Run 16446909

Paper: https://arxiv.org/abs/1901.05375

Formulas:

1-

F F M_{1}

2-

F F M_{2}

3-

i + 1^{t h}

4-

i + 1^{t h}

5-

F F M_{1}

6-

F F M_{2}

7-

F F M_{1}

8-

F F M_{2}

9-

F F M_{1}

10-

F F M_{2}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1104.3421

Formulas:

1-

0, 1, \dots, 20

2-

{(2 s k)}^{s k}

3-

M (x) = M^{'} (x)

4-

{a_{1}, a_{2}, \dots}

5-

2^{n + 1} - 1

6-

2^{n + 1} - 1

7-

n = 0, \dots, 20

8-

s = 1, 2, 5, 10, 21, 42, 85, 170, 341, 682, 1365, 2730, 10 922, 21 845, 43 690, 87 381, 174 762,

9-

349 525, 699 050, 1 398 101, 2 796 202

10-

m o d_{2} (0 + n)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0803.1464

Formulas:

1-

< 1 \times 10^{- 6}

2-

2 e^{2} / h

3-

(2 e^{2} / h)

4-

x_{t h} = 1.6

5-

V_{s d} = 2.0

6-

(2 e^{2} / h)

7-

V_{s d} = 2.0

8-

S_{x} (ω) = \frac{4 ω_{c}^{3} x_{t h}^{2}}{Q} \frac{1}{{(ω_{c}^{2} - ω^{2})}^{2} + {(ω_{c} ω / Q)}^{2}} + S_{x_{n}}

9-

ω_{c} = 2 π ν_{c}

10-

S_{i} (ω) = \frac{4 ω_{c}^{3} i_{t h}^{2}}{Q} \frac{1}{{(ω_{c}^{2} - ω^{2})}^{2} + {(ω_{c} ω / Q)}^{2}} + S_{i_{n}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1402.4152

Formulas:

1-

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

2-

ρ \frac{\partial 𝐯}{\partial t} + ρ (𝐯 \cdot \nabla) 𝐯 = - \nabla p + μ \nabla^{2} 𝐯 + (μ_{b} + \frac{1}{3} μ) \nabla (\nabla \cdot 𝐯),

3-

p = c_{0}^{2} ρ,

4-

= 𝐯_{0} + ε {\tilde{𝐯}}_{1} + ε^{2} {\tilde{𝐯}}_{2} + O (ε^{3}) + \dots,

5-

= p_{0} + ε {\tilde{p}}_{1} + ε^{2} {\tilde{p}}_{2} + O (ε^{3}) + \dots,

6-

= ρ_{0} + ε {\tilde{ρ}}_{1} + ε^{2} {\tilde{ρ}}_{2} + O (ε^{3}) + \dots,

7-

= ϵ {\tilde{𝐯}}_{1},

8-

= ϵ {\tilde{p}}_{1},

9-

= ϵ {\tilde{ρ}}_{1},

10-

= ϵ^{2} {\tilde{𝐯}}_{2},

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0902.0165

Formulas:

1-

n = 1, \dots, N

2-

m {\ddot{x}}_{n} = - U^{'} (x_{n}) + k (x_{n + 1} + x_{n - 1} - 2 x_{n}) - α {\dot{x}}_{n} + ξ_{n} + F_{n} (t),

3-

F_{n} (t)

4-

⟨ ξ_{n} (t) ξ_{m} (t^{'}) ⟩ = 2 D δ_{n m} δ (t - t^{'})

5-

k_{B} T = D / α

6-

U (x) = a x^{2} / 2 + b x^{4} / 4

7-

ω_{ph} \in (\sqrt{U^{''} (0) / m}, \sqrt{[U^{''} (0) + 4 k] / m})

8-

ω_{\max} = \sqrt{(a + 4 k) / m}

9-

F_{n} (t) = {(- 1)}^{n} F_{0} \sin (Ω t) .

10-

x_{N + 1} = x_{1}

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1611.05959

Formulas:

1-

\frac{1}{2 ⌈ w_{M} / w_{m} ⌉}

2-

w_{M} = \max {w}

3-

w_{m} = \min {w}

4-

N = (1, 2, \dots, n)

5-

[x_{i} - \frac{w_{i}}{2}, x_{i} + \frac{w_{i}}{2}]

6-

f (x) = 0

7-

[\frac{w_{i}}{2}, 1 - \frac{w_{i}}{2}]

8-

(x_{1}, x_{2}, \dots, x_{n})

9-

(x_{1}, \dots, x_{i - 1}, x_{i + 1} \dots, x_{n})

10-

c (x, \vec{x})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/2002.00156

Formulas:

1-

F d \bar{3} m

2-

\vec{a} = (l / 2) (0, 1, 1), \vec{b} = (l / 2) (1, 0, 1)

3-

\vec{c} = (l / 2) (1, 1, 0)

4-

32 e (x; x; x)

5-

F \bar{4} 3 m

6-

4 a (0.5; 0; 0)

7-

16 e (0.92954; 0.57046; 0.42954)

8-

16 e (0.32066; 0.17934; 0.32066)

9-

α^{2} F (ω)

10-

λ_{q v} = \frac{γ_{q v}}{π h N (E_{F}) ω_{q v}^{2}},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0701173

Formulas:

1-

w (μ, μ^{'})

2-

P_{u} (μ_{0}, μ)

3-

P_{d} (μ_{0}, μ)

4-

γ (u + μ) \frac{\partial g}{\partial z} = \int \int [- W (μ^{'}, μ, ϕ^{'}, ϕ) g (μ, ϕ) + W (μ, μ^{'}, ϕ, ϕ^{'}) g (μ^{'}, ϕ^{'})] 𝑑 μ^{'} 𝑑 ϕ^{'} + ω \frac{\partial g}{\partial \tilde{ϕ}},

5-

ω = e B / E

6-

W (μ, μ^{'}, ϕ, ϕ^{'})

7-

Δ \equiv ϕ - ϕ^{'}

8-

ω (\partial g / \partial \tilde{ϕ})

9-

γ (u + μ) \frac{\partial g}{\partial z} = \int [- w (μ^{'}, μ) g (μ) + w (μ, μ^{'}) g (μ^{'})] 𝑑 μ^{'},

10-

w (μ, μ^{'})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1006.5591

Formulas:

1-

Z_{k_{F}}^{Na} = 0.58 (7)

2-

n (p) = θ (p_{F} - p)

3-

ζ = n (p_{F}^{-}) - n (p_{F}^{+}) = 1

4-

ϕ_{ν 𝐤} (𝐫) = \sum_{𝐆} {\tilde{ϕ}}_{ν 𝐤}^{𝐆} e^{i (𝐤 + 𝐆) 𝐫}

5-

ζ = {| {\tilde{ϕ}}_{ν = 1, 𝐤_{F}}^{𝐆 = 0} |}^{2} < 1

6-

ζ_{DFT}^{Na} = 0.98 (1)

7-

Σ_{ν, ν} (k, ω)

8-

Z_{𝐤_{F}} = {(1 - {\partial Σ_{1, 1} (𝐤_{F}, ω) / \partial ω |}_{ω = ϵ_{F}})}^{- 1}

9-

Σ = Σ^{e-e} + Σ^{e-p}

10-

ζ_{HEG} = Z_{k_{F}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0804.4872

Formulas:

1-

2 (2 I + 1)

2-

h_{se} = a (r) 𝐬_{1} \cdot 𝐬_{2}

3-

ℋ_{se} = ω_{se} 𝐬_{1} \cdot 𝐬_{2}

4-

ϕ_{se} = ω_{se} τ_{c}

5-

𝐁 = ω_{se} ⟨ 𝐬_{1} ⟩

6-

γ_{se} = n v σ_{se}

7-

ω^{2} / γ_{se} ≪ γ_{se}

8-

d ρ / d t = - i [H, ρ] - k [s_{x}, [s_{x}, ρ]]

9-

H = ω s_{z}

10-

- λ + i Ω

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0503280

Formulas:

1-

\dot{P} \sim 10^{- 11}

2-

0 \overset{''}{.} 058

3-

0 \overset{''}{.} 30

4-

1' \times 1'

5-

0 \overset{''}{.} 32

6-

0 \overset{''}{.} 023

7-

0 \overset{''}{.} 20

8-

0 \overset{''}{.} 22

9-

0 \overset{''}{.} 38

10-

0 \overset{''}{.} 9

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/physics/9807037

Formulas:

1-

𝑭 = \nabla (𝝁 \cdot 𝑩) = - {𝑚𝑔}_{F} μ_{B} \nabla B,

2-

ϕ = 2 π ν

3-

z = z_{0} + a ν

4-

(B_{r}, B_{ϕ}, B_{z})

5-

\frac{μ_{0} I 4 π R}{a^{2}} \sum_{n = 1}^{\infty} n \sin [n (ϕ - \frac{2 π}{a} (z - z_{0}))] K_{n}^{'} (ρ_{0}) I_{n}^{'} (ρ),

6-

\frac{μ_{0} I 2 R}{a r} \sum_{n = 1}^{\infty} n \cos [n (ϕ - \frac{2 π}{a} (z - z_{0}))] K_{n}^{'} (ρ_{0}) I_{n} (ρ),

7-

\frac{μ_{0} I}{a} - \frac{μ_{0} I 4 π R}{a^{2}} \sum_{n = 1}^{\infty} n \cos [n (ϕ - \frac{2 π}{a} (z - z_{0}))] K_{n}^{'} (ρ_{0}) I_{n} (ρ) .

8-

ρ_{0} \equiv 2 π n R / a

9-

ρ \equiv 2 π n r / a

10-

K_{n} (ρ)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1802.02351

Formulas:

1-

G (V, E, L)

2-

L : V \to ℝ^{2}

3-

G_{f} = f (G_{1}, G_{2})

4-

\forall p \in π_{i}, \exists \hat{p} \in π_{f} s.t. d (p, \hat{p}) \leq θ, i \in {1, 2},

5-

d (\cdot, \cdot)

6-

d (p_{0}, p_{1}) = \min_{i = 0, 1} \max_{u \in p_{i}} v (u, p_{1 - i})

7-

v (u, p)

8-

d (p_{0}, p_{1})

9-

\arg \min_{G_{f}} \sum_{p \in π_{f}} ℓ (p) .

