harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{MA}_{\mathbb{C}}(\psi_\phi)(\exp(x))= \frac{1}{2^{r+n}}\mathrm{MA}_{\mathbb{R}}(\psi_\phi)(x)\frac{1}{\mathbf J(x)}\prod_{\alpha \in \Phi_+}\langle\alpha,\nabla \psi_\phi(x)\rangle^2.
\begin{aligned}\sum_{r = -1}^{2p-1}b_r \sum_{w=0}^{p}(-1)^w \tbinom pw \left(\tfrac mn\right)^{p-w}f_r(T_{p-w,w}) = \sum_{w=0}^{p}(-1)^w\tbinom pw \left(\tfrac mn\right)^{p-w}\left[f_0(T_{p-w,0})A_{p-w-1,w}+ f_0(T_{0,w})A_{p-w,w-1}\right].\end{aligned}
\frac{d \, h \left( t , \frac{{D_{B}}^{\frac 1 2}}{\mu}\right)}{d t}= \sum_{k = 0}^{d}( - k ) \frac{1}{2 \mu}\frac{\Gamma \left( \frac{k + 1}2 \right)}{\Gamma \left( \frac 1 2 \right)}a_{d - k}\left( \frac{t}{2 \mu}\right)^{- k - 1}+
-\frac{1}{4n}(h-2n)\big(r(h-2)-r(h+2)\big) &=-\frac{1}{4n}((h-2n)r(h-2)-(h-2n)r(h+2)) &\overset{by (\ref{rh-2 rh+2})}{{=\joinrel=\joinrel=}}-\frac{1}{4n}((h-2n)r(h-2)-(h+2n)r(h-2)) &=r(h-2).
|\phi(k)|\leq e^{24\varepsilon q_n}\frac{e^{\beta_n q_n}}{\max(|\ell|,1)}\max\begin{cases}c_{n,\ell-2}c_{n,\ell-1}e^{-2 q_n L}r_{\ell-2}e^{-q_n L}c_{n,\ell-1}r_{\ell-1}\max((c_{n,\ell-1})^2, (c_{n,\ell})^2) e^{-q_n L}r_{\ell}c_{n,\ell}e^{-q_n L}r_{\ell+1}c_{n,\ell}c_{n,\ell+1}e^{-2 q_n L}r_{\ell+2}\end{cases}
{\cal L}= \frac{1}{g^{2}}\left[ ( D_{\mu}z )^{\dagger}( D^{\mu}z ) + i b \, \epsilon^{\mu \nu}( \partial_{\mu}z )^{\dagger}( \partial_{\nu}z ) - \frac{M^{2}}{2}\, b^{2}- \lambda \, ( z^{\dagger}z - 1 ) \right] ,
\mathrm{i i}) \; \; m_{\alpha}\,{\bf j^{\alpha}}\ne 0 \; \; , \qquad \qquad \frac{\partial}{\partial u}{\bf{\hat{C}}}= ( \log r ) \frac{1}{2}m_{\alpha}\,{\bf j^{\alpha}}( u , \phi ) \quad . \qquad \qquad \qquad \qquad
\frac{(q^2;q^2)_{\infty}}{(q;q^2)_{\infty}}\oint_{C_0}\,\frac{dz}{2{\pi}iz}z^s\sum^{1}_{m=0}\mathcal{B}(z^{-1}a,m;q^2)\mathcal{B}(za,m;q^2)=\mathcal{B}(a^{-1}q^{-\frac{1}{2}},|\tilde{s}|;q^2)\mathcal{B}(a^{-1}q^{-\frac{1}{2}},|\tilde{s}|+1;q^2)\mathcal{B}(a^2,0;q^2) \; .
B(u)=\sum_{\substack{\forall 1 \le i \le r: b_i, y_i \ge 1}}\prod_{i=1}^{r}c(b_i,y_i,\mu_i) y_i u^{b_i}= \prod_{i=1}^{r}(\sum_{b_i,y_i \ge 1}c(b_i,y_i,\mu_i)y_i u^{b_i}).
\sum_{\ell=1}^{2}q^{(2)}_{\ell,e}\cdot f^{(\ell)[e]}_{k-1}x^{[k+e-1]}=-q^{(2)}_{0,d}x^{[d]}\Leftrightarrow \ q^{(2)}_{2,e}\cdot f^{(2)[e]}_{k-1}x^{[k+e-1]}= -(q^{(2)}_{0,d}x^{[d]}+ q^{(2)}_{1,e}\cdot f^{(1)[e]}_{k-1}x^{[k+e-1]}) \Leftrightarrow \ q^{(2)}_{2,e}\cdot f^{(2)[e]}_{k-1}x^{[k+e-1]}= -q^{(2)'}_{0,d}x^{[d]}
- \frac{g^{4}}{2}f^{a b^{\prime}c^{\prime}}f^{a d e}f^{c^{\prime}c g}f^{b b^{\prime}g}\int \frac{d^{4}s}{( 2 \pi )^{4}}( 2 s_{0}+ q_{0}) ( 2 r_{0}- q_{0}) \delta_{k l}\Delta^{- 1}( q ) N D ,
\varphi_{s}^{j}= R_{ss}\cdot U^{j}+ F^{j}= x_{ss}u^{j}+ y_{ss}v^{j}+ z_{ss}w^{j}+ F^{j}, \psi_{t}^{j}= R_{tt}\cdot U^{j}+ H^{j}= x_{tt}u^{j}+ y_{tt}v^{j}+ z_{tt}w^{j}+ H^{j}, \varphi_{t}^{j}+ \psi_{s}^{j}= 2 R_{st}\cdot U^{j}+ G^{j}= 2 \left[x_{st}u^{j}+ y_{st}v^{j}+ z_{st}w^{j}\right] + G^{j}.
