harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
r\to x(r)+\mathrm{i}\cdot y(r)=\C \sqrt[4]{r^{2}+1}\cos\left(\frac{1}{2}\mathrm{arctg}\,r\right)+ &+\mathrm{i}\cdot\left(r+C\sqrt[4]{r^{2}+1}\sin\left(\frac{1}{2}\mathrm{arctg}\,r\right)\right),\r\in\left[0,+\infty\right).
\bar{u}( p + q ) \gamma_{\mu}u ( p ) = \frac{( 2 p + q )_{\mu}}{2 m}\bar{u}( p + q ) u ( p ) + i \epsilon_{\mu \nu \lambda}\frac{q^{\lambda}}{2 m}\bar{u}( p + q ) \gamma^{\nu}u ( p ) \, .
-2 = (R_1)^2 &= ((R_1-A)+A)^2 &=-2(y_1^2+z_1^2+w_1^2+u_1^2+1)+2((R_1 \cdot A)+2) &= -2(y_1^2+z_1^2+w_1^2+u_1^2)+2((R_1 \cdot A)+1).
\int_{D}g(x)\phi(x)dx=\int_{0}^{R}\frac{a_{0}e^{-\alpha r}}{4R^{2}r}\sin \frac{\pi r}{R}dr >\frac{a_{0}}{4R^{3}}\int_{0}^{R}e^{-\alpha r}\sin\frac{\pi r}{R}dr>(\frac{\pi^{2}}{R^{2}}+1)^{\frac{3}{5}},
\Theta\left(f_{1},f_{2},T\right)=d_{Q}\int_{\mathbb{R}^{nd}}f_{2}\left(Q_{0}\left(y_{1}\right)-Q_{0}\left(y_{2}\right),\dots,Q_{0}\left(y_{n-1}\right)-Q_{0}\left(y_{n}\right)\right)f_{1}\left(T^{-1}\left\Vert g_{0}^{-1}y\right\Vert \right)dy,
S(x) = \left(\begin{array}{cccc}0_k K_{--}(x) \Sigma_-(0) B + K_{-+}(x)\Sigma_+ (0) begin{align*}6pt]0_{m, k}K_{+-}(x)\Sigma_-(0) B + K_{++}(x) \Sigma_+(0)\end{array}\right) = \left(\begin{array}{cccc}0_k S_{-+}(x) begin{align*}6pt]0_{m, k}S_{++}(x)\end{array}\right).
G^{[ F ]}= - d x^{0}\otimes d x^{0}+ d x^{1}\otimes d x^{1}+ g_{i^{\prime}j^{\prime}}^{[ F ]}( x , y ) d x^{i^{\prime}}\otimes d x^{j^{\prime}}+ g_{i^{\prime}j^{\prime}}^{[ F ]}( x , y ) \delta y^{i^{\prime}}\otimes \delta y^{j^{\prime}}
\nabla^{\mu}\partial_{\mu}\Phi^{a}= -{\frac{\Phi^{b}( \nabla^{\mu}\partial_{\mu}\Phi^{b})}{\eta^{2}}}\Phi^{a}={\frac{( \partial^{\mu}\Phi^{b}) ( \partial_{\mu}\Phi^{b})}{\eta^{2}}}\Phi^{a}\, .
M_{i j}= \left( \begin{array}{c c}{{\frac{\partial \beta_{( 2 i )}^{\lambda}}{\partial \lambda_{( 2 j )}}}}&{{\frac{\partial \beta_{( 2 i )}^{\lambda}}{\partial \xi_{( 2 j )}}}}\\{{\frac{\partial \beta_{( 2 i )}^{\xi}}{\partial \lambda_{( 2 j )}}}}&{{\frac{\partial \beta_{( 2 i )}^{\xi}}{\partial \xi_{( 2 j )}}}}\\ \end{array}\right) .
\left(\frac{z}{2}\right)^{\nu}\,K_{\nu}(z)=\frac{\sqrt{\pi}}{\Gamma\left(\frac{1}{2}-\nu\right)}\,\intop_{1}^{\infty}e^{-zt}\left(t^{2}-1\right)^{-\nu-\frac{1}{2}}dt,\,\,\,(\nu)<\frac{1}{2},\,\,\,|\arg(z)|<\frac{\pi}{2},
\left\langle \phi_{2}({\bf x}), t_{2}\left| \right. \phi_{1}({\bf x}), t_{1}\right\rangle =\left\langle \phi_{2}({\bf x}) \left| \hat{T}\exp \left[ -\frac{i}{\hbar}\int_{t_{1}}^{t_{2}}\hat{H}dt \right]\right| \phi_{1}({\bf x}) \right\rangle \; ,
\Phi({\bf r}_1^{(1)},\ldots,{\bf r}_{n_1}^{(1)},\ldots,{\bf r}_1^{(M)},\ldots,{\bf r}_{n_M}^{(M)},t) =\left\langle 0\left|\prod_{p=1}^M\frac{1}{\sqrt{n_p !}}\Psi_p ({\bf r}_1^{(p)},t) \cdots\Psi_p({\bf r}_{n_p}^{(p)},t)\right|\Phi\right\rangle
(T')^m(f,g(x):=\sum_{j>N}\int\left|\psi_{aj+m}f*\Phi_{aj+m}\left(x-\frac{t^a+\epsilon_P(t)}{2^{aj}}\right)\right| \left|\psi_{bj+m}g*\Phi_{bj+m}\left(x-\frac{t^b+\epsilon_Q(t)}{2^{bj}}\right)\rho(t)\right| \,dt.
h(1) = \int_{B_{n-j+1}(1)}u^{j}(s_{j})\dots u^n (s_n) d\L^{n-j+1}= I_{n-j+1}(u) , g(1) = \int_{A_{j-1}(1)}u^{j-1}(s_{j-1})\dots u^{1}(s_{1}) d\L^{j-1}= I^\flat_{j-1}(u).
