harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Lambda_{n}^{k}=\left\{(x_{1},x_{2},\dots,x_{n-k+1}):\sum_{j=1}^{n-k+1}x_{j}=k, \ \ \sum_{j=1}^{n-k+1}jx_{j}=n, \x_{j}\in\mathbb{N}\cup \{0\}\right\}.
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\cdot\frac{1}{\sigma^{\prime}\left(\frac{\lambda_{n}^{2}x}{2\pi}\right)e^{\lambda_{n}^{2}x}-1}=\frac{1}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}\frac{\zeta_{p^{\prime}}(1-s)\,\zeta_{p}(2s)}{2\cos\left(\frac{\pi s}{2}\right)}\,\left(\frac{x}{2\pi}\right)^{-s}\,ds.
U_1(t,r) =U_1(t,s) \circ U_1(s,r) U_1(t,r),\ \ \ 0\leq r\leq \sigma\leq t\leq T \ ; U_1(t,t) =\mathbb{I}, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\leq t\leq T; i\partial_t U_1(t,s) = H_1(t)U_2(t,s) \ \ \ \ 0\leq \sigma\leq t\leq T; i\partial_s U_1(t,s) = - U_1(t,s)H_1(s)\quad\ \ 0\leq \sigma\leq t\leq T;
I_{N}^{G}U(t)=\sum_{n=0}^{N}\tilde{U}_{n}^{\lambda}C_{n}^{\lambda}(t),{\rm where}\tilde{U}_{n}^{\lambda}=\frac{1}{\gamma_{n}^{\lambda}}\sum_{j=0}^{N}U(t_{j}^\lambda)C_{n}^{\lambda}(t_j^\lambda)\rho_{j}^\lambda.
R(X_i,X_j)X_k = R^B(X_i,X_j)X_k, & R(X_i,X_j)Y_p = 0, & R(X_i,Y_p)X_j = f^{-1}H^f(X_i,X_j)Y_p, & R(X_i,Y_p)Y_q = -f \,h_{pq}(\nabla^B_{X_i}\,{\rm grad}\: f), & R(Y_p,Y_q)X_i = 0, & R(Y_p,Y_q)Y_r = R^F(Y_p,Y_q)Y_r - ||{\rm grad}\: f ||^2 (h_{qr}Y_p - h_{pr}Y_q),
\hat{\phi}( t , r ) = \frac{1}{2 \pi}\int_{0}^{\infty}\frac{d \omega}{\sqrt{\omega}}\left[ b_{\omega}e^{- i \omega t}\psi_{\omega}( t , r ) + b_{\omega}^{\dagger}e^{i \omega t}\psi_{\omega}^{\ast}( t , r ) \right] .
\begin{array}{l l}{\lefteqn{\! \! J^{P}= 5^{-}:}\vspace{1 m m}}\\{\! \! 0 =}&{\{\{S_{2}, S_{1}\}_{3}, S_{2}\}_{5}; \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad}\\ \end{array}
R + \frac{\gamma}{4}( \frac{(D \Omega )^2}{\Omega^2}- 2 \frac{D \cdot D \Omega}{\Omega}) + (1- \lambda ) \mu \Omega^{- \lambda}-\frac{1}{4}(1 - \frac{\epsilon}{2}) \Omega^{-\epsilon /2}F^2 =T^{X}_{\Omega},
(\pi_X^i\otimes\pi_Y^j)_*\Gamma_{p}= \pi_Y^j\circ \Gamma_p\circ{}^t\pi_X^i & = \frac{1}{N}\Gamma_p\circ\pi_X^j\circ{}^t\Gamma_p\circ \Gamma_p\circ\pi^{2d-i}_X & = \frac{1}{N}\Gamma_p\circ{}^t\Gamma_p\circ \Gamma_p\circ \pi_X^j\circ\pi_X^{2d-i}& = 0.
S(r) < \sum_{k=2r}^{\infty}\dfrac{1}{k\pi^k}\Bigl(\dfrac{k}{2}\Bigr)^r e^{\frac{r(r-1)}{k}}\dfrac{1}{r^k}\leq \dfrac{e^{\frac{r-1}{2}}}{2^r}\sum_{k=2r}^{\infty}\dfrac{k^{r-1}}{(\pi r)^k}\ \ \Bigl(, e^{\frac{r(r-1)}{k}}\leq e^{\frac{r-1}{2}}\forall\k \geq 2r\Bigr).
\boldsymbol{u}_{tt}- (\mu + \kappa)\,\Delta \boldsymbol{u}- (\lambda + \mu)\,\nabla\,{\rm{div}}\,\boldsymbol{u}- \kappa\,(\nabla\times \boldsymbol{\varphi}) + b\,\big( \nabla\theta +\alpha_{0}\,\nabla\theta_{t}\big) +\, \xi_{0}\,\boldsymbol{u}_{t}=0,
\omega(v_\alpha \wedge v_{\beta+n})=\omega(-ie_{\alpha \beta}+ie_{\beta+n,\alpha+n})=-i\sum_{k=1}^{r}(u_{\alpha+n,k}\frac{\partial}{\partial u_{\beta+n,k}}+u_{\alpha,k}\frac{\partial}{\partial u_{\beta,k}})-i r \delta_{\alpha \beta}.
\Theta_1 =\vartheta(s_1)-\tilde\alpha_1-\frac\pi 2=\vartheta(s_2)+\alpha_2-(\tilde\alpha_1+\alpha_2)-\frac\pi 2 &=\vartheta(s_1)+\vartheta(s_2^*)-\pi-(\tilde\alpha_1+\alpha_2) &=\vartheta(s_1)-\vartheta(-s_2^*)-(\tilde\alpha_1+\alpha_2) &=-\int_{s_1}^{s^*_1}\dot\vartheta(s)\,ds-(\tilde\alpha_1+\alpha_2).
