harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta g_{\mu \nu}( x , \tau , \sigma , f ) = ( F \xi )_{\mu \nu}( x ) + 2 [ f^{\star}\delta \sigma \bar{g}_{\mu \nu}] ( x ) + [ f^{\star}\frac{\partial \bar{g}_{\mu \nu}}{\partial \tau_{i}}\delta \tau_{i}] ( x ) .
D_2 \cos_k w = 2 \cos_k\left( \frac{w_1+w_2}2 \right) \left[ \cos_k\left( \frac{w_1-w_2}2 \right) - \cos_k \left( \frac{w_{-1}-w_{-2}}2 + \pi j \right) \right] &= 4 \cos_k \left( \frac{w_1+w_2}2 \right) \sin_k \left( \frac{w_1-w_2 - w_{-1}+ w_{-2}+ 2\pi j}{4}\right) \sin_k \left( \frac{w_1-w_2 + w_{-1}- w_{-2}+ 2\pi j}{4}\right).
\Bigl|\sum_n \psi_N(n) t(n;q)\Bigr| \ll \frac{1}{K}\sum_n \Bigl|\sum_{k=1}^K \psi_N(n+kr) t(n+kr;s) \Bigr| \ll \frac{N^{1/2}}{K}\Bigl(\sum_n \Bigl|\sum_{k=1}^K \psi_N(n+kr) t(n+kr;s)\Bigr|^2\Bigr)^{1/2}\ll \frac{N^{1/2}}{K}\Bigl(\sum_{1\leq k,l\leq K}A(k,l)\Bigr)^{1/2},
\rho(T^{\vec{n}\cdot \epsilon}x_i',x)=\rho(T^{\vec{n}\cdot \epsilon+\vec{m}\cdot \omega_i}x,x)=\rho(T^{\vec{m}\cdot (\hat{\epsilon}+\omega_i)}x,\mathbf{y}(\hat{\epsilon}+\omega_i))<\delta ,
\widetilde{S}_{\ell}\left(z\right)=\frac{\Gamma\left(1/\ell\right)}{\ell z^{1/\ell}}-\frac{1}{\ell}\sum_{\rho}z^{-\rho/\ell}\Gamma\left(\frac{\rho}{\ell}\right)+E_{\ell}\left(a,y\right)
-v_{\hat{\bf i}}(G^{\rm up}_{\lambda}(b)/\tau_\lambda) = \Psi_{\hat{\bf i}}^{(\lambda)}(b)({\rm by\Corollary}\ \ref{the case of longest}) &= (\Psi_{\bf i}^{(\lambda, w)}(b), \underbrace{0, \ldots, 0}_{N-r})({\rm by\Definition}\ \ref{definition2}).
\sum_{i=1}^N \int_{\Omega_i}\varepsilon \nabla u^* \cdot \nabla \varphi\, dx-\sum_{e \in \Gamma}\int_{e}\Big\{\varepsilon \nabla u^* \cdot \nu \Big\}[\varphi]\, dx=\sum_{i=1}^N \int_{\Omega_i}f \varphi\, dx.
\mathbb{P}_{f_0}(|\kappa|> |\kappa^*|) \leq \mathbb{P}_{f_0}(X_{(n)}> s_2) + \mathbb{P}_{f_0}(X_{(1)}< s_1) &\leq \mathbb{P}_{f_{\alpha,s_1,s_2}}(Y_{(n)}> s_2) + \mathbb{P}_{f_{\alpha,s_1,s_2}}(Y_{(1)}< s_1) + d_{\mathrm{KS}}^{(n)}= d_{\mathrm{KS}}^{(n)}.
K_{\mathrm{n p}}= d \left( \frac{S_{R}}{4 \pi}\right)^{p / 2}e^{- b \sqrt{S_{R}}}\left[ 1 +{\cal O}\left( \frac{1}{\pi S_{R}}, \frac{1}{\pi T_{R}}\right) \right] +{\cal O}( e^{- 2 \pi S}, e^{- 2 \pi T}) ,
-2\delta^{i_1i_2}G_{i_1i_2i_3}^{j_1j_2j_3}\gamma_{j_1j_3}\delta_{j_2 l}= -\delta_{i_3l}+\delta^{i_1i_2}G_{i_1i_2i_3}^{\eta j_2j_3}\delta_{j_2j_3}\tilde\gamma_{\eta l}
\tilde{D}_{a}f(x \pm y) = \tilde{B}_{f}(x \pm y,a)+\tilde{D}_{a}f(0) = \tilde{B}_{f}(x,a) \pm \tilde{B}_{f}(y,a) + \tilde{D}_{a}f(0) &= \left(\tilde{D}_{a}f(x) - \tilde{D}_{a}f(0)\right)\pm \left(\tilde{D}_{a}f(y) - \tilde{D}_{a}f(0)\right)+ \tilde{D}_{a}f(0) &= \tilde{D}_a f (x) \pm \tilde{D}_a f (y) \mp \tilde{D}_a f (0).
\frac{\big( q^{h_i}; -\mu_i(h_i) \big)_q}{\big( q^{\mu_j(h_i)}; -\mu_i(h_i) \big)_q}v_k = \frac{q^{-\mu_i(h_i)}q^{h_i}- 1}{q^{-\mu_i(h_i)}q^{\mu_j(h_i)}- 1}v_k\ = \frac{q^{-\mu_i(h_i)}q^{\mu_k(h_i)}- 1}{q^{-\mu_i(h_i)}q^{\mu_j(h_i)}- 1}v_k,
\frac{(1-(p-1)Y)(1+Y)^{p-1}}{(1-(p-1)\gamma)(1+\gamma)^{p-1}}=\frac{Z}{|\zeta|^p},\frac{Y(1+Y)^{p-2}}{\gamma(1+\gamma)^{p-2}}=\left(\frac{|\eta|^qZ}{|\zeta|^pH}\right)^{1/q}.
