Run 11334341

Paper: https://arxiv.org/abs/astro-ph/0602535

Formulas:

1-

β = 8 π p / B^{2}

2-

\frac{\partial 𝐁}{\partial t} + (𝐮 \cdot \nabla) 𝐁 = (𝐁 \cdot \nabla) 𝐮 - 𝐁 \nabla \cdot 𝐮, \nabla \cdot 𝐁 = 0,

3-

ρ \frac{\partial 𝐮}{\partial t} + ρ (𝐮 \cdot \nabla) 𝐮 = - \nabla [p + \frac{B^{2}}{8 π}] + \frac{(𝐁 \cdot \nabla) 𝐁}{4 π},

4-

\frac{\partial ρ}{\partial t} + (𝐮 \cdot \nabla) ρ + ρ \nabla \cdot 𝐮 = 0,

5-

\frac{\partial p}{\partial t} + (𝐮 \cdot \nabla) p + γ p \nabla \cdot 𝐮 = 0,

6-

𝐁 = (B_{0}, 0, 0)

7-

\frac{\partial b_{y}}{\partial t} + u_{x} \frac{\partial b_{y}}{\partial x} + b_{y} \frac{\partial u_{x}}{\partial x} - B_{0} \frac{\partial u_{y}}{\partial x} = 0,

8-

ρ \frac{\partial u_{y}}{\partial t} + ρ u_{x} \frac{\partial u_{y}}{\partial x} - \frac{B_{0}}{4 π} \frac{\partial b_{y}}{\partial x} = 0,

9-

\frac{\partial ρ}{\partial t} + ρ \frac{\partial u_{x}}{\partial x} + u_{x} \frac{\partial ρ}{\partial x} = 0,

10-

ρ \frac{\partial u_{x}}{\partial t} + ρ u_{x} \frac{\partial u_{x}}{\partial x} + \frac{\partial p}{\partial x} + \frac{\partial}{\partial x} \frac{b_{y}^{2}}{8 π} = 0,

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/9803108

Formulas:

1-

00^{h} 45^{m} 5^{s} .62

2-

- 25^{\circ} 33^{'} 40^{''} .2

3-

{}^{2}P_{3 / 2} -^{2} P_{1 / 2}

4-

A_{2.17 μ m} \approx 2.5 \cdot A_{12.8 μ m}

5-

\frac{Δ λ}{λ} \approx 0.015

6-

A_{12.8 μ m} = 0.04 \cdot A_{V}

7-

\frac{L_{Lyc}}{L_{[NeII]}} = 64

8-

00^{h} 45^{m} 5^{s} .62

9-

- 25^{\circ} 33^{'} 40^{''} .2

10-

4, 8, 12, 16, 20, 30, 40 \dots 100

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2002.07180

Formulas:

1-

\sim 5 \times 10^{10} M_{⊙}

2-

\sim 10^{8} M_{⊙}

3-

S_{850 μ m} = 0.81 mJy {(\frac{{SFR}_{obscured}}{100 M_{⊙} {yr}^{- 1}})}^{0.43} {(\frac{M_{d}}{10^{8} M_{⊙}})}^{0.54},

4-

S_{1.1 mm} = 0.35 mJy {(\frac{{SFR}_{obscured}}{100 M_{⊙} {yr}^{- 1}})}^{0.41} {(\frac{M_{d}}{10^{8} M_{⊙}})}^{0.56},

5-

S_{850 μ m}

6-

{SFR}_{obscured} = f_{obscured} {SFR}_{total},

7-

S_{1.1 mm} / S_{1.2 mm} = 1.36

8-

S_{1.1 mm} / S_{1.3 mm} = 1.79

9-

S_{870 μ m} / S_{850 μ m} = 0.92

10-

5 \times 10^{10} M_{⊙}

Formulas (html):

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id="S2.E2.m1.3.3.1.1.3.3" xref="S2.E2.m1.3.3.1.1.3.3.cmml">mJy</mi><mo id="S2.E2.m1.3.3.1.1.3.1a" xref="S2.E2.m1.3.3.1.1.3.1.cmml">⁢</mo><msup id="S2.E2.m1.3.3.1.1.3.4" xref="S2.E2.m1.3.3.1.1.3.4.cmml"><mrow id="S2.E2.m1.3.3.1.1.3.4.2.2" xref="S2.E2.m1.1.1.cmml"><mo id="S2.E2.m1.3.3.1.1.3.4.2.2.1" xref="S2.E2.m1.1.1.cmml">(</mo><mfrac id="S2.E2.m1.1.1" xref="S2.E2.m1.1.1.cmml"><msub id="S2.E2.m1.1.1.2" xref="S2.E2.m1.1.1.2.cmml"><mi id="S2.E2.m1.1.1.2.2" xref="S2.E2.m1.1.1.2.2.cmml">SFR</mi><mi id="S2.E2.m1.1.1.2.3" xref="S2.E2.m1.1.1.2.3.cmml">obscured</mi></msub><mrow id="S2.E2.m1.1.1.3" xref="S2.E2.m1.1.1.3.cmml"><mpadded width="+5pt" id="S2.E2.m1.1.1.3.2" xref="S2.E2.m1.1.1.3.2.cmml"><mn id="S2.E2.m1.1.1.3.2a" xref="S2.E2.m1.1.1.3.2.cmml">100</mn></mpadded><mo id="S2.E2.m1.1.1.3.1" xref="S2.E2.m1.1.1.3.1.cmml">⁢</mo><msub id="S2.E2.m1.1.1.3.3" xref="S2.E2.m1.1.1.3.3.cmml"><mi mathvariant="normal" id="S2.E2.m1.1.1.3.3.2" xref="S2.E2.m1.1.1.3.3.2.cmml">M</mi><mo id="S2.E2.m1.1.1.3.3.3" xref="S2.E2.m1.1.1.3.3.3.cmml">⊙</mo></msub><mo id="S2.E2.m1.1.1.3.1a" xref="S2.E2.m1.1.1.3.1.cmml">⁢</mo><msup id="S2.E2.m1.1.1.3.4" xref="S2.E2.m1.1.1.3.4.cmml"><mi id="S2.E2.m1.1.1.3.4.2" xref="S2.E2.m1.1.1.3.4.2.cmml">yr</mi><mrow id="S2.E2.m1.1.1.3.4.3" xref="S2.E2.m1.1.1.3.4.3.cmml"><mo id="S2.E2.m1.1.1.3.4.3.1" xref="S2.E2.m1.1.1.3.4.3.1.cmml">-</mo><mn id="S2.E2.m1.1.1.3.4.3.2" xref="S2.E2.m1.1.1.3.4.3.2.cmml">1</mn></mrow></msup></mrow></mfrac><mo id="S2.E2.m1.3.3.1.1.3.4.2.2.2" xref="S2.E2.m1.1.1.cmml">)</mo></mrow><mn id="S2.E2.m1.3.3.1.1.3.4.3" xref="S2.E2.m1.3.3.1.1.3.4.3.cmml">0.41</mn></msup><mo id="S2.E2.m1.3.3.1.1.3.1b" xref="S2.E2.m1.3.3.1.1.3.1.cmml">⁢</mo><msup id="S2.E2.m1.3.3.1.1.3.5" xref="S2.E2.m1.3.3.1.1.3.5.cmml"><mrow id="S2.E2.m1.3.3.1.1.3.5.2.2" xref="S2.E2.m1.2.2.cmml"><mo id="S2.E2.m1.3.3.1.1.3.5.2.2.1" xref="S2.E2.m1.2.2.cmml">(</mo><mfrac id="S2.E2.m1.2.2" xref="S2.E2.m1.2.2.cmml"><msub id="S2.E2.m1.2.2.2" xref="S2.E2.m1.2.2.2.cmml"><mi mathvariant="normal" id="S2.E2.m1.2.2.2.2" xref="S2.E2.m1.2.2.2.2.cmml">M</mi><mi mathvariant="normal" id="S2.E2.m1.2.2.2.3" xref="S2.E2.m1.2.2.2.3.cmml">d</mi></msub><mrow id="S2.E2.m1.2.2.3" xref="S2.E2.m1.2.2.3.cmml"><mpadded width="+5pt" id="S2.E2.m1.2.2.3.2" xref="S2.E2.m1.2.2.3.2.cmml"><msup id="S2.E2.m1.2.2.3.2a" xref="S2.E2.m1.2.2.3.2.cmml"><mn id="S2.E2.m1.2.2.3.2.2" xref="S2.E2.m1.2.2.3.2.2.cmml">10</mn><mn id="S2.E2.m1.2.2.3.2.3" xref="S2.E2.m1.2.2.3.2.3.cmml">8</mn></msup></mpadded><mo id="S2.E2.m1.2.2.3.1" xref="S2.E2.m1.2.2.3.1.cmml">⁢</mo><msub id="S2.E2.m1.2.2.3.3" xref="S2.E2.m1.2.2.3.3.cmml"><mi mathvariant="normal" id="S2.E2.m1.2.2.3.3.2" xref="S2.E2.m1.2.2.3.3.2.cmml">M</mi><mo id="S2.E2.m1.2.2.3.3.3" xref="S2.E2.m1.2.2.3.3.3.cmml">⊙</mo></msub></mrow></mfrac><mo id="S2.E2.m1.3.3.1.1.3.5.2.2.2" xref="S2.E2.m1.2.2.cmml">)</mo></mrow><mn id="S2.E2.m1.3.3.1.1.3.5.3" xref="S2.E2.m1.3.3.1.1.3.5.3.cmml">0.56</mn></msup></mrow></mrow><mo id="S2.E2.m1.3.3.1.2" xref="S2.E2.m1.3.3.1.1.cmml">,</mo></mrow></math>, <math><msub id="S2.SS2.p1.2.m1.1.1" xref="S2.SS2.p1.2.m1.1.1.cmml"><mi id="S2.SS2.p1.2.m1.1.1.2" xref="S2.SS2.p1.2.m1.1.1.2.cmml">S</mi><mrow id="S2.SS2.p1.2.m1.1.1.3" xref="S2.SS2.p1.2.m1.1.1.3.cmml"><mpadded width="+5pt" id="S2.SS2.p1.2.m1.1.1.3.2" xref="S2.SS2.p1.2.m1.1.1.3.2.cmml"><mn id="S2.SS2.p1.2.m1.1.1.3.2a" xref="S2.SS2.p1.2.m1.1.1.3.2.cmml">850</mn></mpadded><mo id="S2.SS2.p1.2.m1.1.1.3.1" xref="S2.SS2.p1.2.m1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS2.p1.2.m1.1.1.3.3" xref="S2.SS2.p1.2.m1.1.1.3.3.cmml">μ</mi><mo id="S2.SS2.p1.2.m1.1.1.3.1a" xref="S2.SS2.p1.2.m1.1.1.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S2.SS2.p1.2.m1.1.1.3.4" xref="S2.SS2.p1.2.m1.1.1.3.4.cmml">m</mi></mrow></msub></math>, <math><mrow id="S2.E3.m1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml"><mrow id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml"><msub id="S2.E3.m1.1.1.1.1.2" xref="S2.E3.m1.1.1.1.1.2.cmml"><mi id="S2.E3.m1.1.1.1.1.2.2" xref="S2.E3.m1.1.1.1.1.2.2.cmml">SFR</mi><mi id="S2.E3.m1.1.1.1.1.2.3" xref="S2.E3.m1.1.1.1.1.2.3.cmml">obscured</mi></msub><mo id="S2.E3.m1.1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.cmml">=</mo><mrow id="S2.E3.m1.1.1.1.1.3" xref="S2.E3.m1.1.1.1.1.3.cmml"><msub id="S2.E3.m1.1.1.1.1.3.2" xref="S2.E3.m1.1.1.1.1.3.2.cmml"><mi mathvariant="normal" id="S2.E3.m1.1.1.1.1.3.2.2" xref="S2.E3.m1.1.1.1.1.3.2.2.cmml">f</mi><mi id="S2.E3.m1.1.1.1.1.3.2.3" xref="S2.E3.m1.1.1.1.1.3.2.3.cmml">obscured</mi></msub><mo id="S2.E3.m1.1.1.1.1.3.1" xref="S2.E3.m1.1.1.1.1.3.1.cmml">⁢</mo><msub id="S2.E3.m1.1.1.1.1.3.3" xref="S2.E3.m1.1.1.1.1.3.3.cmml"><mi id="S2.E3.m1.1.1.1.1.3.3.2" xref="S2.E3.m1.1.1.1.1.3.3.2.cmml">SFR</mi><mi id="S2.E3.m1.1.1.1.1.3.3.3" xref="S2.E3.m1.1.1.1.1.3.3.3.cmml">total</mi></msub></mrow></mrow><mo id="S2.E3.m1.1.1.1.2" xref="S2.E3.m1.1.1.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml"><mrow id="S3.p1.3.m3.1.1.2" xref="S3.p1.3.m3.1.1.2.cmml"><msub id="S3.p1.3.m3.1.1.2.2" xref="S3.p1.3.m3.1.1.2.2.cmml"><mi id="S3.p1.3.m3.1.1.2.2.2" xref="S3.p1.3.m3.1.1.2.2.2.cmml">S</mi><mrow id="S3.p1.3.m3.1.1.2.2.3" xref="S3.p1.3.m3.1.1.2.2.3.cmml"><mn id="S3.p1.3.m3.1.1.2.2.3.2" xref="S3.p1.3.m3.1.1.2.2.3.2.cmml">1.1</mn><mo id="S3.p1.3.m3.1.1.2.2.3.1" xref="S3.p1.3.m3.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.p1.3.m3.1.1.2.2.3.3" xref="S3.p1.3.m3.1.1.2.2.3.3.cmml">mm</mi></mrow></msub><mo id="S3.p1.3.m3.1.1.2.1" xref="S3.p1.3.m3.1.1.2.1.cmml">/</mo><msub id="S3.p1.3.m3.1.1.2.3" xref="S3.p1.3.m3.1.1.2.3.cmml"><mi id="S3.p1.3.m3.1.1.2.3.2" xref="S3.p1.3.m3.1.1.2.3.2.cmml">S</mi><mrow id="S3.p1.3.m3.1.1.2.3.3" xref="S3.p1.3.m3.1.1.2.3.3.cmml"><mn id="S3.p1.3.m3.1.1.2.3.3.2" xref="S3.p1.3.m3.1.1.2.3.3.2.cmml">1.2</mn><mo id="S3.p1.3.m3.1.1.2.3.3.1" xref="S3.p1.3.m3.1.1.2.3.3.1.cmml">⁢</mo><mi id="S3.p1.3.m3.1.1.2.3.3.3" xref="S3.p1.3.m3.1.1.2.3.3.3.cmml">mm</mi></mrow></msub></mrow><mo id="S3.p1.3.m3.1.1.1" xref="S3.p1.3.m3.1.1.1.cmml">=</mo><mn id="S3.p1.3.m3.1.1.3" xref="S3.p1.3.m3.1.1.3.cmml">1.36</mn></mrow></math>, <math><mrow id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mrow id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml"><msub id="S3.p1.4.m4.1.1.2.2" xref="S3.p1.4.m4.1.1.2.2.cmml"><mi id="S3.p1.4.m4.1.1.2.2.2" xref="S3.p1.4.m4.1.1.2.2.2.cmml">S</mi><mrow id="S3.p1.4.m4.1.1.2.2.3" xref="S3.p1.4.m4.1.1.2.2.3.cmml"><mn id="S3.p1.4.m4.1.1.2.2.3.2" xref="S3.p1.4.m4.1.1.2.2.3.2.cmml">1.1</mn><mo id="S3.p1.4.m4.1.1.2.2.3.1" xref="S3.p1.4.m4.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.p1.4.m4.1.1.2.2.3.3" xref="S3.p1.4.m4.1.1.2.2.3.3.cmml">mm</mi></mrow></msub><mo id="S3.p1.4.m4.1.1.2.1" xref="S3.p1.4.m4.1.1.2.1.cmml">/</mo><msub id="S3.p1.4.m4.1.1.2.3" xref="S3.p1.4.m4.1.1.2.3.cmml"><mi id="S3.p1.4.m4.1.1.2.3.2" xref="S3.p1.4.m4.1.1.2.3.2.cmml">S</mi><mrow id="S3.p1.4.m4.1.1.2.3.3" xref="S3.p1.4.m4.1.1.2.3.3.cmml"><mn id="S3.p1.4.m4.1.1.2.3.3.2" xref="S3.p1.4.m4.1.1.2.3.3.2.cmml">1.3</mn><mo id="S3.p1.4.m4.1.1.2.3.3.1" xref="S3.p1.4.m4.1.1.2.3.3.1.cmml">⁢</mo><mi id="S3.p1.4.m4.1.1.2.3.3.3" xref="S3.p1.4.m4.1.1.2.3.3.3.cmml">mm</mi></mrow></msub></mrow><mo id="S3.p1.4.m4.1.1.1" xref="S3.p1.4.m4.1.1.1.cmml">=</mo><mn id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml">1.79</mn></mrow></math>, <math><mrow id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><mrow id="S3.p1.5.m5.1.1.2" xref="S3.p1.5.m5.1.1.2.cmml"><msub id="S3.p1.5.m5.1.1.2.2" xref="S3.p1.5.m5.1.1.2.2.cmml"><mi id="S3.p1.5.m5.1.1.2.2.2" xref="S3.p1.5.m5.1.1.2.2.2.cmml">S</mi><mrow id="S3.p1.5.m5.1.1.2.2.3" xref="S3.p1.5.m5.1.1.2.2.3.cmml"><mn id="S3.p1.5.m5.1.1.2.2.3.2" xref="S3.p1.5.m5.1.1.2.2.3.2.cmml">870</mn><mo id="S3.p1.5.m5.1.1.2.2.3.1" xref="S3.p1.5.m5.1.1.2.2.3.1.cmml">⁢</mo><mi id="S3.p1.5.m5.1.1.2.2.3.3" xref="S3.p1.5.m5.1.1.2.2.3.3.cmml">μ</mi><mo id="S3.p1.5.m5.1.1.2.2.3.1a" xref="S3.p1.5.m5.1.1.2.2.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S3.p1.5.m5.1.1.2.2.3.4" xref="S3.p1.5.m5.1.1.2.2.3.4.cmml">m</mi></mrow></msub><mo id="S3.p1.5.m5.1.1.2.1" xref="S3.p1.5.m5.1.1.2.1.cmml">/</mo><msub id="S3.p1.5.m5.1.1.2.3" xref="S3.p1.5.m5.1.1.2.3.cmml"><mi id="S3.p1.5.m5.1.1.2.3.2" xref="S3.p1.5.m5.1.1.2.3.2.cmml">S</mi><mrow id="S3.p1.5.m5.1.1.2.3.3" xref="S3.p1.5.m5.1.1.2.3.3.cmml"><mn id="S3.p1.5.m5.1.1.2.3.3.2" xref="S3.p1.5.m5.1.1.2.3.3.2.cmml">850</mn><mo id="S3.p1.5.m5.1.1.2.3.3.1" xref="S3.p1.5.m5.1.1.2.3.3.1.cmml">⁢</mo><mi id="S3.p1.5.m5.1.1.2.3.3.3" xref="S3.p1.5.m5.1.1.2.3.3.3.cmml">μ</mi><mo id="S3.p1.5.m5.1.1.2.3.3.1a" xref="S3.p1.5.m5.1.1.2.3.3.1.cmml">⁢</mo><mi mathvariant="normal" id="S3.p1.5.m5.1.1.2.3.3.4" xref="S3.p1.5.m5.1.1.2.3.3.4.cmml">m</mi></mrow></msub></mrow><mo id="S3.p1.5.m5.1.1.1" xref="S3.p1.5.m5.1.1.1.cmml">=</mo><mn id="S3.p1.5.m5.1.1.3" xref="S3.p1.5.m5.1.1.3.cmml">0.92</mn></mrow></math>, <math><mrow id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><mrow id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml"><mn id="S3.SS2.p3.1.m1.1.1.2.2" xref="S3.SS2.p3.1.m1.1.1.2.2.cmml">5</mn><mo id="S3.SS2.p3.1.m1.1.1.2.1" xref="S3.SS2.p3.1.m1.1.1.2.1.cmml">×</mo><mpadded width="+1.7pt" id="S3.SS2.p3.1.m1.1.1.2.3" xref="S3.SS2.p3.1.m1.1.1.2.3.cmml"><msup id="S3.SS2.p3.1.m1.1.1.2.3a" xref="S3.SS2.p3.1.m1.1.1.2.3.cmml"><mn id="S3.SS2.p3.1.m1.1.1.2.3.2" xref="S3.SS2.p3.1.m1.1.1.2.3.2.cmml">10</mn><mn id="S3.SS2.p3.1.m1.1.1.2.3.3" xref="S3.SS2.p3.1.m1.1.1.2.3.3.cmml">10</mn></msup></mpadded></mrow><mo id="S3.SS2.p3.1.m1.1.1.1" xref="S3.SS2.p3.1.m1.1.1.1.cmml">⁢</mo><msub id="S3.SS2.p3.1.m1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.3.cmml"><mi mathvariant="normal" id="S3.SS2.p3.1.m1.1.1.3.2" xref="S3.SS2.p3.1.m1.1.1.3.2.cmml">M</mi><mo id="S3.SS2.p3.1.m1.1.1.3.3" xref="S3.SS2.p3.1.m1.1.1.3.3.cmml">⊙</mo></msub></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0506472

