harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\omega_{12}^3 =\frac{\sqrt{3}}{6}-\frac{\cos{(\theta_2-\theta_3)}}{\sin{(\theta_2-\theta_3)}}h_{12}^3, \omega_{23}^1 =\frac{\sqrt{3}}{6}+\frac{\cos{(\theta_1-\theta_3)}}{\sin{(\theta_1-\theta_3)}}h_{12}^3, \omega_{31}^2 =\frac{\sqrt{3}}{6}-\frac{\cos{(\theta_1-\theta_2)}}{\sin{(\theta_1-\theta_2)}}h_{12}^3.
P\left(\omega_{1},\omega_{2}\right)=\left\Vert \mathbf{Q}\left(\omega_{1},\omega_{2},b_{1}^{*}\left(\omega_{1},\omega_{2}\right),b_{2}^{*}\left(\omega_{1},\omega_{2}\right);\mathbf{t}\right)+\bar{p}^{*}\left(\omega_{1},\omega_{2}\right)\mathbf{1}-\mathbf{f}\right\Vert_{2}^{2},
\begin{aligned}C_{1}(\chi, a)=\sum\limits_{h=1}^{q}\overline{\chi}(h)e\left(\frac{ah}{q}\right),C_{1}(q,a)=C_{1}(\chi^{0},a), &C_{3}(\chi, a)=\sum\limits_{h=1}^{q}\overline{\chi}(h)e\left(\frac{ah^{3}}{q}\right),C_{3}(q,a)=C_{3}(\chi^{0},a),\end{aligned}
\varphi (g^{n}_1 g_3) = \varphi \big( g^{i}_1 g_3 \big) \varphi \big( g^{j}_1 \big) \overset{(\ref{eq:5})}{=}\varphi (g_3) \varphi \big( g^{i}_1 \big) \varphi \big( g^{j}_1 \big) \overset{{\bf A3}}{=}\varphi (g_3) \varphi (g^{n}_1) \,,
\forall y\in(-\infty,\kappa],\quad|\varphi'''|+|\varphi''|+|\varphi|+|\psi'''|+|\psi'|+|\psi|\lesssim\varphi'\lesssim\varphi, &\zeta_B'=\begin{cases}3\eta_B' y>B^2, 0 B<y\leq B^2, \varphi_B' y<B.\end{cases}
\langle(b_1,b_2,b_3),\lambda (b_1,b_2,b_3)\rangle=n'\Rightarrow\lambda\cdot|(b_1,b_2,b_3)|^2=n'\Rightarrow\lambda=\frac{n'}{|(b_1,b_2,b_3)|^2}\Rightarrow|Q-P|=|\lambda(b_1,b_2,b_3)|=\frac{n'}{|(b_1,b_2,b_3)|}.
\left[ x \prod_{a = 1}^{n_{c}}( x - \phi_{a}^{2}) + 2 \Lambda m_{1}\cdots m_{n_{f}}\right]^{2}- 4 \Lambda^{2}\prod_{i = 1}^{n_{f}}( x + m_{i}^{2}) = - 4 \Lambda^{2}x \prod_{a = 1}^{n_{c}}( x - \alpha_{a})^{2}.
\gamma^{[\mu_{1}}\gamma^{\mu_{2}}\cdot\cdot\cdot\gamma^{\mu_{r}]}\propto\epsilon^{\mu_{1}\mu_{2}\cdot\cdot\cdot\mu_{d}}\gamma_{[\mu_{r+1}}\gamma_{\mu_{r+2}}\cdot\cdot\cdot\gamma_{\mu_{d}]},r=0,1,...,d-1,
\frac{d}{dt}r(\phi^t(q)) =\left\langle \nabla r,\frac{d}{dt}\phi^{t}(q) \right\rangle=\frac{1}{1-t}\langle \nabla r, \nabla f \rangle \le \frac{1}{1-t}|\nabla f| \le \frac{\sqrt{f}}{1-t},
\phi-\overline{\phi}\in C([0,T]; H^3), \ \phi_t \in C([0,T]; H^2), & u\in C([0,T]; H^{s'})\cap L^\infty([0,T]; H^3), \phi \nabla^4 u\in L^2([0,T] ; L^2), & u_t \in C([0,T]; H^1)\cap L^2([0,T] ; D^2).
L \left( \frac{1}{2},\overline{\chi}\right) = \sum_n \frac{\overline{\chi}(n)}{n^{\frac{1}{2}}}V_1 \left( \frac{n}{q^{\frac{1}{2}}}\right) + \epsilon(\overline{\chi}) \sum_n \frac{\chi(n)}{n^{\frac{1}{2}}}V_1 \left( \frac{n}{q^{\frac{1}{2}}}\right),
{\dot x}^i=v^i,{\dot \theta}^I={\tilde\iota}^J_\mu(q,v^i) K^I_J -{\dot x}^i \Lambda^I_i, \frac{d}{dt}\left(\frac{\partial{\bar R}^\mu}{\partial v^i}\right)=\frac{\partial{\bar R}^\mu}{\partial x^i}- \Lambda^I_i \frac{\partial{\bar R}^\mu}{\partial \theta^I}- \mu_a B_{ij}^a{\dot x}^j.
\mathfrak{Q}_{1}= \mathfrak{B}_{1}; \ \ \mathfrak{Q}_{2}= \mathfrak{B}_{2}; \ \ \mathfrak{Q}_{3}= \mathfrak{B}_{3}; \ \ \mathfrak{Q}_{4}= [\mathfrak{B}_{4}, \mathfrak{B}_{5}] = (x^{4}, x^{3}y, y^{2}, xz, yz, z^{2});
k_w^\Gamma (z) = B(z) \left( \frac{1 - b (z)^{-1}(b (w)^* )^{-1}}{1 - z^{-1}\overline{w}^{-1}}\right) B(w)^*, |z | , |w| > 1 & = B(z) \left( \frac{1 - b ( 1 /\overline{z})^* b (1 / \overline{w})}{1 - z^{-1}\overline{w}^{-1}}\right) B(w)^*, |z | , |w| > 1.
