harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[ \sum_{i=1}^{N}\frac{\partial^2}{\partial{x_i}^2}+ 2 (a + \frac{1}{2}) \sum_{i<j}^N \frac{1}{(x_i - x_j)}(\frac{\partial}{\partial x_i}-\frac{\partial}{\partial x_j}) \right]P_k (x) = 0 \,\,\,.
{{P}_{e}}(|{{I}_{mn}})={{P}_{e}}(|{{I}_{mn}}) &=Q\left( \frac{1}{MN}\sqrt{\frac{{\bar{\gamma}}}{2}\sum\limits_{n=1}^{N}{{{\left( \sum\limits_{m=1}^{M}{{{I}_{mn}}}\right)}^{2}}}}\right)
\mathcal{M}_2(X,q)= C\prod_{p\mid q}\bigg(1+2p^{-1}\bigg)^{-1}X^{1/2}q^{1/2}+O\left(d(q)X^{1/3}q^{2/3}+X^{23/15}q^{-13/15}(\log X)^{15}\right),
\sup_{y \in B_{\delta/\varepsilon}(\zeta_i'))}\omega(y)^{-(2+\nu)}|E(y)|\leq C \varepsilon^{\frac{1}{2}}\bigl|- \mu_i^{\frac{1}{2}}g_\lambda(\zeta_i) + \sum_{j\not=i}\mu_j^{\frac{1}{2}}\,G_{\lambda}(\zeta_i ,\zeta_j)\bigr| + C \varepsilon^2.
R_{| n_{1}, n_{2}, \dots , n_{k}\rangle}^{( n )}( \eta , \vartheta , u ) = R^{( n )}( \eta , \vartheta , u ) \prod_{i \textrm{}\mathrm{o d d}}b_{n_{i}}^{n}( \eta , u ) \prod_{i \textrm{}\mathrm{e v e n}}b_{n_{i}}^{n}( \bar{\eta}, u ) \,
H_1(z')+H_2(z') =\int_{\tilde\gamma_1}\widetilde{d}G_{\Omega}(\cdot,p(z'))+2k_1'\pi+\int_{\tilde\gamma_2}\widetilde{d}G_{\Omega}(\cdot,p(z'))+2k_2'\pi &=\int_{\gamma}\widetilde{d}G_{\Omega}(\cdot,p(z'))+2(k_1'+k_2')\pi,
v\in E^s_x \implies \|Df^k_x(v)\|\geq C\lambda_-^k \|v\| \textrm{for all}k\geq 0, v\in E^u_x \implies \|Df^{k}_x(v)\|\geq C\mu_+^{-k}\|v\| \textrm{for all}k\leq 0.
2e \left( \varepsilon_{\mu \nu}\partial_\mu A^{(1)}_\nu,\bigg[ \frac{-1}{\triangle}\!-\!N \Big[ \frac{e^2 \pi}{(\pi\!+\!gN)^2}\tilde{Q}+\frac{g}{\pi}\tilde{C}\Big]\bigg] \sum_{b=1}^N \sum_{j=1}^{n_b}U_{1 b}\Big[ \delta_n(x_j^{(b)})\!-\!\delta_n(y_j^{(b)}) \Big] \right)
{\cal Z}_{G U}={\displaystyle \int}{\cal D}( A^{\mu}, \pi_{\mu}, \lambda ) e x p{i \displaystyle \int}d^{4}x \left( \pi_{0}{\dot{A}_{0}}+ \pi_{i}{\dot{A}_{i}}-{\widetilde H_{G U}}- \lambda \chi \right) ,
\leq \exp\left(-\frac{(\log y)^2}{2}\right)\sum_{n=1}^{\infty}d_{\ell}(n)e^{-n/(2y)}\leq \exp\left(-\frac{(\log y)^2}{2}\right)\left(e^{1/y}\sum_{a=1}^{\infty}e^{-a/y}\right)^{\ell}\ll \exp\left(-\frac{(\log y)^2}{2}\right) (2y)^{\ell}\ll \exp\left(-\frac{(\log y)^2}{4}\right),
\frac{\partial \ln g_k}{\partial \ln k^2}= - \frac{2.6 \cdot 3g^2_k/(4\pi)^2 \cdot (2.7 - 3 g^2_k/(4\pi)^2)}{(4.8 - 3 g^2_k/(4\pi)^2)(0.81 - 3g^2_k/(4\pi)^2)}\:,
F_{\mu_{0}^{b}\cdots \mu_{d}^{b}}^{( \cal M )}= \mp \sqrt{\sigma}e^{\frac{1}{2}\alpha \phi}E_{\mu_{0}^{b}\cdots \mu_{d}^{b}\alpha}^{b}g^{\alpha \beta}\partial_{\beta}\operatorname{l n}U_{b},
\alpha_{0}( \sigma , \bar{\sigma}) = \frac{1}{2}\left( \mathrm{a r t a n h}\frac{\sigma}{4}+ \bar{\sigma}\right) \, \, \mathrm{a n d}\, \, \, \bar{\alpha}_{0}( \sigma , \bar{\sigma}) = \frac{1}{2}\left( \mathrm{a r t a n h}\frac{\sigma}{4}- \bar{\sigma}\right)
I(d,a,c) ={1 \over 2(a^2+c^2-d^2)}\ln \left[{c^2 \over (d+a)^2}\right] +{a \over (c^2-d^2)^{1/2}(a^2+c^2-d^2)}\arctan \left[ (c^2-d^2)^{1/2}\over d \right].
\mathbf{S}_{a}^{-1}\mathbf{S}_{ab}\mathbf{S}_{b}^{-1}=\left(\mathbf{I}_{a}-\mathbf{\Delta}_{a}\right)\mathbf{S}_{ab}\mathbf{S}_{b}^{-1}=\mathbf{S}_{b}\mathbf{S}_{b}^{-1}-\mathbf{\Delta}\mathbf{S}_{b}\mathbf{S}_{b}^{-1}=\mathbf{I}-\mathbf{\Delta}.
