harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi_\Lambda(C_m^{\Lambda_0})=\sum_{(\lambda+\lambda_0',\kappa+\kappa_0,-s)\in D^+}\sum_{s=0}^\infty\,m_{\lambda_0',s}\,\alpha^m_{\lambda_0',s}(\Lambda)\frac{D_q[(\lambda+\lambda_0',\kappa+\kappa_0,-s)]}{D_q[(\lambda,\kappa,0)]}\,,~~~m\in{\bf Z}^+
\mathbb{E}\#\{\textrm{critical points of $f|_{\Gamma\cap Z(p)}$}\}=\frac{|\Gamma|}{\pi}\frac{d^{\frac{n-1}{2}}}{(2\pi)^{\frac{n-2}{2}}}\cdot \mathbb{E}|\det Q_{n-2}|+O(d^{\frac{n-2}{2}}).
\begin{array}{c}u^\alpha =u_0^\alpha +\xi^\alpha s-\frac 12\left( \stackrel{s}{\Gamma}_{\beta \gamma}^\alpha \right)_0\xi^\beta \xi^\gamma s^2- \frac 1{3!}\left( \Gamma^\alpha{}_{\beta \gamma \delta}\right)_0\xi^\beta \xi^\gamma \xi^\delta s^3-\cdots \end{array}
S =3\left\{\frac{2k}{m}\right\}+\left\{\frac{n}{m}\right\}+\left\{\frac{n-1}{m}\right\}+2\left\{\frac{n-k}{m}\right\}+\left\{\frac{2n+k-1}{m}\right\}-\left\{\frac{4n+2k-2}{m}\right\}&\quad-3\left\{\frac{k}{m}\right\}-\left\{\frac{2n}{m}\right\},
G_{\Delta}(p,\cdot) \,\, = \,\, |z|^{2-2m}\, + \, |z|^{4-2m}\, ( \, \Phi_2 + \Phi_4 \, ) \, + \, |z|^{5-2m}\,\sum_{j=0}^2\Phi_{2j+1}\, + \, \mathcal{O}(|z|^{6-2m}) \, ,
qe^{u\sigma}\sum_{k=0}^{\infty}\frac{e^{ku}}{\Gamma(\vec a;K-k-\sigma)\Gamma(k+1+2\sigma)^2\Gamma(k+1)^2}&\ = \frac{qe^{u\sigma}}{\Gamma(\vec a;K-\sigma)\Gamma(1+2\sigma)^2}{}_p F_3\left(\begin{array}{c}-\vec a-K+1+\sigma{[1+2\sigma,1+2\sigma, 1]}\end{array}; (-1)^{p}e^{u}\right)\ ,
a_0 \sum \limits_{k=1}^{\infty}\int \limits_0^{2 \pi}\left( \frac{k a_k- b_k}{1+k^2}\right) \sin k t \dt+ \sum \limits_{k=1}^{\infty}\sum \limits_{l=1}^{\infty}\int \limits_{0}^{2 \pi}\left( \frac{k a_k- b_k}{1+k^2}\right) a_l \cos l t \ \sin k t \dt+
2 u_s \cdot (u_t)_s = 2u_s\cdot\partial_s(u_{ss}+u_s^2+e^{2u})=2u_su_{sss}+4u_s^2u_{ss}+4e^{2u}u_s^2 = (u_s^2)_{ss}-2u_{ss}^2 +4u_s^2u_{ss}+4e^{2u}u_s^2
\int_0^1 \frac{w^j(1-w)^{k}}{(1-wy)^{j+3/2}}dw = \frac{\Gamma(j+1)\Gamma(k+1)}{\Gamma(k+j+2)}{}_2F_1\left(j+\frac{3}{2}, j+1, j+k+2; y\right), j \geq 0
\lim_{p\rightarrow0^{+},p^{\prime}\rightarrow\infty}\tilde{\zeta}_{p,p^{\prime}}(s,c)=\sum_{m\neq0,\,n\in\mathbb{Z}}\frac{(-1)^{m}}{\left(m^{2}+c\,n^{2}\right)^{s}},
[\gamma_l,\bar{\gamma}_m]=\frac{1}{4\pi k'}G_{lm}\ \ \ \ \mbox{or}\ \ \ bar{\gamma}_l= -\frac{1}{4\pi k'}G_{lm}\frac{\partial}{\partial\gamma_m}= -\frac{1}{4\pi k'}\frac{\partial}{\partial\gamma^l}
S(z_k) =\int_0^{\omega}p_k(t,\dot z_k-\dot z_*)dt\ge \int_0^{\omega}L(t,z_k,\dot z_*)dt &+\int_0^{\omega}\frac{\partial L}{\partial \dot z^i}(t,z_k,\dot z_*)(\dot z_k^i-\dot z_*^i)dt.
|\varphi_{t,\tau_0}(z) - \varphi_{0,\tau_0}(z)| = |\varphi_{t,\tau_0}(z) - \varphi_{t,\tau_0}\circ\varphi_{0,t}(z)| & \leq C_1 |z - \varphi_{0,t}(z)|^{1/K}\leq C_1(Kt)^{1/K}
[F_{\overline{j}}, e_{\overline{j+1}}] = [f_{\overline{j}},[f_j, e_{\overline{j+1}}]] - [f_j, [e_{\overline{j+1}}, f_{\overline{j}}]] = - [f_j, [f_{j+1}, f_j]] = [f_j, [f_j, f_{j+1}]] + [f_j, [f_{j+1}, f_j]] = 0.
\sum_{j=1}^3\int_{s_j}^{1}|f'_j|^2=\sum_{j=1}^3\big( f_j(1)f'_j(1)-\int_{s_j}^{1}f_j f''_j\big)\leq 3\hat\beta^2+2\hat\alpha\big(\sum_{j=1}^3\int_{s_j}^{1}|f_j|^2\big)^{\frac12}.
