harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\|v\|_{L^s(R_1)}^{s/\bar{s}}=\||v|^{s/\bar{s}}\|_{L^{\bar{s}}(R_1)}\le \||v|^{s/\bar{s}}\|^{\bar{a}}_{L^t(R_1)}\||v|^{s/\bar{s}}\|^{1-\bar{a}}_{L^{\bar{q}}(R_1)}= \||v|^{s/\bar{s}}\|^{\bar{a}}_{L^t(R_1)}\|v\|^{(1-\bar{a})s/\bar{s}}_{L^{q}(R_1)}.
p_{n+1}(z) = z [zS_{n-1}(z) - \rho_{n-1}S_{n-1}^{\ast}(z)] + \frac{\rho_{n}}{\rho_{n-1}}[zS_{n-1}(z) - \rho_{n-1}S_{n-1}^{\ast}(z)] , n \geq 1,
\frac1{\omega(B) \ell(B)^\beta}\int_B |g_{21}(y)-(g_{21})_B|d\omega(y)+\frac1{\omega(B) \ell(B)^\beta}\int_B |g_{\sigma_j1}(y)-(g_{\sigma_j1})_B|d\omega(y) &\le C\|b\|_{\Lambda^\beta}\big(M_t(R_{{\rm D},\ell}f_1)(x)+M_t(R_{{\rm D},\ell}f_{\sigma_j})(x)\big)\qquad\mbox{for}j=1,2,\cdots,|G|-1.
\mathcal{M}[e^{-\rho\ln^2(t)}](s)=\mathcal{M}[e^{-\rho\ln^2(t)}](0)\exp\left(\frac{s^2}{4\rho}\right)=\sqrt{\frac{\pi}{\rho}}\exp\left(\frac{s^2}{4\rho}\right)
\tilde{\pi}\left( (g_1,g_2)^{-1}(t_1,t_2) (g_1,g_2) \right) = \Lambda^{-1}\cdot \tilde{\pi}\left( (t_1,t_2) \right) \forall (t_1,t_2) \in \mathfrak{sl}_2(\mathbb{Z}_\ell)^2.
(\partial_t\partial^4_y \tilde u)(t, x, 0)= \bigg(\partial_y u^s(t, 0) + (\partial_y\tilde{u})(t, x, 0)\bigg)\bigg((\partial^3_y\partial_x\tilde{u})(t, x, 0)\bigg) & + \bigg(\partial^3_y u^s(t, 0) + (\partial^3_y\tilde{u})(t, x, 0)\bigg)\bigg((\partial_y\partial_x\tilde{u})(t, x, 0)\bigg).
P \psi = P \left( \begin{array}{c}{{\psi_{0}}}\\{{\psi_{1}}}\\{{\psi_{2}}}\\{{\psi_{3}}}\\{{\psi_{4}}}\\ \end{array}\right) = \left( \begin{array}{c}{{0}}\\{{0}}\\{{0}}\\{{0}}\\{{\psi_{4}}}\\ \end{array}\right) ; \P_{\mu}\psi = \left( \begin{array}{c}{{0}}\\{{0}}\\{{0}}\\{{0}}\\{{\psi_{\mu}}}\\ \end{array}\right) \ \left( \mu = 0 , . . . , 3 \right) .
P_{\rm red}(i\rightarrow f)=\sum_{m=0}^{+\infty}\sum_{\ell=-\infty}^{+\infty}<r_f,\theta_f|m,\ell>\,e^{-\frac{i}{\hbar}(t_f-t_i)E_{m,\ell}}\,<m,\ell|r_i,\theta_i>\ \ \ .
\beta_{k}^{-1}D_{k}u + \sum_{j=0}^{\infty}\beta_{j+k+1}^{-1}C A^{j}B u \lambda^{j+1}= \Theta_{k,0}u + \sum_{j=0}^{\infty}\beta_{j+k+1}^{-1}\beta_{j+k+1}\Theta_{k,j+1}\lambda^{j+1}u = \Theta_{k}(\lambda) u
|\langle\psi_Q,\phi\rangle| =|\langle\psi_Q,\phi_{Q_{0,\mathbf{0}}}\rangle| &\lesssim\|\psi\|_{S_{M+1}}\|\phi\|_{S_{M+1}}\left[\min\left\{[\ell(Q)]^{-1},\ell(Q)\right\}\right]^{M+\frac{n}{2}}\left[1+\frac{|x_Q|}{\ell(Q)\vee 1}\right]^{-(n+M)},
=\mathbb{E}_{X^{n}}\{\frac{1}{M}\sum_{m}Tr\{\Pi_{\rho,\delta}^{n}\rho_{X^{n}(m)}\Pi_{\rho,\delta}^{n}\}\}+\frac{1}{M}\sum_{m}Tr\{\hat{\Pi}_{\rho,\delta}^{n}\mathbb{E}_{X^{n}}\{\rho_{X^{n}(m)}\}\}
\left.\begin{array}{llr}\gamma_{1, 3, (\pm 1, 0, 0)}={\bf 1}_{1}\gamma_{1,{\tilde 7}_{3}, (\pm 1)}= - \alpha{\bf 1}_{1}\gamma_{1, 7_{4}, (0)}= - \alpha{\bf 1}_{2}\gamma_{1, 7_{5}, (0)}= \alpha^2{\bf 1}_{3}\end{array}\right.
g_{M'}^3(\tau,r) \leqslant 4(g_0(r^{-1}))^{-2m'_0}(M'\vee C_{m'_0})+2r +2(C_{m'_0}(g(r^{-1}))+1) \big(\ln\ln(r^{-1}\vee 3)\big)^{-1}+r\exp\big(C\tau\ln\ln (r^{-1}\vee3)\big).
f(t) = f(0) + \int_0^t \left( \int_0^s f''(l) dl \right) ds &= f(0) + \int_0^t \left( \int_0^s \langle x'_l , \nabla^2 u|_{x_l}x'_l\rangle dl \right) ds &\geq f(0) - \int_0^t \left( \int_0^s \langle x'_l , \nabla^2 \phi_0|_{x_l}x'_l\rangle dl \right) ds &\geq f(0) - Ct^2,
\bold{E}_{\alpha}(A\cdot B)=\sum_{\beta,J_1,J_2}(-D)_{J_1}S_{-J_2}\left(\frac{\partial A^{\beta}}{\partial u^{\alpha}_{J_1;J_2}}B^{\beta}+\frac{\partial B^{\beta}}{\partial u^{\alpha}_{J_1;J_2}}A^{\beta}\right).
