Run 11314748

Paper: https://arxiv.org/abs/1807.08806

Formulas:

1-

λ_{j} = \frac{2 t}{j} \sqrt{n_{g}^{2} - n_{w}^{2}},

2-

n_{mode} = ℜ (\frac{k_{z}}{k_{0}})

3-

n_{p m} = \sqrt{n_{core}^{2} - \frac{u_{p m}^{2}}{a^{2} k_{0}^{2}}},

4-

E_{p}^{(ℓ)} (r, ϕ) \sim e^{- r^{2} / w^{2}} {(\frac{r}{w})}^{| ℓ |} L_{p}^{| ℓ |} (\frac{2 r^{2}}{w^{2}}) e^{i ϕ ℓ},

5-

L_{p}^{(| ℓ |)} (\dots)

6-

FP = \frac{\iint {(T (x^{'}, y^{'}) - M (x^{'}, y^{'}))}^{2} 𝑑 x^{'} 𝑑 y^{'}}{\sqrt{\iint T {(x^{'}, y^{'})}^{2} 𝑑 x^{'} 𝑑 y^{'} \iint M {(x^{'}, y^{'})}^{2} 𝑑 x^{'} 𝑑 y^{'}}},

7-

(x^{'}, y^{'})

8-

T (x^{'}, y^{'})

9-

M (x^{'}, y^{'})

10-

θ_{core} = \cos^{- 1} (\frac{n_{p m}}{n_{core}}),

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1910.04839

Formulas:

1-

𝔼 [benefit outgo] = 𝔼 [premium income] .

2-

\frac{d S_{i}}{d t}

3-

= - β (R_{i}) S_{i} I_{i} - \sum_{j \neq i} k_{i j} S_{i} + \sum_{j \neq i} k_{j i} S_{j}

4-

\frac{d I_{i}}{d t}

5-

= β (R_{i}) S_{i} I_{i} - μ (R_{i}) I_{i} - \sum_{j \neq i} l_{i j} I_{i} + \sum_{j \neq i} l_{j i} I_{j}

6-

\frac{d R_{i}}{d t}

7-

= μ (R_{i}) I_{i}

8-

S_{i} (0)

9-

= S_{i, 0}, I_{i} (0) = I_{i, 0}, R_{i} (0) = R_{i, 0}

10-

i = 1, 2, \dots, n

Formulas (html):

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xref="S2.E2Xb.2.1.1.m1.1.1.cmml"><mfrac id="S2.E2Xb.2.1.1.m1.1.1a" xref="S2.E2Xb.2.1.1.m1.1.1.cmml"><mrow id="S2.E2Xb.2.1.1.m1.1.1.2" xref="S2.E2Xb.2.1.1.m1.1.1.2.cmml"><mi mathvariant="normal" id="S2.E2Xb.2.1.1.m1.1.1.2.2" xref="S2.E2Xb.2.1.1.m1.1.1.2.2.cmml">d</mi><mo id="S2.E2Xb.2.1.1.m1.1.1.2.1" xref="S2.E2Xb.2.1.1.m1.1.1.2.1.cmml">⁢</mo><msub id="S2.E2Xb.2.1.1.m1.1.1.2.3" xref="S2.E2Xb.2.1.1.m1.1.1.2.3.cmml"><mi id="S2.E2Xb.2.1.1.m1.1.1.2.3.2" xref="S2.E2Xb.2.1.1.m1.1.1.2.3.2.cmml">R</mi><mi id="S2.E2Xb.2.1.1.m1.1.1.2.3.3" xref="S2.E2Xb.2.1.1.m1.1.1.2.3.3.cmml">i</mi></msub></mrow><mrow id="S2.E2Xb.2.1.1.m1.1.1.3" xref="S2.E2Xb.2.1.1.m1.1.1.3.cmml"><mi mathvariant="normal" id="S2.E2Xb.2.1.1.m1.1.1.3.2" xref="S2.E2Xb.2.1.1.m1.1.1.3.2.cmml">d</mi><mo id="S2.E2Xb.2.1.1.m1.1.1.3.1" xref="S2.E2Xb.2.1.1.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.E2Xb.2.1.1.m1.1.1.3.3" xref="S2.E2Xb.2.1.1.m1.1.1.3.3.cmml">t</mi></mrow></mfrac></mstyle></math>, <math><mrow id="S2.E2Xb.3.2.2.m1.1.1" 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id="S2.SS1.p1.2.m1.3.3" xref="S2.SS1.p1.2.m1.3.3.cmml">…</mi><mo id="S2.SS1.p1.2.m1.4.5.3.2.3" xref="S2.SS1.p1.2.m1.4.5.3.1.cmml">,</mo><mi id="S2.SS1.p1.2.m1.4.4" xref="S2.SS1.p1.2.m1.4.4.cmml">n</mi></mrow></mrow></math>

Correct Categorie: q-bio

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9509177

Formulas:

1-

C e N i S n

2-

C e R h S b

3-

U P t_{3}

4-

u_{i j} (x)

5-

σ^{i j} (x)

6-

H_{I} = - \int d^{3} x σ^{i j} (x) u_{i j} (x) .

7-

σ^{i j} (x)

8-

\nabla_{j} σ^{i j} (x) = - {\dot{P}}_{i} (x)

9-

\dot{E} = - \int d^{3} x σ^{i j} (x) {\dot{u}}_{i j}

10-

σ^{i j} = η^{i j k l} {\dot{u}}_{k l}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1912.12558

Formulas:

1-

Z F + D C +

2-

+ D C_{ℝ} + V = L (ℝ)

3-

Z F + D C +

4-

Z F + D C

5-

(H (κ), \in)

6-

η_{0}, \dots, η_{n - 1} \in l i m (T)

7-

(η_{0}, \dots, η_{n - 1})

8-

{(2^{< ω}, \leq)}^{n}

9-

η \neq ν \in l i m (T)

10-

(M_{1}, M_{2})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0605498

Formulas:

1-

H = v 𝒑 \cdot 𝝈 + Δ_{s o} σ_{z} = [\begin{matrix} Δ_{s o} & - i ℏ v \nabla^{-} \\ - i ℏ v \nabla^{+} & - Δ_{s o} \end{matrix}],

2-

𝝈 = (σ_{x}, σ_{y}, σ_{z})

3-

\nabla^{\pm} = \partial / \partial x \pm i \partial / \partial y

4-

Ψ_{𝒌}^{+} (𝒓) = e^{i 𝒌 \cdot 𝒓} (\begin{matrix} \cos (α_{𝒌} / 2) \\ e^{i ϕ_{𝒌}} \sin (α_{𝒌} / 2) \end{matrix})

5-

E_{𝒌}^{+} = \sqrt{Δ_{s o}^{2} + ℏ^{2} v^{2} k^{2}}

6-

Ψ_{𝒌}^{-} (𝒓) = e^{i 𝒌 \cdot 𝒓} (\begin{matrix} \sin (α_{𝒌} / 2) \\ - e^{i ϕ_{𝒌}} \cos (α_{𝒌} / 2) \end{matrix})

7-

E_{𝒌}^{-} = - \sqrt{Δ_{s o}^{2} + ℏ^{2} v^{2} k^{2}}

8-

\tan ϕ_{𝒌} = k_{y} / k_{x}

9-

\tan α_{𝒌} = ℏ v k / Δ_{s o}

10-

k = \sqrt{k_{x}^{2} + k_{y}^{2}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9803008

Formulas:

1-

\frac{m_{π}}{m ρ} = 0.57 (2)

2-

\frac{m_{π}}{m ρ} = 0.65 (3)

3-

\frac{m_{π}}{m ρ} = 0.75 (1)

4-

a^{- 1} = 2.33 (6)

5-

\frac{m_{π}}{m ρ} = 0.56 (2)

6-

τ_{i n t}

7-

U_{A} (1)

8-

\partial^{μ} J_{μ}^{5} (x) = 2 N_{f} Q (x)

9-

Q (x) = \frac{g^{2}}{64 π^{2}} ϵ^{μ ν ρ σ} F_{μ ν}^{a} (x) F_{ρ σ}^{a} (x) .

10-

Q = m κ_{P} {⟨ T r (γ_{5} G) ⟩}_{U} .

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9401031

Formulas:

1-

α_{c} = π - 2 θ

2-

α_{c} = π - 2 θ,

3-

F = 2 d L γ \tan (α / 2) + 2 d L (γ_{S L} - γ_{S G}) \sec (α / 2),

4-

γ \cos θ = γ_{S G} - γ_{S L},

5-

F = 2 d L γ [\sin (α / 2) - \cos θ] / \cos (α / 2) .

6-

\sin (α / 2) < \cos θ

7-

α < α_{c} = π - 2 θ

8-

V = d^{2} L \tan (α / 2)

9-

z (x, y)

10-

ρ g z = 2 γ H,

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0101416

Formulas:

1-

n = 4.2 \times 10^{11}

2-

n_{0} \sim 3.1 \times 10^{11}

3-

n_{1} \sim 1.1 \times 10^{11}

4-

B_{p e r p}

5-

B_{t o t} = B_{p e r p}

6-

B_{t o t}

7-

n = 4.2 \times 10^{11}

8-

B_{i p} \sim 2.5

9-

B_{i p} \sim 18.5

10-

B_{i p} \sim 2.5

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1103.4544

Formulas:

1-

γ_{J} = \frac{J_{c} (90^{\circ})}{J_{c} (0^{\circ})} = \frac{J_{c}^{∥}}{J_{c}^{⟂}}

2-

γ_{irr} = \frac{B_{irr}^{∥}}{B_{irr}^{⟂}}

3-

B_{c2} (T)

4-

B_{irr} (T)

5-

B_{c2} (T)

6-

B_{c2} (T)

7-

B_{c2} (T)

8-

B_{irr} (T)

9-

B_{c2} (T)

10-

B_{irr} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1712.06827

Formulas:

1-

σ = \sqrt{2 T^{'} d t}

2-

T^{'} = k_{B} T / 3 π η d

3-

v (r) = v_{m} (1 - {(\frac{r}{R})}^{2}),

4-

τ_{B} = d^{2} / D_{0}

5-

U_{H S} = {\begin{matrix} 4 ϵ [{(\frac{d}{r})}^{24} - {(\frac{d}{r})}^{12} + \frac{1}{4}] & , r \leq 2^{\frac{1}{12}} d \\ 0 & , r > 2^{\frac{1}{12}} d \end{matrix}

