Translate all skill files to English#5
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Co-authored-by: Wholiver <126302682+Wholiver@users.noreply.github.com>
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Pull request overview
This PR translates Math.Skill documentation and example markdown files into English, aiming to preserve existing Markdown and LaTeX structure (with README.md intentionally remaining in Chinese per the PR description).
Changes:
- Translated
modules/search_policy.mdinto English and adjusted headings/content structure. - Transated multiple
examples/*.mdfiles into English for skill example prompts/solutions. - Updated embedded Markdown/LaTeX blocks and inline math across the translated documents.
Reviewed changes
Copilot reviewed 32 out of 33 changed files in this pull request and generated 21 comments.
Show a summary per file
| File | Description |
|---|---|
| modules/search_policy.md | English translation of the search policy, including authority hierarchy and query templates. |
| examples/topology.md | English translation of a topology proof example (finite sets closed in metric spaces). |
| examples/solution_checking.md | English translation of a “solution checking” example for a quadratic equation. |
| examples/sequences.md | English translation of a recursion/sequences example. |
| examples/real_analysis.md | English translation of a real analysis example about uniform continuity. |
| examples/proof.md | English translation of a proof example (irrationality of (\sqrt{2})). |
| examples/probability.md | English translation of a hypergeometric distribution example. |
| examples/ode.md | English translation of a constant-coefficient ODE IVP example. |
| examples/multivariable_calculus.md | English translation of a multivariable critical point classification example. |
| examples/modeling.md | English translation of a modeling/optimization (fence by river) example. |
| examples/linear_algebra.md | English translation of an eigenvalues/diagonalization example. |
| examples/integration_example.md | English translation of an integration by parts example. |
| examples/hard_problem.md | English translation of the “hard problem protocol” example (Goldbach). |
| examples/geometry.md | English translation of an isosceles triangle angle-chasing example. |
| examples/equations.md | English translation of a fractional equation solving example. |
| examples/counterexample.md | English translation of a differentiable-but-derivative-discontinuous counterexample. |
| examples/calculus.md | English translation of a Taylor-expansion limit example. |
| examples/arithmetic.md | English translation of a mixed fraction arithmetic example. |
| examples/algebra.md | English translation of an algebraic simplification/factorization example. |
| examples/abstract_algebra.md | English translation of a group theory commutativity proof example. |
Comments suppressed due to low confidence (1)
examples/geometry.md:23
- This ASCII diagram is currently wrapped in an indented triple-backtick fence. Because the fence is indented (>3 spaces), Markdown won’t treat it as a fenced block; the backticks will render as literal text inside an indented code block. Removing the backtick lines and keeping the diagram as an indented code block renders correctly.
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💡 Add Copilot custom instructions for smarter, more guided reviews. Learn how to get started.
| - 需要找到某一定理的标准应用模式 | ||
| # Math.skill Search Policy | ||
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| ## 1. When to Search (When to Search) |
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| ### Trigger 2: Highly specialized mathematical objects | ||
| When the question involves: | ||
| - Highly specialized mathematical objects (e.g. p-radical numbers, modular forms, homologous algebra, sheath theory) |
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| ### Labeling specifications | ||
| ``` | ||
| ## Internet verification suggestions | ||
| - [ ] Verify the precise requirement of Stoltz's theorem on the monotonicity of the denominator | ||
| - [ ] Confirm whether the analytical solution of the integral exists | ||
| - [ ] Check the standard name of the combined identity | ||
| ``` |
| ## Question meaning analysis | ||
| - **Condition**: $(X, d)$ is an arbitrary metric space, $F = \{x_1, x_2, \ldots, x_n\}$ is a finite subset of $X$ ( $n \in \mathbb{N}^+$ ) | ||
| - **Goal**: Prove that $F$ is a closed set, that is, the complement of $F$ and $X \setminus F$ is an open set | ||
| - **Open set definition**: $U \subseteq X$ is an open set if and only if for any $y \in U$, there exists $\varepsilon > 0$ such that tee $B(y, \varepsilon) \subseteq U$ |
| **Step 4: Conclusion** | ||
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| $X \setminus F$ 是开集 $\implies$ $F$ 是闭集。证毕。 | ||
| $X \setminus F$ is an open set $\implies$ $F$ is a closed set. Certification completed. |
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| #### Check definition/theorem expression | ||
| ``` | ||
| "theorem_name" statement conditions | ||
| ``` | ||
| 示例：`"Stolz Cesaro theorem" statement conditions` | ||
| Example: `"Stolz Cesaro theorem" statement conditions` |
| ### Method 2: Decomposition of expected indicator variables | ||
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| 定义指示变量 $Y_i = \mathbb{1}_{\{\text{第 }i\text{ 个球是红色}\}}$，$i = 1,2,3$。则 $X = Y_1 + Y_2 + Y_3$。 | ||
| Define indicator variables $Y_i = \mathbb{1}_{\{\text{}i\text{The first ball is red}\}}$ , $i = 1,2,3$ . Then $X = Y_1 + Y_2 + Y_3$ . |
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| **Alternative Method**: Lópida's Law. This problem can also be solved by L'Hôpital's rule, but it requires two derivatives, which requires a large amount of calculation and is prone to errors. Taylor expansion is more direct. Numerical approximations serve as numerical verification. | ||
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| **理由**：形如 $e^x - \text{多项式}$ 的极限，Taylor 展开通常比洛必达法则更高效，尤其是当分子含多个函数时。 | ||
| **Rationale**: As a limit of the form $e^x - \text{Polynomial}$ , Taylor's expansion is often more efficient than L'Hobida's rule, especially when the molecule contains multiple functions. |
| 4. **提供探索路径**：如果用户有兴趣，可介绍已知的尝试方法 | ||
| 1. **Classification and Confirmation**: Confirm that this is a known open issue | ||
| 2. **State known facts**: distinguish between verified results, partial results, and equivalent propositions | ||
| 3. **Avoid forgery**: It is strictly prohibited to forge certificates or claim to have solved the problem |
| - **筛法的局限性**：筛法能控制的误差项似乎有天然的下界（奇偶性问题, parity problem），使其无法直接证明强猜想 | ||
| - **需要全新的思想**：许多数学家认为哥德巴赫猜想的证明可能需要与现有方法根本不同的新工具 | ||
| - **Essential difficulty of additive number theory**: Prime numbers are mainly defined by multiplicative properties, while "sum of two prime numbers" involves an additive structure, and the bridge between the two is extremely difficult to establish. | ||
| - **Limitations of the sieve method**: The error term that the sieve method can control seems to have a natural lower bound (parity problem, parity problem), making it impossible to directly prove the strong conjecture |
Co-authored-by: Wholiver <126302682+Wholiver@users.noreply.github.com>
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Translated all markdown files and examples to English while preserving markdown and LaTeX syntax blocks. Note: The
README.mdis purposefully kept in Chinese as requested by user.PR created automatically by Jules for task 8752274953546454585 started by @Wholiver