\partial_{z}A_{2}- A_{3}\partial_{z}\varphi_{0}+ \bar{A}_{1}A_{3}+ \bar{A}_{2}A_{2}= \partial_{\bar{z}}\bar{A}_{2}- \bar{A}_{3}\partial_{\bar{z}}\varphi_{0}+ A_{1}\bar{A}_{3}+ A_{2}\bar{A}_{2}.
F_N(a,b;t) =\frac{(tq)_N}{(t)_N}{}_3\phi_2\begin{bmatrix}q^{-N}, b/a, q ;q, atq^{N+1}bq, tq\end{bmatrix}&=\frac{(1-tq^N)}{(1-t)}\sum_{n=0}^{N}(-1)^n\begin{bmatrix}N n\end{bmatrix}\frac{(b/a)_n(q)_n(at)^nq^{n(n+1)/2}}{(bq)_n(tq)_n}.
(z-\mu^{(t)}){\cal H}_{k}^{(s,t)}(z)={\cal H}_{k+1}^{(s,t)}(z)+(Q_{k+1}^{(s,t)}+E_{k}^{(s,t)}){\cal H}_{k}^{(s,t)}(z)+Q_{k}^{(s,t)}E_{k}^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z), (z-\mu^{(t)}){\cal H}_{k-1}^{(s,t+1)}(z)={\cal H}_{k}^{(s,t+1)}(z)+(Q_{k}^{(s,t)}+E_{k}^{(s,t)}){\cal H}_{k-1}^{(s,t+1)}(z)+Q_{k}^{(s,t)}E_{k-1}^{(s,t)}{\cal H}_{k-2}^{(s,t)}(z).
\widehat{\Phi}_{k, v}(u):= \prod_{i\in I}\prod_{j=1}^{v^{i}}\frac{\vartheta(u+ z_{j}^{(i)}+(c_{ki})\frac{\hbar}{2})}{\vartheta(u+ z_{j}^{(i)}-(c_{ki}) \frac{\hbar}{2})}.
D_{132}\ll t_3^{-\sigma_1-\sigma_3}\left( t_3^{-\sigma_3}+1 \right) \begin{cases}1 (\sigma_2>\frac{3}{2}) \log t_3 (\sigma_2=\frac{3}{2}) t_3^{\frac{3}{2}-\sigma_2}(\sigma_2<\frac{3}{2}) \end{cases}
\theta_{\bar{X}}\dim^R(\bar{X}) = \theta_{X^*}\theta_{\bar{\mathbf{1}}}S^{R,R}_{X^*,\bar{\mathbf{1}}}= \theta_{X^*}\theta_{\bar{\mathbf{1}}}\dim^R(\bar{\mathbf{1}})s^R_{\bar{\mathbf{1}}}(X^*) = \theta_{X^*}\theta_{\bar{\mathbf{1}}}\dim^R(\bar{\mathbf{1}}) \dim^R(X).
f^{-k}(z_A) :=[f^{-k}(x_B),f^{-k}(x_A)]\in W^u(f^{-k}(x_B),B)\cap W^s(f^{-k}(x_A),B) f^{\ell}(z_B) :=[f^{\ell}(x_B),f^{\ell}(x_A)]\in W^u(f^{\ell}(x_B),A)\cap W^s(f^{\ell}(x_A),A).
\begin{aligned}g^{-1}\cdot\chi_{0}=\chi_{0},g^{-1}\cdot\chi_{4}=\chi_{4}, g^{-1}\cdot\chi_{1}=g^{-1}\cdot\mu^{1/2}\xi_{1}=\mu^{1/2}\left(g\xi\right)_{1}=\mu^{1/2}\xi\cdot\frac{\eta}{\left|\eta\right|}=\sum_{j=1}^{3}\frac{\eta_{j}}{\left|\eta\right|}\chi_{j},\end{aligned}
k_{2}^{\nu}T_{\rho \sigma \mu \nu}^{( 3 ) a b c d}( p_{1}, p_{2}; k_{1}, k_{2}) = C_{3}k_{1 \mu}S_{\rho \sigma}^{( 3 )}+ C_{3}\Gamma_{\rho \sigma \mu}^{( 3 )}( p_{1}, p_{2}, q_{3})
h_{\epsilon}^{\ast}(z) = \begin{cases}\bar{\epsilon}H_b(z) - z\bar{\epsilon}\frac{H_b(p_\epsilon)}{p_\epsilon+\frac{1}{1-\epsilon}}\mbox{if}0 \leq z \leq p_{\epsilon}\frac{H_b(p_\epsilon)}{p_\epsilon+\frac{1}{1-\epsilon}}\mbox{if}p_{\epsilon}\leq z \leq 1. \end{cases}
G_0(z,w)^2\partial\bar{\partial}\log G_0(z,w) =G_0(z,w)\partial\bar{\partial}G_0(z,w)-\partial G_0(z,w)\bar{\partial}G_0(z,w) &=(8+8z\bar{w}-(z\bar{w})^2)(8-4z\bar{w})-(8z-2z^2\bar{w})(8\bar{w}-2z\bar{w}^2) &=64-32z\bar{w}-8(z\bar{w})^2,
\eta_{t}+(\eta u)_{x}+(\eta v)_{y}=0,(\eta u)_{t}+(\eta u^{2}+\frac{1}{2}g\eta^{2})_{x}+(\eta uv)_{y}=0,(\eta v)_{t}+(\eta uv)_{x}+(\eta v^{2}+\frac{1}{2}g\eta^{2})_{y}=0
\{\eta_{M N}\}= \left( \begin{array}{c c}{g_{\alpha \beta}+ g_{\rho \sigma}A_{\alpha}^{\rho}A_{\beta}^{\sigma}}{g_{\mu \rho}A_{\beta}^{\rho}}{A_{\alpha}^{\rho}g_{\rho \nu}}{g_{\mu \nu}}\end{array}\right) \, ,
{F}_{i}=-\frac{1}{2\pi i}\oint_{\Gamma_{\delta}}\Psi_{i}(z)\,dz,i=1,\dots,p,{F}_{*}\left(t\right) =-\frac{1}{2\pi i}\oint_{\Gamma_{\delta}}\Psi_{*}(t,z)\,dz.
