harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\eta_{\lambda}^{\pm}(x_{0}):=\sqrt{\frac{\lambda_{2}(x_{0})-\lambda^{2}}{\lambda_{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{1}(x_{0})\pm\sqrt{\frac{\lambda^{2}-\lambda_{1}(x_{0})}{\lambda_{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{2}(x_{0}),
\int_{\mathbb{Z}_2}g(t) \binom{t+2^m-1}{2^m-1}^2 \binom{t+2^{m-1}-1}{2^{m-1}-1}^{s}\binom{t+2^m-1}{2^{m-1}-1}^{s}\mathrm{d}t \equiv ~ \frac{1}{2^{m}}\sum_{k=0}^{2^m-1}g(k)\binom{k+2^m-1}{2^m-1}^2 \binom{k+2^{m-1}-1}{2^{m-1}-1}^{s}\binom{k+2^m-1}{2^{m-1}-1}^{s}\pmod{2^{-m+1}\mathbb{Z}_2}.
\left\langle - p_{0 N}\right| h_{\mu \nu}( p_{0}) \alpha_{1}^{\mu}\tilde{\alpha}_{1}^{\nu}= \left. \frac{4}{\alpha^{\prime}}h_{\mu \nu}( p_{0}) \frac{\partial^{2}}{\partial \bar{\psi}_{1 \mu}\partial \psi_{1 \nu}}\left\langle \Omega \left( \psi_{n}, \bar{\psi}_{n}; p_{0}\right) \right| \right|_{\psi , \bar{\psi}= 0}~ ,
+ \; \frac 1{\left( q - 1 \right) !}\left[ \, \left( - \delta_{M}\delta_{N}+ \delta_{N}\delta_{M}\mp C_{M N}^{\; K}\, \cdot \delta_{K}\right) A_{\nu_{2}\ldots \nu_{q}}\right] \Lambda^{\nu_{2}\ldots \nu_{q}}\cdot \delta \left( \sigma - \sigma^{\prime}\right) \quad \cdots \quad ,
\left[\nabla G_{h/2}\ast,\xi\cdot\right] u - G_{h/2}\ast \left(\left(\nabla \cdot \xi\right) u\right) = - \left[ G_{h/2}\ast, (\nabla \cdot \xi ) \right] u + O\Big(\|\nabla \xi\|\int \frac{|z|^2}{h}G_{h/2}(z) |u^{-z}-u| dz\Big).
c^{2}d\tau^{2}=-\frac{1}{4}(3\sqrt{1-\frac{r_{s}}{r_{g}}}-\sqrt{1-\frac{r^{2}r_{s}}{r_{g}^{3}}})^{2}c^{2}dt^{2}+(1-\frac{r^{2}r_{s}}{r_{g}^{3}})^{-1}dr^{2}+r^{2}(d\theta^{2}+sin^{2}\theta d\varphi^{2}),
-\frac{\hbar^{2}}{2\mu}[\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}\frac{\partial\psi}{\partial r})+\frac{1}{r^{2}sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial\psi}{\partial\theta})+\frac{1}{r^{2}sin^{2}\theta}\frac{\partial^{2}\psi}{\partial\varphi^{2}}]-\frac{e^{2}}{4\pi\epsilon_{0}r}\psi=E\psi
\begin{aligned}f(b,c,d) -f(a+b,c,d) +f(a,c,d) = \alpha_1(ac,bc;d) -\alpha_1(a,b;cd) +\alpha_1(a,b;c)\, d -f(a,c,d) +f(a,b+c,d) -f(a,b,d) = a\, \alpha_1(b,c;d) -\alpha_1(ab,ac;d) -\alpha_2(a;bd,cd) +\alpha_2(a;b,c)\, d f(a,b,d) -f(a,b,c+d) +f(a,b,c) = a\, \alpha_2(b;c,d) -\alpha_2(ab;c,d) +\alpha_2(a;bc,bd) \end{aligned}
p_0(t) = \frac{p(e^{\lambda t}-1)}{e^{\lambda t}-p}, p_j(t) = \frac{q^2 e^{\lambda t}}{(e^{\lambda t}-p)^2}\cdot \left(\frac{e^{\lambda t}-1}{e^{\lambda t}-p}\right)^{j-1}, j \geq 1,
L_1 =\frac{1}{6}(2 \lambda_1 +3\lambda_2 +\lambda_3), L_2 =\frac{1}{2}(\lambda_1 +\lambda_2+\lambda_3)+ \frac{1}{2\sqrt{3}}\sqrt{-(\lambda_1 +2\lambda_2)(\lambda_1 +2 \lambda_3)}, L_3 =\frac{1}{4}(2\lambda_1 + \lambda_2 +\lambda_3) +\frac{1}{4\sqrt{3}}\sqrt{-(2\lambda_1+\lambda_2-3\lambda_3) (2\lambda_1-3\lambda_2 +\lambda_3)}.
e(G) \le 6r+9s+\frac{18(|G_{r+s+1}|+\cdots+|G_{r+s+t}|-2t)}7 &=\frac{18(n-2)}7-\frac{(27(r+s+t)+3r+ 9t -36)}7 &<\frac{18(n-2)}7,
\sum^n_{\ell=0}t^{-\ell}\Bigr(\mathrm{Tr\,}\alpha^+_\ell(x)-\mathrm{Tr\,}\alpha^-_\ell(x)\Bigr)dv_X(x) &=\frac{1}{2\pi}\left[\mathrm{Td_b\,}(\nabla^{T^{1,0}X},T^{1,0}X)\wedge e^{-m\frac{d\omega_0}{2\pi}}\wedge\omega_0\right]_{2n+1}(x)+O(t),\ \ \forall x\in X_p.