0 \leq \phi'_2(z_2)=w'(z_2)+Kf'(u(z_2))w(z_2) begin{align*}0.2cm] =cw(z_2)-f(u(z_2))p(u(z_2))+f(u(z_2))v(z_2)+Kf'(u(z_2))w(z_2) begin{align*}0.2cm] =\left[-(K^2f'(u(z_2))+cK)+v(z_2)-p(u(z_2))\right]f(u(z_2))<0,
\frac{d^2}{dx^2}\Re F(x \pm i/4; x_N(k)) = -\Re \psi'(x + N \pm i/4) = -\sum^{\infty}_{n = 0}\frac{(n + N + x)^2 - \frac{1}{16}}{((n + N + x)^2 + \frac{1}{16})^2}< 0.
\mathbb{E}[f_u(X_{[k]})|X_i=x_i] =\sum_{x_{\hat{i}}}p(x_{\hat{i}}|x_i)f_u(x_{[k]}) &=\sum_{x_{\hat{i}}}p(x_{\hat{i}}|x_i)\frac{p(x_{[k]},u)}{p(x_{[k]})p(u)}&=\frac{p(x_i,u)}{p(x_i)p(u)}
G_{k n}\stackrel{k \rightarrow \infty}{\longrightarrow}- \frac{a_{n}}{\pi}\left( - 1 \right)^{k}\left( \frac{1}{A}\right)^{k}k ! k^{p ( n ) - 1}\left[ 1 +{\cal O}\left( 1 / k \right) \right] .
\begin{aligned}I_3(u_0)+\int_{-\infty}^{\infty}\Big(G(u)-\frac{F(u)}{3}\,\Big)(t,x)dx & \geq \frac12 \|\partial_xu(t)\|_2^2 -c\|u_0\|_2^4\,\|\partial_xu(t)\|_2^2\geq \frac14\|\partial_xu(t)\|_2^2.\end{aligned}
\begin{array}{c}{{a ={i \sqrt{2}}\int_{0}^{2 \pi}\sqrt{u + \cos x^{1 0}}d x^{1 0}= i \left( 4 -{\frac{u - 1}{2}}\log{\frac{u - 1}{2}}+ \ldots \right)}}\\{{a_{D}= \sqrt{2}\int_{\cosh x^{6}< u}\sqrt{u - \cosh x^{6}}d x^{6}= 2 \pi \left({\frac{u - 1}{2}}+ \ldots \right)}}\\ \end{array}
\begin{aligned}f(x+yJ) =\frac{1}{2}\left(f(x+yI)+f(x-yI)\right)+\frac{J I}{2}\left(f(x-yI)-f(x+yI)\right) &=\frac{1-JI}{2}f(x+yI)+\frac{1+JI}{2}f(x-yI).\end{aligned}
E [ \mathcal{G}_{i}^2 \mathcal{G}_{j}^2 ] = E \Bigl[ E_{q_i}[ \mathcal{G}_{i}^2 ] \, E_{q_j}[ \mathcal{G}_{j}^2 ] , \| q_i, q_j \|_{\infty}\le \delta_n \Bigr] + o(n^{-2}) \le c'' \frac{(m_{2i+1}- m_{2i}) (m_{2j+1}- m_{2j})}{n^2 \phi(m_{2i}/n) \phi(m_{2j}/n)}+ o(n^{-2}),
B_{kl}= \sum_{j=1}^3x^k_{j}{x^l_j}^\ast \ , \ \C = x_1^1x_3^2-x_3^1x_1^2 \ , \ \C^\ast ={x_3^1}^\ast{x_2^2}^\ast-{x_2^1}^\ast{x_3^2}^\ast \ ,
\partial_{tt}y+\omega y = -\mathcal{F}(\sqrt{\cdot}F(t,\cdot \lambda(t)))(\omega \lambda(t)^{2})+F_{2}(y)(t,\omega) -\mathcal{F}(\sqrt{\cdot}F_{3}(u(y))(t,\cdot \lambda(t)))(\omega \lambda(t)^{2})
T_k\bigg(\frac1{T_k(1)}\bigg)(x) =\int_{\Bbb R^n}\psi(2^k{\bar \rho}(x,y))\frac1{T_k(1)(y)}d\mu(y) \approx\int_{\Bbb R^n}\psi(2^k{\bar \rho}(x,y)) \frac 1{V_k(y)}d\mu(y) \approx \frac 1{V_k(x)}\int_{\Bbb R^n}\psi(2^k{\bar \rho}(x,y))d\mu(y) = \frac 1{V_k(x)}T_k(1)(x)\approx 1.
\Psi_{1}[ R , \Theta ] ={\frac{k_{1}}{\pi}}^{1 / 2}\int d y e^{- i k_{1}y}\Theta ( y ) \Psi_{0}[ \Theta ] \exp \left[ \int d z{\frac{1}{R ( z )}}\partial_{i}\partial_{i}R ( z ) \right] .
\nu(n,r)=\frac{1}{n!}\big|\{\sigma\in\S_n:\; k_j(\sigma)=0\,\ \forall j\in\overline{r+1, n}\}\big|= \sum_{\ell_r(\bar s)=n \atop \ell_{rn}(\bar s)=0}\prod_{j=1}^r\frac{1}{j^{s_j}s_j!},
\langle \varepsilon, \partial_\tau \phi_i \rangle_{L^2_\rho}=\sum_{j=0}^{M}\varepsilon_j \langle \phi_j, \partial_\tau \phi_i \rangle_{L^2_\rho}+ \langle \varepsilon_{-}, \partial_\tau \phi_i \rangle_{L^2_\rho}
F(z+\omega_2\,|\,\overline\omega_1,\omega_2)=\exp\Bigg(\int_{C}\frac{e^{(z+\omega_2)s}}{(e^{\overline\omega_1 s}1-1)(e^{\omega_2 s}-1)}\frac{ds}{s}\Bigg) , 0<(z)<(\overline\omega_1).
