harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
V_{\varphi}(t,x^{(t)},m^{(t)}) &= \inf_{\gamma\in \Gamma}\mathbb{E}_{\varphi}\bigg[f(t,x_t,m_t,\gamma) +V_{\varphi}\left(t,(x^{(t)},\Psi(t,x_t,m_t,\gamma,\xi_{t+1})),(m^{(t)},\mu_{t+1})\right)\bigg|X^{(t)}=x^{(t)},\mu^{(t)}=m^{(t)}\bigg].
i) \psi_{1}(y) \in (0,a) \implies \Gamma_{1}(\psi_{1}(y),y) = 0, y \in \mathcal{S}_{2}ii) \psi_{2}(x) \in (b,1) \implies \Gamma_{2}(x,\psi_{2}(x)) = 0, x \in \mathcal{S}_{1}
\widehat{\psi}_{0,q,\#}(u_0) = \int_{\Lambda_L}dx \psi_{0,q}(x) e^{- 2 \pi i x \cdot u_0}= \int_{\Lambda_L}dx \psi_{0,0}(x) e^{- 2 \pi i ( x \cdot u_0 - x\cdot q)}=: \widehat{\psi}_{0,0,\#}(u_0-q) ,
J_kP_k=\left(\begin{array}{cc}0 0\cr e_1 J_{k-1}\end{array}\right)\left(\begin{array}{cc}1 0\cr Y_{k}P_{k-1}\end{array}\right)=\left(\begin{array}{cc}0 0\cr e_1+J_{k-1}Y_{k}J_{k-1}P_{k-1}\end{array}\right)
\Delta^mH(B^e)\times (2H_1^{-(1-2h/(ep))})^{2h}\times (C_{18}H_1^{2h/(ep)})^{ep-2h}V(ep-2h) = \Delta^mH(B^e)4^hC_{18}^{ep-2h}V(ep-2h) > 2^{pn}\Delta^n,
\sum_{x \in[y_1 - \bar{t}, y_1 + \bar{t}+1]}\max(0, \, \sigma'(x)-f(k)) - \max(0, \sigma(x)-f(k))=-\sigma(y_1 - \bar{t})+f(k) &+ \sum_{x \in [y_1 - \bar{t}+1, y_1]}-m_1 + \sum_{x \in (y_1, y_1 + \bar{t}]}m_1 &+ \sigma(y_1+\bar{t}+1) +m_1 -f(k) - \tau \geq 0,
{\rm Char}_{\cal L}(j_!{\cal F})=-\Bigl(\dfrac{a_X^{\cal L}(j_!{\cal F})}{[P(T^*_XX):X^\vee]}\cdot [T^*_XX]+\sum_{ij\in J}\dfrac{a_{ij}^{\cal L}(j_!{\cal F})}{[P(T_{ij}):T_{ij}^\vee]}\cdot [T_{ij}]+\sum_{x\in \Sigma}a_x^{\cal L}(j_!{\cal F})\cdot [T^*_xX]\Bigr)
P(x) = x_{1}x_{2}x_{3}+ x_{1}x_{4}x_{5}+ x_{2}x_{4}x_{6}+ x_{3}x_{5}x_{6}&= \tfrac{1}{2}(x_{1}+ x_{6})(x_{2}+ x_{5})(x_{3}+ x_{4}) + \tfrac{1}{2}(x_{1}- x_{6})(x_{2}- x_{5})(x_{3}- x_{4}).
U_h|_{I\cap(\Omega_{h,k-1}^{(i)}\cup\Omega_{h,k}^{(i)})}\in O_{\epsilon}({U}_i^{(0)}),i=1,2; |\sigma^{I}_{j}-\sigma_{j0}|<\hat{\epsilon},j=2,3;|\gamma_{4}^{I}|<\hat{\epsilon},
\gamma(k) = \sum_{j=n}^N |A_j| \cdot 2^{-j\alpha + c + e +1}\leq \sum_{j=n}^N 2^{j(\beta-\alpha) + c+ e + 1}\leq \frac{2^{(\beta-\alpha) n + c+ e + 1}}{1-2^{\beta-\alpha}}
\overline{{\alpha}}( Q ) ={\frac{1 2 \pi}{\displaystyle 3 3 \ln{\bigg ({\frac{\mu_{g}^{2}+ Q^{2}}{\kappa^{2}}}\bigg )}- 2 \sum_{f}\ln{\bigg ({\frac{\mu_{f}^{2}+ Q^{2}}{\kappa^{2}}}\bigg )}}},
\left[ -\left({\partial^2 \over \partial\rho_1^2}+{\partial^2 \over \partial\rho^2_2}+{\partial^2 \over \partial\rho^2_3}\right) +{\alpha^2}( \rho_1^2 \rho_2^2 + \rho_2^2 \rho_3^2 + \rho_1^2 \rho_3^2 ) +\frac{{\hat{l}_1}^2}{\rho_1^2}+ \frac{{\hat{l}_2}^2}{\rho_2^2}+ \frac{{\hat{l}_3}^2}{\rho_3^2}\right] \Psi = 2 \, E \Psi
\delta_{\mathrm{d i f f}}\, g_{\alpha \beta}\equiv ({\cal L}_{\vec{v}}g )_{\alpha \beta}= v^{\gamma}( \partial_{\gamma}g_{\alpha \beta}) + g_{\gamma \beta}( \partial_{\alpha}v^{\gamma}) + g_{\gamma \alpha}( \partial_{\beta}v^{\gamma}) ,
\frac{d^2\,}{d\, \tau^2}W(\mathbf{e}+ \tau \, \xi \otimes \eta, \boldsymbol{\mathfrak{K}}+ \tau \, \zeta\otimes \eta)\bigg|_{\tau=0}>0 \forall\ \eta, \xi, \zeta \in \mathbb{R}^3, \ \ \lVert \eta\rVert=\lVert \xi\rVert=\lVert \zeta\rVert=1.
