harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
u_{f}(x,t) =\mathfrak{S}_{t}f(x)-\Pi_{x}\left[\int_{0}^{t}\psi(\xi_{s},u^{t-s}_{f}){\rm d}s\right] &=\mathfrak{S}_{t}f(x)-\Pi_{x}\left[\int_{0}^{t}\phi^{L}(\xi_{s},u_{f}(t-s,\xi_{s})){\rm d}s\right]-\Pi_{x}\left[\int_{0}^{t}\phi^{NL}(\xi_{s},u^{t-s}_{f}){\rm d}s\right].
\begin{cases}T_1(u) + a T(u) - T(u) a = T(a \cdot u -u \cdot a + H (a, Tu) - H (Tu, a)) + T_1'(u), a T_1(u) - T_1(u) a = T_1' (a \cdot u-u \cdot a + H (a, Tu) - H (Tu, a)).\end{cases}
\lim_{q\rightarrow 1}F\left(\frac{1}{2},\frac{5-q}{2(q-1)};\frac{3+q}{2(q-1)};\frac{[\Re(p)]^2}{|p|^2}\right)=F\left(\frac{1}{2},4;4;\frac{[\Re(p)]^2}{|p|^2}\right)
\begin{cases}|K_s(L(\xi, \sigma)y)|^2 = O(|L(\xi, \sigma)y)|^{-1}) e^{-y|L(\xi, \sigma)|}, |K_{1-s}(L(\xi, \sigma)y)|^2 = O(|L(\xi, \sigma)y)|^{-1}) e^{-y|L(\xi, \sigma)|},\end{cases}
\mu(\theta)= - R\theta\int_{\mathbb{R}^{3}}\hat{B}_{ij}(\frac{v-u}{\sqrt{R\theta}})B_{ij}(\frac{v-u}{\sqrt{R\theta}})\,dv>0,i\neq j, \kappa(\theta)= -R^{2}\theta\int_{\mathbb{R}^{3}}\hat{A}_{j}(\frac{v-u}{\sqrt{R\theta}})A_{j}(\frac{v-u}{\sqrt{R\theta}})\,dv>0,
\lim_{k\rightarrow \infty}\ln \left(\int_{\mathbb{S}^n}\tilde{K}e^{n(\psi_k-w_0\circ \pi)}dg_0\right)= \ln \left(\int_{\mathbb{S}^n}\tilde{K}e^{n(\psi_0-w_0\circ \pi)}dg_0\right)
Z_H^k(t)=c_{k,d}\int_{\mathbb{R}^k}^{''}\frac{e^{i(u_1+\ldots +u_k)t}-1}{i(u_1+\ldots +u_k)}|u_1|^{-d}\ldots |u_k|^{-d}\widehat{W}(du_1)\ldots \widehat{W}(du_k);
\pi_{\mathcal{K}}\circ\mathcal{S}_{0}(sU) =\frac{s^3}{8}(a^2+b^2+18c^2+18d^2)(aV_A^{\perp}+b V_B^{\perp}) &\quad+\frac{s^3}{8}(18a^2+18b^2+c^2+d^2)(cV_C^{\perp}+dV_D^{\perp})+O(s^4).
b_{\underline{{{n}}}\sigma}^{\dagger}= \frac{\pi / L_{\perp}}{\sqrt{8 \pi^{3}n}}b_{\underline{{{p}}}\sigma}^{\dagger}\, , \; \; a_{\underline{{{m}}}}^{\dagger}= \frac{\pi / L_{\perp}}{\sqrt{8 \pi^{3}m}}a_{\underline{{{q}}}}^{\dagger}\, , \; \;
H_{k}^{p}(C)=H_{k}^{p}({\bf c}):=\begin{vmatrix}c_p c_{p+1}\cdots c_{p+k-1}c_{p+1}c_{p+2}\cdots c_{p+k}\vdots \vdots \ddots \vdots c_{p+k-1}c_{p+k}\cdots c_{p+2k-2}\end{vmatrix}
\begin{array}{ll}\delta w(t,x)=\int_0^t\int_{{\mathbb R}^n}{\Big (}\sum_{ij}\left(a^{w^2}_{ij}-a^{w^1}_{ij}\right)(s,y) w^1_{,i,j}(s,y)+ +\sum_i(b^{w^2}_i(x)-b^{w^1}_i(x))\frac{\partial w^1}{\partial x_i}{\Big )}p^{w^2}(t,x,s,y)dsdy,\end{array}
\beta(\lambda) = ~i\int_{\real}u(x)e^{-i\lambda x}[(I+R_{\lambda+0i})1](x)~dx ~- i\int_{\real}u(x)e^{-i\lambda x}[(I+R_{\lambda+0i})\widetilde T_{\lambda+0i}(I+R_{\lambda+0i})1](x)~dx +O\left(\frac1{\lambda}\right).
\left( \begin{array}{c c}{0}&{- t \sum_{j}\mathrm{\Large e^{i{\bf k \cdot u}_{j}}}}\\{- t \sum_{j}\mathrm{\Large e^{i{\bf k \cdot v}_{j}}}}&{0}\\ \end{array}\right) \left( \begin{array}{c}{c_{\bullet}}\\{c_{\circ}}\\ \end{array}\right) = E ({\bf k}) \left( \begin{array}{c}{c_{\bullet}}\\{c_{\circ}}\\ \end{array}\right)
\beta_{0,1}(x_0) :=\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_{r}(x_0)}e^{a v_k(x)}\, dx \; \beta_{0,2}(x_0):=\lim_{r\to0}\lim_{k\to+\infty}\frac{1}{2\pi}\int_{B_{r}(x_0)}\varepsilon_{1,k}|x|^{2N}e^{a v_k(x)}\, dx.
