Run 11277762

Paper: https://arxiv.org/abs/gr-qc/0402066

Formulas:

1-

Λ M^{2} > 10^{- 7}

2-

Ω_{Λ} = .73 \pm .04

3-

Λ = 1.3 \times 10^{- 46} k m^{- 2}

4-

g = \frac{ν}{ν_{e}} = \frac{- p \cdot u_{o}}{- p \cdot u_{e}},

5-

u_{o b s}

6-

I (Ω, r, ν_{e})

7-

F_{ν} = \int g^{3} I (Ω, r, ν_{e}) 𝑑 Ω,

8-

(t, r, θ, ϕ)

9-

d s^{2} = - T (r) d t^{2} + R {(r)}^{- 1} d r^{2} + r^{2} d Ω^{2} .

10-

v = \sqrt{\frac{r T^{'}}{2}}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/nucl-th/0601025

Formulas:

1-

d_{A} \equiv ⟨ 𝒅_{A} ⟩

2-

≅ \sum_{n} \frac{1}{E_{0} - E_{n}} (⟨ 0 | e 𝒙 | n ⟩ ⟨ n | {\tilde{H}}_{e N} | 0 ⟩ + c.c.),

3-

{\tilde{H}}_{e N}^{d}

4-

{\tilde{H}}_{e N}^{d}

5-

= [𝒅 \cdot \nabla, H_{0}] + \dots .

6-

{\tilde{H}}_{e N}^{d}

7-

= \frac{e}{10} (⟨ r^{2} 𝒓 ⟩ - \frac{5}{3} \frac{1}{Z} ⟨ {[r^{2} (1 - \frac{4 \sqrt{π}}{5} Y_{2} (\hat{r})) \otimes 𝒓]}_{1} ⟩),

8-

⟨ r^{2} \otimes 𝒓 ⟩

9-

⟨ r^{2} ⟩ \otimes ⟨ 𝒓 ⟩

10-

⟨ 𝒅_{N} ⟩ \propto ⟨ 𝒓 ⟩

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/math/0509212

Formulas:

1-

u : ℝ^{n} \times ℝ_{+} \to ℝ

2-

\dot{u} = Δ u + ⟨ b, \nabla u ⟩ in ℝ^{n} \times ℝ_{+},

3-

b : ℝ^{n} \to ℝ^{n}

4-

b (x) = \frac{x}{| x |} ψ (| x |)

5-

u (x, 0) \to 0

6-

lim_{| x | \to \infty} lim_{t \to \infty} u (x, t) > 0,

7-

⟨ b, x ⟩ \geq 0

8-

⟨ b, x ⟩ \leq 0

9-

b (x) = - \frac{x}{| x |} ψ (| x |)

10-

ψ (r) \approx 1 / r

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0703239

Formulas:

1-

1900 - 2300 km / s

2-

0.11 - 0.12 % {yr}^{- 1}

3-

\sim 3300 \pm 1200 k m / s

4-

\sim 2.3 \pm 0.8 kpc

5-

\sim 0.35 ″ {yr}^{- 1}

6-

\sim 0.14 % {yr}^{- 1}

7-

0.113 % {yr}^{- 1}

8-

0.124 % {yr}^{- 1}

9-

\sim 0.18 % {yr}^{- 1}

10-

R_{s} / R_{c} ≃ 1.11

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0410621

Formulas:

1-

- 1.50 \times 10^{- 6} < w_{d m} < 1.13 \times 10^{- 6}

2-

- 8.78 \times 10^{- 3} < w_{d m} < 1.86 \times 10^{- 3}

3-

w_{d m} < 0

4-

w_{d m} < 0

5-

- 1.50 \times 10^{- 6} < w_{d m} < 1.13 \times 10^{- 6}

6-

- 8.78 \times 10^{- 3} < w_{d m} < 1.86 \times 10^{- 3}

7-

w_{d e} = - 1

8-

- \bar{ρ} (1 + δ Q),

9-

- \bar{ρ} (1 + w) v Q^{i},

10-

\bar{ρ} (1 + w) (v - B) Q_{i},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0506157

Formulas:

1-

Γ_{c} = {(3 {(1 + z)}^{3} E / 32 π n m_{p} c^{5} T^{3})}^{1 / 8} \sim 130 ζ^{3 / 8} E_{52}^{1 / 8} T_{2}^{- 3 / 8} n^{- 1 / 8}

2-

ζ = (1 + z) / 2

3-

E_{52} = E / 10^{52}

4-

T_{2} = T / 10^{2}

5-

t_{d} = (1 + z) {(3 E / 32 π Γ_{0}^{8} n m_{p} c^{5})}^{1 / 3}

6-

N d γ \propto γ^{- p} d γ, γ \geq γ_{m} \sim [(p - 2) / (p - 1)] ϵ_{e} (m_{p} / m_{e}) (Γ_{0} / Γ_{d}) \sim 180 (ϵ_{e} / 0.3) (Γ_{0} / Γ_{d})

7-

R_{d} \sim 2 c Γ_{d}^{2} t_{d} / (1 + z)

8-

τ (R_{d}) = σ_{T} N_{e} / 4 π R_{d}^{2} = (1 / 3) ℛ_{M} σ_{T} n R_{d}

9-

ℛ_{M} = Γ_{d}^{2} / Γ_{0}

10-

h ν_{X}^{'} = h ν_{X} / Γ \sim 50

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1502.00933

Formulas:

1-

Δ T_{c} = T_{c} - T_{c}^{0}

2-

Δ T_{c} / T_{c}^{0} ≃ b_{2}^{*} {(a / λ_{T})}^{2}

3-

b_{2}^{*} = γ^{2} b_{2}

4-

γ = γ (B_{0})

5-

B_{0} ≃ 403 G

6-

Δ T_{c} / T_{c}^{0} ≃ 1.8 (a / λ_{T})

7-

Δ T_{c} \equiv T_{c} - T_{c}^{0}

8-

λ_{T} \equiv \sqrt{2 π ℏ^{2} / m_{a} k_{B} T_{c}^{0}}

9-

Δ T_{c} / T_{c}^{0} ≃ b_{1} (a / λ_{T}) + b_{2} {(a / λ_{T})}^{2} .

10-

N ≃ (2 - 8) \times 10^{5}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/2007.05486

Formulas:

1-

f (x_{0}) = x_{0}

2-

b (r_{0}) = r_{0}

3-

b = b (r)

4-

d s^{2} = - e^{2 Φ (r)} d t^{2} + \frac{d r^{2}}{1 - b (r) / r} + r^{2} (d θ^{2} + {sin}^{2} θ d ϕ^{2}),

5-

Φ = Φ (r)

6-

b = b (r)

7-

b (r_{0}) = r_{0}

8-

b = b (r)

9-

b^{'} (r_{0}) < 1

10-

b (r) < r

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1807.05450

Formulas:

1-

t_2 g

2-

c (2 \times 2)

3-

15 \times 16 \times 17 Å

4-

E_a d s = E_C O / M (111) - (E_C O + E_M (111)),

5-

E_C O / M (111)

6-

E_C O

7-

E_M (111)

8-

d_C O

9-

d_P t - C O

10-

d_P t - C O

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1609.00239

Formulas:

1-

- 2.12 \pm 0.04 m s^{- 1} d^{- 1}

2-

M_{*} > 1.5 M_{⊙}

3-

α = 13^{h} 39^{m} 28^{s}

4-

δ = - 11 ° 17' 43 ″

5-

v \sin i_{*}

6-

- 05^{\circ} 26' 32.88 ″

7-

v \sin i_{*}

8-

u_{+} = u_{a} + u_{b}

9-

u_{-} = u_{a} - u_{b}

10-

M_{p} \sin i = K {(\frac{P}{2 π G})}^{\frac{1}{3}} M_{*}^{\frac{2}{3}} \sqrt{1 - e^{2}},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1307.8255

Formulas:

1-

L_{178 M H z} \sim

2-

α_{4.9 GHz}^{1.4 GHz}

3-

α_{4.9 GHz}^{1.4 GHz}

4-

L_{178 M H z}

5-

km s^{- 1} {Mpc}^{- 1}

6-

S_{5 G H z}

7-

α_{5 GHz}^{1.7 GHz}

8-

\sim 10^{25.5} W {Hz}^{- 1} {sr}^{- 1}

9-

{logP}_{ci5} = 0.62 {logP}_{t} + 8.41,

10-

P_{c i 5}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0409055

Formulas:

1-

| ψ^{+} ⟩ = \frac{| H ⟩ | V ⟩ + | V ⟩ | H ⟩}{\sqrt{2}}

2-

| ϕ^{-} ⟩ = \frac{| 45 ⟩ | 45 ⟩ - | 135 ⟩ | 135 ⟩}{\sqrt{2}}

3-

ρ = | V ⟩ ⟨ V |

4-

ρ = 1 / 2 [(1 + η_{1}) | V ⟩ ⟨ V | + (1 - η_{1}) | H ⟩ ⟨ H |]

5-

N_{2} = N_{0} \cdot α η_{2} [1 - η_{1} c o s (2 θ)]

6-

V = \frac{N_{2}^{V} - N_{2}^{H}}{N_{2}^{V} + N_{2}^{H}} = η_{1}

7-

S_{1} = S_{2} = S_{3} = 0

8-

S_{1} = η_{1} N

9-

S_{2} = S_{3} = 0

10-

P = \sqrt{\frac{S_{1}^{2} + S_{2}^{2} + S_{3}^{2}}{S_{0}^{2}}} .

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0501161

Formulas:

1-

{\hat{E}}_{\pm} = \hat{e} E_{0} {(\frac{ρ}{\sqrt{2} w})}^{| m |} e^{\pm i m ϕ} L_{p}^{| m |} (\frac{ρ^{2}}{2 w^{2}}) e^{- ρ^{2} / 4 w^{2}} e^{i k z} e^{i ω_{L} t},

2-

V_{Ω} (\vec{r}) \propto {| {\hat{E}}_{+} + {\hat{E}}_{-} |}^{2} = \frac{1}{2} ℏ \frac{Ω^{2} (ρ)}{Δ} (1 + \cos [2 m ϕ]),

3-

Δ = ω_{L} - ω_{0}

4-

ℏ Ω (ρ) \equiv 2 E_{0} (\hat{e} \cdot \vec{d}) {(\frac{ρ}{\sqrt{2} w})}^{| m |} L_{p}^{| m |} (\frac{ρ^{2}}{2 w^{2}}) e^{- ρ^{2} / 4 w^{2}}

5-

H = \sum_{σ, =, ↑ ↓} \int d^{3} \vec{r} (ψ_{σ}^{†} (\vec{r}) [- \frac{ℏ^{2}}{2 M} \nabla^{2} - μ_{σ} + V_{σ} (\vec{r}) + V_{Ω} (\vec{r})] ψ_{σ} (\vec{r}) + \frac{g}{2} ψ_{σ}^{†} (\vec{r}) ψ_{- σ}^{†} (\vec{r}) ψ_{- σ} (\vec{r}) ψ_{σ} (\vec{r}))

6-

ψ_{σ}^{†} (\vec{r})

7-

ψ_{σ} (\vec{r})

8-

V_{σ} (\vec{r})

9-

V_{Ω} (\vec{r})

10-

g = \frac{4 π a_{s} ℏ^{2}}{M}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0509836

Formulas:

1-

(2.7 \pm 3.0, 15.2 \pm 4.4)

2-

(1.1 \pm 1.3, 5.3 \pm 2.6)

3-

n (H) = 10^{3}

4-

\int_{Δ ν} I_{ν} 𝑑 ν = Δ ν B_{ν} (T)

5-

E_{2} < E_{1} < E_{0}

6-

g_{J} = 2 J + 1

7-

\frac{g_{0}}{g_{1}} - \frac{n_{0}}{n_{1}} < 0

8-

n_{crit} = A_{01} / q_{01} (T)

9-

g_{0} / g_{1} - n_{0} / n_{1} > 0

10-

g_{0} / g_{1} - n_{0} / n_{1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0207658

Formulas:

1-

{}_{c 2}^{a b}

2-

{}_{c 2}^{a b}

3-

{}_{c 2}^{a b}

4-

{}_{c 2}^{a b}

5-

{}_{c 2}^{a b}

6-

{}_{c 2}^{a b}

7-

{}_{c 2}^{a b}

8-

{}_{c 2}^{a b}

9-

{}_{c 2}^{a b}

10-

{}_{c 2}^{a b}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1708.03050

Formulas:

1-

[O {(2)}_{s} \times O {(2)}_{v}] \times Z_{2}^{d}

2-

(2 + 1) d

3-

(γ_{0}, γ_{1}, γ_{2}) = (σ^{2}, σ^{3}, σ^{1})

4-

ℒ_{QED} = \sum_{j = 1}^{2} {\bar{ψ}}_{j} γ \cdot (\partial - i a) ψ_{j} + m {\bar{ψ}}_{j} ψ_{j} + M \bar{ψ} σ^{3} ψ

5-

ℒ_{CP^{1}} = \sum_{j = 1}^{2} {| (\partial - i b) z_{j} |}^{2} + g {| z_{j} |}^{4} + r {| z_{j} |}^{2} + h z^{†} σ^{3} z

6-

[O {(2)}_{s} \times O {(2)}_{v}] \times Z_{2}^{d}

7-

O {(2)}_{s} = U {(1)}_{s} ⋊ Z_{2}^{s}

8-

{(z_{1}, z_{2})}^{t}

9-

U {(1)}_{s} : z \to e^{i \frac{θ}{2} σ^{3}} z, Z_{2}^{s} : z \to σ^{1} z .

10-

O {(2)}_{v} = U {(1)}_{v} ⋊ Z_{2}^{v}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1706.06736

Formulas:

1-

f_{r f} \sim f_{0}

2-

f_{r f} = 7 - 8

3-

\begin{matrix} \frac{d \hat{m}}{d t} = - γ (\hat{m} \times {\vec{H}}_{e f f}) + α (\hat{m} \times \frac{d \hat{m}}{d t}) \\ - γ \frac{J ℏ P}{2 e M_{s} d (1 + P^{2} \cos ϕ)} [\hat{m} \times (\hat{m} \times {\hat{e}}_{p}) + b_{f} (\hat{m} \times {\hat{e}}_{p})] \end{matrix}

4-

I_{d c} / I_{t h}

5-

{\vec{H}}_{e f f}

6-

{\vec{H}}_{e f f} = {\vec{H}}_{e x t} + {\vec{H}}_{d e m a g} + {\vec{H}}_{T} .

7-

{\vec{H}}_{e x t}

8-

{\vec{H}}_{e x t} = | H_{x} | \hat{x} + | H_{y} | \hat{y} .

9-

{\vec{H}}_{d e m a g}

10-

{\vec{H}}_{d e m a g} = - M_{s} (N_{x} m_{x} \hat{x} + N_{y} m_{y} \hat{y} + N_{z} m_{z} \hat{z}),

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0808.0118

Formulas:

1-

R = \int_{1}^{\infty} (m - M_{R}) ϕ (m) 𝑑 m,

2-

y_{Z} = \frac{1}{1 - R} \int_{1}^{\infty} m p_{Z, m} ϕ (m) 𝑑 m,

3-

Z = y_{Z} \ln (μ^{- 1}),

4-

M_{g} / M_{t o t}

5-

y_{Z} = f (Z)

6-

\int_{0}^{Z} \frac{d w}{f (w)} = \ln (μ^{- 1}) .

7-

W (t) = λ (1 - R) ψ (t),

8-

A (t) = Λ (1 - R) ψ (t) .

