harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|I_{1}(x)| \leq\sum_{k=-\infty}^{-1}\int_{2^{k}R\leq |xy^{-1}|<2^{k+1}R}\frac{|xy^{-1}|^{\alpha-Q}|f(y)|}{(1+|xy^{-1}|)^{\gamma}}dy &\leq \sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{\alpha-Q}}{(1+2^{k}R)^{\gamma}}\int_{2^{k}R\leq |xy^{-1}|<2^{k+1}R}|f(y)|dy &\leq CMf(x)\sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{\alpha-Q+Q/p_{2}}(2^{k}R)^{Q/p_{2}^{'}}}{(1+2^{k}R)^{\gamma}},
\begin{array}{l}\eta_0^*\nabla f(x^*)+\left[\displaystyle \sum_{i=1}^p[y^I]^*_j\nabla^2 c_j(x^*)-A^TA-{\cal J}c_{\gamma}(x^*)^T{\cal J}c_{\gamma}(x^*)-{\cal J}c_{\beta}(x^*)^T{\rm Diag}([v_b]_{\beta}){\cal J}c_{\beta}(x^*)\right]\eta^*_1=0, \end{array}
\frac{s_{1}^{n_{1}}}{n_{1}}=\frac{p-1}{n_{1}(p-1)-\gamma_{1}}\rho_{1}^{\frac{n_{1}(p-1)-\gamma_{1}}{p-1}}\Rightarrow \rho_{1}=\left(\frac{n_{1}(p-1)-\gamma_{1}}{n_{1}(p-1)s_{1}^{-n_{1}}}\right)^{\frac{p-1}{n_{1}(p-1)-\gamma_{1}}},
R_{00}=\frac{{{t}^{2}}}{4{{r}^{4}}}{{e}^{t/r}}, \,R_{11}=-\frac{t}{{{r}^{4}}}( r+\frac{t}{4}), & R_{22}=\frac{t}{2r},\, R_{33}={{R}_{22}}\,{{\sin}^{2}}\theta, \, R = \frac{t^2}{2 r^4}.
\mathcal{V}_\rho((\Phi\star f)_b)(x) :=\sup_{\{t_k\}\downarrow0}\Big(\sum_{k\geq1}\Big|b(x)(\phi_{t_k}\ast f)(x)-\phi_{t_k}\ast (bf)(x) &\qquad\qquad-b(x)(\phi_{t_{k+1}}\ast f)(x)+\phi_{t_{k+1}}\ast (bf)(x)\Big|^\rho\Big)^{1/\rho}.
\Im G_{R}= - \,{\cal C}( \epsilon , h_{L , R}) \sinh{\left({\frac{p_{+}}{2 T_{L}}}+{\frac{p_{-}}{2 T_{R}}}\right)}\left| \Gamma \left( h_{L}+{\frac{i p_{+}}{2 \pi T_{L}}}\right) \Gamma \left( h_{R}+{\frac{i p_{-}}{2 \pi T_{R}}}\right) \right|^{2}\, ,
{\tilde{\rho}^{(1)}}=1+{\frac{1}{\pi}}\arctan\left\{{\frac{(1-s^{2\mu})\cos\mu\pi}{-(1+s^{2\mu})\sin\mu\pi+2s^\mu\cos\mu\beta}}\right\}&~~~~~~~~~~~+{\frac{1}{\pi}}\arctan\left\{{\frac{-(1-s^{2\mu})\cos{\mu\pi}}{(1+s^{2\mu})\sin\mu\pi+2s^{\mu}\cos\mu\beta}}\right\},
b^j = \partial_u g^{jk}\partial_k u - \partial_{(\nabla u)_j}F,{\tilde b}^j = \partial_{\bar u}g^{jk}\partial_k u - \partial_{(\nabla \bar u)_j}F, c =\partial_j (\partial_u g^{jk}) \partial_k u- \partial_{u}F ,{\tilde c}= \partial_j (\partial_{\bar u}g^{jk}) \partial_k u - \partial_{\bar u}F
{\left\|g(x_1+y_1,x_2+y_2)-g(x_1,x_2)\right\|}&\leq{\left\|g(x_1+y_1,x_2+y_2)-g((x_1,x_2)_{a})\right\|}&+{\left\|g((x_1,x_2)_{a})-g(x_1,x_2)\right\|}<\frac{1}{2}\epsilon+\frac{1}{2}\epsilon=\epsilon.
\mathcal{I}^{R}(\theta) =( E^\theta - E, P^\theta - P, R^{-1}(\mathcal C^\theta - \mathcal C^R), R^{-1}(\mathcal J^\theta- \mathcal J^R)) + \mathcal{I}_1^R(\theta)- (\tilde{\lambda}R^{-1}(\log R)^\frac{1}{2},0)
P(\tau,\mu|X)=NormalGamma(\frac{\lambda_{0}\mu_{0}+n\overline{x}}{\lambda_{0}+n},\lambda_{0}+n,\alpha_{0}+\frac{n}{2},\beta_{0}+\frac{1}{2}(ns+\frac{\lambda_{0}n(\overline{x}-\mu_{0})^{2}}{\lambda_{0}+n}))
2 \partial_{\sigma}\left( N g_{s}e^{\Phi_{D}}\rho^{\prime}\right) - \frac{\partial e^{\Phi_{D}}}{\partial \rho}\left( e^{2}- r^{2}e^{2}\omega^{2}+ ( r^{\prime})^{2}+ N g_{s}( \rho^{\prime})^{2}\right) = 0 ,
L_{B}(x^{n},b_{\mathcal{T}}) = \frac{1}{p^{U}_{\mathcal{T}}}\sum_{\pi_{\bar{Y}_{\mathcal{T}}|\bar{X}}\in \mathcal{P}_{n}(\mathcal{Y}_{\mathcal{T}}|\pi_{\bar{X}})}N_{\pi_{\bar{Y}_{\mathcal{T}}|\bar{X}}}(x^n,b_{\mathcal{T}})l_{\pi_{\bar{Y}_{\mathcal{T}}|\bar{X}}}(x^n),
[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0, & [x,\{y,z\}]+[y,\{z,x\}]+[z,\{x,y\}]=0, & [x,\{y,z\}]+\{y,[z,x]\}-\{z,[x,y]\}=0, & [x,[y,z]]+\{y,\{z,x\}\}-\{z,\{x,y\}\}=0.
