harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
z_\star^{(i)}= \begin{cases}2T - i + 1, \textnormal{if}1 \leq i \leq 2T, 0 \textnormal{otherwise}. \end{cases}y_\star^{(i)}= \begin{cases}1, \textnormal{if}1 \leq i \leq 2T, 0 \textnormal{otherwise}. \end{cases}
C_n = \int_{\mathbb{S}^{n-1}}|e_1\cdot\sigma|^2\:\mathcal{H}^{n-1}(\sigma) = \int_{\mathbb{S}^{n-1}}|e_j\cdot\sigma|^2\:\mathcal{H}^{n-1}(\sigma) = \frac{1}{n}\int_{\mathbb{S}^{n-1}}|\sigma|^2\:\mathcal{H}^{n-1}(\sigma) = \frac{\omega_n}{n}.
\mathcal{K}_s([\zeta],[\zeta^{\prime}])(\tau)= \displaystyle{\sum_{i=1}^{\mu}}\psi_s^i(\tau,\omega)\bar{\psi}_s^i(\tau,\omega^{\prime})=\displaystyle{\sum_{r=0}^{\infty}}\mathcal{K}_s^r([\zeta],[\zeta^{\prime}])(\tau).\tau^{-n-r}
K(\Delta\Omega,t)=\frac{|\gamma[A^{2}(t)+(B_{0}-\frac{\omega(t)}{\gamma})^{2}]^{3/2}|}{|\frac{dA(t)}{dt}(\frac{\omega_{0}+\Delta\Omega}{\gamma})-\frac{1}{\gamma}\frac{d\omega(t)}{dt}A(t)|}
\mathcal{F}(D(f_{j})) =\mathcal{F}(U)\otimes_{\mathcal{O}_{U}[F]}\mathcal{O}_{D(f_{j})}[F] & \cong\mathcal{F}(U)\otimes_{\mathcal{O}_{U}[F]}\mathcal{O}_{U}[F]\otimes_{\mathcal{O}_{U}}\mathcal{O}_{U}[\frac{1}{f_{j}}]\cong\mathcal{F}(U)\otimes_{\mathcal{O}_{U}}\mathcal{O}_{U}[\frac{1}{f_{j}}]
\int_{\theta}v^{2}(w) \|(\chi_{w}\oplus \xi_{w})\pi_{F(w)\oplus G(w)}S_{\chi\oplus \xi}^{-1}\pi_{S_{\chi\oplus\xi}^{-1}(F(w)\oplus G(w))}(f,g)\|^{2}d\mu(w) &=\int_{\Theta}v^{2}(w)\|(\chi_{w}\oplus\xi_{w})\pi_{F(w)\oplus G(w)}S_{\chi\oplus \xi}^{-1}(f,g)\|^{2}d\mu(w) &\leq B_{1}\|S_{\chi\oplus \xi}^{-1}\|^{2}\|(f,g)\|^{2}.
\hat{\Omega}\delta_x = x \delta_x \hat{\Omega}\frac{\mathcal{P}}{x - \Omega}1 = - \hat{1}+ x\frac{\mathcal{P}}{x - \Omega}1 \langle \hat{1}|\delta_x ) = x (\hat{1}| \frac{\mathcal{P}}{x - \Omega}|1) =0
\begin{aligned}D(p_{\mu_1,\sigma_1,h}|| p_{\mu_2,\sigma_2,h}) =D(h(x)f_{\mu_1,\sigma_1}(y)|| h(x)f_{\mu_2,\sigma_2}(y)) &\le c(\left\|{\mu_1-\mu_2}\right\|^2_{2,h}+\left\|{\sigma_1-\sigma_2}\right\|^2_{2,h}) &\le cC(\left\|{\mu_1-\mu_2}\right\|^2_{2,h_0}+\left\|{\sigma_1-\sigma_2}\right\|^2_{2,h_0}),\end{aligned}
\delta_B ( f^{(0)}u^4 ) + \delta_F ( f^{(2)}u^3 \left[ \psi^2 \right] )+ u^i u^2 \epsilon \dot{N}^{i(0)}\psi+ \frac{i}{2}u^3 \psi \dot{M}^{(0)}\epsilon=0 ~,
\frac{1}{2^{m-2}C}\mu (B\setminus A_m') = \frac{1}{2^{m-2}C}\mu (B) - \frac{1}{2^{m-2}C}\mu(A_m) & \ge \frac{1}{2^{m-2}C}\mu (B) - \mu(A_m) & \ge \frac{1}{2^{m-2}C}\mu (B) - \frac{1}{2^{m-1}C}\mu (B) & = \frac{1}{2^{m-1}C}\mu (B) \ge \frac{1}{2^{m-1}C}\inf_{\mu' \in M(G)}\mu' (B).
\begin{array}{ll}\left[\sum_{j=e+1}^{k-1}\hat\rho_j \hat\gamma_{i,j}c,\sum_{j=e+1}^{k}\hat\rho_j \hat\gamma_{i,j}c\right] k=e+1,\dots,e+N-1, begin{align*}2ex]\left[0,(\hat\gamma_{e}- \hat\rho_e\hat\gamma_{e,e})c\right] k=e.\end{array}
\sup_{x=(z,y) \in B_{\frac{1}{4}}(0) \times Y}\sup_{x' \in B^{g_{z,t}}(x,\frac{1}{8})}\frac{|{g}_t^\bullet(x) - \mathbf{P}^{g_{z,t}}_{x'x}({g}_t^\bullet(x'))|_{g_{z,t}(x)}}{d^{g_{z,t}}(x,x')^\alpha}\leq C_\alpha
t^{-m(n_1+\cdots+n_N)}\prod_{i=1}^N\sum_{\kappa_1^{(i)}+\cdots+\kappa_m^{(i)}=n_i}\prod_{j=1}^{m}t^{(j-1)\cdot \kappa_{j}^{(i)}}=t^{-m(n_1+\cdots+n_N)}(1+O(t)).
