harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L = \int d^{4}x \varepsilon^{\mu \nu \rho \sigma}\bigg ( C_{1}{^+}{\cal G}_{\mu \nu}^{A}{^+}{\cal G}_{\rho \sigma}^{B}+ C_{2}{^-}{\cal G}_{\mu \nu}^{A}{^-}{\cal G}_{\rho \sigma}^{B}\, + C_{3}{^+}{\cal F}_{\mu \nu}^{A}{^+}{\cal G}_{\rho \sigma}^{B}+ C_{4}{^-}{\cal F}_{\mu \nu}^{A}{^-}{\cal G}_{\rho \sigma}^{B}\bigg ) M_{A B}
I(k)=\frac{1}{(2\pi)^4}\int dp_{\parallel}p_{\parallel}d\varphi_1\int dp_{\perp}p_{\perp}d\varphi_2\frac{e^{i\Theta p_{\parallel}k_{\parallel}\sin\varphi_1}e^{i\Theta p_{\perp}k_{\perp}\sin\varphi_2}}{p^2_{\parallel}+p^2_{\perp}}.
\begin{cases}\underset{0\leq x \leq L}{\sup}\| \partial^k_x \varphi(x,\cdot)\|_{H^{\frac{1-k}3}(0,T)}\le C_T\|\varphi^0\|_{L^2(0,L)}, \underset{0\leq x \leq L}{\sup}\| \partial^k_x \psi(x,\cdot)\|_{H^{\frac{1-k}3}(0,T)}\le C_T\|\psi^0\|_{L^2(0,L)},\end{cases}
\frac{\sum_{|X_i|\le(b/L)^{1/\beta}}g(X_i)}{\sqrt{\sum_{i=1}^nI(|X_i|\le(b/L)^{1/\beta})}}\ge\frac{b}{2}\frac{\sum_{i=1}^n I(|X_i|\le(b/(2L))^{1/\beta})}{\sqrt{\sum_{i=1}^n I(|X_i|\le(b/L)^{1/\beta})}}
L_\mathcal E(h) = \int_0^T \int_\Lambda \chi(\rho_t) v_t\cdot h_t \;\!\mathrm du\;\!\mathrm dt,L_{\mathcal E^\star}(h) = \int_0^T \int_\Lambda \chi(\rho_t)w_t\cdot h_t \;\!\mathrm du\;\!\mathrm dt.
\sum_{1\le k_1<\ldots<k_l\le m}\left|{\textbf E}\left(\prod_{j=1}^l\phi_{n_{k_j}}\cdot \phi_{n_{n_{m+1}}}\right)\right|\le \sqrt{\frac{2^m-1}{8^{m+1}-8^m}}< \frac{1}{2^m}.
dY(t)=\partial_y \varphi(t,W(t))\left(\int_{s}^{t}\phi''(u)W(u)du\right)dW(t)+ \left(\partial_t \varphi(t,W(t))+\frac{1}{2}\partial^2_y \varphi(t,W(t))+\varphi(t,W(t))\phi''(t)W(t)\right)\left(\int_{s}^{t}\phi''(u)W(u)du\right)dt
x^3 + 2wxy + wxy = 11wy + xy + 11z^2 + yz wy^2 + x^2y + yz^2 + xyz = -wy - z^2 -wx^2 + w^2y + wz^2 + wxz = -x^2 - 11wx - wy - wz wxy + z^3 + 2wyz = wx + wz xz - yw = 1.
j_{i_1\ldots i_s}=\sum\limits_{k=0}^s{(-1)^k\over k!\left(k+{D-4\over 2}\right)! (s-k)!\left(s-k+{D-4\over 2}\right)!}\partial_{i_1}\cdots\partial_{i_k}\phi^*\;\partial_{i_{k+1}}\cdots\partial_{i_s}\phi-\mbox{traces}
\begin{aligned}\bar{\nabla}z^{j}(h_{st})(\bar{q}) =\pi_{*}(\nabla y^{j})(h(\bar{\nabla}z^{s},\bar{\nabla}z^{t}))(\pi(q))=\nabla y^{j}(g(\nabla y^{s},\nabla y^{t}))(q) &= y^{s}(\nabla y^{j},\nabla y^{t})(q)+ y^{t}(\nabla y^{s},\nabla y^{j})(q)\end{aligned}
\frac{\partial}{\partial t}\log|\nabla w|^2= (1+\frac{\alpha'}{2}R)\Delta\log|\nabla w|^2+(1+\frac{\alpha'}{2}R)\big|\nabla\log|\nabla w|^2\big|^2-2(1+\frac{\alpha'}{2}R)\frac{|\nabla^2w|^2}{|\nabla w|^2}&-(r+\frac{\alpha'}{4}R^2)
\times \delta (p_{\theta}- p' + \frac{\hbar m}{2}) \delta(p_{\theta}- p'' - \frac{\hbar n}{2}) e^{\frac{2i}{\hbar}[-(p_{\theta}- p') (\theta - \theta '') + (p_{\theta}-p'')(\theta - \theta ')]}f(\theta ', p') g (\theta '', p'').
\mathbb{P}\left(\frac{\log Z_{n}-n \mu}{\sigma \sqrt{n}}\leq x\right)-\Phi(x) =-\left[\mathbb{P}\left(\frac{\log Z_{n}-n \mu}{\sigma \sqrt{n}}> x\right)-(1-\Phi(x))\right] &\geq-\left[(1-\Phi(x))\exp\left\{C\frac{1+x^{3}}{\sqrt{n}}\right\}-(1-\Phi(x))\right] &=-(1-\Phi(x))\left(\exp\left\{C\frac{1+x^{3}}{\sqrt{n}}\right\}-1\right).
