harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|S_0-e^{W_0}| \le e^{|W_0|}\sum_{n=1}^N\frac{1}{n!}\Bigg(\sum_{m=1}^Nc_A^{\frac{m}{2}}\|W_m\|_{L^1}\Bigg)^n &= e^{|W_0|}\left(e^{\sum_{m=1}^Nc_A^{\frac{m}{2}}\|W_m\|_{L^1}}-1\right).
\Psi_E^{(-)}=\left(\begin{array}{l}e^{i \frac{\theta}{2}}\sum f_n e^{i \left(n+\frac{1}{2}\right)\theta}e^{-i \frac{\theta}{2}}\sum g_n e^{i\left(n+\frac{1}{2}\right)\theta}\end{array}\right)
\small w(\lambda,x,y_i) := \left(\sum_{k=1}^{\lfloor c_n-x\rfloor}k^{m-2}\right)\, e^{-\lambda\, x}\, \Pr\left( \sum_{j=1}^n \lfloor Y_{i,j}' \rfloor = \lfloor r_i - y_i\rfloor\right), x \in [0,1].
\Pr_{\rho\sim P_\epsilon}[ \rho\neq \tilde{D}_{1}(H\rho^\intercal)] \leq 2e^{-\frac{l^2}{3\epsilon N}}+(2l+1)\mathop{\max}_{\substack{w,w'\in\{\epsilon N\pm l\}w'\leq w}}\left\{\frac{\Pr_{\rho\sim \lambda_{w,w'}}\big[ |\Omega_\rho^{\{w,w'\}}|>1 \big]}{\Pr_{\rho\sim \lambda_{w,w'}}\big[ |\rho|=w \big]}\right\}.
\begin{aligned}j = -\frac{N}{2}+1,-\frac{N}{2}+2,\dots , \frac{N}{2}\ \ \ (N \even) j = -\frac{N}{2}+\frac{1}{2},-\frac{N}{2}+\frac{3}{2}, \dots , \frac{N}{2}-\frac{1}{2}\ \ \ (N \odd)\end{aligned}
\left( \xi_{\bar{C}}\right)_{A}^{\prime}( x ) \left( \xi_{\bar{D}}\right)_{B}^{\prime}( y ) \hat{I}_{A B}( x ) \delta ( x , y ) = \left( \xi_{\bar{C}}\right)_{A}^{\prime}( x ) \hat{I}_{A B}( x ) \left[ \left( \xi_{\bar{D}}\right)_{B}^{\prime}\right]^{\mathrm{` ` a d j o i n t^{\prime \prime}}}( x ) \delta ( x , y ) ,
\tilde{D}_{\alpha}= e^{i \partial_{i}^{p}D_{i}}e^{- i x_{\mu}p_{\mu}}D_{\alpha}e^{i x_{\nu}p_{\nu}}e^{- i \partial_{j}^{p}D_{j}}, \quad ( \partial_{i}^{p}= \frac{\partial}{\partial p_{i}}) .
a_t = \frac{- \alpha_1 + 3 \alpha_2 + \sqrt{D}}{4 \sqrt{D}}\ , \a_1 = a_2 = \frac{- \alpha_1 - \alpha_2 + \sqrt{D}}{4 \sqrt{D}}\ , \a_{\theta}= \frac{3 \alpha_1 - \alpha_2 + \sqrt{D}}{4 \sqrt{D}},
\widetilde{\psi}_{0}^{(n^{2})}(x;N,M,\lambda_{1})=E_{n}( N,M) \psi_{0}(x;N,M) -\frac{W(\psi_{n},\psi_{0}\,|\, x)}{\Psi_{n}^{(n)}(x;N,M,\lambda_{1})}
\alpha_{1}= \left( \begin{array}{c}{{e^{i n \theta}}}\\{{0}}\\{{0}}\\{{d_{+}\, e^{i ( n + 1 ) \theta}}}\\ \end{array}\right) , \alpha_{3}= \left( \begin{array}{c}{{0}}\\{{d_{+}\, e^{i ( n + 1 ) \theta}}}\\{{e^{i n \theta}}}\\{{0}}\\ \end{array}\right) ,
{\displaystyle\mathcal{W}_h(\Omega)=\{\mathbf{w}_h\in \mathbf{H}(\mathbf{curl};\Omega):\,\,\, \mathbf{w}_h|_e\in R_k,\,\,\mathbf{w}_h\times \mathbf{n}=0 \,\, \hbox{on}\,\,\partial \Omega \},}
a^{2}\nabla_{\nu}b^{2}\leqslant K(h^{\frac{1}{2^{n-1}}},2)^{-r_{n}}a^{2}\sharp_{\nu}b^{2}+(a - b)^2-\sum_{k=0}^{n-1}r_{k}\big[a^{\frac{m_k}{2^k}}b^{1-\frac{m_k}{2^k}}- a^{\frac{m_k+1}{2^k}}b^{1-\frac{m_k+1}{2^k}}\big]^{2},
\hat{H}=t_{1}\sum_{\langle i,j\rangle}(c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.)+t_{2}\sum_{\langle\langle i,j\rangle\rangle}(c_{i\sigma}^{\dagger}c_{j\sigma}+h.c.)+J\sum_{\langle i,j\rangle}(S_{i}\cdot S_{j}-\frac{n_{i}n_{j}}{4})-\mu\sum_{i}n_{i},
z^{A}=\left( A_{i}^{a},B_{a}^{0i},B_{a}^{ij},\pi_{ij}^{a},\varphi_{a},\pi^{a},\varphi_{a}^{\left( 1\right)},\pi^{(1)a},\varphi_{a}^{\left(2\right)},\pi^{(2)a}\right) .
