Run 10695632

Paper: https://arxiv.org/abs/1805.09532

Formulas:

1-

s p^{3} d^{5}

2-

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

3-

d_{3 z 2 - r^{2}}

4-

| Φ_{σ} ⟩ = {(d_{3 z^{2} - r^{2}, σ}, d_{x^{2} - y^{2}, σ}, d_{x y, σ}, p_{x, σ}^{+}, p_{y, σ}^{+}, p_{z, σ}^{-})}^{T}

5-

p_{i}^{ν} = (p_{i}^{t} + ν p_{i}^{b}) / \sqrt{2}

6-

i = x, y, z

7-

H = H_{S K} \otimes 1 + H_{S O},

8-

H_{S K} = \sum_{i, j} \sum_{α, β} (ϵ_{i α} δ_{i j} δ_{α β} + t_{i α, j β}) b_{i α}^{†} b_{j β},

9-

H_{S O} = \sum_{i} \sum_{σ, σ^{'}} \frac{λ_{i}}{2 ℏ} 𝐋_{i} \cdot 𝝉_{σ σ^{'}},

10-

b_{i α}^{†} (b_{i α})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1310.5585

Formulas:

1-

d x / d t = b (x; α, β)

2-

x_{1}^{*} < x_{3}^{*} < x_{2}^{*}

3-

b (x, α, β) = 0

4-

b (x; α, β) = 0

5-

\vec{N} = (n_{1}, n_{2} \dots)

6-

\vec{x} (t) = \vec{N} (t) / V

7-

\vec{N}, V \to \infty

8-

d x (t) = b (x) d t + V^{- \frac{1}{2}} d B_{t}

9-

f_{V}^{s t} (x) = 𝒞_{V} e^{- V ϕ (x)},

10-

f_{V}^{s t} (x)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/2009.11299

Formulas:

1-

M_{⊙} {yr}^{- 1}

2-

\frac{1}{R} \frac{\partial}{\partial R} (R \frac{\partial Φ_{tot}}{\partial R}) + \frac{\partial^{2} Φ_{tot}}{\partial z^{2}} = 4 π G (\sum_{i = 1}^{3} ρ_{i} + ρ_{h})

3-

\frac{\partial}{\partial z} (ρ_{i} {⟨ σ_{z}^{2} ⟩}_{i}) + ρ_{i} \frac{\partial Φ_{tot}}{\partial z} = 0

4-

\begin{matrix} {⟨ σ_{z}^{2} ⟩}_{i} \frac{\partial}{\partial z} (\frac{1}{ρ_{i}} \frac{\partial ρ_{i}}{\partial z}) & = \\ - 4 π G (ρ_{s} + ρ_{HI} + ρ_{H2} + ρ_{h}) \\ + \frac{1}{R} \frac{\partial}{\partial R} (R \frac{\partial Φ_{tot}}{\partial R}) \end{matrix}

5-

{(R \frac{\partial Φ_{tot}}{\partial R})}_{R, z} = {(v_{rot}^{2})}_{R, z}

6-

\begin{matrix} {⟨ σ_{z}^{2} ⟩}_{i} \frac{\partial}{\partial z} (\frac{1}{ρ_{i}} \frac{\partial ρ_{i}}{\partial z}) & = \\ - 4 π G (ρ_{s} + ρ_{HI} + ρ_{H2} + ρ_{h}) \\ + \frac{1}{R} \frac{\partial}{\partial R} (v_{rot}^{2}) \end{matrix}

7-

ρ_{s} (z)

8-

ρ_{H2} (z)

9-

ρ_{HI} (z)

10-

ρ_{h} (R) = \frac{ρ_{0}}{\frac{R}{R_{s}} {(1 + \frac{R}{R_{s}})}^{2}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1912.09085

Formulas:

1-

μ_{L (R) \to (\leftarrow)}

2-

𝕋_{⊳} = (\begin{matrix} t_{\to \to} & t_{\leftarrow \to} \\ t_{\to \leftarrow} & t_{\leftarrow \leftarrow} \end{matrix}), ℝ_{⊳} = (\begin{matrix} r_{\to \to} & r_{\leftarrow \to} \\ r_{\to \leftarrow} & r_{\leftarrow \leftarrow} \end{matrix}),

3-

μ = (μ_{\to} + μ_{\leftarrow}) / 2

4-

μ_{s} = (μ_{\to} - μ_{\leftarrow}) / 2

5-

I = I_{\to} + I_{\leftarrow}

6-

I_{s} = I_{\to} - I_{\leftarrow}

7-

(\begin{matrix} I \\ - I_{s L} \\ I_{s R} \end{matrix}) = - \frac{N e}{h} (\begin{matrix} t & s & s \\ P_{r} r & γ_{r} & γ_{t} \\ P_{t} t & γ_{t} & γ_{r} \end{matrix}) (\begin{matrix} μ_{L} - μ_{R} \\ μ_{s L} \\ μ_{s R} \end{matrix}),

8-

μ_{L} - μ_{R} = - e V

9-

𝒯_{i j} (𝑯, 𝑴) = 𝒯_{j i} (- 𝑯, - 𝑴)

10-

P_{t} t = P_{r} r = s

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1002.3972

Formulas:

1-

|| \hat{𝐞} (𝐫_{⟂}) ||

2-

n_{2} \sim 4 \times 10^{- 19}

3-

{\hat{𝐞}}_{ω} (𝐫_{⟂})

4-

|| \hat{𝐞} || \equiv {[{\hat{e}}_{x}^{2} + {\hat{e}}_{y}^{2} + {\hat{e}}_{z}^{2}]}^{1 / 2}

5-

{\hat{𝐞}}_{ω} (𝐫_{⟂}) = \sum_{j \geq 0} \frac{1}{j!} 𝐟_{ω_{0}}^{(j)} (𝐫_{⟂}) {(\frac{Δ ω}{ω_{0}})}^{j}

6-

Δ ω \equiv ω - ω_{0}

7-

𝐟_{ω_{0}}^{(j)} \equiv {[ω_{0}^{j} \partial^{j} {\hat{𝐞}}_{ω} (𝐫_{⟂}) / \partial ω^{j}]}_{ω = ω_{0}}

8-

j h p v

9-

i \partial_{z} Q + \hat{D} (i \partial_{t}) Q + \sum_{j h p v} γ^{j h p v} {\hat{G}}^{j} (i \partial_{t}) ϕ^{h p v} = 0 .

10-

Q (z, t)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/0805.1701

Formulas:

1-

| 0, 0 ⟩ + r | 1, 1 ⟩ + r^{2} | 2, 2 ⟩ + \dots

2-

2 {(1 - η)}^{2} η^{2} r^{4}

3-

2 {(1 - η)}^{2} r^{2}

4-

⟨ {\hat{a}}_{k}^{†} {\hat{a}}_{l} ⟩

5-

= ⟨ {\hat{b}}_{k}^{†} {\hat{b}}_{l} ⟩ = \frac{δ_{k l}}{2} (\cosh 2 r_{k} - 1)

6-

⟨ {\hat{a}}_{k} {\hat{b}}_{l} ⟩

7-

= {⟨ {\hat{a}}_{k}^{†} {\hat{b}}_{l}^{†} ⟩}^{*} = \frac{δ_{k l}}{2} \sinh 2 r_{k}

8-

⟨ {\hat{a}}_{k} ⟩ = ⟨ {\hat{b}}_{k} ⟩ = 0

9-

\sum_{k} t_{k} {\hat{a}}_{k}

10-

\sum_{k} t_{k}^{'} {\hat{b}}_{k}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0407313

Formulas:

1-

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

2-

k T_{e} = 5 K e V

3-

k T_{e} = 15 K e V

4-

k T_{e} = 15 K e V

5-

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

6-

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

7-

G_{s} (\bar{ν}, ν)

8-

G_{s} (\bar{ν}, ν)

9-

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

10-

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

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1708.08088

Formulas:

1-

δ E / E = 8.1 % / \sqrt{E (G e V)} \oplus 2.1 %

2-

δ E / E = 5.9 % / \sqrt{E (G e V)} \oplus 0.8 %

3-

E > 0.15 G e V

4-

p_{T} > 0.2 G e V / c

5-

m_{e e} \sim 12 M e V / c^{2}

6-

p_{T} = 0.4 G e V / c

7-

\frac{d N}{d (ϕ - Ψ_{n})} \propto 1 + \sum_{n} 2 v_{n} cos (n (ϕ - Ψ_{n}))

8-

v_{2}^{d i r}

9-

v_{3}^{d i r}

10-

v_{n}^{i n c}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: nucl-ex

Result: correct

Paper: https://arxiv.org/abs/1612.02669

Formulas:

1-

v \sin i = 11 \pm 1

2-

B_{l} = - 2.14 \times 10^{11} \frac{\int v V (v) 𝑑 v}{λ g c \int [1 - I (v)] 𝑑 v},

3-

S_{i n d e x} = \frac{a F_{H} + b F_{K}}{c F_{R_{H K}} + d F_{V_{H K}}} + e,

4-

- 1.9 \times 10^{- 2}

5-

v \sin i = 11 \pm 1

6-

v \sin i = 11

7-

H J D = 2454560.52140 + 111 ϕ,

8-

Ω (l) = Ω_{e q} - d Ω \sin^{2} l,

9-

Ω_{eq}, d Ω

10-

Ω_{eq} - d Ω

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1311.6630

Formulas:

1-

R K_{m a x}

2-

K_{m a x}

3-

N (E_{F})

4-

N (E_{F})

5-

N (E_{F})

6-

N (E_{F})

7-

ℏ Ω_{p, x x} = ℏ Ω_{p, y y}

8-

ℏ Ω_{p, z z}

9-

σ_{z z} / σ_{x x} \sim

10-

N (E_{F})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1005.1620

Formulas:

1-

- 1 \leq A^{'} \leq 1, \dots,

2-

- 1 \leq γ^{''} \leq 1 .