-12H\sum_i \mu_i a_{ii}-3 \vert\mathring{a}\vert^2 &=-12H\sum_i \mu_i \mathring{a}_{ii}-3 \vert\mathring{a}\vert^2 &=-3\sum_i \left( \mathring{a}_{ii}^2 -4H \mu_i \mathring{a}_{ii}+4H^2\mu_i^2\right)+12H^2\vert\mathring{A}\vert^2 -3\sum_{i\neq j}\mathring{a}_{ij}^2 &\le 12H^2\vert \mathring{A}\vert^2.
f(n,\mathbf{d})+(d_{i}-1)f(n-1,\mathbf{d})=\begin{cases}d_1-1, \mbox{if}\r=1; d_i f\big(n,(d_1,\dots,\hat{d_i},\dots,d_r)\big)+d_i-1, \mbox{if}\r\geq 2.\end{cases}
\bar \nabla_{U}\bar T(V,W) =\bar \nabla_{\tilde X}\big((T^{\nabla}(Y,Z)^{\sim})+\xi (F^A(Y, Z))\big) &=\big(\nabla_{X}T^{\nabla}(Y,Z)\big)^{\sim}+ \xi( \nabla^A_XF^A(Y, Z)).
\begin{array}{lcl}\phi_a = ( \phi_+,\phi_-, H, \chi) Y_a = ( Y_+,Y_-, Y_H, Y_\chi) \end{array}\quad \begin{array}{lcl}\hat \phi_a = ( \hat \phi_+,\hat \phi_-, \hat H, \hat\chi) q_a = ( q_+,q_-, q_H, q_\chi) \end{array}
\Psi(\{\vec{r}_{n}\})=(\frac{3}{2\pi l^{2}})^{3N/2}exp[-\frac{3}{2l^{2}}\int_{0}^{N}dn(\frac{\partial\vec{R}_{n}}{\partial n})^{2}-\beta\int_{0}^{N}dnU_{e}[\vec{R}(n)]].
\begin{aligned}\|\nabla^2 \varrho_t\|_{L^2}^2 =\|-\nabla^2{\rm div}u- \nabla^2(\varrho{\rm div}u+u \cdot \nabla \varrho)\|_{L^2}^2 &\lesssim \|\nabla^3 u\|_{L^2}^2 +\|\nabla^2 (\varrho,u)\|_{L^3}^2\|\nabla(\varrho,u)\|_{L^6}^2 &\ +\|(\varrho,u)\|_{L^\infty}^2\|\nabla^3 (\varrho,u)\|_{L^2}^2 &\lesssim (1+t)^{-\frac{7}{2}}.\end{aligned}
\frac{\langle\Psi,\Psi\rangle}{\langle\Phi_1,\Phi_2\rangle}\frac{\langle\Phi_1(\underline{\delta s}(\mu_{\underline m})),\Phi_2(\underline{\delta s}(\mu_{-\underline m}))\rangle}{\langle\mathfrak{f},\mathfrak{f}\rangle}\prod_{\sigma\mid\infty}\beta_\sigma\alpha_{\sigma}=\prod_{\sigma\mid\infty}\frac{C_\sigma}{L(1/2,\Pi_\sigma,\chi_\sigma)},
\big\langle f, D_{k'+\ell}D_{k'}(R_{N})^{m-1}D_{k}^{N}D_{k}(g)\big\rangle =\big\langle D_{k'+\ell}(f),D_{k'}(R_{N})^{m-1}D_{k}^{N}D_{k}(g)\big\rangle &=\big\langle D_{k'}D_{k'+\ell}(f),(R_{N})^{m-1}D_{k}^{N}D_{k}(g)\big\rangle.
\frac{1}{|\mathcal F_{N,k}|}\sum_{(p_1,p_2,\dots ,p_t)}^{(1)}\sum_{(m_1,m_2,m_3,\dots,m_t)}^{(a)}\sum_{f \in \mathcal F_{N,k}}a_f(p_1^{2m_1}p_2^{2m_2}\dots p_t^{2m_t}) & = \O\left(\pi_N(x)^{a}(\log \log x)^{t-a}\right)+ \O\left(\frac{\pi_N(x)^t x^{(2M_1 + 2M_2 + \dots + 2M_t)c'}8^{\nu(N)}}{k{N}}\right).
p_iq_{i-1}q_0 =q_i &=q_i \circ g &=(p_iq_{i-1}q_0 )\circ g=(p_i \circ g) (q_{i-1}\circ g) (q_0 \circ g) , i=1,\ldots,r, &=(p_i \circ g)q_{i-1}q_0, i=1,\ldots,r.
(\kappa^*(t), R^*(t)) = \arg \max_{\substack{\kappa(t) \in \mathcal{P}(\mathcal{F}) R(t) \in \mathcal{R}(t)}}\ \sum_{u \in \mathcal{K}_{R(t)}\cap \tau_{\kappa(t)}}\log \left(\frac{R(t)}{N}\right).
\tilde{t}^2 = \frac{2 \vert v_1 \vert^2}{\vert v_1 \vert^2 + 2 \left(1-\frac{1}{m}\right) \vert \hat{v}\vert^2 p_1^2 + \vert v_1 \vert \sqrt{\vert v_1 \vert^2 + 4 \left(1-\frac{1}{m}\right) p_1^2 \vert \hat{v}\vert^2}}.
