Run 6938496

Paper: https://arxiv.org/abs/astro-ph/0602492

Formulas:

1-

\frac{ω_{dyn}}{Ω_{b}} = \frac{q}{{(1 + q)}^{1 / 2}} (\frac{1}{4} {(\frac{r}{a})}^{1 / 2} b_{3 / 2}^{(1)} (\frac{r}{a}))

2-

b_{3 / 2}^{(1)}

3-

ε = \frac{ω_{prec}}{Ω_{b} - ω_{prec}} .

4-

ε ≃ \frac{ω_{prec}}{Ω_{b}} + {(\frac{ω_{prec}}{Ω_{b}})}^{2}

5-

ε = (0.216 \pm 0.018) q .

6-

⟨ r ⟩ ≃ 0.37 a

7-

\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + \frac{v}{r} \frac{\partial u}{\partial ϕ} - \frac{v^{2}}{r} = - \frac{1}{ρ} \frac{\partial p}{\partial r} - \frac{\partial Φ}{\partial r}

8-

\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial r} + \frac{v}{r} \frac{\partial v}{\partial ϕ} + \frac{u v}{r} = - \frac{1}{r ρ} \frac{\partial p}{\partial ϕ} - \frac{1}{r} \frac{\partial Φ}{\partial ϕ}

9-

\frac{\partial ρ}{\partial t} + u \frac{\partial ρ}{\partial r} + \frac{v}{r} \frac{\partial ρ}{\partial ϕ} = - \frac{ρ}{r} (\frac{\partial (r u)}{\partial r} + \frac{\partial v}{\partial ϕ})

10-

\frac{\partial p}{\partial t} + u \frac{\partial p}{\partial r} + \frac{v}{r} \frac{\partial p}{\partial ϕ} = - \frac{γ p}{r} (\frac{\partial (r u)}{\partial r} + \frac{\partial v}{\partial ϕ}) .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1007.0314

Formulas:

1-

| \pm ⟩ = \frac{1}{\sqrt{2}} (| 0 ⟩ \pm | 1 ⟩)

2-

{| Ψ ⟩}_{SP} = \frac{1}{\sqrt{2}} ({| 0 ⟩}_{S} {| 0 ⟩}_{P} \pm {| 1 ⟩}_{S} {| 1 ⟩}_{P})

3-

{| Ψ ⟩}_{SP} = \frac{1}{\sqrt{2}} ({| 0 ⟩}_{S} {| 1 ⟩}_{P} \pm {| 1 ⟩}_{S} {| 0 ⟩}_{P})

4-

{\hat{ρ}}_{S} = \frac{1}{2} ({| 0 ⟩}_{S} ⟨ 0 | + {| 1 ⟩}_{S} ⟨ 1 |)

5-

{| \pm ⟩}_{S} \to {| \mp ⟩}_{S}

6-

{| ψ ⟩}_{S} = \int 𝑑 x ψ (x) {| x ⟩}_{S}

7-

\hat{U} = \exp (i {\hat{x}}_{S} {\hat{p}}_{P} / ℏ)

8-

{| x = 0 ⟩}_{P}

9-

{| Ψ ⟩}_{SP} = \int 𝑑 x ψ (x) {| x ⟩}_{S} {| x ⟩}_{P}

10-

{\hat{ρ}}_{S} = \int 𝑑 x {| ψ (x) |}^{2} {| x ⟩}_{S} ⟨ x |

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1801.08226

Formulas:

1-

p_{⟂} ≫ 2 m_{t}

2-

g g \to H g

3-

q g \to H q

4-

q \bar{q} \to H g

5-

g g \to H g g

6-

q g \to H q g

7-

q \bar{q} \to H g g

8-

\begin{matrix} \frac{m_{t}}{v} \bar{t} t H \to \\ - κ_{g} \frac{α_{s}}{12 π v} G_{μ ν}^{a} G^{μ ν, a} H + κ_{t} \frac{m_{t}}{v} \bar{t} t H . \end{matrix}

9-

σ_{g g \to H} \sim α_{s}^{2} / v^{2} {(κ_{g} + κ_{t})}^{2}

10-

σ_{g g \to H}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/9605025

Formulas:

1-

| t | < \infty .

2-

(R, π_{R})

3-

(0, \infty) \times (- \infty, 0)

4-

(0, \infty) \times (0, \infty),

5-

H (R, π_{R}) = \frac{1}{\sqrt{12}} R | π_{R} | .

6-

L^{2} (0, \infty) \oplus L^{2} (0, \infty)

7-

\hat{H} = {\hat{H}}_{-} \oplus {\hat{H}}_{+}

8-

{\hat{H}}_{\pm} = \mp \frac{i ℏ}{\sqrt{12}} (R \frac{d}{d R} + \frac{1}{2}) .

9-

U : L^{2} (0, \infty) \to L^{2} (𝐑)

10-

(U ψ) (y) = e^{- y / 2} ψ (e^{- y})

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0705.1823

Formulas:

1-

{| ⟨ ψ (0) | ψ (t) ⟩ |}^{2}

2-

⟨ ψ (0) | H^{n} | ψ (0) ⟩

3-

P (t) := {| ⟨ ψ | \exp (- ı H t / ℏ) ψ ⟩ |}^{2} = {| ⟨ \exp (- ı H t / ℏ) ⟩ |}^{2},

4-

Δ E := {⟨ {(H - ⟨ H ⟩)}^{2} ⟩}^{1 / 2}

5-

P (t) ⩽ 1

6-

P (t) = 1 - (h_{2} - h_{1}^{2}) \frac{t^{2}}{ℏ^{2}} + (h_{4} - 4 h_{3} h_{1} + 3 h_{2}^{2}) \frac{t^{4}}{12 ℏ^{4}} - \dots,

7-

h_{n} := ⟨ H^{n} ⟩

8-

| ϕ_{E, κ} ⟩

9-

H | ϕ_{E, κ} ⟩ = E | ϕ_{E, κ} ⟩

10-

K | ϕ_{E, κ} ⟩ = κ | ϕ_{E, κ} ⟩

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1101.1472

Formulas:

1-

α a^{2} N

2-

\frac{F N}{k_{B} T ρ_{0} V} = \frac{F_{h} N}{k_{B} T ρ_{0} V}

3-

- χ N (ϕ_{A} (𝐫) + ϕ_{A2} (𝐫) + ϕ_{S} (𝐫) - {\bar{ϕ}}_{A} - {\bar{ϕ}}_{A2} - {\bar{ϕ}}_{S}) (ϕ_{B} (𝐫) + ϕ_{B2} (𝐫) - {\bar{ϕ}}_{B} - {\bar{ϕ}}_{B2})

4-

- ({\bar{ϕ}}_{A} + {\bar{ϕ}}_{B}) \ln (Q_{AB} / V) - [({\bar{ϕ}}_{A2} + {\bar{ϕ}}_{B2}) / α] \ln (Q_{AB2} / V) - {\bar{ϕ}}_{S} \ln (Q_{S} / V)

5-

ϕ_{i} (𝐫)

6-

Q_{S} [W_{A}] = \int d 𝐫 q_{S} (𝐫, s) q_{S}^{†} (𝐫, s)

7-

\frac{\partial}{\partial s} q_{S} (𝐫, s) = [\frac{1}{6} a^{2} N \nabla^{2} - W_{A} (𝐫)] q_{S} (𝐫, s)

8-

q_{S} (𝐫, 0) = 1

9-

W_{i} (𝐫)

10-

w_{A} (𝐫) - w_{B} (𝐫) = 2 χ N [{\bar{ϕ}}_{A} + {\bar{ϕ}}_{A2} + {\bar{ϕ}}_{S} - ϕ_{A} (𝐫) - ϕ_{A2} (𝐫) - ϕ_{S} (𝐫)]

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1601.08009

Formulas:

1-

PG (2, 𝕂)

2-

PG (2, 𝕂)

3-

PG (2, 𝕂)

4-

PG (2, 𝕂)

5-

PG (2, 𝕂)

6-

PG (2, 𝕂)

7-

PG (2, 𝕂)

8-

PG (2, 𝕂)

9-

PG (2, 𝕂)

10-

(Λ_{1}, Λ_{2}, Λ_{3})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1406.0719

Formulas:

1-

d μ (e^{i (2 π - θ)}) = - d μ (e^{i θ})

2-

{z S_{n} (z) + S_{n}^{*} (z)}

3-

{z S_{n} (z) - S_{n}^{*} (z)}

4-

[- 1, 1]

5-

z S_{n} (z) - S_{n}^{*} (z)

6-

z S_{n} (z) + S_{n}^{*} (z)

7-

μ (ζ) = μ (e^{i θ})

8-

𝒞 = {ζ = e^{i θ} : 0 \leq θ \leq 2 π}

9-

{S_{n} (z)}

10-

\begin{matrix} \int_{𝒞} \bar{S_{m} (ζ)} S_{n} (ζ) 𝑑 μ (ζ) = \int_{0}^{2 π} \bar{S_{m} (e^{i θ})} S_{n} (e^{i θ}) 𝑑 μ (e^{i θ}) = δ_{m n} κ_{n}^{- 2} . \end{matrix}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1108.0337

Formulas:

1-

{F_{n}}_{n = - \infty}^{\infty}

2-

F_{n} = {\begin{matrix} 0, & if n < 0; \\ 1, & if n = 0; \\ \sum_{a = 1}^{k} F_{n - a}, & if n > 0, \end{matrix}

3-

(λ_{1}, λ_{2}, \dots, λ_{m})

4-

\sum_{j = 1}^{m} λ_{j} = n

5-

L = {1, 2, \dots, k}

6-

{1, 2, \dots, k}

7-

{1, 2, \dots, k}

8-

A_{n} = T_{n} / F_{n}

9-

{1, 2, \dots, k}

10-

j \in {1, 2, \dots, k}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1802.09856

Formulas:

1-

{4, 5, \dots, s + 3}

2-

π = π_{1} π_{2} \dots π_{n}

3-

{1, 2, \dots, n}

4-

σ = σ_{1} σ_{2} \dots σ_{r}

5-

{1, 2, \dots, r}

6-

1 \leq i_{1} < i_{2} < \dots < i_{r} \leq n

7-

π_{i_{1}} π_{i_{2}} \dots π_{i_{r}}

8-

σ_{1} σ_{2} \dots σ_{r}

9-

{1, 2, \dots, n}

10-

S_{n} (σ)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1112.1614

Formulas:

1-

[Fe / H] = - 1.83

2-

[Fe / H] = - 2.34

3-

[Fe / H] = - 1.0

4-

14' \times 14'

5-

30 ″ < R < 60 ″

6-

60 ″ < R < 120 ″

7-

0 .^{''} 39

8-

30 ″ < R < 60 ″

9-

60 ″ < R < 120 ″

10-

n (r) = k {\frac{1}{\sqrt{1 + {(r / r_{c})}^{2}}} - \frac{1}{\sqrt{1 + {(r_{t} / r_{c})}^{2}}}}^{2} + b

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0004185

Formulas:

1-

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

2-

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

3-

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

4-

d_{x^{2} - y^{2}} + i s

5-

Φ_{0} / 2 = h c / 4 e = 1.035 \times 10

6-

G - c m^{2}

7-

L ∣ I_{c} ∣ >> Φ_{0}

8-

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

9-

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

10-

Φ = Φ_{0} / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1902.00324

Formulas:

1-

(\nabla \times B) \times B = 0

2-

B_{L} = C_{L} V,

3-

B_{T} = C_{T} {(Q^{2} + U^{2})}^{1 / 4},

4-

θ = a r c t a n (\frac{B_{L}}{B_{T}}),

5-

ϕ = \frac{1}{2} a r c t a n (\frac{U}{Q}),

6-

L = \int_{V} ω (x, y, z) [B^{- 2} {| (\nabla \times 𝐁 \times 𝐁) |}^{2} + {| \nabla \cdot 𝐁 |}^{2}] d^{3} x,