Formulas:

1-

U B V I

2-

m_{U} = 22.7 \pm 0.5

3-

m_{B} = 24.65 \pm 0.15

4-

m_{V} = 25.5 \pm 0.15

5-

m_{I} = 24.4 \pm 0.3

6-

m_{f a r U V} = 23.4

7-

P H O T F L A M = 3.951175 \times 10^{17}

8-

P H O T Z P T = - 21.1

9-

E_{B - V} = 0.012

10-

E_{B - V} = 0.01 \pm 0.04

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1509.09172

Formulas:

1-

\frac{\partial g^{i j}}{\partial y^{k}} \frac{\partial f}{\partial x^{i}} = 0,

2-

M_{1}_{f_{2}} \times_{f_{1}} M_{2}

3-

M_{1}_{f_{2}} \times_{f_{1}} M_{2}

4-

C_{k}^{i j} \frac{\partial f_{1}}{\partial x^{i}} = 0 .

5-

C_{k}^{i j} \frac{\partial f_{2}}{\partial x^{i}} = 0

6-

T M := ⋃_{x \in M} T_{x} M

7-

T M^{0} = T M - {0}

8-

F : T M \to [0, \infty)

9-

y \in T_{x} M

10-

g_{y} (u, v) = {\frac{1}{2} \frac{\partial^{2}}{\partial t \partial s} [F^{2} (y + s u + t v)] |}_{s, t = 0} .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1510.04843

Formulas:

1-

π = (d_{1}, d_{2}, \dots, d_{n})

2-

r_{p o t} (H_{1}, H_{2})

3-

\bar{π} = (N - 1 - d_{N}, \dots, N - 1 - d_{1})

4-

t > 7 (s - 2)

5-

r_{p o t} (K_{s}, T_{t}) = t + s - 2 .

6-

π = (d_{1}, d_{2}, \dots, d_{n})

7-

π = π (G)

8-

r (H_{1}, H_{2})

9-

π = (d_{1}, \dots, d_{n})

10-

r_{p o t} (H_{1}, H_{2})

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1308.3246

Formulas:

1-

M_{B H} - M_{b u l g e}

2-

M_{B H} - σ

3-

z \binom{<}{\sim} 1

4-

1 h^{- 1} G p c

5-

ϵ = 5 h^{- 1}

6-

8.47 \times 10^{8} h^{- 1} M_{⊙}

7-

1.53 \times 10^{8} h^{- 1} M_{⊙}

8-

M_{200} > 1 \times 10^{14} h^{- 1} M_{⊙}

9-

M > 2.5 \times 10^{11} h^{- 1}

10-

M_{B H} - M_{b u l g e}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0811.2205

Formulas:

1-

d_{F e A s}

2-

d_{F e A s} = 8.7 Å

3-

d_{F e A s} = 6.5 Å

4-

d_{F e A s} = 6.4 Å

5-

S c i e n c e

6-

113.30 {(7)}^{o}

7-

107.590 {(32)}^{o}

8-

72.410 {(32)}^{o}

9-

113.30 {(7)}^{o}

10-

Δ ρ / ρ_{0}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/2004.14317

Formulas:

1-

y_{0} \in ℝ^{n},

2-

0 < r_{1} < r_{2} < \infty

3-

A (y_{0}, r_{1}, r_{2}) = {y \in ℝ^{n} : r_{1} < | y - y_{0} | < r_{2}} .

4-

\bar{ℝ^{n}} := ℝ^{n} \cup {\infty} .

5-

Γ (E, F, D)

6-

γ : [a, b] \to \bar{ℝ^{n}}

7-

γ (a) \in E, γ (b) \in F

8-

γ (t) \in D

9-

t \in [a, b] .

10-

x \mapsto f (x) .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0806.0511

Formulas:

1-

ϕ = n_{C} V_{c} ≪ 1 .

2-

N_{C} = n_{C} A_{c} = ℓ^{- 1}

3-

V_{c} ≃ A_{c} R_{c}

4-

ϕ = N_{C} R_{c} ≪ 1, or R_{c} ≪ ℓ .

5-

S_{c, λ} (s^{'}) N_{C} (s^{'}) d s^{'}

6-

𝒩 (s^{'}, s) = \int_{s^{'}}^{s} N_{C} 𝑑 s

7-

P_{esc} (s^{'}, s)

8-

𝒩 (s^{'}, s)

9-

P_{esc} (s^{'}, s) = e^{- t_{λ} (s^{'}, s)}, where t_{λ} (s^{'}, s) = 𝒩 (s^{'}, s) (1 - e^{- τ_{λ}});

10-

t_{λ} (s^{'}, s) ≃ 𝒩 (s^{'}, s) τ_{λ}

Formulas (html):

