harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\alpha \int_{\{\hat{v}\geq \alpha\}}\hat{f}dx'=\alpha \int_{\{\hat{v}\geq \alpha\}}\widetilde{f}dx'=\int_{\{\hat{v}\geq \alpha\}}\widetilde{f}\hat{v}dx' =\int_{\{\hat{v}\geq\alpha\}}\hat{f}\hat{v}dx'\geq\alpha \int_{\{\hat{v}\geq \alpha\}}\hat{f}dx',
\mathbb{E}_{i_1,i_2,\ldots,i_{j}}\langle y,Q_j x\rangle=\mathbb{E}_{i_1,i_2,\ldots,i_{j-1}}\langle \mathcal{L}y,Q_{j-1}x\rangle=\mathbb{E}_{i_1,i_2,\ldots,i_{j-2}}\langle \mathcal{L}^2y,Q_{j-2}x\rangle=\cdots=\langle \mathcal{L}^jy,x\rangle.
\mathbf{E}[Z^*_{n}\mid T_g>n] = \mathbf{E}[S_{n}-g_n\mid T_g>n]=\int_0^\infty \mathbf P(S_n>g_n+x \mid T_g>n ) dx &\ge \int_0^\infty \mathbf P(S_n>g_n+x ) dx = \mathbf{E}(S_{n}-g_n)^+= \mathbf{E}Z_n^+.
\psi = \sum_{m = - N}^{\infty}( - i )^{m + \alpha}J_{m + \alpha}( r^{\prime}) e^{i m \phi}+ \sum_{m = - N - 1}^{- \infty}( - i )^{- ( m + \alpha )}J_{- ( m + \alpha )}( r^{\prime}) e^{i m \phi}.
\|\Phi^{(\mathfrak{s})}\|_{\delta}:=\|\Phi^{(\mathfrak{s})}\|_{\mathcal{L}_2}+\|\Phi^{(\mathfrak{s})}\|_{\gamma(L^2(\mathbb{R};\mathbb{R});W^{\mathfrak{s},2+\delta}(\mathbb{R};\mathbb{C}))}.
\iint_{\mathbb{R}^{2N}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2\alpha}}\,dxdy =\int_{\mathbb{R}^{N}}(-\Delta)^{\frac{\alpha}{2}}u(x)\cdot(-\Delta)^{\frac{\alpha}{2}}v(x)\,dx.
[\begin{matrix}\frac{dx_{1}^{*}}{dp_{1}}\\ \frac{dx_{2}^{*}}{dp_{1}}\\ \frac{dy_{2}^{*}}{dp_{1}}\end{matrix}]=[\begin{matrix}-1&-1&0\\ -1&-3&-1\\ 0&-1&-3\end{matrix}]^{-1}[\begin{matrix}-1\\ 0\\ 0\end{matrix}]=\frac{1}{5}[\begin{matrix}8\\ -3\\ 1\end{matrix}]
\chi_{_H}(P_{n_1,\ldots,n_k}(A))=\sum_{\lambda^{(1)}\vdash n_{1},\ldots,\lambda^{(k)}\vdash n_{k}}m_{\lambda^{(1)},\ldots,\lambda^{(k)}}\chi_{_{\lambda^{(1)},\ldots,\lambda^{(k)}}}.
D_{n}( \eta ) D_{n^{\prime \prime}}( \eta ) = \mathrm{e}^{- \eta^{2}/ 4}\sum_{m = 0}^{\mathrm{m i n}( n , n^{\prime \prime})}\frac{n ! n^{\prime \prime}!}{m ! ( n - m ) ! ( n^{\prime \prime}- m ) !}D_{n + n^{\prime \prime}- 2 m}( \eta ) ,
\begin{array}{ll}u \in X \, [4pt] \int_D \Delta u \, \Delta \omega \, dx = \int_D f \, \omega \, dx + \langle g,\nabla \omega \cdot{n}\rangle_{-\frac12}- \langle h,\omega\rangle_{-\frac32}[4pt] \ \omega \in X \, .\end{array}
\times \exp \left\{\displaystyle 2\sum_{\stackrel{i,j=1}{i \neq j}}^{2M}\sqrt{\displaystyle\frac{\alpha^{\prime}}{2}}\xi_{i}\cdot p_{j}\partial_{z_{i}}G_{o}(z_{i},z_{j})+\displaystyle\sum_{\stackrel{i,j=1}{i \neq j}}^{2M}\xi_{i}\cdot \xi_{j}\partial_{z_{i}}\partial_{z_{j}}G_{o}(z_{i},z_{j})\right\}
T_{0i}= T_{1i}= T_{2i}= 0, T_{3i}= \frac{-\frac{1}{2}\rho t_i^2 (1 + \rho t_i)}{(1 + \frac{\theta}{2}t_i)^2}, T_{4i}= \frac{\frac{1}{2}\rho t_i^2}{(1 + \frac{\theta}{2}t_i)^2}, T_{5i}= \frac{\frac{1}{2}\rho t_i^{3/2}(1-\frac{1-\theta}{2}t_i)}{(1 + \frac{\theta}{2}t_i)^2}.
\log \frak{G}_{E}(\omega; \mathbb{A}) = \log \left[ \exp\left(\sum_{i=1}^{n}w_{i}\log A_{i}\right) \right] & =\sum_{i=1}^{n}w_{i}\log A_{i}& \leq \log \left(\sum_{i=1}^{n}w_{i}A_{i}\right) =\log \frak{A}(\omega; \mathbb{A}).
