harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_{1}( a , b , d ) = - \frac{1}{\left( 2 \pi \right)^{d - 2}a b}\sum_{n = 1}^{\infty}\int d^{d - 2}p \frac{1}{( \vec{p}^{2}+ m^{2}+ ( \frac{n \pi}{a})^{2})}
\sum_{x\in [n]}\sum_{\substack{W_1,W_2\subseteq [k] |W_1|=|W_2|=1}}O(n^{2k-2\l-2}q^{2i})=O(n^{2k-2\l-1}q^{2i})=O\left(q^{2i}\frac{|E_n|^2}{n}\right)\,.
A(0) =a(0)+\dfrac{b(0)}{R(1)}\;,B(0)=\dfrac{b(0)d(1)}{R(1)^2}\;, A(1)=\dfrac{a(1)R(2)+b(1)}{R(1)}\;, A(n) =R(n+1)-r(n-1)\dfrac{d(n)}{d(n-1)}B(n)=b(n)\dfrac{d(n+1)}{d(n)}
a_j = \frac{1}{1 + \rho}\, \frac{1 + \cdots + \rho^j}{1 + \cdots + \rho^{j-1}}= \begin{cases}\displaystyle \frac{1}{1 + \rho}\, \frac{\rho^{j+1}-1}{\rho^j-1}\rho \neq 1 \displaystyle \frac{1}{2}\, \frac{j+1}{j}\rho = 1.\end{cases}
\lVert k_i\rVert_{L_1}=\lVert y_i(\cdot)a_i(\cdot\cdot)\rVert_{L_1}=\lVert y_i(\cdot)\rVert_{L_1}\cdot\lVert a_i(\cdot\cdot)\rVert_{L_1}&\le M^{\frac1{p}}\lVert a_i\rVert_{L_q}\cdot M^{\frac1{q}}\lVert y_i\rVert_{L_p}=M\lVert a_i\rVert_{L_q}\cdot\lVert y_i\rVert_{L_p}.
\begin{aligned}\big\Vert \langle y\rangle^{l+k}\{(D^{\tilde{\alpha}}u\partial_x +D^{\tilde{\alpha}}v\partial_y)D^{\alpha-\tilde{\alpha}}u &-(D^{\tilde{\alpha}}h\partial_x+D^{\tilde{\alpha}}g\partial_y)D^{\alpha-\tilde{\alpha}}h\}\big\Vert_{L^2_y}&\lesssim E_{u,h}^2.\end{aligned}
\nabla_\nu P^\nu_\zeta = \nabla_\nu (\zeta \phi^\nu \phi) - \nabla_\nu \left(\zeta^\nu \frac{\phi^2}{2}\right) &= \zeta (\square \phi)\phi + \zeta \phi^\nu \phi_\nu + \phi^\nu \zeta_\nu \phi - ( \square \zeta) \frac{\phi^2}{2}- \zeta^\nu \phi \phi_\nu &= \zeta \frac{g(\phi)g'(\phi)\phi}{r^2}+ \zeta \phi^\nu \phi_\nu - ( \square \zeta) \frac{\phi^2}{2}
\frac{\partial(\rho\overline{Re_{\theta t}})}{\partial t}+\frac{\partial(\rho U_{j}\overline{Re_{\theta t}})}{\partial x_{j}}=P_{\theta t}+\frac{\partial}{\partial x_{j}}[\sigma_{\theta t}(\mu+\mu_{t})\frac{\partial\overline{Re_{\theta t}}}{\partial x_{j}}]
G(x,y) =H(u,y) &=\sum_i b_i(x-y)^iy^{N-i}&=\sum_{i=0}^N \sum_{j=0}^i (-1)^{i-j}b_i{i\choose j}x^jy^{N-j}&=\sum_{j=0}^N \sum_{i=j}^N (-1)^{i-j}b_i{i\choose j}x^jy^{N-j}.
\frac{1}{2}\frac{d}{dt}\int \left|\nabla_\xi h\right|^2 \varrho\,m_{0}=\int (\nabla_\xi h, \mathcal{L}\nabla_\xi h )\varrho\, m_{0}+ \int (\nabla_\xi h, \left[\nabla_\xi, \mathcal{L}\right]h )\varrho\, m_{0}\,.
\ln R_j(n)^2 \leq \ln R_j(n_0)^2-\sum_{l=n_0}^{n-1}\frac{V(l)}{\sin\pi k_j}\sin2\pi\theta_j(l)+\sum_{l=n_0}^{n-1}\frac{O(1)}{(l-b)^2}&=\ln R_j(n_0)^2+\frac{O(1)}{n_0-b}-\frac{K_1}{\sin\pi k_j}\sum_{l=n_0}^{n-1}\frac{\sin2\pi\theta(l)+\sin2\pi\tilde{\theta}(l)+100}{l-b}\sin2\pi\theta_j(l). &=\ln R_j(n_0)^2+\frac{O(1)}{n_0-b},
\Delta_{g_{\sigma_n}}= \Delta_0 + \sum_{i,j = 1}^k \partial_i \big\{(g^{ij}_{{\sigma_n}}- \delta_{ij}) \partial_j \big\}+ \frac{1}{2D_n}\sum_{i,j=1}^k g_{{\sigma_n}}^{ij}(\partial_i D_n) \partial_j.
\sum_{\pi ( 1 , 2 , . . . n )}\mathrm{T r}\Big [ H^{( 1 )}H^{( l_{\pi ( 1 )})}. . . . . H^{( l_{\pi ( n )})}\Big ] \frac{1}{\tilde{x}_{0}- \tilde{x}_{\pi ( 1 )}}\frac{1}{\tilde{x}_{\pi ( 1 )}- \tilde{x}_{\pi ( 2 )}}. . . . . \frac{1}{\tilde{x}_{\pi ( n )}- \tilde{x}_{0}}\; .