U(t,t') \xi := \xi + (-i) \int_{t'}^t d\tau_1 \, H_1(\tau_1) \xi + (-i)^2 \int_{t'}^t d\tau_1 \int_{t'}^{\tau_1}d\tau_2 \, H_1 (\tau_1) H_1 (\tau_2) \xi + \cdots
\|\varphi\otimes\psi\|_{C^{k}_{{\rm L}^*}(\overline{\Omega}\times \overline{\Omega})}:=\max_{j+l\leq k}\|({\rm L}^*)^{j}\varphi\|_{L^2(\Omega)}\|({\rm L}^*)^{l}\psi\|_{L^2(\Omega)}
\mathrm{Hom}_{E}(\varphi|_{E},\phi'|_{E})=\mathrm{Hom}_{{F'}}(\mathrm{Ind}_{E}^{F'}(\varphi|_{E'}),\phi')=\oplus_{i}\mathrm{Hom}_{{F'}}(\rho_i \otimes \varphi|_{{F'}},\phi')^{\mathrm{deg}(\rho_i)}
S = \int_{M}d^{3}x \; \epsilon^{\sigma \mu \nu}\left[{B_{\sigma}}\!^{a}({F_{\mu \nu}}\!^{a}+ \epsilon^{a b c}{\phi_{\nu}}\!^{b}{\phi_{\mu}}\!^{c}) +{\eta_{\sigma}}\!^{a}D_{[ \mu}{\phi_{\nu ]}}^{a}\right] \; .
= 4 \pi \left({\frac{\ell !}{( 2 \ell + 1 ) ! ( 2 \ell + 1 ) ! !}}\right)^{2}\left|{\frac{\Gamma ( \ell + 1 + 2 i k r_{0})}{\Gamma ( 1 + 2 i k r_{0})}}\right|^{2}\; ( k r_{0})^{2 \ell + 1}
\mathcal M_{\theta_{1234}}(\xi_1,\xi_2,\xi_3,\xi_4)=| -\theta_1\langle\xi_1\rangle_M+\theta_2\langle\xi_2\rangle_M+\theta_3\langle\xi_3\rangle_M-\theta_4\langle\xi_4\rangle_M |.
\lim_{m\rightarrow\infty}\prod_{p\leq m}\frac{D_{\mathrm{k}}^{(p)}(m)}{\left(1-\frac{\nu_p(\mathcal{H}_{\mathrm{k}})}{p}\right)}=\prod_p\left(1-\frac{1}{p}\right)^{{\mathrm{k}}}\left(1-\frac{\nu_p(\mathcal{H}_{\mathrm{k}})}{p}\right)^{-1}\;.
\begin{array}{l l l}{\delta e^{a}= \frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a}\psi}&{}&{\delta \omega^{a b}= - \frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a b}\psi}\\{}&{}&{}\\{\delta \psi = D \epsilon}&{}&{\delta A^{a b c d e}= \frac{1}{8}\bar{\epsilon}{\bf \Gamma}^{a b c d e}\psi .}\\ \end{array}
\frac{\delta S_m}{\delta \phi^A}\frac{\delta S_m}{\delta \phi_{A a}^*}+ V_m^a S_m = 0,\qquad\frac{\delta S_m}{\delta \phi^A}\frac{\delta S_m}{\delta \eta_B}(\sigma_\alpha)_B^{~~\!A}+ V_\alpha S_m = 0,\qquad\frac{\delta S_m}{\delta \phi^A}\frac{\delta S_m}{\delta \eta_B}\gamma^B_A + V_m S_m = 0,
\begin{aligned}d_X(F(x,y),F(u,v))\leq a\d_X(x,F(u,v))+ b\d_X(u, F(x,y))+c\d_X(x,u); & \forall x\geq_{P_1}u,\y\leq_{P_2}v;\ 2b+c<1\end{aligned}
L(x) = \int_{0}^{\infty}dA(t) \int_{0}^{t}dF_n(u)\mathbb{P}(V_n \leq x+t-u, u \geq D_n) = \int_{0}^{\infty}dA(t) \int_{0}^{t}G(u) dF_n(u) V(x+t-u).
\nu_{n}\left( E\right) =\int_{\mathbb{S}(\mathbb{H}_{n}^{1})}\int_{\mathbb{R}}1_{E}\left( r\xi \right) n\rho_{n}^{\alpha}\left( \mathrm{d}r\right) \pi_{n}\left( \mathrm{d}\xi \right) E\in \mathcal{B}\left( \mathbb{H}_{n}\backslash \left \{0\right \}\right)
\mathfrak{T}_h(x\,\,y) =\sum^m_{k=1}\mathfrak{A}_{hk}(x^n)\frac{y^{k-1}-x^{k-1}}{y-x}, \mathfrak{U}_h(x\,\,y) =\sum^m_{k=1}\mathfrak{A}_{hk}(x^n)y^{k-1},
\begin{aligned}\nu\left(a_{s,i_r}\left(\partial_{\frac b{p^{e'}}}Q_{i_r}\right)^{p^{e'}}\right)=\nu\left(a_{s,i_r}\right)+p^{e'}\nu\left(\partial_{\frac b{p^{e'}}}Q_{i_r}\right)= &=\nu\left(a_{s,i_r}Q_{i_r}^s\right)-p^{e'}\left(\beta_{i_r}-\nu\left(\partial_{\frac b{p^{e'}}}Q_{i_r}\right)\right).\end{aligned}
\bar{g}( x ) = \sum_{k = - \infty}^{\infty}\beta_{k}e^{i k x}= \sum_{k = - \infty}^{\infty}\beta^{- k}e^{i k x}= \sum_{m = - \infty}^{\infty}\beta^{m}e^{- i m x}= g^{*}( x )
G_{\mathfrak{m}_{\mathtt{D}(\lambda)}}(z) = \frac{\prod\limits_{i=1}^m (z-\mathtt{y}_{-i})(z-\mathtt{y}_{i})}{(z-\mathtt{x}_0)\prod\limits_{i=1}^m (z-\mathtt{x}_{-i})(z-\mathtt{x}_i)}= \frac{\prod\limits_{i=1}^m \big(z^2-\mathtt{y}_i^2\big)}{z \prod\limits_{i=1}^m \big(z^2-\mathtt{x}_i^2\big)}.
