Run 11299133

Paper: https://arxiv.org/abs/1111.4671

Formulas:

1-

{| 0 ⟩}_{L} \equiv | H ⟩

2-

{| 1 ⟩}_{L} \equiv | V ⟩

3-

{| 0 ⟩}_{L} \equiv {| 1 ⟩}_{U} {| 0 ⟩}_{D}

4-

{| 1 ⟩}_{L} \equiv {| 0 ⟩}_{U} {| 1 ⟩}_{D}

5-

{| 0 ⟩}_{U} {| 1 ⟩}_{D}

6-

{| 0 ⟩}_{U} {| 1 ⟩}_{D}

7-

\cos θ {| 0 ⟩}_{U} {| 1 ⟩}_{D} + \sin θ {| 1 ⟩}_{U} {| 0 ⟩}_{D}

8-

{| 0 ⟩}_{U} {| 1 ⟩}_{D}

9-

{| 0 ⟩}_{U} {| 1 ⟩}_{D}

10-

\cos^{2 N} θ

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/2012.13563

Formulas:

1-

P = (f_{θ}^{0} \circ f_{θ}^{1} \dots f_{θ}^{T}) (B),

2-

f_{θ}^{t} = f_{θ}, \forall t

3-

P^{*} = F_{θ} (P^{*}; B),

4-

h^{k + 1} = f_{θ}^{k} (h^{k}; x), k = 0, 1, 2, \dots, L - 1

5-

f_{θ}^{k} = f_{θ},

6-

lim_{k \to \infty} h^{k + 1} = lim_{k \to \infty} f_{θ} (h^{k}; x) = f_{θ} (h^{*}; x) = h^{*},

7-

h^{*} = f_{θ} (h^{*}; x),

8-

B = {B_{1}, B_{2}, \dots, B_{n}}

9-

P^{0} = {P_{1}^{0}, P_{2}^{0}, \dots, P_{n}^{0}}

10-

Z = {Z_{1}, Z_{2}, \dots, Z_{n}}

Formulas (html):

Correct Categorie: cs

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1706.05569

Formulas:

1-

vec (𝒙, 𝒚)

2-

𝒙_{t} \in SE (3)

3-

𝑹_{t} \in SO (3)

4-

{\dot{𝒑}}_{t} \in ℝ^{3}

5-

ℒ ≜ {𝒍^{[j]} \in ℝ^{3} | j = 1 : n_{l}}

6-

𝒵_{t} \subset ℝ_{\geq 0}

7-

p (z_{t} | 𝒙_{t}, 𝒍)

8-

p (𝐱_{t} | 𝐱_{t - 1}, 𝐮_{t - 1})

9-

z_{1 : t} ≜ {z_{1}, \dots, z_{t}}

10-

p (𝐱_{0})

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1304.2651

Formulas:

1-

2 π c_{1} (M)

2-

\frac{\partial g}{\partial t} = g - Ric (g), g (0) = g_{0} .

3-

κ = κ (g (0)) > 0

4-

{vol}_{g (t)} (B_{g (t)} (x, r)) \geq κ r^{2 n}, \forall t \geq 0, r \leq 1 .

5-

(M, g (t_{i}))

6-

(M_{\infty}, d)

7-

(M, g (t_{i})) \overset{d_{G H}}{⟶} (M_{\infty}, d) .

8-

(M, g (t))

9-

2 π c_{1} (M)

10-

(M_{\infty}, d)

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0905.0519

Formulas:

1-

2 p_{3 / 2}

2-

3 d_{5 / 2}

3-

2 p_{1 / 2}

4-

3 d_{3 / 2}

5-

2 p^{5} 3 l

6-

3 s / 3 d

7-

Ω_{01} = 2 \sum_{k} \sum_{\binom{α_{0} α_{1}}{β_{0} β_{1}}} Q^{k} (α_{0} α_{1}; β_{0} β_{1}) < ψ_{0} | | Z^{k} (α_{0}, α_{1}) | | ψ_{1} > < ψ_{0} | | Z^{k} (β_{0}, β_{1}) | | ψ_{1} >,

8-

Q^{k} (α_{0} α_{1}; β_{0} β_{1}) = \sum_{κ_{0} κ_{1}} {[k]}^{- 1} P^{k} (κ_{0} κ_{1}; α_{0} α_{1}) P^{k} (κ_{0} κ_{1}; β_{0} β_{1}),

9-

P^{k} (κ_{0} κ_{1}; α_{0} α_{1}) = X^{k} (α_{0} κ_{0}; α_{1} κ_{1}) + \sum_{t} {(- 1)}^{k + t} [k] {\binom{j_{α_{0}}}{j_{0}} \binom{j_{1}}{j_{α_{1}}} \binom{t}{k}} X^{t} (α_{0} κ_{0}; κ_{1} α_{1}),

10-

P_{C B}^{k} (κ_{0} κ_{1}; α_{0} α_{1}) = M_{k} (α_{0} α_{1}) R_{k} (κ_{0} κ_{1}),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/2011.11283

Formulas:

1-

\tilde{t} = t^{'} \bar{u} / d = 0

2-

σ_{c}^{2} (t)

3-

D_{m} \approx 10^{- 11}

4-

ℓ \sim \bar{u} t^{'}

5-

r_{D} \sim \sqrt{4 D_{m} t^{'}}

6-

σ_{c}^{2} \sim e^{- γ_{2} \tilde{t}}

7-

\tilde{t} = t^{'} \bar{u} / d

8-

σ_{c}^{2} \approx 10^{- 3}

9-

Pe = \bar{u} d / D_{m}

10-

t^{'} \bar{u} / d

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0301381

Formulas:

1-

\dot{E} = η v^{3} / L,

2-

E \propto v^{2} \propto k^{- 10 / 3}

3-

k = 2 π / ℓ

4-

S_{p} (\vec{r}) = ⟨ {v (\vec{x}) - v (\vec{x} + \vec{r})}^{p} ⟩

5-

ρ v^{2} \propto k^{- 4}

6-

A = v_{rms} / v_{A} = v_{rms} / {(B^{2} / 4 π ρ)}^{- 1 / 2}

7-

τ_{d} = K / \dot{K}

8-

τ_{d} / τ_{ff} = \frac{1}{4 π ξ} {(\frac{32}{3})}^{1 / 2} \frac{κ}{ℳ_{rms}} ≃ 3.9 \frac{κ}{ℳ_{rms}},

9-

ℳ_{rms} = v_{rms} / c_{s}

10-

k - 1 \leq | \vec{k} | \leq k

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1606.06585

Formulas:

1-

𝒓_{i} (t)

2-

𝒖_{i} (t)

3-

U (r_{i j}) = 4 ϵ [{(\frac{σ}{r_{i j}})}^{12} - {(\frac{σ}{r_{i j}})}^{6}],

4-

r_{i j} = | 𝒓_{j} - 𝒓_{i} |

5-

\frac{d 𝒓_{i}}{d t}

6-

= - \frac{1}{η} \sum_{j \neq i} \frac{\partial U (r_{i j})}{\partial 𝒓_{i}} + υ_{0} 𝒖_{i} + \sqrt{2 D_{t r}} 𝝃_{i}^{t r},

7-

\frac{d 𝒖_{i}}{d t}

8-

= \sqrt{2 D_{r}} (𝒖_{i} \times 𝝃_{i}^{r}) .