\beta_{a-1}(I_{u_\ell}) =\sum_{w\in\mathcal{L}_t^f(u)}\binom{n-\min(w)-(d-1)t}{n-2-(d-1)t}-\sum_{\substack{w\in\mathcal{L}_t^f(v) w\ne v}}\binom{\max(w)-(d-1)t-1}{n-2-(d-1)t}&=(d-\ell)\binom{a}{a-1}+m\binom{a-1}{a-1}-(d-\ell)\binom{a}{a-1}=m>0.
V/ N_{f}= (Q_{0}^{2}+ \bar Q_{0}^{2}) \bigl [ Q_{0}\bar Q_{0}\bigr ]^{-2\gamma}+ \frac{g^2}{4\gamma}(Q_{0}^{2}- \bar Q_{0}^{2})^{2}- m_{Q}^{2}Q_{0}^{2}+ m_{\bar Q}^{2}\bar Q_{0}^{2}
\nabla^{2}V=\frac{1}{a^{2}(\zeta^{2}+\xi^{2})}\{\frac{\partial}{\partial\zeta}[(1+\zeta^{2})\frac{\partial V}{\partial\zeta}]+\frac{\partial}{\partial\xi}[(1-\xi^{2})\frac{\partial V}{\partial\xi}]\}+\frac{1}{a^{2}(1+\zeta^{2})(1-\xi^{2})}\frac{\partial^{2}V}{\partial\phi^{2}}
|B_{m+1,v}| \le|B_{m,v}|+|\Gamma_{m+1,v,0}| &\le\frac{ml}{d+m-k}+\frac{d-k}{d+m-k}\frac{l}{d+m+1-k}&=\frac{(m+1)l}{d+m+1-k}=\frac{(m+1)l}{s_{m+1}}.
{\cal L}^{( 1 )}= - \partial_{\mu}\overline{{\psi}}_{1}\gamma_{\mu \nu}\partial_{\nu}\psi_{1}- \partial_{\mu}\overline{{\psi}}_{2}\gamma_{\mu \nu}\partial_{\nu}\psi_{2}- m^{2}\overline{{\psi}}_{1}\psi_{1}+ m^{2}\overline{{\psi}}_{2}\psi_{2}\quad .
\big\langle \varphi_{v^{-1}}\varphi_{v^{-1}}^{v^{-1}}, x \big\rangle = \mu^l (v )^{-2}\mu^l\big(vg^{-1}x'\big)\mu^l\big(g^{-1}x''\big) = \mu^l (v )^{-2}\big\langle \mu^l (v? )\mu^l, g^{-1}x \big\rangle &= \mu^l(v)^{-1}\mu^l(g^{-1}x) = \varphi_{v^{-1}}^{v^{-1}}(x).
K_{0}=\frac{-2}{\rho_{0}^{2}v_{0}}(\omega-\frac{2w_{0}}{\rho_{0}v_{0}})=J_{\theta}(1+J_{\varphi}^{2})\cos\varphi_{0}+(J_{\varphi}J_{\theta}^{\prime}-J_{\theta}J_{\varphi}^{\prime})\sin\varphi_{0},
\begin{array}{rrcll}\min \multicolumn{4}{l}{\max_{c\in U}c^\top x}begin{align*}1.5ex] \textnormal{s.t.}\sum_{(u,v)\in E}x_{u,v}= |N|-1 begin{align*}1ex] \sum_{u,v\in X}x_{u,v}\le |X|-1 \forall \,\emptyset\neq X\subseteq N begin{align*}1ex] x \in \{0,1\}^{E}\end{array}
\sum \limits_{k=0}^{N-1}\partial_\theta c(\theta_k) \cdot p(x_k) \cdot \prod \limits_{j=k+1}^{N}c(\theta_j) \cdot p'(x_j) =\sum \limits_{k=0}^{N-1}\partial_\theta c(\theta_k) \cdot (1-x_k)x_k \cdot \prod \limits_{j=k+1}^{N}c(\theta_j) \cdot p'(x_j).
\oint_{C}\frac{w^{-1}dw}{2\pi\sqrt{-1}}\frac{(1-r\zeta^2)(w^2-q^6)(w-rq^4)}{(w-q^2\zeta^2)(w-q^4\zeta^2)(w-q^4\zeta^{-1})}e^{R(w)+R(q^6 w^{-1})}|0\rangle^{(0)}= (\zeta \leftrightarrow \zeta^{-1}) |0\rangle^{(0)}.
\{f_{i,j}(\beta^{r^t})\}_{j\in[l]}=\{\beta^a:a_i=0\}\bigcup &\Big(\bigcup_{u=0}^{r-2}\{\beta^a:a_i=1,a_{i-1}=\dots=a_{t+2}=0,a_{t+1}=u\}\Big).
2^{-(E-\frac{n}{2})k_-}2^{-(F+\frac{n}{2}-\frac{n}{a})k_+}2^{-(D-\frac{n}{a})l}&\quad=2^{-(E-\frac{n}{2})k_-}2^{-[F+\frac{n}{2}-J-(\frac{n}{a}-J)]k_+}2^{-[D-J-(\frac{n}{a}-J)]l},
\frac{\int_{\Omega\setminus B_{\frac{s_0\rho_0}{2}}(x_0)}|r|^2}{\int_\Omega |r|^2}= 1 - \frac{\int_{B_{\frac{s_0\rho_0}{2}}(x_0)}|r|^2}{\int_\Omega |r|^2}\leq 1- \frac{\int_{B_{\frac{s_0\rho_0}{2}}(x_0)}|r|^2}{\int_{B_{s_0\rho_0}(x_0)}|r|^2}=\frac{15}{16},
g'(r)<0 \quad\mbox{and}w_1'(r)=2r g^{2\tilde k-1}(r)(g(r)+\tilde kr g'(r))>0\quad\forall r\in(0,R_0) \Rightarrow\quad r^2g^{2\tilde k}(r)= w_1(r)> w_1(R_0/2)>0\qquad\qquad\qquad\forall r\in(R_0/2,R_0).
