Run 11331761

Paper: https://arxiv.org/abs/cond-mat/9505016

Formulas:

1-

\frac{ℏ^{2} k_{x}^{2}}{2 m} + Δ_{0} \cos (Q_{x} x + Θ)

2-

+ 2 t_{b} \cos (k_{y} b - q_{x} x + ω_{y} t) .

3-

ℏ = h / 2 π

4-

A_{y} = H x - c E_{y} t

5-

A_{x} = A_{z} = 0

6-

k_{y} \to k_{y} - e A_{y} / c ℏ

7-

q_{x} = e b H / ℏ c

8-

ω_{y} = e b E_{y} / ℏ

9-

Q_{x} = 2 k_{F} - N q_{x}

10-

ω_{y} / q_{x} = c E_{y} / H

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0312558

Formulas:

1-

ν_{e} : ν_{μ} : ν_{τ} = 1 : 2 : 0

2-

\frac{d N}{d E} \propto E^{- (2 \div 2.5)}

3-

E_{\max} \sim β Z (\frac{B}{1 μ G}) (\frac{R}{1 k p c}) 10^{18}

4-

\frac{d Φ}{d E_{ν} d Ω_{ν}}

5-

N_{μ} = \int \int 𝑑 E_{ν} 𝑑 Ω_{ν} A_{ν}^{e f f} (E_{ν}, Ω_{ν}) \frac{d Φ}{d E_{ν} d Ω_{ν}}

6-

E^{- (2 \div 2.5)}

7-

ν_{μ} + {\bar{ν}}_{μ}

8-

E_{μ m i n}

9-

ν_{μ} + {\bar{ν}}_{μ}

10-

E_{ν} \sim 6 - 10^{3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2009.12119

Formulas:

1-

u_{1}, u_{2}, \dots, u_{k} = v

2-

e_{1}, e_{2}, \dots, e_{l}

3-

Q_{1}, Q_{3}, Q_{5}, Q_{7}, Q_{9}

4-

R (c P v)

5-

R (c P v)

6-

Q_{1}, Q_{3}, Q_{5}, \dots, Q_{l - 1}

7-

R (c P v)

8-

V_{odd} (G)

9-

B = {b_{1}, b_{2}, \dots, b_{k}}

10-

C = {c_{1}, c_{2}, \dots, c_{l}}

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0203381

Formulas:

1-

ℛ \equiv D_{L} / a_{B}

2-

1.0 \leq ℛ \sim < 1.4

3-

D_{L}_{1} / D_{L}_{2} = Ω_{p}_{2} / Ω_{p}_{1}

4-

N_{p} \sim > 4 \times 10^{6}

5-

Ω_{p} \sin i = \frac{\int_{- \infty}^{\infty} h (Y) V_{l o s} (X, Y) Σ (X, Y) 𝑑 X 𝑑 Y}{\int_{- \infty}^{\infty} h (Y) X Σ (X, Y) 𝑑 X 𝑑 Y} \equiv \frac{𝒱}{𝒳}

6-

Ω_{p} \sin i = 4.7 \pm 1.7

7-

k m s^{- 1} a r c s e c^{- 1}

8-

D_{L} = 53_{- 15}^{+ 30}^{''}

9-

a_{B} = 69^{''} \pm 5^{''}

10-

ℛ = {0.77}_{- 0.25}^{+ 0.43}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2007.08504

Formulas:

1-

f_{θ^{'}} (I) \equiv (π, 𝐳_{𝐬}, 𝐳_{𝐭})

2-

δ \in ℝ^{| V | \times 3}

3-

ϕ : 𝕊^{2} \to ℝ^{3}

4-

ϕ (\cdot, 𝐳_{𝐬}) : 𝕊^{2} \to ℝ^{3}

5-

ϕ (𝐮, 𝐳_{𝐬}) = \bar{ϕ} (𝐮) + δ (𝐮, 𝐳_{𝐬}); 𝐮 \in 𝕊^{2}; 𝐳_{𝐬} \in ℝ^{d}

6-

\bar{ϕ} (\cdot)

7-

δ (\cdot, \cdot)

8-

\bar{ϕ} (\cdot)

9-

δ (\cdot, 𝐳_{𝐬})

10-

δ (𝐮, 𝐳_{𝐬}) \equiv (δ^{'} (𝐮, 𝐳_{𝐬}) + ℛ (δ^{'} (ℛ (𝐮), 𝐳_{𝐬}))) / 2;

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1010.5110

Formulas:

1-

𝒞_{2 n + 1} (A)

2-

𝒯_{2 n + 1}

3-

𝒯_{2 n + 1} = 𝒞_{2 n + 1} (A) - 𝒞_{2 n + 1} (\bar{A}) - d B_{2 n} (A, \bar{A}) .

4-

𝒯_{2 n + 1} (A, \bar{A}) = (n + 1) \int_{0}^{1} 𝑑 t < Δ A F_{t}^{n} >

5-

A_{t} = t A + (1 - t) \bar{A} = \bar{A} + t Δ A

6-

A, B, C, \dots

7-

d - 1 = 2 n

8-

I, J, K, \dots

9-

d - 1 = 2 n

10-

a, b, c, \dots

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9611100

Formulas:

1-

0.1 \leq Ω_{ν} \leq 0.5

2-

0.3 \leq Ω_{0} \leq 0.5

3-

0.18 \sim < Γ \sim < 0.25

4-

20 h^{- 1} Mpc

5-

- 1.1 \sim < n_{eff} \sim < - 0.9

6-

0.2 \sim < Ω_{ν} \sim < 0.3

7-

0.3 \sim < Ω_{0} \sim < 0.5

8-

z \sim < 5

9-

\sim 10^{11} M_{⊙}

10-

P (k) \propto k

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2002.11138

Formulas:

1-

T_{eff} [K]

2-

\ln ℒ = - \frac{1}{2} [\sum_{i = 1}^{N} \ln (σ_{v r}^{2} + σ_{i}^{2}) + \sum_{i = 1}^{N} \frac{{(v_{i} - < V_{r} >)}^{2}}{σ_{v r}^{2} + σ_{i}^{2}}]

3-

\frac{d v}{d χ}

4-

\ln ℒ = - \frac{1}{2} [\sum_{i = 1}^{N} \ln (σ_{v r}^{2} + σ_{i}^{2}) + \sum_{i = 1}^{N} \frac{{(v_{i} - < V_{r} > - \frac{d v}{d χ} χ_{i})}^{2}}{σ_{v r}^{2} + σ_{i}^{2}}]

5-

χ_{i} = (α_{i} - α_{0}) \cos (δ_{0}) \sin (θ) + (δ_{i} - δ_{0}) \cos (θ)

6-

\sqrt{σ_{v r, i}^{2}} = \sqrt{\frac{\sum_{i = 1}^{N} ρ_{i} V_{r, i}^{2}}{\sum_{i = 1}^{N} ρ_{i}} - {[\frac{\sum_{i = 1}^{N} ρ_{i} V_{r, i}}{\sum_{i = 1}^{N} ρ_{i}}]}^{2} - σ_{i}^{2}}

7-

σ_{v r, i}

8-

σ_{t o t} = \sqrt{k^{2} σ_{i}^{2} + σ_{s}^{2}}

9-

μ_{α} \cos δ

10-

μ_{r} = - 6.1363 \times 10^{- 5} V_{l o s} \frac{R}{d}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0908.0518

Formulas:

1-

P_{C} = 13.1 (1)

2-

T_{C 2} = 92

3-

T_{C 1} = 66

4-

T_{C 1} = 66

5-

𝐪 = (0, \frac{1}{2}, 0)

6-

T_{0} = 6.0 \pm 0.5

7-

a_{0, i}^{- 1} {(\partial a_{i} / \partial P)}_{P = 0}

8-

1.245 (1) \times 10^{- 3}

9-

I \propto {| \vec{Q} \cdot \vec{δ} |}^{2}

10-

2 \pm 0.5, 0, 0

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1405.0217

Formulas:

1-

d_{1} + d_{2} ≪ λ_{0}

2-

θ_{t}^{'} = θ_{inc}

3-

ϵ_{τ} = ϵ_{x} = ϵ_{y}

4-

k_{n}^{2} - (- \frac{ϵ_{τ}}{ϵ_{n}}) k_{τ}^{2} = ϵ_{τ} \frac{ω^{2}}{c^{2}}

5-

d_{1} + d_{2} ≪ λ_{0}

6-

k_{τ} = \sqrt{ϵ_{0}} ω / c

7-

{k_{n} |}_{k_{τ} = \sqrt{ϵ_{0}} \frac{ω}{c}}

8-

= \frac{π}{d_{1} + d_{2}},

9-

{\frac{d^{2} k_{n}}{{dk}_{τ}^{2}} |}_{k_{τ} = 0}

10-

= \frac{1}{\sqrt{ϵ_{0}} \frac{ω}{c}} .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/math/0510248

Formulas:

1-

ℬ =_{d f} 𝒜_{0} \otimes 𝒦

2-

1_{ℳ (ℬ)}

3-

ℬ =_{d f} 𝒜_{0} \otimes 𝒦

4-

{p_{n}}_{n = 1}^{\infty}

5-

P = \sum_{n = 1}^{\infty} p_{n}

6-

P_{N} =_{d f} \sum_{n = 1}^{N} p_{n}

7-

T (𝒜_{0})

8-

{P_{N}}_{N = 1}^{\infty}

9-

T (𝒜_{0})

10-

{P_{N}}_{N = 1}^{\infty}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1901.09305

Formulas:

1-

σ_{d} / d < 0.4

2-

0.07 < σ_{d} / d < 0.15

3-

| Z | < 1.0 kpc

4-

X = 8.34, Y = 0

5-

{\tilde{V}}_{ϕ} - V_{LSR}

6-

σ_{d} / d < 0.1

7-

σ_{d} / d < 0.2

8-

σ_{d} / d < 0.5

9-

σ_{d} / d < 0.1

10-

- 2.5 < Y < 2.5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0401220

Formulas:

1-

4 k \times 4 k

2-

\frac{d N_{t}}{d M_{V}} = 5 \frac{Ω}{4 π} F (M_{V}) \frac{f (M_{V}, r, a)}{0.75 %} {(\frac{a}{a_{0}})}^{- 5 / 2} {(\frac{r}{r_{0}})}^{6} {(\frac{γ}{γ_{0}})}^{3 / 2} {(\frac{Δ χ_{\min}^{2}}{36})}^{- 3 / 2}

3-

F (M_{V})

4-

F (M_{V}) = [\frac{n (M_{V})}{n_{0}}] {[\frac{L (M_{V})}{L_{0}}]}^{3 / 2} {[\frac{R (M_{V})}{R_{0}}]}^{- 7 / 2}

5-

γ_{0} = 1.0 \times 10^{7}

6-

a_{0} = 10 R_{⊙}

7-

r_{0} = 0.10 R_{⊙}

8-

R_{0} = 0.97 R_{⊙}

9-

L_{0} = 0.86 L_{⊙}

10-

n_{0} = 0.0025 {pc}^{- 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

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/0903.5384

Formulas:

1-

θ_{src} > \sqrt{λ / 2 π D}

2-

T_{char} = D θ_{src} / V_{trans}

3-

T_{p e r i o d}

4-

T_{p e r i o d}

5-

T_{p e r i o d}

6-

T_{d e v}^{i} = \frac{T_{i} - T_{p e r i o d}}{T_{p e r i o d}}

7-

T_{p e r i o d}

8-

T_{d e v}

9-

T_{d e v}

10-

T_{d e v}

Formulas (html):

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xref="S1.p4.2.m2.1.1.3.2.4.cmml">D</mi></mrow></msqrt></mrow></math>, <math><mrow id="S1.p4.6.m6.1.1" xref="S1.p4.6.m6.1.1.cmml"><msub id="S1.p4.6.m6.1.1.2" xref="S1.p4.6.m6.1.1.2.cmml"><mi id="S1.p4.6.m6.1.1.2.2" xref="S1.p4.6.m6.1.1.2.2.cmml">T</mi><mi id="S1.p4.6.m6.1.1.2.3" xref="S1.p4.6.m6.1.1.2.3.cmml">char</mi></msub><mo id="S1.p4.6.m6.1.1.1" xref="S1.p4.6.m6.1.1.1.cmml">=</mo><mrow id="S1.p4.6.m6.1.1.3" xref="S1.p4.6.m6.1.1.3.cmml"><mrow id="S1.p4.6.m6.1.1.3.2" xref="S1.p4.6.m6.1.1.3.2.cmml"><mi id="S1.p4.6.m6.1.1.3.2.2" xref="S1.p4.6.m6.1.1.3.2.2.cmml">D</mi><mo id="S1.p4.6.m6.1.1.3.2.1" xref="S1.p4.6.m6.1.1.3.2.1.cmml">⁢</mo><msub id="S1.p4.6.m6.1.1.3.2.3" xref="S1.p4.6.m6.1.1.3.2.3.cmml"><mi id="S1.p4.6.m6.1.1.3.2.3.2" xref="S1.p4.6.m6.1.1.3.2.3.2.cmml">θ</mi><mi id="S1.p4.6.m6.1.1.3.2.3.3" xref="S1.p4.6.m6.1.1.3.2.3.3.cmml">src</mi></msub></mrow><mo id="S1.p4.6.m6.1.1.3.1" xref="S1.p4.6.m6.1.1.3.1.cmml">/</mo><msub id="S1.p4.6.m6.1.1.3.3" xref="S1.p4.6.m6.1.1.3.3.cmml"><mi id="S1.p4.6.m6.1.1.3.3.2" xref="S1.p4.6.m6.1.1.3.3.2.cmml">V</mi><mi id="S1.p4.6.m6.1.1.3.3.3" xref="S1.p4.6.m6.1.1.3.3.3.cmml">trans</mi></msub></mrow></mrow></math>, <math><msub id="S3.SS1.SSS1.p2.4.m4.1.1" xref="S3.SS1.SSS1.p2.4.m4.1.1.cmml"><mi id="S3.SS1.SSS1.p2.4.m4.1.1.2" xref="S3.SS1.SSS1.p2.4.m4.1.1.2.cmml">T</mi><mrow id="S3.SS1.SSS1.p2.4.m4.1.1.3" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.cmml"><mi id="S3.SS1.SSS1.p2.4.m4.1.1.3.2" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.2.cmml">p</mi><mo id="S3.SS1.SSS1.p2.4.m4.1.1.3.1" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.4.m4.1.1.3.3" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.3.cmml">e</mi><mo id="S3.SS1.SSS1.p2.4.m4.1.1.3.1a" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.4.m4.1.1.3.4" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.4.cmml">r</mi><mo id="S3.SS1.SSS1.p2.4.m4.1.1.3.1b" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.4.m4.1.1.3.5" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.5.cmml">i</mi><mo id="S3.SS1.SSS1.p2.4.m4.1.1.3.1c" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.4.m4.1.1.3.6" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.6.cmml">o</mi><mo id="S3.SS1.SSS1.p2.4.m4.1.1.3.1d" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.4.m4.1.1.3.7" xref="S3.SS1.SSS1.p2.4.m4.1.1.3.7.cmml">d</mi></mrow></msub></math>, <math><msub id="S3.SS1.SSS1.p2.6.m6.1.1" xref="S3.SS1.SSS1.p2.6.m6.1.1.cmml"><mi id="S3.SS1.SSS1.p2.6.m6.1.1.2" xref="S3.SS1.SSS1.p2.6.m6.1.1.2.cmml">T</mi><mrow id="S3.SS1.SSS1.p2.6.m6.1.1.3" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.cmml"><mi id="S3.SS1.SSS1.p2.6.m6.1.1.3.2" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.2.cmml">p</mi><mo id="S3.SS1.SSS1.p2.6.m6.1.1.3.1" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.6.m6.1.1.3.3" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.3.cmml">e</mi><mo id="S3.SS1.SSS1.p2.6.m6.1.1.3.1a" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.6.m6.1.1.3.4" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.4.cmml">r</mi><mo id="S3.SS1.SSS1.p2.6.m6.1.1.3.1b" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.6.m6.1.1.3.5" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.5.cmml">i</mi><mo id="S3.SS1.SSS1.p2.6.m6.1.1.3.1c" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.6.m6.1.1.3.6" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.6.cmml">o</mi><mo id="S3.SS1.SSS1.p2.6.m6.1.1.3.1d" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.6.m6.1.1.3.7" xref="S3.SS1.SSS1.p2.6.m6.1.1.3.7.cmml">d</mi></mrow></msub></math>, <math><msub id="S3.SS1.SSS1.p2.8.m8.1.1" xref="S3.SS1.SSS1.p2.8.m8.1.1.cmml"><mi id="S3.SS1.SSS1.p2.8.m8.1.1.2" xref="S3.SS1.SSS1.p2.8.m8.1.1.2.cmml">T</mi><mrow id="S3.SS1.SSS1.p2.8.m8.1.1.3" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.cmml"><mi id="S3.SS1.SSS1.p2.8.m8.1.1.3.2" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.2.cmml">p</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.1.3.1" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.8.m8.1.1.3.3" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.3.cmml">e</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.1.3.1a" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.8.m8.1.1.3.4" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.4.cmml">r</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.1.3.1b" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.8.m8.1.1.3.5" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.5.cmml">i</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.1.3.1c" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.8.m8.1.1.3.6" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.6.cmml">o</mi><mo id="S3.SS1.SSS1.p2.8.m8.1.1.3.1d" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p2.8.m8.1.1.3.7" xref="S3.SS1.SSS1.p2.8.m8.1.1.3.7.cmml">d</mi></mrow></msub></math>, <math><mrow id="S3.Ex1.m1.1.1" xref="S3.Ex1.m1.1.1.cmml"><msubsup id="S3.Ex1.m1.1.1.2" xref="S3.Ex1.m1.1.1.2.cmml"><mi id="S3.Ex1.m1.1.1.2.2.2" xref="S3.Ex1.m1.1.1.2.2.2.cmml">T</mi><mrow id="S3.Ex1.m1.1.1.2.2.3" xref="S3.Ex1.m1.1.1.2.2.3.cmml"><mi id="S3.Ex1.m1.1.1.2.2.3.2" xref="S3.Ex1.m1.1.1.2.2.3.2.cmml">d</mi><mo id="S3.Ex1.m1.1.1.2.2.3.1" xref="S3.Ex1.m1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.2.2.3.3" xref="S3.Ex1.m1.1.1.2.2.3.3.cmml">e</mi><mo id="S3.Ex1.m1.1.1.2.2.3.1a" xref="S3.Ex1.m1.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.2.2.3.4" xref="S3.Ex1.m1.1.1.2.2.3.4.cmml">v</mi></mrow><mi id="S3.Ex1.m1.1.1.2.3" xref="S3.Ex1.m1.1.1.2.3.cmml">i</mi></msubsup><mo id="S3.Ex1.m1.1.1.1" xref="S3.Ex1.m1.1.1.1.cmml">=</mo><mfrac id="S3.Ex1.m1.1.1.3" xref="S3.Ex1.m1.1.1.3.cmml"><mrow id="S3.Ex1.m1.1.1.3.2" xref="S3.Ex1.m1.1.1.3.2.cmml"><msub id="S3.Ex1.m1.1.1.3.2.2" xref="S3.Ex1.m1.1.1.3.2.2.cmml"><mi id="S3.Ex1.m1.1.1.3.2.2.2" xref="S3.Ex1.m1.1.1.3.2.2.2.cmml">T</mi><mi id="S3.Ex1.m1.1.1.3.2.2.3" xref="S3.Ex1.m1.1.1.3.2.2.3.cmml">i</mi></msub><mo id="S3.Ex1.m1.1.1.3.2.1" xref="S3.Ex1.m1.1.1.3.2.1.cmml">-</mo><msub id="S3.Ex1.m1.1.1.3.2.3" xref="S3.Ex1.m1.1.1.3.2.3.cmml"><mi id="S3.Ex1.m1.1.1.3.2.3.2" xref="S3.Ex1.m1.1.1.3.2.3.2.cmml">T</mi><mrow id="S3.Ex1.m1.1.1.3.2.3.3" xref="S3.Ex1.m1.1.1.3.2.3.3.cmml"><mi id="S3.Ex1.m1.1.1.3.2.3.3.2" xref="S3.Ex1.m1.1.1.3.2.3.3.2.cmml">p</mi><mo id="S3.Ex1.m1.1.1.3.2.3.3.1" xref="S3.Ex1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.2.3.3.3" xref="S3.Ex1.m1.1.1.3.2.3.3.3.cmml">e</mi><mo id="S3.Ex1.m1.1.1.3.2.3.3.1a" xref="S3.Ex1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.2.3.3.4" xref="S3.Ex1.m1.1.1.3.2.3.3.4.cmml">r</mi><mo id="S3.Ex1.m1.1.1.3.2.3.3.1b" xref="S3.Ex1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.2.3.3.5" xref="S3.Ex1.m1.1.1.3.2.3.3.5.cmml">i</mi><mo id="S3.Ex1.m1.1.1.3.2.3.3.1c" xref="S3.Ex1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.2.3.3.6" xref="S3.Ex1.m1.1.1.3.2.3.3.6.cmml">o</mi><mo id="S3.Ex1.m1.1.1.3.2.3.3.1d" xref="S3.Ex1.m1.1.1.3.2.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.2.3.3.7" xref="S3.Ex1.m1.1.1.3.2.3.3.7.cmml">d</mi></mrow></msub></mrow><msub id="S3.Ex1.m1.1.1.3.3" xref="S3.Ex1.m1.1.1.3.3.cmml"><mi id="S3.Ex1.m1.1.1.3.3.2" xref="S3.Ex1.m1.1.1.3.3.2.cmml">T</mi><mrow id="S3.Ex1.m1.1.1.3.3.3" xref="S3.Ex1.m1.1.1.3.3.3.cmml"><mi id="S3.Ex1.m1.1.1.3.3.3.2" xref="S3.Ex1.m1.1.1.3.3.3.2.cmml">p</mi><mo id="S3.Ex1.m1.1.1.3.3.3.1" xref="S3.Ex1.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.3.3.3" xref="S3.Ex1.m1.1.1.3.3.3.3.cmml">e</mi><mo id="S3.Ex1.m1.1.1.3.3.3.1a" xref="S3.Ex1.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.3.3.4" xref="S3.Ex1.m1.1.1.3.3.3.4.cmml">r</mi><mo id="S3.Ex1.m1.1.1.3.3.3.1b" xref="S3.Ex1.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.3.3.5" xref="S3.Ex1.m1.1.1.3.3.3.5.cmml">i</mi><mo id="S3.Ex1.m1.1.1.3.3.3.1c" xref="S3.Ex1.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.3.3.6" xref="S3.Ex1.m1.1.1.3.3.3.6.cmml">o</mi><mo id="S3.Ex1.m1.1.1.3.3.3.1d" xref="S3.Ex1.m1.1.1.3.3.3.1.cmml">⁢</mo><mi id="S3.Ex1.m1.1.1.3.3.3.7" xref="S3.Ex1.m1.1.1.3.3.3.7.cmml">d</mi></mrow></msub></mfrac></mrow></math>, <math><msub id="S3.SS1.SSS1.p3.1.m1.1.1" xref="S3.SS1.SSS1.p3.1.m1.1.1.cmml"><mi id="S3.SS1.SSS1.p3.1.m1.1.1.2" xref="S3.SS1.SSS1.p3.1.m1.1.1.2.cmml">T</mi><mrow id="S3.SS1.SSS1.p3.1.m1.1.1.3" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.cmml"><mi id="S3.SS1.SSS1.p3.1.m1.1.1.3.2" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.2.cmml">p</mi><mo id="S3.SS1.SSS1.p3.1.m1.1.1.3.1" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.1.m1.1.1.3.3" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.3.cmml">e</mi><mo id="S3.SS1.SSS1.p3.1.m1.1.1.3.1a" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.1.m1.1.1.3.4" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.4.cmml">r</mi><mo id="S3.SS1.SSS1.p3.1.m1.1.1.3.1b" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.1.m1.1.1.3.5" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.5.cmml">i</mi><mo id="S3.SS1.SSS1.p3.1.m1.1.1.3.1c" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.1.m1.1.1.3.6" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.6.cmml">o</mi><mo id="S3.SS1.SSS1.p3.1.m1.1.1.3.1d" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.1.m1.1.1.3.7" xref="S3.SS1.SSS1.p3.1.m1.1.1.3.7.cmml">d</mi></mrow></msub></math>, <math><msub id="S3.SS1.SSS1.p3.3.m3.1.1" xref="S3.SS1.SSS1.p3.3.m3.1.1.cmml"><mi id="S3.SS1.SSS1.p3.3.m3.1.1.2" xref="S3.SS1.SSS1.p3.3.m3.1.1.2.cmml">T</mi><mrow id="S3.SS1.SSS1.p3.3.m3.1.1.3" xref="S3.SS1.SSS1.p3.3.m3.1.1.3.cmml"><mi id="S3.SS1.SSS1.p3.3.m3.1.1.3.2" xref="S3.SS1.SSS1.p3.3.m3.1.1.3.2.cmml">d</mi><mo id="S3.SS1.SSS1.p3.3.m3.1.1.3.1" xref="S3.SS1.SSS1.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.3.m3.1.1.3.3" xref="S3.SS1.SSS1.p3.3.m3.1.1.3.3.cmml">e</mi><mo id="S3.SS1.SSS1.p3.3.m3.1.1.3.1a" xref="S3.SS1.SSS1.p3.3.m3.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.3.m3.1.1.3.4" xref="S3.SS1.SSS1.p3.3.m3.1.1.3.4.cmml">v</mi></mrow></msub></math>, <math><msub id="S3.SS1.SSS1.p3.4.m4.1.1" xref="S3.SS1.SSS1.p3.4.m4.1.1.cmml"><mi id="S3.SS1.SSS1.p3.4.m4.1.1.2" xref="S3.SS1.SSS1.p3.4.m4.1.1.2.cmml">T</mi><mrow id="S3.SS1.SSS1.p3.4.m4.1.1.3" xref="S3.SS1.SSS1.p3.4.m4.1.1.3.cmml"><mi id="S3.SS1.SSS1.p3.4.m4.1.1.3.2" xref="S3.SS1.SSS1.p3.4.m4.1.1.3.2.cmml">d</mi><mo id="S3.SS1.SSS1.p3.4.m4.1.1.3.1" xref="S3.SS1.SSS1.p3.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.4.m4.1.1.3.3" xref="S3.SS1.SSS1.p3.4.m4.1.1.3.3.cmml">e</mi><mo id="S3.SS1.SSS1.p3.4.m4.1.1.3.1a" xref="S3.SS1.SSS1.p3.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.SS1.SSS1.p3.4.m4.1.1.3.4" xref="S3.SS1.SSS1.p3.4.m4.1.1.3.4.cmml">v</mi></mrow></msub></math>, <math><msub id="S3.F9.3.m1.1.1" xref="S3.F9.3.m1.1.1.cmml"><mi id="S3.F9.3.m1.1.1.2" xref="S3.F9.3.m1.1.1.2.cmml">T</mi><mrow id="S3.F9.3.m1.1.1.3" xref="S3.F9.3.m1.1.1.3.cmml"><mi id="S3.F9.3.m1.1.1.3.2" xref="S3.F9.3.m1.1.1.3.2.cmml">d</mi><mo id="S3.F9.3.m1.1.1.3.1" xref="S3.F9.3.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.F9.3.m1.1.1.3.3" xref="S3.F9.3.m1.1.1.3.3.cmml">e</mi><mo id="S3.F9.3.m1.1.1.3.1b" xref="S3.F9.3.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.F9.3.m1.1.1.3.4" xref="S3.F9.3.m1.1.1.3.4.cmml">v</mi></mrow></msub></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0304454

