harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-a_c(f,g,0)=\begin{pmatrix}(R_3+S_3\mu)\frac{f^3}{3}+S_3\mu\frac{f^2g}{2}S_3\mu\left(\frac{f^3}{3}+\frac{f^2g}{2}\right) -\mu\frac{f^2}{2}\sigma^\prime(0) S_3 \frac{g^3}{3}+(R_3+S_3\mu)\frac{f^2g}{2}+S_3\mu fg^2 S_3 \frac{g^3}{3}+S_3\mu\left(\frac{f^2g}{2}+ fg^2\right) -\left(\mu fg+\frac{g^2}{2}\right) \sigma^\prime(0) 0 0 D \end{pmatrix}.
\eta(1;1) =\zeta(2), \eta(2;1) =\zeta(1,2)+\zeta(3), \eta(3;1) =\zeta(1,1,2)+\zeta(2,2)+\zeta(1,3)+\zeta(4), \eta(2;2) =2\zeta(1,1,2)+\zeta(2,2)+2\zeta(1,3)+\zeta(4).
D^{(n)}_q((d+1)q/2\pm u;t) D^{(n)}_{q,t}(d)\!\!\! \prod_{1\le i\le n}\!\!\! \vartheta(z_i\pm v) = D^{(n)}_q((d+1)q/2\pm v;t) D^{(n)}_{q,t}(d)\!\!\! \prod_{1\le i\le n}\!\!\!\vartheta(z_i\pm u).
v(n) \ =\v(p^{\alpha_1}_1\cdots p^{\alpha_k}_k) \ =\v(p^{\alpha_1}_1) +\cdots+ v(p^{\alpha_k}_k) \ \le \p^{\alpha_1}_1 +\cdots + p^{\alpha_k}_k \ \le \p^{\alpha_1}_1\cdots p^{\alpha_k}_k\ =\n.
S_{( n )}= \int d^{4}x \Bigl [ \frac{1}{6}\, H_{( n )}^{\mu \nu \rho}H_{( n ) \mu \nu \rho}^{\ast}- \frac{2 n^{2}\pi^{2}}{R^{2}}\Bigl ( i B_{( n )}^{\mu \nu}- \frac{R}{2 n \pi}\, F_{( n )}^{\mu \nu}\Bigr ) \Bigl ( - \, i B_{( n ) \mu \nu}^{\ast}- \frac{R}{2 n \pi}\, F_{( n ) \mu \nu}^{\ast}\Bigr ) \Bigr ]
{\frac{e^{2}}{2}}f^{a d c}f^{b d e}\int_{x}\int_{y}\sum_{k}{\frac{\delta^{c e}P_{i j}( k )}{2 \omega_{k}}}e^{i k ( x - y )}G ( x , y ) E_{1 i}^{a}( x ) E_{1 j}^{b}( y )
T_{1,C_1}(V=0) = \frac{q^{2g+1}}{9!\pi \zeta_q(2)}c_8 (2g)^8 \frac{\pi \alpha}{2}+ \frac{q^{2g+1}}{9!\pi \zeta_q(2)}c_9 \Bigg[ \sum_{j=0}^{\alpha-1}\sum_{m=0}^{2g}\frac{\sin (2 \pi \theta_0(2m-j))}{2m-j}(2g-m)^9 + \sum_{j=0}^{\alpha-1}\sum_{m=1}^{2g}\frac{\sin (2 \pi \theta_0(2m+j))}{2m+j}(2g-m)^9 \Bigg] +O(q^{2g+1}g^7 \theta_0^{-1}\alpha).
\overline{\phi}_{h}^{k}= \sum_{j=1}^{N}{a_{j}u_j},\overline{\Psi}_{h}^{k}= \sum_{j=1}^{N}{b_{j}u_j},{\Psi}_{h}^{k-1}= \sum_{j=1}^{N}{c_{j}u_j},
\nu'(0) = -\left(D_vF(\nu(0),0)\right)^{-1}\cdot Dq(x^*)\cdot d_x & = -\left(D_vF(\nu(0),0)\right)^{-1}\cdot \underbrace{{\begin{pmatrix*}[l]\nabla g_{\mathcal{A}(x^*)}(x^*)^T \nabla h(x^*)^T e_{\mathcal{B}(x^*,y^*)}^T\end{pmatrix*}}\cdot d_x}_{=0}\ = 0,
\sum_{B\subset S_8}E^{(S_4,B)}_{n,\ell}= \frac{\ell_1 \left( n-\ell_4 \right) \left( \sum_{j=0}^6 n^j p_j\left( \ell_1,\ell_2,\ell_3,\ell_4 \right) \right)}{n^8 (n-1)(n-2)(n-3)(n-4)(n-5)(n-6)},
\eta_n= \begin{cases}\dfrac{(1)_n(-j)_n(a+c)_n(a-c-j+1)_n}{(-n)_n(n-N)_n}, begin{align*}1.1em] \dfrac{\alpha(1)_n(-j)_j(1)_{n-j-1}(a+c)_n(a-c-j+1)_n}{(-n)_n(n-N)_{N-n}(1)_{2n-1-N}}. \end{cases}
\zeta\left(t(N), \chi_1, s\right) \coloneqq \sum_{\substack{n \ge 1 n = t \bmod N}}\chi_1(n) n^{-s}, \sigma_{k-1}\left(t(N), \chi_1, \chi_2, n\right) \coloneqq\sum_{\substack{uv = n u = t \bmod N}}\chi_1(u) \chi_2(v) v^{k-1}.
{\mathcal J}^{\mu}\; = \; \sum_{i = 1 \atop s = \uparrow , \downarrow}^{N}{\mathcal J}^{( i , s ) \mu}~ , ~ ~ \widehat{{\mathcal J}}_{\ell / r}^{\mu}\; = \; \sum_{i = 1 \atop s = \uparrow , \downarrow}^{N}\widehat{{\mathcal J}}_{\ell , r}^{( i , s ) \mu}~ .
