Run 11332330

Paper: https://arxiv.org/abs/1110.0889

Formulas:

1-

S_{Lag} = - \int d^{4} x λ (\frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ + U_{0}),

2-

\frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ + U_{0} = 0,

3-

(\partial_{μ} ϕ)

4-

(\partial_{μ} ϕ)

5-

ϕ = \sqrt{2 U_{0}} t .

6-

S = \int d^{4} x [\bar{ψ} {γ^{μ} \partial_{μ} + α {(P_{μ}^{ν} γ^{μ} \partial_{ν})}^{2 n + 1}} ψ - λ (\frac{1}{2} \partial_{μ} ϕ \partial^{μ} ϕ + U_{0})] .

7-

P_{μ}^{ν} \equiv δ_{μ}^{ν} + \frac{\partial_{μ} ϕ \partial^{ν} ϕ}{2 U_{0}} .

8-

0 = {γ^{μ} \partial_{μ} + α {(P_{μ}^{ν} γ^{μ} \partial_{ν})}^{2 n + 1}} ψ .

9-

0 = {γ^{0} \partial_{0} + γ^{i} \partial_{i} + α {(γ^{i} \partial_{i})}^{2 n + 1}} ψ .

10-

ω = k \sqrt{1 + α^{2} k^{4 n}} .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1105.3372

Formulas:

1-

P_{r o t}

2-

P_{r o t}

3-

P_{r o t}

4-

P_{r o t}

5-

Δ T_{C 7 c, d}

6-

A c o s (ω t_{i}) + B s i n (ω t_{i}) + C_{i},

7-

C_{i} \in {C_{1}, \dots, C_{27}}

8-

ω = 2 π / P

9-

- K s i n (2 π ϕ)

10-

- 2 π K

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1307.6686

Formulas:

1-

m \frac{d^{2} x_{N}}{d t^{2}}

2-

k (- x_{N} + x_{N - 1} + a) - m g,

3-

m \frac{d^{2} x_{i}}{d t^{2}}

4-

k (x_{i + 1} - 2 x_{i} + x_{i - 1}) - m g,

5-

m \frac{d^{2} x_{1}}{d t^{2}}

6-

k (x_{2} - x_{1} - a) - m g,

7-

x_{i + 1} - x_{i} = a + (m g i / k)

8-

x_{i} = (i - 1) a + \frac{m g i (i - 1)}{2 k} + x_{1},

9-

x_{1} = x_{N 0} - (N - 1) a - m g N (N - 1) / (2 k)

10-

m, k, a, g

Formulas (html):

Correct Categorie: physics

Guessed Categorie: nlin

Result: incorrect

Paper: https://arxiv.org/abs/0802.4461

Formulas:

1-

B, V, R,

2-

B, V, R, or I

3-

B - V = 0.98 \pm 0.10

4-

V - R = 0.49 \pm 0.10

5-

V - I = 1.06 \pm 0.10

6-

T_{\max} = HJD 2, 454, 437.618 (2) + 0.12544 (3) E .

7-

P_{o r b}

8-

B_{7} \sim > 1.5 M_{w d}^{3.1} P_{2}^{2.8},

9-

P_{2} = P_{o r b} / 2

10-

B, V, R, and I

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0908.4155

Formulas:

1-

λ_{i}^{(1)}, i = 1, \dots, N

2-

λ_{j}^{(2)}, j = 1, \dots, M

3-

0^{\circ} \leq λ \leq 360^{\circ}

4-

Δ λ_{k, l} = \min (| λ_{k} - λ_{l} |, 360^{\circ} - | λ_{k} - λ_{l} |) .

5-

D = \frac{Δ}{\bar{Δ}} .

6-

D (n, m) = \frac{Δ_{n, m}}{{\bar{Δ}}_{N, M}} .

7-

\bar{D} = \frac{\sum_{k = 1}^{K - 1} D (k, k + 1)}{K - 1},

8-

\bar{C} (l) = \frac{\sum_{k = 1}^{K - l} D (k, k + l)}{K - l} .

9-

\bar{D} = \bar{C} (1)

10-

\bar{C} (l)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1807.02193

Formulas:

1-

H_{spin} = \sum_{⟨ i j ⟩} J S_{i}^{z} S_{j}^{z} + \sum_{i} t \cdot S_{i}^{x},

2-

S_{i}^{z} = + 1

3-

S_{i}^{z} = - 1

4-

𝒪 (t^{6} / J^{5})

5-

| α_{\vec{r}} ⟩ = S_{\vec{r}}^{x} | α ⟩,

6-

| α_{{\vec{r}}_{i}} ⟩

7-

| α_{A B} ⟩

8-

| m_{α} ⟩ = \sum_{\vec{r}} \frac{{(- 1)}^{r_{x} + r_{y}}}{\sqrt{N}} σ_{\vec{r}}^{x} σ_{\vec{r}}^{z} | α ⟩,

9-

| m_{α} ⟩ = \frac{1}{\sqrt{N}} \sum_{\vec{r}} S_{\vec{r}}^{x} S_{\vec{r}}^{z} | α ⟩ .

10-

t \sum_{\vec{R}} S_{\vec{R}}^{x} | m_{α} ⟩ = \frac{t}{\sqrt{N}} \sum_{\vec{R}, \vec{r}} S_{\vec{R}}^{x} S_{\vec{r}}^{x} S_{\vec{r}}^{z} | α ⟩ =

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0803.1900

Formulas:

1-

N_{H} = 2.05 \times 10^{21}

2-

E / Δ E ≃ 500

3-

≃ 1^{'} - 3^{'}

4-

Γ = {1.29}_{- 0.15}^{+ 0.16}

5-

4.56 \pm 0.48 \times 10^{- 3}

6-

- 2 \ln λ

7-

- 2 \ln λ

8-

- 2 \ln λ \sim 27

9-

Γ = {2.23}_{- 0.27}^{+ 0.36}

10-

N_{H}^{z} \sim 2 \times 10^{22}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9605174

Formulas:

1-

m_{ν_{e}} < \sim 5 eV,

2-

m_{ν_{μ}} < \sim 250 KeV,

3-

m_{ν_{τ}} < \sim 23 MeV

4-

Ω_{ν} = \frac{\sum_{i} m_{ν_{i}}}{94 h^{2} eV},

5-

Ω_{ν} = \frac{ρ_{ν}}{ρ_{crit}}

6-

ρ_{crit} = \frac{3 H_{0}^{2}}{8 π G} ≅ 1.88 \times 10^{- 29} h^{2} \frac{g}{{cm}^{3}}

7-

H_{0} = 100 h \frac{km / s}{Mpc}

8-

\sum_{i = 1} m_{ν_{i}} (\frac{g_{ν_{i}}}{2}) \leq 94 e V,

9-

m_{ν_{i}} < \sim 1

10-

\frac{ℳ}{ℒ} = 5.3 h \frac{ℳ_{⊙}}{ℒ_{⊙}}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/gr-qc/0511126

Formulas:

1-

γ = {(1 - v^{2} / c^{2})}^{- 1 / 2}

2-

\begin{matrix} x = x_{0} - v t_{0}, \\ t = t_{0} - v x_{0} / c^{2}, \end{matrix}

3-

\begin{matrix} x_{0} = x + v t, \\ t_{0} = t + v x / c^{2} . \end{matrix}

4-

(x_{0}, t_{0})

5-

x_{0} (x)

6-

S_{0} (S)

7-

v x_{0} / c^{2}

8-

v x / c^{2}

9-

v Δ x_{0} / c^{2}

10-

\begin{matrix} x^{'} = x_{0} + v t_{0}, \\ t^{'} = t_{0} + v x_{0} / c^{2}, \end{matrix}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/nucl-th/0303070

Formulas:

1-

{}^{233, 235, 238}U

2-

{}^{233, 235, 238}U

3-

Δ_{33} (1232)

4-

Δ_{33} (1232)

5-

{}^{233, 235, 238}U

6-

Δ_{33} (1232)

7-

W_{f} \equiv σ_{f} / σ_{t o t}

8-

E_{γ} = 30 - 140

9-

τ_{c a s}

10-

τ_{c a s} \sim τ_{o}

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0403516

Formulas:

1-

v_{A} = c \sqrt{σ / (1 + σ)}

2-

\cos θ_{up} > β_{up}

3-

Γ = {(1 - β_{up}^{2})}^{- 1 / 2} ≫ 1

4-

\sin θ < 1 / Γ

5-

2 π β r_{L}

6-

r_{L} = c / Ω

7-

E_{y} / B_{z} ≫ 1

8-

c E_{y} / B_{z}

9-

c B_{z} / E_{y}

10-

| E_{y} / B_{x} | > 1

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0201193

Formulas:

1-

\frac{3}{2} α^{- 1} M_{e} c^{2} N^{4}

2-

α = \frac{e^{2}}{ℏ c}

3-

N = 0, 1, 2, \dots

4-

e, μ, t, \dots

5-

M (N) = M_{e} (1 + \frac{3}{2} α^{- 1} \sum_{n = 0}^{n = N} n^{4})

6-

M_{u} = M_{e} / 7.25

7-

s := c t - \vec{x} \cdot \vec{v} / c = s_{0}

8-

\frac{d L}{d V} = \frac{1}{2} \partial_{ν} F \partial^{ν} F - \frac{1}{2} μ^{2} F^{2} - \frac{1}{4 g^{2}} F^{4} .

9-

F = F (ω s)

10-

F = \sqrt{2} ω g k f (ω s)

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1907.09318

Formulas:

1-

B_{n a} / B

2-

B_{n a} / B

3-

B_{n a} / B

4-

G = G_{a} - G_{n a} B = B_{a} - B_{n a},

5-

B_{n a} / B

6-

B_{n a} / B

7-

B_{n a} / B

8-

B_{n a} / B

9-

B_{n a} / B

10-

B_{n a} / B

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1311.5491

Formulas:

1-

ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e}

2-

ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e}

3-

\bar{q} q \to ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e}

4-

q = u, d, c, s

5-

\bar{} bb \to ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e}

6-

γ γ \to ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e}

7-

γ q \to ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e} q

8-

\bar{q} q \to ν_{μ} μ^{+} e^{-} {\bar{ν}}_{e}

9-

e^{+} e^{-} \to 4

10-

α / π \times Γ_{W} / M_{W}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ex

Result: incorrect

Paper: https://arxiv.org/abs/1907.07633

Formulas:

1-

| G | = | H | < 512

2-

ϕ : G \to {GL}_{n} (𝔽)

3-

{GL}_{n} (𝔽)

4-

𝐂𝐓 (G) ≅ 𝐂𝐓 (H)

5-

𝐂𝐓 (G) ≅ 𝐂𝐓 (H)

6-

𝐂𝐓 (G) : Irr (G) \times 𝐂𝐥𝐚𝐬𝐬𝐞𝐬 (G) \to ℂ

7-

𝐂𝐓 (G) (χ, 𝒞) = χ (g) .

8-

Φ : Irr (G) \to Irr (H)

9-

Ψ : 𝐂𝐥𝐚𝐬𝐬𝐞𝐬 (G) \to 𝐂𝐥𝐚𝐬𝐬𝐞𝐬 (H)

10-

C T (G) (χ, 𝒞) = 𝐂𝐓 (H) (Φ (χ), Ψ (𝒞)) .

Formulas (html):

