harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C_{\nu}^{(k)}=\Bigg[\alpha_{\nu-k}\widetilde{\tau}'_{k-1}+\gamma_{\nu-k}\frac{\widetilde{\sigma}''_{k-1}}{2}\Bigg]C_{\nu}^{(k-1)}=\kappa_{\nu+k-1}C_{\nu}^{(k-1)}=\prod_{i=0}^{k-1}\kappa_{\nu+i}C_{\nu},
E\max_{1\leq k\leq2n}\left\vert \sum\nolimits_{j=1}^{k}a_{j}Q^{j}f(\xi_{0})\right\vert^{p}\ll E\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{2j}E^{j}f(\xi_{0})\right\vert^{p}& +E\max_{1\leq k\leq n}\left\vert \sum\nolimits_{j=1}^{k}a_{2j+1}E^{j}f(\xi_{0})\right\vert^{p}.
(2\pi )^4 \frac{\delta}{\delta \hat{\varphi}(q)}L^{0, \, \infty}(\varphi ) \, = \,\hat{L}^{0, \, \infty}_{(1)}(q \, ; \varphi) +\hat{I}^{\, 0, \, \infty}_{(1)}(q ) \, ;
\Psi_{n,t}(x)=\frac{\Phi_{+}(x,\lambda_{n}(t))}{\mid\Phi_{+}(x,\lambda_{n}(t))\mid},\chi_{n,t}(x))=\frac{1}{\overline{\alpha_{n}(t)}}\frac{\overline{\Phi_{-}(x,\lambda_{n}(t))}}{\mid\overline{\Phi_{-}(x,\lambda_{n}(t))}\mid},
\int_{z=0}^{1}\!\int_{y=-2\sqrt{z}}^{2\sqrt{z}}y^j(y+z+1)^k\:dy\:dz = \frac{2^{j+2k+4}}{k+1}\Bigg(\sum_{r=1}^{j+1}\frac{(-2)^{r-1}(j)_{r-1}}{(k+r+1)_{r}}-\sum_{r=1}^{j+1}\frac{(-2)^{r-1}(j)_{r-1}}{(2k+r+2)_{r}}\Bigg)
\int_{Q}\int_{Q}\frac{|\Psi_a(x)-\Psi_a(y)|^{p-2}\,(\Psi_a(x) - \Psi_a(y))}{|x-y|^{N+s\,p}}(\varphi(x)-\varphi(y))\,dx\,dy &\leq \frac{C}{s\,p-1}\,[\varphi]_{W^{s,p}(Q)},
(C^n_{ij}(q))=(C^n_{ji}(q))= \frac{n}{[n]_q}\,\frac{[2]_q^2}{q^{2n}+1+q^{-2n}}\,\left (\begin{array}{cc}q^n+q^{-n}(-1)^n (-1)^n q^n+q^{-n}\end{array}\right )\,;
\frac{\left( L2^{m+2}\right)^{-\left( \frac{\varepsilon}{2}+1\right)}}{2}+\frac{\left( L2^{m+1}\right)^{-\left( \frac{\varepsilon}{2}+1\right)}}{2^{\left( 1+\frac{2}{\varepsilon}\right)}-1}.
|E_0| =\frac{1}{\ell^*}\sum_{e\in K_1\cup K_2}(\ell_e-q_e)+|K_0| &=\frac{m}{\ell^*}-\frac{|K_0|}{\ell^*}-\frac{1}{\ell^*}\sum_{e\in K_1\cup K_2}q_e+|K_0|.
V_{out}=\frac{(R_{f}+R_{1})R_{g}}{(R_{g}+R_{2})R_{1}}V_{2}-\frac{R_{f}}{R_{1}}V_{1}=(\frac{R_{1}+R_{f}}{R_{1}})\cdot(\frac{R_{g}}{R_{g}+R_{2}})V_{2}-\frac{R_{f}}{R_{1}}V_{1}.
\frac{1}{\pi}\iint|h^{\prime}|^{2}dxdy=[\sum_{n\ge1}\frac{1}{n}|\lambda_{-n}|^{2}-\sum_{n\ge1}|\alpha_{n}|^{2}r^{2n}]+[\sum_{n\ge1}\frac{1}{n}|\lambda_{n}|^{2}-\sum_{n\ge1}|\beta_{n}|^{2}R^{-2n}].
\mathcal{K}_{p,q}(k) :=\int_0^{\frac{\pi_{p,q}}{2}}\frac{d\theta}{(1-k^q\sin_{p,q}^q{\theta})^{\frac{1}{p^*}}}=\int_0^1 \frac{dt}{(1-t^q)^\frac1p (1-k^qt^q)^{\frac{1}{p^*}}}, \mathcal{E}_{p,q}(k) :=\int_0^{\frac{\pi_{p,q}}{2}}(1-k^q\sin_{p,q}^q{\theta})^{\frac1p}d\theta=\int_0^1 \left(\frac{1-k^qt^q}{1-t^q}\right)^\frac1p\,dt.
\left[ e^{-\bar{H}_i^2t}\right]_{\mbox{diag}}\simeq{1 \over (4\pi t)^{(D-1)/2}}e^{-m^2\kappa^2\rho^2 t}\left(r_i+\bar{b}_{i,1}t+\bar{b}_{i,2}t^2+...\right)~~~,
\underline{{{\mathrm{d e t}}}}( \mathrm{\bf ~ u}) = \frac{4 q^{2}}{( q^{2}+ 1 )^{2}}\tilde{t}^{2}- q^{2}\tilde{x}^{2}- q^{2}\tilde{y}^{2}- \frac{2 ( q^{4}+ 1 ) q^{2}}{( q^{2}+ 1 )^{2}}\tilde{z}^{2}+ 2 q \left( \frac{q^{2}- 1}{q^{2}+ 1}\right)^{2}\tilde{t}\tilde{z}.