6-

d t = 2 \times 10^{- 5}

7-

P e = (v_{m} d^{2}) / (R D_{0})

8-

L_{x} = 10 d

9-

L_{y} = 20 d

10-

L_{z} = 13.09 d

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0103042

Formulas:

1-

ε_{2} = 2.56 ε_{0}

2-

a_{n} = a_{0} (1 + W)

3-

P_{g a i n} = ε_{0} χ (ω) E = ε_{0} (χ^{'} (ω) + i χ^{''} (ω)) E

4-

χ^{'} (ω)

5-

χ^{''} (ω)

6-

i n d e p e n d e n t

7-

l o g_{10} (T)

8-

f_{a} = ω_{a} / 2 π = 618.56

9-

Δ f_{a} = Δ ω_{a} / 2 π = 15

10-

E_{i n c} = 1

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9512127

Formulas:

1-

N (z) = N_{D} δ (z)

2-

V (z) = - g \exp (- \frac{| z |}{α}),

3-

α = 38.0 {(N_{D} a^{* 2})}^{- 1 / 3}

4-

g = 38.1 {(N_{D} a^{* 2})}^{2 / 3}

5-

N_{D} > 5 \times 10^{10}

6-

[\begin{matrix} E_{g} / 2 - E + V (z) & - i ℏ v \partial \\ - i ℏ v \partial & - E_{g} / 2 - E + V (z) \end{matrix}] (\begin{matrix} f_{c} (z) \\ f_{v} (z) \end{matrix}) = 0,

7-

\partial = d / d z

8-

v = \sqrt{E_{g} / 2 m^{*}}

9-

(\begin{matrix} f_{c} (z) \\ f_{v} (z) \end{matrix}) = [\begin{matrix} E_{g} / 2 + E - V (z) & - i ℏ v \partial \\ - i ℏ v \partial & - E_{g} / 2 + E - V (z) \end{matrix}] (\begin{matrix} ϕ (z) \\ ϕ (z) \end{matrix}),

10-

{- ℏ^{2} v^{2} \frac{d^{2}}{d z^{2}} + \frac{E_{g}^{2}}{4} - {[E - V (z)]}^{2} + i ℏ v \frac{d V (z)}{d z}} ϕ (z) = 0,

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0506330

Formulas:

1-

a = (8.74 \pm 1.33) \times 10^{- 10} m s^{- 2}

2-

a_{r g} = - \frac{G M_{K B}}{2 π (R_{1} + R_{2})} \int_{0}^{2 π} (\sum_{i = 1}^{2} R_{i} g_{r g} (ϕ_{m})) 𝑑 ϕ_{m}

3-

g_{r g} (ϕ_{m}) = \frac{r - R_{i} c o s θ c o s (ϕ_{m} - ϕ)}{{(r^{2} + R_{i}^{2} - 2 r R_{i} c o s θ c o s (ϕ_{m} - ϕ))}^{\frac{3}{2}}}

4-

r = 35 A U

5-

r = 50 A U

6-

R_{m i n} = 30

7-

R_{m a x} = 55

8-

a_{u d} = - \frac{G M_{K B}}{R_{m a x}^{2} - R_{m i n}^{2}} \int_{0}^{2 π} \int_{R_{m i n}}^{R_{m a x}} g_{u d} (r_{m}, ϕ_{m}) 𝑑 r_{m} 𝑑 ϕ_{m}

9-

g_{u d} (r_{m}, ϕ_{m}) = \frac{r_{m} (r - r_{m} c o s θ c o s (ϕ_{m} - ϕ))}{{(r^{2} + r_{m}^{2} - 2 r r_{m} c o s θ c o s (ϕ_{m} - ϕ))}^{\frac{3}{2}}}

10-

r = 35 A U

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/9807061

Formulas:

1-

S_{λ^{'} Σ^{'} λ Σ}^{J}

2-

E_{l a b} = 3.0

3-

K_{0} = l_{α} + L_{α}

4-

l_{α} + L_{α} \leq

5-

K_{0} = 2, 4, 6

6-

J^{Π} = 1 / 2^{+}

7-

N_{c} = 10, 18, 26

8-

j_{m a x}

9-

j_{m a x} = 8

10-

j_{m a x} = 6

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0102224

Formulas:

1-

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

2-

(y_{1}, y_{2}, \dots, y_{l}, \dots)

3-

p_{j} = \tanh (\sum_{i} w_{i, j} x_{i}) .

4-

q_{k} = \tanh (\sum_{j} w_{j, k} p_{j}) .

5-

y_{l} = \sum_{k} w_{k, l} q_{k} .

6-

1 / (1 - \exp [- \sum w x])

7-

e^{{n}} = \frac{1}{2} \sum_{l} β_{l} {(y_{l}^{{n}} - T_{l}^{{n}})}^{2},

8-

\frac{\partial e}{\partial w_{k, l}} = β_{l} (y_{l} - T_{l}) q_{k},

9-

\frac{\partial e}{\partial w_{j, k}} = \sum_{l} β_{l} (y_{l} - T_{l}) \frac{\partial y_{l}}{\partial w_{j, k}} = \frac{\partial q_{k}}{\partial w_{j, k}} \sum_{l} β_{l} (y_{l} - T_{l}) w_{k, l} .

10-

\partial e / \partial 𝐰

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/0909.0845

Formulas:

1-

\hat{H} = \frac{γ}{2} ({\hat{n}}_{1} - {\hat{n}}_{2}) - T ({\hat{a}}_{1}^{†} {\hat{a}}_{2} + {\hat{a}}_{2}^{†} {\hat{a}}_{1}) + \frac{U}{4} {({\hat{n}}_{1} - {\hat{n}}_{2})}^{2},

2-

{\hat{n}}_{μ} = {\hat{a}}_{μ}^{†} {\hat{a}}_{μ}

3-

∣ ψ_{coh} ⟩ = \frac{1}{N!} {(a_{1} {\hat{a}}_{1}^{†} + a_{2} {\hat{a}}_{2}^{†})}^{N} ∣ vac ⟩

4-

{\tilde{ρ}}_{μ ν} (t) = ⟨ {\hat{a}}_{μ}^{†} (t) {\hat{a}}_{ν} (t) ⟩

5-

i \frac{d ρ_{11}}{d t}

6-

- i \frac{d ρ_{22}}{d t} = - T (ρ_{12} - ρ_{21}),

7-

i \frac{d ρ_{12}}{d t}

8-

- γ ρ_{12} + T (ρ_{22} - ρ_{11}) - U N (ρ_{11} - ρ_{22}) ρ_{12},

9-

i \frac{d ρ_{21}}{d t}

10-

γ ρ_{21} - T (ρ_{22} - ρ_{11}) + U N (ρ_{11} - ρ_{22}) ρ_{21},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0204491

Formulas:

1-

N (N - 1) / 2

2-

(0, F_{t h})

3-

(F_{i} \geq F_{t h})

4-

F_{i} \geq F_{t h} \Rightarrow {\begin{matrix} F_{i} \to 0 \\ F_{n n} \to F_{n n} + α F_{i} \end{matrix}

5-

F_{i} < F_{t h}

6-

P_{N} (s)

7-

P_{N} (s) ≃ N^{- β} f (s / N^{D})

8-

P_{N} (s)

9-

P_{N} (s)

10-

P_{N} (s) \sim s^{- τ}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1612.05150

Formulas:

1-

S n O_{2}

2-

i o n s / c m^{2}

3-

S n O_{2}

4-

S n O_{2}

5-

S n O_{2}

6-

S_{e t h}

7-

S n O_{2}

8-

S_{e t h}

9-

S n O_{2}

10-

S n O_{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0902.2896

Formulas:

1-

\hat{E_{y}} = C_{L}^{†} \hat{T_{y}} C_{L}

2-

\hat{T_{y}} = 𝟙 - \hat{𝕋_{𝕟}} = 𝟙 - \sum_{𝕞 = 𝟘}^{θ - 𝟙} | 𝕞 ⟩ ⟨ 𝕞 |

3-

C_{L} = e^{γ (a^{†} c - a c^{†})} {| 0 ⟩}_{c}

4-

{\hat{E}}_{n} = C_{L}^{†} \hat{T_{n}} C_{L}

5-

η = \cos^{2} γ = 0.08

6-

H = i χ a_{H}^{†} a_{V}^{†} + h . c .

7-

a_{H} = \frac{1}{\sqrt{2}} e^{i ϕ} (a_{ϕ} + i a_{ϕ ⟂})

8-

a_{V} = \frac{1}{\sqrt{2}} e^{- i ϕ} (a_{ϕ} - i a_{ϕ ⟂})

9-

H = \frac{i χ}{2} (a_{ϕ}^{†}^{2} + a_{ϕ ⟂}^{†}^{2}) + h . c .

