Run 11312206

Paper: https://arxiv.org/abs/astro-ph/0506649

Formulas:

1-

_{d u s t}

2-

21_{9, 13} \to 21_{8, 14}

3-

21_{9, 12} \to 21_{8, 13}

4-

21_{9, 12} \to 21_{8, 13}

5-

4_{1, 3} \to 4_{0, 4}

6-

21_{9, 13} \to 21_{8, 14}

7-

29_{6, 24} \to 29_{5, 25}

8-

29_{6, 24} \to 29_{4, 26}

9-

4_{1, 3} \to 4_{0, 4}

10-

12_{6, 7} \to 11_{6, 6}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: nucl-ex

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/9907102

Formulas:

1-

| n, t ⟩ = α | n, t + d t ⟩ + β | n + 1, t + d t ⟩ + β | n - 1, t + d t ⟩

2-

α = 1 + α_{1} d t + 𝒪 (d t^{2}) β = β_{1} d t + 𝒪 (d t^{2})

3-

- \frac{d}{d t} | n, t ⟩ = α_{1} | n, t ⟩ + β_{1} | n + 1, t ⟩ + β_{1} | n - 1, t ⟩

4-

⟨ n, t | n^{'}, t ⟩ = δ_{n, n^{'}}

5-

| ψ ⟩ = \sum_{n} ψ (n, t) | n, t ⟩

6-

U = (α_{1} + 2 β_{1}) i ℏ \frac{1}{2 m} = \frac{a^{2} β_{1}}{i ℏ}

7-

i ℏ \frac{d}{d t} ψ (x, t) = U ψ (x, t) - \frac{ℏ^{2}}{2 m} \frac{ψ (x + a, t) + ψ (x - a, t) - 2 ψ (x, t)}{a^{2}}

8-

𝒪 (1 / a^{2})

9-

𝒪 (1 / a)

10-

\vec{x} = a \vec{n}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1411.5756

Formulas:

1-

{\tilde{σ}}_{i j} = 2 \sum_{k = 0}^{i} \sum_{l = 0}^{j} \frac{{(- 1)}^{k + l}}{k + l + 4} (\binom{i}{k}) (\binom{j}{l}) (\frac{2 (k + l + 4)!}{(k + 3)! (l + 3)!} - 1 - \frac{(k + 1) (l + 1)}{(k + 3) (l + 3)}) .

2-

{(x)}_{m} = x (x - 1) \dots (x - (m - 1))

3-

= \frac{16}{{(i + 3)}_{3} {(j + 3)}_{3}} - \frac{24}{{(i + j + 4)}_{4}}, if i \neq j;

4-

= \frac{4}{{(j + 3)}_{3}} + \frac{16}{{(j + 3)}_{3}^{2}} - \frac{24}{{(2 j + 4)}_{4}} .

5-

\sum_{l = 0}^{j} {(- 1)}^{l} (\binom{j}{l}) (\binom{k + l + a}{l + a}) = {\begin{matrix} 0, & if j > k; \\ {(- 1)}^{j}; & if j = k . \end{matrix}

6-

{(1 - 1)}^{j}

7-

\sum_{l = 0}^{j} {(- 1)}^{l} (\binom{j}{l}) (\binom{k + 1 + l + a}{l + a}) = \frac{k + 1 + a}{k + 1} \sum_{l = 0}^{j} {(- 1)}^{l} (\binom{j}{l}) (\binom{k + l + a}{l + a})

8-

+ \sum_{l = 0}^{j} {(- 1)}^{l} \frac{j!}{l! (j - l)!} \frac{l (k + l + a)!}{(k + 1)! (l + a)!} .

9-

\sum_{l = 1}^{j} {(- 1)}^{l} \frac{j!}{(l - 1)! (j - l)!} \frac{(k + l - 1 + a + 1)!}{(k + 1) k! (l - 1 + a + 1)!} = - \frac{j}{k + 1} \sum_{l = 0}^{j - 1} {(- 1)}^{l} (\binom{j - 1}{l}) (\binom{k + l + a + 1}{l + a + 1}) .

10-

{(- 1)}^{j - 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0710.0028

Formulas:

1-

\frac{1}{ℳ_{𝒰}^{k}} O_{B Z} O_{S M},

2-

\frac{C_{𝒰} Λ_{𝒰}^{d_{B Z} - d}}{ℳ_{𝒰}^{k}} O_{B Z} O_{S M},

3-

\frac{d σ}{d T} = \frac{G_{F}^{2} m_{e}}{2 π} [{(g_{V} + g_{A})}^{2} + {(g_{V} - g_{A})}^{2} {(1 - \frac{T}{E_{ν}})}^{2} + [g_{A}^{2} - g_{V}^{2}] \frac{m_{e} T}{E_{ν}}],

4-

g_{V} = 2 \sin^{2} θ_{W} + \frac{1}{2}

5-

g_{A} = - \frac{1}{2}

6-

λ_{0 e} \frac{1}{Λ^{d - 1}} \bar{e} e O_{𝒰} + λ_{0 ν}^{α β} \frac{1}{Λ^{d - 1}} {\bar{ν}}^{α} ν^{β} O_{𝒰} + h . c . .

7-

ℳ_{𝒰_{S}} = \frac{f (d)}{Λ_{𝒰}^{2 d - 2}} [{\bar{ν}}_{β} (k^{'}) ν_{α} (k)] [\bar{e} (p^{'}) e (p)] {[- q^{2} - i ϵ]}^{d - 2},

8-

f (d) = \frac{λ_{0 ν}^{α β} λ_{0 e} A_{d}}{2 \sin (d π)},

9-

A_{d} = \frac{16 π^{5 / 2}}{{(2 π)}^{2 d}} \frac{Γ (d + 1 / 2)}{Γ (d - 1) Γ (2 d)} .

10-

λ_{0} \equiv {(λ_{0 ν} λ_{0 e})}^{1 / 2}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0603086

Formulas:

1-

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

2-

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

3-

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

4-

E_{l p}^{L G} = E_{0} {(\frac{\sqrt{2} r}{w})}^{l} e^{- i l ϕ} e^{- \frac{r^{2}}{w^{2}}} L_{p}^{l} (\frac{2 r^{2}}{w}),

5-

E_{S} = χ^{(3)} E_{F} E_{B} E_{P}^{*},

6-

ω_{F} + ω_{B} - ω_{P} = ω_{s}

7-

{\vec{k}}_{F} + {\vec{k}}_{B} - {\vec{k}}_{P} = {\vec{k}}_{S}

8-

l_{F} + l_{B} - l_{P} = l_{S} .

9-

l_{F} = l_{P} = 0

10-

E (r, ϕ) \propto E_{0} (\frac{\sqrt{2} r}{w}) e^{- \frac{r^{2}}{w^{2}}} L_{0}^{1} (\frac{2 r^{2}}{w}) (e^{- i ϕ} + e^{i ϕ} e^{i φ}) e^{i η r},

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1209.6584

Formulas:

1-

R_{m i n} k_{m a x}

2-

k_{m a x}

3-

R_{m i n}

4-

C m m m

5-

C m m m

6-

N (E_{F})

7-

γ_{b a r e}

8-

γ_{b a r e} (1 + λ)

9-

N (E_{F})

10-

N (E_{F})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1201.2995

Formulas:

1-

P M P^{- 1} = N

2-

θ = 360^{\circ} / n

3-

G = {R, R^{2}, R^{3}, R^{4}, \dots R^{n}, S, S R, S R^{2}, S R^{3}, \dots S R^{n - 1}}

4-

S^{2} = i d; {(S \times R)}^{2} = i d; R^{n} = i d;

5-

(n + 3) / 2

6-

(n + 6) / 2

7-

S \times R^{k} \times S = R^{- k}

8-

R^{n}, R, R^{2}, S R, S R^{2}, S R^{3}, S R^{n - 1}

9-

R, R^{n}, R, R^{2}, S R, S R^{n / 2} and S R^{n / 2 + 1}, S R^{n - 1}

10-

r_{i} = (\begin{matrix} \cos i \times θ & - \sin i \times θ \\ \sin i \times θ & \cos i \times θ \end{matrix})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1804.03098

Formulas:

1-

\bar{F} = 1 - F

2-

X_{1} \leq_{s t} X_{2}

3-

{\bar{F}}_{1} (t) \leq {\bar{F}}_{2} (t)

4-

X_{1} \leq_{h r} X_{2}

5-

r_{1} (t) \geq r_{2} (t)

6-

X_{1} \leq_{l r} X_{2}

7-

f_{1} (t) / f_{2} (t)

8-

Φ^{W} (t)

9-

Φ^{W} (t)

10-

\exp {- \frac{(2 λ_{1} + λ_{2} + μ) t}{2}} [\cosh (\frac{a t}{2}) + \frac{2 λ_{1} + λ_{2} + μ}{a} \sinh (\frac{a t}{2})],

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/2009.11137

Formulas:

1-

P m \bar{3} n

2-

𝒌_{S} = S 𝒌

3-

A_{c} (𝒌, ϵ) = \frac{1}{N_{S}} \sum_{S} \sum_{n} δ_{ξ} (ϵ - ϵ_{c, n 𝒌_{S}}) .