7-

ω (x, y, z)

8-

𝐁 (x, t)

9-

β γ δ / β γ δ

10-

a t a n (B_{z}, B_{t}), B t = \sqrt{B_{x}^{2} + B_{y}^{2}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1306.4817

Formulas:

1-

⟨ T (σ, N) ⟩

2-

σ_{c} (N)

3-

σ_{c} (\infty)

4-

σ_{c} (N) = σ_{c} (\infty) + A N^{- 1 / ν}

5-

⟨ T (σ_{c} (N), N) ⟩

6-

⟨ T (σ_{c} (N), N) ⟩ \sim N^{η}

7-

| Δ σ | \sim N^{- ζ}

8-

1 / N < | Δ σ | < 100 N^{- ζ}

9-

⟨ T (σ, N) ⟩ / N^{η} \sim 𝒢 [{σ_{c} (N) - σ} N^{ζ}]

10-

⟨ T (σ) ⟩ \sim {(σ - σ_{c})}^{- τ}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0411801

Formulas:

1-

ϵ = \frac{1}{2} \frac{ℓ^{2}}{r^{2}} + \frac{1}{2} v^{2} + \frac{1}{γ - 1} a^{2} - \frac{G M}{r - r_{_{S}}},

2-

r_{_{S}} = 2 G M / c^{2}

3-

a = {(γ P / ρ)}^{1 / 2}

4-

K \equiv v a^{(γ + 1) (γ - 1)} r^{3 / 2} (r - r_{_{S}}),

5-

(1 / v) (d v / d r) = N / D

6-

N \equiv - \frac{ℓ^{2}}{r^{3}} + \frac{G M}{{(r - r_{_{S}})}^{2}} + \frac{a^{2} (3 r_{_{S}} - 5 r)}{(γ + 1) r (r - r_{_{S}})}, D \equiv \frac{2 a^{2}}{γ + 1} - v^{2} .

7-

R_{*}^{- 1} \equiv \frac{v_{+}}{v_{-}} = \frac{1}{γ ℳ_{-}^{2}}, Δ ϵ \equiv ϵ_{+} - ϵ_{-} = \frac{v_{+}^{2} - v_{-}^{2}}{2},

8-

ℳ_{-} \equiv v_{-} / a_{-}

9-

{\hat{r}}_{c 3} < r_{*}

10-

L_{jet} \equiv \dot{M} c^{2} △ ϵ,

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0007152

Formulas:

1-

0.1 {\dot{M}}_{in} c^{2} \sim (10^{- 4} - 10^{- 6}) \times L_{x, obs}

2-

T_{eff} (R)

3-

T_{eff} (R) < T_{C} (R) \overset{<}{\sim} few \times 10^{3}

4-

{\dot{M}}_{B} (R)

5-

\dot{M} (R) > {\dot{M}}_{C}

6-

T_{eff} (R) > T_{C} (R)

7-

T_{B} (R)

8-

T_{C} (R)

9-

T_{B} (R)

10-

T_{C} (R)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/9307038

Formulas:

1-

l o c a l

2-

p e r i o d i c

3-

O_{k} / O_{k + 1}

4-

1, 2, 5, 12 \dots O_{k} . .

5-

g i v e n

6-

G_{i i} (z)

7-

G_{i i} (z) = \frac{1}{z - \frac{b_{1}}{z - \frac{b_{2}}{z - \dots}}}

8-

{(z - H)}^{- 1}

9-

b_{\infty} = Δ^{2} / 16

10-

Δ = 2 E_{m a x}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0008149

Formulas:

1-

p_{T} \sim 1 \dots 2

2-

≫ Λ_{Q C D}

3-

{\bar{E}}_{A A}^{T}

4-

T_{A A} (𝟎) σ ⟨ E_{T} ⟩

5-

T_{A A} (𝟎) \sum_{q, \bar{q}, g} \int_{p_{0}, Δ Y} 𝑑 p_{t} 𝑑 y_{1} 𝑑 y_{2} x_{1} f_{i / p} (x_{1}, Q^{2}) x_{2} f_{j / p} (x_{2}, Q^{2}) \frac{d {\hat{σ}}^{i j \to j k}}{d \hat{t}} p_{T},

6-

T_{A A} (𝐛)

7-

σ ⟨ E_{T} ⟩

8-

σ ⟨ E_{T} ⟩

9-

(p_{0}^{2} R_{A}^{2})

10-

K^{'} = (full NLO) / {LO}^{'}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/0912.1754

Formulas:

1-

ϕ (V_{c})

2-

ϕ (V_{c})

3-

V_{c} \approx 100 km s^{- 1}

4-

V_{c} = 30 km s^{- 1}

5-

ϕ (V_{c})

6-

ϕ (V_{c})

7-

ϕ (V_{c})

8-

ϕ (V_{c})

9-

ϕ (V_{c})

10-

5.5 \times 10^{7} M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1004.0126

Formulas:

1-

q_{x} = 2 π / L_{x}

2-

q_{y} = 2 π / L_{y}

3-

a_{S L} = 0.24595

4-

a_{B L} = 0.24583

5-

α_{a} = d \ln a / d T = (- 3.0 \pm 0.7) \cdot 10^{- 6} K^{- 1}

6-

α_{a} = (- 4.8 \pm 1.0) \cdot 10^{- 6} K^{- 1}

7-

α_{c} = d \ln c / d T = (3.5 \pm 0.5) \cdot 10^{- 5} K^{- 1}

8-

α_{c} = 2.7 \cdot 10^{- 5} K^{- 1}

9-

u_{α} (\vec{x}), α = 1, 2

10-

h (\vec{x})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1904.07575

Formulas:

1-

| 𝐓_{S t} | / | 𝐓_{I} | \sim R, with R \equiv \frac{{| 𝐮 - 𝐯 |}^{2}}{ν | 𝛀 - 𝝎 |},

2-

R \approx {(W_{s} / U_{0})}^{2} R e_{f}

3-

R e_{f} = U_{0} L / ν

4-

W_{s} / U_{0} ≳ 1

5-

R e_{f} {(W_{s} / U_{0})}^{2} < 1

6-

W_{s} / U_{0} ≳ 1

7-

R \sim {(W_{s} / U_{0})}^{2} R e_{f}^{1 / 2}

8-

R e_{f} ≫ 1

9-

W_{s} / U_{0} ≳ 1

10-

ν = μ / ρ_{f}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9211066

Formulas:

1-

{\bar{g}}^{2} (L)

2-

{\bar{g}}^{2} (L)

3-

{\bar{g}}^{2} (L)

4-

u = {\bar{g}}^{2} (L)

5-

u^{'} = {\bar{g}}^{2} (s L)

6-

σ (s, u) = u^{'},

7-

u_{0} = {\bar{g}}^{2} (L_{0}) .

8-

u_{i + 1} = σ (2, u_{i}); i = 0, 1, 2, \dots, n - 1 .

9-

{\bar{g}}^{2} (L_{i}) = u_{i},

10-

L_{i} = 2^{i} L_{0} .

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0708.2374

Formulas:

1-

λ^{- 4} \propto ⟨ Δ B^{2} ⟩ \propto σ_{s c}^{2}

2-

σ_{s c} \propto λ^{- 2}

3-

σ_{s c}^{2} = σ^{2} - σ_{n m}^{2}

4-

σ_{a b} \propto λ_{a b}^{- 2}

5-

σ_{s c} \propto λ^{- 2}

6-

i, j, k = a, b, c

7-

λ_{j k}^{- 2} = {(λ_{j} λ_{k})}^{- 1} \propto σ_{j k} = \sqrt{σ_{j} σ_{k}}

8-

σ_{a b} \propto λ_{a b}^{- 2} = {(λ_{a} λ_{b})}^{- 1}

9-

σ_{a b} (T)

10-

σ (T) = σ^{s} (T) + σ^{d} (T)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0711.3166

Formulas:

1-

V = 1 - ϕ_{3}

2-

V = \frac{1}{2} (1 - ϕ_{3}^{2})

3-

\vec{ϕ} = (ϕ_{1}, ϕ_{2}, ϕ_{3})

4-

\vec{ϕ} = \vec{ϕ} (x, y)

5-

\vec{ϕ} \cdot \vec{ϕ} = 1

6-

x^{j} = (x^{1}, x^{2}) = (x, y)

7-

ℰ [ϕ] = ℰ_{2} [ϕ] + μ ℰ_{4} [ϕ] + μ ℰ_{0} [ϕ],

8-

ℰ_{2} = \frac{1}{2} (\partial_{j} ϕ \cdot \partial_{j} ϕ), ℰ_{4} = \frac{1}{2} {(\partial_{1} ϕ \times \partial_{2} ϕ)}^{2}, ℰ_{0} = V (ϕ),

9-

V (\vec{ϕ}) = (1 - ϕ_{3})

10-

V (\vec{ϕ}) = \frac{1}{2} (1 - ϕ_{3}^{2})

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: hep-th

Result: correct

Paper: https://arxiv.org/abs/1110.6332

Formulas:

1-

R_{300 K} / R_{10 K} \approx 6

2-

n_{p} \approx 1.05 μ

3-

P_{trans} (f)

4-

f_{res} (B)

5-

Δ f (B)

6-

Q (B) = f_{res} (B) / Δ f (B)

7-

1 / Q (B)

8-

1 / Q (B)

9-

1 / Q_{v} (B) = 1 / Q (B) - 1 / Q (0)

10-

1 / Q_{v} (B)

Formulas (html):