32\gamma_1 = 16\left(c_5(q,1,p_0)-\frac{c_2(q,1,p_0)c_3(q,2,1,p_0)}{c_1(q,2,p_0)}\right) &=q_1 - \frac{q_2^2}{q_1}&= \frac{1}{q_1}(q_1+q_2)(q_1-q_2)
[E_i,F_j]= \delta_{ij}\frac{K_i-K_i'}{q-q^{-1}}, K_i E_j =q_i^{c_{ij}}E_j K_i, K_i F_j=q_i^{-c_{ij}}F_j K_i, K_i' E_j=q_i^{-c_{ij}}E_j K_i', K_i' F_j=q_i^{c_{ij}}F_j K_i',
a= \frac{\cos\varphi}{\sin \varphi},b = y_1 -\left(\tfrac{\cos\varphi}{\sin \varphi}\right) y_2,c = y_3-\tfrac{1}{2}y_1y_2 + \tfrac{1}{2}\left(\tfrac{\cos \varphi}{\sin \varphi}\right)y_2^2.
A = \left( \begin{matrix}{{D}}&{{E}}\\{{- E^{\ast}}}&{{- D^{\ast}}}\\ \end{matrix}\right)^{2}= \left( \begin{matrix}{{D^{2}- E E^{\ast}}}&{{( D - D^{\ast}) E}}\\{{- ( D - D^{\ast}) E^{\ast}}}&{{( D^{\ast})^{2}- E E^{\ast}}}\\ \end{matrix}\right) ,
j_P^+(E^P_{o_{-,P}\cdot x}(w)) = E_{o_-\cdot x}(w) (w\in W_P^+(1)), j_P^{+*}(E^P_{o_{+,P}\cdot x}(w)) = E_{o_+\cdot x}(w) (w\in W_P^+(1)), j_P^-(E^P_{o_{+,P}\cdot x}(w)) = E_{o_+\cdot x}(w) (w\in W_P^-(1)), j_P^{-*}(E^P_{o_{-,P}\cdot x}(w)) = E_{o_-\cdot x}(w) (w\in W_P^-(1)).
a\nabla_{\nu}b \geqslant a\sharp_{\nu}b+\sum_{k=0}^{n}r_{k}\big[\big(a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}.
\phi( \exp_U (t X_0))\, \phi( \exp_U (s Y)) \, \phi( \exp_U (t X_0))^{-1}\,=\exp_{\widetilde{U}}( \sum_l \{\mu_l (s) + \sum_{i,j}a_{ijl}\lambda_i (t) \mu_j(s)\}\widetilde{Y}_l).
(1-x) \prod_{j=1}^{r-1}\left(1-dxq^{a(j)}\right) F(x) = F(xq^N) + \sum_{j=1}^{r}\left( \sum_{m=0}^{r-j}d^m \sum_{\substack{\alpha < a(r+1) w(\alpha)=j+m}}xq^{\alpha}\left( (-x)^{m-1}{j+m-1 \brack m-1}_{q^N}+ (-x)^m{j+m \brack m}_{q^N}\right) \right) \times \prod_{h=1}^{j-1}\left(1-dxq^{hN+a(r)}\right) F\left(xq^{jN}\right).
u(x_j,t^{n+1})=\sum\limits_{l=0}^{n}(w_{n-l}-w_{n-l+1})u(x_j,t^{l}) + w_n u(x_j,t^{0})+\nu\alpha_0 (u(x_j,t^{n+1}))_{xx}-{}\frac{\alpha_0}{3}\left( u(x_j,t^{n})(u(x_j,t^{n+1}))_x+(u(x_j,t^{n})u(x_j,t^{n+1}))_x\right),
a(t, x, \xi)\geq m_1(t, z)\big(t-\phi(z)\big)^2+\psi(z) =m_1(t, z)\big(t^2+2t|\phi(z)|+\phi^2(z)\big)+\psi(z) \geq c_1t^2+2c_1t|\phi(z)|+m_1(0, z)\phi^2(z)+\psi(z) +\big(m_1(t, z)-m_1(0, z)\big)\phi^2(z).
B_i^\prime := \bigcup \left\{P \in \mathcal{O}(G;A) : P \subseteq B_i (P)\sigma_i \subseteq B_j j <i \right\}, B_i^{\prime \prime}:= \bigcup \left\{P \in \mathcal{O}(G;A) : P \subseteq B_i (P)\sigma_i \subseteq B_i \right\}.
\mbox{var}\left( \#\mathcal{P}_{\Omega}\cap [0, \textbf{x}] \right) = \sum_{i=1}^{N}{\frac{|\Omega_i \cap [0,\textbf{x}]|}{|\Omega_i|}\left(1 - \frac{|\Omega_i \cap [0,\textbf{x}]|}{|\Omega_i|}\right)}= N |[0,\textbf{x}]| - N^2\sum_{i=1}^{N}{\left( |\Omega_i \cap [0,\textbf{x}]| \right)^2}.