A \leq \frac{1}{4 \pi}\int_{0}^{2 \pi}\int_{0}^{2 \pi}\frac{( x ( u ) - x ( v )^{2})}{4 \sin^{2}\left( \frac{f ( u ) - f ( v )}{2}\right)}\frac{\mathrm{d}f ( u )}{\mathrm{d}u}\frac{\mathrm{d}f ( v )}{\mathrm{d}v}\mathrm{d}u \mathrm{d}v
\psi(w)=\sum_{i=1}^r\sum_{n,j\leq J}\int_{F_{n,j}}\langle C_{n,j}(\omega)^*X(\omega)C_{n,j}(\omega)\xi_i(\omega),\eta_i(\omega)\rangle \,d\mu(\omega) +\sum_{i=1}^r \int_F \langle (1-z)\xi_i(\omega),\eta_i(\omega)\rangle \, d\mu(\omega)
\begin{aligned}\left(\frac{d}{dz}+B_n(z)\right)P_n(z) =\beta_n A_n(z)P_{n-1}(z), \left(\frac{d}{dz}-B_n(z)-v'(z)\right)P_{n-1}(z) =-A_{n-1}(z)P_n(z),\end{aligned}
\sum_{m \geq 1}\frac{A(1, m)}{m}U(m^2 p, t) = \frac{1}{2 \pi i}\int_{(1000)}p^{-u}L(1 + 2 u, \tilde f) F(u) \frac{\gamma\Bigl(\dfrac{1}{2}+ u, t\Bigr)}{\gamma\Bigl(\dfrac{1}{2}, t\Bigr)}\frac{d u}{u^2}.
\frac{1}{k}\ln\frac{(2m)^k}{k!}\ge \frac{1}{k}\ln \frac{(2m)^k}{e k^{k+1/2}e^{-k}}=\frac{1}{k}\left(\ln \left(\frac{2m}{k}\right)^{k}+k-1+\ln\sqrt{k}\right)\ge\ln \left(\frac{2m}{k}\right)+1+\frac{\ln(\sqrt{k}/e)}{k}.
K_{1,2}(x,\alpha,t) = \frac{\left( f_2'(x,t) - \Delta_\alpha f_2(x,t) \right) \left( \Delta_\alpha f_1(x,t) + \Delta_\alpha f_2(x,t) \right)}{(1 + (\Delta_\alpha f_1(x,t))^2)(1 + (\Delta_\alpha f_2(x,t))^2)}.
K_{F,1}(\tau) = \frac{2\tau}{3\pi}\left(e+\frac{1}{e}\right)\left(1+\left|\log{\frac{4q\tau}{\pi}}\right|\right) & +\frac{4}{27}\left(1+\frac{1}{e^2}\right)\left(\frac{0.634}{3}\log{2}+0.317\log{(2e^4q^2\tau^2)}+12.802\right).
\phi =\left[ \begin{array}{cccc}a_{11}a_{12}0 0 a_{21}a_{22}0 0 a_{31}a_{32}a_{11}a_{22}+a_{12}a_{21}a_{34}a_{41}a_{42}0 a_{44}\end{array}\right] :a_{44}\left( a_{11}^{2}a_{22}^{2}-a_{12}^{2}a_{21}^{2}\right) \neq 0\mbox{and}a_{11}a_{21}=a_{12}a_{22}=0.
S_{sL}={\rm const}\left[ \sum_{i,j\neq i}\frac{(1-\alpha_i)(1-\alpha_j)}{(2\alpha_i-1)}\ln ({\bf z}_{ij}{\bar{\bf z}}_{ij}) + \frac{\lambda_0}{2}\sum_i \left(2\alpha_i-1 - \frac{1}{2\alpha_i-1}\right) + F[\alpha] \right].
\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\left( \frac{u(x)-u(y)}{\vert x-y\vert^{\alpha}}\right)\frac{v(x)-v(y)}{\vert x-y\vert^{\alpha+d}}dxdy+\int_{\mathbb{R}^{d}}g(u)v\dx =\int_{\mathbb{R}^{d}}K(x)f(x,u)v\dx,\ \W^{\alpha,G}(\mathbb{R}^{d}).
\langle p , \Delta \phi \rangle = \langle \Lambda v^j , v^i \partial_{ij}^2 \phi \rangle + \frac 12 \langle \Lambda v^i , v^i \Delta \phi \rangle+ \langle \partial_i v^j , \left[\Lambda \, , \, \partial_i \phi \right] v^j \rangle.
T_{\alpha \beta}^{X}= \frac{\Omega^{\delta}}{2}( ( \partial_{\alpha}- i e A_{\alpha}) X ( \partial_{\beta}+ i e A_{\beta}) X^{*}- \frac{1}{2}g_{\alpha \beta}g^{\mu \nu}( \partial_{\mu}- i e A_{\mu}) X ( \partial_{\nu}+ i e A_{\nu}) X^{*}) ,
X(t) \leq e^{\int_{0}^{t}{\big(m(s)+ C_{3}n(s)\big)\,ds}}\Big(X(0)+\int_{0}^{t}{\{(C_{1},\alpha,\beta,\gamma)+l(s)+f(s)\}\,ds}\Big) := C(C_{1},l,m,n,f,\alpha,\beta,\gamma,K,t).
{}^{M\!L\!}\textrm{F}_{p}(a+1,b+1;c+1;z) =\frac{1}{B(b+1,c-b)}\int_{0}^{1}t^{b}(1-t)^{c-b-1}(1-zt)^{-a-1}E_{\alpha, \beta}^{\gamma}\left(-\frac{p}{t(1-t)}\right)dt.