\bigtriangleup^{\mu \nu}( x - y , m^{2}) = \int d^{2}z \left[ g^{\mu \nu}\delta^{( 2 )}( x - z ) - \partial_{x}^{\mu}\partial_{x}^{\nu}\bigtriangleup ( x - z ) \right] \bigtriangleup ( z - y ; m^{2}) \; ,
{\cal V}_{1 , 1}^{( i )}\geq 0 , \; \;{\cal V}_{2 , 2}^{( i )}\geq 0 , \; \;{\cal V}_{1 , 1}^{( i )}\geq |{\cal V}_{1 , 2}^{( i )}| , \; \;{\cal V}_{2 , 2}^{( i )}\geq |{\cal V}_{2 , 1}^{( i )}| .
\Gamma_{A=0}= 2V \int_0^\infty \frac{dt}{2t}\left[{\rm Tr}_{\rm NS}\left( \frac{1+(-1)^{\cal F}}{2}\,q^{2L_0-1}\right) -{\rm Tr}_{\rm R}\left( \frac{1+(-1)^{\cal F}}{2}\,q^{2L_0}\right)\right]
\hat{v}_\alpha(0)^q+\left\vert\frac{\nabla \hat{v}_\alpha(0)}{\hat{v}_\alpha(0)}\right\vert^n+\left\vert\frac{\nabla^2\hat{v}_\alpha(0)}{\hat{v}_\alpha(0)}\right\vert^\frac{n}{2}+\vert\mathcal{L}_\xi\hat{Z}_\alpha\vert_\xi(0)=1
\overline X:=\sup\big\{y\in[r_0-\varepsilon_2,\Lambda_0+\varepsilon_2]:\,\max_{[0,t-s]}\,(x+B_{\cdot\wedge\tau^B_{r_0}}-\Lambda_{s+\cdot})<0\big\}\vee (r_0-\varepsilon_2), &\underline X:=\inf\big\{y\in[r_0-\varepsilon_2,\Lambda_0+\varepsilon_2]:\,\max_{[0,t-s]}\,(x+B_{\cdot\wedge\tau^B_{r_0}}-\widetilde\Lambda_{s+\cdot})\ge0\big\}.
E\Bigg[\sum_{j=1}^{\xi_0(t)}e^{-k\alpha\sigma_j}M_k(t-\sigma_j,\mathbf{h})\Bigg] =E\Bigg[\int_0^t e^{-k\alpha s}M_k(t-s,\mathbf{h})\,\xi(\d s)\Bigg] &=\int_0^t e^{-k\alpha s}M_k(t-s,\mathbf{h})\,\mu(\d s).
\mathbb E|M_{\infty}- M^n_{\infty}|^p \eqsim_{p}\mathbb E[M - M^n]_{\infty}^{\frac p2}=\mathbb E \Bigl([M^a - M^n]_{\infty}+ [M^q]_{\infty}\Bigr)^{\frac p2}\geq \mathbb E[M^q]_{\infty}^{\frac p2},
\omega_{\;b\mu}^{a}(x)\longrightarrow \omega_{\;b\mu}^{\prime a}(x)=\Lambda_{c}^{a}(x)\Lambda_{b}^{d}(x)\omega_{\;d\mu}^{c}(x)+\Lambda_{c}^{a}(x)\partial_{\mu}\Lambda_{b}^{c}(x).
(\overline L_0\cdots\overline L_d)_S\geqslant\delta_0\operatorname{\widehat{\mu}_{\max}^{\mathrm{asy}}}(\overline L_0)+\sum_{i=1}^d\delta_i\operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L_i).
\varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right)= 0
\frac{|\delta_1|}{|z-k|^{k+1}}\quad=\Bigg|\frac{(-\|x\|^2/2)^{k+1}}{(k+1)!}\Bigg|\Bigg| \frac{g_1^{(k+1)}(\zeta)}{(-\|x\|^2/2)^{k+1}}\Bigg| &\quad\leq\frac{(M^2/2)^{k+1}}{(k+1)!}\left(1-\frac{\|x\|^2}{d}\zeta\right)^{(d-p-3k-3)/2}e^{\frac{k}{2}\|x\|^2}&\quad\leq\frac{M^{2(k+1)}}{2^{k+1}(k+1)!}e^{\frac{k}{2}M^2},
\Theta_{d,D}(q, \lambda_D)(\widehat{\alpha}) := \left ( \frac{1 - \lambda_D}{\sqrt{\lambda_D}(q-q^{-1})E_D}\right)^{n-1}\left ( \sqrt{\lambda_D}\right)^{\varepsilon(\alpha)}{\rm tr}_{d,D}(\delta ( \alpha ) ) ,
f(x; u, \gamma, \tau) ~\equiv~{{sinh\big({\pi i \over \tau}(1 - \gamma) x \big) ~ sinh\big({2 \pi i \over \tau}\gamma u x\big)~sinh\big({\pi i (N-1) \over \tau}\gamma x \big)}\over{x ~ sinh\big({\pi i \over \tau}x \big)~ sinh\big({\pi i N \over\tau}\gamma x \big)}}\ .
\exp^{\diamond}\big( \int_0^T\int_{\mathbb{R}}\psi(s,\zeta) \tilde{N}(ds,d\zeta)\big)=K \exp \big( \int_0^T\int_{\mathbb{R}}\ln (1+\psi(s,\zeta))\tilde{N}(ds,d\zeta) \big)
\aligned m_\nu(a,b)= \inf_{\mathcal{P}_\nu(a,b)}\Bigl\{\frac{\gamma_p+\gamma_q-2}{2(\gamma_p+\gamma_q)}(|\nabla u|^2_2+|\nabla v|^2_2) &+\frac{22^*_\mu-\gamma_p-\gamma_q}{22^*_\mu(\gamma_p+\gamma_q)}\int_{\mathbb{R}^N}[(I_\mu\ast|u|^{2^*_\mu})|u|^{2^*_\mu}+(I_\mu\ast|v|^{2^*_\mu})|v|^{2^*_\mu}]\Bigr\}>0.\endaligned
\underset{\alpha\rightarrow0}{\lim}\sum_{n=0}^{\infty}\frac{t^{n}}{n!}{\displaystyle\int\limits_{-1}^{0}}w_{n}\left( x;\alpha,\beta,r\right) dx =\frac{\beta t}{e^{\beta t}-1}e^{rt}& =\sum_{n=0}^{\infty}B_{n}\left( \frac{r}{\beta}\right) \beta^{n}\frac{t^{n}}{n!}.