9-

D_{t r} = 1 / (β_{s} η)

10-

⟨ 𝝃_{i}^{t r, r} (t) ⟩

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0305658

Formulas:

1-

(n + 1) s

2-

d_{x z} + d_{y z}

3-

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

4-

d_{x z} + d_{y z}

5-

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

6-

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

7-

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

8-

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

9-

d_{x z} + d_{y z}

10-

d_{x z} + d_{y z}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0603200

Formulas:

1-

\sim 0.6 n m

2-

v_{y} = 200 m / s

3-

v_{y} = 200 m / s

4-

λ_{d b} = 2.8

5-

y = v_{y} t

6-

z = - \frac{1}{2} g t^{2}

7-

ψ (x, t)

8-

ψ_{0} (x) = {(2 π σ_{0}^{2})}^{- \frac{1}{4}} \exp^{- \frac{x^{2}}{4 σ_{0}^{2}}}

9-

σ_{0} = 2 μ m

10-

ψ (x, t) = {(2 π s {(t)}^{2})}^{- \frac{1}{4}} \exp^{- \frac{x^{2}}{4 σ_{0} s (t)}}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0309103

Formulas:

1-

q = M_{2} / M_{1}

2-

q = M_{2} / M_{1}

3-

[F e / H] \overset{>}{\sim} - 1

4-

[F e / H] > - 1.09

5-

f (M) = \frac{{(M_{2} s i n i)}^{3}}{{(M_{1} + M_{2})}^{2}} = \frac{q^{3}}{{(1 + q)}^{2}} M_{1} s i n^{3} i

6-

q_{m i n}

7-

q_{m i n}

8-

M_{1, e s t}

9-

M_{1, e s t}

10-

q_{m i n}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0010040

Formulas:

1-

T_{v i r} \sim 10^{7}

2-

z_{F e} \sim 0.5 - 1

3-

T \sim T_{v i r}

4-

0.1 r_{e} \begin{matrix} < \\ \sim \end{matrix} r \begin{matrix} < \\ \sim \end{matrix} r_{e}

5-

r \begin{matrix} < \\ \sim \end{matrix} 0.1 r_{e}

6-

\sim T_{v i r}

7-

r \begin{matrix} > \\ \sim \end{matrix} 0.1 r_{e}

8-

M (H I) + M (H_{2}) < 10^{7}

9-

M (H I I) \sim 10^{5}

10-

0.1 r_{e} \begin{matrix} < \\ \sim \end{matrix} r \begin{matrix} < \\ \sim \end{matrix} r_{e}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1901.05003

Formulas:

1-

⟨ i | U_{C} | 0 ⟩

2-

(2 N - 1) \times L

3-

{p (i) | i = 0, \dots, 2^{n} - 1}

4-

p (i) = - \ln (1 - i / 2^{n}) / 2^{n}

5-

\min (p (i) / p_{t h}, 1)

6-

{(1, 1)}^{T}

7-

{(1, 1)}^{T}

8-

{p (i) | i = 0, \dots, 2^{n} - 1}

9-

\Pr (p) = e^{- 2^{n} \times p}

10-

p (i) = - \log (1 - i / 2^{n}) / 2^{n}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1204.3480

Formulas:

1-

S_{d} = 5 / 2, g \approx 2

2-

T_{N} = T_{MI} = 8

3-

[0 \bar{1} 1]

4-

χ_{CW} (T)

5-

a^{*} \times b \times c \sim 0.05 \times 3 \times 3

6-

μ_{0} H = 7

7-

μ_{0} H = 7

8-

μ_{0} H = 7

9-

H ⊥ [0 \bar{1} 1]

10-

∠ (H, a^{*}) = 45^{\circ}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1409.4458

Formulas:

1-

ψ (2 S)

2-

2.5 < y_{lab} < 4

3-

- 4 < η < - 2.5

4-

L_{int} = 1.35 \pm 0.07

5-

2.03 < y_{cms} < 3.53

6-

- 4.46 < y_{cms} < - 2.96

7-

L_{int} = 5.01 \pm 0.17

8-

L_{int} = 5.81 \pm 0.18

9-

L_{int} = 68.8 \pm 0.9

10-

σ_{Pb - Pb}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1405.5564

Formulas:

1-

\sim 6 \times 10^{- 14} erg {cm}^{- 2} s^{- 1}

2-

\sim 8 \times 10^{- 14} erg {cm}^{- 2} s^{- 1}

3-

N_{H} = 4.4 \times 10^{21} {cm}^{- 2}

4-

4 e 21 < N_{H} < 6 e 21 {cm}^{- 2}

5-

α_{J2000} = 15^{h} 07^{m} {08.9}^{s} \pm {0.5}^{s}

6-

δ_{J2000} = - 62^{\circ} 16^{'} {44.4}^{''} \pm {0.5}^{''}

7-

N_{H} = 4.4 \times 10^{21} {cm}^{- 2}

8-

F_{2 - 10 k e V} = (1.5 \pm 0.9) \times 10^{- 14} erg {cm}^{- 2} s^{- 1}

9-

F_{2 - 10 k e V} = (6 \pm 2) \times 10^{- 15} erg {cm}^{- 2} s^{- 1}

10-

α_{J2000} = 15^{h} 07^{m} 06^{s} \pm 3^{s}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0902.3226

Formulas:

1-

w_{d e} = p_{d e} / ρ_{d e}

2-

w_{d e} (z) = w_{0} + w_{a} z / (1 + z)

3-

δ p_{d e} = s_{eff} δ ρ_{d e}

4-

δ ρ_{d e}

5-

N_{i} = Δ Ω \int_{z_{i}}^{z_{i + 1}} 𝑑 z \int_{M_{i}}^{M_{i + 1}} 𝑑 M θ [M - M_{m i n} (z)] \frac{d n}{d M d z},

6-

M_{m i n}

7-

s_{e f f}

8-

s_{e f f} = 0

9-

s_{e f f} = - 0.75

10-

s_{e f f}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0906.4803

Formulas:

1-

a = 3 \times 10^{12}

2-

T_{⋆} = 3 \times 10^{4}

3-

{\dot{M}}_{⋆} = 3 \times 10^{- 6} M_{⊙}

4-

v_{w} \sim 2.5 \times 10^{8}

5-

\sim 3 \times (10^{- 3} - 10^{- 2}) a

6-

f = {\dot{M}}_{⋆} / 4 π a^{2} m_{p} v_{c} n_{c} = 0.005

7-

v_{c} = 2.5 \times 10^{8}

8-

R_{j} (z) = 0.1 z

9-

v_{\exp} = 0.1 v_{j}

10-

\sim v_{j} / v_{\exp} \sim 10

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1610.08159

Formulas:

1-

p (z) = a_{0} + a_{1} z + a_{2} z^{2} + a_{3} z^{3} + \dots + a_{n} z^{n}

2-

\max_{| z | = R} | p (z) | \leq (\frac{R^{n} + 1}{2}) \max_{| z | = 1} | p (z) | .

