harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{n_{r}, l_{D}, q}= \pm M c^{2}\left[ 1 + \frac{\alpha^{2}}{\left( \frac 1 2 + \frac 1 2 \sqrt{\left( 2 l_{D}+ 1 \right)^{2}- 4 \left( q^{2}+ \alpha^{2}\right)}+ n_{r}\right)^{2}}\right]^{- 1 / 2}.
\lim_{n\to\infty}\frac{1}{n\pi}\int_{-a_n}^{a_n}\frac{Q(s)}{\sqrt{a_n^2-s^2}}\, ds=\lim_{n\to\infty}\frac{1}{n}\int_{-1}^{1}\frac{Q(a_nt)}{\pi\sqrt{1-t^2}}\, dt=1/\alpha.
T^{\mu\nu}=\hbar^{2}(\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\beta}\eta^{\nu\alpha}-\eta^{\mu\nu}\eta^{\alpha\beta})\partial_{\alpha}\overline{\psi}\partial_{\beta}\psi-\eta^{\mu\nu}M^{2}c^{2}\overline{\psi}\psi.
N_i - N_{i+1}= \sum_{r \in \mathbb{Z}}(n_{rm+i}- n_{rm+i+1}) &= \sum_{r \geq 0}(n_{rm+i}- n_{rm+i+1}) - \sum_{r < 0}(n_{rm+i+1}- n_{rm+i}) &= |\{r\geq 0| M(rm+i) = 0\}| - |\{r < 0| M(rm+i) = 1\}| &= k_i.
\Delta_{\kappa}( \vec{x}, x_{0}) \sim{\frac{- i \kappa \operatorname{c o s h}\left({\frac{\pi \kappa x^{0}}{2}}\right) \Gamma \left( 1 + i \kappa x^{0}\right)}{\left( 2 \pi \right)^{2}\sqrt{1 +{\frac{m^{2}}{\kappa^{2}}}}\, | \vec{x}| \left( \kappa | \vec{x}| \right)^{1 + i \kappa x^{0}}}}
Q(x_{1}, y_{1}, x_{2}, y_{2}) = \tfrac{1}{6}\left(x_{1}^{3}- 3x_{1}y_{1}^{2}- 3x_{1}(x_{2}^{2}- y_{2}^{2}) + 6y_{1}x_{2}y_{2}\right) & = \tfrac{1}{6}x_{1}^{3}- \tfrac{1}{2}x_{1}(x_{2}^{2}+ y_{1}^{2}- y_{2}^{2}) + y_{1}x_{2}y_{2},
M_{0,\rho,\rho'}(qx)=m_{0,\rho,\rho'}(x)M_{0,\rho,\rho'}(x) &M_{\infty,\rho,\rho'}(qx)=m_{\infty,\rho,\rho'}^{-1}(x)M_{\infty,\rho,\rho'}(x).
W ( \Phi ) = \sum_{i = 0}^{N}M_{i}W ( \imath e_{i}) + \frac{1}{2}\alpha \sum_{l = 0}^{N}s_{N - l}\sum_{m = 0}^{l}\mathrm{T r}( \Psi \Phi_{0}^{m}\Psi \Phi_{0}^{2 l - m}) +{\cal O}( \Psi^{3}) \, .
\begin{aligned}R_{ave}= \sum_{m = 1}^{M-1}\int_{0}^{\infty}\log_2(1 + \frac{x \rho \gamma_k}{x \rho \tilde{\gamma_k}+1})f_{|h_{m,k}|^2}(x)dx &+ \int_{0}^{\infty}\log_2(1 + x\rho \gamma_K)f_{|h_{K}|^2}(x)dx,\end{aligned}
q_{\nu,n}(x) := \frac{1}{\sqrt{2\pi}}~ e^{-x^2/2}\sum_{\mu_1 + 2\mu_2 + \ldots + \nu\mu_\nu = \nu}H_{\nu+2s}(x) \prod_{m=1}^\nu \, \frac{1}{\mu_m!}\,\left( \frac{\lambda_{m+2,n}}{(m+2)!}\right)^{\mu_m},
\frac{d\zeta^{h_1,h_2}(t)}{dt}=Df(X^x(t)).\zeta^{h_1,h_2}(t)+D^2f(X^x(t)).(\eta^{h_1}(t),\eta^{h_2}(t))~,\zeta^{h_1,h_2}(0)=0.
\partial_t u_\Gamma -\Delta_\Gamma \mu_\Gamma = -\partial_{\boldsymbol{\nu}}\mu \mbox{on}\Sigma:=\Gamma \times (0,T), \mu_\Gamma = -\Delta_\Gamma u_\Gamma +{\mathcal W}_\Gamma'(u_\Gamma)-f_\Gamma +\partial_{\boldsymbol{\nu}}u\mbox{on}\Sigma,
\delta_{\lambda}\tilde{F}= i [ \lambda , \tilde{F}]_{M}\Longrightarrow \quad \delta_{\lambda}( \frac 1{( \tilde{D}^{2})^{2}}\star \tilde{F}) = i [ \lambda , \frac 1{( \tilde{D}^{2})^{2}}\star \tilde{F}]_{M},
(\bar{\partial}^h\circ\bar{\partial}^{*h}\Phi)_{A_p\overline{B}_q}= -\sum_i(-1)^{i-1}h^{\bar{\varepsilon}\gamma}(\delta_{\bar{\beta}_i}\circ\delta_\gamma)\big(\phi_{A_p\bar{\varepsilon}\bar{\beta}_1\dots\hat{\bar{\beta}}_i\dots\bar{\beta}_q}\big).