T_m \overline{U(z_m ,z_m')}\cdots T_1 \overline{U(z_1 , z_1')}\Psi = \sum_{n_1 , ... , n_m = 0}^\infty T_m U_{n_m}(z_m ,z_m') \cdots T_1 U_{n_1}(z_1 , z_1') \Psi ,
R_{m,n}(z)=\frac{P_{m}(z)}{Q_{n}(z)}=\frac{a_{0}+a_{1}z+a_{2}z^{2}+\cdot\cdot\cdot+a_{m}z^{m}}{b_{0}+b_{1}z+b_{2}z^{2}+\cdot\cdot\cdot+b_{n}z^{n}}
\left.{\rm Disc}\; \left( \int d^4 x{e^{i p \cdot x}\over |x|^{2m+4}}\right) \right|_{-p^2 = \omega^2 -i\epsilon}^{-p^2 \omega^2 +i\epsilon}={2\pi^3i \omega^{2m}\over 4^m m! (m+1)!}
\left[ q ( t , \vec{x}) , \, p ( t , \vec{x}^{\prime}) \right] = \delta ( \vec{x}- \vec{x}^{\prime}) , \quad \left[ \Psi ( t , \vec{x}) , \, \bar{\Psi}( t , \vec{x}^{\prime}) \right] = - i \delta ( \vec{x}- \vec{x}^{\prime})
e^{- \int d^{4}x \frac{\xi}{2 g^{2}}( D^{\mu}V_{\mu})^{2}}= \frac{1}{N}\int{\cal D}a \delta ( D^{\mu}V_{\mu}^{g}- a ) e^{- \int d^{4}x \frac{\xi}{2 g^{2}}a^{2}},
-\nabla_y \cdot \big(D (\nabla_y \tilde{w}_i + \nabla_y (S_0^{-1})^Te_i \big) = 0 \mbox{in}Y^{\ast}(t,x), -D (\nabla_y \tilde{w}_i + \nabla_y (S_0^{-1})^T e_i) \cdot \nu = 0 \mbox{on}\Gamma(t,x), \int_{Y^{\ast}(t,x)}J_0^{-1}(t,x,S_0^{-1}(t,x,y))\tilde{w}_i(t,x,y) dy = 0, \tilde{w}_i \mbox{is}Y\mbox{-periodic}.
\partial^2_\mu \sigma =-f_0(r)\int_0^r\Big[\partial^2_\mu \sigma(Q_\mu-Q)+Q_\mu\Big(\partial_\mu \sigma-\frac{r^2}{2}\Big)^2\Big]f_1\tau d\tau &+f_1(r)\int_0^r \Big[\partial^2_\mu \sigma(Q_\mu-Q)+Q_\mu\Big(\partial_\mu \sigma-\frac{r^2}{2}\Big)^2\Big]f_0\tau d\tau.
U ({\bf r}^{\prime},{\bf r}_{2}) = - \int_{- \infty}^{+ \infty}d \omega 4 \pi \alpha_{1}({\bf r}^{\prime}, i | \omega | ) \alpha_{2}({\bf r}_{2}, i | \omega | ) ( D_{p j}^{( 0 )}({\bf r}^{\prime},{\bf r}_{2}, i \omega ) )^{2}.
u(z) = \log|z| + C +\sum_{j=1}^J \left\{d_j^{}\log|w_j^{}(z)| + \sum_{k=1}^N \left[ a_{jk}^{}\kern 1pt\hbox{Re\kern 1pt}(w_j^{}(z)^{-k}) + b_{jk}^{}\kern 1pt\hbox{Im\kern 1pt}(w_j^{}(z)^{-k})\right] \right\},
\begin{array}{l c l}{g^{a b}\left< T_{a b}\right>}&{=}&{\left( \eta^{a b}+ h^{a b}\right) \left< T_{a b}\right> = 2 \eta^{z \overline{z}}\left< T_{z \overline{z}}\right> + \cal O \left( h^{2}\right)}\\{\& =}&{4 \left< T_{z \overline{z}}\right> + \cal O \left( h^{2}\right) ,}\\ \end{array}
\Delta \phi \leq \int_{t_{0}}^{\infty}\dot{\phi}d t = \dot{\phi}_{0}\int_{t_{0}}^{\infty}\left({\frac{t_{0}}{t}}\right)^{\frac{2}{1 + \alpha}}d t ={\frac{1 + \alpha}{1 - \alpha}}~ \dot{\phi}_{0}t_{0}.
\dot{\varphi}'\cdot\alpha - u_1\cdot d\dot{S}= \dot v_0^{H}-\dot\lambda_0 , d\dot S'\cdot \alpha +\eta d\dot S = \dot{V}_0^{H}- \dot\Lambda_0, -^t D\dot\varphi' \cdot \alpha + \,^t D(u_1\cdot d\dot S) = \dot{V}_1^{H}- \dot\Lambda_1,
M(x) = M(x_0 + \phi(x_0) + x') = df(x_0) + \int_0^1 d M(x_0 + \phi(x_0) + tx')x' \, dt = df(x_0) + Lx' + \int_0^1 \left( d M(x_0 + \phi(x_0) + tx') - dM(0)\right) x'\, dt.
