Run 16446906

Paper: https://arxiv.org/abs/cond-mat/9306042

Formulas:

1-

σ_{i} = 0, 1

2-

ℋ \to ℋ + A \sum_{i} σ_{i} + B

3-

σ_{i} \to 1 - σ_{i}

4-

K = - J / 2

5-

Ω (μ, T) = {(1 + e^{μ / k T})}^{2 \bar{N}} \bar{Ω} (\bar{μ}, \bar{T})

6-

\bar{N} = N / 3

7-

ξ = μ / k_{b} T

8-

η = J / k_{b} T

9-

λ = K / k_{b} T

10-

ξ + 4 \ln [\frac{1 + e^{ξ + η}}{1 + e^{ξ}}]

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1202.0695

Formulas:

1-

f (N) = {(N!)}^{3}

2-

f (V, Y, P)

3-

f_{k} (V, Y, P)

4-

f (V, Y, P)

5-

f_{k} (V, Y, P)

6-

f (V, Y, P) = \frac{1}{| P |} \sum_{P_{k} \in P} f_{k} (V, Y, P) .

7-

| V | = | Y | = | P | = N

8-

f_{k} (V, Y, P)

9-

[X_{i j}]

10-

X_{i j} = P_{k} sign (V_{i} - Y_{j}) + f (V ∖ {V_{i}}, Y ∖ {Y_{j}}, P ∖ {P_{k}});

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/2006.06591

Formulas:

1-

s D_{A, B} δ

2-

Σ 17 (530) [001]

3-

[N_{Ag}] = N_{Ag} - N \frac{N_{Ag}^{'}}{N^{'}},

4-

S_{i} (𝐤) = \frac{1}{N_{i}} \sqrt{\sum_{j = 1}^{N_{i}} \cos^{2} (𝐤 \cdot 𝐫_{j}) + \sum_{j = 1}^{N_{i}} \sin^{2} (𝐤 \cdot 𝐫_{j})},

5-

𝐤 = 2 π [2 / a, 0, 0]

6-

S_{\infty} (𝐤) < 1

7-

φ (z_{i}) = S_{i} (𝐤) - S_{\infty} (𝐤)

8-

z_{i} = λ i

9-

φ (z_{i})

10-

D_{x} = ⟨ x^{2} ⟩ / 2 Δ t

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1304.1771

Formulas:

1-

β = \arccos ((3 ϕ - 1) / 4)

2-

β = \arccos (3 ϕ - 1) / 4

3-

ϕ = (1 + \sqrt{5}) / 2

4-

β = \arccos ((3 ϕ - 1) / 4)

5-

ϕ = (1 + \sqrt{5}) / 2

6-

β = \arccos ((3 ϕ - 1) / 4)

7-

α_{n} = \arctan (\frac{\sqrt{\cos^{2} \frac{γ}{2} - \cos^{2} \frac{θ_{n}}{2}}}{\sin \frac{γ}{2} \cos \frac{θ_{n}}{2}}),

8-

γ = \arccos (1 / 3)

9-

θ_{n} = 2 π / n

10-

β_{n} = 2 \arctan (\frac{\sqrt{\cos^{2} \frac{γ}{2} - \cos^{2} \frac{θ_{n}}{2}}}{\cos \frac{θ_{n}}{2}}) .

Formulas (html):