S_{q,N}(R) =A_{q,N}'(r_{q,N}+R)-A_{q,N}'(r_{q,N})-\alpha_{q,N}R &=N A_{q/N,1}''(Nr_{q,N}+NR)-N A_{q/N,1}''(Nr_{q,N})-N\alpha_{q/N,1}R &=N A_{q/N,1}''(r_{q/N,1}+NR)-N A_{q/N,1}''(r_{q/N,1})-N\alpha_{q/N,1}R &=N S_{q/N,1}(NR).
\sum\limits_{i=1}^{l}m_{i}B_{\sigma ir}=\sum\limits_{i=1}^{l}\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{i}m_{j}(Q_{jr}-\sigma(Q_{ir}\odot Q_{jr})Q_{ir})}{(\sigma-\sigma(Q_{ir}\odot Q_{jr})^{2})^{\frac{3}{2}}}.
K_\nu (z)=\frac{2^\nu \Gamma (\nu +1/2)}{\sqrt{\pi}z^\nu}\int_{0}^{\infty}{\frac{\cos zt\, dt}{(t^2+1)^{\nu +1/2}}},\quad \sum_{k=-\infty}^{+\infty}e^{ikz}=2\pi \sum_{n=-\infty}^{+\infty}\delta (z-2\pi n),
k_{\nu,\lambda,\sigma_k^2}(u,u') = \sigma_k^2 \, \frac{1}{\Gamma(\nu) 2^{\nu-1}}\left(\sqrt{2\nu}\frac{\|u-u'\|}{\lambda}\right)^\nu B_\nu\left(\sqrt{2\nu}\frac{\|u-u'\|}{\lambda}\right) ,
\hat{H}_{0}= \hbar \omega \left[ a^{\dagger}a + \frac{1}{2}\right] \ \ , \ \ \hat{H}_{0}| n > = E_{n}| n > \ \ , \ \E_{n}= \hbar \omega ( n + \frac{1}{2}) \ , \n = 0 , 1 , 2 , \cdots \ .
\int_{0}^{t}\|\partial{_t^k}(\frac{f}{\partial_z\varphi})\|{_{H^{m-k}}^2}d\tau\leq\Lambda(\frac{1}{c_0},\|f\|_{\infty,t}+|h|_{\mathcal{W}{^{1,\infty}_t}})\int_{0}^{t}(\sum_{k=0}^{m}\|\partial{_t^k}f\|{_{H^{m-k}}^2}+|\partial{_t^k}h|{_{m-k+\frac{1}{2}}^2})d\tau.
\frac{1}{2}\delta_{j_1j_2}G_{i_1i_2i_3}^{j_1j_2\eta}\delta^{i_1i_2}\tilde\gamma_{\eta k}+\delta^{i_1i_2}G_{i_1i_2i_3}^{j_1j_2j_3}\gamma_{j_1j_2}\delta_{j_3k}= -\frac{1}{2}\int_0^\infty \frac{d}{dy}\left[\sum_{\alpha,\eta}(e^{-y\tilde\gamma})_{\alpha\eta}(e^{-y\tilde\gamma})_{\alpha\eta}(e^{-y\tilde\gamma})_{i_3 k}\right]dy=\frac{n}{2}\delta_{i_3 k},
c_1 = cov\{\hat{m}_{4y}(\lambda,\lambda,0),\hat{m}_{4y}(\beta,\beta,0)\}\approx \frac{1}{N}\sum_{\tau = -K_N}^{K_N}\big[ \hat{m}_{8y}(\lambda,\lambda,0,\tau,\tau+\beta,\tau+\beta,\tau) \ - \ \hat{m}_{4y}(\lambda,\lambda,0)\hat{m}_{4y}(\beta,\beta,0)\big]
{\delta_{\rho}}{\mathcal{F}_{D}^{0}}={\frac{Q^{2}}{8 \pi}}\int{d^{2}}\xi \sqrt{\hat{g}}\hat{R}\rho +{\frac{Q^{2}}{8 \pi}}\oint{d}\hat{s}( \xi ) \int{d^{2}}\xi^{\prime}\rho ( \xi ){\partial_{\hat{n}}}{\hat{G}_{D}}( \xi , \xi^{\prime}) \sqrt{\hat{g}( \xi^{\prime})}\hat{R}( \xi^{\prime}) .
B_{\varphi}(x,D,\tau) =B^k_{\varphi}(x, D',\tau)+B^{k-1}_{\varphi}(x,D',\tau)(D_d+i\tau\partial_d\varphi) &= \hat{B}^k_{\varphi}(x,D',\tau)+ B^{k-1}_{\varphi}(x,D',\tau)D_d,
F_{1,0}(x,y) =0, \F_{1,1}(x,y) = \frac{2}{3}\pi^2 \Bigl( x^4 + y^4 \Bigr), &\widetilde{F}_{1,0}(x,y) = 0, \ \widetilde{F}_{1,1}(x,y) = \frac{2}{3}\pi^2 \Bigl( x^2 + y^2 \Bigr)^2.
\| \partial_x f^n_t - \partial_x f^{n-1}_t \|_{L^1}\le \int_0^t\!\mathrm{d}s\, \|\left[ G(M^{n-1}(s) - G(M^{n-2}(s) \right] \partial_{xv}f^{n-1}_s \|_{L^1}+ \int_0^t\!\mathrm{d}s\, \| \partial_x G(M^{n-1}(s)) \partial_v f^n_s - \partial_x G(M^{n-2}(s)) \partial_v f^{n-1}_s \|_{L^1}\,.