\left| \langle a\cos\theta, \sum_{k\neq l}a_{k}\cos\theta_{k}+r(t)\rangle \right| < \delta_2 \|a\cos\theta\|_{L^2}\|a_l\cos\theta_l\|_{L^2}\leq \delta_2 \|a\cos\theta- a_l\cos\theta_l\|_{L^2}\cdot \|a_l\cos\theta_l\|_{L^2}+\delta_2 \|a_l\cos\theta_l\|_{L^2}^2 ,
\tilde{\Delta}^{\alpha \beta}= \frac{p^{- \alpha}{}_{\delta}( \tilde{\Gamma}^{n}A_{n})^{\delta \sigma}p^{- \beta}{}_{\sigma}}{2 i \big ( \sqrt{( \pi A )^{2}- \pi^{2}A^{2}}+ ( \pi A ) \big )}, \qquad \tilde{\Delta}\Delta = p^{-}
\frac{d}{dv}(v\omega(v-1)) =\omega(v-2)+\omega'(v-1)\ge\frac{1}{2}-\rho(v-1)\ge\frac{1}{2}-\rho(2)=\log2-\frac{1}{2}, \frac{d}{dv}(v\omega(v-1)) \le 1+\rho(v-1)\le1+\rho(2)=2-\log2,
\int_u^v \frac{x^n(qx/u,qx/v;q)_{\infty}}{(dx,cx;q)_{\infty}}d_qx=\frac{(1-q)v (q,u/v,qv/u,dcuv;q)_{\infty}}{(du,dv,cu,cv;q)_{\infty}}W_n(d,c,u,v).
g_{\alpha \beta}\biggl |_{r \to \infty}= - \nabla_{\alpha}u \nabla_{\beta}u - ( \nabla_{\alpha}u \nabla_{\beta}r + \nabla_{\alpha}r \nabla_{\beta}u ) + \frac{1}{2}( m_{\alpha}m_{\beta}^{*}+ m_{\alpha}^{*}m_{\beta})
\int_0^\infty \left(e^{u\xi}- 1\right)e^{-\xi}\xi^{-\alpha}d\xi = \frac{1}{\alpha - 1}\int_0^\infty \left(w e^{w\xi}+ e^{-\xi}\right)\xi^{1-\alpha}d\xi = &= \frac{1}{\alpha-1}\int_0^\infty e^{-r}r^{1-\alpha}dr \cdot \left(1 - (-w)^{\alpha - 1}\right) = &= \frac{\Gamma(2 - \alpha)}{\alpha-1}\left(1 - (1-u)^{\alpha-1}\right).
|\mu(X_{t_{i-1}}+W_t-W_{t_{i-1}})-\mu(X_t)| & \qquad\leq c\, \bigl(|X_{t_{i-1}}+W_t-W_{t_{i-1}}-X_t|+\sum_{j=1}^k 1_{\{(X_{t_{i-1}}+W_t-W_{t_{i-1}}-\xi_j)\, (X_t-\xi_j)\leq 0\}}\bigr).
\delta_{\bf v}{\cal L}_{0}= \mu \dot{\bf x}\cdot \delta_{\bf v}\dot{\bf x}+ \frac{\mu}{R^{2}}\,{\bf x}\cdot \delta_{\bf v}{\bf x}= \mu{\bf v}\cdot \frac{d}{d t}\left[{\bf x}\, \cosh \left( \frac{t}{R}\right) \right] ,
S [ H \ , B ] = \int d^{4}x \left[{\frac{1}{1 2}}H_{\lambda \mu \nu}H^{\lambda \mu \nu}-{\frac{1}{6}}H^{\lambda \mu \nu}\partial_{[ \lambda}B_{\mu \nu ]}+{\frac{m^{2}}{4}}B_{\mu \nu}B^{\mu \nu}\right]
\mathrm{\Lambda_{\m u}( p^{\prime}, p ) = \frac{- \a l p h a_{s}}{3 \p i}\g a m m a_{\m u}\Gamma ( 2 - \o m e g a - \s i g m a ) \left|^{\a l p h a \to ( 1 / 2 )^{+}}_{\o m e g a \to ( 3 / 2 )^{+}}\right. ~ .}
\setlength{\abovedisplayskip}{5 pt}\setlength{\belowdisplayskip}{5 pt}{\boldsymbol{W}_j^{out}}=\left(\mathbb{E}{\left[{\boldsymbol{v}_{t,j}{\boldsymbol{v}_{t,j}^{\rm T}}}\right]^{- 1}}\mathbb{E}\left[{\boldsymbol{v}_{t,j}m_{t - k}}\right]\right)^{\rm T},
\tilde Y_t^{b}= \ 0-\int_{t}^{T}g^b(t,\xi_s^a,\gamma_s)-D_s+m^b(t,\tilde Y_s^b +\xi_s^a,\xi_s^a)\tilde Y_s^b+\eta^b(t,Z_s^b+\gamma_s,\gamma_s)\cdot \tilde Z_s^bds &-\int_t^T\lambda_s\frac{1-e^{-\alpha \tilde Y_s^{b}}}{\alpha}ds-\int_{t}^{T}\tilde Z_s^{b}\cdot dW_s,
\left|f\left(z\right)\right|\leq S_{f}^{\left(k\right)}=\exp\left[2Ku\left(r^{'}\right)-\frac{2}{K}u\left(r\right)\right]\prod_{n=0}^{\infty}\left[\frac{1+\varphi_{\frac{1}{K}}\left(r_{n}^{'}\right)}{1+\varphi_{K}\left(r_{n}\right)}\right]^{2^{1-n}}
\mathcal{A}u = \sum_{n = 1}^{\infty}\lambda_{n}\langle u, \phi_{n}\rangle_{X}\phi_{n}, \tilde{\mathcal{A}}u = \sum_{n = 1}^{\infty}(\alpha \lambda_{n}+ \beta) \langle u, \phi_{n}\rangle_{X}\phi_{n}u \in D(\mathcal{A}) = D(\tilde{\mathcal{A}})