\bar{q}(n,t)=\sum_{\Omega^{n}_{k}}\prod_{j=1}^{k}\rho^{x_{j}}\binom{r+x_{j}-1}{x_{j}}\left(\frac{\lambda t(1-\rho)^{r}}{1-(1-\rho)^{r}}\right)^{k}\frac{e^{-\lambda t}}{k!},
\delta_{B}( f^{( 0 )}u^{4}) + \delta_{F}( f^{( 2 )}u^{3}\left[ \psi^{2}\right] ) + u^{i}u^{2}\epsilon \dot{N}^{i ( 0 )}\psi + \frac{i}{2}u^{3}\psi \dot{M}^{( 0 )}\epsilon = 0 ,
{\cal L}_{M C S}= - \frac{1}{2}F_{\mu \nu}^{\dagger}F^{\mu \nu}+ \frac{M}{2}\epsilon^{\mu \nu \rho}[ \phi_{\mu}^{\dagger}\partial_{\nu}\phi_{\rho}+ \phi_{\mu}( \partial_{\nu}\phi_{\rho})^{\dagger}] .
\sum_{(d_1,d_2) > 0}\mathsf{N}_{1, d_1 F_1 + d_2 F_2}q_1^{d_1}q_2^{d_2}=12 \sum_{m,n \geq 1}\frac{q_1^{mn}}{n}+ 12 \sum_{m,n \geq 1}\frac{q_2^{mn}}{n}.
\displaystyle \int^{\infty}_{-\infty}\frac{dx}{2\pi}\varphi^{,}_{\pm}(x){\cal Q}_{\pm}(x)=\pm \left( \frac{\pi}{12}-\frac{{\cal Q}^{2}_{\pm}(\mp \infty )}{8\pi (1-\gamma /\pi )}\right) \: .
Q_R(p\otimes b) =Q_R\bigl((\operatorname{id}\otimes\operatorname{id}\otimes\varphi)(\Delta_{13}(a^*)(1\otimes b\otimes x))\bigr) &:=(\operatorname{id}\otimes\operatorname{id}\otimes\varphi)\bigl(\Delta_{13}(a^*)(1\otimes E)(1\otimes b\otimes x)\bigr).
\begin{aligned}\| (\gamma-c) \varphi \|_{W^{\frac{1}{3},\frac{3}{2}}(\partial\Omega)}+ \| c\nabla\varphi \cdot \nu \|_{W^{\frac{1}{3},\frac{3}{2}}(\partial\Omega)}&\le C(\gamma\|\varphi\|_{W^{2,\frac{3}{2}}(\Omega)}+\|\nabla c\|_{L^{2}(\Omega)}\|\varphi\|_{W^{1,6}(\Omega)}).\end{aligned}
{\hat{T}}_{\ \nu}^{\mu}[ C ] ={\frac{1}{2 ( p + 1 ) !}}e^{2 \Phi}G^{\mu \lambda \ldots \sigma}G_{\nu \lambda \ldots \sigma}-{\frac{( p + 1 )}{1 6}}e^{2 \Phi}{\frac{G^{2}}{( p + 2 ) !}}g_{n u}^{\mu}.
\Omega ( p ) = \frac{p_{3}+ i p_{1}}{p^{0}+ p_{2}}, \qquad \qquad \tilde{\Omega}( p ) = \frac{p_{3}- i p_{1}}{p^{0}- p_{2}}, \qquad \qquad C_{n}= \frac{\sqrt{| [ 2 n ]_{\nu}! |}}{2^{n}n !},
c_+^{(h,g)}= \mathrm{pexp}\left( \dfrac{(Q^8-Q^{-8})}{(1 + s^{-\varphi(g)_{2,2}}p^{\varphi(g)_{2,1}})(1 + s^{-\varphi(hg)_{1,2}}p^{\varphi(hg)_{1,1}})(1 + s^{\varphi(hg)_{2,2}}p^{-\varphi(hg)_{2,1}})}\right)\qquad\in\quad\mathbf R,
{\bf \Omega}( x_{1}, \ldots , x_{4 i - 4}, x_{4 i - 3}^{+}, x_{4 i - 2}^{+}, x_{4 i - 1}^{+}, x_{4 i}^{+}, x_{4 i + 1}, \ldots , x_{N}) ={\bf \Omega}( x_{1}, \ldots , x_{N}) , \nonumber
P(x, e^{- 2 \pi ( \alpha -i v) + 2 \pi i h/k)})=e^{2 \pi i n h/k}\omega_{h,k,n}(x)\exp\left(\frac{L_k(x)^3}{8 \pi^2 ( \alpha- iv)^2}+ g_{h,k}(x, 2 \pi ( \alpha - iv)) .\right)
\frac{\partial_G\Delta}{|G_\Delta|}-\frac{\partial_G\Sigma}{n}\le \sum_{m\ge 0}\frac{6(c+1)\log ((4/3)^mt)+9}{(4/3)^mt}&=\frac{6(c+1)\log(4/3)}{t}\sum_{m\ge 0}m(3/4)^m+\frac{6(c+1)\log t+9}{t}\sum_{m\ge 0}(3/4)^m.
\tilde{R}_{\xi}^{+}= \left[ \frac{( L - 2 ) ( L + 1 ) ( L + 4 ) ( L + 2 )}{2 7 \sqrt{3}L^{4}\sin \left( \frac{\pi ( 2 L + 1 )}{3 ( L + 2 )}\right) \sin \left( \frac{\pi ( L - 2 )}{3 L}\right)}\right]_{L = 1}^{\frac{1}{2}}= \left[ \frac{4 0}{2 7 \sqrt{3}}\right]^{\frac{1}{2}}.
