harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
s(c,d)={1\over 4d}\sum_{n=1}^{\vert d\vert -1}\cot\left ({\pi n\over d}\right )\cot\left ({\pi nc\over d}\right )\ \ \ \ \ (c\in{\mathbb Z},\d\in{\mathbb Z}\setminus\{0\},\ \gcd (c,d)=1),
\tilde r^2_{\varepsilon,\sigma}\leq \inf_{D \in \mathbb{N}}\left[\tilde C(\alpha,\beta) \varepsilon^2 \sqrt{\sum_{j=1}^{D \wedge M_1}b_j^{-4}}+ (7+4\sqrt{x_{\alpha/2}})\ \left[\sigma^2 \ln^{3/2}(1/\sigma) \vee a^{-2}_{D \wedge M_0}\right] \right].
u_{1}=\frac{u_{1}^{\prime}+v}{1+u_{1}^{\prime}v/c^{2}},u_{2}=\frac{u_{2}^{\prime}}{\gamma(1+u_{1}^{\prime}v/c^{2})},u_{3}=\frac{u_{3}^{\prime}}{\gamma(1+u_{1}^{\prime}v/c^{2})}.
g(\omega) = \frac{1}{2\sqrt{2\pi\sigma_g}}e^{-\frac{(\omega - \mu)^2}{2\sigma_g}}+ \frac{1}{2\sqrt{2\pi\sigma_g}}e^{-\frac{(\omega + \mu)^2}{2\sigma_g}}\mbox{with}\mu = \frac{\sqrt{2}+ 1}{4\sqrt{2}}\sigma_g = 0.001.
\mathbb{E}(L_{n}) = n \left(\frac{3+\sqrt{5}}{2 \sqrt{5}}\binom{n+\frac{\sqrt{5}}{2}-\frac{3}{2}}{n}- \frac{3-\sqrt{5}}{2 \sqrt{5}}\binom{n-\frac{\sqrt{5}}{2}- \frac{3}{2}}{n}\right) \sim \frac{1+\sqrt{5}}{2 \sqrt{5}}\frac{n^{\frac{\sqrt{5}-1}{2}}}{\Gamma(\frac{\sqrt{5}-1}{2})}.
q_{\alpha, \beta}(t, x, y)=\left(1\wedge\frac{\delta_E(x)}{t^{1/\alpha}}\right)^\beta\left(1\wedge\frac{\delta_E(y)}{t^{1/\alpha}}\right)^\beta\left(t^{-d/\alpha}\wedge\frac{t}{|x-y|^{d+\alpha}}\right).
(\nabla v)^{T}=[\begin{matrix}\partial_{1}v_{1}&\partial_{2}v_{1}&\partial_{3}v_{1}\\ \partial_{1}v_{2}&\partial_{2}v_{2}&\partial_{3}v_{2}\\ \partial_{1}v_{3}&\partial_{2}v_{3}&\partial_{3}v_{3}\end{matrix}]=J.
\frac{( 2 \pi )^{D / 2}}{G N \sqrt{2}}= \frac{\Lambda^{D - 2}( 1 - ( e H / \Lambda^{2})^{D / 2 - 1})}{D / 2 - 1}+ \frac{( e H )^{( D - 1 ) / 2}}{\rho}\int_{\rho^{2}/ e H}^{\infty}\frac{d s}{s^{1 / 2}}e^{- s}.
{\cal R}_{a b c}^{0}= \partial_{b}\Omega_{a c}^{0}- \partial_{c}\Omega_{a b}^{0}+ \Omega_{0 b}^{0}\Omega_{a c}^{0}+ \Omega_{d b}^{0}\Omega_{a c}^{d}- \Omega_{0 c}^{0}\Omega_{a b}^{0}- \Omega_{d c}^{0}\Omega_{a b}^{d},
\left(\int_{x_0}^{1/2}|g(x)|^{L_0}dx\right)^{1/L_0}\leq (\Gamma(L_0+1)\exp(-C_5^*L_0))^{1/L_0}+\|H(x)\|_{L_0}+\|T^{n+1}\mathcal{W}(x)\|_{L_0}&=(\Gamma(L_0+1)\exp(-C_5^*L_0))^{1/L_0}\left(1+O\left(\frac{1}{\Gamma(L_0+1)^{1/L_0}}\right)\right).
\sigma^{2}( \sqrt{s}y , \sqrt{s}y^{\prime}) = s \left( ( y - y^{\prime})^{2}- \frac s 3 R_{\mu \lambda \nu \rho}( p_{0}) ( y - y^{\prime})^{\mu}( y - y^{\prime})^{\nu}y^{\lambda}y^{\rho}+ O ( s^{3 / 2}) \right) ,
T_m\sim_{c_1,c_2}\begin{cases}\frac{m}{\sqrt{q}+m\sqrt{|\xi|}},\ \ \ \ -m|\xi|-\sqrt{q|\xi|}\leq a'\leq \sqrt{q|\xi|}, \sqrt{q}\left(\frac{m|\xi|}{a'(m|\xi|+a')}+\frac{|\gamma|}{\sqrt{a'(m|\xi|+a')}}\right),\ \ .\end{cases}
R_0\leq\min\{I(V_0;Y_1),I(V_0;Y_2)\}, &R_0+R_i\leq\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_i;Y_i|V_0),i=1,2, &R_0+R_1+R_2\leq\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_1;Y_1|V_0)+I(X;Y_2|V_0,V_1), &R_0+R_1+R_2\leq\min\{I(V_0;Y_1),I(V_0;Y_2)\}+I(V_2;Y_2|V_0)+I(X;Y_1|V_0,V_2)
\sum_{n\ge0}^{\infty}(z)_n(y)_n(aq/zy)^n\beta_n=\frac{(aq/z)_{\infty}(aq/y)_{\infty}}{(aq)_{\infty}(aq/zy)_{\infty}}\sum_{n\ge0}^{\infty}\frac{(z)_n(y)_n(aq/zy)^n\alpha_n}{(aq/z)_n(aq/y)_n}.
