harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(T_f)^\prime = T_{f^\prime}+ \sum_{r \in E_{p,q}}(f(r+0) - f(r-0)) \delta_r &= 2(p-q) (\{p \cdot \}- \{q \cdot \}) + \sum_{r \in E_{p,q}}(f(r+0) - f(r-0)) \delta_r
\int_{|\xi|\leq R}{|\xi|^{2s}g(\xi)^{k}e^{-2t|\xi|^{2}g(\xi)}\,d\xi}= \int_{|\xi|\leq R}{|\xi|^{2s}\big(2t|\xi|^{2}\big)^{-k}\big(2t|\xi|^{2}g(\xi)\big)^{k}e^{-2t|\xi|^{2}g(\xi)}\,d\xi}\leq Ct^{-k}\int_{|\xi|\leq R}{|\xi|^{2s-2k}\,d\xi}\leq Ct^{-k}\int_{0}^{R}{r^{2s-2k+1}\,dr}\leq Ct^{-k}R^{2(s-k+1)},
\mathbf{a}+\mathbf{b}=(a_1+\beta_1,a_2+b_1,a_3+b_2,a_4+b_3,\cdots,a_{2k-1}+b_{2k-2},\alpha_1+b_{2k-1},\alpha_2+\beta_2,\alpha_3+\beta_3,\alpha_4+\beta_4),
\sum_{h=1}^H (-1)^{H-h}\sum_{i\in \mathcal{C}_h^{\{1, \dots, H\}}}(c_{i_1}+\dots c_{i_h})^m =\begin{cases}\begin{array}{l l}0, \m<H, \displaystyle \sum_{\substack{n_1,\dots,n_H\ge1 n_1+\dots+n_H=m}}c_1^{n_1}\cdots c_H^{n_H}\binom{m}{n_1,\dots,n_H}, \m \ge H.\end{array}\end{cases}
K(\theta )=\frac{1}{\sqrt{\pi}}\prod^{\infty}_{k=1}\frac{\Gamma \left( k-1/2+\theta /2\pi i\right) \Gamma \left( k-\theta /2\pi i\right)}{\Gamma \left( k+1/2-\theta /2\pi i\right) \Gamma \left( k+\theta /2\pi i\right)}
\partial\zeta_{\Delta}'(0)=\frac{d}{dz}\left.\partial\zeta_{\Delta}(z)\right|_{z=0}=-a_0=-\tau\left(\int\!\!\!\!\!\!-\sigma(\partial\Delta\Delta^{-1})-\frac{1}{2}res(\partial\Delta\Delta^{-1}\log\Delta)\right).
{\bf \Phi}= \left[ \begin{array}{c}{{\, \, \varphi^{+}( x ) \,}}\\{{\, \, \varphi^{-}( x ) \,}}\\ \end{array}\right] , \, \, \,{\bf \Phi}^{T}= \Big [ \, \varphi^{+}( x ) \, \, \, \varphi^{-}( x ) \, \Big ] .
0=\langle J^{'}(\widehat{u}),\widehat{u}\rangle+\lambda_0\langle \varphi^{*}_{\widehat{u}},\widehat{u}\rangle =\lambda_0\langle \varphi^{*}_{\widehat{u}},\widehat{u}\rangle,\ \ \ \varphi^*_{\widehat{u}}\in \partial\varphi(\widehat{u})
P_{m,n}^{(\alpha,\beta)}(x)=\frac{(-1)^m}{\alpha+1+n-m}\bigg(\frac{(x-1)(1+\alpha+\beta+n-m)}{2}P_m^{(-\alpha-1,\beta-1)}(x)P_{n-m-1}^{(\alpha+2,\beta)}(x) \hphantom{P_{m,n}^{(\alpha,\beta)}(x)=}{}+(\alpha+1-m)P_m^{(-\alpha-2,\beta)}(x)P_{n-m}^{(\alpha+1,\beta-1)}(x)\bigg),
|\partial_{t}\langle \chi_{\geq 1}(\frac{2r}{\log^{N}(t)}) F_{0,2}\vert_{r=R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{3}\log^{2b+1-2b\alpha +2N}(t)}+ \frac{C |e'''(t)|}{\log^{b-2b\alpha+2N}(t)}
K_n^{(j)}(x,y)=\frac{1}{4\pi^2}\int_{\gamma}\d u\int_{\tilde\gamma}\d v \frac{F_n^{(j)}(u)}{F_n^{(j)}(v)}\frac{x^{-u}y^{v-1}}{u-v},j=1,2,3,
\sum_{\ell=0}^n \ell^h \, \vartheta_{\mathcal L_{\Gamma}}^{(\ell)}(z) =\sum_{\ell=0}^n \ell^h \, \vartheta_{\mathcal L_{\Gamma'}}^{(\ell)}(z)\qquad0\leq h\leq p_0,
\beta_1(y) = \int_0^y \frac{2i \lambda^3\bar{\psi}- iu^{\prime}e^u}{\lambda^3\bar{\psi}-\lambda^{-3}\psi-e^u u^{\prime}}ds, \beta_2(y) = \int_0^y \frac{2e^u}{\lambda^3\bar{\psi}-\lambda^{-3}\psi-e^u u^{\prime}}ds.
\P(Y(t,x)>R) \geq \P(A_N)\P_N\Biggl(Y_<(\tau_N,\eta_N)\prod_{i=1}^{N}u_<(\tau_{i},\eta_{i};\tau_{i-1},\eta_{i-1}) \zeta_i>R\Biggr) &\geq C^NN^{-N}R^{-1-\frac 2d}\log^{N-1}(R) &= \frac{R^{-1-\frac 2d}(C\log R)^{K\exp(W(\frac 1{dK}\log R))}}{\log R(K\exp(W(\frac 1{dK}\log R)))^{K\exp(W(\frac 1{dK}\log R))}}.