\max_m \left\Vert E^T_m \right\Vert_{\mathbb{L}^{2}(D_m)}=a^{\frac{3}{2}}\max_m \left\Vert \tilde{E}^T_m \right\Vert_{\mathbb{L}^{2}(B)}\lesssim a^{\frac{3}{2}}\cdot a^{1-h}=\mathcal{O}\left(a^{\frac{5}{2}-h}\right),
S_2(t) \ll \sum_{2\log t<k-\delta<\sqrt{\frac{t}{\log t}}}\left(\frac{\exp(i\pi \delta)}{\pi}\frac{(-1)^k}{k-\delta}+\frac{1}{(k-\delta)\log t}+\frac{1}{\sqrt{t\log t}}\right) &\quad+ \sum_{\sqrt{\frac{t}{\log t}}<k-\delta<2\sqrt{t\log t}}\left(\frac{1}{k-\delta}+\frac{1}{\sqrt{t\log t}}\right)=O(\log \log t).
- \frac{1}{2}Q^{i^{\prime}}\sum_{k > 0}\mathrm{e}^{- 2 i k \sigma_{+}^{\prime}}= - \frac{1}{2}Q^{i^{\prime}}\frac{\mathrm{e}^{- 2 i \sigma_{+}^{\prime}}}{1 - \mathrm{e}^{- 2 i \sigma_{+}^{\prime}}}
Q(G^* ) = \left({\begin{array}{*{20}c}{Q(F) + (m - 1)\left({\begin{array}{*{20}c}{I_k}{0^T}0 0 \end{array}}\right)}{\left({\frac{{I_k \otimes \textbf{1}_{m - 1}^T}}{0}}\right)}{(I_k \otimes \textbf{1}_{m - 1}|0)}{I_k \otimes (Q(G_p^{r_1}) + I_{m - 1})}\end{array}}\right)
\tilde{T}:= T\circ h= \sum_{n\geq 0}\tilde{T}^{(n)}z^{n},\ \tilde{T}^{(n)}= \tau_{1}^{(n)}C_{1}+ \tau_{2}^{(n)}C_{2}+ \tau_{3}^{(n)}D + \tau_{4}^{(n)}E
9 9 9 \times 9 9 9 9 + 1 9 9 9 9 = 9 9 9 9^{2}+ 2 \times 9 9 9 9 + 1 = ( 9 9 9 9 + 1 )^{2}= 1 0 0 0 0^{2}= 1 0^{\circ}
\left( p \star f_{0}( N - 1 ) + i N \tanh ({\frac{x}{\hbar}}) \star f_{0}( N - 1 ) \right) \star Q ( N ) = \left( p \star f_{0}( N - 1 ) +{\frac{N}{N - 1}}p \star f_{0}( N - 1 ) \right) \star Q ( N )
\mu(\frac{(1-\lambda)A+\lambda B}{(1-\lambda)\mu(A)^{1/n}+\lambda\mu(B)^{1/n}})=\mu((1-\lambda^{\prime})A^{\prime}+\lambda^{\prime}B)\ge\mu(A^{\prime})^{1-\lambda^{\prime}}\mu(B^{\prime})^{\lambda^{\prime}}=1.
\left|{\bf p}\right|^{v^{\prime}v^{\prime \prime}}= \vert{\bf F}\vert \alpha_{0 v}\int \tilde{\psi}_{v^{\prime}}^{\ast}\tilde{\psi}_{v^{\prime \prime}}d x + \vert{\bf F}\vert \alpha_{v}^{1}\int x \tilde{\psi}_{v^{\prime}}^{\ast}\tilde{\psi}_{v^{\prime \prime}}d x .
\sum_{k=0}^{\lfloor{n}/{2}\rfloor}\binom{n}{2k}\frac{(5F^2_j)^k}{2k+1}\left(\frac{4^{k+1}-1}{k+1}L_{j(n-2k)}{B_{2k+2}}-\frac{L_j^{n-2k}}{2^n}\right) = 0\,.
z = \int \frac{d y}{G ( y )}= \frac{\zeta}{2 ( \alpha^{2}- 1 )}\left[ \frac{1}{\alpha}\ln \left| \frac{\alpha + \hat{y}}{\alpha - \hat{y}}\right| + \ln \left| \frac{1 - \hat{y}}{1 + \hat{y}}\right| \right] .
C_k^{\varphi, \Phi}\left(\frac{2}{3},\frac{1}{3}\right)=\frac{1}{3}(1+k), C_k^{\varphi, \Phi}\left(\frac{1}{3},\frac{2}{3}\right)=\frac{2}{3}\left(\frac{3}{2}+k\right)
T_k(r,s,t) := e^{\alpha_r(s,t)}\sum_{\ell=0}^{L(r,s,t)}\sum_{i=0}^{I(r,s)-\ell}\sum_{j=0}^{J(r,t)-\ell}\frac{\{s-\alpha_r(s,t)\}^i \{t-\alpha_r(s,t)\}^j \alpha_r^\ell(s,t)}{i! j! \ell!}& - \sum_{i=0}^{I(r,s)}\sum_{j=0}^{J(r,t)}\frac{s^it^j}{i! j!},
V_2(\phi,\sigma) =\beta_{1}\frac{(\phi^{\dag}\phi)^2\sigma}{M_{\rm pl}}+\beta_{2}\frac{(\phi^{\dag}\phi)\sigma^2\sigma^{\ast}}{M_{\rm pl}}+\beta_{3}\frac{(\phi^{\dag}\phi)\sigma^3}{M_{\rm pl}}+ \rm{h.c.}.
\tilde A_{h,\tilde h}= \tilde h^m e^{-m\hat h s_h}D_{s_h}^j \hat h^{m-j}P_{m-j}(\hat h x',e^{\hat h s_h}x',y,\hat h^{-1}D_{Y_h})\bigl(\delta(s_h)\delta(Y_h)\bigr)\cdot \hat h^{-n}.
