Run 6938504

Paper: https://arxiv.org/abs/astro-ph/9509064

Formulas:

1-

S U (3) \times S U (2) \times U (1)

2-

Ω_{j} = ρ_{j} / ρ_{c}

3-

ρ_{c} = \frac{3 H^{2} m_{P l}^{2}}{8 π} = 1.88 h_{100}^{2} \times 10^{- 29} g / c m^{3} = 10.5 h_{100}^{2} K e V / c m^{3}

4-

m_{P l} = 1.22 \times 10^{19}

5-

G_{N} = m_{P l}^{- 2}

6-

H = 100 h k m / s e c / M p c

7-

t_{u} = \frac{1}{H} \int_{0}^{1} \frac{d x}{\sqrt{1 - Ω_{t o t} + Ω_{m} x^{- 1} + Ω_{v a c} x^{2}}}

8-

Ω_{v a c}

9-

Ω_{t o t} = Ω_{m} + Ω_{v a c}

10-

1 / H = 9.8 h_{100}^{- 1} \times 10^{9}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1201.1844

Formulas:

1-

δ ν_{n ℓ} / ν_{n ℓ}

2-

\frac{δ ν_{n ℓ}}{ν_{n ℓ}}

3-

\int_{0}^{1} K_{ρ, Γ_{1}}^{n ℓ} (x) \frac{δ ρ}{ρ} d x + \int_{0}^{1} K_{Γ_{1}, ρ}^{n ℓ} (x) \frac{δ Γ_{1}}{Γ_{1}} d x

4-

\frac{F_{surf} (ν_{n ℓ})}{Q_{n ℓ}},

5-

\frac{F_{surf} (ν_{n ℓ})}{Q_{n ℓ}}

6-

K_{ρ, Γ_{1}}^{n ℓ} (r)

7-

K_{Γ_{1}, ρ}^{n ℓ} (r)

8-

(ρ, Γ_{1})

9-

\frac{δ f}{f} = \frac{f_{obs} - f_{ref}}{f_{ref}} .

10-

f_{obs} = f_{ref} (1 + δ f / f)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0605099

Formulas:

1-

H = - t \sum_{< j, l, >, σ} c_{j, σ}^{†} c_{l, σ} + U \sum_{j} n_{j, ↑} n_{j, ↓} .

2-

n_{j, σ} = c_{j σ}^{†} c_{j, σ}

3-

H_{r e l} = - \frac{ℏ^{2} \nabla^{2}}{2 μ} + \frac{1}{2} μ ω^{2} r^{2} + V (r)

4-

V (r) = \frac{4 π ℏ^{2} a}{m} δ^{(3)} (r) \frac{\partial}{\partial r} r .

5-

\frac{a_{h o}}{a} = \sqrt{2} \frac{Γ (- E / 2 ℏ ω + 3 / 4)}{Γ (- E / 2 ℏ ω + 1 / 4)},

6-

a_{h o} = \sqrt{ℏ / m ω}

7-

| a | / a_{h o} ≪ 1

8-

β_{6} = {(2 μ C_{6} / ℏ^{2})}^{1 / 4}

9-

β_{6} / a_{h o} < 0.1

10-

a (E) = {(\frac{1}{a} - \frac{1}{2 a_{h o}} \frac{E}{ℏ ω} \frac{r_{eff}}{a_{h o}})}^{- 1} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1908.04908

Formulas:

1-

χ^{2} / d o f = 1139.0 / 1045

2-

χ^{2} / d o f = 1105.7 / 1042

3-

χ^{2} / d o f = 1047.8 / 1040

4-

n H = {0.07}_{- 0.02}^{+ 0.03} \times 10^{22}

5-

k T_{apec} = {0.96}_{- 0.09}^{+ 0.08}

6-

N o r m_{apec} = {6.4}_{- 2.1}^{+ 2.0}

7-

k T_{BB}^{low} = {0.22}_{- 0.02}^{+ 0.03}

8-

N o r m_{BB}^{low} = {86.3}_{- 47.8}^{+ 90.4}

9-

Γ = {0.73}_{- 0.02}^{+ 0.02}

10-

N o r m_{PL} = {2.66}_{- 0.09}^{+ 0.09} \times 10^{- 3}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0903.3325

Formulas:

1-

\dot{q} = 𝒇 (q, u)

2-

\dot{q} = 𝒇 (q, u), q \in M, u \in U,

3-

\frac{\partial 𝒇 (q, u)}{\partial u} \land \frac{\partial {}^{2}𝒇 (q, u)}{\partial u^{2}} \neq 0, \forall (q, u) \in M \times U .

4-

(q, u) \to (ϕ (q), ψ (q, u))

5-

π : T^{*} M \to M

6-

s_{λ} = λ \circ π_{*}

7-

λ \in T^{*} M

8-

\dot{q} = 𝒇 (q, u), q \in M, u \in U

9-

q (0) \in N_{0}, q (t_{1}) \in N_{1},

10-

t_{1} \to \min (or \max),

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/physics/0401141

Formulas:

1-

Q_{0} \geq 5 \times 10^{9}

2-

2 Nb + 5 {SO}_{4}^{- -} + 5 H_{2} O

3-

{Nb}_{2} O_{5} + 10 H^{+} + 5 {SO}_{4}^{- -} + 10 e^{-}

4-

{Nb}_{2} O_{5} + 6 HF

5-

H_{2} {NbOF}_{5} + {NbOF}_{2} \cdot 0.5 H_{2} O + 1.5 H_{2} O

6-

{NbOF}_{2} \cdot 0.5 H_{2} O + 4 HF

7-

H_{2} {NbF}_{5} + 1.5 H_{2} O

8-

Q_{e x t}

9-

τ = \frac{Q_{L}}{ω_{0}} with Q_{L} = {(\frac{1}{Q_{0}} + \frac{1}{Q_{e x t}})}^{- 1} \approx \frac{Q_{0}}{2} .

10-

Q_{0} (E_{a c c})

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1101.3203

Formulas:

1-

ψ = ψ_{0} \exp (- r / Λ), r ≫ Λ,

2-

σ_{h} = σ_{h 0} \exp [- {(T_{1} / T)}^{1 / ν}], ν = 1, 2, 3 or 4;

3-

a_{B} ⩽ Λ ⩽ \infty .

4-

ψ_{2} = ψ_{20} \exp (- r / Λ_{2}), r ≫ Λ_{2},

5-

a_{2 B} ⩽ Λ_{2} ⩽ \infty .

6-

Δ σ_{h}^{(2)} = Δ σ_{h} \exp (- Δ_{L} / T) .

7-

Δ / δ ε = Δ g_{F} b^{3} ≫ 1,

8-

δ ε = {(g_{F} b^{3})}^{- 1}

9-

(g_{F} Δ) b^{3} ≫ 1 .

10-

Φ (𝐫) = | Φ | \exp (i φ (𝐫)),

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1009.4355

Formulas:

1-

O (\log^{2} k)

2-

O (\log k)

3-

O (\log k)

4-

O (\log k / \log \log k)

5-

O (n^{k + 1})

6-

O (n^{C + 1})

7-

Ω (k^{1 / 3})

8-

O (n \log C)

9-

O (n^{k + 1})

10-

O (1 / ϵ)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0106306

Formulas:

1-

- \infty < x < \infty

2-

G (x, y; t)

3-

G (x, y; t)

4-

\frac{\partial}{\partial t} G (x, y; t) = D (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) G (x, y; t) .

5-

lim_{x ↑ y or y ↓ x} G (x, y; t) = 1,

6-

G (x, y; 0) = \frac{1}{2} + \frac{1}{2} \exp [- 2 c_{0} (y - x)]

7-

lim_{x \to - \infty or y \to \infty} G (x, y; t) = \frac{1}{2} .

8-

G (x, y; t)

9-

ρ (x; t) = - {\frac{\partial}{\partial y} G (x, y; t) |}_{y = x} .

10-

ρ_{n} (x_{1}, x_{2}, \dots, x_{n}; t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1109.3762

Formulas:

1-

(0.5 - 1) \sim 10^{8} M_{⊙}

2-

Δ χ^{2} = 10

3-

{1.57}_{- 0.97}^{+ 0.70} \times 10^{- 5}

4-

c o s θ = 0.5

5-

Δ χ^{2} = 35

6-

(2.7 \pm 1.1) \times 10^{23}

7-

(2.3 \pm 0.2) \times 10^{- 4}

8-

(2.7 - 2.9) \times 10^{- 4}

9-

Δ χ^{2} = 36

10-

N_{H} = 2.6 \times 10^{24}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1706.00502

Formulas:

1-

\frac{\partial Σ}{\partial t} = - \nabla_{p} \cdot (Σ v_{p}),

2-

\frac{\partial}{\partial t} (Σ v_{p}) + {[\nabla \cdot (Σ v_{p} \otimes v_{p})]}_{p} = - \nabla_{p} P + Σ g_{p} + {(\nabla \cdot Π)}_{p},

3-

\frac{\partial e}{\partial t} + \nabla_{p} \cdot (e v_{p}) = - P (\nabla_{p} \cdot v_{p}) - Λ + Γ + {(\nabla \cdot v)}_{{pp}^{'}} : Π_{{pp}^{'}},

4-

P = (γ - 1) e

5-

v_{p} = v_{r} \hat{r} + v_{ϕ} \hat{ϕ}

6-

\nabla_{p} = \hat{r} \partial / \partial r + \hat{ϕ} r^{- 1} \partial / \partial ϕ

7-

g_{p} = g_{r} \hat{r} + g_{ϕ} \hat{ϕ}

8-

α_{visk} = 5 \times 10^{- 3}

9-

Λ = F_{c} σ T_{mp}^{4} \frac{τ}{1 + τ^{2}},

10-

T_{mp} = P μ / R Σ

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0505073

Formulas:

1-

𝒵 = \int D U \exp (- S_{G}) \prod_{f = u, d, s} Det M (m_{f}, μ_{f}) .

2-

μ_{B} = μ_{u} + μ_{d} + μ_{s}

3-

μ_{3} = μ_{u} - μ_{d}

4-

\exp (- Δ S)

5-

n_{i} = \frac{T}{V} \frac{\partial \ln 𝒵}{\partial μ_{i}}, χ_{i j} = \frac{T}{V} \frac{\partial^{2} \ln 𝒵}{\partial μ_{i} \partial μ_{j}} .

6-

χ_{n_{u}, n_{d}}

7-

\frac{Δ P}{T^{4}} \equiv \frac{P (μ, T)}{T^{4}} - \frac{P (0, T)}{T^{4}} = \sum_{n_{u}, n_{d}} χ_{n_{u}, n_{d}} \frac{1}{n_{u}!} {(\frac{μ_{u}}{T})}^{n_{u}} \frac{1}{n_{d}!} {(\frac{μ_{d}}{T})}^{n_{d}}

8-

r_{n + 2} = \sqrt{χ_{B}^{n} / χ_{B}^{n + 2}}

9-

T / T_{c} = 0.9

10-

T / T_{c} = 0.95

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0612532

Formulas:

1-

v_{rot} \sin i = 92.3 \pm 1.5 km s^{- 1}

2-

M_{sdB} = 0.5 M_{⊙}

3-

M \geq 1.47 M_{⊙}

4-

K = 341 \pm 1 km s^{- 1}

5-

P = 0.0950933 \pm 0.0000015 d

6-

T_{eff} = 35 200 \pm 500 K

7-

\log g = 5.61 \pm 0.06 dex

8-

v_{rot} \sin i

9-

v_{rot} \sin i = 92.3 \pm 1.5 km s^{- 1}

10-

f (M_{sdB}, M_{comp}) = \frac{M_{comp}^{3} {(\sin i)}^{3}}{{(M_{sdB} + M_{comp})}^{2}} = \frac{K_{sdB}^{2} P}{2 π G}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/math/0111257

Formulas:

1-

d o m (G)

2-

d o m (G) \geq 2

3-

d o m (G) = 2

4-

d o m (G) \geq (1 + o_{k} (1)) k / (2 \ln k)

5-

d o m (G)

6-

d o m (G) \geq 2

7-

| D | = γ (G)

8-

V (G) ∖ D

9-

d o m (G) > 2

10-

d o m (G)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/nlin/0601040

Formulas:

1-

ω = [0, 1, 11, 1, 1, \dots]

2-

x_{n} + a (1 - y_{n + 1}^{2})

3-

y_{n} - b \sin (2 π x_{n}),

4-

(x, y) \in 𝕋 \times ℝ

5-

\partial x_{n + 1} (x_{n}, y_{n}) / \partial y_{n} \neq 0

6-

S (x, y) = (x + 1 / 2, - y)

7-

ω = [0, 1, 11, 1, 1, \dots]

8-

ω_{i} = x_{i} / i

9-

| ω_{i} - ω_{i - 1} | < 10^{- 7}

10-

ω_{i} < \max_{n < i} ω_{n}

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/2003.04052

Formulas:

1-

D_{t r a i n} = {(X_{i}^{t}, Y_{i}^{t})}_{i = 1}^{N_{t r a i n}}

2-

D_{s u p p o r t} = {(X_{i}^{s}, Y_{i}^{s})}_{i = 1}^{N_{s u p p o r t}}

3-

D_{t e s t} = {(X_{i}^{q})}_{i = 1}^{N_{t e s t}}

4-

X_{i} \in ℝ^{H \times W \times 3}

5-

Y_{i} \in {0, 1}^{H \times W}

6-

{C_{t r a i n}} \cap {C_{s u p p o r t}} = \emptyset

7-

f_{θ} (\cdot)

8-

D_{t r a i n}

9-

c \notin C_{t r a i n}

10-

D_{t e s t}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/hep-th/0509206

Formulas:

1-

π^{2} ϵ_{0} G μ

2-

f^{z} = f^{ϕ} = 0

3-

f^{ρ} \sim (\frac{2.5}{π}) (\frac{G μ}{c^{2}}) (\frac{q^{2}}{4 π ϵ_{0} ρ_{0}^{2}})

4-

f^{ρ} ≃ \frac{π}{4} \frac{G μ}{c^{2}} \frac{e^{2}}{4 π ϵ_{0} ρ_{0}^{2}},

5-

ρ = \sqrt{x^{2} + y^{2}}

6-

r = \sqrt{x^{2} + y^{2} + x^{2}}

7-

\vec{F} = \frac{- e^{2}}{4 π ϵ_{0} r^{2}} \hat{r} .

8-

v_{1} = \frac{e^{2}}{4 π ϵ_{0} ℏ} .

9-

α = \frac{e^{2}}{4 π ϵ_{0} ℏ c} .

10-

{\vec{F}}_{tot} = (- \frac{e^{2}}{4 π ϵ_{0} ρ_{0}^{2}} + \frac{π}{4} \frac{G μ}{c^{2}} \frac{e^{2}}{4 π ϵ_{0} ρ_{0}^{2}}) \hat{ρ} .

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0411213

Formulas:

1-

H = J \sum_{⟨ i j ⟩} 𝐒_{i} \cdot 𝐒_{j},

2-

Θ_{C W} = 390

3-

⟨ 𝐒_{i} \cdot 𝐒_{j} - 𝐒_{k} \cdot 𝐒_{l} ⟩

4-

2 (R_{L T} - R_{H T})

5-

F d \bar{3} m

6-

2 (R_{LT} - R_{HT})

7-

R = {| \frac{1 - \sqrt{ϵ}}{1 + \sqrt{ϵ}} |}^{2}

8-

ϵ (ω, T) = ϵ_{\infty} + \sum_{j} \frac{S_{j} ω_{j}^{2}}{ω_{j}^{2} - ω^{2} - i ω γ_{j}}

9-

ϵ_{\infty} (T) \equiv ϵ (\infty, T)

10-

S_{j} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1003.1033

Formulas:

1-

𝚽_{𝟎} = 𝒉 𝒄 / 𝟐 𝒆

2-

I_{c} (H)

3-

Θ_{v} (x) = \arctan (\frac{x - x_{v}}{z_{v}}) + const .,

4-

Θ_{v} (x)

5-

I_{c} (H)

6-

Θ_{v} (x)

7-

x_{v} = L / 2

8-

I_{c} (Φ)

9-

I_{c} (Φ)

10-

𝑰_{𝒄} (𝚽)

Formulas (html):