D(\widetilde g)=\frac{(4\widetilde c+\widetilde d\,^2)(1-4\widetilde e)}{(\widetilde u+\widetilde d\,\widetilde e\,\widetilde u-\widetilde v-\widetilde c\,\widetilde e\,\widetilde v)^4\widetilde e\,^8}\widetilde\Theta^2,
\overline{\psi}^{(2)}(x) P_+ \psi^{(2)}(x) \; \longleftrightarrow \;\frac{1}{2\pi}c^{(2)}:e^{-ia \varphi^{(1)}(x)}:_{M^{(1)}}:e^{+ib \varphi^{(2)}(x)}:_{M^{(2)}}\; e^{i\frac{\theta}{2}}\; ,
\delta_{\tilde{\partial}^{a}\varphi_{j}, \tilde{\partial}^{b}\varphi_{l}}^{\mu}= ( - 1 )^{| b |}\partial^{a}\partial^{b}\delta_{\varphi_{j}, \varphi_{l}}^{\mu}, \quad \varphi_{j}, \varphi_{l}\in{\cal G}.
\|u_n-u\|^2 = \langle u_n-u,u_n-u\rangle &= \langle u_n, u_n-u\rangle +o(1) &\le \|u_n-\lambda w^*_n\|_*\|u_n-u\|+\lambda w^*_n(u_n-u)+o(1) &\le m_\lambda(u_n)\|u_n-u\|+\lambda\int_a^b |w_n(u_n-u)|\,dx+o(1) &\le \lambda\alpha\int_a^b(1+|u_n|^{s-1})|u_n-u|\,dx+o(1) &\le \lambda C\|u_n-u\|_\infty+o(1),
U_{0}^{n}S t r \left( \ldots \underbrace{\phi^{i_{2 k}^{\prime}}\phi^{i_{2 k - 1}^{\prime}}}\ldots \right) = \frac{n}{n + 2 k}( U_{0}^{n}- U_{0}^{n}|_{i_{n}\leftrightarrow i_{1}^{\prime}}- \ldots - U_{0}^{n}|_{i_{n}\leftrightarrow i_{2 k}^{\prime}}) S t r \left( \ldots \underbrace{\phi^{i_{2 k}^{\prime}}\phi^{i_{2 k - 1}^{\prime}}}\ldots \right) .
\varphi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right) =(c\tau+d)^ke^{2\pi i m \frac{cz^2}{c\tau+d}}\,\varphi(\tau,z), \varphi(\tau,z+x\tau+y) =v_H([x,y;0])^{2m}e^{-2\pi i m (x^2\tau+2xz)}\varphi(\tau,z),
\sigma_{\mu}^{\alpha \beta^{\prime}}= ( \tau^{a}, i ) \ , \qquad \bar{\sigma}_{\mu \, \alpha^{\prime}\beta}= ( \tau^{a}, - i ) \ , \qquad \sigma_{\mu}^{\alpha \alpha^{\prime}}= \bar{\sigma}_{\mu}^{\alpha^{\prime}\alpha}= \epsilon^{\alpha^{\prime}\beta^{\prime}}\bar{\sigma}_{\mu \, \beta^{\prime}\beta}\epsilon^{\alpha \beta}\ ,
\bar\rho_{\frac12}= \frac{\sinh(kb)+\frac12(\bar\rho_0^2-\bar C-1)\sinh(kb)-\bar\rho_0\cosh(kb)}{\cosh(kb)+\frac12(\bar\rho_0^2-\bar C-1)(\cosh(kb)-1)-\bar\rho_0\sinh(kb)},
\Upsilon_\alpha(t):=\sum_{i=0}^{m-1}\int_{I_i}\int_{I_i}\chi_{(0,t)}(s)\phi_\alpha(t-s)[\chi_{(0,t)}(s)\phi_\alpha(t-s)-\chi_{(0,t)}(\tau)\phi_\alpha(t-\tau)]d\tau ds.
F \Big ( m_{N}, H - \sum_{\ell = 1}^{N - 1}{U}_{\ell}\Big ){\cal A}_{k}^{\dagger}{\cal A}_{\ell}= F \Big ( m_{N}, H - \sum_{\ell = 1}^{N - 1}{U}_{\ell}\, + 1 \Big ){\cal A}_{\ell}{\cal A}_{k}^{\dagger}.
M = \frac{i}{\sqrt{{\mathrm{\boldmath ~ \beta ~}}^{2}}}\left( \begin{array}{l l}{E_{\mathrm{\boldmath ~ \beta ~}}}{- E_{-{\mathrm{\boldmath ~ \mu ~}}}}{E_{\mathrm{\boldmath ~ \mu ~}}}{\, \, \, E_{-{\mathrm{\boldmath ~ \beta ~}}}}\end{array}\right) .
\frac{\phi}{\chi^{q_{1}/ q_{2}}}= C \frac{\prod_{r = 1}^{n_{1}}( z - Z_{r})}{\prod_{r = 1}^{n_{2}}( z - Z_{r}^{\prime})^{q_{1}/ q_{2}}}\, , \hspace{4 m m}\mathrm{( ~ C ~ : ~ a ~ c o m p l e x ~ n u m b e r )}.
