harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\beta\|Aa-Ab\|^2 - \|Aa\| \|b\| - \|Ab\| \|b\| = \beta(\|Aa\|^2 - 2\langle Aa,Ab \rangle + \|Ab\|^2) &\ \ \ \ -(\|b\|+\|c\|)\|Aa\| - \|Ab\| \|b\| + \|c\|\|Aa\| &\geq \beta(\|Aa\|^2 - 2\|Aa\| \|Ab\| + \|Ab\|^2) &\ \ \ \ -(\|b\|+\|c\|)\|Aa\| - \|Ab\| \|b\| + \|c\|\|Aa\| &\geq \|c\|\|Aa\|.
Z[\gamma_{1},\gamma_{2}]=\int DA_{\mu}exp(\frac{ik}{4\pi}\int d^{3}x\epsilon^{\lambda\mu\nu}A_{\lambda}\partial_{\mu}A_{\nu}+i\int_{\gamma_{1}}dx^{\mu}A_{\mu}+i\int_{\gamma_{2}}dx^{\mu}A_{\mu})
\|S*g(t)\|^2_{\mathbb H^\mu}\le \sum_{n=1}^\infty \int_0^t \omega(t-\tau,\lambda_1)\lambda_n^{\mu-1}|g_n(\tau)|^2 d\tau & = \int_0^t \omega(t-\tau,\lambda_1)\|g(\tau)\|^2_{\mathbb H^{\mu-1}}d\tau.
\left(\sum_{k=1}^K (\lambda_{k,d}+ \frac{g_k m_k}{\sum_i g_i m_i}\mu_{k,d}\cdot\vec{1}_{\{m_k>0\}}) - q_{dd}\right)p_{\vec{m},d}=\sum_{k=1}^K(\lambda_{k,d}p_{\vec{m-e}_k,d}\cdot\vec{1}_{\{m_k>0\}}+ \frac{g_k (m_k+1)}{\sum_i g_i m_i + g_k}\mu_{k,d}p_{\vec{m+e}_k,d}) + \sum_{\ell\neq d}q_{d\ell}p_{\vec{m},\ell},
X_{\Lambda_1,\Lambda_2}\bigl(a \otimes b \otimes Z_{1,2}^{m |I}\bigr) \in V_{s-1,\nabla}[\Lambda_k]_{k=1,2}\subset V_{s,\nabla}[\Lambda_k]_{k=1,2}, &X_{\Lambda_1,\Lambda_2}\bigl(a \otimes b \otimes Z_{1,2}^{-m-1|I}\bigr) \in V_{s, \nabla}[\Lambda_k]_{k=1,2}
\mathbb{P}\left(\mathcal{A}'(D) \in S \right) =\int_{S}f_{\mathcal{A}'(D)}d\mu & \leq \exp \left(\epsilon d(D,D') \right) \int_{S}f_{\mathcal{A}'(D')}d\mu &= \exp \left(\epsilon d(D,D') \right)\mathbb{P}\left(\mathcal{A}'(D) \in S \right).
F_5(a,b,c,d,e) = \int\limits_0^{\infty}\!dt \, t \, F_3(a,b,t)\!\int\limits_0^{\infty}\!dq \, q F_3(c,d,q) F_3(e,t,q) &= \frac{1}{e^2}\int\limits_0^{\infty}\!dt \, t \,F_3(\tilde{a},\tilde{b},t) \!\int\limits_0^{\infty}\!dq \, q \,F_3(\tilde{c},\tilde{d},q) F_3(1,t,q) \;,
\mathcal{T}_D^n(f)(z) = - \frac{1}{2\pi}\iint_{|\zeta|<1}\left( \frac{f(\zeta)}{\zeta}\,\frac{\zeta + z}{\zeta - z}+ \frac{\overline{f(\zeta)}}{\overline{\zeta}}\,\frac{1+z\overline{\zeta}}{1-z\overline{\zeta}}\right)(\zeta - z+\overline{\zeta - z})^{n-1}\,d\xi\,d\eta,
\begin{aligned}E_{x_1}\big[ E_{x_2}\big[F\big(\mathcal Z_{h_1}(x_1,\cdot)+ \mathcal Z_{h_2}(x_2,\cdot)\big)G\big(\mathcal Z_{h_3}(x_1,\cdot)+ \mathcal Z_{h_4}(x_2,\cdot)\big) \big]\big] &=E_x\big[F\big(\mathcal Z_{\mathbf{s}(h_1,h_2)}(x,\cdot)\big)\big]E_x\big[G\big(\mathcal Z_{\mathbf{s}(h_3,h_4)}(x,\cdot)\big)\big]\end{aligned}
\ \mathrm{I m}{\cal A}_{\mathrm{a n n u l u s}}^{( o o \to o o )}\stackrel{v \to 1_{-}}{\sim}\ \Bigl \vert{\cal A}_{\mathrm{d i s k}}^{( o o \to o o )}( \pi / 2 ) \Bigr \vert^{2}\sim e^{- 2 s \alpha^{\prime}\log 2}\ .