\int_{\Omega}|\partial_{j}(K_{i}\ast\rho)|^{p}=\int_{\Omega}|\partial_{j}K_{i}\ast\rho|^{p}\leq\left\Vert \rho\right\Vert_{L^{p}(\Omega)}^{p}\left\Vert \partial_{j}K_{i}\right\Vert_{L^{\frac{p}{p-1}}(\mathbb{R}^{d})}^{p-1}|\Omega| & =C\left\Vert \rho\right\Vert_{L^{p}(\Omega)}^{p}~\forall j\in\{1,...,d\}
\mathbf{p}B_j^- = \sum_{i=j}^{n}\sum_{s=1}^{i-j+1}\sum_{I\in \mathcal{I}_{ji}(s)}E^{e_I}B_i^-\frac{1}{\prod_{\ell\in I, \ell\neq j}(h_j-h_\ell)},
f_k=\sum_{(i_1,\dots, i_k)\in (\mathbb{Z}^*)^k}\alpha_{i_1,\dots,i_k}^{(k)}:\mathfrak{Re}\langle \mathfrak{e}_{i_1},\cdot \rangle\dots \mathfrak{Re}\langle \mathfrak{e}_{i_k},\cdot \rangle:
\widehat{A}=\begin{bmatrix}a+b+c+d 0 0 0 0 0 0 a-b-c+d 0 0 0 0 0 0 a+b+c+d 0 0 0 0 0 0 a-b-c+d 0 0 0 0 0 0 -a+b-c+d 0 0 0 0 0 0 -a-b+c+d \end{bmatrix}
E = \tfrac{1}{16\pi}\lim_{r\to\infty}B^r_{(g,\pi)}(1,0) P_i =\tfrac{1}{8\pi}\lim_{r\to\infty}B^r_{(g,\pi)}(0, \frac{\partial}{\partial x^i}) \mathcal C_i = \tfrac{1}{16\pi}\lim_{r\rightarrow \infty}B^r_{(g,\pi)}(x^i, 0) \mathcal{J}_{k}= \tfrac{1}{8\pi}\lim_{r\rightarrow \infty}B^r_{(g,\pi)}(0, x\times \frac{\partial}{\partial x^k}),
\tilde{C}(w;b) =\frac{\sum\limits_{l=0}^\infty\mathrm{e}^{-b (w+\lambda) p^{-l}}\frac{\lambda^{l}}{(w+\lambda)^{l}\prod\limits_{i=1}^{l}(1- p^{-i})}}{\sum\limits_{l=0}^\infty\mathrm{e}^{-b(w+\lambda)p^{-l}}\frac{\lambda^{l}}{p^{l}(w+\lambda)^{l}\prod\limits_{i=1}^{l}(1-p^{-i})}}-1.
\overline{\chi}_\lambda^+(x)-\overline{\chi}_\lambda^-(x)=(\overline{\chi}_{\lambda_0}^+(x_0s_0)-\overline{\chi}_{\lambda_0}^-(x_0s_0))(\overline{\chi}_{\lambda\backslash\{0\}}^+(x's')-\overline{\chi}_{\lambda\backslash\{0\}}^-(x's'))
\frac{b_{n-1}}{n}= \sum_{j=1}^n\sum_{k=1}^j \frac{(-1)^{k-1}}{j}\binom{j}{k}a^{(k)}_n= \sum_{k=1}^n \frac{(-1)^{k-1}}{k}\binom{n}{k}a^{(k)}_n.
\begin{array}{l}Y_1(x,\lambda)\!=\!\exp(i\lambda x)\left[\left(\begin{array}{c}1 -i\end{array}\right)+O\left(\frac{1}{\lambda}\right)\right], Y_2(x,\lambda)\!=\!\exp(-i\lambda x)\left[\left(\begin{array}{c}1 i\end{array}\right)+O\left(\frac{1}{\lambda}\right)\right].\end{array}
\hat{w}(f) =\sum_{i=0}^nf ( \rho (x_i) ) \kappa (y_i) =( \kappa \circ \hat{u}\circ \rho^d ) (f) &=\lim_{\lambda}( \kappa \circ \psi_{\lambda}) \circ ( \phi_{\lambda}\circ \rho^d ) (f) .
{\hat A}\left[{T \over 2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu\rho\sigma}\eta^\rho\eta^\sigma ) \right] ~=~\sqrt{\det \left[{\frac{T}{2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu\rho\sigma}\eta^\rho\eta^\sigma) \over\sinh [ \frac{T}{2}({\Omega^{\mu}}_{\nu}+{R^{\mu}}_{\nu\rho\sigma}\eta^\rho\eta^\sigma) ]}\right]}
m(T_i \tilde T_j, A_i \tilde A_j) = \Big\langle T_i \tilde T_j (A_i \tilde A_j), D \Big\rangle= \operatorname{Tr}\Big( \theta_{S_i R_j(e_i),e_1}\Big)= \langle S_i R_j(e_i) , e_1\rangle
I^{q}_{1}= \int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}|G_{\xi, \sigma}(y^{-1}x)u(y)|\,dy\right)^{q}|x|^{-\beta q}dx & \quad\quad\quad\quad\leq C \int_{X}\left(\int_{B\left(0,\frac{|x|}{2}\right)}u(y)dy\right)^{q}\left( G_{\xi, \sigma}\left(\frac{x}{2}\right) \right)^{q}|x|^{-\beta q}dx.