4-

ϵ_{c, n 𝒌}

5-

δ_{ξ} (x) = 1

6-

x \in [- ξ, ξ)

7-

A_{c} (𝒌, ϵ)

8-

A_{T} (𝒌, ϵ)

9-

A_{T} (𝒌, ϵ) \approx \frac{1}{Z_{T}} \sum_{c} A_{c} (𝒌, ϵ) e^{- E_{c} / k_{B} T} .

10-

Z_{T} = \sum_{c} e^{- E_{c} / k_{B} T}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1809.07967

Formulas:

1-

\dot{M} (R_{ν})

2-

\dot{M} (R_{ν}) = 0.015 {(\frac{R_{in}}{R_{s}})}^{1.2} {(1 + \frac{R_{ν}}{10 R_{in}})}^{2} M_{⊙} s^{- 1},

3-

R_{in} ≃ 3 R_{s}

4-

{\dot{M}}_{w} (t)

5-

\dot{M} (R_{ν})

6-

\frac{\dot{M} (R_{ν})}{0.015 M_{⊙} s^{- 1}} (\frac{3 M_{⊙}}{M_{B H}}) = {(\frac{R_{in}}{R_{s}})}^{1.2} {(1 + \frac{R_{ν}}{10 R_{in}})}^{2} .

7-

\dot{M} (R_{ν})

8-

{(α / 0.1)}^{5 / 3}

9-

R_{in} < R_{ν} < R_{d}

10-

\dot{M} (R) = {\dot{M}}_{d} {(\frac{R}{R_{d}})}^{s},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/0609076

Formulas:

1-

{ν 1 h_{9 / 2}, ν 1 i_{13 / 2}}

2-

{π 1 g_{7 / 2}, π 1 h_{11 / 2}}

3-

{ν 1 h_{9 / 2}, ν 1 i_{13 / 2}}

4-

{π 1 g_{7 / 9}, π 1 h_{11 / 2}}

5-

V_{π}^{c} (𝒒)

6-

V_{π}^{T} (𝒒)

7-

V_{π}^{c} (𝒒) = \frac{- 1}{3} {[\frac{f_{π}}{m_{π}}]}^{2}

8-

\frac{𝝈_{1} \cdot 𝝈_{2} 𝒒^{2}}{m_{π}^{2} + 𝒒^{2}} {\vec{τ}}_{1} \cdot {\vec{τ}}_{2},

9-

V_{π}^{T} (𝒒) = \frac{- 1}{3} {[\frac{f_{π}}{m_{π}}]}^{2}

10-

\frac{3 𝝈_{1} \cdot {𝒒 𝝈}_{2} \cdot 𝒒 - 𝝈_{1} \cdot 𝝈_{2} 𝒒^{2}}{m_{π}^{2} + 𝒒^{2}} {\vec{τ}}_{1} \cdot {\vec{τ}}_{2} .

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1303.5091

Formulas:

1-

d = \frac{1}{π} \sin^{- 1} \frac{(R_{l} + R_{p})}{a},

2-

R_{l} = (\frac{0.49 q^{\frac{2}{3}}}{0.6 q^{\frac{2}{3}} + \ln (1 + q^{\frac{1}{3}})}) a

3-

JD (\min) = 2455548.738 + 6.4705315 E,

4-

\pm 1.5 \times 10^{- 6}

5-

15 (1 + β^{'}) d

6-

5 (1 + β^{'}) d

7-

{r_{1}, r_{2}^{'}, r_{2}^{''}, △, i, j, Ω}

8-

{l_{1}, β, β^{'}, μ_{1}, μ_{2}, c, d}

9-

0 \leq r \leq r_{2}^{'}

10-

r_{2}^{'} < r \leq r_{2}^{''}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1209.5804

Formulas:

1-

i n f {d (x, g x) : x \in L}

2-

| g | = inf {d (x, g x) ∣ x \in X}

3-

l^{2} (𝐙)

4-

G = D (P, w)

5-

K (P, w)

6-

K (P, w)

7-

D (P, w)

8-

K (P, w)

9-

G = D (P, w)

10-

K (P, w)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9311056

Formulas:

1-

S_{3} = ⟨ δ^{3} ⟩ / {⟨ {(δ^{(1)})}^{2} ⟩}^{2}

2-

S_{3} = 4, 3, 3.4

3-

S_{3} = 34 / 7

4-

- 3 < n \leq - 1

5-

a (t) \propto t^{2 / 3}

6-

ρ_{0} = 1 / 6 π G t^{2}

7-

△ Φ = 4 π G a^{2} ρ_{0} δ;

8-

\dot{δ} + \frac{1}{a} d i v ((1 + δ) \vec{u}) = 0;

9-

{(a \vec{u})}^{\cdot} + (\vec{u} \nabla) \vec{u} = - \nabla Φ

10-

δ = (ρ - ρ_{0}) / ρ_{o}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0701085

Formulas:

1-

α^{2} F (ω)

2-

{α^{2} F (ω) |}_{ω < ω_{E_{2 g}}}

3-

(𝐞_{i n} 𝐞_{o u t})

4-

𝐞_{o u t}

5-

𝐞_{i n} = (H - i V) / \sqrt{2}

6-

𝐞_{o u t} = 𝐞_{i n}^{*}

7-

\sim {[ω \sqrt{(ω^{2} - {(2 Δ)}^{2})}]}^{- 1}

8-

λ_{L} = 752.5, 647.1, 482.5

9-

σ_{a b} (ω)

10-

H > H_{c 2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/nucl-th/9808077

Formulas:

1-

A (d, p) X

2-

A (d, π) X

3-

{}^{12}C (d, p)

4-

A (d, p) X

5-

A (d, π) X

6-

ψ^{d} (1, 2, \dots, 6) = \hat{A} {φ_{N} (1, 2, 3) φ_{N} (4, 5, 6) χ (𝐫)},

7-

3 q \times 3 q

8-

N^{*} N^{*,'}

9-

χ_{N N} (𝐫)

10-

A (d, p) X

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1405.3560

Formulas:

1-

𝒆 + 𝒑^{↑} \to 𝒆 + 𝑱 / 𝝍 + 𝑿

2-

χ^{2} / d . o . f

3-

e + p^{↑} \to e + J / ψ + X

4-

σ^{e p \to e + J / ψ + X} = \int_{4 m_{c}^{2}}^{4 m_{D}^{2}} 𝑑 M_{c \bar{c}}^{2} \int 𝑑 x_{γ} 𝑑 x_{g} f_{γ / e} (x_{γ}) f_{g / p} (x_{g}) \frac{d {\hat{σ}}^{γ g \to c \bar{c}}}{d M_{c \bar{c}}^{2}} .

5-

f_{g / p} (x)

6-

f_{γ / e} (ξ)

7-

\frac{d σ^{e + p^{↑} \to e + J / ψ + X}}{d M^{2}} = \int 𝑑 x_{γ} 𝑑 x_{g} [d^{2} 𝒌_{⟂ γ} d^{2} 𝒌_{⟂ g}] f_{g / p^{↑}} (x_{g}, 𝒌_{⟂ g}) f_{γ / e} (x_{γ}, 𝒌_{⟂ γ}) \frac{d {\hat{σ}}^{γ g \to c \bar{c}}}{d M^{2}}

8-

M^{2} \equiv M_{c \bar{c}}^{2}

9-

d σ^{↑} - d σ^{↓} = \int 𝑑 x_{γ} 𝑑 x_{g} d^{2} 𝒌_{⟂ γ} d^{2} 𝒌_{⟂ g} Δ^{N} f_{g / p^{↑}} (x_{g}, 𝒌_{⟂ g}) f_{γ / e} (x_{γ}, 𝒌_{⟂ γ}) 𝑑 {\hat{σ}}^{γ g \to c \bar{c}}

10-

\frac{d^{4} σ^{↑}}{d y d M^{2} d^{2} 𝒒_{T}} - \frac{d^{4} σ^{↓}}{d y d M^{2} d^{2} 𝒒_{T}} = \frac{1}{s} \int [d^{2} 𝒌_{⟂ γ} d^{2} 𝒌_{⟂ g}] Δ^{N} f_{g / p^{↑}} (x_{g}, 𝒌_{⟂ g}) f_{γ / e} (x_{γ}, 𝒌_{⟂ γ})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1307.0554

Formulas:

1-

\dot{x} (t) = (A + B) x (t)

2-

\dot{x} (t) = A x (t) + B x (t - τ)

3-

ℝ_{+}^{n} := {x \in ℝ^{n} : x_{i} \geq 0, 1 \leq i \leq n} .

4-

i = 1, \dots, n

5-

x, y \in ℝ^{n}

6-

x_{i} > y_{i}, 1 \leq i \leq n

7-

f : D \to ℝ^{n}

8-

g : D \to ℝ^{n}

9-

\dot{x} (t) = f (x (t)) + g (x (t - τ)), τ \geq 0 .