\mathfrak{k}_s=\mathfrak{k}_s(\xi,\hat{\psi})\in \big\{\mathfrak{u}\in L^2(M\setminus\Sigma;\mathcal{S}\otimes\mathcal{I})\big| B(\mathfrak{u})\in L^2(S^1;\mathbb{C}^2) &\mbox{and}B(\mathfrak{u})\perp \ker(\mathcal{T}_{a^+,a^-})\big\}
\lambda_q^{-2}\|\nabla u_{\leq q}\|_{\infty}\lesssim \l_q^{-2}\sum_{p\leq q}\l_p\|u_p\|_\infty &=\l_q^{-\sigma-1}\sum_{p\leq q}(L\lambda_{q-p})^{\sigma-1}\l_p^\sigma\|u_p\|_\infty &\leq \l_q^{-\sigma-1}\sup_{p\leq q}\l_p^\sigma\|u_p\|_\infty.
\int_{-1/2}^{1/2}F(x + 2 \pi i u , 0 ) \mathrm{d}u = \int_{-1/2}^{1/2}\frac{1}{2}\frac{1 + e^{- x - 2 \pi i u}}{1 - e^{- x - 2 \pi i u}}\mathrm{d}u &= \oint_{|w|=1}\frac{\mathrm{d}w}{2\pi i w}\frac{1}{2}\frac{1 + e^{- x}/w}{1 - e^{-x}/w}\mathrm{d}u &= \frac{1}{2}- \mathbf{1}_{x < 0},
\qquad(f \bullet_i g)(a_1 \otimes \cdots \otimes a_{m+n-1}) = f(a_1 \otimes \cdots \otimes a_{i-1}\otimes g(a_i \otimes \cdots \otimes a_{i+n-1}) \otimes a_{i+n}\otimes \cdots \otimes a_{m+n-1}).
u=c_0(\sinh 2t)^{-1/2}\exp\left[-\frac{\cosh 2t(x^2+y^2)-2xy}{2\sinh 2t}\right] \tilde{u}(\tilde{t},\tilde{x}), &\tilde{t}=-\frac{1}{2}\coth 2t, \tilde{x}=(\sinh 2t)^{-1}(x-y \cosh 2t).
\begin{array}{ll}\int\limits_{\Omega}\varrho^{\gamma}\,\mathrm{d}x \geq \frac{1}{|\Omega|^{\gamma-1}}\left(\int\limits_{\Omega}\varrho \,\mathrm{d}x\right)^{\gamma}= \frac{1}{|\Omega|^{\gamma-1}}\left(\int\limits_{\Omega}\varrho_0 \,\mathrm{d}x\right)^{\gamma}.\end{array}
\vartheta(\tau ,z) =q^{1/8}(\zeta^{1/2}-\zeta^{-1/2})\prod_{n\ge 1}\,(1-q^{n}\zeta)(1-q^n \zeta^{-1})(1-q^n){}=q^{\frac{1}{8}}\zeta^{\frac{1}{2}}\sum_{n\in \Bbb Z}\,(-1)^nq^{\frac{n(n+1)}{2}}\zeta^{n}.
\begin{aligned}\iint \partial_x u_s |u^\epsilon|^2 \leq \Vert\partial_x u_s\Vert_{L^\infty}\Vert u^\epsilon\Vert_{L^2}^2 &\leq L^2 \Vert\partial_x u_s\Vert_{L^\infty}\Vert u^\epsilon_x \Vert_{L^2}^2= L^2\Vert\partial_x u_s\Vert_{L^\infty}\Vert v^\epsilon_y \Vert_{L^2}^2,\end{aligned}
\int_u^t M(s-u,s)\,ds = \int_{0}^{t-u}\int_{[v,v+u]}\mu(dw)\,dv &=\int_{[0,t]}\{(t-u)\wedge w - (w-u)\vee 0\}\mu(dw) &=\int_{[0,u)}\{(t-u)\wedge w \}\mu(dw) + \int_{[u,t]}\{(t-u)\wedge w - (w-u)\}\mu(dw).
\partial_cT_b dx^c\otimes dx^b = \partial_c (x^{-1}\hat{T}_b) dx^c\otimes dx^b = (x\partial_c\hat{T}_b )\frac{dx^c}{x}\otimes \frac{dx^b}{x}, & \Gamma^d_{bc}= \frac{1}{2}g^{dr}( \partial_cg_{rb}+\partial_bg_{rc}-\partial_rg_{bc})= O(1).
g_1=e_1+2a_1e, g_2=e_2+e+2a_2e,g_3=e_2+ 2a_3e, g_4=e_1+e+ 2a_4e , g_5=e+ 2a_5e , g_6=e_1+e_2+2a_6e , g_7=e_1+e_2+e+ 2a_7e,a_1, \ldots , a_7\in [0,n-1].
\begin{cases}\mathrm{d}\nu_t(x) = \alpha\left(\Psi(\nu_t, \mu_t)(x) - \nu_t(x) \right)\mathrm{d}t, \mathrm{d}\mu_t(y) = \alpha\left(\Phi(\nu_t, \mu_t)(y) - \mu_t(y) \right)\mathrm{d}t, t \geq 0,\end{cases}
[ \beta_{q^{\prime}}^{( \sigma )}, \hat{\beta}_{q^{\prime}}^{( \sigma )}] = \sqrt{q^{\prime}}\, e^{- \frac{\epsilon}{2}( \beta^{( \sigma ) 2}- \beta^{( \sigma ) \dagger 2})}\equiv \sqrt{q^{\prime}}{\cal S}_{2}( \epsilon ) , \quad q^{\prime}\equiv e^{- \epsilon}= 1 / q \,{,}
\sqrt{\beta_K}\hat{\mathbf{G}}= \mathbf{Y}_p\boldsymbol{\Psi}_p^{\dagger}(\boldsymbol{\Psi}_{p}\boldsymbol{\Psi}_p^{\dagger})^{-1}= \mathbf{Y}_p\boldsymbol{\Psi}_p^{\dagger}/\left(m_0m_1\rho_p\right).