\int \frac{x p_n(x;-1;q)}{(q^2-x^2)}\left(\,_2\phi_1 ( \frac{x^2}{q^2}, q^{2};x^2;q^2,q)-1 \right)d_qx = & (q^{-n}-1)[n+1]_q \left( p_n(\frac{x}{q};-1;q)+ \frac{x}{q}\,_2\phi_1 ( \frac{x^2}{q^2}, q^{2};x^2;q^2,q)\,_3\phi_2 ( q^{-n}, q^{n+1},x;q;-q;q,q) \right).
\#_{R_{j,k},N_2,n}\lesssim \#\{\lambda_1\in \Lambda^+\cap R_{j,k}\}\cdot\max\#\{\lambda_2\in\Lambda^+:\ |\lambda_2|_\rho\sim N_2,\ |\lambda_2|_\rho^2=n-|\lambda_1|_\rho^2\}&\lesssim MN_2^{r-1}\cdot N_2^{r-2}.
\theta = \left( \begin{array}{c c c c}{{0}}&{{- i e_{1}}}&{{- i e_{2}}}&{{- i e_{3}}}\\{{i e_{1}}}&{{0}}&{{b_{3}}}&{{- b_{2}}}\\{{i e_{2}}}&{{- b_{3}}}&{{0}}&{{b_{1}}}\\{{i e_{3}}}&{{b_{2}}}&{{- b_{1}}}&{{0}}\\ \end{array}\right) ,
\frac{J_{2 i t}(x) - J_{-2 i t}(x)}{\cosh(\pi t)}= - \frac{2 i}{\pi}\tanh(\pi t) \int_{-\infty}^\infty \cos(x \cosh \zeta) e\Bigl( \frac{t \zeta}{\pi}\Bigr) \, d \zeta,
\lim_{n \rightarrow +\infty}\left(\frac{\lambda_{j',n}}{\lambda_{j,n}}+ \frac{\lambda_{j,n}}{\lambda_{j',n}}+ \frac{|x_{j,n}- x_{j',n}|}{\lambda_{j,n}}+ \frac{|t_{j,n}- t_{j',n}|}{\lambda_{j,n}}\right) = + \infty
0=\sum\limits_{j=1, j\neq i}^{n}\frac{m_{j}\sin{(\alpha_{ip}-\alpha_{jp})}}{(1-\cos{(\alpha_{ip}-\alpha_{jp})})^{\frac{3}2}(2-\sigma r_{p}^{2}(1-\cos{(\alpha_{ip}-\alpha_{jp})}))^{\frac{3}{2}}},
\partial_t \rho =-\div_q\left[\frac{c\, p}{\sqrt{m^2c^2+|p|^2}}\rho\right]+\div_p\left[\left(\nabla_q V+\gamma\frac{p}{m}\right)\rho \right] &\qquad+\gamma\theta\div_p\left[\frac{m\,c}{\sqrt{m^2c^2+|p|^2}}\left(I+\frac{p\otimes p}{m^2c^2}\right)\nabla_p \rho\right].
\partial_{\mu}^{z}V_{i}^{\mu}( z , u , v ) = \frac{i g_{1}}{2}\left[ \delta^{4}\left( z - v \right) - \delta^{4}( z - u ) \right] \left( Y_{R}^{i}L + Y_{L}^{i}R \right) \Sigma_{1}^{i}( u - v ) \, ,
H_{n m}^{\{{\bf R}\}}( 0 , 0 ) = \frac{m ! n ! ( - 1 )^{\frac{m + n}2}}{2^{\frac{| m - n |}2}\left( \frac{m + n}2 \right) !}\left[ R_{1 1}^{n}R_{2 2}^{m}\left( r^{2}- 1 \right)^{\mu_{m n}}\right]^{\frac 1 2}P_{\mu_{m n}}^{( \frac{| m - n |}2 , \frac{| m - n |}2 )}\left( \frac r{\sqrt{r^{2}- 1}}\right) ,
\begin{array}{rl}<0|\cdots T(\pi_1\pi_2\pi_5\star e^{\pm ip\cdot x_5}\star\pi_5)\cdots|0> = \displaystyle \sum_{\lambda_1\lambda_2}\tau_{15}^{\lambda_1}\tau_{25}^{\lambda_2}(D_{15}^{\lambda_1}\star e^{\pm ip_+\cdot x_5}\star D_{25}^{\lambda_2}+(1\leftrightarrow 2))<0|\cdots|0>.\end{array}
D(L) =\{f\in B_2 \, : \, \lim_{t\to 0}\frac 1 t(S_t f-f)\}&=\{f\in B_2 \, : \, \lim_{t\to 0}\frac 1 t(e^{-tL_1}U^{-1}f-U^{-1}f)\}&=\{f\in B_2 \, : \, U^{-1}f\in D(L_1)\}&=U D(L_1) &=D(L_2)
\lambda(\hat{x},x^{\star}) = \left(\frac{z}{b}\right)\frac{l_2^{-1}z^{-l_2}-l_1^{-1}z^{-l_1}}{l_1^{-1}z^{-l_1}+l_2^{-1}z^{-l_2}}-1 \kappa(\hat{x},x^{\star}) =z\left[l_1^{-1}+l_2^{-1}-1\right]-z\frac{l_1^{-1}z^{-l_1}-l_2^{-1}z^{-l_2}}{l_1^{-1}z^{-l_1}+l_2^{-1}z^{-l_2}}\left[l_1^{-1}-l_2^{-1}+\ln{\hat{x}}-\ln{x^\star}\right],
\kappa(g)=\kappa(b_0)\mathfrak{s}(b_*)(z,z)_2^{k-1}\mathfrak{h}(h)\mathfrak{h}(h_0)=(z,z)_2^{k-1}(z,\det c_{h_0})_2\mathfrak{s}(b_0b_*)\mathfrak{h}(hh_0),