\Gamma_{e f f}^{I R}[ g^{( 0 )}; \lambda ] = \sum_{n = 0}^{k - 1}{\frac{\lambda^{2 k - 2 n}}{2 k - 2 n}}\int d^{2 k}x \, \sqrt{g^{( 0 )}}\b_{n}( x ) + \log \lambda \int d^{2 k}x \, \sqrt{g^{( 0 )}}\, b_{2 k}( x ) +{\cal O}( \lambda^{0})
\Omega^{const}=\begin{pmatrix}0 (n-2)X_0X_1 (n-4)X_0 X_2 (n-6) X_0X_3 \hdots (2-n)X_0X_{n-1}&0 (n-2)X_1X_2 (n-4)X_1 X_3 \hdots (4-n)X_1X_{n-1}& 0 (n-2)X_2X_3 \hdots (6-n)X_2X_{n-1}&- \vdots & \ddots (n-2)X_{n-2}X_{n-1}& 0\end{pmatrix}
S_{N S}^{\mathrm{q u a d}}= \int A * Q_{B}A = \langle A^{\prime}| b_{0}Q_{B}b_{0}| A^{\prime}\rangle = \langle A^{\prime}| \{b_{0}, Q_{B}\}b_{0}| A^{\prime}\rangle = \langle A^{\prime}| L_{0}^{\mathrm{t o t}}b_{0}| A^{\prime}\rangle .
\int_0^T \int_{\mathbb{R}}[\rho_{3}(s,u)]^2 \rho_{4}(s,u) du ds + \frac{c_{\gamma}}{4}\iint_{\mathbb{R}^2}\frac{[ \int_0^T \rho_{5}(r,u) dr - \int_0^T \rho_{5}(s,v) ds ]^2}{|u-v|^{1 + \gamma}}du dv =0.
\left| \dfrac{\sqrt{s_n^2}}{\sqrt{\sigma_n^2}}- 1 \right| \leq\begin{cases}\dfrac{C'_{\alpha}}{n}(-1<\alpha< 0), begin{align*}4mm]\dfrac{C'_{\alpha}}{n^{1-2\alpha}}(0< \alpha<1/2), begin{align*}4mm]\dfrac{C'_{1/2}}{\log n}(\alpha=1/2).\end{cases}
\displaystyle\int_{\Omega}p\left|\nabla u\right|^{2}\geq \, 2\pi p_{0}\left(\sum_{i=1}^{m}d_{i}^{2}\right) \left( \log \dfrac{R}{R_{0}}- I\left(\dfrac{R}{R_{0}}\right)\right) +2\pi p_{0}\displaystyle\sum_{i\neq j}\left(-\left(1-a^{2}\right)\left|d_{i}\right|\left|d_{j}\right|+d_{i}d_{j}\right)\log \frac{R}{|a_{i}-a_{j}|}-C,
\sum_{k=0}^{n-1}\frac{\binom nk\binom{n+k}k(-1)^k}{2k+1}H_k=-\frac{H_n\binom{2n}n(-1)^n}{2n+1}+\frac{2}{2n+1}\sum_{k=1}^n\frac{(-1)^k}k
+\frac{e_{2}}{h_{3}h_{1}}[\frac{\partial}{\partial x_{3}}(h_{1}\psi_{1})-\frac{\partial}{\partial x_{1}}(h_{3}\psi_{3})]+\frac{e_{3}}{h_{1}h_{2}}[\frac{\partial}{\partial x_{1}}(h_{2}\psi_{2})-\frac{\partial}{\partial x_{2}}(h_{1}\psi_{1})]
H_{k'}^?, H_{\ell'}^?, S_{p_{i'},q_{i'},r_{i'}}, B'_2, \vec{K}, S_{p_m,q_m,r_m}, C_1 = \Psi_1, \Psi_2, \Psi_3; & \vdash \Psi_1 \to H_k^?; & \vdash \Psi_2 \to H_\ell^?; & \vdash \Psi_3 \to S_{p_i,q_i,r_i}.
\phi ( x ) = \frac{\displaystyle c \left( \sqrt{1 + \frac{4 x}{a}}+ 1 \right)}{\displaystyle 2 x}\left( 1 + \frac{\displaystyle c \left( \sqrt{1 + \frac{4 x}{a}}+ 1 \right)}{\displaystyle 2 a x}\right)
\frac{h}{r^{3}}\frac{d}{d r}( h r^{3}\frac{d R}{d r}) + [ \frac{( \omega r_{1}r_{K}r_{5})^{2}}{r^{6}}+ \frac{\omega^{2}( r_{1}^{2}r_{5}^{2}+ r_{1}^{2}r_{K}^{2}+ r_{5}^{2}r_{K}^{2})}{r^{4}}] R = 0 .
|I_2| \leq \int_{-\infty}^0\int_{M}^{M-y}M_1 J(y)dxdy+\int_{-\infty}^0\int_{0}^{-y}M_1J(y)dxdy & = 2M_1\int_0^\infty yJ(y)dy:=C_2<\infty,
\int_{\mathbb{R}^n}\frac{v(x)^{q}}{|x|^{\sigma_1}}(x\cdot \nabla v(x))\,dx={}-(n-\alpha)\int_{\mathbb{R}^n}\frac{v(x)^{q+1}}{|x|^{\sigma_1}}\,dx{}- (n-\alpha)\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(z\cdot (x-z))u(z)^{p}v(x)^{q}}{|x-z|^{n-\alpha + 2}|z|^{\sigma_2}|x|^{\sigma_1}}\,dz dx.
Z^I = Z^I_\infty + \sum_{a= 1}^{a=5}{q^I{}_a \over |\vec x - \vec{\underline{x}}_{a}|}\, \ , \qquad Q^5= \sum_{a= 1}^{a=5}q^5{}_a=Q^7= \sum_{a= 1}^{a=5}q^7{}_a=0 \ ,
\Phi_1(\omega S_1,\omega^2 S_2,\omega^3P) =9(I-P^*P)+(S_1^*S_1-S_2^*S_2)-6\text{Re}\omega(S_1-S_2^*P)\geq 0 \,, \Phi_2(\beta S_1,\beta^2 S_2, \beta^3P) =9(I-P^*P)+(S_2^*S_2-S_1^*S_1)-6\beta^2(S_2-S_1^*P)\geq 0 \,.