\lim_{||\Pi||\to 0}\frac{1}{4}\sum_{[a_i,b_i,c_i]\in\Pi}\Big|\big[s(a_i')-s(a_i)\big] \wedge\big[s(c_i)-s(b_i)\big] \Big|=\int_P\big|\partial_{\ell_1}s(x) \wedge \partial_{\ell_2}s(x) \big|dx\ ,
[u_n]_{s,p}^p \geq S^\sharp\left(\int_{\mathbb{R}^N}(K\ast |u_n|^{p^\sharp}) |u_n|^{p^\sharp}dx \right)^{\frac{p}{2\cdot p^\sharp}}+o(1) &= S^\sharp (p^\sharp /b)^{\frac{p}{2\cdot p^\sharp}}\left([u_n]_{s,p}^p\right)^{\frac{p}{2\cdot p^\sharp}}+o(1),
E_{k-l}^{\textbf{1},\overline{\alpha_N}}= \left(\frac{N}{M}\right)^{\frac{k}{2}-l}\sum_{e\mid N/N_M}\mu(e)\alpha(e)e^{-\frac{k}{2}+l}E_{k-l}^{\textbf{1},\overline{\alpha}}|B_{N/Me}
\lim_{N \to \infty}\kappa_{\chi_\ell}\left(X^{\chi(1)}(N), \ldots, X^{\chi(n)}(N)\right) = \lim_{N \to \infty}\kappa_n(\alpha X(N), \ldots, \alpha X(N)) =\lambda \alpha^n \lim_{N \to \infty}\kappa_{\chi_r}\left(X^{\chi(1)}(N), \ldots, X^{\chi(n)}(N)\right) = \lim_{N \to \infty}\kappa_n(\beta X(N), \ldots, \beta X(N)) =\lambda \beta^n
\left( \begin{array}{c c c}{a}&{{\bf O}_{+}}&{{\bf O}_{v}}\\{}&{}&{}\\{{\bf O}_{+}^{\dagger}}&{b}&{{\bf O}_{-}}\\{}&{}&{}\\{{\bf O}_{v}^{\dagger}}&{{\bf O}_{-}^{\dagger}}&{c}\\ \end{array}\right)
\int_{S^2}|\nabla \hat{\psi}_0| \, dA_{S^2}\leqslant C \left( \int_{S^2}|\nabla \hat{\psi}_0|^2 \, dA_{S^2}\right)^{1/2}= \ 2C \left( \int_{S^2}-\Delta \hat{\psi}_0 \cdot \hat{\psi}_0 \, dA_{S^2}\right)^{1/2}= 2C \|\hat{\psi}_0\|_{L^2(S^2)}.
\boldsymbol{F}= -\ \xi_{0}\,\boldsymbol{u}_{t}\quad\hbox{and}\boldsymbol{G}= -\ \xi\,\boldsymbol{\varphi}_{t}- \int_{0}^{+\infty}g(s)\,\Delta\boldsymbol{\varphi}(\boldsymbol{x},\,t - s)\ds,
2m\left(\gamma\sum_{k=1}^{d}\frac{\partial\ln(\Psi(t,q))}{\partial q_{k}}\right)^2+U(q)=\frac{1}{8m}\left(\frac{h}{\pi}\sum_{k=1}^{d}\frac{\partial\ln(\Psi(t,q))}{\partial q_{k}}\right)^2+U(q)
\partial_p F^{g'}(x, z, p) =: (a^{ij}(x, z, p)p_i + b^j(x, z, p)\big)\partial_j \partial_z F^{g'}(x, z, p) =: -\big(|A_{\Sigma}|^2 + Ric_M(\nu, \nu)\big)z + \frac{1}{2}tr_{\Sigma}\nabla^M_{\nu}\beta + c(x, z, p)
(uv^{-1})\hat{\phi}=(st^{-1})\hat{\phi}\Leftrightarrow (u\phi)(v\phi)^{-1}=(s\phi) (t\phi)^{-1}& \Leftrightarrow u\phi t\phi = s\phi v\phi & \Leftrightarrow ut = sv & \Leftrightarrow uv^{-1}=st^{-1}.
F_{0,\lambda}(R_\lambda) =F(R_{\lambda})+\lambda\mathbb{F}(R_{\lambda}-\Sigma) &\leq\liminf_{i\to\infty}\{F(R_{\delta_i,\lambda})+\lambda|\mathbb{F}(R_{\delta_i,\lambda}-\Sigma)-\eta_{\delta_i}|\}&=\liminf_{i\to\infty}F_{\delta_i,\lambda}(R_{\delta_i,\lambda}).
E\left[T_2(T_2 + 1)\right] = \left(\frac{\nu_2}{\alpha_2}\right)^2 M^2 \left(H_{\nu_2 M}^2 + \sum_{j=1}^{\nu_2 M}\frac{1}{j^2}\right)+ O\left(e^{-\varepsilon M}\right) &= \big(\lambda^{-1}\nu_1 + \nu_2\big)^2 M^2 \left(H_{\nu_2 M}^2 + \sum_{j=1}^{\nu_2 M}\frac{1}{j^2}\right) + O\left(e^{-\varepsilon M}\right)
\int_{\Omega}\frac{|u_k|^{q_k}}{|x|^\alpha}\, dx=\int_{\Omega}\frac{|u_k|^{q_k}}{|x|^{\alpha\frac{q_k}{p^*_\alpha}}}\frac{1}{|x|^{\alpha\frac{p^*_\alpha-q_k}{p^*_\alpha}}}\, dx\leq \left(\int_{\Omega}\frac{|u_k|^{p^*_\alpha}}{|x|^\alpha}\, dx\right)^{\frac{q_k}{p^*_\alpha}}\left(\int_\Omega\frac{1}{|x|^{\alpha}}\, dx\right)^{1-\frac{q_k}{p^*_\alpha}}\leq C
\begin{aligned}\|\zeta\|_{C(\overline{S_T})}\leq c\Bigl(\|\zeta(\cdot,0)\|_{C(\Gamma(0))}+\|f_\zeta^\varepsilon\|_{C(Q_{\varepsilon,T})}+\|\psi_\zeta^\varepsilon\|_{C(\partial_\ell Q_{\varepsilon,T})}\Bigr) +c\varepsilon\Bigl(\|\zeta\|_{C(\overline{S_T})}+\|\zeta_2\|_{C(\overline{S_T})}\Bigr). \end{aligned}
{\tilde{\cal L}}_{W}= - 8 ( D - 2 )^{2}\int d r e^{( D - 1 ) A}\left( \left({\frac{\partial W}{\partial \phi}}\right)^{2}- \kappa^{D - 2}\left({\frac{D - 1}{D - 2}}\right) W^{2}\right) \ .
