harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{j=0}^{\infty}\frac{(dq)_{j}(q^{k}/d)^{j}}{(dq^{k+1})_j(q)_j}=\frac{(q^{k+1})_{\infty}}{(q^{k}/d)_{\infty}}\sum_{j=0}^{\infty}\frac{(dq)_{j}q^{j^2+2jk}}{(dq^{k+1})_j(q^{k+1})_j(q)_j}.
M^{(c)}_{\gamma_2}(x)=\frac{{\rm \mathcal{H}}_{3,4}^{3,3}\Biggl[\frac{\kappa_Im_I\bar{\gamma}}{\kappa m}s\Bigg\vert\{(\chi_2,X_2)\atop(\upsilon_2,\Upsilon_2)}\Biggr]}{s\Gamma(Nm)\Gamma(\kappa)\Gamma(Lm_I)\Gamma(\kappa_I)}.
K = 1 2 \int_{0}^{1 / 2}\left({\frac{1}{4}}- z^{2}\right) \left[ \psi \left({\frac{1}{2}}+ z \right) + \psi \left({\frac{1}{2}}- z \right) \right] ~ d z = - 5 . 9 7 0 1 1 . . . . .
(-1)^j \Delta_{k_1}\dots \Delta_{k_{j-1}}\delta_{k_{j+1}}\delta_{k_{j+2}}\dots \delta_{k_n}\sum_{(l_1,\ldots,l_n)}^{(k_1,\ldots,k_{j-1},k_j,x,k_{j+1},\ldots,k_n)}A(l_1,\ldots,l_n) = - \sum_{(l_j)}^{(k_j,x)}A(k_1,\ldots,k_{j-1},l_j,k_{j+1},\ldots,k_n) ={^x}\Delta_{k_j}^{-1}A(k_1,k_2,\ldots,k_n).
\begin{aligned}\ln_{e_1}\eta =\bigg(e_1(\tilde{b})+\frac{\tilde{a}\,(\tilde{b}\,\lambda_x-\tilde{a}\,\lambda_y)}{\lambda^2}\bigg)\,E_1 &+\bigg(-e_1(\tilde{a})+\frac{\tilde{b}\,(\tilde{b}\,\lambda_x-\tilde{a}\,\lambda_y)}{\lambda^2}\bigg)\,E_2-r\,E_3\end{aligned}
\begin{array}{c}f_{-}=f(-\infty +i\epsilon )=(-1)^{\delta}e^{iZ_{+}(-\infty )}=e^{i{\cal Q}_{+}(-\infty )}f_{+}=f(+\infty +i\epsilon )=(-1)^{\delta}e^{iZ_{+}(+\infty +i\epsilon )}=0\end{array}
\langle P_h^k(\cdot, s), \mu_{j, \ell}\rangle = \frac{\overline{\rho_{j,\ell}(h)}}{(4\pi h)}^{s-1}\frac{\Gamma(s + \frac{\ell}{2}- 1) \Gamma(s - \frac{\ell}{2})}{\Gamma(s - \frac{k}{2})}.
B_{n-1}G_0=P_{n+2,n+1}-\frac{\lambda v^{n+2}}{4\kappa^2}\sum_{m=0}^{n-1}\sum_{l=1}^{n+3-m}\frac{(-1)^{n-m-1+l}(n-m)(n+1-l)(n+2-l)^2}{(l-1)!m!(n+3-m-l)!}\hat{I}_{n,l-1}.
\widetilde{\ell}_0 = 2a_2 \partial_{a_2}+ L_0, \widetilde{\ell}_2 = \tfrac{2}{15}\big(6\gamma_4 + 5 a_2^2\big) \partial_{a_2}- \tfrac{4}{3}a_2 L_0 + L_2 , \widetilde{\ell}_4 = \tfrac{2}{45}\big(27\gamma_6 + 9 a_2 \gamma_4 - 40 a_2^3\big) \partial_{a_2}- \tfrac{1}{3}a_2^2 L_0 - 2 a_2 L_2 .
\tilde{\Gamma}^i_{0k}=\tilde{\Gamma}^i_{k0}=\frac12 \tilde{g}^{im}\left(\tilde{g}_{m0,k}+\tilde{g}_{mk,0}-\tilde{g}_{0k,m}\right) &=\frac12 r^{-2}g^{im}\left(\tilde{g}_{mk,0}\right) &=\frac12 r^{-2}g^{im}\left(2rg_{mk}\right) &=r^{-1}\delta^i_k.
\delta^{-2}{r}^{r}_1(k)e^{2it\theta}|_{k=-k_0}=e^{16itk_0^3-i\zeta^3(144\tau)^{-\frac{1}{2}}}\tau^{i\nu}e^{-2\chi(\xi,-k_0)}{r}^{r}_1(-k_0)4^{2i\nu}e^{\frac{i}{2}\zeta^2}\zeta^{-2i\nu},
(\frac{\omega}{\alpha})_{3}=\omega^{\frac{1-a-b}{3}}=\omega^{-m-n},(\frac{1-\omega}{\alpha})_{3}=\omega^{\frac{a-1}{3}}=\omega^{m},(\frac{3}{\alpha})_{3}=\omega^{\frac{b}{3}}=\omega^{n}.
\delta (d^* H)_{ij}=h_{ml}\nabla_l H_{mij}-\nabla_l(dK_{lij})+(\nabla_l h_{lp}-\frac{1}{2}\nabla_p h_{ll})H_{pij}& \kern2em +\frac{1}{2}(\nabla_l h_{ip}+\nabla_i h_{lp}-\nabla_p h_{li})H_{lpj}+\frac{1}{2}(\nabla_l h_{jp}+\nabla_j h_{lp}-\nabla_p h_{lj})H_{lip}.