10-

a^{†} | 0, 0 ⟩ = | 1, 0 ⟩

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0410010

Formulas:

1-

p (A + B) = p (A) p (B),

2-

S_{q} (A + B) = S_{q} (A) + S_{q} (B) + (1 - q) S_{q} (A) S_{q} (B),

3-

P (H) \propto \exp [- η Tr (H^{†} H)]

4-

P (s) = \frac{π}{2} s e^{- \frac{π}{4} s^{2}}

5-

\exp (- s)

6-

S_{q} [P_{β} (q, H)] = \frac{1 - \int 𝑑 H {[P_{β} (q, H)]}^{q}}{q - 1},

7-

d H = \prod_{m \geq n} d H_{m n}^{(0)} \prod_{γ = 1}^{β - 1} \prod_{m > n} d H_{m n}^{(γ)},

8-

S_{1} [P_{β} (q, H)] = - \int 𝑑 H P_{β} (q, H) \ln P_{β} (q, H)

9-

P_{β} (q, H)

10-

\int 𝑑 H P_{β} (q, H)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/1504.04275

Formulas:

1-

𝐅 = 𝐄 + i 𝐇,

2-

w_{μ} = \frac{1}{2} e_{μ ν σ τ} p^{ν} M^{σ τ}, p_{μ} w^{μ} = 0,

3-

p^{2} = p_{μ} p^{μ} = m^{2}, w^{2} = w_{μ} w^{μ} = - m^{2} s (s + 1), m > 0

4-

w^{2} = p^{2} = p w = 0,

5-

w_{μ} = λ p_{μ}

6-

λ = 𝐤 \cdot 𝐌 / k_{0},

7-

k = (k_{0}, 𝐤)

8-

k^{2} = k_{0}^{2} - 𝐤^{2}

9-

𝐅 = 𝐄 + i 𝐇

10-

𝐅 = 𝐄 + i 𝐇, 𝐆 = 𝐃 + i 𝐁

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1708.08541

Formulas:

1-

Ω_{Λ} = 0.7, Ω_{0} = 0.3,

2-

σ_{Δ v} \sim 90

3-

δ v \sim 45 - 50

4-

- 4 < T_{M} \leq 0

5-

r_{200, X} / r_{200, σ}

6-

r_{200, N} / r_{200, σ}

7-

r_{200, N} / r_{200, X}

8-

v_{z} = c (z - \bar{z}) / (1 + \bar{z})

9-

| v_{z} - {\bar{v}}_{z} | < 2.7 σ_{z}^{NFW} (R)

10-

\log r_{200, σ}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9906410

Formulas:

1-

s i n g l e

2-

p a i r s

3-

M R_{i j} (H) = c_{i j} {(1 - \cos θ_{i j} (H))}^{2}

4-

θ_{m a x, 1, 1}

5-

θ_{m a x, 2, 2}

6-

M R_{i i} = c_{i i} {(\frac{1}{2} θ_{i i}^{2})}^{2} \propto c_{i i} {(θ_{m a x, i i})}^{4}

7-

c o m b i n a t i o n

8-

θ_{m a x, i i}

9-

s i n g l e

10-

θ_{m a x, 1, 1}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1901.06128

Formulas:

1-

{(a^{i j} u_{i})}_{j} = 0 in Q_{6},

2-

{(a^{i j})}_{n \times n}

3-

Q_{r} (x_{0})

4-

Q_{r} := Q_{r} (0)

5-

sup_{Q_{1}} u \leq C inf_{Q_{1}} u,

6-

{|| u ||}_{L^{p_{0}}}

7-

{|| u^{- 1} ||}_{L^{p_{0}}}

8-

{∥ u ∥}_{L^{\infty} (Q_{1})} \leq C {∥ u ∥}_{L^{2} (Q_{3})},

9-

m {x \in Q_{1} : u (x) > c} > c_{0} \Rightarrow u > 1 in Q_{3},

10-

{∥ u ∥}_{L^{\infty} (Q_{1})} \leq C {∥ u ∥}_{L^{p_{0}} (Q_{3})},

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9606145

Formulas:

1-

C u_{2} {(C_{5} H_{12} N_{2})}_{2} C l_{4}

2-

C u_{2} {(C_{5} H_{12} N_{2})}_{2} C l_{4}

3-

(V O_{2}) P_{2} O_{7}

4-

S r C u_{2} O_{3}

5-

L a_{1 - x} S r_{x} C u O_{2.5}

6-

C u_{2} {(C_{5} H_{12} N_{2})}_{2} C l_{4}

7-

C u_{2} {(C_{5} H_{12} N_{2})}_{2} C l_{4}

8-

C u_{2} {(C_{5} H_{12} N_{2})}_{2} C l_{4}

9-

J^{'} / J \sim 5.5

10-

ℋ = J^{'} \sum_{j} 𝐒_{j, 1} \cdot 𝐒_{j, 2} + J \sum_{β, j} 𝐒_{j, β} \cdot 𝐒_{j + 1, β}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1906.12028

Formulas:

1-

B = G (P + 1)

2-

ℬ = {(𝐫_{i}, 𝚠 (𝐫_{i})) | i = 1, 2, \dots, B}

3-

𝚠 (𝐫_{i})

4-

\sum_{i = 1}^{B} 𝚠 (𝐫_{i}) = 1

5-

y \in [1, 2, \dots, C]

6-

𝐛 = \sum_{i = 1}^{B} 𝚠 (𝐫_{i}) \cdot 𝐫_{i} \in ℛ^{d} .

7-

ℒ_{cls} = - \sum_{ℬ \in ℐ} 𝐲^{⊤} \log (f (\sum_{i = 1}^{B} 𝚠 (𝐫_{i}) \cdot 𝐫_{i})),

8-

\bar{𝚠} (𝐫_{i})

9-

𝒦 = {𝐤_{i, j} | i, j = 1, 2, \dots, L}

10-

𝐊 \in ℛ^{d \times L^{2}}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/0706.1453

Formulas:

1-

{Na}_{0.35} {CoO}_{2} - 1.3 H_{2} O

2-

{Na}_{0.7 \pm δ} {CoO}_{2}

3-

θ_{0} = a r c c o s (\frac{1}{\sqrt{3}}) ≃ {54.74}^{\circ}

4-

R_{Co - O} = 1.855

5-

E (| 𝐚_{𝟏 𝐠} ⟩)

6-

4 ε (e_{g}^{'}) + ε (a_{1 g}) + 2 U + 8 U^{'} - 4 J_{H}

7-

E (| 𝐞_{𝐠}^{'} ⟩)

8-

3 ε (e_{g}^{'}) + 2 ε (a_{1 g}) + 2 U + 8 U^{'} - 4 J_{H}

9-

E (| 𝐞_{𝐠}^{'} ⟩) - E (| 𝐚_{𝟏 𝐠} ⟩)

10-

ε (a_{1 g}) - ε (e_{g}^{'})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1411.6480

Formulas:

1-

715 \pm 100 R_{☉}

2-

(5.25 \pm 0.6) \times 10^{50}

3-

13.1 \pm 0.7 M_{☉}

4-

0.012 \pm 0.002 M_{☉}

5-

15 \pm 1 M_{☉}

6-

U B V R I J H K

7-

E (B - V) = {0.037}_{+ 0.008}^{- 0.006}

8-

m - M = 29.96 \pm 0.15

9-

χ = ρ_{c} / ρ

10-

f = μ χ^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0112223

Formulas:

1-

D^{*} π π

2-

ℒ = i \frac{g_{A}^{q}}{2 f_{π}} {\bar{ψ}}_{q} γ_{5} γ_{μ} \partial_{μ} {\vec{ϕ}}_{π} \cdot \vec{τ} ψ_{q},

3-

- i \frac{g_{A}^{q}}{2 f_{π}} \sqrt{\frac{(E^{'} + m_{\bar{q}}) (E + m_{\bar{q}})}{4 E E^{'}}} (1 - \frac{P^{2} - k^{2} / 4}{3 (E^{'} + m_{\bar{q}}) (E + m_{\bar{q}})}) {\vec{σ}}^{q} \cdot \vec{k} τ_{π},

4-

i \frac{g_{A}^{q}}{2 f_{π}} \frac{2 m_{\bar{q}} + E + E^{'}}{\sqrt{4 E E^{'} (E + m_{\bar{q}}) (E^{'} + m_{\bar{q}})}} ω_{π} {\vec{σ}}^{q} \cdot (\frac{{\vec{p}}^{'} + \vec{p}}{2}) τ_{π} .

5-

({\vec{p}}^{'} + \vec{p}) / 2

6-

D^{* \pm} \to D^{\pm} π^{0}

7-

D^{* \pm} \to D^{0} π^{\pm}

8-

D^{* 0} \to D^{0} π^{0}

9-

ℒ_{WT} = - \frac{i}{4 f_{π}^{2}} {\bar{ψ}}_{q} γ_{μ} \vec{τ} \cdot {\vec{ϕ}}_{π} \times \partial_{μ} {\vec{ϕ}}_{π} ψ_{q} .

10-

T = δ_{a b} T^{+} + \frac{1}{2} [τ_{b}, τ_{a}] T^{-},

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1804.04629

Formulas:

1-

α = 3 \frac{ε_{1} - ε_{h}}{ε_{1} + 2 ε_{h}}

2-

ε_{1} = - 2 ε_{h}

3-

d_{1} = 2 r_{1}

4-

ε_{r} (r)

5-

Φ_{0} (r, θ) = (- E_{0} r + \frac{B_{0}}{r^{2}}) \cos θ

6-

Φ_{2} (r, θ) = A_{2} r \cos θ

7-

Φ_{1} (r, θ) = f (r) \cos θ

8-

𝐄 = - \nabla Φ

9-

\nabla \cdot 𝐃_{1} = - \nabla \cdot (ε_{r} (r) \nabla Φ_{1} (r, θ)) = 0

10-

f^{''} (r) + (\frac{2}{r} + \frac{ε_{r}^{'} (r)}{ε_{r} (r)}) f^{'} (r) - \frac{2}{r^{2}} f (r) = 0

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/nucl-th/0603069

Formulas:

1-

B (E 2)

2-

B (E 2)

3-

B (E 2)

4-

B (E 2)

5-

B (E 2)

6-

B (E 2)

7-

\hat{H} = ε J_{0} - \frac{1}{2} V ({\hat{J}}_{+} {\hat{J}}_{+} + {\hat{J}}_{-} {\hat{J}}_{-}),

8-

\frac{1}{2} \sum_{p = 1}^{N} ({\hat{c}}_{+ p}^{†} {\hat{c}}_{+ p} - {\hat{c}}_{- p}^{†} {\hat{c}}_{- p}),

9-

\sum_{p = 1}^{N} {\hat{c}}_{+ p}^{†} {\hat{c}}_{- p},

10-

[{\hat{J}}_{+}, {\hat{J}}_{-}] = 2 {\hat{J}}_{0}, [{\hat{J}}_{0}, {\hat{J}}_{\pm}] = \pm {\hat{J}}_{\pm} .