<math><mrow id="p3.4.m4.1.1" xref="p3.4.m4.1.1.cmml"><mrow id="p3.4.m4.1.1.2" xref="p3.4.m4.1.1.2.cmml"><msub id="p3.4.m4.1.1.2.2" xref="p3.4.m4.1.1.2.2.cmml"><mi id="p3.4.m4.1.1.2.2.2" xref="p3.4.m4.1.1.2.2.2.cmml">R</mi><mrow id="p3.4.m4.1.1.2.2.3" xref="p3.4.m4.1.1.2.2.3.cmml"><mpadded width="+1.7pt" id="p3.4.m4.1.1.2.2.3.2" xref="p3.4.m4.1.1.2.2.3.2.cmml"><mn id="p3.4.m4.1.1.2.2.3.2a" xref="p3.4.m4.1.1.2.2.3.2.cmml">300</mn></mpadded><mo id="p3.4.m4.1.1.2.2.3.1" xref="p3.4.m4.1.1.2.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="p3.4.m4.1.1.2.2.3.3" xref="p3.4.m4.1.1.2.2.3.3.cmml">K</mi></mrow></msub><mo id="p3.4.m4.1.1.2.1" xref="p3.4.m4.1.1.2.1.cmml">/</mo><msub id="p3.4.m4.1.1.2.3" xref="p3.4.m4.1.1.2.3.cmml"><mi id="p3.4.m4.1.1.2.3.2" xref="p3.4.m4.1.1.2.3.2.cmml">R</mi><mrow id="p3.4.m4.1.1.2.3.3" xref="p3.4.m4.1.1.2.3.3.cmml"><mpadded width="+1.7pt" id="p3.4.m4.1.1.2.3.3.2" xref="p3.4.m4.1.1.2.3.3.2.cmml"><mn id="p3.4.m4.1.1.2.3.3.2a" xref="p3.4.m4.1.1.2.3.3.2.cmml">10</mn></mpadded><mo id="p3.4.m4.1.1.2.3.3.1" xref="p3.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mi mathvariant="normal" id="p3.4.m4.1.1.2.3.3.3" xref="p3.4.m4.1.1.2.3.3.3.cmml">K</mi></mrow></msub></mrow><mo id="p3.4.m4.1.1.1" xref="p3.4.m4.1.1.1.cmml">≈</mo><mn id="p3.4.m4.1.1.3" xref="p3.4.m4.1.1.3.cmml">6</mn></mrow></math>, <math><mrow id="p4.2.m2.1.1" xref="p4.2.m2.1.1.cmml"><msub id="p4.2.m2.1.1.2" xref="p4.2.m2.1.1.2.cmml"><mi id="p4.2.m2.1.1.2.2" xref="p4.2.m2.1.1.2.2.cmml">n</mi><mi id="p4.2.m2.1.1.2.3" xref="p4.2.m2.1.1.2.3.cmml">p</mi></msub><mo id="p4.2.m2.1.1.1" xref="p4.2.m2.1.1.1.cmml">≈</mo><mrow id="p4.2.m2.1.1.3" xref="p4.2.m2.1.1.3.cmml"><mpadded width="+1.7pt" id="p4.2.m2.1.1.3.2" xref="p4.2.m2.1.1.3.2.cmml"><mn id="p4.2.m2.1.1.3.2a" xref="p4.2.m2.1.1.3.2.cmml">1.05</mn></mpadded><mo id="p4.2.m2.1.1.3.1" xref="p4.2.m2.1.1.3.1.cmml">⁢</mo><mi id="p4.2.m2.1.1.3.3" xref="p4.2.m2.1.1.3.3.cmml">μ</mi></mrow></mrow></math>, <math><mrow id="p6.6.m6.1.2" xref="p6.6.m6.1.2.cmml"><msub id="p6.6.m6.1.2.2" xref="p6.6.m6.1.2.2.cmml"><mi id="p6.6.m6.1.2.2.2" xref="p6.6.m6.1.2.2.2.cmml">P</mi><mi id="p6.6.m6.1.2.2.3" xref="p6.6.m6.1.2.2.3.cmml">trans</mi></msub><mo id="p6.6.m6.1.2.1" xref="p6.6.m6.1.2.1.cmml">⁢</mo><mrow id="p6.6.m6.1.2.3.2" xref="p6.6.m6.1.2.cmml"><mo stretchy="false" id="p6.6.m6.1.2.3.2.1" xref="p6.6.m6.1.2.cmml">(</mo><mi id="p6.6.m6.1.1" xref="p6.6.m6.1.1.cmml">f</mi><mo stretchy="false" id="p6.6.m6.1.2.3.2.2" xref="p6.6.m6.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.7.m7.1.2" xref="p6.7.m7.1.2.cmml"><msub id="p6.7.m7.1.2.2" xref="p6.7.m7.1.2.2.cmml"><mi id="p6.7.m7.1.2.2.2" xref="p6.7.m7.1.2.2.2.cmml">f</mi><mi id="p6.7.m7.1.2.2.3" xref="p6.7.m7.1.2.2.3.cmml">res</mi></msub><mo id="p6.7.m7.1.2.1" xref="p6.7.m7.1.2.1.cmml">⁢</mo><mrow id="p6.7.m7.1.2.3.2" xref="p6.7.m7.1.2.cmml"><mo stretchy="false" id="p6.7.m7.1.2.3.2.1" xref="p6.7.m7.1.2.cmml">(</mo><mi id="p6.7.m7.1.1" xref="p6.7.m7.1.1.cmml">B</mi><mo stretchy="false" id="p6.7.m7.1.2.3.2.2" xref="p6.7.m7.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.8.m8.1.2" xref="p6.8.m8.1.2.cmml"><mi mathvariant="normal" id="p6.8.m8.1.2.2" xref="p6.8.m8.1.2.2.cmml">Δ</mi><mo id="p6.8.m8.1.2.1" xref="p6.8.m8.1.2.1.cmml">⁢</mo><mi id="p6.8.m8.1.2.3" xref="p6.8.m8.1.2.3.cmml">f</mi><mo id="p6.8.m8.1.2.1a" xref="p6.8.m8.1.2.1.cmml">⁢</mo><mrow id="p6.8.m8.1.2.4.2" xref="p6.8.m8.1.2.cmml"><mo stretchy="false" id="p6.8.m8.1.2.4.2.1" xref="p6.8.m8.1.2.cmml">(</mo><mi id="p6.8.m8.1.1" xref="p6.8.m8.1.1.cmml">B</mi><mo stretchy="false" id="p6.8.m8.1.2.4.2.2" xref="p6.8.m8.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.9.m9.3.4" xref="p6.9.m9.3.4.cmml"><mrow id="p6.9.m9.3.4.2" xref="p6.9.m9.3.4.2.cmml"><mi id="p6.9.m9.3.4.2.2" xref="p6.9.m9.3.4.2.2.cmml">Q</mi><mo id="p6.9.m9.3.4.2.1" xref="p6.9.m9.3.4.2.1.cmml">⁢</mo><mrow id="p6.9.m9.3.4.2.3.2" xref="p6.9.m9.3.4.2.cmml"><mo stretchy="false" id="p6.9.m9.3.4.2.3.2.1" xref="p6.9.m9.3.4.2.cmml">(</mo><mi id="p6.9.m9.1.1" xref="p6.9.m9.1.1.cmml">B</mi><mo stretchy="false" id="p6.9.m9.3.4.2.3.2.2" xref="p6.9.m9.3.4.2.cmml">)</mo></mrow></mrow><mo id="p6.9.m9.3.4.1" xref="p6.9.m9.3.4.1.cmml">=</mo><mrow id="p6.9.m9.3.4.3" xref="p6.9.m9.3.4.3.cmml"><mrow id="p6.9.m9.3.4.3.2" xref="p6.9.m9.3.4.3.2.cmml"><mrow id="p6.9.m9.3.4.3.2.2" xref="p6.9.m9.3.4.3.2.2.cmml"><msub id="p6.9.m9.3.4.3.2.2.2" xref="p6.9.m9.3.4.3.2.2.2.cmml"><mi id="p6.9.m9.3.4.3.2.2.2.2" xref="p6.9.m9.3.4.3.2.2.2.2.cmml">f</mi><mi id="p6.9.m9.3.4.3.2.2.2.3" xref="p6.9.m9.3.4.3.2.2.2.3.cmml">res</mi></msub><mo id="p6.9.m9.3.4.3.2.2.1" xref="p6.9.m9.3.4.3.2.2.1.cmml">⁢</mo><mrow id="p6.9.m9.3.4.3.2.2.3.2" xref="p6.9.m9.3.4.3.2.2.cmml"><mo stretchy="false" id="p6.9.m9.3.4.3.2.2.3.2.1" xref="p6.9.m9.3.4.3.2.2.cmml">(</mo><mi id="p6.9.m9.2.2" xref="p6.9.m9.2.2.cmml">B</mi><mo stretchy="false" id="p6.9.m9.3.4.3.2.2.3.2.2" xref="p6.9.m9.3.4.3.2.2.cmml">)</mo></mrow></mrow><mo id="p6.9.m9.3.4.3.2.1" xref="p6.9.m9.3.4.3.2.1.cmml">/</mo><mi mathvariant="normal" id="p6.9.m9.3.4.3.2.3" xref="p6.9.m9.3.4.3.2.3.cmml">Δ</mi></mrow><mo id="p6.9.m9.3.4.3.1" xref="p6.9.m9.3.4.3.1.cmml">⁢</mo><mi id="p6.9.m9.3.4.3.3" xref="p6.9.m9.3.4.3.3.cmml">f</mi><mo id="p6.9.m9.3.4.3.1a" xref="p6.9.m9.3.4.3.1.cmml">⁢</mo><mrow id="p6.9.m9.3.4.3.4.2" xref="p6.9.m9.3.4.3.cmml"><mo stretchy="false" id="p6.9.m9.3.4.3.4.2.1" xref="p6.9.m9.3.4.3.cmml">(</mo><mi id="p6.9.m9.3.3" xref="p6.9.m9.3.3.cmml">B</mi><mo stretchy="false" id="p6.9.m9.3.4.3.4.2.2" xref="p6.9.m9.3.4.3.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="p6.10.m10.1.2" xref="p6.10.m10.1.2.cmml"><mrow id="p6.10.m10.1.2.2" xref="p6.10.m10.1.2.2.cmml"><mn id="p6.10.m10.1.2.2.2" xref="p6.10.m10.1.2.2.2.cmml">1</mn><mo id="p6.10.m10.1.2.2.1" xref="p6.10.m10.1.2.2.1.cmml">/</mo><mi id="p6.10.m10.1.2.2.3" xref="p6.10.m10.1.2.2.3.cmml">Q</mi></mrow><mo id="p6.10.m10.1.2.1" xref="p6.10.m10.1.2.1.cmml">⁢</mo><mrow id="p6.10.m10.1.2.3.2" xref="p6.10.m10.1.2.cmml"><mo stretchy="false" id="p6.10.m10.1.2.3.2.1" xref="p6.10.m10.1.2.cmml">(</mo><mi id="p6.10.m10.1.1" xref="p6.10.m10.1.1.cmml">B</mi><mo stretchy="false" id="p6.10.m10.1.2.3.2.2" xref="p6.10.m10.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.11.m11.1.2" xref="p6.11.m11.1.2.cmml"><mrow id="p6.11.m11.1.2.2" xref="p6.11.m11.1.2.2.cmml"><mn id="p6.11.m11.1.2.2.2" xref="p6.11.m11.1.2.2.2.cmml">1</mn><mo id="p6.11.m11.1.2.2.1" xref="p6.11.m11.1.2.2.1.cmml">/</mo><mi id="p6.11.m11.1.2.2.3" xref="p6.11.m11.1.2.2.3.cmml">Q</mi></mrow><mo id="p6.11.m11.1.2.1" xref="p6.11.m11.1.2.1.cmml">⁢</mo><mrow id="p6.11.m11.1.2.3.2" xref="p6.11.m11.1.2.cmml"><mo stretchy="false" id="p6.11.m11.1.2.3.2.1" xref="p6.11.m11.1.2.cmml">(</mo><mi id="p6.11.m11.1.1" xref="p6.11.m11.1.1.cmml">B</mi><mo stretchy="false" id="p6.11.m11.1.2.3.2.2" xref="p6.11.m11.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p6.12.m12.3.4" xref="p6.12.m12.3.4.cmml"><mrow id="p6.12.m12.3.4.2" xref="p6.12.m12.3.4.2.cmml"><mrow id="p6.12.m12.3.4.2.2" xref="p6.12.m12.3.4.2.2.cmml"><mn id="p6.12.m12.3.4.2.2.2" xref="p6.12.m12.3.4.2.2.2.cmml">1</mn><mo id="p6.12.m12.3.4.2.2.1" xref="p6.12.m12.3.4.2.2.1.cmml">/</mo><msub id="p6.12.m12.3.4.2.2.3" xref="p6.12.m12.3.4.2.2.3.cmml"><mi id="p6.12.m12.3.4.2.2.3.2" xref="p6.12.m12.3.4.2.2.3.2.cmml">Q</mi><mi id="p6.12.m12.3.4.2.2.3.3" xref="p6.12.m12.3.4.2.2.3.3.cmml">v</mi></msub></mrow><mo id="p6.12.m12.3.4.2.1" xref="p6.12.m12.3.4.2.1.cmml">⁢</mo><mrow id="p6.12.m12.3.4.2.3.2" xref="p6.12.m12.3.4.2.cmml"><mo stretchy="false" id="p6.12.m12.3.4.2.3.2.1" xref="p6.12.m12.3.4.2.cmml">(</mo><mi id="p6.12.m12.1.1" xref="p6.12.m12.1.1.cmml">B</mi><mo stretchy="false" id="p6.12.m12.3.4.2.3.2.2" xref="p6.12.m12.3.4.2.cmml">)</mo></mrow></mrow><mo id="p6.12.m12.3.4.1" xref="p6.12.m12.3.4.1.cmml">=</mo><mrow id="p6.12.m12.3.4.3" xref="p6.12.m12.3.4.3.cmml"><mrow id="p6.12.m12.3.4.3.2" xref="p6.12.m12.3.4.3.2.cmml"><mrow id="p6.12.m12.3.4.3.2.2" xref="p6.12.m12.3.4.3.2.2.cmml"><mn id="p6.12.m12.3.4.3.2.2.2" xref="p6.12.m12.3.4.3.2.2.2.cmml">1</mn><mo id="p6.12.m12.3.4.3.2.2.1" xref="p6.12.m12.3.4.3.2.2.1.cmml">/</mo><mi id="p6.12.m12.3.4.3.2.2.3" xref="p6.12.m12.3.4.3.2.2.3.cmml">Q</mi></mrow><mo id="p6.12.m12.3.4.3.2.1" xref="p6.12.m12.3.4.3.2.1.cmml">⁢</mo><mrow id="p6.12.m12.3.4.3.2.3.2" xref="p6.12.m12.3.4.3.2.cmml"><mo stretchy="false" id="p6.12.m12.3.4.3.2.3.2.1" xref="p6.12.m12.3.4.3.2.cmml">(</mo><mi id="p6.12.m12.2.2" xref="p6.12.m12.2.2.cmml">B</mi><mo stretchy="false" id="p6.12.m12.3.4.3.2.3.2.2" xref="p6.12.m12.3.4.3.2.cmml">)</mo></mrow></mrow><mo id="p6.12.m12.3.4.3.1" xref="p6.12.m12.3.4.3.1.cmml">-</mo><mrow id="p6.12.m12.3.4.3.3" xref="p6.12.m12.3.4.3.3.cmml"><mrow id="p6.12.m12.3.4.3.3.2" xref="p6.12.m12.3.4.3.3.2.cmml"><mn id="p6.12.m12.3.4.3.3.2.2" xref="p6.12.m12.3.4.3.3.2.2.cmml">1</mn><mo id="p6.12.m12.3.4.3.3.2.1" xref="p6.12.m12.3.4.3.3.2.1.cmml">/</mo><mi id="p6.12.m12.3.4.3.3.2.3" xref="p6.12.m12.3.4.3.3.2.3.cmml">Q</mi></mrow><mo id="p6.12.m12.3.4.3.3.1" xref="p6.12.m12.3.4.3.3.1.cmml">⁢</mo><mrow id="p6.12.m12.3.4.3.3.3.2" xref="p6.12.m12.3.4.3.3.cmml"><mo stretchy="false" id="p6.12.m12.3.4.3.3.3.2.1" xref="p6.12.m12.3.4.3.3.cmml">(</mo><mn id="p6.12.m12.3.3" xref="p6.12.m12.3.3.cmml">0</mn><mo stretchy="false" id="p6.12.m12.3.4.3.3.3.2.2" xref="p6.12.m12.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S0.F2.8.m1.1.2" xref="S0.F2.8.m1.1.2.cmml"><mrow id="S0.F2.8.m1.1.2.2" xref="S0.F2.8.m1.1.2.2.cmml"><mn id="S0.F2.8.m1.1.2.2.2" xref="S0.F2.8.m1.1.2.2.2.cmml">1</mn><mo id="S0.F2.8.m1.1.2.2.1" xref="S0.F2.8.m1.1.2.2.1.cmml">/</mo><msub id="S0.F2.8.m1.1.2.2.3" xref="S0.F2.8.m1.1.2.2.3.cmml"><mi id="S0.F2.8.m1.1.2.2.3.2" xref="S0.F2.8.m1.1.2.2.3.2.cmml">Q</mi><mi id="S0.F2.8.m1.1.2.2.3.3" xref="S0.F2.8.m1.1.2.2.3.3.cmml">v</mi></msub></mrow><mo id="S0.F2.8.m1.1.2.1" xref="S0.F2.8.m1.1.2.1.cmml">⁢</mo><mrow id="S0.F2.8.m1.1.2.3.2" xref="S0.F2.8.m1.1.2.cmml"><mo stretchy="false" id="S0.F2.8.m1.1.2.3.2.1" xref="S0.F2.8.m1.1.2.cmml">(</mo><mi id="S0.F2.8.m1.1.1" xref="S0.F2.8.m1.1.1.cmml">B</mi><mo stretchy="false" id="S0.F2.8.m1.1.2.3.2.2" xref="S0.F2.8.m1.1.2.cmml">)</mo></mrow></mrow></math>