{M}= \left( \begin{array}{cc}C \hat 0 \hat 0 \bar C \end{array}\right)\left( \begin{array}{cc}\sigma U \bar \sigma V \bar \sigma V \sigma U \end{array}\right)\left( \begin{array}{cc}D^T \hat 0 \hat 0 \bar D^T \end{array}\! \right).
\delta\circ \kappa(x\otimes [\alpha]) = j_*\circ \partial \circ \bar \alpha_*\circ \theta_{\alpha *}(x\times \kappa_s) &=j_*\circ (\bar{\alpha}|E_\bullet\vert_{\partial \alpha})_* \circ \theta_{\partial \alpha *}(x\times \partial \kappa_{s}),
\rho_M = \begin{pmatrix}1 -m^T \frac{1}{2}\|m\|^2 \eta B-\eta m^T \frac{1}{2}\eta\|m\|^2-Bm \frac{1}{2}\eta_2^2 -\frac{1}{2}\eta_2^2m^T+\eta^TB \frac{1}{4}\eta_2^2\|m\|^2 - \eta^TBm + 1\end{pmatrix},
\widehat{fg}\left( J,\beta\right) = \sum_{\alpha\in\Gamma_d}\sum_{I\in\mathcal{D}}\widehat{f}(I,\alpha) \widehat{g}(I,\alpha) \left\langle h_{E_{\alpha,I}}^1, h_{J}^{\beta}\right\rangle_{L^2}+\widehat{f}\left( J,\beta \right) \left\langle g\right\rangle_{E_{\beta,J}}+\widehat{g}\left( J,\beta\right) \left\langle f\right\rangle_{E_{\beta,J}}.
\textnormal{(II)}= p\int (u^p) (\nabla a,\nabla (\eta^2))\;dx-\int ((p-1)u^p) (\nabla a,\nabla (\eta^2))\;dx \;\;\;\;+\int u \eta^2 ((p-1)u^p) \;dx = \int u^p (\nabla a,\nabla (\eta^2))\;dx+\int u\eta^2 \left ( (p-1)u^p \right )\;dx.
X_m(a,b,c;q) = \sum_{\begin{array}{c}\scriptstyle\sigma_1,\ldots,\sigma_{m-1}=1 \scriptstyle|\sigma_{j+1}-\sigma_j|=1\end{array}}^{r-1}q^{\sum_{k=1}^mk|\sigma_{k+1}-\sigma_{k-1}|/4},\qquad \sigma_0=a, \; \sigma_m=b, \; \sigma_{m+1}=c.
\begin{aligned}\widehat{u^{1}_0}=\frac{\ln N}{2\sqrt{N}}\sum_{N\le k\le(1+\delta)N}2^{\frac{k}{2}}(i\phi(\xi+2^{k}e_1)-i\phi(\xi-2^{k}e_1)), &\widehat{u^{2}_0}=\frac{\ln N}{2\sqrt{N}}\sum_{N\le k\le(1+\delta)N}2^{\frac{k}{2}}(\phi(\xi+2^{k}e_1)+\phi(\xi-2^{k}e_1)).\end{aligned}
\langle E_{1}(l,m,t)E_{2}^{*}(l,m,t)\rangle=\langle A(l,m,t)A^{*}(l,m,t-\frac{R_{2}-R_{1}}{c})\rangle\times\frac{e^{i\omega(\frac{R_{1}-R_{2}}{c})}}{R_{1}R_{2}}
\begin{cases}\partial_t \nu_t(x) = -\frac{\delta V^{\sigma}}{\delta \nu}(\nu_t, \mu_t, x) \nu_t(x), \partial_t \mu_t(y) = \frac{\delta V^{\sigma}}{\delta \mu}(\nu_t, \mu_t, y) \mu_t(y), \end{cases}
\hat{\phi}(x^{0},x^{1}) = \frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}dk^{1}\frac{1}{\sqrt{k^{1}}}\left[ \hat{\Lambda}(k^{1})\,e^{-ik^{1}(x^{0}+x^{1})}\,\,+\,\,\hat{\Lambda}^{\dagger}(k^{1})\,e^{ik^{1}(x^{0}+x^{1})}\right],
H_{B}= 1 +{\frac{r_{B}^{7 - B}}{V_{a}}}{\frac{1}{r^{7 - B - a}}}= 1 +{\frac{g_{s}N_{B}}{V_{a}/ \ell_{s}^{a}}}\left({\frac{\ell_{s}}{| x_{\perp}|}}\right)^{7 - A - b}\, .
f(x)-f(y) = \biggl( \frac{x}{\lvert x\rvert}-\frac{y}{\lvert y\rvert}\biggr) h\biggl(\frac{\lvert x\rvert}{\lVert x\rVert},\lVert x\rVert\biggr) + \frac{y}{\lvert y\rvert}\Biggl(h\biggl(\frac{\lvert x\rvert}{\lVert x\rVert},\lVert x\rVert\biggr)-h\biggl(\frac{\lvert y\rvert}{\lVert y\rVert},\lVert y\rVert\biggr)\Biggr),
\psi_{q}(s)+e^{- s\lambda_l}\phi_l =e^{-A(s- s_0)}\left(\psi_{0,q}(y)+e^{- s_0\lambda_l}\phi_l\right)+\int_{s_0}^{s}e^{-A(s-\tau)}F(\psi(\tau))d\tau =e^{-A(s- s_0)}\widetilde\phi_l+\int_{s_0}^{s}e^{-A(s-\tau)}F(\psi(\tau))d\tau.