\begin{aligned}{H}_{i}(t)= H_{i}(t,x,y,z,z_1,z_2,u_1,u_2;q_i,k_i,k_{1i},k_{2i},p_i)\quad(i=1,2),{H}_{1}^{v_1}(t)= H_{1}(t,x,y,z,z_1,z_2,v_1,u_2;q_1,k_1,k_{11},k_{21},p_1),{H}_{2}^{v_2}(t)= H_{2}(t,x,y,z,z_1,z_2,u_1,v_2;q_2,k_2,k_{12},k_{22},p_2).\end{aligned}
\tilde{\vec{M}}= \Omega_{1 5}^{( 4 )}\vec{M}\, , \hspace{1 c m}\tilde{\vec{N}}= \Omega_{1 5}^{( 4 )}\vec{N}\, , \hspace{1 c m}\tilde{\vec{J}}= \Omega_{1 5}^{( 4 )}\vec{J}\, .
\left|\Psi(A)(u) \right| = \left|\sum_{j=1}^{n}A(x_{j},y_{j}) \right|\leq \sum_{j=1}^{n}\left|A(x_{j},y_{j}) \right| \leq \left\|A \right\|_{X,Y;\ell_{1}}\cdot \left\|(x_{j})_{j=1}^{n}\right\|_{X(E)}\cdot \left\|(y_{j})_{j=1}^{n}\right\|_{Y(F)}.
C = P (C_1 \oplus C_2) =\begin{pmatrix}0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 \end{pmatrix},
4 \frac{\partial^2 r}{\partial z^\alpha\partial z^\beta}= \frac{\partial^2 r}{\partial x^\alpha\partial x^\beta}-\frac{\partial^2 r}{\partial x^{\alpha+n}\partial x^{\beta+n}}-\sqrt{-1}\left(\frac{\partial^2 r}{\partial x^{\alpha+n}\partial x^\beta}+\frac{\partial^2 r}{\partial x^\alpha \partial x^{\beta+n}}\right).
-4b_0^1b_0^0{\frac 1{a^2}}+2b_0^0b_2^2+3b_0^1b_7^2=0,\,\,\,\,\,\,\,4\,b_0^1b_0^0{\frac 1{a^2}}+2b_0^0b_1^2+3b_0^1b_8^2=0
|S_1|=|\{s: \delta_1 \lesssim \delta_2, \tilde{\delta}_1 \lesssim \tilde{\delta}_2 \Delta + \tilde{\Delta}> \lambda^2 \}| \leq |\{s: \delta_1 \lesssim \delta_2 \Delta > \lambda^2/2 \}| + |\{s: \tilde{\delta}_1 \lesssim \tilde{\delta}_2 \tilde{\Delta}> \lambda^2 /2 \}| =: |S_{11}| + |S_{12}|.
\widetilde{c}_2(t; T)= \frac{\prod_{n=1}^t \sin \left( \frac{n \pi}{3T}\right) \prod_{n=1}^{2T-t}\sin \left( \frac{n \pi}{3T}\right)}{\prod_{n=1}^{2T}\sin \left( \frac{n \pi}{3T}\right)}\prod_{n=1}^T\frac{\sin^2 \left( \frac{n \pi}{3T}\right)}{\cos \left( \frac{(2n-T-1) \pi}{6T}\right)}.
\left[ e^{i \gamma \varphi}\right]_{n}( \underline{{{\theta}}}( \lambda ) ) = \left[ e^{i \gamma \varphi}\right]_{m}( \underline{{{\theta}}}^{\prime}) \, \left[ e^{i \gamma \varphi}\right]_{n - m}( \underline{{{\theta}}}^{\prime \prime}) + O ( e^{- \lambda}) \, .
E|D_{n}|=\sum_{k<n/2}(\begin{matrix}2k\\k\end{matrix})2^{-2k}=\sum_{k<n/2}\frac{1}{\sqrt{\pi k}}+O(1)=\frac{1}{\sqrt{\pi}}\int_{1}^{n/2}\frac{dx}{\sqrt{x}}+O(1)=\sqrt{\frac{2n}{\pi}}+O(1).
\#\bigg\{\vec{v}=(v_1,v_2,\ldots, v_N) \in \{1,2, \ldots, n-1\}^N: \Re \bigg(\prod_{j=1}^N h\big(e_n(v_j)\big) \bigg) \geq 1 \bigg\}\leq \big(c+o(1)\big) \big(n-1\big)^N.
