harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\mathcal{C}_{D^*}}y^{1-2\alpha}|\nabla(\xi w^q)|^2\,dxdy \leq \frac C{(R-r)^2}\int_{\mathcal B^+_{R}(z)}y^{1-2\alpha}w^{2q}\,dxdy+C\delta^{\frac{2\alpha}N}\int_{\mathcal{C}_{D^*}}y^{1-2\alpha}|\nabla(\xi w^q)|^2\,dxdy.
\phi ( x^{5}) \ = \ - \, \frac{3}{4}\, \epsilon \, \log \left( \, \frac{\tanh \left[ \, \frac{\omega}{2}( x_{s}- | x^{5}| ) \, \right]}{\tanh \left( \frac{\omega x_{s}}{2}\right)}\right) \, + \, d \ ,
=\sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}[\mathrm{Ind}^G_{P_b}(\delta^{\frac{1}{2}}_{G,P_b}\otimes I^{M_b}_{M_S}(\rho))]\left[ \bigoplus_{(M_S, \mu_S) \in \mathrm{Rel}^{M_b, \mu_b}_{M_S,b'}}r_{-\mu_S}\circ LL(I^{M_b}_{M_S}(\rho))|_{W_{E_{\{\mu_S\}_{M_S}}}}| \cdot|^{S}\right],
(dL_{p})_{Id}\bigr(Z_{1}\bigr)=[\tilde{X}^{0},\,\tilde{X}^{\bf i}]_{p}, &(dL_{p})_{Id}\bigr(Z_{2}\bigr)=\alpha(v)[\tilde{X}^{0},\,\tilde{X}^{\bf j}]_{p}-[\tilde{X}^{\bf i},\,\tilde{X}^{\bf k}]_{p}, &(dL_{p})_{Id}\bigr(Z_{3}\bigr)=\alpha(v)[\tilde{X}^{0},\,\tilde{X}^{\bf k}]_{p}+[\tilde{X}^{\bf i},\,\tilde{X}^{\bf j}]_{p},
\begin{aligned}u(x_{k}) = \frac{u^\prime(x_i)}{2(x_{i}-x_\star)}(x_{k}-x_\star)^2 + u(x_i) + \frac{u^\prime(x_i)}{2}(x_\star-x_i) &= \frac{u^\prime(x_i)}{2\phi_{i}}(\phi_{k})^2 + u(x_i) - \frac{u^\prime(x_i)}{2}(\phi_i). \end{aligned}
\sigma^-(L \mathbf{c}_0)(F)=\left[\begin{array}{cc}\bar{n}_1^2(\lambda^-+\mu^-)+\mu^- \bar{n}_1\bar{n}_2(\lambda^-+\mu^-) \bar{n}_1\bar{n}_2(\lambda^-+\mu^-) \bar{n}_2^2(\lambda^-+\mu^-)+\mu^-\end{array}\right]\left[\begin{array}{c}c^1_0 c^2_0 \end{array}\right]:=K\mathbf{c}_0.
J_{\lambda,\mu}(u,v) =\Big(\frac{1}{p}-\frac{1}{q}\Big)\|(u,v)\|^p+2\Big(\frac{1}{q}-\frac{1}{\alpha+\beta}\Big)\int_\Omega |u|^{\alpha}|v|^\beta dx &<\Big[\Big(\frac{1}{p}-\frac{1}{q}\Big)+\Big(\frac{1}{q}-\frac{1}{\alpha+\beta}\Big)\frac{p-q}{\alpha+\beta -q}\Big]\|(u,v)\|^p=-\frac{(p-q)(\alpha+\beta-p)}{pq(\alpha+\beta)}\|(u,v)\|^p<0.
R_{c}^{2}\, \left( n_{m}- n_{\mathrm{l o o p}}^{\mathrm{u n c}}\right) = R_{m}^{2}\, n_{m}- R_{\mathrm{l o o p}}^{2}\, n_{\mathrm{l o o p}}^{\mathrm{u n c}}\, \, \, \, \, , \, \, \, \, \,
\Bigl (v' + \Bigl( \frac{1}{\alpha}- \frac{2}{\alpha t}- \frac{e^{-t}}{4}\Bigr)v \Bigr )' = \frac{1}{4t}\Bigr ( \underline{v}^\alpha - |v|^\alpha + \frac{4(\alpha -1)}{\alpha^2 t}-\frac{1}{\alpha}e^{-t}+ t e^{-t}\Bigl ) v .
g^{(i)}(X) = \sum_{i' = 0}^i (X^{q})^{(i')}\cdot f^{(i-i')}(X) &= \sum_{i' = 0}^i{q \choose i'}X^{q-i'}\cdot f^{(i-i')}(X) &= X^{q}f^{(i)}(X).
B_1 (s,t) = \frac{\psi^*_0 \left(\frac 1{\sqrt 3}s \right) \psi_0\left(\frac 2{\sqrt 3}s \right)g_z (s) g_z (t) \chi_{\{s <t\}}}{(s^2 +t^2 )}, B_2 (s,t) = \psi^*_0 \left(\frac 1{\sqrt 3}s \right) \psi_0 \left(\frac2{\sqrt 3}s \right)g_z (s) g_z (t) \chi_{\{s \geq t\}}\left[\frac{1}{s^2 +t^2}-\frac{1}{s^2}\right].