\frac{N^{2/3+iv}}{qr^{2/3}}\sum_\pm\sum_{n_1|qr}n_1^{1/3}\sum_{n_2=1}^\infty \frac{\lambda_\pi(n_1,n_2)}{n_2^{1/3}}S(r\bar a, \pm n_2; qr/n_1) \times \int_0^\infty V(z)z^{iv}e\left(\frac{Nxz}{qQ}\pm \frac{3(Nn_1^2n_2z)^{1/3}}{qr^{1/3}}\right)\mathrm{d}z.
\frac{{\rm Li}_{k}(1-e^{-t})}{1-e^{-t}}=\sum_{n=0}^\infty B_n^{(k)}\frac{t^n}{n!}, &\frac{{\rm Li}_{k}(1-e^{-t})}{e^t-1}=\sum_{n=0}^\infty C_n^{(k)}\frac{t^n}{n!}
\begin{aligned}t^5 - t^4 - t^3 u - 1/2 t^2 u^2 + t u^3 - u^4 = 0, t^3 v - t^2 u + 2 t u^2 - 2/3 u^3 = 0, 10t^{10}w^2 - 10t^{10}- 42 t^9 u - 67 t^8 u^2 - 36 t^7 u^3 - 23 t^6 u^4 - 23 t^4 u^6 + 36 t^3 u^7 - 67 t^2 u^8 + 42 t u^9 - 10u^{10}= 0. \end{aligned}
R_{n,1}^{\epsilon,\Delta t}=-\delta\Phi\Bigl(\epsilon(m_{n+1}^{\epsilon,\Delta t}-m^\epsilon(t_{n+1})),\frac{\Delta tm_{n+1}^{\epsilon,\Delta t}}{\epsilon},X_n^\epsilon\Bigr) R_{n,2}^{\epsilon,\Delta t}=\delta\Phi\Bigl(\zeta^\epsilon(t_{n+1})-\zeta^\epsilon(t_n),\epsilon(m_n^{\epsilon,\Delta t}-m^\epsilon(t_n)),X_n^\epsilon\Bigr).
[[b,T_1],T_2]f &=c\|T_1\|_{CZ}\|T_2\|_{CZ}\mathbb{E}_{\omega_1}\mathbb{E}_{\omega_2}\sum_{i_1,j_1=0}^\infty\sum_{i_2,j_2=0}^\infty 2^{-\max(i_1,j_1)\frac{\delta}{2}}2^{-\max(i_2,j_2)\frac{\delta}{2}}[[b,S^{i_1j_1}_{\omega_1}],S^{i_2j_2}_{\omega_2}]f.
\left[\begin{matrix}\widehat{\mathbf{b}}_1 \cdots \widehat{\mathbf{b}}_{2g}\end{matrix}\right]=\left[\begin{matrix}\mathbf{b}_1 \cdots \mathbf{b}_{2g}\end{matrix}\right]\beta\quad\textrm{for some}~\beta=\left[\begin{matrix}P Q R S\end{matrix}\right]\in\mathrm{GL}_{2g}(\mathbb{Z}).
\begin{cases}ua'_1 = a_1+2s u^2a'_2 = a_2-sa_1+3r-s^2 u^3a'_3 = a_3+ra_1+2t u^4a'_4 = a_4-sa_3+2ra_2-(t+rs)a_1+3r^2-2st u^6a'_6 = a_6+ra_4+r^2a_2+r^3-ta_3-t^2-rta_1. \end{cases}
\mathfrak{F}_l^{(k)}=\sum_{m=1}^{k}\left[\displaystyle \frac{\displaystyle\sum_{p=1}^{k}\left(\lambda_m^{k-p}-\displaystyle\sum_{i=1}^{k-p}\lambda_m^{(k-p)-i}\right)t_{p-1}}{\displaystyle \prod_{j=1, j\neq m}^{k}(\lambda_m - \lambda_j)}\right] \lambda_m^l
[D^{\bar{A}}_2] =(c_2c_1-2c_3)(\overline{{\mathcal T}}^{\vee})\cap [C_{U_1}]=(c_2c_1-2c_3)({\mathcal T}^{\vee})\cap [C_{U_1}] =(c_2c_1-2c_3)({\mathcal T}^{\vee})\cap [C_{U_1}] = (16\sigma_1^3-12\sigma_1\sigma_2+12\sigma_3)\cap [C_{U_1}].
S_{P}^{d + 1}= \int_{M_{d + 1}}\Bigl [{\frac{1}{2}}B_{d - 1}\wedge F +{\frac{1}{8}}B_{d - 1}\wedge{}^{\star}B_{d - 1}+{\frac{\mu^{2}}{2}}A \wedge{}^{\star}A \Bigl ] \; ,
s_\beta x = s_\gamma t(a' \gamma^\lor) w z_\mu t(\mu) &= w s_{w^{-1}\gamma}z_\mu t(a' z_\mu^{-1}w^{-1}\gamma^\lor +\mu ) &= w s_{\alpha}z_\mu t(a' z_\mu^{-1}\alpha^\lor +\mu ).
D F(x,p^{\pm})=\left(\begin{array}{cccc}f_{1,u_1}(x,p^{\pm}) f_{1,u_2}(x,p^{\pm}) \cdots f_{1,u_m}(x,p^{\pm}) f_{2,u_1}(x,p^{\pm}) f_{2,u_2}(x,p^{\pm}) \cdots f_{2,u_m}(x,p^{\pm}) \vdots \vdots \ddots \vdots f_{m,u_1}(x,p^{\pm}) f_{m,u_2}(x,p^{\pm}) \cdots f_{m,u_m}(x,p^{\pm})\end{array}\right).
d\widetilde f (j',u)(\mathfrak j,0)= \tfrac{d}{dl_2}|_{l_2=0}\widetilde f (j_{l_2},u) = \tfrac{d}{dl_2}\rho_{s_{+}(l_2)}(s)|_{l_2=0}f(t) \otimes \pi_{j'}(ds-idt) +\rho_{s_{+}(0)}(s) f(t) \otimes \tfrac{d}{dl_2}\pi_{j_{l_2}}(ds-idt)|_{l_2=0}.