(-\Delta)^s|\cdot|^{-\theta}(x) = 2^{2s}\frac{\Gamma\left(\frac{N-\theta}{2}\right) \Gamma(\frac{2s+\theta}{2})}{\Gamma\left(\frac{N-\theta-2s}{2}\right) \Gamma\left(\tfrac{\theta}{2}\right)}\ |x|^{-(\theta + 2s)}, x \neq 0,N>\theta > - 2s,
T(m,s,z)=-\frac{(-1)^{m-1}}{(m-2)!}\frac{d^{m-2}}{dt^{m-2}}[\Gamma(s-t)z^{t-1}]|_{t=0}+\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{s-1+n}}{n!(-s-n)^{m-1}}
\frac{1}{2}m(\frac{dr}{d\tau})^{2}=[\frac{E^{2}}{2mc^{2}}-\frac{1}{2}mc^{2}]+\frac{GMm}{r}-\frac{L^{2}}{2\mu r^{2}}+\frac{G(M+m)L^{2}}{c^{2}\mu r^{3}}
H = \frac{{\bf P}^{2}}{2 m}- \frac{{\bf P}^{2}}{2 m^{2}c^{2}}\left( \frac{{\mathrm{\boldmath \pi}}^{2}}{2 \mu}+ U ( q ) \right) - \frac{{\bf P}^{4}}{8 m^{3}c^{2}}+ \frac{{\mathrm{\boldmath \pi}}^{2}}{2 \mu ( q )}+ v ( q ) .
H ( r , \sigma ) = \left( \frac{R_{0}}{R_{A}}\right)^{6}\Delta^{- 1}( r , \sigma ) e^{- 6 \rho ( r , \sigma )}\, , \qquad \mathcal{Z}( r , \sigma ) = e^{- 2 \rho ( r , \sigma )}\left( e^{2 \rho ( r , \sigma )}- \frac{1}{4}\right) \, .
T^{R}( p^{2}) = T ( p^{2}) - T ( 0 ) - p^{2}\left. \frac{d}{d p^{2}}T ( p^{2}) \right|_{p^{2}= 0}= \frac{p^{4}}{\pi}\int_{4 m^{2}}^{\infty}\! \! \frac{d t}{t - p^{2}}\frac{1}{t^{2}}I m T ( t )
\frac{(\sum_{k}s_{k}h_{j-k})^{2}}{\sum_{k}h_{j-k}^{2}}\le\frac{\sum_{k}s_{k}^{2}\cdot\sum_{k}h_{j-k}^{2}}{\sum_{k}h_{j-k}^{2}}=\sum_{k}s_{k}^{2}=constant.
\int_{B_{N}}|f_{1}f_{2}|^{\frac{s}{2}}\lesssim_{\epsilon}r^{-6+\frac{3s}{2}}N^{{s(\gamma_{lin}+\epsilon)}(\frac{1}{2}-\frac{1}{2^M})}\bigl(\int_{B_{N+r+r^2}}\prod_{i=1}^{2}\|f_{i}\|_{L^2({w}_{B_{r^2}(x+a_{i})})}^{\frac{s}{2}}\,dx \bigr)^{\frac{1}{2}}\prod_{i=1}^{2}\|f_{i}\|_{L^{s,N}(\mathbb{R}^{6})}^{\frac{s}{4}}.
A_i = \left\{(t_1, \dots, t_\ell) \in \Delta^\ell\ | \t_i \geq 1 - \delta_\ell \right\}1 \leq i \leq \ell\ , A_0 = \left\{(t_1, \dots, t_\ell) \in \Delta^\ell \ | \ \sum_{j=1}^n t_j \leq \delta_\ell \right\}\ .
\sum_{i \ge 0}Ce^{-c(t+i)^\gamma \mu(Y)^\gamma}\le C \int_0^\infty e^{- c (t+x)^\gamma \mu(Y)^\gamma}\, dx= \frac{C}{c^{1/\gamma}\mu(Y) \gamma}\int_{ct^\gamma \mu(Y)^\gamma}^\infty e^{-y}y^{\frac{1}{\gamma}-1}\, dy,
(\Delta \Phi )^2 + | \nabla \Psi |^2 \cr = \nabla \cdot \Big((\Psi - c) \nabla \Psi - (\Psi - c)\Delta \Phi \nabla^{\bot}\Phi \Big)+ h^3 \Delta \Phi - (\Psi -c)(\partial_1 h^1+ \partial_2 h^2)
N(m,n) = \mbox{the number of partitions of $n$ with Dyson rank $m$ (\cite{Dy44})}, \overline{N}(m,n) = \mbox{the number of overpartitions of $n$ with Dyson rank $m$ (\cite{Br-Lo-Os09})},{N2}(m,n) = \mbox{the number of partitions of $n$ with distinct odd parts with $M_2$-rank $m$}& \mbox{(\cite{Be-Ga02}, \cite{Lo-Os09})}.
\frac{1}{(2\pi b)^{1/2}}\int_{\delta}^{\infty}e^{-\beta^{2/3}N^{2/3}\frac{a^2}{4b}}da \leq\frac{N^{-1/3}\beta^{-1/3}}{(2\pi b N^{-2/3}\beta^{-2/3})^{1/2}}\int_{-\infty}^{\infty}e^{- \frac{a^2}{4bN^{-2/3}\beta^{-2/3}}}da \leq CN^{-1/3}\beta^{-1/3}.
A_{n}=-i\frac{\omega_{n}}{2n_{n\omega}c}(A_{0}\sqrt{\frac{2}{\pi}})^{n}z_{R}^{2}\int_{-\infty}^{z}\frac{\chi^{(n)}(u)}{q(u)^{2}}duexp(ik_{n}\frac{x^{2}+y^{2}}{2q(z)})
\|\Gamma^r(A)\| = \|\bigcup_{\sigma \in A}\Gamma^r(\{\sigma\})\| \leq \sum_{\sigma \in A}\|\Gamma^r(\{\sigma\})\| = \sum_{\sigma \in A}{r+1 \choose k+1}\cdot w(\sigma) ={r+1 \choose k+1}\cdot \|A\|.