Formulas:

1-

α = a_{1} / a_{2}

2-

λ = L_{1} / L_{2} \approx (m_{1} / m_{2}) α^{1 / 2}

3-

a_{j} / a_{j + 1} ≳ 0.5

4-

a_{j} / a_{j + 1} ≲ 0.1

5-

a_{j} / a_{j + 1} ≲ 1 / 3

6-

α = a_{1} / a_{2}

7-

- [{\dot{r}}_{1} \sin (ω_{1} + f_{1}) + r_{1} {\dot{f}}_{1} \cos (ω_{1} + f_{1})] \sin i_{1}

8-

\frac{- 2 π a_{1}}{P_{1} \sqrt{1 - e_{1}^{2}}} [\cos (ω_{1} + f_{1}) + e_{1} \cos ω_{1}] \sin i_{1},

9-

P_{1} = \frac{2 π a_{1}^{3 / 2}}{\sqrt{G (m_{0} + m_{1})}}

10-

V_{0, r}^{'} = \frac{- m_{1}}{m_{0} + m_{1}} V_{1, r} = K_{1} [\cos (ω_{1} + f_{1}) + e_{1} \cos ω_{1}],

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0809.5092

Formulas:

1-

T ⟨ 0 | O_{U} (x) O_{U}^{†} (0) | 0 ⟩ = i B_{d} {(p^{2})}^{d - 2}

2-

T ⟨ 0 | O_{U} (x) O_{U}^{†} (0) | 0 ⟩ = i \frac{A_{d}}{2 π} \int_{0}^{\infty} 𝑑 M^{2} \frac{ρ (M^{2})}{p^{2} - μ^{2} - M^{2} + i ϵ}

3-

ρ (M^{2}) = {(M^{2})}^{d - 2}

4-

ρ (M^{2}) = {(M^{2})}^{d - 2}

5-

O_{U} O_{S M}

6-

L_{i n t} = Λ^{2 - d} O_{U} H^{2} + c_{ψ} Λ^{1 - d} O_{U} \bar{ψ} ψ + c_{F} Λ^{- d} O_{U} F^{2}

7-

{(g - 2)}_{μ}

8-

Λ^{2 - d} O H^{2}

9-

μ^{4 - d} = Λ^{2 - d} v^{2}

10-

ρ (M^{2})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/0907.1460

Formulas:

1-

I \bar{4} 3 m

2-

χ = C / (T - Θ)

3-

J / k_{B} = - 33.2

4-

- J \sum_{i < j} 𝒔_{i} \cdot 𝒔_{j} - g μ_{B} 𝑩 \cdot \sum_{i} 𝒔_{i}

5-

- \frac{J}{2} (𝑺^{2} - \sum_{i} 𝒔_{i}^{2}) - g μ_{B} 𝑩 \cdot 𝑺,

6-

i = 1, 2, 3, 4

7-

= 𝒔_{1} + 𝒔_{2} + 𝒔_{3} + 𝒔_{4}

8-

E = - \frac{J}{2} {S (S + 1) - 4 s (s + 1)} - g μ_{B} B S_{z},

9-

| J | / (g μ_{B})

10-

- J \sum_{< i j >} 𝒔_{i} \cdot 𝒔_{j} + D \sum_{i} {(𝒔_{i} \cdot 𝒆_{i})}^{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0709.4150

Formulas:

1-

v (r_{ij}) = v_{0} ϵ [{(\frac{σ}{r_{ij}})}^{a} - {(\frac{σ}{r_{ij}})}^{b}],

2-

ρ σ^{2} ≃ 1.16

3-

v_{0} ϵ {(σ / z_{i})}^{a}

4-

δ t = 0.0005 t_{0}

5-

t_{0} = \sqrt{ϵ σ^{2} / m}

6-

N (s) = \frac{1}{s} \int_{s}^{\infty} n (s^{'}) 𝑑 s^{'} .

7-

n (s) \sim s^{- τ}

8-

m_{2} = \sum_{s} s^{2} n (s),

9-

τ = α \ln V + β,

10-

V_{t} \leq V \leq 3

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1712.06167

Formulas:

1-

E (Z, A) = \int d^{3} r e (ρ, α) ρ (r) + \int d^{3} r f_{0} {| \nabla ρ |}^{2} + \frac{e^{2}}{4 π ϵ_{0}} {(4 π)}^{2} \int_{0}^{\infty} 𝑑 r^{'} r^{'} ρ_{p} (r^{'}) \int_{0}^{r^{'}} 𝑑 r r^{2} ρ_{p} (r) .

2-

α = ρ_{I} / ρ

3-

(ρ_{n} - ρ_{p})

4-

e (ρ, α)

5-

e (ρ, α) = e (ρ, 0) + e_{s y m} (ρ) α^{2},

6-

e_{s y m} (ρ)

7-

ρ_{i} (r) = \frac{ρ_{0}}{1 + e^{(r - a_{i}) / c_{i}}},

8-

S_{n} = R_{n} - R_{p},

9-

S_{p} = R_{p} - R_{n},

10-

r . m . s .

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/physics/0008011

Formulas:

1-

i \frac{\partial u}{\partial ξ} + \frac{1}{2} \frac{\partial^{2} u}{\partial τ^{2}} + u {| u |}^{2} = - i Γ u

2-

u (ζ, τ)

3-

\frac{\partial}{\partial τ} \frac{\partial}{\partial u_{τ}^{*}} (e^{Γ ξ} L) + \frac{\partial}{\partial ξ} \frac{\partial}{\partial u_{ξ}^{*}} (e^{Γ ξ} L) - \frac{\partial}{\partial u^{*}} (e^{Γ ξ} L) = 0

4-

L = \frac{1}{2} {| \frac{\partial u}{\partial t} |}^{2} + i (u \frac{\partial u^{*}}{\partial ξ} - u^{*} \frac{\partial u}{\partial ξ}) - \frac{1}{2} {| u |}^{4}

5-

δ \int_{0}^{\infty} \int_{0}^{\infty} e^{Γ ξ} L 𝑑 ξ 𝑑 τ = 0

6-

u (ξ, τ) = A (ξ) sech (\frac{τ}{w (ξ)}) \exp (i ϕ (ξ)),

7-

δ \int_{0}^{\infty} e^{Γ ξ} ⟨ L ⟩ 𝑑 ξ = 0,

8-

⟨ L ⟩ = \frac{{| A |}^{2}}{3 w} + 2 i w (A \frac{d A^{*}}{d ξ} - A^{*} \frac{d A}{d ξ}) + 4 w {| A |}^{2} \frac{d ϕ}{d ξ} - 2 w \frac{{| A |}^{4}}{3}

9-

e^{Γ ξ} ⟨ L ⟩

10-

\frac{d}{d ξ} (w {| A |}^{2}) = - Γ w {| A |}^{2}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/9812029

Formulas:

1-

6 L + 2 N - 6

2-

C_{μ ν ρ}

3-

g_{s} = ⟨ e^{ϕ} ⟩

4-

m_{s}^{2} = 2 π R m_{p}^{3}

5-

g_{s} = 2 π R m_{s} .

6-

T_{F 1} = 2 π R T_{M 2} .

7-

T_{F 1} = 2 π m_{s}^{2}

8-

T_{M 2} = 2 π m_{p}^{3}

9-

T_{D p} = 2 π m_{s}^{p + 1} / g_{s} .

10-

T_{D 2} = 2 π m_{s}^{3} / g_{s} = 2 π m_{p}^{3} = T_{M 2}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/1301.4998

Formulas:

1-

M_{⊙} {yr}^{- 1}

2-

2 \times 10^{5} K

3-

\sim 2 M_{⊙} {yr}^{- 1}

4-

4.3 < \log (T / K) < 5.3

5-

\log (T / K) < 4.3

6-

10 - 20 km s^{- 1}

7-

\sim 30 km s^{- 1}

8-

| v_{LOS} | < 90 km s^{- 1}

9-

| b | < 15^{\circ}

10-

σ = 30 km s^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1909.09032

Formulas:

1-

Z_{s} = R_{s} + i X_{s}

2-

R_{s} (0) \approx 22

3-

R_{s} (0) \approx 0.9

4-

R_{s} (0)

5-

\tilde{σ} = σ_{1} - i σ_{2}

6-

Z_{s} = \sqrt{i μ_{0} ω / \tilde{σ}}

7-

σ_{1} (T)

8-

σ_{1} (0) / σ_{1} (T_{c}) = 0

9-

d_{x^{2} - y^{2}}

10-

d_{x^{2} - y^{2}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0109035

Formulas:

1-

| ψ ⟩ = | o b a ⟩,

2-

| ψ_{f} ⟩ = (\hat{S} \cos γ + \hat{N} \sin γ) \hat{O} (\hat{I} \otimes \hat{B} \otimes \hat{A}) | ψ_{i} ⟩,

3-

γ \in [0, \frac{π}{2}]

4-

\hat{O} = \sum_{i j k ℓ} | ϵ_{i j k} | | n j k ⟩ ⟨ ℓ j k | + \sum_{j ℓ} | m j j ⟩ ⟨ ℓ j j |,

5-

| ϵ_{i j k} | = 1

6-

m = (j + ℓ + 1) (mod 3)

7-

n = (i + ℓ) (mod 3)

8-

\hat{S} = \sum_{i j k ℓ} | ϵ_{i j ℓ} | | i ℓ k ⟩ ⟨ i j k | + \sum_{i j} | i i j ⟩ ⟨ i i j |,

9-

\hat{A} = (a_{i j})

10-

\hat{B} = (b_{i j})

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/math/0504223

Formulas:

1-

(M, \partial_{-} M, \partial_{+} M)

2-

D_{1}^{2} \times I, D_{2}^{2} \times I

3-

\partial D_{1}^{2} \times I, \partial D_{2}^{2} \times I

4-

| q |, | n | \geq 2

5-

| p | \neq | m |

6-

X = S^{1} \times S^{1} \times I

7-

μ = S^{1} \times {*} \times {1}

8-

λ = {*} \times S^{1} \times {0}

9-

Q^{'} \cup X \cup Q^{''}

10-

Q_{p, q} \cup Q

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0102020

Formulas:

1-

ρ^{2} \frac{\partial (E / N)}{\partial ρ}

2-

ρ (E_{F} - E / N),

3-

E_{F} = E / N,

4-

Σ (k) = \frac{δ Φ}{δ G (k)} .

5-

V_{α} (k, p) = λ_{α}^{r} g_{α}^{r} (k) g_{α}^{r} (p) - λ_{α}^{a} g_{α}^{a} (k) g_{α}^{a} (p)

6-

g^{r, a} (p) = \frac{1}{p^{2} + β_{r, a}^{2}}

7-

λ^{r} = 29.6 G e V^{2}

8-

β^{r} = 639 M e V

9-

λ^{a} = 2.91 G e V^{2}

10-

β^{a} = 352 M e V

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1804.10138

Formulas:

1-

Δ n_{e} / n_{e} \approx

2-

Δ I / I \propto 2 Δ n_{e} / n_{e}

3-

Δ n_{e} / n_{e} \approx 5

4-

Δ I / I \approx 10 %

5-

Δ n_{e} / n_{e} \approx 5 %

6-

Δ f = \sqrt{\frac{1}{T} \int_{T} f {(t)}^{2} d t},

7-

Δ f_{signal} = \sqrt{\sum Δ f_{i}^{2}},

8-

Δ I_{meas}^{2} = Δ I^{2} + Δ I_{phot}^{2} + Δ I_{\det}^{2} + Δ I_{scat}^{2} .

9-

e^{-} {DN}^{- 1}

10-

Δ I_{phot} = \sqrt{\frac{I}{N G Δ t}} .