\mu_aC^a_{bc}\tilde{\rm v}^c=\mu_aC^a_{bc}\tilde{\rm u}^c, \tilde{E}_a^{\rm V}(R^\mu) = 0, v^i=\dot{x}^i, X_i^{\rm V}(R^\mu)=p_i, \dot{p}_i= X_i^{\rm C}(R^\mu) - \mu_a B^a_{ij}v^j.
(\frac{Y(\partial X,[\hat{g}])}{Y(\mathbb{S}^n,[g_{\mathbb{S}}])})^{\frac{n}{2}}\leq\lim\limits_{t\rightarrow+\infty}\frac{Vol(\partial B^{g^+}_p(t))}{Vol(\partial B^\mathbb{H}_o(t))}=\frac{\mathcal{A}(p)}{\omega_n}.
\frac{d}{dt}J\vert_\mathcal M=\frac{d}{dt}\big(\sum_{j=1}^n\xi_j(q,t)p_j-\tau(t,q)H(t,q,p)\big)\vert_\mathcal M=\sum_{i=1}^n F_i(\xi_i-\dot q_i\tau)\vert_\mathcal M
\ell(\Phi_1(\underline{\delta s}(\mu_{\underline m})),\chi)\cdot \ell(\Phi_2(\underline{\delta s}(\mu_{-\underline m}),\chi^{-1})=C(\underline k,\underline m)\cdot L(1/2,\Pi,\chi)\cdot\frac{\langle \Phi_1,\Phi_2\rangle}{\langle \Psi,\Psi\rangle}\cdot\prod_{v\nmid\infty}\left(\beta_v(\Phi_{1,v},\Phi_{2,v})\cdot\alpha_{v}(W_{\mathfrak{f},v},W^-_{\mathfrak{f},v})\right),
u^{\mu}( p_{3}, 0 ) \mid_{m \rightarrow 0}= \left( \begin{matrix}{p_{3}}\\{0}\\{0}\\{{\frac{p_{3}^{2}}{E_{p}}}}\\ \end{matrix}\right) \equiv \left( \begin{matrix}{E_{p}}\\{0}\\{0}\\{E_{p}}\\ \end{matrix}\right) \quad ,
\int\limits_Y \left(\int\limits_0^T f(g_t y) \, dt \right)^2\,d\mu(y) = \int\limits_Y \int\limits_0^T \int\limits_0^T f(g_t y) f(g_s y) \, dt \, ds \, d\mu(y) = \int\limits_0^T\int\limits_0^T \left(\int_{Y}f(g_t x) f(g_s x) \, d\mu(x)\right) \, dt \, ds .
\left[\mathbf{F}\right]_{u,t,w}=\frac{\exp\left\{i\frac{2\pi}{n}t\,\left(u-w\right)^{2}\right\}}{\sqrt[3]{n}},\;\left[\mathbf{G}\right]_{u,v,t}=\frac{\exp\left\{i\frac{2\pi}{n}t\,\left(u-v\right)^{2}\right\}}{\sqrt[3]{n}},\:\left[\mathbf{H}\right]_{t,v,w}=\frac{\exp\left\{i\frac{2\pi}{n}t\left(v-w\right)^{2}\right\}}{\sqrt[3]{n}}.
F(g)=(1-\varepsilon)\int_{\widetilde{M}\setminus (\Gamma_{1,A_i}\cup \Gamma_{2,A_i})}\tilde{f}d\mu+\int_{\Gamma_{1,A_i}\cup \Gamma_{2,A_i}}\tilde{g}d\mu>(1-\varepsilon)(1-3\gamma)-2\gamma>1-\varepsilon-5\gamma
\int_{\mathfrak{m}}\big|S(\alpha)\big|d\alpha \le \int_{\mathfrak{m}}\int_{[0,1]^3}\Big|\mathcal{F}(\alpha,\boldsymbol{\beta})\mathcal{H}_1(\alpha,\boldsymbol{\beta})\mathcal{H}_2(\alpha,\boldsymbol{\beta}) \prod_{j=1}^{n-5}f_j(\alpha,\boldsymbol{\beta})\Big|d\boldsymbol{\beta}d\alpha.
F(x)={v_0 x^{k_0}\over{1+u_1(x)x-\displaystyle{\strut v_1 x^{k_0+k_1+\delta}\over{1+u_2(x)x-\displaystyle{\strut v_2 x^{k_1+k_2+\delta}\over{1+u_3(x)x-\displaystyle{\ddots}}}}}}}
|D(s)- E(s)| = s^{2- n}\int_{b\leq s}|\nabla u|^2 (1- |\nabla b|^2) &\leq s^{2- n}\sup_{b(x)\leq s}|\nabla u|^2(x)\cdot \delta \mathrm{Vol}(b\leq s) &\leq C(\delta, n) s^2 \sup_{b(x)\leq s}|\nabla u|^2(x)
v_q\left( \frac{(4k)!(4n-4k)!}{(2k)!(2n-2k)!k!^2(n-k)!^2}\right) = \lfloor 4y \rfloor + \lfloor 4x-4y \rfloor - \lfloor 2y \rfloor - \lfloor 2x-2y \rfloor - 2\lfloor y \rfloor -2\lfloor x-y \rfloor.