<math><mrow id="S2.E1.m1.3.3.1" xref="S2.E1.m1.3.3.1.1.cmml"><mrow id="S2.E1.m1.3.3.1.1" xref="S2.E1.m1.3.3.1.1.cmml"><mi id="S2.E1.m1.3.3.1.1.2" xref="S2.E1.m1.3.3.1.1.2.cmml">ϕ</mi><mo id="S2.E1.m1.3.3.1.1.3" xref="S2.E1.m1.3.3.1.1.3.cmml">=</mo><mrow id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2a.cmml"><msub id="S2.E1.m1.1.1.1.m1.1.1" xref="S2.E1.m1.1.1.1.m1.1.1.cmml"><mi id="S2.E1.m1.1.1.1.m1.1.1.2" xref="S2.E1.m1.1.1.1.m1.1.1.2.cmml">n</mi><mi mathvariant="normal" id="S2.E1.m1.1.1.1.m1.1.1.3" xref="S2.E1.m1.1.1.1.m1.1.1.3.cmml">C</mi></msub><msub id="S2.E1.m1.2.2.2.m1.1.1" xref="S2.E1.m1.2.2.2.m1.1.1.cmml"><mi id="S2.E1.m1.2.2.2.m1.1.1.2" xref="S2.E1.m1.2.2.2.m1.1.1.2.cmml">V</mi><mi mathvariant="normal" id="S2.E1.m1.2.2.2.m1.1.1.3" xref="S2.E1.m1.2.2.2.m1.1.1.3.cmml">c</mi></msub></mrow><mo id="S2.E1.m1.3.3.1.1.4" xref="S2.E1.m1.3.3.1.1.4.cmml">≪</mo><mpadded width="+1.7pt" id="S2.E1.m1.3.3.1.1.5" xref="S2.E1.m1.3.3.1.1.5.cmml"><mn id="S2.E1.m1.3.3.1.1.5a" xref="S2.E1.m1.3.3.1.1.5.cmml">1</mn></mpadded></mrow><mo id="S2.E1.m1.3.3.1.2" xref="S2.E1.m1.3.3.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.E2.m1.3.4" xref="S2.E2.m1.3.4.cmml"><msub id="S2.E2.m1.1.1.1" xref="S2.E2.m1.1.1.1.cmml"><mi id="S2.E2.m1.1.1.1.3" xref="S2.E2.m1.1.1.1.3.cmml">N</mi><mi mathvariant="normal" id="S2.E2.m1.1.1.1.4" xref="S2.E2.m1.1.1.1.4.cmml">C</mi></msub><mo id="S2.E2.m1.3.4.2" xref="S2.E2.m1.3.4.2.cmml">=</mo><mrow id="S2.E2.m1.3.3" xref="S2.E2.m1.3.3a.cmml"><msub id="S2.E2.m1.2.2.1.m1.1.1" xref="S2.E2.m1.2.2.1.m1.1.1.cmml"><mi id="S2.E2.m1.2.2.1.m1.1.1.2" xref="S2.E2.m1.2.2.1.m1.1.1.2.cmml">n</mi><mi mathvariant="normal" id="S2.E2.m1.2.2.1.m1.1.1.3" xref="S2.E2.m1.2.2.1.m1.1.1.3.cmml">C</mi></msub><msub id="S2.E2.m1.3.3.2.m1.1.1" xref="S2.E2.m1.3.3.2.m1.1.1.cmml"><mi id="S2.E2.m1.3.3.2.m1.1.1.2" xref="S2.E2.m1.3.3.2.m1.1.1.2.cmml">A</mi><mi mathvariant="normal" id="S2.E2.m1.3.3.2.m1.1.1.3" xref="S2.E2.m1.3.3.2.m1.1.1.3.cmml">c</mi></msub></mrow><mo id="S2.E2.m1.3.4.3" xref="S2.E2.m1.3.4.3.cmml">=</mo><msup id="S2.E2.m1.3.4.4" xref="S2.E2.m1.3.4.4.cmml"><mi mathvariant="normal" id="S2.E2.m1.3.4.4.2" xref="S2.E2.m1.3.4.4.2.cmml">ℓ</mi><mrow id="S2.E2.m1.3.4.4.3" xref="S2.E2.m1.3.4.4.3.cmml"><mo id="S2.E2.m1.3.4.4.3.1" xref="S2.E2.m1.3.4.4.3.1.cmml">-</mo><mn id="S2.E2.m1.3.4.4.3.2" xref="S2.E2.m1.3.4.4.3.2.cmml">1</mn></mrow></msup></mrow></math>, <math><mrow id="S2.p2.8.m2.3.4" xref="S2.p2.8.m2.3.4.cmml"><msub id="S2.p2.8.m2.1.1.1" xref="S2.p2.8.m2.1.1.1.cmml"><mi id="S2.p2.8.m2.1.1.1.3" xref="S2.p2.8.m2.1.1.1.3.cmml">V</mi><mi mathvariant="normal" id="S2.p2.8.m2.1.1.1.4" xref="S2.p2.8.m2.1.1.1.4.cmml">c</mi></msub><mo id="S2.p2.8.m2.3.4.1" xref="S2.p2.8.m2.3.4.1.cmml">≃</mo><mrow id="S2.p2.8.m2.3.3" xref="S2.p2.8.m2.3.3a.cmml"><msub id="S2.p2.8.m2.2.2.1.m1.1.1" xref="S2.p2.8.m2.2.2.1.m1.1.1.cmml"><mi id="S2.p2.8.m2.2.2.1.m1.1.1.2" xref="S2.p2.8.m2.2.2.1.m1.1.1.2.cmml">A</mi><mi mathvariant="normal" id="S2.p2.8.m2.2.2.1.m1.1.1.3" xref="S2.p2.8.m2.2.2.1.m1.1.1.3.cmml">c</mi></msub><msub id="S2.p2.8.m2.3.3.2.m1.1.1" xref="S2.p2.8.m2.3.3.2.m1.1.1.cmml"><mi id="S2.p2.8.m2.3.3.2.m1.1.1.2" xref="S2.p2.8.m2.3.3.2.m1.1.1.2.cmml">R</mi><mi mathvariant="normal" id="S2.p2.8.m2.3.3.2.m1.1.1.3" xref="S2.p2.8.m2.3.3.2.m1.1.1.3.cmml">c</mi></msub></mrow></mrow></math>, <math><mrow id="S2.E3.m1.5.5.1"><mrow id="S2.E3.m1.5.5.1.1.2" xref="S2.E3.m1.5.5.1.1.3.cmml"><mrow id="S2.E3.m1.5.5.1.1.1.1" xref="S2.E3.m1.5.5.1.1.1.1.cmml"><mi id="S2.E3.m1.5.5.1.1.1.1.2" xref="S2.E3.m1.5.5.1.1.1.1.2.cmml">ϕ</mi><mo id="S2.E3.m1.5.5.1.1.1.1.3" xref="S2.E3.m1.5.5.1.1.1.1.3.cmml">=</mo><mrow id="S2.E3.m1.2.2" xref="S2.E3.m1.2.2a.cmml"><msub id="S2.E3.m1.1.1.1.m1.1.1" xref="S2.E3.m1.1.1.1.m1.1.1.cmml"><mi id="S2.E3.m1.1.1.1.m1.1.1.2" xref="S2.E3.m1.1.1.1.m1.1.1.2.cmml">N</mi><mi mathvariant="normal" id="S2.E3.m1.1.1.1.m1.1.1.3" xref="S2.E3.m1.1.1.1.m1.1.1.3.cmml">C</mi></msub><msub id="S2.E3.m1.2.2.2.m1.1.1" xref="S2.E3.m1.2.2.2.m1.1.1.cmml"><mi id="S2.E3.m1.2.2.2.m1.1.1.2" xref="S2.E3.m1.2.2.2.m1.1.1.2.cmml">R</mi><mi mathvariant="normal" id="S2.E3.m1.2.2.2.m1.1.1.3" xref="S2.E3.m1.2.2.2.m1.1.1.3.cmml">c</mi></msub></mrow><mo id="S2.E3.m1.5.5.1.1.1.1.4" xref="S2.E3.m1.5.5.1.1.1.1.4.cmml">≪</mo><mn id="S2.E3.m1.5.5.1.1.1.1.5" xref="S2.E3.m1.5.5.1.1.1.1.5.cmml">1</mn></mrow><mo rspace="22.5pt" id="S2.E3.m1.5.5.1.1.2.3" xref="S2.E3.m1.5.5.1.1.3a.cmml">,</mo><mrow id="S2.E3.m1.5.5.1.1.2.2" xref="S2.E3.m1.5.5.1.1.2.2.cmml"><mrow id="S2.E3.m1.5.5.1.1.2.2.2.2" xref="S2.E3.m1.5.5.1.1.2.2.2.1.cmml"><mtext id="S2.E3.m1.4.4" xref="S2.E3.m1.4.4a.cmml">or</mtext><mo mathvariant="italic" separator="true" id="S2.E3.m1.5.5.1.1.2.2.2.2.1" xref="S2.E3.m1.5.5.1.1.2.2.2.1.cmml"> </mo><msub id="S2.E3.m1.3.3.1" xref="S2.E3.m1.3.3.1.cmml"><mi id="S2.E3.m1.3.3.1.3" xref="S2.E3.m1.3.3.1.3.cmml">R</mi><mi mathvariant="normal" id="S2.E3.m1.3.3.1.4" xref="S2.E3.m1.3.3.1.4.cmml">c</mi></msub></mrow><mo id="S2.E3.m1.5.5.1.1.2.2.1" xref="S2.E3.m1.5.5.1.1.2.2.1.cmml">≪</mo><mi mathvariant="normal" id="S2.E3.m1.5.5.1.1.2.2.3" xref="S2.E3.m1.5.5.1.1.2.2.3.cmml">ℓ</mi></mrow></mrow><mo id="S2.E3.m1.5.5.1.2">.</mo></mrow></math>, <math><mrow id="S2.SS1.p1.6.m5.6.6" xref="S2.SS1.p1.6.m5.6.6.cmml"><msub id="S2.SS1.p1.6.m5.4.4" xref="S2.SS1.p1.6.m5.4.4.cmml"><mi id="S2.SS1.p1.6.m5.4.4.5" xref="S2.SS1.p1.6.m5.4.4.5.cmml">S</mi><mrow id="S2.SS1.p1.6.m5.4.4.3.2.4" xref="S2.SS1.p1.6.m5.4.4.3.2.3.cmml"><mi mathvariant="normal" id="S2.SS1.p1.6.m5.3.3.2.1.1" xref="S2.SS1.p1.6.m5.3.3.2.1.1.cmml">c</mi><mo id="S2.SS1.p1.6.m5.4.4.3.2.4.1" xref="S2.SS1.p1.6.m5.4.4.3.2.3.cmml">,</mo><mi id="S2.SS1.p1.6.m5.4.4.3.2.2" xref="S2.SS1.p1.6.m5.4.4.3.2.2.cmml">λ</mi></mrow></msub><mo id="S2.SS1.p1.6.m5.6.6.3" xref="S2.SS1.p1.6.m5.6.6.3.cmml">⁢</mo><mrow id="S2.SS1.p1.6.m5.5.5.1.1" xref="S2.SS1.p1.6.m5.5.5.1.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.6.m5.5.5.1.1.2" xref="S2.SS1.p1.6.m5.5.5.1.1.1.cmml">(</mo><msup id="S2.SS1.p1.6.m5.5.5.1.1.1" xref="S2.SS1.p1.6.m5.5.5.1.1.1.cmml"><mi id="S2.SS1.p1.6.m5.5.5.1.1.1.2" xref="S2.SS1.p1.6.m5.5.5.1.1.1.2.cmml">s</mi><mo id="S2.SS1.p1.6.m5.5.5.1.1.1.3" xref="S2.SS1.p1.6.m5.5.5.1.1.1.3.cmml">′</mo></msup><mo stretchy="false" id="S2.SS1.p1.6.m5.5.5.1.1.3" xref="S2.SS1.p1.6.m5.5.5.1.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p1.6.m5.6.6.3a" xref="S2.SS1.p1.6.m5.6.6.3.cmml">⁢</mo><msub id="S2.SS1.p1.6.m5.2.2.1" xref="S2.SS1.p1.6.m5.2.2.1.cmml"><mi id="S2.SS1.p1.6.m5.2.2.1.3" xref="S2.SS1.p1.6.m5.2.2.1.3.cmml">N</mi><mi mathvariant="normal" id="S2.SS1.p1.6.m5.2.2.1.4" xref="S2.SS1.p1.6.m5.2.2.1.4.cmml">C</mi></msub><mo id="S2.SS1.p1.6.m5.6.6.3b" xref="S2.SS1.p1.6.m5.6.6.3.cmml">⁢</mo><mrow id="S2.SS1.p1.6.m5.6.6.2.1" xref="S2.SS1.p1.6.m5.6.6.2.1.1.cmml"><mo stretchy="false" id="S2.SS1.p1.6.m5.6.6.2.1.2" xref="S2.SS1.p1.6.m5.6.6.2.1.1.cmml">(</mo><msup id="S2.SS1.p1.6.m5.6.6.2.1.1" xref="S2.SS1.p1.6.m5.6.6.2.1.1.cmml"><mi id="S2.SS1.p1.6.m5.6.6.2.1.1.2" xref="S2.SS1.p1.6.m5.6.6.2.1.1.2.cmml">s</mi><mo id="S2.SS1.p1.6.m5.6.6.2.1.1.3" xref="S2.SS1.p1.6.m5.6.6.2.1.1.3.cmml">′</mo></msup><mo stretchy="false" id="S2.SS1.p1.6.m5.6.6.2.1.3" xref="S2.SS1.p1.6.m5.6.6.2.1.1.cmml">)</mo></mrow><mo id="S2.SS1.p1.6.m5.6.6.3c" xref="S2.SS1.p1.6.m5.6.6.3.cmml">⁢</mo><mi id="S2.SS1.p1.6.m5.6.6.4" xref="S2.SS1.p1.6.m5.6.6.4.cmml">d</mi><mo id="S2.SS1.p1.6.m5.6.6.3d" xref="S2.SS1.p1.6.m5.6.6.3.cmml">⁢</mo><msup id="S2.SS1.p1.6.m5.6.6.5" xref="S2.SS1.p1.6.m5.6.6.5.cmml"><mi id="S2.SS1.p1.6.m5.6.6.5.2" xref="S2.SS1.p1.6.m5.6.6.5.2.cmml">s</mi><mo id="S2.SS1.p1.6.m5.6.6.5.3" xref="S2.SS1.p1.6.m5.6.6.5.3.cmml">′</mo></msup></mrow></math>, <math><mrow id="S2.SS1.p1.7.m6.4.4" xref="S2.SS1.p1.7.m6.4.4.cmml"><mrow id="S2.SS1.p1.7.m6.4.4.1" xref="S2.SS1.p1.7.m6.4.4.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.7.m6.4.4.1.3" xref="S2.SS1.p1.7.m6.4.4.1.3.cmml">𝒩</mi><mo id="S2.SS1.p1.7.m6.4.4.1.2" xref="S2.SS1.p1.7.m6.4.4.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.7.m6.4.4.1.1.1" xref="S2.SS1.p1.7.m6.4.4.1.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.7.m6.4.4.1.1.1.2" xref="S2.SS1.p1.7.m6.4.4.1.1.2.cmml">(</mo><msup id="S2.SS1.p1.7.m6.4.4.1.1.1.1" xref="S2.SS1.p1.7.m6.4.4.1.1.1.1.cmml"><mi id="S2.SS1.p1.7.m6.4.4.1.1.1.1.2" xref="S2.SS1.p1.7.m6.4.4.1.1.1.1.2.cmml">s</mi><mo id="S2.SS1.p1.7.m6.4.4.1.1.1.1.3" xref="S2.SS1.p1.7.m6.4.4.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p1.7.m6.4.4.1.1.1.3" xref="S2.SS1.p1.7.m6.4.4.1.1.2.cmml">,</mo><mi id="S2.SS1.p1.7.m6.3.3" xref="S2.SS1.p1.7.m6.3.3.cmml">s</mi><mo stretchy="false" id="S2.SS1.p1.7.m6.4.4.1.1.1.4" xref="S2.SS1.p1.7.m6.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p1.7.m6.4.4.2" xref="S2.SS1.p1.7.m6.4.4.2.cmml">=</mo><mrow id="S2.SS1.p1.7.m6.4.4.3" xref="S2.SS1.p1.7.m6.4.4.3.cmml"><msubsup id="S2.SS1.p1.7.m6.4.4.3.1" xref="S2.SS1.p1.7.m6.4.4.3.1.cmml"><mo largeop="true" symmetric="true" id="S2.SS1.p1.7.m6.4.4.3.1.2.2" xref="S2.SS1.p1.7.m6.4.4.3.1.2.2.cmml">∫</mo><msup 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xref="S2.SS1.p1.7.m6.4.4.3.2.2.2.cmml">s</mi></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.SS1.p1.10.m9.3.3" xref="S2.SS1.p1.10.m9.3.3.cmml"><msub id="S2.SS1.p1.10.m9.1.1.1" xref="S2.SS1.p1.10.m9.1.1.1.cmml"><mi id="S2.SS1.p1.10.m9.1.1.1.3" xref="S2.SS1.p1.10.m9.1.1.1.3.cmml">P</mi><mi id="S2.SS1.p1.10.m9.1.1.1.4" xref="S2.SS1.p1.10.m9.1.1.1.4.cmml">esc</mi></msub><mo id="S2.SS1.p1.10.m9.3.3.2" xref="S2.SS1.p1.10.m9.3.3.2.cmml">⁢</mo><mrow id="S2.SS1.p1.10.m9.3.3.1.1" xref="S2.SS1.p1.10.m9.3.3.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.10.m9.3.3.1.1.2" xref="S2.SS1.p1.10.m9.3.3.1.2.cmml">(</mo><msup id="S2.SS1.p1.10.m9.3.3.1.1.1" xref="S2.SS1.p1.10.m9.3.3.1.1.1.cmml"><mi id="S2.SS1.p1.10.m9.3.3.1.1.1.2" xref="S2.SS1.p1.10.m9.3.3.1.1.1.2.cmml">s</mi><mo id="S2.SS1.p1.10.m9.3.3.1.1.1.3" xref="S2.SS1.p1.10.m9.3.3.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p1.10.m9.3.3.1.1.3" xref="S2.SS1.p1.10.m9.3.3.1.2.cmml">,</mo><mi id="S2.SS1.p1.10.m9.2.2" 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id="S2.SS1.p1.13.m12.3.3.1.1.4" xref="S2.SS1.p1.13.m12.3.3.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.E4.m1.9.9.1"><mrow id="S2.E4.m1.9.9.1.1.2" xref="S2.E4.m1.9.9.1.1.3.cmml"><mrow id="S2.E4.m1.9.9.1.1.1.1" xref="S2.E4.m1.9.9.1.1.1.1.cmml"><mrow id="S2.E4.m1.9.9.1.1.1.1.1" xref="S2.E4.m1.9.9.1.1.1.1.1.cmml"><msub id="S2.E4.m1.1.1.1" xref="S2.E4.m1.1.1.1.cmml"><mi id="S2.E4.m1.1.1.1.3" xref="S2.E4.m1.1.1.1.3.cmml">P</mi><mi id="S2.E4.m1.1.1.1.4" xref="S2.E4.m1.1.1.1.4.cmml">esc</mi></msub><mo id="S2.E4.m1.9.9.1.1.1.1.1.2" xref="S2.E4.m1.9.9.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E4.m1.9.9.1.1.1.1.1.1.1" xref="S2.E4.m1.9.9.1.1.1.1.1.1.2.cmml"><mo stretchy="false" id="S2.E4.m1.9.9.1.1.1.1.1.1.1.2" xref="S2.E4.m1.9.9.1.1.1.1.1.1.2.cmml">(</mo><msup id="S2.E4.m1.9.9.1.1.1.1.1.1.1.1" xref="S2.E4.m1.9.9.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E4.m1.9.9.1.1.1.1.1.1.1.1.2" xref="S2.E4.m1.9.9.1.1.1.1.1.1.1.1.2.cmml">s</mi><mo id="S2.E4.m1.9.9.1.1.1.1.1.1.1.1.3" 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id="S2.E4.m1.9.9.1.1.2.2" xref="S2.E4.m1.9.9.1.1.2.2.cmml"><mrow id="S2.E4.m1.9.9.1.1.2.2.1" xref="S2.E4.m1.9.9.1.1.2.2.1.cmml"><msub id="S2.E4.m1.9.9.1.1.2.2.1.3" xref="S2.E4.m1.9.9.1.1.2.2.1.3.cmml"><mi id="S2.E4.m1.9.9.1.1.2.2.1.3.2" xref="S2.E4.m1.9.9.1.1.2.2.1.3.2.cmml">t</mi><mi id="S2.E4.m1.9.9.1.1.2.2.1.3.3" xref="S2.E4.m1.9.9.1.1.2.2.1.3.3.cmml">λ</mi></msub><mo id="S2.E4.m1.9.9.1.1.2.2.1.2" xref="S2.E4.m1.9.9.1.1.2.2.1.2.cmml">⁢</mo><mrow id="S2.E4.m1.9.9.1.1.2.2.1.1.1" xref="S2.E4.m1.9.9.1.1.2.2.1.1.2.cmml"><mo stretchy="false" id="S2.E4.m1.9.9.1.1.2.2.1.1.1.2" xref="S2.E4.m1.9.9.1.1.2.2.1.1.2.cmml">(</mo><msup id="S2.E4.m1.9.9.1.1.2.2.1.1.1.1" xref="S2.E4.m1.9.9.1.1.2.2.1.1.1.1.cmml"><mi id="S2.E4.m1.9.9.1.1.2.2.1.1.1.1.2" xref="S2.E4.m1.9.9.1.1.2.2.1.1.1.1.2.cmml">s</mi><mo id="S2.E4.m1.9.9.1.1.2.2.1.1.1.1.3" xref="S2.E4.m1.9.9.1.1.2.2.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.E4.m1.9.9.1.1.2.2.1.1.1.3" xref="S2.E4.m1.9.9.1.1.2.2.1.1.2.cmml">,</mo><mi id="S2.E4.m1.6.6" 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xref="S2.E4.m1.9.9.1.1.2.2.2.1.2.cmml">,</mo><mi id="S2.E4.m1.7.7" xref="S2.E4.m1.7.7.cmml">s</mi><mo stretchy="false" id="S2.E4.m1.9.9.1.1.2.2.2.1.1.4" xref="S2.E4.m1.9.9.1.1.2.2.2.1.2.cmml">)</mo></mrow><mo id="S2.E4.m1.9.9.1.1.2.2.3.3a" xref="S2.E4.m1.9.9.1.1.2.2.3.3.cmml">⁢</mo><mrow id="S2.E4.m1.9.9.1.1.2.2.3.2.1" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.cmml"><mo stretchy="false" id="S2.E4.m1.9.9.1.1.2.2.3.2.1.2" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.cmml">(</mo><mrow id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.cmml"><mn id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.2" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.2.cmml">1</mn><mo id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.1" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.1.cmml">-</mo><msup id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.cmml"><mi id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.2" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.2.cmml">e</mi><mrow id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.cmml"><mo id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.1" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.1.cmml">-</mo><msub id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.2" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.2.cmml"><mi id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.2.2" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.2.2.cmml">τ</mi><mi id="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.2.3" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.3.3.2.3.cmml">λ</mi></msub></mrow></msup></mrow><mo stretchy="false" id="S2.E4.m1.9.9.1.1.2.2.3.2.1.3" xref="S2.E4.m1.9.9.1.1.2.2.3.2.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><mo id="S2.E4.m1.9.9.1.2">;</mo></mrow></math>, <math><mrow id="S2.SS1.p1.15.m2.5.5" xref="S2.SS1.p1.15.m2.5.5.cmml"><mrow id="S2.SS1.p1.15.m2.4.4.1" xref="S2.SS1.p1.15.m2.4.4.1.cmml"><msub id="S2.SS1.p1.15.m2.4.4.1.3" xref="S2.SS1.p1.15.m2.4.4.1.3.cmml"><mi id="S2.SS1.p1.15.m2.4.4.1.3.2" xref="S2.SS1.p1.15.m2.4.4.1.3.2.cmml">t</mi><mi id="S2.SS1.p1.15.m2.4.4.1.3.3" xref="S2.SS1.p1.15.m2.4.4.1.3.3.cmml">λ</mi></msub><mo id="S2.SS1.p1.15.m2.4.4.1.2" xref="S2.SS1.p1.15.m2.4.4.1.2.cmml">⁢</mo><mrow id="S2.SS1.p1.15.m2.4.4.1.1.1" xref="S2.SS1.p1.15.m2.4.4.1.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.15.m2.4.4.1.1.1.2" xref="S2.SS1.p1.15.m2.4.4.1.1.2.cmml">(</mo><msup id="S2.SS1.p1.15.m2.4.4.1.1.1.1" xref="S2.SS1.p1.15.m2.4.4.1.1.1.1.cmml"><mi id="S2.SS1.p1.15.m2.4.4.1.1.1.1.2" xref="S2.SS1.p1.15.m2.4.4.1.1.1.1.2.cmml">s</mi><mo id="S2.SS1.p1.15.m2.4.4.1.1.1.1.3" xref="S2.SS1.p1.15.m2.4.4.1.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p1.15.m2.4.4.1.1.1.3" xref="S2.SS1.p1.15.m2.4.4.1.1.2.cmml">,</mo><mi id="S2.SS1.p1.15.m2.2.2" xref="S2.SS1.p1.15.m2.2.2.cmml">s</mi><mo stretchy="false" id="S2.SS1.p1.15.m2.4.4.1.1.1.4" xref="S2.SS1.p1.15.m2.4.4.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p1.15.m2.5.5.3" xref="S2.SS1.p1.15.m2.5.5.3.cmml">≃</mo><mrow id="S2.SS1.p1.15.m2.5.5.2" xref="S2.SS1.p1.15.m2.5.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p1.15.m2.5.5.2.3" xref="S2.SS1.p1.15.m2.5.5.2.3.cmml">𝒩</mi><mo id="S2.SS1.p1.15.m2.5.5.2.2" xref="S2.SS1.p1.15.m2.5.5.2.2.cmml">⁢</mo><mrow id="S2.SS1.p1.15.m2.5.5.2.1.1" xref="S2.SS1.p1.15.m2.5.5.2.1.2.cmml"><mo stretchy="false" id="S2.SS1.p1.15.m2.5.5.2.1.1.2" xref="S2.SS1.p1.15.m2.5.5.2.1.2.cmml">(</mo><msup id="S2.SS1.p1.15.m2.5.5.2.1.1.1" xref="S2.SS1.p1.15.m2.5.5.2.1.1.1.cmml"><mi id="S2.SS1.p1.15.m2.5.5.2.1.1.1.2" xref="S2.SS1.p1.15.m2.5.5.2.1.1.1.2.cmml">s</mi><mo id="S2.SS1.p1.15.m2.5.5.2.1.1.1.3" xref="S2.SS1.p1.15.m2.5.5.2.1.1.1.3.cmml">′</mo></msup><mo id="S2.SS1.p1.15.m2.5.5.2.1.1.3" xref="S2.SS1.p1.15.m2.5.5.2.1.2.cmml">,</mo><mi id="S2.SS1.p1.15.m2.3.3" xref="S2.SS1.p1.15.m2.3.3.cmml">s</mi><mo stretchy="false" id="S2.SS1.p1.15.m2.5.5.2.1.1.4" xref="S2.SS1.p1.15.m2.5.5.2.1.2.cmml">)</mo></mrow><mo id="S2.SS1.p1.15.m2.5.5.2.2a" xref="S2.SS1.p1.15.m2.5.5.2.2.cmml">⁢</mo><msub id="S2.SS1.p1.15.m2.5.5.2.4" xref="S2.SS1.p1.15.m2.5.5.2.4.cmml"><mi id="S2.SS1.p1.15.m2.5.5.2.4.2" xref="S2.SS1.p1.15.m2.5.5.2.4.2.cmml">τ</mi><mi id="S2.SS1.p1.15.m2.5.5.2.4.3" xref="S2.SS1.p1.15.m2.5.5.2.4.3.cmml">λ</mi></msub></mrow></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/2007.11341

Formulas:

1-

f_{enc} (𝐗) = (f_{𝜷} (𝐗), f_{𝜽} (𝐗)) = (𝜷, 𝜽)

2-

f_{𝜷} (𝐗) = 𝜷

3-

f_{𝜽} (𝐗) = 𝜽

4-

g_{dec} (𝜷, 𝜽) = \tilde{𝐗}

5-

(𝐗_{1}^{s}, 𝐗_{2}^{s})

6-

(𝜷_{1}^{s}, 𝜽_{1}^{s})

7-

(𝜷_{2}^{s}, 𝜽_{2}^{s})

8-

g_{dec} (𝜷_{2}^{s}, 𝜽_{1}^{s})

9-

g_{dec} (𝜷_{1}^{s}, 𝜽_{2}^{s})

10-

ℒ_{C} = {∥ g_{dec} (f_{𝜷} (𝐗_{2}^{s}), f_{𝜽} (𝒯 (𝐗_{1}^{s}))) - 𝐗_{1}^{s} ∥}_{1},

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0907.5037

Formulas:

1-

\sum_{ξ = 1}^{N} {| ⟨ a | ξ ⟩ |}^{2} = 1

2-

P^{a a} = N \sum_{ξ = 1}^{N} {| ⟨ a | ξ ⟩ |}^{4} = N lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} 𝑑 t {| ⟨ a | a (t) ⟩ |}^{2},

3-

P^{a a} = 1

4-

P^{a a} = N

5-

P^{a b} = N \sum_{ξ = 1}^{N} {| ⟨ a | ξ ⟩ |}^{2} {| ⟨ b | ξ ⟩ |}^{2} = N lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} 𝑑 t {| ⟨ a | b (t) ⟩ |}^{2} .