S_{\mathrm{G F}}\equiv - \int d^{4}x i \mathrm{\boldmath ~ \d e l t a ~}_{\mathrm{B}}\biggl \{\bar{C}^{A}\biggl [ \partial^{\mu}A_{\mu}^{A}+ \frac \alpha 2 B^{A}- \frac \zeta 2 \epsilon^{A B C}\bar{C}^{B}C^{C}\biggr ] \biggr \}.
\hat{H}=\frac{1}{2}\hat{p}_0^2+\frac{\omega_0^2}{2}\hat{q}_0^2+\frac{1}{2}\sum_{k=1}^N\left[\hat{p}_k^2+\omega_k^2 \hat{q}_k^2+2\lambda_k(t)\hat{q}_0\hat{q}_k\right]\;.
\int \frac{d^4q}{(2\pi)^4}\,\Bigg(\frac{(k+q)^2+q^2-k^2}{\Big((k+q)^2+M^2\Big)^2 \Big(q^2+M^2\Big)^2}- \frac{4 M^2}{\Big((k+q)^2+M^2\Big) \Big(q^2+M^2\Big)^3}\Bigg) = 0.
+\infty > c_f +c_g =\left(f+g\circ L\right)^{*}_\Phi\left(\phi\right) \geq \left(f+\psi_0\circ L\right)^{*}_\Phi\left(\phi\right)+g^{*}_\Psi\left(\psi_0\right) & \geq\inf_{\psi\in\Psi}\left(f+\psi\circ L\right)^{*}_\Phi\left(\phi\right)+g^{*}_\Psi\left(\psi\right) & \geq \left(f+g\circ L\right)^{*}_\Phi\left(\phi\right),
\frac{1}{\pi^2 R^4}\int_{B(R)}\int_{B(R)}\int_{\mathbb{S}^1}\int_{\mathbb{S}^1}e(\langle x-y, t- w\rangle)dxdy d\mu_{B}(t)d\mu_{A}(w) = \int_{\mathbb{S}^1}\int_{\mathbb{S}^1}\left(\frac{J_1(R||t-w||)}{R||t-w||}\right)^2d\mu_{B}(t)d\mu_{A}(w)
I_{YM, \theta}={1\over 8 \pi}\int_X d^4 x \sqrt{g}\bigg({4\pi\over g_{YM}^2}{\rm tr}[ F_{\mu\nu}F^{\mu\nu}] +{i \theta \over 4\pi}\varepsilon_{\mu\nu\rho\sigma}{\rm tr}[F^{\mu\nu}F^{\rho\sigma}] \bigg),
V_k=\begin{cases}\{(1,-k), (1,1)\}k>0, \{(1,0)\}k=0,\end{cases}&r_{0,(1,0)}=0,r_{k,(1,-k)}=0,r_{k,(1,1)}=\begin{cases}\frac{1}{k^2}k>0, 0 k=0,\end{cases}
\tilde{u}_{0}=u_{0}\left[ \lambda \delta +\bar{\lambda}\left( 1-\beta_{0}x^{-1}\right) u_{0}-\bar{\lambda}\sum\nolimits_{\nu =0}^{d-2}\gamma_{1}^{d-1-\nu}\left( x^{-1}u_{\nu +1}\right) \right]^{-1}.
\mu_1 =\max_{\gamma:\gamma(h)^2=1}\sum_{v\in V}\biggl(\sum_{h: v}\gamma(h)-\sum_{h: v}\gamma(h)\biggr)^2 &=\big|\big|+\big|\big| &=\big|\big|+\big|\big|-2\cdot\big|\big| &=k+m-2k-2m+2N &=2N-k-m.
\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}(n-\ell-2) (p-1)(p-2)(p-3)p^{n-4}&= (p-1)(p-2)(p-3)p^{n-4}\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}(n-\ell-2) &= (p-1)(p-2)(p-3)p^{n-4}\sum_{k=1}^{n-3}\binom{n-k-2}{2}&= (p-1)(p-2)(p-3)p^{n-4}\binom{n-2}{3}.
0 = \frac{\partial}{\partial x^{i}}\left[ \frac{C^{i}( x )}{\chi ( x )}\right] = \left[ \frac{\partial}{\partial x^{i}}C^{i}( x ) \right] / \chi ( x ) - \frac{\partial}{\partial x^{i}}\chi ( x ) C^{i}( x ) / \chi ( x )^{2}.
\begin{cases}\begin{aligned}\bar h^{1^*}_{11k}+\bar h^{1^*}_{22k}=0, &\bar h^{2^*}_{11k}+\bar h^{2^*}_{22k}=0, &3\bar \lambda_2\bar h^{1^{\ast}}_{22k}+4\bar \lambda \bar h^{2^{\ast}}_{11k}=0.\end{aligned}\end{cases}
\widetilde J_{p,f}(\varphi_M)-\widetilde J_{p,f}(u_0) \le \frac{c_2^{2-q}}{q}\,\int_{\{u_0>M\}}M^{q}\, dx-\frac{c_2^{2-q}}{q}\, \int_{\{u_0> M\}}u_0^{q}\, dx+\int_{\{u_0>M\}}f\, (u_0-M)\, dx &= -\frac{c_2^{2-q}}{q}\, \int_{\{u_0>M\}}\left(u_0^q-M^q\right)\, dx+\int_{\mathbb{R}^N}f\, (u_0-M)_+\, dx.
\big( S ( \phi_a g ) \big) (t) = (S \psi) (t) + \int_0^{t_1}\frac{(\phi_a g - \psi) (s)}{t-s}\, ds + \int_{t_2}^1 \frac{(\phi_a g - \psi) (s)}{t-s}\, ds, t_1 < t < t_2.