D_{0}^{c}= D_{0}^{c}( x_{o u t}, x_{i n}) = \frac{1}{2 ( 2 \pi )^{2}}\int_{0}^{\infty}\frac{d e_{0}}{e_{0}^{2}}\exp \left[ - \frac{i}{2}\left( e_{0}m^{2}+ \frac{\Delta x^{2}}{e_{0}}\right) \right] \; .
c(\tau,\gamma) = \int\limits_{0}^{L}\int\limits_{0}^{L}\int\limits_{\mathcal{S}^{1}}\langle\theta,\dot{\gamma}(t_{1}) \rangle^{2}\langle\theta,\dot{\gamma}(t_{2}) \rangle^{2}d\tau(\vartheta)dt_{1}dt_{2}=\int\limits_{\mathcal{S}^{1}}d\tau(\theta) \left[\int\limits_{0}^{L}\langle \theta,\dot{\gamma}(t) \rangle^{2}dt \right]^{2},
\psi_{k}^{( 2 )}( x ) = \frac{1}{\sqrt{2 \pi ( k^{2}+ 1 ) ( k^{2}+ 4 )}}\left[ \left( 2 - k^{2}- \frac{3}{\cosh^{2}x}\right) \sin k x - 3 k \cos k x \tanh x \right] .
c_\ell =\sum_{i=0}^{\ell-2}\frac{i}{(\ell-i)(i!)^2}&=\sum_{i=0}^{3}\frac{i}{(\ell-i)(i!)^2}+\sum_{i=4}^{\ell-2}\frac{i}{(\ell-i)(i!)^2}&\leq \frac{3}{\ell-3}+\frac{1}{\ell}\sum_{i=4}^{\ell-2}\frac{1}{i!}\mbox{(since $\frac{i}{(\ell-i)i!}\leq \frac{1}{\ell}$ for $\ell\geq 4$}) &\leq \frac{3}{\ell-3}+\frac{e}{\ell},
\langle \bar{P}\rangle_{1}={\frac{\partial}{\partial \bar{t}}}\, \bar{F}_{1}= -{\frac{\langle P \rangle_{1}}{\langle P P \rangle_{0}}}+{\frac{1}{2}}\,{\frac{\langle P P P \rangle_{0}}{\langle P P \rangle_{0}}}\ .
L = \sum_{\alpha}\left( p_{\alpha}^{i}\dot{q}_{i \alpha}+ J_{\alpha}\cos \theta_{\alpha}\dot{\phi}_{\alpha}\right) + \int d^{2}{\bf x}\left({\frac{\kappa}{2}}\epsilon^{i j}\dot{A}_{i}^{a}A_{j}^{a}\right) - H ,
D{\nabla^2}C(\bar r,t|{{\bar r}_{\rm tx}},{t_0}) - \bar v(\bar r)\cdot{\bar \nabla C}(\bar r,t|{{\bar r}_{\rm tx}},{t_0}) -{k_{d}}C(\bar r,t|{{\bar r}_{\rm tx}},{t_0}) + S(\bar r,t,{\bar{r}_{\rm tx}},t_0) = \frac{{\partial C(\bar r,t|{{\bar r}_{\rm tx}},{t_0})}}{{\partial t}},
-\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}(H-\xi)^{l^{*}}H^{k^{*}}= -\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}(H-\xi)^{k^{*}}(H-\xi)^{l^{*}}&-\sum_{i,j,k,l}h_{ij}^{k^{*}}h_{ij}^{l^{*}}\xi^{k^{*}}(H-\xi)^{l^{*}}\leq 2-2\langle H,H-\xi\rangle-\langle H,\xi\rangle = 2-\langle H,H-\xi\rangle-|H|^{2}.
c_2(X) =f^*c_2(Y)-d(C)[F], (-K_X)^2\cdot c_2(X) =(-K_Y)^2\cdot c_2(Y)-4d(C), (-K_X)^4 =(-K_Y)^4-16d(C), \chi(X,-K_X) =\chi(Y,-K_Y)-3d(C).
x^{(m+1)}=(\begin{matrix}0&\frac{1}{3}&\frac{1}{2}\\ \frac{1}{4}&0&\frac{1}{2}\\ \frac{3}{5}&\frac{1}{5}&0\end{matrix})x^{(m)}+(\begin{matrix}\frac{5}{6}\\ -3\\ 2\end{matrix}),m=0,1,2,...
{\cal A}_{\beta}= \frac{1}{8{\pi}^2}[-2(\beta/l^2)l^{-2}e\cdot R\cdot e + (\beta/l^2)^2 R^{ab}{\mbox{\tiny $\wedge$}}R_{ab}+ 2(\beta/l^2)^2 l^{-2}T\cdot T] + O[(\beta/l)^{-2})].
\sum_{m=0}^{N-k}\frac{\left(\frac{cq^{k}}{d}\right)_{m}\left( q^{-(N-k)}\right)_m(dq^{N+1})^m}{(q)_m\left(cq^{k+1}\right)_{m}}=\frac{(dq^{k+1})_{\infty}(\frac{cq^{k}}{d})_{\infty}}{(cq^{k+1})_{\infty}(dq^{N+1})_{\infty}}\;_{2}\phi_{1}\left[\begin{array}{ccc}dq, dq^{N+1};\; \frac{cq^k}{d}dq^{k+1}\end{array}\right].