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xref="S2.p1.4.m4.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.p2.14.m14.4.5" xref="S2.p2.14.m14.4.5.cmml"><mrow id="S2.p2.14.m14.4.5.2.2" xref="S2.p2.14.m14.4.5.2.1.cmml"><mtext id="S2.p2.14.m14.1.1" xref="S2.p2.14.m14.1.1a.cmml">𝐂𝐓</mtext><mo id="S2.p2.14.m14.4.5.2.2a" xref="S2.p2.14.m14.4.5.2.1.cmml">⁡</mo><mrow id="S2.p2.14.m14.4.5.2.2.1" xref="S2.p2.14.m14.4.5.2.1.cmml"><mo stretchy="false" id="S2.p2.14.m14.4.5.2.2.1.1" xref="S2.p2.14.m14.4.5.2.1.cmml">(</mo><mi id="S2.p2.14.m14.2.2" xref="S2.p2.14.m14.2.2.cmml">G</mi><mo stretchy="false" id="S2.p2.14.m14.4.5.2.2.1.2" xref="S2.p2.14.m14.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.14.m14.4.5.1" xref="S2.p2.14.m14.4.5.1.cmml">≅</mo><mrow id="S2.p2.14.m14.4.5.3.2" xref="S2.p2.14.m14.4.5.3.1.cmml"><mtext id="S2.p2.14.m14.3.3" xref="S2.p2.14.m14.3.3a.cmml">𝐂𝐓</mtext><mo id="S2.p2.14.m14.4.5.3.2a" xref="S2.p2.14.m14.4.5.3.1.cmml">⁡</mo><mrow id="S2.p2.14.m14.4.5.3.2.1" xref="S2.p2.14.m14.4.5.3.1.cmml"><mo stretchy="false" id="S2.p2.14.m14.4.5.3.2.1.1" xref="S2.p2.14.m14.4.5.3.1.cmml">(</mo><mi id="S2.p2.14.m14.4.4" xref="S2.p2.14.m14.4.4.cmml">H</mi><mo stretchy="false" id="S2.p2.14.m14.4.5.3.2.1.2" xref="S2.p2.14.m14.4.5.3.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.p2.15.m15.4.5" xref="S2.p2.15.m15.4.5.cmml"><mrow id="S2.p2.15.m15.4.5.2.2" xref="S2.p2.15.m15.4.5.2.1.cmml"><mtext id="S2.p2.15.m15.1.1" xref="S2.p2.15.m15.1.1a.cmml">𝐂𝐓</mtext><mo id="S2.p2.15.m15.4.5.2.2a" xref="S2.p2.15.m15.4.5.2.1.cmml">⁡</mo><mrow id="S2.p2.15.m15.4.5.2.2.1" xref="S2.p2.15.m15.4.5.2.1.cmml"><mo stretchy="false" id="S2.p2.15.m15.4.5.2.2.1.1" xref="S2.p2.15.m15.4.5.2.1.cmml">(</mo><mi id="S2.p2.15.m15.2.2" xref="S2.p2.15.m15.2.2.cmml">G</mi><mo stretchy="false" id="S2.p2.15.m15.4.5.2.2.1.2" xref="S2.p2.15.m15.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S2.p2.15.m15.4.5.1" xref="S2.p2.15.m15.4.5.1.cmml">≅</mo><mrow id="S2.p2.15.m15.4.5.3.2" xref="S2.p2.15.m15.4.5.3.1.cmml"><mtext id="S2.p2.15.m15.3.3" xref="S2.p2.15.m15.3.3a.cmml">𝐂𝐓</mtext><mo id="S2.p2.15.m15.4.5.3.2a" xref="S2.p2.15.m15.4.5.3.1.cmml">⁡</mo><mrow id="S2.p2.15.m15.4.5.3.2.1" xref="S2.p2.15.m15.4.5.3.1.cmml"><mo stretchy="false" id="S2.p2.15.m15.4.5.3.2.1.1" xref="S2.p2.15.m15.4.5.3.1.cmml">(</mo><mi id="S2.p2.15.m15.4.4" xref="S2.p2.15.m15.4.4.cmml">H</mi><mo stretchy="false" id="S2.p2.15.m15.4.5.3.2.1.2" xref="S2.p2.15.m15.4.5.3.1.cmml">)</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.Ex1.m1.4.5" xref="S2.Ex1.m1.4.5.cmml"><mrow id="S2.Ex1.m1.4.5.2.2" xref="S2.Ex1.m1.4.5.2.1.cmml"><mtext id="S2.Ex1.m1.1.1" xref="S2.Ex1.m1.1.1a.cmml">𝐂𝐓</mtext><mo id="S2.Ex1.m1.4.5.2.2a" xref="S2.Ex1.m1.4.5.2.1.cmml">⁡</mo><mrow id="S2.Ex1.m1.4.5.2.2.1" xref="S2.Ex1.m1.4.5.2.1.cmml"><mo stretchy="false" id="S2.Ex1.m1.4.5.2.2.1.1" xref="S2.Ex1.m1.4.5.2.1.cmml">(</mo><mi id="S2.Ex1.m1.2.2" xref="S2.Ex1.m1.2.2.cmml">G</mi><mo stretchy="false" id="S2.Ex1.m1.4.5.2.2.1.2" xref="S2.Ex1.m1.4.5.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.4.5.1" xref="S2.Ex1.m1.4.5.1.cmml">:</mo><mrow id="S2.Ex1.m1.4.5.3" xref="S2.Ex1.m1.4.5.3.cmml"><mrow id="S2.Ex1.m1.4.5.3.2" xref="S2.Ex1.m1.4.5.3.2.cmml"><mrow id="S2.Ex1.m1.4.5.3.2.2" xref="S2.Ex1.m1.4.5.3.2.2.cmml"><mrow id="S2.Ex1.m1.4.5.3.2.2.2" xref="S2.Ex1.m1.4.5.3.2.2.2.cmml"><mtext id="S2.Ex1.m1.4.5.3.2.2.2.2" xref="S2.Ex1.m1.4.5.3.2.2.2.2a.cmml">Irr</mtext><mo id="S2.Ex1.m1.4.5.3.2.2.2.1" xref="S2.Ex1.m1.4.5.3.2.2.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.4.5.3.2.2.2.3.2" xref="S2.Ex1.m1.4.5.3.2.2.2.cmml"><mo stretchy="false" id="S2.Ex1.m1.4.5.3.2.2.2.3.2.1" xref="S2.Ex1.m1.4.5.3.2.2.2.cmml">(</mo><mi id="S2.Ex1.m1.3.3" xref="S2.Ex1.m1.3.3.cmml">G</mi><mo stretchy="false" id="S2.Ex1.m1.4.5.3.2.2.2.3.2.2" xref="S2.Ex1.m1.4.5.3.2.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.4.5.3.2.2.1" xref="S2.Ex1.m1.4.5.3.2.2.1.cmml">×</mo><mtext id="S2.Ex1.m1.4.5.3.2.2.3" xref="S2.Ex1.m1.4.5.3.2.2.3a.cmml">𝐂𝐥𝐚𝐬𝐬𝐞𝐬</mtext></mrow><mo id="S2.Ex1.m1.4.5.3.2.1" xref="S2.Ex1.m1.4.5.3.2.1.cmml">⁢</mo><mrow id="S2.Ex1.m1.4.5.3.2.3.2" xref="S2.Ex1.m1.4.5.3.2.cmml"><mo stretchy="false" id="S2.Ex1.m1.4.5.3.2.3.2.1" xref="S2.Ex1.m1.4.5.3.2.cmml">(</mo><mi id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.4.4.cmml">G</mi><mo stretchy="false" id="S2.Ex1.m1.4.5.3.2.3.2.2" xref="S2.Ex1.m1.4.5.3.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.4.5.3.1" xref="S2.Ex1.m1.4.5.3.1.cmml">→</mo><mi id="S2.Ex1.m1.4.5.3.3" xref="S2.Ex1.m1.4.5.3.3.cmml">ℂ</mi></mrow></mrow></math>, <math><mrow id="S2.Ex2.m1.6.6.1" xref="S2.Ex2.m1.6.6.1.1.cmml"><mrow id="S2.Ex2.m1.6.6.1.1" xref="S2.Ex2.m1.6.6.1.1.cmml"><mrow id="S2.Ex2.m1.6.6.1.1.2" xref="S2.Ex2.m1.6.6.1.1.2.cmml"><mrow id="S2.Ex2.m1.6.6.1.1.2.2.2" xref="S2.Ex2.m1.6.6.1.1.2.2.1.cmml"><mtext id="S2.Ex2.m1.1.1" xref="S2.Ex2.m1.1.1a.cmml">𝐂𝐓</mtext><mo id="S2.Ex2.m1.6.6.1.1.2.2.2a" xref="S2.Ex2.m1.6.6.1.1.2.2.1.cmml">⁡</mo><mrow id="S2.Ex2.m1.6.6.1.1.2.2.2.1" xref="S2.Ex2.m1.6.6.1.1.2.2.1.cmml"><mo stretchy="false" id="S2.Ex2.m1.6.6.1.1.2.2.2.1.1" xref="S2.Ex2.m1.6.6.1.1.2.2.1.cmml">(</mo><mi id="S2.Ex2.m1.2.2" xref="S2.Ex2.m1.2.2.cmml">G</mi><mo stretchy="false" id="S2.Ex2.m1.6.6.1.1.2.2.2.1.2" xref="S2.Ex2.m1.6.6.1.1.2.2.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.6.6.1.1.2.1" xref="S2.Ex2.m1.6.6.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.6.6.1.1.2.3.2" xref="S2.Ex2.m1.6.6.1.1.2.3.1.cmml"><mo stretchy="false" id="S2.Ex2.m1.6.6.1.1.2.3.2.1" xref="S2.Ex2.m1.6.6.1.1.2.3.1.cmml">(</mo><mi id="S2.Ex2.m1.3.3" xref="S2.Ex2.m1.3.3.cmml">χ</mi><mo id="S2.Ex2.m1.6.6.1.1.2.3.2.2" xref="S2.Ex2.m1.6.6.1.1.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.4.4" xref="S2.Ex2.m1.4.4.cmml">𝒞</mi><mo stretchy="false" id="S2.Ex2.m1.6.6.1.1.2.3.2.3" xref="S2.Ex2.m1.6.6.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.6.6.1.1.1" xref="S2.Ex2.m1.6.6.1.1.1.cmml">=</mo><mrow id="S2.Ex2.m1.6.6.1.1.3" xref="S2.Ex2.m1.6.6.1.1.3.cmml"><mi id="S2.Ex2.m1.6.6.1.1.3.2" xref="S2.Ex2.m1.6.6.1.1.3.2.cmml">χ</mi><mo id="S2.Ex2.m1.6.6.1.1.3.1" xref="S2.Ex2.m1.6.6.1.1.3.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.6.6.1.1.3.3.2" xref="S2.Ex2.m1.6.6.1.1.3.cmml"><mo stretchy="false" id="S2.Ex2.m1.6.6.1.1.3.3.2.1" xref="S2.Ex2.m1.6.6.1.1.3.cmml">(</mo><mi id="S2.Ex2.m1.5.5" xref="S2.Ex2.m1.5.5.cmml">g</mi><mo stretchy="false" id="S2.Ex2.m1.6.6.1.1.3.3.2.2" xref="S2.Ex2.m1.6.6.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex2.m1.6.6.1.2" xref="S2.Ex2.m1.6.6.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.p3.7.m3.2.3" xref="S2.p3.7.m3.2.3.cmml"><mi mathvariant="normal" id="S2.p3.7.m3.2.3.2" xref="S2.p3.7.m3.2.3.2.cmml">Φ</mi><mo id="S2.p3.7.m3.2.3.1" xref="S2.p3.7.m3.2.3.1.cmml">:</mo><mrow id="S2.p3.7.m3.2.3.3" xref="S2.p3.7.m3.2.3.3.cmml"><mrow id="S2.p3.7.m3.2.3.3.2" xref="S2.p3.7.m3.2.3.3.2.cmml"><mtext id="S2.p3.7.m3.2.3.3.2.2" xref="S2.p3.7.m3.2.3.3.2.2a.cmml">Irr</mtext><mo id="S2.p3.7.m3.2.3.3.2.1" xref="S2.p3.7.m3.2.3.3.2.1.cmml">⁢</mo><mrow id="S2.p3.7.m3.2.3.3.2.3.2" xref="S2.p3.7.m3.2.3.3.2.cmml"><mo stretchy="false" id="S2.p3.7.m3.2.3.3.2.3.2.1" xref="S2.p3.7.m3.2.3.3.2.cmml">(</mo><mi id="S2.p3.7.m3.1.1" xref="S2.p3.7.m3.1.1.cmml">G</mi><mo stretchy="false" id="S2.p3.7.m3.2.3.3.2.3.2.2" xref="S2.p3.7.m3.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S2.p3.7.m3.2.3.3.1" xref="S2.p3.7.m3.2.3.3.1.cmml">→</mo><mrow id="S2.p3.7.m3.2.3.3.3" xref="S2.p3.7.m3.2.3.3.3.cmml"><mtext id="S2.p3.7.m3.2.3.3.3.2" xref="S2.p3.7.m3.2.3.3.3.2a.cmml">Irr</mtext><mo id="S2.p3.7.m3.2.3.3.3.1" xref="S2.p3.7.m3.2.3.3.3.1.cmml">⁢</mo><mrow id="S2.p3.7.m3.2.3.3.3.3.2" xref="S2.p3.7.m3.2.3.3.3.cmml"><mo stretchy="false" id="S2.p3.7.m3.2.3.3.3.3.2.1" xref="S2.p3.7.m3.2.3.3.3.cmml">(</mo><mi id="S2.p3.7.m3.2.2" xref="S2.p3.7.m3.2.2.cmml">H</mi><mo stretchy="false" id="S2.p3.7.m3.2.3.3.3.3.2.2" xref="S2.p3.7.m3.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.p3.8.m4.2.3" xref="S2.p3.8.m4.2.3.cmml"><mi mathvariant="normal" id="S2.p3.8.m4.2.3.2" xref="S2.p3.8.m4.2.3.2.cmml">Ψ</mi><mo id="S2.p3.8.m4.2.3.1" xref="S2.p3.8.m4.2.3.1.cmml">:</mo><mrow id="S2.p3.8.m4.2.3.3" xref="S2.p3.8.m4.2.3.3.cmml"><mrow id="S2.p3.8.m4.2.3.3.2" xref="S2.p3.8.m4.2.3.3.2.cmml"><mtext id="S2.p3.8.m4.2.3.3.2.2" xref="S2.p3.8.m4.2.3.3.2.2a.cmml">𝐂𝐥𝐚𝐬𝐬𝐞𝐬</mtext><mo id="S2.p3.8.m4.2.3.3.2.1" xref="S2.p3.8.m4.2.3.3.2.1.cmml">⁢</mo><mrow id="S2.p3.8.m4.2.3.3.2.3.2" xref="S2.p3.8.m4.2.3.3.2.cmml"><mo stretchy="false" id="S2.p3.8.m4.2.3.3.2.3.2.1" xref="S2.p3.8.m4.2.3.3.2.cmml">(</mo><mi id="S2.p3.8.m4.1.1" xref="S2.p3.8.m4.1.1.cmml">G</mi><mo stretchy="false" id="S2.p3.8.m4.2.3.3.2.3.2.2" xref="S2.p3.8.m4.2.3.3.2.cmml">)</mo></mrow></mrow><mo id="S2.p3.8.m4.2.3.3.1" xref="S2.p3.8.m4.2.3.3.1.cmml">→</mo><mrow id="S2.p3.8.m4.2.3.3.3" xref="S2.p3.8.m4.2.3.3.3.cmml"><mtext id="S2.p3.8.m4.2.3.3.3.2" xref="S2.p3.8.m4.2.3.3.3.2a.cmml">𝐂𝐥𝐚𝐬𝐬𝐞𝐬</mtext><mo id="S2.p3.8.m4.2.3.3.3.1" xref="S2.p3.8.m4.2.3.3.3.1.cmml">⁢</mo><mrow id="S2.p3.8.m4.2.3.3.3.3.2" xref="S2.p3.8.m4.2.3.3.3.cmml"><mo stretchy="false" id="S2.p3.8.m4.2.3.3.3.3.2.1" xref="S2.p3.8.m4.2.3.3.3.cmml">(</mo><mi id="S2.p3.8.m4.2.2" xref="S2.p3.8.m4.2.2.cmml">H</mi><mo stretchy="false" id="S2.p3.8.m4.2.3.3.3.3.2.2" xref="S2.p3.8.m4.2.3.3.3.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.E1.m1.8.8.1" xref="S2.E1.m1.8.8.1.1.cmml"><mrow id="S2.E1.m1.8.8.1.1" xref="S2.E1.m1.8.8.1.1.cmml"><mrow id="S2.E1.m1.8.8.1.1.4" xref="S2.E1.m1.8.8.1.1.4.cmml"><mi id="S2.E1.m1.8.8.1.1.4.2" xref="S2.E1.m1.8.8.1.1.4.2.cmml">C</mi><mo id="S2.E1.m1.8.8.1.1.4.1" xref="S2.E1.m1.8.8.1.1.4.1.cmml">⁢</mo><mi id="S2.E1.m1.8.8.1.1.4.3" xref="S2.E1.m1.8.8.1.1.4.3.cmml">T</mi><mo id="S2.E1.m1.8.8.1.1.4.1a" xref="S2.E1.m1.8.8.1.1.4.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.4.4.2" xref="S2.E1.m1.8.8.1.1.4.cmml"><mo stretchy="false" id="S2.E1.m1.8.8.1.1.4.4.2.1" xref="S2.E1.m1.8.8.1.1.4.cmml">(</mo><mi id="S2.E1.m1.1.1" xref="S2.E1.m1.1.1.cmml">G</mi><mo stretchy="false" id="S2.E1.m1.8.8.1.1.4.4.2.2" xref="S2.E1.m1.8.8.1.1.4.cmml">)</mo></mrow><mo id="S2.E1.m1.8.8.1.1.4.1b" xref="S2.E1.m1.8.8.1.1.4.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.4.5.2" xref="S2.E1.m1.8.8.1.1.4.5.1.cmml"><mo stretchy="false" id="S2.E1.m1.8.8.1.1.4.5.2.1" xref="S2.E1.m1.8.8.1.1.4.5.1.cmml">(</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">χ</mi><mo id="S2.E1.m1.8.8.1.1.4.5.2.2" xref="S2.E1.m1.8.8.1.1.4.5.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">𝒞</mi><mo stretchy="false" id="S2.E1.m1.8.8.1.1.4.5.2.3" xref="S2.E1.m1.8.8.1.1.4.5.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.8.8.1.1.3" xref="S2.E1.m1.8.8.1.1.3.cmml">=</mo><mrow id="S2.E1.m1.8.8.1.1.2" xref="S2.E1.m1.8.8.1.1.2.cmml"><mrow id="S2.E1.m1.8.8.1.1.2.4.2" xref="S2.E1.m1.8.8.1.1.2.4.1.cmml"><mtext id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4a.cmml">𝐂𝐓</mtext><mo id="S2.E1.m1.8.8.1.1.2.4.2a" xref="S2.E1.m1.8.8.1.1.2.4.1.cmml">⁡</mo><mrow id="S2.E1.m1.8.8.1.1.2.4.2.1" xref="S2.E1.m1.8.8.1.1.2.4.1.cmml"><mo stretchy="false" id="S2.E1.m1.8.8.1.1.2.4.2.1.1" xref="S2.E1.m1.8.8.1.1.2.4.1.cmml">(</mo><mi id="S2.E1.m1.5.5" xref="S2.E1.m1.5.5.cmml">H</mi><mo stretchy="false" id="S2.E1.m1.8.8.1.1.2.4.2.1.2" xref="S2.E1.m1.8.8.1.1.2.4.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.8.8.1.1.2.3" xref="S2.E1.m1.8.8.1.1.2.3.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.2.2.2" xref="S2.E1.m1.8.8.1.1.2.2.3.cmml"><mo stretchy="false" id="S2.E1.m1.8.8.1.1.2.2.2.3" xref="S2.E1.m1.8.8.1.1.2.2.3.cmml">(</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S2.E1.m1.8.8.1.1.1.1.1.1.2" xref="S2.E1.m1.8.8.1.1.1.1.1.1.2.cmml">Φ</mi><mo id="S2.E1.m1.8.8.1.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.1.1.3.2" xref="S2.E1.m1.8.8.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.8.8.1.1.1.1.1.1.3.2.1" xref="S2.E1.m1.8.8.1.1.1.1.1.1.cmml">(</mo><mi id="S2.E1.m1.6.6" xref="S2.E1.m1.6.6.cmml">χ</mi><mo stretchy="false" id="S2.E1.m1.8.8.1.1.1.1.1.1.3.2.2" xref="S2.E1.m1.8.8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.8.8.1.1.2.2.2.4" xref="S2.E1.m1.8.8.1.1.2.2.3.cmml">,</mo><mrow id="S2.E1.m1.8.8.1.1.2.2.2.2" xref="S2.E1.m1.8.8.1.1.2.2.2.2.cmml"><mi mathvariant="normal" id="S2.E1.m1.8.8.1.1.2.2.2.2.2" xref="S2.E1.m1.8.8.1.1.2.2.2.2.2.cmml">Ψ</mi><mo id="S2.E1.m1.8.8.1.1.2.2.2.2.1" xref="S2.E1.m1.8.8.1.1.2.2.2.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.2.2.2.2.3.2" xref="S2.E1.m1.8.8.1.1.2.2.2.2.cmml"><mo stretchy="false" id="S2.E1.m1.8.8.1.1.2.2.2.2.3.2.1" xref="S2.E1.m1.8.8.1.1.2.2.2.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.7.7" xref="S2.E1.m1.7.7.cmml">𝒞</mi><mo stretchy="false" id="S2.E1.m1.8.8.1.1.2.2.2.2.3.2.2" xref="S2.E1.m1.8.8.1.1.2.2.2.2.cmml">)</mo></mrow></mrow><mo stretchy="false" id="S2.E1.m1.8.8.1.1.2.2.2.5" xref="S2.E1.m1.8.8.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.8.8.1.2" xref="S2.E1.m1.8.8.1.1.cmml">.</mo></mrow></math>

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0303366

Formulas:

1-

7 h^{- 1} days

2-

h^{- 1} z_{L} (1 + z_{L})

3-

⟨ 1 month ⟩ \times {⟨ image separation in arcsec ⟩}^{2}

4-

V_{G 2} = 20.4

5-

V_{G 3} = 20.0

6-

V_{G 4} = 21.6

7-

V_{G 5} = 20.3

8-

R_{G 1} = 19.0

9-

R_{G 2} = 20.0

10-

R_{G 3} = 20.5

Formulas (html):