Z = \sqrt{\pi}\int{\cal D}G{\cal D}\bar{A}{\cal D}\Omega{\cal D}\Lambda \, , e x p \bigg ( \int \{G_{a}^{\mu \nu}[ - \alpha G_{\mu \nu}^{a}+ F_{\mu \nu}^{a}( \bar{A}) - 2{\cal D}_{\mu}^{( \bar{A})}\Lambda_{\nu}^{a}] -{M^{- 1}}_{a b}^{\mu \nu}\Omega_{\mu}^{a}\Omega_{\nu}^{b}\}d x \bigg ) ,
{\cal A}_{\mathrm{I R}}={\cal A}( \infty ) -{\cal A}( \Lambda ) \simeq \int_{0}^{\infty}\frac{d S}{S}S^{n / 2 - 1 2}( e^{- p \circ p S / 2 \pi \alpha^{\prime}}- e^{- ( p \circ p + 1 / \Lambda^{2}) S / 2 \pi \alpha^{\prime}}) ~ ,
\Psi_{1}\equiv \pi_{a}\pi^{a}- m^{2}+{\frac{2 e F_{a b}\pi^{a}}{\pi^{2}}}\Theta_{2}^{b}\approx 0 ~ ~ ; ~ ~ \Psi_{2}\equiv \pi . S -{\frac{m j}{2}}+{\frac{e j F_{a b}\pi^{a}}{2 m \pi^{2}}}\Theta_{2}^{b}\approx 0 ,
e^{{\frac{i}{\hbar}}S_{\mathrm{I F}}[ \phi , \phi^{\prime}]}= \int d \varphi \int_{\varphi_{i}}^{\varphi}{\cal D}\varphi \int_{\varphi_{i}^{\prime}}^{\varphi}{\cal D}\varphi^{\prime}e^{{\frac{i}{\hbar}}\int^{\cal C}d^{4}s \left[{\cal L}_{h}^{0}[ \varphi ] +{\cal L}_{\mathrm{i n t}}^{\prime}[ \phi , \varphi ] \right]}\rho_{h}( \varphi_{i}, \varphi_{i}^{\prime}; t_{i}) ,
J_E(\theta,\xi)=J_E\left(\sum_{i=1}^m b_i\cdot\chi_i,\sum_{i=1}^m b_i^* \cdot \gamma_i\right)=\sum_{i,j=1}^mb_i^*(b_j)\int_D \chi_i \wedge \gamma_j=\sum_{i=1}^m b_i^*(b_i)=F([\theta]),
\delta^{New}B^a_{\mu_1\ldots\mu_{p_a}}=\partial_{[\mu_1}\Lambda^a_{\mu_2\ldots\mu_p ]}+g_bp_a!p_bk_{b\mu_1\ldots\mu_{p_a}}^{a\nu_2 \ldots\nu_{p_b}}\Lambda^b_{\nu_2\ldots\nu_{p_b}}.
H(m,n,q,c) = \sum^{n-1}_{j=1}\frac{(1-q^{-jc})(1-q^{-(n-j)c})m(n-j)^{m-1}}{(1-q^{-j})(1-q^{-(n-j)})}-\frac{c(1-q^{-nc})(n^m-(m-1)n^{m-1}-\delta_{1m})}{1-q^{-n}}.
y_1^{n_1}(y_1+y_2)^{n_2}= y_1^{n_1}\{y_2^{n_2}+y_1 \cdot f(y_1, y_2)\}~(\mbox{where}~ f(y_1, y_2) ~\mbox{is a function of}~ y_1~ \mbox{and}~ y_2) &= y_1^{n_1}y_2^{n_2}+y_1^{n_1+1}\cdot f(y_1, y_2) &= y_1^{n_1}y_2^{n_2}~( \mbox{as}~ y_1^{n_1+1}= 0).
S K_{[ p - 2 m , m ]}T = d_{[ p - 2 m , m ]}\sum_{n = m}^{[ p / 2 ]}e_{p - 2 n , n - m}S \bigl ( K^{\otimes ( p - 2 n )}\otimes K_{[ 0 , 1 ]}^{\otimes n}\bigr ) T
\sigma\big(\gamma \tau_t(P_j) K\big) =-i(\tfrac{t}{2})^{1/2}[\sigma(A)-\sigma(A^*)]=2(\tfrac{t}{2})^{1/2}\, (\sigma(A)) =2(\tfrac{t}{2})^{1/2}\,\sigma(\gamma\,(\mathbf R+n)^{-1/2}\,K)\,( \sigma\big(\mathbf \gamma \,\mathbf T_{\partial_j}\, K)\big).
A_{j+\frac{1}{2}}\tilde{U}_{j-\frac{1}{2}}+ B_{j+\frac{1}{2}}\tilde{U}_{j+\frac{1}{2}}+ C_{j+\frac{1}{2}}\tilde{U}_{j+\frac{3}{2}}= D_{j+\frac{1}{2}}\bar{U}_{j-1}+ E_{j+\frac{1}{2}}\bar{U}_{j}+ F_{j+\frac{1}{2}}\bar{U}_{j+1}
\sum_{m=k}^{n}\binom{n}{m}\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}\left( \frac{s}{s+n-m+j}\right)^{2}=f_{n,k}(s) -s f^{(1)}_{n,k}(s) &=\prod_{j=n-k+1}^{n}\left(\frac{j}{s+j}\right)\left[ 1+ \sum_{j=n-k+1}^{n}\frac{s}{s+j}\right]
W^{k+t,p}(\Omega,X) :=\left[W^{k,p}(\Omega,X),W^{k+1,p}(\Omega,X)\right]_{t}, W^{k+t}_p(\Omega,X) :=\left(W^{k,p}(\Omega,X),W^{k+1,p}(\Omega,X)\right)_{t}
S_\lambda S_\lambda^* (f) = W^* t_\lambda t_\lambda^* W(f) = W^* \pi(\chi_{Z(\lambda)}) \pi(f) \xi = W^* \pi( \chi_{Z(\lambda)}\cdot f) \xi = W^* W (\chi_{Z(\lambda)}\cdot f) &= \chi_{Z(\lambda)}\cdot f.
\kappa_{1}a + \kappa_{2}= \int \sqrt{h ( b )}d b \simeq \int \sqrt{\frac{3 \pi}{8}- 2 b^{2}}\, d b ={\frac{4 \, b \,{\sqrt{3 \, \pi - 1 6 \,{b^{2}}}}+ 3 \, \pi \, \operatorname{a r c s i n}({\frac{4 \, b}{{\sqrt{3 \, \pi}}}})}{1 6 \,{\sqrt{2}}}}\quad .