9-

{\begin{matrix} \frac{d M_{t o t}}{d t} = (Λ - λ) (1 - R) ψ (t) \\ \frac{d M_{g}}{d t} = (Λ - λ - 1) (1 - R) ψ (t) \\ \frac{d M_{Z}}{d t} = (1 - R) ψ (t) [Λ Z_{A} + y_{Z} - (λ + 1) Z] \end{matrix}

10-

M_{Z} = Z \cdot M_{g}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1411.5922

Formulas:

1-

Z \in 𝐁 (ℌ_{B}, ℌ_{A})

2-

dom A \subset dom B

3-

∥ A f ∥ \leq c ∥ B f ∥

4-

A, B \in 𝐁 (ℌ, 𝔎)

5-

ran A \subset ran B

6-

W \in 𝐁 (ℌ)

7-

A A^{*} \leq λ B B^{*}

8-

{∥ W ∥}^{2} = inf {μ : A A^{*} \leq μ B B^{*}}

9-

\ker A = \ker W

10-

ran W \subset \bar{ran} B^{*}

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0005298

Formulas:

1-

v_{e} = 1000 km s^{- 1}

2-

v_{\max} = 5 \times 10^{4} km s^{- 1}

3-

T_{b b} = 7500 K

4-

v_{p h} = 16, 000 km s^{- 1}

5-

λ λ 7877, 7896

6-

20, 000 km s^{- 1}

7-

λ λ 4738, 4745

8-

T_{b b} = 7000 K

9-

v_{p h} = 12, 000 km s^{- 1}

10-

19, 000 km s^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0901.2800

Formulas:

1-

p^{↑} {\bar{p}}^{↑} \to l^{+} l^{-} X

2-

d ω \equiv d Q^{2} d x_{F} d Ω

3-

A_{T T} = \frac{d σ^{↑ ↑} / d ω - d σ^{↑ ↓} / d ω}{d σ^{↑ ↑} / d ω + d σ^{↑ ↓} / d ω} = {\hat{a}}_{T T} (θ, ϕ) \frac{\sum_{a} e_{a}^{2} h_{1}^{a} (x_{1}, Q^{2}) h_{1}^{a} (x_{2}, Q^{2})}{\sum_{a} e_{a}^{2} f_{1}^{a} (x_{1}, Q^{2}) f_{1}^{a} (x_{2}, Q^{2})},

4-

Ω = (θ, ϕ)

5-

Q^{2} = {(x_{1} P_{1} + x_{2} P_{2})}^{2} = x_{1} x_{2} s

6-

x_{F} = x_{1} - x_{2}

7-

s = {(P_{1} + P_{2})}^{2}

8-

p^{↑} {\bar{p}}^{↑}

9-

{\hat{a}}_{T T} (θ, ϕ)

10-

0.2 ≲ Q / \sqrt{s} ≲ 0.7

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/2012.14052

Formulas:

1-

\sim 100 {cm}^{2} / Vs

2-

U_{c 1} ≲ U ≲ U_{c 2}

3-

N = 4 \times N_{x}

4-

H = \sum_{i j, σ} t_{i j} c_{i σ}^{†} c_{j σ} + U \sum_{i} (n_{i, ↑} - \frac{1}{2}) (n_{i, ↓} - \frac{1}{2}),

5-

t_{1} = - 1.220 e V

6-

t_{2} = 3.665 e V

7-

t_{3} = - 0.205 e V

8-

t_{4} = - 0.105 e V

9-

t_{5} = - 0.055 e V

10-

U \sum_{i} (⟨ n_{i, ↑} ⟩ n_{i, ↓} + n_{i, ↑} ⟨ n_{i, ↓} ⟩ - ⟨ n_{i, ↑} ⟩ ⟨ n_{i, ↓} ⟩ - \frac{n_{i, ↑} + n_{i, ↓}}{2} + \frac{1}{4})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9412003

Formulas:

1-

R = {(L_{x} L_{y} L_{z})}^{1 / 3}

2-

L = L_{x} = L_{y} = L_{z}

3-

y = 2 c H_{o}^{- 1}

4-

R > 400 h^{- 1}

5-

R > 600 h^{- 1}

6-

\frac{δ T}{T} (θ, ϕ) = - \frac{1}{2} \frac{H_{o}^{2}}{c^{2}} \sum_{𝐤} \frac{δ_{𝐤}}{k^{2}} e^{i 𝐤 \cdot 𝐱}

7-

y \equiv 2 c H_{o}^{- 1}

8-

\frac{δ T}{T} (θ, ϕ) = \sum_{l = 0}^{\infty} \sum_{m = - l}^{l} a_{l m} Y_{l m} (\hat{𝐱}),

9-

a_{l m} = - 2 π i^{l} \frac{H_{o}^{2}}{c^{2}} \sum_{𝐤} \frac{δ_{𝐤}}{k^{2}} j_{l} (k y) Y_{l m}^{*} (\hat{𝐤}),

10-

⟨ {| a_{l m} |}^{2} ⟩ = 16 π \sum_{𝐤} \frac{{| δ_{𝐤} |}^{2}}{{(k y)}^{4}} j_{l}^{2} (k y)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1509.02885

Formulas:

1-

M v_{n} + m c = M v_{n + 1} - m c,

2-

δ v \equiv v_{n + 1} - v_{n} = 2 \frac{m c}{M}

3-

δ v = 2 m c / M

4-

x^{(1)} (0)

5-

- x^{(2)} (0) = \frac{r_{0}}{2},

6-

v^{(1)} (0)

7-

v^{(2)} (0) = 0 .

8-

\frac{d x^{(i)}}{d t}

9-

\frac{d X^{(i)}}{d t}

10-

V^{(1)} = + c

Formulas (html):

Correct Categorie: physics

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1812.06188

Formulas:

1-

\sqrt{s_{NN}} = 5.02 TeV

2-

v_{n} = ⟨ \cos n (φ - Ψ_{n}) ⟩

3-

D^{0} \to K^{-} π^{+}

4-

D^{+} \to K^{-} π^{+} π^{+}

5-

D^{* +} \to D^{0} π^{+} \to K^{-} π^{+} π^{+}

6-

D_{s}^{+} \to ϕ π^{+} \to K^{-} K^{+} π^{+}

7-

\sqrt{s_{NN}} = 5.02 TeV

8-

R_{AA} = (d N_{AA} / d p_{T}) / (⟨ T_{AA} ⟩ d σ_{pp} / d p_{T})

9-

d N_{AA} / d p_{T}

10-

d σ_{pp} / d p_{T}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1002.3012

Formulas:

1-

R_{Bondi} \approx G M / c_{s}^{2}

2-

{\dot{M}}_{Bondi} \approx 4 π R_{Bondi}^{2} ρ_{0} c_{s}

3-

\approx 100 {cm}^{- 3}

4-

R_{Bondi} \approx 0.04 pc \approx 1^{^{''}} \approx 10^{5} R_{s}

5-

{\dot{M}}_{Bondi} \approx 10^{- 5} M_{⊙} {yr}^{- 1}

6-

\dot{M} \approx 3 \times 10^{- 6} M_{⊙} {yr}^{- 1}

7-

\sim 10^{4} R_{s}

8-

L \approx 10^{36} erg s^{- 1} \approx 3 \times 10^{- 9} L_{Edd}

9-

- 5.6 \pm 0.7 \times 10^{5} rad m^{- 2}

10-

2 \times 10^{- 7} M_{⊙} {yr}^{- 1}

Formulas (html):

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xref="S2.SS1.p2.11.m11.1.1.3.2.3.3.2.cmml">6</mn></mrow></msup></mrow><mo id="S2.SS1.p2.11.m11.1.1.3.1" xref="S2.SS1.p2.11.m11.1.1.3.1.cmml">⁢</mo><mpadded width="+5pt" id="S2.SS1.p2.11.m11.1.1.3.3" xref="S2.SS1.p2.11.m11.1.1.3.3.cmml"><msub id="S2.SS1.p2.11.m11.1.1.3.3a" xref="S2.SS1.p2.11.m11.1.1.3.3.cmml"><mi id="S2.SS1.p2.11.m11.1.1.3.3.2" xref="S2.SS1.p2.11.m11.1.1.3.3.2.cmml">M</mi><mo id="S2.SS1.p2.11.m11.1.1.3.3.3" xref="S2.SS1.p2.11.m11.1.1.3.3.3.cmml">⊙</mo></msub></mpadded><mo id="S2.SS1.p2.11.m11.1.1.3.1a" xref="S2.SS1.p2.11.m11.1.1.3.1.cmml">⁢</mo><msup id="S2.SS1.p2.11.m11.1.1.3.4" xref="S2.SS1.p2.11.m11.1.1.3.4.cmml"><mi id="S2.SS1.p2.11.m11.1.1.3.4.2" xref="S2.SS1.p2.11.m11.1.1.3.4.2.cmml">yr</mi><mrow id="S2.SS1.p2.11.m11.1.1.3.4.3" xref="S2.SS1.p2.11.m11.1.1.3.4.3.cmml"><mo id="S2.SS1.p2.11.m11.1.1.3.4.3.1" xref="S2.SS1.p2.11.m11.1.1.3.4.3.1.cmml">-</mo><mn id="S2.SS1.p2.11.m11.1.1.3.4.3.2" xref="S2.SS1.p2.11.m11.1.1.3.4.3.2.cmml">1</mn></mrow></msup></mrow></mrow></math>, <math><mrow id="S2.SS1.p2.12.m12.1.1" xref="S2.SS1.p2.12.m12.1.1.cmml"><mi id="S2.SS1.p2.12.m12.1.1.2" xref="S2.SS1.p2.12.m12.1.1.2.cmml"/><mo id="S2.SS1.p2.12.m12.1.1.1" xref="S2.SS1.p2.12.m12.1.1.1.cmml">∼</mo><mrow id="S2.SS1.p2.12.m12.1.1.3" xref="S2.SS1.p2.12.m12.1.1.3.cmml"><msup id="S2.SS1.p2.12.m12.1.1.3.2" xref="S2.SS1.p2.12.m12.1.1.3.2.cmml"><mn id="S2.SS1.p2.12.m12.1.1.3.2.2" xref="S2.SS1.p2.12.m12.1.1.3.2.2.cmml">10</mn><mn id="S2.SS1.p2.12.m12.1.1.3.2.3" xref="S2.SS1.p2.12.m12.1.1.3.2.3.cmml">4</mn></msup><mo id="S2.SS1.p2.12.m12.1.1.3.1" xref="S2.SS1.p2.12.m12.1.1.3.1.cmml">⁢</mo><msub id="S2.SS1.p2.12.m12.1.1.3.3" xref="S2.SS1.p2.12.m12.1.1.3.3.cmml"><mi id="S2.SS1.p2.12.m12.1.1.3.3.2" xref="S2.SS1.p2.12.m12.1.1.3.3.2.cmml">R</mi><mi id="S2.SS1.p2.12.m12.1.1.3.3.3" xref="S2.SS1.p2.12.m12.1.1.3.3.3.cmml">s</mi></msub></mrow></mrow></math>, <math><mrow id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml"><mi id="S2.SS1.p3.2.m2.1.1.2" xref="S2.SS1.p3.2.m2.1.1.2.cmml">L</mi><mo id="S2.SS1.p3.2.m2.1.1.3" xref="S2.SS1.p3.2.m2.1.1.3.cmml">≈</mo><mrow id="S2.SS1.p3.2.m2.1.1.4" xref="S2.SS1.p3.2.m2.1.1.4.cmml"><mpadded width="+1.7pt" id="S2.SS1.p3.2.m2.1.1.4.2" xref="S2.SS1.p3.2.m2.1.1.4.2.cmml"><msup id="S2.SS1.p3.2.m2.1.1.4.2a" xref="S2.SS1.p3.2.m2.1.1.4.2.cmml"><mn id="S2.SS1.p3.2.m2.1.1.4.2.2" xref="S2.SS1.p3.2.m2.1.1.4.2.2.cmml">10</mn><mn id="S2.SS1.p3.2.m2.1.1.4.2.3" xref="S2.SS1.p3.2.m2.1.1.4.2.3.cmml">36</mn></msup></mpadded><mo id="S2.SS1.p3.2.m2.1.1.4.1" xref="S2.SS1.p3.2.m2.1.1.4.1.cmml">⁢</mo><mpadded width="+1.7pt" id="S2.SS1.p3.2.m2.1.1.4.3" xref="S2.SS1.p3.2.m2.1.1.4.3.cmml"><mi id="S2.SS1.p3.2.m2.1.1.4.3a" xref="S2.SS1.p3.2.m2.1.1.4.3.cmml">erg</mi></mpadded><mo id="S2.SS1.p3.2.m2.1.1.4.1a" xref="S2.SS1.p3.2.m2.1.1.4.1.cmml">⁢</mo><msup id="S2.SS1.p3.2.m2.1.1.4.4" xref="S2.SS1.p3.2.m2.1.1.4.4.cmml"><mi mathvariant="normal" id="S2.SS1.p3.2.m2.1.1.4.4.2" xref="S2.SS1.p3.2.m2.1.1.4.4.2.cmml">s</mi><mrow id="S2.SS1.p3.2.m2.1.1.4.4.3" xref="S2.SS1.p3.2.m2.1.1.4.4.3.cmml"><mo id="S2.SS1.p3.2.m2.1.1.4.4.3.1" xref="S2.SS1.p3.2.m2.1.1.4.4.3.1.cmml">-</mo><mn id="S2.SS1.p3.2.m2.1.1.4.4.3.2" xref="S2.SS1.p3.2.m2.1.1.4.4.3.2.cmml">1</mn></mrow></msup></mrow><mo id="S2.SS1.p3.2.m2.1.1.5" xref="S2.SS1.p3.2.m2.1.1.5.cmml">≈</mo><mrow id="S2.SS1.p3.2.m2.1.1.6" xref="S2.SS1.p3.2.m2.1.1.6.cmml"><mrow id="S2.SS1.p3.2.m2.1.1.6.2" xref="S2.SS1.p3.2.m2.1.1.6.2.cmml"><mn id="S2.SS1.p3.2.m2.1.1.6.2.2" xref="S2.SS1.p3.2.m2.1.1.6.2.2.cmml">3</mn><mo id="S2.SS1.p3.2.m2.1.1.6.2.1" xref="S2.SS1.p3.2.m2.1.1.6.2.1.cmml">×</mo><msup id="S2.SS1.p3.2.m2.1.1.6.2.3" xref="S2.SS1.p3.2.m2.1.1.6.2.3.cmml"><mn id="S2.SS1.p3.2.m2.1.1.6.2.3.2" xref="S2.SS1.p3.2.m2.1.1.6.2.3.2.cmml">10</mn><mrow id="S2.SS1.p3.2.m2.1.1.6.2.3.3" xref="S2.SS1.p3.2.m2.1.1.6.2.3.3.cmml"><mo id="S2.SS1.p3.2.m2.1.1.6.2.3.3.1" xref="S2.SS1.p3.2.m2.1.1.6.2.3.3.1.cmml">-</mo><mn id="S2.SS1.p3.2.m2.1.1.6.2.3.3.2" xref="S2.SS1.p3.2.m2.1.1.6.2.3.3.2.cmml">9</mn></mrow></msup></mrow><mo id="S2.SS1.p3.2.m2.1.1.6.1" xref="S2.SS1.p3.2.m2.1.1.6.1.cmml">⁢</mo><msub id="S2.SS1.p3.2.m2.1.1.6.3" xref="S2.SS1.p3.2.m2.1.1.6.3.cmml"><mi mathvariant="normal" id="S2.SS1.p3.2.m2.1.1.6.3.2" xref="S2.SS1.p3.2.m2.1.1.6.3.2.cmml">L</mi><mi id="S2.SS1.p3.2.m2.1.1.6.3.3" xref="S2.SS1.p3.2.m2.1.1.6.3.3.cmml">Edd</mi></msub></mrow></mrow></math>, <math><mrow id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml"><mrow id="S2.SS1.p4.3.m3.1.1.2" xref="S2.SS1.p4.3.m3.1.1.2.cmml"><mo id="S2.SS1.p4.3.m3.1.1.2.1" xref="S2.SS1.p4.3.m3.1.1.2.1.cmml">-</mo><mn id="S2.SS1.p4.3.m3.1.1.2.2" xref="S2.SS1.p4.3.m3.1.1.2.2.cmml">5.6</mn></mrow><mo id="S2.SS1.p4.3.m3.1.1.1" xref="S2.SS1.p4.3.m3.1.1.1.cmml">±</mo><mrow id="S2.SS1.p4.3.m3.1.1.3" xref="S2.SS1.p4.3.m3.1.1.3.cmml"><mrow id="S2.SS1.p4.3.m3.1.1.3.2" xref="S2.SS1.p4.3.m3.1.1.3.2.cmml"><mn id="S2.SS1.p4.3.m3.1.1.3.2.2" xref="S2.SS1.p4.3.m3.1.1.3.2.2.cmml">0.7</mn><mo id="S2.SS1.p4.3.m3.1.1.3.2.1" xref="S2.SS1.p4.3.m3.1.1.3.2.1.cmml">×</mo><msup id="S2.SS1.p4.3.m3.1.1.3.2.3" xref="S2.SS1.p4.3.m3.1.1.3.2.3.cmml"><mn id="S2.SS1.p4.3.m3.1.1.3.2.3.2" xref="S2.SS1.p4.3.m3.1.1.3.2.3.2.cmml">10</mn><mn id="S2.SS1.p4.3.m3.1.1.3.2.3.3" xref="S2.SS1.p4.3.m3.1.1.3.2.3.3.cmml">5</mn></msup></mrow><mo id="S2.SS1.p4.3.m3.1.1.3.1" xref="S2.SS1.p4.3.m3.1.1.3.1.cmml">⁢</mo><mpadded width="+3.3pt" id="S2.SS1.p4.3.m3.1.1.3.3" xref="S2.SS1.p4.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.p4.3.m3.1.1.3.3a" xref="S2.SS1.p4.3.m3.1.1.3.3.cmml">rad</mi></mpadded><mo id="S2.SS1.p4.3.m3.1.1.3.1a" xref="S2.SS1.p4.3.m3.1.1.3.1.cmml">⁢</mo><msup id="S2.SS1.p4.3.m3.1.1.3.4" xref="S2.SS1.p4.3.m3.1.1.3.4.cmml"><mi mathvariant="normal" id="S2.SS1.p4.3.m3.1.1.3.4.2" xref="S2.SS1.p4.3.m3.1.1.3.4.2.cmml">m</mi><mrow id="S2.SS1.p4.3.m3.1.1.3.4.3" xref="S2.SS1.p4.3.m3.1.1.3.4.3.cmml"><mo id="S2.SS1.p4.3.m3.1.1.3.4.3.1" xref="S2.SS1.p4.3.m3.1.1.3.4.3.1.cmml">-</mo><mn id="S2.SS1.p4.3.m3.1.1.3.4.3.2" xref="S2.SS1.p4.3.m3.1.1.3.4.3.2.cmml">2</mn></mrow></msup></mrow></mrow></math>, <math><mrow id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml"><mrow id="S2.SS1.p4.4.m4.1.1.2" xref="S2.SS1.p4.4.m4.1.1.2.cmml"><mn id="S2.SS1.p4.4.m4.1.1.2.2" xref="S2.SS1.p4.4.m4.1.1.2.2.cmml">2</mn><mo id="S2.SS1.p4.4.m4.1.1.2.1" xref="S2.SS1.p4.4.m4.1.1.2.1.cmml">×</mo><msup id="S2.SS1.p4.4.m4.1.1.2.3" xref="S2.SS1.p4.4.m4.1.1.2.3.cmml"><mn id="S2.SS1.p4.4.m4.1.1.2.3.2" xref="S2.SS1.p4.4.m4.1.1.2.3.2.cmml">10</mn><mrow id="S2.SS1.p4.4.m4.1.1.2.3.3" xref="S2.SS1.p4.4.m4.1.1.2.3.3.cmml"><mo id="S2.SS1.p4.4.m4.1.1.2.3.3.1" xref="S2.SS1.p4.4.m4.1.1.2.3.3.1.cmml">-</mo><mn id="S2.SS1.p4.4.m4.1.1.2.3.3.2" xref="S2.SS1.p4.4.m4.1.1.2.3.3.2.cmml">7</mn></mrow></msup></mrow><mo id="S2.SS1.p4.4.m4.1.1.1" xref="S2.SS1.p4.4.m4.1.1.1.cmml">⁢</mo><msub id="S2.SS1.p4.4.m4.1.1.3" xref="S2.SS1.p4.4.m4.1.1.3.cmml"><mi id="S2.SS1.p4.4.m4.1.1.3.2" xref="S2.SS1.p4.4.m4.1.1.3.2.cmml">M</mi><mo id="S2.SS1.p4.4.m4.1.1.3.3" xref="S2.SS1.p4.4.m4.1.1.3.3.cmml">⊙</mo></msub><mo id="S2.SS1.p4.4.m4.1.1.1a" xref="S2.SS1.p4.4.m4.1.1.1.cmml">⁢</mo><msup id="S2.SS1.p4.4.m4.1.1.4" xref="S2.SS1.p4.4.m4.1.1.4.cmml"><mi id="S2.SS1.p4.4.m4.1.1.4.2" xref="S2.SS1.p4.4.m4.1.1.4.2.cmml">yr</mi><mrow id="S2.SS1.p4.4.m4.1.1.4.3" xref="S2.SS1.p4.4.m4.1.1.4.3.cmml"><mo id="S2.SS1.p4.4.m4.1.1.4.3.1" xref="S2.SS1.p4.4.m4.1.1.4.3.1.cmml">-</mo><mn id="S2.SS1.p4.4.m4.1.1.4.3.2" xref="S2.SS1.p4.4.m4.1.1.4.3.2.cmml">1</mn></mrow></msup></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1307.6693