F =V_{p+1}\, (8\pi^2\alpha')^{-\frac{p+1}{2}}\int_0^\infty \frac{d\tau}{\tau}\tau^{-\frac{p+1}{2}}~e^{-\frac{y^2 \tau}{2\pi\alpha'}}~\left(f_1(e^{-\pi \tau})\right)^{-24}=
P \Big( R_t \leq \frac{\beta}{2}t + y \Big) = E \Big( \prod_{u \in N_s}P^{X^u_s}\big( R_{t - s}\leq \frac{\beta}{2}t + y \big) \Big) &= E \Big( \prod_{u \in N_s}P^{X^u_s}\big( R_{t - s}\leq \frac{\beta}{2}(t - s) + \frac{\beta}{2}s + y \big) \Big)
r=\begin{pmatrix}1 x_1 x_2 x_3 x_4 0 1 \alpha_1(X) \alpha_2(X) \alpha_3(X) 0 0 1 \alpha_4(X) \alpha_5(X) 0 0 0 1 x_1 0 0 0 0 1 \end{pmatrix},X=(x_1,x_2,x_3,x_4).
\psi_{1}'(r_{*}) = \frac{g_{1}'(r_{*}) - f_{1}'(\ell_{*})}{f_{1}''(\ell_{*})(r_{*}- \ell_{*})}, \psi_{2}'(\ell_{*}) = \frac{f_{2}'(r_{*}) - g_{2}'(\ell_{*})}{f_{2}''(r_{*})(r_{*}- \ell_{*})},
\rho C_{p}\frac{T_{n}^{t}-T_{n}^{t-1}}{\Delta t}=k_{n}\frac{T_{n-1}^{t}-T_{n}^{t}}{\Delta x^{2}}+k_{n+1}\frac{T_{n+1}^{t}-T_{n}^{t}}{\Delta x^{2}}+Q_{n}
\begin{array}{c c c c}z: \left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right) \to \left(x^{2,3},x^{4,5},e^{-2\pi iz}x^{6,7},e^{2\pi iz}x^{8,9}\right) y: \left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right) \to \left(e^{-2\pi iy}x^{2,3},e^{2\pi iy}x^{4,5},x^{6,7},x^{8,9}\right)\end{array}
S={1\over{2k^2}}\int dz^{(-4)}du[\bar{q}_1^+D^{++}q_{1}^++\bar{q}_2^+D^{++}q_{2}^+++V^{++}(\bar{q}_1^+q_2^++\bar{q}_2^+q_1^++\lambda^{++})]\qquad.
\mathcal{M}:L^2(\mathbb{R}_+,\,p^2\textstyle{\sqrt{p^2+1}}\,dp)\longrightarrow L^2(\mathbb{R}), \psi\longmapsto \psi^{\sharp}(x)=\frac{1}{\mspace{-4.5mu}\sqrt{\mspace{1.5mu}2\pi}\,}\!\int_{\mathbb{R}}\mspace{-3mu}dt\:e^{-i\mspace{1.5mu}tx}e^{2t}\psi(e^t)
\tilde{\partial}{}_{\alpha \beta}= \frac{\partial}{\partial{\tilde{x}^{\alpha \beta}}},\quad D^{(\tilde{\tau})}_{\alpha}= \frac{\partial}{\partial \tilde{\tau}^{\alpha}}+ \frac{i}{2}\tilde{\tau}^{\beta}\frac{\partial}{\partial{\tilde{x}^{\alpha \beta}}},
\alpha_{i}={\cal{G}}( \alpha_{1}^{\prime}, \ldots , \alpha_{N}^{\prime}) \alpha_{i}^{\prime}, \hspace{1 0 m m}\mathrm{w i t h}\hspace{5 m m}{\cal{G}}( \alpha_{1}^{\prime}, \ldots , \alpha_{N}^{\prime}) = \frac{\sum ( \alpha_{i}^{\prime})^{2}+ 2 \! \begin{array}{c}{{{}}}\\{{{\sum \sum}}}\\{{{}_{j < l}}}\\ \end{array}\! \alpha_{j}^{\prime}\alpha_{l}^{\prime}c_{j l}}{\sum \alpha_{i}^{\prime}}.
0 = \|\mathrm{div}\, u_{1}- \mathrm{div}\, u_{2}\|_{L^{2}(\Omega)}\geq \|\nabla u_{1}- \nabla u_{2}\|_{L^{2}(\Omega)}\geq \tfrac{1}{C_{\mathcal{P}}}\|u_{1}- u_{2}\|_{L^{2}(\Omega)},
\begin{array}{*{20}{cl}}{\mathop{\rm min}\limits_{{P_1},\cdots,{P_K}}}{p_{out,K}}{{\rm{s.}}\,{\rm{t.}}}{{\sum\nolimits_{k = 1}^K{{P_k}{p_{out,k - 1}}}}\le P_{\rm given}}{}{{P_k}\ge 0,\quad{\rm for}0 \le k \le K}, \end{array}
M_{1,Q}(n) = \psi_{\tau Q/N}\ast d \sigma_{\tau}(n) \sum_{Q\leq q \leq 2Q}\sum_{a \in \mathbb Z^{\times}_{q}}e_q (a( \lvert n\rvert^2 - \lambda^2 ) ) = P_{Q, \tau}(n) \cdot \mathsf C_{Q}( \lvert n\rvert^2 - \lambda^2 ) .
\phi(X_1,\tilde Y-(\tilde t-t)X,\tilde t) =\phi(X,Y,t)+\nabla_X\phi(X,Y,t)\cdot(X_1-X) &+\frac 1 2 \langle \nabla_X^2\phi(X,Y,t)(X_1-X), (X_1-X)\rangle &-(\tilde t-t)\bigl (X\cdot\nabla_Y-\partial_t\bigr)\phi(X,Y,t)+{o}(\epsilon^2),
y_{\mathbf{r},r_0}\le \phi_{\omega*}(r_0r)y_{\mathbf{r}}\tilde{y}_{r_0}\sum_{\substack{r_0|f_0 (f_0,W_0)=1}}\sum_{\substack{\mathbf{r}|\mathbf{f}\in\mathcal{D}_k p\nmid (f,f_0)\Rightarrow p|rr_0}}\frac{\mu^2(f_0)}{\phi(f_0)\phi_\omega(f)}.
