Run 10695567

Paper: https://arxiv.org/abs/1412.2516

Formulas:

1-

S = (s_{1}, \dots s_{m})

2-

\underset{D_{U}}{\Pr} [S] = \frac{{| Per (U_{[n], S}) |}^{2}}{s_{1}! \dots s_{m}!},

3-

U_{[n], S}

4-

N = ((\binom{m}{n}))

5-

o (1 / n)

6-

{∥ 𝒟_{\tilde{U}} - 𝒟_{U} ∥}_{1} \leq n {∥ \tilde{U} - U ∥}_{op}

7-

o (\frac{1}{n})

8-

{∥ 𝒟_{\tilde{U}} - 𝒟_{U} ∥}_{1} = o (1)

9-

{∥ \tilde{U} - U ∥}_{op} = o (\frac{1}{n})

10-

O (n \log m)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: stat

Result: incorrect

Paper: https://arxiv.org/abs/1010.2093

Formulas:

1-

r = 2 R_{*}

2-

\exp (- 2.5 A_{V})

3-

4 π J_{λ} = I_{λ} Ω \exp {- \frac{[A_{λ} / A_{V}]}{1.086} A_{V}}

4-

4 π J_{λ} = I_{λ} Ω [(1 - f_{Ω}) \exp {- \frac{[A_{λ} / A_{V}]}{1.086} A_{V}} + f_{Ω}]

5-

{\bar{x}}_{i} (r)

6-

x_{i}^{M} (r)

7-

x_{i}^{m} (r)

8-

x_{i}^{M} (r)

9-

x_{i}^{m} (r)

10-

O + H_{2} \to OH + H

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/hep-th/9611049

Formulas:

1-

S U (N_{c})

2-

\tilde{Q} Φ^{l} Q

3-

l N_{f} \leq 2 N_{c}

4-

N_{c} = 2, N_{f} = 1

5-

S U (N_{c})

6-

Tr Φ^{k + 1}

7-

S U (k N_{f} - N_{c})

8-

\tilde{Q} Φ^{l} Q

9-

l < 2 N_{c} / N_{f}

10-

l = 2 N_{c} / N_{f}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-th

Result: correct

Paper: https://arxiv.org/abs/1707.02143

Formulas:

1-

γ = - {(\frac{\partial \ln ω}{\partial \ln V})}_{T}

2-

γ = V {(\frac{\partial P}{\partial E})}_{V}

3-

γ = \frac{α_{P} B_{T} V}{C_{V}}

4-

E = \frac{3}{2} P V

5-

Z = \frac{P V}{N k_{B} T} = \frac{1 + η + η^{2} - η^{3}}{{(1 - η)}^{3}}

6-

η = \frac{π}{6} ρ σ^{3}

7-

γ = \frac{2}{3} f (ρ)

8-

γ = \frac{2}{3} \times \frac{V}{V - N b}

9-

c_{v} = 2 k_{B}

10-

c_{v} = 2 k_{B}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-th/0612259

Formulas:

1-

ϕ (𝐱, t) ϕ (- 𝐱, t)

2-

⟨ 0 | ϕ (𝐱, t) ϕ (- 𝐱, t) | 0 ⟩

3-

⟨ 0 | U^{- 1} U ϕ (𝐱, t) U^{- 1} U ϕ (- 𝐱, t) U^{- 1} U | 0 ⟩

4-

⟨ 0 | ϕ (- 𝐱, t) ϕ (𝐱, t) | 0 ⟩

5-

⟨ 0 | [ϕ (𝐱, t), ϕ (- 𝐱, t)] | 0 ⟩ = 0

6-

ψ_{a b} (𝐱, t)

7-

ψ_{a b} (𝐲, t)

8-

2 (A + B)

9-

| \dots; 𝐩, s; 𝐩^{'}, s^{'}; \dots ⟩

10-

| \dots; 𝐩^{'}, s^{'}; 𝐩, s; \dots ⟩

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1201.0987

Formulas:

1-

{(U, V, W)}_{⊙}

2-

{(U, V, W)}_{⊙}

3-

G L I M P S E

4-

G L I M P S E

5-

G L I M P S E

6-

km s^{- 1} {kpc}^{- 1}

7-

km s^{- 1} {kpc}^{- 1}

8-

km s^{- 1} {kpc}^{- 1}

9-

km s^{- 1} {kpc}^{- 1}

10-

km s^{- 1} {kpc}^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/9510104

Formulas:

1-

V \equiv \int 𝑑 x^{-} A_{-}

2-

\partial_{-} A_{-} = 0

3-

\partial_{-} + i g V

4-

\partial_{+} (\partial_{-} + i g V) ϕ = U (ϕ)

5-

\partial_{-} ϕ = 0

6-

\int 𝑑 x^{-} A_{-}

7-

\int 𝑑 x^{-} d^{2} x^{r} A_{-}

8-

S U (N)

9-

x_{\mp} = x^{\pm} = (x^{0} \pm x^{3}) / \sqrt{2}

10-

r, s \in {1, 2}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/0607036

Formulas:

1-

R_{c} (m)

2-

0.6 - 0.9 G e V / c^{2}

3-

0.2 - 0.8 G e V / c^{2}

4-

R (m) = \frac{\int 𝑑 y d^{2} p_{t} \frac{d N_{e e}}{d y d^{2} p_{t} d m}}{\int 𝑑 y d^{2} p_{t} \frac{d N_{c h}}{d y d^{2} p_{t}}} .

5-

0.2 - 0.8 G e V / c^{2}

6-

P b A u

7-

0.2 G e V / c

8-

\bar{R} = \frac{\int_{0}^{0.2} R^{P b A u} (m) 𝑑 m}{\int_{0}^{0.2} R^{c} (m) 𝑑 m} = 0.92 \pm 0.17

9-

0.9 G e V / c^{2}

10-

\bar{R} = \frac{\int_{0.2}^{0.9} R^{P b A u} (m) 𝑑 m}{\int_{0.2}^{0.9} R^{c} (m) 𝑑 m} ≃ 2.5

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1607.01019

Formulas:

1-

H_{0} = 70 km s^{- 1} {Mpc}^{- 1}

2-

u g r i z

3-

\log (M / M_{⊙}) < 10.6

4-

\log (M / M_{⊙}) \geq 10.6

5-

Z \approx 0.7 Z_{⊙}

6-

M \sim 10^{10.6} M_{⊙}

7-

M \sim 10^{11} M_{⊙}

8-

u g r i z

9-

p_{edge - on} = 0.5

10-

p_{not edge - on} = 0.5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0602121

Formulas:

1-

⟨ p^{'} | j_{μ}^{em} | p ⟩ = \bar{u} (p^{'}) [F_{1} (t) γ_{μ} + i \frac{F_{2} (t)}{2 M} σ_{μ ν} q^{ν}] u (p),

2-

t = {(p^{'} - p)}^{2}

3-

Q^{2} = - t > 0

4-

F_{1} (t)

5-

F_{2} (t)

6-

F_{1}^{p} (0) = 1, F_{1}^{n} (0) = 0, F_{2}^{p} (0) = κ_{p}, F_{2}^{n} (0) = κ_{n},

7-

F_{i}^{s} = \frac{1}{2} (F_{i}^{p} + F_{i}^{n}), F_{i}^{v} = \frac{1}{2} (F_{i}^{p} - F_{i}^{n}),

8-

G_{E} (t)

9-

F_{1} (t) - τ F_{2} (t),

10-

G_{M} (t)

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/2002.05301

Formulas:

1-

ψ (y, z)

2-

L W_{3, 4, 5}

3-

L W_{1}, L W_{2}, \dots

4-

H_{0}, H_{1}, \dots

5-

L W_{1, 2, \dots}

6-

H_{0, 1, \dots}

7-

𝐁 = (B_{Y}, B_{Z})

8-

Δ k_{x} = e d_{Z} B_{Y} / ℏ

9-

Δ k_{x} = e d_{Y} B_{Z} / ℏ

10-

L W_{2}, L W_{3}, \dots

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0904.1525

Formulas:

1-

ℤ [[A, A^{- 1}, K_{1}, K_{2}, \dots]]

2-

s g n (v)

3-

w (L) = \sum_{v} s g n (v)

4-

ℤ [[A, A^{- 1}, K_{1}, K_{2}, \dots]]

5-

ℤ [[A, A^{- 1}]]

6-

⟨ C ⟩ = K_{n} .

7-

S = \prod_{i = 1}^{n} C_{i}

8-

⟨ S ⟩ = \prod_{i = 1}^{n} ⟨ C_{i} ⟩ .

9-

d = - A^{2} - A^{- 2}

10-

{⟨ L ⟩}_{A} = \sum_{S} A^{α - β} d^{| S | - 1} ⟨ S ⟩

Formulas (html):

Correct Categorie: math

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/1609.02827

Formulas:

1-

x \in [0, 1]

2-

I_{x} (a, b)

3-

I_{x} (a, b) = \frac{1}{B (a, b)} \int_{0}^{x} t^{a - 1} {(1 - t)}^{b - 1} d t,

4-

B (a, b)

5-

B (a, b) = \int_{0}^{1} t^{a - 1} {(1 - t)}^{b - 1} d t = \frac{Γ (a) Γ (b)}{Γ (a + b)}

6-

I_{x} (a, b) = \frac{x^{a} {(1 - x)}^{b - 1}}{a B (a, b)} {}_{2}F_{1} (\begin{matrix} 1, 1 - b \\ a + 1 \end{matrix}; \frac{x}{x - 1}) \sim \frac{x^{a} {(1 - x)}^{b - 1}}{a B (a, b)} \sum_{n = 0}^{\infty} \frac{{(1 - b)}_{n}}{{(a + 1)}_{n}} {(\frac{x}{x - 1})}^{n},

7-

x \in [0, \frac{1}{2})

8-

x \in [0, 1)

9-

x \in (0, 1]

10-

ξ = - \ln x

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1102.1843

Formulas:

1-

n p \to p p π^{-}

2-

n p \to n p π^{+} π^{-}

3-

n p \to d π^{+} π^{-}

4-

n p \to p p π^{-}

5-

p p \to p p π^{0}

6-

n p \to p p π^{-}

7-

n p \to n p π^{+} π^{-}

8-

n p \to d π^{+} π^{-}

9-

g / c m^{2}

10-

p p π^{-}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: nucl-ex

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0503274

Formulas:

1-

C h a n d r a

2-

C h a n d r a

3-

C h a n d r a

4-

A S C A

5-

C h a n d r a

6-

C h a n d r a

7-

C h a n d r a

8-

σ = 0 \overset{''}{.} 98

9-

C h a n d r a

10-

σ = 0 \overset{''}{.} 98

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1809.10417

Formulas:

1-

𝐗 \in ℝ^{H \times W \times C}

2-

ℱ_{d e f o r m}

3-

Θ \in ℝ^{H \times W \times 2}

4-

Θ = ℱ_{d e f o r m} (𝐗),

5-

θ_{i, j} \in Θ

6-

𝐗_{i, j} \in 𝐗

7-

= ℱ_{s a m p l e} (𝐗, [i, j] + θ_{i, j}),

8-

θ_{i, j} \in ℝ^{1 \times 1 \times 2}, θ_{i, j} \in Θ,

9-

1 \leq i \leq H, 1 \leq j \leq W,

10-

[i, j] + θ_{i, j}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/0906.0719

Formulas:

1-

1.5 \cdot 10^{4} K \leq T_{eff} \leq 3 \cdot 10^{4} K

2-

1.2 \cdot 10^{6} L_{⊙} \leq L \leq 1.9 \cdot 10^{6} L_{⊙}

3-

35.7 M_{⊙} \leq M \leq 49.1 M_{⊙}

4-

70 M_{⊙} \leq M_{ZAMS} \leq 90 M_{⊙}

5-

Δ M_{bol} \leq 0.2

6-

Γ_{1} = 4 / 3

7-

6 day \leq Π \leq 31 day

8-

L \sim 10^{6} L_{⊙}

9-

1.5 \cdot 10^{4} K \leq T_{eff} \leq 3 \cdot 10^{4} K

10-

1.2 \cdot 10^{6} L_{⊙} \leq L \leq 1.9 \cdot 10^{6} L_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1610.07229

Formulas:

1-

2^{𝒪 (k)} n^{𝒪 (1)}

2-

𝒪 (k^{4})

3-

f (k) \cdot n^{O (1)}

4-

(G^{'}, k^{'})

5-

| V (G^{'}) | + k^{'} \leq h (k)

6-

(G^{'}, k^{'})

7-

X \subseteq V (G)

8-

M S O_{1}

9-

M S O_{2}

10-

N P \subseteq c o N P / p o l y

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0302603

Formulas:

1-

\sim 10^{13} M_{⊙}

2-

ρ (r) = \frac{σ_{v}^{2}}{2 π G r^{2}},

3-

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

4-

c = r_{200} / r_{s}

5-

C = 3 \times 10^{3}

6-

x \equiv ξ / ξ_{0}

7-

η_{0} = ξ_{0} (D_{s} / D_{l})