\chi_j^\dag\,d\mu\,\chi_j = \begin{cases}\varphi_j^L\,d\mu\,(\varphi_j^L)^\dag, (\varphi_j^R)^\dag d\mu\,\varphi_j^R, \end{cases}x_j^\dag\,d\mu\,x_j = \begin{cases}(\varphi_j^R)^\dag d\mu\,\varphi_j^R j, \varphi_j^L\,d\mu\,(\varphi_j^L)^\dag j. \end{cases}
\partial(\theta_{n}(\cdot, \overline{a}))= c\Longleftrightarrow \bigwedge_{m=1}^{n}[\overline{a}\in U_{n,m}\rightarrow (\exists \; \overline{b}\in M^{\ell_{m}}\; \& \; \forall \; x \; \theta_{n}(x, \overline{a})\leftrightarrow \theta_{m}(x, \overline{b}) \; \& \; c=g_{m}(\overline{b}))]
\begin{cases}\partial_t \bigl(\frac{\omega}r\bigr) + (u\cdot \nabla) \bigl( \frac{\omega}r \bigr)=0, u= -\Delta^{-1}\nabla \times \omega, \omega \Bigr|_{t=0}= \omega_0^{(l)}+ \omega_0^{(h)}; \end{cases}
(1+\tfrac{1-\beta}{\beta})\langle J_Ax-J_Ay,x-y\rangle \geq\tfrac{1}{2\beta}\left(\|x-y\|^2+(\beta-1)^2\|J_Ax-J_Ay\|^2\right) &\quad+(1-\beta)\|J_Ax-J_Ay\|^2 &=\tfrac{1}{2\beta}\|x-y\|^2+\tfrac{(\beta-1)^2+2\beta(1-\beta)}{2\beta}\|J_Ax-J_Ay\|^2 &=\tfrac{1}{2\beta}\|x-y\|^2+\tfrac{1-\beta^2}{2\beta}\|J_Ax-J_Ay\|^2.
G_{-1}=-G_{0,0}vE_{-1}v^*G_{0,0}-G_{0,0}vE_{-2}v^*G_{0,1}&\quad-G_{0,1}vE_{-2}v^*G_{0,0}&=-G_{0,0}v\Bigl(B_0\bigl(C_0+C_0A_1C_0 &\quad\qquad+C_0A_0C_1+C_1A_0C_0\bigr)B_0 &+B_0C_0A_0C_0B_1+B_1C_0A_0C_0B_0\Bigr)v^*G_{0,0}&-G_{0,0}vB_0C_0A_0C_0B_0v^*G_{0,1}&-G_{0,1}vB_0C_0A_0C_0B_0v^*G_{0,0}.
\begin{cases}Q_{i+1,j+1}:=(\min[Q_{i+1,j},W_{i+1,j}]-\min[Q_{i,j},W_{i,j}])+Q_{i,j}, W_{i,j}^+:=(\min[Q_{i+1,j},W_{i+1,j}]-\min[Q_{i,j},W_{i,j}])+W_{i,j},\end{cases}
\begin{array}{c}{{H_{l}s_{l}\left( r , \chi \right) = \left\{2 \sinh{}^{2}\frac{i}{2}\frac{d}{d r}+ \frac{l \left( l + 1 \right)}{2 r \left( r + i \right)}e^{i \frac{d}{d r}}\right\}s_{l}\left( r , \chi \right) =}}\\{{= \left( \cosh \chi - 1 \right) s_{l}\left( r , \chi \right)}}\\ \end{array}
I_{c,\lambda,N}\left(\frac{H}{\ell(x) L(1,\mathbb{X}_D)}\right)=\begin{cases}1 \ell(x)\leq Y, \in[0,1] Y< \ell(x)\leq e^{\lambda N}Y, 0 \ell(x)>e^{\lambda N}Y.\end{cases}
\langle H_2\rangle_{\rho_T}-\langle H_2\rangle_{\rho_g}=-\sum_{i\neq 1}\frac{\lambda^2|\langle 1|\Delta H|i\rangle|^2}{\lambda_i}+\sum_{i\neq 1}\frac{4\lambda^2\sin^2(\lambda_iT/2)|\langle 1|\Delta H|i\rangle|^2}{T^2\lambda_i^3}+O(\lambda^3).
I_{CWZW}[\alpha]= \int dx^0 \left( - \sum_{n\neq 0}\frac{\dot{\alpha}_0^a \alpha_{a\:n}}{ikn}+ \sum_{n\ge 1}\frac{2 \dot{\alpha}_{-n}^a \alpha_{a\:n}}{ikn}+ \tau L_0 - 2\pi \alpha_0^3 \right),
\frac{3}{16 n \log^2n}\left(\begin{matrix}1 -i -i -1\end{matrix}\right) - \frac{3}{16 (n+1) \log^2(n+1)}\left(\begin{matrix}1 -i -i -1\end{matrix}\right) \qquad{}= \frac{3}{16 n^2\log^2n}\left(\begin{matrix}1 -i -i -1\end{matrix}\right) + O\left(\frac{1}{n^2\log^3n}\right).
\sum^{\lfloor n/4\rfloor}_{\substack{r=1 (r,n)=1}}\frac{1}{r^{2}}\equiv(-1)^{\frac{n-1}{2}}4n^{\phi(n)-2}\phi_{J_{4}}^{(2-\phi(n))}(n)E_{\phi(n)-2}\pmod{n}
\begin{array}{c c c c c c}{{\phi_{1}^{1}}}&{{\phi_{1}^{2}}}&{{\ldots}}&{{\phi_{1}^{a}}}&{{\ldots}}&{{\phi_{1}^{m}}}\\{{\vdots}}&{{\vdots}}&{{}}&{{\vdots}}&{{}}&{{\vdots}}\\{{}}&{{}}&{{}}&{{}}&{{}}&{{\phi_{N_{m}}^{m}}}\\{{\vdots}}&{{\phi_{N_{2}}^{2}}}&{{}}&{{\vdots}}\\{{\phi_{N_{1}}^{1}}}\\{{\}}&{{}}&{{}}&{{\phi_{N_{a}}^{a}}}\\ \end{array}
P([x_2,x_3]x_1) = P(x_1[x_2,x_3]) + P(dx_1d[x_2,x_3]) + P([[x_2,x_3],x_1]) P(x_2[x_1,x_3]) = P([x_1,x_3]x_2) + P(d[x_1,x_3]dx_2) + P([x_2,[x_1,x_3]]) P([x_1,x_2]x_3) = P(x_3[x_1,x_2]) + P(dx_3d[x_1,x_2]) + P([[x_1,x_2],x_3]).