H_{2}(t)= C(\|\nabla K(t)\|_{L_{t}^{1}L^{2}}\|\omega\|_{L_{t}^{\infty}L^{2}}^{\frac{1}{2}}\|j\|_{L_{t}^{\infty}L^{2}}^{\frac{1}{2}}\|j\|_{L_{t}^{1}L^{\infty}}^{\frac{1}{2}}+\|\nabla^{2}K(t)\|_{L_{t}^{1}L^{1}}\|u\|_{L_{t}^{\infty}L^{2}}^{\frac{1}{2}}\| j\|_{L_{t}^{2}L^{\infty}}).
I_{MN}= 1 + \sum_{d=1}^\infty (qz)^d \sum_{\substack{\lambda \vdash d \ell(\lambda) \leq N}}s_\lambda(a_1,\dots,a_N)s_\lambda(b_1,\dots,b_N) \Omega_{\frac{q}{N}}^{-1}(\lambda) \Omega_{\frac{1}{N}}^{-1}(\lambda),
\sup_{t\in[0,T]}\| w(t)-w^n(t) \|_{H^{1+\delta}_p}= \sup_{t\in[0,T]}\| w(T-t)-w^n(T-t) \|_{H^{1+\delta}_p}&\leq C \| b - b^n \|_{\infty, H^{-\beta}_q}.
\xi \left( r \right) = 1 + 2 \mathcal{C}_{0}\left| 1 - a \right|^{\frac{a}{a - 1}}r^{\frac{a}{a - 1}}\left( \frac{B}{a}\right)^{\frac{2 - a}{2 \left( a - 1 \right)}}\, ,
\sum_{\lambda =0}^{n-1}\sum_{k=1}^{n-\lambda}(-1)^{k+\lambda+k\lambda +nk+k(\vert a_1\vert +\dots+\vert a_{\lambda}\vert)}\mu_{n-k+1}(a_1,\dots,a_{\lambda},m_k(a_{\lambda +1},\dots ,a_{\lambda +k}),a_{\lambda +k+1}, \dots,a_n)=0.
\lambda=\frac{1}{\nu}\cdot c=\frac{h}{E}\cdot c\approx\frac{4.1357\cdot10^{-15}eV\cdot s}{5.87433\cdot10^{-6}eV}\cdot2.9979\cdot10^{8}m\cdot s^{-1}\approx0.21106m=21.106cm
{\cal L}_{F}= \bar{\psi}i \gamma^{\mu}( \partial_{\mu}- i e A_{\mu}) \psi - m \bar{\psi}\psi +{\frac{g}{2}}\mid \bar{\psi}\gamma^{\mu}\psi \mid^{2}-{\frac{p e^{2}}{4}}\mid A_{\mu \nu}\mid^{2}+{\frac{q e^{2}}{2}}\epsilon_{\mu \nu \lambda}A^{\mu}A^{\nu \lambda}.
\hspace{. 7 i n}+ \sum_{j = 0}^{k - 1}\gamma^{j - k}x_{i}^{-}( - j ) \otimes \gamma^{- ( k + 3 j ) / 2}\varphi_{i}( j - k ) \quad{\it m o d}\; U N_{-}^{2}\otimes U N_{+}
c_{r,s}^{(l)}=q^{\frac{-\hat{c}}{24}}\sum_{m=0}^{N/2}C_{2m}^{l-1}\left\{\sum_{j\in{\bf Z}, m\equiv m_{r,s}(j)}q^{\alpha_{r,s}(j)}-\sum_{j\in{\bf Z}, m\equiv m_{r,-s}(j)}q^{\alpha_{r,-s}(j)}\right\}
w_2 (x_1,x_2,t_1,t_2) = \int\limits_0^{t_1}d\tau_1\int\limits_0^{t_2}d\tau_2 \int q(t_1-\tau_1|x) & \times q(t_2-\tau_2|x) g(x,\tau_1|x_1) g(x,\tau_2|x_2)d^d x .
\mathcal{E}(f) = \left( 1 - \frac{\beta_f}{p \alpha_f}\right), \mathcal{E}^*(f) = \left( 1 - \frac{\beta_f}{\alpha_f}\right), \mathcal{E}(f, g, s) = \left( 1 - \frac{p^{s-1}}{\alpha_f \alpha_g}\right) \left( 1 - \frac{p^{s-1}}{\alpha_f \beta_g}\right) \left( 1 - \frac{\beta_f \alpha_g}{p^s}\right) \left( 1 - \frac{\beta_f \beta_g}{p^s}\right).
\begin{array}{rcl}a_{i,\kappa}= \frac{\kappa}{n-s}+\frac{n-i}{n}-\frac{\gamma}{n(n-s)}\,\,\, (i=1,2,\dots,n), b_{i,\kappa}= \frac{\kappa}{n-s}+\frac{s-i+1}{s}-\frac{\gamma}{s(n-s)}\,\,\, (i=1,2,\dots,s), b_{i,\kappa}= \frac{\kappa}{n-s}+\frac{i-s}{n-s}+\frac{\delta_{i}}{n-s}\,\,\, (i=s+1,\dots,n-1).\end{array}
x^j y^{pq+1-j}= x^{q+1}y^{pq-q}- \sum_{i=1}^{j-2}x^{q+i}y^{pq-q-i}(y-x) - x^j y^{pq-q+2-j}(x^{q-1}-y^{q-1}) &= x^{q+1}y^{pq-q}- \sum_{i=1}^{j-2}t^i x^{q}y^{pq-q}(y-x) - t^{j-1}x y^{pq-q+1}(x^{q-1}-y^{q-1}) &= x^{q+1}y^{pq-q}- \sum_{i=1}^{j-2}t^i x^{q}y^{pq-q}(y-x) - t^{j-1}x^q y^{pq-q+1}+ t^{j-1}x y^{pq}& \in (x^q, y^q)^p = I^{[q]}.