\begin{aligned}\|\sqrt{\rho_{n}}\partial_{x}p_{n}(\rho_{n})\|_{L^{\infty}_{t}L^{2}_{x}}^{2}= \|\sqrt{\rho_{n}}(w_{n}-u_{n})\|_{L^{\infty}_{t}L^{2}_{x}}^{2}\le \left( \|\sqrt{\rho_{n}}w_{n}\|_{L^{\infty}_{t}L^{2}_{x}}^{2}+ \|\sqrt{\rho_{n}}u_{n}\|_{L^{\infty}_{t}L^{2}_{x}}^{2}\right) begin{align*}1ex] \le (T+3)E_{n}, \end{aligned}
\frac{d}{dt}V\left(\mathbf{x}\left(t\right),u_\mathrm{ss}\right) =\dot{\mathbf{x}}^\mathrm{T}\left(t\right)\nabla_xV\left(\mathbf{x}\left(t\right),u_\mathrm{ss}\right) &=-\left\|\nabla_xV\left(\mathbf{x}\left(t\right),u_\mathrm{ss}\right)\right\|^2 &\leqslant0,
\mathcal{I}_{\mathrm{D}}(\delta) = \inf_{\lambda \in \mathbb{R}_{+}^2}\sup_{\pi \in \Pi\left(\mu_1, \mu_2\right)}\left[\langle\lambda, \delta\rangle+\int_{\mathcal{V}}g_\lambda \, d \pi\right] = \sup_{\pi \in \Pi\left(\mu_1, \mu_2\right)}\inf_{\lambda \in \mathbb{R}_{+}^2}\left[\langle\lambda, \delta\rangle+\int_{\mathcal{V}}g_\lambda \, d \pi\right] .
Z_0' =\{(2,1),(1,8), (-3, 15) \}, Z_0'' =\{(2,8), (1, 15)\}, Z_0''' = \{(-3, 1)\}, Z_1' =\{(6,2), (4,14), (0, 17) \}, Z_1'' =\{(6,14), (4,17)\}, Z_1''' = \{(0, 2)\}, Z_2' =\{(9,10), (8,13), (5, 16), (3,19) \}, Z_2'' =\{(9,13), (8,16), (5,19)\}, Z_2''' = \{(3, 10)\}.
d_{n}^{(2)}(\cos\theta,\cos\varphi|q)\allowbreak =\allowbreak\sum_{m=0}^{n}\genfrac{[}{]}{0pt}{}{n}{m}_{q}b_{m}(\cos(\theta+\varphi)|q)b_{n-m}(\cos(\theta-\varphi)|q), f_{n}^{(2)}(\cos\theta,\cos\varphi|q) =\sum_{m=0}^{n}\genfrac{[}{]}{0pt}{}{n}{m}_{q}h_{m}(\cos(\theta+\varphi)|q)h_{n-m}(\cos(\theta-\varphi)|q).
\bar{u}^{\varepsilon,N}_{t}(x)=\frac{1}{N}\sum_{i=1}^N K_{\varepsilon}(x-\bar \xi^{i}_{t}) \bar{V}_t\big (\bar{\xi}^{i},\bar{u}^{\varepsilon,N}(\bar{\xi}^{i}),\nabla \bar u^{\varepsilon,N}(\bar{\xi}^{i}) \big ) ,
q_\eta({\bf{x}}) := p_\theta^{(\alpha)}({\bf{x}}) =\widetilde{N}_{\eta,\alpha}[ 1+ b_\alpha({\bf{x}}- \boldsymbol{\mu})^\top \boldsymbol{\Sigma}^{-1}({\bf{x}}- \boldsymbol{\mu})]_+^{\frac{\alpha}{\alpha-1}},
F_1(B,V) = (1+G_0\, T(B,V))^{-1}\, G_1 -(1+ G_0 \, T(B,V) )^{-1}\, G_1\, T(B,V)\, (1+G_0\, T(B,V))^{-1}\, G_0 & = (1+G_0\, T(B,V))^{-1}\, G_1(1+ T(B,V)\, G_0)^{-1}\big [1+T(B,V)\, G_0 & \qquad\qquad\qquad\qquad\qquad- (1+ T(B,V)\, G_0)\, T(B,V)\, (1+G_0\, T(B,V))^{-1}\, G_0 \big]
- 4 \left( \stackrel{\circ}{P}_{A B C D , a}+ \frac{3}{2}\stackrel{\circ}{P}_{[ C D , a}C_{A B ]}\right) C^{D L}\gamma^{a}\gamma^{0}- 3 \stackrel{\circ}{T}_{[ A B}\delta_{C ]}^{L}\gamma^{a b}= 0 \, .
\mathbb E|X_t^i|^2 = \mathbb E|X_0^i|^2 + 2\int_0^t X_s^i\cdot b^i(X_s)\d s + d \sigma^2 t & = \mathbb E|X_0^i|^2 + 2\int_0^t X_s^i\cdot \big(b(X_s^i) + \gamma^i(X_s)\big)\d s + d\sigma^2 t.
\boldsymbol{E}[\sum_{i=1}^{T-1}\sum_{j=i+1}^T f(j-i) X_i X_j] \leq \boldsymbol{E}[\sum_{i=1}^{T-1}\sum_{j=i+1}^T f(j-i) X_i] &\leq \left(\boldsymbol{E}[\sum_{i=1}^{T-1}X_i^2]\boldsymbol{E}[\sum_{i=1}^{T-1}\left(\sum_{j=i}^T f(j-i)\right)^2 ] \right)^\frac{1}{2}.