\varphi^{\prime \prime}+ \frac{2}{r}\varphi^{\prime}= \frac{e_{0}^{2}}{4 \pi}\; \frac{1}{r^{2}}\sum_{\kappa = 1}^{\infty}\kappa \left( \varrho_{\kappa}+ \varrho_{- \kappa}\right) \, ,
\triangle_{V_s+2\frac{\nabla w}w}(wY) - \left(1+ \left( \frac{\triangle_{V_s}w}{w}- \frac12 \right) - 2 \frac{|\nabla w|^2}{w^2}\right) (wY) + \frac12 \big( dV^\flat (wY, \cdot) \big)^{\sharp}= w U
\mathcal H\left(G, L, \mathcal A\right)=\mathcal H\left(G, L, \mathbb Q\right)\otimes_{\mathbb Q}\mathcal A \mathcal H\left(G, \mathcal A\right)=\mathcal H\left(G, \mathbb Q\right)\otimes_{\mathbb Q}\mathcal A.
P^1=(0.1,0.9)\rightarrow\widetilde{P}^1=(0.1,0.9,1+\varepsilon,1-\varepsilon,1,1,1,1)/7, &P^2=(0.7,0.3)\rightarrow\widetilde{P}^2=(0.7,0.3,1,1,1+\varepsilon,1-\varepsilon,1,1)/7, &P^3=(0.8,0.2)\rightarrow\widetilde{P}^3=(0.8,0.2,1,1,1,1,1+\varepsilon,1-\varepsilon)/7.
{{\mathcal B}_i^{AF}}\left({\pmb \rho}^\prime \right) =\frac{{\sqrt{{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i^\prime}+ 1}\right)}}}}{{\sqrt{\left({{X_i}+{Y_i}+ 1}\right)\left({{W_i^\prime}+ 1}\right)}+ \sqrt{Z_i +W_i^\prime+1}}},
\|\cdot\|=\|\cdot\|^k: C^k \rightarrow [0,1],\|A \| = \sum_{\sigma \in A}w(\sigma)\mbox{where}w(\sigma) = \frac{|\{F \in X(d)\,|\, \sigma \subset F\}|}{{d+1 \choose |\sigma|}\cdot |X(d)|}
\Gamma = -\frac{T^2}{8\pi^2}\int d^4x\,M^2\left[\frac{\,\pi^2}{3}+\frac{1}{8}\frac{M^2}{T^2}\ln\frac{M^2}{T^2}+ \frac{\xi^{(1)}(-2)M^4}{24T^4}+ \cdots\right]\,,
E e^{i \tau \mathcal{G}_s^n + i \nu \mathcal{G}_t^n}= E \left[ E_{q}e^{i \tau (\mathcal{G}_s^n - \mathcal{G}_t^n )}E_{q'}e^{i (\tau + \nu) \mathcal{G}_t^n}\right] = e^{- \frac{\tau^2}{2\beta}\log \frac{\theta(t)}{\theta(s)}}e^{- \frac{(\tau + \nu)^2}{2 \beta}\log \frac{1}{\theta(t)}}+o(1),
Q_v(\varepsilon):= \big[2^*(s_1)-2\big]\Big[\lambda|u|_{2^*(s_1),s_1}^{2^*(s_1)}+\mu|v|_{2^*(s_1),s_1}^{2^*(s_1)}|s|^{2^*(s_1)}\Big]\big[t(\varepsilon)\big]^{2^*(s_1)-3}&+\kappa 2^*(s_2)[2^*(s_2)-2]c(u,v)\big[t(\varepsilon)\big]^{2^*(s_2)-3}|\varepsilon|^\beta
(1+t^{n-1})\binom{[\frac{k}{2}]+[\frac{n-k}{2}]}{[\frac{k}{2}]}_{t^4}=(1+t^{n-1})\frac{\prod_{i=1}^{[\frac{k}{2}]+[\frac{n-k}{2}]}(1-t^{4i})}{\prod_{i=1}^{[\frac{k}{2}]}(1-t^{4i})\prod_{i=1}^{[\frac{n-k}{2}]}(1-t^{4i})}
G \> ' (x,y) \> = \>\> \int_0^{\infty}dT \> N(T) \exp \left ( - \frac{i}{2 \kappa_0}M^2 T \right) \> \left < - \frac{i}{2 \kappa_0}M - i\> \dot x(T) \cdot \zeta (T) \right >_S
0=\mathcal{S}e_{j}\left( z_{j_{\ell}}\right) =\sum_{k=0}^{\ell}b_{j_{k}}\varphi_{z_{j_{k}}}\left( z_{j_{\ell}}\right) +\sum_{i\notin\left\{j_{0},j_{1},...,j_{\ell}\right\}}b_{i}\varphi_{z_{i}}\left( z_{j_{\ell}}\right) .
\|u_1^0(\tau,\xi,\eta)-U(\tau,\xi)\|_{\mathcal{H}^k(\Omega)}^2+\|q^0(\tau,\xi,\eta)-Q(\tau,\xi)\|_{\mathcal{H}^k(\Omega)}^2\leq C_0(1+\tau^k)(M_0+M_e).
D^{+}( x_{1}- x_{2}) = \int \frac{d^{3}k}{( 2 \pi )^{3}2 \omega_{k}}\; \mathrm{e}^{- \mathrm{i}k^{+}( x_{1}- x_{2})}\; , \qquad \omega_{k}= \sqrt{\vec{k}^{2}+ m^{2}}\; ,
\sum_{b=0}^a p^{(a-b)b}= p^{c^2}+ 2 \sum_{b=0}^{c-1}p^{(2c-b)b}= p^{c^2}+ O\bigg( \sum_{b=0}^{c-1}p^{c^2-(c-b)}\bigg) = p^{c^2}+ O\big( p^{c^2-1}\big),
\theta \left( \begin{array}{c}{{\psi_{+}}}\\{{\psi_{-}}}\\ \end{array}\right) = D^{( s )}( i \sigma_{2}) \left( \begin{array}{c}{{\psi_{-}^{*}}}\\{{\psi_{+}^{*}}}\\ \end{array}\right) , \, \, \, \, \, \pm \mathrm{= ( a n t i ) p a r t i c l e \thinspace ~ \thinspace ~ d o u b l i n g}
\mathbb{E}(\langle DY, DY \rangle_\mathcal{H}) =\int_{0}^{1}\int_{0}^{1}g_sg_td\mathbb{E}\langle D_rX_s ,D_rX_t \rangle &=n\int_{0}^{1}\int_{0}^{1}g_sg_td\mathbb{E}(X_s X_t).