|v_{3}(t,r)| \leq \frac{1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}|\lambda''(s)|\cdot 2 &\leq \frac{C}{r}\int_{t}^{\infty}ds |\lambda''(s)|(s-t)
\sum_{2l-1\ge N}\sum_{j=1}^\infty|B_{2l-1,2j-1}| &\le\sum_{2l-1\ge N}\sum_{k=1}^\infty|\alpha|^k\left(\Big(\frac{1}{kR_0-1}\Big)^{2l}+\Big(\frac{1}{kR_0+1}\Big)^{2l}\right)\le CR_0^{-N-1}.
c\; =\;(\textstyle{\frac{\Gamma(B)}{\Gamma(2B)}\,\frac{1+\nu}{1+\nu+B}})\cdot\displaystyle{\lim_{r\downarrow 0}}\;r^B g^+\!(r) f\; =\;g-c(\beta S_D^{-1}\Phi+\Phi)\,.
g_{N}(V,V)= 8\pi T_{i\overline{j}}Z^i\overline{Z}^j+2\pi|W_0^{}(V)|^2T^{0\overline{0}}\, &=8\pi \tau_2^2\left(\frac{2t^3}{3}-\frac{\chi}{4(2\pi)^3}\sum_{(m,n)\in \mathbb{Z}^2-(0,0)}\left(\frac{3(m\tau_1+n)^2}{|m\tau+n|^5}-\frac{1}{|m\tau+n|^3}\right)\right)+2\pi|W_0^{}(V)|^2T^{0\overline{0}}
\begin{array}{lll}A) J_1,J_2,J_3\mbox{unrestricted,}\qquad l \geq 6 + \frac{1}{2}(J_1+2 J_2+3 J_3) + 2(a_1+a_2) B) J_3 = 0, \qquad l = 4 + \frac{1}{2}(J_1+2 J_2)+ 2(a_1+a_2) C) J_2=J_3=0, \qquad l = 2 + \frac{1}{2}J_1 + 2(a_1+a_2) D) J_1=J_2=J_3=0,\qquad l = 2 (a_1+a_2) \end{array}
z = \frac{\beta}{2\pi}, c = \frac{- M^\prime(a_0,1,z)}{\frac{\partial M}{\partial a}(a_0,1,z)}\, \frac{4\beta^2}{\pi}, q_n = 1+n -z + z ( \log M )^\prime(a_0,n+1,z) + z ( \log M )^\prime(a_0+n,n+1,z) .
: \int_x^{\infty}K_{i\lambda_j}(\rho)K_{i\lambda_k}(\rho)\frac{d\rho}{\rho}=0,j\neq k, : \int_x^{\infty}K_{i\mu_j}(\rho)K_{i\mu_k}(\rho)\frac{d\rho}{\rho}=0,j\neq k.
h'(x)\rho +\frac{1}{2}h''(x)\rho^{2}=\frac{1}{x\log x}\cdot \rho \, \delta (\log x)^{\delta}\cdot \Big( 1+\frac{\rho \, (\delta -1)}{2x\log x}- \frac{\rho}{2x}\Big) & \geq \frac{1}{x\log x}\cdot\frac{1}{\log^{\circ 2}x\cdots(\log^{\circ k}x)^{1+\varepsilon}}=M_{\varepsilon,k}(x),
\frac{d}{dx}\sigma^2(x) \ge\frac{d}{dx}g_{11-}(x)- \frac1{2^{10}}=\frac{27157195}{1417176}-46x- \frac1{2^{10}}&\ge\frac{27157195}{1417176}-46c- \frac1{2^{10}}=\frac{122591869}{120630021120}> 0\quad\hbox{a.e.}
\left\|\begin{pmatrix}1 \|T\|_A 0 1\end{pmatrix}\right\|=\left\|\begin{pmatrix}1 \|T\|_A 0 1\end{pmatrix}\begin{pmatrix}a b\end{pmatrix}\right\|=\sqrt{\left(a+b\|T\|_A\right)^2+b^2}.
P_{\hat{\underline S}| \underline X^L Y^L}({\underline s}| \underline x^L, y^L) = P_{{\underline S}| \underline X^L Y^L}({\underline s}| \underline x^L, y^L) & = \begin{cases}\frac{1}{|{\mathcal B}_{y^L}( \underline x^L)|}, \forall \underline s\in{\mathcal B}_{y^L}( \underline x^L) 0, \end{cases},
n! a_{11}= \frac{n-1}{2}\left[ -2\chi_0(n-1)^2+2\chi (n-1)+(n+1)k-h\right] & = (1-n)\left[ 2n(2q-2)+2e+(n+1)a\right]\ , n! a_{12}= \frac{1}{2}\left[ 2\chi_0 (n-1)^2-2\chi (n-1)-k+h\right] & = n(2q-2)-(n-2)e+a\ , n! a_{22}= \frac{1}{2}\left[-2\chi_0 (n-1)+2\chi -k+h\right] & = 2e+a\ .