\theta_{mi}= \langle\psi_m,\phi_i \rangle = \int \cos(\omega_m \pi x) \cos(i \pi x) dx = \frac{i\pi}{(i\pi)^2 - \omega_m^2\pi^2}\cos(\pi \omega_m) \sin(i\pi) - \frac{\omega_m\pi}{(i\pi)^2 - \omega_m^2 \pi^2}\sin(\pi \omega_m) \cos(i\pi) = \frac{\pi \omega_m}{\pi^2 \omega_m^2 - (i\pi)^2}\sin(\pi \omega_m)(-1)^i.
\underset{0\leq t\leq T}{\sup}\Bigg[\frac{1}{N}\underset{i=1}{\overset{N}{\sum}}\mathbb{E}^{\boldsymbol{\xi}}[|\phi^{N,i}(t,\boldsymbol{x}^*_t) -\Psi^{N,i}(t,\boldsymbol{x}^*_t)|^2] \Bigg]\leq \frac{C}{N^2}\Big(1+\frac{1}{N^2}\underset{i,j=1}{\overset{N}{\sum}}|\xi^{i}-\xi^{j}|^2\Big)
p_{\beta,b}(a\cdot x,a\cdot y) = \beta(a,x)\psi_b(p_{\beta,b}(x,y))\beta(a,y)^{-1},\qquad p_{\beta,a}(a\cdot x,a\cdot y) =\beta(a,x)\psi_a(p_{\beta,a}(x,y))\beta(a,y)^{-1}.
\psi(y_{1},y_{2})=A\int_{-\infty}^{\infty}dpe^{-\frac{1}{4}p^{2}\sigma^{2}}e^{-\frac{i}{\hbar}py_{2}}e^{\frac{i}{\hbar}py_{1}}exp[-\frac{(y_{1}+y_{2})^{2}}{16\Omega^{2}}]
\frac{1}{k^{2}( k . n )^{m}}\rightarrow \frac{1}{k^{2}+ i \varepsilon}\left\{\frac{\Theta ( - k^{0})}{( k . n - i \xi )^{m}}+ \frac{\Theta ( k^{0})}{( k . n + i \xi )^{m}}\right\}, \, \, \begin{array}{c}{{\varepsilon \equiv 2 \epsilon k_{0}^{2}\rightarrow 0^{+}}}\\{{\xi \equiv \epsilon | k^{0}| n^{0}\rightarrow 0^{+}}}\\ \end{array},
S \ = \ \delta\left(\frac{m}{(m,M^\infty)},1\right) \frac{\varphi(M)}{M}\lim_{\epsilon \downarrow 0}\int_{-\infty}^{\infty}\phi(x\log R) \left(\frac{k\sqrt{Q}}{4\pi m}\right)^{4\pi i x}\frac{dx}{\epsilon -4\pi ix}+ O_{\epsilon'}\left( \frac{m^{\epsilon'}\log \log M}{\log R}\right).
\sum_{m^{(1)},\cdots, m^{(N)}}\exp \left[ -\pi \vec{m}^T M^{-1}\vec{m}+ 2\pi i \vec{m}^T \vec{x}\right] = \sum_{m^{(1)},\cdots, m^{(N)}}(\det M)^{\frac{1}{2}}\, \exp \left[-\pi(\vec{m}+\vec{x})^{T}M (\vec{m}+\vec{x})\right].
\big\|\{\langle k\rangle^{\frac{R(\mathbf{p},\mathbf{q},\alpha_1,\alpha_2)}{1-\alpha_1\vee \alpha_2}}\}\big\|_{l_{r}^{s_2-s_1,\alpha_1\vee \alpha_2}}=\left(\sum_{k\in \mathbb{Z}^n}\langle k\rangle^{r\big[\frac{s_2-s_1}{1-\alpha_1\vee \alpha_2}+\frac{R(\mathbf{p},\mathbf{q},\alpha_1,\alpha_2)}{1-\alpha_1\vee \alpha_2}\big]}\right)^{1/r}.
\begin{array}{c c}{{\cal Q}^{a}= \left( \begin{array}{c}{{\cal Q}_{+}^{a}}\\{{\cal Q}_{-}^{a}}\\ \end{array}\right)}&{\bar{{\cal Q}}_{a}={{\cal Q}^{a}}^{\dagger}\Gamma^{0}= ( \bar{{\cal Q}}_{a}^{-}, \bar{{\cal Q}}_{a}^{+})}\\ \end{array}
\frac{\partial}{\partial s}I(\alpha,\beta) = I_1(\alpha,\beta):=2\pi i \int\limits_{-\infty}^{\infty}\hat\Gamma\left(z\right)\cdot \left(-\frac{4\beta}{s}\cdot z^3-\frac{2\alpha}{s}\cdot z\right)\cdot e\left(-\beta z^3-\alpha z \right)dz,
t\log t =\frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\left(t-\frac{n}{d}\right)+\frac{1}{2}\xi^{-1}\left(t-\frac{n}{d}\right)^2 \leq \frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\left(t-\frac{n}{d}\right)+\frac{1}{2}\frac{n}{n/d-\sqrt{n}}\leq \frac{n}{d}\log\frac{n}{d}+\left(1+\log\frac{n}{d}\right)\left(t-\frac{n}{d}\right)+d.
\begin{cases}\displaystyle f_t+rf_r=f_{rr}+\frac{f_r}{r}-\frac{\sin(2f)}{2r^2},\ 0<r<\frac12,\t>0, f(r,0)\geq\Phi(r,0),\ 0<r<\frac12, f(0,t)=0,\ \ \f\left(\frac12,t\right)=\gamma, \quad\t>0.\end{cases}
\dot{r}^2+\tilde{V}(r)=0,\;\;\;\;\dot{t}=\frac{-E}{M-\frac{J^2}{4r^2}-\frac{r^2}{l^2}},\;\;\;\;\dot{f}=\frac{E}{\frac{l}{r^2}(M-\frac{J^2}{4r^2})-\frac{1}{l}}.