<math><mrow id="id4.1.m1.1.1" xref="id4.1.m1.1.1.cmml"><mn id="id4.1.m1.1.1.2" xref="id4.1.m1.1.1.2.cmml">7</mn><mo id="id4.1.m1.1.1.1" xref="id4.1.m1.1.1.1.cmml">⁢</mo><mpadded width="+1.7pt" id="id4.1.m1.1.1.3" xref="id4.1.m1.1.1.3.cmml"><msup id="id4.1.m1.1.1.3a" xref="id4.1.m1.1.1.3.cmml"><mi id="id4.1.m1.1.1.3.2" xref="id4.1.m1.1.1.3.2.cmml">h</mi><mrow id="id4.1.m1.1.1.3.3" xref="id4.1.m1.1.1.3.3.cmml"><mo id="id4.1.m1.1.1.3.3.1" xref="id4.1.m1.1.1.3.3.1.cmml">-</mo><mn id="id4.1.m1.1.1.3.3.2" xref="id4.1.m1.1.1.3.3.2.cmml">1</mn></mrow></msup></mpadded><mo id="id4.1.m1.1.1.1a" xref="id4.1.m1.1.1.1.cmml">⁢</mo><mi id="id4.1.m1.1.1.4" xref="id4.1.m1.1.1.4.cmml">days</mi></mrow></math>, <math><mrow id="S1.Ex1.m3.1.1" xref="S1.Ex1.m3.1.1.cmml"><msup id="S1.Ex1.m3.1.1.3" xref="S1.Ex1.m3.1.1.3.cmml"><mi id="S1.Ex1.m3.1.1.3.2" xref="S1.Ex1.m3.1.1.3.2.cmml">h</mi><mrow id="S1.Ex1.m3.1.1.3.3" xref="S1.Ex1.m3.1.1.3.3.cmml"><mo id="S1.Ex1.m3.1.1.3.3.1" xref="S1.Ex1.m3.1.1.3.3.1.cmml">-</mo><mn id="S1.Ex1.m3.1.1.3.3.2" xref="S1.Ex1.m3.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S1.Ex1.m3.1.1.2" xref="S1.Ex1.m3.1.1.2.cmml">⁢</mo><msub id="S1.Ex1.m3.1.1.4" xref="S1.Ex1.m3.1.1.4.cmml"><mi id="S1.Ex1.m3.1.1.4.2" xref="S1.Ex1.m3.1.1.4.2.cmml">z</mi><mi mathvariant="normal" id="S1.Ex1.m3.1.1.4.3" xref="S1.Ex1.m3.1.1.4.3.cmml">L</mi></msub><mo id="S1.Ex1.m3.1.1.2a" xref="S1.Ex1.m3.1.1.2.cmml">⁢</mo><mrow id="S1.Ex1.m3.1.1.1.1" xref="S1.Ex1.m3.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.Ex1.m3.1.1.1.1.2" xref="S1.Ex1.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex1.m3.1.1.1.1.1" xref="S1.Ex1.m3.1.1.1.1.1.cmml"><mn id="S1.Ex1.m3.1.1.1.1.1.2" xref="S1.Ex1.m3.1.1.1.1.1.2.cmml">1</mn><mo id="S1.Ex1.m3.1.1.1.1.1.1" xref="S1.Ex1.m3.1.1.1.1.1.1.cmml">+</mo><msub id="S1.Ex1.m3.1.1.1.1.1.3" xref="S1.Ex1.m3.1.1.1.1.1.3.cmml"><mi id="S1.Ex1.m3.1.1.1.1.1.3.2" xref="S1.Ex1.m3.1.1.1.1.1.3.2.cmml">z</mi><mi mathvariant="normal" id="S1.Ex1.m3.1.1.1.1.1.3.3" xref="S1.Ex1.m3.1.1.1.1.1.3.3.cmml">L</mi></msub></mrow><mo stretchy="false" id="S1.Ex1.m3.1.1.1.1.3" xref="S1.Ex1.m3.1.1.1.1.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S1.E1.m3.2.3" xref="S1.E1.m3.2.3.cmml"><mrow id="S1.E1.m3.2.3.2.2" xref="S1.E1.m3.2.3.2.1.cmml"><mo stretchy="false" id="S1.E1.m3.2.3.2.2.1" xref="S1.E1.m3.2.3.2.1.1.cmml">⟨</mo><mtext id="S1.E1.m3.1.1" xref="S1.E1.m3.1.1a.cmml">1 month</mtext><mo stretchy="false" id="S1.E1.m3.2.3.2.2.2" xref="S1.E1.m3.2.3.2.1.1.cmml">⟩</mo></mrow><mo id="S1.E1.m3.2.3.1" xref="S1.E1.m3.2.3.1.cmml">×</mo><msup id="S1.E1.m3.2.3.3" xref="S1.E1.m3.2.3.3.cmml"><mrow id="S1.E1.m3.2.3.3.2.2" xref="S1.E1.m3.2.3.3.2.1.cmml"><mo stretchy="false" id="S1.E1.m3.2.3.3.2.2.1" xref="S1.E1.m3.2.3.3.2.1.1.cmml">⟨</mo><mtext id="S1.E1.m3.2.2" xref="S1.E1.m3.2.2a.cmml">image separation in arcsec</mtext><mo stretchy="false" id="S1.E1.m3.2.3.3.2.2.2" xref="S1.E1.m3.2.3.3.2.1.1.cmml">⟩</mo></mrow><mn id="S1.E1.m3.2.3.3.3" xref="S1.E1.m3.2.3.3.3.cmml">2</mn></msup></mrow></math>, <math><mrow id="S2.SS2.p1.4.m4.1.1" xref="S2.SS2.p1.4.m4.1.1.cmml"><msub id="S2.SS2.p1.4.m4.1.1.2" xref="S2.SS2.p1.4.m4.1.1.2.cmml"><mi id="S2.SS2.p1.4.m4.1.1.2.2" xref="S2.SS2.p1.4.m4.1.1.2.2.cmml">V</mi><mrow id="S2.SS2.p1.4.m4.1.1.2.3" xref="S2.SS2.p1.4.m4.1.1.2.3.cmml"><mi id="S2.SS2.p1.4.m4.1.1.2.3.2" xref="S2.SS2.p1.4.m4.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.4.m4.1.1.2.3.1" xref="S2.SS2.p1.4.m4.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.4.m4.1.1.2.3.3" xref="S2.SS2.p1.4.m4.1.1.2.3.3.cmml">2</mn></mrow></msub><mo id="S2.SS2.p1.4.m4.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.4.m4.1.1.3" xref="S2.SS2.p1.4.m4.1.1.3.cmml">20.4</mn></mrow></math>, <math><mrow id="S2.SS2.p1.5.m5.1.1" xref="S2.SS2.p1.5.m5.1.1.cmml"><msub id="S2.SS2.p1.5.m5.1.1.2" xref="S2.SS2.p1.5.m5.1.1.2.cmml"><mi id="S2.SS2.p1.5.m5.1.1.2.2" xref="S2.SS2.p1.5.m5.1.1.2.2.cmml">V</mi><mrow id="S2.SS2.p1.5.m5.1.1.2.3" xref="S2.SS2.p1.5.m5.1.1.2.3.cmml"><mi id="S2.SS2.p1.5.m5.1.1.2.3.2" xref="S2.SS2.p1.5.m5.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.5.m5.1.1.2.3.1" xref="S2.SS2.p1.5.m5.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.5.m5.1.1.2.3.3" xref="S2.SS2.p1.5.m5.1.1.2.3.3.cmml">3</mn></mrow></msub><mo id="S2.SS2.p1.5.m5.1.1.1" xref="S2.SS2.p1.5.m5.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.5.m5.1.1.3" xref="S2.SS2.p1.5.m5.1.1.3.cmml">20.0</mn></mrow></math>, <math><mrow id="S2.SS2.p1.6.m6.1.1" xref="S2.SS2.p1.6.m6.1.1.cmml"><msub id="S2.SS2.p1.6.m6.1.1.2" xref="S2.SS2.p1.6.m6.1.1.2.cmml"><mi id="S2.SS2.p1.6.m6.1.1.2.2" xref="S2.SS2.p1.6.m6.1.1.2.2.cmml">V</mi><mrow id="S2.SS2.p1.6.m6.1.1.2.3" xref="S2.SS2.p1.6.m6.1.1.2.3.cmml"><mi id="S2.SS2.p1.6.m6.1.1.2.3.2" xref="S2.SS2.p1.6.m6.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.6.m6.1.1.2.3.1" xref="S2.SS2.p1.6.m6.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.6.m6.1.1.2.3.3" xref="S2.SS2.p1.6.m6.1.1.2.3.3.cmml">4</mn></mrow></msub><mo id="S2.SS2.p1.6.m6.1.1.1" xref="S2.SS2.p1.6.m6.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.6.m6.1.1.3" xref="S2.SS2.p1.6.m6.1.1.3.cmml">21.6</mn></mrow></math>, <math><mrow id="S2.SS2.p1.7.m7.1.1" xref="S2.SS2.p1.7.m7.1.1.cmml"><msub id="S2.SS2.p1.7.m7.1.1.2" xref="S2.SS2.p1.7.m7.1.1.2.cmml"><mi id="S2.SS2.p1.7.m7.1.1.2.2" xref="S2.SS2.p1.7.m7.1.1.2.2.cmml">V</mi><mrow id="S2.SS2.p1.7.m7.1.1.2.3" xref="S2.SS2.p1.7.m7.1.1.2.3.cmml"><mi id="S2.SS2.p1.7.m7.1.1.2.3.2" xref="S2.SS2.p1.7.m7.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.7.m7.1.1.2.3.1" xref="S2.SS2.p1.7.m7.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.7.m7.1.1.2.3.3" xref="S2.SS2.p1.7.m7.1.1.2.3.3.cmml">5</mn></mrow></msub><mo id="S2.SS2.p1.7.m7.1.1.1" xref="S2.SS2.p1.7.m7.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.7.m7.1.1.3" xref="S2.SS2.p1.7.m7.1.1.3.cmml">20.3</mn></mrow></math>, <math><mrow id="S2.SS2.p1.10.m10.1.1" xref="S2.SS2.p1.10.m10.1.1.cmml"><msub id="S2.SS2.p1.10.m10.1.1.2" xref="S2.SS2.p1.10.m10.1.1.2.cmml"><mi id="S2.SS2.p1.10.m10.1.1.2.2" xref="S2.SS2.p1.10.m10.1.1.2.2.cmml">R</mi><mrow id="S2.SS2.p1.10.m10.1.1.2.3" xref="S2.SS2.p1.10.m10.1.1.2.3.cmml"><mi id="S2.SS2.p1.10.m10.1.1.2.3.2" xref="S2.SS2.p1.10.m10.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.10.m10.1.1.2.3.1" xref="S2.SS2.p1.10.m10.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.10.m10.1.1.2.3.3" xref="S2.SS2.p1.10.m10.1.1.2.3.3.cmml">1</mn></mrow></msub><mo id="S2.SS2.p1.10.m10.1.1.1" xref="S2.SS2.p1.10.m10.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.10.m10.1.1.3" xref="S2.SS2.p1.10.m10.1.1.3.cmml">19.0</mn></mrow></math>, <math><mrow id="S2.SS2.p1.11.m11.1.1" xref="S2.SS2.p1.11.m11.1.1.cmml"><msub id="S2.SS2.p1.11.m11.1.1.2" xref="S2.SS2.p1.11.m11.1.1.2.cmml"><mi id="S2.SS2.p1.11.m11.1.1.2.2" xref="S2.SS2.p1.11.m11.1.1.2.2.cmml">R</mi><mrow id="S2.SS2.p1.11.m11.1.1.2.3" xref="S2.SS2.p1.11.m11.1.1.2.3.cmml"><mi id="S2.SS2.p1.11.m11.1.1.2.3.2" xref="S2.SS2.p1.11.m11.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.11.m11.1.1.2.3.1" xref="S2.SS2.p1.11.m11.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.11.m11.1.1.2.3.3" xref="S2.SS2.p1.11.m11.1.1.2.3.3.cmml">2</mn></mrow></msub><mo id="S2.SS2.p1.11.m11.1.1.1" xref="S2.SS2.p1.11.m11.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.11.m11.1.1.3" xref="S2.SS2.p1.11.m11.1.1.3.cmml">20.0</mn></mrow></math>, <math><mrow id="S2.SS2.p1.12.m12.1.1" xref="S2.SS2.p1.12.m12.1.1.cmml"><msub id="S2.SS2.p1.12.m12.1.1.2" xref="S2.SS2.p1.12.m12.1.1.2.cmml"><mi id="S2.SS2.p1.12.m12.1.1.2.2" xref="S2.SS2.p1.12.m12.1.1.2.2.cmml">R</mi><mrow id="S2.SS2.p1.12.m12.1.1.2.3" xref="S2.SS2.p1.12.m12.1.1.2.3.cmml"><mi id="S2.SS2.p1.12.m12.1.1.2.3.2" xref="S2.SS2.p1.12.m12.1.1.2.3.2.cmml">G</mi><mo id="S2.SS2.p1.12.m12.1.1.2.3.1" xref="S2.SS2.p1.12.m12.1.1.2.3.1.cmml">⁢</mo><mn id="S2.SS2.p1.12.m12.1.1.2.3.3" xref="S2.SS2.p1.12.m12.1.1.2.3.3.cmml">3</mn></mrow></msub><mo id="S2.SS2.p1.12.m12.1.1.1" xref="S2.SS2.p1.12.m12.1.1.1.cmml">=</mo><mn id="S2.SS2.p1.12.m12.1.1.3" xref="S2.SS2.p1.12.m12.1.1.3.cmml">20.5</mn></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1509.06470

Formulas:

1-

(𝑿, 𝒚_{𝒔}, 𝒀_{𝒐})

2-

𝑿 \in ℝ^{H \times W \times C}

3-

𝒚_{𝒔} \in ℤ^{1 \times M_{s}}

4-

𝒀_{𝒐} \in ℤ^{H \times W \times M_{o}}

5-

y_{o}^{i j k} = 1

6-

(𝑿, 𝒚_{𝒔})

7-

L_{s c e n e} = - \sum_{k = 1}^{M_{s}} y_{s}^{k} l o g (p_{s}^{k})

8-

p_{s}^{k} = \frac{e^{𝒇^{T} 𝜽_{𝒌}}}{\sum_{i = 1}^{N_{s}} e^{𝒇^{T} 𝜽_{𝒊}}}

9-

p_{o}^{i j k}

10-

L_{o b j e c t} = - \sum_{i} \sum_{j} \sum_{k = 1}^{M_{o}} y_{o}^{i j k} l o g (p_{o}^{i j k})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0703477

Formulas:

1-

e t a l .

2-

e t a l .

3-

M = χ_{H > 2 T} H + M_{0}

4-

χ_{H > 2 T}

5-

χ_{H > 2 T}

6-

χ_{H > 2 T} = \partial M / \partial H_{H > 2 T} = N C / (T - Θ) + χ_{0}

7-

C = {(g μ_{B})}^{2} S (S + 1) / 3 k_{B}

8-

Θ = - 16.3 \pm 3.0

9-

\times 10^{15} μ_{B} / T

10-

(p d σ) = - 1.3 \pm 0.2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1805.03238

Formulas:

1-

𝔽_{q_{1}} \oplus \dots \oplus 𝔽_{q_{r}}

2-

𝐚 = {(a_{n})}_{n = 0}^{\infty}

3-

0 \leq i \leq k - 1

4-

a_{n + k} = \sum_{i = 0}^{k - 1} c_{i} a_{n + i}

5-

(c_{0}, \dots, c_{k - 1}) \in R^{k}

6-

0, 1, 1, 2, 3, 5, 8, 13, \dots

7-

a_{n + 2} = a_{n + 1} + a_{n}

8-

c_{0} = c_{1} = 1

9-

s_{n} = (a_{n}, \dots, a_{n + k - 1}) \in R^{k}

10-

s_{0} = (a_{0}, \dots, a_{k - 1})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1306.3940

Formulas:

1-

a_{eff} < 3 μ m

2-

a_{eff} \sim 10 μ m

3-

\sim 6 μ m

4-

f (a) = \frac{{(a_{eff} ν)}^{2 - 1 / ν}}{Γ (\frac{1 - 2 ν}{ν})} a^{(\frac{1}{ν} - 3)} e^{- \frac{a}{a_{eff} ν}}

5-

a_{eff} = 0.1 μ m

6-

a_{eff} = 1 μ m

7-

a_{eff} = 20.0 μ m

8-

a_{eff} = 200 μ m

9-

λ = 0.1 μ m

10-

a_{eff} = 200 μ m

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1003.0566

Formulas:

1-

R < {0.1, 0.3, 1}

2-

{5, 30, 60} %

3-

Γ \sim < 10^{2.5}

4-

d N_{γ} / d E \propto E^{α}

5-

d N_{γ} / d E \propto E^{β}

6-

< α > ≃ - 0.8

7-

< β > ≃ - 2.3

8-

< E_{p} > ≃ 200

9-

P_{\lim, GBM}^{(50 - 300) keV} \sim

10-

P_{\lim, BATSE}^{(50 - 300) keV} \sim

Formulas (html):