I = \frac{\bar{\beta}}{\sqrt{\pi}}e^{- \bar{m}^{2}\pi}\sum_{k = 1}^{\infty}\frac{\left( \bar{t}_{f}\right)^{2 k - 1}\left( - 1 \right)^{k + 1}}{\left( 2 k - 1 \right) \left( k - 1 \right) !}\int_{0}^{\infty}\frac{d u u^{( 2 k - 5 ) / 2}}{u + \pi}e^{- u \bar{m}^{2}- \frac{\bar{\beta}^{2}}{4 u}}
F(r)=\frac{q}{4\pi\epsilon_{0}}\sum_{i=1}^{N}q_{i}\frac{r-r_{i}}{|r-r_{i}|^{3}}=\frac{q}{4\pi\epsilon_{0}}\sum_{i=1}^{N}q_{i}\frac{\hat{R}_{i}}{|R_{i}|^{2}},
\begin{array}[c]{l}w_{2}\left({\gamma,\alpha,\zeta}\right) -\left({-1}\right)^{m+n}w_{4}\left({\gamma,\alpha,\zeta}\right) =2w_{2}\left({\gamma,\alpha,\zeta}\right) +{O}\left({\gamma^{-2/3}}\right) \left\{{w_{1}\left({\gamma,\alpha,\zeta}\right) +w_{2}\left({\gamma,\alpha,\zeta}\right)}\right\}.\end{array}
\varphi_{m+j}(\mathbf{f})=\frac{l_{j}^{y}}{l_{1}^{y}}\frac{\sum\limits_{i=1}^m \sum\limits_{k=1}^p{e_{i1k}x_{i}z_{k}}}{\sum\limits_{i=1}^m \sum\limits_{k=1}^p{e_{ijk}x_{i}z_{k}}},\thinspace j=1,\thinspace \mathellipsis ,\thinspace n,
\mathbb{E}[x^V;D^{(1)}_2]=\frac{x}{3}\sum_{k=0}^{\infty}\mathbb{P}(V_b=k)\Big{(}\frac{1}{2}\Big{)}^k x^k=\frac{x}{3}\mathcal{L}[f]\Big{(}\frac{x}{2}\Big{)}
\begin{cases}\varphi_x(L,t)=\psi_x(L,t)=0, \,\, (0,T), a\varphi_{xx}(0,\cdot)+\frac{1}{c}\psi_{xx}(0,\cdot)=0, \,\, (0,T), a\varphi_{xx}(L,\cdot)+\frac{1}{c}\psi_{xx}(L,\cdot)=0, \,\, (0,T),\end{cases}
\begin{cases}u_t=\Delta\phi(u) \textrm{in}\Omega\times \mathbb{R}^+ \, , \frac{\partial \phi(u)}{\partial n}=0 \textrm{on}\partial\Omega\times \mathbb{R}^+ \, , u(0)=u_0 \textrm{in}\Omega \, ,\end{cases}
y^d=\nu(2\pi^2t(t-1)^2(t-2)^2+pi^2t^2(2t-2)(t-2)^2+\pi^2t^2(2t-4)(t-1)^2+\pi^4t^2(t-1)^2(t-2)^2-2t^2(t-1)^2 &-2t^2(t-2)^2-2(t-1)^2(t-2)^2-2t^2(2t-2)(2t-4)-4t(2t-2)(t-2)^2-4t(2t-4)(t-1)^2- &2\pi^2t(t-1)^2(t-2)^2-\pi^2t^2(2t-2)(t-2)^2-\pi^2t^2(2t-4)(t-1)^2)sin(\pi x)+t^2(1-t)^2(2-t)^2sin(\pi x).
V\frac{\partial S}{\partial V}=-\sum_{i}\frac{V}{\omega_{i}}\frac{\partial\omega_{i}}{\partial V}k_{B}(\frac{\hbar\omega_{i}}{k_{B}T})^{2}\frac{exp(\frac{\hbar\omega_{i}}{k_{B}T})}{[exp(\frac{\hbar\omega_{i}}{k_{B}T})-1]^{2}}=\sum_{i}\gamma_{i}c_{V,i}
\mathcal{F}(\mathcal{L};q,t)=\sum_{l=1}^L\frac{(-1)^{l-1}}{l}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^l\mathcal{H}(\mathcal{L}_{\mathbf{I}_s};q,t),
(U_1(\lambda)^{-1}\oplus I_{m_1})\begin{bmatrix}0 I_n -X(\lambda)\end{bmatrix}=\begin{bmatrix}\widehat{N}_1(\lambda)^T N_1(\lambda)^T 0 0 0 I_{m_1}\end{bmatrix}\begin{bmatrix}0 I_n -X(\lambda)\end{bmatrix}=\begin{bmatrix}N_1(\lambda)^T -X(\lambda)\end{bmatrix}.
\frac{\partial cb_{3}(c)}{\partial c}|_{c=0}=\frac{b_{0}}{(1-3)_{3-1}}\frac{(\alpha)_{3}(\beta)_{3}}{0!3!}[\frac{1}{1}+\frac{1}{2}+\frac{1}{\alpha}+\frac{1}{\alpha+1}+\frac{1}{\alpha+2}
{\bf E}\bigl[M_{(\tau=1,\lambda_1,\lambda_2)}^q\bigr] = \frac{G(2-q+\lambda_1)}{G(2+\lambda_1)}\frac{G(2-q+\lambda_2)}{G(2+\lambda_2)}\frac{G(1)}{G(-q+1)}\frac{G(2+\lambda_1+\lambda_2)}{G(2-q+\lambda_1+\lambda_2)}.
+ \frac{1}{3 !}g_{0}\left[ \left( \frac{\delta W}{\delta j ( x )}\right)^{3}+ 3 \frac{\delta W}{\delta j ( x )}\frac{\delta^{2}W}{\delta j ( x )^{2}}+ \frac{\delta^{3}W}{\delta j ( x )^{3}}\right] - j ( x ) = 0 .
