harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\boldsymbol{W}_{p}( \nu (t_0), \nu (t_1) ) = \boldsymbol{W}_{p}\left( \nu (t_0), \gamma + \Delta \pi^\prime \right) & \leq (1-\Delta) \boldsymbol{W}_{p}\left( \nu (t_0), (1-\Delta)^{-1}\gamma \right) + \underbrace{\Delta \boldsymbol{W}_{p}\left( \nu (t_0) , \pi^\prime \right)}_{= O(\Delta)}. \end{aligned}
{\cal L}= \frac{p_{1 1}}{2}\left[ \int d p d q \dot{Y}^{i}( p , q ) \dot{Y}^{i}( p , q ) - \frac{1}{P_{1 1}}\int d p d q [ \partial_{q}Y^{i}\partial_{p}Y^{j}- \partial_{q}Y^{j}\partial_{p}Y^{i}] \right] ,
A=A^{}_{\pi_{2}}+a(r)\left(\theta^{2}\otimes e_{3}-\theta^{3}\otimes e_{2}+\theta^{4}\otimes e_{1}\right)+\left(b(r)+1\right)\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right).
T_{2n}= \underbrace{-0 + T_0}_{\Delta T_0}+ \underbrace{- T_0 + T_1}_{\Delta T_1}+ \underbrace{- T_1 + T_2}_{\Delta T_2}+ \dots + \underbrace{- T_{2n-1}+ T_{2n}}_{\Delta T_{2n}}= T_0 + \sum_{i = 0}^{2n}\Delta T_i,
F ( r ) ={\frac{5 r^{8}+ ( l^{2}- 2 8 n^{2}) r^{6}+ 5 n^{2}( 1 4 n^{2}- l^{2}) r^{4}+ 5 ( 3 l^{2}- 2 8 n^{2}) r^{2}- 1 0 m r l^{2}+ 5 n^{6}( l^{2}- 7 n^{2})}{5 l^{2}( r^{2}- n^{2})^{3}}}
\Pi_{h_{\prime}\; \rho \,}^{\rho \; \; \; \; , \, \mu \nu}= - \kappa \frac{a + 4 c + 2 d}{a \left( a + 4 b \right)}\; \Gamma_{h_{\prime}}^{\mu \nu}\equiv - \tilde{\Gamma}_{h_{\prime}}^{\mu \nu},
\frac{\partial t_{\alpha}(\theta)}{\partial\theta}= \biggl\{\frac{\beta' (({\partial F_0( t, \theta )})/({\partial\theta}))-(({\partial F_1( t, \theta )})/({\partial\theta}))}{(({\partial F_1 ( t, \theta )})/({\partial t}))- \beta'(({\partial F_0( t, \theta )})/({\partial t}))}\biggr\}\bigg|_{t=t_{\alpha}(\theta)}.
\Phi ( \omega ; \lambda ) \sim{\frac{2 \omega^{D - 2}}{( 4 \pi )^{( D - 1 ) / 2}}}\sum_{n = 0}^{\infty}{\frac{a_{n}( \lambda )}{\Gamma \left({\frac{D - 1}{2}}- n \right)}}\omega^{- 2 n}.
Z=\lbrace \mathbf{y}=(y_1,z_1,y_2,z_2,...,z_{k-2},y_{k-1}): y_i,z_i \in \mathbb{N}, 1 \leq y_1,...,y_{k-1}\leq 2^{k/3-\frac{k}{2\log k}}, 1 \leq z_i \leq i+1 \rbrace .
D^2u = \begin{pmatrix}f_{xx}+\sum\frac{(n_i^x)^2}{l_i}f_{xy}+\sum\frac{n_i^x n_i^y}{l_i}f_{xy}+\sum\frac{n_i^x n_i^y}{l_i}f_{yy}+ \sum\frac{(n_i^y)^2}{l_i}\end{pmatrix},
n+ n_1+r \leq\begin{cases}1-r+\frac{n^2+n}{2}\textrm{or}2^r\end{cases}\iff \begin{cases}n_1 +2 r\leq 1+\frac{n^2-n}{2}\textrm{or}n+ n_1+r \leq 2^r\end{cases}
Y^{(\mathbb{T})}=\Sigma^{1/2}X^{(\mathbb{T})}, \ \mathcal{G}_{1}^{(\mathbb{T})}=(Y^{(\mathbb{T})}Y^{(\mathbb{T})^*}-zI)^{-1}, \ \mathcal{G}_{2}^{(\mathbb{T})}=(Y^{(\mathbb{T})^{*}}Y^{(\mathbb{T})}-zI)^{-1}.
(6.2)d_{W}(L(W),N(0,1))\le2\sum_{i=1}^{n}(\frac{1}{2}E|X_{i}X_{A_{i}}^{2}|+E|X_{i}X_{A_{i}}X_{B_{i}\backslash A_{i}}|+E|X_{i}X_{A_{i}}|E|X_{B_{i}}|).
\sup_{\psi\in \Psi}-(f+\psi)^*_\Phi (0) - g^*_\Psi (\psi) =-f^{*}_\Phi \left(\phi_\varepsilon\right)-g^{*}_\Psi\left(\psi_\varepsilon\right) =\frac{80}{400}=\frac{4}{5}.
H(z+1) =\begin{vmatrix}f_1(z+1) f_2(z+1) \Delta f_1(z+1) \Delta f_2(z+1)\end{vmatrix}=\begin{vmatrix}f_1(z+1) f_2(z+1) f_1(z+2) f_2(z+2)\end{vmatrix}&=\begin{vmatrix}f_1(z+1) f_2(z+1) (A(z)+2)f_1(z+1)-f_1(z) (A(z)+2)f_2(z+1)-f_2(z)\end{vmatrix}&=H(z).