\mathrm{e}_{\pi_{\lambda,\beta}}(z;\cdot):\mathbb{R}\longrightarrow\mathbb{C},\;x\mapsto\mathrm{e}_{\pi_{\lambda,\beta}}(z;x)={\displaystyle \frac{\mathrm{e}^{xz}}{l_{\pi_{\lambda,\beta}}(z)}}={\displaystyle \frac{\mathrm{e}^{xz}}{E_{\beta}(\lambda(\mathrm{e}^{z}-1))}}.
H_{K[y_{1}, \ldots, y_{n-2}, z_{1}, \ldots, z_{n-2}] / \sum_{i = 1}^{n - 2}(y_{1}, \ldots, y_{i}) (z_{i})}(\lambda) = \frac{1 + (n-2)\lambda}{(1 - \lambda)^{n - 2}}.
\sum\limits_{\substack{i=0 \mbox{\scriptsize $m+i$ even}}}^m (-1)^i{m \choose i}(m+i) \sum\limits_j{\frac{m+i}{2}-1\brace j-1}\gamma_{2j-2}\quad+ \sum\limits_{\substack{i=0 \mbox{\scriptsize $m+i$ odd}}}^m (-1)^i{m \choose i}(m+i) \gamma_{m+i-1}= 0.
||{\bf v}[{\bf u}^a+S_{\theta_i}{\bf u}_{i}]||_{s,\ast,T}\le C\delta^2\begin{cases}\theta_i^{(s+2-\alpha)_+}s+2\not=\alpha, \theta_i s+2=\alpha\end{cases}s\in\{6,\dots,\tilde{\alpha}+6\},
\Phi(f_1,f_2)\ast \Phi(g_1,g_2) =(f_1\ast f_2^{\ast -1})\ast (g_1\ast g_2^{\ast -1})=(f_1*g_1)*(g_2^{*-1}*f_2^{*-1}) &=\Phi(f_1\ast g_1,f_2\ast g_2)=\Phi((f_1,f_2)\cdot_{D}(g_1,g_2)).
{\mathcal V}( z , \bar{z}, q ) = g^{2}\Bigl [ \left( g_{i j^{\star}}k_{\Lambda}^{i}k_{\Sigma}^{j^{\star}}+ 4 h_{u v}k_{\Lambda}^{u}k_{\Sigma}^{v}\right) \bar{L}^{\Lambda}L^{\Sigma}
\begin{aligned}h_1^\beta \equiv \left(N^{-1}\psi(h_1)N\right)^\beta \equiv \prod_{i=-\infty}^{-1}R(c)^{(ab)^{-i}}\cdot \left(R(c)L(d)R(c)^{-1}L(d)^{-1}\right) \cdot \prod_{i=1}^\infty L(d)^{-(ab)^i}\equiv \left( \begin{matrix}\star \star \star 1-cd \end{matrix}\right) \pmod{2^{T-n_1-1}},\end{aligned}
\begin{array}{l}T_{0}|\Omega \rangle = \frac{1}{2(k+\check{g})}S_{a0}^{a}| \Omega \rangle= \frac{p}{2(k+\check{g})}| \Omega \rangle \ J_{0}| \Omega \rangle = \frac{1}{(k+\check{g})}S_{a0}^{a}| \Omega \rangle= \frac{p}{(k+\check{g})}| \Omega \rangle\end{array}
\ddot{z}^{+}- \dot{z}^{+}\frac{\ddot{z}^{-}}{\dot{z}^{-}}- 2 \lambda \dot{z}^{+}( \dot{z}^{+}+ \dot{z}^{-}) + 4 \lambda \dot{z}^{+}\dot{z}^{-}= 0 ,
A_{22}=-\partial_u(a_3)+\partial_x(a_2)+(-a_1a_4+a_2a_2)+r_2=0 A_{23}=\partial_u(a_4)-\partial_x(a_3)-a_3^2+a_2a_4+a_2a_4-a_3^2-r_3=0 A_{32}=-\partial_u(a_2)+\partial_x(a_1)+a_2^2-a_1a_3-a_1a_3+a_2^2+r_1=0 A_{33}=\partial_u(a_3)-\partial_x(a_2)-a_2a_3+a_1a_4-r_2=0.
L_{\uparrow}(w;b,x,b) = \frac{\mathrm{e}^{-(w+\lambda) (x-b/p^k)}}{\frac{1}{L_{\uparrow}(w;b,b/p^k,b)}- \int_0^{x-b/p^k}\frac{\lambda \mathrm{e}^{-(\lambda+w ) t}}{L_{\uparrow}(w;b,b/p^{k-1}+ pt,b)}\mathrm{d}t},
\int_{B_R}f^2d\mu_x(z)-\left(\int_{B_R}f\mbox{}d\mu_x(z)\right)^2= \frac{1}{2}\int_{B_R\times B_R}(f(z)-f(y))^2d\mu_x(z)d\mu_x(y) \leq 2R^2\sup|\nabla f|^2,
\phi'(t_1){}= \varphi(x_1^1 y_1^0 z_1^1) = \varphi(x_1) \varphi(z_1) = \xi \varphi(x_1), \phi'(t_3){}= \varphi(x_1^{-1}y_1^0 z_1^0) = \varphi(x_1)^{-1}= \lambda^{-\ell}\varphi(x_1^{\ell-1}), \phi'(t_4){}= \varphi(x_1^0 y_1^{-1}z_1^0) = \varphi(y_1)^{-1}= \varphi(y_1^{\ell-1}),
-\rho(1-\zeta^2)r\Omega^2+\frac{\partial P}{\partial r}=-\rho\frac{\partial\Phi}{\partial r}, \rho\zeta r^2\Omega^2+\frac{\partial P}{\partial\zeta}=-\rho\frac{\partial\Phi}{\partial\zeta}, \triangle\Phi =4\pi\mathsf{G}\rho,
\sum\limits_{i=1}^{N}|\langle x,\varphi_{i}\rangle||\langle y,\varphi_{i}\rangle| =\sum\limits_{i=M}^{N-M+1}|\langle x,\varphi_{i}\rangle||\langle y,\varphi_{i}\rangle|\leq\sum\limits_{i=M}^{N-M+1}|\langle x,\varphi_{i}\rangle| begin{align*}2 pt] \leq(N-2M+2)\sqrt{\frac{B}{M}}.