10-

H (π_{1}, π_{2}, F)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0204087

Formulas:

1-

H = E (| w ⟩ ⟨ w | + | ψ ⟩ ⟨ ψ |)

2-

H = E (| w ⟩ ⟨ w | + | ψ ⟩ ⟨ ψ |) + ϵ (e^{i ϕ} | w ⟩ ⟨ ψ | + e^{- i ϕ} | ψ ⟩ ⟨ w |)

3-

| ψ ⟩ = x | w ⟩ + \sqrt{1 - x^{2}} | r ⟩

4-

x = ⟨ w | ψ ⟩ (\approx 1 / \sqrt{N})

5-

T = \frac{π}{2} \frac{1}{{[{(E x + ϵ \cos ϕ)}^{2} + (1 - x^{2}) ϵ^{2} \sin^{2} ϕ]}^{1 / 2}}

6-

T y p e 1 .

7-

H_{1} = E (| w ⟩ ⟨ w | + | ψ ⟩ ⟨ ψ |) + ϵ (e^{i ϕ} | w ⟩ ⟨ ψ | + e^{- i ϕ} | ψ ⟩ ⟨ w |)

8-

\frac{π}{2} \frac{1}{E (x c o s ϕ + 1)}

9-

\frac{π}{2 E} (1 + O (x \cos ϕ))

10-

T = \frac{π}{2 E} = O (1)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1112.3542

Formulas:

1-

(μ_{α} \cos δ, μ_{δ})

2-

| 𝐕_{𝐭} | = \frac{A | μ |}{π},

3-

| 𝐕_{𝐫} | = | 𝐕 | \cos λ,

4-

(α_{c p}, δ_{c p})

5-

\cos λ = \sin δ \sin δ_{c p} + \cos δ \cos δ_{c p} \cos (α_{c p} - α) .

6-

\begin{matrix} (\begin{matrix} μ_{∥} \\ μ_{⟂} \end{matrix}) = (\begin{matrix} \sin θ & \cos θ \\ - \cos θ & \sin θ \end{matrix}) (\begin{matrix} μ_{α} \cos δ \\ μ_{δ} \end{matrix}) \end{matrix}

7-

\tan θ = \frac{\sin (α_{c p} - α)}{\cos δ \tan δ_{c p} - \sin δ \cos (α_{c p} - α)} .

8-

(α_{c p}, δ_{c p})

9-

X^{2} = \sum_{j = 1}^{N} t_{⟂_{j}}^{2}

10-

t_{⟂} = μ_{⟂} / σ_{⟂}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1210.1963

Formulas:

1-

W^{1, n} (ℝ^{n})

2-

α_{n} = n ω_{n - 1}^{\frac{1}{n - 1}}

3-

sup_{u \in W^{1, n} (ℝ^{n}), \int_{ℝ^{n}} ({| \nabla u |}^{n} + {| u |}^{n}) 𝑑 x \leq 1} \int_{ℝ^{n}} (e^{α {| u |}^{\frac{n}{n - 1}}} - \sum_{k = 0}^{n - 2} \frac{α^{k} {| u |}^{\frac{n k}{n - 1}}}{k!}) 𝑑 x < \infty .

4-

W^{1, n} (ℝ^{n})

5-

u \in W^{1, n} (ℝ^{n})

6-

ϕ_{i} \in C_{0}^{\infty} (B_{R} (x_{i}))

7-

0 \leq ϕ_{i} \leq 1

8-

B_{R} (x_{i})

9-

B_{R / 2} (x_{i})

10-

\int_{ℝ^{n}} (e^{α {| ϕ_{i} u |}^{\frac{n}{n - 1}}} - \sum_{k = 0}^{n - 2} \frac{α^{k} {| ϕ_{i} u |}^{\frac{n k}{n - 1}}}{k!}) 𝑑 x \leq \int_{ℝ^{n}} {| \nabla (ϕ_{i} u) |}^{n} 𝑑 x

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/math/9807121

Formulas:

1-

A \in C^{n, n}

2-

v^{T} = {[w, 0]}^{T}

3-

v^{*} A v = w^{*} B w = 0

4-

P \in C^{n, n}

5-

A \in C^{n, n}

6-

E = D^{- 1} P^{- 1} A P D

7-

P = (p_{i, j})

8-

p_{n, n} = 1

9-

C = P^{- 1} A P

10-

p_{n, n} = 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1807.01021

Formulas:

1-

L_{k} (G)

2-

V (G), E (G), d i a m (G), Δ (G)

3-

v \in V (G)

4-

N [v] = {v} \cup N (v)

5-

V (G) ∖ D

6-

γ_{t} (G)

7-

| N (v) \cap S | \leq 1

8-

ρ^{0} (G)

9-

| N [v] \cap B | \leq 1

10-

| N [v] \cap B | \leq k

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1510.01345

Formulas:

1-

(+, -, -, -)

2-

8 π G = c = ℏ = 1

3-

e_{a}^{μ} = \partial x^{μ} / \partial ξ^{a}

4-

g_{μ ν} = e_{μ}^{a} e_{ν}^{b} η_{a b}

5-

Γ^{μ} = e_{a}^{μ} γ^{a}

6-

{Γ^{μ}, Γ^{ν}} = 2 g^{μ ν} .

7-

\partial_{μ} ψ \to D_{μ} ψ = \partial_{μ} ψ - Ω_{μ} ψ, \partial_{μ} \bar{ψ} \to D_{μ} \bar{ψ} = \partial_{μ} \bar{ψ} + \bar{ψ} Ω_{μ} .

8-

Ω_{μ} = - \frac{1}{4} g_{ρ σ} [Γ_{μ δ}^{ρ} - e_{b}^{ρ} (\partial_{μ} e_{δ}^{b})] Γ^{δ} Γ^{σ},

9-

L_{f} = \frac{i}{2} [\bar{ψ} Γ^{μ} D_{μ} ψ - (D_{μ} \bar{ψ}) Γ^{μ} ψ] - V (\bar{ψ} ψ) .

10-

\bar{ψ} = ψ^{†} γ^{0}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/1608.02510

Formulas:

1-

C_{2 n} \cdot C_{n + 1}

2-

T_{n} = \frac{2 (4 n - 3)!}{n! (3 n - 1)!} .

3-

f_{- 1} = 1 / 2

4-

f_{n} = (3 n + 2) F_{n}

5-

f_{n} = \frac{4 (3 n + 2)}{n + 1} \sum_{\begin{matrix} i ⩾ - 1, j ⩾ - 1 \\ i + j = n - 2 \end{matrix}} f (i) f (j) .

6-

C_{n} \cdot C_{n + 1}

7-

10 = C_{2} \cdot C_{3}

8-

C_{2 n} \cdot C_{n + 1}

9-

C_{2 n} \cdot C_{n + 1}

10-

C_{4} \cdot C_{3} = 70

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1808.05744

Formulas:

1-

H_{l} (\cdot)

2-

𝐠_{j} = \sum_{i} K_{i, j} 𝐟_{i},

3-

{\hat{𝐟}}_{j | i} = W_{i, j} 𝐟_{i}

4-

𝐠_{j} = \sum_{i} c_{i, j} {\hat{𝐟}}_{j | i} = \sum_{i} W_{i, j} c_{i, j} 𝐟_{i},

5-

c_{i, j} = \frac{\exp (b_{i, j})}{\sum_{k} \exp (b_{i, k})}

6-

b_{i, j} \leftarrow b_{i, j} + {\hat{𝐟}}_{j | i} \cdot squash (𝐠_{j})

7-

{\hat{𝐟}}_{j | i} \cdot squash (𝐠_{j}) = \frac{| 𝐠_{j} |}{1 + {| 𝐠_{j} |}^{2}} {\hat{𝐟}}_{j | i} \cdot 𝐠_{j}

8-

{\hat{𝐟}}_{j | i} \cdot 𝐠_{j}

9-

{\hat{𝐟}}_{j | i} \cdot 𝐠_{j}

10-

= \sum_{l} {\hat{𝐟}}_{j | i} \cdot c_{i, j} {\hat{𝐟}}_{j | l}

Formulas (html):