(\ref{ddd}) \lesssim \sup_{R\in\mathcal{D}_{\mu}}\Big( \frac{1}{|R|}\int_R\sum_{k=\max{(3,\mu-2)}}^{\infty}{2^{(s+m)kq}\big|\Pi^*_kf(x)\big|^q}dx\Big)^{1/q}&\lesssim \sup_{0\leq k\leq \mu-1}{\big\Vert 2^{k(s+m)}\Pi_kf\big\Vert_{L^{\infty}}}+\sup_{R\in\mathcal{D}_{\mu}}\Big(\frac{1}{|R|}\int_R\sum_{k=\mu}^{\infty}{2^{(s+m)kq}\big|\Pi_kf(x) \big|^q}dx\Big)^{1/q}
\sum_{(e_m,rW_m)=1}\frac{y_{\mathbf{r}'}}{\phi_\omega(e_m)}\prod_{p|e_m}\Bigl(\frac{p}{p-1}-\frac{1}{\phi_\omega(p)}\Bigr)^{-1}&=\log{R}\frac{\mathfrak{S}_{WB}(\mathcal{L})W^kB^k}{\phi(WB)^k}\prod_{p|rW_m}\Bigl(1-\frac{1}{p}\Bigr)\prod_{p\nmid rW_m}\Bigl(\frac{p}{p-1}-\frac{1}{\phi_\omega(p)}\Bigr)^{-1}\int_0^\infty H(t_1,\dots, t_k)dt_m,
\sum_{{\bf{k}}}\Lambda_{{\bf{k}}}({\bf{q}}) \frac{- i \langle T a_{{\bf{k}}}^{\dagger n}( -{\bf{q}}) a_{{\bf{k}}^{'}}^{\dagger}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}= G_{2}({\bf{q}},{\bf{k}}^{'},{\bf{q}}^{'}; n )
\Pi_1 (m_0)= \frac{\lambda}{2}~T \sum \limits_n \int \frac{d^3 k}{(2 \pi)^3}\frac{1}{\omega_n^2 + \omega_k^2}= \frac{\lambda}{2}\int \frac{d^3 k}{(2 \pi)^3}\frac{1}{\omega_k(m_0)}\left[ n_B(\omega) + \frac{1}{2}\right],
\begin{aligned}\frac{1}{\tau^k}[f(x)-f(\breve{x}^{k})]-\frac{1}{\tau^{k-1}}[f(x)-f(\breve{x}^{k-1})]+(x-\widetilde{x}^{k})^T[-A^T\widetilde{\lambda}^k &+\tau^k\beta^k(D-A^TA)(\widetilde{x}^{k}-\bar{x}^k)+(2-\gamma)(1-\tau^k) \beta^kA^T(A\breve{x}^{k-1}-b) ]\geq0,~\forall x.\end{aligned}
f^1 :=xu_x+yu_y-xyu_{xy}-u+\frac{1}{6u_{yy}^3}(x^2-3y^2u_{yy}^4), f^2 :=v-xv_x-yv_y+xyv_{xy}+\frac{v_{yy}}{2u_{yy}^4}(x^2+y^2u_{yy}^4), f^3 :=u_x-yu_{xy}+\frac{x}{3u_{yy}^3}, f^4:=xu_{xy}-u_y+yu_{yy}, f^5 :=v_{x}-yv_{xy}-\frac{xv_{yy}}{u_{yy}^4},f^6:=xv_{xy}-v_y+yv_{yy}, f^7 :=u_{xy},~ f^8:=-v_{xy},~f^9:=\log v_{yy}-4\log u_{yy},~f^{10}:=-\log u_{yy}.
\psi_j*\vec f =\left(\widehat{\psi_j}\widehat{\vec f}\right)^\vee=\left[ 2^{-jn}\widehat{\psi}\left( 2^{-j}\cdot\right)\widehat{\vec f}\right]^\vee &=2^{-j\sigma}\left[ 2^{-jn}|\cdot|^\sigma\widehat{\varphi}\left( 2^{-j}\cdot\right)\widehat{\vec f}\right]^\vee=2^{-j\sigma}\left( |\cdot|^\sigma\widehat{\varphi_j}\widehat{\vec f}\right)^\vee,
\overbrace{[x+a, y+b]}=(x+a)\cdot_{\ltimes}(y+b) - (y+b)\cdot_{\ltimes}(x+a) &= x \cdot y+ \ell(x)b+ r(y)a - y \cdot x - \ell(y)a - r(x)b &= [x, y] + (\ell-r)(x)b - (\ell -r)(y)a.
\Vert y - (\bar{R}_j \, \cdot + \bar{c}_j)\Vert^2_{L^2(U)}\le Ck^2 \min_{p=2,4}(1+\vartheta_p) m^{-2}C_m m^{-2}( \gamma + \delta_{p}+ \epsilon\Vert W\Vert_*) & \le C \lambda^2 \min\nolimits_{p=2,4}(1+\vartheta_p) m^{2}C^2_m (\gamma + \delta_p + \epsilon \Vert W \Vert_*)
M_a = M_b \left(\begin{array}{cccc}1 0 \dots 0 (1!)^{-1}(b-a) 1 \dots 0 (2!)^{-1}(b-a)^2 (1!)^{-1}(b-a) \dots 0 \vdots \ddots ((k-1)!)^{-1}(b-a)^{k-1}((k-2)!)^{-1}(b-a)^{k-2}\dots 1\end{array}\right)
\sum_{k=1}^{K}n_{k}\Vert \widehat{\beta}_{-k}(\widehat{\lambda})-\beta\Vert_{2,n,k}^{2}\leq \sum_{k=1}^{K}n_{k}\Vert \widehat{\beta}_{-k}(\bar{\lambda}_{0})-\beta \Vert_{2,n,k}^{2}& +2\sum_{k=1}^{K}\sum_{i\in I_{k}}\varepsilon_{i}X_{i}^{\prime}(\widehat{\beta}_{-k}(\widehat{\lambda})-\widehat{\beta}_{-k}(\bar{\lambda}_{0})).