\ \sum_{n_r=1}^\infty\;\sum_{k_r=1}^{n_r}\;{n_r \choose k_r}\;\frac{(-)^{k_r}}{n_r}\;k_r\;E_r^k\;=\;E_r\;\cdot\;\sum_{n'=0}^\infty\;(E-E_r)^{n'}\;=\;E\;.
A_{1 / 2}({\bf p};{\bf n}) = B \, \left( \begin{array}{c c c}{{( p_{0}+ m - p_{3}) e^{- i \varphi / 2}\cos ({\theta}/ 2 ) - p_{-}e^{i \varphi / 2}\sin ({\theta}/ 2 )}}\\{{( p_{0}+ m + p_{3}) e^{i \varphi / 2}\sin ({\theta}/ 2 ) - p_{+}e^{- i \varphi / 2}\cos ({\theta}/ 2 )}}\\ \end{array}\right) ,
{\rm det\,}\left((a_{j,\ell})^{n-d}_{j,\ell=0}\right)\neq0, &a_{0,\ell}=f_{\ell}(p),\ \ \ell=0,\ldots,n-d, &a_{j,\ell}=(Z_{j}f_{\ell})(p),\ \ \ell=0,\ldots,n-d, j=1,\ldots,n-d.
\sqrt{T}\left(f^{\lambda}_{\hat{\b{z}}}- f^{\lambda}_{\b{z}}\right) =: f_{-1}+f_0, f_{-1}= \sqrt{T}(T_{\hat{\b{x}}}+\lambda)^{-1}(g_{\hat{\b{z}}}- g_{\b{z}}), f_{0}= \sqrt{T}(T_{\hat{\b{x}}}+\lambda)^{-1}(T_{\b{x}}-T_{\hat{\b{x}}}) f^{\lambda}_{\b{z}}.
\frac{\dim H - \dim H_{V_I}}{\dim V_I}= \frac{\dim G_{A,\ell}^\circ - \dim (G_{A,\ell}^\circ )_{V_\ell(A_I)}}{\dim V_\ell(A_I)}= \frac{\dim G_{A_I,\ell}^\circ}{2\dim A_I},
C(L = i+1) &= \max_{P_{{\bf X}_j |Y^{j-1}}\in \mathcal X(B^{\prime}_{a,j}), j\in[i]}\left( \sum_{j=1}^i H(Y_j | Y^{j-1},{\bf S}) \right. & \left. + \max_{P_{{\bf X}_{i+1}| Y^i}\in \mathcal X(B^{\prime}_{a,i+1})}H(Y_{i+1}| Y^{i},{\bf S}) \right),
\bar{F}(x) = \rho_{1}f^*(\gamma) \bar{B}^{r}(x) + \rho_{2}\bar{V}^{r}(x)+\rho_{1}f^{*}(\gamma)\bar{F}_{\gamma}(x) & + \rho_{1}f^{*}(\gamma)\left\lbrace \int_{0}^{\Delta}\bar{F}(x-u)dB^r(u) + \int_{\Delta}^{x-\Delta}\bar{F}(x-u)dB^r(u) + \int_{x-\Delta}^{x}\bar{F}(x-u)dB^r(u)\right\rbrace,
P^{-}= \frac{1}{2 p^{+}}( p^{i})^{2}+ \frac{1}{2 p^{+}}\sum_{n = 1}^{\infty}\left( \alpha_{- n}^{i}\alpha_{n}^{i}+ \tilde{\alpha}_{- n}^{i}\tilde{\alpha}_{n}^{i}+ n \, S_{- n}^{a}S_{n}^{a}+ n \, \tilde{S}_{- n}^{\dot{a}}\tilde{S}_{- n}^{\dot{a}}\right) \nonumber
f_2(h,k) := \Big[\int_{-\infty}^{\infty}G_\nu(t+h)g_\nu(t)dt\Big]^k &= \mathbb{P}\big(\max_{i=1,\ldots,k}\{T_i^{(1)}-T_i^{(2)}\}\leq h\big) &= \mathbb{P}\big(\max_{i=1,\ldots,k}\{T_i^{(1)}+T_i^{(2)}\}\leq h\big) .
\begin{aligned}R^{ub}\left(p, q,\tau\right) +R^{ub}\left(p+\delta, q+\alpha,\tau\right) > &R^{ub}\left(p+\delta, q,\tau\right)+R^{ub}\left(p, q+\alpha,\tau\right),~ \delta,\alpha >0.\end{aligned}
\frac 1{r^{2}\sqrt{A B}}\left[ r^{2}\sqrt{\frac B A}h^{\prime}\right]^{\prime}= \frac 1{\eta^{2}}\frac{\partial{\cal U}}{\partial h}= \frac{2 h u^{2}}{r^{2}}+ \lambda \eta^{2}h ( h^{2}- 1 ) \ ,
\left|\int_{M}a(x)\left(-\Delta_{g}W_{\delta_{\varepsilon}(t),\eta}+hW_{\delta_{\varepsilon}(t),\eta}-f_{\varepsilon}(W_{\delta_{\varepsilon}(t),\eta})\right)\Phi_{\delta_{\varepsilon}(t),\eta}d\mu_{g}\right|=o(|\varepsilon|).
\langle \hat{T}_{\mu}^{\nu}( x ) \rangle_{R e n .}= \langle \hat{T}_{\mu}^{\nu}( x ) \rangle_{R e g .}- \frac 1{4 \pi^{2}\alpha r^{4}\ln ( q r )}( \xi - 1 / 6 ) d i a g ( 2 , - 1 , 3 , 2 ) \ .