- T_{2}({\bf{k}},{\bf{q}}) c_{{\bf{k}}-{\bf{q}}/ 2}\frac{1}{\sqrt{n_{{\bf{k}}-{\bf{q}}/ 2}}}\frac{1}{\sqrt{n_{{\bf{k}}+{\bf{q}}/ 2}}}c_{{\bf{k}}+{\bf{q}}/ 2}^{\dagger}( \frac{n^{\beta}({\bf{k}}+{\bf{q}}/ 2 )}{\langle N \rangle})^{\frac{1}{2}}( \frac{n^{\beta}({\bf{k}}-{\bf{q}}/ 2 )}{\langle N \rangle})^{\frac{1}{2}}
\int_0^\infty\int_{\Omega}|H_t+P(\nu)|d\mu_t =\int_0^\infty\int_{\Omega_0}|H_t+P(\nu)|d\mu_t\le C(\Omega, \Omega_0, K, n), \int_{\Omega_0\times (-1,1)}|H_t|^2d\mu_t \le C(\Omega, \Omega_0, K, n),
X_2 = \sum_{i=0}^K \frac{(\nu_i-t\pi_i)^2}{t\pi_i}\leq \sum_{i=0}^K \frac{\pi_i}{t}\left(\sum_{r=1}^M \binom{M}{r}Q_r(E_N)\right)^2 = \frac{1}{t}\left(\sum_{r=1}^M \binom{M}{r}Q_r(E_N)\right)^2 \sum_{i=0}^K \pi_i =\frac{M}{N}\left(\sum_{r=1}^M \binom{M}{r}Q_r(E_N)\right)^2
\left(bc+q^{-z}\right)M_n\left(q^{-z};\,b,\,{c\over q};q\right)={c(bq^{n+1}-1) \over q^{n+1}}M_{n+1}\left(q^{-z};\,b,\,c;q\right)+{q^{n+1}+c \over q^{n+1}}M_n\left(q^{-z};\,b,\,c;q\right).
g = c I + v^{n}~ t_{n}= \left( \begin{array}{c r}{{c - v_{2}}}&{{- v_{1}- v_{0}}}\\{{- v_{1}+ v_{0}}}&{{c + v_{2}}}\\ \end{array}\right) , ~ ~ \mathrm{w i t h}~ ~ c^{2}+ v^{n}v_{n}= 1 ,
\frac{1}{\sqrt{- \overline{{{g}}}}}\partial_{\mu}[ \pm \sqrt{- \overline{{{g}}}}\enspace \overline{{{g}}}^{\mu \gamma}\overline{{{g}}}^{\nu \delta}\tilde{F}_{\gamma \delta}] - \frac{\tilde{e}^{2}}{8 \omega}\varphi^{2}\overline{{{g}}}^{\alpha \nu}\tilde{A}_{\alpha}= 0
{\cal L}_{\mathrm{Y M}}={\cal L}_{\mathrm{Y M}}^{( i )}+{\cal L}_{\mathrm{Y M}}^{( a )}, \quad{\cal L}_{\mathrm{Y M}}^{( i )}= - \frac 1 4 \left({\cal F}_{\mu \nu}^{i}\right)^{2}, \quad{\cal L}_{\mathrm{Y M}}^{( a )}= - \frac 1 4 \left({\cal F}_{\mu \nu}^{a}\right)^{2}.
\zeta_{0}( \nu ) = -{\frac{\nu \varrho^{- 2 \nu}}{\pi}}\int_{\mu}^{\infty}d \omega \int_{C_{+}}d z{\frac{2 z^{2}}{( z^{2}+ \omega^{2})^{\nu + 1}}}\breve{\Psi}( \omega ; z ) e^{i \epsilon z},
h ( w ) = - \frac{2 v}{\delta_{b}R \mathrm{c o s h}^{2}\frac{w}{\delta_{b}}}+ \frac{1}{2 \pi R^{4}}\sum_{i}\sum_{K M}\frac{K G_{i}}{{\cal N}_{i}}\int_{w}^{\infty}\mathrm{d}w^{\prime}\frac{w^{\prime}}{\mathrm{c o s h}^{2 \gamma_{i}}\frac{w^{\prime}}{\delta_{b}}},
r_{1 2}^{-}( u , v ) = a_{1 2}( u , v ) - s_{1 2}( u ) + s_{2 1}( v ) , \mathrm{~ \ \ \}r_{1 2}^{+}( u , v ) = a_{1 2}( u , v ) + u_{1 2}^{+}+ u_{1 2}^{-},
e^{-itH^a}(I-P^a)h\sim \sum_{a_1<a}e^{-itH_{a_1}^a}P^{a_1}w_{a_1}h=\sum_{a_1<a}(e^{-itT_{a_1}^a}\otimes e^{-itH^{a_1}}P^{a_1})w_{a_1}h
V(D-iI)^{-s}=\sum_j a_j[D,b_j](|D|-iI)^{-s}g(D)^s &=\sum_j a_j(|D|-iI)^{-s}Y_jg(D)^s=\sum_j a_j(D-iI)^{-s}g(D)^{-s}Y_jg(D)^s\in\S^1.
\frac{\mathrm{d}^n}{\mathrm{d}x^n}\bigl((f\circ g)(x)\bigr) = \sum\limits_{k=1}^n \frac{\mathrm{d}^k f}{\mathrm{d}x^k}(g(x)) B_{n,k}\Bigl(\frac{\mathrm{d}g}{\mathrm{d}x}(x),\frac{\mathrm{d}^2 g}{\mathrm{d}x^2}(x),\ldots,\frac{\mathrm{d}^{(n-k+1)}g}{\mathrm{d}x^{(n-k+1)}}(x)\Bigr)
\varepsilon_{\mu \nu \lambda \rho}\partial_{\lambda}\partial_{\rho}\theta_{i}^{\mathrm{s i n g}}( x ) = 2 \pi \Sigma_{\mu \nu}^{i}( x ) \equiv 2 \pi \int_{\Sigma_{i}}^{}d \sigma_{\mu \nu}\left( x^{( i )}( \xi ) \right) \delta \left( x - x^{( i )}( \xi ) \right) .