\mathcal{F}_S(\{E_{j,t}\},B_R) = t^{1-n}\mathcal{F}_S(\{E_j\},B_{Rt}) < \left(\mathcal{H}^{n-1}(\partial B_1) + \frac{1}{2}\omega_{n-1}\right) R^{n-1}\sum_{j=0}^2 \alpha_j.
A_z(n;\chi) = \sum_{0\leq a \leq n}B_z(a, \chi) D_z(n-a, \chi) &\leq q^{n/2}\sum_{0\leq a \leq n}\frac{|B_z(a,\chi)|}{q^{a/2}}\binom{n-a + Am}{n-a}&\leq q^{n/2}\binom{n + Am}{n}\sum_{0\leq a \leq n}\frac{|B_z(a,\chi)|}{q^{a/2}}\leq q^{n/2}\binom{n + Am}{n}n^{c_A}.
-c^0_+\hat{c}+(-c^0+2-\partial_{\beta}W_{v^0}-\sigma^0)\hat{c}_++(-c^0_+-\partial_{\beta}\partial_{\beta}W_{v^0}c_+^0B)\hat{\sigma}=\partial_{\beta}\partial_{\beta}W_{v^0}c^0_+A-f.
\frac{1}{n}{\bf E}\Big\{\sum_{t=1}^{n}||X_t||_{{\mathbb R}^{n_x}}^2\Big\}= \frac{1}{n}\sum_{t=1}^{n}tr \Big(\Gamma_t P_t \Gamma_t^T +D_t K_{Z_t}D_t^T\Big) .
\|D\|_{H^1}\lesssim \sup_{\substack{i\not=j,k\geq 1 2\leq k+\ell\leq 3}}\frac{1}{d^3}\Big(\big\|(\partial_y^k\varphi_{ij})(\partial_y^\ell B_{ij,0})\big\|_{L^2}\Big)+\frac{1}{d^5}\lesssim\frac{1}{d^5}.
\left|\frac{a}{|a|}-\frac{b}{|b|}\right|\le \left|\frac{a}{|a|}-\frac{b}{|a|}\right|+\left|\frac{b}{|a|}-\frac{b}{|b|}\right|= \frac{|a-b|}{|a|}+\frac{|b|-|a|}{|a|}\le \frac{2}{|a|}|a-b|,
\begin{cases}\partial_{t}w_{2}-\Delta w_{2}+w_{2}\cdot\nabla w_{2}+w_{2}\cdot\nabla(u^{c,\gamma}+w_{1})+(u^{c,\gamma}+w_{1})\cdot\nabla w_{2}+\nabla \pi_{2}=0, \mathrm{div}w_{2}=0, w_{2}(x,0)=w_{02}.\end{cases}
\frac{1}{\lambda_{r}}-\frac{1}{(2\pi)^{2}}\left[ 3\left(C_{uv}-\frac{2}{3}\right) -\left(C_{uv}-\ln\frac{m^{2}}{\sigma_{0}^{2}}\right) \frac{m^{2}}{\sigma_{0}^{2}}-\left(C_{uv}-\frac{13}{6}-\ln\frac{m^{2}}{\sigma_{0}^{2}}\right) \frac{1}{r^{2}\sigma_{0}^{2}}\right] =0\, .
\zeta^{k}\colon \Sigma K^{\oplus{k+c\choose c-1}}\to K^{\oplus{k+c-1\choose{c-1}}}(\pi_{u_1\dots u_{k+1}})_{1\leq u_1\leq\dots\leq u_{k+1}\leq c}\mapsto \Big( \sum_{j=1}^c z_j\wedge \pi_{[v_1\dots v_k j]}\Big)_{1\leq v_1\leq\dots\leq v_k\leq c}.
\eta_i(c_i)=\frac{c_i+\sum_{Q_i}{\hat{c}_{ij}}}{1+\frac{1}{q_i}+\sum_{Q_i}{\frac{1}{\hat{q}_{ij}}}}=\frac{c_i+C_i}{1+\frac{1}{q_i}+B_i}
\mathcal L\Big(\sum_{j:\tau_j\leq \sigma_1}(\xi_{\tau_j}-\xi_{\tau_{j}-})\Big)\Longleftrightarrow\sum_{i\not=j}q_{i,j}\,\mathcal L(\Delta_{i,j})+\sum_{i\in E}\Pi_i|_{B(0,1)^c}.
\gamma_{x}= \left[ \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right] \otimes 1 , \gamma_{y}= \left[ \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right] \otimes \left[ \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right] , \gamma_{\bar{x}}= \gamma_{x}^{T}, \gamma_{\bar{y}}= \gamma_{y}^{T}
{\frac{\langle O_{N}^{\dagger}\, O_{N / 2}\, O_{N / 2}\rangle}{\sqrt{O_{N}^{\dagger}\, O_{N}}\, \sqrt{O_{N / 2}^{\dagger}\, O_{N / 2}}\sqrt{O_{N / 2}^{\dagger}\, O_{N / 2}}}}\sim{\frac{1}{2^{N}}}\, .
\begin{array}{ll}y_0=1 y_4= ibx_1 y_1=-(J_1^2 + J_2^2 + \frac{1}{4}+ 2 \alpha /x_3^2) y_5=ibx_2 y_2= -2J_3 y_6= - \frac{1}{2}ib \{x_3, J_1 \}y_3=b^2x_3^2 y_7=-\frac{1}{2}ib \{x_3, J_2 \}\quad ,\end{array}
\left\Vert f^{*n}-g^{*n}\right\Vert_{\mathcal{P}_n}\le\frac{n^n}{n!}\sum_{j=1}^{n}\binom{n}{j}\left\Vert g \right\Vert_{A}^{n-j}\left\Vert f-g \right\Vert_{A}^{j}.
