harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal A}\Phi_k := (-\lambda_k) \Phi_k, D({\cal A}) = \Big\{u \in{\cal H}\Big| \sum^\infty_{k =1}\lambda^2_k |\langle u, \Phi_k \rangle|^2 < \infty\Big\},{\cal A}u = \sum^\infty_{k = 1}(-\lambda_k) \langle u, \Phi_k \rangle \Phi_k.
\beta_N(x^N,B) =\mathbb{P}_{N,m}\circ ({X}^{1,N,m}_1, \dots,{X}^{N,N,m}_1)^{-1}(B) & =\mathbb{P}_{N,m}\bigg(\Big(\Psi(0, x^N_j, \frac{1}{N-1}\sum_{k\neq j}\delta_{x^N_k}, \Phi^{N,m}_j(0,x^N_j), \xi^{j,N,m}_1)\Big)_{j = 1}^N \in B\bigg),
\small\Pr\{\mathcal{H}_j | \hat{\mathcal{H}}_i\}\!=\! \frac{\Pr\{\mathcal{H}_j\}\Pr\{\hat{\mathcal{H}}_i | \mathcal{H}_j\}}{\Pr\{\mathcal{H}_0\}\Pr\{\hat{\mathcal{H}}_i | \mathcal{H}_0\}\!+\!\Pr\{\mathcal{H}_1\}\Pr\{\hat{\mathcal{H}}_i | \mathcal{H}_1\}}\,\, i \! \in\! \{0,1\}. \normalsize
\omega_1 =a_1b_1 + a_2b_2+zc_1+c_2c_3; \omega_2 =-a_1a_2 +b_1b_2-zc_2+c_1c_3; \omega_3 =- a_1b_2 + a_2b_1-zc_3-c_1c_2.
\sum_i (\nabla_{\tilde{e_i}}\xi)_{\tilde{e_i}}-\xi_{S(\tilde{e_i},\tilde{e_i})}=0 \qquad\textrm{and}\sum_iR_{\xi_{\tilde{e_i}}}(\tilde{e_i})-S(\tilde{e_i},\tilde{e_i})=0,
-\sin\psi\cdot\left(\frac{F[p,q](i\mu_{0}+1)w_{2}[p,q](i\mu_{0}+1)w_{3}[p,q](i\mu_{0}+1)}{w_{1}[p,q](i\mu_{0}+1)}\right)^{-\frac{1}{2}}\gamma^{1}\partial_{\eta}u_{n}(\mu)\mid_{\mu=\mu_{0}-i}
J_N^{\ast}(x) =\min_{\boldsymbol{u}}\sum_{k=0}^{N-1}\ell(x_k,u_k) + F(x_N) \{x}_{k+1}=A{x}_{k}+B{u}_{k}, \{x}_0=x \ , x_k \in \mathcal{X}, \k=0,\dots, N-1, \x_N \in \mathcal{X}_f \ , &u_k \in \mathcal{U}, \k=0,\dots, N-1 \ .
\sum_{\mu\in{\cal R}}\widetilde{M}_{\mu\nu}= \sum_{\nu\in{\cal R}}\widetilde{M}_{\mu\nu}= -i{\cal D}\equiv{i\over2}\sum_{\rho\in\Delta_{+}}g_{|\rho|}|\rho|^2\,y (\rho\cdot q) =-{i\over2}\sum_{\rho\in\Delta_+}g_{|\rho|}|\rho|^2\,V (\rho\cdot q).
0.99999997 \sum_{p \geq x}\frac{1}{(p-1)^2}< \sum_{p \geq x}\frac{1}{p^2}= -\frac{\theta(x)}{x^2 \log x}+ \int_x^\infty \frac{\theta(y)}{y^3}\frac{1 + 2 \log y}{(\log y)^2}dy.
\frac{2}{\sqrt{2\pi \epsilon}}e^{-\frac{2}{3}((\epsilon+t)^3-t^3)}\mathbb{E}\left[\left(-2\sqrt{\epsilon^3}\int_{0}^1 \textbf{e}(z)dz \right)\right]=\frac{2}{\sqrt{2\pi \epsilon}}+O(\sqrt{\epsilon}) =\int_{\epsilon}^{\infty}\frac{1}{\sqrt{2\pi u^3}}du+O(\sqrt{\epsilon})
H_{q_{i}}(z,\omega)\equiv\frac{1}{2p_{q}}\int_{0}^{z}\frac{1}{v_{zi}}V_{i}\cdot\tilde{\epsilon}_{q}^{*}(x_{i},y_{i})\times exp[j(\omega\int_{0}^{z}\frac{1}{v_{zi}(z)}dz-k_{zq}z)]dz
\varphi = -|\Omega|_\omega dr^1 \wedge dr^2 \wedge dr^3 + dr^1 \wedge |\Omega|_\omega \omega + dr^2 \wedge \Re \Big( \frac{1}{|\Omega|_\omega}\Omega \Big) + dr^3 \wedge \Im \Big( \frac{1}{|\Omega|_\omega}\Omega \Big).
\int_0^{\frac{\pi}{2}}E_{2n-1}\left( \frac{2}{\pi}x\right) \log\left( \tan x \right) \d x = 4(-1)^n(2n-1)!\pi^{-2n}\int_0^{\frac{\pi}{2}}C_n\left( \frac{2}{\pi}x\right) \log\left( \tan x\right) \d x =\frac{4(-1)^n(2n-1)!}{\pi^{2n}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{2n}}\int_0^{\frac{\pi}{2}}\cos\left(2(2k+1)x\right) \log \left( \tan x\right) \d x.
i_1<...<i_{k-2}, \; \; j_1<...<j_{b-k}, \{i_1,...,i_{k-2}\}\cap \{j_1,...,j_{b-k}\}= \emptyset, \; \{i_1,...,i_{k-2}\}\cup \{j_1,...,j_{b-k}\}= \{2,...,\widehat{t},...,b\}
S^{( 1 )}= \ln \mathrm{D e t}K_{\mathrm{\scriptsize b o s}}- \ln \mathrm{D e t}K_{\mathrm{\scriptsize f e r m i}}- \ln \mathrm{D e t}( 1 - K_{\mathrm{\scriptsize f e r m i}}^{- 1}K_{\mathrm{\scriptsize m i x}}K_{\mathrm{\scriptsize b o s}}^{- 1}K_{\mathrm{\scriptsize m i x}}^{\dagger}) .
