harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{dy}{dt}(t)(y(t))^{-1}= \left(\frac{r}{2}t^{r/2-1}x(\log t)+t^{r/2}\frac{dx}{dt}(\log t)t^{-1}\right)(x(\log t))^{-1}t^{-r/2}&= t^{-1}\left(\frac{r}{2}+\frac{dx}{dt}(\log t)(x(\log t))^{-1}\right)
\begin{array}{l}{\cal W}(x^0_1 -{\rm i}x^4_1,{\bf x_1}; \dots ;x^0_n -{\rm i}x^4_n,{\bf x_n}) = begin{align*}3ex]\quad = \langle\, 0 | \phi(0,{\bf x_1}){\rm e}^{-{\rm i}(x^0_1-x^0_2)H}{\rm e}^{-(x^4_1-x^4_2)H}\phi(0,{\bf x_2})\dots \phi(0,{\bf x_n}) | 0 \,\rangle\,.\end{array}
\mathrm{Find}~ (p_1,\dots,p_K) ~ \rho_i\exp\left(\frac{(2^{\alpha_i\bar{R}}-1)\sigma_i^2}{p_iQ_{ii}}\right) &~\times\prod_{k{\ne}i}\left(1+\frac{(2^{\alpha_i\bar{R}}-1)p_kQ_{ki}}{p_iQ_{ii}}\right)\le1, &p_i\le{P}_i,~i=1,\dots,K,
\P (G_0v_{||E||_1^{1/3}}(d, k ||E||_1^{1/3}, E)) = \P (G_0v (d, (k-1) ||E||_1^{1/3}, E)) \leq \P (G_0v (d, k ||E||_1^{1/3}, 0) + \mathcal{O}_{\upsilon,d}\left( N^{-3/2}\sum_{\ell = 1}^{\infty}(1 + k \ell ||E||^{1/3}_1)^{3}e^{- \frac{\ell^2}{2 d^2 ||E||_1^{1/3}}}\right).
\Omega = 2 i J \left[ \frac{d{\bar{\xi}}\wedge d \xi}{1 + \mid \xi \mid^{2}}- \frac{( \xi d{\bar{\xi}}) \wedge ({\bar{\xi}}d \xi )}{( 1 + \mid \xi \mid^{2})^{2}}\right] , \ \mid \xi \mid^{2}= \sum_{p}\mid \xi_{p}\mid^{2}.
\int_{e}|x|dm^N_t(x) \leq \int_{e}|x|dm^j_0(x) + V_{max}\Delta t^N \sum_{k=0}^{n-1}m^N_{t_{k}}(e_j) &\leq \int_{e}|x|dm^j_0(x) + V_{max}\frac{T}{2^N}\sum_{k=0}^{2^N-1}m^{N}_{t_k}(e_j).
{\bf H}= \frac{i \omega}{2}\int d^{3}p p^{i}\sum_{\lambda}\left[ a^{\dagger}( \vec{p}, \lambda ) \stackrel{\leftrightarrow}{\partial}_{p^{i}}a ( \vec{p}, \lambda ) + b^{\dagger}( \vec{p}, \lambda ) \stackrel{\leftrightarrow}{\partial}_{p^{i}}b ( \vec{p}, \lambda ) \right]
W_{{\bf A , B}}({\bf q , p}) = 2^{N}e x p [ - 2{\bf Z Z^{*}}+ 2{\bf A Z^{*}}+ 2{\bf B Z^{*}}-{\bf A B^{*}}- \frac{\mid{\bf A}\mid^{2}}{2}- \frac{\mid{\bf B}\mid^{2}}{2}] , \nonumber \,
\mathbb{E}(\widetilde{N}_0 /n) - \frac{1-\varepsilon}{2}\{P_0(A)+P_1(A)\}- \frac{\varepsilon}{2}\leq (1-\varepsilon)P_1(A) + \varepsilon - \frac{1-\varepsilon}{2}\{P_0(A)+P_1(A)\}- \frac{\varepsilon}{2}= \frac{\varepsilon}{2}- \frac{1-\varepsilon}{2}\mathrm{TV}(P_0,P_1).
T_{M N}= \frac{8}{3}\partial_{M}\Phi \partial_{N}\Phi - \frac{4}{3}\left( \partial \Phi \right)^{2}G_{M N}- V \! \left( \Phi \right) G_{M N}- \sum_{i}f_{i}\, \delta \! \left( y - y_{i}\right) g_{\mu \nu}\delta_{M}^{\mu}\delta_{N}^{\nu},
2 9 I_{A}[ \sigma ; g_{\mu \nu}] = \int d^{D}x [ \textstyle{\frac{1}{2}}\: \sigma \Delta_{D}\sigma + \sigma \bar{{\cal E}}_{D}] \; , \; \delta I [ \sigma + \phi \, ; \: 2 \phi g_{\mu \nu}] / \delta \phi = \bar{\cal E}_{D}\; .
\left( \begin{array}{c}{t^{\prime}}{z^{\prime}}\end{array}\right) = \left( \begin{array}{c c}{0}{- \frac{2 \lambda}{\sqrt{\mu}}}{\frac{\sqrt{\mu}}{2 \lambda}}{\frac{2 \lambda}{\sqrt{\mu}}}\end{array}\right) \left( \begin{array}{c}{t}{z}\end{array}\right) .
G_{E}( a_{3}, a_{4}, \phi ) = \mu^{4}\sqrt{a_{3}a_{4}}\pi \sum_{s = 0}^{\infty}\frac{( - 1 )^{s + 2}}{2 s + 2}g^{s + 1}\phi^{2 s + 2}\frac{\Gamma ( s )}{\Gamma ( s + 1 )}A_{2}^{c^{2}}( s , a_{3}, a_{4}) .
