPE 2 (Compton-thin obscured),\\\n& 3=TYPE 2 (Compton thick obscured) ,\\\n& -1=Source excluded\\\n29 & Space-density filter flag:\\\n& (-1= filtered out; 0=retained)\\\n30 & $\\Delta$log(L,z) $\\dagger$\\\n\n\\\n$\\star$ Applies to radio-quiet AGN only.\\\n$\\dagger$ Applies to star-forming galaxies only.\\\n\n[^1]: http://s-cubed.physics.ox.ac.uk\n\n[^2]: http://ssc.spitzer.caltech.edu/postbcd/cookbooks/syntheticphotometry.html\n\n[^3]: These SEDs were obtained from http://www.iasf-milano.inaf.it/$\\sim$polletta/templates/swiretemplates.html\n\n[^4]: http://herschel.esac.esa.int/KeyProgrammes.shtml\n\n[^5]: http://herschel.esac.esa.int/Docs/KPOT\n\n[^6]: http://astronomy.sussex.ac.uk/$\\sim$sjo/Hermes\n\n[^7]: http://s-cubed.physics.ox.ac.uk\n"], ["---\nabstract: 'When exposed to the high energy X-ray and ultraviolet radiation of a very active star, water vapor in the upper atmospheres of planets can be photodissociated and rapidly lost to space. In this paper, I study the chemical, thermal, and hydrodynamic processes in the upper atmospheres of terrestrial planets, concentrating on water vapor dominated atmospheres orbiting in the habitable zones of active stars. I consider different stellar activity levels and find very high levels of atmospheric escape in all cases, with the outflowing gas being dominated by atomic hydrogen and oxygen in both their neutral and ion forms. In the lower activity cases, I find that the accumulation of O$_2$ and increases in the D/H ratios in the atmospheres due to mass fractionation are possible, but in the higher activity cases no mass fractionation takes place. Connecting these results to stellar activity evolution tracks for solar mass stars, I show that huge amounts of water vapor can be lost, and both the losses and the amount of O$_2$ that can be accumulated in the atmosphere depend sensitively on the star\u2019s initial rotation rate. For an Earth-mass planet in the habitable zone of a low-mass M-dwarf, my results suggest that the accumulation of atmospheric O$_2$ is unlikely unless water loss can take place after the star\u2019s most active phase.'\nauthor:\n- 'C. P. Johnstone'\nbibliography:\n- 'mybib.bib'\ntitle: Hydrodynamic escape of water vapor atmospheres near very active stars\n---\n\nIntroduction\n============\n\nWater is one of the most important molecules for planets and can influence significantly their surfaces, climates, and potential for sustaining life. In addition to being present in the atmospheres of solar system planets, water has been detected in the atmospheres of many exoplanets (e.g. @Tinetti07; @Wakeford13; @Brogi14; @Tsiaras19). A large number of physical processes take place in the upper atmospheres and exospheres of planets that cause water to be lost to space, most of which are driven by interactions with the central star. For example, non-thermal escape processes include ion pick-up by stellar winds (@Kislyakova14a), cold ion outflows (@Glocer07), and the loss of photochemically produced high-energy particles (@Amerstorfer17). Thermal escape mechanisms include hydrodynamic escape, which takes place when the upper atmosphere is heated to such a temperature that the thermal pressure causes it to accelerate away from the planet at speeds exceeding the escape velocity. This can happen without any outside influences for very low-mass planets (@Stoekl15), in response to the star\u2019s bolometric radiation for planets on short period orbits (@OwenWu16), or most notably in response to the star\u2019s X-ray and ultraviolet emission for planets orbiting active stars (@Tian05; @Erkaev16; @Kubyshkina18). Heating can also take place due to other mechanisms, mostly in response to the star\u2019s wind (@Cohen14; @Lichtenegger16). For example, @Chassefiere96 and @Chassefiere97 suggested for the case of early Venus that energetic neutral atoms (ENAs) created in charge exchange reactions between neutral atmospheric particles and the early solar wind could heat the atmosphere significantly and drive escape, though @Lichtenegger16 found that this mechanism might not lead to significant additional escape.\n\nWater vapor in a planet\u2019s upper atmosphere is photodissociated by the high-energy radiative spectrum of its host star creates a large number of other chemical species such as OH, O$_2$, and O$_3$, and causes most of the thermosphere to be filled neutral and ionized hydrogen and oxygen atoms which can flow away from the planet hydrodynamically (@Kasting83; @Guo19). Due to its lower mass, hydrogen is lost more rapidly than oxygen, though how different these loss rates are depends on several factors, including the total loss rate and which loss process dominates. This preferential loss of H has been suggested as a mechanism for producing significant amounts of O$_2$ in an atmosphere (@WordsworthPierrehumbert14; @LugerBarnes15). In this scenario, a planet with a large amount of atmospheric H$_2$O orbiting a very active star will undergo rapid escape but with much of the O remaining, leading to O$_2$ build-up. For potentially habitable planets, it is expected that this build-up might be more significant for planets orbiting M dwarfs since these stars spend a very long time on the pre-main-sequence, meaning that planets that will eventually be in the habitable zones when the star reaches the main-sequence spend a long time inside the inner edge of the habitable zone where they are too hot for liquid surface water (@RamirezKaltenegger14).\n\nThe build-up of O$_2$ can be prevented if oxygen is absorbed into the surface, which @Wordsworth18 showed could significantly reduce the build-up of O$_2$ during a planet\u2019s early magma ocean phase. In this case the absorbed oxygen can still be released into the atmosphere later in the planet\u2019s lifetime. Alternatively, as the O$_2$ accumulates, the mixing ratio of O in the upper atmosphere will increase, leading to a decrease in the difference between the H and O loss rates (@Tian15). The loss of water can also be inhibited by the presence of a cold-trap if non-condensing species such as N$_2$ or CO$_2$ make up a large fraction of the atmosphere (@WordsworthPierrehumbert13). An initially pure H$_2$O atmosphere cannot have a cold-trap, but a cold-trap can form as O$_2$ builds up, inhibiting H$_2$O escape (@WordsworthPierrehumbert14). If the star\u2019s activity is strong enough to drive rapid atmospheric escape however, we might expect rapid oxygen escape to take place in such a case, possibly removing any accumulated atmospheric O$_2$. Studying the case of GJ\u00a01132b, which is an Earth-mass planet orbiting a low mass M dwarf, @Schaefer16 found that most of the O$_2$ produced by the dissociation and loss of water vapor is lost to space, whereas a much smaller fraction is absorbed into the planet.\n\nMost atmospheric loss processes take place in response to the magnetic activity of the host star, which is responsible for winds and high-energy X-ray and ultraviolet radiation. A star\u2019s X-ray and ultraviolet spectrum (referred to here as \u2018XUV\u2019 and defined as 1 to 400\u00a0nm) is emitted by the photosphere and the magnetically heated chromosphere and corona (@Fontenla16). For a solar mass star, the chromosphere and corona is responsible for the emission at wavelengths shorter than $\\sim$200\u00a0nm and this radiation is absorbed high in the atmospheres of planets, driving heating and photochemistry. This emission depends sensitively on the rotation rate of the star (@Reiners14). Since rotation rates decline with age due to angular momentum removal by stellar winds, there is also a corresponding decline in XUV emission (@Guedel97). This means that when the solar system was young the upper atmospheres of solar system planets were likely significantly hotter and more expanded than currently (@Kulikov07; @Tian08; @Johnstone18). However, the situation is complicated by the fact that stars can be born with different rotation rates, and stars born as slow rotators will evolve very differently to stars born as fast rotators (@Johnstone15; @Tu15). A solar mass star born as a rapid rotator will remain highly active for much longer than a star born as a slow rotator, which has important consequences for the subsequent evolution of planetary atmospheric escape (@Johnstone15letter).\n\nIn this paper, I study the reaction of a water vapor atmosphere to the high XUV spectrum of a very active star. The aim is to use a newly developed state-of-the-art physical upper atmosphere model to study hydrodynamically outflowing water vapor atmospheres and to understand the loss rates of hydrogen, deuterium, and oxygen. In Section\u00a0\\[sect:model\\], I describe the physical upper atmosphere model used here, in Section\u00a0\\[sect:results\\], I present the modelling results, in Section\u00a0\\[sect:evo\\], I use these results to study the long term evolution of water losses, and in Section\u00a0\\[sect:conclusions\\], I discuss the significance of the results.\n\n![ Evolution of stellar X-ray flux at 1\u00a0AU for solar mass stars with different initial rotation rates as calculated by @Tu15. The red, green, and blue tracks correspond to the cases of slow, median, and fast rotators and the dashed black line shows the case of a star that remains at the saturation threshold for its entire lifetime. The grey shades area shows the range of activity levels that I study in this paper. []{data-label=\"fig:XUVtracks\"}](XrayTracks.pdf){width=\"45.00000%\"}\n\nModel {#sect:model}\n=====\n\n![ Hydrodynamic simulation of a rapidly outflowing water vapor atmosphere. The panels show as functions of altitude the outflow speed (*upper-panel*) and the densities of selected neutral (*middle-panel*) and ion (*lower-panel*) species. In the upper panel, the sound and escape speeds are shown and all three lines cross at the same location, as required for a pressure driven Parker wind. []{data-label=\"fig:Case5\"}](VelCase5.pdf \"fig:\"){width=\"45.00000%\"} ![ Hydrodynamic simulation of a rapidly outflowing water vapor atmosphere. The panels show as functions of altitude the outflow speed (*upper-panel*) and the densities of selected neutral (*middle-panel*) and ion (*lower-panel*) species. In the upper panel, the sound and escape speeds are shown and all three lines cross at the same location, as required for a pressure driven Parker wind. []{data-label=\"fig:Case5\"}](NeutralDensityCase5.pdf \"fig:\"){width=\"45.00000%\"} ![ Hydrodynamic simulation of a rapidly outflowing water vapor atmosphere. The panels show as functions of altitude the outflow speed (*upper-panel*) and the densities of selected neutral (*middle-panel*) and ion (*lower-panel*) species. In the upper panel, the sound and escape speeds are shown and all three lines cross at the same location, as required for a pressure driven Parker wind. []{data-label=\"fig:Case5\"}](IonDensityCase5.pdf \"fig:\"){width=\"45.00000%\"}\n\n[ccccccccccccc]{} 1 & 18.4 & 98.7 & 231.3 & 256.2 & 310-840 & 3.71 & 6.21 & 2.25 & 0.075 & 0.272 & 0.861 & 0.402\\\n2 & 22.9 & 124.5 & 252.5 & 302.6 & 180-790 & 4.78 & 6.87 & 2.00 & 0.091 & 0.367 & 0.853 & 0.494\\\n3 & 30.0 & 166.0 & 280.9 & 373.4 & 93-730 & 6.38 & 7.73 & 1.72 & 0.12 & 0.529 & 0.868 & 0.613\\\n4 & 42.3 & 241.2 & 321.5 & 494.2 & 26-640 & 8.98 & 8.96 & 1.43 & 0.16 & 0.829 & 0.903 & 0.765\\\n5 & 68.5 & 409.8 & 391.6 & 746.9 & 22-550 & 13.3 & 11.0 & 1.11 & 0.24 & 1.46 & 0.953 & 0.925\\\n6 & 77.5 & 470.2 & 412.8 & 833.9 & 22-530 & 14.1 & 11.5 & 1.04 & 0.27 & 1.66 & 0.962 & 0.951\\\n7 & 89.0 & 548.2 & 438.0 & 944.2 & 20-510 & 15.3 & 12.1 & 0.964 & 0.29 & 1.92 & 0.971 & 0.972\\\n8 & 104.0 & 651.8 & 468.5 & 1088.0 & $<$490 & 16.3 & 12.8 & 0.875 & 0.32 & 2.45 & 0.980 & 0.990\\\n9 & 124.2 & 794.7 & 506.6 & 1282.6 & $<$460 & 17.4 & 13.5 & 0.794 & 0.37 & 2.67 & 0.986 & 1.000\\\n10 & 152.8 & 1001.7 & 555.5 & 1558.6 & $<$440 & 18.3 & 14.3 & 0.729 & 0.43 & 3.25 & 0.993 & 1.000\\\n11 & 195.7 & 1321.9 & 621.4 & 1976.0 & $<$400 & 19.3 & 15.3 & 0.650 & 0.48 & 3.76 & 0.994 & 1.000\\\n12 & 266.3 & 1866.0 & 715.8 & 2667.5 & $<$360 & 19.5 & 16.3 & 0.560 & 0.59 & 4.95 & 0.998 & 1.000\\\n13 & 399.4 & 2933.9 & 865.2 & 3987.3 & $<$320 & 19.0 & 17.5 & 0.524 & 0.75 & 7.03 & 1.000 & 1.000\\\n14 & 719.0 & 5632.0 & 1146.9 & 7217.5 & $<$250 & 15.1 & 19.1 & 0.482 & 0.93 & 11.3 & 1.000 & 1.000 \\[table:grid\\]\n\nThe system that I study in this paper consists of an Earth-mass planet with a pure water vapor atmosphere orbiting at 1\u00a0AU around an active solar mass star. Using my atmosphere model, I calculate the 1D atmosphere structure between altitudes of 50 and 100,000\u00a0km assuming a zenith angle of 0$^{\\circ}$. The zero zenith angle assumption is motivated by the assumptions made in Section\u00a0\\[sect:evo\\] to calculate total atmospheric loss rates from the model results. The lower boundary altitude is largely arbitrary and our results are not sensitive to this choice since it is anyway negligible compared to the radius of the planet. At the lower boundary, I assume a gas composed of H$_2$O and HDO, with a temperature of 250\u00a0K, corresponding approximately to the planet\u2019s effective temperature, and a number density of . The mixing ratio of HDO at the lower boundary is , corresponding approximately to the value in the Earth\u2019s oceans (@Eberhardt95) and is typical for values in chondritic meteorites (@Marty16), though since the abundance of deuterium is too small to influence the model, any sufficiently small base HDO mixing ratio could have been chosen. The base number density is mostly arbitrary and experiments with other values have shown that my results are not sensitive to this value as long as it is large enough that all of the absorption of the XUV spectrum takes place within the computational domain. The upper boundary altitude is also arbitrary and only chosen to be far enough from the planet that the hydrodynamic outflow is supersonic within the simulation domain; at this boundary, standard zero-gradient outflow conditions are used.\n\nFor the stellar XUV spectra in the wavelength range 1 to 400\u00a0nm, I use the method and codes developed for solar mass stars by @Claire12. It is important to use realistic stellar spectra since the shape of a star\u2019s XUV spectrum depends on its activity level: more active stars have hotter coronae (@JohnstoneGuedel15), meaning that larger fractions of their X-ray and extreme ultraviolet emission are at the higher energy parts of the spectrum (@Guedel04; @SanzForcada11). In Fig.\u00a0\\[fig:XUVtracks\\], I show evolutionary tracks for the X-ray emission of solar mass stars with different initial rotation rates, where the shades region shows the range of activity levels that I study in this paper. In this paper, I use the X-ray flux, $F_\\mathrm{X}$, at the planet\u2019s orbit as the measure of stellar activity because the X-ray luminosity of a star is easily measured and often available for exoplanet hosts.\n\nTo calculate the atmospheric structures and loss rates, I use The Kompot Code, recently developed by @Johnstone18 and @Johnstone19a. This is a sophisticated general-purpose first-principles physical model for the upper atmospheres of planets, developed to take into account the range of physical processes taking place in the upper atmosphere. The model has been designed such that it can be applied to any type of planet with arbitrary atmospheric compositions. In @Johnstone18, detailed descriptions of the physical model (Section\u00a02) and numerical methods (Appendices) can be found. As in @Johnstone19a, I have simplied slightly the model presented in @Johnstone18 by removing the assumption that the neutrals, ions, and electrons have separate temperatures and instead only using a single temperature. I have tested this simplification by rerunning the model for the modern Earth\u2019s upper atmosphere and find good agreement with the models presented in @Johnstone18. Another difference to the model is the chemical network which I have changed in two ways. Firstly, all chemical reactions and species that contain elements other than hydrogen and oxygen have been removed since I assume a purely H$_2$O gas at the lower boundary. Secondly, I have added chemical reactions and species involving deuterium; the reactions included in the model are taken from @LiangYung09 and @GarciaMunoz07, with additional XUV photoreactions taken from the PHIDRATES database (@HuebnerMukherjee15).\n\nThe model starts from a set of arbitrary initial conditions and evolves the state of the system forward in time by a large number of small timesteps until it reaches a steady state. The basic set of equations describing the evolution of the atmospheric properties is $$\\label{eqn:main_speciescontinuity}\n\\frac{\\partial n_j}{ \\partial t} \n+ \\frac{1}{r^2} \\frac{\\partial \\left[ r^2 ( n_j v + \\Phi_{\\mathrm{d},j}) \\right] }{\\partial r} \n=\nS_j ,$$ $$\\label{eqn:main_momentum}\n\\frac{\\partial ( \\rho v ) }{ \\partial t} \n+ \\frac{1}{r^2} \\frac{\\partial \\left[ r^2 \\left( \\rho v^2 + p \\right) \\right] }{\\partial r} \n=\n- \\rho g \n+ \\frac{2 p }{r} ,$$ $$\\label{eqn:main_energy}\n\\begin{aligned}\n\\frac{\\partial e }{ \\partial t} &\n+ \\frac{1}{r^2} \\frac{\\partial \\left[ r^2 v \\left( e + p \\right) \\right] }{\\partial r} \n= - \\rho v g + Q_{\\mathrm{h}} - Q_{\\mathrm{c}} \\\\\n & + \\frac{1}{r^2} \\frac{\\partial}{\\partial r} \\left[ r^2 \\kappa_\\mathrm{cond} \\frac{\\partial T}{\\partial r} + r^2 \\kappa_\\mathrm{eddy} \\left( \\frac{\\partial T}{\\partial r} + \\frac{g}{c_\\mathrm{P}} \\right) \\right] ,\n\\end{aligned}$$ where $r$ is the radius, $n_j$ is the number density of the $j$th species, $\\rho$ is the total mass density, $v$ is the bulk advection speed, is the momentum density, $e$ is the energy density, $T$ and $p$ are the temperature and thermal pressure, $\\Phi_{\\mathrm{d},j}$ and $S_j$ are the diffusive particle flux and chemical source term of the $j$th species, $g$ is the gravitational acceleration, $Q_{\\mathrm{h}}$ and $Q_{\\mathrm{c}}$ are the volumetric heating and cooling rates, $\\kappa_\\mathrm{cond}$ is the thermal conductivity, $\\kappa_\\mathrm{eddy}$ is the eddy conductivity, and $c_\\mathrm{P}$ is the specific heat at constant pressure. Since chemistry and diffusion do not change the total mass density of the gas, Eqn.\u00a0\\[eqn:main\\_speciescontinuity\\] implies the standard mass continuity equation.\n\nThe evolution of the chemical structure of the atmosphere is described by Eqn.\u00a0\\[eqn:main\\_speciescontinuity\\]. The processes taken into account are hydrodynamic advection, eddy and molecular diffusion, and chemistry, including neutral and ion chemistry and XUV driven photochemistry. In total, 223 chemical reactions, including 32 photoreactions, are considered, and the gas is composed of 33 chemical species of which 14 are ions. The chemistry is solved in a time-dependent way with no assumption of chemical equilibrium. The full set of hydrodynamic equations are solved using the explicit scheme described in @Kaeppeli16. This scheme is \u2018well-balanced\u2019, and can therefore accurately calculate the structure of an atmosphere that is in hydrostatic equilibrium. The evolution of the thermal structure of the atmosphere is described by Eqn.\u00a0\\[eqn:main\\_energy\\]. The processes taken into account are hydrodynamic advection, heating by stellar XUV and infrared radiation, cooling by the emission of radiation to space, and thermal conduction. The effects of adiabatic cooling are present in the model. A strength of my model is the calculation of the heating rate, which is done from first-principles without the use of arbitrary free parameters such as the heating efficiency. The XUV heating model includes the effects of direct heating by absorbed XUV photons, energy released by exothermic chemical reactions, and heating of thermal electrons by collisions with high-energy photoelectrons created by photoionization reactions.\n\nThe cooling model includes the emission of radiation to space by H$_2$O, H, and O. For H$_2$O cooling, I use the method by @Hollenbach79 and summarized by @Kasting83 who used this method for calculating cooling from water vapor in planetary atmospheres; this model considers excitation of water molecules by atomic hydrogen and takes into account non-local thermodynamic equilibrium effects. For H cooling, I consider the effects of Ly-$\\alpha$ emission to space using the method given by @MurrayClay09 and @Guo19. For O cooling, I consider emission at 63\u00a0$\\mu$m and 147\u00a0$\\mu$m using the parameterisations derived by @Bates51 and commonly used for calculating cooling in the upper thermospheres of planets. In future work, I intend to implement in the model a more sophisticated treatment of radiative cooling.\n\nAnother change to the model that I make for this study is the way that the molecular diffusion coefficients are calculated. Molecular diffusion coefficients are different for different chemical species and depend on the composition of the background gas. In previous models, I used measured diffusion coefficients for several common species (see Eqn.\u00a025 of @Johnstone18) assuming a background atmosphere composed of N$_2$, and then made reasonable guesses for the values for all other species. As in @Hobbs19, I use in this paper the equation for the molecular diffusion coefficients for the $i$th species, $D_i$, derived by [@ChapmanCowling70], given by $$D_i = \\frac{3}{8 N \\left[ 0.5 ( d_i + \\bar{d} ) \\right]^2} \\left[ \\frac{k_\\mathrm{B} T \\left( m_i + \\bar{m} \\right) }{ 2 \\pi m_i \\bar{m} } \\right]^{\\frac{1}{2}}$$ where $N$ is the total number density of the gas, $d_i$ is the particle diameter of the $i$th species, $\\bar{d}$ is the average particle diameter of the entire gas, $k_\\mathrm{B}$ is the Boltzmann constant, $T$ is the gas temperature, $m_i$ is the molecular mass of the $i$th species, and $\\bar{m}$ is the average molecular mass of the entire gas. For both $\\bar{d}$ and $\\bar{m}$, the number density weighted averages are used. The motivation for using this approach for diffusion coefficients is that in the atmospheres considered in this paper, molecular diffusion is especially important in a small region immediately above where H$_2$O is photodissociated and the gas composition changes rapidly from H$_2$O to H and O. In this region, which typically extends a few hundred km in altitude, the composition of the background gas changes from being dominated by water molecules to being dominated by atomic hydrogen and oxygen, with several other molecules being abundant at different altitudes. Therefore, I consider a more general and theoretical approach for diffusion coefficients to be more appropriate here than using measured values that are only appropriate for specific background gases. However, the two approaches lead to very similar diffusion coefficients with values that are typically within a factor of $\\sim$1.5 at the altitudes at which molecular diffusion is important.\n\nIn Fig.\u00a0\\[fig:Case5\\], I show an example simulation for a water vapor atmosphere of an Earth-mass planet orbiting an active solar mass star. This is Case\u00a05 presented in the following section. The strong stellar XUV field heats the gas to such a high temperature that it accelerates away from the planet in the form of a transonic Parker wind, with the point where the atmosphere becomes supersonic being also where the outflow speed exceeds the escape speed. The dissociation of the water molecules takes place very low in the atmosphere, at an altitude of $\\sim$100\u00a0km, and the upper atmosphere is dominated by atomic hydrogen and oxygen, and their ion equivalents.