\|g\|_2^2= \int_G |f|^2 dx +\sum_l\int_{K_l}(\frac{1}{|K_l|}\int_{K_l}f)^2 dx \le \int_G |f|^2 dx +\sum_l\frac{1}{|K_l|}(\int_{K_l}f dx)^2 \le \int_G |f|^2 dx +\sum_l\int_{K_l}|f|^2 dx=\|f\|_2^2.
|H(u(1),p(1))-\,H(u(2),p(2))|\big|_{{K}'}\leq \,\tilde{C}_U\,\Big(\sum^{8N}_{k=1}\big|(u^k(1)-u^k(2))\big| &+\,\sum^{8N}_{k=1}\,\,\big|\,(p_k(1)-p_k(2))\big|\Big) + 1/2\,\sum^{8N}_{k=1}\big|(u^k(1)-u^k(2))\big|+ 1/2\,\sum^{8N}_{k=1}\,\big|(p_k(1)-p_k(2))\big|
\mathcal{M}_\Sigma(n,r,d)\,:=\, \{(A,u): \bar \partial^A_{j_\Sigma}u\,=\,0\,=\,F_A +e^2 (\mu_\tau^\kappa \circ u) \omega_\Sigma, [c_1(A)]\,=\,d[\Sigma] \}/\mathcal{G}\, .
\int_{0}^{\infty}e^{-\omega x}x^{-\alpha}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|\eta x)dx=\omega^{\alpha-1}G_{p+1,q}^{m,n+1}(\begin{matrix}\alpha,a_{p}\\b_{q}\end{matrix}|\frac{\eta}{\omega}),
0 = \left\langle \phi^{\prime}, \left( ( 5 h_{0}+ 1 ) \; h_{0}\; \Delta \, ( Q_{- 2}) - 1 2 q_{0}\; \Delta \, ( L_{- 1})^{2}+ 6 q_{0}\, ( h_{0}+ 1 ) \; \Delta \, ( L_{- 2}) \right) \left( \phi \otimes \phi_{0}\right) \right\rangle .
\lim\limits_{x\rightarrow 0^+}x \, \coth(x) = \lim\limits_{x\rightarrow 0^+}x+\lim\limits_{x\rightarrow 0^+}\Big(\frac{e^{2x}-e^0}{2x-0}\Big)^{-1}= 0+\frac{1}{\exp'(0)}= 1.
\int_0^{11m-\sigma+\epsilon}\frac{n_{H_1}(r,\chi)}{r}dr =\frac{1}{2\pi}\int_0^{2\pi}\log{|H_1( 11m + (11m-\sigma+\epsilon)e^{i\theta}, \chi )|}d\theta &\quad\quad- \log{|H_1(11m, \chi)|}.
f(z,\alpha,\beta)=2\Gamma(\alpha)^{-1}\Gamma(\beta)^{-1}z^{\frac{\alpha+\beta}{2}-1}K_{\alpha-\beta}(2\sqrt{z})=\frac{1}{\alpha\beta}f_{K}(\frac{z}{\alpha\beta};1,\alpha,\beta),z\ge0
\cdot\left[(2n-1)(4n-1){\cal G}_n+2\sum_{j=0}^{n-1}{\cal G}_{n-j}\left({\cal G}_{j+1}{\cal G}_0c(j,n)-\sum_{k=0}^{j+1}{\cal G}_{j+1-k}{\cal G}_{k}d(j,k,n)\right)\right],\; n\ge 0,
\begin{cases}\U(t+1) <\frac{1}{2}, \sigma^{\eta}_{i}(t+1)=+1,\ \sigma^{\varsigma}_{i}(t+1)=+1; \U(t+1) \geq \frac{1}{2}, \sigma^{\eta}_{i}(t+1)=-1,\ \sigma^{\varsigma}_{i}(t+1)=-1.\end{cases}
Z^{k}_{H,\lambda}(t) =C_{H,k}\int_{\mathbb{R}^k}^{''}\frac{e^{it(\omega_1+\cdots+\omega_k)}-1}{i(\omega_1+\cdots+\omega_k)}&\quad\times\prod_{j=1}^{k}(\lambda+i\omega_j)^{- (\frac{1}{2}-\frac{1-H}{k})}\widehat{B}(d\omega_1)\ldots\widehat{B}(d\omega_k),
\varepsilon\frac{dh_{2}(t,\varepsilon)}{dt}=-{\mathcal{A}}_{2}^{T}(\varepsilon)h_{1}(t,\varepsilon)-{\mathcal{A}}_{4}^{T}(\varepsilon)h_{2}(t,\varepsilon)- \varepsilon \big(P_{2}^{*}(\varepsilon)\big)^{T}f_{1}(t)-\varepsilon P_{3}^{*}(\varepsilon)f_{2}(t),
\left( \begin{array}{c}x_0 x_1 x_2 x_3 \end{array}\right) = \frac{1}{2\sqrt{6}}\sqrt{3-\frac{3}{\sqrt{5}}}\left ( \begin{array}{c}\omega+1 i \omega-1 i \end{array}\right) + \frac{1}{2\sqrt{2}}\sqrt{1+\frac{3}{\sqrt{5}}}\left ( \begin{array}{c}0 \omega 0 -\omega \end{array}\right)
\left( P_{1}+ i P_{2}\right) \psi_{1}\left( x^{1}, x^{2}\right) = \hbar k_{1}\psi_{2}\, \left( x^{1}, x^{2}\right) , \; \left( P_{1}- i P_{2}\right) \psi_{2}\left( x^{1}, x^{2}\right) = \hbar k_{1}\psi_{1}\left( x^{1}, x^{2}\right) \, .
g_{t,l,v}\left(\begin{matrix}0 1 p^{n}0 \end{matrix}\right) = n(-v^{-1}p^{t+l})z(vp^{n-l})g_{t-n+2l,0,v^2}\left( \begin{matrix}1 1+v^{-1}p^{l-n}0 -v^{-2}\end{matrix}\right).