G_{\mu \nu}= \left( \begin{array}{c c c}{{- \frac{2 \pi \alpha^{\prime}}{\theta_{0}}}}&{{0}}&{{- \frac{2 \pi \alpha^{\prime}}{\theta_{1}}}}\\{{0}}&{{\frac{2 \pi \alpha^{\prime}}{\theta_{0}}+ \biggl ( \frac{2 \pi \alpha^{\prime}}{\theta_{1}}\biggr )^{2}}}&{{0}}\\{{- \frac{2 \pi \alpha^{\prime}}{\theta_{1}}}}&{{0}}&{{1}}\\ \end{array}\right) ,
T_{\nu\rho}= \frac{1}{8\pi}\left(F^2_{\nu\rho}- \frac 14 g_{\nu\rho}F^2 +G_{IJ}\partial_{\nu}X^I\partial_{\rho}X^J - \frac 12 g_{\nu\rho}G_{IJ}\partial_{\lambda}X^I\partial^{\lambda}X^J\right)
\int_0^r \left[(\phi''+\frac{z}{2}\phi'-\frac{z}{2}e_\sigma\psi')\cdot \phi'' +(\psi''+\frac{z}{2}\psi'-\frac{z}{2}e_\sigma\phi')\cdot \psi''\right]{\rm{d}}z =\int_0^r |(\phi'',\psi'')|^2+\frac{1}{2}\int_0^r z(\phi'\phi''+\psi'\psi'')-\frac{1}{2}\int_0^r ze_\sigma(\psi'\phi''+\phi'\psi'').
\bigcap_{m\geq 0}(\mathbb{Q}(m) \Rightarrow \ominus_D \bigcup_{\ell\geq 0}(\mathbb{Q}(\ell) \wedge \|m=\ell\|)) \leq \ominus_D \bigcup_{\ell\geq 0}(\mathbb{Q}(n) \wedge \|n=\ell\|)
\int_{\Omega}(\epsilon_{0}^{-1}\nabla\times\widetilde{H_{\eta,\ell}})\cdot(\nabla\times\overline{\widetilde{H_{\eta,\ell}}})dx-k^{2}\int_{\Omega}\mu|\widetilde{H_{\eta,\ell}}|^{2}dx +\int_{\Omega}((\epsilon^{-1}(x)-\epsilon_{0}^{-1}(x))\nabla\times H)\cdot(\nabla\times\overline{H})dx = -\tau^{-1}I_{\rho}^{\eta,\ell}.
x=\frac{1}{a_{1}}+\frac{1}{a_{1}a_{2}}+\frac{1}{a_{1}a_{2}a_{3}}+\cdot\cdot\cdot=\frac{1}{a_{1}}(1+\frac{1}{a_{2}}(1+\frac{1}{a_{3}}(1+\cdot\cdot\cdot)))
c_\infty^{(\widetilde{\rm cJ})}(\tau;\beta,p,q) = \sum_{j=0}^\infty h_j(\beta,p,q) \tau^{j}{\rm sgn}\, \tau + \sum_{j=0}^\infty \tilde{h}_j(\beta,p,q) \tau^{j}, |\tau| < |\tau^*|,
R_{l}( r ) = \frac{u_{l}( r )}{r^{( D_{0}- 1 ) / 2}}\stackrel{( r \rightarrow 0 )}{\sim}\mathrm{\boldmath \large ~ \left\{~ \right. ~}\! \! \! r^{\alpha_{+ , l}}\mathrm{\boldmath \large ~ , ~}\! r^{\alpha_{- , l}}\mathrm{\boldmath \large ~ \left. ~ \right\}~}\! \! \; ,
h_P^2(t)\ = \t\sum_{V}x^{2}+t^2\left(\sum_{E}(y+z)^{2}- \sum_{V}x^{2}\right)+t^3\left(\sum_{E^o}(y+z)^{2}- \sum_{V^o}x^{2}\right)+t^4\sum_{V^o}x^{2}.
S^{i,1}(\psi_1,\psi_2,\psi_3) =\textstyle\sum_\ell(d(x^\ell\psi_\ell)-x^\ell d\psi_\ell)=\sum_\ell dx^\ell\wedge\psi_\ell, S^{i,2}\omega =(dx^1\wedge\omega,dx^2\wedge\omega,dx^3\wedge\omega).
\Delta_2(|X_2|^{2m}K_{s-m}) ={}(4(s-m)p_2|X|^2+8(s-m)(2s+q-2m-1)|X_2|^2)|X_2|^{2m}K_{s-m-1}+2m(2m+p_2-2)|X_2|^{2m-2}K_{s-m}+16m(s-m)|X|^2|X_2|^{2m}K_{s-m-1}
A = \left( \begin{array}{c c}{\alpha}&{- \overline{{\beta}}}\\{\beta}&{\overline{{\alpha}}}\\ \end{array}\right) , \quad \alpha = \cos{\frac{\theta}{2}}e^{i ( \psi + \phi ) / 2}, \quad \beta = \sin{\frac{\theta}{2}}e^{i ( \psi - \phi ) / 2},
\big\Vert \big(\hat{\Pi}_{V_1}-\sum_{j=1}^k(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^j(1-\hat{\Pi}_{V_1})\big)(f_2-\hat{\Pi}_{V_2}f_2)\big\Vert_n &\leq\frac{1}{1-\rho_{0,\delta}^2}\Vert \hat{\Pi}_{V_1}(f_2-\hat{\Pi}_{V_2}f_2)\Vert_n.
