Run 6249155

Paper: https://arxiv.org/abs/cond-mat/0205186

Formulas:

1-

10^{- 4} - 10^{- 6}

2-

⟨ n | \hat{O} | m ⟩ = O_{0} δ_{m n} + o (\exp (- L))

3-

(a b), (c d) \dots

4-

Φ = (n_{s} + \frac{1}{2}) Φ_{0}

5-

E_{J} = \frac{ℏ}{2 e} I_{c}

6-

E_{C} = \frac{e 2}{2 C}

7-

ℒ = \sum_{(i j)} \frac{1}{16 E_{C}} (\dot{ϕ_{i}} - \dot{ϕ_{j}}) 2 + E_{J} \cos (ϕ_{i} - ϕ_{j} - a_{i j})

8-

ϕ_{i j} = ϕ_{i} - ϕ_{j} - a_{i j}

9-

\sum_{Γ} ϕ_{i j} = 2 π Φ_{Γ} / Φ_{0} + 2 π n

10-

\frac{Φ_{0}}{2} (\frac{1}{2} + m)

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/cs/0101001

Formulas:

1-

\min {f (x) : x_{l} \leq x \leq x_{u}},

2-

f : ℝ^{n} \mapsto ℝ

3-

\nabla f (x)

4-

\nabla^{2} f (x)

5-

\nabla^{2} f (x)

6-

\min {f (x) : x_{l} \leq x \leq x_{u}, c_{l} \leq c (x) \leq c_{u}},

7-

c : ℝ^{n} \mapsto ℝ^{m}

8-

c^{'} (x)

9-

L (x, u) = f (x) + ⟨ u, c (x) ⟩

10-

f : ℝ^{n} \mapsto ℝ

Formulas (html):

Correct Categorie: cs

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0610827

Formulas:

1-

l e f t

2-

F W H M

3-

F W H M

4-

r / R = 0.101 \pm 0.002

5-

a / R = 3.84 \pm 0.16

6-

Δ χ^{2} = χ_{m i n}^{2} + 1

7-

Δ χ^{2} = χ_{m i n}^{2} + 1 + Δ χ_{s y s t}^{2}

8-

Δ χ_{s y s t}^{2} = n σ_{r}^{2} / σ_{w}^{2}

9-

r / R = 0.110 \pm 0.002

10-

a / R = {8.07}_{- 0.69}^{+ 0.44}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0405093

Formulas:

1-

≃ 0.3 - 0.4 ″

2-

M_{C2} = 7.7 \times 10^{5} - 2.6 \times 10^{6} M_{⊙}

3-

0.1 - 100 M_{⊙}

4-

T > 35, 000 - 40, 000

5-

M ≃ 40 - 60 M_{⊙}

6-

λ_{rest} = 1.87 μ

7-

λ = 1.9 - 2.4 μ

8-

λ = 3 - 4.1 μ

9-

1.5 ″ \times 1.6 ″

10-

3 ″ \times 1.6 ″

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1709.07290

Formulas:

1-

C_{1} = (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}) and C_{2} = (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}),

2-

{(c, d) : A (c, d) = 1}

3-

(\begin{matrix} 1 & 𝟏 & 0 & 𝟎 & 𝟎 & 𝟏 \\ 1 & 𝟎 & 0 & 𝟏 & 𝟏 & 𝟎 \end{matrix}),

4-

(\begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \end{matrix})

5-

2 \times (u + l)

6-

(\binom{u + l}{u})

7-

(\begin{matrix} 0 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}),

8-

(\begin{matrix} 1 & 𝟎 & 0 & 𝟏 & 𝟏 & 𝟎 \\ 1 & 𝟏 & 0 & 𝟎 & 𝟎 & 𝟏 \end{matrix}) .

9-

P : Ω \times Ω \to [0, 1]

10-

π_{a}^{'} P (a, b) = π_{b}^{'} P (b, a) \forall a, b \in Ω

Formulas (html):

Correct Categorie: cs

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1612.02186

Formulas:

1-

G = \frac{Q_{T O T}}{q} = \frac{C_{p i x e l} (V_{o p} - V_{B D})}{q}

2-

Q_{T O T}

3-

C_{p i x e l}

4-

Δ G (V_{o p}, T)

5-

Δ V_{o p}

6-

Δ G (V_{o p}, T) = \frac{\partial G}{\partial V_{o p}} Δ V_{o p} + \frac{\partial G}{\partial T} Δ T

7-

\partial G / \partial V

8-

\partial G / \partial T

9-

\partial G / \partial V

10-

\partial G / \partial T

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1910.05526

Formulas:

1-

Q_{C D W}

2-

T_{C D W} \sim

3-

Q_{C D W}

4-

Q_{C D W}

5-

Q_{C D W}

6-

T_{C D W}

7-

T_{C D W}

8-

Q_{C D W}

9-

T_{C D W}

10-

Q_{C D W}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1703.08652

Formulas:

1-

Γ = (V, E)

2-

H (n, q)

3-

H (n, q)

4-

Γ = (V, E)

5-

H (n, q)

6-

Cay (G, X)

7-

v u^{- 1} \in X

8-

Cay (ℤ_{n}, S)

9-

SL (2, 2^{f})

10-

Cay (ℤ_{n}, S)

Formulas (html):

Correct Categorie: math

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1404.7547

Formulas:

1-

𝑿 = {(X_{1}, \dots, X_{d})}^{T} \in ℝ^{d}

2-

G = (V, E)

3-

V = {1, \dots, d}

4-

E \subseteq {(j, k) : 1 \leq j, k \leq d}

5-

X_{1}, \dots, X_{d}

6-

X_{∖ {j, k}}

7-

X_{j} ⟂ X_{k} | X_{∖ {j, k}}

8-

(j, k) \notin E

9-

g (\cdot) : ℝ^{d} \to ℝ^{K}

10-

𝒀 = {(Y_{1}, \dots, Y_{K})}^{T} = g (𝑿)

Formulas (html):

Correct Categorie: stat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1012.4646

Formulas:

1-

E = E_{kin} + E_{Coul} + E_{Sky},

2-

E_{Sky} = \int d^{3} 𝒓 ℰ_{Sky} (𝒓)

3-

ρ_{q} (𝒓, σ; 𝒓^{'}, σ^{'})

4-

σ, σ^{'} = 1, - 1

5-

ℰ_{Sky} (𝒓)

6-

ρ_{q} (𝒓) = \sum_{σ = \pm 1} ρ_{q} (𝒓, σ; 𝒓, σ),

7-

τ_{q} (𝒓) = \sum_{σ = \pm 1} \int d^{3} 𝒓^{'} δ (𝒓 - 𝒓^{'}) \nabla \cdot \nabla^{'} ρ_{q} (𝒓, σ; 𝒓^{'}, σ),

8-

𝒋_{𝒒} (𝒓) = - \frac{i}{2} \sum_{σ = \pm 1} \int d^{3} 𝒓^{'} δ (𝒓 - 𝒓^{'}) (\nabla - \nabla^{'}) ρ_{q} (𝒓, σ; 𝒓^{'}, σ),

9-

𝒔_{𝒒} (𝒓) = \sum_{σ, σ^{'} = \pm 1} ρ_{q} (𝒓, σ; 𝒓, σ^{'}) ⟨ σ^{'} | \hat{𝝈} | σ ⟩,

10-

T_{q μ} (𝒓) = \sum_{σ, σ^{'} = \pm 1} \int d^{3} 𝒓^{'} δ (𝒓 - 𝒓^{'}) \nabla \cdot \nabla^{'} ρ_{q} (𝒓, σ; 𝒓^{'}, σ^{'}) ⟨ σ^{'} | {\hat{σ}}_{μ} | σ ⟩,

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/9608058

Formulas:

1-

R_{p} / r_{*} - e

2-

⟨ R_{p} / r_{*} ⟩ ≃ 100

3-

R_{p} / r_{*} < 5

4-

q \equiv M_{2} / M_{1}

5-

g (q, A)

6-

g (q, A) = \frac{0.075}{A} 0.04 q^{- 2.7} \exp (- \int_{A \cdot {(6.3)}^{- 2 / 3}}^{A \cdot {(6.3)}^{2 / 3}} \int_{q}^{1} 0.075 A^{', - 1} 0.04 q^{', - 2.7} 𝑑 A^{'} 𝑑 q^{'}) .

7-

Φ_{P} (\log M_{d}, \log P_{X})

8-

{< V_{k}^{2} >}^{1 / 2}

9-

{< V_{k}^{2} >}^{1 / 2} = 300

10-

α_{C E} = 1

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/astro-ph/0606101

Formulas:

1-

- \frac{G M_{d}}{\sqrt{R^{2} + {(a + \sqrt{z^{2} + b^{2}})}^{2}}}; Φ_{b} = - \frac{G M_{b}}{r + c};

2-

- \frac{G M_{h} \ln (1 + r / r_{s})}{r} {[\ln (1 + c_{200}) - \frac{c_{200}}{1 + c_{200}}]}^{- 1} .

3-

M_{d} = 8.4 \times 10^{10} M_{⊙}

4-

M_{b} = 3.3 \times 10^{10} M_{⊙}

5-

M_{h} (r_{200}) = 6.8 \times 10^{11} M_{⊙}

6-

c_{200} = r_{200} / r_{s} = 22

7-

c = r_{vir} / r_{s} = 20

8-

ϵ = Ψ - v^{2} / 2

9-

Ψ = - Φ_{N F W} + Φ_{N F W} (\infty)

10-

f_{⋆} (ϵ) / f_{N F W} (ϵ) \times N (ϵ)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/hep-lat/9609027

Formulas:

1-

S U (2)

2-

ρ^{2} \equiv β_{t} / β_{s}

3-

\int D U U_{α β} U_{γ δ}^{†} χ_{j} (U P_{\vec{x} + i} U^{†} P_{\vec{x}}^{†}) \equiv δ_{α δ} δ_{γ β} 𝒞_{α β}^{(j)} .

4-

{| U_{α β} |}^{2}

5-

𝒞_{11}^{(j)} = 𝒞_{22}^{(j)}

6-

𝒞_{12}^{(j)} = 𝒞_{21}^{(j)}

7-

𝒞_{α β}^{(j)}

8-

K^{(j)} (θ_{\vec{x}}, θ_{\vec{x} + i})

9-

\int D U χ_{j} (U P_{2} U^{†} P_{1}^{†})

10-

\frac{1}{d_{j}} χ_{j} (P_{2}) χ_{j} (P_{1}^{†})

Formulas (html):

Correct Categorie: hep-lat

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/0709.1671

Formulas:

1-

ψ \to ψ^{U} \equiv U^{- 1} ψ

2-

h^{- 1} [A]

3-

Ψ := h^{- 1} [A] ψ,

4-

h^{- 1} [A] \to h^{- 1} [A^{U}] = h^{- 1} [A] U,

5-

A \to A^{U} \equiv U^{- 1} A U + U^{- 1} \partial U .

6-

χ (A) = 0

7-

χ (A^{h [A]}) = 0 .

8-

χ (A^{U h [A^{U}]}) = 0 .

9-

A^{h [A]} = A^{U h [A^{U}]}

10-

h [A] = U h [A^{U}] ⟹ h^{- 1} [A^{U}] = h^{- 1} [A] U .

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/0810.0942

Formulas:

1-

𝝆_{M} = ρ^{\otimes M} = \underset{M times}{\underset{⏟}{ρ \otimes ρ \otimes \dots \otimes ρ}} .

2-

M = n_{+} + n_{-}

3-

α, β \in {+, -}

4-

\begin{matrix} C H = - P_{+}^{A} (A_{1}) - P_{+}^{B} (B_{1}) + P_{+ +} (A_{1}, B_{1}) \\ + P_{+ +} (A_{1}, B_{2}) + P_{+ +} (A_{2}, B_{1}) - P_{+ +} (A_{2}, B_{2}) \leq 0 \end{matrix}

5-

P_{+ +} (A_{i}, B_{j})

6-

\begin{matrix} 1 . & Majority voting: if n_{+} \geq n_{-} \to “ + ”, otherwise “ - ” \\ 2 . & Unanimous voting: if n_{+} = M \to “ + ”, otherwise “ - ” \end{matrix}

7-

N = ⌈ M / 2 ⌉, ⌈ 2 M / 3 ⌉, \dots, M

8-

| ψ ⟩ = \cos θ | 00 ⟩ + \sin θ | 11 ⟩

9-

ρ = | ψ ⟩ ⟨ ψ |

10-

| Ψ_{M} ⟩ = {| ψ ⟩}^{\otimes M}

Formulas (html):