\n\nResults {#sect:results}\n=======\n\n![ Vertical structures of the hydrodynamically outflowing atmospheres for several cases with different input stellar XUV fluxes, showing outflow velocity (*upper-panel*), temperature (*middle-panel*), and ionization fraction (*lower-panel*) as functions of altitude throughout the simulation domain. In all panels, the crosses on each line show where the atmospheres become supersonic. []{data-label=\"fig:hydrogrid\"}](TgasGrid.pdf \"fig:\"){width=\"45.00000%\"} ![ Vertical structures of the hydrodynamically outflowing atmospheres for several cases with different input stellar XUV fluxes, showing outflow velocity (*upper-panel*), temperature (*middle-panel*), and ionization fraction (*lower-panel*) as functions of altitude throughout the simulation domain. In all panels, the crosses on each line show where the atmospheres become supersonic. []{data-label=\"fig:hydrogrid\"}](VelGrid.pdf \"fig:\"){width=\"45.00000%\"} ![ Vertical structures of the hydrodynamically outflowing atmospheres for several cases with different input stellar XUV fluxes, showing outflow velocity (*upper-panel*), temperature (*middle-panel*), and ionization fraction (*lower-panel*) as functions of altitude throughout the simulation domain. In all panels, the crosses on each line show where the atmospheres become supersonic. []{data-label=\"fig:hydrogrid\"}](XionGrid.pdf \"fig:\"){width=\"45.00000%\"}\n\n![ Several properties of the outflowing atmospheres as functions of the input stellar X-ray flux. []{data-label=\"fig:hydrogridFx\"}](TgasFxGrid.pdf \"fig:\"){width=\"45.00000%\"} ![ Several properties of the outflowing atmospheres as functions of the input stellar X-ray flux. []{data-label=\"fig:hydrogridFx\"}](altSonicFxGrid.pdf \"fig:\"){width=\"45.00000%\"} ![ Several properties of the outflowing atmospheres as functions of the input stellar X-ray flux. []{data-label=\"fig:hydrogridFx\"}](XionFxGrid.pdf \"fig:\"){width=\"45.00000%\"}\n\n![ *Upper-panel:* outward hydrodynamic mass flux at $10^5$\u00a0km altitude as a function of input stellar X-ray flux. *Middle-panel:* outward hydrodynamic particle fluxes of oxygen and hydrogen atoms (including those contained in molecules and ions), with the solid lines and circles showing the simulation results and the dotted red line showing half the hydrogen flux, indicating the oxygen loss rate expected if the dissociation products of H$_2$O were lost with the same efficiency. *Lower-panel:* fractionation factors for H and D, $f_\\mathrm{H,D}$, and for H and O, $f_\\mathrm{H,O}$, as a function of the mass flux. In all panels, the circles show the results of individual simulations []{data-label=\"fig:MdotgridFx\"}](MfluxFxGridlog.pdf \"fig:\"){width=\"43.00000%\"} ![ *Upper-panel:* outward hydrodynamic mass flux at $10^5$\u00a0km altitude as a function of input stellar X-ray flux. *Middle-panel:* outward hydrodynamic particle fluxes of oxygen and hydrogen atoms (including those contained in molecules and ions), with the solid lines and circles showing the simulation results and the dotted red line showing half the hydrogen flux, indicating the oxygen loss rate expected if the dissociation products of H$_2$O were lost with the same efficiency. *Lower-panel:* fractionation factors for H and D, $f_\\mathrm{H,D}$, and for H and O, $f_\\mathrm{H,O}$, as a function of the mass flux. In all panels, the circles show the results of individual simulations []{data-label=\"fig:MdotgridFx\"}](NfluxGrid.pdf \"fig:\"){width=\"43.00000%\"} ![ *Upper-panel:* outward hydrodynamic mass flux at $10^5$\u00a0km altitude as a function of input stellar X-ray flux. *Middle-panel:* outward hydrodynamic particle fluxes of oxygen and hydrogen atoms (including those contained in molecules and ions), with the solid lines and circles showing the simulation results and the dotted red line showing half the hydrogen flux, indicating the oxygen loss rate expected if the dissociation products of H$_2$O were lost with the same efficiency. *Lower-panel:* fractionation factors for H and D, $f_\\mathrm{H,D}$, and for H and O, $f_\\mathrm{H,O}$, as a function of the mass flux. In all panels, the circles show the results of individual simulations []{data-label=\"fig:MdotgridFx\"}](fractionationMfluxGrid.pdf \"fig:\"){width=\"43.00000%\"}\n\nTo understand the hydrodynamic outflow of water vapor atmosphere, I calculate fourteen models for an Earth-mass planet with a fully H$_2$O atmosphere orbiting a solar mass star at 1\u00a0AU. The cases considered differ in the input stellar XUV spectrum calculated using the method and codes presented by @Claire12. Specifically, I use the their solar spectra corresponding to solar ages of 4.25, 4.30, 4.35, 4.40, 4.45, 4.46, 4.47, 4.48, 4.49, 4.50, 4.51, 4.52, 4.53, and 4.54\u00a0Gyr ago. Note however that the correspondence between age and activity presented in @Claire12 is likely incorrect given that the early activity of the Sun is not known and would have depended sensitively on the Sun\u2019s initial rotation rate (@Johnstone15; @Tu15). In Fig.\u00a0\\[fig:XUVtracks\\], three possible evolutionary tracks for the Sun\u2019s rotation are shown, with the grey shaded area showing the range of solar X-ray fluxes at 1\u00a0AU that I consider.\n\nIn Table.\u00a0\\[table:grid\\], I give a summary for each considered case of the input XUV spectrum fluxes and the expected age ranges that the early Sun, or any solar-mass star, would have these fluxes. For the age ranges, I use the slow and rapid rotator tracks for X-ray emission shown in Fig.\u00a0\\[fig:XUVtracks\\]. These tracks give the expected evolutions of solar mass stars born at the 10th and 90th percentiles of the distribution of rotation rates. Therefore, for each input spectrum, the lower limit on the age is the age at which the slow rotator reaches the corresponding activity level of the spectrum and the upper limit is the age at which the fast rotator reaches this activity level. This is meant as an approximate measure of the expected age for a given X-ray flux and does not take into account the fact that stars may lie below 10th or above the 90th percentiles of the rotation distribution and does not take into account the large spread in X-ray luminosities for stars with a given mass, rotation rate, and luminosity (e.g. see Fig.\u00a03 of @Reiners14).\n\n![image](DensityCase2.pdf){width=\"43.00000%\"} ![image](DensityCase14.pdf){width=\"43.00000%\"} ![image](MolAtomCase2.pdf){width=\"43.00000%\"} ![image](MolAtomCase14.pdf){width=\"43.00000%\"} ![image](HDOCase2.pdf){width=\"43.00000%\"} ![image](HDOCase14.pdf){width=\"43.00000%\"}\n\nIn Fig.\u00a0\\[fig:hydrogrid\\], the vertical structures of the gas temperature, outflow speed, and ionization fraction are shown for several of the cases calculated. In all cases, the absorption of the stellar XUV spectrum heats the gas to very large temperatures, causing the atmosphere to flow away from the planet hydrodynamically in the form of a transonic wind. At altitudes of $\\sim$5000-20,000\u00a0km, the outflow speeds exceeds the escape velocity; this point is also the sonic point as required in a thermal pressure driven Parker wind and is marked by the crosses in Fig.\u00a0\\[fig:hydrogrid\\]. In no case is the exobase found within the simulation domain.\n\nThe dependence on the input X-ray flux of the maximum gas temperature, the sonic point altitude, and the maximum ionization fraction are shown in Fig.\u00a0\\[fig:hydrogridFx\\]. In the higher activity cases, the atmospheres are heated to higher temperatures, accelerated to larger outflow speeds, and become more ionized. In the highest activity cases, the temperature at the upper boundary of the simulation stops increasing with increasing activity and instead decreases due to adiabatic cooling, which can cause a negative temperature gradient in hydrodynamic outflows (@Tian08; see also Fig.\u00a01 of @Johnstone15letter). This can be best seen in the upper panel of Fig.\u00a0\\[fig:hydrogridFx\\], where the maximum temperature of $\\sim$20,000\u00a0K is reached at an input X-ray flux of $\\sim$250\u00a0erg\u00a0s$^{-1}$\u00a0cm$^{-2}$. The sonic point altitude, shown in the lower panel of Fig.\u00a0\\[fig:hydrogridFx\\], gets closer to the planet as the input XUV flux increases due to the more rapid acceleration of the wind, but at very high activity cases appears to become approximately independent of activity with a value of $\\sim$500\u00a0km. The ionization fraction at the upper boundary continues to increase with increasing activity, even at very high activity level, and in the most active case considered is 93% meaning that as it flows away from the planet, the gas is almost entirely ionized.\n\nIn Fig.\u00a0\\[fig:MdotgridFx\\], I show the outward hydrodynamic mass flux as a function of input stellar X-ray flux. As expected, higher stellar activity cases have significantly higher atmospheric losses. In and near the range of input X-ray fluxes considered, the mass flux at the upper boundary of the simulation domain (at an altitude of $10^5$\u00a0km) can be described by $$\\label{eqn:Mdot}\n\\log F_\\mathrm{mass} = \n-0.239 \\left( \\log F_\\mathrm{X} \\right)^2 \n+ 1.982 \\log  F_\\mathrm{X}\n- 3.681 ,$$ where the units of $F_\\mathrm{X}$ are erg\u00a0s$^{-1}$\u00a0cm$^{-2}$ and $F_\\mathrm{mass}$ are . As a note of caution, it is not trivial to estimate total atmospheric mass loss rates from these fluxes and this problem is discussed in more detail in Section\u00a0\\[sect:evo\\]. Since I calculate 1D hydrodynamic models for the direction pointing in the direction of the star, simply multiplying the mass flux by , where $R$ is the Earth\u2019s radius plus $10^5$\u00a0km, would overestimate the mass loss rates by a factor of a few. The outward particle fluxes for hydrogen and oxygen and the hydrogen-oxygen and hydrogen-deuterium fractionation factors are also shown in Fig.\u00a0\\[fig:MdotgridFx\\] and discussed below.\n\nFor Case\u00a02 and Case\u00a014, representing moderate and high activity cases, information about the chemical structures of the upper atmospheres is shown in Fig.\u00a0\\[fig:chem\\] for five neutral and two ion species. At the base of the simulation, the gas is composed entirely of H$_2$O, which is then quickly photodissociated, creating several molecular and atomic species including H$_2$, O$_2$, H, and O. At slightly higher altitudes, the H$_2$ and O$_2$ are then also photodissociated and the gas becomes completely dominated by H and O. Between altitudes of approximately 100 and 1000\u00a0km, H$^+$ and O$^+$ start to become important, and in the high activity case, H$^+$, O$^+$, and free electrons are the most abundant species by the upper boundary of the simulation domain. It is important to note that at no point in the atmosphere is the gas dominated by atomic hydrogen: while H has the highest number density throughout most of the atmosphere, the heavier O has the highest mass density of all species. In Fig.\u00a0\\[fig:chem\\], I also show the total mixing ratios of atoms and molecules in both simulations showing where in the atmosphere most of the dissociation of molecules takes place.\n\n![image](MatLossEvo.pdf){width=\"45.00000%\"} ![image](MatLossEvoLog.pdf){width=\"45.00000%\"}\n\n![ Cumulative evolution of hydrogen (*upper-panel*) and oxygen (*middle-panel*) losses for the evolutionary cases shown in Fig.\u00a0\\[fig:MlossGrid\\], with the amount of oxygen that builds up in the atmosphere as a result of the different loss efficiencies of hydrogen and oxygen (*lower-panel*). In the lower panel, I assume 1 Earth ocean is 270\u00a0bar. []{data-label=\"fig:NlossGrid\"}](NlossEvo1.pdf \"fig:\"){width=\"45.00000%\"} ![ Cumulative evolution of hydrogen (*upper-panel*) and oxygen (*middle-panel*) losses for the evolutionary cases shown in Fig.\u00a0\\[fig:MlossGrid\\], with the amount of oxygen that builds up in the atmosphere as a result of the different loss efficiencies of hydrogen and oxygen (*lower-panel*). In the lower panel, I assume 1 Earth ocean is 270\u00a0bar. []{data-label=\"fig:NlossGrid\"}](NlossEvo2.pdf \"fig:\"){width=\"45.00000%\"} ![ Cumulative evolution of hydrogen (*upper-panel*) and oxygen (*middle-panel*) losses for the evolutionary cases shown in Fig.\u00a0\\[fig:MlossGrid\\], with the amount of oxygen that builds up in the atmosphere as a result of the different loss efficiencies of hydrogen and oxygen (*lower-panel*). In the lower panel, I assume 1 Earth ocean is 270\u00a0bar. []{data-label=\"fig:NlossGrid\"}](O2buldupEvo.pdf \"fig:\"){width=\"45.00000%\"}\n\nIn the upper-left panel of Fig.\u00a0\\[fig:chem\\], the dotted red line shows half of the hydrogen mixing ratio, indicating approximately the values expected if the dissociation products of H$_2$O were lost equally for Case\u00a02. The lower values for O relative to this line in the upper thermosphere show that H is lost more efficiently than O, as expected given its lower mass. The reason for the different loss efficiencies of H, D, and O in the atmosphere is that in the region where the H$_2$O is photodissociated, molecular diffusion is very important and separates the particles by mass. This is shown more effectively by the red line in the lower-left panel of Fig.\u00a0\\[fig:chem\\], which shows the ratio of the densities of oxygen atoms to hydrogen atoms (including atoms held in molecules and ions) divided by the base value. In this region around an altitude of $\\sim$100\u00a0km, initially H becomes less abundant relative to O because it is more rapidly transported upwards by molecular diffusion and then becomes more abundant for the same reason. A similar trend is seen in the D/H ratio, shown as the blue line in the lower-left panel of Fig.\u00a0\\[fig:chem\\], though the separation by mass is much less significant since the masses of D and H are much more similar.\n\nThis mass fractionation is much lower than we would expect for a fully hydrostatic atmosphere because molecular diffusion is only able to separate the species by mass in a small range of altitudes above the homopause. The homopause is defined as the altitude at which the effect of molecular diffusion exceeds the effect of eddy diffusion, and below the homopause, eddy diffusion forces all the species that are not quickly created and destroyed by chemical reactions to have mixing ratios that are uniform with altitude. This is typically at the base of the thermosphere since this is where the rapid increase in temperature due to XUV heating causes a rapid increase in the molecular diffusion coefficients. Above the homopause, molecular diffusion separates the species by mass; however, in a hydrodynamically outflowing atmosphere, the hydrodynamic advection of the gas also has the effect of making the mixing ratios of each species uniform with altitude, and since this is the dominant effect in the upper thermosphere, very little mass separation takes place. The mass fractionation is quite small because the distance between the homopause and where the advection of species dominates is very small such that molecular diffusion does not have much of an opportunity to separate the species by mass. This is consistent with the results presented by @Gillmann09 for mass fractionation of noble gases in the atmosphere of early Venus.\n\nAs can be seen in the lower right panel of Fig.\u00a0\\[fig:chem\\], for our highest activity case, almost no mass fractionation takes place. This is due to the much stronger hydrodynamic escape that takes place in this case. The fractionation factor between the two escaping species, $f_{i,j}$, is defined by , where $F_i$ and $N_i$ are the escape flux and total atmospheric abundance of the $i$th particle (@Mandt09). It is reasonable to assume that the term is equal to the ratio of the densities () of the two particles at the upper boundary of the simulation and that is equal to the ratio () at the base of the simulation, where in both cases the densities include also atoms contained within molecules. The fractionation factors between O and H, $f_\\mathrm{H,O}$, and between D and H, $f_\\mathrm{H,D}$, as functions of the input X-ray flux are shown in Fig.\u00a0\\[fig:MdotgridFx\\]. The value for $f_\\mathrm{H,O}$ is 0.4 for the lowest activity case and quickly increases to 1.0 as the input XUV flux increases. Similarly, the value for $f_\\mathrm{H,D}$ increases from $\\sim$0.9 to 1.0. Therefore, for extreme hydrodynamic escape driven by very active stars, no mass fractionation takes place, leading to no oxygen accumulation and no increase in the D/H ratio.\n\nEvolution of losses and accumulation of O$_2$ {#sect:evo}\n=============================================\n\nIn this section, I study the evolution of losses from an Earth-mass planet orbiting at 1\u00a0AU around a solar mass star with a fully water vapor atmosphere between ages of 10\u00a0Myr to 1\u00a0Gyr. For all calculations, I assume that there is an infinite reservoir of water vapor available to the atmosphere to feed these losses. To calculate the mass loss rates, I use Eqn.\u00a0\\[eqn:Mdot\\] to get the outward mass flux as a function of the input stellar XUV flux. These fluxes are calculated using the stellar activity evolution tracks calculated by @Tu15. However, care must be taken to ensure that the mass loss rates are not overestimated since the simulations from which Eqn.\u00a0\\[eqn:Mdot\\] is derived give the mass flux in the direction of the star, which is likely a factor of a few higher than the average mass flux in all directions. As in @Erkaev13, I assume that the loss takes place over 3$\\pi$ steradians, which is based on the assumption that losses do not take place where the atmosphere is not irradiated by the star due to the planet\u2019s shadow. As in @Johnstone15letter, I then average the incoming stellar XUV energy over this 3$\\pi$ steradians, which is implemented in Eqn.\u00a0\\[eqn:Mdot\\] by inputting the in place of the XUV flux and then calculating the mass loss rate as , where $R$ is the radius of the upper boundary of the simulation domain. Although this is a simple way to calculate the total atmospheric losses from 1D simulations, it has a good advantage that it makes sense from an energy conservation point of view if a zero zenith angle is assumed in the simulations. The energy available to drive atmospheric losses is no greater than the total energy input into the atmosphere and the XUV energy that does not directly drive losses is only lost because of real atmospheric processes that decrease how efficiently the absorbed XUV energy is converted into heat and how much of that heat drives losses, as opposed to energy being lost due to geometric assumptions. To get $f_\\mathrm{H,O}$ for this mass flux, I use a simple polynomial interpolation using the relation between the mass flux and $f_\\mathrm{H,O}$ shown in the lower panel of Fig.\u00a0\\[fig:MdotgridFx\\]. The loss rates for oxygen and hydrogen atoms, $\\dot{N}_\\mathrm{O}$ and $\\dot{N}_\\mathrm{H}$, are given by and , where $m_\\mathrm{O}$ and $m_\\mathrm{H}$ are the masses of oxygen and hydrogen atoms.\n\nIn Fig.\u00a0\\[fig:MlossGrid\\], the total amount of atmosphere lost as a function of age in the first Gyr is shown for the cases of the slow, medium, and fast stellar rotator tracks, and for the case of a star that remains at the saturation threshold for the first billion years (dashed black line). For the slow stellar rotator case, the total atmospheric mass loss is $\\sim$1\u00a0Earth oceans (). For the fast stellar rotator case, the total atmospheric mass loss is $\\sim$40\u00a0Earth oceans, which is much larger since the initially rapidly rotating star remains highly active for much longer. Due to the decay in activity after approximately 300\u00a0Myr, this is much less than the 120\u00a0Earth oceans lost if the star remains at the saturation threshold for the entire first Gyr as might be the case for a low mass M star. In both the first 300\u00a0Myr for the fast rotator case and in the constantly saturated case, the rapid hydrodynamic escape means that the planet is unlikely to be able to hold onto any water vapor atmosphere, and water outgassed from the interior or delivered to the planet from external sources will be lost very rapidly. A water vapor atmosphere could survive if the planet is formed with a significant fraction of its mass as water and therefore a sufficiently large enough reservoir of water if available to compensate for the losses to space. Also, a water vapor atmosphere could form by outgassing after the star\u2019s activity has decayed to more moderate levels.\n\nThe total losses of hydrogen and oxygen in Earth ocean equivalents for each stellar rotator case is shown in Fig.\u00a0\\[fig:NlossGrid\\]. I define an Earth ocean equivalent for a given atom to be the number of those atoms in one Earth ocean. The differences between the dashed and solid lines shown in the upper and middle panels is due to the different loss efficiencies of H and O. The resulting build-up of O$_2$ molecules due to the slightly lower loss rates of O is shown in the lower panel of Fig.\u00a0\\[fig:NlossGrid\\]. For the fast rotator case, no O$_2$ is able to build-up in the atmosphere for the first $\\sim$400\u00a0Myr despite the very large atmospheric losses. This is due to the weak fractionation for atmospheres with very high loss rates meaning that hydrogen and oxygen atoms are lost at a 2:1 ratio. After this phase, the mass loss decreases leading to a large amount of O$_2$ remaining in the atmosphere, and by 1\u00a0Gyr it is possible that $\\sim$300\u00a0bar of O$_2$ is accumulated. The mass fractionation in the slow rotator case is much more efficient early on due to the higher fractionation factors, but due to the much weaker overall mass loss, approximately the same amount of O$_2$ can be accumulated. Interestingly, the medium rotator case leads to the most significant accumulation of O$_2$ since this case the star\u2019s activity spends much of the first Gyr at levels high enough to cause high escape rates while still low enough to allow fractionation factors below unity. The assumption that the star remains saturated for the first Gyr leads to no O$_2$ build-up in the atmosphere since the mass loss rate remains high for the entire time. In all cases, the mass loss and the accumulation of O$_2$ can be limited if the reservoir of H$_2$O available to the atmosphere is exhausted. This is most likely to happen for the fast rotator case and we might expect a much smaller accumulation of O$_2$ to take place; in this case, no O$_2$ accumulation might take place if the water in the atmosphere is entirely removed within the first 400\u00a0Myr and no significant amount of water is released into to the atmosphere afterwards.\n\n![image](HfluxFxGrid.pdf){width=\"45.00000%\"} ![image](OfluxFxGrid.pdf){width=\"45.00000%\"}\n\nDiscussion {#sect:conclusions}\n==========\n\nIn this paper, I use a sophisticated physical model for the upper atmospheres of planets to study the hydrodynamic losses of water vapor atmospheres on Earth mass planets orbiting active stars with a range of activity levels. In all cases studied, the heating of the atmosphere is strong enough to cause strong hydrodynamic escape leading to rapid water loss from the atmosphere. Since the water molecules are dissociated by XUV photoreactions, the outflowing material consists of atomic hydrogen and oxygen, which in the most active cases and mostly ionized. For the strongest outflows, I find that there is no separation of the particles by mass in the outflow, meaning that the losses will not lead to a build-up of oxygen or a change in the atmosphere\u2019s D/H ratio.