\begin{array}{rcl}\hat{Q}= S_+ \hat{C}^+ + S_- \hat{C}^- +A_1(\hat{S}_3) \hat{C}^3+ A_2(\hat{S}_3) \hat{\bar{C}}_- \hat{C}^- \hat{C}^3+A_3(\hat{S}_3) \hat{\bar{C}}_+ \hat{C}^+ \hat{C}^3 begin{align*}3mm] +A_4(\hat{S}_3) \hat{\bar{C}}_3 \hat{C}^+ \hat{C}^- +A_5(\hat{S}_3) \hat{\bar{C}}_+ \hat{\bar{C}}_- \hat{C}^+ \hat{C}^- \hat{C}^3{}~,\end{array}
f_i'(x) = (F_{\eta_i}'\circ F^{-1}_{\eta_i})'(x) = \frac{F_{\eta_i}''\circ F^{-1}_{\eta_i}(x)}{F_{\eta_i}'\circ F^{-1}_{\eta_i}(x)}= - V'_i(F^{-1}_{\eta_i}(x)).
c({\mathcal T^{\vee}})= \; c(\mathcal{O}_{\mathrm{G}(3,V)}(1))/(\Omega_{\mathrm{G}(3,V)}(1)) = \; 1+4\sigma_1+8\sigma_1^2+(8\sigma_1^2+6\sigma_1\sigma_2-6\sigma_3)+(24\sigma_1^2\sigma_2-24\sigma_1\sigma_3) &+(30\sigma_1\sigma_2^2-30\sigma_2\sigma_3)+(10\sigma_2^3+24\sigma_1\sigma_2\sigma_3-24\sigma_3^2)+18\sigma_2^2\sigma_3+12\sigma_2\sigma_3^2+4\sigma_3^3
\widehat D_n(x|a_1,\dots,a_r) =\prod_{j=1}^r\left(\frac{t e^{a_j t}}{e^{a_j t}-1}\right)(x)_n &=\sum_{m=0}^n S_1(n,m)\prod_{j=1}^r\left(\frac{t e^{a_j t}}{e^{a_j t}-1}\right)x^m &=\sum_{m=0}^n S_1(n,m)e^{(a_1+\cdots+a_r)t}\prod_{j=1}^r\left(\frac{t}{e^{a_j t}-1}\right)x^m &=\sum_{m=0}^n S_1(n,m)B_m(x+a_1+\cdots+a_r|a_1,\dots,a_r)\,.
\mathbf{z}^\top = ( v_{\infty}^{-1}, v_{\infty}^{-1}\phi_{1}, v_{\infty}^{-1}\phi_{2}, \dots) \mathbf{z}_{n+1}^\top = ( v_{n+1}^{-1}, v_{n+1}^{-1}\phi_{n,1}, \dots, v_{n+1}^{-1}\phi_{n,n}),
I_{\psi ,{\widetilde \psi}}\equiv{\frac{1}{2}}\int_{M}\epsilon^{\mu \nu \rho \sigma}\left[{\widetilde \psi}_{\mu}^{A^{\prime}}\; e_{A A^{\prime}\nu}\; D_{\rho}\; \psi_{\; \; \sigma}^{A}- \psi_{\; \; \mu}^{A}\; e_{A A^{\prime}\nu}\; D_{\rho}\;{\widetilde \psi}_{\sigma}^{A^{\prime}}\right] d^{4}x \; ,
T_{\mu \nu}= \phi_{, \mu}\phi_{, \nu}- \frac 1 2 g_{\mu \nu}\left( \phi_{, \rho}\phi^{, \rho}+ m^{2}\phi^{2}\right) + \xi \left[ ( R_{\mu \nu}- \frac 1 2 g_{\mu \nu}R ) \phi^{2}+ g_{\mu \nu}( \phi^{2})_{~ ; \rho}^{, \rho}- ( \phi^{2})_{; \mu \nu}\right] \, .
R^{[i,j]}= \log_2 \left( 1+ \frac{\gamma^{[i,j]}}{\frac{S+\Delta^{[i,j]}}{\mathsf{SNR}}+ \sum_{k\neq i}^{K}\sum_{s=1}^{S}\left|{\mathbf{f}_{k}^{[i,j]}}^{\mathsf{H}}\mathbf{v}^{[k,s]}\right|^2}\right),
\|W_x(t)\|_1^2 + \|W_{xt}(t)\|^2+ \int_0^t \big(\|W_{xx}\|^2 + \|W_{x\tau}\|^2\big) d\tau \leqslant O(1)\delta^{-1/4}\big( \|W(0)\|_2^2 + \|W_t(0)\|_1^2\big)+ O(1) \delta_0^{{1}/{4}}.
\chi\left(\widetilde{C}_{Hom_p(K_r, G(n,\frac{1}{2}))}\right) \leq\left(1+\binom{r}{2}\right)\frac{r}{\binom{r}{2}}(1 +b\epsilon)\log_2 n\quad & = \left(r+\frac{2}{r-1}+ \epsilon\right)\log_2 n
\boldsymbol{\eta}=(\boldsymbol{\nu}_{1}, \ldots , \boldsymbol{\nu}_{a-1}, \boldsymbol{\nu}_{a}^{b, -}, \boldsymbol{\nu}_{a+1}^{b, +}, \boldsymbol{\nu}_{a+2}, \ldots , \boldsymbol{\nu}_{r})
\mathbb{T}\left( \mathcal{H}_{\ast},\{\mathbf{h}_p\}_{p=0}^{11},\{0\}_{p=0}^{11}\right) =\prod_{p=0}^{11}\left[\mathbf{h}'_p, \mathbf{h}_p\right]^{(-1)^{(p+1)}}.