\frac{i \theta^{i m}}{2 \pi}\frac{i \theta^{j n}}{2 \pi}\int_{0}^{1}d \tau^{\prime}e^{- 2 \pi i \tau^{\prime}}l^{m^{\prime}}F_{m m^{\prime}}( x + l \tau^{\prime}) \int_{0}^{1}d \tau^{\prime \prime}e^{2 \pi i \tau^{\prime \prime}}l^{n^{\prime}}F_{n n^{\prime}}( x + l \tau^{\prime \prime}) .
H_{2,n}^{(\{1,1\})}(x) =8n(n-1)(n-2)\int_0^xH_{\{1,1\}}(t)H_{n-3}(t)\,{\rm d}t+16(n-1)(n-2)H_{n-2}(0) &=\int_0^xH_{\{1,1\}}(t)H'''_{n}(t)\,{\rm d}t+8(n-2)H_{n-1}'(0).
\sum_{(i,j)\in S_1}\pi_1^*(i,j)P^*((i,j),(k,l)) =\sum_{j\in\Sigma_2}\sum_{i\in\Sigma_1}\pi_1(i)P_{ij}P_{jk}P_{kl}&=\sum_{j\in\Sigma_2}\pi_2(j)P_{jk}P_{kl}=\pi_1(k)P_{kl}=\pi_1^*(k,l).
\int_X-\chi(u_{k}-\phi)\theta_{u_{k}}^n =\int_X-\chi(-k)\theta_{\phi}^n+\int_{\{u>\phi-k\}}\chi(-k)\theta_{u}^n+\int_{\{u>\phi-k\}}-\chi(u-\phi)\theta_{u}^n &=-\chi(-k)d_{\theta}(\phi, u)+\int_{\{u\leq\phi-k\}}-\chi(-k)\theta_{u}^n+\int_{\{u>\phi-k\}}-\chi(u-\phi)\theta_{u}^n &=-\chi(-k)d_{\theta}(\phi, u)+\int_{X}-\chi(u-\phi)\theta_{u}^n.
r_x \leq A_{ess(pg(f))}+ \displaystyle \sum_{t=0}^{s-1}2 \times S((0,0)-(n+a_{i_t},m+b_{i_t})-(n+a_{i_{t+1}},m+b_{i_{t+1}})) = 2 \times S((0,0)-(n,m)-(n+a_k,m)-(n+a_k,m+b_k)) \leq (n+a_k) \cdot b_k + (-m) \cdot a_k.
V_{st}=\sigma\iint dxdy[\sqrt{1+(\frac{\partial\eta}{\partial x})^{2}+(\frac{\partial\eta}{\partial y})^{2}}-1]\approx\frac{1}{2}\sigma\iint dxdy[(\frac{\partial\eta}{\partial x})^{2}+(\frac{\partial\eta}{\partial y})^{2}],
( \vec{X}( x_{1}) , \vec{X}( x_{2}) ) = \sum_{\vec{k}\in K}\left\{R ( k , | \tilde{t}_{k}( x_{1}) - \tilde{t}_{k}( x_{2}) | ) + b ( k )^{2}\right\}\mathrm{e}^{i k_{\mu}( x_{1}^{\mu}- x_{2}^{\mu})}\; , \hspace{1 c m}x_{1}, x_{2}\in{\sf M}^{4}\; .
2N(x) = \int u_1^{1/p_1}(x-y)u_2^{1/p_2}(y) u_1^{1/p_1}(x-z)u_2^{1/p_2}(z) \times & |g_1(x-y) + g_2(y) - (g_1(x-z) + g_2(z))|^2 \, \mathrm{d}y\mathrm{d}z
\chi_\lambda^+(gx)=\begin{cases}\overline{\chi}_{\lambda_0}^+(gx_0s_0)\chi_{\lambda\backslash\{0\}}^+(x's')+\overline{\chi}_{\lambda_0}^-(gx_0s_0)\chi_{\lambda\backslash\{0\}}^-(x's') x_0s_0\in\tilde{A}_{p(t_0-q)}, 0 \end{cases}
[d^1_{r,s}(m,i)G_{m+r,i+s},G_{n,j}]+[G_{m,i},d^1_{r,s}(n,j)G_{n+r,j+s}] &\quad=2qd^1_{r,s}(m,i)L_{m+n+r,i+j+s}+2qd^1_{r,s}(n,j)L_{m+n+r,i+j+s},
\int_{R}^{\infty}\frac{|\log r|^{p}}{r^{2}}dr =\int_{R}^{\infty}\frac{(\log r)^{p}}{r^{2}}dr=\int_{R}^{\infty}\frac{(r^{-\frac{1}{2p}}\log r)^{p}}{r^{\frac{3}{2}}}dr &\leq 2^{p+1}\left(\frac{p}{e}\right)^{p}\frac{1}{\sqrt{R}}.
f_{X^2}(\mathcal{B}) = \left(\begin{array}{cccc}{t_U}^2 +{t_W}^2 0 t_U t_W t_U t_V 0{t_U}^2 +{t_W}^2 t_U t_V t_U t_W t_U t_W t_U t_V{t_U}^2 +{t_V}^2 0 t_U t_V t_U t_W 0{t_U}^2 +{t_V}^2 \end{array}\right).