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9909488

Formulas:

1-

\sim 2.8 \times 10^{- 8}

2-

N_{H} \sim 5 \times 10^{21}

3-

k T = 0.74 \pm 0.02

4-

(4.0 \pm 0.36) \times 10^{21}

5-

k T_{b b}

6-

(6.8 \pm 0.2) \times 10^{- 10}

7-

(5.9 \pm 0.2) \times 10^{- 10}

8-

χ_{ν}^{2} \sim 0.6 - 1.2

9-

T_{e f f}

10-

T_{b b} / T_{e f f}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1706.08857

Formulas:

1-

e \in E (G)

2-

Ψ_{G} = \sum_{T} \prod_{e \notin T} a_{e}

3-

a_{1} + a_{2} + a_{3}

4-

\int_{a_{e} \geq 0} \frac{Ω}{Ψ_{G}^{2}}

5-

Ω = \sum_{i = 1}^{| E (G) |} {(- 1)}^{i} d a_{1} \dots d a_{i - 1} d a_{i + 1} \dots d a_{| E (G) |}

6-

c_{2}^{(p)} (G) = \frac{{[Ψ_{G}]}_{p}}{p^{2}} mod p

7-

c_{2}^{(p)} (G) = 0

8-

c_{2}^{(p)} = 0

9-

36 ζ {(3)}^{2}

10-

c_{2}^{(p)} (G)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0407083

Formulas:

1-

2.0 \pm 0.1 \to 2.1 \pm 0.2

2-

2.2 \pm 0.1 \to 1.7 \pm 0.1

3-

2.4 \pm 0.1 \to 1.5 \pm 0.3

4-

3.1 \pm 0.3 \to 3.0 \pm 0.8

5-

2.4 \pm 0.5 \to 1.9 \pm 0.2

6-

(0.8 - 1.5) \times 10^{- 8}

7-

(3 - 10) \times 10^{41}

8-

g = (G M / R^{2}) (1 + z) = 2.45 \times 10^{14} cm s^{- 2}

9-

F_{Edd} = c g / κ

10-

κ = 0.2 (1 + X) {cm}^{2} g^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1905.01065

Formulas:

1-

P (x, y, φ)

2-

r_{m i n}

3-

r_{m a x}

4-

g r i d m a p, P (x, y, φ)

5-

P (x, y, φ)

6-

r_{m i n}

7-

r_{m a x}

8-

c e l l

9-

c i r c l e

10-

c e l l

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/2008.11210

Formulas:

1-

M_{B H} - σ

2-

2 \overset{''}{.} 5

3-

12 \overset{''}{.} 5

4-

32 \overset{''}{.} 5

5-

6 \overset{''}{.} 25

6-

W 1 - W 2 > 0.486 \exp {0.092 {(W 2 - 13.07)}^{2}}

7-

W 1 - W 2 > 0.486

8-

L_{b o l} / L_{14 - 195 keV} = 8.47

9-

\log (M_{*} / M_{⊙})

10-

L_{b o l} \sim 10^{43}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1512.06740

Formulas:

1-

E_{j} (\bar{𝐫}, t) = F_{j} (x, y) A_{j} (z, t) e^{i (k_{j} z - ω_{j} t)}

2-

A_{j} (z, t)

3-

ω_{j} = ω_{0} \mp j Ω_{B}

4-

A_{+} (z, t)

5-

A_{-} (z, t)

6-

Ω_{B} / 2 π \approx

7-

ω_{0} / 2 π \approx

8-

E_{+} (\bar{𝐫}, t)

9-

= F_{p} (x, y) A_{+} (z, t) e^{i (k_{0} z - ω_{0} t)} + c . c .

10-

E_{-} (\bar{𝐫}, t)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0512449

Formulas:

1-

[0, δ r]

2-

l_{0, /} c

3-

⟨ 𝐫^{2} ⟩ = 4 D t

4-

⟨ 𝐫^{2} ⟩ \propto t^{ν}

5-

δ r = 10^{- 6}

6-

P (𝐫, t)

7-

P (𝐫, t)

8-

P (𝐫, t)

9-

P (𝐫, t)

10-

P (𝐫, t) = \frac{1}{t^{ν}} P (\frac{𝐫}{t^{ν / 2}}, 1) .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0609598

Formulas:

1-

M ≳ 5 \times 10^{9} M_{⊙}

2-

L_{Edd} = \frac{4 π G M c}{κ} ≃ 10^{47} M_{9} erg s^{- 1}

3-

M = 10^{9} M_{9} M_{⊙}

4-

L_{Edd} = ϵ c^{2} {\dot{M}}_{acc}

5-

\dot{M} = (1 - ϵ) {\dot{M}}_{acc} .

6-

\dot{M} = \frac{1 - ϵ}{ϵ} \frac{M}{t_{Edd}}

7-

t_{Edd} = \frac{κ c}{4 π G} = 4.5 \times 10^{8} yr .

8-

\frac{M}{M_{0}} < \exp [\frac{1}{ϵ_{m i n}} - 1] (\frac{t}{t_{Edd}}) .

9-

M / M_{0} ≲ 20

10-

\sim 5 \times 10^{9} M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0004092

Formulas:

1-

\sqrt{s_{e^{+} e^{-}}} = 500

2-

α^{2} α_{s}^{5}

3-

2 j e t s

4-

\sqrt{s_{e e}} \leq 500

5-

E_{γ γ}^{m a x} \sim 0.8 \sqrt{s_{e e}}

6-

M_{S} = 200 - 400

7-

M_{n} = 2 m_{\tilde{t}} + E_{n}

8-

M_{S} = 200 - 600

9-

\sim {| E_{n} |}^{- 1}

10-

b + c h a r g i n o

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1411.1494

Formulas:

1-

[\begin{matrix} E_{x}^{o}^{*} & E_{y}^{o}^{*} \\ E_{x}^{i} & E_{y}^{i} \end{matrix}] [\begin{matrix} T_{x x} \\ T_{y x} \end{matrix}] = [\begin{matrix} E_{x}^{i}^{*} \\ E_{x}^{o} \end{matrix}]

2-

i, j = x, y

3-

T_{x y} = T_{y x}

4-

T_{y y} = - \exp (2 i ∠ T_{y x}) T_{x x}^{*}

5-

𝐓 = 𝐕 [\begin{matrix} e^{i ϕ_{1}} & 0 \\ 0 & e^{i ϕ_{2}} \end{matrix}] 𝐕^{T} = 𝐑 (θ) 𝚫 𝐑 (- θ),

6-

𝐕^{T} = 𝐕^{- 1}

7-

\exp (i ϕ)

8-

T = 1 - \sqrt{{| t_{x} - e^{i ϕ_{x}} |}^{2} + {| t_{y} - e^{i ϕ_{y}} |}^{2}} .