\Pr_{\rho\sim P_\epsilon}[ d(c+\rho)\neq c] \leq 2e^{-\frac{\sqrt{N}}{3\epsilon}}+N \mathop{\max}_{\substack{S\subseteq \{\epsilon N\pm N^{3/4}\}1\leq|S|\leq 2}}\Big\{\frac{1}{\binom{N}{S}}\sum_{j=0}^N \Pr_{c\sim C^\perp}\big[|c|=j\big] K_S(j)^2-1 \Big\},
I_{r,s,q}=\frac{1}{s!}\prod_{j=0}^{s-1}\frac{q!}{j!(j+q)!}\sum_{m_1=0}^{r-1}\cdots \sum_{m_s=0}^{r-1}\prod_{1\leq j<k \leq s}(m_j-m_k)^2\prod_{j=1}^{s}\binom{q+m_j}{q}\alpha^{m_j}.
\nu_{n}(c) =\sum\limits_{x=0}^{\infty}(x-n+1)_{n}\frac{( \vec{\alpha})_{x}}{( \vec{\beta}+1)_{x}}\frac{c^{x}}{x!}=\frac{(\vec{\alpha})_{k}}{( \vec{\beta}+1)_{k}}\,{}_{p}F_{q}\left[\begin{matrix}\vec{\alpha}+k \vec{\beta}+1+k\end{matrix};c\right] .
\int d^{2}\zeta_{S S}d^{2}\xi_{S S}\int d^{4}x_{0}\mathrm{T r}( \phi^{3}) = \frac{3}{2}\left( - \frac{g^{2}}{2^{4}\pi^{2}}\right) \left( \mathrm{T r}( A_{1}) - \frac{i g}{4 \sqrt{2}\pi^{2}\rho^{2}}\sum_{k = 3}^{N_{c}}( \xi_{M k}\zeta_{N k}+ \xi_{N k}\zeta_{M k}) \right) .
r^{2}( \lambda ) ={\frac{1}{( \widetilde{N}- \widetilde{M})^{2}}}\sum_{m , n = \widetilde{M}}^{\widetilde{N}}\left[{\frac{W_{\lambda m n}( N = 1 6 )}{W_{\lambda m n}( N = 2 0 )}}- 1 \right]^{2},
\bigl[(\bar{\nu}-\varepsilon\bar{\tau}_\varepsilon^i)\cdot\nabla\rho^\varepsilon\bigr](x,t)+k_d^{-1}\left[\Bigl(\overline{V_\Gamma}+\varepsilon\,\overline{\partial^\circ g_i}+\varepsilon^2\bar{g}_i\bar{\tau}_\varepsilon^i\cdot\overline{\nabla_\Gamma V_\Gamma}\Bigr)\rho^\varepsilon\right](x,t) = 0.
(n_1, \dots ,n_{\frac{d(d+1)}{2}})= (0,\underbrace{1,1}_{2},\underbrace{2,2,2}_{3},\ldots, \underbrace{d-2,d-2,\dots,d-2}_{d-1},\underbrace{d-1,d-1,\ldots,d-1}_{d})
B_{c o v}= i ( - \pi )^{\omega}p^{-}( p^{2})^{\omega - 2}\sum_{n = 0}^{\infty}\frac{\Gamma ( 3 - \omega + n ) \Gamma ( \omega - 1 + n )}{\Gamma ( \omega + n ) \Gamma ( n + 2 )}( 1 - \eta^{n + 1})
\Phi_0\left( T^{\psi}_{\varphi_i(h)}u - u, T^{\psi}_{\varphi_i(h)}u - u \right) = \Phi_0 (T^{\psi}_{\varphi_i(h)}u, T^{\psi}_{\varphi_i(h)}u) - \Phi_0(u,u) + 2 \Phi_0(u, T^{\psi}_{\varphi_i(h)}u - u).
D_1(w) :=2f'_0f'_1[\lambda f_0-(rw-c)f'_0]+[\lambda f_1-(rw-c)f'_1][(f'_0)^2+f''_1]+R(f'_1)^2, D_2(w) :=2f'_0f'_1[\lambda f_1-(rw-c)f'_1]+(f'_1)^2[\lambda f_0-(rw-c)f'_0], D_3(w) :=(f'_1)^2[\lambda f_1-(rw-c)f'_1].
\nabla g(\overline{x})\Big[D\mathcal{N}_{K}(g(\overline{x})|\lambda)(g'(\overline{x})d) - \frac{1}{2}\nabla\Upsilon(g'(\overline{x})d)\Big] \!=\mathcal{N}_{\mathcal{C}_{\Gamma}(\overline{x},\overline{v})}(d) \!=\nabla g(\overline{x})\mathcal{N}_{\mathcal{C}_{K}(g(\overline{x}),\lambda)}(g'(\overline{x})d)
{\cal G}({\bf x}) \approx{\frac{\beta m}{4 \pi^{2}\vert{\bf x}\vert}}\; K_{1}( m \vert{\bf x}\vert ) \approx{\frac{\beta m^{1 / 2}}{2^{5 / 2}\pi^{3 / 2}\vert{\bf x}\vert^{3 / 2}}}\; e^{- m \vert{\bf x}\vert}\; ,
\mathcal B_{n,k}\left(\sum_{1\leq i\leq a_1}\Psi_{\{[\{1\},i]\}}, \sum_{1\leq i\leq a_2}\Psi_{\{[\{1,2\},i]\}},\dots,\sum_{1\leq i\leq a_m}\Psi_{\{[\{1,\dots,m\},i]\}},\dots\right)=\sum_{\Pi\vDash n\atop \#\Pi=k}\Psi_\Pi.
