harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\widehat{\eta}(d\gamma)= \int_{\overline \Omega}\Big(\lambda_1\eta_{1,x}+\lambda_2 \eta_{2,x}\Big)(d\gamma) dm_0(x), supp(\lambda_1\eta_{1,x}+\lambda_2 \eta_{2,x})\subset \Gamma^\eta[x] \ \ \x\in \overline{\Omega}\setminus(A_1\cup A_2),\end{cases}
{\rm Re}\frac{\zeta''}{\zeta'}(s) =\frac{\zeta''}{\zeta'}(0)-2+\sum_{\rho_1}\frac{1}{\rho_1}+\sum_{\rho_1}\frac{\sigma-\beta_1}{(\sigma-\beta_1)^2+(t-\gamma_1)^2}-\sum_{n\geq1}\frac{a_n\sigma+|s|^2}{a_n|s+a_n|^2}&\leq\frac{\zeta''}{\zeta'}(0)-2+\sum_{\rho_1}\frac{1}{\rho_1}-\sum_{n\geq1}\frac{\sigma}{|s+a_n|^2}-\sum_{n\leq |s|}\frac{|s|^2}{a_n|s+a_n|^2}.
D_\lambda(p_{W_{T}, \hat W_{T\times \{d\}}}\|p_{W_{T}}s_{\hat W_{T\times \{d\}}}) \le \sum_{k=1}^n D_\lambda( q_{Y_{T^c}|X_{\mathcal{I}}}\|s_{Y_{T^c,k}|X_{T^c,k}}|p_{X_{\mathcal{I},k}}^{(\lambda, T)}).
F(f) = \mathbb E \sum_{n\geq 0}\langle f_n, g_n\rangle,\;\;\; f=(f_n)_{n\geq 0}\in Q_q^p, \|(g_n)_{n\geq 0}\|_{Q_{p'}^{q'}}\lesssim_{p, q}\|F\|_{(Q_q^p)^*}.
\small{n_{+}\left( r,\frac{\textup{\textbf{w}}(p \otimes \tau) \textup{\textbf{w}}^{\ast}}{2}\right) = n_{+}\left( r,\frac{(p \otimes c) e_{+}\textup{\textbf{w}}^{\ast}\textup{\textbf{w}}e_{+}(p \otimes c^{\ast})}{2}\right), r > 0,}
\mu'(h) = 2h ( \Delta u, u )_X - h(u, \partial_\nu u)_{\partial X}. \intertext{Applying Gauss-Green again, we get}\mu'(h) = h ( \Delta u, u )_X + h \| \nabla u\|_X^2= h^{-1}\Bigl( \| u \|^2 + \| h\nabla u\|^2 \Bigr) > 0.
g(X,Z) = -\Phi({f}X, Z) -g(X,\widetilde{Q}Z) +\sum\nolimits_{i}\big(\eta^i(X)\,\eta^i(Z) +\eta^i(X)\,\eta^i(\widetilde{Q}Z)\big) = -\Phi({f}X, Z) + \sum\nolimits_{i}\eta^i(X)\,\eta^i(Z) - g(X, \widetilde{Q}Z).
\textrm{GMT}(\mathbf{k}) := \bigsqcup_{\mu^{(n)}\in \textrm{AR}_n}\bigsqcup_{\mathbf{k}^{(n-1)}\in \mu^{(n)}(\mathbf{k})}\bigsqcup_{\mu^{(n-1)}\in \textrm{AR}_{n-1}}\bigsqcup_{\mathbf{k}^{(n-2)}\in \mu^{(n-1)}(\mathbf{k}^{(n-1)})}\cdots \bigsqcup_{\mu^{(2)}\in \textrm{AR}_{2}}\bigsqcup_{\mathbf{k}^{(1)}\in \mu^{(2)}(\mathbf{k}^{(2)})}\textrm{AR}_1
S_A=-\frac{2}{\kappa_{11}^2}\int d^{11}x \left\{\frac{\sqrt{|g|}}{2\cdot 2\cdot 4!}F^{\mu\nu\lambda\xi}F_{\mu\nu\lambda\xi}+\frac{1}{12}\frac{1}{3!(4!)^2}\epsilon^{\mu_1\ldots \mu_{11}}A_{\mu_1\mu_2\mu_3}F_{\mu_4\ldots \mu_7}F_{\mu_8\ldots \mu_{11}}\right\}
\begin{bmatrix}1 - x . . . . . 1 -y -1 . . . . 1 x . . . . . 1 -y -3 . . . . 1 x . . . . . 1\end{bmatrix}s =\begin{bmatrix}. . . . . 1\end{bmatrix},s =\begin{bmatrix}x(1-yx)(3-yx) (1-yx)(3-yx) -x(3-yx) 3-yx -x 1\end{bmatrix}
\mathbb{W}_{s,t}(\omega)=\begin{pmatrix}-w^{3,3}_{s,t}(\omega) -w^{2,2}_{s,t}(\omega) w^{1,2}_{s,t}(\omega) w^{1,3}_{s,t}(\omega) w^{2,1}_{s,t}(\omega) -w^{3,3}_{s,t}(\omega) -w^{1,1}_{s,t}(\omega) w^{2,3}_{s,t}(\omega) w^{3,1}_{s,t}(\omega) w^{3,2}_{s,t}(\omega) -w^{2,2}_{s,t}(\omega)-w^{1,1}_{s,t}(\omega) \end{pmatrix}\, .