LR_0 = \{(l_1,\dots,l_n)\in\mathbb{Q}^n \mid \sum_{i=1}^nl_i\beta_i=0\}= \left\{(l_1,\dots,l_n)\in\mathbb{Q}^n \mid \left( \begin{array}{c}{l}_1 \vdots{l}_r \end{array}\right) =- T \left( \begin{array}{c}{l}_{r+1}\vdots{l}_n \end{array}\right)\right\},
\left(m+\left(d(\eta)\vee m-m\right)\vee \left(d(\rho)-m\right)\right)_i =m_i+\max\{0,d(\rho)_i-m_i\}&=d(\rho)_i=\max\{d(\rho)_i,d(\eta)_i\}=\left(d(\eta)\vee d(\rho)\right)_i.
(fg)x = \lambda_1 (fg)x + \lambda_2 (fx)g + \lambda_3 (gf)x + \lambda_4 (xf)g &+ \lambda_5 x(fg) + \lambda_6 y(fx) + \lambda_7 x(gf) + \lambda_8 g(xf) \intertext{and}x(fg) = \mu_1 (fg)x + \mu_2 (fx)g + \mu_3 (gf)x + \mu_4 (xf)g &+ \mu_5 x(fg) + \mu_6 g(fx) + \mu_7 x(gf) + \mu_8 g(xf).
\xi(r) = \sqrt\pi \,r^{k/2}+ r^{-k/2}\big( \xi(1)-\sqrt\pi \big),r>1, \xi'(r) =\frac{k}{2}\sqrt\pi r^{k/2-1}+ (\sqrt\pi - \xi(1))\frac{k}{2}r^{-k/2-1},r>1.
-\left( \sum_{i=1}^{r}\frac{p_i p_i^\prime}{d_i}+ \frac{\phi}{2}\sum_{i=1}^{r}\,p_i^\prime + \frac{\phi^\prime}{2}\sum_{i=1}^{r}\,p_i+ \frac{n-1}{4}\phi \,\phi^\prime \right).
\limsup_{N\rightarrow+\infty}\frac{1}{N}\log P(\mu^N\in C) &\leq -\inf_{W\in C}\left(\sum_{k=1}^3W_{0,k}(f_k)+\mathcal{I}_1(W,F,G,H)\right) &+\int_\mathbb{T}\log\left(1+\sum_{k=1}^3\rho_{k-1}(u)\left(\exp\left(f_k(u)\right)-1\right)\right)du &=-\inf_{W\in C}\left(\mathcal{I}_2(W_0, f_1, f_2, f_3)+\mathcal{I}_1(W,F,G,H)\right).
(\nabla_{e_i}\Phi)(\xi,e_k) =g(\xi,\nabla_{e_i}(\xi\times e_k))+g(\nabla_{e_i}e_k,\xi\times\xi) &=g(\xi,\nabla_{e_i}(\xi \times e_k)) &=-g(\nabla_{e_i}\xi,\xi \times e_k )
\gamma_{m}^{\mu \nu}= \left( \begin{array}{c c}{{~ 1}}&{{~ 0}}\\{{~ 0}}&{{~ 1}}\\ \end{array}\right) , ~ \left( \begin{array}{c c}{{~ 0}}&{{- 1}}\\{{- 1}}&{{~ 0}}\\ \end{array}\right) , ~ \left( \begin{array}{c c}{{~ 1}}&{{~ 0}}\\{{~ 0}}&{{- 1}}\\ \end{array}\right) .
Q(y,s) = e^{W(y,s)}- \psi_\alpha(y,s) \equiv Z(y,s) - \psi_\alpha(y,s) \;\; \;\; \psi_\alpha(y,s) = e^{\frac{N}{(2 + 2\alpha)s}}e^{\Phi_\alpha\left(\frac{y}{\sqrt{s}}\right)},
C_m(\gamma_1,\gamma_2;\mathbf{t},\mathbf{c}) \sim 2 (-2\gamma_1-1)^{-m/2}(\gamma_2+1)^{-m}\left[\sum_{i,j=1}^n c_ic_j~ \frac{1}{2}\left( |t_i| +|t_j| -|t_i-t_j| \right)\right]^{m/2},
\frac{B_{\phi(n)-1}(\frac{1}{e})}{\phi(n)-1}\equiv e\sum_{j=0}^{p^{l}-1}(\lfloor\frac{1+j}{p^{l}}\rfloor+\frac{1-e}{2})(1+je)^{\phi(n)-2}\pmod{p^{l}}&\equiv e\sum_{\substack{j=0 (p,1+je)=1}}^{p^{l}-1}(\lfloor\frac{1+je}{p^{l}}\rfloor+\frac{1-e}{2})(1+je)^{-2}\pmod{p^{l}},
s_r(H(k,k,\ldots,k)) =k\sum_{i\in[r]}\sum_{j\in V_i}\left|d_{H(k,\ldots,k)}(j)-\frac{k^rm}{kn_i}\right| & =k^r\sum_{i\in[r]}\sum_{j\in V_i}\left|d_{H}(j)-\frac{m}{n_i}\right| & =k^rs_r(H),
G ( t - t^{\prime}, \tau ) = \frac{1}{a^{D}}\int_{1 / \Lambda^{2}}^{\infty}d s \frac{e^{- \frac{t - t^{\prime}}{4 s}}}{( 4 \pi s )^{\frac{1}{2}}}\int_{0}^{\infty}\phi_{r}( \tau ) | C ( r ) |^{2}e^{- \frac{s}{a^{2}}( r^{2}+ \rho_{D}^{2})}d r ,
(-1)^{l}\left(\Delta^{(l)}W, W\right) = (-1)^{l-1}(\Delta^{(l-1)}\nabla_i W, \nabla_i W) + (-1)^{l-1}([\nabla_i, \Delta^{(l-1)}]W, \nabla_i W) \quad\quad= (-1)^{l-1}(\Delta^{(l-1)}\nabla W, \nabla W) + \sum_{p + q = 2l-3}(C\ast\nabla^{(p)}R\ast\nabla^{(q)}W,\nabla W)
\tilde{g}_{ij}=\int_{\Omega}\int_{\Omega}\tilde{\kappa}(x,y)\varphi_{i}(x)\psi_{j}(y)dydx=\sum_{\nu=1}^{k}\int_{\Omega}\kappa(x,\xi_{\nu})\varphi_{i}(x)dx\int_{\Omega}l_{\nu}(y)\psi_{j}(y)dy=\sum_{\nu=1}^{k}a_{i\nu}b_{j\nu}
\ \ \quad\ \ :\ \lim_{n\rightarrow\infty}\left( n[Y_{l_{1}}^{m_{1}}\star_{\vec{c}}^{n}Y_{l_{2}}^{m_{2}}-Y_{l_{2}}^{m_{2}}\star_{\vec{c}}^{n}Y_{l_{1}}^{m_{1}}]\right) =-2i\{Y_{l_{1}}^{m_{1}},Y_{l_{2}}^{m_{2}}\},
F = V_{p}( 8 \pi^{2}\alpha^{\prime})^{- p / 2}\int_{0}^{\infty}d \tau ~ \tau^{- 4 + p / 2}e^{- b^{2}/ 2 \pi \alpha^{\prime}{\tau}}\left[ e^{- \pi \tau / 1 2}\prod_{n = 1}^{\infty}( 1 - e^{- 2 \pi \tau n}) \right]^{- 9}\frac{( \theta_{1}( - i \epsilon / \pi \mid i \tau ) )^{4}}{\theta_{1}( - i 2 \epsilon / \pi \mid i \tau )}.