\begin{array}{c}min\Big\{max\Big[rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f^{ij}(x);\qquad\qquad\qquad\qquad V^{ij}(x)-N^{ij}[V](x)\Big];V^{ij}(x)-M^{ij}[V](x)\Big\}=0.\end{array}
S_2 =\sum_{m=1}^kS_2^{(m)}&=\sum_{m=1}^k\sum_{\substack{x\leq n<2x n\equiv \nu_0\pmod{W^\prime}}}\chi_\mathbb{P}(nM+h_m)\left(\sum_{d_i\mid nM+h_i}\lambda_{d_1,\ldots,d_k}\right)^2,
0<\int_C f(u)h_\varepsilon (|\nabla u|)\psi^2_\varepsilon \,dx = \, \int_C |\nabla u|^{p-2}(\nabla u,\nabla |\nabla u|) h_{\varepsilon}'(|\nabla u|) \psi_\varepsilon^2 \,dx\, &+ 2\int_C |\nabla u|^{p-2}(\nabla u, \nabla \psi_\varepsilon)h_{\varepsilon}(|\nabla u|)\psi_\varepsilon\,dx.
\mu'(t)=\psi'_{p,q}(s+t) - \psi'_{p,q}(t) = (\ln q)^{2}\sum_{n=1}^{p}\left[ \frac{nq^{n(s+t)}}{1-q^{n}}- \frac{nq^{nt}}{1-q^{n}}\right] &= (\ln q)^{2}\sum_{n=1}^{p}\frac{nq^{nt}(q^{ns}-1)}{1-q^{n}}\leq0.
q^{{n-k+1 \choose 3}-(k-1){n-k \choose 2}}\operatorname{DET}\big|S_{n+k-i-j}\big(q; q^{i-1}\beta\big) \big|_{1 \le i,j \le k}{}=q^{k(k-1)(4k+1)/6}\prod_{a=1}^{k-1}\big(1+q^{a-1}\beta\big) \mathfrak{G}_{\pi_{k}^{(n)}}\big(1,q,q^2,\ldots\big).
h = g(d^*e_n,\hat{y},h) - g(0,\hat{y},h) = d^*g_{x_n}(\tau d^*e_n,\hat{y},h) \leq d^* |g_x(0,\hat{y},h)|+Cd^*(d^*+h)|\hat{y}| + Kh(h+d^*).
f_{3,k}:= -f^*_{3,k}-\frac{1}{2}\left.\{f_1,G_2^0 \}_k\right|_{p={p}^0_k}-\frac{1}{2}\left.\{f_2,G_1\}_k\right|_{p={p}^0_k}-\frac{1}{12}\left.\{\{f_1,G_1 \},G_1 \}_k\right|_{p={p}^0_k}.
\varphi^\alpha_\nu =-\log||s_\alpha||^2_h\ast\rho_\nu=-(\log|s_\alpha|^2)\ast\rho_\nu-(\log h)\ast\rho_\nu =-\log|s_\alpha|^2-(\log h)\ast\rho_\nu, h^\alpha_\nu =h_\alpha e^{\log|s_\alpha|^2+(\log h)\ast\rho_\nu}=\exp\{(\log h)\ast\rho_\nu\}=h_\beta e^{\log|s_\beta|^2+(\log h)\ast\rho_\nu}=h^\beta_\nu,
L(f) = \tilde{g}^{i\overline{j}}\partial\overline{\partial}f(e_{i},\overline{e}_{j}) & = \tilde{g}^{i\overline{j}}\left(e_{i}\overline{e}_{j}(f)-[e_{i},\overline{e}_{j}]^{(0,1)}(f)\right).
\|f\|_{Y_0}^2= \|f\|_{L^{\infty}L^2}^2+\nu\|\nabla f\|_{L^{2}L^2}^2, \|f\|_{X_a}^2= \|e^{a\nu^{1/3}t}f\|_{L^{\infty}L^2}^2+\|e^{a\nu^{1/3}t}\nabla\Delta^{-1}\partial_x f\|_{L^{2}L^2}^2 &+\nu^{1/3}\|e^{a\nu^{1/3}t}f\|_{L^{2}L^2}^2+\nu\|e^{a\nu^{1/3}t}\nabla f\|_{L^{2}L^2}^2.
\mathbb{H}_{\Phi}^{d}f\left( x\right) ={\displaystyle\sum\limits_{k_{1}=1}^{\infty}}...{\displaystyle\sum\limits_{k_{n}=1}^{\infty}}\Phi\left( r_{k_{1}}^{\left( 1\right)},...,r_{k_{n}}^{\left( n\right)}\right) f\left( r_{k_{1}}^{\left( 1\right)}x_{1},...,r_{k_{n}}^{\left(n\right)}x_{n}\right)
1_{B}\cdot u = 1_{B}\cdot \sum_{i=1}^{l}h_{i}= 1_{B}\cdot\sum_{i=1}^{l}\sum_{j=1}^{s_{i}}\lambda_{ij}u_{ij}= \sum_{i=1}^{l}\sum_{j=1}^{s_{i}}\lambda_{ij}1_{B}\cdot u_{ij}=0.
V = V_{1}+\frac{1}{2}V_{2}=g^{2}\frac{m_{H}}{8m^{2}}\left(H^{2}+\phi^{2}+\frac{2m}{g}H\right)^{2}-\frac{m_{H}^{2}}{2}H^{2}, S^{(2)}= 1+gi\int{\mathcal L}_{\rm tot}d^{4}x+i^{2}\frac{g^{2}}{2}\int d^{4}x\int d^{4}x^{\prime}T{\mathcal L}{\mathcal L}^{\prime},g{\mathcal L}_{\rm tot}=g{\mathcal L}+V.
\begin{array}{r c l}{}&{}&{\delta ( i_{h}i_{k}N^{( 8 )}) = 6 \partial ( i_{h}i_{k}\Omega^{( 7 )})}\\{}&{}&{}\\{}&{}&{\delta ( i_{h}i_{k}{\cal N}^{( 8 )}) = 6 \partial ( i_{h}i_{k}{\tilde{\Omega}}^{( 7 )}) .}\\ \end{array}
\mathbb{P}( \widehat{Q}_0 < \widehat{Q}_1 < \cdots) \stackrel{1}{=}\prod_{j=1}^\infty \mathbb{P}(G_j \le 1) \stackrel{1}{=}\prod_{j=1}^\infty \frac{j(j+2)}{(j+1)^2}= \frac{1}{2},
( a \nabla_{\nu}b)^{2}\leqslant ( a \sharp_{\nu}b )^{2}+ (1-r_{0})^{2}(a-b)^{2}- \sum_{k=1}^{\infty}r_{k}\big[ a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}- a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big]^{2}.