<math><mrow id="id1.1.m1.1.1" xref="id1.1.m1.1.1.cmml"><msub id="id1.1.m1.1.1.2" xref="id1.1.m1.1.1.2.cmml"><mi id="id1.1.m1.1.1.2.2" xref="id1.1.m1.1.1.2.2.cmml">𝚽</mi><mn id="id1.1.m1.1.1.2.3" xref="id1.1.m1.1.1.2.3.cmml">𝟎</mn></msub><mo mathvariant="bold" id="id1.1.m1.1.1.1" xref="id1.1.m1.1.1.1.cmml">=</mo><mrow id="id1.1.m1.1.1.3" xref="id1.1.m1.1.1.3.cmml"><mrow id="id1.1.m1.1.1.3.2" xref="id1.1.m1.1.1.3.2.cmml"><mrow id="id1.1.m1.1.1.3.2.2" xref="id1.1.m1.1.1.3.2.2.cmml"><mi id="id1.1.m1.1.1.3.2.2.2" xref="id1.1.m1.1.1.3.2.2.2.cmml">𝒉</mi><mo id="id1.1.m1.1.1.3.2.2.1" xref="id1.1.m1.1.1.3.2.2.1.cmml">⁢</mo><mi id="id1.1.m1.1.1.3.2.2.3" xref="id1.1.m1.1.1.3.2.2.3.cmml">𝒄</mi></mrow><mo mathvariant="bold" id="id1.1.m1.1.1.3.2.1" xref="id1.1.m1.1.1.3.2.1.cmml">/</mo><mn id="id1.1.m1.1.1.3.2.3" xref="id1.1.m1.1.1.3.2.3.cmml">𝟐</mn></mrow><mo id="id1.1.m1.1.1.3.1" xref="id1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="id1.1.m1.1.1.3.3" xref="id1.1.m1.1.1.3.3.cmml">𝒆</mi></mrow></mrow></math>, <math><mrow id="p1.3.m3.1.2" xref="p1.3.m3.1.2.cmml"><msub id="p1.3.m3.1.2.2" xref="p1.3.m3.1.2.2.cmml"><mi id="p1.3.m3.1.2.2.2" xref="p1.3.m3.1.2.2.2.cmml">I</mi><mi id="p1.3.m3.1.2.2.3" xref="p1.3.m3.1.2.2.3.cmml">c</mi></msub><mo id="p1.3.m3.1.2.1" xref="p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="p1.3.m3.1.2.3.2" xref="p1.3.m3.1.2.cmml"><mo stretchy="false" id="p1.3.m3.1.2.3.2.1" xref="p1.3.m3.1.2.cmml">(</mo><mi id="p1.3.m3.1.1" xref="p1.3.m3.1.1.cmml">H</mi><mo stretchy="false" id="p1.3.m3.1.2.3.2.2" xref="p1.3.m3.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.E1.m1.4.4.1" xref="S0.E1.m1.4.4.1.1.cmml"><mrow id="S0.E1.m1.4.4.1.1" xref="S0.E1.m1.4.4.1.1.cmml"><mrow id="S0.E1.m1.4.4.1.1.2" xref="S0.E1.m1.4.4.1.1.2.cmml"><msub id="S0.E1.m1.4.4.1.1.2.2" xref="S0.E1.m1.4.4.1.1.2.2.cmml"><mi mathvariant="normal" id="S0.E1.m1.4.4.1.1.2.2.2" xref="S0.E1.m1.4.4.1.1.2.2.2.cmml">Θ</mi><mi id="S0.E1.m1.4.4.1.1.2.2.3" xref="S0.E1.m1.4.4.1.1.2.2.3.cmml">v</mi></msub><mo id="S0.E1.m1.4.4.1.1.2.1" xref="S0.E1.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S0.E1.m1.4.4.1.1.2.3.2" xref="S0.E1.m1.4.4.1.1.2.cmml"><mo stretchy="false" id="S0.E1.m1.4.4.1.1.2.3.2.1" xref="S0.E1.m1.4.4.1.1.2.cmml">(</mo><mi id="S0.E1.m1.1.1" xref="S0.E1.m1.1.1.cmml">x</mi><mo stretchy="false" id="S0.E1.m1.4.4.1.1.2.3.2.2" xref="S0.E1.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S0.E1.m1.4.4.1.1.1" xref="S0.E1.m1.4.4.1.1.1.cmml">=</mo><mrow id="S0.E1.m1.4.4.1.1.3" xref="S0.E1.m1.4.4.1.1.3.cmml"><mrow id="S0.E1.m1.4.4.1.1.3.2.2" xref="S0.E1.m1.4.4.1.1.3.2.1.cmml"><mi id="S0.E1.m1.2.2" xref="S0.E1.m1.2.2.cmml">arctan</mi><mo id="S0.E1.m1.4.4.1.1.3.2.2a" xref="S0.E1.m1.4.4.1.1.3.2.1.cmml">⁡</mo><mrow id="S0.E1.m1.4.4.1.1.3.2.2.1" xref="S0.E1.m1.4.4.1.1.3.2.1.cmml"><mo id="S0.E1.m1.4.4.1.1.3.2.2.1.1" xref="S0.E1.m1.4.4.1.1.3.2.1.cmml">(</mo><mfrac id="S0.E1.m1.3.3" xref="S0.E1.m1.3.3.cmml"><mrow id="S0.E1.m1.3.3.2" xref="S0.E1.m1.3.3.2.cmml"><mi id="S0.E1.m1.3.3.2.2" xref="S0.E1.m1.3.3.2.2.cmml">x</mi><mo id="S0.E1.m1.3.3.2.1" xref="S0.E1.m1.3.3.2.1.cmml">-</mo><msub id="S0.E1.m1.3.3.2.3" xref="S0.E1.m1.3.3.2.3.cmml"><mi id="S0.E1.m1.3.3.2.3.2" xref="S0.E1.m1.3.3.2.3.2.cmml">x</mi><mi id="S0.E1.m1.3.3.2.3.3" xref="S0.E1.m1.3.3.2.3.3.cmml">v</mi></msub></mrow><msub id="S0.E1.m1.3.3.3" xref="S0.E1.m1.3.3.3.cmml"><mi id="S0.E1.m1.3.3.3.2" xref="S0.E1.m1.3.3.3.2.cmml">z</mi><mi id="S0.E1.m1.3.3.3.3" xref="S0.E1.m1.3.3.3.3.cmml">v</mi></msub></mfrac><mo id="S0.E1.m1.4.4.1.1.3.2.2.1.2" xref="S0.E1.m1.4.4.1.1.3.2.1.cmml">)</mo></mrow></mrow><mo id="S0.E1.m1.4.4.1.1.3.1" xref="S0.E1.m1.4.4.1.1.3.1.cmml">+</mo><mi id="S0.E1.m1.4.4.1.1.3.3" xref="S0.E1.m1.4.4.1.1.3.3.cmml">const</mi></mrow></mrow><mo id="S0.E1.m1.4.4.1.2" xref="S0.E1.m1.4.4.1.1.cmml">.</mo><mo id="S0.E1.m1.4.4.1.3" xref="S0.E1.m1.4.4.1.1.cmml">,</mo></mrow></math>, <math><mrow id="p2.5.m4.1.2" xref="p2.5.m4.1.2.cmml"><msub id="p2.5.m4.1.2.2" xref="p2.5.m4.1.2.2.cmml"><mi mathvariant="normal" id="p2.5.m4.1.2.2.2" xref="p2.5.m4.1.2.2.2.cmml">Θ</mi><mi id="p2.5.m4.1.2.2.3" xref="p2.5.m4.1.2.2.3.cmml">v</mi></msub><mo id="p2.5.m4.1.2.1" xref="p2.5.m4.1.2.1.cmml">⁢</mo><mrow id="p2.5.m4.1.2.3.2" xref="p2.5.m4.1.2.cmml"><mo stretchy="false" id="p2.5.m4.1.2.3.2.1" xref="p2.5.m4.1.2.cmml">(</mo><mi id="p2.5.m4.1.1" xref="p2.5.m4.1.1.cmml">x</mi><mo stretchy="false" id="p2.5.m4.1.2.3.2.2" xref="p2.5.m4.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p3.1.m1.1.2" xref="p3.1.m1.1.2.cmml"><msub id="p3.1.m1.1.2.2" xref="p3.1.m1.1.2.2.cmml"><mi id="p3.1.m1.1.2.2.2" xref="p3.1.m1.1.2.2.2.cmml">I</mi><mi id="p3.1.m1.1.2.2.3" xref="p3.1.m1.1.2.2.3.cmml">c</mi></msub><mo id="p3.1.m1.1.2.1" xref="p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="p3.1.m1.1.2.3.2" xref="p3.1.m1.1.2.cmml"><mo stretchy="false" id="p3.1.m1.1.2.3.2.1" xref="p3.1.m1.1.2.cmml">(</mo><mi id="p3.1.m1.1.1" xref="p3.1.m1.1.1.cmml">H</mi><mo stretchy="false" id="p3.1.m1.1.2.3.2.2" xref="p3.1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.F1.8.m2.1.2" xref="S0.F1.8.m2.1.2.cmml"><msub id="S0.F1.8.m2.1.2.2" xref="S0.F1.8.m2.1.2.2.cmml"><mi mathvariant="normal" id="S0.F1.8.m2.1.2.2.2" xref="S0.F1.8.m2.1.2.2.2.cmml">Θ</mi><mi id="S0.F1.8.m2.1.2.2.3" xref="S0.F1.8.m2.1.2.2.3.cmml">v</mi></msub><mo id="S0.F1.8.m2.1.2.1" xref="S0.F1.8.m2.1.2.1.cmml">⁢</mo><mrow id="S0.F1.8.m2.1.2.3.2" xref="S0.F1.8.m2.1.2.cmml"><mo stretchy="false" id="S0.F1.8.m2.1.2.3.2.1" xref="S0.F1.8.m2.1.2.cmml">(</mo><mi id="S0.F1.8.m2.1.1" xref="S0.F1.8.m2.1.1.cmml">x</mi><mo stretchy="false" id="S0.F1.8.m2.1.2.3.2.2" xref="S0.F1.8.m2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.F1.10.m4.1.1" xref="S0.F1.10.m4.1.1.cmml"><msub id="S0.F1.10.m4.1.1.2" xref="S0.F1.10.m4.1.1.2.cmml"><mi id="S0.F1.10.m4.1.1.2.2" xref="S0.F1.10.m4.1.1.2.2.cmml">x</mi><mi id="S0.F1.10.m4.1.1.2.3" xref="S0.F1.10.m4.1.1.2.3.cmml">v</mi></msub><mo id="S0.F1.10.m4.1.1.1" xref="S0.F1.10.m4.1.1.1.cmml">=</mo><mrow id="S0.F1.10.m4.1.1.3" xref="S0.F1.10.m4.1.1.3.cmml"><mi id="S0.F1.10.m4.1.1.3.2" xref="S0.F1.10.m4.1.1.3.2.cmml">L</mi><mo id="S0.F1.10.m4.1.1.3.1" xref="S0.F1.10.m4.1.1.3.1.cmml">/</mo><mn id="S0.F1.10.m4.1.1.3.3" xref="S0.F1.10.m4.1.1.3.3.cmml">2</mn></mrow></mrow></math>, <math><mrow id="p5.8.m8.1.2" xref="p5.8.m8.1.2.cmml"><msub id="p5.8.m8.1.2.2" xref="p5.8.m8.1.2.2.cmml"><mi id="p5.8.m8.1.2.2.2" xref="p5.8.m8.1.2.2.2.cmml">I</mi><mi id="p5.8.m8.1.2.2.3" xref="p5.8.m8.1.2.2.3.cmml">c</mi></msub><mo id="p5.8.m8.1.2.1" xref="p5.8.m8.1.2.1.cmml">⁢</mo><mrow id="p5.8.m8.1.2.3.2" xref="p5.8.m8.1.2.cmml"><mo stretchy="false" id="p5.8.m8.1.2.3.2.1" xref="p5.8.m8.1.2.cmml">(</mo><mi mathvariant="normal" id="p5.8.m8.1.1" xref="p5.8.m8.1.1.cmml">Φ</mi><mo stretchy="false" id="p5.8.m8.1.2.3.2.2" xref="p5.8.m8.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p5.12.m12.1.2" xref="p5.12.m12.1.2.cmml"><msub id="p5.12.m12.1.2.2" xref="p5.12.m12.1.2.2.cmml"><mi id="p5.12.m12.1.2.2.2" xref="p5.12.m12.1.2.2.2.cmml">I</mi><mi id="p5.12.m12.1.2.2.3" xref="p5.12.m12.1.2.2.3.cmml">c</mi></msub><mo id="p5.12.m12.1.2.1" xref="p5.12.m12.1.2.1.cmml">⁢</mo><mrow id="p5.12.m12.1.2.3.2" xref="p5.12.m12.1.2.cmml"><mo stretchy="false" id="p5.12.m12.1.2.3.2.1" xref="p5.12.m12.1.2.cmml">(</mo><mi mathvariant="normal" id="p5.12.m12.1.1" xref="p5.12.m12.1.1.cmml">Φ</mi><mo stretchy="false" id="p5.12.m12.1.2.3.2.2" xref="p5.12.m12.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.F2.12.1.m1.1.2" xref="S0.F2.12.1.m1.1.2.cmml"><msub id="S0.F2.12.1.m1.1.2.2" xref="S0.F2.12.1.m1.1.2.2.cmml"><mi id="S0.F2.12.1.m1.1.2.2.2" xref="S0.F2.12.1.m1.1.2.2.2.cmml">𝑰</mi><mi id="S0.F2.12.1.m1.1.2.2.3" xref="S0.F2.12.1.m1.1.2.2.3.cmml">𝒄</mi></msub><mo mathvariant="sans-serif" id="S0.F2.12.1.m1.1.2.1" xref="S0.F2.12.1.m1.1.2.1.cmml">⁢</mo><mrow id="S0.F2.12.1.m1.1.2.3.2" xref="S0.F2.12.1.m1.1.2.cmml"><mo mathvariant="bold" stretchy="false" id="S0.F2.12.1.m1.1.2.3.2.1" xref="S0.F2.12.1.m1.1.2.cmml">(</mo><mi id="S0.F2.12.1.m1.1.1" xref="S0.F2.12.1.m1.1.1.cmml">𝚽</mi><mo mathvariant="bold" stretchy="false" id="S0.F2.12.1.m1.1.2.3.2.2" xref="S0.F2.12.1.m1.1.2.cmml">)</mo></mrow></mrow></math>

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1201.2735

Formulas:

1-

A (e, e^{'} ρ^{0})

2-

r_{⟂} \sim 1 / Q

3-

l_{c} = 2 ν / (Q^{2} + M_{q \bar{q}}^{2})

4-

< M_{π^{+} π^{-}} <

5-

z_{ρ} = E_{ρ} / ν

6-

e p \to e Δ^{+ +} π^{-}

7-

e p \to e Δ^{0} π^{+}

8-

e p \to e p π^{+} π^{-}

9-

T_{A} = (N_{A}^{ρ} / ℒ_{A}^{i n t}) / (N_{D}^{ρ} / ℒ_{D}^{i n t}),

10-

ℒ_{A, D}^{i n t}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0404419

Formulas:

1-

S (A, B) = f (S (A), S (B)),

2-

f (S (A), S (B)) = f (S (B), S (A)) .

3-

f (0, S (B)) = S (B), f (S (A), 0) = S (A),

4-

f (0, 0) = 0 .

5-

S (A, B) = S (A) + S (B),

6-

S (A, B) = S (A) + S (B) + ω S (A) S (B),

7-

E (A, B) = E (A) + E (B),

8-

d S (A, B)

9-

d E (A, B)

10-

\frac{\partial S (A, B)}{\partial S (A)} \frac{\partial S (A)}{\partial E (A)} = \frac{\partial S (A, B)}{\partial S (B)} \frac{\partial S (B)}{\partial E (B)} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1404.4293

Formulas:

1-

Δ g (x)

2-

Δ G \equiv \int_{0}^{1} 𝑑 x Δ g (x)

3-

\frac{1}{2} = \frac{1}{2} Δ Σ + Δ G + L_{q} + L_{g},

4-

Δ g (x)

5-

5 ≲ p_{T} ≲ 30 GeV

6-

x Δ g (x, Q_{0}^{2}) = N_{g} x^{α_{g}} {(1 - x)}^{β_{g}} (1 + η_{g} x^{κ_{g}}),

7-

N_{g}, α_{g}, β_{g}, η_{g}

8-

| Δ f | / f \leq 1

9-

f (x, Q^{2})

10-

Δ g (x, Q^{2})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0211089

Formulas:

1-

S \sim {(t_{0} - t)}^{2 / 3}

2-

S \sim (t_{0} - t)

3-

S \sim {(t_{0} - t)}^{2 / 3}

4-

S \sim (t_{0} - t)

5-

120 - 150 μ m

6-

F_{0} = 1.36 R^{2} E^{2}

7-

E_{1} < E < E_{2}

8-

S \sim {(t_{0} - t)}^{β}

9-

N \sim 1 / t^{α}

10-

Γ = 4 π^{2} f^{2} A_{0} / g

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9711075

Formulas:

1-

Φ_{axi} (R) = - \frac{c^{2}}{a} \frac{1}{{(R^{2} + a^{2})}^{1 / 2}},

2-

c = v_{\max} {(27 / 4)}^{1 / 4} a

3-

Φ_{bar} (R, θ) = ϵ (R) Φ_{axi} \cos 2 θ,

4-

ϵ (R) = ϵ_{0} \frac{a R^{2}}{{(R^{2} + a^{2})}^{3 / 2}},

5-

Φ_{BH} (R) = - f_{BH} \frac{c^{2}}{a} \frac{1}{{(R^{2} + a_{BH}^{2})}^{1 / 2}},

6-

c^{2} / a = 48.7

7-

f_{BH} = 0.01, 0.001

8-

Ω_{bar} = 7.0, 3.5

9-

Ω_{bar} = Ω (R) - \frac{κ (R)}{2},

10-

κ^{2} = R \frac{d Ω^{2}}{d R} + 4 Ω^{2} .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0901.4706

Formulas:

1-

E_{0} δ (t)

2-

σ (ω) = \frac{4 π}{c} ω ℑ 𝔪 α (ω),

3-

E_{Δ SCF} = E (N, e–h) - E (N)

4-

E (N, e–h)

5-

σ (ω) = 9 \frac{ω}{c} V_{obj} \frac{ℑ 𝔪 ϵ (ω)}{{[ℜ 𝔢 ϵ (ω) + 2]}^{2} + {[ℑ 𝔪 ϵ (ω)]}^{2}},

6-

ℑ 𝔪 ϵ (ω)

7-

ℜ 𝔢 ϵ (ω)

8-

{\tilde{σ}}_{abs} (ω)

9-

σ_{lum} (ω) \sim ω^{2} \frac{1}{e^{ℏ ω / k_{b} T} - 1} {\tilde{σ}}_{abs} (ω),

10-

{\tilde{σ}}_{abs} (ω)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0812.1669

Formulas:

1-

ω_{l} (𝐤)

2-

E^{Q H A}

3-

E^{Q H A} (V) = E^{c l} (V) + \frac{ℏ}{2} \sum_{l = 1}^{3 N} \int_{B Z} g (𝐤) ω_{l} (𝐤, V) 𝑑 𝐤

4-

E^{Q H A}

5-

ζ_{l}^{m} ζ_{l^{'}}^{n}

6-

E^{Q H A} (a) = E^{c l} (a) + \frac{ℏ}{2} \sum_{i = 1}^{N - 1} ω_{i} (a),

7-

E^{c l} (a)

8-

ω_{i} (a)

9-

ω (k, a) = 2 \sqrt{\frac{V^{''} (a)}{m}} | \sin (\frac{k a}{2}) |,

10-

k = π n / (N - 1) a

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/q-bio/0605035

Formulas:

1-

H_{GG} = \sum_{(i, j) \sim (i^{'}, j^{'})} J (τ (σ_{i, j}), τ (σ_{i^{'}, j^{'}})) (1 - δ_{σ_{i, j}, σ_{i^{'}, j^{'}}}) + λ \sum_{σ} {(a (σ) - A_{τ (σ)})}^{2} Γ (A_{τ (σ)}),

2-

τ (σ_{i j})

3-

(1 - δ_{σ_{i, j}, σ_{i^{'}, j^{'}}})

4-

A_{τ (σ)}

5-

i = 1, \dots, n

6-

| Dir (x_{i}) |

7-

H_{GG} = \sum_{σ \sim σ^{'}} | σ \cap σ^{'} | J (τ (σ), τ (σ^{'})) + λ \sum_{σ} {(a (σ) - A_{τ (σ)})}^{2} Γ (A_{τ (σ)})

8-

| σ \cap σ^{'} |

9-

| σ \cap σ^{'} |

10-

| Dir (x_{i} \cap x_{j}) |

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0108193

Formulas:

1-

0 .^{''} 2 - 0 .^{''} 4

2-

1 - 2 \times 10^{16}

3-

0 .^{''} 1 - 0 .^{''} 6

4-

\sim 0 .^{''} 3

5-

1^{''} \approx 5 \times 10^{16}

6-

λ λ 4959, 5007

7-

λ λ 9824, 9850

8-

0 .^{''} 1

9-

1 .^{''} 5

10-

\sim 0 .^{''} 2 - 0 .^{''} 6

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0603176

Formulas:

1-

\log (L_{R} / L_{X}) \sim - 15.5

2-

f_{c} = 19 \pm 6 %

3-

T_{b} = 2 \times 10^{9} F_{ν, mJy} ν_{GHz}^{- 2} d_{pc}^{2} {(R / R_{J})}^{- 2} K

4-

R_{J} \approx R_{s} \approx 7 \times 10^{9}

5-

R \sim (1 - 2) R_{J}

6-

T_{b} \approx 10^{8} - 10^{9}

7-

R \approx 0.035 R_{s} \approx 2.5 \times 10^{8}

8-

N (γ) \propto γ^{- p}

9-

ν_{m} \approx 1.66 \times 10^{4} n_{e}^{0.23} R^{0.23} B^{0.77} Hz,

10-

F_{ν, m} \approx 1.54 \times 10^{- 4} B^{- 0.76} R^{2} d^{- 2} ν_{m}^{2.76} μ Jy,

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0804.2202

Formulas:

1-

ρ = - 0.88, - 0.88, - 0.82, - 0.73

2-

p = 4.8 \times 10^{- 11}, 7.3 \times 10^{- 12}, 2.9 \times 10^{- 9}, 7.8 \times 10^{- 7}

3-

h \in (0.5, 1]

4-

k = \min [N - 1, 8]

5-

D (N) = {⟨ Θ (1 - \frac{\max (A_{f}, N - A_{f})}{N}) ⟩}_{A_{i}},

6-

(0, 1, \dots N)

7-

N = 15, 25, 35

8-

5 > h k = 0.6

9-

4 / 7 \approx 0.57 < h

10-

D (8) = \frac{1}{9} = 0 . \bar{1}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9407057

Formulas:

1-

- L \leq x \leq L

2-

P_{n} (x, y)

3-

P_{n + 1} (x, y)

4-

p_{x} (y - 1) p_{y} (y - 1) P_{n} (x - 1, y - 1)

5-

p_{x} (y + 1) [1 - p_{y} (y + 1)] P_{n} (x - 1, y + 1)

6-

[1 - p_{x} (y - 1)] p_{y} (y - 1) P_{n} (x + 1, y - 1)

7-

[1 - p_{x} (y + 1)] [1 - p_{y} (y + 1)] P_{n} (x + 1, y + 1)

8-

δ_{n + 1, 0} δ_{x, x_{o}} δ_{y, y_{o}},

9-

p_{x} (p_{y})

10-

p_{x / y} (y) = {\begin{matrix} p_{x / y}^{u} & if y \geq 1 \\ p_{x / y}^{d} & otherwise, \end{matrix}

Formulas (html):