3-

S (0, K) := {z : | z | = K}, 0 < K \leq 1,

4-

{(\frac{\max_{| z | = R} | p (z) |}{\max_{| z | = 1} | p (z) |})}^{s}

5-

p (z) = \sum_{j = 0}^{n} a_{j} z^{j}

6-

M (p, r) := \max_{| z | = r} | p (z) |, r > 0,

7-

|| p || := \max_{| z | = 1} | p (z) |,

8-

D (0, K) := {z : | z | < K}, K > 0 .

9-

p (z) = \sum_{j = 0}^{n} a_{j} z^{j}

10-

M (p, R) \leq R^{n} || p || .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1410.1782

Formulas:

1-

H = \frac{⟨ x, N ⟩}{2}

2-

H = \frac{⟨ x, N ⟩}{2} + λ,

3-

d V_{μ} = \exp (- \frac{{| x |}^{2}}{4}) d V

4-

d A_{μ} = \exp (- \frac{{| x |}^{2}}{4}) d A

5-

F : Σ \times [- ϵ, ϵ] \to ℝ^{n + 1}

6-

⟨ \partial_{t} F (x, 0), N (x) ⟩ = u

7-

Σ_{t} = F (σ, t)

8-

\begin{matrix} \partial_{t} V_{μ} (Ω_{t}) & = \int_{Σ} u 𝑑 A_{μ}, \\ \partial_{t} A_{μ} (Σ_{t}) & = \int_{Σ} u (H - \frac{⟨ x, N ⟩}{2}) 𝑑 A_{μ} . \end{matrix}

9-

\int_{Σ} u 𝑑 A_{μ} = 0

10-

k = - ⟨ x, N ⟩ + λ .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1409.2946

Formulas:

1-

\dot{M} = 0.1 \times 10^{- 8} M_{⊙} {yr}^{- 1}

2-

\dot{M} = 2 \times 10^{- 8} M_{⊙} {yr}^{- 1}

3-

R \propto M^{0.226 - 0.262}

4-

Σ_{g} = \frac{\dot{M}}{3 π α (a) c_{s} h} \exp (\frac{- t}{τ_{dep}}) η,

5-

\dot{M} ≃ 2.5 \times 10^{- 8} {(\frac{M_{*}}{M_{⊙}})}^{1.3 \pm 0.3} M_{⊙} {yr}^{- 1} .

6-

\sim 2.5 \times 10^{- 8}

7-

M_{⊙} {yr}^{- 1}

8-

M_{⊙} {yr}^{- 1}

9-

α_{eff} (a) = \frac{α_{dead} - α_{mri}}{2} [\erf (\frac{a - a_{crit}}{0.1 a_{crit}}) + 1] + α_{mri},

10-

η = 0.5 [\erf (\frac{a - a_{mstr}}{0.1 a_{mstr}}) + 1],

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1710.11152

Formulas:

1-

R_{p} = 1 - 4 R_{\oplus}

2-

𝒟 = \sum_{i = 1}^{N_{sys}} \sum_{\begin{matrix} j = 1 \\ P_{j} < P_{j + 1} \end{matrix}}^{N_{pl} - 1} {[{(\log \frac{R_{j + 1}}{R_{j}})}^{2} + {(\log \frac{M_{j + 1}}{M_{j}})}^{2}]}^{1 / 2} .

3-

ℳ = (M_{1}, M_{2}, \dots, M_{N})

4-

ℛ = (R_{1}, R_{2}, \dots, R_{N})

5-

{1, 2, \dots, N}

6-

{𝒟 (X) = 𝒟 [ℳ (X), ℛ (X)] ∣ X \in S_{N}}

7-

{𝒟 (X) ∣ X \in S_{N}}

8-

{𝒟 (X) ∣ X \in S_{N}}

9-

𝒟_{R} = \sum_{i = 1}^{N_{sys}} \sum_{\begin{matrix} j = 1 \\ P_{j} < P_{j + 1} \end{matrix}}^{N_{pl} - 1} | \log \frac{R_{j + 1}}{R_{j}} |,

10-

𝒪_{R} = \sum_{i = 1}^{N_{sys}} \sum_{\begin{matrix} j = 1 \\ P_{j} < P_{j + 1} \end{matrix}}^{N_{pl} - 1} \log \frac{R_{j + 1}}{R_{j}},

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1804.10294

Formulas:

1-

2.7 \pm 0.5 (s t a t) \pm 0.5 (s y s t) \times 10^{- 37} {cm}^{2} / Ar

2-

8.4 \pm 0.9 {(s t a t)}_{- 0.8}^{+ 1.0} (s y s t) \times 10^{- 38} {cm}^{2} / Ar

3-

d E / d x

4-

d E / d x

5-

\frac{d σ}{d X} = \frac{N - N_{b}}{Δ X N_{A r} Φ ϵ_{UF}},

6-

d σ / d p_{μ}

7-

d σ / d θ_{μ}

8-

d σ / d θ_{π}

9-

d σ / d θ_{μ π}

10-

d σ / d p_{μ}

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0406215

Formulas:

1-

(0, 0, 1.25)

2-

(0, 0, 1.5)

3-

(0, 0, 2)

4-

(0, 0, 2.2)

5-

(0, 0, 3.8)

6-

(0, 0, 4.8)

7-

3 \times 0.5 \times 2 {mm}^{3}

8-

(0, 0, 2)

9-

(h, k, l, m)

10-

(h, k, l, 0)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1609.00526

Formulas:

1-

I (V) \neq I (- V)

2-

I (V) \neq I (- V)

3-

H = H_{el} + H_{ch} + H_{tn}

4-

H_{ch} = \sum_{i = 1} ϵ_{i} c_{i}^{†} c_{i} + \sum_{i = 1} t (c_{i + 1}^{†} c_{i} + c_{i}^{†} c_{i + 1})

5-

ϵ_{i} = ϵ_{i}^{0} + ϵ_{i} (V)

6-

ϵ_{i} (V)

7-

ϵ_{i} (V) = V / 2 - i V / (N + 1)

8-

A_{n} = e^{i k n} + r e^{- i k n}

9-

B_{n} = τ e^{i k n}

10-

T (E) = {| B_{1} |}^{2} = {| τ |}^{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1906.00289

Formulas:

1-

H_{D C} = \sum_{i = 1}^{N} (c 𝜶_{𝒊} \cdot 𝒑_{𝒊} + (β_{i} - 1) c^{2} + V_{i}) + \sum_{i > i}^{N} \frac{1}{r_{i j}},

2-

Ψ (Γ P J)

3-

Φ (γ_{i} P J)

4-

Ψ (γ P J) = \sum_{i = 1}^{M} c_{i} Φ (γ_{i} P J),

5-

ϕ (𝐫) = \frac{1}{r} (\begin{matrix} P_{n κ} (r) χ_{κ m} (θ, ϕ) \\ i Q_{n κ} (r) χ_{- κ m} (θ, ϕ) \end{matrix}) .