\forall w\in\mathbb{T},DG(\lambda,b,\Omega,0,0)(h_{1},h_{2})(w)=\sum_{n=1}^{\infty}n\mathbf{m}M_{n\mathbf{m}}(\lambda,b,\Omega)\left(\begin{array}{c}a_{n}b_{n}\end{array}\right)e_{n\mathbf{m}}(w).
M_b(f)(x) =\sup_{Q\ni{x}}\frac1{|Q|}\int_Q|b(x)-b(y)||f(y)|dy \le{C}\|b\|_{\dot{\Lambda}_{\beta}}\sup_{Q\ni{x}}\frac1{|Q|^{1-\beta/n}}\int_Q |f(y)|dy ={C}\|b\|_{\dot{\Lambda}_{\beta}}\mathfrak{M}_{\beta}(f)(x).
{\cal Q}_{\bar{i}\bar{j}}(\psi,\bar{\psi})\equiv c\alpha\beta_{\bar{i}\bar{j}}+\bar{c}\alpha'\beta'_{\bar{i}\bar{j}}=c(\psi^{\dag}\zeta)(\zeta^{\dag}\gamma_{\bar{i}\bar{j}}\psi)+\bar{c}(\psi^{T}\bar{\zeta})(\zeta^{\dag}\gamma_{\bar{i}\bar{j}}\bar{\psi})
\varphi^{A}(t,x,\omega)=\alpha_{t-\sum_{j=1}^{i-1}t_{j}}^{\omega_{i}}(\alpha_{t_{i-1}}^{\omega_{i-1}}(\cdots(\alpha_{t_{1}}^{\omega_{1}}(x))\cdots)),\; \; \;t\in \left( \sum_{j=0}^{i-1}t_{j},\sum_{j=0}^{i}t_{j}\right] .
P(Z_V = z_V) = \sum_{B \subseteq V}P(U_B = 1, U_{V \setminus B}= 0, X_B = z_B, Y_{V \setminus B}= z_{V \setminus B}) &= \frac{1}{2^{|V|}}\sum_{B \subseteq V}P(X_B = z_B, Y_{V \setminus B}= z_{V \setminus B}).
\tilde\Gamma_s=\left\{z=z^{(s)}+\vartheta^{(s)}\varepsilon_s(z^{(s)})\,,\quad\vartheta=\vartheta^{(s)}\left(1+\frac{\varepsilon_s\varepsilon'_s}2\right)+\varepsilon_s(z^{(s)})\right\}\,,
{\pi}^+=\frac{{\delta}L}{{\delta}{\partial}_+A_+}=0,\;{\pi}^-=\frac{{\delta}L}{{\delta}{\partial}_+A_-}=F_{+-},\;{\pi}^i=\frac{{\delta}L}{{\delta}{\partial}_+A_i}=F^+_{\;\;i},\;{\pi}_B=\frac{{\delta}L}{{\delta}{\partial}_+B}=0,
z_{tt}(t, x) - \bar{a}_{ij}(t, x) \partial_{x_{i}}\partial_{x_{j}}z(t, x) = \bar{f}(t, x)\phantom{0}(t, x) \in (0, T) \times \Omega, z(t, x) = 0\phantom{\bar{f}(t, x)}(t, x) \in [0, T] \times \partial \Omega, z(0, x) = z^{0}(x), z_{t}(0, x) = z^{1}(x)\phantom{00}x \in \Omega.
\sum_{p\in \mathcal{T}}h_{qp}(u)v_{{\mathcal{R}},{\mathcal{T}},p,n}^i(u)=0, \forall q\in \bar{\mathcal{R}}_i,\forall n\in[tN^{(N_R-r-t)(N_T-t+1)}],\forall u\in[S],
\Bigl|\mathrm{dist}\Bigl(v, \int_s^rF(x(t),m(t)dt\Bigr)- \mathrm{dist}\Bigl(v',\int_s^rF(x'(t),m'(t)dt\Bigr)\Bigr| \leq \|v-v'\|+L\int_s^r(\|x(t)-x'(t)\|+W_1(m(t),m'(t)))dt.
\frac{\mathrm{I m}R_{i i}^{( 1 )}}{\mathrm{I m}R_{i i}^{( 0 )}}= \operatorname{e x p}\left[ - \frac{1}{\hbar}\int_{- \infty}^{\infty}d t \Gamma^{( 1 )}( t ) \right] \; .