\psi_{2^{j}+\ell -1}(x) = \begin{cases}+2^{\tfrac{j}{2}}, ~ x \in \left[\frac{\ell-1}{2^j}, \frac{\ell-1/2}{2^j}\right] -2^{\tfrac{j}{2}}, x \in \left[\frac{\ell-1/2}{2^j}, \frac{\ell}{2^j}\right] 0, \end{cases}.
\begin{aligned}(\Theta_1\!-\!\Theta_2)^T \nabla^2\eta(\hat{\Theta})(\Theta_1\!-\!\Theta_2) &\!=\!|V_1\!-\!V_2|^2\! +\!(\Xi_1\!-\!\Xi_2)^T\nabla^2R(\hat{\Xi})(\Xi_1\!-\!\Xi_2)\!+\!(F_1\!-\!F_2)^T\nabla^2H(\hat{F})(F_1\!-\!F_2).\end{aligned}
[X \xleftarrow p V \xrightarrow s Y; E] \circ_{\otimes}[Y \xleftarrow q W \xrightarrow t Z; F] &: = [(X \xleftarrow p V \xrightarrow s Y) \circ (Y \xleftarrow q W \xrightarrow s Z);\widetilde{q}^*E \otimes \widetilde{s}^*F ],
2\sum_{k=0}^a\frac{(-a-p)_{p+k}(\frac12)_{p+k}}{(1)_{p+k}^2}\cdot \bigg(\frac{4z}{(1+z)^2}\bigg)^k \equiv -\frac{\Phi(0)}{4^{p-1}(1+z)^{2a-1}}-\frac{p}{(1+z)^{2a}}\cdot\frac{\Psi_1'(0)+z\cdot\Psi_2'(0)}{2}\pmod{p^2}.
-( x_{N}\mathcal{D}_{N}f(x))^{\ast}K(x) g(x) +f(x)^{\ast}K(x)x_{N}\mathcal{D}_{N}g(x) \qquad{}=x_{N}\partial_{N}\{f(x)^{\ast}K(x)g(x) \}-\kappa\sum_{1\leq j\leq N-1}\frac{1}{x_{N}-x_{j}}\big\{x_{j}f(x(j,N))^{\ast}\tau((j,N))K(x) g(x) \qquad\quad{}+x_{N}f(x)^{\ast}K(x) \tau((j,N)) g( x( j,N)) \big\}.
S_{t h e r m o}= 2 \pi \left[{\frac{4}{( D - 3 )^{2}}}Q_{1}^{( 1 )}Q_{1}^{( 2 )}( 2 m )^{\frac{2}{D - 3}}-{\frac{2}{( D - 3 )}}\sum_{i = 1}^{[{\frac{D - 1}{2}}]}J_{i}^{2}\right]^{\frac{1}{2}}.
F'(t)\leq -\lambda_0\xi(t)F(t)+\mu E'(t)+\xi(t)\dfrac{\mu}{2}\int_0^\infty g(s)\|\partial_x^{k}\eta^t\|^2ds \leq -\lambda_0\xi(t)F(t)+\mu E'(t)-\mu E'(t)+c_1\dfrac{\mu}{2}\xi(t)\int_t^\infty g(s)h(t,s)ds.
\Pi(x)=\sum_{n\le x}\Lambda(n)\int_{n}^{x}\frac{dt}{tlog^{2}t}+\frac{1}{logx}\sum_{n\le x}\Lambda(n)=\int_{2}^{x}\frac{\psi(t)dt}{tlog^{2}t}+\frac{\psi(x)}{logx}.
\mathcal{H}_{f}(u) = \sum_{i=1}^A\zeta^{k_i}\sum_{x\in U_i}(-1)^{\langle u,x\rangle}- \sum_{i=1}^A\zeta^{k_i}+ \zeta^\rho - 2^m+A-1 &= 2^m\zeta^{k_r}- A + 2^m+1-\zeta^\rho + \zeta^\rho - 2^m+A-1 = 2^m\zeta^{k_r}.
\sum_{\pi ( 1 , 2 , . . . n )}\mathrm{T r}\Big [ \tau^{( l_{0})}\tau^{( l_{\pi ( 1 )})}. . . . . \tau^{( l_{\pi ( n )})}\Big ] \frac{1}{\tilde{x}_{0}- \tilde{x}_{\pi ( 1 )}}\frac{1}{\tilde{x}_{\pi ( 1 )}- \tilde{x}_{\pi ( 2 )}}. . . . . \frac{1}{\tilde{x}_{\pi ( n )}- \tilde{x}_{0}}\; ,
C^{\uparrow}_{2n+1}= \frac{1}{2n+1}\log_2 \left[\left(2 + 2\cdot2^{-2}\right) 1^T_{2n-1}\exp_2\left(\omega_{2n-1|0}\right)\right] = \frac{1}{2n+1}\left[\log_2\left(\frac{5}{4}\right) + n\cdot\log_2\left(\frac{5}{2}\right)\right].
0=<\Omega{\cal O}>_J=<{\partial_r{\cal O}\over \partial \Phi^A}{\partial_l\Sigma\over\partial K_A}-{\partial_r{\cal O}\over \partial K_A}{\partial_l\Sigma\over\partial \Phi^A}-i\hbar (-1)^{\varepsilon_A+1}{\partial_r\over \partial\Phi^A}{\partial_r{\cal O}\over \partial K_A}>_J.
S_{E}={\frac{1}{2 \kappa_{D}^{2}}}\int d^{D}x \sqrt{- g^{E}}\left[{\cal R}_{g^{E}}-{\frac{4}{D - 2}}( \partial \phi )^{2}-{\frac{1}{2 \cdot ( p + 2 ) !}}e^{2 a_{p}\phi}F_{p + 2}^{2}-{\frac{1}{2 \cdot D !}}e^{2 a \phi}F_{D}^{2}\right] ,
\Omega= \left(\begin{array}{cccc}J_{11}\cdots J_{1N}Q_1 \vdots \ddots \vdots \vdots J_{N1}\cdots J_{NN}Q_N -{\overline{Q}}_1 \cdots -{\overline{Q}}_N \sum_{\mu=1}^{N}J_{\mu\mu}\end{array}\right)
\begin{aligned}\partial_tU \cdot \overline{h(U-u)}+\nabla U :\overline{hu\otimes(U-u)}&=\partial_tU\cdot \overline{h(U-u)}+\nabla U :\overline{hU\otimes(U-u)}+\nabla U :\overline{h(u-U)\otimes(U-u)},\end{aligned}
\psi_p(s) - \psi_p(r) + \psi_p'(r) (r-s) = |s|^p - |r|^p + p \frac{|r|^p}{r}(r-s) \geq p (p-1)\begin{cases}\frac{(r-s)^2}{2 |r|^{2-p}}, 2 |r| \geq |s| \frac{|r-s|^p}{4}, |s| \geq 2 |r|.\end{cases}
|A|\le |A^{-1}|^{d-1}\det A\le |A^{-1}|^{d-1}\frac{1}{1+\epsilon_{\rho}}\rho \le \frac{C_A^{d-1}}{1-\epsilon_{\rho}}\rho^{max}_d\le C_{|A|}:= \frac{8C_A^{d}}{7}\rho^{max}_d\quad.