Formulas (html):

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xref="S1.p5.5.m5.1.1.2.2.3.cmml">I</mi></mrow><mo id="S1.p5.5.m5.1.1.2.1" xref="S1.p5.5.m5.1.1.2.1.cmml">/</mo><mi id="S1.p5.5.m5.1.1.2.3" xref="S1.p5.5.m5.1.1.2.3.cmml">I</mi></mrow><mo id="S1.p5.5.m5.1.1.1" xref="S1.p5.5.m5.1.1.1.cmml">≈</mo><mrow id="S1.p5.5.m5.1.1.3" xref="S1.p5.5.m5.1.1.3.cmml"><mn id="S1.p5.5.m5.1.1.3.2" xref="S1.p5.5.m5.1.1.3.2.cmml">10</mn><mo id="S1.p5.5.m5.1.1.3.1" xref="S1.p5.5.m5.1.1.3.1.cmml">%</mo></mrow></mrow></math>, <math><mrow id="S1.p5.6.m6.1.1" xref="S1.p5.6.m6.1.1.cmml"><mrow id="S1.p5.6.m6.1.1.2" xref="S1.p5.6.m6.1.1.2.cmml"><mrow id="S1.p5.6.m6.1.1.2.2" xref="S1.p5.6.m6.1.1.2.2.cmml"><mi mathvariant="normal" id="S1.p5.6.m6.1.1.2.2.2" xref="S1.p5.6.m6.1.1.2.2.2.cmml">Δ</mi><mo id="S1.p5.6.m6.1.1.2.2.1" xref="S1.p5.6.m6.1.1.2.2.1.cmml">⁢</mo><msub id="S1.p5.6.m6.1.1.2.2.3" xref="S1.p5.6.m6.1.1.2.2.3.cmml"><mi id="S1.p5.6.m6.1.1.2.2.3.2" xref="S1.p5.6.m6.1.1.2.2.3.2.cmml">n</mi><mi mathvariant="normal" id="S1.p5.6.m6.1.1.2.2.3.3" xref="S1.p5.6.m6.1.1.2.2.3.3.cmml">e</mi></msub></mrow><mo id="S1.p5.6.m6.1.1.2.1" xref="S1.p5.6.m6.1.1.2.1.cmml">/</mo><msub id="S1.p5.6.m6.1.1.2.3" xref="S1.p5.6.m6.1.1.2.3.cmml"><mi id="S1.p5.6.m6.1.1.2.3.2" xref="S1.p5.6.m6.1.1.2.3.2.cmml">n</mi><mi mathvariant="normal" id="S1.p5.6.m6.1.1.2.3.3" xref="S1.p5.6.m6.1.1.2.3.3.cmml">e</mi></msub></mrow><mo id="S1.p5.6.m6.1.1.1" xref="S1.p5.6.m6.1.1.1.cmml">≈</mo><mrow id="S1.p5.6.m6.1.1.3" xref="S1.p5.6.m6.1.1.3.cmml"><mn id="S1.p5.6.m6.1.1.3.2" xref="S1.p5.6.m6.1.1.3.2.cmml">5</mn><mo id="S1.p5.6.m6.1.1.3.1" xref="S1.p5.6.m6.1.1.3.1.cmml">%</mo></mrow></mrow></math>, <math><mrow id="S3.E1.m1.2.2.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1" xref="S3.E1.m1.2.2.1.1.cmml"><mrow id="S3.E1.m1.2.2.1.1.2" xref="S3.E1.m1.2.2.1.1.2.cmml"><mi mathvariant="normal" id="S3.E1.m1.2.2.1.1.2.2" xref="S3.E1.m1.2.2.1.1.2.2.cmml">Δ</mi><mo id="S3.E1.m1.2.2.1.1.2.1" xref="S3.E1.m1.2.2.1.1.2.1.cmml">⁢</mo><mi id="S3.E1.m1.2.2.1.1.2.3" xref="S3.E1.m1.2.2.1.1.2.3.cmml">f</mi></mrow><mo id="S3.E1.m1.2.2.1.1.1" xref="S3.E1.m1.2.2.1.1.1.cmml">=</mo><msqrt id="S3.E1.m1.1.1" xref="S3.E1.m1.1.1.cmml"><mrow id="S3.E1.m1.1.1.1" xref="S3.E1.m1.1.1.1.cmml"><mfrac id="S3.E1.m1.1.1.1.3" xref="S3.E1.m1.1.1.1.3.cmml"><mn id="S3.E1.m1.1.1.1.3.2" xref="S3.E1.m1.1.1.1.3.2.cmml">1</mn><mi id="S3.E1.m1.1.1.1.3.3" xref="S3.E1.m1.1.1.1.3.3.cmml">T</mi></mfrac><mo id="S3.E1.m1.1.1.1.2" xref="S3.E1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S3.E1.m1.1.1.1.4" xref="S3.E1.m1.1.1.1.4.cmml"><msub id="S3.E1.m1.1.1.1.4.1" xref="S3.E1.m1.1.1.1.4.1.cmml"><mo largeop="true" symmetric="true" id="S3.E1.m1.1.1.1.4.1.2" xref="S3.E1.m1.1.1.1.4.1.2.cmml">∫</mo><mi id="S3.E1.m1.1.1.1.4.1.3" xref="S3.E1.m1.1.1.1.4.1.3.cmml">T</mi></msub><mrow id="S3.E1.m1.1.1.1.4.2" xref="S3.E1.m1.1.1.1.4.2.cmml"><mi id="S3.E1.m1.1.1.1.4.2.2" xref="S3.E1.m1.1.1.1.4.2.2.cmml">f</mi><mo id="S3.E1.m1.1.1.1.4.2.1" xref="S3.E1.m1.1.1.1.4.2.1.cmml">⁢</mo><mpadded width="-1.7pt" id="S3.E1.m1.1.1.1.4.2.3" xref="S3.E1.m1.1.1.1.4.2.3.cmml"><msup id="S3.E1.m1.1.1.1.4.2.3a" xref="S3.E1.m1.1.1.1.4.2.3.cmml"><mrow id="S3.E1.m1.1.1.1.4.2.3.2.2" xref="S3.E1.m1.1.1.1.4.2.3.cmml"><mo stretchy="false" id="S3.E1.m1.1.1.1.4.2.3.2.2.1" xref="S3.E1.m1.1.1.1.4.2.3.cmml">(</mo><mi id="S3.E1.m1.1.1.1.1" xref="S3.E1.m1.1.1.1.1.cmml">t</mi><mo stretchy="false" id="S3.E1.m1.1.1.1.4.2.3.2.2.2" xref="S3.E1.m1.1.1.1.4.2.3.cmml">)</mo></mrow><mn id="S3.E1.m1.1.1.1.4.2.3.3" xref="S3.E1.m1.1.1.1.4.2.3.3.cmml">2</mn></msup></mpadded><mo id="S3.E1.m1.1.1.1.4.2.1a" xref="S3.E1.m1.1.1.1.4.2.1.cmml">⁢</mo><mrow id="S3.E1.m1.1.1.1.4.2.4" xref="S3.E1.m1.1.1.1.4.2.4.cmml"><mo rspace="0pt" id="S3.E1.m1.1.1.1.4.2.4.1" xref="S3.E1.m1.1.1.1.4.2.4.1.cmml">d</mo><mi id="S3.E1.m1.1.1.1.4.2.4.2" xref="S3.E1.m1.1.1.1.4.2.4.2.cmml">t</mi></mrow></mrow></mrow></mrow></msqrt></mrow><mo id="S3.E1.m1.2.2.1.2" xref="S3.E1.m1.2.2.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S3.E2.m1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml"><mrow id="S3.E2.m1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.cmml"><mrow id="S3.E2.m1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.2.cmml"><mi mathvariant="normal" id="S3.E2.m1.1.1.1.1.2.2" xref="S3.E2.m1.1.1.1.1.2.2.cmml">Δ</mi><mo id="S3.E2.m1.1.1.1.1.2.1" xref="S3.E2.m1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S3.E2.m1.1.1.1.1.2.3" xref="S3.E2.m1.1.1.1.1.2.3.cmml"><mi id="S3.E2.m1.1.1.1.1.2.3.2" xref="S3.E2.m1.1.1.1.1.2.3.2.cmml">f</mi><mi id="S3.E2.m1.1.1.1.1.2.3.3" xref="S3.E2.m1.1.1.1.1.2.3.3.cmml">signal</mi></msub></mrow><mo id="S3.E2.m1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.cmml">=</mo><msqrt id="S3.E2.m1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.3.cmml"><mrow id="S3.E2.m1.1.1.1.1.3.2" xref="S3.E2.m1.1.1.1.1.3.2.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S3.E2.m1.1.1.1.1.3.2.1" xref="S3.E2.m1.1.1.1.1.3.2.1.cmml">∑</mo><mrow id="S3.E2.m1.1.1.1.1.3.2.2" xref="S3.E2.m1.1.1.1.1.3.2.2.cmml"><mi mathvariant="normal" id="S3.E2.m1.1.1.1.1.3.2.2.2" xref="S3.E2.m1.1.1.1.1.3.2.2.2.cmml">Δ</mi><mo id="S3.E2.m1.1.1.1.1.3.2.2.1" xref="S3.E2.m1.1.1.1.1.3.2.2.1.cmml">⁢</mo><msubsup id="S3.E2.m1.1.1.1.1.3.2.2.3" xref="S3.E2.m1.1.1.1.1.3.2.2.3.cmml"><mi id="S3.E2.m1.1.1.1.1.3.2.2.3.2.2" xref="S3.E2.m1.1.1.1.1.3.2.2.3.2.2.cmml">f</mi><mi id="S3.E2.m1.1.1.1.1.3.2.2.3.2.3" xref="S3.E2.m1.1.1.1.1.3.2.2.3.2.3.cmml">i</mi><mn id="S3.E2.m1.1.1.1.1.3.2.2.3.3" xref="S3.E2.m1.1.1.1.1.3.2.2.3.3.cmml">2</mn></msubsup></mrow></mrow></msqrt></mrow><mo id="S3.E2.m1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S3.E3.m1.1.1.1" xref="S3.E3.m1.1.1.1.1.cmml"><mrow id="S3.E3.m1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.cmml"><mrow id="S3.E3.m1.1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.2.cmml"><mi mathvariant="normal" id="S3.E3.m1.1.1.1.1.2.2" xref="S3.E3.m1.1.1.1.1.2.2.cmml">Δ</mi><mo id="S3.E3.m1.1.1.1.1.2.1" xref="S3.E3.m1.1.1.1.1.2.1.cmml">⁢</mo><msubsup id="S3.E3.m1.1.1.1.1.2.3" xref="S3.E3.m1.1.1.1.1.2.3.cmml"><mi id="S3.E3.m1.1.1.1.1.2.3.2.2" xref="S3.E3.m1.1.1.1.1.2.3.2.2.cmml">I</mi><mi id="S3.E3.m1.1.1.1.1.2.3.2.3" xref="S3.E3.m1.1.1.1.1.2.3.2.3.cmml">meas</mi><mn id="S3.E3.m1.1.1.1.1.2.3.3" xref="S3.E3.m1.1.1.1.1.2.3.3.cmml">2</mn></msubsup></mrow><mo id="S3.E3.m1.1.1.1.1.1" xref="S3.E3.m1.1.1.1.1.1.cmml">=</mo><mrow id="S3.E3.m1.1.1.1.1.3" xref="S3.E3.m1.1.1.1.1.3.cmml"><mrow id="S3.E3.m1.1.1.1.1.3.2" xref="S3.E3.m1.1.1.1.1.3.2.cmml"><mi mathvariant="normal" id="S3.E3.m1.1.1.1.1.3.2.2" xref="S3.E3.m1.1.1.1.1.3.2.2.cmml">Δ</mi><mo id="S3.E3.m1.1.1.1.1.3.2.1" xref="S3.E3.