\mbox{\textbf{P1-3:}}\;\; \max_{\mathbf{x}}\mathbf{c}^T \mathbf{x}\mbox{s.t.}\;\; \sum_{k=1}^K x_{(j-1)K+k}\leq L_j,~j=1,2,\cdots,J \;\; \sum_{j=1}^J x_{k+(j-1)K}\leq 1,~k=1,2,\cdots,K \;\; \mbox{Constraints (\ref{eqn:mmhn95}),~(\ref{eqn:mmhn80})}.
(P_k)_j=\begin{bmatrix}\overbrace{0 \ldots 0}^{(k-2)}U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{1}{2}}U_1(\alpha_r) \ldots \overbrace{0 \ldots 0}^{(k-1)}\mu^{\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) 0\end{bmatrix}^{T}
\Psi\left(\frac{x+s}{2}\right) \leq \frac{1}{2}[\lambda_1(x)+\lambda_1(s)]+\frac{1}{4}[\lambda_2(x)+\lambda_2(s)+\lambda_4(x)+\lambda_4(s] = \frac{1}{2}\Psi(x)+\frac{1}{2}\Psi(s)
p^{-(k+1)}= \ \sum_{l=0}^{\infty}\binom{-(k+1)}{l}(-(1-p))^l = \ \sum_{l=0}^{\infty}(-1)^l \binom{(k+1) + l - 1}{l}(-(1-p))^l =\sum_{l=0}^{\infty}\binom{k+l}{k}(1-p)^l.
A_{m+1}(t,q,y,u):=\left(\begin{matrix}q_{n-m,n-m}\begin{matrix}a_{n-m}(t,y,u) 0 \cdots 0 \end{matrix}\begin{matrix}q_{n-m+1,n-m}\vdots q_{n,n-m}\end{matrix}\boxed{A_{m}(t,q,y,u)}\end{matrix}\right)
\rho(h,\omega)=\sum_{d|h,\, d|P(\omega)}\mu(d)=\sum_{\substack{d|h,\, d|P(\omega) d<x^\varepsilon}}\mu(d)+O\Bigl(\sum_{\substack{d|h,\,d|P(\omega) x^\varepsilon<d<\omega x^\varepsilon}}1\Bigr).
S_{\rm em}= -{1\over 2}\int\! d^2x\,\Bigl( \partial_0\phi\partial_1\tilde\phi +\partial_0\tilde\phi\partial_1\phi +\bigl( \partial_1\phi\bigr)^2 + \bigl( \partial_1\tilde\phi\bigr)^2+ F_e\, \cos \bigl( 2g_{\rm eff}\phi\bigr) + F_m\,\cos\bigl( 2\pi\tilde\phi /g_{\rm eff}\bigr)\Bigr)
{\rm Re}\W_{\vec{k}, \vec{l}}(\vec{x}, \vec{y}) -{\rm Re}\W_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = 0,{\rm Im}\W_{\vec{k}, \vec{l}}(\vec{x}, \vec{y}) +{\rm Im}\W_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = 0,
\underset{\tau\to+0}{\mbox{{\rm ess liminf}}}\int_{B(0,R)}\int_{{\bf R}^N}G(x-y,t-\tau)u(y,\tau)\eta(x)\,dx\,dy \qquad\ge\int_{{\bf R}^N}\eta(y)\,d\mu'(y)-C\|\eta(t)-\eta\|_{L^\infty(B(0,R))}
2\mathsf{Re}(P^*u,iB u)=\sum_{j=1}^N\underset{(\ref{GP1}.1)}{\underbrace{2\mathsf{Re}(X_j^*f X_ju,iB u)}}+\underset{(\ref{GP1}.2)}{\underbrace{2\mathsf{Re}\big((X^*_{N+1}-iX^*_0)u,iB u\big)}}.
1=\frac{\mathrm{vol}(\varphi_{t}^{\vec{v}}(K))+\mathrm{vol}(\varphi_{t}^{\vec{v}}(K^{c}))}{\mathrm{vol}(\mathbb{S}^{3}(\sqrt{1+t^{2}}))}<\frac{\mathrm{vol}(K)+\mathrm{vol}(K^{c})}{\mathrm{vol}(\mathbb{S}^{3})}=1
(g *^\prime h)(n) = \sum_{\substack{d \,\mid\, n 1 < d < n}}g(d) \, h(n/d) \ll n^s \sum_{\substack{d \,\mid\, n 1 < d < n}}4^{d+n/d}\ll 2^n n^s \sum_{\substack{d \,\mid\, n 1 < d < n}}1 &\ll 2^n n^{s+1}\ll 3^n ,
\sum_{r=2}^k \left\{\sum_{p=0}^{r-2}(-1)^p \left[a_{k-r+2+p}(n-p)(n-p-1) + b_{k-r+1+p}(n-p) + c_{k-r+p}\right] e_p\right\}z^{k+n-r}+ = 0.
f(X_1,\dots, X_n) :=\left(\langle S_r(E), P_r \rangle +\frac{1}{2}\|L_r(E)\|_2^2 \right) \varphi\Bigl(\frac{\|E\|_{\infty}}{\delta}\Bigr) &= \sum_{k\geq 3}\sum_{L\in{\cal L}_k}(-1)^{m_L-1}\sum_{\nu\in V_L}f_{\nu,L}(X_1,\dots, X_n),
\frac{1}{2t_n}\left(u_n \Phi(x) u_n^{-1}- \Phi(x)\right) =-\Phi(x_n)+\frac{1}{2t_n}\sum\limits_{k=1}^{n-1}\left(u_n \Phi(t_k x_k)u_n^{-1}- \Phi(t_k x_k)\right).