A^{main,\pm}_{j,<0}(t,x,\xi) = \Pi^{\omega,\pm}_{> |\eta|^\delta}\Pi^{\omega,\pm}_{cone}A_{j,<0}A^{pert,\pm}_{j,<0}(t,x,\xi) = (\Pi^{\omega,\pm}_{< |\eta|^\delta}\Pi^{\omega,\pm}_{cone}+ \Pi^{\omega,\pm}_{null}+ \Pi^{\omega,\pm}_{out})A_{j,<0}
\mathbf{D}_r=\left[\frac{\partial[\mathbf{A}_r]_{:,1}}{\partial\mu^{\mathrm{r}}_1}\quad\frac{\partial[\mathbf{A}_r]_{:,2}}{\partial\mu^{\mathrm{r}}_2}\quad\cdots\quad\frac{\partial[\mathbf{A}_r]_{:,Z}}{\partial\mu^{\mathrm{r}}_Z}\right]
\begin{aligned}\big\|\varphi_i^L(u^{(\delta)})-\varphi_i^K(u^{(\delta)})\big\|_{L^1(Q_T)}\le \big\|u_i^{(\delta)}\big(\varphi(v_\delta^L)-\varphi(v_\delta^K)\big)\big\|_{L^1(Q_T)}&\phantom{xx}{}+ 2\big\|L\big(1-\varphi(v_\delta^L)\big) - K\big(1-\varphi(v_\delta^K)\big)\|_{L^1(Q_T)}&=: I_1 + I_2.\end{aligned}
\begin{array}{lll}\delta_{2}= r_{a}^{2}r_{x}-r_{x}r_{a}^{2}=r_{a}\left(r_{a}r_{x}\right)-r_{x}r_{a}^{2}= r_{a}\left(r_{x}r_{a}+\delta_{1}\right)-r_{x}r_{a}^{2}= \left(r_{a}r_{x}\right)r_{a}+r_{a}\delta_{1}-r_{x}r_{a}^{2}= \left(r_{x}r_{a}+\delta_{1}\right)r_{a}+r_{a}\delta_{1}-r_{x}r_{a}^{2}=\delta_{1}r_{a}+r_{a}\delta_{1}\in r_A^{2}.\end{array}
\hat{P}(C_{1}= C_{2}) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\hat{f}_{XYZS}(x,y,z,s) \delta(x-y) \delta(z-s) \mathrm{d}x\mathrm{d}y \mathrm{d}z\mathrm{d}s
(\mathbf{z}^w)^* =(e^{(u\cos \theta +v\sin \theta )/r},(v\cos \theta -u\sin \theta )/r) =\mathbf{e}^{\frac{1}{r}(u\cos \theta +v\sin \theta )+\frac{i}{r}(v\cos \theta -u\sin \theta )}.
\partial_t S(t)v(x)=\int_{\mathbb{R}^n}e^{-\frac{|\xi|^2 t}{2(1+|\xi|^2)}}\left[ \cos\left(\tfrac{|\xi|^2 \sqrt{3+4|\xi|^2}t}{2(1+|\xi|^2)}\right)- \tfrac{1}{\sqrt{3+4|\xi|^2}}\sin\left(\tfrac{|\xi|^2 \sqrt{3+4|\xi|^2}t}{2(1+|\xi|^2)}\right) \right] \widehat{v}(\xi)e^{ix\cdot\xi}d\xi.
H_0(\mathbf{y})=-6y_1y_2y_3, &H_1(\mathbf{y})=y_1(y_1^2+3y_2^2+3y_3^2), &H_2(\mathbf{y})=y_2(y_1^2+3y_2^2+9y_3^2), &H_3(\mathbf{y})=3y_3(y_1^2+y_2^2+3y_3^2).
\sum_{1 \le k \le L}\frac{R^{k}}{k}\min\{q^{k/2},d(\chi)\}= \sum_{1 \le k \le L}\frac{(R\sqrt{q})^k}{k}\le \sum_{1 \le k \le L}(R\sqrt{q})^k \le \frac{(R\sqrt{q})^L}{1-(R\sqrt{q})^{-1}}\le 6d(\chi) R^L
\lambda^{-2n}H,E_{-\alpha}, \lambda^{-2n}E_{\pm\alpha}, F_{-\alpha_1}, \lambda^{-2n}F_{\pm\alpha_1}, \lambda^{-2n}F_{\alpha_2},F_{-\alpha_2},\lambda^{-2n}F_{-\alpha_2}, \lambda^{-1}H', \lambda^{-(2n+1)}H',~~n>0
\sigma^{-1}_{\sigma^{-1}_{(a,i)}((b,j))}((c,k))=\sigma^{-1}_{(\alpha^h(b)+\alpha^i(g),j+h)}((c,k))=(\alpha^h(c)+\alpha^{j+h}(g),k+h)=\sigma^{-1}_{\sigma^{-1}_{(a',i')}((b,j))}((c,k)).