10-

ϕ \in C ([- τ, 0], ℝ^{n})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1008.0343

Formulas:

1-

R e_{δ} \to \infty

2-

δ (x) \propto x / \ln^{2} R e_{x}

3-

λ = \frac{2 τ_{w}}{ρ U_{0}^{2}} \propto 1 / \ln^{2} δ (x)

4-

\frac{d δ (x)}{d x} \to 0

5-

u_{*}^{2} = ν {| \frac{\partial U (y)}{\partial y} |}_{w a l l}

6-

y_{+} = \frac{y u_{*}}{ν} = O (1)

7-

U_{+} (y_{+}) = U (y_{+}) / u_{*} \propto y_{+}

8-

U (x, y) = U (\frac{y}{δ (x)})

9-

λ \propto \frac{d δ}{d x} \propto {(\frac{u_{*}}{U_{0}})}^{2}

10-

λ_{p i p e} \propto R e_{D}^{- \frac{1}{4}}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1401.8088

Formulas:

1-

\begin{matrix} \frac{d 𝐦}{d t} = & - γ 𝐦 \times 𝐇 - γ H_{s} 𝐦 \times (𝐩 \times 𝐦) \\ - γ β H_{s} 𝐦 \times 𝐩 + α 𝐦 \times \frac{d 𝐦}{d t}, \end{matrix}

2-

𝐇 = (H_{K} - 4 π M) m_{z} 𝐞_{z}

3-

E = - M (H_{K} - 4 π M) m_{z}^{2} / 2

4-

𝐇 = - \partial E / \partial (M 𝐦)

5-

H_{s} = \frac{ℏ η I}{2 e (1 + λ 𝐦 \cdot 𝐩) M V},

6-

V = π \times 60 \times 60 \times 2

7-

γ = 1.732 \times 10^{7}

8-

I_{c} = [4 α e M V / (ℏ η λ)] (H_{K} - 4 π M)

9-

β > 0 (< 0)

10-

m_{x} (t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1406.3212

Formulas:

1-

Tr (𝐀^{(j)}) > 0

2-

j = 1, \dots, n

3-

A (\begin{matrix} α \\ β \end{matrix}) A (\begin{matrix} β \\ α \end{matrix}) \leq 0

4-

α, β \subset {1, \dots, n},

5-

| α | = | β |

6-

𝐀 = (\begin{matrix} 1 & 2 \\ - 1 & 5 \end{matrix}) .

7-

𝐀^{(2)} = det (𝐀) = 7 .

8-

𝐃 = diag {d_{11}, d_{22}}

9-

𝐃𝐀 = (\begin{matrix} d_{11} & 2 d_{11} \\ - d_{22} & 5 d_{22} \end{matrix});

10-

{(𝐃𝐀)}^{2} = (\begin{matrix} d_{11}^{2} - 2 d_{11} d_{22} & 2 d_{11}^{2} + 10 d_{11} d_{22} \\ - d_{11} d_{22} - 5 d_{22}^{2} & - 2 d_{11} d_{22} + 25 d_{22}^{2} \end{matrix});

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1601.01855

Formulas:

1-

d s = (d ε - p_{th} / ρ d \ln ρ) / T

2-

[Fe / H] = - 2

3-

Δ s = s_{ad} - s_{\min}

4-

\frac{s - s_{\min}}{Δ s} .

5-

\sqrt{\frac{ρ}{ρ ({d s |}_{\max})}} .

6-

⟨ τ_{Ross} ⟩ = 1

7-

F_{rad}^{*} = F_{rad} / F_{tot}

8-

\partial_{t} s + u \cdot \nabla s

9-

- \frac{1}{T} \nabla \cdot F_{rad} + q_{visc},

10-

2 \bar{⟨ u_{x} \partial_{x} s ⟩} + \bar{⟨ u_{z} \partial_{z} s ⟩}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2009.07019

Formulas:

1-

N = N_{0} + N_{mag},

2-

N_{0} = \dot{M} Ω_{K} (R_{0}) R_{0}^{2},

3-

(R, ϕ, z)

4-

N_{mag} = - \int_{R_{0}}^{\infty} B_{ϕ} B_{z} R^{2} 𝑑 R .

5-

N = N_{0} n (ω),

6-

ω \equiv \frac{Ω_{S}}{Ω_{K} (R_{0})} \equiv {(\frac{R_{0}}{R_{c}})}^{3 / 2} .

7-

R_{c} = {(\frac{G M}{Ω_{S}^{2}})}^{1 / 3}

8-

n \approx 1.39 {1 - ω [4.03 {(1 - ω)}^{0.173} - 0.878]} {(1 - ω)}^{- 1} .

9-

(x, ϕ, z)

10-

\frac{B_{ϕ}}{B_{z}} = {\begin{matrix} γ (Ω_{S} - Ω_{K}) / Ω_{K}, & Ω_{S} \leq Ω_{K} \\ γ (Ω_{S} - Ω_{K}) / Ω_{S}, & Ω_{S} \geq Ω_{K}, \end{matrix}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1706.10219

Formulas:

1-

σ = \frac{\sqrt{\sum_{i = 1}^{N} σ_{i}^{2}}}{N} \approx \frac{σ^{t y p i c a l}}{\sqrt{N}},

2-

σ^{t y p i c a l}

3-

f_{A F D}^{'} (x, h)

4-

= \frac{1}{2} [\frac{f (x + h) - f (x)}{h} + \frac{f (x) - f (x - h)}{h}],

5-

= \frac{f (x + h) - f (x - h)}{2 h} .

6-

f_{R E}^{'} (x, h) = \frac{f (x - 2 h) - 8 f (x - h) + 8 f (x + h) - f (x + 2 h)}{12 h} .

7-

f_{L D I}^{'} (x, h) = \frac{3}{2 h^{3}} \int_{x - h}^{x + h} (x - t) f (t) 𝑑 t .

8-

{x_{i}}_{i = 1}^{i = 16}

9-

x_{1} = x - h

10-

x_{16} = x + h,

Formulas (html):

Correct Categorie: math

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1906.08995

Formulas:

1-

{| α ⟩}_{A} {| 0 ⟩}_{B}

2-

{| α / \sqrt{2} ⟩}_{A} {| i α / \sqrt{2} ⟩}_{B}

3-

{\hat{U}}_{k} (φ) = \exp [i φ {({\hat{a}}^{†} \hat{a})}^{k}]

4-

| ψ ⟩ = \exp [i φ {({\hat{a}}^{†} \hat{a})}^{2}] \exp (i θ {\hat{b}}^{†} \hat{b}) {| \frac{α}{\sqrt{2}} ⟩}_{A} {| \frac{i α}{\sqrt{2}} ⟩}_{B},

5-

{\hat{X}}_{B} = \hat{b} + {\hat{b}}^{†}

6-

⟨ {\hat{X}}_{B} ⟩ = \frac{1}{\sqrt{2}} ⟨ ψ | [(\hat{b} + {\hat{b}}^{†}) + i (\hat{a} - {\hat{a}}^{†})] | ψ ⟩ .

7-

{\hat{U}}_{BS}^{†} \hat{b} {\hat{U}}_{BS} = (\hat{b} + i \hat{a}) / \sqrt{2}

8-

{\hat{U}}_{BS} = \exp [i π ({\hat{a}}^{†} \hat{b} + {\hat{b}}^{†} \hat{a}) / 4]

9-

\frac{1}{\sqrt{2}} ⟨ \hat{b} + {\hat{b}}^{†} ⟩ = - | α | \sin θ,

10-

\hat{a} g ({\hat{a}}^{†} \hat{a}) = g ({\hat{a}}^{†} \hat{a} + 1) \hat{a}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9405369

Formulas:

1-

m_{t} = 174 \pm 10_{- 12}^{+ 13}

2-

m_{t} = 165_{- 14 - 19}^{+ 13 + 18}

3-

q \bar{q} \to t \bar{t}

4-

g g \to t \bar{t}

5-

W \to l ν, q {\bar{q}}^{'}

6-

p \bar{p} \to t \bar{t} \to l ν j j j j,

7-

l = e^{\pm}, μ^{\pm}

8-

p \bar{p} \to W j j j j \to l ν j j j j

9-

p \bar{p} \to W W (W Z) j j \to l ν j j j j

10-

O (α_{s}^{3})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0404129

Formulas:

1-

(H_{c 1} << H_{e} << H_{c 2})

2-

c \nabla \times 𝐁 / 4 π

3-

δ = λ {(B / 4 π M)}^{1 / 2}

4-

4 π M \approx Φ_{0} {(8 π λ^{2})}^{- 1} \ln (H_{c 2} / B)

5-

(λ^{2} / δ^{2}) \equiv p ≪ 1

6-

𝐁_{0} = c o n s t

7-

E_{M} \approx (\frac{p}{2}) {(\frac{Φ_{0}}{4 π δ})}^{2} \ln (\frac{1}{p})

8-

{(4 π M / B)}^{2} \equiv p^{2} << 1

9-

{(1 - t)}^{3}

10-

t \equiv T / T_{c 0}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0005110

Formulas:

1-

𝒟 (A B)

2-

𝒟 (A B)

3-

𝒟 (A B)

4-

d (A B)

5-

𝒟 (A B)

6-

𝒟 (A B)

7-

d (A B)

8-

𝒟 (A B)

9-

𝒟 (A C)

10-

𝒟 (C B)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0409127

Formulas:

1-

z = \frac{k T_{e}}{m c^{2}}

2-

I (ν) = \int_{0}^{\infty} I_{o} (\bar{ν}) G_{s} (\bar{ν}, ν) 𝑑 \bar{ν}

3-

I_{o} (ν) = \frac{2 h ν^{3}}{c^{2}} {(e x p (\frac{h ν}{k T_{R}}) - 1)}^{- 1}

4-

G_{s} (\bar{ν}, ν)

5-

G_{s} (\bar{ν}, ν)

6-

G_{s} (\bar{ν}, ν) = (1 - τ) δ (\bar{ν} - ν) + τ G (\bar{ν}, ν)

7-

G (\bar{ν}, ν) = \frac{1}{\sqrt{π} W (ν)} e^{- {(\frac{\bar{ν} - (1 - 2 z) ν}{W (ν)})}^{2}} .