| \frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}- \frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime}|^{2 - n}= \sum_{N = 0}^{N_{0}}\sum_{\alpha = 0}^{N}\frac{1}{\lambda_{N}}Y_{\alpha}^{N}( \frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}) Y_{\alpha}^{N}( \frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime})
\begin{aligned}\lim_{n\to\infty}P_{n,k}(W_{\mathcal{S}(n,k;M_n)})=-(1-c)^k\sum_{l=1}^{k-1}\binom kl(\frac c{1-c})^l\frac l{k-l}+ k\sum_{l=1}^{k-1}\binom klc^l\int_0^{1-c}\frac{y^{k-l-1}}{1-y^k}dy-c^k\log(1-(1-c)^k). \end{aligned}
e_0(U_{n+j}) \le C\Big(e_1(W_{n+j}) + e_0(W_{n-1}) + e_0(W_n) +\sum_{k=1}^{j-1}|w_{n+k}|^2\Big) & \le C \Big(\sum_{k=1}^j e_1(W_{n+k}) + e_0(W_{n-1}) + e_0(W_n)\Big) \,.
{}^b \widetilde{A}^\epsilon (M):= (\mathcal{A}(G)\hat{\otimes}\,{}^b \Psi^{-\infty,\epsilon}(S))^{K\times K}\,, \widetilde{A}^\epsilon (M):= (\mathcal{A}(G)\hat{\otimes}\,\Psi^{-\infty,\epsilon}(S))^{K\times K}
\lim_{r \rightarrow 0}G_{i}( r ) \sim \frac{q_{i}}{\pi r}\frac{1}{1 + \exp \varepsilon_{i}( 0 , r , 0 )}\left. \frac{d \varepsilon_{i}( 0 , r , \mu_{i}/ 2 )}{d \mu_{i}}\right\vert_{\mu_{i}= 0}\, .
\displaystyle \mathbb{E}\sum_{\substack{j\in J_{K}}}\Phi(B_{t_{j}},t_{j}) (B_{t_{j+1}}-B_{t_{j}})=\displaystyle \sum_{\substack{j\in J_{K}}}\mathbb{E}\Phi(B_{t_{j}},t_{j}) \mathbb{E}(B_{t_{j+1}}-B_{t_{j}})=0
r=x+e_{5}=(\begin{matrix}0\\ 1\\ 1\\ 0\\ 0\\ 1\\ 1\end{matrix})+(\begin{matrix}0\\ 0\\ 0\\ 0\\ 1\\ 0\\ 0\end{matrix})=(\begin{matrix}0\\ 1\\ 1\\ 0\\ 1\\ 1\\ 1\end{matrix})
\| K_1(z)- K_2(z) \| \le | z^{-1}-\chi(z)^{-1}| \| B_1(z)\| + |\chi(z)|^{-1}\|B_1(z)-B_2(z)\| \le \frac{|z-\chi(z)|}{|z\chi(z)|}+ c|z|^{-1}\|B_1(z)-B_2(z)\| \le c \tau + c|z|^{-1}\|B_1(z)-B_2(z)\| .
\lambda^*\beta w - \Delta(\beta w) = \beta F - \beta \nabla_H\Pi- 2(\partial_z\beta) (\partial_z w) - (\partial_z^2\beta) w \quad\Omega', \partial_z(\beta w) = 0 \Gamma_u', \beta w = 0 \Gamma_b', \partial_z(\beta w) \Gamma_l'.
x^1=\varphi^1\in\mathcal S^1,x^2=(\theta_1^2,\varphi^2)\in\mathcal S^2,\quad\dots, &x^{n-1}=(\theta_1^{n-1},\dots,\theta_{n-2}^{n-1},\varphi^{n-1})\in\mathcal S^{n-1},x^n=(\theta_1^{n},\dots,\theta_{n-1}^{n},\varphi^{n})\in\mathcal S^n.
g_{i,j}g_{k,l}=g_{k,l}g_{i,j}{\rm if}(i-k) (i-l) (j-k) (j-l) > 0, g_{i,j}g_{i,k}g_{j,k}=g_{i,k}g_{j,k}g_{i,j}=g_{j,k}g_{i,j}g_{i,k}{\rm if}1 \le i < j < k \le n, g_{i,k}g_{i,l}g_{j,l}g_{k,l}=g_{i,l}g_{j,l}g_{k,l}g_{i,k}{\rm if}1 \le i < j <k < l \le n.
a^{c}_{\Lambda}(t):=\frac{1}{c}\left(a(t)-1_{c\in(0,1)}\int_{c<|y|\leq1}y\Lambda_t(dy)+1_{c\in(1,\infty)}\int_{1<|y|\leq c}y\Lambda_t(dy)\right),~~ A^c(t):=\frac{1}{c^2}A(t).
H_{k}\left( \theta,\theta^{\ast}\right) :=\sum_{\mathbf{b}=\left(b_{1},\dots,b_{k}\right) \in\left\{\theta,\theta^{\dagger}\right\}^{k}}C_{k}\left( \mathbf{b}\right) b_{1}\dots b_{k}\in\mathbb{\mathbb{R}}\left\langle \theta,\theta^{\ast}\right\rangle .