F_{3}^{\mathrm{e q u i v}}\simeq \vert B \vert^{2}7 A \, , \qquad A = \frac{2 \alpha^{2}}{4 5}\frac{( \hbar / m c )^{3}}{m c^{2}}= 0 . 2 6 5 \frac{\mathrm{f m}^{3}}{\mathrm{M e V}}\, .
\Gamma^{+}( \partial_{+}- \frac{1}{2}A \partial_{-}) \psi_{I}+ \Gamma^{-}\partial_{-}\rho_{I}+ \Gamma^{J}\partial_{J}( \psi_{I}+ \rho_{I}) = J_{I}- \frac{1}{8}\Gamma_{I}\Gamma^{J}J_{J},
\mathcal{I}\left(a, m, 0, \frac{\pi}{4}\right) = \frac{\sqrt{\pi}\Gamma\left(m + \frac{1}{2}\right) \,_{2}F_{1}\left( \frac{1}{2}, m, m+1, \frac{1}{1+a}\right)}{2(1+a)^{m}\Gamma(m+1)}- \frac{F_{1}\left(\frac{1}{2}, \frac{1}{2}- m, m, \frac{3}{2}, \frac{1}{2}, \frac{1}{2(1+a)}\right)}{\sqrt{2}(1+a)^{m}}.
\begin{array}{l}\mu^{N,1}_{\boldsymbol{\alpha}^*_t}+h_1^N(\mu^{N,1}_{\boldsymbol{\alpha}^*_t},\ldots,\mu^{N,N}_{\boldsymbol{\alpha}^*_t})=\Delta^{N,1}_{*,t}, \quad\vdots \qquad\qquad~~\vdots\qquad~~~~\vdots \mu^{N,N}_{\boldsymbol{\alpha}^*_t}+h_N^N(\mu^{N,1}_{\boldsymbol{\alpha}^*_t},\ldots,\mu^{N,N}_{\boldsymbol{\alpha}^*_t})=\Delta^{N,N}_{*,t},\end{array}
\lim_n\rho_n =\lim_{n\to\infty}\sum_{i,j=1}^n\sqrt{\lambda_i\lambda_j}E_{ij}\otimes E_{ij}=\lim_{n,m\to\infty}\sum_{i=1}^n\sum_{j=1}^m\sqrt{\lambda_i\lambda_j}E_{ij}\otimes E_{ij}=\sum_{i,j}\sqrt{\lambda_i\lambda_j}E_{ij}\otimes E_{ij}
D_{U_\gamma}= \left( \mathbf{I}-{U_\gamma}^*{U_\gamma}\right)^{1/2}= \left( 1 -|\gamma|^2\right)^{1/2}b_1b_1^*, D_{{U_\gamma}^*}= \left( \mathbf{I}-{U_\gamma}{U_\gamma}^*\right)^{1/2}= \left( 1 -|\gamma|^2\right)^{1/2}b b^*
\int_{X}\Psi\bigg(\frac{\chi_S}{\Psi^{-1}(\mu(S)^{-1})^{-1}}\bigg)d\mu=\int_S\Psi\bigg(\frac{1}{\Psi^{-1}(\mu(S)^{-1})^{-1}}\bigg)d\mu=\int_S\frac{1}{\mu(S)}d\mu=1.
\begin{aligned}\lambda_n^{(1)}= b(\psi_n,\psi_n), \lambda_n^{(2)}= \sum_{j=1,j\neq n}^{\infty}\frac{|b(\psi_n,\psi_j)|^2}{\mu_n-\mu_j}, \phi_n^{(1)}= \sum_{j=1,j\neq n}^{\infty}\frac{b(\psi_n,\psi_j)}{\mu_n-\mu_j}\psi_j.\end{aligned}
\delta S=\delta\int_{A}^{B}nds=\delta\int_{x_{3A}}^{x_{3B}}n\frac{ds}{dx_{3}}dx_{3}=\delta\int_{x_{3A}}^{x_{3B}}L(x_{1},x_{2},\dot{x}_{1},\dot{x}_{2},x_{3})dx_{3}=0
L_{\lambda_1}(\mathbf{\rho},\mathbf{S},\mathbf{u}) = \left( ||\rho||_*+ I_{\mathcal{C}}(\rho) \right) + ||\mathbf{S}||_1 &+ \frac{\lambda_1}{2}||\mathbf{A}(\mathbf{\rho}) + \mathbf{A}(\mathbf{S})- \mathbf{y}+ \mathbf{u}||_2^2,
\bar{\cal X}^{\mu}_{,i}\bar{\cal X}^{\nu}_{,j}{\cal G}_{\mu\nu}=\bar{g}_{ij},\;\;\bar{\cal X}^{\mu}_{,i}\bar\eta^{\nu}{\cal G}_{\mu\nu}=0,\;\;\bar{\eta}^{\mu}\bar{\eta}^{\nu}{\cal G}_{\mu\nu}= \varepsilon
C_1(x)==1-\int \d y g(y)h(y)\frac{\mathcal{P}}{x-y}C_2(x)=g(x)h(x) & A(x)=\frac{\mathcal{P}}{x-\Omega}| g \rangle A'(x)= \langle h |\frac{\mathcal{P}}{x-\Omega}& B(x) = g(x) | \delta_x \rangle B'(x)=h(x) \langle \delta_x |
\cdots \stackrel{a(l-1)^\dagger}{\longrightarrow}{\cal E}(l-1) \stackrel{a(l)^\dagger}{\longrightarrow}{\cal E}(l) \stackrel{a(l+1)^\dagger}{\longrightarrow}{\cal E}(l+1) \stackrel{a(l+2)^\dagger}{\longrightarrow}\cdots
\Omega^{\mathcal{L}_1}_{n-1,w}\cong\bigoplus_{r+s=n-1}H_r(W;{\mathbb Z}_2)\otimes_{{\mathbb Z}_2}\Omega_s\cong\Omega^{\mathcal{L}_1}_{n-1}/K^{\mathcal{L}_1}_{n-1,w}\cong\Omega^{\mathcal{L}_1}_{n-1,s}/K^{\mathcal{L}_1}_{n-1,s,w}.