\|f(t, u)- f(t,v)\|_{H^{1+\delta}_p}= \|ue^{-t|\cdot|}-ve^{-t|\cdot|}\|_{H^{1+\delta}_p}= \|(u -v)e^{-t|\cdot|}\|_{H^{1+\delta}_p}\leq c \|u -v \|_{H^{1+\delta}_p}\|e^{-t|\cdot|}\|_{H^{1+\delta}_p}.
\nu[A] \; = \; \lim_{V_4 \rightarrow \infty}\; \frac{g^2}{8 \pi^2}\;\int_{V_4}\; d^4x \; \partial_\mu K_\mu \; = \;\lim_{V_4 \rightarrow \infty}\; \frac{g^2}{8 \pi^2}\;\int_{\partial V_4}\; d^3 \sigma \; \hat{n}_\mu K_\mu \; .
S_k(y)-S_k(\gamma) \stackrel{\mbox{}^{(\star)}}{\geq}\ \sqrt{2k}\cdot \Big [ l(y)-l(\gamma)\Big ] + \int_{C(y)}\sigma - \int_{C(\gamma)}\sigma &\stackrel{\mbox{}^{(\star\star)}}{=}\sqrt{2k}\cdot \Big [ l(y)-l(\gamma)\Big ] + \int_{\Pi_y\setminus \Pi}\!\! \sigma\ = \ \mathcal T_k(\Pi_y)-\mathcal T_k(\Pi)\, ,
ip=e=F(i')F(p'),\, pi=1_x,\, F(p')F(i')=F(p'i')=F(1_{x'})=1_{F(x')}, F(p')i\circ pF(i')= F(p')F(i')F(p')F(i')=1_{F(x')}pF(i')\circ F(p')i=pipi=1_x.
\left\{\psi_{u\mathbf{j}_{|u|}}:=\tilde{\psi}_{u\mathbf{j}_{|u|}}/(\tilde{\psi}_{u\mathbf{j}_{|u|}},\tilde{\psi}_{u\mathbf{j}_{|u|}})_{\phi_{u}},\,\mathbf{j}_{|u|}\in\mathbb{N}_{0}^{|u|}\right\}
\frac{d^2}{dt^2}\mathcal{F}(\rho(t))= \frac{2}{\Delta x^4}\sum_{(i,j)\in E}\sum_{(k,l)\in E}h_{ij, kl}(F_i-F_j)_+\rho_i(F_k-F_l)_+\rho_k &+o(\frac{d}{dt}\mathcal{F}(\rho(t))).
(\begin{matrix}j_{1}&j_{2}&j_{3}\\m_{1}&m_{2}&m_{3}\end{matrix})=(\begin{matrix}j_{2}&j_{3}&j_{1}\\m_{2}&m_{3}&m_{1}\end{matrix})=(\begin{matrix}j_{3}&j_{1}&j_{2}\\m_{3}&m_{1}&m_{2}\end{matrix}).
\sigma((x\otimes x)T+T(x\otimes x)) =\bigcap_{\alpha>0}\sigma_{\frac{\delta}{\alpha}}((x\otimes x)T+T(x\otimes x)) &=\bigcap_{\alpha>0}\sigma_{\frac{\delta}{\alpha}}((x\otimes x)\psi(T)+\psi(T)(x\otimes x)) &=\sigma((x\otimes x)\psi(T)+\psi(T)(x\otimes x)).
2\Big|\sum_{n=1}^{n_1}\frac{e^{it\log n}}{\sqrt{n}}\Big| \le 2\sum_{n=1}^5 \frac{1}{\sqrt{n}}+ 2\int_5^{n_1}\frac{1}{\sqrt{x}}\,dx &= 2\sum_{n=1}^5 \frac{1}{\sqrt{n}}+ 4\sqrt{n_1}-4\sqrt{5}.
\frac{D_2}{D_1}=e^{-2i\pi \mu}e^{4aL \mu}\frac{\Gamma(2\mu)\Gamma(1-\mu-\nu-\lambda)\Gamma(-\mu-\nu+\lambda)}{\Gamma(-2\mu)\Gamma(1+\mu-\nu-\lambda)\Gamma(\mu-\nu+\lambda)}
m = \frac{d}{d\sigma}A_N ( \sigma) = \frac{1}{N}\frac{\int_{\mathbb{R}^N}\sum_{i=1}^{N}x_i \exp \left( \sigma \sum_{i=1}^{N}x_i - H(x) \right)dx}{\int_{\mathbb{R}^N}\exp \left( \sigma \sum_{i=1}^{N}x_i - H(x) \right) dx}= \frac{1}{N}\sum_{i=1}^N m_i .
S_{u,c}':=\frac{{\displaystyle \sum_{{\textstyle{\emptyset\ne v\subseteq\{1,\cdots,N\}\atop v\neq u}}}}\mathbb{E}\left[y_{u,G}(\mathbf{X}_{u})y_{v,G}(\mathbf{X}_{v})\right]}{\sigma^{2}},
L_d(R_k,R_{k+1}) \approx hL(R_k,\frac{R_{k+1}-R_k}{h}) = \frac{h}{2}tr(\frac{R_k^T(R_{k+1}-R_k)}{h}J_d \frac{(R_{k+1}-R_k)^T R_k}{h}) = \frac{1}{2h}tr((I_{3 \times 3}-F_k)J_d)
= \delta^{\textrm{np}}_0\left[\rho_0\sum_{j=1}^{N}\eta_j^{\frac{\alpha}{2}}\left(\frac{P_j}{P_i}\right)\right]^{-\frac{2}{\alpha}}\frac{(\mathbb{E}[I^{-1}_0(1)])^{\frac{2}{\alpha}}}{\mathbb{E}\left[I^{-\frac{2}{\alpha}}_0(1)\right]}.