D^{(3)}\equiv\left|\begin{array}{c}{\; 1\;\; \;\; c_{12}\; \;\; c_{13}\;}{c_{12}\; \;\; \; 1\;\; \;\; c_{23}\;}{c_{13}\; \;\; c_{23}\; \;\; \; 1\;\;}\end{array}\right|=1-c_{12}^2-c_{13}^2-c_{23}^2+2c_{12}c_{13}c_{23}.
I := \sum_{k=1}^n\sum_{z\leq |y|/2},J_{\leq n/2}:= \sum_{k=1}^{n/2}\sum_{z > |y|/2}\mbox{and}J_{>n/2}:= \sum_{k=1+n/2}^n\sum_{z > |y|/2}\mbox{(say)}.
{}\displaystyle \sum_{n=1}^{\infty}\frac{1}{16}\frac{n}{{S_n\sqrt{S_n}}}|A_{n}|^{2}<\displaystyle\sum_{n=1}^{\infty}V_{n}|A_{n}|^{2}\leq\displaystyle \sum_{n=1}^{\infty}{\sqrt{S_n}}\frac{|A_{n}-A_{n-1}|^{2}}{n}.
\sum_{n\ge0}(X)_n(Y)_n(aq/XY)^n\beta_n=\frac{(aq/X)_{\infty}(aq/Y)_{\infty}}{(aq)_{\infty}(aq/XY)_{\infty}}\sum_{n\ge0}\frac{(X)_n(Y)_n(aq/XY)^n\alpha_n}{(aq/X)_n(aq/Y)_n}.
\begin{aligned}\|f\|_{L^{2/(\alpha-1)}}\leq C^1_{SE}(\alpha)\|\Lambda^{1-\alpha/2}f\|_{L^2}, \|g\|_{L^{\infty}}\leq C^2_{SE}(\alpha)\|g\|_{H^\frac{\alpha}{2}}, \|f\|_{L^{2+\frac{2\alpha-2}{2-\alpha}}}\leq C^3_{SE}(\alpha)\|\Lambda^{(\alpha-1)/2}f\|_{L^2}, \|f\|_{W^{\frac{\alpha}{2},\infty}}\leq C^4_{SE}(\alpha)\|f\|_{\dot{H}^{\alpha}}.\end{aligned}
\begin{aligned}|\nabla_x \varphi(d(z, m(x, y)))| = \limsup_{x' \to x}{|\varphi(d(z, m(x', y)))-\varphi(d(z, m(x, y)))| \over d(x', x)}&\le{1 \over 2}|\varphi'(d(z, m(x, y)))|.\end{aligned}
[c_2,d_2] - c_2 = \binom{c_2}{2}+ d_2 - c_2 = \binom{c_2}{2}- \binom{c_2-1}{1}+ d_2-1 = \binom{c_2-1}{2}+ d_2 -1 = [c_2-1, d_2 -1].
E=(\frac{2e^{2}}{L_{B}})\int_{0}^{\infty}\frac{kdk}{k^{2}+k_{B}^{2}r_{l}^{2}}J_{0}(k)J_{0}(\sqrt{\frac{l^{\prime}}{l}}k)J_{0}(k\frac{r_{12}}{r_{l}}).
R^0\widehat{S}(D_A(a_*\omega_X))=R\widehat{S}(D_A(a_*\omega_X)) = R\widehat{S}(D_A\circ Ra_*(\omega_X))=R\widehat{S}(Ra_*\circ D_X(\omega_X)) = R\widehat{S}(Ra_*R\mathcal{H}om(\omega_X, \omega_X[2]))=R\hat{S}Ra_*\mathcal{O}_X[2],
\varphi(b) = \min\limits_{\begin{array}{c}\phi_1,\dots, \phi_{T}\in \mathbb{R}^{\widehat{r}_u}\end{array}}\left|\left(Y-\sum_{k=1}^K b_kX_{k}-\left(\widehat{\lambda}_1,\dots,\widehat{\lambda}_N\right)^\top \left(\phi_1,\dots,\phi_T\right)\right)\widehat{M}_v\right|^2_2.
B_{1}^{\prime}(m_{1},m_{2},m_{3})=\left( h_{2}(\mathbf{z}_{1},\mathbf{z}_{2})h_{2}(\mathbf{z}_{2},\mathbf{z}_{3})h_{2}(\mathbf{z}_{3},\mathbf{z}_{1})\right)^{n},
U^* \left(\!\!\begin{array}{cccc}\sigma_1 x_1 \sigma_2 x_2 \sigma_3 x_3 x_4 \end{array}\!\!\right) Q \left(\!\!\begin{array}{cccc}0 0 1 0 0 0 0 i 1 0 0 0 0 -i 0 0 \end{array}\!\!\right)Q \left(\!\!\begin{array}{c}\sigma_1 x_1 \sigma_2 x_2 \sigma_3 x_3 x_4 \end{array}\!\!\right) U = 0,
\langle \theta_{\varphi^{\vee}}f , \theta_{\varphi^{\vee}}f \rangle = \int_{[G \times G]}\left( \int_{[H]}\theta_{\varphi}(g_1,h) \, \overline{\theta_{\varphi}(g_2,h)}\, dh \right) \, \overline{f(g_1)}\, f(g_2) \, dg_1 \, dg_2 \ .