\partial^\alpha \tilde{F}_\lambda (\partial v) = \lambda \sum_{\alpha_1 + \cdots + \alpha_l = \alpha}C_{\alpha_1,\cdots, \alpha_l}\sum_{0 \leq j_1, \cdots, j_l \leq d \atop 1 \leq i_1, \cdots, i_l \leq d}D_{i_1j_1}\cdots D_{i_lj_l}\tilde{F}(\partial v) \partial^{\alpha_1}\partial_{j_1}v^{i_1}\cdots \partial^{\alpha_l}\partial_{j_l}v^{i_l},
\begin{aligned}\sigma(t)= \, \alpha(\tau + t)^{\alpha-1}+ \frac{C^{m-1}m (N-1) k\coth(k)}{a(m-1)}(\tau +t)^{\alpha m -\beta}-\frac{\beta}{m-1}(\tau+t)^{\alpha-1}\leq \, \frac{2 C^{m-1}m (N-1) k \coth(k)}{a(m-1)}(\tau+t)^{\alpha m -\beta}\, .\end{aligned}
\int_V Q_\lambda (v) dv = \int_V \frac{r M(v)}{r + \lambda - v \cdot p}dv + \int_V \left[ \frac{p \cdot \nabla D}{\left( r + \lambda - v \cdot p \right)^2}+ \frac{2 \vert p \vert^2 D(v)}{\left( r + \lambda - v \cdot p \right)^3}\right] Q_\lambda dv,
cos\alpha=\frac{\hat{n}_{1}\cdot\hat{n}_{2}}{|\hat{n}_{1}||\hat{n}_{2}|}=\frac{a_{1}a_{2}+b_{1}b_{2}+c_{1}c_{2}}{\sqrt{a_{1}^{2}+b_{1}^{2}+c_{1}^{2}}\sqrt{a_{2}^{2}+b_{2}^{2}+c_{2}^{2}}}.
\widehat{R}_{\mu \nu}^{ab}=R_{\mu \nu}^{ab}+i\theta^{\rho \tau}R_{\mu\nu \rho \tau}^{ab}+\theta^{\rho \tau}\theta^{\kappa \sigma}R_{\mu \nu\rho \tau \kappa \sigma}^{ab}+O(\theta^{3})
var_{P_{\alpha}}\left[ \chi_{P_{\alpha},a,eff}^{1}(\mathbf{V};\mathcal{G})\right]=var_{P_{\alpha}}\left[ \psi_{P_{\alpha},a}(\mathbf{O};\mathcal{G})\right]-var_{P_{\alpha}}\left[ \Delta_{P_{\alpha}}\left( \mathbf{O}\right) \right]
(i) \quad\mu_\alpha\sim\mu_{inv}, (ii) \,\,\,H(\mu_\alpha,\mu_{inv})=\prod_{n=2}^\infty\sum_{r\in{\mathbb F}_p}\sqrt{\frac{\alpha_{1n}(r)}{p}}>0, (iii) S_{11}^L(\mu_\alpha)<\infty.
\sum_{k\le2}\;\sum_{(\epsilon_1,\cdots,\epsilon_k)\in\{-1,0,+1\}^k}\!\!\!\!\!F_{k;\epsilon_1\cdots\epsilon_k}(n_1,n_2;\phi)=1-\frac{\phi^2}{2!}\;\frac{n_1n_2}{3}+{\cal O}(\phi^4)\; .
||F_n||^2 = \sum\limits_{k=1}^n \bigg[f_{k,1}+f_{k,2}\bigg]^2\leq 2\sum\limits_{k=1}^n f_{k,1}^2+ 2\sum\limits_{k=1}^n f_{k,2}^2\equiv 2 I_{1,n}+2 I_{2,n}.
\mathcal{B}= \left\{(u, q, p_1,\ldots,p_r)\in (\mathcal{O}_S^\times\cap k)\times\mathbb{Z}^{r+1}\ : \ 0<|\alpha_i qu-p_i|<\frac{1}{H^\varepsilon(u)|q|^{\frac{d}{r}+\epsilon}}\mbox{for all}1\leq i\leq r\right\}
G \left( a , r_{0}\right) = \lim_{R \rightarrow \infty}\left( \left\langle \infty | S \left( R , r_{0}\right) | 0 \right\rangle_{a}+ \int_{r_{0}}^{R}\left\langle \infty | S \left( R , r \right) H_{i n t}S \left( r , r_{0}\right) | 0 \right\rangle_{a}\frac{d r}{r}\right) ,
\int_{[0,1)^2}uf_\ell \zeta\, dx =\int_{[0,1)^2}\big(uf_\ell-(uf)_\ell\big) \zeta\, dx+\int_{[0,1)^2}(uf)_\ell \zeta\, dx &\stackrel{\eqref{w23}}{\lesssim}[u]_\alpha [f]_\beta \ell^{\alpha+\beta}\|\zeta\|_{L^1}+ \langle uf, \zeta*\phi_\ell \rangle_{{\cal D}',{\cal D}}\longrightarrow \langle uf, \zeta \rangle_{{\cal D}',{\cal D}}\textrm{as}\ell \to 0,
\theta:\Omega\times[0,3\cdot2^{-k}]\to\mathbb{R}, \theta((x_1,x_2),t) = 2\left\|f_0\right\|_{L^\infty}C_0^{-1}\left(\mathrm{erf}\left(\frac{-x_1}{\sqrt{4\nu t}}\right) + \mathrm{erf}\left(\frac{x_1-a}{\sqrt{4\nu t}}\right)\right).
z(s) =h^2\beta_h(s)+\frac{h^2}{2}\overline{\beta_h(s)}+\frac{\sigma^2}{2}\overline{\beta_\sigma(s)}, \zeta(s) =i\beta_h(s)(K_1 + K_3)-i\overline{\beta_h(s)}K_2-i\overline{\beta_\sigma(s)}K, \gamma(s) =-is\beta_h(s)(|K_1|^2+|K_3|^2)+is\overline{\beta_h(s)}|K_2|^2+is\overline{\beta_\sigma(s)}|K|^2.