\mathcal{F}\{p(t,\cdot)\}(\xi) =\frac{2^{d/2}}{|\xi|^{\frac{d}{2}-1}}\int_{0}^{\infty}\rho^{-\frac{d}{2}}\textnormal{H}_{12}^{20}\left[\frac{1}{4}t^{-\alpha}\rho^{2}\ \Big|\begin{array}{cc}(1,\alpha) (\frac{d}{2},1) (1,1)\end{array}\right]J_{\frac{d}{2}-1}(|\xi|\rho)d\rho,
\begin{array}{l}\lim\limits_{n\rightarrow \infty}T_{s,t}^{\tilde e}f(m_0^n)=\lim\limits_{n\rightarrow \infty}\mathbb{E}^{m_0^n}[f(\mu_{t-s}^{+s,\tilde e})]=\mathbb{E}^{m_0}[f(\mu_{t-s}^{+s,\tilde e})]=T_{s,t}^{\tilde e}f(m_0).\end{array}
\frac{I}{2\pi p}= \frac{\alpha^2-1}{2 - p + 2\alpha}+ \frac{2}{2 - p + \alpha}- \frac{1}{2 - p}= \frac{\alpha^2 (\alpha(2 - p) + p^2 - 4 p + 2)}{(2 - p + 2\alpha)(2 - p + \alpha)(2 - p)}
r^{2}\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{\partial^{2}\psi}{\partial\phi^{2}}=r^{2}\frac{\partial^{2}\psi}{\partial x^{2}}+r^{2}\frac{\partial^{2}\psi}{\partial y^{2}}-r(cos\phi\frac{\partial\psi}{\partial x}+sin\phi\frac{\partial\psi}{\partial y})
(217t^6+920t^5+972t^4+1436t^3+205t^2-172t+6)(-t+1)^2 < (217t^6+920t^5+972t^4+1436t^3+205t^2-172t+6)(-t+1)^2\biggr\rvert_{t=0.6263716633 + 10^{-10}}< 81.3892822381 .
{\hat{Q}}\frac{\partial^{2}{\hat{H}}^{( n )}}{\partial{\hat{Q}}^{2}}= \frac{1}{2}\frac{\partial^{2}{\hat{H}}^{( n )}}{\partial{\hat{S}}\partial{\hat{P}}}- \frac{1}{2}\frac{\partial^{2}({\hat{P}}{\hat{H}}^{( n )})}{\partial{\hat{P}}\partial{\hat{Q}}}\, .
i) a_{1}\le \ell^{1}\le \ell^{2}\le a_{2}, ii) \hat{U}_{1}(x,\ell^{2},r)\le \hat{U}_{1}(\ell^{1},\ell^{2},r),\forall x \in (0,r), iii) \hat{U}_{2}(\ell^{1},y,r)\le \hat{U}_{2}(\ell^{1},\ell^{2},r),\forall y \in [0,r).
\lim_{q\downarrow 0}{P}^{x,i}(A, t<{\mathbf{e}_q}\,|\,{\tau}^-_0>{\mathbf{e}_q})=\lim_{q\downarrow 0}{E}^{x,i}\left[\mathbf{1}_{(A, \, t<{\tau}^-_0 < \mathbf{e}_q)}\frac{{P}^{\xi_t, J_t}({\tau}^-_0>{\mathbf{e}_q})}{{P}^{x,i}({\tau}^-_0>{\mathbf{e}_q})}\right].
{\cal L}= -{\scriptstyle \frac{1}{2}}\partial_{\mu}\varphi \partial^{\mu}\varphi -{\scriptstyle \frac{1}{2}}m^{2}\varphi^{2}-{\scriptstyle \frac{1}{4}}\lambda \left( \partial_{\mu}\varphi \partial^{\mu}\varphi \right)^{2}
G_{n}^{( N )}( L , r , l ) = \widetilde{\sum}_{\tilde{m}\in{\bf Z}^{N + 1}}q^{m C^{- 1}m}( q )_{L - 1}\prod_{i = 0}^{N}\frac{1}{( q )_{m_{i}}}\left\{\sum_{k = n}^{N}q^{- e_{n - l}C^{- 1}m}B^{( N )}( L , r , l | k ) \right\}
\begin{cases}u_1^{\prime\prime}+ A u_1 = 0 \,, u_i^{\prime\prime}+ A u_i+ C_{i i-1}u_{i-1}= 0 \,, 2 \le i \le n \,, \displaystyle{u_{i}^{\prime\prime}+ A u_{i}+ \sum_{k=n-1}^{i-1}C_{i k}\,u_k =0 \,, n+1 \le i \le n+p \,,}(u_i,u_i^{\prime})(0)=(u_i^0,u_i^1) \mbox{for}i=1, \ldots n+p\,,\end{cases}
|u(z)| \lesssim_{d,r}\int_{|y-z|\le 1+|z|}|u(y)| dy + \sum_{i=1}^{\infty}(2^i(1+|z|))^{-10d}\int_{|y-z| \sim 2^{i}(1+|z|)}|u(y) | dy & \lesssim_{d,r}(\mathcal M(|u|^r)(0) )^{\frac 1 r}\cdot (1+|z|)^{\frac dr}.
\Lambda_1 = \Lambda_2=\Lambda_3 =0, \Lambda_4 = \sqrt 2 (b^3_2-b^7_5+b^6_2-b^7_1), \Lambda_5 =2(b^1_1-b^6_6)+b^2_2-b^7_7+b^3_3-b^5_5-3(b^5_1-b^6_3)+b^6_2-b^7_1, \Lambda_6 = 4(b^5_1-b^6_3), \Lambda_7 = b^1_2-b^7_6-\sqrt 2(b^3_4+b^4_1+b^4_5+b^6_4)-b^5_2+b^7_3-b^6_2+b^7_1.
f_1 = -B_{31}B_{22}B_{13}y^3 - C_{12}C_{21}A_{33}x z^2 + (\,\,\cdots) y^2 z f_2 = A_{33}z (B_{31}C_{12}x z - C_{21}y ( B_{22}y + B_{32}z)) f_3 = A_{33}z (B_{13}C_{21}x z - C_{12}y ( B_{22}y + B_{23}z))
\hat{P}^{\hat{a}\hat{b}\hat{c}}\equiv \hat{\cal H}^{\hat{a}\hat{b}\hat{d}}\hat{C}_{\hat{d}}^{\hat{c}}={\frac{1}{K}}\, \left[ ( 1 +{\frac{1}{1 2}}\hat{{\cal H}}^{2}) \, \hat{\cal H}^{\hat{a}\hat{b}\hat{c}}- \,{\frac{1}{4}}( \hat{{\cal H}}^{3})^{\hat{a}\hat{b}\hat{c}}\right] \, .
|D\widehat\psi_s|_{g_o}^2 =(g_{0})_{ab}|_{\widehat\psi_s}\frac{\partial \widehat\psi_s^a}{\partial x^i}\frac{\partial \widehat\psi_s^b}{\partial x^j}g_o^{ij}={\rm tr}_{g_o}(\widehat\psi_s^*g_o).