\limsup_{x\to \infty}\frac{g^{\ast 2}(x)}{g(x)}\le (2/\lambda^2) e^\lambda \Big(\lambda -e^{-\lambda}\sum_{n\neq 2}^\infty \liminf_{x\to\infty}\frac{\lambda^n}{n!}\frac{g^{\ast n}(x)}{g(x)}\Big)=2,
\mathbf{\dot{C}}_{0}=\left. \tfrac{d}{dt}\right\vert_{t=0}(W_{t}^{\mathbf{P}})^{\ast}(\mathrm{pr}^{\ast}\mathbf{A)}=L_{Y}(\mathrm{pr}^{\ast}\mathbf{A)})=-sL_{\overline{X}}(\mathrm{pr}^{\ast}\mathbf{A)})-(\mathbb{\iota}_{\overline{X}}\mathrm{pr}^{\ast}\mathbf{A)})ds & =-\mathrm{pr}^{\ast}(v_{\mathbf{A}}(X))ds.
\left[ \frac{a(v_j) a^{\iota_2}(\bar v_j)}{b(v_j) b^{\iota_2}(\bar v_j)}\right]^{N}= - \prod_{i \neq j}^{n}\frac{a(v_j-v_i)}{a(v_i-v_j)},\qquad j=1,...n.
\mathbb{I}^\omega(N\ell)(Z_1,\dots,Z_n) = \omega^{-N^2\sum_{i<j}\mu_i\mu_j\varepsilon_{ij}}Z_1^{N\mu_1}\dots Z_n^{N\mu_n}&= Z_1^{N\mu_1}\dots Z_n^{N\mu_n}&= \mathbb{I}^1(\ell)(Z_1^N,\dots,Z_n^N).
\frac{d}{dt}\oint_C\sqrt{\mu}d\sigma =\;\oint_C\left(\frac{\mu_t}{2\sqrt{\mu}}+\sqrt{\mu}\frac{\bar{g}_t}{\bar{g}}\right)d\sigma =\;\oint_C\frac{1}{6\sqrt{\mu}}\mu_{\sigma^2}d\sigma =\;\frac{1}{12}\oint_C\mu^{-3/2}\mu_{\sigma}^2d\sigma>0,
\begin{array}{lcl}L_{\varepsilon,R,a}(v) = L_{\varepsilon,R,a}(v)- L_{g}^{u_{\varepsilon,R,a}}(v)- Q_{\varepsilon,R,a}(r^{-\overline\gamma}|x|^{\overline\gamma}v_\phi+h+v) & + H_\delta(u_{\varepsilon,R,a})-H_g(u_{\varepsilon,R,a})- L_g^{u_{\varepsilon,R,a}}( r^{-\overline\gamma}|x|^{\overline\gamma}v_\phi+h),\end{array}
|g.s.>=|0,0,-3J,0>=\frac{1}{\sqrt{12}}(2|\uparrow\downarrow\uparrow\downarrow>+2|\downarrow\uparrow\downarrow\uparrow>-|\uparrow\uparrow\downarrow \downarrow> -|\uparrow\downarrow \downarrow\uparrow>-|\downarrow \downarrow\uparrow\uparrow>-|\downarrow\uparrow\uparrow\downarrow>)\quad .
e^1_\tau =\int_{0}^{t}e^{(t-s)A}\bigl(F_1(X_{s}^{\tau})-F_1(X_{s}^{0})\bigr)ds+( \int_{0}^{t}e^{(t-s+\tau)A}\bigl(\sigma(X_{s}^{\tau})-\sigma(X_{s}^{0})\bigr)dW(s) & +\int_{0}^{t}\bigl(e^{\tau A}-I\bigr)e^{(t-s)A}\sigma(X_{s}^{0})dW(s),
\frac{d}{dt}\Big(\frac{|z(t)|}{z(t)}\Big)^k r_k(z(t)) = r_k(z(t)) \frac{d}{dt}\Big(\frac{|z(t)|}{z(t)}\Big)^k + \Big(\frac{|z(t)|}{z(t)}\Big)^k r_k'(z(t)) z'(t).
\int_a^b \int_\Omega \chi^2(x)u(s,x)\overline{\phi(s,x)}\eta'(s)\,{\rm d}x\,{\rm d}s = \int_{a}^{b}\eta(s) \int_{\Omega}\Bigl(\bigl(A(s,x)\nabla u(s,x)\cdot\nabla \chi^{2}(x)\bigr) \overline{\phi(s,x)}-u(s,x)\bigl(\nabla\chi^{2}(x)\cdot \overline{A(s,x)^{*}\nabla \phi(s,x)}\bigr)\Bigr)\,{\rm d}x\,{\rm d}s.