\begin{aligned}\frac{xy}{xy-uv}\exp\left(-\frac{xy\left(xuv-xy(u+v)+yuv\right)}{xy-uv}\right)-\frac{xy}{xy-uv}\exp\left(-\frac{uv(x^{2}+y^{2})}{xy-uv}\right)I_{0}\left(2\frac{\sqrt{uv(xy)^{3/2}}}{xy-uv}\right) =\sum_{0\le j<k<\infty}\frac{\left(u^{j}v^{k}+u^{k}v^{j}\right)}{j!k!}C_{j}\left(k;x\right)C_{j}\left(k;y\right). \end{aligned}
\tau_{a}. (\tau_{a}\cdots \tau_{b-1}\tau_{b+1}\cdots \tau_{c}Q(z) ) = (x_a - x_{a+1}) \tau_{a+1}\cdots \tau_{b-1}\tau_{b+1}\cdots \tau_{c}Q(z) = \tau_{a+1}\cdots \tau_{b-1}x_{b}\tau_{b+1}\cdots \tau_{c}Q(z)
I_{1}:I_{2}:I_{3}:\cdot\cdot\cdot:I_{n+2}=(\begin{matrix}n+1\\ 0\end{matrix}):(\begin{matrix}n+1\\ 1\end{matrix}):(\begin{matrix}n+1\\ 2\end{matrix}):\cdot\cdot\cdot:(\begin{matrix}n+1\\n+1\end{matrix})
\Psi^{\alpha_1^1....\alpha_1^m,\alpha_2^1....\alpha_2^m}(X_1,X_2)=\sum_{R_1 f_1,R_2 f_2}\psi^{\alpha_1^1 \alpha_1^2,\alpha_2^1 \alpha_2^2}_{R_1 f_1,R_2 f_2}(x_1,x_2)\phi^{\alpha_1^3....\alpha_1^m}_{R_1f_1}(y_1)\phi^{\alpha_2^3....\alpha_2^m}_{R_2f_2}(y_2).
K_Ec_i\pi_\mu c_j^{-1}K_E =K_E\sigma(a_i)\sigma(\pi_\mu)\sigma(a_j^{-1})K_E &= K_E\sigma(a_i)\pi_\mu \sigma(s_\mu a_j^{-1})K_E &= K_E\sigma(a_i)\pi_\mu \sigma(d_j^{-1})K_E,
\lim_{j\rightarrow+\infty}\int_{\Pi^{\xi_{i}}}\mathcal{F}_{\varepsilon_{j}}^{\xi_{i}}((u_{j})_{z}^{\xi_{i}},\Omega_{z}^{\xi_{i}})~d\mathcal{H}^{n-1}(z)=:M_{i}<+\infty~,
\frac{\partial}{\partial x}\lambda_E(x,y) =\frac{1}{\frac{\partial x}{\partial \lambda}}=\frac{1}{\frac{\partial \phi_{-}(\alpha k+\alpha-1,y,\lambda, E)}{\partial \lambda}}&=\frac{R_{-}^{2}(\alpha k+\alpha -1,y,\lambda,E)}{\sum_{i=0}^{\alpha-1}f_iu_{-}^2(\alpha k+i,y,\lambda,E)}.
M^2_{scalar\,\pm}~ ~=~~ M^2_{\pm}~-~{3 \over 4}\, \alpha \gamma\, \mu \,\widetilde{m}_\lambda \Bigg (1~ \pm ~ \sqrt{1+x}~ \Bigg ) \Bigg (2~\pm~{1\over \sqrt{1+x}}~\Bigg),
\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}q^{n^{2}/2}}{\left(q,c;q\right)_{n}}\left(-\frac{c}{aq^{1/2}}\right)^{n}=\frac{\left(c/a;q\right)_{\infty}}{\left(c;q\right)_{\infty}}
g_{C,x_0}(x) =\inf\left\{\lambda>0\,|\,x-x_0\in \lambda(C-x_0) \right\}=\sup\left\{t>0\,|\, x-x_0\in \frac{1}{t}(C-x_0) \right\}^{-1}=\sup\left\{t>0 \,|\, x_0+t(x-x_0)\in C \right\}^{-1}=\frac{1}{\tau(x_0,x)}.
\partial_t u - (1+D_t^{\{m\}})\Delta u = g(x)p(t) + f_1(u) \Omega, t\in (0,T), u = 0 \partial\Omega,\; t\ge 0, u(0) = \xi \Omega, \int_\Omega \kappa(x)u(t,x)dx = \psi(t),\; t\in [0,T],
\hat{A}\geq \frac{\alpha+1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\beta}{\alpha}\vert L_{A\Omega_1}\vert -2\vert U_{A\Omega_1}\vert \geq \frac{\alpha+1-\vert 1-\alpha \vert}{\alpha}D_{A\Omega_1}-\frac{2\beta}{\alpha}\vert B_{A\Omega_1}\vert:= T.
E_{1,l}= \lambda[2(4a+3)z_1 + 8(a+1)b_1 + 2(a+1)(2l+d) - 2b_1 - l(l+d-1)], \psi_{1,l}(r) \propto r^l (1+\lambda r^2)^{-a-1}[1-z_1(1+\lambda r^2)] \times \exp{[- \lambda(b_1+2b_2+3b_3)r^2 - \lambda^2 (b_2+3b_3) r^4 - \lambda^3 b_3 r^6]},
\sum_{m=j}^{M}(\kappa_m-\alpha_{m-j+1})^+=\sum_{m=j}^{M}(\lambda_{m+1}-\alpha_{m-j+1})^+=\sum_{m=j}^{M-1}(\lambda_{m+1}-\alpha_{m-j+1})^+=\sum_{m=j+1}^{M}(\lambda_m-\alpha_{m-j})^+,p+1\leq j\leq M.