\bar{V}_1 :=V_1+(y+z+1)V_5+e^{u_{xx}}V_6+(y-z+1)V_7+(e^{u_{xx}}+1)V_8, \bar{V}_2 :=V_2+(y+z+1)V_5+e^{u_{xx}}V_6+(y-z+1)V_7+(e^{u_{xx}}+1)V_8, \bar{V}_3 :=V_3+(y-z+1)V_5+(e^{u_{xx}}+1)V_6-(y+z)V_7-e^{u_{xx}}V_8.
\Delta_n(x,\xi_0):= W(\xi_0,n,x,\mathsf{L}^n)-W(\xi_0,n,x,\mathsf{R}^n) = E_{\xi_0}\biggl[\varphi\biggl(x+\sum_{i=1}^{n}Z_i^{\mathsf{L}^n}\biggr)\biggr] - E_{\xi_0}\biggl[\varphi\biggl(x+\sum_{i=1}^{n}Z_i^{\mathsf{R}^n}\biggr)\biggr].
\left<Z_{i},\mathcal{B}_{ku}^{j}\left(Z_{j}\right)\right>\left<Z_{k}, Z_{u}\right>=\displaystyle\sum_{l'=1}^{p-q}\varphi_{il'}^{ik}\left(Z_{i},Z_{k}\right)\overline{\varphi_{jl'}^{ju}\left(Z_{j},Z_{u}\right)},
a_{P}T_{P}=a_{E}[\frac{T_{E}+{T_{E}}^{0}}{2}]+a_{W}[\frac{T_{W}+{T_{W}}^{0}}{2}]+[{a_{P}}^{0}-\frac{a_{E}}{2}-\frac{a_{W}}{2}]{T_{P}}^{0}+b
\mathcal{N}_\ell(P) := \{y(\lambda)^T\in\mathbb{F}(\lambda)^{1\times m}\mbox{such that}y(\lambda)^TP(\lambda) = 0\}, \mathcal{N}_r(P) := \{x(\lambda)\in\mathbb{F}(\lambda)^{n \times 1}\mbox{such that}P(\lambda)x(\lambda) = 0\},
\sum_{y\in Y}(f(y)^2+df(y)) = \sum_{i=1}^{n+1}(f(i)^2+df(i)) + n \sum_{i=0}^{d-1}(i^2+di) - \sum_{i=1}^{n}(k_{i-1}(f)^2+dk_{i-1}(f)).
\Gamma ( \Phi_{0}+ \delta \Phi ) = - V ( \Phi_{0}+ \delta \Phi ) + \frac{1}{2}\Gamma_{\mu \nu}^{( 2 )}( \Phi_{0}) \partial^{\mu}\delta \Phi \partial^{\nu}\delta \Phi +{\cal O}( \partial \delta \Phi )^{4}
E_{0}= \Bigl ({\frac{4 R T \lambda^{2}}{\eta}}\Bigr )^{{\frac{1}{2}}}\sum_{n \neq 0 , 1}{\frac{1}{| n |^{{\frac{1}{2}}}}}= \Bigl ({\frac{4 R T \lambda^{2}}{\eta}}\Bigr )^{{\frac{1}{2}}}\Bigl [ 2 \zeta ( - 1 / 2 ) - 1 \Bigr ] .
\frac{d}{dt}\left(\int_{\Omega}u_j+ \int_{M}v_i\right) =d\int_{\Omega}\Delta u_j +\int_{\Omega}H_j(u) +\tilde d\int_{M}\Delta v_i + \int_{M}F_j(u,v) \leq \beta\int_{\Omega}(u_j+1)+\int_{M}( G_j(u,v)+F_i(u,v) ) \leq \beta\int_{\Omega}(u_j+1) + \alpha \int_{M}(u_j+v_i+1)
4 \Lambda^{4}\prod_{a = 1}^{n_{c}- 2}( x - \phi_{a})^{2}- 4 \Lambda^{4}\prod_{i = 1}^{2 n_{c}- 4}( x + m_{i}) = - 4 \Lambda^{2}\alpha_{n_{c}- 1}^{4}\prod_{a = 1}^{n_{c}- 3}( x - \alpha_{a})^{2}.
A \begin{bmatrix}y_1y_2\dots y_{n-1}x_n y_1y_2\dots x_{n-1}y_n \vdots x_1y_2\dots y_{n-1}y_n y_1y_2\dots y_{n-1}y_n \end{bmatrix}=\begin{bmatrix}0 0 \vdots 0 0 \end{bmatrix}.
f_{\mu}(z_1,\dots,z_n;q,t)=\prod_{i=1}^{r-1}(1-q^i)\times{\rm Tr}\left(A_{\mu_1}(z_1) A_{\mu_2}(z_2) \dots A_{\mu_n}(z_n)k^{u(r-1)}\otimes k^{u(r-2)}\otimes \cdots \otimes k^{u}\right),
\int_{\mathbb{R}^{N}}|u|^{\alpha(x)}|v|^{\beta(x)}\,dx \leqslant c_{2}\max \Big\{\|u\|^{\alpha^{-}}\|v\|^{\beta^{-}}, \|u\|^{\alpha^{-}}\|v\|^{\beta^{+}}, \|u\|^{\alpha^{+}}\|v\|^{\beta^{-}},\|u\|^{\alpha^{+}}\|v\|^{\beta^{+}}\Big\}\leqslant c_{2}\max \Big\{\|(u,v)\|^{q^{-}}, \|(u,v)\|^{q^{+}}\Big\}.