\delta \upsilon^k = - \frac{\Lambda}{12}\epsilon^k +\rho^\sigma\omega^k_\sigma \ ,\qquad \delta \theta_{\mu}^k = D_{\mu}\epsilon^k + 2 \rho^{\nu}\Sigma_{\nu \mu}^k \ ,\qquad \delta \xi^{\mu}= - \rho^{\mu}\ ,
\min\{\rho_1,\rho_2\}=\sigma, begin{align*}2mm] \alpha =\begin{cases}\dfrac{\rho_2-\rho_1}{\rho_2+\rho_1}\ \max\{\rho_1,\rho_2\}<\infty, begin{align*}2ex]+1 \ \rho_2=\infty, -1 \ \rho_1=\infty,\end{cases}
+\sum\limits_{k=0}^{N-1}\frac{4\mu^2_n|\Delta_{k,n}|}{T|P_n|e^{4\mu_nT}\prod\limits_{j=1}^{N-1}e^{2q_{j,n}T}\left(1-|\bar{\xi}_n(T)|-|\bar{\Lambda}_n(T)|-|\bar{\beta}_n(T)|\right)}e^{q_{k,n}T},t\in[0,T].
\Bigl\{(f,\phi,\lambda,\delta) \Bigm| \check{b}\bigl(V(\lambda)\bigr) \leq\phi \leq \hat{b}\bigl(V(\lambda)\bigr),\f = A(\lambda),\ (\lambda_i, \delta_i) \in (\ref{eq:SOS2-log}), \; i = 1,2\Bigr\}.
L_{e f f}^{( 0 )}= \int d^{2}x [ ( \partial_{t}\mid \phi \mid )^{2}+ ( \partial_{t}\mid \chi \mid )^{2}+ \frac{\kappa}{q_{1}}B_{2}\partial_{t}\omega_{1}+ \frac{\kappa}{q_{2}}B_{1}\partial_{t}\omega_{2}] .
\dot{V}(x(t))= u(t)x_1(t)x_2(t)+x_2(t)g(t,x_1,x_2,u)-x_2^{q+1}(t) \leq (\bar{u}(t)+\alpha(t)+\beta(t))V(x(t))+\beta(t), \forall t\ge 0,
\mathcal{G}\ =\ \{x_{6}t_{u_2}-y_{6}t_{u_1},\,\,x_{5}t_{u_3}-y_{5}t_{u_1},\,\,x_{6}t_{u_4}-y_{6}t_{u_3},\,\,x_{5}t_{u_3}-y_{5}t_{u_1},\,\,x_{4}t_{u_5}-y_{4}t_{u_4},\,\, &x_{3}t_{u_6}-y_{3}t_{u_5},\,\,x_{2}t_{u_7}-y_{2}t_{u_1},\,\,x_{1}t_{u_8}-y_{1}t_{u_1},\,\,t_{u_1}t_{u_4}-t_{u_2}t_{u_3}\}.
H = \frac12\,g^{ij}p_ip_j\,, I = \left(\frac{\det(g)}{\det(\bar{g})}\right)^{\frac23}\,\bar{g}^{ij}p_ip_j\,, J = \left(\frac{\det(g)}{\det(\hat{g})}\right)^{\frac23}\,\hat{g}^{ij}p_ip_j\,,
\mathcal{J}(\zeta u,\zeta v)\leqslant \zeta^{\overline{p}}d_2\|(u, v)\|^{\overline{p}}- \frac{\zeta^{q^{-}}}{q^{+}}\int_{\mathbb{R}^{N}}|u|^{\alpha(x)}|v|^{\beta(x)}\,dx ,
\begin{array}{lll}f_L^M((L',R')\star (L,R))\circ u = f_L^M(L'\circ L,R\circ R')\circ f_L(s)\circ t = f_L(L'\circ L\circ s)\circ t = f_L^M(L',R')\circ f_L(L\circ s)\circ t = f_L^M(L',R')\circ f_L^M(L,R)\circ f_L(s)\circ t = f_L^M(L',R')\circ f_L^M(L,R)\circ u. \end{array}
\left[ \begin{array}{c c}{a_{0}}&{1}\\{1}&{0}\\ \end{array}\right] \left[ \begin{array}{c c}{a_{1}}&{1}\\{1}&{0}\\ \end{array}\right] \cdots \left[ \begin{array}{c c}{a_{i}}&{1}\\{1}&{0}\\ \end{array}\right] = \left[ \begin{array}{c c}{p_{i}}&{p_{i - 1}}\\{q_{i}}&{q_{i - 1}}\\ \end{array}\right] .
\int_{\{\Psi<-t\}}\left\vert\frac{\tilde f_1+\tilde f_2}{2}\right\vert^2+\int_{\{\Psi<-t\}}\left\vert\frac{\tilde f_1-\tilde f_2}{2}\right\vert^2 =\frac{\int_{\{\Psi<-t\}}|\tilde f_1|^2+\int_{\{\Psi<-t\}}|\tilde f_2|^2}{2}=G(t),
G_j=(m-j)(m-j-1) \Big((\ell-1)\frac y\Delta+2i\frac{\partial}{\partial\tau'}\Big)F_j +(m-j-1)(j+1) \Big((\ell-1)\frac{2v}\Delta-2i\frac{\partial}{\partial z}\Big)F_{j+1}+(j+2)(j+1) \Big((\ell-1)\frac{y'}\Delta+2i\frac{\partial}{\partial\tau}\Big)F_{j+2}
\begin{array}{lllll}\dot{z}= (D^{\mathrm{T}}\otimes\mathbb{I})h(\mathbf{0}) \mathbf{0}= f(\mathbf{0},\hat u) \dot{\hat{z}}= 0 \dot{\phi}= -\Sigma^{-1}\left(\mathbf{1}+ \Phi(z)\right).\end{array}
D: =\{\phi: [-\beta,0]\rightarrow \mathbb{X}: \phi \ \mbox{is piecewise continuous having jump discontinuity at finite number}\ \qquad\mbox{of fixed points}\ \{\beta_0,\beta_1,........\beta_{l+1}\}\subset[-\beta,0], \ \mbox{such that}\ -\beta=\beta_0\leq\beta_1<\beta_2......<\beta_l &\qquad<\beta_{l+1}=0 \},
\delta_{\epsilon}b = ( \bar{\epsilon}_{1}\Gamma_{m}d \theta_{1}- \bar{\epsilon}_{2}\Gamma_{m}d \theta_{2}) d X^{m}+ \bar{\epsilon}_{1}\Gamma_{m}d \theta_{1}\bar{\theta}_{1}\Gamma^{m}d \theta_{1}- \bar{\epsilon}_{2}\Gamma_{m}d \theta_{2}\bar{\theta}_{2}\Gamma^{m}d \theta_{2}.