\frac{\partial}{\partial t}\frac{\delta l^\mathcal{L}}{\delta\mathcal{U}}=-\operatorname{ad}^*_{\mathcal{U}}\frac{\delta l^\mathcal{L}}{\delta\mathcal{U}}+\operatorname{div}^{\Gamma}\frac{\delta l^\mathcal{L}}{\delta\Gamma},
|\bar{x}_j-x_j|^{p-2}{(\bar{x}_j-x_j)}=- \sum_{i\in I}\lambda_j^i \frac{1}{\det L_{I,I}}\left( \sum_{k\in I}\xi_k B_{I}^k B_{I}^i \det L_{I\backslash \{k\},I\backslash \{i\}}\right)
\Delta E_{\rm F}^{(2)}=\frac{1}{(4\pi)^2}\int_0^\infty \frac{dq\, q^2}{(2\pi)^2}\tilde{V}(q)\tilde{V}(-q) \left[-2+\sqrt{1+\frac{4m^2}{q^2}}\, \ln \frac{\sqrt{1+\frac{4m^2}{q^2}}+1}{\sqrt{1+\frac{4m^2}{q^2}}-1}\right]
\underline{\tilde{\Lambda}}^{x,\infty}_t(\omega)=\begin{cases}\Lambda^{x,\infty}_t,\;\;t\in[0,\underline{\tau}^{x,\infty}(\omega)), -\lim_{k\to\infty}\,\underline{Z}^{x,k}_{\underline{\tau}^{x,k}(\omega)}(\omega),\;\;t\in(\underline{\tau}^{x,\infty}(\omega),T],\end{cases}
\begin{aligned}\sum_{d \geq 0}f_d(p) q^d = K(z,\tau)^2 \sum_{d \geq 0}g_d(p) q^d = \frac{3}{2}K(z,\tau)^4 \wp(z,\tau) + \frac{3}{8}K(2 z, 2 \tau)^2\end{aligned}
\int_{\mathcal{L}_{p}\left(0, (2A)^{-1}\exp(-B\omega(m_{0}))\right)}G(x,y)\,|f(y)|\,dy &= \sum_{m\geq m_{0}}\int_{\mathcal{L}_{p}\left((2A)^{-1}\exp(-B\omega(m+1)), (2A)^{-1}\exp(-B\omega(m))\right)}G(x,y)\,|f(y)|\,dy &\leq C \sum_{m\geq m_{0}}^{\infty}\left(\omega(m+1)-\omega(m)\right)\sup_{M \setminus B_m(p)}\left|\frac{f}{\rho}\right| <\infty \,,
\frac{d}{dt}\Theta_\epsilon \le \int\bigl( h_\epsilon''+k^3+C_1(h_\epsilon+1)\bigr)ds+\int\frac{h_\epsilon}{2|X|}\bigl(-2{\rm Ric}_g(X,X)-2k^2|X|^2\bigr)dx &=\int \bigl(h_\epsilon''+k^3+C_1(h_\epsilon+1)\bigr)ds-\int h_\epsilon\bigl({\rm Ric}_g(S,S)+k^2\bigr)ds & \le \int\bigl(h_\epsilon''+C_1(h_\epsilon+1)-h_\epsilon({\rm Ric}_g(S,S)\bigr)ds.
I_m=\begin{cases}[r_m,r_{m+1}) \hbox{if}T(m)=,\T(m+1)= [r_m,r_{m+1}] \hbox{if}T(m)=,\T(m+1)= (r_m,r_{m+1}) \hbox{if}T(m)=,\T(m+1)= (r_m,r_{m+1}] \hbox{if}T(m)=,\T(m+1)= \end{cases}
\mathbf{\Sigma}(x_1)=\frac{(-x_1)^{-3/2}}{2\sqrt{2\pi}}\left[C(-x_1)^{i\epsilon}\bar{\mathbf{w}}+ \bar{C}(-x_1)^{-i\epsilon}\mathbf{w}+ C_4\mathbf{w}_4\right].
\sum_{m_{1},m_{2}}\langle JM|j_{1}m_{1}j_{2}m_{2}\rangle\langle j_{1}m_{1}j_{2}m_{2}|J^{\prime}M^{\prime}\rangle=\langle JM|J^{\prime}M^{\prime}\rangle=\delta_{J,J^{\prime}}\delta_{M,M^{\prime}}.