9-

T_{x x} E_{x}^{i} + T_{y x} E_{y}^{i} = E_{x}^{o},

10-

T_{y x} E_{x}^{i} - \frac{T_{y x}}{T_{y x}^{*}} T_{x x}^{*} E_{y}^{i} = E_{y}^{o},

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1009.2578

Formulas:

1-

{Al}_{x} {Ga}_{1 - x} As

2-

{Al}_{x} {Ga}_{1 - x} As

3-

n_{S} = 4.9 \times 10^{15} m^{- 2}

4-

μ = 33 m^{2} / Vs

5-

V_{TIP} \sim - 30 V

6-

Δ x \sim 50 nm

7-

Δ x \sim 10 nm

8-

V ≲ - 0.6 V

9-

V_{2 D E G}

10-

T_{el} \approx 100 mK

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1905.06769

Formulas:

1-

A \equiv (c_{T f}^{2} \frac{k^{2}}{ω^{2}} - 1) / η_{m 0} = \frac{1}{1 - i ω τ_{d}},

2-

τ_{d} = \frac{m_{p}}{6 π μ_{f} a} = \frac{2 a^{2}}{9 ν_{f} δ},

3-

δ = ρ_{f} / ρ_{p}

4-

F_{f, z} = (ρ_{f} - ρ_{p}) V_{p} a_{c} F,

5-

F = 1 + \frac{4 (1 - τ)}{4 τ - (9 i G / β^{2})}

6-

τ = 1 / δ = \frac{ρ_{p}}{ρ_{f}}

7-

β = \frac{a}{δ} = a \sqrt{\frac{ω ρ_{f}}{2 μ_{f}}},

8-

G = 1 + λ + \frac{λ^{2}}{9} - \frac{{(1 + λ)}^{2} f (λ)}{κ [λ^{3} - λ^{2} \tanh λ - 2 f (λ)] + (λ + 3) f (λ)},

9-

λ = (1 + i) β

10-

f (λ) = λ^{2} \tanh λ - 3 λ + 3 \tanh λ

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1205.0544

Formulas:

1-

ν_{n, l} \approx Δ ν (n + \frac{l}{2} + ϵ) - δ ν_{0 l} .

2-

δ ν_{0 l}

3-

Δ ν / 2 - δ ν_{01}

4-

[Fe / H] = - 0.2 dex

5-

g r i z

6-

P (T_{eff} | ϵ_{A}, Γ)

7-

P (T_{eff} | ϵ_{B}, Γ)

8-

P (ϵ_{A} | T_{eff}, Γ)

9-

P (ϵ_{B} | T_{eff}, Γ)

10-

P (ϵ_{pref} | T_{eff}^{phot}, Γ)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0307480

Formulas:

1-

n_{H} d_{H} = n_{L} d_{L} = λ / 4

2-

35 - 55 % (n_{H})

3-

60 - 85 % (n_{L})

4-

λ_{i + 1} - λ_{i} = 2 + i

5-

λ_{i + 1} - λ_{i} = 2 + i

6-

n_{L} / n_{H} = 1.4 / 2.2

7-

λ_{i + 1} - λ_{i} = 3

8-

n_{L} / n_{H} = 1.3 / 2.3

9-

n_{L} / n_{H} = 1.6 / 2

10-

i = 1, 2, \dots r

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/0107001

Formulas:

1-

ψ_{0} [h_{i j}] = N \int δ g e x p (- I_{E} [g]),

2-

ψ_{0} [h_{i j}]

3-

S = \frac{1}{4} \int d^{4} X \sqrt{- g} [A C_{i j k l}^{2} + B {(R - 4 Λ)}^{2}],

4-

C_{i j k l}

5-

C_{i j k l}

6-

k = o, \pm 1

7-

S = - \frac{1}{2 π^{2}} \int d^{4} X \sqrt{- g} R .

8-

d s^{2} = e x p 2 α (η) [d η^{2} - d χ^{2} - s i n^{2} χ (d θ^{2} + s i n^{2} θ d ϕ^{2})],

9-

R = - 6 e x p [- 2 α] (1 + {\dot{α}}^{2} + \ddot{α})

10-

\dot{α} = \frac{d α}{d η}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/1110.6779

Formulas:

1-

R (n, k)

2-

{1, 2, \dots, n}

3-

R (n, k)

4-

[n] = {1, 2, \dots, n}

5-

π = π (1) π (2) \dots π (n) \in 𝒮_{n}

6-

π (i - 1) < π (i) > π (i + 1)

7-

π (i - 1) > π (i) < π (i + 1)

8-

i \in {2, 3, \dots, n - 1}

9-

R (n, k)

10-

R (n, k) = k R (n - 1, k) + 2 R (n - 1, k - 1) + (n - k) R (n - 1, k - 2)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1205.1035

Formulas:

1-

𝑴 = 𝝁 \times 𝑩,

2-

\oint d ℓ \times 𝑿 = \int d 𝝈 \times (\nabla \times 𝑿) + \int (d 𝝈 \cdot \nabla) 𝑿 - \int d 𝝈 (\nabla \cdot 𝑿),

3-

\int (f 𝒀 \cdot \nabla g + g 𝒀 \cdot \nabla f + f g \nabla \cdot 𝒀) d^{3} 𝒓 = 0

4-

d 𝑭 = I d ℓ \times 𝑩,

5-

𝑴 = I \oint 𝒓 \times (d ℓ \times 𝑩) .

6-

t \in [0, 1]

7-

d ℓ = \dot{𝒔} d t

8-

𝑴 = I \int_{0}^{1} 𝒔 \times (\dot{𝒔} \times 𝑩) d t .

9-

𝑿 \times (𝒀 \times 𝒁) = 𝒀 (𝑿 \cdot 𝒁) - 𝒁 (𝑿 \cdot 𝒀)

10-

𝑴 = I \int_{0}^{1} {\dot{𝒔} (𝒔 \cdot 𝑩) - 𝑩 (\dot{𝒔} \cdot 𝒔)} d t

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1905.06981

Formulas:

1-

M_{s c o r e}

2-

R e f_{i m g} = \max_{r} {{(\sum_{i = 1}^{N} {(M_{s c o r e})}_{i})}_{r}}

3-

M_{s c o r e}

4-

X = {x_{1}, x_{2}, \dots, x_{T}}

5-

Y = {y_{1}, y_{2}, \dots, y_{T}}

6-

L_{t r i p l e t} = \max {d (a, p) - d (a, n) + m a r g i n, 0}

7-

d (a, p)

8-

d (a, n)

9-

P_{1} \in ℝ^{n \times l}

10-

P_{2} \in ℝ^{m \times l}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1707.00097

Formulas:

1-

T x \equiv O x

2-

⟨ M_{p}, M ⟩

3-

P T = W (-) M T

4-

P T = W (-) M T

5-

(c o r)

6-

G_{ν} \equiv A_{p ν}

7-

A_{p ν} \subset G_{ν} \subset W_{ν}

8-

(c o r)

9-

T_{P R μ}

10-

T_{P R μ}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1205.0194

Formulas:

1-

ω_{L} > ω_{TAP} > ω_{Trap} .

2-

ξ = ω_{L} / ω_{Trap}

3-

{\vec{B}}_{S} (\vec{r})

4-

B_{S}^{', x y}

5-

B_{S}^{', x} = B_{S}^{', y} = - B_{S}^{', z} / 2

6-

\vec{r} (t) = r_{D} \cos (ω_{TAP} t) \hat{x} + r_{D} \sin (ω_{TAP} t) \hat{y} .

7-

U = m_{F} g_{F} μ_{B} \frac{ω_{TAP}}{2 π} \int_{0}^{2 π / ω_{TAP}} | {\vec{B}}_{S} (\vec{r} (t)) | d t .

8-

B_{S}^{', x y}

9-

B_{S}^{', x y}

10-

B_{S}^{', x y} \approx 80

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/nucl-th/9507020

Formulas:

1-

3 \times 3 \times 2 \times 2 = 36

2-

d (e, e^{'} N_{1}) N_{2}

3-

d (e, e^{'} \vec{n} \vec{p})

4-

d (e, e^{'} \vec{n} \vec{p})

5-

⟨ λ_{p} λ_{n} | J_{λ_{γ}} | λ_{D} ⟩

6-

⟨ λ_{p} λ_{n} | J \cdot ϵ_{λ_{γ}} | λ_{D} ⟩

7-

⟨ λ_{p} λ_{n} | J_{λ_{γ}} | λ_{D} ⟩

8-

{(- 1)}^{(λ_{p} - λ_{n}) - (λ_{γ} - λ_{D})} ⟨ - λ_{p} - λ_{n} | J_{- λ_{γ}} | - λ_{D} ⟩,

9-

J \cdot ϵ_{λ} = J_{μ} ϵ_{λ}^{μ}

10-

t_{s m_{s} λ m}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1908.04983

Formulas:

1-

x f_{i}^{p / A} (x, Q_{0}^{2}) = c_{0} x^{c_{1}} {(1 - x)}^{c_{2}} (1 + c_{3} x + c_{4} x^{2})

2-

i = g, u_{v} d_{v} \bar{u}, \bar{d}, s, \bar{s}

3-

i = 0, \dots, 4

4-

c_{k} \to c_{k} (A) = c_{k, 0} + c_{k, 1} (1 - A^{- c_{k, 2}})

5-

k = 0, \dots, 4

6-

(1 - A^{- c_{k, 2}})

7-

f_{i}^{N / A} (x, Q^{2}) = \frac{Z \cdot f_{i}^{p / A} + (A - Z) \cdot f_{i}^{n / A}}{A},

8-

Z = (A - Z) = \frac{A}{2}

9-

\bar{u} = \bar{d} = s = \bar{s}

10-

σ (A_{1}) / σ (A_{2})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1701.06502

Formulas:

1-

ℛ = \cup_{v \in 𝒩} ℛ_{v}

2-

𝒫 = {0, 1, 2, \dots, | 𝒫 |}

3-

{1, \dots, Λ}

4-

γ_{l} = \frac{g_{l} p_{l}}{I_{l} + σ^{2}} = \frac{g_{l} p_{l}}{\sum_{l^{'} \in ℒ ∖ l} g_{l^{'} l} p_{l^{'}} + σ^{2}}

5-

r_{l} = B_{l} \log_{2} (1 + γ_{l})

6-

g_{l} = ρ (d_{l})

7-

g_{l l^{'}} = ρ (d_{l l^{'}})

8-

g_{l l^{'}} = 0

9-

g_{l l^{'}} = \infty

10-

g_{l l^{'}} ≜ {\begin{matrix} ρ (d_{l l^{'}}), & if l and l^{'} use different radios \\ and are in the same channel; \\ 0, & if l and l^{'} use different radios \\ and are in different channels; \\ \infty, & if l and l^{'} share one or more radios. \end{matrix}

Formulas (html):