I_{1}( B ) = - \frac{e B}{{\pi}^{\frac{3}{2}}\beta}\sum_{k = 1}^{\infty}\frac{( - 1 )^{k + 1}}{k}\int_{0}^{\infty}d x e^{- \frac{x^{2}}{4}- \frac{1}{x^{2}}\beta^{2}k^{2}m^{2}}\frac{1}{e^{\frac{2 \beta^{2}e B k^{2}}{x^{2}}}- 1}.
G_{m}( D ; \alpha_{1}, . . . , \alpha_{m}) = [ \prod_{i = 1}^{m}\Gamma ( \alpha_{i}) ]^{- 1}( \prod_{i = 1}^{m}\int_{0}^{\infty}d t_{i}\, t_{i}^{\alpha_{i}- 1}) \, T^{- \frac{D}{2}}e^{- \frac{1}{T}\sum_{i < j}t_{i}t_{j}x_{i j}^{2}}
\pi_{(D,b)}= \pi_{(diag(D_1,\ldots,D_l),b)}= = \pi_{(diag(P_1U_1L_1,\ldots,P_lU_lL_l),b)}= = \pi_{(PUL,b)}= = \pi_{(L,b)}\circ \pi_{(U,0)}\circ \pi_{(P,0)},
f_{AB}{}^{C}=C_{AB}{}^{C}, f_{AB}{}^{C+m}=0, f_{A(B+m)}{}^{C}=f_{(A+m)B}{}^{C}=0, f_{A(B+m)}{}^{C+m}=f_{(A+m)B}^{C+m}=C_{AB}{}^{C}, f_{(A+m)(B+m)}{}^{C}=-C_{AB}{}^{C}, f_{(A+m)(B+m)}{}^{C+m}=0.
C=2\sqrt{2}mN \max_{1\leq k\leq m}C_k=6\sqrt{2}mN \frac{\max_{1\leq k\leq m}r_k(r_k-1)}{\min_{1\leq k\leq m}\min_{1\leq j,l\leq r_k, j\neq l}|\alpha_{k,j}-\alpha_{k,l}|}
p({x_m})=\sin\bigg({\frac{{x-{x_m}}}{2}}\bigg), &q({x_m})=\sin\bigg({\frac{{{x_m}-x}}{2}}\bigg), &\chi =\sin \bigg({\frac{h}{2}}\bigg)\sin(h)\sin\bigg({\frac{{3h}}{2}}\bigg).
{\cal W}_{I J}= \frac{8 d_{I}d_{J}}{| G |}\sum_{m = 0}^{N - 1}\prod_{i = 1}^{3}( - 1 )^{m}\cos \left( \frac{\pi m a_{i}}{N}\right) \bar{\chi}^{I}( m ) \chi^{J}( m ) .
\left( k_{3}- k_{2}\right)_{\lambda}T_{\lambda \mu \nu}^{A V V}= - 2 m i T_{\mu \nu}^{P V V}+ T_{\mu \nu}^{A V}\left( k_{1}, k_{2}; m \right) - T_{\nu \mu}^{A V}\left( k_{3}, k_{1}; m \right) .
({\sum}^{\prime}- \int_{0}^{\infty}d n ) f ( n ) = 2 \int_{0}^{\infty}\frac{d t}{e^{2 \pi t}- 1}[ \tan^{- 1}\frac{1 / 2}{b + t}- \tan^{- 1}\frac{1 / 2}{b - t}]
\Phi_{0}\vert_{V_{\ell}}( \tau , z , \nu ) = - \sum_{g = 0}^{\infty}x^{2 g - 2}\varphi_{2 g - 2 , m}\vert_{V_{\ell}}( \tau , z ) , \quad ( \ell = 0 , 1 , 2 , \ldots ) .
G_{r e n}( x , y ) = \mu^{a}\int_{}^{}\frac{d^{4}p}{( 2 \pi )^{4}}\, e^{- i p \cdot ( x - y )}\; \frac{\Gamma ( 1 + a )}{\left( i \! \! \not \! p + m_{r}\right)^{1 + a}}\; ,
\mathfrak{M}(-n\,|\,\tau,\lambda_1,\lambda_2) = \prod_{k=0}^{n-1}\frac{\Gamma\bigl(2+\lambda_1+\lambda_2+(n+2+k)/\tau\bigr)\Gamma\bigl(1-1/\tau\bigr)}{\Gamma\bigl(1+\lambda_1+(k+1)/\tau\bigr)\Gamma\bigl(1+\lambda_2+(k+1)/\tau\bigr)\Gamma\bigl(1+k/\tau\bigr)}.