6-

| a ⟩ ⟨ a |

7-

Tr \hat{α} = 1

8-

P^{α β} = N \sum_{ξ = 1}^{N} ⟨ ξ | \hat{α} | ξ ⟩ ⟨ ξ | \hat{β} | ξ ⟩ = N lim_{T \to \infty} \frac{1}{2 T} \int_{- T}^{T} 𝑑 t Tr \hat{α} \hat{β} (t),

9-

\bar{P^{a b}} \approx \bar{\sum_{- τ}^{τ} P^{a b} (t)},

10-

P^{a b} (t) = {| ⟨ a | b (t) ⟩ |}^{2} + ⟨ a | a (t) ⟩ ⟨ b (t) | b ⟩,

Formulas (html):

Correct Categorie: nlin

Guessed Categorie: nlin

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0507528

Formulas:

1-

L_{x} / λ = 38.4

2-

L_{z} / λ = 32

3-

L_{z} / λ = \pm 12

4-

E_{d r i} / E_{0} = α_{1} / \cosh^{2} (x / L_{d r i}) + α_{2}

5-

L_{d r i} / λ = 2

6-

E_{0} = V_{A} B_{0} / c

7-

α_{1} = α_{2} = 0.1

8-

α_{1} = α_{2} = 0.2

9-

N_{x} \times N_{z} = 1536 \times 1280

10-

T_{i, ps} / T_{e, ps} = 4

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/0909.5488

Formulas:

1-

ρ_{t o t a l} (𝐠) = \frac{1}{Ω} \int_{Ω} ρ_{t o t a l} (𝐫) e^{- i 𝐠 . 𝐫} d^{3} 𝐫

2-

ρ_{t o t a l} (𝐫)

3-

ρ_{t o t a l}

4-

\sum_{i} [α_{𝐬_{i}}^{r e c o n} (𝐠) - α_{𝐬_{i}}^{p s e u d o} (𝐠)]

5-

α^{r e c o n}

6-

α^{p s e u d o}

7-

α_{𝐬_{i}} (𝐠) = \int_{| 𝐫 - 𝐬_{i} | < r_{s}} ρ (𝐫) e^{- i 𝐠 . 𝐫} d^{3} 𝐫

8-

α_{𝟎} (𝐠) = 4 π \sum_{L} {(- i)}^{l} Y_{L} (\hat{𝐠}) \int_{0}^{r_{s}} ρ_{L} (r) j_{l} (g r) r^{2} 𝑑 r

9-

e^{i 𝐪 . 𝐫} = 4 π \sum_{L} i^{l} j_{l} (q r) Y_{L}^{*} (\hat{𝐪}) Y_{L} (\hat{𝐫})

10-

(\frac{1}{4}, \frac{1}{4}, \frac{1}{4})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/0804.1792

Formulas:

1-

{}^{12}C {(α, γ)}^{16} O

2-

17.5 < m_{F 814 W} < 21.0

3-

f_{bin}^{q > 0.5} = N_{OBS}^{B} / N_{OBS}^{A} - N_{ART}^{B} / N_{ART}^{A}

4-

N_{OBS}^{A (B)}

5-

N_{ART}^{A (B)}

6-

f_{bin}^{q > 0.5} = 0.16 \pm 0.02

7-

f_{bin}^{0.5 < q < 0.6}

8-

(f_{bin}^{q > 0.5} - f_{bin}^{q > 0.6})

9-

f_{bin}^{tot} ≃ 32 \pm 3 %

10-

f_{bin}^{tot} = 25 %

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1802.05052

Formulas:

1-

Z = - 2.99 cm

2-

{\bar{n}}_{e} / n_{G} = 0.3

3-

n_{G} = (I_{p} / π a^{2}) 10^{20} m^{- 3}

4-

Φ_{K} (t) = Φ_{0} + \sum_{k = 1}^{K (T)} A_{k} φ (\frac{t - t_{k}}{τ_{d}})

5-

t \in [0, T]

6-

φ (x) = {\begin{matrix} \exp (- x / (1 - λ)), & x \geq 0, \\ \exp (x / λ), & x < 0, \end{matrix}

7-

λ \in [0, 1]

8-

γ = τ_{d} / τ_{w}

9-

P_{Φ} (ϕ) = \frac{{(ϕ - Φ_{0})}^{γ - 1}}{{⟨ A ⟩}^{γ} Γ (γ)} \exp (- \frac{ϕ - Φ_{0}}{⟨ A ⟩}), ϕ > Φ_{0} .

10-

⟨ Φ ⟩ = Φ_{0} + γ ⟨ A ⟩

Formulas (html):

Correct Categorie: physics

Guessed Categorie: physics

Result: correct

Paper: https://arxiv.org/abs/2002.07452

Formulas:

1-

f_{k} (𝐱_{k, u})

2-

f_{k} (𝐱_{k, u}) = {\begin{matrix} \frac{1}{π {(R_{k})}^{2}}, & if ∥ 𝐱_{k, u} ∥ \leq R_{k}, \\ 0, & otherwise, \end{matrix}

3-

𝐡_{u} = \sqrt{M} \frac{α_{u, 0} 𝐚 (θ_{u, 0})}{(1 + d_{u}^{η_{u, L O S}})} + \sum_{l = 1}^{L} \sqrt{M} \frac{α_{u, l} 𝐚 (θ_{u, l})}{(1 + d_{u}^{η_{u, N L O S}})},

4-

𝐚 (θ_{u, l})

5-

𝐚 (θ_{u, l}) = \frac{1}{\sqrt{M}} {[\begin{matrix} 1, e^{- j π θ_{u, l}}, \dots, e^{- j π (M - 1) θ_{u, l}} \end{matrix}]}^{T} .

6-

θ_{u, l} = 2 \frac{D}{λ} \sin (ϕ_{u, l})

7-

ϕ_{u, l} \in [0, 2 π]

8-

α_{u, l} \sim 𝒞 𝒩 (0, σ_{u, l}^{2})

9-

η_{u, L O S}

10-

η_{u, N L O S}

Formulas (html):