\frac{\partial f_{\mu}}{\partial \dot{x}^{\nu}}+ \frac{\partial f_{\nu}}{\partial \dot{x}^{\mu}}= 0 \qquad \Longrightarrow \qquad \frac{\partial^{2}f_{\mu}}{\partial \dot{x}^{\nu}\partial \dot{x}^{\lambda}}+ \frac{\partial^{2}f_{\nu}}{\partial \dot{x}^{\mu}\partial \dot{x}^{\lambda}}= 0 \ .
\begin{cases}\overline{B}_r \phi_1 \leq u \leq \phi_2 \overline{B}_r u=g \partial B_r F(D^2 u) \leq 0 \{u < \phi_2 \}\cap B_r F(D^2 u) \geq 0 \{u > \phi_1 \}\cap B_r\end{cases}
\sum_{\alpha \in \Phi_H^{+}}\alpha = 2(n-1)t_1+ 2(n-2) t_2 + \dots + 2 t_{n-1}= \sum_{k=1}^{n-2}k(2n-k-1)\alpha_k + \frac{n(n-1)}{2}(\alpha_{n-1}+\alpha_n).
\Vert f \Vert_{B^r_{p,\infty}}:=\sup_{m\in\mathbb{N}}2^{mr}\Vert \sum_{2^m\leq \langle \xi\rangle < 2^{m+1}}d_{\xi}[\xi(x)\widehat{f}(\xi)]\Vert_{L^p(G)}<\infty.
\int_0^T \vartheta^{\prime}\langle u , v \rangle_{[L^{p^{\prime}}_{\sigma}]^* , L^{p^{\prime}}_{\sigma}}\; d t = \int_{\Omega}\int_0^T \mathcal{U}\vartheta^{\prime}\cdot \overline{A_{p^{\prime}}^{\frac{1}{2}}v}\; d t \; d x = - \int_{\Omega}\int_0^T \mathcal{U}^{\prime}\vartheta \cdot \overline{A_{p^{\prime}}^{\frac{1}{2}}v}\; d t \; d x.
C = - \frac{1}{3 \pi}\frac{1}{\beta^{4}}( 4 + \beta^{2}m^{2})^{3 / 2}+ \frac{{\vert e \vert}^{2}H^{2}}{1 2 \pi}\frac{1}{( 4 + \beta^{2}m^{2})^{1 / 2}},
D_{\lambda}^{n,\gamma}\varphi_{v_{j,b,d}}(z)=2^{v_{j}}\Gamma \left( v_{j}+\frac{b+1}{2}\right) D_{\lambda}^{n,\gamma}\left[z^{1-v_{j}/2}w_{v_{j},b,d}(\sqrt{z})\right]
S_{2}={1\over 2}\int dp_{1}dp_{2}dp_{3}dp_{4}\big[{-g^{2}\over 4(2\pi )^{d}}\delta (p_{1}-p_{2}+p_{3}-p_{4}) +i\delta (p_{1}-p_{2})\delta (p_{3}-p_{4}){\sigma^{0}}^{-1}(p_{1}){\sigma^{0}}^{-1}(p_{3})\big]\eta (p_{4},p_{1})\eta(p_{2},p_{3})
\zeta_{\mathrm{fml}}(1-m,\mathbf a,\mathbf x):=(-1)^r(m-1)!\sum_{|\mathbf l|=m+r-1}\prod_{i=1}^r \frac{B_{l_i}(x_i)a_i^{l_i-1}}{l_i!}(\mathbf a \in (\mathbb R^\times)^r,\ \mathbf x \in \mathbb R^r).
f(x,y) = \Re\left[\frac{2(x+iy)\overline{b}+(x+iy)^2\overline{d}+\overline{b^2}c}{1-c\overline{d}}-\frac{|x+iy|^2+(x+iy)^2\overline{c}}{1-|c|^2}-\frac{|b|^2+\overline{b^2}d}{1-|d|^2}\right].
M_{T_w}(x,v) = \frac{c(x)}{(2 \pi T_w(x))^{d/2}}e^{\frac{-|v|^2}{2T_w(x)}}, c (x) = \left ( \int_{\{v \cdot n_x >0\}}\frac{1}{(2 \pi T_w(x))^{d/2}}e^{\frac{-|v|^2}{2T_w(x)}}|v \cdot n_x| \d v \right )^{-1},
\frac{\mbox{d}}{\mbox{d}u}D_u Y(t) = \frac{\ 1 \}{t}\, \int_0^t f^\prime (X(s)) \, \frac{\mbox{d}}{\mbox{d}u}D_u X(s) \, \mbox{d}s & = \frac{\ 1 \}{t}\, \int_u^t f^\prime (X(s)) \, Z(s,u) \, A_1 (u, X_u) \, \mbox{d}s,
P_{m,n}^{(\alpha,\beta)}(x) =\frac{(-1)^m}{\alpha+1+n-m}\left[\frac{1}{2}( \alpha+\beta +n-m+1)(x-1)P_{m}^{(-\alpha-1,\beta-1)}(x)P_{n-m-1}^{(\alpha+2,\beta)}(x)\right. &\qquad\qquad\qquad\qquad+\left.(\alpha -m +1)P_{m}^{(-\alpha-2,\beta)}(x)P_{n-m}^{(\alpha+1,\beta-1)}(x) \right].