3-

| A^{'} + i A^{''} | = 1, \dots,

4-

| γ^{'} + i γ^{''} | = 1,

5-

A = A^{'} + i A^{''}, \dots,

6-

γ = γ^{'} + i γ^{''},

7-

⟨ A B C ⟩ = ⟨ (A^{'} + i A^{''}) (B^{'} + i B^{''}) (C^{'} + i C^{''}) ⟩

8-

⟨ A B C ⟩

9-

(A^{'} + i A^{''}) (B^{'} + i B^{''}) (C^{'} + i C^{''})

10-

| ⟨ A B C ⟩ + ⟨ a b c ⟩ + ⟨ α β γ ⟩ + ⟨ A a α ⟩ + ⟨ B b β ⟩ - ⟨ C c γ ⟩ | \leq 3 \sqrt{3},

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1806.05270

Formulas:

1-

𝒩^{+} (ℋ)

2-

M u l t (ℋ)

3-

ℋ \subset 𝒩^{+} (ℋ)

4-

b / a \in 𝒩^{+} (ℋ)

5-

𝒩_{l e f t}^{+}

6-

N^{+} (𝔻)

7-

{a f : f \in H^{2}}

8-

H^{2} (𝔻)

9-

f (z) \to z f (z)

10-

H^{p} (𝔻)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0804.4172

Formulas:

1-

\ddot{ϕ_{b}} + 3 \frac{\dot{a}}{a} \dot{ϕ_{b}} + \frac{d V}{d ϕ_{b}} + 3 \frac{Ω_{m 0} H_{0}^{2}}{8 π G} \frac{1}{ϕ_{b}} a^{- 3} = 0,

2-

\dot{𝐯} + (2 \frac{\dot{a}}{a} + \frac{\dot{ϕ_{b}}}{ϕ_{b}}) 𝐯 = - (𝟏 + \frac{𝟏}{𝟒 π 𝐆} \frac{𝟏}{ϕ_{𝐛}^{𝟐}}) \nabla 𝚽 .

3-

\nabla^{2} Φ = \frac{4 π G}{a^{3}} (ρ - \bar{ρ}),

4-

V (ϕ_{b, 0}) = Ω_{Λ 0} ρ_{c r i t} .

5-

\frac{d ϕ_{b}}{d t} = - \frac{H_{0}^{2}}{G} \frac{3 Ω_{m 0}}{8 π ϕ_{0}} \frac{1}{a^{3}} t,

6-

m_{D M} = m_{D M, 0} \frac{ϕ_{b}}{ϕ_{b, 0}} .

7-

{(\frac{\dot{a}}{a})}^{2} = H_{0}^{2} Ω_{m 0} \frac{ϕ_{b}}{ϕ_{b, 0}} a^{- 3} + \frac{8 π G}{3} [\frac{1}{2} {(\frac{d ϕ_{b}}{d t})}^{2} + V (ϕ_{b})] .

8-

V (ϕ) = K / ϕ^{α},

9-

K (G^{1 + α / 2} / H_{0}^{2})

10-

ϕ_{init} (G^{1 / 2})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2003.00406

Formulas:

1-

ℒ_{c l s}

2-

ℒ_{r e g}

3-

ℒ_{O I M}

4-

ℒ_{r p n} = ℒ_{c l s} + ℒ_{r e g} + ℒ_{O I M}

5-

ℒ_{d e t} = ℒ_{c l s} + ℒ_{r e g}

6-

ℒ_{c l s}

7-

ℒ_{r e g}

8-

ℒ_{r e - i d} = ℒ_{1} + ℒ_{O I M}

9-

ℒ_{1} = - \log a_{i}

10-

ℒ_{O I M} = - l o g p_{i}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1906.10887

Formulas:

1-

\tilde{P} \mapsto B \tilde{P}

2-

P \mapsto C (P + F)

3-

P = {𝐩_{i}}_{i = 1}^{N}, P \in 𝐑^{3 \times N}

4-

A \in 𝐑^{3 \times 3}

5-

𝐩_{i} \mapsto A 𝐩_{i} + 𝐛

6-

A \in 𝐑^{4 \times 4}

7-

{\tilde{𝐩}}_{i} \mapsto A {\tilde{𝐩}}_{i}

8-

P \in 𝐑^{3 \times N}

9-

ℱ^{(t - 1)} \in 𝐑^{𝔣^{(t - 1)} \times N}

10-

𝐠_{i, j}^{(t)} = A_{i}^{(t)} 𝐩_{j} + 𝐛_{i}^{(t)}, i = 1, 2, \dots, k^{(t)} .

Formulas (html):

Correct Categorie: cs

Guessed Categorie: eess

Result: incorrect

Paper: https://arxiv.org/abs/1801.00494

Formulas:

1-

v_{b}^{'} = v_{b} e^{λ t_{d}} / 2

2-

x = \ln (v_{b} / v_{0})

3-

x^{'} = x + λ t_{d} - \ln 2 .

4-

t_{d} = \ln 2 / λ

5-

\frac{d x}{d t} = - \frac{1}{γ} V^{'} (x) + σ ξ .

6-

⟨ ξ (t^{'}) ξ (t^{'} + t) ⟩ = δ (t)

7-

V (x) = 0

8-

V (x) = k x^{2} / 2

9-

\frac{d x}{d t} = - \frac{k}{γ} x + σ ξ,

10-

p (x, t)

Formulas (html):

Correct Categorie: q-bio

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1503.00272

Formulas:

1-

\frac{1}{3} | 00 ⟩ ⟨ 00 | + \frac{1}{3} | 01 ⟩ ⟨ 01 | + \frac{1}{3} | 11 ⟩ ⟨ 11 |

2-

a_{j} \in {- 1, + 1}

3-

| ℬ_{C H S H}^{L V M} (λ) | = | a_{1} b_{1} + a_{1} b_{2} + a_{2} b_{1} - a_{2} b_{2} | \leq 2

4-

ℬ_{C H S H}^{L V M} \equiv \sum_{λ} ℬ_{C H S H}^{L V M} (λ) μ (λ) = E (A_{1}, B_{1}) + E (A_{1}, B_{2}) + E (A_{2}, B_{1}) - E (A_{2}, B_{2})

5-

| ℬ_{C H S H}^{L V M} | \leq 2 .

6-

𝐀_{𝐣} (𝐁_{𝐣}) = 𝐚_{𝐣} (𝐛_{𝐣}) \cdot σ

7-

𝐚_{𝐣} (𝐛_{𝐣})

8-

σ = (σ_{x}, σ_{y}, σ_{z})

9-

ℬ_{C H S H}

10-

𝐀_{𝟏} \otimes 𝐁_{𝟏} + 𝐀_{𝟏} \otimes 𝐁_{𝟐} + 𝐀_{𝟐} \otimes 𝐁_{𝟏} - 𝐀_{𝟐} \otimes 𝐁_{𝟐} .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0603785

Formulas:

1-

δ V_{N T}

2-

B_{p o s}

3-

B_{p o s} = Q \sqrt{4 π ρ} \frac{δ V_{N T}}{δ ϕ} \approx 9.3 \sqrt{n (H_{2})} \frac{Δ V_{N T}}{δ ϕ} μ G,

4-

ρ = m n (H_{2})

5-

n (H_{2})

6-

Δ V_{N T} = δ V_{N T} \sqrt{8 \ln 2}

7-

{(M / Φ)}_{o b s}

8-

{(M / Φ)}_{c r i t}

9-

λ = \frac{{(M / Φ)}_{o b s}}{{(M / Φ)}_{c r i t}} = \frac{(m N A / B A)}{(1 / 2 π \sqrt{G})} = 7.6 \times 10^{- 21} \frac{N (H_{2})}{B},

10-

m = 2.8 m_{H}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1411.0842

Formulas:

1-

ω / Ω_{c, edge} \sim 1.3 - 1.4

2-

δ ω / ω \sim 0.3

3-

ω = ℓ Ω_{e} + k_{∥} v_{∥},

4-

v_{B} = {(2 T_{e} / m_{e})}^{1 / 2}

5-

ω_{p e} ≃ 2 π \times 5.8

6-

ω_{p e} ≃ 2 π \times 15.2

7-

ω_{UH} = {(ω_{p e}^{2} + Ω_{e}^{2})}^{1 / 2}

8-

k \sim 0.1 ω_{p e} / v_{B}

9-

k_{∥} \sim 0.1 ω_{p e} / (\sqrt{2} v_{B})

10-

ω = 1.2 ω_{p e}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1911.04854

Formulas:

1-

= \sum_{k} ω_{k} a_{k}^{†} a_{k} + \sum_{j = 1}^{M} Ω_{j} | e_{j} ⟩ ⟨ e_{j} |

2-

+ \sum_{j, k} V_{k, j} (σ_{j}^{+} a_{k} + σ_{j}^{-} a_{k}^{†})

3-

σ_{j}^{+} = | e_{j} ⟩ ⟨ g_{j} |

4-

σ_{j}^{-} = | g_{j} ⟩ ⟨ e_{j} |

5-

i \frac{\partial}{\partial t} | ψ (t) ⟩ = H | ψ (t) ⟩

6-

| ψ (t) ⟩

7-

N = \sum_{k} ω_{k} a_{k}^{†} a_{k} + \sum_{j = 1}^{M} | e_{j} ⟩ ⟨ e_{j} |

8-

| ψ (t) ⟩

9-

| ψ (t) ⟩ = \sum_{j} A_{j} (t) | g_{1} g_{2} \dots e_{j} \dots g_{M}, 0 ⟩ + \sum_{k} C_{k} (t) | g_{1} g_{2} \dots g_{M}, 1_{k} ⟩

10-

A_{j} (t)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0609083

Formulas:

1-

Δ μ_{i} (T, p_{i})