\bigg\{{\bf x}= (x_{k, l})_{2 \le k \le r, 1 \le l \le N}\in \widehat{\Delta}_{{\bf i}, \lambda_1, \ldots, \lambda_r}\cap \mathbb{Z}^{(r-1) N}~\bigg| &\lambda_1 + \cdots + \lambda_r - \sum_{2 \le k \le r, 1 \le l \le N}x_{k, l}\alpha_{i_{k, l}}= \nu \bigg\},
\mathrm{d}\left( \sqrt[N_k]{\alpha'_k}\underset{E_k/F}{\sqcup}N_k \Lambda_k\right) = \prod_{v \in S_k}\mathrm{ES}^2_{R'_{k+i,v}}\left( \mathrm{d}\left(\sqrt[N_k]{\alpha_{k,v}}\underset{E_{k, \dot{v}}/F_v}{\sqcup}N_kl_{k,v}(\Lambda_k) \right) \right) = \prod_{v \in S_k}\mathrm{ES}^2_{R'_{k+i,v}}(\mathrm{loc}_{k,v}(\xi_{k,v}))
m_1=\lim_{n\rightarrow +\infty}J(w_n) \geq \liminf_{n\rightarrow +\infty}J(t_{\widehat{w}^+}w^+_n+s_{\widehat{w}^-}w^-_n) & \geq J(t_{\widehat{w}^+}\widehat{w}^++s_{\widehat{w}^-}\widehat{w}^-) & \geq m_1.
\int_{{\cal M}_{\alpha}}R^{\mu \nu \lambda \rho}R_{\mu \nu \lambda \rho}= \alpha \int_{{\cal M}_{\alpha = 1}}R^{\mu \nu \lambda \rho}R_{\mu \nu \lambda \rho}+ 8 \pi ( 1 - \alpha ) \int_{\Sigma}R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}+ O ( ( 1 - \alpha )^{2}) ,
2\, \Im m\, \int \frac{dx}{(2\pi )^{2}i}\varphi^{,}_{\pm}(x+i\eta )\log \left( 1+(-)^{\delta}e^{iZ_{\pm}(x+i\eta )}\right) =\pm \left( \frac{1}{24}-\frac{{\cal Q}^{2}_{\pm}(\mp \infty )}{16\pi^{2}}\frac{p+1}{p}\right)
|sF(s)-\alpha|\le s\int_{0}^{A}|f(t)-\alpha|e^{-st}dt+s\int_{A}^{\infty}|f(t)-\alpha|e^{-st}dt\le2s\int_{0}^{A}e^{-st}dt+\epsilon s\int_{A}^{\infty}e^{-st}dt=I+II.
\|\varphi\|^2_{\mathcal{H}_0}:= \|u\|^2_1 + \|u\|^2_{L^2(\Gamma)}+ \|v\|^2 & = \left( \|\nabla u\|^2 + \|u\|^2 \right) + \|u\|^2_{L^2(\Gamma)}+ \|v\|^2.
\begin{array}{c}{{Z_{N}( + \infty ) = N \pi + \chi_{\infty}\left( N_{H}- 2 N_{S}- M_{C}- 2 \theta ( p - 1 ) M_{W}\right) +}}\\{{+ \pi \, \textrm{s i g n}( p - 1 ) \, M_{W}+ \displaystyle \int d x \, G ( x ) \,{\cal Q}_{N}( + \infty ) + \alpha .}}\\ \end{array}
\nabla_v f(x) = \left\langle \nabla f(x), v\right\rangle = \lim_{\varepsilon \to 0}\frac{f(x+\varepsilon v)-f(x)}{\varepsilon}, \nabla_{v_2}\nabla_{v_1}f(x) = \left\langle \nabla^2 f(x), v_1 v^\top_2\right\rangle_{\mathrm{HS}}= \lim_{\varepsilon \to 0}\frac{\nabla_{v_1}f(x+\varepsilon v_2)-\nabla_{v_1}f(x)}{\varepsilon},
\begin{aligned}\sum_{n=L_m+1}^M\mathbf{1}_{[0,y)}(x_n)\leq \sum_{n=L_m+1}^M\mathbf{1}_{U_1}(q_n)\cdot \mathbf{1}_{V_1}(r_n) +\sum_{h=1}^{H}\sum_{n=L_m+1}^M\left(\mathbf{1}_{U_{h+1}\backslash U_h}(q_n)\cdot \mathbf{1}_{V_{h+1}}(r_n)+\mathbf{1}_{U_h}(q_n)\cdot \mathbf{1}_{V_{h+1}\backslash V_h}(r_n)\right). \end{aligned}
\sum_{i=1}^{k}q^{d_{i}^{-}(x)}T_{i}^{(-)}(x)T_{i}^{(+)}(x).f(x)=\sum_{i=1}^{k}q^{d_{i}^{-}(x)}d_{i}^{+}(x)f(x)=\frac{1}{1-q}\sum_{a=1}^{r}(c_{a}-[c_{a}]_{q})f(x).
\begin{array}{r c l c r c l c r c l c r c l}{{f_{1}}}&{{=}}&{{- 6 i}}&{{;}}&{{f_{3}}}&{{=}}&{{- 6 i}}&{{;}}&{{f_{2}}}&{{=}}&{{- \frac{4}{3}a_{5}}}&{{;}}&{{f_{4}}}&{{=}}&{{- 8 i a_{5}}}\\ \end{array}
G(x,\tau|0,0)=\frac{1}{Z}\sum_{\alpha,\alpha^{\prime}}e^{-\beta E_{\alpha^{\prime}}}e^{\tau(E_{\alpha^{\prime}}-E_{\alpha})}\langle\alpha^{\prime}|\psi(x)|\alpha\rangle\langle\alpha|\psi^{\dagger}(0)|\alpha^{\prime}\rangle.