\sum_{m\in\mathbb{Z},\left\vert m\right\vert\leq k\delta}\mathrm{dim\,}H^0_{b,m}(X,L^k\otimes E) &=r(2\pi)^{-n-1}\frac{1}{n!}k^{n+1}\int_X\int_{[-\delta,\delta]}(i\mathcal{R}^L_x-sd\omega_0(x))^{n}\wedge\omega_0(x)ds+o(k^{n+1}),
{\cal A}_{A A \phi \phi}= - \frac{G_{o}^{2}}{\alpha_{e}^{\prime}}\frac{\Gamma ( - t ) \Gamma ( - s )}{\Gamma ( 1 - t - s )}\frac{1}{\sin ( \pi u )}\zeta \cdot \zeta^{\prime}\Big ( s u e \cdot e^{\prime}- 2 \alpha_{e}^{\prime}u e \cdot p_{2}e^{\prime}\cdot p_{3}- 2 \alpha_{e}^{\prime}s e \cdot p_{3}e^{\prime}\cdot p_{2}\Big ) \Phi_{+}
h_{2n+2|0}= \begin{bmatrix}h_{2n|0}\frac{1}{2}h_{2n|0}+ \frac{1}{2}\tilde{I}_{2n}h_{2n|0}+ 1_{2n}\frac{3}{4}h_{2n|0}+ \frac{1}{4}\tilde{I}_{2n}h_{2n|0}+ \frac{3}{2}1_{2n}\frac{1}{4}h_{2n|0}+ \frac{3}{4}\tilde{I}_{2n}h_{2n|0}+ \frac{3}{2}1_{2n}\end{bmatrix}.
\hskip -2mm \sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(2i-3)(i+j+1)}&=\sum_{i=0}^{p-1}\sum_{j=0}^{p-1}\frac{p^2(-1)^{i+j}}{(2i-1)(i+j+1)}+\sum_{j=1}^{p-1}\left(\frac{p^2(-1)^j}{3j}-\frac{p^2(-1)^j}{(2p-3)(j+p)}\right) &\quad{}+\sum_{i=0}^{p-1}\left(\frac{p^2(-1)^i}{i(2i-3)}+\frac{p^2(-1)^i}{(2i-3)(i+p)}\right),
T_\epsilon(V)|_H \;\cong\; \bigoplus_{r=0}^s \; T_{r,\epsilon}\textnormal{with}T_{r,\epsilon}\;\; = \!\!\! \bigoplus_{\epsilon_1 \cdots \epsilon_s = \epsilon \atop \# \{i \mid \epsilon_i = + 1\}= r}T_{\epsilon_1}(W) \boxtimes \cdots \boxtimes T_{\epsilon_s}(W)
d ( Y_{t}^{2}) = d Y_{t}\, Y_{t}+ Y_{t}\, d Y_{t}= 2 \, d Y_{t}\, Y_{t}+ \sigma^{2}\, d t = 2 \, \sigma \, d X_{t}\, Y_{t}+ d t \, ( \sigma^{2}- 2 k \, Y_{t}^{2})
\frac{1}{N^{s+1}}\int\limits_{\mathbf{w}\in\mathbb{R}^{s+1}}^* g_1(\psi_1(\mathbf{w})) \prod\limits_{k=1}^{s+1}b_k^\prime(\mathbf{w}) b_k(\mathbf{w})\, d\mathbf{w}.
g ( \tau ) = \frac{\vartheta_{2}^{4}( \tau )}{\vartheta_{3}^{4}( \tau ) + \vartheta_{4}^{4}( \tau )}, f ( \tau ) = \frac{\vartheta_{2}^{4}( \tau ) \vartheta_{4}^{4}( \tau )}{( \vartheta_{2}^{4}( \tau ) - \vartheta_{4}^{4}( \tau ) )^{2}},
(4s^{2}-1)M_{N}^{(1)\prime\prime}+(16s+2(\alpha-1)+4N)M_{N}^{(1)\prime}+(9-\alpha^{2})M_{N}^{(1)}&+(1-s^{2})M_{N-1}^{(2)\prime\prime}-(5s+2\alpha-1+4N)M_{N-1}^{(2)\prime}+(\alpha^{2}-4)M_{N-1}^{(2)}=0\ .
\frac{\partial^2 C^2}{\partial z^i_{\epsilon}\partial z^j_{\eta}}=\frac{\partial}{\partial z^j_{\eta}}\left( 2C\frac{\partial C}{\partial z^i_{\epsilon}}\right)=2\frac{\partial C}{\partial z^i_{\epsilon}}\frac{\partial C}{\partial z^j_{\eta}}+2C\frac{\partial^2 C}{\partial z^i_{\epsilon}\partial z^j_{\eta}}.
{N_{{\rm{S,}}k}}= \left\lceil \frac{{16{\pi^2}{d_{{\rm T},k}}d_{{k,{\rm R}}}^{\rm{r}}}}{{{\lambda^2}}}\sqrt{\frac{{\left({{\kappa_{{\rm{T}},k}}+ 1}\right){L_{{k,{\rm{R}}}}}{\xi_0^\star}}}{{{\kappa_{{\rm{T}},k}}{N_{\rm{T}}}{N_{\rm{R}}}}}}\right\rceil,
{I_{D}}^{\mathrm r}\equiv-VT{{\mathrm{tr}}{\bf1}\over2(4\pi)^{D/2}}\int_{1/\Lambda^{2}}^{\infty}\!d\tau\,\tau^{-{D\over2}-1}\left\{{\mathrm e}^{-\tau(m^{2}+\epsilon)}G_{D}(\tau F)-1\right\}\ .