S_{G B}^{v o l u m e}= 2 \left( 1 - ( e^{U}R^{\prime})^{2}\right) |_{r = r_{h}}- 2 ( e^{2 U})^{\prime}|_{r = r_{h}})^{- 1}\left[ ( e^{2 U})^{\prime}\left( 1 - ( e^{U}R^{\prime})^{2}\right) \right]_{r = r_{0}}\ .
\begin{array}{llll}\phi(out)_g = Tr(W_{out}U_{g^{-1}}) = Tr(SW_{in}S^{-1}U_{g^{-1}}) = Tr(W_{in}S^{-1}U_{g^{-1}}S) = Tr(W_{in}U_{g^{-1}}) = \phi(in)_g \quad . \end{array}
\begin{cases}u_{tt}= u_{xx}, (x,t)\in (0,1)\times (0,\infty), u(0,t) = 0, t\in (0,\infty), u_x(1,t) = - au_t(1,t) + b p(0,t), t\in (0,\infty), \varepsilon p_t = p_{xx}, (x,t)\in (0,1)\times (0,\infty), p_x(0,t) = c p(0,t), t\in (0,\infty), p_x(1,t) = d u_t(1,t), t\in (0,\infty), \end{cases}
[ V_{A}^{B}, \bar{\Omega}] ( - 1 )^{\varepsilon_{B}}+ [ \bar{V}_{A}^{B}, \Omega ] ( - 1 )^{\varepsilon_{B}}+ i \hbar \delta_{A}^{B}= i \hbar V_{A}^{C}\bar{V}_{C}^{B}+ i \hbar \bar{V}_{A}^{C}V_{C}^{B}.
\begin{aligned}G_1(k) =\frac{y^{2k}-1}{y-1}+x,G_2(k)=y^{2k-1}+x\cdot\frac{y^{2k}-1}{y-1},G_3(k)=y+\frac{x^{2k}-1}{x-1},G_4(k)=y\cdot\frac{x^{2k}-1}{x-1}+x^{2k-1}.\end{aligned}
a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}=a_2' \frac{d z_1}{z_1}\wedge \frac{d b_3}{b_3}= \frac{d z_1}{z_1}\wedge \frac{d z_2}{z_2}+ l_2\frac{d z_1}{z_1}\wedge \frac{d b_2}{b_2}
Q_a(g(X\circ Y),gZ) = c(g)^2 Q_e(X\circ Y,Z) = c(g)^2{\mathrm T}_e(X,Y,Z)= c(g)^{-1}{\mathrm T}_a(gX,gY,gZ) & = \det(a)^{-1}{\mathrm T}_a(gX,gY,gZ) = Q_a(gX\circ_a gY,gZ).
T_{B}^{( i )}( z ) = \displaystyle \sum_{\mu = 0}^{n - 1}G_{a + \bar{\varepsilon}_{\mu}}\phi^{* ( a , a + \bar{\varepsilon}_{\mu})}( z ) K^{( i )}\left( \left. a + \bar{\varepsilon}_{\mu}\begin{array}{c}{{a}}\\{{a}}\\ \end{array}\right| z \right) \phi^{( a + \bar{\varepsilon}_{\mu}, a )}( z ) .
T_{(\cdot)}(t) = g^{-1}\bigl(t, X(t)\bigr) = \begin{pmatrix}\begin{array}{l l}1 -1^T 0 I_D \end{array}\end{pmatrix}\begin{pmatrix}\begin{array}{c}t X(t) \end{array}\end{pmatrix}= \begin{pmatrix}\begin{array}{c}t-\sum_{j=1}^D X_j(t) X_1(t) \cdot X_D(t) \end{array}\end{pmatrix}.
{\cal E}[ \phi ] = \int d x \left[ \frac{1}{2}\left( \frac{d \phi_{1}}{d x}\right)^{2}+ \frac{1}{2}\left( \frac{d \phi_{2}}{d x}\right)^{2}+ \left( 4 \phi_{1}^{2}+ 2 \sigma \phi_{2}^{2}- 1 \right)^{2}+ 1 6 \sigma^{2}\phi_{1}^{2}\phi_{2}^{2}\right]
M(\lambda_0)\begin{pmatrix}v u\end{pmatrix}=\begin{pmatrix}M_{\overline S\overline S}(\lambda_0) M_{\overline SS}(\lambda_0) M_{S\overline S}(\lambda_0) M_{SS}(\lambda_0)\end{pmatrix}\begin{pmatrix}v u\end{pmatrix}=\lambda_0\begin{pmatrix}v u\end{pmatrix}.
N \to \infty \ , \quad \mathrm{~ g_{\mathrm{~ s}}~ , ~ \a l p h a^{\prime}~ , a n d ~ s ~ f i x e d \ ,}\quad R = ( 4 \pi \alpha^{2}g_{\mathrm{s}}N )^{1 / 4}\ , \quad \omega = \frac{1}{2}R s^{1 / 2}\ .
\left\langle H \right\rangle_{\Omega}= \left\langle \Omega \right| \int{d x_{1}}\left({- \frac{1}{2}\Pi_{1}\left({1 - \frac{{e^{2}}}{\pi}\frac{1}{{\partial_{1}^{2}}}}\right)^{- 1}\Pi^{1}}\right) \left| \Omega \right\rangle .
A_{0}( \theta , \phi ) = \frac{2 \pi \gamma}{\beta}\left( \begin{matrix}{{\cos \theta}}&{{\sin \theta e^{- i n \phi}}}\\{{\sin \theta e^{i n \phi}}}&{{- \cos \theta}}\\ \end{matrix}\right) , \ \mathrm{\boldmath ~ A ~}= 0 ;
\Delta ( Q_{0}^{a}) = 1 \otimes Q_{0}^{a}+ Q_{0}^{a}\otimes 1 \mathrm{~ a n d ~}\Delta ( Q_{1}^{a}) = 1 \otimes Q_{1}^{a}+ Q_{1}^{a}\otimes 1 +{f_{b c}}^{a}Q_{0}^{b}\otimes Q_{0}^{c}
\sum_{i=1}^{i^*}P_{X^nY^n}(x_i^n,y_i^n) \geq 1-\varepsilon-\gamma\intertext{and}\sum_{i=1}^{i^*-1}P_{X^nY^n}(x_i^n,y_i^n) < 1-\varepsilon-\gamma.