8-

y \equiv η / η_{0}

9-

ψ (x) = | x |,

10-

ξ_{0} = 4 π {(σ_{v} / c)}^{2} (D_{l} D_{ls} / D_{s})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/quant-ph/9812049

Formulas:

1-

f = O (g)

2-

f = Ω (g)

3-

f = Θ (g)

4-

V_{1}, \dots, V_{n}

5-

{V_{1} = false, V_{2} = false, V_{3} = true}

6-

{V_{1} = false, V_{2} = false, V_{3} = true, V_{4} = true}

7-

{V_{1} = false, V_{2} = false, V_{3} = false, V_{4} = true}

8-

{V_{1} = false, V_{2} = false, V_{3} = true, V_{4} = false}

9-

d (r, s)

10-

d (r, s) = | r | + | s | - 2 | r \land s |

Formulas (html):

<math><mrow id="S1.p10.1.m1.1.2" xref="S1.p10.1.m1.1.2.cmml"><mi id="S1.p10.1.m1.1.2.2" xref="S1.p10.1.m1.1.2.2.cmml">f</mi><mo id="S1.p10.1.m1.1.2.1" xref="S1.p10.1.m1.1.2.1.cmml">=</mo><mrow id="S1.p10.1.m1.1.2.3" xref="S1.p10.1.m1.1.2.3.cmml"><mi id="S1.p10.1.m1.1.2.3.2" xref="S1.p10.1.m1.1.2.3.2.cmml">O</mi><mo id="S1.p10.1.m1.1.2.3.1" xref="S1.p10.1.m1.1.2.3.1.cmml">⁢</mo><mrow id="S1.p10.1.m1.1.2.3.3.2" xref="S1.p10.1.m1.1.2.3.cmml"><mo id="S1.p10.1.m1.1.2.3.3.2.1" xref="S1.p10.1.m1.1.2.3.cmml">(</mo><mi id="S1.p10.1.m1.1.1" xref="S1.p10.1.m1.1.1.cmml">g</mi><mo id="S1.p10.1.m1.1.2.3.3.2.2" xref="S1.p10.1.m1.1.2.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.p10.6.m6.1.2" xref="S1.p10.6.m6.1.2.cmml"><mi id="S1.p10.6.m6.1.2.2" xref="S1.p10.6.m6.1.2.2.cmml">f</mi><mo id="S1.p10.6.m6.1.2.1" xref="S1.p10.6.m6.1.2.1.cmml">=</mo><mrow id="S1.p10.6.m6.1.2.3" xref="S1.p10.6.m6.1.2.3.cmml"><mi mathvariant="normal" id="S1.p10.6.m6.1.2.3.2" xref="S1.p10.6.m6.1.2.3.2.cmml">Ω</mi><mo id="S1.p10.6.m6.1.2.3.1" xref="S1.p10.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S1.p10.6.m6.1.2.3.3.2" xref="S1.p10.6.m6.1.2.3.cmml"><mo id="S1.p10.6.m6.1.2.3.3.2.1" xref="S1.p10.6.m6.1.2.3.cmml">(</mo><mi id="S1.p10.6.m6.1.1" xref="S1.p10.6.m6.1.1.cmml">g</mi><mo id="S1.p10.6.m6.1.2.3.3.2.2" xref="S1.p10.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.p10.9.m9.1.2" xref="S1.p10.9.m9.1.2.cmml"><mi id="S1.p10.9.m9.1.2.2" xref="S1.p10.9.m9.1.2.2.cmml">f</mi><mo id="S1.p10.9.m9.1.2.1" xref="S1.p10.9.m9.1.2.1.cmml">=</mo><mrow id="S1.p10.9.m9.1.2.3" xref="S1.p10.9.m9.1.2.3.cmml"><mi mathvariant="normal" id="S1.p10.9.m9.1.2.3.2" xref="S1.p10.9.m9.1.2.3.2.cmml">Θ</mi><mo id="S1.p10.9.m9.1.2.3.1" xref="S1.p10.9.m9.1.2.3.1.cmml">⁢</mo><mrow id="S1.p10.9.m9.1.2.3.3.2" xref="S1.p10.9.m9.1.2.3.cmml"><mo id="S1.p10.9.m9.1.2.3.3.2.1" xref="S1.p10.9.m9.1.2.3.cmml">(</mo><mi id="S1.p10.9.m9.1.1" xref="S1.p10.9.m9.1.1.cmml">g</mi><mo id="S1.p10.9.m9.1.2.3.3.2.2" xref="S1.p10.9.m9.1.2.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.p1.2.m2.3.3.2" xref="S2.p1.2.m2.3.3.3.cmml"><msub id="S2.p1.2.m2.2.2.1.1" xref="S2.p1.2.m2.2.2.1.1.cmml"><mi id="S2.p1.2.m2.2.2.1.1.2" xref="S2.p1.2.m2.2.2.1.1.2.cmml">V</mi><mn id="S2.p1.2.m2.2.2.1.1.3" xref="S2.p1.2.m2.2.2.1.1.3.cmml">1</mn></msub><mo id="S2.p1.2.m2.3.3.2.3" xref="S2.p1.2.m2.3.3.3.cmml">,</mo><mi mathvariant="normal" id="S2.p1.2.m2.1.1" xref="S2.p1.2.m2.1.1.cmml">…</mi><mo id="S2.p1.2.m2.3.3.2.4" xref="S2.p1.2.m2.3.3.3.cmml">,</mo><msub id="S2.p1.2.m2.3.3.2.2" xref="S2.p1.2.m2.3.3.2.2.cmml"><mi id="S2.p1.2.m2.3.3.2.2.2" xref="S2.p1.2.m2.3.3.2.2.2.cmml">V</mi><mi id="S2.p1.2.m2.3.3.2.2.3" xref="S2.p1.2.m2.3.3.2.2.3.cmml">n</mi></msub></mrow></math>, <math><mrow id="S2.p1.13.m13.1.1.1" xref="S2.p1.13.m13.1.1.2.cmml"><mo stretchy="false" id="S2.p1.13.m13.1.1.1.2" xref="S2.p1.13.m13.1.1.2.cmml">{</mo><mrow id="S2.p1.13.m13.1.1.1.1.2" xref="S2.p1.13.m13.1.1.1.1.3.cmml"><mrow id="S2.p1.13.m13.1.1.1.1.1.1" xref="S2.p1.13.m13.1.1.1.1.1.1.cmml"><msub id="S2.p1.13.m13.1.1.1.1.1.1.2" xref="S2.p1.13.m13.1.1.1.1.1.1.2.cmml"><mi id="S2.p1.13.m13.1.1.1.1.1.1.2.2" xref="S2.p1.13.m13.1.1.1.1.1.1.2.2.cmml">V</mi><mn id="S2.p1.13.m13.1.1.1.1.1.1.2.3" xref="S2.p1.13.m13.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.p1.13.m13.1.1.1.1.1.1.1" xref="S2.p1.13.m13.1.1.1.1.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.p1.13.m13.1.1.1.1.1.1.3" xref="S2.p1.13.m13.1.1.1.1.1.1.3a.cmml">false</mtext></mrow><mo id="S2.p1.13.m13.1.1.1.1.2.3" xref="S2.p1.13.m13.1.1.1.1.3a.cmml">,</mo><mrow id="S2.p1.13.m13.1.1.1.1.2.2.2" xref="S2.p1.13.m13.1.1.1.1.2.2.3.cmml"><mrow id="S2.p1.13.m13.1.1.1.1.2.2.1.1" xref="S2.p1.13.m13.1.1.1.1.2.2.1.1.cmml"><msub id="S2.p1.13.m13.1.1.1.1.2.2.1.1.2" xref="S2.p1.13.m13.1.1.1.1.2.2.1.1.2.cmml"><mi id="S2.p1.13.m13.1.1.1.1.2.2.1.1.2.2" xref="S2.p1.13.m13.1.1.1.1.2.2.1.1.2.2.cmml">V</mi><mn id="S2.p1.13.m13.1.1.1.1.2.2.1.1.2.3" xref="S2.p1.13.m13.1.1.1.1.2.2.1.1.2.3.cmml">2</mn></msub><mo id="S2.p1.13.m13.1.1.1.1.2.2.1.1.1" xref="S2.p1.13.m13.1.1.1.1.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.p1.13.m13.1.1.1.1.2.2.1.1.3" xref="S2.p1.13.m13.1.1.1.1.2.2.1.1.3a.cmml">false</mtext></mrow><mo id="S2.p1.13.m13.1.1.1.1.2.2.2.3" xref="S2.p1.13.m13.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S2.p1.13.m13.1.1.1.1.2.2.2.2" xref="S2.p1.13.m13.1.1.1.1.2.2.2.2.cmml"><msub id="S2.p1.13.m13.1.1.1.1.2.2.2.2.2" xref="S2.p1.13.m13.1.1.1.1.2.2.2.2.2.cmml"><mi id="S2.p1.13.m13.1.1.1.1.2.2.2.2.2.2" xref="S2.p1.13.m13.1.1.1.1.2.2.2.2.2.2.cmml">V</mi><mn id="S2.p1.13.m13.1.1.1.1.2.2.2.2.2.3" xref="S2.p1.13.m13.1.1.1.1.2.2.2.2.2.3.cmml">3</mn></msub><mo id="S2.p1.13.m13.1.1.1.1.2.2.2.2.1" xref="S2.p1.13.m13.1.1.1.1.2.2.2.2.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.p1.13.m13.1.1.1.1.2.2.2.2.3" xref="S2.p1.13.m13.1.1.1.1.2.2.2.2.3a.cmml">true</mtext></mrow></mrow></mrow><mo stretchy="false" id="S2.p1.13.m13.1.1.1.3" xref="S2.p1.13.m13.1.1.2.cmml">}</mo></mrow></math>, <math><mrow id="S2.Ex1.m1.1.1.1" xref="S2.Ex1.m1.1.1.2.cmml"><mo stretchy="false" id="S2.Ex1.m1.1.1.1.2" xref="S2.Ex1.m1.1.1.2.cmml">{</mo><mrow id="S2.Ex1.m1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.3.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.cmml"><msub id="S2.Ex1.m1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.1.2.2" xref="S2.Ex1.m1.1.1.1.1.1.1.2.2.cmml">V</mi><mn id="S2.Ex1.m1.1.1.1.1.1.1.2.3" xref="S2.Ex1.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.Ex1.m1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex1.m1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex1.m1.1.1.1.1.2.3" xref="S2.Ex1.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S2.Ex1.m1.1.1.1.1.2.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.3.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.2.2.1.1" xref="S2.Ex1.m1.1.1.1.1.2.2.1.1.cmml"><msub id="S2.Ex1.m1.1.1.1.1.2.2.1.1.2" xref="S2.Ex1.m1.1.1.1.1.2.2.1.1.2.cmml"><mi id="S2.Ex1.m1.1.1.1.1.2.2.1.1.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.1.1.2.2.cmml">V</mi><mn id="S2.Ex1.m1.1.1.1.1.2.2.1.1.2.3" xref="S2.Ex1.m1.1.1.1.1.2.2.1.1.2.3.cmml">2</mn></msub><mo id="S2.Ex1.m1.1.1.1.1.2.2.1.1.1" xref="S2.Ex1.m1.1.1.1.1.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex1.m1.1.1.1.1.2.2.1.1.3" xref="S2.Ex1.m1.1.1.1.1.2.2.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex1.m1.1.1.1.1.2.2.2.3" xref="S2.Ex1.m1.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.3.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.cmml"><msub id="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.2" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.2.cmml"><mi id="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.2.2.cmml">V</mi><mn id="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.2.3" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.2.3.cmml">3</mn></msub><mo id="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.1" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.3" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.1.1.3a.cmml">true</mtext></mrow><mo id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.3" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.3a.cmml">,</mo><mrow id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.cmml"><msub id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.2.2" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.2.2.cmml">V</mi><mn id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.2.3" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.2.3.cmml">4</mn></msub><mo id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.1" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.3" xref="S2.Ex1.m1.1.1.1.1.2.2.2.2.2.2.3a.cmml">true</mtext></mrow></mrow></mrow></mrow><mo stretchy="false" id="S2.Ex1.m1.1.1.1.3" xref="S2.Ex1.m1.1.1.2.cmml">}</mo></mrow></math>, <math><mrow id="S2.Ex2.m1.1.1.1" xref="S2.Ex2.m1.1.1.2.cmml"><mo stretchy="false" id="S2.Ex2.m1.1.1.1.2" xref="S2.Ex2.m1.1.1.2.cmml">{</mo><mrow id="S2.Ex2.m1.1.1.1.1.2" xref="S2.Ex2.m1.1.1.1.1.3.cmml"><mrow id="S2.Ex2.m1.1.1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><msub id="S2.Ex2.m1.1.1.1.1.1.1.2" xref="S2.Ex2.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex2.m1.1.1.1.1.1.1.2.2" xref="S2.Ex2.m1.1.1.1.1.1.1.2.2.cmml">V</mi><mn id="S2.Ex2.m1.1.1.1.1.1.1.2.3" xref="S2.Ex2.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.Ex2.m1.1.1.1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex2.m1.1.1.1.1.1.1.3" xref="S2.Ex2.m1.1.1.1.1.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex2.m1.1.1.1.1.2.3" xref="S2.Ex2.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S2.Ex2.m1.1.1.1.1.2.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.3.cmml"><mrow id="S2.Ex2.m1.1.1.1.1.2.2.1.1" xref="S2.Ex2.m1.1.1.1.1.2.2.1.1.cmml"><msub id="S2.Ex2.m1.1.1.1.1.2.2.1.1.2" xref="S2.Ex2.m1.1.1.1.1.2.2.1.1.2.cmml"><mi id="S2.Ex2.m1.1.1.1.1.2.2.1.1.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.1.1.2.2.cmml">V</mi><mn id="S2.Ex2.m1.1.1.1.1.2.2.1.1.2.3" xref="S2.Ex2.m1.1.1.1.1.2.2.1.1.2.3.cmml">2</mn></msub><mo id="S2.Ex2.m1.1.1.1.1.2.2.1.1.1" xref="S2.Ex2.m1.1.1.1.1.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex2.m1.1.1.1.1.2.2.1.1.3" xref="S2.Ex2.m1.1.1.1.1.2.2.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex2.m1.1.1.1.1.2.2.2.3" xref="S2.Ex2.m1.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.3.cmml"><mrow id="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.cmml"><msub id="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.2" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.2.cmml"><mi id="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.2.2.cmml">V</mi><mn id="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.2.3" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.2.3.cmml">3</mn></msub><mo id="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.1" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.3" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.3" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.3a.cmml">,</mo><mrow id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.cmml"><msub id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.2.2" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.2.2.cmml">V</mi><mn id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.2.3" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.2.3.cmml">4</mn></msub><mo id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.1" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.3" xref="S2.Ex2.m1.1.1.1.1.2.2.2.2.2.2.3a.cmml">true</mtext></mrow></mrow></mrow></mrow><mo stretchy="false" id="S2.Ex2.m1.1.1.1.3" xref="S2.Ex2.m1.1.1.2.cmml">}</mo></mrow></math>, <math><mrow id="S2.Ex3.m1.1.1.1" xref="S2.Ex3.m1.1.1.2.cmml"><mo stretchy="false" id="S2.Ex3.m1.1.1.1.2" xref="S2.Ex3.m1.1.1.2.cmml">{</mo><mrow id="S2.Ex3.m1.1.1.1.1.2" xref="S2.Ex3.m1.1.1.1.1.3.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.1.1.cmml"><msub id="S2.Ex3.m1.1.1.1.1.1.1.2" xref="S2.Ex3.m1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.1.1.2.2" xref="S2.Ex3.m1.1.1.1.1.1.1.2.2.cmml">V</mi><mn id="S2.Ex3.m1.1.1.1.1.1.1.2.3" xref="S2.Ex3.m1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S2.Ex3.m1.1.1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex3.m1.1.1.1.1.1.1.3" xref="S2.Ex3.m1.1.1.1.1.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex3.m1.1.1.1.1.2.3" xref="S2.Ex3.m1.1.1.1.1.3a.cmml">,</mo><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.3.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.2.2.1.1" xref="S2.Ex3.m1.1.1.1.1.2.2.1.1.cmml"><msub id="S2.Ex3.m1.1.1.1.1.2.2.1.1.2" xref="S2.Ex3.m1.1.1.1.1.2.2.1.1.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.2.1.1.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.1.1.2.2.cmml">V</mi><mn id="S2.Ex3.m1.1.1.1.1.2.2.1.1.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.1.1.2.3.cmml">2</mn></msub><mo id="S2.Ex3.m1.1.1.1.1.2.2.1.1.1" xref="S2.Ex3.m1.1.1.1.1.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex3.m1.1.1.1.1.2.2.1.1.3" xref="S2.Ex3.m1.1.1.1.1.2.2.1.1.3a.cmml">false</mtext></mrow><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.3a.cmml">,</mo><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.3.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.cmml"><msub id="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.2.2.cmml">V</mi><mn id="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.2.3.cmml">3</mn></msub><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.1" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.1.1.3a.cmml">true</mtext></mrow><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.3a.cmml">,</mo><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.cmml"><msub id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.cmml">V</mi><mn id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.3.cmml">4</mn></msub><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.1" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.1.cmml">=</mo><mtext class="ltx_font_smallcaps" mathvariant="normal" id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.3a.cmml">false</mtext></mrow></mrow></mrow></mrow><mo stretchy="false" id="S2.Ex3.m1.1.1.1.3" xref="S2.Ex3.m1.1.1.2.cmml">}</mo></mrow></math>, <math><mrow id="S2.p3.9.m9.2.3" xref="S2.p3.9.m9.2.3.cmml"><mi id="S2.p3.9.m9.2.3.2" xref="S2.p3.9.m9.2.3.2.cmml">d</mi><mo id="S2.p3.9.m9.2.3.1" xref="S2.p3.9.m9.2.3.1.cmml">⁢</mo><mrow id="S2.p3.9.m9.2.3.3.2" xref="S2.p3.9.m9.2.3.3.1.cmml"><mo stretchy="false" id="S2.p3.9.m9.2.3.3.2.1" xref="S2.p3.9.m9.2.3.3.1.cmml">(</mo><mi id="S2.p3.9.m9.1.1" xref="S2.p3.9.m9.1.1.cmml">r</mi><mo id="S2.p3.9.m9.2.3.3.2.2" xref="S2.p3.9.m9.2.3.3.1.cmml">,</mo><mi id="S2.p3.9.m9.2.2" xref="S2.p3.9.m9.2.2.cmml">s</mi><mo stretchy="false" id="S2.p3.9.m9.2.3.3.2.3" xref="S2.p3.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E1.m1.5.5" xref="S2.E1.m1.5.5.cmml"><mrow id="S2.E1.m1.5.5.3" xref="S2.E1.m1.5.5.3.cmml"><mi id="S2.E1.m1.5.5.3.2" xref="S2.E1.m1.5.5.3.2.cmml">d</mi><mo id="S2.E1.m1.5.5.3.1" xref="S2.E1.m1.5.5.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.5.5.3.3.2" xref="S2.E1.m1.5.5.3.3.1.cmml"><mo stretchy="false" id="S2.E1.m1.5.5.3.3.2.1" xref="S2.E1.m1.5.5.3.3.1.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">r</mi><mo id="S2.E1.m1.5.5.3.3.2.2" xref="S2.E1.m1.5.5.3.3.1.cmml">,</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">s</mi><mo stretchy="false" id="S2.E1.m1.5.5.3.3.2.3" xref="S2.E1.m1.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.5.5.2" xref="S2.E1.m1.5.5.2.cmml">=</mo><mrow id="S2.E1.m1.5.5.1" xref="S2.E1.m1.5.5.1.cmml"><mrow id="S2.E1.m1.5.5.1.3" xref="S2.E1.m1.5.5.1.3.cmml"><mrow id="S2.E1.m1.5.5.1.3.2.2" xref="S2.E1.m1.5.5.1.3.2.1.cmml"><mo stretchy="false" id="S2.E1.m1.5.5.1.3.2.2.1" xref="S2.E1.m1.5.5.1.3.2.1.1.cmml">|</mo><mi id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">r</mi><mo stretchy="false" id="S2.E1.m1.5.5.1.3.2.2.2" xref="S2.E1.m1.5.5.1.3.2.1.1.cmml">|</mo></mrow><mo id="S2.E1.m1.5.5.1.3.1" xref="S2.E1.m1.5.5.1.3.1.cmml">+</mo><mrow id="S2.E1.m1.5.5.1.3.3.2" xref="S2.E1.m1.5.5.1.3.3.1.cmml"><mo stretchy="false" id="S2.E1.m1.5.5.1.3.3.2.1" xref="S2.E1.m1.5.5.1.3.3.1.1.cmml">|</mo><mi id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml">s</mi><mo stretchy="false" id="S2.E1.m1.5.5.1.3.3.2.2" xref="S2.E1.m1.5.5.1.3.3.1.1.cmml">|</mo></mrow></mrow><mo id="S2.E1.m1.5.5.1.2" xref="S2.E1.m1.5.5.1.2.cmml">-</mo><mrow id="S2.E1.m1.5.5.1.1" xref="S2.E1.m1.5.5.1.1.cmml"><mn id="S2.E1.m1.5.5.1.1.3" xref="S2.E1.m1.5.5.1.1.3.cmml">2</mn><mo id="S2.E1.m1.5.5.1.1.2" xref="S2.E1.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.5.5.1.1.1.1" xref="S2.E1.m1.5.5.1.1.1.2.cmml"><mo stretchy="false" id="S2.E1.m1.5.5.1.1.1.1.2" xref="S2.E1.m1.5.5.1.1.1.2.1.cmml">|</mo><mrow id="S2.E1.m1.5.5.1.1.1.1.1" xref="S2.E1.m1.5.5.1.1.1.1.1.cmml"><mi id="S2.E1.m1.5.5.1.1.1.1.1.2" xref="S2.E1.m1.5.5.1.1.1.1.1.2.cmml">r</mi><mo id="S2.E1.m1.5.5.1.1.1.1.1.1" xref="S2.E1.m1.5.5.1.1.1.1.1.1.cmml">∧</mo><mi id="S2.E1.m1.5.5.1.1.1.1.1.3" xref="S2.E1.m1.5.5.1.1.1.1.1.3.cmml">s</mi></mrow><mo stretchy="false" id="S2.E1.m1.5.5.1.1.1.1.3" xref="S2.E1.m1.5.5.1.1.1.2.1.cmml">|</mo></mrow></mrow></mrow></mrow></math>