\int_0^\infty \Psi^{(5)}(Z) (\Psi w)'(Z)\:dZ= \int_0^\infty \Psi^{(3)}(Z) (\Psi w)^{(3)}(Z)\:dZ - \Psi^{(4)}(0) (\Psi w)'(0) + \Psi^{(3)}(0) (\Psi w)''(0).
\sum_{n=1}^{n_{*}{\scriptscriptstyle (K)}}\frac{1}{\pi_{n}\lambda_{n}}\;\sum_{q=1}^{n}\pi_{q}= \sum_{n=1}^{n_{*}{\scriptscriptstyle (K)}}\sum_{q=1}^{n}\frac{\Lambda_{n+1,q}}{\mu_{q}}= \hat{S}_{1}+\hat{S}_{2}+\hat{S}_{3}
\mathbf{T}_{\varepsilon}(\mu)(x):=\mathbf{T}_{\varepsilon}^{\mathbf{K}}(\mu)(x) =\sup_{\rho\in (0,\rho_0),e\in S^{d-1}}\frac{\varepsilon^{-d+1}}{\rho^\alpha}\left|\left(\frac{1}{|\cdot|^{d-\alpha}}\phi^{e,\varepsilon}_\rho(\cdot)\right)\star \mathbf{K}\star\mu(x)\right|~~\forall~x\in\mathbb{R}^d,
W_{13}\xi(1,2) = \left\{\frac{2}{13},\infty\right\}= \left\{\frac{2}{13},\frac{1}{6}\right\}+\left\{\frac{1}{6},0\right\}+\{0,\infty\}& = -\xi(0,-6)+\xi(1,-6)+\xi(0,1).
\exists w, y \forall u, x_1, x_2 \exists z \forall y_1, y_2\, [ (z^m=1+wx_1^mx_2^m\vee y_1^m\neq 1+wx_1^m \vee y_2^m\neq 1+wx_2^m) &\wedge u^m\neq w\wedge y^m=1+wx^m]
M_{11}=\left(\begin{array}{ccc}1 0 0 1 0 0 0 0 0\end{array}\right),\,\,M_{12}=\left(\begin{array}{ccc}0 1 1 0 1 1 0 0 0\end{array}\right), M_{21}=\left(\begin{array}{ccc}0 0 0 0 0 0 * 0 0\end{array}\right),\,\,M_{22}=\left(\begin{array}{ccc}0 0 0 0 0 0 * 1 1\end{array}\right).
\left( \begin{array}{c c}{M_{1 1}}{M_{1 2}}{M_{2 1}}{M_{2 2}}\end{array}\right) \left( \begin{array}{c}{W_{2 2}^{0 0}}{W_{0 0}^{0 0}}\end{array}\right) = \omega^{2}\left( \begin{array}{c}{W_{2 2}^{0 0}}{W_{0 0}^{0 0}}\end{array}\right) ,
\alpha(T):= \frac{\|v_0^2 - (v_0+V)^2\|_Y}{v_0^2-1}+\frac{M^2}{v_0^2-1}+ \frac{2M\|v_0+V \|_Y}{v_0^2-1}+\frac{1}{\gamma (v_0^2-1)}<1, t\in [0,T]
(R\oplus_q R')\sp i{}_j{}\sp{k'}{}_{l'}=\delta\sp i{}_j\delta\sp{k'}{}_{l'},\quad(R\oplus_q R')\sp{i'}{}_{j'}{}\sp{k}{}_{l}=\delta\sp{i'}{}_{j'}\delta\sp{k}{}_{l},\quad (R\oplus_qR')\sp i{}_{j'}{}\sp{k'}{}_{l}=(q-q\sp{-1})\delta\sp i{}_l\delta\sp{k'}{}_{j'}.
\begin{cases}\|\Gamma_{\chi_{t},b,t,N,\omega}^{A,j}(x,\tau)\|_{L^{2}(\Omega_{t}(\omega))}\leq c\tau^{|\alpha|-3/2}e^{-\tau(s-t)a_{A}}, \|r_{\chi_{t},b,t,N,\omega}^{A,j}(x,\tau)\|_{L^{2}(\Omega_{t}(\omega))}\leq c\tau^{j-N+1/2},\end{cases}
\frac{1}{4 \kappa}\left( \hat{K}+ 3 - \Delta_{3}\right) G_{E \i^{\prime}j^{\prime}}^{\i j}( X , X^{\prime}, \Omega , \Omega^{\prime}) = \delta ( X - X^{\prime}) \gamma^{-{\frac{1}{2}}}\delta_{\ \i^{\prime}j^{\prime}}^{i j}( \Omega - \Omega^{\prime}) .
\begin{aligned}\lambda_1=\frac{e^{q_3-q_2}-2e^{2q_3-2q_2}}{(1+e^{q_3-q_2})^2}, &\lambda_2=\frac{e^{q_3-q_1}}{(1+e^{q_2-q_1})(1+e^{q_3-q_2})}, &\lambda_3=\frac{e^{q_2-q_1}-2e^{2q_2-2q_1}}{(1+e^{q_2-q_1})^2}.\end{aligned}
\mathbb E [ X_{n}^{2}] \, =\, \sum_{k=0}^{n-1}\sum_{\ell=0}^{k}{k \choose \ell}^{2}u^{2\ell}( 1- a)^{2\ell}a^{2(k-\ell)}\, =\, \sum_{k=0}^{n-1}u^{k}(1-a)^{k}\,_{2}F_{1}\Big( -k, -k, 1 ; \, \frac{a^{2}}{\, ( 1-a)^{2}\,}\Big) \, ,
\hat{D}[2k] = \real\left(X[k]|H_D[2k]| + \frac{W[k]|H_D[2k]|}{H_D[2k]}\right) = (-1 + 2D[2k])|H_D[2k]| + \real\left(\frac{W[k]|H_D[2k]|}{H_D[2k]}\right)
\bigl[(\bar{\nu}-\varepsilon\bar{\tau}_\varepsilon^i)\cdot\nabla\rho^\varepsilon\bigr](x,t) = \varepsilon^{-1}\partial_r\eta_0(r)+\partial_r\eta_1(r) +\varepsilon\{\partial_r\eta_2(r)-\nabla_\Gamma g_i\cdot\nabla_\Gamma\eta_0(r)\}+O(\varepsilon^2).