\tilde U(X) = 40X^{12}+ 16X^{11}- 43X^{10}- 26 X^9 + 9X^8 - 34X^7 \qquad\qquad+ 34X^6 - 9X^5 - 26 X^4 - 43X^3 + 16X^2 - 40X + 65, \tilde V(X) = -18X^{12}+ 45X^{11}- 32X^{10}+ 63X^9 - 51X^8 + 52X^7 \qquad\qquad+ 52X^6 - 51X^5 + 63X^4 - 32X^3 + 45X^2 - 18X + 1.
C^{mn}_{rs}= i[\pi^{m}_{rt}, F^{n}_{ts}] = i[\pi^{n}_{ts}, F^{m}_{rt}] = [A^{m}_{rt},(A^{n \dagger})_{ts}] \stackrel{_{\textstyle =}}{_{_{N}}}A^{m}_{rt}(A^{n \dagger})_{ts}
\cos(z) \le \mathsf{e}^{\frac{-z^2}{2}}\; \; (0 \le z \le \frac{\pi}{2}), |\sin(z)| \leq |z|, & \frac{n^2}{2}n \sum_{j=1}^{\lfloor m/2 \rfloor}\frac{\mathsf{e}^{-\theta_j^2 \frac{\ell}{2}}}{n^2}16 \, \theta_j^4 = 8 \frac{(2\pi)^4}{n^6}n^3 \sum_{j=1}^{\lfloor m/2 \rfloor}j^4\mathsf{e}^{-\theta_j^2 \frac{\ell}{2}}.
S_{1 2}= \frac{3 ( \mathrm{\boldmath ~ \sigma ~}^{( 1 )}\cdot{\bf r}) ( \mathrm{\boldmath ~ \sigma ~}^{( 2 )}\cdot{\bf r})}{r^{2}}- \mathrm{\boldmath ~ \sigma ~}^{( 1 )}\cdot \mathrm{\boldmath ~ \sigma ~}^{( 2 )}
h_{1-L}^{\mathcal{L}}(t)-h_{0}^{\mathcal{L}_{1,L+1}}(t)\prod_{\alpha=2}^Lh_{-1}^{\mathcal{K}_\alpha}(t)=h_{1-L}^{\mathcal{L}^{(1)(L+1)}}(t)-h_{0}^{\mathcal{K}_1\otimes\mathcal{K}_{L+1}}(t)\prod_{\alpha=2}^Lh_{-1}^{\mathcal{K}_\alpha}(t)
\tilde a_i =\frac{f(he_i+x_i)-f(x_i)}{h}=\frac{g(\|a-he_i-x_i\|_{l_2^d}^2)-g(\|a-x_i\|_{l_2^d}^2)}{h}&=g'(\xi_{h,i})\frac{\|a-he_i-x_i\|_{l_2^d}^2-\|a-x_i\|_{l_2^d}^2}{h}
\begin{aligned}\int_{[0,T]^2}R(s-u,\omega(s)-\omega(u))dsdu=\sum_{k,m=0}^{N+1}Q_{km},~~Q_{km}=\int_{\tau_k}^{\tau_{k+1}}\int_{\tau_m}^{\tau_{m+1}}R(s-u,\omega(s)-\omega(u))dsdu. \end{aligned}
{\cal M}( \Sigma ) = \{\Sigma^{k}\mid \Sigma^{k}\in s u ( 2 ) \otimes \wedge^{2}~ ~ ,{}^{t . f .}\Sigma^{i}\wedge \Sigma^{j}= 0 , ~ d \Sigma^{k}= 0 \}/ \{d i f f e o . \}.
\sum_{k=1}^n \sum_{\sigma\in Sh(k,n-k)}\epsilon(\sigma) Q_{n-k+1}^1(Q_k^1(x_{\sigma(1)}\vee\cdots\vee x_{\sigma(k)})\vee x_{\sigma(k+1)}\vee \dots \vee x_{\sigma(n)})=0.
\frac{\mathtt{h}_{a, \nu}'(z)}{\mathtt{h}_{a, \nu}(z)}= \frac{1+a-a \nu}{2z}+\frac{a^{a/2}}{2 \sqrt{z}}\frac{{}_a\mathtt{B}_{2a-1, a \nu-a+1, 1}'(a^{a/2}\sqrt{z})}{{}_a\mathtt{B}_{2a-1, a \nu-a+1, 1}(a^{a/2}\sqrt{z})},
{{\bf{R}}_{{{\bf{n}}_{\sf q}}{{\bf{n}}_{\sf q}}}}={\mathbb E}\!\left\{{{{\bf{n}}_{\sf q}}{\bf{n}}_{\sf q}^H \big|{\bf{G}}}\right\}= \alpha \left({1 - \alpha}\right){\sf diag}{\left({p_u{\bf{G}}{{\bf{R}}_{x}}{{\bf{G}}^H}+{\bf{I}}}\right)},
\int_0^x \partial_y f(y) dy = f(x)-f(0) \Rightarrow f(x) = \int_0^x \partial_y f(y) dy + f(0) \Rightarrow \left|f(x)\right| \le \int_0^x \left|\partial_y f(y)\right| dy + \left|f(0)\right| \le \int_0^1 \left|\partial_y f(y)\right| dy + \left|f(0)\right|.
\langle (P^{\nu_\alpha}_t)^*(f)-f;g \rangle_{L^2(\mu_\alpha)}= \int_{0}^t \left\langle (D^{\alpha-1})^{*}(f) + \frac{1}{p_\alpha}(R^{\alpha})^*(p_\alpha , f); \nabla P_s^{\nu_\alpha}(g) \right\rangle_{L^2(\mu_\alpha)}ds.