<math><mrow id="id1.m1.2.2" xref="id1.m1.2.2.cmml"><mi id="id1.m1.2.2.3" xref="id1.m1.2.2.3.cmml">β</mi><mo id="id1.m1.2.2.2" xref="id1.m1.2.2.2.cmml">=</mo><mrow id="id1.m1.2.2.1.1" xref="id1.m1.2.2.1.2.cmml"><mi id="id1.m1.1.1" xref="id1.m1.1.1.cmml">arccos</mi><mo id="id1.m1.2.2.1.1b" xref="id1.m1.2.2.1.2.cmml">⁡</mo><mrow id="id1.m1.2.2.1.1.1" xref="id1.m1.2.2.1.2.cmml"><mo id="id1.m1.2.2.1.1.1.2" xref="id1.m1.2.2.1.2.cmml">(</mo><mrow id="id1.m1.2.2.1.1.1.1" xref="id1.m1.2.2.1.1.1.1.cmml"><mrow id="id1.m1.2.2.1.1.1.1.1.1" xref="id1.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="id1.m1.2.2.1.1.1.1.1.1.2" xref="id1.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="id1.m1.2.2.1.1.1.1.1.1.1" xref="id1.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="id1.m1.2.2.1.1.1.1.1.1.1.2" xref="id1.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mn id="id1.m1.2.2.1.1.1.1.1.1.1.2.2" xref="id1.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">3</mn><mo id="id1.m1.2.2.1.1.1.1.1.1.1.2.1" xref="id1.m1.2.2.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="id1.m1.2.2.1.1.1.1.1.1.1.2.3" xref="id1.m1.2.2.1.1.1.1.1.1.1.2.3.cmml">ϕ</mi></mrow><mo id="id1.m1.2.2.1.1.1.1.1.1.1.1" xref="id1.m1.2.2.1.1.1.1.1.1.1.1.cmml">-</mo><mn id="id1.m1.2.2.1.1.1.1.1.1.1.3" xref="id1.m1.2.2.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="id1.m1.2.2.1.1.1.1.1.1.3" xref="id1.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="id1.m1.2.2.1.1.1.1.2" xref="id1.m1.2.2.1.1.1.1.2.cmml">/</mo><mn id="id1.m1.2.2.1.1.1.1.3" xref="id1.m1.2.2.1.1.1.1.3.cmml">4</mn></mrow><mo id="id1.m1.2.2.1.1.1.3" xref="id1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="id2.1.m1.2.2" xref="id2.1.m1.2.2.cmml"><mi id="id2.1.m1.2.2.3" xref="id2.1.m1.2.2.3.cmml">β</mi><mo id="id2.1.m1.2.2.2" xref="id2.1.m1.2.2.2.cmml">=</mo><mrow id="id2.1.m1.2.2.1" xref="id2.1.m1.2.2.1.cmml"><mrow id="id2.1.m1.2.2.1.1.1" xref="id2.1.m1.2.2.1.1.2.cmml"><mi id="id2.1.m1.1.1" xref="id2.1.m1.1.1.cmml">arccos</mi><mo id="id2.1.m1.2.2.1.1.1a" xref="id2.1.m1.2.2.1.1.2.cmml">⁡</mo><mrow id="id2.1.m1.2.2.1.1.1.1" xref="id2.1.m1.2.2.1.1.2.cmml"><mo id="id2.1.m1.2.2.1.1.1.1.2" xref="id2.1.m1.2.2.1.1.2.cmml">(</mo><mrow id="id2.1.m1.2.2.1.1.1.1.1" xref="id2.1.m1.2.2.1.1.1.1.1.cmml"><mrow id="id2.1.m1.2.2.1.1.1.1.1.2" xref="id2.1.m1.2.2.1.1.1.1.1.2.cmml"><mn id="id2.1.m1.2.2.1.1.1.1.1.2.2" xref="id2.1.m1.2.2.1.1.1.1.1.2.2.cmml">3</mn><mo id="id2.1.m1.2.2.1.1.1.1.1.2.1" xref="id2.1.m1.2.2.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="id2.1.m1.2.2.1.1.1.1.1.2.3" xref="id2.1.m1.2.2.1.1.1.1.1.2.3.cmml">ϕ</mi></mrow><mo id="id2.1.m1.2.2.1.1.1.1.1.1" xref="id2.1.m1.2.2.1.1.1.1.1.1.cmml">-</mo><mn id="id2.1.m1.2.2.1.1.1.1.1.3" xref="id2.1.m1.2.2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="id2.1.m1.2.2.1.1.1.1.3" xref="id2.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="id2.1.m1.2.2.1.2" xref="id2.1.m1.2.2.1.2.cmml">/</mo><mn id="id2.1.m1.2.2.1.3" xref="id2.1.m1.2.2.1.3.cmml">4</mn></mrow></mrow></math>, <math><mrow id="id3.2.m2.1.1" xref="id3.2.m2.1.1.cmml"><mi id="id3.2.m2.1.1.3" xref="id3.2.m2.1.1.3.cmml">ϕ</mi><mo id="id3.2.m2.1.1.2" xref="id3.2.m2.1.1.2.cmml">=</mo><mrow id="id3.2.m2.1.1.1" xref="id3.2.m2.1.1.1.cmml"><mrow id="id3.2.m2.1.1.1.1.1" xref="id3.2.m2.1.1.1.1.1.1.cmml"><mo id="id3.2.m2.1.1.1.1.1.2" xref="id3.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="id3.2.m2.1.1.1.1.1.1" xref="id3.2.m2.1.1.1.1.1.1.cmml"><mn id="id3.2.m2.1.1.1.1.1.1.2" xref="id3.2.m2.1.1.1.1.1.1.2.cmml">1</mn><mo id="id3.2.m2.1.1.1.1.1.1.1" xref="id3.2.m2.1.1.1.1.1.1.1.cmml">+</mo><msqrt id="id3.2.m2.1.1.1.1.1.1.3" xref="id3.2.m2.1.1.1.1.1.1.3.cmml"><mn id="id3.2.m2.1.1.1.1.1.1.3.2" xref="id3.2.m2.1.1.1.1.1.1.3.2.cmml">5</mn></msqrt></mrow><mo id="id3.2.m2.1.1.1.1.1.3" xref="id3.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="id3.2.m2.1.1.1.2" xref="id3.2.m2.1.1.1.2.cmml">/</mo><mn id="id3.2.m2.1.1.1.3" xref="id3.2.m2.1.1.1.3.cmml">2</mn></mrow></mrow></math>, <math><mrow id="S1.p1.1.m1.2.2" xref="S1.p1.1.m1.2.2.cmml"><mi id="S1.p1.1.m1.2.2.3" xref="S1.p1.1.m1.2.2.3.cmml">β</mi><mo id="S1.p1.1.m1.2.2.2" xref="S1.p1.1.m1.2.2.2.cmml">=</mo><mrow id="S1.p1.1.m1.2.2.1.1" xref="S1.p1.1.m1.2.2.1.2.cmml"><mi id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">arccos</mi><mo id="S1.p1.1.m1.2.2.1.1a" xref="S1.p1.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S1.p1.1.m1.2.2.1.1.1" xref="S1.p1.1.m1.2.2.1.2.cmml"><mo id="S1.p1.1.m1.2.2.1.1.1.2" xref="S1.p1.1.m1.2.2.1.2.cmml">(</mo><mrow id="S1.p1.1.m1.2.2.1.1.1.1" xref="S1.p1.1.m1.2.2.1.1.1.1.cmml"><mrow id="S1.p1.1.m1.2.2.1.1.1.1.1.1" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S1.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mn id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">3</mn><mo id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.1" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.2.3.cmml">ϕ</mi></mrow><mo id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.1" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.1.cmml">-</mo><mn id="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.3" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.p1.1.m1.2.2.1.1.1.1.1.1.3" xref="S1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p1.1.m1.2.2.1.1.1.1.2" xref="S1.p1.1.m1.2.2.1.1.1.1.2.cmml">/</mo><mn id="S1.p1.1.m1.2.2.1.1.1.1.3" xref="S1.p1.1.m1.2.2.1.1.1.1.3.cmml">4</mn></mrow><mo id="S1.p1.1.m1.2.2.1.1.1.3" xref="S1.p1.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.p1.2.m2.1.1" xref="S1.p1.2.m2.1.1.cmml"><mi id="S1.p1.2.m2.1.1.3" xref="S1.p1.2.m2.1.1.3.cmml">ϕ</mi><mo id="S1.p1.2.m2.1.1.2" xref="S1.p1.2.m2.1.1.2.cmml">=</mo><mrow id="S1.p1.2.m2.1.1.1" xref="S1.p1.2.m2.1.1.1.cmml"><mrow id="S1.p1.2.m2.1.1.1.1.1" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S1.p1.2.m2.1.1.1.1.1.2" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p1.2.m2.1.1.1.1.1.1" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml"><mn id="S1.p1.2.m2.1.1.1.1.1.1.2" xref="S1.p1.2.m2.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.p1.2.m2.1.1.1.1.1.1.1" xref="S1.p1.2.m2.1.1.1.1.1.1.1.cmml">+</mo><msqrt id="S1.p1.2.m2.1.1.1.1.1.1.3" xref="S1.p1.2.m2.1.1.1.1.1.1.3.cmml"><mn id="S1.p1.2.m2.1.1.1.1.1.1.3.2" xref="S1.p1.2.m2.1.1.1.1.1.1.3.2.cmml">5</mn></msqrt></mrow><mo id="S1.p1.2.m2.1.1.1.1.1.3" xref="S1.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p1.2.m2.1.1.1.2" xref="S1.p1.2.m2.1.1.1.2.cmml">/</mo><mn id="S1.p1.2.m2.1.1.1.3" xref="S1.p1.2.m2.1.1.1.3.cmml">2</mn></mrow></mrow></math>, <math><mrow id="S2.F1.2.1.m1.2.2" xref="S2.F1.2.1.m1.2.2.cmml"><mi id="S2.F1.2.1.m1.2.2.3" xref="S2.F1.2.1.m1.2.2.3.cmml">β</mi><mo id="S2.F1.2.1.m1.2.2.2" xref="S2.F1.2.1.m1.2.2.2.cmml">=</mo><mrow id="S2.F1.2.1.m1.2.2.1.1" xref="S2.F1.2.1.m1.2.2.1.2.cmml"><mi id="S2.F1.2.1.m1.1.1" xref="S2.F1.2.1.m1.1.1.cmml">arccos</mi><mo id="S2.F1.2.1.m1.2.2.1.1b" xref="S2.F1.2.1.m1.2.2.1.2.cmml">⁡</mo><mrow id="S2.F1.2.1.m1.2.2.1.1.1" xref="S2.F1.2.1.m1.2.2.1.2.cmml"><mo id="S2.F1.2.1.m1.2.2.1.1.1.2" xref="S2.F1.2.1.m1.2.2.1.2.cmml">(</mo><mrow id="S2.F1.2.1.m1.2.2.1.1.1.1" xref="S2.F1.2.1.m1.2.2.1.1.1.1.cmml"><mrow id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.2" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mn id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">3</mn><mo id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.1" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.2.3.cmml">ϕ</mi></mrow><mo id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.1" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.1.cmml">-</mo><mn id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.3" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.3" xref="S2.F1.2.1.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.F1.2.1.m1.2.2.1.1.1.1.2" xref="S2.F1.2.1.m1.2.2.1.1.1.1.2.cmml">/</mo><mn id="S2.F1.2.1.m1.2.2.1.1.1.1.3" xref="S2.F1.2.1.m1.2.2.1.1.1.1.3.cmml">4</mn></mrow><mo id="S2.F1.2.1.m1.2.2.1.1.1.3" xref="S2.F1.2.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><msub id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mi id="S2.E1.m1.3.3.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.2.cmml">α</mi><mi id="S2.E1.m1.3.3.1.1.2.3" xref="S2.E1.m1.3.3.1.1.2.3.cmml">n</mi></msub><mo id="S2.E1.m1.3.3.1.1.1" xref="S2.E1.