Xu=0_{F^{n}}\Rightarrow(\begin{matrix}-B\\I_{n-r}\end{matrix})u=0_{F^{n}}\Rightarrow(\begin{matrix}-Bu\\u\end{matrix})=(\begin{matrix}0_{F^{r}}\\ 0_{F^{n-r}}\end{matrix})\Rightarrow u=0_{F^{n-r}}.
\begin{array}{c}{f_{-}= f ( - \infty + i \epsilon ) = ( - 1 )^{\delta}e^{i Z_{+}( - \infty )}= e^{i{\cal Q}_{+}( - \infty )}}\\{f_{+}= f ( + \infty + i \epsilon ) = ( - 1 )^{\delta}e^{i Z_{+}( + \infty + i \epsilon )}= 0}\\ \end{array}
\mathcal{E}_i(X)=\frac{1}{h}\int_{b_i\cap T^{s'}}E_i(P,X)ds(P), ~~ \mathcal{F}_i(X)=\frac{1}{h}\int_{b_i\cap T^{s'}}F_i(P,X)ds(P), ~~ \textit{if}~X\in T_{non}^s.
\begin{aligned}\Bigg|\int\limits_{|x-e|+|x|=2\tau}\frac{q(x)}{|2\tau x-|x|e|}dS_{x}\Bigg|\leq K \int\limits_{|x-e|+|x|\leq 2\tau}\frac{|q(x)|}{|x||x-e|}dx,\ \forall\tau\in(1/2,T/2].\end{aligned}
\begin{aligned}\sum_{n=1}^{\infty}|r_n(2N)|\leq \sum_{l_1=1}^{\infty}\sum_{l_2=1}^{\infty}\ldots \sum_{l_N=1}^{\infty}\frac{|\lambda|^N}{p^{2nN}(1-p^{-2})^N}({p^{-(2N-2)l_1}}-p^{-2Nl_1})(p^{-(2N-4)l_2}-p^{-(2N-2)l_2}) &\ldots (1-p^{-2l_N})|f_{n+l_1+l_2+\ldots+l_N}|. \end{aligned}
( 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ - \frac{7}{2}) \rightarrow ( 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ - \frac{7}{2}) ( \frac{7}{2}) \oplus ( 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ 0 ~ - \frac{5}{2}) ( - \frac{7}{2}) .
i_{[a_f,a_g]}\mathrm{d}\eta= [\mathcal{L}_{a_f},i_{a_g}]\mathrm{d}\eta = \mathrm{d}i_{a_f}i_{a_g}\mathrm{d}\eta+i_{a_f}\mathrm{d}i_{a_g}\mathrm{d}\eta-i_{a_g}\mathrm{d}i_{a_f}\mathrm{d}\eta-i_{a_g}i_{a_f}\mathrm{d}\mathrm{d}\eta = \mathrm{d}\eta(a_f,a_g) = \{f,g\}.
f(\omega) =\omega (Q(\Phi)-E(\Phi))-(E(U_0)+\omega Q(U_0)+\sqrt{\omega}\ \mathbf{c}_0\cdot \mathbf{P}(U_0)) &=\omega (Q(\Phi)-E(\Phi)-Q(U_0))-\sqrt{\omega}\ \mathbf{c}_0\cdot \mathbf{P}(U_0)-E(U_0).
\forall\,i\in\mathbb{Z}_{n},\;f^{(0)}\left(i\right)\,:=i,\mbox{and}\forall\,k\ge0,\;f^{(k+1)}\left(i\right)=f^{(k)}\left(f\left(i\right)\right)=f\left(f^{(k)}\left(i\right)\right).
\lim_{n\to\infty}\limsup_{m\to\infty}\frac{1}{m}\#\left\{0 \le k\le m: \frac{\log\|\rho(\xi_k^{-1}\xi_{k+n})\| - \Lambda n}{\sqrt{n}}< t \right\}=\frac{1}{\sigma_j\sqrt{2\pi}}\int_{-\infty}^t e^{-\frac{s^2}{2\sigma_j^2}}ds.
f_{\Omega^{(N)}}(x)= &\frac{N}{\mathcal{B}(M,K)}\left[\frac{(-1)^{M}(M-1)!}{\mathcal{B}(M,K)\prod_{m=1}^{M}(m-M-K)}\right]^{N-1}\sum_{k=N-1}^{(N-1)(M+K-1)}q_{k-(N-1)}\frac{x^{M+k-1}}{(1+x)^{N(M+K-1)+1}}
\hat{X}_{l}=\begin{cases}\sum^{N_{0}}_{i=1}\frac{1}{N_{0}}X_{0}^{i}, l=0, \sum^{N_{l}}_{i=1}\frac{1}{N_{l}}\big(X_{l}^{i}-X_{l-1}^{i}\big), l>0,\end{cases}
\left(\frac{q}{p}\right) =\left(\frac{-1}{p}\right)\left(\frac{|q|}{p}\right)=(-1)^{\frac{p-1}{2}}\left(\frac{p}{|q|}\right)\cdot (-1)^{\frac{p-1}{2}\frac{|q|-1}{2}}=\left(\frac{p}{|q|}\right)\cdot(-1)^{\frac{p-1}{2}\frac{|q|+1}{2}}&=\left(\frac{p}{|q|}\right)\cdot(-1)^{\frac{p-1}{2}\frac{-q+1}{2}}=\left(\frac{p}{|q|}\right)\cdot (-1)^{\frac{p-1}{2}\frac{q-1}{2}}
||T||_{L(E)}=\sup\{\frac{||T(x)||_{E}}{||x||_{E}},x\in E\}& \leq\sup\{\frac{||T(x)||_{E}}{||x||_{F}},x\in F\}=\sup\{\frac{||\Lambda(T)(x)||_{F}}{||x||_{F}},x\in F\}.
\partial_{t}u_{i}\left( t,x\right) \sum\limits_{\left\vert \alpha\right\vert =2l}a_{\alpha}D^{\alpha}u_{i}\left( t,x\right)+\sum\limits_{j=1}^{\infty}a_{ij}u_{j}\left( t,x\right) =f_{i}\left(t,x\right) x\in R^{n}t\in \left( 0,\infty \right) ,
\left| \sum_{n=0}^N (n+1) a_n(\rho) \rho^{n}r^n e^{in\theta}\right|\le\left(\sum_{n=0}^\infty |a_n(\rho)|^2 \right)^{1/2}\left(\sum_{n=0}^\infty (n+1)^2 r^{2n}\rho^{2n}\right)^{1/2}.