\prod\limits_{k=0}^L \eta_{M-1,N}(q\,|\,\hat{a}_i, b_0+ka_i) =\exp\Bigl(\int\limits_0^\infty \frac{dt}{t}(e^{-qt}-1)\frac{e^{-b_0t}\prod_{j=1}^N (1-e^{-b_j t})}{\prod_{j\neq i}^M (1-e^{-a_j t})}\frac{1-e^{-a_i t(L+1)}}{1-e^{-a_i t}}\Bigr).
\sum_{n=1}^{N}\left[\begin{array}{c}N n\end{array}\right] \frac{(q)_n\left(\frac{b}{a}\right)_n(a)_{N-n}a^n}{(b)_n(1-q^n)(a)_{N}}=\displaystyle\sum_{m=1}^{\infty}\frac{(a^m-b^m)(1-q^{mN})}{1-q^m}.
f_2(m,a,b,c,d) = (a+b-c-d-1)(a+b-c-d-2) \quad\cdot [c^3(a+b-c-d)+(a-c)(3c^2+3c+1)(b-m-c)] + (a-c)(a-c-1)(b-m-c)(b-m-c-1) \quad\cdot [3(a+b-c-d-2)(c+1)+(a-c-2)(b-m-c-2)]
\underline{\underline R}_{a 0}=- \frac 1{\ell^2}\underline{\Theta}_{a}\wedge \underline{\Theta}_{0}-{1\over r}\left(d\underline{\gamma}_{a}+ \underline{\omega}_{a b}\wedge \underline{\gamma}^{b}\right) + O(r^{-3}) \qquad ,
y_q(u) =\sum_{\mathcal{R}:|\mathcal{R}|=r+1}\sum_{\mathcal{T}:|\mathcal{T}|=t}\sum_{i=1}^t\left[\sum_{p\in \mathcal{T}}h_{qp}(u)\left(\mathbf{v}_{{\mathcal{R}},{\mathcal{T}},p}^i(u)\right)^T\right]\mathbf{x}_{{\mathcal{R}},{\mathcal{T}}}^i,
\left\| \, \left|\sum_{i=1}^n B^*_iA_i\right|^{\alpha}\, \right\|^2 \le \left\| \,\left|\sum_{i=1}^n B^*_iB_i\right|^{\alpha}\, \right\| \left\| \, \left|\sum_{i=1}^n A^*_iA_i\right|^{\alpha}\, \right\|
|au^2-bv^2| =av^2\left|\left(\frac{u}{v}+\sqrt{\frac{b}{a}}\right)\left(\frac{u}{v}-\sqrt{\frac{b}{a}}\right)\right| \leqslant av^2\left(2\sqrt{\frac{b}{a}}+\frac{\varepsilon}{B^2}\right)\frac{\varepsilon}{B^2}\leqslant 2\varepsilon \sqrt{ab}+\frac{a\varepsilon^2}{B^2}=2\varepsilon\sqrt{ab}+o(1).
\Lambda (df_1, \ldots, df_{n}) = \{f_1, \ldots, f_n\}_{\Pi}= \Pi (d_A f_1, \ldots, d_A f_n) & = \Pi (\rho^*df_1, \ldots, \rho^*df_n)= (\wedge^n\rho)(\Pi)(df_1, \ldots ,df_n).
U_{k_{1}\, \, k_{2}}^{\, \, l_{1}\, \, l_{2}}\left( \begin{array}{c c c}{{\lambda}}&{{\mu}}&{{\rho}}\\{{k_{1}}}&{{k_{2}}}&{{m}}\\ \end{array}\right) \left( \begin{array}{c c c}{{\lambda}}&{{\mu}}&{{\rho^{\prime}}}\\{{l_{1}}}&{{l_{2}}}&{{m^{\prime}}}\\ \end{array}\right)^{\ast}
\begin{array}{ll}d(|z(\cdot,t)|^2_{L^2(0,1)})=(\sigma^2-2c)|z(\cdot,t)|^2_{L^2(0,1)}dt-2|z_x(\cdot,t)|^2_{L^2(0,1)}dt\cr \ \ \ +2z(1,t)\widetilde{w}(t)dt+2\sigma|z(\cdot,t)|_{L^2(0,1)}^2dB(t).\end{array}
\begin{array}[c]{c}i\int_{M}(A_{\alpha\beta}f^{\alpha\beta}-A_{\overline{\alpha}\overline{\beta}}f^{\overline{\alpha}\overline{\beta}})=i\int_{M}(A_{\alpha\beta},^{\alpha\beta}-A_{\overline{\alpha}\overline{\beta}},^{\overline{\alpha}\overline{\beta}})f\leq2||\operatorname{Im}(A_{\alpha\beta},^{\alpha\beta})||_{L^{2}}\left\Vert f\right\Vert_{L^{2}}.\end{array}
B_{[ i j}{B^{\prime}}_{k l ]}= 0 \hspace{0 . 1 5 i n}\Leftrightarrow \hspace{0 . 1 5 i n}B_{i j}= u_{[ i}v_{j ]}\hspace{0 . 1 i n}\mathrm{a n d}\hspace{0 . 1 i n}{B^{\prime}}_{i j}= v_{[ i}w_{j ]}.
h_t(r,z)=\sum_{n={-\infty}}^{+\infty}\sum_{k=0}^{+\infty}(2k+|n|+1) e^{-(4k(k+|n|+1)+2|n|)t}e^{inz}(\cos r)^{|n|}P_k^{0,|n|}(\cos 2r).