<math><mrow id="id8.7.m4.3.4" xref="id8.7.m4.3.4.cmml"><mi id="id8.7.m4.3.4.2" xref="id8.7.m4.3.4.2.cmml">R</mi><mo id="id8.7.m4.3.4.1" xref="id8.7.m4.3.4.1.cmml"><</mo><mrow id="id8.7.m4.3.4.3.2" xref="id8.7.m4.3.4.3.1.cmml"><mo stretchy="false" id="id8.7.m4.3.4.3.2.1" xref="id8.7.m4.3.4.3.1.cmml">{</mo><mn id="id8.7.m4.1.1" xref="id8.7.m4.1.1.cmml">0.1</mn><mo id="id8.7.m4.3.4.3.2.2" xref="id8.7.m4.3.4.3.1.cmml">,</mo><mn id="id8.7.m4.2.2" xref="id8.7.m4.2.2.cmml">0.3</mn><mo id="id8.7.m4.3.4.3.2.3" xref="id8.7.m4.3.4.3.1.cmml">,</mo><mn id="id8.7.m4.3.3" xref="id8.7.m4.3.3.cmml">1</mn><mo stretchy="false" id="id8.7.m4.3.4.3.2.4" xref="id8.7.m4.3.4.3.1.cmml">}</mo></mrow></mrow></math>, <math><mrow id="id9.8.m5.3.4" xref="id9.8.m5.3.4.cmml"><mrow id="id9.8.m5.3.4.2.2" xref="id9.8.m5.3.4.2.1.cmml"><mo stretchy="false" id="id9.8.m5.3.4.2.2.1" xref="id9.8.m5.3.4.2.1.cmml">{</mo><mn id="id9.8.m5.1.1" xref="id9.8.m5.1.1.cmml">5</mn><mo id="id9.8.m5.3.4.2.2.2" xref="id9.8.m5.3.4.2.1.cmml">,</mo><mn id="id9.8.m5.2.2" xref="id9.8.m5.2.2.cmml">30</mn><mo id="id9.8.m5.3.4.2.2.3" xref="id9.8.m5.3.4.2.1.cmml">,</mo><mn id="id9.8.m5.3.3" xref="id9.8.m5.3.3.cmml">60</mn><mo stretchy="false" id="id9.8.m5.3.4.2.2.4" xref="id9.8.m5.3.4.2.1.cmml">}</mo></mrow><mo id="id9.8.m5.3.4.1" xref="id9.8.m5.3.4.1.cmml">%</mo></mrow></math>, <math><mrow id="id19.18.m2.2.3" xref="id19.18.m2.2.3.cmml"><mi mathvariant="normal" id="id19.18.m2.2.3.2" xref="id19.18.m2.2.3.2.cmml">Γ</mi><mo id="id19.18.m2.2.3.1" xref="id19.18.m2.2.3.1.cmml">⁢</mo><mrow id="id19.18.m2.2.2" xref="id19.18.m2.2.2c.cmml"><mpadded depth="+3.2pt" height="-3.2pt" voffset="-3.2pt" id="id19.18.m2.2.2a" xref="id19.18.m2.2.2c.cmml"><mo id="id19.18.m2.1.1.1.1.m1.1.1" xref="id19.18.m2.1.1.1.1.m1.1.1.cmml">∼</mo></mpadded><mpadded depth="-1.2pt" height="+1.2pt" voffset="1.2pt" width="0.0pt" id="id19.18.m2.2.2b" xref="id19.18.m2.2.2c.cmml"><mo id="id19.18.m2.2.2.2.1.1.m1.1.1" xref="id19.18.m2.2.2.2.1.1.m1.1.1.cmml"><</mo></mpadded></mrow><mo id="id19.18.m2.2.3.1a" xref="id19.18.m2.2.3.1.cmml">⁢</mo><msup id="id19.18.m2.2.3.3" xref="id19.18.m2.2.3.3.cmml"><mn id="id19.18.m2.2.3.3.2" xref="id19.18.m2.2.3.3.2.cmml">10</mn><mn id="id19.18.m2.2.3.3.3" xref="id19.18.m2.2.3.3.3.cmml">2.5</mn></msup></mrow></math>, <math><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mrow id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml"><mrow id="S2.SS1.p3.1.m1.1.1.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.cmml"><mrow id="S2.SS1.p3.1.m1.1.1.2.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.2.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2.2.2.2" xref="S2.SS1.p3.1.m1.1.1.2.2.2.2.cmml">d</mi><mo id="S2.SS1.p3.1.m1.1.1.2.2.2.1" xref="S2.SS1.p3.1.m1.1.1.2.2.2.1.cmml">⁢</mo><msub id="S2.SS1.p3.1.m1.1.1.2.2.2.3" xref="S2.SS1.p3.1.m1.1.1.2.2.2.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.2.2.2.3.2" xref="S2.SS1.p3.1.m1.1.1.2.2.2.3.2.cmml">N</mi><mi id="S2.SS1.p3.1.m1.1.1.2.2.2.3.3" xref="S2.SS1.p3.1.m1.1.1.2.2.2.3.3.cmml">γ</mi></msub></mrow><mo id="S2.SS1.p3.1.m1.1.1.2.2.1" xref="S2.SS1.p3.1.m1.1.1.2.2.1.cmml">/</mo><mi id="S2.SS1.p3.1.m1.1.1.2.2.3" xref="S2.SS1.p3.1.m1.1.1.2.2.3.cmml">d</mi></mrow><mo id="S2.SS1.p3.1.m1.1.1.2.1" xref="S2.SS1.p3.1.m1.1.1.2.1.cmml">⁢</mo><mi id="S2.SS1.p3.1.m1.1.1.2.3" xref="S2.SS1.p3.1.m1.1.1.2.3.cmml">E</mi></mrow><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">∝</mo><msup id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml">E</mi><mi id="S2.SS1.p3.1.m1.1.1.3.3" xref="S2.SS1.p3.1.m1.1.1.3.3.cmml">α</mi></msup></mrow></math>, <math><mrow id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml"><mrow id="S2.SS1.p3.3.m3.1.1.2" xref="S2.SS1.p3.3.m3.1.1.2.cmml"><mrow id="S2.SS1.p3.3.m3.1.1.2.2" xref="S2.SS1.p3.3.m3.1.1.2.2.cmml"><mrow id="S2.SS1.p3.3.m3.1.1.2.2.2" xref="S2.SS1.p3.3.m3.1.1.2.2.2.cmml"><mi id="S2.SS1.p3.3.m3.1.1.2.2.2.2" xref="S2.SS1.p3.3.m3.1.1.2.2.2.2.cmml">d</mi><mo id="S2.SS1.p3.3.m3.1.1.2.2.2.1" xref="S2.SS1.p3.3.m3.1.1.2.2.2.1.cmml">⁢</mo><msub id="S2.SS1.p3.3.m3.1.1.2.2.2.3" xref="S2.SS1.p3.3.m3.1.1.2.2.2.3.cmml"><mi id="S2.SS1.p3.3.m3.1.1.2.2.2.3.2" xref="S2.SS1.p3.3.m3.1.1.2.2.2.3.2.cmml">N</mi><mi id="S2.SS1.p3.3.m3.1.1.2.2.2.3.3" xref="S2.SS1.p3.3.m3.1.1.2.2.2.3.3.cmml">γ</mi></msub></mrow><mo id="S2.SS1.p3.3.m3.1.1.2.2.1" xref="S2.SS1.p3.3.m3.1.1.2.2.1.cmml">/</mo><mi id="S2.SS1.p3.3.m3.1.1.2.2.3" xref="S2.SS1.p3.3.m3.1.1.2.2.3.cmml">d</mi></mrow><mo id="S2.SS1.p3.3.m3.1.1.2.1" xref="S2.SS1.p3.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S2.SS1.p3.3.m3.1.1.2.3" xref="S2.SS1.p3.3.m3.1.1.2.3.cmml">E</mi></mrow><mo id="S2.SS1.p3.3.m3.1.1.1" xref="S2.SS1.p3.3.m3.1.1.1.cmml">∝</mo><msup id="S2.SS1.p3.3.m3.1.1.3" xref="S2.SS1.p3.3.m3.1.1.3.cmml"><mi id="S2.SS1.p3.3.m3.1.1.3.2" xref="S2.SS1.p3.3.m3.1.1.3.2.cmml">E</mi><mi id="S2.SS1.p3.3.m3.1.1.3.3" xref="S2.SS1.p3.3.m3.1.1.3.3.cmml">β</mi></msup></mrow></math>, <math><mrow id="S2.SS1.p3.5.m5.1.2" xref="S2.SS1.p3.5.m5.1.2.cmml"><mrow id="S2.SS1.p3.5.m5.1.2.2.2" xref="S2.SS1.p3.5.m5.1.2.2.1.cmml"><mo id="S2.SS1.p3.5.m5.1.2.2.2.1" xref="S2.SS1.p3.5.m5.1.2.2.1.1.cmml"><</mo><mi id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml">α</mi><mo id="S2.SS1.p3.5.m5.1.2.2.2.2" xref="S2.SS1.p3.5.m5.1.2.2.1.1.cmml">></mo></mrow><mo id="S2.SS1.p3.5.m5.1.2.1" xref="S2.SS1.p3.5.m5.1.2.1.cmml">≃</mo><mrow id="S2.SS1.p3.5.m5.1.2.3" xref="S2.SS1.p3.5.m5.1.2.3.cmml"><mo id="S2.SS1.p3.5.m5.1.2.3.1" xref="S2.SS1.p3.5.m5.1.2.3.1.cmml">-</mo><mn id="S2.SS1.p3.5.m5.1.2.3.2" xref="S2.SS1.p3.5.m5.1.2.3.2.cmml">0.8</mn></mrow></mrow></math>, <math><mrow id="S2.SS1.p3.6.m6.1.2" xref="S2.SS1.p3.6.m6.1.2.cmml"><mrow id="S2.SS1.p3.6.m6.1.2.2.2" xref="S2.SS1.p3.6.m6.1.2.2.1.cmml"><mo id="S2.SS1.p3.6.m6.1.2.2.2.1" xref="S2.SS1.p3.6.m6.1.2.2.1.1.cmml"><</mo><mi id="S2.SS1.p3.6.m6.1.1" xref="S2.SS1.p3.6.m6.1.1.cmml">β</mi><mo id="S2.SS1.p3.6.m6.1.2.2.2.2" xref="S2.SS1.p3.6.m6.1.2.2.1.1.cmml">></mo></mrow><mo id="S2.SS1.p3.6.m6.1.2.1" xref="S2.SS1.p3.6.m6.1.2.1.cmml">≃</mo><mrow id="S2.SS1.p3.6.m6.1.2.3" xref="S2.SS1.p3.6.m6.1.2.3.cmml"><mo id="S2.SS1.p3.6.m6.1.2.3.1" xref="S2.SS1.p3.6.m6.1.2.3.1.cmml">-</mo><mn id="S2.SS1.p3.6.m6.1.2.3.2" xref="S2.SS1.p3.6.m6.1.2.3.2.cmml">2.3</mn></mrow></mrow></math>, <math><mrow id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml"><mrow id="S2.SS1.p3.7.m7.1.1.1.1" xref="S2.SS1.p3.7.m7.1.1.1.2.cmml"><mo id="S2.SS1.p3.7.m7.1.1.1.1.2" xref="S2.SS1.p3.7.m7.1.1.1.2.1.cmml"><</mo><msub id="S2.SS1.p3.7.m7.1.1.1.1.1" xref="S2.SS1.p3.7.m7.1.1.1.1.1.cmml"><mi id="S2.SS1.p3.7.m7.1.1.1.1.1.2" xref="S2.SS1.p3.7.m7.1.1.1.1.1.2.cmml">E</mi><mi id="S2.SS1.p3.7.m7.1.1.1.1.1.3" xref="S2.SS1.p3.7.m7.1.1.1.1.1.3.cmml">p</mi></msub><mo id="S2.SS1.p3.7.m7.1.1.1.1.3" xref="S2.SS1.p3.7.m7.1.1.1.2.1.cmml">></mo></mrow><mo id="S2.SS1.p3.7.m7.1.1.2" xref="S2.SS1.p3.7.m7.1.1.2.cmml">≃</mo><mn id="S2.SS1.p3.7.m7.1.1.3" xref="S2.SS1.p3.7.m7.1.1.3.cmml">200</mn></mrow></math>, <math><mrow id="S2.SS1.p3.8.m8.3.4" xref="S2.SS1.p3.8.m8.3.4.cmml"><msubsup id="S2.SS1.p3.8.m8.3.4.2" xref="S2.SS1.p3.8.m8.3.4.2.cmml"><mi id="S2.SS1.p3.8.m8.3.4.2.2.2" xref="S2.SS1.p3.8.m8.3.4.2.2.2.cmml">P</mi><mrow id="S2.SS1.p3.8.m8.3.3.2.4" xref="S2.SS1.p3.8.m8.3.3.2.3.cmml"><mi id="S2.SS1.p3.8.m8.2.2.1.1" xref="S2.SS1.p3.8.m8.2.2.1.1.cmml">lim</mi><mo id="S2.SS1.p3.8.m8.3.3.2.4.1" xref="S2.SS1.p3.8.m8.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p3.8.m8.3.3.2.2" xref="S2.SS1.p3.8.m8.3.3.2.2.cmml">GBM</mi></mrow><mrow id="S2.SS1.p3.8.m8.1.1.1" xref="S2.SS1.p3.8.m8.1.1.1.cmml"><mrow id="S2.SS1.p3.8.m8.1.1.1.1.1" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p3.8.m8.1.1.1.1.1.2" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p3.8.m8.1.1.1.1.1.1" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.cmml"><mn id="S2.SS1.p3.8.m8.1.1.1.1.1.1.2" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.2.cmml">50</mn><mo id="S2.SS1.p3.8.m8.1.1.1.1.1.1.1" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.1.cmml">-</mo><mn id="S2.SS1.p3.8.m8.1.1.1.1.1.1.3" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.3.cmml">300</mn></mrow><mo stretchy="false" id="S2.SS1.p3.8.m8.1.1.1.1.1.3" xref="S2.SS1.p3.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p3.8.m8.1.1.1.2" xref="S2.SS1.p3.8.m8.1.1.1.2.cmml">⁢</mo><mi id="S2.SS1.p3.8.m8.1.1.1.3" xref="S2.SS1.p3.8.m8.1.1.1.3.cmml">keV</mi></mrow></msubsup><mo id="S2.SS1.p3.8.m8.3.4.1" xref="S2.SS1.p3.8.m8.3.4.1.cmml">∼</mo><mi id="S2.SS1.p3.8.m8.3.4.3" xref="S2.SS1.p3.8.m8.3.4.3.cmml"/></mrow></math>, <math><mrow id="S2.SS1.p4.2.m2.3.4" xref="S2.SS1.p4.2.m2.3.4.cmml"><msubsup id="S2.SS1.p4.2.m2.3.4.2" xref="S2.SS1.p4.2.m2.3.4.2.cmml"><mi id="S2.SS1.p4.2.m2.3.4.2.2.2" xref="S2.SS1.p4.2.m2.3.4.2.2.2.cmml">P</mi><mrow id="S2.SS1.p4.2.m2.3.3.2.4" xref="S2.SS1.p4.2.m2.3.3.2.3.cmml"><mi id="S2.SS1.p4.2.m2.2.2.1.1" xref="S2.SS1.p4.2.m2.2.2.1.1.cmml">lim</mi><mo id="S2.SS1.p4.2.m2.3.3.2.4.1" xref="S2.SS1.p4.2.m2.3.3.2.3.cmml">,</mo><mi id="S2.SS1.p4.2.m2.3.3.2.2" xref="S2.SS1.p4.2.m2.3.3.2.2.cmml">BATSE</mi></mrow><mrow id="S2.SS1.p4.2.m2.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.cmml"><mrow id="S2.SS1.p4.2.m2.1.1.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p4.2.m2.1.1.1.1.1.2" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.p4.2.m2.1.1.1.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml"><mn id="S2.SS1.p4.2.m2.1.1.1.1.1.1.2" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.2.cmml">50</mn><mo id="S2.SS1.p4.2.m2.1.1.1.1.1.1.1" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.1.cmml">-</mo><mn id="S2.SS1.p4.2.m2.1.1.1.1.1.1.3" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.3.cmml">300</mn></mrow><mo stretchy="false" id="S2.SS1.p4.2.m2.1.1.1.1.1.3" xref="S2.SS1.p4.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p4.2.m2.1.1.1.2" xref="S2.SS1.p4.2.m2.1.1.1.2.cmml">⁢</mo><mi id="S2.SS1.p4.2.m2.1.1.1.3" xref="S2.SS1.p4.2.m2.1.1.1.3.cmml">keV</mi></mrow></msubsup><mo id="S2.SS1.p4.2.m2.3.4.1" xref="S2.SS1.p4.2.m2.3.4.1.cmml">∼</mo><mi id="S2.SS1.p4.2.m2.3.4.3" xref="S2.SS1.p4.2.m2.3.4.3.cmml"/></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1108.2699

Formulas:

1-

ρ_{S} (t)

2-

ρ_{S}^{'} (t)

3-

ρ^{'} = (Φ \otimes I) ρ

4-

ρ_{S} (t)

5-

ρ_{S}^{'} (t)

6-

ρ_{S} = {Tr}_{E} ρ

7-

Φ (A) = \sum_{μ} π_{μ} A π_{μ} .

8-

ρ ⟶ ρ^{'} = (Φ \otimes I) ρ = \sum_{μ} Π_{μ} ρ Π_{μ},

9-

Π_{μ} = π_{μ} \otimes I

10-

ρ_{S}^{'} = {Tr}_{E} ρ^{'} = ρ_{S}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/2009.07003

Formulas:

1-

Ha \in [27.5, 40]

2-

m \in {2, 3, 4}

3-

Re = Ω r_{i}^{2} / ν

4-

Ha = B_{0} r_{i} / μ_{0} ρ ν η^{1 / 2}

5-

𝐁_{0} = B_{0} \cos (θ) {\hat{e}}_{r} - B_{0} \sin (θ) {\hat{e}}_{θ}

6-

\partial_{t} 𝐯 + 𝐯 \cdot \nabla v = - \nabla p + {Re}^{- 1} \nabla^{2} 𝐯 + {Ha}^{2} {Re}^{- 1} (\nabla \times 𝐛) \times {\hat{𝐞}}_{z}

7-

0 = \nabla^{2} 𝐛 + \nabla \times (𝐯 \times {\hat{𝐞}}_{z}) .