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xref="S2.Ex1.m3.4.4.4.2.2.2.cmml">y</mi><mo id="S2.Ex1.m3.4.4.4.2.2.1" xref="S2.Ex1.m3.4.4.4.2.2.1.cmml">-</mo><mn id="S2.Ex1.m3.4.4.4.2.2.3" xref="S2.Ex1.m3.4.4.4.2.2.3.cmml">1</mn></mrow><mo stretchy="false" id="S2.Ex1.m3.4.4.4.2.5" xref="S2.Ex1.m3.4.4.4.3.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.Ex2.m3.4.4" xref="S2.Ex2.m3.4.4.cmml"><msub id="S2.Ex2.m3.4.4.6" xref="S2.Ex2.m3.4.4.6.cmml"><mi id="S2.Ex2.m3.4.4.6.2" xref="S2.Ex2.m3.4.4.6.2.cmml">p</mi><mi id="S2.Ex2.m3.4.4.6.3" xref="S2.Ex2.m3.4.4.6.3.cmml">x</mi></msub><mo id="S2.Ex2.m3.4.4.5" xref="S2.Ex2.m3.4.4.5.cmml">⁢</mo><mrow id="S2.Ex2.m3.1.1.1.1" xref="S2.Ex2.m3.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.Ex2.m3.1.1.1.1.2" xref="S2.Ex2.m3.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex2.m3.1.1.1.1.1" xref="S2.Ex2.m3.1.1.1.1.1.cmml"><mi id="S2.Ex2.m3.1.1.1.1.1.2" xref="S2.Ex2.m3.1.1.1.1.1.2.cmml">y</mi><mo id="S2.Ex2.m3.1.1.1.1.1.1" xref="S2.Ex2.m3.1.1.1.1.1.1.cmml">+</mo><mn id="S2.Ex2.m3.1.1.1.1.1.3" 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id="S2.E1.m3.7.7.1.2" xref="S2.E1.m3.7.7.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p1.7.m1.1.1" xref="S2.SS1.p1.7.m1.1.1.cmml"><mpadded width="+3.3pt" id="S2.SS1.p1.7.m1.1.1.3" xref="S2.SS1.p1.7.m1.1.1.3.cmml"><msub id="S2.SS1.p1.7.m1.1.1.3a" xref="S2.SS1.p1.7.m1.1.1.3.cmml"><mi id="S2.SS1.p1.7.m1.1.1.3.2" xref="S2.SS1.p1.7.m1.1.1.3.2.cmml">p</mi><mi id="S2.SS1.p1.7.m1.1.1.3.3" xref="S2.SS1.p1.7.m1.1.1.3.3.cmml">x</mi></msub></mpadded><mo id="S2.SS1.p1.7.m1.1.1.2" xref="S2.SS1.p1.7.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.7.m1.1.1.1.1" xref="S2.SS1.p1.7.m1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.7.m1.1.1.1.1.2" xref="S2.SS1.p1.7.m1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS1.p1.7.m1.1.1.1.1.1" xref="S2.SS1.p1.7.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.7.m1.1.1.1.1.1.2" xref="S2.SS1.p1.7.m1.1.1.1.1.1.2.cmml">p</mi><mi id="S2.SS1.p1.7.m1.1.1.1.1.1.3" xref="S2.SS1.p1.7.m1.1.1.1.1.1.3.cmml">y</mi></msub><mo stretchy="false" id="S2.SS1.p1.7.m1.1.1.1.1.3" xref="S2.SS1.p1.7.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E2.m1.3.4" xref="S2.E2.m1.3.4.cmml"><mrow id="S2.E2.m1.3.4.2" xref="S2.E2.m1.3.4.2.cmml"><msub id="S2.E2.m1.3.4.2.2" xref="S2.E2.m1.3.4.2.2.cmml"><mi id="S2.E2.m1.3.4.2.2.2" xref="S2.E2.m1.3.4.2.2.2.cmml">p</mi><mrow id="S2.E2.m1.3.4.2.2.3" xref="S2.E2.m1.3.4.2.2.3.cmml"><mi id="S2.E2.m1.3.4.2.2.3.2" xref="S2.E2.m1.3.4.2.2.3.2.cmml">x</mi><mo id="S2.E2.m1.3.4.2.2.3.1" xref="S2.E2.m1.3.4.2.2.3.1.cmml">/</mo><mi id="S2.E2.m1.3.4.2.2.3.3" xref="S2.E2.m1.3.4.2.2.3.3.cmml">y</mi></mrow></msub><mo id="S2.E2.m1.3.4.2.1" xref="S2.E2.m1.3.4.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.4.2.3.2" xref="S2.E2.m1.3.4.2.cmml"><mo stretchy="false" id="S2.E2.m1.3.4.2.3.2.1" xref="S2.E2.m1.3.4.2.cmml">(</mo><mi id="S2.E2.m1.3.3" xref="S2.E2.m1.3.3.cmml">y</mi><mo stretchy="false" id="S2.E2.m1.3.4.2.3.2.2" xref="S2.E2.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.3.4.1" xref="S2.E2.m1.3.4.1.cmml">=</mo><mrow id="S2.E2.m1.3.4.3.2" xref="S2.E2.m1.3.4.3.1.cmml"><mo id="S2.E2.m1.3.4.3.2.1" xref="S2.E2.m1.3.4.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml"><mtr id="S2.E2.m1.2.2a" xref="S2.E2.m1.2.2.cmml"><mtd columnalign="left" id="S2.E2.m1.2.2b" xref="S2.E2.m1.2.2.cmml"><msubsup id="S2.E2.m1.1.1.1.2.1" xref="S2.E2.m1.1.1.1.2.1.cmml"><mi id="S2.E2.m1.1.1.1.2.1.2.2" xref="S2.E2.m1.1.1.1.2.1.2.2.cmml">p</mi><mrow id="S2.E2.m1.1.1.1.2.1.2.3" xref="S2.E2.m1.1.1.1.2.1.2.3.cmml"><mi id="S2.E2.m1.1.1.1.2.1.2.3.2" xref="S2.E2.m1.1.1.1.2.1.2.3.2.cmml">x</mi><mo id="S2.E2.m1.1.1.1.2.1.2.3.1" xref="S2.E2.m1.1.1.1.2.1.2.3.1.cmml">/</mo><mi id="S2.E2.m1.1.1.1.2.1.2.3.3" xref="S2.E2.m1.1.1.1.2.1.2.3.3.cmml">y</mi></mrow><mi id="S2.E2.m1.1.1.1.2.1.3" xref="S2.E2.m1.1.1.1.2.1.3.cmml">u</mi></msubsup></mtd><mtd columnalign="left" id="S2.E2.m1.2.2c" xref="S2.E2.m1.2.2.cmml"><mrow id="S2.E2.m1.1.1.1.1.1" 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xref="S2.E2.m1.2.2.cmml"><mrow id="S2.E2.m1.2.2.2.1.1.3" xref="S2.E2.m1.2.2.2.1.1.1a.cmml"><mtext id="S2.E2.m1.2.2.2.1.1.1" xref="S2.E2.m1.2.2.2.1.1.1.cmml">otherwise</mtext><mo id="S2.E2.m1.2.2.2.1.1.3.1" xref="S2.E2.m1.2.2.2.1.1.1a.cmml">,</mo></mrow></mtd></mtr></mtable><mi id="S2.E2.m1.3.4.3.2.2" xref="S2.E2.m1.3.4.3.1.1.cmml"/></mrow></mrow></math>

Correct Categorie: cond-mat

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/2010.16031

Formulas:

1-

{F_{i}}_{t = 1}^{T}

2-

(x_{i}^{t}, y_{i}^{t})

3-

{\hat{B}}_{j}^{t + 1} = H (B_{j}^{t})

4-

{\hat{B}}_{j}^{t + 1} = B_{j}^{t}

5-

{a_{j k}}_{k = 1}^{K}

6-

w_{j k} = A (a_{j k}, {\hat{B}}_{j}^{t})

7-

{B_{j k}^{t + 1}}

8-

{c_{j k}^{t + 1}}

9-

B_{j}^{t + 1} = \sum_{k = 1}^{K} \frac{w_{j k}^{t + 1} B_{j k}^{t + 1}}{w_{j k}^{t + 1}}, c_{j}^{t + 1} = \sum_{k = 1}^{K} \frac{w_{j k}^{t + 1} c_{j k}^{t + 1}}{w_{j k}^{t + 1}},

10-

σ_{a c t i v e}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1405.7475

Formulas:

1-

ω_{i} := ⟨ V_{i}, E_{i}, l_{i} ⟩,

2-

e = ⟨ v_{s}, v_{t} ⟩

3-

l_{i} (v)

4-

l_{i} (v)

5-

r := ⟨ v_{r}, ω_{r} ⟩

6-

ω_{a} \overset{r}{⟹} ω_{b}

7-

ω_{a} \overset{r}{⟹} ω_{b} = ⟨ V_{a} \cup V_{r}, E_{a} \cup E_{r}, l_{b} ⟩

8-

l_{b} (v) = {\begin{matrix} l_{r} (v) & if v \in (V_{r} ∖ V_{a}) \cup {v_{r}} \\ l_{a} (v) & otherwise. \end{matrix}

9-

ω_{a} \overset{r}{⟹} ω_{b}

10-

γ := ⟨ m_{γ}, f_{γ} ⟩,

Formulas (html):