\max\bigg\{\max_{1\leq j \leq n}\bigg(\bigg( KL_{\nu_{j}}+KL_{q}\kappa_{b}\gamma b\bigg) \bigg(1+\frac{M^{2}K^{2}b}{\delta^{j}}\bigg)\bigg), \qquad\qquad\bigg(1+\frac{M^{2}K^{2}b}{\delta^{0}}\bigg)KL_{q}\kappa_{b}\gamma b,\ \max_{1\leq j \leq n}L_{\nu_{j}}\bigg\}<1, \mbox{where}\ \gamma=\frac{b}{\beta}.
N_\varphi(w)=\sum_{z\in \varphi^{-1}\{w\}}\log\frac{1}{|z|}\,\,\,and\,\,\,N_{\varphi,\gamma}(w)=\sum_{z\in \varphi^{-1}\{w\}}\big(\log\frac{1}{|z|}\big)^\gamma,
e^{- S^{free}_{b}(m;\Lambda,\lambda)}=\int{\cal D}A{\cal D}\psi{\cal D}\overline\psi\quad J(A)\quad e^{\overline\psi\gamma^\mu\partial_\mu\psi +m\overline\psi e^{-2i\varphi\gamma_5}\psi +(A,d\lambda+i\ast d\Lambda)}.
{E}^{0,i}\left[e^{-\int_0^u \Phi^+_{J^+_s}(q,0)ds}e^{-\alpha \mathcal{L}^{-1}_u-\beta\mathcal{H}_u}\mathbf{1}_{(J^+_u= j)}\right] = [e^{-\kappa^+(q+\alpha, \beta)}]_{i,j}.
\Phi^{<}( \eta ) = \Phi_{\mathrm{i}}+ \frac{1}{3}\eta_{\mathrm{i}}\Phi_{\mathrm{i}}^{\prime}- \frac{1}{3}\eta_{\mathrm{i}}\Phi_{\mathrm{i}}^{\prime}\frac{\eta_{\mathrm{i}}^{3}}{\eta^{3}}.
\Gamma_{\pm}^{( p )}= \frac{- i}{2 \omega_{\pm}}[ ( 1 + N_{\pm}) \Sigma_{1 2 ( 2 1 )}- N_{\pm}\Sigma_{2 1 ( 1 2 )}] \raisebox{- 2 . 8 5 m m}{\scriptsize{p_{0}= \pm \omega_{\pm}}},
\mathcal{A}^h({\rm e}^{{\rm i}\theta}) = a_0+2\sum_{j=1}^\infty a_j \cos j \omega, \mathcal{B}^h({\rm e}^{{\rm i}\theta}) =2 (\sin\tfrac\theta2 + |\alpha| \cos\tfrac\theta2) \sum_{j=1}^\infty b_j \frac{\sin j\omega}{\sin \frac{\omega}{2}},
\mu X_{j-1}(u) e^{-\int_{\tau_{j+1}}^u G_j(v) \: dv}\leq (1 + 2 \delta) k_N^2 s e^{\int_{\tau_j}^u G_{j-1}(v) \: dv}e^{-\int_{\tau_{j+1}}^u G_j(v) \: dv}&= (1 + 2 \delta) k_N^2 s e^{\int_{\tau_j}^{\tau_{j+1}}G_j(v) \: dv}e^{-s(u - \tau_j)}&\leq \frac{2 k_N^2 s^2}{\mu}e^{-s(u - \tau_j)}.
e_k = |\upsilon_k|^2 \big( \sum_{\ell \in \mathcal{K}}|\boldsymbol{\theta}_i^T \mathrm{diag}(\mathbf{h}_k^H) \mathbf{G}\mathbf{w}_\ell|^2 + \sigma_k^2\big) begin{align*}-0.5em] - 2 \mathrm{Re}\{\upsilon_k^* \boldsymbol{\theta}_i^T \mathrm{diag}(\mathbf{h}_k^H) \mathbf{G}\mathbf{w}_k \}+ 1,
\left|\langle f(t)-\sum\limits_{n=1}^{k}h_{n}(t)x_{n}^{*},x\rangle\right| =\left|\langle\sum\limits_{n=k+1}^{\infty}h_{n}(t)x_{n}^{*},x\rangle\right| &\leq \left|\langle h_{k_{0}}(t)x_{k_{0}}^{*},x\rangle\right|\leq \left|\langle x_{k}^{*},x\rangle\right|<\epsilon
\sum_{\beta\in I_1|\alpha|=k}|\partial^{\alpha}f_1^*(z_\beta)|^2\le C_2\|f_1^*\|_{\partial M,\rho}^2\le C_1C_2\sum_{\beta\in I_1,\alpha\in L_{k-1}}|a_{\beta,\alpha}|^2,
\begin{aligned}\omega =-\frac{1}{\kappa}\sum_{i=1}^n \frac{da_i \wedge db_i}{G_i}+\frac{1}{2\kappa}\sum_{i=1}^n \sum_{s>i}\frac{1}{G_i G_{i+1}}\left(b_i da_i- a_i db_i \right) \wedge \left(b_s da_s+a_s db_s \right)\,, \end{aligned}
{\rm Ind}^{(2n)}(D_a,P_{\rm F})=\frac{1}{2}\;{\rm Tr}\left(\begin{matrix}(P_{\rm F}-\mathfrak{D}_a^\ast P_{\rm F}\mathfrak{D}_a)^{2n+1}0\cr 0 -(P_{\rm F}-\mathfrak{D}_a P_{\rm F}\mathfrak{D}_a^\ast)^{2n+1}\cr\end{matrix}\right).
wt(\pi^{\ast m}\ast f_{\alpha}(\pi))=m\lambda^{\prime}-\alpha=lm\delta-\alpha-m\alpha_{2}-m\sum_{d(\alpha^{\prime})<d(\alpha_{2})}k_{\alpha^{\prime}}\alpha^{\prime}=lm\delta-\alpha-\sum_{d(\alpha^{\prime})<d(\alpha)}k_{\alpha^{\prime}}^{\prime}\alpha^{\prime},
Z_{gh}=\int D[C,\tilde{\cal P};{\cal P},\tilde C]\cdot\exp\{i\int\limits_{\tau_i}^{\tau_f}(\tilde{\cal P}_a\dot C^a -\dot{\tilde C}_a{\cal P}^a-\tilde{\cal P}_a{\cal P}^a)d\tau\}\, .