G_0^0(x) = ~-\frac1{2\pi}\int_{-1}^x \frac1{\xi}~d\xi+ \frac1{2\pi}\int_0^{-1}\frac{e^{i\xi}-1}{\xi}~d\xi+\frac{1}{2\pi}\int_{-1}^{-\infty}\frac{e^{i\xi}}{\xi}~d\xi &-\frac1{2\pi}\int_1^{2}\frac{\chi(\xi)}{\xi}~d\xi = ~-\frac1{2\pi}\log|x|+\frac{c_2}{2\pi}.
- \nabla \cdot \left(\varepsilon \nabla u^*\right) + e^{u^* - v^*}- e^{w^* - u^*}= k_1, - \nabla \cdot (\mu_n e^{u^* - v^*}\nabla v^*) - Q(u^*,v^*,w^*) (e^{w^*-v^*}- 1) = 0, - \nabla \cdot (\mu_p e^{w^* - u^*}\nabla w^*) + Q(u^*,v^*,w^*) (e^{w^*-v^*}- 1) = 0.
\Delta =\{\alpha_1= e_1-e_2, \ldots, \alpha_{n-1}= e_{n-1}-e_n, \alpha_n= e_{n-1}+e_n\}, \Delta^\vee =\{\alpha_1^{\vee}= e_1^*-e_2^*, \ldots, \alpha_{n-1}^\vee= e_{n-1}^* -e_n^*, \alpha_n^\vee= e_{n-1}^* + e_n^* - e_0^*\},
f(e^{i\theta_{1}}e^{i\theta_{2}})=[\begin{matrix}cos(\theta_{1}+\theta_{2})&-sin(\theta_{1}+\theta_{2})\\sin(\theta_{1}+\theta_{2})&cos(\theta_{1}+\theta_{2})\end{matrix}]=f(e^{i\theta_{1}})\times f(e^{i\theta_{2}}),
\prod_{i = 1}^{k}\bigl ( 1 + \eta_{i}n ( x_{i}) \bigr ) - \prod_{i = 1}^{k}\eta_{i}n ( x_{i}) ={\frac{\prod_{i = 1}^{k}n ( x_{i})}{\tilde{n}( \sum_{i = 1}^{k}x_{i})}}.
\hat{\psi}(\theta)=\hat{g}(\theta)+\hat{\phi},\hat{g}(\theta)=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}g(W_{i},\hat{\gamma}_{\ell},\theta),\hat{\phi}=\frac{1}{n}\sum_{\ell=1}^{L}\sum_{i\in I_{\ell}}\phi(W_{i},\hat{\gamma}_{\ell},\hat{\alpha}_{\ell},\tilde{\theta}_{\ell}).
\int_{0}^{\infty}d\sigma^{2}(\frac{1}{\sigma^{2}})^{\alpha+1/2+1}exp(-\frac{2\beta+(x-\mu)^{2}}{2\sigma^{2}})=\Gamma(\alpha+1/2)(\frac{2\beta+(x-\mu)^{2}}{2})^{-(\alpha+1/2)}
q_{(k+m)n}|\lambda_{kn}^2| \le \prod_{i=1}^{(k+m)n}{q_{i}\over q_{i-1}}\ \prod_{l=0}^{kn-1}\theta_l^2 \le \left(\prod_{i=kn+1}^{(k+m)n}(r_{i-1}+ 1) \right) \ \prod_{j=1}^{kn}{q_j \over q_{j-1}}\theta_{j-1}^2 \le A_{m,n}\ \prod_{j=1}^{kn}{q_j \over q_{j-1}}\theta_{j-1}^2,
\sum_{j=1}^{g(0)}2^{j-1}\sum_{k=0}^{|\mathbf{A}_p|_j^{p-1}-1}q^k = \sum_{j=1}^{g(P_p-2)}2^{j-1}\sum_{k=0}^{P_p-2}q^k + \sum_{j=g(P_p-2)+1}^{g(P_p-3)}2^{j-1}\sum_{k=0}^{P_p-3}q^k + &~ \cdots + \sum_{j=g(1)+1}^{g(0)}2^{j-1}&= \sum_{k=0}^{P_p - 2}q^k \sum_{j=1}^{g(k)}2^{j-1}= \sum_{k=0}^{P_p - 2}q^k(2^{g(k)}-1).
\left|\sum_{n=1}^{n_1}\frac{\cos(t\log n-\theta(t))}{\sqrt{n}}\right| \le \frac{1}{2}\left|\sum_{n=1}^{n_1}\frac{e^{it\log n-i\theta(t)}}{\sqrt{n}}\right| + \frac{1}{2}\left|\sum_{n=1}^{n_1}\frac{e^{i\theta(t)-it\log n}}{\sqrt{n}}\right| &= \left|\sum_{n=1}^{n_1}n^{-1/2+it}\right|.
\delta_h(t_n,\zeta)= x(t_n+\zeta h)-\{x(t_n)+(1-\theta)\int_{t_n}^{t_n+\zeta h}f(t_n,x(t_n),x(qt_n)){\rm d}t &+\theta\int_{t_n}^{t_n+\zeta h}f(t_{n+1},x(t_{n+1}),x(qt_{n+1})){\rm d}t &+\int_{t_n}^{t_n+\zeta h}g(t_n,x(t_n),x(qt_n)){\rm d}B(t)\}.
0 =\det\left(J-\mu-e_{k}e_{k}^{\top}J\right) =\det\left(J-\mu\right)\det\left(I_{n}-e_{k}e_{k}^{\top}JG_{J}\left(\mu\right)\right) =\det\left(J-\mu\right)\left(1-e_{k}^{\top}JG_{J}\left(\mu\right)e_{k}\right).