<math><mrow id="S4.Ex1.m1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml"><mrow id="S4.Ex1.m1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml"><msub id="S4.Ex1.m1.1.1.1.1.5" xref="S4.Ex1.m1.1.1.1.1.5.cmml"><mi id="S4.Ex1.m1.1.1.1.1.5.2" xref="S4.Ex1.m1.1.1.1.1.5.2.cmml">ω</mi><mi id="S4.Ex1.m1.1.1.1.1.5.3" xref="S4.Ex1.m1.1.1.1.1.5.3.cmml">i</mi></msub><mo id="S4.Ex1.m1.1.1.1.1.4" xref="S4.Ex1.m1.1.1.1.1.4.cmml">:=</mo><mrow id="S4.Ex1.m1.1.1.1.1.3.3" xref="S4.Ex1.m1.1.1.1.1.3.4.cmml"><mo id="S4.Ex1.m1.1.1.1.1.3.3.4" xref="S4.Ex1.m1.1.1.1.1.3.4.cmml">⟨</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.1.2.cmml">V</mi><mi id="S4.Ex1.m1.1.1.1.1.1.1.1.3" xref="S4.Ex1.m1.1.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S4.Ex1.m1.1.1.1.1.3.3.5" xref="S4.Ex1.m1.1.1.1.1.3.4.cmml">,</mo><msub id="S4.Ex1.m1.1.1.1.1.2.2.2" xref="S4.Ex1.m1.1.1.1.1.2.2.2.cmml"><mi id="S4.Ex1.m1.1.1.1.1.2.2.2.2" 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xref="S4.SS1.p2.5.m4.1.1.1.1.1.cmml"><mi id="S4.SS1.p2.5.m4.1.1.1.1.1.2" xref="S4.SS1.p2.5.m4.1.1.1.1.1.2.cmml">v</mi><mi id="S4.SS1.p2.5.m4.1.1.1.1.1.3" xref="S4.SS1.p2.5.m4.1.1.1.1.1.3.cmml">s</mi></msub><mo id="S4.SS1.p2.5.m4.2.2.2.2.4" xref="S4.SS1.p2.5.m4.2.2.2.3.cmml">,</mo><msub id="S4.SS1.p2.5.m4.2.2.2.2.2" xref="S4.SS1.p2.5.m4.2.2.2.2.2.cmml"><mi id="S4.SS1.p2.5.m4.2.2.2.2.2.2" xref="S4.SS1.p2.5.m4.2.2.2.2.2.2.cmml">v</mi><mi id="S4.SS1.p2.5.m4.2.2.2.2.2.3" xref="S4.SS1.p2.5.m4.2.2.2.2.2.3.cmml">t</mi></msub><mo id="S4.SS1.p2.5.m4.2.2.2.2.5" xref="S4.SS1.p2.5.m4.2.2.2.3.cmml">⟩</mo></mrow></mrow></math>, <math><mrow id="S4.SS1.p2.8.m7.1.2" xref="S4.SS1.p2.8.m7.1.2.cmml"><msub id="S4.SS1.p2.8.m7.1.2.2" xref="S4.SS1.p2.8.m7.1.2.2.cmml"><mi id="S4.SS1.p2.8.m7.1.2.2.2" xref="S4.SS1.p2.8.m7.1.2.2.2.cmml">l</mi><mi id="S4.SS1.p2.8.m7.1.2.2.3" xref="S4.SS1.p2.8.m7.1.2.2.3.cmml">i</mi></msub><mo id="S4.SS1.p2.8.m7.1.2.1" xref="S4.SS1.p2.8.m7.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.8.m7.1.2.3.2" xref="S4.SS1.p2.8.m7.1.2.cmml"><mo stretchy="false" id="S4.SS1.p2.8.m7.1.2.3.2.1" xref="S4.SS1.p2.8.m7.1.2.cmml">(</mo><mi id="S4.SS1.p2.8.m7.1.1" xref="S4.SS1.p2.8.m7.1.1.cmml">v</mi><mo stretchy="false" id="S4.SS1.p2.8.m7.1.2.3.2.2" xref="S4.SS1.p2.8.m7.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S4.SS1.p2.10.m9.1.2" xref="S4.SS1.p2.10.m9.1.2.cmml"><msub id="S4.SS1.p2.10.m9.1.2.2" xref="S4.SS1.p2.10.m9.1.2.2.cmml"><mi id="S4.SS1.p2.10.m9.1.2.2.2" xref="S4.SS1.p2.10.m9.1.2.2.2.cmml">l</mi><mi id="S4.SS1.p2.10.m9.1.2.2.3" xref="S4.SS1.p2.10.m9.1.2.2.3.cmml">i</mi></msub><mo id="S4.SS1.p2.10.m9.1.2.1" xref="S4.SS1.p2.10.m9.1.2.1.cmml">⁢</mo><mrow id="S4.SS1.p2.10.m9.1.2.3.2" xref="S4.SS1.p2.10.m9.1.2.cmml"><mo stretchy="false" id="S4.SS1.p2.10.m9.1.2.3.2.1" xref="S4.SS1.p2.10.m9.1.2.cmml">(</mo><mi id="S4.SS1.p2.10.m9.1.1" xref="S4.SS1.p2.10.m9.1.1.cmml">v</mi><mo stretchy="false" id="S4.SS1.p2.10.m9.1.2.3.2.2" xref="S4.SS1.p2.10.m9.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S4.Ex2.m1.2.2" xref="S4.Ex2.m1.2.2.cmml"><mi id="S4.Ex2.m1.2.2.4" xref="S4.Ex2.m1.2.2.4.cmml">r</mi><mo id="S4.Ex2.m1.2.2.3" xref="S4.Ex2.m1.2.2.3.cmml">:=</mo><mrow id="S4.Ex2.m1.2.2.2.2" xref="S4.Ex2.m1.2.2.2.3.cmml"><mo id="S4.Ex2.m1.2.2.2.2.3" xref="S4.Ex2.m1.2.2.2.3.cmml">⟨</mo><msub id="S4.Ex2.m1.1.1.1.1.1" xref="S4.Ex2.m1.1.1.1.1.1.cmml"><mi id="S4.Ex2.m1.1.1.1.1.1.2" xref="S4.Ex2.m1.1.1.1.1.1.2.cmml">v</mi><mi id="S4.Ex2.m1.1.1.1.1.1.3" xref="S4.Ex2.m1.1.1.1.1.1.3.cmml">r</mi></msub><mo id="S4.Ex2.m1.2.2.2.2.4" xref="S4.Ex2.m1.2.2.2.3.cmml">,</mo><msub id="S4.Ex2.m1.2.2.2.2.2" xref="S4.Ex2.m1.2.2.2.2.2.cmml"><mi id="S4.Ex2.m1.2.2.2.2.2.2" xref="S4.Ex2.m1.2.2.2.2.2.2.cmml">ω</mi><mi id="S4.Ex2.m1.2.2.2.2.2.3" xref="S4.Ex2.m1.2.2.2.2.2.3.cmml">r</mi></msub><mo id="S4.Ex2.m1.2.2.2.2.5" xref="S4.Ex2.m1.2.2.2.3.cmml">⟩</mo></mrow></mrow></math>, <math><mrow id="S4.SS1.p5.1.m1.1.1" xref="S4.SS1.p5.1.m1.1.1.cmml"><msub id="S4.SS1.p5.1.m1.1.1.2" xref="S4.SS1.p5.1.m1.1.1.2.cmml"><mi id="S4.SS1.p5.1.m1.1.1.2.2" xref="S4.SS1.p5.1.m1.1.1.2.2.cmml">ω</mi><mi id="S4.SS1.p5.1.m1.1.1.2.3" xref="S4.SS1.p5.1.m1.1.1.2.3.cmml">a</mi></msub><mover id="S4.SS1.p5.1.m1.1.1.1" xref="S4.SS1.p5.1.m1.1.1.1.cmml"><mo movablelimits="false" id="S4.SS1.p5.1.m1.1.1.1.2" xref="S4.SS1.p5.1.m1.1.1.1.2.cmml">⟹</mo><mi id="S4.SS1.p5.1.m1.1.1.1.3" xref="S4.SS1.p5.1.m1.1.1.1.3.cmml">r</mi></mover><msub id="S4.SS1.p5.1.m1.1.1.3" xref="S4.SS1.p5.1.m1.1.1.3.cmml"><mi id="S4.SS1.p5.1.m1.1.1.3.2" xref="S4.SS1.p5.1.m1.1.1.3.2.cmml">ω</mi><mi id="S4.SS1.p5.1.m1.1.1.3.3" xref="S4.SS1.p5.1.m1.1.1.3.3.cmml">b</mi></msub></mrow></math>, <math><mrow id="S4.Ex3.m1.3.3" xref="S4.Ex3.m1.3.3.cmml"><msub id="S4.Ex3.m1.3.3.5" xref="S4.Ex3.m1.3.3.5.cmml"><mi id="S4.Ex3.m1.3.3.5.2" xref="S4.Ex3.m1.3.3.5.2.cmml">ω</mi><mi id="S4.Ex3.m1.3.3.5.3" xref="S4.Ex3.m1.3.3.5.3.cmml">a</mi></msub><mover id="S4.Ex3.m1.3.3.6" xref="S4.Ex3.m1.3.3.6.cmml"><mo movablelimits="false" id="S4.Ex3.m1.3.3.6.2" xref="S4.Ex3.m1.3.3.6.2.cmml">⟹</mo><mi id="S4.Ex3.m1.3.3.6.3" xref="S4.Ex3.m1.3.3.6.3.cmml">r</mi></mover><msub id="S4.Ex3.m1.3.3.7" xref="S4.Ex3.m1.3.3.7.cmml"><mi id="S4.Ex3.m1.3.3.7.2" xref="S4.Ex3.m1.3.3.7.2.cmml">ω</mi><mi id="S4.Ex3.m1.3.3.7.3" xref="S4.Ex3.m1.3.3.7.3.cmml">b</mi></msub><mo id="S4.Ex3.m1.3.3.8" xref="S4.Ex3.m1.3.3.8.cmml">=</mo><mrow id="S4.Ex3.m1.3.3.3.3" xref="S4.Ex3.m1.3.3.3.4.cmml"><mo id="S4.Ex3.m1.3.3.3.3.4" xref="S4.Ex3.m1.3.3.3.4.cmml">⟨</mo><mrow id="S4.Ex3.m1.1.1.1.1.1" xref="S4.Ex3.m1.1.1.1.1.1.cmml"><msub id="S4.Ex3.m1.1.1.1.1.1.2" xref="S4.Ex3.m1.1.1.1.1.1.2.cmml"><mi id="S4.Ex3.m1.1.1.1.1.1.2.2" xref="S4.Ex3.m1.1.1.1.1.1.2.2.cmml">V</mi><mi id="S4.Ex3.m1.1.1.1.1.1.2.3" xref="S4.Ex3.m1.1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S4.Ex3.m1.1.1.1.1.1.1" xref="S4.Ex3.m1.1.1.1.1.1.1.cmml">∪</mo><msub id="S4.Ex3.m1.1.1.1.1.1.3" xref="S4.Ex3.m1.1.1.1.1.1.3.cmml"><mi id="S4.Ex3.m1.1.1.1.1.1.3.2" xref="S4.Ex3.m1.1.1.1.1.1.3.2.cmml">V</mi><mi id="S4.Ex3.m1.1.1.1.1.1.3.3" xref="S4.Ex3.m1.1.1.1.1.1.3.3.cmml">r</mi></msub></mrow><mo id="S4.Ex3.m1.3.3.3.3.5" xref="S4.Ex3.m1.3.3.3.4.cmml">,</mo><mrow id="S4.Ex3.m1.2.2.2.2.2" xref="S4.Ex3.m1.2.2.2.2.2.cmml"><msub id="S4.Ex3.m1.2.2.2.2.2.2" xref="S4.Ex3.m1.2.2.2.2.2.2.cmml"><mi id="S4.Ex3.m1.2.2.2.2.2.2.2" xref="S4.Ex3.m1.2.2.2.2.2.2.2.cmml">E</mi><mi id="S4.Ex3.m1.2.2.2.2.2.2.3" xref="S4.Ex3.m1.2.2.2.2.2.2.3.cmml">a</mi></msub><mo id="S4.Ex3.m1.2.2.2.2.2.1" xref="S4.Ex3.m1.2.2.2.2.2.1.cmml">∪</mo><msub id="S4.Ex3.m1.2.2.2.2.2.3" xref="S4.Ex3.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex3.m1.2.2.2.2.2.3.2" xref="S4.Ex3.m1.2.2.2.2.2.3.2.cmml">E</mi><mi id="S4.Ex3.m1.2.2.2.2.2.3.3" xref="S4.Ex3.m1.2.2.2.2.2.3.3.cmml">r</mi></msub></mrow><mo id="S4.Ex3.m1.3.3.3.3.6" xref="S4.Ex3.m1.3.3.3.4.cmml">,</mo><msub id="S4.Ex3.m1.3.3.3.3.3" xref="S4.Ex3.m1.3.3.3.3.3.cmml"><mi id="S4.Ex3.m1.3.3.3.3.3.2" xref="S4.Ex3.m1.3.3.3.3.3.2.cmml">l</mi><mi id="S4.Ex3.m1.3.3.3.3.3.3" xref="S4.Ex3.m1.3.3.3.3.3.3.cmml">b</mi></msub><mo id="S4.Ex3.m1.3.3.3.3.7" xref="S4.Ex3.m1.3.3.3.4.cmml">⟩</mo></mrow></mrow></math>, <math><mrow id="S4.Ex4.m1.5.6" xref="S4.Ex4.m1.5.6.cmml"><mrow id="S4.Ex4.m1.5.6.2" xref="S4.Ex4.m1.5.6.2.cmml"><msub id="S4.Ex4.m1.5.6.2.2" xref="S4.Ex4.m1.5.6.2.2.cmml"><mi id="S4.Ex4.m1.5.6.2.2.2" xref="S4.Ex4.m1.5.6.2.2.2.cmml">l</mi><mi id="S4.Ex4.m1.5.6.2.2.3" xref="S4.Ex4.m1.5.6.2.2.3.cmml">b</mi></msub><mo id="S4.Ex4.m1.5.6.2.1" xref="S4.Ex4.m1.5.6.2.1.cmml">⁢</mo><mrow id="S4.Ex4.m1.5.6.2.3.2" xref="S4.Ex4.m1.5.6.2.cmml"><mo stretchy="false" id="S4.Ex4.m1.5.6.2.3.2.1" xref="S4.Ex4.m1.5.6.2.cmml">(</mo><mi id="S4.Ex4.m1.5.5" xref="S4.Ex4.m1.5.5.cmml">v</mi><mo stretchy="false" id="S4.Ex4.m1.5.6.2.3.2.2" xref="S4.Ex4.m1.5.6.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex4.m1.5.6.1" xref="S4.Ex4.m1.5.6.1.cmml">=</mo><mrow id="S4.Ex4.m1.4.4" xref="S4.Ex4.m1.5.6.3.1.cmml"><mo id="S4.Ex4.m1.4.4.5" xref="S4.Ex4.m1.5.6.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S4.Ex4.m1.4.4.4" xref="S4.Ex4.m1.5.6.3.1.cmml"><mtr id="S4.Ex4.m1.4.4.4a" xref="S4.Ex4.m1.5.6.3.1.cmml"><mtd columnalign="left" id="S4.Ex4.m1.4.4.4b" xref="S4.Ex4.m1.5.6.3.1.cmml"><mrow id="S4.Ex4.m1.1.1.1.1.1.1" xref="S4.Ex4.m1.1.1.1.1.1.1.cmml"><msub id="S4.Ex4.m1.1.1.1.1.1.1.3" xref="S4.Ex4.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex4.m1.1.1.1.1.1.1.3.2" xref="S4.Ex4.m1.1.1.1.1.1.1.3.2.cmml">l</mi><mi id="S4.Ex4.m1.1.1.1.1.1.1.3.3" xref="S4.Ex4.m1.1.1.1.1.1.1.3.3.cmml">r</mi></msub><mo id="S4.Ex4.m1.1.1.1.1.1.1.2" xref="S4.Ex4.m1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex4.m1.1.1.1.1.1.1.4.2" xref="S4.Ex4.m1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.Ex4.m1.1.1.1.1.1.1.4.2.1" xref="S4.Ex4.m1.1.1.1.1.1.1.cmml">(</mo><mi id="S4.Ex4.m1.1.1.1.1.1.1.1" xref="S4.Ex4.m1.1.1.1.1.1.1.1.cmml">v</mi><mo stretchy="false" id="S4.Ex4.m1.1.1.1.1.1.1.4.2.2" xref="S4.Ex4.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mtd><mtd columnalign="left" id="S4.Ex4.m1.4.4.4c" xref="S4.Ex4.m1.5.6.3.1.cmml"><mrow id="S4.Ex4.m1.2.2.2.2.2.1" xref="S4.Ex4.m1.2.2.2.2.2.1.cmml"><mrow id="S4.Ex4.m1.2.2.2.2.2.1.4" xref="S4.Ex4.m1.2.2.2.2.2.1.4.cmml"><mtext id="S4.Ex4.m1.2.2.2.2.2.1.4.2" xref="S4.Ex4.m1.2.2.2.2.2.1.4.2a.cmml">if </mtext><mo id="S4.Ex4.m1.2.2.2.2.2.1.4.1" xref="S4.Ex4.m1.2.2.2.2.2.1.4.1.cmml">⁢</mo><mi id="S4.Ex4.m1.2.2.2.2.2.1.4.3" xref="S4.Ex4.m1.2.2.2.2.2.1.4.3.cmml">v</mi></mrow><mo id="S4.Ex4.m1.2.2.2.2.2.1.3" xref="S4.Ex4.m1.2.2.2.2.2.1.3.cmml">∈</mo><mrow id="S4.Ex4.m1.2.2.2.2.2.1.2" xref="S4.Ex4.m1.2.2.2.2.2.1.2.cmml"><mrow id="S4.Ex4.m1.2.2.2.2.2.1.1.1.1" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.2" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.1" xref="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.1.cmml"><msub id="S4.Ex4.m1.2.2.2.2.2.1.1.1.1.1.2" 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id="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.1" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.1.cmml"><mi id="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.1.2" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.1.2.cmml">v</mi><mi id="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.1.3" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.1.3.cmml">r</mi></msub><mo stretchy="false" id="S4.Ex4.m1.2.2.2.2.2.1.2.2.1.3" xref="S4.Ex4.m1.2.2.2.2.2.1.2.2.2.cmml">}</mo></mrow></mrow></mrow></mtd></mtr><mtr id="S4.Ex4.m1.4.4.4d" xref="S4.Ex4.m1.5.6.3.1.cmml"><mtd columnalign="left" id="S4.Ex4.m1.4.4.4e" xref="S4.Ex4.m1.5.6.3.1.cmml"><mrow id="S4.Ex4.m1.3.3.3.3.1.1" xref="S4.Ex4.m1.3.3.3.3.1.1.cmml"><msub id="S4.Ex4.m1.3.3.3.3.1.1.3" xref="S4.Ex4.m1.3.3.3.3.1.1.3.cmml"><mi id="S4.Ex4.m1.3.3.3.3.1.1.3.2" xref="S4.Ex4.m1.3.3.3.3.1.1.3.2.cmml">l</mi><mi id="S4.Ex4.m1.3.3.3.3.1.1.3.3" xref="S4.Ex4.m1.3.3.3.3.1.1.3.3.cmml">a</mi></msub><mo id="S4.Ex4.m1.3.3.3.3.1.1.2" xref="S4.Ex4.m1.3.3.3.3.1.1.2.cmml">⁢</mo><mrow id="S4.Ex4.m1.3.3.3.3.1.1.4.2" xref="S4.Ex4.m1.3.3.3.3.1.1.cmml"><mo stretchy="false" id="S4.Ex4.m1.3.3.3.3.1.1.4.2.1" xref="S4.Ex4.m1.3.3.3.3.1.1.cmml">(</mo><mi id="S4.Ex4.m1.3.3.3.3.1.1.1" xref="S4.Ex4.m1.3.3.3.3.1.1.1.cmml">v</mi><mo stretchy="false" id="S4.Ex4.m1.3.3.3.3.1.1.4.2.2" xref="S4.Ex4.m1.3.3.3.3.1.1.cmml">)</mo></mrow></mrow></mtd><mtd columnalign="left" id="S4.Ex4.m1.4.4.4f" xref="S4.Ex4.m1.5.6.3.1.cmml"><mtext id="S4.Ex4.m1.4.4.4.4.2.1" xref="S4.Ex4.m1.4.4.4.4.2.1a.cmml">otherwise.</mtext></mtd></mtr></mtable></mrow></mrow></math>, <math><mrow id="S4.F3.11.4.m4.1.1" xref="S4.F3.11.4.m4.1.1.cmml"><msub id="S4.F3.11.4.m4.1.1.2" xref="S4.F3.11.4.m4.1.1.2.cmml"><mi id="S4.F3.11.4.m4.1.1.2.2" xref="S4.F3.11.4.m4.1.1.2.2.cmml">ω</mi><mi id="S4.F3.11.4.m4.1.1.2.3" xref="S4.F3.11.4.m4.1.1.2.3.cmml">a</mi></msub><mover id="S4.F3.11.4.m4.1.1.1" xref="S4.F3.11.4.m4.1.1.1.cmml"><mo movablelimits="false" id="S4.F3.11.4.m4.1.1.1.2" xref="S4.F3.11.4.m4.1.1.1.2.cmml">⟹</mo><mi id="S4.F3.11.4.m4.1.1.1.3" xref="S4.F3.11.4.m4.1.1.1.3.cmml">r</mi></mover><msub id="S4.F3.11.4.m4.1.1.3" xref="S4.F3.11.4.m4.1.1.3.cmml"><mi id="S4.F3.11.4.m4.1.1.3.2" xref="S4.F3.11.4.m4.1.1.3.2.cmml">ω</mi><mi id="S4.F3.11.4.m4.1.1.3.3" xref="S4.F3.11.4.m4.1.1.3.3.cmml">b</mi></msub></mrow></math>, <math><mrow id="S4.Ex5.m1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.cmml"><mrow id="S4.Ex5.m1.1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.cmml"><mi id="S4.Ex5.m1.1.1.1.1.4" xref="S4.Ex5.m1.1.1.1.1.4.cmml">γ</mi><mo id="S4.Ex5.m1.1.1.1.1.3" xref="S4.Ex5.m1.1.1.1.1.3.cmml">:=</mo><mrow id="S4.Ex5.m1.1.1.1.1.2.2" xref="S4.Ex5.m1.1.1.1.1.2.3.cmml"><mo id="S4.Ex5.m1.1.1.1.1.2.2.3" xref="S4.Ex5.m1.1.1.1.1.2.3.cmml">⟨</mo><msub id="S4.Ex5.m1.1.1.1.1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex5.m1.1.1.1.1.1.1.1.2" xref="S4.Ex5.m1.1.1.1.1.1.1.1.2.cmml">m</mi><mi id="S4.Ex5.m1.1.1.1.1.1.1.1.3" xref="S4.Ex5.m1.1.1.1.1.1.1.1.3.cmml">γ</mi></msub><mo id="S4.Ex5.m1.1.1.1.1.2.2.4" xref="S4.Ex5.m1.1.1.1.1.2.3.cmml">,</mo><msub id="S4.Ex5.m1.1.1.1.1.2.2.2" xref="S4.Ex5.m1.1.1.1.1.2.2.2.cmml"><mi id="S4.Ex5.m1.1.1.1.1.2.2.2.2" xref="S4.Ex5.m1.1.1.1.1.2.2.2.2.cmml">f</mi><mi id="S4.Ex5.m1.1.1.1.1.2.2.2.3" xref="S4.Ex5.m1.1.1.1.1.2.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex5.m1.1.1.1.1.2.2.5" xref="S4.Ex5.m1.1.1.1.1.2.3.cmml">⟩</mo></mrow></mrow><mo id="S4.Ex5.m1.1.1.1.2" xref="S4.Ex5.m1.1.1.1.1.cmml">,</mo></mrow></math>

Correct Categorie: cs

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1203.2841

Formulas:

1-

Δ = μ_{f} / r_{g}

2-

P (Δ, p)

3-

P (Δ, p)

4-

P (Δ, p)

5-

P (Δ, p)

6-

P (Δ, p) = \min {1, Δ p} = \min {1, \frac{μ_{f}}{r_{g}} \cdot p} .

7-

P (Δ, p)

8-

n P (Δ, p)

9-

n_{b s} = n P (Δ, p) \cdot \max {\frac{r_{t}}{R_{\max}}, \frac{1}{N_{\max}}} .

10-

\max {\frac{r_{t}}{R_{\max}}, \frac{1}{N_{\max}}}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1706.02912

Formulas:

1-

N_{E} = 0.8 N

2-

N_{I} = 0.2 N

3-

τ_{m} \frac{d V_{i}}{d t} = - V_{i} + τ_{m} \sum_{j} J_{i j} s_{j} (t - d),

4-

V_{0} = 0 mV

5-

J_{E} = J = 0.1 mV

6-

J_{I} = - g J = - 0.8 mV

7-

s_{j} (t) = \sum_{k} δ (t - t_{j}^{k})

8-

ν_{e x t}

9-

V_{th} = 20 mV

10-

V_{r} = 10 mV

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1401.6276

Formulas:

1-

P (X, H, Θ) = P (X | H, Θ) P (H | Θ) P (Θ)

2-

P (X, Θ)

3-

= \int P (X, H, Θ) 𝑑 H

4-

= argmax P (X, Θ),

5-

= \nabla^{2} \log P (X, \hat{Θ})

6-

P (Θ | X) \approx 𝒩 (Θ | \hat{Θ}, - 𝚲^{- 1})

7-

\int P (H | X, Θ^{'}) 𝑑 H = 1

8-

\begin{matrix} \log P (X, Θ) & = \int P (H | X, Θ^{'}) \log P (X, Θ) 𝑑 H \\ = \int P (H | X, Θ^{'}) \log P (X | Θ) 𝑑 H + \log P (Θ) \\ = \int P (H | X, Θ^{'}) \log \frac{P (X, H | Θ)}{P (H | X, Θ)} d H + \log P (Θ) \\ = \int P (H | X, Θ^{'}) \log [\frac{P (X, H | Θ)}{P (H | X, Θ)} \times \frac{P (H | X, Θ^{'})}{P (H | X, Θ^{'})}] 𝑑 H + \log P (Θ) \\ = A (Θ^{'}, Θ) + D (Θ^{'}, Θ) \end{matrix}

9-

A (Θ^{'}, Θ)

10-

A (Θ^{'}, Θ)

Formulas (html):