{f_{\left.{{\Upsilon_{{T_1},D}}}\right|{\Gamma_{{T_1}}}}}\left( z \right) = \frac{{{\Gamma_{{T_1}}}+ 1}}{{{{\bar \gamma}_0}}}\exp \left({- \frac{{{\Gamma_{{T_1}}}+ 1}}{{{{\bar \gamma}_0}}}z}\right)
H_n(v)=\sum^{2k_n+1}_{r=1}\sum_{(i,j)\in{\cal I}_r}g_{i,(j-1)(2k_n+1)+r}(\tilde{Y}_{i,(j-1)(2k_n+1)+r},v):= \sum^{2k_n+1}_{r=1}H_n(v,r).
Z_{\mathrm{R M T}}= \int_{N \times N \, \mathrm{\scriptsize ~ m a t r i c e s}}\! \! \! \! \! \! \! \! \! \! d \mu ( W ) \ \prod_{f}\det{\left( \begin{array}{c c}{{m_{f}}}&{{W^{\dagger}}}\\{{W}}&{{m_{f}}}\\ \end{array}\right)}.
\langle|H_{\rho=-1,m,N_{s}=1,M_{\sigma}=1}^{\rho=1,n,N_{s}=0,M_{\sigma}=0}|^{2}\rangle_{avg}=\frac{g^{2}}{4\Omega}|\int v_{m}^{*}u_{n}d\tau|^{2}(\tilde{\psi}_{s}\psi_{s}-\frac{\mu c^{2}}{K_{\sigma}}\tilde{\psi}_{s}\beta\psi_{s}),
\int_{-\infty}^\infty\frac{\left(bq^x,aq^{-x};p\right)_\infty}{\left(-q^x,-q^{1-x};q\right)_\infty}\,e^{ixy}\,dx &=\frac{2\pi i/\log q}{\sinh\frac{\pi y}{\log q}}\frac{(-q,-q,e^{iy},qe^{-iy};q)_\infty}{(q,q,-e^{iy},-qe^{-iy};q)_\infty}\sum_{n=-\infty}^\infty \frac{(bq^n,aq^{-n};p)_\infty}{(-q^n,-q^{1-n};q)_\infty}\, e^{iny}.
F_1 = E - \frac{E \times H}{\lvert E \times H \rvert}\times H + i\left(H + \frac{E \times H}{\lvert E \times H \rvert}\times E \right) F_2 = E + \frac{E \times H}{\lvert E \times H \rvert}\times H + i\left(H - \frac{E \times H}{\lvert E \times H \rvert}\times E \right)
S_{gWZ}[A^{h}, h\cdot g] = S_{gWZ}[A,g] - S_{gWZ}[A,h] S^{1,0}_{gWZW}[(A^{1,0})^{h}, h\cdot g] = S^{1,0}_{gWZ}[A^{1,0},g] - S^{1,0}_{gWZ}[A^{1,0},h].
\mathbf{J}_\nu(\, \underline{r}\,\underline{\smash{j}}_0^\intercal ) = \mathbf{J}_\nu(\, \underline{r}\,(\tilde{\underline{\omega}}+\underline{b})^\intercal ) = \sum_{s=-\infty}^\infty \sum_{t=0}^\infty \frac{(-1)^t2^{-2t-s}}{t!(t+s)!}D_{\underline{r}}^{2t+s}\mathbf{J}_{\nu-s}(\, \underline{r}\,\tilde{\underline{\omega}}^\intercal ) D_{\underline{b}}^{2t+s}.
\mathbb{X}^{N}_t=1+\int_0^t\mathbb{X}^{N}_{s^-}\otimes d\mathbf{X}_s+\sum_{0<s\leq t}\mathbb{X}^{N}_{s^-}\otimes\left(-\frac 1 2 (\Delta X_s)^{\otimes 2}+\sum_{k=2}^N \frac{1}{k!}(\Delta X_s, Anti(\Delta \mathbb{X}_s))^{\otimes k}\right).
E ( x ) = \left( \begin{array}{c c}{{I_{R}}}&{{2 B ( x )}}\\{{0}}&{{I_{L}}}\\ \end{array}\right) \; \; F_{R}( x ) = \left( \begin{array}{c}{{G_{R}( x )}}\\{{0}}\\ \end{array}\right) \; \; F_{L}( x ) = \left( \begin{array}{c c}{{0}}&{{G_{L}( x )}}\\ \end{array}\right) .
c_{01}^1 = 1,c_{02}^2=1,c_{07}^7=-1,c_{08}^8=-1\ , c_{13}^1 = 1,c_{15}^2=1,c_{17}^3=1,c_{17}^0=-1, c_{18}^4=1\ , c_{24}^1 = 1,c_{26}^2=1,c_{27}^5=1,c_{28}^6=1, c_{28}^0=-1\ , c_{34}^4 = -1,c_{35}^5=1,c_{37}^7=1\ , c_{45}^3 = -1,c_{45}^6=1,c_{46}^4=-1, c_{47}^8=1\ , c_{56}^5 = 1,c_{58}^7=1\ , c_{68}^8 = 1 \ .
\int_0^1 x^{\lambda - 1}\left(1 - x\right)^{\nu_1 M}dx = \frac{\Gamma(\lambda) \, \Gamma(\nu_1 M + 1)}{\Gamma(\lambda + \nu_1 M + 1)}\sim \frac{\Gamma(\lambda)}{\nu_1^{\lambda}}\cdot \frac{1}{M^{\lambda}},M \to \infty.