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1901.11339

Formulas:

1-

χ_{□} (G) = χ (G^{2})

2-

χ_{□} (G) \leq 7

3-

χ_{□} (G) \leq 8

4-

G P (2 k + 1, k)

5-

{u_{1}, \dots, u_{2 k + 1}}

6-

{v_{1}, \dots, v_{2 k + 1}}

7-

{u_{1}, \dots, u_{2 k + 1}}

8-

1 \leq i \leq 2 k + 1

9-

v_{i} v_{i + k}

10-

1 \leq i \leq 2 k + 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/hep-ex/0309041

Formulas:

1-

H \to c \bar{c}

2-

H \to c \bar{c}

3-

H \to c \bar{c}

4-

H \to c \bar{c}

5-

H \to b \bar{b}

6-

H \to c \bar{c}

7-

e^{+} e^{-} \to Z^{*} \to Z^{\circ} H

8-

Z^{\circ} \to q \bar{q}

9-

H \to q \bar{q}

10-

Z^{\circ} \to ν \bar{ν}

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0505419

Formulas:

1-

w_{i j} = w_{j i}

2-

s_{i} = \sum_{j \in Γ (i)} w_{i j}

3-

P (k) \sim k^{- γ}

4-

P (w) \sim w^{- θ}

5-

P (s) \sim s^{- α}

6-

Π_{i \to j} = \frac{s_{j} + A}{\sum_{k (\neq i)} (s_{k} + A)} .

7-

P (s) \sim s^{- α}

8-

α = 2 + A / m^{2}

9-

P (w) \sim w^{- θ}

10-

t = N - N_{0}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1105.5125

Formulas:

1-

{\dot{M}}_{a c c}

2-

a c i s r e a d c o r r

3-

_{a s y m m}

4-

_{a s y m m}

5-

v a p e c

6-

_{2, f l a r e}

7-

v a p e c

8-

v a p e c

9-

T_{s} = 1.5 \times 10^{5} {[\frac{v_{s}}{100 km s^{- 1}}]}^{2} K .

10-

v_{j e t}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ex/0302018

Formulas:

1-

⟨ m_{ν} ⟩ \leq 300 - 1300

2-

o (t) = \int_{- \infty}^{+ \infty} i (τ) h (τ - t) 𝑑 τ

3-

i (t) = i_{e} (t) + i_{h} (t)

4-

i_{e (h)} = - q_{e (h)} {\vec{E}}_{w} \vec{v_{e (h)}} (\vec{E})

5-

v (r) = μ E (r)

6-

v (E) = \frac{μ E}{{(1 + {(\frac{E}{E_{sat}})}^{β})}^{1 / β}}

7-

\vec{E} (\vec{r}) = \vec{\nabla} ϕ (\vec{r})

8-

\nabla^{2} ϕ (\vec{r}) = - \frac{ρ (\vec{r})}{ϵ}, depleted

9-

\nabla^{2} ϕ (\vec{r}) = 0, non - depleted

10-

ϕ ({\vec{r}}_{i n t}) - ϕ ({\vec{r}}_{e x t}) = V_{0}

Formulas (html):