\mathcal{K}^{s,p}(\mathbb{R}^{n}_+):=\mathrm{H}^{s,p}(\mathbb{R}_+,\mathrm{L}^p(\mathbb{R}^{n-1}))\cap \mathrm{H}^{s-1,p}(\mathbb{R}_+,\mathrm{H}^{1,p}(\mathbb{R}^{n-1}))
\begin{array}{r c l}{g}&{=}&{1 + \sigma^{I}T_{I},}\\{}&{}&{}\\{h}&{=}&{1 - \sigma^{I}W_{I}{}^{i}M_{i},}\\{}&{}&{}\\{x^{\mu \prime}}&{=}&{x^{\mu}+ \sigma^{I}k_{( I )}{}^{\mu}.}\\ \end{array}
g(w)=\frac{r(\frac{a_{1}}{a_{2}})^{d_{2}}}{B(\frac{d_{1}}{r},\frac{d_{2}}{r})}\frac{w^{-d_{2}-1}}{(1+(\frac{a_{2}}{a_{1}})^{-r}w^{-r})^{\frac{d_{1}+d_{2}}{r}}},w>0
\begin{cases}(-\Delta)^{\frac{n}{2}}h(x,y)=0, y\in \Omega, (-\Delta)^i h(x,y) =(-\Delta)^i \left(C_n \ln\left(\frac{1}{|x-y|}\right)\right),\,\, i=0, 1, \cdots, \frac{n}{2}-1, y\in \partial \Omega.\end{cases}
\frac{Q(\zeta_{b}^{(0)})\tau (\zeta_{b}^{(0)})}{\mathbf{A}(-\zeta_{b}^{(0)})}=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{c=1 c\neq a}}^{\mathsf{N}}\frac{\cosh 2\zeta_{b}^{(1)}-\cosh 2\zeta_{c}^{(0)}}{\cosh2\zeta_{a}^{(0)}-\cosh 2\zeta_{c}^{(0)}}Q(\zeta_{a}^{(0)})
\Gamma_{\mu \nu \rho}^{1 , s i n g}( p , q ) \sim \{\frac{\tilde{p}_{\mu}\tilde{p}_{\nu}\tilde{p}_{\rho}}{\xi^{2}| \tilde{p}|^{4}}+ ( p \rightarrow q ) + ( p \rightarrow - ( p + q ) ) \}
\alpha_{\tau_{n}}^{i_{n}}\circ \cdots \circ \alpha_{\tau_{1}}^{i_{1}}=L_{\alpha_{\tau_{n}}^{i_{n}}\left( \cdots \left( \alpha_{\tau_{1}}^{i_{1}}(e)\right) \cdots \right)}\circ \psi_{\tau_{n}}^{i_{n}}\circ \cdots \circ\psi_{\tau_{1}}^{i_{1}}.
f^*\beta_\alpha(U)(s) = f^*\alpha(U)[\{U\rightarrow U\},\bar s] &= \alpha(U)f^*[U\rightarrow U,\bar s] &= \alpha(V)[U\times_U V \rightarrow V, p^*\bar s] &= \alpha(V)[V\rightarrow V,\overline{f^*s}] &= \beta_\alpha(V)f^*s
\left\langle \exp \left\{i g \int_{0}^{T}d \tau \,{\dot{x}}_{\mu}( \tau ) A_{\mu}\left[ x ( \tau ) \right] \right\}\right\rangle_{A}^{r e g}= \exp \left( - \alpha \, \frac{T}{2 b}+ a \ln \frac{T}{b}+ \mathrm{f i n i t e}\; \, \mathrm{t e r m s}\right)
R_{m}^{2 , \alpha}\Phi_{2 \beta_{1}, 2 \beta_{2}}^{2}( z ; q^{1 / 2}) \ = \frac{q^{m \beta_{1}}}{q^{1 / 2}+ q^{- 1 / 2}}\Phi_{2 \beta_{1}+ 1 - \alpha , 2 \beta_{2}- 1 + \alpha}^{2}( z ; q^{1 / 2}) \
J^{\pm}(u)=\begin{pmatrix}1 0 f_{1}^{\pm}(u) 1\end{pmatrix}\begin{pmatrix}k_{1}^{\pm}(u) 0 0 k_{2}^{\pm}(u)\end{pmatrix}\begin{pmatrix}1 e_{1}^{\pm}(u) 0 1\end{pmatrix}.
\frac{K_n'(t)}{\sinh(t)^{n-1}}= 1 -\frac{((n+1)s + n)((n-3)s + n+2) + 2(n-3)s(s+1)}{(n-1)(n-3)s^2 + 2n(n-1)s + n(n+2)}&+ 4(n-1) \frac{s(s+1)((n-3)s + n+2)((n-3)s +n)}{((n-1)(n-3)s^2 + 2n(n-1)s + n(n+2))^2}&= \frac{24 s^2}{((n-1)(n-3)s^2 + 2n(n-1)s + n(n+2))^2}&\geq 0.
\int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s+1}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda,j\in[k:\eta(n)+\ell(n)-p-1], \int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s+1}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda-1,j\in[\eta(n)+\ell(n)-p:\ell(n)], \int_{a_{k}}^{b_{k}}G_{n,k}(\tau)\tau^{s}d\mu_{k,j}(\tau) =0,0\leq s\leq \lambda,j\in[\ell(n):p-1],
2^{2m}Kh_{00}^{2m+4}W_0^{m+1}h^i_l = 2^{2m+2}h_{00}^{2m+2}W_0^{2m+2}+ &^{h}{{{R_0}^i}_{0l}}2^{2m+1}h_{00}^{2m+1}W_0^{m+1}.2^{2m}W_0^{m+1}\Phi^i_{\parallel l}- 2^{2m}h_{00}^{m+1}W_0.2^{2m}h_{00}^{2m}W_0^{2m+1}\Phi^i_{l \parallel 0}+ 2^{2m+3}h_{00}^{2m+2}W_0^{2m+2}\Phi^r{{{\Phi_r}^i}_l},
\int_0^M \int_{E_t^-}\partial_t{\Psi}(x,t)dxdt =\int_{E_0^-}\int_0^{\min(v(x),M)}\partial_t{\Psi}(x,t)dtdx &=\int_{E_0^-}\big({\Psi}(x,\min(v(x),M))-{\Psi}(x,0)\big)dx
\displaystyle\|X_0u\|^2_{L^2(\mathbb{R}^2)}=\frac{1}{L^2}\|\varphi_L-\varphi\|^2_{L^2(\mathbb{R}^2)}=\frac{1}{L^2}\Big(\underbrace{\|\varphi_L\|^2_{L^2(\mathbb{R}^2)}}_{=1}+\underbrace{\|\varphi\|^2_{L^2(\mathbb{R}^2)}}_{=1}-2\int_{\mathbb{R}^2}\varphi_L\varphi\; dqdp\Big) &=\frac{2}{L^2}\Big(1-\int_{\mathbb{R}^2}\varphi_L\varphi\;dqdp\Big){~.}
\delta T_{N}^{M}= \left( \begin{array}{c c c}{0}&{0}&{0}\\{0}&{- \delta \rho}&{- ( \rho + p ) e^{\alpha_{0}}v_{, i}}\\{0}&{( \rho + p ) e^{- \alpha_{0}}v_{, i}}&{\delta p \: \delta_{i j}}\\ \end{array}\right) \: \delta ( y ) ,
{\cal A}_{f i}^{( 1 )}= \int d{\bf x \,}\mathrm{e}^{\, - i \,{\bf p}^{^{\prime}}{\bf \cdot \, x}}\, \left[ \frac{m}{w_{{\bf p}}}\, H_{1}({\bf x},{\bf \nabla}) + V ({\bf x )}\right] \, \mathrm{e}^{\, i \,{\bf p \, \cdot \, x}}\; .