\|g\|_{p,\varphi_p}^p=\int^1_0r^{1-pk_p}\int^{2\pi}_0|g(re^{i\theta})|^pd\theta{dr}\geq2\pi|g(0)|^p\int^1_0r^{1-pk_p}dr=\frac{2\pi}{2-pk_p}|g(0)|^p,
\begin{pmatrix}0 0 0 0 0 0 \epsilon^6 0 -\epsilon^{8}0 0 0 0 0 0 0 -\epsilon^4 0 \epsilon^6 0 0 0 0 0 0 \end{pmatrix},\begin{pmatrix}0 0 0 0 0 0 0 0 0 0 -\epsilon^3 \epsilon^8 \epsilon^9 0 0 -\epsilon^2 \epsilon^7 \epsilon^8 0 0 \epsilon^{-3}-\epsilon^2 -\epsilon^3 0 0 \end{pmatrix}.
\mathcal{T}(G)=\{\tau\geq 0\ |\ \forall (u,v)\in E(G):\ \sum_{j\in [q]}\tau_{u,v}(i,j)=\tau_u(i)\ \mbox{and}\ \forall u\in V(G):\ \sum_{i\in [q]}\tau_u(i)=1\}
{\left.{\frac{{{\rm{d}}\rho (\varepsilon )}}{{{\rm{d}}\varepsilon}}}\right|_{\varepsilon = 0}}= \int_0^A{\eta (x)\left\{\ln ({f_X(x)}) + 1+ \frac{1}{2}\ln (1 +{\varsigma^2}x) \right\}{\rm{d}}x}= 0.
{\bar{X}}_{k}(t) =p_{k}-{\bar{B}}_{k}(t)\geq 0,k\in \mathbb{N},{\bar{Y}}(t) =\sum_{k=0}^{\infty}(k-2){\bar{B}}_{k}(t),{\bar{X}}_{0}(t) ={\bar{Y}}(t)+{\bar{\eta}}(t)\geq 0.
\begin{aligned}J_{2}(t) \leq \int_{0}^{t}|\bar{X}_{s}^{h}|^{2}ds+C_{H,T}K(t^{2}){(\kappa(|\bar{X}_{t}^{0}|)^{2}+1)}\cdot\|u_{t}\|^{2}_{\mathcal{H}_{H}}.\end{aligned}
\frac{e^{4A_0}}{2 M^2_{Pl}}\left(3 p_0-\rho_0 + e^{-2A_0}(3 p_{*}- \rho_{*}) \right) = \kappa^2({\phi '}_0 \varphi ' -2{\phi^{'}}^2_0 F-\frac{\partial V}{\partial \phi}\varphi -4 A' \phi'_0 \varphi).
(n-6)\int_{B_\rho}\Delta\sigma_1(A)|\nabla\varphi|_g^2d\mu_g = (n-6)\int_{B_\rho}(\Delta\sigma_1(A)(p)+O(r))(|\varphi'|^2+O(r^2)|\varphi|^2)dx = -(n-6)^3\frac{|W(p)|^2}{12(n-1)}\int_{B_\rho}\frac{u_\epsilon^2r^2}{(\epsilon^2+r^2)^2}dx+\int_{B_\rho}\frac{O(r^3)u_\epsilon^2}{(\epsilon^2+r^2)^2}dx.
( \; \psi=\psi_{1}+{\cal I}\psi_{2}+{\cal J}\psi_{3}+{\cal K}\psi_{4}\; \; \; ;\; \; \; \psi_{1}, \; \psi_{2}, \; \psi_{3}, \; \psi_{4}\; \in\;{\cal C}(1, \; i) \; ) \; \; ,
A= \frac{1}{4n \sigma_{2n}}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}\frac{1}{(x_1^2 +y_1^2+ \cdots + 1/4+y_n^2)^{2n}}\,dx_1 dy_1\cdots \widehat{dx_{n}}dy_n=\frac{1}{4n}X = \frac{1}{2n}
\left\{{\frac{D e t^{\prime}\left[ - ( d^{2}/ d \tau^{2}) + V^{\prime \prime}[ x_{c 1}( \tau ) ] \right]}{D e t \left[ - ( d^{2}/ d \tau^{2}) + \nu^{2}\right]}}\right\}^{- 1 / 2}\, s q r t{{\frac{S_{1}}{2 \pi}}}\d \tau_{c}
\max_{\{\theta_m\}_{m=1}^{M}}\!\!\!\!\mathcal{H}\!\left(\!\{\theta_m\}_{m=1}^{M}\!\right)\!\!:=\!\!\sum_{t}\!R_{t}\!\left(\!\{\theta_m\!\}_{m=1}^{M}\!\right)\!+\!\sum_{r}\!R_{r}\!\left(\!\{\theta_m\}_{m=1}^{M}\!\right)\!,
\int_{0}^{\infty}\frac{d \tau}{\tau^{2}}\mathrm{e x p}\left( - \frac{p}{2 \tau}\right) \Gamma \left( q , \frac{c \tau}2 \right) = \left( \frac p 4 \right)^{\frac{q - 2}2}c^{\frac q 2}K_{q}( \sqrt{p c}) .
F[x,y,z]=\frac{\sqrt{(z^{\prime\prime}y^{\prime}-z^{\prime}y^{\prime\prime})^{2}+(x^{\prime\prime}z^{\prime}-z^{\prime\prime}x^{\prime})^{2}+(y^{\prime\prime}x^{\prime}-x^{\prime\prime}y^{\prime})^{2}}}{(x^{\prime2}+y^{\prime2}+z^{\prime2})^{3/2}}
P(M_t<t^a)= P(\max_{0\le u\le t^a}B(H(u))>\max_{t^a\le u \le t}\tilde{B}(H(u))-\tilde{B}(H(t^a))) \ge Q(t^{\frac{a-\epsilon}{2}},t^{\frac{1-a+2\epsilon}{2}})-\epsilon = V(\frac{a-\epsilon}{1-a+2\epsilon})-\epsilon,
x_{1}(z)_{-1}x(z)=\frac{1}{4z^{2}}\left(\frac{(z_1^{-1/2}+z^{-1/2})^{2}}{(z_{1}z)^{-1/2}}X_{1}(z_1^{-1/2})X(z^{-1/2})\right)\Bigg|_{z_1=z}.