S^2 \longrightarrow \mathbb{CP}^1, \mathbf{u}= (x_1, x_2, x_3) \mapsto\begin{cases}\displaystyle{\zeta = - \frac{x_1+ix_2}{1+x_3}}\textrm{on $N$}begin{align*}2ex]\displaystyle{\tilde{\zeta}= - \frac{x_1-ix_2}{1-x_3}}\textrm{on $S$}\rlap{.}\end{cases}
\mathbf u(t) = P_\mu\left( \mathbf S(t) \mathbf u_0 + i \int_0^t \mathbf S(t-s) \mathbf N\left( \Theta_R^{\mathbf u}(s) P_\mu \mathbf u(s) \right) \, ds + i \int_0^t \mathbf S(t-s) \mathbf M\left( P_\mu \mathbf u(s) \right) \, dW(s) \right),
M_\omega T_x f(t) = f(t-x) \, e^{2 \pi i \omega \cdot t}= f(t-x) \, e^{2 \pi i \omega \cdot (t-x)}e^{2 \pi i \omega \cdot x}= e^{2 \pi i \omega \cdot x}\T_x M_\omega f(t).
P\cdot P_1^{a_{1p}}\cdot\ldots\cdot P_n^{a_{np}}\in E,~~\mbox{and hence}~~P^2\cdot P_1^{2a_{1p}}\cdot\ldots\cdot P_n^{2a_{np}}= \left\langle \frac{u+v\sqrt{-m}}{2^{1-\delta}}\right\rangle
\mathbf{D}_\mathrm{x}=\! \sqrt{\frac{1}{\lambda}\mathbf{\Lambda}_\mathbf{\tilde{A}}^{+}\!+\!\frac{1}{4}{\mathbf{\Lambda}_{\mathbf{\tilde{A}}}^{+}}^2}\!-\! \frac{1}{2}\mathbf{\Lambda}_\mathbf{\tilde{A}}^{+}.
\begin{array}{l}{\dot{o}^{A}= \frac{1}{2}\dot{m}o^{A}- \i \phi \iota^{A},}\\{\dot{\iota}^{A}= \i \mu o^{A}- \frac{1}{2}\dot{\overline{{m}}}\iota^{A}+ \acute{o}^{A}.}\\ \end{array}
\theta_k(x) ={a}_k |x|^\frac{2k(p-q)}{1-q}\theta_0 +{\sf h}_k(x) ({a}_k\not=0),{\sf h}_k(x) = |x|^\frac{2k(p-q)}{1-q}\theta_0 \sum_{i=1}^{N'}b_i |x|^\frac{2(p-q)i}{1-q}.
\mu_{1,j}:= \left| \frac{\partial^{l+j}}{\partial n^{l+j}}(qf(mn+mh) - qf(mn)) \right|= qm^{l+j}\cdot mh \, \left|f^{(l+j+1)}(t_0) \right| &= \frac{qh}{n^{l+j+1}}t^{l+j+1}\left| f^{(l+j+1)}(t_0) \right|,
\hat{f}(x_1,\dots,x_k) := |I|^{-k(k-1)}\sum_{a_1,\dots,a_k \in I^{k-1}}f(x_1 + \sum_{j=2}^k a_{1,j-1}x_j, x_2 + \sum_{j=2}^k a_{2,j-1}x_j, \dots, x_k + \sum_{j=2}^k a_{k,j-1}x_j)
r = \frac{- 1}{1 - \frac{1}{\sqrt{3}}\cos{\varphi}}\ln \left( 1 - \frac{t}{\tilde{t}}\right) , \hspace{. 2 i n}\tilde{t}= \frac{t_{o}}{1 - \frac{1}{\sqrt{3}}\cos{\varphi}}.
\delta_{\epsilon}V^{( 1 ) \mu_{1}\ldots \mu_{p - 2 k - 3}}= \partial^{\left[ \mu_{1}\right.}\bar{\epsilon}^{\left. \mu_{2}\ldots \mu_{p - 2 k - 3}\right]}+ M \epsilon^{\mu_{1}\ldots \mu_{p - 2 k - 3}}+ \partial_{\mu}\bar{\epsilon}^{\mu \mu_{1}\ldots \mu_{p - 2 k - 3}},
\begin{array}{rcl}{\cal R}=\bar{\cal C}{\cal K}~, & \bar{\cal C}=\displaystyle\sum_{m=0}^{p-1}\frac{q^{-m(m-1)/2}}{[m]!}\left(-(q-q^{-1})\right)^m e^m\otimes f^m~, &{\cal K}=\displaystyle\frac{1}{p}\sum_{n,n'=0}^{p-1}q^{2nn'}t^n \otimes\bar{t}^{n'}~.\end{array}
\small \det P =\det \left[ \left(x-1-\dfrac{c_2^2n_1}{x-\dfrac{1}{r_1+1}}+\dfrac{r}{2r+n_1}-\dfrac{c_1^22r}{x-1-c_3^2\dfrac{n_2}{x-\frac{1}{r_2+1}}}\right)I_n +\dfrac{c_1^2r\mathcal{L}(G)}{x-1-c_3^2\dfrac{n_2}{x-\frac{1}{r_2+1}}}-\dfrac{r\mathcal{L}(G)}{2r+n_1}\right].
E_u=\Pi\circ\left(\widetilde{\mathbb{B}}_{u^2}'\right)^2\circ \Pi,F_u=\Pi\circ \left(\widetilde{\mathbb{B}}_{u^2}'\right)^2\circ \Pi^{\bot}, &G_u=\Pi^{\bot}\circ \left(\widetilde{\mathbb{B}}_{u^2}'\right)^2\circ \Pi,H_u=\Pi^{\bot}\circ\left(\widetilde{\mathbb{B}}_{u^2}'\right)^2\circ \Pi^{\bot}.
