harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x_d \equiv x_{\infty}\underset{j=1}{\stackrel{d-1}{\textstyle{\prod}}}x_j^{\frac{\ell_j -1}{2}}\mod{[F,F]}, \eta_i \equiv x_0^{\mathrm{lk}(\ell_i,\ell_0)}x_{\infty}^{\mathrm{lk}(\ell_i,\ell_d)}\underset{j=1}{\stackrel{d-1}{\textstyle{\prod}}}x_j^{\mathrm{lk}(\ell_i,\ell_j)+\frac{\ell_j -1}{2}\mathrm{lk}(\ell_i,\ell_d)}\mod{[F,F]},
{\bf E}\Bigl[\Bigl(M_\mu^{(c=2)}\bigl(0,\frac{1}{2}\bigr)\Bigr)^{n}\Bigl( M_\mu^{(c=2)}\bigl(\frac{1}{2}, 1\bigr)\Bigr)^{m}\Bigr] = n!m! \int\limits_{\substack{0<t_1<\cdots<t_{n}<1/2 1/2<t_{n+1}<\cdots<t_{n+m}<1}}\prod\limits_{k<p}^{n+m}|t_p-t_k|^{-\mu\,2^{p-k-1}}\,dt,
M' = \left[\begin{array}{c c c c c c c c c c c c c c c c}0 0 0 0 0 0 0 0 1 2 3 4 4 3 2 1 0 0 0 0 1 2 3 4 0 0 0 0 4 3 2 1 0 0 1 2 0 0 3 4 0 0 4 3 0 0 2 1 0 1 0 2 0 3 0 4 0 4 0 3 0 2 0 1 \end{array}\right]\,.
P_{r}^{2}= P_{1}^{2}+ P_{2}^{2}= \hbar^{2}\gamma ( a_{1}^{+}a_{1}+ a_{1}a_{1}^{+}) \, , \; 2 L_{z}= \hbar ( a_{1}^{+}a_{1}+ a_{1}a_{1}^{+}- a_{2}^{+}a_{2}- a_{2}a_{2}^{+}) .
c_1^{\rm in}(x) =2+12(x-0.5)^2, c_2^{\rm in}(x)= 1+2x, &{\partial_x c_i}+q_ic_i{\partial_x \psi}=0,x=0, 1, & \partial_x\psi(t, 0)=\partial_x \psi(t, 1)=0,
\int_M \Big(\langle C(K_1),h_2 \rangle+\langle K_2,D(h_1) \rangle\Big) e^{-f}dV_g=\int_M \Big(\langle C(K_2),h_1 \rangle+\langle K_1,D(h_2) \rangle\Big) e^{-f}dV_g .
\frac{(2n+|x|_{1})!}{\prod_{i=1}^{d}q_{i}!(q_{i}+|x_{i}|)!}\leq 2^{2n+|x|_{1}}\frac{(n+|x|_{1})!}{\prod_{i=1}^{d}(q_{i}+|x_{i}|)!}C_{d}\frac{d^{n}}{n^{\frac{d-1}{2}}}.
m_{w\lambda}= \bigl( \underbrace{r_{p_{M-L+1}}\cdots r_{p_M}}_{=w}\bigr) \bigl( \underbrace{\pi r_{i_{M+1}}\cdots r_{i_{\ell}}}_{=m_{\lambda}}\bigr) = \pi r_{i_{M-L+1}'}\cdots r_{i_{M}'}r_{i_{M+1}}\cdots r_{i_{\ell}};
S_1=\frac{X}{V}\sideset{}{^\prime}\sum_{\substack{d_1,\ldots,d_k e_1,\ldots,e_k}}\frac{\lambda_{d_1,\ldots,d_k}\lambda_{e_1,\ldots,e_k}}{\prod_{i=1}^k[d_i,e_i]}+O\Bigl(\sideset{}{^\prime}\sum_{\substack{d_1,\ldots,d_k e_1,\ldots,e_k}}\lvert\lambda_{d_1,\ldots,d_k}\lambda_{e_1,\ldots,e_k}\rvert\Bigr),
\mathcal{L}(x, \beta, N)=\big[ B(x)(T_x-1)+E(x)\big(T_x^{-1}-1\big)\big]R_n(x), B(x)=\frac{(x+\beta_1-\beta_0)(x+\beta_1)(x+\beta_2+N)(N-x)}{(2x+\beta_1)(2x+\beta_1+1)}, E(x)=\frac{x(x+\beta_0)(N-x-\beta_1+\beta_2)(N+x+\beta_1)}{(2x+\beta_1)(2x+\beta_1-1)}, \kappa(n, c)= (n+c ) (n+c-1 ).
0 =\int_{K \backslash G /H}F(\ddot{x})d\mu(\ddot{x}) =\int_{K \backslash G /H}Q(f)(\ddot{x})d\mu(\ddot{x}) &=\int_{K \backslash G /H}\int_K \int_H f(k^{-1}x h) dh dk d\mu(\ddot{x}) &=\int_G f(x) \rho(x) dx>0
\Vert \mathcal{M}_f \cdot \mathcal{M}_g \Vert_F^2 = \sum_{\mathcal{S}\in [n]^{d'-r}}\Vert \mathcal{M}_f \cdot \mathcal{M}_{g, \mathcal{S}}\Vert_F^2 \le \sum_{\mathcal{S}\in [n]^{d'-r}}\kappa^2 \cdot \Vert \mathcal{M}_{g, \mathcal{S}}\Vert_F^2 = \kappa^2 \cdot \Vert \mathcal{M}_g \Vert_F^2.