<\psi_{qG}|\psi_{qG}>= \int\limits_0^{\infty}\left[1+\frac{2(q-1)|\Im(p)|x}{\hbar\sqrt{2(q+1)}}+ \frac{(q-1)^2|p|^2x^2}{\hbar^22(q+1)}\right]^{\frac{2}{1-q}}\;dx.
\sigma^{( 1 )}= \frac{m^{2}}{8 \sqrt{3}\pi}\eta^{- 1 / 2}\int_{0}^{1}d{\tilde{h}}\; ({\tilde{h}}- 1 ) ( 1 7 + 1 8{\tilde{h}}) = - \frac{2 3 \; m^{2}}{1 6 \sqrt{3}\pi}\eta^{- 1 / 2}.
\int_{\Omega}\sum_{i=1}^m \Psi(u^i(t_2))+\delta\int_{t_1}^{t_2}\int_{\Omega}\sum_{i=1}^m\left|\nabla u^i\right|^2 +\delta_0\int_{t_1}^{t_2}\int_{\Omega}\sum_{i=1}^m\left|\nabla u^i\right|^2\leq\int_{\Omega}\sum_{i=1}^m\Psi(u^i_0),
\frac{A_{4}}{A_{0}}=\frac{M_{0}}{M_{4}}\frac{p_{t0}}{p_{t4}}(\frac{1+\frac{\gamma-1}{2}M_{0}^{2}}{1+\frac{\gamma-1}{2}M_{4}^{2}})^{-\frac{\gamma+1}{2(\gamma-1)}}
n_{2}=\sum_{\alpha_{l}=1}^{\chi_{c}}(c_{\alpha_{l-1}\alpha_{l}})^{2}\cdot({\lambda^{\prime}}_{\alpha_{l}}^{[l]})^{2}=\sum_{\alpha_{l}=1}^{\chi_{c}}(c_{\alpha_{l-1}\alpha_{l}})^{2}\frac{(\lambda_{\alpha_{l}}^{[l]})^{2}}{R}=\frac{S_{1}}{R}
\begin{array}{l}{{{\mathcal D}_{V}f = V^{a}\nabla_{a}f + \frac{n - 2}{2 n}( \nabla_{a}V^{a}) f}}\\{{\delta_{V}f = V^{a}\nabla_{a}f + \frac{n + 2}{2 n}( \nabla_{a}V^{a}) f .}}\\ \end{array}
\widehat{K}=M^{-1}\begin{bmatrix}A^T \|x_S\|_2A^T\left(I_m-\frac{1}{\|r\|_2^2}rr^T\right) \|r\|_2\left(I_n-\frac{1}{\|r\|_2^2}A^Trx_S^T\right) \end{bmatrix}.
\left|D(|Du|^{\frac{p-2+s}2}Du)\right| =\frac{1}{2}\alpha^{\frac{p+s}{2}}(\alpha-1)(p+s)|x_1|^{\frac{(\alpha-1)(p+s)}{2}-1}&=C(p,s,\gamma)|x_1|^{\frac{p+s}{2(\gamma+1)}-1}.
f^{2}(r)=\int_{0}^{r}\left(2 f(s) f'(s)\right) ds =2 \int_{0}^{r}\frac{f(s)}{\sqrt{s}}(f'(s) \sqrt{s}) ds \leq 2 ||\frac{f}{r}||_{L^{2}(r dr)}||f'||_{L^{2}(r dr)}
\dfrac{\langle x_{i,r}^-[\mu], [\lambda] \rangle_{H}}{\langle [\lambda], [\lambda] \rangle_{H}}=\delta((\mu, \lambda) \in \mathfrak{P}_i({\bf v})) (-1)^{v_i(\lambda) - v_{i+1}(\lambda)}c_1(\mathcal{L}_i({\bf v}))^r \dfrac{e(N_{\mu \lambda})}{e(T_{\lambda})}.
Y(q x) = B(x)Y(x) \mapsto \widehat{Y}(q x) = F(x)\widehat{Y}(x), & B(x) = B_{\infty}+ \sum^{N}_{i = 1}\frac{B_{i}}{1 - x/b_{i}}, F(x) = F_{\infty}+ \sum^{N}_{i = 1}\frac{F_{i}}{1 - x/b_{i}},
\tilde{\mathcal{G}}(x^{-1})=\lim_{n\to\infty}\, x^{-n}\tilde{F}_{n}(a,p;x)=\lim_{n\to\infty}\,(px)^{-n}F_{n}(a,p;px)=\mathcal{G}(p^{-1}x^{-1}).