<math><mrow id="S2.p2.2.m2.1.2" xref="S2.p2.2.m2.1.2.cmml"><msub id="S2.p2.2.m2.1.2.2" xref="S2.p2.2.m2.1.2.2.cmml"><mi id="S2.p2.2.m2.1.2.2.2" xref="S2.p2.2.m2.1.2.2.2.cmml">H</mi><mi id="S2.p2.2.m2.1.2.2.3" xref="S2.p2.2.m2.1.2.2.3.cmml">l</mi></msub><mo id="S2.p2.2.m2.1.2.1" xref="S2.p2.2.m2.1.2.1.cmml">⁢</mo><mrow id="S2.p2.2.m2.1.2.3.2" xref="S2.p2.2.m2.1.2.cmml"><mo stretchy="false" id="S2.p2.2.m2.1.2.3.2.1" xref="S2.p2.2.m2.1.2.cmml">(</mo><mo id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml">⋅</mo><mo stretchy="false" id="S2.p2.2.m2.1.2.3.2.2" xref="S2.p2.2.m2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><msub id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mi id="S2.E1.m1.3.3.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.2.cmml">𝐠</mi><mi id="S2.E1.m1.3.3.1.1.2.3" xref="S2.E1.m1.3.3.1.1.2.3.cmml">j</mi></msub><mo id="S2.E1.m1.3.3.1.1.1" xref="S2.E1.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml"><munder id="S2.E1.m1.3.3.1.1.3.1" xref="S2.E1.m1.3.3.1.1.3.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S2.E1.m1.3.3.1.1.3.1.2" xref="S2.E1.m1.3.3.1.1.3.1.2.cmml">∑</mo><mi id="S2.E1.m1.3.3.1.1.3.1.3" xref="S2.E1.m1.3.3.1.1.3.1.3.cmml">i</mi></munder><mrow id="S2.E1.m1.3.3.1.1.3.2" xref="S2.E1.m1.3.3.1.1.3.2.cmml"><msub id="S2.E1.m1.3.3.1.1.3.2.2" xref="S2.E1.m1.3.3.1.1.3.2.2.cmml"><mi id="S2.E1.m1.3.3.1.1.3.2.2.2" xref="S2.E1.m1.3.3.1.1.3.2.2.2.cmml">K</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">i</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></msub><mo id="S2.E1.m1.3.3.1.1.3.2.1" xref="S2.E1.m1.3.3.1.1.3.2.1.cmml">⁢</mo><msub id="S2.E1.m1.3.3.1.1.3.2.3" xref="S2.E1.m1.3.3.1.1.3.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.3.2.3.2" xref="S2.E1.m1.3.3.1.1.3.2.3.2.cmml">𝐟</mi><mi id="S2.E1.m1.3.3.1.1.3.2.3.3" xref="S2.E1.m1.3.3.1.1.3.2.3.3.cmml">i</mi></msub></mrow></mrow></mrow><mo id="S2.E1.m1.3.3.1.2" xref="S2.E1.m1.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p1.6.m2.2.3" xref="S2.SS1.p1.6.m2.2.3.cmml"><msub id="S2.SS1.p1.6.m2.2.3.2" xref="S2.SS1.p1.6.m2.2.3.2.cmml"><mover accent="true" id="S2.SS1.p1.6.m2.2.3.2.2" xref="S2.SS1.p1.6.m2.2.3.2.2.cmml"><mi id="S2.SS1.p1.6.m2.2.3.2.2.2" xref="S2.SS1.p1.6.m2.2.3.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.SS1.p1.6.m2.2.3.2.2.1" xref="S2.SS1.p1.6.m2.2.3.2.2.1.cmml">^</mo></mover><mrow id="S2.SS1.p1.6.m2.2.3.2.3" xref="S2.SS1.p1.6.m2.2.3.2.3.cmml"><mi id="S2.SS1.p1.6.m2.2.3.2.3.2" xref="S2.SS1.p1.6.m2.2.3.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.SS1.p1.6.m2.2.3.2.3.1" xref="S2.SS1.p1.6.m2.2.3.2.3.1.cmml">|</mo><mi id="S2.SS1.p1.6.m2.2.3.2.3.3" xref="S2.SS1.p1.6.m2.2.3.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.SS1.p1.6.m2.2.3.1" xref="S2.SS1.p1.6.m2.2.3.1.cmml">=</mo><mrow id="S2.SS1.p1.6.m2.2.3.3" xref="S2.SS1.p1.6.m2.2.3.3.cmml"><msub id="S2.SS1.p1.6.m2.2.3.3.2" xref="S2.SS1.p1.6.m2.2.3.3.2.cmml"><mi id="S2.SS1.p1.6.m2.2.3.3.2.2" xref="S2.SS1.p1.6.m2.2.3.3.2.2.cmml">W</mi><mrow id="S2.SS1.p1.6.m2.2.2.2.4" xref="S2.SS1.p1.6.m2.2.2.2.3.cmml"><mi id="S2.SS1.p1.6.m2.1.1.1.1" xref="S2.SS1.p1.6.m2.1.1.1.1.cmml">i</mi><mo id="S2.SS1.p1.6.m2.2.2.2.4.1" xref="S2.SS1.p1.6.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS1.p1.6.m2.2.2.2.2" xref="S2.SS1.p1.6.m2.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S2.SS1.p1.6.m2.2.3.3.1" xref="S2.SS1.p1.6.m2.2.3.3.1.cmml">⁢</mo><msub id="S2.SS1.p1.6.m2.2.3.3.3" xref="S2.SS1.p1.6.m2.2.3.3.3.cmml"><mi id="S2.SS1.p1.6.m2.2.3.3.3.2" xref="S2.SS1.p1.6.m2.2.3.3.3.2.cmml">𝐟</mi><mi id="S2.SS1.p1.6.m2.2.3.3.3.3" xref="S2.SS1.p1.6.m2.2.3.3.3.3.cmml">i</mi></msub></mrow></mrow></math>, <math><mrow id="S2.E2.m1.7.7.1" xref="S2.E2.m1.7.7.1.1.cmml"><mrow id="S2.E2.m1.7.7.1.1" xref="S2.E2.m1.7.7.1.1.cmml"><msub id="S2.E2.m1.7.7.1.1.2" xref="S2.E2.m1.7.7.1.1.2.cmml"><mi id="S2.E2.m1.7.7.1.1.2.2" xref="S2.E2.m1.7.7.1.1.2.2.cmml">𝐠</mi><mi id="S2.E2.m1.7.7.1.1.2.3" xref="S2.E2.m1.7.7.1.1.2.3.cmml">j</mi></msub><mo id="S2.E2.m1.7.7.1.1.3" xref="S2.E2.m1.7.7.1.1.3.cmml">=</mo><mrow id="S2.E2.m1.7.7.1.1.4" xref="S2.E2.m1.7.7.1.1.4.cmml"><munder id="S2.E2.m1.7.7.1.1.4.1" xref="S2.E2.m1.7.7.1.1.4.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S2.E2.m1.7.7.1.1.4.1.2" xref="S2.E2.m1.7.7.1.1.4.1.2.cmml">∑</mo><mi id="S2.E2.m1.7.7.1.1.4.1.3" xref="S2.E2.m1.7.7.1.1.4.1.3.cmml">i</mi></munder><mrow id="S2.E2.m1.7.7.1.1.4.2" xref="S2.E2.m1.7.7.1.1.4.2.cmml"><msub id="S2.E2.m1.7.7.1.1.4.2.2" xref="S2.E2.m1.7.7.1.1.4.2.2.cmml"><mi id="S2.E2.m1.7.7.1.1.4.2.2.2" xref="S2.E2.m1.7.7.1.1.4.2.2.2.cmml">c</mi><mrow id="S2.E2.m1.2.2.2.4" xref="S2.E2.m1.2.2.2.3.cmml"><mi id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml">i</mi><mo id="S2.E2.m1.2.2.2.4.1" xref="S2.E2.m1.2.2.2.3.cmml">,</mo><mi id="S2.E2.m1.2.2.2.2" xref="S2.E2.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S2.E2.m1.7.7.1.1.4.2.1" xref="S2.E2.m1.7.7.1.1.4.2.1.cmml">⁢</mo><msub id="S2.E2.m1.7.7.1.1.4.2.3" xref="S2.E2.m1.7.7.1.1.4.2.3.cmml"><mover accent="true" id="S2.E2.m1.7.7.1.1.4.2.3.2" xref="S2.E2.m1.7.7.1.1.4.2.3.2.cmml"><mi id="S2.E2.m1.7.7.1.1.4.2.3.2.2" xref="S2.E2.m1.7.7.1.1.4.2.3.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.E2.m1.7.7.1.1.4.2.3.2.1" xref="S2.E2.m1.7.7.1.1.4.2.3.2.1.cmml">^</mo></mover><mrow id="S2.E2.m1.7.7.1.1.4.2.3.3" xref="S2.E2.m1.7.7.1.1.4.2.3.3.cmml"><mi id="S2.E2.m1.7.7.1.1.4.2.3.3.2" xref="S2.E2.m1.7.7.1.1.4.2.3.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E2.m1.7.7.1.1.4.2.3.3.1" xref="S2.E2.m1.7.7.1.1.4.2.3.3.1.cmml">|</mo><mi id="S2.E2.m1.7.7.1.1.4.2.3.3.3" xref="S2.E2.m1.7.7.1.1.4.2.3.3.3.cmml">i</mi></mrow></msub></mrow></mrow><mo id="S2.E2.m1.7.7.1.1.5" xref="S2.E2.m1.7.7.1.1.5.cmml">=</mo><mrow id="S2.E2.m1.7.7.1.1.6" xref="S2.E2.m1.7.7.1.1.6.cmml"><munder id="S2.E2.m1.7.7.1.1.6.1" xref="S2.E2.m1.7.7.1.1.6.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S2.E2.m1.7.7.1.1.6.1.2" xref="S2.E2.m1.7.7.1.1.6.1.2.cmml">∑</mo><mi id="S2.E2.m1.7.7.1.1.6.1.3" xref="S2.E2.m1.7.7.1.1.6.1.3.cmml">i</mi></munder><mrow id="S2.E2.m1.7.7.1.1.6.2" xref="S2.E2.m1.7.7.1.1.6.2.cmml"><msub id="S2.E2.m1.7.7.1.1.6.2.2" xref="S2.E2.m1.7.7.1.1.6.2.2.cmml"><mi id="S2.E2.m1.7.7.1.1.6.2.2.2" xref="S2.E2.m1.7.7.1.1.6.2.2.2.cmml">W</mi><mrow id="S2.E2.m1.4.4.2.4" xref="S2.E2.m1.4.4.2.3.cmml"><mi id="S2.E2.m1.3.3.1.1" xref="S2.E2.m1.3.3.1.1.cmml">i</mi><mo id="S2.E2.m1.4.4.2.4.1" xref="S2.E2.m1.4.4.2.3.cmml">,</mo><mi id="S2.E2.m1.4.4.2.2" xref="S2.E2.m1.4.4.2.2.cmml">j</mi></mrow></msub><mo id="S2.E2.m1.7.7.1.1.6.2.1" xref="S2.E2.m1.7.7.1.1.6.2.1.cmml">⁢</mo><msub id="S2.E2.m1.7.7.1.1.6.2.3" xref="S2.E2.m1.7.7.1.1.6.2.3.cmml"><mi id="S2.E2.m1.7.7.1.1.6.2.3.2" xref="S2.E2.m1.7.7.1.1.6.2.3.2.cmml">c</mi><mrow id="S2.E2.m1.6.6.2.4" xref="S2.E2.m1.6.6.2.3.cmml"><mi id="S2.E2.m1.5.5.1.1" xref="S2.E2.m1.5.5.1.1.cmml">i</mi><mo id="S2.E2.m1.6.6.2.4.1" xref="S2.E2.m1.6.6.2.3.cmml">,</mo><mi id="S2.E2.m1.6.6.2.2" xref="S2.E2.m1.6.6.2.2.cmml">j</mi></mrow></msub><mo id="S2.E2.m1.7.7.1.1.6.2.1a" xref="S2.E2.m1.7.7.1.1.6.2.1.cmml">⁢</mo><msub id="S2.E2.m1.7.7.1.1.6.2.4" xref="S2.E2.m1.7.7.1.1.6.2.4.cmml"><mi id="S2.E2.m1.7.7.1.1.6.2.4.2" xref="S2.E2.m1.7.7.1.1.6.2.4.2.cmml">𝐟</mi><mi id="S2.E2.m1.7.7.1.1.6.2.4.3" xref="S2.E2.m1.7.7.1.1.6.2.4.3.cmml">i</mi></msub></mrow></mrow></mrow><mo id="S2.E2.m1.7.7.1.2" xref="S2.E2.m1.7.7.