\rho_{ij}=\max\left(\left| \sigma_{ij}^{(0)}\right|, \left|\sigma_{ij}^{(1)}\right|\right),A_{ij}^{(r)}= \int_{\sigma_{ij}^{(0)}}^{\sigma_{ij}^{(1)}}\frac{\left(1+ |h|\right)^{(n-r)/2}}{(1-h^2)^{\hat r/2}}\, dh.
\sum_{n = 1}^{\infty}\frac{1}{t^{2}+ n^{2}z^{2}/ 4}= \frac 1 2 \left[ \frac{2 \pi}{z t}\, \frac{1}{\mathrm{t h}( 2 \pi t / z )}- \frac{1}{t^{2}}\right] ,
\eta(p)a(p^rI)-\eta(p^{r+1})a(I) &=-\epsilon a\begin{pmatrix}p^{r-1}m &p^{r+1}m\end{pmatrix}-\epsilon \sum_{\substack{1\le u<p/2 u^2\not\equiv-1\,(p)}}a\left(p^rm\begin{pmatrix}(1+u^2)/p u u p\end{pmatrix}\right).
K_{\mathcal{U}_\eta}((\overline{w}_{\i})_{\i=1}^{\eta},(\overline{v}_{\i})_{\i=1}^{\eta})=~d(\mathcal{U}((\overline{w}_{\i})_{\i=1}^{\eta}),\overline{w}_\eta)+d(\mathcal{U}((\overline{v}_{\i})_{\i=1}^{\eta}),\overline{v}_\eta)
(a_{3,1}r^2 +a_{3,2}r)y^{(n+2)}+(a_{2,0}r^2+[a_{2,1}+2na_{3,1}]r+na_{3,2}+a_{2,2})y^{(n+1)}&+\left((2na_{2,0}-\tau_{1,0})r+n(n-1)a_{3,1}+na_{2,1}-\tau_{1,1}\right)y^{(n)}+\left(n(n-1)a_{2,0}-n\tau_{1,0}\right)y^{(n-1)}=0,n=0,1,\dots.
1- \sum_{k=0}^\infty \big(q^{\Delta_{4k+1}}+q^{\Delta_{4k+2}}\big) +\sum_{k=0}^\infty \big(q^{\Delta_{4k+3}}+q^{\Delta_{4k+4}}\big) = \Big(1+ \sum_{n=1}^\infty (-1)^n (q^{P_{5,n}}+ q^{Q_{5,n}})\Big)\cdot \Big( \sum_{n=1}^\infty q(n) q^{2n}\Big).
\begin{array}{ll}G_{i_0}^0(X(T)) = \int_{0}^{T}f_{i_0}^0(s,x(s),u(s))ds+g_{i_0}^0(x(T)) \null > G_{i_0}^0(X_0(T))=\int_{0}^{T}f_{i_0}^0(s,x_0(s),u_0(s))ds+g_{i_0}^0(x_0(T)).\end{array}
d s^{2}= g_{\alpha \beta}^{( 2 )}( t , r ) d x^{\alpha}d x^{\beta}+ r^{2}d \Omega^{2}+ g_{a b}^{( p - 1 )}( x^{c}) d x^{a}d x^{b}+ g_{k l}^{( 7 - p )}( x^{m}) d x^{k}d x^{l}\; ,
{d{f_1}}=\nabla.(D_{1}\nabla{{f_1}})dt-\nabla.(F_1f_1)dt+R_1(f_1,f_2,\beta_1)dt+R(f_1,f_2,s_1,s_2)dt+f_1\sum_{i=0}^nh_{i,1}(x,y)dB_i,
\{h_{\sigma_i},h_{\sigma_j}\}= \langle \Omega_{M^{[0,1]}}^{-1},d h_{\sigma_i}\wedge d h_{\sigma_j}\rangle = \int_0^1 (\frac{\delta h_{\sigma_i}}{\delta X^\mu}\frac{\delta h_{\sigma_j}}{\delta \eta_\mu}-\frac{\delta h_{\sigma_j}}{\delta X^\mu}\frac{\delta h_{\sigma_i}}{\delta \eta_\mu})dt= h_{[\sigma_i,\sigma_j]}\;.
K_{m+\alpha}((m+\alpha)z')\ \sim\ \sqrt{\frac{\pi}{2(m+\alpha)}}\frac{e^{-(m+\alpha)\eta '}}{(1+z'^2)^{1/4}}\left\{1+\sum_{j=1}^{\infty}(-1)^j \frac{u_j([1+z'^2]^{-1/2})}{(m+\alpha)^j}\right\}\ ,
U_\lambda(o,\, o\, |\, z) =\sum_{n=1}^\infty f^{(2n)}_\lambda(o,\, o)z^{2n}= \sum_{n=1}^\infty c_{n-1}\left(\frac{d-1}{d-1+\lambda}\right)^{n-1}\left(\frac{\lambda}{d-1+\lambda}\right)^n z^{2n}=\frac{\lambda}{d-1+\lambda}z^2 \, \mathcal{C}\left(\frac{\lambda(d-1)z^2}{(d-1+\lambda)^2}\right) ,
\sum_{s=1}^{n}l_{s}((j+1)\alpha_{s}+t_{s}\alpha_{s})\allowbreak+\allowbreak\sum_{s=n+1}^{n+k}l_{s}(j\alpha_{s}+t_{s}\alpha_{s})\allowbreak =\allowbreak j\sum_{s=1}^{n+k}l_{s}\alpha_{s}+\sum_{s=1}^{n}l_{s}(t_{s}+1)\alpha_{s}+\sum_{s=n+1}^{n+k}l_{s}t_{s}\alpha_{s}.