(-q;q^2)_\infty\sum_{k=0}^{\infty}\frac{q^{2kr+\ell}}{1+q^{2kr+\ell}}= (-q;q^2)_\infty\sum_{k=0}^{\infty}\frac{q^{2kr+\ell}}{1-q^{2(2kr+\ell)}}-(-q;q^2)_\infty\sum_{k=0}^{\infty}\frac{q^{2(2kr+\ell)}}{1-q^{2(2kr+\ell)}}=: \sum_{n=0}^{\infty}c_{\ell,r}(n)q^n,
D(x):=\det \left[ \begin{array}{ccccc}\alpha_1 \alpha_1^q \alpha_1^{q^2}\cdots \alpha_1^{q^{n}}\alpha_2 \alpha_2^q \alpha_2^{q^2}\cdots \alpha_2^{q^{n}}\vdots \vdots \vdots \ddots \vdots \alpha_n \alpha_n^q \alpha_n^{q^2}\cdots \alpha_n^{q^{n}}x x^q x^{q^2}\cdots x^{q^{n}}\end{array}\right]
{\Theta_\delta}\left({a,b}\right)= \Gamma \left( a \right)U\left({a,a + \delta + 1;b}\right) &={e^b}\sum\limits_{k = 0}^{a - 1}{\left({\begin{array}{*{20}{c}}{a - 1}k\end{array}}\right)}\frac{{{{\left({- 1}\right)}^{a - k - 1}}}}{{{b^{\delta + k + 1}}}}\Gamma \left({\delta + k + 1,b}\right)
g(T^\psi,\delta T^\psi) =T_{ab}\nabla^aU^b-2U^mT_a^{\phantom{a}n}\varphi^{\phantom{m}}_{mnb}=-\nabla^a(T_{ab})U^b+\nabla^a(T_{ab}U^b)-2U^mT_a^{\phantom{a}n}\varphi^{\phantom{A}}_{mnb}.
V_FN(b,c)=\int_{\mathbb{R}}g_+\frac{1}{\sqrt{2\pi c}}e^{-\frac{(g-b)^2}{2c}}dg=\int_{\mathbb{R}}(g+b)_+\frac{1}{\sqrt{2\pi c}}e^{-\frac{g^2}{2c}}=\int_{\mathbb{R}}(\sqrt{c}g+b)_+\frac{1}{\sqrt{2\pi}}e^{-\frac{g^2}{2}}dg.
0 = \Big \Vert f_{l}- \sum_{\substack{n \in \{1 , \ldots , N \}n \not = l}}\sum_{k \in \mathbb{Z}}c_{n,k}f_{n}( \cdot - k ) \Big \vert L_{2}(\mathbb{R}) \Big \Vert^{2}& \geq C_{1}\Big ( 1 + \sum_{\substack{n \in \{1 , \ldots , N \}n \not = l}}\sum_{k \in \mathbb{Z}}| c_{n,k}|^{2}\Big ) & \geq C_{1}
< \bar{\psi}\psi > ={\frac{1}{2}}\lim_{\beta \rightarrow \infty}{\frac{1}{m_{2}( \beta ) - m_{1}( \beta )}}\int_{m_{1}( \beta )}^{m_{2}( \beta )}d m \lim_{h \rightarrow 0}\lim_{V \rightarrow \infty}\langle i ( \bar{\psi}_{1}\gamma_{5}\psi_{1}- \bar{\psi}_{2}\gamma_{5}\psi_{2}) \rangle
\left\lbrace\begin{array}{lll}\displaystyle \phi_r'' -\Delta\phi_r +\sum_{s=1}^p\alpha_{rs}\phi_s=0 \hbox{in}(0,+\infty) \times \Omega, \displaystyle \partial_\nu\phi_r +\sum_{s=1}^p\beta_{rs}\phi_s=0 \hbox{on}(0,+\infty) \times \Gamma, t=0:\phi_r=(E_r,\widehat{U}_0), \phi_r'=(E_r, \widehat{U}_1) \Omega. \end{array}\right.
A_{0}= ( 4 \pi t )^{d / 2}K ( t ) \sum_{i}\mathrm{t r}\left( T_{i}\right) e^{- m_{i}^{2}t}\left( \delta_{i 1}\mathrm{t r}P + \delta_{i 2}+ \delta_{i 3}+ \delta_{i 1}\xi^{d / 2}\right)
\zeta^h_N(s)=\sum_{\lambda\in Spec\Delta_N}\left(\lambda+h^2\right)^{-s}=\frac{1}{\Gamma(s)}\sum_{\lambda\in Spec\Delta_N}\int_0^\infty t^{s-1}e^{-t(\lambda+h^2)}dt = \frac{1}{\Gamma(s)}\sum_{\lambda\in Spec\Delta_N}\int_0^1 t^{s-1}e^{-t\lambda}e^{-th^2}dt+\frac{1}{\Gamma(s)}\sum_{\lambda\in Spec\Delta_N}\int_1^\infty t^{s-1}e^{-t(\lambda+h^2)}dt.
\begin{cases}J_\phi ''(t) + b(t) J'_\phi (t)\geq 2^{-1}\| \phi^{\ell}\|_{L^1(\mathbb R^n)}^{1-p}J_{\phi}(t)^p \mbox{for}t \in [0,t_0), J_\phi (0) = I_\phi (0) - A(n,p,\ell,\phi ), J_\phi '(0) = A_1 J_\phi (0), \end{cases}
\overline{x_d}(w_{01}w_{02}^{-1})\overline{x_d^{-1}}= (w_{01}w_{02}^{-1})^{-1}= [\overline{x_d},w_{01}^{-1}], \overline{x_d}w_i \overline{x_d^{-1}}= w_i^{-1}
|\gamma_{i}^{k}| = \|f_{i}^{k}\|_{L^{\infty}}|B(y_{i},2^{k}r)|^{1/p}&\leq \frac{|B(y_{i},r)|}{|B(y_{i},2^{k-1}r)|}\|f_{i}\|_{L^{\infty}}|B(y_{i},2^{k}r)|^{1/p}&\lesssim C_{i}(2^{(k-1)})^{n}(2^{kn}|B(y_{i},r)|)^{1/p}&\lesssim C_{i}2^{kn(\frac{1}{p}-1)}|B(y_{i},r)|^{1/p}.