\beta = \frac{1 6 \sigma^{2}\eta^{2}}{\alpha}\int_{0}^{\infty}\mathrm{d}\xi \xi^{2}\phi_{\sigma}^{2}\left( \xi \right) , \quad \phi_{\sigma}\left( \xi \right) = \mathrm{e}^{- 2 \sigma \xi}\int_{0}^{\infty}\mathrm{d}s \mathrm{e}^{- 2 \xi s}{\cal V}_{\sigma}\left( \eta , \zeta , s \right)
\begin{cases}\partial_{t}v_{k}-\Delta v_{k}=\mathrm{div}(-v_{k}\otimes v_{k}+v_{k}\otimes w+w\otimes v_{k}) +\mathrm{div}(u^{c,\gamma}\otimes v_{k}+v_{k}\otimes u^{c,\gamma})+\nabla\pi_{k}, \mathrm{div}v_{k}=0, v_{k}(x,0)=w_{0}(x)-w_{0,k}(x),\end{cases}
\begin{array}{l}I_{\epsilon}=\int_{m}^{M}\left[ \int_{\Omega}a_{\epsilon,A(y)}(x)\,|\nabla u|^{p(x)-2}\nabla u\nabla \phi_{\epsilon}dx\right]dy =\int_{m}^{M}\left[ \int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla \Phi_{\epsilon,y}dx\right]dy =\int_{m}^{M}\left[ \int_{\Omega}h(x)\Phi_{\epsilon,y}dx\right] dy\geq 0.\end{array}
I_\rho(v,w)= \langle d\gamma_x(\rho)(v),d\gamma_x(\rho)(w) \rangle = \langle \cos(\rho)d\sigma_x(v)+\sin(\rho)d(N\circ\sigma)_x(v),\cos(r)d\sigma_x(w)+\sin(\rho)d(N\circ\sigma)_x(w)\rangle = I(\cos(\rho)v+\sin(\rho)B(v),\cos(\rho)w+\sin(\rho)B(w))\,.
\tilde{C}_{\alpha}^{\, \alpha_{0}}( p ) : = \int_{\alpha_{0}}^{\alpha}d \alpha^{\prime}\, e^{\, - \alpha^{\prime}( \, p \eta p + ( \varepsilon + i ) m^{2})}\quad , \qquad \varepsilon > 0 \ , \ 0 < \alpha_{0}\leq \alpha < \infty \quad ,
C = \sup_{F_X(x)}I(X;Y) =\sup_{F_X(x)}\int{D\left(f_{Y|X}(.|x)||f_Y(.)\right)dF_X(x)}\leq \sup_{F_X(x)}\int{D\left(f_{Y|X}(.|x)||q_Y(.)\right)dF_X(x)}\leq \sup_{x\in\mathbb{S}}D\left(f_{Y|X}(.|x)||q_Y(.)\right)
\sum_{\substack{k=0 n - k \equiv 0 \,(\mathrm{mod}\,2)}}^n \binom{n}{k}\big ( 2^k L_{jk}+ 2 L_j^k \big ) (\sqrt{5}F_j)^{n-k}\frac{2^{n-k+2}-1}{n-k+2}B_{n-k+2}= L_j^n,
\|u_h^{n}\|^2+\Delta t\sum_{k=2}^n\|\nabla u_h^{k-\theta}\|^2+\gamma\Delta t\sum_{k=2}^n\|\sigma_h^{k-\theta}\|^2\leq C(\|u_h^{0}\|^2+\Delta t\sum_{k=1}^n\|g^{k}\|^2).
\left\lbrace \begin{array}{lllll}\alpha_i = \alpha_1\left(\frac{\alpha_{m+1}}{\alpha_1}\right)^{\frac{\gamma_i-\gamma_1}{\gamma_{m+1}-\gamma_1}}1\leqslant i \leqslant r, \sum\limits_{i=1}^r \alpha_i = 1, \sum_{i=1}^r \gamma_i \alpha_i = 0. \end{array}\right.
\lim_n\langle B_nRA\xi, C\xi\rangle =\lim_n\langle RA\xi, B_nC\xi\rangle=\lim_n\langle RA\xi, CB_n\xi\rangle &=\langle RA\xi, CT\xi\rangle=\langle RA\xi, TC\xi\rangle=\langle TRA\xi, C\xi\rangle\,,
{\cal M}\displaystyle \int_{- \infty}^{\infty}\displaystyle \frac{d x}{2 \pi}\displaystyle \frac{1}{2}e^{- x}{\cal Q}( x ) \doteq \displaystyle \frac{1}{L}\displaystyle \int_{- \infty}^{\infty}\displaystyle \frac{d x}{2 \pi}e^{- x}{\cal Q}_{-}( x ) .
\psi^{'}_u(t) = (n-1-q) t^{n-2-q}m(\|tu\|^n) \|u\|^n + n t^{2n-2-q}m{'}(\|tu\|^n) \|u\|^{2n}+ \frac{q}{t^{q+1}}\int_{\Omega}(|x|^{-\mu}* F(tu)) f(tu).u ~dx - t^{-q}\bigg[ \int_{\Omega}(|x|^{-\mu}* f(tu).u) f(tu).u ~dx + \int_{\Omega}(|x|^{-\mu}* F(tu)) f^{'}(tu).u^2 ~dx \bigg].