\vert J_3\vert \leqslant C_{\Omega ,\rho}^2\sum \limits_{j=1}^d \Vert a_j\Vert^2_{L_\rho (\Omega)}\Vert u\Vert^2_{H^1(\mathbb{R}^d)}+\int_{\mathbb{R}^d}\vert \mathbf{w}_k(\mathbf{x})\vert^2\vert \mathbf{D}u(\mathbf{x})\vert^2\,d\mathbf{x}.
\overset{n}{\underset{i=1}{\sum}}\frac{M^{\#}\left(\epsilon_{i}\right)}{\sigma_{i}^{2}}=\overset{n}{\underset{i=1}{\sum}}\frac{M^{\#}\left(\epsilon_{i}\right)}{\sigma_{i}^{2}}\cdot\frac{1}{n}\overset{n}{\underset{i=1}{\sum}}\sigma_{i}^{2}\geq\frac{1}{n}\left(\overset{n}{\underset{i=1}{\sum}}\sqrt{M^{\#}\left(\epsilon_{i}\right)}\right)^{2},
[r_{12}, r_{13}] = \sum_{i,j}[a_i, a_j] \otimes b_i \otimes b_j, [r_{13}, r_{23}] = \sum_{i,j}a_i \otimes a_j \otimes [b_i, b_j], [r_{12},r_{23}] = \sum_{i,j}a_i \otimes [b_i, a_j] \otimes b_j.
{I}_{l}^{k, \pm}=\int (\psi^{k, \pm}_l \psi_j\mathfrak a)(s)e^{i\lambda\mathcal P(x,y,s)}ds, \ \{I}_{\ast}=\int (\psi_\ast\psi_j\mathfrak a)(s)e^{i\lambda\mathcal P(x,y,s)}ds.
\sum_{i=0}^{\infty}m(A_{h_i}\cap (B^\delta_{h_i,h_0,M})^c) \le \sum_{i=0}^{\infty}\frac{2C_6(M+1)\sqrt{\lambda}}{\delta (K_1 +i K_2)^{\mathcal{K}\frac{\log \lambda}{2}}}< \infty.
\frac{\partial h_{rs}}{\partial \bar{z}_v}+\sum_{c=1}^n (h_{cs}\frac{\partial \varphi^{r}_v}{\partial z_c}-\varphi^c_v\frac{\partial h_{rs}}{\partial z_c}+h_{r c}\frac{\partial \varphi^{s}_v}{\partial z_c})=0\,\,\,\,\,.
\begin{array}{ll}{\mathcal E}'_{(V^{\pm}, \Phi^{\pm})}(W^{\pm},\Theta^{\pm})= (u^{a,\pm}+V_1^\pm)\partial_x\Theta^\pm+(w^{a,\pm}+V_3^\pm)\partial_z\Theta^\pm begin{align*}2mm] -W^\pm_2+\partial_{x}(\Psi^{a,\pm}+\Phi^{\pm})W_1^{\pm}+\partial_{z}(\Psi^{a,\pm}+\Phi^{\pm})W_3^{\pm}\end{array}
\sup_{s>0}\ \dfrac{\max_{[-s,s]}\, \tilde{f}}{\max\big\{\tilde{f}(-s),\tilde{f}(s)\big\}}\, \le \, c_0.\footnote{In particular, if $\tilde{f}(s)$ is monotone, then (H1)' is satisfied with $c_0=1$.}
v_{1}^{1}=y_{1,[2]}^{1,d}-a_{1}^{1,1}(\varphi_{1,[1]}^{1}-y_{1,[1]}^{1,d})-a_{1}^{1,0}(\varphi_{1}^{1}-y_{1}^{1,d}) v_{1,[1]}^{1}=y_{1,[3]}^{1,d}-a_{1}^{1,1}(v_{1}^{1}-y_{1,[2]}^{1,d})-a_{1}^{1,0}(\varphi_{1,[1]}^{1}-y_{1,[1]}^{1,d}) v_{2}^{1}=y_{2,[2]}^{1,d}-a_{2}^{1,1}(\varphi_{2,[1]}^{1}-y_{2,[1]}^{1,d})-a_{2}^{1,0}(\varphi_{2}^{1}-y_{2}^{1,d})
\begin{array}{c}\omega \epsilon E_B^3-\partial_1H_A^2+\partial_2H_A^1+\partial_3E_A^0=0, \omega \epsilon H_B^0+\partial_1H_A^1+\partial_2H_A^2+\partial_3H_A^3=0, -\omega \mu E_A^0+\partial_1E_B^1+\partial_2E_B^2+\partial_3E_B^3=0, \omega \mu H_A^3-\partial_1E_B^2+\partial_2E_B^1-\partial_3H_B^0=0,\end{array}
0= m(\cot \vartheta)^{4}-\frac{3}{2}(m+n)(\cot \vartheta)^{2}+n = (m(\cot \vartheta)^{2}-n)(m(\cot \vartheta)^{2}-1)-\left( \frac{m+n}{2}\right)(\cot \vartheta)^{2}.