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msup id="S3.E3.m1.1.1.1.1.3.2.3" xref="S3.E3.m1.1.1.1.1.3.2.3.cmml"><mi id="S3.E3.m1.1.1.1.1.3.2.3.2" xref="S3.E3.m1.1.1.1.1.3.2.3.2.cmml">I</mi><mn id="S3.E3.m1.1.1.1.1.3.2.3.3" xref="S3.E3.m1.1.1.1.1.3.2.3.3.cmml">2</mn></msup></mrow><mo id="S3.E3.m1.1.1.1.1.3.1" xref="S3.E3.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="S3.E3.m1.1.1.1.1.3.3" xref="S3.E3.m1.1.1.1.1.3.3.cmml"><mi mathvariant="normal" id="S3.E3.m1.1.1.1.1.3.3.2" xref="S3.E3.m1.1.1.1.1.3.3.2.cmml">Δ</mi><mo id="S3.E3.m1.1.1.1.1.3.3.1" xref="S3.E3.m1.1.1.1.1.3.3.1.cmml">⁢</mo><msubsup id="S3.E3.m1.1.1.1.1.3.3.3" xref="S3.E3.m1.1.1.1.1.3.3.3.cmml"><mi id="S3.E3.m1.1.1.1.1.3.3.3.2.2" xref="S3.E3.m1.1.1.1.1.3.3.3.2.2.cmml">I</mi><mi id="S3.E3.m1.1.1.1.1.3.3.3.2.3" xref="S3.E3.m1.1.1.1.1.3.3.3.2.3.cmml">phot</mi><mn id="S3.E3.m1.1.1.1.1.3.3.3.3" xref="S3.E3.m1.1.1.1.1.3.3.3.3.cmml">2</mn></msubsup></mrow><mo id="S3.E3.m1.1.1.1.1.3.1a" xref="S3.E3.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="S3.E3.m1.1.1.1.1.3.4" xref="S3.E3.m1.1.1.1.1.3.4.cmml"><mi mathvariant="normal" id="S3.E3.m1.1.1.1.1.3.4.2" xref="S3.E3.m1.1.1.1.1.3.4.2.cmml">Δ</mi><mo id="S3.E3.m1.1.1.1.1.3.4.1" xref="S3.E3.m1.1.1.1.1.3.4.1.cmml">⁢</mo><msubsup id="S3.E3.m1.1.1.1.1.3.4.3" xref="S3.E3.m1.1.1.1.1.3.4.3.cmml"><mi id="S3.E3.m1.1.1.1.1.3.4.3.2.2" xref="S3.E3.m1.1.1.1.1.3.4.3.2.2.cmml">I</mi><mi id="S3.E3.m1.1.1.1.1.3.4.3.2.3" xref="S3.E3.m1.1.1.1.1.3.4.3.2.3.cmml">det</mi><mn id="S3.E3.m1.1.1.1.1.3.4.3.3" xref="S3.E3.m1.1.1.1.1.3.4.3.3.cmml">2</mn></msubsup></mrow><mo id="S3.E3.m1.1.1.1.1.3.1b" xref="S3.E3.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="S3.E3.m1.1.1.1.1.3.5" xref="S3.E3.m1.1.1.1.1.3.5.cmml"><mi mathvariant="normal" id="S3.E3.m1.1.1.1.1.3.5.2" xref="S3.E3.m1.1.1.1.1.3.5.2.cmml">Δ</mi><mo id="S3.E3.m1.1.1.1.1.3.5.1" xref="S3.E3.m1.1.1.1.1.3.5.1.cmml">⁢</mo><msubsup id="S3.E3.m1.1.1.1.1.3.5.3" xref="S3.E3.m1.1.1.1.1.3.5.3.cmml"><mi id="S3.E3.m1.1.1.1.1.3.5.3.2.2" xref="S3.E3.m1.1.1.1.1.3.5.3.2.2.cmml">I</mi><mi id="S3.E3.m1.1.1.1.1.3.5.3.2.3" xref="S3.E3.m1.1.1.1.1.3.5.3.2.3.cmml">scat</mi><mn id="S3.E3.m1.1.1.1.1.3.5.3.3" xref="S3.E3.m1.1.1.1.1.3.5.3.3.cmml">2</mn></msubsup></mrow></mrow></mrow><mo id="S3.E3.m1.1.1.1.2" xref="S3.E3.m1.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S3.SS1.SSS1.p1.4.m4.1.1" xref="S3.SS1.SSS1.p1.4.m4.1.1.cmml"><mpadded width="+1.7pt" id="S3.SS1.SSS1.p1.4.m4.1.1.2" xref="S3.SS1.SSS1.p1.4.m4.1.1.2.cmml"><msup id="S3.SS1.SSS1.p1.4.m4.1.1.2a" xref="S3.SS1.SSS1.p1.4.m4.1.1.2.cmml"><mi id="S3.SS1.SSS1.p1.4.m4.1.1.2.2" xref="S3.SS1.SSS1.p1.4.m4.1.1.2.2.cmml">e</mi><mo id="S3.SS1.SSS1.p1.4.m4.1.1.2.3" xref="S3.SS1.SSS1.p1.4.m4.1.1.2.3.cmml">-</mo></msup></mpadded><mo id="S3.SS1.SSS1.p1.4.m4.1.1.1" xref="S3.SS1.SSS1.p1.4.m4.1.1.1.cmml">⁢</mo><msup id="S3.SS1.SSS1.p1.4.m4.1.1.3" xref="S3.SS1.SSS1.p1.4.m4.1.1.3.cmml"><mi id="S3.SS1.SSS1.p1.4.m4.1.1.3.2" xref="S3.SS1.SSS1.p1.4.m4.1.1.3.2.cmml">DN</mi><mrow id="S3.SS1.SSS1.p1.4.m4.1.1.3.3" xref="S3.SS1.SSS1.p1.4.m4.1.1.3.3.cmml"><mo id="S3.SS1.SSS1.p1.4.m4.1.1.3.3.1" xref="S3.SS1.SSS1.p1.4.m4.1.1.3.3.1.cmml">-</mo><mn id="S3.SS1.SSS1.p1.4.m4.1.1.3.3.2" xref="S3.SS1.SSS1.p1.4.m4.1.1.3.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S3.E4.m1.1.1.1" xref="S3.E4.m1.1.1.1.1.cmml"><mrow id="S3.E4.m1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.cmml"><mrow id="S3.E4.m1.1.1.1.1.2" xref="S3.E4.m1.1.1.1.1.2.cmml"><mi mathvariant="normal" id="S3.E4.m1.1.1.1.1.2.2" xref="S3.E4.m1.1.1.1.1.2.2.cmml">Δ</mi><mo id="S3.E4.m1.1.1.1.1.2.1" xref="S3.E4.m1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S3.E4.m1.1.1.1.1.2.3" xref="S3.E4.m1.1.1.1.1.2.3.cmml"><mi id="S3.E4.m1.1.1.1.1.2.3.2" xref="S3.E4.m1.1.1.1.1.2.3.2.cmml">I</mi><mi id="S3.E4.m1.1.1.1.1.2.3.3" xref="S3.E4.m1.1.1.1.1.2.3.3.cmml">phot</mi></msub></mrow><mo id="S3.E4.m1.1.1.1.1.1" xref="S3.E4.m1.1.1.1.1.1.cmml">=</mo><msqrt id="S3.E4.m1.1.1.1.1.3" xref="S3.E4.m1.1.1.1.1.3.cmml"><mfrac id="S3.E4.m1.1.1.1.1.3.2" xref="S3.E4.m1.1.1.1.1.3.2.cmml"><mi id="S3.E4.m1.1.1.1.1.3.2.2" xref="S3.E4.m1.1.1.1.1.3.2.2.cmml">I</mi><mrow id="S3.E4.m1.1.1.1.1.3.2.3" xref="S3.E4.m1.1.1.1.1.3.2.3.cmml"><mi id="S3.E4.m1.1.1.1.1.3.2.3.2" xref="S3.E4.m1.1.1.1.1.3.2.3.2.cmml">N</mi><mo id="S3.E4.m1.1.1.1.1.3.2.3.1" xref="S3.E4.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E4.m1.1.1.1.1.3.2.3.3" xref="S3.E4.m1.1.1.1.1.3.2.3.3.cmml">G</mi><mo id="S3.E4.m1.1.1.1.1.3.2.3.1a" xref="S3.E4.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S3.E4.m1.1.1.1.1.3.2.3.4" xref="S3.E4.m1.1.1.1.1.3.2.3.4.cmml">Δ</mi><mo id="S3.E4.m1.1.1.1.1.3.2.3.1b" xref="S3.E4.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><mi id="S3.E4.m1.1.1.1.1.3.2.3.5" xref="S3.E4.m1.1.1.1.1.3.2.3.5.cmml">t</mi></mrow></mfrac></msqrt></mrow><mo id="S3.E4.m1.1.1.1.2" xref="S3.E4.m1.1.1.1.1.cmml">.</mo></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9307254