\bigl\| f_0^{-1}-g_{n,0}^{-1}\bigr\|_{\mathbb{W}^{1,p^*}(\Omega_1,\mathbb{R}^d)}=\bigl\|f_0^{-1}-g_{n,0}^{-1}\bigr\|_{C^0(\Omega_1)}+\left(\int_{\Omega_1}\left|Df_0^{-1}-Dg_{n,0}^{-1}\right|^{p^*}\,d\mu\right)^{\frac{1}{p^*}}\ .
- \frac{1}{r^{2}\cosh \tau}\partial_{\tau}( \cosh \tau \partial_{\tau}\phi ) + \frac{1}{r^{2}}\partial_{r}( r^{2}f ( r ) \partial_{r}\phi ) + f ( r )^{- 1}\partial_{\chi}^{2}\phi + \frac{1}{r^{2}\cosh^{2}\tau}\partial_{\psi}^{2}\phi - m^{2}\phi = 0 .
\|(u_\beta(s_0), \partial_t u_\beta(s_0))\|_{(\dot H^1\times L^2)(\Omega_{A_\delta}(s_0))}= \|(u^{\rm ext}_\beta(s_0), \partial_t u^{\rm ext}_\beta(s_0))\|_{(\dot H^1\times L^2)(\Omega_{A_\delta}(s_0))}& \leq \|(u_\beta^{\rm ext}(s_0), \partial_t u_\beta^{\rm ext}(s_0))\|_{\dot H^1\times L^2}\lesssim \delta,
H^{-1}B H = a A - a^{2}B^{-1}, \;\;H^{-1}K H = a^{-1}A - a^{-2}K^{-1}, &H^{-1}B^\downarrow H = a A - a^{2}(B^\downarrow)^{-1}, H^{-1}K^\downarrow H = a^{-1}A - a^{-2}(K^\downarrow)^{-1},
\sigma_{\mathrm{ess}}(\hat{H}^\sharp_{\mathrm{loc}}(a, b)) = [- \sqrt{b^2 + (\lvert a \rvert + 1)^2}, - \sqrt{b^2 + (\lvert a \rvert - 1)^2}] & \cup[\sqrt{b^2 + (\lvert a \rvert - 1)^2}], \sqrt{b^2 + (\lvert a \rvert + 1)^2}].
\langle\langle\Delta_h u,\,u\rangle\rangle_\omega = h^{2(n-p)}\,\langle\langle\Delta_{\omega_h}u,\,u\rangle\rangle_{\omega_h}= h^{2(n-p)}\,(||du||^2_{\omega_h}+ ||d^\star_{\omega_h}u||^2_{\omega_h}).
F_{Q}=\frac{\hbar^{2}}{4m}[\frac{\nabla(\nabla^{2}\rho)}{\rho}-\frac{\nabla(\nabla\rho\cdot\nabla\rho)}{2\rho^{2}}-(\frac{\nabla^{2}\rho}{\rho}-\frac{\nabla\rho\cdot\nabla\rho}{\rho^{2}})\frac{\nabla\rho}{\rho}]
(2a)^{-s}s\sum_{k=1}^{\infty}\frac{(-1)^{k}}{(k\pi)^{2}}\int_{0}^{\infty}du\,e^{-u}\left[\zeta\left(s+1, \frac32 +\frac{c}{2a}-\frac{iu}{2k\pi}\right) + \zeta\left(s+1, \frac32 +\frac{c}{2a}+\frac{iu}{2k\pi}\right)\right]=
\mathcal{I}(\delta) = \inf_{\lambda \in \mathbb{R}^2_+}\left\{\langle \lambda, \delta \rangle + \sup_{\pi \in \widetilde{\mathit{\Pi}}}\int_{\mathbb{R}^{2d+2}}\left(f_{\mathcal{S}}\right)_\lambda \, d \pi \right\}
\pi^j\gamma(\widetilde{0})= \pi^{j'}\gamma'(\widetilde{0})\;\Leftrightarrow\;(\gamma')^{-1}\pi^{j-j'}\gamma(\widetilde{0})= \widetilde{0}\;\Leftrightarrow\;(\gamma')^{-1}\pi^{j-j'}\gamma\in \mathcal{G}(X)_{\widetilde{0}}.
\begin{cases}\partial_{tt}u - \partial_{x}(a\partial_{x}u) + b_{1}\partial_{t}u - \partial_{x}(b_{2}\partial_{x}\partial_{t}u)= 0 \mbox{in}(0, \infty) \times (0,\pi), u(t,0) = u(t,\pi) = 0 \mbox{on}(0, \infty), u(0,x) = u_{0}(x) \mbox{in}(0,\pi) u_{t}(0,x) = u_{1}(x) \mbox{in}(0,\pi),\end{cases}
\sup_{m \ge 1}\sum^d_{i=1}\left\|\frac{\partial}{\partial y_i}\phi_{m,r}\right\|_\infty \,<\, c_1\, r^{-1}\sup_{m \ge 1}\sum^d_{i,j=1}\left\|\frac{\partial^2}{\partial y_i\partial y_j}\phi_{m,r}\right\|_\infty \,<\, c_1\, r^{-2}.