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xref="S2.E1.m1.1.1.1.1.6.2.cmml"><mrow id="S2.E1.m1.1.1.1.1.6.2.2" xref="S2.E1.m1.1.1.1.1.6.2.2.cmml"><mi id="S2.E1.m1.1.1.1.1.6.2.2.2" xref="S2.E1.m1.1.1.1.1.6.2.2.2.cmml">ρ</mi><mo movablelimits="false" id="S2.E1.m1.1.1.1.1.6.2.2.1" xref="S2.E1.m1.1.1.1.1.6.2.2.1.cmml">⊗</mo><mi id="S2.E1.m1.1.1.1.1.6.2.2.3" xref="S2.E1.m1.1.1.1.1.6.2.2.3.cmml">ρ</mi><mo movablelimits="false" id="S2.E1.m1.1.1.1.1.6.2.2.1a" xref="S2.E1.m1.1.1.1.1.6.2.2.1.cmml">⊗</mo><mi mathvariant="normal" id="S2.E1.m1.1.1.1.1.6.2.2.4" xref="S2.E1.m1.1.1.1.1.6.2.2.4.cmml">⋯</mi><mo movablelimits="false" id="S2.E1.m1.1.1.1.1.6.2.2.1b" xref="S2.E1.m1.1.1.1.1.6.2.2.1.cmml">⊗</mo><mi id="S2.E1.m1.1.1.1.1.6.2.2.5" xref="S2.E1.m1.1.1.1.1.6.2.2.5.cmml">ρ</mi></mrow><mo movablelimits="false" id="S2.E1.m1.1.1.1.1.6.2.1" xref="S2.E1.m1.1.1.1.1.6.2.1.cmml">⏟</mo></munder><mrow id="S2.E1.m1.1.1.1.1.6.3" xref="S2.E1.m1.1.1.1.1.6.3.cmml"><mi id="S2.E1.m1.1.1.1.1.6.3.2" xref="S2.E1.m1.1.1.1.1.6.3.2.cmml">M</mi><mo id="S2.E1.m1.1.1.1.1.6.3.1" xref="S2.E1.m1.1.1.1.1.6.3.1.cmml">⁢</mo><mtext id="S2.E1.m1.1.1.1.1.6.3.3" xref="S2.E1.m1.1.1.1.1.6.3.3a.cmml"> times</mtext></mrow></munder></mrow><mo id="S2.E1.m1.1.1.1.2" xref="S2.E1.m1.1.1.1.1.cmml">.</mo></mrow></math>, <math><mrow id="S2.p3.7.m7.1.1" xref="S2.p3.7.m7.1.1.cmml"><mi id="S2.p3.7.m7.1.1.2" xref="S2.p3.7.m7.1.1.2.cmml">M</mi><mo id="S2.p3.7.m7.1.1.1" xref="S2.p3.7.m7.1.1.1.cmml">=</mo><mrow id="S2.p3.7.m7.1.1.3" xref="S2.p3.7.m7.1.1.3.cmml"><msub id="S2.p3.7.m7.1.1.3.2" xref="S2.p3.7.m7.1.1.3.2.cmml"><mi id="S2.p3.7.m7.1.1.3.2.2" xref="S2.p3.7.m7.1.1.3.2.2.cmml">n</mi><mo id="S2.p3.7.m7.1.1.3.2.3" xref="S2.p3.7.m7.1.1.3.2.3.cmml">+</mo></msub><mo id="S2.p3.7.m7.1.1.3.1" xref="S2.p3.7.m7.1.1.3.1.cmml">+</mo><msub id="S2.p3.7.m7.1.1.3.3" xref="S2.p3.7.m7.1.1.3.3.cmml"><mi id="S2.p3.7.m7.1.1.3.3.2" xref="S2.p3.7.m7.1.1.3.3.2.cmml">n</mi><mo id="S2.p3.7.m7.1.1.3.3.3" xref="S2.p3.7.m7.1.1.3.3.3.cmml">-</mo></msub></mrow></mrow></math>, <math><mrow id="S2.SS1.p1.3.m3.4.5" xref="S2.SS1.p1.3.m3.4.5.cmml"><mrow id="S2.SS1.p1.3.m3.4.5.2.2" xref="S2.SS1.p1.3.m3.4.5.2.1.cmml"><mi id="S2.SS1.p1.3.m3.3.3" xref="S2.SS1.p1.3.m3.3.3.cmml">α</mi><mo id="S2.SS1.p1.3.m3.4.5.2.2.1" xref="S2.SS1.p1.3.m3.4.5.2.1.cmml">,</mo><mi id="S2.SS1.p1.3.m3.4.4" xref="S2.SS1.p1.3.m3.4.4.cmml">β</mi></mrow><mo id="S2.SS1.p1.3.m3.4.5.1" xref="S2.SS1.p1.3.m3.4.5.1.cmml">∈</mo><mrow id="S2.SS1.p1.3.m3.4.5.3.2" xref="S2.SS1.p1.3.m3.4.5.3.1.cmml"><mo stretchy="false" id="S2.SS1.p1.3.m3.4.5.3.2.1" xref="S2.SS1.p1.3.m3.4.5.3.1.cmml">{</mo><mo id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">+</mo><mo id="S2.SS1.p1.3.m3.4.5.3.2.2" xref="S2.SS1.p1.3.m3.4.5.3.1.cmml">,</mo><mo id="S2.SS1.p1.3.m3.2.2" xref="S2.SS1.p1.3.m3.2.2.cmml">-</mo><mo stretchy="false" id="S2.SS1.p1.3.m3.4.5.3.2.3" xref="S2.SS1.p1.3.m3.4.5.3.1.cmml">}</mo></mrow></mrow></math>, <math><mtable displaystyle="true" rowspacing="0pt" id="S2.E2.m1.81.81.20" xref="S2.E2.m1.71.71.10.cmml"><mtr id="S2.E2.m1.81.81.20a" xref="S2.E2.m1.71.71.10.cmml"><mtd columnalign="right" id="S2.E2.m1.81.81.20b" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.75.75.14.65.33.33" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.75.75.14.65.33.33.34" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.1.1.1.1.1.1" xref="S2.E2.m1.1.1.1.1.1.1.cmml">C</mi><mo id="S2.E2.m1.75.75.14.65.33.33.34.1" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mi id="S2.E2.m1.2.2.2.2.2.2" xref="S2.E2.m1.2.2.2.2.2.2.cmml">H</mi></mrow><mo id="S2.E2.m1.3.3.3.3.3.3" xref="S2.E2.m1.3.3.3.3.3.3.cmml">=</mo><mrow id="S2.E2.m1.75.75.14.65.33.33.33" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.73.73.12.63.31.31.31.2" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.72.72.11.62.30.30.30.1.1" xref="S2.E2.m1.71.71.10.cmml"><mo id="S2.E2.m1.4.4.4.4.4.4" xref="S2.E2.m1.4.4.4.4.4.4.cmml">-</mo><mrow id="S2.E2.m1.72.72.11.62.30.30.30.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><msubsup id="S2.E2.m1.72.72.11.62.30.30.30.1.1.1.3" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.5.5.5.5.5.5" xref="S2.E2.m1.5.5.5.5.5.5.cmml">P</mi><mo id="S2.E2.m1.6.6.6.6.6.6.1" xref="S2.E2.m1.6.6.6.6.6.6.1.cmml">+</mo><mi id="S2.E2.m1.7.7.7.7.7.7.1" xref="S2.E2.m1.7.7.7.7.7.7.1.cmml">A</mi></msubsup><mo id="S2.E2.m1.72.72.11.62.30.30.30.1.1.1.2" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mrow id="S2.E2.m1.72.72.11.62.30.30.30.1.1.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mo stretchy="false" id="S2.E2.m1.8.8.8.8.8.8" xref="S2.E2.m1.71.71.10a.cmml">(</mo><msub id="S2.E2.m1.72.72.11.62.30.30.30.1.1.1.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.9.9.9.9.9.9" xref="S2.E2.m1.9.9.9.9.9.9.cmml">A</mi><mn id="S2.E2.m1.10.10.10.10.10.10.1" xref="S2.E2.m1.10.10.10.10.10.10.1.cmml">1</mn></msub><mo stretchy="false" id="S2.E2.m1.11.11.11.11.11.11" xref="S2.E2.m1.71.71.10a.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.12.12.12.12.12.12" xref="S2.E2.m1.12.12.12.12.12.12.cmml">-</mo><mrow id="S2.E2.m1.73.73.12.63.31.31.31.2.2" xref="S2.E2.m1.71.71.10.cmml"><msubsup id="S2.E2.m1.73.73.12.63.31.31.31.2.2.3" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.13.13.13.13.13.13" xref="S2.E2.m1.13.13.13.13.13.13.cmml">P</mi><mo id="S2.E2.m1.14.14.14.14.14.14.1" xref="S2.E2.m1.14.14.14.14.14.14.1.cmml">+</mo><mi id="S2.E2.m1.15.15.15.15.15.15.1" xref="S2.E2.m1.15.15.15.15.15.15.1.cmml">B</mi></msubsup><mo id="S2.E2.m1.73.73.12.63.31.31.31.2.2.2" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mrow id="S2.E2.m1.73.73.12.63.31.31.31.2.2.1.1" xref="S2.E2.m1.71.71.10.cmml"><mo stretchy="false" id="S2.E2.m1.16.16.16.16.16.16" xref="S2.E2.m1.71.71.10a.cmml">(</mo><msub id="S2.E2.m1.73.73.12.63.31.31.31.2.2.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.17.17.17.17.17.17" xref="S2.E2.m1.17.17.17.17.17.17.cmml">B</mi><mn id="S2.E2.m1.18.18.18.18.18.18.1" xref="S2.E2.m1.18.18.18.18.18.18.1.cmml">1</mn></msub><mo stretchy="false" id="S2.E2.m1.19.19.19.19.19.19" xref="S2.E2.m1.71.71.10a.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.20.20.20.20.20.20" xref="S2.E2.m1.20.20.20.20.20.20.cmml">+</mo><mrow id="S2.E2.m1.75.75.14.65.33.33.33.4" xref="S2.E2.m1.71.71.10.cmml"><msub id="S2.E2.m1.75.75.14.65.33.33.33.4.4" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.21.21.21.21.21.21" xref="S2.E2.m1.21.21.21.21.21.21.cmml">P</mi><mrow id="S2.E2.m1.22.22.22.22.22.22.1" xref="S2.E2.m1.22.22.22.22.22.22.1.cmml"><mi id="S2.E2.m1.22.22.22.22.22.22.1.2" xref="S2.E2.m1.22.22.22.22.22.22.1.2.cmml"/><mo id="S2.E2.m1.22.22.22.22.22.22.1.1" xref="S2.E2.m1.22.22.22.22.22.22.1.1.cmml">+</mo><mo id="S2.E2.m1.22.22.22.22.22.22.1.3" xref="S2.E2.m1.22.22.22.22.22.22.1.3.cmml">+</mo></mrow></msub><mo id="S2.E2.m1.75.75.14.65.33.33.33.4.3" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mrow id="S2.E2.m1.75.75.14.65.33.33.33.4.2.2" xref="S2.E2.m1.71.71.10.cmml"><mo stretchy="false" id="S2.E2.m1.23.23.23.23.23.23" xref="S2.E2.m1.71.71.10a.cmml">(</mo><msub id="S2.E2.m1.74.74.13.64.32.32.32.3.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.24.24.24.24.24.24" xref="S2.E2.m1.24.24.24.24.24.24.cmml">A</mi><mn id="S2.E2.m1.25.25.25.25.25.25.1" xref="S2.E2.m1.25.25.25.25.25.25.1.cmml">1</mn></msub><mo id="S2.E2.m1.26.26.26.26.26.26" xref="S2.E2.m1.71.71.10a.cmml">,</mo><msub id="S2.E2.m1.75.75.14.65.33.33.33.4.2.2.2" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.27.27.27.27.27.27" xref="S2.E2.m1.27.27.27.27.27.27.cmml">B</mi><mn id="S2.E2.m1.28.28.28.28.28.28.1" xref="S2.E2.m1.28.28.28.28.28.28.1.cmml">1</mn></msub><mo stretchy="false" id="S2.E2.m1.29.29.29.29.29.29" xref="S2.E2.m1.71.71.10a.cmml">)</mo></mrow></mrow></mrow></mrow></mtd></mtr><mtr id="S2.E2.m1.81.81.20c" xref="S2.E2.m1.71.71.10.cmml"><mtd columnalign="right" id="S2.E2.m1.81.81.20d" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.81.81.20.71.38.38" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.81.81.20.71.38.38.38" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.79.79.18.69.36.36.36.4" xref="S2.E2.m1.71.71.10.cmml"><mrow id="S2.E2.m1.77.77.16.67.34.34.34.2.2" xref="S2.E2.m1.71.71.10.cmml"><mo id="S2.E2.m1.30.30.30.1.1.1" xref="S2.E2.m1.71.71.10a.cmml">+</mo><mrow id="S2.E2.m1.77.77.16.67.34.34.34.2.2.2" xref="S2.E2.m1.71.71.10.cmml"><msub id="S2.E2.m1.77.77.16.67.34.34.34.2.2.2.4" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.31.31.31.2.2.2" xref="S2.E2.m1.31.31.31.2.2.2.cmml">P</mi><mrow id="S2.E2.m1.32.32.32.3.3.3.1" xref="S2.E2.m1.32.32.32.3.3.3.1.cmml"><mi id="S2.E2.m1.32.32.32.3.3.3.1.2" xref="S2.E2.m1.32.32.32.3.3.3.1.2.cmml"/><mo id="S2.E2.m1.32.32.32.3.3.3.1.1" xref="S2.E2.m1.32.32.32.3.3.3.1.1.cmml">+</mo><mo id="S2.E2.m1.32.32.32.3.3.3.1.3" xref="S2.E2.m1.32.32.32.3.3.3.1.3.cmml">+</mo></mrow></msub><mo id="S2.E2.m1.77.77.16.67.34.34.34.2.2.2.3" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mrow id="S2.E2.m1.77.77.16.67.34.34.34.2.2.2.2.2" xref="S2.E2.m1.71.71.10.cmml"><mo stretchy="false" id="S2.E2.m1.33.33.33.4.4.4" xref="S2.E2.m1.71.71.10a.cmml">(</mo><msub id="S2.E2.m1.76.76.15.66.33.33.33.1.1.1.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.34.34.34.5.5.5" xref="S2.E2.m1.34.34.34.5.5.5.cmml">A</mi><mn id="S2.E2.m1.35.35.35.6.6.6.1" xref="S2.E2.m1.35.35.35.6.6.6.1.cmml">1</mn></msub><mo id="S2.E2.m1.36.36.36.7.7.7" xref="S2.E2.m1.71.71.10a.cmml">,</mo><msub id="S2.E2.m1.77.77.16.67.34.34.34.2.2.2.2.2.2" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.37.37.37.8.8.8" xref="S2.E2.m1.37.37.37.8.8.8.cmml">B</mi><mn id="S2.E2.m1.38.38.38.9.9.9.1" xref="S2.E2.m1.38.38.38.9.9.9.1.cmml">2</mn></msub><mo stretchy="false" id="S2.E2.m1.39.39.39.10.10.10" xref="S2.E2.m1.71.71.10a.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.40.40.40.11.11.11" xref="S2.E2.m1.71.71.10a.cmml">+</mo><mrow id="S2.E2.m1.79.79.18.69.36.36.36.4.4" xref="S2.E2.m1.71.71.10.cmml"><msub id="S2.E2.m1.79.79.18.69.36.36.36.4.4.4" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.41.41.41.12.12.12" xref="S2.E2.m1.41.41.41.12.12.12.cmml">P</mi><mrow id="S2.E2.m1.42.42.42.13.13.13.1" xref="S2.E2.m1.42.42.42.13.13.13.1.cmml"><mi id="S2.E2.m1.42.42.42.13.13.13.1.2" xref="S2.E2.m1.42.42.42.13.13.13.1.2.cmml"/><mo id="S2.E2.m1.42.42.42.13.13.13.1.1" xref="S2.E2.m1.42.42.42.13.13.13.1.1.cmml">+</mo><mo id="S2.E2.m1.42.42.42.13.13.13.1.3" xref="S2.E2.m1.42.42.42.13.13.13.1.3.cmml">+</mo></mrow></msub><mo id="S2.E2.m1.79.79.18.69.36.36.36.4.4.3" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mrow id="S2.E2.m1.79.79.18.69.36.36.36.4.4.2.2" xref="S2.E2.m1.71.71.10.cmml"><mo stretchy="false" id="S2.E2.m1.43.43.43.14.14.14" xref="S2.E2.m1.71.71.10a.cmml">(</mo><msub id="S2.E2.m1.78.78.17.68.35.35.35.3.3.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.44.44.44.15.15.15" xref="S2.E2.m1.44.44.44.15.15.15.cmml">A</mi><mn id="S2.E2.m1.45.45.45.16.16.16.1" xref="S2.E2.m1.45.45.45.16.16.16.1.cmml">2</mn></msub><mo id="S2.E2.m1.46.46.46.17.17.17" xref="S2.E2.m1.71.71.10a.cmml">,</mo><msub id="S2.E2.m1.79.79.18.69.36.36.36.4.4.2.2.2" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.47.47.47.18.18.18" xref="S2.E2.m1.47.47.47.18.18.18.cmml">B</mi><mn id="S2.E2.m1.48.48.48.19.19.19.1" xref="S2.E2.m1.48.48.48.19.19.19.1.cmml">1</mn></msub><mo stretchy="false" id="S2.E2.m1.49.49.49.20.20.20" xref="S2.E2.m1.71.71.10a.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.50.50.50.21.21.21" xref="S2.E2.m1.50.50.50.21.21.21.cmml">-</mo><mrow id="S2.E2.m1.81.81.20.71.38.38.38.6" xref="S2.E2.m1.71.71.10.cmml"><msub id="S2.E2.m1.81.81.20.71.38.38.38.6.4" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.51.51.51.22.22.22" xref="S2.E2.m1.51.51.51.22.22.22.cmml">P</mi><mrow id="S2.E2.m1.52.52.52.23.23.23.1" xref="S2.E2.m1.52.52.52.23.23.23.1.cmml"><mi id="S2.E2.m1.52.52.52.23.23.23.1.2" xref="S2.E2.m1.52.52.52.23.23.23.1.2.cmml"/><mo id="S2.E2.m1.52.52.52.23.23.23.1.1" xref="S2.E2.m1.52.52.52.23.23.23.1.1.cmml">+</mo><mo id="S2.E2.m1.52.52.52.23.23.23.1.3" xref="S2.E2.m1.52.52.52.23.23.23.1.3.cmml">+</mo></mrow></msub><mo id="S2.E2.m1.81.81.20.71.38.38.38.6.3" xref="S2.E2.m1.71.71.10a.cmml">⁢</mo><mrow id="S2.E2.m1.81.81.20.71.38.38.38.6.2.2" xref="S2.E2.m1.71.71.10.cmml"><mo stretchy="false" id="S2.E2.m1.53.53.53.24.24.24" xref="S2.E2.m1.71.71.10a.cmml">(</mo><msub id="S2.E2.m1.80.80.19.70.37.37.37.5.1.1.1" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.54.54.54.25.25.25" xref="S2.E2.m1.54.54.54.25.25.25.cmml">A</mi><mn id="S2.E2.m1.55.55.55.26.26.26.1" xref="S2.E2.m1.55.55.55.26.26.26.1.cmml">2</mn></msub><mo id="S2.E2.m1.56.56.56.27.27.27" xref="S2.E2.m1.71.71.10a.cmml">,</mo><msub id="S2.E2.m1.81.81.20.71.38.38.38.6.2.2.2" xref="S2.E2.m1.71.71.10.cmml"><mi id="S2.E2.m1.57.57.57.28.28.28" xref="S2.E2.m1.57.57.57.28.28.28.cmml">B</mi><mn id="S2.E2.m1.58.58.58.29.29.29.1" xref="S2.E2.m1.58.58.58.29.29.29.1.cmml">2</mn></msub><mo stretchy="false" id="S2.E2.m1.59.59.59.30.30.30" xref="S2.E2.m1.71.71.10a.