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.E3.m1.8.9" xref="S2.E3.m1.8.9.cmml"><msub id="S2.E3.m1.8.9.2" xref="S2.E3.m1.8.9.2.cmml"><mi id="S2.E3.m1.8.9.2.2" xref="S2.E3.m1.8.9.2.2.cmml">c</mi><mrow id="S2.E3.m1.2.2.2.4" xref="S2.E3.m1.2.2.2.3.cmml"><mi id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml">i</mi><mo id="S2.E3.m1.2.2.2.4.1" xref="S2.E3.m1.2.2.2.3.cmml">,</mo><mi id="S2.E3.m1.2.2.2.2" xref="S2.E3.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S2.E3.m1.8.9.1" xref="S2.E3.m1.8.9.1.cmml">=</mo><mfrac id="S2.E3.m1.8.8" xref="S2.E3.m1.8.8.cmml"><mrow id="S2.E3.m1.5.5.3" xref="S2.E3.m1.5.5.3.cmml"><mi id="S2.E3.m1.5.5.3.5" xref="S2.E3.m1.5.5.3.5.cmml">exp</mi><mo id="S2.E3.m1.5.5.3.4" xref="S2.E3.m1.5.5.3.4.cmml">⁢</mo><mrow id="S2.E3.m1.5.5.3.3.1" xref="S2.E3.m1.5.5.3.3.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.5.5.3.3.1.2" xref="S2.E3.m1.5.5.3.3.1.1.cmml">(</mo><msub id="S2.E3.m1.5.5.3.3.1.1" xref="S2.E3.m1.5.5.3.3.1.1.cmml"><mi id="S2.E3.m1.5.5.3.3.1.1.2" xref="S2.E3.m1.5.5.3.3.1.1.2.cmml">b</mi><mrow id="S2.E3.m1.4.4.2.2.2.4" xref="S2.E3.m1.4.4.2.2.2.3.cmml"><mi id="S2.E3.m1.3.3.1.1.1.1" xref="S2.E3.m1.3.3.1.1.1.1.cmml">i</mi><mo id="S2.E3.m1.4.4.2.2.2.4.1" xref="S2.E3.m1.4.4.2.2.2.3.cmml">,</mo><mi id="S2.E3.m1.4.4.2.2.2.2" xref="S2.E3.m1.4.4.2.2.2.2.cmml">j</mi></mrow></msub><mo stretchy="false" id="S2.E3.m1.5.5.3.3.1.3" xref="S2.E3.m1.5.5.3.3.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.E3.m1.8.8.6" xref="S2.E3.m1.8.8.6.cmml"><msub id="S2.E3.m1.8.8.6.4" xref="S2.E3.m1.8.8.6.4.cmml"><mo largeop="true" symmetric="true" id="S2.E3.m1.8.8.6.4.2" xref="S2.E3.m1.8.8.6.4.2.cmml">∑</mo><mi id="S2.E3.m1.8.8.6.4.3" xref="S2.E3.m1.8.8.6.4.3.cmml">k</mi></msub><mrow id="S2.E3.m1.8.8.6.3" xref="S2.E3.m1.8.8.6.3.cmml"><mi id="S2.E3.m1.8.8.6.3.3" xref="S2.E3.m1.8.8.6.3.3.cmml">exp</mi><mo id="S2.E3.m1.8.8.6.3.2" xref="S2.E3.m1.8.8.6.3.2.cmml">⁢</mo><mrow id="S2.E3.m1.8.8.6.3.1.1" xref="S2.E3.m1.8.8.6.3.1.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.8.8.6.3.1.1.2" xref="S2.E3.m1.8.8.6.3.1.1.1.cmml">(</mo><msub id="S2.E3.m1.8.8.6.3.1.1.1" xref="S2.E3.m1.8.8.6.3.1.1.1.cmml"><mi id="S2.E3.m1.8.8.6.3.1.1.1.2" xref="S2.E3.m1.8.8.6.3.1.1.1.2.cmml">b</mi><mrow id="S2.E3.m1.7.7.5.2.2.4" xref="S2.E3.m1.7.7.5.2.2.3.cmml"><mi id="S2.E3.m1.6.6.4.1.1.1" xref="S2.E3.m1.6.6.4.1.1.1.cmml">i</mi><mo id="S2.E3.m1.7.7.5.2.2.4.1" xref="S2.E3.m1.7.7.5.2.2.3.cmml">,</mo><mi id="S2.E3.m1.7.7.5.2.2.2" xref="S2.E3.m1.7.7.5.2.2.2.cmml">k</mi></mrow></msub><mo stretchy="false" id="S2.E3.m1.8.8.6.3.1.1.3" xref="S2.E3.m1.8.8.6.3.1.1.1.cmml">)</mo></mrow></mrow></mrow></mfrac></mrow></math>, <math><mrow id="S2.E4.m1.5.5" xref="S2.E4.m1.5.5.cmml"><msub id="S2.E4.m1.5.5.3" xref="S2.E4.m1.5.5.3.cmml"><mi id="S2.E4.m1.5.5.3.2" xref="S2.E4.m1.5.5.3.2.cmml">b</mi><mrow id="S2.E4.m1.2.2.2.4" xref="S2.E4.m1.2.2.2.3.cmml"><mi id="S2.E4.m1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.cmml">i</mi><mo id="S2.E4.m1.2.2.2.4.1" xref="S2.E4.m1.2.2.2.3.cmml">,</mo><mi id="S2.E4.m1.2.2.2.2" xref="S2.E4.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S2.E4.m1.5.5.2" xref="S2.E4.m1.5.5.2.cmml">←</mo><mrow id="S2.E4.m1.5.5.1" xref="S2.E4.m1.5.5.1.cmml"><msub id="S2.E4.m1.5.5.1.3" xref="S2.E4.m1.5.5.1.3.cmml"><mi id="S2.E4.m1.5.5.1.3.2" xref="S2.E4.m1.5.5.1.3.2.cmml">b</mi><mrow id="S2.E4.m1.4.4.2.4" xref="S2.E4.m1.4.4.2.3.cmml"><mi id="S2.E4.m1.3.3.1.1" xref="S2.E4.m1.3.3.1.1.cmml">i</mi><mo id="S2.E4.m1.4.4.2.4.1" xref="S2.E4.m1.4.4.2.3.cmml">,</mo><mi id="S2.E4.m1.4.4.2.2" xref="S2.E4.m1.4.4.2.2.cmml">j</mi></mrow></msub><mo id="S2.E4.m1.5.5.1.2" xref="S2.E4.m1.5.5.1.2.cmml">+</mo><mrow id="S2.E4.m1.5.5.1.1" xref="S2.E4.m1.5.5.1.1.cmml"><mrow id="S2.E4.m1.5.5.1.1.3" xref="S2.E4.m1.5.5.1.1.3.cmml"><msub id="S2.E4.m1.5.5.1.1.3.2" xref="S2.E4.m1.5.5.1.1.3.2.cmml"><mover accent="true" id="S2.E4.m1.5.5.1.1.3.2.2" xref="S2.E4.m1.5.5.1.1.3.2.2.cmml"><mi id="S2.E4.m1.5.5.1.1.3.2.2.2" xref="S2.E4.m1.5.5.1.1.3.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.E4.m1.5.5.1.1.3.2.2.1" xref="S2.E4.m1.5.5.1.1.3.2.2.1.cmml">^</mo></mover><mrow id="S2.E4.m1.5.5.1.1.3.2.3" xref="S2.E4.m1.5.5.1.1.3.2.3.cmml"><mi id="S2.E4.m1.5.5.1.1.3.2.3.2" xref="S2.E4.m1.5.5.1.1.3.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E4.m1.5.5.1.1.3.2.3.1" xref="S2.E4.m1.5.5.1.1.3.2.3.1.cmml">|</mo><mi id="S2.E4.m1.5.5.1.1.3.2.3.3" xref="S2.E4.m1.5.5.1.1.3.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.E4.m1.5.5.1.1.3.1" xref="S2.E4.m1.5.5.1.1.3.1.cmml">⋅</mo><mi id="S2.E4.m1.5.5.1.1.3.3" xref="S2.E4.m1.5.5.1.1.3.3.cmml">squash</mi></mrow><mo id="S2.E4.m1.5.5.1.1.2" xref="S2.E4.m1.5.5.1.1.2.cmml">⁢</mo><mrow id="S2.E4.m1.5.5.1.1.1.1" xref="S2.E4.m1.5.5.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E4.m1.5.5.1.1.1.1.2" xref="S2.E4.m1.5.5.1.1.1.1.1.cmml">(</mo><msub id="S2.E4.m1.5.5.1.1.1.1.1" xref="S2.E4.m1.5.5.1.1.1.1.1.cmml"><mi id="S2.E4.m1.5.5.1.1.1.1.1.2" xref="S2.E4.m1.5.5.1.1.1.1.1.2.cmml">𝐠</mi><mi id="S2.E4.m1.5.5.1.1.1.1.1.3" xref="S2.E4.m1.5.5.1.1.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="S2.E4.m1.5.5.1.1.1.1.3" xref="S2.E4.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.E5.m1.3.3" xref="S2.E5.m1.3.3.cmml"><mrow id="S2.E5.m1.3.3.1" xref="S2.E5.m1.3.3.1.cmml"><mrow id="S2.E5.m1.3.3.1.3" xref="S2.E5.m1.3.3.1.3.cmml"><msub id="S2.E5.m1.3.3.1.3.2" xref="S2.E5.m1.3.3.1.3.2.cmml"><mover accent="true" id="S2.E5.m1.3.3.1.3.2.2" xref="S2.E5.m1.3.3.1.3.2.2.cmml"><mi id="S2.E5.m1.3.3.1.3.2.2.2" xref="S2.E5.m1.3.3.1.3.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.E5.m1.3.3.1.3.2.2.1" xref="S2.E5.m1.3.3.1.3.2.2.1.cmml">^</mo></mover><mrow id="S2.E5.m1.3.3.1.3.2.3" xref="S2.E5.m1.3.3.1.3.2.3.cmml"><mi id="S2.E5.m1.3.3.1.3.2.3.2" xref="S2.E5.m1.3.3.1.3.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.3.3.1.3.2.3.1" xref="S2.E5.m1.3.3.1.3.2.3.1.cmml">|</mo><mi id="S2.E5.m1.3.3.1.3.2.3.3" xref="S2.E5.m1.3.3.1.3.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.E5.m1.3.3.1.3.1" xref="S2.E5.m1.3.3.1.3.1.cmml">⋅</mo><mi id="S2.E5.m1.3.3.1.3.3" xref="S2.E5.m1.3.3.1.3.3.cmml">squash</mi></mrow><mo id="S2.E5.m1.3.3.1.2" xref="S2.E5.m1.3.3.1.2.cmml">⁢</mo><mrow id="S2.E5.m1.3.3.1.1.1" xref="S2.E5.m1.3.3.1.1.1.1.cmml"><mo stretchy="false" id="S2.E5.m1.3.3.1.1.1.2" xref="S2.E5.m1.3.3.1.1.1.1.cmml">(</mo><msub id="S2.E5.m1.3.3.1.1.1.1" xref="S2.E5.m1.3.3.1.1.1.1.cmml"><mi id="S2.E5.m1.3.3.1.1.1.1.2" xref="S2.E5.m1.3.3.1.1.1.1.2.cmml">𝐠</mi><mi id="S2.E5.m1.3.3.1.1.1.1.3" xref="S2.E5.m1.3.3.1.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="S2.E5.m1.3.3.1.1.1.3" xref="S2.E5.