\tilde{\omega}_{\theta,n}(p_1,...,p_n):=\omega_\theta(\tilde{\phi}_\theta(p_1)\cdots \tilde{\phi}_\theta(p_n))=\prod_{1\leq l<r\leq n}e^{ip_l \cdot \theta p_r}\cdot\tilde{\omega}_{0,n}(p_1,...,p_n).
\tilde S^{(p)}_{j,j'}= \Theta(\mu_j,\mu_{j'}) \tilde F_{j,j'}, \Theta(\mu_j,\mu_{j'}) = \left[1 + \frac{(\mu_j -\mu_{j'})^2}{8\gamma^2 (\mu_j + \mu_{j'})}\right]^{-1},
\mathcal{I}_1\ \ll \L^{10}\sum_{\substack{|\mathbf{x}'|,|z_1|,|z_1'|,|z_2|,|z_2'|\ll X (z_1 \gamma_1^{T}+z_2 \gamma_2^T)B^{-1}A_1\mathbf{x}'+d_1z_1z_1'+d_2z_2z_2'=0}}1.
\partial_t u_1 - \omega_3 u_2 = -\partial_1 \left(p + \frac{1}{2}|u|^2\right) + \Delta u_1 \partial_t u_2 + \omega_3 u_1 = -\partial_2 \left(p + \frac{1}{2}|u|^2\right) + \Delta u_2 \partial_3 p = 0 \partial_1 u_1 + \partial_2 u_2 = 0.
\left\langle u(\tau), z^{\tau}\right\rangle_{\mathcal{H}^{-s}, \mathcal{H}^{s}}=\left\langle u_{0}, S(\tau)z^\tau\right\rangle_{\mathcal{H}^{-s}, \mathcal{H}^{s}}- \int_{0}^{\tau}f(t)\lim_{x\rightarrow 1^-}S(\tau-t) z^{\tau}_{x}(x) \mathrm{d}t,
R_{a,b}=\bigcup\limits_{k=r(a,b)+1}^{r(a,b)+D}\big\{{a^kb=[\alpha(a,k,b,1)}]^{\kappa(a,k,b,1)},\a^{k+D}b=[{\alpha(a,k+D,b,1)}]^{\kappa(a,k+D,b,1)}\big\},
\frac{1}{|b_i|}\int_{b_i}u(P)ds(P)= u^{s}(X)+\nabla u^{s}(X)\cdot (M_i-X) + \mathcal{R}_i(X) &+\frac{1}{|b_i|}\int_{b_i\cap T^{s'}}\left(\left( M^{s}(\widetilde{Y}_i)-I \right)\nabla u^{s}(X)\right) \cdot (P-\widetilde{Y}_i) ds(P), ~ i\in \mathcal{I}^{int},
-\frac{\hbar^2}{2m}\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR(r)}{dr}\right)+\left[\frac{\hbar^2\ell(\ell+1)}{2mr^2}+\frac{1}{2}m\omega^2r^2\right]R(r)=ER(r).
D_{s_{j}}(z) = \left( \frac{(\sqrt{z+x_{1}}-i\sqrt{x_{j-1}-x_{1}})(\sqrt{z+x_{1}}+i\sqrt{x_{j}-x_{1}})}{(\sqrt{z+x_{1}}-i\sqrt{x_{j}-x_{1}})(\sqrt{z+x_{1}}+i\sqrt{x_{j-1}-x_{1}})}\right)^{\frac{\log s_{j}}{2\pi i}}.
\lim_{n \to \infty}\Sigma_n( z_1, z_2) = \frac{(z_1z_2)^{-\frac{s}{2}}\Big\{A_{II}^{(s,v)}(1, z_1) B_{II}^{(s,v)}(1, z_2) - A_{II}^{(s,v)}(1, z_2) B_{II}^{(s,v)}(1, z_1)\Big\}}{2 \big[C_{II}^{(s,v)}(1)\big]^2 ( z_1 - z_2)}.
w ( m , m^{\prime}) = \frac{1}{\sqrt{2 \pi N | \Phi_{m}^{( 2 )}( t_{+}) |}}\frac{1}{\sqrt{2 \pi N | \Phi_{m^{\prime}}^{( 2 )}( s_{+}) |}}\mathrm{e}^{- N ( \Phi_{m}( t_{+}) + \Phi_{m^{\prime}}( s_{+}) )}\mathrm{d e t}M ( t_{+}, s_{+}) ,
F_1(X,Y,Z) =(X^m-Y^m)(Z^m-\omega^2 X^m)(Y^m-\omega Z^m) F_2(X,Y,Z) =(X^m-\omega Y^m)(Z^m-X^m)(Y^m-\omega^2 Z^m) F_3(X,Y,Z) =(X^m-\omega^2 Y^m)(Z^m-\omega X^n)(Y^m-Z^m),
\left[ I_{\mu}, \widetilde{I}\right] = - 2 \widetilde{I}_{\mu}, \hspace{0 . 3 i n}\left[ \widetilde{I}_{\mu}, \widetilde{I}\right] = - 2 I_{\mu}, \hspace{0 . 3 i n}\left[ I_{\alpha \beta}, \widetilde{I}\right] = 0 .