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id="S2.SS1.p1.3.m3.3.3.1.1.3" xref="S2.SS1.p1.3.m3.3.3.1.1.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.Ex1.m1.7.7" xref="S2.Ex1.m1.7.7.cmml"><mrow id="S2.Ex1.m1.7.7.1" xref="S2.Ex1.m1.7.7.1.cmml"><msub id="S2.Ex1.m1.7.7.1.3" xref="S2.Ex1.m1.7.7.1.3.cmml"><mi id="S2.Ex1.m1.7.7.1.3.2" xref="S2.Ex1.m1.7.7.1.3.2.cmml">f</mi><mi id="S2.Ex1.m1.7.7.1.3.3" xref="S2.Ex1.m1.7.7.1.3.3.cmml">k</mi></msub><mo id="S2.Ex1.m1.7.7.1.2" xref="S2.Ex1.m1.7.7.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.7.7.1.1.1" xref="S2.Ex1.m1.7.7.1.1.1.1.cmml"><mo stretchy="false" id="S2.Ex1.m1.7.7.1.1.1.2" xref="S2.Ex1.m1.7.7.1.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.7.7.1.1.1.1" xref="S2.Ex1.m1.7.7.1.1.1.1.cmml"><mi id="S2.Ex1.m1.7.7.1.1.1.1.2" xref="S2.Ex1.m1.7.7.1.1.1.1.2.cmml">𝐱</mi><mrow id="S2.Ex1.m1.6.6.2.4" xref="S2.Ex1.m1.6.6.2.3.cmml"><mi id="S2.Ex1.m1.5.5.1.1" xref="S2.Ex1.m1.5.5.1.1.cmml">k</mi><mo id="S2.Ex1.m1.6.6.2.4.1" xref="S2.Ex1.m1.6.6.2.3.cmml">,</mo><mi id="S2.Ex1.m1.6.6.2.2" xref="S2.Ex1.m1.6.6.2.2.cmml">u</mi></mrow></msub><mo stretchy="false" id="S2.Ex1.m1.7.7.1.1.1.3" xref="S2.Ex1.m1.7.7.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex1.m1.7.7.2" xref="S2.Ex1.m1.7.7.2.cmml">=</mo><mrow id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.7.7.3.1.cmml"><mo id="S2.Ex1.m1.4.4.5" xref="S2.Ex1.m1.7.7.3.1.1.cmml">{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt" id="S2.Ex1.m1.4.4.4" xref="S2.Ex1.m1.7.7.3.1.cmml"><mtr id="S2.Ex1.m1.4.4.4a" xref="S2.Ex1.m1.7.7.3.1.cmml"><mtd columnalign="left" id="S2.Ex1.m1.4.4.4b" xref="S2.Ex1.m1.7.7.3.1.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.cmml"><mfrac id="S2.Ex1.m1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.cmml"><mn id="S2.Ex1.m1.1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.3.cmml">1</mn><mrow id="S2.Ex1.m1.1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.3.cmml">π</mi><mo id="S2.Ex1.m1.1.1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><msup id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">R</mi><mi id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.3.cmml">k</mi></msub><mo stretchy="false" id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mn id="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex1.m1.1.1.1.1.1.1.1.1.1.3.cmml">2</mn></msup></mrow></mfrac><mo id="S2.Ex1.m1.1.1.1.1.1.1.3.1" xref="S2.Ex1.m1.1.1.1.1.1.1.1.cmml">,</mo></mrow></mtd><mtd columnalign="left" id="S2.Ex1.m1.4.4.4c" xref="S2.Ex1.m1.7.7.3.1.cmml"><mrow id="S2.Ex1.m1.2.2.2.2.2.1.3" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.cmml"><mrow id="S2.Ex1.m1.2.2.2.2.2.1.3.1" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.cmml"><mrow id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.cmml"><mpadded width="+3.3pt" id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.3" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.3.cmml"><mi id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.3a" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.3.cmml">if</mi></mpadded><mo id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.2" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.2.cmml">⁢</mo><mrow id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.2.cmml"><mo id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1.2" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.2.1.cmml">∥</mo><msub id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1.1" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1.1.cmml"><mi id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1.1.2" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1.1.2.cmml">𝐱</mi><mrow id="S2.Ex1.m1.2.2.2.2.2.1.2.2.4" xref="S2.Ex1.m1.2.2.2.2.2.1.2.2.3.cmml"><mi id="S2.Ex1.m1.2.2.2.2.2.1.1.1.1" xref="S2.Ex1.m1.2.2.2.2.2.1.1.1.1.cmml">k</mi><mo id="S2.Ex1.m1.2.2.2.2.2.1.2.2.4.1" xref="S2.Ex1.m1.2.2.2.2.2.1.2.2.3.cmml">,</mo><mi id="S2.Ex1.m1.2.2.2.2.2.1.2.2.2" xref="S2.Ex1.m1.2.2.2.2.2.1.2.2.2.cmml">u</mi></mrow></msub><mo id="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.1.3" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.1.1.2.1.cmml">∥</mo></mrow></mrow><mo id="S2.Ex1.m1.2.2.2.2.2.1.3.1.2" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.2.cmml">≤</mo><msub id="S2.Ex1.m1.2.2.2.2.2.1.3.1.3" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.3.cmml"><mi id="S2.Ex1.m1.2.2.2.2.2.1.3.1.3.2" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.3.2.cmml">R</mi><mi id="S2.Ex1.m1.2.2.2.2.2.1.3.1.3.3" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.3.3.cmml">k</mi></msub></mrow><mo id="S2.Ex1.m1.2.2.2.2.2.1.3.2" xref="S2.Ex1.m1.2.2.2.2.2.1.3.1.cmml">,</mo></mrow></mtd></mtr><mtr id="S2.Ex1.m1.4.4.4d" xref="S2.Ex1.m1.7.7.3.1.cmml"><mtd columnalign="left" id="S2.Ex1.m1.4.4.4e" xref="S2.Ex1.m1.7.7.3.1.cmml"><mrow id="S2.Ex1.m1.3.3.3.3.1.1.3" xref="S2.Ex1.m1.7.7.3.1.cmml"><mn id="S2.Ex1.m1.3.3.3.3.1.1.1" xref="S2.Ex1.m1.3.3.3.3.1.1.1.cmml">0</mn><mo id="S2.Ex1.m1.3.3.3.3.1.1.3.1" xref="S2.Ex1.m1.7.7.3.1.cmml">,</mo></mrow></mtd><mtd columnalign="left" id="S2.Ex1.m1.4.4.4f" xref="S2.Ex1.m1.7.7.3.1.cmml"><mrow id="S2.Ex1.m1.4.4.4.4.2.1.3" xref="S2.Ex1.m1.7.7.3.1.cmml"><mi id="S2.Ex1.m1.4.4.4.4.2.1.1" xref="S2.Ex1.m1.4.4.4.4.2.1.1.cmml">otherwise</mi><mo id="S2.Ex1.m1.4.4.4.4.2.1.3.1" xref="S2.Ex1.m1.7.7.3.1.cmml">,</mo></mrow></mtd></mtr></mtable></mrow></mrow></math>, <math><mrow id="S2.Ex2.m1.17.17.1" xref="S2.Ex2.m1.17.17.1.1.cmml"><mrow id="S2.Ex2.m1.17.17.1.1" xref="S2.Ex2.m1.17.17.1.1.cmml"><msub id="S2.Ex2.m1.17.17.1.1.2" xref="S2.Ex2.m1.17.17.1.1.2.cmml"><mi id="S2.Ex2.m1.17.17.1.1.2.2" xref="S2.Ex2.m1.17.17.1.1.2.2.cmml">𝐡</mi><mi id="S2.Ex2.m1.17.17.1.1.2.3" xref="S2.Ex2.m1.17.17.1.1.2.3.cmml">u</mi></msub><mo id="S2.Ex2.m1.17.17.1.1.1" xref="S2.Ex2.m1.17.17.1.1.1.cmml">=</mo><mrow id="S2.Ex2.m1.17.17.1.1.3" xref="S2.Ex2.m1.17.17.1.1.3.cmml"><mrow id="S2.Ex2.m1.17.17.1.1.3.2" xref="S2.Ex2.m1.17.17.1.1.3.2.cmml"><msqrt id="S2.Ex2.m1.17.17.1.1.3.2.2" xref="S2.Ex2.m1.17.17.1.1.3.2.2.cmml"><mi id="S2.Ex2.m1.17.17.1.1.3.2.2.2" xref="S2.Ex2.m1.17.17.1.1.3.2.2.2.cmml">M</mi></msqrt><mo id="S2.Ex2.m1.17.17.1.1.3.2.1" xref="S2.Ex2.m1.17.17.1.1.3.2.1.cmml">⁢</mo><mfrac id="S2.Ex2.m1.8.8" xref="S2.Ex2.m1.8.8.cmml"><mrow id="S2.Ex2.m1.5.5.5" xref="S2.Ex2.m1.5.5.5.cmml"><msub id="S2.Ex2.m1.5.5.5.7" xref="S2.Ex2.m1.5.5.5.7.cmml"><mi id="S2.Ex2.m1.5.5.5.7.2" xref="S2.Ex2.m1.5.5.5.7.2.cmml">α</mi><mrow id="S2.Ex2.m1.2.2.2.2.2.4" xref="S2.Ex2.m1.2.2.2.2.2.3.cmml"><mi id="S2.Ex2.m1.1.1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.1.1.cmml">u</mi><mo id="S2.Ex2.m1.2.2.2.2.2.4.1" xref="S2.Ex2.m1.2.2.2.2.2.3.cmml">,</mo><mn id="S2.Ex2.m1.2.2.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.2.2.cmml">0</mn></mrow></msub><mo id="S2.Ex2.m1.5.5.5.6" xref="S2.Ex2.m1.5.5.5.6.cmml">⁢</mo><mi id="S2.Ex2.m1.5.5.5.8" xref="S2.Ex2.m1.5.5.5.8.cmml">𝐚</mi><mo id="S2.Ex2.m1.5.5.5.6a" xref="S2.Ex2.m1.5.5.5.6.cmml">⁢</mo><mrow id="S2.Ex2.m1.5.5.5.5.1" xref="S2.Ex2.m1.5.5.5.5.1.1.cmml"><mo stretchy="false" id="S2.Ex2.m1.5.5.5.5.1.2" xref="S2.Ex2.m1.5.5.5.5.1.1.cmml">(</mo><msub id="S2.Ex2.m1.5.5.5.5.1.1" xref="S2.Ex2.m1.5.5.5.5.1.1.cmml"><mi id="S2.Ex2.m1.5.5.5.5.1.1.2" xref="S2.Ex2.m1.5.5.5.5.1.1.2.cmml">θ</mi><mrow id="S2.Ex2.m1.4.4.4.4.2.4" xref="S2.Ex2.m1.4.4.4.4.2.3.cmml"><mi id="S2.Ex2.m1.3.3.3.3.1.1" xref="S2.Ex2.m1.3.3.3.3.1.1.cmml">u</mi><mo id="S2.Ex2.m1.4.4.4.4.2.4.1" xref="S2.Ex2.m1.4.4.4.4.2.3.cmml">,</mo><mn id="S2.Ex2.m1.4.4.4.4.2.2" xref="S2.Ex2.m1.4.4.4.4.2.2.cmml">0</mn></mrow></msub><mo stretchy="false" id="S2.Ex2.m1.5.5.5.5.1.3" xref="S2.Ex2.m1.5.5.5.5.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.Ex2.m1.8.8.8.3" xref="S2.Ex2.m1.8.8.8.3.1.cmml"><mo stretchy="false" id="S2.Ex2.m1.8.8.8.3.2" xref="S2.Ex2.m1.8.8.8.3.1.cmml">(</mo><mrow id="S2.Ex2.m1.8.8.8.3.1" xref="S2.Ex2.m1.8.8.8.3.1.cmml"><mn id="S2.Ex2.m1.8.8.8.3.1.2" xref="S2.Ex2.m1.8.8.8.3.1.2.cmml">1</mn><mo id="S2.Ex2.m1.8.8.8.3.1.1" xref="S2.Ex2.m1.8.8.8.3.1.1.cmml">+</mo><msubsup id="S2.Ex2.m1.8.8.8.3.1.3" xref="S2.Ex2.m1.8.8.8.3.1.3.cmml"><mi id="S2.Ex2.m1.8.8.8.3.1.3.2.2" xref="S2.Ex2.m1.8.8.8.3.1.3.2.2.cmml">d</mi><mi id="S2.Ex2.m1.8.8.8.3.1.3.2.3" xref="S2.Ex2.m1.8.8.8.3.1.3.2.3.cmml">u</mi><msub id="S2.Ex2.m1.7.7.7.2.2" xref="S2.Ex2.m1.7.7.7.2.2.cmml"><mi id="S2.Ex2.m1.7.7.7.2.2.4" xref="S2.Ex2.m1.7.7.7.2.2.4.cmml">η</mi><mrow id="S2.Ex2.m1.7.7.7.2.2.2.2.2" xref="S2.Ex2.m1.7.7.7.2.2.2.2.3.cmml"><mi id="S2.Ex2.m1.6.6.6.1.1.1.1.1" xref="S2.Ex2.m1.6.6.6.1.1.1.1.1.cmml">u</mi><mo id="S2.Ex2.m1.7.7.7.2.2.2.2.2.2" xref="S2.Ex2.m1.7.7.7.2.2.2.2.3.cmml">,</mo><mrow id="S2.Ex2.m1.7.7.7.2.2.2.2.2.1" xref="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.cmml"><mi id="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.2" xref="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.2.cmml">L</mi><mo id="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.1" xref="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.3" xref="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.3.cmml">O</mi><mo id="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.1a" xref="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.4" xref="S2.Ex2.m1.7.7.7.2.2.2.2.2.1.4.cmml">S</mi></mrow></mrow></msub></msubsup></mrow><mo stretchy="false" id="S2.Ex2.m1.8.8.8.3.3" xref="S2.Ex2.m1.8.8.8.3.1.cmml">)</mo></mrow></mfrac></mrow><mo id="S2.Ex2.m1.17.17.1.1.3.1" xref="S2.Ex2.m1.17.17.1.1.3.1.cmml">+</mo><mrow id="S2.Ex2.m1.17.17.1.1.3.3" xref="S2.Ex2.m1.17.17.1.1.3.3.cmml"><munderover id="S2.Ex2.m1.17.17.1.1.3.3.1" xref="S2.Ex2.m1.17.17.1.1.3.3.1.cmml"><mo largeop="true" movablelimits="false" symmetric="true" id="S2.Ex2.m1.17.17.1.1.3.3.1.2.2" xref="S2.Ex2.m1.17.17.1.1.3.3.1.2.2.cmml">∑</mo><mrow id="S2.Ex2.m1.17.17.1.1.3.3.1.2.3" xref="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.cmml"><mi id="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.2" xref="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.2.cmml">l</mi><mo id="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.1" xref="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.1.cmml">=</mo><mn id="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.3" xref="S2.Ex2.m1.17.17.1.1.3.3.1.2.3.3.cmml">1</mn></mrow><mi id="S2.Ex2.m1.17.17.1.1.3.3.1.3" xref="S2.Ex2.m1.17.17.1.1.3.3.1.3.cmml">L</mi></munderover><mrow id="S2.Ex2.m1.17.17.1.1.3.3.2" xref="S2.Ex2.m1.17.17.1.1.3.3.2.cmml"><msqrt id="S2.Ex2.m1.17.17.1.1.3.3.2.2" xref="S2.Ex2.m1.17.17.1.1.3.3.2.2.cmml"><mi id="S2.Ex2.m1.17.17.1.1.3.3.2.2.2" xref="S2.Ex2.m1.17.17.1.1.3.3.2.2.2.cmml">M</mi></msqrt><mo id="S2.Ex2.m1.17.17.1.1.3.3.2.1" xref="S2.Ex2.m1.17.17.1.1.3.3.2.1.cmml">⁢</mo><mfrac id="S2.Ex2.m1.16.16" xref="S2.Ex2.m1.16.16.cmml"><mrow id="S2.Ex2.m1.13.13.5" xref="S2.Ex2.m1.13.13.5.cmml"><msub id="S2.Ex2.m1.13.13.5.7" xref="S2.Ex2.m1.13.13.5.7.cmml"><mi id="S2.Ex2.m1.13.13.5.7.2" xref="S2.Ex2.m1.13.13.5.7.2.cmml">α</mi><mrow id="S2.Ex2.m1.10.10.2.2.2.4" xref="S2.Ex2.m1.10.10.2.2.2.3.cmml"><mi id="S2.Ex2.m1.9.9.1.1.1.1" xref="S2.Ex2.m1.9.9.1.1.1.1.cmml">u</mi><mo id="S2.Ex2.m1.10.10.2.2.2.4.1" xref="S2.Ex2.m1.10.10.2.2.2.3.cmml">,</mo><mi id="S2.Ex2.m1.10.10.2.2.2.2" xref="S2.Ex2.m1.10.10.2.2.2.2.cmml">l</mi></mrow></msub><mo id="S2.Ex2.m1.13.13.5.6" xref="S2.Ex2.m1.13.13.5.6.cmml">⁢</mo><mi id="S2.Ex2.m1.13.13.5.8" xref="S2.Ex2.m1.13.13.5.8.cmml">𝐚</mi><mo id="S2.Ex2.m1.13.13.5.6a" xref="S2.Ex2.m1.13.13.5.6.cmml">⁢</mo><mrow id="S2.Ex2.m1.13.13.5.5.1" xref="S2.Ex2.m1.13.13.5.5.1.1.cmml"><mo stretchy="false" id="S2.Ex2.m1.13.13.5.5.1.2" xref="S2.Ex2.m1.13.13.5.5.1.1.cmml">(</mo><msub id="S2.Ex2.m1.13.13.5.5.1.1" xref="S2.Ex2.m1.13.13.5.5.1.1.cmml"><mi id="S2.Ex2.m1.13.13.5.5.1.1.2" xref="S2.Ex2.m1.13.13.5.5.1.1.2.cmml">θ</mi><mrow id="S2.Ex2.m1.12.12.4.4.2.4" xref="S2.Ex2.m1.12.12.4.4.2.3.cmml"><mi id="S2.Ex2.m1.11.11.3.3.1.1" xref="S2.Ex2.m1.11.11.3.3.1.1.cmml">u</mi><mo id="S2.Ex2.m1.12.12.4.4.2.4.1" xref="S2.Ex2.m1.12.12.4.4.2.3.cmml">,</mo><mi id="S2.Ex2.m1.12.12.4.4.2.2" xref="S2.Ex2.m1.12.12.4.4.2.2.cmml">l</mi></mrow></msub><mo stretchy="false" id="S2.Ex2.m1.13.13.5.5.1.3" xref="S2.Ex2.m1.13.13.5.5.1.1.cmml">)</mo></mrow></mrow><mrow id="S2.Ex2.m1.16.16.8.3" xref="S2.Ex2.m1.16.16.8.3.1.cmml"><mo stretchy="false" id="S2.Ex2.m1.16.16.8.3.2" xref="S2.Ex2.m1.16.16.8.3.1.cmml">(</mo><mrow id="S2.Ex2.m1.16.16.8.3.1" xref="S2.Ex2.m1.16.16.8.3.1.cmml"><mn id="S2.Ex2.m1.16.16.8.3.1.2" xref="S2.Ex2.m1.16.16.8.3.1.2.cmml">1</mn><mo id="S2.Ex2.m1.16.16.8.3.1.1" xref="S2.Ex2.m1.16.16.8.3.1.1.cmml">+</mo><msubsup id="S2.Ex2.m1.16.16.8.3.1.3" xref="S2.Ex2.m1.16.16.8.3.1.3.cmml"><mi id="S2.Ex2.m1.16.16.8.3.1.3.2.2" xref="S2.Ex2.m1.16.16.8.3.1.3.2.2.cmml">d</mi><mi id="S2.Ex2.m1.16.16.8.3.1.3.2.3" xref="S2.Ex2.m1.16.16.8.3.1.3.2.3.cmml">u</mi><msub id="S2.Ex2.m1.15.15.7.2.2" xref="S2.Ex2.m1.15.15.7.2.2.cmml"><mi id="S2.Ex2.m1.15.15.7.2.2.4" xref="S2.Ex2.m1.15.15.7.2.2.4.cmml">η</mi><mrow id="S2.Ex2.m1.15.15.7.2.2.2.2.2" xref="S2.Ex2.m1.15.15.7.2.2.2.2.3.cmml"><mi id="S2.Ex2.m1.14.14.6.1.1.1.1.1" xref="S2.Ex2.m1.14.14.6.1.1.1.1.1.cmml">u</mi><mo id="S2.Ex2.m1.15.15.7.2.2.2.2.2.2" xref="S2.Ex2.m1.15.15.7.2.2.2.2.3.cmml">,</mo><mrow id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.cmml"><mi id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.2" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.2.cmml">N</mi><mo id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.1" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.3" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.3.cmml">L</mi><mo id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.1a" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.4" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.4.cmml">O</mi><mo id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.1b" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.5" xref="S2.Ex2.m1.15.15.7.2.2.2.2.2.1.5.cmml">S</mi></mrow></mrow></msub></msubsup></mrow><mo stretchy="false" id="S2.Ex2.m1.16.16.8.3.3" xref="S2.Ex2.m1.16.16.8.3.1.cmml">)</mo></mrow></mfrac></mrow></mrow></mrow></mrow><mo id="S2.Ex2.m1.17.17.1.2" xref="S2.Ex2.m1.17.17.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S2.SS2.p3.2.m2.3.3" xref="S2.SS2.p3.2.m2.3.3.cmml"><mi id="S2.SS2.p3.2.m2.3.3.3" xref="S2.SS2.p3.2.m2.3.3.3.cmml">𝐚</mi><mo id="S2.SS2.p3.2.m2.3.3.2" xref="S2.SS2.p3.2.m2.3.3.2.cmml">⁢</mo><mrow id="S2.SS2.p3.2.m2.3.3.1.1" xref="S2.SS2.p3.2.m2.3.3.1.1.1.cmml"><mo stretchy="false" id="S2.SS2.p3.2.m2.3.3.1.1.2" xref="S2.SS2.p3.2.m2.3.3.1.1.1.cmml">(</mo><msub id="S2.SS2.p3.2.m2.3.3.1.1.1" xref="S2.SS2.p3.2.m2.3.3.1.1.1.cmml"><mi id="S2.SS2.p3.2.m2.3.3.1.1.1.2" xref="S2.SS2.p3.2.m2.3.3.1.1.1.2.cmml">θ</mi><mrow id="S2.SS2.p3.2.m2.2.2.2.4" xref="S2.SS2.p3.2.m2.2.2.2.3.cmml"><mi id="S2.SS2.p3.2.m2.1.1.1.1" xref="S2.SS2.p3.2.m2.1.1.1.1.cmml">u</mi><mo id="S2.SS2.p3.2.m2.2.2.2.4.1" xref="S2.SS2.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p3.2.m2.2.2.2.2" xref="S2.SS2.p3.2.m2.2.2.2.2.cmml">l</mi></mrow></msub><mo stretchy="false" id="S2.SS2.p3.2.m2.3.3.1.1.3" xref="S2.SS2.p3.2.m2.3.3.1.1.1.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S2.Ex3.m1.4.4.1" xref="S2.Ex3.m1.4.4.1.1.cmml"><mrow id="S2.Ex3.m1.4.4.1.1" xref="S2.Ex3.m1.4.4.1.1.cmml"><mrow id="S2.Ex3.m1.4.4.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.cmml"><mi id="S2.Ex3.m1.4.4.1.1.1.3" xref="S2.Ex3.m1.4.4.1.1.1.3.cmml">𝐚</mi><mo id="S2.Ex3.m1.4.4.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex3.m1.4.4.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S2.Ex3.m1.4.4.1.1.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.cmml">(</mo><msub id="S2.Ex3.m1.4.4.1.1.1.1.1.1" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S2.Ex3.m1.4.4.1.1.1.1.1.1.2" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.2.cmml">θ</mi><mrow id="S2.Ex3.m1.3.3.2.4" xref="S2.Ex3.m1.3.3.2.3.cmml"><mi id="S2.Ex3.m1.2.2.1.1" xref="S2.Ex3.m1.2.2.1.1.cmml">u</mi><mo id="S2.Ex3.m1.3.3.2.4.1" xref="S2.Ex3.m1.3.3.2.3.cmml">,</mo><mi id="S2.Ex3.m1.3.3.2.2" xref="S2.Ex3.m1.3.3.2.2.cmml">l</mi></mrow></msub><mo stretchy="false" id="S2.Ex3.m1.4.4.1.1.1.1.1.3" xref="S2.Ex3.m1.4.4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex3.m1.4.4.1.1.2" xref="S2.Ex3.m1.4.4.1.1.2.cmml">=</mo><mrow id="S2.Ex3.m1.4.4.1.1.3" xref="S2.Ex3.m1.4.4.1.1.3.cmml"><mstyle displaystyle="true" id="S2.Ex3.m1.4.4.1.1.3.2" xref="S2.Ex3.m1.4.4.1.1.3.2.cmml"><mfrac id="S2.Ex3.m1.4.4.1.1.3.2a" xref="S2.Ex3.m1.4.4.1.1.3.2.cmml"><mn id="S2.Ex3.m1.4.4.1.1.3.2.2" xref="S2.Ex3.m1.4.4.1.1.3.2.2.cmml">1</mn><msqrt id="S2.Ex3.m1.4.4.1.1.3.2.3" xref="S2.Ex3.m1.4.4.1.1.3.2.3.cmml"><mi id="S2.Ex3.m1.4.4.1.1.3.2.3.2" xref="S2.Ex3.m1.4.4.1.1.3.2.3.2.cmml">M</mi></msqrt></mfrac></mstyle><mo id="S2.Ex3.m1.4.4.1.1.3.1" xref="S2.Ex3.m1.4.4.1.1.3.1.cmml">⁢</mo><msup id="S2.Ex3.m1.4.4.1.1.3.3" xref="S2.Ex3.m1.4.4.1.1.3.3.cmml"><mrow id="S2.Ex3.m1.1.1a.3" xref="S2.Ex3.m1.1.1a.2.cmml"><mo id="S2.Ex3.m1.1.1a.3.1" xref="S2.Ex3.m1.1.1a.2.1.cmml">[</mo><mtable id="S2.Ex3.m1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.cmml"><mtr id="S2.Ex3.m1.1.1.1.1a" xref="S2.Ex3.m1.1.1.1.1.cmml"><mtd columnalign="center" id="S2.Ex3.m1.1.1.1.1b" xref="S2.Ex3.m1.1.1.1.1.cmml"><mrow id="S2.Ex3.m1.1.1.1.1.9.9.9.9.9" xref="S2.Ex3.m1.1.1.1.1.9.9.9.9.10.cmml"><mn id="S2.Ex3.m1.1.1.1.1.6.6.6.6.6" xref="S2.Ex3.m1.1.1.1.1.6.6.6.6.6.cmml">1</mn><mo id="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.3" xref="S2.Ex3.m1.1.1.1.1.9.9.9.9.10.cmml">,</mo><msup id="S2.Ex3.m1.1.1.1.1.8.8.8.8.8.1" xref="S2.Ex3.m1.1.1.1.1.8.8.8.8.8.1.cmml"><mi id="S2.Ex3.m1.1.1.1.1.8.8.8.8.8.1.2" xref="S2.Ex3.m1.1.1.1.1.8.8.8.8.8.1.2.cmml">e</mi><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.cmml"><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.3.cmml">-</mo><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.2.cmml">j</mi><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.1" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.1.cmml">⁢</mo><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.3" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.3.cmml">π</mi><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.1a" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.1.cmml">⁢</mo><msub id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.4" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.4.cmml"><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.4.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.4.4.2.cmml">θ</mi><mrow id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.4" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">u</mi><mo id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.4.1" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.3.cmml">,</mo><mi id="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.2.2.2.2.2.2.2.2.2.cmml">l</mi></mrow></msub></mrow></mrow></msup><mo id="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.4" xref="S2.Ex3.m1.1.1.1.1.9.9.9.9.10.cmml">,</mo><mi mathvariant="normal" id="S2.Ex3.m1.1.1.1.1.7.7.7.7.7" xref="S2.Ex3.m1.1.1.1.1.7.7.7.7.7.cmml">⋯</mi><mo id="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.5" xref="S2.Ex3.m1.1.1.1.1.9.9.9.9.10.cmml">,</mo><msup id="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.2" xref="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.2.cmml"><mi id="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.2.2" xref="S2.Ex3.m1.1.1.1.1.9.9.9.9.9.2.2.cmml">e</mi><mrow id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.cmml"><mo id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.4" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.4.cmml">-</mo><mrow id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.3" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.3.cmml">j</mi><mo id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.2" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.2.cmml">⁢</mo><mi id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.4" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.4.cmml">π</mi><mo id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.2a" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.2.cmml">⁢</mo><mrow id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.cmml"><mo stretchy="false" id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.2" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.cmml">(</mo><mrow id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.cmml"><mi id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.2" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.2.cmml">M</mi><mo id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.1.cmml">-</mo><mn id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.3" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.3.cmml">1</mn></mrow><mo stretchy="false" id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.3" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.2b" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.2.cmml">⁢</mo><msub id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.5" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.5.cmml"><mi id="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.5.2" xref="S2.Ex3.m1.1.1.1.1.5.5.5.5.5.3.3.5.2.cmml">θ</mi><mrow id="S2.Ex3.m1.1.1.1.1.4.4.4.4.4.2.2.2.4" xref="S2.Ex3.m1.1.1.1.1.4.4.4.4.4.2.2.2.3.cmml"><mi id="S2.Ex3.m1.1.1.1.1.3.3.3.3.3.1.1.1.1" xref="S2.Ex3.m1.1.1.1.1.3.3.3.3.3.1.1.1.1.cmml">u</mi><mo id="S2.Ex3.m1.1.1.1.1.4.4.4.4.4.2.2.2.4.1" xref="S2.Ex3.m1.1.1.1.1.4.4.4.4.4.2.2.2.3.cmml">,</mo><mi id="S2.Ex3.m1.1.1.1.1.4.4.4.4.4.2.2.2.2" xref="S2.Ex3.m1.1.1.1.1.4.4.4.4.4.2.2.2.2.cmml">l</mi></mrow></msub></mrow></mrow></msup></mrow></mtd></mtr></mtable><mo id="S2.Ex3.m1.1.1a.3.2" xref="S2.Ex3.m1.1.1a.2.1.cmml">]</mo></mrow><mi id="S2.Ex3.m1.4.4.1.1.3.3.2" xref="S2.Ex3.m1.4.4.1.1.3.3.2.cmml">T</mi></msup></mrow></mrow><mo id="S2.Ex3.m1.4.4.1.2" xref="S2.Ex3.m1.4.4.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.SS2.p4.1.m1.6.6" xref="S2.SS2.p4.1.m1.6.6.cmml"><msub id="S2.SS2.p4.1.m1.6.6.3" xref="S2.SS2.p4.1.m1.6.6.3.cmml"><mi id="S2.SS2.p4.1.m1.6.6.3.2" xref="S2.SS2.p4.1.m1.6.6.3.2.cmml">θ</mi><mrow id="S2.SS2.p4.1.m1.2.2.2.4" xref="S2.SS2.p4.1.m1.2.2.2.3.cmml"><mi id="S2.SS2.p4.1.m1.1.1.1.1" xref="S2.SS2.p4.1.m1.1.1.1.1.cmml">u</mi><mo id="S2.SS2.p4.1.m1.2.2.2.4.1" xref="S2.SS2.p4.1.m1.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p4.1.m1.2.2.2.2" xref="S2.SS2.p4.1.m1.2.2.2.2.cmml">l</mi></mrow></msub><mo id="S2.SS2.p4.1.m1.6.6.2" xref="S2.SS2.p4.1.m1.6.6.2.cmml">=</mo><mrow id="S2.SS2.p4.1.m1.6.6.1" xref="S2.SS2.p4.1.m1.6.6.1.cmml"><mn id="S2.SS2.p4.1.m1.6.6.1.3" xref="S2.SS2.p4.1.m1.6.6.1.3.cmml">2</mn><mo id="S2.SS2.p4.1.m1.6.6.1.2" xref="S2.SS2.p4.1.m1.6.6.1.2.cmml">⁢</mo><mfrac id="S2.SS2.p4.1.m1.6.6.1.4" xref="S2.SS2.p4.1.m1.6.6.1.4.cmml"><mi id="S2.SS2.p4.1.m1.6.6.1.4.2" xref="S2.SS2.p4.1.m1.6.6.1.4.2.cmml">D</mi><mi id="S2.SS2.p4.1.m1.6.6.1.4.3" xref="S2.SS2.p4.1.m1.6.6.1.4.3.cmml">λ</mi></mfrac><mo id="S2.SS2.p4.1.m1.6.6.1.2a" xref="S2.SS2.p4.1.m1.6.6.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.1.m1.6.6.1.1.1" xref="S2.SS2.p4.1.m1.6.6.1.1.2.cmml"><mi id="S2.SS2.p4.1.m1.5.5" xref="S2.SS2.p4.1.m1.5.5.cmml">sin</mi><mo id="S2.SS2.p4.1.m1.6.6.1.1.1a" xref="S2.SS2.p4.1.m1.6.6.1.1.2.cmml">⁡</mo><mrow id="S2.SS2.p4.1.m1.6.6.1.1.1.1" xref="S2.SS2.p4.1.m1.6.6.1.1.2.cmml"><mo stretchy="false" id="S2.SS2.p4.1.m1.6.6.1.1.1.1.2" xref="S2.SS2.p4.1.m1.6.6.1.1.2.cmml">(</mo><msub id="S2.SS2.p4.1.m1.6.6.1.1.1.1.1" xref="S2.SS2.p4.1.m1.6.6.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.1.m1.6.6.1.1.1.1.1.2" xref="S2.SS2.p4.1.m1.6.6.1.1.1.1.1.2.cmml">ϕ</mi><mrow id="S2.SS2.p4.1.m1.4.4.2.4" xref="S2.SS2.p4.1.m1.4.4.2.3.cmml"><mi id="S2.SS2.p4.1.m1.3.3.1.1" xref="S2.SS2.p4.1.m1.3.3.1.1.cmml">u</mi><mo id="S2.SS2.p4.1.m1.4.4.2.4.1" xref="S2.SS2.p4.1.m1.4.4.2.3.cmml">,</mo><mi id="S2.SS2.p4.1.m1.4.4.2.2" xref="S2.SS2.p4.1.m1.4.4.2.2.cmml">l</mi></mrow></msub><mo stretchy="false" id="S2.SS2.p4.1.m1.6.6.1.1.1.1.3" xref="S2.SS2.p4.1.m1.6.6.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.SS2.p4.4.m4.4.4" xref="S2.SS2.p4.4.m4.4.4.cmml"><msub id="S2.SS2.p4.4.m4.4.4.3" xref="S2.SS2.p4.4.m4.4.4.3.cmml"><mi id="S2.SS2.p4.4.m4.4.4.3.2" xref="S2.SS2.p4.4.m4.4.4.3.2.cmml">ϕ</mi><mrow id="S2.SS2.p4.4.m4.2.2.2.4" xref="S2.SS2.p4.4.m4.2.2.2.3.cmml"><mi id="S2.SS2.p4.4.m4.1.1.1.1" xref="S2.SS2.p4.4.m4.1.1.1.1.cmml">u</mi><mo id="S2.SS2.p4.4.m4.2.2.2.4.1" xref="S2.SS2.p4.4.m4.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p4.4.m4.2.2.2.2" xref="S2.SS2.p4.4.m4.2.2.2.2.cmml">l</mi></mrow></msub><mo id="S2.SS2.p4.4.m4.4.4.2" xref="S2.SS2.p4.4.m4.4.4.2.cmml">∈</mo><mrow id="S2.SS2.p4.4.m4.4.4.1.1" xref="S2.SS2.p4.4.m4.4.4.1.2.cmml"><mo stretchy="false" id="S2.SS2.p4.4.m4.4.4.1.1.2" xref="S2.SS2.p4.4.m4.4.4.1.2.cmml">[</mo><mn id="S2.SS2.p4.4.m4.3.3" xref="S2.SS2.p4.4.m4.3.3.cmml">0</mn><mo id="S2.SS2.p4.4.m4.4.4.1.1.3" xref="S2.SS2.p4.4.m4.4.4.1.2.cmml">,</mo><mrow id="S2.SS2.p4.4.m4.4.4.1.1.1" xref="S2.SS2.p4.4.m4.4.4.1.1.1.cmml"><mn id="S2.SS2.p4.4.m4.4.4.1.1.1.2" xref="S2.SS2.p4.4.m4.4.4.1.1.1.2.cmml">2</mn><mo id="S2.SS2.p4.4.m4.4.4.1.1.1.1" xref="S2.SS2.p4.4.m4.4.4.1.1.1.1.cmml">⁢</mo><mi id="S2.SS2.p4.4.m4.4.4.1.1.1.3" xref="S2.SS2.p4.4.m4.4.4.1.1.1.3.cmml">π</mi></mrow><mo stretchy="false" id="S2.SS2.p4.4.m4.4.4.1.1.4" xref="S2.SS2.p4.4.m4.4.4.1.2.cmml">]</mo></mrow></mrow></math>, <math><mrow id="S2.SS2.p4.6.m6.6.6" xref="S2.SS2.p4.6.m6.6.6.cmml"><msub id="S2.SS2.p4.6.m6.6.6.3" xref="S2.SS2.p4.6.m6.6.6.3.cmml"><mi id="S2.SS2.p4.6.m6.6.6.3.2" xref="S2.SS2.p4.6.m6.6.6.3.2.cmml">α</mi><mrow id="S2.SS2.p4.6.m6.2.2.2.4" xref="S2.SS2.p4.6.m6.2.2.2.3.cmml"><mi id="S2.SS2.p4.6.m6.1.1.1.1" xref="S2.SS2.p4.6.m6.1.1.1.1.cmml">u</mi><mo id="S2.SS2.p4.6.m6.2.2.2.4.1" xref="S2.SS2.p4.6.m6.2.2.2.3.cmml">,</mo><mi id="S2.SS2.p4.6.m6.2.2.2.2" xref="S2.SS2.p4.6.m6.2.2.2.2.cmml">l</mi></mrow></msub><mo id="S2.SS2.p4.6.m6.6.6.2" xref="S2.SS2.p4.6.m6.6.6.2.cmml">∼</mo><mrow id="S2.SS2.p4.6.m6.6.6.1" xref="S2.SS2.p4.6.m6.6.6.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.6.m6.6.6.1.3" xref="S2.SS2.p4.6.m6.6.6.1.3.cmml">𝒞</mi><mo id="S2.SS2.p4.6.m6.6.6.1.2" xref="S2.SS2.p4.6.m6.6.6.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p4.6.m6.6.6.1.4" xref="S2.SS2.p4.6.m6.6.6.1.4.cmml">𝒩</mi><mo id="S2.SS2.p4.6.m6.6.6.1.2a" xref="S2.SS2.p4.6.m6.6.6.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.6.m6.6.6.1.1.1" xref="S2.SS2.p4.6.m6.6.6.1.1.2.cmml"><mo stretchy="false" id="S2.SS2.p4.6.m6.6.6.1.1.1.2" xref="S2.SS2.p4.6.m6.6.6.1.1.2.cmml">(</mo><mn id="S2.SS2.p4.6.m6.5.5" xref="S2.SS2.p4.6.m6.5.5.cmml">0</mn><mo id="S2.SS2.p4.6.m6.6.6.1.1.1.3" xref="S2.SS2.p4.6.m6.6.6.1.1.2.cmml">,</mo><msubsup id="S2.SS2.p4.6.m6.6.6.1.1.1.1" xref="S2.SS2.p4.6.m6.6.6.1.1.1.1.cmml"><mi id="S2.SS2.p4.6.m6.6.6.1.1.1.1.2.2" xref="S2.SS2.p4.6.m6.6.6.1.1.1.1.2.2.cmml">σ</mi><mrow id="S2.SS2.p4.6.m6.4.4.2.4" xref="S2.SS2.p4.6.m6.4.4.2.3.cmml"><mi id="S2.SS2.p4.6.m6.3.3.1.1" xref="S2.SS2.p4.6.m6.3.3.1.1.cmml">u</mi><mo id="S2.SS2.p4.6.m6.4.4.2.4.1" xref="S2.SS2.p4.6.m6.4.4.2.3.cmml">,</mo><mi id="S2.SS2.p4.6.m6.4.4.2.2" xref="S2.SS2.p4.6.m6.4.4.2.2.cmml">l</mi></mrow><mn id="S2.SS2.p4.6.m6.6.6.1.1.1.1.2.3" xref="S2.SS2.p4.6.m6.6.6.1.1.1.1.2.3.cmml">2</mn></msubsup><mo stretchy="false" id="S2.SS2.p4.6.m6.6.6.1.1.1.4" xref="S2.SS2.p4.6.m6.6.6.1.1.2.cmml">)</mo></mrow></mrow></mrow></math>, <math><msub id="S2.SS2.p5.3.m3.2.3" xref="S2.SS2.p5.3.m3.2.3.cmml"><mi id="S2.SS2.p5.3.m3.2.3.2" xref="S2.SS2.p5.3.m3.2.3.2.cmml">η</mi><mrow id="S2.SS2.p5.3.m3.2.2.2.2" xref="S2.SS2.p5.3.m3.2.2.2.3.cmml"><mi id="S2.SS2.p5.3.m3.1.1.1.1" xref="S2.SS2.p5.3.m3.1.1.1.1.cmml">u</mi><mo id="S2.SS2.p5.3.m3.2.2.2.2.2" xref="S2.SS2.p5.3.m3.2.2.2.3.cmml">,</mo><mrow id="S2.SS2.p5.3.m3.2.2.2.2.1" xref="S2.SS2.p5.3.m3.2.2.2.2.1.cmml"><mi id="S2.SS2.p5.3.m3.2.2.2.2.1.2" xref="S2.SS2.p5.3.m3.2.2.2.2.1.2.cmml">L</mi><mo id="S2.SS2.p5.3.m3.2.2.2.2.1.1" xref="S2.SS2.p5.3.m3.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.3.m3.2.2.2.2.1.3" xref="S2.SS2.p5.3.m3.2.2.2.2.1.3.cmml">O</mi><mo id="S2.SS2.p5.3.m3.2.2.2.2.1.1a" xref="S2.SS2.p5.3.m3.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.3.m3.2.2.2.2.1.4" xref="S2.SS2.p5.3.m3.2.2.2.2.1.4.cmml">S</mi></mrow></mrow></msub></math>, <math><msub id="S2.SS2.p5.4.m4.2.3" xref="S2.SS2.p5.4.m4.2.3.cmml"><mi id="S2.SS2.p5.4.m4.2.3.2" xref="S2.SS2.p5.4.m4.2.3.2.cmml">η</mi><mrow id="S2.SS2.p5.4.m4.2.2.2.2" xref="S2.SS2.p5.4.m4.2.2.2.3.cmml"><mi id="S2.SS2.p5.4.m4.1.1.1.1" xref="S2.SS2.p5.4.m4.1.1.1.1.cmml">u</mi><mo id="S2.SS2.p5.4.m4.2.2.2.2.2" xref="S2.SS2.p5.4.m4.2.2.2.3.cmml">,</mo><mrow id="S2.SS2.p5.4.m4.2.2.2.2.1" xref="S2.SS2.p5.4.m4.2.2.2.2.1.cmml"><mi id="S2.SS2.p5.4.m4.2.2.2.2.1.2" xref="S2.SS2.p5.4.m4.2.2.2.2.1.2.cmml">N</mi><mo id="S2.SS2.p5.4.m4.2.2.2.2.1.1" xref="S2.SS2.p5.4.m4.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.4.m4.2.2.2.2.1.3" xref="S2.SS2.p5.4.m4.2.2.2.2.1.3.cmml">L</mi><mo id="S2.SS2.p5.4.m4.2.2.2.2.1.1a" xref="S2.SS2.p5.4.m4.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.4.m4.2.2.2.2.1.4" xref="S2.SS2.p5.4.m4.2.2.2.2.1.4.cmml">O</mi><mo id="S2.SS2.p5.4.m4.2.2.2.2.1.1b" xref="S2.SS2.p5.4.m4.2.2.2.2.1.1.cmml">⁢</mo><mi id="S2.SS2.p5.4.m4.2.2.2.2.1.5" xref="S2.SS2.p5.4.m4.2.2.2.2.1.5.cmml">S</mi></mrow></mrow></msub></math>