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xref="S1.E1.m1.6.6.cmml">Θ</mi><mo stretchy="false" id="S1.E1.m1.8.8.2.7.2.2" xref="S1.E1.m1.8.8.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.E2.m1.2.3" xref="S1.E2.m1.2.3.cmml"><mi id="S1.E2.m1.2.3.2" xref="S1.E2.m1.2.3.2.cmml">P</mi><mo id="S1.E2.m1.2.3.1" xref="S1.E2.m1.2.3.1.cmml">⁢</mo><mrow id="S1.E2.m1.2.3.3.2" xref="S1.E2.m1.2.3.3.1.cmml"><mo stretchy="false" id="S1.E2.m1.2.3.3.2.1" xref="S1.E2.m1.2.3.3.1.cmml">(</mo><mi id="S1.E2.m1.1.1" xref="S1.E2.m1.1.1.cmml">X</mi><mo id="S1.E2.m1.2.3.3.2.2" xref="S1.E2.m1.2.3.3.1.cmml">,</mo><mi mathvariant="normal" id="S1.E2.m1.2.2" xref="S1.E2.m1.2.2.cmml">Θ</mi><mo stretchy="false" id="S1.E2.m1.2.3.3.2.3" xref="S1.E2.m1.2.3.3.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.E2.m2.3.4" xref="S1.E2.m2.3.4.cmml"><mi id="S1.E2.m2.3.4.2" xref="S1.E2.m2.3.4.2.cmml"/><mo id="S1.E2.m2.3.4.1" xref="S1.E2.m2.3.4.1.cmml">=</mo><mstyle displaystyle="true" id="S1.E2.m2.3.4.3" xref="S1.E2.m2.3.4.3.cmml"><mrow id="S1.E2.m2.3.4.3a" xref="S1.E2.m2.3.4.3.cmml"><mo largeop="true" symmetric="true" id="S1.E2.m2.3.4.3.1" xref="S1.E2.m2.3.4.3.1.cmml">∫</mo><mrow id="S1.E2.m2.3.4.3.2" xref="S1.E2.m2.3.4.3.2.cmml"><mi id="S1.E2.m2.3.4.3.2.2" xref="S1.E2.m2.3.4.3.2.2.cmml">P</mi><mo id="S1.E2.m2.3.4.3.2.1" xref="S1.E2.m2.3.4.3.2.1.cmml">⁢</mo><mrow id="S1.E2.m2.3.4.3.2.3.2" xref="S1.E2.m2.3.4.3.2.3.1.cmml"><mo stretchy="false" id="S1.E2.m2.3.4.3.2.3.2.1" xref="S1.E2.m2.3.4.3.2.3.1.cmml">(</mo><mi id="S1.E2.m2.1.1" xref="S1.E2.m2.1.1.cmml">X</mi><mo id="S1.E2.m2.3.4.3.2.3.2.2" xref="S1.E2.m2.3.4.3.2.3.1.cmml">,</mo><mi id="S1.E2.m2.2.2" xref="S1.E2.m2.2.2.cmml">H</mi><mo id="S1.E2.m2.3.4.3.2.3.2.3" xref="S1.E2.m2.3.4.3.2.3.1.cmml">,</mo><mi mathvariant="normal" id="S1.E2.m2.3.3" xref="S1.E2.m2.3.3.cmml">Θ</mi><mo rspace="4.2pt" stretchy="false" id="S1.E2.m2.3.4.3.2.3.2.4" xref="S1.E2.m2.3.4.3.2.3.1.cmml">)</mo></mrow><mo id="S1.E2.m2.3.4.3.2.1a" xref="S1.E2.m2.3.4.3.2.1.cmml">⁢</mo><mrow id="S1.E2.m2.3.4.3.2.4" xref="S1.E2.m2.3.4.3.2.4.cmml"><mo rspace="0pt" id="S1.E2.m2.3.4.3.2.4.1" xref="S1.E2.m2.3.4.3.2.4.1.cmml">𝑑</mo><mi id="S1.E2.m2.3.4.3.2.4.2" xref="S1.E2.m2.3.4.3.2.4.2.cmml">H</mi></mrow></mrow></mrow></mstyle></mrow></math>, <math><mrow id="S1.E3.m2.3.3.1" xref="S1.E3.m2.3.3.1.1.cmml"><mrow id="S1.E3.m2.3.3.1.1" xref="S1.E3.m2.3.3.1.1.cmml"><mi id="S1.E3.m2.3.3.1.1.2" xref="S1.E3.m2.3.3.1.1.2.cmml"/><mo id="S1.E3.m2.3.3.1.1.1" xref="S1.E3.m2.3.3.1.1.1.cmml">=</mo><mrow id="S1.E3.m2.3.3.1.1.3" xref="S1.E3.m2.3.3.1.1.3.cmml"><mrow id="S1.E3.m2.3.3.1.1.3.2" xref="S1.E3.m2.3.3.1.1.3.2.cmml"><mo movablelimits="false" id="S1.E3.m2.3.3.1.1.3.2.1" xref="S1.E3.m2.3.3.1.1.3.2.1.cmml">argmax</mo><mo id="S1.E3.m2.3.3.1.1.3.2a" xref="S1.E3.m2.3.3.1.1.3.2.cmml">⁡</mo><mi id="S1.E3.m2.3.3.1.1.3.2.2" xref="S1.E3.m2.3.3.1.1.3.2.2.cmml">P</mi></mrow><mo id="S1.E3.m2.3.3.1.1.3.1" xref="S1.E3.m2.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S1.E3.m2.3.3.1.1.3.3.2" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml"><mo stretchy="false" id="S1.E3.m2.3.3.1.1.3.3.2.1" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml">(</mo><mi id="S1.E3.m2.1.1" xref="S1.E3.m2.1.1.cmml">X</mi><mo id="S1.E3.m2.3.3.1.1.3.3.2.2" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml">,</mo><mi mathvariant="normal" id="S1.E3.m2.2.2" xref="S1.E3.m2.2.2.cmml">Θ</mi><mo stretchy="false" id="S1.E3.m2.3.3.1.1.3.3.2.3" xref="S1.E3.m2.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo id="S1.E3.m2.3.3.1.2" xref="S1.E3.m2.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S1.E3.m5.2.3" xref="S1.E3.m5.2.3.cmml"><mi id="S1.E3.m5.2.3.2" xref="S1.E3.m5.2.3.2.cmml"/><mo id="S1.E3.m5.2.3.1" xref="S1.E3.m5.2.3.1.cmml">=</mo><mrow id="S1.E3.m5.2.3.3" xref="S1.E3.m5.2.3.3.cmml"><mrow id="S1.E3.m5.2.3.3.2" xref="S1.E3.m5.2.3.3.2.cmml"><mrow id="S1.E3.m5.2.3.3.2.1" xref="S1.E3.m5.2.3.3.2.1.cmml"><msup id="S1.E3.m5.2.3.3.2.1.1" xref="S1.E3.m5.2.3.3.2.1.1.cmml"><mo id="S1.E3.m5.2.3.3.2.1.1.2" xref="S1.E3.m5.2.3.3.2.1.1.2.cmml">∇</mo><mn id="S1.E3.m5.2.3.3.2.1.1.3" xref="S1.E3.m5.2.3.3.2.1.1.3.cmml">2</mn></msup><mo id="S1.E3.m5.2.3.3.2.1a" xref="S1.E3.m5.2.3.3.2.1.cmml">⁡</mo><mi id="S1.E3.m5.2.3.3.2.1.2" xref="S1.E3.m5.2.3.3.2.1.2.cmml">log</mi></mrow><mo id="S1.E3.m5.2.3.3.2a" xref="S1.E3.m5.2.3.3.2.cmml">⁡</mo><mi id="S1.E3.m5.2.3.3.2.2" xref="S1.E3.m5.2.3.3.2.2.cmml">P</mi></mrow><mo id="S1.E3.m5.2.3.3.1" xref="S1.E3.m5.2.3.3.1.cmml">⁢</mo><mrow id="S1.E3.m5.2.3.3.3.2" xref="S1.E3.m5.2.3.3.3.1.cmml"><mo stretchy="false" id="S1.E3.m5.2.3.3.3.2.1" xref="S1.E3.m5.2.3.3.3.1.cmml">(</mo><mi id="S1.E3.m5.1.1" xref="S1.E3.m5.1.1.cmml">X</mi><mo id="S1.E3.m5.2.3.3.3.2.2" xref="S1.E3.m5.2.3.3.3.1.cmml">,</mo><mover accent="true" id="S1.E3.m5.2.2" xref="S1.E3.m5.2.2.cmml"><mi mathvariant="normal" id="S1.E3.m5.2.2.2" xref="S1.E3.m5.2.2.2.cmml">Θ</mi><mo stretchy="false" id="S1.E3.m5.2.2.1" xref="S1.E3.m5.2.2.1.cmml">^</mo></mover><mo stretchy="false" id="S1.E3.m5.2.3.3.3.2.3" xref="S1.E3.m5.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.E4.m1.3.3" xref="S1.E4.m1.3.3.cmml"><mrow id="S1.E4.m1.2.2.1" xref="S1.E4.m1.2.2.1.cmml"><mi id="S1.E4.m1.2.2.1.3" xref="S1.E4.m1.2.2.1.3.cmml">P</mi><mo id="S1.E4.m1.2.2.1.2" xref="S1.E4.m1.2.2.1.2.cmml">⁢</mo><mrow id="S1.E4.m1.2.2.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.cmml"><mo stretchy="false" id="S1.E4.m1.2.2.1.1.1.2" xref="S1.E4.m1.2.2.1.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.2.2.1.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.cmml"><mi mathvariant="normal" id="S1.E4.m1.2.2.1.1.1.1.2" xref="S1.E4.m1.2.2.1.1.1.1.2.cmml">Θ</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S1.E4.m1.2.2.1.1.1.1.1" xref="S1.E4.m1.2.2.1.1.1.1.1.cmml">|</mo><mi id="S1.E4.m1.2.2.1.1.1.1.3" xref="S1.E4.m1.2.2.1.1.1.1.3.cmml">X</mi></mrow><mo stretchy="false" id="S1.E4.m1.2.2.1.1.1.3" xref="S1.E4.m1.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.E4.m1.3.3.3" xref="S1.E4.m1.3.3.3.cmml">≈</mo><mrow id="S1.E4.m1.3.3.2" xref="S1.E4.m1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.E4.m1.3.3.2.3" xref="S1.E4.m1.3.3.2.3.cmml">𝒩</mi><mo id="S1.E4.m1.3.3.2.2" xref="S1.E4.m1.3.3.2.2.cmml">⁢</mo><mrow id="S1.E4.m1.3.3.2.1.1" xref="S1.E4.m1.3.3.2.1.1.1.cmml"><mo stretchy="false" id="S1.E4.m1.3.3.2.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.cmml">(</mo><mrow id="S1.E4.m1.3.3.2.1.1.1" xref="S1.E4.m1.3.3.2.1.1.1.cmml"><mi mathvariant="normal" id="S1.E4.m1.3.3.2.1.1.1.3" xref="S1.E4.m1.3.3.2.1.1.1.3.cmml">Θ</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S1.E4.m1.3.3.2.1.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.2.cmml">|</mo><mrow id="S1.E4.m1.3.3.2.1.1.1.1.1" xref="S1.E4.m1.3.3.2.1.1.1.1.2.cmml"><mover accent="true" id="S1.E4.m1.1.1" xref="S1.E4.m1.1.1.cmml"><mi mathvariant="normal" id="S1.E4.m1.1.1.2" xref="S1.E4.m1.1.1.2.cmml">Θ</mi><mo stretchy="false" id="S1.E4.m1.1.1.1" xref="S1.E4.m1.1.1.1.cmml">^</mo></mover><mo id="S1.E4.m1.3.3.2.1.1.1.1.1.2" xref="S1.E4.m1.3.3.2.1.1.1.1.2.cmml">,</mo><mrow 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id="S2.SS1.p1.2.m2.2.2.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.2.2.1.1.3" xref="S2.SS1.p1.2.m2.2.2.1.1.3.cmml">P</mi><mo id="S2.SS1.p1.2.m2.2.2.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.2.m2.2.2.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.cmml"><mi id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.3" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.3.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.2.cmml">|</mo><mrow id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml"><mi id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml">X</mi><mo id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.2" xref="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.2.cmml">,</mo><msup id="S2.SS1.p1.2.m2.2.2.1.1.1.1.1.1.1.1" 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xref="S2.E5.m1.123.123.8.cmml"><mtr id="S2.E5.m1.131.131.16a" xref="S2.E5.m1.123.123.8.cmml"><mtd columnalign="right" id="S2.E5.m1.131.131.16b" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.7.7.7.7.7a" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.7.7.7.7.7a.9" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.1.cmml">log</mi><mo id="S2.E5.m1.7.7.7.7.7a.9a" xref="S2.E5.m1.123.123.8a.cmml">⁡</mo><mi id="S2.E5.m1.2.2.2.2.2.2" xref="S2.E5.m1.2.2.2.2.2.2.cmml">P</mi></mrow><mo id="S2.E5.m1.7.7.7.7.7a.8" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.7.7.7.7.7a.10" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.3.3.3.3.3.3" xref="S2.E5.m1.123.123.8a.cmml">(</mo><mi id="S2.E5.m1.4.4.4.4.4.4" xref="S2.E5.m1.4.4.4.4.4.4.cmml">X</mi><mo id="S2.E5.m1.5.5.5.5.5.5" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.6.6.6.6.6.6" xref="S2.E5.m1.6.6.6.6.6.6.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.7.7.7.7.7.7" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow></mtd><mtd columnalign="left" id="S2.E5.m1.131.131.16c" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.124.124.9.116.28.21" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.124.124.9.116.28.21.22" xref="S2.E5.m1.123.123.8a.cmml"/><mo id="S2.E5.m1.8.8.8.8.1.1" xref="S2.E5.m1.8.8.8.8.1.1.cmml">=</mo><mstyle displaystyle="true" id="S2.E5.m1.124.124.9.116.28.21.21" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.124.124.9.116.28.21.21a" xref="S2.E5.m1.123.123.8.cmml"><mo largeop="true" symmetric="true" id="S2.E5.m1.9.9.9.9.2.2" xref="S2.E5.m1.9.9.9.9.2.2.cmml">∫</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.10.10.10.10.3.3" xref="S2.E5.m1.10.10.10.10.3.3.cmml">P</mi><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.2" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.11.11.11.11.4.4" xref="S2.E5.m1.123.123.8a.cmml">(</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.12.12.12.12.5.5" xref="S2.E5.m1.12.12.12.12.5.5.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.13.13.13.13.6.6" xref="S2.E5.m1.13.13.13.13.6.6.cmml">|</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.1.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.14.14.14.14.7.7" xref="S2.E5.m1.14.14.14.14.7.7.cmml">X</mi><mo id="S2.E5.m1.15.15.15.15.8.8" xref="S2.E5.m1.123.123.8a.cmml">,</mo><msup id="S2.E5.m1.124.124.9.116.28.21.21.1.1.1.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi mathvariant="normal" id="S2.E5.m1.16.16.16.16.9.9" xref="S2.E5.m1.16.16.16.16.9.9.cmml">Θ</mi><mo id="S2.E5.m1.17.17.17.17.10.10.1" xref="S2.E5.m1.17.17.17.17.10.10.1.cmml">′</mo></msup></mrow></mrow><mo stretchy="false" id="S2.E5.m1.18.18.18.18.11.11" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.2a" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.3" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.19.19.19.19.12.12" xref="S2.E5.m1.19.19.19.19.12.12.cmml">log</mi><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.3a" xref="S2.E5.m1.123.123.8a.cmml">⁡</mo><mi id="S2.E5.m1.20.20.20.20.13.13" xref="S2.E5.m1.20.20.20.20.13.13.cmml">P</mi></mrow><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.2b" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.4" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.21.21.21.21.14.14" xref="S2.E5.m1.123.123.8a.cmml">(</mo><mi id="S2.E5.m1.22.22.22.22.15.15" xref="S2.E5.m1.22.22.22.22.15.15.cmml">X</mi><mo id="S2.E5.m1.23.23.23.23.16.16" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.24.24.24.24.17.17" xref="S2.E5.m1.24.24.24.24.17.17.cmml">Θ</mi><mo rspace="4.2pt" stretchy="false" id="S2.E5.m1.25.25.25.25.18.18" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow><mo id="S2.E5.m1.124.124.9.116.28.21.21.1.2c" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.124.124.9.116.28.21.21.1.5" xref="S2.E5.m1.123.123.8.cmml"><mo rspace="0pt" id="S2.E5.m1.26.26.26.26.19.19" xref="S2.E5.m1.26.26.26.26.19.19.cmml">𝑑</mo><mi id="S2.E5.m1.27.27.27.27.20.20" xref="S2.E5.m1.27.27.27.27.20.20.cmml">H</mi></mrow></mrow></mrow></mstyle></mrow></mtd></mtr><mtr id="S2.E5.m1.131.131.16d" xref="S2.E5.m1.123.123.8.cmml"><mtd id="S2.E5.m1.131.131.16e" xref="S2.E5.m1.123.123.8a.cmml"/><mtd columnalign="left" id="S2.E5.m1.131.131.16f" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.126.126.11.118.28.28" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.126.126.11.118.28.28.29" xref="S2.E5.m1.123.123.8a.cmml"/><mo id="S2.E5.m1.28.28.28.1.1.1" xref="S2.E5.m1.28.28.28.1.1.1.cmml">=</mo><mrow id="S2.E5.m1.126.126.11.118.28.28.28" xref="S2.E5.m1.123.123.8.cmml"><mstyle displaystyle="true" 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id="S2.E5.m1.125.125.10.117.27.27.27.1.1.1.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.34.34.34.7.7.7" xref="S2.E5.m1.34.34.34.7.7.7.cmml">X</mi><mo id="S2.E5.m1.35.35.35.8.8.8" xref="S2.E5.m1.123.123.8a.cmml">,</mo><msup id="S2.E5.m1.125.125.10.117.27.27.27.1.1.1.1.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi mathvariant="normal" id="S2.E5.m1.36.36.36.9.9.9" xref="S2.E5.m1.36.36.36.9.9.9.cmml">Θ</mi><mo id="S2.E5.m1.37.37.37.10.10.10.1" xref="S2.E5.m1.37.37.37.10.10.10.1.cmml">′</mo></msup></mrow></mrow><mo stretchy="false" id="S2.E5.m1.38.38.38.11.11.11" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow><mo id="S2.E5.m1.126.126.11.118.28.28.28.2.2.3a" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.126.126.11.118.28.28.28.2.2.4" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.39.39.39.12.12.12" xref="S2.E5.m1.39.39.39.12.12.12.cmml">log</mi><mo id="S2.E5.m1.126.126.11.118.28.28.28.2.2.4a" xref="S2.E5.m1.123.123.8a.cmml">⁡</mo><mi id="S2.E5.m1.40.40.40.13.13.13" 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id="S2.E5.m1.88.88.88.14.14.14.2.4" xref="S2.E5.m1.88.88.88.14.14.14.2.4.cmml">P</mi><mo id="S2.E5.m1.88.88.88.14.14.14.2.3" xref="S2.E5.m1.88.88.88.14.14.14.2.3.cmml">⁢</mo><mrow id="S2.E5.m1.88.88.88.14.14.14.2.2.1" xref="S2.E5.m1.88.88.88.14.14.14.2.2.2.cmml"><mo stretchy="false" id="S2.E5.m1.88.88.88.14.14.14.2.2.1.2" xref="S2.E5.m1.88.88.88.14.14.14.2.2.2.cmml">(</mo><mi id="S2.E5.m1.88.88.88.14.14.14.1.1" xref="S2.E5.m1.88.88.88.14.14.14.1.1.cmml">X</mi><mo id="S2.E5.m1.88.88.88.14.14.14.2.2.1.3" xref="S2.E5.m1.88.88.88.14.14.14.2.2.2.cmml">,</mo><mrow id="S2.E5.m1.88.88.88.14.14.14.2.2.1.1" xref="S2.E5.m1.88.88.88.14.14.14.2.2.1.1.cmml"><mi id="S2.E5.m1.88.88.88.14.14.14.2.2.1.1.2" xref="S2.E5.m1.88.88.88.14.14.14.2.2.1.1.2.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.88.88.88.14.14.14.2.2.1.1.1" xref="S2.E5.m1.88.88.88.14.14.14.2.2.1.1.1.cmml">|</mo><mi mathvariant="normal" id="S2.E5.m1.88.88.88.14.14.14.2.2.1.1.3" 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id="S2.E5.m1.90.90.90.16.16.16.2.2.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml"><mo stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.2.2.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml">(</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.3" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.3.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.2.cmml">|</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.2.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.1.1" xref="S2.E5.m1.90.90.90.16.16.16.1.1.cmml">X</mi><mo id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.2.cmml">,</mo><msup id="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.2.2.1.1.1.1.1.cmml"><mi mathvariant="normal" 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xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.3.cmml">H</mi><mo lspace="2.5pt" rspace="2.5pt" stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.2.cmml">|</mo><mrow id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.2.cmml"><mi id="S2.E5.m1.90.90.90.16.16.16.3.1" xref="S2.E5.m1.90.90.90.16.16.16.3.1.cmml">X</mi><mo id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.2.cmml">,</mo><msup id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.2" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.2.cmml">Θ</mi><mo id="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.3" xref="S2.E5.m1.90.90.90.16.16.16.4.2.1.1.1.1.1.3.cmml">′</mo></msup></mrow></mrow><mo stretchy="false" id="S2.E5.m1.90.90.90.16.16.16.4.2.1.3" 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id="S2.E5.m1.131.131.16.123.18.18.18" xref="S2.E5.m1.123.123.8.cmml"><mrow id="S2.E5.m1.130.130.15.122.17.17.17.1" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.101.101.101.2.2.2" xref="S2.E5.m1.101.101.101.2.2.2.cmml">A</mi><mo id="S2.E5.m1.130.130.15.122.17.17.17.1.2" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.130.130.15.122.17.17.17.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.102.102.102.3.3.3" xref="S2.E5.m1.123.123.8a.cmml">(</mo><msup id="S2.E5.m1.130.130.15.122.17.17.17.1.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi mathvariant="normal" id="S2.E5.m1.103.103.103.4.4.4" xref="S2.E5.m1.103.103.103.4.4.4.cmml">Θ</mi><mo id="S2.E5.m1.104.104.104.5.5.5.1" xref="S2.E5.m1.104.104.104.5.5.5.1.cmml">′</mo></msup><mo id="S2.E5.m1.105.105.105.6.6.6" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.106.106.106.7.7.7" xref="S2.E5.m1.106.106.106.7.7.7.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.107.107.107.8.8.8" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow><mo id="S2.E5.m1.108.108.108.9.9.9" xref="S2.E5.m1.108.108.108.9.9.9.cmml">+</mo><mrow id="S2.E5.m1.131.131.16.123.18.18.18.2" xref="S2.E5.m1.123.123.8.cmml"><mi id="S2.E5.m1.109.109.109.10.10.10" xref="S2.E5.m1.109.109.109.10.10.10.cmml">D</mi><mo id="S2.E5.m1.131.131.16.123.18.18.18.2.2" xref="S2.E5.m1.123.123.8a.cmml">⁢</mo><mrow id="S2.E5.m1.131.131.16.123.18.18.18.2.1.1" xref="S2.E5.m1.123.123.8.cmml"><mo stretchy="false" id="S2.E5.m1.110.110.110.11.11.11" xref="S2.E5.m1.123.123.8a.cmml">(</mo><msup id="S2.E5.m1.131.131.16.123.18.18.18.2.1.1.1" xref="S2.E5.m1.123.123.8.cmml"><mi mathvariant="normal" id="S2.E5.m1.111.111.111.12.12.12" xref="S2.E5.m1.111.111.111.12.12.12.cmml">Θ</mi><mo id="S2.E5.m1.112.112.112.13.13.13.1" xref="S2.E5.m1.112.112.112.13.13.13.1.cmml">′</mo></msup><mo id="S2.E5.m1.113.113.113.14.14.14" xref="S2.E5.m1.123.123.8a.cmml">,</mo><mi mathvariant="normal" id="S2.E5.m1.114.114.114.15.15.15" xref="S2.E5.m1.114.114.114.15.15.15.cmml">Θ</mi><mo stretchy="false" id="S2.E5.m1.115.115.115.16.16.16" xref="S2.E5.m1.123.123.8a.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr></mtable></math>, <math><mrow id="S2.SS1.p1.3.m1.2.2" xref="S2.SS1.p1.3.m1.2.2.cmml"><mi id="S2.SS1.p1.3.m1.2.2.3" xref="S2.SS1.p1.3.m1.2.2.3.cmml">A</mi><mo id="S2.SS1.p1.3.m1.2.2.2" xref="S2.SS1.p1.3.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p1.3.m1.2.2.1.1" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.3.m1.2.2.1.1.2" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">(</mo><msup id="S2.SS1.p1.3.m1.2.2.1.1.1" xref="S2.SS1.p1.3.m1.2.2.1.1.1.cmml"><mi mathvariant="normal" id="S2.SS1.p1.3.m1.2.2.1.1.1.2" xref="S2.SS1.p1.3.m1.2.2.1.1.1.2.cmml">Θ</mi><mo id="S2.SS1.p1.3.m1.2.2.1.1.1.3" xref="S2.SS1.p1.3.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p1.3.m1.2.2.1.1.3" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">,</mo><mi mathvariant="normal" id="S2.SS1.p1.3.m1.1.1" xref="S2.SS1.p1.3.m1.1.1.cmml">Θ</mi><mo stretchy="false" id="S2.SS1.p1.3.m1.2.2.1.1.4" xref="S2.SS1.p1.3.m1.2.2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E6.m1.2.2" xref="S2.E6.m1.2.2.cmml"><mi id="S2.E6.m1.2.2.3" xref="S2.E6.m1.2.2.3.cmml">A</mi><mo id="S2.E6.m1.2.2.2" xref="S2.E6.m1.2.2.2.cmml">⁢</mo><mrow id="S2.E6.m1.2.2.1.1" xref="S2.E6.m1.2.2.1.2.cmml"><mo stretchy="false" id="S2.E6.m1.2.2.1.1.2" xref="S2.E6.m1.2.2.1.2.cmml">(</mo><msup id="S2.E6.m1.2.2.1.1.1" xref="S2.E6.m1.2.2.1.1.1.cmml"><mi mathvariant="normal" id="S2.E6.m1.2.2.1.1.1.2" xref="S2.E6.m1.2.2.1.1.1.2.cmml">Θ</mi><mo id="S2.E6.m1.2.2.1.1.1.3" xref="S2.E6.m1.2.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.E6.m1.2.2.1.1.3" xref="S2.E6.m1.2.2.1.2.cmml">,</mo><mi mathvariant="normal" id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml">Θ</mi><mo stretchy="false" id="S2.E6.m1.2.2.1.1.4" xref="S2.E6.m1.2.2.1.2.cmml">)</mo></mrow></mrow></math>

Correct Categorie: stat

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0607343

Formulas:

1-

Ψ (t) = e^{i ω_{0} t} e^{- t^{2} / 2 σ^{2}} .