H^{1}( d P_{9},{\cal O}_{d P_{9}}( 6 \sigma |_{d P_{9}}- 5 F ) ) |_{b_{- 8}}\stackrel{- m_{1}g_{2}}{\rightarrow}H^{1}( d P_{9},{\cal O}_{d P_{9}}( 9 \sigma |_{d P_{9}}- F ) ) |_{c_{- 3}}.
\begin{array}{ll}c \!\! \!\! \displaystyle = \frac{2[C(b+\frac{1}{a})+(\epsilon_1 M_2(R)+C_{\epsilon_1}) ]}{\sigma}+ b \!\! \!\! \displaystyle \le C [(\epsilon_1+a)M_2(R)+a(M^\prime_2(R))^2 +\frac{1}{a}] + C_{\epsilon_1}+\frac{C^\prime}{R^2},\end{array}
H(\nabla v)-H(\nabla \sigma_t)\ge \frac{d}{ds}H(\nabla\widetilde\sigma_s)_{|s=0}=\frac{d}{ds}H(\nabla \sigma_{t+s\,(1-t})_{|s=0}&=\frac{d}{ds}H(\nabla \sigma_{s})_{|s=t}\, (1-t)
= \sum_{n=1}^{\infty}n^4 \mathbb{E}\int_0^T ||\Phi_{n+1}(s)- \Phi_{n}(s)||^2_{\mathcal{L}(K_Q,H)}dE_s \leq \frac{T^{\beta}}{\Gamma(\beta+1)}\sum_{n=1}^{\infty}\frac{n^4}{2^n}< \infty.
(B\rtimes (h \ltimes (b_{ij})_{i \rho j})) \cdot (A\rtimes (g \ltimes (a_{ij})_{i \rho j}))= (B \;\;^{h^{-1}}((b_{ij})_{i \rho j}\cdot A)^{h^{-1}}) \rtimes (hg \ltimes(b_{\tilde{g}(i) \tilde{g}(j)}a_{ij})_{i \rho j}).
\lim_{k \to \infty}(t_{k_0+k}- t_{k_0}) = \lim_{k \to \infty}\sum_{i=k_0+1}^{k_0+k}h_i \le \lim_{k \to \infty}\frac{r}{c}\sum_{i=k_0+1}^{k_0+k}\frac{1}{\dot{\alpha}(t_{i-1})}.
\theta_{\mu}\left( \frac{z}{\tau}; -\frac{1}{\tau}\right) = \frac{(-i\tau)^{\lambda +\frac{r}{2}}}{\sqrt{|L'/L|}}e^{\pi i Q(A^{-1}A^{*})}\sum_{\nu \in L'/L}e^{-2 \pi i B(\mu, \nu) + 2 \pi i \frac{Q(z)}{\tau}}\theta_{\nu}(z; \tau) \theta_{\mu}(z; \tau + 1) = e^{2 \pi i Q\left(\mu + \frac{1}{2}A^{-1}A^{*}\right)}\theta_{\mu}(z; \tau),
-T_0^3T_2 - 2T_0^2T_1T_2 + 6T_0^2T_2^2 - 2T_0T_1^3 + 9T_0T_1^2T_2 + 12T_0T_1T_2^2 - 3T_0T_2^3 + T_1^4 + 6T_1^3T_2{}+ 5T_1^2T_2^2 - 4T_1T_2^3 - T_2^4 = 0
A=\int_{\Sigma}d\xi^1\wedge d\xi^2\left\{\big(\mathrm{det}[\partial_a (X+\delta y).\partial_b(X+\delta y)]\big)^{1/2}+\big(\mathrm{det}[\partial_a(X-\delta y).\partial_b(X-\delta y)]\big)^{1/2}\right\}
J[\xi_1^{y,s,\varepsilon},p](\hat{x},\hat{t})+K_{(0,\hat{x})}[\xi_1^{y,s,\varepsilon},p](\hat{x},\hat{t}) &=-(M_1+M_2+\|f\|_\infty)(J[\rho,\rho'(\hat{x})](\hat{x},\hat{t})+K_{(0,\hat{x})}[\rho,\rho'(\hat{x})](\hat{x},\hat{t})) &=-(M_1+M_2+\|f\|_\infty)(D_x^\alpha\rho)_x(\hat{x}) &=M_1+M_2+\|f\|_\infty.
(a_0,\dots, a_{N_{k-1}})(S^{[k]})^{-1}(\mathcal Q(\boldsymbol J))^{[k,m]}\hat S^m_k+(a_0,\dots, a_{N_{k-1}})(S^{[k]})^{-1}(\mathcal Q(\boldsymbol J))^{[k]}\big(S^{[k]}\big)^{-1}=(c_1,\dots,c_{r_{k,m}})\hat S^m_k, (a_0,\dots, a_{N_{k-1}})(S^{[k]})^{-1}(\mathcal Q(\boldsymbol J))^{[k,m]}\tilde S^m_k =(c_1,\dots,c_{r_{k,m}})\tilde S^m_k.
c_{b}= \frac{2}{n+1-2b}\left(z_b - \frac{1}{n-1-2b}\sum_{i=b+1}^{n-b-1}z_i\right) =\frac{\sum_{i=b}^{n-b}z_i}{n+1-2b}- \frac{\sum_{i=b+1}^{n-b-1}z_i}{n-1-2b}.
\theta_1 =\frac{1}{2}(\arg\alpha_1-\arg\alpha_2+\arg\beta_1-\arg\beta_2), \delta_1 =u(\theta_1) \theta_2 =\frac{1}{2}(\arg\alpha_1-\arg\alpha_2-\arg\beta_1+\arg\beta_2), \delta_2 =u(\theta_2)
\delta_{i_1 i_2}[\delta_{\pi(i_n i_1}D_{i_2 \cdots i_{n-1}) k}] = \delta_{\pi(i_n 1}D_{1 \cdots i_{n-1}) k}+ \delta_{\pi(i_n 2}D_{2 \cdots i_{n-1}) k}+ \delta_{\pi(i_n 3}D_{3 \cdots i_{n-1}) k}= D_{\pi(i_n i_3 \cdots i_{n-1}) k}.