Q(m,z,p) = \sum_{j=0}^{p-1}\frac{(-2)^j}{m^{j+2}}\sum_{k = 1}^\infty \frac{(- 1)^k \zeta (2k) z^{2k}}{k^{p-j}}& - 2 \sum_{j=0}^{p-1}\frac{(-2)^j}{m^{j+2}}P(m,z,p-1-j) + \left (- \frac{2}{m}\right )^{p}S_{3,m}(z),
\Sigma(s,t) \subseteq\bigcup_{n=1}^N{[s,s+t]^{\leftthreetimes{2n-1}}\times [0,s]^{\leftthreetimes(2(N-n)+1)}}=[s,s+t]\leftthreetimes\left(\bigcup_{n=1}^N [s,s+t]^{\leftthreetimes2(n-1)}\times[0,s]^{\leftthreetimes2(N-n)}\right)\leftthreetimes[0,s].
\left|\frac{A+A}{A+A}\right|\geq{\left|\bigcup_{i=1}^{|A/A|-1}\left(\frac{A+A}{A+A}\cap{M_{(l_i,m_i)}}\right)\right|}=\sum_{i=1}^{|A/A|-1}\left|\frac{A+A}{A+A}\cap{M_{(l_i,m_i)}}\right|.
\int_{B_r(x_0)}\Big(\left|\nabla u(x)\right|^p-\left|\nabla v(x)\right|^p\Big)\,dx =\int_{B_r(x_0)}\left(\int_0^1\frac{d}{ds}\left|\nabla u_s(x)\right|^p\,ds\right)\,dx &=\int_{B_r(x_0)}\left(\int_0^1 p\left|\nabla u_s(x)\right|^{p-2}\nabla u_s(x)\cdot \nabla(u-v)(x)\,ds\right)\,dx.
\begin{cases}f: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid f(x,y) \mapsto (y,x) g: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid g(x,y) \mapsto (-x,-y) \varphi: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid \varphi(x,y) \mapsto (x,x-y) I: \ \mathbb{Z}^2 \rightarrow \mathbb{Z}^2 \mid I(x,y) \mapsto (x,y)\end{cases}
2\delta \,\acute{}\, \kappa v^{3}+\left[ \delta \,\acute{}\,^{2}+4\delta^{2}\left( 1+\kappa \lambda \right) \right] v^{2}-2\delta^{2}\delta \,\acute{}\left( \kappa-\lambda \right) v-\delta^{2}\delta \,\acute{}\,^{2}=0,
E_3^q=I_3-\sum_{j_3,j_2,j_1=0}^q C_{j_3j_2j_1}^2-\sum_{j_3,j_2,j_1=0}^q C_{j_3j_1j_2}C_{j_3j_2j_1}\ \ \ (i_1=i_2\ne i_3),
\frac{|\zeta||\eta|}{Y^{p-1}}\left[Y^p+(p-1)^{-p}\left(\frac{\bar{Z}}{|\zeta|^p}-1\right)\right]-|\zeta||\eta|Y =\frac{|\zeta||\eta|(p-1)^{-p}(\bar{Z}/|\zeta|^p-1)}{Y^{p-1}}.
\hat{G}^{D}( z , w ) = G^{D}( z_{i}, z_{j}) - 2 \sum_{l , m}\oint_{a_{l}}\partial{w}G^{D}( z_{i}, w ) I m ( \tau )_{l m}\oint_{a_{m}}\partial_{w^{\prime}}G^{D}( z_{j}, w^{\prime})
\{e_{a_0}, e_{a_i}, e_{bc_i}, e_{ij}, e_{c}\}=\biggl\{1,\ \frac{p^{i}}{p^{0}},\ \Big(1+\frac{\mathcal{I}}{mc^2}\Big)p^{i},\ \Big(1+\frac{\mathcal{I}}{mc^2}\Big)\frac{p^{i}p^{j}}{p^{0}},\ \Big(1+\frac{\mathcal{I}}{mc^2}\Big)p^{0}\biggl\}\sqrt{F_E^0}
f(x) = \lambda g^*\left(\frac{1}{\lambda n}A x\right)+\frac{\gamma}{2n}\|x\|_2^2,\Psi(x) = \frac{1}{n}\sum_{i=1}^n \left(\phi^*(-x_i)-\frac{\gamma}{2}\|x_i\|_2^2\right).
\pi_{t,k}(\varphi) \, =\, \pi_{0, k}(\varphi) + \int^{t}_{0}(\pi_{s, k}(\varphi b) - \pi_{s,k}(b) \pi_{s,k}(\varphi) ) \big({\mathrm d}X_{s}- \pi_{s,k}(b){\mathrm d}s \big) + \int^{t}_{0}\pi_{s,k+1}( \widetilde{\mathcal A}_{s}\varphi){\mathrm d}s
\left\{\phi_{a}^{(1)},\phi^{(1)}\right\}=\left\{\phi_{a}^{(1)},T\right\}=\left. \left\{T,\phi_{3,n}^{(1)}\right\}\right|_{\phi =T=0}=\left. \left\{T,T\right\}\right|_{\phi =T=0}=0\;,\;\;a=1,2\,.
l=\int d\tau=\int\frac{d\tau}{d\phi}d\phi=\int\sqrt{\frac{(d\tau)^{2}}{(d\phi)^{2}}}d\phi=\int\sqrt{\frac{-g_{\mu\nu}dx^{\mu}dx^{\nu}}{d\phi d\phi}}d\phi=\int fd\phi
\begin{cases}\tau u_{ttt}+ \alpha u_{tt}-c^2 \Delta u -b \Delta u_t =0 \textrm{on $(0,T)\times\Omega$}\frac{\partial u}{\partial \nu}=g \textrm{on $(0,T)\times\Gamma_0$}\frac{\partial u}{\partial \nu}+\frac{1}{c}u_t= 0 \textrm{on $(0,T)\times\Gamma_1$}u(0,x)=u_0(x)\,,\; u_t(0,x)=u_1(x)\,; u_{tt}(0,x)=u_2(x) \textrm{on $\Omega$}\end{cases}
\det{\cal L}_{{\tilde{x}}^{\prime}}= 2 \epsilon_{1}^{2}\epsilon_{2}\epsilon_{2 1}a_{2 1}\left( a_{2 1}- \epsilon_{1}\right) \left( a_{1 2}+ \epsilon \right) \left( a_{1 2}+ 2 \epsilon_{1}+ \epsilon_{2}\right) .