6-

H^{Z F} = (\begin{matrix} H^{(P P)} & H^{(P Q)} \\ H^{(Q P)} & H^{(Q Q)} \end{matrix}),

7-

H^{(Q Q)}

8-

B_{i j} = - \sum_{i < j}^{N} \frac{1}{2 r_{i j}} [(𝜶_{i} \cdot 𝜶_{j}) + \frac{(𝜶_{i} \cdot 𝒓_{i j}) (𝜶_{j} \cdot 𝒓_{i j})}{r_{i j}^{2}}],

9-

N_{C S F}

10-

N_{C S F}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1902.09224

Formulas:

1-

# {n \leq x : f (n) = k} \sim A_{k} x

2-

# {n \leq x : f (n) = ω (n) - k} \sim \frac{B x {(\log \log x)}^{k}}{k! \log x},

3-

A_{k}, B > 0

4-

n = p_{1}^{a_{1}} \dots p_{s}^{a_{s}}

5-

p_{1} < \dots < p_{s}

6-

a_{1}, \dots, a_{s}

7-

a_{1}, \dots, a_{s}

8-

f (1) := 0

9-

f (n) := # {a_{1}, \dots, a_{s}}

10-

{(A_{k})}_{k \geq 1}

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0606017

Formulas:

1-

{| Ψ ⟩}_{k} = {| x ⟩}_{A_{k}} \otimes {| y ⟩}_{B_{k}},

2-

P_{a}^{A} = P_{a}^{B} = 0

3-

C = 2 \sqrt{P_{a}^{A}} .

4-

ρ_{1, 2} = \sum_{i} p_{i} ρ_{1}^{(i)} \otimes ρ_{2}^{(i)},

5-

{| Ψ ⟩}_{1} \otimes {| Ψ ⟩}_{2}

6-

P_{a}^{A} = P_{a}^{B} = 1 / 4

7-

ρ_{1, 2} = \sum_{i} p_{i} {| Ψ^{(i)} ⟩}_{1} ⟨ Ψ^{(i)} | \otimes {| Ψ^{(i)} ⟩}_{2} ⟨ Ψ^{(i)} |,

8-

{| Ψ^{(i)} ⟩}_{k}

9-

H^{A_{k}} \otimes H^{B_{k}}

10-

P_{a}^{A} = {[\sum_{i} p_{i} C (| Ψ^{(i)} ⟩)]}^{2} / 4,

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/1511.03788

Formulas:

1-

N := dim Π_{n} = (\binom{n + 2}{2}) .

2-

𝒳_{s} = {(x_{1}, y_{1}), (x_{2}, y_{2}), \dots, (x_{s}, y_{s})}

3-

{c_{1}, \dots, c_{s}}

4-

p (x_{i}, y_{i}) = c_{i}, i = 1, 2, \dots s .

5-

A = (x_{k}, y_{k}) \in 𝒳_{s}

6-

p (x_{k}, y_{k}) = 1 and p (x_{i}, y_{i}) = 0, for all 1 \leq i \leq s, i \neq k .

7-

ℓ (x, y) = 0 .

8-

p = ℓ r, 𝑤ℎ𝑒𝑟𝑒 r \in Π_{n - 1} .

9-

ℓ_{1}, ℓ_{2}, ℓ_{3}

10-

ℓ_{1}^{^{'}}, ℓ_{2}^{^{'}}, ℓ_{3}^{^{'}}

Formulas (html):