\frac{\partial\bar\alpha(x,y;\alpha)}{\partial\ln x}=\beta\left(\frac{y}{x},\bar\alpha(x,y;\alpha)\right)\,\,;\,\,\,\frac{\partial s(x,y;\alpha)}{\partial\ln x}=\gamma\left(\frac{y}{x},\bar\alpha(x,y;\alpha)\right)s(x,y;\alpha)~,
U_{t}^{H_{1}\circ S_{t}}=\exp\left( -\frac{i}{2\hbar}\int_{0}^{t}\sigma(z(t^{\prime}),\dot{u}(t))dt^{\prime}\right) \widehat{T}(u(t)) & =\exp\left( -\frac{i}{2\hbar}\int_{0}^{t}\sigma(S_{t^{\prime}}z(t^{\prime}),\dot{z}(t))dt^{\prime}\right) \widehat{T}(u(t))
\pi=6sin^{-1}(\frac{1}{2})=\sum_{n=0}^{\infty}\frac{3\cdot(\begin{matrix}2n\\n\end{matrix})}{16^{n}(2n+1)}=\frac{3}{16^{0}\cdot1}+\frac{6}{16^{1}\cdot3}+\frac{18}{16^{2}\cdot5}+\frac{60}{16^{3}\cdot7}+\cdot\cdot\cdot
A_1 = V_1\cap V_2 , A_2 = V_2\backslash U_1\cap V_2 , A_3 = X\backslash U_2 , A_{12}= U_1\cap V_2 \backslash A_1 , A_{13}= V_1\backslash A_1 , A_{23}= (U_2\backslash U_1) \backslash A_2 , A_{123}= (U_1\backslash V_1) \backslash A_{12}
\|u_t(\cdot,t)\|_{H^1}^2= \|u_t(\cdot,0)\|_{H^1}^2+\int_0^t \frac{d}{d\tau}\|u_t(\cdot,\tau)\|_{H^1}^2 d\tau \leq \|u_t(\cdot,0)\|_{H^1}^2+T_0C.
\left( \begin{array}{c c c}{2 x_{1}^{2}- 1}&{2 x_{1}x_{2}}&{2 x_{1}x_{3}}\\{2 x_{1}x_{2}}&{2 x_{2}^{2}- 1}&{2 x_{2}x_{3}}\\{2 x_{1}x_{3}}&{2 x_{2}x_{3}}&{2 x_{3}^{2}- 1}\\ \end{array}\right)
= ( - 1 )^{m}\, \left( \frac{\omega}{\pi \hbar}\right)^{1 / 2}\, \left( \frac{1}{2 \pi \hbar}\right)^{1 / 2}\, \left( \frac{m !}{( m + | \ell | ) !}\right)^{1 / 2}\, e^{i \ell \theta}\, e^{i p z / \hbar}\, u^{| \ell |}\, e^{- u^{2}/ 2}\, L_{m}^{| \ell |}( u^{2}) \ \ \ ,
\mathbb{E}\left[\left(\sum_{j=1}^{V_i}Z_{ij}\right)^2\right] = \frac{\mathbb{E}\left[ Z^2 \right]}{\epsilon_{k,n}(\delta)}+\frac{2-2\epsilon_{k,n}(\delta)}{\epsilon_{k,n}^2(\delta)}\mathbb{E}\left[ Z \right]^2
d(\sigma^2\mu) =i_0\sigma\mu-i_1\sigma\mu+\sigma^2\mu\{\mu\}+\sum_{p,q\geq0}\mu\{i_0\sigma\mu,\stackrel{p}{\dots},i_0\sigma\mu,\sigma^2\mu,i_1\sigma\mu,\stackrel{q}{\dots},i_1\sigma\mu\}.
W_{2}( R ) ={\frac{1}{\sqrt{2}R}}B J_{1}B^{- 1}= \left( \begin{matrix}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- i / R^{2}}}&{{i}}\\{{0}}&{{i}}&{{0}}&{{0}}\\{{0}}&{{i / R^{2}}}&{{0}}&{{0}}\\ \end{matrix}\right) ,
u_{y}=\frac{u_{y}^{\prime}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1+\frac{v}{c^{2}}u_{x}^{\prime}},u_{y}^{\prime}=\frac{u_{y}\sqrt{1-\frac{v^{2}}{c^{2}}}}{1-\frac{v}{c^{2}}u_{x}},
\int_\Omega f^{'}_0(U_{\mu_i,\xi_i})\psi^l_{\mu_i,\xi_i}( P\psi^h_{\mu_j,\xi_j}-\psi^h_{\mu_j,\xi_j})dx \leq |f^{'}_0(U_{\mu_i,\xi_i})|_{\frac{n}{2}}|\psi^l_{\mu_i,\xi_i}|_{\frac{2n}{n-2}}|P\psi^h_{\mu_j,\xi_j}-\psi^h_{\mu_j,\xi_j}|_{\frac{2n}{n-2}}=o\bigg(\Big(\frac{\epsilon}{|\ln\epsilon|^2}\Big) \bigg).
1\leq \|\boldsymbol{h}\|_2\leq\frac{N^{1/d}(v,w)}{s\max(|v|,|w|)}:=\max \boldsymbol{h}, \frac{N^{1/d}(v,w)}{2\pi s\max(|v|,|w|)}\leq \|\boldsymbol{h}\|_2\leq\frac{N^{1/d}(v,w)}{2\pi s\min(|v|,|w|)}:=\min \boldsymbol{h}, &\|\boldsymbol{h}\|_2\geq\frac{N^{1/d}(v,w)}{2\pi s\min(|v|,|w|)}.
\int_I|\sigma'|^2 E^2=\int_I\frac{E^2}{|{v}|^2}\left({v}'-({v}'\cdot \frac{{v}}{|{v}|}) \frac{{v}}{|{v}|}\right)^2\leq 4\int_I|{v}'|^2\leq 8\mathfrak{E}_{W}({v})<+\infty.
\log (|\tilde{g}|^t w) = (t/t_*)\log(|\tilde{g}|^{t_*}w^{t^*/t}) & = (t/t_*)\log(|\tilde{g}|^{t_*}w) + (t/t_*)\log(w^{t_*/t - 1}) & = (t/t_*)\log(|\tilde{g}|^{t_*}w) + (t/t_*)(t_*/t - 1)\log(w).