Correct Categorie: quant-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1307.7755

Formulas:

1-

0 \overset{''}{.} 02

2-

M_{F 555 W} \sim > - 5.7

3-

0 \overset{''}{.} 17

4-

M_{F 555 W} = - 9.02 \pm 0.28

5-

α = 12^{h} 01^{m} 52 \overset{s}{.} 72 \pm 0 \overset{\circ}{.} 2

6-

δ = - 18 ° 52' 18 \overset{''}{.} 3 \pm 0 \overset{\circ}{.} 2

7-

E {(B - V)}_{G} = 0.04 \pm 0.01

8-

E {(B - V)}_{h o s t}

9-

E {(B - V)}_{tot} = 0.24 \pm 0.02

10-

E (B - V)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0704.0843

Formulas:

1-

A / γ^{2} ≃ 1 \times 10^{- 5} μ Ω

2-

ρ = A T^{2}

3-

\frac{A}{γ^{2}} = \frac{9 α F}{8 π e^{2}} = 21.3 α F a [μ Ω {(mol K/mJ)}^{2}],

4-

v_{p μ}^{i} = {(π T)}^{2} \sum_{p^{'}, k} W_{p p^{'} k}^{i j} ρ_{p^{'}}^{j} ρ_{p - k}^{i} ρ_{p^{'} + k}^{j} (l_{p μ}^{i} + l_{p^{'} μ}^{j} - l_{p^{'} + k μ}^{j} - l_{p - k μ}^{i}),

5-

ρ_{p}^{i} = δ (μ - ε_{p}^{i})

6-

μ = x, y, z

7-

σ \equiv σ_{μ} = 2 e^{2} \sum_{p, i} ρ_{p}^{i} v_{p μ}^{i} l_{p μ}^{i},

8-

l_{p μ}^{i} \propto e_{p}^{i} \equiv \frac{v_{p μ}^{i}}{| v_{p μ}^{i} |},

9-

α F \equiv \frac{\sum_{i, j} α^{i j} c_{i, j}}{{(ρ^{2} | v_{x} |)}^{2}},

10-

c_{i, j} = \sum_{\binom{k_{1}, k_{2}, k_{3}, k_{4}}{k_{1} + k_{2} = k_{3} + k_{4}}} ρ_{k_{1}}^{i} ρ_{k_{2}}^{j} ρ_{k_{3}}^{j} ρ_{k_{4}}^{i} {(e_{k_{1}}^{i} + e_{k_{2}}^{j} - e_{k_{3}}^{j} - e_{k_{4}}^{i})}^{2} / 4 ρ_{i} ρ_{j},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0704.2685

Formulas:

1-

0 \overset{''}{.} 34

2-

\geq 4 \times 10^{- 8} - 6 \times 10^{- 5}

3-

0 \overset{''}{.} 34

4-

0 \overset{''}{.} 4

5-

0 \overset{''}{.} 2

6-

0 \overset{''}{.} 34

7-

0 \overset{''}{.} 35

8-

0 \overset{''}{.} 4

9-

0 \overset{''}{.} 78 \times 0 \overset{''}{.} 34

10-

1 \overset{''}{.} 89 \times 0 \overset{''}{.} 94

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1810.01153

Formulas:

1-

\frac{d^{2} x^{μ}}{d τ^{2}} + Γ_{λ σ}^{μ} \frac{d x^{λ}}{d τ} \frac{d x^{σ}}{d τ} = q F^{μ ν} u_{ν},

2-

μ = 0, 1, 2, 3

3-

g^{μ ν} u_{ν} := d x^{μ} / d τ

4-

F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}

5-

(-, +, +, +)

6-

g_{μ ν} = (\begin{matrix} - α^{2} + β_{k} β^{k} & β_{i} \\ β_{j} & γ_{i j} \end{matrix}),

7-

u_{i} = g_{i μ} u^{μ}

8-

\frac{d x^{i}}{d t} = γ^{i j} \frac{u_{j}}{u^{0}} - β^{i},

9-

\frac{d u_{i}}{d t} = - α u^{0} \partial_{i} α + u_{k} \partial_{i} β^{k} - \frac{u_{j} u_{k}}{2 u^{0}} \partial_{i} γ^{j k} + q (F_{i 0} + F_{i j} \frac{u^{j}}{u^{0}}),

10-

u^{0} = {(γ^{j k} u_{j} u_{k} + ϵ)}^{1 / 2} / α,

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0307537

Formulas:

1-

P (k) \sim k^{- γ}

2-

N_{t} = \frac{3 (3^{t} + 1)}{2}

3-

P_{c u m} (k)

4-

P_{c u m} (k) \sim k^{1 - γ}

5-

γ = 1 + \ln 3 / \ln 2

6-

\bar{C} = 4 / 5

7-

\bar{l} \sim \ln N

8-

P (k) \sim k^{- γ}

9-

P (k) \sim k^{1 - γ}

10-

γ = 1 + \ln 3 / \ln 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0702702

Formulas:

1-

\sim 0.88 S_{q s}

2-

< 3 λ / D

3-

λ = 2.2 μ m

4-

= 4 λ / D

5-

\tilde{r} = r D / λ

6-

M_{H} (r)

7-

circ (\tilde{r} / 4)

8-

M_{G} (r)

9-

1 - e^{- {(\tilde{r} / 3.6097)}^{2}},

10-

M_{{BL}_{4 t h}} (r)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/0903.0071

Formulas:

1-

x_{m} (t)

2-

x_{m} (0) = 0

3-

v_{f} = - v_{s} + 2 v_{m} .

4-

v_{m} = v_{s} / 2

5-

x (t) = v t

6-

x_{m} (t)

7-

x_{m} (t_{c}) = v t_{c}

8-

\frac{d x_{m}}{d t} (t_{c}) = \frac{v}{2} .

9-

v = x_{m} (t_{c}) / t_{c}

10-

\frac{d x_{m}}{d t} (t_{c}) = \frac{x_{m} (t_{c})}{2 t_{c}} .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1504.08085

Formulas:

1-

{\ddot{x}}_{1} + γ_{1} {\dot{x}}_{1} + ω_{1}^{2} x_{1}

2-

- k x_{2},

3-

{\ddot{x}}_{2} + γ_{2} {\dot{x}}_{2} + ω_{2}^{2} x_{2}

4-

- k x_{1},

5-

(i = 1, 2)

6-

ω_{1} = ω_{2} = 1.0

7-

ω_{1} = ω_{2} = 1.0

8-

- γ_{1} y_{1} - ω_{1}^{2} x_{1} - k x_{2},

9-

- γ_{2} y_{2} - ω_{2}^{2} x_{2} - k x_{1} .

10-

\dot{\vec{z}} (t) = M \vec{z} (t)

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/0803.4059

Formulas:

1-

L n_{2 - x}

2-

L n_{2 - x}

3-

ϵ - μ = ε_{0} \pm \sqrt{Δ E^{2} + 4 t^{2} {(\cos k_{x} a + \cos k_{y} a)}^{2}}

4-

- 4 t^{'} \cos k_{x} a \cos k_{y} a - 2 t^{''} (\cos 2 k_{x} a + \cos 2 k_{y} a),

5-

- t^{''} / t^{'} = 0.50

6-

- t^{'} / t =

7-

- t^{'} / t =

8-

- t^{''} / t^{'} = 0.50

9-

\sim (π, 0)

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/0311158

Formulas:

1-

ρ = p_{1} ρ_{1} + p_{2} ρ_{2}

2-

ρ (P_{1} P_{2})

3-

tr (ρ P_{1} P_{2})

4-

ρ_{1} = P_{1} ρ P_{1} / ρ (P_{1})

5-

ρ_{12} = P_{2} ρ_{1} P_{2} / ρ_{1} (P_{2}) = P_{2} P_{1} ρ P_{1} P_{2} / ρ (P_{1} P_{2})

6-

P_{2} P_{1} ρ P_{1} P_{2} / ρ (P_{1} P_{2})

7-

{(P_{1}^{n})}_{n = 1}^{N}

8-

\sum_{n = 1}^{N} P_{1}^{n} = 1

9-

ρ (P_{1}^{n})

10-

\sum_{n = 1}^{N} ρ (P_{1}^{n}) P_{1}^{n} ρ P_{1}^{n} / ρ (P_{1}^{n}) = \sum_{n = 1}^{N} P_{1}^{n} ρ P_{1}^{n}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1812.09356

Formulas:

1-

ξ = (R / Q) \sqrt{Q} / f

2-

V (n t_{b} + t_{r}) = I_{0} \frac{e}{c} t_{b} T_{12} \sum_{m = 0}^{\infty} W (m t_{b}) V ([n - m] t_{b}),

3-

W (τ) = {(\frac{R}{Q})}_{λ} \frac{ω_{λ}^{2}}{2 c} e^{- (ω_{λ} / 2 Q_{λ}) τ} \sin (ω_{λ} τ) .

4-

δ = (t_{r} / t_{b} - n_{r}) \in [0, 1)

5-

{(R / Q)}_{λ}

6-

w (δ, ω)

7-

ϵ = (ω_{λ} / 2 Q_{λ}) t_{b}

8-

κ = t_{b} (e / c^{2}) {(R / Q)}_{λ} (ω_{λ}^{2} / 2)

9-

V (n t_{b})

10-

{\tilde{V}}^{Σ} (ω) = t_{b} \sum_{n = - \infty}^{\infty} V (n t_{b}) e^{i ω n t_{b}}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0410567

Formulas:

1-

Y_{n}^{l} = f_{l} (\sum_{j = 1}^{N} W_{n, j}^{l} Y_{j}^{l - 1} + β_{n}^{l}) .

2-

f_{l} (x)

3-

f_{l} (x)

4-

x (for linear layers),

5-

f_{l} (x)

6-

a \tanh (b x) (for non - linear layers) .

7-

𝐨 = 𝐅 (𝐱),

8-

N_{N} (l)

9-

f_{l} (x)

10-

χ^{2} = \sum_{j} {(o_{j} - o_{j}^{t})}^{2}

Formulas (html):