I_{2^{k-1}}=\sum_{\substack{m\geq 1 r_{l}\leq m_{l}^{-}l=1,...,k-1}}\int_{\Delta_{0,t}^{m_{1}+...+m_{k}}}\mathcal{H}_{m_{1}+...+m_{k}}^{X}(u)du_{m_{1}+m_{2}}...du_{1}
N ( x , y , a , b , d ) = \frac{1}{( 2 \pi )^{d - 2}a b}\sum_{n , n^{\prime}= 1}^{\infty}\int d^{d - 2}p \frac{\left( \cos ( \frac{2 n \pi x}{a}) \cos ( \frac{2 n^{\prime}\pi y}{b}) \right)}{\left( \vec{p}^{\, 2}+ ( \frac{n \pi}{a})^{2}+ ( \frac{n^{\prime}\pi}{b})^{2}+ m^{2}\right)}.
\frac{d\tilde{\mathbb{P}}_{\mathcal{G},\vec{H}}}{d\tilde{\mathbb{P}}_{\mathcal{G},0}}=\frac{\prod_{\mathcal{C}}\cosh(H(\mathcal{C}))}{\tilde{\mathbb{E}}_{\mathcal{G},0}\left(\prod_{\mathcal{C}}\cosh\left(H(\mathcal{C})\right)\right)},
L_d (q_{m},q_{m+1}) =\frac{h}{2}[ \alpha(q_{m}) \cdot \frac{q_{m+1}-q_{m}}{h}-H(q_{m})+\alpha(q_{m+1})\cdot \frac{q_{m+1}-q_{m}}{h}-H(q_{m+1})], L_d (q_{m-1},q_{m}) =\frac{h}{2}[ \alpha(q_{m-1})\cdot \frac{q_{m}-q_{m-1}}{h}-H(q_{m-1})+\alpha(q_{m})\cdot \frac{q_{m}-q_{m-1}}{h}-H(q_{m})].
\frac{1}{\phi_0 m^2}\frac{\partial^2\phi}{\partial t^2}-\frac{1}{\phi_0 m^2}\frac{\partial^2\phi}{\partial x^2}+\frac{\eta_a}{\phi_0 m^2}\frac{\partial\phi}{\partial t}=-\frac{\phi}{\phi_0}+\frac{\alpha\phi_o}{m^2}\frac{\phi^2}{\phi_o^2}-\frac{\lambda\phi_0^2}{m^2}\frac{\phi^3}{\phi_0^3}+\frac{\xi_a}{\phi_0 m^2}.
e(G^*) = (t(k-2)+\varepsilon)+t(2(k-2)-3)+\max\{2\varepsilon-3,0\}&=3(t(k-2)+\varepsilon)-2\varepsilon-3t+\max\{2\varepsilon-3,0\}&=3(n-1)-3t+\max\{-3,-2\varepsilon\}&= \begin{cases}3n-9 \, k+1\le n\le2k-3 3n-11 \, n=2k-2 3n-12 \,2k-1\le n\le 2k \end{cases}&\ge e(G).
C_\vartheta(\theta) \theta_t + \theta p_\theta u_y = \left(\frac{\kappa\theta_y}{v}\right)_y +\frac{\varepsilon u_y^2}{v}+ \frac{\mu |\mathbf{w}_y|^2}{v}+ \frac{\nu |\mathbf{h}_y|^2}{v},
\nabla^{\mathbb{P}}_{X,J}:\mathcal{M}^{2}_{\mathbb{P}}([X],\mu) \to \mathcal{L}^2_{\mathbb{P}}([X],\mu), F(t,X_t,J_t) := \int_0^t \phi(s) dX(s)+\int_0^t \int_{\mathbb{R}^d_0}\psi(s,y) \widetilde{J}_X(dsdy) \mapsto (\nabla_X F,(\nabla F)) = (\phi, \psi).
{\Phi}+ r{\Phi}^{\prime}+{\varphi}_{A}+{\varphi}_{B}= 0 \; \mathrm{o r , ~ e q u i v a l e n t l y ,}\; \frac{\overline{{{\cal{N}}}}^{\, \prime}}{g}+{\varphi}_{A}+{\varphi}_{B}= 0 \, ,
c_{n}= \frac{-\mathcal{I}m (\rho_{n-1}^{(2)}\alpha_{n-1})}{1-\mathcal{R}e(\rho_{n-1}^{(2)}\alpha_{n-1})}\mbox{and}g_{1, n-1}= \frac{1}{2}\frac{\big|1 - \rho_{n-1}^{(2)}\alpha_{n-1}\big|^2}{\big[1 - \mathcal{R}e(\rho_{n-1}^{(2)}\alpha_{n-1})\big]}, n \geq 1,
E(\sup\limits_{n-1\leq t\leq n}|x(t)|^2) \leq E(|x(n-1)|^2)+(b+2d^2)E\int^n_{n-1}|x(qs)|^2{\rm d}s+2c^2E\int^n_{n-1}|x(s)|^2{\rm d}s ~~~+2E(\sup\limits_{n-1\leq t\leq n}\int^t_{n-1}x^T(s)\cdot g(s,x(s),x(qs)){\rm d}B(s)).