H_{1}= 1 + \sum_{i}{\frac{Q_{i}}{[ | \vec{x}- \vec{x}_{0 \, i}|^{2}+{\frac{4 Q}{( p - 3 )^{2}}}| \vec{z}- \vec{z}_{0}|^{p - 3}]^{\frac{2}{p - 3}}}}, \ \ \H_{2}={\frac{Q}{| \vec{z}- \vec{z}_{0}|^{5 - p}}}.
f_x^0(m_i^x) w_{1,2}\hat{u}_{0,x}= f^0_0 (m_i^x w_{1,2}\hat{u}_{0,x}) + f^1(d_e(w_{1,2}\hat{u}_{0,x})) (m_i^x); \sum_{t=1}^{d_x}n_t^x h_{t,i}^x w_{1,2}\hat{u}_{0,x}= f^0_0 \left( \sum_{j=1}^{d_0}m_j^0 M_{j,i}^x \right) + f^1(d_e(w_{1,2}\hat{u}_{0,x})) (m_i^x);
\liminf_{j\rightarrow+\infty}\int_{\Pi^{\xi_{1}}}\mathcal{F}_{\varepsilon_{j}}^{\xi_{1}}((u_{j})_{z}^{\xi_{1}},\Omega_{z}^{\xi_{1}})~d\mathcal{H}^{n-1}(z)=:M_{1}<+\infty~,
\left( \left( \left( b_{n + 1}[ d_{n}> < d_{n}] \right) + \left( b_{n + 1}[ d^{\prime \prime}> < d^{\prime \prime}] \right) \right) \; \left/ \bigoplus_{d^{\prime}\in R_{d}^{\prime}}b_{n + 1}[ d_{n}^{\prime}> < d_{n}^{\prime}] \right. \right)
A(\lambda):=\int_0^\infty\{t\}\{\lambda t\}\frac{dt}{t^2}\:, &F(x):=\frac{x+1}{2}A(1)-A(x)-\frac{x}{2}\log x\:, &H(x):=2\sum_{j\geq 0}(-1)^j\beta_{j-1}(x)F(\alpha_j(x))\:, &B_1(t):=t-\lfloor t\rfloor -1/2,\ \:, &B_2(t):=\{t\}^2-\{t\}+1/6,\ (t\in\mathbb{R}) \ \:.
\begin{gathered}\prod_{\alpha \in \Delta^+_\gamma}H_\alpha \left( \xi_{-s(\Lambda(\sigma) + \delta_M)- i e_1 \nu}\right)= \left\lbrace \prod_{\alpha \in \Delta^+_\gamma}\langle-s(\Lambda + \delta_M) - i \nu e_1, \alpha\rangle \right\rbrace \left( \xi_{-s(\Lambda(\sigma) + \delta_M)- i e_1 \nu}\right).\end{gathered}
\sum_{i=1}^{3}a_i z_i^{2}+ \sum_{1 \leq j < k \leq 3}b_{jk}z_j z_k + \sum_{i=1}^{3}c_i z_i = (K-t \eta_{e})^{2}- r^{2}t^{2}|P_{V}(e)|^{2},
+3c_{2}^{1}a_{1}^{2}a_{2}^{2}+c_{1}^{0}a_{1}^{2}a_{1}^{1}+c_{1}^{1}a_{2}^{1}a_{2}^{3}+c_{1}^{0}a_{2}^{1}a_{2}^{4}-c_{2}^{0}a_{1}^{2}a_{2}^{3}+c_{2}^{0}a_{1}^{1}a_{2}^{4}-3a_{1}^{0}c_{2}^{2}a_{2}^{3}
\mathbf{P}(Y_{t}\le a\mid\mathcal{F}_{u}) =\mathbf{P}\left(Y_{v}-(W_{t}-W_{v})\le aY_{t}^{(v)}\le a\,\middle|\,\mathcal{F}_{v}\right)=\mathbf{P}\left(Y_{v}-(W_{t}-W_{v})\le aY_{t}^{(v)}\le a\,\middle|\:Y_{v}\right).
\frac{g_1^{(j)}(k)}{(-\|x\|^2 /2)^j}\quad\leq\left(1-\frac{k\|x\|^2}{d}\right)^{-(p+k+1+2j)/2}\quad\leq\left(1-\frac{kM^2}{d}\right)^{-(p+3k+1)}=\exp\left[ -(p+3k+1)\log \left( 1 -\frac{kM^2}{d}\right)\right] \quad\leq\exp\left[ (p+3k+1)\frac{4}{3}\frac{kM^2}{d}\right] & \leq \exp\left[ \frac{pM^2}{d}\gamma_1(k)\right]
- f^{a}{}_{cb}C^{c}{}_{d}{}^{\nu}A^{\prime b}{}_{\mu}A^{\prime d \mu}+ \left[-\partial_{\mu}C^{a}{}_{d}{}^{\mu}+ C^{a}{}_{c\mu}C^{c}{}_{d}{}^{\mu}\right] A^{\prime d \nu}.
\int_{\Omega}\mathcal{A}(x,\nabla u_n)\cdot\nabla(\phi^{p}u_n)\,dx = p\,\int_{\Omega}\phi^{p-1}\,u_n\,\mathcal{A}(x,\nabla u_n)\cdot\nabla\phi\,dx +\int_{\Omega}\phi^{p}\,\mathcal{A}(x,\nabla u_n)\cdot\nabla u_n\,dx,
u\cdot (v\cdot (x,\xi))-v\cdot (u\cdot (x,\xi)) =([u,[v,x]], t\mapsto t \cdot (u\cdot \xi(v)) )- ([v,[u,x]], t\mapsto t \cdot (v\cdot \xi(u)) ) &=([u,[v,x]]-[v,[u,x]], t\mapsto (t \cdot (u\cdot \xi(v))- t \cdot (v\cdot \xi(u))) &=([[u,v],x], t\mapsto \xi([u,v])
\dot{\phi}^N_i= \omega_i^N(t)+\frac{1}{N}\sum_{j=1}^NW^N_{ij}g(\phi^N_j-\phi^N_i),0<t\le T^*, \dot{W}^N_{ij}= -\varepsilon(W^N_{ij}+h(\phi_j^N-\phi^N_i)),0<t\le T^* \phi^N_i(0)= \varphi^N_{i},W^N_{ij}(0)=W^N_{i,j,0},i,j=1,\ldots,N,
\psi_{K}= \prod_{I < J}\prod_{i \leq j}( z_{i}^{I}- z_{j}^{J})^{K_{I , J}}\prod_{I}\prod_{i < j}( z_{i}^{I}- z_{j}^{I})^{K_{I , I}}\exp ( -{\frac{1}{2}}( \sum_{i , I}\mid \! z_{i}^{I}\! \mid^{2}) \, ,
\rho_{r}( t_{*}) \simeq \rho_{_\mathrm{E W}}\left({\frac{a_{_\mathrm{E W}}}{a ( t_{*})}}\right)^{3 ( 1 + \omega )}= 1 0^{- 6 8}M_{p}^{4}\times 1 0^{4 1 ( 3 ( 1 + \omega ) )}\ .