<math><mrow id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml"><mrow id="S1.p1.1.m1.1.1.1.1" xref="S1.p1.1.m1.1.1.1.2.cmml"><mo stretchy="false" id="S1.p1.1.m1.1.1.1.1.2" xref="S1.p1.1.m1.1.1.1.2.1.cmml">⟨</mo><msub id="S1.p1.1.m1.1.1.1.1.1" xref="S1.p1.1.m1.1.1.1.1.1.cmml"><mi id="S1.p1.1.m1.1.1.1.1.1.2" xref="S1.p1.1.m1.1.1.1.1.1.2.cmml">m</mi><mi id="S1.p1.1.m1.1.1.1.1.1.3" xref="S1.p1.1.m1.1.1.1.1.1.3.cmml">ν</mi></msub><mo stretchy="false" id="S1.p1.1.m1.1.1.1.1.3" xref="S1.p1.1.m1.1.1.1.2.1.cmml">⟩</mo></mrow><mo id="S1.p1.1.m1.1.1.2" xref="S1.p1.1.m1.1.1.2.cmml">≤</mo><mrow id="S1.p1.1.m1.1.1.3" xref="S1.p1.1.m1.1.1.3.cmml"><mn id="S1.p1.1.m1.1.1.3.2" xref="S1.p1.1.m1.1.1.3.2.cmml">300</mn><mo id="S1.p1.1.m1.1.1.3.1" xref="S1.p1.1.m1.1.1.3.1.cmml">-</mo><mn id="S1.p1.1.m1.1.1.3.3" xref="S1.p1.1.m1.1.1.3.3.cmml">1300</mn></mrow></mrow></math>, <math><mrow id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml"><mrow id="S2.E1.m1.3.3.3" xref="S2.E1.m1.3.3.3.cmml"><mi id="S2.E1.m1.3.3.3.2" xref="S2.E1.m1.3.3.3.2.cmml">o</mi><mo id="S2.E1.m1.3.3.3.1" xref="S2.E1.m1.3.3.3.1.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.3.3.2" xref="S2.E1.m1.3.3.3.cmml"><mo stretchy="false" id="S2.E1.m1.3.3.3.3.2.1" xref="S2.E1.m1.3.3.3.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">t</mi><mo stretchy="false" id="S2.E1.m1.3.3.3.3.2.2" xref="S2.E1.m1.3.3.3.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.3.3.2" xref="S2.E1.m1.3.3.2.cmml">=</mo><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.cmml"><msubsup id="S2.E1.m1.3.3.1.2" xref="S2.E1.m1.3.3.1.2.cmml"><mo largeop="true" symmetric="true" id="S2.E1.m1.3.3.1.2.2.2" xref="S2.E1.m1.3.3.1.2.2.2.cmml">∫</mo><mrow id="S2.E1.m1.3.3.1.2.2.3" xref="S2.E1.m1.3.3.1.2.2.3.cmml"><mo id="S2.E1.m1.3.3.1.2.2.3.1" xref="S2.E1.m1.3.3.1.2.2.3.1.cmml">-</mo><mi mathvariant="normal" id="S2.E1.m1.3.3.1.2.2.3.2" xref="S2.E1.m1.3.3.1.2.2.3.2.cmml">∞</mi></mrow><mrow id="S2.E1.m1.3.3.1.2.3" xref="S2.E1.m1.3.3.1.2.3.cmml"><mo id="S2.E1.m1.3.3.1.2.3.1" xref="S2.E1.m1.3.3.1.2.3.1.cmml">+</mo><mi mathvariant="normal" id="S2.E1.m1.3.3.1.2.3.2" xref="S2.E1.m1.3.3.1.2.3.2.cmml">∞</mi></mrow></msubsup><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml">i</mi><mo id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.4.2" xref="S2.E1.m1.3.3.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.3.3.1.1.4.2.1" xref="S2.E1.m1.3.3.1.1.cmml">(</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">τ</mi><mo stretchy="false" id="S2.E1.m1.3.3.1.1.4.2.2" xref="S2.E1.m1.3.3.1.1.cmml">)</mo></mrow><mo id="S2.E1.m1.3.3.1.1.2a" xref="S2.E1.m1.3.3.1.1.2.cmml">⁢</mo><mi id="S2.E1.m1.3.3.1.1.5" xref="S2.E1.m1.3.3.1.1.5.cmml">h</mi><mo id="S2.E1.m1.3.3.1.1.2b" xref="S2.E1.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.3.3.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.3.3.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.1.1.1.2" xref="S2.E1.m1.3.3.1.1.1.1.1.2.cmml">τ</mi><mo id="S2.E1.m1.3.3.1.1.1.1.1.1" xref="S2.E1.m1.3.3.1.1.1.1.1.1.cmml">-</mo><mi id="S2.E1.m1.3.3.1.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.3.cmml">t</mi></mrow><mo stretchy="false" id="S2.E1.m1.3.3.1.1.1.1.3" xref="S2.E1.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E1.m1.3.3.1.1.2c" xref="S2.E1.m1.3.3.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.3.3.1.1.6" xref="S2.E1.m1.3.3.1.1.6.cmml"><mo rspace="0pt" id="S2.E1.m1.3.3.1.1.6.1" xref="S2.E1.m1.3.3.1.1.6.1.cmml">𝑑</mo><mi id="S2.E1.m1.3.3.1.1.6.2" xref="S2.E1.m1.3.3.1.1.6.2.cmml">τ</mi></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.E2.m1.3.4" xref="S2.E2.m1.3.4.cmml"><mrow id="S2.E2.m1.3.4.2" xref="S2.E2.m1.3.4.2.cmml"><mi id="S2.E2.m1.3.4.2.2" xref="S2.E2.m1.3.4.2.2.cmml">i</mi><mo id="S2.E2.m1.3.4.2.1" xref="S2.E2.m1.3.4.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.4.2.3.2" xref="S2.E2.m1.3.4.2.cmml"><mo stretchy="false" id="S2.E2.m1.3.4.2.3.2.1" xref="S2.E2.m1.3.4.2.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">t</mi><mo stretchy="false" id="S2.E2.m1.3.4.2.3.2.2" xref="S2.E2.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.3.4.1" xref="S2.E2.m1.3.4.1.cmml">=</mo><mrow id="S2.E2.m1.3.4.3" xref="S2.E2.m1.3.4.3.cmml"><mrow id="S2.E2.m1.3.4.3.2" xref="S2.E2.m1.3.4.3.2.cmml"><msub id="S2.E2.m1.3.4.3.2.2" xref="S2.E2.m1.3.4.3.2.2.cmml"><mi id="S2.E2.m1.3.4.3.2.2.2" xref="S2.E2.m1.3.4.3.2.2.2.cmml">i</mi><mi mathvariant="normal" id="S2.E2.m1.3.4.3.2.2.3" xref="S2.E2.m1.3.4.3.2.2.3.cmml">e</mi></msub><mo id="S2.E2.m1.3.4.3.2.1" xref="S2.E2.m1.3.4.3.2.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.4.3.2.3.2" xref="S2.E2.m1.3.4.3.2.cmml"><mo stretchy="false" id="S2.E2.m1.3.4.3.2.3.2.1" xref="S2.E2.m1.3.4.3.2.cmml">(</mo><mi id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml">t</mi><mo stretchy="false" id="S2.E2.m1.3.4.3.2.3.2.2" xref="S2.E2.m1.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.3.4.3.1" xref="S2.E2.m1.3.4.3.1.cmml">+</mo><mrow id="S2.E2.m1.3.4.3.3" xref="S2.E2.m1.3.4.3.3.cmml"><msub id="S2.E2.m1.3.4.3.3.2" xref="S2.E2.m1.3.4.3.3.2.cmml"><mi id="S2.E2.m1.3.4.3.3.2.2" xref="S2.E2.m1.3.4.3.3.2.2.cmml">i</mi><mi mathvariant="normal" id="S2.E2.m1.3.4.3.3.2.3" xref="S2.E2.m1.3.4.3.3.2.3.cmml">h</mi></msub><mo id="S2.E2.m1.3.4.3.3.1" xref="S2.E2.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S2.E2.m1.3.4.3.3.3.2" xref="S2.E2.m1.3.4.3.3.cmml"><mo stretchy="false" id="S2.E2.m1.3.4.3.3.3.2.1" xref="S2.E2.m1.3.4.3.3.cmml">(</mo><mi id="S2.E2.m1.3.3" xref="S2.E2.m1.3.3.cmml">t</mi><mo stretchy="false" id="S2.E2.m1.3.4.3.3.3.2.2" xref="S2.E2.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.E3.m1.4.5" xref="S2.E3.m1.4.5.cmml"><msub id="S2.E3.m1.4.5.2" xref="S2.E3.m1.4.5.2.cmml"><mi id="S2.E3.m1.4.5.2.2" xref="S2.E3.m1.4.5.2.2.cmml">i</mi><mrow id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.cmml"><mi mathvariant="normal" id="S2.E3.m1.1.1.1.3" xref="S2.E3.m1.1.1.1.3.cmml">e</mi><mo id="S2.E3.m1.1.1.1.2" xref="S2.E3.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.1.1.1.4.2" xref="S2.E3.m1.1.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.1.1.1.4.2.1" xref="S2.E3.m1.1.1.1.cmml">(</mo><mi mathvariant="normal" id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml">h</mi><mo stretchy="false" id="S2.E3.m1.1.1.1.4.2.2" xref="S2.E3.m1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.E3.m1.4.5.1" xref="S2.E3.m1.4.5.1.cmml">=</mo><mrow id="S2.E3.m1.4.5.3" xref="S2.E3.m1.4.5.3.cmml"><mo id="S2.E3.m1.4.5.3.1" xref="S2.E3.m1.4.5.3.1.cmml">-</mo><mrow id="S2.E3.m1.4.5.3.2" xref="S2.E3.m1.4.5.3.2.cmml"><msub id="S2.E3.m1.4.5.3.2.2" xref="S2.E3.m1.4.5.3.2.2.cmml"><mi id="S2.E3.m1.4.5.3.2.2.2" xref="S2.E3.m1.4.5.3.2.2.2.cmml">q</mi><mrow id="S2.E3.m1.2.2.1" xref="S2.E3.m1.2.2.1.cmml"><mi mathvariant="normal" id="S2.E3.m1.2.2.1.3" xref="S2.E3.m1.2.2.1.3.cmml">e</mi><mo id="S2.E3.m1.2.2.1.2" xref="S2.E3.m1.2.2.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.2.2.1.4.2" xref="S2.E3.m1.2.2.1.cmml"><mo stretchy="false" id="S2.E3.m1.2.2.1.4.2.1" xref="S2.E3.m1.2.2.1.cmml">(</mo><mi mathvariant="normal" id="S2.E3.m1.2.2.1.1" xref="S2.E3.m1.2.2.1.1.cmml">h</mi><mo stretchy="false" id="S2.E3.m1.2.2.1.4.2.2" xref="S2.E3.m1.2.2.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.E3.m1.4.5.3.2.1" xref="S2.E3.m1.4.5.3.2.1.cmml">⁢</mo><msub id="S2.E3.m1.4.5.3.2.3" xref="S2.E3.m1.4.5.3.2.3.cmml"><mover accent="true" id="S2.E3.m1.4.5.3.2.3.2" xref="S2.E3.m1.4.5.3.2.3.2.cmml"><mi id="S2.E3.m1.4.5.3.2.3.2.2" xref="S2.E3.m1.4.5.3.2.3.2.2.cmml">E</mi><mo id="S2.E3.m1.4.5.3.2.3.2.1" xref="S2.E3.m1.4.5.3.2.3.2.1.cmml">→</mo></mover><mi mathvariant="normal" id="S2.E3.m1.4.5.3.2.3.3" xref="S2.E3.m1.4.5.3.2.3.3.cmml">w</mi></msub><mo id="S2.E3.m1.4.5.3.2.1a" xref="S2.E3.m1.4.5.3.2.1.cmml">⁢</mo><mover accent="true" id="S2.E3.m1.3.3" xref="S2.E3.m1.3.3.cmml"><msub id="S2.E3.m1.3.3.1" xref="S2.E3.m1.3.3.1.cmml"><mi id="S2.E3.m1.3.3.1.3" xref="S2.E3.m1.3.3.1.3.cmml">v</mi><mrow id="S2.E3.m1.3.3.1.1.1" xref="S2.E3.m1.3.3.1.1.1.cmml"><mi mathvariant="normal" id="S2.E3.m1.3.3.1.1.1.3" xref="S2.E3.m1.3.3.1.1.1.3.cmml">e</mi><mo id="S2.E3.m1.3.3.1.1.1.2" xref="S2.E3.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.1.1.1.4.2" xref="S2.E3.m1.3.3.1.1.1.cmml"><mo stretchy="false" id="S2.E3.m1.3.3.1.1.1.4.2.1" xref="S2.E3.m1.3.3.1.1.1.cmml">(</mo><mi mathvariant="normal" id="S2.E3.m1.3.3.1.1.1.1" xref="S2.E3.m1.3.3.1.1.1.1.cmml">h</mi><mo stretchy="false" id="S2.E3.m1.3.3.1.1.1.4.2.2" xref="S2.E3.m1.3.3.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.E3.m1.3.3.2" xref="S2.E3.m1.3.3.2.cmml">→</mo></mover><mo id="S2.E3.m1.4.5.3.2.1b" xref="S2.E3.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S2.E3.m1.4.5.3.2.4.2" xref="S2.E3.m1.4.4.cmml"><mo stretchy="false" id="S2.E3.m1.4.5.3.2.4.2.1" xref="S2.E3.m1.4.4.cmml">(</mo><mover accent="true" id="S2.E3.m1.4.4" xref="S2.E3.m1.4.4.cmml"><mi id="S2.E3.m1.4.4.2" xref="S2.E3.m1.4.4.2.cmml">E</mi><mo id="S2.E3.m1.4.4.1" xref="S2.E3.m1.4.4.1.cmml">→</mo></mover><mo stretchy="false" id="S2.E3.m1.4.5.3.2.4.2.2" xref="S2.E3.m1.4.4.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.p2.5.m4.2.3" xref="S2.p2.5.m4.2.3.cmml"><mrow id="S2.p2.5.m4.2.3.2" xref="S2.p2.5.m4.2.3.2.cmml"><mi id="S2.p2.5.m4.2.3.2.2" xref="S2.p2.5.m4.2.3.2.2.cmml">v</mi><mo id="S2.p2.5.m4.2.3.2.1" xref="S2.p2.5.m4.2.3.2.1.cmml">⁢</mo><mrow id="S2.p2.5.m4.2.3.2.3.2" xref="S2.p2.5.m4.2.3.2.cmml"><mo stretchy="false" id="S2.p2.5.m4.2.3.2.3.2.1" xref="S2.p2.5.m4.2.3.2.cmml">(</mo><mi id="S2.p2.5.m4.1.1" xref="S2.p2.5.m4.1.1.cmml">r</mi><mo stretchy="false" id="S2.p2.5.m4.2.3.2.3.2.2" xref="S2.p2.5.m4.2.3.2.cmml">)</mo></mrow></mrow><mo id="S2.p2.5.m4.2.3.1" xref="S2.p2.5.m4.2.3.1.cmml">=</mo><mrow id="S2.p2.5.m4.2.3.3" xref="S2.p2.5.m4.2.3.3.cmml"><mi id="S2.p2.5.m4.2.3.3.2" xref="S2.p2.5.m4.2.3.3.2.cmml">μ</mi><mo id="S2.p2.5.m4.2.3.3.1" xref="S2.p2.5.m4.2.3.3.1.cmml">⁢</mo><mi id="S2.p2.5.m4.2.3.3.3" xref="S2.p2.5.m4.2.3.3.3.cmml">E</mi><mo id="S2.p2.5.m4.2.3.3.1a" xref="S2.p2.5.m4.2.3.3.1.cmml">⁢</mo><mrow id="S2.p2.5.m4.2.3.3.4.2" xref="S2.p2.5.m4.2.3.3.cmml"><mo stretchy="false" id="S2.p2.5.m4.2.3.3.4.2.1" xref="S2.p2.5.m4.2.3.3.cmml">(</mo><mi id="S2.p2.5.m4.2.2" xref="S2.p2.5.m4.2.2.cmml">r</mi><mo stretchy="false" id="S2.p2.5.m4.2.3.3.4.2.2" xref="S2.p2.5.m4.2.3.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.E4.m1.3.4" xref="S2.E4.m1.3.4.cmml"><mrow id="S2.E4.m1.3.4.2" xref="S2.E4.m1.3.4.2.cmml"><mi id="S2.E4.m1.3.4.2.2" xref="S2.E4.m1.3.4.2.2.cmml">v</mi><mo id="S2.E4.m1.3.4.2.1" xref="S2.E4.m1.3.4.2.1.cmml">⁢</mo><mrow id="S2.E4.m1.3.4.2.3.2" xref="S2.E4.m1.3.4.2.cmml"><mo stretchy="false" id="S2.E4.m1.3.4.2.3.2.1" xref="S2.E4.m1.3.4.2.cmml">(</mo><mi id="S2.E4.m1.3.3" xref="S2.E4.m1.3.3.cmml">E</mi><mo stretchy="false" id="S2.E4.m1.3.4.2.3.2.2" xref="S2.E4.m1.