K_{w}( t_{1}, t_{2}, \cdots , t_{l}) = \prod_{j = 1}^{l}\Gamma ( - \beta_{j})^{- 1}\times \{\exp ( - t_{l}F_{m_{l}}) \cdots \exp ( - t_{1}F_{m_{1}}) \prod_{j = 1}^{l}t_{j}^{- \beta_{j}- 1}1 \},
\left[ A\omega_{1},\omega_{2}\right] =\left\langle LJL\omega_{1},\omega_{2}\right\rangle =\left\langle JL\omega_{1},L\omega_{2}\right\rangle=-\left\langle L\omega_{1},JL\omega_{2}\right\rangle & =-\left[ \omega_{1},A\omega_{2}\right] ,
{\mathcal D}_x \biggl[ \Pi(x;y)V(y) \widehat{f}(y;s)\biggr]_{1,y}=\sum_{i=1}^m \langle \alpha t^{n-1}q^{\lambda_i};\alpha t^{n-i}\rangle \biggl[ \Pi(x;y)V(y) \widehat{f}(y;s)\biggr]_{1,y},
f_{1}(x) = \begin{cases}x(a-x), x \in [0,a] (1-x)(a-x), x \in (a,1],\end{cases}g_{1}(x) = \begin{cases}x(b-x), x \in [0,b) (1-x)(b-x), x \in [b,1],\end{cases}f_{2}= -g_{1}, g_{2}= -f_{1}, h_{1}= -h_{2}.
0 = 3swx^3 + 3cwx^3 - 3wx^3 + 3x^3 + \frac{1}{2}x^3wx^3 + \frac{1}{2}uwx^3wx^3 - \frac{1}{2}wx^3wx^3 & = 3 (u-1)wx^3 + 3x^3 + \frac{1}{2}x^3wx^3 + \frac{1}{2}(u-1)wx^3wx^3
\begin{aligned}x^0(u,v) =\frac{1}{2}\int(1+q^2)fdu-(1+r^2)gdv, x^1(u,v) =-\frac{1}{2}\int(1-q^2)fdu+(1-r^2)gdv, x^2(u,v) =-\int qfdu+rgdv. \end{aligned}
\|\nabla u_R\|^p_{L^p(B_{1/2})}\le\frac{C}{(\gamma_{i_0}-1)^p}\,\left(C_5\,\frac{i_0^{4\,(N+p)}}{(1-\gamma_{i_0-1})^p}\right)^{i_0+1}\,(1-s)\,\mathcal{A}_1,
H_K(\beta)\begin{cases}\infty, \beta\in[0,\beta_P(d)], \in(0,\infty), \beta\in (\beta_P(d),\beta_P(d)+\epsilon_d], \in[0,\infty), \beta \in (\beta_P(d)+\epsilon_d, \beta_c(d)), 0, \beta\geq \beta_c(d). \end{cases}
C(FX)=\sum_{i=1}^n \sum_{j=1}^n e(C(v_j) ,F(v_i) ) X_{ij}&=\sum_{i=1}^n \sum_{j=i+1}^n (e(C(v_j) ,F(v_i) ) +e(C(v_i) ,F(v_j) ) )X_{ij}+\sum_{i=1}^{n}e(C(v_i) ,F(v_i) ) X_{ii}.
b\left(4\frac{C^1_{2,1}}{C^1_1}-3\frac{C^1_1}{\mu}\right)\omega_1\wedge\omega_2=-c\omega_2\wedge\theta_{56}, &c\left(4\frac{C^1_{2,1}}{C^1_1}-3\frac{C^1_1}{\mu}\right)\omega_1\wedge\omega_2=-(a\omega_1+b\omega_2)\wedge\theta_{56}, &-b\omega_1\wedge\theta_{56}+a\omega_2\wedge\theta_{56}=0.
\frac{1}{1-tE(yx)H(x)}=\sum_{n=0}^{\infty}t^{n}\exp\Big(n\sum_{k=1}^{\infty}\frac{p_{k}}{k}(-1)^{k-1}y^{k}x^{k}\Big)\exp\Big(n\sum_{k=1}^{\infty}\frac{p_{k}}{k}x^{k}\Big) =\sum_{n=0}^{\infty}t^{n}\exp\Big(n\sum_{k=1}^{\infty}\frac{p_{k}}{k}(1-(-y)^{k})x^{k}\Big).
\bar{H}(\omega,\phi_{\partial},\Sigma)=\frac{1}{2}\int_{\Omega}\beta\wedge\ast\omega+\int_{\Omega}\Phi v_{\Omega}+\frac{1}{2}\int_{\partial\Omega}\mathrm{tr}(\phi)\wedge\ast\boldsymbol{n}(d\phi)+\frac{\tau}{\rho}\int_{\Sigma}v_{\Sigma}.
\mathcal{A}\partial_1{\mathbf V}=\mathcal{F}-\mathcal{A}_0\partial_t{\mathbf V}-\mathcal H_{(0)}\sigma\partial_1{\mathbf V}-\mathcal{A}_2\partial_2{\mathbf V}-\mathcal{A}_3{\mathbf V}\quad\mbox{in}\,\,\,\Omega_T\,,
\begin{array}{c}\underset{s\rightarrow 0}{\lim}h^{-1}\left( s\right) =\underset{s\rightarrow 0}{\lim}\left( h^{-1}\right)^{\prime}\left( s\right) =\underset{s\rightarrow 0}{\lim}s\left( h^{-1}\right)^{\prime \prime}\left( s\right) =\underset{s\rightarrow 0}{\lim}s^{2}\left( h^{-1}\right)^{\prime \prime \prime}\left( s\right) =0,\end{array}
~E_x^{U, \omega}(U_k - U_{k-1}| U_{k-1}= m) = \delta_k ~,~\mbox{for each $k \geq 1$ and $m \geq M$, and}& \left|E_x^{U, \omega}(U_k - U_{k-1}| U_{k-1}= m)\right| \leq M ~,~ \mbox{for each $k \geq 1$ and $m \geq 0$}.
f ( \zeta ,{\bar{\zeta}}) = \langle \zeta | \hat{O}_{f}( b , b^{+}) | \zeta \rangle = \int{\frac{d^{2}p}{( 2 \pi )^{2}}}\tilde{f}( p ,{\bar{p}}) e^{- i ( \bar{p}\zeta + p \bar{\zeta})}.