L(x) := \mathbb{P}(W_n + V_{\ell_{n+1}}- \tau_n \leq x, W_n \geq D_n, W_n - \tau_n \leq 0,N_{n+1}>0,0 \leq \ell_{n+1}< n) = \int_{0}^{\infty}dA(t) \int_{0}^{t}dF_n(u)\mathbb{P}(V_{\ell_{n+1}}\leq x+t-u, u \geq D_n,N_{n+1}>0,0 \leq \ell_{n+1}< n).
a\ast_{\hbar}b(z)=\left( \tfrac{1}{4\pi\hbar}\right)^{2n}\int_{\mathbb{R}^{4n}}e^{-\frac{2i}{\hbar}\partial\sigma(z,z^{\prime},z^{\prime\prime})}a(z^{\prime})b(z^{\prime\prime})dz^{\prime}dz^{\prime\prime};
{\bf{Cov}}\left[V_{\varepsilon}(\psi), \,V_{\varepsilon}(\xi)\right] = \begin{cases}-2 \, \log|e^{2\pi i\psi}-e^{2\pi i\xi}|, \, |\xi-\psi|> \varepsilon, -2\log\varepsilon, \psi=\xi,\end{cases}
= \; \frac{1}{\pi^2}\int d^2p \frac{1}{p^2 + m_d^2}\; \frac{1}{p^2}\;\sin^2(p_1L) \;\bigg[ \frac{p_2}{p_1}e^{ip_2 L}-\frac{p_1}{p_2}\Big( 1 - e^{ip_2 L}\Big) \bigg] \; = \; 0 \; .
r ( n , - 2 t ) \sim{\frac{1 ! 2 ! \cdots ( n - 1 ) !}{\pi^{n / 2}\, 2^{( 5 n^{2}+ n ) / 2}}}t^{- 3 n^{2}/ 4}\exp \left({\frac{8}{3}}n t^{3 / 2}\right) \, .
\sum_{n=0}^{\infty}p_{-7}\left(5^{2j-1}+\dfrac{13\times5^{2j-1}+7}{24}\right)q^{n}=\sum_{l=1}^{\infty}c(2j-1,l)q^{l-1}\dfrac{E_{5}^{6l-1}}{E_{1}^{6l+6}}, \sum_{n=0}^{\infty}p_{-7}\left(5^{2j}+\dfrac{17\times5^{2j}+7}{24}\right)q^{n}=\sum_{l=1}^{\infty}c(2j,l)q^{l-1}\dfrac{E_{5}^{6l}}{E_{1}^{6l+7}}.
\zeta(k_1,k_2)\zeta(\ell)= \sum_{\substack{i_1\ge1,i_2\ge2 i_1+i_2=k_2+\ell}}\binom{i_2-1}{\ell-1}\zeta(k_1,i_1,i_2) +\sum_{\substack{i_1,i_2\ge1,i_3\ge2 i_1+i_2+i_3=k_1+k_2+\ell}}\binom{i_3-1}{k_2-1}\left[\binom{i_2-1}{k_1-i_1}+\binom{i_2-1}{k_1-1}\right]\zeta(i_1,i_2,i_3).
\theta^{(4m+2)}(1)(\delta_4^m\gamma_2')=k^{2m}\delta_4^m\gamma_2 + k^{2m}\delta_4^m\gamma_2'=\delta_4^mk^{2m}\gamma_2+\delta_4^mk^{2m}\gamma_2' = \delta_4^m\gamma_2 + \delta_4^m\gamma_2',
E_{1}(\mbox{\mathversion{bold}$v$}(t))=E_{1}(\mbox{\mathversion{bold}$v$}^{R})+E_{1}(\mbox{\mathversion{bold}$\varphi$}(t))+R_{1}(\mbox{\mathversion{bold}$v$}^{R},\mbox{\mathversion{bold}$\varphi$}(t)),
\rho\left(q^{1/2}s_{k}(q)\right)=\frac{C(q^{1/2}s_{k}(q);q)}{D'(q^{1/2}s_{k}(q);q)}=\frac{\mathcal{C}_{q}(q^{1/2}s_{k}(q))}{\mathcal{S}_{q}'(s_{k}(q))}=-\frac{\mathcal{S}_{q}(qs_{k}(q))}{s_{k}(q)\mathcal{S}_{q}'(s_{k}(q))}
\begin{aligned}\nu \int_0^\infty \partial_y^2 D^\alpha u\cdot \langle y \rangle^{2(l+k)}D^\alpha u\mathrm{d}y = -\nu \Vert \langle y\rangle^{l+k}\partial_y D^\alpha u\Vert_{L^2_y}^2 &+\nu (\partial_y D^\alpha u\cdot D^\alpha u)|_{y=0}+2\nu (l+k)\int_0^\infty \langle y \rangle^{2(l+k)-2}y\partial_y D^\alpha u\cdot D^\alpha u\mathrm{d}y.\end{aligned}
\mu_{K}= \sum_{k\in \mathcal{K}}\sum_{\xi\in \mathcal{E}^{(k)}}\mu_{f}(I_k) \delta_{\zeta^k}\Big/ \sum_{k\in \mathcal{K}}\sum_{\xi\in \mathcal{E}^{(k)}}\mu_{f}(I_k) .