\mathcal{B}(z,\{\overline{\alpha}\}) =\widetilde{B}(z,\{\alpha \}){}_a \langle 0| K(z,t,\alpha_0)|0 \rangle_a A(z)+\widetilde{A}(z,\{\alpha \}){}_a \langle 1| K(z,t,\alpha_0)|1 \rangle_a B(z,t,\{\alpha \}) &=(tz-\alpha_0) \widetilde{B}(z,\{\alpha \})A(z,\{\alpha \})+(z^{-1}+\alpha_0) \widetilde{A}(z,\{\alpha \}) B(z,\{\alpha \}).
L_0 =\sum_{i=1}^\ell \sum_{m =1}^\infty x_{i,m}\bigg(m\partial_{x_{i,m}}-(1-\delta_{i,1})\partial_{x_{i-1,m}}\bigg) & +\sum_{i=1}^\ell\sum_{m =1}^\infty x_{\ell+i,m}\bigg(m\partial_{x_{\ell+i,m}}+(1-2\delta_{i,1})\partial_{x_{\ell+i-1,m}}\bigg) &+\sum_{i=2}^\ell x_{\ell+i,0}\partial_{x_{\ell+i-1,0}}+ \frac12 x_{\ell+1,0}^2-\frac{\ell}{8}I.
\lim_{x\to+\infty}\frac{f_{\mathcal G_I^0}(x+\varepsilon)}{f_{\mathcal G_I^0}(x)}=\lim_{x\to+\infty}\Big(\frac{x+\varepsilon}{x}\Big)^{L-1}\Big(\frac{Lx+\gamma}{L(x+\varepsilon)+\gamma}\Big)^{L-\alpha}& = 1,
I_3 =\int_0^r \left[(\bar\phi+e_\delta-\delta)(2-\delta+\bar\phi+e_\delta)(\bar\phi'',\bar\psi'')\cdot (\phi,\psi)\right]{\rm{d}}z \leq\Vert (\phi,\psi)\Vert_{L^2}\cdot \Vert (\bar\phi'',\bar\psi'')\Vert_{L^2}\cdot\Vert (\bar\phi+e_\delta-\delta)(2-\delta+\bar\phi+e_\delta)\Vert_{L^\infty},
\Theta_{\lambda, T}(\tau, z, t) = e^{2 \pi int}\sum_{\gamma \in \frac{\bar{\lambda}}{n}+M}^{}\epsilon^\pm (t_\gamma) \frac{q^{n \frac{(\gamma | \gamma)}{2}}e^{2 \pi i n \gamma (z)}}{\prod_{\beta \in T}^{}(1 \pm q^{-(\gamma| \beta)}e^{-2\pi i \beta (z)})}.
\mathrm{mod}_p(d\varphi,\iota;l)=\left(\frac{2}{t}\sum_{\{\alpha,\beta\}\in\mathcal{S}^+_{(d\varphi,\iota)}}+\frac{1}{t}\sum_{\{\alpha,\beta\}\in\mathcal{S}^0_{(d\varphi,\iota)}}\right)\left(e^{\frac{1}{2}(\ell_{\alpha}(\phi)+\ell_{\beta}(\phi))}+(-1)^{\alpha\cdot\beta}\right)^{-1}- \frac{k_l\cdot\mathrm{gcd}(s,t)}{t}.
\begin{gathered}A_0(a_i)^2\subseteq A_0(a_i),A_{\eta}(a_i)^2\subseteq A_0(a_i)+A_1(a_i), (A_0(a_i)+A_1(a_i))A_{\eta}(a_i)\subseteq A_{\eta}(a_i),A_0(a_i)A_1(a_i) = (0).\end{gathered}
f_{1}= f_{2}= -{\frac{d}{2}}{\frac{1}{\mathrm{s i n h}( d x^{9})}}\ , \quad f_{3}= -{\frac{d}{2}}{\frac{\mathrm{c o s h}( d x^{9})}{\mathrm{s i n h}( d x^{9})}}\ .
\begin{aligned}\frac{u^2_1 \psi_u}{w^2 \ln w}- \frac{w^2 + 1}{u w^2 \ln w}\sum_{i,j}\frac{\partial F}{\partial a_{ij}}a_{ij}+ \frac{1}{u w^2}\sum_{i,j}\frac{\partial F}{\partial a_{ij}}a_{ij}\geq \left( 1 - \frac{2}{\ln w}\right)\frac{1}{u w^2}\sum_{i,j}\frac{\partial F}{\partial a_{ij}}a_{ij}\geq 0.\end{aligned}
\Gamma(x;y) = \frac{c\,\sqrt{2}}{\sqrt[4]{(x_1^2+y_1^2)^2+4\,(x_2-y_2)^2}}\cdot \mathrm{K}\left(\frac{1}{2}+ \frac{x_1y_1}{\sqrt[4]{(x_1^2+y_1^2)^2+4\,(x_2-y_2)^2}}\right),
\begin{array}{l}{{2 ( n - n_{1}) - 2 ( n_{2}+ \cdots + n_{j}) + \beta_{i - 1 j - 1}^{( k - 1 )}\geq a_{1}^{( j )}\geq \cdots \geq a_{n_{j}- n_{j + 1}}^{( j )}\geq 0 ,}}\\{{n - n_{1}- ( n_{2}+ \cdots + n_{k - 1}) \geq b_{1}\geq \cdots \geq b_{n_{k - 1}}\geq 0 ,}}\\ \end{array}
F_3\cup F_4 =(F_1\setminus F_2)\cup (F_2\setminus F_1)\cup\bigl[(F_3\cup F_4)\cap F_1\cap F_2\bigr]\cup\bigl[(F_3\cup F_4)\setminus( F_1\cup F_2)\bigr] &=(F_1\setminus F_2)\cup (F_2\setminus F_1)\cup\bigl[(F_3\cup F_4)\cap F_1\cap F_2\bigr]\cup\bigl[(F_3\cap F_4)\setminus( F_1\cup F_2)\bigr]\,,
\sup_{K_j}\frac{1}{|x- \lambda - i\delta|}= \frac{1}{\sqrt{\frac{(M\rho (0) + Y )^2}{t^2}+ 2^{-2j}\frac{1}{M\rho(0)^4 t^2}}}\leq 2^{j+1}t M\rho (0)^2 \sim |K_j|^{-1}.