\frac{1}{n}\sum_{k\in N}e_{k,j+n-2}^n+\frac{1}{n}\sum_{\tau=0}^{n-s-2}e^{n-\tau-1}_{i,j+\tau}&+\left(1-\frac{1}{n}\right)\left[\sum_{\tau=0}^{n-s-2}\frac{1}{n^{n-\tau-s-1}}e^{s}_{i,j+\tau}\right]+\frac{1}{n}\sum_{\tau=1}^{s-1}e_{i,j+n-s-2+\tau}^{s-\tau}
\times\;\prod_{i=1}^{n^\prime_b}\overline{\psi}^{(b)}({x^\prime}^{(b)}_i)P_+ \psi^{(b)}({x^\prime}^{(b)}_i)\prod_{i=1}^{m^\prime_b}\overline{\psi}^{(b)}({y^\prime}^{(b)}_i)P_- \psi^{(b)}({y^\prime}^{(b)}_i) \Big\rangle_0
d^{A}\theta^{\alpha}=\Sigma_{b<c}A_{bc}^{\alpha}\theta^{b}\wedge \theta^{c}+\Sigma_{b,\gamma}B_{b\gamma}^{\alpha}\theta^{b}\wedge \theta^{\gamma}+\Sigma_{\beta <\gamma}C_{\beta \gamma}^{\alpha}\theta^{\beta}\wedge \theta^{\gamma},
0 = ({\cal M}^{2})^{2}- \frac 1{4 k_{q}}\tilde{\gamma}\tilde{g}^{2}( q_{0}^{2}+ \bar{q}_{0}^{2}){\cal M}^{2}+ \frac 1{4 k_{T}k_{q}}( \tilde{\gamma}\tilde{g}^{2}- \frac 1{k_{T}}) q_{0}^{2}\bar{q}_{0}
\frac{C^{i}-C^{j}}{K_{i}}\ln \frac{1}{\lambda_{i}}+\frac{C^{j}-C^{i}}{K_{j}}\ln \frac{1}{\lambda_{j}}& +\kappa(C^{j}- C^{i})\theta_{i}+\kappa (C^{i}-C^{j})\theta_{j}\leq 0.
V = \left( \begin{array}{c | c}{\begin{array}{c c}{0}&{B}\\{- B}&{0}\\ \end{array}}&{M}\\ \hline{- M^{T}}&{\begin{array}{c c}{0}&{\tilde{B}}\\{- \tilde{B}}&{0}\\ \end{array}}\\ \end{array}\right) .
\mathcal{F}^{\Psi}_{\left[2\ell_{[L,M,T]},M,\gamma_{[L,M,T]}\right]}~=~\Big\{g\in BV^{\Psi}\Big(\left[0,2\ell_{[L,M,T]}\right],[-M,M]\Big)~\big|~ TV^{\Psi}(g,[0, 2\ell_{[L,M,T]}])\leq \gamma_{[L,M,T]}\Big\}
V = ( \nabla \phi_{3})^{2}+ \phi_{3}^{2}\left[ ( \partial_{i}{\mathbf{n}})^{2}+ ( \mathbf{n}\cdot \mathrm{r o t}{\ \mathbf{n}})^{2}\right] - ( \mathbf{n}\cdot \nabla \phi_{3})^{2}+ ( [ \mathbf{n}\times \mathrm{r o t \}\mathbf{n}] \cdot \nabla \phi_{3}^{2}) ~ ,
- M^{3}\int d^{5}x \sqrt{g}{\cal R}+ \int d^{5}x \sqrt{g}\left( \frac{1}{2}\nabla \phi \nabla \phi - V ( \phi ) \right) - \int d^{4}x \sqrt{g_{4}}\lambda_{P}( \phi ) - \int d^{4}x \sqrt{g_{4}}\lambda_{T}( \phi ) ,
J'_G(t,r,\xi) :=\gamma_0 (\gamma_0 - 1) t \int_0^1 \big( \mathbf 1_{\gamma(\cdot) = \gamma_0}\kappa \cdot (\phi \eta_{ut})^{\gamma_0 - 1}G\big( ru^{\frac{1}{\gamma_0 - 1}}\big)^{\gamma_0 - 1}\big) (\xi_{(1-u)t}) du
\ \ \ \det S(z) &= \prod_{i=(k,n)\in B_*^+}(z+k\cdot\omega+\mu_n)\prod_{i=(k,n)\in B_*^-}(-z-k\cdot\omega+\mu_n) &\ \ \ +O((\varepsilon+\delta)^{3/4}) &\sim\prod_{l=1}^{\# B_*}(z-\sigma_l)+O((\varepsilon+\delta)^{3/4}).
\rho_{A}(\delta_{1/r}^{A}\delta_{r}^{B}x) = r^{-1}\rho_{A}(\delta_{r}^{B}x) &\leq C r^{-1}\rho_{B}(\delta_{r}^{B}x) = C \rho_{B}(x) \leq C \sup\{\rho_{B}(z) \colon \|z\|=1\}\leq D,
y_{_k}=\sqrt{\eta_k}\bigg(h_{kk}x_k+\sum_{\substack{j=1 j\neq k}}^{K}h_{kj}x_j+w_k\bigg)+n_k,\forall k\in\mathcal{K}, y_{_k}=\sqrt{1-\eta_k}\bigg(\sum_{j=1}^{K}h_{kj}x_j+w_k\bigg),\forall k\in\mathcal{K},
- \frac{d}{d t}\delta ( t - t^{\prime}) g_{\sigma 0}+ \varepsilon \int d^{3}\mathbf{x}a_{\nu 0}\frac{\partial}{\partial t}\frac{\delta^{2}W}{\delta J_{\nu}^{\alpha}( \mathbf{x}, t ) \delta J_{\sigma}^{\alpha}( \mathbf{y}, t^{\prime})}\mid_{_{_{0 = J = . . .}}}= 0
{\cal Z}^{(N_f)}_\nu(\mu)=\int\limits_0^{2\pi}\prod_{j=1}^{N_f}\left(\frac{d\phi_j}{2\pi}\exp\left[\mu\cos(\phi_j)-i\nu\phi_j-ij\phi_j \right]\right)\varepsilon_{i_1\ldots i_{N_f}}e^{i\phi_{i_1}}\ldots e^{iN_f\phi_{i_{N_f}}}\;,
(e_1,f_1)=-1+\frac{1}{\alpha},(e_2,f_2)=-1+\alpha,(e_3,f_3)=1, &(\psi_{\alpha_1,\beta_1,\gamma_1},\psi_{\alpha_2,\beta_2,\gamma_2})=-2\epsilon_{\alpha_1,\alpha_2}\epsilon_{\beta_1,\beta_2}\epsilon_{\gamma_1,\gamma_2}.