4S= (2n-1) \sum_{k=1}^{n-1}\zeta(2k,2n-2k)+\frac{3-5n}4\zeta(2n)+(2n-3)\zeta(2)\zeta(2n-2) = \frac{n}4\zeta(2n)+(2n-3)\zeta(2)\zeta(2n-2)
\frac{1}{2}(1+\vec{a}\cdot\vec{\sigma})=(\begin{matrix}cos^{2}(\frac{\theta}{2})&e^{-i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})\\e^{+i\phi}sin(\frac{\theta}{2})cos(\frac{\theta}{2})&sin^{2}(\frac{\theta}{2})\end{matrix})
\varphi_x=-\frac{2iq}{|\Lambda|}\sum_{k\in\mathcal{M}_\epsilon^K}\frac{e^{ik\cdot x}e^{-ik\cdot(u+u')/2}}{\Bigl[\sum_{i=1}^D(1-\cos k^{(i)})\Bigr]^2}\sin[k\cdot(u-u')/2].
|H_kf(x)| =\bigg|\int_{\Bbb R^N}H_k(x,y)[f(y)-f(x)]d\omega(y)\bigg| \leqslant C\int_{|x-y|\leqslant r^{6-(k-M)}}r^{-M}(V_{k-M-1}(x))^{-1}\|x-y\|^s\|f\|_sd\omega(y) \leqslant Cr^{-M}r^{-(k-M)s}\|f\|_s.
n_{\rm eff}^2(n,\eta )\simeq n^2-n^2\biggl(\frac{n}{n_{\rm inf}}\biggr)^2(1-n_{\rm b}^2\eta^2), \quad \frac{a''}{a}\simeq n_{\rm b}^2\biggl[1-6\biggl(\frac{\eta}{\eta_0}\biggr)^2\biggr].
\begin{aligned}\mathcal{K}_{L}^{n}=\{\mathbf{v}\in{{\mathbf{R}}^{n}}:\mathbf{v}\ge 0\}, n\ge 1, &\mathcal{K}_{S}^{n}=\{\mathbf{v}\in{{\mathbf{R}}^{n}}:{{\mathbf{v}}_{1}}\ge \left\|{{\mathbf{v}}_{2:n}}\right\|\}, n\ge2,\end{aligned}
=C_{k}h^{(k-l)}y_{0}^{-(k-l)}\left\langle D_{y}^{m}u\right\rangle_{B}^{(\gamma)}\leq C_{k}\varepsilon^{(k-l)}\left\langle D_{y}^{m}u\right\rangle_{B}^{(\gamma)},i\leq m,
{\cal L}={\cal L}_0 (C_{\mu\nu\alpha}) +{\cal L}_0 (B_{\mu\nu}) - 2\sqrt{- 3 \Omega (d-3)}C_{\mu\nu\alpha}D^\mu B^{\nu\alpha}-\frac{\Omega (d-3)}{2}C^{\mu\nu\alpha}C_{\mu\nu\alpha}
\lambda_{m+1}x_{0}= \frac{3}{2}(2x_{0}-y_{0,1}(x_{0},x_{1},x_{2},\lambda_{m+1},r)-y_{0,2}(x_{0},x_{1},x_{2},\lambda_{m+1},r)) &+\frac{3}{2}(2x_{0}-y_{2,0}(x_{1}',x_{2}',x_{0},\lambda_{m+1},r)-y_{2,1}(x_{1}',x_{2}',x_{0},\lambda_{m+1},r))
O(z_1, z_2, \dots, z_g)= \P(G=0) \prod_{i=1}^g Y_i(z_i)+ \sum_{j=1}^{g-1}\P(G=j)\prod_{i=1}^{j}z_i\, \prod_{i=j+1}^g Y_i(z_i) + \P(G=g) \prod_{i=1}^g z_i.
E_{r e g}^{i n}= \frac{1}{2}\zeta_{i n}( s - 1 / 2 ) \mu^{2 s}, \hspace{1 c m}E_{r e g}^{o u t}= \frac{1}{2}\zeta_{o u t}( s - 1 / 2 ) \mu^{2 s}.
\begin{array}{l}{S_{e f f}= \sum_{x y}\operatorname{l o g}I ( \phi_{x}, \phi_{y}) +}{\sum_{x}\left( \operatorname{l o g}\Delta^{2}( \phi_{x}) - N \sum_{k = 1}^{N}V ( \phi_{x}^{k}) \right)}\end{array}
\bigg[(Ad(gX)^\sim f\bigg](x) = \bigg\{\frac{d}{dt}f(x\exp t Ad(g)X)\bigg\}_{t = 0}=\bigg\{\frac{d}{dt}f(xg\exp t X g^{-1})\bigg\}= \bigg\{\frac{d}{dt}f^{R_g}(xg\exp tX)\bigg\}_{t = 0}=\bigg(\tilde{X}f^{R_g}\bigg)(xg = \bigg(\tilde{X}f^{R_g}\bigg)^{R_{g^{-1}}}(x)
[D_{\lambda},S_{i}]f(x,n,y) = (\psi_{\lambda}(x,n,y)-\psi_{\lambda}(\sigma(x),n-1,y))\chi_{C_i}(x)f(\sigma(x),n-1,y) &=(\psi_{\lambda}(x,n,y)-\psi_{\lambda}(\sigma(x),n-1,y))(S_{i}f)(x,n,y).