E_{0,l}= \lambda [2a(2l+d) + 2b_1(4a-2l-d+3) - l(l+d-1)], \psi_{0,l}(r) \propto r^l (1+\lambda r^2)^{-a}\exp\left(- \frac{b_1}{1+\lambda r^2}- \frac{b_2}{(1+\lambda r^2)^2}\right),
\bar{M}<-n \Leftrightarrow \forall i\in\{-n,\ldots,0\},\,\exists j\in\{i,\ldots,0\}:\,\,U_{j}> \alpha_{i-j}&\Leftrightarrow \forall i\in\{-n,\ldots,0\},\,\exists j\in\{i,\ldots,0\}:\,\,H^{(i-1)}_{j}=0 &\Leftrightarrow \forall i\in\{-n,\ldots,0\},\,\,H^{(i-1)}_{0}=0 &\Leftrightarrow H^{(-n-1)}_{0}=0,
\|\Phi(u)\|_{L^2(|V|)}\leq C[V]_{\mathcal{R}}\|Vu\|_{L^2(|V|^{-1})}&=C[V]_{\mathcal{R}}\|u\|_{L^2(|V|)}&<\frac12\|u\|_{L^2(|V|)}.
I_4 =-\int_{B_{\frac{|x|}{2}}(0)}\dfrac{(\Upsilon(y)-\Upsilon(x))^{p-1}}{|x-y|^{N+ps}}dy &\le -\left(\dfrac{2}3\right)^{N+ps}\dfrac1{|x|^{N+ps}}\int_{B_{\frac{|x|}{2}}(0)}(\Upsilon(y)-\Upsilon(x))^{p-1}dy.
\int_{-\infty}^\infty \![\widehat{h}(k)+\widehat{h}(-k)S(k)]\widehat{h}(k)^\dag dk+2\pi\sum_{j=1}^N [\widehat{h}(-ik_j)C_j][\widehat{h}(-ik_j)C_j]^\dag\!=0.
\mathcal{L}_{F}=\sum_{\sigma=1}^{N}\overline{\psi}_{\sigma}v_{\sigma, \mu}\left( \partial_{\mu}-ia_{\mu}\right)\gamma_{\mu}\psi_{\sigma}+\left|\left(\partial_{\mu}-ia_{\mu}\right)b\right|^{2}+V(\left|b\right|^{2}).
\frac{1}{T}\sum_{t=2}^{T+1}\Delta_{t}\le48R_{1}\max\left\{26^{\frac{1}{p}}T^{\frac{1-p}{p}}\sigma\gamma^{\frac{p-1}{p}};2\left(2LR_{1}+LR_{0}+\mu\sigma+\left\Vert g_{0}\right\Vert_{*}\right)T^{-1}\gamma\right\}=O\left(T^{\frac{1-p}{p}}\right).
\frac{d}{{dx}}\left({\frac{{- \left({\cos_{p,q}x}\right)^{p - 1}}}{{\left({1 - k^q \left({\sin_{p,q}x}\right)^q}\right)^{1 - \frac{1}{p}}}}}\right) = \frac{{(p - 1)q}}{p}\frac{{\left({\sin_{p,q}x}\right)^{q - 1}\left({1 - k^q}\right)}}{{\left({1 - k^q \left({\sin_{p,q}x}\right)^q}\right)^{2 - \frac{1}{p}}}},
\widetilde\phi_{n,l}(zr)\cdot \widetilde\phi_{n,l}(z^\prime s)=\frac{1}{\kappa_p}\int_{D}\widetilde\phi_{n,l}(zz^\prime (rs-w\sqrt{1- r^2}\sqrt{1- s^2}))\cdot(1-|w|^2)^{p-2}\> dw
\sum_{n\geq0}U_{2k+1,2a}(n)q^n= \frac{(-q^2;q)_\infty(q^{2a+1},q^{4k-2a+1},q^{4k+2};q^{4k+2})_\infty}{(q^2;q^2)_\infty}&+\frac{xq(-q^2;q)_\infty(q^{2a-1},q^{4k-2a+3},q^{4k+2};q^{4k+2})_\infty}{(q^2;q^2)_\infty}.
\tilde{f}_i (x) := \sum_{j \geq 0, \j \geq -\langle \lambda, \alpha_i^\vee \rangle}\frac{1}{[j+1]!_i}F_i^{j+1}x_j, \tilde{e}_i (x) := \sum_{j > 0, \j \geq -\langle \lambda, \alpha_i^\vee \rangle}\frac{1}{[j-1]!_i}F_i^{j-1}x_j
\int_0^t \int_0^t \frac{\|I_1(\theta',\theta)\|_{L^2(\Omega)}^2}{|\theta'-\theta|^{1+2\beta}}d\theta d\theta' = \int_0^t \int_0^t \frac{|K_H(t,\theta')-K_H(t,\theta)|^2}{|\theta'-\theta|^{1+2\beta}}d\theta d\theta'<\infty
\sum_{n\geq 0}\overline{U}_{k,a}(n)q^n=\sum_{n\geq 0}\overline{U}_{k,a-1}(n)q^n=\frac{(-q^2;q^2)^2_\infty(q^{a},q^{2k-a},q^{2k};q^{2k})_\infty}{(q^{2};q^2)_\infty}.