t_2(2^k-4)= t_2(2^{k-1}-2)+t_2(2^{k-1}-3)=t_2(2^{k-2}-1)+t_2(2^{k-2}-2)-2t_2(2^{k-2}-2) = t_2(2^{k-2}-1)-t_2(2^{k-2}-2)=\frac{1}{3}\left((-2)^k-1)\right), t_2(2^k-5)= -2t_2(2^{k-1}-3)=4t_2(2^{k-2}-2)=\frac{1}{3}\left(4-(-2)^k\right)=1-t_2(2^k-4), t_2(2^k-3)= -2t_2(2^{k-1}-2)=\frac{1}{3}\left(-2-(-2)^k\right)=-1-t_2(2^k-4).
\Pr(X = k) \Pr(Y = \ell) = (1+o(1)) \frac{1}{n \sigma_p \sigma_q}\int_{\Delta(k,\ell)}\phi \left( \frac{y-nq}{\sqrt{n}\sigma_q}\right)\phi \left( \frac{x-np}{\sqrt{n}\sigma_p}\right) dx dy,
\chi_2(s) = \sum_{s^{-1/2}< m \leq s^{-1}}-\phi'(ms) & \leq O(s^{-u-1}) \sum_{s^{-1/2}< m < s^{-1}}\frac{1}{m^{u+1}}& = O(s^{-u-1}) O(s^{u/2}) & = O(s^{-u/2-1}),
\begin{aligned}s\int_{Q_T}\Theta (w_x)^2dxdt + s^3\int_{Q_T}\Theta^3 \left(\frac{x-x_0}{a}\right)^2 w^2 dxdt &\le C\left(\int_{Q_T}h^2 \frac{e^{2s\varphi(t,x)}}{a}dxdt+sd_1\int_0^T\Theta(t)\Big[(x-x_0)e^{R(x-x_0)^2}(w_x)^2\Big]_{x=0}^{x=1}dt \right). \end{aligned}
|f(x)-f(y)| \leq \sum_{n=1}^{M}|f\left( (n\delta+r) \cdot z +y\right)-f \left(((n-1)\delta+r ) \cdot z +y \right)| + |f(r \cdot z+y)-f(y)| & \leq\varepsilon \cdot ( M+1) \leq \varepsilon \cdot (\|x-y \| \cdot \delta^{-1}+1).
\bar{f}(t,x,\bar{u}(t,x),\nabla\bar{u}(t,x),\bar{v}(t,x)) =e^{Lt}f(t,x,e^{-Lt}\bar{u}(t,x),e^{-Lt}\nabla\bar{u}(t,x),e^{-Lt}\bar{v}(t,x))-L\bar{u}(t,x) \bar{g}(t,x,\bar{u}(t,x),\nabla\bar{u}(t,x),\bar{v}(t,x)) =e^{Lt}g(t,x,e^{-Lt}\bar{u}(t,x),e^{-Lt}\nabla\bar{u}(t,x),e^{-Lt}\bar{v}(t,x)).
\overline{U}_{2k,2a}(x;q)-\overline{U}_{2k,2a-2}(x;q) &=(xq)^{2a}U_{2k,2k-2a}(xq;q)+(xq)^{2a-2}U_{2k,2k-2a+2}(xq;q) &=(xq)^{2a}(-xq^2;q^2)_\infty \overline{Q}_{k,k-a}(x^2q^2;q^2)+(xq)^{2a-2}(-xq^2;q^2)_\infty \overline{Q}_{k,k-a+1}(x^2q^2;q^2).
\sigma_{}((I+\tilde{F}\tilde{F}^\top)^{-\frac{1}{2}}) = \frac{1}{\sigma_{}((I+\tilde{F}\tilde{F}^\top)^{\frac{1}{2}})}&= \frac{1}{\sqrt{\sigma_{}(I+\tilde{F}\tilde{F}^\top)}}&= \frac{1}{\sqrt{\lVert(I+\tilde{F}\tilde{F}^\top)\rVert_2}},
\|(fg+gf)^{2N}\| = \|P_n(fg) + P_n(gf) + Q_n(fgf) + Q_n(gfg)\| &\le 2 \|P_n(fg)\| + 2\|Q_n(fgf)\| &\le 2 \|fg\|^{2N-1}A_N(\|fg\|) + 2\|fg\|^{2N}B_N(\|fg\|) &= 2 a^{2N-1}A_N(a) + 2 a^{2N}B_N(a)
\alpha(y) = -\sum_{n=1}^{N(x,h)}\frac{v(f^{n-1}(y))}{Df^n(y)}-\frac{1}{Df^{N(x,h)}(y)}\sum_{n> N(x,h)}^{\infty}\frac{v(f^{n-1}(y))}{Df^{n-N(x,h)}(f^{N(x,h)}(y))}&= -\sum_{n=1}^{N(x,h)}\frac{v(f^{n-1}(y))}{Df^n(y)}+ O(h),
H[\varphi,\pi]=\frac{1}{2}\sum_{p}\left[ \pi_p \pi_{-p}+ p^2 \varphi_p \varphi_{-p}\right]+{\lambda\over 4!V}\sum_{p_1 p_2 p_3 p_4}\varphi_{p_1}\varphi_{p_2}\varphi_{p_3}\varphi_{p_4}~\delta_{p_1+p_2+p_3+p_4,0}~,
\langle 0 | T_{\mu \nu}| 0 \rangle_{\mathrm{\tiny R e n .}}= \mathrm{d i a g}\left[ \langle 0 | T_{t t}| 0 \rangle , 0 , \langle 0 | T_{l l}| 0 \rangle , \langle 0 | T_{l l}| 0 \rangle \right] .