K_p \equiv 2 \sum_{i=1}^{k}\dfrac{1}{2i-1}\binom{2k}{2i-2}b^{2(i-1)}a^{2(k-i)+1}& - \sum_{j=1}^{2k}\dfrac{(-1)^j}{j}\binom{2k}{j-1}b^{2k-j}a^{j-1}\pmod{p},
\beta'(h,h') =\alpha'(h+h')-\alpha'(h)-\alpha'(h')= &=\alpha(h+h')-\psi(h+h')-(\alpha(h)-\psi(h))-(\alpha(h')-\psi(h'))= &=\beta(h,h')-(\psi(h+h')-\psi(h)-\psi(h'))= &=\beta(h,h')-\psi^{*}(h,h')
{\cal E}_{[0,n]}^{FB}(\kappa) \triangleq \Big\{g_t:{\cal M}_n \times{\cal Y}^{t-1}\longmapsto{\cal X}_t,~~ x_0=g_0(w, y^{-1}), x_t=e_t(w, y^{i-1}),~~ w\in{\cal M}_n, ~t=0, \ldots, n: & \frac{1}{n+1}{\bf E}^g\Big(c_{0,n}(X^n,Y^{n-1})\Big)\leq \kappa \Big\}.
\sum_{i=2m+1}^{n-1}\sum_{j=1}^{i}jV_{i,j}\psi^{n}_i(s)\psi^{n}_j(s) =\sum_{i=2m+1}^{n-1}\sum_{j=1}^{2m}jV_{i,j}\psi^{n}_i(s)\psi^{n}_j(s) + \sum_{i=2m+1}^{n-1}\sum_{j=i}^{n-1}iV_{j,i}\psi^{n}_j(s)\psi^{n}_i(s).
S_{Q 2 f}( M_{n , i}) = \sum_{i = 0}^{L - 1}\frac{1}{N}\mathrm{T r}\, ( 2 M_{1 , i}^{2}+ 2 M_{2 , i}^{2}) + \frac{2 g}{N^{2}}\mathrm{T r}\, M_{1 , i}^{2}\mathrm{T r}\, M_{2 , i}^{2}
t\frac{dw_{*}}{dt}=-w_{*}\left(1+t\left(\frac{2t}{1+t^{2}}+\frac{t\left(|M|^{*}\right)^{2}}{2}\right)\right)+\frac{(R_{g(t)}-t^{2}\overline{R})_{*}}{2(1+t)^{2}}.
\breve{H}_\lambda(t)\;=\;\begin{pmatrix}\widehat{H}(t) \lambda\, k(t) \lambda \,k(t)^* \widehat{H}'(t)\end{pmatrix}\;=\;\begin{pmatrix}\widehat{H}(t) 0 0 \widehat{H}'(t)\end{pmatrix}\;+\;\lambda\,K(t)\;,
\delta m \sim g_{I i}M_{P}^{( 4 )}\sim M_{P}^{( 4 )}g_{s}\frac{\left| k_{i}^{1}+ k_{i}^{2}\right|}{M_{s}}\left( \frac{t_{s}}{g_{s}t_{\mathrm p h y s}}\right)^{1 / 2}
(0,0)_e = \eta^2 - (e-3)\eta - 1, (0,i)_e = (i,0)_e = (i,i)_e = \eta^2 + \eta \textrm{for}i \ne 0, (i,j)_e = \eta^2 \textrm{for}i \ne j \textrm{and}i,j \ne 0,
I_{sp}= \int_{M}d^4 x \ \left[ \overline{\psi}_{+}\gamma^{\mu}\left( \partial_{\mu}+ A_{\mu}\right) \psi_{+}+ \overline{\psi}_{+}\gamma^{\mu}\sigma_{\mu}\psi_{+}+ \left({\rm terms ~ with~~}+ \rightarrow - \right) \right].
\begin{aligned}(T_{q, k_1}(\Phi_u)*T_{q, k_2}(\Phi_v) )_{q}^{(g_1,g_2;h_3,h_4)}(y) =T_{q,\mathbf{s}_3}(\Phi_u) (\mathcal Z_{g_1}(y,\cdot) ) T_{q,\mathbf{s}_4}(\Phi_v) (\mathcal Z_{g_2}(y,\cdot) ) \end{aligned}
T^\mu_{~\mu}= -\frac{T_3 V(T)}{\sqrt{-X}}\left[ (4 + 3\partial_\rho T \partial^\rho T) +\left((1+\frac12 \partial_\rho T\partial^\rho T)\delta^\mu_{~\sigma}- \partial^\mu T \partial_\sigma T\right) F_{\mu\nu}F^{\sigma\nu}\right].
C_{\infty}A = \big\{f \in C^{\infty}([0, 1], \, A) \colon{\mbox{$f^{(n)}(0) = 0$ for $n = 1, 2, \ldots$ and $f^{(n)}(1) = 0$ for $n = 0, 1, 2, \ldots$}}\big\},
E_*(t, \tau) = -i e^{-i \tau A(t)^{1/2}}\sum_{{1 \le j, l \le p: \; j \ne l}}J_{jl}(t, \tau), J_{jl}(t, \tau) := t |t| \int_0^{\tau}e^{i \tilde{\tau}A(t)^{1/2}}F(t) P_j G_* P_l e^{-i \tilde{\tau}(t^2 S)^{1/2}P}P \, d \tilde{\tau}.
\epsilon \left( C_{a}^{\alpha_{1}}\right) = \epsilon \left( l^{\alpha_{1}}\right) = \epsilon \left( f_{\alpha_{1}}\right) , \; \epsilon \left( p_{a}^{\alpha_{1}}\right) = \epsilon \left( f_{\alpha_{1}}\right) + 1 , \; a = 1 , 2 , 3 .
\sup_{x \in\left( -\infty, x_{0}\right)}\left\| \langle x\rangle \mathcal{P}^{+}\left(z^{-i}\bar{r}_1(z) e^{2i z x}\right)\right\|_{L_z^2}\leq\left\|z^{-i}\bar r_1(z)\right\|_{H^1},i=0,1, &\sup_{x \in \left( -\infty, x_{0}\right)}\left\|\langle x\rangle \mathcal{P}^-\left(r_2(z) e^{-2iz x}\right)\right\|_{L_{z}^2}\leq\left\|r_2(z)\right\|_{H^1}.