{\widehat T}{}_l[\varphi]\equiv\frac{2l-1}{2\pi i}\int\limits_{|z|<1}\varphi(z,\bar z)(1-z\bar z)^{2l-2}dzd\bar z=\varphi(0,0)+\frac{1}{2l}\left.\bigtriangleup\varphi(z,\bar z)\right|_{\scriptstyle z=\bar z=0}+{\cal O}(l^{-2})
\langle y\rangle_{T^{*3}}= ~\{y, T^{*3}y ,\ldots, T^{*3(j-1-r)}y\}~,~~~\langle y\rangle_{T^{*3}}=j-r, \langle z\rangle_{T^{*3}}= ~\{z, T^{*3}z ,\ldots, T^{*3(n-2j-2+r)}z\}~,~~~\langle z\rangle_{T^{*3}}=n-2j-1+r.
\pi ( x ) ~ = ~ \frac{\delta \,{\cal L}}{\delta \, \dot{\varphi}\, ( x )}~ = ~ \dot{\varphi}+ \lambda \, \frac{\delta \,{\cal L}_{i n t}}{\delta \, \dot{\varphi}}\, \left( \varphi , \vec{\bigtriangledown}\, \varphi , \dot{\varphi}\right) ~ = ~ \dot{\varphi}+ \lambda \, \dot{\varphi}\, \left[ \left( \vec{\bigtriangledown}\, \varphi \right)^{2}- \dot{\varphi}^{2}\right] ,
\rho_{i+\frac{1}{2}}^{L}=\rho_{i}+0.5\phi(r_{i})(\rho_{i}-\rho_{i-1}),\rho_{i+\frac{1}{2}}^{R}=\rho_{i+1}-0.5\phi(r_{i+1})(\rho_{i+1}-\rho_{i}),
G_{0}^{( 2 )}( \lambda , \vec{r}_{1}, z_{1}, \vec{r}_{3}, z_{3}) = \frac{\lambda}{2}\int d^{d - 1}r_{2}\int_{0}^{L}d z_{2}G_{0}^{( 2 )}( \vec{r}_{1}- \vec{r}_{2}; z_{1}, z_{2}) G_{0}^{( 2 )}( \vec{0}, z_{2}) G_{0}^{( 2 )}( \vec{r}_{2}- \vec{r}_{3}; z_{2}, z_{3}) .
J= \begin{pmatrix}b_0 \sqrt{u}_1 0 0 \sqrt{u}_1 \ddots \ddots 0 0 \ddots \ddots \sqrt{u_j}0 0 \sqrt{u_j}b_j 1 b_{j+1}\sqrt{u_{j+2}}0 0 \sqrt{u_{j+2}}\ddots \ddots 0 0 \ddots \ddots \sqrt{u_N}0 0 \sqrt{u_N}b_N \end{pmatrix}.
\forall\theta\in[0,2\pi], \mathbb{E}\bigl[H_{X}(\theta) \bigr]=\frac{1}{\pi}\int_0^{\pi}\mathbb{E}\bigl[H_{X'}(\theta) \bigr]\,d\theta, &\forall(s,t)\in[0,2\pi]^2, \mathbb{E}\bigl[H_{X}(s)H_{X}(t)\bigr]=\frac{1}{\pi}\int_0^{\pi}\mathbb{E}\bigl[H_{X'}(\theta)H_{X'}(\theta+s-t) \bigr]\,d\theta.
\max\{n+1,\xi/\varepsilon\}^n = \max\{(n+1)^n,\frac{1}{n!}(\xi/\varepsilon)^n n!\}\leq \max\{(n+1)^n,n! e^{\xi/\varepsilon}\}\leq n! e^{\xi/\varepsilon}\frac{(n+1)^n}{n!}&\leq C n! e^n e^{\xi/\varepsilon},
E(x)=const\,exp(-\frac{im\tau}{2})f(r)\left\|\begin{array}{c}-exp(i(\varphi-\tau))r_{\bot}/\sqrt{R^2+r^2}iexp(i(\varphi-\tau))r_{\bot}/\sqrt{R^2+r^2}1 i 0\end{array}\right\|
\begin{aligned}d(p_0,y) \leq d(y,(t_0,w,\theta))+d((t_0,w,\theta),(t_0,w_0,\theta))+d((t_0,w_0,\theta),p_0) &\leq t-t_0+\sinh t_0|w-w_0|+\lambda\cosh t_0|\theta-\theta_0|\end{aligned}
K \sum_{r=1}^{\lceil \log_2 K \rceil}N_r/2^{r-1}\le 2 K + \frac{64 \beta K}{\epsilon^2}\sum_{r=1}^\infty \left(\frac{8}{9}\right)^{r-1}\log \frac{2^r 3}{\delta}&= O \left(\frac{K}{\epsilon^2}\log \frac{1}{\delta}\right).