\left( 2H - \sum_{i=11}^{16}s_i E_i - \sum_{j=1}^{16}r_jE_j \right) - \frac{1}{2}\left( 3H - \sum_{i=11}^{16}E_i \right) \equiv \frac{1}{2}\left( H - \sum_{i=11}^{16}(2s_i-1) E_i - \sum_{j=1}^{16}2r_jE_j\right)
\begin{array}{c}\Omega=d\left(p_\mu dx^\mu +\theta^\mu_i g_{\mu\nu}D\theta^\nu_i\right)= =dp_\mu\wedge dx^\mu+\frac 12 R_{\mu\nu\lambda\rho}\theta^\mu_i\theta^\nu_i dx^\lambda\wedge dx^\rho +g_{\mu\nu}D\theta^\mu_i\wedge D\theta^\nu_i,\end{array}
\Big\langle \:\chi_{jk}\,\boxtimes\,\big(u_0^{\otimes(N_1-j)}\otimes v_0^{\otimes(N_2-k)}\big)\;,\;\chi_{\ell r}\,\boxtimes\,\big(u_0^{\otimes(N_1-\ell)}\otimes v_0^{\otimes(N_2-r)}\big)\;\Big\rangle\;= =\|\chi_{jk}\|_2^2\;\delta_{j\ell}\,\delta_{kr}\,,
\dot{\theta}= \left(\frac{x}{2}- \frac{m-1}{x}\right)\sin \theta + \left(\frac{n-1}{y}- \frac{y}{2}\right)\cos \theta + \lambda &\geq - \frac{m-1}{x}\dot{x}\tan \theta - \left(\frac{n-1}{y_b}- \frac{y_b}{2}\right) + \lambda &\geq - \delta \frac{\dot{x}}{x}- \left(\frac{n-1}{y_b}- \frac{y_b}{2}\right) + \lambda,
\begin{aligned}\| \partial_t \widetilde e^{\; 0}_{\tau,h}(t)\|_1 \le \| \partial_t u^0 (t) - \partial_t L_\tau u^0_{\tau,h}(t) \|_1 + \|-\partial_t\theta^0(t)\|_1 \lesssim \tau^{k+1}+ h^{r}\qquad\; t\in\bar I\,,\end{aligned}
I_1(s) =\int_{x}^{x+2s}\frac{f(x+s)dt}{(x+s-t)^2+v(s)^2}, I_2(s) =\int_{x+s}^{x+2s}\frac{(f(t)-f(x+s))dt}{(x+s-t)^2+v(s)^2}-\int_{x}^{x+s}\frac{(f(x+s)-f(t))dt}{(x+s-t)^2+v(s)^2}
\bar{C}(\mathcal G) = \max_{q(x_1)}\min_{p(y_2|x_1)\in \mathcal{P}}\log\left(\sum_{y_2}\underset{x_1: q(x_1)>0}{\max}\;p(y_2|x_1)\right) &=\min_{p(y_2|x_1)\in \mathcal{P}}\log\left(\sum_{y_2}\underset{x_1}{\max}\;p(y_2|x_1)\right).
e(N(1)) = (-1)^{n-1}\frac{p^{(n-1)}(1)}{(n-1)!}& = (-1)^{n-1}\left[\sum_{t=1}^{n-1}(-1)^t\cdot t\cdot \sum_{1\leq j_1\leq\cdots\leq j_{t+1}\leq n}{d_{j_1}+\cdots + d_{j_{t+1}}-1\choose n-1}\right] + (-1)^{n-1}& = \sum_{t=1}^{n-1}\left[(-1)^{n-1+t}\cdot t\cdot \sum_{1\leq j_1\leq\cdots\leq j_{t+1}\leq n}{d_{j_1}+\cdots + d_{j_{t+1}}-1\choose n-1}\right] + 1,
\|u(t,x)-u(s,y)\|_{L^p}\le \sum_{n=1}^\infty \|I_n(h_n(\cdot,t,x))-I_n(h_n(\cdot,s,y))\|_{L^p}&\le \sum_{n=1}^\infty (p-1)^{n/2}\|I_n(h_n(\cdot,t,x))-I_n(h_n(\cdot,s,y))\|_{L^2}&= \sum_{n=1}^\infty (p-1)^{n/2}\sqrt{n!}\|h_n(\cdot,t,x)-h_n(\cdot,s,y)\|_{\mathcal H^{\otimes n}},
\int_{\Omega}\nabla u : \nabla v \dx +\int_0^{2\pi}\left(\partial_{x_1}\eta\right)\left(\partial_{x_1}\zeta\right) \dx_1=\int_{\Omega}f\cdot v \dx+\int_0^{2\pi}g \zeta \dx_1 ([v,\zeta]\in \mathcal{V}).
\|{\tilde T}_{\bf p}\|_{L^2}\ \lesssim \ \begin{cases}2^{- q/2}\ \ \ |c_{\alpha_{*}}\, 2^{p j_{*}+ q k_{*}}|^{-\delta}\mbox{\rm if}\p\le 0 2^{-(p+q)/2}\ |c_{\alpha_{*}}\, 2^{p(j_{*}-1) + q k_{*}}|^{-\delta}\mbox{\rm if}\p\ge 0 \end{cases}
\left. Z ( \Omega , D ) = \int \, d \mu_{I_{\beta}}\, ( A ) \, \sum_{\tau}\, c_{\tau}\, \left[ z ( \beta ) c_{\beta}( A , D - \Omega ) + z ( \beta , \rho^{\tau}) \, c_{\beta}( A , D + \rho^{\Omega_{\tau}}) \right] \times \prod_{\rho \in{\cal E}_{\tau}}\, \left[ 1 + z ( \beta , \rho ) \, c_{\beta}( A , \rho ) \right] \ . \right.
(\delta_\mathrm{Hoch}f) (a_1, \ldots, a_{n+1}) =~ (-1)^{n+1}~ a_1 \cdot f (a_2, \ldots, a_{n+1}) + f(a_1, \ldots, a_n) \cdot a_{n+1}~ + \sum_{i=1}^n (-1)^{i+n+1}~ f (a_1, \ldots, a_i \cdot a_{i+1}, \ldots , a_{n+1}),
\sum_{t = 1}^\infty f_\delta(t) = \delta + \sum_{t = 2}^{K_\delta+1}f_\delta(t) + \sum_{t = K_\delta + 2}^\infty f(t) = \delta + x + f(K_\delta + 1) + 1 - F(K_\delta + 1) = 1.