8-

W^{2} (ν) = 4 \frac{k T_{e}}{m_{e} c^{2}} ν^{2} = 4 z ν^{2}

9-

G_{s} (\bar{ν}, ν) = (1 - 2 τ) δ (\bar{ν} - ν) + \frac{τ}{\sqrt{π} W (ν)} e^{- {(\frac{\bar{ν} - ν}{W (ν)})}^{2}} + τ δ (\bar{ν} - (1 - 2 z) ν)

10-

α = \frac{\bar{ν} - ν}{W (ν)}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1409.1555

Formulas:

1-

Q_{i j} = S (n_{i} n_{j} - δ_{i j} / 2)

2-

ρ \frac{D v_{i}}{D t} = η \nabla^{2} v_{i} - \partial_{i} p + \partial_{j} σ_{i j},

3-

\frac{D Q_{i j}}{D t} = λ S u_{i j} + Q_{i k} ω_{k j} - ω_{i k} Q_{k j} + γ^{- 1} H_{i j} .

4-

D / D t = \partial_{t} + 𝒗 \cdot \nabla

5-

u_{i j} = (\partial_{i} v_{j} + \partial_{j} v_{i}) / 2

6-

ω_{i j} = (\partial_{i} v_{j} - \partial_{j} v_{i}) / 2

7-

H_{i j} = - δ F_{LdG} / δ Q_{i j}

8-

F_{LdG} = \frac{1}{2} \int d^{2} r [K {| \nabla 𝑸 |}^{2} + C tr 𝑸^{2} (tr 𝑸^{2} - 1)],

9-

σ_{i j} = σ_{i j}^{e} + σ_{i j}^{a}

10-

σ_{i j}^{e} = - λ H_{i j} + Q_{i k} H_{k j} - H_{i k} Q_{k j}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0012340

Formulas:

1-

l = 1, 2, 3

2-

G = (\begin{matrix} 1 + g & g & g \\ g & 1 + g & g \\ g & g & 1 + g \end{matrix}) {(1 + 3 g)}^{- 1 / 3} .

3-

H = \sum_{N N} J_{1} 𝐒_{i} \cdot 𝐒_{j} + \sum_{N N N} J_{2} 𝐒_{i} \cdot 𝐒_{j} .

4-

H^{FM} = (6 J_{1} + 3 J_{2}) S^{2},

5-

H^{AFI} = (- 2 J_{1} + 3 J_{2}) S^{2},

6-

H^{AFII} = - 3 J_{2} S^{2},

7-

J_{1} \to J_{1}^{'}

8-

J_{1} \to J_{1}^{''}

9-

H^{FM} = (3 J_{1}^{'} + 3 J_{1}^{''} + 3 J_{2}) S^{2},

10-

H^{AFI} = (- J_{1}^{'} - J_{1}^{''} + 3 J_{2}) S^{2},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0701213

Formulas:

1-

ρ (T) χ (T) = c o n s t

2-

ρ (T) χ (T) = c o n s t

3-

ρ (T) = ρ_{0} + A \exp (Δ / k_{B} T)

4-

10^{2} μ Ω

5-

χ (T) = χ_{0} + n C / (T - Θ)

6-

χ_{0} = 5.6 \cdot 10^{- 4}

7-

C = 1.86 \cdot 10^{- 3}

8-

\sim k_{B} T ρ (ϵ_{F})

9-

ρ (ϵ_{F})

10-

10^{- 4} - 10^{- 3}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0311158

Formulas:

1-

ρ = p_{1} ρ_{1} + p_{2} ρ_{2}

2-

ρ (P_{1} P_{2})

3-

tr (ρ P_{1} P_{2})

4-

ρ_{1} = P_{1} ρ P_{1} / ρ (P_{1})

5-

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

6-

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

7-

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

8-

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

9-

ρ (P_{1}^{n})

10-

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

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1805.06985

Formulas:

1-

ℏ ω \approx 1.8 eV

2-

ℏ ω \approx 1.95 eV

3-

ℏ ω \approx 2.7 eV

4-

{| χ^{(2)} |}^{2}

5-

{| χ^{(3)} |}^{2}

6-

- ({\vec{k}}_{1} + 2 {\vec{k}}_{2})

7-

ω_{m} = ω_{Σ} \equiv ω_{1} + 2 ω_{2}

8-

ℏ ω_{2} = ℏ ω_{A} / 2 \approx 0.9 eV

9-

ℏ ω_{1} = ℏ (ω_{C} - ω_{A}) \approx 0.9 eV

10-

χ^{(1)} (- ω_{1}, ω_{1}) = \sum_{a, g} \sum_{𝐤} \frac{μ_{g a 𝐤}^{2}}{Δ_{g a 𝐤}^{1}},

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0204484

Formulas:

1-

ℱ = \int 𝑑 𝐱 [a {| ψ |}^{2} + \frac{b}{2} {| ψ |}^{4} + \frac{ℏ^{2}}{2 m^{*}} {| (\nabla - i \frac{e^{*}}{ℏ c} 𝐀) ψ |}^{2}],

2-

a = a_{0} (T - T_{c})

3-

(τ + i τ^{'}) (\frac{\partial}{\partial t} + i \frac{e^{*}}{ℏ} ϕ) ψ = - \frac{δ ℱ}{δ ψ^{*}} + ζ .

4-

⟨ ζ^{*} (𝐫, t) ζ (𝐫^{'}, t^{'}) ⟩ = 2 T τ δ (𝐫 - 𝐫^{'}) δ (t - t^{'}) .

5-

ψ (𝐫, t) = \int 𝑑 𝐫^{'} 𝑑 t^{'} G (𝐫, t; 𝐫^{'}, t^{'}) ζ (𝐫^{'}, t^{'})

6-

𝐣^{e} = - i \frac{e^{*} ℏ}{2 m^{*}} ⟨ ψ^{*} (\nabla - i \frac{e^{*}}{ℏ c} 𝐀) ψ ⟩ + c.c.,

7-

𝐣^{Q} = - \frac{ℏ^{2}}{2 m^{*}} ⟨ (\frac{\partial}{\partial t} - i \frac{e^{*}}{ℏ} ϕ) ψ^{*} (\nabla - \frac{i e^{*}}{ℏ c} 𝐀) ψ ⟩ + c.c.

8-

⟨ 𝐣 ⟩ = \frac{\int D ψ D \bar{ψ} {⟨ 𝐣 ⟩}_{ψ \bar{ψ} ϕ} e^{- S_{eff} (ψ, \bar{ψ}, ϕ)}}{\int D ψ D \bar{ψ} e^{- S_{eff} (ψ, \bar{ψ}, ϕ)}},

9-

S_{eff} (ψ, \bar{ψ}, ϕ)

10-

{⟨ 𝐣 ⟩}_{ψ \bar{ψ} ϕ}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1301.2004

Formulas:

1-

1.5 \pm 0.3 M_{s o l}

2-

39.4 \pm 1.0 p c

3-

M_{J u p i t e r}

4-

1.9 \times 10^{- 2} M_{\oplus}

5-

1.2 \times 10^{- 1} M_{\oplus}

6-

1.1 \times 10^{- 6} M_{\oplus}

7-

d n (m) = A m^{- α} d m

8-

d n (a) = C a^{2 - 3 α} d a

9-

M_{T} = \int_{a_{m i n}}^{a_{m a x}} M (a) 𝑑 n (a) .