\frac{\partial}{\partial r}h(r,s) =\frac{d^{-1}(\log^{(N)}(r\exp^{(N)}(s^d)))^{1/d-1}}{r\prod_{p=1}^{N-1}\log^{(p)}(r\exp^{(N)}(s^d))}, \frac{\partial}{\partial s}h(r,s) =\frac{s^{d-1}(\log^{(N)}(r\exp^{(N)}(s^d)))^{1/d-1}\prod_{p=1}^{N-1}\exp^{(p)}(s^d)}{\prod_{p=1}^{N-1}\log^{(p)}(r\exp^{(N)}(s^d))}.
\left( A^{\prime}e^{n A}\right)^{\prime}={\frac{2 \pi G_{5}}{3}}e^{n A}\left( T_{\mu}{}^{\mu}+ ( 2 n - 4 ) T_{5}{}^{5}\right) -{\frac{1 - n}{1 2}}W^{n - 2}R_{g}.
I_2 =\sum_{\substack{(m_{ij})\in SSYT(\lambda) (m)\in SSYT(e^1)}}\frac{1}{m(m+m_{11})^2(m+m_{21})^2m_{12}^{3}}, I_3=I_4 =\sum_{\substack{(m_{ij})\in SSYT(\lambda) (m)\in SSYT(e^1)}}\frac{1}{m(m+m_{11})(m+m_{21})^3m_{12}^{3}}.
\Lambda : = [N] = \{1, 2, \cdots, N \}, \Lambda_{1}: = \bigcup_{l \in \mathbb{Z}}\Lambda \cap \left( \left[1, K-R \right] + (l-1)K \right) = \bigcup_{l =1}^{M}\Lambda_1^{(l)}, \Lambda_{2}: = \bigcup_{l \in \mathbb{Z}}\Lambda \cap \left( \left[ K-R+1 , K \right] + (l-1)K \right) = \bigcup_{l =1}^{M}\Lambda_2^{(l)},
d s^{2}= a^{2}( d \psi + p \cos \theta_{1}d \phi_{1}+ q \cos \theta_{2}d \phi_{2})^{2}+ b^{2}( d \theta_{1}^{2}+ \sin^{2}\theta_{1}d \phi_{1}^{2}) + c^{2}( d \theta_{2}^{2}+ \sin^{2}\theta_{2}d \phi_{2}^{2}) ,
q_i(t_1, \dots, t_n) = \det \begin{pmatrix}F(t_1)t_1^i F(t_2)t_2^i \dots F(t_n)t_n^i t_1^{n-2}t_2^{n-2}\dots t_n^{n-2}\vdots \vdots \ddots \vdots t_1 t_2 \dots t_n 1 1 \dots 1\end{pmatrix}
a_n^{\lambda}=\frac{\left(\lambda\right)_{n}}{\left(1\right)_{n}}\frac{\left(\lambda+\frac{1}{2}\right)_{n}}{\left(\frac{1}{2}\right)_{n}}, b_n^{\lambda}=\frac{2\lambda\left(\lambda+1\right)_{n}}{\left(1\right)_{n}}\frac{\left(\lambda+\frac{1}{2}\right)_{n}}{\left(\frac{3}{2}\right)_{n}}.
sNE\left( X_{1},...,X_{k};\pi_{1},...,\pi_{k}\right) =\left\{z\in \underset{i=1}{\overset{k}{\prod}}\left[ 0,d_{S_{i}}\right] :\overset{k}{\underset{i=1}{\sum}}z_{i}=r\ \ \forall i\in \left\{1,...,k\right\},\z_{i}>0\right\}.
f\big(\rho(\eta_t(z))\big) =\varphi(f)(\eta_t(z))=\beta_{t^{-1}}(\varphi(f))(z)=v_{t^{-1}}\varphi(f)v_{t^{-1}}^*=\varphi(\alpha_{t^{-1}}(f))(z) &=\alpha_{t^{-1}}(f)(\rho(z))=f\big(\theta_t(\rho(z))\big).
|\phi(k)| \leq e^{24\varepsilon q_n}\frac{e^{\beta_n q_n}}{\max(|\ell|,1)}\max\begin{cases}c_{n,\ell-1}g_{|x_2-k|, \ell}e^{-q_n L}r_{\ell-1}g_{|x_2-k|, \ell}e^{-q_n L}r_{\ell}c_{n,\ell}(c_{n,\ell+1})^2 e^{-q_n L}r_{\ell+1}c_{n,\ell}c_{n,\ell+1}e^{-2q_n L}r_{\ell+2}\end{cases}
[\begin{matrix}c_{3}c_{1}+s_{3}s_{2}s_{1}&-c_{3}s_{1}+s_{3}s_{2}c_{1}&s_{3}c_{2}\\c_{2}s_{1}&c_{2}c_{1}&-s_{2}\\ -s_{3}c_{1}+c_{3}s_{2}s_{1}&s_{3}s_{1}+c_{3}s_{2}c_{1}&c_{3}c_{2}\end{matrix}]
G(\gamma(T),m_1(T))-G(\gamma(T),m_2(T))\leq u_1(0,x_0)-u_2(0,x_0) &-\int_0^{T}\Big[L(\gamma(s),\dot \gamma(s))+ F(\gamma(s), m_1(s))\Big]\ds + \int_0^{T}\Big[L(\gamma(s),\dot \gamma(s))+ F(\gamma(s), m_2(s))\Big]\ds &=u_1(0,x_0))-u_2(0,x_0)+\int_0^{T}F(\gamma(s), m_2(s))-F(\gamma(s), m_1(s))\ds.