\frac{n}{2}\iint_\Omega |\nabla \phi|^2~d\mathbf{x}dy=\frac{n+1}{2}g\int_{\mathbb{R}^{n-1}}\eta^2~d\mathbf{x}+\frac12T\int_{\mathbb{R}^{n-1}}(\sqrt{1+|\nabla\eta|^2}-1)~d\mathbf{x}.
D^{-\frac{1}{2}}MD^{-\frac{1}{2}}=[\begin{matrix}A^{-\frac{1}{2}}&0\\ 0&C^{-\frac{1}{2}}\end{matrix}][\begin{matrix}A&B\\B^{*}&C\end{matrix}][\begin{matrix}A^{-\frac{1}{2}}&0\\ 0&C^{-\frac{1}{2}}\end{matrix}]=[\begin{matrix}I_{p}&A^{-\frac{1}{2}}BC^{-\frac{1}{2}}\\C^{-\frac{1}{2}}B^{*}A^{-\frac{1}{2}}&I_{q}\end{matrix}]
\int_{x_0}^x t^\alpha \frac{L(t)}{t}\mathrm{d}t\sim \alpha^{-1}x^\alpha L(x)\intertext{and this result remains true also for $\alpha=0$ in the sense that}\frac{\int_{x_0}^x \frac{L(t)}{t}\mathrm{d}t}{L(x)}\to\infty\qquad\mbox{as}x\to\infty,
y_{(2k-1)m+i,j}+ y_{(2(r_0-k+1)-1)m +(m+1-i),n+1-j}= 2MN -\tfrac{1}{2}mnr +\tfrac{3}{2}, \mbox{\and}y_{(2k)m+i,j}+ y_{2(r_0-k)m +(m+1-i),n+1-j}= \tfrac{1}{2}mnr +\tfrac{3}{2}.
\bigl((f_2^n)^*H_2\bigr)\cdot H_2^{d_2-1}= \left(\sum_{j=0}^{r-1}c_{r,j}(n) f_2^j\right)^*H_2 \cdot H_2^{d_2-1}= \sum_{j=0}^{r-1}\bigl(c_{r,j}(n) f_2^j\bigr)^*H_2 \cdot H_2^{d_2-1}= \sum_{j=0}^{r-1}c_{r,j}(n)^2 (f_2^j)^*H_2 \cdot H_2^{d_2-1}\le C(A_2,H_2,f_2)n^{2r-2},
\begin{array}{c}{{\nabla{\bf K}= \frac 1{\sqrt{2 \epsilon}}\{\left( - \frac 1{\sqrt{2 \epsilon}}{\bf K}+ \frac 1{2 \epsilon}{\bf M}_{1}\right) \left( \vec{R}- i \frac 1 2 \vec{\Gamma}\right) + \frac 1{2 \Gamma}{\bf M}_{2}\vec{\Gamma}}}\\{{+ i \frac 1{\Gamma \left( \epsilon^{2}- \eta^{2}\right)}\frac{\xi^{*}}{\left| \xi \right|}M_{3}\left( \vec{\Gamma}\times \vec{R}\right) \}}}\\ \end{array}
\| h_n - h_m \|_{\sup}\leqslant \sum_{i=m}^{n-1}\frac{1}{d^{i+1}}\|((f^{\mathrm{an}})^i)^*(\lambda) \|_{\sup}= \frac{\| \lambda \|_{\sup}}{d^{m+1}}\sum_{i=0}^{n-m-1}\frac{1}{d^{i}}& \leqslant \frac{\| \lambda \|_{\sup}}{d^{m+1}}\sum_{i=0}^{\infty}\frac{1}{d^{i}}=\frac{\| \lambda \|_{\sup}}{d^m(d - 1)}.
\widetilde{\varphi}_{x,y}\left( k\right) =\varphi_{x,y}\left( k\right)\left( \prod\limits_{j=1}^{m_{1}}\frac{1}{k-a_{j}}\right) \left(\prod\limits_{j=1}^{m_{2}}\frac{k-\overline{b}_{j}}{k-b_{j}}\right) .
Q(C_i,C_j) \ = \ 3 \cdot Q(A_{i},A_j) \ \i,j \in J \ \i,j \in J' Q(C_i,C_j) \ = \Q(A_{i},A_j) \ \i \in J \ \j \in J'. Q(C_i,C_{n+1}) \ = \N^2 \ \i \in J', Q(C_{n+1},C_i) \ = \N^2 \ \i \in J.
{\mathrm{\boldmath ~ \alpha ~}}_{1}^{*}\cdot{\bf h}- \frac{{\mathrm{\boldmath ~ \alpha ~}}_{1}^{*}\cdot{\mathrm{\boldmath ~ \alpha ~}}_{2}^{*}}{4 \pi r_{1 2}}\sqrt{1 -{\bf v}_{2}^{2}}\sqrt{g_{2}^{2}+ q_{2}^{2}}.
-\frac{i}{2}\left( \frac{n+2}{n^{2}}\right) \Delta^{ab}\left[ n\tau_{1\cdots nab}+\left( n-1\right) \tau_{1\cdots anb}+\cdots +0\cdot \tau_{a1\cdots nb}\right] +\mathrm{cyc}_{1\cdots n}
|U_i| =|U_i'|+1=\ell'+1 +e_{C'}(U_{i+1}', U_i')+e_{C'}(U_i', U_{i-1}') &=\ell+2 +e_{C'}(U_{i+1}', U_i')+e_{C'}(U_i', U_{i-1}')=\ell+e_C(U_{i+1}, U_i)+e_C(U_i, U_{i-1}),
H^1(G_{\beta}(O_r),\Bbb C^{\times}) =(G_{\beta}(O_r),\Bbb C^{\times}), H^1(K_1(O_r)\cap G_{\beta}(O_r),\Bbb C^{\times})^{G_{\beta}(\Bbb F)}=(K_1(O_r)\cap G_{\beta}(O_r),\Bbb C^{\times})
\Phi(e_\mu e_{3\mu}-e_{2\mu}^2-\mu e_{4\mu}) =\left((a+3\mu b)(a+\mu b+3\mu)-(a+2\mu b)(a+2\mu b+2\mu)-\mu(a+4\mu b)\right)t^{4\mu}&=\mu^2b(1-b)t^{4 \mu}.