\n\nConnecting these results to evolutionary tracks for stellar XUV evolution, I study how much water vapor can be lost and how this might change the atmosphere\u2019s composition. The results depend sensitively on the star\u2019s activity evolution which is determined by its mass and initial rotation rate of the star. For solar mass stars, I find that a fast rotator can remove several tens of Earth oceans of water in the first billion years if a large enough reservoir of water is available, whereas a slow rotator can remove $\\sim$1\u00a0Earth ocean. The two cases however lead to very similar total amounts of oxygen accumulated, though if there is only a limited reservoir of atmospheric water vapor, then this build-up will be limited for the fast rotator case. In both cases, $\\sim$300\u00a0bar of O$_2$ can be accumulated, whereas for the medium rotator case, a large amount of $\\sim$ 450\u00a0bar is possible. This O$_2$ will not necessary remain in the atmosphere, but can be absorbed into the planet\u2019s surface (@Wordsworth18), lost hydrodynamically when it starts to become abundant enough in the upper atmosphere (@Tian15), or lost later by other processes such as stellar wind pick-up (@Kulikov07). I also study the case of a star that remains at the saturation threshold, which is possible for fully convective M-dwarfs which remain highly active for very long time periods, and in some cases even several billion years (@West08). In this case, very large amounts of water can be lost from their atmospheres, but no build-up of O$_2$ takes place since the loss takes place rapidly enough that there is no significant mass fractionation in the outflow. This means that the very large amount of O$_2$ accumulation that has been suggested for planets orbiting in the habitable zones of M-dwarfs (e.g. @LugerBarnes15) could be unreaslistic.\n\nIt is important to consider the possible sources of error in my results. While the model is the most sophisticated general purpose aeronomy model that has applied to problems of exoplanetary upper atmospheres, there are a very large number of improvements that are possible. An obvious improvement would be the extension of the model to 3D geometries. While 1D models are able to accurately reproduce the upper atmospheres of solar system terrestrial planets, suggesting that considering more dimensions might by unnecessary, 3D models would be able to take into account the effects of different planetary rotation rates and planetary magnetic fields. Given that in the most active cases that I consider in this paper, the gas becomes almost entirely ionized, strong interactions with the planet\u2019s intrinsic magnetic field (if present) might be possible, which has been shown to be able to drastically reduce mass loss rates for hot Jupiters (e.g. @Khodachenko15). Another planned improvement to the model is the implementation of a more sophisticated model for atmospheric cooling which could change the mass loss rates. Finally, it is important to note that I have only considered the reaction of an atmosphere to stellar spectra of active solar mass stars. While planets in the habitable zones of active lower mass stars, such as fully convective M stars, will receive similar amounts of total X-ray and EUV radiation, they will have different spectral shapes (@Fontenla16). The most important difference is that the photospheric spectrum, which for a G star dominates the XUV emission at wavelengths longer than $\\sim$160\u00a0nm, is shifted to much longer wavelengths for M stars. This could be important since this radiation is strongly absorbed by O$_2$ and O$_3$ molecules, which are created by the photodissociation of H$_2$O.\n\nIt is often stated in the literature that the photodisociation and heating of a water vapor outflow leads to a hydrodynamic outflow of H that drags O with it. This is a misleading interpretation of what is taking place and implies that thermal pressure gradients only accelerate the hydrogen, while the oxygen atoms are only accelerated because they collide with the already accelerated hydrogen atoms. In reality, the acceleration takes place where the atmosphere is still dense and collisional, and both the hydrogen and oxygen atoms are accelerated by the thermal pressure of the surrounding gas. Atomic oxygen accelerates away from the planet not because it is \u2018dragged\u2019 by hydrogen, but because it is accelerated by the same pressure gradients that cause the acceleration of hydrogen. It should be noted that at no point in the atmosphere is the gas dominated by atomic hydrogen: although H is the most numerous species above the lower thermosphere, O contributes most of the mass.\n\nIn all cases considered, the XUV heating of the gas, the acceleration of the hydrodynamic flow, the dissociation of H$_2$O, and the separation of the various species by mass by molecular diffusion happen all within a very small range of altitudes. A major difference between hydrostatic and hydrodynamically outflowing atmospheres is that above this region in hydrodynamic atmospheres, the outward advection flow dominates over molecular diffusion. This drastically limits how much molecular diffusion can separate particles by mass, leading to weak fractionation, and for the most active cases, there is no mass fractionation. The weak isotopic fractionation of hydrogen caused by extreme hydrodynamic escape is consistent with the results of @Kasting83. While they did not study such high solar activity levels, they found decreasing mass fractionation of hydrogen with increasing activity level (see their Fig.\u00a013).\n\nThe strong outflows that I have calculated will lead to planets that are surrounded by a large cloud of particles (@Kislyakova13; @Bourrier15). If this cloud of particles is transiting the star, the escape might lead to observable signatures that could be used to constrain the properties of the outflow and of the atmosphere. The most obvious of these is transit observations in the host star\u2019s Ly-$\\alpha$ line (@Kislyakova19), and such signatures have already been seen in several systems (e.g. @Ehrenreich08; @Kislyakova14b; @Lavie17). In this case, the star can be transited by neutral hydrogen atoms flowing out of the planet\u2019s atmosphere and by energetic neutral atoms (ENAs) created by charge exchange between neutral atmospheric particles and stellar wind protons. In both cases, a supply of neutral hydrogen from the planet\u2019s atmosphere is needed. In Fig.\u00a0\\[fig:HOlossGrid\\], I show the outflow rates of neutral hydrogen and oxygen atoms. For low input XUV fluxes, the neutral hydrogen and oxygen outflows increase rapidly with increasing XUV flux; however, this breaks down at very high stellar activities due to the increasing ionization of the gas. For the most active cases, higher stellar activity leads to fewer neutral atoms flowing away from the planet, suggesting that the very high ionization fractions for the highest stellar activities might limit our ability to observe the outflow using Ly-$\\alpha$ absorption.\n\nGiven how weak the isotopic fractionation of hydrogen by rapid hydrodynamic escape is, it is interesting to consider how much water must have been lost in order to produce the very large D/H ratio found on Venus (e.g. @Grinspoon93; @Johnson19). The total initial inventory of deuterium needed to produce the current D/H ratio in the atmosphere of Venus is given by , where $N_\\mathrm{D}$ and $N_\\mathrm{D}^0$ are the current and initial inventories of D respectively, and $R$ and $R_0$ are the current and initial D/H ratios respectively. Although the D/H ratio of the source of water in Venus\u2019 atmosphere is not known, it was likely much smaller than the current value, meaning that the term should be very large. Assuming an $R_0$ similar to that of the modern Earth gives . In the lowest activity case calculated here, the value for $f_\\mathrm{H,D}$ is 0.86, which gives and means that an unrealistically large initial inventory of water must have been lost from Venus\u2019 atmosphere if extreme hydrodynamic escape was the primary mechanism for D/H enrichment. Instead, it is likely that the large D/H ratio is a consequence of water loss that took place after the Sun\u2019s activity had declined significantly, which could have been either after the first few hundred million years if the Sun was a slow rotator or the first billion years if it was a fast rotator. The dominant loss process was likely either a much weaker hydrodynamic escape or non-thermal escape processes. This does not rule out the possibility that rapid hydrodynamic water loss took place early in Venus\u2019 evolution and if so, it could have resulted in massive losses of water without any build-up of O$_2$ in the atmosphere.\n\nAcknowledgments\n===============\n\nThis study was carried out with the support by the FWF NFN project S11601-N16 \u201cPathways to Habitability: From Disk to Active Stars, Planets and Life\u201d and the related subproject S11604-N16.\n"], ["---\nauthor:\n- 'B. Barbuy'\n- 'A. C. S. Fria\u00e7a'\n- 'C. R. da Silveira'\n- 'V. Hill'\n- 'M. Zoccali'\n- 'D. Minniti'\n- 'A. Renzini'\n- 'S. Ortolani'\n- 'A. G\u00f3mez'\ntitle: ' Zinc abundances in Galactic bulge field red giants: Implications for DLA systems [^1] '\n---\n\n= ==1=1=0pt =2=2=0pt =2=2=0pt\n\n[Zinc in stars is an important reference element because it is a proxy to Fe in studies of damped Lyman-$\\alpha$ systems, permitting a comparison of chemical evolution histories of bulge stellar populations and DLAs. In terms of nucleosynthesis, it behaves as an alpha element because it is enhanced in metal-poor stars. Abundance studies in different stellar populations can give hints to the Zn production in different sites. ]{} [The aim of this work is to derive the iron-peak element Zn abundances in 56 bulge giants from high resolution spectra. These results are compared with data from other bulge samples, as well as from disk and halo stars, and damped Lyman-$\\alpha$ systems, in order to better understand the chemical evolution in these environments. ]{} [High-resolution spectra were obtained using FLAMES+UVES on the Very Large Telescope. We computed the Zn abundances using the lines at 4810.53 and 6362.34 [\u00c5]{}. We considered the strong depression in the continuum of the 6362.34 [\u00c5]{} line, which is caused by the wings of the 6361.79 [\u00c5]{} line suffering from autoionization. CN lines blending the 6362.34 [\u00c5]{} line are also included in the calculations. ]{} [We find \\[Zn/Fe\\]=+0.24$\\pm$0.02 in the range $-$1.3 $<$ \\[Fe/H\\] $<$ $-$0.5 and \\[Zn/Fe\\]=+0.06$\\pm$0.02 in the range $-$0.5 $<$ \\[Fe/H\\] $<$ $-$0.1, whereas for \\[Fe/H\\]$\\geq$ $-$0.1, it shows a spread of $-$0.60 $<$ \\[Zn/Fe\\] $<$ +0.15, with most of these stars having low \\[Zn/Fe\\]$<$0.0. These low zinc abundances at the high metallicity end of the bulge define a decreasing trend in \\[Zn/Fe\\] with increasing metallicities. A comparison with Zn abundances in DLA systems is presented, where a dust-depletion correction was applied for both Zn and Fe. When we take these corrections into account, the \\[Zn/Fe\\] vs. \\[Fe/H\\] of the DLAs fall in the same region as the thick disk and bulge stars. Finally, we present a chemical evolution model of Zn enrichment in massive spheroids, representing a typical classical bulge evolution.]{}\n\nIntroduction\n============\n\nZinc is an interesting element to study because it can be observed in damped Lyman-$\\alpha$ systems (DLAs), where it is assumed as a proxy for Fe and where it provides most of our knowledge of the chemical evolution of the Universe at high redshift, through abundances in DLAs (Pettini et al. 1999; Prochaska & Wolfe 2002).\n\nZinc abundance derivation is important given its production in different nucleosynthesis processes and environments: weak s-process in hydrostatic phases of He and C burning in massive stars, complete and incomplete Si-burning, explosive burning in core-collapse SNe, and the main s-process in low and intermediate mass stars (e.g. Umeda & Nomoto 2002; Bisterzo et al. 2004). Zinc is in the so-called upper iron group with atomic masses in the range 57$\\leq$A$\\leq$ 66, which includes species up to $^{66}$Zn (Woosley & Weaver 1995). Umeda & Nomoto (2002) show that in massive stars, the iron-peak elements Cr, Mn, Co, and Zn are produced in complete Si-burning with a peak temperature T$_{\\rm peak}$ $>$ 5$\\times$10$^{9}$ K and in incomplete Si-burning at temperatures in the range 4$\\times$10$^{9}$ $<$ T$_{\\rm peak}$ $<$ 5$\\times$10$^{9}$ K. They also found that \\[Zn/Fe\\] is greater for deeper mass cuts in the explosion process, smaller neutron excess, and higher explosion energies and that a higher Zn abundance results from deep mixing of complete Si-burning products and a fallback. At lower metallicities, $^{64}$Zn is produced in complete Si-burning (cf. Umeda & Nomoto 2003, 2005). In addition, hypernovae, defined as supernovae with high explosion energies (E$_{51}$ $\\simgreat$ 2 for M$\\sim$13 M$_{\\odot}$ and E$_{51}$ $\\simgreat$ 20 for M $\\simgreat$ 20 M$_{\\odot}$), give rise to ejecta with \\[Zn/Fe\\] as high as $\\sim$0.5 (Umeda & Nomoto 2002; Nomoto et al. 2013). Therefore, Kobayashi et al. (2006) have invoked hypernovae to explain the high \\[Zn/Fe\\] ratios in metal-poor stars.\n\nThe majority of the Fe-peak elements show solar abundance ratios in most objects for all metallicities. The elements Sc, Mn, Cu, and Zn, however, show different trends (e.g. Sneden et al. 1991; Nissen et al. 2000; Ishigaki et al. 2013; Barbuy et al. 2013). Nissen & Schuster (2011) find that Zn behaves like alpha elements, with high-alpha halo stars and thick disk stars displaying also high Zn abundances, whereas the low-alpha halo stars show a lower Zn enhancement that decreases with metallicity. This distinct behaviour is explained if both these elements (Zn and alphas) are produced by core-collapse supernovae. The \\[Zn/Fe\\] decrease with metallicity in low-alpha stars is then expected, characterizing a system with slower chemical enrichment.\n\nA useful means of better understanding the nucleosynthesis processes yielding iron-peak elements is their abundance derivation in different stellar populations. In the present work, we derive Zn abundances for a sample of 56 bulge field stars, observed at high resolution with the FLAMES-UVES spectrograph (Lecureur et al. 2007; Zoccali et al. 2006; Hill et al. 2011), where we adopt the stellar parameters effective temperature T$_{\\rm eff}$, gravity log g, metallicity \\[Fe/H\\][^2], and microturbulence velocity from these previous determinations.\n\nWe compare our results with previous Zn abundance determinations for bulge, disk, and halo stars from the literature. Bensby et al. (2013, and references therein) derived Zn for microlensed bulge dwarfs. Prochaska et al. (2000), Reddy et al. (2006), Nissen & Schuster (2011), and Bensby et al. (2014) derived Zn abundances for thick disk stars. A comparison with thin disk abundances is also given, based on work by Allende-Prieto et al. (2004), Bensby et al. (2014), and Pomp\u00e9ia (2003).\n\nWe also compare the present results to literature abundances for damped Lyman-alpha systems, where Zn is considered as a proxy for Fe. We included comparisons with analyses by Akerman et al. (2005), Cooke et al. (2011, 2013), Kulkarni et al. (2007), and Vladilo et al. (2011).\n\nFinally, we also present a model of Zn enrichment in massive spheroid systems, which is based on the code described in Lanfranchi & Fria\u00e7a (2003). It would represent the early bulge enrichment and its chemical evolution.\n\nIn Sect. 2 the observations are summarized and the adopted atomic constants and solar abundances given. In Sect. 3 the basic stellar parameters are listed, and the abundance derivation of Zn is described. The results are compared with literature data and discussed in Sect. 4. A summary is given in Sect. 5.\n\nObservations, atomic line parameters, and solar abundances\n==========================================================\n\nThe present UVES data were obtained using the UVES-FLAMES instrument at the 8.2 m Kueyen ESO telescope, as described in Zoccali et al. (2006), Lecureur et al. (2007), and Hill et al. (2011). The spectra cover the wavelength range 4800-6800 [\u00c5]{} with a resolution of R $\\sim$ 45 000 and a pixel scale of 0.0147 [\u00c5]{}/pix. Targets are bulge K giants, with magnitudes $\\sim$0.5 above the red clump, in four fields, including Baade\u2019s Window. Zinc lines were checked against oscillator strengths log gf in the literature, using in particular the data bases from Kur\u00facz (1995) website[^3], NIST[^4], and VALD (Piskunov et al. 1995). Table \\[lines2\\] reports excitation potential, log gf values from the literature with their references, and adopted log gf values. The lines were fitted to the solar high resolution observations using the same UVES spectrograph [^5] as the present sample of spectra, to spectra from Arcturus (Hinkle et al. 2000), and from the metal-rich giant $\\mu$ Leo, with spectra observed with the ESPaDOns/CFHT spectrograph, at a resolution of R $\\sim$ 80 000 and a S/N $\\sim$ 500 (Lecureur et al. 2007). The fits of 4810.529 and 6362.339 [\u00c5]{} lines in these reference stars are presented in Fig. \\[sun\\].\n\nAutoionization of lines\n-----------------------\n\nThe measurement of the 6362.350 line has to take a continuum lowering in the range $\\sim$6360.8 - 6363.1 [\u00c5\u00a0 into account]{}, owing to the 6361.940 auto-ionization line. Mitchell & Mohler (1965) identified depressions that are $\\sim$2.6 [\u00c5]{} wide at the locations of the multiplet lines at 6318, 6343, and 6363 [\u00c5]{}, which are caused by auto-ionizing transitions 3d4p 3F$^{\\circ}$ - 3d4d 3G. Following the recipe that Lecureur et al. (2007) used for the 6318 [\u00c5]{} line, we adjusted the radiative broadening factor for the 6363 [\u00c5]{} line to match the line profile to standard stars (Sun, Arcturus, $\\mu$ Leo), as well as to sample stars, thus taking the much reduced lifetime of the level suffering auto-ionization into account. The best-fitting value is 32\u00a0000 higher than the standard radiative broadening ($\\gamma$$_{\\rm rad}$ = 0.21/$\\lambda$$^{2}$). Taking this effect into account, we derived an astrophysical log gf value for the 6361.940 line, as reported in Table \\[lines2\\].\n\n[lrrrrrrrrrrrrrrrr]{} Species & [$\\lambda$]{} ([\u00c5]{}) & & & & & &\\\nZnI & 4810.529 & 4.0782 & $-$0.137 & \u2014 & $-$0.137 &$-$0.17$^1$,$-$0.16$^2$,$-$0.31$^3$ &$-$0.25\\\n& 6362.339 & 5.7961 & +0.150 & +0.158 & 0.150 &+0.14$^{1,2}$ & +0.05\\\nCaI & 6361.786 & 4.4510 & +0.954 & \u2014 & +0.317 &$-$0.2$^{4}$ & $-$0.20\\\nCrI & 4810.509 & 2.9870 & \u2014 & \u2014 & $-$3.142/$-$2.899& $-$2.90$^{4}$ & $-$2.90\\\nTiI & 4810.705 & 2.4870 & \u2014 & \u2014 & $-$2.576/$-$2.563& $-$1.00$^{4}$ & $-$1.00\\\nV1 & 4810.730 & 3.1310 & \u2014 & \u2014 & $-$1.246/$-$2.534&$-$1.25$^{4}$ & $-$1.25\\\nCrI & 4810.732 & 3.0790 & \u2014 & $-$1.30 & $-$1.300/$-$0.644& $-$1.90$^{4}$ & $-$1.90\\\n\nSolar abundances\n----------------\n\nTable \\[sol\\] gives literature abundances for Fe and Zn for the Sun, Arcturus, and $\\mu$ Leo. The adopted solar abundances for Fe and Zn are from Grevesse & Sauval (1998), whereas for Arcturus they are from the present fits, when adopting stellar parameters from Mel\u00e9ndez et al. (2003). For the metal-rich star $\\mu$ Leo, the stellar parameters and C, N, O abundances from Lecureur et al. (2007) were adopted: T$_{\\rm eff}$ = 4540 K, log g = 2.3, \\[Fe/H\\] = +0.3, v$_{\\rm t}$ = 1.3 km s$^{-1}$. The C, N, O abundances for $\\mu$ Leo were revised from $\\epsilon$(C) = 8.85, $\\epsilon$(N) = 8.55, $\\epsilon$(O) = 9.12, to \\[C/Fe\\]=$-$0.3, \\[N/Fe\\]=+0.6, \\[O/Fe\\]=$-$0.1 or $\\epsilon$(C) = 8.55, $\\epsilon$(N) = 8.83, $\\epsilon$(O) = 8.97. Also for better fitting the line on the red side of the 4810.5 [\u00c5]{} line, which is not well fitted with literature log gf values, we refitted them with the astrophysical log gf values reported in Table \\[lines2\\], assuming \\[Cr/Fe\\]=\\[Ti/Fe\\]=0.0 for $\\mu$ Leo. Finally, for $\\mu$ Leo, the resulting Zn abundance of \\[Zn/Fe\\]=$-$0.1 is derived from both ZnI lines.\n\n$$\\begin{array}{lcccccccccccc}\n\\hline\\hline\n\\noalign{\\smallskip}\n%\\hbox{El.} & \\hbox{Z} & \\multispan4  \\hbox{log $\\epsilon(X)_{\\odot}$}&& \\multis%pan2 \n%  \\hbox{log $\\epsilon(X)_{\\rm Arcturus}$}  \n%&  &  \\hbox{log $\\epsilon$(X)$_{\\rm \\mu Leo}$} &  \\\\\n%\\hbox{El.} & \\hbox{Z} & \\multispan4  \\hbox{log $\\epsilon(X)_{\\odot}$}&& \\multis%pan2   \\hbox{log $\\epsilon(X)$}  &  &  \\hbox{log $\\epsilon$(X)} &  \\\\\n\\hbox{El.} & \\hbox{Z} & \\multispan6  \\hbox{log $\\epsilon(X)$} & & & & \\\\\n  \\cline{3-11}  \\\\\n%& \\multispan2   \\hbox{log $\\epsilon(X)$}  &  &  \\hbox{log $\\epsilon$(X)} &  \\\\\n\\hbox{} & \\hbox{} & \\multispan4  Sun && \\multispan2 \n  \\hbox{\\rm Arcturus}  \n&  &  \\hbox{\\rm $\\mu$ Leo} &  \\\\\n  \\cline{3-6}  \\cline{8-9} \\cline{11-11}  \\\\\n&  & (1) & (2)  & (3) & (4) & & (5)  & (6)& & (7) &  \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\noalign{\\hrule\\vskip 0.1cm}\n\\hbox{Fe} & 26  &~7.67 &  ~7.50  & ~7.50  & ~7.46 & &6.98 \n&6.95 & & 7.80 &  \\\\\n%\\hbox{Fe} & 26  &~7.67 &  ~7.50  & ~7.50  & ~7.46 & &6.98(-0.52) \n%&6.95 (-0.54) & 7.80 &  \\\\\n%\\hbox{Cu} & 29  &~4.21 &  ~4.21  & ~4.19 & ~4.27 &  & &-- & ... & \\\\\n\\hbox{Zn} & 30  &~4.60 &  ~4.60  & ~4.56 & ~4.65 & & 4.26 & 4.06 & & 4.80 &  \\\\\n\\noalign{\\vskip 0.1cm}\n\\hline\n\\noalign{\\smallskip}\n\\end{array}$$\n\nAbundance analysis\n==================\n\nElemental abundances were obtained through line-by-line spectrum synthesis calculations, carried out using the code described in Barbuy et al. (2003) and Coelho et al. (2005). The molecular lines present in the region, namely the CN B$^2$$\\Sigma$-X$^2$$\\Sigma$ blue system, CN A$^2$$\\Pi$-X$^2$$\\Sigma$ red system, C$_2$ Swan A$^3$$\\Pi$-X$^3$$\\Pi$, MgH A$^3$$\\Pi$-X$^3$$\\Sigma^{+}$, and TiO A$^3$$\\Phi$-X$^3$$\\Delta$ $\\gamma$ and B$^3$$\\Pi$-X$^3$$\\Delta$ $\\gamma$\u2019 systems were taken into account. The atmospheric models were obtained by interpolation in the grid of MARCS LTE models (Gustafsson et al. 2008). The stellar parameters, reported in Table \\[cncorrected\\], were adopted from the detailed analyses by Zoccali et al. (2006) and Lecureur et al. (2007) for 43 bulge field red giants. Another 13 field red clump stars were analysed by Hill et al. (2011) based on both the UVES and the GIRAFFE spectra. We adopted the parameters derived from the UVES spectra, which were not given in Hill et al. (2011) but already reported in Barbuy et al. (2013).\n\nC, N, O abundances and blending CN lines\n----------------------------------------\n\nThe 6362.339 [\u00c5]{} line is blended with the CN lines reported in Table \\[cnlines\\], corresponding to the laboratory measurements by Davis & Phillips (1963). We initially adopted the C, N, and O abundances derived in Lecureur et al. (2007). Based on the observed profile of the 6362 [\u00c5]{} line, however, we realized that in some cases the CN blend was overestimated, producing a spurious asymmetry in the red wing of the 6362 [\u00c5]{} line. In these cases (16 stars marked with \\*\\* in Table \\[cncorrected\\], we recomputed C, N, and O abundances. We finally decided to recompute C, N, and O for all sample stars, given the influence of the CN blend in the Zn abundance derivation from the 6362 [\u00c5]{} line. Oxygen was also derived for the two most metal-poor stars of the sample BW-f4 and BW-f8, for which no previous derivation was available.