Q(\xi )=\frac{1}{16\pi G}\int d\varphi \left\{\xi^{\perp}\left[ \frac{l}{r}+\frac{r^{3}}{l^{3}}\left( g_{rr}-\frac{l^{2}}{r^{2}}\right) \right] + \frac{1}{lr}\left( \xi^{\perp}+r\xi^{\perp},_{r}\right) \left[ g_{\varphi \varphi}-r^{2}\right] +2\xi^{\varphi}\pi_{\varphi}^{r}\right\}.
\int_{G}k_l(x)dx =\int_{G}\sum_{2^l\leq \langle \xi\rangle^2<2^{l+1}}\sum_{i,j=1}^{d_\xi}d_\xi\xi_{ij}(x)dx &=\sum_{2^l\leq \langle \xi\rangle^2<2^{l+1}}\sum_{i,j=1}^{d_\xi}d_\xi \int_{G}\xi_{ij}(x)dx &=\sum_{2^l\leq \langle \xi\rangle^2<2^{l+1}}\sum_{i,j=1}^{d_\xi}d_\xi \int_{G}\xi_{ij}(x)\xi_{0}(x)dx=0,
\mathbb{P}[\mathbf{G}(m_1)=x,\mathbf{G}(m_2)=y]=\mathbb{P}[\mathbf{G}(m_2)=y\,|\,\mathbf{G}(m_1)=x]\mathbb{P}[\mathbf{G}(m_1)=x] &=\det \left(\Delta^{j-i}w_{m_2-m_1}(y_j-x_i)\right)_{1\le i,j\le n}\det\left(\Delta^{j-i}w_{m_1}(x_j)\right)_{1\le i,j\le n}.
0= \int_{\mathbb{R}^3}\varphi(x_1, x_2) e^{i\zeta\cdot x}\dx=\int_{B_R}\varphi(x_1, x_2) e^{i\zeta\cdot x}\dx = \int_{\mathbb{R}^2}e^{i\xi'\cdot x'}\varphi(x')\dx'\cdot \int_{\{x_3; (x', x_3)\in B_R\}}e^{-\xi_3^\perp\cdot x_3}\dx_3,
\int_{\mathbf{S}^{N-1}\times \mathbf{S}^{N-1}}\frac{d\omega_1\, d\omega_2}{|\omega_1-\xi\,\omega_2|^{N+s\,p}}=|\mathbf{S}^{N-1}|\int_{\mathbf{S}^{N-1}}\frac{d\omega_2}{|{\bf e}_1-\xi\,\omega_2|^{N+s\,p}},
{\hat \Pi}_{{\hat \mu}{\hat \nu}}{\cal D}_{\hat \imath}{\hat X}^{\hat \mu}{\cal D}_{\hat \jmath}{\hat X}^{\hat \nu}={\hat g}_{{\hat \mu}{\hat \nu}}D_{\hat \imath}{\hat X}^{\hat \mu}D_{\hat \jmath}{\hat X}^{\hat \nu}\, ,
{\chi}_U(\mathcal{F})=\frac{[H^0_{et}(U,\mathcal{F})][H^2_c(U,\mathcal{F})]}{[H^3_c(U,\mathcal{F})][\ker(\Psi^1(\mathcal{F}))]\prod_{v\in S}[H^0(K_v,\mathcal{F}_v)]}.
E_f={\frac{R_\natural}{\ell_{11}^2}}=E_f^\prime={\frac{R_\star}{(\ell_{11}^\prime)^2}}\quad , \qquad\rho_i\equiv{\frac{R_i}{\ell_{11}}}=\rho_i^\prime={\frac{R_i^\prime}{\ell_{11}^\prime}}\quad .
(\nabla_X A)Y-(\nabla_Y A)X= c\{\eta(X)\phi Y-\eta(Y)\phi X-2g(\phi X,Y)\xi\}&+c\sum_{a=1}^3 \{\eta_a(X)\phi_a Y-\eta_a(Y)\phi_a X -2g(\phi_a X,Y)\xi_a &+\eta_a(\phi X)\theta_a Y-\eta_a(\phi Y)\theta_a X\}.
R(e^{-2\pi t})=\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}t^{-2k-1}}{(2k+1)\zeta(2k+1)}+\frac{1}{2}\sum_{\rho}\frac{t^{-\rho}}{\rho cos(\pi\rho/2)\zeta^{\prime}(\rho)}
\displaystyle Z_{\scriptscriptstyle C P I}^{\varphi^{4}}= \int{\cal{D}}\xi^{a}{\cal{D}}\lambda_{a}{\cal{D}}c^{a}{\cal{D}}\bar{c}_{a}\; e^{i \int d^{4}x \tilde{\cal{L}}^{\varphi^{4}}}
\frac{\mu[\pmb{a}_1^n]}{\mu[\pmb{a}_2^n]}= \sum_{\pmb{a}_{n+1}^m\in A^{m-n}}\frac{\mu[\pmb{a}_1^m]}{\mu[\pmb{a}_2^m]}\times\frac{\mu[\pmb{a}_2^m]}{\mu[\pmb{a}_2^n]}=\mathbb{E}_{p}\left(\frac{\mu[\pmb{a}_1^m]}{\mu[\pmb{a}_2^m]}\right),
c_{m}^{i}( \xi_{1}, \xi_{2}) = K_{m}^{i p_{1}, \ldots , p_{\alpha}q_{1}, \ldots , q_{\beta}}\xi_{1 p_{1}}, \ldots , \xi_{1 p_{\alpha}}\xi_{2 q_{1}}, \ldots , \xi_{2 q_{\beta}}.