\theta_{E}=\sum_{i=1}x_{i}\frac{\partial}{\partial x_{i}}+z\frac{\partial}{\partial z}, \theta_{k}=\sum_{i \in C_{\geq k}}\left(\prod_{j \in E_{< k}}(x_{j}-x_{i})\prod_{a \in \psi (v_{k})}(x_{i}-az)\right)\frac{\partial}{\partial x_{i}}\ \ (1 \leq k \leq \ell)
K(\rho_{-},\rho_{+},t)=H''(\rho_{+}) \left (-\frac{t}{2}\int_{\rho_{-}}^{\rho_{+}}\rho_{*}^2 H''(\rho_{*})d\rho_{*}\right) \times \mathbb{E}\left[ \left(-\int_{\rho_{-}}^{\rho_{+}}\textbf{e}(tH'(\rho_{*}))d\rho_{*}\right)\right]
c(K_{\lfloor n/2\rfloor, \lceil n/2\rceil})=(1+o(1))\begin{cases}I_0(2)\pi \left(\frac{n}{2e}\right)^n \mbox{if $n$ is even}I_1(2) \pi \left(\frac{n}{2e}\right)^n \mbox{if $n$ is odd}\end{cases}.
I_{21}\ll \sum_{\ell}\frac{\sqrt{z}}{\log_2 (kN)\cdot \sqrt{2^{\ell-1}z}}\ll \frac{1}{\log_2(kN)}, I_{22}=\sum_{\substack{z\leq p\leq y_1 p\mid N}}\frac{\lambda_f(p^\nu)}{p}\ll \frac{1}{\sqrt{z}}\ll \frac{1}{\log_2(kN)}.
dV_t=V_{t-}\Big( u_t dt+ dM_t+ \sum_{0 \leq j_1+\ldots+j_l \leq Q}c_{j_1,\ldots,j_l}(V_t) \prod_{i=1}^{l}\left(\Delta L^{(i)}_t\right)^{j_i}\Big)+ (V_0-V_t)dN_t,
f_{k+1}= A_{2-n}A_{4-n}A_{6-n}\Big(\sum_{i=1}^{k+1}\psi_i\Big)+K_{6-n}\Big(\sum_{i=1}^{k+1}\psi_i\Big)+f = f_k+A_{2-n}A_{4-n}A_{6-n}\psi_{k+1}+K_{6-n}\psi_{k+1}= O(r^{k+2}).
( A_{i}+ \partial_{i}\varphi ) ( \vec{x}) \rightarrow i D_{i}( \vec{x}) \equiv i \delta_{i}( \vec{x}) + \frac{1}{2 \pi k}\sum_{a}s_{a}\varepsilon_{i j}\frac{( x - x_{a})^{j}}{| \vec{x}- \vec{x}_{a}|^{2}},
\Gamma(2\eta + 1/2; 2\tau, 2\eta) = \frac{(2\tau + 1/2; 2\tau, 2\eta)}{(2\eta+1/2; 2\tau, 2\eta)}= \frac{(2\tau + 1/2; 2\tau)}{(2\eta+1/2; 2\eta)}\Gamma(\tau) = (\tau; 2\tau)^{-1}\Gamma(12\eta) = \frac{\theta_0(4\eta; 2\tau)}{(4\eta; 8\eta)}\Gamma(\tau + 1/2) = (\tau + 1/2; 2\tau)^{-1}\Gamma(\tau + 12\eta) = \theta_0(\tau + 4\eta; 2\tau).
{\eta}_{b}^{( i )}D_{b b^{'}}^{( i )}| b^{'^{\prime}}; \hat{g}\rangle ; \ \b^{'}= ( 1 / 2 , ( 3 - 2 \alpha ) / 2 , 1 ) ; \ \i = N_{1}, N_{2}, \ \N_{1}- N_{2}= 0 ~ m o d \, \, 3 .
\frac{1}{(2\pi)^2}\int_{BZ}\frac{\L^\alpha}{\omega-\omega^\alpha}d\alpha = \L^*\frac{1}{(2\pi)^2}\int_{BZ}\frac{1}{\omega - \omega^*+\frac{1}{2}c_\delta |\alpha-\alpha^*|^2}d\alpha + B_1(\omega) &=\L^*\frac{1}{(2\pi)^2}\int_{|\alpha-\alpha^*|<1}\frac{}{\omega - \omega^*+\frac{1}{2}c_\delta |\alpha-\alpha^*|^2}d\alpha + B_2(\omega).
\Omega_{i}^{( 2 ; 4 - 2 \varepsilon )}= \int_{0}^{\tau_{0 i}}\frac{\mathrm{d}\theta}{\cos^{2 - 2 \varepsilon}\theta}= 2 \tan \tau_{0 i}\; \;_{2}F_{1}\left( \left. \begin{array}{c}{{1 / 2 , \; \varepsilon}}\\{{3 / 2}}\\ \end{array}\right| - \tan^{2}\tau_{0 i}\right) ,
\operatorname{Rep}_A(\operatorname{Aut}_{\omega}^{\otimes}) \simeq \varprojlim QC(\operatorname{\check{C}ech}(\mathsf{Spec}\,A \to \mathsf{M}_{\mathcal{C}^{\otimes}})) &\simeq \varprojlim \operatorname{Cobar}(\omega)
c_{G_{k}/ U ( 1 )^{\ell}}= \frac{k - 1}{k + h}h \ell , \, \quad \quad \Delta ( \Lambda , \lambda ) = \frac{( \Lambda \cdot ( \Lambda + 2 \rho ) )}{2 ( k + h )}- \frac{( \lambda \cdot \lambda )}{2 k}\; .