A(t) := \frac{c_p}{c_\mu}\left(\gamma -2 (\alpha+1) \right)\rho(x_t)^{\gamma-\alpha}B(t) := -\frac{1}{c_\mu}(\alpha+1)\rho(x_t)^{-\alpha}C(t) :=\frac{c^2_p}{c_\mu}\left(\gamma - (\alpha+1) \right)\rho(x_t)^{2\gamma-\alpha}.
\sum_{n=1}^\infty \mathbb{P}\left[\phi^{-1}(n)\log \Lambda_{a_n,\phi(n)}(s,\omega)\geq\varepsilon\right] = \sum_{n=1}^\infty\mathbb{P}\left[\Lambda_{a_n,\phi(n)}(s,\omega)\geq\exp(\phi(n)\varepsilon)\right] \leq \sum_{n=1}^\infty 1\cdot\exp(-\phi(n)\varepsilon)<\infty.
|J_s^N(t)| = |Y_1^{N}(t)| + |Y_2^{N}(1-t)| \fallingdotseq t|\dot\gamma_3(t)^N|+ (1-t)|\dot\gamma_2(t)^N| =t \sin \beta_s a_3 + (1-t)\sin\alpha_s a_2,
-\frac{\partial L_{d}D_{\alpha}(p\|q)}{\partial q_{j}}= -\frac{1}{\alpha}\left[\frac{a-1}{a-b}A^{a-2}-\frac{b-1}{a-b}A^{b-2}\right] &+\frac{1}{\alpha}\left[\frac{a-1}{a-b}B^{a-2}-\frac{b-1}{a-b}B^{b-2}\right]p^{\alpha}_{j}q^{-\alpha}_{j}
\phi(x)=\begin{pmatrix}\phi_1(x) \phi_2(x) \phi_3(x) \phi_4(x)\end{pmatrix}=\begin{pmatrix}\chi_1(x) \chi_2(x) \bar \chi^{\bar 1}(x) \bar \chi^{\bar 2}(x)\end{pmatrix}=\begin{pmatrix}\chi_1(x) \chi_2(x) \chi_2^*(x) -\chi_1^*(x) \end{pmatrix}
L^{Kr}=\sum_{i}p_iE_{ii}-\frac{\nu}{2\pi i}\sum_{i\neq j}\frac{\sigma(z+\frac{\alpha_{ij}(D)}{\pi k}{\mbox Im \tau})}{\sigma(z)\sigma(\frac{\alpha_{ij}(D)}{\pi k}{\mbox Im \tau})}E_{ij}
L ( t ) : = \sum_{l = 1}^{\infty}\frac{1}{2}\frac{1}{1 + \cosh l t}\frac{1}{l}, \quad \widetilde{K}( s_{1}) : = s_{1}^{3 N^{2}/ 2 - 1}K_{3 N^{2}- 4}( 2 \sqrt{s_{1}}) ,
\|h-T_{G,w}QT_{F,v}^{\dag}(h)\| = \|T_{F,v}T_{F,v}^{\dag}(h)-T_{G,w}QT_{F,v}^{\dag}(h)\| & \leq \lambda\|h\|+\mu\|T_{F,v}^{\dag}h\| & \leq \left(\lambda+\mu\sqrt{\frac{B_{F,v}}{A_{F,v}}}\right)\|h\| & < \|h\|.
\frac{t_s}{t_r}= \sum_{i=0}^{s-r}\frac{a^i q^{i(d-2r)}(\theta_s - \theta_r)(\theta_s - \theta_{r+1})\cdots (\theta_s - \theta_{r+i-1})}{(q^2;q^2)_i}, \frac{t_r}{t_s}= \sum_{i=0}^{s-r}\frac{a^{-i}q^{i(2r-d)}(\theta_s - \theta_r)(\theta_s - \theta_{r+1})\cdots (\theta_s - \theta_{r+i-1})}{(q^{-2};q^{-2})_i}.
\Psi_i(\widetilde{\Psi}(x))=[M:N]^{-1}\Psi_i(\Psi(z^{-1/2}xz^{-1/2}))=[M:N](\sigma_i^{1/2}u_i^*z^{-1/2})x(z^{-1/2}u_i\sigma_i^{1/2})=u_i^*xu_i
\sum_{I_1 I_2 \cdots I_n = (I)}X^{I_1, I_2, \ldots, I_n}\equiv \sum_{I_1, I_2, \ldots, I_n}\delta^{I_1 I_2 \cdots I_n}_{(I)}X^{I_1, I_2, \ldots, I_n}.
\mathcal{L}= 7 \nabla_{E}^{2}M - \frac{6 3}{2}( \nabla_{E}M )^{2}- 4 2 \kappa^{2}e^{- 9 M}- \frac{1}{2 \cdot 4 !}e^{2 1 M}F_{4}^{2}\, .
\tilde{C}': =\max\left\{\max_{i\in \{\alpha,\beta,r\}}\sup_{\Omega_\delta}|\partial_i\tilde{A}|,\max_{i\in \{\alpha,\beta,r\}}\sup_{\Omega_\delta}|\partial_i\tilde{B}|\right\}, \tilde{\tilde{C}}': =\max\left\{\max_{i\in \{\alpha,\beta\}}\sup_{R_\delta}|\partial_i^nc_+|,\max_{i\in \{\alpha,\beta\}}\sup_{R_\delta}|\partial_i^nc_-|\right\},n\leq 2.