f_{|H^R|}(x) = \frac{2^{M+1}(K^R_h+1)^M(K_g+1)^M(K_d+1)}{(\beta^R)^{2M}\Omega_h^{M}\Omega^M_g\Omega_d}\cdot e^{-MK^R_h-MK_g-K_d}\cdot \frac{x^{2M+1}}{(2M+1)!}.
dist(u,d) = pos(u) +pos(d)-n -1 ~~(-1) = wd(u)+shift(u)+wd(d)+shift(d)-n -1 ~~(-1) = wd(u)+wd(d)+shift(u)+n-(lenSlope(u)+L(u))-n -1 ~~(-1) = wd(d)+wd(u)-h_{val}-lenSlope(u) -1 ~~(-1) \leqslant h_{peak}-h_val-lenSlope(u) -1 ~~(-1) = -1 ~~(-1)~.
\left<\,f,\, U\,\left\{\,a_{i}\,\right\}\,|\, a_{\,2},\, \cdots,\, a_{\,n}\,\right> \,=\, \left<\,f,\, \sum\limits_{i \,=\, 1}^{\,\infty}\,a_{i}\,C\,f_{\,i}\,|\, a_{\,2},\, \cdots,\, a_{\,n}\,\right> &=\,\,\sum\limits_{i \,=\, 1}^{\,\infty}\,\overline{\,a_{i}}\,\left<\,C\,f,\, f_{\,i}\,|\, a_{\,2},\, \cdots,\, a_{\,n}\,\right>.
\Omega_{\beta_{l}}^{R^{k}}\equiv E_{[ E ]}\otimes \rho_{l \mathrm{~ m o d ~}x}~ \Omega_{\beta_{l}}^{S}\equiv \mathrm{g c d}( k , 2 m + 1 ) E_{[ S ]}\otimes \rho_{l \mathrm{~ m o d ~}x}
\tilde{Z}= \left( \begin{array}{c c c c}{{\tilde{z}_{+ +}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\tilde{z}_{--}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\tilde{z}_{Z Z}}}&{{\tilde{z}_{Z A}}}\\{{0}}&{{0}}&{{\tilde{z}_{A Z}}}&{{\tilde{z}_{A A}}}\\ \end{array}\right)
\frac{\partial^2 u}{\partial t\partial s}(t,s) = \langle (1,\frac{\partial}{\partial t}\mu_{\hat X^1_t}(\omega)), (1,\frac{\partial}{\partial s}\mu_{\hat Y^1_s}(\omega^\prime)) \rangle_{\mathbb{R}\oplus \mathcal{H}_{\mathcal{S}}}u(t,s).
\tilde{\phi}S_{\hspace{. 3 t r u e c m}\alpha \beta}^{\mu}= 8 \pi \Sigma_{\hspace{. 3 t r u e c m}\alpha \beta}^{\mu}+ \frac{1}{2}( \delta_{\alpha}^{\mu}\tilde{\phi}_{, \beta}- \delta_{\beta}^{\mu}\tilde{\phi}_{, \alpha}) ,
I_{\mathrm{b2u}}\left( k \right) = \frac{P_{\mathrm{bs}}a_{\mathrm{b2u},k}}{N_{\mathrm{bs}}}\left(\max \{\|\mathcal{H}^{\dagger}_{\mathrm{b2s},i}\| \}_{i=1}^{N_{\mathrm{bs}}}\right)^{-2}\,\,\, \times \| \mathbf{H}_{\mathrm{b2u},k}\mathcal{H}^{\dagger}_{\mathrm{b2s}}\|_\mathrm{F}^2
\dfrac{(\Tilde{\mathcal{N}}(f) \circ \rho_{\eta}(X))'}{\Tilde{\mathcal{N}}(f) \circ \rho_{\eta}(X)}= \eta \dfrac{(\Tilde{\mathcal{N}}(f))'}{\Tilde{\mathcal{N}}(f)}\circ \rho_{\eta}(X).
\int_0^1 \Psi(\gamma(t)) \, dt = \frac{1}{2\pi}\int_0^{2\pi}\Psi\left( \gamma\left(\frac{x}{2\pi}\right)\right) \, dx = \frac{1}{2\pi}\left(|I|-(2\pi-|I|) \right) = 2\cdot \mathbb{P}(a,b,c)-1.
\frac{d^2R_{n}}{dT^2}=-\frac{1}{\log(2)}\times &\frac{a_n\left(2b_n\left(b_n\left(1-T\right)+T\right)+a_n\left(2b_n\left(1-T\right)+T\right)\right)}{\left(a_n\left(-1+T\right)+b_n\left(-1+T\right)-T\right)^2\left(b_n+T-b_nT\right)^2}<0,
V ( \zeta ) = \zeta_{\mu \nu}\left( \partial X^{\mu}+ i k_{\rho}\psi^{\rho}\psi^{\mu}\right) \left( \bar{\partial}X^{\nu}+ i k_{\lambda}\bar{\psi}^{\lambda}\bar{\psi}^{\nu}\right) e^{i k X}( z , \bar{z}) .