I = \int d^{2}\theta \int d^{4}x_{1}{\bar{W}}^{'^{\prime \prime}2}{\Big \{\frac{W^{\prime \prime}( x_{1}, \theta )}{K_{\Phi \bar{\Phi}}^{2}( x_{1}, \theta )}\Big \}}^{3}S ( p_{1}, p_{2}) |_{p_{1}, p_{2}= 0}
\left(a(y)D^{2}h(y)\right) = \frac{1}{\|y\|}\left( \, a(y) - \sum\limits_{i,j=1}^{m}a_{ij}(y)\frac{y_{i}y_{j}}{\|y\|^{2}}\right) \leq \frac{m\overline{\Lambda}}{\|y\|}.
4I_{\alpha}^{\ \ \mu}I_{\nu}^{\ \ \beta}= \sum_{(a_1, \cdots, a_k) \in \mathfrak{B}}(-1)^{k(k+1)/2+s}(\gamma_{a_1\dots a_k})_{\alpha}^{\ \ \beta}(\gamma_{a_1\dots a_k})_{\nu}^{\ \ \mu}.
A^{k}(i_{1},\dots,i_{l})=\frac{1}{\binom{k-l}{k-2-l}}\sum_{p=1}^{\binom{n-l}{k-2-l}}A^{k}(i_{1},\dots,i_{l},i_{l+1}^{p},\dots,i_{k-2}^{p})=0.
\mathrel{\phantom{=}}d \left( x^k \left( \frac{4}{l+4}y^{l+4}+\frac{2}{l+2}g(x) y^{l+2}\right) \right) - x^k \left( 4y^3+2g(x)y \right) y^l dy &= \left( \frac{4}{l+4}k x^{k-1}y^{l+4}+ \frac{2}{l+2}\left( k x^{k-1}g(x) + x^k g'(x) \right) y^{l+2}\right) dx,
\mathcal{J}_i(\tau_i,\tau_{-i})=\mathbb{E}\left\{\frac{e^{-\beta\tau_i}x_i(\tau_i)}{\frac{1}{N}\sum\limits_{j=1, j \neq i}^Ne^{-\beta\tau_j}x_j(\tau_j)}- e^{-\beta\tau_i}K\right\}.
g \left( \sum_{i=1}^n \i< \eta_i, \pi_Y (a) \eta_i >_B \right) =\sum_{i=1}^n \i< \xi_f, \pi_f ( \i< \eta_i, \pi_Y (a) \eta_i >_B ) x_g \xi_f > &=\sum_{i=1}^n \i< \eta_i \otimes \xi_f, \pi_Y (a) \eta_i \otimes x_g \xi_f > &=\sum_{i=1}^n \i< \eta_i \otimes \xi_f, \theta_Y^K (a \otimes x_g ) ( \eta_i \otimes \xi_f) >.
X^{m}( \sigma^{\pm 2}, \eta^{\pm}) = X^{m}( \sigma^{\pm 2}) + \frac{i}{\sqrt{8}}\eta^{+}\psi_{+}^{m}( \sigma^{\pm 2}) + \frac{i}{\sqrt{8}}\eta^{-}\psi_{-}^{m}( \sigma^{\pm 2}) + i \eta^{+}\eta^{-}F^{m}( \sigma^{\pm 2}) ,
S [ x^{\mu}, p_{\mu}; \lambda ] = \int_{\tau_{i}}^{\tau_{f}}d \tau \left[ \dot{x}^{\mu}P_{\mu}- \lambda \phi \right] = \int_{\tau_{i}}^{\tau_{f}}d \tau \left[ \dot{x}^{\mu}P_{\mu}- \frac{1}{2}\lambda \left[ P^{2}+ m^{2}\right] \right] \ .
d s^{2}= \frac{1}{1 6}( d \psi - \sum_{i = 1}^{3}\cos{\theta_{i}}\, d \phi_{i})^{2}+ \frac{1}{8}\sum_{i = 1}^{3}( d \theta_{i}^{2}+ \sin^{2}{\theta_{i}}\, d \phi_{i}^{2}) \ ,
T^{(m)}_{1,e'_1}(i) = T^{(m)}_{1,e''_1}(i), T^{(m)}_{1,e'_2}(i) = T^{(m)}_{1,e''_2}(i), f^{(m),i}_{1,e'_1}\overline{f^{(m),i}_{1,e'_2}}= -f^{(m),i}_{1,e''_1}\overline{f^{(m),i}_{1, e''_2}}\quad e'_1-e'_2 =e''_1-e''_2.
\Vert M_{\widetilde{\phi}}(a_1, \ldots, a_n ) \Vert_{S_p^N}\leq \Vert T_{\phi}^{(N)}\Vert \Vert a_1 \Vert_{S_{p_1}^N}\ldots \Vert a_n \Vert_{S_{p_n}^N}.
D_{k}=\frac{1}{a_{0}^{(n+\gamma)/ \beta}}\sum_{j=0}^{n}{b_{n-j}\sum_{i=0}^{j}{-\frac{n+\gamma}{\beta}\choose i}\frac{1}{a_{0}}{B}_{j,i}(a_{1},a_{2},...a_{j-i+1})},
\mathcal{G}^{4s}_{1}(\vec{a};\vec{b}) =\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}dx_1dx_2\cdots dx_n &=\prod_{i=1}^{n}|b_i|-|a_i|.