\inf_{x\in D_{k+2}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}\ge \inf_{x\in D_{k+1}}\frac{u^k_{k+1}(x)}{v^k_{k+1}(x)}\ge \inf_{x\in D_k}\frac{u(x)}{v(x)}\ge \inf_{x\in D_i}\frac{u(x)}{v(x)}, i=0, \dots k.
\lambda_{112}\lambda_{qrr}[a_{(1\,q)},a_{(2\,r)}]_c=\lambda_{112}\lambda_{qrr}[[a_{(1\,q)},[a_2,a_{(3\,r)}]_c]_c &=\lambda_{112}\lambda_{qrr}[[a_{(1\,q)},a_2]_c,a_{(3\,r)}]_c -\lambda_{112}\lambda_{qrr}\chi_{(3\,r)}(g_{2})[a_{(1\,q)},a_{(3\,r)}]_ca_2 &+\lambda_{112}\lambda_{qrr}\chi_{2}(g_{(1\,q)})a_2[a_{(1\,q)},a_{(3\,r)}]_c.
U_{2}U_{1}P^{i}U_{1}^{\dagger}U_{2}^{\dagger}= - \int d^{3}x \left( \partial_{-}A_{j}\partial_{i}A_{j}+ i \sqrt{2}\psi_{+}^{\dagger}\partial_{i}\psi_{+}\right) \equiv P_{{f i n}}^{i},
\beta_{2}(k)=(\begin{matrix}2k\\k\end{matrix})\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})^{-1}(\begin{matrix}k\\j\end{matrix})\sum_{m=0}^{j}(\begin{matrix}j\\m\end{matrix})^{4}=1,4,36,424,5716,...
c_j \xi^j = \ \ \ \ \b_j \xi^{j-1}+ a_j \eta^{j-1}\j=1,\dots,\nu c_j \eta^j \ge |-a_j \xi^{j-1}+ b_j \eta^{j-1}| j=1,\dots,\nu b_\nu \eta^\nu \le a_\nu \xi^\nu ,
H = \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) \quad , \quad E = \left( \begin{array}{c c}{0}&{1}\\{0}&{0}\\ \end{array}\right) \quad , \quad F = \left( \begin{array}{c c}{0}&{0}\\{1}&{0}\\ \end{array}\right) .
\begin{aligned}\left | e^{it(\varphi(x_1 + y, x_2) - \varphi(x_1, x_2))}- 1 \right | \leq |t(\varphi(x_1 + y, x_2) - \varphi(x_1, x_2))| = |t| |y| \left | \int_0^1 \partial_{x_1}\varphi(x_1 + sy,x_2)\, ds \right | . \end{aligned}
\left| \mathcal{B}_{phys}\right\rangle =\int\limits_{-\infty}^{\infty}d\rho \sum_{m\in Z}\ \sum_{r\in Z_{+}}\,e^{-i\rho \Theta}\,\left|B_{p^{+}=m+\alpha ,\,p^{-}=-\frac{r}{m+\alpha},\,p_{2}=\rho}\right\rangle .
E\{C_{i_1}\cdots C_{i_k}\}= \rho^{-m}\frac{n!}{q_{\phi}(n)}\frac{q_{\phi}(n-m)}{(n-m)!}\prod_{j=1}^{k}\left(\frac{\phi m_{i_j}\rho^{i_j}}{i_j!}\right).
{\hat p}= (\, 0.15293342, \, 0.089760679, \, 0.021266977, \, 0.015778191, &0.12976986, \, 0.076165372, \, 0.020853199, \, 0.015471205, &0.13533793, \, 0.11789409, \, 0.018820142, \, 0.0207235, &0.083859917, \, 0.073051125, \, 0.01347576, \, 0.014838619 \, ).
\displaystyle\sum_{i=0}^{n}\dfrac{1}{\binom{n+1}{i}}\sum_{\mathrm{card}(I)=i}\left(\prod_{j\in I}x_{j}\right)\cdot f_{n+1-i}\left(\prod_{k\in \left\{1, \ldots, n+1\right\}\setminus I}x_{k}\right)=0 \; \left(x_1, \ldots, x_{n+1}\in R\right)
R_{x}^{\alpha}f(x)\ =-\frac{1}{2\Gamma(2-\alpha)\cos\alpha\pi/2}\times\left[ \frac{d^{2}}{dx^{2}}\int_{-\infty}^{x}(x-x^{\prime})^{-\alpha+1}f(x^{\prime})dx^{\prime}+\frac{d^{2}}{dx^{2}}\int_{x}^{\infty}(x^{\prime}-x)^{-\alpha+1}f(x^{\prime})dx^{\prime}\right] ,
U_n \subset \{\, x \in \P^1 \mid \lvert x - d_0 \rvert \leq \lvert b_2 \rvert \ \ \lvert x - d_{\nu(i)}\rvert \geq \lvert b_2 \rvert \,\}& = \{\,x \in \P^1 \mid \lvert x - a_i \rvert = \lvert b_2 \rvert \,\}& = \{\,x \in \P^1 \mid \lvert x - a_i \rvert = \lvert d_0 -a_i \rvert\, \}.
D_+F = \Big(-\frac{2l(2l-1)}{\Delta}- \frac{4i(2l-1)}{\Delta}\big(y\frac{\partial}{\partial\tau}+v\frac{\partial}{\partial z}+y'\frac{\partial}{\partial\tau'}\big)+4\big(4\frac{\partial^2}{\partial \tau \partial \tau'}-\frac{\partial^2}{\partial z^2}\big)\Big)F.