\iint F(x,z)W^{\rho_\sigma}_v(u)((x,z)^{-1}) \,dz dx =\iint F(x,z)W^{\rho_\sigma}_v(u)((x^{-1},\overline{z}\sigma(x^{-1},x) )) \,dz dx =\iint \overline{z}f(x) W^{\rho}_v(u)(x^{-1}) z \overline{\sigma(x,x^{-1})}\,dzdx = f\#W^\rho_v(u)(1).
\mu_{\{v,u\}}\mu_{\{u,H\}}-\mu_{u}\mu{\{v,u,H\}}=\sum_{\substack{P\in \mathcal{P}(v,H) P\, in\, G_w-u}}(W(P))^{2}\mu(G_w-u-P,x)\mu(G_w-\{H,u,P\},x).
\left(\sqrt{2\ln \tfrac{e}{\gamma}}+\sqrt{\tfrac{\alpha \epsilon^2}{2\gamma}}\right)^2=2\ln \frac{e}{\gamma}+2\sqrt{\tfrac{\alpha \epsilon^2}{2\gamma}}\sqrt{2\ln \tfrac{e}{\gamma}}+\frac{\alpha \epsilon^2}{2\gamma}\le \frac{1-\sqrt{\alpha}}{\gamma}\epsilon\le 1-\epsilon+\frac{1-\sqrt{\alpha}}{\gamma}\epsilon.
\mathbb P[T_{N}=x]=\mathbb P[T_{N+1}+z_{N}=x]=\int_{-\infty}^\infty \mathbb P\left[T_{N+1}+z_{N}=x\biggl| T_{N+1}=y\right]dF_{N+1}(y) &=\int_{-\infty}^\infty \mathbb P\left[z_{N}=x-y\biggl| T_{N+1}=y\right]dF_{N+1}(y)=\int_{-\infty}^\infty \mathbb P\left[z_{N}=x-y\right]dF_{N+1}(y)=0,
|B_{n,k}| \le k \, 2^k \sum_{\substack{n_1 + \cdots + n_k = n n_i \ge 1,\, 2\le i\le k, \; n_1 \ge n/k}}C_{n_1-1}C_{n_2-1}\cdots C_{n_k-1}= k \, 2^k \sum_{\substack{n_2 + \cdots + n_k \le n-(n/k) n_i \ge 1,\, 2\le i\le k}}C_{n-n_2-\cdots-n_k-1}C_{n_2-1}\cdots C_{n_k-1}\, .
\frac{d}{d \Lambda}e^{- \frac{1}{\hbar}\left( L^{\Lambda , \Lambda_{0}}( \varphi ) + I^{\Lambda , \Lambda_{0}}\right)}= \frac{\hbar}{2}\langle \frac{\delta}{\delta \varphi}, \dot{C}^{\Lambda , \Lambda_{0}}\frac{\delta}{\delta \varphi}\rangle e^{- \frac{1}{\hbar}\left( L^{\Lambda , \Lambda_{0}}( \varphi ) + I^{\Lambda , \Lambda_{0}}\right)}.
{\cal L}^{(0)}=-\frac{e^2 B^2}{24\pi^{2}}\ln \frac{\Lambda^{2}}{m^{2}}-\frac{e^2 B^2}{8\pi^{2}}\int_{0}^{\infty}\frac{ds}{s^{2}}\;e^{-m^2s/(eB)}(\coth s-\frac{1}{s}-\frac{s}{3})
\lim_{k\to \infty}\int_{\mathbb{R}^{N}}\Phi_{R}^{q(x)}|u_{k}(x)|^{\alpha(x)}|v_{k}(x)|^{\beta(x)}\,dx = \lim_{k\to \infty}\int_{\mathbb{R}^{N}}\Phi_{R}^{q(x)}|u_{k}(x)|^{\alpha(x)}|v_{k}(x)|^{\beta(x)}\,dx > \frac{\nu_{\infty}}{2}
P_t(x,y)= \frac1{2\pi^{(n+1)/2}}\int_0^\infty\frac{t}{s^{3/2}}\,e^{-\frac{t^2}{4s}}\,\frac{e^{-\frac{|y-e^{-s}x|^2}{1-e^{-2s}}}}{(1-e^{-2s})^{n/2}}\,ds.
\frac{1}{|Q|}\int_Q{\sum_{k=\mu+1}^{\infty}{\big|T_{[d_k]}f(x)\big|}}dx \leq \frac{1}{|Q|}\sum_{P\in\mathcal{B}_Q}{\int_P{\sum_{k=\mu+1}^{\infty}{\big|T_{[d_k]}f(x) \big|}}dx}&\lesssim \sup_{P\in\mathcal{D}_{\mu}}\frac{1}{|P|}\int_P{\sum_{k=\mu+1}^{\infty}{\big|T_{[d_k]}f(x)\big|}}dx
\left(\partial_{0,\nu}\left(\begin{array}{cc}1 0 0 1\end{array}\right)+\left(\begin{array}{cc}0 -\partial_{1}-\partial_{1}0\end{array}\right)\right)\left(\begin{array}{c}u v\end{array}\right)=\left(\begin{array}{c}\partial_{0,\nu}^{-1}f 0\end{array}\right).