ds^2 = H_p^{-1/2}[-dt^2 + \sum_{i=1}^{p-1}(dx^i)^2 + H_5^{NS}(dx^p)^2]+ H_p^{1/2}[\sum_{i=p+1}^6(dx^i)^2 + H_5^{NS}(dr^2 + r^2d\Omega_2^2)]
\pi(s,y)\sigma_0(s,y) = \Phi(s,y) +\frac{b_0(s,y)}{\sigma_0(s,y)}=\frac{\mathbb{E}[D_s\delta_Y(y)|\mathcal{F}_s]}{\mathbb{E}[\delta_Y(y)|\mathcal{F}_s]}+\frac{b_0(s,y)}{\sigma_0(s,y)}
\int_{\mathbb{R}^{6}}|\sum_{a=1}^{\infty}\sum_{b=1}^{\infty}u_{a}v_{b}T_{1,a}(x)T_{2,b}(x)|^{\frac{p}{4}}dx \lesssim_{\nu}(\sum_{a=1}^{\infty}\sum_{b=1}^{\infty}u_{a}v_{b})^{\frac{p}{4}}
\{i_1,i'_1,\ldots,i_{\lambda_1},i'_{\lambda_1}\}= \{1,\ldots,\lambda_1,s+1,\ldots,\lambda_1+s\}\{i_{\lambda_1+1},i'_{\lambda_1+1},\ldots,i_{s},i'_{s}\}= \{\lambda_1+1,\ldots,s,\lambda_1+1+s,\ldots,2s\}\{j_1,\ldots,j_k\}= \{2s+1,\ldots,r\},
Y = 2 [ ( x_{1 2}x_{3 4}) - 2 \frac{( x_{1 2}x_{2 4}) ( x_{3 4}x_{2 4})}{x_{2 4}^{2}}] \frac{1}{x_{2 4}^{2}}( 1 + \mathcal{O}( x_{1 2}, x_{3 4}) )
\begin{array}{l}\alpha_{11}(r,\theta) ={\left(z_{tt}- z_{ss}\right) \sin \theta \cos \theta + 2 z_{st}\cos^{2}\theta}, \alpha_{12}(r,\theta) ={- z_{ss}}, \alpha_{21}(r,\theta) ={z_{tt}}, \alpha_{22}(r,\theta) ={\left(z_{tt}- z_{ss}\right) \sin \theta \cos \theta - 2 z_{st}\sin^{2}\theta},\end{array}
\hat{X}_t^{\kappa, \delta}= e^{t A}\hat{X}^{\kappa, \delta}_0 + \int_0^t e^{( \kappa + t - s ) A}F( \hat{X}_s^{\kappa, \delta}) \, ds + \int_0^t e^{( \kappa + t - s ) A}B( \hat{X}_s^{\kappa, \delta}) \, dW_s ,
\sum_{\substack{d \, \le \, y (d,\,a_{2})\,=\,1}}\rho_{\lambda}(d) \Delta(x;\l(d),1) \le (\log x)^{\lambda}\sum_{\substack{d \, \le \, y (d,\,a_{2})\,=\,1}}|\Delta(x;\l(d),1)| & \le(\log x)^{\lambda}\sum_{d\,\leq\, \sqrt{x}/(\log x)^{B}}\gamma(d)\,|\Delta(x; d, 1)| &\ll\frac{x}{(\log x)^{A-\lambda}}.
\left(\frac{p}{c(d_x)c(d_{x^*})}\right) =(-1)^{\frac{(p-1)(c(d_x)c(d_{x^*})-1)}4}\left(\frac{c(d_x) c(d_{x^*})}p\right) =(-1)^{\frac{(p-1)(d_x d_{x^*}-1)}4}\left(\frac{-1}p\right) =\left(\frac{p}{d_xd_{x^*}}\right).
c_{n}=\frac{1}{\alpha a_{0}^{(n+\beta)/\alpha}}\sum_{k=0}^{n}b_{n-k}\sum_{j=0}^{k}(\begin{matrix}-\frac{n+\beta}{\alpha}\\j\end{matrix})\frac{1}{a_{0}^{j}}\hat{B}_{k,j}(a_{1},a_{2},...,a_{k-j+1}).
\partial_{a_0}^m \partial_1^k G(a_0, x_1 - y, x_2) = \partial_{a_0}^m \partial_1^k \left( \frac{e^{-x_2}}{( 4 \pi a_0 x_2)^{\frac{1}{2}}}e^{-z^2}\right) = e^{-x_2}\partial_{a_0}^m \left( (a_0 x_2)^{-\frac{1 +k}{2}}P_k(z) e^{-z^2}\right) = e^{-x_2}P_{k + 2m}(z, a^{-\frac{1}{2}}_0) e^{-z^2}x_2^{-\frac{1 +k}{2}}.
E_1^3E_2-(q^2+1+q^{-2})E_1^2 E_2 E_1 + (q^2+1+q^{-2})E_1 E_2 E_1^2-E_2 E_1^3{}= 0 E_2^2 E_1 - (q^2+q^{-2})E_2 E_1 E_2 + E_1 E_2^2{}= 0.
\widetilde{\sigma}=\begin{cases}k_1, -d\leq r\leq 0, \sigma, 0<r<\delta, k_2, \delta \leq r <d, \end{cases}\widetilde{\mu}=\begin{cases}k_1, -d\leq r\leq 0, \mu, 0<r<\delta, k_2, \delta \leq r <d. \end{cases}
f_k(z, n_1, \dots, n_j)=\frac{\partial}{\partial y_1^{n_1}\partial y_2^{n_2}\dots \partial y_j^{n_j}}\, \frac{Y(y_1,y_2,\dots,y_j,\overbrace{z,\dots,z}^{k-j},1,\dots,1)}{n_1!n_2!\dots n_j!\,z^{k-j}}\Big|_{y_1=\dots=y_j=0}.