\hat{\nu}_j^{**}=\bigl(\hat{\phi}_{XY}^{j+1}+\hat{\phi}_{XY}^{j},\hat{\phi}_{XY}^{j+2}+\hat{\phi}_{XY}^{j},\ldots,\hat{\phi}_{XY}^p+\hat{\phi}_{XY}^{j}\bigr),1\leq j\leq j^*, p=4,
\left\{{\left({{\alpha_{{k,{\rm{R}}},i}},\Theta_{{k,{\rm{R}}},i}^{\rm{A}},\Theta_{{k,{\rm{R}}},i}^{\rm{D}}}\right),i \in{{\mathcal L}_{{k,{\rm{R}}}}}}\right\}_{k = 1}^K.
\left\{\not \! \partial ( \phi + \psi / 2 ) + e^{3 \psi / 8 + \phi / 2}\gamma_{7}\left({\bf F}^{1}+ \kappa{\bf F}^{2}\right) - i s_{z}e^{\phi}\gamma_{7}\not \! \partial \kappa \right\}\epsilon_{6}= 0 ,
D_k(R_N)^{m+1}D^N_{k'}=D_k\sum_{|k_0-\ell_0|>N}D_{k_0}D_{\ell_0}\sum_{|k_1-\ell_1|>N}D_{k_1}D_{\ell_1}\cdots\sum_{|k_m-\ell_m|>N}D_{k_m}D_{\ell_m}D^N_{k'}&=\sum_{|k_0-\ell_0|>N}\cdots \sum_{|k_m-\ell_m|>N}D_kD_{k_0}D_{\ell_0}D_{k_1}D_{\ell_1}\cdots D_{k_m}D_{\ell_m}D^N_{k'}.
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(96n+12)q^n\equiv \sum_{n=0}^{\infty}\textup{PD}_\textup{t}(288n+36)q^n\equiv 2f_1^3\equiv 2\left(f_3a(q^3)-3qf_9^3\right)~(\textup{mod}~4),
\Gamma(F) \lesssim_\lambda \int_1^\infty \Big( \sum_{j=1}^m \sum_{l=k_{j-1}}^{k_j-1}\int_{\mathbb{R}^5}F(y,x') F(x,x')F(y,y')F(x,y') & g_{\alpha2^{k_j}}(x'-y') \omega_{2^l}(x-q) \omega_{2^l}(y-q) \,dx dy dx' dy' dq \, \Big) \alpha^{-\lambda}d\alpha.
\frac{k!}{|G|}\sum_{r_{1}+r_{2}+...+r_{n}=k}[z^{r_{1}}]g(z)[z^{r_{2}}]g(z)\cdot\cdot\cdot[z^{r_{n}}]g(z)=\frac{k!}{|G|}[z^{k}]g(z)^{n}
\phi_1 \phi_2 = a_3 \phi_2\phi_1 \phi_i \psi_j = a_j \psi_j \phi_i \textrm{if}i\not= j \phi_i \psi_i = \psi_i \phi_i \psi_i \psi_j = \psi_j \psi_i \ \forall \i, j \in \{1,2\}.
C_{i,u}=\big ({\widetilde{D}}_{[1,i-1]}^{X},{\widetilde{D}}_{i1}^{\prime},\dots,{\widetilde{D}}_{i,u-1}^{\prime},{\mathbf{0}},{D^{\prime}}_{i,u+1}^{\ast},\dots,{D^{\prime}}_{i,u_{k_{i}}}^{\ast},{D}_{[i+1,n]}^{Y}\big )\,,
\lefteqn{H(\bar{X}) + I(\bar{W}\wedge \bar{Y}|\bar{X})}&= I(\bar{W}\wedge \bar{X},\bar{Y}) + H(\bar{X}|\bar{W}) &\le r_{0,n}+ r_{1,n}&\le \frac{1}{n}\log |{\cal T}_{\bar{X}}^n| &~~~+ \frac{3\log n + \log \log |{\cal X}| + \log |{\cal X}| + 3 + |{\cal X}||{\cal Y}| \log (n+1)}{n}&\le H(\bar{X}) + \delta_n.
\sigma_{\!\operatorname{ess}}(B_{+,0}) = \emptyset \Leftrightarrow \lim_{x\to b_+}R_+(x) \int_{x}^{b_+}w_+(\xi) d\xi = 0, 0\not\in\sigma_{\!\operatorname{ess}}(B_{+,0}) \Leftrightarrow \sup_{x\in(0,b_+)}R_+(x) \int_{x}^{b_+}w_+(\xi) d\xi < +\infty
w^{-2\delta}\langle P(\chi,\chi;-z^\ast)|V(\psi,\rho)|P\left(\overline\chi,\overline\chi;-{1\over w}\right)\rangle=z^{-2\delta}\langle P(\chi,\chi;-w^\ast)|V(\psi,\rho)|P\left(\overline\chi,\overline\chi;-{1\over z}\right)\rangle\,,
\begin{aligned}I_{2,2}\leq \frac{c\lambda^{\tilde{p}-1}}{\mathcal{M}^{\frac{2s}{n}}}\sum_{\mathcal{K}\times I\in \mathcal{A}}\int_{\lambda_{0}}^{N\mathcal{M}^{-1}}\mu_{\tau,t}\left(\mathcal{K}\times I\right)\,d\lambda\end{aligned}
\int_0^\infty \int_r^\infty |J_\mu(r)|^q |H^{(1)}_\mu(r)|^q (rr')^{-q(\nu-2)/2 + \nu-1}dr\,dr' & = \int_0^{\mu-\mu^{1/3}}\int_r^{\mu-\mu^{1/3}}|J_\mu(r)|^q |H^{(1)}_\mu(r)|^q (rr')^{-q(\nu-2)/2 + \nu-1}dr\,dr' & + \int_0^\infty \int_{\max\{r,\mu-\mu^{1/3}\}}^\infty |J_\mu(r)|^q |H^{(1)}_\mu(r)|^q (rr')^{-q(\nu-2)/2 + \nu-1}dr\,dr'
\sum_{m=0}^{\infty}\sum_{\ell=0}^{\min\{2p+1,2m\}}A_{m,\ell}=\sum_{\ell=0}^{2p}\sum_{m=\lceil\ell/2\rceil}^{\infty}A_{m,\ell}+\sum_{m=p+1}^{\infty}A_{m,2p+1}.