Formulas:

1-

S U {(2)}_{L}

2-

R_{o b s} / R_{M C} = 0.60 \pm 0.07 \pm 0.05

3-

R_{o b s} / R_{M C} = 0.54 \pm 0.05 \pm 0.12

4-

R = N (ν_{μ}) / N (ν_{e})

5-

P \to e^{+} ν ν

6-

P \to e^{+} ν_{L} ν_{L} or P \to e^{+} ν_{L}^{c} ν_{L} or P \to e^{+} ν_{L}^{c} ν_{L}^{c} .

7-

P \to e^{+} π^{0}

8-

τ_{P} > 5 \times 10^{32}

9-

M^{(n - 4)}

10-

S U {(3)}_{c} \otimes S U {(2)}_{L} \otimes U {(1)}_{Y}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0601199

Formulas:

1-

λ = E_{J} / E_{C}

2-

E_{n}, n = 0, 1, 2, \dots,

3-

q = C_{G} V_{G}

4-

E_{n} (q, φ)

5-

ν_{10} = (E_{1} - E_{0}) / h

6-

λ = E_{J} / E_{C} \approx 30

7-

E_{C} = e^{2} / 2 C

8-

C = C_{J1} + C_{J2} + C_{G}

9-

E_{C} \approx 50 - 80 μ

10-

E_{J0} = (Φ_{0} / 2 π) I_{c0} \approx 100 μ

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/gr-qc/0402009

Formulas:

1-

E > \sqrt{m E_{p}}

2-

L_{p} \sim 10^{- 35} m

3-

{[m_{p r o t o n} / E_{g u t}]}^{4} \sim 10^{- 64}

4-

E_{p} \equiv 1 / L_{p} \sim 10^{28} e V

5-

E_{T O T}^{2} ≃ N^{2} E^{2} = N^{2} {\vec{p}}^{2} + N^{2} m^{2} + η N^{n} E^{n} N^{2} {\vec{p}}^{2} / (N^{n} E_{p}^{n}) ≃ {\vec{p}}_{T O T}^{2} + m_{T O T}^{2} + η E_{T O T}^{n} {\vec{p}}_{T O T}^{2} / (N^{n} E_{p}^{n})

6-

0 = f (E, {\vec{p}}^{2}, m; E_{p}) ≃ E^{2} - {\vec{p}}^{2} - m^{2} - η \frac{E^{n}}{E_{p}^{n}} {\vec{p}}^{2},

7-

v = d E / d p

8-

v = d E / d p

9-

v = d E / d p

10-

v = d E / d p

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/0901.1041

Formulas:

1-

i ℏ \frac{\partial Ψ_{1}}{\partial t} = [- \frac{ℏ^{2}}{2 M} \nabla^{2} + V_{ext} (𝐫) + g_{11} {| Ψ_{1} |}^{2} + g_{12} {| Ψ_{2} |}^{2}] Ψ_{1},

2-

i ℏ \frac{\partial Ψ_{2}}{\partial t} = [- \frac{ℏ^{2}}{2 M} \nabla^{2} + V_{ext} (𝐫) + g_{21} {| Ψ_{1} |}^{2} + g_{22} {| Ψ_{2} |}^{2}] Ψ_{2},

3-

V_{ext} (𝐫) = M ω_{⟂}^{2} (x^{2} + y^{2}) / 2

4-

\nabla^{2} = \partial^{2} / \partial x^{2} + \partial^{2} / \partial y^{2}

5-

g_{i j} = 4 π ℏ^{2} a_{i j} / M

6-

a_{i j} = a_{j i}

7-

(x, y) \to (x / l, y / l)

8-

t \to ω_{⟂} t

9-

Ψ_{j} \to Ψ_{j} \sqrt{ℏ ω_{⟂} / (2 | g_{11} |)}

10-

B_{12} = - g_{12} / | g_{11} |

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0304015

Formulas:

1-

B^{3} Σ_{u}^{-}

2-

X^{3} Σ_{g}^{-}

3-

B^{3} Σ_{u}^{-}

4-

X^{3} Σ_{g}^{-}

5-

{\dot{n}}_{i} = - n_{i} \sum_{j} P_{i j} + \sum_{j} n_{j} P_{j i} + Q_{i} i = 1, \dots, N

6-

B_{i j} ρ_{i j}

7-

X^{3} Σ_{g}^{-}

8-

B^{3} Σ_{u}^{-}

9-

\sum_{i} n_{i} = 1

10-

{\bar{F}}_{λ} (t, ρ) = \int_{- \frac{π}{2}}^{\frac{π}{2}} F_{λ} (t) n (t) \frac{ρ}{c o s^{2} θ} 𝑑 θ

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1211.1507

Formulas:

1-

(N - 1) \times (N - 1)

2-

{x_{i}}_{i = 1}^{n}

3-

{y_{j}}_{j = 1}^{n - 1}

4-

x_{1} \geq y_{1} \geq x_{2} \geq \dots \geq x_{n - 1} \geq y_{n - 1} \geq x_{n} .

5-

{x_{i}}_{i = 1}^{n}

6-

{y_{j}}_{j = 1}^{n - 1}

7-

z_{0} = \sum_{i = 1}^{n} x_{i} - \sum_{j = 1}^{n - 1} y_{j} .

8-

w^{{x_{i}}, {y_{j}}} (x)

9-

w^{{x_{i}}, {y_{j}}} (x) : ℝ \to ℝ

10-

\frac{\partial}{\partial x} w^{{x_{i}}, {y_{j}}} (x) = \pm 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1108.0250

Formulas:

1-

F_{1} (Q^{2})

2-

F_{2} (Q^{2})

3-

G_{E} (Q^{2})

4-

G_{M} (Q^{2})

5-

⟨ B | J_{had}^{μ} (0) | B ⟩ = \bar{u} (p^{'}) (γ^{μ} F_{1} (Q^{2}) + i \frac{σ^{μ ν}}{2 M} q_{ν} F_{2} (Q^{2})) u (p),

6-

u (p^{'})

7-

A^{'} \sum_{i = 1}^{3} e_{i} 𝟏 - B^{'} \sum_{i \neq j}^{3} e_{i} [2 σ_{𝐢} \cdot σ_{𝐣} - (𝟑 σ_{𝐢𝐳} σ_{𝐣𝐳} - σ_{𝐢} \cdot σ_{𝐣})] - C^{'} \sum_{𝐢 \neq 𝐣 \neq 𝐤}^{𝟑} 𝐞_{𝐢} [𝟐 σ_{𝐣} \cdot σ_{𝐤} - (𝟑 σ_{𝐣𝐳} σ_{𝐤𝐳} - σ_{𝐣} \cdot σ_{𝐤})] .

8-

\hat{r^{2}} = A \sum_{i = 1}^{3} e_{i} 𝟏 + B \sum_{i \neq j}^{3} e_{i} σ_{𝐢} \cdot σ_{𝐣} + C \sum_{𝐢 \neq 𝐣 \neq 𝐤}^{𝟑} 𝐞_{𝐢} σ_{𝐣} \cdot σ_{𝐤},

9-

\hat{Q} = B^{'} \sum_{i \neq j}^{3} e_{i} (3 σ_{i z} σ_{j z} - σ_{𝐢} \cdot σ_{𝐣}) + C^{'} \sum_{i \neq j \neq k}^{3} e_{i} (3 σ_{j z} σ_{k z} - σ_{𝐣} \cdot σ_{𝐤}) .

10-

B = - 2 B^{'}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/nucl-th/0211086

Formulas:

1-

F_{2} (x_{B}, Q^{2})

2-

ℓ (E) + h (p) ⟹ ℓ (E^{'}) + X

3-

- q^{2} = Q^{2}

4-

x_{B} = \frac{Q^{2}}{2 p \cdot q} = \frac{Q^{2}}{2 m_{h} ν}

5-

ν = E - E^{'}

6-

Q^{2} = 4 E E^{'} \sin^{2} (θ / 2)

7-

\frac{d σ^{DIS}}{d x_{B} d Q^{2}} = \frac{σ_{Mott}}{E E^{'}} \frac{π F_{2} (x_{B}, Q^{2})}{ϵ x_{B}} [\frac{1 + ϵ R (x_{B}, Q^{2})}{1 + R (x_{B}, Q^{2})}],

8-

R = σ_{L} / σ_{T} = [(1 + 4 x_{B}^{2} m^{2} / Q^{2}) F_{2} - 2 x_{B} F_{1}] / [2 x_{B} F_{1}]

9-

F_{i} (x_{B}, Q^{2}) \equiv e_{i}^{μ ν} W_{μ ν} (x_{B}, Q^{2})

10-

W_{μ ν} (x_{B}, Q^{2})

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0503005

Formulas:

1-

R \geq q {- Q_{μ} f (E_{μ}) H_{2} (E_{μ}) + Q_{1} [1 - H_{2} (e_{1})]},

2-

f (x) \geq 1

3-

f (x) = 1

4-

H_{2} (x)

5-

H_{2} (x) = - x \log_{2} (x) - (1 - x) \log_{2} (1 - x) .

6-

Δ \leq \frac{μ}{μ^{'} - μ} (\frac{μ e^{- μ} Q_{μ^{'}}}{μ^{'} e^{- μ^{'}} Q_{μ}} - 1) + \frac{μ e^{- μ} Y_{0}}{μ^{'} Q_{μ}},

7-

R \geq q Q_{μ} {- H_{2} (E_{μ}) + (1 - Δ) [1 - H_{2} (\frac{E_{μ}}{1 - Δ})]} .