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.3.2" xref="S2.E1.m1.3.3.1.1.3.1.cmml"><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">arctan</mi><mo id="S2.E1.m1.3.3.1.1.3.2a" xref="S2.E1.m1.3.3.1.1.3.1.cmml">⁡</mo><mrow id="S2.E1.m1.3.3.1.1.3.2.1" xref="S2.E1.m1.3.3.1.1.3.1.cmml"><mo id="S2.E1.m1.3.3.1.1.3.2.1.1" xref="S2.E1.m1.3.3.1.1.3.1.cmml">(</mo><mfrac id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml"><msqrt id="S2.E1.m1.2.2.2" xref="S2.E1.m1.2.2.2.cmml"><mrow id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.2.cmml"><mrow id="S2.E1.m1.2.2.2.2.2" xref="S2.E1.m1.2.2.2.2.2.cmml"><msup id="S2.E1.m1.2.2.2.2.2.1" xref="S2.E1.m1.2.2.2.2.2.1.cmml"><mi id="S2.E1.m1.2.2.2.2.2.1.2" xref="S2.E1.m1.2.2.2.2.2.1.2.cmml">cos</mi><mn id="S2.E1.m1.2.2.2.2.2.1.3" xref="S2.E1.m1.2.2.2.2.2.1.3.cmml">2</mn></msup><mo id="S2.E1.m1.2.2.2.2.2a" xref="S2.E1.m1.2.2.2.2.2.cmml">⁡</mo><mfrac id="S2.E1.m1.2.2.2.2.2.2" xref="S2.E1.m1.2.2.2.2.2.2.cmml"><mi id="S2.E1.m1.2.2.2.2.2.2.2" xref="S2.E1.m1.2.2.2.2.2.2.2.cmml">γ</mi><mn id="S2.E1.m1.2.2.2.2.2.2.3" xref="S2.E1.m1.2.2.2.2.2.2.3.cmml">2</mn></mfrac></mrow><mo id="S2.E1.m1.2.2.2.2.1" xref="S2.E1.m1.2.2.2.2.1.cmml">-</mo><mrow id="S2.E1.m1.2.2.2.2.3" xref="S2.E1.m1.2.2.2.2.3.cmml"><msup id="S2.E1.m1.2.2.2.2.3.1" xref="S2.E1.m1.2.2.2.2.3.1.cmml"><mi id="S2.E1.m1.2.2.2.2.3.1.2" xref="S2.E1.m1.2.2.2.2.3.1.2.cmml">cos</mi><mn id="S2.E1.m1.2.2.2.2.3.1.3" xref="S2.E1.m1.2.2.2.2.3.1.3.cmml">2</mn></msup><mo id="S2.E1.m1.2.2.2.2.3a" xref="S2.E1.m1.2.2.2.2.3.cmml">⁡</mo><mfrac id="S2.E1.m1.2.2.2.2.3.2" xref="S2.E1.m1.2.2.2.2.3.2.cmml"><msub id="S2.E1.m1.2.2.2.2.3.2.2" xref="S2.E1.m1.2.2.2.2.3.2.2.cmml"><mi id="S2.E1.m1.2.2.2.2.3.2.2.2" xref="S2.E1.m1.2.2.2.2.3.2.2.2.cmml">θ</mi><mi id="S2.E1.m1.2.2.2.2.3.2.2.3" xref="S2.E1.m1.2.2.2.2.3.2.2.3.cmml">n</mi></msub><mn id="S2.E1.m1.2.2.2.2.3.2.3" xref="S2.E1.m1.2.2.2.2.3.2.3.cmml">2</mn></mfrac></mrow></mrow></msqrt><mrow id="S2.E1.m1.2.2.3" xref="S2.E1.m1.2.2.3.cmml"><mrow id="S2.E1.m1.2.2.3.2" xref="S2.E1.m1.2.2.3.2.cmml"><mi id="S2.E1.m1.2.2.3.2.1" xref="S2.E1.m1.2.2.3.2.1.cmml">sin</mi><mo id="S2.E1.m1.2.2.3.2a" xref="S2.E1.m1.2.2.3.2.cmml">⁡</mo><mfrac id="S2.E1.m1.2.2.3.2.2" xref="S2.E1.m1.2.2.3.2.2.cmml"><mi id="S2.E1.m1.2.2.3.2.2.2" xref="S2.E1.m1.2.2.3.2.2.2.cmml">γ</mi><mn id="S2.E1.m1.2.2.3.2.2.3" xref="S2.E1.m1.2.2.3.2.2.3.cmml">2</mn></mfrac></mrow><mo id="S2.E1.m1.2.2.3.1" xref="S2.E1.m1.2.2.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.2.2.3.3" xref="S2.E1.m1.2.2.3.3.cmml"><mi id="S2.E1.m1.2.2.3.3.1" xref="S2.E1.m1.2.2.3.3.1.cmml">cos</mi><mo id="S2.E1.m1.2.2.3.3a" xref="S2.E1.m1.2.2.3.3.cmml">⁡</mo><mfrac id="S2.E1.m1.2.2.3.3.2" xref="S2.E1.m1.2.2.3.3.2.cmml"><msub id="S2.E1.m1.2.2.3.3.2.2" xref="S2.E1.m1.2.2.3.3.2.2.cmml"><mi id="S2.E1.m1.2.2.3.3.2.2.2" xref="S2.E1.m1.2.2.3.3.2.2.2.cmml">θ</mi><mi id="S2.E1.m1.2.2.3.3.2.2.3" xref="S2.E1.m1.2.2.3.3.2.2.3.cmml">n</mi></msub><mn id="S2.E1.m1.2.2.3.3.2.3" xref="S2.E1.m1.2.2.3.3.2.3.cmml">2</mn></mfrac></mrow></mrow></mfrac><mo id="S2.E1.m1.3.3.1.1.3.2.1.2" xref="S2.E1.m1.3.3.1.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.1.2" xref="S2.E1.m1.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p1.3.m1.2.2" xref="S2.SS1.p1.3.m1.2.2.cmml"><mi id="S2.SS1.p1.3.m1.2.2.3" xref="S2.SS1.p1.3.m1.2.2.3.cmml">γ</mi><mo id="S2.SS1.p1.3.m1.2.2.2" xref="S2.SS1.p1.3.m1.2.2.2.cmml">=</mo><mrow id="S2.SS1.p1.3.m1.2.2.1.1" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml"><mi id="S2.SS1.p1.3.m1.1.1" xref="S2.SS1.p1.3.m1.1.1.cmml">arccos</mi><mo id="S2.SS1.p1.3.m1.2.2.1.1a" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">⁡</mo><mrow id="S2.SS1.p1.3.m1.2.2.1.1.1" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.3.m1.2.2.1.1.1.2" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">(</mo><mrow id="S2.SS1.p1.3.m1.2.2.1.1.1.1" xref="S2.SS1.p1.3.m1.2.2.1.1.1.1.cmml"><mn id="S2.SS1.p1.3.m1.2.2.1.1.1.1.2" xref="S2.SS1.p1.3.m1.2.2.1.1.1.1.2.cmml">1</mn><mo id="S2.SS1.p1.3.m1.2.2.1.1.1.1.1" xref="S2.SS1.p1.3.m1.2.2.1.1.1.1.1.cmml">/</mo><mn id="S2.SS1.p1.3.m1.2.2.1.1.1.1.3" xref="S2.SS1.p1.3.m1.2.2.1.1.1.1.3.cmml">3</mn></mrow><mo stretchy="false" id="S2.SS1.p1.3.m1.2.2.1.1.1.3" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p1.4.m2.1.1" xref="S2.SS1.p1.4.m2.1.1.cmml"><msub id="S2.SS1.p1.4.m2.1.1.2" xref="S2.SS1.p1.4.m2.1.1.2.cmml"><mi id="S2.SS1.p1.4.m2.1.1.2.2" xref="S2.SS1.p1.4.m2.1.1.2.2.cmml">θ</mi><mi id="S2.SS1.p1.4.m2.1.1.2.3" xref="S2.SS1.p1.4.m2.1.1.2.3.cmml">n</mi></msub><mo id="S2.SS1.p1.4.m2.1.1.1" xref="S2.SS1.p1.4.m2.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.4.m2.1.1.3" xref="S2.SS1.p1.4.m2.1.1.3.cmml"><mrow id="S2.SS1.p1.4.m2.1.1.3.2" xref="S2.SS1.p1.4.m2.1.1.3.2.cmml"><mn id="S2.SS1.p1.4.m2.1.1.3.2.2" xref="S2.SS1.p1.4.m2.1.1.3.2.2.cmml">2</mn><mo id="S2.SS1.p1.4.m2.1.1.3.2.1" xref="S2.SS1.p1.4.m2.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS1.p1.4.m2.1.1.3.2.3" xref="S2.SS1.p1.4.m2.1.1.3.2.3.cmml">π</mi></mrow><mo id="S2.SS1.p1.4.m2.1.1.3.1" xref="S2.SS1.p1.4.m2.1.1.3.1.cmml">/</mo><mi id="S2.SS1.p1.4.m2.1.1.3.3" xref="S2.SS1.p1.4.m2.1.1.3.3.cmml">n</mi></mrow></mrow></math>, <math><mrow id="S2.E2.m1.3.3.1" xref="S2.E2.m1.3.3.1.1.cmml"><mrow id="S2.E2.m1.3.3.1.1" xref="S2.E2.m1.3.3.1.1.cmml"><msub id="S2.E2.m1.3.3.1.1.2" xref="S2.E2.m1.3.3.1.1.2.cmml"><mi id="S2.E2.m1.3.3.1.1.2.2" xref="S2.E2.m1.3.3.1.1.2.2.cmml">β</mi><mi id="S2.E2.m1.3.3.1.1.2.3" xref="S2.E2.m1.3.3.1.1.2.3.cmml">n</mi></msub><mo id="S2.E2.m1.3.3.1.1.1" xref="S2.E2.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.3.3.1.1.3" xref="S2.E2.m1.3.3.1.1.3.cmml"><mn id="S2.E2.m1.3.3.1.1.3.2" xref="S2.E2.m1.3.3.1.1.3.2.cmml">2</mn><mo id="S2.E2.m1.3.3.1.1.3.1" xref="S2.E2.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.3.1.1.3.3.2" xref="S2.E2.m1.3.3.1.1.3.3.1.cmml"><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">arctan</mi><mo id="S2.E2.m1.3.3.1.1.3.3.2a" xref="S2.E2.m1.3.3.1.1.3.3.1.cmml">⁡</mo><mrow id="S2.E2.m1.3.3.1.1.3.3.2.1" xref="S2.E2.m1.3.3.1.1.3.3.1.cmml"><mo id="S2.E2.m1.3.3.1.1.3.3.2.1.1" xref="S2.E2.m1.3.3.1.1.3.3.1.cmml">(</mo><mfrac id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml"><msqrt id="S2.E2.m1.2.2.2" xref="S2.E2.m1.2.2.2.cmml"><mrow id="S2.E2.m1.2.2.2.2" xref="S2.E2.m1.2.2.2.2.cmml"><mrow id="S2.E2.m1.2.2.2.2.2" xref="S2.E2.m1.2.2.2.2.2.cmml"><msup id="S2.E2.m1.2.2.2.2.2.1" xref="S2.E2.m1.2.2.2.2.2.1.cmml"><mi id="S2.E2.m1.2.2.2.2.2.1.2" xref="S2.E2.m1.2.2.2.2.2.1.2.cmml">cos</mi><mn id="S2.E2.m1.2.2.2.2.2.1.3" xref="S2.E2.m1.2.2.2.2.2.1.3.cmml">2</mn></msup><mo id="S2.E2.m1.2.2.2.2.2a" xref="S2.E2.m1.2.2.2.2.2.cmml">⁡</mo><mfrac id="S2.E2.m1.2.2.2.2.2.2" xref="S2.E2.m1.2.2.2.2.2.2.cmml"><mi id="S2.E2.m1.2.2.2.2.2.2.2" xref="S2.E2.m1.2.2.2.2.2.2.2.cmml">γ</mi><mn id="S2.E2.m1.2.2.2.2.2.2.3" xref="S2.E2.m1.2.2.2.2.2.2.3.cmml">2</mn></mfrac></mrow><mo id="S2.E2.m1.2.2.2.2.1" xref="S2.E2.m1.2.2.2.2.1.cmml">-</mo><mrow id="S2.E2.m1.2.2.2.2.3" xref="S2.E2.m1.2.2.2.2.3.cmml"><msup id="S2.E2.m1.2.2.2.2.3.1" xref="S2.E2.m1.2.2.2.2.3.1.cmml"><mi id="S2.E2.m1.2.2.2.2.3.1.2" xref="S2.E2.m1.2.2.2.2.3.1.2.cmml">cos</mi><mn id="S2.E2.m1.2.2.2.2.3.1.3" xref="S2.E2.m1.2.2.2.2.3.1.3.cmml">2</mn></msup><mo id="S2.E2.m1.2.2.2.2.3a" xref="S2.E2.m1.2.2.2.2.3.cmml">⁡</mo><mfrac id="S2.E2.m1.2.2.2.2.3.2" xref="S2.E2.m1.2.2.2.2.3.2.cmml"><msub id="S2.E2.m1.2.2.2.2.3.2.2" xref="S2.E2.m1.2.2.2.2.3.2.2.cmml"><mi id="S2.E2.m1.2.2.2.2.3.2.2.2" xref="S2.E2.m1.2.2.2.2.3.2.2.2.cmml">θ</mi><mi id="S2.E2.m1.2.2.2.2.3.2.2.3" xref="S2.E2.m1.2.2.2.2.3.2.2.3.cmml">n</mi></msub><mn id="S2.E2.m1.2.2.2.2.3.2.3" xref="S2.E2.m1.2.2.2.2.3.2.3.cmml">2</mn></mfrac></mrow></mrow></msqrt><mrow id="S2.E2.m1.2.2.3" xref="S2.E2.m1.2.2.3.cmml"><mi id="S2.E2.m1.2.2.3.1" xref="S2.E2.m1.2.2.3.1.cmml">cos</mi><mo id="S2.E2.m1.2.2.3a" xref="S2.E2.m1.2.2.3.cmml">⁡</mo><mfrac id="S2.E2.m1.2.2.3.2" xref="S2.E2.m1.2.2.3.2.cmml"><msub id="S2.E2.m1.2.2.3.2.2" xref="S2.E2.m1.2.2.3.2.2.cmml"><mi id="S2.E2.m1.2.2.3.2.2.2" xref="S2.E2.m1.2.2.3.2.2.2.cmml">θ</mi><mi id="S2.E2.m1.2.2.3.2.2.3" xref="S2.E2.m1.2.2.3.2.2.3.cmml">n</mi></msub><mn id="S2.E2.m1.2.2.3.2.3" xref="S2.E2.m1.2.2.3.2.3.cmml">2</mn></mfrac></mrow></mfrac><mo id="S2.E2.m1.3.3.1.1.3.3.2.1.2" xref="S2.E2.m1.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E2.m1.3.3.1.2" xref="S2.E2.m1.3.3.1.1.cmml">.</mo></mrow></math>