\n\nThe derivation of C, N, and O abundances was carried out by fitting the Swan C$_2$ (0,1) A$^3$$\\Pi$-X$^3$$\\Pi$ bandhead at 5635 [\u00c5]{}, the red CN (5,1) A$^2$$\\Pi$-X$^2$$\\Sigma$ bandhead at 6332.18 [\u00c5]{}, and the forbidden oxygen \\[OI\\]6300.311 [\u00c5]{} line. Fits to these C,N,O abundance indicators are illustrated in Fig. \\[cno\\] for star B6-b6. These features have to be fitted iteratively, given that a change in the abundance of any of them has an impact on the molecule dissociative equilibrium. A further check of the CN line intensity was done for the asymmetry of the 6362 [\u00c5]{} line.\n\nIn Table \\[cncorrected\\] we report the Lecureur et al. (2007) C, N, O abundances, and in the next column, we assign the letter [**c**]{} when the abundance is confirmed; otherwise, we give the newly revised abundances. A few comments on individual stars are given in the last column of Table \\[cncorrected\\]: Telluric means that such lines mask the \\[OI\\]6300 and \\[OI\\]6363 lines, thus preventing any possibility of deriving its oxygen abundance, such as in BL-7; by the comment \u201cCN-strong\u201d, we mean that CN lines that are too strong blend the 6362 [\u00c5]{} line, leading to this line being discarded in 15 stars: B6-b2, B6-f1, B6-f8, BW-b5, BW-f1, BW-f7, B3-b3, B3-b5, B3-b7, B3-f5, BWc-2, BWc-3, BWc-5, BWc-6, and BWc-8.\n\nNo Zn lines were useful for stars B3-b3, B3-f5, and BWc-2, but they were kept in the line list for reporting their C, N, O abundance derivation. This means that we derived Zn abundances for 53 (and not 56) stars. In about a third of the sample stars, an asymmetry due to the CN lines was clear, permitting a confirmation of the CNO abundances derived, such as for the star B3-b1.\n\nThese corrected C, N, O abundances were adopted here for the purpose of the present paper, which is to correct the derivation of the Zn abundances. A more thorough discussion of these revised oxygen abundances will be deferred to elsewhere. The Zn abundances were recomputed for the stars where the CNO abundances were revised, taking into account the newly derived C, N, O abundances, given in Column 11 of Table \\[cncorrected\\], and the final Zn abundances are reported in Column 14.\n\n$$\\begin{array}{lccrrrr}\n\\hline\\hline\n\\noalign{\\smallskip}\n\\hbox{v',v''} & \\hbox{$\\lambda$({\\rm \\AA})} & \\hbox{Branch}&& \n  \\hbox{J}    &   \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\noalign{\\hrule\\vskip 0.1cm}\n\\hbox{(4,0)} & 6362.743 &R1 & 46  & \\\\\n\\hbox{(4,0} & 6362.450  &Q1 & 39  & \\\\\n\\hbox{(10,5)} & 6362.765  &P2 & 21 &  \\\\\n\\hbox{(10,5)} & 6362.548  &P2 & 10 &  \\\\\n\\noalign{\\vskip 0.1cm}\n\\hline\n\\noalign{\\smallskip}\n\\end{array}$$\n\nFor the fit of the 4810 [\u00c5]{} line, we adopted a procedure of balancing the continua points at the pseudo-continuum regions 4808.25, 4811.55, and 4812.6, with more weight for the 4811.55 [\u00c5]{} that appears to be a better defined continuum. For the 6362.34 [\u00c5]{} line, with a few exceptions, the local continuum is affected by the depression due to the line and, despite its being considered in the calculations, some extra depression remains. We gave priority to the continuum in the region 6361.5-6362.1 [\u00c5]{} on the blue side of the Zn line. Examples of fitting ZnI lines in sample spectra are given in Figs. \\[b6f1zn\\], \\[b6f8zn\\], \\[bwf8zn\\], and \\[bwc4zn\\] for stars B6-f1, B6-f8, BW-f8, and BWc-4. In these figures we also show the calculations with no Zn (synthetic spectra in blue), showing that the 4810 [\u00c5]{} line is free of blends, and the 6362 [\u00c5]{} line shows a blend with CN lines. These figures show examples of the cases of no inteference by CN lines (BW-f8), a moderate or negligible presence of CN lines (BWc-4), and strong contaminating CN lines (B6-f1, B6-f8), leading us not to consider the line for such stars. Star B6-f8 shows a detectable good fit to the CN line on the red side of the 6362 [\u00c5]{} line, but this indicator was discarded as well.\n\nAs regards non-LTE corrections, Takeda et al. (2005) have computed non-LTE effects on both the 4810 and 6362 [\u00c5]{} lines used here. The corrections are below 0.1 dex for the metal-poor stars and below 0.05 for stars with \\[Fe/H\\] $>$ $-$1.0. Consequently, in the present work, we do not correct the abundances for this effect.\n\nErrors\n------\n\nWe have adopted uncertainties in the atmospheric parameters of $\\pm$ 150 K for effective temperature, $\\pm$ 0.20 for surface gravity, and $\\pm$ 0.10 in \\[Fe/H\\] and $\\pm$ 0.10 kms$^{-1}$ for microturbulent velocity, as explained in Barbuy et al. (2013).\n\nThe errors in \\[Zn/Fe\\] are computed by using model atmospheres with parameters changed by these uncertainties, applied to the stars B6-f1 and BW-f8. The errors are estimated from the differences in \\[Zn/Fe\\], derived using the modified models relative to the adopted model. We adopt a mean of the uncertainty on the 4810 and 6362 [\u00c5]{} lines. These uncertainties are given in Table \\[errors2\\] for star B6-f1, for which the 6362 line is strongly affected by the CN blending, and BW-f8 with negligible CN blending. The higher sensitivity to effective temperature in B6-f1 is due to the CN lines blending the 6362.339 [\u00c5]{} line: it is the CN lines that are more sensitive to temperature. Since the stellar parameters are covariant, the sum of these errors is an upper limit. On the other hand, a continuum location uncertainty introduces a further uncertainty in \\[Zn/Fe\\] of $\\pm$0.1.\n\n$$\\begin{array}{lccccc}\n\\hline\\hline\n\\noalign{\\smallskip}\n\\hbox{Star} & \\hbox{$\\Delta$T$_{\\rm eff}$} & \\hbox{$\\Delta$$\\log$ g}\n& \\hbox{$\\Delta$[Fe/H]} & \\hbox{$\\Delta$v$_{t}$} & \\hbox{($\\sum$x$^{2}$)$^{1/2}$} \\\\\n& \\hbox{(+150 K)} & \\hbox{(+0.2)} & \\hbox{(+0.1)} & \\hbox{(+0.1 kms$^{-1}$})&\\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\noalign{\\vskip 0.1cm}\n\\noalign{\\hrule\\vskip 0.1cm}\n\\hbox{B6-f1}  &  $-$0.17  &   +0.05 & $-$0.06 & $-$0.01 & 0.19 \\\\\n\\hbox{BW-f8}  &     0.00  &   $-$0.05 & +0.10 & $-$0.02 & 0.11 \\\\\n\\noalign{\\vskip 0.1cm}\n\\hline\n\\noalign{\\smallskip}\n\\end{array}$$\n\nResults\n=======\n\nThe derived Zn abundances are given in Table \\[cncorrected\\]. In Fig. \\[plotzn\\], the \\[Zn/Fe\\] vs. \\[Fe/H\\] behaviour is shown for the present sample, where different symbols represent the four different fields, and the red clump stars. In Fig. \\[plotzn\\]a, we overplotted the Zn abundances of dwarf bulge stars observed thanks to microlensing magnification, by Bensby et al. (2013). There is good agreement between the present results and those by Bensby et al. at metallicities $-$1.4 $<$ \\[Fe/H\\] $<$ 0.0. On the other hand, for our metal-rich giants with \\[Fe/H\\] $>$ 0.0, we find a wide spread of $-$0.6 $<$ \\[Zn/Fe\\] $<$ +0.15 and a decreasing trend with metallicity. Further comments on the Bensby et al. sample of dwarf bulge stars are given in Sect. 4.3. In Fig. \\[plotzn\\]b we show the same stars as in Fig. \\[plotzn\\]a, adding metal-poor halo stars analysed by Cayrel et al. (2004) and halo and thick disk stars results from Ishigaki et al. (2013) and Nissen & Schuster (2011). Ishigaki et al. results seem to show lower \\[Zn/Fe\\] in the -3.0 $<$ \\[Fe/H $<$ $-$1.0 metallicity range with respect to the Nissen & Schuster and Cayrel et al. results.\n\nComparison with thick disk Zn abundances\n----------------------------------------\n\nIn Fig. \\[plotzn1\\] we compare the present bulge Zn abundances with those for thick disk stars in the literature. The results by Nissen & Schuster (2011) are maintained in all panels in Fig. \\[plotzn1\\], showing good agreement with other samples of thick disk stars. Given the narrow spread and the metallicity range covered by these data, we adopted the Nissen & Schuster (2011) results as a reference for the thick disk. Panels a, b, c, d show the Zn abundances derived respectively by Bensby et al. (2014), Mishenina et al. (2011), Prochaska et al. (2000), and Reddy et al. (2006). All thick disk samples show very good agreement with the present data, in the metallicity range of thick disk stars (-1.2 $\\simless$ \\[Fe/H\\] $\\simless$ 0.0). The Bensby et al. (2014) thick disk stars were selected based on kinematical criteria TD/D $>$ 2, with these numbers corresponding to a probability defined in Bensby et al. (2003). Thick disk stars from Bensby et al. (2014) displayed in Fig. \\[plotzn1\\]a include an old metal-poor thick disk, whereas the more metal-rich thick disk stars are younger ($<$8Gyr). The Bensby et al. old metal-poor thick disk stars, reaching down to \\[Fe/H\\]$\\approx$-2.0, show a moderate enhancement of \\[Zn/Fe\\]$\\approx$+0.15, compatible with our results for the metal-poor bulge stars. Their more metal-rich thick disk component with \\[Fe/H\\] $>$ $-$0.3 show Zn-to-iron essentially solar, whereas our sample includes stars with low Zn-to-iron. The metallicity limits for the thick disk often give \\[Fe/H\\]=$-$0.3 as an upper limit; however, Bensby et al. classify some of their more metal-rich stars as probable thick-disk members, based on kinematics, as explained above, and they tend to be younger than $\\simless$ 8 Gyr. Comments on the Bensby et al. sample consisting of dwarfs are given below in Sect. 4.3.\n\nIn the metallicity range occupied by the old thick disk stars, the bulge appears compatible with the thick disk abundances for the samples from Bensby et al. (2014), Mishenina et al. (2011), Prochaska et al. (2000), and Reddy et al. (2006). A distinction between the present data and thick disk stars may be present for the metal-rich stars, where most stars in the present sample show Zn under-abundances, and the Bensby et al. data show \\[Zn/Fe\\]$\\sim$0.0. It is important to note that there are not as many such old metal-rich stars in the thick disk population studied by Bensby et al. (2013).\n\nComparison with thin disk Zn abundances\n---------------------------------------\n\nIn Fig. \\[plotzn3\\] we compare the present Zn abundances with those for thin disk stars derived by Bensby et al. (2014), and Allende-Prieto et al. (2004). The data are also compared with the metal-rich dwarf stars from Pomp\u00e9ia (2003). Thick-disk stars by Nissen & Schuster (2011) are also kept in all plots, as a reference for the thick disk. The thin disk Zn abundances from Bensby et al. (2014) are consistent with a mean \\[Zn/Fe\\]$\\sim$0.0, with a fraction of stars showing higher \\[Zn/Fe\\] up to $\\sim$+0.4. The Allende-Prieto et al. (2004) stars show a mean \\[Zn/Fe\\]$>$0 and a clear and an increasing trend with metallicity. The dwarf stars studied by Pomp\u00e9ia (2003) overlap with the present bulge stars data, except for the more metal-rich ones, which show no under-abundance. This difference may indicate that the Pomp\u00e9ia (2003) sample would be rather old thin disk stars and not bulge-like stars, as concluded in Trevisan et al. (2011) and again discussed in Trevisan & Barbuy (2014).\n\nInspecting differences with literature samples\n----------------------------------------------\n\nThe comparisons with literature data in the previous sections include several samples of dwarf stars: the microlensed dwarf bulge stars from Bensby et al. (2013), the thin and thick disk stars from Bensby et al. (2014), the thin disk stars from Allende-Prieto et al. (2004) and Pomp\u00e9ia (2003), and the halo and thick disk stars from Nissen & Schuster (2011). Bensby et al. (2013, 2014) and Allende-Prieto et al. (2004) used the 4810.5 and 6362.2 [\u00c5]{} lines, and Nissen & Schuster (2011) used the 4722.1 and 4810.5 [\u00c5]{} lines. (Pomp\u00e9ia (2003) does not report the lines used.)\n\nThe lines used are, in most cases, shared with the present work. The dwarfs are all hotter than the present sample, such that their samples have spectra with essentially no molecular lines. This could explain any difference in abundances from the 6362.2 [\u00c5]{} line, that has a blend with CN lines affecting our determinations. On the other hand, the 4810.5 [\u00c5]{} line has no molecular lines, and it shows the same Zn deficiencies as the 6362.2 [\u00c5]{} line for many of the most metal-rich giant stars, but differently from the dwarfs.\n\nThe differences for metal-rich stars could stem from other blends, possibly unknown. At present we cannot identify a reason not to confirm the Zn-under-abundance in some of the present sample of metal-rich bulge giants. The flat or decreasing \\[Zn/Fe\\] at high metallicities has the important consequence of indicating (or not) the contribution of SNIa (see Sect. 4.5).\n\nComparison with damped Lyman-$\\alpha$ systems\n---------------------------------------------\n\nA comparison of the present bulge data with DLAs is possible thanks to the availability of a large data base of Zn abundances over a wide range of metallicities for these objects. Such a comparison can shed light not only on the nature of DLAs, but also on the formation process of bulges.\n\nThe high H I column densities that characterize the DLAs indicate that they belong to environments with relatively high gas density, which could give rise to components of present day massive galaxies. The presence of metals in their spectra is also a clue that star formation has already taken place at significant rates inside them or in their neighbourhood. In this way, two favoured candidates for sites hosting DLAs are proto-disks and young star-forming spheroids. Lanfranchi and Fria\u00e7a (2003) have investigated the evolution of the metallicity of DLAs in order to set constraints on the nature of these objects and also to shed light on the connections between DLAs and galaxy formation. The comparison of the observed trends of \\[$\\alpha$/Fe\\] and \\[N/$\\alpha$\\] with the predictions of a chemo-dynamical model is consistent with a scenario in which the DLA population is dominated by disks at low redshifts and by young spheroids at high redshifts. With the data base available for that work, nearly the totality of DLAs at $z > 1.5$ are explained as spheroid systems formed in the redshift range 1.7$<$z$<$4.5, with masses between 10$^9$ and 10$^{10}$ M$_{\\odot}$ and typical specific star formation rates of 1 to 3 Gyr$^{-1}$, where the definition of specific star formation is given in the footnote below[^6]. These objects might have taken part in merger processes that then lead to bulges. It is possible that the most massive of them, those with N$_{\\rm HI} \\approx 10^{22}$ cm$^{-2}$, could be proto-bulges.\n\nZinc has played a special role in estimations of the metallicity of DLAs (Pettini et al. 1990, 1994, 1997; Akerman et al. 2005). That Zn is non-refractory guarantees that it is not heavily depleted in the ISM. On the observational side, Zn has two strong transitions at 2026 and 2062 [\u00c5]{}, which almost always lie outside the Ly-$\\alpha$ forest and are rarely saturated owing to their low oscillator strengths and low Zn abundances. Given that Zn is only a trace element that contributes $\\approx 10^{-4}$ of the mass density for the heavy elements, and because of the weakness of the Zn transitions, it is impossible to measure its abundance in low-metallicity DLAs, and besides that, the large rest-frame wavelengths of the transitions at 2026 and 2062 [\u00c5]{} make them difficult to measure at high redshift.\n\nStudies of Zn in damped Lyman-$\\alpha$ systems with different redshifts have shown that \\[Zn/H\\] tends to show no correlation with distance and age. Akerman et al. (2005), for example, suggest that \\[Zn/H\\] in DLAs shows a constant value of \\[Zn/H\\] = $-$0.88$\\pm$0.21 in the redshift range 1.86 $<$ z$_{abs}$ $<$ 3.45. Comparisons of stellar data with abundances in DLAs were carried out by several authors (Pettini et al. 1994; Pettini et al 1997a; Prochaska et al. 2000; Wolfe et al. 2005; Nissen et al. 2007).\n\nIn Figs. \\[DLA\\]a,b, we plot \\[Zn/H\\] vs. \\[Fe/H\\] for the present sample and for the DLA samples by Akerman et al. (2005), Cooke et al. (2011, 2013), Kulkarni et al. (2007), and Vladilo et al. (2011). For the Akerman et al. (2005) data, we did not consider the cases with upper limits alone. For the DLA samples of Kulkarni et al. (2007) and Vladilo et al. (2011), a tranformation \\[Fe/H\\] vs. redshift from the literature (Pei & Fall 1995) was adopted. Other such relations are given by Cen & Ostriker (1999) and Madau & Pozzetti (2000), for example. This figure shows that a comparison of Zn in DLAs and in bulge stars benefit from an overlap only at metallicities $-$1.5 $<$ \\[Fe/H\\] $<$ $-$0.1, with most sample bulge stars being more metal-rich than this.\n\n### Selected DLAs with Fe abundance measurements\n\n$$\\begin{array}{lccr@{}r@{}r@{}r@{}r@{}}\n\\hline\\hline\n\\noalign{\\smallskip}\n\\hbox{QSO} & \\hbox{z$_{\\rm abs}$} & \\hbox{[Zn/H]}\n& \\hbox{[Fe/H]} & \\phantom{-}\\hbox{[Fe/H]$_{\\rm c}$} & \n\\phantom{-}\\hbox{[Zn/Fe]} \n& \\phantom{-}\\hbox{[Zn/Fe]$_{\\rm c}$}    &   \\\\\n\\noalign{\\smallskip}\n\\hline\n\\noalign{\\smallskip}\n\\noalign{\\hrule\\vskip 0.1cm}\n\\noalign{\\smallskip}\n\\noalign{Akerman et al. 2005}\n\\noalign{\\smallskip}    \n\\noalign{\\hrule\\vskip 0.1cm}\n\\noalign{\\smallskip}\n\\hbox{B0438-436} & 2.34736 & $-$0.68 & $-$1.30& $-$0.87& 0.62 &  0.24&\\\\        \n\\hbox{B0458-020} & 2.03950 & $-$1.15 & $-$1.61& $-$1.34& 0.46 &  0.21&\\\\        \n\\hbox{B0528-250} & 2.14100 & $-$1.45 & $-$1.57& $-$1.34& 0.12 & $-$0.09&\\\\      \n\\hbox{B0528-250} & 2.81100 & $-$0.47 & $-$1.11& $-$0.57& 0.64 &  0.18&\\\\        \n\\hbox{B1055-301} & 1.90350 & $-$1.26 & $-$1.57& $-$1.32& 0.31 &  0.08&\\\\        \n\\hbox{B1230-101} & 1.93136 & $-$0.17 & $-$0.63&  0.10& 0.46 & $-$0.15&\\\\        \n\\hbox{B2314-409} & 1.85730 & $-$1.01 & $-$1.29& $-$1.01& 0.28 &  0.02&\\\\        \n\\noalign{\\smallskip}\n\\noalign{\\hrule\\vskip 0.1cm}\n\\noalign{\\smallskip}\n\\noalign{Vladilo et al. 2011}\n\\noalign{\\smallskip}\n\\noalign{\\hrule\\vskip 0.1cm}\n\\noalign{\\smallskip}\n\\hbox{0216+080}  & 2.2930 & $-$0.63 &$-$1.12&$-$0.67 &0.49 &0.10 &\\\\\n\\hbox{2206-199A} & 1.9200 & $-$0.33 &$-$0.87&$-$0.25 &0.54 &0.02 &\\\\\n%\\hbox{2359-0216} & 2.0900 &\n\\noalign{\\vskip 0.1cm}\n\\hline\n\\noalign{\\smallskip}\n\\end{array}$$\n\nFor a few DLAs studied by Akerman et al. (2005) and Cooke et al. (2013), iron abundances (\\[Fe/H\\]) were measured, in addition to zinc abundances. We selected these objects to analyse the relation between the abundances of iron and zinc in DLAs. Vladilo et al. (2011) quote only \\[Zn/H\\] values for most of their DLA sample. However, they were able to derive, based on archive UVES spectra, the column densities for two DLAs of their + sample , 0216+080 at $z_{abs}$=2.2930, and 2206-199A at $z_{abs}$=1.9200, which we also include in our list of measured Fe abundances in DLAs. An important concern is that Fe is a refractory element, and its depletion from the gas phase (observed through the absorption lines) into dust has to be taken into account. Therefore we applied a dust correction to the \\[Fe/H\\] values of the Akerman et al. (2005) and Vladilo et al. (2011) subsamples. No dust correction was needed for the Cooke et al. data, because their systems have \\[Fe/H\\] $<$ $-$2, and at these low metallicities dust depletion becomes negligible (Pettini et al. 1997a). The systems for which we apply dust-depletion corrections for Fe are listed in Table \\[DLA-fe\\].\n\nApplying dust corrections to the chemical abundances of a gas system, either a DLA or the Galactic ISM, is a very complex task. We considered a variety of dust correction models, following Lanfranchi & Fria\u00e7a (2003), to obtain the depletion $\\delta_X$ of a given element X in the DLA, thus recovering the intrinsic abundance \\[X/H\\]$_i$ of that element from the observed one \\[X/H\\]: $$\\delta_X \\;=\\; [X/H] \\; -\\; [X/H]_i\n.$$\n\nIn the Galactic ISM, the suprasolar \\[Zn/Fe\\], \\[Zn/Cr\\], and \\[Zn/Si\\] ratios provide evidence of dust depletion by the refractories Fe, Cr, and Si. In addition, the dust depletion pattern depends on the type of environment through which the line of sight passes. Savage & Sembach (1996) considered four different dust depletion patterns corresponding to four types of Galactic ISM environments: (1) cool clouds in the Galactic disk (CD), (2) warm clouds in the disk (WD), (3) disk plus halo clouds (WHD), and (4) warm halo clouds (WH). On the other hand, we should consider a nucleo-synthetic contribution to the Zn/Fe enhancement.\n\nIn our dust corrections to the Fe abundances, we followed Lanfranchi & Fria\u00e7a (2003). We applied four distinct dust models to 16 DLAs of Lanfranchi & Fria\u00e7a (2003) with small uncertainties for the abundances of zinc, iron, and one more refractory element (Cr, Si, or Mg). We assumed a range for the intrinsic \\[Zn/Fe\\] ratio of 0.0 (solar), 0.1, 0.2, and 0.3. The suprasolar values of \\[Zn/Fe\\] are suggested by determinations of low metallicity objects and could be checked [**a posteriori**]{}. We take the WD and the WH as reference media. It is not appropriate to consider the cool clouds as a reference for the DLA environment, because the cool clouds exhibit levels of dust depletion (\\[Fe/Zn\\], \\[Cr/Zn\\], \\[Si/Zn\\]) that are much higher than those observed in DLAs, and the molecular hydrogen fractions are typically low in DLAs, in contrast to the large number of molecules in the cool clouds. In addition, by using models of chemical evolution, Vladilo et al. (2011) have shown that, for most of the DLAs in their sample, the iron depletion is within the range delimited by the values typical of WD and WH Galactic clouds.\n\nThe degree of depletion should increase with metallicity. We cannot use \\[Fe/H\\] directly to obtain the metallicity because iron itself is highly depleted into dust. Therefore, \\[Zn/H\\] is taken as a metallicity indicator because it is a volatile element. We derive a correction for dust depletion as a function of metallicity from the trend of the iron depletion predicted by the dust models with respect to the observed \\[Zn/H\\] values, applying the four distinct dust models to 16 DLAs. Then, a quadratic fit to the resulting total of 64 points gives the dust correction as a function of \\[Zn/H\\]. The inserted plot Fig. \\[DLAfe\\] shows the trend of the dust depletion correction for the abundance of iron with the observed \\[Zn/H\\] derived from applying our dust models to the DLAs selected from Lanfranchi & Fria\u00e7a (2003). The curve is the quadratic fit to the trend that is used for dust correction.\n\nFinally, although less sensitive to dust depletion, the zinc abundance is also corrected in the way described in Lanfranchi & Fria\u00e7a (2003). Figure \\[DLAfe\\] shows \\[Zn/Fe\\] vs. \\[Fe/H\\] for the DLA samples with determination of both \\[Zn/H\\] and \\[Fe/H\\], compared with the present results for bulge stars, thick disk, and halo stars from Nissen & Schuster (2011) and Ishigaki et al. (2011) and for metal-poor halo stars from Cayrel et al. (2004). The Akerman et al. (2005) and Vladilo et al. (2011) data have been corrected for dust depletion as explained above.\n\nThe Zn vs. Fe enrichment indicated by the DLA data from Akerman et al. (2005) and Vladilo et al. (2011) shows an overlap not only with thick disk and halo stars data from Nissen & Schuster (2011) and Ishigaki et al. (2013), but also with the pattern of the bulge stars, including a few subsolar \\[Zn/Fe\\] values.\n\nChemical evolution models of zinc in massive spheroids\n------------------------------------------------------\n\nSince bulge stars are probes of bulge formation and evolution, Fig. \\[amancioznfe\\] shows the comparison of the \\[Zn/Fe\\] vs. \\[Fe/H\\] in bulge stars as derived in the present paper with a chemo-dynamical model describing a classical bulge. The computed models assume a specific star formation rate of $\\nu_{\\rm SF}=$ 3 Gyr$^{-1}$, a baryonic mass of 2$\\times$10$^9$ M$_{\\odot}$, and a dark halo mass $M_{H}$= 1.3$\\times$10$^{10}$ M$_{\\odot}$ (total mass of 1.5$\\times$10$^{10}$ M$_{\\odot}$, thus reproducing the cosmological proportion of baryonic mass, $\\Omega_b = 0.04$, to total matter mass, $\\Omega_m = 0.3$). In the present calculations, we adopt an $\\Omega_m = 0.3$, $\\Omega_{\\Lambda}= 0.7$, $H_0=70$ km s$^{-1}$ Mpc$^{-1}$ cosmology, with the corresponding age of the universe of 13.47 Gyr. (An age of 13.799$\\pm$0.038 Gyr is the most recent and updated value from the Planck satellite data as given by the Planck collaboration: Adam et al. (2015).)