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xref="S1.p3.3.m3.1.1.1.1.1.3.cmml">k</mi></msub><mo id="S1.p3.3.m3.2.2.2.2.4" xref="S1.p3.3.m3.2.2.2.3.cmml">,</mo><msub id="S1.p3.3.m3.2.2.2.2.2" xref="S1.p3.3.m3.2.2.2.2.2.cmml"><mi id="S1.p3.3.m3.2.2.2.2.2.2" xref="S1.p3.3.m3.2.2.2.2.2.2.cmml">y</mi><mi id="S1.p3.3.m3.2.2.2.2.2.3" xref="S1.p3.3.m3.2.2.2.2.2.3.cmml">k</mi></msub><mo stretchy="false" id="S1.p3.3.m3.2.2.2.2.5" xref="S1.p3.3.m3.2.2.2.3.cmml">)</mo></mrow><mo id="S1.p3.3.m3.2.2.6" xref="S1.p3.3.m3.2.2.6.cmml">∈</mo><msub id="S1.p3.3.m3.2.2.7" xref="S1.p3.3.m3.2.2.7.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.3.m3.2.2.7.2" xref="S1.p3.3.m3.2.2.7.2.cmml">𝒳</mi><mi id="S1.p3.3.m3.2.2.7.3" xref="S1.p3.3.m3.2.2.7.3.cmml">s</mi></msub></mrow></math>, <math><mrow id="S1.Ex2.m1.5.5.1"><mrow id="S1.Ex2.m1.5.5.1.1.2" xref="S1.Ex2.m1.5.5.1.1.3.cmml"><mrow id="S1.Ex2.m1.5.5.1.1.1.1.2" xref="S1.Ex2.m1.5.5.1.1.1.1.3.cmml"><mrow id="S1.Ex2.m1.5.5.1.1.1.1.1.1" xref="S1.Ex2.m1.5.5.1.1.1.1.1.1.cmml"><mrow 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id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.cmml"><mi id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.4" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.4.cmml">p</mi><mo id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.3" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.2" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.3.cmml"><mo stretchy="false" id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.2.3" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.3.cmml">(</mo><msub id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.1.1.1.1" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.1.1.1.1.cmml"><mi id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.1.1.1.1.2" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.1.1.1.1.2.cmml">x</mi><mi id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.1.1.1.1.3" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.2.4" xref="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.3.cmml">,</mo><msub id="S1.Ex2.m1.5.5.1.1.1.1.2.2.1.1.2.2.2.2" 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xref="S1.p4.3.m3.3.3.1.1.cmml"><mrow id="S1.p4.3.m3.3.3.1.1" xref="S1.p4.3.m3.3.3.1.1.cmml"><mrow id="S1.p4.3.m3.3.3.1.1.2" xref="S1.p4.3.m3.3.3.1.1.2.cmml"><mi mathvariant="normal" id="S1.p4.3.m3.3.3.1.1.2.2" xref="S1.p4.3.m3.3.3.1.1.2.2.cmml">ℓ</mi><mo id="S1.p4.3.m3.3.3.1.1.2.1" xref="S1.p4.3.m3.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S1.p4.3.m3.3.3.1.1.2.3.2" xref="S1.p4.3.m3.3.3.1.1.2.3.1.cmml"><mo stretchy="false" id="S1.p4.3.m3.3.3.1.1.2.3.2.1" xref="S1.p4.3.m3.3.3.1.1.2.3.1.cmml">(</mo><mi id="S1.p4.3.m3.1.1" xref="S1.p4.3.m3.1.1.cmml">x</mi><mo id="S1.p4.3.m3.3.3.1.1.2.3.2.2" xref="S1.p4.3.m3.3.3.1.1.2.3.1.cmml">,</mo><mi id="S1.p4.3.m3.2.2" xref="S1.p4.3.m3.2.2.cmml">y</mi><mo stretchy="false" id="S1.p4.3.m3.3.3.1.1.2.3.2.3" xref="S1.p4.3.m3.3.3.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="S1.p4.3.m3.3.3.1.1.1" xref="S1.p4.3.m3.3.3.1.1.1.cmml">=</mo><mn id="S1.p4.3.m3.3.3.1.1.3" xref="S1.p4.3.m3.3.3.1.1.3.cmml">0</mn></mrow><mo id="S1.p4.3.m3.3.3.1.2" xref="S1.p4.3.m3.3.3.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S1.Ex3.m1.2.2.1"><mrow id="S1.Ex3.m1.2.2.1.1.2" xref="S1.Ex3.m1.2.2.1.1.3.cmml"><mrow id="S1.Ex3.m1.2.2.1.1.1.1" xref="S1.Ex3.m1.2.2.1.1.1.1.cmml"><mi id="S1.Ex3.m1.2.2.1.1.1.1.3" xref="S1.Ex3.m1.2.2.1.1.1.1.3.cmml">p</mi><mo id="S1.Ex3.m1.2.2.1.1.1.1.2" xref="S1.Ex3.m1.2.2.1.1.1.1.2.cmml">=</mo><mrow id="S1.Ex3.m1.2.2.1.1.1.1.1.1" xref="S1.Ex3.m1.2.2.1.1.1.1.1.2.cmml"><mrow id="S1.Ex3.m1.2.2.1.1.1.1.1.1.1" xref="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.cmml"><mi mathvariant="normal" id="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.2" xref="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.2.cmml">ℓ</mi><mo id="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.1" xref="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.3" xref="S1.Ex3.m1.2.2.1.1.1.1.1.1.1.3.cmml">r</mi></mrow><mo rspace="12.5pt" id="S1.Ex3.m1.2.2.1.1.1.1.1.1.2" xref="S1.Ex3.m1.2.2.1.1.1.1.1.2.cmml">,</mo><mtext id="S1.Ex3.m1.1.1" xref="S1.Ex3.m1.1.1a.cmml">𝑤ℎ𝑒𝑟𝑒</mtext></mrow></mrow><mo mathvariant="italic" separator="true" id="S1.Ex3.m1.2.2.1.1.2.3" xref="S1.Ex3.m1.2.2.1.1.3a.cmml"> </mo><mrow id="S1.Ex3.m1.2.2.1.1.2.2" xref="S1.Ex3.m1.2.2.1.1.2.2.cmml"><mi id="S1.Ex3.m1.2.2.1.1.2.2.2" xref="S1.Ex3.m1.2.2.1.1.2.2.2.cmml">r</mi><mo id="S1.Ex3.m1.2.2.1.1.2.2.1" xref="S1.Ex3.m1.2.2.1.1.2.2.1.cmml">∈</mo><msub id="S1.Ex3.m1.2.2.1.1.2.2.3" xref="S1.Ex3.m1.2.2.1.1.2.2.3.cmml"><mi mathvariant="normal" id="S1.Ex3.m1.2.2.1.1.2.2.3.2" xref="S1.Ex3.m1.2.2.1.1.2.2.3.2.cmml">Π</mi><mrow id="S1.Ex3.m1.2.2.1.1.2.2.3.3" xref="S1.Ex3.m1.2.2.1.1.2.2.3.3.cmml"><mi id="S1.Ex3.m1.2.2.1.1.2.2.3.3.2" xref="S1.Ex3.m1.2.2.1.1.2.2.3.3.2.cmml">n</mi><mo id="S1.Ex3.m1.2.2.1.1.2.2.3.3.1" xref="S1.Ex3.m1.2.2.1.1.2.2.3.3.1.cmml">-</mo><mn id="S1.Ex3.m1.2.2.1.1.2.2.3.3.3" xref="S1.Ex3.m1.2.2.1.1.2.2.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S1.Ex3.m1.2.2.1.2">.</mo></mrow></math>, <math><mrow id="S1.Thmlemma2.p1.1.1.m1.3.3.3" xref="S1.Thmlemma2.p1.1.1.m1.3.3.4.cmml"><msub id="S1.Thmlemma2.p1.1.1.m1.1.1.1.1" xref="S1.Thmlemma2.p1.1.1.m1.1.1.1.1.cmml"><mi mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.1.1.1.1.2" xref="S1.Thmlemma2.p1.1.1.m1.1.1.1.1.2.cmml">ℓ</mi><mn mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.1.1.1.1.3" xref="S1.Thmlemma2.p1.1.1.m1.1.1.1.1.3.cmml">1</mn></msub><mo mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.3.3.3.4" xref="S1.Thmlemma2.p1.1.1.m1.3.3.4.cmml">,</mo><msub id="S1.Thmlemma2.p1.1.1.m1.2.2.2.2" xref="S1.Thmlemma2.p1.1.1.m1.2.2.2.2.cmml"><mi mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.2.2.2.2.2" xref="S1.Thmlemma2.p1.1.1.m1.2.2.2.2.2.cmml">ℓ</mi><mn mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.2.2.2.2.3" xref="S1.Thmlemma2.p1.1.1.m1.2.2.2.2.3.cmml">2</mn></msub><mo mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.3.3.3.5" xref="S1.Thmlemma2.p1.1.1.m1.3.3.4.cmml">,</mo><msub id="S1.Thmlemma2.p1.1.1.m1.3.3.3.3" xref="S1.Thmlemma2.p1.1.1.m1.3.3.3.3.cmml"><mi mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.3.3.3.3.2" xref="S1.Thmlemma2.p1.1.1.m1.3.3.3.3.2.cmml">ℓ</mi><mn mathvariant="normal" id="S1.Thmlemma2.p1.1.1.m1.3.3.3.3.3" xref="S1.Thmlemma2.p1.1.1.m1.3.3.3.3.3.cmml">3</mn></msub></mrow></math>, <math><mrow id="S1.Thmlemma2.p1.2.2.m2.3.3.3" xref="S1.Thmlemma2.p1.2.2.m2.3.3.4.cmml"><msubsup id="S1.Thmlemma2.p1.2.2.m2.1.1.1.1" xref="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.cmml"><mi mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.2" xref="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.2.cmml">ℓ</mi><mn mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.3" xref="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.3.cmml">1</mn><msup id="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.3" xref="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.3.cmml"><mi id="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.3a" xref="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.3.cmml"/><mo mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.3.1" xref="S1.Thmlemma2.p1.2.2.m2.1.1.1.1.2.3.1.cmml">′</mo></msup></msubsup><mo mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.3.3.3.4" xref="S1.Thmlemma2.p1.2.2.m2.3.3.4.cmml">,</mo><msubsup id="S1.Thmlemma2.p1.2.2.m2.2.2.2.2" xref="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.cmml"><mi mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.2" xref="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.2.cmml">ℓ</mi><mn mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.3" xref="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.3.cmml">2</mn><msup id="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.3" xref="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.3.cmml"><mi id="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.3a" xref="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.3.cmml"/><mo mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.3.1" xref="S1.Thmlemma2.p1.2.2.m2.2.2.2.2.2.3.1.cmml">′</mo></msup></msubsup><mo mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.3.3.3.5" xref="S1.Thmlemma2.p1.2.2.m2.3.3.4.cmml">,</mo><msubsup id="S1.Thmlemma2.p1.2.2.m2.3.3.3.3" xref="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.cmml"><mi mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.2" xref="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.2.cmml">ℓ</mi><mn mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.3" xref="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.3.cmml">3</mn><msup id="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.3" xref="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.3.cmml"><mi id="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.3a" xref="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.3.cmml"/><mo mathvariant="normal" id="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.3.1" xref="S1.Thmlemma2.p1.2.2.m2.3.3.3.3.2.3.1.cmml">′</mo></msup></msubsup></mrow></math>