T^{(k)}_{V/N^2}(e_{q,N})=N\frac{d^k}{de^k}_{\mid_{e=e_{q,N}}}\!\!\!\!\big(T_V(N^2e)\big)=N^{2k+1}T_V^{(k)}(N^2 e_{q,N}) =N^{2k+1}T^{(k)}_{V}(e_{q/N,1})
\big|\partial_\xi\Psi[\lambda,\xi,\dot{\lambda},\dot{\xi},\phi][\bar{\xi}](x, t)\big|\lesssim \frac{t_0^{-\varepsilon}}{R^{a-2s}}\|\bar{\xi}(t)\|_{1+\sigma}\left(\sum_{j=1}^k\frac{\mu_0^{\frac{n-2s}{2}-1}(t)}{1+|y_j|^{a-2s}}\right),
\eta : = \left( \begin{array}{c c c c}{{\varepsilon - i}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\varepsilon + i}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\varepsilon + i}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{\varepsilon + i}}\\ \end{array}\right) \qquad .
f_{1/4}(x|\mu) = \int_0^\infty f_{1/2}(x|y) f_{1/2}(y|\mu) dy = \int_0^\infty \frac{y}{2\sqrt{\pi x^3}}\, e^{-y^2/4x}\,\frac{\mu}{2\sqrt{\pi y^3}}\, e^{-\mu^2/4y}\, dy = \frac{\mu}{4\pi\sqrt{x^3}}\int_0^\infty \frac{1}{\sqrt{y}}\, e^{-\mu^2/4y-y^2/4x}\, dy
x_{+,-}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}=\frac{(-b\pm\sqrt{b^{2}-4ac})(-b\mp\sqrt{b^{2}-4ac})}{2a(-b\mp\sqrt{b^{2}-4ac})}=\frac{4ac}{2a(-b\mp\sqrt{b^{2}-4ac})}
\langle f(1 - i x_1, \dots, 1 - i x_N) \rangle^{(\rm Cy)}= \langle f(2x_1,\dots, 2x_N) \rangle^{(\rm J)}\Big |_{\lambda_1 = - \beta(N+p-1)/2 - 1 + iq\atop\lambda_2 = - \beta(N+p-1)/2 - 1 - i q},
P_N (vu_1) = P_N (P_{\ll N}v \cdot u_1) + P_N(P_{\gtrsim N}v \cdot u_1) &\sim P_N(P_{\ll N}v \cdot P_N u_1) + P_N(P_{\gtrsim N}v \cdot P_{\gtrsim N}u_1).
C_{\bar k}:= \prod_{i=1}^n \dfrac{(1-qt^{n-i})}{(1-q^{k+1}t^{n-i})}\prod_{i=1}^n \dfrac{(q^{1+k}t^{n-i})_{k}}{(qt^{n-i})_{k}}=\prod_{i=1}^n \dfrac{(q^{2+k}t^{n-i})_{k-1}}{(q^2t^{n-i})_{k-1}}
\begin{aligned}m,\d,\ \alpha_0,\ \alpha_1,\ \Vert g\Vert_{L_\infty},\ \Vert g^{-1}\Vert_{L_\infty}; &\textnormal{the parameters of the lattice}\ \Gamma; \ \ \textnormal{the domain}\ \mathcal{O}.\end{aligned}
I_{\pm0}(x^*,y^*) :=\ \{i=1,\dots,n \mid x_i^*\not=0,\,y^*_i=0\}, I_{00}(x^*,y^*) :=\ \{i=1,\dots,n \mid x_i^*=0,\,y^*_i=0\}, I_{0+}(x^*,y^*) :=\ \{i=1,\dots,n \mid x_i^*=0,\,y^*_i\in(0,1)\}, I_{01}(x^*,y^*) :=\ \{i=1,\dots,n \mid x_i^*=0,\,y^*_i=1\}.
\langle I(x)\rangle = \{f(x)I(x)|f(x)\in{\mathbb{F}_{q}[x]}/{\langle x^n-1\rangle}\}= \{f(x)I(x)|f(x)\in \mathbb{F}_{q}[x], f(x)=0 0\leq deg(f(x))< n-deg(I(x))\},
\mathcal{B}(s_1,s_2) = \sum_{g \mid h}\frac{1}{g^{s_1+s_2+1}}\sum_{d=1}^{\infty}\frac{\mu(d) \sigma_{k}(gd,s_1+1) \sigma_{\ell}(gd,s_2+1)}{d^{s_1+s_2+2}}
C_{N,p}\, \int_{\mathbb{R}^N}u^{p-1}\, F_*(x)^{\gamma-p}\,\varphi\, dx &=\int_{\mathbb{R}^N}F_*(x)^\gamma\,F(\nabla u)^{p-1}\,\langle \nabla F(\nabla u),\nabla\varphi\rangle\, dx,\varphi\in C^1_0(\mathbb{R}^N\setminus\{0\}).
\begin{aligned}\frac{\partial^2r}{\partial v\partial u}+\frac{\mu_1(1-q^2)+2\mu_2qr}{-\mu_1(1-q^2)r+\mu_2q(1-r^2)}\frac{\partial r}{\partial u}\frac{\partial r}{\partial v}+\frac{(\mu_1^2-\mu_2^2)(1+q^2)(1-r^2)r}{[-\mu_1(1-q^2)r+\mu_2q(1-r^2)][-\mu_1q(1-r^2)+\mu_2(1-q^2)r]}\frac{\partial r}{\partial u}\frac{\partial q}{\partial v}=0. \end{aligned}
\{Q_A, Q_B \}= Z_{AB}= \left(C\Gamma^{\widetilde{\mu}_1\widetilde{\mu}_2}\right)_{AB}\, Z_{[\widetilde{\mu}_1\widetilde{\mu}_2]}+ \left(C\Gamma^{\widetilde{\mu}_1 \ldots \widetilde{\mu}_6}\right)_{AB}\, Z_{[\widetilde{\mu}_1 \ldots \widetilde{\mu}_6]}\, ,
\mathrm{ad}_{V_q}(X_k)=\sum_{j=1}^nv_j\left[X_j,\overline{X_k}\right]-\overline{X_k}(v_j)X_j = \sum_{j=1}^nv_j\left(H_{\mathcal{L}}\right)_{j,k}-\overline{X_k}(v_j)X_j =- \sum_{j=1}^n\overline{X_k}(v_j)X_j & =- \sum_{j=1}^n \frac{\partial v_j}{\partial x_k}X_j.