\lim_{q\rightarrow\infty}\frac{q^{1-1/p}}{(\Gamma(q/p'+2))^{\frac{1}{q}}}=\lim_{q\rightarrow\infty}\frac{q^{1-1/p}}{(1+o(1))\left(\frac{q}{ep'}\right)^{1/p'}}=(ep')^{1/p'},
\mathrm{d i m}_{q}{\tilde{\rho}}_{\ell}={\frac{( [ N ] [ N + 1 ] \ldots [ N + 2 \ell - 1 ] ) \prod_{a = 1}^{N - 2}( [ N - a ] [ N - a + 1 ] \ldots [ N - a + \ell - 1 ] )}{[ \ell ] ! ( [ N + 2 \ell - 2 ] [ N + 2 \ell - 3 ] \ldots [ N + \ell - 1 ] ) \prod_{a = 1}^{N - 2}( [ a ] [ a + 1 ] \ldots [ a + \ell - 1 ] )}}
d M_\eta(t) =e^{-\eta t}\frac{\sqrt{2\gamma}}{m}v(t)\, dW_0(t)+\frac{e^{-\eta t}}{m}\sum_{k=1}^N \sqrt{2\lambda_k}z_k(t)\, dW_k(t) &+e^{-\eta t}\sum_{k>N}\sqrt{2\lambda_k}k^{-2s}z_k(t)\, dW_k(t).
\xi_t=\frac{A_g(M)}{A_f(M)}\xi_r+\frac{A_h(M)}{A_f(M)}\xi_s=\frac{A_g(M)}{A_f(M)}\xi_r+\frac{A_g(M)}{A_f(M)}\frac{A_h(M)}{A_g(M)}\xi_s.
\mu\Big(E_f(Q_{\rho_k}, c_0 \lambda_0 M^k)\Big) \leq \alpha^k \mu\big(E_f(Q_{2 R}, c_0 \lambda_0 )\big) + \sum_{i=0}^{k-1}\alpha^{k - i}\Big[\nu\Big(E_g(Q_{\rho_i}, c_0 \lambda_0 M^i)\Big)+ \hat\nu\Big(E_{\hat g}(Q_{\rho_i}, c_0 \lambda_0 M^i)\Big)^{\hat p}\Big].
\int \frac{d^{4}q}{( 2 \pi )^{4}}\: \hat{V}( q ) \: e^{- i \: \frac{q}{2}\: ( x + y )}\: \int_{- \infty}^{\infty}d \tau \epsilon ( q^{2}) \: \delta^{( \alpha )}( z ) \: H_{z}^{( \beta )}( x , y ) \:{\mathcal{P}}( \tau ) e^{- i q \tau ( y - x )}
\mathbb{E}[ \|x^{k+1}- x^*\|^2] \leq q \mathbb{E}[\|x^k-x^*\|^2] + 4p_{\max}\alpha_{\max}^2 C^2N +2p_{\max}\alpha_{\max}B \sum_{j=1}^N \mathbb{E}[\|v^k_j-y^k\|],
\partial^{\mu}\left( \partial^{\nu}\tilde{\partial}_{+}\bar{\phi}\, \, \varepsilon^{\prime}\sigma_{\nu \mu}\tilde{\partial}_{-}\psi \, \, \right) + \tilde{\partial}_{+}\left( \tilde{\partial}_{-}\bar{V}_{+}\, \bar{\varepsilon}_{\dot{\alpha}}\bar{\sigma}_{\mu}^{\dot{\alpha}\beta}\partial^{\mu}\tilde{\partial}_{-}\psi_{\beta}\right) .
\begin{array}{l}g_0(x)F_1(x)+g_1(x)F_2(x) =((x-1)^r+u(x-1)^{k_1}p_1(x)+u^2(x-1)^{k_2}p_2(x)+u^3(x-1)^{k_3}p_3(x))F_1(x) +(u(x-1)^{r_1}+u^2(x-1)^{k_4}p_4(x)+u^3(x-1)^{k_5}p_5(x))F_2(x), \end{array}
\mathcal{N}_\varepsilon= \{u\in \mathcal{H}(\Omega)\setminus\{0\}\mid \langle I'_\varepsilon(u),u\rangle =0\}= \left\{u\in \mathcal{H}(\Omega)\setminus\{0\}\mid \int_{\Omega}(K_H(x)\nabla u|\nabla u)dx=\frac{1}{\varepsilon^2}\int_{\Omega}\left(u-q\ln\frac{1}{\varepsilon}\right)^{p}_+udx\right\}.
e^{- J_G(r)}- \| (\phi^{-1}f)(\xi_t)^{\gamma_0 - 1}e^{-J_G(r)}\|_{\Pi_x^{(\phi)};\frac{1}{\gamma_0 - 1}}=e^{- J_G(r)}\Big( 1 - \Pi_x^{(\phi)}[ (\phi^{-1}f)(\xi_t) ]^{\gamma_0 - 1}\Big) \xrightarrow[t\to \infty]{}0, x\in E, r\geq 0.
P''''_\ell(\cos \phi)= \frac{\ell+1}{\sin^8 \phi}\sum_{u=0}^3 \ell^u \sum_{n=0}^2 \sum_{v=0}^4{_4}\omega_{u,v}(\cos \phi) p_{n,\ell+v}(\phi) +\phi^{-5}O(\ell).
b''_l = \sum_{i\geq 1}\frac{1}{(i-1)!}(l-1)^{\underline{i-1}}\sum_{\substack{k_1,\dots,k_i\geq 2 k_1+\cdots+k_i=l}}b'_{k_1}a'_{k_2}\cdots a'_{k_i}.
F ( k_{0}) = \frac{k_{0}}{2}\int_{0}^{1}d u \frac{\sqrt{\sqrt{1 + 4 k_{0}^{2}( 1 - u )}- 1}}{\sqrt{u \left( 2 k_{0}^{2}+ 1 - \sqrt{1 + 4 k_{0}^{2}( 1 - u )}~ \right) \left( 1 + 4 k_{0}^{2}( 1 - u ) \right)}}= \frac{1}{2}\ln \frac{R_{2}}{R_{1}}.
\sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k\equiv0\pmod{[n]}.
\frac{(-p)^{-\lfloor\frac{1}{2}+\frac{j}{p-1}\rfloor}}{\Gamma_p(\frac{1}{2})}~\Gamma_p\left(\left\langle\frac{1}{2}+\frac{j}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle1-\frac{j}{p-1}\right\rangle\right) &=\frac{1}{p}\sum_{t\in\mathbb{F}_p^\times}\overline{\omega}^j(-t)\varphi(t(t-1)).
\frac{1}{N}\sum_{c=1}^{m}\sum_{i\in I_{c}^{N}}\Biggl[g^{(c)}(T,\,z_{i}^{N}(T))-g^{(c)}(0,\,z_{i}^{N}(0))-\int_{0}^{T}g_{t}^{(c)}(t,\,z_{i}^{N}(t))dt -\frac{1}{2}\int_{0}^{T}g_{xx}^{(c)}(t,\,z_{i}^{N}(t))d\left\langle z_{i}^{N},\,z_{i}^{N}\right\rangle_{t}\Biggr]
d_{\Phi_{T}}(\alpha) = \mu\left\{(x,\chi) \in G\times\widehat{G}: |\Phi_T(x,\chi)| > \alpha \right\}\leq \left( \frac{\|T\|_{\mathcal{B}_2(L^2(G))}}{\alpha}\right)^2, &\mu\left\{(x,\chi) \in G\times\widehat{G}: |\Phi_T(x,\chi)| > \alpha \right\}\lesssim \frac{M_\psi \|T\|_{\mathcal{B}_1(L^1(G))}}{\alpha}.
\left\langle H^\square_{\epsilon, q}(\theta) u, u \right\rangle = \epsilon^2 q^2 \Bigl( \left\langle H^\square_{0}(\theta) \varphi, \varphi \right\rangle + 2 \Re \left\langle \psi, V^\square \varphi \right \rangle \Bigl) + \epsilon^3 q^3 \left\langle V^\square \varphi, \varphi\right\rangle .
\tilde C = \tilde{C}(\mu \delta^{-1},c_A) &= \frac{1 + \sqrt{2}}{2}\mu \delta^{-1}\left(1\vee \mu \delta^{-1}\right)+ \left[2 + \sqrt{\mu \delta^{-1}}\left(1\vee \sqrt{\mu \delta^{-1}}\right)\right]\left[c_A + \frac{3}{2}(1-\mu) \delta^2\right].
\frac{d}{dt}F_{N}(t,x) =\frac{1}{N^{2}}\stackrel[l=1]{(i-1)^{\ast}(j)}{\sum}\stackrel[r=1]{N}{\sum}m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t)) &=\frac{1}{N^{2}}\stackrel[r=1]{N}{\sum}\stackrel[l=1]{(i-1)^{\ast}(j)}{\sum}m_{l}(t)m_{r}(t)S(x_{l}(t)-x_{r}(t))=\mathbf{S}[F_{N}].
S ( \sigma ) = \frac{1}{4}\int_{0}^{2 \pi}d \sigma^{\prime}d \sigma^{\prime \prime}e^{\frac{1}{4}\gamma \eta p \left( \epsilon ( \sigma - \sigma^{\prime}) - \epsilon ( \sigma - \sigma^{\prime \prime}) \right)}e^{\psi^{+}( \sigma^{\prime})}e^{\psi^{-}( \sigma^{\prime \prime})}
\begin{array}{l}{}^{A}(\lambda\Phi +\mu\Psi)^{(n)}=\lambda{}^{A}\Phi^{(n)}+ \mu{}^{A}\Psi^{(n)}, \quad \forall \lambda, \mu \in C, ({}^{A}\Phi,{}^{A}\Psi)=\S^{\infty}_{n=0}({}^{A}\Phi^{(n)},{}^{A}\Psi^{(n)}).\end{array}
c_{b_{k}}=\frac{1}{2}\|(\widetilde{u}_{k},\widetilde{v}_{k})\|_{E}^{2}+\frac{1}{2}\Big(B(\widetilde{u}_{k})+B(\widetilde{v}_{k})\Big)-\frac{1}{2p}F(\widetilde{u}_{k},\widetilde{v}_{k}) &\geq\Big(\frac{1}{2}-\frac{\alpha}{2(p\alpha-1)}\Big)\|(\widetilde{u}_{k},\widetilde{v}_{k})\|_{E}^{2},
\lim_{c\rightarrow \infty}\left( k(j+c)^{N(j)}(k(j+c)^n-1)\right) = \left(\lim_{c\rightarrow \infty}k(j+c)\right)^{N(j)}\left((\lim_{c\rightarrow\infty}k(j+c))^n-1\right) = k^{N(j)}(k^n-1) \in \hat{I}_{\mathfrak{p}},
\int_{\P^n}e(T\P^n)=\sum_{j=0}^n\frac{e(i_j^*T\P^n)}{e(N_{p_j/X})}=\sum_{j=0}^n\frac{e(i_j^*T\P^n)}{e(i_j^*T\P^n)}=n+1.
\begin{aligned}\mathbb{P}\left(||(P_{n_1}^\pi\hat{r}_{n_1,n_2}^\pi)^2-(P^\pi\hat{r}^\pi)^2||_\infty\leq\max_{x\in\mathcal{X}}\alpha_1(x)\alpha_2(x)\right)\geq1-16\delta_1(n_1).\end{aligned}
w(t_1)=\frac{1}{15},w(t_2)=\frac{1}{3},w(t_3)=1, g_1 = -\frac{1}{33}t_1\big(3t_1^{10}t_2 + 11t_2^3 - 33t_3\big), g_2 = \frac{1}{76}\big({-}5t_1^{20}+ 114t_1^{10}t_2^2 + 19t_2^4 + 76t_2t_3\big), g_3 = \frac{1}{870}\big(100t_1^{30}+ 1740t_1^{20}t_2^2 - 5220t_1^{10}t_2^4 + 116t_2^6 + 435t_3^2\big).