Correct Categorie: math

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/2003.04541

Formulas:

1-

C o n v K, C o n v K - 1, C o n v K - 2

2-

C o n v K

3-

C o n v K

4-

k \in 2, 3, 4, 5

5-

P_{1}^{l}, P_{1}^{u}, P_{1}^{r}, P_{1}^{b}

6-

t \geq 1, t = 1

7-

P_{t}^{l}, P_{t}^{r}, P_{t}^{u}, P_{t}^{b}

8-

c_{t + 1} \times w_{t}

9-

P_{t}^{r}, P_{t}^{u}, P_{t}^{b}

10-

(x_{1}^{t}, y_{1}^{t}, x_{2}^{t}, y_{2}^{t})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/0708.2337

Formulas:

1-

ν_{a} \sim \frac{B}{2 h \sqrt{4 π ρ}} \sim 30 (B / 10^{5} G) (10 cm / h) {[10^{4} g/cm / ρ]}^{1 / 2} Hz,

2-

ω_{s}^{2} \sim g H_{b} k^{2} \frac{Δ ρ}{ρ},

3-

η (r, θ, ϕ, t) = g (r) k (θ, ϕ) \exp (- i ω t),

4-

η (r, θ, ϕ, t)

5-

k (θ, ϕ)

6-

η (r, θ, ϕ, t) = g (r) f (θ) \exp i (m ϕ - ω t),

7-

{\hat{L}}_{q, m} [f_{q, m} (θ)] = - λ (q, m) f_{q, m} (θ)

8-

q = 2 Ω / ω

9-

{\hat{L}}_{q, m} = \frac{\partial}{\partial ζ} (\frac{1 - ζ^{2}}{1 - q^{2} ζ^{2}} \frac{\partial}{\partial ζ}) - \frac{m^{2}}{(1 - ζ^{2}) (1 - q^{2} ζ^{2})}

10-

- \frac{q m (1 + q^{2} ζ^{2})}{{(1 - q^{2} ζ^{2})}^{2}},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0509726

Formulas:

1-

_{d u s t}

2-

_{d u s t}

3-

_{d u s t}

4-

_{d u s t}

5-

_{d u s t}

6-

Σ_{d u s t}

7-

ρ (z) \propto s e c^{2} (z / z_{0})

8-

M_{⊙} y r^{- 1}

9-

σ_{R m s}

10-

σ_{R m s}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1001.2706

Formulas:

1-

[x^{μ}, x^{ν}] = i θ^{μ ν}

2-

(f ⋆ g) (x) = {e^{\frac{i}{2} θ^{μ ν} \partial_{y_{μ}} \partial_{z_{ν}}} f (y) g (z) |}_{x = y = z}

3-

x^{μ} ⋆ x^{ν} - x^{ν} ⋆ x^{μ} = {[x^{μ}, x^{ν}]}_{⋆} = i θ^{μ ν}

4-

\frac{\partial}{\partial x^{μ}} f = i {(θ)}_{μ ν}^{- 1} [x^{ν}, f]

5-

F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ} - i {[A_{μ}, A_{ν}]}_{⋆}

6-

S = - \frac{1}{4} \int d x F_{μ ν} ⋆ F^{μ ν}

7-

F \to U ⋆ F ⋆ U^{†}

8-

U ⋆ U^{†} = 1

9-

A_{μ} = A_{μ}^{α} λ^{α}

10-

X^{μ} = x^{μ} + θ^{μ ν} A_{ν}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-th

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0507662

Formulas:

1-

B_{z} (x, y, 0, t)

2-

β \sim 10^{- 3} \dots 10^{- 2}

3-

ρ_{0} \propto {| 𝐁_{0} (𝐱) |}^{3 / 2}

4-

τ_{A} = h_{0} / v_{A0}

5-

| 𝐁_{0} (h_{0}) |

6-

10^{31 – 32}

7-

η (z) = - z d \ln B_{ex} (0, 0, z, 0) / d z

8-

Φ = - 5.0 π

9-

B_{z} (x, y, 0, t)

10-

\sim (10^{31} – 10^{32})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/physics/0511136

Formulas:

1-

d 𝐫 := {d k_{t}, d v_{t}}

2-

d k_{t} = k_{j} - k_{i}

3-

d v_{t} = v_{j} - v_{i}

4-

𝐫_{i} := {k_{i}, v_{i}}

5-

𝐕_{m} (𝐫) = ⟨ d 𝐫 ⟩ / d t

6-

D d t = \sqrt{{⟨ | d 𝐫 | ⟩}^{2} - {⟨ d 𝐫 ⟩}^{2}} .

7-

η := D / {| 𝐕 |}_{m}

8-

𝐕_{m} (𝐫)

9-

𝐕_{m} (𝐫)

10-

𝐕_{m} (𝐫)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/physics/0608149

Formulas:

1-

C = \frac{k^{2}}{4 π ε_{h} ε_{0}} \sum_{m, n = - \infty}^{\infty} [1 + \frac{j}{k R_{m n}} + \frac{1}{k^{2} R_{m n}^{2}}]

2-

\times \frac{e^{- j k R_{m n}} - j 𝐤_{t} \cdot 𝐑_{m n}}{R_{m n}},

3-

k_{t} = \sqrt{k_{x}^{2} + k_{y}^{2}}

4-

\frac{1}{α} = \frac{ε_{m} + 2 ε_{h}}{3 ε_{0} ε_{h} \frac{4 π r_{0}^{3}}{3} (ε_{m} - ε_{h})} + j \frac{k^{3}}{6 π ε_{0} ε_{h}},

5-

ε_{m} = 1 - \frac{ω_{p}^{2}}{ω (ω - j ω_{D})},

6-

Re (\frac{1}{α}) = Re (C) .

7-

ω_{r} = ω_{p} / \sqrt{1 + 2 ε_{h}} .

8-

q = k_{x} / \sqrt{2} = k_{y} / \sqrt{2}

9-

z = 2 h + d

10-

𝐏_{m} = α 𝐄_{loc} = α [𝐄_{m}^{ext} + \sum_{n \neq m} 𝐄_{n}],

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0103203

Formulas:

1-

m λ = P (x_{p} - x_{p 0}) / (R \cos θ)

2-

| E_{A C I S} * m λ / (h c) | - 1 < Δ

3-

E_{A C I S}

4-

E_{A C I S}

5-

E_{A C I S}

6-

(h c) / (m λ)

7-

E_{A C I S}

8-

(h c) / (m λ)

9-

k T = 1.590 \pm 0.016

10-

1.81 \pm 0.03 \times 10^{22}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0610342

Formulas:

1-

G = M = - 4 E_{T} = 1

2-

t_{d} = G M^{5 / 2} / {(- 4 E_{T})}^{3 / 2} \equiv 1 .

3-

t_{r h} = \frac{0.138 N r_{h}^{3 / 2}}{\sqrt{G M} \ln (0.11 N)},

4-

r_{c} = \sqrt{\frac{\sum_{n = 1, N} r_{n}^{2} ρ_{n}^{2}}{\sum_{n = 1, N} ρ_{n}^{2}}},

5-

(3 / 2) k T

6-

N = 3 \cdot 10^{5}

7-

M = 3 \cdot 10^{5} M_{☉}

8-

t_{r h} \approx 8.5 \cdot 10^{8} y r

9-

\approx 10 A U

10-

W_{0} = 3, 5, 7, 11

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0401209

Formulas:

1-

M = 1.6 M_{⊙}

2-

M = 1.4 M_{⊙}

3-

k T_{bb} = 0.66

4-

\sim 50^{o} - 60^{o}

5-

μ I (μ)

6-

M = 1.4 M_{⊙}

7-

R = 2 r_{g} = 8.4

8-

μ I (μ)

9-

\int μ I (μ) 𝑑 μ = 1

10-

I_{bb} (μ) \propto \exp (- τ / μ)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1704.04264

Formulas:

1-

2 f - 3 f

2-

(χ^{(2)})

3-

(χ^{(3)})

4-

χ^{(2)} - χ^{(3)}

5-

(χ^{(3)})

6-

ω_{j} = ω_{0} + d_{1} j + d_{2} j^{2} / 2

7-

Ω_{j} = Ω_{0} + D_{1} j + D_{2} j^{2} / 2

8-

ω_{j, comb} = ω_{0} + d_{1} j

9-

Ω_{j, comb} = 2 ω_{0} + d_{1} j

10-

D_{int, IR} = ω_{j} - ω_{j, comb}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0706.2905

Formulas:

1-

S = M_{*}^{3} \int d^{5} x \sqrt{- g} R + M_{P}^{2} \int d^{4} x \sqrt{- \tilde{g}} \tilde{R}

2-

r_{c} = m_{c}^{- 1} = M_{P}^{2} / (2 M_{*}^{3})

3-

d s_{(5)}^{2} = - e^{- λ} d t^{2} + e^{λ} d r^{2} + r^{2} d Ω^{2} + γ d r d y + e^{σ} d y^{2},

4-

U_{z}^{2} - 4 (1 + U) U_{z} - 8 U (2 + U) = 0,

5-

z = \ln (r / r_{0})

6-

U = - 2 \exp (- 2 z) P_{r} / 3 m_{c}^{2} r_{0}^{2}

7-

P_{r} = d P / d r

8-

P (r) = {r (1 - \exp (- λ)) |}_{y = 0}

9-

R (U) = {[- \frac{(1 + 3 U + f)}{U^{2} {(3 + 3 U + \sqrt{3} f)}^{2 \sqrt{3}} (- 5 - 3 U + f)}]}^{\frac{1}{8}},

10-

A (U) = {[- \frac{(- 5 - 3 U + f) {(- 3 - 3 U - \sqrt{3} f)}^{2 \sqrt{3}}}{{(U + 2)}^{2} (1 + 3 U + f)}]}^{\frac{1}{8}},

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/1505.05447

Formulas:

1-

\frac{\partial ρ}{\partial t} + \nabla \cdot (ρ v) = 0

2-

\frac{\partial (ρ v)}{\partial t} + \nabla \cdot (ρ v v) = - \nabla p - ρ \nabla Φ + X

3-

\frac{\partial (ρ s)}{\partial t} + \nabla \cdot (ρ v s) = Q,

4-

\frac{\partial (ρ h)}{\partial t} + \nabla \cdot (ρ h v) = - \frac{\partial p}{\partial ϕ} - ρ \frac{\partial Φ}{\partial ϕ} + ϖ X_{ϕ},

5-

h = ϖ v_{ϕ} = ϖ^{2} Ω

6-

ϖ = r \sin θ

7-

A = \bar{A} + A^{'},

8-

\bar{A} = \frac{1}{2 π} \int_{0}^{2 π} A d ϕ .