\nabla\varphi\cdot\nabla\psi=\frac{\partial\varphi}{\partial x}\frac{\partial\psi}{\partial x}+\frac{\partial\varphi}{\partial y}\frac{\partial\psi}{\partial y}=\frac{\partial\psi}{\partial y}\frac{\partial\psi}{\partial x}-\frac{\partial\psi}{\partial x}\frac{\partial\psi}{\partial y}=0.
{{{m}\choose k}{{n}\choose l}-{\overline S_{k,l}}}= \sum_{i\in I_{k,m}}\sum_{j\in J_{l,n}}P((A_{i_{1}}\cup A_{i_{2}}\cup ...\cup A_{i_{k}})\cap ( B_{j_{1}}\cup B_{j_{2}}\cup ...\cup B_{j_{l}})).
\begin{aligned}- F^{ij}u_{m\tau}R^m_{i\tau j}= - F^{ij}W_{m\tau}R^m_{i\tau j}+ F^{ij}(\chi_{m\tau}+ \tilde{W}_{m\tau}) R^m_{i\tau j}&= - F^{\alpha\beta}W_{\tau\tau}R_{\tau\alpha\tau\beta}+ F^{ij}(\chi_{m\tau}+ \tilde{W}_{m\tau}) R^m_{i\tau j}& \leq C (1 + |\nabla u|^2) \sum_i F^{ii},\end{aligned}
S_{E}= - \frac{1}{1 6 \pi G_{d - 1}}\int_{\Sigma}d^{d - 1}x \sqrt{g}\left[ R - m e^{\frac{4 \phi}{\sqrt{d - 2}( d - 3 )}}- \frac{4}{d - 3}( \partial \phi )^{2}- \frac{1}{4}e^{\frac{- 4 \sqrt{d - 2}}{d - 3}\phi}F^{2}\right] ,
c_k(s) = \alpha(s) \pi^{\frac{n(n-1)}{2}}\frac{\Gamma(s+k-2n+3) \Gamma(s+k-2n+5) \ldots \Gamma(s+k-1)}{\Gamma(s+k-n+2) \Gamma(s+k-n+3) \ldots \Gamma(s+k)},
\log{F(s,\chi)}= \log{\epsilon(\chi)}+ \frac{\pi}{2}\left( \frac{1}{2}- \kappa \right)i - 1+ \left( \frac{1}{2}- s \right) \left( \log{(q(1-s))}- \log{2\pi}+ \frac{\pi i}{2}\right) &\quad\quad+ s + \int_0^\infty \frac{[u] - u + \frac{1}{2}}{u + 1 - s}du- \sum_{n=1}^\infty \frac{e^{\pi i(s+\kappa)n}}{n}
\mathcal{F}_N(\mathbf{h},\mathbf{g})=\mathbb{E}\left[\int_0^T \mathcal{L}\left(t,\mathbf{Y}(t),\frac{1}{N}\sum_{n=1}^{N}\delta_{X^n(t)}\right)dt+\int_0^T \Psi\left(\mathbf{h}(t),\mathbf{g}\left(\frac{1}{N}\sum_{n=1}^{N}\delta_{X^n(t)}\right)\right)dt\right],
{\psi_0}^{\pm}=e^{i\theta}\Big[\,{\psi_{01}}^{\pm}\,+\,i{\psi_{02}}^{\pm}\,\Big], begin{align*}1mm]{\psi_j^1}^{\pm}=e^{i\theta}\Big[\,{\psi_{j1}^1}^{\pm}\sin{j\theta}\,+\,i{\psi_{j2}^1}^{\pm}\cos{j\theta}\,\Big], begin{align*}1mm]{\psi_j^2}^{\pm}=e^{i\theta}\Big[\,{\psi_{j1}^2}^{\pm}\cos{j\theta}\,+\,i{\psi_{j2}^2}^{\pm}\sin{j\theta}\,\Big].
\widetilde{\Xi}_c=\{\lambda=(\lambda_1,\lambda_2,\lambda_3)\in \sqrt{-1}\frak{t}^*: \lambda_j\equiv\frac{1}{2}\textrm{mod}\mathbb{Z}, ~\lambda\textrm{is}\Delta\textrm{-regular and}\Delta_c^+\textrm{-dominant}\}.
f(p^{a_{3}}) = 1 + \sum_{i=1}^{a_{1}}\frac{p^{3i}p^{\min\{i-1, a_{3}- i-1\}}(p-1)}{p^{2i}p^{\min \{2i, 2a_{3}- 2i\}}}+ \sum_{i=a_{1}+1}^{a_{2}}\frac{p^{2i}p^{\min\{i-1, a_{3}- i - 1\}}(p-1)}{p^{2i}p^{\min \{2i, 2a_{3}- 2i\}}}+ \sum_{i=a_{2}+1}^{a_{3}}\frac{p^{i}p^{\min\{i-1, a_{3}- i - 1\}}(p-1)}{p^{2i}p^{\min \{2i, 2a_{3}- 2i\}}}.
\begin{aligned}F( z+ \lambda \tau + \mu, \tau) = (-1)^{\lambda + \mu}q^{-\lambda/2}p^{-\lambda}K(z,\tau) \Theta(z + \lambda \tau + \mu, \tau) = (-1)^{\lambda + \mu}q^{-\lambda/2}p^{-\lambda}\Theta(z,\tau) \,.\end{aligned}
0 \geq e^{\kappa \phi}a^{ij}D_{ij}w(x_0) = a^{ij}D_{ij}v - 2\kappa a^{ij}D_i\phi D_j v - \kappa v a^{ij}D_{ij}\phi + \kappa^2 v a^{ij}D_i\phi D_j \phi.