\langle \phi | x^{ml}\rangle =N^{1/2}(n) ( \beta /2 \pi)^{n/4}\int_{I_n}d^n \rho~ (1- \beta \rho^2/2)^{1/2+ \sqrt{1/4 + n/8}}e^{i x \cdot \rho/\hbar}\phi(\rho)
\varphi^{(3,2)}(\xi)= I_{0}+\sum_{j=0}^{\infty}I_{j+1}\xi^{j}+I_{0,1}^{2}(1-\xi)+(1-\xi)\sum_{j=0}^{\infty}I_{j+1,2}^{2}\xi^{j}+(1-\xi)^{2}I_{0,1}^{3}+(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+1,2}^{3}\xi^{j},
\Omega(\epsilon)=\bigg\{z\in \mathbb{R}^{n+2m+p}|\epsilon\leq\|H(z)\|\leq\|H(z^{(0)})\|, \textrm{min}\left(S\Lambda\hat{e}\right)\geq \frac{\bar{\tau}_1}{m}\frac{1}{2}s^T\lambda,\s^T\lambda\geq\bar{\tau}_2\frac{1}{2}\|R(z)\|\bigg\},
e^{-\delta S_{eff}}\;=\; \int{\cal D}\vec a \exp\left\{-\sum_{n=-\infty}^{+\infty}\frac{1}{4}\int \frac{d^2k}{(2\pi)^2}{\tilde F}_{jk}(-n,-k)[1+\Pi (n,k)]{\tilde F}_{jk}(n,k) \right\}\;.
b_A(\{8'\}) = b_2^{(0,01,01,0)}= a_2^0 + 2a_2^1 + a_2^{2''}= 6+2(9)+12=36, b_B(\{8'\}) = b_1^{(0,0,1,0)}= a_1^1 = 3, b_C(\{8'\}) = b_2^{(0,01,0,0)}= a_2^0 + a_2^1 = 6+9=15, b_D(\{8'\}) = b_4^{(1,0,0,0)}= a_4^1 = 89,
\begin{aligned}u_t(r, t) =C \zeta'(t) F^{\frac 1{m-1}}- \frac C{m-1}\zeta(t) F^{\frac 1{m-1}-1}\frac{\eta'(t)}{\eta(t)}\frac{r}{a}\eta(t) &= C \zeta'(t) F^{\frac 1{m-1}}- \frac{C}{m-1}\zeta(t)\frac{\eta'(t)}{\eta(t)}F^{\frac 1{m-1}-1}+\frac C{m-1}\zeta(t)\frac{\eta'(t)}{\eta(t)}F^{\frac 1{m-1}}\,;\end{aligned}
\psi_{1}= \frac{3}{2 \sqrt{2}}( f_{1}+ f_{2}) - \frac{1}{\sqrt{2}}f \ , p s i_{2}= \frac{2}{3}\bar{\phi}+ \frac{1}{4 \sqrt{2}}( f_{1}+ f_{2}) + \frac{1}{2 \sqrt{2}}f .
2 \, d = \sum_{i = 1}^r m_i \, \dim_k V_{m_i}+ \sum_{i = 1}^r ( - m_{r - i + 1}) \, \dim_k V_{m_{r - i + 1}}= \sum_{i = 1}^r m_i \, \dim_k V_{m_i}- \sum_{i = 1}^r m_i \, \dim_k V_{m_i}= 0 ,
r^{*,\pi}_n(x_n|y^{n-1}_{n-M},y_n)=\Big(\frac{q_n(y_n|y^{n-1}_{n-M},x_n)}{\nu^{\pi}_{n}(y_n|y^{n-1}_{n-J})}\Big){\pi}_{n}(x_n|y^{n-1}_{n-J})
\hat{T}_{\mu \nu}( k ) = \frac{1}{2 \pi}\delta_{\mu \nu}+ \frac{1}{2 \pi}\Big ( 2 \frac{k_{\mu}k_{\nu}}{k^{2}}\! - \! \delta_{\mu \nu}\Big ) \Bigg ( - 1 \! + \! \Big ( \sqrt{\frac{m^{2}}{k^{2}}\! + \! \frac{1}{4}}\! - \! \frac{1}{4}\frac{1}{\sqrt{\frac{m^{2}}{k^{2}}\! + \! \frac{1}{4}}}\Big ) l ( k , m ) \Bigg )
\sum_{n=0}^\infty \sum_{j=0}^{k-1}\binom{n}{j}x^n = \sum_{j=0}^{k-1}\sum_{n=j}^\infty \binom{n}{j}x^n &= \sum_{j=0}^{k-1}\frac{x^j}{(1-x)^{j+1}}&= \frac{1}{1-x}\left( \frac{1-\left(\frac{x}{1-x}\right)^k}{1-\frac{x}{1-x}}\right) &= \frac{(1-x)^k-x^k}{(1-2x)(1-x)^k}.
\int_{0}^{T}- \psi'(t) (\tilde{\textbf{u}}^{\star}, \phi)_{0}~ dt + \sum_{j=1}^{d}\int_{0}^{T}\psi(t) (A_{j}(\tilde{\textbf{u}}^{\star}+\bar{\textbf{u}})\partial_{x_{j}}\tilde{\textbf{u}}^{\star}, \phi)_{0}~ dt = 0.
{\sigma}_{2}( | z | \to \infty ) \approx \frac{2{\hbar}c e^{- \frac{\pi}{2}{\beta}}}{|{\Gamma}( 1 + i{\beta}) |}z^{- 1 / 4}\cos \left( \frac{z}{2{\hbar}c}+ \beta \mathrm{l n}\frac{z}{{\hbar}c}+ \delta \right) .