A(v, v) = A_1(v, v) +\frac{\beta_0}{h}(v_{1/2}^+)^2 + (vv_x)^+_{1/2}\geq \gamma \|v\|_E^2 + \frac{1}{2}\int_{I_1}v_x^2dx + \frac{\beta_0}{2h}(v_{1/2}^+)^2 -\frac{h}{2\beta_0}(v_{x}^+)^2_{1/2}\geq \gamma \|v\|_E^2 +\frac{\beta_0}{2h}(v_{1/2}^+)^2,
\beta_k = \alpha_k \frac{3 a^2 + a (k + 3 b k - 3 m) +b (k^2 - (2 + b) k m + 3 (1 - b) m^2)}{3 a^2 - k^2 + k m + 3 (1 - b) b m^2 - a (2 k + 3 m)}.
|-\int_{t}^{t+\frac{1}{2}}\frac{ds}{(s-t)}\int_{0}^{s-t}\rho d\rho \frac{\lambda''(s)}{r}\partial_{r}\left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}\right)| &\leq \frac{C}{t^{2}\log^{b+1}(t)}
\sigma_N (X \eta ' ) = \sigma_N (g_m Y \eta ' ) &= [u_1 T_{\zeta '_1}, \ldots , u_{L}T_{\zeta '_{L}}, r_i u_{L+1}T_{\zeta '_{L+1}}, \ldots , r_i u_{K}T_{\zeta '_K}, u_{K+1}T_{\zeta '_{K+1}}, \ldots , u_N T_{\zeta '_N}] .
\|u_n\|_{\alpha+2}^{\alpha+2}=(\alpha+2)(-m_c+o_n(1)+\frac{1}{2}\|\Delta_{x,y}u_n\|_2^2+\frac{1}{2}\|\nabla_{x,y}u_n\|_2^2)\geq (\alpha+2)(-m_c+o_n(1)).
A_{s_o}=\{t\in(0,\frac T2]:\ \exists\, x\in B_{\frac \rho2},\u(x,t)<\frac L{2^{s_o}}\}A(t) =\{x\in B_{\frac \rho2}:\u(x,t)<L(1-\frac 1{2^{s_o}})\},t\in(0,\frac T2] A =\left\{(x,t)\in B_{\frac \rho2}\times(0,\frac T2]:\u(x,t)<L(1-\frac 1{2^{s_o}})\right\}.
a_1 =\sum_{k=3}^{g+1}\tbinom{k-1}{2}\tfrac{k!(k-1)!}{12}(-1)^{g+1-k}(g+1-k)! \tfrac{g!}{(k-1)!}\tbinom{g}{k-1}\tbinom{g+1}{k}&=\tfrac{g!(g+1)!}{12}(-1)^g\sum_{k=2}^{g}(-1)^{k}\tbinom{k}{2}\tbinom{g}{k}=0,
\gamma\left( a+2,a\right) =\left( a+1\right) \gamma\left(a+1,a\right) -a^{a+1}\exp\left( -a\right) , \gamma\left( a+3,a\right) =\left( a+2\right) \left( a+1\right)\gamma\left( a+1,a\right) -2\left( a+1\right) a^{a+1}\exp\left(-a\right) .
\Xi^{\nu \rho}(x)=\int d^3x_2....d^3x_nF_{\mu_2\nu_2}(x_2)....F_{\mu_n\nu_n}(x_n)\Omega^{\nu \rho \mu_2\nu_2...\mu_n\nu_n}(x,x_2,...,x_n)\;.
\mathcal{H}^{2}_{app}(\mathcal{A},X^{**}) =\mathcal{H}^{1}_{app}(\mathcal{A},\mathcal{BL}(\mathcal{A},X^{**})) &=\mathcal{H}^{1}_{app}(\mathcal{A},(\mathcal{A}\hat{\otimes}X^*)^*) &=\{0\}.
g_{M N}= \left( \begin{matrix}{e^{- \gamma}e^{- 2 \sigma}g_{\mu \nu}}&{0}&{0}\\{0}&{e^{2 \gamma}e^{- 2 \sigma}}&{0}\\{0}&{0}&{e^{\sigma}\delta_{i \overline{j}}}\\ \end{matrix}\right) .
\mathcal{TS}^{op}=\{\pi^*(B_1),\ldots,\pi^*(B_{k-2}),\pi^*(B_{k-1})-E,E,\pi^*(B_{k})-E,\pi^*(B_{k+1}),\ldots,\pi^*(B_n)\}=\{\pi^*(A_{n-1}),\ldots,\pi^*(A_{n-k+2}),\pi^*(A_{n-k+1})-E,E,\pi^*(A_{n-k})-E,\pi^*(A_{n-k-1}),\ldots,\pi^*(A_1),\pi^*(A_n)\}
\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p-2}\,(u(x)-u(y))}{|x-y|^{N+s\,p}}\,\Big(\varphi(x)-\varphi(y)\Big)\,dx\,dy=\int_\Omega f\, \varphi,\,dx\mbox{for every}\varphi,
\hat{\cal L}_0 = - \int d^{\, 3}u \, d^{\, 3}v \sum_{\tau = \pm}\hat{\phi}^{(\tau) \dagger}({\bf x}; x_0) \hat{\tau}_3 D_\tau (x;{\bf u},{\bf v}) \hat{\phi}^{(\tau)}({\bf v}; x_0) ,
K_N(r)=\begin{cases}\frac{p}{4(p-1)}r^{p}0\leq r\leq N &\frac{N^{p-2}}{2}(r-N)^2+\frac{p^2}{4(p-1)}N^{p-1}(r-N)+\frac{p}{4(p-1)}N^{p}r> N.\end{cases}
Rp^{}_{1,1*}\,p^{\,!}_{1,2}((Rp^{}_{E',1*}\,p^{\,!}_{E',2}(F^{\wedge_{E'}}))^{\wedge^{}_1}) & =Rp^{}_{1,1*}\,p^{\,!}_{1,2}\,Rp^{}_{E',1*}((p^{\,!}_{E',2}(F^{\wedge_{E'}}) )^{\wedge^{}_1}) & =Rp^{}_{1,1*}\,p^{\,!}_{1,2}\,Rp^{}_{E',1*}\,p^{\,!}_{E',2}(F^{\wedge_{E'}\wedge^{}_1})= Rp^{}_{1*}p^{!}_2(F^{\wedge_E}).