\displaystyle \gamma^*(0,\mathbf{1}_E,\psi'_E)=\lim\limits_{s\to 0^+}\zeta_E(s) \gamma(s,\mathbf{1}_E,\psi'_E)=(\frac{\log(q_F)}{\log(q_E)})\lim\limits_{s\to 0^+}\zeta_F(s) \gamma(s,\mathbf{1}_E,\psi'_E)
Q^{(l_{0}^{o},l_{1}^{o},l_{2}^{o},l_{3}^{o})}(E)= Q^{(l_{1}^{o},l_{0}^{o},l_{3}^{o},l_{2}^{o})}(E)= Q^{(l_{2}^{o},l_{3}^{o},l_{0}^{o},l_{1}^{o})}(E)= Q^{(l_{3}^{o},l_{2}^{o},l_{1}^{o},l_{0}^{o})}(E) & =Q^{(l_{0},l_{1},l_{2},l_{3})}(E)= Q^{(l_{1},l_{0},l_{3},l_{2})}(E)=Q^{(l_{2},l_{3},l_{0},l_{1})}(E)=Q^{(l_{3},l_{2},l_{1},l_{0})}(E) .
\|Vx\|=\left\| \sum_{j=1}^{n}f_j(x)\omega_j \right\|=\left(\sum_{j=1}^n|f_j(x)|^p\right)^\frac{1}{p}=\left\| \sum_{j=1}^{n}f_j(x)\tau_j \right\|=\|x\|.
\begin{array}{c}{A^{b a}= ( - )^{\epsilon_{a}+ \epsilon_{b}+ \epsilon_{a}\epsilon_{b}}A^{a b}, ( A^{a b})^{*}= A^{b a},}\\{( B^{a})^{*}= ( - )^{\epsilon_{a}}B^{a}, C^{*}= C .}\\ \end{array}
n = \begin{cases}1, 3, 5, 9, 11, 17, 29, 41, 2\cdot4^m, 6\cdot4^m, 14\cdot4^m ~(m \ge 0) k = 4, 33 k = 5, 1, 2, \dots, k-1, k+1, k+2, k+4, k+5, k+7, k+10, k+13 k \ge 5.\end{cases}
I_{{00}_{\tau_{p+1},\tau_p}}^{(i_1 i_2)q}=\frac{\Delta}{2}\left(\zeta_0^{(i_1)}\zeta_0^{(i_2)}+\sum_{i=1}^{q}\frac{1}{\sqrt{4i^2-1}}\left(\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right) -{\bf 1}_{\{i_1=i_2\}}\right),
D_{m n}^{j}( u ) = \sqrt{\frac{( j + m ) ! ( j - m ) !}{( j + n ) ! ( j - n ) !}}\sum_{\mu}\left( \begin{array}{l}{j + n}\\{\mu}\\ \end{array}\right) \left( \begin{array}{l}{j - n}\\{j - \mu - m}\\ \end{array}\right) \cdot
\begin{aligned}\Psi(n) =\int_I \left(\varphi'\circ \ss_n(y)-\varphi'\circ \ss_\infty(y)\right)^2\left(\frac{d\ss_\infty}{d\ss}\right)^2d\ss =\int_W \left(\varphi'\circ \ss_n(y)-\varphi'\circ \ss_\infty(y)\right)^2d\ss_\infty.\end{aligned}
\sigma_{\hat{h}_{1,k}}^2 = \beta_{1,k}+\frac{1}{(\tau \lambda)^2}\Big(\sum_{j=1}^{L}\sum_{k'=1}\beta_{j,k}|\textbf{s}_{j,k'}\textbf{p}_{1,k}^H |^2 + \|\textbf{p}_{1,k}\|^2 \sigma_n^2 \Big) &=\beta_{1,k}+\frac{\sum_{j=1}^{L}\sum_{k'=1}^{K}\alpha(1-\lambda)\beta_{j,k'}+\sigma_n^2}{\tau\lambda}.
\|u_{\lambda}\|\geq\min\bigg{\{}\frac{\lambda(1-{\rho_{\mathcal{B}}}^2)}{1-{\lambda}^2{\rho_{\mathcal{B}}}^2},\frac{\lambda\sqrt{1-{\rho_{\mathcal{B}}}^2}}{\sqrt{1-{\lambda}^2{\rho_{\mathcal{B}}}^2}}\bigg{\}}\|u\|=\frac{\lambda(1-{\rho_{\mathcal{B}}}^2)}{1-{\lambda}^2{\rho_{\mathcal{B}}}^2}\|u\|
\frac{i}{2}\int_{0}^{\infty}\frac{I_{i \omega}( k \xi_{1}) I_{- i \omega}( k \xi_{2}) - I_{- i \omega}( k \xi_{1}) I_{i \omega}( k \xi_{2})}{I_{i \omega}( k \xi_{2}) K_{i \omega}( k \xi_{1}) - I_{i \omega}( k \xi_{1}) K_{i \omega}( k \xi_{2})}
M=\left[\begin{array}[c]{cc}P Q R S\end{array}\right] =\left[\begin{array}[c]{cc}\mathbf{1}QS^{-1}0 \mathbf{1}\end{array}\right] \left[\begin{array}[c]{cc}P-QS^{-1}R 0 0 S\end{array}\right] \left[\begin{array}[c]{cc}\mathbf{1}0 S^{-1}R \mathbf{1}\end{array}\right] ,
\begin{aligned}\langle \mu + \rho, \epsilon_{1}-\epsilon_{j_1-1}\rangle = \langle \mu + \rho, \epsilon_1-\epsilon_{i_1}\rangle + \langle \mu + \rho, \epsilon_{i_1}- \epsilon_{j_1-1}\rangle = (i_1-1)a + p-b_{j_1-2}&< p,\end{aligned}
0 \in W( U^*(\|B\|_{\infty}A + \lambda \|A\|_{\infty}B)V) = W(\|B\|_{\infty}\|A\|_{\infty}V^*V + \lambda \|A\|_{\infty}U^*BV) &=W(\|B\|_{\infty}\|A\|_{\infty}I + \lambda \|A\|_{\infty}U^*BV).