\chi^{\mu}= -{\frac{C}{2 \gamma \Phi}}\nabla^{\mu}\varphi +{\frac{\Theta}{2 \gamma \Phi}}( \nabla^{\mu}\Phi ) \varphi +{\frac{C}{4 \gamma \Phi}}( \nabla^{\mu}\Phi ) h - \nabla^{\nu}\bar{h}_{\mu \nu}-{\frac{1}{\Phi}}( \nabla^{\nu}\Phi ) \bar{h}_{\nu}^{\mu}.
F ({\cal A}, G , q ) = \sum_{k = 1}^{\infty}F_{k}^{( \mathrm{c o v )}}( G ) \sum_{n = - \infty}^{+ \infty}\mathrm{e}^{- k{\cal A}v ( n )}q^{k}= \sum_{n = - \infty}^{+ \infty}F^{( \mathrm{c o v )}}( G , \mathrm{e}^{-{\cal A}v ( n )}q )
\Delta E=\frac{\sinh (\pi \sqrt{E})}{2\pi^{3}}~\int dz\,K_{i\sqrt{E}}(e^{z})\,\left( \int dx\, e^{2nx}\exp \left( -\frac{1}{2}\,e^{2x-z}\right) \right) \,\left( \int dpK_{ip}\left( e^{z}\right)\, e^{isp/2}\right) .
\varphi(N-3) =\sum_{k=0}^{N-1}s_{N-1-k}\ \varphi (k+1)-a_{N-2}b_0c_0\varphi(2)-a_{N-2}b_0c_1\varphi(1) &-c_0\varphi(1)\sum_{k=0}^{N-1}a_kb_{N-1-k}.
\sum_{n=1}^{N}\frac{q^n}{1- q^{2n}}= \sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right] (-1)^{n-1}\frac{(-q)_{n-1}(-q)_{N-n}}{(-q)_{N}}\frac{q^n}{1-q^n}.
(\mathbf{G}^{(t)}\mathbf{x}^{(t)}-\mathbf{b}^{(t)})^{T}(\hat{\mathbf{x}}^{(t)}-\mathbf{x}^{(t)}) = \left(\sum_{n=1}^{N}(\mathbf{G}_{n}^{(t)}\mathbf{x}^{(t)}-\mathbf{b}_{n}^{(t)})\right)^{T}(\hat{\mathbf{x}}^{(t)}-\mathbf{x}^{(t)}),
\sum \limits_{k=0}^\infty \theta_{B_{R_x}(x)}(r_{x,k}) \leq \frac{C^{\frac 1 2}}{C^{\frac 1 2}-1}\cdot \sum \limits_{k=0}^\infty \int \limits_{R_x/ C^{\frac{k+1}2}}^{R_x/ C^{\frac{k}2}}\frac{\theta_{B_{R_x}(x)}(r)}r \dr \leq \frac{C^{\frac 1 2}}{C^{\frac 1 2}-1}\int \limits_{0}^{R_x}\frac{\theta_{B_{R_x}(x)}(r)}r \dr <\infty.
\frac{\exp\left\{\sum_{j=1}^r{x^j j^{-1}}\right\}}{x^n\sqrt{2\pi \sum_{j=1}^rjx^j}}\left(1+O\left(\frac{1}{u}\right)\right) =\varrho(u)\left(1+O\left(\frac{u\log(u+1)}{r}\right)\right)
\mathcal{C}(\overline{\mathcal{X}}):=\big\{\mathcal{Z}\in\mathbb{X}\ |\h'(\overline{\mathcal{X}})\mathcal{Z}=0,\,G'(\overline{\mathcal{X}})\mathcal{Z}\in\mathcal{T}_{K}(G(\overline{\mathcal{X}})),\, \langle f'(\overline{\mathcal{X}}),\mathcal{Z}\rangle=0\big\}.
\lim\sup_{n}\frac{\mathbb{E}^{x}(\max_{1\leq k\leq n}(\bar{R}_{k}^{m})^{2})}{n}\leq24\mathbb{E}_{\pi}[\sup_{k\geq0}|\sum_{j=0}^{k}f_{m}Q^{j}f_{m}|]\pi-a.s.
z\;\frac{d}{dz}{}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(z) =\frac{z}{a}{}_{a}\mathtt{B}_{2a-1, a \nu-a,1}(z) -\left(\nu(2-a)+a-1\right){}_{a}\mathtt{B}_{2a-1, a \nu-a+1,1}(z).
\max \{0, x_i - \overline{b}_i, \max \{0,x_i - \overline{b}_i + \Delta b_i\}- \theta^k(x)\}= \max\{0, x_i - \overline{b}_i, x_i - \overline{b}_i + \Delta b_i - x_k + \overline{b}_k - \Delta b_k\}.
\mathbb{E}\mathrm{Tr}\left(\frac{R}{ks}-\mathrm{I}_{nk}\right)^2 =\frac{1}{k^2s^2}\mathbb{E}\mathrm{Tr}(R^2)-\frac{2}{ks}\mathbb{E}\mathrm{Tr}R+nk &=\frac{2n^2}{s}-\frac{2n^2}{ks}+\frac{n}{k},
F(k,j) =\frac{2k+2j+1}{16^k}\binom{2k}{k}^2\frac{\binom{2k+2j}{k+j}\binom{2k+2j}{2j}}{\binom{2j}{j}}, G(k,j) =-\frac{2(2k-1)}{16^{k-1}}\binom{2k-2}{k-1}^2\frac{\binom{2k+2j-2}{k+j-1}\binom{2k+2j-2}{2j}}{\binom{2j}{j}}.