(X \smallsetminus{\downarrow}^{\mathbb{X}}p^{-1}[Y \smallsetminus U]) \cup (X \smallsetminus{\downarrow}^{\mathbb{X}}p^{-1}[Y \smallsetminus V]) = X \smallsetminus{\downarrow}^{\mathbb{X}}p^{-1}[Y \smallsetminus (U \cup V)].
{\bf T}_{V_\omega}\big( \lambda_q(k) \big) = \pm \omega \frac{\sqrt{\vert W \vert}{\bf P}_q^\# \sqrt{\vert W \vert}}{k}\pm \omega \sqrt{\vert W \vert}{\bf Q}_q^\# \big( \mathcal{H}_0(b) - \lambda_q(k) \big)^{-1}\sqrt{\vert W \vert}.
I_{h,k,n}(x)=\omega_{h,k,n}(x) \,\int_{- \frac{1}{k (k+ k')}}^{\frac{1}{k(k+k'')}}\exp\left(\frac{L_k(x)^3}{8 \pi^2 ( \alpha - iv)^2}+ 2 \pi n ( \alpha - iv)\right)\exp\left(g_{h,k}(x, 2 \pi ( \alpha - iv)) \right) \, dv .
S^3 \setminus N(K) = (\Sigma_g^1 \times [0,1])/ ((x, 1) \sim (\phi_K(x), 0)) &= (f(\Sigma_g^1) \times [0,1])/ ((x, 1) \sim (f\circ\phi_K\circ f^{-1}(x), 0)).
C_{\mathbb{R}^{3}}(v,\varphi)\overset{{\rm def}}{=}\left\{u\in\mathbb{R}^{3}\::\: \left\langle u, v\right\rangle \geq\cos\varphi\left\Vert u\right\Vert \left\Vert v\right\Vert \right\}.
\ \ \left. \left. + \frac{3}{2}\gamma \varphi_0^2 \right )\dot{\phi}\left ( \frac{\dot{a}}{a}\right ) \right ] + e^{-2 k \phi}\left [ \frac{\gamma}{2}\varphi_0^2 k^2 \dot{\phi}^2 - V_0 \varphi_0^{\alpha}- e^{\psi (2 - \alpha)}\right ] \ .
\|w\|^2=\lambda\int_{-1}^1 f(x)|w_1(x, 0)|^{p}|w_2(x,0)|^q dx +\int_{-1}^1 g_1(w_1(x, 0),w_2(x,0))w_1(x, 0)dx &+\int_{-1}^1 g_2(w_1(x, 0),w_2(x,0))w_2(x, 0)dx.
\Delta(\sigma_t\circ\sigma'_s) =(\tau_t\otimes\sigma_t)\bigl(\Delta(\sigma'_s)\bigr)=\bigl((\tau_t\circ\sigma'_s)\otimes(\sigma_t\circ\tau_{-s})\bigr)\circ\Delta &=\bigl((\sigma'_s\circ\tau_t)\otimes(\tau_{-s}\circ\sigma_t)\bigr)\circ\Delta=(\sigma'_s\otimes\tau_{-s})\bigl(\Delta(\sigma_t)\bigr)=\Delta(\sigma'_s\circ\sigma_t).
T_gT_l^nh_1 = T_l^nT_g T_{[g, l^n]}h_1 = T_l^n T_g T_q^n T_{r_n}h_1 = T_l^n T_g \gamma(q^{-1})^n h_1 = \gamma(q)^{-n}h_1(g^{-1}x_0) T_l^n h_1,
\widetilde{Q}_{\lambda}^{-{\alpha_{\Re}}}({w^+}'):=\int_{W_a^+}e^{-\widetilde{\mathrm{Q}}({w^+}',w^+)}(w^+)^{\alpha_{\Re}}\;\mathcal{D}_\lambda w^+:=\int_{W_a^+}e^{2\pi i\langle w'^+,w^+\rangle}e^{-\mathrm{Q}(w^+)}(w^+)^{\alpha_{\Re}}\;\mathcal{D}_\lambda w^+\;,
\sum_{i=1}^t(n+1-i)a_i =(n+1)\sum_{i=1}^ta_i-\sum_{i=1}^tia_i &=\begin{cases}\frac{1}{2}n^2t+O\left(n^{19/8}\right),~ ~t\geqslant n^{1/2}-n^{1/4}\frac{1}{2}n^2t+O\left(n^{9/4}\sqrt{n^{1/2}-t}\right),~ t<n^{1/2}-n^{1/4}.\end{cases}
\Vert u-Jz_{\mathrm{nc}}\Vert_{L^2(\Omega)}^2 = a(u,{w}-JI{w})+a_{\mathrm{pw}}({w}, z_{\mathrm{nc}}-Jz_{\mathrm{nc}})+a(u, JI{w})-a_{\mathrm{pw}}({w},z_{\mathrm{nc}}).
\begin{array}{c}h=\left\{A^{-},A^{+}\right\}_{q}=2\left\{\left( \frac{1}{\cos\frac{\omega x}{2}}\cdot \cosh \frac{i}{2}\frac{d}{dx}\right)^{2}-\cosh \frac{\omega}{4}\right\}= =\frac{\stackrel{\wedge}{P}}{2}+V\left( x\right)\end{array}
\{(m_{1}+2m_{2}) (\cot \langle 2\alpha_{1}, H\rangle )^{2}-n_{3}\}((\cot \langle 2\alpha_{1}, H\rangle )^{2}-1) = -(m_{1}+2m_{2})(\cot \langle 2\alpha_{1}, H\rangle )^{2}.