\sum_{k=0}^\infty(-1)^k\frac{(q;q^2)_k(-q;q^2)_k^2}{(q^4;q^4)_k(-q^4;q^4)_k^2}[6k+1]q^{3k^2}=\frac{(q^3;q^4)_\infty (q^5;q^4)_\infty}{(-q^4;q^4)_\infty^2}, \sum_{k=0}^\infty\frac{(q^2;q^4)_k(-q;q^2)_k^2}{(q^4;q^4)_k(-q^4;q^4)_k^2}[6k+1]q^{k^2}=\frac{(-q^2;q^4)_\infty^2}{(1-q)(-q^4;q^4)_\infty^2},
G(a,x) = \lim_{N \to \infty}\sum_{0 \ne n =N}^{-N}\biggl( \frac{e^{-2\pi i na}}{x-2\pi in}- \frac{e^{2\pi i na}}{2\pi in}\biggr) =\sum_{n=1}^\infty \frac{x e^{-2\pi i na}}{2\pi i n(x-2\pi i n)}- \sum_{n=1}^\infty \frac{x e^{2\pi i na}}{2\pi i n(x+2\pi i n)}.
| e^{i \delta_{n}}|^{2}= \frac{( \operatorname{c o s h}\frac{\pi r_{n}^{2}}{v}- \operatorname{c o s h}\frac{\pi c}{v})^{4}}{\operatorname{c o s h}^{6}\frac{\pi r_{n}^{2}}{2 v}( \operatorname{c o s h}\frac{\pi r_{n}^{2}}{v}+ \operatorname{c o s h}\frac{\pi 2 c}{v})}
(e^{\tau x_1}(-\Delta_g)e^{-\tau x_1}w| T^*_\tau e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}v)_{L^2(M)}&=(e^{\tau x_1}(-\Delta_g)e^{-\tau x_1}w|v)_{L^2(M)}&=(w|e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}v)_{L^2(M)}.
S_R =(\{-R_1,R_1\}\times[-R_2,R_2]\times\cdots\times[-R_n,R_n]) &\cup([-R_1,R_1]\times\{-R_2,R_2\}\times\cdots\times[-R_n,R_n]) &\cup\cdots\cup([-R_1,R_1]\times[-R_2,R_2]\times\cdots\times\{-R_n,R_n\}).
\displaystyle\min_{j=0,1,\ldots,N-1}\|\mathcal{G}_t(x_j)\|^2\leq \frac{1}{N}\sum^{N-1}_{j=0}\|\mathcal{G}_t(x_j)\|^2\leq\frac{4t^{-1}(F(x_0)-F^*+4L\sum^{N}_{j=1}\varepsilon_{j})}{N},
H(Y_3|Y_2)=H(Y_3|Y_2=y_2)= H(Y_4|Y_1)= H(Y_4|Y_1=y_1) = \frac{d+2}{2}\cdot\frac{1}{d}\log \frac{2d}{d+2}+\frac{d-2}{2}\cdot\frac{1}{d}\log d &H(Y_3|Y_1)=H(Y_3|Y_1=y_1)= H(Y_4|Y_2)= H(Y_4|Y_2=y_2) = \frac{1}{2}\log 2+\frac{1}{2}\log d
C^{( d )}\; = \; d \, - \, \frac 3 2 \ \alpha^{\prime}\biggl [ r \, - \, \frac{1}{1 2}\H_{i j k}H^{i j k}\, - \, 4 \, \nabla^{j}\Phi \, \nabla_{j}\Phi \, + \, 4 \, \triangle_{g}\Phi \biggr ] \ ,
\sum_{j = 0}^{n_y-l_y}{\vphantom{\sum}}'\sum_{i = n_x-l_x+1}^{\infty}\left|c_{i,j}\right|\le (n_y-l_y+1)\dfrac{4V_k}{2k\pi^2}\begin{cases}\Pi_{1}[s](n_x-l_x),~~~~~~~~~\mbox{if}k=2s, \Pi_{0}[s](n_x-l_x),~~~~~~~~~\mbox{if}k=2s+1,\end{cases}
\left(\frac{A_n^{(2)}(0)}{n^6 z}\right)^2 = \frac{A_n^{(1)}(0)(A_n^{(1)}(z) - A_n^{(1)}(0))A_n^{(1)}(0)(A_n^{(1)}(z) - A_n^{(1)}(0))}{n^{12}z^4}= \mathcal{O}\left(\frac{1}{n}\right)
N^{{\cal P}_{0}}~ = ~ \sum_{m_{1}= - 1 / 2}^{1 / 2}\cdots \sum_{m_{p_{0}}= - 1 / 2}^{1 / 2}\left[ ~ \delta_{( \sum_{n = 1}^{p_{0}}m_{n}) , 0}~ - ~ \frac 1 2 ~ \delta_{( \sum_{n = 1}^{p_{0}}m_{n}) , 1}~ - ~ \frac 1 2 ~ \delta_{( \sum_{n = 1}^{p_{0}}m_{n}) , - 1}~ \right]
\sum_{w=1}^{cm(k)}\binom{\frac{k(k-1)}{2}}{w}\frac{{(q-1)}^w}{q^{c'kw}}\leq \sum_{w=1}^{cm(k)}\left(\frac{k(k-1)}{2}\right)^w\frac{{(q-1)}^w}{q^{c'kw}}\leq &\leq\sum_{w=1}^{cm(k)}q^{w(-c'k+o(k))}\leq q^{-c'k+o(k)}
P_{\nu_{e}\rightarrow \nu_{e}}= \vert < \nu_{e}( 0 ) \vert \nu_{e}( t ) > \vert^{2}= 1 - s i n^{2}2 \theta_{\nu}s i n^{2}\left[{\frac{1}{2}}( E_{2}- E_{1}) t \right] \quad .