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E2.m1.60.60.60.31.31.31" xref="S2.E2.m1.60.60.60.31.31.31.cmml">≤</mo><mn id="S2.E2.m1.61.61.61.32.32.32" xref="S2.E2.m1.61.61.61.32.32.32.cmml">0</mn></mrow></mtd></mtr></mtable></math>, <math><mrow id="S2.SS1.p2.1.m1.2.2" xref="S2.SS1.p2.1.m1.2.2.cmml"><msub id="S2.SS1.p2.1.m1.2.2.4" xref="S2.SS1.p2.1.m1.2.2.4.cmml"><mi id="S2.SS1.p2.1.m1.2.2.4.2" xref="S2.SS1.p2.1.m1.2.2.4.2.cmml">P</mi><mrow id="S2.SS1.p2.1.m1.2.2.4.3" xref="S2.SS1.p2.1.m1.2.2.4.3.cmml"><mi id="S2.SS1.p2.1.m1.2.2.4.3.2" xref="S2.SS1.p2.1.m1.2.2.4.3.2.cmml"/><mo id="S2.SS1.p2.1.m1.2.2.4.3.1" xref="S2.SS1.p2.1.m1.2.2.4.3.1.cmml">+</mo><mo id="S2.SS1.p2.1.m1.2.2.4.3.3" xref="S2.SS1.p2.1.m1.2.2.4.3.3.cmml">+</mo></mrow></msub><mo id="S2.SS1.p2.1.m1.2.2.3" xref="S2.SS1.p2.1.m1.2.2.3.cmml">⁢</mo><mrow id="S2.SS1.p2.1.m1.2.2.2.2" xref="S2.SS1.p2.1.m1.2.2.2.3.cmml"><mo stretchy="false" id="S2.SS1.p2.1.m1.2.2.2.2.3" xref="S2.SS1.p2.1.m1.2.2.2.3.cmml">(</mo><msub id="S2.SS1.p2.1.m1.1.1.1.1.1" xref="S2.SS1.p2.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.1.m1.1.1.1.1.1.2" xref="S2.SS1.p2.1.m1.1.1.1.1.1.2.cmml">A</mi><mi id="S2.SS1.p2.1.m1.1.1.1.1.1.3" xref="S2.SS1.p2.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo id="S2.SS1.p2.1.m1.2.2.2.2.4" xref="S2.SS1.p2.1.m1.2.2.2.3.cmml">,</mo><msub id="S2.SS1.p2.1.m1.2.2.2.2.2" xref="S2.SS1.p2.1.m1.2.2.2.2.2.cmml"><mi id="S2.SS1.p2.1.m1.2.2.2.2.2.2" xref="S2.SS1.p2.1.m1.2.2.2.2.2.2.cmml">B</mi><mi id="S2.SS1.p2.1.m1.2.2.2.2.2.3" xref="S2.SS1.p2.1.m1.2.2.2.2.2.3.cmml">j</mi></msub><mo stretchy="false" id="S2.SS1.p2.1.m1.2.2.2.2.5" xref="S2.SS1.p2.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow></math>, <math><mtable columnspacing="0pt" displaystyle="true" rowspacing="0pt" id="S2.E3.m1.18.18"><mtr id="S2.E3.m1.18.18a"><mtd columnalign="right" id="S2.E3.m1.18.18b"><mrow id="S2.E3.m1.4.4.4.3.3"><mn id="S2.E3.m1.3.3.3.2.1.1" xref="S2.E3.m1.3.3.3.2.1.1.cmml">1</mn><mo id="S2.E3.m1.4.4.4.3.2.2" xref="S2.E3.m1.20.20.2.3.cmml">.</mo></mrow></mtd><mtd columnalign="left" id="S2.E3.m1.18.18c"><mrow id="S2.E3.m1.10.10.10.9.7"><mrow id="S2.E3.m1.1.1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.1c.cmml"><mtext id="S2.E3.m1.1.1.1.1.1.1a" xref="S2.E3.m1.1.1.1.1.1.1c.cmml">Majority voting: if </mtext><mrow id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.cmml"><msub id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.2" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.2.cmml"><mi id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.2.2" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.2.2.cmml">n</mi><mo id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.2.3" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.2.3.cmml">+</mo></msub><mo id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.1" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.1.cmml">≥</mo><msub id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.3" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.3.cmml"><mi id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.3.2" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.3.2.cmml">n</mi><mo id="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.3.3" xref="S2.E3.m1.1.1.1.1.1.1.1.m1.1.1.3.3.cmml">-</mo></msub></mrow><mtext id="S2.E3.m1.1.1.1.1.1.1b" xref="S2.E3.m1.1.1.1.1.1.1c.cmml"> </mtext></mrow><mo id="S2.E3.m1.5.5.5.4.2.2" xref="S2.E3.m1.5.5.5.4.2.2.cmml">→</mo><mrow id="S2.E3.m1.10.10.10.9.7.8"><mrow id="S2.E3.m1.10.10.10.9.7.8.1"><mtext id="S2.E3.m1.6.6.6.5.3.3" xref="S2.E3.m1.6.6.6.5.3.3a.cmml">“</mtext><mo id="S2.E3.m1.7.7.7.6.4.4" xref="S2.E3.m1.7.7.7.6.4.4.cmml">+</mo><mtext id="S2.E3.m1.8.8.8.7.5.5" xref="S2.E3.m1.8.8.8.7.5.5a.cmml">”, otherwise “</mtext></mrow><mo id="S2.E3.m1.9.9.9.8.6.6" xref="S2.E3.m1.9.9.9.8.6.6.cmml">-</mo><mtext id="S2.E3.m1.10.10.10.9.7.7" xref="S2.E3.m1.10.10.10.9.7.7a.cmml">”</mtext></mrow></mrow></mtd></mtr><mtr id="S2.E3.m1.18.18d"><mtd columnalign="right" id="S2.E3.m1.18.18e"><mrow id="S2.E3.m1.12.12.12.3.3"><mn id="S2.E3.m1.11.11.11.2.1.1" xref="S2.E3.m1.11.11.11.2.1.1.cmml">2</mn><mo id="S2.E3.m1.12.12.12.3.2.2" xref="S2.E3.m1.20.20.2.3.cmml">.</mo></mrow></mtd><mtd columnalign="left" id="S2.E3.m1.18.18f"><mrow id="S2.E3.m1.18.18.18.9.7"><mrow id="S2.E3.m1.2.2.2.1.1.1" xref="S2.E3.m1.2.2.2.1.1.1c.cmml"><mtext id="S2.E3.m1.2.2.2.1.1.1a" xref="S2.E3.m1.2.2.2.1.1.1c.cmml">Unanimous voting: if </mtext><mrow id="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1" xref="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.cmml"><msub id="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.2" xref="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.2.cmml"><mi id="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.2.2" xref="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.2.2.cmml">n</mi><mo id="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.2.3" xref="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.2.3.cmml">+</mo></msub><mo id="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.1" xref="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.1.cmml">=</mo><mi id="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.3" xref="S2.E3.m1.2.2.2.1.1.1.1.m1.1.1.3.cmml">M</mi></mrow><mtext id="S2.E3.m1.2.2.2.1.1.1b" xref="S2.E3.m1.2.2.2.1.1.1c.cmml"> </mtext></mrow><mo id="S2.E3.m1.13.13.13.4.2.2" xref="S2.E3.m1.13.13.13.4.2.2.cmml">→</mo><mrow id="S2.E3.m1.18.18.18.9.7.8"><mrow id="S2.E3.m1.18.18.18.9.7.8.1"><mtext id="S2.E3.m1.14.14.14.5.3.3" xref="S2.E3.m1.14.14.14.5.3.3a.cmml">“</mtext><mo id="S2.E3.m1.15.15.15.6.4.4" xref="S2.E3.m1.15.15.15.6.4.4.cmml">+</mo><mtext id="S2.E3.m1.16.16.16.7.5.5" xref="S2.E3.m1.16.16.16.7.5.5a.cmml">”, otherwise “</mtext></mrow><mo id="S2.E3.m1.17.17.17.8.6.6" xref="S2.E3.m1.17.17.17.8.6.6.cmml">-</mo><mtext id="S2.E3.m1.18.18.18.9.7.7" xref="S2.E3.m1.18.18.18.9.7.7a.cmml">”</mtext></mrow></mrow></mtd></mtr></mtable></math>, <math><mrow id="S2.SS1.p2.9.m2.4.4" xref="S2.SS1.p2.9.m2.4.4.cmml"><mi id="S2.SS1.p2.9.m2.4.4.4" xref="S2.SS1.p2.9.m2.4.4.4.cmml">N</mi><mo id="S2.SS1.p2.9.m2.4.4.3" xref="S2.SS1.p2.9.m2.4.4.3.cmml">=</mo><mrow id="S2.SS1.p2.9.m2.4.4.2.2" xref="S2.SS1.p2.9.m2.4.4.2.3.cmml"><mrow id="S2.SS1.p2.9.m2.3.3.1.1.1.1" xref="S2.SS1.p2.9.m2.3.3.1.1.1.2.cmml"><mo stretchy="false" id="S2.SS1.p2.9.m2.3.3.1.1.1.1.2" xref="S2.SS1.p2.9.m2.3.3.1.1.1.2.1.cmml">⌈</mo><mrow id="S2.SS1.p2.9.m2.3.3.1.1.1.1.1" xref="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.cmml"><mi id="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.2" xref="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.2.cmml">M</mi><mo id="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.1" xref="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.1.cmml">/</mo><mn id="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.3" xref="S2.SS1.p2.9.m2.3.3.1.1.1.1.1.3.cmml">2</mn></mrow><mo stretchy="false" id="S2.SS1.p2.9.m2.3.3.1.1.1.1.3" xref="S2.SS1.p2.9.m2.3.3.1.1.1.2.1.cmml">⌉</mo></mrow><mo id="S2.SS1.p2.9.m2.4.4.2.2.3" xref="S2.SS1.p2.9.m2.4.4.2.3.cmml">,</mo><mrow id="S2.SS1.p2.9.m2.4.4.2.2.2.1" xref="S2.SS1.p2.9.m2.4.4.2.2.2.2.cmml"><mo stretchy="false" id="S2.SS1.p2.9.m2.4.4.2.2.2.1.2" xref="S2.SS1.p2.9.m2.4.4.2.2.2.2.1.cmml">⌈</mo><mrow id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.cmml"><mrow id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.cmml"><mn id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.2" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.2.cmml">2</mn><mo id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.1" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.1.cmml">⁢</mo><mi id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.3" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.2.3.cmml">M</mi></mrow><mo id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.1" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.1.cmml">/</mo><mn id="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.3" xref="S2.SS1.p2.9.m2.4.4.2.2.2.1.1.3.cmml">3</mn></mrow><mo stretchy="false" id="S2.SS1.p2.9.m2.4.4.2.2.2.1.3" xref="S2.SS1.p2.9.m2.4.4.2.2.2.2.1.cmml">⌉</mo></mrow><mo id="S2.SS1.p2.9.m2.4.4.2.2.4" xref="S2.SS1.p2.9.m2.4.4.2.3.cmml">,</mo><mi mathvariant="normal" id="S2.SS1.p2.9.m2.1.1" xref="S2.SS1.p2.9.m2.1.1.cmml">…</mi><mo id="S2.SS1.p2.9.m2.4.4.2.2.5" xref="S2.SS1.p2.9.m2.4.4.2.3.cmml">,</mo><mi id="S2.SS1.p2.9.m2.2.2" xref="S2.SS1.p2.9.m2.2.2.cmml">M</mi></mrow></mrow></math>, <math><mrow id="S2.E4.m1.6.7" xref="S2.E4.m1.6.7.cmml"><mrow id="S2.E4.m1.4.4.4" xref="S2.E4.m1.4.4.3.cmml"><mo fence="true" stretchy="false" id="S2.E4.m1.4.4.4.1" xref="S2.E4.m1.4.4.3.1.cmml">|</mo><mi id="S2.E4.m1.4.4.2" xref="S2.E4.m1.4.4.2.cmml">ψ</mi><mo stretchy="false" id="S2.E4.m1.4.4.4.2" xref="S2.E4.m1.4.4.3.1.cmml">⟩</mo></mrow><mo id="S2.E4.m1.6.7.1" xref="S2.E4.m1.6.7.1.cmml">=</mo><mrow id="S2.E4.m1.6.7.2" xref="S2.E4.m1.6.7.2.cmml"><mrow id="S2.E4.m1.6.7.2.2" xref="S2.E4.m1.6.7.2.2.cmml"><mi id="S2.E4.m1.6.7.2.2.1" xref="S2.E4.m1.6.7.2.2.1.cmml">cos</mi><mo id="S2.E4.m1.6.7.2.2a" xref="S2.E4.m1.6.7.2.2.cmml">⁡</mo><mrow id="S2.E4.m1.6.7.2.2.2" xref="S2.E4.m1.6.7.2.2.2.cmml"><mi id="S2.E4.m1.6.7.2.2.2.2" xref="S2.E4.m1.6.7.2.2.2.2.cmml">θ</mi><mo id="S2.E4.m1.6.7.2.2.2.1" xref="S2.E4.m1.6.7.2.2.2.1.cmml">⁢</mo><mrow id="S2.E4.m1.5.5.4" xref="S2.E4.m1.5.5.3.cmml"><mo fence="true" stretchy="false" id="S2.E4.m1.5.5.4.1" xref="S2.E4.m1.5.5.3.1.cmml">|</mo><mn id="S2.E4.m1.5.5.2" xref="S2.E4.m1.5.5.2.cmml">00</mn><mo stretchy="false" id="S2.E4.m1.5.5.4.2" xref="S2.E4.m1.5.5.3.1.cmml">⟩</mo></mrow></mrow></mrow><mo id="S2.E4.m1.6.7.2.1" xref="S2.E4.m1.6.7.2.1.cmml">+</mo><mrow id="S2.E4.m1.6.7.2.3" xref="S2.E4.m1.6.7.2.3.cmml"><mi id="S2.E4.m1.6.7.2.3.1" xref="S2.E4.m1.6.7.2.3.1.cmml">sin</mi><mo id="S2.E4.m1.6.7.2.3a" xref="S2.E4.m1.6.7.2.3.cmml">⁡</mo><mrow id="S2.E4.m1.6.7.2.3.2" xref="S2.E4.m1.6.7.2.3.2.cmml"><mi id="S2.E4.m1.6.7.2.3.2.2" xref="S2.E4.m1.6.7.2.3.2.2.cmml">θ</mi><mo id="S2.E4.m1.6.7.2.3.2.1" xref="S2.E4.m1.6.7.2.3.2.1.cmml">⁢</mo><mrow id="S2.E4.m1.6.6.4" xref="S2.E4.m1.6.6.3.cmml"><mo fence="true" stretchy="false" id="S2.E4.m1.6.6.4.1" xref="S2.E4.m1.6.6.3.1.cmml">|</mo><mn id="S2.E4.m1.6.6.2" xref="S2.E4.m1.6.6.2.cmml">11</mn><mo stretchy="false" id="S2.E4.m1.6.6.4.2" xref="S2.E4.m1.6.6.3.1.cmml">⟩</mo></mrow></mrow></mrow></mrow></mrow></math>, <math><mrow id="S2.SS2.p1.1.m1.2.3" xref="S2.SS2.p1.1.m1.2.3.cmml"><mi id="S2.SS2.p1.1.m1.2.3.2" xref="S2.SS2.p1.1.m1.2.3.2.cmml">ρ</mi><mo id="S2.SS2.p1.1.m1.2.3.1" xref="S2.SS2.p1.1.m1.2.3.1.cmml">=</mo><mrow id="S2.SS2.p1.1.m1.2.2" xref="S2.SS2.p1.1.m1.2.2a.cmml"><mrow id="S2.SS2.p1.1.m1.1.1.1.m1.1.2.2" xref="S2.SS2.p1.1.m1.1.1.1.m1.1.2.2.cmml"><mo fence="true" stretchy="false" id="S2.SS2.p1.1.m1.1.1.1.m1.1.2.2.1" xref="S2.SS2.p1.1.m1.1.1.1.m1.1.2.2.cmml">|</mo><mi id="S2.SS2.p1.1.m1.1.1.1.m1.1.1" xref="S2.SS2.p1.1.m1.1.1.1.m1.1.1.cmml">ψ</mi><mo stretchy="false" id="S2.SS2.p1.1.m1.1.1.1.m1.1.2.2.2" xref="S2.SS2.p1.1.m1.1.1.1.m1.1.2.2.cmml">⟩</mo></mrow><mrow id="S2.SS2.p1.1.m1.2.2.2.m1.1.2.2" xref="S2.SS2.p1.1.m1.2.2.2.m1.1.2.2.cmml"><mo stretchy="false" id="S2.SS2.p1.1.m1.2.2.2.m1.1.2.2.1" xref="S2.SS2.p1.1.m1.2.2.2.m1.1.2.2.cmml">⟨</mo><mi id="S2.SS2.p1.1.m1.2.2.2.m1.1.1" xref="S2.SS2.p1.1.m1.2.2.2.m1.1.1.cmml">ψ</mi><mo fence="true" stretchy="false" id="S2.SS2.p1.1.m1.2.2.2.m1.1.2.2.2" xref="S2.SS2.p1.1.m1.2.2.2.m1.1.2.2.cmml">|</mo></mrow></mrow></mrow></math>, <math><mrow id="S2.SS2.p1.2.m2.4.5" xref="S2.SS2.p1.2.m2.4.5.cmml"><mrow id="S2.SS2.p1.2.m2.3.3.2" xref="S2.SS2.p1.2.m2.3.3.3.cmml"><mo fence="true" stretchy="false" id="S2.SS2.p1.2.m2.3.3.2.2" xref="S2.SS2.p1.2.m2.3.3.3.1.cmml">|</mo><msub id="S2.SS2.p1.2.m2.3.3.2.1" xref="S2.SS2.p1.2.m2.3.3.2.1.cmml"><mi mathvariant="normal" id="S2.SS2.p1.2.m2.3.3.2.1.2" xref="S2.SS2.p1.2.m2.3.3.2.1.2.cmml">Ψ</mi><mi id="S2.SS2.p1.2.m2.3.3.2.1.3" xref="S2.SS2.p1.2.m2.3.3.2.1.3.cmml">M</mi></msub><mo stretchy="false" id="S2.SS2.p1.2.m2.3.3.2.3" xref="S2.SS2.p1.2.m2.3.3.3.1.cmml">⟩</mo></mrow><mo id="S2.SS2.p1.2.m2.4.5.1" xref="S2.SS2.p1.2.m2.4.5.1.cmml">=</mo><msup id="S2.SS2.p1.2.m2.4.5.2" xref="S2.SS2.p1.2.m2.4.5.2.cmml"><mrow id="S2.SS2.p1.2.m2.4.4.4" xref="S2.SS2.p1.2.m2.4.4.3.cmml"><mo fence="true" stretchy="false" id="S2.SS2.p1.2.m2.4.4.4.1" xref="S2.SS2.p1.2.m2.4.4.3.1.cmml">|</mo><mi id="S2.SS2.p1.2.m2.4.4.2" xref="S2.SS2.p1.2.m2.4.4.2.cmml">ψ</mi><mo stretchy="false" id="S2.SS2.p1.2.m2.4.4.4.2" xref="S2.SS2.p1.2.m2.4.4.3.1.cmml">⟩</mo></mrow><mrow id="S2.SS2.p1.2.m2.4.5.2.2" xref="S2.SS2.p1.2.m2.4.5.2.2.cmml"><mi id="S2.SS2.p1.2.m2.4.5.2.2.2" xref="S2.SS2.p1.2.m2.4.5.2.2.2.cmml"/><mo id="S2.SS2.p1.2.m2.4.5.2.2.1" xref="S2.SS2.p1.2.m2.4.5.2.2.1.cmml">⊗</mo><mi id="S2.SS2.p1.2.m2.4.5.2.2.3" xref="S2.SS2.p1.2.m2.4.5.2.2.3.cmml">M</mi></mrow></msup></mrow></math>