C_T \frac{\Gamma(c) \Gamma(b-a)}{\Gamma(c-a) \Gamma(b)}=\frac{\Gamma(c) \Gamma(a-b)}{\Gamma(c-b) \Gamma(a)}\left( C_R + \frac{\Gamma(2-c)\Gamma(c-b)}{\Gamma(c) \Gamma(2-c-b)}\right)(\alpha-1)^{b-a}2^{a-b}
\prod_{k=0}^\infty(1+\alpha X^k) =\sum_{k=0}^\infty\frac{\alpha^kX^{\binom{k}{2}}}{X^k!}, \prod_{k=1}^\infty\frac{1}{1-\alpha X^k}=\sum_{k=0}^\infty\frac{\alpha^kX^k}{X^k!}.
x y = \det A \, \det B = a_{0}\, \Pi_{i = 1}^{k - 1}( a_{0}+ s_{i}) = a_{0}^{k}\, \det \left( 1 +{\frac{1}{a}}\Sigma \right) = \sum_{i = 0}^{k - 1}a_{0}^{k - i}S_{i}( \Sigma ) .
{\rm (i)}\,\,\,\,\, \tau_{A}(x, \xi) =\int_{\Omega}\widetilde{k}_{A}(x,y)\overline{u_{\xi}(y)}dy;{\rm (ii)}\,\,\,\,\,\tau_{A}(x,\xi) =v_{\xi}^{-1}(x)\int_{\Omega}\widetilde{K}_{A}(x,y)v_{\xi}(y)dy.
2 \pi{P}_{j}( \beta ) = \delta_{j 0}m \cosh \beta + \sum_{l = 0}^{\infty}{\cal I}_{j l}^{( \infty )}\int_{- \infty}^{\infty}d \beta^{\prime}\frac{1}{\cosh ( \beta )}\rho_{l}( \beta^{\prime}) .
0\le\int \left(-\log(|1-w|/2)\right)\mu(dw)= \int\liminf_{r\nearrow1}\left(-\log(|1-rw|/2)\right)\mu(dw)\le \liminf_{r\nearrow 1}\int\left(-\log(|1-rw|/2)\right)\mu(dw)<\infty.
\nu_{(a_k^-, i_k)}^* = \nu_{(1, i_k)}^* = \nu_{(1, i_k)}^{\hat{S}^{k-1}}= \nu_{(0, i_k)}^{\hat{S}^{k-1}}+ c_i / w_{(1, i_k)}^{\hat{S}^{k-1}}\geq \nu_{(0, i_k)}^{\hat{S}^{k-1}}= \nu_{(a_{k+1}^-, i_{k+1})}^{\hat{S}^{k-1}},
\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}0 -1 1 0\end{pmatrix}, z\in \Sigma_2, &\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}1 0 -e^{\pi i\alpha}1\end{pmatrix}, z\in \Sigma_1, &\Upsilon_+(z)=\Upsilon_-(z)\begin{pmatrix}1 0 -e^{-\pi i\alpha}1\end{pmatrix}, z\in \Sigma_3.
\Gamma(\mathcal{B}\lvert\mathcal{C})=\sum_{\ell=0}^k\sum_{1\leqslant i_1<\cdots<i_\ell\leqslant k}(-1)^\ell\Gamma_0\big(\mathcal{B}\lvert\bigcup_{i=1}^\ell\big((\mathcal{C}\setminus\{c_{i_j}\})\cup(\{c_{i_j}-1\})\big)\big).
\left\langle A_{\mu}(x,e)~A_{\nu}(x^{\prime},e^{\prime})\right\rangle =\frac{1}{(2\pi)^{3/2}}\int e^{-i(x-x^{\prime})p}M_{\mu\nu}(p;e,e^{\prime})\frac{d^{3}p}{2p_{0}}& M_{\mu\nu}=-g_{\mu\nu}+\frac{p_{\mu}p_{\nu}}{pe_{-}pe_{+}^{\prime}}-\frac{p_{\mu}e_{\nu}}{pe_{-}}-\frac{p_{\nu}e_{\mu}^{\prime}}{pe_{+}^{\prime}}
\dfrac{\omega^{(g)}_{m,n+1}}{\prod_{j=1}^m dx_j\prod_{i=m+1}^{m+n+1}dy_i}= \left(W^{(g)}_{m,n+1}+\delta_{(g,m,n),(0,0,0)}\right)\prod_{j=1}^m \left(-\dfrac{1}{x_i}\right) \prod_{i=m+1}^{m+n+1}\left(-\dfrac{1}{y_i}\right).
\begin{aligned}P (y_r(n) |x_{a\oplus b}(n)) = \sum_{x_a(n) \in \mathcal{M}, x_b(n) \in \mathcal{M}}& P(x_a(n),x_b(n)) P(y_r(n)|x_a(n) \oplus x_b(n)).\end{aligned}
\sum_{l = - \infty}^{\infty}e^{- 4 \pi^{2}( l + w_{i})^{2}\beta^{- 2}t}={\frac{\beta}{\sqrt{4 \pi t}}}\sum_{k = - \infty}^{\infty}\eta_{i}^{k}e^{-{\frac{\beta^{2}k^{2}}{4 t}}}.
-\frac{\alpha}{2}\int_0^{+\infty}|u|^2v_tdx =-\frac{\alpha}{2}\int_0^{+\infty}\Big(\int_0^x(|u|^2)_x(x',t)dx'+|u(0,t)|^2\Big)v_tdx &=-\frac{\alpha}{2}|u(0,t)|^2\int_{0}^{+\infty}v_tdx-\frac{\alpha}{2}\int_0^{+\infty}\Big[\int_0^x(|u|^2)_x(x',t)dx'\Big]v_tdx &:=I(u,v) + II(u,v).
\frac{\partial u}{\partial t}= \gamma_u\, \Delta u - k_{eq}\,u\,v + l_{eq}\,w, \frac{\partial v}{\partial t}= \gamma_v\, \Delta v - k_{eq}\,u\,v + l_{eq}\,w, \frac{\partial w}{\partial t}= \gamma_w\, \Delta w + k_{eq}\,u\,v - l_{eq}\,w.