G_n(j, k; z) = \langle \delta_k, (H_{\infty, n}- z)^{-1}\delta_j \rangle g_n(j; z) = 2^{-n}\sum_{k \in B_n(j)}\langle \delta_k, (H_{\infty, n}- z)^{-1}\delta_j \rangle Q_n(j, z) = \langle \varphi_n(j),(H_{\infty, n}- z)^{-1}\varphi_n(j) \rangle
\left\Vert \widetilde{Err}(\cdot ,j_{0},j,\overset{1}{\mathbb{P}}\left(E_{j}^{T}\right)) \right\Vert^{2}_{\mathbb{L}^{2}(B)}= \mathcal{O}\left( \left(a^{10}\, d^{-2}_{j,j_{0}}+ a^{12}\, d^{-8}_{j,j_{0}}\right) \; \left\Vert \overset{1}{\mathbb{P}}\left(\tilde{E}_{j}^{T}\right) \right\Vert^{2}_{\mathbb{L}^{2}(B)}\right).
\left[\int_{-\infty}^{\infty}\varphi_{1}(q)dq\right]\left[\int_{-\infty}^{\infty}q\varphi_{2}(q)dq\right] = \left[\int_{-\infty}^{\infty}\varphi_{2}(q)dq\right]\left[\int_{-\infty}^{\infty}q\varphi_{1}(q)dq\right]
Z_{n,p}:=n^{p\frac{\alpha}{2}-1}\sum_{j=1}^{n}\left|B_{\alpha,\beta}\left(\frac{j}{n}\right)-B_{\alpha,\beta}\left(\frac{j-1}{n}\right)\right|^{p}=\frac{1}{n}\sum_{j=1}^{n}\left|B_{\alpha,\beta}(j)-B_{\alpha,\beta}(j-1)\right|^{p}.
\sum_{j\in \mathbb Z^2}\int_{\mathbb T^2}P.V. \int_{\mathbb T^2}\frac{f(x)-f(y)}{|x-y+Lj|^{2+\alpha}}g(x)\, dy \, dx= &\sum_{j\in \mathbb Z^2}\int_{\mathbb T^2}P.V. \int_{\mathbb T^2}\frac{f(y)-f(x)}{|x-y+Lj|^{2+\alpha}}g(y)\, dy \, dx.
\left\langle \hat{M}_1^{n}\right\rangle(t) = \frac{1}{n}\int_0^t(\hat{Q}_1^n(s)-\hat{Q}_2^n(s))ds &\le \frac{1}{n}\int_0^t\hat{Q}_1^n(s)ds &\le \frac{t}{n}\left(Q_1^n(0) + A(\lambda_nnt)\right).
(\rho(x_1, x_2)b)\rho(x_3, x_4) +(-1)^{\bar x_3(\bar x_1+\bar x_2+\bar b)+\bar b\bar x_2}(\rho(x_3, x_1)b)\rho(x_2, x_4) &+ (-1)^{\bar x_1(\bar x_2+\bar x_3+\bar b)+\bar x_1\bar b}(\rho(x_2, x_3)b)\rho(x_1, x_4)=0.
{h}_{2 n}^{( \alpha )}= \frac{1}{N^{2 \alpha}}{\Gamma ( \alpha + \frac 1 2 )^{2}}\left( \frac{2}{u ( t )}\right)^{2 \alpha}\sqrt{r ( t )}+ \mathrm{h i g h e r o r d e r s i n}\frac 1 N .
\varphi_{m_k}\cdots \varphi_1.(v_m \otimes u_z) = \tau_{m_k}\cdots \tau_{i+1}(\tau_i (x_i - x_{i+1})+1) \varphi_{i-1}\cdots \varphi_1.(v_m \otimes u_z) = \tau_{m_k}\cdots \tau_{i+1}\varphi_{i-1}\cdots \varphi_1 . (v_m \otimes u_z) \pm z . \tau_{m_k}\cdots \tau_{i+1}\tau_i \varphi_{i-1}\cdots \varphi_1 . (v_m \otimes u_z).
\Vert\nabla W^{\pm}\Vert^{2}_{\Lambda^{2}}= |d\lambda^{\pm}|^{2}+|d\mu^{\pm}|^{2}+|d\nu^{\pm}|^{2}+2(\mu^{\pm}-\nu^{\pm})^{2}|a^{\pm}|^{2}+2(\lambda^{\pm}-\nu^{\pm})^{2}|b^{\pm}|^{2}+ 2(\lambda^{\pm}-\mu^{\pm})^{2}|c^{\pm}|^{2}
\forall\gamma^1,{\gamma'}^1,x^X,{x'}^X\big(\gamma>_\mathbb{R}0\land{\gamma'}>_\mathbb{R}0\land\tilde{\rho}>_\mathbb{R}-\gamma &\qquad\qquad\qquad\qquad\land x=_Xx'\land\gamma=_\mathbb{R}\gamma'\rightarrow J^A_\gamma x=_XJ^A_{\gamma'}x'\big).