F_{( J \neq 0 )}\approx - \frac{l^{3}\zeta ( 3 )}{\sqrt{2}\beta^{3}}\left( \frac{r_{+}}{r_{+}^{2}- r_{-}^{2}}\right)^{3 / 2}\frac{r_{+}}{\sqrt{\epsilon}}+ \frac{l \Omega_{H}}{\pi}\frac{r_{-}}{r_{+}^{2}- r_{-}^{2}}\left[ \frac{1}{2 \epsilon_{m}^{3}}- \frac{\zeta ( 3 )}{( \beta \Omega_{H})^{3}}\frac{r_{+}^{2}}{r_{+}^{2}- r_{-}^{2}}\right] \ln \left( \frac{r_{+}}{\epsilon}\right) ~ ,
\int_a^x f(\hat{x})\,(d\hat{x})^n\,\equiv\,\underbrace{\int^{x}_{\!a}\!\int^{x_n}_{\!a}\!\! \int^{x_{n-1}}_{\!a}\!\!\!\!\!\!\!\!\cdots\!\int^{x_3}_{\!a}\!\!\int^{x_2}_{\!a}\!}_{n \mbox{\footnotesize-times}}f(x_1) \;dx_1\,dx_2 \cdots dx_{n-1}\,dx_n
\Delta_{-}( x^{\prime \prime}- x^{\prime}) = - i \int_{\Sigma ( \tilde{x}^{0})}d^{3}{\bf \tilde{x}}\Delta_{-}( x^{\prime \prime}- \tilde{x}) \stackrel{\leftrightarrow}{\partial_{0}}\Delta_{-}( \tilde{x}- x^{\prime}) .
\limsup_{r \rightarrow 0}\left| \frac{\log \mu ( \Sigma_r ( \xi , x ) )}{\log r}- a \right| =\limsup_{N \rightarrow \infty}\left| \frac{\log \mu ( \Sigma_{\lambda^{M N}(x)}( \xi , x ) )}{\log \lambda^{M N}(x)}- a \right|
G_{F S}( \lambda ) = [ a \sin{( 1 . 2 3 \lambda )}+ b \cos{( 1 . 2 3 \lambda )}] e^{0 . 5 5 3 \lambda}+ c e^{- 0 . 3 3 2 \lambda}+ d e^{- 2 . 5 7 \lambda}+ 2 . 8 9 ,
\lim_{\epsilon\to0}\Big\langle \Lambda^TE(\cdot,\frac{1+\epsilon + ir}{2}) , \Lambda^TE(\cdot,\frac{1+\epsilon + ir}{2}) \Big\rangle &=2\log T + m\Big(\frac{1-ir}{2}\Big)\frac{T^{ir}}{ir}- m\Big(\frac{1+ir}{2}\Big)\frac{T^{-ir}}{ir}- \frac12\left(\frac{m'}{m}\Big(\frac{1-ir}{2}\Big) + \frac{m'}{m}\Big(\frac{1+ir}{2}\Big)\right).
\frac 1x \#\left\{n \le x: \frac{\log_2 P^{(\alpha)}(n)-\alpha\log_2 x}{\sqrt{\log_2 x}}< t\right\}= \Phi\left(\frac{t}{\sqrt{\alpha(1-\alpha)}}\right) + O_\alpha\left(\frac{1}{\sqrt{\log_3 x}}\right)
A(t)v(\widehat x)=\frac{1}{(2\pi)^{2n}}\iint\!\! e^{i(\Psi(t,\widehat x,\widehat\eta)-\langle\widehat y,\widehat \eta\rangle)}a(t,\widehat x,\widehat\eta)v(\widehat y)d\widehat yd\widehat\eta
G^-_{1,\varepsilon}(x)={I}+\sum\limits_{r=1}^{\infty}\varepsilon^r N_{r,\varphi}^{-}(x) (\Lambda^{+}_{0}(x))^{-1}, G^+_{1,\varepsilon}(x)={I}+\sum\limits_{r=1}^{\infty}\varepsilon^r N_{r,\varphi}^{+}(x).
\langle \psi^{-}|{\mathsf s}^{-}\rangle \langle^{+}{\mathsf s}| \phi^{+}\rangle = \langle \psi^{o u t}|{\mathsf s}\rangle \langle{\mathsf s}| \phi^{i n}\rangle \in{\cal S}\cap{\cal H}_{-}^{2},
\Big<\sum_{l \leq j \epsilon}T_a^{j,l}f, g\Big> \lesssim_{\epsilon}\sum_{j\ge 0}2^{(-j\rho+j\epsilon)n(1/r-1/s)}2^{jm - jn(\frac{1}{s}- \frac{1}{r})}& \times\sum_{\substack{Q: \ell(Q)= 2^{\lfloor{-j\rho+j\epsilon+10}\rfloor}}}|Q|\left<f\right>_{r, Q}\left<g\right>_{s', Q}.
\rho_L = \begin{pmatrix}u_3 \hat{u}^T u_0\|\hat{u}\|^2 u_3\xi u_3\xi\cdot v^T+A u_0\xi\|\hat{u}\|^2+Av \frac{u_3}{2}\|\xi\|^2 \frac{u_3}{2}\|\xi\|^2v^T+\xi^TA \frac{u_0}{2}\|\xi\|^2\|\hat{u}\|^2+\xi^TAv+u_3^{-1}\end{pmatrix}
\lim_{\xi_{1}\rightarrow 0}\langle 0_D| T_{i}^{k}| 0_D\rangle=\langle 0_{R}| T_{i}^{k}| 0_{R}\rangle +\langle T_{i}^{k}\rangle^{(1b)}_D(\xi_{2},\xi ),\quad \xi <\xi_{2},
\int_{({\mathbb R}^d)^{k-1}}V(K_1\cap K_2^{x_2}\cap \dots \cap K_k^{x_k}[j], M_{j+1},\dots,M_d)\, \lambda^{k-1}(d(x_2,\dots ,x_k)) & = \sum_{{m_1,\dots,m_k=j}\atop{m_1+\dots +m_k=(k-1)d+j}}^dV^{(j)}_{m_1,\dots,m_k}(K_1,\dots,K_k;M_{j+1},\dots,M_d),
\Pr_{c\sim\mathcal{D}(n,d)}[|c|\leq 2^{-j}\cdot 2^n] \leq2^{-\big( 1-17(\frac{j}{1-\frac{d}{n}}+\frac{2-\frac{d}{n}}{(1-\frac{d}{n})^2})(\frac{d}{n})^{j-1}\big)\binom{n}{\leq d}+ O(n^4)}.