2-

Ψ_{(α, β)} (t) = \frac{1}{α} Ψ (\frac{t - β}{α}), β \in ℛ, α > 0 .

3-

T (α, β) = \int_{- \infty}^{\infty} f (t) {\bar{Ψ}}_{(α, β)} (t) 𝑑 t .

4-

β \in [0, t]

5-

T (α, t)

6-

T (α, t)

7-

ω = \frac{ω_{0}}{α} .

8-

T (ω, t)

9-

T (ω, t)

10-

χ (t) = \frac{1}{n (t)} \sum_{i = 1}^{n (t)} | s_{i} | .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1811.11116

Formulas:

1-

v (H) / α (H)

2-

χ_{f} (G)

3-

χ_{f} (G) \geq v (G) / α (G)

4-

χ_{f} (G) \geq χ_{f} (H)

5-

χ_{f} (G) \geq \frac{v (H)}{α (H)} for every non-null H \subseteq G .

6-

ρ (G) := \max_{\emptyset \neq H \subseteq G} \frac{v (H)}{α (H)} .

7-

χ_{f} (G)

8-

χ_{f} (G) / ρ (G)

9-

χ_{f} (G) \leq C \cdot ρ (G)

10-

C \geq 343 / 282 \sim 1.216

Formulas (html):

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id="id2.2.m2.2.3.3.2.2" xref="id2.2.m2.2.3.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p2.1.m1.1.2" xref="S1.p2.1.m1.1.2.cmml"><msub id="S1.p2.1.m1.1.2.2" xref="S1.p2.1.m1.1.2.2.cmml"><mi id="S1.p2.1.m1.1.2.2.2" xref="S1.p2.1.m1.1.2.2.2.cmml">χ</mi><mi id="S1.p2.1.m1.1.2.2.3" xref="S1.p2.1.m1.1.2.2.3.cmml">f</mi></msub><mo id="S1.p2.1.m1.1.2.1" xref="S1.p2.1.m1.1.2.1.cmml">⁢</mo><mrow id="S1.p2.1.m1.1.2.3.2" xref="S1.p2.1.m1.1.2.cmml"><mo stretchy="false" id="S1.p2.1.m1.1.2.3.2.1" xref="S1.p2.1.m1.1.2.cmml">(</mo><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">G</mi><mo stretchy="false" id="S1.p2.1.m1.1.2.3.2.2" xref="S1.p2.1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p3.1.m1.3.4" xref="S1.p3.1.m1.3.4.cmml"><mrow id="S1.p3.1.m1.3.4.2" xref="S1.p3.1.m1.3.4.2.cmml"><msub id="S1.p3.1.m1.3.4.2.2" xref="S1.p3.1.m1.3.4.2.2.cmml"><mi id="S1.p3.1.m1.3.4.2.2.2" xref="S1.p3.1.m1.3.4.2.2.2.cmml">χ</mi><mi id="S1.p3.1.m1.3.4.2.2.3" xref="S1.p3.1.m1.3.4.2.2.3.cmml">f</mi></msub><mo id="S1.p3.1.m1.3.4.2.1" xref="S1.p3.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="S1.p3.1.m1.3.4.2.3.2" xref="S1.p3.1.m1.3.4.2.cmml"><mo stretchy="false" id="S1.p3.1.m1.3.4.2.3.2.1" xref="S1.p3.1.m1.3.4.2.cmml">(</mo><mi id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml">G</mi><mo stretchy="false" id="S1.p3.1.m1.3.4.2.3.2.2" xref="S1.p3.1.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S1.p3.1.m1.3.4.1" xref="S1.p3.1.m1.3.4.1.cmml">≥</mo><mrow id="S1.p3.1.m1.3.4.3" xref="S1.p3.1.m1.3.4.3.cmml"><mrow id="S1.p3.1.m1.3.4.3.2" xref="S1.p3.1.m1.3.4.3.2.cmml"><mrow id="S1.p3.1.m1.3.4.3.2.2" xref="S1.p3.1.m1.3.4.3.2.2.cmml"><mi id="S1.p3.1.m1.3.4.3.2.2.2" xref="S1.p3.1.m1.3.4.3.2.2.2.cmml">v</mi><mo id="S1.p3.1.m1.3.4.3.2.2.1" xref="S1.p3.1.m1.3.4.3.2.2.1.cmml">⁢</mo><mrow id="S1.p3.1.m1.3.4.3.2.2.3.2" xref="S1.p3.1.m1.3.4.3.2.2.cmml"><mo stretchy="false" id="S1.p3.1.m1.3.4.3.2.2.3.2.1" xref="S1.p3.1.m1.3.4.3.2.2.cmml">(</mo><mi id="S1.p3.1.m1.2.2" xref="S1.p3.1.m1.2.2.cmml">G</mi><mo stretchy="false" id="S1.p3.1.m1.3.4.3.2.2.3.2.2" xref="S1.p3.1.m1.3.4.3.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p3.1.m1.3.4.3.2.1" xref="S1.p3.1.m1.3.4.3.2.1.cmml">/</mo><mi id="S1.p3.1.m1.3.4.3.2.3" xref="S1.p3.1.m1.3.4.3.2.3.cmml">α</mi></mrow><mo id="S1.p3.1.m1.3.4.3.1" xref="S1.p3.1.m1.3.4.3.1.cmml">⁢</mo><mrow id="S1.p3.1.m1.3.4.3.3.2" xref="S1.p3.1.m1.3.4.3.cmml"><mo stretchy="false" id="S1.p3.1.m1.3.4.3.3.2.1" xref="S1.p3.1.m1.3.4.3.cmml">(</mo><mi id="S1.p3.1.m1.3.3" xref="S1.p3.1.m1.3.3.cmml">G</mi><mo stretchy="false" id="S1.p3.1.m1.3.4.3.3.2.2" xref="S1.p3.1.m1.3.4.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.p3.5.m5.2.3" xref="S1.p3.5.m5.2.3.cmml"><mrow id="S1.p3.5.m5.2.3.2" xref="S1.p3.5.m5.2.3.2.cmml"><msub id="S1.p3.5.m5.2.3.2.2" xref="S1.p3.5.m5.2.3.2.2.cmml"><mi id="S1.p3.5.m5.2.3.2.2.2" xref="S1.p3.5.m5.2.3.2.2.2.cmml">χ</mi><mi id="S1.p3.5.m5.2.3.2.2.3" xref="S1.p3.5.m5.2.3.2.2.3.cmml">f</mi></msub><mo id="S1.p3.5.m5.2.3.2.1" xref="S1.p3.5.m5.2.3.2.1.cmml">⁢</mo><mrow id="S1.p3.5.m5.2.3.2.3.2" xref="S1.p3.5.m5.2.3.2.cmml"><mo stretchy="false" id="S1.p3.5.m5.2.3.2.3.2.1" xref="S1.p3.5.m5.2.3.2.cmml">(</mo><mi id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml">G</mi><mo stretchy="false" id="S1.p3.5.m5.2.3.2.3.2.2" xref="S1.p3.5.m5.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.p3.5.m5.2.3.1" xref="S1.p3.5.m5.2.3.1.cmml">≥</mo><mrow id="S1.p3.5.m5.2.3.3" xref="S1.p3.5.m5.2.3.3.cmml"><msub id="S1.p3.5.m5.2.3.3.2" xref="S1.p3.5.m5.2.3.3.2.cmml"><mi id="S1.p3.5.m5.2.3.3.2.2" xref="S1.p3.5.m5.2.3.3.2.2.cmml">χ</mi><mi id="S1.p3.5.m5.2.3.3.2.3" xref="S1.p3.5.m5.2.3.3.2.3.cmml">f</mi></msub><mo id="S1.p3.5.m5.2.3.3.1" xref="S1.p3.5.m5.2.3.3.1.cmml">⁢</mo><mrow id="S1.p3.5.m5.2.3.3.3.2" xref="S1.p3.5.m5.2.3.3.cmml"><mo stretchy="false" id="S1.p3.5.m5.2.3.3.3.2.1" xref="S1.p3.5.m5.2.3.3.cmml">(</mo><mi id="S1.p3.5.m5.2.2" xref="S1.p3.5.m5.2.2.cmml">H</mi><mo stretchy="false" id="S1.p3.5.m5.2.3.3.3.2.2" xref="S1.p3.5.m5.2.3.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.Ex1.m1.4.4.1"><mrow id="S1.Ex1.m1.4.4.1.1.2" xref="S1.Ex1.m1.4.4.1.1.3.cmml"><mrow id="S1.Ex1.m1.4.4.1.1.1.1" xref="S1.Ex1.m1.4.4.1.1.1.1.cmml"><mrow id="S1.Ex1.m1.4.4.1.1.1.1.2" xref="S1.Ex1.m1.4.4.1.1.1.1.2.cmml"><msub id="S1.Ex1.m1.4.4.1.1.1.1.2.2" xref="S1.Ex1.m1.4.4.1.1.1.1.2.2.cmml"><mi id="S1.Ex1.m1.4.4.1.1.1.1.2.2.2" xref="S1.Ex1.m1.4.4.1.1.1.1.2.2.2.cmml">χ</mi><mi id="S1.Ex1.m1.4.4.1.1.1.1.2.2.3" xref="S1.Ex1.m1.4.4.1.1.1.1.2.2.3.cmml">f</mi></msub><mo id="S1.Ex1.m1.4.4.1.1.1.1.2.1" xref="S1.Ex1.m1.4.4.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.4.4.1.1.1.1.2.3.2" xref="S1.Ex1.m1.4.4.1.1.1.1.2.cmml"><mo stretchy="false" id="S1.Ex1.m1.4.4.1.1.1.1.2.3.2.1" xref="S1.Ex1.m1.4.4.1.1.1.1.2.cmml">(</mo><mi id="S1.Ex1.m1.3.3" xref="S1.Ex1.m1.3.3.cmml">G</mi><mo stretchy="false" id="S1.Ex1.m1.4.4.1.1.1.1.2.3.2.2" xref="S1.Ex1.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.4.4.1.1.1.1.1" xref="S1.Ex1.m1.4.4.1.1.1.1.1.cmml">≥</mo><mfrac id="S1.Ex1.m1.2.2" xref="S1.Ex1.m1.2.2.cmml"><mrow id="S1.Ex1.m1.1.1.1" xref="S1.Ex1.m1.1.1.1.cmml"><mi id="S1.Ex1.m1.1.1.1.3" xref="S1.Ex1.m1.1.1.1.3.cmml">v</mi><mo id="S1.Ex1.m1.1.1.1.2" xref="S1.Ex1.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.1.1.1.4.2" xref="S1.Ex1.m1.1.1.1.cmml"><mo stretchy="false" id="S1.Ex1.m1.1.1.1.4.2.1" xref="S1.Ex1.m1.1.1.1.cmml">(</mo><mi id="S1.Ex1.m1.1.1.1.1" xref="S1.Ex1.m1.1.1.1.1.cmml">H</mi><mo stretchy="false" id="S1.Ex1.m1.1.1.1.4.2.2" xref="S1.Ex1.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S1.Ex1.m1.2.2.2" xref="S1.Ex1.m1.2.2.2.cmml"><mi id="S1.Ex1.m1.2.2.2.3" xref="S1.Ex1.m1.2.2.2.3.cmml">α</mi><mo id="S1.Ex1.m1.2.2.2.2" xref="S1.Ex1.m1.2.2.2.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.2.2.2.4.2" xref="S1.Ex1.m1.2.2.2.cmml"><mo stretchy="false" id="S1.Ex1.m1.2.2.2.4.2.1" xref="S1.Ex1.m1.2.2.2.cmml">(</mo><mi id="S1.Ex1.m1.2.2.2.1" xref="S1.Ex1.m1.2.2.2.1.cmml">H</mi><mo stretchy="false" id="S1.Ex1.m1.2.2.2.4.2.2" xref="S1.Ex1.m1.2.2.2.cmml">)</mo></mrow></mrow></mfrac></mrow><mo mathvariant="italic" separator="true" id="S1.Ex1.m1.4.4.1.1.2.3" xref="S1.Ex1.m1.4.4.1.1.3a.cmml"> </mo><mrow id="S1.Ex1.m1.4.4.1.1.2.2" xref="S1.Ex1.m1.4.4.1.1.2.2.cmml"><mrow id="S1.Ex1.m1.4.4.1.1.2.2.2" xref="S1.Ex1.m1.4.4.1.1.2.2.2.cmml"><mtext id="S1.Ex1.m1.4.4.1.1.2.2.2.2" xref="S1.Ex1.m1.4.4.1.1.2.2.2.2a.cmml"> for every non-null </mtext><mo id="S1.Ex1.m1.4.4.1.1.2.2.2.1" xref="S1.Ex1.m1.4.4.1.1.2.2.2.1.cmml">⁢</mo><mi id="S1.Ex1.m1.4.4.1.1.2.2.2.3" xref="S1.Ex1.m1.4.4.1.1.2.2.2.3.cmml">H</mi></mrow><mo id="S1.Ex1.m1.4.4.1.1.2.2.1" xref="S1.Ex1.m1.4.4.1.1.2.2.1.cmml">⊆</mo><mi id="S1.Ex1.m1.4.4.1.1.2.2.3" xref="S1.Ex1.m1.4.4.1.1.2.2.3.cmml">G</mi></mrow></mrow><mo id="S1.Ex1.m1.4.4.1.2">.</mo></mrow></math>, <math><mrow id="S1.Ex2.m1.4.4.1" xref="S1.Ex2.m1.4.4.1.1.cmml"><mrow id="S1.Ex2.m1.4.4.1.1" xref="S1.Ex2.m1.4.4.1.1.cmml"><mrow id="S1.Ex2.m1.4.4.1.1.2" xref="S1.Ex2.m1.4.4.1.1.2.cmml"><mi id="S1.Ex2.m1.4.4.1.1.2.2" xref="S1.Ex2.m1.4.4.1.1.2.2.cmml">ρ</mi><mo id="S1.Ex2.m1.4.4.1.1.2.1" xref="S1.Ex2.m1.4.4.1.1.2.1.cmml">⁢</mo><mrow id="S1.Ex2.m1.4.4.1.1.2.3.2" xref="S1.Ex2.m1.4.4.1.1.2.cmml"><mo stretchy="false" id="S1.Ex2.m1.4.4.1.1.2.3.2.1" xref="S1.Ex2.m1.4.4.1.1.2.cmml">(</mo><mi id="S1.Ex2.m1.3.3" xref="S1.Ex2.m1.3.3.cmml">G</mi><mo stretchy="false" id="S1.Ex2.m1.4.4.1.1.2.3.2.2" xref="S1.Ex2.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S1.Ex2.m1.4.4.1.1.1" xref="S1.Ex2.m1.4.4.1.1.1.cmml">:=</mo><mrow id="S1.Ex2.m1.4.4.1.1.3" xref="S1.Ex2.m1.4.4.1.1.3.cmml"><mpadded width="+2.8pt" id="S1.Ex2.m1.4.4.1.1.3.1" xref="S1.Ex2.m1.4.4.1.1.3.1.cmml"><munder id="S1.Ex2.m1.4.4.1.1.3.1a" xref="S1.Ex2.m1.4.4.1.1.3.1.cmml"><mi id="S1.Ex2.m1.4.4.1.1.3.1.2" xref="S1.Ex2.m1.4.4.1.1.3.1.2.cmml">max</mi><mrow id="S1.Ex2.m1.4.4.1.1.3.1.3" xref="S1.Ex2.m1.4.4.1.1.3.1.3.cmml"><mi mathvariant="normal" id="S1.Ex2.m1.4.4.1.1.3.1.3.2" xref="S1.Ex2.m1.4.4.1.1.3.1.3.2.cmml">∅</mi><mo id="S1.Ex2.m1.4.4.1.1.3.1.3.3" xref="S1.Ex2.m1.4.4.1.1.3.1.3.3.cmml">≠</mo><mi id="S1.Ex2.m1.4.4.1.1.3.1.3.4" xref="S1.Ex2.m1.4.4.1.1.3.1.3.4.cmml">H</mi><mo id="S1.Ex2.m1.4.4.1.1.3.1.3.5" xref="S1.Ex2.m1.4.4.1.1.3.1.3.5.cmml">⊆</mo><mi id="S1.Ex2.m1.4.4.1.1.3.1.3.6" xref="S1.Ex2.m1.4.4.1.1.3.1.3.6.cmml">G</mi></mrow></munder></mpadded><mo id="S1.Ex2.m1.4.4.1.1.3a" xref="S1.Ex2.m1.4.4.1.1.3.cmml">⁡</mo><mpadded width="+1.7pt" id="S1.Ex2.m1.2.2" xref="S1.Ex2.m1.2.2.cmml"><mfrac id="S1.Ex2.m1.2.2a" xref="S1.Ex2.m1.2.2.cmml"><mrow id="S1.Ex2.m1.1.1.1" xref="S1.Ex2.m1.1.1.1.cmml"><mi id="S1.Ex2.m1.1.1.1.3" xref="S1.Ex2.m1.1.1.1.3.cmml">v</mi><mo id="S1.Ex2.m1.1.1.1.2" xref="S1.Ex2.m1.1.1.1.2.cmml">⁢</mo><mrow id="S1.Ex2.m1.1.1.1.4.2" xref="S1.Ex2.m1.1.1.1.cmml"><mo stretchy="false" id="S1.Ex2.m1.1.1.1.4.2.1" xref="S1.Ex2.m1.1.1.1.cmml">(</mo><mi id="S1.Ex2.m1.1.1.1.1" xref="S1.Ex2.m1.1.1.1.1.cmml">H</mi><mo stretchy="false" id="S1.Ex2.m1.1.1.1.4.2.2" xref="S1.Ex2.m1.1.1.1.cmml">)</mo></mrow></mrow><mrow id="S1.Ex2.m1.2.2.2" xref="S1.Ex2.m1.2.2.2.cmml"><mi id="S1.Ex2.m1.2.2.2.3" xref="S1.Ex2.m1.2.2.2.3.cmml">α</mi><mo id="S1.Ex2.m1.2.2.2.2" xref="S1.Ex2.m1.2.2.2.2.cmml">⁢</mo><mrow id="S1.Ex2.m1.2.2.2.4.2" xref="S1.Ex2.m1.2.2.2.cmml"><mo stretchy="false" id="S1.Ex2.m1.2.2.2.4.2.1" xref="S1.Ex2.m1.2.2.2.cmml">(</mo><mi id="S1.Ex2.m1.2.2.2.1" xref="S1.Ex2.m1.2.2.2.1.cmml">H</mi><mo stretchy="false" id="S1.Ex2.m1.2.2.2.4.2.2" xref="S1.Ex2.m1.2.2.2.cmml">)</mo></mrow></mrow></mfrac></mpadded></mrow></mrow><mo id="S1.Ex2.m1.4.4.1.2" xref="S1.Ex2.m1.4.4.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S1.p4.2.m2.1.2" xref="S1.p4.2.m2.1.2.cmml"><msub id="S1.p4.2.m2.1.2.2" xref="S1.p4.2.m2.1.2.2.cmml"><mi id="S1.p4.2.m2.1.2.2.2" xref="S1.p4.2.m2.1.2.2.2.cmml">χ</mi><mi id="S1.p4.2.m2.1.2.2.3" xref="S1.p4.2.m2.1.2.2.3.cmml">f</mi></msub><mo id="S1.p4.2.m2.1.2.1" xref="S1.p4.2.m2.1.2.1.cmml">⁢</mo><mrow id="S1.p4.2.m2.1.2.3.2" xref="S1.p4.2.m2.1.2.cmml"><mo stretchy="false" id="S1.p4.2.m2.1.2.3.2.1" xref="S1.p4.2.m2.1.2.cmml">(</mo><mi id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">G</mi><mo stretchy="false" id="S1.p4.2.m2.1.2.3.2.2" xref="S1.p4.2.m2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.p4.4.m4.2.3" xref="S1.p4.4.m4.2.3.cmml"><mrow id="S1.p4.4.m4.2.3.2" xref="S1.p4.4.m4.2.3.2.cmml"><mrow id="S1.p4.4.m4.2.3.2.2" xref="S1.p4.4.m4.2.3.2.2.cmml"><msub id="S1.p4.4.m4.2.3.2.2.2" xref="S1.p4.4.m4.2.3.2.2.2.cmml"><mi id="S1.p4.4.m4.2.3.2.2.2.2" xref="S1.p4.4.m4.2.3.2.2.2.2.cmml">χ</mi><mi id="S1.p4.4.m4.2.3.2.2.2.3" xref="S1.p4.4.m4.2.3.2.2.2.3.cmml">f</mi></msub><mo id="S1.p4.4.m4.2.3.2.2.1" xref="S1.p4.4.m4.2.3.2.2.1.cmml">⁢</mo><mrow id="S1.p4.4.m4.2.3.2.2.3.2" xref="S1.p4.4.m4.2.3.2.2.cmml"><mo stretchy="false" id="S1.p4.4.m4.2.3.2.2.3.2.1" xref="S1.p4.4.m4.2.3.2.2.cmml">(</mo><mi id="S1.p4.4.m4.1.1" xref="S1.p4.4.m4.1.1.cmml">G</mi><mo stretchy="false" id="S1.p4.4.m4.2.3.2.2.3.2.2" xref="S1.p4.4.m4.2.3.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p4.4.m4.2.3.2.1" xref="S1.p4.4.m4.2.3.2.1.cmml">/</mo><mi id="S1.p4.4.m4.2.3.2.3" xref="S1.p4.4.m4.2.3.2.3.cmml">ρ</mi></mrow><mo id="S1.p4.4.m4.2.3.1" xref="S1.p4.4.m4.2.3.1.cmml">⁢</mo><mrow id="S1.p4.4.m4.2.3.3.2" xref="S1.p4.4.m4.2.3.cmml"><mo stretchy="false" id="S1.p4.4.m4.2.3.3.2.1" xref="S1.p4.4.m4.2.3.cmml">(</mo><mi id="S1.p4.4.m4.2.2" xref="S1.p4.4.m4.2.2.cmml">G</mi><mo stretchy="false" id="S1.p4.4.m4.2.3.3.2.2" xref="S1.p4.4.m4.2.3.cmml">)</mo></mrow></mrow></math>, <math><mrow id="Thmthm1.p1.2.2.m2.2.3" xref="Thmthm1.p1.2.2.m2.2.3.cmml"><mrow id="Thmthm1.p1.2.2.m2.2.3.2" xref="Thmthm1.p1.2.2.m2.2.3.2.cmml"><msub id="Thmthm1.p1.2.2.m2.2.3.2.2" xref="Thmthm1.p1.2.2.m2.2.3.2.2.cmml"><mi id="Thmthm1.p1.2.2.m2.2.3.2.2.2" xref="Thmthm1.p1.2.2.m2.2.3.2.2.2.cmml">χ</mi><mi id="Thmthm1.p1.2.2.m2.2.3.2.2.3" xref="Thmthm1.p1.2.2.m2.2.3.2.2.3.cmml">f</mi></msub><mo mathvariant="italic" id="Thmthm1.p1.2.2.m2.2.3.2.1" xref="Thmthm1.p1.2.2.m2.2.3.2.1.cmml">⁢</mo><mrow id="Thmthm1.p1.2.2.m2.2.3.2.3.2" xref="Thmthm1.p1.2.2.m2.2.3.2.cmml"><mo mathvariant="normal" stretchy="false" id="Thmthm1.p1.2.2.m2.2.3.2.3.2.1" xref="Thmthm1.p1.2.2.m2.2.3.2.cmml">(</mo><mi id="Thmthm1.p1.2.2.m2.1.1" xref="Thmthm1.p1.2.2.m2.1.1.cmml">G</mi><mo mathvariant="normal" stretchy="false" id="Thmthm1.p1.2.2.m2.2.3.2.3.2.2" xref="Thmthm1.p1.2.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo mathvariant="normal" id="Thmthm1.p1.2.2.m2.2.3.1" xref="Thmthm1.p1.2.2.m2.2.3.1.cmml">≤</mo><mrow id="Thmthm1.p1.2.2.m2.2.3.3" xref="Thmthm1.p1.2.2.m2.2.3.3.cmml"><mrow id="Thmthm1.p1.2.2.m2.2.3.3.2" xref="Thmthm1.p1.2.2.m2.2.3.3.2.cmml"><mi id="Thmthm1.p1.2.2.m2.2.3.3.2.2" xref="Thmthm1.p1.2.2.m2.2.3.3.2.2.cmml">C</mi><mo mathvariant="normal" id="Thmthm1.p1.2.2.m2.2.3.3.2.1" xref="Thmthm1.p1.2.2.m2.2.3.3.2.1.cmml">⋅</mo><mi id="Thmthm1.p1.2.2.m2.2.3.3.2.3" xref="Thmthm1.p1.2.2.m2.2.3.3.2.3.cmml">ρ</mi></mrow><mo mathvariant="italic" id="Thmthm1.p1.2.2.m2.2.3.3.1" xref="Thmthm1.p1.2.2.m2.2.3.3.1.cmml">⁢</mo><mrow id="Thmthm1.p1.2.2.m2.2.3.3.3.2" xref="Thmthm1.p1.2.2.m2.2.3.3.cmml"><mo mathvariant="normal" stretchy="false" id="Thmthm1.p1.2.2.m2.2.3.3.3.2.1" xref="Thmthm1.p1.2.2.m2.2.3.3.cmml">(</mo><mi id="Thmthm1.p1.2.2.m2.2.2" xref="Thmthm1.p1.2.2.m2.2.2.cmml">G</mi><mo mathvariant="normal" stretchy="false" id="Thmthm1.p1.2.2.m2.2.3.3.3.2.2" xref="Thmthm1.p1.2.2.m2.2.3.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mi id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml">C</mi><mo id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml">≥</mo><mrow id="S1.p5.2.m2.1.1.4" xref="S1.p5.2.m2.1.1.4.cmml"><mn id="S1.p5.2.m2.1.1.4.2" xref="S1.p5.2.m2.1.1.4.2.cmml">343</mn><mo id="S1.p5.2.m2.1.1.4.1" xref="S1.p5.2.m2.1.1.4.1.cmml">/</mo><mn id="S1.p5.2.m2.1.1.4.3" xref="S1.p5.2.m2.1.1.4.3.cmml">282</mn></mrow><mo id="S1.p5.2.m2.1.1.5" xref="S1.p5.2.m2.1.1.5.cmml">∼</mo><mn id="S1.p5.2.m2.1.1.6" xref="S1.p5.2.m2.1.1.6.cmml">1.216</mn></mrow></math>