\Gamma_{g}^{\alpha \beta \gamma}\; ( x , y , z ) = \Gamma_{r}^{\alpha \beta \gamma}\; ( x , y , z ) -{\frac{i}{4 \pi^{2}}}\; \varepsilon^{\beta i \gamma j}{\frac{\partial}{\partial x_{\alpha}}}{\frac{\partial}{\partial y^{i}}}G ( x , y ){\frac{\partial}{\partial z^{j}}}\delta ( y - z ) .
{\cal F}_{\bar{\mu}\bar{\nu}}^{i}= \partial_{\bar{\mu}}{\cal A}_{\bar{\nu}}^{i}- \partial_{\bar{\nu}}{\cal A}_{\bar{\mu}}^{i}, \ \ \H_{\bar{\mu}\bar{\nu}\bar{\rho}}= ( \partial_{\bar{\mu}}B_{\bar{\nu}\bar{\rho}}+ 2{\cal A}_{\bar{\mu}}^{i}L_{i j}{\cal F}_{\bar{\nu}\bar{\rho}}^{j}) + \mathrm{c y c . \p e r m s .}.
\hat{T}_{CM}[\rho_{CM}(t)]=(\frac{1}{\pi r_{C}^{2}})^{\frac{3}{2}}\int_{\infty}^{\infty}d^{3}xe^{-\frac{(\hat{Q}-x)^{2}}{2r_{C}^{2}}}\rho_{CM}(t)e^{-\frac{(\hat{Q}-x)^{2}}{2r_{C}^{2}}}
A(u_h^*,\varphi_h)+ B(u_h^*,\varphi_h) + D(u_h^*,\varphi_h)+ E(u_h^*,\varphi_h)+ J(u_h^*,\varphi_h) =C(\varphi_h)+ F(\varphi_h)+ I(\varphi_h)\forall\varphi_h \in X_h.
\sum_{i=1}^m \,\frac{b_i}{a_i +b_i}= \sum_{i=1}^m \,\frac{2n_i q_i^2 \xi_i^2}{2(p_i \xi_i-(n_i+1)q_i^2 \xi_i^2) +2n_i q_i^2 \xi_i^2}& = \sum_{i=1}^m \,\frac{2 n_i q_i^2 \xi_i^2}{(\sum_{j=1}^m \,n_j q_j^2 \xi_j^2) -q_i^2 \xi_i^2},
= \frac{p(p-1)^2x^2}{(1-px)(1-p^2x)}\bigg(\frac{x^3}{1-x}- \frac{p^8x^7}{1-p^2x^2}- \frac{px^4}{1-px^2}+ \frac{p^9x^8}{1-p^3x^3}\bigg).
P\cdot Q :=\sum_{\underline{k}\in \Sigma}\sum_{\underline{l}\in \Sigma}\sum_{\underline{0}\, \le \, \underline{i}\, \le \, \underline{k}}\binom{k_1}{i_1}\dots\binom{k_n}{i_n}a_{\underline{k}}\frac{\partial^{|\underline{i}|}\, b_{\underline{l}}}{\partial x_1^{i_1}\dots \partial x_n^{i_n}}\partial^{\underline{k}+ \underline{l}- \underline{i}}.
\frac{1}{q ( q (p_1 , p_2) , p_3)}= \frac{1}{q (p_1 , p_2)}+ \frac{1}{p_3}- 1 = \frac{1}{p_1}+ \frac{1}{p_2}+ \frac{1}{p_3}- 2 = \frac{1}{q (p_1,p_2,p_3)},
{\left( \gamma^{a}\right)_{\alpha}}^{\beta}= \left[ \left( \begin{array}{c c}{i}&{0}\\{0}&{- i}\\ \end{array}\right) , \left( \begin{array}{c c}{0}&{i}\\{i}&{0}\\ \end{array}\right) , \left( \begin{array}{c c}{0}&{i}\\{- i}&{0}\\ \end{array}\right) \right] .
\begin{array}{c c}{{\Delta^{2}= 0 \; , \; \; U^{2}= 0 \; , \; \; V^{2}= 0 ,}}&{{}}\\{{}}&{{}}\\{{\Delta U + U \Delta = 0 , \Delta V + V \Delta = 0 , \; \; U V + V U = 0 .}}&{{}}\\ \end{array}
\widehat\Gamma'\widehat\Sigma_{\varepsilon\varepsilon}^{-1}\bigl[(I_N\otimes\widehat B)M_{z z}\bigl(I_N\otimes\widehat B{}'\bigr)-\widehat\Sigma_{z z}\bigr]\widehat\Sigma_{\varepsilon\varepsilon}^{-1}\widehat\Gamma=0
-\phi_ka_h\nabla v+\sigma_i\nabla\big((x_k+\phi_k)\partial_iv\big)-(x_k+\phi_k)\phi_ia\nabla\partial_iv+\phi_i\partial_iva(e_k+\nabla\phi_k)-\sigma_k \nabla v =\partial_iv(\phi_ia+\sigma_i)(e_k+\nabla\phi_k)-(\phi_ka_h+\sigma_k)\nabla v-(x_k+\phi_k)(\phi_ia-\sigma_i)\nabla\partial_iv.