I_{\delta,\theta}\geq(1-\theta)\frac{\rho}{2}\big\{(1-\nu)\ln \det\Theta + \nu\ln\det\Theta-3 \ln T_{\delta}+\delta\big(\ln T_{int}-\ln T_{\delta}\big)\big\}\cr =(1-\theta)\frac{\rho}{2}\big\{\ln\det\Theta+\delta\ln T_{int}-(3+\delta)\ln T_{\delta}\big\}.
\Psi(0)={}_3F_2\bigg[\begin{matrix}-a 1+a \beta &1 2\beta\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(\frac12)\Gamma(\frac12+\beta)\Gamma(\beta)}{\Gamma(\frac12-\frac12a)\Gamma(1+\frac12a)\Gamma(\beta-\frac12a)\Gamma(\frac12+\beta+\frac12a)}.
(A^{*m}|A^p|^{2k}A^m)^\frac{m}{pk+m}= (U_m^*|A^{*m}||A^p|^{2k}|A^{*m}|U_m)^\frac{m}{pk+m}&=U_m^*(|A^{*m}||A^p|^{2k}|A^{*m}|)^\frac{m}{pk+m}U_m.
\hat{\cal M}= \left( \hat{\cal M}_{i j}\right) = \frac{1}{\Im \mathrm{m}\lambda}\left( \begin{array}{r r}{| \hat{\lambda}|^{2}}&{- \Re \mathrm{e}\hat{\lambda}}\\{- \Re \mathrm{e}\hat{\lambda}}&{1}\\ \end{array}\right) ,
\sum_{x=0}^{\infty}{\dfrac{(N_x+1)^{-\rho/p + (1-2\rho)/q}}{x+1}}\leq \left(\sum_{x=0}^{\infty}\dfrac{(N_x+1)^{-\rho}}{(x+1)^{\alpha}}\right)^{1/p}\left(\sum_{x=0}^{\infty}\dfrac{(N_x+1)^{1-2\rho}}{(x+1)^{2\alpha}}\right)^{1/q}<+\infty.
\P_N\Biggl(\prod_{i=1}^{N}(\Delta \tau_i)^{-\frac d2}>C^{-N}R \Biggr) \geq C^NN^{(1+\frac d2)N}R^{-1-\frac 2d}\biggl(\log \frac{tR^{\frac2{dN}}}{CN}\biggr)^{N-1}.
\int_{B_R \cap \{|v| > k\}}|v|^q \,dx = \int_0^\infty q t^{q - 1}|\{|v| \geq \max(t,k) \}| \,dt \leq |\{|v| \geq k \}| k^{q}+ q \int_k^\infty t^{q - 1}|\{|v| \geq t \}| \,dt,
0 \leq c_m = \| \left(\mathbf{W}^{\mathrm{RF}}\right)^{\mathrm{H}}\mathbf{p}_m\|^2_2 = \|\mathbf{U}_\mathrm{p}^{\mathrm{H}}\mathbf{p}_m\|^2_2 \leq \|\mathbf{U}^\mathrm{H}\mathbf{p}_m\|^2_2 = 1,
\int_{\gamma_k}\frac{\omega_T}{V_T}=-\int_{\gamma_k}dH_T=-\int_{\gamma_k}\Big[d\left(\ln A_T\right)+d\left(\ln B\right)+d\left(\ln C\right)\Big] &=2\sum_{j=1}^r\int_{\gamma_k}d\theta_j=\pm 4\pi s_k\neq 0\,,
m_{\pi_1}=\frac{1}{2}\left([5]_q!-[4]_q!(1+2(-1)^i+2(-1)^j) \right) &=\frac{1}{2}(q+1)^2(q^2+1)(q^2+q+1)\left(q(q+1)(q^2+1)-2(-1)^i-2(-1)^j \right).
D_{0}=\frac{2D_{1}^{\prime}\sigma_{\mathcal{F}}^{2}}{M},D_{1}=N_{0}2^{N_{0}}M_{N_{0}},D_{2}=\frac{D_{2}^{\prime\prime}P_{N_{0}}^{2}}{(1+M+D_{0})^{2N_{0}}},
\left\|\sum_{j\in \mathbb{S}}\widehat{F}_jA_jx\right\|^p=\left\|\sum_{j\in \mathbb{S}}\widehat{F}_j\widehat{L}_j\left(\sum_{k\in \mathbb{S}}L_kA_kx\right)\right\|^p=\sum_{j\in \mathbb{S}}\left\|\widehat{L}_j\left(\sum_{k\in \mathbb{S}}L_kA_kx\right)\right\|^p =\sum_{j\in \mathbb{S}}\|A_jx\|^p.
\Vert\partial_2\varphi(t)\Vert^2_{L^2(\mathbb R)}\le C_2\{\Vert{\mathbf V}_n(t)\vert_{x_1=0}\Vert^2_{L^2(\mathbb R)}+\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}\}&\le C_2\{I(t)+I_{1,n}(t)+\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}\}\,,
U(p)\left(\sum_{n \in \mathbb{Z}}C(n)q^{n}\right):=\sum_{n \in \mathbb{Z}}C(pn)q^{n}& V(p)\left(\sum_{n \in \mathbb{Z}}C(n)q^{n}\right):=\sum_{n \in \mathbb{Z}}C(n/p)q^{n}& D^{k-1}\left(\sum_{n \in \mathbb{Z}}C(n)q^{n}\right):=\sum_{n \in \mathbb{Z}}n^{k-1}C(n)q^{n}.