\overline{\mathbf{H}}_k^{[ j]}={\left[{\mathbf{H}}_k^{[ 1],T}, \ldots ,{\mathbf{H}}_k^{[{j - 1}],T},{\mathbf{H}}_k^{[{j + 1}],T}, \ldots,{\mathbf{H}}_k^{[ J],T}\right]^T},
f(z) =\sqrt{\left( a_{11}x_{1}+a_{21}x_{2}\right)^{2}+\left(a_{11}y_{1}+a_{21}y_{2}\right)^{2}}& +\sqrt{\left( a_{12}x_{1}+a_{22}x_{2}\right)^{2}+\left( a_{12}y_{1}+a_{22}y_{2}\right)^{2}}
f^{}=\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{n\in \mathbb{Z}-\{0\}}\left(\frac{1}{|n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},0,n}}}{n^2}
u(x)=O\left( \left(\frac{1}{|x|}\right)^{n-2}\right) + o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}((n-2)\lambda+\alpha)}\right) v(x)=O\left( \left(\frac{1}{|x|}\right)^{n-2}\right).
A^{n,q}_{F\otimes L,h_F\otimes h_\nu,\omega}=[i\Theta_{F,h_F}\otimes\mathrm{id}_L,\Lambda_\omega]+[i\Theta_{L,h_\nu}\otimes\mathrm{id}_F,\Lambda_\omega] \geq B_{h_F,\omega}+(2c-\beta_\nu)[\omega\otimes\mathrm{id}_F,\Lambda_\omega] \geq q(2c-\beta_\nu)>qc>0
\tilde E_1(p,q) = (p\,i,0), \tilde F_1(p,q) = (0,q\,i), \tilde E_2(p,q) = (p\,j,0), \tilde F_2(p,q) = (0,q\,j), \tilde E_3(p,q) = -(p\,k,0), \tilde F_3(p,q) = -(0,q\,k),
{}_{2}\phi_{1}\left[ \begin{matrix}a, b c \end{matrix}\, ; q, z \right] = \frac{(b, a z; q)_{\infty}}{(c, z; q)_{\infty}}{}_{2}\phi_{1}\left[ \begin{matrix}\frac{c}{b}, z a z \end{matrix}\, ; q, b \right],
\lbrack K ]_{q \bar{q}}= \left( \begin{array}{l l}{q \bar{q}+ q^{- 1}\bar{q}^{- 1}}{- 1}{- ( 1 + \bar{q}^{2}+ \bar{q}^{- 2})}{q \bar{q}^{3}+ q^{- 1}\bar{q}^{- 3}}\end{array}\right)
T_{\phi \phi}= \frac{1}{(k - 2)}g_{ab}{\large :}J_{\phi}^a (z) J_{\phi}^b (z){\large :}= \frac{1}{k - 2}\Sigma{\large :}J^a_{n-m}J_m^a{\large :}z^{-n-2}= \Sigma L_n z^{-n-2},
Z_{R}(\tau)=Z_{\frac{1}{R}}(\tau)=\frac{1}{|\eta(\tau)|^{2}}\sum_{n,w\in\mathbb{Z}}q^{\frac{1}{4}[\frac{n}{R}+Rw]^{2}}\overline{q}^{\frac{1}{4}[\frac{n}{R}-Rw]^{2}}
\lim_{x\to x_0, x\in\Omega}\frac{u(x)}{h_1(x)}= \lim_{x\to x_0, x\in\Omega}\frac{\mathbb{G}_{|\Omega}^s\mu(x)+\mathbb{P}_{|\Omega}^s\zeta(x)}{h_1(x)}= \zeta(x_0).
m:=(\frac{\partial log_{10}\eta}{\partial(T_{g}/T)})_{T=Tg}=\frac{1}{ln10}(\frac{\partial ln\eta}{\partial(T_{g}/T)})_{T=Tg}=\frac{T_{g}}{ln10}(\frac{-\partial ln\eta}{\partial T})_{T=Tg}
\left\lbrace^{\, A_{c}}_{A_{2c}}\right\rbrace = \frac{4}{\pi}\left(1 - \frac{1}{\sqrt{M}}\right) \int_{0}^{\pi{/}2}\sin^{\left\lbrace^{2m_{c}}_{4m_{c}}\right\rbrace}{\rm d}\theta - \frac{4}{\pi}\left(1 - \frac{1}{\sqrt{M}}\right)^{2}\int_{0}^{\pi{/}4}\sin^{\left\lbrace^{2m_{c}}_{4m_{c}}\right\rbrace}{\rm d}\theta
d(i(x),i(y))\geq \sum_{i=0}^m d(i(y_i),i(y_{i+1}))-2m(\log{(\csc{\frac{\pi}{72}})}+1) & \geq \frac{1}{2}\sum_{i=0}^m d(i(y_i),i(y_{i+1}))\geq \frac{1}{4}\sum_{i=0}^m d_0(y_i,y_{i+1})\geq \frac{1}{4}d_0(x,y).
P = (a - (2 + \frac{u+v}{2u.v}).b)(a - (1 + \frac{u+w}{2u.w}).b))(a - (\frac{v+w}{2v.w}).b) \ + & - 4\lambda^{-2}.(a - (\frac{3}{2}+ \frac{u.v+v.w+w.u}{2u.v.w}).b))(a - \frac{u.v+v.w+w.u}{2u.v.w}).b)).