\n\nThe evolution of the model is followed until 13 Gyr, and it gives the evolution of the average Zn chemical abundance of the stellar population for several radii. As can be seen in Fig. \\[amancioznfe\\], its end point falls in the locus of the data for the present-day stars in the bulge of the Galaxy. The trend towards decreasing \\[Zn/Fe\\] ratio with increasing \\[Fe/H\\] for higher metallicities is reproduced well by the model. This is because although the bulge is formed rapidly in the classical scenario, the star formation goes on for a few Gyr. In the present case, the stellar mass is built up during at least $\\approx 3$ Gyr, which allows the contribution of type Ia supernovae (SNIa) to be relevant, increasing the Fe abundance. The ejecta of SNIa exhibit a very low \\[Zn/Fe\\] ratio. The present model uses the SNIa yields of Iwamoto et al (1999), \\[Zn/Fe\\] $\\approx -1.2$ for a zero initial metallicity, and \\[Zn/Fe\\] $\\approx -1.6$ for a solar initial metallicity. Therefore, as a result of the continuing star formation, the classical bulge model predicts subsolar \\[Zn/Fe\\] ratios for higher metallicities, as observed in the present sample of bulge stars.\n\nThe nucleo-synthetic nature of Zn is complex; it is neither an $\\alpha$-element nor a Fe peak element. Theoretical work on the nucleo-synthesis of Zn often predicts that it may originate in massive stars, but a complete network for production of Zn is not yet available. For instance, the classical core-collapse SN II yield calculations of Woosley and Weaver (1995) are known to underestimate the Zn abundance. One solution to this problem could be $ \\alpha$-rich freeze-out neutrino winds, as predicted by Woosley and Hoffman (1992). On the other hand, Umeda & Nomoto (2002) have produced nucleo-synthesis calculations in core-collapse explosions of massive low metallicity stars that do show large \\[Zn/Fe\\] for deeper mass cuts, smaller neutron excesses, and higher explosion energies. In the last case, the supernova would be classified as a hypernova as defined by Nomoto et al. (2013, and references therein). Therefore, in the nucleosynthesis input of our chemical calculations, we consider the core-collapse SN II models of Woosley and Weaver (1995), but, for lower metallicities, we use the results of high explosion-energy hypernovae (Umeda & Nomoto 2002; 2003; 2005; Nomoto et al. 2006; 2013).\n\nIn summary, most of the bulge star data obtained in this work can be explained by a classical scenario of bulge formation. The trend towards a decreasing \\[Zn/Fe\\] ratio with increasing \\[Fe/H\\] seems to be reproduced well by the model. In addition, the model also accounts for the abundances of the halo stars, which can be thought of as a relic of the same galaxy formation sequence of events that gave rise to the bulge. The high \\[Zn/Fe\\] ratio of our calculations results from including hypernovae. The sensitivity of our results to the nucleo-synthesis prescription is shown in the lower panel of \\[amancioznfe\\], in which the Woosley & Weaver (1995) yields are used at low metallicities. The resulting \\[Zn/Fe\\] is very low, which favours hypernovae at low metallicities, as given in the upper panel. Finally, the decreasing trend of \\[Zn/Fe\\] at high metallicities is due to Fe enrichment from SNIa.\n\nZn and alpha elements\n---------------------\n\nThe Zn enhancement in metal-poor stars suggests that \\[Zn/Fe\\] behaves like alpha elements. For this reason, in Fig. \\[alpha\\] we compare the Zn abundances with those for the alpha elements O, Mg, Si, Ca, and Ti, derived in Lecureur et al. (2007), Zoccali et al. (2006), and Gonzalez et al. (2011). For oxygen the revised values given in Table \\[cncorrected\\] for selected stars are plotted instead of the previous values. The trend shown by Zn appears similar to that of the alpha elements and, more strikingly, of oxygen and calcium. The low \\[Zn/Fe\\] for high metallicity stars is compatible with the oxygen abundances.\n\nThe variation in Zn in lockstep with alpha elements is made evident further in Fig. \\[zno\\], where \\[Zn/O\\] shows essentially no trend with \\[Fe/H\\] or with \\[O/H\\].\n\nConclusions\n===========\n\nThe iron-peak elements Sc, Mn, Cu, and Zn show a different chemical enrichment pattern than do the even-Z iron-peak elements Fe and Ni. In Barbuy et al. (2013), we confirmed that Mn behaves as a secondary element with low \\[Mn/Fe\\] in metal-poor stars, by increasing with increasing metallicity. In the present work we show that, in the metal-rich bulge stars, Zn-to-Fe decreases with increasing metallicity, complementing the long established high Zn abundances in metal-poor stars (e.g. Sneden et al. 1991; Nissen & Schuster 2011).\n\nOur main comments on the comparison between the data points and models are the following: a) The most metal-rich bulge dwarfs from Bensby et al. (2011) show a constant \\[Zn/Fe\\], which implies that there is no contribution of SNe type I in the bulge, whereas a decrease in \\[Zn/Fe\\] in the present sample of giants, as derived here, implies that there is enrichment from Type I, SNe as predicted by the models. The bulge stars are a complex mix of stellar populations of different ages and formation processes. b) The high \\[Zn/Fe\\] in very metal-poor stars favours enrichment from hypernovae, as defined by Nomoto et al. (2013 and references therein) acting at these low metallicities. It is interesting to point out that the hypernovae as defined by Nomoto et al. (2013 and references therein) might be related to the spinstars as defined by Frischknecht et al. (2012) and Meynet et al. (2006) and discussed in terms of early enrichment of the bulge in Chiappini et al. (2011). c) The drop in \\[Zn/Fe\\] for moderately metal-poor stars ($-$2.2 $\\simless$ \\[Fe/H\\] $\\simless$ $-$1.6) corresponds to the normal metal-poor supernovae, here using the yields from Woosley & Weaver (1995).\n\nFor the DLA systems with measured Fe abundances, it was crucial to correct for dust depletion. The Zn abundances were also corrected for dust depletion, even if these corrections are smaller. The chemical evolution models predict subsolar \\[Zn/Fe\\] values at relatively high metallicities (\\[FeH\\]$\\simgreat$-1.0), as confirmed for a few systems.\n\n[l@r@r@r@r@r@r@r@r@r@r@r@r@r@r@r@r@r@]{} Star & OGLE n$^{\\circ}$ & $V$ & $\\phantom{-}$$\\phantom{-}$T$_{\\rm eff}$ & $\\phantom{-}$log\u00a0$g$ & $\\phantom{-}$v$_{\\rm t}$ & \\[Fe/H\\] & \\[C/Fe\\] & \\[N/Fe\\] & \\[O/Fe\\] & \\[C,N,O/Fe\\]$_{\\rm corr}$ & \\[Zn1/Fe\\] & \\[Zn2/Fe\\] & \\[Zn/Fe\\] & Comments\\\nB6-b1& 29280c3 &16.14& 4400 & 1.8 & 1.6 & 0.07 &$-$0.16 & 0.39 & 0.04 & c,c,c & 0.00 & $-$0.40 & $-$0.20 &\\\nB6-b2& 83500c6 &16.40&4200 &1.5 &1.4 &$-$0.01 & \u2014 & \u2014 & \u2014 & $-$0.2,0.6,$-$0.1 & $-$0.15 & \u2014 & $-$0.15 & CN-strong\\\nB6-b3& 31220c2 &16.09& 4700 & 2.0 & 1.6 & 0.10 & $-$0.16& 0.11 & 0.19 & c,0.3,$-$0.15 & $-$0.25 & $-$0.30 & $-$0.27 &\\\nB6-b4& 60208c7 &16.12& 4400 & 1.9 & 1.7 & $-$0.41 & $-$0.14& 0.53 & 0.53 & c,c,0.3 & 0.00 & 0.00 & 0.00 &\\\nB6-b5& 31090c2 &16.09&4600 & 1.9 & 1.8 & $-$0.37 & $-$0.11& 0.56 & 0.33 & c,0.3,0.15 & +0.20 & 0.00 & +0.10 &\\\nB6-b6\\*\\*& 77743c7 &16.09&4600 & 1.9 & 1.8 & 0.11 & $-$0.03& 0.57 & 0.01 & $\\leq$$-$0.1,0.7,0.0 & $-$0.30 & $-$0.50 & $-$0.40 &\\\nB6-b8& 108051c7 &16.29& 4100 & 1.6 & 1.3 & 0.03 & 0.08& 0.05 & 0.10 & 0,c,0 & $-$0.15 & 0.00 & $-$0.08 &\\\nB6-f1\\*\\*& 23017c3 &15.96& 4200 & 1.6 & 1.5 & $-$0.01 & 0.05& 0.55 & 0.18 & 0.0,0.35,0.07 & $-$0.30 & \u2014 & $-$0.30 & CN-strong\\\nB6-f2& 90337c7 &15.91& 4700 & 1.7 & 1.5 & $-$0.51 & $-$0.04& 0.56 & 0.39 & 0,0.05,0.18 & +0.05 & +0.05 & +0.05 &\\\nB6-f3& 21259c2 &15.71& 4800 & 1.9 & 1.3 & $-$0.29 & $-$0.09& 0.53 & 0.18 & c,0.3,0.05 & +0.20 & 0.00 & +0.10 &\\\nB6-f5\\*\\*& 33058c2 &15.90& 4500 & 1.8 & 1.4 & $-$0.37 & 0.37& 0.53 &\u2014& $-$0.1,0.15,0.0 & 0.00 & +0.2 & +0.10 &\\\nB6-f7\\*\\* & 100047c6 &15.95& 4300 & 1.7& 1.6 & $-$0.42 & 0.42& 0.57 &\u2014& 0,0.3,0.1 & $-$0.30 & 0.00 & $-$0.15 &\\\nB6-f8 & 11653c3 &15.65&4900 & 1.8 & 1.6 & 0.04 & $-$0.11& 0.51 &$-$0.17 & c,0.35,$-$0.3 & $-$0.60 & \u2014 & $-$0.60 & CN-strong\\\nBW-b2 & 214192 &16.58& 4300 & 1.9 & 1.5 & 0.22 & 0.05& 0.25 & 0.23 & $-$0.1,0.1,$-$0.1 & $-$0.30 & 0.00 & $-$0.15 &\\\nBW-b4& 545277 &16.95& 4300 & 1.4 & 1.4 & 0.07 & \u2014 & \u2014 & \u2014 &$-$0.1,0,$-$0.1 & \u2014 & 0.00 & 0.00 &\\\nBW-b5& 82760 &16.64& 4000 & 1.6 & 1.2 & 0.17 & 0.06 & 0.56 & 0.09 & c,0,0 & $-$0.30 & \u2014 & $-$0.30 & CN-strong\\\nBW-b6& 392931 &16.42& 4200 & 1.7 & 1.3 & $-$0.25 & 0.05 & 0.56 & 0.09 &$-$0.3,0.9,0.25 & 0.00 & 0.00 & 0.00 &\\\nBW-b7 & 554694 &16.69& 4200 & 1.4 & 1.2 & 0.10 & \u2014 & \u2014 & \u2014 &$-$0.23,$-$0.1,$-$0.2 & $-$0.30 & $-$0.30 & $-$0.30&\\\nBW-f1 & 433669 &16.14& 4400 & 1.8 & 1.6 & 0.32 & $-$0.26& 0.24 &$-$0.02 & $-$0.2,0.5,$-$0.15 & $-$0.35 & \u2014 & $-$0.35 & CN-strong\\\nBW-f4& 537070 &16.07&4800 & 1.9 & 1.7 & $-$1.21 & 0.04& 0.54 &\u2014 &c,c,0.3 & +0.30 & +0.30 & +0.30 &\\\nBW-f5\\*\\*& 240260 &15.88& 4800 & 1.9 & 1.3 & $-$0.59 & 0.03& 0.53 & 0.31 & c,0.2,0.1 & +0.30 & 0.00 & +0.15 &\\\nBW-f6& 392918 &16.37& 4100 & 1.7 & 1.5 & $-$0.21 & 0.08& 0.58 & 0.46 & c,0.4,0.18 & 0.00 & +0.30 & +0.15 &\\\nBW-f7 & 357480 &16.31& 4400 & 1.9 & 1.7 & 0.11 & $-$0.10& 0.30 &0.03 & $-$0.2,0.6,$-$0.1 & $-$0.20 & \u2014 & $-$0.20 & CN-strong\\\nBW-f8 & 244598 &16.00& 5000 & 2.2 & 1.8 & $-$1.27 & 0.03 & 0.53 & \u2014 & 0,+0.2,+0.5 & +0.30 & +0.30 & +0.30 &\\\nBL-1& 1458c3 & 15.37 & 4500 & 2.1 & 1.5 & $-$0.16 & 0.03 & 0.18 & 0.26 & 0.15,0.4,0.3 & +0.10 & 0.00 & +0.05 &\\\nBL-3& 1859c2 & 15.53 & 4500 & 2.3 & 1.4 & $-$0.03 & $-$0.07 & 0.18 & 0.22 &c,0.3,0.1 & +0.20 & 0.00 & +0.10 &\\\nBL-4\\*\\*& 3328c6 & 14.98 & 4700 & 2.0 & 1.5 & 0.13 & $-$0.04 & 0.41 & 0.02 & $-$0.1,0.3,$-$0.2 & $-$0.30 & $-$0.30 & $-$0.30 &\\\nBL-5\\*\\*& 1932c2 & 15.39 & 4500 & 2.1 & 1.6 & 0.16 & 0.04 & 0.33 & 0.07 & $-$0.02,c,$-$0.05 & $-$0.25 & $-$0.30 & $-$0.27 &\\\nBL-7& 6336c7 & 15.33 & 4700 & 2.4 & 1.4 & $-$0.47 & $-$0.17 & 0.38 & 0.46 & c,c,0.3$^t$ & +0.30 & +0.30 & +0.30 & telluric\\\nB3-b1 & 132160C4 &16.35& 4300 & 1.7 & 1.5 & $-$0.78 &$-$0.10 & 0.45 & 0.55 & 0.1,0.6,0.4 & \u2014 & +0.30 & +0.30 &\\\nB3-b2& 262018C7 &16.63& 4500 & 2.0 & 1.5 & 0.18 &$-$0.13 & 0.42 & 0.12 & c,0,$-$0.1 & $-$0.30 & +0.10 & $-$0.10 &\\\nB3-b3\\*\\*& 90065C3 &16.59& 4400 & 2.0 & 1.5 & 0.18 &$-$0.09 & 0.46 &$-$0.19 & 0,0,$-$0.15 & \u2014 & \u2014 & \u2014 & CN-strong\\\nB3-b4& 215681C6 &16.36& 4500 & 2.1 & 1.7 & 0.17 &$-$0.16 & 0.39 &$-$0.06 &c,c,c & 0.00 & \u2014 & 0.00 &\\\nB3-b5& 286252C7 &16.23& 4600 & 2.0 & 1.5 & 0.11 &$-$0.15 & 0.40 & 0.00 & $-$0.2,0,$-$0.3 & 0.00 & \u2014 & 0.00 & CN-strong\\\nB3-b7& 282804C7 &16.36& 4400 & 1.9 & 1.3 & 0.20 &$-$0.16 & 0.39 &$-$0.06 &c,0,$-$0.25 & $-$0.50 & \u2014 & $-$0.50 & CN-strong\\\nB3-b8& 240083C6 &16.49& 4400 & 1.8 & 1.4 & $-$0.62 &$-$0.16 & 0.39 & 0.52 &c,0.3,0.25 & +0.30 & +0.30 & +0.30 &\\\nB3-f1\\*\\*& 129499C4 &16.32& 4500 & 1.9 & 1.6 & 0.04 & 0.09 & 0.44 & 0.19 & 0,0.4,0.1 & 0.00 & 0.00 & 0.00 &\\\nB3-f2& 259922C7 &16.54& 4600 & 1.9 & 1.8 & $-$0.25 &$-$0.15 & 0.40 &\u2014 & c,c,(0.2) & 0.00 & 0.00 & 0.00 & telluric\\\nB3-f3\\*\\*& 95424C3 &16.32& 4400 & 1.9 & 1.7 & 0.06 &$-$0.08 & 0.47 & $-$0.08& 0,0,c & +0.2 & $-$0.15 & 0.03 &\\\nB3-f4& 208959C6 &16.51& 4400 & 2.1 & 1.5 & 0.09 & 0.10 & 0.45 & 0.43 & c,0.1,0.1 & 0.00 & +0.30 & +0.15 &\\\nB3-f5& 49289C2 &16.61& 4200 & 2.0 & 1.8 & 0.16 &$-$0.06 & 0.49 & 0.09 &c,c,$-$0.05 & \u2014 & \u2014 & \u2014 & CN-strong\\\nB3-f7& 279577C7 &16.28& 4800 & 2.1 & 1.7 & 0.16 &$-$0.02 & 0.33 & 0.05 & c,c,c & $-$0.60 & $-$0.60 & $-$0.60 &\\\nB3-f8& 193190C5 &16.26& 4800 & 1.9 & 1.5 & 0.20 &$-$0.17 & 0.38 & 0.00 &c,0.28,$-$0.25 & $-$0.30 & $-$0.60 & $-$0.45 &\\\nBWc-1\\*\\*& 393125 &16.84& 4476 & 2.1 & 1.5 & 0.09 & 0.12 & 0.50 & 0.19 & 0.05,0.3,0.1 & $-$0.30 & $-$0.10 & $-$0.20 &\\\nBWc-2 & 545749 &17.19& 4558 & 2.2 & 1.2 & 0.18 &$-$0.09 & 0.52 & 0.07 & $-$0.2,0.2,$-$0.15 & \u2014 & \u2014 & \u2014 & CN-strong\\\nBWc-3& 564840 &16.91& 4513 &2.1 &1.3 & 0.28 & 0.04 & 0.51 & 0.19 & $-$0.1,0.4,0 & $-$0.30 & \u2014 & $-$0.30 & CN-strong\\\nBWc-4 & 564857 &16.76& 4866 & 2.2& 1.3 & 0.06 & 0.36 & 0.28 & 0.05 & $-$0.1,0.05,$-$0.05 & $-$0.30 & $-$0.40 & $-$0.35 &\\\nBWc-5\\*\\* & 575542 &16.98& 4535 &2.1 & 1.5 &0.42 &$-$0.01& 0.59 & 0.04 & $-$0.05,0.4,$-$0.1 & 0.00 & \u2014 & 0.00 & CN-strong\\\nBWc-6& 575585 &16.74& 4769 &2.2 &1.3 &$-$0.25 &$-$0.20 & 0.69 & 0.43 & c,c,0.25 & 0.00 & \u2014 & 0.00 & CN-strong\\\nBWc-7 & 67577 & 17.01 & 4590 &2.2 &1.1 &$-$0.25 &$-$0.20 & 0.50 & 0.44 &c,0.3,0.25 & 0.00 & 0.00 & 0.00 &\\\nBWc-8& 78255 & 16.97 & 4610 &2.2 &1.3 & 0.37 &$-$0.22 & 0.47 &$-$0.07 & c,0.1,$-$0.35 & 0.00 & \u2014 & 0.00 & CN-strong\\\nBWc-9\\*\\*& 78271 & 16.90 & 4539 &2.1 &1.5 & 0.15 &$-$0.13 & 0.77 & 0.11 & $-$0.1,0.2,$-$0.05 & 0.00 & $-$0.10 & $-$0.05 &\\\nBWc-10& 89589 & 16.70 & 4793 &2.2& 1.3 & 0.07 &$-$0.15 & 0.54 & 0.15 & $-$0.2,0.3,0 & 0.00 & 0.00 & 0.00 &\\\nBWc-11\\*\\*& 89735 & 16.69 & 4576 &2.1 & 1.0 & 0.17 &$-$0.14 & 0.56 &\u2014 & $-$0.2,0,$-$0.2 & $-$0.10 & 0.00 & $-$0.05 &\\\nBWc-12\\*\\*& 89832 & 16.92 & 4547 &2.1&1.3 & 0.23 & 0.19 & 0.29 & 0.24 & $-$0.1,0.1,$-$0.05 & $-$0.30 & $-$0.60 & $-$0.45 &\\\nBWc-13\\*\\*& 89848 & 16.73 & 4584 &2.1 &1.1 & 0.36 & 0.12 & 0.51 & 0.13 & 0,$-$0.15,$-$0.1 & $-$0.10 & $-$0.30 & $-$0.20 &\\\n\nBB acknowledges partial financial support by CNPq, CAPES, and FAPESP. 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A. 1995, ApJS, 101, 181\n\nZoccali, M., Lecureur, A., Barbuy, B. et al. 2006, A&A, 457, L1\n\n[^1]: Observations collected both at the European Southern Observatory, Paranal, Chile (ESO programmes 71.B-0617A, 73.B0074A, and GTO 71.B-0196)\n\n[^2]: We adopted here the usual spectroscopic notation that \\[A/B\\] = log(N$_{\\rm A}$/N$_{\\rm B}$)$_{\\star}$ $-$ log(N$_{\\rm A}$/N$_{\\rm B}$)$_{\\odot}$ and $\\epsilon$(A) = log(N$_{\\rm\n    A}$/N$_{\\rm B}$) + 12 for the elements A and B.\n\n[^3]: http://www.pmp.uni-hannover.de/cgi-bin/ssi/test/kurucz/sekur.html\n\n[^4]: http://physics.nist.gov/PhysRefData/ASD/lines$_-$form.html\n\n[^5]: http://www.eso.org/observing/dfo/quality/UVES/pipeline/solar[$_{-}$]{}spectrum.html\n\n[^6]: Specific star formation rate (SFR) (in Gyr$^{-1}$) is the ratio of the SFR in M$_{\\odot}$ Gyr$^{-1}$ over the gas mass in M$_{\\odot}$ available for star formation.\n"], ["---\nabstract: |\n    A timed network consists of an arbitrary number of initially identical 1-clock timed automata, interacting via hand-shake communication. In this setting there is no unique central controller, since all automata are initially identical. We consider the universal safety problem for such controller-less timed networks, i.e., verifying that a bad event (enabling some given transition) is impossible regardless of the size of the network.\n\n    This universal safety problem is dual to the existential coverability problem for timed-arc Petri nets, i.e., does there exist a number $m$ of tokens, such that starting with $m$ tokens in a given place, and none in the other places, some given transition is eventually enabled.\n\n    We show that these problems are PSPACE-complete.\nauthor:\n- Parosh Aziz Abdulla\n- Mohamed Faouzi Atig\n- Radu Ciobanu\n- Richard Mayr\n- Patrick Totzke\nbibliography:\n- 'main.bib'\ntitle: 'Universal Safety for Timed Petri Nets is PSPACE-complete'\n---\n\nIntroduction {#sec:intro}\n============\n\n##### Background.\n\nTimed-arc Petri nets (TPN) [@AA2001; @S2005; @AMM07; @BHR2006; @JJMS2011] are an extension of Petri nets where each token carries one real-valued clock and transitions are guarded by inequality constraints where the clock values are compared to integer bounds (via strict or non-strict inequalities). The known models differ slightly in what clock values newly created tokens can have, i.e., whether newly created tokens can inherit the clock value of some input token of the transition, or whether newly created tokens always have clock value zero. We consider the former, more general, case.\n\nDecision problems associated with the reachability analysis of (extended) Petri nets include *Reachability* (can a given marking reach another given marking?) and *Coverability* (can a given marking ultimately enable a given transition?).\n\nWhile Reachability is undecidable for all these TPN models [@RVCE1999], Coverability is decidable using the well-quasi ordering approach of [@ACJT2000; @FS2001] and complete for the hyper-Ackermannian complexity class $F_{\\omega^{\\omega^\\omega}}$ [@HSS2012]. With respect to Coverability, TPN are equivalent [@BFHR2010] to (linearly ordered) data nets [@LNORW2008].\n\nThe *Existential Coverability* problem for TPN asks, for a given place $p$ and transition $t$, whether there exists a number $m$ such that the marking $M(m) {{\\overset{\\text{\\tinydef}}{=}}}m\\cdot\\{(p,{\\boldsymbol{0}})\\}$ ultimately enables $t$. Here, $M(m)$ contains exactly $m$ tokens on place $p$ with all clocks set to zero and *no other tokens*. This problem corresponds to checking safety properties in distributed networks of arbitrarily many (namely $m$) initially identical timed processes that communicate by handshake. A negative answer certifies that the \u2018bad event\u2019 of transition $t$ can never happen regardless of the number $m$ of processes, i.e., the network is safe for any size. Thus by checking existential coverability, one solves the dual problem of *Universal Safety*. (Note that the $m$ timed tokens/processes are only initially identical. They can develop differently due to non-determinacy in the transitions.)\n\nThe corresponding problem for timed networks studied in [@ADM2004] does not allow the dynamic creation of new timed processes (unlike the TPN model which can increase the number of timed tokens), but considers multiple clocks per process (unlike our TPN with one clock per token).\n\nThe TPN model above corresponds to a distributed network without a central controller, since initially there are no tokens on other places that could be used to simulate one. Adding a central controller would make *Existential Coverability* polynomially inter-reducible with normal *Coverability* and thus complete for $F_{\\omega^{\\omega^\\omega}}$ [@HSS2012] (and even undecidable for $>1$ clocks per token [@ADM2004]).\n\nAminof et.\u00a0al.\u00a0[@ARZS2015] study the model checking problem of $\\omega$-regular properties for the controller-less model and in particular claim an [$\\mathsf{EXPSPACE}$]{}\u00a0upper bound for checking universal safety. However, their result only holds for discrete time (integer-valued clocks) and they do not provide a matching lower bound.\n\n##### Our contribution.\n\nWe show that *Existential Coverability* (and thus universal safety) is decidable and [$\\mathsf{PSPACE}$]{}-complete. This positively resolves an open question from [@ADM2004] regarding the decidability of universal safety in the controller-less networks. Moreover, a symbolic representation of the set of coverable configurations can be computed (using exponential space).\n\nThe [$\\mathsf{PSPACE}$]{}\u00a0lower bound is shown by a reduction from the iterated monotone Boolean circuit problem. (It does not follow directly from the [$\\mathsf{PSPACE}$]{}-completeness of the reachability problem in timed automata of [@AD:1994], due to the lack of a central controller.)\n\nThe main ideas for the [$\\mathsf{PSPACE}$]{}\u00a0upper bound are as follows. First we provide a logspace reduction of the Existential Coverability problem for TPN to the corresponding problem for a syntactic subclass, non-consuming TPN. Then we perform an abstraction of the real-valued clocks, similar to the one used in [@AMM07]. Clock values are split into integer parts and fractional parts. The integer parts of the clocks can be abstracted into a finite domain, since the transition guards cannot distinguish between values above the maximal constant that appears in the system. The fractional parts of the clock values that occur in a marking are ordered sequentially. Then every marking can be abstracted into a string where all the tokens with the $i$-th fractional clock value are encoded in the $i$-th symbol in the string. Since token multiplicities do not matter for existential coverability, the alphabet from which these strings are built is finite. The primary difficulty is that the length of these strings can grow dynamically as the system evolves, i.e., the space of these strings is still infinite for a given TPN. We perform a forward exploration of the space of reachable strings. By using an acceleration technique, we can effectively construct a symbolic representation of the set of reachable strings in terms of finitely many regular expressions. Finally, we can check existential coverability by using this symbolic representation.\n\nTimed Petri Nets\n================\n\nWe use ${{\\mathbb N}}$ and ${{\\mathbb R}_{\\geq 0}}$ to denote the sets of nonnegative integers and reals, respectively. For $n\\in{{\\mathbb N}}$ we write ${[{n}]}$ for the set ${{\\mathopen{}\\mathclose\\bgroup\\originalleft}\\{0,\\ldots,n{\\aftergroup\\egroup\\originalright}\\}}$.\n\nFor a set ${{\\it A}}$, we use ${{{{\\it A}}}^*}$ to denote the set of words, i.e.\u00a0finite sequences, over ${{\\it A}}$, and write ${\\varepsilon}$ for the empty word. If $R$ is a regular expression over ${{\\it A}}$ then ${{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(R{\\aftergroup\\egroup\\originalright})}\\subseteq {{\\it A}}^*$ denotes its language.\n\nA *multiset* over a set $X$ is a function $M:X\\to{\\mathbb{N}}$. The set ${{X}^\\oplus}$ of all (finitely supported) multisets over $X$ is partially ordered pointwise (by $\\le$). The multiset union of $M,M'\\in{{X}^\\oplus}$ is $(M\\oplus M')\\in{{X}^\\oplus}$ with $(M\\oplus M')(\\alpha){{\\overset{\\text{\\tinydef}}{=}}}M(\\alpha)+M'(\\alpha)$ for all $\\alpha\\in X$. If $M\\ge M'$ then the multiset difference $(M\\ominus M')$ is the unique $M''\\in{{X}^\\oplus}$ with $M=M'\\oplus M''$. We will use a monomial representation and write for example $(\\alpha + \\beta^3)$ for the multiset $(\\alpha\\mapsto1, \\beta\\mapsto 3)$. For a multiset $M$ and a number $m\\in{\\mathbb{N}}$ we let $m\\cdot M$ denote the $m$-fold multiset sum of $M$. We further lift this to sets of numbers and multisets on the obvious fashion, so that in particular ${\\mathbb{N}}\\cdot S {{\\overset{\\text{\\tinydef}}{=}}}\\{n\\cdot M\\mid n\\in {\\mathbb{N}}, M\\in S\\}$.\n\n\\[sec:model\\] *Timed Petri nets* are place/transition nets where each token carries a real value, sometimes called its *clock value* or *age*. Transition firing depends on there being sufficiently many tokens whose value is in a specified interval. All tokens produced by a transition either have age $0$, or inherit the age of an input-token of the transition. To model time passing, all token ages can advance simultaneously by the same (real-valued) amount.\n\n\\[def:MTPN\\] A *timed Petri net* (TPN) ${\\mathcal{N}} = (P,T, {\\mathit{Var}}, G, {\\mathit{Pre}},{\\mathit{Post}})$ consists of finite sets of *places* $P$, *transitions* $T$ and *variables* ${\\mathit{Var}}$, as well as functions $G,{\\mathit{Pre}},{\\mathit{Post}}$ defining transition *guards*, *pre*\u2013 and *postconditions*, as follows.