(\sum_{k=1}^{n}|a_{k}|^{2})(\sum_{k=1}^{n}|b_{k}|^{2})-|\sum_{k=1}^{n}a_{k}b_{k}|^{2}=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}|a_{i}\overline{b}_{j}-a_{j}\overline{b}_{i}|^{2}
{\sf adiam}(D)_x:=\inf\left\{\frac{(R_{1}-R_{0})R_{1}}{R_{0}}: \exists a\in \mathbb{R}^d \mbox{such that}x\in\partial A(a,R_0,R_1) \mbox{and}D\subset A(a,R_0,R_1) \right\},
(T_{w_{\lambda + \nu}}f)((\lambda + \nu)_+) =(T_{w_{\lambda + s'_j\nu}}T_jf)((\lambda + s'_j\nu)_+) =(T_j f) (\lambda + s'_j\nu) - d_{\lambda,s'_j\nu}(1-t_{\vartheta}^{-1})(T_jf)(\lambda)
\mathrm{Var}(\hat\beta|\hat\tau>0) = \Sigma_{\beta\beta}(\delta_0) - \Sigma_{\beta\tau}(\delta_0) \Sigma^{-1}_{\tau\tau}(\delta_0)\Sigma_{\tau\beta}(\delta_0) + \Sigma_{\beta\tau}(\delta_0) \Sigma^{-2}_{\tau\tau}(\delta_0)\Sigma_{\tau\beta}(\delta_0)\mathrm{Var}(\hat\tau-\tau_0|\hat\tau>0)
\tilde{\varphi}\left(\left[\left(\begin{smallmatrix}a \ast 0 a^{-1}\end{smallmatrix}\right), \epsilon \right] g \right) = \epsilon \chi_{\psi}(a)\mu(a)|a|_p\tilde{\varphi}(g) = \epsilon \chi_{\psi}(a)\chi_{\delta}(a)|a|^{3/2}_p\tilde{\varphi}(g)
\sum_{k=0}^{M}[4k-1]\frac{(q^{-1};q^2)_k^6}{(q^2;q^2)_k^6}q^{7k}\equiv\frac{\Omega_q(n)[3][5]}{[2]^4[4]^2}\sum_{k=0}^{(n-3)/2}\frac{(q^3;q^2)_k^3(q^7;q^2)_k}{(q^2;q^2)_k(q^6;q^2)_k^3}q^{2k}.
|x_{n+1}| \le |x_n(1-h_nx_n^2)|+|u_{n+1}| &\le h_n|x_n|^3 \left |1-\frac 1{h_nx_n^2}\right|+|u_{n+1}| &\le h_n|x_n|^3 +|u_{n+1}|.
\begin{array}{r@{}l}d\left( M_{t_2}(p_3,p_2),M_{t_1}(p_2,p_1), \right){}\le d(M_{t_2}(p_3,p_2),p_2) + d(p_2,M_{t_1}(p_2,p_1)){}= (1-t_2)d(p_2,p_3) + t_1 d(p_1,p_2) . \end{array}
\Vert L_{g_{0}}(\upsilon-\alpha^{\frac{n+2}{n-2}}\omega)\Vert_{L^{\frac{2n}{n+2}}}\,,\Vert L_{g_{0}}(\upsilon_{j}-\alpha^{\frac{n+2}{n-2}}\mathrm{e}_{j})\Vert_{L^{\frac{2n}{n+2}}}=o_{\varepsilon}(1).
\mathrm{~ e}^{- S_{\mathrm{e f f}}[ A_{0}]}\equiv \sum_{B_{M}}\int [ d \vec{a}] < \vec{a}\vert \mathrm{~ e}^{- H / T}\mathrm{~ e}^{i \frac{\kappa}{T}\int A_{0}( B_{M}+ B / 2 )}\vert \vec{a}+ \vec{\nabla}A_{0}/ T >
{\langle O_{N}^{\dagger}\, O_{N/2}\, O_{N/2}\rangle\over\sqrt{O_{N}^{\dagger}\, O_{N}}\,\sqrt{O_{N/2}^{\dagger}\, O_{N/2}}\sqrt{O_{N/2}^{\dagger}\, O_{N/2}}}\sim{1 \over 2^{N}}\, .
u = \frac{x_{1 2}^{2}x_{3 4}^{2}}{x_{1 3}^{2}x_{2 4}^{2}}\, , \, \, \, \, v = \frac{x_{1 2}^{2}x_{3 4}^{2}}{x_{1 4}^{2}x_{2 3}^{2}}\, .
d\omega((f,x), (g,y) (h,z)) = d\omega(f,g,h) + d\omega(f,g,z) + d\omega(f,y,h) + d\omega(f,y,z) + d\omega(x,g,h) + d\omega(x,g,z) + d\omega(x,y,h) + d\omega(x,y,z) = 0.
\tilde{\cal S}^{(2n)}= \sum\limits_{j=1}^{n-1}\frac{(-1)^{n+j}2 ((\ell+n)!)^2}{(\ell+j)! (\ell+2n-j)!}P_{\ell-2n+j}P_{\ell-j}+ P_{\ell-n}^2.