Formulas:

1-

\sum_{i + j = n} (\binom{2 i}{i}) (\binom{2 j}{j}) = 4^{n}

2-

\sum_{i + j = n} (\binom{2 i}{i}) (\binom{2 j}{j}) = 4^{n} .

3-

{(1 - 4 x)}^{- 1 / 2} = \sum_{n \geq 0} (\binom{2 n}{n}) x^{n},

4-

\sum_{i + j = n} (\binom{2 i - ℓ}{i}) (\binom{2 j + ℓ}{j}) = 4^{n} .

5-

\sum_{i = 0}^{p} {(- 1)}^{i} (\binom{ℓ - i}{p}) (\binom{p}{i}) = 1 .

6-

{1, \dots, ℓ}

7-

j = 1, \dots, p

8-

{p + 1, \dots, n} \in 𝒜

9-

(\binom{ℓ}{ℓ - p}) - 1 = | 𝒜_{1} \cup \dots \cup 𝒜_{p} |

10-

= \sum_{i = 1}^{p} {(- 1)}^{i + 1} (\sum_{1 \leq j_{1} < \dots < j_{i} \leq p} | 𝒜_{j_{1}} \cap \dots \cap 𝒜_{j_{i}} |)

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0908.1852

Formulas:

1-

n = 1, 2, \dots, N

2-

{\vec{Y}}_{n o r m}

3-

{\vec{Y}}_{m o d}

4-

\vec{Y} = {\vec{Y}}_{m o d} - {\vec{Y}}_{n o r m}

5-

{\vec{R}}_{n + 1} - R_{n} \cdot {\vec{Y}}_{n}

6-

T = \sum_{n = 1}^{N} [\frac{m_{n}}{2} {(\dot{{\vec{Y}}_{n}})}^{2}]

7-

U_{d e v} = \sum_{n = 1}^{N} \frac{ϵ_{n}}{2} {({\vec{Y}}_{n})}^{2}

8-

U_{r o t} = \sum_{n = 1}^{N} \frac{k_{n}}{2} {({\vec{R}}_{n + 1} - R \cdot {\vec{Y}}_{n})}^{2}

9-

U = U_{r o t} + U_{d e v}

10-

U_{d e v}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1903.02232

Formulas:

1-

{T_{0}, \dots, T_{i}}

2-

{C_{0}, \dots, C_{i}}

3-

{M_{i}^{0}, \dots, M_{i}^{m}}

4-

{T_{i}^{t}}_{W}

5-

t = {\begin{matrix} 1, T_{i} is always visible within W (e.g., T_{5} and T_{13} in Fig. 2) \\ 0, T_{i} is partially visible within W (e.g., T_{25} in Fig. 2) \\ - 1, T_{i} is invisible within W \end{matrix}

6-

{T_{i}^{t}}_{C_{0}}^{t = 1}

7-

\frac{| {T_{i}^{t}}_{C_{0}}^{t = 1} |}{| {T_{i}^{t}}_{C_{0}}^{t = 1} | + | {T_{i}^{t}}_{C_{0}}^{t = 0} |} \geq 0.8

8-

{T_{i}^{t}}_{C}^{t = 0, 1}

9-

{F^{(j, k)}}_{C}^{0 < | j - k | \leq r}

10-

F^{(j, k)}

Formulas (html):