((n_1, \ldots, n_{c+e}) \circ \alpha_{L'})( j_B(\lambda)) =((n_1, \ldots, n_{c+e}) \circ \alpha_{L'})( j_A(l_{c+1})) &= ((n_1, \ldots,n_c, p N) \circ \alpha_L)(l_{c+1}) &= n_1 \mathrm{lk}(L_1,L_{c+1}) + \ldots + n_c \mathrm{lk}(L_c,L_{c+1}) + 0= \ell.
u_{\pm}=\frac{1}{4\sqrt{2}}\Big((F_{\mu\nu}F_{\mu\nu}+ F_{\mu\nu}^{*}F_{\mu\nu})^{\frac{1}{2}}\pm ( F_{\mu\nu}F_{\mu\nu}-F_{\mu\nu}^{*}F_{\mu\nu})^{\frac{1}{2}}\Big)
C_f(w) = \frac{\varGamma((r-p-q)w+1-a-b)\, \varGamma(1-r w)}{\varGamma(1-a-p w)\, \varGamma(1-b-q w)}, begin{align*}1mm] C_g(w) = \frac{\varGamma((r-p-q)w+1-a-b) \, \varGamma(r w-1)}{\varGamma((r-p)w-a) \varGamma((r-q)w-b)}.
u(s) =e^{-(t-s)A(s)}u(t) +\int_{s}^{t}e^{-(r-s)A(s)}(\mathcal{A}(s)+B(r)A(r))u(r) dr -\int_{s}^{t}e^{-(r-s)A(s)}[f(r)-P(r)u(r)] dr.
\begin{aligned}I\!I_3 \lesssim (\| u\|_{L^3}\|\nabla B\|_{L^6}+\|B\|_{L^3}\|\nabla u\|_{L^6})\|\nabla^2 B\|_{L^2}& +(\|\nabla^2 \varrho\|_{L^2}\|\nabla B\|_{L^2}+\|\nabla B\|_{L^3}+\|B\|_{H^2}) \|\nabla^2 B\|_{L^2}^2 &\lesssim \delta (\|\nabla^2 u\|_{L^2}^2+\|\nabla^2 B\|_{L^2}^2).\end{aligned}
\widetilde{\Pi}(\alpha \mathbf{z})=\alpha \mathbf{z}\overline{\eta}(\alpha \mathbf{z})+\overline{\alpha \mathbf{z}}\eta(\alpha \mathbf{z})= \alpha \mathbf{z}\overline{\alpha}\,\overline{\eta}(\mathbf{z})+\overline{\alpha}\, \mathbf{\overline{z}}\alpha \eta(\mathbf{z})=\lvert \alpha \rvert^2 \widetilde{\Pi}(\mathbf{z}).
M_0(k,0):=\left(\Pi_\xi L_\xi\Pi_\xi\right)(k,0)=\left(\begin{array}{ccc}\frac{i}{2}(\omega_{1,\xi}+\omega_{-1,\xi}) \frac{1}{2}(\omega_{1,\xi}-\omega_{-1,\xi}) 0 - \frac{1}{2}(\omega_{1,\xi}-\omega_{-1,\xi}) \frac{i}{2}(\omega_{1,\xi}+\omega_{-1,\xi}) 0 0 0 i\omega_{0,\xi}\end{array}\right).
\sum_{\nu = 1}^N{\frac{1}{{\nu^s (\mu - \nu )^s}}}= \sum_{i = 1}^{s}{\binom{2s-i-1}{s-1}\frac{{H_{N,i}+ H_{\mu - 1,i}- H_{\mu - N - 1,i}}}{{\mu^{2s - i}}}}\,.
E_{1}= \sum_{j=1}^{\infty}\lambda_{j}^{2}a_{j}^{2}&= \sum_{j=1}^{\infty}\lambda_{j}^{2}\Pi_{l}(g_{j}^{2}, h_{j,1}^{2}, h_{j,2}^{2}) + \sum_{j=1}^{\infty}\lambda_{j}^{2}(a_{j}^{2}- \Pi_{l}(g_{j}^{2}, h_{j,1}^{2}, h_{j,2}^{2}) &=: M_{2}+ E_{2},
(\bar{\varepsilon}^{a}_+)^{\dagger}(\bar{\varepsilon}^{a}_-) = 0;\hskip 1.0cm (\bar{\varepsilon}^{a}_+)^{\dagger}(\bar{\varepsilon}^{a}_+) =(\bar{\varepsilon}^{a}_-)^{\dagger}(\bar{\varepsilon}^{a}_-) = 1.
\|{F}^{(\delta)}_{1,N}v_0\|_p^p\le \sum_{|k|\le N}\int_{\eta_{k-1}}^{\eta_k}v_0^{p}(x)\biggl(\int_{\eta_{k-1}}^x v_1^{-p'}(t)\bigl|G^{(\delta)}_{1,k}(t)\bigr|^{p'-1}\,dt\biggr)^p\,dx \lesssim \sum_{|k|\le N}\int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}\bigl|G_{1,k}^{(\delta)}\bigr|^{p'}=:\bigl[\mathbf{G}^{(\delta)}_{1,N}(g)\bigr]^{p'}.
\mathrm{ad}_{R}^{\circ}(X)(\mathsf{M}_{m}^{n}) =\pi(K_{2\rho}^{-1}X_{(1)}K_{2\rho})(S^{-2}(X_{(2)})\triangleright\mathsf{M}_{m}^{n})\pi(K_{2\rho}^{-1}S(X_{(3)})K_{2\rho}).
\mathcal{(N\vee Y)\wedge(X\vee Y)}=(\mathcal{N\vee N}_2\vee\mathcal{SL)\wedge(X\vee N}_2\vee\mathcal{SL}) &=\bigl((\mathcal{N\vee N}_2)\wedge(\mathcal{X\vee N}_2)\bigr)\vee\mathcal{SL}\ldotp
I_2(k) = O \Big( k^{\frac{\beta+2}{\beta+1}}\Big)^{-1/2}\cdot O\big(x_0^\alpha \big) &= O\Big( k^{-\frac{\beta/2+1}{\beta+1}}\Big) \cdot O\Big( k^{-\frac{\alpha}{\beta+1}}\Big) &= O\Big( k^{-\frac{1 + \alpha +\beta/2}{\beta+1}}\Big).