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1603.01678

Formulas:

1-

\frac{1}{2 m_{s}} {(ℏ 𝐤 - e 𝐀)}^{2} + α {[𝝈 \times (ℏ 𝐤 - e 𝐀)]}_{z}

2-

β [𝝈 \cdot (ℏ 𝐤 - e 𝐀)],

3-

𝝈 = (σ_{x}, σ_{y}, σ_{z})

4-

𝐤 = (k_{x}, k_{y})

5-

𝐀 = (0, [E / ω] \cos ω t)

6-

{\hat{ℋ}}_{e} = {\hat{ℋ}}_{0} + {\hat{ℋ}}_{k}

7-

{\hat{ℋ}}_{0} = \frac{e^{2} E^{2}}{2 m_{s} ω^{2}} \cos^{2} ω t - (α σ_{x} + β σ_{y}) \frac{e E}{ω} \cos ω t

8-

{\hat{ℋ}}_{k} = \frac{ℏ^{2} k_{x}^{2}}{2 m_{s}} + (β σ_{x} - α σ_{y}) ℏ k_{x}

9-

i ℏ \partial Ψ_{0} / \partial t = {\hat{ℋ}}_{0} Ψ_{0}

10-

Ψ_{0}^{\pm} = \frac{1}{\sqrt{2}} (\begin{matrix} 1 \\ \pm \frac{α + i β}{γ} \end{matrix}) e^{- i \frac{e^{2} E^{2}}{4 m_{s} ℏ ω^{2}} (t + \frac{\sin 2 ω t}{2 ω})} e^{\pm i \frac{e E γ}{ℏ ω^{2}} \sin ω t},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0610643

Formulas:

1-

Γ^{*} (r)

2-

Γ^{*} (r) \propto r^{- γ}

3-

20 - 25 h^{- 1}

4-

\sim 60 h^{- 1}

5-

M_{a b s}

6-

M_{a b s} \sim - 21.4

7-

\sim 100 h^{- 1}

8-

\sim 30 h^{- 1}

9-

\sim 70 h^{- 1}

10-

α_{2000} = 275^{\circ}, δ_{2000} = 0^{\circ}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/cond-mat/0609174

Formulas:

1-

M_{Z F C}

2-

S_{Z F C} (t) = (1 / H) d M_{Z F C} (t) / d \ln t

3-

S_{Z F C} (t)

4-

M_{Z F C} (t)

5-

M_{Z F C} (t)

6-

M_{Z F C} (t)

7-

M_{Z F C} (t)

8-

M_{Z F C}

9-

M_{Z F C} (t)

10-

S_{Z F C} (t)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0009339

Formulas:

1-

t_{e x p}

2-

2.6 \times 10^{6} M_{⊙}

3-

(3 - 6) \times 10^{- 6}

4-

a_{3 - D} = G \times M / r_{3 - D}^{2}

5-

M c o s^{3} (θ) = a_{2 - D} \times r_{2 - D}^{2} / G

6-

2.3 - 3.3 \times 10^{6} M_{⊙}

7-

8 \times 10^{12} M_{⊙} / p c^{3}

8-

\sim 3 \times 10^{6} M_{⊙}

9-

2.3 - 3.3 \times 10^{6} M_{⊙}

10-

2.6 \times 10^{6} M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1409.0353

Formulas:

1-

J / ψ π π

2-

ψ (2 S)

3-

\sqrt{(s)} = 7 T e V

4-

X (3872) \to γ ψ (2 S)

5-

γ ψ (2 S)

6-

2.7 f b^{- 1}

7-

M e V / c^{2}

8-

M (J / ψ ϕ) - M (J / ψ)

9-

M (J / ψ ϕ) - M (J / ψ)

10-

M (J / ψ ϕ) - M (J / ψ)

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/9706426

Formulas:

1-

7 \times 10^{- 6} {eV}^{2},

2-

δ m_{32}^{2} \approx 0.3 {eV}^{2},

3-

U_{CF, sma} \approx (\begin{matrix} 0.994 & 0.044 & 0.100 \\ - 0.108 & 0.530 & 0.841 \\ - 0.015 & - 0.847 & 0.532 \end{matrix}) .