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E5.m1.3.3.2" xref="S2.E5.m1.3.3.2.cmml">=</mo><mrow id="S2.E5.m1.3.3.3" xref="S2.E5.m1.3.3.3.cmml"><mrow id="S2.E5.m1.3.3.3.2" xref="S2.E5.m1.3.3.3.2.cmml"><mfrac id="S2.E5.m1.2.2" xref="S2.E5.m1.2.2.cmml"><mrow id="S2.E5.m1.1.1.1.1" xref="S2.E5.m1.1.1.1.2.cmml"><mo stretchy="false" id="S2.E5.m1.1.1.1.1.2" xref="S2.E5.m1.1.1.1.2.1.cmml">|</mo><msub id="S2.E5.m1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.cmml"><mi id="S2.E5.m1.1.1.1.1.1.2" xref="S2.E5.m1.1.1.1.1.1.2.cmml">𝐠</mi><mi id="S2.E5.m1.1.1.1.1.1.3" xref="S2.E5.m1.1.1.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="S2.E5.m1.1.1.1.1.3" xref="S2.E5.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S2.E5.m1.2.2.2" xref="S2.E5.m1.2.2.2.cmml"><mn id="S2.E5.m1.2.2.2.3" xref="S2.E5.m1.2.2.2.3.cmml">1</mn><mo id="S2.E5.m1.2.2.2.2" xref="S2.E5.m1.2.2.2.2.cmml">+</mo><msup id="S2.E5.m1.2.2.2.1" xref="S2.E5.m1.2.2.2.1.cmml"><mrow id="S2.E5.m1.2.2.2.1.1.1" xref="S2.E5.m1.2.2.2.1.1.2.cmml"><mo stretchy="false" id="S2.E5.m1.2.2.2.1.1.1.2" xref="S2.E5.m1.2.2.2.1.1.2.1.cmml">|</mo><msub id="S2.E5.m1.2.2.2.1.1.1.1" xref="S2.E5.m1.2.2.2.1.1.1.1.cmml"><mi id="S2.E5.m1.2.2.2.1.1.1.1.2" xref="S2.E5.m1.2.2.2.1.1.1.1.2.cmml">𝐠</mi><mi id="S2.E5.m1.2.2.2.1.1.1.1.3" xref="S2.E5.m1.2.2.2.1.1.1.1.3.cmml">j</mi></msub><mo stretchy="false" id="S2.E5.m1.2.2.2.1.1.1.3" xref="S2.E5.m1.2.2.2.1.1.2.1.cmml">|</mo></mrow><mn id="S2.E5.m1.2.2.2.1.3" xref="S2.E5.m1.2.2.2.1.3.cmml">2</mn></msup></mrow></mfrac><mo id="S2.E5.m1.3.3.3.2.1" xref="S2.E5.m1.3.3.3.2.1.cmml">⁢</mo><msub id="S2.E5.m1.3.3.3.2.2" xref="S2.E5.m1.3.3.3.2.2.cmml"><mover accent="true" id="S2.E5.m1.3.3.3.2.2.2" xref="S2.E5.m1.3.3.3.2.2.2.cmml"><mi id="S2.E5.m1.3.3.3.2.2.2.2" xref="S2.E5.m1.3.3.3.2.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.E5.m1.3.3.3.2.2.2.1" xref="S2.E5.m1.3.3.3.2.2.2.1.cmml">^</mo></mover><mrow id="S2.E5.m1.3.3.3.2.2.3" xref="S2.E5.m1.3.3.3.2.2.3.cmml"><mi id="S2.E5.m1.3.3.3.2.2.3.2" xref="S2.E5.m1.3.3.3.2.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.3.3.3.2.2.3.1" xref="S2.E5.m1.3.3.3.2.2.3.1.cmml">|</mo><mi id="S2.E5.m1.3.3.3.2.2.3.3" xref="S2.E5.m1.3.3.3.2.2.3.3.cmml">i</mi></mrow></msub></mrow><mo id="S2.E5.m1.3.3.3.1" xref="S2.E5.m1.3.3.3.1.cmml">⋅</mo><msub id="S2.E5.m1.3.3.3.3" xref="S2.E5.m1.3.3.3.3.cmml"><mi id="S2.E5.m1.3.3.3.3.2" xref="S2.E5.m1.3.3.3.3.2.cmml">𝐠</mi><mi id="S2.E5.m1.3.3.3.3.3" xref="S2.E5.m1.3.3.3.3.3.cmml">j</mi></msub></mrow></mrow></math>, <math><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><msub id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml"><mover accent="true" id="S2.SS1.p3.1.m1.1.1.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.SS1.p3.1.m1.1.1.2.2.1" xref="S2.SS1.p3.1.m1.1.1.2.2.1.cmml">^</mo></mover><mrow id="S2.SS1.p3.1.m1.1.1.2.3" xref="S2.SS1.p3.1.m1.1.1.2.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2.3.2" xref="S2.SS1.p3.1.m1.1.1.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.SS1.p3.1.m1.1.1.2.3.1" xref="S2.SS1.p3.1.m1.1.1.2.3.1.cmml">|</mo><mi id="S2.SS1.p3.1.m1.1.1.2.3.3" xref="S2.SS1.p3.1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">⋅</mo><msub id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml">𝐠</mi><mi id="S2.SS1.p3.1.m1.1.1.3.3" xref="S2.SS1.p3.1.m1.1.1.3.3.cmml">j</mi></msub></mrow></math>, <math><mrow id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml"><msub id="S2.Ex1.m1.1.1.2" xref="S2.Ex1.m1.1.1.2.cmml"><mover accent="true" id="S2.Ex1.m1.1.1.2.2" xref="S2.Ex1.m1.1.1.2.2.cmml"><mi id="S2.Ex1.m1.1.1.2.2.2" xref="S2.Ex1.m1.1.1.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.Ex1.m1.1.1.2.2.1" xref="S2.Ex1.m1.1.1.2.2.1.cmml">^</mo></mover><mrow id="S2.Ex1.m1.1.1.2.3" xref="S2.Ex1.m1.1.1.2.3.cmml"><mi id="S2.Ex1.m1.1.1.2.3.2" xref="S2.Ex1.m1.1.1.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.Ex1.m1.1.1.2.3.1" xref="S2.Ex1.m1.1.1.2.3.1.cmml">|</mo><mi id="S2.Ex1.m1.1.1.2.3.3" xref="S2.Ex1.m1.1.1.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.Ex1.m1.1.1.1" xref="S2.Ex1.m1.1.1.1.cmml">⋅</mo><msub id="S2.Ex1.m1.1.1.3" xref="S2.Ex1.m1.1.1.3.cmml"><mi id="S2.Ex1.m1.1.1.3.2" xref="S2.Ex1.m1.1.1.3.2.cmml">𝐠</mi><mi id="S2.Ex1.m1.1.1.3.3" xref="S2.Ex1.m1.1.1.3.3.cmml">j</mi></msub></mrow></math>, <math><mrow id="S2.Ex1.m2.2.3" xref="S2.Ex1.m2.2.3.cmml"><mi id="S2.Ex1.m2.2.3.2" xref="S2.Ex1.m2.2.3.2.cmml"/><mo id="S2.Ex1.m2.2.3.1" xref="S2.Ex1.m2.2.3.1.cmml">=</mo><mrow id="S2.Ex1.m2.2.3.3" xref="S2.Ex1.m2.2.3.3.cmml"><mstyle displaystyle="true" id="S2.Ex1.m2.2.3.3.1" xref="S2.Ex1.m2.2.3.3.1.cmml"><munder id="S2.Ex1.m2.2.3.3.1a" xref="S2.Ex1.m2.2.3.3.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S2.Ex1.m2.2.3.3.1.2" xref="S2.Ex1.m2.2.3.3.1.2.cmml">∑</mo><mi id="S2.Ex1.m2.2.3.3.1.3" xref="S2.Ex1.m2.2.3.3.1.3.cmml">l</mi></munder></mstyle><mrow id="S2.Ex1.m2.2.3.3.2" xref="S2.Ex1.m2.2.3.3.2.cmml"><mrow id="S2.Ex1.m2.2.3.3.2.2" xref="S2.Ex1.m2.2.3.3.2.2.cmml"><msub id="S2.Ex1.m2.2.3.3.2.2.2" xref="S2.Ex1.m2.2.3.3.2.2.2.cmml"><mover accent="true" id="S2.Ex1.m2.2.3.3.2.2.2.2" xref="S2.Ex1.m2.2.3.3.2.2.2.2.cmml"><mi id="S2.Ex1.m2.2.3.3.2.2.2.2.2" xref="S2.Ex1.m2.2.3.3.2.2.2.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.Ex1.m2.2.3.3.2.2.2.2.1" xref="S2.Ex1.m2.2.3.3.2.2.2.2.1.cmml">^</mo></mover><mrow id="S2.Ex1.m2.2.3.3.2.2.2.3" xref="S2.Ex1.m2.2.3.3.2.2.2.3.cmml"><mi id="S2.Ex1.m2.2.3.3.2.2.2.3.2" xref="S2.Ex1.m2.2.3.3.2.2.2.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.Ex1.m2.2.3.3.2.2.2.3.1" xref="S2.Ex1.m2.2.3.3.2.2.2.3.1.cmml">|</mo><mi id="S2.Ex1.m2.2.3.3.2.2.2.3.3" xref="S2.Ex1.m2.2.3.3.2.2.2.3.3.cmml">i</mi></mrow></msub><mo id="S2.Ex1.m2.2.3.3.2.2.1" xref="S2.Ex1.m2.2.3.3.2.2.1.cmml">⋅</mo><msub id="S2.Ex1.m2.2.3.3.2.2.3" xref="S2.Ex1.m2.2.3.3.2.2.3.cmml"><mi id="S2.Ex1.m2.2.3.3.2.2.3.2" xref="S2.Ex1.m2.2.3.3.2.2.3.2.cmml">c</mi><mrow id="S2.Ex1.m2.2.2.2.4" xref="S2.Ex1.m2.2.2.2.3.cmml"><mi id="S2.Ex1.m2.1.1.1.1" xref="S2.Ex1.m2.1.1.1.1.cmml">i</mi><mo id="S2.Ex1.m2.2.2.2.4.1" xref="S2.Ex1.m2.2.2.2.3.cmml">,</mo><mi id="S2.Ex1.m2.2.2.2.2" xref="S2.Ex1.m2.2.2.2.2.cmml">j</mi></mrow></msub></mrow><mo id="S2.Ex1.m2.2.3.3.2.1" xref="S2.Ex1.m2.2.3.3.2.1.cmml">⁢</mo><msub id="S2.Ex1.m2.2.3.3.2.3" xref="S2.Ex1.m2.2.3.3.2.3.cmml"><mover accent="true" id="S2.Ex1.m2.2.3.3.2.3.2" xref="S2.Ex1.m2.2.3.3.2.3.2.cmml"><mi id="S2.Ex1.m2.2.3.3.2.3.2.2" xref="S2.Ex1.m2.2.3.3.2.3.2.2.cmml">𝐟</mi><mo stretchy="false" id="S2.Ex1.m2.2.3.3.2.3.2.1" xref="S2.Ex1.m2.2.3.3.2.3.2.1.cmml">^</mo></mover><mrow id="S2.Ex1.m2.2.3.3.2.3.3" xref="S2.Ex1.m2.2.3.3.2.3.3.cmml"><mi id="S2.Ex1.m2.2.3.3.2.3.3.2" xref="S2.Ex1.m2.2.3.3.2.3.3.2.cmml">j</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.Ex1.m2.2.3.3.2.3.3.1" xref="S2.Ex1.m2.2.3.3.2.3.3.1.cmml">|</mo><mi id="S2.Ex1.m2.2.3.3.2.3.3.3" xref="S2.Ex1.m2.2.3.3.2.3.3.3.cmml">l</mi></mrow></msub></mrow></mrow></mrow></math>