<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"><msubsup id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml"><mi id="S2.E1.m1.3.3.1.1.3.2.2" xref="S2.E1.m1.3.3.1.1.3.2.2.cmml">Y</mi><mi id="S2.E1.m1.3.3.1.1.3.3" xref="S2.E1.m1.3.3.1.1.3.3.cmml">n</mi><mi id="S2.E1.m1.3.3.1.1.3.2.3" xref="S2.E1.m1.3.3.1.1.3.2.3.cmml">l</mi></msubsup><mo id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.1" xref="S2.E1.m1.3.3.1.1.1.cmml"><msub id="S2.E1.m1.3.3.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.3.cmml"><mi id="S2.E1.m1.3.3.1.1.1.3.2" xref="S2.E1.m1.3.3.1.1.1.3.2.cmml">f</mi><mi id="S2.E1.m1.3.3.1.1.1.3.3" xref="S2.E1.m1.3.3.1.1.1.3.3.cmml">l</mi></msub><mo id="S2.E1.m1.3.3.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.3.3.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.cmml"><munderover id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.2.cmml">∑</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.2.cmml">j</mi><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.1.cmml">=</mo><mn id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.2.3.3.cmml">1</mn></mrow><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.1.3.cmml">N</mi></munderover><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.cmml"><msubsup id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.2.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.2.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.2.2.2.cmml">W</mi><mrow id="S2.E1.m1.2.2.2.4" xref="S2.E1.m1.2.2.2.3.cmml"><mi id="S2.E1.m1.1.1.1.1" xref="S2.E1.m1.1.1.1.1.cmml">n</mi><mo id="S2.E1.m1.2.2.2.4.1" xref="S2.E1.m1.2.2.2.3.cmml">,</mo><mi id="S2.E1.m1.2.2.2.2" xref="S2.E1.m1.2.2.2.2.cmml">j</mi></mrow><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.2.2.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.2.2.3.cmml">l</mi></msubsup><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.1.cmml">⁢</mo><msubsup id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.2.cmml">Y</mi><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.3.cmml">j</mi><mrow id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.2.cmml">l</mi><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.1.cmml">-</mo><mn id="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.2.2.3.2.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><mo id="S2.E1.m1.3.3.1.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.1.cmml">+</mo><msubsup id="S2.E1.m1.3.3.1.1.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.3.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.3.2.2" xref="S2.E1.m1.3.3.1.1.1.1.1.1.3.2.2.cmml">β</mi><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.3.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.3.3.cmml">n</mi><mi id="S2.E1.m1.3.3.1.1.1.1.1.1.3.2.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.3.2.3.cmml">l</mi></msubsup></mrow><mo rspace="4.2pt" id="S2.E1.m1.3.3.1.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.1.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.p3.9.m9.1.2" xref="S2.p3.9.m9.1.2.cmml"><msub id="S2.p3.9.m9.1.2.2" xref="S2.p3.9.m9.1.2.2.cmml"><mi id="S2.p3.9.m9.1.2.2.2" xref="S2.p3.9.m9.1.2.2.2.cmml">f</mi><mi id="S2.p3.9.m9.1.2.2.3" xref="S2.p3.9.m9.1.2.2.3.cmml">l</mi></msub><mo id="S2.p3.9.m9.1.2.1" xref="S2.p3.9.m9.1.2.1.cmml">⁢</mo><mrow id="S2.p3.9.m9.1.2.3.2" xref="S2.p3.9.m9.1.2.cmml"><mo stretchy="false" id="S2.p3.9.m9.1.2.3.2.1" xref="S2.p3.9.m9.1.2.cmml">(</mo><mi id="S2.p3.9.m9.1.1" xref="S2.p3.9.m9.1.1.cmml">x</mi><mo stretchy="false" id="S2.p3.9.m9.1.2.3.2.2" xref="S2.p3.9.m9.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.Ex1.m1.1.2" xref="S2.Ex1.m1.1.2.cmml"><msub id="S2.Ex1.m1.1.2.2" xref="S2.Ex1.m1.1.2.2.cmml"><mi id="S2.Ex1.m1.1.2.2.2" xref="S2.Ex1.m1.1.2.2.2.cmml">f</mi><mi id="S2.Ex1.m1.1.2.2.3" xref="S2.Ex1.m1.1.2.2.3.cmml">l</mi></msub><mo id="S2.Ex1.m1.1.2.1" xref="S2.Ex1.m1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.1.2.3.2" xref="S2.Ex1.m1.1.2.cmml"><mo stretchy="false" id="S2.Ex1.m1.1.2.3.2.1" xref="S2.Ex1.m1.1.2.cmml">(</mo><mi id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml">x</mi><mo stretchy="false" id="S2.Ex1.m1.1.2.3.2.2" xref="S2.Ex1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.Ex1.m3.1.1.1" xref="S2.Ex1.m3.1.1.1.1.cmml"><mrow id="S2.Ex1.m3.1.1.1.1" xref="S2.Ex1.m3.1.1.1.1.cmml"><mpadded width="+5pt" id="S2.Ex1.m3.1.1.1.1.3" xref="S2.Ex1.m3.1.1.1.1.3.cmml"><mi id="S2.Ex1.m3.1.1.1.1.3a" xref="S2.Ex1.m3.1.1.1.1.3.cmml">x</mi></mpadded><mo id="S2.Ex1.m3.1.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m3.1.1.1.1.1.1" xref="S2.Ex1.m3.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.Ex1.m3.1.1.1.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex1.m3.1.1.1.1.1.1.1" xref="S2.Ex1.m3.1.1.1.1.1.1.1.cmml"><mpadded width="+1.7pt" id="S2.Ex1.m3.1.1.1.1.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex1.m3.1.1.1.1.1.1.1.2a" xref="S2.Ex1.m3.1.1.1.1.1.1.1.2.cmml">for</mi></mpadded><mo id="S2.Ex1.m3.1.1.1.1.1.1.1.1" xref="S2.Ex1.m3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.Ex1.m3.1.1.1.1.1.1.1.3" xref="S2.Ex1.m3.1.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex1.m3.1.1.1.1.1.1.1.3a" xref="S2.Ex1.m3.1.1.1.1.1.1.1.3.cmml">linear</mi></mpadded><mo id="S2.Ex1.m3.1.1.1.1.1.1.1.1a" xref="S2.Ex1.m3.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.Ex1.m3.1.1.1.1.1.1.1.4" xref="S2.Ex1.m3.1.1.1.1.1.1.1.4.cmml">layers</mi></mrow><mo rspace="4.2pt" stretchy="false" id="S2.Ex1.m3.1.1.1.1.1.1.3" xref="S2.Ex1.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m3.1.1.1.2" xref="S2.Ex1.m3.1.1.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.E2.m1.1.2" xref="S2.E2.m1.1.2.cmml"><msub id="S2.E2.m1.1.2.2" xref="S2.E2.m1.1.2.2.cmml"><mi id="S2.E2.m1.1.2.2.2" xref="S2.E2.m1.1.2.2.2.cmml">f</mi><mi id="S2.E2.m1.1.2.2.3" xref="S2.E2.m1.1.2.2.3.cmml">l</mi></msub><mo id="S2.E2.m1.1.2.1" xref="S2.E2.m1.1.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.1.2.3.2" xref="S2.E2.m1.1.2.cmml"><mo stretchy="false" id="S2.E2.m1.1.2.3.2.1" xref="S2.E2.m1.1.2.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">x</mi><mo stretchy="false" id="S2.E2.m1.1.2.3.2.2" xref="S2.E2.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E2.m3.2.2.1" xref="S2.E2.m3.2.2.1.1.cmml"><mrow id="S2.E2.m3.2.2.1.1" xref="S2.E2.m3.2.2.1.1.cmml"><mi id="S2.E2.m3.2.2.1.1.4" xref="S2.E2.m3.2.2.1.1.4.cmml">a</mi><mo id="S2.E2.m3.2.2.1.1.3" xref="S2.E2.m3.2.2.1.1.3.cmml">⁢</mo><mrow id="S2.E2.m3.2.2.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.2.cmml"><mi id="S2.E2.m3.1.1" xref="S2.E2.m3.1.1.cmml">tanh</mi><mo id="S2.E2.m3.2.2.1.1.1.1a" xref="S2.E2.m3.2.2.1.1.1.2.cmml">⁡</mo><mrow id="S2.E2.m3.2.2.1.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.2.cmml"><mo stretchy="false" id="S2.E2.m3.2.2.1.1.1.1.1.2" xref="S2.E2.m3.2.2.1.1.1.2.cmml">(</mo><mrow id="S2.E2.m3.2.2.1.1.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.1.1.1.cmml"><mi id="S2.E2.m3.2.2.1.1.1.1.1.1.2" xref="S2.E2.m3.2.2.1.1.1.1.1.1.2.cmml">b</mi><mo id="S2.E2.m3.2.2.1.1.1.1.1.1.1" xref="S2.E2.m3.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E2.m3.2.2.1.1.1.1.1.1.3" xref="S2.E2.m3.2.2.1.1.1.1.1.1.3.cmml">x</mi></mrow><mo rspace="7.5pt" stretchy="false" id="S2.E2.m3.2.2.1.1.1.1.1.3" xref="S2.E2.m3.2.2.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m3.2.2.1.1.3a" xref="S2.E2.m3.2.2.1.1.3.cmml">⁢</mo><mrow id="S2.E2.m3.2.2.1.1.2.1" xref="S2.E2.m3.2.2.1.1.2.1.1.cmml"><mo stretchy="false" id="S2.E2.m3.2.2.1.1.2.1.2" xref="S2.E2.m3.2.2.1.1.2.1.1.cmml">(</mo><mrow id="S2.E2.m3.2.2.1.1.2.1.1" xref="S2.E2.m3.2.2.1.1.2.1.1.cmml"><mrow id="S2.E2.m3.2.2.1.1.2.1.1.2" xref="S2.E2.m3.2.2.1.1.2.1.1.2.cmml"><mpadded width="+1.7pt" id="S2.E2.m3.2.2.1.1.2.1.1.2.2" xref="S2.E2.m3.2.2.1.1.2.1.1.2.2.cmml"><mi id="S2.E2.m3.2.2.1.1.2.1.1.2.2a" xref="S2.E2.m3.2.2.1.1.2.1.1.2.2.cmml">for</mi></mpadded><mo id="S2.E2.m3.2.2.1.1.2.1.1.2.1" xref="S2.E2.m3.2.2.1.1.2.1.1.2.1.cmml">⁢</mo><mi id="S2.E2.m3.2.2.1.1.2.1.1.2.3" xref="S2.E2.m3.2.2.1.1.2.1.1.2.3.cmml">non</mi></mrow><mo id="S2.E2.m3.2.2.1.1.2.1.1.1" xref="S2.E2.m3.2.2.1.1.2.1.1.1.cmml">-</mo><mrow id="S2.E2.m3.2.2.1.1.2.1.1.3" xref="S2.E2.m3.2.2.1.1.2.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.E2.m3.2.2.1.1.2.1.1.3.2" xref="S2.E2.m3.2.2.1.1.2.1.1.3.2.cmml"><mi id="S2.E2.m3.2.2.1.1.2.1.1.3.2a" xref="S2.E2.m3.2.2.1.1.2.1.1.3.2.cmml">linear</mi></mpadded><mo id="S2.E2.m3.2.2.1.1.2.1.1.3.1" xref="S2.E2.m3.2.2.1.1.2.1.1.3.1.cmml">⁢</mo><mi id="S2.E2.m3.2.2.1.1.2.1.1.3.3" xref="S2.E2.m3.2.2.1.1.2.1.1.3.3.cmml">layers</mi></mrow></mrow><mo rspace="4.2pt" stretchy="false" id="S2.E2.m3.2.2.1.1.2.1.3" xref="S2.E2.m3.2.2.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E2.m3.2.2.1.2" xref="S2.E2.m3.2.2.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.E3.m1.2.2.1" xref="S2.E3.m1.2.2.1.1.cmml"><mrow id="S2.E3.m1.2.2.1.1" xref="S2.E3.m1.2.2.1.1.cmml"><mi id="S2.E3.m1.2.2.1.1.2" xref="S2.E3.m1.2.2.1.1.2.cmml">𝐨</mi><mo id="S2.E3.m1.2.2.1.1.1" xref="S2.E3.m1.2.2.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.2.2.1.1.3" xref="S2.E3.m1.2.2.1.1.3.cmml"><mi id="S2.E3.m1.2.2.1.1.3.2" xref="S2.E3.m1.2.2.1.1.3.2.cmml">𝐅</mi><mo id="S2.E3.m1.2.2.1.1.3.1" xref="S2.E3.m1.2.2.1.1.3.1.cmml">⁢</mo><mrow id="S2.E3.m1.2.2.1.1.3.3.2" xref="S2.E3.m1.2.2.1.1.3.cmml"><mo stretchy="false" id="S2.E3.m1.2.2.1.1.3.3.2.1" xref="S2.E3.m1.2.2.1.1.3.cmml">(</mo><mi id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml">𝐱</mi><mo rspace="4.2pt" stretchy="false" id="S2.E3.m1.2.2.1.1.3.3.2.2" xref="S2.E3.m1.2.2.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E3.m1.2.2.1.2" xref="S2.E3.m1.2.2.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.p6.2.m2.1.2" xref="S2.p6.2.m2.1.2.cmml"><msub id="S2.p6.2.m2.1.2.2" xref="S2.p6.2.m2.1.2.2.cmml"><mi id="S2.p6.2.m2.1.2.2.2" xref="S2.p6.2.m2.1.2.2.2.cmml">N</mi><mi id="S2.p6.2.m2.1.2.2.3" xref="S2.p6.2.m2.1.2.2.3.cmml">N</mi></msub><mo id="S2.p6.2.m2.1.2.1" xref="S2.p6.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.p6.2.m2.1.2.3.2" xref="S2.p6.2.m2.1.2.cmml"><mo stretchy="false" id="S2.p6.2.m2.1.2.3.2.1" xref="S2.p6.2.m2.1.2.cmml">(</mo><mi id="S2.p6.2.m2.1.1" xref="S2.p6.2.m2.1.1.cmml">l</mi><mo stretchy="false" id="S2.p6.2.m2.1.2.3.2.2" xref="S2.p6.2.m2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.p6.4.m4.1.2" xref="S2.p6.4.m4.1.2.cmml"><msub id="S2.p6.4.m4.1.2.2" xref="S2.p6.4.m4.1.2.2.cmml"><mi id="S2.p6.4.m4.1.2.2.2" xref="S2.p6.4.m4.1.2.2.2.cmml">f</mi><mi id="S2.p6.4.m4.1.2.2.3" xref="S2.p6.4.m4.1.2.2.3.cmml">l</mi></msub><mo id="S2.p6.4.m4.1.2.1" xref="S2.p6.4.m4.1.2.1.cmml">⁢</mo><mrow id="S2.p6.4.m4.1.2.3.2" xref="S2.p6.4.m4.1.2.cmml"><mo stretchy="false" id="S2.p6.4.m4.1.2.3.2.1" xref="S2.p6.4.m4.1.2.cmml">(</mo><mi id="S2.p6.4.m4.1.1" xref="S2.p6.4.m4.1.1.cmml">x</mi><mo stretchy="false" id="S2.p6.4.m4.1.2.3.2.2" xref="S2.p6.4.m4.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml"><msup id="S2.SS1.p4.5.m5.1.1.3" xref="S2.SS1.p4.5.m5.1.1.3.cmml"><mi id="S2.SS1.p4.5.m5.1.1.3.2" xref="S2.SS1.p4.5.m5.1.1.3.2.cmml">χ</mi><mn id="S2.SS1.p4.5.m5.1.1.3.3" xref="S2.SS1.p4.5.m5.1.1.3.3.cmml">2</mn></msup><mo id="S2.SS1.p4.5.m5.1.1.2" xref="S2.SS1.p4.5.m5.1.1.2.cmml">=</mo><mrow id="S2.SS1.p4.5.m5.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.cmml"><msub id="S2.SS1.p4.5.m5.1.1.1.2" xref="S2.SS1.p4.5.m5.1.1.1.2.cmml"><mo largeop="true" symmetric="true" id="S2.SS1.p4.5.m5.1.1.1.2.2" xref="S2.SS1.p4.5.m5.1.1.1.2.2.cmml">∑</mo><mi id="S2.SS1.p4.5.m5.1.1.1.2.3" xref="S2.SS1.p4.5.m5.1.1.1.2.3.cmml">j</mi></msub><msup id="S2.SS1.p4.5.m5.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.cmml"><mrow id="S2.SS1.p4.5.m5.1.1.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p4.5.m5.1.1.1.1.1.1.2" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml"><msub id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.2" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.2.cmml"><mi id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.2.2" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.2.2.cmml">o</mi><mi id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.2.3" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.2.3.cmml">j</mi></msub><mo id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.1.cmml">-</mo><msubsup id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.cmml"><mi id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.2.2" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.2.2.cmml">o</mi><mi id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.3" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.3.cmml">j</mi><mi id="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.2.3" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.3.2.3.cmml">t</mi></msubsup></mrow><mo stretchy="false" id="S2.SS1.p4.5.m5.1.1.1.1.1.1.3" xref="S2.SS1.p4.5.m5.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.SS1.p4.5.m5.1.1.1.1.3" xref="S2.SS1.p4.5.m5.1.1.1.1.3.cmml">2</mn></msup></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1102.3314

Formulas:

1-

k = 0, \dots, 6

2-

r_{i, l} = {\dot{z}}_{k} (t_{i}) + γ_{l} + ϵ_{i} + ϵ_{J},

3-

P ({\dot{z}}_{k} | r) = \frac{P (r | {\dot{z}}_{k}) P ({\dot{z}}_{k})}{\sum_{j = 0}^{p} P (r | {\dot{z}}_{j}) P ({\dot{z}}_{j})},

4-

P (r | {\dot{z}}_{k}) = \int f (r | θ_{k}, {\dot{z}}_{k}) p (θ_{k} | {\dot{z}}_{k}) 𝑑 θ_{k},

5-

f (r | θ_{k}, {\dot{z}}_{k})

6-

p (θ_{k} | {\dot{z}}_{k})

7-

P ({\dot{z}}_{k})

8-

P ({\dot{z}}_{k}) ≫ P ({\dot{z}}_{k - 1})

9-

k = 0, \dots, 6

10-

m_{p} \sin i

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0301386

Formulas:

1-

C_{w} = [- 0.26; 0.16]

2-

C_{c} = [- 0.09; 0.63]

3-

C_{b} = [- 0.45; - 0.02]

4-

C_{d} = [- 0.576; - 0.17]

5-

C_{p} = [- 0.32; 0.28]

6-

C_{t} = [- 1.27; 1.96]

7-

p_{i} \approx \frac{\bar{E}}{{\bar{t}}_{i}} .