\sum_{n=0}^\infty a_n \frac{t^{n+1}}{(n+1)!}= \frac{3}{2}P \left(\frac{1}{4}, \frac{2}{3}t \right) = \frac{6 e^{t/2}\left(\arcsin(e^{t/2}/2) - \arcsin(1/2) \right)}{(4- e^t)^{1/2}}.
0 ={}Q^{20}y_{10}+ Q^{18}y_{12}+ Q^{17}y_{13}+ x^4(Q^{12}y_{10}) + y_9^2 (Q^4 x)^2 + y_7^2 Q^9Q^5 x + y_8^2 Q^8 Q^4 x + (Q^9 y_9) (Q^4x)^2 + (Q^{10}y_8) (Q^4 x)^2 + y_5^2 (Q^{11}Q^7 x + Q^{10}Q^8 x + x^4 Q^6 Q^4 x)
A^{(2)}= \frac{1}{y^{-\frac{1}{2}}+ y^\frac{1}{2}}A^{(2)}_{\underline{\chi(\O_S)}}(y)\,, B^{(2)}= A^{(2)}_{\underline{K_S^2}}(y)\,, C^{(2)}_{11}= - q^{-\frac{1}{4}}\frac{1}{y^{-\frac{1}{2}}+ y^\frac{1}{2}}A^{(2)}_{\underline{\beta^1\beta^1}}(y) A^{(2)}_{\underline{\beta^1K_S}}(y)\,,
\begin{aligned}\int_{0}^{\left(u^m-k\right)_-}F(\xi)\,d\xi \geq (u^m-k)_-^{2-\alpha}-\frac{m}{m+1}(u^m-k)_-^{2-\alpha}=\frac{1}{m+1}(u^m-k)_-^{2-\alpha}\end{aligned}
(V)' \hspace{7mm}\Psi_{\bf D}=\left(\begin{array}{c}a_1 +{\rm j}b_1 a_2 +{\rm j}b_2 a_3 +{\rm j}b_3 a_4 +{\rm j}b_4\end{array}\right) \hspace{5mm}\leftrightarrow \hspace{5mm}\Psi^{SR}_{\bf C}=\left(\begin{array}{c}b_3+{\rm i}b_1 b_4+{\rm i}b_2 a_3+{\rm i}a_1 a_4+{\rm i}a_2\end{array}\right)
\sum_{k=0}^\infty \int_{[0,T] \times [0,1]}\ell(\varphi_k^\varepsilon(s,y)) \, ds\,dy \le \sum_{k=0}^\infty \int_{[0,T] \times [0,1]}\ell(\varphi_k(s,y)) \, ds\,dy + \frac{\sigma}{2}\le I_T(\boldsymbol{\zeta},\psi ) + \sigma.
C_{m_{1},m_{2},m_{3}}^{j_{1},j_{2},j_{3}}\ =\ (-1)^{j_{1}+j_{2}-j_{3}}C_{-m_{1},-m_{2},-m_{3}}^{j_{1},j_{2},j_{3}}\ =\ (-1)^{j_{1}+j_{2}-j_{3}}C_{m_{2},m_{1},m_{3}}^{j_{2},j_{1},j_{3}}
(I-\mathbb{H})F_z\circ Z Z_{\alpha}D_tZ= [D_tZ,\mathbb{H}](\partial_{\alpha}D_t\bar{Z}-\sum_{j=1}^N\frac{\lambda_j i Z_{\alpha}}{2\pi(Z(\alpha,t)-z_j(t))^2}) = [D_tZ,\mathbb{H}]\partial_{\alpha}D_t\bar{Z}-\sum_{j=1}^N[D_tZ,\mathbb{H}]\frac{\lambda_j i Z_{\alpha}}{2\pi(Z(\alpha,t)-z_j(t))^2})
V(X(t)) = V(x) + \int_0^t 2 f'(1 + |X(s)|^2)^{}X(s) \cdot b(X(s)) ds + 2 \int_0^t f'(1 + |X(s)|^2) \; X(s) \cdot \sigma(X(s)) dW(s) + 2\int_0^t f''(1 + |X(s)|^2)^{}\, |\sigma^*(X(s)) X(s)|^2 ds + \, \int_0^t f'(1 + |X(s)|^2)^{}\| \sigma(X(s)) \|^2 ds.
\frac{\pi}{2}=\sum_{k=0}^{\infty}\frac{k!}{(2k+1)!!}=\sum_{k=0}^{\infty}\frac{2^{k}k!^{2}}{(2k+1)!}=1+\frac{1}{3}(1+\frac{2}{5}(1+\frac{3}{7}(1+\cdot\cdot\cdot)))
\sigma_{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \sigma_{2}= \left( \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right) , \sigma_{3}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) ,
\begin{cases}y_{1,tt}- \Delta y_1 +c(x) y_2 =0 \mbox{in}(0,T) \times \Omega \,, y_{2,tt}- \Delta y_2 =b v \mbox{in}(0,T) \times \Omega \,, y_1=y_2=0 \mbox{in}(0,T) \times \Gamma \,, (y_1, y_{1,t})(0,.)=(y_1^0, y_1^1) , (y_2, y_{2,t})(0,.)=(y^0, y^1) \mbox{in}\Omega\,,\end{cases}
\mathbf{\lambda}_1 = \mathcal{M}(\mathbf{c}^1_1,\ldots,\mathbf{c}^L_1) + q \mathbf{\zeta}_1, \mathbf{\lambda}_2 = \mathcal{M}(\mathbf{c}^1_2,\ldots,\mathbf{c}^L_2) + q \mathbf{\zeta}_2,
\frac1{\omega(P)^{\frac2p-1}}\sum\limits_{Q\subseteq P}\mu(Q) |\psi_Q(q_{j}(\cdot,y))(x_Q)|^2 \lesssim \sup_{x\in P}\frac 1{\omega(B(x,r^{-j}))^{\frac2p}}\sum\limits_{k=k_0}^{\infty}r^{-2|k-j|\varepsilon}\leqslant C_j \sup_{x\in P}\frac 1{\omega(B(x,r^{-j}))^{\frac2p}}.