<math><mrow id="S3.p1.1.m1.3.3.2" xref="S3.p1.1.m1.3.3.3.cmml"><mo id="S3.p1.1.m1.3.3.2.3" xref="S3.p1.1.m1.3.3.3.cmml">{</mo><msub id="S3.p1.1.m1.2.2.1.1" xref="S3.p1.1.m1.2.2.1.1.cmml"><mi id="S3.p1.1.m1.2.2.1.1.2" xref="S3.p1.1.m1.2.2.1.1.2.cmml">T</mi><mn id="S3.p1.1.m1.2.2.1.1.3" xref="S3.p1.1.m1.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.p1.1.m1.3.3.2.4" xref="S3.p1.1.m1.3.3.3.cmml">,</mo><mi mathvariant="normal" id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml">…</mi><mo id="S3.p1.1.m1.3.3.2.5" xref="S3.p1.1.m1.3.3.3.cmml">,</mo><msub id="S3.p1.1.m1.3.3.2.2" xref="S3.p1.1.m1.3.3.2.2.cmml"><mi id="S3.p1.1.m1.3.3.2.2.2" xref="S3.p1.1.m1.3.3.2.2.2.cmml">T</mi><mi id="S3.p1.1.m1.3.3.2.2.3" xref="S3.p1.1.m1.3.3.2.2.3.cmml">i</mi></msub><mo id="S3.p1.1.m1.3.3.2.6" xref="S3.p1.1.m1.3.3.3.cmml">}</mo></mrow></math>, <math><mrow id="S3.p2.1.m1.3.3.2" xref="S3.p2.1.m1.3.3.3.cmml"><mo id="S3.p2.1.m1.3.3.2.3" xref="S3.p2.1.m1.3.3.3.cmml">{</mo><msub id="S3.p2.1.m1.2.2.1.1" xref="S3.p2.1.m1.2.2.1.1.cmml"><mi id="S3.p2.1.m1.2.2.1.1.2" xref="S3.p2.1.m1.2.2.1.1.2.cmml">C</mi><mn id="S3.p2.1.m1.2.2.1.1.3" xref="S3.p2.1.m1.2.2.1.1.3.cmml">0</mn></msub><mo id="S3.p2.1.m1.3.3.2.4" xref="S3.p2.1.m1.3.3.3.cmml">,</mo><mi mathvariant="normal" id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml">…</mi><mo id="S3.p2.1.m1.3.3.2.5" xref="S3.p2.1.m1.3.3.3.cmml">,</mo><msub id="S3.p2.1.m1.3.3.2.2" xref="S3.p2.1.m1.3.3.2.2.cmml"><mi id="S3.p2.1.m1.3.3.2.2.2" xref="S3.p2.1.m1.3.3.2.2.2.cmml">C</mi><mi id="S3.p2.1.m1.3.3.2.2.3" xref="S3.p2.1.m1.3.3.2.2.3.cmml">i</mi></msub><mo id="S3.p2.1.m1.3.3.2.6" xref="S3.p2.1.m1.3.3.3.cmml">}</mo></mrow></math>, <math><mrow id="S3.p2.3.m3.3.3.2" xref="S3.p2.3.m3.3.3.3.cmml"><mo stretchy="false" id="S3.p2.3.m3.3.3.2.3" xref="S3.p2.3.m3.3.3.3.cmml">{</mo><msubsup id="S3.p2.3.m3.2.2.1.1" xref="S3.p2.3.m3.2.2.1.1.cmml"><mi id="S3.p2.3.m3.2.2.1.1.2.2" xref="S3.p2.3.m3.2.2.1.1.2.2.cmml">M</mi><mi id="S3.p2.3.m3.2.2.1.1.2.3" xref="S3.p2.3.m3.2.2.1.1.2.3.cmml">i</mi><mn id="S3.p2.3.m3.2.2.1.1.3" xref="S3.p2.3.m3.2.2.1.1.3.cmml">0</mn></msubsup><mo id="S3.p2.3.m3.3.3.2.4" xref="S3.p2.3.m3.3.3.3.cmml">,</mo><mi mathvariant="normal" id="S3.p2.3.m3.1.1" xref="S3.p2.3.m3.1.1.cmml">…</mi><mo id="S3.p2.3.m3.3.3.2.5" xref="S3.p2.3.m3.3.3.3.cmml">,</mo><msubsup id="S3.p2.3.m3.3.3.2.2" xref="S3.p2.3.m3.3.3.2.2.cmml"><mi id="S3.p2.3.m3.3.3.2.2.2.2" xref="S3.p2.3.m3.3.3.2.2.2.2.cmml">M</mi><mi id="S3.p2.3.m3.3.3.2.2.2.3" xref="S3.p2.3.m3.3.3.2.2.2.3.cmml">i</mi><mi id="S3.p2.3.m3.3.3.2.2.3" xref="S3.p2.3.m3.3.3.2.2.3.cmml">m</mi></msubsup><mo stretchy="false" id="S3.p2.3.m3.3.3.2.6" xref="S3.p2.3.m3.3.3.3.cmml">}</mo></mrow></math>, <math><msub id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml"><mrow id="S4.p1.3.m3.1.1.1.1" xref="S4.p1.3.m3.1.1.1.2.cmml"><mo id="S4.p1.3.m3.1.1.1.1.2" xref="S4.p1.3.m3.1.1.1.2.cmml">{</mo><msubsup id="S4.p1.3.m3.1.1.1.1.1" xref="S4.p1.3.m3.1.1.1.1.1.cmml"><mi id="S4.p1.3.m3.1.1.1.1.1.2.2" xref="S4.p1.3.m3.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.p1.3.m3.1.1.1.1.1.2.3" xref="S4.p1.3.m3.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.p1.3.m3.1.1.1.1.1.3" xref="S4.p1.3.m3.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S4.p1.3.m3.1.1.1.1.3" xref="S4.p1.3.m3.1.1.1.2.cmml">}</mo></mrow><mi id="S4.p1.3.m3.1.1.3" xref="S4.p1.3.m3.1.1.3.cmml">W</mi></msub></math>, <math><mrow id="S4.E1.m1.9.10" xref="S4.E1.m1.9.10.cmml"><mi id="S4.E1.m1.9.10.2" xref="S4.E1.m1.9.10.2.cmml">t</mi><mo id="S4.E1.m1.9.10.1" xref="S4.E1.m1.9.10.1.cmml">=</mo><mrow id="S4.E1.m1.9.9" xref="S4.E1.m1.9.10.3.1.cmml"><mo id="S4.E1.m1.9.9.10" xref="S4.E1.m1.9.10.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S4.E1.m1.9.9.9" xref="S4.E1.m1.9.10.3.1.cmml"><mtr id="S4.E1.m1.9.9.9a" xref="S4.E1.m1.9.10.3.1.cmml"><mtd columnalign="left" id="S4.E1.m1.9.9.9b" xref="S4.E1.m1.9.10.3.1.cmml"><mrow id="S4.E1.m1.4.4.4.4.4.4" xref="S4.E1.m1.4.4.4.4.4.4.cmml"><mn id="S4.E1.m1.4.4.4.4.4.4.5" xref="S4.E1.m1.4.4.4.4.4.4.5.cmml">1</mn><mo rspace="12.5pt" id="S4.E1.m1.4.4.4.4.4.4.6" xref="S4.E1.m1.4.4.4.4.4.4.6.cmml">,</mo><mrow id="S4.E1.m1.4.4.4.4.4.4.4.4" xref="S4.E1.m1.4.4.4.4.4.4.4.4h.cmml"><msub id="S4.E1.m1.1.1.1.1.1.1.1.1.m1.1.1" xref="S4.E1.m1.1.1.1.1.1.1.1.1.m1.1.1.cmml"><mi id="S4.E1.m1.1.1.1.1.1.1.1.1.m1.1.1.2" xref="S4.E1.m1.1.1.1.1.1.1.1.1.m1.1.1.2.cmml">T</mi><mi id="S4.E1.m1.1.1.1.1.1.1.1.1.m1.1.1.3" xref="S4.E1.m1.1.1.1.1.1.1.1.1.m1.1.1.3.cmml">i</mi></msub><mtext id="S4.E1.m1.4.4.4.4.4.4.4.4a" xref="S4.E1.m1.4.4.4.4.4.4.4.4h.cmml"> is always visible within </mtext><mi id="S4.E1.m1.2.2.2.2.2.2.2.2.m2.1.1" xref="S4.E1.m1.2.2.2.2.2.2.2.2.m2.1.1.cmml">W</mi><mtext id="S4.E1.m1.4.4.4.4.4.4.4.4b" xref="S4.E1.m1.4.4.4.4.4.4.4.4h.cmml"> (e.g., </mtext><msub id="S4.E1.m1.3.3.3.3.3.3.3.3.m3.1.1" xref="S4.E1.m1.3.3.3.3.3.3.3.3.m3.1.1.cmml"><mi id="S4.E1.m1.3.3.3.3.3.3.3.3.m3.1.1.2" xref="S4.E1.m1.3.3.3.3.3.3.3.3.m3.1.1.2.cmml">T</mi><mn id="S4.E1.m1.3.3.3.3.3.3.3.3.m3.1.1.3" xref="S4.E1.m1.3.3.3.3.3.3.3.3.m3.1.1.3.cmml">5</mn></msub><mtext id="S4.E1.m1.4.4.4.4.4.4.4.4c" xref="S4.E1.m1.4.4.4.4.4.4.4.4h.cmml"> and </mtext><msub id="S4.E1.m1.4.4.4.4.4.4.4.4.m4.1.1" xref="S4.E1.m1.4.4.4.4.4.4.4.4.m4.1.1.cmml"><mi id="S4.E1.m1.4.4.4.4.4.4.4.4.m4.1.1.2" xref="S4.E1.m1.4.4.4.4.4.4.4.4.m4.1.1.2.cmml">T</mi><mn id="S4.E1.m1.4.4.4.4.4.4.4.4.m4.1.1.3" xref="S4.E1.m1.4.4.4.4.4.4.4.4.m4.1.1.3.cmml">13</mn></msub><mtext id="S4.E1.m1.4.4.4.4.4.4.4.4d" xref="S4.E1.m1.4.4.4.4.4.4.4.4h.cmml"> in Fig.</mtext><mtext id="S4.E1.m1.4.4.4.4.4.4.4.4e" xref="S4.E1.m1.4.4.4.4.4.4.4.4h.cmml"><a href="#S4.F2" title="Figure 2 ‣ 4 Motion Estimation in Video Clips ‣ Robust Video Background Identification by Dominant Rigid Motion EstimationThis work was supported by the Singapore PSF grant 1521200082, and partly done when Kaimo, Nianjuan and Jiangbo were with Advanced Digital Sciences Center (ADSC), Singapore." class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a></mtext></mrow><mo stretchy="false" id="S4.E1.m1.4.4.4.4.4.4.7" xref="S4.E1.m1.4.4.4.4.4.4.7.cmml">)</mo></mrow></mtd><mtd id="S4.E1.m1.9.9.9c" xref="S4.E1.m1.9.10.3.1.1.cmml"/></mtr><mtr id="S4.E1.m1.9.9.9d" xref="S4.E1.m1.9.10.3.1.cmml"><mtd columnalign="left" id="S4.E1.m1.9.9.9e" xref="S4.E1.m1.9.10.3.1.cmml"><mrow id="S4.E1.m1.7.7.7.7.3.3.6" xref="S4.E1.m1.7.7.7.7.3.3.5.cmml"><mn id="S4.E1.m1.7.7.7.7.3.3.4" xref="S4.E1.m1.7.7.7.7.3.3.4.cmml">0</mn><mo rspace="12.5pt" id="S4.E1.m1.7.7.7.7.3.3.6.1" xref="S4.E1.m1.7.7.7.7.3.3.5.cmml">,</mo><mrow id="S4.E1.m1.7.7.7.7.3.3.3.3" xref="S4.E1.m1.7.7.7.7.3.3.3.3h.cmml"><msub id="S4.E1.m1.5.5.5.5.1.1.1.1.m1.1.1" xref="S4.E1.m1.5.5.5.5.1.1.1.1.m1.1.1.cmml"><mi id="S4.E1.m1.5.5.5.5.1.1.1.1.m1.1.1.2" xref="S4.E1.m1.5.5.5.5.1.1.1.1.m1.1.1.2.cmml">T</mi><mi id="S4.E1.m1.5.5.5.5.1.1.1.1.m1.1.1.3" xref="S4.E1.m1.5.5.5.5.1.1.1.1.m1.1.1.3.cmml">i</mi></msub><mtext id="S4.E1.m1.7.7.7.7.3.3.3.3a" xref="S4.E1.m1.7.7.7.7.3.3.3.3h.cmml"> is partially visible within </mtext><mi id="S4.E1.m1.6.6.6.6.2.2.2.2.m2.1.1" xref="S4.E1.m1.6.6.6.6.2.2.2.2.m2.1.1.cmml">W</mi><mtext id="S4.E1.m1.7.7.7.7.3.3.3.3b" xref="S4.E1.m1.7.7.7.7.3.3.3.3h.cmml"> (e.g., </mtext><msub id="S4.E1.m1.7.7.7.7.3.3.3.3.m3.1.1" xref="S4.E1.m1.7.7.7.7.3.3.3.3.m3.1.1.cmml"><mi id="S4.E1.m1.7.7.7.7.3.3.3.3.m3.1.1.2" xref="S4.E1.m1.7.7.7.7.3.3.3.3.m3.1.1.2.cmml">T</mi><mn id="S4.E1.m1.7.7.7.7.3.3.3.3.m3.1.1.3" xref="S4.E1.m1.7.7.7.7.3.3.3.3.m3.1.1.3.cmml">25</mn></msub><mtext id="S4.E1.m1.7.7.7.7.3.3.3.3c" xref="S4.E1.m1.7.7.7.7.3.3.3.3h.cmml"> in Fig.</mtext><mtext id="S4.E1.m1.7.7.7.7.3.3.3.3d" xref="S4.E1.m1.7.7.7.7.3.3.3.3h.cmml"><a href="#S4.F2" title="Figure 2 ‣ 4 Motion Estimation in Video Clips ‣ Robust Video Background Identification by Dominant Rigid Motion EstimationThis work was supported by the Singapore PSF grant 1521200082, and partly done when Kaimo, Nianjuan and Jiangbo were with Advanced Digital Sciences Center (ADSC), Singapore." class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a></mtext><mtext id="S4.E1.m1.7.7.7.7.3.3.3.3g" xref="S4.E1.m1.7.7.7.7.3.3.3.3h.cmml">)</mtext></mrow></mrow></mtd><mtd id="S4.E1.m1.9.9.9f" xref="S4.E1.m1.9.10.3.1.1.cmml"/></mtr><mtr id="S4.E1.m1.9.9.9g" xref="S4.E1.m1.9.10.3.1.cmml"><mtd columnalign="left" id="S4.E1.m1.9.9.9h" xref="S4.E1.m1.9.10.3.1.cmml"><mrow id="S4.E1.m1.9.9.9.9.2.2.3" xref="S4.E1.m1.9.9.9.9.2.2.4.cmml"><mrow id="S4.E1.m1.9.9.9.9.2.2.3.1" xref="S4.E1.m1.9.9.9.9.2.2.3.1.cmml"><mo id="S4.E1.m1.9.9.9.9.2.2.3.1.1" xref="S4.E1.m1.9.9.9.9.2.2.3.1.1.cmml">-</mo><mn id="S4.E1.m1.9.9.9.9.2.2.3.1.2" xref="S4.E1.m1.9.9.9.9.2.2.3.1.2.cmml">1</mn></mrow><mo rspace="12.5pt" id="S4.E1.m1.9.9.9.9.2.2.3.2" xref="S4.E1.m1.9.9.9.9.2.2.4.cmml">,</mo><mrow id="S4.E1.m1.9.9.9.9.2.2.2.2" xref="S4.E1.m1.9.9.9.9.2.2.2.2b.cmml"><msub id="S4.E1.m1.8.8.8.8.1.1.1.1.m1.1.1" xref="S4.E1.m1.8.8.8.8.1.1.1.1.m1.1.1.cmml"><mi id="S4.E1.m1.8.8.8.8.1.1.1.1.m1.1.1.2" xref="S4.E1.m1.8.8.8.8.1.1.1.1.m1.1.1.2.cmml">T</mi><mi id="S4.E1.m1.8.8.8.8.1.1.1.1.m1.1.1.3" xref="S4.E1.m1.8.8.8.8.1.1.1.1.m1.1.1.3.cmml">i</mi></msub><mtext id="S4.E1.m1.9.9.9.9.2.2.2.2a" xref="S4.E1.m1.9.9.9.9.2.2.2.2b.cmml"> is invisible within </mtext><mi id="S4.E1.m1.9.9.9.9.2.2.2.2.m2.1.1" xref="S4.E1.m1.9.9.9.9.2.2.2.2.m2.1.1.cmml">W</mi></mrow></mrow></mtd><mtd id="S4.E1.m1.9.9.9i" xref="S4.E1.m1.9.10.3.1.1.cmml"/></mtr></mtable></mrow></mrow></math>, <math><msubsup id="S4.SS1.p1.4.m4.1.1" xref="S4.SS1.p1.4.m4.1.1.cmml"><mrow id="S4.SS1.p1.4.m4.1.1.1.1.1" xref="S4.SS1.p1.4.m4.1.1.1.1.2.cmml"><mo id="S4.SS1.p1.4.m4.1.1.1.1.1.2" xref="S4.SS1.p1.4.m4.1.1.1.1.2.cmml">{</mo><msubsup id="S4.SS1.p1.4.m4.1.1.1.1.1.1" xref="S4.SS1.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S4.SS1.p1.4.m4.1.1.1.1.1.1.2.2" xref="S4.SS1.p1.4.m4.1.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS1.p1.4.m4.1.1.1.1.1.1.2.3" xref="S4.SS1.p1.4.m4.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.SS1.p1.4.m4.1.1.1.1.1.1.3" xref="S4.SS1.p1.4.m4.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S4.SS1.p1.4.m4.1.1.1.1.1.3" xref="S4.SS1.p1.4.m4.1.1.1.1.2.cmml">}</mo></mrow><msub id="S4.SS1.p1.4.m4.1.1.1.3" xref="S4.SS1.p1.4.m4.1.1.1.3.cmml"><mi id="S4.SS1.p1.4.m4.1.1.1.3.2" xref="S4.SS1.p1.4.m4.1.1.1.3.2.cmml">C</mi><mn id="S4.SS1.p1.4.m4.1.1.1.3.3" xref="S4.SS1.p1.4.m4.1.1.1.3.3.cmml">0</mn></msub><mrow id="S4.SS1.p1.4.m4.1.1.3" xref="S4.SS1.p1.4.m4.1.1.3.cmml"><mi id="S4.SS1.p1.4.m4.1.1.3.2" xref="S4.SS1.p1.4.m4.1.1.3.2.cmml">t</mi><mo id="S4.SS1.p1.4.m4.1.1.3.1" xref="S4.SS1.p1.4.m4.1.1.3.1.cmml">=</mo><mn id="S4.SS1.p1.4.m4.1.1.3.3" xref="S4.SS1.p1.4.m4.1.1.3.3.cmml">1</mn></mrow></msubsup></math>, <math><mrow id="S4.E2.m1.3.4" xref="S4.E2.m1.3.4.cmml"><mfrac id="S4.E2.m1.3.3" xref="S4.E2.m1.3.3.cmml"><mrow id="S4.E2.m1.1.1.1.1" xref="S4.E2.m1.1.1.1.2.cmml"><mo stretchy="false" id="S4.E2.m1.1.1.1.1.2" xref="S4.E2.m1.1.1.1.2.1.cmml">|</mo><msubsup id="S4.E2.m1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.cmml"><mrow id="S4.E2.m1.1.1.1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.1.1.2.cmml"><mo stretchy="false" id="S4.E2.m1.1.1.1.1.1.1.1.1.2" xref="S4.E2.m1.1.1.1.1.1.1.1.2.cmml">{</mo><msubsup id="S4.E2.m1.1.1.1.1.1.1.1.1.1" xref="S4.E2.m1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.1.1.1.1.1.1.1.1.1.2.2" xref="S4.E2.m1.1.1.1.1.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.E2.m1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.E2.m1.1.1.1.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.E2.m1.1.1.1.1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo stretchy="false" id="S4.E2.m1.1.1.1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.1.1.2.cmml">}</mo></mrow><msub id="S4.E2.m1.1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.1.3.cmml"><mi id="S4.E2.m1.1.1.1.1.1.1.3.2" xref="S4.E2.m1.1.1.1.1.1.1.3.2.cmml">C</mi><mn id="S4.E2.m1.1.1.1.1.1.1.3.3" xref="S4.E2.m1.1.1.1.1.1.1.3.3.cmml">0</mn></msub><mrow id="S4.E2.m1.1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.1.1.3.cmml"><mi id="S4.E2.m1.1.1.1.1.1.3.2" xref="S4.E2.m1.1.1.1.1.1.3.2.cmml">t</mi><mo id="S4.E2.m1.1.1.1.1.1.3.1" xref="S4.E2.m1.1.1.1.1.1.3.1.cmml">=</mo><mn id="S4.E2.m1.1.1.1.1.1.3.3" xref="S4.E2.m1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msubsup><mo stretchy="false" id="S4.E2.m1.1.1.1.1.3" xref="S4.E2.m1.1.1.1.2.1.cmml">|</mo></mrow><mrow id="S4.E2.m1.3.3.3" xref="S4.E2.m1.3.3.3.cmml"><mrow id="S4.E2.m1.2.2.2.1.1" xref="S4.E2.m1.2.2.2.1.2.cmml"><mo stretchy="false" id="S4.E2.m1.2.2.2.1.1.2" xref="S4.E2.m1.2.2.2.1.2.1.cmml">|</mo><msubsup id="S4.E2.m1.2.2.2.1.1.1" xref="S4.E2.m1.2.2.2.1.1.1.cmml"><mrow id="S4.E2.m1.2.2.2.1.1.1.1.1.1" xref="S4.E2.m1.2.2.2.1.1.1.1.1.2.cmml"><mo stretchy="false" id="S4.E2.m1.2.2.2.1.1.1.1.1.1.2" xref="S4.E2.m1.2.2.2.1.1.1.1.1.2.cmml">{</mo><msubsup id="S4.E2.m1.2.2.2.1.1.1.1.1.1.1" xref="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.2.2" xref="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.2.3" xref="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.3" xref="S4.E2.m1.2.2.2.1.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo stretchy="false" id="S4.E2.m1.2.2.2.1.1.1.1.1.1.3" xref="S4.E2.m1.2.2.2.1.1.1.1.1.2.cmml">}</mo></mrow><msub id="S4.E2.m1.2.2.2.1.1.1.1.3" xref="S4.E2.m1.2.2.2.1.1.1.1.3.cmml"><mi id="S4.E2.m1.2.2.2.1.1.1.1.3.2" xref="S4.E2.m1.2.2.2.1.1.1.1.3.2.cmml">C</mi><mn id="S4.E2.m1.2.2.2.1.1.1.1.3.3" xref="S4.E2.m1.2.2.2.1.1.1.1.3.3.cmml">0</mn></msub><mrow id="S4.E2.m1.2.2.2.1.1.1.3" xref="S4.E2.m1.2.2.2.1.1.1.3.cmml"><mi id="S4.E2.m1.2.2.2.1.1.1.3.2" xref="S4.E2.m1.2.2.2.1.1.1.3.2.cmml">t</mi><mo id="S4.E2.m1.2.2.2.1.1.1.3.1" xref="S4.E2.m1.2.2.2.1.1.1.3.1.cmml">=</mo><mn id="S4.E2.m1.2.2.2.1.1.1.3.3" xref="S4.E2.m1.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msubsup><mo stretchy="false" id="S4.E2.m1.2.2.2.1.1.3" xref="S4.E2.m1.2.2.2.1.2.1.cmml">|</mo></mrow><mo id="S4.E2.m1.3.3.3.3" xref="S4.E2.m1.3.3.3.3.cmml">+</mo><mrow id="S4.E2.m1.3.3.3.2.1" xref="S4.E2.m1.3.3.3.2.2.cmml"><mo stretchy="false" id="S4.E2.m1.3.3.3.2.1.2" xref="S4.E2.m1.3.3.3.2.2.1.cmml">|</mo><msubsup id="S4.E2.m1.3.3.3.2.1.1" xref="S4.E2.m1.3.3.3.2.1.1.cmml"><mrow id="S4.E2.m1.3.3.3.2.1.1.1.1.1" xref="S4.E2.m1.3.3.3.2.1.1.1.1.2.cmml"><mo stretchy="false" id="S4.E2.m1.3.3.3.2.1.1.1.1.1.2" xref="S4.E2.m1.3.3.3.2.1.1.1.1.2.cmml">{</mo><msubsup id="S4.E2.m1.3.3.3.2.1.1.1.1.1.1" xref="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.cmml"><mi id="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.2.2" xref="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.2.2.cmml">T</mi><mi id="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.2.3" xref="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.2.3.cmml">i</mi><mi id="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.3" xref="S4.E2.m1.3.3.3.2.1.1.1.1.1.1.3.cmml">t</mi></msubsup><mo stretchy="false" id="S4.E2.m1.3.3.3.2.1.1.1.1.1.3" xref="S4.E2.m1.3.3.3.2.1.1.1.1.2.cmml">}</mo></mrow><msub id="S4.E2.m1.3.3.3.2.1.1.1.3" xref="S4.E2.m1.3.3.3.2.1.1.1.3.cmml"><mi id="S4.E2.m1.3.3.3.2.1.1.1.3.2" xref="S4.E2.m1.3.3.3.2.1.1.1.3.2.cmml">C</mi><mn id="S4.E2.m1.3.3.3.2.1.1.1.3.3" xref="S4.E2.m1.3.3.3.2.1.1.1.3.3.cmml">0</mn></msub><mrow id="S4.E2.m1.3.3.3.2.1.1.3" xref="S4.E2.m1.3.3.3.2.1.1.3.cmml"><mi id="S4.E2.m1.3.3.3.2.1.1.3.2" xref="S4.E2.m1.3.3.3.2.1.1.3.2.cmml">t</mi><mo id="S4.E2.m1.3.3.3.2.1.1.3.1" xref="S4.E2.m1.3.3.3.2.1.1.3.1.cmml">=</mo><mn id="S4.E2.m1.3.3.3.2.1.1.3.3" xref="S4.E2.m1.3.3.3.2.1.1.3.3.cmml">0</mn></mrow></msubsup><mo stretchy="false" id="S4.E2.m1.3.3.3.2.1.3" xref="S4.E2.m1.3.3.3.2.2.1.cmml">|</mo></mrow></mrow></mfrac><mo id="S4.E2.m1.3.4.1" xref="S4.E2.m1.3.4.1.cmml">≥</mo><mn id="S4.E2.m1.3.4.2" xref="S4.E2.m1.3.4.2.cmml">0.8</mn></mrow></math>, <math><msubsup id="S4.SS2.p1.2.m2.3.3" xref="S4.SS2.p1.2.m2.3.3.cmml"><mrow id="S4.SS2.p1.2.m2.3.3.1.1.1" xref="S4.SS2.p1.2.m2.3.3.1.1.2.cmml"><mo id="S4.SS2.p1.2.m2.3.3.1.1.1.2" xref="S4.SS2.p1.2.m2.3.3.1.1.2.cmml">{</mo><msubsup id="S4.SS2.p1.2.m2.3.3.1.1.1.1" xref="S4.SS2.p1.2.m2.3.3.1.1.1.1.cmml"><mi id="S4.SS2.p1.2.m2.3.3.1.1.1.1.2.2" xref="S4.SS2.p1.2.m2.3.3.1.1.1.1.2.2.cmml">T</mi><mi id="S4.SS2.p1.2.m2.3.3.1.1.1.1.2.3" xref="S4.SS2.p1.2.m2.3.3.1.1.1.1.2.3.cmml">i</mi><mi id="S4.SS2.p1.2.m2.3.3.1.1.1.1.3" xref="S4.SS2.p1.2.m2.3.3.1.1.1.1.3.cmml">t</mi></msubsup><mo id="S4.SS2.p1.2.m2.3.3.1.1.1.3" xref="S4.SS2.p1.2.m2.3.3.1.1.2.cmml">}</mo></mrow><mi id="S4.SS2.p1.2.m2.3.3.1.3" xref="S4.SS2.p1.2.m2.3.3.1.3.cmml">C</mi><mrow id="S4.SS2.p1.2.m2.2.2.2" xref="S4.SS2.p1.2.m2.2.2.2.cmml"><mi id="S4.SS2.p1.2.m2.2.2.2.4" xref="S4.SS2.p1.2.m2.2.2.2.4.cmml">t</mi><mo id="S4.SS2.p1.2.m2.2.2.2.3" xref="S4.SS2.p1.2.m2.2.2.2.3.cmml">=</mo><mrow id="S4.SS2.p1.2.m2.2.2.2.5.2" xref="S4.SS2.p1.2.m2.2.2.2.5.1.cmml"><mn id="S4.SS2.p1.2.m2.1.1.1.1" xref="S4.SS2.p1.2.m2.1.1.1.1.cmml">0</mn><mo id="S4.SS2.p1.2.m2.2.2.2.5.2.1" xref="S4.SS2.p1.2.m2.2.2.2.5.1.cmml">,</mo><mn id="S4.SS2.p1.2.m2.2.2.2.2" xref="S4.SS2.p1.2.m2.2.2.2.2.cmml">1</mn></mrow></mrow></msubsup></math>, <math><msubsup id="S4.SS2.p1.3.m3.4.4" xref="S4.SS2.p1.3.m3.4.4.cmml"><mrow id="S4.SS2.p1.3.m3.4.4.1.1.1" xref="S4.SS2.p1.3.m3.4.4.1.1.2.cmml"><mo id="S4.SS2.p1.3.m3.4.4.1.1.1.2" xref="S4.SS2.p1.3.m3.4.4.1.1.2.cmml">{</mo><msup id="S4.SS2.p1.3.m3.4.4.1.1.1.1" xref="S4.SS2.p1.3.m3.4.4.1.1.1.1.cmml"><mi id="S4.SS2.p1.3.m3.4.4.1.1.1.1.2" xref="S4.SS2.p1.3.m3.4.4.1.1.1.1.2.cmml">F</mi><mrow id="S4.SS2.p1.3.m3.2.2.2.4" xref="S4.SS2.p1.3.m3.2.2.2.3.cmml"><mo stretchy="false" id="S4.SS2.p1.3.m3.2.2.2.4.1" xref="S4.SS2.p1.3.m3.2.2.2.3.cmml">(</mo><mi id="S4.SS2.p1.3.m3.1.1.1.1" xref="S4.SS2.p1.3.m3.1.1.1.1.cmml">j</mi><mo id="S4.SS2.p1.3.m3.2.2.2.4.2" xref="S4.SS2.p1.3.m3.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p1.3.m3.2.2.2.2" xref="S4.SS2.p1.3.m3.2.2.2.2.cmml">k</mi><mo stretchy="false" id="S4.SS2.p1.3.m3.2.2.2.4.3" xref="S4.SS2.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></msup><mo id="S4.SS2.p1.3.m3.4.4.1.1.1.3" xref="S4.SS2.p1.3.m3.4.4.1.1.2.cmml">}</mo></mrow><mi id="S4.SS2.p1.3.m3.4.4.1.3" xref="S4.SS2.p1.3.m3.4.4.1.3.cmml">C</mi><mrow id="S4.SS2.p1.3.m3.3.3.1" xref="S4.SS2.p1.3.m3.3.3.1.cmml"><mn id="S4.SS2.p1.3.m3.3.3.1.3" xref="S4.SS2.p1.3.m3.3.3.1.3.cmml">0</mn><mo id="S4.SS2.p1.3.m3.3.3.1.4" xref="S4.SS2.p1.3.m3.3.3.1.4.cmml"><</mo><mrow id="S4.SS2.p1.3.m3.3.3.1.1.1" xref="S4.SS2.p1.3.m3.3.3.1.1.2.cmml"><mo stretchy="false" id="S4.SS2.p1.3.m3.3.3.1.1.1.2" xref="S4.SS2.p1.3.m3.3.3.1.1.2.1.cmml">|</mo><mrow id="S4.SS2.p1.3.m3.3.3.1.1.1.1" xref="S4.SS2.p1.3.m3.3.3.1.1.1.1.cmml"><mi id="S4.SS2.p1.3.m3.3.3.1.1.1.1.2" xref="S4.SS2.p1.3.m3.3.3.1.1.1.1.2.cmml">j</mi><mo id="S4.SS2.p1.3.m3.3.3.1.1.1.1.1" xref="S4.SS2.p1.3.m3.3.3.1.1.1.1.1.cmml">-</mo><mi id="S4.SS2.p1.3.m3.3.3.1.1.1.1.3" xref="S4.SS2.p1.3.m3.3.3.1.1.1.1.3.cmml">k</mi></mrow><mo stretchy="false" id="S4.SS2.p1.3.m3.3.3.1.1.1.3" xref="S4.SS2.p1.3.m3.3.3.1.1.2.1.cmml">|</mo></mrow><mo id="S4.SS2.p1.3.m3.3.3.1.5" xref="S4.SS2.p1.3.m3.3.3.1.5.cmml">≤</mo><mi id="S4.SS2.p1.3.m3.3.3.1.6" xref="S4.SS2.p1.3.m3.3.3.1.6.cmml">r</mi></mrow></msubsup></math>, <math><msup id="S4.SS2.p1.8.m8.2.3" xref="S4.SS2.p1.8.m8.2.3.cmml"><mi id="S4.SS2.p1.8.m8.2.3.2" xref="S4.SS2.p1.8.m8.2.3.2.cmml">F</mi><mrow id="S4.SS2.p1.8.m8.2.2.2.4" xref="S4.SS2.p1.8.m8.2.2.2.3.cmml"><mo stretchy="false" id="S4.SS2.p1.8.m8.2.2.2.4.1" xref="S4.SS2.p1.8.m8.2.2.2.3.cmml">(</mo><mi id="S4.SS2.p1.8.m8.1.1.1.1" xref="S4.SS2.p1.8.m8.1.1.1.1.cmml">j</mi><mo id="S4.SS2.p1.8.m8.2.2.2.4.2" xref="S4.SS2.p1.8.m8.2.2.2.3.cmml">,</mo><mi id="S4.SS2.p1.8.m8.2.2.2.2" xref="S4.SS2.p1.8.m8.2.2.2.2.cmml">k</mi><mo stretchy="false" id="S4.SS2.p1.8.m8.2.2.2.4.3" xref="S4.SS2.p1.8.m8.2.2.2.3.cmml">)</mo></mrow></msup></math>