\left|\int_{[0,1]}u h\right| =\left|\int_{[0,1]}u h+u'h'-u'h' \right|=\big| \langle u,h\rangle_{H^1}-\langle u',h'\rangle_{L^2}\big| \leq \big|\langle u,h\rangle_{H^1}\big|+\big|\langle u',h'\rangle_{L^2}\big| \leq \|u\|_{H^1}\|h\|_{H^1}+\|u'\|_{L^2}\|h'\|_{L^2}\leq2\|u\|_{H^1}\|h\|_{H^1}\,,
\sum_{\genfrac{}{}{0pt}{}{n\geq0}{m|12n+1}}b_3(n)q^{\frac{24n+m(3a+b)+2}{24m}}\equiv_m\frac{\eta^6(z)\eta^6(3z)g(m;z)}{\prod_{n=1}^\infty(1-q^{3n})^a(1-q^{n})^b}.
| \lambda_{-}|^{2}\: \overline{{\lambda_{-}}}\: T_{[ 0 ]}^{( - 1 )}\; = \; \left( T_{[ 0 ]}^{( - 1 )}\: T_{[ 0 ]}^{( 0 )}\: \overline{{T_{[ 0 ]}^{( - 1 )}}}\right)^{2}\overline{{T_{[ 0 ]}^{( 0 )}}}\; .
\begin{aligned}\Lambda(\mathbf{c,c^\prime})(Pj+k)= \Lambda(\mathbf{a,a^\prime})(j)\Lambda(\mathbf{b},\mathbf{b'})(k) &+\Lambda(\mathbf{a,a^\prime})(j+1)\Lambda(\mathbf{b},\mathbf{b'})(-P+k), 0\le k < P -N< j< N.\end{aligned}
f(x_0, x_1, \ldots , x_{n-1}) & := (f_0(x_0, \ldots, x_{n_0-1}), f_0(x_{n_0}, \ldots, x_{2n_0 - 1}), \ldots, f_0(x_{(\ell-1)n_0}, \ldots, x_{\ell n_0 - 1}), x_{\ell n_0}, \ldots, x_{\ell n_0 + m-1}).
\langle\mathcal{L}_0,\dots,\mathcal{L}_q\rangle(\mathcal{X}_g^{q+1}/\mathcal{X}_g) = \langle\mathcal{L}_0,\dots,\mathcal{L}_{q-2},\langle \mathcal{L}_{q-1},\mathcal{L}_q\rangle(\mathcal{X}_g^{q+1}/\mathcal{X}^q_g)\rangle (\mathcal{X}_g^q/\mathcal{X}_g),
X_t = S_t\, X_0 + \int_0^t Y^t_s \, ds + \int_0^t Z^t_s \, dW_s, \bar{X}_t = S_t\, \bar{X}_0 + \int_0^t \bar{Y}^t_s \, ds + \int_0^t \bar{Z}^t_s \, dW_s.
P'_{\ell}(\cos \phi)=\frac{\ell}{\sin^2 \phi}\big[ h_0(0) \, \sin \phi \sin \psi_{0,\ell}\, s_{0,0}(\ell,\phi) \big] + \phi^{-2-1/2}O(\ell^{-1/2}) + O(\phi^{-1}),
\tilde \Omega_\psi(k)\tilde x^-_mb =\delta_{k,-m}b+\sum_{r\geq 0}g_{q^{-1}}(r)\tilde x^-_{m+r}\Omega_\psi(k-r)\tilde x_{i_1}^-v_\lambda & =\delta_{k,-m}b-\tilde x^-_{m+1}\Omega_\psi(k-1)\tilde x_{i_1}^-v_\lambda & =\delta_{k,-m}b-\delta_{k-1,-i_1}\tilde x^-_{m+1}v_\lambda .
\pi_r\circ\mathrm{Fl}^{Y_{-1}^\prime}_{t}(z)=z_r+\delta_{r,n}it,\pi_r\circ\mathrm{Fl}^{Y_{-j}}_{t}(z)=z_r+\delta_{r,j}it\quad\quad\forall\, j\in\{1,\ldots, n\},
\bar{\theta}(t, \cdot) = \sum_{l = 0}^{s - 2}\frac{t^{l}}{l!}\bar{\theta}^{l}+ \int_{0}^{t}\int_{0}^{t_{1}}\dots \int_{0}^{t_{s-2}}\tilde{\theta}(\tau, \cdot) \mathrm{d}\tau \mathrm{d}t_{s-2}\dots \mathrm{d}t_{1},
|-\frac{1}{2\pi}\int_{t}^{\infty}ds \int_{B_{s-t}(0)}\frac{dA(y)}{\sqrt{(s-t)^{2}-|y|^{2}}}_{v_{4,2}^{1}}| \leq \frac{C}{t^{3}\log^{3b-1+2N-2\alpha b}(t)}
A_0 = \left\{\begin{bmatrix}0 0 \end{bmatrix}\right\},\, A_1 = \left\{\begin{bmatrix}0 1 \end{bmatrix}\right\},\, A_2 = \left\{\begin{bmatrix}1 -1 \end{bmatrix}, \begin{bmatrix}-1 -1 \end{bmatrix}\right\}.