u=-2\frac{\theta^1_x}{\theta^1}, v=\left(-2\frac{\theta^1_x}{\theta^1}y+\frac{\theta^0}{\theta^1}\right)_x\quad\mbox{with}\theta^i=\theta^i(t,x)\colon\ \theta^i_t=\theta^i_{xx},i=0,1.
\begin{aligned}\left| \phi_N^{(s)}(t,Z_s) \right| \leq \left| \phi_N^{(s)}(t,Z_s^*) \right| + N \left| \phi_N^{(s-1)}(t,Z_s^{(i)}) \right| +N \left| \phi_N^{(s-1)}(t,Z_s^{(j)}) \right| + & + N \left| \phi_N^{(s-1)}(t,(Z_s^*)^{(i)}) \right| +N \left| \phi_N^{(s-1)}(t,(Z_s^*)^{(j)}) \right|\end{aligned}
h \; = \; \frac{L^{2}}{4}\cdot \frac{u}{u + v}, \qquad \gamma \; = \; 1 - \frac{u + v}{L^{2}u}, \qquad \ell_{0}+ \nu_{0}\; = \; \frac{v}{L u}\; = \; \frac{v}{n},
\alpha ={s(v)\big(ns(w)-e^{w\over2}(p-v)\big)\over e^{v\over2}s(w)(m-w)-e^{w\over2}s(v)(p-v)}\ , \cr\tilde u = m \ , \cr\tilde v = p \ , \cr\tilde c = \alpha\Big({dv-cm\over 1-e^{-v}}+{cm\over v}\Big) + \beta\Big({cw-dp\over 1-e^{-w}}+{dp\over w}\Big)+e \ .
P\{\max_{0\le s\le t}\mathcal{T}(s)=0\ |\V(0) = -c_2,\N(t) = 2k\}\ =\ \binom{2k}{k}\frac{(c_1c_2)^k}{(c_1+c_2)^{2k}}+ \Bigl(1-\frac{c_1}{c_2}\Bigr)\sum_{j=0}^{k-1}\binom{2k}{j}\frac{c_1^{j}\,c_2^{2k-j}}{(c_1+c_2)^{2k}}
\#(I \cap [w(j), n+j])) = \#(L(w) \cap [j]) + \#(R(w) \cap [j]) + \#(I \cap Z(A) \cap [w(j), n+j]) &\geq \#(L(w) \cap [j]) + \#(R(w) \cap [j]) + \#(A \cap [j]) = j.
R_{m n}{}^{p q}\rightarrow W_{m n}{}^{p q}- \frac{1 6}{d - 2}\delta_{[ m}{}^{[ p}( \bar{\psi}_{| r |}\Gamma^{| r |}\psi_{n ]}{}^{q ]}- \bar{\psi}^{| r |}\Gamma^{q ]}\psi_{n ] r})
{\mathrm Re}\partial_{p^2}\Gamma_{+-}(p^2)\big|_{p^2 = \mu_W^2}= 1 \quad{\mathrm Re}\partial_{p^2}\Gamma_{HH}(p^2)\big|_{p^2 = \mu_H^2}= 1 \quad{\mathrm Re}\partial_{p^2}\Gamma_{\chi\chi}(p^2)\big|_{p^2 = \mu_\chi^2}= 1
\{(\mathcal{H}_{i})_{A_{N-1}},(\mathcal{H}_{j})_{A_{N-1}}\}=\{(\mathcal{H}_{i}^{\varepsilon})_{A_{N-1}},(\mathcal{H}_{j}^{\varepsilon^{^{\prime}}})_{A_{N-1}}\}=0,\ \ \ \ \ \ \varepsilon ,\varepsilon^{^{\prime}}=\pm , \ \i,j=1,\ldots ,N.
\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{\delta}_{ij}\widetilde{m}_{\Lambda_j}dy+\int_{\mathcal{Y}^d}(\Lambda_i+\widetilde{w}_{y_i})\widetilde{w}_{y_i\Lambda_j}\widetilde{m}_{\Lambda_j}dy=\int_{\mathcal{Y}^d}\frac{\widetilde{m}_{\Lambda_j}^2}{\widetilde{m}}dy.
(R1)^{1}=-\frac{1}{\tau}\int\limits^{t_1}_{t_0}\int\limits^{t_1}_{t}\Delta u_s(\cdot,s)dsdt=-\frac{1}{\tau}\int\limits^{t_1}_{t_0}\int\limits^{s}_{t_0}\Delta u_s(\cdot,s)dtds=-\frac{1}{\tau}\int\limits^{t_1}_{t_0}s\Delta u_s(\cdot,s)ds.
\widehat{\xi}(S) = \frac{1}{2^n}\sum_{T\subseteq N}(-1)^{|S\cap T|}\xi(T) & = \frac{1}{2^n}\sum_{T\subseteq N}(-1)^{|S\cap T|}\sum_{K\subseteq T}m^\xi(K) = \frac{1}{2^n}\sum_{K\subseteq N}m^\xi(K)\sum_{T\supseteq K}(-1)^{|S\cap T|}.
|\Pi_n(134/2)| = 1+ \sum_{k=1}^{\lfloor n/2 \rfloor}\sum_{f=0}^{n-2k}|\Pi_{n,k,f}(134/2)| &= 1+ \sum_{k=1}^{\lfloor n/2 \rfloor}\sum_{f=0}^{n-2k}{n-f-k-1\choose k-1}{2k+f\choose f}(2k)!!.