\psi(y)= y^{-\frac{3}{4}}g\Big(\frac{m^2 y}{N}\Big)e\Big(\frac{2\sqrt{y}}{c}-\frac{T^2c}{4\pi^2\sqrt{y}}\Big) \widehat{k^*}\Big( \frac{MTc}{2\pi^2 \sqrt{y}}\Big).
1 \leq \frac{E_{}}{E_{}}begin{align*}0.1cm] = \frac{1}{Q_{\min}}\cdot \frac{\log e}{2 \, \varphi(\pi_Q)}\cdot\frac{\log \left(1 + \frac{Q_{\min}\, \delta^2}{2}\right)}{\frac{\log e\; \cdot \; Q_{\min}\, \delta^2}{2}}begin{align*}0.1cm] \leq \frac{1}{Q_{\min}}
\lim_{n\rightarrow\infty}q^{n(n-1)}\varphi_{2n}(x;q^{-1})=\mathcal{S}h_{q}(x) \quad\mbox{and}\lim_{n\rightarrow\infty}q^{n^{2}}\varphi_{2n+1}(x;q^{-1})=\mathcal{C}h_{q}(x)
f(x;k,\sigma_{1}^{2},...,\sigma_{k}^{2})=\sum_{i=1}^{k}\frac{e^{-\frac{x}{\sigma_{i}^{2}}}}{\sigma_{i}^{2}\prod_{j=1,j\ne i}^{k}(1-\frac{\sigma_{j}^{2}}{\sigma_{i}^{2}})}forx\ge0.
{\widehat T}{}_{l}[ \varphi ] \equiv \frac{2 l - 1}{2 \pi i}\int_{| z | < 1}\varphi ( z , \bar{z}) ( 1 - z \bar{z})^{2 l - 2}d z d \bar{z}= \varphi ( 0 , 0 ) + \frac{1}{2 l}\left. \bigtriangleup \varphi ( z , \bar{z}) \right|_{\scriptstyle z = \bar{z}= 0}+{\cal O}( l^{- 2})
T_2T_1^*(z_1^{m_1}z_2^{m_2}\eta) = \begin{cases}M_{z_2}^{\alpha_2}D[U] (\bar{\alpha_1}z_1^{m_1-1}z_2^{m_2}\eta) 0 \end{cases}&=\begin{cases}\bar{\alpha_1}\alpha_2 z_1^{m_1-1}z_2^{m_2+1}U^{m_1-1}\eta 0 .\end{cases}
{\cal E}_{f}\ = \ - \frac 1 4 C_{s}\sum_{l = - \infty}^{\infty}\int_{m}^{\infty}d k [ k^{2}- m^{2}]^{1 - s}\frac{\partial}{\partial k}[ \ln f_{l}( i k ) - \ln f_{l}^{a s}( i k ) ]
A^{0}_{1}(U)=\rho,A^{0}_{3}(U)=\left(\begin{array}{cc}\frac{\rho e_{\theta}}{\theta}&\frac{\tau}{\kappa\theta}\mathbb{I}_{3\times 3}\end{array}\right)\quad\mbox{and}A^{0}_{2}(U)=\rho\mathbb{I}_{3\times 3}.
(f \star g)_\delta(x) -f_\delta(x) \star g_\delta(x) =\int_{B_\delta(0)}\big(f(x-y)-f(x) \big)\star \big(g(x-y)-g(x) \big) \rho_\delta(y) \, dy & - \int_{B_\delta(0)}\big(f(x-y)-f(x) \big)\rho_\delta(y) \, dy\star \int_{B_\delta(0)}\big(g(x-y)-g(x) \big)\rho_\delta(y) \, dy ,
{\frac{X_{1}^{2}}{X_{3}^{2}}}={\frac{1}{s}}\; , \quad{\frac{X_{2}^{2}}{X_{3}^{2}}}={\frac{t}{s}}\; , \quad{\frac{2 X_{1}\cdot X_{3}}{X_{3}^{2}}}={\frac{1 + s - t}{s}}\; , \quad{\frac{2 X_{2}\cdot X_{3}}{X_{3}^{2}}}={\frac{1 - s - t}{s}}\; .
g(r) = \int_{0}^{\infty}\phi(r,\xi) \mathcal{F}(g)(\xi) \rho(\xi) d\xi = \int_{0}^{\infty}\frac{\sqrt{r}}{f(\sqrt{\xi})^{2}}\mathcal{F}_{H}(\frac{g(\cdot)}{\sqrt{\cdot}})(\sqrt{\xi}) \rho(\xi) \tilde{\phi}_{\sqrt{\xi}}(r) d\xi
K^{\alpha}_{R,1}(z,w) = \frac{1}{\pi}\left( 1 + z \overline{w}\,{_2F_1}\left( \begin{array}{c}1 , 1 2 \end{array}\bigg | z \overline{w}\right)\right)= \frac{1}{\pi}\left( 1 + \ln\left( \frac 1{1-z}\right)\right)
\mathbb{P}(a_j(x) = 1) = \frac{M^n - (M-2)^n}{M^n}\mathbb{P}(a_j(x) = M-3) = \frac{(M-2)^n - (M-2n)^n}{M^n}\mathbb{P}(a_j(x) = 1/M) = \frac{(M-2n)^n}{M^n}.
g : ( \Gamma_{1}^{1 , 9}, \Gamma_{2}^{1 , 9}, \Gamma_{1}^{1 , 1}, \Gamma_{2}^{1 , 1}, \Gamma^{2 , 2}) \mapsto ( \Gamma_{2}^{1 , 9}, \Gamma_{1}^{1 , 9}, \Gamma_{1}^{1 , 1}, - \Gamma_{2}^{1 , 1}, - \Gamma^{2 , 2}) .