\nu_{\mathbf{f}}\left( \left \{x\in \mathbb{R}^{\mathbb{N}}:\int_{\mathbb{R}^{\mathbb{N}}}\pi^{\star}d\nu_{\mathbf{f}}\leq \lim_{n}\frac{1}{n}\sum_{k=1}^{n}x_{k}=\pi^{\star}\left( x\right) \leq \int_{\mathbb{R}^{\mathbb{N}}}\pi^{\star}d\bar{\nu}_{\mathbf{f}}\right \}\right) =1.
e^{i S_{\Lambda - \delta \Lambda}( \phi_{+}, \phi_{-})}\equiv \int \prod_{\Lambda_{0}> |{\vec{q}}| > \Lambda - \delta \Lambda}{\cal D}[ \phi_{+}({\vec{q}}, t ) ]{\cal D}[ \phi_{-}({\vec{q}}, t ) ] e^{i S_{c l}[ \phi_{+}, \phi_{-}]}
\hat{\chi}= \sum_{M = 1}^{n}\left( \bar{z}_{+}^{M}\, \, \bar{z}_{-}^{M}\right) \Big ( \begin{array}{c c}{{\mathrm{e}^{V}}}\\ \end{array}\Big ) \left( \begin{array}{c}{{z_{+}^{M}}}\\{{z_{-}^{M}}}\\ \end{array}\right) \ ,
\begin{aligned}\mathbf{x}(\alpha) \triangleq \mathbf{x}+ \alpha \Delta \mathbf{x},\mathbf{y}(\alpha) \triangleq \mathbf{y}+ \alpha \Delta \mathbf{y},\mathbf{s}(\alpha) \triangleq \mathbf{s}+ \alpha \Delta \mathbf{s}.\end{aligned}
Ric^{\prime}(X,W)=Ric(X,W)+\sum_{j=1}^{k}\langle R^{\prime}(X,e_{j})e_{j},W\rangle+\sum_{j=1}^{k}(\sum_{i=1}^{m}\alpha_{j}(X,E_{i})\alpha_{j}(E_{i},W)-H_{j}\alpha_{j}(X,W)).
\int_D \widetilde{f}\hat{v}dx'=\int_{\{\hat{v}\geq \alpha\}}\widetilde{f}\hat{v}dx' +\int_{\{\hat{v}<\alpha\}}\widetilde{f}\hat{v}dx'=\int_{\{\hat{v}\geq\alpha\}}\hat{f}\hat{v}dx'+\int_{\{\hat{v}<\alpha\}}\hat{f}\hat{v}dx'=\int_D\hat{f}\hat{v}dx'.
\begin{array}{l}{( 1 ) \quad{\displaystyle{\frac{\delta}{\delta b_{a}}}}\Delta = 0 , \qquad ( 2 ) \quad \displaystyle{\int}d^{3}x{\displaystyle{\frac{\delta}{\delta c^{a}}}}\Delta = 0 , \qquad ( 3 ) \quad{\cal G}^{a}\Delta = 0 ,}\\{( 4 ) \quad{\cal W}_{X}\Delta = 0 , \qquad X = \mathrm{d i f f . , L o r e n t z , r i g i d}.}\\ \end{array}
p_{t}(x,y)\leq\begin{cases}c'\delta^{-\beta_{2}\alpha_{F}}F_{t}(x,y)\exp\bigl(-\Phi(\gamma'_{\delta}d(x,y),t)\bigr) \textrm{if}t<\Psi(R), c'\delta^{-\beta_{2}\alpha_{F}}(\inf_{M\times M}F_{\Psi(R)}) \textrm{if}t\geq\Psi(R)\end{cases}
(f_1(x)+2f_2(x)+2uf_{14}(x))^* = f_1^*(x)+2x^if_2^*(x)+2ux^jf_{14}^*(x) = (f_1(x)+2f_2(x)+2uf_{14}(x))+2ux^jf_{14}^*(x)+2uf_{14}(x) =(f_1(x)+2f_2(x)+2uf_{14}(x))+uf_4(x)b(x) \in C
a_3=\frac{1}{2(1+2\alpha)k_3}\left[-q_2+\frac{4(1+2\alpha)k_3-(1+3\alpha)k_2^2}{k_2^2(1+\alpha)^2}p_1^2\right],\quad\quad\alpha\in\mathbb{C}\setminus\left\{-1;-\frac{1}{2}\right\}.
\hat{G}_{\mu}^{( n )}= \left( \Gamma_{\mu}\otimes 1 \otimes \cdots \otimes 1 + 1 \otimes \Gamma_{\mu}\otimes \cdots \otimes 1 + \cdots + 1 \otimes 1 \otimes \cdots \otimes \Gamma_{\mu}\right)_{S y m},
g_w(z)\equiv \frac{d^3}{dz^3}f_w(z) = \sum_b\frac{\delta w_b}{\epsilon w_b}\left[ \frac{(\alpha_b-\beta_b)}{(z-\alpha_b)(z-\beta_b)}\right]^2\left[ \bar{\cal P}_{bc}\left( u(z)\right) -\bar{\cal P}''\left( u(z)\right) \right] \ ,
H_{3}([1,6])=h_{6}(h_{24}+2 h_{25}+ 2 h_{34}+4 h_{35}+6 h_{45}), H_4([1,8])=h_{8}\big(h_{246}+2 h_{247}+ 2 h_{256}+4 h_{257}+6 h_{267}+2 h_{346}+ 4 h_{347}+ 4 h_{356}\hphantom{H_4([1,8])=}{}+8 h_{357}+12 h_{367}+ 6 h_{456}+ 12 h_{457}+ 18 h_{467}+24 h_{567}\big).