2-

T, p_{O_{2}}, p_{CO}

3-

(\sqrt{5} \times \sqrt{5}) R 27^{\circ}

4-

T, p_{O_{2}}, p_{CO}

5-

(Δ μ_{O}, Δ μ_{CO})

6-

p (2 \times 2)

7-

(Δ μ_{O}, Δ μ_{CO})

8-

(T, p_{i})

9-

(T, p_{O_{2}}, p_{CO})

10-

T = 300, 400, 600

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1303.2263

Formulas:

1-

u, v \in V (H)

2-

d_{H} (u, v) = 2

3-

\max {d (u), d (v)} \geq n / 2

4-

{K_{1, 3}, P_{7}, D}

5-

{K_{1, 3}, P_{7}, H}

6-

x y \in E (G)

7-

x, y \in V (G^{'})

8-

u, v \in V (G^{'})

9-

d_{G^{'}} (u, v) = 2

10-

m a x {d (u), d (v)} \geq n / 2

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1203.6883

Formulas:

1-

- O - Mo - O -

2-

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

3-

B_{s a t}

4-

d M / d B

5-

B_{s a t} / 3

6-

B_{s a t} / 3

7-

d M / d B

8-

B_{s a t} / 3

9-

H_{b a n d} = J \frac{D}{N} [{\vec{S}}^{2} - γ ({\vec{S}}_{A}^{2} + {\vec{S}}_{B}^{2} + {\vec{S}}_{C}^{2})],

10-

E (S, S_{A}, S_{B}, S_{C})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0111290

Formulas:

1-

σ_{m i n} (a) = 2.7 {(a / 5 A U)}^{- 3 / 2} g / c m^{2} .

2-

5 \times 10^{5} - 10^{6}

3-

σ (a) = σ_{0} {(a / 5 A U)}^{- α} .

4-

M = {(2 π)}^{3 / 2} {(\frac{2}{3 M_{⊙}})}^{1 / 2} p^{3 / 2} n^{3 / 2} σ^{3 / 2} a^{3}

5-

[a - n r_{H} / 2, a + n r_{H} / 2]

6-

σ_{0} = 10 g / c m^{2}

7-

\frac{d e_{M}^{2}}{d t} = C (m ⟨ e_{m}^{2} ⟩ - M e_{M}^{2}) K_{e}

8-

{⟨ e_{m}^{2} ⟩}^{1 / 2}

9-

C = \frac{16 G^{2} ρ_{s w}_{1} l n Λ}{v_{K}^{3} {(⟨ e_{1}^{2} ⟩ + ⟨ e_{2}^{2} ⟩)}^{3 / 2}},

10-

e_{M}^{2} M ≃ ⟨ e_{m}^{2} ⟩ m

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1510.00062

Formulas:

1-

(m \frac{d^{2}}{d t^{2}} - η_{1} \frac{d}{d t}) x - k (X - x) = f_{0} \exp [i ω t]

2-

(M \frac{d^{2}}{d t^{2}} + η_{2} \frac{d}{d t} + K) X - k (X - x) = 0,

3-

f_{0} \exp [i ω t]

4-

ω_{0}^{2} = K / M = k / m

5-

K X_{0} / f_{0}

6-

\bar{η} = η / (M ω_{0})

7-

η_{1} = η_{2} = η

8-

\bar{η} = η / (M ω_{0})

9-

η d X / d t

10-

\bar{η} K X_{0}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0503020

Formulas:

1-

(x_{0}, - a / 2, 0)

2-

(x_{0}, a / 2, 0)

3-

(x_{0}, - a / 2, 0)

4-

(x_{0} - a, a / 2, 0)

5-

{\tilde{u}}_{1} = a {(2 π)}^{- 1} [\tan^{- 1} (y / x) + x y / (2 (1 - ν) (x^{2} + y^{2}))]

6-

[u_{1}] = a

7-

x = x_{1} = l a

8-

y = x_{2} = m a

9-

z = x_{3} = n a

10-

{\tilde{u}}_{i} (x, y, z, t) = a u_{i} (l, m, n; t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1104.1686

Formulas:

1-

α ⪯_{k - 1}^{ℬ} β ⪯_{k - 1}^{ℬ} γ and α ⪯_{k}^{ℬ} γ ⟹ α ⪯_{k}^{ℬ} β

2-

ℬ ↾ α ⪯_{k}^{\infty} ℬ ↾ β

3-

⪯_{2}^{ℬ}, \subseteq, ⪯_{1}^{ℬ}, \subseteq, ⪯_{0}^{ℬ}

4-

h (α) < α

5-

φ (h (a)) < h^{+} (b)

6-

{(- \infty, a)}^{𝐏} ≺^{𝐏^{+}} b ≺^{𝐏^{+}} a

7-

h (a) ≺_{1}^{ℬ} h (b)

8-

h (a) ≺_{1}^{\infty} h (b)

9-

h [{(- \infty, a)}^{𝐏}] \cup {φ (h (a))} < \tilde{X} < h (b)

10-

h [{(- \infty, a)}^{𝐏}] \cup \tilde{X}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1105.0944

Formulas:

1-

B - V = 0.63 \pm 0.02

2-

V - R = 0.35 \pm 0.02

3-

R - I = 0.31 \pm 0.04

4-

m (1, 1, 0) = 15.35 \pm 0.05

5-

M_{d} \sim (7.2 \pm 3.6) \times 10^{4}

6-

P_{r o t} = 22.23 \pm 0.01

7-

1.8 < b / a < 2.1

8-

ν_{o} = 20 \pm 20 °

9-

180 ° ≲ α_{j e t} ≲ 120 °

10-

δ_{j e t} \approx - 60 °

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0902.2292

Formulas:

1-

10^{33} {cm}^{- 2} s^{- 1}

2-

E_{e} = 70 GeV

3-

10^{33} {cm}^{- 2} s^{- 1}

4-

5 \times 10^{31} {cm}^{- 2} s^{- 1}

5-

s_{_{LHeC}} / s_{_{HERA}} \sim 20

6-

Q^{2} \sim 10^{6} {GeV}^{2}

7-

x \overset{<}{\sim} 10^{- 4}

8-

2 - 200 {fb}^{- 1}

9-

E_{e} = 20 GeV

10-

E_{e} = 140 GeV

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ex

Result: correct

Paper: https://arxiv.org/abs/1811.10235

Formulas:

1-

k_{x}^{2} + k_{y}^{2} \leq {(ω / c)}^{2}

2-

(k_{x}, k_{y})

3-

k_{x}^{2} + k_{y}^{2} \leq {(ω / c)}^{2}

4-

k_{x}^{2} + k_{y}^{2} > {(ω / c)}^{2}

5-

Δ d / d = (d^{'} - d) / d

6-

Δ l / l = (l^{'} - l) / l

7-

Δ d / d = (d^{'} - d) / d

8-

2 π / 2 a = π / a

9-

E_{xt} = 1 - R

10-

k_{x}^{2} + k_{y}^{2} = {(ω / c)}^{2}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1408.2526

Formulas:

1-

\sim 10^{53} erg s^{- 1}

2-

θ_{jet} = 0.1 °

3-

12 + \log (O / H) > 8.7

4-

< 10^{8} M_{⊙}

5-

12 + \log (O / H) > 8.7

6-

10^{53} erg s^{- 1}

7-

3 < j / 10^{16} {cm}^{2} s^{- 1} < 20

8-

Z < 0.5 Z_{⊙}

9-

n_{H} = 10 {cm}^{- 3}

10-

Z_{crit} = 10^{- 4}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1609.00623

Formulas:

1-

w (r) \neq δ (r)

2-

w (r) = δ (r)

3-

f (r) e^{- r / ξ}

4-

g (r) e^{- r / ξ}

5-

ϵ \nabla^{2} ψ (𝐫) = - \sum_{i = 1}^{K} q_{i} ρ_{i} (𝐫)

6-

ρ_{i} (𝐫)

7-

w_{i} (𝐫 - 𝐫^{'})

8-

\int 𝑑 𝐫^{'} w_{i} (𝐫 - 𝐫^{'}) = 1

9-

ϵ \nabla^{2} ψ (𝐫) = - \sum_{i = 1}^{K} q_{i} \int 𝑑 𝐫^{'} ρ_{i} (𝐫^{'}) w_{i} (𝐫 - 𝐫^{'}),

10-

w_{i} (𝐫 - 𝐫^{'}) = δ (𝐫 - 𝐫^{'})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1408.4506

Formulas:

1-

M_{out} \sim 1.9 \times 10^{- 6} M_{⊙}

2-

\sim 2.9 \times 10^{- 5} M_{⊙}

3-

P = 8.7 \times 10^{- 6}

4-

E = 2.0 \times 10^{- 5}

5-

F = 1.2 \times 10^{- 8}

6-

L = 4.7 \times 10^{- 6}

7-

{\dot{M}}_{mol} = M_{out} / t_{dyn}

8-

M \geq v^{2} l / 2 G

9-

P = M_{out} v_{\max}

10-

{\dot{M}}_{wind} t_{dyn} v_{wind}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0709.2661

Formulas:

1-

M_{n, r} (𝐱)

2-

M_{n, r} (𝐪, 𝐱) = {(\sum_{i = 1}^{n} q_{i} x_{i}^{r})}^{\frac{1}{r}}

3-

𝐪 = (q_{1}, q_{2}, \dots, q_{n})

4-

𝐱 = (x_{1}, x_{2}, \dots, x_{n})

5-

q_{i} > 0, 1 \leq i \leq n

6-

\sum_{i = 1}^{n} q_{i} = 1

7-

M_{n, 0} (𝐪, 𝐱)

8-

M_{n, r} (𝐪, 𝐱)

9-

x_{i} > 0, 1 \leq i \leq n

10-

M_{n, r} (𝐪, 𝐱)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1612.09320

Formulas:

1-

\log M_{H} / M_{WD}

2-

\log M_{cvz} / M_{WD}

3-

M_{WD} = 0.632 M_{⊙}

4-

M_{WD} = 0.837 M_{⊙}

5-

\log (d M / d t) \approx 5.5

6-

d M / d t

7-

\log (d M / d t) \approx 8

8-

[X / H] = \log n (X) / n (H)

9-

\log (d M / d t) \approx 7.8

10-

\log (d M / d t) \approx 9.5

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1307.1981

Formulas:

1-

n ≢ 3, 7 (\mod 10)

2-

H H^{T} = n I

3-

H H^{T} \equiv n I (\mod m)

4-

MH (n, m)

5-

MH (n, 0)

6-

MH (n, m)

7-

m = 2, 3, 4

8-

MH (n, m)

9-

m \equiv 0 (\mod 4)

10-

n \equiv 0 (\mod 4)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1806.07034

Formulas:

1-

\frac{d^{2} X}{d s^{2}} + (\frac{p_{0}^{'}}{p_{0}}) \frac{d X}{d s} + k_{x}^{2} (s) X - \frac{K u_{0} π λ_{3}}{l^{2}} G_{311} (X, Y, u_{0} T) X -

2-

{(\frac{δ}{l p_{0}})}^{2} \frac{ϵ_{x}^{2}}{X^{3}} = 0

3-

\frac{d^{2} Y}{d s^{2}} + (\frac{p_{0}^{'}}{p_{0}}) \frac{d Y}{d s} + k_{y}^{2} (s) Y - \frac{K u_{0} π λ_{3}}{l^{2}} G_{131} (X, Y, u_{0} T) Y -

4-

{(\frac{δ}{l p_{0}})}^{2} \frac{ϵ_{y}^{2}}{Y^{3}} = 0

5-

\frac{d^{2} T}{d s^{2}} + 3 (\frac{p_{0}^{'}}{p_{0}}) \frac{d T}{d s} + k_{t}^{2} (s) T - \frac{K u_{0} π λ_{3}}{l^{2}} G_{113} (X, Y, u_{0} T) T -

6-

{(\frac{δ}{l p_{0} u_{0}^{2}})}^{2} \frac{ϵ_{t}^{2}}{T^{3}} = 0

7-

G_{l m n} (x, y, z)

8-

\frac{3}{2} \int_{0}^{\infty} \frac{d s}{{(x^{2} + s)}^{l / 2} {(y^{2} + s)}^{m / 2} {(z^{2} + s)}^{n / 2}}

9-

T = ω Δ t

10-

p_{0} = m c γ_{0} β_{0}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/0708.0664

Formulas:

1-

{\hat{ρ}}_{A} {\hat{ρ}}_{B}

2-

{\hat{ρ}}_{B} (λ) {\hat{ρ}}_{B} (λ)

3-

{\hat{ρ}}_{B} (λ) {\hat{ρ}}_{B} (λ)

4-

{\hat{ρ}}_{A B} = M [{\hat{ρ}}_{A} (λ) {\hat{ρ}}_{A} (λ)]

5-

M [\dots] = \int \dots p (λ) d λ .

6-

{\hat{ρ}}_{A} (𝐚) = \frac{1}{2} ({\hat{1}}_{A} + 𝐚 {\hat{𝝈}}_{A}), {\hat{ρ}}_{A} (𝐛) = \frac{1}{2} ({\hat{1}}_{A} + 𝐛 {\hat{𝝈}}_{B}),

7-

𝐚 = (a_{1}, a_{2}, a_{3})

8-

𝐛 = (b_{1}, b_{2}, b_{3})

9-

𝐚^{2}, 𝐛^{2} \leq 1

10-

λ = (𝐚, 𝐛)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/0807.4189

Formulas:

1-

V_{OMC} = - 40 km s^{- 1}

2-

- 60 km s^{- 1}

3-

V_{☉} = - 28 km s^{- 1}

4-

V_{T} = - 48 km s^{- 1}

5-

V_{T} = - 44 km s^{- 1}

6-

V_{☉} = - 8 km s^{- 1}

7-

V_{☉} = - 36 km s^{- 1}

8-

V_{OMC} = - 62 km s^{- 1}

9-

7.1 ″ \times 7.1 ″

10-

V_{T} = 48 km s^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0302110

Formulas:

1-

Δ E_{rot} = α G M_{*}^{2} Ω_{*} / 2 c \approx 10^{53} ergs

2-

t_{sd} = 4 (\frac{α}{0.5}) {(\frac{M_{*}}{2 M_{⊙}})}^{2} {(\frac{R_{*}}{15 km})}^{- 6} {(\frac{Ω_{*}}{10^{4} s^{- 1}})}^{- 3} {(\frac{B_{*}}{10^{12} G})}^{- 2} yr,

3-

\sim 10^{12} - 10^{13} G

4-

\sim > 10^{14} G

5-

1.2 - 1.6 M_{⊙}

6-

σ_{w} = B_{w}^{2} / 4 π ρ_{w} c^{2}

7-

M_{ej} \approx 10 M_{⊙}

8-

v_{b} \sim {(Δ E_{rot} / M_{ej})}^{1 / 2} \approx 0.1 c

9-

n_{H, equiv} = \frac{4 p + {(B + E)}^{2} / 4 π}{m_{p} c^{2}},

10-

n_{H, equiv} (r) \propto r^{- k}

Formulas (html):