\mu_{u}^{A}= \left( \mu_{1}^{A}, \dots , \mu_{N - 2}^{A}, 0 , 0 \right) , \qquad ( x^{\alpha})^{v}= \left( 0 , \dots , 0 , x^{\mu}\sigma_{\mu}^{\alpha \beta^{\prime}}\right) \quad \mathrm{w i t h}\quad N - 2 + \beta^{\prime}= v .
I_1=\bar M^{\bar g}M_{\bar g}\,,\;\, I_2=\frac 1 2 \bar B^{\bar g f}B_{\bar gf}\,,\;\, I_3=\bar M_{\bar g}\bar B^{\bar g f}B_{\bar k f}M^{\bar k}\,,\;\,I_4=\frac 1 2 \bar B^{\bar k h}\bar B^{\bar g f}B_{\bar g h}B_{\bar k f}\,.
\left( \frac{i \phi}{{2 \pi}}\right)^{2 n}\int_{M}\mathrm{e}^{i \phi H}\frac{\omega^{n}}{n !}= \sum_{d H = 0}\frac{\mathrm{e}^{i \phi H}{{\sqrt{\operatorname*{d e t}\omega}}}}{\sqrt{\operatorname*{d e t}\mathrm{P f}H}}.
\begin{array}{l}{\xi_{v}= \eta_{u}\mathrm{( i n t e g r a b l e c o n d i t i o n )}}\\{( q u + u^{2}) \xi_{u}+ ( a + 2 u v - p u + q v ) \xi_{v}- ( p v - v^{2}) \eta_{v}= 0}\\ \end{array}
\begin{aligned}Q_1(x,y,q,k):= \left(l_1x^{l_1}+l_2x^{l_2}\right)\left(y-x-q+b(1+k)[\ln{x}-\ln{y}]\right) &-(x-b(1+k))(y^{l_1}-x^{l_1}+y^{l_2}-x^{l_2}),\end{aligned}&\begin{aligned}Q_2(x,y,q,k):= \left(l_1y^{l_1}+l_2y^{l_2}\right)\left(y-x-q+b(1+k)[\ln{x}-\ln{y}]\right) &-(y-b(1+k))(y^{l_1}-x^{l_1}+y^{l_2}-x^{l_2}).\end{aligned}
\frac{\partial}{\partial t_{0}}R_{n+1}[u] = \frac{n}{2n+1}\left( \frac{1}{2}\lambda^{2}\frac{\partial^{3}}{\partial t_{0}^{3}}+ 2 u(t_{0}) \frac{\partial}{\partial t_{0}}+ \frac{\partial}{\partial t_{0}}u(t_{0}) \right) R_{n}[u].
\sum_{\alpha=0}^{n}(-1)^{\alpha+1}(\sigma_3 x_3)^{\alpha}\,(\sigma_1 T_1+\sigma_2 T_2)p_{\alpha}(x_1,x_2)= \sum_{\alpha=0}^{n-1}(\sigma_3 x_3)^{\alpha}[\alpha+\mu_3(1+(-1)^{\alpha})]p_{\alpha+1}(x_1,x_2),
\left.\begin{array}{lc}\mbox{tachyon}|\Omega\rangle_{NS}\cong c e^{-\phi}, \mbox{massless NS}\psi^{\mu}_{-1/2}|\Omega\rangle_{NS}\cong \psi^{\mu}ce^{-\phi}, \mbox{massless R}|\vec{s}\rangle_R\cong ce^{-\phi/2}\Theta_{\vec{s}},\end{array}\right.
{\rm Tr}(X\cdot T_z(s)) ={\rm Tr}([BA^{\frac12},A^{\frac12+is}]Y^{-is}XB^{is}B^{z-1}A^{z-1}) +{\rm Tr}(A^{is}Y^{-is}XB^{is}B^{z-1}[BA,A^{z-1}])+{\rm Tr}(Y^{-is}XB^{is}[BA^{\frac12},A^{\frac12+is}]Y^{z-1}).
\begin{aligned}p_j = m_j \cup x^+ \setminus x^-{\bar p_j}={\bar m_j}\cup{\bar x}^+ \setminus ({\bar x}^- \cup x) m_j = p_j \cup x^- \setminus x^+ \bar m_j = \bar p_j \cup{\bar x}^- \cup x \setminus{\bar x}^+\rlap{.}\end{aligned}
\begin{pmatrix}\dot{y}_{1}\dot{y}_{2}\dot{\zeta}\end{pmatrix}= \begin{pmatrix}m(l_s) 1 0 0 -[n-2-m(l_s)] 0 0 0 -\varpi \end{pmatrix}\begin{pmatrix}y_1 y_2 \zeta \end{pmatrix}+ \begin{pmatrix}0 - g(y_1,-\frac{\ln(\zeta)}{\varpi};l_s) 0 \end{pmatrix}
\lim_{t\rightarrow 0^{+}}\left\{\langle 0|\hat{\psi}_a^{+}\left( t\right) \hat{\psi}_b\left( 0\right) |0\rangle +\langle 0|\hat{\psi}_b\left( 0\right) \hat{\psi}_a^{+}\left( t\right) |0\rangle \right\}=\delta_{ab,}^{}
\bigg[\frac{1}{3}\underline{R}\dot{R}(T)- \bigg(\frac{M^{2}}{6\underline{R}K}-\frac{M^{2}}{4K}\bigg)\bigg]e^{-4Kd}-\frac{1}{4}\frac{M^{2}}{K}-\frac{M^{2}}{2\underline{R}K}=0.