B_i= \begin{cases}B_{i-1}\cup \{E^i_j\}_{1\le j\le 2(i+1)},\ \ \mbox{if\; \}i \mbox{is even}; B_{i-1}\cup \{E^i_j\}_{1\le j\le 2i-1},\ \ \mbox{if\; \}i \mbox{is odd}. \end{cases}
[\begin{matrix}Y^{\prime}\\C_{B}\\C_{R}\end{matrix}]=\frac{1}{256}[\begin{matrix}77&150&29\\ -43&-84&127\\ 127&-106&-21\end{matrix}][\begin{matrix}R^{\prime}\\G^{\prime}\\B^{\prime}\end{matrix}]+[\begin{matrix}0\\ 128\\ 128\end{matrix}]
R_{\varepsilon}= v^{(2,m)}(x,t) + \varepsilon\Bigl( Y_{2,m}(\tfrac{x_1}{\varepsilon})\, \partial_{x_1}v^{(2,m)}(x,t) +\chi_2(x_2){\cal N}^{(2)}_{2,m}\bigl(\tfrac{x_1}{\varepsilon}, \tfrac{x_2 + l_1 + l_2}{\varepsilon}, x_1, t \bigr) \Bigr), begin{align*}2mm] (x,t) \in G^{(2,m)}_\varepsilon\times (0,T), m = 1, 2, 3, 4.
\begin{aligned}0\rightarrow C^{\bullet}(W^{u}_{Z_{2}}/W^{u}_{Y}, F)\otimes \mathbb{C}^{-}\overset{\psi^{-}_{2}}\longrightarrow C^{\bullet}(W_{\overline{Z_{2}}}^{u}, \overline{F})^{-}\rightarrow 0,\end{aligned}
\bar{\alpha}^-|\Phi_s\rangle=\Bigl(-\frac{\partial^I}{\partial^+}\bar{\alpha}^I+\frac{s+d-2}{\hat{\partial}^+}\bar{\alpha}^z-\frac{2(\hat{\partial}^+-\alpha^+\bar{\alpha}^z)}{\hat{\partial}^+(\hat{\partial}^+-2\alpha^+\bar{\alpha}^z)}\bar{\alpha}^z\Bigr)|\Phi_s\rangle\,.
\begin{gathered}\det\left(I_{N}+H\right)^{-1}\exp\left(W^{*}H\left(I_{N}+H\right)^{-1}W\right) =\sum_{K}\prod_{m=1}^{N}\prod_{n=1}^{N}\left(-h_{m,n}\right)^{k_{m,n}}\binom{c_{m}}{k_{1,m},\dotsc,k_{N,m}}\left(\rho_{m}e^{i\theta_{m}}\right)^{r_{m}-c_{m}}L_{c_{m}}^{\left(r_{m}-c_{m}\right)}\left(\rho_{m}^2\right),\end{gathered}
Z(G_{4,3}\vee B^{(2)}\vee G_{4,3})= \begin{cases}Z(Q_3(5,1,2))=18 (B^{(2)}=Q_3(\ddot1,1,2)) Z(Q_3(4,1,3))=20 (B^{(2)}=Q_3(\dot2,1,\dot1)) Z(C(3,3,2))=23 (B^{(2)}=C(\dot1,\dot1,2)), \end{cases}
\frac{1}{4}+Y_0^2 =s.(1-s) &=\frac{-s.\Gamma (s)}{\underset{k=0}{\overset{K}{\sum}}x_0^{-2.(k+1)}.\alpha_{2.k+1}.\Gamma (s+2.k+1)-\Gamma (s).\underset{n=1}{\overset{n=N}{\sum}}\frac{n^{-s}}{x_0^{1-s}}}
\begin{aligned}\alpha_{d-\mu_1}= -(q_{0,d-\mu}B_{0,\mu_2}+ q_{1,d-\mu}B_{1,\mu_2}+ q_{2,d-\mu}B_{2,\mu_2}) \beta_{d-\mu_2}= q_{0,d-\mu}A_{0,\mu_1}+ q_{1,d-\mu}A_{1,\mu_1}+ q_{2,d-\mu}A_{2,\mu_1}.\end{aligned}
U_1=\{z\in \mathbb R^N : \|z-x\|>5{\mathfrak R}^k, \|z-\sigma_1(x)\|\le 5{\mathfrak R}^k\}, &U_{j+1}=\{z\in \mathbb R^N :\|z-x\|>5{\mathfrak R}^k, \|z-\sigma_{j+1}(x)\|\le 5{\mathfrak R}^k\}\setminus \Big(\bigcup_{i=1}^j U_i\Big)\quad\mbox{for}1\leq j\leq |G|-1.
\int_{0}^{t}\beta_{u}^{1}\left( g_{u}\right) dg_{u}\otimes\int_{0}^{t}\beta_{u}^{2}\left( g_{u}\right) dg_{u}=\int_{0}^{t}\beta_{u}\left( g_{u}\right) dg_{u}\forall t\in \left[ 0,T\right]
\left[{\frac{d n_{d}( \omega | D )}{d \omega}}\right]_{\tiny \mathrm{d i v}}= r_{d}\,{\frac{\Gamma \left( 1 - \frac D 2 \right)}{( 4 \pi )^{D / 2}}}\, \,{\frac{m^{D - 4}}{\kappa}}\, \, \int_{\Sigma}\left[ 2 \left( m^{2}+{\frac{R}{1 2}}-{\frac{\cal Q}{8}}\right) -{\frac{\omega^{2}}{\kappa^{2}}}{\cal P}\right] .
D^{(n)}_{q,t}(d) D^{(n)}_q(u_0,u_1,u_2,u_3;t) = D^{(n)}_q(u_0+dq/2,u_1+dq/2,u_2+dq/2,u_3+dq/2;t) D^{(n)}_{q,t}(d).
T_{\varepsilon,B}=\frac{6\varepsilon}{\pi^2 \alpha^2}B^{\frac{2}{3}+\delta}+O(B^{\frac{2}{3}+\frac{2}{3}\varepsilon+(1-\varepsilon)\frac{\delta}{2}})=\frac{6\varepsilon}{\pi^2 \alpha^2}(B^{\frac{2}{3}+\delta}+o(1)).