10-

a_{m a x}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0710.3751

Formulas:

1-

W_{i} \in [W_{m i n}, W_{m a x}]

2-

i \in [1, N]

3-

H_{i} \in [H_{m i n}, H_{m a x}]

4-

H_{m i n}

5-

H_{m i n}

6-

H_{m i n}

7-

H_{m i n}

8-

H_{i} = H_{m i n}

9-

N_{C} (0)

10-

N - N_{C} (0)

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0010034

Formulas:

1-

{𝐁𝐢}_{𝟐} {𝐒𝐫}_{𝟐} {𝐂𝐚𝐂𝐮}_{𝟐} 𝐎_{𝟖 + δ}

2-

{Bi}_{2} {Sr}_{2} {CaCu}_{2} O_{8 + δ}

3-

{Bi}_{2} {Sr}_{2} {CaCu}_{2} O_{8 + δ}

4-

ω_{p} (0, T) = \frac{c}{\sqrt{ϵ} λ_{c} (T)} = \frac{c}{\sqrt{ϵ} γ λ_{a b} (T)},

5-

λ_{a b, c}

6-

{(c ϕ_{0} / 8 π^{2} {j_{a b}, j_{c}} s)}^{1 / 2}

7-

{(h c^{2} / 16 π^{2} e {j_{a b}, j_{c}} s)}^{1 / 2}

8-

h c / 2 e

9-

ω_{p} (H, T)

10-

ω_{p}^{2} (H, T) = ω_{p}^{2} (0, T) ⟨ \cos φ_{n, n + 1} ⟩ (H, T),

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0710.0989

Formulas:

1-

e (D) \cdot D

2-

ℐ (D) \subseteq 𝒪_{X}

3-

κ (X, D) \geq 0

4-

| e (D) \cdot D |

5-

ℐ (n || e (D) \cdot D ||) = 𝒪_{X}

6-

n = n (ϵ, x)

7-

D_{n} \in | n D |

8-

{mult}_{x} (D_{n}) < n ϵ

9-

(D \cdot C) \geq 0

10-

ν (X, D) := sup {ν \in ℕ : D^{ν} ≢ 0} .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1406.3936

Formulas:

1-

u v b y β

2-

σ \bar{\log g}

3-

u v b y β

4-

u v b y β

5-

m_{1} + 0.18 (b - y),

6-

c_{1} - 0.20 (b - y) .

7-

- 0.01 (2) + 0.32 (9) [Ca / H], N = 95,

8-

- 0.25 (4) + 0.26 (9) [Ca / H], N = 26,

9-

- 0.68 (13) + 0.40 (11) [Ca / H], N = 24,

10-

- 0.32 (11) + 0.83 (12) [Ca / H], N = 23 .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1003.4710

Formulas:

1-

10^{44} erg {yr}^{- 1} {Mpc}^{- 3}

2-

ϵ_{e} E_{TOT}^{iso} \sim

3-

ϵ_{B} E_{TOT}^{iso}

4-

ϵ_{p} E_{TOT}^{iso}

5-

ρ_{em, GRB} \approx {\dot{n}}_{0} ϵ_{e} E_{TOT}^{iso} = 5 \times 10^{42} - 10^{44} {ergs Mpc}^{- 3} {yr}^{- 1}

6-

ρ_{h a d, G R B}

7-

ρ_{had, GRB} = {\dot{n}}_{0} ϵ_{p} E_{TOT}^{iso} \approx 10^{44} {ergs Mpc}^{- 3} {yr}^{- 1}

8-

ϵ_{p} / ϵ_{e} \approx 1 - 20

9-

τ_{γ γ} (E) < 1

10-

10^{- 4} ergs {cm}^{- 2} s^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1311.4441

Formulas:

1-

Φ = - \frac{G M_{*}}{\sqrt{x^{2} + y^{2} + z^{2}}} - \frac{G M_{p l}}{\sqrt{{(x - A)}^{2} + y^{2} + z^{2}}} - \frac{1}{2} 𝛀^{2} ({(x - A \frac{M_{p l}}{M_{p l} + M_{*}})}^{2} + y^{2}),

2-

ρ_{1} v_{1}^{2} + p_{1} = ρ_{2} v_{2}^{2} + p_{2},

3-

\begin{matrix} ρ_{a t m} (r) & = & ρ_{0} \cdot \exp {- \frac{G M_{p l}}{R_{g a s} T_{a t m}} (\frac{1}{r_{0}} - \frac{1}{r})} \\ p_{a t m} (r) & = & ρ_{a t m} R_{g a s} T_{a t m} \end{matrix},

4-

ρ_{a t m}

5-

R_{g a s}

6-

T_{a t m}

7-

p_{a t m}

8-

p_{a t m} (r) = ρ_{w} 𝐯_{𝐰}^{2} \cos^{2} (𝐧, 𝐯_{𝐰}) + p_{w},

9-

\cos (𝐧, 𝐯_{𝐰}) = 1

10-

p_{s} = ρ_{s} \cdot R_{g a s} \cdot T_{a t m}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0902.1468

Formulas:

1-

I \bar{4} 3 d

2-

4 ν_{Q} = 38.13

3-

\pm \frac{7}{2} \leftrightarrow \pm \frac{9}{2}

4-

\begin{matrix} 1 - \frac{M (t)}{M (\infty)} & = \frac{16}{715} \exp (- \frac{36 t}{T_{1}}) + \frac{49}{165} \exp (- \frac{21 t}{T_{1}}) \\ + \frac{80}{143} \exp (- \frac{10 t}{T_{1}}) + \frac{4}{33} \exp (- \frac{3 t}{T_{1}}) . \end{matrix}

5-

A_{hf} = 94 (5)

6-

𝒦_{0} = - 9 (0.3)

7-

A_{hf} = 94 (5)

8-

𝒦_{0} = - 9 (0.3)

9-

χ_{c f} = ⟨ S_{c} S_{f} ⟩

10-

𝒦_{c f} = 𝒦 - 𝒦_{0} - A χ

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/gr-qc/9605056

Formulas:

1-

N_{r} (t)

2-

N_{r} = n_{r} R^{3}

3-

N_{r} {(t)}^{- \frac{1}{3}} T R = c o n s t .

4-

{[N_{r} (t)]}^{- \frac{1}{3}} \frac{T}{ν} = c o n s t .

5-

N_{r} (t) = c o n s t

6-

N_{r} {(t)}^{- \frac{1}{3}} λ T = c o n s t . .

7-

N_{r} (t)

8-

ρ_{T_{1}} (λ_{1}) Δ λ_{1}

9-

ρ_{T_{2}} (λ_{2}) Δ λ_{2}

10-

\frac{Δ λ_{2}}{Δ λ_{1}} = {(\frac{N_{r} (t_{2})}{N_{r} (t_{1})})}^{\frac{1}{3}} \frac{T_{1}}{T_{2}},

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1401.7051

Formulas:

1-

W_{L} (χ_{L}, χ_{G}) = \frac{3}{2} Ω_{m} (1 + z_{L}) χ_{L} (1 - \frac{χ_{L}}{χ_{G}})

2-

κ (\hat{θ}) = \int_{0}^{χ_{G}} δ (χ_{L}, \hat{θ}) W_{L} (χ_{L}, χ_{G}) 𝑑 χ_{L}

3-

⟨ \tilde{κ} (ℓ_{𝟏}) \tilde{κ} (ℓ_{𝟐}) ⟩ = {(2 π)}^{2} δ^{D} (ℓ_{𝟏} + ℓ_{𝟐}) P (ℓ_{1}),

4-

δ^{D} (ℓ)

5-

C_{i j}^{α β} (ℓ) = \int_{0}^{χ} \frac{W_{i}^{α} (χ^{'}) W_{j}^{β} (χ^{'})}{χ^{', 2}} P (ℓ; χ^{'}) 𝑑 χ^{'},

6-

α, β \in G, ϕ

7-

W_{i}^{G} (χ) = \int_{0}^{χ} W_{L} (χ^{'}, χ) f_{i} (χ^{'}) 𝑑 χ^{'},

8-

f_{i} (χ)

9-

f_{i} (χ) \approx δ^{D} (χ - χ^{*})

10-

W^{ϕ} (χ) = \frac{3}{2} Ω_{m} (1 + z) χ (1 - \frac{χ}{χ^{*}}),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0110427

Formulas:

1-

v (x, y)

2-

u (x, y)

3-

y \sim x^{2} / d

4-

v_{s} (s, z)

5-

s_{i}, i = 1 - 4

6-

v_{s} (s_{i}, z)

7-

v_{s} (s, z) = v_{0} \exp (- α_{1} z / d - α_{2} {(z / d)}^{2})

8-

v_{s} (s, z) = v_{0} \exp (- α_{1} z / d)

9-

Q \propto {(W - α d)}^{5 / 2}

10-

W \times \int v_{s} (s, z) 𝑑 z

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0901.3085

Formulas:

1-

Δ σ_{z} ≲ 10^{- 3}

2-

12 (σ_{z} / (1 + z) / 0.03)

3-

Ω_{m} = 0.27; Ω_{Λ} = 0.73; Ω_{b} = 0.046; n_{s} = 0.96; σ_{8} = 0.79

4-

100 h^{- 1} Mpc

5-

P (k) = P_{linear} (k) e^{- \frac{k^{2} σ^{2}}{2}},

6-

σ = s_{0} D,

7-

D \propto δ (z) / δ (0)

8-

10.9 h^{- 1} Mpc

9-

12.4 h^{- 1} Mpc

10-

ξ (r) = \frac{1}{2 π^{2}} \int P (k) \frac{k}{r} \sin (k r) d k .