\frac{1}{8}\mathrm{T r}^{2}\left[ O + \left( \partial \! \! \! / B \! \! \! \! / \right) \right] - \frac{1}{2}\mathrm{T r}\left[ O + \left( \partial \! \! \! / B \! \! \! \! / \right) \right]^{2}= ( \partial_{\lbrack \mu}B_{\nu ]}) ( \partial^{\lbrack \mu}B^{\nu ]})
\delta{\cal E}_{2}^{1 , 0}= \frac{8 K}{a_{0}^{6}}\left\{\left[ \left( \frac{a_{0}}{\Lambda}\right)^{2}- 2 \left( \frac{a_{0}}{\Lambda}\right) \right] e^{- 2 \Lambda / a_{0}}+ 4 E_{1}\left( \frac{2 \Lambda}{a_{0}}\right) \right\}\,{.}
\tilde{C_{2}}(n, K, R, \lambda_{1}, \lambda_{2}, b, p)= \max\left\{\begin{aligned}\frac{nKp}{2(p-1)};\min_{C_3\in(0,\frac{2(2-p)}{np})}\left\{\frac{((A+2K)R^{2}C_{3}+C_{1}^{2})np}{(2(2-p)-C_{3}np)R^{2}C_{3}}\right\}\end{aligned}\right\}
\sum_{0\le j\le k-1}C^j_{k-1}\|\tau^j\partial^j_\tau f\|_{L^\infty([0,t];L^2)}\|\tau^{k-1-j}\partial^{k-1-j}_\tau f\|_{L^2([0,t];L^2_A)}\le\sum_{0\le j\le k-1}C^j_{k-1}B^{j-1}(j-2)!B^{k-2-j}(k-3-j)! \le B^{k-3}(k-3!)\left(\sum_{2\le j\le k-3}\frac{k(k-1)}{j(j-1)(k-1-j)(k-2-j)}+6\right).
Z(G_{3,2}\vee C^{(2)}\vee G_{3,2}) = \begin{cases}Z(C(8)) =8 (C^{(2)}=C(\ddot6)) Z(C(6,2)) =13 (C^{(2)}= C(\dot5,\dot1)) Z(C(5,3)) =16 (C^{(2)}=C(5,\ddot 1)) Z(C(2,4,2)) =20 (C^{(2)}=C(\dot1,4,\dot1)) . \end{cases}
\int_0^1 dt\int_{L_t}h_t (e^{-i\hat{\theta}}\Omega)= \int_{L_t}f_{L_t}(e^{-i\hat{\theta}}\Omega) |^{t=1}_{t=0}- \int_{\cup_t L_t}\lambda\wedge (e^{-i\hat{\theta}}\Omega).
\varepsilon_{u}= \eta_{1}\gamma^{1 3}\varepsilon_{l}, \, v a r e p s i l o n_{u}= \eta_{2}\gamma^{2 4}\varepsilon_{l}, \, v a r e p s i l o n_{u}= \eta_{5}\gamma^{5 6}\varepsilon_{l}, \, v a r e p s i l o n_{u}= i \eta_{e}\eta_{q}\varepsilon_{l},
p\alpha = (a_0a_1^{n_1}a_2^{n_2}\cdots a_m^{n_m})(pa_{m+1}^{n_{m+1}}) \in A p\beta = (pa_1^{n_1}a_2^{n_2}\cdots a_m^{n_m})(pa_{m+1}^{n_{m+1}}) \in A
\partial_{R,\Omega}:=\partial_{N,\Omega}+\tau \gamma_{0,\Omega}:H^s_{}(\Omega)\to H^{s-3/2}(\partial \Omega)\quad \partial_{R,K}:=\partial_{N,K}+\tau \gamma_{0,K}:H^s(K)\to H^{s-3/2}(\partial \Omega).
[h_\chi(z), H^{\gamma}(w^2)] = [h_\chi(z), V^+(w)\gamma (w^2)w^{2h_0}V^-(w)] & =\Big(-\sum_{n>0}\frac{z^{2n-2}}{w^{2n}}+i_{z, w}\frac{1}{z^2-w^2}+i_{w, z}\frac{1}{w^2-z^2}-\sum_{n>0}\frac{w^{2n}}{z^{2n+2}}\Big)H^{\gamma}(w^2)=\frac{1}{z^2}H^{\gamma}(w^2).
S_{T M}^{4}= \int d^{4}x \Bigl [ -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}-{\frac{1}{1 2 \mu^{2}}}H_{\mu \nu \lambda}H^{\mu \nu \lambda}-{\frac{1}{4}}\varepsilon^{\mu \nu \lambda \rho}B_{\mu \nu}F_{\lambda \rho}\Bigl ] ,
\int_{0}^{1}\left(x^{\alpha+\beta}u_{x}v_{x}-\frac{\mu}{x^{2-\alpha-\beta}}u v\right) \mathrm{d}x=-\int_{0}^{1}x^{\beta}\left(\left(x^{\alpha}u_{x}\right)_{x}+\beta x^{\alpha-1}u_{x}+\frac{\mu}{{x^{2-\alpha}}}u\right) v \mathrm{d}x
T_q(t)P_q^s - T_{\bar{q}}(t)P_{\bar{q}}^s = (T_q(t-n) - T_{\bar{q}}(t-n)) T_q(n) P_q^s+ T_{\bar{q}}(t-n) T_q(n) P_q^s(P_q^s-\! P_{\bar{q}}^s) + T_{\bar{q}}(t-n) \sum_{k=0}^{n-1}T_q(n-k-1) P_q^s (T_q(1) - T_{\bar{q}}(1)) T_{\bar{q}}(k)P_{\bar{q}}^s + T_{\bar{q}}(t-n) (P_q^s- P_{\bar{q}}^s) T_{\bar{q}}(n)P_{\bar{q}}^s.
d_{\Omega_n}(p,q) = d_{\Omega_n}(\varphi_{\delta}(0), \varphi_{\delta}(\zeta_0/\delta)) \leq d_{\Delta}(0,\zeta_0/\delta) \leq (1+\epsilon)d_{\Delta}(0,\zeta_0) = (1+\epsilon)d_{\Omega}(p,q).