{\Pr}\bigg(g_{T,R}^\theta ~\bigg|~{x_T}= 0\bigg) \sim \mathcal{N}\bigg({{\mu}_{{{0}_{T,R}}}},{\sigma}_{{0}_{T,R}}^2\bigg),{\Pr}\bigg(g_{T,R}^\theta ~\bigg|~{x_T}= 1\bigg) \sim \mathcal{N}\bigg({\mu_{{1}_{T,R}}},\sigma_{{1}_{T,R}}^2\bigg),
\left( \sum_{j_{1},\cdots ,j_{m}=1}^{\infty}\left\vert T(e_{j_{1}},\cdots,e_{j_{m}})\right\vert^{\frac{2mp}{mp+p-2m}}\right)^{\frac{mp+p-2m}{2mp}}\leq C_{m,p}^{\mathbb{K}}\left\Vert T\right\Vert ,
\frac{A_n}{F_n(1)}=\frac{\left(1-q^n\right) \left(1-a q^n\right) \left(1-a b q^{n-\gamma +1}\right)}{a q^{n-\gamma}\left(1-b q^{n+1}\right) \left(1-q^{\gamma +n}\right) \left(1-a b q^{n+1}\right)}
|| \leq 4 \int_{t}^{2t}ds \frac{||\lambda_{0,0}'''||_{L^{\infty}(t,2t)}(s-t)}{1+s-t}+ \frac{4}{1+t}\int_{2t}^{\infty}ds |\lambda_{0,0}''(s)| &\leq \frac{C}{t^{2}\log^{b+1}(t)}
R_{Cayley}\left(\sum_{k=1}^{d-1}(k-1)!i_kn^{k-1}, \sum_{k=1}^{d-1}j_{d-k}\frac{(d-1)!}{(d-k)!}n^{k-1}\right) = A_{i_1,\ldots,i_{d-1},j_1,\ldots,j_{d-1}}.
\max \quad\sum_{m=1}^{2}(p_2^mr+w_dp_1^md_{1,0})x_1^m + (d_{1,0}-d_{1,2})w_dp_1^mx_2^m & - w_dp_1^md_{1,0}s.t. \ 0 \leq x_1^1+x_1^2 \leq 1
\Phi_{\alpha_{0}}^{*(1)}=N_{\alpha_{1}\alpha_{0}}n_{1}^{\alpha_{1}}+\cdots ,\;\varphi_{\alpha_{1}}^{*(1)}=\left( -\right)^{\epsilon_{\alpha_{0}}\left( \epsilon \left( f_{\alpha_{1}}\right) +1\right)}N_{\alpha_{1}\alpha_{0}}\nu_{1}^{\alpha_{0}}+\cdots ,
M=(\begin{matrix}x_{1}&x_{2}&y_{1}&y_{2}&y_{3}\end{matrix})=(\begin{matrix}0&1&0&0&0\\ 3&-15&0&0&0\\ -9&30&0&0&1\\ 9&-1&0&3&-2\\ -3&-45&9&0&0\end{matrix}),
\frac{\mu_{0}}{4\pi}\int_{0}^{2\pi}\int_{0}^{R}\frac{r^{4}(-sin\phi\hat{x}+cos\phi\hat{y})}{(x^{2}-2xrcos\phi+y^{2}-2yrsin\phi+r^{2}+z^{2})^{\frac{1}{2}}}drd\phi
\mathbb{E}\Big[\big([(A^{(2)}- \frac{\delta}{4}M)P_{\hat{V}}]_{ij}\big)^2\Big] = \mathbb{E}\Big[\big(\sum_{v=1}^n[A^{(2)}- \frac{\delta}{4}M]_{iv}[P_{\hat{V}}]_{vj}\big)^2\Big]\cr= \sum_{v=1}^{n}[P_{\hat{V}}]_{vj}^2 \mathbb{E}[ ([A^{(2)}- \frac{\delta}{4}M]_{iv})^2] \le \frac{\delta}{4}\sum_{v=1}^{n}[P_{\hat{V}}]_{vj}^2.
ds^2=\left(1+\frac{\tilde k_3}{\hat r^5}\right)^{-2/3}dx^{\mu}dx^{\nu}\eta_{\mu\nu}+\left(1+\frac{\tilde k_3}{\hat r^5}\right)^{1/3}(dz^2+dy^{\tilde{m}}dy^{\tilde{m}}).
I_{m n}^{\prime \prime}= \frac{2}{A^{4}}\int \frac{d^{d}k}{{( 2 \pi )}^{d}}\frac{k_{m}k_{n}d x d y d z \delta ( 1 - x - y - z )}{{( k^{2}+ 2 k p ( x + y ) + a_{2}x + a_{3}( y + z ) )}^{3}}
Y_{\pm}v_{\pm}= \sum_{i=0}^n \left(\kappa\sqrt{\pm 1}\, (q^{n/2-i}\mp q^{n}q^{i-n/2}) \lambda_i v_i \, + \, s [n-i] \l_i v_{i+1}\right) = \sum_{i=0}^n \left( - q^{n/2}\sqrt{\pm 1}\, [i] \lambda_i \, + \, s [n-i+1] \l_{i-1}\right) v_{i}= 0.
n(n-1)\left(\frac{n-d}{d(n-1)}\right)^\frac{p}{2}+n =\frac{1}{(n^2-n)^{\frac{p}{2}-1}}\left(\frac{n^2}{d}-n\right)^\frac{p}{2}+n &\leq \sum_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)f_k(\tau_j)|^\frac{p}{2}+\sum_{j=1}^{n}|f_j(\tau_j)f_j(\tau_j)|^\frac{p}{2}&=\sum_{1\leq j,k \leq n}|f_j(\tau_k)f_k(\tau_j)|^\frac{p}{2}.