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1303.6459

Formulas:

1-

ρ_{h a r d} - ρ_{e a s y}

2-

η = (ρ_{h a r d} - ρ_{e a s y}) / (ρ_{h a r d} + ρ_{e a s y})

3-

H_{1} < H < H_{2}

4-

H_{2} < H < H_{3}

5-

ρ_{h a r d} (H)

6-

ρ_{h a r d}

7-

d_{x z, y z}

8-

d_{x z, y z}

9-

H_{2} < H < H_{3}

10-

H_{1} < H < H_{3}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9701047

Formulas:

1-

\log (a g e)

2-

{}_{R}^{t o t}

3-

{}_{R}^{b u l}

4-

\nabla (B - R)

5-

\nabla (U - R)

6-

\nabla (R - K)

7-

\nabla (R - I)

8-

\nabla (J - K)

9-

Δ (C o l o u r) / Δ (\log r)

10-

\frac{Δ (B - R)}{Δ (h_{K})}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0602165

Formulas:

1-

n (r) = \frac{n_{0}}{{[1 + {(r / r_{0})}^{2}]}^{3 / 4}} .

2-

n_{jet} = n_{0} / 100

3-

r_{0} = 100 kpc

4-

c_{ISM} = 1000 km s^{- 1}

5-

n_{0} = 0.01 {cm}^{- 3}

6-

T_{0} = 4.4 \times 10^{7}

7-

9.3 \times 10^{45} erg s^{- 1}

8-

r_{0} = 10 kpc

9-

c_{I S M} = 500 km s^{- 1}

10-

n_{0} = 0.05 {cm}^{- 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1106.2160

Formulas:

1-

^{3, 67, 70, *}

2-

^{20, 21, 69, 71}

3-

m_{p} = 10.4 \pm 1.7 M_{\oplus}

4-

M_{⋆} = 0.56 \pm 0.09 M_{⊙}

5-

a = 3.2 \binom{+ 1.9}{- 0.5}

6-

P = 7.6 \binom{+ 7.7}{- 1.5}

7-

10 - 100 M_{\oplus}

8-

10 - 100 M_{\oplus}

9-

\sim 6 \times 10^{- 5}

10-

(RA, DEC) = (17^{h} 48^{m} {05.95}^{s}, - 35^{\circ} 00^{'} {19.48}^{''})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-lat/0310036

Formulas:

1-

S_{g} = β \sum_{x, μ, ν} c_{μ ν} [1 - \frac{1}{2 N_{c}} Tr (P_{μ ν} (x) + P_{μ ν}^{†} (x))],

2-

ξ = a_{c} / a_{f}

3-

c_{μ ν} = {\begin{matrix} ξ^{2} (1 + η_{f f}^{(1)} g^{2} + O (g^{4})) & f-f; \\ 1 + η_{c f}^{(1)} g^{2} + O (g^{4}) & c-f; \\ \frac{1}{ξ^{2}} (1 + η_{c c}^{(1)} g^{2} + O (g^{4})) & c-c; \end{matrix}

4-

η_{μ ν}^{(1)}

5-

c_{c c} = 1 / ξ^{2}, c_{c f} = 1, and c_{f f} = ξ^{2} .

6-

V (R) = lim_{T \to \infty} - \frac{1}{T} \log < W (R, T) > .

7-

η_{μ ν}^{(1)}

8-

η_{c c}^{(1)}

9-

c_{c f} / c_{f f}

10-

V (n, 0, 0) = V (0, 0, ξ n)

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/1410.6129

Formulas:

1-

a \equiv c J / G M^{2} > 1

2-

a = c J / G M^{2} < 1

3-

ν_{L} = ν_{K} - ν_{r}

4-

ν_{U} = ν_{K} + ν_{r}

5-

ν_{L} = 2 (ν_{K} - ν_{r})

6-

ν_{U} = 2 ν_{K} - ν_{r}

7-

ν_{L} = ν_{K} - ν_{r}

8-

ν_{L} = ν_{K} - ν_{r}

9-

ν_{U} = 2 ν_{K} - ν_{θ}

10-

M = 10 M_{☉}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0307730

Formulas:

1-

Γ = π N ν {| V |}^{2}

2-

g_{dot} = Γ / Δ

3-

Δ + e^{2} / C

4-

H = H_{dot} + \sum_{k = 1}^{N} (H_{L e a d}^{k} + H_{C o u p l i n g}^{k}) .

5-

H_{dot} = \sum_{i = 1}^{N_{dot}} (ϵ_{i} - μ) a_{i}^{†} a_{i} + U \sum_{i > j}^{N_{dot}} a_{i}^{†} a_{i} a_{j}^{†} a_{j}

6-

U = e^{2} / C

7-

H_{L e a d}^{k} = μ \sum_{j = 1}^{\infty} c_{j}^{k, †} c_{j}^{k} - t \sum_{j = 1}^{\infty} c_{j}^{k, †} c_{j + 1}^{k} + h . c .,

8-

H_{C o u p l i n g}^{k} = \sum_{i = 1}^{N_{dot}} V_{k, i} a_{i}^{†} c_{1}^{k} + h . c .,

9-

- t \sum_{k = 1}^{N - 1} \sum_{j = 1}^{\infty} c_{j}^{k, †} c_{j}^{k + 1} + h . c .

10-

n = \sum_{m = 1}^{ε_{m} < 0} \sum_{i = 1}^{N_{dot}} {| ⟨ m | a_{i}^{†} a_{i} | m ⟩ |}^{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1111.6597

Formulas:

1-

Γ \leftrightarrow (Λ, A) \leftrightarrow X

2-

P_{U | V = v} = P_{U | V = v, W = w}

3-

P_{W | V = v} = P_{W | V = v, U = u}

4-

P_{X | Ψ = ψ, A = a} (x) = ⟨ ψ | Π_{x}^{a} | ψ ⟩

5-

P_{A | Γ = γ} = P_{A}

6-

Λ \leftrightarrow (Ψ, A) \leftrightarrow X .

7-

P_{X | Λ = λ, Ψ = ψ, A = a} = P_{X | Ψ = ψ, A = a}

8-

P_{Λ Ψ A} (λ, ψ, a) > 0

9-

P_{X | Λ = λ, Ψ = ψ, A = a} = P_{X | Λ = λ, A = a} .