\sum_{i = 1}^{r}\Lambda ( H_{i}) \Lambda^{\prime}( H^{i}) = ( \Lambda_{0}, \Lambda_{0}^{\prime}) \, , ~ \forall \Lambda = ( \Lambda_{0}, \kappa , \sigma ) \, , \, \Lambda^{\prime}= ( \Lambda_{0}^{\prime}, \kappa^{\prime}, \sigma^{\prime}) \in{\cal H}^{*}
\delta_{\alpha,\beta}( \delta_{\alpha,\beta}^*g) =\big( (1-x^2)^{1/2}((1-x^2)^{1/2})' \big)'+xg' &-\Big(\frac{\alpha}{2}+\frac{\beta}{2}\Big)g -\frac{[(\frac{\beta}{2}- \frac{\alpha}{2})-(\frac{\alpha}{2}+\frac{\beta}{2}+1)x]^2}{1-x^2},
\lim_{s\to 0}\frac{\phi_{s+t}^*g(s+t)(v_1,v_2)-\phi_{s+t}^*g(t)(v_1,v_2)}{s}=\lim_{s\to 0}\frac{g(s+t)_{ij}-g(t)_{ij}}{s}\lim_{s\to 0}[((\phi_{s+t})_*v_1)^i((\phi_{s+t})_*v_2)^j] & =(\partial_t g)_{ij}(\phi_{t*}v_1)^i(\phi_{t*}v_2)^j & =\phi_t^*\partial_t g(v_1,v_2).
\sum_{\{k_{i}\}}\prod_{i=1}^{4}\frac{1}{k_{i}!}\left( \partial_{i}^{\mu_{i}}\right)^{k_{i}}u(0)\int \prod_{i=1}^{k}{\rm d}^{4}x_{i}~\left(x_{i}^{\mu_{i}}\right)^{k_{i}}~\Gamma_{x_{1}\cdots x_{k}0}.
\left( \widetilde{\mathcal{X}^{K\textnormal{th}}}\right)_{0i}(q,Q) = \sum_{\substack{a + b = i 0 \leq a, b \leq N}}(-1)^a \binom{\ell_q \left( Q \right)}{a}J_b \left( q,\left( \frac{1-q}{z}\right)^{N+1}Q \right)
mod\sigma_{y}^{2}(n\tau_{0})=\frac{1}{2n^{4}(M-3n+2)}\sum_{j=0}^{M-3n+1}\{\sum_{i=j}^{j+n-1}(\sum_{k=i}^{i+n-1}\overline{y}_{k+n}-\overline{y}_{k})\}^{2}
P(E_{1}^{k}; J \times \mathbb{R}^{n-1}) =P(E_{2}^{k}; J \times \mathbb{R}^{n-1})= \cdots = P(E_{N_{k}}^{k}; J \times \mathbb{R}^{n-1}) &= P (F_{\ell^{k}}; J \times \mathbb{R}^{n-1}).
\delta_{\bullet}^{i}=f^{i},\delta_{[t_{1},...,t_{n}]}^{i}=\sum_{j_{1},...,j_{n}=1}^{N}(\delta_{t_{1}}^{j_{1}}\cdot\cdot\cdot\delta_{t_{n}}^{j_{n}})\partial_{j_{1}}\cdot\cdot\cdot\partial_{j_{n}}f^{i}.
S_{a,F}(\mathbf{Q})=\bigsqcup_{(\boldsymbol{\varepsilon},\mathbf{m}) \in \Sigma\times M}\pi_{\mathbf{m},\boldsymbol{\varepsilon}}\left(\mathcal{T}_{\mathbf{m},\boldsymbol{\varepsilon}}(\mathbf{Z})\right).
A_7(0,1) = \frac{14}{3}Jq^7-\frac{14}{3}Jq^6Jq^1+\frac{7}{3}Jq^5Jq^2 -\frac{7}{15}Jq^4Jq^3-\frac{1}{3}Jq^3Jq^4+Jq^1Jq^6=0.
\mathbb{E}_{\mathbb{P}}\left[\left(\int_{T_{0}}^{\left(T_{0}+\delta\right) \wedge T}\left|\triangle b_{t}^{i, n, \infty}\right|^{2}d t\right)^{p}\right] \leq \frac{p ! \beta^{p}\delta^{p}}{n^{p}},i=1,\cdots,n.
f^N(e)\approx \left (f(e_0)+\frac{(e-e_0)^2}{2}f^{\prime \prime}(e_0)\right )^N\approx f^N(e_0) \mathrm{e}^{-N(e-e_0)^2/2 \sigma^2},\sigma^2=-\frac{f(e_0)}{f^{\prime \prime}(e_0)}\ .
\Lambda_d:=\frac{1}{(4\pi)^{\frac{d}{8}}}\left( \frac{\Gamma(\frac{d}{4})}{\Gamma(\frac{3d}{4})}\right)^{\frac{1}{2}}\left( \frac{\Gamma(d)}{\Gamma(\frac{d}{2})}\right)^{\frac{1}{4}},
[A_{0}(\textbf{u}), \textbf{P}]=\Bigg[\left( \begin{array}{cccc}\frac{f(\textbf{u})}{\rho}0 0 Id \end{array}\right), \left( \begin{array}{cccc}1 0 0 \mathbb{P}\end{array}\right) \Bigg] = 0.
\eta_{{\bf i}, \mathcal I}^{\ast}\mathcal{L}_{\mathcal I,\lambda_1,\dots,\lambda_r}= \bigg\{(\left[(p_{k,l})_{1\leq k \leq r, 1 \leq l \leq N_k}], [p_1,\dots,p_r,w] \right) ~\bigg|~ & \left[\prod_{l=1}^{N_1}p_{1,l}, \dots, \prod_{l=1}^{N_{r}}p_{r,l}\right]= [p_1,\dots,p_r] Z_{\mathcal I}\bigg\}.