\nu_{\zeta}(r,s) = c_{\zeta,\alpha}\, (rs)^{-\zeta}\, \left(\frac{r\wedge s}{r\vee s}\right)^{\zeta}\, \frac{1}{(r\vee s)^{\alpha+1}}\,_2\tilde F_1\left(\zeta+\frac{\alpha+1}{2},\frac{\alpha+2}{2};\zeta+\frac12;\frac{s^2}{r^2}\right).
-\Delta v_\delta = f_0(w_\delta)=f_0(P(U_{\delta,e_n}-U_{\delta,-e_n})){B}, \partial_\nu v_\delta=0\partial{B}, \int_{B}v_\delta = 0, -\Delta w_\delta = f_0(U_{\delta,e_n})-f_0(U_{\delta,-e_n}){B}, \partial_\nu w_\delta=0\partial{B}, \int_{B}w_\delta = 0.
\begin{array}{r l}{U^{\prime}V^{\dag}}&{= - \Delta^{\prime}f K^{\dag}( 1 - K f K^{\dag})_{\Pi}^{- 1 / 2}( 1 - K f K^{\dag})^{1 / 2}}\\{}&{= - \Delta^{\prime}f K^{\dag}( 1 - \Pi_{0}) = - \Delta^{\prime}f K^{\dag}.}\\ \end{array}
m\geq \frac{1}{2\log\Big(\frac{\sum_{i=1}^g \sqrt{2\bar{k}_i(1+\delta_{\bar{k}_i})}+\max_i(\sqrt{\bar{k}_i(1-\delta_{\bar{k}_i})/8})}{\min_i(\sqrt{\bar{k}_i(1-\delta_{\bar{k}_i})/8})}\Big)}\bar{k}\log\Big(\frac{n}{\bar{k}}\Big)
\begin{aligned}A_{1}X=C_{1},\XB_{1}=D_{1}, A_{2}Y=C_{2},\YB_{2}=D_{2}, A_{3}Z=C_{3},\ZB_{3}=D_{3}, E_{1}XF_{1}+E_{2}YF_{2}+E_{3}ZF_{3}=C.\end{aligned}
\begin{array}{r c l}{{E_{n}^{( 1 )}= \hbar g \omega \left( 2 n + \frac{9}{2}\right) \ \ \}}&{{,}}&{{\ \ \d_{n}^{( 1 )}= \frac{1}{1 2 0}( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) ( n + 5 ) \ \ \ ,}}\\{{E_{n}^{( 2 )}= \hbar g \omega \left( 2 n + \frac{1 5}{2}\right) \ \ \}}&{{,}}&{{\ \ \d_{n}^{( 2 )}= \frac{1}{1 2 0}( n + 1 ) ( n + 2 ) ( n + 3 ) ( n + 4 ) ( n + 5 ) \ \ \ .}}\\ \end{array}
d^2(X_t,X_{\sigma_k})=2\int_{\sigma_k}^td(X_s,X_{\sigma_k})\d (d(X_s,X_{\sigma_k}))+\langle d(X_{\cdot},X_{\sigma_k})\rangle_t-\langle d(X_{\cdot},X_{\sigma_k})\rangle_{\sigma_k}.
0<p<1-e^{-2s}\;\;\;\Rightarrow\;\;\;\exists \beta_1>1:\; \begin{cases}\phi_1(\beta)>0,\;\;\;{\rm if}\;\;\; 0<\beta<\beta_1, \phi_1(\beta)=0,\;\;\;{\rm if}\;\;\; \beta=\beta_1, \phi_1(\beta)<0,\;\;\;{\rm if}\;\;\;\beta>\beta_1,\end{cases}
x_i = \frac{\overrightarrow{S_2 S_i}(1)}{U_{1 1}}\Big[1+ \frac{U_{1 2}L_{2 1}}{U_{2 2}}\Big]-\frac{U_{1 2}}{U_{1 1}}\overrightarrow{S_2 S_i}(2),\;y_i = \frac{\overrightarrow{S_2 S_i}(2)- L_{2 1}\overrightarrow{S_2 S_i}(1)}{U_{22}}.
L_n (\theta) = \sum_{i=1}^n \log \left(\frac{n^{1/ \alpha}}{\delta X_{\frac{i-1}{n}}^{1/\alpha}}\varphi_\alpha \left( n^{1/\alpha}\frac{X_{\frac{i}{n}}- X_{\frac{i-1}{n}}- \frac{a}{n}+ \frac{b}{n}X_{\frac{i-1}{n}}}{\delta X_{\frac{i-1}{n}}^{1/ \alpha}}\right) \right),
\begin{aligned}0\longrightarrow P(n-N)^{\oplus\alpha_{N-n}}\longrightarrow P(n-N+1)^{\oplus\alpha_{N-n-1}}\longrightarrow\dots &\longrightarrow P(-1)^{\oplus\alpha_1}\longrightarrow P^{\oplus\alpha_0}\longrightarrow E\longrightarrow0.\end{aligned}
\int\psi^2\left(y(\partial_y\bar{w}_k)^2+(\partial_x\bar{w}_k)^2\right) &\leq C\int(\psi^2+y\psi_y^2+\psi_x^2+|\psi_y|)\bar{w}_k^2+C\theta_k^{-2}\int\psi^2|f_k|^2\leq C_\psi\frac 1{\theta^2_k}.