\lim_{n}\frac{1}{\phi(n)}[S_{n,\phi(n)}(j;\omega)-\sum_{i=1}^bS_{a_n,\phi(n)}(i;\omega)p(i,j)] = \lim_{n}\frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}\sum_{i=1}^b\mathbf{1}_{\{i\}}(\xi_{k-1})[p_k(i,j)-p(i,j)] = 0, \ \a.e. \ \ \forall j\in \mathbf{X}.
t H(L_t)= \frac{t_0^2}t \big\langle \Lambda_{t_0}, L_{t_0}\big\rangle+ 2 \frac{t_0 (t-t_0)}t \big\langle \Lambda_{t_0}, L_{t_0,t}\big\rangle+ \frac{(t-t_0)^2}t H(L_{t_0,t}).
\left(xyz + (1-\alpha^3)^{-1}x^3\right)^{|\alpha^3|}= \alpha^{-\frac{(|\alpha^3|-1) |\alpha^3|}{2}}x^{|\alpha^3|}y^{|\alpha^3|}z^{|\alpha^3|}+ (1-\alpha^3)^{-|\alpha^3|}x^{3|\alpha^3|}.
\sum_{j = - \infty}^{\infty}\left( j^{2}+ a^{2}\right)^{- 2 s}= \int_{- \infty}^{\infty}d j \left( j^{2}+ a^{2}\right)^{- 2 s}+ 2 \int_{- \infty + i \epsilon}^{\infty + i \epsilon}\left( e^{- 2 \pi i z}- 1 \right)^{- 1}\left( z^{2}+ a^{2}\right)^{- 2 s}d z
0 \le\sum_{l\in \Omega_1}(| l\cap W|-1)(| l\cap W|-2) &=\sum_{l\in \Omega_1}| l\cap W|(| l\cap W|-1)-2\sum_{l\in \Omega_1}| l\cap W|+2|\Omega_1| &\le|W|(|W|-1)-2\sum_{l\in \Omega_1}| l\cap W|+2|\Omega_1|,
M\left(t_{i},T_{j}\right):=\Pi\left(t_{0},T_{j}\right)+\sum_{k=1}^{i}\left(\Pi\left(t_{k},T_{j}\right)-\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{k},T_{j}\right)\left|\mathcal{F}_{k-1}\right.\right]\right)
P_{\alpha}= \frac{( \lfloor (2n-1)^{4d}\rfloor )(\lfloor (2n-1)^{4d}\rfloor - 1) \dots (\lfloor (2n-1)^{4d}\rfloor - k)}{|W'_n| (|W'_n| - 1) \dots (|W'_n| - k)}< \frac{(2n-1)^{k 4 d}}{|W'_n|^{k}}
\mathcal{D}= \frac{L(1, \tilde f) \bigl(A(p, 1) p - 1\bigr)}{p^{\frac{3}{2}}\pi}\int_0^{\infty}k(t) \tanh(\pi t) t \, d t + O(M T^{\frac{1}{7}+ \varepsilon}p^{\varepsilon})
H_3^n(\alpha) =h\left(\frac{1}{3}*\alpha\right)+\left(\frac{2}{3}*\alpha\right)h\left(\frac{1-\alpha^2}{2-\alpha}\right) &+\left(\frac{1}{3}*\alpha\right)h\left(\frac{1-\alpha+\alpha^2}{1+\alpha}\right)+(n-2)h(\alpha) &\geq h\left(\frac{1}{3}*\alpha\right)+\frac{1}{3}h(\alpha)+\frac{2}{3}+(n-2)h(\alpha)
{\rm Re}\, \chi^{\rm D}_T(\omega) = -\frac{\hbar A}{4\pi^2}\left(\frac{k_B T}{\hbar}\right)^5\int_0^{\Omega}dv\sqrt{v(\Omega-v)}\int_{0}^{\infty}du\, \frac{u^2}{e^{\frac{u}{2}}-1}\left[1+{\cal O}(\Omega^2)\right]
\begin{array}{r l l}\nabla \psi^i_{\Pi}\displaystyle =\mbox{argmax}_{W_\Pi}\left[- \int_{\Pi}j_i^2~dx+2\int_{\partial \mathcal{B}_i}(j_i\cdot n)~ds\right], begin{align*}12pt]W_{\Pi}\displaystyle= \left\{j\in L^2(\Pi;\mathbb{R}^d):~\nabla\cdot j=0~\mbox{in}\Pi,~j\cdot n=0~\mbox{on}\partial\Pi\right\}.\end{array}
a_2 = z + c \mathbb{E}[\delta_2(z)] + O(v_0^{-24}n^{-2}) a_1 = z^2 - c + 1 + (c-c_{n,m}) + z c \mathbb{E}[\delta_2(z)] - z \mathbb{E}[\delta_1(z)] + O(v_0^{-24}n^{-2}) a_0 = z - z^2\mathbb{E}[\delta_1(z)] +c \mathbb{E}[\delta_1(z)] + O(v_0^{-24}n^{-2})
\mathcal{A}^\mathrm{uni}= \left\{A \in \mathcal{M}\;:\; \exists L \;\; \lim_{D \rightarrow \infty}\sup_{x\in(1,\infty)}\left| \frac{1}{\log(D)}\int_x^{xD}\frac{1_A(y)}{y}\mathrm{d}y - L \right| = 0 \right\}
{\rm Pr}(E|b_0,b_1,\cdots,b_M)= &\sum_{y \underset{b_0=0}{\overset{b_0=1}{\lessgtr}}{\rm Thr}}{\frac{e^{-\mathbb{E}(\textbf{y}_R| b_0,b_1,\cdots,b_M)}(\mathbb{E}(\textbf{y}_R| b_0, b_1,\cdots,b_M))^y}{y!}}.