Formulas (html):

<math><mrow id="id2.2.m2.1.1" xref="id2.2.m2.1.1.cmml"><mrow id="id2.2.m2.1.1.2" xref="id2.2.m2.1.1.2.cmml"><mi mathvariant="normal" id="id2.2.m2.1.1.2.2" xref="id2.2.m2.1.1.2.2.cmml">Δ</mi><mo id="id2.2.m2.1.1.2.1" xref="id2.2.m2.1.1.2.1.cmml">⁢</mo><msub id="id2.2.m2.1.1.2.3" xref="id2.2.m2.1.1.2.3.cmml"><mi id="id2.2.m2.1.1.2.3.2" xref="id2.2.m2.1.1.2.3.2.cmml">σ</mi><mi id="id2.2.m2.1.1.2.3.3" xref="id2.2.m2.1.1.2.3.3.cmml">z</mi></msub></mrow><mo id="id2.2.m2.1.1.1" xref="id2.2.m2.1.1.1.cmml">≲</mo><msup id="id2.2.m2.1.1.3" xref="id2.2.m2.1.1.3.cmml"><mn id="id2.2.m2.1.1.3.2" xref="id2.2.m2.1.1.3.2.cmml">10</mn><mrow id="id2.2.m2.1.1.3.3" xref="id2.2.m2.1.1.3.3.cmml"><mo id="id2.2.m2.1.1.3.3.1" xref="id2.2.m2.1.1.3.3.1.cmml">-</mo><mn id="id2.2.m2.1.1.3.3.2" xref="id2.2.m2.1.1.3.3.2.cmml">3</mn></mrow></msup></mrow></math>, <math><mrow id="S1.p1.5.m5.1.1" xref="S1.p1.5.m5.1.1.cmml"><mn id="S1.p1.5.m5.1.1.3" xref="S1.p1.5.m5.1.1.3.cmml">12</mn><mo id="S1.p1.5.m5.1.1.2" xref="S1.p1.5.m5.1.1.2.cmml">⁢</mo><mrow id="S1.p1.5.m5.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.p1.5.m5.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S1.p1.5.m5.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.cmml"><msub id="S1.p1.5.m5.1.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.1.3.cmml"><mi id="S1.p1.5.m5.1.1.1.1.1.3.2" xref="S1.p1.5.m5.1.1.1.1.1.3.2.cmml">σ</mi><mi id="S1.p1.5.m5.1.1.1.1.1.3.3" xref="S1.p1.5.m5.1.1.1.1.1.3.3.cmml">z</mi></msub><mo id="S1.p1.5.m5.1.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.1.2.cmml">/</mo><mrow id="S1.p1.5.m5.1.1.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.p1.5.m5.1.1.1.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.p1.5.m5.1.1.1.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.cmml"><mn id="S1.p1.5.m5.1.1.1.1.1.1.1.1.2" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.p1.5.m5.1.1.1.1.1.1.1.1.1" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S1.p1.5.m5.1.1.1.1.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.3.cmml">z</mi></mrow><mo stretchy="false" id="S1.p1.5.m5.1.1.1.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p1.5.m5.1.1.1.1.1.2a" xref="S1.p1.5.m5.1.1.1.1.1.2.cmml">/</mo><mn id="S1.p1.5.m5.1.1.1.1.1.4" xref="S1.p1.5.m5.1.1.1.1.1.4.cmml">0.03</mn></mrow><mo stretchy="false" id="S1.p1.5.m5.1.1.1.1.3" xref="S1.p1.5.m5.1.1.1.1.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.p1.1.m1.2.2.2" xref="S2.p1.1.m1.2.2.3.cmml"><mrow id="S2.p1.1.m1.1.1.1.1" xref="S2.p1.1.m1.1.1.1.1.cmml"><msub id="S2.p1.1.m1.1.1.1.1.2" xref="S2.p1.1.m1.1.1.1.1.2.cmml"><mi mathvariant="normal" id="S2.p1.1.m1.1.1.1.1.2.2" xref="S2.p1.1.m1.1.1.1.1.2.2.cmml">Ω</mi><mi id="S2.p1.1.m1.1.1.1.1.2.3" xref="S2.p1.1.m1.1.1.1.1.2.3.cmml">m</mi></msub><mo id="S2.p1.1.m1.1.1.1.1.1" xref="S2.p1.1.m1.1.1.1.1.1.cmml">=</mo><mn id="S2.p1.1.m1.1.1.1.1.3" xref="S2.p1.1.m1.1.1.1.1.3.cmml">0.27</mn></mrow><mo id="S2.p1.1.m1.2.2.2.3" xref="S2.p1.1.m1.2.2.3a.cmml">;</mo><mrow id="S2.p1.1.m1.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.3.cmml"><mrow id="S2.p1.1.m1.2.2.2.2.1.1" xref="S2.p1.1.m1.2.2.2.2.1.1.cmml"><msub id="S2.p1.1.m1.2.2.2.2.1.1.2" xref="S2.p1.1.m1.2.2.2.2.1.1.2.cmml"><mi mathvariant="normal" id="S2.p1.1.m1.2.2.2.2.1.1.2.2" xref="S2.p1.1.m1.2.2.2.2.1.1.2.2.cmml">Ω</mi><mi mathvariant="normal" id="S2.p1.1.m1.2.2.2.2.1.1.2.3" xref="S2.p1.1.m1.2.2.2.2.1.1.2.3.cmml">Λ</mi></msub><mo id="S2.p1.1.m1.2.2.2.2.1.1.1" xref="S2.p1.1.m1.2.2.2.2.1.1.1.cmml">=</mo><mn id="S2.p1.1.m1.2.2.2.2.1.1.3" xref="S2.p1.1.m1.2.2.2.2.1.1.3.cmml">0.73</mn></mrow><mo id="S2.p1.1.m1.2.2.2.2.2.3" xref="S2.p1.1.m1.2.2.2.2.3a.cmml">;</mo><mrow id="S2.p1.1.m1.2.2.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.3.cmml"><mrow id="S2.p1.1.m1.2.2.2.2.2.2.1.1" xref="S2.p1.1.m1.2.2.2.2.2.2.1.1.cmml"><msub id="S2.p1.1.m1.2.2.2.2.2.2.1.1.2" xref="S2.p1.1.m1.2.2.2.2.2.2.1.1.2.cmml"><mi mathvariant="normal" id="S2.p1.1.m1.2.2.2.2.2.2.1.1.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.1.1.2.2.cmml">Ω</mi><mi id="S2.p1.1.m1.2.2.2.2.2.2.1.1.2.3" xref="S2.p1.1.m1.2.2.2.2.2.2.1.1.2.3.cmml">b</mi></msub><mo id="S2.p1.1.m1.2.2.2.2.2.2.1.1.1" xref="S2.p1.1.m1.2.2.2.2.2.2.1.1.1.cmml">=</mo><mn id="S2.p1.1.m1.2.2.2.2.2.2.1.1.3" xref="S2.p1.1.m1.2.2.2.2.2.2.1.1.3.cmml">0.046</mn></mrow><mo id="S2.p1.1.m1.2.2.2.2.2.2.2.3" xref="S2.p1.1.m1.2.2.2.2.2.2.3a.cmml">;</mo><mrow id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.3.cmml"><mrow id="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.cmml"><msub id="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.2" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.2.cmml"><mi id="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.2.2.cmml">n</mi><mi id="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.2.3" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.2.3.cmml">s</mi></msub><mo id="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.1" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.1.cmml">=</mo><mn id="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.3" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.1.1.3.cmml">0.96</mn></mrow><mo id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.3" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.3a.cmml">;</mo><mrow id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.cmml"><msub id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.2.cmml"><mi id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.2.2" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.2.2.cmml">σ</mi><mn id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.2.3" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.2.3.cmml">8</mn></msub><mo id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.1" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.1.cmml">=</mo><mn id="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.3" xref="S2.p1.1.m1.2.2.2.2.2.2.2.2.2.2.3.cmml">0.79</mn></mrow></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.p2.2.m2.1.1" xref="S2.p2.2.m2.1.1.cmml"><mpadded width="+1.7pt" id="S2.p2.2.m2.1.1.2" xref="S2.p2.2.m2.1.1.2.cmml"><mn id="S2.p2.2.m2.1.1.2a" xref="S2.p2.2.m2.1.1.2.cmml">100</mn></mpadded><mo id="S2.p2.2.m2.1.1.1" xref="S2.p2.2.m2.1.1.1.cmml">⁢</mo><msup id="S2.p2.2.m2.1.1.3" xref="S2.p2.2.m2.1.1.3.cmml"><mi id="S2.p2.2.m2.1.1.3.2" xref="S2.p2.2.m2.1.1.3.2.cmml">h</mi><mrow id="S2.p2.2.m2.1.1.3.3" xref="S2.p2.2.m2.1.1.3.3.cmml"><mo id="S2.p2.2.m2.1.1.3.3.1" xref="S2.p2.2.m2.1.1.3.3.1.cmml">-</mo><mn id="S2.p2.2.m2.1.1.3.3.2" xref="S2.p2.2.m2.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.p2.2.m2.1.1.1a" xref="S2.p2.2.m2.1.1.1.cmml">⁢</mo><mi id="S2.p2.2.m2.1.1.4" xref="S2.p2.2.m2.1.1.4.cmml">Mpc</mi></mrow></math>, <math><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mi id="S2.E1.m1.3.3.1.1.2.2" xref="S2.E1.m1.3.3.1.1.2.2.cmml">P</mi><mo id="S2.E1.m1.3.3.1.1.2.1" xref="S2.E1.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.2.3.2" xref="S2.E1.m1.3.3.1.1.2.cmml"><mo stretchy="false" id="S2.E1.m1.3.3.1.1.2.3.2.1" xref="S2.E1.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">k</mi><mo stretchy="false" id="S2.E1.m1.3.3.1.1.2.3.2.2" xref="S2.E1.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.3.3.1.1.1" xref="S2.E1.m1.3.3.1.1.1.cmml">=</mo><mrow id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml"><msub id="S2.E1.m1.3.3.1.1.3.2" xref="S2.E1.m1.3.3.1.1.3.2.cmml"><mi id="S2.E1.m1.3.3.1.1.3.2.2" xref="S2.E1.m1.3.3.1.1.3.2.2.cmml">P</mi><mi id="S2.E1.m1.3.3.1.1.3.2.3" xref="S2.E1.m1.3.3.1.1.3.2.3.cmml">linear</mi></msub><mo id="S2.E1.m1.3.3.1.1.3.1" xref="S2.E1.m1.3.3.1.1.