R_{12}=(db^{-1}a-c)^{-1}\left(ad-da+cb-bc+da-db^{-1}ab\right) (\Delta')^{-1}& =(db^{-1}a-c)^{-1}\left(ad-bc +cb-db^{-1}ab\right) (\Delta')^{-1}& =(db^{-1}a-c)^{-1}\left(\Delta' +(c -db^{-1}a)b\right) (\Delta')^{-1}& =\left(-b+ (db^{-1}a-c)^{-1}\Delta'\right)(\Delta')^{-1}
\mu_d\bigl(B_x(r_{n,v}) \cap B_y(r_{n,v})\bigr) = V_d r_{n,v}^d I_{\frac{d+1}{2},\frac{1}{2}}\biggl( 1- \frac{\|x-y\|^2}{4r_{n,v}^2}\biggr) &= \frac{ve^{\Psi(k)}}{n-1}I_{\frac{d+1}{2},\frac{1}{2}}\biggl(1- \frac{\|z\|^2}{4\{vf(x)\}^{2/d}}\biggr)
\frac{(-1)^{p}}{\prod_{i=0}^q (w_i-v)^{\beta-1}\prod_{i=1}^{q-1}(\xi_i-v)}\frac{d^p}{dv^p}\left[ \prod_{i=0}^q (w_i-v)^{\beta-1}\prod_{i=1}^{q-1}(\xi_i-v) \right] .
\xi \equiv \zeta \mathrm{e}^{\frac{g_{m}^{2}}{2}D_{m_{H}}( 0 )}= \frac{m_{W}^{7 / 2}}{g}\delta \left( \frac{\lambda}{g^{2}}\right) \mathrm{e}^{\frac{2 \pi m_{W}}{g^{2}}\left( - 2 \epsilon + \mathrm{e}^{- c}\right)}
z = \Lambda^{2}\exp \{- \frac{\pi}{2}\frac{\Lambda}{g^{2}}( 1 + \frac{4 \theta^{2}g^{2}}{\eta^{2}}) \}= \Lambda^{2}\exp \{- \frac{\pi}{2}( \frac{\Lambda}{g^{2}}+ \frac{4 \theta^{2}g^{2}}{\Lambda}) \}
E_{\tau}^{\mu}\equiv \left( 1 , 0 , 0 \right) \quad ; \quad E_{\sigma}^{\mu}\equiv \left( 0 , 1 , 0 \right) \quad ; \quad n^{\mu}= \left( 0 , 0 , \frac{1}{b_{0}^{2}+ l^{2}}\right)
H_f(G) =\frac{1}{3}\Big(4f(1) - f(4)\Big)n + \frac{2}{3}\Big(f(4) - f(1)\Big)m begin{align*}4mm] \ + \left(f(2) -\frac{2}{3}f(1) -\frac{1}{3}f(4)\right) n_2 + \left(f(3) -\frac{1}{3}f(1) -\frac{2}{3}f(4)\right) n_3 \,.
\partial_{J}^{I}u_{i}^{K}= \delta_{J}^{K}u_{i}^{I}-{\frac{1}{N}}\delta_{J}^{I}u_{i}^{K}\; , \qquad \partial_{J}^{I}u_{K}^{i}= - \delta_{K}^{I}u_{J}^{i}+{\frac{1}{N}}\delta_{J}^{I}u_{K}^{i}\; .
I_{2}= ( - 1 ) \int \delta^{3}\! \left({\vec{r}_{1}+ \vec{r}_{2}- \vec{r}}\right) \delta ( r_{1}+ r_{2}- t ) \frac{d^{3}r_{1}}{4 \pi r_{1}}\frac{d^{3}r_{2}}{4 \pi r_{2}{\ell_{0}}^{2}}
\Sigma_{2a}= \sum \limits_{\delta = \pm 1}\sum \limits_{\substack{D_1 | D_2 m_2 D_1^2 = n_1 D_2}}\frac{\tilde{S}(\delta m_1, n_1, n_2, D_1, D_2)}{D_1 D_2}\tilde{\mathcal{J}}_{\delta}\left(\sqrt{\frac{m_1 n_1 n_2}{D_1 D_2}}\right),
\hat{Q}_{1}=r_{1}+i\vartheta\frac{\partial}{\partial r_{2}},\;\; \hat{Q}_{2}=r_{2}, \hat{P}_{1}=-i\hbar\frac{\partial}{\partial r_{1}},\;\; \hat{P}_{2}=-i\hbar\frac{\partial}{\partial r_{2}},
I=\frac{M}{2}\int_0^{s}d\tau \dot{x}_i^2+\frac{M}{2}\int_0^{s'}d\tau '\dot{y}_j^2+\frac{M}{2}(s+s')+\frac{K_R}{2}\int_0^s d\tau \epsilon_{ij}x_i\dot{x}_j+\frac{K_R}{2}\int_0^{s'}d\tau ' \epsilon_{ij}y_i\dot{y}_j.
0> x_\infty\cdot f(0,x_\infty,\eta(0)) =v(0)\cdot f(0,v(0),\eta(0))= u^*(kr)\cdot f(0,u^*(kr),u^*(kr-\tau(0,u^*(kr)))) & =u^*(kr)\cdot \dot{u}^*(kr) = \tfrac12\tfrac{d}{dt}|u^*(kr)|.
\Gamma_2^{-1}(z\,|\,a_1,a_2) = (2\pi)^{-z/2a_1}a_2^{1+{}_2 S_0 (z\,|\,a_1, a_2)}G\Bigl(\frac{z}{a_1}\,\Big|\,\frac{a_2}{a_1}\Bigr).