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1510.01314

Formulas:

1-

(H, ⟨ \cdot, \cdot ⟩) .

2-

A \nabla_{ν} B := (1 - ν) A + ν B, the weighted
arithmetic mean

3-

A ♯_{ν} B := A^{1 / 2} {(A^{- 1 / 2} B A^{- 1 / 2})}^{ν} A^{1 / 2}, the weighted geometric mean.

4-

ν \in [0, 1],

5-

a^{1 - ν} b^{ν} \leq (1 - ν) a + ν b

6-

S (h) := {\begin{matrix} \frac{h^{\frac{1}{h - 1}}}{e \ln (h^{\frac{1}{h - 1}})} if h \in (0, 1) \cup (1, \infty) \\ 1 if h = 1 . \end{matrix}

7-

{lim}_{h \to 1} S (h) = 1,

8-

S (h) = S (\frac{1}{h}) > 1

9-

(1, \infty) .

10-

S ({(\frac{a}{b})}^{r}) a^{1 - ν} b^{ν} \leq (1 - ν) a + ν b \leq S (\frac{a}{b}) a^{1 - ν} b^{ν},

Formulas (html):

<math><mrow id="S1.p1.3.m3.4.4.1"><mrow id="S1.p1.3.m3.4.4.1.1.1" xref="S1.p1.3.m3.4.4.1.1.2.cmml"><mo id="S1.p1.3.m3.4.4.1.1.1.2" xref="S1.p1.3.m3.4.4.1.1.2.cmml">(</mo><mi id="S1.p1.3.m3.3.3" xref="S1.p1.3.m3.3.3.cmml">H</mi><mo id="S1.p1.3.m3.4.4.1.1.1.3" xref="S1.p1.3.m3.4.4.1.1.2.cmml">,</mo><mrow id="S1.p1.3.m3.4.4.1.1.1.1.2" xref="S1.p1.3.m3.4.4.1.1.1.1.1.cmml"><mo id="S1.p1.3.m3.4.4.1.1.1.1.2.1" xref="S1.p1.3.m3.4.4.1.1.1.1.1.cmml">⟨</mo><mo id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml">⋅</mo><mo id="S1.p1.3.m3.4.4.1.1.1.1.2.2" xref="S1.p1.3.m3.4.4.1.1.1.1.1.cmml">,</mo><mo id="S1.p1.3.m3.2.2" xref="S1.p1.3.m3.2.2.cmml">⋅</mo><mo id="S1.p1.3.m3.4.4.1.1.1.1.2.3" xref="S1.p1.3.m3.4.4.1.1.1.1.1.cmml">⟩</mo></mrow><mo id="S1.p1.3.m3.4.4.1.1.1.4" xref="S1.p1.3.m3.4.4.1.1.2.cmml">)</mo></mrow><mo id="S1.p1.3.m3.4.4.1.2">.</mo></mrow></math>, <math><mrow id="S1.Ex1.m1.2.2" xref="S1.Ex1.m1.2.2.cmml"><mrow id="S1.Ex1.m1.2.2.3" xref="S1.Ex1.m1.2.2.3.cmml"><mi id="S1.Ex1.m1.2.2.3.2" xref="S1.Ex1.m1.2.2.3.2.cmml">A</mi><mo id="S1.Ex1.m1.2.2.3.1" xref="S1.Ex1.m1.2.2.3.1.cmml">⁢</mo><mrow id="S1.Ex1.m1.2.2.3.3" xref="S1.Ex1.m1.2.2.3.3.cmml"><msub id="S1.Ex1.m1.2.2.3.3.1" xref="S1.Ex1.m1.2.2.3.3.1.cmml"><mo id="S1.Ex1.m1.2.2.3.3.1.2" xref="S1.Ex1.m1.2.2.3.3.1.2.cmml">∇</mo><mi id="S1.Ex1.m1.2.2.3.3.1.3" xref="S1.Ex1.m1.2.2.3.3.1.3.cmml">ν</mi></msub><mo id="S1.Ex1.m1.2.2.3.3a" xref="S1.Ex1.m1.2.2.3.3.cmml">⁡</mo><mi id="S1.Ex1.m1.2.2.3.3.2" xref="S1.Ex1.m1.2.2.3.3.2.cmml">B</mi></mrow></mrow><mo id="S1.Ex1.m1.2.2.2" xref="S1.Ex1.m1.2.2.2.cmml">:=</mo><mrow id="S1.Ex1.m1.2.2.1.1" xref="S1.Ex1.m1.2.2.1.2.cmml"><mrow id="S1.Ex1.m1.2.2.1.1.1" xref="S1.Ex1.m1.2.2.1.1.1.cmml"><mrow id="S1.Ex1.m1.2.2.1.1.1.1" xref="S1.Ex1.m1.2.2.1.1.1.1.cmml"><mrow id="S1.Ex1.m1.2.2.1.1.1.1.1.1" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S1.Ex1.m1.2.2.1.1.1.1.1.1.2" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex1.m1.2.2.1.1.1.1.1.1.1" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.cmml"><mn id="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.2" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.1" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.1.cmml">-</mo><mi id="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.3" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.3.cmml">ν</mi></mrow><mo id="S1.Ex1.m1.2.2.1.1.1.1.1.1.3" xref="S1.Ex1.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.Ex1.m1.2.2.1.1.1.1.2" xref="S1.Ex1.m1.2.2.1.1.1.1.2.cmml">⁢</mo><mi id="S1.Ex1.m1.2.2.1.1.1.1.3" xref="S1.Ex1.m1.2.2.1.1.1.1.3.cmml">A</mi></mrow><mo id="S1.Ex1.m1.2.2.1.1.1.2" xref="S1.Ex1.m1.2.2.1.1.1.2.cmml">+</mo><mrow id="S1.Ex1.m1.2.2.1.1.1.3" xref="S1.Ex1.m1.2.2.1.1.1.3.cmml"><mi id="S1.Ex1.m1.2.2.1.1.1.3.2" xref="S1.Ex1.m1.2.2.1.1.1.3.2.cmml">ν</mi><mo id="S1.Ex1.m1.2.2.1.1.1.3.1" xref="S1.Ex1.m1.2.2.1.1.1.3.1.cmml">⁢</mo><mi id="S1.Ex1.m1.2.2.1.1.1.3.3" xref="S1.Ex1.m1.2.2.1.1.1.3.3.cmml">B</mi></mrow></mrow><mo id="S1.Ex1.m1.2.2.1.1.2" xref="S1.Ex1.m1.2.2.1.2.cmml">,</mo><mrow id="S1.Ex1.m1.1.1" xref="S1.Ex1.m1.1.1c.cmml"><mtext id="S1.Ex1.m1.1.1a" xref="S1.Ex1.m1.1.1c.cmml"> the </mtext><mtext mathvariant="italic" id="S1.Ex1.m1.1.1b" xref="S1.Ex1.m1.1.1c.cmml">weighted arithmetic mean</mtext></mrow></mrow></mrow></math>, <math><mrow id="S1.Ex2.m1.2.2" xref="S1.Ex2.m1.2.2.cmml"><mrow id="S1.Ex2.m1.2.2.3" xref="S1.Ex2.m1.2.2.3.cmml"><mi id="S1.Ex2.m1.2.2.3.2" xref="S1.Ex2.m1.2.2.3.2.cmml">A</mi><mo id="S1.Ex2.m1.2.2.3.1" xref="S1.Ex2.m1.2.2.3.1.cmml">⁢</mo><msub id="S1.Ex2.m1.2.2.3.3" xref="S1.Ex2.m1.2.2.3.3.cmml"><mi mathvariant="normal" id="S1.Ex2.m1.2.2.3.3.2" xref="S1.Ex2.m1.2.2.3.3.2.cmml">♯</mi><mi id="S1.Ex2.m1.2.2.3.3.3" xref="S1.Ex2.m1.2.2.3.3.3.cmml">ν</mi></msub><mo id="S1.Ex2.m1.2.2.3.1a" xref="S1.Ex2.m1.2.2.3.1.cmml">⁢</mo><mi id="S1.Ex2.m1.2.2.3.4" xref="S1.Ex2.m1.2.2.3.4.cmml">B</mi></mrow><mo id="S1.Ex2.m1.2.2.2" xref="S1.Ex2.m1.2.2.2.cmml">:=</mo><mrow id="S1.Ex2.m1.2.2.1.1" xref="S1.Ex2.m1.2.2.1.2.cmml"><mrow id="S1.Ex2.m1.2.2.1.1.1" xref="S1.Ex2.m1.2.2.1.1.1.cmml"><msup id="S1.Ex2.m1.2.2.1.1.1.3" xref="S1.Ex2.m1.2.2.1.1.1.3.cmml"><mi id="S1.Ex2.m1.2.2.1.1.1.3.2" xref="S1.Ex2.m1.2.2.1.1.1.3.2.cmml">A</mi><mrow id="S1.Ex2.m1.2.2.1.1.1.3.3" xref="S1.Ex2.m1.2.2.1.1.1.3.3.cmml"><mn id="S1.Ex2.m1.2.2.1.1.1.3.3.2" xref="S1.Ex2.m1.2.2.1.1.1.3.3.2.cmml">1</mn><mo id="S1.Ex2.m1.2.2.1.1.1.3.3.1" xref="S1.Ex2.m1.2.2.1.1.1.3.3.1.cmml">/</mo><mn id="S1.Ex2.m1.2.2.1.1.1.3.3.3" xref="S1.Ex2.m1.2.2.1.1.1.3.3.3.cmml">2</mn></mrow></msup><mo id="S1.Ex2.m1.2.2.1.1.1.2" xref="S1.Ex2.m1.2.2.1.1.1.2.cmml">⁢</mo><msup id="S1.Ex2.m1.2.2.1.1.1.1" xref="S1.Ex2.m1.2.2.1.1.1.1.cmml"><mrow id="S1.Ex2.m1.2.2.1.1.1.1.1.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.cmml"><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.cmml"><msup id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.cmml"><mi id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.2.cmml">A</mi><mrow id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.cmml"><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.1.cmml">-</mo><mrow id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.cmml"><mn id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.2.cmml">1</mn><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.1.cmml">/</mo><mn id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.3" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.2.3.2.3.cmml">2</mn></mrow></mrow></msup><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.3" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.3.cmml">B</mi><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.1a" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.1.cmml">⁢</mo><msup id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.cmml"><mi id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.2.cmml">A</mi><mrow id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.cmml"><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.1.cmml">-</mo><mrow id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.cmml"><mn id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.2" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.2.cmml">1</mn><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.1" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.1.cmml">/</mo><mn id="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.3" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.4.3.2.3.cmml">2</mn></mrow></mrow></msup></mrow><mo id="S1.Ex2.m1.2.2.1.1.1.1.1.1.3" xref="S1.Ex2.m1.2.2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S1.Ex2.m1.2.2.1.1.1.1.3" xref="S1.Ex2.m1.2.2.1.1.1.1.3.cmml">ν</mi></msup><mo id="S1.Ex2.m1.2.2.1.1.1.2a" xref="S1.Ex2.m1.2.2.1.1.1.2.cmml">⁢</mo><msup id="S1.Ex2.m1.2.2.1.1.1.4" xref="S1.Ex2.m1.2.2.1.1.1.4.cmml"><mi id="S1.Ex2.m1.2.2.1.1.1.4.2" xref="S1.Ex2.m1.2.2.1.1.1.4.2.cmml">A</mi><mrow id="S1.Ex2.m1.2.2.1.1.1.4.3" xref="S1.Ex2.m1.2.2.1.1.1.4.3.cmml"><mn id="S1.Ex2.m1.2.2.1.1.1.4.3.2" xref="S1.Ex2.m1.2.2.1.1.1.4.3.2.cmml">1</mn><mo id="S1.Ex2.m1.2.2.1.1.1.4.3.1" xref="S1.Ex2.m1.2.2.1.1.1.4.3.1.cmml">/</mo><mn id="S1.Ex2.m1.2.2.1.1.1.4.3.3" xref="S1.Ex2.m1.2.2.1.1.1.4.3.3.cmml">2</mn></mrow></msup></mrow><mo id="S1.Ex2.m1.2.2.1.1.2" xref="S1.Ex2.m1.2.2.1.2.cmml">,</mo><mrow id="S1.Ex2.m1.1.1" xref="S1.Ex2.m1.1.1c.cmml"><mtext id="S1.Ex2.m1.1.1a" xref="S1.Ex2.m1.1.1c.cmml"> the </mtext><mtext mathvariant="italic" id="S1.Ex2.m1.1.1b" xref="S1.Ex2.m1.1.1c.cmml">weighted geometric mean.</mtext></mrow></mrow></mrow></math>, <math><mrow id="S1.p2.2.m2.3.3.1" xref="S1.p2.2.m2.3.3.1.1.cmml"><mrow id="S1.p2.2.m2.3.3.1.1" xref="S1.p2.2.m2.3.3.1.1.cmml"><mi id="S1.p2.2.m2.3.3.1.1.2" xref="S1.p2.2.m2.3.3.1.1.2.cmml">ν</mi><mo id="S1.p2.2.m2.3.3.1.1.1" xref="S1.p2.2.m2.3.3.1.1.1.cmml">∈</mo><mrow id="S1.p2.2.m2.3.3.1.1.3.2" xref="S1.p2.2.m2.3.3.1.1.3.1.cmml"><mo stretchy="false" id="S1.p2.2.m2.3.3.1.1.3.2.1" xref="S1.p2.2.m2.3.3.1.1.3.1.cmml">[</mo><mn id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">0</mn><mo id="S1.p2.2.m2.3.3.1.1.3.2.2" xref="S1.p2.2.m2.3.3.1.1.3.1.cmml">,</mo><mn id="S1.p2.2.m2.2.2" xref="S1.p2.2.m2.2.2.cmml">1</mn><mo stretchy="false" id="S1.p2.2.m2.3.3.1.1.3.2.3" xref="S1.p2.2.m2.3.3.1.1.3.1.cmml">]</mo></mrow></mrow><mo id="S1.p2.2.m2.3.3.1.2" xref="S1.p2.2.m2.3.3.1.1.cmml">,</mo></mrow></math>, <math><mrow id="S1.E1.m1.1.1" xref="S1.E1.m1.1.1.cmml"><mrow id="S1.E1.m1.1.1.3" xref="S1.E1.m1.1.1.3.cmml"><msup id="S1.E1.m1.1.1.3.2" xref="S1.E1.m1.1.1.3.2.cmml"><mi id="S1.E1.m1.1.1.3.2.2" xref="S1.E1.m1.1.1.3.2.2.cmml">a</mi><mrow id="S1.E1.m1.1.1.3.2.3" xref="S1.E1.m1.1.1.3.2.3.cmml"><mn id="S1.E1.m1.1.1.3.2.3.2" xref="S1.E1.m1.1.1.3.2.3.2.cmml">1</mn><mo id="S1.E1.m1.1.1.3.2.3.1" xref="S1.E1.m1.1.1.3.2.3.1.cmml">-</mo><mi id="S1.E1.m1.1.1.3.2.3.3" xref="S1.E1.m1.1.1.3.2.3.3.cmml">ν</mi></mrow></msup><mo id="S1.E1.m1.1.1.3.1" xref="S1.E1.m1.1.1.3.1.cmml">⁢</mo><msup id="S1.E1.m1.1.1.3.3" xref="S1.E1.m1.1.1.3.3.cmml"><mi id="S1.E1.m1.1.1.3.3.2" xref="S1.E1.m1.1.1.3.3.2.cmml">b</mi><mi id="S1.E1.m1.1.1.3.3.3" xref="S1.E1.m1.1.1.3.3.3.cmml">ν</mi></msup></mrow><mo id="S1.E1.m1.1.1.2" xref="S1.E1.m1.1.1.2.cmml">≤</mo><mrow id="S1.E1.m1.1.1.1" xref="S1.E1.m1.1.1.1.cmml"><mrow id="S1.E1.m1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.cmml"><mrow id="S1.E1.m1.1.1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.1.1.1.cmml"><mo id="S1.E1.m1.1.1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S1.E1.m1.1.1.1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.1.1.1.cmml"><mn id="S1.E1.m1.1.1.1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S1.E1.m1.1.1.1.1.1.1.1.1" xref="S1.E1.m1.1.1.1.1.1.1.1.1.cmml">-</mo><mi id="S1.E1.m1.1.1.1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.1.1.1.3.cmml">ν</mi></mrow><mo id="S1.E1.m1.1.1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.E1.m1.1.1.1.1.2" xref="S1.E1.