8-

P_{i} (t) d t

9-

P^{'} (τ_{i}) d τ_{i} \approx N^{'} \exp (- α / τ_{i} - τ_{i} / β) d τ_{i}

10-

τ_{i} = t / {\bar{t}}_{i}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1106.2667

Formulas:

1-

d n_{e} / d t = - Γ_{f} n_{e}

2-

d n_{i} / d t = (Γ_{f} / 2) n_{e} - Γ_{s} n_{i}

3-

n_{e} (t) = n_{0} e^{- Γ_{f} t}

4-

n_{i} (t) = n_{0} Γ_{f} (e^{- Γ_{f} t} - e^{- Γ_{s} t}) / 2 (Γ_{s} - Γ_{f})

5-

n_{e} (t) + n_{i} (t)

6-

n_{0} e^{- 1} = n_{e} (τ) + n_{i} (τ)

7-

\frac{2 (Γ_{s} - Γ_{f})}{e} = (2 Γ_{s} - Γ_{f}) \exp (- Γ_{f} τ) - Γ_{f} \exp (- Γ_{s} τ)

8-

n_{e} (t)

9-

n_{i} (t)

10-

n_{e} (t) + N_{f} (t) = n_{0}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0903.1012

Formulas:

1-

c_{i} = c_{i} (𝐫 - 𝐫_{i}, t)

2-

\frac{\partial c_{i}}{\partial t} - D^{2} △ c_{i} + k c_{i} = J_{i} (𝐫, t) .

3-

c_{i} (𝐫 - 𝐫_{i}, t) = \int G_{d} (𝐫 - 𝐫^{'}, t) c_{i}^{0} (𝐫^{'}) 𝑑 𝐫^{'} + \int_{0}^{t} 𝑑 t^{'} \int G_{d} (𝐫 - 𝐫^{'}, t - t^{'}) J_{i} (𝐫^{'}, t^{'}) 𝑑 𝐫^{'},

4-

c_{i}^{0} (𝐫)

5-

G_{d} (𝐫, t) = \frac{1}{{(4 π t D^{2})}^{d / 2}} e^{- k t - \frac{𝐫^{2}}{4 t D^{2}}} .

6-

J_{i} (𝐫, t) = a δ^{(d)} (𝐫 - 𝐫_{i}) j_{i} (t),

7-

j_{i} (t)

8-

j_{i} (t) \leq 1

9-

c_{i} (𝐫 - 𝐫_{i}, t) = a \int_{0}^{t} 𝑑 t^{'} G_{d} (𝐫 - 𝐫_{i}, t - t^{'}) j_{i} (t^{'}) .

10-

j_{i} (t)

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: q-bio

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0504233

Formulas:

1-

L \propto {(1 + z)}^{1.8 \pm 0.3}

2-

M_{500} \propto E {(z)}^{- α}

3-

M \propto T^{3 / 2}

4-

d T / d r

5-

r = 7^{'} - 9^{'}

6-

T_{500}^{s p e c}

7-

T_{500}^{e m w}

8-

(10^{14} M_{☉})

9-

70 kpc < r < r_{500}

10-

70 kpc < r < r_{500}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/9710120

Formulas:

1-

- L_{1} < ξ^{1} < L_{2}

2-

(ξ^{2}, ξ^{3})

3-

{t, x^{i}}

4-

d s_{e x t}^{2} = d t^{2} - γ_{i j} d x^{i} d x^{j},

5-

{τ, ξ^{j}}

6-

d s_{i n t}^{2} = a^{2} (τ, ξ) d τ^{2} - 2 b_{i} (τ, ξ) d τ d ξ^{i} - {\tilde{γ}}_{i j} (τ, ξ) d ξ^{i} d ξ_{j},

7-

- \infty < t, τ < \infty

8-

a^{2} (τ, ξ) > 0

9-

t_{l e f t} = τ, x_{l e f t}^{i} = x_{l e f t}^{i} (ξ^{1}, ξ^{2}, ξ^{3}), t_{r i g h t} = t_{r} (τ), x_{r i g h t}^{i} = x_{r i g h t}^{i} (τ, ξ^{1}, ξ^{2}, ξ^{3})

10-

(t_{1} (τ), M_{e x t 1}^{3})

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1302.1831

Formulas:

1-

\partial_{t} A = ϵ A + (b_{1} + i c_{1}) \nabla_{⊥}^{2} A - (b_{3} - i c_{3}) {| A |}^{2} A - (b_{5} - i c_{5}) {| A |}^{4} A,

2-

\nabla_{⊥}^{2} = \frac{1}{r} \frac{\partial}{\partial r} (r \frac{\partial}{\partial r})

3-

\nabla_{⊥}^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}

4-

A (r; t)

5-

u (x, y)

6-

ℱ (u) (k_{x}, k_{y}) = \hat{u} (k_{x}, k_{y}) = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{- i (k_{x} x + k_{y} y)} u (x, y) 𝑑 x 𝑑 y,

7-

ℱ^{- 1} (\hat{u}) (x, y) = u (x, y) = \frac{1}{2 π} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} e^{i (k_{x} x + k_{y} y)} \hat{u} (k_{x}, k_{y}) 𝑑 k_{x} 𝑑 k_{y} .

8-

\hat{u} (k_{x}, k_{y})

9-

\hat{A_{t}} = [ϵ - (b_{1} + i c_{1}) (k_{x}^{2} + k_{y}^{2})] \hat{A} - (b_{3} - i c_{3}) \hat{{| A |}^{2} A} - (b_{5} - i c_{5}) \hat{{| A |}^{4} A},

10-

\hat{A_{t}} = α (k_{x}, k_{y}) \hat{A} + β \hat{{| A |}^{2} A} + γ \hat{{| A |}^{4} A},

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/1904.12788

Formulas:

1-

(r, θ, ϕ)

2-

\vec{B} = B_{ϕ} (θ) \sin θ {\hat{e}}_{ϕ}

3-

Ω = Ω_{0} (1 - s_{2} \cos^{2} θ - s_{4} \cos^{4} θ) = Ω_{0} + Ω_{1} (θ),

4-

\frac{\partial u_{θ}}{\partial t} + Ω_{1} \frac{\partial u_{θ}}{\partial ϕ} - 2 Ω \cos θ u_{ϕ} = - \frac{g}{R} \frac{\partial h}{\partial θ} + \frac{B_{ϕ}}{4 π ρ R} \frac{\partial b_{θ}}{\partial ϕ} - 2 \frac{B_{ϕ} \cos θ}{4 π ρ R} b_{ϕ},

5-

\frac{\partial u_{ϕ}}{\partial t} + Ω_{1} \frac{\partial u_{ϕ}}{\partial ϕ} + \frac{u_{θ}}{\sin θ} \frac{\partial [Ω \sin^{2} θ]}{\partial θ} = - \frac{g}{R \sin θ} \frac{\partial h}{\partial ϕ} + \frac{B_{ϕ}}{4 π ρ R} \frac{\partial b_{ϕ}}{\partial ϕ} + \frac{b_{θ}}{4 π ρ R \sin θ} \frac{\partial [B_{ϕ} \sin^{2} θ]}{\partial θ},

6-

\frac{\partial b_{θ}}{\partial t} + Ω_{1} \frac{\partial b_{θ}}{\partial ϕ} = \frac{B_{ϕ}}{R} \frac{\partial u_{θ}}{\partial ϕ},

7-

\frac{\partial}{\partial θ} (\sin θ b_{θ}) + \frac{\partial b_{ϕ}}{\partial ϕ} + \frac{B_{ϕ} \sin θ}{H} \frac{\partial h}{\partial ϕ} = 0,

8-

\frac{\partial h}{\partial t} + Ω_{1} \frac{\partial h}{\partial ϕ} + \frac{H}{R \sin θ} \frac{\partial}{\partial θ} (\sin θ u_{θ}) + \frac{H}{R \sin θ} \frac{\partial u_{ϕ}}{\partial ϕ} = 0,

9-

ϵ = \frac{Ω_{0}^{2} R^{2}}{g H},

10-

G = 1 / ϵ = g H / (R^{2} Ω_{0}^{2})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0303192

Formulas:

1-

u v b y

2-

u v b y

3-

E (B - V)

4-

E (B - V)

5-

u v b y

6-

u v b y

7-

u v b y

8-

E (b - y)

9-

U B V R I

10-

u v b y

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0808.0846

Formulas:

1-

M \approx 3 \cdot 10^{6} M_{⊙}

2-

[Z] < - 2

3-

\frac{d M_{g}}{d t} = - ψ (t) + \int_{M_{m i n}}^{M_{m a x}} ψ (t - τ_{_{M}}) φ (M, t - τ_{_{M}}) (M - M_{r}) 𝑑 M - {\dot{M}}_{g}^{o u t} + {\dot{M}}_{g}^{i n},

4-

M_{r} (M)

5-

φ (M, t)

6-

M_{m i n}

7-

M_{m a x}

8-

{\dot{M}}_{g}^{o u t}

9-

{\dot{M}}_{g}^{i n}

10-

ψ (t) = f ρ^{n} V,

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1705.01377

Formulas:

1-

τ = C_{A (B C)}^{2} - C_{A B}^{2} - C_{A C}^{2},

2-

C_{A (B C)}

3-

λ_{0} | 000 ⟩ + λ_{1} e^{i θ} | 100 ⟩ + λ_{2} | 101 ⟩ + λ_{3} | 110 ⟩

4-

+ λ_{4} | 111 ⟩,

5-

\sum_{i} λ_{i}^{2} = 1

6-

θ \in [0, π]

7-

{| 0 ⟩, | 1 ⟩}

8-

τ_{ψ} = 4 λ_{0}^{2} λ_{4}^{2} .

9-

O = 2 (σ_{x} \otimes σ_{x} \otimes σ_{x}),

10-

{\hat{a}}_{1} . \vec{σ} \otimes {\hat{a}}_{2} . \vec{σ} \otimes {\hat{a}}_{3} . \vec{σ} - {\hat{a}}_{1} . \vec{σ} \otimes {\hat{b}}_{2} . \vec{σ} \otimes {\hat{b}}_{3} . \vec{σ} -

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0306004

Formulas:

1-

δ_{i}^{'} + 3 (c_{s i}^{2} - w_{i}) \frac{a^{'}}{a} δ_{i}

2-

- (1 + w_{i}) (k v_{i} + \frac{h_{L}^{'}}{2}) - 3 w_{i} \frac{a^{'}}{a} Γ_{i},

3-

v_{i}^{'} + (1 - 3 c_{s i}^{2}) \frac{a^{'}}{a} v_{i}

4-

\frac{c_{s i}^{2}}{1 + w_{i}} k δ_{i} + \frac{w_{i}}{1 + w_{i}} k Γ_{i},

5-

h_{L}^{''} + \frac{a^{'}}{a} h_{L}^{'}

6-

- \sum_{i} (1 + 3 c_{s i}^{2}) 8 π G ρ_{i} a^{2} δ_{i} - 24 π G a^{2} \sum_{i} p_{i} Γ_{i},

7-

c_{s i}^{2} = p_{i}^{'} / ρ_{i}^{'}

8-

w_{i} = p_{i} / ρ_{i}

9-

p_{c h} = - M^{4 (α + 1)} / ρ_{c h}^{α} .