\lambda_{1}\lambda_{2}+2c\nu^{2}+\displaystyle\frac{c(\lambda_{1}b_{1}^{2}-\lambda_{2}b_{2}^{2})}{\lambda_{2}-\lambda_{1}}+\displaystyle\frac{2\nu^{2}(b_{1}^{2}+b_{2}^{2})}{(\lambda_{2}-\lambda_{1})^{2}}=0.
|\mu_{1}(\check{l_1},\check{l_2})| = 2\frac{\sin{\frac{\pi (4k+r+1)}{2(2k^2+(r+3)k+c_r)}}}{\sin{\frac{\pi}{2k^2+(r+3)k+c_r}}}\cos{\frac{\pi(r-2)}{2(2k^2+(r+3)k+c_r)}}&=4k+r+1-\frac{2\pi^2}{3}\frac{1}{k}+O(k^{-2}), \mathrm{RB}(\hat{l}+1) =2\sqrt{2(2k^2+(r+3)k+c_r)-4k-r-2}&=4k+r+1-\frac{(r+3)^2-8c_r}{8}\frac{1}{k}+O(k^{-2}).
\left( \begin{array}{c}{\vec{F}^{( 1 )}}\\{- \vec{F}^{( 2 )}}\\ \end{array}\right) = \frac{1}{\sqrt{2}}\left( \begin{array}{c c}{1}&{1}\\{- 1}&{1}\\ \end{array}\right) \left( \begin{array}{c}{\vec{F^{R}}}\\{\vec{F}^{L}}\\ \end{array}\right) .
\frac{\partial}{\partial\tau}(\frac{x_{H}}{x_{L}})=\frac{x_{H}}{x_{L}}(\frac{1}{x_{H}}(\frac{\partial x_{H}}{\partial p_{H}}+\frac{\partial x_{H}}{\partial p_{L}})-\frac{1}{x_{L}}(\frac{\partial x_{L}}{\partial p_{H}}+\frac{\partial x_{L}}{\partial p_{L}})).
F_N(1) =\frac{1}{N}\sum^{N-1}_{n=1}\frac{1-\cos\phi_n}{(1-\cos\phi_n)^2}&=\frac{1}{2N}\sum^{N-1}_{n=1}\frac{1}{\sin^2 \phi_n/2}=\frac{N}{3}-\frac{1}{3N}
\quad\Pr\left(>\theta|\Phi,\Phi(\mathcal{A})>0 \right) \approx \sum_{m=1}^{M}C_{M}^m(-1)^{m+1}\prod\limits_{x_{i}\in\Phi\backslash\{x_1\}\cap{\mathcal{A}}}\frac{1}{\left(1+\frac{m\eta\theta r_1^{\alpha}}{Mr_i^{\alpha}}\right)^M}.
\partial_{t}^{2}f - r^{- 1}\partial_{r^{*}}\left( r^{2}\partial_{r^{*}}{\frac{f}{r}}\right) + \left( 1 -{\frac{2 M}{r}}\right) \left( \frac{l \left( l + 1 \right)}{r^{2}}+ m^{2}\right) f = 0 .
\int_{I^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k\geq \int_{I_1^k}\phi_{k,n}(y_1,\cdots,y_k)dy_1\cdots dy_k = S(I)^k(\ln n)^{-k}\int_{(-l_n,l_n)^k}\phi_{k,n}\big(\gamma_n(u_1),\cdots,\gamma_n(u_k)\big)\prod_{j=1}^k\beta_n(u_j)du_1\cdots du_k.
B(q, N+1) = \sum_{n = 1}^{N}\frac{(-1)^{n - 1}nq^{n(n + 1)/2}}{(q)_n (1 - q^n) (q)_{N-n}}\frac{1}{(1 - q^{N+1-n})}+ \frac{(-1)^N (N+1) q^{(N+1)(N+2)/2}}{(q)_{N+1}(1- q^{N+1})}.
d\sum\limits_{i=0}^{q-2}\binom{q}{i}(lx+1)^{q-i}B_{i}\equiv d\sum\limits_{i=0}^{q-2}\binom{q}{i}B_{i}\equiv d\sum\limits_{i=0}^{\frac{q-2}{2}}\binom{q}{2i}B_{2i}\pmod{4}.
R_{k,i}=\log_{2}\bigg(1+\frac{P_{k,i}{\lvert \mathbf{d}_{k,i}^{H}\mathbf{h}_{k}\rvert}^{2}}{\sum_{\pi_{k',i'}>\pi_{k,i}}P_{k',i'}{\lvert \mathbf{d}_{k',i'}^{H}\mathbf{h}_{k'}\rvert}^{2}+\sigma_{\textrm{ul}}^{2}}\bigg),
\dot{r}= \eta - \frac{\sqrt{M_{1}\pi}}{\lambda e^{\lambda r}}\: \: \: \: ; \: \: \: \: ( e^{\lambda r}- \eta \sqrt{M_{1}\pi}/ \lambda ) = e^{\eta \lambda t}( e^{\lambda r ( 0 )}- \eta \sqrt{M_{1}\pi}/ \lambda )
\begin{array}{c c c c c}{{| I ( k ) | \simeq | \vec{k}|^{- 3 + \epsilon},}}&{{| \phi ( k ) | \simeq | \vec{k}|^{\tau},}}&{{\tau > - \epsilon}}&{{\mathrm{\a s \}}}&{{| \vec{k}| \rightarrow 0 ,}}\\ \end{array}
F_{n+1}= \left[\begin{array}{cccc|c}F_n 0 \hline0 \cdots 0 U I_{n+1}C_{n+1}\end{array}\right] :{\begin{array}{lll}R(-1)^{\binom{n+2}{2}}\xrightarrow{F_n}R^{\binom{n+1}{2}}\;\; \bigoplus \searrow \;\bigoplus R(-1)^{n+2}\longrightarrow R^{n+1}\end{array}}
\mbox{Prod}\left(\mathbf{U},\overline{\mathbf{U}}^{\top^{\left(2m-1\right)}},\cdots\overline{\mathbf{U}}^{\top^{2k+1}},\mathbf{U}^{\top^{2k}},\cdots,\mathbf{U}^{\top^{2}},\overline{\mathbf{U}}^{\top}\right)=\boldsymbol{\Delta}.