\geq \Lambda^{-2}\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|tw_{0}(x)-tw_{0}(y)|^{p}}{|x-y|^{N+ps}}dx dy-t^{p_{s}^{\ast}}\int_\Omega |w_{0}|^{p^{\ast}_{s}}dx-\lambda \int_{\Omega}f(x,tw_{0})tw_{0}dx &=\Lambda^{-2}t^{p}[w_{0}]_{s,p}^{p}-t^{p_{s}^{\ast}}\int_\Omega |w_{0}|^{p^{\ast}_{s}}dx-\lambda \int_{\Omega}f(x,tw_{0})tw_{0}dx
s(x_n\vert\cdots\vert x_0) \quad= \sum_{\substack{0\leq i\leq n+1 i}}d_i(x_n\vert\cdots\vert x_0) \quad+a t(u_0\vert\cdots\vert u_n) \quad= \sum_{\substack{0\leq i\leq n+1 i}}d_i(x_n\vert\cdots\vert x_0) \quad+b \:,
\begin{aligned}\mathcal{I}(\mathbf{F}_R, \mathbf{F}_B) \approx \log_2 \left( \left| \mathbf{I}_{N_s}+ \frac{\gamma}{N_s}\mathbf{\Sigma}_1^2 \right| \right) - N_s + ||\mathbf{V}_1^H\mathbf{F}_R\mathbf{F}_B||_F^2,\end{aligned}
A^{g}~=~\int_{M_g}\frac{1}{(\det \rm{Im}\tau)^{2}}\int\prod_{i=1}^{2g}d^{2}x_{i}|\det{\omega_{i}(y_j)}\det{\omega_{i}(u_{j})}|^2\langle \prod_{k=1}^{3g-3}|(\mu_k G^{-})|^2\rangle_{\rm top},
f_{2 2}^{>}( L , m , z , d ) = - \frac{1}{4 L^{d - 2}}\sum_{k \geq \frac{d - 3}{2}}^{\infty}C^{( 1 )}( d , k ) ( L m )^{2 k}\int_{L m}^{\infty}d u \, u^{d - 3 - 2 k}( \coth u - 1 ) \cosh ( \frac{2 u z}{L}) .
u_{n+1}^{(c)}= u_{n+1}& - \Delta t \frac{5}{6}F_n + \Delta t \frac{2}{3}F_{n+\frac{1}{2}}+ \Delta t \frac{1}{6}F_{n+1}& - \frac{\Delta t}{2}V_m V_m^T ( F_{n+\frac{1}{2}}- F_n ).
G_{lm,l^{\prime}m^{\prime}}\left( \mathbf{R}\left( R,\theta,\phi\right)\right) =\sum_{m_{1},m_{1}^{\prime}}D_{m,m_{1}}^{\left( l\right)}\left(0,-\theta,-\phi\right) \left[ D_{m^{\prime},m_{1}^{\prime}}^{\left(l^{\prime}\right)}\left( 0,-\theta,-\phi\right) \right]^{\ast}G_{lm_{1},l^{\prime}m_{1}^{\prime}}\left( R\mathbf{\hat{e}}_{z}\right).
N_{1 1}= \left( \begin{matrix}{{\frac{{\partial^{2}}V}{\partial{T_{1}}^{2}}}}\\ \end{matrix}\right)_{0}, N_{1 2}= \left( \begin{matrix}{{\frac{{\partial^{2}}V}{\partial{T_{1}}\partial{T_{2}}}}}\\ \end{matrix}\right)_{0}, N_{2 2}= \left( \begin{matrix}{{\frac{{\partial^{2}}V}{\partial{T_{2}}^{2}}}}\\ \end{matrix}\right)_{0}
S=\frac{1}{4}A_{\rm EH}=\frac{1}{4}\left[(r^2 C)^\frac{\tilde d+1}{2}D^\frac{d-1}{2}\right]_{r=r_+}=\frac{1}{4}\Omega_{\tilde d+1}r_+^{\tilde d+1}\left[1-\left(\frac{r_-}{r_+}\right)^{\tilde d}\right]^\frac{\tilde d+2}{2 \tilde d}.
\mathcal{O^+}= \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}\sum_{c > 0}\frac{S(n, p; c)}{c}H_{m, n}^+ \Bigl (\frac{4 \pi \sqrt{n p}}{c}\Bigr ),
\widetilde{J}_1^N(m_0^{\otimes N}, \gamma^N,\widetilde{\gamma}^N):= \mathbb{E}_N\left[ \sum_{t=0}^T f(t,\widetilde X_t^{1,N},\mu_t,\widetilde u^{1,N}_t ) + F(\widetilde X_T^{1,N},\mu_T)\right],
L_{\omega,\mathbf{c}}(V) =\sum_{j=1}^3\left(C_{1,j}\|\nabla v_j\|_{L^2}^2+C_{2,j}\|v_j\|_{L^2}^2+\sum_{k=1}^dC_{3,j,k}\|\partial_kv_j+c_{j,k}v_j\|_{L^2}^2\right) &=\sum_{j=1}^3\left(C_{1,j}\sum_{k=1}^d\|\partial_k v_j\|_{L^2}^2+C_{2,j}\|v_j\|_{L^2}^2+\sum_{k=1}^dC_{3,j,k}\|\partial_kv_j+c_{j,k}v_j\|_{L^2}^2\right)
\operatorname{tr}_{\mathcal K}\left( V^n(A\otimes\omega)(V^\dagger)^n \right)= \operatorname{tr}_{\tilde{\mathcal K}}(\operatorname{tr}_{\ell_2}(U_n( A\otimes\tilde\omega ) U_n^\dagger\otimes e_ne_n^\dagger))= \operatorname{tr}_{\tilde{\mathcal K}}( U_n( A\otimes\tilde\omega )U_n^\dagger ) = T^n( A)
\mathcal{B}_p[\mathcal{H};X]:= \left[\mathcal{B}_1[\mathcal{H};X],\mathcal{B}_\infty[\mathcal{H};X]\right]_{1/p}\mbox{(in the sense of complex interpolation)}\mathcal{B}_{p,r}[\mathcal{H};X]:= \left[\mathcal{B}_1[\mathcal{H};X],\mathcal{B}_\infty[\mathcal{H};X]\right]_{1/p,r}\mbox{(in the sense of real interpolation)}.