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/quant-ph/9710043

Formulas:

1-

ν_{⟂} \leq \frac{E_{max}}{h}

2-

ν_{⟂} \leq \frac{2 E}{h},

3-

| ψ_{0} ⟩ = \sum_{n} c_{n} | E_{n} ⟩

4-

{| E_{n} ⟩}

5-

τ_{⟂} \geq \frac{h}{4 E} .

6-

τ_{⟂} \geq \frac{h}{4 Δ E}

7-

| ψ_{t} ⟩ = \sum_{n} c_{n} e^{- i \frac{E_{n} t}{ℏ}} | E_{n} ⟩

8-

S (t) = ⟨ ψ_{0} | ψ_{t} ⟩ = \sum_{n = 0}^{\infty} {| c_{n} |}^{2} e^{- i \frac{E_{n} t}{ℏ}}

9-

\sum_{n = 0}^{\infty} {| c_{n} |}^{2} \cos (\frac{E_{n} t}{ℏ})

10-

\sum_{n = 0}^{\infty} {| c_{n} |}^{2} (1 - \frac{2}{π} (\frac{E_{n} t}{ℏ} + \sin (\frac{E_{n} t}{ℏ})))

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0010213

Formulas:

1-

d_{x} = - 0.1 J_{FM}

2-

d_{z} = 0.1 J_{FM}

3-

B_{SW} = 2 d_{z}

4-

J_{FM} \sum_{⟨ i, j ⟩} {\underline{S}}_{i} \cdot {\underline{S}}_{j} - \sum_{i} (d_{z} S_{i z}^{2} + d_{x} S_{i x}^{2} + {\underline{S}}_{i} \cdot \underline{B})

5-

J_{AFM} \sum_{⟨ i, j ⟩} ϵ_{i} ϵ_{j} σ_{i} σ_{j} - \sum_{i} B ϵ_{i} σ_{i}

6-

J_{INT} \sum_{⟨ i, j ⟩} ϵ_{i} σ_{i} S_{j z} .

7-

\underline{B} = B z_{¯}^{^} + B_{y} y_{¯}^{^}

8-

ϵ_{i} = 0, 1

9-

J_{AFM} = - J_{FM} / 2

10-

J_{INT} = - J_{AFM}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1207.3599

Formulas:

1-

T^{F r a m e}

2-

T^{F r a m e}

3-

T_{n, n + 1}^{G B} = \frac{F}{100} \times \frac{1}{2} [T S_{n} + T S_{n + 1}]

4-

T_{1}^{G B} = \frac{F \times T S_{1}}{100}

5-

T_{n}^{G B} = \frac{F \times T S_{n}}{100}

6-

D = M i n (T S_{1} \dots \dots T S_{n}) \times \frac{F}{100}

7-

△ T = E x p e c t e d A r r i v a l T i m e - C u r r e n t A r r i v a l T i m e

8-

D V = {\begin{matrix} 0 & if ∣ △ T ∣ < D \\ △ T & if ∣ △ T ∣ > D \end{matrix}

9-

E_{T o t a l} = \sum_{k = 1}^{N} E_{S l e e p_{k}} + \sum_{k = 1}^{N} E_{A c t i v e_{k}}

10-

T_{S l e e p} = T_{F r a m e} - T_{A c t i v e}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9911039

Formulas:

1-

κ_{s e a} = 0.1375

2-

m_{P S} / m_{V}

3-

m_{PS} / m_{V} \approx 0.8 - 0.5

4-

O (1 / M)

5-

E (\vec{p}) - E (0) = \sqrt{{\vec{p}}^{2} + M^{2}} - M

6-

O (α / M)

7-

(1 + α ρ_{0}) J_{4, l a t}^{(0)} + (1 + α ρ_{1}) J_{4, l a t}^{(1)}

8-

α ρ_{2} J_{4, l a t}^{(2)} .

9-

α_{\bar{M S}}^{T I} (1 / a)

10-

f_{B_{s}} / f_{B}

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0009490

Formulas:

1-

M_{B} (M a x)

2-

Δ m_{15} (B)

3-

Δ m_{15} (B)

4-

Δ m_{15} (B)

5-

τ_{LC} \propto κ_{opt}^{1 / 2} M_{ej}^{3 / 4} E_{K}^{- 1 / 4} .

6-

ρ_{DDT} = 1.3 - {3 10}^{7}

7-

κ_{opt} = 0.25 X_{Fe - gp} + 0.025 (1 - X_{Fe - gp}) [{cm}^{2} g^{- 1}] .

8-

Δ m (15) (B o l)

9-

Δ m (15) (B o l)

10-

κ_{opt} = [0.25 X_{Fe - gp} + 0.025 (1 - X_{Fe - gp})] {(\frac{t_{d}}{17})}^{- \frac{3}{2}} [{cm}^{2} g^{- 1}],

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1105.3073

Formulas:

1-

A_{V} = 5.348 \times 10^{- 22} (N_{H, tot} / 1 {cm}^{- 2})

2-

f_{pe} = Γ_{pe} (A_{V}) / Γ_{pe} (0)

3-

Γ_{pe} (A_{V})

4-

Γ_{pe} (0)

5-

Γ_{pe} = ⟨ f_{pe} ⟩ Γ_{pe} (0)

6-

⟨ f_{H_{2}, H_{2}} ⟩

7-

⟨ f_{CO, CO} ⟩

8-

⟨ f_{H_{2}, CO} ⟩

9-

f_{H_{2}, H_{2}}

10-

r_{acc} = 0.0026 pc

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9908031

Formulas:

1-

W [C] = ⟨ tr U [C] ⟩

2-

S U (2)

3-

A_{p} (U) = β_{A} | tr U_{p} | + β tr U_{p},

4-

W [C] \leq \exp (- ρ (β) | A |)

5-

(Const) \ln [1 + {⟨ θ_{\partial^{*} p} ⟩}^{(+)} \tanh K_{0}]

6-

(Const) e^{- 4 β} {⟨ θ_{\partial^{*} p} ⟩}^{(+)} .

7-

K_{0} \equiv \frac{1}{2} \ln \coth (2 β)

8-

{⟨ θ_{\partial^{*} p} ⟩}^{(+)}

9-

η_{c} = \prod_{p \in \partial c} sign tr U_{p} .

10-

θ_{\partial^{*} p} = 1

Formulas (html):