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xref="S2.SS1.p2.4.m4.1.1.4.2.3.cmml">2</mn></mrow><mo id="S2.SS1.p2.4.m4.1.1.4.1" xref="S2.SS1.p2.4.m4.1.1.4.1.cmml">⁢</mo><mi id="S2.SS1.p2.4.m4.1.1.4.3" xref="S2.SS1.p2.4.m4.1.1.4.3.cmml">c</mi></mrow><mo id="S2.SS1.p2.4.m4.1.1.5" xref="S2.SS1.p2.4.m4.1.1.5.cmml">≈</mo><mrow id="S2.SS1.p2.4.m4.1.1.6" xref="S2.SS1.p2.4.m4.1.1.6.cmml"><mpadded width="+5pt" id="S2.SS1.p2.4.m4.1.1.6.2" xref="S2.SS1.p2.4.m4.1.1.6.2.cmml"><msup id="S2.SS1.p2.4.m4.1.1.6.2a" xref="S2.SS1.p2.4.m4.1.1.6.2.cmml"><mn id="S2.SS1.p2.4.m4.1.1.6.2.2" xref="S2.SS1.p2.4.m4.1.1.6.2.2.cmml">10</mn><mn id="S2.SS1.p2.4.m4.1.1.6.2.3" xref="S2.SS1.p2.4.m4.1.1.6.2.3.cmml">53</mn></msup></mpadded><mo id="S2.SS1.p2.4.m4.1.1.6.1" xref="S2.SS1.p2.4.m4.1.1.6.1.cmml">⁢</mo><mi id="S2.SS1.p2.4.m4.1.1.6.3" xref="S2.SS1.p2.4.m4.1.1.6.3.cmml">ergs</mi></mrow></mrow></math>, <math><mrow id="S2.Ex1.m1.6.6.1" xref="S2.Ex1.m1.6.6.1.1.cmml"><mrow id="S2.Ex1.m1.6.6.1.1" xref="S2.Ex1.m1.6.6.1.1.cmml"><msub id="S2.Ex1.m1.6.6.1.1.2" xref="S2.Ex1.m1.6.6.1.1.2.cmml"><mi id="S2.Ex1.m1.6.6.1.1.2.2" xref="S2.Ex1.m1.6.6.1.1.2.2.cmml">t</mi><mi id="S2.Ex1.m1.6.6.1.1.2.3" xref="S2.Ex1.m1.6.6.1.1.2.3.cmml">sd</mi></msub><mo id="S2.Ex1.m1.6.6.1.1.1" xref="S2.Ex1.m1.6.6.1.1.1.cmml">=</mo><mrow id="S2.Ex1.m1.6.6.1.1.3" xref="S2.Ex1.m1.6.6.1.1.3.cmml"><mn id="S2.Ex1.m1.6.6.1.1.3.2" xref="S2.Ex1.m1.6.6.1.1.3.2.cmml">4</mn><mo id="S2.Ex1.m1.6.6.1.1.3.1" xref="S2.Ex1.m1.6.6.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.6.6.1.1.3.3.2" xref="S2.Ex1.m1.1.1.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.3.2.1" xref="S2.Ex1.m1.1.1.cmml">(</mo><mfrac id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.2" xref="S2.Ex1.m1.1.1.2.cmml">α</mi><mn id="S2.Ex1.m1.1.1.3" xref="S2.Ex1.m1.1.1.3.cmml">0.5</mn></mfrac><mo id="S2.Ex1.m1.6.6.1.1.3.3.2.2" xref="S2.Ex1.m1.1.1.cmml">)</mo></mrow><mo id="S2.Ex1.m1.6.6.1.1.3.1a" xref="S2.Ex1.m1.6.6.1.1.3.1.cmml">⁢</mo><msup id="S2.Ex1.m1.6.6.1.1.3.4" xref="S2.Ex1.m1.6.6.1.1.3.4.cmml"><mrow id="S2.Ex1.m1.6.6.1.1.3.4.2.2" xref="S2.Ex1.m1.2.2.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.4.2.2.1" xref="S2.Ex1.m1.2.2.cmml">(</mo><mfrac id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml"><msub id="S2.Ex1.m1.2.2.2" xref="S2.Ex1.m1.2.2.2.cmml"><mi id="S2.Ex1.m1.2.2.2.2" xref="S2.Ex1.m1.2.2.2.2.cmml">M</mi><mo id="S2.Ex1.m1.2.2.2.3" xref="S2.Ex1.m1.2.2.2.3.cmml">*</mo></msub><mrow id="S2.Ex1.m1.2.2.3" xref="S2.Ex1.m1.2.2.3.cmml"><mpadded width="+1.7pt" id="S2.Ex1.m1.2.2.3.2" xref="S2.Ex1.m1.2.2.3.2.cmml"><mn id="S2.Ex1.m1.2.2.3.2a" xref="S2.Ex1.m1.2.2.3.2.cmml">2</mn></mpadded><mo id="S2.Ex1.m1.2.2.3.1" xref="S2.Ex1.m1.2.2.3.1.cmml">⁢</mo><msub id="S2.Ex1.m1.2.2.3.3" xref="S2.Ex1.m1.2.2.3.3.cmml"><mi id="S2.Ex1.m1.2.2.3.3.2" xref="S2.Ex1.m1.2.2.3.3.2.cmml">M</mi><mo id="S2.Ex1.m1.2.2.3.3.3" xref="S2.Ex1.m1.2.2.3.3.3.cmml">⊙</mo></msub></mrow></mfrac><mo id="S2.Ex1.m1.6.6.1.1.3.4.2.2.2" xref="S2.Ex1.m1.2.2.cmml">)</mo></mrow><mn id="S2.Ex1.m1.6.6.1.1.3.4.3" xref="S2.Ex1.m1.6.6.1.1.3.4.3.cmml">2</mn></msup><mo id="S2.Ex1.m1.6.6.1.1.3.1b" xref="S2.Ex1.m1.6.6.1.1.3.1.cmml">⁢</mo><msup id="S2.Ex1.m1.6.6.1.1.3.5" xref="S2.Ex1.m1.6.6.1.1.3.5.cmml"><mrow id="S2.Ex1.m1.6.6.1.1.3.5.2.2" xref="S2.Ex1.m1.3.3.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.5.2.2.1" xref="S2.Ex1.m1.3.3.cmml">(</mo><mfrac id="S2.Ex1.m1.3.3" xref="S2.Ex1.m1.3.3.cmml"><msub id="S2.Ex1.m1.3.3.2" xref="S2.Ex1.m1.3.3.2.cmml"><mi id="S2.Ex1.m1.3.3.2.2" xref="S2.Ex1.m1.3.3.2.2.cmml">R</mi><mo id="S2.Ex1.m1.3.3.2.3" xref="S2.Ex1.m1.3.3.2.3.cmml">*</mo></msub><mrow id="S2.Ex1.m1.3.3.3" xref="S2.Ex1.m1.3.3.3.cmml"><mpadded width="+5pt" id="S2.Ex1.m1.3.3.3.2" xref="S2.Ex1.m1.3.3.3.2.cmml"><mn id="S2.Ex1.m1.3.3.3.2a" xref="S2.Ex1.m1.3.3.3.2.cmml">15</mn></mpadded><mo id="S2.Ex1.m1.3.3.3.1" xref="S2.Ex1.m1.3.3.3.1.cmml">⁢</mo><mi id="S2.Ex1.m1.3.3.3.3" xref="S2.Ex1.m1.3.3.3.3.cmml">km</mi></mrow></mfrac><mo id="S2.Ex1.m1.6.6.1.1.3.5.2.2.2" xref="S2.Ex1.m1.3.3.cmml">)</mo></mrow><mrow id="S2.Ex1.m1.6.6.1.1.3.5.3" xref="S2.Ex1.m1.6.6.1.1.3.5.3.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.5.3.1" xref="S2.Ex1.m1.6.6.1.1.3.5.3.1.cmml">-</mo><mn id="S2.Ex1.m1.6.6.1.1.3.5.3.2" xref="S2.Ex1.m1.6.6.1.1.3.5.3.2.cmml">6</mn></mrow></msup><mo id="S2.Ex1.m1.6.6.1.1.3.1c" xref="S2.Ex1.m1.6.6.1.1.3.1.cmml">⁢</mo><msup id="S2.Ex1.m1.6.6.1.1.3.6" xref="S2.Ex1.m1.6.6.1.1.3.6.cmml"><mrow id="S2.Ex1.m1.6.6.1.1.3.6.2.2" xref="S2.Ex1.m1.4.4.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.6.2.2.1" xref="S2.Ex1.m1.4.4.cmml">(</mo><mfrac id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.4.4.cmml"><msub id="S2.Ex1.m1.4.4.2" xref="S2.Ex1.m1.4.4.2.cmml"><mi mathvariant="normal" id="S2.Ex1.m1.4.4.2.2" xref="S2.Ex1.m1.4.4.2.2.cmml">Ω</mi><mo id="S2.Ex1.m1.4.4.2.3" xref="S2.Ex1.m1.4.4.2.3.cmml">*</mo></msub><mrow id="S2.Ex1.m1.4.4.3" xref="S2.Ex1.m1.4.4.3.cmml"><mpadded width="+5pt" id="S2.Ex1.m1.4.4.3.2" xref="S2.Ex1.m1.4.4.3.2.cmml"><msup id="S2.Ex1.m1.4.4.3.2a" xref="S2.Ex1.m1.4.4.3.2.cmml"><mn id="S2.Ex1.m1.4.4.3.2.2" xref="S2.Ex1.m1.4.4.3.2.2.cmml">10</mn><mn id="S2.Ex1.m1.4.4.3.2.3" xref="S2.Ex1.m1.4.4.3.2.3.cmml">4</mn></msup></mpadded><mo id="S2.Ex1.m1.4.4.3.1" xref="S2.Ex1.m1.4.4.3.1.cmml">⁢</mo><msup id="S2.Ex1.m1.4.4.3.3" xref="S2.Ex1.m1.4.4.3.3.cmml"><mi mathvariant="normal" id="S2.Ex1.m1.4.4.3.3.2" xref="S2.Ex1.m1.4.4.3.3.2.cmml">s</mi><mrow id="S2.Ex1.m1.4.4.3.3.3" xref="S2.Ex1.m1.4.4.3.3.3.cmml"><mo id="S2.Ex1.m1.4.4.3.3.3.1" xref="S2.Ex1.m1.4.4.3.3.3.1.cmml">-</mo><mn id="S2.Ex1.m1.4.4.3.3.3.2" xref="S2.Ex1.m1.4.4.3.3.3.2.cmml">1</mn></mrow></msup></mrow></mfrac><mo id="S2.Ex1.m1.6.6.1.1.3.6.2.2.2" xref="S2.Ex1.m1.4.4.cmml">)</mo></mrow><mrow id="S2.Ex1.m1.6.6.1.1.3.6.3" xref="S2.Ex1.m1.6.6.1.1.3.6.3.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.6.3.1" xref="S2.Ex1.m1.6.6.1.1.3.6.3.1.cmml">-</mo><mn id="S2.Ex1.m1.6.6.1.1.3.6.3.2" xref="S2.Ex1.m1.6.6.1.1.3.6.3.2.cmml">3</mn></mrow></msup><mo id="S2.Ex1.m1.6.6.1.1.3.1d" xref="S2.Ex1.m1.6.6.1.1.3.1.cmml">⁢</mo><mpadded width="+5pt" id="S2.Ex1.m1.6.6.1.1.3.7" xref="S2.Ex1.m1.6.6.1.1.3.7.cmml"><msup id="S2.Ex1.m1.6.6.1.1.3.7a" xref="S2.Ex1.m1.6.6.1.1.3.7.cmml"><mrow id="S2.Ex1.m1.6.6.1.1.3.7.2.2" xref="S2.Ex1.m1.5.5.