K_n(x, y) = \frac{1}{2\pi i}\left(\frac{y}{x}\right)^\nu \oint_\Sigma\frac{ds}{s}\int_{0}^{\infty}\frac{dt}{t}\, k_n(s,t)\left(\frac{s}{t}\right)^\nu e^{\frac{x}{s}-\frac{y}{t}}.
\|u\|_{L^\infty((0,T),L^2)}^2+\|u\|_{L^2((0,T),H^2)}^2+\|u\|_{L^4((0,T),L^4)}^4 &\quad\le C(1+Te^{\omega T})\left(\|u_0\|_2^2+\|f\|_{L^2((0,T),L^2)}^2\right)
g(z)=g[A_{w}^{-1}]_{2m+2}(\zeta)j(A_{w}^{-1},\zeta)^{2m+2}=\frac{(1-\zeta)^{2m+2}}{(2it)^{m+1}}\sum_{n\gg-\infty}a_{n}(w)\zeta^{n},
G(z_{out},z_{in})=\frac 1{16 \sqrt 2}e^{\gamma^+ \frac \partial{\partial\mu}}e^{\gamma^-\frac \partial{\partial\nu}}\langle z_{out}|{({\hat p}^2)}^{-1}\left(\mu{\hat{p'_{\theta}}}^2{\hat{p'_{\bar\theta}}}^2+ \nu{\hat{p'_{\bar\theta}}}^2{\hat{p'_{\theta}}}^2 \right) |z_{in}\rangle |_{\mu=\nu=0},
\hat{h}=\left( \begin{array}{ccccc}1 0 0 \cdots 0 0 \omega 0 \cdots 0 0 0 \omega^{2}\ddots \vdots \vdots \vdots \ddots \ddots 0 0 0 \cdots 0 \omega^{n-1}\end{array}\right) .
e^{-(t-t_{k+1})L_{k+1}}(e^{-(t_{k+1}-t_k)L_k}w) = e^{-(t-t_k)L_k}w-\int_{t_{k+1}}^te^{-(t-\sigma)L_{k+1}}{\rm div}\,(A_{k+1}-A_k)\nabla e^{-(\sigma-t_k)L_k}w\,{\rm d}\sigma.
\begin{cases}\partial_t \mathbb{U}(A_t) + \inf_{u \in \mathrm{U}}\sup_{v \in \mathrm{V}}\mathcal{H}(A_t,u,v,(\mathbb{U}, \partial_x \mathbb{U}, \partial_{xx}\mathbb{U})(A_t) ) = 0 , ~~~~~~~~~~ t \in [0,T),~ A_t \in \Lambda \mathbb{U}(A_T) = m(A_T),~ A_T \in \Lambda_T.\end{cases}
\int_{\{H_N(\vec p, \vec u)=E_N\}\cap S}\max\left( \vec p \cdot \hat n(\vec u) ,0\right) \frac{d\hat{\sigma}(\vec p, \vec u)}{|\nabla H_N(\vec p, \vec u)|}\sim \frac{2^{N-2}N^{N}S_{2N-3}}{(N-1) \prod_{j=2}^N \sqrt{\lambda^s_j}}\left( \frac{N}{\beta}- E_N^s \right)^{N-1}
\limsup_{n \rightarrow \infty}\mathbb{P}(Y(n)>2N_\varepsilon K_\varepsilon) &~~~~~\leq \limsup_{n \rightarrow \infty}\mathbb{P}(\Omega(n,\varepsilon)) +\limsup_{n \rightarrow \infty}\sum_{l=1,2}\mathbb{P}\big(\sum_{i:t_{i,n}^{(l)}\leq T}\big(\Delta_{i,n}^{(l)}X^{(l)}\big)^2> K_\varepsilon\big)<\delta.
\Xi_{st}- \tilde{\Xi}_{st}= \sum_{n=1}^p Y_s^{(n)}X_{st}^{(n)}- \sum_{n=1}^p \tilde{Y}_s^{(n)}\tilde{X}_{st}^{(n)}& = \sum_{n=1}^p Y_s^{(n)}(X_{st}^{(n)}- \tilde{X}_{st}^{(n)}) + \sum_{n=1}^p (Y_s^{(n)}- \tilde{Y}_s^{(n)})\tilde{X}_{st}^{(n)}
F_{ST}=\frac{\overline{p}(1-\overline{p})-\sum c_{i}p_{i}(1-p_{i})}{\overline{p}(1-\overline{p})}=\frac{\overline{p}(1-\overline{p})-\overline{p(1-p)}}{\overline{p}(1-\overline{p})}
\mathbf{R}_{\bar{1}2\bar{1}2}=0,~~\mathbf{R}_{1\bar{1}1\bar{1}}+\mathbf{R}_{2\bar{2}2\bar{2}}-2\mathbf{R}_{1\bar{1}2\bar{2}}+2\mathbf{R}_{1\bar{2}\bar{1}2}=0.
L=\frac{1}{c_{2}(H)}\, \int_{0}^{\tau}[ D_{+}^{H-1/2}g]_u \, dW_u+ \frac{1}{2 (c_{2}(H))^{2}}\int_{0}^{\tau}[ D_{+}^{H-1/2}g]_u^2 \, du ,
[\![T_{q,kq+p}\#-T_{p,q}\#-kT_{q,q+1}]\!] = [\![T_{q,kq+p}]\!]-[\![T_{p,q}]\!]-k[\![T_{q,q+1}]\!] = (k[1,q-1,q-1,1]+O_1)-(\lfloor q/p\rfloor[1,p-1,p-1,1]+O_2)-k([1,q-1,q-1,1]+O_3) = O_1-\lfloor q/p\rfloor[1,p-1,p-1,1]-O_2-kO_3.