10-

ρ_{b} = ρ_{b 0} a^{- 3},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0109462

Formulas:

1-

\frac{Δ α (θ)}{α} = [(2 + γ) (v_{0} / c)] \cos θ

2-

(F - A) / A

3-

A c c e p t

4-

A c c e p t = [1 - 0.537 \tan ϕ_{x}] [1 - 0.537 \tan ϕ_{y}] \cos^{3} ϕ

5-

[1 - 0.537 \tan ϕ_{x}] [1 - 0.537 \tan ϕ_{y}]

6-

c o s^{2} ϕ

7-

y = A + B \cos [15 (x - C)] + D \cos [30 (x - E)]

8-

- 3.35 \times 10^{- 3}

9-

- 3.78 \times 10^{- 3}

10-

- 2.77 \times 10^{- 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1708.03698

Formulas:

1-

\min_{𝐰} {∥ 𝐗𝐰 - 𝐲 ∥}_{2}^{2} + λ {∥ 𝐰 ∥}_{2}^{2}

2-

𝒘 = {(𝑿^{T} 𝑿 + λ 𝑰)}^{- 1} 𝑿^{T} 𝒚

3-

𝑿 = 𝑭 d i a g (\hat{𝒙}) 𝑭^{H}

4-

𝑿^{H} 𝑿 = 𝑭 d i a g ({\hat{𝒙}}^{*} ⊙ \hat{𝒙}) 𝑭^{H}

5-

\hat{𝒘} = \frac{\hat{𝒙} ⊙ \hat{𝒚}}{\hat{𝒙} ⊙ \hat{𝒙} + 𝝀}

6-

\hat{𝜶} = \frac{\hat{𝒚}}{\hat{𝒙} ⊙ \hat{𝒚} + 𝝀}

7-

\hat{𝜶} = \frac{\hat{𝒚}}{{\hat{𝒌}}^{𝒙 𝒙} + λ}

8-

\min_{𝒘, 𝒓} {∥ 𝑿 𝒘 - 𝒚 ∥}_{2}^{2} + λ_{1} {∥ 𝒘 ∥}_{2}^{2} + {∥ 𝑨 𝒓 - 𝒈 ∥}_{2}^{2} + λ_{2} {∥ 𝒓 ∥}_{2}^{2}

9-

𝐰 \in ℝ^{m \times n}

10-

𝐗 \in ℝ^{m n \times m n}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1611.05418

Formulas:

1-

\frac{\partial h}{\partial X}

2-

A_{0}, A_{1}, \dots, A_{L}

3-

a_{i} b_{i} f_{i}

4-

A_{1}, \dots, A_{L}

5-

\min (z_{v}, b_{v})

6-

a_{i} b_{i} f_{i}

7-

f (X) \prod_{e \in P} w_{e}

8-

ϕ^{𝒩} (X) = v (X) \sum_{P \in 𝒫} \prod_{e \in P} w_{e},

9-

a (X) \frac{f_{0}}{\sum_{e} f (e)}

10-

e = (n_{1}, n_{2})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0512473

Formulas:

1-

H = H_{d} + H_{L} + H_{R} + H_{d m} + H_{d p s} + H_{s p}

2-

- \sum_{σ, n = 1}^{N} ε_{d}^{σ} C_{n, σ}^{†} C_{n, σ} - \sum_{σ, n = 1}^{N - 1} t_{d}^{σ} (C_{n, σ}^{†} C_{n + 1, σ} + C_{n + 1, σ}^{†} C_{n, σ}),

3-

- \sum_{σ, n \leq 0} ε_{m L}^{σ} C_{n, σ}^{†} C_{n, σ} - \sum_{σ, n \leq 0} t_{m L}^{σ} (C_{n, σ}^{†} C_{n + 1, σ} + C_{n + 1, σ}^{†} C_{n, σ}),

4-

- \sum_{σ, n \geq N + 1} ε_{m R}^{σ} C_{n, σ}^{†} C_{n, σ} - \sum_{σ, n \geq N + 1} t_{m R}^{σ} (C_{n, σ}^{†} C_{n + 1, σ} + C_{n + 1, σ}^{†} C_{n, σ}),

5-

- \sum_{σ} t_{d m} (C_{0, σ}^{†} C_{1, σ} + C_{1, σ}^{†} C_{0, σ}) - \sum_{σ} t_{d m} (C_{N, σ}^{†} C_{N + 1, σ} + C_{N + 1, σ}^{†} C_{N, σ}),

6-

H_{d p s}

7-

- \sum_{σ, n = 1}^{N} Σ_{n}^{σ} (E) C_{n, σ}^{†} C_{n, σ},

8-

- \sum_{σ, n = 1}^{N} t_{d}^{s o} C_{n, σ}^{†} C_{n, \bar{σ}} - \sum_{σ, n = 1}^{N - 1} t_{d}^{s} (C_{n, σ}^{†} C_{n + 1, \bar{σ}} + C_{n + 1, \bar{σ}}^{†} C_{n, σ}) .

9-

= 1, \dots, N

10-

4 t_{m X}^{σ}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1704.05188

Formulas:

1-

{𝐯_{𝐢}}_{i = 1}^{n}

2-

{v_{i}^{j}}_{j = 1}^{c}

3-

v^{j} = \max (v_{1}^{j}, v_{2}^{j}, \dots, v_{n}^{j}) .

4-

G = (V, E)

5-

E = {e (v_{i}, v_{j})}

6-

G = (V, E)

7-

v_{\max} = \arg \max_{i} d_{i}

8-

d_{i} = \sum_{j \in V} e (v_{i}, v_{j})

9-

V_{neighbor} = {v | e (v, v_{\max}) = 1}

10-

V^{'} = V^{'} \cup {v_{\max}}

Formulas (html):