8-

r_{i} / r_{o} = 0.5

9-

Ha \in [0, 80]

10-

Ha \in [0, 80]

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-lat/9707028

Formulas:

1-

Z_{M, N} = \sum_{q_{1}, \dots, q_{M}} p (q_{1}) \dots p (q_{M}) δ_{q_{1} + \dots + q_{M}, N}

2-

Z_{M, N} \sim \exp (M f (ρ) + \dots)

3-

Z_{M, N} \sim \exp (N f (r) + \dots)

4-

1 / N \leq r \leq 1

5-

K (μ) = - \log \sum_{q} p (q) e^{- μ q}

6-

K^{'} (μ_{c r})

7-

p (q) = q^{- α} e^{- μ_{c r} q}

8-

f (r) = {\begin{matrix} - r K (μ_{*} (r)) + μ_{*} (r) & for & r > r_{c r} \\ - r κ_{c r} + μ_{c r} & for & r \leq r_{c r} \end{matrix}

9-

μ_{*} (r)

10-

\frac{1}{r} = K^{'} (μ_{*})

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/9606447

Formulas:

1-

S O (10)

2-

S U {(2)}_{L} \times S U {(2)}_{R} \times S U {(4)}_{c} \equiv G_{224}

3-

{[S U (3)]}^{3}

4-

λ_{R, /} = M_{B - L} / M_{G U T}

5-

S O (10)

6-

{[S U (3)]}^{3}

7-

R = {(- 1)}^{3 (B - L) + 2 S}

8-

S O (10)

9-

{[S U (3)]}^{3}

10-

{[S U (3)]}^{3}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1902.00125

Formulas:

1-

L (I, c, l, s) = L_{c o n f} (I, c) + α L_{l o c} (I, l) + β L_{s e g} (I, s)

2-

L_{s e g} (I, s)

3-

L_{s e g} (I, s) = \frac{1}{M} \sum || s_{(x, y)} - {\hat{s}}_{(x, y)} ||

4-

L_{c o n f}

5-

L_{c o n f} (I, c) = η_{\hat{s} c} {(1 - \hat{c})}^{γ} l o g (\hat{c})

6-

η_{\hat{s} c} = \sqrt{\frac{S_{a l l}}{\hat{S_{c}}}}

7-

S_{a l l}

8-

L_{l o c} (I, l)

9-

L_{l o c} (I, l) = \frac{1}{N} \sum_{i = 1}^{N} \sum_{p \in (x, y, w, h)} s m o o t h_{L 1} (l_{i}^{p} - {\hat{l}}_{i}^{p})

10-

s m o o t h_{L 1} = {\begin{matrix} 0.5 x^{2} & | x | < 1 \\ | x | - 0.5 & | x | \geq 1 \end{matrix}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1111.3628

Formulas:

1-

P_{i} = 4.21 \cdot 10^{- 18}

2-

ν = 0.4, μ = 0.55

3-

ν = 0.4, μ = 0.41

4-

Y = 2 \cdot 10^{8}

5-

MPF (D, t) = MPF (D, 1 y r) \cdot \frac{N_{1} (t)}{N_{1} (1 y r)}

6-

N_{1} (t)

7-

N_{1} (t) = a (e^{b t} - 1) + c t

8-

a = 1.23 \times 10^{- 15}

9-

c = 1.30 \times 10^{- 3}

10-

Y \sim 3 \cdot 10^{4}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1511.06614

Formulas:

1-

L_{X} \approx 10^{34} - 10^{36}

2-

k T = 0.1 - 0.5

3-

B = 10^{14} - 10^{15}

4-

Δ ϕ = 0.5 - 2.0

5-

B = 3 \times 10^{14}

6-

B = 3 \times 10^{14}

7-

F_{P} \sim 5.1 \times 10^{- 11}

8-

F_{S} \sim 3.6 \times 10^{- 11}

9-

(1.56 \pm 0.05) \times 10^{22}

10-

(1.17 \pm 0.02) \times 10^{22}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0309204

Formulas:

1-

𝐯_{sb} = 𝛀 \times 𝐫

2-

Ω \approx 0.97 ω_{⟂}

3-

L = \frac{1}{2} i ℏ \int 𝑑 V (Ψ^{*} \frac{\partial Ψ}{\partial t} - Ψ \frac{\partial Ψ^{*}}{\partial t}) - E [Ψ],

4-

ω_{⟂} / 2 π = 8.3

5-

ω_{z} / 2 π = 5.4

6-

d_{⟂} \approx 3.73 μ

7-

d_{z} \approx 4.62 μ

8-

R_{z} (0) / R_{⟂} (0) = ω_{⟂} / ω_{z} \approx 1.54

9-

\frac{R_{⟂} (\bar{Ω})}{R_{⟂} (0)} = {(1 - {\bar{Ω}}^{2})}^{- 3 / 10}, and \frac{R_{z} (\bar{Ω})}{R_{z} (0)} = {(1 - {\bar{Ω}}^{2})}^{1 / 5},

10-

\bar{Ω} = Ω / ω_{⟂}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1210.5645

Formulas:

1-

ρ (t) = \sum_{i j} (E_{i}^{(1)} \otimes E_{j}^{(2)}) ρ (0) (E_{i}^{(1), †} \otimes E_{j}^{(2), †}) = Λ_{t} (ρ (0)),

2-

E_{i}^{(a)} = E_{i}^{(a)} (t)

3-

\sum_{i} E_{i}^{(a), †} E_{i}^{(a)} = 𝕀^{(a)}

4-

E_{0} (q) = \sqrt{1 - 3 q / 4} 𝕀

5-

E_{i} (q) = \sqrt{q / 4} σ_{i}

6-

i = 1, 2, 3,

7-

E_{0} = | 0 ⟩ ⟨ 0 | + \sqrt{1 - q} | 1 ⟩ ⟨ 1 |

8-

E_{1} = \sqrt{q} | 0 ⟩ ⟨ 1 |

9-

E_{0} = \sqrt{1 - q} 𝕀

10-

E_{1} = \sqrt{q} | 0 ⟩ ⟨ 0 |

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/0709.0468

Formulas:

1-

ν i_{13 / 2}

2-

ν h_{9 / 2}

3-

ℓ = 5 and 6

4-

0_{core}^{+} \otimes ν i_{13 / 2}

5-

3_{core}^{-} \otimes ν f_{7 / 2}

6-

0_{core}^{+} \otimes ν h_{9 / 2}

7-

2_{core}^{+} \otimes ν f_{7 / 2}

8-

π h_{11 / 2} - π g_{7 / 2}

9-

π 1 g_{7 / 2}

10-

π 2 d_{5 / 2}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9807351

Formulas:

1-

G_{N S} = \frac{4 e^{2}}{h} \sum_{n} \frac{T_{n}^{2}}{{(2 - T_{n})}^{2}},

2-

G_{N S} = \frac{4 e^{2}}{h} \frac{T_{Q D}^{2}}{{(2 - T_{Q D})}^{2}} .

3-

G_{N} = \frac{2 e^{2}}{h} T_{Q D} .

4-

U \sim e^{2} / C

5-

Im Σ (ε_{F}) = - Γ / 2,

6-

Γ = Γ_{L} + Γ_{R}

7-

⟨ n ⟩ = \frac{2}{π} Im [\log G^{r} (ε_{F})] .

8-

T_{Q D} = Γ_{L} Γ_{R} {| G^{r} (ε_{F}) |}^{2},

9-

T_{Q D} = \frac{4 Γ_{L} Γ_{R}}{Γ^{2}} \sin^{2} φ

10-

φ = π ⟨ n ⟩ / 2

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0801.0420

Formulas:

1-

V (x, t) = \frac{V_{0} (t)}{\cosh^{2} [(x - x_{c} (t)) / L]},

2-

V_{0} (t) = A_{0} [1 + κ \sin (ω t + Δ)],

3-

x_{c} (t) = x_{0} \sin (ω t) .

4-

v_{c} (t) \equiv {\dot{x}}_{c} (t) = ω x_{0} \cos (ω t) .

5-

L = x_{0} / 2

6-

r (k_{F}, V_{0}) = \frac{Γ (1 + ν - i k_{F} L) Γ (- ν - i k_{F} L) Γ (i k_{F} L)}{Γ (1 + ν) Γ (- ν) Γ (- i k_{F} L)},

7-

t (k_{F}, V_{0}) = \frac{Γ (1 + ν - i k_{F} L) Γ (- ν - i k_{F} L)}{Γ (1 - i k_{F} L) Γ (- i k_{F} L)},

8-

ν = \frac{1}{2} [- 1 + \sqrt{1 - 8 V_{0} L^{2}}] .

9-

x_{c} (t)

10-

k = \pm k_{F} - v_{c}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1703.01883

Formulas:

1-

w, h = \frac{f_{x, y} R}{Z},

2-

x_{m i n}

3-

x_{m a x}

4-

x_{s r c} = x / (w - 1) * (x_{m a x} - x_{m i n})

5-

x_{1} = ⌊ x_{s r c} ⌋

6-

x_{2} = x_{1} + 1

7-

λ = x_{2} - x_{s r c}

8-

r o w_{o u t} [x] = r o w [x_{1}] * λ + r o w [x_{2}] * (1 - λ)

9-

r o w_{o u t} [x] = r o w [x_{1}]

10-

[- \infty, + \infty] \to [- 1, + 1]

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1408.6216

Formulas:

1-

l (γ) / 2

2-

γ : S^{1} \to M

3-

l (γ) / k

4-

d (γ (t), γ (t + 2 π / k)) = l (γ) / k \forall t \in S^{1} .

5-

l (γ) / 2

6-

L \leq 2 diam (M)

7-

γ : (- ϵ, ϵ) \to X_{n}

8-

a \in (- ϵ, 0)

9-

b \in (0, ϵ)

10-

γ (a), γ (0), γ (b) \in X_{n}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1802.09967

Formulas:

1-

Υ_{disc}^{m a x} < 0.7 M_{⊙} / L_{⊙}

2-

ℱ (R) =

3-

\frac{V_{obs} (R) - V_{bar} (R)}{δ V_{obs} (R)} \cdot w (R) .

4-

V_{bar}^{2} (R) = Υ_{disc} V_{disc}^{2} + Υ_{bul} V_{bulge}^{2} + V_{gas}^{2},

5-

f \cdot w (R) \cdot ℱ (R)

6-

Γ_{disc} = 0.88 \pm 0.02

7-

Γ_{disc} = 4.07 \pm 0.17

8-

Γ_{disc} = 1.47 \pm .02

9-

Γ_{disc} = 1.31 \pm 0.08

10-

Γ_{disc} = 0.70 \pm 0.01

Formulas (html):

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id="id8.1.m1.1.1.3.2.1" xref="id8.1.m1.1.1.3.2.1.cmml">⁢</mo><msub id="id8.1.m1.1.1.3.2.3" xref="id8.1.m1.1.1.3.2.3.cmml"><mi mathvariant="normal" id="id8.1.m1.1.1.3.2.3.2" xref="id8.1.m1.1.1.3.2.3.2.cmml">M</mi><mo id="id8.1.m1.1.1.3.2.3.3" xref="id8.1.m1.1.1.3.2.3.3.cmml">⊙</mo></msub></mrow><mo id="id8.1.m1.1.1.3.1" xref="id8.1.m1.1.1.3.1.cmml">/</mo><msub id="id8.1.m1.1.1.3.3" xref="id8.1.m1.1.1.3.3.cmml"><mi mathvariant="normal" id="id8.1.m1.1.1.3.3.2" xref="id8.1.m1.1.1.3.3.2.cmml">L</mi><mo id="id8.1.m1.1.1.3.3.3" xref="id8.1.m1.1.1.3.3.3.cmml">⊙</mo></msub></mrow></mrow></math>, <math><mrow id="S2.E1.m1.1.2" xref="S2.E1.m1.1.2.cmml"><mrow id="S2.E1.m1.1.2.2" xref="S2.E1.m1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.1.2.2.2" xref="S2.E1.m1.1.2.2.2.cmml">ℱ</mi><mo id="S2.E1.m1.1.2.2.1" xref="S2.E1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.1.2.2.3.2" xref="S2.E1.m1.1.2.2.cmml"><mo stretchy="false" id="S2.E1.m1.1.2.2.3.2.1" xref="S2.E1.m1.1.2.2.cmml">(</mo><mi 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xref="S2.E2.m1.2.2.1.1.2.cmml">(</mo><mi id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml">R</mi><mo stretchy="false" id="S2.E2.m1.2.2.1.1.2.3.2.2" xref="S2.E2.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.E2.m1.2.2.1.1.1" xref="S2.E2.m1.2.2.1.1.1.cmml">=</mo><mrow id="S2.E2.m1.2.2.1.1.3" xref="S2.E2.m1.2.2.1.1.3.cmml"><mrow id="S2.E2.m1.2.2.1.1.3.2" xref="S2.E2.m1.2.2.1.1.3.2.cmml"><msub id="S2.E2.m1.2.2.1.1.3.2.2" xref="S2.E2.m1.2.2.1.1.3.2.2.cmml"><mi mathvariant="normal" id="S2.E2.m1.2.2.1.1.3.2.2.2" xref="S2.E2.m1.2.2.1.1.3.2.2.2.cmml">Υ</mi><mi id="S2.E2.m1.2.2.1.1.3.2.2.3" xref="S2.E2.m1.2.2.1.1.3.2.2.3.cmml">disc</mi></msub><mo id="S2.E2.m1.2.2.1.1.3.2.1" xref="S2.E2.m1.2.2.1.1.3.2.1.cmml">⁢</mo><msubsup id="S2.E2.m1.2.2.1.1.3.2.3" xref="S2.E2.m1.2.2.1.1.3.2.3.cmml"><mi id="S2.E2.m1.2.2.1.1.3.2.3.2.2" xref="S2.E2.m1.2.2.1.1.3.2.3.2.2.cmml">V</mi><mi id="S2.E2.m1.2.2.1.1.3.2.3.2.3" xref="S2.E2.m1.2.2.1.1.3.2.3.2.3.cmml">disc</mi><mn id="S2.E2.m1.2.2.1.1.3.2.3.3" xref="S2.E2.m1.2.2.1.1.3.2.3.3.cmml">2</mn></msubsup></mrow><mo id="S2.E2.m1.2.2.1.1.3.1" xref="S2.E2.m1.2.2.1.1.3.1.cmml">+</mo><mrow id="S2.E2.m1.2.2.1.1.3.3" xref="S2.E2.m1.2.2.1.1.3.3.cmml"><msub id="S2.E2.m1.2.2.1.1.3.3.2" xref="S2.E2.m1.2.2.1.1.3.3.2.cmml"><mi mathvariant="normal" id="S2.E2.m1.2.2.1.1.3.3.2.2" xref="S2.E2.m1.2.2.1.1.3.3.2.2.cmml">Υ</mi><mi id="S2.E2.m1.2.2.1.1.3.3.2.3" xref="S2.E2.m1.2.2.1.1.3.3.2.3.cmml">bul</mi></msub><mo id="S2.E2.m1.2.2.1.1.3.3.1" xref="S2.E2.m1.2.2.1.1.3.3.1.cmml">⁢</mo><msubsup id="S2.E2.m1.2.2.1.1.3.3.3" xref="S2.E2.m1.2.2.1.1.3.3.3.cmml"><mi id="S2.E2.m1.2.2.1.1.3.3.3.2.2" xref="S2.E2.m1.2.2.1.1.3.3.3.2.2.cmml">V</mi><mi id="S2.E2.m1.2.2.1.1.3.3.3.2.3" xref="S2.E2.m1.2.2.1.1.3.3.3.2.3.cmml">bulge</mi><mn id="S2.E2.m1.2.2.1.1.3.3.3.3" xref="S2.E2.m1.2.2.1.1.3.3.3.3.cmml">2</mn></msubsup></mrow><mo id="S2.E2.m1.2.2.1.1.3.1a" xref="S2.E2.m1.2.2.1.1.3.1.cmml">+</mo><msubsup id="S2.E2.m1.2.2.1.1.3.4" xref="S2.E2.m1.2.2.1.1.3.4.cmml"><mi id="S2.E2.m1.2.2.1.1.3.4.2.2" xref="S2.E2.m1.2.2.1.1.3.4.2.2.cmml">V</mi><mi id="S2.E2.m1.2.2.1.1.3.4.2.3" xref="S2.E2.m1.2.2.1.1.3.4.2.3.cmml">gas</mi><mn id="S2.E2.m1.2.2.1.1.3.4.3" xref="S2.E2.m1.2.2.1.1.3.4.3.cmml">2</mn></msubsup></mrow></mrow><mo id="S2.E2.m1.2.2.1.2" xref="S2.E2.m1.2.2.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS1.p3.2.m2.2.3" xref="S2.SS1.p3.2.m2.2.3.cmml"><mrow id="S2.SS1.p3.2.m2.2.3.2" xref="S2.SS1.p3.2.m2.2.3.2.cmml"><mrow id="S2.SS1.p3.2.m2.2.3.2.2" xref="S2.SS1.p3.2.m2.2.3.2.2.cmml"><mrow id="S2.SS1.p3.2.m2.2.3.2.2.2" xref="S2.SS1.p3.2.m2.2.3.2.2.2.cmml"><mi id="S2.SS1.p3.2.m2.2.3.2.2.2.2" xref="S2.SS1.p3.2.m2.2.3.2.2.2.2.cmml">f</mi><mo id="S2.SS1.p3.2.m2.2.3.2.2.2.1" xref="S2.SS1.p3.2.m2.2.3.2.2.2.1.cmml">⋅</mo><mi id="S2.SS1.p3.2.m2.2.3.2.2.2.3" xref="S2.SS1.p3.2.m2.2.3.2.2.2.3.cmml">w</mi></mrow><mo id="S2.SS1.p3.2.m2.2.3.2.2.1" xref="S2.SS1.p3.2.m2.2.3.2.2.1.cmml">⁢</mo><mrow id="S2.SS1.p3.2.m2.2.3.2.2.3.2" xref="S2.SS1.p3.2.m2.2.3.2.2.cmml"><mo stretchy="false" id="S2.SS1.p3.2.m2.2.3.2.2.3.2.1" xref="S2.SS1.p3.2.m2.2.3.2.2.cmml">(</mo><mi id="S2.SS1.p3.2.m2.1.1" xref="S2.SS1.p3.2.m2.1.1.cmml">R</mi><mo stretchy="false" id="S2.SS1.p3.2.m2.2.3.2.2.3.2.2" xref="S2.SS1.p3.2.m2.2.3.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p3.2.m2.2.3.2.1" xref="S2.SS1.p3.2.m2.2.3.2.1.cmml">⋅</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.2.m2.2.3.2.3" xref="S2.SS1.p3.2.m2.2.3.2.3.cmml">ℱ</mi></mrow><mo id="S2.SS1.p3.2.m2.2.3.1" xref="S2.SS1.p3.2.m2.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.2.m2.2.3.3.2" xref="S2.SS1.p3.2.m2.2.3.cmml"><mo stretchy="false" id="S2.SS1.p3.2.m2.2.3.3.2.1" xref="S2.SS1.p3.2.m2.2.3.cmml">(</mo><mi id="S2.SS1.p3.2.m2.2.2" xref="S2.SS1.p3.2.m2.2.2.cmml">R</mi><mo stretchy="false" id="S2.SS1.p3.2.m2.2.3.3.2.2" xref="S2.SS1.p3.2.m2.2.3.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S3.SS1.SSSx1.p1.2.m2.1.1" xref="S3.SS1.SSSx1.p1.2.m2.1.1.cmml"><msub id="S3.SS1.SSSx1.p1.2.m2.1.1.2" xref="S3.SS1.SSSx1.p1.2.m2.1.1.2.cmml"><mi mathvariant="normal" id="S3.SS1.SSSx1.p1.2.m2.1.1.2.2" xref="S3.SS1.SSSx1.p1.2.m2.1.1.2.2.cmml">Γ</mi><mi id="S3.SS1.SSSx1.p1.2.m2.1.1.2.3" xref="S3.SS1.SSSx1.p1.2.m2.1.1.2.3.cmml">disc</mi></msub><mo id="S3.SS1.SSSx1.p1.2.m2.1.1.1" xref="S3.SS1.SSSx1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSSx1.p1.2.m2.1.1.3" xref="S3.SS1.SSSx1.p1.2.m2.1.1.3.cmml"><mn id="S3.SS1.SSSx1.p1.2.m2.1.1.3.2" xref="S3.SS1.SSSx1.p1.2.m2.1.1.3.2.cmml">0.88</mn><mo id="S3.SS1.SSSx1.p1.2.m2.1.1.3.1" xref="S3.SS1.SSSx1.p1.2.m2.1.1.3.1.cmml">±</mo><mn id="S3.SS1.SSSx1.p1.2.m2.1.1.3.3" xref="S3.SS1.SSSx1.p1.2.m2.1.1.3.3.cmml">0.02</mn></mrow></mrow></math>, <math><mrow id="S3.SS1.SSSx2.p1.10.m10.1.1" xref="S3.SS1.SSSx2.p1.10.m10.1.1.cmml"><msub id="S3.SS1.SSSx2.p1.10.m10.1.1.2" xref="S3.SS1.SSSx2.p1.10.m10.1.1.2.cmml"><mi mathvariant="normal" id="S3.SS1.SSSx2.p1.10.m10.1.1.2.2" xref="S3.SS1.SSSx2.p1.10.m10.1.1.2.2.cmml">Γ</mi><mi id="S3.SS1.SSSx2.p1.10.m10.1.1.2.3" xref="S3.SS1.SSSx2.p1.10.m10.1.1.2.3.cmml">disc</mi></msub><mo id="S3.SS1.SSSx2.p1.10.m10.1.1.1" xref="S3.SS1.SSSx2.p1.10.m10.1.1.1.cmml">=</mo><mrow id="S3.SS1.SSSx2.p1.10.m10.1.1.3" xref="S3.SS1.SSSx2.p1.10.m10.1.1.3.cmml"><mn id="S3.SS1.SSSx2.p1.10.m10.1.1.3.2" xref="S3.SS1.SSSx2.p1.10.m10.1.1.3.2.cmml">4.07</mn><mo id="S3.SS1.SSSx2.p1.10.m10.1.1.3.1" xref="S3.SS1.SSSx2.p1.10.m10.1.1.3.1.cmml">±</mo><mn id="S3.SS1.SSSx2.p1.10.m10.1.1.3.3" xref="S3.SS1.SSSx2.p1.10.m10.1.1.3.3.cmml">0.17</mn></mrow></mrow></math>, <math><mrow id="S3.SS2.SSSx1.p1.2.m2.1.1" xref="S3.SS2.SSSx1.p1.2.m2.1.1.cmml"><msub id="S3.SS2.SSSx1.p1.2.m2.1.1.2" xref="S3.SS2.SSSx1.p1.2.m2.1.1.2.cmml"><mi mathvariant="normal" id="S3.SS2.SSSx1.p1.2.m2.1.1.2.2" xref="S3.SS2.SSSx1.p1.2.m2.1.1.2.2.cmml">Γ</mi><mi id="S3.SS2.SSSx1.p1.2.m2.1.1.2.3" xref="S3.SS2.SSSx1.p1.2.m2.1.1.2.3.cmml">disc</mi></msub><mo id="S3.SS2.SSSx1.p1.2.m2.1.1.1" xref="S3.SS2.SSSx1.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS2.SSSx1.p1.2.m2.1.1.3" xref="S3.SS2.SSSx1.p1.2.m2.1.1.3.cmml"><mn id="S3.SS2.SSSx1.p1.2.m2.1.1.3.2" xref="S3.SS2.SSSx1.p1.2.m2.1.1.3.2.cmml">1.47</mn><mo id="S3.SS2.SSSx1.p1.2.m2.1.1.3.1" xref="S3.SS2.SSSx1.p1.2.m2.1.1.3.1.cmml">±</mo><mn id="S3.SS2.SSSx1.p1.2.m2.1.1.3.3" xref="S3.SS2.SSSx1.p1.2.m2.1.1.3.3.cmml">.02</mn></mrow></mrow></math>, <math><mrow id="S3.SS2.SSSx2.p1.3.m3.1.1" xref="S3.SS2.SSSx2.p1.3.m3.1.1.cmml"><msub id="S3.SS2.SSSx2.p1.3.m3.1.1.2" xref="S3.SS2.SSSx2.p1.3.m3.1.1.2.cmml"><mi mathvariant="normal" id="S3.SS2.SSSx2.p1.3.m3.1.1.2.2" xref="S3.SS2.SSSx2.p1.3.m3.1.1.2.2.cmml">Γ</mi><mi id="S3.SS2.SSSx2.p1.3.m3.1.1.2.3" xref="S3.SS2.SSSx2.p1.3.m3.1.1.2.3.cmml">disc</mi></msub><mo id="S3.SS2.SSSx2.p1.3.m3.1.1.1" xref="S3.SS2.SSSx2.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S3.SS2.SSSx2.p1.3.m3.1.1.3" xref="S3.SS2.SSSx2.p1.3.m3.1.1.3.cmml"><mn id="S3.SS2.SSSx2.p1.3.m3.1.1.3.2" xref="S3.SS2.SSSx2.p1.3.m3.1.1.3.2.cmml">1.31</mn><mo id="S3.SS2.SSSx2.p1.3.m3.1.1.3.1" xref="S3.SS2.SSSx2.p1.3.m3.1.1.3.1.cmml">±</mo><mn id="S3.SS2.SSSx2.p1.3.m3.1.1.3.3" xref="S3.SS2.SSSx2.p1.3.m3.1.1.3.3.cmml">0.08</mn></mrow></mrow></math>, <math><mrow id="S3.SS2.SSSx3.p1.2.m2.1.1" xref="S3.SS2.SSSx3.p1.2.m2.1.1.cmml"><msub id="S3.SS2.SSSx3.p1.2.m2.1.1.2" xref="S3.SS2.SSSx3.p1.2.m2.1.1.2.cmml"><mi mathvariant="normal" id="S3.SS2.SSSx3.p1.2.m2.1.1.2.2" xref="S3.SS2.SSSx3.p1.2.m2.1.1.2.2.cmml">Γ</mi><mi id="S3.SS2.SSSx3.p1.2.m2.1.1.2.3" xref="S3.SS2.SSSx3.p1.2.m2.1.1.2.3.cmml">disc</mi></msub><mo id="S3.SS2.SSSx3.p1.2.m2.1.1.1" xref="S3.SS2.SSSx3.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS2.SSSx3.p1.2.m2.1.1.3" xref="S3.SS2.SSSx3.p1.2.m2.1.1.3.cmml"><mn id="S3.SS2.SSSx3.p1.2.m2.1.1.3.2" xref="S3.SS2.SSSx3.p1.2.m2.1.1.3.2.cmml">0.70</mn><mo id="S3.SS2.SSSx3.p1.2.m2.1.1.3.1" xref="S3.SS2.SSSx3.p1.2.m2.1.1.3.1.cmml">±</mo><mn id="S3.SS2.SSSx3.p1.2.m2.1.1.3.3" xref="S3.SS2.SSSx3.p1.2.m2.1.1.3.3.cmml">0.01</mn></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1404.4251