A(t)^{1/2}F(t) = |t| S^{1/2}P + t |t| \bigl( Z S^{1/2}P + S^{1/2}P Z^* \bigr)+ \frac{1}{2}t |t| N_0 S^{-1/2}P+ t |t| S^{-1/2}P N_* + I_*(t) + \Phi(t).
\sum_{\l \in S \L}| \langle f, \rho(\l) \widehat{S}g \rangle|^2 = \sum_{\l \in \L}| \langle f, \rho(S \l) \widehat{S}g \rangle|^2 & = \sum_{\l \in \L}| \langle f, \widehat{S}\rho(\l) \widehat{S}^{-1}\widehat{S}g \rangle|^2 & = \sum_{\l \in \L}| \langle \widehat{S}^{-1}f, \rho(\l) g \rangle|^2 .
g^a = g (a^{-1})^g a = x.\big( x^{2\alpha}y^{2\beta}z^{2\gamma}\cdot (y^{-2\lambda}z^{-2\mu})^x \cdot y^{2\lambda}z^{2\mu}\big) = x.(x^{2\alpha}y^{2\beta + 4\lambda}z^{2\gamma + 4\mu});
\nabla_k P_t[\phi](x) = \lim_{s \to 0}\int_H \frac{\phi(e^{tA}x + s e^{tA}k +y) - \phi(e^{tA}x +y)}{s}\mu_t(dy) = \int_H \nabla_{e^{tA}k}\, \phi(e^{tA}x+y) \, \mu_t(dy) = \int_H\langle \nabla^{K}\, \phi(e^{tA}x+y), e^{tA}k \rangle_K \, \mu_t(dy)
\widehat\Gamma'\widehat\Sigma_{\varepsilon\varepsilon}^{-1}\bigl[(I_N\otimes\widehat B)M_{z z}\bigl(I_N\otimes\widehat B{}'\bigr)-\widehat\Sigma_{z z}\bigr]\widehat\Sigma_{\varepsilon\varepsilon}^{-1}\widehat\Gamma=0.
\mathbb{E}\left(\left[\sum\limits_{i=1}^{N(t)}X_i+t\right]^+\right) = \sum_{j=0}^{\infty}p_j(t)\int_{s=0}^{\infty}\sum_{k=0}^{j-1}\frac{1}{k!}e^{-\mu (s-t)}(\mu(s-t))^{k}ds,
\hat{Q}^{+}( \xi_{\alpha}) = \xi_{\alpha}^{2}\frac{\partial}{\partial \xi_{\alpha}}- J_{\alpha}\xi_{\alpha}, \ \ \hat{Q}^{-}( \xi_{\alpha}) = - \frac{\partial}{\partial \xi_{\alpha}}, \ \ \hat{Q}^{3}( \xi_{\alpha}) = \xi_{\alpha}\frac{\partial}{\partial \xi_{\alpha}}- \frac{J_{\alpha}}{2}.
I_{E C S}\sim \int d^{3}x \left[ f_{0}\left( \partial_{1}f_{2}- \partial_{2}f_{1}\right) - f_{1}\left( \partial_{0}f_{2}- \partial_{2}f_{0}\right) + f_{2}\left( \partial_{0}f_{1}- \partial_{1}f_{0}\right) \right] \ .
\sum_{d\le y}f(d)\mathfrak{S}(\{0, d\}) =\int_{2}^yf(x)\,d\left(\sum_{d\le x}\mathfrak{S}(\{0, d\})\right) &=\int_{2}^yf(x)\,dx+O(1)+O(f(y)\log y)+O\left(\int_{2}^y\frac{f(x)}{x}\,dx\right).
m \geq\frac{1+\varepsilon}{(\sqrt{\delta+1}-1)^2}\left( \sum_{i =1}^{k_m}\sum_{\stackrel{j=1}{j\neq i}}^{k_m}\frac{\lambda_i \lambda_j}{(\lambda_i-\lambda_j)^2}+ \sum_{i =1}^{k_m}(n-k_m)\frac{\lambda_i\sigma^2}{(\sigma^2-\lambda_i)^2}\right),
\begin{array}{lcr}\sum\limits_{\nu=1}^{8}X_{m_1n_1}^{\mu}(\sigma)X_{m_2n_2}^{\nu}(\sigma)X_{m_3n_3}^{\nu}(\sigma)=-8{\gamma}^{\mu}X_{m_1n_1}(\sigma)X_{m_2n_2}(\sigma)X_{m_3n_3}(\sigma) .\end{array}
a_{1}a_{2}a_{3}\cdot\cdot\cdot a_{n}=a_{1}(a_{1}+d)(a_{1}+2d)...(a_{1}+(n-1)d)=\prod_{k=0}^{n-1}(a_{1}+kd)=d^{n}\frac{\Gamma(\frac{a_{1}}{d}+n)}{\Gamma(\frac{a_{1}}{d})}
\widetilde{\psi}_{k}^{(n^{2})}\big(x;\omega ,N,- \Vert \psi_{n}\Vert^{2}\big) = [ E_{n}(\omega )-E_{k}(\omega ) ] \psi_{k}(x;\omega,N) \hphantom{\widetilde{\psi}_{k}^{(n^{2})}\big(x;\omega ,N,- \Vert \psi_{n}\Vert^{2}\big) =}{}-2\omega z^{( 3N+5/2) /2}\exp ( -z/2 ) \frac{L_{n,k}^{N}(z) L_{n}^{N}(z)}{Q_{n}^{N}(z)},
\leq \sum_{n=s+1}^\infty2^{-2n}\left[\int\int_T \left|\sum_{k,\nu=N_{s-1}}^{\overline{n}, \overline{m}}c_{k,\nu}^{(s)}W_k(x) \cdot W_\nu(y)\right| h_s^{-1}dxdy\right]<{1 \over 3}2^{-2s}.