\vec{e}_{0}=\frac{1}{\sqrt{1-2m/r}}\partial_{t},\vec{e}_{1}=\sqrt{1-2m/r}\partial_{r},\vec{e}_{2}=\frac{1}{r}\partial_{\theta},\vec{e}_{3}=\frac{1}{rsin(\theta)}\partial_{\phi}
= e^{- \frac{i}{2 R}\frac{\partial}{\partial t_{0}}}\sum_{n}\frac{1}{( \frac{1}{2 \sqrt{\alpha^{\prime}}}( 2 n + 1 ) i - t_{0})^{2}}= e^{- \frac{i}{2 R}\frac{\partial}{\partial t_{0}}}\frac{\partial}{\partial t_{0}}\sum_{n}\frac{1}{\frac{1}{2 \sqrt{\alpha^{\prime}}}( 2 n + 1 ) i - t_{0}}.
\hat c(\gamma) := C_s \mathrm{P.V.}\int_{-\infty}^{+\infty}[|1 + \tau|^{-\gamma}- 1] |\tau|^{-(1 + 2s)}d\tau c^\perp(\gamma) := 2 C_s \int_{0}^{+\infty}[(1 + \tau^2)^{-\gamma/2}- 1] \tau^{-(1 + 2s)}d\tau.
\Phi=\left( \begin{array}{cc}-HC 0 0 - HC \end{array}\right),\,\,\,\,\,\,{\cal K}=\left( \begin{array}{cc}K^{11}K^{12}K^{21}K^{22}\end{array}\right),\,\,\,\,\,\,\chi^{\mu}=\left( \begin{array}{c}K^{13}\psi^{3\mu\dagger}K^{23}\psi^{3\mu\dagger}\end{array}\right).
- \ln \frac{|\xi-z|}{|\xi-z^*||z|}\leq \frac{1}{2}\left( \frac{\frac{(1-|\xi|^2)(1-|z|^2)}{|\xi-z^*|^2|z|^2}}{\frac{|\xi-z|^2}{|\xi-z^*|^2|z|^2}}\right)^{\frac{1}{2}}\leq \frac{(1-|\xi|)^{\frac{1}{2}}(1-|z|)^{\frac{1}{2}}}{|\xi-z|}.
[\mathfrak{h}^{s_0,p_0}(\Omega),\mathfrak{h}^{s_1,p_1}(\Omega)]_{\theta}=\mathfrak{h}^{s,p}(\Omega) [\mathfrak{h}^{-s_0,p_0'}(\Omega),\mathfrak{h}^{-s_1,p_1'}(\Omega)]_{\theta}=\mathfrak{h}^{-s,p'}(\Omega)
\Delta_\phi ( L(\psi) - \tilde L (\tilde \psi) ) = f - \tilde f - \Delta_\phi \tilde L(\tilde \psi) + \Delta_{\tilde \phi}\tilde L (\tilde \psi) &= f - \tilde f + (g_{\tilde \phi}^{i \bar \jmath}- g_\phi^{i \bar \jmath}) (\tilde L(\tilde \psi))_{i \bar \jmath}
\sum_{k=2^n}^{2^{n+1}-1}a_kk^{it}=\sum_{k=2^n}^{2^{n+1}-1}(S_{k,n}-S_{k-1,n})k^{it}= \sum_{k=2^n}^{2^{n+1}-1}S_{k,n}(k^{it}-(k+1)^{it}) + 2^{(n+1)it}S_{2^{n+1}-1,n}\, .
S_{M}(\omega_{m},T)=\int_{-\infty}^{\infty}\tilde{E}_{in}^{*}(\omega)\tilde{E}_{in}(\omega+\omega_{m})e^{j\omega_{m}[\frac{\phi(\omega+\omega_{m})-\phi(\omega)}{\omega_{m}}]}e^{j\omega T}d\omega
V=\vert \Omega \vert^{-1}\int_{\Omega}(b(\mathbf{D})\Lambda (\mathbf{x}))^*g(\mathbf{x})(b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x}))\,d\mathbf{x}, &W=\vert \Omega \vert^{-1}\int_{\Omega}(b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x}))^*g(\mathbf{x})(b(\mathbf{D})\widetilde{\Lambda}(\mathbf{x}))\,d\mathbf{x}.
\mathbf{E}_{r\delta_{x}}\exp\left\{-\int_{B}\mathrm{d}y\X_{s}(y)p_{s}(y-x)\right\}=\exp\left\{-r\,v_{s,x}^{n}(s,x)\right\}\leq\exp\left\{-h_{\beta}(s)v_{s,x}^{n}(s,x)\right\},
\boldsymbol{K}_{j-1}:=\sum_{r=1}^n \frac{\tilde{S}_{jr}}{\tilde{S}_{1r}}\left( \frac{\partial}{\partial q_r}\otimes \mathrm{d}q_r+ \frac{\partial}{\partial p_r}\otimes \mathrm{d}p_r\right), j=1,\ldots,n .
U_{1 .}U_{2 .}( U_{1 2}U_{2 1}U_{1 .}U_{2 .})^{p_{1}}= \alpha ( p_{1}, p_{2}) \; \; U_{1 .}U_{2 .}( U_{1 2}U_{2 1}U_{1 .}U_{2 .})^{p_{2}}.
x_{n}=\sum_{m=1}^{n}\frac{\left(-1\right)^{m}}{n!}\sum_{p=1}^{m}\left(-1\right)^{m-p}\binom{m}{p}S_{p,n}=\sum_{p=1}^{n}\frac{\left(-1\right)^{p}}{n!}S_{p,n}\sum_{m=p}^{n}\binom{m}{p}.