\lambda_l(x'^\rho_i,y'^\rho_i;x^\nu_a,y^\nu_a)\,x'^\mu_k\,-\lambda_l(x'^\nu_i,y'^\nu_i;x^\nu_a,y^\nu_a)\,x^\mu_k &= \,\tilde{\lambda}_l(x'^\rho_i,y'^\rho_i;x^\nu_a,y^\nu_a)\,\tilde{x}'^\mu_k -\tilde{\lambda}_l(x'^\nu_i,y'^\nu_i;x^\nu_a,y^\nu_a)\,\tilde{x}^\mu_k .
B_{k,\ell}\left(x_1,x_2,\cdots,x_{k-\ell+1}\right)=\sum_{\left\{j_i\right\}}\frac{k!}{j_1!j_2!\cdots j_{k-\ell+1}!}\left(\frac{x_1}{1!}\right)^{j_1}\left(\frac{x_2}{2!}\right)^{j_2}\cdots \left(\frac{x_{k-\ell+1}}{(k-\ell+1)!}\right)^{j_{k-\ell+1}},
g(t) = \frac{2\arcsin(1/r)}{\pi^2 \sqrt{1-t^2}}- \frac{1}{\pi^2 t}\log\left|\frac{(1/r-t)(1+t/r+\sqrt{1-t^2}\sqrt{1-r^{-2}})}{(1/r+t)(1-t/r+\sqrt{1-t^2}\sqrt{1-r^{-2}})}\right|.
\left| \nabla_{y'}^\beta \partial_{y_d}r'_\lambda(y',y_d,z_d)\right|+\left| \nabla_{y'}^\beta \partial_{y_d}r_{d,\lambda}(y',y_d,z_d)\right| & \leq \frac{C}{(y_d + z_d +|y'|)^{d-1+\beta}}\frac{e^{-c|\lambda|^{\frac 12}z_d}}{\big (1+|\lambda|^{\frac 12}(y_d+z_d + |y'|) \big ) \big ( 1+ |\lambda|^\frac12 ( y_d +z_d) \big )},
[{\cal F},{\cal H}_{A}] = \frac{\delta{\cal F}}{\delta g_{i j}}\frac{\delta{\cal H_{A}}}{\delta p^{i j ( A )}}- \frac{\delta{\cal F}}{\delta p^{i j ( A )}}\frac{\delta{\cal H_{A}}}{\delta g_{i j}}= \frac{\delta{\cal F}}{\delta s^{A}},
I_{k}\cap B_{k-1}=\big(\bigcap_{i=1}^{k-1}A_{k-1}^{(i)}\big)\cap B_{k-1}=\bigcap_{i=1}^{k-1}(A_{k-1}^{(i)}\cap B_{k-1}) &=\big(\bigcap_{i=1}^{k-2}A_{k-2}^{(i)}\big)\cap B_{k-2}=I_{k-1}\cap B_{k-2}.
M_N(a, b, \lambda) \triangleq_{\{t_1\cdots t_N\}}\prod\limits_{j=1}^N (1-t_j)^a(1-1/t_j)^b \prod\limits_{k<l}^N \Bigl(1-\frac{t_k}{t_l}\Bigr)^\lambda\Bigl(1-\frac{t_l}{t_k}\Bigr)^\lambda,
\left(\frac12 a_1 + \frac12 b_1\right)^3 - r_1^3 = \left(\frac12 a_1 + \frac12 b_1\right)^3 - \frac14 a_1^3 - \frac34a_1b_1^2 &= \left( \frac12 b_1 - \frac12 a_1\right)^3 > 0
g{h_{2}}^{2}v+{h_{2}}^{2}vg=({h_{2}}^{2}-{h_{1}}^{2})ug+h_{2}(u-v)v^{2}+\frac{2U}{\rho\Delta t}+0.25h_{1}\frac{g^{2}({h_{2}}^{2}-{h_{1}}^{2})}{u}
{\tilde{\mu}}= \frac{- 4 e^{G / 2}\partial_{T}\{\log \eta ( T ) ) \log \eta ( \frac{T}{3}) \}\partial_{U}\{\log \eta ( \frac{U}{3}) \log \eta ( U ) \}}{9 \eta^{2}( T ) \eta^{2}( \frac{U}{3}) \eta^{- 2}( U ) \eta^{- 2}( \frac{T}{3})}.
W'_1 = \{k \in \{1,\ldots, n\}\, \mid \, k \prec_\chi \alpha_1 \}, W'_2 = \{k \in \{1,\ldots, n\}\, \mid \, \beta_1 \prec_\chi k\}, W''_1 = \{k \in \{1,\ldots, n\}\, \mid \, \alpha_2 \prec k \preceq_\chi \theta\}, \mbox{and}W''_2 = \{k \in \{1,\ldots, n\}\, \mid \, \gamma \preceq_\chi k \prec_\chi \beta_2\}.
\frac{\lambda_\mathrm{max}(\Phi_J^*\Phi_J)}{\lambda_\mathrm{min}(\Phi_J^*\Phi_J)}\geq\frac{\|\Phi_J u\|_2^2}{\|\Phi_J v\|_2^2}>\frac{1}{4}\cdot\frac{\lambda_\mathrm{max}(\Sigma_{JJ})}{\lambda_\mathrm{min}(\Sigma_{JJ})}=\frac{J+1}{4}
- \frac{1}{2}\sum_{m , n = - \infty}^{\infty}( m - n ) : ( \tilde{c}_{m}\tilde{c}_{n}\tilde{b}_{m + n}- \tilde{c}_{m}\tilde{c}_{n}b_{- ( m + n )}) : | B ; p , 0 \rangle = 0 \, ,
\frac{\partial}{\partial \theta_{k-1}}\phi_{-}(\alpha k+\alpha-1,\theta_{k-1},\omega_k,E)=\frac{\sin^2\phi_{-}(\alpha k+\alpha-1,\theta_{k-1},\omega_k,E)}{\cos^2\phi_{-}(\alpha k+\alpha-2,\theta_{k-1},\omega_k,E)}\frac{\partial}{\partial \theta_{k-1}}\phi_{-}(\alpha k+\alpha-2,\theta_{k-1},\omega_k,E).