E_{in}^{-1}(y_n) \mathcal{O}\left((x-y) n^{-\frac{5}{2}})\right) E_{in}(x_n) = \left(\mathbb{I}+ C_n(y_n) \right)^{-1}E_n^{-1}(y_n) \mathcal{O}\left((x-y) n^{- \frac{5}{2}}\right)E_n(x_n) \left( \mathbb{I}+ C_n(x_n) \right) & = E_n^{-1}(y_n) \mathcal{O}\left((x-y) n^{- \frac{5}{2}}\right)E_n(x_n)
G_{\mu\nu}=\left(\begin{array}{ccc}-\frac{2\pi\alpha^{\prime}}{\theta_0}0 -\frac{2\pi\alpha^{\prime}}{\theta_1}0 \frac{2\pi\alpha^{\prime}}{\theta_0}+\biggl(\frac{2\pi\alpha^{\prime}}{\theta_1}\biggr)^2 0 -\frac{2\pi\alpha^{\prime}}{\theta_1}0 1\end{array}\right),
\Big\|{u\choose \dot{u}}-V_0(t){u_0^+\choose u_1^+}\Big\|_{\dot H^1\times L^2}=\Big\|\int_t^\infty V_0(t-s){0\choose F(u(s))}ds\Big\|_{\dot H^1\times L^2}&\lesssim\|(|u|^{\frac{4}{n-2}})u\|_{L_t^{2(n-2)/(n+2)}L_z^{2n(n-2)/(n-3)(n+2)}}&\lesssim\|u\|_{L_t^{2}L_z^{2n/(n-3)}((t,\infty)\times X)}^{\frac{4}{n-2}}&\rightarrow 0,t\rightarrow\infty.
K ( \theta ) = \frac{s_{\frac{1}{2}}( \theta ) s_{\frac{1 + a}{2}}( \theta ) s_{\frac{2 - a}{2}}( \theta )}{s_{\frac{1 - E}{2}}( \theta ) s_{\frac{1 + E}{2}}( \theta ) s_{\frac{1 - F}{2}}( \theta ) s_{\frac{1 + F}{2}}( \theta )}\, \, \, ,
a_k(x) =x^2\delta_{k=3}+xa_{k-1}(v)+xb_{k-1}(v)+xc_{k-1}(v), b_k(x) =x^2\delta_{k=3}+v^2b_2(x)+x(b_k(x)+b_{k+1}(x)+\cdots)+x(a_{k-1}(x)+a_k(x)+\cdots), c_k(x) =xb_k(x)+x(c_k(x)+c_{k+1}(x)+\cdots), b_2(x) =xb_2(x)+x(c_3(x)+c_4(x)+\cdots),
\sum_{s=1}^T \frac{w_s}{\sum_{t=s}^Tb(t)+1}\le \sum_{s=1}^T \frac{\Delta M(T-s+1)}{\frac{1}{2}b(T)(T-s+1)+1}\le\frac{2}{b(T)}\sum_{t=1}^T\frac{\Delta M(t)}{t}
( \dot{T}^{M})^{\dot{A}\dot{B}}( \Gamma^{A})^{\dot{B}\ddot{A}}- ( \Gamma^{A})^{\dot{A}\ddot{B}}( \ddot{T}^{M})^{\ddot{B}\ddot{A}}= ( T^{M})^{A B}( \Gamma^{B})^{\dot{A}\ddot{A}}.
\Delta_{\lambda,i,\sigma,\mathfrak D}(z):=\frac{\prod_{\langle a \rangle=\langle \lambda_{i}\rangle, a\leq 0}U_{i}(\sigma)+az}{\prod_{\langle a \rangle=\langle \lambda_{i}\rangle, a\leq \lambda_{i}-c_{1}(L_{i})\cdot\mathfrak D}U_{i}(\sigma)+az}
{|\langle+X|\psi(t)\rangle|}^{2}={|\frac{\langle0|+\langle1|}{\sqrt{2}}(\frac{1}{\sqrt{2}}exp[\frac{-iE_{+}t}{\hbar}]|0\rangle+\frac{1}{\sqrt{2}}exp[\frac{-iE_{-}t}{\hbar}]|1\rangle)|}^{2}=cos^{2}(\frac{\omega t}{2}),
R_k \mapsto \sum_{\substack{0 \leq s_1, \dots, s_q \leq m+1 0 \leq s_{q+1}\leq r}}\binom{r}{s_{q+1}}\left(\prod_{i=1}^q \binom{m+1}{s_i}\right) \left(\prod_{i=1}^{q+1}(-\lambda_i)^{s_i}\right)R_{k-\sum_{i=1}^{q+1}s_i},
\frac{\mathbb{P}\Big\{W_j > 0 \ \Big|\ |W_j|, \boldsymbol{W}_{-j}\Big\}}{\mathbb{P}\Big\{W_j < 0 \ \Big|\ |W_j|, \boldsymbol{W}_{-j}\Big\}}= \frac{\mathbb{P}\Big\{W_j > 0 \ \Big|\ |W_j| \Big\}}{\mathbb{P}\Big\{W_j < 0 \ \Big|\ |W_j|\Big\}}a.s.
\left[ \begin{array}{c}w_{1}\left( t\right) w_{0}\left( t\right) w_{3}\left( t\right) w_{2}\left( t\right)\end{array}\right] =\left[ \begin{array}{cccc}0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0\end{array}\right] \left[ \begin{array}{c}w_{0}\left( t\right) w_{1}\left( t\right) w_{2}\left( t\right) w_{3}\left( t\right)\end{array}\right]
\int_0^{\infty}dkk^{{\mu}-1}{\rm sin}(ak+\frac{b}{k})={\pi}\left(\frac{b}{a}\right)^{\frac{\mu}{2}}\{{\rm cos}\frac{{\mu}{\pi}}{2}J_{\mu}(2\sqrt{ab})-{\rm sin}\frac{{\mu}{\pi}}{2}N_{\mu}(2\sqrt{ab}) \}