\|K_{\nu,0}^{-1}\|_{\mathcal{L}(L^2(\mathbb{R}^2))}=\|\displaystyle\int_0^{+\infty}e^{-tK_{\nu,0}}dt\|_{\mathcal{L}(L^2(\mathbb{R}^2))}\le \int_0^{+\infty}\|e^{-tK_{\nu,0}}\|_{\mathcal{L}(L^2(\mathbb{R}^2))}dt{~,}
{\cal H}^{( 0 )}={\frac{1}{2}}\left[{n_{i}}^{\mu}( \Xi ) \Pi_{\mu}\right] b ( \Xi ) \left[{n_{i}}^{\nu}( \Xi ) \Pi_{\nu}\right] -{\frac{i}{4}}{n_{i}}^{\mu}( \Xi ) b ( \Xi ){n_{i}}^{\nu}( \Xi ) \varepsilon_{\mu \nu}
=\sum_{n=1}^{\infty}\frac{1}{n^{2}}-\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=\frac{\pi^{2}}{6}-\sum_{n=1}^{\infty}(\frac{1}{n}-\frac{1}{n+1})=\frac{\pi^{2}}{6}-1.
\limsup_{i\to+\infty}I_u(x+r_{j_i}y_j, 0^+) \leq\inf_{s>0}\limsup_{i\to+\infty}I_u(x+r_{j_i}y_j, r_{j_i}\,s) &= \inf_{s>0}\limsup_{i\to+\infty}I_{u_{x,r_{j_i}}}(y_j, s)= \inf_{s>0}I_{w}(y, s) = I_w(y, 0^+).
\varOmega_3(1)={}\det \begin{pmatrix}D^{0}\mu^1_{2,1}(1) D^{0}\mu^1_{2,2}(1) D^{0}\mu^1_{2,3}(1) D^{1}\mu^1_{2,1}(1) D^{1}\mu^1_{2,2}(1) D^{1}\mu^1_{2,3}(1) D^{2}\mu^1_{2,1}(1) D^{2}\mu^1_{2,2}(1) D^{2}\mu^1_{2,3}(1) \end{pmatrix}.
{\bf P}_e^{(n)}\triangleq \frac{1}{M_n}\sum_{w \in{\cal M}_n}{\bf P}^g \Big\{d_{0,n}(Y^{n}) \neq w | W=w\Big\}\equiv{\bf P}^g\Big\{d_{0,n}(Y^n) \neq W \Big\}\leq \epsilon_n
(\xi, w, v) :=\, \lim\limits_{i\to\infty}\frac{\Psi(k_i,y_i) - \Psi(h_i,x_i)}{\| \Psi(k_i,y_i) - \Psi(h_i,x_i) \|}\, \in \mathfrak{m}\times \overline{W_{K}}\times V^H \ .
\begin{aligned}\Big({\|x-x'\|\over \|x-y\|}\Big)^\varepsilon\frac{1}{\omega(B(x,\|x-y\|))}\leqslant C \Big(\frac{\|x-x'\|}{\|x-y\|}\Big)^\varepsilon \frac1{\omega(B(x,d(x,y)))}\Big(\frac{d(x,y)}{\|x- y\|}\Big)^{N}&\leqslant C\Big(\frac{\|x-x'\|}{\|x-y\|}\Big)^\varepsilon\frac1{\omega(B(x,d(x,y)))}.\end{aligned}
\frac{M(B_t)}{M(x)}\mathbf{1}_{\{t<\tau^\Gamma\}}={\rm e}^{p(\xi_{\varphi(t)}-\log\|x\|)}\frac{m_1(\Theta_{\varphi(t)})}{m_1(\arg(x))}\mathbf{1}_{\{t<\tau^\Gamma\}}, t\geq 0.
\| \left( A(x^{1}_{j},\ldots,x^{n}_{j})\right)_{j=1}^{m}\|_{Rad(F)}\le C\prod_{i=1}^{n}\Vert(x_{j}^{i})_{j=1}^{m}\Vert_{\ell_{2}^{w}(E_{i})}
\mathcal{R}_{}\! =\! \bigcup_{q\in[0,0.5]}\!\Big\{\left(R_s,R_\ell,R_w\right)\!\colon\! &0\leq R_s\leq 1- H_b(q*p_A), &R_\ell\geq H_b(q*p_A)- H_b(q), &R_w\geq H_b(q*p_A)- H_b(q)\Big\}.
{\cal V}= ( U^{\Lambda \Sigma}- 3 \bar{L}^{\Lambda}L^{\Sigma}) P_{\Lambda}^{x}P_{\Sigma}^{x}+ \frac 1 3 \bar{L}^{\Lambda}L^{\Sigma}h^{u v}\nabla_{u}P_{\Lambda}^{x}\nabla_{v}P_{\Sigma}^{x},
\|v\|^s r_s(\bar v) = r_s(v) \mbox{and}\|f\|^s \int_{\mathcal{S}}\varphi(u) \langle \bar f, u \rangle^s \nu_s(du) = \int_{\mathcal{S}}\varphi(u) \langle f, u \rangle^s \nu_s(du),
d s^{2}= \frac{\Delta - \alpha^{2}\sin^{2}\theta}{\Sigma}\left( d t - \omega d \varphi \right)^{2}- \Sigma \left( \frac{d \hat{\rho}^{2}}{\Delta}+ d \theta^{2}+ \frac{\Delta \sin^{2}\theta d \varphi^{2}}{\Delta - \alpha^{2}\sin^{2}\theta}\right) \, ,
\delta_{\varepsilon_{2}}\left( x \right) \star \delta_{\varepsilon_{1}}\left( p \right) = \frac{1}{\pi \sqrt{\theta^{2}+ \varepsilon_{2}\varepsilon_{1}}}\exp \left( - \left[ \frac{x^{2}\varepsilon_{1}- 2 i x p \theta + p^{2}\varepsilon_{2}}{\theta^{2}+ \varepsilon_{2}\varepsilon_{1}}\right] \right) ,