L_{i}=\int_{x}\psi^{\dagger}(x_{1})\psi^{\dagger}(x_{2})\cdot\cdot\cdot\psi^{\dagger}(x_{n})V(x_{1},x_{2},...,x_{n})\psi(x_{1})\psi(x_{2})\cdot\cdot\cdot\psi(x_{n}).
\widetilde{A_n}^{(j)}(-k) = \left( \widetilde{A_n}(t) \cdot U(t)\right)^{(j-1)}\big|_{t=-k}&= \sum_{v=0}^{j-1}\binom{j-1}{v}\widetilde{A_n}^{(j-1-v)}(-k) \cdot U^{(v)}(-k).
I_<(\xi,d) = 2 \int_0^{1}du \frac{(1-u/2)^{1-d}}{u^{d/2}}\exp (-d\xi^2/u) \times \int_0^{u/(2-u)}\frac{dv}{(1-v^2)^{d/2}}+ \frac{\sqrt{\pi}~ \Gamma(1-d/2)}{\Gamma(3/2-d/2)}\times \int_1^2 du~ u^{-d/2}(1-u/2)^{1-d}\exp (-d\xi^2/u).
\begin{array}{rl}(1-H)^N < e^{-1}(1+ 4q^{-b/2}+4bq^{-b}+ \frac{3}{4}(4q^{-b/2}+4bq^{-b})^2) &= e^{-1}+ 4e^{-1}q^{-b/2}+ 4e^{-1}(b+3)q^{-b}+ 24e^{-1}bq^{-3b/2}+ 12e^{-1}b^2 q^{-2b}&< e^{-1}+ 4be^{-1}q^{-b/2},\end{array}
\lim_{\lambda\downarrow 0}i_{F_\lambda}\ge \frac{i_F}{C}\, \lim_{\lambda\downarrow 0}\frac{1}{\eta_H(\lambda)\, \lambda}=\infty \lim_{\lambda\uparrow \infty}i_{F_\lambda}\le C\, i_F\, \lim_{\lambda\uparrow \infty}\frac{\eta_H(1/\lambda)}{\lambda}=0.
\frac{W_{n}(x^{2};\alpha,\beta,\gamma,\delta)}{(\alpha+\beta)_{n}(\alpha+\gamma)_{n}(\alpha+\delta)_{n}}=\,_{4}F_{3}(-n,n+\alpha+\beta+\gamma+\delta-1,\alpha+ix,\alpha-ix;\alpha+\beta,\alpha+\gamma,\alpha+\delta;1).
\alpha_{h',g',f'}\bullet (h' \rhd \Theta) \bullet \alpha^{-1}_{h',g'',f}\bullet (\Sigma \lhd f) \bullet \alpha_{h,g,f}\bullet (h \rhd \eta) = ( \eta' \lhd f') \bullet \Pi ( (\eta')^{-1}\lhd f')\bullet \alpha_{h',g',f'}\bullet (h' \rhd \Theta) \bullet \alpha^{-1}_{h',g'',f}\bullet (\Sigma \lhd f) \bullet \alpha_{h,g,f}\bullet (h \rhd \eta) = \Pi
j^{0}= - \frac{J^{0}}{| \psi |^{2}}= \frac{1}{2}i \partial_{\tau}\ln \left( \frac{\psi}{\psi^{*}}\right) + \varphi \ \ , \ \ \vec{j}= - \frac{\vec{J}}{| \psi |^{2}}= - \frac{1}{2}i \vec{\partial}\ln \left( \frac{\psi}{\psi^{*}}\right) + \vec{a}.
\xi_{k}^{1}( z )^{d}= \frac{k}{\frac{1}{2}+ M}z^{\frac{1 + M}{2}}( 1 - z )^{\frac{M}{2}}F ( \frac{1}{2}+ M - | k | , \frac{1}{2}+ M + | k | , \frac{3}{2}+ M ; z )
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_1}y_4+\theta_2y_5, [y_2, y_1]=\alpha_6y_5, [y_2, y_2]=\beta_2y_5, [y_1, y_3]=\frac{\alpha_1\beta_4}{\gamma_1}y_5, [y_2, y_3]=y_4=-[y_3, y_2].
\gamma^0 = \left( \begin{array}{cc}\sigma_3 0 0 -\sigma_3 \end{array}\right),\ \ \ \ \ \gamma^1 = \left( \begin{array}{cc}i\sigma_1 0 0 -i\sigma_1 \end{array}\right),\ \ \ \ \ \gamma^2 = \left( \begin{array}{cc}i\sigma_2 0 0 -i\sigma_2 \end{array}\right)
\log A_{\ell_0}(\tfrac{1}{5}) \leq \log A_{\ell_0}(\tfrac{1}{4}) - \tfrac{\log 2}{20K_0}|\log \varepsilon| \leq 2(\nu-1) |\log \varepsilon| - \tfrac{\log 2}{20K_0}|\log \varepsilon| < -2 |\log \varepsilon| = \log \varepsilon^2,
\lbrack (K+1)d\boldsymbol{x}^{k}]\boldsymbol{I}_{ij}^{-}\boldsymbol{P}_{i}^{\pm}=2d\boldsymbol{x}^{k}\boldsymbol{I}_{ij}^{-}\boldsymbol{P}_{k}^{\pm}-\frac{1}{2}d\boldsymbol{x}^{ijk}.
\begin{cases}dx_i =( Ax_i +B\hat u_i +G m_i+\gamma p_i) dt +D dW_i, \dot{m_i}= (A + G)m_i +B\bar u +\gamma p_i, \dot{p_i}= -(A+G)^T p_i -(I-\Gamma)^T Q [\mathbb{E}x_i- (\Gamma m_i+\eta)],\end{cases}
\int \phi \partial^\zeta_x \nabla \phi_t \cdot Q(u)\cdot \partial^\zeta_x u_t \leq C\Big(|\nabla \phi|_\infty|\phi_t|_{D^2}|\nabla u|_\infty|u_t|_{D^2}+|\phi_t|_{D^2}|\nabla^2u|_{3}|\phi \nabla^2 u_t|_6+|\phi\nabla^3 u_t|_2|\nabla u|_\infty\|\phi_t\|_2\Big) \leq C| u_t|^2_{D^2}+\frac{\alpha}{20}|\phi\nabla^3 u_t|^2_2.