\Vert p_{1}^{\varphi_{t}}f(x_1-x_2)\Vert = \left\Vert p_{1}^{\varphi_{t}}|f(x_1-x_2)|^2 p_{1}^{\varphi_{t}}\right\Vert^{1/2}\leq C(t)\Lambda^{-1/2}\Vert f\Vert_2
\textbf{c}=\left[\begin{array}{cccc}c_{00}c_{01}\dots c_{0,\ell-1}c_{10}c_{11}\ldots c_{1,\ell-1}\vdots \vdots \ddots \vdots c_{n-1,0}c_{n-1,1}\ldots c_{n-1,\ell-1}\end{array}\right]\in C,
\overline{c}_{X^{\ast}}(g_1, g_2) =m_{X^{\ast}}(g_1g_2)^{-1}m_{X^{\ast}}(g_1) m_{X^{\ast}}(g_2) \widetilde{c}_{{X^{\ast}}}(g_1,g_2) &=[ \widetilde{\beta}(g_1g_2)m_{X^{\ast}}(g_1g_2)^{-1}] [ \widetilde{\beta}(g_1)m_{X^{\ast}}(g_1)^{-1}]^{-1}[\widetilde{\beta}(g_2)m_{X^{\ast}}(g_2)^{-1}]^{-1}.
S(z)=\frac{1}{f(z)-a}-b=\frac{h(z)-bg(z)+abh(z)}{g(z)-ah(z)}=\frac{\sum_{j=0}^n \tilde G_j(z)e^{w_jz^q}}{\sum_{j=0}^n \tilde H_j(z)e^{w_jz^q}}
J_{\nu-1}^{2}(ix)-e^{2 \theta}J_{\nu}^{2}(ix)={J_{\nu}'}^{2}(ix)-\left({\nu^{2}\over x^{2}}+e^{2\theta}\right)J_{\nu}^{2}(ix)+{2\nu \over ix}J_{\nu}(ix)J_{\nu}'(ix).
\hat{\rho}_{ABM}(t) := \left(\mathcal{N}_A\left(\frac{\lambda\, t}{\eta}\right)\otimes\mathcal{N}_B\left(\frac{(1-\lambda)\,t}{|1-\eta|}\right)\otimes\mathbb{I}_M\right)(\hat{\rho}_{ABM})\;, \hat{\rho}_{CM}(t) := (\mathcal{B}_\eta\otimes\mathbb{I}_M)(\hat{\rho}_{ABM}(t))\;.
\hphantom{{}={}}(x^i,(a^i{}_j,a^i{}_{j\alpha}a^\alpha{}_k),(b^i{}_j,b^i{}_{jk})).(g,X) * =(x^i,(a^i{}_\alpha g^\alpha{}_j,a^i{}_{\beta\alpha}g^\beta{}_ja^\alpha{}_\beta g^\beta{}_k),(h^i{}_\alpha b^\alpha{}_\beta g^\beta{}_j+X^i{}_j,h^i{}_\alpha b^\alpha{}_{\beta k}g^\beta{}_j)).
|\varphi(z)| \leq C_{\varepsilon}\mathrm{e}^{\varepsilon|z|}\left(\sum_{n=0}^{\infty}(n!)^{2}2^{nq}|\varphi_{n}|^{2}\right)^{1/2}\left(\sum_{n=0}^{\infty}2^{-nq}\sigma_{\varepsilon}^{-2n}\right)^{1/2}=C_{\varepsilon}(1-2^{-q}\sigma_{\varepsilon}^{-2})^{-1/2}\|\varphi\|_{q,1,\pi_{\lambda,\beta}}\mathrm{e}^{\varepsilon|z|},
P_{i+1}=A_{i}(P_{i}-P_{i}C_{i}^{T}(C_{i}P_{i}C_{i}^{T}+W_{i})^{-1}C_{i}P_{i})A_{i}^{T}+V_{i},P_{0}=\mathbb{E}[(x_{0}-\hat{x}_{0})(x_{0}-\hat{x}_{0})^{T}].
lim_{x\rightarrow\infty}x(\frac{A}{x-1}+\frac{Bx+C}{x^{2}+x+1})=lim_{x\rightarrow\infty}\frac{Ax}{x-1}+lim_{x\rightarrow\infty}\frac{Bx^{2}+Cx}{x^{2}+x+1}=A+B,
V_1 : = \{F_0^{r, r + 2\varepsilon r}(q, s), q\in U_2, s \in [0, 1]\}V_2 : = \{\Psi^{r}_0(x, \varepsilon r, 7 \varepsilon r)(q, s), q \in F_0^{r, r + 2\varepsilon r}(U_2, 1), s \in [0, 1]\}V : = V_1 \cup V_2.
\begin{cases}s\in [0,1-\frac{1}{\alpha}), \textrm{if}\;\; \bar{s}= \frac{N}{2}+1-\alpha, s\geq 0, \;\bar{s}+ s\alpha \leq \frac{N}{2}+1-\alpha, \textrm{if}\;\; \bar{s}= -s_0 \in (-\frac{N}{2},\frac{N}{2}+1-\alpha).\end{cases}
f\Big(s^{(m)}_{\beta,0},\dots,s^{(m)}_{\beta,j-1}\Big)+c(s^{(m)}_{\beta,j})= f\Big(s^{(m-1)}_{\alpha_\beta,0},\dots,s^{(m-1)}_{\alpha_\beta,j-1}\Big)+c(s^{(m-1)}_{\alpha_\beta,j})
Q^1_{i\bar j}=g^{k\bar l}g^{m\bar n}T_{ik\bar n}T_{\bar j\bar l m}\,, Q^2_{i\bar j}=g^{\bar l k}g^{\bar n m}T_{\bar l\bar n i}T_{km \bar j}\,, &Q^3_{i\bar j}=g^{\bar l k}g^{\bar n m}T_{ik\bar l}T_{\bar j\bar n m}\,, Q^4_{i\bar j}=\frac12g^{\bar l k}g^{\bar n m}(T_{mk\bar l}T_{\bar n\bar ji}+T_{\bar n\bar lk}T_{mi\bar j})\,,
\nabla^{\ast}\nabla h=-h_{00}^{\prime\prime}e^{0}\otimes e^{0}-h^{\prime\prime}_{0i}e^{0}\otimes e^{i}-h^{\prime\prime}_{ij}e^{i}\otimes e^{j}&-(m-1)\coth(r)(h_{00}^{\prime}e^{0}\otimes e^{0}+h^{\prime}_{0i}e^{0}\otimes e^{i}+h^{\prime}_{ij}e^{i}\otimes e^{j}) +\mathcal{D}h,
\tilde{R}\boldsymbol{u}+ \frac{1}{2}\tilde{R}^{-1}B^{T}(P+P^T)\boldsymbol{x}-\tilde{R}^{-1}B^{T}\boldsymbol{g}= \boldsymbol{0}\implies \boldsymbol{u}=- \frac{1}{2}{R}^{-1}B^T\left( P^T +P \right)\boldsymbol{x}+R^{-1}B^T\boldsymbol{g}
S(a;b_1,\ldots,b_{k-1})=\sum_{n=1}^\infty\frac{({\rm sign}\,a)^{n}}{n^{|\,a|}}\prod_{i=1}^{k-1}\sum_{m_i=1}^{n-1}\frac{({\rm sign}\,b_i)^{m_i}}{m_i^{|\,b_i|}}\,.