Y(a_{i_{h+1}}, \dots , a_{i_{h+k-1}}, bc) = Y(a_{i_{h+1}}, \dots , a_{i_{h+k-1}}, b)c + (-1)^{p(b)p(c)}Y(a_{i_{h+1}}, \dots , a_{i_{h+k-1}}, c)b - Y(a_{i_{h+1}}, \dots , a_{i_{h+k-1}}, 1)bc.
\lambda_{n,1}^{(2)}=\frac{\partial\overline{H}_0}{\partial q_{n,0}}\ ,\ lambda_{n,j_n}^{(2)}=\frac{\partial\overline{H}_0}{\partial q_{n,j_n-1}}-\mu_{n,j_n-1}\ ,\ \j_n=2,3,\cdots,m_n-1\ .
y=\mp\prod\limits_{j=1}^{2n+2}(\lambda-\lambda_{j})^{\frac{1}{2}}\mathop{=}\limits_{\zeta\rightarrow 0}\mp\zeta^{-n-1}\left(1+\alpha_{0}\zeta+O(\zeta^{2})\right),~~as~~P\rightarrow P_{\infty\pm}.
I_{\alpha \rho}= - \int_{0}^{1}d h \int_{0}^{h}d t \frac{\partial c_{t}^{\mu}}{\partial x^{\rho}}\frac{\partial c_{t}^{\nu}}{\partial t}\frac{\partial c_{h}^{\delta}}{\partial x^{\alpha}}\frac{\partial c_{h}^{\beta}}{\partial h}[ G_{\beta \delta}( c_{h}) , G_{\mu \nu}( c_{t}) ]
{\cal E}[ A_{0}, A_{1}= 0 ] = \int_{0}^{2 \pi R}d y \left[ \bigg | \frac{d A_{0}}{d y}\bigg |^{2}+ \frac{\Lambda^{4}}{g^{2}}\right] \geq \frac{2 \pi R \Lambda^{4}}{g^{2}}.
B_{1}= \left[ \begin{matrix}{{}}&{{}}&{{\vdots}}&{{}}\\{{0}}&{{}}&{{\vdots}}&{{0}}\\{{\cdots}}&{{\cdots}}&{{\cdots}}&{{}}\\{{0}}&{{}}&{{\vdots}}&{{v t}}\\ \end{matrix}\right] , \ \B_{2}= \left[ \begin{matrix}{{}}&{{}}&{{\vdots}}&{{}}\\{{0}}&{{}}&{{\vdots}}&{{0}}\\{{\cdots}}&{{\cdots}}&{{\cdots}}&{{}}\\{{0}}&{{}}&{{\vdots}}&{{b}}\\ \end{matrix}\right] , \ \B_{3}= 0 .
\Big|\sum_{\widetilde\alpha}(-1)^{\widetilde{q}(\widetilde\alpha)}Z_{\widetilde\alpha,\widetilde{D}_0}(\widetilde{G})\Big|=\Big(\sum_{w_1(\alpha)=0}(-1)^{\frac{q(\alpha)}{2}}Z_{\alpha,D_0}(G)\Big)^2+\Big(\sum_{w_1(\alpha)=1}(-1)^{\frac{q(\alpha)-1}{2}}Z_{\alpha,D_0}(G)\Big)^2\,.
\|\mu_k(t)\|_2^2\leq \sum_{j=\ell_{k-1}+1}^{\ell_k}\delta_n \psi_{j_n,k}(t_j)^2\leq \|\psi\|_{\infty}^2\delta_n2^{j_n}(\ell_k-\ell_{k-1})\leq 2 n\delta^2/(md)
x^{m + 1 , n + 1 , M N}= \frac{q^{m + n + 1}}{\sqrt{( 1 - q^{2 ( m + 1 )}) ( 1 - q^{2 ( n + 1 )})}}\sum_{K , L}c_{\xi K}^{* M}c_{\eta L}^{N}x^{m n K L}
h(\bar{x})=\begin{cases}\displaystyle \delta+\frac{x^2}{\alpha}, \alpha:=\frac{2\alpha_1\alpha_2}{\alpha_1+\alpha_2}, d=2 begin{align*}8pt]\displaystyle \delta+\frac{x^2}{a}+\frac{y^2}{b}, a:=\frac{2a_1a_2}{a_1+a_2},~ b:=\frac{2b_1b_2}{b_1+b_2}, d=3\end{cases}
(u_{0,n},u_{1,n})=\sum_{j=1}^J\left(\frac{1}{\lambda_{jn}^2}U_L^j(\frac{x}{\lambda_{jn}},\frac{-t_{jn}}{\lambda_{jn}}),\frac{1}{\lambda_{jn}^3}\partial_tU_L^j(\frac{x}{\lambda_{jn}},\frac{-t_{jn}}{\lambda_{jn}})\right)+(w^J_{0,n},w^J_{1,n}),\,\,\,\forall J\ge 1,
X := E^\infty \cup \{\mu \in{\rm Path}(E) \mid r_E(\mu)\}\cup \{\mu \in{\rm Path}(E) \ \vert \r_E(\mu) \in \operatorname{Inf}(E)\}G_E := \{(\alpha x, |\alpha| - |\beta|, \beta x) \ \vert \ \alpha,\beta \in{\rm Path}(E), x \in X, r_E(\alpha) = r_E(\beta) = s_E(x)\}.