\sum_{n=1}^{\frac{m-1}{2}}(1+q^n+q^{2n})\frac{(1-q)(q;q^2)_{n}(q^{2};q^2)_{n-1}q^{-{n\choose 2}-\frac{m-1}{2}}}{(1+q^n)(q;q)_{n}^2}begin{align*}5pt] =\sum_{n=1}^{\frac{m-1}{2}}\frac{1+q^n+q^{2n}}{(1+q^n)[2n]}{2n\brack n}q^{-{n\choose 2}-\frac{m-1}{2}}\equiv 0\pmod{\Phi_m(q)},
\hat{\hat{T}\,}_{\hat{\hat{\mu}}\hat{\hat{\nu}}}{}^{\hat{\hat{\rho}}}= - \left( i_{\hat{\hat{k}}_{( n )}}\hat{\hat{C}\,}\right)_{\hat{\hat{\mu}}\hat{\hat{\nu}}}Q^{n m}\hat{\hat{k}}_{( m )}{}^{\hat{\hat{\rho}}}\, .
\frac{d}{ds}\Big|_{s=0}E_h(\tilde u_s) = \frac1h \int G_h(z) \int ( u-u^{-z}) \cdot \left( (Id-u\otimes u) \varphi -(Id-u^{-z}\otimes u^{-z}) \varphi^{-z}\right) dx\,dz,
r^{\prime}_{\xi}(\omega) = \frac{-(1 + \omega)^{\alpha}- (\xi - \omega) \alpha (1 + \omega)^{\alpha - 1}}{(1 + \omega)^{2\alpha}}& = - \frac{1}{(1 + \omega)^{\alpha}}\left( 1 + \alpha \frac{\xi - \omega}{1 + \omega}\right).
\mathcal{K}_{j,\mathbf{v}}(\alpha)=\sum_{\substack{1\le z_j \le X 1-v_j\le z_j\le X-v_j}}\Lambda(z_j)\Lambda(z_j+v_j)e\big(2\alpha\sum_{k=1}^5a_{j+4,k+4}v_k\big).
X_0(T) = \big\{\, \lambda\in X^\star(T) \langle\lambda,\alpha^\vee\rangle = 0 \alpha\in R^+ \,\big\}X_1(T) = \big\{\, \lambda\in X^\star(T) 0\leq \langle\lambda,\alpha^\vee \rangle\leq p-1 \alpha\in R^+ \,\big\}
a_{l}=\frac{\sqrt{d!}}{\sqrt{j_{1}!\cdot\cdot\cdot j_{n}!j_{n+1}!}}x_{1}^{j_{1}}\cdot\cdot\cdot x_{n}^{j_{n}}\sqrt{c}^{j_{n+1}}|j_{1}+j_{2}+...+j_{n}+j_{n+1}=d
\partial_t\ln h+\frac{1}{4}\partial_t\ln k^* =\frac{1}{t}+O(\frac{1}{|t|^{1+\delta/2}})-\frac{1}{4}(2+\frac{\delta}{2})\frac{1}{t}&=\frac{1}{2t}(1-\frac{\delta}{4})+O(\frac{1}{|t|^{1+\delta/2}})
\mathbf{A}=\mathbf{U}\cdot\mbox{diag}\left(\begin{array}{c}\lambda_{0}\vdots \lambda_{n-1}\end{array}\right)\cdot\mathbf{V}^{\top},\,\mbox{subject to}\quad\mathbf{I}_{n}=\mathbf{U}\cdot\mathbf{V}^{\top},
f ( s , t ; u , v ) = \sum_{i = 3}^{I}s^{i}t^{8 - i}f_{8 + n ( 4 - i )}( u , v ) , ~ g ( s , t ; u , v ) = \sum_{j = 4}^{J}s^{j}t^{1 2 - j}g_{1 2 + n ( 6 - j )}( u , v ) .
(\frac{N_{p}}{C}\frac{E}{A}-\frac{L_{p}}{A})\frac{d^{2}\beta}{dt^{2}}+(\frac{L_{p}}{A}\frac{N_{r}}{C}-\frac{N_{p}}{C}\frac{L_{r}}{A})\frac{d\beta}{dt}-(\frac{L_{p}}{A}\frac{N_{\beta}}{C}-\frac{L_{\beta}}{A}\frac{N_{p}}{C})\beta=0
K_{( h )}^{d}\simeq{\frac{1}{( d - 1 )}}{\frac{1}{\Sigma_{d}}}\left({\frac{1}{\rho^{d - 1}}}+{\frac{b_{0}^{( d )}h}{\rho^{d - 2}}}+ . . . +{\frac{b_{k}^{( d )}h^{k}}{\rho^{d - k - 1}}}+ . . . + b_{d - 1}^{( d )}h^{d - 1}\ln ( \rho h ) \right) ~ ~ ,
F_{\tau^b}^{(2)}(0)=-2F_{\tau^1 \tau^{b-1}}^{(1)}(0)F^{(2)}(0)+\sum_{e=0}^n \sum_{f=0}^n F_{\tau^1 \tau^e}^{(1)}(0)\eta^{ef}F_{\tau^f \tau^{b-1}}^{(1)}(0).
\begin{cases}i\partial_{t}S + ic_{1}\partial_{x}S + \partial^{2}_{x}S = c_{2}SL +c_{3}|u|^2u, \partial_{t}L + c_{4}\partial_{x}L+ P(\partial_{x})L + c_{5}L\partial_{x}L = c_{6}|S|^2,\end{cases}
\int_{\overline{M}}\overline{g}( \mathrm{div}^{\nabla'}(T_{r}),\overline{\nabla}_{\widehat{W}}\widehat{W})dV=\int_{\overline{M}}((-1)^{r}(r+2)S_{r+2}-c\mathrm{tr}(T_{r}))dV.