\sum_{t=1}^{T}\sum_{i=1}^n\frac{(G_1+G_2\|q_{i,t+1}\|)\|\epsilon^z_{i,t}\|}{\gamma_{t}}\le \sum_{t=1}^{T}\sum_{i=1}^n\Big(\frac{2\gamma_0(G_1^2+G_2^2\|q_{i,t+1}\|^2)}{\sigma\gamma_{t}^2}+\frac{\sigma\|\epsilon^z_{i,t}\|^2}{4\gamma_0}\Big).
(\overline{\partial}_{E,p}+ \overline{\partial}^t_{E,p})_{\max}(\phi_i\omega)=\phi_i(\overline{\partial}_{E,p}+ \overline{\partial}^t_{E,p})\omega+(\overline{\partial}_{\max}\phi_i)\wedge \omega-{\rm Int}(\overline{\partial}_{\max}\phi_i)\omega
\stackrel{\mathrm{H W}}{S^{\alpha \beta}}= \int \Psi^{\dagger}S_{\mathrm{o p}}^{\alpha \beta}\Psi d^{3}x , \quad S_{\mathrm{o p}}^{\alpha \beta}= \frac{1}{2}\sigma^{\alpha \beta}+ \frac{i}{2 m}\left( p^{\alpha}\gamma^{\beta}- p^{\beta}\gamma^{\alpha}\right) .
\mathbb{E}\left[ \left( \frac{1}{m}\sum_{j=1}^{m}(\Delta^{n,j}_{t-})^{p}\right)^{k}\right] \leq C \left( \sum_{k'=1}^{k-1}m^{k'-k}\varepsilon^{(k',pk)}_{n}(t) + \varepsilon^{(k,p)}_{n}(t) \right),
\int_{\mathbb{R}^N}\phi(t_R|\nabla w_R|)(t_R|\nabla w_R|)^2dx+\int_{\mathbb{R}^N}\phi(t_R|w_R|)(t_R|w_R|)^2 dx=\int_{\mathbb{R}^N}f(t_Rw_R)(t_Rw_R) dx.
M_{n}=(-1)^nM_n^{(-1,1)}=(-1)^nM_n^{(2-3,1)}&=(-1)^n\sum_{k=0}^n\binom{n}{k}M_k^{(2,1)}(-3)^{n-k}=\sum_{k=0}^n(-1)^k3^{n-k}\binom{n}{k}C_{k+1}.
P\left(\frac{n-A_n^*}{f_2(n)}\leq\varepsilon\right) =P(A_n(t)>t,\quad\forall t=a_n,\ldots,\lfloor n-\varepsilon f_2(n)\rfloor) &=P(B_4^{(n)}\cap B_5^{(n)}),
\langle [\nabla_r,A_q]_- u^{},w_r|w|^{q-2}\rangle=c\left\langle \bigl[(\nabla_r \mathsf{f}^i)\mathsf{f}^l + \mathsf{f}^i\nabla_r \mathsf{f}^l \bigr]w_l,\nabla_i(w_r|w|^{q-2})\right\rangle =:S_{1}+S_{2},
\begin{cases}a_0^B=1, &a_1^B=\alpha_1, &a_2^B=2^{-1}(a_1^B\alpha_1-\alpha_2), &a_3^B=3^{-1}(a_2^B\alpha_1-a_1^B\alpha_2+\alpha_3), &\vdots &a_{4t-1}^B=(4t-1)^{-1}(a_{4t-2}^B\alpha_1-a_{4t-3}^B\alpha_2+\cdots+\alpha_{4t-1}).\end{cases}
\Gamma(\mathcal O(k))=\begin{cases}\sum_{l=0}^{\infty}\mathcal H_{n+1}^{k+2l,2l}, \,\,k \geqq 0, \sum_{l=0}^{\infty}\mathcal H_{n+1}^{2l,|k|+2l}, \,\,k \leqq 0.\end{cases}
I(b)= \frac{2^{1-2\alpha}-2}{1-2\alpha}-2^{-\alpha}(f(1) + \mu_\alpha) +2b^{1-2\alpha}\left( f\left(\frac 1b \right) + f(1) - \frac{2^{-1-\alpha}}{1-2\alpha}+ 2^{-1-\alpha}\mu_\alpha \right)
g^{(1)}=\left[\begin{array}{cc}\dfrac{e^{t}Q_c(x-vt)}{Q_c(x-vt-x_0)}-\dfrac{1}{c}Q_c(x-vt) -\dfrac{1}{c}Q_c(x-vt) \dfrac{e^{-t}Q_c(x-vt)}{Q_c(x-vt+x_0)}\end{array}\right],
\left[\frac{E(x^7b,x^{60})E(x^{13}b,x^{60})E(x^{37}/b,x^{60})E(x^{43}/b,x^{60})}{E(x^7/b,x^{60})E(x^{13}/b,x^{60})E(x^{37}b,x^{60})E(x^{43}b,x^{60})}\right]^N=b^{2N}.
\sum \delta_j^{\beta}= \sum \left( \delta_j^{\beta}\, j^a \right) \, j^{-a}\leq \left( \sum \delta_j^2 \, j^{\frac{2a}{\beta}}\right)^{\frac{\beta}{2}}\left( \sum j^{- \frac{2a}{2-\beta}}\right)^{\frac{2-\beta}{2}}\, .
\Delta E_{n,m;k}=\frac{2 \lambda^2 N}{(1+\lambda^2)^2}\left [ \left (\frac{2k-n}{3}+2k-m\right ) \frac{1}{k}\, C_{n-1}^{k-1}\, C_{m-1}^{k-1}\right .+\left .\frac{2}{3}\left ( C_{n-2}^{k-2}\, C_{m-1}^{k-1}- C_{n-1}^{k-1}\, C_{m-2}^{k-2}\right ) \right ]\ .