9-

\frac{\partial \bar{ρ}}{\partial t} + \nabla_{⊥} \cdot (\bar{ρ} {\bar{v}}_{⊥}) = 𝒟

10-

\bar{ρ} \frac{\partial \bar{h}}{\partial t} + \bar{ρ} ({\bar{v}}_{⊥} \cdot \nabla_{⊥}) \bar{h} = - \nabla_{⊥} \cdot (ϖ \bar{ρ} \bar{v_{ϕ}^{'} v_{⊥}^{'}}) + ϖ {\bar{X}}_{ϕ} + ℋ

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1002.3495

Formulas:

1-

n n^{'} n

2-

ℋ = \sum_{i} V_{i} | i ⟩ ⟨ i | + \sum_{⟨ i, j ⟩} τ_{i j} | i ⟩ ⟨ j |

3-

V_{i} = V (𝐫_{i})

4-

τ_{0} \sim 3 eV

5-

τ_{i j} = - τ_{0} \exp (\frac{2 π i}{Φ_{0}} \int_{𝐫_{i}}^{𝐫_{j}} 𝐀 (𝐫) d 𝐫)

6-

Φ_{0} = h / e

7-

𝐁 (𝐫) = (0, 0, B)

8-

𝐀 (𝐫) = (- B y, 0, 0)

9-

for y \geq - w / 2,

10-

for y \leq - w / 2 - w_{s},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-ex/0202030

Formulas:

1-

c m^{- 2} s^{- 1}

2-

V_{d r i f t} ≃

3-

μ m / n s

4-

x = c o n s t

5-

V_{d r i f t}

6-

m m / μ s

7-

∣ \cos θ ∣ <

8-

σ_{v t x} =

9-

\pm V_{d r i f t} \times Δ t

10-

\pm V_{d r i f t} \times Δ t

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0404249

Formulas:

1-

L_{bol} \approx 5.6 L_{☉}

2-

α_{2000} = 5^{h} 46^{m} 13 \overset{s}{.} 13

3-

δ_{2000} = - 0^{\circ} 6^{'} 4 \overset{''}{.} 8

4-

α_{2000} = 5^{h} 46^{m} 13 \overset{s}{.} 13

5-

δ_{2000} = - 0^{\circ} 6^{'} 4 \overset{''}{.} 8

6-

{(J - H)}_{0} = 1.15

7-

{(H - K)}_{0} = 1.08

8-

E (J - H) = 1.43

9-

{(J - H)}_{0}

10-

{(H - K)}_{0}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2005.10129

Formulas:

1-

{ϕ_{i} (𝐫)}

2-

{ϕ_{a} (𝐫)}

3-

i, j = 1, \dots, N_{o}

4-

a, b = N_{o} + 1, \dots, N

5-

β \frac{{\bar{K}}_{a i}}{ϵ_{a} - ϵ_{i}},

6-

γ \sum_{i} w_{a i} w_{b i},

7-

K_{a i} = (ϕ_{a} ϕ_{i} | ϕ_{a} ϕ_{i})

8-

{\bar{K}}_{a i} = \frac{\sum_{b j} δ_{l_{a} l_{b}} δ_{l_{i} l_{j}} δ_{n_{a} n_{b}} δ_{n_{i} n_{j}} K_{b j}}{(2 l_{a} + 1) (2 l_{i} + 1)},

9-

(p | q) = \int \frac{p (𝐫_{1}) q (𝐫_{2})}{| 𝐫_{1} - 𝐫_{2} |} d^{3} 𝐫_{1} d^{3} 𝐫_{2}

10-

ℰ_{0} = \frac{1}{2} \sum_{i j} w_{i j} K_{i j} + \sum_{a i} w_{a i} K_{a i} + \frac{1}{2} \sum_{a b} w_{a b} K_{a b}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9510351

Formulas:

1-

S \bar{p} p S

2-

\equiv E \sin θ

3-

η \equiv - \log \tan (θ / 2)

4-

△ R \equiv {(△ η^{2} + △ ϕ^{2})}^{1 / 2}

5-

m > m_{m i n}

6-

\cos θ < {(\cos θ)}_{m a x}

7-

m_{m i n}

8-

{(\cos θ)}_{m a x}

9-

σ_{E} = 0.1 E .

10-

p_{T}^{h a r d}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1705.05223

Formulas:

1-

ℛ (𝒙, t) = \bar{n_{i} n_{j}}

2-

ℛ (𝒙, t)

3-

𝒖 (𝒙, t)

4-

\partial_{t} u_{i} + u_{k} \partial_{k} u_{i}

5-

= - \partial_{i} p + ν \partial^{2} u_{i} + \partial_{k} σ_{i k} + f_{i},

6-

\partial_{t} ℛ_{i j} + u_{j} \partial_{j} ℛ_{i j}

7-

= (\partial_{k} u_{i}) ℛ_{k j} + ℛ_{i k} (\partial_{k} u_{j}) -

8-

2 ℛ_{i j} (\partial_{l} u_{k}) ℛ_{k l},

9-

\partial_{k} = \partial / \partial x_{k}

10-

p (𝒙, t)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9609185

Formulas:

1-

H_{I} = - \sum_{i} (\sum_{ℓ > 0} J_{ℓ} σ_{i}^{z} σ_{i + ℓ}^{z} + g σ_{i}^{x})

2-

N_{0} \equiv ⟨ σ^{z} ⟩ \neq 0

3-

T \sim | m c^{2} |

4-

ϵ_{- p} = ϵ_{p}

5-

ϵ_{p} = {({(m c^{2})}^{2} + c^{2} p^{2} + 𝒪 (p^{4}))}^{1 / 2}

6-

| m | \sim | g - g_{c} |

7-

J_{ℓ > 1} = 0

8-

ℏ = k_{B} = 1

9-

N_{0} = {(1 - {(g / J_{1})}^{2})}^{1 / 8}

10-

ϵ_{p} = 2 {(J_{1}^{2} + g^{2} - 2 J_{1} g \cos (p a))}^{1 / 2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0301322

Formulas:

1-

{\tilde{r}}_{E} \approx 3.8 AU

2-

θ_{E} \approx 0.1 mas

3-

M \approx {0.050}_{- 0.011}^{+ 0.016} M_{⊙}

4-

RA = 17 : 47 : 23.29

5-

DEC = - 34 : 59 : 52.4

6-

λ = 267 .^{\circ} 364

7-

β = - 11 .^{\circ} 587

8-

l = 355 .^{\circ} 0084

9-

b = - 3 .^{\circ} 4640

10-

I_{base} = 15.10 \pm 0.02

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1009.1202

Formulas:

1-

Δ l_{c o a l} \approx 22000 km

2-

Δ τ_{c o a l} \approx 420 s

3-

τ_{c o a l} = \frac{1}{q_{c o a l}} . \frac{l_{c o a l}}{v_{A}},

4-

q_{c o a l} = \frac{u_{c o a l}}{v_{A}},

5-

_{c o a l}

6-

_{c o a l}

7-

Δ u_{c o a l} = \frac{Δ l_{c o a l}}{Δ τ_{c o a l}},

8-

_{c o a l}

9-

ℰ_{c} \approx \frac{L B^{2} a^{2}}{2} \ln \frac{L}{a}

10-

ℰ_{c} \approx 1.0 \times 10^{31} ergs .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1009.0115

Formulas:

1-

V > 17 m a g

2-

V > 17 m a g

3-

P_{o r b}

4-

M i n I (HJD) = 2 452 487.9876 (37) + 0^{d} .1366319 (6) \times E .

5-

Δ ϕ \approx 0.040 \pm 0.005

6-

M i n I (HJD) = 2 453 796.248245 + 0^{d} .062959041 \times E .

7-

f (m_{3}) = 2.673 \pm 0.002 M_{⊙}

8-

M i n I = T_{0} + P \times E + A_{m o d} \sin [2 π (E - T_{s}) / P_{m o d}]

9-

T_{0} (B J D)

10-

P_{o r b} (d)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1901.05099

Formulas:

1-

\sum_{\vec{k}, σ} (ϵ_{\vec{k}}^{c} c_{\vec{k}, σ}^{†} c_{\vec{k}, σ} + ϵ_{\vec{k}}^{f} f_{\vec{k}, σ}^{†} f_{\vec{k}, σ}) + \sum_{i} U f_{i, ↑}^{†} f_{i, ↑} f_{i, ↓}^{†} f_{i, ↓} +

2-

\sum_{i, σ} (μ_{f} f_{i, σ}^{†} f_{i, σ} + μ_{c} c_{i, σ}^{†} c_{i, σ}) + H_{Hyb},

3-

\sum_{\vec{k}} 2 V ({(σ_{x})}_{ρ_{1} ρ_{2}} \sin k_{x} c_{\vec{k}, ρ_{1}}^{†} f_{\vec{k}, ρ_{2}} + h.c.) +

4-

\sum_{\vec{k}} 2 V ({(σ_{y})}_{ρ_{1} ρ_{2}} \sin k_{y} c_{\vec{k}, ρ_{1}}^{†} f_{\vec{k}, ρ_{2}} + h.c.) .

5-

ϵ_{\vec{k}}^{c / f} = 2 t_{c / f} (\cos k_{x} + \cos k_{y})

6-

t_{f} / t = 0.2

7-

\vec{k} = (0, 0)

8-

\vec{B} = B (0, 0, 1)

9-

\vec{A} = B (- y, 0, 0)

10-

\int A_{x} (x, y) 𝑑 x, ϕ_{x, y}^{y} = \int A_{y} (x, y) 𝑑 y,

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1112.5258

Formulas:

1-

pp / p \bar{} p \to HW / Z \to H + 2

2-

- (5 - 10) %

3-

p_{T, H} \overset{>}{\sim} 200 GeV

4-

M_{H} = 120 GeV

5-

pp / p \bar{} p \to HW / Z \to H + 2

6-

H \to b \bar{} b

7-

pp \to V^{*} \to H V

8-

- (5 - 10) %

9-

7 (14) TeV

10-

1 - 2 (3 - 4) %

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1204.0362

Formulas:

1-

ℓ_{V} (Γ, x)

2-

ℓ_{V} (Γ) = (ℓ_{0}, ℓ_{1}, \dots, ℓ_{n})

3-

ℓ_{V} (Γ, x) = ℓ_{0} + ℓ_{1} x + \dots + ℓ_{n} x^{n}

4-

ℓ_{V} (Γ, x)

5-

ξ_{V} (Γ, x) = ξ_{0} + ξ_{1} x + \dots + ξ_{⌊ n / 2 ⌋} x^{⌊ n / 2 ⌋}

6-

ℓ_{V} (Γ, x) = \sum_{i = 0}^{⌊ n / 2 ⌋} ξ_{i} x^{i} {(1 + x)}^{n - 2 i} .

7-

ξ_{V} (Γ, x)

8-

ξ_{V} (Γ) = (ξ_{0}, ξ_{1}, \dots, ξ_{⌊ n / 2 ⌋})

9-

ξ_{V} (Γ, x)

10-

ξ_{V} (Γ, x)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1811.08076

Formulas:

1-

\begin{matrix} λ_{1} (t) & = & μ_{1} (t) + \int_{0}^{t} ϕ_{11} (t - s) 𝑑 N_{1} (s) + \int_{0}^{t} ϕ_{12} (t - s) 𝑑 N_{2} (s), \\ λ_{2} (t) & = & μ_{2} (t) + \int_{0}^{t} ϕ_{21} (t - s) 𝑑 N_{1} (s) + \int_{0}^{t} ϕ_{22} (t - s) 𝑑 N_{2} (s), \end{matrix}

2-

ϕ (t - s)

3-

ϕ_{i j} (u) = k_{i j} e^{b_{i j} v_{j}} [e^{- α_{i j} u} - e^{- β_{i j} u}],

4-

ϕ_{i j} (0) = 0

5-

ϕ_{i j} (u) = k_{i j} e^{b_{i j} v_{j}} [e^{- α_{i j} u} + e^{- β_{i j} u}] .