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.E4.m1.3.4.1" xref="S2.E4.m1.3.4.1.cmml">=</mo><mfrac id="S2.E4.m1.2.2" xref="S2.E4.m1.2.2.cmml"><mrow id="S2.E4.m1.2.2.4" xref="S2.E4.m1.2.2.4.cmml"><mi id="S2.E4.m1.2.2.4.2" xref="S2.E4.m1.2.2.4.2.cmml">μ</mi><mo id="S2.E4.m1.2.2.4.1" xref="S2.E4.m1.2.2.4.1.cmml">⁢</mo><mi id="S2.E4.m1.2.2.4.3" xref="S2.E4.m1.2.2.4.3.cmml">E</mi></mrow><msup id="S2.E4.m1.2.2.2" xref="S2.E4.m1.2.2.2.cmml"><mrow id="S2.E4.m1.2.2.2.2.1" xref="S2.E4.m1.2.2.2.2.1.1.cmml"><mo stretchy="false" id="S2.E4.m1.2.2.2.2.1.2" xref="S2.E4.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S2.E4.m1.2.2.2.2.1.1" xref="S2.E4.m1.2.2.2.2.1.1.cmml"><mn id="S2.E4.m1.2.2.2.2.1.1.2" xref="S2.E4.m1.2.2.2.2.1.1.2.cmml">1</mn><mo id="S2.E4.m1.2.2.2.2.1.1.1" xref="S2.E4.m1.2.2.2.2.1.1.1.cmml">+</mo><msup id="S2.E4.m1.2.2.2.2.1.1.3" xref="S2.E4.m1.2.2.2.2.1.1.3.cmml"><mrow id="S2.E4.m1.2.2.2.2.1.1.3.2.2" xref="S2.E4.m1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E4.m1.2.2.2.2.1.1.3.2.2.1" xref="S2.E4.m1.1.1.1.1.cmml">(</mo><mfrac id="S2.E4.m1.1.1.1.1" xref="S2.E4.m1.1.1.1.1.cmml"><mi id="S2.E4.m1.1.1.1.1.2" xref="S2.E4.m1.1.1.1.1.2.cmml">E</mi><msub id="S2.E4.m1.1.1.1.1.3" xref="S2.E4.m1.1.1.1.1.3.cmml"><mi id="S2.E4.m1.1.1.1.1.3.2" xref="S2.E4.m1.1.1.1.1.3.2.cmml">E</mi><mi id="S2.E4.m1.1.1.1.1.3.3" xref="S2.E4.m1.1.1.1.1.3.3.cmml">sat</mi></msub></mfrac><mo stretchy="false" id="S2.E4.m1.2.2.2.2.1.1.3.2.2.2" xref="S2.E4.m1.1.1.1.1.cmml">)</mo></mrow><mi id="S2.E4.m1.2.2.2.2.1.1.3.3" xref="S2.E4.m1.2.2.2.2.1.1.3.3.cmml">β</mi></msup></mrow><mo stretchy="false" id="S2.E4.m1.2.2.2.2.1.3" xref="S2.E4.m1.2.2.2.2.1.1.cmml">)</mo></mrow><mrow id="S2.E4.m1.2.2.2.4" xref="S2.E4.m1.2.2.2.4.cmml"><mn id="S2.E4.m1.2.2.2.4.2" xref="S2.E4.m1.2.2.2.4.2.cmml">1</mn><mo id="S2.E4.m1.2.2.2.4.1" xref="S2.E4.m1.2.2.2.4.1.cmml">/</mo><mi id="S2.E4.m1.2.2.2.4.3" xref="S2.E4.m1.2.2.2.4.3.cmml">β</mi></mrow></msup></mfrac></mrow></math>, <math><mrow id="S2.p4.1.m1.2.3" xref="S2.p4.1.m1.2.3.cmml"><mrow id="S2.p4.1.m1.2.3.2" xref="S2.p4.1.m1.2.3.2.cmml"><mover accent="true" id="S2.p4.1.m1.2.3.2.2" xref="S2.p4.1.m1.2.3.2.2.cmml"><mi id="S2.p4.1.m1.2.3.2.2.2" xref="S2.p4.1.m1.2.3.2.2.2.cmml">E</mi><mo id="S2.p4.1.m1.2.3.2.2.1" xref="S2.p4.1.m1.2.3.2.2.1.cmml">→</mo></mover><mo id="S2.p4.1.m1.2.3.2.1" xref="S2.p4.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S2.p4.1.m1.2.3.2.3.2" xref="S2.p4.1.m1.1.1.cmml"><mo stretchy="false" id="S2.p4.1.m1.2.3.2.3.2.1" xref="S2.p4.1.m1.1.1.cmml">(</mo><mover accent="true" id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml"><mi id="S2.p4.1.m1.1.1.2" xref="S2.p4.1.m1.1.1.2.cmml">r</mi><mo id="S2.p4.1.m1.1.1.1" xref="S2.p4.1.m1.1.1.1.cmml">→</mo></mover><mo stretchy="false" id="S2.p4.1.m1.2.3.2.3.2.2" xref="S2.p4.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.p4.1.m1.2.3.1" xref="S2.p4.1.m1.2.3.1.cmml">=</mo><mrow id="S2.p4.1.m1.2.3.3" xref="S2.p4.1.m1.2.3.3.cmml"><mover accent="true" id="S2.p4.1.m1.2.3.3.2" xref="S2.p4.1.m1.2.3.3.2.cmml"><mo id="S2.p4.1.m1.2.3.3.2.2" xref="S2.p4.1.m1.2.3.3.2.2.cmml">∇</mo><mo id="S2.p4.1.m1.2.3.3.2.1" xref="S2.p4.1.m1.2.3.3.2.1.cmml">→</mo></mover><mo id="S2.p4.1.m1.2.3.3.1" xref="S2.p4.1.m1.2.3.3.1.cmml">⁢</mo><mi id="S2.p4.1.m1.2.3.3.3" xref="S2.p4.1.m1.2.3.3.3.cmml">ϕ</mi><mo id="S2.p4.1.m1.2.3.3.1a" xref="S2.p4.1.m1.2.3.3.1.cmml">⁢</mo><mrow id="S2.p4.1.m1.2.3.3.4.2" xref="S2.p4.1.m1.2.2.cmml"><mo stretchy="false" id="S2.p4.1.m1.2.3.3.4.2.1" xref="S2.p4.1.m1.2.2.cmml">(</mo><mover accent="true" id="S2.p4.1.m1.2.2" xref="S2.p4.1.m1.2.2.cmml"><mi id="S2.p4.1.m1.2.2.2" xref="S2.p4.1.m1.2.2.2.cmml">r</mi><mo id="S2.p4.1.m1.2.2.1" xref="S2.p4.1.m1.2.2.1.cmml">→</mo></mover><mo stretchy="false" id="S2.p4.1.m1.2.3.3.4.2.2" xref="S2.p4.1.m1.2.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.E5.m1.4.4" xref="S2.E5.m1.4.4.cmml"><mrow id="S2.E5.m1.4.4.3" xref="S2.E5.m1.4.4.3.cmml"><mrow id="S2.E5.m1.4.4.3.2" xref="S2.E5.m1.4.4.3.2.cmml"><msup id="S2.E5.m1.4.4.3.2.1" xref="S2.E5.m1.4.4.3.2.1.cmml"><mo id="S2.E5.m1.4.4.3.2.1.2" xref="S2.E5.m1.4.4.3.2.1.2.cmml">∇</mo><mn id="S2.E5.m1.4.4.3.2.1.3" xref="S2.E5.m1.4.4.3.2.1.3.cmml">2</mn></msup><mo id="S2.E5.m1.4.4.3.2a" xref="S2.E5.m1.4.4.3.2.cmml">⁡</mo><mi id="S2.E5.m1.4.4.3.2.2" xref="S2.E5.m1.4.4.3.2.2.cmml">ϕ</mi></mrow><mo id="S2.E5.m1.4.4.3.1" xref="S2.E5.m1.4.4.3.1.cmml">⁢</mo><mrow id="S2.E5.m1.4.4.3.3.2" xref="S2.E5.m1.2.2.cmml"><mo stretchy="false" id="S2.E5.m1.4.4.3.3.2.1" xref="S2.E5.m1.2.2.cmml">(</mo><mover accent="true" id="S2.E5.m1.2.2" xref="S2.E5.m1.2.2.cmml"><mi id="S2.E5.m1.2.2.2" xref="S2.E5.m1.2.2.2.cmml">r</mi><mo id="S2.E5.m1.2.2.1" xref="S2.E5.m1.2.2.1.cmml">→</mo></mover><mo stretchy="false" id="S2.E5.m1.4.4.3.3.2.2" xref="S2.E5.m1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.E5.m1.4.4.2" xref="S2.E5.m1.4.4.2.cmml">=</mo><mrow id="S2.E5.m1.4.4.1.1" xref="S2.E5.m1.4.4.1.2.cmml"><mrow id="S2.E5.m1.4.4.1.1.1" xref="S2.E5.m1.4.4.1.1.1.cmml"><mo id="S2.E5.m1.4.4.1.1.1.1" xref="S2.E5.m1.4.4.1.1.1.1.cmml">-</mo><mfrac id="S2.E5.m1.1.1" xref="S2.E5.m1.1.1.cmml"><mrow id="S2.E5.m1.1.1.1" xref="S2.E5.m1.1.1.1.cmml"><mi id="S2.E5.m1.1.1.1.3" xref="S2.E5.m1.1.1.1.3.cmml">ρ</mi><mo id="S2.E5.m1.1.1.1.2" xref="S2.E5.m1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E5.m1.1.1.1.4.2" xref="S2.E5.m1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E5.m1.1.1.1.4.2.1" xref="S2.E5.m1.1.1.1.1.cmml">(</mo><mover accent="true" id="S2.E5.m1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.cmml"><mi id="S2.E5.m1.1.1.1.1.2" xref="S2.E5.m1.1.1.1.1.2.cmml">r</mi><mo id="S2.E5.m1.1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.1.cmml">→</mo></mover><mo stretchy="false" id="S2.E5.m1.1.1.1.4.2.2" xref="S2.E5.m1.1.1.1.1.cmml">)</mo></mrow></mrow><mi id="S2.E5.m1.1.1.3" xref="S2.E5.m1.1.1.3.cmml">ϵ</mi></mfrac></mrow><mo rspace="4.2pt" id="S2.E5.m1.4.4.1.1.2" xref="S2.E5.m1.4.4.1.2.cmml">,</mo><mi id="S2.E5.m1.3.3" xref="S2.E5.m1.3.3.cmml">depleted</mi></mrow></mrow></math>, <math><mrow id="S2.E6.m1.3.3" xref="S2.E6.m1.3.3.cmml"><mrow id="S2.E6.m1.3.3.3" xref="S2.E6.m1.3.3.3.cmml"><mrow id="S2.E6.m1.3.3.3.2" xref="S2.E6.m1.3.3.3.2.cmml"><msup id="S2.E6.m1.3.3.3.2.1" xref="S2.E6.m1.3.3.3.2.1.cmml"><mo id="S2.E6.m1.3.3.3.2.1.2" xref="S2.E6.m1.3.3.3.2.1.2.cmml">∇</mo><mn id="S2.E6.m1.3.3.3.2.1.3" xref="S2.E6.m1.3.3.3.2.1.3.cmml">2</mn></msup><mo id="S2.E6.m1.3.3.3.2a" xref="S2.E6.m1.3.3.3.2.cmml">⁡</mo><mi id="S2.E6.m1.3.3.3.2.2" xref="S2.E6.m1.3.3.3.2.2.cmml">ϕ</mi></mrow><mo id="S2.E6.m1.3.3.3.1" xref="S2.E6.m1.3.3.3.1.cmml">⁢</mo><mrow id="S2.E6.m1.3.3.3.3.2" xref="S2.E6.m1.1.1.cmml"><mo stretchy="false" id="S2.E6.m1.3.3.3.3.2.1" xref="S2.E6.m1.1.1.cmml">(</mo><mover accent="true" id="S2.E6.m1.1.1" xref="S2.E6.m1.1.1.cmml"><mi id="S2.E6.m1.1.1.2" xref="S2.E6.m1.1.1.2.cmml">r</mi><mo id="S2.E6.m1.1.1.1" xref="S2.E6.m1.1.1.1.cmml">→</mo></mover><mo stretchy="false" id="S2.E6.m1.3.3.3.3.2.2" xref="S2.E6.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E6.m1.3.3.2" xref="S2.E6.m1.3.3.2.cmml">=</mo><mrow id="S2.E6.m1.3.3.1.1" xref="S2.E6.m1.3.3.1.2.cmml"><mn id="S2.E6.m1.2.2" xref="S2.E6.m1.2.2.cmml">0</mn><mo rspace="4.2pt" id="S2.E6.m1.3.3.1.1.2" xref="S2.E6.m1.3.3.1.2.cmml">,</mo><mrow id="S2.E6.m1.3.3.1.1.1" xref="S2.E6.m1.3.3.1.1.1.cmml"><mi id="S2.E6.m1.3.3.1.1.1.2" xref="S2.E6.m1.3.3.1.1.1.2.cmml">non</mi><mo id="S2.E6.m1.3.3.1.1.1.1" xref="S2.E6.m1.3.3.1.1.1.1.cmml">-</mo><mi id="S2.E6.m1.3.3.1.1.1.3" xref="S2.E6.m1.3.3.1.1.1.3.cmml">depleted</mi></mrow></mrow></mrow></math>, <math><mrow id="S2.E7.m1.2.2" xref="S2.E7.m1.2.2.cmml"><mrow id="S2.E7.m1.2.2.2" xref="S2.E7.m1.2.2.2.cmml"><mrow id="S2.E7.m1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.cmml"><mi id="S2.E7.m1.1.1.1.1.3" xref="S2.E7.m1.1.1.1.1.3.cmml">ϕ</mi><mo id="S2.E7.m1.1.1.1.1.2" xref="S2.E7.m1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E7.m1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E7.m1.1.1.1.1.1.1.2" xref="S2.E7.m1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.E7.m1.1.1.1.1.1.1.1" xref="S2.E7.m1.1.1.1.1.1.1.1.cmml"><mover accent="true" id="S2.E7.m1.1.1.1.1.1.1.1.2" xref="S2.E7.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S2.E7.m1.1.1.1.1.1.1.1.2.2" xref="S2.E7.m1.1.1.1.1.1.1.1.2.2.cmml">r</mi><mo id="S2.E7.m1.1.1.1.1.1.1.1.2.1" xref="S2.E7.m1.1.1.1.1.1.1.1.2.1.cmml">→</mo></mover><mrow id="S2.E7.m1.1.1.1.1.1.1.1.3" xref="S2.E7.m1.1.1.1.1.1.1.1.3.cmml"><mi mathvariant="normal" id="S2.E7.m1.1.1.1.1.1.1.1.3.2" xref="S2.E7.m1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S2.E7.m1.1.1.1.1.1.1.1.3.1" xref="S2.E7.m1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E7.m1.1.1.1.1.1.1.1.3.3" xref="S2.E7.m1.1.1.1.1.1.1.1.3.3.cmml">n</mi><mo id="S2.E7.m1.1.1.1.1.1.1.1.3.1a" xref="S2.E7.m1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E7.m1.1.1.1.1.1.1.1.3.4" xref="S2.E7.m1.1.1.1.1.1.1.1.3.4.cmml">t</mi></mrow></msub><mo stretchy="false" id="S2.E7.m1.1.1.1.1.1.1.3" xref="S2.E7.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E7.m1.2.2.2.3" xref="S2.E7.m1.2.2.2.3.cmml">-</mo><mrow id="S2.E7.m1.2.2.2.2" xref="S2.E7.m1.2.2.2.2.cmml"><mi id="S2.E7.m1.2.2.2.2.3" xref="S2.E7.m1.2.2.2.2.3.cmml">ϕ</mi><mo id="S2.E7.m1.2.2.2.2.2" xref="S2.E7.m1.2.2.2.2.2.cmml">⁢</mo><mrow id="S2.E7.m1.2.2.2.2.1.1" xref="S2.E7.m1.2.2.2.2.1.1.1.cmml"><mo stretchy="false" id="S2.E7.m1.2.2.2.2.1.1.2" xref="S2.E7.m1.2.2.2.2.1.1.1.cmml">(</mo><msub id="S2.E7.m1.2.2.2.2.1.1.1" xref="S2.E7.m1.2.2.2.2.1.1.1.cmml"><mover accent="true" id="S2.E7.m1.2.2.2.2.1.1.1.2" xref="S2.E7.m1.2.2.2.2.1.1.1.2.cmml"><mi id="S2.E7.m1.2.2.2.2.1.1.1.2.2" xref="S2.E7.m1.2.2.2.2.1.1.1.2.2.cmml">r</mi><mo id="S2.E7.m1.2.2.2.2.1.1.1.2.1" xref="S2.E7.m1.2.2.2.2.1.1.1.2.1.cmml">→</mo></mover><mrow id="S2.E7.m1.2.2.2.2.1.1.1.3" xref="S2.E7.m1.2.2.2.2.1.1.1.3.cmml"><mi mathvariant="normal" id="S2.E7.m1.2.2.2.2.1.1.1.3.2" xref="S2.E7.m1.2.2.2.2.1.1.1.3.2.cmml">e</mi><mo id="S2.E7.m1.2.2.2.2.1.1.1.3.1" xref="S2.E7.m1.2.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E7.m1.2.2.2.2.1.1.1.3.3" xref="S2.E7.m1.2.2.2.2.1.1.1.3.3.cmml">x</mi><mo id="S2.E7.m1.2.2.2.2.1.1.1.3.1a" xref="S2.E7.m1.2.2.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E7.m1.2.2.2.2.1.1.1.3.4" xref="S2.E7.m1.2.2.2.2.1.1.1.3.4.cmml">t</mi></mrow></msub><mo stretchy="false" id="S2.E7.m1.2.2.2.2.1.1.3" xref="S2.E7.m1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E7.m1.2.2.3" xref="S2.E7.m1.2.2.3.cmml">=</mo><msub id="S2.E7.m1.2.2.4" xref="S2.E7.m1.2.2.4.cmml"><mi id="S2.E7.m1.2.2.4.2" xref="S2.E7.m1.2.2.4.2.cmml">V</mi><mn id="S2.E7.m1.2.2.4.3" xref="S2.E7.m1.2.2.4.3.cmml">0</mn></msub></mrow></math>