Z_{T M}^{r e d}= \int{\cal D}p{\cal D}q d e t{\bigl \{\Theta_{a}, \Upsilon_{b}\bigr \}}\delta ( \theta ) \delta ({\theta}_{i}^{T}) \delta ( \Upsilon ) \delta ({\Upsilon}_{i}^{T}) e^{i \int ( p \dot{q}-{\cal H}_{T M})}\; ,
\phi^0-\overline{\phi}\in C([0,T^*];H^{3}),\phi^0 \nabla^4 u^0 \in L^2([0,T^*];L^2), & u^0\in C([0,T^*];H^{s'})\cap L^\infty([0,T^*];H^3) \s' \in [2,3).
\left( \frac{C_{18}}{\lambda C_{19}C_{20}^{\kappa_1 + \kappa_2 + 2}}\right)^{\frac{p}{\kappa_1 + \kappa_2 + 2}}\leq \left( \|{y_1}_n\|_{m_1}^{p}+ \|{y_2}_n\|_{m_2}^{p}\right).
-264\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(20)}=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{L_R\otimes C}-\widetilde{L_R\otimes C}\otimes \widetilde{L_R\otimes C}+\widetilde{L_R\otimes C})\right\}^{(20)}.
\mathcal{O}_{c,x}^{\alpha ,\theta}f(x):=\sum_{n=0}^{\infty}\frac{c^{n}}{n!}\underbrace{\mathcal{D}_{x}^{\alpha ,\theta}...\mathcal{D}_{x}^{\alpha,\theta}}_{n-times}f(x),c\in \mathbb{R},,\alpha \in(0,2],|\theta |\leq \min \{\alpha ,2-\alpha \}.
\|\psi^*(g)x\|^2 = \|(\psi^*(g)x \mid \psi^*(g)x)_A\|_{\infty}= \sup_{v \in E^0}\sum_{s(e)=v}|g(\psi(e))x(e)|^2 \le \|g\|_{\infty}^2 \|x\|^2,
S^{\prime}= - \int d^{5}\sigma \left( e^{- \phi}\sqrt{- G}\sqrt{1 + e^{2 \phi}z_{1}+ e^{4 \phi}( \frac{z_{1}^{2}}{2}- z_{2})}+ \frac{\epsilon_{\mu \nu \lambda \sigma \tau}e^{\phi}C^{\mu}\tilde{{\cal H}}^{\nu \lambda}\tilde{{\cal H}}^{\sigma \tau}}{8 ( 1 + C_{1}^{2})}\right) + \int \Omega^{\prime},
\vert F(z+h)-F(z) \vert = \vert L(h) + e \vert \leq \vert L(h) \vert + \vert e \vert \leq \frac 6 5 \vert h \vert + \frac 6 5 w_x(\vert F(z+h)-F(z)\vert) \cdot \vert F(z+h)-F(z)\vert.
I:= \left\| \sum_{1 \le n \le k_1}\frac{w_n \mathbf{1}_{B^{cl}(x_{n}, t_n r_{n})}}{\mu B^{cl}(x_{n}, t_n r_{n})}\right\|_p\le C_1\left\| \sum_{1 \le n \le N}\frac{w_n \mathbf{1}_{B^{cl}(x_{n}, r_{n})}}{\mu B^{cl}(x_{n}, r_{n})}\right\|_p,
\frac{\lambda_2^{K_{\ell}}}{\lambda_2^L}=\Bigl\{2^{(\dim M_2^{K_{\ell}}-m)/2}\cdot|M_2^{K_{\ell}}|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}\Bigl\{2^{(\dim M_2^L-m)/2}\cdot|M_2^L|^{-1}\cdot\prod_{i=1}^{m}(2^{2i}-1)\Bigr\}^{-1}\leq 2^{1/2}(2^{(n_{2,\nu_2}+1)/2}+1)(2^{(n_{2,\nu_2}-1)/2}+1).
\rho (\eta )\sim \biggl(\frac{\eta_1}{\eta_2}\biggr)^3\int_{\cal K}{\rm d}k_{\rm phys}\,k^2_{\rm phys}\omega_{\rm phys}(k)=\biggl(\frac{\eta_1}{\eta_2}\biggr)^3{\cal O}(k_{\rm c}^4),
0=\lim_{t\rightarrow \infty}\displaystyle \int_{M}e^{4\lambda}[Q-e^{-4\lambda}(Q_{0})_{\ker}]^{2}d\mu =\displaystyle \int_{M}e^{4\lambda_{\infty}}[Q_{\infty}-e^{-4\lambda_{\infty}}(Q_{0})_{\ker}]^{2}d\mu_{\infty}
\|f-m_Q-s_n\|_q= \left\{\sum\limits_{Q'\in\P[D]_n(Q)}\|f-m_{Q'}\|_{L_q(Q')}^q \right\}^{1/q}= \left\{\sum\limits_{Q'\in\P[D]_n(Q)}E_k(f;Q';L_q)\right\}^{1/q}.
\Gamma_{d i v}^{m a t}= \frac{1}{\varepsilon}\int \left( \gamma_{1}\frac{1}{2}g^{\mu \nu}\partial_{\mu}\varphi \partial_{\nu}\varphi - \gamma_{2}\frac{\varphi^{4}}{4 !}+ \gamma_{3}\frac{1}{2}R \varphi^{2}\right) \sqrt{- g}~ d^{4}x .