6-

ϕ_{i j} (0) = 0

7-

𝒪 (N^{2})

8-

q_{i j} = \int_{0}^{+ \infty} | ϕ_{i j} (u) | 𝑑 u

9-

Q = [\begin{matrix} k_{11} e^{b_{11} v_{1}} (\frac{1}{α_{11}} - \frac{1}{β_{11}}) & k_{12} e^{b_{12} v_{2}} (\frac{1}{α_{12}} - \frac{1}{β_{12}}) \\ k_{21} e^{b_{21} v_{1}} (\frac{1}{α_{21}} - \frac{1}{β_{21}}) & k_{22} e^{b_{22} v_{2}} (\frac{1}{α_{22}} - \frac{1}{β_{22}}) \end{matrix}] .

10-

Q = [\begin{matrix} k_{11} e^{b_{11} v_{1}} (\frac{1}{α_{11}} + \frac{1}{β_{11}}) & k_{12} e^{b_{12} v_{2}} (\frac{1}{α_{12}} + \frac{1}{β_{12}}) \\ k_{21} e^{b_{21} v_{1}} (\frac{1}{α_{21}} + \frac{1}{β_{21}}) & k_{22} e^{b_{22} v_{2}} (\frac{1}{α_{22}} + \frac{1}{β_{22}}) \end{matrix}] .

Formulas (html):

Correct Categorie: q-fin

Guessed Categorie: q-fin

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0507072

Formulas:

1-

p r o t o n + c o r e

2-

1 n s - 1 s

3-

1 M e V

4-

(Γ ≲ 10^{- 12} M e V)

5-

t_{1 / 2} ≳ 1 n s

6-

M e V s

7-

| ψ^{+} ⟩ = | u ⟩ ⟨ u | ψ^{+} ⟩ + P | ψ^{+} ⟩,

8-

P = 1 - | u ⟩ ⟨ u |

9-

(E - H) | ψ^{+} ⟩ = 0

10-

P | ψ^{+} ⟩ = | φ^{+} ⟩ + G_{P P}^{+} (E) P H | u ⟩ ⟨ u | ψ^{+} ⟩,

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/1010.1051

Formulas:

1-

ρ_{s} (r) = \frac{M_{s}}{2 π a^{3}} \frac{a^{4}}{r {(r + a)}^{3}},

2-

ϕ (r) = - \frac{G M}{(r + a)} .

3-

R_{e} = 1.8513 a

4-

ρ_{d} (r) = \frac{ρ_{0}}{(r / r_{d}) {(1 + r / r_{d})}^{2}},

5-

ρ_{0} = \frac{1}{3} ρ_{c r i t} Ω_{m} Δ_{v i r} \frac{c^{3}}{\ln (1 + c) - \frac{c}{(1 + c)}},

6-

ρ_{c r i t}

7-

M_{v i r}

8-

r_{v i r}

9-

Δ_{v i r}

10-

ρ_{c r i t} Ω_{m}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0608003

Formulas:

1-

Δ_{\vec{k}} = - \sum_{\vec{k^{'}}} v (\vec{k}, {\vec{k}}^{'}) \frac{1}{2 E_{{\vec{k}}^{'}}} Δ_{{\vec{k}}^{'}},

2-

v (\vec{k}, {\vec{k}}^{'})

3-

E_{\vec{k}} = \sqrt{{(ϵ_{\vec{k}} - ϵ_{F})}^{2} + Δ_{\vec{k}}^{2}}

4-

v (\vec{k}, {\vec{k}}^{'})

5-

G (ρ, β; ω) = v + v \sum_{k_{1} k_{2}} \frac{| k_{1} k_{2} ⟩ Q (k_{1}, k_{2}) ⟨ k_{1} k_{2} |}{ω - ϵ (k_{1}) - ϵ (k_{2}) + i η} G (ρ, β; ω),

6-

k_{i} \equiv ({\vec{k}}_{i}, σ_{i}, τ_{i})

7-

Q (k_{1}, k_{2})

8-

β = (ρ_{n} - ρ_{p}) / ρ

9-

ϵ (k) = ℏ^{2} k^{2} / (2 m) + U (k)

10-

Δ_{F}^{n} = Δ (k_{F}^{n})

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0506157

Formulas:

1-

Γ_{c} = {(3 {(1 + z)}^{3} E / 32 π n m_{p} c^{5} T^{3})}^{1 / 8} \sim 130 ζ^{3 / 8} E_{52}^{1 / 8} T_{2}^{- 3 / 8} n^{- 1 / 8}

2-

ζ = (1 + z) / 2

3-

E_{52} = E / 10^{52}

4-

T_{2} = T / 10^{2}

5-

t_{d} = (1 + z) {(3 E / 32 π Γ_{0}^{8} n m_{p} c^{5})}^{1 / 3}

6-

N d γ \propto γ^{- p} d γ, γ \geq γ_{m} \sim [(p - 2) / (p - 1)] ϵ_{e} (m_{p} / m_{e}) (Γ_{0} / Γ_{d}) \sim 180 (ϵ_{e} / 0.3) (Γ_{0} / Γ_{d})

7-

R_{d} \sim 2 c Γ_{d}^{2} t_{d} / (1 + z)

8-

τ (R_{d}) = σ_{T} N_{e} / 4 π R_{d}^{2} = (1 / 3) ℛ_{M} σ_{T} n R_{d}

9-

ℛ_{M} = Γ_{d}^{2} / Γ_{0}

10-

h ν_{X}^{'} = h ν_{X} / Γ \sim 50

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/0101027

Formulas:

1-

p_{g} \sim m_{s}^{2}

2-

L ≅ k . m_{s}^{3}

3-

k ≅ 5 \times 10^{- 65} W . k g^{- 3}

4-

m_{s (m a x)}

5-

m_{m a x} ≅ 20 m_{S u n}

6-

m_{S u n}

7-

m_{s} ≅ 60 m_{S u n}

8-

ε_{g} = - \frac{R . c^{4}}{8 π . G} = - \frac{3 m . c^{2}}{4 π . a . r^{2}}

9-

k . m_{m a x}^{3} \leq \frac{d}{d t} \int \frac{R . c^{4}}{8 π . G} 𝑑 V

10-

k . m_{m a x}^{3} \leq \frac{m_{m a x} . c^{2}}{t_{c}}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/0810.4506

Formulas:

1-

\frac{d σ}{d x} = \frac{α}{4 π} \int \frac{d Q^{2}}{Q^{2}} [σ_{T} \frac{1 + x^{2}}{2} + σ_{L} (1 - x^{2})],

2-

Q \cdot ϵ_{L} = 0

3-

ϵ_{L}^{2} = - 1

4-

ϵ_{L}^{μ} = {\tilde{X}}^{μ} / \sqrt{- {\tilde{X}}^{2}}, {\tilde{X}}^{μ} = (- g^{μ ν} + Q^{μ} Q^{ν} / Q^{2}) X_{ν},

5-

X^{μ} = a P_{1}^{μ} + b P_{2}^{μ},

6-

\hat{𝑿} = (a 𝑷_{1} + b 𝑷_{2}) / | a 𝑷_{1} + b 𝑷_{2} |

7-

Q \equiv {(Q^{2})}^{1 / 2}

8-

q \bar{q} \to γ^{*}

9-

q \bar{q} \to γ_{L}^{*}

10-

{\hat{σ}}_{L} = \frac{8 π^{2} e_{q}^{2} α ⟨ k_{⟂}^{2} ⟩ (a^{2} x_{2}^{2} + b^{2} x_{1}^{2})}{3 Q^{2} {(a x_{2} - b x_{1})}^{2}} δ (x_{1} x_{2} s - Q^{2}),

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0008368

Formulas:

1-

\dot{M} \sim > 10^{- 2} M_{⊙} {yr}^{- 1}

2-

α_{cold} \approx 0.01 << α_{hot} \approx 0.1

3-

R e_{M} \approx 3 \times 10^{3}

4-

M = 10^{7} M_{⊙}

5-

α_{hot} = α_{cold} = 0.1

6-

α_{hot} = 0.1 ≫ α_{cold} = 0.01

7-

Q^{+} = (9 / 8) ν Σ Ω^{2} = Q^{-} = 4 σ T_{c}^{4} / (3 Σ κ)

8-

α_{cold} \approx α_{hot} \approx 0.1

9-

R e_{M} \approx 3 \times 10^{3}

10-

R e_{M} = \frac{c_{s} H}{η},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2009.13140

Formulas:

1-

{Q_{k}^{(n)}}

2-

| ϕ_{trial} (\vec{θ}) ⟩

3-

⟨ Q_{k}^{(n)} ⟩

4-

⟨ Q_{k}^{(n)} ⟩

5-

⟨ H^{n} ⟩ \equiv ⟨ ϕ_{trial} (\vec{θ}) | H^{n} | ϕ_{trial} (\vec{θ}) ⟩

6-

c_{n} (\vec{θ})

7-

E_{0}^{(\inf)} (\vec{θ})

8-

E_{0}^{(\inf)} (\vec{θ})

9-

\min {⟨ H ⟩}_{\vec{θ}}

10-

{I, X, Y, Z}

Formulas (html):