\frac{\partial\Phi}{\partial n}=\frac{1}{\sqrt{1+(\frac{\partial h}{\partial x})^{2}+(\frac{\partial h}{\partial y})^{2}}}\{\frac{\partial\Phi}{\partial z}+\frac{\partial h}{\partial x}\frac{\partial\Phi}{\partial x}+\frac{\partial h}{\partial y}\frac{\partial\Phi}{\partial y}\}=0,atz=-h(x,y),
{_2F}_{1}\left(\begin{array}{cc}\frac{1}{4}, \frac{3}{4}1+\frac{1}{2}\end{array}\mid1\right)= \frac{\Gamma(1+\frac{1}{2})\Gamma(\frac{1}{2})}{\Gamma(1+\frac{1}{4})\Gamma(\frac{3}{4})}.
E ={\frac{1}{2 i}}\int_{- \infty}^{\infty}{\frac{d \omega}{2 \pi}}e^{- i \omega \tau}\sum_{l = 1}^{\infty}( 2 l + 1 ) \int_{0}^{\infty}r^{2}\, d r \, 2 k^{2}[ F_{l}( r , r ) + G_{l}( r , r ) ] .
\Big \langle : e^{i \varphi ( f )}:_{M}e^{i \varphi ( g )}:_{M}\Big \rangle_{C_{m}}\; = \; \exp \left( - \frac{1}{2}( f , C_{m}f ) - ( f , C_{m}g ) - \frac{1}{2}( g , C_{m}g ) \; + \right.
\frac{1}{\Sigma_n^2}\overbrace{\left(\frac{\lambda+l}{\lambda A_l^0}\right)^2}^{=N(n,l)}[l(2\lambda+l)]^\mathfrak d \int_0^\infty\rho^{\mathfrak d}\,e^{-\frac{\rho l(2\lambda+l)}{2\lambda}}\,\frac{d\rho}{\rho}&\qquad=\frac{N(n,l)}{\Sigma_n^2}\,[l(2\lambda+l)]^\mathfrak d\left[\frac{2\lambda}{l(2\lambda+l)}\right]^\mathfrak d\Gamma(\mathfrak d)=\frac{N(n,l)}{C},
{\bf y}_1=\sum_{j=1}^{J-1}{\bf G}_{1j}{\bf x}_j+{\bf G}_{1J}{\bf x}_J+{\bf z}_1,{\bf y}_2={\bf G}_{2J}{\bf x}_J+\sum_{j=1}^{J-1}{\bf G}_{2j}{\bf x}_j+{\bf z}_2,
S^{\mu\nu}\equiv{{\partial L}\over{\partial \sigma_{\mu\nu}}}=L^{-1}[J^2\sigma^{\mu\nu}+{{MJ}\over{({{a_1a_2}\over2}+a_3})^{1\over 2}}(a_1\sigma^{\mu\nu}+(u^\mu\sigma^{\nu\lambda}- u^\nu\sigma^{\mu\lambda})u_\lambda )].
\mathbf{K}_{\nu}(x)=\mathbf{H}_{\nu}(x)-Y_{\nu}(x)=\frac{2(\frac{x}{2})^\nu}{\sqrt{\pi}\Gamma(\nu+\frac{1}{2})}\int_0^{\infty}(1+t^2)^{\nu-\frac{1}{2}}e^{-xt}dt.
\begin{aligned}||D^2 w||_{L^p(\Omega_{1/{12}})}^{p}\leq \sum\limits_{s_0=2}^{\infty}\sum\limits_{Q_k\in\mathcal{F}^{s_0}}\left(\sum\limits_{j=0}^{s_0}\sum\limits_{s=s_0-j-6}^{{\infty}}||D^2 w^{s,j}_{k}||_{L^p(Q_k)}\right)^p.\end{aligned}
\left|\left\{g\in\mathcal G:\big|\left[b, R_j\right]\varphi(g)\big|>{A_1\over 2}\right\}\right| \leq\theta_b\int_{B(0,1)}{2|\varphi(g)|\over A_1}\left(1+\log^+\left({2|\varphi(g)|\over A_1}\right)\right)dg &={2\theta_b\over A_1}\left(1+\log^+\left({2\over A_1}\right)\right).
\langle \mathcal{A}(\mathbf{H}) \mathbf{u}, \tilde{\mathbf{u}}\rangle_{\mathcal{V}'; \mathcal{V}}:= a(\mathbf{u}, \tilde{\mathbf{u}}; \mathbf{H}) \mathbf{u}, \tilde{\mathbf{u}}\in \mathcal{V}.
\sum_{L=0}^M \frac{(\rho_1)_L (\rho_2)_L (aq/\rho_1\rho_2)^L}{(aq/\rho_1)_L (aq/\rho_2)_L}\frac{\alpha_L}{(q)_{M-L}(aq)_{M+L}}=\sum_{L=0}^M \frac{(\rho_1)_L (\rho_2)_L (aq/\rho_1\rho_2)^L (aq/\rho_1\rho_2)_{M-L}}{(aq/\rho_1)_M (aq/\rho_2)_M (q)_{M-L}}\beta_L,
\Psi_{\bf C}\rightarrow \left( \begin{array}{c c}{e^{\frac{\mathrm{i}}{2}\mathrm{~{\bf ~ \sigma ~ \cdot}~}( \mathrm{~{\bf ~ \theta}~}- \mathrm{i}\mathrm{~{\bf ~ \phi}~})}}{0}{0}{e^{\frac{\mathrm{i}}{2}\mathrm{~{\bf ~ \sigma ~ \cdot}~}( \mathrm{~{\bf ~ \theta}~}+ \mathrm{i}\mathrm{~{\bf ~ \phi}~})}}\end{array}\right) \cdot \Psi_{\bf C},
\left\langle \mathcal F^m_{\mu \nu}\mathcal F^{m \mu \nu}\right\rangle = -20 \left[ \partial_{\mu y}\partial^\mu_x \mathcal G (y,x) + g^2 a_\mu (x) a^\mu (x) \mathcal G (x,x) - i g a^\mu (x) \partial_{\mu x}\mathcal P^* (y,x) + i g a^\mu (x) \partial_{\mu x}\mathcal P (y,x) \right] \Bigr |_{y=x}
\alpha\lesssim \left\|\sum_{j=-\infty}^\infty\chi_{{\mathcal O}_j \setminus{\mathcal O}_{j+1}}{\mathcal M}f\right\|_{{\mathcal M}_{p,\phi}}=\|{\mathcal M}f\|_{{\mathcal M}_{p,\phi}}=\|f\|_{H{\mathcal M}_{p,\phi}}.