\widetilde{\Lambda}(p,q,r) = \lambda_1(p)^{\frac{r-q}{r-p}}(r-p) \left(\frac{(p-q)^{p-q}}{(r-q)^{r-q}}\right)^{\frac{1}{r-p}}c(q,t_q\lambda_1(p))^{q-p},
\hat{F}_e\left({x}\right)=\sum\limits_{\ell,n \in \left\{{{},{}}\right\}}\left({\rm{\mathbf{1}}}\left( d < \eta\left(G_\ell,G_n^e,x\right)\right) \frac{d^2}{2}+ \frac{\varrho^2-d^2}{2}\right){{{\Pr}_{\ell n}}}
f_m(z)\mid_{-1}\gamma=\sum_{\substack{n=0 n\equiv 0\pmod{\ell^m}}}^{\infty}a_0(n)q_{24\ell^m}^n-\sum_{\substack{n=0 n\equiv 0\pmod{\ell^{m+1}}}}^{\infty}a_0(n)q_{24\ell^m}^n,
\partial_{3}F_{3}(r,\rho,z) = \frac{-4(-1+\alpha) r^{2}z^{-3+2\alpha}(1+(r^{2}-\rho^{2})z^{2\alpha-2})}{((1+(r^{2}-\rho^{2})z^{2\alpha-2})^{2}+4\rho^{2}z^{2\alpha-2})^{3/2}}
\widetilde{\mathcal{E}}[C,Q] = \sup \left\{\int_{\mathbb{S}_+^1}\int_\Omega (2 c_0^2 a + 1) \d C(x,\theta)\right. + \int_{\mathbb{S}_+^1}\int_\Omega b \d Q(x,\theta);\; \left. (a,b)\in C_b(\Omega\times\mathbb{S}_+^1, K_2) \right\}.
(b^{g|_0}, (a^{\mathbf{e}_{{1}}})^{g|_0}, \dots, (a^{\mathbf{e}_{{p-1}}})^{g|_0})^{a^k}= \psi_1(b^g) = \psi_1(b^{\underline{c}^jh}) &= (b^{\underline{c}^j|_0h|_0}, (a^{\mathbf{e}_{{1}}})^{h|_1}, \dots, (a^{\mathbf{e}_{{p-1}}})^{h|_{p-1}})^{h|^\varnothing}.
\left. \hat{F}_{\alpha \beta}^I\right|_{{\rm reg}}=\frac{(\partial \Pi )^2}{2\Pi^2}e_\alpha^{a\,}\,\hat{\xi}_{ab}^{+}\,e_\beta^{b\,}-\frac{D_\alpha\,\partial_\gamma \Pi}{2\Pi}e^{a\,\gamma}\,\hat{\xi}_{ab}^{+}\,e_\beta^b-e_\alpha^{a\,}\,\hat{\xi}_{ab}^{+}\,e^{b\,\delta}\frac{D_\beta\,\partial_\delta \Pi}{2\Pi}
\sum_{i=1}^{\left \lfloor{\frac{y^{1/2}}{a}}\right \rfloor}(y-a^{2}i^{2})^{n/2}\leq \frac{\sqrt{\pi}}{2a}\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+3}{2}\right)}y^{(n+1)/2}- \frac12 y^{n/2}+\frac{(2an)^{n/2}}{(n+2)^{(n+2)/2}}y^{n/4}.
R < -\left(\log\frac{1-p}{2}+\beta^*(p)\right) & = 1-\log(1-p) -\frac{\sqrt{p}}{\sqrt{p}+\sqrt{1-p}}\log\frac{p}{1-p}- 2h\left(\frac{\sqrt{p}}{\sqrt{p}+\sqrt{1-p}}\right) & = 1-h_{\frac{1}{2}}(p),
d_{\rm TV, low}= 1 - \left( \frac{4 \gamma_e}{(\gamma_e+1)^2}\right)^{\frac{M}{4}}, & d_{\rm TV, up}= \sqrt{1 - \left( \frac{4 \gamma_e}{(\gamma_e+1)^2}\right)^{\frac{M}{2}}},
G({\bf s}_{\alpha, \delta}) ={\bf s}_{\frac{\alpha}{1+\alpha}, \frac{\delta}{1+\alpha}}, \widetilde{G}({\bf s}_{\alpha, \delta}) ={\bf s}_{\frac{\alpha}{1+\alpha}, \frac{\alpha+ \delta}{1+\alpha}}E({\bf s}_{\alpha, \delta}) ={\bf s'}_{1-\alpha, 1-\delta}.
\frac{d}{dv}\left(hvf(v)-\kappa\int_{\beta-1}^{v-1}F(u)\,du\right)=h(f(v)+vf'(v))-\kappa F(v-1) &=(h-\kappa)F(v-1)+h(1-\kappa^{-1})vf'(v)>0\quad\mbox{for}v>\beta,
\begin{array}{ll}[D,p^{\mu}]=-ip^{\mu}~~~ ~~~[D,k^{\mu}]=ik^{\mu}{}{}{[D,q_{\pm a}]=-i\textstyle{\frac{1}{2}}q_{\pm a}}~~~ ~~~[D,s^{\pm a}]=i\textstyle{\frac{1}{2}}s^{\pm a}{}{}\multicolumn{2}{c}{[D,D]=[D,R]=[D,M^{\mu\nu}]=[D,{{\cal A}_{\pm a}}^{b}]=0}\end{array}
S=\int dt \left[ n^{-1}e^{-\varphi}\left( 3\dot{\alpha}^2 -\dot{\varphi}^2 +6k n^2 e^{-2\alpha}+6\dot{\beta}^2 \right) +\frac{1}{2n}e^{\varphi+ 6\alpha}\dot{\sigma}^2 -\frac{n}{2}Q^2 e^{-6\beta +3\alpha}\right] ,
|\mathrm{cm}(x,x+y)|= \frac{\left|[x+y,\beta x - (x+y)]\right|}{||\beta x -(x+y)||_a||x+y||}= \frac{\beta\left|[y,x]\right|}{||\beta x -(x+y)||_a||x+y||}; \ \mathrm{and}\left|\mathrm{cm}(y,x+y)\right| = \frac{\left|[x+y,\alpha y - (x+y)]\right|}{||\alpha y - (x+y)||_a||x+y||}= \frac{\alpha\left|[x,y]\right|}{||\alpha y - (x+y)||_a||x+y||}.