\lim_{n\to \infty}\int_{\mathbb{R}^N}|\nabla v_n|^2dx=\int_{\mathbb{R}^N}|\nabla \varphi|^2dx, \lim_{n \to \infty}\int_{\mathbb{R}^N}A(|v_n(x)|)dx=\int_{\mathbb{R}^N}A(|\varphi(x)|)dx.
\mathcal{E}_{s,\Lambda}^{\rm cp}(m^d)\leq E_{s,\Lambda}^{\rm cp}(\omega^m)=\sum_{\substack{x_j,x_k\in \frac{1}{m}\Lambda\cap\Omega_\Lambda x_j\neq x_k}}\zeta_\Lambda(s;x_j-x_k)=m^d(m^s-1)\zeta_\Lambda(s),
{\cal L}_1^m{}[\Lambda] = -{1 \over 4}m^2 (\partial_\mu \Lambda_\nu - \partial_\nu \Lambda_\mu)^2+{\alpha_1 m^2 \over 2}(\Lambda_\mu)^2+{\alpha_1 m^2 \over 2}(\partial^\mu \Lambda_\mu)\Delta^{-1}(\partial^\nu \Lambda_\nu) .
4 c_{b}\lambda(t)\int_{0}^{\infty}d\xi \frac{\xi \cos(t\xi)}{t^{2}}\partial_{2}\psi_{v_{2}}\lambda'(t) =-4 c_{b}\lambda(t) \int_{0}^{\infty}d\xi \frac{\cos(t\xi)}{t^{4}}\partial_{\xi}^{2}\left(\xi \partial_{2}\psi_{v_{2}}\right) \lambda'(t)
\frac{\prod_{j=1}^{s}\det\left( \Sigma_{C_{j}}\right)^{\alpha_{j}+\frac{c_{j}+1}{2}}}{\prod_{j=2}^4 \det\left( \Sigma_{S_{j}}\right)^{\beta_{j}+\frac{s_{j}+1}{2}}}\propto \prod_{j=1}^{8}D_{jj}^{\gamma_j}\Sigma_{66}^{\eta},
\eta_{(2)}(\theta) = \begin{cases}\hat u +D_1\delta\theta, \theta\in\bigl[-\frac{\delta+\hat u}{D_1\delta},0 \bigr], - \delta, \theta\in\bigl[-a-c\delta,-\frac{\delta+\hat u}{D_1\delta}\bigr],\quad\frac{\delta+\hat u}{D_1\delta}<r_+=a+c\delta.\end{cases}
\lambda_{A}^{i}= \epsilon^{A B}\, \mathcal{C}\, \left( \overline{{{\lambda}}}^{i B}\right)^{T}\, \quad ; \quad \overline{{{\lambda}}}^{i B}= \left( \lambda_{B}^{i}\right)^{\dagger}\gamma_{0}\quad ; \quad A , B = 1 , \dots , 2 \, .
y_{1}= \rho_{0}^{2}\mathrm{e x p}\left( 1 -{\frac{A}{B}}\right) \, , \, y_{\chi}= \sqrt{e}y_{1}\, , \, \chi = 1 +{\frac{B}{2 \rho_{0}^{4}}}\mathrm{e x p}\left({\frac{2 A}{B}}- 3 \right) ~ .
\frac{d^{n-1}M_1}{dv^{n-1}}=P_1-\frac{v}{6}(n-1)!\sum_{m=0}^{n-1}\sum_{l=1}^{n+3-m}\frac{(-1)^{n+1-l+m}(n-m)(n+1-l)(n+2-l)}{m!(l-1)!(n+3-m-l)!}\hat{I}_{n,l-1}.
\sum_{m=0}^N\alpha^m c_A^{\frac{m}{2}}\|S_m\|_{L^1}&\le e^{|W_0|}\Bigg(1+\sum_{n=1}^N\frac{1}{n!}\prod_{l=1}^n\Bigg(\sum_{m_l=1}^Nc_A^{\frac{m_l}{2}}\|W_{m_l}\|_{L^1}\sum_{k_l=0}^{m_l}\left(\begin{array}{c}m_l k_l\end{array}\right)\alpha^{k_l}\Bigg)\Bigg) &\le e^{\sum_{m=0}^N(\alpha+1)^mc_A^{\frac{m}{2}}\|W_m\|_{L^1}}.
\begin{cases}\partial_t\tilde{\phi}= -(-\Delta)^{s}\phi + p(u^*_{\mu, \xi})^{p-1}\tilde{\phi}+ \tilde{N}(\tilde{\phi}) + S(\mu^*_{\mu, \xi}), \Omega\times (t_0, \infty), \tilde{\phi}= -u^*_{\mu, \xi}, (\mathbb{R}^n\setminus \Omega)\times (t_0, \infty).\end{cases}
\Delta_{2 k}\left( f(x),g(x) \right)= \det \left( \begin{array}{ccccc}a_0 a_1 a_2 \dots a_{2k-1}b_0 b_1 b_2 \dots b_{2k-1}0 a_0 a_1 \dots a_{2k-2}0 b_0 b_1 \dots b_{2k-2}\vdots \vdots \vdots \vdots 0 0 0 \dots b_{k}\end{array}\right)_{2k \times 2k}.
\tau(\varphi)^{\alpha}=\sum\limits_{i,j=1}^m g^{ij}\left( \frac{\partial^2\varphi^{\alpha}}{\partial x_i \partial x_j}-\sum\limits_{k=1}^m{^M\Gamma}_{ij}^k \frac{\partial \varphi^{\alpha}}{\partial x_k}+\sum\limits_{\beta,\gamma=1}^n{^NL}_{\beta \gamma}^\alpha\frac{\partial \varphi^\beta}{\partial x_i}\frac{\partial \varphi^\gamma}{\partial x_j}\right).