R_{1212}=-\frac{7}{3}\left(\frac{B^1_{11,3}}{\mu}\right)^2,~~~R^{\bot}_{1212}=\frac{2}{3}\left(\frac{B^1_{11,3}}{\mu}\right)^2-\left(\frac{B^3_{22,1}}{\mu}\right)^2+\left(\frac{B^3_{11,2}}{\mu}\right)^2+\left(\frac{B^4_{11,2}}{\mu}\right)^2.
\begin{aligned}(i) \left| w^\varepsilon (x,t) \right| \leq K, \medskip (ii) \left| \partial_x w^\varepsilon (x,t) \right| \leq K, \medskip (iii) \left| w^\varepsilon (x,t) - w^\varepsilon (x,s) \right| \leq K \left(\left| t-s \right|^{1/2}+ |t-s| \right). \end{aligned}
\frac{1}{2}\left(C \otimes I_2\right) = \frac{1}{2}\left[ \begin{array}{cccc}c_{X,X}0 c_{X, Y}0 0 c_{X,X}0 c_{X, Y}c_{Y, X}0 c_{Y, Y}0 0 c_{Y, X}0 c_{Y, Y}\end{array}\right],
y_{i+1}= \left( 2 y_i - y_{i-1}\left( 1 - 2\zeta\omega\frac{\Delta t}{2}\right) + f\left(y_i,x_i\right) \Delta t^2 \right) \left( 1 + 2\zeta\omega\frac{\Delta t}{2}\right)^{-1}, y_1 = y_0 + \Delta t v_0 \left( 1 - 2\zeta\omega\frac{\Delta t}{2}\right) + f\left(y_0,x_0\right) \frac{\Delta t^2}{2},
v(n) = \frac{(-1)^{g-1}}{(2 \pi i)^{g-1}}\int_{\Gamma_g}\cdots \int_{\Gamma_2}\psi_1(n_1; \lambda_1) \psi_2(n_2; \lambda_2) \cdots \psi_g(n_g; \lambda_g) \,\frac{d\lambda_2 \cdots d\lambda_g}{\lambda_2 \cdots \lambda_g},
E(\eta_m) + \sum_{k = 1}^m \frac{\tau}{2}R\left(\eta_{k - 1}, \tfrac{\eta_k - \eta_{k - 1}}{\tau}\right) \le E(\eta_0) + \frac{1}{2c}\sum_{k = 1}^m \tau \|f_k\|_{L^2}^2.
\sigma_{m,rm;n,qn}:= \left( 1+\frac{1}{r}\right)\left( 1+\frac{1}{q}\right)\sigma_{m(r+1)-1,n(q+1)-1}-\left( 1+\frac{1}{r}\right)\frac{1}{q}\sigma_{m(r+1)-1,n-1}& -\frac{1}{r}\left( 1+\frac{1}{q}\right)\sigma_{m-1,n(q+1)-1}+\frac{1}{rq}\sigma_{m-1,n-1},
\delta_t \boldsymbol{u}^n :=\boldsymbol{u}^n-\boldsymbol{u}^{n-1}\widetilde{D}_t \boldsymbol{u}^n :=\frac{\delta_t \boldsymbol{u}^n}{\Delta t}=\frac{\boldsymbol{u}^n-\boldsymbol{u}^{n-1}}{\Delta t}
F=\frac{1}{(m|g|)^2}\sum_{\substack{\ell=1 (\ell\neq k)}}^n\sum_{i,h,h'=0}^{m-1}[\tilde{v}_i\tilde{w}_i]_{\ell,j}[\tilde{v}_{i+h+1}\tilde{w}_{i+h+1}]_{k,\ell}[\tilde{v}_{i+h'+1}\tilde{w}_{i+h'+1}]_{j,k}\frac{\lambda_\ell^{m-h-1}\lambda_k^{m-h'-1+h}\lambda_j^{h'}}{(\lambda_\ell^m-\lambda_k^m)(\lambda_k^m-\lambda_j^m)}.
\left[ \begin{array}{c}{{{}}}\\{{{\sum \sum}}}\\{{{}_{j < l}}}\\ \end{array}\alpha_{j}\alpha_{l}k_{j l}^{2}- \sum \alpha_{i}m_{i}^{2}\right] \Rightarrow - \left[ \sum \alpha_{i}^{2}m_{i}^{2}\! + \! 2 \! \begin{array}{c}{{{}}}\\{{{\sum \sum}}}\\{{{}_{j < l}}}\\ \end{array}\alpha_{j}\alpha_{l}m_{j}m_{l}c_{j l}\right] ,
L^{2}=-\left(\frac{d{\sf t}}{d\lambda}\right)^2+e^{2{\sf t}/l}\left(\frac{d{\sf r}}{d\lambda}\right)^2+e^{2{\sf t}/l}{\sf r}^2 \sum_{b=1}^{d-2}\left( \prod_{a=1}^{b-1}\sin^2 \theta_{a}\right)\left( \frac{d\theta_b}{d\lambda}\right)^2 ,
-\frac{\partial}{\partial t}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\left\vert \nabla_{x}w\right\vert^{2}\,d\gamma(x)-z^{2-1/s}\int_{\left\{x:\,\left\vert w(x,z)\right\vert >t\right\}}\frac{\partial^{2}w}{\partial z^{2}}\,d\gamma(x)=0.