\langle \sigma_{I}| = \lim_{\widetilde{\xi}^{1}\rightarrow 0}\left( \frac{1}{\widetilde{\xi}^{1}}\right)^{2 h_{\sigma_{I}}}\langle 0 | \sigma_{I}^{-}\left( - \frac{1}{\widetilde{\xi}^{1}}\right) ,
\mu^{i\rightarrow j}(\vec{\sigma}_{i\rightarrow j},\vec{\sigma}_{j\rightarrow i})\propto\sum_{\{\vec{\sigma}_{k\rightarrow i}\}_{k\in\partial i\backslash j}}W_{i}(\vec{\sigma}_{i\rightarrow j};\{\vec{\sigma}_{k\rightarrow i}\}_{k\in\partial i})\prod_{k\in\partial i\backslash j}\mu^{k\rightarrow i}(\vec{\sigma}_{k\rightarrow i},\vec{\sigma}_{i\rightarrow k}),
\frac{1}{2}\frac{d}{dt}\Vert v_i^{-}(\cdot,t)\Vert^2_{2,M}+\frac{1}{2}\frac{d}{dt}\Vert u_j^{-}(\cdot,t)\Vert^2_{2,\Omega}+\tilde d_i \int_M |\nabla v_i^{-}|^2\d\sigma+ d_j \int_\Omega |\nabla u_j^{-}|^2\dx &= -\int_\Omega u_j^{-}H_j(u^{+})\dx-\int_M u_j^{-}G_j(u^{+},v^{+})\d\sigma -\int_Mv_i^{-}F_i(u^{+}, v^{+})\d\sigma
\partial_s\Psi =\partial_su\left(0,\cos\theta,\sin\theta\right), \partial_x\Psi =\left(1,\partial_xu\cos\theta,\partial_xu\sin\theta\right), \partial_\theta\Psi =\left(u+\sqrt{2}\right)\left(0,-\sin\theta,\cos\theta\right)+\partial_\theta u\left(0,\cos\theta,\sin\theta\right).
K^* =\{u\in B:\; s_K(u)\leq 1\}=\{u\in B:\; s_K(u)+s_{tB}(u)\leq 1+t\|u\|\}\subset \{u\in B:\; s_K(u)+s_{tB}(u)\leq 1+t\}=(1+t)\{u\in (1+t)^{-1}B:\; s_K(u)+s_{tB}(u)\leq 1\}\subset (1+t) \{u\in B:\; s_{K+tB}(u)\leq 1\}= (1+t)(K+tB)^*\,.
< \lambda | \tilde{\Phi}_{\mu}^{V_{2}^{( 1 )}\lambda}( z_{2}) \tilde{\Phi}_{\lambda}^{V_{1}^{( 1 )}\mu}( z_{1}) | \lambda > = \sum_{m_{1}, m_{2}= 0 , 1 \atop m_{1}+ m_{2}= 1}v_{m_{1}}^{( 1 )}\otimes v_{m_{2}}^{( 1 )}< \lambda | \tilde{\Phi}_{\mu , m_{2}}^{V_{2}^{( 1 )}\lambda}( z_{2}) \tilde{\Phi}_{\lambda , m_{1}}^{V_{1}^{( 1 )}\mu}( z_{1}) | \lambda > .
b_{n,1}:= \tau(p_{r(0^{(n)}0^{(n)})}), b_{n,2}:= \tau(p_{r(0^{(n)}1^{(n)})}), b_{n,3}:= \tau(p_{r(1^{(n)}0^{(n)})}), b_{n,4}:= \tau(p_{r(1^{(n)}1^{(n)})})
T_{n \times p}=\left(\begin{array}{ccccc}a_{0}a_{-1}a_{-2}\cdots a_{1-p}a_{1}a_{0}a_{-1}\cdots a_{2-p}a_{2}a_{1}a_{0}\cdots a_{3-p}\vdots \vdots \vdots \cdots \vdots a_{n-1}a_{n-2}a_{n-3}\cdots a_{n-p}\end{array}\right).
\mathcal{ME}[u_0]:=\frac{\mathcal E[u_0]}{\mathcal E[\varphi]},\mathcal{MG}[u_0]:=\frac{\|\sqrt{\mathcal K_\lambda}u_0\|}{\|\sqrt{\mathcal K_\lambda}\varphi\|},\mathcal{MP}[u_0]:=\frac{\mathcal P[u_0]}{\mathcal P[\varphi]}.
\sum_{i=1}^K \phi(\eta_i \alpha_{i,y+1}) = \lambda_y \sum_{i=1}^K \phi( \eta'_i \alpha_{i,y+1}) \Longrightarrow \sum_{i=1}^K \eta_i \alpha_{i,y+1}= \lambda_y \sum_{i=1}^K \eta'_i \alpha_{i,y+1},
R =\frac{1}{NB}\left[|\mathcal{M}_2|\times|\mathcal{A}| +|\mathcal{M}_1|\times|\mathcal{A}^{*c}|+|\mathcal{M}_3|\times(|\mathcal{A}|-|\mathcal{A}^*|)\right] =[H(p_1^*)-H(p_1)]\times q_1 +[H(p_2^*)-H(p_2)]\times q_2^*+[H(p_2)-H(p_1^*)]\times(q_1- q_1^*).