I_1 - \frac{(k-2)!}{(4\pi m)^{k-1}}\leq \frac{1}{(4\pi)^{k-1}}\sum_{n=\ell}^\infty |A_k^{(1)}(m,n)|^2 e^{-2\pi \sqrt{3}n}\frac{(k-2)!}{\ell^{k-1}}\sum_{i=0}^\infty \frac{(2\pi \sqrt{3}\ell)^i}{i!}\leq \frac{(k-2)! e^{2\pi \sqrt{3}\ell}}{(4\pi\ell)^{k-1}}\sum_{n=\ell}^\infty |A_k^{(1)}(m,n)|^2 e^{-2\pi \sqrt{3}n}.
\frac{\lambda}{12}\sum_{\{x_i\in G\}}\sum_{k_i=1}^{\mu}\; \phi^{k_1k_2k_3}_{x_1x_2x_3}\phi^{k_1k_4k_5}_{x_1x_4x_5}\phi^{k_2k_5k_6}_{x_2x_5x_6}\phi^{k_3k_6k_4}_{x_3x_6x_4}=
S=\int d^3x\left[\frac12(\nabla\vec\chi)^2+\frac12(\nabla\psi)^2+\frac{m_H^2}{2}\psi^2-2\zeta{\rm e}^{g_m\psi}\sum\limits_{i=1}^{N(N-1)/2}\cos\left(g_m\vec q_i\vec\chi\right)\right].
\delta(P\searrow Q) =\frac{1}{|min(Q)|}\left( \sum \limits_{\underset{}{v\in X_2,\, v\in \hbox{\tiny{min}}(Q)}}P \otimes Q + \sum \limits_{\underset{J\circledcirc Q}{}}(P\otimes \mathbf 1)\searrow (J \otimes Q\backslash J) + \sum \limits_{\underset{J\circledcirc Q}{}}(\mathbf 1\otimes P)\searrow (J \otimes Q\backslash J)\right).
\alpha ={s(v)}{e^{w\over2}v(w-m)+nws(w)\over (m-w)(e^{v\over2}ws(w)-e^{w\over2}vs(v))}\cr\tilde u = m \cr\tilde v ={mv\over w}\cr\tilde c = \Big[\Big({\alpha\over 1-e^{-v}}-{\alpha\over v}\Big)(w-m) +\Big({\beta\over 1-e^{-w}}-{\beta\over w}\Big)(v-m)+\Big[1+\alpha\Big({w\over v}-1\Big)\Big]c+e
\mathfrak{N}_{\alpha}=(\iota\circ \mathcal{L})^{\dag}\circ\mathfrak{M}_{\alpha}\circ (\iota\circ \mathcal{L}) = (\iota\circ \mathcal{L})^{\dag}\circ\mathfrak{M}_{\alpha,1}\circ (\iota\circ \mathcal{L})+ (\iota\circ \mathcal{L})^{\dag}\circ\mathfrak{M}_{\alpha,2}\circ (\iota\circ \mathcal{L})= \mathfrak{N}_{\alpha,1}+\mathfrak{N}_{\alpha,2}.
\div\, W (x) = \phi'\big(\textstyle{\frac{|x|}{t}}\big)\cdot\,\frac{x}{t\,|x|}\Big(\frac{|\nabla u|^2}{2}x-(\nabla u\cdot x)\nabla u\Big)|x_{n+1}|^a + \phi\big(\textstyle{\frac{|x|}{t}}\big)\,\frac{n+a-1}{2}\,|\nabla u(x)|^2|x_{n+1}|^a.
[ b_{{\bf{0}}}, b_{{\bf{0}}}^{\dagger}] = b_{{\bf{0}}}b_{{\bf{0}}}^{\dagger}- b_{{\bf{0}}}^{\dagger}b_{{\bf{0}}}= e x p ( - i \mathrm{}X_{0}) \mathrm{}N_{0}\mathrm{}e x p ( i \mathrm{}X_{0}) - N_{0}= 1
\alpha_{i_{1}}= \gamma_{1}, \; \; \alpha_{i_{2}}= s_{\alpha_{i_{1}}}^{- 1}( \gamma_{2}) , \; \; \alpha_{i_{3}}= s_{\alpha_{i_{2}}}^{- 1}s_{\alpha_{i_{1}}}^{- 1}( \gamma_{3}) , \; \; \ldots \; \; ,
\bar{u}^{g,\epsilon}(T,s_{\epsilon_{n}},x_{\epsilon_{n}},y_{\epsilon_n}) =\frac{U_2(z_{\epsilon_{n}}-g(s_{\epsilon_{n}}))-U_2(\ell^{\epsilon_{n}}(x_{\epsilon_{n}},y_{\epsilon_{n}})-g(s_{\epsilon_{n}}))}{\epsilon^2},
\sum_j (-1)^{\tfrac{s-j}{2}}\binom{j\tfrac{(p-1)}{2}}{\tfrac{s-j}{2}}Q_{(r+ps-pj)(p-1)}Q_{j(p-1)}&=q^{\tfrac{p-1}{2}}\sum_i (-1)^{\tfrac{r+s(p-1)-i}{2}}\binom{\tfrac{i(p-1)}{2}}{\tfrac{r+s(p-1)-i}{2}}Q_{(r+ps-pi)(p-1)}Q_{i(p-1)},
\varphi \big( g^{j}_1 g_3 \big) \varphi \big( g^{i}_1 g_2 \big) = \varphi ((g^{n}_1 g_3) g_2) \mbox{or}\varphi \big( g^{j}_1 g_3 \big) \varphi \big( g^{i}_1 g_2 \big) = \varphi (g_2 (g^{n}_1 g_3)) \,.