b_{n-1}= q_{n-1}a_{n-1,n-1}+ a_{n,n-1}b_{n-2}- q_{n-1}a_{n-1,n-2}= q_{n-2}a_{n-2,n-2}+ a_{n,n-2}&\vdots b_1 - q_{n-1}a_{n-1,1}- \cdots - q_2a_{2,1}= q_1a_{1,1}+ a_{n,1}
\varepsilon^{ad}(P_\pm)^{B b}_{A c}+ \varepsilon^{bd}(P_\pm)^{B a}_{A c}- (\sigma^\alpha)^{ab}(\sigma_\alpha)^e_{~c}(P_\mp)^{B d}_{A e}=-(\sigma^\alpha)^{ab}\bigr( (\sigma_\alpha)^d_{~c}\delta^B_A + \delta^d_c (\sigma_\alpha)^B_{~A}\bigr)
\iint_{Q_t}\mathbf{b}_t\cdot \mathbf{b}_{xx}\omega^2 dxdt \leq C-\frac{1}{2}\int_\Omega|\mathbf{b}_x|^2\omega^2dx+\frac{\nu}{4}\iint_{Q_t}|\mathbf{b}_{xx}|^2\omega^2dxds & +C\int_{0}^t\|u_x\|_{L^\infty(\Omega)}\int_\Omega|\mathbf{b}_{x}|^2\omega^2dxds +C\iint_{Q_t}|\mathbf{w}_{x}|^2\omega^2 dxds.
\Omega^{C_2}_{*-n\sigma}= \dfrac{\Omega^{C_2}_*\{1,\dots,u^n\}}{\begin{matrix}u^{k-1}(d_{i,j}-c_{i,j}) = u^k d_{i,j+1}u^{k-1}(q_j-p_j) = u^kq_{j+1}\end{matrix}}
\displaystyle{\left( \frac{d^{2}}{d r^{2}}+ \frac{2}{r}\frac{d}{d r}- \frac{l ( l + 1 )}{r^{2}}+ \frac{\omega^{2}}{c^{2}}\right)}f_{l}( r ) = 0 , \ \mathrm{f o r}\r < R ,
\mathbb{P}(\mathbb{M}_n(\theta_0) < \mathbb{M}_n(\theta_0 + \varepsilon)) \ge \mathbb{P}\left(\sum_{i=1}^n \left\{\log\frac{p_{\theta_0 + \varepsilon}(X_i)}{p_{\theta_0}(X_i)}- \mathbb{E}\left[\log\frac{p_{\theta_0 + \varepsilon}(X_i)}{p_{\theta_0}(X_i)}\right]\right\}\le 0\right).
A_{Ti}^{a}({\bf{r}}) = \sum_{{\bf{k}}; s=1,2}\frac{\epsilon_{i}^{s}({\bf{k}})}{\sqrt{2k}}\,[a_{s}^{a}({\bf{k}})e^{+i{\bf{k}}\cdot{\bf{r}}}+a_{s}^{a\dagger}({\bf{k}})e^{-i{\bf{k}}\cdot{\bf{r}}}]\;,
\frac{e^{- 4 \pi \tau}}{1 - e^{- 4 \pi \tau}}\sim - \frac 1 4 \frac{q^{2}}{p^{( 2 ) 2}}\; , \; \; \frac{e^{- 4 \pi l^{\prime}}}{1 - e^{- 4 \pi l^{\prime}}}\sim - \frac 1 4 \frac{k^{2}}{p^{( 1 ) 2}}\; .
= 4 \pi \frac{\partial \beta_{n}}{\partial M}- P_{n z}\sum_{B}\frac{\partial \beta_{B}}{\partial M}\frac{{\cal P}( z_{B})}{( z_{n}- z_{B})^{2}\prod_{C \neq B}( z_{B}- z_{C})}+ P_{n z}~ p_{1}^{\prime}( z_{n}) .
\frac{\binom{m-cm}{t}}{\binom{m-k}{t}}=\frac{(m-cm)\cdots(m-cm-t+1)}{(m-k)\cdots(m-k-t+1)}\leq &\leq{\left(\frac{m-cm}{m-k}\right)}^t={(1-c)}^t{\left(\frac{k+1}{k-1}\right)}^t
|v(t,z)|_{L^{2}(\mathcal Z,\tilde{\mu}_{t})}= \sqrt{\int_{\mathcal Z}|v(t,z)|^{2}d\tilde{\mu}_{t}(z)}&\leq M r\sqrt{\sum_{j=2}^{r}\int_{\mathcal Z}|z|^{2(2j-1)}d\tilde{\mu}_{t}}(z) \in L^{1}([-T,T]).
I ={\frac{i g f_{a c b}}{( 2 \pi )^{6}}}\int d u_{4}d^{4}x_{4_{A}}d^{2}\theta_{4}^{+}d^{2}\bar{\theta}_{4}^{+}\{\frac{( 1 4 )}{\hat{x}_{1 4}^{2}}}\;{\frac{( 4 2 )}{\hat{x}_{4 2}^{2}}}\;{\frac{( \theta_{3 4})^{2}}{\tilde{x}_{3 4}^{2}}}\; .
\begin{array}{rcl}f-D_{4}= 0 5\,D_{4}+g+D_{3}= 0 -10\,D_{4}-4\,D_{3}-D_{2}= 0 10\,D_{4}+6\,D_{3}+3\,D_{2}+D_{1}= 0 -5\,D_{4}-4\,D_{3}-3\,D_{2}-2\,D_{1}+h = 0. \end{array}
\mathcal{N}(n, 2c+1, \{1,2c\}) =\mathcal{N}(2cq+r+2c, 2c+1, \{1,2c\}) = \mathcal{N}(2cq+ r+6,7,\{1,2c\}) \ref{two}&=\mathcal{N}(6q+r'+6,7,\{1,6\}) \ref{three}&=\mathcal{N}(\phi_1(n),7,\{1,6\})