\Delta_{\bar{\xi}_i}u^{(i)}_{\alpha -1}= 0 \ \ \ \ \Upsilon_i, \partial_{\bar{\nu}_{\bar{\xi}_i}}u^{(i)}_{\alpha -1}= 0 \ \ \ \ \partial \Upsilon_i, \langle u_{\alpha -1}^{(i)}\rangle_{\Upsilon_i}= 0,
|f |_{\mathcal{H}^\beta}= \max_{1 \leq j <\beta}\| |f^{(j)}|^\beta/|f|^{\beta-j}\|_\infty^{1/j}= \max_{1\leq j <\beta}\left( \sup_{x\in[0,1]}\frac{|f^{(j)}(x)|^\beta}{|f(x)|^{\beta-j}}\right)^{1/j}
\frac{1}{NT^{2}}\sum_{i=1}^{N}\P\biggl((\alpha+C)T^{1/4}+\sup_{s \in [0,T]}\vert W_{s}^{i}\vert\geq\frac{\varepsilon_{0}}{4}\biggr) \leq\frac{1}{NT^{2}}\sum_{i=1}^{N}\P\biggl(\sup_{s \in [0,T]}\vert W_{s}^{i}\vert\geq\frac{\varepsilon_{0}}{8}\biggr) \leq cT^{-2}\exp\bigl(-c^{-1}T^{-1}\bigr),
\mathcal{E}^{Z,d}(u,v) :=\int\int_{Z\times Z}\frac{(u(\xi)-u(\eta))(v(\xi)-v(\eta))}{d(\xi,\eta)^{2q}}d\mathcal{H}_{d}(\xi)d\mathcal{H}_{d}(\eta), B_2(d) :=\{u\in L^2(Z,\mathcal{H}_{d})\ ;\ \mathcal{E}^{Z,d}(u,u)<\infty\},
\beta_i:=\begin{cases}(\log m_i)^{-1}, \mbox{if}n=8, m_i^{-\frac{2}{n-4}}, \mbox{if}n=9, m_i^{-\frac{4}{n-4}}\log m_i, \mbox{if}n=10, m_i^{-\frac{4}{n-4}}, \mbox{if}n\ge 11.\end{cases}
\omega_n^i(e_n)=\frac{\langle\nabla_{e_n}\nabla u,e_i\rangle}{|\nabla u|}=\frac{\langle\nabla_{e_i}\nabla u,e_n\rangle}{|\nabla u|}=\frac{\langle\nabla_{e_i}\nabla u,\nabla u\rangle}{|\nabla u|^2}=\frac{|\nabla u|_i}{|\nabla u|}, \quad,
\widetilde{q}_1:=a^{1-p}\xi^p(q) =a^{1-p}\Bigl(\xi^p(c^p)-(\xi^p)_1(c^p){\cdot}(ab)^{p-1}c+\sum_{i\ge 2}(\xi^p)_i(c^p){\cdot}(-(ab)^{p-1}c)^i\Bigr) &\in aw^p-\xi '(c)^p{\cdot}b^{p-1}c+a^{p-1}b^{2(p-1)}c^2S[ab,c]\subset B.
\rho_L (q)=\frac{1}{q^{L-1}}\left[\ 1 + \sum_{1\leq i<j\leq L}(q-1)^{j-i}B_{ij}\ + \ \sum_{1\leq i<j<k<l \leq L}(q-1)^{j-i +l-k}\B^{(2)}_{i,j,k,l}+ \cdots\right]
a\sqrt{2t}+ 2t b = r \geq \frac12 r + \frac{br^2}{2 a^2}= a \sqrt{2 \frac{r^2}{8a^2}}+ 2b\frac{r^2}{4 a^2}\geq a \sqrt{2 \frac{r^2}{8a^2}}+ 2b \frac{r^2}{8 a^2}.
H^{1}( \Omega^{2}) = \frac{\Omega^{2}}{4 ( 2 \pi )^{4}}\int_{1}^{\infty}\! d \xi ( n_{\xi}^{+}- n_{\xi}^{-}) \int_{1}^{\xi}\! d \eta ( n_{\eta}^{+}- n_{\eta}^{-}) \ln \left| \frac{( \xi^{2}- 1 ) ( \eta^{2}- 1 )}{\xi^{2}- \eta^{2}}\right| ,
E(M_{\alpha})=\frac{1}{16\pi}lim_{r\rightarrow\infty}\int_{|x|=r}\sum_{p=1}^{3}\sum_{q=1}^{3}(\partial_{q}(\Phi_{\alpha}^{*}g)_{pq}-\partial_{p}(\Phi_{\alpha}^{*}g)_{qq})\frac{x^{p}}{|x|}dH^{2}(x),
\delta W=\sum_{k=1}^{f}[(\sum_{i=1}^{m}F_{i}\cdot\frac{\partial r_{i}}{\partial q_{k}}+\sum_{j=1}^{n}M_{j}\cdot\frac{\partial\phi_{j}}{\partial q_{k}})\delta q_{k}]=\sum_{k=1}^{f}Q_{k}\delta q_{k},
\theta_{1}^{\pm}= q^{- 1 / 6}\prod_{n = 0}^{\infty}( 1 + q^{n + \frac{1}{2}}y^{- 2}) ( 1 + q^{n + \frac{1}{2}})^{6}( 1 + q^{n + \frac{1}{2}}y^{2})
1-24\sum^{\infty}_{n=1}\frac{nq^{2pn}}{1-q^{2pn}}=\frac{3}{\pi\sqrt{r}p}+\frac{4K[r]^2}{\pi^2\sqrt{r}pM^2_p}\left[-3\alpha(p^2r)+p\sqrt{r}(1+k^2_{p^2r})\right],
r_{m+1/2}=\frac{1}{2}(r_{m+1}+r_m), r_{m,\theta}= \theta r_{m+1}+(1-2\theta)r_m+\theta r_{m-1}, \partial_t r_{m+1/2}=(r_{m+1}-r_m)/\Delta t, \partial^2_t r_m=(r_{m+1}-2 r_m+r_{m-1})/ (\Delta t)^2, \delta_t r_m = (r_{m+1}-r_{m-1})/ (2 \Delta t).