\widehat{g}^{t}(k,\omega) = \int f(p,k) \left[\int_{-\infty}^{\infty}e^{-2\pi i \left(\frac{k}{L}\cdot v(p)\right)t}e^{-2 \pi i \omega t}dt\right]d^3p &=\int f(p,k) \; \delta\left(\omega + \frac{k}{L}\cdot \frac{p}{\sqrt{1+\lvert p \rvert^2}}\right)d^3p,
b_{2,1}=b_{1,1}^{\ast}-b_{1,1}\beta_{2,2}=\varphi_{z_{0}}\left(z_{2}\right) -\varphi_{z_{0}}\left( z_{1}\right) \varphi_{z_{1}}\left(z_{2}\right) , b_{2,2}=\beta_{2,2}=\varphi_{z_{1}}\left( z_{2}\right) .
V(\hat{f_1}^h,\ldots,\hat{f_n}^h) \cap \{X_0\neq 0\}\subset V(\hat{f_1},\ldots,\hat{f_n}), V(\hat{f_1}^h,\ldots,\hat{f_n}^h) \cap \{X_0=0\}\subset V(X_{i_1},\ldots,X_{i_n}).
\frac{|\hat\omega_\infty^\bullet(\hat{x}_\infty) - \mathbf{P}^{g_P}_{\hat{\gamma}_\infty}(\hat\omega_\infty^\bullet(\hat{x}'_\infty))|_{g_P(\hat{x}_\infty)}}{d^{g_P}(\hat{x}_\infty,\hat{x}')^\alpha}= 1,
\!\!\!\!S_0(f)\!= \!\frac{N_0}{\alpha_{\rm c}}\! \left( \!\frac{1}{\frac{\mu}{2}\int\limits_0^{\pi}G(\cos \theta \cdot f) \sin \theta{\rm d}\theta}- \frac{1}{G(\cos \theta_{\rm c}\cdot f)}\!\right)_{\!\!\!\!+}\!\!,
\int\limits_{S^{n-1}}\int\limits_0^\pi Cf(u,\beta,\psi)\sin\psi d\psi d\beta= \frac{\pi}{|S^{n-1}|}\int\limits_{S^{n-1}}Rf(\omega,\omega\cdot u)\int\limits_{S^{n-1}}|\omega \cdot \beta|d\beta d\omega.
Y_t^k=\xi^k-\sum_{l=1}^m\int_t^T Z_s^{k,l}dB_s^l+\frac{1}{2}\sum_{l=1}^m\int_t^T \sum_{i,j=1}^n \Gamma_{ij}^k (Y_s)Z_s^{i,l}Z_s^{j,l}ds.
\int\limits_GF(g)\,dg =2^{n(n+3)/2}\,\pi^{n(n+1)/2}\prod_{m=1}^n\frac{\Gamma(k-(n+m)/2)}{\Gamma(k-(n-m)/2)}=2^{n(n+2)}\,\pi^{n(n+1)/2}\prod_{m=1}^n\,\prod_{j=1}^m\frac{1}{2k-n-m-2+2j}.
\varepsilon_{1 0}^{\mu \kappa r_{1}\cdots r_{8}}\varepsilon_{1 0}^{\nu \lambda s_{1}\cdots s_{8}}\times \eta_{\mu \nu}\times \zeta_{\kappa \lambda}^{( 5 )}\times \prod_{i = 1}^{4}\zeta_{r_{2 i - 1}s_{2 i - 1}}^{( i )}k_{r_{2 i}}^{( i )}k_{s_{2 i}}^{( i )}\ .
\int e^{i \vec{q}\cdot \vec{\varrho}}\frac{\vec{q}}{\vec{q}^{2}+ \mu^{2}}\frac{d^{2}\vec{q}}{( 2 \pi )^{2}}= i \frac{| \mu |}{2 \pi}K_{1}( | \mu | \varrho ) \hat{\varrho}~ ~ ,
\Pr(Y \geq X) =(1+o(1)) \int_{np - b_p}^{np + b_p}\int_{nq - b_p}^{nq + b_q}1{\{x \le y\}}\phi \left( \frac{y-nq}{\sqrt{n}\sigma_q}\right)\phi \left( \frac{x-np}{\sqrt{n}\sigma_p}\right) dy dx + o(n^{-1}),
\partial_i v_{\delta}[\hat{\imath}]=\sum_{j\in S}\sqrt{\lvert j \rvert}\sqrt{\xi_j+\varepsilon^{2(b-1)}(y_{\delta})_j}e^{\mathrm{i}(\theta_0)_j}\left(\mathrm{i}\hat{\Theta}_j+\varepsilon^{2(b-1)}\frac{\hat{y}_j+(\partial_i G(i_0(\varphi))[\hat{\imath}])_j}{2\,\lvert j \rvert\,(\xi_j+\varepsilon^{2(b-1)}(y_{\delta})_j)}\right)\,e^{\mathrm{i}j x}
\left\{{\frac{D e t^{\prime}\left[ - ( d^{2}/ d \tau^{2}) + V^{\prime \prime}[ x_{c}( \tau ) ] \right]}{D e t \left[ - ( d^{2}/ d \tau^{2}) + \omega^{2}\right]}}\right\}^{- 1 / 2}s q r t{{\frac{S_{e o}}{2 \pi}}}\d \tau_{c}
\begin{cases}a_{31}^{\prime}(t) = -2\pi i [ a_{31}(t) + e^{6\pi i}a_{10}^3 (t) b_{01}(t)] b_{22}^{\prime}(t) = -2\pi i [ b_{22}(t) - e^{6\pi i}a_{10}^2 (t) b_{01}^2 (t)]\end{cases}