Correct Categorie: cs

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1510.03744

Formulas:

1-

f_{ℓ}^{1 μ m} = ℓ f_{R} + f_{0}^{1 μ m},

2-

f_{m}^{2 μ m} = m f_{R} + f_{0}^{2 μ m},

3-

f_{n}^{4 μ m} = n f_{R} + f_{0}^{4 μ m},

4-

f_{0}^{1 μ m}

5-

f_{0}^{2 μ m}

6-

f_{0}^{4 μ m}

7-

f_{0}^{2 μ m} = \frac{f_{0}^{1 μ m}}{2} or f_{0}^{2 μ m} = \frac{f_{0}^{1 μ m}}{2} + \frac{f_{R}}{2},

8-

f_{0}^{4 μ m} = \frac{f_{0}^{2 μ m}}{2} or f_{0}^{4 μ m} = \frac{f_{0}^{2 μ m}}{2} + \frac{f_{R}}{2} .

9-

f_{0}^{2 μ m}

10-

f_{0}^{700 n m} = f_{0}^{1 μ m} + f_{0}^{2 μ m} .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/2001.09176

Formulas:

1-

V (ℋ) = {x_{1}, \dots, x_{n}}

2-

S \in ℰ (ℋ)

3-

W \subseteq V (ℋ)

4-

S \in ℰ (ℋ)

5-

ℰ (ℋ) ∖ {S}

6-

R : R = K [x_{1}, \dots, x_{n}]

7-

\deg x_{i} = 1

8-

\prod_{x_{i} \in S} x_{i}

9-

{x_{i_{1}}, \dots, x_{i_{k}}}

10-

{x_{1}, \dots, x_{n}}

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0409707

Formulas:

1-

0.4 < R / R_{vir} < 0.7

2-

R < 0.4 R_{vir}

3-

0.4 \sim < R / R_{vir} \sim < 0.7

4-

T \sim 10^{6} K

5-

Ω_{m} = 0.3, Ω_{Λ} = 0.7

6-

H_{0} = 100 h

7-

192 h^{- 1} Mpc

8-

m_{DM} = 4.6 \times 10^{9} h^{- 1} M_{⊙}

9-

m_{gas} = 6.9 \times 10^{8} h^{- 1} M_{⊙}

10-

0.4 < R / R_{vir} < 0.7

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/0101241

Formulas:

1-

\tilde{E} (𝐩) \to 0

2-

{(λ Φ^{4})}_{4}

3-

f (𝐩^{2}) \equiv ⟨ N_{𝐩} ⟩

4-

f (𝐩^{2})

5-

f_{δ} (𝐩^{2})

6-

f_{δ} (𝐩^{2})

7-

{\tilde{E}}_{Λ} (𝐩)

8-

f_{δ} (𝐤^{2})

9-

\sqrt{𝐩^{2} + M_{h}^{2}}

10-

\sqrt{𝐩^{2} + M_{h}^{2}}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0910.2375

Formulas:

1-

(x_{0}, y_{0})

2-

(x_{0}, y_{0})

3-

I (x, y) = a + b x + c y + d x^{2} + e x y + f y^{2}

4-

(x_{s}, y_{s})

5-

(x_{s}, y_{s}) = (\frac{c e - b f}{2 d f - 2 e^{2}}, \frac{b e - c d}{2 d f - 2 e^{2}})

6-

\sqrt{{(x_{s} - x_{0})}^{2} + {(y_{s} - y_{0})}^{2}}

7-

S = A \times 2 π σ^{2}

8-

A e^{\frac{- {(x - x_{s})}^{2} - {(y - y_{s})}^{2}}{2 σ^{2}}} + μ_{s k y}

9-

μ_{s k y}

10-

(x_{0}, y_{0})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1006.0598

Formulas:

1-

Δ (t) = ⟨ {[r (t) - ⟨ r ⟩ (t)]}^{2} ⟩ = 2 D t

2-

Δ (t) = 2 F t^{1 / 2}

3-

r_{j} (t)

4-

u_{j} (t) = r_{j} (t) - j L

5-

r_{j} = j L

6-

Γ_{u, u + 1} = Γ_{u, u - 1} = Γ = 1 / 2

7-

r_{j - 1} (t) < r_{j} (t) < r_{j + 1} (t)

8-

u_{j - 1} (t) - L < u_{j} (t) < u_{j + 1} (t) + L .

9-

Δ = ⟨ {[r_{j} (t) - ⟨ r_{j} (t) ⟩]}^{2} ⟩ \sim t

10-

Δ \sim t^{1 / 2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0104377

Formulas:

1-

f (x, t) + g (x) η (t)

2-

- \frac{1}{τ} \frac{d}{d η} V_{q} (η) + \frac{1}{τ} ξ (t)

3-

< ξ (t) ξ (t^{'}) > = 2 D δ (t - t^{'})

4-

V_{q} (η)

5-

V_{q} (η) = \frac{1}{β (q - 1)} \ln [1 + β (q - 1) \frac{η^{2}}{2}]

6-

f (x, t)

7-

U (x, t)

8-

S (t) \sim F \cos (ω t)

9-

f (x, t) = - \frac{\partial U}{\partial x} = - U_{0}^{'} + S (t)

10-

U_{0} (x)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1309.5325

Formulas:

1-

F (ρ_{1}, ρ_{2}) = {(Tr \sqrt{\sqrt{ρ_{1}} ρ_{2} \sqrt{ρ_{1}}})}^{2}

2-

D_{B} (ρ_{1}, ρ_{2}) = \sqrt{2 [1 - F (ρ_{1}, ρ_{2})]}

3-

1 - \sqrt{F (ρ_{1}, ρ_{2})} \leq \frac{1}{2} {|| ρ_{1} - ρ_{2} ||}_{1} \leq \sqrt{1 - F (ρ_{1}, ρ_{2})}

4-

ρ = \frac{1}{4} (𝕀 \otimes 𝕀 + \sum_{j = 1}^{3} c_{j} σ_{j} \otimes σ_{j})

5-

ρ^{A} = ρ^{B} = 𝕀 / 2

6-

λ_{0} = \frac{1}{4} (1 - c_{1} - c_{2} - c_{3})

7-

λ_{1} = \frac{1}{4} (1 - c_{1} + c_{2} + c_{3})

8-

λ_{2} = \frac{1}{4} (1 + c_{1} - c_{2} + c_{3})

9-

λ_{3} = \frac{1}{4} (1 + c_{1} + c_{2} - c_{3})

10-

F (ρ_{1}, ρ_{2}) = \frac{1}{4} \sum_{i = 0}^{3} \sqrt{λ_{i, 1} λ_{i, 2}}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1605.06825

Formulas:

1-

{1, 2, \dots, s t}

2-

σ_{1}, \dots, σ_{k}

3-

σ_{1}, \dots, σ_{k}

4-

π (i) ⇑ π (j)

5-

([n], ⇑)

6-

σ_{1}, \dots, σ_{k}

7-

N E_{4, 3}

8-

E N_{4, 3}

9-

{1, 2, \dots, n}

10-

P = (X, ⇑)

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1104.0236

Formulas:

1-

f_{0} \pm δ f

2-

| 2 δ f | > 20 MHz

3-

w_{0} = 55 μ m

4-

f_{0} \pm δ f

5-

R + R \cos (Ω t + ϕ)

6-

Ω = 4 π δ f

7-

2 π \times 60 MHz

8-

R \cos^{2} (Ω t)

9-

\frac{1}{2} N \cos ϕ

10-

\frac{1}{2} N \sin ϕ

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/math/0204047

Formulas:

1-

E_{T} = f^{*} E

2-

φ : R^{q} \to R^{p}

3-

R^{q} \overset{φ}{\to} R^{p} \to E \to 0

4-

E / 𝔪^{n} E

5-

R^{m} \oplus {(R / I)}^{n}

6-

R^{q} \overset{φ}{\to} R^{p} \to E \to 0

7-

p = {dim}_{R / 𝔪} (E / 𝔪 E)

8-

φ : R^{q} \to R^{p}

9-

φ : R^{q} \to R^{p}

10-

(\begin{matrix} 1_{m \times m} & 0_{m \times (q - m)} \\ 0_{(p - m) \times m} & 0_{(p - m) \times (q - m)} \end{matrix})

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/2007.12660

Formulas:

1-

{\dot{𝐫}}_{i} = 𝝍_{i} + 𝐅_{i}

2-

τ {\dot{𝝍}}_{i} = - 𝝍_{i} + 𝜼_{i}

3-

𝐅_{i} = \sum_{j \neq i} 𝐟_{i j}

4-

𝐟_{i j} = - \nabla_{𝐫_{i}} ϕ (r_{i j})

5-

r_{i j} = | 𝐫_{i} - 𝐫_{j} |

6-

ϕ (r) = {(r / σ)}^{- 12} / 12

7-

⟨ η_{i}^{α} (t) ⟩ = 0

8-

⟨ η_{i}^{α} (t) η_{j}^{β} (s) ⟩ = 2 D δ_{i j} δ^{α β} δ (t - s)

9-

D = v^{2} τ

10-

Δ t = 10^{- 3}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1608.02323

Formulas:

1-

m (r) = {\begin{matrix} m_{0} + b {(r - r_{c})}^{β_{m}} & for r \geq r_{c}, \\ 0 & for r < r_{c}, \end{matrix}

2-

{γ_{m}, {\bar{ν}}_{m}}

3-

p_{s} \sim s^{- τ_{a}}

4-

τ_{a} = 3 / 2

5-

{τ_{a}, σ_{a}, γ_{a}, {\bar{ν}}_{a}}

6-

1 - β_{m} = γ_{a}

7-

P_{\infty} (r)

8-

P_{\infty} (r)

9-

{β_{p}, γ_{p}, {\bar{ν}}_{p}}

10-

p_{s} (r) \sim s^{- τ_{a}} \exp (- s / s^{*})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1211.6756

Formulas:

1-

β \equiv v / c \sim 10^{- 3}

2-

𝒪 (10^{- 5})

3-

{\hat{𝒏}}^{'} = (\frac{\cos θ + β}{1 + β \cos θ}) \hat{𝒗} + \frac{\hat{𝒏} - \hat{𝒗} \cos θ}{γ (1 + β \cos θ)},

4-

\cos θ \equiv \hat{𝒏} \cdot \hat{𝒗}

5-

ν^{'} = γ ν (1 + β \cos θ),

6-

γ = {(1 - β^{2})}^{- 1 / 2}

7-

\cos θ^{'} = \frac{\cos θ + β}{1 + β \cos θ} .

8-

Y_{ℓ m} (θ^{'}, ϕ) \propto P_{ℓ}^{m} (\cos θ^{'})

9-

P_{ℓ} (\cos θ)

10-

C_{ℓ}^{'} \approx C_{ℓ} (1 + 4 β^{2} + 𝒪 (β^{3})) .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1504.08266

Formulas:

1-

{\tilde{φ}}_{trunc} = \frac{v_{ADC}}{v_{s}} \frac{2^{- N}}{\sqrt{3 f_{s}}} .

2-

v_{s} = v_{ADC} / 2

3-

[{\tilde{φ}}_{\tilde{τ}}] = rad / \sqrt{Hz}

4-

[\tilde{τ_{}}] = s / \sqrt{Hz}

5-

{\tilde{φ}}_{\tilde{τ}} = 2 π f_{s} \cdot \tilde{τ_{}} .

6-

φ_{i, c} = φ_{i, s} - φ_{i, p} \frac{f_{i, s}}{f_{p}} .

7-

\arg (T (f_{s}))

8-

φ_{Δ} = φ_{s} - φ_{v}

9-

φ_{err} = \frac{v_{v}}{v_{s}} \sin (φ_{s} - φ_{v}) .