N(n) - \sum_{i=1}^2 (-1)^{i+1}\binom{2}{i}N(n-24i) = 576n-12672 = 576(24k+r)-12672 &= (9216k + 384r - 12768)+(4608k + 192r + 96) &= 2\left(48\left(96k+4r-133\right)\right) + 96\left(2 \left(24k+r\right)+1\right) &= 2\cdot 48\cdot t(n-24)+96\left(2n + 1\right).
\langle \kappa^{\prime}, s | L_{0}| \kappa , s \rangle = \left[ \frac{s}{2}+ \frac{i \kappa}{4}\right] \delta ( \kappa - 2 i - \kappa^{\prime}) + \left[ \frac{s}{2}- \frac{i \kappa}{4}\right] \delta ( \kappa + 2 i - \kappa^{\prime}) .
a+b+2x_0 =\frac{2(\lambda+1)}{\alpha}, ab+2x_0(a+b)+x_0^2 =\frac{(\lambda-1)^2-2(\alpha+2\alpha_{-1})\beta}{\alpha^2}, 2abx_0+(a+b)x_0^2 =-\frac{2\beta(\lambda-1)}{\alpha^2}, abx_0^2 =\frac{\beta^{2}}{\alpha^2}.
M_{e^{-i\xi\left( \cdot\right)}}LM_{e^{i\xi\left( \cdot\right)}}=\sum_{l=0}^{m}\left( -1\right)^{l}\left( \partial+i\xi\right)^{l}M_{b_{l}\left( \cdot\right)}\left( \partial+i\xi\right)^{l}.
\det(A_n) = \prod_{\substack{1 \leqslant d_1 < d_2 \leqslant n d_1, d_2 \mid n}}\mathcal{R}(\Phi_{d_2}, \Phi_{d_1}) &= (-1)^{n-1}\prod_{i=1}^r p_i^{\frac{n(1-p_i^{-\alpha_i})}{(p_i-1)}}.
3\intop_\Omega\bigg|{1\over r}\psi_{1,r}\bigg|^2dx=-\intop_\Omega\psi_{1,rr}{1\over r}\psi_{1,r}dx-\intop_\Omega\psi_{1,zz}{1\over r}\psi_{1,r}dx-\intop_\Omega\omega_1{1\over r}\psi_{1,r}dx.
\rho^{*}= \int{\frac{d \rho}{f^{2}}}={\frac{R^{2}}{2 ( \rho_{+}^{2}- \rho_{-}^{2})}}\left[ \rho_{+}\ln \left({\frac{\rho - \rho_{+}}{\rho + \rho_{-}}}\right) - \rho_{-}\ln \left({\frac{\rho - \rho_{-}}{\rho + \rho_{-}}}\right) \right] .
\langle F^{1}, \phi\rangle_{L^{2}(\Omega)}= \langle F, \Phi\rangle_{(L^{2}(\Omega))^{9}}\leftarrow \langle AV_{n}, \Phi\rangle_{(L^{2}(\Omega))^{9}}=\tfrac{1}{\rho_{1}}\langle V^{3}_{n}, \phi\rangle_{L^{2}(\Omega)}\to \tfrac{1}{\rho_{1}}\langle V^{3}, \phi\rangle_{L^{2}(\Omega)}.
V_t^{(n)}(\theta):=(1-\theta)X_t+\theta X_t^{(n)}. &=x_0+\int_{0}^{t}\left\{(1-\theta)b(X_s) + \theta b(X_{\eta_n(s)}^{(n)}) \right\}ds+\int_{0}^{t}\left\{(1-\theta)\sigma(X_s) + \theta \sigma(X_{\eta_n(s)}^{(n)}) \right\}dW_s.
\int\limits_{[0,\,1]^n}\prod_{i=1}^n s_i^{\lambda_1}(1-s_i)^{\lambda_2}\, \prod\limits_{i<j}^n |s_i-s_j|^{-2/\tau}ds_1\cdots ds_n = \prod_{k=0}^{n-1}\frac{\Gamma(1-(k+1)/\tau)\Gamma(1+\lambda_1-k/\tau)\Gamma(1+\lambda_2-k/\tau)}{\Gamma(1-1/\tau)\Gamma(2+\lambda_1+\lambda_2-(n+k-1)/\tau)},
\dot{\Lambda}^{[\frac{d}{p}]-1}e^{(t - \tau)\Delta}\mathbb{P}\nabla .\big(u(\tau) \otimes v(\tau)\big) =\frac{1}{(t - \tau)^{\frac{d+1}{2}}}K\Big(\frac{.}{\sqrt{t - \tau}}\Big)*\Big(\dot{\Lambda}^{[\frac{d}{p}]-1}\big(u(\tau) \otimes v(\tau)\big)\Big),
\sum_{n , m , l = 0}^{\infty}[ n + m + l + c ]^{- s}= \frac{1}{2}\zeta ( s - 2 ; c ) + ( \frac{3}{2}- c ) \zeta ( s - 1 ; c ) + \frac{1}{2}( c - 1 ) ( c - 2 ) \zeta ( s ; c ) .
\Psi_T(U^1,U^2) = \left| \int_{0}^{T}(G(\tau), Z_t(\tau) ) d\tau\right|+ \left| \int_{0}^{T}(G(t), Z(t) ) dt\right| + \left| \int_{0}^{T}dt \int_{t}^{T}(G(\tau), Z_t(\tau) ) d\tau \right|
i{\cal{M}}_1 = 16i\alpha g^4\sin({q\tilde{l}\over 2})\epsilon_{\nu}q_{\mu}\left[g^{\mu\nu}(q - p)^{\rho}+ g^{\nu\rho}(p - l)^{\mu}+ g^{\rho\mu}(l-q)^{\nu}\right]{\tilde{l}_{\rho}(\tilde{l}j) \over l^4 \tilde{l}^4}.