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xref="S0.E2.m1.4.4.cmml">exp</mi><mo id="S0.E2.m1.5.5.1.1a" xref="S0.E2.m1.5.5.1.2.cmml">⁡</mo><mrow id="S0.E2.m1.5.5.1.1.1" xref="S0.E2.m1.5.5.1.2.cmml"><mo stretchy="false" id="S0.E2.m1.5.5.1.1.1.2" xref="S0.E2.m1.5.5.1.2.cmml">(</mo><mrow id="S0.E2.m1.5.5.1.1.1.1" xref="S0.E2.m1.5.5.1.1.1.1.cmml"><mo lspace="4.2pt" id="S0.E2.m1.5.5.1.1.1.1.1" xref="S0.E2.m1.5.5.1.1.1.1.1.cmml">-</mo><mrow id="S0.E2.m1.5.5.1.1.1.1.2" xref="S0.E2.m1.5.5.1.1.1.1.2.cmml"><mi id="S0.E2.m1.5.5.1.1.1.1.2.2" xref="S0.E2.m1.5.5.1.1.1.1.2.2.cmml">ρ</mi><mo id="S0.E2.m1.5.5.1.1.1.1.2.1" xref="S0.E2.m1.5.5.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S0.E2.m1.5.5.1.1.1.1.2.3.2" xref="S0.E2.m1.5.5.1.1.1.1.2.cmml"><mo stretchy="false" id="S0.E2.m1.5.5.1.1.1.1.2.3.2.1" xref="S0.E2.m1.5.5.1.1.1.1.2.cmml">(</mo><mi id="S0.E2.m1.2.2" xref="S0.E2.m1.2.2.cmml">β</mi><mo rspace="4.2pt" stretchy="false" id="S0.E2.m1.5.5.1.1.1.1.2.3.2.2" xref="S0.E2.m1.5.5.1.1.1.1.2.cmml">)</mo></mrow><mo id="S0.E2.m1.5.5.1.1.1.1.2.1a" xref="S0.E2.m1.5.5.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S0.E2.m1.5.5.1.1.1.1.2.4.2" xref="S0.E2.m1.5.5.1.1.1.1.2.4.1.cmml"><mo stretchy="false" id="S0.E2.m1.5.5.1.1.1.1.2.4.2.1" xref="S0.E2.m1.5.5.1.1.1.1.2.4.1.1.cmml">|</mo><mi id="S0.E2.m1.3.3" xref="S0.E2.m1.3.3.cmml">A</mi><mo rspace="4.2pt" stretchy="false" id="S0.E2.m1.5.5.1.1.1.1.2.4.2.2" xref="S0.E2.m1.5.5.1.1.1.1.2.4.1.1.cmml">|</mo></mrow></mrow></mrow><mo stretchy="false" id="S0.E2.m1.5.5.1.1.1.3" xref="S0.E2.m1.5.5.1.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S0.Ex1.m3.4.4" xref="S0.Ex1.m3.4.4.cmml"><mrow id="S0.Ex1.m3.4.4.3.2" xref="S0.Ex1.m3.2.2a.cmml"><mo stretchy="false" id="S0.Ex1.m3.4.4.3.2.1" xref="S0.Ex1.m3.2.2a.cmml">(</mo><mtext id="S0.Ex1.m3.2.2" xref="S0.Ex1.m3.2.2.cmml">Const</mtext><mo rspace="4.2pt" stretchy="false" id="S0.Ex1.m3.4.4.3.2.2" xref="S0.Ex1.m3.2.2a.cmml">)</mo></mrow><mo id="S0.Ex1.m3.4.4.2" xref="S0.Ex1.m3.4.4.2.cmml">⁢</mo><mrow id="S0.Ex1.m3.4.4.1.1" xref="S0.Ex1.m3.4.4.1.2.cmml"><mi id="S0.Ex1.m3.3.3" xref="S0.Ex1.m3.3.3.cmml">ln</mi><mo id="S0.Ex1.m3.4.4.1.1a" xref="S0.Ex1.m3.4.4.1.2.cmml">⁡</mo><mrow id="S0.Ex1.m3.4.4.1.1.1" xref="S0.Ex1.m3.4.4.1.2.cmml"><mo stretchy="false" id="S0.Ex1.m3.4.4.1.1.1.2" xref="S0.Ex1.m3.4.4.1.2.cmml">[</mo><mrow id="S0.Ex1.m3.4.4.1.1.1.1" xref="S0.Ex1.m3.4.4.1.1.1.1.cmml"><mn id="S0.Ex1.m3.4.4.1.1.1.1.3" xref="S0.Ex1.m3.4.4.1.1.1.1.3.cmml"> 1</mn><mo id="S0.Ex1.m3.4.4.1.1.1.1.2" xref="S0.Ex1.m3.4.4.1.1.1.1.2.cmml">+</mo><mrow id="S0.Ex1.m3.4.4.1.1.1.1.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.cmml"><msup id="S0.Ex1.m3.4.4.1.1.1.1.1.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.cmml"><mrow id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.2.cmml"><mo maxsize="120%" minsize="120%" id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.2.1.cmml">⟨</mo><msub id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.2.cmml">θ</mi><mrow id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.cmml"><msup id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.1.cmml"><mo id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.1.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.1.2.cmml">∂</mo><mo id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.1.3" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.1.3.cmml">∗</mo></msup><mo id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3a" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.1.3.2.cmml">p</mi></mrow></msub><mo maxsize="120%" minsize="120%" id="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.1.3" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.1.2.1.cmml">⟩</mo></mrow><mrow id="S0.Ex1.m3.1.1.1.3" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S0.Ex1.m3.1.1.1.3.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.cmml">(</mo><mo id="S0.Ex1.m3.1.1.1.1" xref="S0.Ex1.m3.1.1.1.1.cmml">+</mo><mo stretchy="false" id="S0.Ex1.m3.1.1.1.3.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></msup><mo id="S0.Ex1.m3.4.4.1.1.1.1.1.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S0.Ex1.m3.4.4.1.1.1.1.1.3" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.cmml"><mi id="S0.Ex1.m3.4.4.1.1.1.1.1.3.1" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.1.cmml">tanh</mi><mo id="S0.Ex1.m3.4.4.1.1.1.1.1.3a" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.cmml">⁡</mo><mpadded width="+1.7pt" id="S0.Ex1.m3.4.4.1.1.1.1.1.3.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.2.cmml"><msub id="S0.Ex1.m3.4.4.1.1.1.1.1.3.2a" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.2.cmml"><mi id="S0.Ex1.m3.4.4.1.1.1.1.1.3.2.2" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.2.2.cmml">K</mi><mn id="S0.Ex1.m3.4.4.1.1.1.1.1.3.2.3" xref="S0.Ex1.m3.4.4.1.1.1.1.1.3.2.3.cmml">0</mn></msub></mpadded></mrow></mrow></mrow><mo stretchy="false" id="S0.Ex1.m3.4.4.1.1.1.3" xref="S0.Ex1.m3.4.4.1.2.cmml">]</mo></mrow></mrow></mrow></math>, <math><mrow id="S0.E3.m3.3.3.1" xref="S0.E3.m3.3.3.1.1.cmml"><mrow id="S0.E3.m3.3.3.1.1" xref="S0.E3.m3.3.3.1.1.cmml"><mrow id="S0.E3.m3.3.3.1.1.3.2" xref="S0.E3.m3.2.2a.cmml"><mo stretchy="false" id="S0.E3.m3.3.3.1.1.3.2.1" xref="S0.E3.m3.2.2a.cmml">(</mo><mtext id="S0.E3.m3.2.2" xref="S0.E3.m3.2.2.cmml">Const</mtext><mo rspace="5.3pt" stretchy="false" id="S0.E3.m3.3.3.1.1.3.2.2" xref="S0.E3.m3.2.2a.cmml">)</mo></mrow><mo id="S0.E3.m3.3.3.1.1.2" xref="S0.E3.m3.3.3.1.1.2.cmml">⁢</mo><mpadded width="+1.7pt" id="S0.E3.m3.3.3.1.1.4" xref="S0.E3.m3.3.3.1.1.4.cmml"><msup id="S0.E3.m3.3.3.1.1.4a" xref="S0.E3.m3.3.3.1.1.4.cmml"><mi id="S0.E3.m3.3.3.1.1.4.2" xref="S0.E3.m3.3.3.1.1.4.2.cmml">e</mi><mrow id="S0.E3.m3.3.3.1.1.4.3" xref="S0.E3.m3.3.3.1.1.4.3.cmml"><mo id="S0.E3.m3.3.3.1.1.4.3.1" xref="S0.E3.m3.3.3.1.1.4.3.1.cmml">-</mo><mrow id="S0.E3.m3.3.3.1.1.4.3.2" xref="S0.E3.m3.3.3.1.1.4.3.2.cmml"><mn id="S0.E3.m3.3.3.1.1.4.3.2.2" xref="S0.E3.m3.3.3.1.1.4.3.2.2.cmml">4</mn><mo id="S0.E3.m3.3.3.1.1.4.3.2.1" xref="S0.E3.m3.3.3.1.1.4.3.2.1.cmml">⁢</mo><mi id="S0.E3.m3.3.3.1.1.4.3.2.3" xref="S0.E3.m3.3.3.1.1.4.3.2.3.cmml">β</mi></mrow></mrow></msup></mpadded><mo id="S0.E3.m3.3.3.1.1.2a" xref="S0.E3.m3.3.3.1.1.2.cmml">⁢</mo><mpadded width="+2.8pt" id="S0.E3.m3.3.3.1.1.1" xref="S0.E3.m3.3.3.1.1.1.cmml"><msup id="S0.E3.m3.3.3.1.1.1a" xref="S0.E3.m3.3.3.1.1.1.cmml"><mrow id="S0.E3.m3.3.3.1.1.1.1.1" xref="S0.E3.m3.3.3.1.1.1.1.2.cmml"><mo maxsize="120%" minsize="120%" id="S0.E3.m3.3.3.1.1.1.1.1.2" xref="S0.E3.m3.3.3.1.1.1.1.2.1.cmml">⟨</mo><msub id="S0.E3.m3.3.3.1.1.1.1.1.1" xref="S0.E3.m3.3.3.1.1.1.1.1.1.cmml"><mi id="S0.E3.m3.3.3.1.1.1.1.1.1.2" xref="S0.E3.m3.3.3.1.1.1.1.1.1.2.cmml">θ</mi><mrow id="S0.E3.m3.3.3.1.1.1.1.1.1.3" xref="S0.E3.m3.3.3.1.1.1.1.1.1.3.cmml"><msup id="S0.E3.m3.3.3.1.1.1.1.1.1.3.1" xref="S0.E3.m3.3.3.1.1.1.1.1.1.3.1.cmml"><mo id="S0.E3.m3.3.3.1.1.1.1.1.1.3.1.2" xref="S0.E3.m3.3.3.1.1.1.1.1.1.3.1.2.cmml">∂</mo><mo id="S0.E3.m3.3.3.1.1.1.1.1.1.3.1.3" xref="S0.E3.m3.3.3.1.1.1.1.1.1.3.1.3.cmml">∗</mo></msup><mo id="S0.E3.m3.3.3.1.1.1.1.1.1.3a" xref="S0.E3.m3.3.3.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S0.E3.m3.3.3.1.1.1.1.1.1.3.2" xref="S0.E3.m3.3.3.1.1.1.1.1.1.3.2.cmml">p</mi></mrow></msub><mo maxsize="120%" minsize="120%" id="S0.E3.m3.3.3.1.1.1.1.1.3" xref="S0.E3.m3.3.3.1.1.1.1.2.1.cmml">⟩</mo></mrow><mrow id="S0.E3.m3.1.1.1.3" xref="S0.E3.m3.3.3.1.1.1.cmml"><mo stretchy="false" id="S0.E3.m3.1.1.1.3.1" xref="S0.E3.m3.3.3.1.1.1.cmml">(</mo><mo id="S0.E3.m3.1.1.1.1" xref="S0.E3.m3.1.1.1.1.cmml">+</mo><mo stretchy="false" id="S0.E3.m3.1.1.1.3.2" xref="S0.E3.m3.3.3.1.1.1.cmml">)</mo></mrow></msup></mpadded></mrow><mo id="S0.E3.m3.3.3.1.2" xref="S0.E3.m3.3.3.1.1.cmml">.</mo></mrow></math>, <math><mrow id="p2.6.m1.2.2" xref="p2.6.m1.2.2.cmml"><msub id="p2.6.m1.2.2.3" xref="p2.6.m1.2.2.3.cmml"><mi id="p2.6.m1.2.2.3.2" xref="p2.6.m1.2.2.3.2.cmml">K</mi><mn id="p2.6.m1.2.2.3.3" xref="p2.6.m1.2.2.3.3.cmml">0</mn></msub><mo id="p2.6.m1.2.2.2" xref="p2.6.m1.2.2.2.cmml">≡</mo><mrow id="p2.6.m1.2.2.1" xref="p2.6.m1.2.2.1.cmml"><mfrac id="p2.6.m1.2.2.1.3" xref="p2.6.m1.2.2.1.3.cmml"><mn id="p2.6.m1.2.2.1.3.2" xref="p2.6.m1.2.2.1.3.2.cmml">1</mn><mn id="p2.6.m1.2.2.1.3.3" xref="p2.6.m1.2.2.1.3.3.cmml">2</mn></mfrac><mo id="p2.6.m1.2.2.1.2" xref="p2.6.m1.2.2.1.2.cmml">⁢</mo><mrow id="p2.6.m1.2.2.1.1" xref="p2.6.m1.2.2.1.1.cmml"><mi id="p2.6.m1.2.2.1.1.2" xref="p2.6.m1.2.2.1.1.2.cmml">ln</mi><mo id="p2.6.m1.2.2.1.1a" xref="p2.6.m1.2.2.1.1.cmml">⁡</mo><mrow id="p2.6.m1.2.2.1.1.1.1" xref="p2.6.m1.2.2.1.1.1.2.cmml"><mi id="p2.6.m1.1.1" xref="p2.6.m1.1.1.cmml">coth</mi><mo id="p2.6.m1.2.2.1.1.1.1a" xref="p2.6.m1.2.2.1.1.1.2.cmml">⁡</mo><mrow id="p2.6.m1.2.2.1.1.1.1.1" xref="p2.6.m1.2.2.1.1.1.2.cmml"><mo stretchy="false" id="p2.6.m1.2.2.1.1.1.1.1.2" xref="p2.6.m1.2.2.1.1.1.2.cmml">(</mo><mrow id="p2.6.m1.2.2.1.1.1.1.1.1" xref="p2.6.m1.2.2.1.1.1.1.1.1.cmml"><mn id="p2.6.m1.2.2.1.1.1.1.1.1.2" xref="p2.6.m1.2.2.1.1.1.1.1.1.2.cmml">2</mn><mo id="p2.6.m1.2.2.1.1.1.1.1.1.1" xref="p2.6.m1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="p2.6.m1.2.2.1.1.1.1.1.1.3" xref="p2.6.m1.2.2.1.1.1.1.1.1.3.cmml">β</mi></mrow><mo stretchy="false" id="p2.6.m1.2.2.1.1.1.1.1.3" xref="p2.6.m1.2.2.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></mrow></math>, <math><msup id="p3.1.m1.2.2" xref="p3.1.m1.2.2.cmml"><mrow id="p3.1.m1.2.2.1.1" xref="p3.1.m1.2.2.1.2.cmml"><mo maxsize="120%" minsize="120%" id="p3.1.m1.2.2.1.1.2" xref="p3.1.m1.2.2.1.2.1.cmml">⟨</mo><msub id="p3.1.m1.2.2.1.1.1" xref="p3.1.m1.2.2.1.1.1.cmml"><mi id="p3.1.m1.2.2.1.1.1.2" xref="p3.1.m1.2.2.1.1.1.2.cmml">θ</mi><mrow id="p3.1.m1.2.2.1.1.1.3" xref="p3.1.m1.2.2.1.1.1.3.cmml"><msup id="p3.1.m1.2.2.1.1.1.3.1" xref="p3.1.m1.2.2.1.1.1.3.1.cmml"><mo id="p3.1.m1.2.2.1.1.1.3.1.2" xref="p3.1.m1.2.2.1.1.1.3.1.2.cmml">∂</mo><mo id="p3.1.m1.2.2.1.1.1.3.1.3" xref="p3.1.m1.2.2.1.1.1.3.1.3.cmml">∗</mo></msup><mo id="p3.1.m1.2.2.1.1.1.3a" xref="p3.1.m1.2.2.1.1.1.3.cmml">⁡</mo><mi id="p3.1.m1.2.2.1.1.1.3.2" xref="p3.1.m1.2.2.1.1.1.3.2.cmml">p</mi></mrow></msub><mo maxsize="120%" minsize="120%" id="p3.1.m1.2.2.1.1.3" xref="p3.1.m1.2.2.1.2.1.cmml">⟩</mo></mrow><mrow id="p3.1.m1.1.1.1.3" xref="p3.1.m1.2.2.cmml"><mo stretchy="false" id="p3.1.m1.1.1.1.3.1" xref="p3.1.m1.2.2.cmml">(</mo><mo id="p3.1.m1.1.1.1.1" xref="p3.1.m1.1.1.1.1.cmml">+</mo><mo stretchy="false" id="p3.1.m1.1.1.1.3.2" xref="p3.1.m1.2.2.cmml">)</mo></mrow></msup></math>, <math><mrow id="S0.E4.m1.1.1.1" xref="S0.E4.m1.1.1.1.1.cmml"><mrow id="S0.E4.m1.1.1.1.1" xref="S0.E4.m1.1.1.1.1.cmml"><msub id="S0.E4.m1.1.1.1.1.2" xref="S0.E4.m1.1.1.1.1.2.cmml"><mi id="S0.E4.m1.1.1.1.1.2.2" xref="S0.E4.m1.1.1.1.1.2.2.cmml">η</mi><mi id="S0.E4.m1.1.1.1.1.2.3" xref="S0.E4.m1.1.1.1.1.2.3.cmml">c</mi></msub><mo id="S0.E4.m1.1.1.1.1.1" xref="S0.E4.m1.1.1.1.1.1.cmml">=</mo><mrow id="S0.E4.m1.1.1.1.1.3" xref="S0.E4.m1.1.1.1.1.3.cmml"><mpadded width="+1.7pt" id="S0.E4.m1.1.1.1.1.3.1" xref="S0.E4.m1.1.1.1.1.3.1.cmml"><munder id="S0.E4.m1.1.1.1.1.3.1a" xref="S0.E4.m1.1.1.1.1.3.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S0.E4.m1.1.1.1.1.3.1.2" xref="S0.E4.m1.1.1.1.1.3.1.2.cmml">∏</mo><mrow id="S0.E4.m1.1.1.1.1.3.1.3" xref="S0.E4.m1.1.1.1.1.3.1.3.cmml"><mi id="S0.E4.m1.1.1.1.1.3.1.3.2" xref="S0.E4.m1.1.1.1.1.3.1.3.2.cmml">p</mi><mo id="S0.E4.m1.1.1.1.1.3.1.3.1" xref="S0.E4.m1.1.1.1.1.3.1.3.1.cmml">∈</mo><mrow id="S0.E4.m1.1.1.1.1.3.1.3.3" xref="S0.E4.m1.1.1.1.1.3.1.3.3.cmml"><mo id="S0.E4.m1.1.1.1.1.3.1.3.3.1" xref="S0.E4.m1.1.1.1.1.3.1.3.3.1.cmml">∂</mo><mo id="S0.E4.m1.1.1.1.1.3.1.3.3a" xref="S0.E4.m1.1.1.1.1.3.1.3.3.cmml">⁡</mo><mi id="S0.E4.m1.1.1.1.1.3.1.3.3.2" xref="S0.E4.m1.1.1.1.1.3.1.3.3.2.cmml">c</mi></mrow></mrow></munder></mpadded><mrow id="S0.E4.m1.1.1.1.1.3.2" xref="S0.E4.m1.1.1.1.1.3.2.cmml"><mpadded width="+1.7pt" id="S0.E4.m1.1.1.1.1.3.2.2" xref="S0.E4.m1.1.1.1.1.3.2.2b.cmml"><mtext id="S0.E4.m1.1.1.1.1.3.2.2a" xref="S0.E4.m1.1.1.1.1.3.2.2b.cmml">sign</mtext></mpadded><mo id="S0.E4.m1.1.1.1.1.3.2.1" xref="S0.E4.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S0.E4.m1.1.1.1.1.3.2.3" xref="S0.E4.m1.1.1.1.1.3.2.3.cmml">tr</mi><mo id="S0.E4.m1.1.1.1.1.3.2.1a" xref="S0.E4.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mpadded width="+2.8pt" id="S0.E4.m1.1.1.1.1.3.2.4" xref="S0.E4.m1.1.1.1.1.3.2.4.cmml"><msub id="S0.E4.m1.1.1.1.1.3.2.4a" xref="S0.E4.m1.1.1.1.1.3.2.4.cmml"><mi id="S0.E4.m1.1.1.1.1.3.2.4.2" xref="S0.E4.m1.1.1.1.1.3.2.4.2.cmml">U</mi><mi id="S0.E4.m1.1.1.1.1.3.2.4.3" xref="S0.E4.m1.1.1.1.1.3.2.4.3.cmml">p</mi></msub></mpadded></mrow></mrow></mrow><mo id="S0.E4.m1.1.1.1.2" xref="S0.E4.m1.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="p6.3.m3.1.1" xref="p6.3.m3.1.1.cmml"><msub id="p6.3.m3.1.1.2" xref="p6.3.m3.1.1.2.cmml"><mi id="p6.3.m3.1.1.2.2" xref="p6.3.m3.1.1.2.2.cmml">θ</mi><mrow id="p6.3.m3.1.1.2.3" xref="p6.3.m3.1.1.2.3.cmml"><msup id="p6.3.m3.1.1.2.3.1" xref="p6.3.m3.1.1.2.3.1.cmml"><mo id="p6.3.m3.1.1.2.3.1.2" xref="p6.3.m3.1.1.2.3.1.2.cmml">∂</mo><mo id="p6.3.m3.1.1.2.3.1.3" xref="p6.3.m3.1.1.2.3.1.3.cmml">∗</mo></msup><mo id="p6.3.m3.1.1.2.3a" xref="p6.3.m3.1.1.2.3.cmml">⁡</mo><mi id="p6.3.m3.1.1.2.3.2" xref="p6.3.m3.1.1.2.3.2.cmml">p</mi></mrow></msub><mo id="p6.3.m3.1.1.1" xref="p6.3.m3.1.1.1.cmml">=</mo><mn id="p6.3.m3.1.1.3" xref="p6.3.m3.1.1.3.cmml">1</mn></mrow></math>

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0405295

Formulas:

1-

A_{1 - x} B_{x}

2-

g_{e f f}

3-

𝐇_{𝐏} = H_{P} \hat{𝐲}

4-

𝐇_{𝐁} = H_{B} \hat{𝐱}

5-

𝐇_{𝐓} = H_{T} \hat{𝐲}

6-

σ_{p h} = 25 ps

7-

σ_{t j} = 30 ps

8-

\sqrt{σ_{p h}^{2} + σ_{t j}^{2}} = 39 ps

9-

0 < H_{B} < 100 Oe

10-

H_{P} \sim 7 𝐎𝐞

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1505.03838

Formulas:

1-

V_{L J} (r) = 4 ε [{(\frac{σ}{r})}^{12} - {(\frac{σ}{r})}^{6}],

2-

r_{c} = 2.5 σ

3-

ρ = 0.75 σ^{- 3}

4-

1.1 ε / k_{B}

5-

△ t_{M D} = 0.005 τ

6-

τ = σ \sqrt{m / ε}

7-

ε_{pf} = 1.0 ε

8-

0.1 ε ⩽ ε_{pf} ⩽ 1.0 ε

9-

3 \times 10^{6} τ

10-

ε_{pf} = 1.0 ε

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1309.6605

Formulas:

1-

{\dot{𝐱}}_{i} (t) = 𝐅 (𝐱_{i} (t)) + σ \sum_{j} A_{i j} 𝐇 (𝐱_{j}), i = 1, \dots, N,

2-

{𝐬_{1}, \dots, 𝐬_{M}}

3-

B = T A T^{- 1}

4-

\dot{𝜼} (t) = [\sum_{m = 1}^{M} E^{(m)} \otimes D 𝐅 (𝐬_{m} (t)) + σ B \otimes I_{n} \sum_{m = 1}^{M} J^{(m)} \otimes D 𝐇 (𝐬_{m} (t))] 𝜼 (t),

5-

{𝐬_{1}, \dots, 𝐬_{M}}

6-

ℐ (x) = (1 - \cos x) / 2

7-

x_{i}^{t + 1} = [β ℐ (x_{i}^{t}) + σ \sum_{j} A_{i j} ℐ (x_{j}^{t}) + δ] mod 2 π

8-

Δ x_{RMS} \equiv {({\bar{< {(x_{i}^{t} - {\bar{x}}^{t})}^{2} >}}_{T})}^{1 / 2}

9-

x_{i}^{t} - {\bar{x}}^{t}

10-

𝒢 = ℋ_{1} \times \dots \times ℋ_{ν}

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1304.0823

Formulas:

1-

l = 0, 1, 2 \dots

2-

p (s | Θ) = \sum_{k = 1}^{K} ω_{k} 𝓝 (s; μ_{k}, Σ_{k})

3-

Θ \equiv {ω_{k}, μ_{k}, Σ_{k}}_{k = 1, 2, \dots, K}

4-

= [α_{k} n_{k} / T + (1 - α_{k}) {\bar{𝝎}}_{k}] γ

5-

= α_{k} 𝐄_{k} (s) + (1 - α_{k}) {\bar{𝝁}}_{k}

6-

= α_{k} 𝐄_{k} (s^{2}) + (1 - α_{k}) ({\bar{𝝈}}_{k}^{2} + {\bar{𝝁}}_{k}^{2}) - μ_{k}^{2}

7-

𝐄_{k} (s)

8-

𝐄_{k} (s^{2})

9-

= \sum_{t = 1}^{T} 𝐏𝐫 (k | s_{t})

10-

𝐄_{k} (s)

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9801031

Formulas:

1-

D γ_{5} D = D γ_{5} + γ_{5} D

2-

X (m) = (\begin{matrix} B + m & C \\ - C^{†} & B + m \end{matrix}),

3-

\begin{matrix} {(C)}_{x α i, y β j} & = \frac{1}{2} \sum_{μ = 1}^{4} σ_{μ}^{α β} [δ_{y, x + \hat{μ}} {(U_{μ} (x))}_{i j} - δ_{x, y + \hat{μ}} {(U_{μ}^{†} (y))}_{i j}], \\ {(B)}_{x α i, y β j} & = \frac{1}{2} δ_{α β} \sum_{μ = 1}^{4} [2 δ_{x y} δ_{i j} - δ_{y, x + \hat{μ}} {(U_{μ} (x))}_{i j} - δ_{x, y + \hat{μ}} {(U_{μ}^{†} (y))}_{i j}], \end{matrix}

4-

(\begin{matrix} 0 & σ_{μ} \\ σ_{μ}^{†} & 0 \end{matrix}) = γ_{μ} .

5-

U_{μ} (x)

6-

(- 1, 0)

7-

V = X \frac{1}{\sqrt{X^{†} X}} .

8-

det (X) = 0

9-

1 + e^{i θ}, θ \in [0, 2 π]

10-

γ_{5} X γ_{5} = X^{†},

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-lat

Result: correct

Run 6938504 (Agent767)