\frac{1}{y^{\sigma_1}}\sum_{n \leq x}\frac{1}{n^{\sigma_2}(y+n)^{\sigma_3}}\ll \frac{1}{y^{\sigma_1+\sigma_3}}\sum_{n \leq x}\frac{1}{n^{\sigma_2}}&\ll \begin{cases}x^{-\sigma_1-\sigma_3}(\sigma_2>1) x^{-\sigma_1-\sigma_3}(\log x) (\sigma_2=1) x^{1-\sigma_1-\sigma_2-\sigma_3}(\sigma_2<1), \end{cases}
|S||K| \Pr(S,K)\leq \underset{S \in \frac{K}{I}}{\sum}|C_{\frac{S}{I}}(P)| \underset{u\in I}{\sum}|C_I(u)| = \left|\frac{S}{I}\right| \left|\frac{K}{I}\right| \Pr \left(\frac{S}{I},\frac{K}{I}\right)|I|^2\Pr(I) = |S||K|\Pr \left(\frac{S}{I},\frac{K}{I}\right)\Pr(I).
\int_{- \infty}^{\infty}{\frac{d p}{2 \pi}}e^{- \alpha_{1}p^{2}}e^{- \alpha_{2}( p + k )^{2}}={\frac{1}{\sqrt{4 \pi ( \alpha_{1}+ \alpha_{2})}}}e x p \left[ - k^{2}\left({\frac{\alpha_{1}\alpha_{2}}{\alpha_{1}+ \alpha_{2}}}\right) \right]
\frac{d z_2}{d q}= \frac{\xi''(q)(1-q)-[\xi'(1)-\xi'(q) +(1-q)\xi''(q)]\frac{1-\xi(q)-\xi'(q)(1-q)}{(1-q)[\xi'(1)-\xi'(q)]}}{(1-q)[\xi'(1)-\xi'(q)](\frac{2+z_2}{z_2^3}\log(1+z_2)-\frac2{z_2^2})}.
\Phi^{\alpha}\rightarrow V \Phi^{\alpha}= \left( \begin{array}{c}{- f^{\alpha}( r ) e^{i \theta}+ f^{\beta}( r )}\\{f^{\alpha}( r ) e^{i \theta}+ f^{\beta}( r )}\\ \end{array}\right) = \left( \begin{array}{c}{\phi^{1}( r , \theta )}\\{\phi^{2}( r , \theta )}\\ \end{array}\right) .
M(\boldsymbol{a})=\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}T^{j}\sigma_{1}^{( a_1 )}\dots\sigma_{l}^{( a_l )}T^{-j}=\frac{1}{\sqrt{n}}\sum_{j=0}^{n-1}\sigma_{1+j}^{( a_1 )}\dots\sigma_{l+j}^{( a_l )}
I_{D}= \frac{1}{1 6 \pi G_{D}}\int d^{D}x \sqrt{- g}\left( R - \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{2 ( p + 2 ) !}e^{a \phi}( F_{( p + 2 )})^{2}\right)
\mathcal{J}_{\mathrm{D}}(\delta) = \inf_{\substack{\lambda \in \mathbb{R}_{+}^L, \ \psi_\ell > - \infty (\psi_\ell)_{\ell \in [L]}\in \prod_{\ell \in [L]}L^1(\mu_\ell)}}\Bigg\{\langle \lambda, \delta \rangle + \sum_{\ell=1}^L \int \psi_\ell d \mu_\ell : \sum_{\ell=1}^L \psi_\ell(s_\ell) \ge g_{\lambda, [L]}(s), \forall s \Bigg \}.
\binom{2p}{p}(T\xi^{p}v_1,T\xi^{p}v_2) = -\sum_{r=0}^{p-1}\binom{2p}{r}(\xi^r v_1, \xi^{2p-r}v_2)-\sum_{r=p+1}^{2p}\binom{2p}{r}(\xi^{r}v_1, \xi^{2p-r}v_2) = \binom{2p}{p}(\xi^{p}v_1,\xi^{p}v_2),
II \lesssim \bigg( \frac{1}{\mu(B(x,\varepsilon))}\int_{B(x,\varepsilon)}|T_{\delta}(1_{B(x,2\varepsilon)}\nu_1, 1_{B(x,2\varepsilon)^c}\nu_2)(w)|^{1/4}\,d\mu(w) \bigg)^4 &+ \bigg( \frac{1}{\mu(B(x,\varepsilon))}\int_{B(x,\varepsilon)}|T_{\delta}(1_{B(x,2\varepsilon)}\nu_1, 1_{B(x,2\varepsilon)}\nu_2)(w)|^{1/4}\,d\mu(w) \bigg)^4 = II' + II''.
\lim_{\Delta T \rightarrow \infty}R_{\beta}( E , \Delta T ) = \frac{| E |}{2 \pi}\left[ \Theta ( - E ) \left( 1 + \frac{1}{e^{\beta | E |}- 1}\right) + \Theta ( E ) \frac{1}{e^{\beta E}- 1}\right] .