((\omega_H*\delta_x*\omega_H) *(\omega_H*\delta_y*\omega_H))(HgH)= &=\frac{1}{\omega_{Hx^{-1}H}\cdot \omega_{HyH}}\sum_{k=1}^{\omega_{Hx^{-1}H}}\sum_{l=1}^{\omega_{HyH}}(\omega_H*\delta_{\tilde x_k^{-1}}*\delta_{y_l}*\omega_H)(HgH).
\lim_{N\rightarrow\infty}{\displaystyle\int\limits_{1}^{2N}}e^{i\pi x}x^{1/x}dx =0.0707760393...-(0.6840003894...)i \ =-\frac{2}{\pi}i+\lim_{N\rightarrow\infty}{\displaystyle\int\limits_{1}^{2N+1}}e^{i\pi x}x^{1/x}dx
Z_n^*= (2s^3_n|\tau_n|)^{-1}\Big[2\tau_n s_n^2 \Big(\dfrac{1}{n}\sum_{i=1}^{n}\psi''(\bar{\epsilon}_i) [\mathbf{x_i}'((\mathbf{E_*}L_n^*)^{-1}\Delta_n^*)]\Big) - \tau^2_n\Big(\dfrac{2}{n}\sum_{i=1}^{n}\psi(\bar{\epsilon}_i) \psi'(\bar{\epsilon}_i)[\mathbf{x_i}'((\mathbf{E_*}L_n^*)^{-1}\Delta_n^*)]\Big)\Big]
e_i(x_1, \dotsc, x_i, x_{i+1}, \dotsc, x_n) =\begin{cases}(x_1, \dotsc, x_i + 1, x_{i+1}- 1, \dotsc, x_n) x_{i+1}> 0, 0 ,\end{cases}f_i(x_1, \dotsc, x_i, x_{i+1}, \dotsc, x_n) =\begin{cases}(x_1, \dotsc, x_i - 1, x_{i+1}+ 1, \dotsc, x_n) x_i > 0, 0 ,\end{cases}
(1-s\mu_X)(1-2d_n\rho s) = (1-s \mu_X)(1-s(1-\mu_X)/3) = (1+2(1-\mu_X)s/3)(1-s) + s^2(1-\mu_X)(\mu_X+2)/3 \ge (1+2(1-\mu_X)s/3)(1-s) = (1+4d_n\rho s)(1-s).
P\left(\omega_{1},\omega_{2}\right)=\left\Vert \mathbf{Q}\left(\omega_{1},\omega_{1},a_{1,i}^{*}\left(\omega_{1},\omega_{2}\right),b_{1,i}^{*}\left(\omega_{1},\omega_{2}\right),b_{2,i}^{*}\left(\omega_{1},\omega_{2}\right);\mathbf{t}\right)+\bar{p}_{,i}^{*}\left(\omega_{1},\omega_{2}\right)\mathbf{1}-\mathbf{f}\right\Vert_{2}^{2}.
\Pr\left( \{Y_{i,n}'\}\in dx,\ \sum_{j=1}^n \lfloor Y_{i,j}' \rfloor = \lfloor r_i - y_i\rfloor\right) = \frac{\sum_{k=1}^{\lfloor c_n-x\rfloor}k^{m-2}}{c^{m-1}}\ \frac{\lambda e^{-\lambda\, x}}{1-e^{-\lambda}}\ \Pr\left( \sum_{j=1}^n \lfloor Y_{i,j}' \rfloor = \lfloor r_i - y_i\rfloor\right).
\sum_{k=1}^m{1 \over \bar{T}_i \bar{T}_k}\left( \bar{T}_i \delta_{ik}+ X^{(m)}_{ik}\right) A_{kj}\ = \ \delta_{ij}+{1\over{\bar T}_i}\left(X^{(m)}_{ij}-X_{ij}-\sum_{k=1}^m{1\over{\bar T}_k}X^{(m)}_{ik}X_{kj}\right)=\delta_{ij}\ .
\psi(\alpha) \star \psi(\beta) = \lfloor \phi(p_{i}); h_{i}\rceil \star \lfloor \phi(p_{j}); h_{j}\rceil = \lfloor \gamma_{2}\phi(p_{i})\phi(p_{j}); h_{i}+ h_{j}\rceil = \lfloor \gamma_{2}\phi(p_{i}p_{j}); h_{i}+ h_{j}\rceil.
\rho_{m,n}^{(2)}\big(\rho_{n,j}^{(2)}(d_j^*d_j)-\rho_{n,j}^{(2)}(d_j)^*\rho_{n,j}^{(2)}(d_j)\big)=\rho_{m,j}^{(2)}(d_j^*d_j)-\rho_{m,n}^{(2)}\big(\rho_{n,j}^{(2)}(d_j)^*\rho_{n,j}^{(2)}(d_j)\big)\geq 0.
T_{51}= s \lambda^3 \iint_Q \xi u A \nabla u \cdot \nabla \eta \left( A \nabla \eta \cdot \nabla \eta\right) dx dt \geq -Cs^2 \lambda^4 \iint_Q \xi\left|A \nabla \eta \cdot \nabla \eta\right|^2|u|^2 dx dt-C\lambda^2 \iint_Q \xi| A\nabla u \cdot\nabla \eta|^2 dx dt,
m'_{ij}= c_{\lambda_i}(i)\,c_{\lambda_j}(j) + \sum_{m>0}\beta^m c_{\lambda_i+m}(i)\,c_{\lambda_j}(j) + \sum_{\ell>0}\sum_{m\geq 0}(-1)^\ell \left( \binom{\ell+m-1}{m}+ \binom{\ell+m}{m}\right) \beta^{m}\, c_{\lambda_i+\ell+m}(i)\, c_{\lambda_j-\ell}(j) ,
\lvert b(s, X_{s}, F_{s}^{(u)}) \rvert \, =\, \Big \lvert \int_{\mathbb R}\widetilde{b}(s, X_{s}, y ){\mathrm d}F_{s}^{(u)}(y) \Big \rvert \le C_{T}\big ( 1 + u (\lvert X_{s}\rvert + \lvert \widetilde{X}_{s}\rvert) + (1-u) ( \lvert X_{s}\rvert + \mathbb E [ \lvert X_{s}\rvert ] ) \big ) \,
B=\frac{1}{2}(1-\zeta^2)\partial_\zeta^2+(\zeta\partial_\zeta)^2+\frac{c^\prime+d-5}{2}\zeta\partial_\zeta-m^{ij}\zeta^i\partial_{\zeta^j}-\lambda(\lambda+\frac{c^\prime+d-3}{2})\,.