Correct Categorie: eess

Guessed Categorie: eess

Result: correct

Paper: https://arxiv.org/abs/1706.08055

Formulas:

1-

N_{b} \equiv N_{x} N_{y}

2-

l_{x} = l_{y} \equiv l

3-

K_{i n t} = 3 / 8 E l_{z}

4-

K_{i n t}

5-

K_{i n t} / 2

6-

𝐅_{i n t}^{(i j)} = k_{i j} (r_{i j} - l_{i j}) (𝐫_{j} - 𝐫_{i}) / r_{i j}

7-

𝐯 = (v, 0)

8-

𝐅_{s}^{(i)} = K_{s} (𝐯 t + 𝐫_{i}^{0} - 𝐫_{i})

9-

𝐅_{m o t}^{(i)} = \sum_{j} 𝐅_{i n t}^{(i j)} + 𝐅_{s}^{(i)}

10-

G = 3 / 8 E

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/hep-ph/0008010

Formulas:

1-

- ℒ_{m a s s} = {\bar{l}}_{i L} {(M_{l})}_{i j} l_{j R} + {\bar{ν}}_{i L} {(M_{D})}_{i j} ν_{j R} + \frac{1}{2} ν_{i R}^{T} C {(M_{R})}_{i j} ν_{j R} + h . c .,

2-

S U {(2)}_{L} \times U (1)

3-

M_{k} = c_{k} [Δ + P_{k}],

4-

Δ = (\begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix}), P_{k} = (\begin{matrix} 0 & 0 & 0 \\ 0 & a_{k} & 0 \\ 0 & 0 & b_{k} \end{matrix}), k = l, D, R

5-

| a_{k} |, | b_{k} | ≪ 1

6-

{(a, b, c)}_{D}

7-

M_{eff} = - M_{D} M_{R}^{- 1} M_{D}^{T} = - c_{eff} [Δ + P_{D}] Z {[Δ + P_{D}]}^{T},

8-

c_{eff} = c_{D}^{2} / c_{R}

9-

Z \equiv {[Δ + P_{R}]}^{- 1} = \frac{1}{a_{R} b_{R}} . (\begin{matrix} a_{R} + b_{R} + a_{R} b_{R} & - b_{R} & - a_{R} \\ - b_{R} & b_{R} & 0 \\ - a_{R} & 0 & a_{R} \end{matrix}) .