\left<T^\mu{}_\mu(x)\right>={a^{-N}\over 2^{N-1}\pi^{N/2}\Gamma \left(\frac N2\right)}\sum_{n=0}^{\frac N2-1}c^N_{2n+1}\left[ \frac{(-1)^{n+1}}{2(n+1)}4^{-n-1}-2H_n(0)\right].
\begin{aligned}-144 y^7 z^{16}y'' = 48A^2(z^{12}y^2 - z^{10}y^4) +2A\sqrt{6}\,(10 z^8 y - 11 z^6 y^3 - 8 z^4 y^5) &+ 12 z^4 - 4 z^2 y^2 -7y^4. \end{aligned}
det(\lambda I-U)=(\lambda-1)^{\lfloor\frac{N+4}{4}\rfloor}(\lambda+1)^{\lfloor\frac{N+2}{4}\rfloor}(\lambda+i)^{\lfloor\frac{N+1}{4}\rfloor}(\lambda-i)^{\lfloor\frac{N-1}{4}\rfloor}.
\tilde{f}_{i_k}^{n+1}(\pi^{s_{i_{k+1}}\cdots s_{i_l}\lambda}\otimes \pi^{w\mu}) = \tilde{f}_{i_k}^{n}(\pi^{s_{i_{k+1}}\cdots s_{i_l}\lambda}) \otimes \tilde{f}_{i_k}(\pi^{w\mu}) \not= 0.
\lim_{\lambda \rightarrow \infty}\bigg\{\frac{N[\mathfrak{e},\lambda V]}{N_{sc}[\mathfrak{e},\lambda V]}\bigg\}\ =\lim_{\lambda \rightarrow \infty}\bigg\{\frac{N[\mathfrak{e},\lambda V]}{1+N_{sc}[\mathfrak{e},\lambda|x|^{^{\alpha_{d}}}V]}\bigg\}\ =\ 1.
\big\| \big(\tfrac{\partial^2}{\partial x^2}\psi\big)( x, y ) \, ( v_1, v_2 ) \big\|_V= \left\| u_{0,3}( t, x ) \big( F( y ), v_1, v_2 \big) \right\|_V\leq \tfrac{c_{-\vartheta, 0, 0}}{( T - t )^{\vartheta}}\, \| F( y ) \|_{H_{-\vartheta}},
\int_{\vert x\vert \leq R}u^{1+p}(t,\cdot)=B_N R^{N}\int_{\vert x\vert \leq R}\frac{1}{B_N R^{N}}u^{1+p}(t,\cdot)\geq \frac{1}{B_N^{p}R^{Np}}m^{1+p}(t),
\begin{aligned}{\mathcal P}_{\leq0}[F_{\Lambda}]:= \{q^{-1}+2(16-r(\Lambda))\}\,{\bf e}_{0}+2^{g(\Lambda)+1}\{16-r(\Lambda)\}\,{\bf v}_{0}&+2^{g(\Lambda)}q^{-\frac{1}{4}}{\bf v}_{3}-2^{16-r(\Lambda)}q^{\frac{12-r(\Lambda)}{4}}\{1+(28-r(\Lambda))q^{2}\}\,{\bf e}_{{\bf 1}_{\Lambda}}.\end{aligned}
\mathbb P\Big(\inf_{i > m}\inf_{s \in [0,t]}\{Y_i(s)\}\le \Gamma\Big) \le \sum_{i>m}\mathbb P \Big(\inf_{s \in [0,t]}\{W_i(s)\}\le \Gamma - X_i(0)\Big) &= 2 \sum_{i>m}\mathbb P\Big( W(1) \le \frac{\Gamma-X_i(0)}{\sqrt t}\Big)=2 \sum_{i>m}\Big(\frac{X_i(0)- \Gamma}{\sqrt t}\Big)\,,
m_1 = F(1,1,1)F(-1,1,1)\equiv (\alpha_1+\alpha_2+\gamma_1+\gamma_2)^2 , m_2 = F(1,-1,1)F(-1,-1,1)\equiv (\alpha_1+\alpha_2-\gamma_1-\gamma_2)^2, m_3 =F(1,1,-1)F(-1,1,-1)\equiv (\alpha_1-\alpha_2+\gamma_1-\gamma_2)^2 , m_4 = F(1,-1,-1)F(-1,-1,-1)\equiv (\alpha_1-\alpha_2-\gamma_1+\gamma_2)^2 .
\|\xi^{m_0,N_0}\|_{\ell^p_\phi}= \sup_{m\in\mathbb{Z},N\in\omega}\frac{1}{\phi(2N+1)}\left(\frac{1}{|S_{m,N}|}\sum_{k\in S_{m,N}}|\xi^{m_0,N_0}_k|^p\right)^\frac1p \geq \frac{1}{\phi(2N_0+1)}\left(\frac{|S_{m_0,N_0}|}{|S_{m_0,N_0}|}\right)^\frac1p = \frac{1}{\phi(2N_0+1)}.
{\cal Z}={\cal N}\int D Q \exp \left\{\int d^{4}x{\cal L}_{\mathrm{e f f}}[ Q_{i}^{A}, Q_{i}^{3}, B_{\mu}^{3}] \right\}= \exp \left\{- \beta V U_{\mathrm{e f f}}( B , \beta , g ) \right\}
g_{t,\ell,-1,m}=\sum_{j=0}^{\ell+1}H(m-2\ell-tj+(t-1)) g_{t+1,\ell+1,j,m}&=\sum_{j=0}^{\ell+1}H(m-2(\ell+1)-tj)g_{t+1,\ell+1,j,m}&=g_{t,\ell+1,0,m}.