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id="S3.SS1.p1.5.m5.1.1.3.2.3" xref="S3.SS1.p1.5.m5.1.1.3.2.3.cmml">v</mi></msub></mrow></mrow></math>, <math><mrow id="S3.SS1.p1.7.m7.6.6" xref="S3.SS1.p1.7.m7.6.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.7.m7.6.6.3" xref="S3.SS1.p1.7.m7.6.6.3.cmml">𝒫</mi><mo id="S3.SS1.p1.7.m7.6.6.2" xref="S3.SS1.p1.7.m7.6.6.2.cmml">=</mo><mrow id="S3.SS1.p1.7.m7.6.6.1.1" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml"><mo stretchy="false" id="S3.SS1.p1.7.m7.6.6.1.1.2" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml">{</mo><mn id="S3.SS1.p1.7.m7.1.1" xref="S3.SS1.p1.7.m7.1.1.cmml">0</mn><mo id="S3.SS1.p1.7.m7.6.6.1.1.3" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml">,</mo><mn id="S3.SS1.p1.7.m7.2.2" xref="S3.SS1.p1.7.m7.2.2.cmml">1</mn><mo id="S3.SS1.p1.7.m7.6.6.1.1.4" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml">,</mo><mn id="S3.SS1.p1.7.m7.3.3" xref="S3.SS1.p1.7.m7.3.3.cmml">2</mn><mo id="S3.SS1.p1.7.m7.6.6.1.1.5" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml">,</mo><mi mathvariant="normal" id="S3.SS1.p1.7.m7.4.4" xref="S3.SS1.p1.7.m7.4.4.cmml">…</mi><mo id="S3.SS1.p1.7.m7.6.6.1.1.6" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml">,</mo><mrow id="S3.SS1.p1.7.m7.6.6.1.1.1.2" xref="S3.SS1.p1.7.m7.6.6.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS1.p1.7.m7.6.6.1.1.1.2.1" xref="S3.SS1.p1.7.m7.6.6.1.1.1.1.1.cmml">|</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.p1.7.m7.5.5" xref="S3.SS1.p1.7.m7.5.5.cmml">𝒫</mi><mo stretchy="false" id="S3.SS1.p1.7.m7.6.6.1.1.1.2.2" xref="S3.SS1.p1.7.m7.6.6.1.1.1.1.1.cmml">|</mo></mrow><mo stretchy="false" id="S3.SS1.p1.7.m7.6.6.1.1.7" xref="S3.SS1.p1.7.m7.6.6.1.2.cmml">}</mo></mrow></mrow></math>, <math><mrow id="S3.SS1.p2.1.m1.3.4.2" xref="S3.SS1.p2.1.m1.3.4.1.cmml"><mo stretchy="false" id="S3.SS1.p2.1.m1.3.4.2.1" xref="S3.SS1.p2.1.m1.3.4.1.cmml">{</mo><mn id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">1</mn><mo id="S3.SS1.p2.1.m1.3.4.2.2" xref="S3.SS1.p2.1.m1.3.4.1.cmml">,</mo><mi mathvariant="normal" id="S3.SS1.p2.1.m1.2.2" xref="S3.SS1.p2.1.m1.2.2.cmml">⋯</mi><mo id="S3.SS1.p2.1.m1.3.4.2.3" xref="S3.SS1.p2.1.m1.3.4.1.cmml">,</mo><mi mathvariant="normal" id="S3.SS1.p2.1.m1.3.3" xref="S3.SS1.p2.1.m1.3.3.cmml">Λ</mi><mo stretchy="false" id="S3.SS1.p2.1.m1.3.4.2.4" xref="S3.SS1.p2.1.m1.3.4.1.cmml">}</mo></mrow></math>, <math><mrow id="S3.E1.m1.1.1" xref="S3.E1.m1.1.1.cmml"><msub id="S3.E1.m1.1.1.2" xref="S3.E1.m1.1.1.2.cmml"><mi id="S3.E1.m1.1.1.2.2" xref="S3.E1.m1.1.1.2.2.cmml">γ</mi><mi id="S3.E1.m1.1.1.2.3" xref="S3.E1.m1.1.1.2.3.cmml">l</mi></msub><mo id="S3.E1.m1.1.1.3" xref="S3.E1.m1.1.1.3.cmml">=</mo><mfrac id="S3.E1.m1.1.1.4" xref="S3.E1.m1.1.1.4.cmml"><mrow id="S3.E1.m1.1.1.4.2" xref="S3.E1.m1.1.1.4.2.cmml"><msub id="S3.E1.m1.1.1.4.2.2" xref="S3.E1.m1.1.1.4.2.2.cmml"><mi id="S3.E1.m1.1.1.4.2.2.2" xref="S3.E1.m1.1.1.4.2.2.2.cmml">g</mi><mi id="S3.E1.m1.1.1.4.2.2.3" xref="S3.E1.m1.1.1.4.2.2.3.cmml">l</mi></msub><mo id="S3.E1.m1.1.1.4.2.1" xref="S3.E1.m1.1.1.4.2.1.cmml">⁢</mo><msub id="S3.E1.m1.1.1.4.2.3" xref="S3.E1.m1.1.1.4.2.3.cmml"><mi id="S3.E1.m1.1.1.4.2.3.2" xref="S3.E1.m1.1.1.4.2.3.2.cmml">p</mi><mi id="S3.E1.m1.1.1.4.2.3.3" xref="S3.E1.m1.1.1.4.2.3.3.cmml">l</mi></msub></mrow><mrow id="S3.E1.m1.1.1.4.3" xref="S3.E1.m1.1.1.4.3.cmml"><msub id="S3.E1.m1.1.1.4.3.2" xref="S3.E1.m1.1.1.4.3.2.cmml"><mi id="S3.E1.m1.1.1.4.3.2.2" xref="S3.E1.m1.1.1.4.3.2.2.cmml">I</mi><mi id="S3.E1.m1.1.1.4.3.2.3" xref="S3.E1.m1.1.1.4.3.2.3.cmml">l</mi></msub><mo id="S3.E1.m1.1.1.4.3.1" xref="S3.E1.m1.1.1.4.3.1.cmml">+</mo><msup id="S3.E1.m1.1.1.4.3.3" xref="S3.E1.m1.1.1.4.3.3.cmml"><mi id="S3.E1.m1.1.1.4.3.3.2" xref="S3.E1.m1.1.1.4.3.3.2.cmml">σ</mi><mn id="S3.E1.m1.1.1.4.3.3.3" xref="S3.E1.m1.1.1.4.3.3.3.cmml">2</mn></msup></mrow></mfrac><mo id="S3.E1.m1.1.1.5" xref="S3.E1.m1.1.1.5.cmml">=</mo><mfrac id="S3.E1.m1.1.1.6" xref="S3.E1.m1.1.1.6.cmml"><mrow id="S3.E1.m1.1.1.6.2" xref="S3.E1.m1.1.1.6.2.cmml"><msub id="S3.E1.m1.1.1.6.2.2" xref="S3.E1.m1.1.1.6.2.2.cmml"><mi id="S3.E1.m1.1.1.6.2.2.2" xref="S3.E1.m1.1.1.6.2.2.2.cmml">g</mi><mi id="S3.E1.m1.1.1.6.2.2.3" xref="S3.E1.m1.1.1.6.2.2.3.cmml">l</mi></msub><mo id="S3.E1.m1.1.1.6.2.1" xref="S3.E1.m1.1.1.6.2.1.cmml">⁢</mo><msub id="S3.E1.m1.1.1.6.2.3" xref="S3.E1.m1.1.1.6.2.3.cmml"><mi id="S3.E1.m1.1.1.6.2.3.2" xref="S3.E1.m1.1.1.6.2.3.2.cmml">p</mi><mi id="S3.E1.m1.1.1.6.2.3.3" xref="S3.E1.m1.1.1.6.2.3.3.cmml">l</mi></msub></mrow><mrow id="S3.E1.m1.1.1.6.3" xref="S3.E1.m1.1.1.6.3.cmml"><mrow id="S3.E1.m1.1.1.6.3.2" xref="S3.E1.m1.1.1.6.3.2.cmml"><munder id="S3.E1.m1.1.1.6.3.2.1" xref="S3.E1.m1.1.1.6.3.2.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S3.E1.m1.1.1.6.3.2.1.2" xref="S3.E1.m1.1.1.6.3.2.1.2.cmml">∑</mo><mrow id="S3.E1.m1.1.1.6.3.2.1.3" xref="S3.E1.m1.1.1.6.3.2.1.3.cmml"><msup id="S3.E1.m1.1.1.6.3.2.1.3.2" xref="S3.E1.m1.1.1.6.3.2.1.3.2.cmml"><mi id="S3.E1.m1.1.1.6.3.2.1.3.2.2" xref="S3.E1.m1.1.1.6.3.2.1.3.2.2.cmml">l</mi><mo id="S3.E1.m1.1.1.6.3.2.1.3.2.3" xref="S3.E1.m1.1.1.6.3.2.1.3.2.3.cmml">′</mo></msup><mo id="S3.E1.m1.1.1.6.3.2.1.3.1" xref="S3.E1.m1.1.1.6.3.2.1.3.1.cmml">∈</mo><mrow id="S3.E1.m1.1.1.6.3.2.1.3.3" xref="S3.E1.m1.1.1.6.3.2.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E1.m1.1.1.6.3.2.1.3.3.2" xref="S3.E1.m1.1.1.6.3.2.1.3.3.2.cmml">ℒ</mi><mo id="S3.E1.m1.1.1.6.3.2.1.3.3.1" xref="S3.E1.m1.1.1.6.3.2.1.3.3.1.cmml">∖</mo><mi id="S3.E1.m1.1.1.6.3.2.1.3.3.3" xref="S3.E1.m1.1.1.6.3.2.1.3.3.3.cmml">l</mi></mrow></mrow></munder><mrow id="S3.E1.m1.1.1.6.3.2.2" xref="S3.E1.m1.1.1.6.3.2.2.cmml"><msub id="S3.E1.m1.1.1.6.3.2.2.2" xref="S3.E1.m1.1.1.6.3.2.2.2.cmml"><mi id="S3.E1.m1.1.1.6.3.2.2.2.2" xref="S3.E1.m1.1.1.6.3.2.2.2.2.cmml">g</mi><mrow id="S3.E1.m1.1.1.6.3.2.2.2.3" xref="S3.E1.m1.1.1.6.3.2.2.2.3.cmml"><msup id="S3.E1.m1.1.1.6.3.2.2.2.3.2" xref="S3.E1.m1.1.1.6.3.2.2.2.3.2.cmml"><mi id="S3.E1.m1.1.1.6.3.2.2.2.3.2.2" xref="S3.E1.m1.1.1.6.3.2.2.2.3.2.2.cmml">l</mi><mo id="S3.E1.m1.1.1.6.3.2.2.2.3.2.3" xref="S3.E1.m1.1.1.6.3.2.2.2.3.2.3.cmml">′</mo></msup><mo id="S3.E1.m1.1.1.6.3.2.2.2.3.1" xref="S3.E1.m1.1.1.6.3.2.2.2.3.1.cmml">⁢</mo><mi id="S3.E1.m1.1.1.6.3.2.2.2.3.3" xref="S3.E1.m1.1.1.6.3.2.2.2.3.3.cmml">l</mi></mrow></msub><mo id="S3.E1.m1.1.1.6.3.2.2.1" xref="S3.E1.m1.1.1.6.3.2.2.1.cmml">⁢</mo><msub id="S3.E1.m1.1.1.6.3.2.2.3" xref="S3.E1.m1.1.1.6.3.2.2.3.cmml"><mi id="S3.E1.m1.1.1.6.3.2.2.3.2" xref="S3.E1.m1.1.1.6.3.2.2.3.2.cmml">p</mi><msup id="S3.E1.m1.1.1.6.3.2.2.3.3" xref="S3.E1.m1.1.1.6.3.2.2.3.3.cmml"><mi id="S3.E1.m1.1.1.6.3.2.2.3.3.2" xref="S3.E1.m1.1.1.6.3.2.2.3.3.2.cmml">l</mi><mo id="S3.E1.m1.1.1.6.3.2.2.3.3.3" xref="S3.E1.m1.1.1.6.3.2.2.3.3.3.cmml">′</mo></msup></msub></mrow></mrow><mo id="S3.E1.m1.1.1.6.3.1" xref="S3.E1.m1.1.1.6.3.1.cmml">+</mo><msup id="S3.E1.m1.1.1.6.3.3" xref="S3.E1.m1.1.1.6.3.3.cmml"><mi id="S3.E1.m1.1.1.6.3.3.2" xref="S3.E1.m1.1.1.6.3.3.2.cmml">σ</mi><mn id="S3.E1.m1.1.1.6.3.3.3" xref="S3.E1.m1.1.1.6.3.3.3.cmml">2</mn></msup></mrow></mfrac></mrow></math>, <math><mrow id="S3.E2.m1.2.2" xref="S3.E2.m1.2.2.cmml"><msub id="S3.E2.m1.2.2.4" xref="S3.E2.m1.2.2.4.cmml"><mi id="S3.E2.m1.2.2.4.2" xref="S3.E2.m1.2.2.4.2.cmml">r</mi><mi id="S3.E2.m1.2.2.4.3" xref="S3.E2.m1.2.2.4.3.cmml">l</mi></msub><mo id="S3.E2.m1.2.2.3" xref="S3.E2.m1.2.2.3.cmml">=</mo><mrow id="S3.E2.m1.2.2.2" xref="S3.E2.m1.2.2.2.cmml"><msub id="S3.E2.m1.2.2.2.4" xref="S3.E2.m1.2.2.2.4.cmml"><mi id="S3.E2.m1.2.2.2.4.2" xref="S3.E2.m1.2.2.2.4.2.cmml">B</mi><mi id="S3.E2.m1.2.2.2.4.3" xref="S3.E2.m1.2.2.2.4.3.cmml">l</mi></msub><mo id="S3.E2.m1.2.2.2.3" xref="S3.E2.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.E2.m1.2.2.2.2.2" xref="S3.E2.m1.2.2.2.2.3.cmml"><msub id="S3.E2.m1.1.1.1.1.1.1" xref="S3.E2.m1.1.1.1.1.1.1.cmml"><mi id="S3.E2.m1.1.1.1.1.1.1.2" xref="S3.E2.m1.1.1.1.1.1.1.2.cmml">log</mi><mn id="S3.E2.m1.1.1.1.1.1.1.3" xref="S3.E2.m1.1.1.1.1.1.1.3.cmml">2</mn></msub><mo id="S3.E2.m1.2.2.2.2.2a" xref="S3.E2.m1.2.2.2.2.3.cmml">⁡</mo><mrow id="S3.E2.m1.2.2.2.2.2.2" xref="S3.E2.m1.2.2.2.2.3.cmml"><mo stretchy="false" id="S3.E2.m1.2.2.2.2.2.2.2" xref="S3.E2.m1.2.2.2.2.3.cmml">(</mo><mrow id="S3.E2.m1.2.2.2.2.2.2.1" xref="S3.E2.m1.2.2.2.2.2.2.1.cmml"><mn id="S3.E2.m1.2.2.2.2.2.2.1.2" xref="S3.E2.m1.2.2.2.2.2.2.1.2.cmml">1</mn><mo id="S3.E2.m1.2.2.2.2.2.2.1.1" xref="S3.E2.m1.2.2.2.2.2.2.1.1.cmml">+</mo><msub id="S3.E2.m1.2.2.2.2.2.2.1.3" xref="S3.E2.m1.2.2.2.2.2.2.1.3.cmml"><mi id="S3.E2.m1.2.2.2.2.2.2.1.3.2" xref="S3.E2.m1.2.2.2.2.2.2.1.3.2.cmml">γ</mi><mi id="S3.E2.m1.2.2.2.2.2.2.1.3.3" xref="S3.E2.m1.2.2.2.2.2.2.1.3.3.cmml">l</mi></msub></mrow><mo stretchy="false" id="S3.E2.m1.2.2.2.2.2.2.3" xref="S3.E2.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S3.SS1.p6.2.m2.1.1" xref="S3.SS1.p6.2.m2.1.1.cmml"><msub id="S3.SS1.p6.2.m2.1.1.3" xref="S3.SS1.p6.2.m2.1.1.3.cmml"><mi id="S3.SS1.p6.2.m2.1.1.3.2" xref="S3.SS1.p6.2.m2.1.1.3.2.cmml">g</mi><mi id="S3.SS1.p6.2.m2.1.1.3.3" xref="S3.SS1.p6.2.m2.1.1.3.3.cmml">l</mi></msub><mo id="S3.SS1.p6.2.m2.1.1.2" xref="S3.SS1.p6.2.m2.1.1.2.cmml">=</mo><mrow id="S3.SS1.p6.2.m2.1.1.1" xref="S3.SS1.p6.2.m2.1.1.1.cmml"><mi id="S3.SS1.p6.2.m2.1.1.1.3" xref="S3.SS1.p6.2.m2.1.1.1.3.cmml">ρ</mi><mo id="S3.SS1.p6.2.m2.1.1.1.2" xref="S3.SS1.p6.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.p6.2.m2.1.1.1.1.1" xref="S3.SS1.p6.2.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS1.p6.2.m2.1.1.1.1.1.2" xref="S3.SS1.p6.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.p6.2.m2.1.1.1.1.1.1" xref="S3.SS1.p6.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p6.2.m2.1.1.1.1.1.1.2" xref="S3.SS1.p6.2.m2.1.1.1.1.1.1.2.cmml">d</mi><mi id="S3.SS1.p6.2.m2.1.1.1.1.1.1.3" xref="S3.SS1.p6.2.m2.1.1.1.1.1.1.3.cmml">l</mi></msub><mo stretchy="false" id="S3.SS1.p6.2.m2.1.1.1.1.1.3" xref="S3.SS1.p6.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S3.SS1.p6.7.m7.1.1" xref="S3.SS1.p6.7.m7.1.1.cmml"><msub id="S3.SS1.p6.7.m7.1.1.3" xref="S3.SS1.p6.7.m7.1.1.3.cmml"><mi id="S3.SS1.p6.7.m7.1.1.3.2" xref="S3.SS1.p6.7.m7.1.1.3.2.cmml">g</mi><mrow id="S3.SS1.p6.7.m7.1.1.3.3" xref="S3.SS1.p6.7.m7.1.1.3.3.cmml"><mi id="S3.SS1.p6.7.m7.1.1.3.3.2" xref="S3.SS1.p6.7.m7.1.1.3.3.2.cmml">l</mi><mo id="S3.SS1.p6.7.m7.1.1.3.3.1" xref="S3.SS1.p6.7.m7.1.1.3.3.1.cmml">⁢</mo><msup id="S3.SS1.p6.7.m7.1.1.3.3.3" xref="S3.SS1.p6.7.m7.1.1.3.3.3.cmml"><mi id="S3.SS1.p6.7.m7.1.1.3.3.3.2" xref="S3.SS1.p6.7.m7.1.1.3.3.3.2.cmml">l</mi><mo id="S3.SS1.p6.7.m7.1.1.3.3.3.3" xref="S3.SS1.p6.7.m7.1.1.3.