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S1.E1.m1.1.1.1.1.3" xref="S1.E1.m1.1.1.1.1.3.cmml">a</mi></mrow><mo id="S1.E1.m1.1.1.1.2" xref="S1.E1.m1.1.1.1.2.cmml">+</mo><mrow id="S1.E1.m1.1.1.1.3" xref="S1.E1.m1.1.1.1.3.cmml"><mi id="S1.E1.m1.1.1.1.3.2" xref="S1.E1.m1.1.1.1.3.2.cmml">ν</mi><mo id="S1.E1.m1.1.1.1.3.1" xref="S1.E1.m1.1.1.1.3.1.cmml">⁢</mo><mi id="S1.E1.m1.1.1.1.3.3" xref="S1.E1.m1.1.1.1.3.3.cmml">b</mi></mrow></mrow></mrow></math>, <math><mrow id="S1.E2.m1.8.9" xref="S1.E2.m1.8.9.cmml"><mrow id="S1.E2.m1.8.9.2" xref="S1.E2.m1.8.9.2.cmml"><mi id="S1.E2.m1.8.9.2.2" xref="S1.E2.m1.8.9.2.2.cmml">S</mi><mo id="S1.E2.m1.8.9.2.1" xref="S1.E2.m1.8.9.2.1.cmml">⁢</mo><mrow id="S1.E2.m1.8.9.2.3.2" xref="S1.E2.m1.8.9.2.cmml"><mo id="S1.E2.m1.8.9.2.3.2.1" xref="S1.E2.m1.8.9.2.cmml">(</mo><mi id="S1.E2.m1.8.8" xref="S1.E2.m1.8.8.cmml">h</mi><mo id="S1.E2.m1.8.9.2.3.2.2" xref="S1.E2.m1.8.9.2.cmml">)</mo></mrow></mrow><mo id="S1.E2.m1.8.9.1" xref="S1.E2.m1.8.9.1.cmml">:=</mo><mrow id="S1.E2.m1.8.9.3" xref="S1.E2.m1.8.9.3.cmml"><mrow id="S1.E2.m1.8.9.3.2.2" xref="S1.E2.m1.8.9.3.2.1.cmml"><mo id="S1.E2.m1.8.9.3.2.2.1" xref="S1.E2.m1.8.9.3.2.1.1.cmml">{</mo><mtable displaystyle="true" rowspacing="0pt" id="S1.E2.m1.7.7" xref="S1.E2.m1.7.7.cmml"><mtr id="S1.E2.m1.7.7a" xref="S1.E2.m1.7.7.cmml"><mtd columnalign="left" id="S1.E2.m1.7.7b" xref="S1.E2.m1.7.7.cmml"><mrow id="S1.E2.m1.6.6.6.6.6" xref="S1.E2.m1.6.6.6.6.6.cmml"><mrow id="S1.E2.m1.6.6.6.6.6.8" xref="S1.E2.m1.6.6.6.6.6.8.cmml"><mstyle displaystyle="false" id="S1.E2.m1.2.2.2.2.2.2" xref="S1.E2.m1.2.2.2.2.2.2.cmml"><mfrac id="S1.E2.m1.2.2.2.2.2.2a" xref="S1.E2.m1.2.2.2.2.2.2.cmml"><msup id="S1.E2.m1.2.2.2.2.2.2.4" xref="S1.E2.m1.2.2.2.2.2.2.4.cmml"><mi id="S1.E2.m1.2.2.2.2.2.2.4.2" xref="S1.E2.m1.2.2.2.2.2.2.4.2.cmml">h</mi><mfrac id="S1.E2.m1.2.2.2.2.2.2.4.3" xref="S1.E2.m1.2.2.2.2.2.2.4.3.cmml"><mn id="S1.E2.m1.2.2.2.2.2.2.4.3.2" xref="S1.E2.m1.2.2.2.2.2.2.4.3.2.cmml">1</mn><mrow id="S1.E2.m1.2.2.2.2.2.2.4.3.3" xref="S1.E2.m1.2.2.2.2.2.2.4.3.3.cmml"><mi id="S1.E2.m1.2.2.2.2.2.2.4.3.3.2" xref="S1.E2.m1.2.2.2.2.2.2.4.3.3.2.cmml">h</mi><mo id="S1.E2.m1.2.2.2.2.2.2.4.3.3.1" xref="S1.E2.m1.2.2.2.2.2.2.4.3.3.1.cmml">-</mo><mn id="S1.E2.m1.2.2.2.2.2.2.4.3.3.3" xref="S1.E2.m1.2.2.2.2.2.2.4.3.3.3.cmml">1</mn></mrow></mfrac></msup><mrow id="S1.E2.m1.2.2.2.2.2.2.2" xref="S1.E2.m1.2.2.2.2.2.2.2.cmml"><mi id="S1.E2.m1.2.2.2.2.2.2.2.4" xref="S1.E2.m1.2.2.2.2.2.2.2.4.cmml">e</mi><mo id="S1.E2.m1.2.2.2.2.2.2.2.3" xref="S1.E2.m1.2.2.2.2.2.2.2.3.cmml">⁢</mo><mrow id="S1.E2.m1.2.2.2.2.2.2.2.2.1" xref="S1.E2.m1.2.2.2.2.2.2.2.2.2.cmml"><mi id="S1.E2.m1.1.1.1.1.1.1.1.1" xref="S1.E2.m1.1.1.1.1.1.1.1.1.cmml">ln</mi><mo id="S1.E2.m1.2.2.2.2.2.2.2.2.1a" 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xref="S1.p3.2.m2.2.3.2.cmml">)</mo></mrow></mrow><mo id="S1.p3.2.m2.2.3.3" xref="S1.p3.2.m2.2.3.3.cmml">=</mo><mrow id="S1.p3.2.m2.2.3.4" xref="S1.p3.2.m2.2.3.4.cmml"><mi id="S1.p3.2.m2.2.3.4.2" xref="S1.p3.2.m2.2.3.4.2.cmml">S</mi><mo id="S1.p3.2.m2.2.3.4.1" xref="S1.p3.2.m2.2.3.4.1.cmml">⁢</mo><mrow id="S1.p3.2.m2.2.3.4.3.2" xref="S1.p3.2.m2.2.2.cmml"><mo id="S1.p3.2.m2.2.3.4.3.2.1" xref="S1.p3.2.m2.2.2.cmml">(</mo><mfrac id="S1.p3.2.m2.2.2" xref="S1.p3.2.m2.2.2.cmml"><mn id="S1.p3.2.m2.2.2.2" xref="S1.p3.2.m2.2.2.2.cmml">1</mn><mi id="S1.p3.2.m2.2.2.3" xref="S1.p3.2.m2.2.2.3.cmml">h</mi></mfrac><mo id="S1.p3.2.m2.2.3.4.3.2.2" xref="S1.p3.2.m2.2.2.cmml">)</mo></mrow></mrow><mo id="S1.p3.2.m2.2.3.5" xref="S1.p3.2.m2.2.3.5.cmml">></mo><mn id="S1.p3.2.m2.2.3.6" xref="S1.p3.2.m2.2.3.6.cmml">1</mn></mrow></math>, <math><mrow id="S1.p3.6.m6.3.3.1"><mrow id="S1.p3.6.m6.3.3.1.1.2" xref="S1.p3.6.m6.3.3.1.1.1.cmml"><mo id="S1.p3.6.m6.3.3.1.1.2.1" xref="S1.p3.6.m6.3.3.1.1.1.cmml">(</mo><mn id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml">1</mn><mo id="S1.p3.6.m6.3.3.1.1.2.2" xref="S1.p3.6.m6.3.3.1.1.1.cmml">,</mo><mi mathvariant="normal" id="S1.p3.6.m6.2.2" xref="S1.p3.6.m6.2.2.cmml">∞</mi><mo id="S1.p3.6.m6.3.3.1.1.2.3" xref="S1.p3.6.m6.3.3.1.1.1.cmml">)</mo></mrow><mo id="S1.p3.6.m6.3.3.1.2">.</mo></mrow></math>, <math><mrow id="S1.E3.m1.3.3.1" xref="S1.E3.m1.3.3.1.1.cmml"><mrow id="S1.E3.m1.3.3.1.1" xref="S1.E3.m1.3.3.1.1.cmml"><mrow id="S1.E3.m1.3.3.1.1.1" xref="S1.E3.m1.3.3.1.1.1.cmml"><mi id="S1.E3.m1.3.3.1.1.1.3" xref="S1.E3.m1.3.3.1.1.1.3.cmml">S</mi><mo id="S1.E3.m1.3.3.1.1.1.2" xref="S1.E3.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S1.E3.m1.3.3.1.1.1.1.1" xref="S1.E3.m1.3.3.1.1.1.1.1.1.cmml"><mo id="S1.E3.m1.3.3.1.1.1.1.1.2" xref="S1.E3.m1.3.3.1.1.1.1.1.1.cmml">(</mo><msup id="S1.E3.m1.3.3.1.1.1.1.1.1" xref="S1.E3.m1.3.3.1.1.1.1.1.1.cmml"><mrow id="S1.E3.m1.3.3.1.1.1.1.1.1.2.2" xref="S1.E3.m1.1.1.cmml"><mo id="S1.E3.m1.3.3.1.1.1.1.1.1.2.2.1" xref="S1.E3.m1.1.1.cmml">(</mo><mfrac id="S1.E3.m1.1.1" xref="S1.E3.m1.1.1.cmml"><mi id="S1.E3.m1.1.1.2" xref="S1.E3.m1.1.1.2.cmml">a</mi><mi id="S1.E3.m1.1.1.3" xref="S1.E3.m1.1.1.3.cmml">b</mi></mfrac><mo id="S1.E3.m1.3.3.1.1.1.1.1.1.2.2.2" xref="S1.E3.m1.1.1.cmml">)</mo></mrow><mi id="S1.E3.m1.3.3.1.1.1.1.1.1.3" xref="S1.E3.m1.3.3.1.1.1.1.1.1.3.cmml">r</mi></msup><mo id="S1.E3.m1.3.3.1.1.1.1.1.3" xref="S1.E3.m1.3.3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S1.E3.m1.3.3.1.1.1.2a" xref="S1.E3.m1.3.3.1.1.1.2.cmml">⁢</mo><msup id="S1.E3.m1.3.3.1.1.1.4" xref="S1.E3.m1.3.3.1.1.1.4.cmml"><mi id="S1.E3.m1.3.3.1.1.1.4.2" xref="S1.E3.m1.3.3.1.1.1.4.2.cmml">a</mi><mrow id="S1.E3.m1.3.3.1.1.1.4.3" xref="S1.E3.m1.3.3.1.1.1.4.3.cmml"><mn id="S1.E3.m1.3.3.1.1.1.4.3.2" xref="S1.E3.m1.3.3.1.1.1.4.3.2.cmml">1</mn><mo id="S1.E3.m1.3.3.1.1.1.4.3.1" xref="S1.E3.m1.3.3.1.1.1.4.3.1.cmml">-</mo><mi id="S1.E3.m1.3.3.1.1.1.4.3.3" xref="S1.E3.m1.3.3.1.1.1.4.3.3.cmml">ν</mi></mrow></msup><mo id="S1.E3.m1.3.3.1.1.1.2b" xref="S1.E3.m1.3.3.1.1.1.2.cmml">⁢</mo><msup id="S1.E3.m1.3.3.1.1.1.5" xref="S1.E3.m1.3.3.1.1.1.5.cmml"><mi id="S1.E3.m1.3.3.1.1.1.5.2" xref="S1.E3.m1.3.3.1.1.1.5.2.cmml">b</mi><mi id="S1.E3.m1.3.3.1.1.1.5.3" xref="S1.E3.m1.3.3.1.1.1.5.3.cmml">ν</mi></msup></mrow><mo id="S1.E3.m1.3.3.1.1.4" xref="S1.E3.m1.3.3.1.1.4.cmml">≤</mo><mrow id="S1.E3.m1.3.3.1.1.2" xref="S1.E3.m1.3.3.1.1.2.cmml"><mrow id="S1.E3.m1.3.3.1.1.2.1" xref="S1.E3.m1.3.3.1.1.2.1.cmml"><mrow id="S1.E3.m1.3.3.1.1.2.1.1.1" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.cmml"><mo id="S1.E3.m1.3.3.1.1.2.1.1.1.2" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S1.E3.m1.3.3.1.1.2.1.1.1.1" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.cmml"><mn id="S1.E3.m1.3.3.1.1.2.1.1.1.1.2" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.2.cmml">1</mn><mo id="S1.E3.m1.3.3.1.1.2.1.1.1.1.1" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.1.cmml">-</mo><mi id="S1.E3.m1.3.3.1.1.2.1.1.1.1.3" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.3.cmml">ν</mi></mrow><mo id="S1.E3.m1.3.3.1.1.2.1.1.1.3" xref="S1.E3.m1.3.3.1.1.2.1.1.1.1.cmml">)</mo></mrow><mo id="S1.E3.m1.3.3.1.1.2.1.2" xref="S1.E3.m1.3.3.1.1.2.1.2.cmml">⁢</mo><mi id="S1.E3.m1.3.3.1.1.2.1.3" xref="S1.E3.m1.3.3.1.1.2.1.3.cmml">a</mi></mrow><mo id="S1.E3.m1.3.3.1.1.2.2" xref="S1.E3.m1.3.3.1.1.2.2.cmml">+</mo><mrow id="S1.E3.m1.3.3.1.1.2.3" xref="S1.E3.m1.3.3.1.1.2.3.cmml"><mi id="S1.E3.m1.3.3.1.1.2.3.2" xref="S1.E3.m1.3.3.1.1.2.3.2.cmml">ν</mi><mo id="S1.E3.m1.3.3.1.1.2.3.1" xref="S1.E3.m1.3.3.1.1.2.3.1.cmml">⁢</mo><mi id="S1.E3.m1.3.3.1.1.2.3.3" xref="S1.E3.m1.3.3.1.1.2.3.3.cmml">b</mi></mrow></mrow><mo id="S1.E3.m1.3.3.1.1.5" xref="S1.E3.m1.3.3.1.1.5.cmml">≤</mo><mrow id="S1.E3.m1.3.3.1.1.6" xref="S1.E3.m1.3.3.1.1.6.cmml"><mi id="S1.E3.m1.3.3.1.1.6.2" xref="S1.E3.m1.3.3.1.1.6.2.cmml">S</mi><mo id="S1.E3.m1.3.3.1.1.6.1" xref="S1.E3.m1.3.3.1.1.6.1.cmml">⁢</mo><mrow id="S1.E3.m1.3.3.1.1.6.3.2" xref="S1.E3.m1.2.2.cmml"><mo id="S1.E3.m1.3.3.1.1.6.3.2.1" xref="S1.E3.m1.2.2.cmml">(</mo><mfrac id="S1.E3.m1.2.2" xref="S1.E3.m1.2.2.cmml"><mi id="S1.E3.m1.2.2.2" xref="S1.E3.m1.2.2.2.cmml">a</mi><mi id="S1.E3.m1.2.2.3" xref="S1.E3.m1.2.2.3.cmml">b</mi></mfrac><mo id="S1.E3.m1.3.3.1.1.6.3.2.2" xref="S1.E3.m1.2.2.cmml">)</mo></mrow><mo id="S1.E3.m1.3.3.1.1.6.1a" xref="S1.E3.m1.3.3.1.1.6.1.cmml">⁢</mo><msup id="S1.E3.m1.3.3.1.1.6.4" xref="S1.E3.m1.3.3.1.1.6.4.cmml"><mi id="S1.E3.m1.3.3.1.1.6.4.2" xref="S1.E3.m1.3.3.1.1.6.4.2.cmml">a</mi><mrow id="S1.E3.m1.3.3.1.1.6.4.3" xref="S1.E3.m1.3.3.1.1.6.4.3.cmml"><mn id="S1.E3.m1.3.3.1.1.6.4.3.2" xref="S1.E3.m1.3.3.1.1.6.4.3.2.cmml">1</mn><mo id="S1.E3.m1.3.3.1.1.6.4.3.1" xref="S1.E3.m1.3.3.1.1.6.4.3.1.cmml">-</mo><mi id="S1.E3.m1.3.3.1.1.6.4.3.3" xref="S1.E3.m1.3.3.1.1.6.4.3.3.cmml">ν</mi></mrow></msup><mo id="S1.E3.m1.3.3.1.1.6.1b" xref="S1.E3.m1.3.3.1.1.6.1.cmml">⁢</mo><msup id="S1.E3.m1.3.3.1.1.6.5" xref="S1.E3.m1.3.3.1.1.6.5.cmml"><mi id="S1.E3.m1.3.3.1.1.6.5.2" xref="S1.E3.m1.3.3.1.1.6.5.2.cmml">b</mi><mi id="S1.E3.m1.3.3.1.1.6.5.3" xref="S1.E3.m1.3.3.1.1.6.5.3.cmml">ν</mi></msup></mrow></mrow><mo id="S1.E3.m1.3.3.1.2" xref="S1.E3.m1.3.3.1.1.cmml">,</mo></mrow></math>