8-

ρ_{A} = \sum_{i = 0}^{\infty} \frac{μ^{i}}{i!} e^{- μ} | i ⟩ ⟨ i |,

9-

| 0 ⟩ ⟨ 0 |

10-

| i ⟩ ⟨ i |

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/physics/0204082

Formulas:

1-

Δ E_{i} = \frac{3 π c^{2} γ}{2 ω_{0}^{3}} I \times \sum \frac{c_{i j}^{2}}{δ_{i j}}

2-

δ_{i j} = ω - ω_{i j}

3-

Δ_{H F} \sim 3

4-

Δ_{F}^{'} ≫ Δ_{H F} ≫ Δ_{H F}^{'}

5-

Δ E (𝐫) = \frac{π c^{2} γ}{ω_{0}^{3}} \frac{I (𝐫)}{δ_{F}}

6-

δ_{F} = ω - ω_{F}

7-

| 5 S_{1 / 2}, F ⟩ \to | 5 P_{3 / 2} ⟩

8-

{\tilde{Δ}}_{H F} (𝐫) - Δ_{H F} = \frac{π c^{2} γ}{ω_{0}^{3}} \frac{Δ_{H F}}{δ^{2}} [\frac{1}{1 - {(\frac{Δ_{H F}}{2 δ})}^{2}}] I (𝐫)

9-

{\tilde{Δ}}_{H F} (𝐫)

10-

δ ≜ (δ_{F = 2} + δ_{F = 3}) / 2

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0311308

Formulas:

1-

(M + 1, N)

2-

(M, N + 1)

3-

0.5 k_{B} T_{e} ≲ t < Δ

4-

N, M = 0, 1, 2 \dots

5-

2 \times 10^{11} {cm}^{- 2}

6-

2 \times 10^{5} {cm}^{2} / Vs

7-

Δ \sim 80 μ eV

8-

E_{C} = e^{2} / C_{o} \sim 500 μ eV

9-

C_{o} \sim 320 aF

10-

δ g_{ls (rs)}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9606288

Formulas:

1-

𝒪 (α_{s})

2-

𝒪 (α_{s}^{2})

3-

𝒪 (M^{4} / s^{2})

4-

𝒪 (α_{s}^{2})

5-

𝒪 (α_{s}^{2})

6-

R_{F \bar{F}} = r^{(0)} + (\frac{α}{π}) r^{(1)} + {(\frac{α}{π})}^{2} r^{(2)} + \dots,

7-

\frac{- i}{p^{2} + i ϵ} (g^{μ σ} - ξ \frac{p^{μ} p^{σ}}{p^{2}}) (- Π_{σ ρ} (m^{2}, p^{2})) \frac{- i}{p^{2} + i ϵ} (g^{ρ ν} - ξ \frac{p^{ρ} p^{ν}}{p^{2}})

8-

\frac{1}{3} (\frac{α}{π}) \int_{4 m^{2}}^{\infty} \frac{d λ^{2}}{λ^{2}} [\frac{- i g^{μ ν}}{p^{2} - λ^{2} + i ϵ}] R_{f \bar{f}} (λ^{2}) + \dots,

9-

r_{f}^{(2)} (M^{2}, s) = \frac{1}{3} \int_{4 m^{2}}^{\infty} \frac{d λ^{2}}{λ^{2}} r^{(1)} (M^{2}, s, λ^{2}) R_{f \bar{f}} (m^{2}, λ^{2}),

10-

r^{(1)} (M^{2}, s, λ^{2})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1610.03340

Formulas:

1-

G_{q, r} (x)

2-

p = q n + r

3-

G_{q, r} (x) < φ (q) \log^{2} x

4-

G_{q, r} (x) \sim a (\log (li x / φ (q)) + b)

5-

a = φ (q) x / li x

6-

p = q n + r \leq x

7-

b = O ({(\log q)}^{1 / 2} \log \log x)

8-

G_{q, r} (x)

9-

G_{q, r} (x)

10-

\gcd (q, r) = 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0303115

Formulas:

1-

f_{0.5 - 2} \sim 10^{- 14}

2-

f_{2 - 10} \sim {3 10}^{- 11}

3-

f_{0.5 - 2} \sim 10^{- 15}

4-

f_{2 - 10} \sim 10^{- 13}

5-

N_{H} = 22 - 23

6-

l e f t

7-

r i g h t

8-

f_{2 - 10} = {4 10}^{- 14}

9-

f_{0.5 - 2} \sim {5.5 10}^{- 17}

10-

f_{2 - 10} \sim {4.5 10}^{- 16}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1506.03576

Formulas:

1-

κ - {(ET)}_{2} {Cu}_{2} {(CN)}_{3}

2-

{EtMe}_{3} Sb {[Pd {(dmit)}_{2}]}_{2}

3-

{CuZn}_{3} {(OH)}_{6} {Cl}_{2}

4-

S (𝐪, ω)

5-

S (𝐪, ω)

6-

S (𝐪, ω)

7-

κ - {(ET)}_{2} {Cu}_{2} {(CN)}_{3}

8-

{EtMe}_{3} Sb {[Pd {(dmit)}_{2}]}_{2}

9-

{CuZn}_{3} {(OH)}_{6} {Cl}_{2}

10-

{[Zn {(OH)}_{6}]}^{4 -}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0604100

Formulas:

1-

⟨ v \sin i ⟩ \sim

2-

⟨ v \sin i ⟩ \sim

3-

v \sin i ≲ 25

4-

E (b - y)

5-

Δ a = - 7

6-

v \sin i = 60.5

7-

[N_{El} / N_{tot}]

8-

[N_{El} / N_{tot}]

9-

[N_{El} / N_{tot}]

10-

[N_{El} / N_{tot}]

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1904.02173

Formulas:

1-

M e r c u r y 6

2-

M e r c u r y 6

3-

a_{o u t}

4-

M_{t o t}

5-

N_{e m b}

6-

N_{p l n}

7-

A M D_{M V E M}

8-

A M D = \frac{\sum_{i} m_{i} \sqrt{a_{i}} [1 - \sqrt{(1 - e_{i}^{2})} \cos i_{i}]}{\sum_{i} m_{i} \sqrt{a_{i}}}

9-

M_{M e r c} <

10-

M_{V e n, E a r} >

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1604.00310

Formulas:

1-

2 (k - 1 + 1 / k)

2-

\sum_{e \in M \cap δ (v)} d_{e} \leq b_{v}

3-

d_{e} = b_{v} = 1

4-

H = (V, ℰ)

5-

\sum_{S \in ℰ} c_{S} x_{S}

6-

\sum_{S | v \in S} d_{S} x_{S}

7-

\in {0, 1}

8-

x_{S} \in {0, 1}

9-

k - 1 + 1 / k

10-

k - 1 + 1 / k

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9706107

Formulas:

1-

2.7 \times 10^{42} erg s^{- 1}

2-

\sim (3.8 \times 10^{39} erg s^{- 1}) {\dot{M}}_{100}

3-

10^{4} M_{⊙} {yr}^{- 1}

4-

{\dot{M}}_{X} \sim 10 - 1000 M_{⊙} {yr}^{- 1}

5-

1.1 \times 10^{- 16} erg {cm}^{- 2} s^{- 1}

6-

1.2 - 1.5 \times 10^{- 16} erg {cm}^{- 2} s^{- 1}

7-

V_{LSR} = - 13.5 km s^{- 1}

8-

R_{O III} \equiv \frac{F_{4363}}{F_{4959} + F_{5007}} \approx 0.05 - 0.07

9-

R_{O III} \approx 0.005

10-

R_{O III} < 0.02 (3 σ)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1803.09722

Formulas:

1-

d (\cdot, \cdot)

2-

d (𝐳_{i}, 𝐳_{j}) = {[Δ x, Δ y, Δ z, Δ x^{2}, Δ y^{2}, Δ z^{2}]}^{T},

3-

𝐳_{i} = (x_{i}, y_{i}, z_{i})

4-

𝐳_{j} = (x_{j}, y_{j}, z_{j})

5-

Δ x = x_{i} - x_{j}, Δ y = y_{i} - y_{j}

6-

Δ z = z_{i} - z_{j}

7-

Δ x^{2} = {(x_{i} - x_{j})}^{2}, Δ y^{2} = {(y_{i} - y_{j})}^{2}

8-

Δ z^{2} = {(z_{i} - z_{j})}^{2}

9-

𝐈 = {(I_{n}, 𝐳_{n})}_{n = 1}^{N}

10-

N = {N_{2 D}, N_{3 D}}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0606141

Formulas:

1-

≪ m_{e} c^{2}

2-

\leq m_{e} c^{2} / γ \approx 2

3-

l_{p h} \equiv L_{p h} σ_{T} / (R m_{e} c^{3})

4-

n_{N T} (γ) d γ = n_{o} {(γ - 1)}^{- α} d γ

5-

l_{p - p} \equiv L_{p - p} σ_{T} / (R m_{e} c^{3})

6-

N_{+} (γ)

7-

\frac{\partial}{\partial γ} ({\dot{γ}}_{C} N_{+} (γ)) + N_{+} (γ) \int_{1}^{γ} 𝑑 γ^{'} P (γ, γ^{'}) -

8-

\int_{γ}^{\infty} 𝑑 γ^{'} P (γ^{'}, γ) N_{+} (γ^{'}) + {\dot{N}}_{+} (γ) = Q_{+, p p} (γ) + Q_{+, γ γ} (γ)

9-

N_{-} (γ)

10-

\frac{\partial}{\partial γ} ({\dot{γ}}_{C} N_{-} (γ)) + N_{-} (γ) \int_{1}^{γ} 𝑑 γ^{'} P (γ, γ^{'}) -

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1003.2511

Formulas:

1-

u_{t} + u u_{x} - u_{x x} - u_{y y} = 0 .

2-

u_{t} + u u_{x} + α (t) u - u_{x x} - u_{y y} = 0 .

3-

x_{i}^{*} = x_{i} + ϵ ξ_{i} (x, y, t, u) + O (ϵ^{2}), i = 1, 2, 3, 4,

4-

ξ_{1} = ξ, ξ_{2} = η, ξ_{3} = τ, ξ_{4} = ϕ

5-

p r^{(4)} V

6-

V = τ (x, y, t; u) \partial_{t} + ξ (x, y, t; u) \partial_{x} + η (x, y, t; u) \partial_{y} + ϕ (x, y, t; u) \partial_{u},

7-

{p r^{(4)} V Ω (x, y, t; u) |}_{Ω (x, y, t; u) = 0} = 0,

8-

{(ξ_{1})}_{u} = 0,

9-

{(ξ_{2})}_{u} = 0,

10-

{(ξ_{3})}_{u} = 0,

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Run 11331761 (Agent397)