\left\|Y_1\left(\begin{array}{cc}1 0 0 1\end{array}\right)Y_2\right\|^2 < |c_{11}d_{11}|^2+|c_{11}d_{12}|^2+|c_{22}d_{22}|^2 &=|a_{11}b_{11}|^2+|a_{11}b_{22}|^2 &=\left\|X_1\left(\begin{array}{cc}1 1 0 0\end{array}\right)X_2\right\|^2
\theta^{AA^{\prime}}=(\mathbf{L}^{-1}\mathbf{)}_{B^{\prime}}^{A^{\prime}}(\mathbf{L}^{-1}\mathbf{)}_{B}^{A}\mathbf{dx}^{BB^{\prime}}=(\delta_{B^{\prime}}^{A^{\prime}}+\omega_{B^{\prime}}^{A^{\prime}}\overline{\mathbf{m}}\mathbf{)}(\delta_{B}^{A}+\omega_{B}^{A}\mathbf{m)dx}^{BB^{\prime}}
[\begin{matrix}1\\ 0\\ 0\\ 0\\ 0\\ \vdots\end{matrix}],[\begin{matrix}1\\ -1\\ 0\\ 0\\ 0\\ \vdots\end{matrix}],[\begin{matrix}5\\ -14\\ 21\\ 0\\ 0\\ \vdots\end{matrix}]
\tilde u_+(\xi)=\sum\limits_{k=1}^n\frac{\tilde c_k(\xi')}{\sigma_+(\xi)\Lambda^{k-\ae}_{+}(\xi',\xi_m)}+\frac{1}{\sigma_+(\xi)\Lambda^{n-\ae}_{+}(\xi',\xi_m)}(\Pi_{\pm}\Lambda^{n}_{+}\tilde h)(\xi',\xi_m),
L^{int}= - e (\overline\psi_1 \gamma^\mu \gamma^5 \psi_1 -\overline \psi_2 \gamma^\mu\gamma^5\psi_2 ) C_\mu + ie (\overline \psi_1 \gamma^\mu \psi_1 +\overline \psi_2\gamma^\mu \psi_2 ) B_\mu
E[J,B]=\frac{1}{2}\int_{D}(I(\vec{x})-J(\vec{x}))^{2}d\vec{x}+\lambda\frac{1}{2}\int_{D/B}\vec{\nabla}J(\vec{x})\cdot\vec{\nabla}J(\vec{x})d\vec{x}+\nu\int_{0}^{1}(\frac{d}{ds}B(s))^{2}ds
Q e^{i k X ( \pi / 2 ) / 2}| \tilde{I}\rangle = Q \exp \left( \sqrt{2 \alpha^{\prime}}k \left({\frac{a_{0}}{2}}+ \sum_{n \ge 1}\sqrt{{\frac{1}{n}}}( - 1 )^{n}a_{2 n}\right) \right) | \tilde{I}\rangle = 0 .
{\cal L}_q(x,x')=\bar{q}_{\alpha\alpha'}(x,x')\left(i\hbar c\gamma^\mu\partial_\mu-mc^2 \right)_{\alpha\beta}\left( i\hbar{c}'\gamma^{\mu'}\partial_{\mu'}-m{c'}^2\right)_{\alpha'\beta'}q_{\beta\beta'}(x,x')
M_{ab,c}^{\hspace{.18in}d}\equiv{\delta_{(a}}^d\hat{G}_{b)c}+\frac{1}{2}f_{e(a}^{\ \ \d}f_{b)c}^{\ \ \e}\hspace{.3in}N_{ab,c}^{\hspace{.18in}de}\equiv\frac{i}{2}\delta_{(a}^{(d}f_{b)c}^{\ \ \e)}.
-264\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right\}^{(20)}=\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{V_C}-\widetilde{V_C}\otimes \widetilde{V_C}+\widetilde{V_C})\right\}^{(20)}.
M_{ii}=\frac{1}{kf_k(B)^{k-1}}\sum_{\mbox{\tiny$\begin{array}{c}i\leq j_1<j_2<...<j_{k-1}\leq n, j_1,...,j_{k-1}\neq i \end{array}$}}\det{B_{(j_1,...,j_{k-1})}},
f(p)=\sup_{x\in\langle H_o\cdot p\rangle}f(x)\geq \sup_{x\in \langle H_o\cdot p\rangle\cap C}f(x)\geq\inf_{x\in \langle H_o\cdot p\rangle\cap C}f(x)\geq \inf_{\gamma\in C}f(\gamma).
{\rm d}({\rm L}_n) \ge{\rm d}\bigg(\bigcup_{a\in A}\bigcup_{c\in C_a^v}{\rm L}_{ac}\bigg) = \sum_{a\in A}\sum_{c\in C_a^v}{\rm d}({\rm L}_{ac}) = \sum_{a\in A}{\rm d}({\rm L}_a)\,\sum_{c\in C_a}\frac{1}{c},
\omega(\sigma^{j})\leq \frac{1}{\tilde{\delta}}\sum^{j/2}_{l=0}\omega_{1}(\sigma^{j-l})\omega_{2}(\sigma^{l})+\frac{1}{\tilde{\delta}}\sum^{j}_{l=j/2}\omega_{1}(\sigma^{j-l})\omega_{2}(\sigma^{l})+\sigma^{j\alpha}.