\n\nFor every transition $t\\in T$, the guard $G(t)$ maps variables to (open, half-open or closed) intervals with endpoints in ${\\mathbb{N}}\\cup\\{\\infty\\}$, restricting which values variables may take. All numbers are encoded in unary. The precondition ${\\mathit{Pre}}(t)$ is a finite multiset over $(P{\\times}{\\mathit{Var}})$. Let ${\\mathit{Var}}(t)\\subseteq{\\mathit{Var}}$ be the subset of variables appearing positively in ${\\mathit{Pre}}(t)$. The postcondition ${\\mathit{Post}}(t)$ is then a finite multiset over $(P{\\times}(\\{0\\}\\cup{\\mathit{Var}}(t)))$, specifying the locations and clock values of produced tokens. Here, the symbolic clock value is either $0$ (demanding a reset to age $0$), or a variable that appeared already in the precondition.\n\nA *marking* is a finite multiset over $P{\\times}{{\\mathbb R}_{\\geq 0}}$.\n\n\\[ex-mtpn\\] The picture below shows a place/transition representation of an TPN with four places and one transition. ${\\mathit{Var}}(t)=\\{x,y\\}$, ${\\mathit{Pre}}(t) = (p,x)^2 + (q,y)$, $G(t)(x)=[0,5]$, $G(t)(y)=]1,2]$ and ${\\mathit{Post}}(t)= (r,y)^3 + (s,0)$.\n\n\\(t) \\[transition, label=above:[$t$]{},align=left\\] [$0\\le x \\le 5$\\\n$1 < y \\le 2$ ]{}; (p) \\[above left=of t,place, label=left:[$p$]{}\\] ; (q) \\[below left=of t,place, label=left:[$q$]{}\\] ; (r) \\[above right=of t,place, label=right:[$r$]{}\\] ; (s) \\[below right=of t,place, label=right:[$s$]{}\\] ;\n\n\\(p) edge node\\[above right, pos=0.3\\][$x^2$]{} (t); (q) edge node\\[below right, pos=0.3\\][$y$]{} (t); (t) edge node\\[above\\][$y^3$]{} (r); (t) edge node\\[above\\][$0$]{} (s);\n\nThe transition $t$ consumes two tokens from place $p$, both of which have the same clock value $x$ (where $0 \\le x \\le 5$) and one token from place $q$ with clock value $y$ (where $1 < y \\le 2$). It produces three tokens on place $r$ who all have the same clock value $y$ (where $y$ comes from the clock value of the token read from $q$), and another token with value $0$ on place $s$.\n\nThere are two different binary step relations on markings: *discrete* steps ${\\longrightarrow_{t}}$ which fire a transition $t$ as specified by the relations $G,{\\mathit{Pre}}$, and ${\\mathit{Post}}$, and *time passing* steps ${\\longrightarrow_{d}}$ for durations $d\\in{{\\mathbb R}_{\\geq 0}}$, which simply increment all clocks by $d$.\n\n\\[def:mtpn:dsteps\\] For a transition $t\\in T$ and a variable evaluation $\\pi:{\\mathit{Var}}\\to {{\\mathbb R}_{\\geq 0}}$, we say that *$\\pi$ satisfies $G(t)$* if $\\pi(x)\\in G(t)(x)$ holds for all $x\\in {\\mathit{Var}}$. By lifting $\\pi$ to multisets over $(P{\\times}{\\mathit{Var}})$ (respectively, to multisets over $(P{\\times}(\\{0\\}\\cup{\\mathit{Var}}))$ with $\\pi(0)=0$) in the canonical way, such an evaluation translates preconditions ${\\mathit{Pre}}(t)$ and ${\\mathit{Post}}(t)$ into markings $\\pi({\\mathit{Pre}}(t))$ and $\\pi({\\mathit{Post}}(t))$, where for all $p\\in P$ and $c\\in{{\\mathbb R}_{\\geq 0}}$, $$\\begin{aligned}\n        \\pi({\\mathit{Pre}}(t))(p,c){{\\overset{\\text{\\tinydef}}{=}}}\\sum_{\\pi(v)=c} {\\mathit{Pre}}(t)(p,v)\n        \\qquad\n        \\text{and}\n        \\qquad\n        \\pi({\\mathit{Post}}(t))(p,c){{\\overset{\\text{\\tinydef}}{=}}}\\sum_{\\pi(v)={c}} {\\mathit{Post}}(t)(p,{v}).\n    \\end{aligned}$$ A transition $t\\in T$ is called *enabled* in marking $M$, if there exists an evaluation $\\pi$ that satisfies $G(t)$ and such that $\\pi({\\mathit{Pre}}(t))\\le M$. In this case, there is a discrete step $M{\\longrightarrow_{t}}M'$ from marking $M$ to $M'$, defined as $\n        M' = M \\ominus \\pi({\\mathit{Pre}}(t)) \\oplus \\pi({\\mathit{Post}}(t)).\n        $\n\n\\[def:mtpn:tsteps\\] Let $M$ be a marking and $d\\in {{\\mathbb R}_{\\geq 0}}$. There is a time step $M{\\longrightarrow_{d}}M'$ to the marking $M'$ with $M'(p,{c}){{\\overset{\\text{\\tinydef}}{=}}}M(p,{c}-{d})$ for ${c}\\ge{d}$, and $M'(p,{c}){{\\overset{\\text{\\tinydef}}{=}}}0$, otherwise. We also refer to $M'$ as $(M+d)$.\n\nWe write ${\\xrightarrow{}_{\\textit{Time}}}$ for the union of all timed steps, ${{\\xrightarrow{}_{\\textit{Disc}}}}$ for the union of all discrete steps and simply ${\\xrightarrow{}}$ for ${{\\xrightarrow{}_{\\textit{Disc}}}}\\cup{{\\xrightarrow{}_{\\textit{Time}}}}{}$. The transitive and reflexive closure of ${\\xrightarrow{}}$ is ${\\xrightarrow{*}}$. ${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(M{\\aftergroup\\egroup\\originalright})}$ denotes the set of markings $M'$ for which there is an $M''\\ge M'$ with $M{\\xrightarrow{*}}M''$.\n\nWe are interested in the *existential coverability problem* ([$\\exists$COVER ]{}\u00a0for short), as follows.\n\n[ ]{}\n\nWe show that this problem is [$\\mathsf{PSPACE}$]{}-complete. Both lower and upper bound will be shown (w.l.o.g., see \\[lem:mtpn:wlog\\]) for the syntactic subclass of *non-consuming* TPN, defined as follows.\n\n\\[def:nonconsuming\\] A *timed Petri net* $(P,T, {\\mathit{Var}}, G, {\\mathit{Pre}},{\\mathit{Post}})$ is *non-consuming* if for all $t\\in T$, $p\\in P$ and $x\\in{\\mathit{Var}}$ it holds that both 1) ${\\mathit{Pre}}(t)(p,x) \\le 1$, and 2) ${\\mathit{Pre}}(t) \\le {\\mathit{Post}}(t)$.\n\nIn a non-consuming TPN, token multiplicities are irrelevant for discrete transitions. Intuitively, having one token $(p,{c})$ is equivalent to having an inexhaustible supply of such tokens.\n\nThe first condition is merely syntactic convenience. It asks that each transition takes at most one token from each place. The second condition in \\[def:nonconsuming\\] implies that for each discrete step $M{\\longrightarrow_{t}}M'$ we have $M' \\ge M$. Therefore, once a token $(p,{c})$ is present on a place $p$, it will stay there unchanged (unless time passes), and it will enable transitions with $(p,{c})$ in their precondition.\n\nWherever possible, we will from now on therefore allow ourselves to use the set notation for markings, that is simply treat markings $M\\in{{(P{\\times}{{\\mathbb R}_{\\geq 0}})}^\\oplus}$ as sets $M\\subseteq (P{\\times}{{\\mathbb R}_{\\geq 0}})$.\n\nLower Bound {#sec:lowerbound}\n===========\n\n[$\\mathsf{PSPACE}$]{}-hardness of [$\\exists$COVER ]{}does not follow directly from the [$\\mathsf{PSPACE}$]{}-completeness of the reachability problem in timed automata of [@AD:1994]. The non-consuming property of our TPN makes it impossible to fully implement the control-state of a timed automaton. Instead our proof uses multiple timed tokens and a reduction from the iterated monotone Boolean circuit problem [@GMST2016].\n\nA depth-1 monotone Boolean circuit is a function $F:\\{0,1\\}^n \\to \\{0,1\\}^n$ represented by $n$ constraints: For every $0\\le i < n$ there is a constraint of the form $i'=j \\otimes k,$ where $0\\le j,k< n$ and $\\otimes \\in \\{\\wedge, \\vee \\}$, which expresses how the next value of bit $i$ depends on the current values of bits $j$ and $k$. For every bitvector ${\\boldsymbol{v}}\\in\\{0,1\\}^n$, the function $F$ then satisfies $F({\\boldsymbol{v}})[i]{{\\overset{\\text{\\tinydef}}{=}}}{\\boldsymbol{v}}[j]\\otimes {\\boldsymbol{v}}[k]$. It is [$\\mathsf{PSPACE}$]{}-complete to check whether for a given vector ${\\boldsymbol{v}}\\in\\{0,1\\}^n$ there exists a number $m\\in{\\mathbb{N}}$ such that $F^m({\\boldsymbol{v}})[0]=1$.\n\nTowards a lower bound for [$\\exists$COVER ]{}\u00a0(\\[thm:LB\\]) we construct a non-consuming TPN as follows, for a given circuit. The main idea is to simulate circuit constraints by transitions that reset tokens of age $1$ (encoding ${\\boldsymbol{v}}$) to fresh ones of age $0$ (encoding $F({\\boldsymbol{v}})$), and let time pass by one unit to enter the next round.\n\n(Tj) \\[place, label=right:[$\\mathit{True}_j$]{}\\] ; (Ti) \\[place, below=of Tj, label=right:[$\\mathit{True}_{i}$]{}\\] ; (Tk) \\[place, below=of Ti, label=right:[$\\mathit{True}_{k}$]{}\\] ; (Fj) \\[place, right=of Tj, label=left:[$\\mathit{False}_j$]{}\\] ; (Fi) \\[place, right=of Ti, label=left:[$\\mathit{False}_{i}$]{}\\] ; (Fk) \\[place, right=of Tk, label=left:[$\\mathit{False}_{k}$]{}\\] ;\n\n(iB) \\[transition, left=of Ti, label=left:[$i.B$]{}\\] [$x=y=1$]{}; (iL) \\[transition, right=of Fj, label=right:[$i.L$]{}\\] [$x=1$]{}; (iR) \\[transition, right=of Fk, label=right:[$i.R$]{}\\] [$x=1$]{}; (Tj) edge node\\[above\\][$x$]{} (iB); (Tk) edge node\\[above\\][$y$]{} (iB); (iB) edge node\\[above\\][$0$]{} (Ti);\n\n(Fj) edge node\\[above\\][$x$]{} (iL); (iL) edge node\\[above\\][$0$]{} (Fi);\n\n(iL) edge node\\[above\\] (Fj); (iR) edge node\\[above\\][ ]{}(Fk); (iB) edge node\\[above\\](Tj); (iB) edge node\\[above\\](Tk);\n\n(Fk) edge node\\[above\\][$x$]{} (iR); (iR) edge node\\[above\\][$0$]{} (Fi);\n\nFor every bit $0\\le i<n$, the net contains two places ${\\mathit{True}_{i}}$ and ${\\mathit{False}_{i}}$. A marking $M_{{\\boldsymbol{v}}} \\le P{\\times}{{\\mathbb R}_{\\geq 0}}$ is an *encoding* of a vector ${\\boldsymbol{v}}\\in\\{0,1\\}^n$ if for every $0 \\leq i < n$ the following hold.\n\n1.  $({\\mathit{True}_{i}},0) \\in M_{{\\boldsymbol{v}}}\\iff {\\boldsymbol{v}}[i]=1$.\n\n2.  $({\\mathit{False}_{i}},0) \\in M_{{\\boldsymbol{v}}}\\iff {\\boldsymbol{v}}[i]=0$.\n\n3.  If $(p,c)\\in M_{{\\boldsymbol{v}}} $ then $c=0$ or $c\\geq 1$.\n\nNote that in particular one cannot have both $({\\mathit{True}_{i}},0)$ and $({\\mathit{False}_{i}},0)$ in $M_{{\\boldsymbol{v}}}$. For every constraint $i'=j\\land k$ we introduce three transitions, $i.L, i.R$, and $i.B$, where $$\\begin{aligned}\n        {\\mathit{Pre}}(i.B) &{{\\overset{\\text{\\tinydef}}{=}}}{({\\mathit{True}_{j}},x) + ({\\mathit{True}_{k}},y)}\n        &&\n        {\\mathit{Post}}(i.B) {{\\overset{\\text{\\tinydef}}{=}}}{\\mathit{Pre}}(i.B) + {({\\mathit{True}_{i}},0)}\\\\\n        {\\mathit{Pre}}(i.L) &{{\\overset{\\text{\\tinydef}}{=}}}{({\\mathit{False}_{j}},x)}\n        &&\n        {\\mathit{Post}}(i.L) {{\\overset{\\text{\\tinydef}}{=}}}{\\mathit{Pre}}(i.L) + ({\\mathit{False}_{i}},0)\\\\\n        {\\mathit{Pre}}(i.R) &{{\\overset{\\text{\\tinydef}}{=}}}{({\\mathit{False}_{k}},x)}\n        &&\n        {\\mathit{Post}}(i.R) {{\\overset{\\text{\\tinydef}}{=}}}{\\mathit{Pre}}(i.R) + ({\\mathit{False}_{i}},0)\n    \\end{aligned}$$ and the guard for all transitions is $G(x)=G(y)=1$. See \\[fig:LB\\] for an illustration. For disjunctions $i'=j\\lor k$ the transitions are defined analogously, with ${\\mathit{True}_{}}$ and ${\\mathit{False}_{}}$ inverted. The correctness proof of our construction rests on the following simple observation.\n\n[lemma]{}[claimone]{}\\[claim1\\] If $F({\\boldsymbol{v}})={\\boldsymbol{v}}'$ then for every encoding $M_{{\\boldsymbol{v}}}$ of ${\\boldsymbol{v}}$, there exists an encoding $M_{{\\boldsymbol{v'}}}$ of ${\\boldsymbol{v}}'$ such that $M_{{\\boldsymbol{v}}}{\\longrightarrow_{1}}{{\\xrightarrow{*}_{\\textit{Disc}}}}M_{{\\boldsymbol{v}}'}$. Conversely, if $M_{{\\boldsymbol{v}}}{\\longrightarrow_{1}}{{\\xrightarrow{*}_{\\textit{Disc}}}}M_{{\\boldsymbol{v}}'}$ for encodings $M_{{\\boldsymbol{v}}}$ and $M_{{\\boldsymbol{v'}}}$ of ${\\boldsymbol{v}}$ and ${\\boldsymbol{v}}'$ respectively, then $F({\\boldsymbol{v}}) = {\\boldsymbol{v'}}$.\n\nFor the first part, we construct a sequence $M_0 {{\\xrightarrow{}_{\\textit{Disc}}}} M_1 {{\\xrightarrow{}_{\\textit{Disc}}}} \\dots {\\xrightarrow{}_{\\textit{Disc}}} M_{n-1} $ where $M_0 {{\\overset{\\text{\\tinydef}}{=}}}(M_{{\\boldsymbol{v}}}+1)$ and every step $M_{i-1} {{\\xrightarrow{}_{\\textit{Disc}}}} M_i$ adds tokens simulating the $i$th constraint of $F$. Since the TPN is non-consuming, we will have that $M_i\\ge (M_{{\\boldsymbol{v}}}+1)$, for all $i<n$. Consider now constraint $i'$, and assume w.l.o.g.\u00a0that $i' = j \\land k$ (the other case is analogous). There are two cases depending on ${\\boldsymbol{v'}}[i]$.\n\n1.  Case ${\\boldsymbol{v'}}[i]=1$. By our assumption that $F({\\boldsymbol{v}})={\\boldsymbol{v'}}$ we know that ${\\boldsymbol{v}}[j] =1$ and ${\\boldsymbol{v}}[k] =1$. So $({\\mathit{True}_{j}},1) \\in (M_{{\\boldsymbol{v}}}+1) \\le M_{i-1}$ and $({\\mathit{True}_{k}},1) \\in (M_{{\\boldsymbol{v}}}+1) \\le M_{i-1}$. By construction of the net, there is a transition $i.B$ with ${\\mathit{Pre}}(i.B) = {({\\mathit{True}_{j}},1) + ({\\mathit{True}_{k}},1)}$ and ${\\mathit{Post}}(i.B) = {\\mathit{Pre}}(i.B) + {({\\mathit{True}_{i}},0)} $. This justifies step $M_{i-1} {\\longrightarrow_{i.B}} M_{i}$ and therefore that $(True_i, 0) \\in M_i\\le M_{n-1}$. Also notice that no marking reachable from $M_0$ using only discrete steps can contain the token $({\\mathit{False}_{i}},0)$. This is because these can only be produced by transitions requiring either $({\\mathit{False}_{j}},1)$ or $({\\mathit{False}_{k}},1)$, which are not contained in $M_0$ by assumption that $M_{{\\boldsymbol{v}}}$ encodes ${\\boldsymbol{v}}$. Therefore $({\\mathit{False}_{i}},0)\\notin M_{n-1}$.\n\n2.  Case ${\\boldsymbol{v'}}[i]=0$. W.l.o.g., ${\\boldsymbol{v}}[j] =0$. Therefore, $({\\mathit{False}_{j}},1) \\in (M_{{\\boldsymbol{v}}}+1) \\le M_{i-1}$. By construction of the net, there exists transition $i.L$ with ${\\mathit{Pre}}(i.L) = {({\\mathit{False}_{j}},1)}$ and ${\\mathit{Post}}(i.L) = {\\mathit{Pre}}(i.L) + {({\\mathit{False}_{i}},0)}$. This justifies the step $M_{i-1} {\\longrightarrow_{i.L}} M_{i}$, with $(False_i, 0) \\in M_i\\le M_{n-1}$. Notice again that no marking reachable from $M_0$ using only discrete steps can contain the token $({\\mathit{True}_{i}},0)$. This is because these can only be produced by transitions $i.B$, requiring both $({\\mathit{True}_{j}},1),({\\mathit{True}_{k}},1)\\in M_{0}$, contradicting our assumptions. Hence, $({\\mathit{True}_{i}},0)\\notin M_{n-1}$.\n\nWe conclude that the constructed marking $M_{n-1}$ is an encoding of ${\\boldsymbol{v'}}$.\n\nFor the other part of the claim, assume that there exist markings $M_{{\\boldsymbol{v}}}$ and $M_{{\\boldsymbol{v'}}}$ which are encodings of vectors ${\\boldsymbol{v}}$ and ${\\boldsymbol{v'}}$, respectively, with $M_{{\\boldsymbol{v}}}{\\longrightarrow_{1}}{{\\xrightarrow{*}_{\\textit{Disc}}}}M_{{\\boldsymbol{v'}}}$. We will show that $F({\\boldsymbol{v}})={\\boldsymbol{v}}'$. Recall that $F({\\boldsymbol{v}})[i]{{\\overset{\\text{\\tinydef}}{=}}}{\\boldsymbol{v}}[j]\\otimes {\\boldsymbol{v}}[k]$, where $0\\le j,k< n$ and $\\otimes \\in \\{\\wedge, \\vee \\}$. We will show for each $i <  n$ that ${\\boldsymbol{v}}'[i] = {\\boldsymbol{v}}[j]\\otimes {\\boldsymbol{v}}[k]$. Again, consider the constraint $i'$, and assume w.l.o.g.\u00a0that $i' = j \\land k$ (the other case is analogous). There are two cases.\n\n1.  Case ${\\boldsymbol{v'}}[i]=1$. By definition of a marking encoding, we have that $({\\mathit{True}_{i}},0)\\in M_{{\\boldsymbol{v}}}$. By construction, there is a transition $i.B$ with ${\\mathit{Pre}}(i.B) = {({\\mathit{True}_{j}},1) + ({\\mathit{True}_{k}},1)}$ and ${\\mathit{Post}}(i.B) = {\\mathit{Pre}}(i.B) + {({\\mathit{True}_{i}},0)}$. By assumption, it holds that $(M_{{\\boldsymbol{v}}}+1) {{\\xrightarrow{*}_{\\textit{Disc}}}} M_{{\\boldsymbol{v}}}'$, where $M_{{\\boldsymbol{v}}}{\\longrightarrow_{1}} (M_{{\\boldsymbol{v}}}+1)$. Note that $({\\mathit{True}_{j}},1)\\in (M_{{\\boldsymbol{v}}}+1)$ and $({\\mathit{True}_{k}},1)\\in (M_{{\\boldsymbol{v}}}+1)$. Hence, we have that ${\\boldsymbol{v}}[j]=1$ and ${\\boldsymbol{v}}[k]=1$, and therefore that $F({\\boldsymbol{v}})[i] = {\\boldsymbol{v'}}[i] = {\\boldsymbol{v}}[j] \\land {\\boldsymbol{v}}[k]$.\n\n2.  Case ${\\boldsymbol{v'}}[i]=0$. Then $({\\mathit{False}_{i}},0)\\in M_{{\\boldsymbol{v}}}$ and, since this token can only be produced by transitions $i.L$ or $i.R$, either $({\\mathit{False}_{j}},1)\\in (M_{{\\boldsymbol{v}}}+1)$ or $({\\mathit{False}_{k}},1)\\in (M_{{\\boldsymbol{v}}}+1)$.\n\n    Therefore $({\\mathit{False}_{j}},0)\\in (M_{{\\boldsymbol{v}}})$ or $({\\mathit{False}_{k}},0)\\in (M_{{\\boldsymbol{v}}})$ and because $M_{{\\boldsymbol{v}}}$ is an encoding of ${\\boldsymbol{v}}$, this means that either ${\\boldsymbol{v}}[j]=0$ or ${\\boldsymbol{v}}[k]=0$. Therefore, $F({\\boldsymbol{v'}})[i]= {\\boldsymbol{v}}[j]\\land {\\boldsymbol{v}}[k] =0$.\n\n\\[thm:LB\\] [$\\exists$COVER ]{} is [$\\mathsf{PSPACE}$]{}-hard for non-consuming TPN.\n\nFor a given monotone Boolean circuit, define a non-consuming TPN as above. By induction on $m\\in{\\mathbb{N}}$ using \\[claim1\\], we derive that there exists $m\\in{\\mathbb{N}}$ with $F^m({\\boldsymbol{v}})={\\boldsymbol{v}}'$ and ${\\boldsymbol{v}}'[0]=1$ if, and only if, there exists encodings $M_{{\\boldsymbol{v}}}$ of ${\\boldsymbol{v}}$ and $M_{{\\boldsymbol{v'}}}$ of ${\\boldsymbol{v'}}$, with $M_{{\\boldsymbol{v}}}{\\xrightarrow{*}} M_{{\\boldsymbol{v}}'}$. Moreover, if there is a marking $M$ such that $M_{{\\boldsymbol{v}}}{\\xrightarrow{*}} M$ and $0\\in {{\\it frac}(M)}$, where $M$ contains a token of age $0$, then $M\\le M_{{\\boldsymbol{v'}}}$ for some encoding $M_{{\\boldsymbol{v}}'}$ of a vector ${\\boldsymbol{v'}}=F^m({\\boldsymbol{v}})$. This means that it suffices to add one transition $t$ with ${\\mathit{Pre}}(t)=({\\mathit{True}_{0}},0)$ whose enabledness witnesses the existence of a reachable encoding $M_{{\\boldsymbol{v}}'}$ containing a token $({\\mathit{True}_{0}},0)$. By the properties above, there exists $m\\in{\\mathbb{N}}$ with $F^m({\\boldsymbol{v}})={\\boldsymbol{v}}'$ and ${\\boldsymbol{v}}'[0]=1$ iff $M_{{\\boldsymbol{v}}}{\\xrightarrow{*}}M_{{\\boldsymbol{v}}'}{\\xrightarrow{t}}$.\n\nThis lower bound holds even for discrete time TPN, e.g.\u00a0[@EFR2000], because the proof uses only timed steps with duration $d=1$.\n\nUpper Bound {#sec:upperbound}\n===========\n\nWe start by observing that we can restrict ourselves, without loss of generality, to non-consuming TPN (\\[def:nonconsuming\\]) for showing the upper bound. Intuitively, since we start with an arbitrarily high number of tokens anyway, it does not matter how many of them are consumed by transitions during the computation, since some always remain.\n\n[lemma]{}[lemnonconsuming]{} \\[lem:mtpn:wlog\\] \\[lem:nonconsuming\\] The [$\\exists$COVER ]{}\u00a0problem for TPN logspace-reduces to the [$\\exists$COVER ]{}problem for non-consuming TPN. That is, for every TPN ${\\mathcal{N}}$ and for every place $p$ and transition $t$ of ${\\mathcal{N}}$, one can construct, using logarithmic space, a non-consumimg TPN ${\\mathcal{N}}'$ together with a place $p'$ and transition $t'$ of ${\\mathcal{N}}'$, so that there exists $M\\in{\\mathit{Cover}_{{\\mathcal{N}}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({\\mathbb{N}}\\cdot\\{(p,{0})\\}{\\aftergroup\\egroup\\originalright})}$ enabling $t$ in ${\\mathcal{N}}$ if and only if there exists $M'\\in{\\mathit{Cover}_{{\\mathcal{N}}'}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({\\mathbb{N}}\\cdot\\{(p',0)\\}{\\aftergroup\\egroup\\originalright})}$ that enables $t'$ in ${\\mathcal{N}}'$.\n\nFirst notice that the first condition in \\[def:nonconsuming\\], that asks that every transition takes at most one token each place, is merely a syntactic convenience. A net satisfying this condition can be constructed by adding a few extra places and intermediate transitions to first distribute tokens to those extra places for the original transition to consume.\n\nSo let\u2019s assume w.l.o.g., that ${\\mathcal{N}}$ satisfies this condition and let ${\\mathcal{N}}'$ be the non-consuming variant derived from ${\\mathcal{N}}$ where for all transitions $T$, ${\\mathit{Post}}_{{\\mathcal{N}}'}(t){{\\overset{\\text{\\tinydef}}{=}}}{\\mathit{Post}}_{\\mathcal{N}}(t)\\oplus{\\mathit{Pre}}_{\\mathcal{N}}(t)$. Notice that then, for every discrete step $M {\\longrightarrow_{t}}M'$ we have that $M \\leq M'$. We prove the following claim.\n\n*For every place $p$ and transition $t$ of ${\\mathcal{N}}$ there exists $M\\in{\\mathit{Cover}_{{\\mathcal{N}}}({\\mathbb{N}}\\cdot\\{(p,{0}\\})}$ enabling $t$ in ${\\mathcal{N}}$ if, and only if there exists $M'\\in{\\mathit{Cover}_{{\\mathcal{N}}'}({\\mathbb{N}}\\cdot\\{(p,0)\\})}$ that enables $t$ in ${\\mathcal{N}}'$.*\n\nThe \u201c${\\mathcal{N}}\\to{\\mathcal{N}}'$\u201d direction follows from the observation that the pointwise ordering $\\le$ on markings, is a simulation: If $M{\\xrightarrow{}}N$ and $M'\\ge M$ then there exists an $N'\\ge N$ with $M'{\\xrightarrow{}}N'$. For the other direction, suppose there exists a witnessing path $$m\\cdot\\{(p,{0})\\}~=~M_0{\\xrightarrow{}}M_1{\\xrightarrow{}}M_2{\\xrightarrow{}}\\cdots{\\xrightarrow{}}M_k{\\xrightarrow{t}}$$ of length $k$ in ${\\mathcal{N}}'$. We can inductively derive a witnessing path in ${\\mathcal{N}}$ backwards, again using the fact that $\\le$ is a simulation. First note that if $M'$ enables $t$, then every $m'\\cdot M'$ with $m'>0$ enables $t$, (in both nets). Suppose $M_i{\\xrightarrow{\\rho}}$ is a path of length $(k-i)$ that ends in a $t$-transition. By the simulation property, there is such a path from every $m\\cdot M_i$, $m>0$. Further, there must exist markings $M'_{i-1}\\in\\ {\\downarrow\\!{(}}{\\mathbb{N}}\\cdot M_{i-1})$ and $M'_{i}\\in\\ {\\downarrow\\!{(}}{\\mathbb{N}}\\cdot M_{i})$ such that $M'_{i-1}{\\xrightarrow{}} M'_{i}$. It suffices to pick $M'_{i-1}{{\\overset{\\text{\\tinydef}}{=}}}B\\cdot M_{i-1}$, where $B\\in{\\mathbb{N}}$ is the maximal cardinality of any multiset ${\\mathit{Pre}}(t)$ (This number is itself bounded by ${{\\lvertP\\rvert}}\\cdot {{\\lvert{\\mathit{Var}}\\rvert}}$ by our assumption on ${\\mathit{Pre}}(t)$). We conclude that in ${\\mathcal{N}}$ there is a path ending in a $t$-transition and starting in marking $(B\\cdot k) \\cdot M_0$, which is in ${\\mathbb{N}}\\cdot\\{(p,{0})\\}$.\n\nRegion Abstraction {#sec:regionabs}\n------------------\n\nWe recall a constraint system called regions defined for timed automata [@AD:1994]. The version for TPN used here is similar to the one in [@AMM07].\n\nConsider a fixed, nonconsuming TPN ${\\mathcal{N}}=(P,T,{\\mathit{Var}},G,{\\mathit{Pre}},{\\mathit{Post}})$. Let ${c_\\mathit{max}}$ be the largest finite value appearing in transition guards $G$. Since different tokens with age $>{c_\\mathit{max}}$ cannot be distinguished by transition guards, we consider only token ages below or equal to ${c_\\mathit{max}}$ and treat the integer parts of older tokens as equal to ${c_\\mathit{max}}+1$. Let ${{\\it int}(c)}{{\\overset{\\text{\\tinydef}}{=}}}\\min\\{{c_\\mathit{max}}+1, \\lfloor{c}\\rfloor\\}$ and ${{\\it frac}(c)} {{\\overset{\\text{\\tinydef}}{=}}}c-\\lfloor{c}\\rfloor$ for a real value $c\\in{{\\mathbb R}_{\\geq 0}}$. We will work with an abstraction of TPN markings as words over the alphabet $\\Sigma {{\\overset{\\text{\\tinydef}}{=}}}2^{P{\\times}{[{{c_\\mathit{max}}+1}]}}$. Each symbol $X\\in\\Sigma$ represents the places and integer ages of tokens for a particular fractional value.\n\n\\[oneclock:abstr\\] Let $M\\subseteq P {\\times}{{\\mathbb R}_{\\geq 0}}$ be a marking and let ${{\\it frac}(M)} {{\\overset{\\text{\\tinydef}}{=}}}\\{{{\\it frac}(c)}\\mid (p,c)\\in M\\}$ be the set of fractional clock values that appear in $M$.\n\nLet $S\\subset [0,1[$ be a finite set of real numbers with $0\\in S$ and ${{\\it frac}(M)}\\subseteq S$ and let $f_0,f_1,\\dots,f_n$, be an enumeration of $S$ so that $f_{i-1} < f_i$ for all $i\\le n$. The *$S$-abstraction* of $M$ is $${\\mathit{abs}_{S}(M)} {{\\overset{\\text{\\tinydef}}{=}}}x_0x_1\\dots x_n \\in \\Sigma^*$$ where $x_i {{\\overset{\\text{\\tinydef}}{=}}}\\{(p,{{\\it int}(c)}) \\mid (p,c)\\in M \\land {{\\it frac}(c)}=f_i\\}$ for all $i\\le n$. We simply write ${\\mathit{abs}_{}(M)}$ for the shortest abstraction, i.e. with respect to $S=\\{0\\}\\cup{{\\it frac}(M)}$.