10-

τ_{c} \approx l_{c} / (2 / 3 c)

Formulas (html):

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xref="S2.E1.m1.1.1.1.1.3.2.3.cmml"><mi id="S2.E1.m1.1.1.1.1.3.2.3.2" xref="S2.E1.m1.1.1.1.1.3.2.3.2.cmml">v</mi><mtext id="S2.E1.m1.1.1.1.1.3.2.3.3" xref="S2.E1.m1.1.1.1.1.3.2.3.3a.cmml">s</mtext></msub></mfrac><mo id="S2.E1.m1.1.1.1.1.3.1" xref="S2.E1.m1.1.1.1.1.3.1.cmml">⁢</mo><mfrac id="S2.E1.m1.1.1.1.1.3.3" xref="S2.E1.m1.1.1.1.1.3.3.cmml"><msup id="S2.E1.m1.1.1.1.1.3.3.2" xref="S2.E1.m1.1.1.1.1.3.3.2.cmml"><mn id="S2.E1.m1.1.1.1.1.3.3.2.2" xref="S2.E1.m1.1.1.1.1.3.3.2.2.cmml">2</mn><mrow id="S2.E1.m1.1.1.1.1.3.3.2.3" xref="S2.E1.m1.1.1.1.1.3.3.2.3.cmml"><mo id="S2.E1.m1.1.1.1.1.3.3.2.3.1" xref="S2.E1.m1.1.1.1.1.3.3.2.3.1.cmml">-</mo><mi id="S2.E1.m1.1.1.1.1.3.3.2.3.2" xref="S2.E1.m1.1.1.1.1.3.3.2.3.2.cmml">N</mi></mrow></msup><msqrt id="S2.E1.m1.1.1.1.1.3.3.3" xref="S2.E1.m1.1.1.1.1.3.3.3.cmml"><mrow id="S2.E1.m1.1.1.1.1.3.3.3.2" xref="S2.E1.m1.1.1.1.1.3.3.3.2.cmml"><mn id="S2.E1.m1.1.1.1.1.3.3.3.2.2" xref="S2.E1.m1.1.1.1.1.3.3.3.2.2.cmml">3</mn><mo id="S2.E1.m1.1.1.1.1.3.3.3.2.1" xref="S2.E1.m1.1.1.1.1.3.3.3.2.1.cmml">⁢</mo><msub id="S2.E1.m1.1.1.1.1.3.3.3.2.3" xref="S2.E1.m1.1.1.1.1.3.3.3.2.3.cmml"><mi id="S2.E1.m1.1.1.1.1.3.3.3.2.3.2" xref="S2.E1.m1.1.1.1.1.3.3.3.2.3.2.cmml">f</mi><mi id="S2.E1.m1.1.1.1.1.3.3.3.2.3.3" xref="S2.E1.m1.1.1.1.1.3.3.3.2.3.3.cmml">s</mi></msub></mrow></msqrt></mfrac></mrow></mrow><mo id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.SS1.p1.4.m1.1.1" xref="S2.SS1.p1.4.m1.1.1.cmml"><msub id="S2.SS1.p1.4.m1.1.1.2" xref="S2.SS1.p1.4.m1.1.1.2.cmml"><mi id="S2.SS1.p1.4.m1.1.1.2.2" xref="S2.SS1.p1.4.m1.1.1.2.2.cmml">v</mi><mtext id="S2.SS1.p1.4.m1.1.1.2.3" xref="S2.SS1.p1.4.m1.1.1.2.3a.cmml">s</mtext></msub><mo id="S2.SS1.p1.4.m1.1.1.1" xref="S2.SS1.p1.4.m1.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.4.m1.1.1.3" xref="S2.SS1.p1.4.m1.1.1.3.cmml"><msub id="S2.SS1.p1.4.m1.1.1.3.2" xref="S2.SS1.p1.4.m1.1.1.3.2.cmml"><mi id="S2.SS1.p1.4.m1.1.1.3.2.2" xref="S2.SS1.p1.4.m1.1.1.3.2.2.cmml">v</mi><mtext id="S2.SS1.p1.4.m1.1.1.3.2.3" xref="S2.SS1.p1.4.m1.1.1.3.2.3a.cmml">ADC</mtext></msub><mo id="S2.SS1.p1.4.m1.1.1.3.1" xref="S2.SS1.p1.4.m1.1.1.3.1.cmml">/</mo><mn id="S2.SS1.p1.4.m1.1.1.3.3" xref="S2.SS1.p1.4.m1.1.1.3.3.cmml">2</mn></mrow></mrow></math>, <math><mrow id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml"><mrow id="S2.SS3.p1.2.m2.1.1.1.1" xref="S2.SS3.p1.2.m2.1.1.1.2.cmml"><mo stretchy="false" id="S2.SS3.p1.2.m2.1.1.1.1.2" xref="S2.SS3.p1.2.m2.1.1.1.2.1.cmml">[</mo><msub id="S2.SS3.p1.2.m2.1.1.1.1.1" xref="S2.SS3.p1.2.m2.1.1.1.1.1.cmml"><mover accent="true" id="S2.SS3.p1.2.m2.1.1.1.1.1.2" xref="S2.SS3.p1.2.m2.1.1.1.1.1.2.cmml"><mi id="S2.SS3.p1.2.m2.1.1.1.1.1.2.2" xref="S2.SS3.p1.2.m2.1.1.1.1.1.2.2.cmml">φ</mi><mo id="S2.SS3.p1.2.m2.1.1.1.1.1.2.1" xref="S2.SS3.p1.2.m2.1.1.1.1.1.2.1.cmml">~</mo></mover><mover accent="true" id="S2.SS3.p1.2.m2.1.1.1.1.1.3" xref="S2.SS3.p1.2.m2.1.1.1.1.1.3.cmml"><mi id="S2.SS3.p1.2.m2.1.1.1.1.1.3.2" xref="S2.SS3.p1.2.m2.1.1.1.1.1.3.2.cmml">τ</mi><mo id="S2.SS3.p1.2.m2.1.1.1.1.1.3.1" xref="S2.SS3.p1.2.m2.1.1.1.1.1.3.1.cmml">~</mo></mover></msub><mo stretchy="false" id="S2.SS3.p1.2.m2.1.1.1.1.3" xref="S2.SS3.p1.2.m2.1.1.1.2.1.cmml">]</mo></mrow><mo id="S2.SS3.p1.2.m2.1.1.2" xref="S2.SS3.p1.2.m2.1.1.2.cmml">=</mo><mrow id="S2.SS3.p1.2.m2.1.1.3" xref="S2.SS3.p1.2.m2.1.1.3.cmml"><mtext id="S2.SS3.p1.2.m2.1.1.3.2" xref="S2.SS3.p1.2.m2.1.1.3.2a.cmml">rad</mtext><mo id="S2.SS3.p1.2.m2.1.1.3.1" xref="S2.SS3.p1.2.m2.1.1.3.1.cmml">/</mo><msqrt id="S2.SS3.p1.2.m2.1.1.3.3" xref="S2.SS3.p1.2.m2.1.1.3.3.cmml"><mtext id="S2.SS3.p1.2.m2.1.1.3.3.2" xref="S2.SS3.p1.2.m2.1.1.3.3.2a.cmml">Hz</mtext></msqrt></mrow></mrow></math>, <math><mrow id="S2.SS3.p1.4.m4.1.2" xref="S2.SS3.p1.4.m4.1.2.cmml"><mrow id="S2.SS3.p1.4.m4.1.2.2.2" xref="S2.SS3.p1.4.m4.1.2.2.1.cmml"><mo stretchy="false" id="S2.SS3.p1.4.m4.1.2.2.2.1" xref="S2.SS3.p1.4.m4.1.2.2.1.1.cmml">[</mo><mover accent="true" id="S2.SS3.p1.4.m4.1.1" xref="S2.SS3.p1.4.m4.1.1.cmml"><msub id="S2.SS3.p1.4.m4.1.1.2" xref="S2.SS3.p1.4.m4.1.1.2.cmml"><mi id="S2.SS3.p1.4.m4.1.1.2.2" xref="S2.SS3.p1.4.m4.1.1.2.2.cmml">τ</mi><mrow id="S2.SS3.p1.4.m4.1.1.2.3" xref="S2.SS3.p1.4.m4.1.1.2.3a.cmml"/></msub><mo id="S2.SS3.p1.4.m4.1.1.1" xref="S2.SS3.p1.4.m4.1.1.1.cmml">~</mo></mover><mo stretchy="false" id="S2.SS3.p1.4.m4.1.2.2.2.2" xref="S2.SS3.p1.4.m4.1.2.2.1.1.cmml">]</mo></mrow><mo id="S2.SS3.p1.4.m4.1.2.1" xref="S2.SS3.p1.4.m4.1.2.1.cmml">=</mo><mrow id="S2.SS3.p1.4.m4.1.2.3" xref="S2.SS3.p1.4.m4.1.2.3.cmml"><mtext id="S2.SS3.p1.4.m4.1.2.3.2" xref="S2.SS3.p1.4.m4.1.2.3.2a.cmml">s</mtext><mo id="S2.SS3.p1.4.m4.1.2.3.1" xref="S2.SS3.p1.4.m4.1.2.3.1.cmml">/</mo><msqrt id="S2.SS3.p1.4.m4.1.2.3.3" xref="S2.SS3.p1.4.m4.1.2.3.3.cmml"><mtext id="S2.SS3.p1.4.m4.1.2.3.3.2" xref="S2.SS3.p1.4.m4.1.2.3.3.2a.cmml">Hz</mtext></msqrt></mrow></mrow></math>, <math><mrow id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><mrow id="S2.E2.m1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.cmml"><msub id="S2.E2.m1.1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.2.cmml"><mover accent="true" id="S2.E2.m1.1.1.1.1.2.2" xref="S2.E2.m1.1.1.1.1.2.2.cmml"><mi id="S2.E2.m1.1.1.1.1.2.2.2" xref="S2.E2.m1.1.1.1.1.2.2.2.cmml">φ</mi><mo id="S2.E2.m1.1.1.1.1.2.2.1" xref="S2.E2.m1.1.1.1.1.2.2.1.cmml">~</mo></mover><mover accent="true" id="S2.E2.m1.1.1.1.1.2.3" xref="S2.E2.m1.1.1.1.1.2.3.cmml"><mi id="S2.E2.m1.1.1.1.1.2.3.2" xref="S2.E2.m1.1.1.1.1.2.3.2.cmml">τ</mi><mo id="S2.E2.m1.1.1.1.1.2.3.1" xref="S2.E2.m1.1.1.1.1.2.3.1.cmml">~</mo></mover></msub><mo id="S2.E2.m1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.1.1.1.1.3" xref="S2.E2.m1.1.1.1.1.3.cmml"><mrow id="S2.E2.m1.1.1.1.1.3.2" xref="S2.E2.m1.1.1.1.1.3.2.cmml"><mn id="S2.E2.m1.1.1.1.1.3.2.2" xref="S2.E2.m1.1.1.1.1.3.2.2.cmml">2</mn><mo id="S2.E2.m1.1.1.1.1.3.2.1" xref="S2.E2.m1.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.E2.m1.1.1.1.1.3.2.3" xref="S2.E2.m1.1.1.1.1.3.2.3.cmml">π</mi><mo id="S2.E2.m1.1.1.1.1.3.2.1a" xref="S2.E2.m1.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.E2.m1.1.1.1.1.3.2.4" xref="S2.E2.m1.1.1.1.1.3.2.4.cmml"><mi id="S2.E2.m1.1.1.1.1.3.2.4.2" xref="S2.E2.m1.1.1.1.1.3.2.4.2.cmml">f</mi><mi id="S2.E2.m1.1.1.1.1.3.2.4.3" xref="S2.E2.m1.1.1.1.1.3.2.4.3.cmml">s</mi></msub></mrow><mo id="S2.E2.m1.1.1.1.1.3.1" xref="S2.E2.m1.1.1.1.1.3.1.cmml">⋅</mo><mover accent="true" id="S2.E2.m1.1.1.1.1.3.3" xref="S2.E2.m1.1.1.1.1.3.3.cmml"><msub id="S2.E2.m1.1.1.1.1.3.3.2" xref="S2.E2.m1.1.1.1.1.3.3.2.cmml"><mi id="S2.E2.m1.1.1.1.1.3.3.2.2" xref="S2.E2.m1.1.1.1.1.3.3.2.2.cmml">τ</mi><mrow id="S2.E2.m1.1.1.1.1.3.3.2.3" xref="S2.E2.m1.1.1.1.1.3.3.2.3a.cmml"/></msub><mo id="S2.E2.m1.1.1.1.1.3.3.1" xref="S2.E2.m1.1.1.1.1.3.3.1.cmml">~</mo></mover></mrow></mrow><mo id="S2.E2.m1.1.1.1.2" xref="S2.E2.m1.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.E3.m1.9.9.1" xref="S2.E3.m1.9.9.1.1.cmml"><mrow id="S2.E3.m1.9.9.1.1" xref="S2.E3.m1.9.9.1.1.cmml"><msub id="S2.E3.m1.9.9.1.1.2" xref="S2.E3.m1.9.9.1.1.2.cmml"><mi id="S2.E3.m1.9.9.1.1.2.2" xref="S2.E3.m1.9.9.1.1.2.2.cmml">φ</mi><mrow id="S2.E3.m1.2.2.2.4" xref="S2.E3.m1.2.2.2.3.cmml"><mi id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml">i</mi><mo id="S2.E3.m1.2.2.2.4.1" xref="S2.E3.m1.2.2.2.3.cmml">,</mo><mi id="S2.E3.m1.2.2.2.2" xref="S2.E3.m1.2.2.2.2.cmml">c</mi></mrow></msub><mo id="S2.E3.m1.9.9.1.1.1" xref="S2.E3.m1.9.9.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.9.9.1.1.3" xref="S2.E3.m1.9.9.1.1.3.cmml"><msub id="S2.E3.m1.9.9.1.1.3.2" xref="S2.E3.m1.9.9.1.1.3.2.cmml"><mi id="S2.E3.m1.9.9.1.1.3.2.2" xref="S2.E3.m1.9.9.1.1.3.2.2.cmml">φ</mi><mrow id="S2.E3.m1.4.4.2.4" xref="S2.E3.m1.4.4.2.3.cmml"><mi id="S2.E3.m1.3.3.1.1" xref="S2.E3.m1.3.3.1.1.cmml">i</mi><mo id="S2.E3.m1.4.4.2.4.1" xref="S2.E3.m1.4.4.2.3.cmml">,</mo><mi id="S2.E3.m1.4.4.2.2" xref="S2.E3.m1.4.4.2.2.cmml">s</mi></mrow></msub><mo id="S2.E3.m1.9.9.1.1.3.1" xref="S2.E3.m1.9.9.1.1.3.1.cmml">-</mo><mrow id="S2.E3.m1.9.9.1.1.3.3" xref="S2.E3.m1.9.9.1.1.3.3.cmml"><msub id="S2.E3.m1.9.9.1.1.3.3.2" xref="S2.E3.m1.9.9.1.1.3.3.2.cmml"><mi id="S2.E3.m1.9.9.1.1.3.3.2.2" xref="S2.E3.m1.9.9.1.1.3.3.2.2.cmml">φ</mi><mrow id="S2.E3.m1.6.6.2.4" xref="S2.E3.m1.6.6.2.3.cmml"><mi id="S2.E3.m1.5.5.1.1" xref="S2.E3.m1.5.5.1.1.cmml">i</mi><mo id="S2.E3.m1.6.6.2.4.1" xref="S2.E3.m1.6.6.2.3.cmml">,</mo><mi id="S2.E3.m1.6.6.2.2" xref="S2.E3.m1.6.6.2.2.cmml">p</mi></mrow></msub><mo id="S2.E3.m1.9.9.1.1.3.3.1" xref="S2.E3.m1.9.9.1.1.3.3.1.cmml">⁢</mo><mfrac id="S2.E3.m1.8.8" xref="S2.E3.m1.8.8.cmml"><msub id="S2.E3.m1.8.8.2" xref="S2.E3.m1.8.8.2.cmml"><mi id="S2.E3.m1.8.8.2.4" xref="S2.E3.m1.8.8.2.4.cmml">f</mi><mrow id="S2.E3.m1.8.8.2.2.2.4" xref="S2.E3.m1.8.8.2.2.2.3.cmml"><mi id="S2.E3.m1.7.7.1.1.1.1" xref="S2.E3.m1.7.7.1.1.1.1.cmml">i</mi><mo id="S2.E3.m1.8.8.2.2.2.4.1" xref="S2.E3.m1.8.8.2.2.2.3.cmml">,</mo><mi id="S2.E3.m1.8.8.2.2.2.2" xref="S2.E3.m1.8.8.2.2.2.2.cmml">s</mi></mrow></msub><msub id="S2.E3.m1.8.8.4" xref="S2.E3.m1.8.8.4.cmml"><mi id="S2.E3.m1.8.8.4.2" xref="S2.E3.m1.8.8.4.2.cmml">f</mi><mi id="S2.E3.m1.8.8.4.3" xref="S2.E3.m1.8.8.4.3.cmml">p</mi></msub></mfrac></mrow></mrow></mrow><mo id="S2.E3.m1.9.9.1.2" xref="S2.E3.m1.9.9.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S3.SS2.SSS1.p2.2.m2.2.2.1" xref="S3.SS2.SSS1.p2.2.m2.2.2.2.cmml"><mi id="S3.SS2.SSS1.p2.2.m2.1.1" xref="S3.SS2.SSS1.p2.2.m2.1.1.cmml">arg</mi><mo id="S3.SS2.SSS1.p2.2.m2.2.2.1a" xref="S3.SS2.SSS1.p2.2.m2.2.2.2.cmml">⁡</mo><mrow id="S3.SS2.SSS1.p2.2.m2.2.2.1.1" xref="S3.SS2.SSS1.p2.2.m2.2.2.2.cmml"><mo stretchy="false" id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.2" xref="S3.SS2.SSS1.p2.2.m2.2.2.2.cmml">(</mo><mrow id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.cmml"><mi id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.3" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.3.cmml">T</mi><mo id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.2" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.2" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.cmml"><mi id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.2" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.2.cmml">f</mi><mi id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.3" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.3.cmml">s</mi></msub><mo stretchy="false" id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.3" xref="S3.SS2.SSS1.p2.2.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S3.SS2.SSS1.p2.2.m2.2.2.1.1.3" xref="S3.SS2.SSS1.p2.2.m2.2.2.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S4.p3.3.m3.1.1" xref="S4.p3.3.m3.1.1.cmml"><msub id="S4.p3.3.m3.1.1.2" xref="S4.p3.3.m3.1.1.2.cmml"><mi id="S4.p3.3.m3.1.1.2.2" xref="S4.p3.3.m3.1.1.2.2.cmml">φ</mi><mi mathvariant="normal" id="S4.p3.3.m3.1.1.2.3" xref="S4.p3.3.m3.1.1.2.3.cmml">Δ</mi></msub><mo id="S4.p3.3.m3.1.1.1" xref="S4.p3.3.m3.1.1.1.cmml">=</mo><mrow id="S4.p3.3.m3.1.1.3" xref="S4.p3.3.m3.1.1.3.cmml"><msub id="S4.p3.3.m3.1.1.3.2" xref="S4.p3.3.m3.1.1.3.2.cmml"><mi id="S4.p3.3.m3.1.1.3.2.2" xref="S4.p3.3.m3.1.1.3.2.2.cmml">φ</mi><mtext id="S4.p3.3.m3.1.1.3.2.3" xref="S4.p3.3.m3.1.1.3.2.3a.cmml">s</mtext></msub><mo id="S4.p3.3.m3.1.1.3.1" xref="S4.p3.3.m3.1.1.3.1.cmml">-</mo><msub id="S4.p3.3.m3.1.1.3.3" xref="S4.p3.3.m3.1.1.3.3.cmml"><mi id="S4.p3.3.m3.1.1.3.3.2" xref="S4.p3.3.m3.1.1.3.3.2.cmml">φ</mi><mtext id="S4.p3.3.m3.1.1.3.3.3" xref="S4.p3.3.m3.1.1.3.3.3a.cmml">v</mtext></msub></mrow></mrow></math>, <math><mrow id="S4.E4.m1.2.2.1" xref="S4.E4.m1.2.2.1.1.cmml"><mrow id="S4.E4.m1.2.2.1.1" xref="S4.E4.m1.2.2.1.1.cmml"><msub id="S4.E4.m1.2.2.1.1.3" xref="S4.E4.m1.2.2.1.1.3.cmml"><mi id="S4.E4.m1.2.2.1.1.3.2" xref="S4.E4.m1.2.2.1.1.3.2.cmml">φ</mi><mtext id="S4.E4.m1.2.2.1.1.3.3" xref="S4.E4.m1.2.2.1.1.3.3a.cmml">err</mtext></msub><mo id="S4.E4.m1.2.2.1.1.2" xref="S4.E4.m1.2.2.1.1.2.cmml">=</mo><mrow id="S4.E4.m1.2.2.1.1.1" xref="S4.E4.m1.2.2.1.1.1.cmml"><mfrac id="S4.E4.m1.2.2.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.3.cmml"><msub id="S4.E4.m1.2.2.1.1.1.3.2" xref="S4.E4.m1.2.2.1.1.1.3.2.cmml"><mi id="S4.E4.m1.2.2.1.1.1.3.2.2" xref="S4.E4.m1.2.2.1.1.1.3.2.2.cmml">v</mi><mtext id="S4.E4.m1.2.2.1.1.1.3.2.3" xref="S4.E4.m1.2.2.1.1.1.3.2.3a.cmml">v</mtext></msub><msub id="S4.E4.m1.2.2.1.1.1.3.3" xref="S4.E4.m1.2.2.1.1.1.3.3.cmml"><mi id="S4.E4.m1.2.2.1.1.1.3.3.2" xref="S4.E4.m1.2.2.1.1.1.3.3.2.cmml">v</mi><mtext id="S4.E4.m1.2.2.1.1.1.3.3.3" xref="S4.E4.m1.2.2.1.1.1.3.3.3a.cmml">s</mtext></msub></mfrac><mo id="S4.E4.m1.2.2.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S4.E4.m1.2.2.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml"><mi id="S4.E4.m1.1.1" xref="S4.E4.m1.1.1.cmml">sin</mi><mo id="S4.E4.m1.2.2.1.1.1.1.1a" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">⁡</mo><mrow id="S4.E4.m1.2.2.1.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml"><mo stretchy="false" id="S4.E4.m1.2.2.1.1.1.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">(</mo><mrow id="S4.E4.m1.2.2.1.1.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.cmml"><msub id="S4.E4.m1.2.2.1.1.1.1.1.1.1.2" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mi id="S4.E4.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">φ</mi><mtext id="S4.E4.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.2.3a.cmml">s</mtext></msub><mo id="S4.E4.m1.2.2.1.1.1.1.1.1.1.1" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.1.cmml">-</mo><msub id="S4.E4.m1.2.2.1.1.1.1.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.3.cmml"><mi id="S4.E4.m1.2.2.1.1.1.1.1.1.1.3.2" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.3.2.cmml">φ</mi><mtext id="S4.E4.m1.2.2.1.1.1.1.1.1.1.3.3" xref="S4.E4.m1.2.2.1.1.1.1.1.1.1.3.3a.cmml">v</mtext></msub></mrow><mo stretchy="false" id="S4.E4.m1.2.2.1.1.1.1.1.1.3" xref="S4.E4.m1.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S4.E4.m1.2.2.1.2" xref="S4.E4.m1.2.2.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S4.SS2.p1.13.m13.1.1" xref="S4.SS2.p1.13.m13.1.1.cmml"><msub id="S4.SS2.p1.13.m13.1.1.3" xref="S4.SS2.p1.13.m13.1.1.3.cmml"><mi id="S4.SS2.p1.13.m13.1.1.3.2" xref="S4.SS2.p1.13.m13.1.1.3.2.cmml">τ</mi><mtext id="S4.SS2.p1.13.m13.1.1.3.3" xref="S4.SS2.p1.13.m13.1.1.3.3a.cmml">c</mtext></msub><mo id="S4.SS2.p1.13.m13.1.1.2" xref="S4.SS2.p1.13.m13.1.1.2.cmml">≈</mo><mrow id="S4.SS2.p1.13.m13.1.1.1" xref="S4.SS2.p1.13.m13.1.1.1.cmml"><msub id="S4.SS2.p1.13.m13.1.1.1.3" xref="S4.SS2.p1.13.m13.1.1.1.3.cmml"><mi id="S4.SS2.p1.13.m13.1.1.1.3.2" xref="S4.SS2.p1.13.m13.1.1.1.3.2.cmml">l</mi><mtext id="S4.SS2.p1.13.m13.1.1.1.3.3" xref="S4.SS2.p1.13.m13.1.1.1.3.3a.cmml">c</mtext></msub><mo id="S4.SS2.p1.13.m13.1.1.1.2" xref="S4.SS2.p1.13.m13.1.1.1.2.cmml">/</mo><mrow id="S4.SS2.p1.13.m13.1.1.1.1.1" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.SS2.p1.13.m13.1.1.1.1.1.2" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.p1.13.m13.1.1.1.1.1.1" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.cmml"><mrow id="S4.SS2.p1.13.m13.1.1.1.1.1.1.2" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.cmml"><mn id="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.2" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.2.cmml">2</mn><mo id="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.1" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.1.cmml">/</mo><mn id="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.3" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.2.3.cmml">3</mn></mrow><mo id="S4.SS2.p1.13.m13.1.1.1.1.1.1.1" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.p1.13.m13.1.1.1.1.1.1.3" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.3.cmml">c</mi></mrow><mo stretchy="false" id="S4.SS2.p1.13.m13.1.1.1.1.1.3" xref="S4.SS2.p1.13.m13.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></math>