H_{n}(x)=(-1)^{n}e^{x^{2}}\frac{d^{n}}{dx^{n}}(\frac{1}{2\sqrt{\pi}}\int e^{isx-\frac{s^{2}}{4}}ds)=(-1)^{n}e^{x^{2}}\frac{1}{2\sqrt{\pi}}\int(is)^{n}e^{isx-\frac{s^{2}}{4}}ds.
\phi(\psi(0)) = \frac{a_{d}(a'_0)^{d}+ \ldots + a_1 a'_0(b'_0)^{d-1}+ a_{0}(b'_0)^d}{b_{d}(a'_0)^{d}+ \ldots + b_1a'_0 (b'_0)^{d-1}+ b_{0}(b'_0)^d}.
\frac{R_{12}}{-i-\omega(2,2)}u_2+\frac{R_{14}}{-i-\omega(4,4)}u_4=i(iR_{12}-iR_{14})=-R_{12}+R_{14}=-1=u_1; \frac{R_{32}}{-i-\omega(2,2)}u_2+\frac{R_{34}}{-i-\omega(4,4)}u_4=-R_{32}+R_{34}=1=u_3.
S_{B}[ A ] \; = \; i \, \frac{1}{\eta}\, S_{C S}[ A ] \; + \; R [ \widetilde{F}] + \frac{i}{\theta}\int d^{3}x d^{3}y j_{\mu}^{T}( x ){\mathcal K}_{\mu \nu}( x - y ) j_{\nu}^{T}( y ) \; ,
\inf_{(\mu,\omega) \in \mathbb{B}}\sup_{\nu \in \mathbb{A}}\mathcal{L}\bigl((\mu,\omega),\nu \bigr) = \max_{\nu \in \mathbb{A}}\inf_{(\mu,\omega) \in \mathbb{B}}\mathcal{L}\bigl((\mu,\omega),\nu \bigr).
\sigma_0(\xi_i,\xi_{i+1},\dots,\xi_{i+m}) =1, \sigma_\alpha(\xi_i,\xi_{i+1},\dots,\xi_{i+m}) =\sum_{i_0+i_1+\dots+i_m=\alpha}\xi_i^{i_0}\xi_{i+1}^{i_1}\dots\xi_{i+m}^{i_m}.
\frac{\Gamma(k-1)}{( 4 \pi)^{k-1}}\sum_{g\in H_k(N,\chi)}\rho_g(n)\overline{\rho_g(m)}= \delta_{n=m}+ 2\pi i^{-k}\sum_{\substack{q\geq 1 N\mid q}}\frac{S_{\chi}(m, n; q)}{q}J_{k-1}\!\left(\frac{4\pi \sqrt{mn}}{q}\right),
V^{[n]}_{z_1z_1z_1}- V^{[n]}_{z_1}+U_{z_1}=0, & V^{[j]}_{z_1z_1}-V^{[j]}+U\,\Omega^{(j+1)}=0, j=1\dots n-1, & \left(\Omega^{[1]}\,U\right)_{z_1}+U_{z_2}=0, & \Omega^{[j]}_{z_1}=UV^{[j]}_{z_1}, j=1\dots n.
\lambda_{b; b_1, \ldots, b_a ; s +1}= \lambda_{b; b_1, \ldots, b_a ; s}- \sum_{j=1}^{a}\lambda_{b +1 ; b_1, \ldots, b_j +1 , \ldots, b_a ;s}- s\lambda_{b+1; b_1, \ldots, b_a, 2 ; s-1}.
0\le \widetilde \eta\le 1, \widetilde \eta\equiv 1 \mbox{on}\widetilde B, \widetilde \eta\equiv 0 \mbox{on}\mathbb{R}^N\setminus B_\frac{r_{i_0}+r_{i_0-1}}{2}, |\nabla \widetilde\eta|\le \frac{c'_N}{r_{i_0-1}-r_{i_0}}\mbox{and}|D^2\widetilde\eta|\leq \frac{c'_N}{(r_{i_0-1}-r_{i_0})^2}.
I(G-N[v_0]) = x \cdot (2x+1)^{2k+4}&= x \cdot \bigg[\sum_{i=0}^{2k+4}\binom{2k+4}{i}(2x)^{i}\bigg] &= x \cdot [(2k)^{2k+4}+\dots] &= 2^{2k+4}x^{2k+5}+\dots
\begin{aligned}x_t'-x_t\equiv\left(m_2^{-1}m_1-n_2^{-1}n_1\right)\frac{r(r-1)(r-2)}{3K}+\sum_{\ell=1}^{r-2}\ell\frac{r}{K}(k_\ell'-k_\ell) \pmod{r}\end{aligned}
\begin{aligned}g''(r_0)= \frac{(14+g(r_0))g'(r_0)}{3r_0}+\frac{(g(r_0)-1)(g(r_0)+2)(g(r_0)+3)}{3r_0^2}&+\frac{5g'(r_0)^2}{3(3+g(r_0))}\end{aligned}
\hat{H}=-\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}+\omega^2r^2+\frac{f_1(\Omega_1)}{x_1^2+\cdots+x_{n_1}^2}+\frac{f_2(\Omega_2)}{x_{n_1+1}^2+\cdots+x_{n_2}^2}+\cdots+\frac{f_N(\Omega_N)}{x_{n_{N-1}+1}^2+\cdots+x_{n_{N}}^2},
g_{j_1}h(J\setminus(D\cup J_1)) =g(I_1)g(I_2)h(J\setminus(D\cup J_1)) &=g(J_1\setminus I_1)g(J_2\setminus I_2)h(J\setminus(D\cup J_2))=g_{j_2}h(J\setminus(D\cup J_2)).