10-

M_{0} \equiv - M_{eff} / c_{eff}

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1707.00136

Formulas:

1-

f (t, x, p)

2-

\frac{\partial f}{\partial t} + u \frac{\partial f}{\partial x} = \frac{\partial}{\partial x} [D \frac{\partial f}{\partial x}] + \frac{1}{3} \frac{d u}{d x} p \frac{\partial f}{\partial p} + Q

3-

Q (t, x, p) = \frac{η n_{1} u_{1}}{4 π p_{i}^{2}} δ (p - p_{i}) δ (x) Q_{t} (t),

4-

Q_{t} (t)

5-

t_{1} = 4 D_{1} / u_{1}^{2}

6-

t_{2} = 4 D_{2} / u_{2}^{2}

7-

Q_{t} (t)

8-

f_{o} (t, p) = f_{o} (p) \int_{0}^{τ} Q_{t} (τ - τ^{'}) φ_{o} (τ^{'}) 𝑑 τ^{'} .

9-

f_{o} (p) = \frac{η n_{1}}{4 π p_{i}^{3}} \frac{3 σ}{σ - 1} {(\frac{p}{p_{i}})}^{- s_{f}},

10-

s_{f} = 3 σ / (σ - 1),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: nucl-th

Result: incorrect

Paper: https://arxiv.org/abs/2009.10705

Formulas:

1-

V_{X} (𝐫, 𝐫^{'}, [{ψ_{i}}]) = - \sum_{i = 1}^{N_{occ}} \frac{ψ_{i}^{*} (𝐫) ψ_{i} (𝐫^{'})}{| 𝐫 - 𝐫^{'} |}

2-

N_{G} ≫ N_{B} > N_{occ}

3-

V_{X} [{ψ_{i}}]

4-

W_{j} (𝐫) = [V_{X} (𝐫, 𝐫^{'}, [{ψ_{i}}]) ψ_{j} (𝐫^{'})] (𝐫)

5-

N_{occ} N_{e} N_{G} \log N_{G}

6-

M_{j k} = \int 𝑑 𝐫 ψ_{j}^{*} (𝐫) W_{k} (𝐫)

7-

N_{e}^{2} N_{G}

8-

- M = L L^{†}

9-

ξ_{i} (𝐫)

10-

= W_{j} (𝐫) {(L^{†})}_{j i}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/1401.0392

Formulas:

1-

v_{out} \sim 1000 km s^{- 1}

2-

\times 1000 M_{⊙} {yr}^{- 1}

3-

M_{σ} \sim - \frac{f_{c} κ σ^{4}}{π G^{2}} \sim - 3.67 \times 10^{8} σ_{200}^{4} M_{⊙},

4-

σ \equiv 200 σ_{200}

5-

R_{IC} \sim 0.5 M_{8} / σ_{200}

6-

v \sim 1000 km s^{- 1}

7-

v_{w} \sim 0.1 c

8-

v_{e} = {(\frac{2 η σ^{2} c}{3 f})}^{1 / 3} \sim - 925 σ_{200}^{2 / 3} f^{- 1 / 3} {kms}^{- 1},

9-

f \equiv f_{g} / f_{c}

10-

v_{out} = 4 / 3 v_{e}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1809.04502

Formulas:

1-

Λ_{N C} > 655

2-

[{\hat{x}}^{μ}, {\hat{x}}^{ν}] = i θ^{μ ν} = \frac{i C^{μ ν}}{Λ_{N C}^{2}},

3-

{[m a s s]}^{- 2}

4-

(f * g) (x) = {\exp (\frac{1}{2} i θ^{μ ν} \frac{\partial}{\partial x^{μ}} \frac{\partial}{\partial y^{ν}}) f (x) g (y) |}_{y \to x}

5-

Λ_{N C} \sim \sqrt{s}

6-

Λ_{N C} ≃ 1

7-

Λ_{N C} = 700

8-

p p / p \bar{p}

9-

Λ_{N C} ≳ 0.4

10-

q \bar{q} ⟶ γ^{*} / Z ⟶ l^{-} l^{+} .

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: hep-ph

Result: correct

Paper: https://arxiv.org/abs/1805.10329

Formulas:

1-

u_{l} \sim l_{⟂}^{1 / 3}

2-

l_{∥}^{- 1} V_{A} \approx l_{⟂}^{- 1} u_{l} .

3-

Ω \sim l_{⟂}^{- 2 / 3}

4-

< v_{L S R} <

5-

p p = \sqrt{{⟨ Q ⟩}^{2} + {⟨ U ⟩}^{2}},

6-

\nabla^{S} (𝐗, v_{z})

7-

\nabla^{S} (𝐗, v_{z}) = \frac{| S (𝐗, v_{z}) - S (𝐗 + 𝐗^{'}, v_{z}) |}{| 𝐗^{'} |},

8-

V = (0.186 (\frac{R}{R_{⊙}}) + 0.887) V_{⊙},

9-

V_{r} = R_{⊙} sin (l) (\frac{V}{R} - \frac{V_{⊙}}{R_{⊙}}) .

10-

Q (𝐗, v_{z}) = I (𝐗, v_{z}) cos (2 θ (𝐗, v_{z})),

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0101260

Formulas:

1-

\sim 0.7 v_{c r i t}

2-

v_{c r i t}

3-

v_{ϕ} \propto 1 / r

4-

v \sin (i)

5-

E {(H - K)}_{i s}

6-

E {(H - K)}_{c s}

7-

E {(B - V)}_{{(H - K)}_{i s}}

8-

\exp [- τ_{e x t} (λ)]

9-

τ_{e x t} (λ)

10-

{(M_{λ_{1}} - M_{λ_{2}})}_{o b s}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0509516

Formulas:

1-

\sim 250, 000 M_{⊙}

2-

Δ N_{tr} / N_{bin} = 0.05 f_{bin} ⟨ m_{b} ⟩ ⟨ a ⟩ per Gyr .

3-

⟨ m_{b} ⟩ \approx 1 M_{⊙}

4-

⟨ a ⟩ \approx 10 R_{⊙}

5-

Δ N_{tr} = 600 T_{9}^{- 1 / 3} per Gyr .

6-

M_{3} / (M_{1} + M_{2}) \approx 0.5 \pm 0.1

7-

M_{1} + M_{2} \approx 1.3 \pm 0.3 M_{⊙}

8-

P_{out} / P_{in} \approx 1000

9-

a_{out} / a_{in} \approx 100

10-

e_{out} \approx 0.95 \pm 0.05

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1702.02724

Formulas:

1-

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

2-

\sqrt{D_{V} \times J_{A - V}}

3-

| J_{A - V} |

4-

J_{V - V}^{in} = J_{V - V}^{out}

5-

\vec{H} = H \hat{z}

6-

θ = \tan^{- 1} [M_{a b}^{V} / M_{z}^{V}]

7-

3 p^{6} 3 d^{5} 4 s^{2}

8-

3 d^{8} 4 s^{1}

9-

3 s^{2} 3 p^{6} 3 d^{4} 4 s^{1}

10-

2 s^{2} 2 p^{4}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/0809.0508

Formulas:

1-

E (V - B) = 0.059

2-

B V R I

3-

0.15 - 120 M_{⊙}

4-

SFR = {\begin{matrix} 5.52 \times 10^{- 29} L_{1.4 GHz} & L_{1.4 GHz} > L_{c} \\ \frac{5.52 \times 10^{- 29} L_{1.4 GHz}}{0.1 + 0.9 {(L_{1.4 GHz} / L_{c})}^{0.3}} & L_{1.4 GHz} < L_{c}, \end{matrix}

5-

L_{c} = 6.4 \times 10^{28}

6-

L_{1.4 GHz} = \frac{4 π \times 10^{- 23} D_{L}^{2} F_{ν}}{1 + z} \times {[\frac{ν_{obs} (1 + z)}{1.4}]}^{- α} .

7-

M_{d} = \frac{F_{ν} D_{L}^{2}}{(1 + z) κ (ν_{r}) B (ν_{r}, T)},

8-

200 GHz ≲ ν_{r} ≲ 5000 GHz

9-

B (ν_{r}, T)

10-

κ (ν_{r}) = 0.067 {[\frac{ν_{r} (GHz)}{250}]}^{β} m^{2} {kg}^{- 1}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0304058

Formulas:

1-

Γ_{r} \equiv Γ_{s} Γ_{s | d} (1 - β_{s} β_{s | d})

2-

β_{s | d} \to 1 / 3

3-

Γ_{r} \to Γ_{s} / \sqrt{2}

4-

𝒫 (μ^{i}; μ^{e})

5-

𝒫_{u} (μ_{u}^{i}; μ_{u}^{e})

6-

𝒫_{d} (μ_{d}^{i}; μ_{d}^{e})

7-

f_{d}^{2 n + 1} (μ_{d}, ϵ_{d})

8-

f_{u}^{2 n} (μ_{u}, ϵ_{u})

9-

N = \int d μ_{u} d ϵ_{u} f_{u}^{2 n} (μ_{u}, ϵ_{u})

10-

N = \int d μ_{d} d ϵ_{d} f_{d}^{2 n + 1} (μ_{d}, ϵ_{d})

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0207389

Formulas:

1-

E_{d e t} = 2.25

2-

E_{d e t} = 2.25

3-

E_{d e t} = 2.00

4-

E_{d e t} = 2.03

5-

E_{d e t}

6-

s i n g l e

7-

d I / d V

8-

d I / d V

9-

e x p (i π m z / L)

10-

n, L_{z}, m

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: cond-mat

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0210181

Formulas:

1-

ψ (x_{1}, x_{2})

2-

ψ (x_{1}, x_{2}) = ϕ (x_{1}) χ (x_{2})

3-

ϕ (x_{1})

4-

χ^{*} (x_{2})

5-

ϕ (x_{1}) = \int ψ (x_{1}, x_{2}) χ^{*} (x_{2}) 𝑑 x_{2}

6-

χ^{*} (x_{2})

7-

ϕ (x_{1})

8-

ψ (x_{1}, x_{2})

9-

Ψ_{A} (x_{1}, x_{2}) = \frac{1}{\sqrt{2}} [ϕ (x_{1}) χ (x_{2}) - ϕ (x_{2}) χ (x_{1})]

10-

ψ (x_{1}, x_{2}) = \sum_{i . j} C_{i j} ϕ_{i} (x_{1}) ϕ_{j} (x_{2})

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0511418

Formulas:

1-

F^{α β} \equiv - ⟨ \frac{\partial^{2} \ln ℒ}{\partial p_{α} \partial p_{β}} ⟩,

2-

ℒ (p_{α} | data)

3-

⟨ Δ {\hat{p}}_{α}^{2} ⟩ \geq 1 / F_{α α},

4-

⟨ Δ {\hat{p}}_{α} Δ {\hat{p}}_{β} ⟩ ≃ {(F^{- 1})}_{α β} .

5-

⟨ δ_{𝒌} δ_{𝒌^{'}} ⟩ = {(2 π)}^{3} δ_{3 D} (𝒌 + 𝒌^{'}) P (k),

6-

δ (𝒓) \equiv ρ (𝒓) / \bar{ρ} - 1

7-

δ_{3 D} (𝒌)

8-

P (k) / 2

9-

⟨ Δ \hat{P} {(k)}^{2} ⟩ = 2 P {(k)}^{2} / N_{k},

10-

I \equiv - ⟨ \frac{\partial^{2} \ln ℒ}{\partial \ln A^{2}} ⟩ .

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1205.5210

Formulas:

1-

| u_{ε} (x) - u_{0} (x) | \leq C_{κ} ε^{(d - 1) / 2} \frac{1}{d {(x)}^{κ}}

2-

\forall κ > d - 1

3-

{∥ u_{ε} - u_{0} ∥}_{L^{p}} \leq C_{κ} ε^{κ}

4-

\forall κ \in (0, 1 / 2 p)

5-

g (x / ε)

6-

u_{ε} (x) = \int_{\partial D} P (x, y) g (y / ε) 𝑑 σ (y),

7-

P (x, y) = n (y) A (y) \nabla G (x, y)

8-

| \partial D | \bar{g}

9-

A (x / ϵ)

10-

{|| u_{ε} - u_{0} ||}_{L^{2} (D)} \leq C_{α} ε^{κ}, \forall κ < (d - 1) / (3 d + 5) .

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/0808.2193

Formulas:

1-

B V R I

2-

B V R I

3-

B V R I

4-

B V R I

5-

U V W 1

6-

U V M 1

7-

U V W 2

8-

B V R I

9-

U V W 1

10-

U V M 2

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0509416

Formulas:

1-

0.6 ρ_{*} < ρ < 1.5 ρ_{*}

2-

ρ_{*} = 2.6 \times 10^{14}

3-

5 km \overset{<}{\sim} r \overset{<}{\sim} 9 km

4-

0.6 ρ_{*} < ρ < ρ_{*}

5-

ρ_{*} < ρ < 1.5 ρ_{*}

6-

ω_{s} = \nabla \times 𝐯_{s}

7-

\nabla \times 𝐯_{s} = 0

8-

\frac{d_{n} 𝐯_{n}}{d t} = - \frac{\nabla p_{n}}{ρ} + ν_{n} \nabla^{2} 𝐯_{n} + \frac{ρ_{s}}{ρ} 𝐅 - \frac{ρ_{s} ν_{s}}{ρ} \nabla | ω_{s} |,

9-

\frac{d_{s} 𝐯_{s}}{d t} = - \frac{\nabla p_{s}}{ρ} + ν_{s} 𝐓 - \frac{ρ_{n}}{ρ} 𝐅 - \frac{ρ_{s} ν_{s}}{ρ} \nabla | ω_{s} |,

10-

d_{n, s} / d t = \partial / \partial t + 𝐯_{n, s} \cdot \nabla

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1609.05590

Formulas:

1-

P_{1} \dots P_{n}

2-

L_{c l s}

3-

L_{l o c}

4-

L_{p o s e}

5-

I o U > 0.5

6-

L_{t o t} = \frac{1}{N} (L_{c l s} + α_{1} L_{l o c} + α_{2} L_{p o s e}),

7-

L_{c l s}

8-

L_{p o s e}

9-

L_{l o c}

10-

I o U > 0.5

Formulas (html):

Correct Categorie: cs

Guessed Categorie: cs

Result: correct

Paper: https://arxiv.org/abs/1605.08193

Formulas:

1-

| Δ MI | / | Δ \bar{σ} |

2-

| Δ MI | / | Δ \bar{σ} |

3-

λ_{s} \approx + 26 \times 10^{- 6}

4-

2 θ = {44.25}^{\circ}

5-

2 θ \sim 44^{\circ}

6-

t_{F M} ≪ t_{S}

7-

t_{F M} \approx 0.0046 t_{S}

8-

\bar{σ} = \frac{Y}{6 (1 - ν_{s}) r} \frac{t_{S}^{2}}{t_{F M}},

9-

MI = Z (H) - Z (H_{m a x})

10-

Z (H_{m a x})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1911.11163

Formulas:

1-

R_{vir} = 210 kpc

2-

10^{- 3} μ G

3-

2 \times 10^{12} M_{⊙}

4-

6 \times 10^{3} M_{⊙}

5-

4 \times 10^{4} M_{⊙}

6-

1.2 \times R_{200, mean}

7-

R_{vir} = R_{200, crit}

8-

\propto k^{- 5 / 3}

9-

p_{kin} = \frac{1}{2 V} \int ρ 𝐯^{2} 𝑑 V

10-

10^{- 2} μ G

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/math/0311015

Formulas:

1-

{X_{α} : α \in A}

2-

{α \in A : x_{α} \neq x_{α}^{*}}

3-

f g : Y \to Y

4-

{X_{α}; p_{α}^{β} : α < β < κ}

5-

\lim_{\leftarrow} {X_{α}; p_{α}^{β} : α < β < δ}

6-

H^{n} (X, G)

7-

H^{n} (X, G)

8-

𝐓 = {z \in ℂ : | z | = 1}

9-

ϕ : A \to H^{1} (X, ℤ)

10-

χ_{a} : X \to 𝐓

Formulas (html):

Correct Categorie: math

Guessed Categorie: math

Result: correct

Paper: https://arxiv.org/abs/1202.5577

Formulas:

1-

i \frac{d 𝐀 (z)}{d z} = 𝐇 (z) 𝐀 (z),

2-

𝐀 = {[a_{i} (z), a_{m} (z), a_{1} (z), a_{2} (z), \dots, a_{N} (z)]}^{T}

3-

𝐇 (z) = [\begin{matrix} 0 & Ω_{p} (z) & 0 & \dots & 0 \\ Ω_{p} (z) & 0 & Ω_{1} (z) & \dots & Ω_{N} (z) \\ 0 & Ω_{1} (z) & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & Ω_{N} (z) & 0 & \dots & 0 \end{matrix}] .

4-

a_{k} (z)

5-

k = i, m, 1, 2, \dots, N

6-

I_{k} = {| a_{k} (z) |}^{2}

7-

Ω_{p} (z)

8-

Ω_{k} (z)

9-

k = 1, 2, \dots, N

10-

Ω_{p} (z)

Formulas (html):

Correct Categorie: physics

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0906.0928

Formulas:

1-

{(1 - x)}^{3}

2-

g_{1} (E, E^{'}, θ) = \frac{M Q^{2} ν}{4 α_{e}^{2}} \frac{E}{E^{'}} \frac{1}{E + E^{'}} [Δ σ_{∥} (E, E^{'}, θ) + \tan \frac{θ}{2} Δ σ_{⟂} (E, E^{'}, θ)]

3-

g_{1}^{{}^{3}{He}} (x, Q^{2})

4-

g_{1}^{{}^{3}{He}} = P_{n} g_{1}^{n} + 2 P_{p} g_{1}^{p}

5-

P_{n} = 0.86 \pm 0.02

6-

P_{p} = - 0.028 \pm 0.004

7-

{\tilde{Γ}}_{1} (Q^{2}) = \int_{x_{m i n}}^{x_{m a x}} 𝑑 x g_{1} (x, Q^{2})

8-

x_{m a x}

9-

x_{m i n}

10-

A_{1} = \frac{A_{∥}}{D (1 + η ξ)} - \frac{η A_{⟂}}{d (1 + η ξ)}

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1004.3418

Formulas:

1-

δ S = 2 π k_{B} \frac{m c}{ℏ} δ x .