\begin{array}{c}{A^{b a}= ( - )^{\epsilon_{a}+ \epsilon_{b}+ \epsilon_{a}\epsilon_{b}}A^{a b}, \hspace{3 m m}( A^{a b})^{*}= A^{b a},}{( B^{a})^{*}= ( - )^{\epsilon_{a}}B^{a}, \hspace{3 m m}C^{*}= C .}\end{array}
( \nabla_{\beta r}v_{\alpha q}^{~ \underline{{\beta}}}) v_{\underline{{\beta}}}^{~ \gamma q^{\prime}}={\frac{1}{4}}( \nabla_{\beta r}u_{a}^{~ \underline{{a}}}) u_{\underline{{a}}}^{~ i}( \gamma^{a})_{\alpha}^{~ \gamma}( \gamma_{i})_{q}^{~ q^{\prime}}\equiv{\frac{1}{4}}\hat{\Omega}_{\beta r , a}^{~ i}( \gamma^{a})_{\alpha}^{~ \gamma}( \gamma_{i})_{q}^{~ q^{\prime}}.
\sum_{j=1}^t \frac{1}{j}\leq 1 + \int_1^t \frac{1}{x}dx = 1 + \ln x|_1^t = 1+ \ln t, \sum_{j=1}^t \frac{1}{\sqrt{j}}\geq \int_1^{t+1}\frac{1}{\sqrt{x}}dx = 2\sqrt{x}|_{1}^{t+1}= 2 (\sqrt{t+1}- 1).
\begin{aligned}-\iint g_s \partial_y u^\epsilon h^\epsilon \leq L \Vert g_{s}\Vert_{L^\infty}\Vert h^\epsilon_x\Vert_{L^2}\Vert u^\epsilon_y \Vert_{L^2}= L \Vert g_{s}\Vert_{L^\infty}\Vert g^\epsilon_y\Vert_{L^2}\Vert u^\epsilon_y \Vert_{L^2},\end{aligned}
E_r (z_1,\ldots ,z_n;t,t_0) P_\lambda(z_1,\ldots ,z_n;q,t,\mathbf{t}) = \sum_{\substack{\mu\in\Lambda_n \mu\sim_r\lambda}}C^{\mu,n}_{\lambda , r}(q,t,\mathbf{t})P_\mu (z_1,\ldots ,z_n;q,t,\mathbf{t}) ,
d s_{P}^{2}={\frac{\left|{\partial_{a}^{3}{\cal F}}\right|^{2}}{\left( \mathrm{I m}\, \tau \right)^{2}}}| d a |^{2}={\frac{\left| \partial_{u}\tau \right|^{2}}{\left( \mathrm{I m}\, \tau \right)^{2}}}| d u |^{2}= e^{\varphi}| d u |^{2},
d_{e}L=\partial^{\mu}Q_{\mu},~Q_{\mu}=d_{e}V_{\mu}=j_{\mu}u,u:=d_{e}\phi & ~(d_{e}+d_{e^{\prime}})TLL^{\prime}=\partial^{\mu}TQ_{\mu}L^{\prime}+\partial^{\mu}TLQ_{\mu}^{\prime}
I_j=\frac{1}{|{\mathcal E}_j|}\frac{1}{\sqrt{\delta}}\int \frac{d\xi}{(1+\xi^2)^\frac{3}{4}}\int \frac{d\eta}{(1+4\eta^2)^\frac{3}{4}}\exp\left( - \frac{6b_j^2\eta^2}{1 + 4\eta^2}\right)= \frac{C_1}{|{\mathcal E}_j|(1+|b_j|)\sqrt{\delta}}\le C_2,
\frac{d^\ell}{dt^\ell}\phi_X(t) = \frac{d}{dt}\left(\frac{d^{\ell-1}}{dx^{\ell-1}}\phi_X(t)\right) &= \frac{d}{dt}\left(\int_\mathbb{R}x^{\ell-1}f_X(x)e^{\mu tx}dx\mu^{\ell-1}\right) &= \int_\mathbb{R}x^{\ell-1}f_X(x)\frac{d}{dt}\left(e^{\mu tx}\right)dx\mu^{\ell-1}&= \int_\mathbb{R}x^\ell f_X(x)e^{\mu tx}dx\mu^\ell .
\begin{cases}w_{\chi_{t},b,t,N,\omega}^{A}=\chi_{t}(x')Q_{t}e^{i\tau x\cdot\xi}e^{-\tau(x\cdot\omega-t)A_{t}^{A}(x')}b+\Gamma_{\chi_{t},b,t,N,\omega}^{A}(x,\tau) w_{\chi_{t},b,t,N,\omega}^{B}=\chi_{t}(x')Q_{t}e^{i\tau x\cdot\xi}e^{-\tau(x\cdot\omega-t)A_{t}^{B}(x')}b+\Gamma_{\chi_{t},b,t,N,\omega}^{B}(x,\tau)\end{cases}
~\Phi(u_i((L_i/R)^3x)+\eta(R,s,1;x)) \leq ~\Phi(u_i)(R/L_i)^{3N}+ C_0R^{3N-1}+\left(\frac{1}{2}bC_3L^{3N-5}_i\int_{\mathbb{R}^N}|\nabla u_i|^2\right) (R/L_i)^{6N-11}.