{\bf w}_t^{(i)}-\frac{t}{\Delta}{\bf w}_{\Delta}^{(i)}=\frac{1}{2}a_{i,0}+\sum_{r=1}^{\infty}\left(a_{i,r}{\rm cos}\frac{2\pi rt}{\Delta}+b_{i,r}{\rm sin}\frac{2\pi rt}{\Delta}\right),
\int_{\Omega}\int_{\Omega_{1}\backslash \Omega}\sum_{{i,j}_{j \neq i}}\overline{u}_{i}(x) \underline{u}_{j}(y) K(x,y) \, dy dx = \int_{\Omega}\int_{\Omega_{1}\backslash \Omega}\sum_{{i,j}_{j \neq i}}\overline{u}_{i}(x)\phi_{j}(y) K(x,y) \, dy \, dx
D^{j_1}_{o_ao_b}(U_1)D^{j_2}_{-p_c,-p_b}(U_2^\dagger)D^{j_3}_{-q_c,-q_d}(U_3^\dagger)D^{j_4}_{r_ar_d}(U_4)D^{j_5}_{-s_f,-s_e}(U_5^\dagger)D^{j_6}_{t_ft_g}(U_6)
d s^{2}= \left[ \left( 1 - \frac{q^{2}}{r_{1}^{2}}\right) \left( 1 - \frac{p^{2}}{r_{2}^{2}}\right) \right]^{- \frac{1}{2}}d t^{2}- \left[ \left( 1 - \frac{q^{2}}{r_{1}^{2}}\right) \left( 1 - \frac{p^{2}}{r_{2}^{2}}\right) \right]^{\frac{1}{2}}d \vec{x}^{2}.
\Delta_n \leqslant\frac{A}{n \bar{B}_n^{3/2}}\sum\limits_{i,j=1}^n\left(|\mu_{ij}+\bar{a}_{ij}^\ast|^{3}+ E|\bar{X}_{ij}-\bar{a}_{ij}|^3 \right)+\frac{1}{n}\sum\limits_{i,j=1}^n P(|X_{ij}-\mu_{ij}| \geqslant b_n)+ \Theta_n + \Upsilon_n
\delta g_{\mu \nu}= h_{\mu \nu}( z = 0 ) = h_{\mu \nu}( z = \ell ) ={\frac{1 6 \pi k \ell_{P}^{2}\ell^{2}}{8 \delta^{4}}}\left( \eta_{\mu \nu}- 4 n_{\mu}n_{\nu}\right) .
\pi=lim_{n\rightarrow\infty}\frac{2^{4n}n!^{4}}{n(2n)!^{2}}=lim_{n\rightarrow\infty}\frac{2^{4n}}{n{(\begin{matrix}2n\\n\end{matrix})}^{2}}=lim_{n\rightarrow\infty}\frac{1}{n}(\frac{(2n)!!}{(2n-1)!!})^{2}
\langle A_{\{\alpha \}}( x ) A_{\{\beta \}}( 0 ) \rangle ={\frac{\Gamma \left({\frac{n}{2}}\right)}{2 ( n - 2 ) \pi^{\frac{n}{2}}}}{\frac{{\cal A}_{\{\alpha \}, \{\beta \}}}{| x |^{n - 2}}}.
\sigma_i : V_i \,\,\to\,\, \bigoplus_{h'=i}V_{h''},{}\!\!\!\!\!\!\tau_i: \bigoplus_{h''=i}V_{h'}\,\,\to\,\, V_i , v_i \,\,\mapsto\,\, (f_h(v_i))_{h'=i},{}(v_{h'}) \,\,\mapsto\,\, \sum_{h''=i}f_h(v_{h'}).
d s^{2}= \left( \frac{\rho^{2}}{X e^{4 A}}\right)^{- 1 / 2}d x_{/ /}^{2}+ \left( \frac{\rho^{2}}{X e^{4 A}}\right)^{1 / 2}\sum_{i = 1}^{6}( d u_{i})^{2}
\begin{aligned}E^T({\rm Id}_q \otimes P^M)f(X^M)({\rm Id}_q \otimes P^M)E =E^T\left(\sum_{i=1,j=1}^{m}A_{ij}\otimes P^M{X^M_i}{X^M_j}P^M\right) E &=E^T\left(\sum_{i=1,j=1}^{m}A_{ij}\otimes M^s_{ij}\right) E<0.\end{aligned}
2[q_{\infty}\wedge\ast\mathcal{N}]\Big(\frac{\partial}{\partial z},\frac{\partial}{\partial \bar{z}}\Big)\xi =2i\big((q_{\infty})_{z}(\mathcal{N}_{\bar{z}}\xi)+(q_{\infty})_{\bar{z}}(\mathcal{N}_{z}\xi)-\mathcal{N}_{z}((q_{\infty})_{\bar{z}}\xi)-\mathcal{N}_{\bar{z}}((q_{\infty})_{z}\xi)\big).
\Phi([a\cos(2\pi \ell\phi) + b \sin(2\pi \ell \phi)] u_\ell + [-a\sin(2\pi \ell \phi) + b \cos(2\pi \ell \phi)]w_\ell,p) = [c\cos(2\pi \ell\phi) + d \sin(2\pi \ell \phi)] u_\ell + [-c\sin(2\pi \ell \phi) + d \cos(2\pi \ell \phi)]w_\ell.
E_{P}\left[ \left. \frac{1}{\lambda_{\overline{\mathbf{a}}_{k}}\left( \overline{\mathbf{G}}_{k},\overline{\mathbf{B}}_{k}\mathbf{;}P\right)}\right\vert \overline{\mathbf{G}}_{k}\mathbf{,}\overline{\mathbf{A}}_{k}\mathbf{=}\overline{\mathbf{a}}_{k}\right] =\frac{1}{\lambda_{\overline{\mathbf{a}}_{k}}\left( \overline{\mathbf{G}}_{k}\mathbf{;}P\right)}.