Formulas:

1-

0.1 c ≲ v ≲ 0.5 c

2-

r_{L} = γ β m_{e} c^{2} / e {⟨ B_{⟂} ⟩}^{1 / 2}

3-

{⟨ B_{⟂} ⟩}^{1 / 2}

4-

\sim F_{L} τ_{λ}

5-

𝐅_{L} = (e / c) 𝐯 \times 𝐁

6-

τ_{λ} \sim λ_{B} / v_{⟂}

7-

Δ p_{⟂} \sim F_{L} τ_{λ} \sim e (B / c) λ_{B}

8-

p_{⟂} \sim γ m_{e} v_{⟂}

9-

α_{λ} \approx Δ p_{⟂} / p_{⟂} \sim e (B / c) λ_{B} / γ m_{e} v_{⟂}

10-

⟨ α^{2} ⟩ = D_{α α} t .

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1511.07741

Formulas:

1-

G = (V, A)

2-

T = (V, A_{T})

3-

G^{'} = (V, A_{T} \cup A^{'})

4-

O (m \log n)

5-

t (v) = d (v)

6-

t (v) \neq d (v)

7-

(d (v), v) \in A

8-

(d (v), v)

9-

(d (v), v) \notin A

10-

t (v) > v

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0601241

Formulas:

1-

Δ θ_{d i s k} ≃ {18.5}^{\circ}

2-

θ_{d i s k} ≃ 70^{\circ}

3-

0 \leq θ < 360^{\circ}

4-

10^{- 11} {ergs cm}^{- 2} s^{- 1}

5-

70^{\circ} < θ < 110^{\circ}

6-

250^{\circ} < θ < 290^{\circ}

7-

Δ θ = θ - θ_{d i s k}

8-

θ_{d i s k}

9-

Δ θ_{d i s k} = {18.5}^{\circ}

10-

70^{\circ} < θ < 110^{\circ}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0608145

Formulas:

1-

(1 - γ) k

2-

(1 - γ) k

3-

\begin{matrix} \frac{# (pure output copies) - # (pure input copies)}{# (noisy input copies)} \\ = \frac{k - (1 - γ) k}{k} = γ . \end{matrix}

4-

\begin{matrix} σ_{00} & = & I_{2} & = & [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], \\ σ_{01} & = & σ_{x} & = & [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], \\ σ_{10} & = & σ_{z} & = & [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], \\ σ_{11} & = & σ_{y} & = & [\begin{matrix} 0 & - i \\ i & 0 \end{matrix}] . \end{matrix}

5-

v, w \in ℤ_{2}^{n}

6-

a = [\begin{matrix} v \\ w \end{matrix}]

7-

σ_{a} = σ_{v_{1} w_{1}} \otimes \dots \otimes σ_{v_{n} w_{n}} .

8-

{1, i, - 1, - i}

9-

σ_{a} σ_{b} = {(- 1)}^{a^{T} P b} σ_{b} σ_{a}, where P = [\begin{matrix} 0 & I_{n} \\ I_{n} & 0 \end{matrix}] .

10-

{(- 1)}^{b_{i}} σ_{s_{i}}

Formulas (html):