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.7.2.2.1" xref="S2.Ex1.m1.5.5.cmml">(</mo><mfrac id="S2.Ex1.m1.5.5" xref="S2.Ex1.m1.5.5.cmml"><msub id="S2.Ex1.m1.5.5.2" xref="S2.Ex1.m1.5.5.2.cmml"><mi id="S2.Ex1.m1.5.5.2.2" xref="S2.Ex1.m1.5.5.2.2.cmml">B</mi><mo id="S2.Ex1.m1.5.5.2.3" xref="S2.Ex1.m1.5.5.2.3.cmml">*</mo></msub><mrow id="S2.Ex1.m1.5.5.3" xref="S2.Ex1.m1.5.5.3.cmml"><mpadded width="+5pt" id="S2.Ex1.m1.5.5.3.2" xref="S2.Ex1.m1.5.5.3.2.cmml"><msup id="S2.Ex1.m1.5.5.3.2a" xref="S2.Ex1.m1.5.5.3.2.cmml"><mn id="S2.Ex1.m1.5.5.3.2.2" xref="S2.Ex1.m1.5.5.3.2.2.cmml">10</mn><mn id="S2.Ex1.m1.5.5.3.2.3" xref="S2.Ex1.m1.5.5.3.2.3.cmml">12</mn></msup></mpadded><mo id="S2.Ex1.m1.5.5.3.1" xref="S2.Ex1.m1.5.5.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.Ex1.m1.5.5.3.3" xref="S2.Ex1.m1.5.5.3.3.cmml">G</mi></mrow></mfrac><mo id="S2.Ex1.m1.6.6.1.1.3.7.2.2.2" xref="S2.Ex1.m1.5.5.cmml">)</mo></mrow><mrow id="S2.Ex1.m1.6.6.1.1.3.7.3" xref="S2.Ex1.m1.6.6.1.1.3.7.3.cmml"><mo id="S2.Ex1.m1.6.6.1.1.3.7.3.1" xref="S2.Ex1.m1.6.6.1.1.3.7.3.1.cmml">-</mo><mn id="S2.Ex1.m1.6.6.1.1.3.7.3.2" xref="S2.Ex1.m1.6.6.1.1.3.7.3.2.cmml">2</mn></mrow></msup></mpadded><mo id="S2.Ex1.m1.6.6.1.1.3.1e" xref="S2.Ex1.m1.6.6.1.1.3.1.cmml">⁢</mo><mpadded width="+5pt" id="S2.Ex1.m1.6.6.1.1.3.8" xref="S2.Ex1.m1.6.6.1.1.3.8.cmml"><mi id="S2.Ex1.m1.6.6.1.1.3.8a" xref="S2.Ex1.m1.6.6.1.1.3.8.cmml">yr</mi></mpadded></mrow></mrow><mo id="S2.Ex1.m1.6.6.1.2" xref="S2.Ex1.m1.6.6.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p2.9.m4.1.1" xref="S2.SS1.p2.9.m4.1.1.cmml"><mi id="S2.SS1.p2.9.m4.1.1.2" xref="S2.SS1.p2.9.m4.1.1.2.cmml"/><mo id="S2.SS1.p2.9.m4.1.1.1" xref="S2.SS1.p2.9.m4.1.1.1.cmml">∼</mo><mrow id="S2.SS1.p2.9.m4.1.1.3" xref="S2.SS1.p2.9.m4.1.1.3.cmml"><msup id="S2.SS1.p2.9.m4.1.1.3.2" xref="S2.SS1.p2.9.m4.1.1.3.2.cmml"><mn id="S2.SS1.p2.9.m4.1.1.3.2.2" xref="S2.SS1.p2.9.m4.1.1.3.2.2.cmml">10</mn><mn id="S2.SS1.p2.9.m4.1.1.3.2.3" xref="S2.SS1.p2.9.m4.1.1.3.2.3.cmml">12</mn></msup><mo id="S2.SS1.p2.9.m4.1.1.3.1" xref="S2.SS1.p2.9.m4.1.1.3.1.cmml">-</mo><mrow id="S2.SS1.p2.9.m4.1.1.3.3" xref="S2.SS1.p2.9.m4.1.1.3.3.cmml"><mpadded width="+5pt" id="S2.SS1.p2.9.m4.1.1.3.3.2" xref="S2.SS1.p2.9.m4.1.1.3.3.2.cmml"><msup id="S2.SS1.p2.9.m4.1.1.3.3.2a" xref="S2.SS1.p2.9.m4.1.1.3.3.2.cmml"><mn id="S2.SS1.p2.9.m4.1.1.3.3.2.2" xref="S2.SS1.p2.9.m4.1.1.3.3.2.2.cmml">10</mn><mn id="S2.SS1.p2.9.m4.1.1.3.3.2.3" xref="S2.SS1.p2.9.m4.1.1.3.3.2.3.cmml">13</mn></msup></mpadded><mo id="S2.SS1.p2.9.m4.1.1.3.3.1" xref="S2.SS1.p2.9.m4.1.1.3.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.SS1.p2.9.m4.1.1.3.3.3" xref="S2.SS1.p2.9.m4.1.1.3.3.3.cmml">G</mi></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p3.1.m1.2.3" xref="S2.SS1.p3.1.m1.2.3.cmml"><mi id="S2.SS1.p3.1.m1.2.3.1" xref="S2.SS1.p3.1.m1.2.3.1.cmml"/><mrow id="S2.SS1.p3.1.m1.2.2.2.2" xref="S2.SS1.p3.1.m1.2.2.2.2b.cmml"><mpadded depth="+4.0pt" height="-4.0pt" voffset="-4.0pt" width="0.0pt" id="S2.SS1.p3.1.m1.2.2.2.2a" xref="S2.SS1.p3.1.m1.2.2.2.2b.cmml"><mo id="S2.SS1.p3.1.m1.1.1.1.1.1.1.1.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.1.1.1.1.1.m1.1.1.cmml">∼</mo></mpadded><mo id="S2.SS1.p3.1.m1.2.2.2.2.2.m1.1.1" xref="S2.SS1.p3.1.m1.2.2.2.2.2.m1.1.1.cmml">></mo></mrow><mrow id="S2.SS1.p3.1.m1.2.3.2" xref="S2.SS1.p3.1.m1.2.3.2.cmml"><mpadded width="+5pt" id="S2.SS1.p3.1.m1.2.3.2.2" xref="S2.SS1.p3.1.m1.2.3.2.2.cmml"><msup id="S2.SS1.p3.1.m1.2.3.2.2a" xref="S2.SS1.p3.1.m1.2.3.2.2.cmml"><mn id="S2.SS1.p3.1.m1.2.3.2.2.2" xref="S2.SS1.p3.1.m1.2.3.2.2.2.cmml">10</mn><mn id="S2.SS1.p3.1.m1.2.3.2.2.3" xref="S2.SS1.p3.1.m1.2.3.2.2.3.cmml">14</mn></msup></mpadded><mo id="S2.SS1.p3.1.m1.2.3.2.1" xref="S2.SS1.p3.1.m1.2.3.2.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.SS1.p3.1.m1.2.3.2.3" xref="S2.SS1.p3.1.m1.2.3.2.3.cmml">G</mi></mrow></mrow></math>, <math><mrow id="S2.SS2.p1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.cmml"><mn id="S2.SS2.p1.1.m1.1.1.2" xref="S2.SS2.p1.1.m1.1.1.2.cmml">1.2</mn><mo id="S2.SS2.p1.1.m1.1.1.1" xref="S2.SS2.p1.1.m1.1.1.1.cmml">-</mo><mrow id="S2.SS2.p1.1.m1.1.1.3" xref="S2.SS2.p1.1.m1.1.1.3.cmml"><mpadded width="+1.7pt" id="S2.SS2.p1.1.m1.1.1.3.2" xref="S2.SS2.p1.1.m1.1.1.3.2.cmml"><mn id="S2.SS2.p1.1.m1.1.1.3.2a" xref="S2.SS2.p1.1.m1.1.1.3.2.cmml">1.6</mn></mpadded><mo id="S2.SS2.p1.1.m1.1.1.3.1" xref="S2.SS2.p1.1.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.SS2.p1.1.m1.1.1.3.3" xref="S2.SS2.p1.1.m1.1.1.3.3.cmml"><mi id="S2.SS2.p1.1.m1.1.1.3.3.2" xref="S2.SS2.p1.1.m1.1.1.3.3.2.cmml">M</mi><mo id="S2.SS2.p1.1.m1.1.1.3.3.3" xref="S2.SS2.p1.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><msub id="S3.p2.4.m4.1.1.2" xref="S3.p2.4.m4.1.1.2.cmml"><mi id="S3.p2.4.m4.1.1.2.2" xref="S3.p2.4.m4.1.1.2.2.cmml">σ</mi><mi id="S3.p2.4.m4.1.1.2.3" xref="S3.p2.4.m4.1.1.2.3.cmml">w</mi></msub><mo id="S3.p2.4.m4.1.1.1" xref="S3.p2.4.m4.1.1.1.cmml">=</mo><mrow id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml"><mrow id="S3.p2.4.m4.1.1.3.2" xref="S3.p2.4.m4.1.1.3.2.cmml"><msubsup id="S3.p2.4.m4.1.1.3.2.2" xref="S3.p2.4.m4.1.1.3.2.2.cmml"><mi id="S3.p2.4.m4.1.1.3.2.2.2.2" xref="S3.p2.4.m4.1.1.3.2.2.2.2.cmml">B</mi><mi id="S3.p2.4.m4.1.1.3.2.2.2.3" xref="S3.p2.4.m4.1.1.3.2.2.2.3.cmml">w</mi><mn id="S3.p2.4.m4.1.1.3.2.2.3" xref="S3.p2.4.m4.1.1.3.2.2.3.cmml">2</mn></msubsup><mo id="S3.p2.4.m4.1.1.3.2.1" xref="S3.p2.4.m4.1.1.3.2.1.cmml">/</mo><mn id="S3.p2.4.m4.1.1.3.2.3" xref="S3.p2.4.m4.1.1.3.2.3.cmml">4</mn></mrow><mo id="S3.p2.4.m4.1.1.3.1" xref="S3.p2.4.m4.1.1.3.1.cmml">⁢</mo><mi id="S3.p2.4.m4.1.1.3.3" xref="S3.p2.4.m4.1.1.3.3.cmml">π</mi><mo id="S3.p2.4.m4.1.1.3.1a" xref="S3.p2.4.m4.1.1.3.1.cmml">⁢</mo><msub id="S3.p2.4.m4.1.1.3.4" xref="S3.p2.4.m4.1.1.3.4.cmml"><mi id="S3.p2.4.m4.1.1.3.4.2" xref="S3.p2.4.m4.1.1.3.4.2.cmml">ρ</mi><mi id="S3.p2.4.m4.1.1.3.4.3" xref="S3.p2.4.m4.1.1.3.4.3.cmml">w</mi></msub><mo id="S3.p2.4.m4.1.1.3.1b" xref="S3.p2.4.m4.1.1.3.1.cmml">⁢</mo><msup id="S3.p2.4.m4.1.1.3.5" xref="S3.p2.4.m4.1.1.3.5.cmml"><mi id="S3.p2.4.m4.1.1.3.5.2" xref="S3.p2.4.m4.1.1.3.5.2.cmml">c</mi><mn id="S3.p2.4.m4.1.1.3.5.3" xref="S3.p2.4.m4.1.1.3.5.3.cmml">2</mn></msup></mrow></mrow></math>, <math><mrow id="S3.p3.6.m6.1.1" xref="S3.p3.6.m6.1.1.cmml"><msub id="S3.p3.6.m6.1.1.2" xref="S3.p3.6.m6.1.1.2.cmml"><mi id="S3.p3.6.m6.1.1.2.2" xref="S3.p3.6.m6.1.1.2.2.cmml">M</mi><mi id="S3.p3.6.m6.1.1.2.3" xref="S3.p3.6.m6.1.1.2.3.cmml">ej</mi></msub><mo id="S3.p3.6.m6.1.1.1" xref="S3.p3.6.m6.1.1.1.cmml">≈</mo><mrow id="S3.p3.6.m6.1.1.3" xref="S3.p3.6.m6.1.1.3.cmml"><mn id="S3.p3.6.m6.1.1.3.2" xref="S3.p3.6.m6.1.1.3.2.cmml">10</mn><mo id="S3.p3.6.m6.1.1.3.1" xref="S3.p3.6.m6.1.1.3.1.cmml">⁢</mo><msub id="S3.p3.6.m6.1.1.3.3" xref="S3.p3.6.m6.1.1.3.3.cmml"><mi id="S3.p3.6.m6.1.1.3.3.2" xref="S3.p3.6.m6.1.1.3.3.2.cmml">M</mi><mo id="S3.p3.6.m6.1.1.3.3.3" xref="S3.p3.6.m6.