E^{\bar{M}}_{tot}(\Sigma)={\lambda^2\over48\pi}\left[\int_{-\infty}^{v_1}\left(1-\partial_v\bar{v}\right)^2+\int_{v_1}^{v_2}\left(1-\partial_v\bar{v}\right)^2+\int_{v_2}^{\infty}\left(1-\partial_v\bar{v}\right)^2\right]
\begin{aligned}F(u) =\frac1{\sigma(u)}\mbox{for}u\in(0,\alpha], f(u) =0 \mbox{for}u\in(0,\beta], \frac{d}{du}\bigl(u^\kappa F(u)\bigr) =\kappa u^{\kappa-1}f(u-1) \mbox{for}u>\alpha, \frac{d}{du}\bigl(u^\kappa f(u)\bigr) =\kappa u^{\kappa-1}F(u-1) \mbox{for}u>\beta\end{aligned}
\left(\langle\underline t,\partial_{\underline x}\rangle^2+\Delta_{\underline y}\right)\big(g(\theta,\rho)\underline\nu P_{\ell}(\underline y)\big)=\left(\partial_{\theta}^2g+\partial_{\rho}^2g+(2\ell+q-1)\partial_\rho\left(\frac{g}{\rho}\right)\right)\underline\nu P_{\ell}.
\begin{aligned}Z_n^{a,s}= Z_n^{a,s}(\tilde\tau_1 > n) + \sum_{k=1}^n Z_n^{a,s}(\tilde\tau_1 = k) &\leq e^{n (F(0,h) + o(1))}+ \sum_{k=1}^n e^{k F(0,h)}Z^a_{n - k}.\end{aligned}
|\sqrt{\rho}u_t(t)|^2_2 +\int_{\tau}^{t}|\nabla u_t|^2_{2}s\leq |\sqrt{\rho}u_t(\tau)|^2_2+Cc^{8}_1\int_{\tau}^{t}|\sqrt{\rho}u_t|^2_2 s+Cc^{8}_1.
\left(\begin{array}{cc}(1-G^{-1}U^TZ^{-1}U)^{-1}G^{-1}-(1-G^{-1}U^TZ^{-1}U)^{-1}G^{-1}U^{T}Z^{-1}-(1- Z^{-1}UG^{-1}U^T)^{-1}Z^{-1}U G^{-1}(1- Z^{-1}U G^{-1}U^T)^{-1}Z^{-1}\end{array}\right)
\begin{array}{l}{{\mathcal D}_{V}f = V^{a}\nabla_{a}f + \frac{n - 2}{2 n}( \nabla_{a}V^{a}) f}{\delta_{V}f = V^{a}\nabla_{a}f + \frac{n + 2}{2 n}( \nabla_{a}V^{a}) f .}\end{array}
\widehat{\deg}(\overline L) \leqslant\widehat{\mu}_{\max}(\overline F^\vee)+\sum_{i=1}^d\widehat{\mu}_{\max}(\overline E_d)=\widehat{\mu}(\overline F^\vee)+\sum_{i=1}^d\widehat{\mu}_{\max}(\overline E_d) & =-\widehat{\mu}(\overline F)+\sum_{i=1}^d\widehat{\mu}_{\max}(\overline E_d)
\begin{cases}\tilde{u}_{\tilde{t}}=A\tilde{u}+\tilde{f}(x,\tilde{t}), \ \tilde{Q}_1, \frac{\partial \tilde{u}}{\partial \nu_{A}}=0, \ \partial\Omega\times(0,2\delta_1), \tilde{u}(x,0)=u(x,T-\delta_1), \ \Omega.\end{cases}
\begin{pmatrix}\varphi^{j}\psi^{j}\varphi_{s}^{j}- F^{j}\end{pmatrix}=\begin{pmatrix}x_{s}y_{s}z_{s}x_{t}y_{t}z_{t}x_{ss}y_{ss}z_{ss}\end{pmatrix}\begin{pmatrix}u^{j}v^{j}w^{j}\end{pmatrix}=M(s,t)\begin{pmatrix}u^{j}v^{j}w^{j}\end{pmatrix}\, .
e^{is\Delta}(\overline{e^{is\Delta}g_{K,\sigma}}e^{is\Delta}g_{K_1,h}\overline{e^{is\Delta}g_{K_2,h}}) =\frac{|\beta_h(s)|^2\overline{\beta_\sigma(s)}}{(2\pi)^6\big(1+4isz(s)\big)}&\quad\times e^{-\frac{z(s)}{1 + 4isz(s)}|x|^2 + \frac{1}{1 + 4i sz(s)}x \cdot \zeta(s) + \frac{is}{1 + 4i sz(s)}\zeta(s) \cdot \zeta(s)}e^{\gamma(s)}
f_{\beta}\left( T_{a}( z ) \right) = T_{a}^{\prime}( z ) \left[ f_{\beta}( z ) + \frac{( 1 - w_{a})}{w_{a}( \alpha_{a}- \beta_{a})^{2}}( z - \alpha_{a})^{2}\frac{\delta \beta_{a}}{\epsilon}\right] \ .
\Gamma\lbrack A\rbrack = \! -2\! \int_0^{\infty}{dT\over T}{\rm e}^{-m^2T}\int{\cal D}X{\rm exp}\left\{- \int_0^T \!\!\!\!d\tau\int\!\!d\theta \, \left[\!-{1\over 4}X\,D^3X \!-\!ie DX^{\mu}A_{\mu}(X) \right] \right\}
\sqrt{\int_{\widetilde{X}}|\widetilde{u}|_{\pi^*h}^2\,dV_{\pi^*\omega}}\leq M \sqrt{\int_{\widetilde{X}}|\pi^*v|_{\pi^*h, \pi^*\omega}^2\,dV_{\pi^*\omega}}=M \sqrt{\ell\cdot \int_{X}|v|_{h, \omega}^2\,dV_{\omega}}
\Gamma^{\hspace{0.2cm}\mu}_{abc}(p_{1}, k, -p_{2})_{GV{\overline{G}}}= f_{abc}\left[ p^{\mu}_{1}\tilde{F}_{(f)0}+ k^{\mu}\tilde{F}_{(f) 1}\right]+ d_{abc}\left[ p^{\mu}_{1}\tilde{F}_{(d)0}+ k^{\mu}\tilde{F}_{(d)1}\right] \, ,