4-

δ m_{j i}^{2} = m_{j}^{2} - m_{i}^{2}

5-

m_{j}^{2} > m_{i}^{2}

6-

ν_{α} = \sum_{i} U_{α i} ν_{i}

7-

α = e, μ,

8-

i = 1, 2, 3

9-

2 \times 10^{- 5} {eV}^{2},

10-

δ m_{32}^{2} \approx 0.3 {eV}^{2},

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1801.03764

Formulas:

1-

\hat{U} = e^{i ϕ} [\begin{matrix} r & i t \\ i t^{*} & r^{*} \end{matrix}],

2-

{| r |}^{2} + {| t |}^{2} = 1

3-

\hat{U} = {\hat{U}}^{T}

4-

{\hat{U}}_{FULL} = e^{i ϕ_{H}} [\begin{matrix} r_{H} e^{- i θ_{H}} & i t_{H} & 0 & 0 \\ i t_{H} & r_{H} e^{i θ_{H}} & 0 & 0 \\ 0 & 0 & r_{V} e^{i (Φ - θ_{V})} & i t_{V} e^{i Φ} \\ 0 & 0 & i t_{V} e^{i Φ} & r_{V} e^{i (Φ + θ_{V})} \end{matrix}],

5-

Φ = ϕ_{V} - ϕ_{H}

6-

𝑬 = {[E_{1 H}, E_{2 H}, E_{1 V}, E_{2 V}]}^{T}

7-

= e^{i ϕ_{H}} [\begin{matrix} r_{H} e^{- i θ_{H}} & 0 \\ 0 & r_{V} e^{i (Φ - θ_{V})} \end{matrix}],

8-

= {\hat{J}}_{2 \to 1} = e^{i ϕ_{H}} [\begin{matrix} i t_{H} & 0 \\ 0 & t_{V} e^{i Φ} \end{matrix}],

9-

= e^{i ϕ_{H}} [\begin{matrix} r_{H} e^{i θ_{H}} & 0 \\ 0 & r_{V} e^{i (Φ + θ_{V})} \end{matrix}] .

10-

ψ = θ_{H} - θ_{V}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/1302.0593

Formulas:

1-

E_{P N C} = \sum_{n} [\frac{⟨ b | {\hat{d}}_{E 1} | n ⟩ ⟨ n | {\hat{h}}_{W} | a ⟩}{E_{a} - E_{n}} + \frac{⟨ b | {\hat{h}}_{W} | n ⟩ ⟨ n | {\hat{d}}_{E 1} | a ⟩}{E_{b} - E_{n}}],

2-

δ E_{n} = ⟨ n | \hat{L} | n ⟩ .

3-

({\hat{H}}_{0} + {\hat{Σ}}^{(2)} - E_{n}) ψ_{n}^{(BO)} = 0,

4-

λ {\hat{Σ}}^{(2)}

5-

n p_{1 / 2}

6-

n p_{3 / 2}

7-

{\hat{d}}_{E 1} \to {\hat{d}}_{E 1} + δ V_{E 1}

8-

{\hat{h}}_{W} \to {\hat{h}}_{W} + δ V_{W}

9-

({\hat{H}}_{0} - ε_{c}) δ ψ_{c} = - (\hat{f} + δ V_{f}) ψ_{c} .

10-

{\hat{H}}_{0} + {\hat{Σ}}^{(2)}

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1101.2855

Formulas:

1-

H_{0} = 71 km s^{- 1} {Mpc}^{- 1}

2-

Δ (g - i)

3-

\sim 10 μ m

4-

A_{V} / τ_{9.7 μ m} = 18.0 \pm 1.0

5-

A_{V} = 1.086 τ_{V}

6-

τ_{9.7 μ m} \geq 1

7-

λ ≳ 30 μ m

8-

\sim 1 μ m

9-

R_{o} ≲ a few pc

10-

E = \frac{1}{N} \sqrt{\sum_{i = 1}^{N} (\frac{F_{A G N} \cdot f_{i}^{m} - f_{i}^{obs}}{σ_{i}})}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1610.00117

Formulas:

1-

G (n, p)

2-

G (n, p)

3-

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

4-

G (n, p)

5-

(\binom{n}{2}) p

6-

G (n, p)

7-

G (n, p)

8-

G (n, p)

9-

G (n, p)

10-

h (G) > 0

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/hep-ph/0310026

Formulas:

1-

n K^{+} K_{s}^{0}

2-

γ p \to p K^{+} K^{-}

3-

S U_{f} (3)

4-

V_{s u r f a c e} (r) = - V_{0}^{s} [\frac{4 e^{(r - R) / a}}{{(1 + e^{(r - R) / a})}^{2}}] .

5-

L = 2, 3, 4

6-

V_{0}^{s} = A [α + β F (L)],

7-

c_{l} = (l - 1) (l + 2) ρ_{0}^{2} σ,

8-

\sim r d V^{v o l} / d r

9-

V_{0}^{S} = A [α + β (L - 1) (L + n)],

10-

L = 2, 3, 4

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/0803.1403

Formulas:

1-

E_{sn} = 1.2 \times 10^{51}

2-

E_{sn} = 1.2 \times 10^{51}

3-

F_{γ} \propto d^{- 7}

4-

B = 10^{2} - 10^{3} μ

5-

M_{ej} = 1.4 M_{⊙}

6-

E_{sn} = 0.27 \times 10^{51}

7-

p \approx m_{p} c

8-

B_{d} \approx 240 μ

9-

B_{d} < 30 μ

10-

B_{d} \approx 300 \pm 60 μ

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-ex/0511046

Formulas:

1-

d N_{g} / d y \sim

2-

C (Δ ϕ) \propto \frac{d N_{r e a l} / d Δ ϕ}{d N_{m i x} / d Δ ϕ} \propto \frac{1}{N_{t r i g}} \frac{d N}{d Δ ϕ}

3-

d N_{r e a l} / d Δ ϕ

4-

d N_{m i x} / d Δ ϕ

5-

1 / N_{t r i g} d N / d Δ ϕ

6-

Δ ϕ_{F} = Δ ϕ_{N} + ϕ_{j j}

7-

⟨ \sin^{2} ϕ_{jj} ⟩ = \frac{⟨ \sin^{2} Δ ϕ_{F} ⟩ - ⟨ \sin^{2} Δ ϕ_{N} ⟩}{1 - 2 ⟨ \sin^{2} Δ ϕ_{N} ⟩}

8-

N_{p a r t}

9-

p_{T, t r i g}

10-

p_{T t r i g} <

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1504.06697

Formulas:

1-

- \sum_{i, j} J_{i j} 𝐒_{i} \cdot 𝐒_{j} - \sum_{i} D_{i} {(S_{i}^{z})}^{2},

2-

𝐒 = (g_{J} - 1) 𝐉

3-

g_{J} \sqrt{J (J + 1)}

4-

200 [K] \overset{<}{\sim} T \overset{<}{\sim} 500 [K]

5-

ℋ_{T} + ℋ_{R} + ℋ_{RT},

6-

- \sum_{i, j \in T} J_{i j}^{TT} 𝐒_{i} \cdot 𝐒_{j} - \sum_{i \in T} D_{i}^{T} {(S_{i}^{z})}^{2},