{\cal G}_{\omega}( r , r^{\prime}, \theta ) = \sum_{n = 0}^{\infty}{\frac{2 \nu \Gamma \left({\frac{D}{2}}- 1 \right)}{8 ( \pi r r^{\prime})^{D / 2 - 1}\sin \pi \nu}}C_{n}^{( D / 2 - 1 )}( \cos \theta ) \left[ J_{\nu}( k r_{<}) J_{- \nu}( k r_{>}) - \beta J_{\nu}( k r ) J_{\nu}( k r^{\prime}) \right] ,
\delta_{\varepsilon_{2}}\left( x\right) \star \delta_{\varepsilon_{1}}\left( p\right) =\frac{1}{\pi \sqrt{\theta^{2}+\varepsilon_{2}\varepsilon_{1}}}\exp \left( -\left[ \frac{x^{2}\varepsilon_{1}-2ixp\theta +p^{2}\varepsilon_{2}}{\theta^{2}+\varepsilon_{2}\varepsilon_{1}}\right] \right) ,
|a-\alpha|+|x-\chi| < 4 \eta |\chi-\alpha|, |x-a| \leq (1+4\eta)|\chi-\alpha|, \begin{aligned}|f(\chi)-f(\alpha)| \leq ( \varepsilon/2 + \kappa ) ( |a-\alpha|+|x-\chi| ) + \kappa |x-a| \leq ( 8 \eta ( \varepsilon + \kappa ) + \kappa ) |\chi-\alpha| = ( \varepsilon/2 + \kappa) |\chi-\alpha|\end{aligned}
\tilde{U}_{N_{s}N_{r}}( \{\sigma_{p}\}) = \int_{C_{v_{r}}}\Psi_{\sigma , N_{s}}( t ) \frac{d \theta d z}{2 \pi i}P_{N_{r}}( t ) + \int_{\tilde{C}_{r}}\Psi_{\sigma , N_{s}}( t ) \frac{d \theta d z}{2 \pi i}P_{N_{r}}^{( a )}( t )
\left|I(w,m)\right| \; =\; \binom{3m+\ell}{m}- \binom{3m+\ell}{<m}+ \sum\limits_{e=1}^m 4e\left[\binom{3m+\ell}{m-e}- \binom{3m+\ell}{<m-e}\right] \; =\; \binom{3m+\ell}{m}-\sum\limits_{e=1}^m \left(2e^2-6e+1\right) \binom{3m+\ell}{m-e}.
\langle \nabla_V U, A\rangle{}=\frac{c+4(\beta-\gamma)\langle \nabla_U V, A \rangle}{4(\alpha-\gamma)}, \langle \nabla_A U, V\rangle{}=-\frac{(a^2-2b^2)c-4(\beta-\gamma)\langle \nabla_U V, A\rangle}{4(\alpha-\beta)}.
\Big(\int_{\{\tau<\infty\}}\prod\limits_{i=1}^{\infty}E_{\tau}(f_i)^pvd\mu\Big)^{\frac{1}{p}}\leq C\prod\limits_{i=1}^{\infty}\|f_i\|_{L^{p_i}(\omega_i)}, ~\forall \tau\in\mathcal{T},~ f_i\in L^{p_i}(\omega_i),~i\in N,
\langle \Phi_{1}\rangle = \left( \frac{1}{\sqrt{2}}, 0, 0 \right)^T,\quad \langle \Phi_{2}\rangle = \left( 0, 0, 0 \right)^T,\quad \langle \Phi_{3}\rangle = \left( 0, 0, 0 \right)^T.
\lambda = - \alpha \gamma_{1 , 3 , ( 0 , 0 , 0 )}\lambda \gamma_{1 , 7_{3}, ( 0 )}^{- 1}= - \alpha \gamma_{1 , 7_{3}, ( 0 )}\lambda \gamma_{1 , 3 , ( 0 , 0 , 0 )}^{- 1}
\lim_{s \rightarrow \infty}\frac{e^{f}}{b^{2n-1}}\int_{s_0}^{s}b^{2n-1}e^{-f}\left(Q^3-Q\right) \; \mathrm{d}s \geq n\sqrt{n+1}\lim_{s \rightarrow \infty}e^{f}\int_{s_0}^s e^{-f}\; \mathrm{d}s &= - \frac{n\sqrt{n+1}}{f'_{\infty}}> 0,
\begin{array}{c}u(\sigma^{+},\sigma^{-})=u_0+\frac{p^{+}}2\left[ (\sigma^{+}+f(\sigma^{+}))+(\sigma^{-}-g(\sigma^{-}))\right] v(\sigma^{+},\sigma^{-})=v_0+\frac{p^{-}}2\left[ (\sigma^{+}-f(\sigma^{+}))+(\sigma^{-}+g(\sigma^{-}))\right]\end{array}
A^\circ + B^\circ = \{x^* \in X^* \ | \ \mbox{there exist}a^*,\, b^*\in X^* \mbox{s.t.}x^*=a^*+b^* \mbox{and}&\langle a^*, a \rangle \le 1 \mbox{and}\langle b^*, b \rangle \le 1 \mbox{for all}a\in A, \, b\in B \}\, ,
\sum_{j=1}^m |x_j|^2 \leqslant\Big( \sum_{j=1}^m |x_j| \big\lVert v_j-w_j \big\rVert \Big)^2 \leqslant\Big( \sum_{j=1}^m |x_j|^2 \Big)\Big( \sum_{j=1}^m \big\lVert v_j-w_j \big\rVert^2 \Big).