\begin{aligned}c(x',y,t) ={}\sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x)\vert}\bigg\{c(x,y,t) & +(1-2x^{2})h(x)\left[\frac{1}{2}\frac{\partial}{\partial y}\beta_{1}(y,t)-\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\alpha(y,t)\right]\bigg\}\end{aligned}
\begin{array}{l l}{{{\bf ( 3 , 3 , 1 ) , ( 1 , 1 , 1 ) , ( 3 , 1 , 2 4 8 )}\qquad}}&{{\mathrm{w i t h}\; \Delta = 4 \; ( r e l e v a n t )}}\\{{{\bf ( 4 , 4 , 1 ) , ( 2 , 2 , 1 ) , ( 4 , 2 , 2 4 8 )}\qquad}}&{{\mathrm{w i t h}\; \Delta = 6 \; ( m a r g i n a l ) .}}\\ \end{array}
d^1(a,b,c,d) =(a+b+c+d,a+g_2b+g_2g_3c+g_2g_3g_4d) &=(a+b+c+d,a+b+c+d)+(0,(g_2-1)b)+(0,(g_2g_3-1)c)+(0,(g_2g_3g_4-1)d) &=(a+b+c+d)(1,1)+((g_2-1)b+(g_2g_3-1)c+(g_2g_3g_4-1)d)(0,1)
x^{-1}\cdot \dfrac{\prod\limits_{i=0}^1(\theta_x-\theta_y-i)\prod\limits_{i=0}^1(\theta_x-\theta_z-i)}{\prod\limits_{i=0}^1(\theta_x-i)}, x^{-1}\cdot \dfrac{(\theta_x-\theta_y)\theta_y(\theta_x-\theta_z)}{\theta_x},x^{-1}\cdot \dfrac{(\theta_x-\theta_y)(\theta_x-\theta_z)\theta_z}{\theta_x},
\Phi(x,t)=\Phi(x,0)+\int \limits_{0}^{t}\int \limits_{\mathbb{H}^{n}}\mathcal{K}(x,y,T-s)e^{\mu s}u(y,s)\big( e^{\beta u^{p}(y,s)}-1\big)~{\rm d}v_{\mathbb{H}^{n}}(y) \,{\rm d}s.
-c_1^1l_0^1+ c_0^1l_1^1-c_1^3l_0^3 +c_0^3l_1^3 = -c_1^0l_0^0+c_0^0l_1^0-c_1^2l_0^2+c_0^2l_1^2-c_1^4l_0^4+c_0^4l_1^4.
\lim_{u\rightarrow0}\Bigr(u^{-n}[b_n(0,\sqrt{u}x)]_{2n}+u^{-n+1}[b_{n-1}(0,\sqrt{u}x)]_{2n}+\cdots\Bigr)=\hat g(x)\in C^\infty(\mathbb{C}^n,{\mathbb{C}}T^{*\bullet}\mathbb{C}^n)
(u^3+8tu^2+12t^2u+7u^2+38tu+12t^2+22u+30t+16)c_2 & =(u^3+4tu^2+6t^2u+11u^2+28tu+6t^2+34u+24t+24)b_2 & +(u^3+4tu^2+4tu+7u^2+6u)c_3-(u^3+2tu^2+2tu+5u^2+4u)b_3
\widetilde{J}(z,u,k) = \sum_{j=k}^{K-1}\Big(r(\tilde{z}^{u}(t_j), \nu_j) +\int_{t_j}^{t_{j+1}}\phi(\tilde{z}^{u}(s), \nu_j)\, ds\Big) + \psi\big(\tilde{z}^{u}(T)\big) \,,u \in \mathcal{U}_{o, k}\,, \widetilde{U}(z,k) = \inf_{u \in \mathcal{U}_{o,k}}\widetilde{J}(z,u,k)\,,
\lim_{( p_{1}+ p_{2})^{2}\rightarrow m_{b_{n}}^{2}}Z_{n}( \theta_{1}, \theta_{2}) \equiv \lim_{\theta_{1 2}\rightarrow \theta + \theta_{1 2}^{b_{n}}}Z_{n}( \theta_{1}, \theta_{2}) = Z_{n}( \theta ) .
F(T)\simeq-0.22924\,\frac{T}{R}-\frac{3\,T}{64\,R}\,\ln\frac{R\,T}{2\,\hbar\,c}-\frac{51}{65536}\,\frac{\hbar^2\,c^2}{R^3\,T}+{\cal O}(T^{-3}).
D_{z}(T;N) =\zeta_N(1+{ia})T+\mathcal{O}(T^{1/2+\varepsilon}) -\; \frac{1}{2\pi i}\bigg[\int_{1+\varepsilon_T+iT^{1/2}}^{\frac{1}{2}+iT^{1/2}}+\int_{\frac{1}{2}+iT^{1/2}}^{\frac{1}{2}-iT^{1/2}}+\int_{\frac{1}{2}-iT^{1/2}}^{1+\varepsilon_T-iT^{1/2}}\bigg] \zeta(s)\zeta_N(s+{ia})T^s\frac{ds}{s}.
f_{\mathcal{C}_n}(\mathbf{x}|m,\mathbf{s})=\mspace{-12mu}\sum_{(i,j)\in\mathcal{I}_n\times\mathcal{J}_n}\mspace{-18mu}\hat{P}^{(\mathcal{C}_n)}(i,j|m,\mathbf{s})p_{X|U,V,S}^n\big(\mathbf{x}\big|\mathbf{u}(i),\mathbf{v}(i,j,m),\mathbf{s}\big),\mspace{-12mu}\quad\forall(m,\mathbf{s},\mathbf{x})\in\mathcal{M}_n\times\mathcal{S}^n\times\mathcal{X}^n.