<math><mrow id="S1.p3.3.m3.1.1" xref="S1.p3.3.m3.1.1.cmml"><mrow id="S1.p3.3.m3.1.1.1.1" xref="S1.p3.3.m3.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.p3.3.m3.1.1.1.1.2" xref="S1.p3.3.m3.1.1.1.1.1.cmml">(</mo><mrow id="S1.p3.3.m3.1.1.1.1.1" xref="S1.p3.3.m3.1.1.1.1.1.cmml"><mn id="S1.p3.3.m3.1.1.1.1.1.2" xref="S1.p3.3.m3.1.1.1.1.1.2.cmml">1</mn><mo id="S1.p3.3.m3.1.1.1.1.1.1" xref="S1.p3.3.m3.1.1.1.1.1.1.cmml">-</mo><mi id="S1.p3.3.m3.1.1.1.1.1.3" xref="S1.p3.3.m3.1.1.1.1.1.3.cmml">γ</mi></mrow><mo stretchy="false" id="S1.p3.3.m3.1.1.1.1.3" xref="S1.p3.3.m3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p3.3.m3.1.1.2" xref="S1.p3.3.m3.1.1.2.cmml">⁢</mo><mi id="S1.p3.3.m3.1.1.3" xref="S1.p3.3.m3.1.1.3.cmml">k</mi></mrow></math>, <math><mrow id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml"><mrow id="S1.p3.5.m5.1.1.1.1" xref="S1.p3.5.m5.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.p3.5.m5.1.1.1.1.2" xref="S1.p3.5.m5.1.1.1.1.1.cmml">(</mo><mrow id="S1.p3.5.m5.1.1.1.1.1" xref="S1.p3.5.m5.1.1.1.1.1.cmml"><mn id="S1.p3.5.m5.1.1.1.1.1.2" xref="S1.p3.5.m5.1.1.1.1.1.2.cmml">1</mn><mo id="S1.p3.5.m5.1.1.1.1.1.1" xref="S1.p3.5.m5.1.1.1.1.1.1.cmml">-</mo><mi id="S1.p3.5.m5.1.1.1.1.1.3" xref="S1.p3.5.m5.1.1.1.1.1.3.cmml">γ</mi></mrow><mo stretchy="false" id="S1.p3.5.m5.1.1.1.1.3" xref="S1.p3.5.m5.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p3.5.m5.1.1.2" xref="S1.p3.5.m5.1.1.2.cmml">⁢</mo><mi id="S1.p3.5.m5.1.1.3" xref="S1.p3.5.m5.1.1.3.cmml">k</mi></mrow></math>, <math><mtable displaystyle="true" rowspacing="0pt" id="S1.Ex1.m1.5.5" xref="S1.Ex1.m1.5.5.cmml"><mtr id="S1.Ex1.m1.5.5a" xref="S1.Ex1.m1.5.5.cmml"><mtd columnalign="center" id="S1.Ex1.m1.5.5b" xref="S1.Ex1.m1.5.5.cmml"><mstyle displaystyle="false" id="S1.Ex1.m1.3.3.3.3.3" xref="S1.Ex1.m1.3.3.3.3.3.cmml"><mfrac id="S1.Ex1.m1.3.3.3.3.3a" xref="S1.Ex1.m1.3.3.3.3.3.cmml"><mrow id="S1.Ex1.m1.2.2.2.2.2.2.2" xref="S1.Ex1.m1.2.2.2.2.2.2.2.cmml"><mrow id="S1.Ex1.m1.2.2.2.2.2.2.2.4" xref="S1.Ex1.m1.2.2.2.2.2.2.2.4.cmml"><mi mathvariant="normal" id="S1.Ex1.m1.2.2.2.2.2.2.2.4.2" xref="S1.Ex1.m1.2.2.2.2.2.2.2.4.2.cmml">#</mi><mo id="S1.Ex1.m1.2.2.2.2.2.2.2.4.1" xref="S1.Ex1.m1.2.2.2.2.2.2.2.4.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.2.2.2.2.2.2.2.4.3.2" xref="S1.Ex1.m1.1.1.1.1.1.1.1.1a.cmml"><mo stretchy="false" id="S1.Ex1.m1.2.2.2.2.2.2.2.4.3.2.1" xref="S1.Ex1.m1.1.1.1.1.1.1.1.1a.cmml">(</mo><mtext id="S1.Ex1.m1.1.1.1.1.1.1.1.1" xref="S1.Ex1.m1.1.1.1.1.1.1.1.1.cmml">pure output copies</mtext><mo stretchy="false" id="S1.Ex1.m1.2.2.2.2.2.2.2.4.3.2.2" xref="S1.Ex1.m1.1.1.1.1.1.1.1.1a.cmml">)</mo></mrow></mrow><mo id="S1.Ex1.m1.2.2.2.2.2.2.2.3" xref="S1.Ex1.m1.2.2.2.2.2.2.2.3.cmml">-</mo><mrow id="S1.Ex1.m1.2.2.2.2.2.2.2.5" xref="S1.Ex1.m1.2.2.2.2.2.2.2.5.cmml"><mi mathvariant="normal" id="S1.Ex1.m1.2.2.2.2.2.2.2.5.2" xref="S1.Ex1.m1.2.2.2.2.2.2.2.5.2.cmml">#</mi><mo id="S1.Ex1.m1.2.2.2.2.2.2.2.5.1" xref="S1.Ex1.m1.2.2.2.2.2.2.2.5.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.2.2.2.2.2.2.2.5.3.2" xref="S1.Ex1.m1.2.2.2.2.2.2.2.2a.cmml"><mo stretchy="false" id="S1.Ex1.m1.2.2.2.2.2.2.2.5.3.2.1" xref="S1.Ex1.m1.2.2.2.2.2.2.2.2a.cmml">(</mo><mtext id="S1.Ex1.m1.2.2.2.2.2.2.2.2" xref="S1.Ex1.m1.2.2.2.2.2.2.2.2.cmml">pure input copies</mtext><mo stretchy="false" id="S1.Ex1.m1.2.2.2.2.2.2.2.5.3.2.2" xref="S1.Ex1.m1.2.2.2.2.2.2.2.2a.cmml">)</mo></mrow></mrow></mrow><mrow id="S1.Ex1.m1.3.3.3.3.3.3.3" xref="S1.Ex1.m1.3.3.3.3.3.3.3.cmml"><mi mathvariant="normal" id="S1.Ex1.m1.3.3.3.3.3.3.3.3" xref="S1.Ex1.m1.3.3.3.3.3.3.3.3.cmml">#</mi><mo id="S1.Ex1.m1.3.3.3.3.3.3.3.2" xref="S1.Ex1.m1.3.3.3.3.3.3.3.2.cmml">⁢</mo><mrow id="S1.Ex1.m1.3.3.3.3.3.3.3.4.2" xref="S1.Ex1.m1.3.3.3.3.3.3.3.1a.cmml"><mo stretchy="false" id="S1.Ex1.m1.3.3.3.3.3.3.3.4.2.1" xref="S1.Ex1.m1.3.3.3.3.3.3.3.1a.cmml">(</mo><mtext id="S1.Ex1.m1.3.3.3.3.3.3.3.1" xref="S1.Ex1.m1.3.3.3.3.3.3.3.1.cmml">noisy input copies</mtext><mo stretchy="false" id="S1.Ex1.m1.3.3.3.3.3.3.3.4.2.2" xref="S1.Ex1.m1.3.3.3.3.3.3.3.1a.cmml">)</mo></mrow></mrow></mfrac></mstyle></mtd></mtr><mtr id="S1.Ex1.m1.5.5c" xref="S1.Ex1.m1.5.5.cmml"><mtd columnalign="center" id="S1.Ex1.m1.5.5d" xref="S1.Ex1.m1.5.5.cmml"><mrow id="S1.Ex1.m1.5.5.5.2.2.2" xref="S1.Ex1.m1.5.5.5.2.2.2.1.cmml"><mrow id="S1.Ex1.m1.5.5.5.2.2.2.1" xref="S1.Ex1.m1.5.5.5.2.2.2.1.cmml"><mi id="S1.Ex1.m1.5.5.5.2.2.2.1.2" xref="S1.Ex1.m1.5.5.5.2.2.2.1.2.cmml"/><mo id="S1.Ex1.m1.5.5.5.2.2.2.1.3" xref="S1.Ex1.m1.5.5.5.2.2.2.1.3.cmml">=</mo><mstyle displaystyle="false" id="S1.Ex1.m1.4.4.4.1.1.1" xref="S1.Ex1.m1.4.4.4.1.1.1.cmml"><mfrac id="S1.Ex1.m1.4.4.4.1.1.1a" xref="S1.Ex1.m1.4.4.4.1.1.1.cmml"><mrow id="S1.Ex1.m1.4.4.4.1.1.1.1" xref="S1.Ex1.m1.4.4.4.1.1.1.1.cmml"><mi id="S1.Ex1.m1.4.4.4.1.1.1.1.3" xref="S1.Ex1.m1.4.4.4.1.1.1.1.3.cmml">k</mi><mo id="S1.Ex1.m1.4.4.4.1.1.1.1.2" xref="S1.Ex1.m1.4.4.4.1.1.1.1.2.cmml">-</mo><mrow id="S1.Ex1.m1.4.4.4.1.1.1.1.1" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.cmml"><mrow id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.2" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.cmml"><mn id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.2" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.1" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.1.cmml">-</mo><mi id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.3" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.3.cmml">γ</mi></mrow><mo stretchy="false" id="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.3" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.Ex1.m1.4.4.4.1.1.1.1.1.2" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.2.cmml">⁢</mo><mi id="S1.Ex1.m1.4.4.4.1.1.1.1.1.3" xref="S1.Ex1.m1.4.4.4.1.1.1.1.1.3.cmml">k</mi></mrow></mrow><mi id="S1.Ex1.m1.4.4.4.1.1.1.3" xref="S1.Ex1.m1.4.4.4.1.1.1.3.cmml">k</mi></mfrac></mstyle><mo id="S1.Ex1.m1.5.5.5.2.2.2.1.4" xref="S1.Ex1.m1.5.5.5.2.2.2.1.4.cmml">=</mo><mi id="S1.Ex1.m1.5.5.5.2.2.2.1.5" xref="S1.Ex1.m1.5.5.5.2.2.2.1.5.cmml">γ</mi></mrow><mo id="S1.Ex1.m1.5.5.5.2.2.2.2" xref="S1.Ex1.m1.5.5.5.2.2.2.1.cmml">.</mo></mrow></mtd></mtr></mtable></math>, <math><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.Ex2.m1.8.8" xref="S2.Ex2.m1.8.8.cmml"><mtr id="S2.Ex2.m1.8.8a" xref="S2.Ex2.m1.8.8.cmml"><mtd columnalign="center" id="S2.Ex2.m1.8.8b" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.2.2.2.3.1" xref="S2.Ex2.m1.2.2.2.3.1.cmml"><mi id="S2.Ex2.m1.2.2.2.3.1.2" xref="S2.Ex2.m1.2.2.2.3.1.2.cmml">σ</mi><mn id="S2.Ex2.m1.2.2.2.3.1.3" xref="S2.Ex2.m1.2.2.2.3.1.3.cmml">00</mn></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8c" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.2.2.2.4.1" xref="S2.Ex2.m1.2.2.2.4.1.cmml">=</mo></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8d" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.2.2.2.5.1" xref="S2.Ex2.m1.2.2.2.5.1.cmml"><mi id="S2.Ex2.m1.2.2.2.5.1.2" xref="S2.Ex2.m1.2.2.2.5.1.2.cmml">I</mi><mn id="S2.Ex2.m1.2.2.2.5.1.3" xref="S2.Ex2.m1.2.2.2.5.1.3.cmml">2</mn></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8e" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.2.2.2.6.1" xref="S2.Ex2.m1.2.2.2.6.1.cmml">=</mo></mtd><mtd columnalign="left" id="S2.Ex2.m1.8.8f" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.2.2.2.2.2.2" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.2.2.2.2.2.2.1.2" xref="S2.Ex2.m1.2.2.2.2.2.2.1.1.cmml"><mo id="S2.Ex2.m1.2.2.2.2.2.2.1.2.1" xref="S2.Ex2.m1.2.2.2.2.2.2.1.1.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.Ex2.m1.1.1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mtr id="S2.Ex2.m1.1.1.1.1.1.1a" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.1.1.1.1.1.1b" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mn id="S2.Ex2.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.1.1.1.cmml">1</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.1.1.1.1.1.1c" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mn id="S2.Ex2.m1.1.1.1.1.1.1.1.2.1" xref="S2.Ex2.m1.1.1.1.1.1.1.1.2.1.cmml">0</mn></mtd></mtr><mtr id="S2.Ex2.m1.1.1.1.1.1.1d" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.1.1.1.1.1.1e" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mn id="S2.Ex2.m1.1.1.1.1.1.1.2.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.2.1.1.cmml">0</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.1.1.1.1.1.1f" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml"><mn id="S2.Ex2.m1.1.1.1.1.1.1.2.2.1" xref="S2.Ex2.m1.1.1.1.1.1.1.2.2.1.cmml">1</mn></mtd></mtr></mtable><mo id="S2.Ex2.m1.2.2.2.2.2.2.1.2.2" xref="S2.Ex2.m1.2.2.2.2.2.2.1.1.1.cmml">]</mo></mrow><mo id="S2.Ex2.m1.2.2.2.2.2.2.2" xref="S2.Ex2.m1.8.8.cmml">,</mo></mrow></mtd></mtr><mtr id="S2.Ex2.m1.8.8g" xref="S2.Ex2.m1.8.8.cmml"><mtd columnalign="center" id="S2.Ex2.m1.8.8h" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.4.4.4.3.1" xref="S2.Ex2.m1.4.4.4.3.1.cmml"><mi id="S2.Ex2.m1.4.4.4.3.1.2" xref="S2.Ex2.m1.4.4.4.3.1.2.cmml">σ</mi><mn id="S2.Ex2.m1.4.4.4.3.1.3" xref="S2.Ex2.m1.4.4.4.3.1.3.cmml">01</mn></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8i" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.4.4.4.4.1" xref="S2.Ex2.m1.4.4.4.4.1.cmml">=</mo></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8j" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.4.4.4.5.1" xref="S2.Ex2.m1.4.4.4.5.1.cmml"><mi id="S2.Ex2.m1.4.4.4.5.1.2" xref="S2.Ex2.m1.4.4.4.5.1.2.cmml">σ</mi><mi id="S2.Ex2.m1.4.4.4.5.1.3" xref="S2.Ex2.m1.4.4.4.5.1.3.cmml">x</mi></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8k" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.4.4.4.6.1" xref="S2.Ex2.m1.4.4.4.6.1.cmml">=</mo></mtd><mtd columnalign="left" id="S2.Ex2.m1.8.8l" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.4.4.4.2.2.2" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.4.4.4.2.2.2.1.2" xref="S2.Ex2.m1.4.4.4.2.2.2.1.1.cmml"><mo id="S2.Ex2.m1.4.4.4.2.2.2.1.2.1" xref="S2.Ex2.m1.4.4.4.2.2.2.1.1.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.Ex2.m1.3.3.3.1.1.1" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mtr id="S2.Ex2.m1.3.3.3.1.1.1a" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.3.3.3.1.1.1b" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mn id="S2.Ex2.m1.3.3.3.1.1.1.1.1.1" xref="S2.Ex2.m1.3.3.3.1.1.1.1.1.1.cmml">0</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.3.3.3.1.1.1c" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mn id="S2.Ex2.m1.3.3.3.1.1.1.1.2.1" xref="S2.Ex2.m1.3.3.3.1.1.1.1.2.1.cmml">1</mn></mtd></mtr><mtr id="S2.Ex2.m1.3.3.3.1.1.1d" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.3.3.3.1.1.1e" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mn id="S2.Ex2.m1.3.3.3.1.1.1.2.1.1" xref="S2.Ex2.m1.3.3.3.1.1.1.2.1.1.cmml">1</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.3.3.3.1.1.1f" xref="S2.Ex2.m1.3.3.3.1.1.1.cmml"><mn id="S2.Ex2.m1.3.3.3.1.1.1.2.2.1" xref="S2.Ex2.m1.3.3.3.1.1.1.2.2.1.cmml">0</mn></mtd></mtr></mtable><mo id="S2.Ex2.m1.4.4.4.2.2.2.1.2.2" xref="S2.Ex2.m1.4.4.4.2.2.2.1.1.1.cmml">]</mo></mrow><mo id="S2.Ex2.m1.4.4.4.2.2.2.2" xref="S2.Ex2.m1.8.8.cmml">,</mo></mrow></mtd></mtr><mtr id="S2.Ex2.m1.8.8m" xref="S2.Ex2.m1.8.8.cmml"><mtd columnalign="center" id="S2.Ex2.m1.8.8n" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.6.6.6.3.1" xref="S2.Ex2.m1.6.6.6.3.1.cmml"><mi id="S2.Ex2.m1.6.6.6.3.1.2" xref="S2.Ex2.m1.6.6.6.3.1.2.cmml">σ</mi><mn id="S2.Ex2.m1.6.6.6.3.1.3" xref="S2.Ex2.m1.6.6.6.3.1.3.cmml">10</mn></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8o" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.6.6.6.4.1" xref="S2.Ex2.m1.6.6.6.4.1.cmml">=</mo></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8p" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.6.6.6.5.1" xref="S2.Ex2.m1.6.6.6.5.1.cmml"><mi id="S2.Ex2.m1.6.6.6.5.1.2" xref="S2.Ex2.m1.6.6.6.5.1.2.cmml">σ</mi><mi id="S2.Ex2.m1.6.6.6.5.1.3" xref="S2.Ex2.m1.6.6.6.5.1.3.cmml">z</mi></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8q" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.6.6.6.6.1" xref="S2.Ex2.m1.6.6.6.6.1.cmml">=</mo></mtd><mtd columnalign="left" id="S2.Ex2.m1.8.8r" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.6.6.6.2.2.2" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.6.6.6.2.2.2.1.2" xref="S2.Ex2.m1.6.6.6.2.2.2.1.1.cmml"><mo id="S2.Ex2.m1.6.6.6.2.2.2.1.2.1" xref="S2.Ex2.m1.6.6.6.2.2.2.1.1.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.Ex2.m1.5.5.5.1.1.1" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mtr id="S2.Ex2.m1.5.5.5.1.1.1a" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.5.5.5.1.1.1b" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mn id="S2.Ex2.m1.5.5.5.1.1.1.1.1.1" xref="S2.Ex2.m1.5.5.5.1.1.1.1.1.1.cmml">1</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.5.5.5.1.1.1c" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mn id="S2.Ex2.m1.5.5.5.1.1.1.1.2.1" xref="S2.Ex2.m1.5.5.5.1.1.1.1.2.1.cmml">0</mn></mtd></mtr><mtr id="S2.Ex2.m1.5.5.5.1.1.1d" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.5.5.5.1.1.1e" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mn id="S2.Ex2.m1.5.5.5.1.1.1.2.1.1" xref="S2.Ex2.m1.5.5.5.1.1.1.2.1.1.cmml">0</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.5.5.5.1.1.1f" xref="S2.Ex2.m1.5.5.5.1.1.1.cmml"><mrow id="S2.Ex2.m1.5.5.5.1.1.1.2.2.1" xref="S2.Ex2.m1.5.5.5.1.1.1.2.2.1.cmml"><mo id="S2.Ex2.m1.5.5.5.1.1.1.2.2.1.1" xref="S2.Ex2.m1.5.5.5.1.1.1.2.2.1.1.cmml">-</mo><mn id="S2.Ex2.m1.5.5.5.1.1.1.2.2.1.2" xref="S2.Ex2.m1.5.5.5.1.1.1.2.2.1.2.cmml">1</mn></mrow></mtd></mtr></mtable><mo id="S2.Ex2.m1.6.6.6.2.2.2.1.2.2" xref="S2.Ex2.m1.6.6.6.2.2.2.1.1.1.cmml">]</mo></mrow><mo id="S2.Ex2.m1.6.6.6.2.2.2.2" xref="S2.Ex2.m1.8.8.cmml">,</mo></mrow></mtd></mtr><mtr id="S2.Ex2.m1.8.8s" xref="S2.Ex2.m1.8.8.cmml"><mtd columnalign="center" id="S2.Ex2.m1.8.8t" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.8.8.8.3.1" xref="S2.Ex2.m1.8.8.8.3.1.cmml"><mi id="S2.Ex2.m1.8.8.8.3.1.2" xref="S2.Ex2.m1.8.8.8.3.1.2.cmml">σ</mi><mn id="S2.Ex2.m1.8.8.8.3.1.3" xref="S2.Ex2.m1.8.8.8.3.1.3.cmml">11</mn></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8u" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.8.8.8.4.1" xref="S2.Ex2.m1.8.8.8.4.1.cmml">=</mo></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8v" xref="S2.Ex2.m1.8.8.cmml"><msub id="S2.Ex2.m1.8.8.8.5.1" xref="S2.Ex2.m1.8.8.8.5.1.cmml"><mi id="S2.Ex2.m1.8.8.8.5.1.2" xref="S2.Ex2.m1.8.8.8.5.1.2.cmml">σ</mi><mi id="S2.Ex2.m1.8.8.8.5.1.3" xref="S2.Ex2.m1.8.8.8.5.1.3.cmml">y</mi></msub></mtd><mtd columnalign="center" id="S2.Ex2.m1.8.8w" xref="S2.Ex2.m1.8.8.cmml"><mo id="S2.Ex2.m1.8.8.8.6.1" xref="S2.Ex2.m1.8.8.8.6.1.cmml">=</mo></mtd><mtd columnalign="left" id="S2.Ex2.m1.8.8x" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.8.8.8.2.2.2" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.8.8.8.2.2.2.1.2" xref="S2.Ex2.m1.8.8.8.2.2.2.1.1.cmml"><mo id="S2.Ex2.m1.8.8.8.2.2.2.1.2.1" xref="S2.Ex2.m1.8.8.8.2.2.2.1.1.1.cmml">[</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.Ex2.m1.7.7.7.1.1.1" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mtr id="S2.Ex2.m1.7.7.7.1.1.1a" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.7.7.7.1.1.1b" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mn id="S2.Ex2.m1.7.7.7.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.7.1.1.1.1.1.1.cmml">0</mn></mtd><mtd columnalign="right" id="S2.Ex2.m1.7.7.7.1.1.1c" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mrow id="S2.Ex2.m1.7.7.7.1.1.1.1.2.1" xref="S2.Ex2.m1.7.7.7.1.1.1.1.2.1.cmml"><mo id="S2.Ex2.m1.7.7.7.1.1.1.1.2.1.1" xref="S2.Ex2.m1.7.7.7.1.1.1.1.2.1.1.cmml">-</mo><mi id="S2.Ex2.m1.7.7.7.1.1.1.1.2.1.2" xref="S2.Ex2.m1.7.7.7.1.1.1.1.2.1.2.cmml">i</mi></mrow></mtd></mtr><mtr id="S2.Ex2.m1.7.7.7.1.1.1d" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mtd columnalign="right" id="S2.Ex2.m1.7.7.7.1.1.1e" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mi id="S2.Ex2.m1.7.7.7.1.1.1.2.1.1" xref="S2.Ex2.m1.7.7.7.1.1.1.2.1.1.cmml">i</mi></mtd><mtd columnalign="right" id="S2.Ex2.m1.7.7.7.1.1.1f" xref="S2.Ex2.m1.7.7.7.1.1.1.cmml"><mn id="S2.Ex2.m1.7.7.7.1.1.1.2.2.1" xref="S2.Ex2.m1.7.7.7.1.1.1.2.2.1.cmml">0</mn></mtd></mtr></mtable><mo id="S2.Ex2.m1.8.8.8.2.2.2.1.2.2" xref="S2.Ex2.m1.8.8.8.2.2.2.1.1.1.cmml">]</mo></mrow><mo id="S2.Ex2.m1.8.8.8.2.2.2.2" xref="S2.Ex2.m1.8.8.cmml">.</mo></mrow></mtd></mtr></mtable></math>, <math><mrow id="S2.SS1.SSS1.p1.1.m1.2.3" xref="S2.SS1.SSS1.p1.1.m1.2.3.cmml"><mrow id="S2.SS1.SSS1.p1.1.m1.2.3.2.2" xref="S2.SS1.SSS1.p1.1.m1.2.3.2.1.cmml"><mi id="S2.SS1.SSS1.p1.1.m1.1.1" xref="S2.SS1.SSS1.p1.1.m1.1.1.cmml">v</mi><mo id="S2.SS1.SSS1.p1.1.m1.2.3.2.2.1" xref="S2.SS1.SSS1.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S2.SS1.SSS1.p1.1.m1.2.2" xref="S2.SS1.SSS1.p1.1.m1.2.2.cmml">w</mi></mrow><mo id="S2.SS1.SSS1.p1.1.m1.2.3.1" xref="S2.SS1.SSS1.p1.1.m1.2.3.1.cmml">∈</mo><msubsup id="S2.SS1.SSS1.p1.1.m1.2.3.3" xref="S2.SS1.SSS1.p1.1.m1.2.3.3.cmml"><mi id="S2.SS1.SSS1.p1.1.m1.2.3.3.2.2" xref="S2.SS1.SSS1.p1.1.m1.2.3.3.2.2.cmml">ℤ</mi><mn id="S2.SS1.SSS1.p1.1.m1.2.3.3.2.3" xref="S2.SS1.SSS1.p1.1.m1.2.3.3.2.3.cmml">2</mn><mi id="S2.SS1.SSS1.p1.1.m1.2.3.3.3" xref="S2.SS1.SSS1.p1.1.m1.2.3.3.3.cmml">n</mi></msubsup></mrow></math>, <math><mrow id="S2.SS1.SSS1.p1.2.m2.1.2" xref="S2.SS1.SSS1.p1.2.m2.1.2.cmml"><mi id="S2.SS1.SSS1.p1.2.m2.1.2.2" xref="S2.SS1.SSS1.p1.2.m2.1.2.2.cmml">a</mi><mo id="S2.SS1.SSS1.p1.2.m2.1.2.1" xref="S2.SS1.SSS1.p1.2.m2.1.2.1.cmml">=</mo><mrow id="S2.SS1.SSS1.p1.2.m2.1.2.3.2" xref="S2.SS1.SSS1.p1.2.m2.1.2.3.1.cmml"><mo id="S2.SS1.SSS1.p1.2.m2.1.2.3.2.1" xref="S2.SS1.SSS1.p1.2.m2.1.2.3.1.1.cmml">[</mo><mtable columnspacing="5pt" rowspacing="0pt" id="S2.SS1.SSS1.p1.2.m2.1.1" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"><mtr id="S2.SS1.SSS1.p1.2.m2.1.1a" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"><mtd columnalign="center" id="S2.SS1.SSS1.p1.2.m2.1.1b" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.SSS1.p1.2.m2.1.1.1.1.1" xref="S2.SS1.SSS1.p1.2.m2.1.1.1.1.1.cmml">v</mi></mtd><mtd id="S2.SS1.SSS1.p1.2.m2.1.1c" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"/></mtr><mtr id="S2.SS1.SSS1.p1.2.m2.1.1d" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"><mtd columnalign="center" id="S2.SS1.SSS1.p1.2.m2.1.1e" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.SSS1.p1.2.m2.1.1.2.1.1" xref="S2.SS1.SSS1.p1.2.m2.1.1.2.1.1.cmml">w</mi></mtd><mtd id="S2.SS1.SSS1.p1.2.m2.1.1f" xref="S2.SS1.SSS1.p1.2.m2.1.1.cmml"/></mtr></mtable><mo id="S2.SS1.SSS1.p1.2.m2.1.2.3.2.2" xref="S2.SS1.SSS1.p1.2.m2.1.2.3.1.1.cmml">]</mo></mrow></mrow></math>, <math><mrow id="S2.Ex3.m1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml"><mrow id="S2.Ex3.m1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml"><msub id="S2.Ex3.m1.1.1.1.1.2" xref="S2.Ex3.m1.1.1.1.1.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.cmml">σ</mi><mi id="S2.Ex3.m1.1.1.1.1.2.3" xref="S2.Ex3.m1.1.1.1.1.2.3.cmml">a</mi></msub><mo id="S2.Ex3.m1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.Ex3.m1.1.1.1.1.3" xref="S2.Ex3.m1.1.1.1.1.3.cmml"><msub id="S2.Ex3.m1.1.1.1.1.3.2" xref="S2.Ex3.m1.1.1.1.1.3.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.3.2.2" xref="S2.Ex3.m1.1.1.1.1.3.2.2.cmml">σ</mi><mrow id="S2.Ex3.m1.1.1.1.1.3.2.3" xref="S2.Ex3.m1.1.1.1.1.3.2.3.cmml"><msub id="S2.Ex3.m1.1.1.1.1.3.2.3.2" xref="S2.Ex3.m1.1.1.1.1.3.2.3.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.3.2.3.2.2" xref="S2.Ex3.m1.1.1.1.1.3.2.3.2.2.cmml">v</mi><mn id="S2.Ex3.m1.1.1.1.1.3.2.3.2.3" xref="S2.Ex3.m1.1.1.1.1.3.2.3.2.3.cmml">1</mn></msub><mo id="S2.Ex3.m1.1.1.1.1.3.2.3.1" xref="S2.Ex3.m1.1.1.1.1.3.2.3.1.cmml">⁢</mo><msub id="S2.Ex3.m1.1.1.1.1.3.2.3.3" xref="S2.Ex3.m1.1.1.1.1.3.2.3.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.3.2.3.3.2" xref="S2.Ex3.m1.1.1.1.1.3.2.3.3.2.cmml">w</mi><mn id="S2.Ex3.m1.1.1.1.1.3.2.3.3.3" xref="S2.Ex3.m1.1.1.1.1.3.2.3.3.3.cmml">1</mn></msub></mrow></msub><mo id="S2.Ex3.m1.1.1.1.1.3.1" xref="S2.Ex3.m1.1.1.1.1.3.1.cmml">⊗</mo><mi mathvariant="normal" id="S2.Ex3.m1.1.1.1.1.3.3" 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id="S2.SS1.SSS1.p1.5.m3.4.4.2.2.1" xref="S2.SS1.SSS1.p1.5.m3.4.4.2.2.1.cmml">-</mo><mi id="S2.SS1.SSS1.p1.5.m3.4.4.2.2.2" xref="S2.SS1.SSS1.p1.5.m3.4.4.2.2.2.cmml">i</mi></mrow><mo stretchy="false" id="S2.SS1.SSS1.p1.5.m3.4.4.2.7" xref="S2.SS1.SSS1.p1.5.m3.4.4.3.cmml">}</mo></mrow></math>, <math><mrow id="S2.E1.m1.2.2.1"><mrow id="S2.E1.m1.2.2.1.1.2" xref="S2.E1.m1.2.2.1.1.3.cmml"><mrow id="S2.E1.m1.2.2.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.cmml"><mrow id="S2.E1.m1.2.2.1.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.1.3.cmml"><msub id="S2.E1.m1.2.2.1.1.1.1.3.2" xref="S2.E1.m1.2.2.1.1.1.1.3.2.cmml"><mi id="S2.E1.m1.2.2.1.1.1.1.3.2.2" xref="S2.E1.m1.2.2.1.1.1.1.3.2.2.cmml">σ</mi><mi id="S2.E1.m1.2.2.1.1.1.1.3.2.3" xref="S2.E1.m1.2.2.1.1.1.1.3.2.3.cmml">a</mi></msub><mo id="S2.E1.m1.2.2.1.1.1.1.3.1" xref="S2.E1.m1.2.2.1.1.1.1.3.1.cmml">⁢</mo><msub id="S2.E1.m1.2.2.1.1.1.1.3.3" xref="S2.E1.m1.2.2.1.1.1.1.3.3.cmml"><mi id="S2.E1.m1.2.2.1.1.1.1.3.3.2" xref="S2.E1.m1.2.2.1.1.1.1.3.3.2.cmml">σ</mi><mi id="S2.E1.m1.2.2.1.1.1.1.3.3.3" xref="S2.E1.m1.2.2.1.1.1.1.3.3.3.cmml">b</mi></msub></mrow><mo id="S2.E1.m1.2.2.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.2.cmml">=</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.cmml"><msup id="S2.E1.m1.2.2.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.cmml"><mrow id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.1.cmml">-</mo><mn id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo stretchy="false" id="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.E1.m1.2.2.1.1.1.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.cmml"><msup id="S2.E1.m1.2.2.1.1.1.1.1.1.3.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.2.cmml"><mi id="S2.E1.m1.2.2.1.1.1.1.1.1.3.2.2" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.2.2.cmml">a</mi><mi id="S2.E1.m1.2.2.1.1.1.1.1.1.3.2.3" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.2.3.cmml">T</mi></msup><mo id="S2.E1.m1.2.2.1.1.1.1.1.1.3.1" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E1.m1.2.2.1.1.1.1.1.1.3.3" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.3.cmml">P</mi><mo id="S2.E1.m1.2.2.1.1.1.1.1.1.3.1a" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E1.m1.2.2.1.1.1.1.1.1.3.4" xref="S2.E1.m1.2.2.1.1.1.1.1.1.3.4.cmml">b</mi></mrow></msup><mo id="S2.E1.m1.2.2.1.1.1.1.1.2" xref="S2.E1.m1.2.2.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.E1.m1.2.2.1.1.1.1.1.3" xref="S2.E1.m1.2.2.1.1.1.1.1.3.cmml"><mi id="S2.E1.m1.2.2.1.1.1.1.1.3.2" xref="S2.E1.m1.2.2.1.1.1.1.1.3.2.cmml">σ</mi><mi id="S2.E1.m1.2.2.1.1.1.1.1.3.3" xref="S2.E1.m1.2.2.1.1.1.1.1.3.3.cmml">b</mi></msub><mo id="S2.E1.m1.2.2.1.1.1.1.1.2a" 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xref="S2.E1.m1.1.1.2.1.1.3.cmml">n</mi></msub></mtd><mtd columnalign="center" id="S2.E1.m1.1.1f" xref="S2.E1.m1.1.1.cmml"><mn id="S2.E1.m1.1.1.2.2.1" xref="S2.E1.m1.1.1.2.2.1.cmml">0</mn></mtd></mtr></mtable><mo id="S2.E1.m1.2.2.1.1.2.2.3.2.2" xref="S2.E1.m1.2.2.1.1.2.2.3.1.1.cmml">]</mo></mrow></mrow></mrow><mo id="S2.E1.m1.2.2.1.2">.</mo></mrow></math>, <math><mrow id="S2.SS1.SSS2.p1.3.m3.1.1" xref="S2.SS1.SSS2.p1.3.m3.1.1.cmml"><msup id="S2.SS1.SSS2.p1.3.m3.1.1.1" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.cmml"><mrow id="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.2" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.cmml"><mo id="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.1" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.1.cmml">-</mo><mn id="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.2" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo stretchy="false" id="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.3" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><msub id="S2.SS1.SSS2.p1.3.m3.1.1.1.3" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.3.cmml"><mi id="S2.SS1.SSS2.p1.3.m3.1.1.1.3.2" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.3.2.cmml">b</mi><mi id="S2.SS1.SSS2.p1.3.m3.1.1.1.3.3" xref="S2.SS1.SSS2.p1.3.m3.1.1.1.3.3.cmml">i</mi></msub></msup><mo id="S2.SS1.SSS2.p1.3.m3.1.1.2" xref="S2.SS1.SSS2.p1.3.m3.1.1.2.cmml">⁢</mo><msub id="S2.SS1.SSS2.p1.3.m3.1.1.3" xref="S2.SS1.SSS2.p1.3.m3.1.1.3.cmml"><mi id="S2.SS1.SSS2.p1.3.m3.1.1.3.2" xref="S2.SS1.SSS2.p1.3.m3.1.1.3.2.cmml">σ</mi><msub id="S2.SS1.SSS2.p1.3.m3.1.1.3.3" xref="S2.SS1.SSS2.p1.3.m3.1.1.3.3.cmml"><mi id="S2.SS1.SSS2.p1.3.m3.1.1.3.3.2" xref="S2.SS1.SSS2.p1.3.m3.1.1.3.3.2.cmml">s</mi><mi id="S2.SS1.SSS2.p1.3.m3.1.1.3.3.3" xref="S2.SS1.SSS2.p1.3.m3.1.1.3.3.3.cmml">i</mi></msub></msub></mrow></math>