\n\nThe abstraction of marking $M = \\{(p,2.1),(q,2.2),(p,5.1),(q,5.1)\\}$ is ${\\mathit{abs}_{}(M)}=\\emptyset~\\{(p,2),(p,5),(q,5)\\}~\\{(q,2)\\}$. The first symbol is $\\emptyset$, because $M$ contains no token with an integer age (i.e., no token whose age has fractional part $0$). The second and third symbols represent sets of tokens with fractional values $0.1$ and $0.2$, respectively.\n\nClocks with integer values play a special role in the behavior of TPN, because the constants in the transition guards are integers. Thus we always include the fractional part $0$ in the set $S$ in \\[oneclock:abstr\\].\n\nWe use a special kind of regular expressions over $\\Sigma$ to represent coverable sets of TPN markings as follows.\n\n\\[expression:def\\] A regular expression $E$ over $\\Sigma$ represents the downward-closed set of TPN markings covered by one that has an abstraction in the language of $E$: $${[\\![E]\\!]}{{\\overset{\\text{\\tinydef}}{=}}}\\{N\\mid \\exists M \\exists S.~\n    M\\ge N \\land {\\mathit{abs}_{S}(M)}\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(E{\\aftergroup\\egroup\\originalright})}\\}.$$\n\nAn expression is *simple* if it is of the form $E = x_0x_1\\dots x_{k}$ where for all $i\\le k$ either $x_i\\in\\Sigma$ or $x_i={y_i}^*$ for some $y_i\\in\\Sigma$. In the latter case we say that $x_i$ *carries a star*. That is, a simple expression is free of Boolean combinators and uses only concatenation and Kleene star. We will write $\\hat{x}_i$ to denote the symbol in $\\Sigma$ at position $i$: it is $x_i$ if $x_i\\in\\Sigma$ and $y_i$ otherwise.\n\nNotice that for all simple expressions $\\alpha,\\beta$ so that ${{\\lvert\\alpha\\rvert}}>0$, we have that ${[\\![\\alpha\\emptyset\\beta]\\!]} ={[\\![\\alpha\\beta]\\!]}$. However, unless $\\alpha$ has length $0$ or is of the form $\\alpha=\\emptyset\\alpha'$, we have ${[\\![\\emptyset\\alpha]\\!]} \\neq{[\\![\\alpha]\\!]}$. This is because a marking $M$ that contains a token $(p,c)$ with ${{\\it frac}(c)}=0$ has the property that all abstractions ${\\mathit{abs}_{S}(M)}=x_0\\dots x_k$ of $M$ have $x_0\\neq \\emptyset$.\n\nThe following lemmas express the effect of TPN transitions at the level of the region abstraction. state that maximally firing of discrete transitions (the relation ${\\xrightarrow{*}_{\\textit{Disc}}}$) is computable and monotone. state how to represent timed-step successor markings.\n\n[lemma]{}[lemdiscall]{} \\[lem:discall\\] \\[lem:monotonicity\\] For every non-consuming TPN ${\\mathcal{N}}$ there are polynomial time computable functions $f: \\Sigma \\times \\Sigma \\times \\Sigma \\to \\Sigma$ and $g: \\Sigma \\times \\Sigma \\times \\Sigma \\to \\Sigma$ with the following properties.\n\n1.  \\[lem:discall-ad1\\] $f$ and $g$ are monotone (w.r.t.\u00a0subset ordering) in each argument.\n\n2.  \\[lem:discall-ad2\\] $f(\\alpha, \\beta, x) \\supseteq x$ and $g(\\alpha, \\beta, x) \\supseteq x$ for all $\\alpha,\\beta,x \\in \\Sigma$.\n\n3.  \\[lem:discall-ad3\\] Suppose that $E=x_0x_1\\dots x_k$ is a simple expression, $\\alpha {{\\overset{\\text{\\tinydef}}{=}}}x_0$ and $\\beta {{\\overset{\\text{\\tinydef}}{=}}}\\bigcup_{i>0} \\hat{x}_i$, and $E' = x'_0x'_1\\dots x'_k$ is the derived expression defined by conditions:\n\n    1.  $x_0' {{\\overset{\\text{\\tinydef}}{=}}}f(\\alpha,\\beta,x_0)$,\n\n    2.  $x_i' {{\\overset{\\text{\\tinydef}}{=}}}g(\\alpha,\\beta,\\hat{x}_i)^*$ for $i>0$,\n\n    3.  $x_i'$ carries a star iff $x_i$ does.\n\n    Then ${[\\![E']\\!]} = \\{M'' \\mid \\exists M \\in {[\\![E]\\!]} \\land M{{\\xrightarrow{*}_{\\textit{Disc}}}}M' \\ge M''\\}$.\n\nA proof of this statement is in the appendix. It is essentially due to the monotonicity of discrete transition firing in TPN and the fact that iteratively firing transitions must saturate due to the nonconsuming semantics. We first prove it only for star-free expressions $E$ in condition 3 (\\[lem:discall-starfree\\]) and then generalize to all simple expressions by induction.\n\n\\[def:discall\\] We will write ${\\mathit{SAT}(E)} {{\\overset{\\text{\\tinydef}}{=}}}E'$ for the successor expression $E'$ of $E$ guaranteed by \\[lem:discall\\]. I.e., ${\\mathit{SAT}(E)}$ is the saturation of $E$ by maximally firing discrete transitions.\n\nNotice that by definition it holds that ${[\\![E]\\!]} \\subseteq {[\\![{\\mathit{SAT}(E)}]\\!]}\\subseteq{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E]\\!]}{\\aftergroup\\egroup\\originalright})}$, and consequently also that ${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![{\\mathit{SAT}(E)}]\\!]}{\\aftergroup\\egroup\\originalright})}={\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E]\\!]}{\\aftergroup\\egroup\\originalright})}$.\n\n\\[lem:simulation\\] Suppose that $X=x_0x_1\\dots x_k$ is a simple expression of length $k+1$ with ${\\mathit{SAT}(X)}=x'_0x'_1\\dots x'_k$ and $x_0,x'_0\\in\\Sigma$. Let $Y=y_0\\alpha_1y_1\\alpha_2\\dots \\alpha_k y_k$ be a simple expression with ${\\mathit{SAT}(Y)}=y'_0\\alpha'_1y'_1\\alpha'_2\\dots \\alpha'_k y'_k$ and $y_0,y'_0\\in\\Sigma$.\n\nIf $\\hat{x}_i \\subseteq \\hat{y}_i$ for all $i\\le k$ then $\\hat{x}'_i \n    \\subseteq\n    \\hat{y}'_i \n    $ for all $i\\le k$.\n\nThe assumption of the lemma provides that $\\alpha_x{{\\overset{\\text{\\tinydef}}{=}}}x_0 \\subseteq \\alpha_y{{\\overset{\\text{\\tinydef}}{=}}}y_0$ and $\\beta_x{{\\overset{\\text{\\tinydef}}{=}}}\\bigcup_{k\\ge i>0}\\hat{x}_i \\subseteq \\beta_y{{\\overset{\\text{\\tinydef}}{=}}}\\bigcup_{k\\ge i>0}\\hat{y}_i$. Therefore, by \\[lem:discall-ad1\\] of \\[lem:discall\\], we get that $$x'_0 = f(\\alpha_x,\\beta_x,x_0)\n    \\quad\\subseteq\\quad\n    f(\\alpha_y,\\beta_y,y_0)\n    =y'_0$$ and similarly, for all $k\\ge i\\ge 0$, that $\\hat{x}'_i = g(\\alpha_x,\\beta_x,\\hat{x}_i)\n    ~\\subseteq~\n    g(\\alpha_y,\\beta_y,\\hat{y}_i)\n    = \\hat{y}'_i.\n    $\n\nFor $x\\in\\Sigma$ we write $(x+1){{\\overset{\\text{\\tinydef}}{=}}}\\{(p,{{\\it int}(n+1)})\\mid (p,n)\\in x\\}$ for the symbol where token ages are incremented by $1$.\n\n[lemma]{}[lemepsilonsteps]{} \\[lem:epsilonsteps\\] $\n    {[\\![\\emptyset E]\\!]} = \n    \\{M' \\mid \\exists M \\in {[\\![E]\\!]} \\land M{\\longrightarrow_{d}}M' \\land d < 1-\\max(frac(M))\\}\n    $.\n\n*\u201c$\\supseteq$\u201d*: Suppose that $M$ is a non-empty marking in ${[\\![E]\\!]}$, $d<1-\\max({{\\it frac}(M)})$ and $M{\\longrightarrow_{d}}M'$. The assumption on $d$ implies that for every token $(p,c) \\in M$ we have ${{\\it int}(c)}={{\\it int}(c+d)}$. In other words, the integral part of the token age remained the same. Therefore $(p,{{\\it int}(c)}) = (p,{{\\it int}(c+d)}) \\in M'$. Also from the assumption on $d$ we get that $${{\\it frac}(M')}= \\{x+d \\mid x \\in {{\\it frac}(M)}\\}$$ Recall that ${\\mathit{abs}_{}(M)}={\\mathit{abs}_{S}(M)}$ and ${\\mathit{abs}_{}(M')}={\\mathit{abs}_{S'}(M')}$ for the sets $S{{\\overset{\\text{\\tinydef}}{=}}}\\{0\\}\\cup {{\\it frac}(M)}$ and $S'{{\\overset{\\text{\\tinydef}}{=}}}\\{0\\}\\cup {{\\it frac}(M')}$. Clearly, $0\\notin {{\\it frac}(M')}$. There are two cases:\n\n1.  $0\\in{{\\it frac}(M)}$. Then ${\\mathit{abs}_{}(M')}=\\emptyset{\\mathit{abs}_{}(M)}\\in {{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\emptyset E{\\aftergroup\\egroup\\originalright})}$, and consequently, $M'\\in{[\\![\\emptyset E]\\!]}$.\n\n2.  $0\\notin{{\\it frac}(M)}$. Then ${\\mathit{abs}_{}(M')}={\\mathit{abs}_{}(M)}=\\emptyset w\\in {{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(E{\\aftergroup\\egroup\\originalright})}$. Suppose that $E=x_0\\alpha$, i.e., $E$ has $x_0\\in\\Sigma$ as its leftmost symbol, and $w\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\alpha{\\aftergroup\\egroup\\originalright})}$. If $x_0=\\emptyset$ then ${[\\![E]\\!]}={[\\![\\emptyset E]\\!]}$ and thus ${\\mathit{abs}_{}(M')}\\in{[\\![\\emptyset E]\\!]}$. Otherwise, if $x_0\\neq\\emptyset$ then $x_0w\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(E{\\aftergroup\\egroup\\originalright})}$ and $x_0w={\\mathit{abs}_{}(M'')}$ for some marking $M''\\ge M'$. So again, $M'\\in{[\\![\\emptyset E]\\!]}$.\n\n*\u201c$\\subseteq$\u201d*: W.l.o.g., pick a non-empty marking $M'\\in {[\\![\\emptyset E]\\!]}$. If $E$ has $\\emptyset$ as its leftmost symbol, then ${[\\![\\emptyset E]\\!]}={[\\![E]\\!]}$ and the claim follows using $d=0$, since then $M'\\in{[\\![E]\\!]}$. So suppose that $E$ does not start with $\\emptyset$. Note that by \\[oneclock:abstr\\], there are no tokens in the marking $M'$ whose clocks have fractional value zero. Let $$d{{\\overset{\\text{\\tinydef}}{=}}}\\min({{\\it frac}(M')})$$ be the minimal fractional clock value among the tokens of $M'$ and based on this, define $M {{\\overset{\\text{\\tinydef}}{=}}}\\{ (p,c-d)\\mid (p,c)\\in N' \\} $. By construction of $M$ we get $M{\\longrightarrow_{d}}M'$ and also that $\\max({{\\it frac}(M)}) = \\max({{\\it frac}(M')})-d < 1$. Therefore that $1-\\max({{\\it frac}(M)}) < 1 -d$. Finally, observe that ${{\\it frac}(M)}= \\{x-d \\mid x \\in {{\\it frac}(M')} \\}$ and $0\\in{{\\it frac}(M)}$. It follows that ${\\mathit{abs}_{}(M')}=\\emptyset{\\mathit{abs}_{}(M)}$ and therefore that ${\\mathit{abs}_{}(M)}\\in {{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(E{\\aftergroup\\egroup\\originalright})}$ and $M\\in{[\\![E]\\!]}$. This means that $M'$ is included in the set on the right in the claim.\n\n[lemma]{}[lemnonstarrotation]{}\\[oneclock:rota\\] \\[lem:nonstarrotation\\] Let $\\alpha z$ be a simple expression where $\\hat{z}=z\\in\\Sigma$ (the rightmost symbol is not starred). Then, ${[\\![(z+1)\\alpha]\\!]}$ contains a marking $N$ if, and only if, there exists markings $N'\\ge N$ and $M$, and a set $S\\subseteq [0,1[$ so that\n\n1.  ${{\\lvertS\\rvert}} = {{\\lvert\\alpha z\\rvert}}$\n\n2.  ${\\mathit{abs}_{S}(M)}\\in {{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\alpha z{\\aftergroup\\egroup\\originalright})}$\n\n3.  $M{\\longrightarrow_{d}}N'$ for $d = 1-\\max(S)$.\n\nSuppose markings $N,N',M$, a set $S\\subseteq [0,1[$ and $d\\in{{\\mathbb R}_{\\geq 0}}$ so that the conditions 1 to 3 are satisfied. Let $S'{{\\overset{\\text{\\tinydef}}{=}}}\\{0\\}\\cup \\{s+d\\mid s\\in S\\setminus\\{d\\}\\}$. Then, ${{\\lvertS'\\rvert}}={{\\lvertS\\rvert}}$ and ${\\mathit{abs}_{S'}(N')} \\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}((z+1)\\alpha{\\aftergroup\\egroup\\originalright})}$, which witnesses that $N\\in{[\\![(z+1)\\alpha]\\!]}$.\n\nConversely, let $N\\in{[\\![(z+1)\\alpha]\\!]}$ be a non-empty marking. If ${{\\lvert\\alpha\\rvert}} = 0$, then $N\\in{[\\![(z+1)]\\!]}$ and so ${\\mathit{abs}_{S}(N)}\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}((z+1){\\aftergroup\\egroup\\originalright})}$ for $S{{\\overset{\\text{\\tinydef}}{=}}}{{\\it frac}(N)}=\\{0\\}$. This means that $M{\\longrightarrow_{1}}N=(M+1)$ for a marking $M$ with ${\\mathit{abs}_{S}(M)}\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(z{\\aftergroup\\egroup\\originalright})}={{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\alpha z{\\aftergroup\\egroup\\originalright})}$.\n\nIf ${{\\lvert\\alpha\\rvert}} >0$, pick some marking $N'\\ge N$ and set $S'$ so that ${\\mathit{abs}_{S'}(N')}=(z+1)w$, for some word $w\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\alpha{\\aftergroup\\egroup\\originalright})}$. Then we must have that ${{\\lvertS'\\rvert}}={{\\lvert(z+1)\\alpha\\rvert}}>1$ and so $d{{\\overset{\\text{\\tinydef}}{=}}}\\min(S'\\setminus\\{0\\})$ exists. Let $S{{\\overset{\\text{\\tinydef}}{=}}}\\{s-d\\mid s\\in S'\\}\\cup\\{1-d\\}$ and $M$ be the unique marking with $M{\\longrightarrow_{d}}N'$. Notice that $1-d=\\max(S)$. It follows that ${\\mathit{abs}_{S}(M)}=wz\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\alpha z{\\aftergroup\\egroup\\originalright})}$.\n\nWe will often use the following simple fact, which is a direct consequence of \\[lem:nonstarrotation\\].\n\n\\[cor:nonstarrotation\\] ${[\\![(z+1)\\alpha]\\!]}\n    \\subseteq {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha z]\\!]}{\\aftergroup\\egroup\\originalright})}$.\n\nFinally, the following lemma will be the basis for our exploration algorithm.\n\n[lemma]{}[lemaccstep]{} \\[lem:accstep\\] Let $\\alpha x_0^*$ be a simple expression with ${\\mathit{SAT}(\\alpha x_0^*)}=\\alpha x_0^*$. Then $\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})} = \n    {[\\![\\alpha x_0^*]\\!]}\n    \\cup {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![(x_0+1)\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}$.\n\nFor the right to left inclusion notice that ${[\\![\\alpha x_0^*]\\!]} \\subseteq {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}$ trivially holds. For the rest, we have $\n    {[\\![(x_0+1)\\alpha x_0^*]\\!]}\n    \\subseteq\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n    $ by \\[cor:nonstarrotation\\], and therefore ${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![(x_0+1)\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n    ~\\subseteq~\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}{\\aftergroup\\egroup\\originalright})}\n    =\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n    $. For the left to right inclusion, we equivalently show that $${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})} \\setminus\n    {[\\![\\alpha x_0^*]\\!]}\n    \\subseteq\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![(x_0+1)\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}$$ Using the assumption that ${\\mathit{SAT}(\\alpha x_0^*)} = \\alpha x_0^*$, the set on the left contains everything coverable from ${[\\![\\alpha x_0^*]\\!]}$ by a sequence that starts with a (short) time step. It can therefore be written as $${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\n    \\{N_1\\mid \\exists N_0\\in{[\\![\\alpha x_0^*]\\!]} \\land N_0{\\longrightarrow_{d}}N_1\n        \\land 0<d< 1-\\max(frac(N_0))\n    \\}\n    {\\aftergroup\\egroup\\originalright})}.$$ By \\[lem:epsilonsteps\\] and because ${[\\![\\emptyset\\alpha]\\!]}\\subseteq {[\\![X\\alpha]\\!]}$ for all $X\\in \\Sigma$ and $\\alpha\\in\\Sigma^*$, we conclude that indeed, $\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})} \\setminus\n    {[\\![\\alpha x_0^*]\\!]}\n    ~\\subseteq~\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![\\emptyset\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n    \\subseteq\n    {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![(x_0+1)\\alpha x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n    $.\n\nAcceleration\n------------\n\nWe propose an acceleration procedure based on unfolding expressions according to \\[lem:accstep\\] (interleaved with saturation steps to guarantee its premise) and introducing new Kleene stars to keep the length of intermediate expressions bounded. This procedure (depicted in \\[alg:inner\\_loop\\]), is used to characterize an initial subset of the coverability set.\n\na simple expression $S_0 = x_1x_0^*$ (of length 2 and with last symbol starred) simple expressions $S_1, S_i$ and $R$, of lengths 2, 4, and 2, respectively.\n\n$S_1 {{\\overset{\\text{\\tinydef}}{=}}}x_1^1(x_0^1)^*= {\\mathit{SAT}(x_1x_0^*)}$ $S_2 {{\\overset{\\text{\\tinydef}}{=}}}x_2^2x_1^2(x_0^2)^*= {\\mathit{SAT}((x_0^1+1)S_1)}$ $S_3 {{\\overset{\\text{\\tinydef}}{=}}}x_3^3x_2^3x_1^3(x_0^3)^*= {\\mathit{SAT}((x_0^2+1)S_2)}$ $i\\gets 3$\n\n$x_{i+1}^{i+1}x_{i}^{i+1}x_{i-1}^{i+1}x_1^{i+1}(x_0^{i+1})^*{{\\overset{\\text{\\tinydef}}{=}}}{\\mathit{SAT}((x_0^{i}+1)S_{i})}$ $S_{i+1} {{\\overset{\\text{\\tinydef}}{=}}}x_{i+1}^{i+1}(x_{i}^{i+1})^*x_1^{i+1}(x_0^{i+1})^*$ $i\\gets i+1$ $R {{\\overset{\\text{\\tinydef}}{=}}}(x_1^i+1)(x_{i-1}^i)^*$ $S_1, S_i, R$\n\n= \\[ align=center\\] = \\[graphnode, red\\] = \\[graphnode, black\\] = \\[graphnode, black\\] = \\[graphnode, black\\] = \\[align=left,text width=3.5cm\\] = \\[-&gt;, lightgray\\] = \\[-, lightgray\\] = \\[-&gt;, red\\] = \\[draw,-&gt;, blue\\]\n\n(S0-0) \\[snode\\] [$x_0^{*}$]{}; (S0-1)\\[snode, left=of S0-0\\] [$x_1$]{}; (S0-label) \\[label,right=of S0-0\\] [start]{};\n\n(S1-0) \\[snode, below =of S0-0\\] [$(x_0^1)^{*}$]{}; (S1-1) \\[ snode, left =of S1-0\\] [$x_1^1$]{};\n\n(S1-label) \\[below=of S0-label\\] [$S_1 = {\\mathit{SAT}(x_1x_0^*)} $]{} ;\n\n(S1-rot-0) \\[snode, below =of S1-0\\] [$(x_0^1)^{*} $]{}; (S1-rot-1) \\[snode, left =of S1-rot-0\\] [$x_1^1$]{}; (S1-rot-2) \\[newnode, left =of S1-rot-1\\] [ $(x_0^1+1)$]{}; (S1-rot-label) \\[label,below =of S1-label\\] [$(x_0^1+1)S_1 $]{};\n\n(S2-0) \\[snode, below =of S1-rot-0\\] [$(x_0^2)^{*}$]{}; (S2-1) \\[snode, left =of S2-0 \\] [$x_1^2$]{}; (S2-2) \\[nnewnode, left = of S2-1\\] [$x_2^2$]{}; (S2-label)\\[label, below=of S1-rot-label\\] [$S_2={\\mathit{SAT}((x_0^1+1)S_1)}$]{};\n\n(S2-rot-0) \\[snode, below =of S2-0 \\] [$(x_0^2)^{*} $]{}; (S2-rot-1) \\[snode, left = of S2-rot-0\\] [$x_1^2$]{}; (S2-rot-2) \\[mnode, left = of S2-rot-1\\] [ $x_2^2$]{}; (S2-rot-3)\\[newnode, left =of S2-rot-2\\][ $(x_0^2+1)$]{}; (S2-rot-label) \\[label,below =of S2-label\\] [$(x_0^2+1)S_2 $]{};\n\n(S3-0) \\[snode, below =of S2-rot-0\\][$(x_0^3)^{*}$]{}; (S3-1) \\[snode, left =of S3-0 \\] [$x_1^3$]{}; (S3-2) \\[mnode, left =of S3-1 \\][$x_2^3$]{}; (S3-3) \\[nnewnode, left =of S3-2 \\][$x_3^3$]{}; (S3-label)\\[label, below=of S2-rot-label\\] [$S_3={\\mathit{SAT}((x_0^2+1)S_2)}$]{};\n\n(S3-rot-0) \\[snode, below =of S3-0 \\] [$(x_0^3)^{*} $]{}; (S3-rot-1) \\[snode, left =of S3-rot-0\\] [$x_1^3$]{}; (S3-rot-2) \\[mnode, left =of S3-rot-1\\] [ $x_2^3$]{}; (S3-rot-3) \\[mnode,left =of S3-rot-2\\] [$x_3^3$]{}; (S3-rot-4)\\[newnode, left =of S3-rot-3\\][ $(x_0^3+1)$]{}; (S3-rot-label) \\[label,below =of S3-label\\] [$(x_0^3+1)S_3 $]{};\n\n(L4-0) \\[snode,below =of S3-rot-0\\][$(x_0^4)^{*}$]{}; (L4-1) \\[snode, left =of L4-0\\][$x_1^4$]{}; (L4-2) \\[mnode,left =of L4-1\\] [$x_2^4$]{}; (L4-3) \\[mnode,left =of L4-2\\] [$x_3^4$]{}; (L4-4) \\[nnewnode,left =of L4-3\\] [$x_4^4$]{}; (L4-label)\\[label, below=of S3-rot-label\\] [${\\mathit{SAT}((x_0^3+1)S_3)}$]{};\n\n(S4-0) \\[snode, below=of L4-0\\] [$(x_0^4)^{*}$]{}; (S4-1) \\[snode, left =of S4-0\\][$x_1^4$]{}; (S4-2) \\[mnode, left=of S4-1\\]; (S4-3) \\[mnode, left=of S4-2\\][$(x_3^4)^{*}$]{}; (S4-4) \\[nnewnode, left =of S4-3\\][$x_4^4$]{}; (S4-label)\\[label, below=of L4-label\\] [$S_4$]{};\n\n(S4-rot-0) \\[snode, below =of S4-0\\][$(x_0^4)^{*}$]{}; (S4-rot-1) \\[snode, left = of S4-rot-0\\] [$x_1^4$ ]{}; (S4-rot-2) \\[mnode, left=of S4-rot-1\\]; (S4-rot-3) \\[mnode, left=of S4-rot-2\\][$(x_3^4)^{*}$ ]{}; (S4-rot-4)\\[mnode, left =of S4-rot-3\\][$x_4^4$]{}; (S4-rot-5) \\[newnode, left =of S4-rot-4\\][$(x_0^4+1)$ ]{}; (S4-rot-label) \\[label,below =of S4-label\\] [$(x_0^4+1)S_4 $]{};\n\n(L5-0) \\[snode, below=of S4-rot-0\\] [$(x_0^5)^{*} $]{}; (L5-1) \\[snode, left=of L5-0\\] [$x_1^5$]{}; (L5-2) \\[mnode, left=of L5-1\\] ; (L5-3) \\[mnode, left=of L5-2\\] [$(x_3^5)^*$]{}; (L5-4) \\[mnode, left=of L5-3\\] [$x_4^5$]{}; (L5-5) \\[nnewnode, left=of L5-4\\] [$x_5^5$]{}; (L5-label)\\[label, below=of S4-rot-label\\] [${\\mathit{SAT}((x_0^4+1)S_4)}$]{};\n\n(S5-0) \\[snode, below=of L5-0\\] [$(x_0^5)^{*}$]{}; (S5-1) \\[snode, left=of S5-0\\] [$x_1^5$]{}; (S5-2) \\[snode, left=of S5-1\\] ; (S5-3) \\[snode, left=of S5-2\\] ; (S5-4) \\[mnode, left=of S5-3\\] [$(x_4^5)^{*}$]{}; (S5-5) \\[nnewnode,left=of S5-4\\] [$x_5^5$]{}; (S5-label)\\[label, below=of L5-label\\] [$S_5$]{};\n\n(S5-rot-0) \\[snode, below=1cm of S5-0\\][$\\vdots$]{}; (S5-rot-1) \\[snode, left = of S5-rot-0\\] [$\\vdots$]{}; (S5-rot-2) \\[mnode, left=of S5-rot-1\\]; (S5-rot-3) \\[mnode, left=of S5-rot-2\\]; (S5-rot-4)\\[mnode, left =of S5-rot-3\\][$\\vdots$]{}; (S5-rot-5)\\[mnode, left =of S5-rot-4\\][$\\vdots$]{}; (S5-rot-6) \\[newnode, left =of S5-rot-5\\][$\\vdots$]{}; (S5-rot-label) \\[label,right=of S5-rot-0\\] [$\\vdots$]{};\n\n(S0-0) -&gt; (S1-0); (S0-1) -&gt; (S1-1); \\[evaluate=[=int(-1);=int(-2);]{}\\] in [2,...,5]{}[ iin [0,...,]{}[ ]{} ]{} (S1-rot-2) -&gt; (S2-rot-3); (S2-rot-3) -&gt; (S3-rot-4); (S3-rot-4) -&gt; (S4-rot-5); (S4-rot-5) -&gt; (S5-rot-6);\n\n(S2-2) -&gt; (S3-3); (S3-3) -&gt; (S4-4); (S4-4) -&gt; (S5-5);\n\n(S3-2) -&gt; (S4-3); (S4-3) -&gt; (S5-4);\n\n\\[evaluate=[=int(-1);=int(-2);]{}\\] in [1,...,5]{}[ iin [0,...,]{}[ ]{} ]{}\n\n; ; ; ; ; ; ; ;\n\nGiven a length-2 simple expression $S_0$ where the rightmost symbol is starred, the algorithm will first saturate (\\[def:discall\\], in line 1), and then alternatingly rotate a copy of the rightmost symbol (\\[oneclock:rota\\]), and saturate the result (see lines 2, 3, 6). Since each such round extends the length of the expression by one, we additionally collapse them (in line 7) by adding an extra Kleene star to the symbol at the second position. The crucial observation for the correctness of this procedure is that the subsumption step in line 7 does not change the cover sets of the respective expressions.\n\nObserve that \\[alg:inner\\_loop\\] is well defined because the ${\\mathit{SAT}(S_i)}$ are computable by \\[lem:discall\\]. Termination is guaranteed by the following simple observation.\n\n\\[lem:termobs\\] Let $x_j^i\\in\\Sigma$ be the symbols computed by \\[alg:inner\\_loop\\]. Then\n\n1.  $x_{j}^{i+1}\\supseteq x_{j}^{i}$, for all $i>j\\ge 0$.\n\n2.  $x_{i}^{i}\\supseteq x_{i-1}^{i-1}$ and $x_{i}^{i+1}\\supseteq x_{i-1}^{i}$, for all $i\\ge 3$.\n\nThe first item is guaranteed by Point\u00a02 of \\[lem:discall\\]. In particular this means that $x_0^{i+1}\\supseteq x_0^{i}$ and therefore that $(x_0^{i+1}+1)\\supseteq (x_0^{i}+1)$ for all $i\\ge 0$ (indicated as red arrows in \\[fig:algorithm1\\]). The second item now follows from this observation by \\[lem:simulation\\].\n\n\\[lem:termination\\] \\[alg:inner\\_loop\\] terminates with $i \\le 4\\cdot {{\\lvertP\\rvert}}\\cdot({c_\\mathit{max}}+1)$.\n\nFrom \\[lem:termobs\\] we deduce that for all $i\\ge2$, the expression $S_{i+1}$ is point-wise larger than or equal to $S_i$ with respect to the subset ordering on symbols. The claim now follows from the observation that all expressions $S_{i\\ge 3}$ have length $4$ and that every symbol $x_i\\in\\Sigma$ can only increase at most ${{\\lvertP\\rvert}}\\cdot ({c_\\mathit{max}}+1)$ times.\n\n\\[lem:acc:correctness\\] Suppose that $S_1,S_\\ell, R$ be the expressions computed by \\[alg:inner\\_loop\\] applied to the simple expression $x_1x_0^*$. Then ${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![x_1x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n    = {[\\![S_1]\\!]}\n    \\cup {[\\![S_\\ell]\\!]}\n    \\cup {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![R]\\!]}{\\aftergroup\\egroup\\originalright})}\n    $.\n\nLet $S_1,\\ldots S_\\ell$ denote the expressions defined in lines 1,2,3, and 7 of the algorithm. That is, $\\ell$ is the least index $i$ such that $S_{i+1}=S_{i}$. We define a sequence $E_i$ of expressions inductively, starting with $E_1{{\\overset{\\text{\\tinydef}}{=}}}S_1$ and if $E_i=e_i^ie_{i-1}^i\\dots e_0^i$, we let $\nE_{i+1} {{\\overset{\\text{\\tinydef}}{=}}}e_{i+1}^{i+1}e_i^{i+1}e_{i-1}^{i+1}\\dots e_0^{i+1}\n{{\\overset{\\text{\\tinydef}}{=}}}{\\mathit{SAT}((\\hat{e}_0^i+1)E_i)}\n$. Here, the superscript indicates the position of a symbol and not iteration. This is the sequence of expressions resulting from unfolding \\[lem:accstep\\], interleaved with saturation steps, just in line 6 of the algorithm. That is, the expressions $E_i$ are *not* collapsed (line 7) and instead grow in length with $i$. Still, $E_1=S_1$, $E_2=S_2$ and $E_2=S_3$, but $E_4\\neq S_4$, because the latter is the result of applying the subsumption step of line $7$ in our algorithm. Notice that ${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![x_1x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})} = {\\mathopen{}\\mathclose\\bgroup\\originalleft}(\\bigcup_{k-1\\ge i\\ge 1}{[\\![E_i]\\!]}{\\aftergroup\\egroup\\originalright}) \\cup {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E_k]\\!]