\left[ \mu \frac{\partial}{\partial \mu}~+~ \sum_{\kappa = 0}^\infty \tilde{\beta}_{(\kappa)}(g_\alpha) \frac{\partial}{\partial g_\kappa}~+~ \tilde{\gamma}_m(g_\alpha) m \frac{\partial}{\partial m}~+~ \frac{n}{2}\tilde{\gamma}(g_\alpha) \right] G^{(n)}(p,m,\mu,g_\beta) ~=~ 0
\begin{aligned}T_{f}\left(A^{{'E}_{2}}_{2}, h^{{'E}_{2}}\right)-T_{f}\left(A^{{'E}_{2}}_{2}, h^{{'E}_{2}}_{L^{2}}\right)=\widetilde{f}\left(\nabla^{{'E}_{2}},h^{{'E}_{2}}_{L^{2}}, h^{{'E}_{2}}\right).\end{aligned}
\lim_{m \rightarrow 0^{+}}\lim_{\beta \rightarrow \infty}\langle{\bar{\psi}}\psi{\rangle}_{\beta}= 0 \lim_{m \rightarrow 0^{-}}\lim_{\beta \rightarrow \infty}\langle{\bar{\psi}}\psi{\rangle}_{\beta}= \frac{e B}{2 \pi},
\lim_{r\to0+}\mathop{\sup_{s\le r, \theta\in[-\infty,+\infty]}}_{w\in\{0,1\}, x,x+y\in I}\frac{|X(x+y(s,\theta, w))-X(x)|}{s\sqrt{\log(1+s^{-1})}}=C_{17}\qquad\mbox{a.s.},
\mathbb{E}\bigl[H_Y^{(N)}\bigr]= \begin{pmatrix}\frac{4}{\pi}begin{align*}3pt]\frac{4}{\pi}\end{pmatrix}C\bigl[H_Y^{(N)}\bigr]=\biggl(1+\frac{2}{\pi}\biggr) \begin{pmatrix}1 1 1 1 \end{pmatrix}.
\int_{X}\int_{X}\left [\int_{(0,\infty)}\frac{e^{-\gamma(x,y)r}- e_{\ell}(r)\omega_{\ell,\infty}(\gamma(x,y)r)}{r^{\ell}}d\lambda(r)\right ]d\mu(x)d\mu(y) = \int_{(0,\infty)}\frac{1}{r^{\ell}}\left [\int_{X}\int_{X}e^{-\gamma(x,y)r}d\mu(x)d\mu(y)\right ]d\lambda(r) \geq 0,
\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)+\frac{\kappa_1}{x+\lambda_1}}-U(x)\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1+\lambda_1}}-V(x)\right) = \max_{\lambda}\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)+\frac{\kappa_1}{x+\lambda}}-U(x)\frac{c_0}{\Phi_0(\nu;x+1)+\frac{\kappa_1}{x+1+\lambda}}-V(x)\right).
f=x_1x_3^p+\sum_{i=0}^q\lambda_ix_2^ix_3^{w_1+p-iw_2}=\Bigl(x_1+\sum_{i=0}^{q-1}\lambda_ix_2^ix_3^{w_1-iw_2}\Bigr)x_3^p+\lambda_qx_2^qx_3^{w_1+p-qw_2}
B(e_{xy},e_{uv}) =B(e_x,e_{ux})=-B(e_{ux},e_x)=-B(e_ue_{ux}e_x,e_x)=-B(e_u,e_ue_{ux}e_x) &=-e_uB(e_u,e_{ux})e_x=-B(e_u,e_{ux})(u,x)e_{ux}
\mathbb{C}_t^{\pi}(r_t,r_{t+1},\ldots,r_n;y_{t-J}^{t-1})\triangleq{\bf E}^{\pi}\bigg\{\sum_{i=t}^n\log\Big(\frac{r_i(X_i|y_{i-M}^{i-1},Y_i)}{{\pi}_{i}(X_i|y_{i-J}^{i-1})}\Big)\Big{|}Y_{t-J}^{t-1}=y_{t-J}^{t-1}\bigg\},~t\in\mathbb{N}_0^n
\delta(w_{(j,r_s)})=\sum_{}^{}\lambda_{(i,t)}[w_{(i,r_p)},w_{(t,r_q)}],\,\,\,\,\,\,\,\,\,\,\,\,\, r_p\leq r_q\,\,,\,\,r_p+r_q=r_s-1,
\mathcal{J}:=\begin{cases}(\{1,2,\ldots,n\}\setminus \mathcal{I})\cup{\{n_2\}}\textrm{if $n_2-n_1$ is an even number}\,, \{1,2,\ldots,n\}\setminus \mathcal{I}\textrm{if $n_2-n_1$ is an odd number}\,.\end{cases}
\|S_n^{-1/2}g\|_n^2 = \langle S_n^{-1/2}g,S_n^{-1/2}g\rangle_n = \langle g,S_n^{-1}g\rangle_n = \langle g,g\rangle=\|g\|^2\leq a_n^{-1}\|g\|_n^2 \, .
\frac{1}{8/P+\frac{1}{2}(x^2+l^2-n^2-1)}=\begin{array}{ccccccccccc}1 1^2-n^2 1^2-l^2 3^2-n^2 3^2-l^2 \cline{1-1}\cline{3-3}\cline{5-5}\cline{7-7}\cline{9-9}\cline{11-11}x^2-1 + 1 + x^2-1 + 1 + x^2-1 +\cdots\end{array}.
\widehat{A}\cdot \widehat{x}=(-u_1 v_1 + u_3 v_3)\cdot x_1 + (u_1v_2+u_2v_3)\cdot x_1x_2 + (-u_2v_1-u_3v_2)\cdot x_2 + (u_3v_1) 1\in \mathcal{A}.