\Delta_{\alpha,\alpha}(\rho)=\rho^{2\alpha-2}\left(\rho \sin\rho-W_{\alpha,\alpha}(a,\rho)+\int\limits_{0}^{1}(U_{\alpha,\alpha}(t)\cos \rho t+V_{\alpha,\alpha}(t)\sin \rho t)\,dt\right),
\tilde{a}(X)=\begin{pmatrix}(R_3+S_3)\frac{f^3}{3}+S_3\mu\frac{f^2g}{2}S_3\mu\left(\frac{f^3}{3}+\frac{f^2g}{2}\right) S_3\frac{g^3}{3}+(R_3+S_3)\frac{f^2g}{2}+S_3 fg^2 S_3 \frac{g^3}{3}+S_3\mu\left(\frac{f^2g}{2}+ fg^2\right)\end{pmatrix}\mbox{in}\; \Omega=(0,L)
\left|\Phi(T)(u) \right| \leq \sum_{j=1}^{n}\left|\psi_{j}(T(x_{j})) \right| \leq \left\|T \right\|_{X;Y^{\rm dual}}\cdot \left\|(x_{j})_{j=1}^{n}\right\|_{X(E)}\cdot \left\|(\psi_{j})_{j=1}^{n}\right\|_{Y(F^*)}.
{\big(I\odot\bar{I}\big)_o}^{A}{}_{B}= \,\,{\mathbb Y}_{r}{}^A{}_B\big(v^s(\nabla_s\bar{v}^r)+\bar{v}^s(\nabla_s{v}^r)-\tfrac{2}{3}v^r(\nabla_t\bar{v}^t)-\tfrac{2}{3}\bar{v}^r(\nabla_t{v}^t)\big) & + \mathrm{"lower\,\, adjoint\,\, tractor \,\, slots"}.
T_{k}\mathbf{u}_{T}=\sum_{\ell=1}^{\infty}\frac{z_{\ell}\left( k\right)+1}{\operatorname*{i}k}\sum_{m\in\iota_{\ell}}u_{\ell}^{m}\mathbf{T}_{\ell}^{m}+\sum_{\ell=1}^{\infty}\frac{\operatorname*{i}k}{z_{\ell}\left(k\right) +1}\sum_{m\in\iota_{\ell}}U_{\ell}^{m}\nabla_{\Gamma}Y_{\ell}^{m},
h_n=g_{n,1}x_{n,1}^{\varepsilon_{n,1}}y_{n,1}^{-\varepsilon_{n,1}}g_{n,1}^{-1}g_{n,2}x_{n,2}^{\varepsilon_{n,2}}y_{n,2}^{-\varepsilon_{n,2}}g_{n,2}^{-1}\cdots g_{n,m_n}x_{n,m_n}^{\varepsilon_{n,m_n}}y_{n,m_n}^{-\varepsilon_{n,m_n}}g_{n,m_n}^{-1}\sum\limits_{i=1}^{m_n}\rho_{g_{n,i}}(x_{n,i},y_{n,i})<\delta.
\sum_{s=1}^{A}\left(\left|2m+1-2^{j+1}\varphi\left(2^j\varphi(m)+2^js\right)\right|-1\right) =\sum_{s=1}^{A}\left(\left|2m+1-2^{j+1}\left(\frac{m}{2^j}+\varphi(2^js)\right)\right|-1\right) =\sum_{s=1}^{A}\left(|1-2^{j+1}\varphi(2^js)|-1\right)=\sum_{s=0}^{A}\left(|1-2\varphi(s)|-1\right).
\langle T_{z z}\rangle^{\mathrm{i n f}}= - \int \frac{d \kappa \kappa^{3}}{4 \pi^{2}}, \quad \langle T_{0 0}\rangle^{\mathrm{i n f}}= - \int \frac{d \kappa \kappa^{3}}{1 2 \pi^{2}}.
\begin{cases}E_2(q^2)\longleftrightarrow \cfrac{\pi^2}{24\cdot h^2}-\cfrac{1}{4h}E_2(q^4)\longleftrightarrow\cfrac{\pi^2}{4\cdot 24\cdot h^2}-\cfrac{1}{8h}E_4(q^4)\longleftrightarrow\cfrac{\pi^4}{16\cdot 240 \cdot h^4}\end{cases}
\delta(X)=\frac{1}{2}(\mathrm{min}\{m\equiv 2\mu(X)+1 \bmod{2}\mid \exists x \in H^m_{\mathbb{Z}/4}(X),\; U^\ell x\neq 0 \; \mathrm{for \; all}\; \ell\geq 0, \; Qx=0\}-1).
\langle\nabla_{X_{1}}\nabla_{X_{2}}X_{2}-\nabla_{X_{2}}\nabla_{X_{1}}X_{2},X_{1}\rangle =-c\nu^{2}+\displaystyle\frac{c(\lambda_{2}b_{2}^{2}-\lambda_{1}b_{1}^{2})}{\lambda_{2}-\lambda_{1}}-\displaystyle\frac{\nu^{2}(b_{1}^{2}+b_{2}^{2})}{(\lambda_{2}-\lambda_{1})^{2}}.
\liminf_{n\rightarrow \infty}E(V^{j_0}(\cdot-x^{j_0}_n)) \leq \liminf_{n\rightarrow \infty}E(V^{j_0}(\cdot-x^{j_0}_n)) + \liminf_{n\rightarrow \infty}E(v^{j_0}_n) &\leq \liminf_{n\rightarrow \infty}\left(E(V^{j_0}(\cdot-x^{j_0}_n)) + E(v^{j_0}_n) \right) &=\liminf_{n\rightarrow \infty}E(v_n) = d_M.
