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The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving.
4. Maxwell's Equations
\[ \nabla \cdot E = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot B = 0 \]
\[ \nabla \times E = -\frac{\partial B}{\partial t} \]
\[ \nabla \times B = \mu_0 J + \mu_0 \varepsilon_0 \frac{\partial E}{\partial t} \]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[ \int_a^b f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving.
4. Maxwell's Equations
\[
\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \\
\nabla \cdot \mathbf{B} = 0 \\
\n\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \\
\n\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}
\]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
4. Maxwell's Equations
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot \mathbf{B} = 0 \]
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
| 1. Euler’s Identity | 2. Pythagorean Theorem |
|---------------------|------------------------|
| \( e^{i\pi} + 1 = 0 \) | \( a^2 + b^2 = c^2 \) |
| Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry. | In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture. |
| 3. The Fundamental Theorem of Calculus | 4. Maxwell’s Equations |
|----------------------------------------|------------------------|
| \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \) | \(\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \)
| Establishes that differentiation and integration are inverse operations. If \( F \) is an antiderivative of \( f \), the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving. | \( \nabla \cdot \mathbf{B} = 0 \)
| \( \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \) | \( \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \) |
| Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones. |
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
3. The Fundamental Theorem of Calculus
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
4. Maxwell's Equations
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot \mathbf{B} = 0 \]
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. Euler’s Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
3. The Fundamental Theorem of Calculus
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
4. Maxwell’s Equations
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot \mathbf{B} = 0 \]
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. Euler's Identity
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. Pythagorean Theorem
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. The Fundamental Theorem of Calculus
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals \( F(b) - F(a) \). Revolutionized mathematical problem-solving.
4. Maxwell's Equations
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot \mathbf{B} = 0 \]
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
The 20 Most Important Mathematical Equations
A journey through the most elegant and influential formulas in mathematics
1. **Euler’s Identity**
\[ e^{i\pi} + 1 = 0 \]
Connects five fundamental constants (e, i, π, 1, 0), revealing the profound relationship between exponential functions and trigonometry.
2. **Pythagorean Theorem**
\[ a^2 + b^2 = c^2 \]
In right triangles, the hypotenuse squared equals the sum of the squares of the other sides. Cornerstone of geometry with applications in navigation and architecture.
3. **The Fundamental Theorem of Calculus**
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]
Establishes that differentiation and integration are inverse operations. If F is an antiderivative of f, the definite integral equals F(b) - F(a). Revolutionized mathematical problem-solving.
4. **Maxwell’s Equations**
\[ \nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0} \]
\[ \nabla \cdot \mathbf{B} = 0 \]
\[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
\[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]
Unified electricity and magnetism as manifestations of the same force. Describes electromagnetic field behavior, predicting waves traveling at light speed. Enabled technologies from radio to smartphones.
\ No newline at end of file
February Flow
Data Components:
**Code:**
- The-stack-V2
**Codetext:**
- SE, whatever we’ve scraped
**Webtext:**
- HQ OCLM
**DATA MIXES**
- ~85% Source code
- ~15% Codetext
- ~0% Webtext
\[ \text{Deepseek} \quad \text{Starcoder} \quad \text{Arctic} \]
\ No newline at end of file
February Flow
Data Components:
Code:
The-Stack-V2
Codetext:
SE, whatever we've scraped
Webtext:
HQOCLM
DATA MIXES
~85% Source Code
~10% Codetext
~5% Webtext
~85% The-stack-V2
~15% Codetext
~0% Webtext
Deepseek
Cider
Jstaceder
2
loose source code
Arctic
\ No newline at end of file
Data Components:
Code:
The-stack-V2
Codetext:
SE, whatever we've scraped
Webtext:
HQ, OCLM
DATA MIXES
~ 85% Source Code
~ 10% Codetext
~ 5% Webtext
~ 85% The-stack-V2
~ 15% Codetext
~ 0% Webtext
Deepseek
Cider
Deepsea
Arctic
\ No newline at end of file
February Flow
Data Components:
Code:
- The-stack-V2
Codetext:
- SE, whatever we've scraped
Webtext:
- HQ OCLM
DATA MIXES
- ~85% Source Code
- ~10% Codetext
- ~5% Webtext
Deepseek Coder
- ~85% The-stack-V2
- ~15% Codetext
- ~0% Webtext
Jstaceder
Source Code
Arctic
\ No newline at end of file
V-February Flow
Data Components:
Code:
- The-stack-V2
CodeText:
- SE, whatever we’ve scraped
WebText:
- HQ, OCLM
DATA MIXES
- ~85% Source Code
- ~10% CodeText
- ~5% Webtext
Deepseek
- 85% The-stack-V2
- 15% CodeText
- 0% Webtext
Cider
- 100% Source Code
Arctic
\ No newline at end of file
P1: 100% Source code
P2: 80% code
20% language
Code Data Recipe [Staceder]
1) Order by Repo ✓
2) Call Heuristic Filters ×
3) Group by Repo, lang -> minhash ✓
4) Pack into Repo-level docs Δ
5) Select PL's Δ
i) Pack into FIM tokens X
✓: Eng Done
X: Eng definitively NOT done
D: so so easy
Use Preprocessed codetext, web text
\ No newline at end of file
P1: 100% Source code
P2: 80% code 20% language
Granite
Code Data Recipe [staceder]
1) Order by Repo ✓
2) Call heuristic filters X
3) Group by Repo, lang → minhash ✓
4) Pack into Repo-level docs Δ
5) Select PL's Δ
i) Pack into FIM tokens X
Eng: Done
Eng: Definitely NOT done
D: So-so easy
Use preprocessed codetext, webtext
\ No newline at end of file
P1: 100% Source code
P2: 80% code
20% language
Code Data Recipe [Staceder]
1) Order by Repo ✓
2) Call Heuristic Filters x
3) Group by Repo, lang → minhash ✓
4) Pack into Repo-level docs △
5) Select PL’s △
• Pack into FIM tokens X
✓: Eng Done
x: Eng definitely NOT done
△: so so easy
Use Preprocessed codetext, webtext
\ No newline at end of file
Granite
Code Data Recipe [staged]
1) Order by Repo ✓
2) Call Heuristic Filters ✗
3) Group by Repo, lang → minhash ✓
4) Pack into Repo-level docs ▽
5) Select PL's △
i) Pack into FIM tokens ✗
Eng: Done
Eng: Definitely NOT done
D: So so easy
Use Preprocessed codetext, webtext
\ No newline at end of file
P1: 100% Source Code → Granite
P2: 80% code
20% language
Code Data Recipe [staceder]
1) Order by Repo ✓
2) Call Heuristic Filters x
3) Group by Repo, lang → minhash ✓
4) Pack into Repo-level docs △
5) Select PL's △
D) Pack into FIM tokens x
Eng: Done
X: Eng definitively NOT done
D: so so easy
Use Preprocessed codetext, webtext
\ No newline at end of file
ARCH + TRAINING
- Pick Arch like OLMO 18
- OR replicate a 30 model
- Follow standard LR flow
Eval:
Hacky nonsense for now
\ No newline at end of file
ARCH + TRAINING
- Pick arch like OLMO-18
- or replicate a 3D model
- Follow standard LR flow
Eval:
Hacky nonsense for now
\ No newline at end of file
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