\gamma(x)t\gamma(x^{-1})-xt=([\nu]^x-1)-xt \in t^2k[[t]], \gamma(x)t^n\gamma(x^{-1})-(xt)^n \in t^{n+1}k[[t]], \gamma(x)t^{np^{rm}}\gamma(x^{-1})-(xt)^{np^{rm}}=(\gamma(x)t^{n}\gamma(x^{-1})-(xt)^{n})^{p^{rm}}\in t^{(n+1)p^{rm}}k[[t]].
\frac{(x_{\infty}^{+}-x_{0}^{-})(x_{0}^{+}-x_{\infty}^{-})}{(x_{\infty}^{+}-x_{\infty}^{-})(x_{0}^{+}-x_{0}^{-})}=\frac{(s^{2}-st+\frac{3-\sqrt{5}}{2}t^{2})(s^{2}+\frac{3+\sqrt{5}}{2}st+\frac{3+\sqrt{5}}{2}t^{2})}{\sqrt{5}st(s^{2}-st-t^{2})}
\frac{1}{n}\sum_{i=b+1}^{n-1}x_{i-k_1}x_{i-k_2}\alpha_{c_1}(\frac{i}{n}) \alpha_{c_2}(\frac{i}{n})=\int_0^1 \widetilde{\gamma}(t, k_2-k_1, c_1, c_2)+O(n^{-\alpha_3}),
\|\theta^n\cdot\tilde Q^n\|_T \le3\|\theta^n\cdot Y^n\|_T\le4\theta_{\max}\Big(\|y\|_1T+\sum_{j=1}^d(\|\tilde A^n_j\|_T+\|\tilde S^n_j\|_T)\Big) &\le4\theta_{\max}\Big(\|y\|_1T+d(M+1)\Big)=L,
\Gamma=a+a_{\mu}\gamma^{\mu}+\frac{1}{2}a_{\mu\nu}\gamma^{\mu}\gamma^{\nu}+\frac{1}{3!}a_{\mu\nu\rho}\gamma^{\mu}\gamma^{\rho}\gamma^{\sigma}+\frac{1}{4!}a_{\mu\nu\rho\sigma}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma},
P_{ed, ps2}=\frac{2\pi\lambda_m\int\limits_{0}^{\infty}e^{-\pi\lambda_m r^2}e^{-\pi\lambda_m r^2\rho(T,\delta)}\exp\{-\pi (\frac{P_p T}{P_m})^{\frac{2}{\alpha}}r^2C(L_m(r),\alpha)\lambda_p \}(1-e^{-\pi\lambda_m r^2\rho( S_{tm},1)})r}{1-\frac{1}{1+\rho( S_{tm},1)}}
N_{4(r+2)}- N_{4(r+1)}+N_{4r}-1 &=\left(5N_{4(r+1)}-2N_{4r}+N_{4(r-1)}\right) -N_{4(r+1)}+N_{4r}-1 &=4N_{4(r+1)}-N_{4r}+N_{4(r-1)}-1 &=3N_{4(r+1)}+N_{4(r+1)}-N_{4r}+N_{4(r-1)}-1 &=3N_{4(r+1)}+3\sum_{k=0}^{r}N_{4k}=3\sum_{k=0}^{r+1}N_{4k}.
=(-1)^m\,\left(\frac{\omega}{\pi\hbar}\right)^{1/2}\,\left(\frac{1}{2\pi\hbar}\right)^{1/2}\,\left(\frac{m!}{(m+|\ell|)!}\right)^{1/2}\,e^{i\ell\theta}\,e^{ipz/\hbar}\,u^{|\ell|}\,e^{-u^2/2}\,L^{|\ell|}_m(u^2)\ \ \ ,
\tilde{t}_{ij}(a): =t_{ij}(a)\oplus0, \tilde{u}_{kl}(a): =u_{kl}(a)\oplus0, \tilde{v}_{kl}(a): =v_{kl}(a)\oplus0, \tilde{w}_{kl}(a): =w_{kl}(a)\oplus0, \tilde{f}_{i'k'}(a): =f_{i'k'}(a)\oplus0, \tilde{g}_{k'i'}(a): =g_{k'i'}(a)\oplus0,
g_{11}=\epsilon_1 f_1(x_1,x_2)\,e^{\,-2\,y_1\int\sqrt{|g_{00}|}\,{\rm d}\,t}, g_{22}=\epsilon_2 f_2(x_1,x_2)\,e^{\,-2\,y_2\int\sqrt{|g_{00}|}\,{\rm d}\,t}.
(u,v)\longmapsto \left(\frac{b(\frac{u}{v}-1)}{\frac{u}{v}(b-\frac{u}{v}a)},\frac{\frac{u}{v}a(\frac{u}{v}-1)}{b-\frac{u}{v}a}\right)=\left(\frac{bv(u-v)}{u(bv-ua)},\frac{ua(u-v)}{v(bv-ua)}\right).
\begin{aligned}w_{11}(x,z) = \exp\Bigl(\frac{\kappa iz}{\rho_3}x^{\rho_3}\Bigr)M\Bigl(a_-,\,b_-,\,-\frac{2\kappa iz}{\rho_3}x^{\rho_3}\Bigr), begin{align*}1ex]w_{21}(x,z) = -\frac{\kappa_2}{\rho_2}zx^{\rho_2}\exp\Bigl(\frac{\kappa iz}{\rho_3}x^{\rho_3}\Bigr)M\Bigl(a_+,\,b_+,\,-\frac{2\kappa iz}{\rho_3}x^{\rho_3}\Bigr),\end{aligned}
T_x = \frac{z^2+1}{z^4}\left(\frac{z^4}{z^2+1}, x^{-1}\left(x^4+\frac{z^4}{z^2+1}x^2+1\right), x^4+1, x^4+1, x \left(x^4+\frac{z^4}{z^2+1}x^2+1\right), \frac{z^4}{z^2+1}x^4\right),
\frac{g^{2}N}{4 \pi^{2}}\mapsto \frac{g^{2}N}{4 \pi^{2}}- \frac{g^{4}N^{2}\Gamma^{2}( \omega - 1 )}{1 2 8 \, \pi^{2 \omega}( 2 - \omega ) ( 2 \omega - 3 )}.