[\,x_{1}^-(0),\,z_{6}^-(1)\,]_{(rs)^{-1}}=0, [\,x_{2}^-(0),\,z_{7}^-(1)\,]_{r^{-1}s^{-2}}=0, [\,x_{3}^-(0),\,z_{7}^-(1)\,]_{(rs)^{2}}=0, [\,x_{4}^-(0),\,z_{6}^-(1)\,]_{(rs)^{2}}=0.
4\partial_{s_{1}}u_{1}\left( x,t,y_{1},s_{1}\right) -\nabla_{y_{1}}\cdot\int_{S_{2}}\int_{S_{3}}\int_{Y_{2}}a\left( y^{2},s^{3},\nabla u\left(x,t\right) \right. +\left. \nabla_{y_{1}}u_{1}\left( x,t,y_{1},s_{1}\right) +\nabla_{y_{2}}u_{2}\left( x,t,y^{2},s^{3}\right) \right) dy_{2}ds_{3}ds_{2}=0
{\cal K}={\frac{1}{2}}\int_{0}^{\infty}{\frac{d \tau_{2}}{\tau_{2}^{1 4 - d / 2}}}{\frac{1}{\eta^{2 4 - d}( 2 i \tau_{2})}}\sum_{\beta = 1}^{\cal N}\gamma_{\beta}( 2 i \tau_{2}) \ .
n_{\gamma}=\frac{1}{\pi^{2}}{(\frac{k_{B}T}{\hbar c})}^{3}\int_{0}^{\infty}\frac{x^{2}}{e^{x}-1}dx=\frac{2\zeta(3)}{\pi^{2}}{(\frac{k_{B}T}{\hbar c})}^{3}\approx20.3(\frac{T}{1K})^{3}cm^{-3}
\sum_{|\lambda/\mu|=n+1}{f'}_{\lambda/\mu}g(\lambda) =\sum_{|\lambda/\mu|=n}{f'}_{\lambda/\mu}g(\lambda)+\sum_{|\lambda/\mu|=n}{f'}_{\lambda/\mu}Dg(\lambda) &=\sum_{k=0}^n\binom{n}{k}D^kg(\mu)+\sum_{k=0}^n\binom{n}{k}D^{k+1}g(\mu) &=\sum_{k=0}^{n+1}\binom{n+1}{k}D^kg(\mu).
E[X_1 \mid X_1 <x] = \frac{1}{\Phi(x)}\int_{-\infty}^x \xi \phi(\xi) d\xi = \frac{1}{\Phi (x)}(- \phi(\xi))|_{-\infty}^x = - \frac{\phi(x)}{\Phi(x)}, E[X_1 \mid x < X_1 < -x] = 0.
\bar{X}_{L,t_{k}}=\Big(\sum^{N_{0}}_{i=1}\tilde{w}_{0}(X_{0,t_{k}}^{i})X_{0,t_{k}}^{i}\Big)+\sum^{L}_{l=1}\Big(\sum^{N_{l}}_{i=1}\big(\tilde{w}_{l}(X_{l,t_{k}}^{i})X_{l,t_{k}}^{i}-\tilde{w}_{l-1}(X_{l-1,t_{k}}^{i})X_{l-1,t_{k}}^{i}\big)\Big)
\begin{array}{rl}m\sp2(\wp(v)-\wp(z)) =\frac{M\sp2}{2}\sum_{k=-\infty}\sp{\infty}\left(\frac{e\sp{-i\pi\tau}}{\cosh(\phi+i\pi (2k-1)\tau)-1}-\frac{e\sp{-i\pi\tau}}{\cosh(z+2i\pi k\tau)-1}\right) &{\longrightarrow \atop{Im(\tau)\rightarrow\infty}}2 M\sp2\cosh\phi-M\sp2\left(w+\frac{1}{w}\right).\end{array}
\Psi_{\underline{{{o}}}}^{\prime}( \vec{\sigma}_{2 o}) \sum_{\underline{{{n}}}}\, c_{\underline{{{n}}}}\Psi_{\underline{{{n}}}}^{\prime}( \vec{\sigma}_{1 o}) = - \Psi_{\underline{{{o}}}}^{\prime}( \vec{\sigma}_{2 o}) \frac{R}{\cal N}= - \frac{n_{o}( \vec{\sigma}_{2 o})}{\cal N}.
{}d^{n-4}\cdot \hat{h}_{\phi_P}(\beta_P) =\hat{h}_{\phi_P}(\phi_P^{n-4}(\beta_P)) begin{align*}3pt] \leq h(\phi_P^{n-4}(\beta_P))+\kappa_3\cdot h_X(P)+\kappa_4 begin{align*}3pt] \leq (\kappa_1+\kappa_3)\,h_X(P)+(\kappa_2+\kappa_4).
\left\|\left\{g_j\right\}_{j\in\mathbb Z}\right\|_{L^p\ell^q(\widehat P)}\leq\left\|\left\{\gamma_jE_j\left(f_j\right)\right\}_{j\in\mathbb Z}\right\|_{L^p\ell^q(\widehat P)}&\lesssim\left\|\left\{E_j\left(f_j\right)\right\}_{j\in\mathbb Z}\right\|_{L^p\ell^q(\widehat P)}=\left\|\left\{f_j\right\}_{j\in\mathbb Z}\right\|_{L^p\ell^q(\widehat P)},
2\langle\nabla_wv,\partial_t\rangle=d\xi(v,w)=-u^2d\theta(v,w) &\xi([v,w])=\langle \nabla_vw-\nabla_wv,\partial_t\rangle=2\langle\nabla_vw,\partial_t\rangle=u^2d\theta(v,w).