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S3.p3.7.m7.1.1" xref="S3.p3.7.m7.1.1.cmml"><msub id="S3.p3.7.m7.1.1.3" xref="S3.p3.7.m7.1.1.3.cmml"><mi id="S3.p3.7.m7.1.1.3.2" xref="S3.p3.7.m7.1.1.3.2.cmml">v</mi><mi id="S3.p3.7.m7.1.1.3.3" xref="S3.p3.7.m7.1.1.3.3.cmml">b</mi></msub><mo id="S3.p3.7.m7.1.1.4" xref="S3.p3.7.m7.1.1.4.cmml">∼</mo><msup id="S3.p3.7.m7.1.1.1" xref="S3.p3.7.m7.1.1.1.cmml"><mrow id="S3.p3.7.m7.1.1.1.1.1" xref="S3.p3.7.m7.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.p3.7.m7.1.1.1.1.1.2" xref="S3.p3.7.m7.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.p3.7.m7.1.1.1.1.1.1" xref="S3.p3.7.m7.1.1.1.1.1.1.cmml"><mrow id="S3.p3.7.m7.1.1.1.1.1.1.2" xref="S3.p3.7.m7.1.1.1.1.1.1.2.cmml"><mi mathvariant="normal" id="S3.p3.7.m7.1.1.1.1.1.1.2.2" xref="S3.p3.7.m7.1.1.1.1.1.1.2.2.cmml">Δ</mi><mo id="S3.p3.7.m7.1.1.1.1.1.1.2.1" xref="S3.p3.7.m7.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S3.p3.7.m7.1.1.1.1.1.1.2.3" xref="S3.p3.7.m7.1.1.1.1.1.1.2.3.cmml"><mi id="S3.p3.7.m7.1.1.1.1.1.1.2.3.2" xref="S3.p3.7.m7.1.1.1.1.1.1.2.3.2.cmml">E</mi><mi id="S3.p3.7.m7.1.1.1.1.1.1.2.3.3" xref="S3.p3.7.m7.1.1.1.1.1.1.2.3.3.cmml">rot</mi></msub></mrow><mo id="S3.p3.7.m7.1.1.1.1.1.1.1" xref="S3.p3.7.m7.1.1.1.1.1.1.1.cmml">/</mo><msub id="S3.p3.7.m7.1.1.1.1.1.1.3" xref="S3.p3.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S3.p3.7.m7.1.1.1.1.1.1.3.2" xref="S3.p3.7.m7.1.1.1.1.1.1.3.2.cmml">M</mi><mi id="S3.p3.7.m7.1.1.1.1.1.1.3.3" xref="S3.p3.7.m7.1.1.1.1.1.1.3.3.cmml">ej</mi></msub></mrow><mo stretchy="false" id="S3.p3.7.m7.1.1.1.1.1.3" xref="S3.p3.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.p3.7.m7.1.1.1.3" xref="S3.p3.7.m7.1.1.1.3.cmml"><mn id="S3.p3.7.m7.1.1.1.3.2" xref="S3.p3.7.m7.1.1.1.3.2.cmml">1</mn><mo id="S3.p3.7.m7.1.1.1.3.1" xref="S3.p3.7.m7.1.1.1.3.1.cmml">/</mo><mn id="S3.p3.7.m7.1.1.1.3.3" xref="S3.p3.7.m7.1.1.1.3.3.cmml">2</mn></mrow></msup><mo id="S3.p3.7.m7.1.1.5" xref="S3.p3.7.m7.1.1.5.cmml">≈</mo><mrow id="S3.p3.7.m7.1.1.6" xref="S3.p3.7.m7.1.1.6.cmml"><mn id="S3.p3.7.m7.1.1.6.2" xref="S3.p3.7.m7.1.1.6.2.cmml">0.1</mn><mo id="S3.p3.7.m7.1.1.6.1" xref="S3.p3.7.m7.1.1.6.1.cmml">⁢</mo><mi id="S3.p3.7.m7.1.1.6.3" xref="S3.p3.7.m7.1.1.6.3.cmml">c</mi></mrow></mrow></math>, <math><mrow id="S3.Ex2.m1.4.4.1" xref="S3.Ex2.m1.4.4.1.1.cmml"><mrow id="S3.Ex2.m1.4.4.1.1" xref="S3.Ex2.m1.4.4.1.1.cmml"><msub id="S3.Ex2.m1.4.4.1.1.2" xref="S3.Ex2.m1.4.4.1.1.2.cmml"><mi id="S3.Ex2.m1.4.4.1.1.2.2" xref="S3.Ex2.m1.4.4.1.1.2.2.cmml">n</mi><mrow id="S3.Ex2.m1.2.2.2.4" xref="S3.Ex2.m1.2.2.2.3.cmml"><mi mathvariant="normal" id="S3.Ex2.m1.1.1.1.1" xref="S3.Ex2.m1.1.1.1.1.cmml">H</mi><mo id="S3.Ex2.m1.2.2.2.4.1" xref="S3.Ex2.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex2.m1.2.2.2.2" xref="S3.Ex2.m1.2.2.2.2.cmml">equiv</mi></mrow></msub><mo id="S3.Ex2.m1.4.4.1.1.1" xref="S3.Ex2.m1.4.4.1.1.1.cmml">=</mo><mpadded width="+1.7pt" id="S3.Ex2.m1.3.3" xref="S3.Ex2.m1.3.3.cmml"><mfrac id="S3.Ex2.m1.3.3a" xref="S3.Ex2.m1.3.3.cmml"><mrow id="S3.Ex2.m1.3.3.1" xref="S3.Ex2.m1.3.3.1.cmml"><mrow id="S3.Ex2.m1.3.3.1.3" xref="S3.Ex2.m1.3.3.1.3.cmml"><mn id="S3.Ex2.m1.3.3.1.3.2" xref="S3.Ex2.m1.3.3.1.3.2.cmml">4</mn><mo id="S3.Ex2.m1.3.3.1.3.1" xref="S3.Ex2.m1.3.3.1.3.1.cmml">⁢</mo><mi id="S3.Ex2.m1.3.3.1.3.3" xref="S3.Ex2.m1.3.3.1.3.3.cmml">p</mi></mrow><mo id="S3.Ex2.m1.3.3.1.2" xref="S3.Ex2.m1.3.3.1.2.cmml">+</mo><mrow id="S3.Ex2.m1.3.3.1.1" xref="S3.Ex2.m1.3.3.1.1.cmml"><mrow id="S3.Ex2.m1.3.3.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.cmml"><msup id="S3.Ex2.m1.3.3.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.cmml"><mrow id="S3.Ex2.m1.3.3.1.1.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.Ex2.m1.3.3.1.1.1.1.1.1.2" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex2.m1.3.3.1.1.1.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.2" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.2.cmml">B</mi><mo id="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.1" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.3" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.3.cmml">E</mi></mrow><mo stretchy="false" id="S3.Ex2.m1.3.3.1.1.1.1.1.1.3" xref="S3.Ex2.m1.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S3.Ex2.m1.3.3.1.1.1.1.3" xref="S3.Ex2.m1.3.3.1.1.1.1.3.cmml">2</mn></msup><mo id="S3.Ex2.m1.3.3.1.1.1.2" xref="S3.Ex2.m1.3.3.1.1.1.2.cmml">/</mo><mn id="S3.Ex2.m1.3.3.1.1.1.3" xref="S3.Ex2.m1.3.3.1.1.1.3.cmml">4</mn></mrow><mo id="S3.Ex2.m1.3.3.1.1.2" xref="S3.Ex2.m1.3.3.1.1.2.cmml">⁢</mo><mi id="S3.Ex2.m1.3.3.1.1.3" xref="S3.Ex2.m1.3.3.1.1.3.cmml">π</mi></mrow></mrow><mrow id="S3.Ex2.m1.3.3.3" xref="S3.Ex2.m1.3.3.3.cmml"><msub id="S3.Ex2.m1.3.3.3.2" xref="S3.Ex2.m1.3.3.3.2.cmml"><mi id="S3.Ex2.m1.3.3.3.2.2" xref="S3.Ex2.m1.3.3.3.2.2.cmml">m</mi><mi mathvariant="normal" id="S3.Ex2.m1.3.3.3.2.3" xref="S3.Ex2.m1.3.3.3.2.3.cmml">p</mi></msub><mo id="S3.Ex2.m1.3.3.3.1" xref="S3.Ex2.m1.3.3.3.1.cmml">⁢</mo><msup id="S3.Ex2.m1.3.3.3.3" xref="S3.Ex2.m1.3.3.3.3.cmml"><mi id="S3.Ex2.m1.3.3.3.3.2" xref="S3.Ex2.m1.3.3.3.3.2.cmml">c</mi><mn id="S3.Ex2.m1.3.3.3.3.3" xref="S3.Ex2.m1.3.3.3.3.3.cmml">2</mn></msup></mrow></mfrac></mpadded></mrow><mo id="S3.Ex2.m1.4.4.1.2" xref="S3.Ex2.m1.4.4.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S3.p4.6.m4.3.4" xref="S3.p4.6.m4.3.4.cmml"><mrow id="S3.p4.6.m4.3.4.2" xref="S3.p4.6.m4.3.4.2.cmml"><msub id="S3.p4.6.m4.3.4.2.2" xref="S3.p4.6.m4.3.4.2.2.cmml"><mi id="S3.p4.6.m4.3.4.2.2.2" xref="S3.p4.6.m4.3.4.2.2.2.cmml">n</mi><mrow id="S3.p4.6.m4.2.2.2.4" xref="S3.p4.6.m4.2.2.2.3.cmml"><mi mathvariant="normal" id="S3.p4.6.m4.1.1.1.1" xref="S3.p4.6.m4.1.1.1.1.cmml">H</mi><mo id="S3.p4.6.m4.2.2.2.4.1" xref="S3.p4.6.m4.2.2.2.3.cmml">,</mo><mi id="S3.p4.6.m4.2.2.2.2" xref="S3.p4.6.m4.2.2.2.2.cmml">equiv</mi></mrow></msub><mo id="S3.p4.6.m4.3.4.2.1" xref="S3.p4.6.m4.3.4.2.1.cmml">⁢</mo><mrow id="S3.p4.6.m4.3.4.2.3.2" xref="S3.p4.6.m4.3.4.2.cmml"><mo stretchy="false" id="S3.p4.6.m4.3.4.2.3.2.1" xref="S3.p4.6.m4.3.4.2.cmml">(</mo><mi id="S3.p4.6.m4.3.3" xref="S3.p4.6.m4.3.3.cmml">r</mi><mo stretchy="false" id="S3.p4.6.m4.3.4.2.3.2.2" xref="S3.p4.6.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S3.p4.6.m4.3.4.1" xref="S3.p4.6.m4.3.4.1.cmml">∝</mo><msup id="S3.p4.6.m4.3.4.3" xref="S3.p4.6.m4.3.4.3.cmml"><mi id="S3.p4.6.m4.3.4.3.2" xref="S3.p4.6.m4.3.4.3.2.cmml">r</mi><mrow id="S3.p4.6.m4.3.4.3.3" xref="S3.p4.6.m4.3.4.3.3.cmml"><mo id="S3.p4.6.m4.3.4.3.3.1" xref="S3.p4.6.m4.3.4.3.3.1.cmml">-</mo><mi id="S3.p4.6.m4.3.4.3.3.2" xref="S3.p4.6.m4.3.4.3.3.2.cmml">k</mi></mrow></msup></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/quant-ph/9505006