Correct Categorie: physics

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1209.3654

Formulas:

1-

\sqrt{2 Γ k_{B} T}

2-

e^{- m v^{2} / 2 k_{B} T}

3-

Γ_{1} (z) {\dot{x}}_{1} = - \frac{\partial U (z)}{\partial x_{1}} + \sqrt{2 D_{1} (z)} η_{1} (t)

4-

Γ_{2} (z) {\dot{x}}_{2} = - \frac{\partial U (z)}{\partial x_{2}} + \sqrt{2 D_{2} (z)} η_{2} (t) .

5-

Γ_{1} (z)

6-

Γ_{2} (z)

7-

z = x_{1} - x_{2}

8-

< η_{i} (t) > = 0

9-

< η_{1} (t_{1}) η_{2} (t_{2}) > = δ_{12} δ (t_{1} - t_{2})

10-

x = (x_{1} + x_{2}) / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1703.10613

Formulas:

1-

t_{ff} \approx 1 / \sqrt{G ρ_{disk}}

2-

t_{cool} > 3 Ω_{kep}^{- 1}

3-

Σ_{disk} < 1 / 3 Σ_{cloud}

4-

Σ_{disk} > 1 / 3 Σ_{cloud}

5-

t_{cool} < 3 Ω_{kep}^{- 1}

6-

Q_{disk} = σ_{v, disk} Ω_{disk} / π G Σ_{disk}

7-

Σ_{cloud} \approx Σ_{cloud, crit}

8-

t_{kep} \approx 3 Ω_{kep}^{- 1}

9-

t_{ff, disk} = \sqrt{1 / G ρ_{disk}}

10-

σ_{v, thermal} = 0.5 km / s

Formulas (html):