\prod_{n = 1 \atop n \equiv 1 - j ( \mathrm{m o d}2 )}^{\infty}( 1 + q^{n}) = \displaystyle \sum_{r = 0}^{\infty}\frac{q^{r^{2}+ j r}}{( q )_{r}( - q )_{r}}= \displaystyle \sum_{r = 0}^{\infty}\frac{q^{r^{2}+ j r}}{( q^{2}; q^{2})_{r}},{}~ j = 0 , 1 .
L^{n,3}_\tau = n^{2\alpha+ 1}\int^{\tau}_0\gamma_s (\sigma'\sigma)^2(X_{\eta_n(s)}) (s-\eta_n(s))^\alpha \times \left[ (W_s-W_{\eta_n(s)}) \left( \int_{\eta_n(s)}^s \psi_{n,1}(u,s) dW_u \right) -\int_{\eta_n(s)}^s \psi_{n ,1}(u,s) du \right] ds.
\mu(z) = \frac{u(1+\frac{h}{2})^2}{u(1+h)}= \frac{u(\frac{h}{2})^2}{u(h)}&= p^{\frac{-12}{d}}\cdot \frac{(u\circ w_p)(h)}{(u\circ w_p)(\frac{h}{2})^2}
A_{M}\tilde{A}_{N}= R_{M N , M^{\prime}N^{\prime}}\tilde{A}_{N^{\prime}}A_{M^{\prime}}, A_{M}^{\dagger}\tilde{A}_{N}^{\dagger}= \tilde{A}_{N^{\prime}}^{\dagger}A_{M^{\prime}}^{\dagger}R_{M^{\prime}N^{\prime}, M N}
dq_t= \tilde\gamma^{-1}(t,q_t)\left(-\partial_t\psi(t,q_t)-\nabla_qV(t,q_t)+\tilde F(t,q_t)\right) dt & +\tilde S(t,q_t)dt+\tilde\gamma^{-1}(t,q_t)\sigma(t,q_t)\circ dW_t = \tilde\gamma^{-1}(t,q_t)F(t,q_t)dt+S(t,q_t)dt+\tilde\gamma^{-1}(t,q_t)\sigma(t,q_t)dW_t.
W=\sum_{n\ge 1}\sum_{\begin{subarray}{c}\{1, \ldots, n+1\}\stackrel{\psi}{\to}V(Q), \psi(n+1)=\psi(1)\end{subarray}}\sum_{e_i \in E_{\psi(i), \psi(i+1)}}a_{\psi, e_{\bullet}}\cdot e_1 e_2\ldots e_{n}
b=(a_{s-1}(1-T)^{p},\ldots,a_{1}(1-T)^{p^{s-1}}, a_{0}(1-T)^{p^{s}})+(a_{s-1}'T^{p},\ldots,a_{1}'T^{p^{s-1}}, a_{0}T^{p^{s}}).
\int d^{4}x^{\prime}\,d^{4}y^{\prime}\,\Gamma_{ik,i^{\prime}k^{\prime}}^{\left( 2\right)}\left( x,y;x^{\prime},y^{\prime}\right) \psi^{i^{\prime}k^{\prime}}\left( x^{\prime},y^{\prime}\right) =0
\begin{aligned}\mathbf{V}_i \triangleq \sum_{k \in \mathcal{K}_i}|\tau_{k}|^2 \sum_{p\in\mathcal{K}}\mathbf{v}_{k,p}\mathbf{v}_{k,p}^H, \tilde{\mathbf{v}}_i \triangleq \sum_{k \in \mathcal{K}_i}\sqrt{1 + \iota_{k}}\mathbf{v}_{k,k}\tau_{k}, \forall i \in \{\mathrm{t,r}\}. \end{aligned}
\xi_{M_0}(i+1)-\xi_{M_0}(i) \leq\left|\textstyle\bigcup_{j\in J}C_{M_0}(j)\right|-\left|\textstyle\bigcup_{j\in J'}C_{M_0}(j)\right| &\leq|C_{M_0}(j_0)|=|S_{j_0}|=\alpha &\leq\beta.
u(x,0)=u_{0}(x),\;\;\;\;{\partial_{t}}u(x,0)=v_{0}(x),\;\;\;\;\sigma(x,0)=0,\;\;\;\;\varepsilon (x,0)=0,\;\;x\in \mathbb{R}, \lim_{x\rightarrow \pm \infty}u(x,t)=0,\;\;\;\;\lim_{x\rightarrow \pm\infty}\sigma (x,t)=0,\;\;t>0.
X_i = \textstyle\sum_{j \in I}(k_{ji}D_{c_{ji}}+ c_{ij}D_{e_{ij}}) Y_i = \textstyle\sum_{j\in I}(c_{ji}D_{c_{ji}}+ e_{ij}D_{e_{ij}}) - h Z = \textstyle\sum_{i,j\in I}(k_{ij}D_{k_{ij}}+ c_{ij}D_{c_{ij}}) - s
\big| B(x_1,x_1,x_1,x_1)+B(x_2,x_2,x_2,x_2)+6B(x_1,x_1,x_2,x_2)-4B(x_1,x_1,x_1,x_2)-4B(x_1,x_2,x_2,x_2)\big| = 3\big| A(x_1,x_1)+A(x_2,x_2)-2A(x_1,x_2) \big|^2\le C | x_1-x_2|^4,