\Sigma ( p ) ={\frac{i}{\theta}}\int_{0}^{1}d x \; \int{\frac{d^{3}q}{( 2 \pi )^{3}}}{\frac{\left( q_{\lambda}+ x p_{\lambda}\right) \left[ \left( p_{\rho}( 1 - x ) - q_{\rho}\right) \; 2 \; g_{\lambda}^{\rho}- 2 M T^{3}\gamma_{\lambda}\right]}{\left[ q^{2}+ p^{2}x ( 1 - x ) - M^{2}x \right]^{2}}}
\lambda_{B_R,p}^{(n)}= \sqrt{\frac{\pi}{k_p}}\frac{t_p^{2n}}{2^{2n-1}(n-1)! (n-2)! (t_p^2 + t_s^2)}\left(1+\mathcal{O}(\frac{1}{n})\right), \ \varphi_{B_R,p}^{(n)}=^{in\theta_x}.
h_n(t_1,x_1,...,t_n,x_n,t,x)=\frac{1}{n!}\prod_{i=1}^n p_D(t_{\tau(i+1)}-t_{\tau(i)},x_{\tau(i+1)}, x_{\tau(i)}) (\mathcal{G}u_0)_{t_{\tau(1)}}(x_{\tau(1)}),
S_{\mathrm{q u a d}}= - \tau_{2 3}V_{2 4}+ \tau_{2 3}\int d^{2 4}\! x \left( - \frac{1}{2}\partial^{a}\varphi \partial_{a}\varphi - \frac{1}{2}V^{\prime \prime}( \phi_{*}) \varphi^{2}\right) .
\bigg|\int_{B_R}(u\ast h-u)dV\bigg| =\bigg|\int_{B_R}\int_{B_{\epsilon}}h(w)(u(z-w)-u(z))dV_wdV_z\bigg| &=\bigg|\int_{B_{\epsilon}}h(w)\int_{B_R}(u(z-w)-u(z))dV_wdV_z\bigg| &\leq \int_{B_{\epsilon}}h(w)|\hat{u}(-w)-\hat{u}(0)|dV_w,
J[\psi^{(1)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p_1\to \infty}}$\cr}}}\sum_{j_1=0}^{p_1}C_{j_1}\zeta_{j_1}^{(i_1)},
N_{(2,7)}(n) = ~ 8\sigma(\frac{n}{2}) - 32\sigma(\frac{n}{8}) + 8\sigma(\frac{n}{7}) - 32\sigma(\frac{n}{28}) + 64\, W_{(2,7)}(n) + 1024\, W_{(2,7)}(\frac{n}{4}) - 256\, \biggl( W_{(7,8)}(n) + W_{(1,14)}(\frac{n}{2}) \biggr).
\displaystyle{\langle \Psi^{\rho r}( z_{1}) \Psi^{\sigma s}( z_{2}) \rangle = C_{\Psi}\frac{I^{\rho \sigma}( \hat{{\cal X}}_{1 2}) I^{r s}( V ( z_{1 2}) )}{( \operatorname*{d e t}{\cal X}_{1 2})^{\frac{1}{2}\eta}}}
\|\theta(\varepsilon) \|^2 +\| v_h(0)\|_V^2 - 2\int_0^{\varepsilon}\int_0^t K(t-s) a(\theta(s),v_h(t))dsdt \leq \|\theta(0) \|^2+2\max_{0 \leq t \leq \varepsilon}\|\theta (t)\| \int_0^{\varepsilon}\|\dot \omega\|dt.
V_{\beta \alpha}\equiv V^{\mu}\left( \widetilde{\sigma}_{\mu}\right)_{\beta \alpha}\mathrm{}; \mathrm{}f_{\alpha \beta}\equiv \left( \tilde{\sigma}_{\mu}\right)_{\alpha \beta}f^{\mu}\mathrm{}; \mathrm{}f^{\mu}= - \frac i 2 \varepsilon^{\mu \nu \rho}F_{\nu \rho}\mathrm{},
b_{4}= \frac{2}{3}\frac{\omega_{t}^{2}}{\omega_{y}^{2}}\frac{1}{\epsilon_{y}^{2}}+ \frac{3 \omega_{y}^{2}- \omega_{t}^{2}}{2 \omega_{y}^{2}+ 2 \omega_{t}^{2}}\, , \qquad c_{4}= - 2 \frac{\omega_{t}^{2}}{\omega_{y}^{2}}\frac{1}{\epsilon_{y}^{2}}\, .
\rho_\mu^* \circ \varphi_\mu^* \, \Omega_{\mbox{\tiny{$\overline{Q}$}}}= \iota_\mu^*(\Omega_{\mbox{\tiny{$\widetilde{Q}$}}}- B_{\mbox{\tiny{$\!\langle\! J\mathcal{K}\!\rangle$}}}+\widetilde{\bf B}) - \iota_\mu^*\widetilde{\mathcal{B}}.
0\to H^0(ka\L-2(j_2+1)B_2)\to H^0(ka\L-2j_2B_2)\to \bigoplus_{i=1}^{s_2}M_{ka+4j_2}(\Gamma_i), 0\to H^0(ka\L-\frac{k}{2}B_2-(j_4+1)B_4)\to H^0(ka\L-\frac{k}{2}B_2-j_4B_4)\to \bigoplus_{i=1}^{s_4}M_{ka+4j_4}(\Gamma_i).
\beta^r(\xi)=\frac{\sqrt{2\pi}e^{\frac{i\pi}{4}}e^{-\frac{\pi \nu(-k_0)}{2}}}{{q}^r_{1}(-k_0)\Gamma(-i\nu(-k_0))}, \gamma^r(\xi)=\frac{\sqrt{2\pi}e^{-\frac{i\pi}{4}}e^{-\frac{\pi \nu(-k_0)}{2}}}{{q}^r_{2}(-k_0)\Gamma(i\nu(-k_0))}
R_{\mathrm{sym}}^{(1)}= \frac{F}{T}= \left [ \sum_{\substack{S \subset [1:K] |S|=t+1}}\frac{1/{K \choose t}}{\log \left(1+\min_{k \in S}|\mathbf{h}_k^H \mathbf{w}_S^*|^2 SNR\right)}\right ]^{-1}
\gamma_{k}^{2}=(\sum_{i=1}^{k}\omega_{\Sigma_{i}})^{2}-2k\frac{\phi(\tau)}{1+\tau}(\sum_{i=1}^{k}\omega_{\Sigma_{i}})\wedge\frac{\sqrt{-1}dw\wedge d\bar{w}}{\lvert w\rvert^{2}}.