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.3.3.2" xref="S2.E1.m1.3.3.1.1.3.cmml"><mo stretchy="false" id="S2.E1.m1.3.3.1.1.3.3.2.1" xref="S2.E1.m1.3.3.1.1.3.cmml">(</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">k</mi><mo rspace="4.2pt" stretchy="false" id="S2.E1.m1.3.3.1.1.3.3.2.2" xref="S2.E1.m1.3.3.1.1.3.cmml">)</mo></mrow><mo id="S2.E1.m1.3.3.1.1.3.1a" xref="S2.E1.m1.3.3.1.1.3.1.cmml">⁢</mo><msup id="S2.E1.m1.3.3.1.1.3.4" xref="S2.E1.m1.3.3.1.1.3.4.cmml"><mi id="S2.E1.m1.3.3.1.1.3.4.2" xref="S2.E1.m1.3.3.1.1.3.4.2.cmml">e</mi><mrow id="S2.E1.m1.3.3.1.1.3.4.3" xref="S2.E1.m1.3.3.1.1.3.4.3.cmml"><mo id="S2.E1.m1.3.3.1.1.3.4.3.1" xref="S2.E1.m1.3.3.1.1.3.4.3.1.cmml">-</mo><mfrac id="S2.E1.m1.3.3.1.1.3.4.3.2" xref="S2.E1.m1.3.3.1.1.3.4.3.2.cmml"><mrow id="S2.E1.m1.3.3.1.1.3.4.3.2.2" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.cmml"><msup id="S2.E1.m1.3.3.1.1.3.4.3.2.2.2" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.2.cmml"><mi id="S2.E1.m1.3.3.1.1.3.4.3.2.2.2.2" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.2.2.cmml">k</mi><mn id="S2.E1.m1.3.3.1.1.3.4.3.2.2.2.3" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.2.3.cmml">2</mn></msup><mo id="S2.E1.m1.3.3.1.1.3.4.3.2.2.1" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.1.cmml">⁢</mo><msup id="S2.E1.m1.3.3.1.1.3.4.3.2.2.3" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.3.cmml"><mi id="S2.E1.m1.3.3.1.1.3.4.3.2.2.3.2" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.3.2.cmml">σ</mi><mn id="S2.E1.m1.3.3.1.1.3.4.3.2.2.3.3" xref="S2.E1.m1.3.3.1.1.3.4.3.2.2.3.3.cmml">2</mn></msup></mrow><mn id="S2.E1.m1.3.3.1.1.3.4.3.2.3" xref="S2.E1.m1.3.3.1.1.3.4.3.2.3.cmml">2</mn></mfrac></mrow></msup></mrow></mrow><mo id="S2.E1.m1.3.3.1.2" xref="S2.E1.m1.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.2.cmml">σ</mi><mo id="S2.E2.m1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.1.1.1.1.3" xref="S2.E2.m1.1.1.1.1.3.cmml"><msub id="S2.E2.m1.1.1.1.1.3.2" xref="S2.E2.m1.1.1.1.1.3.2.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.2" xref="S2.E2.m1.1.1.1.1.3.2.2.cmml">s</mi><mn id="S2.E2.m1.1.1.1.1.3.2.3" xref="S2.E2.m1.1.1.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.E2.m1.1.1.1.1.3.1" xref="S2.E2.m1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.3.3" xref="S2.E2.m1.1.1.1.1.3.3.cmml">D</mi></mrow></mrow><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.p5.1.m1.2.3" xref="S2.p5.1.m1.2.3.cmml"><mi id="S2.p5.1.m1.2.3.2" xref="S2.p5.1.m1.2.3.2.cmml">D</mi><mo id="S2.p5.1.m1.2.3.1" xref="S2.p5.1.m1.2.3.1.cmml">∝</mo><mrow id="S2.p5.1.m1.2.3.3" xref="S2.p5.1.m1.2.3.3.cmml"><mrow id="S2.p5.1.m1.2.3.3.2" xref="S2.p5.1.m1.2.3.3.2.cmml"><mrow id="S2.p5.1.m1.2.3.3.2.2" xref="S2.p5.1.m1.2.3.3.2.2.cmml"><mi id="S2.p5.1.m1.2.3.3.2.2.2" xref="S2.p5.1.m1.2.3.3.2.2.2.cmml">δ</mi><mo id="S2.p5.1.m1.2.3.3.2.2.1" xref="S2.p5.1.m1.2.3.3.2.2.1.cmml">⁢</mo><mrow id="S2.p5.1.m1.2.3.3.2.2.3.2" xref="S2.p5.1.m1.2.3.3.2.2.cmml"><mo stretchy="false" id="S2.p5.1.m1.2.3.3.2.2.3.2.1" xref="S2.p5.1.m1.2.3.3.2.2.cmml">(</mo><mi id="S2.p5.1.m1.1.1" xref="S2.p5.1.m1.1.1.cmml">z</mi><mo stretchy="false" id="S2.p5.1.m1.2.3.3.2.2.3.2.2" xref="S2.p5.1.m1.2.3.3.2.2.cmml">)</mo></mrow></mrow><mo id="S2.p5.1.m1.2.3.3.2.1" xref="S2.p5.1.m1.2.3.3.2.1.cmml">/</mo><mi id="S2.p5.1.m1.2.3.3.2.3" xref="S2.p5.1.m1.2.3.3.2.3.cmml">δ</mi></mrow><mo id="S2.p5.1.m1.2.3.3.1" xref="S2.p5.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.p5.1.m1.2.3.3.3.2" xref="S2.p5.1.m1.2.3.3.cmml"><mo stretchy="false" id="S2.p5.1.m1.2.3.3.3.2.1" xref="S2.p5.1.m1.2.3.3.cmml">(</mo><mn id="S2.p5.1.m1.2.2" xref="S2.p5.1.m1.2.2.cmml">0</mn><mo stretchy="false" id="S2.p5.1.m1.2.3.3.3.2.2" xref="S2.p5.1.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.p5.4.m4.1.1" xref="S2.p5.4.m4.1.1.cmml"><mpadded width="+1.7pt" id="S2.p5.4.m4.1.1.2" xref="S2.p5.4.m4.1.1.2.cmml"><mn id="S2.p5.4.m4.1.1.2a" xref="S2.p5.4.m4.1.1.2.cmml">10.9</mn></mpadded><mo id="S2.p5.4.m4.1.1.1" xref="S2.p5.4.m4.1.1.1.cmml">⁢</mo><msup id="S2.p5.4.m4.1.1.3" xref="S2.p5.4.m4.1.1.3.cmml"><mi id="S2.p5.4.m4.1.1.3.2" xref="S2.p5.4.m4.1.1.3.2.cmml">h</mi><mrow id="S2.p5.4.m4.1.1.3.3" xref="S2.p5.4.m4.1.1.3.3.cmml"><mo id="S2.p5.4.m4.1.1.3.3.1" xref="S2.p5.4.m4.1.1.3.3.1.cmml">-</mo><mn id="S2.p5.4.m4.1.1.3.3.2" xref="S2.p5.4.m4.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.p5.4.m4.1.1.1a" xref="S2.p5.4.m4.1.1.1.cmml">⁢</mo><mi id="S2.p5.4.m4.1.1.4" xref="S2.p5.4.m4.1.1.4.cmml">Mpc</mi></mrow></math>, <math><mrow id="S2.p5.5.m5.1.1" xref="S2.p5.5.m5.1.1.cmml"><mpadded width="+1.7pt" id="S2.p5.5.m5.1.1.2" xref="S2.p5.5.m5.1.1.2.cmml"><mn id="S2.p5.5.m5.1.1.2a" xref="S2.p5.5.m5.1.1.2.cmml">12.4</mn></mpadded><mo id="S2.p5.5.m5.1.1.1" xref="S2.p5.5.m5.1.1.1.cmml">⁢</mo><msup id="S2.p5.5.m5.1.1.3" xref="S2.p5.5.m5.1.1.3.cmml"><mi id="S2.p5.5.m5.1.1.3.2" xref="S2.p5.5.m5.1.1.3.2.cmml">h</mi><mrow id="S2.p5.5.m5.1.1.3.3" xref="S2.p5.5.m5.1.1.3.3.cmml"><mo id="S2.p5.5.m5.1.1.3.3.1" xref="S2.p5.5.m5.1.1.3.3.1.cmml">-</mo><mn id="S2.p5.5.m5.1.1.3.3.2" xref="S2.p5.5.m5.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S2.p5.5.m5.1.1.1a" xref="S2.p5.5.m5.1.1.1.cmml">⁢</mo><mi id="S2.p5.5.m5.1.1.4" xref="S2.p5.5.m5.1.1.4.cmml">Mpc</mi></mrow></math>, <math><mrow id="S2.E3.m1.4.4.1" xref="S2.E3.m1.4.4.1.1.cmml"><mrow id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.1.cmml"><mrow id="S2.E3.m1.4.4.1.1.3" xref="S2.E3.m1.4.4.1.1.3.cmml"><mi id="S2.E3.m1.4.4.1.1.3.2" xref="S2.E3.m1.4.4.1.1.3.2.cmml">ξ</mi><mo id="S2.E3.m1.4.4.1.1.3.1" xref="S2.E3.m1.4.4.1.1.3.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.3.3.2" xref="S2.E3.m1.4.4.1.1.3.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.3.3.2.1" xref="S2.E3.m1.4.4.1.1.3.cmml">(</mo><mi id="S2.E3.m1.1.1" xref="S2.E3.m1.1.1.cmml">r</mi><mo stretchy="false" id="S2.E3.m1.4.4.1.1.3.3.2.2" xref="S2.E3.m1.4.4.1.1.3.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.4.4.1.1.2" xref="S2.E3.m1.4.4.1.1.2.cmml">=</mo><mrow id="S2.E3.m1.4.4.1.1.1" xref="S2.E3.m1.4.4.1.1.1.cmml"><mfrac id="S2.E3.m1.4.4.1.1.1.3" xref="S2.E3.m1.4.4.1.1.1.3.cmml"><mn id="S2.E3.m1.4.4.1.1.1.3.2" xref="S2.E3.m1.4.4.1.1.1.3.2.cmml">1</mn><mrow id="S2.E3.m1.4.4.1.1.1.3.3" xref="S2.E3.m1.4.4.1.1.1.3.3.cmml"><mn id="S2.E3.m1.4.4.1.1.1.3.3.2" xref="S2.E3.m1.4.4.1.1.1.3.3.2.cmml">2</mn><mo id="S2.E3.m1.4.4.1.1.1.3.3.1" xref="S2.E3.m1.4.4.1.1.1.3.3.1.cmml">⁢</mo><msup id="S2.E3.m1.4.4.1.1.1.3.3.3" xref="S2.E3.m1.4.4.1.1.1.3.3.3.cmml"><mi id="S2.E3.m1.4.4.1.1.1.3.3.3.2" xref="S2.E3.m1.4.4.1.1.1.3.3.3.2.cmml">π</mi><mn id="S2.E3.m1.4.4.1.1.1.3.3.3.3" xref="S2.E3.m1.4.4.1.1.1.3.3.3.3.cmml">2</mn></msup></mrow></mfrac><mo id="S2.E3.m1.4.4.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.cmml"><mo largeop="true" symmetric="true" id="S2.E3.m1.4.4.1.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.1.2.cmml">∫</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.cmml"><mi id="S2.E3.m1.4.4.1.1.1.1.1.3" xref="S2.E3.m1.4.4.1.1.1.1.1.3.cmml">P</mi><mo id="S2.E3.m1.4.4.1.1.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.4.2" xref="S2.E3.m1.4.4.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.4.2.1" xref="S2.E3.m1.4.4.1.1.1.1.1.cmml">(</mo><mi id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2.cmml">k</mi><mo rspace="4.2pt" stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.4.2.2" xref="S2.E3.m1.4.4.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E3.m1.4.4.1.1.1.1.1.2a" xref="S2.E3.m1.4.4.1.1.1.1.1.2.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.E3.m1.4.4.1.1.1.1.1.5" xref="S2.E3.m1.4.4.1.1.1.1.1.5.cmml"><mfrac id="S2.E3.m1.4.4.1.1.1.1.1.5a" xref="S2.E3.m1.4.4.1.1.1.1.1.5.cmml"><mi id="S2.E3.m1.4.4.1.1.1.1.1.5.2" xref="S2.E3.m1.4.4.1.1.1.1.1.5.2.cmml">k</mi><mi id="S2.E3.m1.4.4.1.1.1.1.1.5.3" xref="S2.E3.m1.4.4.1.1.1.1.1.5.3.cmml">r</mi></mfrac></mpadded><mo id="S2.E3.m1.4.4.1.1.1.1.1.2b" xref="S2.E3.m1.4.4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.1.2.cmml"><mi id="S2.E3.m1.3.3" xref="S2.E3.m1.3.3.cmml">sin</mi><mo id="S2.E3.m1.4.4.1.1.1.1.1.1.1a" xref="S2.E3.m1.4.4.1.1.1.1.1.1.2.cmml">⁡</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.1.2.cmml"><mo stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.1.1.1.2.cmml">(</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.2" xref="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.1" xref="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.3" xref="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo rspace="5.8pt" stretchy="false" id="S2.E3.m1.4.4.1.1.1.1.1.1.1.1.3" xref="S2.E3.m1.4.4.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E3.m1.4.4.1.1.1.1.1.2c" xref="S2.E3.m1.4.4.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.4.4.1.1.1.1.1.6" xref="S2.E3.m1.4.4.1.1.1.1.1.6.cmml"><mo rspace="0pt" id="S2.E3.m1.4.4.1.1.1.1.1.6.1" xref="S2.E3.m1.4.4.1.1.1.1.1.6.1.cmml">d</mo><mi id="S2.E3.m1.4.4.1.1.1.1.1.6.2" xref="S2.E3.m1.4.4.1.1.1.1.1.6.2.cmml">k</mi></mrow></mrow></mrow></mrow></mrow><mo id="S2.E3.m1.4.4.1.2" xref="S2.E3.m1.4.4.1.1.cmml">.</mo></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9912491