Correct Categorie: math

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0209345

Formulas:

1-

Δ_{π} = 1 - 3

2-

Δ = 1.76 k_{B} T_{c} = 5.9

3-

R_{t} = 0.2 - 0.4

4-

T_{c} = 37 - 38.6

5-

Δ T_{c} = 0.4 - 0.6

6-

H_{c2} = ϕ_{0} / (2 π ξ^{2})

7-

ϕ_{0} = h / 2 e

8-

ξ_{GL}^{c} = ξ_{GL}^{ab} \times H_{c2}^{c} / H_{c2}^{ab} ≃ 1 - 2

9-

Δ (0) = 2.3

10-

N (ϵ, Γ) = N_{0} | Re \frac{ϵ - i Γ}{\sqrt{{(ϵ - i Γ)}^{2} - Δ^{2}}} | .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1403.2078

Formulas:

1-

L a_{0.3} S r_{0.7} A l_{0.65} T a_{0.35} O_{3}

2-

L a A l O_{3}

3-

S i O_{2} / S i

4-

L a A l O_{3}

5-

L a_{0.3} S r_{0.7} A l_{0.65} T a_{0.35} O_{3}

6-

Q_{t o t}

7-

Q_{e x t}

8-

Q_{i n t}

9-

Q_{t o t} = {[Q_{e x t}^{- 1} + Q_{i n t}^{- 1}]}^{- 1}

10-

Q_{i n t}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1204.5650

Formulas:

1-

\begin{matrix} {\dot{ρ}}_{𝐤}^{λ} & = 2 ℑ [Ω_{𝐤}^{v c, *} p_{𝐤}] + {{\dot{ρ}}_{𝐤}^{λ} |}_{s c a t}, \\ {\dot{p}}_{𝐤} & = [i Δ ω_{𝐤} - i Ω_{𝐤}^{λ λ} - γ_{𝐤}] p_{𝐤} - i Ω_{𝐤}^{v c} [ρ_{𝐤}^{c} - ρ_{𝐤}^{v}] + 𝒰_{𝐤}, \\ {\dot{n}}_{𝐪}^{j} & = - γ_{p h} [n_{𝐪}^{j} - n_{B}] + Γ_{j, 𝐪}^{e m} [1 + n_{𝐪}^{j}] - Γ_{j, 𝐪}^{a b s} n_{𝐪}^{j} . \end{matrix}

2-

Γ_{j, 𝐪}^{e m}

3-

Γ_{j, 𝐪}^{a b s}

4-

p_{𝐤} (t)

5-

γ_{𝐤} (t)

6-

𝒰_{𝐤} (t)

7-

{{\dot{ρ}}_{𝐤}^{λ} |}_{s c a t} = \sum_{λ_{1}, 𝐤^{'}} 𝒲_{𝐤^{'} \to 𝐤}^{λ_{1} \to λ} ρ_{𝐤^{'}}^{λ_{1}} (1 - ρ_{𝐤}^{λ}) - 𝒲_{𝐤^{'} \leftarrow 𝐤}^{λ_{1} \leftarrow λ} (1 - ρ_{𝐤^{'}}^{λ_{1}}) ρ_{𝐤}^{λ} .

8-

(λ_{1}, 𝐤^{'})

9-

𝒲_{𝐤^{'} \to 𝐤}^{λ_{1} \to λ} = \sum_{λ_{2}, λ_{3}, 𝐪} 𝒱_{𝐤, 𝐤^{'}, 𝐪}^{λ, λ_{1}, λ_{2}, λ_{3}} (1 - ρ_{𝐤^{'} + 𝐪}^{λ_{3}}) ρ_{𝐤 + 𝐪}^{λ_{2}},

10-

𝒱_{𝐤, 𝐤^{'}, 𝐪}^{λ, λ_{1}, λ_{2}, λ_{3}}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1405.6897

Formulas:

1-

𝓞 (𝜶_{𝐬} 𝜶)

2-

𝒪 (α_{s} α)

3-

𝒪 (α_{s} α)

4-

𝒪 (α_{s} α)

5-

𝒪 (α_{s} α)

6-

𝒪 (α_{s} α)

7-

𝒪 (α_{s} α)

8-

𝒪 (α_{s} α)

9-

𝒪 (α_{s} α)

10-

1 / (p^{2} - M_{V}^{2} + i M_{V} Γ_{V})

Formulas (html):

Correct Categorie: hep-ph

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1609.01612

Formulas:

1-

π_{f} (G)

2-

π_{f} (G) \leq 3

3-

π_{f} (K_{m, m}) = π_{f} (K_{m + 1, m}) = \frac{3 m}{m + 1}

4-

π_{f} (G)

5-

π_{f} (G) < \sqrt{2}

6-

π^{\circ} (G)

7-

π_{f}^{\circ} (G)

8-

π^{\circ} (G) = 1

9-

π^{\circ} (G) \leq 2

10-

π^{\circ} (K_{n}) \geq \log_{2} \log_{3} (n - 1)

Formulas (html):

Correct Categorie: math

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/1806.06966

Formulas:

1-

c \in ⋃_{v \in V (G)} L (v)

2-

⌊ η (c) / k ⌋

3-

⌈ η (c) / k ⌉

4-

k \geq Δ (G) + ⌈ | V (G) | / 2 ⌉

5-

⌈ | V (G) | / k ⌉

6-

⌊ | V (G) | / k ⌋

7-

K_{2 m + 1, 2 m + 1}

8-

(2 m + 1)

9-

k \geq 2 m + 2 = Δ (K_{2 m + 1, 2 m + 1}) + 1

10-

k \geq Δ (G) + 1

Formulas (html):

Correct Categorie: math

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9712112

Formulas:

1-

τ_{i + 1} = 0

2-

1 - (1 - β) ϕ_{i + 1}

3-

ρ_{c} = 1 - β

4-

u (x) = f (x) + β (1 - f (x))

5-

f (x) = \exp (- λ x / v)

6-

u (x) = β + (1 - β) \exp (- λ x / v) for x > 0

7-

u (0) = 0

8-

\dot{x} = u (k L) - u (x) + η (t) \equiv - \frac{d Φ}{d x} + η (t) .

9-

x^{*} = k L

10-

\exp [Φ (x^{*}) - Φ (0)]

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/1207.4501

Formulas:

1-

𝒗 (𝒒) = - \nabla Φ (𝒒)

2-

x_{i} (q, t) = q_{i} + D v_{i}, v_{i} \equiv \frac{d x_{i}}{d D} = v_{i} (q) .

3-

D = D (t)

4-

λ_{1} (q), λ_{2} (q), λ_{3} (q)

5-

d_{i k} \equiv - \partial v_{i} / \partial q_{k}

6-

V (q, t) = V_{0} (1 - D λ_{1}) (1 - D λ_{2}) (1 - D λ_{3}) .

7-

λ_{1} (q) \geq λ_{2} (q)

8-

λ_{2} (q) \geq λ_{3} (q)

9-

λ_{1} (q)

10-

λ_{1} (q) = λ_{2} (q)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0706.3883

Formulas:

1-

A_{0} := π {(D_{0} / 2)}^{2}

2-

d F / d ℓ

3-

d F / d ℓ

4-

\dot{ℓ} = 1 μ

5-

\dot{ℓ} = 0.03 μ

6-

𝒟 = 0.01 \partial F / \partial λ

7-

ℓ (t) = v t + Δ_{ℓ} \sin (ω t)

8-

\dot{ℓ} = 0.1 μ

9-

\dot{ℓ} = 1 μ

10-

d F / d ℓ

Formulas (html):

Correct Categorie: physics

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1612.01951

Formulas:

1-

V = e^{- r T} θ {F Φ [θ (\frac{\log \frac{F}{K}}{σ \sqrt{T}} + \frac{1}{2} σ \sqrt{T})] - K Φ [θ (\frac{\log \frac{F}{K}}{σ \sqrt{T}} - \frac{1}{2} σ \sqrt{T})]},

2-

F := S_{0} e^{(r - d) T}

3-

Φ (x) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x} \exp (- \frac{u^{2}}{2}) d u,

4-

= \frac{1}{2} + \frac{1}{\sqrt{2 π}} \int_{0}^{x} \exp (- \frac{u^{2}}{2}) d u

5-

= \frac{1}{2} + \frac{1}{2} \erf \frac{x}{\sqrt{2}} .

6-

V = f (σ)

7-

σ_{n + 1} = σ_{n} - \frac{f (σ_{n}) - V}{f^{'} (σ_{n})},

8-

ℂ_{*} = ℂ ∖ {0}

9-

\exp (- z^{2} / 2)

10-

Φ (z) = \frac{1}{2} + \frac{1}{\sqrt{2 π}} \int_{0}^{z} \exp (- \frac{u^{2}}{2}) d u

Formulas (html):

Correct Categorie: q-fin

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/1608.00023

Formulas:

1-

Γ_{S} / Γ_{L} = 16

2-

Γ_{R} / Γ_{L} = {0.5, 1, 2, 4}

3-

H = H_{Q D} + \sum_{α = L, R, S} H_{α} + H_{T},

4-

H_{Q D} = ϵ_{0} \sum_{σ} d_{σ}^{†} d_{σ} + U n_{↑} n_{↓}

5-

n_{σ} \equiv d_{σ}^{†} d_{σ}

6-

H_{α} = \sum_{k, σ} ϵ_{α k} c_{α k σ}^{†} c_{α k σ}

7-

c_{α k σ}^{†}

8-

c_{α k σ}

9-

α = {L, R}

10-

H_{S} = \sum_{k, σ} ϵ_{S k} c_{S k σ}^{†} c_{S k σ} + \sum_{k} (Δ c_{S - k ↑}^{†} c_{S k ↓}^{†} + Δ^{*} c_{S k ↓} c_{S - k ↑})

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/1606.06131

Formulas:

1-

{| ϕ ⟩}_{C} = \sum_{j = 0}^{n - 1} α_{j} {| j ⟩}_{C}

2-

U_{T} = \sum_{j = 0}^{n - 1} α_{j} V_{j}

3-

| 0 ⟩ {⟨ 0 |}_{C}^{\otimes k}

4-

{| Ψ_{e x t} ⟩}_{T}

5-

U_{T} {| Ψ_{e x t} ⟩}_{T}

6-

U_{T} = \sum_{j = 0}^{n - 1} α_{j} V_{j},

7-

\sum_{j = 0}^{n - 1} {| α_{j} |}^{2} = 1 .

8-

{| ϕ ⟩}_{C} = \sum_{j = 0}^{n - 1} α_{j} {| j ⟩}_{C},

9-

\exp (i θ) V_{j}

10-

{{| j d ⟩}_{T}, \dots, {| (j + 1) d - 1 ⟩}_{T}}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1805.08178

Formulas:

1-

l k (T_{1}, T_{2}) = \frac{1}{2} \sum_{d \in T_{1} \cap T_{2}} s g n (d) = \sum_{d \in T_{1} over T_{2}} s g n (d) = \sum_{d \in T_{2} over T_{1}} s g n (d)

2-

l k (T_{1}, T_{2}) = l k (T_{2}, T_{1})

3-

v l k (T_{1}, T_{2}) = \sum_{d \in T_{1} over T_{2}} s g n (d) .