\frac{1}{| \vec{r}_{N 1}|}\quad \rightarrow \quad \frac{1}{| \vec{r}_{N 1}- 2 \pi \vec{\zeta}/ R |}= \frac{1}{| \vec{x}_{N}- \vec{x}_{1}- 2 \pi \tilde{R}\vec{\zeta}/ \alpha^{\prime}|},
f_{( \eta )}^{\vec{\alpha}\beta \gamma}= f^{\alpha [ 1 ] \beta b [ 1 ]}f^{b [ 1 ] \alpha [ 2 ] b [ 2 ]}f^{b [ 2 ] \alpha [ 3 ] b [ 3 ]}\cdots f^{b [ \eta - 2 ] \alpha [ \eta - 1 ] b [ \eta - 1 ]}f^{b [ \eta - 1 ] \alpha [ \eta ] \gamma},
0 = fg-qgf =\sum_{i \geq n}f_i x^i \left(\lambda y + \sum_{j \geq 1}g_j x^j\right) - q \left(\lambda y + \sum_{j \geq 1}g_j x^j\right)\sum_{i \geq n}f_i x^i &= \sum_{i \geq n}\lambda(q^i-q)yf_i x^i + \sum_{i \geq n+1}\left(\sum_{k=n}^{i-1}f_k \alpha^k(g_{i-k}) \right)x^i,
G=\frac{1}{64}\sum_{k=1}^{\infty}\frac{256^{k}(580k^{2}-184k+15)}{k^{3}(2k-1)(\begin{matrix}6k\\ 3k\end{matrix})(\begin{matrix}6k\\ 4k\end{matrix})(\begin{matrix}4k\\ 2k\end{matrix})}
H_{S}= \sum_{n}\frac{1}{2}\left[ e^{- 3 H t}L^{3}\pi_{\phi}({\bf k}_{n}, t ) \pi_{\phi}( -{\bf k}_{n}, t ) + \frac{e^{3 H t}}{L^{3}}( e^{- 2 H t}{\bf k}_{n}^{2}+ m^{2}) \phi ({\bf k}_{n}, t ) \phi ( -{\bf k}_{n}, t ) \right] + \frac{e^{3 H t}}{L^{3}}V_{F},
J_2= \int_0^r -\frac{z^3}{2}(\bar\psi \bar\psi'\cdot\phi+\bar\psi \bar\phi'\cdot\psi){\rm{d}}z \leq \Vert z(\phi,\psi)\Vert_{L^2}\cdot \Vert z(\bar\phi',\bar\psi')\Vert_{L^2}\cdot\frac{1}{2}\Vert z\bar\psi\Vert_{L^\infty},
\phi_{( p_{2}, q_{2}; r_{2}, s_{2})}\otimes \phi_{( p_{1}, q_{1}; r_{1}, s_{1})}= \sum_{( p , q ) \in \Lambda ( p_{1}, q_{1}, p_{2}, q_{2})}\sum_{( r , s ) \in \Lambda ( r_{1}, s_{1}, r_{2}, s_{2})}\left[ \phi_{( p , q ; r , s )}\right] ,
\Omega_4= G_m \sum_{z\in \mathbb{F}_p^*}\sum_{y\in \mathbb{F}_p^*}\zeta_p^{-y}\sum_{\delta \in \mathbb{F}_p^*}\zeta_p^{- \frac{A}{4z}\delta^2 -\big( \frac{By}{2z}+\rho \big)\delta}.
\delta_{c_3}\circ \delta_{c_3}= \eta_1\delta_{c_1}+ \eta_2 \delta_{c_2}+ \eta_3\delta_{c_3}+ \eta_4\delta_{c_4}, &\delta_{c_4}\circ \delta_{c_4}= \eta_1\delta_{c_1}+ \eta_2 \delta_{c_2}+ \eta_4\delta_{c_3}+ \eta_3\delta_{c_4},
\{Q^{+}_{\epsilon},Q^{-}_{\epsilon}\}=a^{+2}\alpha\gamma^{*}+a^{-2}\alpha^{*}\gamma+\frac{1}{2}\left(\{a^{+},a^{-}\}-\epsilon K[a^{-},a^{+}]\right)(\alpha\alpha^{*}+\gamma\gamma^{*}).
\Sigma_{- ; 1 ; 3}^{- ; 2}+ \Sigma_{- ; 2 ; 3}^{- ; 1}, \quad \Sigma_{- ; 1 ; 2}^{- ; 3}+ 2 \Sigma_{- ; 3 ; 2}^{- ; 1}+ 3 \Sigma_{- ; 1 , 2 ; 2}^{- ; 1 , 1}, \quad \Sigma_{1 ; 1 ; 2}^{1 ; 2}+ \Sigma_{1 ; 2 ; 2}^{1 ; 1}.