V_{6}=-\varepsilon\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})(\mu^{2})\sum_{i=1}^{d}\left[\Theta_{p_{i}q}(t,x,\mu^{1},Dw^{1})\frac{\partial\mu^{1}}{\partial x_{i}}-\Theta_{p_{i}q}(t,x,\mu^{2},Dw^{2})\frac{\partial\mu^{1}}{\partial x_{i}}\right]\dx,
\zeta_{2}\left(-n_{1},-n_{2}\right)=\frac{1}{n_{2}+1}\sum_{k_{2}=0}^{n_{2}+1}\sum_{l_{1}=0}^{n_{1}+n_{2}+2-k_{2}}\sum_{l_{2}=0}^{k_{2}}\frac{\binom{n_{2}+1}{k_{2}}\binom{n_{1}+n_{2}+2-k_{2}}{l_{1}}\binom{k_{2}}{l_{2}}}{n_{1}+n_{2}+2-k_{2}}B_{l_{1}}B_{l_{2}}.
\int_{|y|\leq(t-s)^{1/2}}\frac{X_{s}(\mathrm{d}y)}{\max\bigl\{(t-s)^{1/2},|y|\bigr\}}& =\,(t-s)^{-1/2}X_{s}\bigl((-(t-s)^{1/2},\,(t-s)^{1/2})\bigr)\,\leq\ \theta^{-1}
M =-A+T^{R}D_{1}-T^{I}D_{2}+i\left( T^{R}D_{2}+T^{I}D_{1}\right) = & =-A+\left( T^{R}+iT^{I}\right) \left( D_{1}+iD_{2}\right) = & =-A+T\mathbf{g}^{\prime}(\mathbf{z}^{\ast}),
a_{63ER}(t)= \, 256t^8-2304t^7-3232t^6+1008t^5+2337t^4-504t^3-808t^2+288t+16, b_{63ER}(t)= \, 27648(16t^4-11t^2+4)(2t-1)^3(t+1)^3t^3,
\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\frac{(-1)^{n-1}z^nq^{n^2}(q^2;q^2)_n}{(zq;q^2)_n(1-zq^{2n})}=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\frac{(q;q^2)_{n-1}(q^2;q^2)_{n}(zq^2;q^2)_{N-n}z^{n}q^{2n-1}}{(zq;q^2)_{n}(zq^2;q^2)_N}.
S_{3/2}= \int_{M}d^{4}x \sqrt{g}\biggr[{1\over 2}\tilde{\kappa}_{\mu}{\widehat \nabla}\kappa^{\mu}+\tilde{\epsilon}{\widehat \nabla}\epsilon+{1\over 2}\tilde{\eta}{\widehat \nabla}\eta \biggr] \; ,
X_{i_1,i_2,\dots,i_d}= \sum_{j_1=1}^{r_1}\sum_{j_2=1}^{r_2}\cdots \sum_{j_{d-1}=1}^{r_{d-1}}T^{(1)}_{1,i_1,j_1}\, T^{(2)}_{j_1,i_2,j_2}\, \cdots \, T^{(d)}_{j_{d-1},i_d,1}.
\Phi_{1_{\chi|_{\setminus q}}}(T_1, \ldots, T_{q-1}, T_qT_{q+1}, T_{q+2},\ldots, T_n) = \Phi_{1_\chi}(T_1, \ldots, T_n) + \sum_{\substack{\pi \in BNC(\chi) |\pi| =2, q\nsim_\pi q+1}}\Phi_\pi(T_1, \ldots, T_n)
r_n:= \sum_{k=0}^{n-1}\ell_k = \sum_{k=0}^{n-1}\frac{1}{(1+k)^s}=(1+o(1))\times \begin{cases}\frac{n^{1-s}}{1-s}, s\in[0,1), begin{align*}1mm]\log(n), s=1,\end{cases}
{\cal L}_{{\rm kin}}^{(0)}= e^{-\phi}k^{-{1\over 2}}\sqrt{\dot X^\mu \dot X^\nu \bigl (g_{\mu\nu}-k^2A_\mu^{(2)}A_\nu^{(2)}\bigr )-k^2{\dot S}^2 -2 k^2 \dot S{\dot X}^\mu A_\mu^{(2)}}\, .
\widetilde{A}_{i, r}^{-1}\widetilde{A}_{j, s}^{-1}=t^{\alpha(i, r; j, s)}\widetilde{A}_{j, s}^{-1}\widetilde{A}_{i, r}^{-1}, \ \alpha(i, r; j, s) =\begin{cases}2(-\delta_{r-s, 2r_i}+\delta_{r-s, -2r_i}) \i=j, 2(-\delta_{r-s, -2}+\delta_{r-s, 2}) \ |i-j|=1, 0 .\end{cases}
\sum\limits_{\ell=0}^{2}\frac{d}{dt}\langle\nabla^{\ell}u^{\epsilon},\epsilon\nabla^{\ell+1}\phi^{\epsilon}\rangle+\frac{3}{4}\|\nabla\phi^{\epsilon}\|_{H^2_{\frac{P'(\rho^{\epsilon})}{\rho^{\epsilon}}}}^2\lesssim D( u^{\epsilon},\eta^{\epsilon},\tau^{\epsilon})+\|\mathrm{div}u^{\epsilon}\|_{H^2}^2.
\mathrm{div}(\nabla V \, V)=\partial_j\Big(V^i\,\partial_i V^j \Big)=\partial_j V^i\,\partial_i V^j+V^i\,\partial_j\partial_i V^j =\mathrm{tr}((\nabla V)^2)+V^i\,\partial_j\partial_iV^j\,,
\int_{0}^{1}\frac{x^{n}(logx)^{n}}{n!}dx=\frac{1}{n!}\frac{1^{n+1}}{n+1}(-1)^{n}\frac{(n)_{n}}{(n+1)^{n}}=(-1)^{n}(n+1)^{-(n+1)}.