S = \sum_{t = 1}^{T}\left\{\sum_{m > 0}\left[ - \phi_{m}^{*}( t ) \phi_{m}( t ) - x^{2}\phi_{m}^{*}( t ) \phi_{m}( t - 1 ) \right] - g_{_P}\tau \Phi^{*}( t ) \Phi ( t - 1 ) \right\},
\begin{array}{l}\displaystyle{\inf}\; \mathbb{E}_{\xi_2,\ldots,\xi_{T_{\max}, D_2,\ldots,D_{T_{\max}}}}\Big[ \displaystyle{\sum_{t=1}^{T_{\max}}}\; D_{t-1}f_{t}(x_{t}, x_{t-1}, \xi_t)\Big] x_t \in X_t( x_{t-1}, \xi_t )\;\mbox{a.s.},\;x_{t}\;{\overline{\mathcal{F}}}_t \mbox{-measurable,}t=1,\ldots,T_{\max},\end{array}
\frac{\mathcal{R_{\phi}}(t+h_n)-\mathcal{R_{\phi}}(t)}{S_t \sigma_t \ell_tJ_t h_n \sqrt{-\log |h_n|}}= \frac{1}{\ell_t \sigma_t \sqrt{-\log |h_n|}}\sum_{j=1}^{N_3(t,h_n)}\left(\phi(f_t^{j}(c))-\int \phi d\mu_t\right) +r'(t,h_n),
\frac{1}{3^{b+1}\alpha}\max\{\vert 12z_3\vert,\vert z_1+z_2\vert\}=\max\{\vert 4\beta\vert,\vert 3^{2b}\alpha^2\vert\}\geq e^{h(\tilde{x})}>e^{0.5N^2-4.444}.
\nabla P_{E_{{\boldsymbol w}_j}}[i] = \frac{1}{M\sqrt{2\pi}\sqrt{{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]}}\sum^{M}_{j=1}\exp\big(-\frac{c^2_j[i]}{2}\big)sgn(s_j) \frac{{\boldsymbol{\bar{r}}}[i] - \frac{1}{2}\bar{s}_j[i]{\boldsymbol w}_j[i]}{\sigma_{n}{\boldsymbol w}^\emph{H}_j[i]{\boldsymbol w}_j[i]},
\varepsilon(h,y)\varepsilon(x,y+z)[[\alpha(y),h\cdot z],\alpha^2(x)]= 2\varepsilon(x,y+z)[\alpha(h)\cdot [y,z],\alpha^2(x)] &-\varepsilon(x,y+z)[[h\cdot y,\alpha(z)],\alpha^2(x)]=0.
p\left({{d_{pF}}\leq \delta}\right) = \frac{{M{\delta^{2nk -{2k^2}}}}}{{k!}}\int_{{{\mathbf{\Omega}}_2}}{\prod\limits_{i = 1}^k{{x_i}^{n - 2k}}\prod\limits_{1 \leq i < j \leq k}{{{\left({{x_i}-{x_j}}\right)}^2}}d{x_1}\cdots d{x_k}}
\Phi ({\bf r}_{1}^{( 1 )}, \ldots ,{\bf r}_{n_{M}}^{( M )}, t ) = e^{- \frac{i}{\hbar c}\Lambda}\bar{\Phi}({\bf r}_{1}^{( 1 )}, \ldots ,{\bf r}_{n_{M}}^{( M )}, t ) ,
[ \delta_{\kappa_{1}}, \delta_{\kappa_{2}}] z^{M}= \delta_{b}z^{M}= - \mathrm{i}( b_{\nu}{\widetilde B}^{\nu}) z^{M}, \qquad b^{\mu}= - \mathrm{i}{\overline{\kappa}}_{1}\Gamma^{\mu}\kappa_{2},
K_n^{\lambda,\theta,T}(y,y') := - \frac{1}{n}\sum_{z^n = (-1)^{\theta_2}}\frac{z^{h-h'}e^{- (\lambda+\theta_1 \pi i+Tz)([t'-t]+\mathrm{1}_{t'=t})}}{1 - e^{- (\lambda+\theta_1 \pi i +Tz)}}.
Jf(\theta, s) = \sqrt{\det[( \nabla f) (\nabla f)^T]}=\sqrt{\det\left(\begin{array}{cc}\partial_\theta f \cdot \partial_\theta f \partial_\theta f\cdot \partial_s f \partial_sf \cdot \partial_\theta f \partial_s f\cdot \partial_s f\end{array}\right).}
\frac{\partial u_\lambda}{\partial t}\, - \, \triangle u_\lambda \, + \, \lambda^{1-\kappa}\bigl( | u_\lambda |^2 \, - \, 1 \bigr) u_\lambda \; = \; 0, &\frac 12 |\nabla u_\lambda|^2 \, + \, \frac{\lambda^{1-\kappa}}4 \chi ((| u_\lambda |^2 \, - \, 1 )^2).
\mathbf{v}= \mathbf{v}_{0}+ \nabla \varphi, \mathbf{v}_{0}\in \mathbf{H}^{1}_{t,0}(\Omega),\varphi \in H_{0}^{1}(\Omega)\cap H^{2}(\Omega), \Vert \mathbf{v}_{0}\Vert_{\mathbf{H}^{1}}\leq C\Vert \mathbf{v}\Vert_{\mathbf{H}^{1}}, \;\; \Vert\varphi\Vert_{H^{2}}\leq C \Vert \mathbf{v}\Vert_{\mathbf{H}^{1}}.
\pi_\sigma(a_1\otimes a_2)\lambda^\sigma(t,t)=\pi_{\varphi_1\bar{\otimes}\varphi_2}(a_1\otimes a_2)\lambda^{\varphi_1\bar{\otimes}\varphi_2}(t), a \in M_1, b \in M_2, (t,t) \in H