\left|\sum_{k=1}^\infty\alpha^k\left(\Big(\frac{1}{kR_0-1}\Big)^{2j-1}-\Big(\frac{1}{kR_0+1}\Big)^{2j-1}\right)\right| &<\sum_{k=1}^\infty\left(\Big(\frac{1}{2k-1}\Big)^{2j-1}-\Big(\frac{1}{2k+1}\Big)^{2j-1}\right)=1.
\alpha^N_X:C^NX \rightarrow \Omega^N \Sigma^N X \left[d_{\underline{n}},x^{\underline{n}}\right] \mapsto \left(t\mapsto\begin{cases}[x^i,d_i^{-1}(t)], t\in d_i(I^N) \ast, t\not\in d_{\underline n}\left(\coprod_{\underline n}I^N\right) \end{cases}\right)
\frac{A}{2 \pi}\int d^{2}x ~ \epsilon^{\mu \nu}~ \partial_{\mu}\omega_{\nu}+ \frac{B}{2 \pi}\int d^{2}x ~ \epsilon^{\mu \nu}~ \partial_{\mu}a_{\nu}= ( 2 - 2 g ) A + \frac{B V}{2 \pi}= N ,
{\cal L}={\cal R}\sqrt{{\cal-G}}= -(k^{ij}-hg^{ij})\sqrt{g}\frac{d g_{ij}}{ds}-(N{\cal H}_{0}+ N^{i}{\cal H}_{i})- 2\frac{d}{ds}(h\sqrt{g})-\nabla_{i}\varphi^{i}.
h_{\lambda z_1+\mu z_2}(x) =\Phi(\langle a_{z_1,z_2},x\rangle+\lambda b_{z_1,z_2}+\mu c_{z_1,z_2}), f_{z_1}(x) =\Phi(\langle a_{z_1,z_2},x\rangle+b_{z_1,z_2}), g_{z_2}(x) =\Phi(\langle a_{z_1,z_2},x\rangle+c_{z_1,z_2}).
{F_X}\left( x \right) = \frac{{{m^m}}}{{\Gamma \left( m \right)}}\sum\limits_{j = 0}^M{\frac{{{K^j}{\alpha_j}}}{{j!}}}\frac{1}{{\Gamma \left({j + 1}\right)}}\gamma \left({j + 1,\frac{x}{{2{\sigma^2}}}}\right),
\left| \begin{array}{l}\displaystyle c=1+\frac{2}{n_1-\ell}\,\sum_{k=\ell}^{n_1-1}\nu_k\,. begin{align*}0.5cm] \displaystyle c=1+\frac{2}{n_2-\ell}\,\sum_{k=\ell}^{n_2-1}\nu_k\,. \end{array}\right.
M^+(u,v) =\frac{(-1)^{u+v}}{\det(M+xy^*)}\det ((M+ xy^*)_{\bar{v},\bar{u}}) -xy^*(v,u) =\frac{(-1)^{u+v}}{\det(M+xy^*)}\sum_{C\subset [n]\setminus u}\det ((M\xleftarrow{C}xy^*)_{\bar{v},\bar{u}})-xy^*(v,u).
r^{m_-}\left\langle\left. \prod_{i=1}^m\tau_{a_{i}}(\delta_{i})\prod_{i=1}^n\tau_{a_{m+i}}(\gamma_{m+i})\right|\mathbf \mu\right\rangle^{(Y_{D_0,r},D_\infty)}_{g,\vec k,n,\vec\mu,d}
F = \tilde{F}((x_i-x_{j_1})x_{j_1}, (x_i-x_{j_1})x_{j_2}, (x_i - x_{j_2})x_{j_1}, (x_i-x_{j_2})x_{j_2}, (x_{j_1}- x_{j_2}), (x_{j_1}- x_{j_2}) x_i).
m(t,x)\leq\frac{C_{1}\vert x\vert^{2}}{16C_{2}^{2}t}+\frac{C_{1}\Lambda^{\mu}\vert x\vert^{\mu}}{4^{\mu}C_{2}^{\mu}t^{\mu-\nu}}-\frac{\vert x\vert^{2}}{4C_{2}t}=\frac{C_{1}\vert x\vert^{2}}{16C_{2}^{2}t}+\frac{C_{1}\Lambda^{\mu}\vert x\vert^{2}}{4^{\mu}C_{2}^{\mu}t^{1}}\cdot\frac{\vert x\vert^{\mu-2}}{t^{\mu-\nu-1}}-\frac{\vert x\vert^{2}}{4C_{2}t}\leq-\frac{\vert x\vert^{2}}{8C_{2}t}
\bar{P}_{1}V \bar{P}_{2}= \bar{P}_{2}V \bar{P}_{1}\quad , \quad \bar{P}_{1}V Z_{2}= Z_{2}V \bar{P}_{1}\quad , \quad D_{1}\bar{P}_{2}= V \bar{P}_{2}D_{1}V^{- 1}
T_k(f;g_1,...,g_{k-1})=\sum_{\substack{0\leq i_1+... + i_{k-1}\leq n-k+1 i_s\geq 0}}\pm \gamma(f;1^{i_1},g_1,1^{i_2},g_2,...,g_{k-1},1^{n-i_1+...+i_{k-1}}).