\lim_{\zeta,\xi\to\chi}(\zeta/\xi)^2 (1+\xi/\zeta)^{4}D_{\zeta}R_{p'}(\zeta)X(\xi)=p'\lim_{\zeta,\xi\to\chi}(\zeta/\xi)^2 (1+\xi/\zeta)^{4}R_{p'}(\zeta)D_{\xi}X(\xi).
c_f^{(g),+,(m)}(x,x') =\Big(\prod_{k=0}^{m-1}f(g^kx)^{-1}\Big)\Big(\prod_{k=0}^{m-1}f(g^kx')^{-1}\Big)^{-1}, c_f^{(g),-,(m)}(x,x') =\Big(\prod_{k=1}^{m-1}f(g^{-k}x)\Big)\Big(\prod_{k=1}^{m-1}f(g^{-k}x')\Big)^{-1}.
\int_0^{\infty}d\omega_a^R \int_0^{\infty}d \omega_b^R =\int_0^{\infty}d\omega_-^R \int_{\frac{\omega_-^R}{2}}^{\infty}d \omega_+^R+\int_{-\infty}^0 d\omega_-^R \int_{\frac{-\omega_-^R}{2}}^{\infty}d\omega_+^R\equiv \int d{\tilde \omega}_-^R \int d{\tilde \omega}_+^R.
X_1(t)=\alpha(t)(e_1(t)+e_5(t))+\beta(t)(e_2(t)+e_4(t)), &X_2(t)=\tfrac{1}{\sqrt{\sigma(t)}}(\tfrac{\beta(t)}{\sqrt{3}}(e_1(t)-e_5(t))-\sqrt{3}\alpha(t)(e_2(t)-e_4(t)), &Z_1(t)=\tfrac{1}{\sqrt{\sigma(t)}}(\sqrt{3}\alpha(t)(e_1(t)-e_5(t))+\tfrac{\beta(t)}{\sqrt{3}}(e_2(t)-e_4(t)), &Z_2(t)=\beta(t)(e_1(t)+e_5(t))-\alpha(t)(e_2(t)+e_4(t))
(b-ma)^2- (b-ma)^2 \bigg(\frac{\beta-m2^{1-s}g(a)}{g(b)-mg(a)}\bigg)^{\frac{1}{s}}- (b-ma)^2 \bigg(\frac{\beta-m2^{1-s}f(a)}{f(b)-mf(a)}\bigg)^{\frac{1}{s}}+(b-ma)^2 \bigg(\frac{\beta-m2^{1-s}f(a)}{f(b)-mf(a)}\bigg)^{\frac{1}{s}}.\bigg(\frac{\beta-m2^{1-s}g(a)}{g(b)-mg(a)}\bigg)^{\frac{1}{s}}=\beta.
\sigma(x_{[5]}) = \sum_{i \in [q], j \in [t]}\lambda_{ij}\Big(\sum_{\{a,b,c\}\in \binom{[5]}{3}}\tau_{i}(x_a, x_b, x_c) \tilde{\gamma}_i(x_{[5] \setminus \{a,b,c\}})\Big).
\left|\frac{b_{t,i}^{2-\epsilon}}{a_{t,i}}\right|=\left|\frac{b_{t,i}^x}{a_{t,i}}\right|\cdot |b_{t,i}^{2-x-\epsilon}| \le D^{(t-i)(2-x-\epsilon)}\cdot D^{-2^{j+1}}.
\tilde{R}= \frac{\tilde{a}M^{2}( \gamma - 1 )^{2}e^{\phi_{0}}}{\rho^{4}}\left( 1 - \frac{\rho_{0}}{\rho}\right)^{- \frac{1 + 3 k^{2}- 2 l}{1 + k^{2}}}.
\limsup_{n \to + \infty , \mathrm{t y p e 1}}\left| \frac{1 - q^{2 n}}{1 + q^{2 n}}\right|^{1 / ( 2 n )}\ \le \ \limsup_{n \to + \infty , \mathrm{t y p e 1}}1 \ = \ 1 \ .
E_{n}^{\prime}\approx E_{n}+\frac{1}{2}\int_{-1}^{1}\psi_{n}(U^{\prime}-U)\psi_{n}^{*}dx+\frac{1}{4}\sum_{k\ne n}\frac{|\int_{-1}^{1}\psi_{n}(U^{\prime}-U)\psi_{k}^{*}dx|^{2}}{E_{n}-E_{k}}
\begin{array}{cl}\langle s_{z_o, 0}, s_{z_o, 0}\rangle = \frac 12, \langle s_{z_o,0}, s_{z_o, i}\rangle = 0, \,\, (i = 1, \ldots, 4) \langle s_{z_o, i}, s_{z_o, i}\rangle = \frac 34, \langle s_{z_o, i}, s_{z_o, j}\rangle = -\frac 14, \, (i\neq j)\,\, (i, j = 1, \ldots, 4) \end{array}
\Delta^{a}= ( - 1 )^{\epsilon_{A}}\frac{\delta_{L}}{\delta \phi^{A}}\frac{\delta}{\delta \phi_{A a}^{*}}, \qquad \Delta_{\alpha}= ( - 1 )^{\epsilon_{A}}( \sigma_{\alpha})_{B}^{\! A}\frac{\delta_{L}}{\delta \phi^{A}}\frac{\delta}{\delta \eta_{B}}.