K ( w \prime_{n}, w \prime_{m}^{\ast}; w \prime_{r}, w \prime_{s}^{\ast}; t ) = \sum_{\kappa > 0}f_{\kappa}( w \prime_{n}, w \prime_{m}^{\ast}) f_{\kappa}( w \prime_{r}, w \prime_{s}^{\ast}) e^{i E_{\kappa}t}
( P_{1 2})_{a b}= \langle V_{a}^{+ +}( 1 ) \vert V_{b}^{+ +}( 2 ) \rangle = ( D_{1}^{+})^{4}\left({\frac{\delta^{8}( \theta_{1}- \theta_{2})}{x_{1 2}^{2}}}\right) \delta^{( - 2 , 2 )}( u_{1}, u_{2}) \delta_{a b}\ .
\Pi_{i k}^{+}\left( L_{n}^{1 / 2}( x ) \right)_{k l}{\cal L}_{n}( x ) \Pi_{l j}^{+}= ( x + i / 2 ) \rho_{i}{\cal L}_{n}( x - i ) \rho_{j}^{+}( \rho^{+}\rho + 1 )^{- 1},
c_{4a}=8\nu^4+40\nu^3-1395\nu^2+4354\nu+4757>0 c_{4b}=4(1-\nu) (4\nu^5+21\nu^4-712\nu^3+2011\nu^2+3102\nu+3050) +(1-\mu)(2\nu^6+11\nu^5-360\nu^4+912\nu^3+1705\nu^2+3655\nu+543)\ge 0.
\sum_{l=2}^{k}x(l-1)\cdot d(d-1)^{l-1}= xd\sum_{l=2}^{k}\left(l(d-1)^{l-1}-(d-1)^{l-1}\right) &= xd \left(f_{d-1}(k)-1-\frac{(d-1)^k-(d-1)}{d-2}\right) &\leq xd\left(\frac{k(d-1)^{k}}{(d-2)}-1-\frac{(d-1)^k-(d-1)}{d-2}\right) &\leq \frac{xd}{d-2}\left((k-1)(d-1)^k+1\right) &\leq \left(\frac{d+1}{d-2}\right)\cdot x(k-1)(d-1)^k.
+\frac{b_0^{(3)}}{L_0^{(3)}}Q^{(3)}|\Lambda^e_{a_1 a_2}\rangle_3 +\frac{b_0^{(3)}}{L_0^{(3)}}[\delta \varepsilon_1 |\Omega_{1 a_1 a_2}\rangle +\delta \varepsilon_2 |\Omega_{2 a_1 a_2}\rangle ]
\langle ( \Delta \alpha )^{2}\rangle^{1 / 2}\simeq 1 . 6^{0}\left( \frac{B ( t_{d e c})}{B_{c}}\right) \left( \frac{\omega_{M}}{\omega}\right)^{2}, B_{c}= 1 0^{- 3}~ G a u s s , \omega_{M}\simeq 3 \times 1 0^{1 0}~ H z
\sum^n_{j=0}(-1)^j\mathrm{dim\,}H^j_{b,m}(X)=\int_X\Bigr(\mathrm{Tr\,}e^{-t\Box^+_{b,m}}(x,x)-\mathrm{Tr\,}e^{-t\Box^-_{b,m}}(x,x)\Bigr)dv_X.
V_\delta(E_\infty) = \begin{cases}\displaystyle \log\frac{2}{r}+ \frac{2}{\pi}\int_1^r \frac{\log{x}\,dx}{\sqrt{r^2-x^2}}+ 2 \int_1^r \log{x}\,d\mu_{E_\infty}(x) r\geq 1, r/2 \end{cases}
\Phi_f(\alpha^\ast \mathcal{L}) = \frac{1}{2}f^{-1}\alpha^\vee \phi_\mathcal{L}\alpha &= f^{-1}\alpha^\vee f \left(\frac{1}{2}f^{-1}\phi_{\mathcal{L}}\right) \alpha &= \alpha' \Phi_f(\mathcal{L}) \alpha.
\begin{array}{c}\left\{L_1^n,L_2^n\right\}=\left( a-\alpha \right)L_1^nL_2^n-L_1^nL_2^n\left( a+\alpha \right) , \left\{L_1^n,L_2^{n+1}\right\}=L_1^n\left( \alpha -s\right) L_2^{n+1}, \left\{L_1^n,L_2^m\right\}=0,\left| n-m\right| \geq 2.\end{array}
U(t,t') = \prod_{n=1}^{N}\exp \left( -\frac{i}{\hbar}\int_{t'}^{t}V(\widehat{x}+ ct''- x_n) dt'' \left[ \sigma_{+}^{(n)}\exp \left( -i \frac{\omega}{c}\widehat{x}\right) + \mbox{h.c.}\right] \right) .
\max\left\{\prod_{i=0}^{n-1}\sup_{t\in D^+}\left[\frac{m_i(pt)}{m_i(t)^p}\right],\prod_{i=0}^{n-1}\sup_{t\in D^-}\left[\frac{m_i(pt)}{m_i(t)^p}\right]\right\}<a_D^{-n}a.s.