\frac{1}{N_{F}}\sum^{N_{F}}_{j=1}\tilde{X}_{F}^{j, n}=\sum^{N_{F}}_{j=1}\sum^{N_{F}}_{i=1}\frac{1}{N_{F}}T_{i,j}N_{F}X_{F}^{i, n}=\sum^{N_{F}}_{i=1}w_{F}^{i, n}X_{F}^{i, n},
d s_{( 1 0 )}^{2}= e^{\frac 2 3 \phi}\left[ d x^{a}\, d x^{b}\, \eta_{a b}+{\frac{1}{\sqrt{2 \kappa | u |^{2}+ u_{0}}}}\left( d u^{i}+ \epsilon^{i j k}\tilde{A}^{j}u^{k}\right)^{2}+ \sqrt{2 \kappa | u |^{2}+ u_{0}}\; d \tilde{s}_{3}\right]
\Vert u - ( g \otimes a ) u \Vert_{\epsilon}\leq \Vert u - ( g \otimes a ) u \Vert_{\pi}&\leq \sum_{j=1}^{n}( \Vert g_j -gg_j \Vert_{\infty}\, \Vert a_j \Vert_{\mathcal{A}}+ \Vert g_j \Vert_{\infty}\, \Vert a_j -aa_j \Vert_{\mathcal{A}}+ \Vert g_j - gg_j \Vert_{\infty}\, \Vert a_j - aa_j \Vert_{\mathcal{A}}&< \epsilon
=b_{m}^{*}(0)b_{m}(0)N\int d^{3}r\varphi_{m}^{*}(r)H_{at}(r)\varphi_{m}(r)+b_{m}^{*}(0)\sum_{R_{n}}e^{-ik\cdot R_{n}}\int d^{3}r\varphi_{m}^{*}(r-R_{n})\Delta U(r)\psi_{m}(r)
\int_{{[0,2\pi]}^m}\frac{dx_1dx_2\cdots dx_m}{\displaystyle\sum_{j=1}^{m}\left|\cos\left(\left(\frac{\alpha-\beta}{2}\right)x_j\right)-\cos \left(\left(n-\frac{\alpha+\beta}{2}\right)x_j\right)\right|^{2s}}& =2^{m-2s}\int_{[0,\pi]^m}\frac{dx_1dx_2\cdots dx_m}{\displaystyle\sum_{j=1}^m\sin^{2s}((n-\alpha) x_j)\sin^{2s}((n-\beta) x_j)}.
(\cot \langle 2\alpha_{2}, H \rangle )^{2}= \frac{m_{2}+m_{3}+n_{3}\pm \sqrt{(m_{2}+m_{3}+n_{3})^{2}-(2m_{1}+m_{2}+2m_{3})(m_{2}+2n_{2})}}{2m_{1}+m_{2}+2m_{3}}
\Big[\frac{\partial}{\partial\omega_1}\,,\frac{\partial}{\partial\omega_2}\Big]=\frac{\partial}{\partial\omega_3}\,, \Big[\frac{\partial}{\partial\omega_2}\,,\frac{\partial}{\partial\omega_3}\Big]=\frac{\partial}{\partial\omega_1}\,,\Big[\frac{\partial}{\partial\omega_3}\,,\frac{\partial}{\partial\omega_1}\Big]=\frac{\partial}{\partial\omega_2}\,.
( Ran(T)+Ran(L) ) \oplus \{0\}=Ran(M) &=Ran(M M^{\ast})^{1 / 2}&=Ran\left(\begin{array}{cc}\left(TT^{*}+L L^{*}\right)^{1 / 2}0 0 0\end{array}\right) &=Ran\left(\sqrt{T T^{*}+LL^{*}}\right) \oplus\{0\}.
< J_{\alpha}( x ) ~{\cal J}_{\mu \nu \rho}^{( 3 )}( 0 ) > = ( \pi_{\mu \nu}\pi_{\rho \alpha}+ \pi_{\mu \rho}\pi_{\nu \alpha}+ \pi_{\nu \rho}\pi_{\mu \alpha}) \left( \frac{f_{3 1}( t )}{| x |^{4}}\right)
(\theta_rw)_2-(\phi_rw)_2=[(\theta_r)_2-(\phi_r)_2]w_0+[(\theta_r)_1-(\phi_r)_]w_1=[(\theta_{r-1})_1+(\phi_{r-1})_1]w_0+[(\theta_{r-1})_0+(\phi_{r-1})_0]w_1
P_1^{[4]}\stackrel{(a)}\leq\sum_{\substack{(\tilde{m}_p,\tilde{m}_1,\tilde{w})\neq (1,1,1), \tilde{i}\in\mathcal{I}}}2^{-n\big(I(U_0,U_1;Y_1)-\tau^{[4]}_1(\delta)\big)}\leq2^{n(R_p+R_1+\tilde{R}+R')}2^{-n\big(I(U_0,U_1;Y_1)-\tau^{[4]}_1(\delta)\big)}=2^{n\big(R_p+R_1+\tilde{R}+R'-I(U_0,U_1;Y_1)+\tau^{[4]}_1(\delta)\big)}
\frac{\partial\overline{H}_0}{\partial q_{n,\alpha_n}}\left(q_{n,\alpha_n},p_{n,m_n-1}\right)=-\frac{\partial L_0}{\partial q_{n,\alpha_n}}\left(q_{n,\alpha_n},\dot{q}_{n,m_n-1}(q_{n,\alpha_n},p_{n,m_n-1})\right)\ ,
\mathbb{E}\left[ \Vert (B_n-I)x\Vert_2^2\right] =\mathbb{E}\Big[\sum_{j=1}^{d}(\langle b_j,\Pi_V f\rangle_n-\langle b_j,\Pi_V f\rangle)^2 \Big] &=\frac{1}{n}\sum_{j=1}^{d_1}\operatorname{Var}(b_j(X)\Pi_Vf(X))\leq \frac{\varphi^2d}{n}\Vert \Pi_Vf\Vert^2.