4-

v l k (T_{1}, T_{2}) \neq v l k (T_{2}, T_{1})

5-

(T_{1}, T_{2})

6-

v l k (T_{1}, T_{2}) - v l k (T_{2}, T_{1})

7-

W (T_{1}, T_{2})

8-

W (T_{2}, T_{1}) = - W (T_{1}, T_{2})

9-

| W (T_{1}, T_{2}) |

10-

v l k (L_{1}, L_{2}), v l k (L_{2}, L_{1})

Formulas (html):

Correct Categorie: math

Guessed Categorie: gr-qc

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/9406008

Formulas:

1-

T_{c} \approx 227 K

2-

f_{n} = \frac{d}{4 π \sqrt{3}} {(\frac{β_{n}}{L})}^{2} \sqrt{\frac{E}{ρ}}

3-

\cos β_{n} \cosh β_{n} + 1 = 0

4-

χ \sim {| T - T_{c} |}^{- γ}

5-

E ≃ 3.5 \times 10^{11}

6-

Q_{t h}^{- 1}

7-

T E_{T} α_{p}^{2} / C_{p} \frac{ω τ}{1 + ω^{2} τ^{2}}

8-

τ = {(d / π)}^{2} C_{p} / κ

9-

E_{T}, α_{p}, κ

10-

Q_{t h}^{- 1} \sim 6.5 \times 10^{- 8} .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/1304.7695

Formulas:

1-

{\tilde{ξ}}^{\pm} = \frac{\sum (E^{i} \pm p_{z}^{i})}{\sqrt{s}},

2-

- \infty < η < 4.9

3-

- 4.9 < η < + \infty

4-

Δ η \times Δ ϕ = 0.0174 \times 0.0174

5-

Δ η \times Δ ϕ = 0.087 \times 0.087

6-

3.0 < | η | < 5.0

7-

- \ln [\tan (\frac{θ}{2})]

8-

η^{j1, j2} \to - η^{j1, j2}

9-

\frac{d σ_{jj}}{d \tilde{ξ}} = \frac{N_{jj}^{i}}{L \cdot ϵ \cdot A^{i} \cdot Δ {\tilde{ξ}}^{i}}

10-

p_{T}^{j1, j2} > 20

Formulas (html):

Correct Categorie: hep-ex

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/0712.0718

Formulas:

1-

R_{*} ≃ 2 R_{⊙}

2-

15 R_{*} \leq H \leq 26 R_{*}

3-

α_{J 2000.0} = 04^{h} 03^{m} 05 .^{s} 586056

4-

δ_{J 2000.0} = 26^{\circ} 00^{'} 01 .^{''} 50288

5-

α_{J 2000.0} = 04^{h} 14^{m} 12 .^{s} 919833

6-

δ_{J 2000.0} = 28^{\circ} 12^{'} 12 .^{''} 19953

7-

𝒜 ℐ 𝒫 𝒮

8-

15 R_{*} \leq H \leq 26 R_{*}

9-

θ > \arcsin {(B_{s} / B_{H})}^{1 / 2}

10-

R_{*} / c ≃ 85

Formulas (html):

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xref="S2.p3.3.m3.1.1.3.cmml"><msup id="S2.p3.3.m3.1.1.3.2" xref="S2.p3.3.m3.1.1.3.2.cmml"><mn id="S2.p3.3.m3.1.1.3.2.2" xref="S2.p3.3.m3.1.1.3.2.2.cmml">04</mn><mi mathvariant="normal" id="S2.p3.3.m3.1.1.3.2.3" xref="S2.p3.3.m3.1.1.3.2.3.cmml">h</mi></msup><mo id="S2.p3.3.m3.1.1.3.1" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><msup id="S2.p3.3.m3.1.1.3.3" xref="S2.p3.3.m3.1.1.3.3.cmml"><mn id="S2.p3.3.m3.1.1.3.3.2" xref="S2.p3.3.m3.1.1.3.3.2.cmml">14</mn><mi mathvariant="normal" id="S2.p3.3.m3.1.1.3.3.3" xref="S2.p3.3.m3.1.1.3.3.3.cmml">m</mi></msup><mo id="S2.p3.3.m3.1.1.3.1a" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><mn id="S2.p3.3.m3.1.1.3.4" xref="S2.p3.3.m3.1.1.3.4.cmml">12</mn><mo id="S2.p3.3.m3.1.1.3.1b" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><msup id="S2.p3.3.m3.1.1.3.5" xref="S2.p3.3.m3.1.1.3.5.cmml"><mpadded width="0.0pt" id="S2.p3.3.m3.1.1.3.5.2" xref="S2.p3.3.m3.1.1.3.5.2b.cmml"><mtext mathvariant="bold-sans-serif" id="S2.p3.3.m3.1.1.3.5.2a" xref="S2.p3.3.m3.1.1.3.5.2b.cmml">.</mtext></mpadded><mi mathvariant="normal" id="S2.p3.3.m3.1.1.3.5.3" xref="S2.p3.3.m3.1.1.3.5.3.cmml">s</mi></msup><mo id="S2.p3.3.m3.1.1.3.1c" xref="S2.p3.3.m3.1.1.3.1.cmml">⁢</mo><mn id="S2.p3.3.m3.1.1.3.6" xref="S2.p3.3.m3.1.1.3.6.cmml">919833</mn></mrow></mrow></math>, <math><mrow id="S2.p3.4.m4.1.1" xref="S2.p3.4.m4.1.1.cmml"><msub id="S2.p3.4.m4.1.1.2" xref="S2.p3.4.m4.1.1.2.cmml"><mi id="S2.p3.4.m4.1.1.2.2" xref="S2.p3.4.m4.1.1.2.2.cmml">δ</mi><mrow id="S2.p3.4.m4.1.1.2.3" xref="S2.p3.4.m4.1.1.2.3.cmml"><mi id="S2.p3.4.m4.1.1.2.3.2" xref="S2.p3.4.m4.1.1.2.3.2.cmml">J</mi><mo id="S2.p3.4.m4.1.1.2.3.1" xref="S2.p3.4.m4.1.1.2.3.1.cmml">⁢</mo><mn id="S2.p3.4.m4.1.1.2.3.3" xref="S2.p3.4.m4.1.1.2.3.3.cmml">2000.0</mn></mrow></msub><mo id="S2.p3.4.m4.1.1.1" xref="S2.p3.4.m4.1.1.1.cmml">=</mo><mrow id="S2.p3.4.m4.1.1.3" xref="S2.p3.4.m4.1.1.3.cmml"><msup id="S2.p3.4.m4.1.1.3.2" xref="S2.p3.4.m4.1.1.3.2.cmml"><mn id="S2.p3.4.m4.1.1.3.2.2" xref="S2.p3.4.m4.1.1.3.2.2.cmml">28</mn><mo id="S2.p3.4.m4.1.1.3.2.3" xref="S2.p3.4.m4.1.1.3.2.3.cmml">∘</mo></msup><mo id="S2.p3.4.m4.1.1.3.1" xref="S2.p3.4.m4.1.1.3.1.cmml">⁢</mo><msup id="S2.p3.4.m4.1.1.3.3" xref="S2.p3.4.m4.1.1.3.3.cmml"><mn id="S2.p3.4.m4.1.1.3.3.2" xref="S2.p3.4.m4.1.1.3.3.2.cmml">12</mn><mo id="S2.p3.4.m4.1.1.3.3.3" xref="S2.p3.4.m4.1.1.3.3.3.cmml">′</mo></msup><mo id="S2.p3.4.m4.1.1.3.1a" xref="S2.p3.4.m4.1.1.3.1.cmml">⁢</mo><mn id="S2.p3.4.m4.1.1.3.4" xref="S2.p3.4.m4.1.1.3.4.cmml">12</mn><mo id="S2.p3.4.m4.1.1.3.1b" xref="S2.p3.4.m4.1.1.3.1.cmml">⁢</mo><msup id="S2.p3.4.m4.1.1.3.5" xref="S2.p3.4.m4.1.1.3.5.cmml"><mpadded width="0.0pt" id="S2.p3.4.m4.1.1.3.5.2" xref="S2.p3.4.m4.1.1.3.5.2b.cmml"><mtext mathvariant="bold-sans-serif" id="S2.p3.4.m4.1.1.3.5.2a" xref="S2.p3.4.m4.1.1.3.5.2b.cmml">.</mtext></mpadded><mo id="S2.p3.4.m4.1.1.3.5.3" xref="S2.p3.4.m4.1.1.3.5.3.cmml">′′</mo></msup><mo id="S2.p3.4.m4.1.1.3.1c" xref="S2.p3.4.m4.1.1.3.1.cmml">⁢</mo><mn id="S2.p3.4.m4.1.1.3.6" xref="S2.p3.4.m4.1.1.3.6.cmml">19953</mn></mrow></mrow></math>, <math><mrow id="S2.p4.1.m1.1.1" xref="S2.p4.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.p4.1.m1.1.1.2" xref="S2.p4.1.m1.1.1.2.cmml">𝒜</mi><mo id="S2.p4.1.m1.1.1.1" xref="S2.p4.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p4.1.m1.1.1.3" xref="S2.p4.1.m1.1.1.3.cmml">ℐ</mi><mo id="S2.p4.1.m1.1.1.1a" xref="S2.p4.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p4.1.m1.1.1.4" xref="S2.p4.1.m1.1.1.4.cmml">𝒫</mi><mo id="S2.p4.1.m1.1.1.1b" xref="S2.p4.1.m1.1.1.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.p4.1.m1.1.1.5" xref="S2.p4.1.m1.1.1.5.cmml">𝒮</mi></mrow></math>, <math><mrow id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><mrow id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml"><mn id="S3.SS2.p1.3.m3.1.1.2.2" xref="S3.SS2.p1.3.m3.1.1.2.2.cmml">15</mn><mo id="S3.SS2.p1.3.m3.1.1.2.1" xref="S3.SS2.p1.3.m3.1.1.2.1.cmml">⁢</mo><msub id="S3.SS2.p1.3.m3.1.1.2.3" xref="S3.SS2.p1.3.m3.1.1.2.3.cmml"><mi id="S3.SS2.p1.3.m3.1.1.2.3.2" xref="S3.SS2.p1.3.m3.1.1.2.3.2.cmml">R</mi><mo id="S3.SS2.p1.3.m3.1.1.2.3.3" xref="S3.SS2.p1.3.m3.1.1.2.3.3.cmml">*</mo></msub></mrow><mo id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml">≤</mo><mi id="S3.SS2.p1.3.m3.1.1.4" xref="S3.SS2.p1.3.m3.1.1.4.cmml">H</mi><mo id="S3.SS2.p1.3.m3.1.1.5" xref="S3.SS2.p1.3.m3.1.1.5.cmml">≤</mo><mrow id="S3.SS2.p1.3.m3.1.1.6" xref="S3.SS2.p1.3.m3.1.1.6.cmml"><mn id="S3.SS2.p1.3.m3.1.1.6.2" xref="S3.SS2.p1.3.m3.1.1.6.2.cmml">26</mn><mo id="S3.SS2.p1.3.m3.1.1.6.1" xref="S3.SS2.p1.3.m3.1.1.6.1.cmml">⁢</mo><msub id="S3.SS2.p1.3.m3.1.1.6.3" xref="S3.SS2.p1.3.m3.1.1.6.3.cmml"><mi id="S3.SS2.p1.3.m3.1.1.6.3.2" xref="S3.SS2.p1.3.m3.1.1.6.3.2.cmml">R</mi><mo id="S3.SS2.p1.3.m3.1.1.6.3.3" xref="S3.SS2.p1.3.m3.1.1.6.3.3.cmml">*</mo></msub></mrow></mrow></math>, <math><mrow id="S3.SS2.p4.4.m4.1.1" xref="S3.SS2.p4.4.m4.1.1.cmml"><mi id="S3.SS2.p4.4.m4.1.1.3" xref="S3.SS2.p4.4.m4.1.1.3.cmml">θ</mi><mo id="S3.SS2.p4.4.m4.1.1.2" xref="S3.SS2.p4.4.m4.1.1.2.cmml">></mo><mrow id="S3.SS2.p4.4.m4.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.cmml"><mi id="S3.SS2.p4.4.m4.1.1.1.3" xref="S3.SS2.p4.4.m4.1.1.1.3.cmml">arcsin</mi><mo id="S3.SS2.p4.4.m4.1.1.1.2" xref="S3.SS2.p4.4.m4.1.1.1.2.cmml">⁢</mo><msup id="S3.SS2.p4.4.m4.1.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.1.cmml"><mrow id="S3.SS2.p4.4.m4.1.1.1.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.p4.4.m4.1.1.1.1.1.1.2" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.cmml"><msub id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.2" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.2.cmml"><mi id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.2.2" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.2.2.cmml">B</mi><mi id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.2.3" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.2.3.cmml">s</mi></msub><mo id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.1" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.1.cmml">/</mo><msub id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.3" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.3.cmml"><mi id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.3.2" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.3.2.cmml">B</mi><mi id="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.3.3" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.3.3.cmml">H</mi></msub></mrow><mo id="S3.SS2.p4.4.m4.1.1.1.1.1.1.3" xref="S3.SS2.p4.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS2.p4.4.m4.1.1.1.1.3" xref="S3.SS2.p4.4.m4.1.1.1.1.3.cmml"><mn id="S3.SS2.p4.4.m4.1.1.1.1.3.2" xref="S3.SS2.p4.4.m4.1.1.1.1.3.2.cmml">1</mn><mo id="S3.SS2.p4.4.m4.1.1.1.1.3.1" xref="S3.SS2.p4.4.m4.1.1.1.1.3.1.cmml">/</mo><mn id="S3.SS2.p4.4.m4.1.1.1.1.3.3" xref="S3.SS2.p4.4.m4.1.1.1.1.3.3.cmml">2</mn></mrow></msup></mrow></mrow></math>, <math><mrow id="S3.SS2.p4.7.m7.1.1" xref="S3.SS2.p4.7.m7.1.1.cmml"><mrow id="S3.SS2.p4.7.m7.1.1.2" xref="S3.SS2.p4.7.m7.1.1.2.cmml"><msub id="S3.SS2.p4.7.m7.1.1.2.2" xref="S3.SS2.p4.7.m7.1.1.2.2.cmml"><mi id="S3.SS2.p4.7.m7.1.1.2.2.2" xref="S3.SS2.p4.7.m7.1.1.2.2.2.cmml">R</mi><mo id="S3.SS2.p4.7.m7.1.1.2.2.3" xref="S3.SS2.p4.7.m7.1.1.2.2.3.cmml">*</mo></msub><mo id="S3.SS2.p4.7.m7.1.1.2.1" xref="S3.SS2.p4.7.m7.1.1.2.1.cmml">/</mo><mi mathvariant="normal" id="S3.SS2.p4.7.m7.1.1.2.3" xref="S3.SS2.p4.7.m7.1.1.2.3.cmml">c</mi></mrow><mo id="S3.SS2.p4.7.m7.1.1.1" xref="S3.SS2.p4.7.m7.1.1.1.cmml">≃</mo><mn id="S3.SS2.p4.7.m7.1.1.3" xref="S3.SS2.p4.7.m7.1.1.3.cmml">85</mn></mrow></math>

Correct Categorie: astro-ph

Guessed Categorie: astro-ph

Result: correct

Paper: https://arxiv.org/abs/quant-ph/0411120

Formulas:

1-

| F = 4, m_{F} = 4 ⟩

2-

| F = 3, m_{F} = 3 ⟩

3-

6 S_{1 / 2}

4-

ω_{at} / 2 π

5-

\partial_{x} ω_{at} / 2 π = 3.2

6-

ω_{at} / 2 π

7-

F = 4 \leftrightarrow F^{'} = 4

8-

F = 3 \leftrightarrow F^{'} = 4

9-

\begin{matrix} Ω_{R} (t) & = Ω_{\max} \sin^{2} (π t / t_{p}) \\ δ (t) & = δ_{c} + sign (t - t_{p} / 2) \cdot δ_{\max} \sqrt{1 - \sin^{4} (π t / t_{p})} \end{matrix}

10-

0 \leq t \leq t_{p}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0410313

Formulas:

1-

M = 1.05 M_{⊙}

2-

M_{env} = 0.4 \times 10^{- 4} M_{⊙}

3-

T_{eff} = 14000 K

4-

T_{eff} \sim (11 - 12) \times 10^{3} K

5-

T_{eff} \sim (21 - 24) \times 10^{3} K

6-

\sim 10^{- 4} M_{⊙}

7-

L \sim ⟨ \dot{M} ⟩ k_{B} T_{core} / m_{p}

8-

ω = N k_{h} / k_{r}

9-

k_{h}^{2} = l (l + 1) / r^{2}

10-

M = 1.05 M_{⊙}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/quant-ph/0505217

Formulas:

1-

{| Ψ ⟩}_{12} = \frac{1}{\sqrt{2}} ({| ↑ ⟩}_{1} {| ↓ ⟩}_{2} + {| ↓ ⟩}_{1} {| ↑ ⟩}_{2}) .