g_{n-1}=1, (w_++w_-)g_{n-1}-g_{n-2}=1, w_+w_- g_{p+2}-( w_++w_- )g_{p+1}+g_p =0, \;\;\; (0 \leq p \leq n-3), w_+w_- g_1 - (w_++w_-)g_0 =0, w_+w_- g_0 =K.
(n-\alpha)p +\sigma_2 - n +{}p(q(n-\alpha) +\sigma_1 - n) ={}(n-\alpha)p + \sigma_2 - n + pq(n-\alpha) + \sigma_1 p - np ={}(pq-1)(n-\alpha) - \lbrace \alpha(1+p) - (\sigma_2 + \sigma_1 p)\rbrace ={}(pq-1)(n-\alpha - M) ={}0.
Y^{(n)}(t_{n, i+1})=Y^{(n)}(t_{n, i})+ \int_{t_{n, i}}^{t_{n, i+1}}g(t_{n, i}, W_0^{t_{n, i}}, Y_0^{(n), t_{n, i}}) ds + B(t_{n, i+1})-B(t_{n, i}),
t^{\sigma}\left\|\int_0^t e^{-(t-\tau)H^{\beta}}(f(u)-f(v)) \, d\tau\right\|_{L^{a}}\leq C \sum_{k=0}^{\infty}\frac{\lambda^k}{k!}t^{\sigma}\int_0^t (t-\tau)^{-\frac d{2\beta}(\frac 1r-\frac 1a)}\|(u-v)(|u|^{pk+m-1}+|v|^{pk+m-1})\|_{L^r}\, d\tau.
(t+ \Delta t)^p( f(X_{t + \Delta t}) - f(x)) - t^p(f(X_t) - f(x)) = t^p (f(X_{t + \Delta t}) - f(X_t)) & + pt^{p-1}(f(X_{t+\Delta t}) - f(x))\Delta t + o(\Delta t) &= t^p \langle G_f(X_t, \dot X_t), \dot X_t\rangle \Delta t & + pt^{p-1}(f(X_{t+\Delta t}) - f(x))\Delta t + o(\Delta t),
\frac{\int_0^L \frac{|v'|^2 V_{\gamma_0}}{|\gamma_0'|}+ \frac{k^2 |v|^2 | \gamma_0'|}{V_{\gamma_0}}\,dt}{\int_0^L |v|^2 V_{\gamma_0}|\gamma_0'| \,dt}= \frac{\int_0^{L_0}\frac{|w'|^2 V_{\chi_0}}{|\chi_0'|}+ \frac{k^2 |w|^2 | \chi_0'|}{V_{\chi_0}}\,dt}{\int_0^{L_0}|w|^2 V_{\chi_0}|\chi_0'| \,dt}
\begin{aligned}\int_{Q_T}\partial_t\rho^{-1}\psi dxdt=\lim_{k\rightarrow+\infty}\int_{Q_T}\partial_t\rho^{-1}_k\psi dxdt=\lim_{k\rightarrow+\infty}\int_{Q_T}(r^2_ku_k)_x\psi dxdt = -\lim_{k\rightarrow+\infty}\int_{Q_T}r_k^2u_k\psi_xdxdt=-\lim_{k\rightarrow+\infty}\int_{Q_T}r^2u\psi_xdxdt=\int_{Q_T}(r^2u)_x\psi dxdt.\end{aligned}
\sum_{\ell=k/8}^k\binom{k}{\ell}\beta^{\ell pm^2/16}\big(1-\beta^{pm^2/32}\big)^{k-\ell}\le k\cdot\binom{k}{k/8}\beta^{kpm^2/1000}\le 2^k\beta^{pkm^2/1000}
\begin{array}{ll}\bar{p}= \left( \frac{1}{|\Omega|}\int\limits_{\Omega}p_0^{\frac{1}{\gamma}}\,\mathrm{d}x \right)^{\gamma},\bar{S}= \frac{1}{\Omega}\int\limits_{\Omega}S_0 \,\mathrm{d}x, \bar{\varrho}= \frac{1}{\Omega}\int\limits_{\Omega}\varrho_0 \,\mathrm{d}x,\end{array}
\min_{\mathbf{\Phi}}~~ P(R_s), ~~ \mathbf{\Omega}= \mathbf{W}\mathbf{\Phi}\mathbf{W}^{\dag}, ~~(\mathbf{\Phi}) = N_t-1, & \mathbf{\Phi}= \left[\phi_1, \phi_2, \dots, \phi_{N_t-1}\right],