I(\zeta)=\frac{\mathrm{Re}(\zeta)}{2\rho_\beta^2 \rho_\gamma^2}(\rho_\beta^2 + \rho_\gamma^2 - K)+\frac{1}{4\rho_\beta^2 \rho_\gamma^2}( K^2 - 2K + 2\rho_\beta^2 + 2\rho_\gamma^2) - 1,
\int{\cal D}X^{1}( \sigma ){\cal D}X^{2}( \sigma ){\cal D}X^{3}( \sigma ) \, \Delta ( X^{1}( \sigma ) + X^{2}( \sigma ) - X^{3}( \sigma ) ) \psi ( X^{1}) \psi ( X^{2}) \psi^{*}( X^{3}) .
\nabla \cdot [\frac{1}{b^\alpha}{(\mathbf{f}a + \mathbf{gc})}] =\frac{1}{b^{\alpha+1}}[(1+\alpha)b\nabla \cdot (\mathbf{f}a + \mathbf{gc}) - \alpha \nabla \cdot (b\mathbf{f}a + b\mathbf{gc})].\raisetag{30pt}
\lefteqn{\psi_{U}\Big ( \sum_{{i},{j}}{f}_{{i}}{g}_{{j}}\otimes f_{{i}}^{\prime}g_{{j}}^{\prime}\Big ) \: = \: \sum_{{i},{j}}{f}_{{i}}{g}_{{j}}\otimes f_{{i}}^{\prime}g_{{j}}^{\prime}\otimes 1}.
2\Delta x^{\top}H_P(x,V)\Delta x =\Delta F\bullet \left(\mathcal{L}_{F_P(x)}^{-1}\mathcal{L}_{{V}_P}+\mathcal{L}_{V_P}\mathcal{L}_{F_P(x)}^{-1}\right)\Delta F =\mathcal{L}_{F_P(x)}^{-1}(\Delta F)\bullet \left(\mathcal{L}_{V_P}\mathcal{L}_{F_P(x)}+\mathcal{L}_{F_P(x)}\mathcal{L}_{V_P}\right)\mathcal{L}_{F_P(x)}^{-1}(\Delta F)>0
\left(p^{\,(\varphi_2,\psi_2,\rho_2)}\right)^{(\varphi_1,\psi_1,\rho_1)}_i=\psi_1 \left(p^{\,(\varphi_2,\psi_2,\rho_2)}\right)_{\varphi_1^{-1}(i)}\rho_1=\psi_1\psi_2 p_{\varphi_2^{-1}(\varphi_1^{-1}(i))}\rho_2 \rho_1=\psi_1\psi_2 p_{(\varphi_1\varphi_2)^{-1}(i)}\rho_1\rho_2.
\begin{cases}\frac{Du}{Dt}+\hat{f}w-fv=-\frac{1}{\rho_{0}}P_{x}, \frac{Dv}{Dt}+fu=-\frac{1}{\rho_{0}}P_{y}, \frac{Dw}{Dt}-\hat{f}u=-\frac{1}{\rho_{0}}P_{z}-g, \end{cases}
\sum_{\substack{\{\sigma \in{D}^{I}_{n}: \sigma(a)=n \}}}(-1)^{\ell (\sigma )}x^{L(\sigma )}= \sum_{\substack{\{\sigma \in{D}^{I}_n: \sigma(a)=n, \sigma (a-1) < \sigma (a+1) \}}}\left( (-1)^{\ell (\sigma )}x^{L(\sigma )}+ (-1)^{\ell (\sigma^a )}x^{L(\sigma^a )}\right) =0.
x^-_{\theta}(1)=y_{1\,2}^-(1) &=[\,x_2^-(0),\ldots,x_{n-1}^-(0),\, x_n^-(0),\,\ldots,\,x_{1}^-(1)\,]_{(s,\,\ldots,\, s,\,s^2,\,r^{-1},\,\ldots,r^{-1})}, &x_{\theta}^+(-1)=y_{1\,2}^+(-1) &=[\,x_{1}^+(-1),\,\ldots,\,x_n^+(0),\,x_{n-1}^+(0),\,\ldots, x_2^+(0)\,]_{\langle r,\,\ldots,\, r,\, r^{2},\,s^{-1},\,\ldots,\, s^{-1}\rangle}.
\vert\vert f\vert\vert_{L^2(m)}^2=\sum_{k=0}^{+\infty}k!\sum_{(i_1,\dots, i_k)\in (\{-N,\dots, N\}\setminus\{0\})^k}\big\vert\alpha_{i_1,\dots ,i_k}^{(k)}\big\vert^2\sigma_{i_1}^2\dots \sigma_{i_k}^2.
y^{2}= x^{3}+ \sum_{k = - 4}^{4}\sum_{\ell = 0}^{8 - n k}f_{k \ell}\, z^{\prime}{}^{\ell}z^{4 - k}u^{\ell - k}x + \sum_{k = - 6}^{6}\sum_{\ell = 0}^{1 2 - n k}g_{k \ell}\, z^{\prime}{}^{\ell}z^{6 - k}u^{\ell - k}.
S_1(a,\,n,\,d) =2na, S_2(a,\,n,\,d) =2na^2+2n(4n^2-1)d^2/3, S_3(a,\,n,\,d) =2na^3+2n(4n^2-1)ad^2, S_4(a,\,n,\,d) =2na^4+4n(4n^2-1)a^2d^2 & +2n(4n^2-1)(12n^2-7)d^4/15.
\|u\|_{H^m}^2=\sum\limits_{l \in \mathbb{Z}}(1+\mu_l^2)^m|\widehat{u}_l|^2,\ \mathrm{for}\u(x)=\sum\limits_{l\in \mathbb{Z}}\widehat{u}_l e^{i\mu_l(x-a)},\ \mu_l=\frac{2\pi l}{b - a},