2-

{| Ψ ⟩}_{A_{1} A_{2}} = α {| B_{1} ⟩}_{A_{1}} {| B_{2} ⟩}_{A_{2}} + β {| B_{2} ⟩}_{A_{1}} {| B_{1} ⟩}_{A_{2}},

3-

{| Ψ ⟩}_{A_{1} A_{2}}

4-

{| Ψ ⟩}_{A_{1} A_{2}} = (α c_{A_{1}, B_{1}}^{†} c_{A_{2}, B_{2}}^{†} + β c_{A_{1}, B_{2}}^{†} c_{A_{2}, B_{1}}^{†}) | 0 ⟩,

5-

c_{A_{i}, B_{j}}^{†}

6-

| A_{i}, B_{j} ⟩

7-

{| Ψ ⟩}_{A_{1} A_{2}}

8-

(α c_{A_{1}, B_{1}}^{†} c_{A_{2}, B_{2}}^{†} \pm β c_{A_{2}, B_{1}}^{†} c_{A_{1}, B_{2}}^{†}) | 0 ⟩

9-

α {| A_{1} ⟩}_{B_{1}} {| A_{2} ⟩}_{B_{2}} \pm β {| A_{2} ⟩}_{B 1} {| A_{1} ⟩}_{B 2},

10-

{| Ψ ⟩}_{A_{1} A_{2}}

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: cond-mat

Result: incorrect

Paper: https://arxiv.org/abs/2002.07360

Formulas:

1-

{}^{13}C (α, n)

2-

{}^{13}C (α, n)

3-

{}^{13}C (α, n)

4-

{}^{13}C (α, n)

5-

{}^{13}C (α, n)

6-

{}^{13}C (α, n)

7-

{}^{13}C (α, n)

8-

{}^{13}C (α, n)

9-

{}^{13}C (α, n)

10-

{}^{13}C (α, n)

Formulas (html):

Correct Categorie: nucl-ex

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1612.08729

Formulas:

1-

n = 6.3 \times 10^{11}

2-

n = 6.1 \times 10^{11}

3-

n = 5.5 \times 10^{12}

4-

n_{p} = n_{0} + C e^{- b / T}

5-

σ (n) = A {(n - n_{p})}^{p}

6-

n_{p} = n_{0} + C e^{- b / T}

7-

n_{p} = 0.83 + 1.46 e^{- 2.25 / T}

8-

n_{p} = 0.95 + 0.88 e^{- 3.00 / T}

9-

V_{t h} = 0.14

10-

n_{c o n f}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/cond-mat/0412126

Formulas:

1-

τ = \frac{ℏ}{m α^{2} c^{2}}

2-

\frac{1}{v^{2}} \frac{\partial^{2} T}{\partial t^{2}} + \frac{m_{e} γ}{ℏ} \frac{\partial T}{\partial t} + \frac{2 V m_{e} γ}{ℏ^{2}} T - \frac{\partial^{2} T}{\partial x^{2}} = F (x, t) .

3-

γ = {(1 - α^{2})}^{- \frac{1}{2}}

4-

F (x, t)

5-

T (x, t) = e^{- \frac{t}{2 τ}} u (x, t),

6-

\frac{1}{v^{2}} \frac{\partial^{2} u}{\partial t^{2}} - \frac{\partial^{2} u}{\partial x^{2}} + q u (x, t) = e^{\frac{t}{2 τ}} F (x, t)

7-

q = \frac{2 V m}{ℏ^{2}} - {(\frac{m v}{2 ℏ})}^{2}

8-

m = m_{e} γ .

9-

F (x, t) \to 0

10-

\frac{1}{v^{2}} \frac{\partial^{2} u}{\partial t^{2}} - \frac{\partial^{2} u}{\partial x^{2}} - {(\frac{m v}{2 ℏ})}^{2} u (x, t) = 0 .

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/1107.3397

Formulas:

1-

L_{s u n}

2-

σ_{m} - ϵ_{m} > σ_{i} + ϵ_{i}

3-

{(ϵ_{s t}^{2} + ϵ_{i}^{2})}^{0.5}

4-

{(σ_{m}^{2} - σ_{i}^{2})}^{0.5}

5-

l o g (M_{BH}) = α l o g (σ) + β l o g (L) + γ,

6-

l o g (M_{BH}) = 2 l o g (σ) + l o g (R) + l o g (f) - l o g (G),

7-

σ^{2} R G^{- 1} + a^{'} L

8-

σ_{[O III]}

9-

σ_{[O III]}

10-

σ_{[O III]}

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: hep-ph

Result: incorrect

Paper: https://arxiv.org/abs/1412.3471

Formulas:

1-

V (r) = D / r^{3}

2-

\hat{H} = - \frac{1}{2} \sum_{i = 1}^{N} \nabla_{i}^{2} + \sum_{i < j} \frac{1}{{| 𝐫_{i} - 𝐫_{j} |}^{3}}

3-

a \equiv m D / ℏ^{2}

4-

ϵ_{0} \equiv ℏ^{2} / m a^{2} = D / a^{3}

5-

V (r) = D / r^{3}

6-

R = d \exp (\frac{γ_{b}}{γ_{d}})

7-

γ_{d} = ϵ_{0} a^{3} {(ρ_{S} - ρ_{L})}^{2}

8-

Ψ (Λ) = \exp (- Λ \hat{H}) Ψ_{T}

9-

Ψ_{T} = \exp [- \sum_{i < j} u (r_{i j})] \times \exp [- α \sum_{i = 1}^{N} ({| 𝐫_{i} - 𝐛_{i} |}^{2})]

10-

𝐛_{1}, \dots 𝐛_{N}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/0810.0207

Formulas:

1-

(e^{+} - H_{2})

2-

(e^{+} - H_{2})

3-

(e^{+} - H_{2})

4-

(λ_{j}, μ_{j}, ϕ_{j})

5-

j \in {1, 2, 3}

6-

Ψ_{t} = (S + a_{t} T + p_{0} χ_{0}) ψ_{G} + \sum_{i = 1}^{M} p_{i} χ_{i},

7-

T = S + ⅈ C,

8-

S = \frac{N}{λ_{3} - 1} \sin [c (λ_{3} - 1)]

9-

C = \frac{N}{λ_{3} - 1} \cos [c (λ_{3} - 1)] {1 - \exp [- γ (λ_{3} - 1)]} .

10-

c = k R / 2

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1008.3496

Formulas:

1-

α_{s} (Q_{s}^{2})

2-

Q_{s} \sim 1 - 2

3-

\frac{1}{2} ⟨ 𝐩_{T}^{2} ⟩ - ⟨ p_{L}^{2} ⟩ \approx \frac{4 π^{2}}{3} \frac{η}{s} \frac{T}{τ} .

4-

η = \frac{1}{3} n \bar{p} λ_{f},

5-

Δ p = g Q^{a} B^{a} r_{m} .

6-

λ_{f} = r_{m} ⟨ {\bar{p}}^{2} / {(Δ p)}^{2} ⟩ = \frac{{\bar{p}}^{2}}{g^{2} Q^{2} ⟨ B^{2} ⟩ r_{m}} .

7-

η_{A} = \frac{n {\bar{p}}^{3}}{3 g^{2} Q^{2} ⟨ B^{2} ⟩ r_{m}} \approx \frac{\frac{9}{4} s T^{3}}{g^{2} Q^{2} ⟨ B^{2} ⟩ r_{m}},

8-

\hat{q} = \frac{⟨ {(Δ p_{⟂} (L))}^{2} ⟩}{L} .

9-

{\hat{q}}_{A} = \frac{⟨ {(Δ p_{⟂})}^{2} ⟩}{r_{m}} = g^{2} Q^{2} ⟨ B^{2} ⟩ r_{m} .

10-

\frac{η_{A}}{s} \approx c \frac{T^{3}}{{\hat{q}}_{A}},

Formulas (html):

Correct Categorie: nucl-th

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1810.09095

Formulas:

1-

\hat{D} (α)

2-

n_{S} = N_{S} / M ≫ 1

3-

δ α_{η = 1}^{E} ≃ 1 / (4 M \sqrt{n_{S}})

4-

δ α_{η = 1}^{P} ≃ 1 / (4 \sqrt{M n_{S}})

5-

\hat{D} (α)

6-

{\hat{ρ}}_{in}^{(i)}

7-

{\hat{ρ}}_{in}^{(i)}

8-

{\hat{ρ}}_{in}^{(i)} = | α_{i} ⟩ ⟨ α_{i} |

9-

| α_{i} ⟩ = | α / \sqrt{N} ⟩ ≃ | 0 ⟩ + α_{i} | 1 ⟩

10-

{| α_{i} |}^{2} ≪ 1

Formulas (html):

Correct Categorie: quant-ph

Guessed Categorie: physics

Result: incorrect

Paper: https://arxiv.org/abs/1304.5461

Formulas:

1-

⟨ {\hat{L}}_{z} ⟩ = ℏ

2-

⟨ {\hat{L}}_{z} ⟩ = - ℏ

3-

[- \frac{ℏ^{2}}{2 m} \nabla^{2} - e U (x, y, z)] ψ (x, y, z) = E ψ (x, y, z),

4-

ψ (x, y, z)

5-

ϕ (x, y, z)

6-

ψ (x, y, z) = ϕ (x, y, z) \exp (\frac{2 π i z}{λ}),

7-

E = h^{2} / 2 m λ^{2}

8-

k_{x}^{2} + k_{y}^{2} + k_{z}^{2} = \frac{1}{λ^{2}} .

9-

\frac{\partial ϕ (x, y, z)}{\partial z} = [\frac{i λ}{4 π} \nabla_{x y}^{2} + i σ U (x, y, z)] ϕ (x, y, z),

10-

σ = 2 π m e λ / h^{2}

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: hep-th

Result: incorrect

Paper: https://arxiv.org/abs/0807.4094

Formulas:

1-

l_{f} = 100 μ

2-

w_{f} = 3.3 μ

3-

s_{f} = 3.3 μ

4-

20 \log_{10} | S_{21} |

5-

w_{g} = 10 μ m

6-

f_{0} = \frac{c}{\sqrt{ϵ_{eff}}} \frac{1}{2 l} .

7-

c / \sqrt{ϵ_{eff}} = v_{ph}

8-

2 l = λ_{0}

9-

v_{ph} = 1 / \sqrt{L_{ℓ} C_{ℓ}}

10-

\frac{μ_{0}}{4} \frac{K (k_{0}^{'})}{K (k_{0})},

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: astro-ph

Result: incorrect

Paper: https://arxiv.org/abs/1304.3319

Formulas:

1-

s (s_{k} / t_{k})

2-

\sum_{j = 0}^{\infty} \frac{1}{b^{2^{j}}}, b \geq 3 .

3-

s (s_{k} / t_{k})

4-

(m, n) = 1

5-

s (m / n)

6-

s (m / n) = \sum_{k = 1}^{n} ((k / n)) ((m k / n))

7-

S (m / n) = 12 s (m / n)

8-

S (s_{k} / t_{k})

9-

S (s_{k} / t_{k})

10-

x (b) = \sum_{j = 0}^{\infty} \frac{1}{b^{2^{j}}}, b \geq 3 .

Formulas (html):

Correct Categorie: math

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1702.08442

Formulas:

1-

𝕋^{2} = ℝ^{2} / ℤ^{2}

2-

| 𝒫^{'} - 𝒫 | < ϵ

3-

δ (𝒫^{'}) \leq δ (𝒫)

4-

δ (𝒫) = \sum_{i} A (D_{i}) / A (𝕋^{2})

5-

𝐫 = (r_{1}, \dots, r_{n})

6-

𝒫 = (D_{1}, \dots, D_{n})

7-

t 𝐫, t \geq 1

8-

𝒫 (t_{1})

9-

𝒫 (t_{1})

10-

𝐩^{'} = (𝐩_{1}^{'}, \dots, 𝐩_{n}^{'})

Formulas (html):

Correct Categorie: math

Guessed Categorie: hep-lat

Result: incorrect

Paper: https://arxiv.org/abs/astro-ph/0503607

Formulas:

1-

3 - 4 r_{eff}

2-

⟨ Fe ⟩ = 0.5 (Fe5270 + Fe5335)

3-

{[MgFe]}^{'} = \sqrt{Mg b (0.72 \times Fe5270 + 0.28 \times Fe5335)}

4-

[Z / H] \sim 0.2

5-

[Z / H] \sim - 0.2

6-

[Z / H] \sim - 0.5

7-

E (B - V) = 0.03

8-

[Z / H] = - 0.1

9-

⟨ Fe ⟩ = 0.5 (Fe5270 + Fe5335)

10-

⟨ Fe ⟩ = 0.5 (Fe5270 + Fe5335)

Formulas (html):

Correct Categorie: astro-ph

Guessed Categorie: math

Result: incorrect

Paper: https://arxiv.org/abs/hep-th/9604110

Formulas:

1-

S = \int d^{3} ζ (\frac{1}{2} \sqrt{- g} g^{i j} E_{i}^{a} E_{j}^{b} η_{a b} - \frac{1}{6} ϵ^{i j k} E_{i}^{A} E_{j}^{B} E_{k}^{C} A_{C B A} - \frac{1}{2} \sqrt{- g}) .

2-

A_{C B A}

3-

S = \int d^{3} ζ (\frac{1}{2} \sqrt{- g} g^{i j} \partial_{i} x^{m} \partial_{j} x^{n} G_{m n} - \frac{1}{6} ϵ^{i j k} \partial_{i} x^{m} \partial_{j} x^{n} \partial_{k} x^{p} A_{p n m} - \frac{1}{2} \sqrt{- g}) .

4-

x^{m}, m = 1, \dots, 11

5-

x^{I} = x^{I} + 2 π R^{I}, I = 10, 11

6-

R^{11} R^{10} = 2 .

7-

z^{i} i = 1, 2

8-

g^{i j} = ϕ^{- 2 / 3} (\begin{matrix} {\hat{g}}^{i j} + ϕ V_{i} V_{j} & ϕ^{2} V_{i} \\ ϕ^{2} V_{j} & ϕ^{2} \end{matrix}) .

9-

S = \int d^{2} z \sqrt{- \hat{g}} {\hat{g}}^{i j} \partial_{i} {\hat{x}}^{m} \partial_{j} {\hat{x}}^{n} {\hat{G}}_{m n} + \frac{1}{2} ϵ^{i j} \partial_{i} {\hat{x}}^{m} \partial_{j} {\hat{x}}^{n} {\hat{A}}_{n m}, i, j = 1, 2 .

10-

{\hat{x}}^{m}, m = 1, \dots, 10

Formulas (html):

Correct Categorie: hep-th

Guessed Categorie: cs

Result: incorrect

Paper: https://arxiv.org/abs/1705.02765

Formulas:

1-

l_{p} / D ≫ 1

2-

R ≃ l_{p} {(L / l_{p})}^{1 / 2}

3-

l_{p} / a \sim 1

4-

l_{p} / a ≫ 1

5-

l_{p} / D ≪ 1

6-

l_{p} / D ≫ 1

7-

U^{L J} (r) = {\begin{matrix} 4 ϵ [{\frac{σ}{r}}^{12} - {\frac{σ}{r}}^{6} + \frac{1}{4}] & r < 2^{1 / 6} σ \\ 0 & r > 2^{1 / 6} σ \end{matrix},

8-

U^{s} (r) = - \frac{1}{2} k_{s} r_{0}^{2} \ln [1 - {(\frac{r}{r_{0}})}^{2}] .

9-

U^{b} (θ) = \frac{κ}{σ} (1 - \cos θ),

10-

l_{p} = κ / k_{B} T

Formulas (html):

Correct Categorie: cond-mat

Guessed Categorie: gr-qc

Result: incorrect

Run 6249155 (TestAgent)