d(\theta_n u(t))=-\theta_n\nu A u(t)dt-\theta_n\lambda(r(t))u(t)dt+\theta_nf(t, u(t))dt &\quad+\theta_ng(t)dt+\sum_{k=1}^\infty(\theta_nh_k(t)+\theta_n\sigma_k( t, u(t)))dW_k(t),t>s.
\begin{aligned}\nu_1(A) = \int_A \lim_{n \rightarrow \infty}f_n(x) \nu_1(\mathrm{d}x) & \leq \liminf_{n \rightarrow \infty}\frac{1}{h_2(r_n)}\int_A \nu_2(A \cap B(x,r_n)) \nu_1(\mathrm{d}x) & \leq \liminf_{n \rightarrow \infty}\frac{1}{h_2(r_n)}\int_X \int_A 1_{B(x,r_n)}(y) \nu_2(\mathrm{d}y) \nu_1(\mathrm{d}x).\end{aligned}
{\mathbb E}(\|U_{[n/q]+1}\|_H^2)=\sum_{q[n/q] +1}^{n}{\mathbb E}(\|Y_j\|_H^2) +2\sum_{j=q[n/q] +1}^{n}\sum_{\ell=q[n/q] +1}^{j-1}{\mathbb E}(<Y_j, Y_\ell>)\, .
c_d ( f_1 , f_2 ) = d\left\{m_1 \lambda_2 - m_2 \lambda_1 + \frac{1}{4 \pi}\int_0^{2 \pi}\biggl( \frac{d \tilde{f}_1}{d \sigma}\tilde{f}_2 - \frac{d \tilde{f}_2}{d \sigma}\tilde{f}_1 \biggr) d \sigma \right\}.
\eta_v\left(B(x,r)\setminus v u^{-1}B(x,r)\right) =\eta_{x^{-1}v}\left(B(0,r)\setminus x^{-1}vu^{-1}B(x,r)\right) &=\eta_{x^{-1}v}\left(B(0,r)\setminus z B(0,r)\right)\,,
ds^{2}=-\left( r^{2}/l^{2}-\frac{\mu}{r}-1\right) dt^{2}+ \frac{dr^{2}}{\left(r^{2}/l^{2}-\frac{\mu}{r}-1\right)}+\frac{r^{2}}{z^{2}}\left( dz^{2}+d\varphi^{2}\right),
{\cal F}_{\mu \nu}^{a b}= F_{\mu \nu}^{a b}+ \Sigma_{\mu \nu}^{a b}+ \Theta_{\mu \nu}^{a b}, \ \ \{\cal F}_{\mu \nu}^{i}= F_{\mu \nu}^{i}+ \Sigma_{\mu \nu}^{i}, \ \ \{\cal F}_{\mu \nu}^{a}= F_{\mu \nu}^{a}+ \Sigma_{\mu \nu}^{a},
\mathbf{P}_{m}(\tau) = -\mathbf{R}_{\mathbf{D}}(im)k_{1}^{\boldsymbol{\delta}_{\mathbf{D}}}\Pi_{l=0}^{(\boldsymbol{\delta}_{\mathbf{D}}-1)k_{1}-1}(\tau - \mathbf{q}_{l}(m))
\gamma(a)\delta(b)(c)+(-1)^{|a||b|}\gamma(b)\delta(a)(c) =\gamma(a)\{b,c\}+(-1)^{|a||b|}\gamma(b)\{a, c\}&=a\{b, c\}+(-1)^{|a||b|}b\{a,c\}&=\delta(ab)(c),
\int_{-\infty}^{\infty}|\frac{1-cos(x)}{x^{2}}|dx\le\int_{-\infty}^{-\epsilon}\frac{2}{x^{2}}dx+\int_{-\epsilon}^{\epsilon}e^{|x|}dx+\int_{\epsilon}^{\infty}\frac{2}{x^{2}}dx\le K,
J^{w_1}(\textbf{X}_{SRS}^{(n)}) =\frac{-1}{2}\prod_{i=1}^{n}\left(\int_{-\infty}^{\infty}w_1(x_i)f^2(x_i)dx_i\right) &=\frac{-1}{2}\left(-2J^{w_1}(X)\right)^n &=\frac{-1}{2}\left(E(\Lambda_X^{w_1}(U))\right)^n
\rho_s(x) = \frac{(x + s + \tfrac12 )^{x + s}}{(x+ \tfrac12 )^x}e^{-s}\left( 1 + O(x^{-1}) \right) & = (x/e)^{s}\left( 1+s/x \right)^{x + s}\left( 1 + O\left( x^{-1}\right) \right)
\|\left(\Phi(X^*AY)\Phi(Y^*AY)^{-1}\Phi(Y^*AX)\right)^{\frac{p}{2}}\Phi(X^*AX)^{-\frac{p}{2}}\| &\le \left(\frac{M-m}{M+m}\right)^p\left(\frac{M}{m}\right)^{\frac{p}{2}}, p\ge 1.
L= \frac{1}{2(k+4)}\Big({-2}(W_{11}^{(2)}+W_{22}^{(2)})+W_{12}^{(1)}W_{21}^{(1)}+\frac{3}{4}(W_{11}^{(1)}W_{11}^{(1)}+W_{22}^{(1)}W_{22}^{(1)}) begin{align*}0.5em] -\frac{1}{2}W_{11}^{(1)}W_{22}^{(1)}-(k+2)(W_{11}^{(1)}+W_{22}^{(1)})'-(W_{11}^{(1)}-W_{22}^{(1)})'\Big),
\Delta(\{ab, c\}) =\Delta(a\{b, c\}+(-1)^{|a||b|}b\{a, c\}) &=\Delta(a)\Delta(\{b, c\})+(-1)^{|a||b|}\Delta(b)\Delta(\{a, c\}) &=\Delta(a)\{\Delta(b), \Delta(c)\}+(-1)^{|a||b|}\Delta(b)\{\Delta(a), \Delta(c)\}&=\{\Delta(ab), \Delta(c)\}.