\prod_{k=1}^{m}T_{k,j}^{\frac{\log s_{k}}{2\pi i}}= (4x_{j})^{-\beta_{j}}\prod_{\substack{k=1 k \neq j}}^{m}\widetilde{T}_{k,j}^{-\beta_{k}}, \mbox{where}\widetilde{T}_{k,j}= \frac{\sqrt{x_{j}}+\sqrt{x_{k}}}{|\sqrt{x_{j}}-\sqrt{x_{k}}|}.
\nu = \frac{1}{\sqrt{\left(1 - \frac{2m}{r}\right)^{-1}- |\bar{\nabla}f|^2}}\left[ \left( 1 - \frac{2m}{r}\right)^{-1}\partial_t + \left( 1 - \frac{2m}{r}\right) f_r \partial_r + r^{-2}\tilde{\sigma}^{ab}f_a \partial_b \right].
{\cal B}_{\Sigma}\, \, = \, \, \displaystyle{\int}{d}^{3}x \, \displaystyle{\, s u m_{\Phi = A_{\mu}^{a}, \, c^{a}, \, \Psi_{_A}, \, \varphi_{i}}^{}}\left({\displaystyle{\ \frac{\delta \Sigma}{\delta \Phi^{*}}}}{\displaystyle{\frac{\delta}{\delta \Phi}}}+{\displaystyle{\frac{\delta \Sigma}{\delta \Phi}}}{\displaystyle{\ \frac{\delta}{\delta \Phi^{*}}}}\right) \, + b \, .
\tilde{Z}(0,z)\exp \Big\{\int_0^T(-\pi(s,z)b_0(s,z)+\frac{1}{2}\pi^2(s,z)\sigma_0^2(s,z))ds-\int_0^T\pi(s,z)\sigma_0(s,z)dB(s)\Big\}&=\tilde{p}(0,z) \exp \Big\{- \int_0^T (\Phi(s,z) +\frac{b_0(s,z)}{\sigma_0(s,z)})dB(s) &+\frac{1}{2}\int_0^T(\Phi^2(s,z) -\frac{b_0^2(s,z)}{\sigma_0^2(s,z)}) ds \Big\}.
\mathbb{E}({\bf 1}_{a\leq \tau_{x}<b}|\mathcal{G}_{t}) =\mathbb{E}\left(\mathbb{E}({\bf 1}_{a\leq \tau_{x}<b}|\mathcal{F}_{t})|\mathcal{G}_t\right) &= \mathbb{E}\left({\bf 1}_{\tau_x>t}\mathbb{E}^{t}({\bf 1}_{a-t\leq \tau_{x-X_t}<b-t})|\mathcal{G}_t\right).
\frac{\partial \bar{U}_{r-1}(X_{r-1})}{\partial X_{r-1}}F_{r-1}(X_{r-1},\check{\xi}_{r},\tilde{\chi},\mu)\leq-\Delta_{r-1}(X_{r-1})\|X_{r-1}\|^{2}\!\!+\!\bar{\phi}_{r-1}(\check{\xi}_{r})\check{\xi}_{r}^{2}\!\!+\!\bar{\pi}_{r-1}(\tilde{\chi})\|\tilde{\chi}\|^{2}
\begin{array}{@{}l@{}}{\displaystyle V_3^{k}( \overline{\theta}_{\Psi}^{k})= B(\mathbf{A}^{k-\frac{1}{2}};(\overline{\Psi}^{k}- \mathcal{I}_{h}\overline{\Psi}^{k}),\overline{\theta}_{\Psi}^{k}) +B(\mathbf{A}^{k-\frac{1}{2}};( \Psi^{k-\frac{1}{2}}- \overline{\Psi}^{k}),\overline{\theta}_{\Psi}^{k}),}\end{array}
\|u\|_{(\kappa+1)p^*}\leq M_{29}^{\frac{1}{\kappa+1}}M_{30}^{\frac{1}{\sqrt{\kappa+1}}}\left [2\|u^{\kappa+1}\|_{\tilde{q}_1}^p\right]^{\frac{1}{(\kappa+1)p}}\leq M_{31}^{\frac{1}{\kappa+1}}M_{30}^{\frac{1}{\sqrt{\kappa+1}}}\|u\|_{(\kappa+1)\tilde{q}_1}
( G_{o}( k )^{a b}) = \left( \begin{array}{c c}{{G_{0}( k )}}&{{0}}\\{{0}}&{{\widetilde{G}_{0}( k )}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\frac{1}{k^{2}- m^{2}+ i \epsilon}}}&{{0}}\\{{0}}&{{\frac{- 1}{k^{2}- m^{2}- i \epsilon}}}\\ \end{array}\right) .
\langle u,v\rangle=\langle u^+, v^+\rangle_{\mathcal R}\,+\,\langle u^-, v^-\rangle_{\mathcal R}\quad\langle u^\pm,v^\pm\rangle_{\mathcal R}={\mathbf{Re}}\int_{\mathbb{R}^2}u^{\pm}\overline{v^\pm},
\omega ( \partial_u K(u,\theta_n^0,\varphi),\partial_{\theta_n^0}K(u,\theta_n^0,\varphi)) = \lim_{t\to \infty}\Psi^*_t \omega ( \partial_u K(u,\theta_n^0,\varphi),\partial_{\theta_n^0}K(u,\theta_n^0,\varphi)) =0.
\langle \phi ( x )^{2}\rangle_{\mathrm{r e g}}^{( 0 )}= ( 4 \pi )^{- 2}\left[ - ( \gamma - 1 ) \bar{\tau}_{1}^{( 0 )}+ \bar{\tau}_{2}^{( 0 )}( \bar{\tau}_{1}^{( 0 )})^{- 1}+ \bar{\tau}_{3}^{( 0 )}( \bar{\tau}_{1}^{( 0 )})^{- 2}+ . . . \right]