\begin{array}{c}{{\mathcal{E}( x ) \rightarrow \mathcal{E}^{V}( x ) = \Lambda \mathcal{E}( \Lambda^{- 1}( x - a ) ) ,}}\\{{\mathcal{E}( x ) \rightarrow \mathcal{E}^{T S}( x ) = F ( \Lambda ) \mathcal{E}( \Lambda^{- 1}( x - a ) ) ,}}\\{{\mathcal{E}( x ) \rightarrow \mathcal{E}^{S}( x ) = S ( \Lambda ) \mathcal{E}( \Lambda^{- 1}( x - a ) ) ,}}\\ \end{array}
\frac{\mu[0,x)+\mu[0,x]}{2}-x=\lim_{h\to 0}\frac{f(x+h)+f(x-h)}{2}&=\frac{1}{\pi}\sum_{n=1}^\infty\Im(\frac{\hat{\mu}(n)}{n}[e(nx)-1])+\lim_{n\to \infty}\frac{1}{2n}\sum_{j=1}^n \hat{\mu}(j)
\left\|Q_{L}\circ T_{\varphi}\circ I_{M}\right\|_{X;Y^{\rm dual}}= \|\Psi^{-1}(T)\|= \left\|\varphi|_{M \otimes_{\alpha_{X;Y}}L^\perp}\circ (Id_M \otimes Q_L^*)\right\| \leq \left\| \varphi\right\|\cdot \left\| I_{M}\otimes Q_{L}^*\right\| \leq \left\| \varphi\right\|.
(x-1)^{\kappa(\Gamma)}{\rm Tutte}(\Gamma,x,y)={\widetilde{T}}_n({\boldsymbol{m}},x,y)\Big |_{m_{ij}=0 \ \ \ (i,j) \notin \Gamma \atop m_{ij}=\ell_{ij}\ \ (i,j) \in \Gamma}.
\left\|\Box_k^{\alpha_1}|~M_1\rightarrow M_2\right\|\gtrsim\lim_{N\rightarrow \infty}\frac{\|\Box_k^{\alpha_1}G_{k,N}\|_{M_2}}{\|G_{k,N}\|_{M_1}}\sim2^{j\widetilde{A_3}(\mathbf{p},q,\alpha_1,\alpha_2)}.
\frac{\mathrm{d}}{\mathrm{d}\omega}\left(\frac{2d}{\pi^{2}+4d^{2}\omega^{2}}\frac{\omega}{\tanh(d\, \omega)}\right) & =\frac{d\, [ -2d \omega (\pi^{2}+ 4d^{2}\omega^{2}) + (\pi^{2}- 4d^{2}\omega^{2}) \sinh(d\, \omega)]}{(\pi^{2}+ 4d^{2}\omega^{2})^{2}\sinh^{2}(d\, \omega)}\leq 0
R_i(\mathbf{V},\mathbf{\Omega}) =\log\big|\mathbf{I}+\mathbf{H}_i(\mathbf{V}\mathbf{V}^H+\mathbf{\Omega})\mathbf{H}^H_i\big| &-\bigg|\mathbf{I}+\mathbf{H}_i(\sum_{j\ne i}\mathbf{V}_j\mathbf{V}_j^H+\mathbf{\Omega})\mathbf{H}^H_i\bigg|.
\log |{\cal M}_0| \le \log |\tilde{{\cal M}}_0| + \log n + \log \log |{\cal X}| + 2, \log |{\cal M}_0| |{\cal M}_1| \le \log |{\cal T}^n_{\bar{X}}| + \log n + \log \log |{\cal X}| + 2, \log |{\cal M}_2| = \log |\tilde{{\cal M}}_2|,
R\int_{A_1}D\Phi(\bar{g}^R, \bar{\pi}^R)\cdot (x^k,0) \, dx = R \left(B^2_{((g^\theta)^R,(\pi^\theta)^R)}(x^k,0) - B^1_{(g^R,\pi^R)}(x^k,0)\right) &= R^{-1}\left(B^{2R}_{(g^\theta, \pi^\theta)}(x^k,0) - B^R_{(g,\pi)}(x^k,0)\right),
Q_1^{k,m}(1-q^{2k+m}r^2) &=\sum_{i=0}^{m}{{m+1}\choose{i}}_{\!\!q}q^{i(2k+i-1)/2}\prod_{j=0}^{i-1}(q^{k+j}r^2-r) \prod_{j=0}^{m-i}(1-q^{2k+m-j}r^2) &=\sum_{i=0}^{m+1}{{m+1}\choose{i}}_{\!\!q}q^{i(2k+i-1)/2}\prod_{j=0}^{i-1}(q^{k+j}r^2-r) \prod_{j=1}^{m+1-i}(1-q^{2k+m+1-j}r^2) -q^{(m+1)(2k+m)/2}\prod_{j=0}^{m}(q^{k+j}r^2-r).
x_0=1-\frac{x_0\sum_{l \in \mathbb Z_0}{\lambda_lz_l}}{1+\sum_{l \in \mathbb Z_0}{\lambda_lz_l}}, \ \ \x_j=\frac{x_0\lambda_{j}z_{j}}{1+\sum_{l \in \mathbb Z_0}{\lambda_lz_l}}, \ \ \j\in \mathbb Z_0.