L_{F}= \exp \left( \begin{array}{c c}{{}}&{{( P \Xi )_{\alpha N}}}\\{{( \overline{{{P \Xi}}})^{\beta M}}}&{{}}\\ \end{array}\right) \exp \left( \begin{array}{c c}{{}}&{{( Q \Xi )_{\alpha N}}}\\{{( \overline{{{Q \Xi}}})^{\beta M}}}&{{}}\\ \end{array}\right)
\bigg|\frac{2\eta}{\pi d^3}\sum_{m=1}^\aleph|V_m|^2\bigg| \lesssim a^{-2}d^{-3}\sum_{m=1}^\aleph|V_m|^2=a^{-2}d^{-3}\sum_{m=1}^\aleph\big|[P_{D_m}]\cdot Q_{m}\big|^2 = a^{-2}\, d^{-3}\, \left\Vert [P_{0}] \right\Vert^{2}\, \sum_{m=1}^\aleph\big| Q_{m}\big|^2.
\int_{\mathcal{H}}\vert\langle e_{i_1},x\rangle\vert^{2\ell_1}\dots\vert\langle e_{i_r},x \rangle\vert^{2\ell_r}\, dm(x)=\prod_{t=1}^{r}\int_{\mathcal{H}}\vert \langle e_{i_t},x\rangle \vert^{2\ell_t}\,dm(x) =\prod_{t=1}^{r}\big(\ell_t!\, 2^{\ell_t}\sigma_{i_t}^{2\ell_t}\big) &=\Bigg(\prod_{t=1}^{r}\ell_t!\Bigg)2^k\sigma_{i_1}^{2\ell_1}\dots\sigma_{i_r}^{2\ell_r},
\eta_{m\kappa}\left(0\right)+\eta_{-m\kappa}\left(0\right)=\left(N_\kappa^++N_\kappa^-\right)\pi+\left(-1\right)^l\frac{\pi}{2}\left[\sin^2\eta_{m\kappa}\left(0\right)+\sin^2\eta_{-m\kappa}\left(0\right)\right]\,,
L(b_{k,n}|{\bf{y}}) = \ln\frac{\sum\limits_{\forall a_m\in\mathbb{A}_n^{(1)}}\mathcal{P}(x_k = a_m|{\bf{y}})}{\sum\limits_{\forall a_m\in\mathbb{A}_n^{(0)}}\mathcal{P}(x_k = a_m|{\bf{y}})}.
P_{p}^{t_{i},T_{j}}\left(v_{p},\pi_{p}\right)=-\Pi\left(t_{i},T_{j}\right)V_{p}\left(t_{i},T_{j}\right)-O_{p}\left(t_{i},T_{j}\right)G_{em}\left(t_{i},T_{j}\right)-\sum_{l\in L}G_{l}\left(t_{i},T_{j}\right)F_{p,l}\left(t_{i},T_{j}\right).
E_j\cdots E_{i+1}(I-A_{i})\ = \ (I-B)(F_1F_2\cdots F_k) (G_1)^{-1}\cdots (G_i)^{-1}\textnormal{if}i<j (I-A_{j})\ = \ (I-B)(F_1F_2\cdots F_k) (G_1)^{-1}\cdots (G_j)^{-1}\textnormal{if}i=j \ .
x_{n}=\sum_{i=1}^{n}N(n,i)k^{i-1}=\sum_{i=1}^{n}\frac{1}{n}(\begin{matrix}n\\i\end{matrix})(\begin{matrix}n\\i-1\end{matrix})k^{i-1}={}_{2}F_{1}(1-n,-n;2;k).
H = \frac{1}{(2 \pi)^3}\sum_{\mu=0}^{3}\int \frac{d^3 k}{2 k^0}\, \omega(\vec{k}) a_\mu^\dagger (\vec{k}) a_\mu (\vec{k})= \frac{1}{2(2 \pi)^3}\sum_{\mu=0}^{3}\int d^3 k \, a_\mu^\dagger (\vec{k}) a_\mu (\vec{k}) \, ,
f(x) = \sum_{j \geq -1}\sum_{k \in \mathbb{Z}}\beta_{jk}\psi_{jk}(x) = \sum_{j \geq -1}\sum_{k \in \mathbb{Z}}\left\{\int_{\mathbb{R}}f(x')\psi_{jk}(x')\, \mathrm{d}x'\right\}\psi_{jk}(x).
\int_{\big\{x \,:\, x^{\eta - 1}\| f \|_{L^1 (x^{-\eta}dx)}> \lambda \big\}}x^{-\eta}\, dx=\int_{\big\{x \,:\, x > \big(\lambda \| f \|_{L^1 (x^{-\eta}dx)}^{-1}\big)^{1/(\eta - 1)}\big\}}x^{-\eta}\, dx\simeq\frac{\| f \|_{L^1 (x^{-\eta}dx)}}{\lambda}.
g(\epsilon_{\{\sigma(1)\}}) = \epsilon_{\{\sigma'(1)\}}, & g(\epsilon_{\{\sigma(1),\sigma(2)\}}) = \epsilon_{\{\sigma'(1),\sigma'(2)\}}, &\dots & g(\epsilon_{\{\sigma(1),\dots,\sigma(n)\}}) = \epsilon_{\{\sigma'(1),\dots,\sigma'(n)\}}
V(x)=\sup_{\theta \in\mathcal{A}}K(\theta) = \sup_{\theta \in\mathcal{A}}E_x\left[ \int_{0}^{\infty}\frac{1}{1-\alpha}\theta^{1-\alpha}_t e^{X_t}e^{-rt}\,dt - \int_{0}^{\infty}e^{Y_t-rt}\,d\theta_{t}\right] .