C_{\varepsilon}( \beta ) = \frac{\prod_{j \in I_{- \varepsilon}}( u_{j}+ 1 )^{[ \beta^{\vee}: \alpha_{j}^{\vee}]}}{\prod_{i \in I_{\varepsilon}}u_{i}^{\max ( [ \beta^{\vee}: \alpha_{i}^{\vee}] , 0 )}}
L\left(\frac{2\eta}{\sigma^2}\right) = \begin{cases}\frac{-2(2\eta - a)(\eta - \eta_2)(\eta - \eta_2')}{(a-2\eta)^2 - 1}, \eta \in \left( 0, \frac{a-1}{2}\right)\cup \left( \frac{a+1}{2}, +\infty\right) \frac{\eta}{1 - (a-2\eta)^2}, \eta\in \left( \frac{a-1}{2}, \frac{a+1}{2}\right),\end{cases}
E_{y}(\lambda )=y+\sum_{l=1}^{r}\dfrac{1}{\omega_{l}}\left\langle(\lambda -A)y,e_{l}\right\rangle\,\biggl(e_{r+l}+\sum_{p\ge 2}\dfrac{\lambda^{p-1}}{\omega_{(p-1)r+l}\dots\omega_{r+l}}\,e_{pr+l}\biggr)\!
(P-Q)^2 x_i = \lambda_i (2 - \lambda_i) x_i, 1 \leq i \leq m, (P-Q)^2 x_{m+i}= \lambda_i ( 2 - \lambda_i) x_{m+i}, 1 \leq i \leq m, (P-Q)^2 x_i = x_i, 2m < i \leq q, (P-Q)^2 x_i = 0, q < i \leq d.
d l^{2}= \frac{1}{2 f^{2}}\left| d E + \frac{2 \mathrm{I m}\Phi}{\mathrm{I m}z}d \Phi - \left( \frac{\mathrm{I m}\Phi}{\mathrm{I m}z}\right)^{2}d z \right|^{2}- \frac{1}{f \mathrm{I m}z}\left| d \Phi - \frac{\mathrm{I m}\Phi}{\mathrm{I m}z}d z \right|^{2}+ \frac{| d z |^{2}}{2 ( \mathrm{I m}z )^{2}}.
\mu_T(e_1,e_1)=e_1,\mu_T(e_1,e_2)= e_1,\mu_T(e_2,e_1)=a e_1+(1-a)e_2, \mu_T(e_2,e_2)=a e_1+(1-a)e_2, \alpha_T(e_1)=e_1, \alpha_T(e_2)=a e_1+(1-a) e_2, \beta_T(e_1)=e_1, \beta_T(e_2)= e_1.
\left|\frac{1}{\rho_{M_0}(M)}-\frac{1}{\rho_{M_0}(M_{k,n-2})}\right| =\left|\frac{1}{\|M_0 M\|}-\frac{1}{\|M_0 M_{k,n-2}\|}\right|\leq &\leq\frac{C\Lambda_{n-2}}{{\|M_0 M_{k,n-2}\|}^2}\leq \frac{C\Lambda_{n-2}}{{|k-k_0|}^2 \Lambda^{2}_{n-2}}\leq\frac{C}{\Lambda_n{|k-k_0|}^2}.
\begin{array}{c c c}{s A_{\mu}= - D_{\mu}c ,}&{s c = c^{2},}&{s \phi = \left[ c , \phi \right] ,}\\{s \psi_{\mu}= \left\{c , \psi_{\mu}\right\},}&{s \chi_{\mu \nu}= \left\{c , \chi_{\mu \nu}\right\},}&{s \eta = \left\{c , \eta \right\},}\\{s \overline{{\phi}}= \left[ c , \overline{{\phi}}\right] ,}&{}&{}\\ \end{array}
{\cal R}( x ) = \sum_{m = 0}^{\infty}\frac{b_{m}}{m !}\Lambda^{m}( x ) = \left( \frac{e^{\Lambda ( x )}- 1}{\Lambda ( x )}\right)^{- 1}\, , \qquad \Lambda_{j}^{i}( x ) = x^{k}f_{k j}^{i}\, ,
S_{Q 2}= \frac{1}{N}\mathrm{T r}\, \left( \sum_{i = 0}^{L - 1}( M_{1 , i + 1}- M_{1 , i})^{2}+ ( M_{2 , i + 1}- M_{2 , i})^{2}+ g ( i [ M_{1 , i}, M_{2 , i}] )^{2}\right)
\sin^{2}\theta g_{1}([X,Y],W) =g_{2}((\nabla \pi_{\ast})(Y,\mathcal{C}X)-(\nabla\pi_{\ast})(X,\mathcal{C}Y),\pi_{\ast}\omega W)+g_{1}(\mathcal{A}_{X}\mathcal{B}Y+\mathcal{A}_{Y}\mathcal{B}X,\omega W) &+\eta (Y)g_{1}(X,\omega W)-\eta (X)g_{1}(Y,\omega W),
Spec~(A_\alpha(G))=\left\{\left[\frac{2m\alpha}{n}+h\right]^{lr}, \left[\frac{2m\alpha}{n}-h\right]^{l}, \left[\frac{2m\alpha}{n}+k\right]^{a}, \left[\frac{2m\alpha}{n}-k\right]^{ar}\right\},
\begin{aligned}( \Phi'_{1,-1}\circ\rho)(L_{-1}) \,=\, - \partial + z, ( \Phi'_{1,-1}\circ\rho)(L_{0}) \,=\, -\partial^2+z \partial + \frac{1-d}2 , ( \Phi'_{1,-1}\circ\rho)(L_{1}) \,=\, -\partial^3 + z \partial^2 +(1-d)\partial. \end{aligned}