Correct Categorie: hep-ex

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/physics/0507123

Formulas:

1-

n_{z}^{3} r_{33} - n_{y}^{3} r_{23}

2-

V_{o} \sin ω t

3-

L_{k} = l_{o} + n_{k} l,

4-

l_{o} + l = 10

5-

Δ_{k} = \frac{c}{2 L_{k}}

6-

ν_{k} = m Δ_{k}

7-

\frac{n_{y}^{3}}{2} r_{23} E_{z}

8-

\frac{n_{z}^{3}}{2} r_{33} E_{z}

9-

δ ν_{k} ≃ - \frac{2 m l}{c} Δ_{k}^{2} δ n_{k} .

10-

\frac{c}{n_{y}^{3} l Δ_{y} ν_{y}} \frac{δ ν_{y}}{E_{z}}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0701577

Formulas:

1-

\begin{matrix} H_{0} = & - \sum_{⟨ i j ⟩, σ} t_{i j} (c_{i σ}^{†} c_{j σ} + c_{j σ}^{†} c_{i σ}) + U \sum_{i} n_{i ↑} n_{i ↓} \\ + \frac{1}{2} \sum_{i \neq j} V_{i j} (n_{i} - 1) (n_{j} - 1) . \end{matrix}

2-

n_{i σ} = c_{i σ}^{†} c_{i σ}

3-

n_{i} = \sum_{σ} n_{i σ}

4-

H = H_{0} + e F z = H_{0} + μ F,

5-

α (ω; 0)

6-

α (ω; F)

7-

Δ α (ω; F) = α (ω; F) - α (ω; 0)

8-

Δ E_{E x 1}

9-

Δ E_{E x 1} = \sum_{j} \frac{{| ⟨ E x 1 | μ | j A_{g} ⟩ |}^{2} F^{2}}{E_{E x 1} - E_{j A_{g}}},

10-

Δ E_{E x 1}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1608.07520

Formulas:

1-

𝔻 := {z \in ℂ : | z | < 1}

2-

f (0) = 0 = f^{'} (0) - 1

3-

\log \frac{f (z)}{z} = 2 \sum_{n = 1}^{\infty} γ_{n} z^{n}

4-

𝔻 := {z \in ℂ : | z | < 1}

5-

f (z) = z + \sum_{n = 2}^{\infty} a_{n} z^{n} .