S_{\tau_{n}(i)}>0,\S_{\tau_{n+m}(i)}-S_{\tau_{n}(i)}>-q_{n},\S_{\tau_{(s+1)n+m}(i)}-S_{\tau_{sn+m}(i)}>0, &\quad\min_{0\le j\le n}\big(S_{\tau_{sn+m}(i)+j}-S_{\tau_{sn+m}(i)}\big)\ \ge\q_{n},s=1,\ldots,r
[ W^{g e n}, \underline{{{m}}}\partial_{\underline{{{m}}}}] = z \int \left\{\hat{\xi}_{A}\frac{m}{e}\frac{\delta}{\delta \varphi_{2}}+ \xi_{A}\frac{m}{e}\frac{\delta}{\delta \hat{\varphi}_{2}}\right\}+ ( \chi \partial_{\xi}+ \chi_{A}\partial_{\xi_{A}}) z \xi_{A}\frac{m}{e}\int \frac{\delta}{\delta q_{2}}
\operatorname{p}^{(OW)}_{f_K}(\phi)=\sup_{\eta\in \mathbb{U}_X}\operatorname{p}^{(OW)}_{f_K}[\phi,\eta_k]\leq\sup_{\eta\in \mathbb{U}_X}\operatorname{p}^{(OW)}_{f_K}[\phi,\eta_K]\leq \sup_{\epsilon\in \mathbb{U}_X}\operatorname{p}^{(OW)}_{f_K}[\phi,\epsilon]=\operatorname{p}^{(OW)}_{f_K}(\phi)
\big(I+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}\big)(A+UV^{\ast}) =A+UV^{\ast}+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}A+(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}A^{\dagger}UV^{\ast}&=A+UV^{\ast}-(A^{\dagger})^{\ast}VE_{S_{A}}V^{\ast}F_{A}+(A^{\dagger})^{\ast}VE_{S_{A}}S_{A}V^{\ast}&=A+UV^{\ast}.
{\cal L}^{\prime}_{vec}\, = \,{\rm i}\, \Bigl [{\cal F}^{-T}\, \bigl ( A + B{\bar{\cal N}}\bigr )^T{\bar{\cal N}}^\prime( A + B{\bar{\cal N}}\bigr ){\cal F}^{-}\, - \,{\cal F}^{+T}\, \bigl ( A + B{\cal N}\bigr )^T{\cal N}^\prime( A + B{\cal N}\bigr ){\cal F}^{+}\Bigr ]
\frac{1}{2}\frac{d}{dt}\|U\|^2_P + \epsilon (\|U_x\|^2_P+\|U_y\|^2_P) +\frac{1}{2}\oint\limits_{\partial\Omega}U^T (n_1 A+n_2 B) U - 2\epsilon U^T(n_1 PU_x+n_2PU_y) \ds= 0.
d_i=x_i\frac{\partial}{\partial x_i}+\beta \sum_{j<i}\frac{x_j}{x_i-x_j}\left( 1 - K_{ij}\right)+\beta \sum_{i<j}\frac{x_i}{x_i-x_j}\left( 1 - K_{ij}\right)+\beta\left(i-1\right)
=\frac{p_{1}V_{1}}{1-n}((\frac{V_{1}}{V_{2}})^{n-1}-1)=\frac{p_{1}V_{1}}{1-n}((\frac{p_{2}}{p_{1}})^{\frac{n-1}{n}}-1)=\frac{p_{1}V_{1}}{1-n}(\frac{T_{2}}{T_{1}}-1)
d\Pi(\boldsymbol{\theta}_{-(\boldsymbol{j},\boldsymbol{k})})=\prod_{(\boldsymbol{x,y})\neq(\boldsymbol{j},\boldsymbol{k})}[(1-\omega_{\boldsymbol{x},n})d\delta_0(\theta_{\boldsymbol{x,y}})+\omega_{\boldsymbol{x},n}p(\theta_{\boldsymbol{x,y}})d\theta_{\boldsymbol{x,y}}].
\begin{aligned}z_i \in \Delta_i,\ \delta_{it}\in \{0,1\}, z_{i\tau(i,t)}\geq \delta_{it}\geq z_{i\tau(i,t)+1}i = 1, \ldots,d ,\t = 1, \ldots,l_i-1,\end{aligned}&f_i = a_{i0}z_{i0}+ \sum_{j = 1}^{n}(a_{ij}- a_{ij-1})z_{ij}=: F_i(z_i) i =1, \ldots, d .
f(x_0) =(\chi_rf)(x_0)=(B\nabla^4_b\chi_rf)(x_0)+\sum^3_{j=0}(B\nabla^j_b\chi_rf)(x_0) &=\int(B\nabla_b)(x_0,y)\nabla^3_b(\chi_rf)(y)dv_M(y)+\sum^3_{j=0}\int B(x_0,y)\nabla^j_b(\chi_rf)(y)dv_M(y),
[a_j^{-1}S_j , G_{t_k}]_{samp}= \bigg[\frac{S_j}{a_j}, \overline{a_k}\frac{G_{t_k}}{\overline{a_k}}\bigg]_{samp}= \frac{a_k}{a_j}[S_j , M_k]_{samp}= \frac{a_k}{a_j}\delta_{j,k}= \delta_{j,k}
\langle \phi_{n , m}\left( z_{1}\right) \phi_{1 , 2}\left( z_{2}\right) \phi_{1 , 2}\left( z_{3}\right) \phi_{n , m}\left( z_{4}\right) \rangle = \left( \prod_{i < j}z_{i j}^{\gamma_{i j}}\right) \eta^{2 \alpha_{n , m}\alpha_{1 , 2}- \gamma_{1 2}}\left( 1 - \eta \right)^{2 \alpha_{1 , 2}^{2}- \gamma_{2 3}}Y \left( \eta \right)