Z_{a b}^{S}= \left( \begin{array}{c c c c}{{0}}&{{z_{+}^{2}}}&{{0}}&{{0}}\\{{z_{+}^{2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{z_{H}^{2}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{z_{\chi}^{2}}}\\ \end{array}\right) \qquad \begin{array}{c c c}{{Z_{f_{i}}^{R}}}&{{=}}&{{\tilde{z}_{f_{i}}^{2}}}\\{{Z_{f_{i}}^{L}}}&{{=}}&{{z_{f_{i}}^{2}}}\\ \end{array}
{\cal S}_{\sigma}= \frac{1}{4 \pi \alpha^{\prime}}\int_{\Sigma}d^{2}\sigma \sqrt{\gamma}\gamma^{\alpha \beta}\partial_{\alpha}X^{M}\partial_{\beta}X_{M}+ \frac{1}{4 \pi}\int_{\Sigma}d^{2}\sigma \sqrt{\gamma}\Phi ( X ) R^{( 2 )}( \sigma , \tau )
\lim_{\epsilon\to 0}\left( \int_0^{x-\epsilon}\frac{f(y)}{x-y}dy + \int_{x+\epsilon}^{x_0}\frac{f(y)}{x-y}dy \right) = f(x) \cdot \left[ \log x - \log\left( x_0 - x \right) \right] + \int_0^{x_0}\frac{f(y) - f(x)}{x-y}dy,
\begin{cases}\begin{array}{ll}\dot{v}_m=\log\det(v_m)_{\alpha\bar{\beta}}-Av_m+f(z,t_m+t)+2^{-m+1}\mbox{on}\Omega\times (0,T-t_k), v_m(z,0)=u(z,t_m)+2^{-m+1}, \mbox{on}\bar{\Omega}, v_m(z,t)=\varphi (z,t_m+t)+2^{-m+1}, \mbox{on}\partial\Omega\times [0,T-t_k). \end{array}\end{cases}
\frac 1{2 \Gamma \left( - \frac 1 2 \right)}\sum_{j = 0}^{d}\frac{2 a_{j}}{j - d - 1}+ \frac 1{2 \Gamma \left( - \frac 1 2 \right)}r_{d + 1}\left( - \frac 1 2 \right) \, \, + \left. \frac{\mu^{s + 1}}{\Gamma \left( \frac s 2 \right)}\frac{a_{d + 1}}{s + 1}\right\rfloor_{s = - 1}=
f_{\omega}(x) = \sum_{i=1}^{d}\bigg[\Phi_{\omega_{i}}^{1}(x_{i})\mathbb{1}_{(-\infty, 0)}(x_{i})+ \Phi_{\omega_{i}}^{2}(x_{i})\mathbb{1}_{[0, a]}(x_{i})+\Phi_{\omega_{i}}^{3}(x_{i})\mathbb{1}_{(a, \infty)}(x_{i}) \bigg],
\phi(s_\alpha) = (S_{i_1}S_{i_2}S_{i_2}^*) (S_{i_2}S_{i_3}S_{i_3}^*) \cdots (S_{i_{k-1}}S_{i_k}S_{i_k}^*) = S_{i_1}(S_{i_2}S_{i_2}^* S_{i_2}) (S_{i_3}S_{i_3}^* S_{i_3}) \cdots (S_{i_{k-1}}S_{i_{k-1}}^* S_{i_{k-1}}) S_{i_k}S_{i_k}^* = S_{i_1}S_{i_2}\cdots S_{i_k}S_{i_k}^*.
g_1:=2[z_{tt}, \mathfrak{H}]\frac{\bar{z}_{t\alpha}}{z_{\alpha}}+2[z_t, \mathfrak{H}] \frac{\bar{z}_{tt\alpha}}{z_{\alpha}}-\frac{1}{\pi i}\int \Big(\frac{z_t(\alpha,t)-z_t(\beta,t)}{z(\alpha,t)-z(\beta,t)}\Big)^2 \bar{z}_{t\beta}d\beta.
\| \tilde{g}_i \|_{L_\nu^1}=\int_{[0,\frac{1}{2}]}|2 g_{\max}+ 2 g_i(2t)|\nu(dt) + \int_{[\frac{1}{2},1]}|2 g_{\max}- 2 g_i(2t-1)|\nu(dt) &= \int_{[0,1]}g_{\max}+ g_i(t)\nu(dt) + \int_{[0,1]}g_{\max}- g_i(t)\nu(dt) &= 2 g_{\max}>0
\begin{aligned}\dot z_1 = g_1(z_1, z_2) = \frac{p_1 z_2}{1/p_2 + z_2}- p_3 z_1 + u, \dot z_2 = g_2(z_1, z_2) = \frac{- z_2^2 p_4}{1 + z_1/p_5}+ p_{6}z_2, \end{aligned}
F_0(z)=\lim_{N\to\infty}A(z)^2\sum_{|k|\leq N}\bigg\{\frac{F(s_k)}{A'(s_k)^2(z-s_k)^2}+ \frac{F'(s_k)A'(s_k)-F(s_k)A''(s_k)}{A'(s_k)^2(z-s_k)}\bigg\}\,,
X^\bot_{\mu\nu}X^\bot_{\rho\sigma}= \frac{1}{2}\left(\,\pi_{\mu\nu}\pi_{\sigma\rho}- \pi_{\mu\sigma}\pi_{\nu\rho}- \pi_{\mu\rho}\pi_{\nu\sigma}\, \right) Q_0^2 + \frac{i}{4}\left(\, \pi_{\mu\sigma}\varepsilon_{\nu\rho\lambda}+ \pi_{\nu\rho}\varepsilon_{\mu\sigma\lambda}\, \right) e^{(0)\lambda}Q_0.
R_{M N}- \frac{1}{2}G_{M N}R = \left( \begin{array}{c c c}{{\frac{\nabla^{2}\log ( f ( r ) )}{2 f ( r )}~ b^{2}~ \eta_{\mu \nu}}}&{{~}}&{{~ \}}&{{0}}&{{~ \}}&{{~}}&{{0}}\\ \end{array}\right)
T_\ell \left( \sum_{n=1}^\infty a_n \underline{q}^{n-1}d \underline{q}\right) = \sum_{n = 1}^\infty a_{\ell n}\underline{q}^{n-1}d \underline{q}+ \sum_{n=1}^\infty a_n \underline{q}^{\ell n-1}d\underline{q}.