<math><msubsup id="S3.SS1.SSS1.p2.1.m1.1.1" xref="S3.SS1.SSS1.p2.1.m1.1.1.cmml"><mrow id="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.1.2.cmml"><mo stretchy="false" id="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.2" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.1.2.cmml">{</mo><msub id="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.1" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.1.2" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.1.2.cmml">𝐯</mi><mi id="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.1.3" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.1.3.cmml">𝐢</mi></msub><mo stretchy="false" id="S3.SS1.SSS1.p2.1.m1.1.1.1.1.1.3" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.SSS1.p2.1.m1.1.1.1.3" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.3.cmml"><mi id="S3.SS1.SSS1.p2.1.m1.1.1.1.3.2" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.3.2.cmml">i</mi><mo id="S3.SS1.SSS1.p2.1.m1.1.1.1.3.1" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.3.1.cmml">=</mo><mn id="S3.SS1.SSS1.p2.1.m1.1.1.1.3.3" xref="S3.SS1.SSS1.p2.1.m1.1.1.1.3.3.cmml">1</mn></mrow><mi id="S3.SS1.SSS1.p2.1.m1.1.1.3" xref="S3.SS1.SSS1.p2.1.m1.1.1.3.cmml">n</mi></msubsup></math>, <math><msubsup id="S3.SS1.SSS1.p2.3.m3.1.1" xref="S3.SS1.SSS1.p2.3.m3.1.1.cmml"><mrow id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.2.cmml"><mo stretchy="false" id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.2" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.2.cmml">{</mo><msubsup id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.2.2" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.2.2.cmml">v</mi><mi id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.2.3" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.3" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.1.3.cmml">j</mi></msubsup><mo stretchy="false" id="S3.SS1.SSS1.p2.3.m3.1.1.1.1.1.3" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.1.2.cmml">}</mo></mrow><mrow id="S3.SS1.SSS1.p2.3.m3.1.1.1.3" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.3.cmml"><mi id="S3.SS1.SSS1.p2.3.m3.1.1.1.3.2" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.3.2.cmml">j</mi><mo id="S3.SS1.SSS1.p2.3.m3.1.1.1.3.1" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.3.1.cmml">=</mo><mn id="S3.SS1.SSS1.p2.3.m3.1.1.1.3.3" xref="S3.SS1.SSS1.p2.3.m3.1.1.1.3.3.cmml">1</mn></mrow><mi id="S3.SS1.SSS1.p2.3.m3.1.1.3" xref="S3.SS1.SSS1.p2.3.m3.1.1.3.cmml">c</mi></msubsup></math>, <math><mrow id="S3.Ex1.m1.3.3.1" xref="S3.Ex1.m1.3.3.1.1.cmml"><mrow id="S3.Ex1.m1.3.3.1.1" xref="S3.Ex1.m1.3.3.1.1.cmml"><msup id="S3.Ex1.m1.3.3.1.1.5" xref="S3.Ex1.m1.3.3.1.1.5.cmml"><mi id="S3.Ex1.m1.3.3.1.1.5.2" xref="S3.Ex1.m1.3.3.1.1.5.2.cmml">v</mi><mi id="S3.Ex1.m1.3.3.1.1.5.3" xref="S3.Ex1.m1.3.3.1.1.5.3.cmml">j</mi></msup><mo id="S3.Ex1.m1.3.3.1.1.4" xref="S3.Ex1.m1.3.3.1.1.4.cmml">=</mo><mrow id="S3.Ex1.m1.3.3.1.1.3.3" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml"><mi id="S3.Ex1.m1.1.1" xref="S3.Ex1.m1.1.1.cmml">max</mi><mo id="S3.Ex1.m1.3.3.1.1.3.3a" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml">⁡</mo><mrow id="S3.Ex1.m1.3.3.1.1.3.3.3" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml"><mo stretchy="false" id="S3.Ex1.m1.3.3.1.1.3.3.3.4" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml">(</mo><msubsup id="S3.Ex1.m1.3.3.1.1.1.1.1.1" xref="S3.Ex1.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex1.m1.3.3.1.1.1.1.1.1.2.2" xref="S3.Ex1.m1.3.3.1.1.1.1.1.1.2.2.cmml">v</mi><mn id="S3.Ex1.m1.3.3.1.1.1.1.1.1.2.3" xref="S3.Ex1.m1.3.3.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S3.Ex1.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex1.m1.3.3.1.1.1.1.1.1.3.cmml">j</mi></msubsup><mo id="S3.Ex1.m1.3.3.1.1.3.3.3.5" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml">,</mo><msubsup id="S3.Ex1.m1.3.3.1.1.2.2.2.2" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.cmml"><mi id="S3.Ex1.m1.3.3.1.1.2.2.2.2.2.2" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.2.2.cmml">v</mi><mn id="S3.Ex1.m1.3.3.1.1.2.2.2.2.2.3" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.2.3.cmml">2</mn><mi id="S3.Ex1.m1.3.3.1.1.2.2.2.2.3" xref="S3.Ex1.m1.3.3.1.1.2.2.2.2.3.cmml">j</mi></msubsup><mo id="S3.Ex1.m1.3.3.1.1.3.3.3.6" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml">,</mo><mi mathvariant="normal" id="S3.Ex1.m1.2.2" xref="S3.Ex1.m1.2.2.cmml">…</mi><mo id="S3.Ex1.m1.3.3.1.1.3.3.3.7" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml">,</mo><msubsup id="S3.Ex1.m1.3.3.1.1.3.3.3.3" xref="S3.Ex1.m1.3.3.1.1.3.3.3.3.cmml"><mi id="S3.Ex1.m1.3.3.1.1.3.3.3.3.2.2" xref="S3.Ex1.m1.3.3.1.1.3.3.3.3.2.2.cmml">v</mi><mi id="S3.Ex1.m1.3.3.1.1.3.3.3.3.2.3" xref="S3.Ex1.m1.3.3.1.1.3.3.3.3.2.3.cmml">n</mi><mi id="S3.Ex1.m1.3.3.1.1.3.3.3.3.3" xref="S3.Ex1.m1.3.3.1.1.3.3.3.3.3.cmml">j</mi></msubsup><mo stretchy="false" id="S3.Ex1.m1.3.3.1.1.3.3.3.8" xref="S3.Ex1.m1.3.3.1.1.3.4.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex1.m1.3.3.1.2" xref="S3.Ex1.m1.3.3.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S3.SS1.SSS2.p2.1.m1.2.3" xref="S3.SS1.SSS2.p2.1.m1.2.3.cmml"><mi id="S3.SS1.SSS2.p2.1.m1.2.3.2" xref="S3.SS1.SSS2.p2.1.m1.2.3.2.cmml">G</mi><mo id="S3.SS1.SSS2.p2.1.m1.2.3.1" xref="S3.SS1.SSS2.p2.1.m1.2.3.1.cmml">=</mo><mrow id="S3.SS1.SSS2.p2.1.m1.2.3.3.2" xref="S3.SS1.SSS2.p2.1.m1.2.3.3.1.cmml"><mo stretchy="false" id="S3.SS1.SSS2.p2.1.m1.2.3.3.2.1" xref="S3.SS1.SSS2.p2.1.m1.2.3.3.1.cmml">(</mo><mi id="S3.SS1.SSS2.p2.1.m1.1.1" xref="S3.SS1.SSS2.p2.1.m1.1.1.cmml">V</mi><mo id="S3.SS1.SSS2.p2.1.m1.2.3.3.2.2" xref="S3.SS1.SSS2.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS1.SSS2.p2.1.m1.2.2" xref="S3.SS1.SSS2.p2.1.m1.2.2.cmml">E</mi><mo stretchy="false" id="S3.SS1.SSS2.p2.1.m1.2.3.3.2.3" xref="S3.SS1.SSS2.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S3.SS1.SSS2.p2.4.m4.1.1" xref="S3.SS1.SSS2.p2.4.m4.1.1.cmml"><mi id="S3.SS1.SSS2.p2.4.m4.1.1.3" xref="S3.SS1.SSS2.p2.4.m4.1.1.3.cmml">E</mi><mo id="S3.SS1.SSS2.p2.4.m4.1.1.2" xref="S3.SS1.SSS2.p2.4.m4.1.1.2.cmml">=</mo><mrow id="S3.SS1.SSS2.p2.4.m4.1.1.1.1" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.2.cmml"><mo stretchy="false" id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.2" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.2.cmml">{</mo><mrow id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.4" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.4.cmml">e</mi><mo id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.3" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.3.cmml"><mo stretchy="false" id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.3" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.3.cmml">(</mo><msub id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.1" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.1.cmml"><mi id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.1.2" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.1.2.cmml">v</mi><mi id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.1.3" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.4" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.3.cmml">,</mo><msub id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.2" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.2.cmml"><mi id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.2.2" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.2.2.cmml">v</mi><mi id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.2.3" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.2.5" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S3.SS1.SSS2.p2.4.m4.1.1.1.1.3" xref="S3.SS1.SSS2.p2.4.m4.1.1.1.2.cmml">}</mo></mrow></mrow></math>, <math><mrow id="alg1.l1.m1.2.3" xref="alg1.l1.m1.2.3.cmml"><mi id="alg1.l1.m1.2.3.2" xref="alg1.l1.m1.2.3.2.cmml">G</mi><mo id="alg1.l1.m1.2.3.1" xref="alg1.l1.m1.2.3.1.cmml">=</mo><mrow id="alg1.l1.m1.2.3.3.2" xref="alg1.l1.m1.2.3.3.1.cmml"><mo stretchy="false" id="alg1.l1.m1.2.3.3.2.1" xref="alg1.l1.m1.2.3.3.1.cmml">(</mo><mi id="alg1.l1.m1.1.1" xref="alg1.l1.m1.1.1.cmml">V</mi><mo id="alg1.l1.m1.2.3.3.2.2" xref="alg1.l1.m1.2.3.3.1.cmml">,</mo><mi id="alg1.l1.m1.2.2" xref="alg1.l1.m1.2.2.cmml">E</mi><mo stretchy="false" id="alg1.l1.m1.2.3.3.2.3" xref="alg1.l1.m1.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="alg1.l4.m1.1.1" xref="alg1.l4.m1.1.1.cmml"><msub id="alg1.l4.m1.1.1.2" xref="alg1.l4.m1.1.1.2.cmml"><mi id="alg1.l4.m1.1.1.2.2" xref="alg1.l4.m1.1.1.2.2.cmml">v</mi><mi id="alg1.l4.m1.1.1.2.3" xref="alg1.l4.m1.1.1.2.3.cmml">max</mi></msub><mo id="alg1.l4.m1.1.1.1" xref="alg1.l4.m1.1.1.1.cmml">=</mo><mrow id="alg1.l4.m1.1.1.3" xref="alg1.l4.m1.1.1.3.cmml"><mi id="alg1.l4.m1.1.1.3.1" xref="alg1.l4.m1.1.1.3.1.cmml">arg</mi><mo id="alg1.l4.m1.1.1.3a" xref="alg1.l4.m1.1.1.3.cmml">⁡</mo><mrow id="alg1.l4.m1.1.1.3.2" xref="alg1.l4.m1.1.1.3.2.cmml"><msub id="alg1.l4.m1.1.1.3.2.1" xref="alg1.l4.m1.1.1.3.2.1.cmml"><mi id="alg1.l4.m1.1.1.3.2.1.2" xref="alg1.l4.m1.1.1.3.2.1.2.cmml">max</mi><mi id="alg1.l4.m1.1.1.3.2.1.3" xref="alg1.l4.m1.1.1.3.2.1.3.cmml">i</mi></msub><mo id="alg1.l4.m1.1.1.3.2a" xref="alg1.l4.m1.1.1.3.2.cmml">⁡</mo><msub id="alg1.l4.m1.1.1.3.2.2" xref="alg1.l4.m1.1.1.3.2.2.cmml"><mi id="alg1.l4.m1.1.1.3.2.2.2" xref="alg1.l4.m1.1.1.3.2.2.2.cmml">d</mi><mi id="alg1.l4.m1.1.1.3.2.2.3" xref="alg1.l4.m1.1.1.3.2.2.3.cmml">i</mi></msub></mrow></mrow></mrow></math>, <math><mrow id="alg1.l4.m2.2.2" xref="alg1.l4.m2.2.2.cmml"><msub id="alg1.l4.m2.2.2.4" xref="alg1.l4.m2.2.2.4.cmml"><mi id="alg1.l4.m2.2.2.4.2" xref="alg1.l4.m2.2.2.4.2.cmml">d</mi><mi id="alg1.l4.m2.2.2.4.3" xref="alg1.l4.m2.2.2.4.3.cmml">i</mi></msub><mo id="alg1.l4.m2.2.2.3" xref="alg1.l4.m2.2.2.3.cmml">=</mo><mrow id="alg1.l4.m2.2.2.2" xref="alg1.l4.m2.2.2.2.cmml"><msub id="alg1.l4.m2.2.2.2.3" xref="alg1.l4.m2.2.2.2.3.cmml"><mo largeop="true" symmetric="true" id="alg1.l4.m2.2.2.2.3.2" xref="alg1.l4.m2.2.2.2.3.2.cmml">∑</mo><mrow id="alg1.l4.m2.2.2.2.3.3" xref="alg1.l4.m2.2.2.2.3.3.cmml"><mi id="alg1.l4.m2.2.2.2.3.3.2" xref="alg1.l4.m2.2.2.2.3.3.2.cmml">j</mi><mo id="alg1.l4.m2.2.2.2.3.3.1" xref="alg1.l4.m2.2.2.2.3.3.1.cmml">∈</mo><mi id="alg1.l4.m2.2.2.2.3.3.3" xref="alg1.l4.m2.2.2.2.3.3.3.cmml">V</mi></mrow></msub><mrow id="alg1.l4.m2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.cmml"><mi id="alg1.l4.m2.2.2.2.2.4" xref="alg1.l4.m2.2.2.2.2.4.cmml">e</mi><mo id="alg1.l4.m2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.2.3.cmml">⁢</mo><mrow id="alg1.l4.m2.2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.3.cmml"><mo stretchy="false" id="alg1.l4.m2.2.2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.2.2.3.cmml">(</mo><msub id="alg1.l4.m2.1.1.1.1.1.1.1" xref="alg1.l4.m2.1.1.1.1.1.1.1.cmml"><mi id="alg1.l4.m2.1.1.1.1.1.1.1.2" xref="alg1.l4.m2.1.1.1.1.1.1.1.2.cmml">v</mi><mi id="alg1.l4.m2.1.1.1.1.1.1.1.3" xref="alg1.l4.m2.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="alg1.l4.m2.2.2.2.2.2.2.4" xref="alg1.l4.m2.2.2.2.2.2.3.cmml">,</mo><msub id="alg1.l4.m2.2.2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.2.2.cmml"><mi id="alg1.l4.m2.2.2.2.2.2.2.2.2" xref="alg1.l4.m2.2.2.2.2.2.2.2.2.cmml">v</mi><mi id="alg1.l4.m2.2.2.2.2.2.2.2.3" xref="alg1.l4.m2.2.2.2.2.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" id="alg1.l4.m2.2.2.2.2.2.2.5" xref="alg1.l4.m2.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="alg1.l5.m1.3.3" xref="alg1.l5.m1.3.3.cmml"><msub id="alg1.l5.m1.3.3.3" xref="alg1.l5.m1.3.3.3.cmml"><mi id="alg1.l5.m1.3.3.3.2" xref="alg1.l5.m1.3.3.3.2.cmml">V</mi><mtext id="alg1.l5.m1.3.3.3.3" xref="alg1.l5.m1.3.3.3.3a.cmml">neighbor</mtext></msub><mo id="alg1.l5.m1.3.3.2" xref="alg1.l5.m1.3.3.2.cmml">=</mo><mrow id="alg1.l5.m1.3.3.1.1" xref="alg1.l5.m1.3.3.1.2.cmml"><mo stretchy="false" id="alg1.l5.m1.3.3.1.1.2" xref="alg1.l5.m1.3.3.1.2.1.cmml">{</mo><mi id="alg1.l5.m1.2.2" xref="alg1.l5.m1.2.2.cmml">v</mi><mo stretchy="false" id="alg1.l5.m1.3.3.1.1.3" xref="alg1.l5.m1.3.3.1.2.1.cmml">|</mo><mrow id="alg1.l5.m1.3.3.1.1.1" xref="alg1.l5.m1.3.3.1.1.1.cmml"><mrow id="alg1.l5.m1.3.3.1.1.1.1" xref="alg1.l5.m1.3.3.1.1.1.1.cmml"><mi id="alg1.l5.m1.3.3.1.1.1.1.3" xref="alg1.l5.m1.3.3.1.1.1.1.3.cmml">e</mi><mo id="alg1.l5.m1.3.3.1.1.1.1.2" xref="alg1.l5.m1.3.3.1.1.1.1.2.cmml">⁢</mo><mrow id="alg1.l5.m1.3.3.1.1.1.1.1.1" xref="alg1.l5.m1.3.3.1.1.1.1.1.2.cmml"><mo stretchy="false" id="alg1.l5.m1.3.3.1.1.1.1.1.1.2" xref="alg1.l5.m1.3.3.1.1.1.1.1.2.cmml">(</mo><mi id="alg1.l5.m1.1.1" xref="alg1.l5.m1.1.1.cmml">v</mi><mo id="alg1.l5.m1.3.3.1.1.1.1.1.1.3" xref="alg1.l5.m1.3.3.1.1.1.1.1.2.cmml">,</mo><msub id="alg1.l5.m1.3.3.1.1.1.1.1.1.1" xref="alg1.l5.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="alg1.l5.m1.3.3.1.1.1.1.1.1.1.2" xref="alg1.l5.m1.3.3.1.1.1.1.1.1.1.2.cmml">v</mi><mi id="alg1.l5.m1.3.3.1.1.1.1.1.1.1.3" xref="alg1.l5.m1.3.3.1.1.1.1.1.1.1.3.cmml">max</mi></msub><mo stretchy="false" id="alg1.l5.m1.3.3.1.1.1.1.1.1.4" xref="alg1.l5.m1.3.3.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="alg1.l5.m1.3.3.1.1.1.2" xref="alg1.l5.m1.3.3.1.1.1.2.cmml">=</mo><mn id="alg1.l5.m1.3.3.1.1.1.3" xref="alg1.l5.m1.3.3.1.1.1.3.cmml">1</mn></mrow><mo stretchy="false" id="alg1.l5.m1.3.3.1.1.4" xref="alg1.l5.m1.3.3.1.2.1.cmml">}</mo></mrow></mrow></math>, <math><mrow id="alg1.l6.m1.1.1" xref="alg1.l6.m1.1.1.cmml"><msup id="alg1.l6.m1.1.1.3" xref="alg1.l6.m1.1.1.3.cmml"><mi id="alg1.l6.m1.1.1.3.2" xref="alg1.l6.m1.1.1.3.2.cmml">V</mi><mo id="alg1.l6.m1.1.1.3.3" xref="alg1.l6.m1.1.1.3.3.cmml">′</mo></msup><mo id="alg1.l6.m1.1.1.2" xref="alg1.l6.m1.1.1.2.cmml">=</mo><mrow id="alg1.l6.m1.1.1.1" xref="alg1.l6.m1.1.1.1.cmml"><msup id="alg1.l6.m1.1.1.1.3" xref="alg1.l6.m1.1.1.1.3.cmml"><mi id="alg1.l6.m1.1.1.1.3.2" xref="alg1.l6.m1.1.1.1.3.2.cmml">V</mi><mo id="alg1.l6.m1.1.1.1.3.3" xref="alg1.l6.m1.1.1.1.3.3.cmml">′</mo></msup><mo id="alg1.l6.m1.1.1.1.2" xref="alg1.l6.m1.1.1.1.2.cmml">∪</mo><mrow id="alg1.l6.m1.1.1.1.1.1" xref="alg1.l6.m1.1.1.1.1.2.cmml"><mo stretchy="false" id="alg1.l6.m1.1.1.1.1.1.2" xref="alg1.l6.m1.1.1.1.1.2.cmml">{</mo><msub id="alg1.l6.m1.1.1.1.1.1.1" xref="alg1.l6.m1.1.1.1.1.1.1.cmml"><mi id="alg1.l6.m1.1.1.1.1.1.1.2" xref="alg1.l6.m1.1.1.1.1.1.1.2.cmml">v</mi><mi id="alg1.l6.m1.1.1.1.1.1.1.3" xref="alg1.l6.m1.1.1.1.1.1.1.3.cmml">max</mi></msub><mo stretchy="false" id="alg1.l6.m1.1.1.1.1.1.3" xref="alg1.l6.m1.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow></math>

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1308.2024

Formulas:

1-

(F - F_{m i n}) / (F_{m a x} - F_{m i n})

2-

F_{m i n}

3-

F_{m a x}

4-

M p c^{- 1}

5-

d = \frac{{(z + 1)}^{2} - 1}{{(z + 1)}^{2} + 1} (\frac{c}{H_{0}}),

6-

i < r \leq i + 1

7-

i = 0, 1, 2, \dots

8-

Σ_{H} (R) = \frac{M_{d}}{2 π h^{2}} \exp (\frac{- R}{h}) .

9-

Σ_{f} (R) = α \frac{M_{d}}{2 π h^{2}} \exp (\frac{- β R}{h}),

10-

χ^{2} = \sum_{i = 1}^{k} \frac{{(Q_{i} - E_{i})}^{2}}{E_{i}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Run 10695567 (Agent093)