\frac{-\log \mu_Y(\tau_r > t)}{s\mu(U_r)^{1-\alpha}}= \frac{- \log \mu_Y(\tau_r > t \wedge A_u^c)}{s\mu(U_r)^{1-\alpha}}-\frac{\log[1 + \frac{\mu_Y(\tau_r > t \wedge A_u)}{\mu_Y(\tau_r > t \wedge A_u^c)}]}{s\mu(U_r)^{1-\alpha}}.
\left( P_{\cal E}\right)_{\mu \nu}= \frac{1}{2}\left( \bar{n}_{\mu}n_{\nu}+ n_{\mu}\bar{n}_{\nu}\right) , \qquad \left( P_{\cal H}\right)_{\mu \nu}= \frac{1}{2}\left( \bar{m}_{\mu}m_{\nu}+ m_{\mu}\bar{m}_{\nu}\right)
\int \sum_{k=0}^\infty 2^{ks}L_k P_Ef (x)G_k(x) dx\, &=\sum_{k=0}^\infty 2^{ks}\sum_{j=0}^\infty\int L_k P_E L_j \widetilde L_jf (x) G_k(x) dx &=\int \sum_{j=0}^\infty \widetilde L_j f (y) L_j'P_E'\big[{\scriptstyle\sum}_k 2^{ks}L_k'G_k\big](y) dy.
{\bar{g}}_{0}( r , x ) = \frac{{\kappa}^{2}}{{\pi}r x}\int_{- \infty}^{\infty}\frac{{\operatorname{s i n}}({\alpha}r ){\operatorname{s i n}}({\alpha}x ) d{\alpha}}{{\alpha}^{4}\left({\alpha}^{2}+ 2{\kappa}\right)}\, ,
\Psi =\left(\begin{array}{c}P^2-P^1 \vdots P^m-P^1\end{array}\right) &=\left(\begin{array}{ccc}P^2_1-P^1_1 \cdots P^2_n-P^1_n \vdots \vdots P^m_1-P^1_1 \cdots P^m_n-P^1_n\end{array}\right).
\tilde{\check{y}}_{\mathrm{d},i}[n] =\sum_{k\in\mathcal{K}}\tilde{h}_{i,k}[n]\tilde{x}_{\mathrm{d},k}[n]+\tilde{z}_{\mathrm{d},i}[n]+\tilde{e}_{\mathrm{d},i}[n],
\sum\limits_{\left\{i,j\right\}\in E\left( G^{\prime}\right)}x_{i}x_{j}=\sum\limits_{\left\{i,j\right\}\in E\left( G\right)}x_{i}x_{j}-x_{k+s-1}x_{k+s}-x_{k+s+2}x_{k+s+3}+x_{k+s}x_{k+s+2}+x_{k+s-1}x_{k+s+3}& =\sum\limits_{\left\{i,j\right\}\in E\left( G\right)}x_{i}x_{j}.
< \sigma_{0}\sigma_{1}>_{1}\left({\frac{1}{\lambda_{1}\lambda_{2}^{3}}}+{\frac{1}{\lambda_{1}^{3}\lambda_{2}}}\right) ={\frac{1}{4}}\Gamma_{1}+{\frac{1}{4}}( \Gamma_{2}+ \Gamma_{3}+ \lambda_{1}\leftrightarrow \lambda_{2}) ={\frac{1}{2}}{\frac{\lambda_{1}^{2}+ \lambda_{2}^{2}}{\lambda_{1}^{3}\lambda_{2}^{3}}}.
K(x) = P_n(0)\int_{z=x}^{\infty}\int_{s=0}^{z-x}V(z-s)f_{B}(s) \lambda e^{-\lambda(z-x)}ds dz + \int_{z=x}^{\infty}\int_{u=0}^{z-x}\int_{0}^{z-x-u}\bar{G}(u)V(z-u-s)f_{B}(s)f_n(u)\lambda e^{-\lambda(z-x)}ds du dz =: K_{1}(x) + K_{2}(x).
f(s,x,t,y) =\lim_{\epsilon \rightarrow 0}\frac{1}{2\epsilon}\int_{y-\epsilon}^{y+\epsilon}\mathbb{E}[Z^{\partial}(t) \vert W^{\partial}(t)=y,W^{\partial}(s)=x]p_{t-s}^{\partial}(x,z)dz &=p_{t-s}^{\partial}(x,y)\mathbb{E}[Z^{\partial}(t) \vert W^{\partial}(t)=y, W^{\partial}(s)=x]
(a\phi_{d,d_1})^*\hat{\cal L}_{1,{\cal E}_d}= [a]^* \phi_{\gcd(d,d_1),d_1}^* \phi_{d,\gcd(d,d_1)}^*\hat{\cal L}_{1,{\cal E}_d}= \hat{\cal L}_{1,{\cal E}_{d_1}}^{a^2 \gcd(d,d_1)^2/d_1d},
qw(q)^3 =q\frac{\phi(-q)}{\phi(-q^3)^3}\left(\frac{1}{8q}\left(\frac{\phi(-q^3)^3}{\phi(-q)}-\frac{\phi(-q)^3}{\phi(-q^3)}\right)\right) &=\frac{1}{8}\left(1-\frac{\phi(-q)^4}{\phi(-q^3)^4}\right).
\hat{h}= \left( \begin{array}{c c c c c}{1}&{0}&{0}&{\cdots}&{0}\\{0}&{\omega}&{0}&{\cdots}&{0}\\{0}&{0}&{\omega^{2}}&{\ddots}&{\vdots}\\{\vdots}&{\vdots}&{\ddots}&{\ddots}&{0}\\{0}&{0}&{\cdots}&{0}&{\omega^{n - 1}}\\ \end{array}\right) .