Formulas:

1-

M_{L} = M_{D} M_{R}^{- 1} M_{D}^{T},

2-

| m_{3} | ≫ | m_{1, 2} |

3-

m_{1} ≃ - m_{2}

4-

S O (10)

5-

S U (4) \times S U (2) \times S U (2)

6-

M_{L} = (\begin{matrix} 2 ϵ & δ & δ \\ δ & σ & ρ \\ δ & ρ & σ \end{matrix})

7-

\frac{1}{2} (m_{1} c^{2} + m_{2} s^{2})

8-

\frac{1}{\sqrt{2}} (m_{1} - m_{2}) c s

9-

ϵ_{2} + \frac{m_{3}}{2}

10-

ϵ_{2} - \frac{m_{3}}{2}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9903122

Formulas:

1-

t_{e x p}

2-

λ_{L y α}^{o b s}

3-

r_{n e b}

4-

v_{n e b}

5-

t_{d y n}

6-

t_{d y n} \sim

7-

t_{d y n} \sim

8-

\leq \frac{d E}{d t} \leq

9-

\frac{d E}{d t}

10-

\frac{d E}{d t}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9912170

Formulas:

1-

\sim 10^{- 3} - 10^{- 2}

2-

\frac{d ρ}{d t} + ρ \nabla \vec{v} = 0,

3-

ρ \frac{d \vec{v}}{d t} = - \nabla P + ρ \nabla Φ + \nabla 𝚷,

4-

ρ \frac{d e}{d t} = - P \nabla \vec{v} + Q,

5-

Φ = - G M / r

6-

P = (γ - 1) ρ e

7-

ν = α \frac{c_{s}^{2}}{Ω_{K}},

8-

c_{s} = \sqrt{P / ρ}

9-

Ω_{K} = \sqrt{G M / r^{3}}

10-

N_{r} \times N_{θ} = 130 \times 50

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Run 11312206 (Agent336)