2-

N = \frac{A c^{3}}{G ℏ} .

3-

F δ x = T δ S .

4-

S = \frac{1}{8 π G} \int \sqrt{- g} d^{4} x \nabla_{i} a^{i}

5-

S = k_{B} \ln Ω

6-

S = \frac{1}{4} \frac{k_{B} c^{3}}{G ℏ} A,

7-

\frac{d S}{d A} = \frac{1}{4} \frac{k_{B} c^{3}}{G ℏ},

8-

d \tilde{E} = d K + d U .

9-

d E = T d S - d U .

10-

δ U = \frac{\partial U}{\partial x} δ x = - F δ x .

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/9901007

Formulas:

1-

\sum_{m, n} J_{m n} {\hat{c}}_{m}^{†} {\hat{c}}_{n},

2-

\sum_{K} ℏ ω_{K} {\hat{a}}_{K}^{†} {\hat{a}}_{K},

3-

\sum_{k} ℏ Ω_{k} {\hat{b}}_{k}^{†} {\hat{b}}_{k},

4-

{\hat{c}}_{m}^{†} = {\hat{d}}_{m}^{†} {\hat{e}}_{m}

5-

{\hat{H}}_{e - f} = \sum_{m, K} ℏ ω_{K_{0}} F_{K}^{m} ({\hat{a}}_{K} {\hat{c}}_{m}^{†} + {\hat{a}}_{- K}^{†} {\hat{c}}_{m}),

6-

F_{K}^{m} = - \frac{1}{ℏ ω_{K_{0}}} \sqrt{\frac{ℏ}{2 ϵ_{0} V ω_{K}}} \frac{e}{m_{e}} ϵ_{K} \cdot ⟨ m | \hat{𝐩} | 0 ⟩ \exp (i 𝐊 \cdot 𝐫_{m}) .

7-

{\hat{H}}_{e - p h}

8-

{\hat{H}}_{e - p h} = \frac{1}{\sqrt{N}} \sum_{m, k} ℏ Ω_{k} G_{k}^{m} {\hat{c}}_{m}^{†} {\hat{c}}_{m} ({\hat{b}}_{k} + {\hat{b}}_{- k}^{†})

9-

\hat{ρ} (t)

10-

\hat{ρ} (t)

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0312142

Formulas:

1-

E : ℱ \to ℒ (ℋ)

2-

E (X) \geq O

3-

E (Ω) = I

4-

E {(X)}^{2} = E (X)

5-

E : ℱ \to ℒ (ℋ)

6-

E : ℱ \to ℒ (ℋ)

7-

p_{ψ}^{E} (X) = ⟨ ψ | E (X) ψ ⟩, X \in ℱ .

8-

P^{A} : ℬ (ℝ) \to ℒ (ℋ)

9-

P : ℱ \to ℒ (ℋ)

10-

ν : Ω \times ℱ \to [0, 1]

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: quant-ph

Result: correct

Paper: https://arxiv.org/abs/physics/0607144

Formulas:

1-

k_{B} T_{U H} = \frac{ℏ a}{2 π c},

2-

a = g = 9.8 m / s^{2}

3-

4 \times 10^{- 20} K

4-

n = n_{0} + n_{2} I

5-

d n_{0} / d ω < 0

6-

d n_{0} / d ω < 0

7-

k_{B} T_{U H} = \frac{ℏ}{2 π} ∣ \frac{d c^{*}}{d x} ∣,

8-

(ω_{0}, ω_{0} + Δ ω)

9-

Δ ω = \frac{1}{2 π} ∣ \frac{d c^{*}}{d x} ∣

10-

Δ ω = \frac{c}{2 π n^{2}} ∣ \frac{d n}{d x} ∣ = \frac{1}{2 π n} ∣ \frac{d n}{d t} ∣ = \frac{n_{2}}{2 π n} ∣ \frac{d I}{d t} ∣,

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1307.5939

Formulas:

1-

F_{ν} \propto ν^{- β_{O}}

2-

t = 10^{2}, 10^{3}

3-

t = 10^{2}, 10^{3}

4-

t = 10^{2}, 10^{3}

5-

M_{R, c} = 16.1 \pm 1.8

6-

M_{R} = 14 - 22

7-

M_{R, c} = 18.4 \pm 1.6

8-

M_{R, c} = 15.6 \pm 2.2

9-

M_{R, c} = 18.3

10-

z \sim z + d z

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/1003.1033

Formulas:

1-

𝚽_{𝟎} = 𝒉 𝒄 / 𝟐 𝒆

2-

I_{c} (H)

3-

Θ_{v} (x) = \arctan (\frac{x - x_{v}}{z_{v}}) + const .,

4-

Θ_{v} (x)

5-

I_{c} (H)

6-

Θ_{v} (x)

7-

x_{v} = L / 2

8-

I_{c} (Φ)

9-

I_{c} (Φ)

10-

𝑰_{𝒄} (𝚽)

Formulas (html):

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<math><mrow id="p1.3.m3.1.2" xref="p1.3.m3.1.2.cmml"><msub id="p1.3.m3.1.2.2" xref="p1.3.m3.1.2.2.cmml"><mi id="p1.3.m3.1.2.2.2" xref="p1.3.m3.1.2.2.2.cmml">I</mi><mi id="p1.3.m3.1.2.2.3" xref="p1.3.m3.1.2.2.3.cmml">c</mi></msub><mo id="p1.3.m3.1.2.1" xref="p1.3.m3.1.2.1.cmml">⁢</mo><mrow id="p1.3.m3.1.2.3.2" xref="p1.3.m3.1.2.cmml"><mo stretchy="false" id="p1.3.m3.1.2.3.2.1" xref="p1.3.m3.1.2.cmml">(</mo><mi id="p1.3.m3.1.1" xref="p1.3.m3.1.1.cmml">H</mi><mo stretchy="false" id="p1.3.m3.1.2.3.2.2" xref="p1.3.m3.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.E1.m1.4.4.1" xref="S0.E1.m1.4.4.1.1.cmml"><mrow id="S0.E1.m1.4.4.1.1" xref="S0.E1.m1.4.4.1.1.cmml"><mrow id="S0.E1.m1.4.4.1.1.2" xref="S0.E1.m1.4.4.1.1.2.cmml"><msub id="S0.E1.m1.4.4.1.1.2.2" xref="S0.E1.m1.4.4.1.1.2.2.cmml"><mi mathvariant="normal" id="S0.E1.m1.4.4.1.1.2.2.2" xref="S0.E1.m1.4.4.1.1.2.2.2.cmml">Θ</mi><mi id="S0.E1.m1.4.4.1.1.2.2.3" xref="S0.E1.m1.4.4.1.1.2.2.3.cmml">v</mi></msub><mo 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xref="S0.E1.m1.3.3.2.2.cmml">x</mi><mo id="S0.E1.m1.3.3.2.1" xref="S0.E1.m1.3.3.2.1.cmml">-</mo><msub id="S0.E1.m1.3.3.2.3" xref="S0.E1.m1.3.3.2.3.cmml"><mi id="S0.E1.m1.3.3.2.3.2" xref="S0.E1.m1.3.3.2.3.2.cmml">x</mi><mi id="S0.E1.m1.3.3.2.3.3" xref="S0.E1.m1.3.3.2.3.3.cmml">v</mi></msub></mrow><msub id="S0.E1.m1.3.3.3" xref="S0.E1.m1.3.3.3.cmml"><mi id="S0.E1.m1.3.3.3.2" xref="S0.E1.m1.3.3.3.2.cmml">z</mi><mi id="S0.E1.m1.3.3.3.3" xref="S0.E1.m1.3.3.3.3.cmml">v</mi></msub></mfrac><mo id="S0.E1.m1.4.4.1.1.3.2.2.1.2" xref="S0.E1.m1.4.4.1.1.3.2.1.cmml">)</mo></mrow></mrow><mo id="S0.E1.m1.4.4.1.1.3.1" xref="S0.E1.m1.4.4.1.1.3.1.cmml">+</mo><mi id="S0.E1.m1.4.4.1.1.3.3" xref="S0.E1.m1.4.4.1.1.3.3.cmml">const</mi></mrow></mrow><mo id="S0.E1.m1.4.4.1.2" xref="S0.E1.m1.4.4.1.1.cmml">.</mo><mo id="S0.E1.m1.4.4.1.3" xref="S0.E1.m1.4.4.1.1.cmml">,</mo></mrow></math>, <math><mrow id="p2.5.m4.1.2" xref="p2.5.m4.1.2.cmml"><msub id="p2.5.m4.1.2.2" xref="p2.5.m4.1.2.2.cmml"><mi mathvariant="normal" id="p2.5.m4.1.2.2.2" xref="p2.5.m4.1.2.2.2.cmml">Θ</mi><mi id="p2.5.m4.1.2.2.3" xref="p2.5.m4.1.2.2.3.cmml">v</mi></msub><mo id="p2.5.m4.1.2.1" xref="p2.5.m4.1.2.1.cmml">⁢</mo><mrow id="p2.5.m4.1.2.3.2" xref="p2.5.m4.1.2.cmml"><mo stretchy="false" id="p2.5.m4.1.2.3.2.1" xref="p2.5.m4.1.2.cmml">(</mo><mi id="p2.5.m4.1.1" xref="p2.5.m4.1.1.cmml">x</mi><mo stretchy="false" id="p2.5.m4.1.2.3.2.2" xref="p2.5.m4.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p3.1.m1.1.2" xref="p3.1.m1.1.2.cmml"><msub id="p3.1.m1.1.2.2" xref="p3.1.m1.1.2.2.cmml"><mi id="p3.1.m1.1.2.2.2" xref="p3.1.m1.1.2.2.2.cmml">I</mi><mi id="p3.1.m1.1.2.2.3" xref="p3.1.m1.1.2.2.3.cmml">c</mi></msub><mo id="p3.1.m1.1.2.1" xref="p3.1.m1.1.2.1.cmml">⁢</mo><mrow id="p3.1.m1.1.2.3.2" xref="p3.1.m1.1.2.cmml"><mo stretchy="false" id="p3.1.m1.1.2.3.2.1" xref="p3.1.m1.1.2.cmml">(</mo><mi id="p3.1.m1.1.1" xref="p3.1.m1.1.1.cmml">H</mi><mo stretchy="false" id="p3.1.m1.1.2.3.2.2" xref="p3.1.m1.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.F1.8.m2.1.2" xref="S0.F1.8.m2.1.2.cmml"><msub id="S0.F1.8.m2.1.2.2" xref="S0.F1.8.m2.1.2.2.cmml"><mi mathvariant="normal" id="S0.F1.8.m2.1.2.2.2" xref="S0.F1.8.m2.1.2.2.2.cmml">Θ</mi><mi id="S0.F1.8.m2.1.2.2.3" xref="S0.F1.8.m2.1.2.2.3.cmml">v</mi></msub><mo id="S0.F1.8.m2.1.2.1" xref="S0.F1.8.m2.1.2.1.cmml">⁢</mo><mrow id="S0.F1.8.m2.1.2.3.2" xref="S0.F1.8.m2.1.2.cmml"><mo stretchy="false" id="S0.F1.8.m2.1.2.3.2.1" xref="S0.F1.8.m2.1.2.cmml">(</mo><mi id="S0.F1.8.m2.1.1" xref="S0.F1.8.m2.1.1.cmml">x</mi><mo stretchy="false" id="S0.F1.8.m2.1.2.3.2.2" xref="S0.F1.8.m2.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.F1.10.m4.1.1" xref="S0.F1.10.m4.1.1.cmml"><msub id="S0.F1.10.m4.1.1.2" xref="S0.F1.10.m4.1.1.2.cmml"><mi id="S0.F1.10.m4.1.1.2.2" xref="S0.F1.10.m4.1.1.2.2.cmml">x</mi><mi id="S0.F1.10.m4.1.1.2.3" xref="S0.F1.10.m4.1.1.2.3.cmml">v</mi></msub><mo id="S0.F1.10.m4.1.1.1" xref="S0.F1.10.m4.1.1.1.cmml">=</mo><mrow id="S0.F1.10.m4.1.1.3" xref="S0.F1.10.m4.1.1.3.cmml"><mi id="S0.F1.10.m4.1.1.3.2" xref="S0.F1.10.m4.1.1.3.2.cmml">L</mi><mo id="S0.F1.10.m4.1.1.3.1" xref="S0.F1.10.m4.1.1.3.1.cmml">/</mo><mn id="S0.F1.10.m4.1.1.3.3" xref="S0.F1.10.m4.1.1.3.3.cmml">2</mn></mrow></mrow></math>, <math><mrow id="p5.8.m8.1.2" xref="p5.8.m8.1.2.cmml"><msub id="p5.8.m8.1.2.2" xref="p5.8.m8.1.2.2.cmml"><mi id="p5.8.m8.1.2.2.2" xref="p5.8.m8.1.2.2.2.cmml">I</mi><mi id="p5.8.m8.1.2.2.3" xref="p5.8.m8.1.2.2.3.cmml">c</mi></msub><mo id="p5.8.m8.1.2.1" xref="p5.8.m8.1.2.1.cmml">⁢</mo><mrow id="p5.8.m8.1.2.3.2" xref="p5.8.m8.1.2.cmml"><mo stretchy="false" id="p5.8.m8.1.2.3.2.1" xref="p5.8.m8.1.2.cmml">(</mo><mi mathvariant="normal" id="p5.8.m8.1.1" xref="p5.8.m8.1.1.cmml">Φ</mi><mo stretchy="false" id="p5.8.m8.1.2.3.2.2" xref="p5.8.m8.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="p5.12.m12.1.2" xref="p5.12.m12.1.2.cmml"><msub id="p5.12.m12.1.2.2" xref="p5.12.m12.1.2.2.cmml"><mi id="p5.12.m12.1.2.2.2" xref="p5.12.m12.1.2.2.2.cmml">I</mi><mi id="p5.12.m12.1.2.2.3" xref="p5.12.m12.1.2.2.3.cmml">c</mi></msub><mo id="p5.12.m12.1.2.1" xref="p5.12.m12.1.2.1.cmml">⁢</mo><mrow id="p5.12.m12.1.2.3.2" xref="p5.12.m12.1.2.cmml"><mo stretchy="false" id="p5.12.m12.1.2.3.2.1" xref="p5.12.m12.1.2.cmml">(</mo><mi mathvariant="normal" id="p5.12.m12.1.1" xref="p5.12.m12.1.1.cmml">Φ</mi><mo stretchy="false" id="p5.12.m12.1.2.3.2.2" xref="p5.12.m12.1.2.cmml">)</mo></mrow></mrow></math>, <math><mrow id="S0.F2.12.1.m1.1.2" xref="S0.F2.12.1.m1.1.2.cmml"><msub id="S0.F2.12.1.m1.1.2.2" xref="S0.F2.12.1.m1.1.2.2.cmml"><mi id="S0.F2.12.1.m1.1.2.2.2" xref="S0.F2.12.1.m1.1.2.2.2.cmml">𝑰</mi><mi id="S0.F2.12.1.m1.1.2.2.3" xref="S0.F2.12.1.m1.1.2.2.3.cmml">𝒄</mi></msub><mo mathvariant="sans-serif" id="S0.F2.12.1.m1.1.2.1" xref="S0.F2.12.1.m1.1.2.1.cmml">⁢</mo><mrow id="S0.F2.12.1.m1.1.2.3.2" xref="S0.F2.12.1.m1.1.2.cmml"><mo mathvariant="bold" stretchy="false" id="S0.F2.12.1.m1.1.2.3.2.1" xref="S0.F2.12.1.m1.1.2.cmml">(</mo><mi id="S0.F2.12.1.m1.1.1" xref="S0.F2.12.1.m1.1.1.cmml">𝚽</mi><mo mathvariant="bold" stretchy="false" id="S0.F2.12.1.m1.1.2.3.2.2" xref="S0.F2.12.1.m1.1.2.cmml">)</mo></mrow></mrow></math>

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0405022

Formulas:

1-

0.5 < M < 1.2 M_{⊙}

2-

M < 0.4 M_{⊙}

3-

m_{r e l} = - 2.5 \log_{10} (1 + f_{rel})

4-

σ_{c a n d}

5-

σ_{c a n d}

6-

2 \dots 3 σ

7-

T_{e f f} > 2800

8-

τ_{1 / 2} = 0.5 \times 11 / 7 \times 18

9-

v \sin i =

10-

P = (105 \pm 61) (M / M_{⊙}) - (1.5 \pm 10)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-ex/9508007

Formulas:

1-

λ_{a t t}

2-

σ_{C C T}

3-

σ_{C C T}

4-

Q E (λ)

5-

N_{p e}^{m e a s} = \frac{N_{0} s i n^{2} Θ_{c} d}{c o s θ} G

6-

G = \frac{N_{p e} (M C)}{\int \frac{d N}{d λ} Q E (λ) 𝑑 λ} \frac{c o s θ}{d}

7-

N_{p e} (M C)

8-

\frac{d N}{d λ}

9-

Q E (λ)

10-

ϵ_{r e f}

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: hep-ex

Result: correct

Paper: https://arxiv.org/abs/1901.05059

Formulas:

1-

α_{S}^{FSR} (M_{Z})

2-

Δ Φ < 60 °

3-

120 ° < Δ Φ < 60 °

4-

Δ Φ > 120 °

5-

α_{S}^{FSR} (M_{Z})

6-

α_{S}^{FSR} (M_{Z})

7-

α_{S}^{FSR} (M_{Z})

8-

α_{S}^{FSR} (M_{Z})

9-

α_{S}^{FSR} (M_{Z})

10-

α_{S}^{FSR} (M_{Z}) = {0.120}_{- 0.006}^{+ 0.006}

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: physics

Result: incorrect

Run 11334341 (Agent708)