6-

Re (z f^{'} (z) / f (z)) > 0

7-

Re (1 + (z f^{''} (z) / f^{'} (z))) > 0

8-

z f^{'} \in 𝒮^{*}

9-

α \in (- π / 2, π / 2)

10-

Re (e^{i α} \frac{z f^{'} (z)}{g (z)}) > 0, z \in 𝔻 .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1210.8383

Formulas:

1-

\hat{H} = \sum_{i = 1}^{m} ε_{i} {\hat{n}}_{i} + g \sum_{i, j = 1}^{m} {\hat{S}}_{i}^{†} S_{j},

2-

{i, j = 1 \dots m}

3-

{\hat{n}}_{i} = a_{i}^{†} a_{i} + a_{\bar{i}}^{†} a_{\bar{i}},

4-

{\hat{S}}_{i}^{†} = a_{i}^{†} a_{\bar{i}}^{†}, {\hat{S}}_{i} = {({\hat{S}}_{i}^{†})}^{†} = a_{\bar{i}} a_{i},

5-

s u (2)

6-

[{\hat{S}}_{i}^{0}, {\hat{S}}_{j}^{†}] = δ_{i j} S_{i}^{†}, [{\hat{S}}_{i}^{0}, {\hat{S}}_{j}] = - δ_{i j} S_{i}, [{\hat{S}}_{i}^{†}, {\hat{S}}_{j}] = 2 δ_{i j} S_{i}^{0},

7-

{\hat{S}}_{i}^{0} = \frac{1}{2} ({\hat{n}}_{i} - 1)

8-

s u (2)

9-

| ψ ({x}) ⟩ = \prod_{α = 1}^{N} (\sum_{i = 1}^{m} \frac{{\hat{S}}_{i}^{†}}{2 ε_{i} - x_{α}}) | θ ⟩,

10-

{x} = {x_{1}, x_{2}, \dots x_{N}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1806.00756

Formulas:

1-

N_{1} = L \times L

2-

M_{i n t e r}

3-

M_{i n t e r}

4-

M_{i n t e r}

5-

S (p, r)

6-

[p, r; L]

7-

S (p, r)

8-

S (p, r)

9-

S (p, r)

10-

M_{i n t e r} = 2 \times L \times L

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0608506

Formulas:

1-

τ_{e} \sim N^{1 + 2 ν}

2-

τ \sim N^{2 ν}

3-

τ \sim N^{1 + ν}

4-

τ \sim N^{1 + ν}

5-

τ_{e q u i l} \sim N^{1 + 2 ν}

6-

τ_{e} \sim N^{1 + 2 ν}

7-

τ_{e} \sim N^{1 + 2 ν}

8-

τ \sim N^{2 ν}

9-

τ \sim N^{1 + ν}

10-

U_{L J} (r) = {\begin{matrix} 4 ε [{(\frac{σ}{r})}^{12} - {(\frac{σ}{r})}^{6}] + ε, & r \leq 2^{1 / 6} σ; \\ 0, & r > 2^{1 / 6} σ, \end{matrix}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1808.06317

Formulas:

1-

R_{V_{G} = 3.5 V} (T)

2-

A E_{x} M_{y}

3-

R (30 K)

4-

R (50 K)

5-

R_{HT} (T)

6-

R_{LT} (T)

7-

R_{HT} (T)

8-

R_{HT} (T)

9-

R_{LT} (T)

10-

R_{HT} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/0507068

Formulas:

1-

\ln (p_{T}^{2} / m^{2})

2-

\ln (p_{T}^{2} / m^{2})

3-

𝒪 (m^{2} / p_{T}^{2})

4-

d = 4 - 2 ϵ

5-

\ln (p_{T}^{2} / m^{2})

6-

m^{2} / p_{T}^{2}

7-

X_{c} = D^{0}, D^{⋆ +}, D^{+}, D_{s}^{+}

8-

(X_{c} + {\bar{X}}_{c}) / 2

9-

μ_{R} = μ_{F} = μ_{F}^{'} = m_{T}

10-

0.5 \leq μ_{F} / μ_{R}, μ_{F}^{'} / μ_{R}, μ_{F} / μ_{F}^{'} \leq 2

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/0302120

Formulas:

1-

d s^{2} = - d t^{2} + a^{2} (t) (\frac{1}{1 - k r^{2}} d r^{2} + r^{2} d θ^{2} + r^{2} \sin^{2} θ d ϕ^{2})

2-

- 1, 0, 1

3-

V (ϕ) = m^{2} ϕ^{2} / 2

4-

- 3 \frac{k + {\dot{a}}^{2}}{a^{2}} + ρ_{ϕ} = 0

5-

\frac{k + {\dot{a}}^{2} + 2 a \ddot{a}}{a^{2}} + p_{ϕ} = 0

6-

\ddot{ϕ} + m^{2} ϕ + \frac{3 \dot{a}}{a} ϕ = 0

7-

ρ_{ϕ} = \frac{1}{2} ({\dot{ϕ}}^{2} + m^{2} ϕ^{2}) \equiv \frac{1}{2} {\dot{ϕ}}^{2} + V (ϕ),

8-

p_{ϕ} = \frac{1}{2} ({\dot{ϕ}}^{2} - m^{2} ϕ^{2}) \equiv \frac{1}{2} {\dot{ϕ}}^{2} - V (ϕ) .

9-

G_{0}^{0} = T_{0}^{0}

10-

G_{1}^{1} = T_{1}^{1}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/0905.3579

Formulas:

1-

E_{γ} = h ν

2-

W = W_{0} . ν / ν_{0} = W_{0} (z_{0} + 1)

3-

U ≃ 4 \times 10^{- 14} J m^{- 3}

4-

T = {2.7}^{\circ} K

5-

N ≃ 4 \times 10^{8} m^{- 3}

6-

T_{o b s} / T_{e q u i}

7-

T_{o b s}

8-

T_{e q u i}

9-

\approx 0.3 m^{- 3}

10-

η_{γ} \equiv n_{γ} / n_{B} ≃ 1

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1102.0548

Formulas:

1-

(ℳ^{n}, g, f)

2-

R_{i j} + \nabla_{i} \nabla_{j} f - \frac{1}{2} g_{i j} = 0

3-

R + {| \nabla f |}^{2} - f = 0

4-

R (x) ⩾ C_{ϵ} d {(x, O)}^{- 2 - ϵ}

5-

d (x, O)

6-

(ℳ^{n}, g, f)

7-

R (x) d {(x, O)}^{2} ⩾ C_{0}^{- 1}

8-

d (x, O) ⩾ C_{0}

9-

\frac{1}{4} {[{(d (x, O) - C_{1})}_{+}]}^{2} \leq f (x) \leq \frac{1}{4} {(d (x, O) + 2 f {(O)}^{\frac{1}{2}})}^{2},

10-

c_{+} ≑ \max (c, 0)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1311.1240

Formulas:

1-

𝐅 = [f_{i j}], \forall i \in ℳ, j \in 𝒩

2-

f_{i j} = 0

3-

f_{i j} = 1

4-

f_{i j} = - 1

5-

j \in ℒ ∖ 𝒲_{i}

6-

(ℳ ∖ 𝒯_{κ_{ρ}}) \cup ℛ

7-

j \in ℋ_{k} and l \in ℋ_{i} and ℳ_{j} \cap ℳ_{l} = \emptyset

8-

ℳ_{j} \cap ℳ_{l} = \emptyset

9-

{(\frac{| 𝒲_{i} |}{1 - p_{i}})}^{n}

10-

κ_{g} (κ_{s}^{*})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1305.5154

Formulas:

1-

χ_{T} (T)

2-

χ_{T} = \frac{𝒞 (T)}{T} .

3-

¯ d W = μ_{0} \int [𝐇_{0} (𝐫) + 𝐇_{1} (𝐫)] \cdot 𝑑 𝐌 (𝐫) 𝑑 𝐫,

4-

¯ d W = μ_{0} V H_{int} d M,

5-

H_{int} = H_{0} - 𝒟 M

6-

χ_{T} = \frac{M}{H_{int}} .

7-

μ_{0} H_{0} = 0.005

8-

= 1.5 \times 10^{- 3}

9-

= 1.2 \times 10^{- 3}

10-

χ_{T} T / C

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0411457

Formulas:

1-

G a A s

2-

j_{j}^{i} = σ_{s} ϵ_{i j k} E_{k}

3-

ϵ_{i j k}

4-

\vec{S} = 3 / 2

5-

H = (γ_{1} + \frac{5}{2} γ_{2}) \frac{k^{2}}{2 m} - \frac{γ_{2}}{m} {(\vec{k} \cdot \vec{S})}^{2} + α (\vec{S} \times \vec{k}) \cdot \hat{z}

6-

< k_{z} > = 0, ⟨ k_{z}^{2} ⟩ \approx {(π ℏ / a)}^{2}

7-

E_{\pm}^{H H} = \frac{γ_{1}}{2 m} k^{2} \pm \frac{1}{2} α k - \sqrt{α^{2} k^{2} \pm \frac{α γ_{2}}{m} k (k^{2} + ⟨ k_{z}^{2} ⟩) + \frac{γ_{2}^{2}}{m^{2}} (k^{4} + {⟨ k_{z}^{2} ⟩}^{2} - k^{2} ⟨ k_{z}^{2} ⟩)}

8-

E_{\pm}^{L H} = \frac{γ_{1}}{2 m} k^{2} \pm \frac{1}{2} α k + \sqrt{α^{2} k^{2} \pm \frac{α γ_{2}}{m} k (k^{2} + ⟨ k_{z}^{2} ⟩) + \frac{γ_{2}^{2}}{m^{2}} (k^{4} + {⟨ k_{z}^{2} ⟩}^{2} - k^{2} ⟨ k_{z}^{2} ⟩)}

9-

Δ = 2 γ_{2} ⟨ k_{z}^{2} ⟩ / m

10-

k << ⟨ k_{z} ⟩

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

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