10-

P_{X | Ψ = ψ, A = a} = P_{X | Λ = λ, A = a},

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0607271

Formulas:

1-

L_{bol} \sim 3 \times 10^{11}

2-

37 \times 512^{2} \times {0.8}^{'', 2}

3-

S_{1.4 GHz} \geq 2, 000

4-

S_{1.4 GHz} = 50 μ Jy

5-

Ω_{m} = 0.27, Ω_{Λ} = 0.73

6-

S_{ν} \propto ν^{- 0.8}

7-

σ_{1.4 GHz} > 30 μ Jy {beam}^{- 1}

8-

L_{1.4 GHz} < 10^{24} W {Hz}^{- 1}

9-

L_{far - IR} / L_{UV}

10-

R - K_{s} > 5.3

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1305.4488

Formulas:

1-

m \in {1, 2, \dots, M}

2-

ω = m_{1} m_{2} \dots m_{l}

3-

m_{j} \in {1, 2, \dots, M}

4-

m = m_{l + 1}

5-

m_{1} m_{2} \dots m_{l} + m ⇌_{W (ω m, ω)}^{W (ω, ω m)} m_{1} m_{2} \dots m_{l} m

6-

W (ω, ω^{'})

7-

W (ω^{'}, ω)

8-

ω^{'} = ω m

9-

P_{t} (ω)

10-

\frac{d P_{t} (ω)}{d t} = \sum_{ω^{'}} [P_{t} (ω^{'}) W (ω^{'}, ω) - P_{t} (ω) W (ω, ω^{'})] .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/1802.10076

Formulas:

1-

N_{e} (t) = N_{0} e^{α t},

2-

E C = δ_{1} \times δ_{2} \times \dots \times δ_{n},

3-

δ_{1}, δ_{2}, \dots, δ_{n}

4-

1^{s t}, 2^{n d}, \dots, n^{t h}

5-

V_{m i n}

6-

V_{m a x}

7-

V_{m a x} = \frac{m ω^{2} x^{2}}{2 e},

8-

V_{m i n} = \frac{m ω^{2} x^{2}}{e} \frac{1}{\sqrt{4 + {(2 n + 1)}^{2} π^{2}}},

9-

V_{c} = V_{0} (1 - e^{- \frac{t}{2 τ}}) e^{i ω_{0} t}

10-

{\begin{matrix} \frac{d | V_{c} |}{d t} = \frac{1}{2 τ} (| V_{0} | - | V_{c} |) - f_{0} δ V_{m p} \frac{e N_{e} (t)}{2 Q_{0} | V_{c} |} ω_{0} R_{s h} \\ \frac{d N_{e}}{d t} = f_{0} α (| V_{c} |) N_{e} \end{matrix}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/nucl-ex/0303015

Formulas:

1-

{}^{3}{\vec{He}} (\vec{e}, e^{'} n)

2-

{}^{3}{\vec{He}} (\vec{e}, e^{'} n)

3-

{(GeV / c)}^{2}

4-

(e, e^{'} n)

5-

G_{e n} \cdot G_{m n}

6-

A (θ^{*}, ϕ^{*}) = \frac{1}{P_{e} \cdot P_{t}} \cdot \frac{N^{+} - N^{-}}{N^{+} + N^{-}}

7-

{}^{3}{\vec{He}} (\vec{e}, e^{'} n)

8-

N^{+} (N^{-})

9-

A_{∥} = A (0^{o}, 0^{o})

10-

A_{⟂} = A (90^{o}, 0^{o})

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: nucl-ex

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0402553

Formulas:

1-

r_{S} = 2 G M_{B H} / c^{2}

2-

c r_{S} = 2 G M_{B H} / c

3-

ℰ = \frac{ϑ_{e}^{2}}{2} + \frac{a_{e}^{2}}{γ - 1} + \frac{λ^{2}}{2 x^{2}} + g (x)

4-

g (x) = - \frac{1}{2 (x - 1)}

5-

a_{e} = \sqrt{γ p / ρ}

6-

P = K ρ^{γ}

7-

\dot{M} = ϑ_{e} ρ a^{ζ} x^{β} {(x - 1)}^{δ} .

8-

β = 3 / 2, ζ = 1, δ = 1

9-

β = 2, ζ = 0, δ = 0

10-

β = 1, ζ = 0, δ = 0

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0905.4269

Formulas:

1-

E^{2} (z) = Ω_{m} {(1 + z)}^{3} + Ω_{Λ} - 2 \sqrt{Ω_{r_{c}}} [1 + Ω_{α} E^{2} (z)] E (z),

2-

E (z) = H / H_{0}

3-

Ω_{m} = \frac{κ_{4}^{2} ρ_{m_{0}}}{3 H_{0}^{2}}, Ω_{Λ} = \frac{κ_{4}^{2} Λ}{3 H_{0}^{2}}, Ω_{r_{c}} = \frac{1}{4 r_{c}^{2} H_{0}^{2}},

4-

Ω_{α} = \frac{8}{3} α H_{0}^{2} .

5-

Ω_{m} + Ω_{Λ} = 1 + 2 \sqrt{Ω_{r_{c}}} (1 + Ω_{α}) .

6-

Ω_{m} + Ω_{Λ} < 1

7-

3 Ω_{m} < 2 + 2 (1 + Ω_{α}) \sqrt{Ω_{r_{c}}} .

8-

H^{2} = \frac{κ_{4}^{2}}{3} (ρ_{m} + ρ_{eff})

9-

= \frac{3 H_{0}^{2}}{κ_{4}^{2}} [Ω_{Λ} - 2 \sqrt{Ω_{r_{c}}} (1 + Ω_{α} E^{2} (z)) E (z)] .

10-

{\dot{ρ}}_{eff} + 3 H (1 + w_{eff}) ρ_{eff} = 0 .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1401.6827

Formulas:

1-

n = 1, 2, 3, \dots

2-

x = \log_{10} P / 1 d

3-

f_{1} = 1 - f_{M}

4-

f_{C} = f_{2} + 2 f_{3} + 3 f_{4} + \dots

5-

f_{H} = \sum_{n = 3}^{\infty} f_{n} = 1 - f_{1} - f_{2}

6-

q = M_{B} / M_{A} < 1

7-

q_{sys} = (M_{B} + M_{C}) / M_{A}

8-

q = M_{B} / M_{A}

9-

q = M_{C} / M_{A}

10-

q_{sys} = M_{C} / (M_{A} + M_{B})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/0308160

Formulas:

1-

U {(1)}_{Q}

2-

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

3-

S U {(3)}_{c}

4-

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

5-

S U {(3)}_{c}

6-

U (y) = - C y^{2} + B y^{4}

7-

f_{0} \cos γ t

8-

l_{0} = l + d

9-

F = - 2 k (s - l_{0}) \sin θ = - 2 k (s - l - d) \sin θ .

10-

s = {(l^{2} + y^{2})}^{1 / 2}

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1911.04623

Formulas:

1-

𝒟_{base} = {(𝐈_{1}, y_{1}), \dots, (𝐈_{N}, y_{N})}

2-

y_{n} \in {1, \dots, A}

3-

f_{θ} (𝐈)

4-

f_{θ} (𝐈)

5-

𝐖 \in ℝ^{D \times A}

6-

{\arg \min}_{θ, 𝐖} \sum_{(𝐈, y) \in 𝒟_{base}} ℓ (𝐖^{⊤} f_{θ} (𝐈), y),

7-

𝒙 = f_{θ} (𝐈)

8-

d (𝒙, 𝒙^{'}) \in ℝ_{0}^{+}

9-

𝒟_{support} = {({\hat{𝒙}}_{1}, 1), \dots, ({\hat{𝒙}}_{C}, C)}

10-

y (\hat{𝒙}) = {\arg \min}_{c \in {1, \dots, C}} d (\hat{𝒙}, {\hat{𝒙}}_{c}) .

Formulas (html):

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id="S2.E1.m1.6.6.1.1.1.1.1.2" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml">(</mo><mover accent="true" id="S2.E1.m1.5.5" xref="S2.E1.m1.5.5.cmml"><mi id="S2.E1.m1.5.5.2" xref="S2.E1.m1.5.5.2.cmml">𝒙</mi><mo stretchy="false" id="S2.E1.m1.5.5.1" xref="S2.E1.m1.5.5.1.cmml">^</mo></mover><mo id="S2.E1.m1.6.6.1.1.1.1.1.3" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml">,</mo><msub id="S2.E1.m1.6.6.1.1.1.1.1.1" xref="S2.E1.m1.6.6.1.1.1.1.1.1.cmml"><mover accent="true" id="S2.E1.m1.6.6.1.1.1.1.1.1.2" xref="S2.E1.m1.6.6.1.1.1.1.1.1.2.cmml"><mi id="S2.E1.m1.6.6.1.1.1.1.1.1.2.2" xref="S2.E1.m1.6.6.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mo stretchy="false" id="S2.E1.m1.6.6.1.1.1.1.1.1.2.1" xref="S2.E1.m1.6.6.1.1.1.1.1.1.2.1.cmml">^</mo></mover><mi id="S2.E1.m1.6.6.1.1.1.1.1.1.3" xref="S2.E1.m1.6.6.1.1.1.1.1.1.3.cmml">c</mi></msub><mo stretchy="false" id="S2.E1.m1.6.6.1.1.1.1.1.4" xref="S2.E1.m1.6.6.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.6.6.1.2" xref="S2.E1.m1.6.6.1.1.cmml">.</mo></mrow></math>

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/gr-qc/9611025

Formulas:

1-

d n (t) / d t = σ_{cap} ρ (t) c,

2-

d n / d t

3-

S / (4 π r_{-}^{2})

4-

r_{-} = M - {(M^{2} + Q_{*}^{2})}^{1 / 2}

5-

d t / d τ

6-

r = r_{0} ≫ M

7-

d t / d τ \approx {(κ_{-} τ)}^{- 1}

8-

κ_{-} \equiv \frac{1}{2} {| d f / d r |}_{r = r_{-}}

9-

f = 1 - 2 M / r + Q_{*}^{2} / r^{2}

10-

d t / d τ

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1001.3939

Formulas:

1-

J H K_{s}

2-

58' \times 58'

3-

U B V R I

4-

U B V R I

5-

U B V R I

6-

B V R I

7-

J H K_{s}

8-

J H K_{s}

9-

J H K_{s}

10-

U B V R I

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1002.3545

Formulas:

1-

{⟨ Q_{t}^{n} ⟩}_{c}

2-

lim_{t \to \infty} \frac{1}{t} {⟨ Q_{t}^{2} ⟩}^{ring} ≃ \frac{1}{L} A (T); lim_{t \to \infty} \frac{1}{t} {⟨ Q_{t}^{2 n} ⟩}_{c}^{ring} ≃ \frac{1}{L^{2}} B_{n} (T) for n \geq 2; lim_{t \to \infty} \frac{1}{t} {⟨ Q_{t}^{n} ⟩}_{c}^{open} ≃ \frac{1}{L} C_{n} (T_{a}, T_{b}) .