\mu_k(\tau,\theta)=\frac{1}{\tau^2}e^{\frac{k}{|\tau|}}(\cos(k\theta),\operatorname{sgn}(\tau)\sin(k\theta)), \lambda_k(\tau,\theta)=\frac{1}{\tau^2}e^{\frac{k}{|\tau|}}(-\sin(k\theta),\operatorname{sgn}(\tau)\cos(k\theta))\qquad(k\in\mathbb N).
\iota(a\star b) = ab+ \sum_{k=1}^\infty \left(\frac{i}{2}\right)^k (-1)^k\rho^{2k}P_k(\iota a,\iota b) = ab+ \sum_{k=1}^\infty \left(\frac{i}{2}\right)^k \rho^{2k}P_k(\iota b,\iota a)=\iota(b)\star \iota(a)
\Sigma+\eta > \frac{1}{M}\sum\limits_{l=1}^M\Big((h^{D_1}(k_0)\phi_l,\phi_l) \|\psi_l\|_{L^2}^2+(h^{D_2}(K-k_0)\psi_l,\psi_l) \|\phi_l\|_{L^2}^2\Big) \ge (z_1(k_0) +z_2(K-k_0)) \,\, \frac{\sum\limits_{l=1}^M \|\phi_l\|_{L^2}^2 \|\psi_l\|_{L^2}^2}{M}=z_1(k_0) +z_2(K-k_0).
\Sigma ( p , \beta ) ={\frac{C}{f ( p , \beta ) ( 2 f ( p , \beta ) - m_{2})}}\biggl [ c h ( 2 g ( p , \beta ) ) +{\frac{\int{\frac{d p c h^{2}( 2 g ( p , \beta ) )}{f^{2}( p , \beta ) ( 2 f ( p , \beta ) - m_{2})}}}{1 +{\frac{\nu^{4}( \varphi , \beta )}{1 6 ( 2 \pi )^{D}}}\int{\frac{d p c h ( 2 g ( p , \beta ) )}{f^{2}( p , \beta ) ( 2 f ( p , \beta ) - m_{2})}}}}\biggr ]
\sum_{j_1=0}^{N_1}\sum_{j_2=0}^{N_2}w_2(j_1,j_2,k_1,k_2) P_{n_1(k_1),n_2(k_2)}(m_1(j_1),m_2(j_2))\overline{P_{n_1(k_1'),n_2(k_2')}(m_1(j_1),m_2(j_2))}\qquad{}=\delta_{k_1,k_1'}\delta_{k_2,k_2'},
M_{\beta}^{2}( \phi ) = m_{0}^{2}+ \frac{\lambda_{0}}{6}\left( \frac{\phi^{2}}{N}+ F_{\beta}( \phi ) \right) + \frac{\eta_{0}}{5 !}\left( \frac{\phi^{2}}{N}+ F_{\beta}( \phi ) \right)^{2}.
\begin{array}{r c l}{{}}&{{}}&{{Z_{m}^{n}Z_{m}^{n}\propto Z_{m}^{n}~ ,}}\\{{}}&{{}}&{{S_{m}w S_{m}\propto S_{m}~ , \forall w \in A_{2 m}~ ,}}\\{{}}&{{}}&{{Y_{m}w Y_{m}\propto Y_{m}~ , \forall w \in A_{n}~ ,}}\\{{}}&{{}}&{{Y_{m}Y_{m}\not = 0 ~ .}}\\ \end{array}
\begin{array}{rcl}\mathrm{d}x^{1}\mathrm{d}x^{2}\mathrm{d}x^{3}\mathrm{d}\varphi^{1}= \mathrm{d}x^{1}\mathrm{d}\varphi^{2}= \mathrm{d}x^{2}\mathrm{d}\delta(\varphi^{1}) = \mathrm{d}x^{1}+\mathrm{d}u^{1}\mathrm{d}\delta(\varphi^{2}) = \frac{1}{u^{1}+1}\mathrm{d}x^{3}-\frac{x^{3}}{(u^{1}+1)^{2}}\mathrm{d}u^{1}\end{array}
f_{X'}(t) = \lim_{\epsilon\to0}\frac{\mathbb{P}(t\leq F<t+\epsilon|F<G)}{\epsilon}= \lim_{\epsilon\to0}\frac{\mathbb{P}(t\leq F<t+\epsilon)\mathbb{P}(F<G|t\leq F<t+\epsilon)}{\epsilon\mathbb{P}(F<G)}= \frac{\lambda e^{-t\lambda}\mathbb{P}(G>t)}{1-p},
H(z)= ~ z \left(\frac 1{1-2 z(1+ M(z))}\right)^2=~z \left(\frac 1{1- 2z\left(1+ \psi_M(L_M(z))\right)}\right)^2=-\frac{t (-1 + 3 t)^3}{(1 - 5 t + 8 t^2)^2}~;
|F(t)| \leq \kappa_2 \max \lbrace |x_i(t)-z_i|, \: i\in \lbrace -N,-N+1,N,N+1 \rbrace \rbrace, \kappa_2 = 8 \max \lbrace |V_1(x,y)|,|V_2(x,y)|, \: |x-y| \leq 2 \rbrace \cdot \max \lbrace |V_{12}(x,y)|,|x-y| \leq 2 \rbrace.
K_{E}( x , b^{\prime}) ={\frac{( 1 - \bar{X}^{2}- \bar{Y}^{2})^{2 h_{+}}}{[ 2 \bar{X}\bar{X}^{\prime}+ 2 \bar{Y}\bar{Y}^{\prime}- ( 1 + \bar{X}^{2}+ \bar{Y}^{2}) \cosh^{2}r_{+}\Delta \bar{\phi}]^{2 h_{+}}}}.