\gamma_e(t) = \bmatrix a_1\cos(\alpha_1 t) a_1 \sin( \alpha_1 t) \vdots a_{k}\cos(\alpha_k t) a_k \sin( \alpha_k t))\endbmatrix \hskip .3in{\rm or}\hskip .3in \gamma_o(t) = \bmatrix a_1\cos(\alpha_1 t) a_1 \sin( \alpha_1 t) \vdots a_{k}\cos(\alpha_k t) a_k \sin( \alpha_k t)) bt \endbmatrix
2\Phi_{\rm C}(c_1) (a_0, a_1) =\int_M f_0 \,{\rho_0}^*( f_1) \left( - \frac{\rho''_0}{\rho'_0}\right) dx - \int_M f_1 \,{\rho_1}^*( f_0) \left(- \frac{\rho''_1}{\rho'_1}\right) dx &= -\int_M f_0 \,(f_1\circ \phi) \, \frac{\phi''}{\phi'}\, dx + \int_M f_1 \,( f_0 \circ \phi^{-1}) \, \frac{({\phi}^{-1})''}{({\phi}^{-1})'}\, dx .
A_{C}^{N S N S}={\frac{G_{C}}{4}}\int_{0}^{\infty}{\frac{d t}{2 t}}( 8 \pi^{2}t )^{- 5}{\frac{f_{3}( e^{- \pi t})^{8}}{f_{1}( e^{- \pi t})^{8}}}\, .
\sum_{n=0}^{\infty}\mathcal{E}_{n}^{\left( s\right)}\left( \alpha;x\right) \frac{t^{n}}{n!}=\left( \frac{2}{\left( 1+\alpha t\right)^{1/\alpha}+1}\right)^{s}\left( 1+\alpha t\right)^{x/\alpha}.
K^aF(t,x) = \int a(\xi) e^{\pm i(t-s)|\xi|}e^{i \xi(x-y)}(O(t,x,\xi)O^{-1}(s,y,\xi)) F(s,y) (O(t,x,\xi) O^{-1}(s,y,\xi))^{-1}d\xi
\Omega_{C_2}= MU_{C_2}^{\mathbf{C}^{\infty}}\subset MU_{C_2}^{\mathbf{C}^{\infty+\sigma}}\subset MU_{C_2}^{\mathbf{C}^{\infty+2\sigma}}\subset \cdots \subset MU_{C_2}.^{\mathbf{C}^{\infty+\infty\sigma}}= MU_{C_2}
C_k =\bigl(R(t)(t+k)\bigr)|_{t=-k}&=(-1)^{d+b_4+k}\binom{k-b_1}{k-a_1}\binom{k-b_2}{k-a_2}\binom{k-b_3}{k-a_3}\binom{b_4-a_4-1}{k-a_4}\in\mathbb Z
Z^{( 2 )}( J , B ) = \{\det \tilde{K}_{i k}) \}^{- 1}\exp \left\{- \frac{i g^{2}}{2}\int d^{4}x ( L_{i}( J ) \tilde{K}_{i k}^{- 1}L_{k}( J ) + J_{+}( K_{+ +})^{- 1}J_{+}) \right\}
I(X_{<t}; B_t|L,B_{<t}) \le I(X_{<t}; B_t,B_{>t}|L,B_{<t}) &= H(X_{<t}|L,B_{<t}) - H(X_{<t}|L,B^n) &\le H(X_{<t}|B_{<t}) - H(X_{<t}|L,B^n) &= H(X_{<t}|B_{<t}) - H(X_{<t}|B^n) &= H(X_{<t}|B_{<t}) - H(X_{<t}|B_{<t}) &= 0.
\sum_{i\in\mathcal{I}}\left( (A_i-X)^T \otimes( \partial_{x_j}\Phi^{s}_i(X)) \right) + \sum_{i\in\mathcal{I}^{s'}}\left( (A_i-\overline{X}_i)^T\otimes( \partial_{x_j}\Phi^{s}_i(X)) \right) (\overline{M}^s -I_4) = I^j,
\tilde{\alpha}^{(A)}_{-m}\rightarrow\frac{1}{\sqrt{m}}\tilde{\alpha}^{(A)}_{-m}\,\,,\,\,\tilde{A}_{-r}\rightarrow\frac{1}{\sqrt{r}\gamma}\tilde{A}_{-r}\,\,,\,\,\tilde{B}_{-s}\rightarrow\frac{1}{\sqrt{s}\gamma}\tilde{B}_{-s}\, .
S = \int \! \! d^{4}x \int \! \! d y \Bigl [{\cal L}_{\mathrm{5 D}}^{U ( 1 )}+ \delta ( y ){\cal L}_{\mathrm{4 D}}^{U ( 1 )}+ \delta ( y - \pi R ){\cal L}_{\mathrm{4 D}}^{\times}\Bigr ] .
\Im m_{\mu_A\boxplus\mu_B}(z)\sim\begin{cases}\sqrt{\kappa+\eta}\,,\quad E\in \mathrm{supp}\,\mu_A\boxplus\mu_B\,, \frac{\eta}{\sqrt{\kappa+\eta}}\,, E\not\in \mathrm{supp}\,\mu_A\boxplus\mu_B\,,\end{cases}
\int_{\mathbb{D}}|f_{r}'(z)-f'(z)|^{2}(\omega(z))^{-1}(1-|z|^{2})dm(z) =\int_{|z|\leq\delta}|f_{r}'(z)-f'(z)|^{2}(\omega(z))^{-1}(1-|z|^{2})dm(z) +\int_{1-\delta<|z|\leq 1}|f_{r}'(z)-f'(z)|^{2}(\omega(z))^{-1}(1-|z|^{2})dm(z) =Int_{1}+Int_{2}.
\frac{u(x)}{v(x)}\le (1+\epsilon^{k+1})\frac{\sum^k_{i=0}u^{k-i}_{k+1}(x)}{\sum^k_{i=0}v^{k-i}_{k+1}(x)}\le \tau\cdot \inf_{y\in D_{k+2}}\frac{u^k_{k+1}(y)}{v^k_{k+1}(y)}.