3-

B_{n} (T)

4-

C_{n} (T_{a}, T_{b})

5-

{⟨ Q_{t} ⟩}^{open} / t

6-

{⟨ Q_{t}^{2} ⟩}^{ring} / t

7-

{⟨ Q_{t}^{2} ⟩}^{open} / t

8-

lim_{t \to \infty} \frac{1}{t} {⟨ Q_{t} ⟩}^{open} \sim L^{α - 1} C_{1} (T_{a}, T_{b}) .

9-

⟨ Q_{t}^{2} ⟩ / t

10-

{⟨ Q_{t}^{n} ⟩}_{c} / t

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1410.4222

Formulas:

1-

N_{n b} \to \infty

2-

N_{n b} \propto N^{1 / 2}

3-

O (N^{1 + δ})

4-

h \propto N^{- 1 / 3}

5-

N_{n b} \approx 30 - 100

6-

\propto N_{n b}^{- 1 / 2}

7-

\sim (d - 1) N_{n b}^{- 1} \log N_{n b}

8-

\sim N_{n b}^{- 1} \log N_{n b}

9-

N_{n b}^{- 1} <

10-

< N_{n b}^{- 1 / 2}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0508272

Formulas:

1-

10^{12} h^{- 1} M_{⊙}

2-

ρ (r) = \frac{4 ρ_{s}}{(r / R_{s}) {(1 + r / R_{s})}^{2}},

3-

10^{12} h^{- 1} M_{⊙}

4-

100 km s^{- 1} {Mpc}^{- 1}

5-

N \sim 2.1 \times 10^{6}

6-

8 h^{- 1} Mpc

7-

10^{12} h^{- 1} M_{⊙}

8-

10^{12} h^{- 1} M_{⊙}

9-

10^{13} h^{- 1} M_{⊙}

10-

5 \times 10^{11} h^{- 1} M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1712.09422

Formulas:

1-

α = \frac{G M_{⊙}}{b c^{2}}

2-

α \sim 2 \times 10^{- 6}

3-

\sqrt{c^{2} + 2 Φ}

4-

η = c^{2} d t^{2} - d 𝐱^{2}

5-

c^{2} + 2 Φ

6-

d 𝐱^{2} = d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ \cdot d φ^{2})

7-

d r^{2} / (1 + 2 \frac{Φ}{c^{2}})

8-

Φ = - G M_{⊙} / r

9-

Λ = \frac{m c^{2}}{h}

10-

A_{Δ} (𝐩)

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-lat/0703023

Formulas:

1-

⟨ \bar{ψ} ψ ⟩

2-

S_{QED} = β \sum_{x} \sum_{μ, ν} (1 - Re P_{μ ν} (x))

3-

U_{μ} (x) \equiv e^{i θ_{μ} (x)} \in U (1)

4-

P_{μ ν} (x) \equiv U_{μ} (x) U_{ν} (x + \hat{μ}) U_{μ}^{†} (x + \hat{ν}) U_{ν}^{†} (x)

5-

θ_{μ} (x)

6-

θ_{μ ν} (x) \equiv θ_{μ} (x) + θ_{ν} (x + \hat{μ}) - θ_{μ} (x + \hat{ν}) - θ_{ν} (x) \in (- 4 π, 4 π]

7-

S_{QED} = β \sum_{x} \sum_{μ, ν} (1 - \cos θ_{μ ν} (x)) .

8-

β_{c} = 1.0111331 (21)

9-

{\bar{θ}}_{μ ν} \in (- π, π]

10-

2 π n_{μ ν}

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/1906.00289

Formulas:

1-

H_{D C} = \sum_{i = 1}^{N} (c 𝜶_{𝒊} \cdot 𝒑_{𝒊} + (β_{i} - 1) c^{2} + V_{i}) + \sum_{i > i}^{N} \frac{1}{r_{i j}},

2-

Ψ (Γ P J)

3-

Φ (γ_{i} P J)

4-

Ψ (γ P J) = \sum_{i = 1}^{M} c_{i} Φ (γ_{i} P J),

5-

ϕ (𝐫) = \frac{1}{r} (\begin{matrix} P_{n κ} (r) χ_{κ m} (θ, ϕ) \\ i Q_{n κ} (r) χ_{- κ m} (θ, ϕ) \end{matrix}) .

6-

H^{Z F} = (\begin{matrix} H^{(P P)} & H^{(P Q)} \\ H^{(Q P)} & H^{(Q Q)} \end{matrix}),

7-

H^{(Q Q)}

8-

B_{i j} = - \sum_{i < j}^{N} \frac{1}{2 r_{i j}} [(𝜶_{i} \cdot 𝜶_{j}) + \frac{(𝜶_{i} \cdot 𝒓_{i j}) (𝜶_{j} \cdot 𝒓_{i j})}{r_{i j}^{2}}],

9-

N_{C S F}

10-

N_{C S F}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1811.04728

Formulas:

1-

(0, \pm 1)

2-

{0, 1, - 1}

3-

I \subseteq {1, \dots, n}

4-

{1, \dots, n} ∖ I

5-

{0, 1, - 1}

6-

(\begin{matrix} A_{1} & A_{2} \\ A_{3} & A_{4} \end{matrix})

7-

A / A_{1} := A_{4} - A_{3} A_{1}^{- 1} A_{2}

8-

r (A) = r (A_{1}) + r (A / A_{1})

9-

(\begin{matrix} 0 & - c & - 1 \\ c & 0 & 0 & - 1 \\ 1 & 0 & ⋱ & ⋱ & ⋱ \\ 1 & ⋱ & ⋱ & ⋱ & - 1 \\ ⋱ & ⋱ & ⋱ & 0 & - 1 \\ 1 & 0 & 0 & b \\ 1 & a & 0 \end{matrix}),

10-

a, b, c \in {- 1, 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1906.03448

Formulas:

1-

U_{W C A} (r) = {\begin{matrix} 4 ϵ [{(\frac{σ}{r})}^{12} - {(\frac{σ}{r})}^{6}] + ϵ; r < 2^{1 / 6} σ \\ 0; r \geq 2^{1 / 6} σ \end{matrix} .

2-

U_{50, 49} (r) = {\begin{matrix} 50 {(\frac{50}{49})}^{49} ϵ [{(\frac{σ}{r})}^{50} - {(\frac{σ}{r})}^{49}] + ϵ; r < \frac{50}{49} σ \\ 0; r \geq \frac{50}{49} σ \end{matrix} .

3-

U_{50, 49} (r)

4-

T^{*} = \frac{ϵ}{k_{B} T} = 2 / 3

5-

T^{*} = 2 / 3

6-

t^{*} = t \sqrt{\frac{k_{B} T}{σ^{2} m}}

7-

ρ^{*} = ρ σ^{3}

8-

P^{*} = \frac{P σ^{3}}{k_{B} T}

9-

η^{*} = η (\frac{σ^{2}}{\sqrt{m k_{B} T}})

10-

δ t^{*} = 0.0011

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0906.1826

Formulas:

1-

\begin{matrix} 𝒰_{L} & = \frac{ϵ_{0}}{2} \sum_{< i j >} {(| 𝐫_{i j} | - | 𝐞_{i j} |)}^{2} + ϵ_{1} \sum_{i} {(\sum_{< j >} {\hat{𝐫}}_{i j})}^{2} \\ + ϵ_{2} \sum_{< i j >} (1 - {\hat{𝐧}}_{i} \cdot {\hat{𝐧}}_{j}) + ϵ_{3} \sum_{< i j >} ({\hat{𝐧}}_{i} \cdot {\hat{𝐫}}_{i j}) ({\hat{𝐧}}_{j} \cdot {\hat{𝐫}}_{j i}) \end{matrix}

2-

{\hat{𝐫}}_{i j} = \frac{𝐫_{i j}}{| 𝐫_{i j} |}

3-

𝐫_{i j} = 𝐞_{i j}

4-

𝐫_{i j} = 𝐞_{i j} + 𝐮_{j}^{Y} - 𝐮_{i}^{X}

5-

𝒰_{L} = 𝒰_{L}^{ϵ_{0}} + 𝒰_{L}^{ϵ_{1}} + 𝒰_{L}^{ϵ_{2}} + 𝒰_{L}^{ϵ_{3}}

6-

𝒰_{L}^{ϵ_{0}} = \sum_{X \neq Y} \sum_{⟨ i j ⟩} \frac{ϵ_{0}}{2} {[𝐞_{i j} \cdot (𝐮_{j}^{Y} - 𝐮_{i}^{X})]}^{2} .

7-

𝒰_{L}^{ϵ_{1}} = ϵ_{1} \sum_{X \neq Y} \sum_{n} [\sum_{α (n)} (𝐮_{α (n)}^{X} - 𝐮_{n}^{Y}) ({\hat{𝐣}}_{α}^{2} + {\hat{𝐳}}^{2}) (𝐮_{α (n)}^{X} - 𝐮_{n}^{Y}) + \sum_{α (n) \neq β (n)} (𝐮_{α (n)}^{X} - 𝐮_{n}^{Y}) (I d - \frac{{\hat{𝐢}}_{α} {\hat{𝐢}}_{β}}{2} - 2 {\hat{𝐢}}_{α}^{2}) (𝐮_{β (n)}^{X} - 𝐮_{n}^{Y})] .

8-

𝐮_{α (n)}^{X}

9-

𝐧_{n}^{A} = (𝐫_{n n 2} - 𝐫_{n n}) \times (𝐫_{n n 3} - 𝐫_{n n}),

10-

𝐧_{α (n)}^{B}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/2011.12933

Formulas:

1-

μ_{e f f}

2-

μ_{e f f}

3-

D i s c o v e r

4-

R \bar{3} c

5-

μ_{e f f}

6-

μ_{e f f}

7-

μ_{e f f}

8-

μ_{e f f}

9-

μ_{e f f}

10-

H_{E B} = | H_{1} + H_{2} | / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Run 16446909 (Agent672)