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xref="S1.F1.23.m9.3.3.1.2.1.cmml">⟩</mo></mrow><mo id="S1.F1.23.m9.6.6.5" xref="S1.F1.23.m9.6.6.5.cmml">≡</mo><mrow id="S1.F1.23.m9.6.6.4.3" xref="S1.F1.23.m9.6.6.4.4.cmml"><mo stretchy="false" id="S1.F1.23.m9.6.6.4.3.4" xref="S1.F1.23.m9.6.6.4.4.1.cmml">⟨</mo><mrow id="S1.F1.23.m9.4.4.2.1.1" xref="S1.F1.23.m9.4.4.2.1.1.cmml"><msub id="S1.F1.23.m9.4.4.2.1.1.2" xref="S1.F1.23.m9.4.4.2.1.1.2.cmml"><mi id="S1.F1.23.m9.4.4.2.1.1.2.2" xref="S1.F1.23.m9.4.4.2.1.1.2.2.cmml">ϕ</mi><mi id="S1.F1.23.m9.4.4.2.1.1.2.3" xref="S1.F1.23.m9.4.4.2.1.1.2.3.cmml">trial</mi></msub><mo id="S1.F1.23.m9.4.4.2.1.1.1" xref="S1.F1.23.m9.4.4.2.1.1.1.cmml">⁢</mo><mrow id="S1.F1.23.m9.4.4.2.1.1.3.2" xref="S1.F1.23.m9.1.1.cmml"><mo stretchy="false" id="S1.F1.23.m9.4.4.2.1.1.3.2.1" xref="S1.F1.23.m9.1.1.cmml">(</mo><mover accent="true" id="S1.F1.23.m9.1.1" xref="S1.F1.23.m9.1.1.cmml"><mi id="S1.F1.23.m9.1.1.2" xref="S1.F1.23.m9.1.1.2.cmml">θ</mi><mo stretchy="false" id="S1.F1.23.m9.1.1.1" xref="S1.F1.23.m9.1.1.1.cmml">→</mo></mover><mo stretchy="false" id="S1.F1.23.m9.4.4.2.1.1.3.2.2" xref="S1.F1.23.m9.1.1.cmml">)</mo></mrow></mrow><mo fence="true" stretchy="false" id="S1.F1.23.m9.6.6.4.3.5" xref="S1.F1.23.m9.6.6.4.4.1.cmml">|</mo><msup id="S1.F1.23.m9.5.5.3.2.2" xref="S1.F1.23.m9.5.5.3.2.2.cmml"><mi id="S1.F1.23.m9.5.5.3.2.2.2" xref="S1.F1.23.m9.5.5.3.2.2.2.cmml">H</mi><mi id="S1.F1.23.m9.5.5.3.2.2.3" xref="S1.F1.23.m9.5.5.3.2.2.3.cmml">n</mi></msup><mo fence="true" stretchy="false" id="S1.F1.23.m9.6.6.4.3.6" xref="S1.F1.23.m9.6.6.4.4.1.cmml">|</mo><mrow id="S1.F1.23.m9.6.6.4.3.3" xref="S1.F1.23.m9.6.6.4.3.3.cmml"><msub id="S1.F1.23.m9.6.6.4.3.3.2" xref="S1.F1.23.m9.6.6.4.3.3.2.cmml"><mi id="S1.F1.23.m9.6.6.4.3.3.2.2" xref="S1.F1.23.m9.6.6.4.3.3.2.2.cmml">ϕ</mi><mi id="S1.F1.23.m9.6.6.4.3.3.2.3" xref="S1.F1.23.m9.6.6.4.3.3.2.3.cmml">trial</mi></msub><mo id="S1.F1.23.m9.6.6.4.3.3.1" xref="S1.F1.23.m9.6.6.4.3.3.1.cmml">⁢</mo><mrow id="S1.F1.23.m9.6.6.4.3.3.3.2" xref="S1.F1.23.m9.2.2.cmml"><mo stretchy="false" id="S1.F1.23.m9.6.6.4.3.3.3.2.1" xref="S1.F1.23.m9.2.2.cmml">(</mo><mover accent="true" id="S1.F1.23.m9.2.2" xref="S1.F1.23.m9.2.2.cmml"><mi id="S1.F1.23.m9.2.2.2" xref="S1.F1.23.m9.2.2.2.cmml">θ</mi><mo stretchy="false" id="S1.F1.23.m9.2.2.1" xref="S1.F1.23.m9.2.2.1.cmml">→</mo></mover><mo stretchy="false" id="S1.F1.23.m9.6.6.4.3.3.3.2.2" xref="S1.F1.23.m9.2.2.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S1.F1.23.m9.6.6.4.3.7" xref="S1.F1.23.m9.6.6.4.4.1.cmml">⟩</mo></mrow></mrow></math>, <math><mrow id="S1.F1.24.m10.1.2" xref="S1.F1.24.m10.1.2.cmml"><msub id="S1.F1.24.m10.1.2.2" xref="S1.F1.24.m10.1.2.2.cmml"><mi id="S1.F1.24.m10.1.2.2.2" xref="S1.F1.24.m10.1.2.2.2.cmml">c</mi><mi id="S1.F1.24.m10.1.2.2.3" xref="S1.F1.24.m10.1.2.2.3.cmml">n</mi></msub><mo id="S1.F1.24.m10.1.2.1" xref="S1.F1.24.m10.1.2.1.cmml">⁢</mo><mrow id="S1.F1.24.m10.1.2.3.2" xref="S1.F1.24.m10.1.1.cmml"><mo stretchy="false" id="S1.F1.24.m10.1.2.3.2.1" xref="S1.F1.24.m10.1.1.cmml">(</mo><mover accent="true" id="S1.F1.24.m10.1.1" xref="S1.F1.24.m10.1.1.cmml"><mi id="S1.F1.24.m10.1.1.2" xref="S1.F1.24.m10.1.1.2.cmml">θ</mi><mo stretchy="false" id="S1.F1.24.m10.1.1.1" xref="S1.F1.24.m10.1.1.1.cmml">→</mo></mover><mo stretchy="false" id="S1.F1.24.m10.1.2.3.2.2" xref="S1.F1.24.m10.1.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.F1.25.m11.2.3" xref="S1.F1.25.m11.2.3.cmml"><msubsup id="S1.F1.25.m11.2.3.2" xref="S1.F1.25.m11.2.3.2.cmml"><mi id="S1.F1.25.m11.2.3.2.2.2" xref="S1.F1.25.m11.2.3.2.2.2.cmml">E</mi><mn id="S1.F1.25.m11.2.3.2.2.3" xref="S1.F1.25.m11.2.3.2.2.3.cmml">0</mn><mrow id="S1.F1.25.m11.1.1.1.3" xref="S1.F1.25.m11.2.3.2.cmml"><mo stretchy="false" id="S1.F1.25.m11.1.1.1.3.1" xref="S1.F1.25.m11.2.3.2.cmml">(</mo><mi id="S1.F1.25.m11.1.1.1.1" xref="S1.F1.25.m11.1.1.1.1.cmml">inf</mi><mo stretchy="false" id="S1.F1.25.m11.1.1.1.3.2" xref="S1.F1.25.m11.2.3.2.cmml">)</mo></mrow></msubsup><mo id="S1.F1.25.m11.2.3.1" xref="S1.F1.25.m11.2.3.1.cmml">⁢</mo><mrow id="S1.F1.25.m11.2.3.3.2" xref="S1.F1.25.m11.2.2.cmml"><mo stretchy="false" id="S1.F1.25.m11.2.3.3.2.1" xref="S1.F1.25.m11.2.2.cmml">(</mo><mover accent="true" id="S1.F1.25.m11.2.2" xref="S1.F1.25.m11.2.2.cmml"><mi id="S1.F1.25.m11.2.2.2" xref="S1.F1.25.m11.2.2.2.cmml">θ</mi><mo stretchy="false" id="S1.F1.25.m11.2.2.1" xref="S1.F1.25.m11.2.2.1.cmml">→</mo></mover><mo stretchy="false" id="S1.F1.25.m11.2.3.3.2.2" xref="S1.F1.25.m11.2.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.F1.26.m12.2.3" xref="S1.F1.26.m12.2.3.cmml"><msubsup id="S1.F1.26.m12.2.3.2" xref="S1.F1.26.m12.2.3.2.cmml"><mi id="S1.F1.26.m12.2.3.2.2.2" xref="S1.F1.26.m12.2.3.2.2.2.cmml">E</mi><mn id="S1.F1.26.m12.2.3.2.2.3" xref="S1.F1.26.m12.2.3.2.2.3.cmml">0</mn><mrow id="S1.F1.26.m12.1.1.1.3" xref="S1.F1.26.m12.2.3.2.cmml"><mo stretchy="false" id="S1.F1.26.m12.1.1.1.3.1" xref="S1.F1.26.m12.2.3.2.cmml">(</mo><mi id="S1.F1.26.m12.1.1.1.1" xref="S1.F1.26.m12.1.1.1.1.cmml">inf</mi><mo stretchy="false" id="S1.F1.26.m12.1.1.1.3.2" xref="S1.F1.26.m12.2.3.2.cmml">)</mo></mrow></msubsup><mo id="S1.F1.26.m12.2.3.1" xref="S1.F1.26.m12.2.3.1.cmml">⁢</mo><mrow id="S1.F1.26.m12.2.3.3.2" xref="S1.F1.26.m12.2.2.cmml"><mo stretchy="false" id="S1.F1.26.m12.2.3.3.2.1" xref="S1.F1.26.m12.2.2.cmml">(</mo><mover accent="true" id="S1.F1.26.m12.2.2" xref="S1.F1.26.m12.2.2.cmml"><mi id="S1.F1.26.m12.2.2.2" xref="S1.F1.26.m12.2.2.2.cmml">θ</mi><mo stretchy="false" id="S1.F1.26.m12.2.2.1" xref="S1.F1.26.m12.2.2.1.cmml">→</mo></mover><mo stretchy="false" id="S1.F1.26.m12.2.3.3.2.2" xref="S1.F1.26.m12.2.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.F1.28.m14.1.2" xref="S1.F1.28.m14.1.2.cmml"><mi id="S1.F1.28.m14.1.2.2" xref="S1.F1.28.m14.1.2.2.cmml">min</mi><mo id="S1.F1.28.m14.1.2.1" xref="S1.F1.28.m14.1.2.1.cmml">⁢</mo><msub id="S1.F1.28.m14.1.2.3" xref="S1.F1.28.m14.1.2.3.cmml"><mrow id="S1.F1.28.m14.1.2.3.2.2" xref="S1.F1.28.m14.1.2.3.2.1.cmml"><mo stretchy="false" id="S1.F1.28.m14.1.2.3.2.2.1" xref="S1.F1.28.m14.1.2.3.2.1.1.cmml">⟨</mo><mi id="S1.F1.28.m14.1.1" xref="S1.F1.28.m14.1.1.cmml">H</mi><mo stretchy="false" id="S1.F1.28.m14.1.2.3.2.2.2" xref="S1.F1.28.m14.1.2.3.2.1.1.cmml">⟩</mo></mrow><mover accent="true" id="S1.F1.28.m14.1.2.3.3" xref="S1.F1.28.m14.1.2.3.3.cmml"><mi id="S1.F1.28.m14.1.2.3.3.2" xref="S1.F1.28.m14.1.2.3.3.2.cmml">θ</mi><mo stretchy="false" id="S1.F1.28.m14.1.2.3.3.1" xref="S1.F1.28.m14.1.2.3.3.1.cmml">→</mo></mover></msub></mrow></math>, <math><mrow id="S2.p1.2.m2.4.5.2" xref="S2.p1.2.m2.4.5.1.cmml"><mo stretchy="false" id="S2.p1.2.m2.4.5.2.1" xref="S2.p1.2.m2.4.5.1.cmml">{</mo><mi id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">I</mi><mo id="S2.p1.2.m2.4.5.2.2" xref="S2.p1.2.m2.4.5.1.cmml">,</mo><mi id="S2.p1.2.m2.2.2" xref="S2.p1.2.m2.2.2.cmml">X</mi><mo id="S2.p1.2.m2.4.5.2.3" xref="S2.p1.2.m2.4.5.1.cmml">,</mo><mi id="S2.p1.2.m2.3.3" xref="S2.p1.2.m2.3.3.cmml">Y</mi><mo id="S2.p1.2.m2.4.5.2.4" xref="S2.p1.2.m2.4.5.1.cmml">,</mo><mi id="S2.p1.2.m2.4.4" xref="S2.p1.2.m2.4.4.cmml">Z</mi><mo stretchy="false" id="S2.p1.2.m2.4.5.2.5" xref="S2.p1.2.m2.4.5.1.cmml">}</mo></mrow></math>