Q(x) = \frac1{\log x}\bigg( 1+O\bigg( \frac{\log\log x}{\log x}\bigg) \bigg) \bigg( \frac{x^2}{2\log x}+ O\bigg( \frac{x^2}{\log^2x}\bigg) \bigg) + O\bigg( \frac{x^2}{\log^3x}\bigg) = \frac{x^2}{2\log^2 x}+ O\bigg( \frac{x^2\log\log x}{\log^3x}\bigg),
\left| S \left( \left(h_{j,\omega}\right)_{j,\omega}\right) \right| \leq \left[{K \choose 0}+{K \choose 2}+ \ldots \right] (1+\delta) -\left[{K \choose 1}+{K \choose 3}+ \ldots \right] (1-\delta) = 2^K \delta.
\varphi_{ij}= \frac{(|Dw|^2)_{ij}}{|Dw|^2\log|Dw|^2}+h'u_{ij}+\big[h''-(1+\log|Dw|^2)h'^2\big]u_iu_j &+\big[g''-(1+\log|Dw|^2)g'^2\big]\gamma^i\gamma^j-(1+\log|Dw|^2)h'g'(\gamma^iu_j+\gamma^ju_i)+g'(\gamma^i)_j.
T_\lambda T_\mu = s_\lambda s_\mu = c_{\Lambda}( \lambda , \mu ) s_{\lambda \mu}= (-1)^1 s_{\lambda \mu}= - s_{\lambda \mu}T_{\mu'}T_{\lambda'}= s_{\mu'}s_{\lambda'}= c_{\Lambda}( \mu' , \lambda' ) s_{\mu'\lambda'}= (-1)^0 s_{\lambda \mu}= s_{\lambda \mu}.
Q_1 = \frac{\mathrm{P}\left( |h_n|^2 >|g_{n,2}|^2,|g_{n,2}|^2 < y, |h_n|^2>\xi_1, |g_{n,2}|^2>\xi_1 \right)}{\mathrm{P}\left( |h_n|^2>\xi_1, |g_{n,2}|^2>\xi_1 \right)}= \frac{\mathrm{P}\left( |h_n|^2>\max\left\{\xi_1, |g_{n,2}|^2 \right\},\xi_1<|g_{n,2}|^2< y \right)}{\mathrm{P}\left( |h_n|^2>\xi_1, |g_{n,2}|^2>\xi_1 \right)}
c_{0,r}= \sup_{x \in V}\bigg[ \frac{\|\varphi(x)\|_W}{|\!\max\{1,\|x\|_{V}\}|^r}\bigg] \qquad c_{1,r}= \sup_{x \in V}\bigg[ \frac{\|\varphi'(x)\|_{L(V,W)}}{|\!\max\{1,\|x\|_{V}\}|^r}\bigg]
\overrightarrow{{\bf X}^{( n )}}( x ) = \overrightarrow{{\bf X}^{( 0 )}}( x ) +{\bf Y}( x ) \int_{x_{0}}^{x}{\bf Y}^{( - 1 )}( t ) \overrightarrow{{\bf F}}( t , \overrightarrow{{\bf X}^{( n - 1 )}}( t ) ) d t
e^{-\int_{t^*}^u G_j(v) \: dv}X_{j-1}(u) \leq (1 + \delta) X_{j-1}(t^*) e^{\int_{t^*}^u G_{j-1}(v) \: dv}e^{-\int_{t^*}^u G_j(v) \: dv}&= (1 + \delta) X_{j-1}(t^*) e^{-s(u - t^*)}.
\left( L_{EH}\right)_{rs}^{ij}=\tfrac{1}{2-\delta_{ij}}\frac{\partial L_{EH}}{\partial y_{rs,ij}}& =\tfrac{1}{1+\delta_{rs}}\rho \left(y^{ir}y^{js}+y^{jr}y^{is}-2y^{rs}y^{ij}\right) ,
\begin{aligned}\left| \int_{\Omega}f(x,v_n)(v_n-v)dx\right| \leqslant c_1\int_{\Omega}\left| v_n\right|^{q(x)-1}\left| v_n-v\right| dx \leqslant c_1\left| \left| |v_n|^{q(x)-1}\right| \right|_{\frac{q(x)}{q(x)-1}}\left| \left| v_n-v\right| \right|_{q(x)}\longrightarrow 0. \end{aligned}
\;\; f(x\textbf*)=f(x)\textbf* \Leftrightarrow (f_1e_1+f_2e_2)(x_1\textbf*e_1+x_2\textbf*e_2)=(f_1(x_1)e_1+f_2(x_2)e_2)\textbf* \Leftrightarrow f_1(x_1\textbf*)= f_1(x_1)\textbf*\;\; f_2(x_2\textbf*)= f_2(x_2)\textbf*.
\sum_{m=-\infty}^\infty I_{r,n}(w+2\pi im) = I_{r,n}(w) \left( 1 + \sum_{m\not= 0}\, \frac{I_{r,n}(w+2\pi im)}{I_{r,n}(w)}\right) *[0.3cm] =: I_{r,n}(w) \big( 1 + R_{r,n}(w)\big),
\times \prod_{I=2}^N \;\Bigg\langle\prod_{b=1}^N \; \prod_{j=1}^{n_b}: e^{- i 2 \sqrt{\pi}U_{I b}\Phi^{(I)}(x^{(b)}_j)}:_{M^{(I)}}\;\prod_{j=1}^{m_b}: e^{+ i 2 \sqrt{\pi}U_{I b}\Phi^{(I)}(y^{(b)}_j)}:_{M^{(I)}}\Bigg\rangle_{K^{(I)}}
(1\otimes \epsilon_g\cdot a\otimes 1)(m) =\sum a_{(0)}\otimes (a_{(1)}\cdot \epsilon_g)(m) &=\sum a_{(0)}\left( a_{(1)}\cdot \epsilon_g(m_{(1)}) \right) m_{(0)}&=\sum a_{(0)}\epsilon_g(m_{(1)}a_{(1)}) m_{(0)}&=\sum a_{(0)}m_{(1)}(g)a_{(1)}(g) m_{(0)}&= (ga)(gm).