\langle \delta^{2}k_{R}\rangle = \langle k_{R}^{2}\rangle - \langle k_{R}\rangle^{2}= \lambda^{2}\left( \frac{n + 2}{N n}\right)^{2}\left\{\left\langle \left( \sum_{\bf i}\Vert \varphi_{\bf i}\Vert^{2}\right)^{2}\right\rangle - \left( \sum_{\bf i}\langle \Vert \varphi_{\bf i}\Vert^{2}\rangle \right)^{2}\right\}
\widetilde{\psi}(i_1,i_2)= e^{\frac{k}{2}\log^2 \frac{i_1+\sqrt{i_1^2-4i_2}}{i_1-\sqrt{i_1^2-4i_2}}}\ \ \ \ \ (i_1,i_2)\in{D(i_1,i_2)}\cup\gamma_2\,
D_k = \max_{\substack{(x_\ell,y_\ell) \in \{(r,s), (s,u) \}, \ell= 1, \dots, 4}}4n^2 \sum \prod_{\ell = 1}^4 | Z_{2n;i_\ell}(y_\ell) - Z_{2n;i_\ell}(x_\ell) |,
\m^{2}( \beta ) = m_{0}^{2}+ \lambda_{0}F ( D ) \left( m ( \beta ) \right)^{D - 2}\int_{m ( \beta ) \beta}^{\infty}d \tau \left( ( \frac{\tau}{m ( \beta ) \beta})^{2}- 1 \right)^{\frac{D - 3}{2}}\frac{1}{e^{\tau}- 1}.
\frac{\gamma_2(\Gamma_{n,c})}{\gamma_3(\Gamma_{n,c})}=\left< \overline{a}^{p_1^{y_1}-1}\gamma_3(\Gamma_{n,c}),...,\overline{a}^{p_r^{y_r}-1}\gamma_3(\Gamma_{n,c}) \right>=\left< \overline{a}^m \gamma_3(\Gamma_{n,c}) \right>
Z : = \int{\cal D}A_{\mu \nu}{\cal D}\Lambda_{\mu}{\cal D}B_{\mu}{\cal D}C_{\mu}{\cal D}\bar{C}_{\mu}{\cal D}d{\cal D}\bar{d}{\cal D}N{\cal D}P{\cal D}B^{( 1 )}{\cal D}C^{\prime}{\cal D}\bar{C}^{\prime}{\cal D}B^{\prime}\exp \left[ i \int d^{4}x{\cal L}_{t o t}^{\prime}\right] .
m_{\nu}=P^{-1}m_{0,\nu}P=\mbox{\scriptsize $\left(\!\begin{array}{cccc}u_{+}u_{-}v_{-}v_{+}u_{-}u_{+}v_{+}v_{-}w_{-}w_{+}u_{+}u_{-}w_{+}w_{-}u_{-}u_{+}\end{array}\!\right)$},
e^{\sum_{j = 1}^d x_j}\det((-1)^{j-1}\phi^{(1-j)}_i(x_j))_{i, j = 1}^d = e^{\sum_{j = 1}^d x_j}\det((-1)^{d-j}\phi^{(d-j)}_i(x_j))_{i, j = 1}^d.
W_0(z) =\int_{B_1}|\nabla z|^2\,dx-\|\bar c\|^2_2=\int_0^1r\,dr\int_{0}^{\pi+\delta}\left(|\partial_r z|^2+\frac{|\partial_\theta z|^2}{r^2}\right)\,d\theta-\|\bar c\|^2_2 &=\int_0^1r\,dr\int_{0}^{\pi+\delta}\left(|\bar c|^2+|\bar c'|^2\right)\,d\theta-\|\bar c\|^2_2 &=\frac12 \left(\|\bar c'\|^2_2-\|\bar c\|^2_2\right)
\begin{aligned}\left(S_{n - 1;i}(X) - \frac{c}{C^m_n}S_{m - 1;i}(X) \right) X_{i\bar i}\leq\sum_l \left(S_{n - 1;l}(X) - \frac{c}{C^m_n}S_{m - 1;l}(X) \right) X_{l\bar l}&\leq n S_n (X),\end{aligned}
C_{2,\ell}:=\left(\begin{array}{cccccccc}\frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}0 0 0 \dots 0 0 0 0 \frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}0 \dots 0 0 \vdots \vdots \vdots \vdots \vdots \vdots \vdots 0 0 0 0 0 \dots \frac{1}{\sqrt{2}}\frac{1}{\sqrt{2}}\end{array}\right)\,.
\lim_{s\rightarrow0}\Gamma(s)\left( \frac{1}{\left( cz+d\right)^{s}}G(Vz,s;A,B)-G(z,s;B_{-b},A_{-c})\right) & =-2if^{\ast}(0)\lim_{s\rightarrow0}\Gamma(s)\sin(\pi s)L(s,A_{c};0)+\sum\limits_{j=1}^{c}\sum\limits_{\mu=0}^{k-1}\sum\limits_{v=0}^{k-1}f^{\ast}(b(\mu c+j))f(c(\left[ dj/c\right]-v))I(z,0,c,d,0,0).
A_{ji}^{-1}\frac{|\alpha_{i}|^{2}}{2}=\left\langle \lambda_{j},\lambda_{i}\right\rangle =\sum_{x=1}^{h}\frac{x}{h}\,\left\langle \lambda_{j},\sigma^{x+\frac{c_{i}-1}{2}}\gamma_{i}\right\rangle \;,
F_m^-\big((x,t),\xi,\nu,M\big) = \inf_{(a,c)\in \mathcal H_m}\sup_{(b,d)\in \mathcal H_m}\Phi\big(a,b,c,d;(x,t),\nu,M\big)+r\xi, F_m^+\big((x,t),\xi,\nu,M\big) =\sup_{(b,d)\in \mathcal H_m}\inf_{(a,c)\in \mathcal H_m}\Phi\big(a,b,c,d;(x,t),\nu,M\big)+r\xi