\mathcal{L}(y(v),v,p(v)):= \mathcal{J}(y(v),v) + \int_{\Omega}\big( (\nu(\boldsymbol{x}) \nabla y(\boldsymbol{x})) + f(\boldsymbol{x}) + v(\boldsymbol{x})\big) p(\boldsymbol{x}) \, d\boldsymbol{x},
\P\big( \cap_{j=1}^k \{T_{s_{j-1}+1}\le t_{j1},\ldots,T_{s_j}\le t_{j\ell_j}\}\mid \cap_{j=1}^k N(t_{j-1},t_j] =\ell_j \big) = \prod_{j=1}^k \P\big( T_{1}\le t_{j1}-t_{j-1},\ldots,T_{\ell_j}\le t_{j\ell_j}-t_{j-1}\mid N(0,t_j-t_{j-1}]=\ell_j \big).
\lim\limits_{x \rightarrow 1-}{Q\big(z+(1-z)x\big)-\big(z+(1-z)x\big) \over (1-x)(1-Q'(x))}={1 \over 1-Q'(1)}\lim\limits_{x \rightarrow 1-}{Q\big(z+(1-z)x\big)-\big(z+(1-z)x\big) \over 1-x}&={1-z \over 1-Q'(1)}\lim\limits_{x \rightarrow 1-}{Q\big(1-(1-z)(1-x)\big)-Q(1)+(1-z)(1-x) \over (1-z)(1-x)}~=1-z.
a\left( z,w,k\right) =\left[ \dfrac{1}{\pi^{2}}\int\frac{1}{z-\zeta}e_{-k}\left( \zeta\right) u(\zeta)\frac{1}{\overline{\zeta}-\overline{w}}~d\zeta\right] e_{k}(w)\overline{u}(w).
\Pr_{a_1,\ldots,a_{n-k}}\left[-L \le -a_0 \le L \right] = \Pr\left[\frac{-L}{\sqrt{n-k}}\le g \le \frac{L}{\sqrt{n-k}}\right] + O\left(\frac{1}{\sqrt{n}}\right) = \Omega\left(\frac{1}{\sqrt{n}}\right) \; ,
S^{(1)}_2(n) = \sum_{\nu=1}^n \nu^2 = \psi^{(1)}_2(n) = \left[ 1 + \frac{2}{3}(n - 1) \right] B_{2, n-1}= \frac{1}{6}n (n+1) (2n+1)\,. & S^{(1)}_3(n) = \sum_{\nu=1}^n \nu^3 = \psi^{(1)}_3(n) = B_{2,n-1}+ \frac{3}{2}(n - 1) B_{3,n-1}= \left[\frac{n(n+1)}{2}\right]^2\,.
\beta(D_{L}\oplus\delta_{R})=[\begin{matrix}0&\delta_{R}\\D_{L}&0\end{matrix}]=i\beta\partial_{t}+i[\begin{matrix}0&\sigma^{k}\\ -\sigma^{k}&0\end{matrix}]\nabla_{k}-m[\begin{matrix}0&\eta\omega K\\ -\eta\omega K&0\end{matrix}]
\|\mathfrak{S}_{t}f-\mathfrak{S}_{s}f\|_{L^{2}(E,m)}=\|\mathfrak{S}_{s}\left(\mathfrak{S}_{t-s}f-f\right)\|_{L^{2}(E,m)}&\le\|\mathfrak{S}_{t-s}f-f\|_{L^{2}(E,m)}\to 0s\to t.
{\cal Z}_{\rm 3D{\,}AHM}=\int DA_\mu D\theta^{\rm sing.}D\theta^{\rm reg.}\exp\left\{-\int d^3x\left[\frac{1}{4e^2}{\cal F}_{\mu\nu}^2+\frac{\eta^2}{2}\left(\partial_\mu\theta-A_\mu\right)^2\right]\right\}.
\alpha = (1+c_T) \sqrt{\frac{1+\delta}{1-\delta}}\sqrt{1 -\alpha_0^2}\, , \beta = (1+c_T) \left[\sqrt{\frac{1+\delta}{1-\delta}}\left(\frac{\beta_0}{\alpha_0}+ \frac{\alpha_0 \beta_0}{\sqrt{1-\alpha_0^2}}\right) + \frac{2}{\sqrt{1-\delta}}\right] \, , \alpha_0 = c_H (1 - \delta) - \delta \, , \beta_0 = (1+c_H)\sqrt{1+\delta}\, .
\mathbb{P}[\omega_M \in dt \vert S(0)=x]=\int_{x}^{+\infty}\mathbb{P}[_{\omega \ge 0}S(\omega) \in dt, \max\limits_{\omega \ge 0}S(\omega) \in dz, \max\limits_{\omega \le 0}S(\omega) < z \vert S(0)=x] =\int_{x}^{+\infty}\frac{1}{2}j(t)\partial_y f(0,x-z,t,0)\mathbb{P}[S(\omega)<z \omega \le 0 \vert S(0)=x]dzdt
S_{\sigma}={1\over{4\pi\alpha^{\prime}}}\int d^2\sigma [\sqrt{\gamma}\gamma^{\alpha\beta}\partial_{\alpha}X^M \partial_{\beta}X^N\hat{G}_{MN}+{\epsilon}^{\alpha\beta}\partial_{\alpha}X^M \partial_{\beta}X^N\hat{B}_{MN}-{1\over 2}\sqrt{\gamma}\hat{\Phi}{\cal R}^{(2)}].
\left\|\begin{pmatrix}1 \|T\|_A 0 1\end{pmatrix}\right\|^2 =r\left[\begin{pmatrix}1 0 \|T\|_A 1\end{pmatrix}\begin{pmatrix}1 \|T\|_A 0 1\end{pmatrix}\right] =r\left[\begin{pmatrix}1 \|T\|_A \|T\|_A \|T\|_A^2+1\end{pmatrix}\right] =\frac{1}{2}\left(2+\|T\|_A^2+\sqrt{\|T\|_A^4+4\|T\|_A^2}\right).
