add some formulas

[g0del-esch3r-bach]

i propose to add the following formulae:
Skein Relation of the Jones Polynomial
Jones' Definition of the HOMFLY Polynomial
Fine Structure Constant
King's Rule of Integration
Inner Product on 4-Momenta
Lagrange's Four Square Theorem
thx!

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Important

Add six new mathematical formulas to problems.js, including the Skein Relation of the Jones Polynomial and Lagrange's Four Square Theorem.

• Additions:
  • Added "Skein Relation of the Jones Polynomial" with formula \frac{V_{L_+}}{t}-tV_{L_-}=\left(\sqrt t-\frac1{\sqrt t}\right)V_{L_0}.
  • Added "Jones' Definition of the HOMFLY Polynomial" with formula P_L(t,x)=\left(\frac{1-t^2}{tx}\right)^{n-1}\left(\frac{2t}{x+\sqrt{x^2+4}}\right)^{e}\operatorname{tr}(\pi(\alpha)).
  • Added "Fine Structure Constant" with formula \alpha=\frac{e^2}{4\pi\epsilon_0\hbar c}\approx\frac{1}{137}.
  • Added "King's Rule of Integration" with formula \int_a^b f(x)\mathrm dx=\int_a^b f(a+b-x)\mathrm dx.
  • Added "Inner Product on 4-Momenta" with formula \mathbf{P}_1\cdot\mathbf{P}_2=E_1E_2-\mathbf{p}_1\cdot\mathbf{p}_2.
  • Added "Lagrange's Four Square Theorem" with formula 4\{n^2\mid n\in\mathbb{N}\}=\mathbb{N}.

^{This description was created by for 4e730be. It will automatically update as commits are pushed.}

[ellipsis-dev]

👍 Looks good to me! Reviewed everything up to 4e730be in 1 minute and 57 seconds

More details

• Looked at 40 lines of code in 1 files
• Skipped 0 files when reviewing.
• Skipped posting 13 drafted comments based on config settings.

1. public/assets/js/problems.js:915

• Draft comment:
  Skein relation formula: verify that the notation (e.g. using \sqrt{t}) is consistent with standard forms.
• Reason this comment was not posted:
  Comment did not seem useful. Confidence is useful = 0% <= threshold 50%
  The comment is asking the PR author to verify the consistency of notation, which falls under the rule of not asking the author to confirm or double-check things. It doesn't provide a specific suggestion or point out a clear issue with the code.

2. public/assets/js/problems.js:920

• Draft comment:
  Jones' HOMFLY Polynomial: Double-check the formula and reference—it’s less standard and may need clarification.
• Reason this comment was not posted:
  Comment did not seem useful. Confidence is useful = 0% <= threshold 50%
  The comment is asking the PR author to double-check the formula and reference, which violates the rule against asking the author to confirm or double-check things. It doesn't provide a specific suggestion or point out a clear issue with the code.

3. public/assets/js/problems.js:925

• Draft comment:
  Fine Structure Constant formula appears standard; ensure consistency of spacing in denominators.
• Reason this comment was not posted:
  Confidence changes required: 20% <= threshold 50%
  None

4. public/assets/js/problems.js:930

• Draft comment:
  King's Rule of Integration: Looks good; consider adding a brief note on its symmetry if needed.
• Reason this comment was not posted:
  Confidence changes required: 10% <= threshold 50%
  None

5. public/assets/js/problems.js:935

• Draft comment:
  Inner Product on 4-Momenta: Confirm that the metric signature (E1E2 - p1·p2) is intended and document if necessary.
• Reason this comment was not posted:
  Confidence changes required: 30% <= threshold 50%
  None

6. public/assets/js/problems.js:940

• Draft comment:
  Lagrange's Four Square Theorem: The notation using Minkowski set addition (4{n^2 | n in N}) may be non-standard; consider clarifying or using conventional wording.
• Reason this comment was not posted:
  Comment looked like it was already resolved.

7. public/assets/js/problems.js:920

• Draft comment:
  The HOMFLY polynomial formula uses an exponent 'e' and trace operator that might be non‐standard or ambiguous. Verify if the intended notation (and variables, e.g. 'n-1' and 'e') is correct.
• Reason this comment was not posted:
  Comment did not seem useful. Confidence is useful = 0% <= threshold 50%
  The comment is asking the PR author to verify if the notation and variables used in the HOMFLY polynomial formula are correct. This falls under asking the author to confirm their intention or to double-check things, which is against the rules.

8. public/assets/js/problems.js:940

• Draft comment:
  The notation for Lagrange's Four Square Theorem using '4{n^2 | n in ℕ}' (Minkowski sum) is non-standard and might confuse readers. Consider using a clearer formulation.
• Reason this comment was not posted:
  Decided after close inspection that this draft comment was likely wrong and/or not actionable: usefulness confidence = 20% vs. threshold = 50%
  The comment makes a reasonable point - the Minkowski sum notation is less common and could be harder to understand. The suggested alternative using quantifiers is more explicit and clearer. However, the current notation is mathematically valid and concise, and the description already explains what the notation means. This seems more like a style preference than a clear issue.
  The current notation, while terse, is mathematically correct and the description explains it. The suggested change would make the formula longer and potentially harder to read in LaTeX.
  While the suggested notation might be more familiar to some readers, the current notation is valid and explained in the description. The conciseness of the current form may actually be preferable for this context.
  The comment should be deleted since it suggests a style change rather than fixing a clear issue, and the current notation is valid and explained.

9. public/assets/js/problems.js:480

• Draft comment:
  Typographical error: In the title "Mertens' therorem", 'therorem' should be corrected to 'theorem'.
• Reason this comment was not posted:
  Comment was not on a location in the diff, so it can't be submitted as a review comment.

10. public/assets/js/problems.js:725

• Draft comment:
  Typographical error: In the title "Defintion of the Quasi-Stationary Distribution", 'Defintion' should be corrected to 'Definition'.
• Reason this comment was not posted:
  Comment was not on a location in the diff, so it can't be submitted as a review comment.

11. public/assets/js/problems.js:900

• Draft comment:
  Typographical error: In the title "Definition of a Psuedorandom Generator", 'Psuedorandom' should be corrected to 'Pseudorandom'.
• Reason this comment was not posted:
  Comment was not on a location in the diff, so it can't be submitted as a review comment.

12. public/assets/js/problems.js:855

• Draft comment:
  Typographical error: In the title "Condtional Entropy", 'Condtional' should be corrected to 'Conditional'.
• Reason this comment was not posted:
  Comment was not on a location in the diff, so it can't be submitted as a review comment.

13. public/assets/js/problems.js:941

• Draft comment:
  Typographical error: In the description of "Lagrange's Four Square Theorem", 'minkowski' should be capitalized to 'Minkowski' as it is a proper noun.
• Reason this comment was not posted:
  Comment did not seem useful. Confidence is useful = 0% <= threshold 50%
  This comment is purely informative as it points out a typographical error in a description, which is not directly related to the code functionality or logic. It does not suggest a code change or improvement.

Workflow ID: wflow_Mz1Ll51lLvJERPF7

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