D_{\alpha}(\theta,\epsilon) = -\i\nu \int_{\theta}^{\theta+2\epsilon}d \phi\left(1- \frac{\left|\theta + \epsilon - \phi \right|}{\epsilon}\right) P'_{\alpha}(\phi) = -\i\nu \int_{-1}^{1}d \mu \mbox{sgn}(\mu) P_{\alpha}\left(\theta + \epsilon (1+ \mu) \right)
\rho^n_{n, ne_0 + 1}(V) \ = \n^2(g-1) + 1 - n(n - rne_0 - r + rn(g-1)) \ = \n^2 g - n^2 + 1 - n^2 ( 1 - re_0 + r(g-1) ) + nr \ = \n^2 \cdot \hbox{\Large (}g - (1 - re_0 + r(g-1) ) \hbox{\Large )}+ nr - n^2 + 1 \ = \n^2 \cdot \rho^1_{1,e_0}(V) + n(r-n) + 1 \ = \n(r-n) + 1 .
\sharp E(a,b,\varepsilon,B,r)=\frac{3}{2\pi^2}\frac{\Phi(b)\Phi(a)\Psi(b-a)}{ba^{\frac{1}{2}}}\varepsilon B^{\frac{1}{2}-\frac{1}{r}}+O_{\tau_i,\varepsilon,\delta}\left(b^{\frac{23}{8}+\delta}B^{\frac{3}{4}(\frac{1}{2}-\frac{1}{r})}\log B \right)
\sum_{n\ge0}R_n(x)y^n = \sum_{m\ge 0}A_m(x)y^{2m}+ \sum_{m\ge 0}B_m(x)y^{2m+1}&= \sum_{m\ge 0}A_m(x)y^{2m}+ y\sum_{m\ge 0}B_m(x)y^{2m}&= \frac{1-xy^4-x^2y^4+x^2y^6}{1-(1+x+x^2)y^2+x^2y^4}+ y\frac{1+x-xy^2-x^2y^2}{1-(1+x+x^2)y^2+x^2y^4}&= \frac{1+(1+x)y-(x+x^2)y^3-(x +x^2)y^4+x^2y^6}{1-(1+x+x^2)y^2+x^2y^4}.
e \left( b \right) \equiv \sum_{a \in q \left( b \right) - \left\{b \right\}}\left[ m \left( a \right) - 1 \right] = \left[ \sum_{a \in q \left( b \right) - \left\{b \right\}}m \left( a \right) \right] - o \left( b \right) + 1
\overline\Delta_m(a_1 a_2 a_3) = a_2 a_3 \otimes a_1 + a_1 a_2 \otimes a_3 + a_1 a_3 \otimes a_2 &+ a_1 \otimes a_2a_3 + a_2 \otimes a_1 | a_3 + a_3 \otimes a_1 a_2.
F_{6}\big(3^{-2/3}t\big) = \frac{(q_2-1)}{2q_2}\exp\left(\frac{1}{3}\int_{t}^{\infty}\omega(s)ds -\frac{2}{3}\int_{t}^{\infty}\frac{u_s(s)}{u(s)}\frac{1+q_2(s)}{q_2(s)}ds\right).
\psi_{1}= \left( \begin{array}{c}{| 0 \rangle \! \rangle \langle \! \langle 0 |}\\{| 0 \rangle \! \rangle \langle \! \langle 1 |}\\{\vdots}\\{| 0 \rangle \! \rangle \langle \! \langle k - 1 |}\\ \end{array}\right) , \quad \psi_{2}= \left( \begin{array}{c}{0}\\{0}\\{\vdots}\\{0}\\ \end{array}\right) , \quad \xi = S_{k},
V=\left( v_{\alpha}^{\beta}\right)_{1\leq\alpha\leq qN}^{1\leq \beta\leq qN}\in\mathcal{M}_{qN\times qN}\left(\mathbb{C}\right),A=\left(a^{ij}_{kl}\right)_{1\leq k,l\leq q}^{1\leq i,j\leq q}\in\mathcal{M}_{q^{2}\times q^{2}}\left(\mathbb{C}\right),
\begin{cases}\Box u + u = P(u, \partial_t\partial_xu, \partial^2_xu\,; \partial_tu,\partial_xu) & u(1,x)=\varepsilon u_0(x) & \partial_tu(1,x)=\varepsilon u_1(x)\end{cases}t\ge 1 \, , x \in \mathbb{R}\, , \varepsilon \in ]0,1[ \,.
\mathcal{D}^{(m)}_n(x,y;a,b,c)=\sum_{k_1+k_2+\ldots +k_r=m}\binom{m}{k_1,k_2,\ldots k_r}\frac{{E}^{(-k_1,-k_2,\ldots -k_{r-1})}_n(x;a,b,c)}{(\ln a+\ln b)^n}\left(\frac{(r-1)y\ln c+\ln a}{\ln a +\ln b}\right)^{k_r}.
\sum_{k+\ell\leq s}\left[ |z^{(k,\ell)}(t)|^2+|\bar{z}^{(k,\ell)}(t)|^2+\sum_{i\in I_1}\left(|\xi^{(k,\ell)}_{i,1}(t)|^2+|\xi^{(k,\ell)}_{i,2}(t)|^2 \right)+\right. \left.\sum_{i\neq 0}|\xi^{(k,\ell)}_i(t)|^2 \right] &=(1+o(1))|z(t)|^2.
A ( \Lambda_{1}, K_{1}; \ldots ; \Lambda_{N}, K_{N}) = g^{( N - 2 )}\int \prod_{i = 1}^{N}d^{2}z_{i}\biggl \langle \prod_{j = 1}^{N}e^{i / 2 \K_{j}\cdot X}W_{\Lambda}( z_{j},{\bar{z}}_{j}) e^{i / 2 \K_{j}\cdot \bar{X}}\biggl \rangle \ .
\beta = \Sigma^{-1}\zeta , \zeta^{\top}= \pi_t\left(X_t d\mathbf{Y}_t^{\top}\right) - \pi_t\left(X\right) \pi_t \left(d\mathbf{Y}_t^{\top}\right) + \pi_t \left(\left[\mathbf{L}^{\dagger}_t,X_t\right]\mathbf{S}_td\mathbf{A}d\mathbf{Y}_t^{\top}\right), \Sigma = \pi_t \left(d\mathbf{Y}_td\mathbf{Y}_t^{\top}\right),