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0207159

Formulas:

1-

1 < Γ < Γ_{j}

2-

B = 10^{- 5} G

3-

T_{a c c} \sim ζ r_{g} c V_{A}^{- 2}

4-

V_{A} \sim 0.01 c

5-

n_{p} \sim 10^{- 4} {cm}^{- 3}

6-

T_{l o s s} \sim 3 m_{e} c / 4 σ_{T} γ u

7-

u = u_{B} + u_{c m b} + u_{a g n}

8-

u_{c m b}

9-

u_{a g n}

10-

u_{c m b} = a T_{c m b}^{4} Γ^{2} < a T_{c m b}^{4} Γ_{j}^{2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0101494

Formulas:

1-

^{3, 4, 5, 6}

2-

\approx 1.3 \times 10^{- 16}

3-

\approx 6.5 \times 10^{- 16}

4-

\overset{<}{\sim} 3 \times 10^{- 16}

5-

ℛ - K_{s} > 5.0

6-

\approx 1.9 \times 10^{- 17}

7-

\approx 17^{'} \times 19^{'}

8-

\approx 1.3 \times 10^{- 16}

9-

\approx 6.5 \times 10^{- 16}

10-

(1.6 \pm 0.4) \times 10^{20}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/9611072

Formulas:

1-

Δ X Δ T ≳ α^{'}

2-

g_{s}^{1 / 3} ℓ_{s}

3-

Δ T \sim \frac{1}{E}

4-

Δ X \sim ℓ_{s}^{2} E

5-

Δ X Δ T ≳ ℓ_{s}^{2},

6-

\frac{Δ X + Δ T}{2} \geq ℓ_{s}

7-

\frac{ℓ_{s}^{2}}{Δ T} \sim ℓ_{s}^{2} E

8-

α (0) = 2

9-

A (E) \sim E^{α (t) - 1}

10-

Δ T \sim \frac{1}{Δ X} \to 0

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1511.03111

Formulas:

1-

\dot{P} = 2.28 \times 10^{- 12} s s^{- 1}

2-

2 \times 10^{17} g s^{- 1}

3-

10^{- 16} - 10^{- 14} s s^{- 1}

4-

\dot{P} = 2.28 \times 10^{- 12} s s^{- 1}

5-

n = Ω \ddot{Ω} / {\dot{Ω}}^{2} = 0.9 \pm 0.2

6-

Ω = 2 π / P

7-

ω = Ω / Ω_{m}

8-

R_{m} = 1.6 \times 10^{8} {(\frac{B}{10^{12} G})}^{4 / 7} {(\frac{\dot{M}}{10^{18} g s^{- 1}})}^{- 2 / 7} cm .

9-

R_{co} = {(G M / Ω^{2})}^{1 / 3}

10-

N_{mag} = \frac{\dot{M} \sqrt{2 G M R_{m}}}{3} (\frac{2}{3 ω} - 1),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0502534

Formulas:

1-

I (ν, t)

2-

U (ν, t)

3-

I (ν, t) = U^{*} (ν, t) U (ν, t)

4-

| U (ν, t) | = \sqrt{I (ν, t)}

5-

U (ν, t)

6-

u_{j} (ν, t)

7-

U (ν, t) = \sum_{j} u_{j} (ν, t) = \sum_{j} {\tilde{u}}_{j} \exp [2 π i (ν τ_{j} + ω_{j} t)] .

8-

(ν, t) \to (τ, ω)

9-

\tilde{U} (τ, ω) = \sum_{j} {\tilde{u}}_{j} δ (τ - τ_{j}) δ (ω - ω_{j}),

10-

I (ν, t) = U^{*} (ν, t) U (ν, t)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/2004.10272

Formulas:

1-

\frac{\partial f_{s}}{\partial t} + v \frac{\partial f_{s}}{\partial x} + \frac{q_{s}}{m_{s}} E \frac{\partial f_{s}}{\partial v}

2-

\frac{\partial E}{\partial x} - \sum_{s} 4 π q_{s} n_{s} \int 𝑑 v f_{s}

3-

ω = ω_{r} + i ω_{i}

4-

| ω_{i} | ≪ | ω_{r} |

5-

= 1 + \sum_{s} D_{r, s}

6-

= \sum_{s} (i ω_{i} \frac{\partial D_{r, s}}{\partial ω_{r}} + D_{i, s}) .

7-

\equiv - \frac{ω_{p s}^{2}}{k^{2}} P V \int 𝑑 v \frac{\partial f_{s 0} / \partial v}{v - ω_{r} / k}

8-

\equiv - {π \frac{ω_{p s}^{2}}{k^{2}} \frac{\partial f_{s 0}}{\partial v} |}_{ω_{r} / k},

9-

\equiv ⟨ \frac{E^{2}}{8 π} ⟩ = \frac{E_{0}^{2}}{16 π} .

10-

= W [ω_{r} \frac{\partial}{\partial ω_{r}} (\sum_{s} D_{r, s})] .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0512225

Formulas:

1-

G (𝐫) \equiv ⟨ {[h (𝐫) - h (0)]}^{2} ⟩

2-

G (𝐫) \sim \ln (r)

3-

\sqrt{G} (r) \propto r^{α}

4-

1 / {| 𝐪 |}^{2}

5-

𝐡_{⟂} (𝐫)

6-

𝐡_{⟂} (𝐫)

7-

P 4_{2} / m n m

8-

G (x, T)

9-

G (x, T)

10-

G (x, T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1811.02905

Formulas:

1-

τ_{f} = ρ_{p} / ρ_{g} \cdot d / c

2-

ρ_{p} = 10^{3} kg / m^{3}

3-

ρ_{g} = 10^{- 6} kg / m^{3}

4-

c = 1000 m / s

5-

k = a - b \exp (T - c)

6-

Δ t_{h e a t}

7-

\propto Δ t_{h e a t}^{1 / 4}

8-

T_{s} = 1200 K

9-

T_{C} = 1400 K

10-

T_{m} = 1500 K

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1312.4286

Formulas:

1-

\hat{H} = {\hat{H}}_{e} + {\hat{H}}_{ph} + g (| 1 ⟩ ⟨ 1 | - | 2 ⟩ ⟨ 2 |) \otimes ({\hat{b}}^{†} + \hat{b}),

2-

{\hat{H}}_{ph} = ℏ ω {\hat{b}}^{†} \hat{b}

3-

\hat{ρ} (0) = {\hat{ρ}}_{e} (0) \otimes e^{- β {\hat{H}}_{ph}} / Tr [e^{- β {\hat{H}}_{ph}}]

4-

{\hat{ρ}}_{e} (0)

5-

{\hat{ρ}}_{e} (t) = {Tr}_{ph} [\hat{ρ} (t)]

6-

\hat{ρ} (t)

7-

{\hat{ρ}}_{e} (t)

8-

\hat{ℋ} = {\hat{H}}_{e} + {\hat{ℋ}}_{ph} + ℏ \sqrt{2} g | 1 ⟩ ⟨ 1 | \otimes ({\hat{c}}_{1}^{†} + {\hat{c}}_{1}) + ℏ \sqrt{2} g | 2 ⟩ ⟨ 2 | \otimes ({\hat{c}}_{2}^{†} + {\hat{c}}_{2}),

9-

{\hat{ℋ}}_{ph} = ℏ ω {\hat{c}}_{1}^{†} {\hat{c}}_{1} + ℏ ω {\hat{c}}_{2}^{†} {\hat{c}}_{2}

10-

\hat{ρ} (0) = {\hat{ρ}}_{e} (0) \otimes e^{- β {\hat{ℋ}}_{ph}} / Tr [e^{- β {\hat{ℋ}}_{ph}}]

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1303.4051

Formulas:

1-

M = {0; 1}

2-

\begin{matrix} \bar{x} = 1 - x, \\ x \land y = x \cdot y, \\ x \lor y = x + y - x \cdot y . \end{matrix}

3-

x \land y = \bar{\bar{x} \lor \bar{y}}

4-

\bar{\bar{x} \lor \bar{y}} = 1 - \bar{x} \lor \bar{y} = 1 - (1 - x + 1 - y - (1 - x) \cdot (1 - y)) = x \cdot y = x \land y .

5-

h (t) = {\begin{matrix} 1, t \geq 0, \\ 0, t < 0 . \end{matrix}

6-

\prod_{i = 1}^{n} h (t - τ_{i}) = h (t - \max_{i = \bar{1, n}} τ_{i}),

7-

y_{c} = f_{1} (t - τ) \cdot f_{2} (t - τ),

8-

f_{1} (t)

9-

f_{2} (t)

10-

Y = X_{1} X_{2} \lor \bar{X_{3}} X_{4}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/1708.02599

Formulas:

1-

p_{x} \times p_{y} \times p_{z}

2-

p_{x} \times p_{y} \times p_{z} = 46 \times 46 \times 7

3-

P_{x} \times P_{y} \times P_{z} = 318 \times 318 \times 33

4-

p_{x} \times p_{y} \times p_{z}

5-

(P_{x}, P_{y}, P_{z})

6-

v (x, y, z)

7-

(x, y, z)

8-

M (x, y, z) = \exp (- {|| v (x, y, z) - v (P_{x} / 2, P_{y} / 2, P_{z} / 2) ||}^{2})

9-

v (P_{x} / 2, P_{y} / 2, P_{z} / 2)

10-

3.6 nm \times 3.6 nm \times 40 nm

Formulas (html):

Correct Categorie: cs

Guessed Categorie: quant-ph

Result: incorrect

Run 16446906 (Agent672)