Formulas:

1-

W^{2} - d W / d ρ = U_{e f f}^{D O}

2-

W_{1} = \frac{l}{ρ} - \frac{(2 l + 1)}{ρ (1 + ρ^{2 κ})}

3-

W = 𝒱^{- 1} + W_{1}

4-

\frac{d 𝒱}{d ρ} + 2 W_{1} 𝒱 = - 1 .

5-

\frac{d}{d ρ} [𝒱 \exp (2 \int W_{1} 𝑑 ρ)] = - \exp (2 \int W_{1} 𝑑 ρ),

6-

𝒱 = - \exp (- 2 \int W_{1} 𝑑 ρ) \cdot \int \exp (2 \int W_{1} 𝑑 ρ) 𝑑 ρ .

7-

W_{1} = - \frac{d}{d ρ} \ln f (ρ)

8-

\int W_{1} 𝑑 ρ = - \ln f (ρ)

9-

𝒱 = - f^{2} (ρ) \int f^{- 2} (ρ) 𝑑 ρ .

10-

\int \frac{{(ρ^{- κ} + ρ^{κ})}^{\frac{(2 l + 1)}{κ}}}{ρ} 𝑑 ρ .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/2012.13857

Formulas:

1-

I n t r o d u c t i o n

2-

C a l c u l a t i o n m e t h o d

3-

C r y s t a l s t u r c t u r e

4-

T o p o l o g i c a l s t a t e s

5-

S_{Γ_{1, 2, 3}}

6-

α^{2} F (ω)

7-

N_{S} (ω)

8-

S u p e r c o n d u c t i v i t y

9-

α^{2} F (ω)

10-

α^{2} F (ω)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9903294

Formulas:

1-

300 p h o t o n s

2-

s^{- 1} c m^{- 2} s r^{- 1} n m^{- 1}

3-

e r g s

4-

s^{- 1} c m^{- 2} s r^{- 1} H z^{- 1}

5-

p h o t o n s

6-

s^{- 1} c m^{- 2} s r^{- 1} n m^{- 1}

7-

c m^{- 2} s^{- 1} s r^{- 1}

8-

h = 6.6261 \times 10^{- 27}

9-

c = 2.9979 \times 10^{10}

10-

p h o t o n s

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/9610039

Formulas:

1-

{\hat{O}}_{λ μ} = r^{λ} Y_{λ μ} (\hat{r})

2-

(N - Z) / A

3-

{((N - Z) / A)}^{2}

4-

B (E 2)

5-

B (E 2)

6-

B {(E 2)}_{p}

7-

B {(E 2)}_{n}

8-

B {(E 2)}_{i s}

9-

B {(E 2)}_{i v}

10-

e^{2} f m^{4}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: nucl-th

Result: correct

Paper: https://arxiv.org/abs/2007.13779

Formulas:

1-

< 10 M y r

2-

w_{d} = k R_{H}

3-

R_{H} = {(M_{P} / 3 M_{*})}^{1 / 3} R_{0}

4-

M_{P} = {(\frac{w_{d}}{k R_{0}})}^{3} 3 M_{*} .

5-

(0.1 M_{J} - 2 M_{J})

6-

M_{P} = 0.0021 \times {(\frac{w_{g}}{R_{0}})}^{2} {(\frac{h_{0}}{0.05})}^{\frac{3}{2}} {(\frac{α}{10^{- 3}})}^{\frac{1}{2}} M_{*}

7-

G = R_{0} = M_{*} = 1

8-

P_{0} = 2 π / Ω_{K0}

9-

Ω_{K} = \sqrt{G M_{*} / R^{3}}

10-

Σ_{g} (R) = Σ_{g0} {(\frac{R}{R_{0}})}^{- σ},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0302181

Formulas:

1-

r_{eff} = c t / L^{z}

2-

r = L_{∥} / L_{⟂}

3-

ξ = \hat{ξ} Δ^{- ν}, τ = \hat{τ} Δ^{- z ν},

4-

r = t / L^{z}

5-

(x \pm 1 / 2, t + 1 / 2)

6-

p_{c}^{bond} = 0.644700185 (5)

7-

p_{c}^{site} = 0.70548522 (4)

8-

z = 1.580745 (10)

9-

P_{t} (L, t, l)

10-

r = t / L^{z}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1110.4998

Formulas:

1-

p_{I R} = p_{3.4} = p_{4.6}

2-

p_{I R} / p_{V}

3-

p_{I R} / p_{V}

4-

p_{I R} / p_{V}

5-

p_{I R} / p_{V}

6-

p_{I R} / p_{V}

7-

p_{I R} / p_{V}

8-

p_{W 1} = p_{W 2}

9-

p_{I R} / p_{V}

10-

p_{I R} / p_{V}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1112.2308

Formulas:

1-

ℱ = {[Tr (\sqrt{\sqrt{{\hat{ρ}}_{1}} {\hat{ρ}}_{2} \sqrt{{\hat{ρ}}_{1}}})]}^{2} .

2-

ℱ = {| ⟨ α | β ⟩ |}^{2} = \exp (- {| α - β |}^{2})

3-

{| α ⟩, | β ⟩}

4-

{| α + A ⟩, | β + A ⟩}

5-

| α - β | = 1

6-

| α - β | = 0.325

7-

| χ ⟩ = \sqrt{F} | ψ ⟩ + \sqrt{1 - F} | ψ_{⟂} ⟩

8-

ℱ = {| ⟨ ψ | χ ⟩ |}^{2} = F

9-

ψ_{j} (x) = {(a_{j} / π)}^{1 / 4} \exp [- \frac{1}{2} (a_{j} + i b_{j}) {(x - x_{j})}^{2} + i p_{j} x],

10-

ℱ = {| ⟨ ψ_{1} | ψ_{2} ⟩ |}^{2}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1301.0998

Formulas:

1-

I_{m}, I_{n} ϵ I

2-

2 r \times 2 r

3-

η ϵ [0, min (m_{1}, n_{1})]

4-

γ_{k} = (ϕ_{j} - Θ_{i})

5-

n o b i n s

6-

n o b i n s

7-

R i n t e r

8-

γ = (ϕ_{j} - θ_{i}) mod 360 °

9-

ψ = \frac{d_{2}}{d_{1}}

10-

R i n t e r \subseteq R

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0711.1541

Formulas:

1-

𝐏𝐃 (g, \underline{𝐱}) = \int_{- \infty}^{\infty} 𝑑 τ 𝑑 s g (τ) g (s) G^{i j} (τ - s) {⟨ E_{i} (τ, \underline{𝐱}) E_{j} (s, \underline{𝐱}) ⟩}_{𝒮},

2-

E_{i} (τ, \underline{𝐱})

3-

G^{i j} (τ - s)

4-

{⟨ J ⟩}_{𝒮} = 𝐏𝐃 (g, \underline{𝐱}) - 𝐏𝐃 (g, \underline{𝐲}) .

5-

{⟨ P [E_{i} (t, 𝐱)] ⟩}_{(𝒮, F)} = {⟨ P [F_{i} (t, 𝐱) + E_{i} (t, 𝐱)] ⟩}_{𝒮},

6-

F_{i} (t, 𝐱) = {⟨ E_{i} (t, 𝐱) ⟩}_{F}

7-

E_{i} (t, \underline{𝐱}) = \int 𝑑 ν (p^{a}) [e^{- i ω_{p} t} ψ_{i} (p^{a}, \underline{𝐱}) b (p^{a}) + e^{i ω_{p} t} \bar{ψ_{i} (p^{a}, \underline{𝐱})} b^{*} (p^{a})] .

8-

ψ_{i} (p^{a}, \underline{𝐱})

9-

d ν (p^{a})

10-

p^{a} = (\vec{p}, α)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Run 10695632 (Agent093)