\rho_{134}(S_1)= S_2S_1S_1^*+S_1S_2S_2^* &\rho_{134}(S_2)= S_2^2S_1^*+S_1^2S_2^* &\rho_{142}(S_1)= S_2^2S_1^*+S_1^2S_2^* &\rho_{142}(S_2)= S_2S_1S_1^*+S_1 S_2S_2^*
X =-2\binom m3-\frac 14(a-3b+M+N+4)\binom m2-\frac 14\,ab(2c+M-N) &\quad-\left(\frac 12\,b(a-b)+\frac 1{16}(a-3b+M+N)(a+b+4c-3M+N+2)\right)m,
{\cal F}^{W_{3}}_{W_{1}W_{2}}: W_{1}\otimes W_{2}\to\mbox{\rm Hom}({\cal M}_{W_{1}W_{2}}^{W_{3}}, \overline{W}_{3})=({\cal M}^{W_{3}}_{W_{1}W_{2}})^{*}\otimes \overline{W}_{3}.
o_\lambda(\mathbf{x}^{\pm};z)=\sum_{\substack{\varepsilon_i\in\{0,1\}1\leq i\leq n+1}}(-z^{-1})^{|\varepsilon|}o_{\lambda_1-\varepsilon_1,\dots,\lambda_{n}-\varepsilon_{n}}(\mathbf{\bar{x}}^{\pm}).
\left. \left( \frac{\partial}{\partial ~t}\right) ~\psi_{N}\left(z,t\right) \right\vert_{z=z_{n}\left( t\right)}=-\dot{z}_{n}\left(t\right) ~\prod\limits_{\ell =1,~\ell \neq n}^{N}\left[ z_{n}\left( t\right)-z_{\ell}\left( t\right) \right] ~.
{\cal Z}_{_I}= \int D \overline{{{\xi}}}D \xi D \overline{{{\chi}}}D \chi \exp \left\{i \int d^{2}x \left[ - i \overline{{{\chi}}}\partial \! \! \! / \chi + i \overline{{{\chi}}}\partial \! \! \! / \gamma^{0}\partial \! \! \! / \xi - i \overline{{{\xi}}}\partial \! \! \! / \gamma^{0}\partial \! \! \! / \chi + m \overline{{{\xi}}}\partial \! \! \! / \partial \! \! \! / \xi \right] \right\}.
\mathbb E\left[\sup_{s\leq t}\|\int_0^t \sigma_s dW_s\|^m\right] \leq C_m \mathbb E\left[\left(\int_0^t (\Sigma_s) ds\right)^{\frac m2}\right] &=C_m\mathbb E\left[\left(\int_0^t \|\sigma_s\|_{L_{}(U,H)}^2 ds\right)^{\frac m2}\right],
h_n([n]) \cdot \tilde{f}_n(\tilde{\pi}) = h_n([n])\cdot \prod_{B \in \tilde{\pi}}h_n(B) &= h_n(B_1) \cdot h_n \left([n] \cdot \prod_{B\neq B_1 \in \pi}B\right) &= h_n(B_1) \cdot h_n\left(\prod_{B\in \pi}B\right) &= h_n(B_1) h_n(B_1 \cup \{1\}) h_n \left(\prod_{B\neq B_1 \in \pi}B\right) &=0
\liminf_{k\to+\infty}\Vert u_x(t_{n_k},\cdot+x(t_{n_k}))\Vert_{L^2(K)}\leq \lim_{k\to+\infty}\Vert u(t_{n_k},\cdot+x(t_{n_k}))\Vert_{L^2(K)}=\Vert \varphi_{c^\star}\Vert_{L^2(K)}
\tilde{\delta}_n(\frac{1}{2}) - \delta_0^{1/2}= \frac{1}{m_{\frac{1}{2}}(\alpha_0)}\frac{1}{n}\sum_{i=2}^n A_i - \delta_0^{1/2}+ \frac{1}{m_{\frac{1}{2}}(\alpha_0)}\epsilon_n^2(\tilde{\alpha}_n) \frac{1}{n}\sum_{i=2}^n A_i+ R_n
{p}(t,y) := - [A^{(4)}(t,y) \nabla(\mathrm{det}D\Phi^{-1}(t,y))+ \partial_t ({q}(t,y) \mathrm{det}D\Phi^{-1}(t,y) ) ]\mathrm{det}D \Phi(t,\Phi^{-1}(t,y))\,,{q}(t,y) := - \dot{\Phi}(t, \Phi^{-1}(t,y))\,,{g}(t,y) := f(t, \Phi^{-1}(t,y))\,.
c_6:=\left(1+\frac{2}{p\,(p+1)}\,\left(\frac{2}{(p+1)\,c_2^{2}}\right)^p\right)^{-\frac{1}{p+1}}\, \left(\frac{c_2^2}{p\, 4^{p+1}}\right)^\frac{1}{p+1}.
\begin{multlined}[t][12.5cm]\left|\widetilde{\Pi}^{P}_{\nu_{G},\nu_{G}+3b}\left((z_{0},z_{1}),(z_{0},z_{1})\right)\right|\sim 2\sqrt{\frac{b}{\pi}}\left(\frac{x}{y}\right)^{\nu_{G}}e^{bF(1/2,y_{b})}\leq 2\sqrt{\frac{b}{\pi}}\left(\frac{1}{2y_{b}}\right)^{\nu_{G}}e^{b\delta(b)}\end{multlined}