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S3.SS1.p6.7.m7.1.1.2" xref="S3.SS1.p6.7.m7.1.1.2.cmml">=</mo><mrow id="S3.SS1.p6.7.m7.1.1.1" xref="S3.SS1.p6.7.m7.1.1.1.cmml"><mi id="S3.SS1.p6.7.m7.1.1.1.3" xref="S3.SS1.p6.7.m7.1.1.1.3.cmml">ρ</mi><mo id="S3.SS1.p6.7.m7.1.1.1.2" xref="S3.SS1.p6.7.m7.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.p6.7.m7.1.1.1.1.1" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS1.p6.7.m7.1.1.1.1.1.2" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.p6.7.m7.1.1.1.1.1.1" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.cmml"><mi id="S3.SS1.p6.7.m7.1.1.1.1.1.1.2" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.2.cmml">d</mi><mrow id="S3.SS1.p6.7.m7.1.1.1.1.1.1.3" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.cmml"><mi id="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.2" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.2.cmml">l</mi><mo id="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.1" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.3" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.3.cmml"><mi id="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.3.2" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.3.2.cmml">l</mi><mo id="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.3.3" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><mo stretchy="false" id="S3.SS1.p6.7.m7.1.1.1.1.1.3" xref="S3.SS1.p6.7.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S3.SS1.p6.13.m13.1.1" xref="S3.SS1.p6.13.m13.1.1.cmml"><msub id="S3.SS1.p6.13.m13.1.1.2" xref="S3.SS1.p6.13.m13.1.1.2.cmml"><mi id="S3.SS1.p6.13.m13.1.1.2.2" xref="S3.SS1.p6.13.m13.1.1.2.2.cmml">g</mi><mrow id="S3.SS1.p6.13.m13.1.1.2.3" xref="S3.SS1.p6.13.m13.1.1.2.3.cmml"><mi id="S3.SS1.p6.13.m13.1.1.2.3.2" xref="S3.SS1.p6.13.m13.1.1.2.3.2.cmml">l</mi><mo id="S3.SS1.p6.13.m13.1.1.2.3.1" xref="S3.SS1.p6.13.m13.1.1.2.3.1.cmml">⁢</mo><msup id="S3.SS1.p6.13.m13.1.1.2.3.3" xref="S3.SS1.p6.13.m13.1.1.2.3.3.cmml"><mi id="S3.SS1.p6.13.m13.1.1.2.3.3.2" xref="S3.SS1.p6.13.m13.1.1.2.3.3.2.cmml">l</mi><mo id="S3.SS1.p6.13.m13.1.1.2.3.3.3" xref="S3.SS1.p6.13.m13.1.1.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S3.SS1.p6.13.m13.1.1.1" xref="S3.SS1.p6.13.m13.1.1.1.cmml">=</mo><mn id="S3.SS1.p6.13.m13.1.1.3" xref="S3.SS1.p6.13.m13.1.1.3.cmml">0</mn></mrow></math>, <math><mrow id="S3.SS1.p7.4.m4.1.1" xref="S3.SS1.p7.4.m4.1.1.cmml"><msub id="S3.SS1.p7.4.m4.1.1.2" xref="S3.SS1.p7.4.m4.1.1.2.cmml"><mi id="S3.SS1.p7.4.m4.1.1.2.2" xref="S3.SS1.p7.4.m4.1.1.2.2.cmml">g</mi><mrow id="S3.SS1.p7.4.m4.1.1.2.3" xref="S3.SS1.p7.4.m4.1.1.2.3.cmml"><mi id="S3.SS1.p7.4.m4.1.1.2.3.2" xref="S3.SS1.p7.4.m4.1.1.2.3.2.cmml">l</mi><mo id="S3.SS1.p7.4.m4.1.1.2.3.1" xref="S3.SS1.p7.4.m4.1.1.2.3.1.cmml">⁢</mo><msup id="S3.SS1.p7.4.m4.1.1.2.3.3" xref="S3.SS1.p7.4.m4.1.1.2.3.3.cmml"><mi id="S3.SS1.p7.4.m4.1.1.2.3.3.2" xref="S3.SS1.p7.4.m4.1.1.2.3.3.2.cmml">l</mi><mo id="S3.SS1.p7.4.m4.1.1.2.3.3.3" xref="S3.SS1.p7.4.m4.1.1.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S3.SS1.p7.4.m4.1.1.1" xref="S3.SS1.p7.4.m4.1.1.1.cmml">=</mo><mi mathvariant="normal" id="S3.SS1.p7.4.m4.1.1.3" xref="S3.SS1.p7.4.m4.1.1.3.cmml">∞</mi></mrow></math>, <math><mrow id="S3.Ex1.m1.13.14" xref="S3.Ex1.m1.13.14.cmml"><msub id="S3.Ex1.m1.13.14.2" xref="S3.Ex1.m1.13.14.2.cmml"><mi id="S3.Ex1.m1.13.14.2.2" xref="S3.Ex1.m1.13.14.2.2.cmml">g</mi><mrow id="S3.Ex1.m1.13.14.2.3" xref="S3.Ex1.m1.13.14.2.3.cmml"><mi id="S3.Ex1.m1.13.14.2.3.2" xref="S3.Ex1.m1.13.14.2.3.2.cmml">l</mi><mo id="S3.Ex1.m1.13.14.2.3.1" xref="S3.Ex1.m1.13.14.2.3.1.cmml">⁢</mo><msup id="S3.Ex1.m1.13.14.2.3.3" xref="S3.Ex1.m1.13.14.2.3.3.cmml"><mi id="S3.Ex1.m1.13.14.2.3.3.2" xref="S3.Ex1.m1.13.14.2.3.3.2.cmml">l</mi><mo id="S3.Ex1.m1.13.14.2.3.3.3" xref="S3.Ex1.m1.13.14.2.3.3.3.cmml">′</mo></msup></mrow></msub><mo id="S3.Ex1.m1.13.14.1" xref="S3.Ex1.m1.13.14.1.cmml">≜</mo><mrow id="S3.Ex1.m1.13.13" xref="S3.Ex1.m1.13.14.3.1.cmml"><mo id="S3.Ex1.m1.13.13.14" xref="S3.Ex1.m1.13.14.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S3.Ex1.m1.13.13.13" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtr id="S3.Ex1.m1.13.13.13a" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtd columnalign="left" id="S3.Ex1.m1.13.13.13b" xref="S3.Ex1.m1.13.14.3.1.cmml"><mrow id="S3.Ex1.m1.7.7.7.7.3.1.1" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.cmml"><mrow id="S3.Ex1.m1.7.7.7.7.3.1.1.1" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.cmml"><mi id="S3.Ex1.m1.7.7.7.7.3.1.1.1.3" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.3.cmml">ρ</mi><mo id="S3.Ex1.m1.7.7.7.7.3.1.1.1.2" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.2" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.cmml"><mi id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.2" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.2.cmml">d</mi><mrow id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.2" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.2.cmml">l</mi><mo id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.1" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.1.cmml">⁢</mo><msup id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.3" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.3.cmml"><mi id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.3.2" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.3.2.cmml">l</mi><mo id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.3.3" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.3.3.3.cmml">′</mo></msup></mrow></msub><mo stretchy="false" id="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.3" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex1.m1.7.7.7.7.3.1.1.2" xref="S3.Ex1.m1.7.7.7.7.3.1.1.1.cmml">,</mo></mrow></mtd><mtd columnalign="left" id="S3.Ex1.m1.13.13.13c" xref="S3.Ex1.m1.13.14.3.1.cmml"><mrow id="S3.Ex1.m1.2.2.2.2.2.2" xref="S3.Ex1.m1.2.2.2.2.2.2d.cmml"><mtext id="S3.Ex1.m1.2.2.2.2.2.2a" xref="S3.Ex1.m1.2.2.2.2.2.2d.cmml">if </mtext><mi id="S3.Ex1.m1.1.1.1.1.1.1.1.1.m1.1.1" xref="S3.Ex1.m1.1.1.1.1.1.1.1.1.m1.1.1.cmml">l</mi><mtext id="S3.Ex1.m1.2.2.2.2.2.2b" xref="S3.Ex1.m1.2.2.2.2.2.2d.cmml"> and </mtext><msup id="S3.Ex1.m1.2.2.2.2.2.2.2.2.m2.1.1" xref="S3.Ex1.m1.2.2.2.2.2.2.2.2.m2.1.1.cmml"><mi id="S3.Ex1.m1.2.2.2.2.2.2.2.2.m2.1.1.2" xref="S3.Ex1.m1.2.2.2.2.2.2.2.2.m2.1.1.2.cmml">l</mi><mo id="S3.Ex1.m1.2.2.2.2.2.2.2.2.m2.1.1.3" xref="S3.Ex1.m1.2.2.2.2.2.2.2.2.m2.1.1.3.cmml">′</mo></msup><mtext id="S3.Ex1.m1.2.2.2.2.2.2c" xref="S3.Ex1.m1.2.2.2.2.2.2d.cmml"> use different radios</mtext></mrow></mtd></mtr><mtr id="S3.Ex1.m1.13.13.13d" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtd columnalign="left" id="S3.Ex1.m1.13.13.13e" xref="S3.Ex1.m1.13.14.3.1.cmml"><mi id="S3.Ex1.m1.8.8.8.8.1.1" xref="S3.Ex1.m1.8.8.8.8.1.1.cmml"/></mtd><mtd columnalign="left" id="S3.Ex1.m1.13.13.13f" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtext id="S3.Ex1.m1.9.9.9.9.2.1" xref="S3.Ex1.m1.9.9.9.9.2.1a.cmml"> and are in the same channel;</mtext></mtd></mtr><mtr id="S3.Ex1.m1.13.13.13g" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtd columnalign="left" id="S3.Ex1.m1.13.13.13h" xref="S3.Ex1.m1.13.14.3.1.cmml"><mrow id="S3.Ex1.m1.10.10.10.10.3.1.3" xref="S3.Ex1.m1.13.14.3.1.cmml"><mn id="S3.Ex1.m1.10.10.10.10.3.1.1" xref="S3.Ex1.m1.10.10.10.10.3.1.1.cmml">0</mn><mo id="S3.Ex1.m1.10.10.10.10.3.1.3.1" xref="S3.Ex1.m1.13.14.3.1.cmml">,</mo></mrow></mtd><mtd columnalign="left" id="S3.Ex1.m1.13.13.13i" xref="S3.Ex1.m1.13.14.3.1.cmml"><mrow id="S3.Ex1.m1.4.4.4.4.2.2" xref="S3.Ex1.m1.4.4.4.4.2.2d.cmml"><mtext id="S3.Ex1.m1.4.4.4.4.2.2a" xref="S3.Ex1.m1.4.4.4.4.2.2d.cmml">if </mtext><mi id="S3.Ex1.m1.3.3.3.3.1.1.1.1.m1.1.1" xref="S3.Ex1.m1.3.3.3.3.1.1.1.1.m1.1.1.cmml">l</mi><mtext id="S3.Ex1.m1.4.4.4.4.2.2b" xref="S3.Ex1.m1.4.4.4.4.2.2d.cmml"> and </mtext><msup id="S3.Ex1.m1.4.4.4.4.2.2.2.2.m2.1.1" xref="S3.Ex1.m1.4.4.4.4.2.2.2.2.m2.1.1.cmml"><mi id="S3.Ex1.m1.4.4.4.4.2.2.2.2.m2.1.1.2" xref="S3.Ex1.m1.4.4.4.4.2.2.2.2.m2.1.1.2.cmml">l</mi><mo id="S3.Ex1.m1.4.4.4.4.2.2.2.2.m2.1.1.3" xref="S3.Ex1.m1.4.4.4.4.2.2.2.2.m2.1.1.3.cmml">′</mo></msup><mtext id="S3.Ex1.m1.4.4.4.4.2.2c" xref="S3.Ex1.m1.4.4.4.4.2.2d.cmml"> use different radios</mtext></mrow></mtd></mtr><mtr id="S3.Ex1.m1.13.13.13j" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtd columnalign="left" id="S3.Ex1.m1.13.13.13k" xref="S3.Ex1.m1.13.14.3.1.cmml"><mi id="S3.Ex1.m1.11.11.11.11.1.1" xref="S3.Ex1.m1.11.11.11.11.1.1.cmml"/></mtd><mtd columnalign="left" id="S3.Ex1.m1.13.13.13l" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtext id="S3.Ex1.m1.12.12.12.12.2.1" xref="S3.Ex1.m1.12.12.12.12.2.1a.cmml"> and are in different channels;</mtext></mtd></mtr><mtr id="S3.Ex1.m1.13.13.13m" xref="S3.Ex1.m1.13.14.3.1.cmml"><mtd columnalign="left" id="S3.Ex1.m1.13.13.13n" xref="S3.Ex1.m1.13.14.3.1.cmml"><mrow id="S3.Ex1.m1.13.13.13.13.3.1.3" xref="S3.Ex1.m1.13.14.3.1.cmml"><mi mathvariant="normal" id="S3.Ex1.m1.13.13.13.13.3.1.1" xref="S3.Ex1.m1.13.13.13.13.3.1.1.cmml">∞</mi><mo id="S3.Ex1.m1.13.13.13.13.3.1.3.1" xref="S3.Ex1.m1.13.14.3.1.cmml">,</mo></mrow></mtd><mtd columnalign="left" id="S3.Ex1.m1.13.13.13o" xref="S3.Ex1.m1.13.14.3.1.cmml"><mrow id="S3.Ex1.m1.6.6.6.6.2.2" xref="S3.Ex1.m1.6.6.6.6.2.2d.cmml"><mtext id="S3.Ex1.m1.6.6.6.6.2.2a" xref="S3.Ex1.m1.6.6.6.6.2.2d.cmml">if </mtext><mi id="S3.Ex1.m1.5.5.5.5.1.1.1.1.m1.1.1" xref="S3.Ex1.m1.5.5.5.5.1.1.1.1.m1.1.1.cmml">l</mi><mtext id="S3.Ex1.m1.6.6.6.6.2.2b" xref="S3.Ex1.m1.6.6.6.6.2.2d.cmml"> and </mtext><msup id="S3.Ex1.m1.6.6.6.6.2.2.2.2.m2.1.1" xref="S3.Ex1.m1.6.6.6.6.2.2.2.2.m2.1.1.cmml"><mi id="S3.Ex1.m1.6.6.6.6.2.2.2.2.m2.1.1.2" xref="S3.Ex1.m1.6.6.6.6.2.2.2.2.m2.1.1.2.cmml">l</mi><mo id="S3.Ex1.m1.6.6.6.6.2.2.2.2.m2.1.1.3" xref="S3.Ex1.m1.6.6.6.6.2.2.2.2.m2.1.1.3.cmml">′</mo></msup><mtext id="S3.Ex1.m1.6.6.6.6.2.2c" xref="S3.Ex1.m1.6.6.6.6.2.2d.cmml"> share one or more radios.</mtext></mrow></mtd></mtr></mtable></mrow></mrow></math>