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1612.08993

Formulas:

1-

β_{2} = - 20 {ps}^{2} / km

2-

n_{2} = 0.9 \times 10^{- 20} m^{2} W^{- 1}

3-

A_{eff} = 85 μ m^{2}

4-

f_{o f f}

5-

κ / (2 π)

6-

n_{2} = 0.9 \times 10^{- 20} m^{2} W^{- 1}

7-

A_{eff} = 85 μ m^{2}

8-

β_{2} = - 20 {ps}^{2} / km

9-

Δ \cdot {(dFSR / d ν)}^{- 1}

10-

Δ \cdot {(dFSR / d ν)}^{- 1}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0107105

Formulas:

1-

H = \sum_{i j} t_{i j} d_{i}^{+} d_{j} + U \sum_{i} f_{i}^{+} f_{i} d_{i}^{+} d_{i} + E_{f} \sum_{i} f_{i}^{+} f_{i},

2-

t_{i j} = - t

3-

t_{i j} = 0

4-

(f_{i}^{+} f_{i})

5-

{\tilde{t}}_{i j} = t_{i j} + t_{i j}^{'} (f_{i}^{+} f_{i} + f_{j}^{+} f_{j}),

6-

H = \sum_{< i j >} t_{i j} d_{i}^{+} d_{j} + \sum_{< i j >} t_{i j}^{'} (f_{i}^{+} f_{i} + f_{j}^{+} f_{j}) d_{i}^{+} d_{j} + U \sum_{i} f_{i}^{+} f_{i} d_{i}^{+} d_{i} + E_{f} \sum_{i} f_{i}^{+} f_{i} .

7-

n_{d} = \frac{1}{L} \sum_{i} d_{i}^{+} d_{i}

8-

n_{f} = \frac{1}{L} \sum_{i} f_{i}^{+} f_{i}

9-

f_{i}^{+} f_{i}

10-

H = \sum_{< i j >} h_{i j} (w) d_{i}^{+} d_{j} + E_{f} \sum_{i} w_{i},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/0709.0461

Formulas:

1-

y_{m} = \int_{𝒜} ϕ_{m}^{*} (\hat{𝛀}) E (\hat{𝛀}) d Ω,

2-

ϕ_{m} (\hat{𝛀})

3-

E (\hat{𝛀})

4-

z_{p} = \sum_{m = 1}^{M} S_{p m} y_{m}, p = 1 \to P,

5-

ψ_{p} (\hat{𝛀})

6-

ψ_{p} (\hat{𝛀}) = \sum_{m = 1}^{M} S_{p m}^{*} ϕ_{m} (\hat{𝛀}),

7-

z_{p} = \int_{𝒜} ψ_{p}^{*} (\hat{𝛀}) E (\hat{𝛀}) d Ω

8-

⟨ z_{p} z_{p^{'}}^{*} ⟩ = \int_{𝒜} \int_{𝒜} ψ_{p}^{*} ({\hat{𝛀}}_{1}) ψ_{p^{'}} ({\hat{𝛀}}_{2}) Γ ({\hat{𝛀}}_{1}, {\hat{𝛀}}_{2}) d Ω_{1} d Ω_{2},

9-

Γ ({\hat{𝛀}}_{1}, {\hat{𝛀}}_{2}) \equiv ⟨ E ({\hat{𝛀}}_{1}) E^{*} ({\hat{𝛀}}_{2}) ⟩

10-

Γ ({\hat{𝛀}}_{1}, {\hat{𝛀}}_{2})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: quant-ph

Result: incorrect

Paper: https://arxiv.org/abs/1902.03748

Formulas:

1-

T_{o b s}

2-

T_{o b s + 1}

3-

T_{p r e d}

4-

T_{p r e d}

5-

T_{o b s} \times d

6-

T_{o b s} \times d

7-

T_{o b s} \times h \times w

8-

ℝ^{T_{o b s} \times C}

9-

ℝ^{T_{o b s} \times d}

10-

(x_{b}, y_{b}, w_{b}, h_{b})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1005.2341

Formulas:

1-

c^{'} = f λ

2-

(4 - 6) \cdot 10^{7} m / s

3-

\tilde{P} (r, t) = A g_{m n} (r) \sin (ω t + φ)

4-

g_{m n} (r) = J_{m} (χ_{m n}^{c} ρ / a) \cos (m ϕ)

5-

c^{'} χ_{m n}^{c} / 2 π a

6-

f = N V / Φ_{0}

7-

c^{'} \approx 7 \cdot 10^{7}

8-

c^{'} χ_{m n} / 2 π a

9-

a_{i} / a < 0.3

10-

a_{i} / a << 1

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1806.09381

Formulas:

1-

Γ_{i} = γ_{i} \min (\frac{E_{i} - β E}{E - β E}, 0)

2-

𝒟 = {d_{1}, d_{2}, \dots, d_{N}}

3-

𝒮 = {s_{1}, s_{2}, \dots, s_{M}}

4-

G = (V, W)

5-

P_{l, m i}

6-

P_{s, i j}

7-

S N R_{L R}

8-

S N R_{S R}

9-

P_{l, m i}

10-

P_{s, i j}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1606.02660

Formulas:

1-

ϕ : V (G) \to V (H)

2-

ϕ (x) ϕ (y) \in E (H)

3-

x y \in E (G)

4-

Hom (G, H)

5-

hom (G, H) = | Hom (G, H) |

6-

hom (G, H_{ind})

7-

Hom (G, K_{q})

8-

hom (G, H)

9-

i (G) = hom (G, H_{ind})

10-

i (G) \leq i {(K_{r, r})}^{n / 2 r} .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/hep-ph/9410202

Formulas:

1-

S U {(3)}_{L} \times S U {(3)}_{R}

2-

S U {(3)}_{V}

3-

F_{2} (Q^{2})

4-

ℒ = ℒ_{χ} + ℒ_{I} + ℒ_{C} .

5-

ℒ_{χ} = \bar{ψ} (i \partial^{μ} γ_{μ} - \hat{m}) ψ + ℒ_{4} + ℒ_{6} .

6-

U_{A} (1)

7-

m_{u} = m_{u}^{0} - g_{4} ⟨ u u ⟩ - g_{6} ⟨ \bar{d} d ⟩ ⟨ \bar{s} s ⟩

8-

r_{q} = ξ / m_{q}

9-

N (s, b) \propto N (s) \cdot D_{c}^{A} \otimes D_{c}^{B},

10-

N (s) ≃ κ \frac{(1 - {⟨ x ⟩}_{q}) \sqrt{s}}{m_{q}},

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9705185

Formulas:

1-

Σ_{c r i t} = (c^{2} / 4 π G) (D_{s} / D_{d} D_{d s})

2-

Σ^{*} \approx 0.2 Σ_{c r i t}

3-

Ω_{0} + λ_{0} = 1

4-

2.2 \times 10^{12} Ω_{0}

5-

W (r, s)

6-

Σ \geq 0.1 Σ^{*}

7-

𝜽 = (θ_{x}, θ_{y})

8-

𝜶 = \frac{4 G}{c^{2}} D_{d} \sum_{i} σ_{i} \int_{s_{i}} \frac{𝜽 - 𝜽_{i}}{{| 𝜽 - 𝜽_{i} |}^{2}} d^{2} 𝜽_{i},

9-

𝜷 - 𝜽 = \frac{D_{d s}}{D_{s}} 𝜶 .

10-

\partial 𝜷 / \partial 𝜽

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0802.2926

Formulas:

1-

V_{i j} (R)

2-

V (r) = \frac{C_{1}}{r^{η_{1}}} + \frac{C_{2}}{r^{η_{2}}} \cos (k_{*} r + ϕ_{*})

3-

χ^{2} \equiv \sum Δ E_{i}^{2} / σ_{E}^{2} + \sum {| Δ 𝐅_{j} |}^{2} / σ_{F}^{2}

4-

{Δ E_{i}}

5-

{Δ 𝐅_{j}}

6-

σ_{E} / σ_{F} \sim 10^{- 3}

7-

I m \bar{3}

8-

I m \bar{3}

9-

I m m m

10-

Δ E_{VASP, pair}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Run 11332330 (Agent397)