k_p([\omega])\big(\big[H^{(2)}\big]-f_p([\omega])\big)+\prod\limits_{i=1}^{p}([\omega]- h_{i,1})\prod\limits_{i=1}^{p-1}([\omega]-h_{mp+p+i,1}) \prod\limits_{i=1}^{m^2p}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg)=0
|G_i| \le \Big( \frac{i-1}{d+i-1-k}+ \frac{1}{d+i-k}\Big) \prod_{j=1}^i s_j p_j- \Big( \frac{i-1}{d+i-1-k}- \frac{i-1}{d+i-k}\Big) \prod_{j=1}^i s_jp_j & = \frac{i}{d+i-k}\prod_{j=1}^i s_jp_j.
\int_0^1 t^n v(t) \, dt = \int_0^r t^n v(t) \, dt + \int_r^1 t^n v(t) \, dt \le r^n \int_0^1 v(t) \, dt + \sup_{x \in [r,1]}\frac{v(x)}{w(x)}\int_r^1 t^n w(t) \, dt.
P(\cdot)=\sup_h\frac{{\rm Rm}_g(h,h)}{|h|^2}=\sup_h\frac{{\rm R}_{ijkl}(\cdot)h^{ik}h^{jl}}{h_{ik}h^{ik}}, ~\forall~{\rm symmetric}~{\rm 2-tensor}~ h\neq 0.
w_{n}( z ) = \left. \frac{1}{( n - 1 ) !}\frac{\partial^{n - 1}}{\partial w^{n - 1}}S ( z , w ) \right|_{w = 0}= \frac{1}{z^{n}}+ \sum_{m = 1}^{\infty}B_{n m}z^{m - 1},
L\cap(-m,m)\subseteq\bigcup_{q=2}^{\infty}V_{n,q}\cap(-m,m)\subseteq\bigcup_{q=2}^{\infty}\bigcup_{p=-mq}^{mq}(\frac{p}{q}-\frac{1}{q^{n}},\frac{p}{q}+\frac{1}{q^{n}}).
|u|^{2}\equiv|v\oplus u^{\prime}|^{2}=\frac{1}{(1+\frac{v\cdot u^{\prime}}{c^{2}})^{2}}[(v+u^{\prime})^{2}-\frac{1}{c^{2}}(v\times u^{\prime})^{2}]=|u^{\prime}\oplus v|^{2}.
p_{2}= \int_{0}^{\infty}\frac{d \omega}{2 \pi}\int_{0}^{\infty}\frac{d \omega^{\prime \prime}}{2 \pi}\vert \langle o u t \omega \omega^{\prime \prime}\vert i n \rangle \vert^{2}= \int_{0}^{\infty}\frac{d \omega}{2 \pi}p_{2 \omega}.
\frac{L(x;s,t)-L(x;t,s)}{s-t}=\frac1{s-t}\left[\frac{x}{1-sx-tx^2}-\frac{x}{1-tx-sx^2}\right] &=\frac1{s-t}\left[\frac{(s-t)(1-x)x^2}{(1-sx-tx^2)(1-tx-sx^2)}\right].
K_1(z)= \frac{1}{N+2}+\frac{(t+1)(1-z)}{N+2}\int_0^1\frac{(1-u)^{N+2}}{(1-u+zu)^{t+2}}\,du = \frac{1}{N+2}+\frac{(t+1)(1-z)}{N+2}K_2(z).
\{x_{1},x_{2}\}=\{(\begin{matrix}0\\ 3\\ -9\\ 9\\ -3\end{matrix})(\begin{matrix}1\\ -15\\ 30\\ -1\\ -45\end{matrix})\},\{y_{1},y_{2},y_{3}\}=\{(\begin{matrix}0\\ 0\\ 0\\ 0\\ 9\end{matrix})(\begin{matrix}0\\ 0\\ 0\\ 3\\ 0\end{matrix})(\begin{matrix}0\\ 0\\ 1\\ -2\\ 0\end{matrix})\}.
m_2 =-\frac{1}{3}\left[\mu+\frac{-1+i\sqrt{3}}{2}\left(\mu+\sqrt{3}\mu^{1/2}-\frac{3}{2}+O(\mu^{-1/2})\right)+\frac{-1-i\sqrt{3}}{2}\left(\mu-\sqrt{3}\mu^{1/2}-\frac{3}{2}+O(\mu^{-1/2})\right)\right] &=-\frac{1}{3}\left[\frac{3}{2}+3i\mu^{1/2}+O(\mu^{-1/2})\right] &=-\frac{1}{2}-i\mu^{1/2}+O(\mu^{-1/2}),
\mathbb E\left[\|\Delta_i \mathcal S B^{\mathfrak H}\|^4\right] \geq \left(\int_0^{1-i\Delta_n}\mathbb E\left[\left(B^{\mathfrak H}_{i\Delta_n+y}-B^{\mathfrak H}_{(i-1)\Delta_n+y}\right)^2\right]dy\right)^2 = (n-i)^2 \Delta_n^{2+4\mathfrak H}.
\langle{\rm sol}|\frac{1}{2}\frac{(\delta m^{2})^{2}}{\lambda}\int_{-\infty}^{+\infty}[1-\cos \frac{\sqrt{\lambda}}{m}\phi_{\rm sol}(x)]dx=\frac{\lambda}{8m L^{2}}(\sum\frac{1}{\tilde{\omega}})^{2}