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1812.11156

Formulas:

1-

\sum p_{k} | ψ_{k} ⟩ ⟨ ψ_{k} | \otimes ρ_{k}

2-

= \frac{m}{m - 1} \min_{χ \in Ω_{0}} {∥ ρ - χ ∥}^{2}

3-

= \frac{m}{m - 1} \min_{Π^{A}} {∥ ρ - Π^{A} (ρ) ∥}^{2},

4-

{∥ C ∥}^{2} = Tr (C^{†} C)

5-

Π^{A} = {Π_{k}^{A}}

6-

Π^{A} (ρ) := \sum_{k} (Π_{k}^{A} \otimes I^{B}) ρ (Π_{k}^{A} \otimes I^{B})

7-

ℂ^{m} \otimes ℂ^{n},

8-

𝜻 = {(ζ_{1}, ζ_{2}, \dots, ζ_{d^{2} - 1})}^{T}

9-

ρ = \frac{1}{m n} [I_{m} \otimes I_{n} + 𝒙^{T} 𝜻 \otimes I_{n} + I_{m} \otimes 𝒚^{T} 𝜻 + \sum T_{i j} ζ_{i} \otimes ζ_{j}] .

10-

𝒟 (ρ) ⩾ \frac{2}{m (m - 1) n} [{∥ 𝒙 ∥}^{2} + \frac{2}{n} {∥ T ∥}^{2} - \sum_{k = 1}^{m - 1} λ_{k}^{↓} (G)]

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/hep-th/9407065

Formulas:

1-

U_{q} (s u_{2})

2-

U_{q} (s u_{1, 1})

3-

U_{q} (s u_{2})

4-

U_{q} (s u_{1, 1})

5-

s p (8, ℝ)

6-

U_{q} (s u_{2})

7-

U_{q} (s u_{1, 1})

8-

U_{q} (s u_{2})

9-

U_{q} (s u_{1, 1})

10-

s p (8, ℝ)

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cond-mat

Result: incorrect

Run 6938496 (Agent767)