}{\\aftergroup\\egroup\\originalright})}$ holds for all $k\\in{\\mathbb{N}}$. We will use that $$\\label{eq:acc:EisS}\n\\bigcup_{i\\ge 2}{[\\![E_i]\\!]}\n= \\bigcup_{i\\ge 2}{[\\![S_i]\\!]}\n={[\\![S_\\ell]\\!]}.$$ We start by observing that for all $i,j\\in{\\mathbb{N}}$ it holds that $e_j^i = x_j^i$. For $i\\le 3$ this holds trivially by definition of $E_i=S_i$. For larger $i$, this can be seen by induction using \\[lem:discall\\]. Towards the first equality in \\[eq:acc:EisS\\], let $S_i^j$ be the expression resulting from $S_i = x_{i}^{i}({x_{i-1}^{i}})^{*}x_{1}^{i}({x_0^{i}})^*$ by unfolding the first star $j$ times. That is, $S_i^j{{\\overset{\\text{\\tinydef}}{=}}}x_{i}^{i}({x_{i-1}^{i}})^{(j)}x_{1}^{i}(x_0^{i})^*$, where the superscript $(j)$ denotes $j$-fold concatenation. Clearly, ${[\\![S_i]\\!]}=\\bigcup_{j\\ge 0}{[\\![S_i^j]\\!]}$ and so the $\\supseteq$-direction of the first equality in \\[eq:acc:EisS\\] follows by $$\\begin{aligned}\n{[\\![S_i^j]\\!]}= {[\\![x_{i}^{i}({x_{i-1}^{i}})^{(j)}x_{1}^{i}(x_0^{i})^*]\\!]}\n&\\subseteq\n{[\\![x_{i+j}^{i+j}\n{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\n{x_{i+j-1}^{i+j}}\n{x_{i+j-2}^{i+j}}\n\\ldots\n{x_{i}^{i+j}}\n{\\aftergroup\\egroup\\originalright})\nx_{1}^{i+1}(x_0^{i+1})^*]\\!]}\\\\\n&\\subseteq\n{[\\![x_{i+j}^{i+j}\n{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\n{x_{i+j-1}^{i+j}}\n{x_{i+j-2}^{i+j}}\n\\ldots\n{x_{i}^{i+j}}\n{\\aftergroup\\egroup\\originalright})\n{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\n{x_{i-1}^{i+j}}\n\\ldots\n{x_{2}^{i+j}}\n{\\aftergroup\\egroup\\originalright})\nx_{1}^{i+1}(x_0^{i+j})^*]\\!]}\\\\\n&={[\\![E_{i+j}]\\!]},\\end{aligned}$$ where the first inclusion is due to \\[lem:termobs\\]. The same helps for the other direction: $${[\\![E_i]\\!]} \n= {[\\![x_{i}^{i}x_{i-1}^{i}x_{i-2}^{i}\\dots x_2^ix_1^i x_0^{i}]\\!]}\n\\subseteq {[\\![x_{i}^{i}{(x_{i-1}^{i})}^{(i-2)}x_{1}^{i}x_0^{i}]\\!]}\n={[\\![S_i^{i-2}]\\!]}\n={[\\![S_i]\\!]},$$ which completes the proof of the first equality in \\[eq:acc:EisS\\]. The second equality holds because ${[\\![S_i]\\!]}\\subseteq{[\\![S_{i+1}]\\!]}$ for all $i\\ge 2$, by \\[lem:termobs\\], and by definition of $S_\\ell=S_{\\ell+1}$. As a next step we show that $$\\label{eq:acc:Rconnection}\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_\\ell]\\!]}{\\aftergroup\\egroup\\originalright})}\n= \n{[\\![S_\\ell]\\!]}\n\\cup\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![R]\\!]}{\\aftergroup\\egroup\\originalright})}$$ First observe that $\n{[\\![R]\\!]}\n={[\\![(x_1^\\ell+1){(x_{\\ell-1}^\\ell)}^*]\\!]}\n={[\\![(x_1^\\ell+1)x_\\ell^\\ell {(x_{\\ell-1}^\\ell)}^*]\\!]}\n$ and consequently,\n\n$$\\begin{aligned}\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![R]\\!]}{\\aftergroup\\egroup\\originalright})}\n&=\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![(x_1^\\ell+1)x_\\ell^\\ell {(x_{\\ell-1}^\\ell)}^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n\\\\\n&\\subseteq\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![x_\\ell^\\ell {(x_{\\ell-1}^\\ell)}^*x_1^\\ell]\\!]}{\\aftergroup\\egroup\\originalright})}\\\\\n&\\subseteq\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![x_\\ell^\\ell {(x_{\\ell-1}^\\ell)}^*x_1^\\ell {(x_0^\\ell)}^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n={\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_\\ell]\\!]}{\\aftergroup\\egroup\\originalright})}\\end{aligned}$$\n\nwhere the first equation follows by \\[cor:nonstarrotation\\] and the second because ${{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(x_\\ell^\\ell {(x_{\\ell-1}^\\ell)}^*x_1^\\ell{\\aftergroup\\egroup\\originalright})}\\subseteq {{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(x_\\ell^\\ell {(x_{\\ell-1}^\\ell)}^*x_1^\\ell {(x_0^\\ell)}^*{\\aftergroup\\egroup\\originalright})}$. For the left to right inclusion in \\[eq:acc:Rconnection\\], consider a marking $M\\in{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_\\ell]\\!]}{\\aftergroup\\egroup\\originalright})}\\setminus {[\\![S_\\ell]\\!]}$. We show that $M\\in{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![R]\\!]}{\\aftergroup\\egroup\\originalright})}$. Recall that ${\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_\\ell]\\!]}{\\aftergroup\\egroup\\originalright})}$ consists of all those markings $M$ so that there exists a finite path $$M_0{\\xrightarrow{*}_{\\textit{Disc}}}M'_0{\\xrightarrow{d_1}_{\\textit{Time}}}M_1{\\xrightarrow{*}_{\\textit{Disc}}}M'_1{\\xrightarrow{d_2}_{\\textit{Time}}}M_2\\dots M'_{k-1}{\\xrightarrow{*}_{\\textit{Disc}}}M_k$$ alternating between timed and (sequences of) discrete transition steps, with $M_0\\in{[\\![S_\\ell]\\!]}$, $M_k\\ge M$ and all $d_i\\le \\max({{\\it frac}(M'_i)})$.\n\nBy our choice of $M$, there must be a first expression in the sequence which is not a member of ${[\\![S_\\ell]\\!]}$. Since ${[\\![{\\mathit{SAT}(S_\\ell)}]\\!]} = {[\\![S_\\ell]\\!]}$, we can assume an index $i>0$ so that $M_i\\notin {[\\![S_\\ell]\\!]}$ but $M'_{i-1}\\in {[\\![S_\\ell]\\!]}$ that is, the step that takes us out of ${[\\![S_\\ell]\\!]}$ is a timed step.\n\nBecause ${[\\![S_\\ell]\\!]}=\\bigcup_{i\\ge 2}{[\\![S_i]\\!]}$, it must hold that $M'_{i-1} \\in {[\\![S_j]\\!]} ={[\\![x_j^j(x_{j-1}^j)^* x_1^j(x_0^j)^*]\\!]}$ for some index $j\\ge 2$. We claim that it already holds that $$\\label{eq:acc:almostR}\n    M'_{i-1} \\in {[\\![x_j^j{(x_{j-1}^j)}^*x_1^j]\\!]}.$$ Suppose not. If $d_i<\\max({{\\it frac}(M'_{i-1})})$ then $M_i\\in{[\\![\\emptyset S_j]\\!]}\\subseteq {[\\![S_j]\\!]}$ by \\[lem:epsilonsteps\\], contradiction. Otherwise, if $d_i=\\max({{\\it frac}(M'_{i-1})})$, notice that every abstraction ${\\mathit{abs}_{S}(M'_{i-1})}\\in{{\\mathcal{L}}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(S_j{\\aftergroup\\egroup\\originalright})}$ must have ${{\\lvertS\\rvert}}=4$. So by \\[lem:nonstarrotation\\], $M_i\\in {[\\![(x_0^j+1)S_j]\\!]}$. But then again $${[\\![(x_0^j+1)S_j]\\!]}\n\\subseteq\n{[\\![{\\mathit{SAT}((x_0^j+1)S_j)}]\\!]}\n\\subseteq\n{[\\![S_{j+1}]\\!]},$$ contradicting our assumption that $M_i\\notin {[\\![S_\\ell]\\!]}$. Therefore \\[eq:acc:almostR\\] holds. By \\[lem:nonstarrotation\\] we derive that $M_i\\in{[\\![(x_1^j+1)x_j^j(x_{j-1}^j)^*]\\!]}\n={[\\![(x_1^j+1)(x_{j-1}^j)^*]\\!]}\n\\subseteq{[\\![(x_1^\\ell+1)(x_{\\ell-1}^\\ell)^*]\\!]}\n={[\\![R]\\!]}$. This concludes the proof of \\[eq:acc:Rconnection\\].\n\nNotice that by \\[lem:accstep\\] we have that $$\\label{eq:acc:EisSunr1}\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![x_1 x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n={[\\![{\\mathit{SAT}(x_1 x_0^*)}]\\!]}\n\\cup{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![{\\mathit{SAT}(x_1 x_0^*)}]\\!]}{\\aftergroup\\egroup\\originalright})}\n={[\\![S_1]\\!]}\n\\cup{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_1]\\!]}{\\aftergroup\\egroup\\originalright})}.$$ Analogously, we get for every $i\\ge 1$ that $$\\begin{aligned}\n    \\label{eq:acc:2}\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E_i]\\!]}{\\aftergroup\\egroup\\originalright})}\n= \n{[\\![{\\mathit{SAT}(E_i)}]\\!]}\n\\cup {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![{\\mathit{SAT}((x^i_0+1)E_i)}]\\!]}{\\aftergroup\\egroup\\originalright})}\n=\n  {[\\![E_i]\\!]}\n  \\cup {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E_{i+1}]\\!]}{\\aftergroup\\egroup\\originalright})}\\end{aligned}$$ This used \\[lem:accstep\\] and the fact that ${\\mathit{SAT}(E_i)}=E_i$ by construction. Using \\[eq:acc:2\\] and that ${[\\![E_i]\\!]}\\subseteq{[\\![E_{i+1}]\\!]}$ for $i\\ge 2$, we deduce $$\\label{eq:acc:3}\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_1]\\!]}{\\aftergroup\\egroup\\originalright})}\n={\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E_1]\\!]}{\\aftergroup\\egroup\\originalright})}\n=\n{[\\![E_1]\\!]}\n\\cup\n{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\n\\bigcup_{i\\ge 2}{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![E_i]\\!]}{\\aftergroup\\egroup\\originalright})}\n{\\aftergroup\\egroup\\originalright}).$$ Finally we can conclude the desired result as follows. $$\\begin{aligned}\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![x_1 x_0^*]\\!]}{\\aftergroup\\egroup\\originalright})}\n&{\\overset{\\text{\\tiny(\\ref{eq:acc:EisSunr1})}}{=}}\n{[\\![S_1]\\!]}\n\\cup\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_1]\\!]}{\\aftergroup\\egroup\\originalright})}\n{\\overset{\\text{\\tiny(\\ref{eq:acc:3})}}{=}}\n{[\\![S_1]\\!]}\n\\cup\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}(\n\\bigcup_{i\\ge 2}{[\\![E_i]\\!]}\n{\\aftergroup\\egroup\\originalright})}\\\\\n&{\\overset{\\text{\\tiny(\\ref{eq:acc:EisS})}}{=}}\n{[\\![S_1]\\!]}\n\\cup\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![S_\\ell]\\!]}{\\aftergroup\\egroup\\originalright})}\\\\\n&{\\overset{\\text{\\tiny(\\ref{eq:acc:Rconnection})}}{=}}\n{[\\![S_1]\\!]}\n\\cup\n{[\\![S_\\ell]\\!]}\n\\cup\n{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({[\\![R]\\!]}{\\aftergroup\\egroup\\originalright})}\n\\qedhere\\end{aligned}$$\n\nMain Result {#sec:central}\n-----------\n\nThe following theorem summarizes our main claims regarding the [$\\exists$COVER ]{}problem.\n\n\\[thm:central\\] Consider an instance of [$\\exists$COVER ]{}with ${\\cal N} = (P,T, {\\mathit{Var}}, G, {\\mathit{Pre}},{\\mathit{Post}})$ a non-consuming TPN where ${c_\\mathit{max}}$ is the largest constant appearing in the transition guards $G$ encoded in unary, and let $p$ be an initial place and $t$ be a transition.\n\n1.  \\[thm:cent:1\\] The number of different simple expressions of length $m$ is $B(m) {{\\overset{\\text{\\tinydef}}{=}}}2^{({{\\lvertP\\rvert}} \\cdot ({c_\\mathit{max}}+2) \\cdot m)+m}$.\n\n2.  \\[thm:cent:2\\] It is possible to compute a symbolic representation of the set of markings coverable from some marking in the initial set ${\\mathbb{N}}\\cdot \\{(p,{0})\\}$, as a finite set of simple expressions. I.e., one can compute simple expressions $S_1,\\dots,S_\\ell$ s.t. $\\bigcup_{1\\le i\\le \\ell}{[\\![S_i]\\!]} =  {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({\\mathbb{N}}\\cdot \\{(p,{0})\\}{\\aftergroup\\egroup\\originalright})}$ and where $\\ell \\le 3\\cdot B(2)$. Each of the $S_i$ has length either $2$ or $4$.\n\n3.  \\[thm:cent:3\\] Checking if there exists $M\\in{\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({\\mathbb{N}}\\cdot \\{(p,0)\\}{\\aftergroup\\egroup\\originalright})}$ with $M{\\longrightarrow_{t}}$ can be done in ${\\mathcal{O}}({{\\lvertP\\rvert}}\\cdot{c_\\mathit{max}})$ deterministic space.\n\nFor \\[thm:cent:1\\] note that a simple expression is described by a word where some symbols have a Kleene star. There are ${{\\lvert\\Sigma\\rvert}}^m$ different words of length $m$ and $2^m$ possibilities to attach stars to symbols. Since the alphabet is $\\Sigma {{\\overset{\\text{\\tinydef}}{=}}}2^{P{\\times}{[{{c_\\mathit{max}}+1}]}}$ and ${{\\lvert{[{{c_\\mathit{max}}+1}]}\\rvert}}={c_\\mathit{max}}+2$, the result follows.\n\nTowards \\[thm:cent:2\\], we can assume w.l.o.g.\u00a0that our TPN is non-consuming by \\[lem:mtpn:wlog\\], and thus the region abstraction introduced in \\[sec:regionabs\\] applies. In particular, the initial set of markings ${\\mathbb{N}}\\cdot \\{(p,{0})\\}$ is represented exactly by the expression $S_0 {{\\overset{\\text{\\tinydef}}{=}}}\\{(p,0)\\}\\emptyset^*$ where $\\emptyset \\in \\Sigma$ is the symbol corresponding to the empty set. That is, we have ${[\\![S_0]\\!]} = {\\mathbb{N}}\\cdot \\{(p,{0})\\}$ and thus ${\\mathit{Cover}_{}({[\\![S_0]\\!]})} = {\\mathit{Cover}_{}({\\mathbb{N}}\\cdot \\{(p,{0})\\})}$.\n\nThe claimed expressions $S_i$ are the result of iterating \\[alg:inner\\_loop\\] until a previously seen expression is revisited. Starting at $i=0$ and $S_0 {{\\overset{\\text{\\tinydef}}{=}}}\\{(p,0)\\}\\emptyset^*$, each round will set $S_{i+1},S_{i+2}$ and $S_{i+3}$ to the result of applying \\[alg:inner\\_loop\\] to $S_i$, and increment $i$ to $i+3$.\n\nNotice that then all $S_i$ are simple expressions of length $2$ or $4$ and that in particular, all expressions with index divisible by $3$ are of the form $ab^*$ for $a,b\\in\\Sigma$. Therefore after at most $B(2)$ iterations, an expression $S_\\ell$ is revisited (with $\\ell\\le 3B(2)$). Finally, an induction using \\[lem:acc:correctness\\] provides that $\\bigcup_{1\\le i\\le \\ell}{[\\![S_i]\\!]} =  {\\mathit{Cover}_{}{\\mathopen{}\\mathclose\\bgroup\\originalleft}({\\mathbb{N}}\\cdot \\{(p,{0})\\}{\\aftergroup\\egroup\\originalright})}$.\n\nTowards \\[thm:cent:3\\], we modify the above algorithm for the [$\\exists$COVER ]{}problem with the sliding window technique. The algorithm is the same as above where instead of recording all the expressions $S_1,\\dots,S_\\ell$, we only store the most recent ones and uses them to decide whether the transition $t$ is enabled. If the index $i$ reaches the maximal value of $3\\cdot B(2)$ we return unsuccessfully.\n\nThe bounded index counter uses ${\\mathcal{O}}(\\log(B(2)))$ space; uses space ${\\mathcal{O}}(\\log(B(5)))$ because it stores only simple expressions of length $\\le 5$. The space required to store the three expressions resulting from each application of \\[alg:inner\\_loop\\] is ${\\mathcal{O}}(3 \\cdot \\log(B(4)))$. For every encountered simple expression we can check in logarithmic space whether the transition $t$ is enabled by some marking in its denotation. Altogether the space used by our new algorithm is bounded by ${\\mathcal{O}}(\\log(B(5)))$. By \\[thm:cent:1\\], this is ${\\mathcal{O}}(|P|\\cdot ({c_\\mathit{max}}+2))={\\mathcal{O}}({{\\lvertP\\rvert}}\\cdot{c_\\mathit{max}})$.\n\nThe [$\\exists$COVER ]{}problem for TPN is [$\\mathsf{PSPACE}$]{}-complete.\n\nThe [$\\mathsf{PSPACE}$]{}\u00a0lower bound was shown in \\[thm:LB\\]. The upper bound follows from \\[lem:mtpn:wlog\\] and \\[thm:cent:3\\] of \\[thm:central\\].\n\nConclusion and Future Work {#sec:conclusion}\n==========================\n\nWe have shown that *Existential Coverability* (and its dual of universal safety) is [$\\mathsf{PSPACE}$]{}-complete for TPN with one real-valued clock per token. This implies the same complexity for checking safety of arbitrarily large timed networks without a central controller. The absence of a central controller makes a big difference, since the corresponding problem *with* a central controller is complete for $F_{\\omega^{\\omega^\\omega}}$ [@HSS2012].\n\nIt remains an open question whether these positive results for the controller-less case can be generalized to multiple real-valued clocks per token. In the case *with* a controller, safety becomes undecidable already for two clocks per token [@ADM2004].\n\nAnother question is whether our results can be extended to more general versions of timed Petri nets. In our version, clock values are either inherited, advanced as time passes, or reset to zero. However, other versions of TPN allow the creation of output-tokens with new non-deterministically chosen non-zero clock values, e.g., the timed Petri nets of [@AMM07; @AA2001] and the read-arc timed Petri nets of [@BHR2006].\n\n\\[sec:appendix\\]\n\nProof of \\[lem:discall\\]\n========================\n\n\\[lem:discall-starfree\\] For every non-consuming TPN ${\\mathcal{N}}$ there are polynomial time computable functions $f: \\Sigma \\times \\Sigma \\times \\Sigma \\to \\Sigma$ and $g: \\Sigma \\times \\Sigma \\times \\Sigma \\to \\Sigma$ with the following properties.\n\n1.  \\[lem:discall-starfree-ad1\\] $f$ and $g$ are monotone (w.r.t.\u00a0subset ordering) in each argument.\n\n2.  \\[lem:discall-starfree-ad2\\] $f(\\alpha, \\beta, x) \\supseteq x$ and $g(\\alpha, \\beta, x) \\supseteq x$ for all $\\alpha,\\beta,x \\in \\Sigma$.\n\n3.  \\[lem:discall-starfree-ad3\\] For every word $w=x_0x_1\\dots x_k$ over $\\Sigma$, $\\alpha {{\\overset{\\text{\\tinydef}}{=}}}x_0$ and $\\beta {{\\overset{\\text{\\tinydef}}{=}}}\\bigcup_{i>0} x_i$, and\\\n    $w' {{\\overset{\\text{\\tinydef}}{=}}}f(\\alpha,\\beta,x_0)g(\\alpha,\\beta,x_1)\\dots g(\\alpha,\\beta,x_k)$ we have ${[\\![w']\\!]} = \n        \\{M'' \\mid \\exists M \\in {[\\![w]\\!]} \\land M{{\\xrightarrow{*}_{\\textit{Disc}}}}M' \\ge M''\\}\n        $.\n\n(Sketch). It suffices to show the existence of such functions $f_t$ and $g_t$ for individual transitions $t \\in T$ and ${\\longrightarrow_{t}}$ instead of ${{\\xrightarrow{*}_{\\textit{Disc}}}}$. The functions $f$ and $g$ can then be obtained by iterated applications of $f_t$ and $g_t$ (for all transitions $t$) until convergence. (In addition to expanding $x$, the results of each application $f_t$ and $g_t$ are also added to $\\alpha$ and $\\beta$, respectively.) This works, because the functions $f_t$ and $g_t$ are monotone and operate on the finite domain/range $\\Sigma$. Since we have a polynomial number of transitions, and each symbol in $\\Sigma$ can increase (by strict subset ordering) at most ${{\\lvertP\\rvert}} \\cdot ({c_\\mathit{max}}+1)$ times, the number of iterations is polynomial. Moreover, the properties of \\[lem:discall-starfree-ad1\\], \\[lem:discall-starfree-ad2\\] and \\[lem:discall-starfree-ad3\\] carry over directly from $f_t$ and $g_t$ to $f$ and $g$, respectively.\n\nNow we consider the definitions and properties of the functions $f_t$ and $g_t$ for a particular transition $t$. Given a variable evaluation $\\pi:{\\mathit{Var}}\\to {{\\mathbb R}_{\\geq 0}}$, we define the functions $\\pi_0$ and $\\pi_{>0}$ from sets over $(P{\\times}{\\mathit{Var}})$ to sets over $(P{\\times}{\\mathbb{N}})$ as follows. Intuitively, they cover the parts of the assignment $\\pi$ with zero/nonzero fractional values, respectively. Let $\\pi_0(S) {{\\overset{\\text{\\tinydef}}{=}}}\\{(p,c) \\,|\\, (p,y) \\in S\\ \\wedge\\ \\pi(y)=c \\in {\\mathbb{N}}\\}$ and $\\pi_{>0}(S) {{\\overset{\\text{\\tinydef}}{=}}}\\{(p,c) \\,|\\, (p,y) \\in S\\ \\wedge\\ \\lfloor\\pi(y)\\rfloor=c\n\\ \\wedge\\ {{\\it frac}}(\\pi(y)) >0\\}$. The definitions are lifted to multisets in the straightforward way.\n\nNow let $t$ be a transition. We say that $(\\alpha,\\beta)$ enables $t$ iff $\\exists \\pi:{\\mathit{Var}}\\to {{\\mathbb R}_{\\geq 0}}$ such that $\\pi(y)\\in G(t)(y)$ for all variables $y$ and $\\pi_0({\\mathit{Pre}}(t)) \\subseteq \\alpha$ and $\\pi_{>0}({\\mathit{Pre}}(t)) \\subseteq \\beta$. Thus if ${\\mathit{abs}_{}(M)} = x_0x_1\\dots x_n$ then $M$ enables $t$ iff $(x_0, \\bigcup_{i>0} x_i)$ enables $t$, since all transition guards in $G(t)$ are intervals bounded by integers (i.e., $t$ cannot distinguish between different nonzero fractional values). Moreover, enabledness can be checked in polynomial time (choose integers for the part in $\\alpha$ and rationals with fractional part $1/2$ for the part in $\\beta$).\n\nIn the case where $(\\alpha,\\beta)$ does not enable $t$ we just let $g_t(\\alpha,\\beta,x) {{\\overset{\\text{\\tinydef}}{=}}}x$ and $f_t(\\alpha,\\beta,x) {{\\overset{\\text{\\tinydef}}{=}}}x$. The conditions above are trivially satisfied in this case.\n\nIn the case where $(\\alpha,\\beta)$ enables $t$, let $g_t(\\alpha,\\beta,x) {{\\overset{\\text{\\tinydef}}{=}}}x \\cup \\gamma$ where $\\gamma$ is defined as follows. We have $(p,c) \\in \\gamma$ iff there is a $(p,y) \\in {\\mathit{Post}}(t)$ and $(q, y) \\in {\\mathit{Pre}}(t)$ such that $(q,c) \\in x$. Similarly, let $f_t(\\alpha,\\beta,x) {{\\overset{\\text{\\tinydef}}{=}}}x \\cup \\gamma$ where $\\gamma$ is defined as follows. We have $(p,c) \\in \\gamma$ iff either (1) there is a $(p,y) \\in {\\mathit{Post}}(t)$ and $(q, y) \\in {\\mathit{Pre}}(t)$ such that $(q,c) \\in x$, or (2) $c=0$ and there is a $(p,0) \\in {\\mathit{Post}}(t)$. All these conditions can be checked in polynomial time. \\[lem:discall-starfree-ad1\\] and \\[lem:discall-starfree-ad2\\] follow directly from the definition.\n\nTowards \\[lem:discall-starfree-ad3\\], we show ${[\\![w']\\!]} \\supseteq \\{M'' \\mid \\exists M \\in {[\\![w]\\!]} \\land M\n{\\longrightarrow_{t}} M' \\ge M''\\}$. (The proof of the reverse inclusion $\\subseteq$ is similar.) Let $w=x_0x_1\\dots x_k$, $\\alpha {{\\overset{\\text{\\tinydef}}{=}}}x_0$, $\\beta {{\\overset{\\text{\\tinydef}}{=}}}\\bigcup_{i>0} x_i$ such that $(\\alpha,\\beta)$ enables $t$ and $w' {{\\overset{\\text{\\tinydef}}{=}}}f_t(\\alpha,\\beta,x_0)g_t(\\alpha,\\beta,x_1)\\dots g_t(\\alpha,\\beta,x_k)$. If $M \\in {[\\![w]\\!]}$ and $M {\\longrightarrow_{t}} M'$ then $M' \\ge M$ since ${\\mathcal{N}}$ is non-consuming. We show that every additional token $(p,u) \\in M' \\ominus M$ is included in ${[\\![w']\\!]}$. (This implies the inclusion above, since $M' \\ominus M \\ge M'' \\ominus M$.) For every additional token $(p,u) \\in M' \\ominus M$ there are two cases.\n\n-   Assume ${{\\it frac}}(u) >0$. Then the token $(p,u)$ must have inherited its clock value from some token $(q,u) \\in M$ via a variable $y$ specified in the Pre/Post of $t$ (since discrete transitions cannot create new fractional parts of clock values). This case is covered by $\\gamma$ in the definition of $g_t$ above. In particular, if $(q,u) \\in M$ was abstracted to $x_i$ in $w$ then $(p,u) \\in M'$ is abstracted to $g_t(\\alpha,\\beta,x_i)$ in $w'$.\n\n-   Assume ${{\\it frac}}(u)=0$. Then there are two cases. In the first case the token $(p,u)$ inherited its clock value from some token $(q,u) \\in M$ via a variable $y$ specified in the Pre/Post of $t$. This case is covered by part (1) of $\\gamma$ in the definition of $f_t$ above. In particular, $(q,u) \\in M$ was abstracted to $x_0$ in $w$, because ${{\\it frac}}(u)=0$. Thus $(p,u) \\in M'$ is abstracted to $f_t(\\alpha,\\beta,x_0)$ in $w'$. In the second case the token $(p,u)$ got its clock value via a clock-reset to zero. This case is covered by part (2) of $\\gamma$ in the definition of $f_t$ above. In particular, in this case we must have $u=0$, and $(p,0) \\in M'$ was abstracted to $f_t(\\alpha,\\beta,x_0)$ in $w'$.\n\nIt follows that ${\\mathit{abs}_{}(M')} \\le w'$, i.e., by the ordering on symbols in $\\Sigma$, every letter in ${\\mathit{abs}_{}(M')}$ is smaller than the corresponding letter in $w'$. Thus $M' \\in {[\\![w']\\!]}$. Since $M' \\ge M''$ and ${[\\![w']\\!]}$ is downward closed, we also have $M'' \\in {[\\![w']\\!]}$ as required.\n\nLet $f$ and $g$ be the functions from \\[lem:discall-starfree\\], which immediately yields \\[lem:discall-ad1\\] and \\[lem:discall-ad2\\]. Towards \\[lem:discall-ad3\\], consider all words $w$ in ${\\mathcal{L}}(E)$ that contain each starred symbol in $E$ at least once. (The other cases are irrelevant for ${[\\![E]\\!]}$ since they are subsumed by monotonicity.) For each such word $w$, the $\\alpha, \\beta$ derived from $w$ in \\[lem:discall-starfree\\] are the same as the $\\alpha, \\beta$ derived from $E$ in \\[lem:discall-ad3\\]. If $x_i$ in $E$ carries a star then $w$ contains a corresponding nonempty subsequence $x_i\\dots x_i$. We apply \\[lem:discall-starfree\\] to each such $w$ to obtain the corresponding $w'$. The word $w'$ then contains the corresponding subsequence $g(\\alpha,\\beta,x_i)\\dots g(\\alpha,\\beta,x_i)$. Let $E'$ then be defined as in \\[lem:discall-ad3\\], i.e., by applying functions to the symbols and keeping the stars at the same symbols as in $E$. By \\[lem:discall-starfree\\], this is computable in polynomial time. We have ${\\mathcal{L}}(E') = \\bigcup_{w \\in {\\mathcal{L}}(E)} \\{w'\\}$. Thus ${[\\![E']\\!]} = \\bigcup_{w \\in {\\mathcal{L}}(E)} {[\\![w']\\!]}\n=\n\\bigcup_{w \\in {\\mathcal{L}}(E)} \\{M'' \\mid \\exists M \\in {[\\![w]\\!]} \\land\nM{{\\xrightarrow{*}_{\\textit{Disc}}}}M' \\ge M''\\}\n=\n\\{M'' \\mid \\exists M \\in {[\\![E]\\!]} \\land M{{\\xrightarrow{*}_{\\textit{Disc}}}}M' \\ge M''\\}\n$ for \\[lem:discall-ad3\\] as required.\n"]]