<math><mrow id="id4.2.m2.1.1" xref="id4.2.m2.1.1.cmml"><msub id="id4.2.m2.1.1.2" xref="id4.2.m2.1.1.2.cmml"><mi id="id4.2.m2.1.1.2.2" xref="id4.2.m2.1.1.2.2.cmml">t</mi><mi id="id4.2.m2.1.1.2.3" xref="id4.2.m2.1.1.2.3.cmml">ff</mi></msub><mo id="id4.2.m2.1.1.1" xref="id4.2.m2.1.1.1.cmml">≈</mo><mrow id="id4.2.m2.1.1.3" xref="id4.2.m2.1.1.3.cmml"><mn id="id4.2.m2.1.1.3.2" xref="id4.2.m2.1.1.3.2.cmml">1</mn><mo id="id4.2.m2.1.1.3.1" xref="id4.2.m2.1.1.3.1.cmml">/</mo><msqrt id="id4.2.m2.1.1.3.3" xref="id4.2.m2.1.1.3.3.cmml"><mrow id="id4.2.m2.1.1.3.3.2" xref="id4.2.m2.1.1.3.3.2.cmml"><mi id="id4.2.m2.1.1.3.3.2.2" xref="id4.2.m2.1.1.3.3.2.2.cmml">G</mi><mo id="id4.2.m2.1.1.3.3.2.1" xref="id4.2.m2.1.1.3.3.2.1.cmml">⁢</mo><msub id="id4.2.m2.1.1.3.3.2.3" xref="id4.2.m2.1.1.3.3.2.3.cmml"><mi id="id4.2.m2.1.1.3.3.2.3.2" xref="id4.2.m2.1.1.3.3.2.3.2.cmml">ρ</mi><mi id="id4.2.m2.1.1.3.3.2.3.3" xref="id4.2.m2.1.1.3.3.2.3.3.cmml">disk</mi></msub></mrow></msqrt></mrow></mrow></math>, <math><mrow id="id9.7.m7.1.1" xref="id9.7.m7.1.1.cmml"><msub id="id9.7.m7.1.1.2" xref="id9.7.m7.1.1.2.cmml"><mi id="id9.7.m7.1.1.2.2" xref="id9.7.m7.1.1.2.2.cmml">t</mi><mi id="id9.7.m7.1.1.2.3" xref="id9.7.m7.1.1.2.3.cmml">cool</mi></msub><mo id="id9.7.m7.1.1.1" xref="id9.7.m7.1.1.1.cmml">></mo><mrow id="id9.7.m7.1.1.3" xref="id9.7.m7.1.1.3.cmml"><mn id="id9.7.m7.1.1.3.2" xref="id9.7.m7.1.1.3.2.cmml">3</mn><mo id="id9.7.m7.1.1.3.1" xref="id9.7.m7.1.1.3.1.cmml">⁢</mo><msubsup id="id9.7.m7.1.1.3.3" xref="id9.7.m7.1.1.3.3.cmml"><mi mathvariant="normal" id="id9.7.m7.1.1.3.3.2.2" xref="id9.7.m7.1.1.3.3.2.2.cmml">Ω</mi><mi id="id9.7.m7.1.1.3.3.2.3" xref="id9.7.m7.1.1.3.3.2.3.cmml">kep</mi><mrow id="id9.7.m7.1.1.3.3.3" xref="id9.7.m7.1.1.3.3.3.cmml"><mo id="id9.7.m7.1.1.3.3.3.1" xref="id9.7.m7.1.1.3.3.3.1.cmml">-</mo><mn id="id9.7.m7.1.1.3.3.3.2" xref="id9.7.m7.1.1.3.3.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow></math>, <math><mrow id="id10.8.m8.1.1" xref="id10.8.m8.1.1.cmml"><msub id="id10.8.m8.1.1.2" xref="id10.8.m8.1.1.2.cmml"><mi mathvariant="normal" id="id10.8.m8.1.1.2.2" xref="id10.8.m8.1.1.2.2.cmml">Σ</mi><mi id="id10.8.m8.1.1.2.3" xref="id10.8.m8.1.1.2.3.cmml">disk</mi></msub><mo id="id10.8.m8.1.1.1" xref="id10.8.m8.1.1.1.cmml"><</mo><mrow id="id10.8.m8.1.1.3" xref="id10.8.m8.1.1.3.cmml"><mrow id="id10.8.m8.1.1.3.2" xref="id10.8.m8.1.1.3.2.cmml"><mn id="id10.8.m8.1.1.3.2.2" xref="id10.8.m8.1.1.3.2.2.cmml">1</mn><mo id="id10.8.m8.1.1.3.2.1" xref="id10.8.m8.1.1.3.2.1.cmml">/</mo><mn id="id10.8.m8.1.1.3.2.3" xref="id10.8.m8.1.1.3.2.3.cmml">3</mn></mrow><mo id="id10.8.m8.1.1.3.1" xref="id10.8.m8.1.1.3.1.cmml">⁢</mo><msub id="id10.8.m8.1.1.3.3" xref="id10.8.m8.1.1.3.3.cmml"><mi mathvariant="normal" id="id10.8.m8.1.1.3.3.2" xref="id10.8.m8.1.1.3.3.2.cmml">Σ</mi><mi id="id10.8.m8.1.1.3.3.3" xref="id10.8.m8.1.1.3.3.3.cmml">cloud</mi></msub></mrow></mrow></math>, <math><mrow id="id12.10.m10.1.1" xref="id12.10.m10.1.1.cmml"><msub id="id12.10.m10.1.1.2" xref="id12.10.m10.1.1.2.cmml"><mi mathvariant="normal" id="id12.10.m10.1.1.2.2" xref="id12.10.m10.1.1.2.2.cmml">Σ</mi><mi id="id12.10.m10.1.1.2.3" xref="id12.10.m10.1.1.2.3.cmml">disk</mi></msub><mo id="id12.10.m10.1.1.1" xref="id12.10.m10.1.1.1.cmml">></mo><mrow id="id12.10.m10.1.1.3" xref="id12.10.m10.1.1.3.cmml"><mrow id="id12.10.m10.1.1.3.2" xref="id12.10.m10.1.1.3.2.cmml"><mn id="id12.10.m10.1.1.3.2.2" xref="id12.10.m10.1.1.3.2.2.cmml">1</mn><mo id="id12.10.m10.1.1.3.2.1" xref="id12.10.m10.1.1.3.2.1.cmml">/</mo><mn id="id12.10.m10.1.1.3.2.3" xref="id12.10.m10.1.1.3.2.3.cmml">3</mn></mrow><mo id="id12.10.m10.1.1.3.1" xref="id12.10.m10.1.1.3.1.cmml">⁢</mo><msub id="id12.10.m10.1.1.3.3" xref="id12.10.m10.1.1.3.3.cmml"><mi mathvariant="normal" id="id12.10.m10.1.1.3.3.2" xref="id12.10.m10.1.1.3.3.2.cmml">Σ</mi><mi id="id12.10.m10.1.1.3.3.3" xref="id12.10.m10.1.1.3.3.3.cmml">cloud</mi></msub></mrow></mrow></math>, <math><mrow id="id13.11.m11.1.1" xref="id13.11.m11.1.1.cmml"><msub id="id13.11.m11.1.1.2" xref="id13.11.m11.1.1.2.cmml"><mi id="id13.11.m11.1.1.2.2" xref="id13.11.m11.1.1.2.2.cmml">t</mi><mi id="id13.11.m11.1.1.2.3" xref="id13.11.m11.1.1.2.3.cmml">cool</mi></msub><mo id="id13.11.m11.1.1.1" xref="id13.11.m11.1.1.1.cmml"><</mo><mrow id="id13.11.m11.1.1.3" xref="id13.11.m11.1.1.3.cmml"><mn id="id13.11.m11.1.1.3.2" xref="id13.11.m11.1.1.3.2.cmml">3</mn><mo id="id13.11.m11.1.1.3.1" xref="id13.11.m11.1.1.3.1.cmml">⁢</mo><msubsup id="id13.11.m11.1.1.3.3" xref="id13.11.m11.1.1.3.3.cmml"><mi mathvariant="normal" id="id13.11.m11.1.1.3.3.2.2" xref="id13.11.m11.1.1.3.3.2.2.cmml">Ω</mi><mi id="id13.11.m11.1.1.3.3.2.3" xref="id13.11.m11.1.1.3.3.2.3.cmml">kep</mi><mrow id="id13.11.m11.1.1.3.3.3" xref="id13.11.m11.1.1.3.3.3.cmml"><mo id="id13.11.m11.1.1.3.3.3.1" xref="id13.11.m11.1.1.3.3.3.1.cmml">-</mo><mn id="id13.11.m11.1.1.3.3.3.2" xref="id13.11.m11.1.1.3.3.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow></math>, <math><mrow id="S1.p2.2.m2.2.3" xref="S1.p2.2.m2.2.3.cmml"><msub id="S1.p2.2.m2.2.3.2" xref="S1.p2.2.m2.2.3.2.cmml"><mi id="S1.p2.2.m2.2.3.2.2" xref="S1.p2.2.m2.2.3.2.2.cmml">Q</mi><mi id="S1.p2.2.m2.2.3.2.3" xref="S1.p2.2.m2.2.3.2.3.cmml">disk</mi></msub><mo id="S1.p2.2.m2.2.3.1" xref="S1.p2.2.m2.2.3.1.cmml">=</mo><mrow id="S1.p2.2.m2.2.3.3" xref="S1.p2.2.m2.2.3.3.cmml"><mrow id="S1.p2.2.m2.2.3.3.2" xref="S1.p2.2.m2.2.3.3.2.cmml"><mrow id="S1.p2.2.m2.2.3.3.2.2" xref="S1.p2.2.m2.2.3.3.2.2.cmml"><msub id="S1.p2.2.m2.2.3.3.2.2.2" xref="S1.p2.2.m2.2.3.3.2.2.2.cmml"><mi id="S1.p2.2.m2.2.3.3.2.2.2.2" xref="S1.p2.2.m2.2.3.3.2.2.2.2.cmml">σ</mi><mrow id="S1.p2.2.m2.2.2.2.4" xref="S1.p2.2.m2.2.2.2.3.cmml"><mi mathvariant="normal" id="S1.p2.2.m2.1.1.1.1" xref="S1.p2.2.m2.1.1.1.1.cmml">v</mi><mo id="S1.p2.2.m2.2.2.2.4.1" xref="S1.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S1.p2.2.m2.2.2.2.2" xref="S1.p2.2.m2.2.2.2.2.cmml">disk</mi></mrow></msub><mo id="S1.p2.2.m2.2.3.3.2.2.1" xref="S1.p2.2.m2.2.3.3.2.2.1.cmml">⁢</mo><msub id="S1.p2.2.m2.2.3.3.2.2.3" xref="S1.p2.2.m2.2.3.3.2.2.3.cmml"><mi mathvariant="normal" id="S1.p2.2.m2.2.3.3.2.2.3.2" xref="S1.p2.2.m2.2.3.3.2.2.3.2.cmml">Ω</mi><mi id="S1.p2.2.m2.2.3.3.2.2.3.3" xref="S1.p2.2.m2.2.3.3.2.2.3.3.cmml">disk</mi></msub></mrow><mo id="S1.p2.2.m2.2.3.3.2.1" xref="S1.p2.2.m2.2.3.3.2.1.cmml">/</mo><mi mathvariant="normal" id="S1.p2.2.m2.2.3.3.2.3" xref="S1.p2.2.m2.2.3.3.2.3.cmml">π</mi></mrow><mo id="S1.p2.2.m2.2.3.3.1" xref="S1.p2.2.m2.2.3.3.1.cmml">⁢</mo><mi id="S1.p2.2.m2.2.3.3.3" xref="S1.p2.2.m2.2.3.3.3.cmml">G</mi><mo id="S1.p2.2.m2.2.3.3.1a" xref="S1.p2.2.m2.2.3.3.1.cmml">⁢</mo><msub id="S1.p2.2.m2.2.3.3.4" xref="S1.p2.2.m2.2.3.3.4.cmml"><mi mathvariant="normal" id="S1.p2.2.m2.2.3.3.4.2" xref="S1.p2.2.m2.2.3.3.4.2.cmml">Σ</mi><mi id="S1.p2.2.m2.2.3.3.4.3" xref="S1.p2.2.m2.2.3.3.4.3.cmml">disk</mi></msub></mrow></mrow></math>, <math><mrow id="S1.T1.4.4.2.m1.2.3" xref="S1.T1.4.4.2.m1.2.3.cmml"><msub id="S1.T1.4.4.2.m1.2.3.2" xref="S1.T1.4.4.2.m1.2.3.2.cmml"><mi mathvariant="normal" id="S1.T1.4.4.2.m1.2.3.2.2" xref="S1.T1.4.4.2.m1.2.3.2.2.cmml">Σ</mi><mi id="S1.T1.4.4.2.m1.2.3.2.3" xref="S1.T1.4.4.2.m1.2.3.2.3.cmml">cloud</mi></msub><mo id="S1.T1.4.4.2.m1.2.3.1" xref="S1.T1.4.4.2.m1.2.3.1.cmml">≈</mo><msub id="S1.T1.4.4.2.m1.2.3.3" xref="S1.T1.4.4.2.m1.2.3.3.cmml"><mi mathvariant="normal" id="S1.T1.4.4.2.m1.2.3.3.2" xref="S1.T1.4.4.2.m1.2.3.3.2.cmml">Σ</mi><mrow id="S1.T1.4.4.2.m1.2.2.2.4" xref="S1.T1.4.4.2.m1.2.2.2.3.cmml"><mi id="S1.T1.4.4.2.m1.1.1.1.1" xref="S1.T1.4.4.2.m1.1.1.1.1.cmml">cloud</mi><mo id="S1.T1.4.4.2.m1.2.2.2.4.1" xref="S1.T1.4.4.2.m1.2.2.2.3.cmml">,</mo><mi id="S1.T1.4.4.2.m1.2.2.2.2" xref="S1.T1.4.4.2.m1.2.2.2.2.cmml">crit</mi></mrow></msub></mrow></math>, <math><mrow id="S1.T1.20.20.2.m1.1.1" xref="S1.T1.20.20.2.m1.1.1.cmml"><msub id="S1.T1.20.20.2.m1.1.1.2" xref="S1.T1.20.20.2.m1.1.1.2.cmml"><mi id="S1.T1.20.20.2.m1.1.1.2.2" xref="S1.T1.20.20.2.m1.1.1.2.2.cmml">t</mi><mi id="S1.T1.20.20.2.m1.1.1.2.3" xref="S1.T1.20.20.2.m1.1.1.2.3.cmml">kep</mi></msub><mo id="S1.T1.20.20.2.m1.1.1.1" xref="S1.T1.20.20.2.m1.1.1.1.cmml">≈</mo><mrow id="S1.T1.20.20.2.m1.1.1.3" xref="S1.T1.20.20.2.m1.1.1.3.cmml"><mn id="S1.T1.20.20.2.m1.1.1.3.2" xref="S1.T1.20.20.2.m1.1.1.3.2.cmml">3</mn><mo id="S1.T1.20.20.2.m1.1.1.3.1" xref="S1.T1.20.20.2.m1.1.1.3.1.cmml">⁢</mo><msubsup id="S1.T1.20.20.2.m1.1.1.3.3" xref="S1.T1.20.20.2.m1.1.1.3.3.cmml"><mi mathvariant="normal" id="S1.T1.20.20.2.m1.1.1.3.3.2.2" xref="S1.T1.20.20.2.m1.1.1.3.3.2.2.cmml">Ω</mi><mi id="S1.T1.20.20.2.m1.1.1.3.3.2.3" xref="S1.T1.20.20.2.m1.1.1.3.3.2.3.cmml">kep</mi><mrow id="S1.T1.20.20.2.m1.1.1.3.3.3" xref="S1.T1.20.20.2.m1.1.1.3.3.3.cmml"><mo id="S1.T1.20.20.2.m1.1.1.3.3.3.1" xref="S1.T1.20.20.2.m1.1.1.3.3.3.1.cmml">-</mo><mn id="S1.T1.20.20.2.m1.1.1.3.3.3.2" xref="S1.T1.20.20.2.m1.1.1.3.3.3.2.cmml">1</mn></mrow></msubsup></mrow></mrow></math>, <math><mrow id="S1.T1.21.21.1.m1.2.3" xref="S1.T1.21.21.1.m1.2.3.cmml"><msub id="S1.T1.21.21.1.m1.2.3.2" xref="S1.T1.21.21.1.m1.2.3.2.cmml"><mi id="S1.T1.21.21.1.m1.2.3.2.2" xref="S1.T1.21.21.1.m1.2.3.2.2.cmml">t</mi><mrow id="S1.T1.21.21.1.m1.2.2.2.4" xref="S1.T1.21.21.1.m1.2.2.2.3.cmml"><mi id="S1.T1.21.21.1.m1.1.1.1.1" xref="S1.T1.21.21.1.m1.1.1.1.1.cmml">ff</mi><mo id="S1.T1.21.21.1.m1.2.2.2.4.1" xref="S1.T1.21.21.1.m1.2.2.2.3.cmml">,</mo><mi id="S1.T1.21.21.1.m1.2.2.2.2" xref="S1.T1.21.21.1.m1.2.2.2.2.cmml">disk</mi></mrow></msub><mo id="S1.T1.21.21.1.m1.2.3.1" xref="S1.T1.21.21.1.m1.2.3.1.cmml">=</mo><msqrt id="S1.T1.21.21.1.m1.2.3.3" xref="S1.T1.21.21.1.m1.2.3.3.cmml"><mrow id="S1.T1.21.21.1.m1.2.3.3.2" xref="S1.T1.21.21.1.m1.2.3.3.2.cmml"><mrow id="S1.T1.21.21.1.m1.2.3.3.2.2" xref="S1.T1.21.21.1.m1.2.3.3.2.2.cmml"><mn id="S1.T1.21.21.1.m1.2.3.3.2.2.2" xref="S1.T1.21.21.1.m1.2.3.3.2.2.2.cmml">1</mn><mo id="S1.T1.21.21.1.m1.2.3.3.2.2.1" xref="S1.T1.21.21.1.m1.2.3.3.2.2.1.cmml">/</mo><mi id="S1.T1.21.21.1.m1.2.3.3.2.2.3" xref="S1.T1.21.21.1.m1.2.3.3.2.2.3.cmml">G</mi></mrow><mo id="S1.T1.21.21.1.m1.2.3.3.2.1" xref="S1.T1.21.21.1.m1.2.3.3.2.1.cmml">⁢</mo><msub id="S1.T1.21.21.1.m1.2.3.3.2.3" xref="S1.T1.21.21.1.m1.2.3.3.2.3.cmml"><mi id="S1.T1.21.21.1.m1.2.3.3.2.3.2" xref="S1.T1.21.21.1.m1.2.3.3.2.3.2.cmml">ρ</mi><mi id="S1.T1.21.21.1.m1.2.3.3.2.3.3" xref="S1.T1.21.21.1.m1.2.3.3.2.3.3.cmml">disk</mi></msub></mrow></msqrt></mrow></math>, <math><mrow id="S2.SS1.p3.7.m7.2.3" xref="S2.SS1.p3.7.m7.2.3.cmml"><msub id="S2.SS1.p3.7.m7.2.3.2" xref="S2.SS1.p3.7.m7.2.3.2.cmml"><mi id="S2.SS1.p3.7.m7.2.3.2.2" xref="S2.SS1.p3.7.m7.2.3.2.2.cmml">σ</mi><mrow id="S2.SS1.p3.7.m7.2.2.2.4" xref="S2.SS1.p3.7.m7.2.2.2.3.cmml"><mi mathvariant="normal" id="S2.SS1.p3.7.m7.1.1.1.1" xref="S2.SS1.p3.7.m7.1.1.1.1.cmml">v</mi><mo id="S2.SS1.p3.7.m7.2.2.2.4.1" xref="S2.SS1.p3.7.m7.2.2.2.3.cmml">,</mo><mi id="S2.SS1.p3.7.m7.2.2.2.2" xref="S2.SS1.p3.7.m7.2.2.2.2.cmml">thermal</mi></mrow></msub><mo id="S2.SS1.p3.7.m7.2.3.1" xref="S2.SS1.p3.7.m7.2.3.1.cmml">=</mo><mrow id="S2.SS1.p3.7.m7.2.3.3" xref="S2.SS1.p3.7.m7.2.3.3.cmml"><mrow id="S2.SS1.p3.7.m7.2.3.3.2" xref="S2.SS1.p3.7.m7.2.3.3.2.cmml"><mn id="S2.SS1.p3.7.m7.2.3.3.2.2" xref="S2.SS1.p3.7.m7.2.3.3.2.2.cmml">0.5</mn><mo id="S2.SS1.p3.7.m7.2.3.3.2.1" xref="S2.SS1.p3.7.m7.2.3.3.2.1.cmml">⁢</mo><mi id="S2.SS1.p3.7.m7.2.3.3.2.3" xref="S2.SS1.p3.7.m7.2.3.3.2.3.cmml">km</mi></mrow><mo id="S2.SS1.p3.7.m7.2.3.3.1" xref="S2.SS1.p3.7.m7.2.3.3.1.cmml">/</mo><mi mathvariant="normal" id="S2.SS1.p3.7.m7.2.3.3.3" xref="S2.SS1.p3.7.m7.2.3.3.3.cmml">s</mi></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/0911.2437

Formulas:

1-

2.39 \begin{matrix} + 0.42 \\ - 0.33 \end{matrix} \times 10^{- 3} {eV}^{2},

2-

7.67 \begin{matrix} + 0.52 \\ - 0.53 \end{matrix} \times 10^{- 5} {eV}^{2},

3-

0.466 \begin{matrix} + 0.178 \\ - 0.135 \end{matrix},

4-

0.312 \begin{matrix} + 0.063 \\ - 0.049 \end{matrix},

5-

0.046, (0.016 \pm 0.010),

6-

U_{P M N S} = (\begin{matrix} 2 / \sqrt{6} & 1 / \sqrt{3} & 0 \\ - 1 / \sqrt{6} & 1 / \sqrt{3} & - 1 / \sqrt{2} \\ - 1 / \sqrt{6} & 1 / \sqrt{3} & 1 / \sqrt{2} \end{matrix}),

7-

\sin^{2} θ_{23} = 0.5, \sin^{2} θ_{12} = 0.33

8-

\sin^{2} θ_{13} = 0

9-

\sum_{i} m_{i} \leq 0.17 - 1.2 eV,

10-

B (ν_{α L}, N_{α L}^{c})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9708264

Formulas:

1-

M > 40 M_{⊙}

2-

10^{- 2} < τ < 10^{2}

3-

Y_{\max} = g_{rad, \max} / g_{grav}

4-

g_{Edd} = \frac{4 π σ T_{eff}^{4} G}{L_{Edd} / M_{⋆}} \approx 6.55 \times 10^{- 16} T_{eff}^{4} {(\frac{μ_{e}}{1.15})}^{- 1} cm s^{- 2},

5-

\sim 13, 000 - 30, 000

6-

T_{eff} > 40, 000

7-

M_{b o l}

8-

\log T_{e f f}

9-

Y_{\max} = g_{rad, \max} / g_{grav}

10-

M_{bol} = - 10 for T_{e f f} ≳ 25, 000

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/gr-qc/9501007

Formulas:

1-

0.11 m_{p l}

2-

d s^{2} = d t^{2} - a^{2} (t) d r^{2} - b^{2} (t) (d θ^{2} + \sin^{2} θ d φ^{2}) .

3-

(a = 1, 2, 3),

4-

L = \sqrt{- g} [\frac{1}{2} g^{μ ν} \partial_{μ} Φ^{a} \partial_{ν} Φ^{b} δ_{a b} - V (Φ)]

5-

V (Φ) \equiv \frac{1}{4} λ {(δ_{a b} Φ^{a} Φ^{b} - η^{2})}^{2} .

6-

Φ \equiv (Φ^{a}) = η ϕ (t) [\sin ϑ \cos φ, \sin ϑ \sin φ, \cos ϑ] .

7-

8 π G = 1

8-

2 \frac{\dot{a}}{a} \frac{\dot{b}}{b} + {(\frac{\dot{b}}{b})}^{2} + \frac{1}{b^{2}}

9-

\frac{η^{2}}{2} ({\dot{ϕ}}^{2} + 2 \frac{ϕ^{2}}{b^{2}} + \frac{λ η^{2}}{2} {(ϕ^{2} - 1)}^{2})

10-

2 \frac{\ddot{b}}{b} + {(\frac{\dot{b}}{b})}^{2} + \frac{1}{b^{2}}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/math/0112266

Formulas:

1-

p = r b .

2-

G (F) .

3-

G = G (F) .

4-

C o l (G)

5-

C o l (G)

6-

Δ (G, C)

7-

π (G, C)

8-

Δ (G, C)

9-

Δ (F) = R + B + A l t

10-

π (C^{'}) = π (C) .

Formulas (html):

Correct Categorie: math

Guessed Categorie: quant-ph

Result: incorrect

Run 11277762 (Agent389)