F^{\circ 1}\left( v\right) \left( r\right) \left( u\right) :=\left(Dr\right) \left( F\left( v\right) \left( u\right) \right) \left( u\right) \forall u\in \mathcal{U}\forall r\in C^{1,loc}\left( \mathcal{U},\mathcal{U}\right)
E = \int d^{2}x \left({\frac{\kappa^{2}- e^{2}| \phi |^{2}}{2 e^{2}| \phi |^{2}}}B^{2}+{\frac{1}{2}}\left[ | \partial_{i}\phi |^{2}+{\frac{\kappa^{2}e^{2}| \phi |^{2}}{\kappa^{2}- e^{2}| \phi |^{2}}}\tilde{A}_{i}^{2}\right] + V ( \phi ) \right) \, .
\begin{array}{r | r | r | r}{{n}}&{{m}}&{{N}}&{{\bar{N}}}\\ \hline{{1}}&{{1}}&{{1}}&{{0}}\\{{1}}&{{- 1}}&{{0}}&{{1}}\\{{- 1}}&{{1}}&{{0}}&{{1}}\\{{- 1}}&{{- 1}}&{{1}}&{{0}}\\{{2}}&{{0}}&{{0}}&{{0}}\\{{- 2}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{2}}&{{0}}&{{0}}\\{{0}}&{{- 2}}&{{0}}&{{0}}\\ \end{array}
\int_{B_{\sqrt{\theta}\,R}\setminus B_{\theta\, R}}\frac{h(x, y)}{|x-y|^{N+s\,p}}\, dy \leq -\frac{c\,(M-m)^{p-1}}{|x|^{N+s\,p}}\int_{B_{\sqrt{\theta}\,R}\setminus B_{\theta\, R}}|y|^{s\,p-N}\, dy &\leq -c\,\theta^\frac{s\,p}{2}\,\Big(1-\theta^\frac{s\,p}{2}\Big)\,(M-m)^{p-1}\,\frac{R^{s\,p}}{|x|^{N+s\,p}},
\begin{aligned}\ d^s(\Delta(x))=d^s(\sum_{a+b=n}x_{t_1,\dots,t_a}\otimes x_{t_{a+1},\dots,t_n})= &\sum_{a+b=n}\Bigl(d^s(x_{t_1,\dots,t_a})\otimes x_{t_{a+1},\dots,t_n}+(-1)^{\deg x_{t_1,\dots,t_a}}x_{t_1,\dots,t_a}\otimes d^s(x_{t_{a+1},\dots,t_n})\Bigr)\end{aligned}
z_{ki}= \binom{n}{k}\frac{(2i+\sigma)(\alpha+l+i+1-k)_{n-i}(k+l-n)_{i-k-l}}{(i+k+l+\sigma)_{n-k-l+1}}, begin{align*}1ex] z_{hi}= \frac{(n+1-h)(\beta+k+h)}{h(\alpha+l+n+1-h)}z_{h-1,i}(h=k+1,k+2,\ldots,n-l);
d_n=d_{n-1}+b_{n-1}-b_{n-2}+\binom{n-3}{2}+\sum_{a=3}^{n-3}\sum_{\ell=0}^{n-3-a}\sum_{m=1}^{n-2-a-\ell}\binom{n-5-\ell-m}{a-3}m, n \geq 5,
E ={\frac{1}{2}}m_{f}\dot{x}^{2}+ V ( x ) ; V ( x ) = E [ 1 -{\frac{1 +{\frac{E}{2 m_{f}}}H_{F}}{( 1 +{\frac{E}{m_{f}}}H_{F})^{2}}}] +{\frac{J_{\parallel}^{2}}{2 m_{f}x^{2}}}{\frac{1}{( 1 +{\frac{E}{m_{f}}}H_{F})^{2}}}.
\bigg(\sum_{m=m_1}^{m_2}\frac{1}{\varphi(p^m)}\, (p^m)^{\alpha\lambda}\bigg)^{1/\lambda}= \bigg(\frac{p}{p-1}\sum_{m=m_1}^{m_2}(p^{\alpha\lambda-1})^m \bigg)^{1/\lambda},
(I+A)^{-1}B_0 = c^2 \begin{pmatrix}N_0 0 0 \end{pmatrix}\,, (I+A)^{-1}B_1 = \frac{b}{c^2}(I+A)^{-1}B_0=bc^{-2}\begin{pmatrix}N_0 0 0 \end{pmatrix}
\left\langle \prod_{b=1}^N \left[\psi_{1}^{(b)}(-L,0) \; U\Big({\cal C}(L)\Big)\;\overline{\psi}_{2}^{(b)}(+L,0) \right]\prod_{c=1}^l \left[\psi_{1}^{(c)}(-L,0) \; U\Big({\cal C}(L)\Big)\;\overline{\psi}_{2}^{(c)}(+L,0) \right]\right\rangle_0^\theta
M_t = \sum_{s\in \mathcal T \cap [0, t]}\Delta M_s = \sum_{s\in \mathcal T \cap [0, t]}\Bigl(\Delta M^1_s + \Delta M^2_s + \Delta M^3_s\Bigr) =\sum_{s\in \mathcal T \cap [0, t]}\Delta M^1_s + \sum_{s\in \mathcal T \cap [0, t]}\Delta M^2_s + \sum_{s\in \mathcal T \cap [0, t]}\Delta M^3_s = \widetilde M^1_t + \widetilde M^2_t + \widetilde M^3_t.
\begin{cases}\frac{\partial}{\partial s}\psi(s) + D_{A(s)}\psi(s) =0 \frac{\partial}{\partial s}A(s) + *_3F_A= r(q_3(\psi )-i*_3\omega)- \frac{1}{2}*_3F_{A_{K^{-1}}}- \frac{1}{2}i*_3 \wp_3 \end{cases}