\begin{aligned}\phi_{n}''(z) +\left(\frac{1}{z}+\frac{1}{z-t}-\frac{1}{z-t+t R_n(t)}\right)\phi_{n}'(z) &+\left(\frac{-\frac{1}{4}\alpha^2}{z^2}+\frac{\frac{1}{2}(2n+\alpha+1)-\kappa_1(t)-\kappa_2(t)}{z}+\frac{\kappa_1(t)}{z-t}+\frac{\kappa_2(t)}{z-t+t R_n(t)}-\frac{1}{4}\right)\phi_{n}(z)=0,\end{aligned}
S_{E}^{v o l}= \sum_{a}\frac{V_{a}}{4 G_{d}}+ \frac{\beta}{1 6 \pi G_{d}}\int_{\Sigma}F \wedge \bar{G}- - \frac{\beta}{8 \pi G_{d}}\int_{\partial \Sigma_{\infty}}( \partial^{i}\phi ) d \sigma_{i},
\int_G\int_G\mu^{-1/2}(g')\mu^{-1/2}(g)\langle\sqrt{\mathfrak{c}}Qg(e),\sqrt{\mathfrak{c}}Q(g'(e))\rangle_{H^0}\cdot(g(h),g'(h))_H\,dg\,dg' &=\int_G\int_G\mu^{-1/2}(g)\mu^{-1/2}(g')\langle Q\mathfrak{c}Qg(e),g'(e)\rangle_{H^0}\cdot(g(h),g'(h))_H\,dg'\,dg.
|\partial_{\xi_i}\hat{V}(\xi + \frac{{\rm i}t}{\zeta}\xi)| = |\widehat{x_i V}(\xi + \frac{{\rm i}t}{\zeta}\xi)|\lesssim e^{\frac{|\xi|}{\zeta}}\|V\|_{L^\infty(B_R)},0<t<1,
C_1 \> \left[J_{\lambda}\left(\frac{\sqrt{4\alpha c_{\epsilon}}}{\epsilon\> x^{\epsilon \over 2}}\right)+\frac{\sqrt{4\alpha c_{\epsilon}}}{x^{\epsilon \over 2}}J_{\lambda}^\prime\left(\frac{\sqrt{4\alpha c_{\epsilon}}}{\epsilon\> x^{\epsilon \over 2}}\right)\right]_{x=a}\> =\> 0\;\;\;.
H(\psi^{\prime})=\frac{p}{\rho_{0}}+\frac{\rho}{2\rho_{0}}(v_{r}^{\prime2}+v_{\theta}^{\prime2}+v_{z}^{\prime2})+\frac{\rho}{\rho_{0}}gz,\Gamma(\psi^{\prime})=rv_{\theta}^{\prime}
-\frac{2\tau^4}{(1+\tau^2)^2}f'(u)u_y \le \tau u_{xxx}={}- 3\tau u_{xy}\rho'' - \tau u_y \rho''' ={}\frac{3\rho''}{2}(-H + u_{xx}+ \tau^2 u_{yy}) - \tau u_y \rho'''.
\begin{cases}J\left(z,0\right)<0 \,z\in\,\Omega, J\left(z,0\right)>0 \,z\in\left(\mathbb{R}^{n}\setminus\Omega\right), J\left(z,0\right)=0 \,z\in\left(\Omega\setminus\,\Omega\right),\end{cases}
D_{\mu}\delta q^{i}\equiv{\cal D}_{\mu}\delta q^{i}-{( j_{u})_{\mu}}^{\nu}{\cal D}_{\nu}\delta q^{j}{( J_{u})_{j}}^{i}-{( j_{u})_{\mu}}^{\nu}\partial_{\nu}q^{j}{\cal D}_{k}{( J_{u})_{j}}^{i}\delta q^{k}= 0 .
\sigma(S)= \sum_{i=1}^{r+1}ix_i=\sum_{i=1}^{r}ix_i+(r+1)x_{r+1}=\sum_{i=1}^{r}ix_i+(r+1)(n-\sum_{i=1}^{r}x_i)=(r+1)n+\sum_{i=1}^{r}(i-r-1)x_i &\leq (r+1)n-2\sum_{i=1}^{r}i-(r-1)=(r+1)n-r(r+1)-r+1
0-f^{\prime}r^{m-1}e^{-\frac{r^2}{4}}=\int^{+\infty}_{r}r^{m-1}e^{-\frac{r^2}{4}}(-\lambda e^{-\frac{r^2}{2(m-2)}}+r^{-2}\lambda_k)f dr &\le \int^{+\infty}_{r}r^{m-3}e^{-\frac{r^2}{4}}\lambda_kf dr &\le \int^{+\infty}_{r}r^{m-3}e^{-\frac{r^2}{4}}\lambda_kdr f.
\begin{tabular}[c]{l}$D_{n}=\phi^{2}C_{n}\partial_{x}^{2}-2x\phi\left[ \left( \phi-1\right)C_{n}+\phi\right] \partial_{x}$ $+\left( n-2\gamma_{n}\right) \left[ 2x^{2}\phi-\left( \phi-2x^{2}\phi+nx^{2}-2x^{2}\gamma_{n}\right) C_{n}\right] +4\gamma_{n}C_{n-1}C_{n}^{2}.$\end{tabular}
\Vert \rho_{t_n}(x,y) \xi_n (x) - \xi_n (y) \Vert^2 = \Vert (\rho(x,y)(0))_{t_n}- 0_{t_n}\Vert^2 & = \Vert (\rho(x,y)(0))_{t_n}\Vert^2 + \Vert 0_{t_n}\Vert^2 - 2 \langle (\rho(x,y)(0))_{t_n}, 0_{t_n}\rangle & = 2 \left(1 - \exp(-t_n \Vert \rho(x,y)(0) \Vert^2) \right)