\frac{1}{x}\exp\{- \int_0^T\pi(s,y)\sigma_0(s,y)dB(s)-\int_0^T[\pi(s,y)b_0(s,y)-\frac{1}{2}\pi^2(s,y)\sigma_0^2(s,y)]ds \}&= p(0,y) \exp\{- \int_0^T \{\Phi(s,y) +\frac{b_0(s,y}{\sigma_0(s,y)}\}dB(s)+\frac{1}{2}\int_0^T\int_0^t \{\Phi^2(s,y) -\frac{b_0^2(s,y)}{\sigma_0^2(s,y)}\}ds\}
e_R = -d^2e_L-cdh_L+c^2f_L, &h_R = -2bde_L-(ad+bc)h_L+2acf_L, &f_R = b^2e_L +abh_L-a^2f_L, &\omega_e = d \cdot (db)-b \cdot (dd), &\omega_h = \frac{1}{2}(d \cdot (da)+c \cdot (db)-b \cdot (dc)-a \cdot (dd)), &\omega_f =-c \cdot (da) + a \cdot (dc).
\mathcal T_*P^t=\varepsilon\tilde P^t,\mathcal T_*P^x=a\tilde P^x+c\tilde P^y,\mathcal T_*P^y=b\tilde P^x+d\tilde P^y, \mathcal T_*G^x=\varepsilon a\tilde G^x+\varepsilon c\tilde G^y,\mathcal T_*G^y=\varepsilon b\tilde G^x+\varepsilon d\tilde G^y,\mathcal T_*D =\tilde D,\mathcal T_*\Pi=\varepsilon\tilde\Pi,
E\oplus B_r:=\bigcup_{x\in E}B_r(x)=(\partial E\oplus B_r)\cup E=(\partial E\oplus B_r)\cup(E\ominus B_r),{\mbox{where}}E\ominus B_r:=E\setminus \left(\bigcup_{x\in\partial E}B_r(x)\right)=E\setminus\big(\partial E)\oplus B_r\big).
\alpha\lambda_{\lambda_{\alpha\lambda_{x}(y)}^{-1}\alpha\lambda_{\alpha(x)}\lambda_{y}(z)}^{-1}\lambda_{\lambda_{x}(y)}^{-1}=\lambda_{\lambda_{\alpha\lambda_{\alpha(x)}\lambda_{y}(z)}^{-1}\alpha\lambda_{x}(y)}^{-1}\lambda_{\lambda_{\alpha(x)}\lambda_{y}(z)}^{-1}\alpha.
\begin{array}{ll}(\delta_{BiH}^nf)(y ; a_1,\cdots,a_{n+1}) =\varphi^{n-1}(a_1)\bullet_0^yf(d_o(y) ; a_2\cdots,a_{n+1}) &+\sum^n_{i=1}(-1)^if(d_i(y);\alpha(a_1),\cdots, a_i\bullet_i^ya_{i+1}, \psi(a_{i+2}), \cdots, \psi(a_{n+1})) &+(-1)^{n+1}f(d_{n+1}(y) ; a_1,\cdots,a_n)\bullet_{n+1}^y\psi^{n-1}(a_{n+1}),\end{array}
Q^{-}_{\rho}(z_0) :=B_\rho(x_0)\times (t_0-\rho^p,t_0), begin{align*}2mm]Q^{+}_{\rho}(z_0) :=B_\rho(x_0)\times (t_0,t_0+\rho^p), begin{align*}2mm]Q_{\rho}(z_0) :=B_\rho(x_0)\times (t_0-\rho^p,t_0+\rho^p),
\delta_m^j \; \frac{\partial X_{}^\sigma}{\partial p\>\!_i^\mu}\, \epsilon_{\nu \sigma \rho_1 \ldots \rho_{n-2}}^{}~ =~\delta_m^i \; \frac{\partial X_{}^\sigma}{\partial p\>\!_j^\nu}\, \epsilon_{\mu \sigma \rho_1 \ldots \rho_{n-2}}^{}~.
( 1 + 1 ) \mathrm{d i m}\stackrel{T}{\longrightarrow}( 2 + 1 ) \mathrm{d i m}\stackrel{S}{\longrightarrow}( 2 + 1 ) \mathrm{d i m}\stackrel{T}{\longrightarrow}( 1 + 1 ) \mathrm{d i m}
\prod_{a=1}^{\tilde{n}_{c}}(\Lambda \omega^{2\ell-1}+ \delta_{\ell}-\phi_{a})^{2}= (\Lambda \omega^{2\ell-1})^{\tilde{n}_{c}}\left[ 1 + 2 \sum_{a=1}^{\tilde{n}_{c}}\frac{\delta_{\ell}-\phi_{a}}{\Lambda\omega^{2\ell-1}}\right].
M_{(\tau, \lambda_1, \lambda_2)}= \frac{2\pi \tau^{1/\tau}}{\Gamma(1-1/\tau)}\,\beta^{-1}_{22}(\tau, b_0=\tau,\,b_1=1+\tau \lambda_1, \,b_2=1+\tau \lambda_2) \times \times\beta_{1,0}^{-1}(\tau, b_0=\tau(\lambda_1+\lambda_2+1)+1),
\frac{1}{2\left( N\kappa+m+1\right)}\left( T_{1}-T_{2}\right)h_{2m+1}^{-}& =\sum_{j=0}^{m}2^{-j}\frac{g_{j}^{o}\left( N\kappa-\kappa+m\right)}{\left( N\kappa+m+1\right)_{j+1}}\left( N\kappa+m+j+1\right)\phi_{2m,2j}& =\sum_{j=0}^{m}2^{-j}\frac{g_{j}^{o}\left( N\kappa-\kappa+m\right)}{\left( N\kappa+m+1\right)_{j}}\phi_{2m,2j}=h_{2m}^{+}