\det \begin{pmatrix}\alpha \; \epsilon_{ij}|\beta| \; \epsilon_{ik}|\beta| \cr \epsilon_{ij}|\beta| \; \alpha \; \epsilon_{jk}|\beta| \cr \epsilon_{ik}|\beta| \; \epsilon_{jk}|\beta| \; \alpha \end{pmatrix}= \alpha^3+ 2 \epsilon_{ij}\epsilon_{ik}\epsilon_{jk}\beta^2|\beta| - 3\alpha\beta^2.
J_{L}[ u ] = \det \left| \! \left| \frac{\delta \left\{x_{0}\sigma^{\prime}- u ( x_{0}e^{\sigma}, b ) e^{- \sigma}\right\}e^{2 \sigma}}{\delta \sigma}\right| \! \right| \det \left| \! \left| \frac{\delta \sigma}{\delta b}\right| \! \right| .
\int d^{3}\vec{y}S ( \vec{x}, \vec{y}) S^{- 1}( \vec{y}- \vec{z}) = \left( \begin{array}{c c c}{1}&{0}&{0}\\{0}&{\delta_{\mu}^{\rho}}&{0}\\{0}&{0}&{\delta_{j}^{k}}\\ \end{array}\right) \delta ( \vec{x}- \vec{z}) .
g(z) = (1-z^2)f_t(z) = (1-z^2)\sum_{n=0}^\infty \xi_n z^n = \sum_{n=0}^\infty (\xi_n z^n- \xi_n z^{n+2}) = 1 +\xi_1 z + \sum_{n=2}^\infty (\xi_n-\xi_{n-2})z^n.
\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r^{2}}\frac{\partial^{2}\psi}{\partial\phi^{2}}=\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}}-\frac{1}{r}\frac{\partial\psi}{\partial r}
\frac{1}{2}\int_{M_1(t)}|\nabla u|^2\d x- \frac{1}{2}\int_{M_2(t)}|\nabla u|^2\d x \le \frac{1}{2}\int_{M_1(t)}(|\nabla u|^2-c^2)\d x- \frac{1}{2}\int_{M_2(t)}(|\nabla u|^2-c^2)\d x=o(t).
\delta_{I}\mathcal{P}_{2i}^{a}=-\frac{1}{2}\epsilon_{0ijk}F^{jka}+\partial_{i}\pi^{a},\;\delta_{I}\mathcal{P}_{1}^{a}=-\pi^{a}+\pi^{(1)a},\;\delta_{I}\mathcal{P}_{2}^{a}=\pi^{(2)a},
< T_{J , J}W_{J_{1}, m_{1}}\prod_{i = 2}^{N}T_{J_{i}, J_{i}}> = \frac{4 J ( m_{1}- J_{1})}{p - 2 J}< W_{J + J_{1}- 1 , J + m_{1}}\prod_{i = 2}^{N}T_{J_{i}, J_{i}}> .
\sum_{n=1}^\infty e^{-h_\mu\tilde{\psi}(n)+(1+\varepsilon)\sqrt{2\rho_\mu\tilde{\psi}(n)\log\log\tilde{\psi}(n)}}\leq \sum_{n=1}^\infty e^{-h_\mu\psi(n)+(1+\varepsilon)\sqrt{2\rho_\mu\psi(n)\log\log\psi(n)}}+\sum_{n=1}^\infty e^{-h_\mu n+(1+\varepsilon)\sqrt{2\rho_\mu n\log\log n}}< \infty.
\Phi_i^{-1}(0) \subset \left( \bigcup_{j=1}^l B_{cm_i^{-1/2}}(x_j) \right) \cup \left( \bigcup_{j=l+1}^k B_{cm_i^{-1/2}}(m_i x_j) \right) , \ \ \mathrm{deg}( \Phi_i \Big\vert_{\partial B_{cm_i^{-1/2}}(x_j)}) =1 .
\delta_{B}( f^{( 2 )}u^{3}\left[ \psi^{2}\right] ) + \delta_{F}( f^{( 4 )}u^{2}\left[ \psi^{4}\right] ) + u^{i}u \epsilon \dot{N}^{i ( 2 )}\left[ \psi^{2}\right] \psi + \frac{i}{2}u^{2}\psi \dot{M}^{( 2 )}\left[ \psi^{2}\right] \epsilon = 0 ~ ,
\lim_{t\to\infty}e^{-\lambda t}S_{j,+}(t) = \nu Y \int_0^\infty e^{-\lambda s}\left(\sum_{k=1}^\infty p_{j,+}^k(s)\right) ds, & \lim_{t\to\infty}e^{-\lambda t}S_{j,-}(t) = \nu Y \int_0^\infty e^{-\lambda s}\left(\sum_{k=1}^\infty p_{j,-}^k(s)\right) ds,
D_1 =\underset{g}{\sum}\:\:\underset{h\neq 1}{\sum}(w_1(g,(gh)^{-1})-w_2(gh,g^{-1})) = \underset{g}{\sum}\frac{1}{z_i-g\cdot z_k}\:\underset{h\neq 1}{\sum}[X_{ik}(g),X_{ik}(gh)]( \frac{1}{z_k-h\cdot z_k}+\frac{1}{z_k-h^{-1}\cdot z_k}).
{D_k}= \frac{{{\delta_k}}}{{{\lambda^{mM + k}}\Gamma \left( U \right)}}{\left({1 +{\lambda^{- 1}}}\right)^{- \left({mM + k}\right)}},{\delta_{k+1}}=\frac{m}{{k + 1}}\sum\limits_{q = 1}^{k + 1}{\sum\limits_{i = 1}^M{{{\left({1 - \frac{\lambda}{{{\lambda_i}}}}\right)}^q}}}{\delta_{k + 1 - i}},{\delta_0}= 1.