7-

- \sum_{m \in R} D_{m}^{R} {(J_{m}^{z})}^{2},

8-

\sum_{i \in T, m \in R} J_{i m}^{RT} 𝐒_{i} \cdot (g_{J} - 1) 𝐉_{m}

9-

200 [K] \overset{<}{\sim} T \overset{<}{\sim} 500 [K]

10-

{(5 d)}^{m}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1607.03170

Formulas:

1-

k_{1} \times k_{2} \times k_{3}

2-

k_{i} = b_{i} / (2 π \times 0.06)

3-

i = 1, 2, 3

4-

k_{i} = b_{i} / (2 π \times 0.04)

5-

k_{i} = b_{i} / (2 π \times 0.02)

6-

Δ H (P) = \frac{H_{{Li}_{m} C_{n}} (P) - m H_{Li} (P) - n H_{C} (P)}{m + n}

7-

H_{{Li}_{m} C_{n}}

8-

x_{C} = \frac{n}{m + n}

9-

5 {Li}_{8} C_{3} + 2 {Li}_{3} C_{4} \overset{40GPa,-2.82eV}{\to} 23 {Li}_{2} C

10-

{Li}_{4} C + 2 {Li}_{2} C \overset{40GPa,-0.38eV}{\to} {Li}_{8} C_{3}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0309017

Formulas:

1-

V (\hat{A} + \hat{B})

2-

V (\hat{A}) + V (\hat{B})

3-

V (\hat{A} \hat{B})

4-

V (\hat{A}) V (\hat{B})

5-

\forall \hat{A}, \hat{B} \in 𝒦 such that [\hat{A}, \hat{B}] = 0,

6-

A, B, C, \dots

7-

\hat{A}, \hat{B}, \hat{C}, \dots

8-

A, B, C, \dots

9-

\hat{A}, \hat{B}, \hat{C}, \dots

10-

{d^{1}, d^{2}, \dots}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/0907.1105

Formulas:

1-

𝒪 (α_{s})

2-

m_{1 / 2} < 180

3-

m_{1 / 2} \leq 360

4-

𝒪 (α_{s})

5-

p \bar{p}, p p \to q {\bar{q}}^{'} + X \to {\tilde{χ}}_{i} {\tilde{χ}}_{j} + X

6-

m_{{\tilde{χ}}_{i, j}^{0, \pm}}

7-

𝒪 (α_{s})

8-

α_{s}^{n} / p_{T}^{2} \ln^{m} (M^{2} / p_{T}^{2})

9-

m \leq 2 n - 1

10-

\frac{d σ^{RES}}{d p_{T}^{2}} (p_{T}, M, s) = \frac{M^{2}}{s} \int_{0}^{\infty} d b \frac{b}{2} J_{0} (b p_{T}) 𝒲 (b, M, s),

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/nucl-th/9905045

Formulas:

1-

S_{i j} = 1

2-

T_{i j} = 1

3-

E_{l a b} = 648

4-

E_{l a b} = 10

5-

E_{l a b} = 2.5

6-

E_{l a b} = 3.0

7-

E_{l a b} = 648

8-

E_{l a b} = 10

9-

S_{i j} = 1

10-

T_{i j} = 1

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0307212

Formulas:

1-

Ψ_{i j} \equiv \frac{\partial (δ θ_{i})}{\partial θ_{j}} \equiv (\begin{matrix} κ + γ_{1} & γ_{2} \\ γ_{2} & κ - γ_{1} \end{matrix}),

2-

δ θ_{i} (θ)

3-

Ψ_{i j} = \int_{0}^{χ_{h}} 𝑑 χ g (χ) \partial_{i} \partial_{j} Φ,

4-

g (χ) = 2 \int_{χ}^{χ_{h}} 𝑑 χ^{'} n (χ^{'}) \frac{r (χ) r (χ^{'} - χ)}{r (χ^{'})},

5-

r = a^{- 1} D_{A}

6-

\int 𝑑 χ n (χ) = 1

7-

\sum_{i = 1}^{2} ⟨ \tilde{γ_{i}} (𝐥) \tilde{γ_{i}} (𝐥^{'}) ⟩ = {(2 π)}^{2} δ (𝐥 - 𝐥^{'}) C_{𝐥}

8-

P (k, χ)

9-

C_{l} = \frac{9}{16} {(\frac{H_{0}}{c})}^{4} Ω_{m}^{2} \int_{0}^{χ_{h}} 𝑑 χ {[\frac{g (χ)}{a r (χ)}]}^{2} P (\frac{l}{r}, χ),

10-

P (k, χ)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1012.0405

Formulas:

1-

S_{J F} = \frac{1}{2 k} \int d^{4} x \sqrt{- g} f (R) + S_{m} (g_{μ ν}, ψ)

2-

k \equiv 8 π G

3-

{\bar{g}}_{μ ν} = Ω g_{μ ν}

4-

ϕ = \frac{1}{2 β \sqrt{k}} \ln Ω

5-

Ω \equiv \frac{d f}{d R} = f^{^{'}} (R)

6-

S_{E F} = \frac{1}{2} \int d^{4} x \sqrt{- \bar{g}} {\frac{1}{k} \bar{R} - {\bar{g}}^{μ ν} \nabla_{μ} ϕ \nabla_{ν} ϕ - 2 V (ϕ)} + S_{m} ({\bar{g}}_{μ ν} e^{2 β \sqrt{k} ϕ}, ψ)

7-

V (ϕ (R)) = \frac{R f^{'} (R) - f (R)}{2 k f^{', 2} (R)}

8-

{\bar{G}}_{μ ν} = k ({\bar{T}}_{μ ν}^{ϕ} + {\bar{T}}_{μ ν}^{m})

9-

{\bar{T}}_{μ ν}^{ϕ} = \nabla_{μ} ϕ \nabla_{ν} ϕ - \frac{1}{2} {\bar{g}}_{μ ν} \nabla^{γ} ϕ \nabla_{γ} ϕ - V (ϕ) {\bar{g}}_{μ ν}

10-

{\bar{T}}_{μ ν}^{m} = \frac{- 2}{\sqrt{- \bar{g}}} \frac{δ S_{m} ({\bar{g}}_{μ ν}, ψ)}{δ {\bar{g}}^{μ ν}}

Formulas (html):

Correct Categorie: gr-qc

Guessed Categorie: gr-qc

Result: correct

Paper: https://arxiv.org/abs/1605.06997

Formulas:

1-

a b i n i t i o

2-

a b i n i t i o

3-

F m \bar{3} m

4-

d_{x z, y z}

5-

\bar{M} - \bar{Γ} - \bar{M}

6-

\bar{M} - \bar{Γ} - \bar{M}

7-

\bar{Γ} - \bar{M} - \bar{Γ}

8-

\bar{X} - \bar{M} - \bar{X}

9-

\bar{Γ} - \bar{M} - \bar{Γ}

10-

\bar{X} - \bar{M} - \bar{X}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

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