harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_{\Lambda \Pi}^{( 4 + n )}\equiv \left[ \begin{matrix}{{\mathrm{e}^{-{\frac{1}{\alpha}}\varphi}g_{\mu \nu}+ \mathrm{e}^{\frac{2 \varphi}{n \alpha}}\rho_{i j}A_{\mu}^{i}A_{\nu}^{j}}}&{{\mathrm{e}^{\frac{2 \varphi}{n \alpha}}\rho_{i j}A_{\lambda}^{j}}}\\{{\mathrm{e}^{\frac{2 \varphi}{n \alpha}}\rho_{i j}A_{\pi}^{i}}}&{{\mathrm{e}^{\frac{2 \varphi}{n \alpha}}\rho_{i j}}}\\ \end{matrix}\right] ,
Conv \left( \frac{K-t_p}{t_p+1}\right) \leq R^{*}_{\textrm{uncoded}}(M_s,M_p) \leq Conv \left( \frac{\sum_{n=1}^{\Lambda-t_s}\mathcal{L}_n\binom{\Lambda-n}{t_s}}{\binom{\Lambda}{t_s}}\right)
E_{0,1}(\lambda(t),\lambda'(t),\lambda''(t)) = 2\frac{\lambda'(t)^{2}}{\lambda(t)^{2}}+ \frac{2 \lambda''(t)}{\lambda(t)}-\frac{4 \alpha \lambda''(t) \log(\lambda(t))}{\lambda(t)}\left(\frac{1}{-1+\lambda(t)^{2\alpha}}+1\right)
\mathcal{E}(t_s) = -\frac{\pi}{6L}+ \frac{\mathcal{B}-\mathcal{C}}{L}+\frac{\mathcal{C}}{2L}\int_{0}^{L}\sum_{n=-\infty}^\infty\left[\delta\left(t_s-x-nL\right)+ \delta\left(t_s+x-nL\right)\right] dx.
\Omega_{2}= \hat{R}_{M N P Q}\hat{R}^{M N P Q}- 4 \hat{R}_{M N}\hat{R}^{M N}+ \hat{R}^{2}\stackrel{\mathrm{4 D}}{=}\frac{1}{4}\hat{R}_{M N}^{A B}\hat{R}_{P Q}^{C D}\eta^{M N P Q}\eta_{A B C D}.
A^{( 1 )}= \left( \begin{array}{c c c}{A^{( 1 ) i j , k l}}&{A^{( 1 ) i j , 0}}&{A^{( 1 ) i j , b}}\\{A^{( 1 ) 0 , k l}}&{A^{( 1 ) 0 0}}&{A^{( 1 ) 0 b}}\\{A^{( 1 ) a , k l}}&{A^{( 1 ) a 0}}&{A^{( 1 ) a b}}\\ \end{array}\right)
n! =\Gamma(n+1),(2n-1)!!\sqrt{\pi}=2^{n}\Gamma\left(n+\tfrac{1}{2}\right), n! \sqrt{\pi}=2^{n}\Gamma\left(\tfrac{n}{2}+1\right)\Gamma\left(\tfrac{n+1}{2}\right) , \int_0^\infty r^se^{-\frac{1}{2}r^2}dr = 2^{\frac{s-1}{2}}\Gamma\left(\tfrac{s+1}{2}\right).
f - q|_g = \L(x L(p|_{L^2(yz)})) + x L^2(p|_{L^2(yz)}) - L^2(x p|_{L(yL(z))}) - L^2(x p|_{yL^2(z)}) \equiv \ -L(x L(p|_{L(yL(z))})) - L(x L(p|_{yL^2(z)})) - x L^2(p|_{L(yL(z))}) - x L^2(p|_{yL^2(z)}) &\ + L(x L(p|_{L(yL(z))})) + x L^2(p|_{L(yL(z))}) + L(x L(p|_{yL^2(z)})) + x L^2(p|_{yL^2(z)}) \equiv \ 0,
T(r)\geqslant \sum_{i=s,t,l}(N(r,\min\{\nu_{(f^u,H_i),\leqslant k_i};1\leqslant u\leqslant 3\}) &-N^{(1)}_{(f,H_i),\le k_i}(r))+ 2\sum_{i=1}^qN^{(1)}_{(f,H_i),\le k_i}(r)+o(T(r)),
S = \frac{1}{( 2 \pi )^{7}g_{s}^{2}}\int d^{1 0}x \: \sqrt{- g_{1 0}}\left[ e^{- 2 \phi}\left( R_{1 0}+ 4 ( \nabla \phi )^{2}\right) - \frac{1}{4}F_{\mu \nu}^{2}\right] ,
p_{e,w_{00}}=v^{\ell(w_0)-1}+v^{\ell(w_0)-5}+\cdots +v^{\ell(w_0)-2n+1}p_{e,u_{00}}=v^{\ell(w_0)-3}+v^{\ell(w_0)-7}+\cdots +v^{\ell(w_0)-2n-1}.
\sup_{j\ge 0}\ \|P_{n}T^{\,j}P_{l}\,x\| \le\Bigl(\prod_{r=1}^{M_{0}-1}C_{\varphi^{r-1}(l)}\Bigr)\,\cdot\,C_{\varphi^{M_{0}-1}(l)}\,\cdot\,\Bigl\|\sum_{k=b_l}^{b_{l+1}-1}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\,x_{k}e_{k}\ \Bigr\|_{1}&\le C_{l}\,\cdot\,\Bigl\|\sum_{k=b_{l}}^{b_{l+1}-1}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\,x_{k}e_{k}\Bigr\|_{1},
c_{\delta^{\prime}}:={\displaystyle \int \limits_{\delta^{\prime}}^{\infty}}\left( 1+\ln \frac{t}{r}\right) \sup_{x\in{\mathbb{R}^{n}}}\varphi_{1}(x,t)\frac{dt}{t^{\frac{n}{q}-\frac{n}{s}+1}}<\infty
2B_s(z) = -A_s'(z)+\int_0^z\left(f(t)A_s(t)-\frac{2\mu+1}{t}A_s'(t)\right) dt, 2A_{s+1}(z) = \frac{2\mu+1}{z}B_s(z)- B_s'(z)+\int f(z)B_s(z) dz.
\Phi^{[m;s]}_{\hbox{add}}(\tau, z_1, z_2, t) = \frac{1}{2}e^{2\pi i (m+1)t}\sum\limits^{-s+2m+1}_{j=-s}{R_{j;\,m+1}\left(\tau, \frac{z_1-z_2}{2}\right) (\Theta_{j,\,m+1}-\Theta_{-j,\,m+1}) (\tau, z_1+z_2)}
\left( \begin{array}{l}{\dot{b}( t )}\\{\dot{c}^{\dagger}( t )}\\ \end{array}\right) = \left( \begin{array}{l l}{0}&{1}\\{1}&{0}\\ \end{array}\right) \left( \begin{array}{l}{b ( t )}\\{c^{\dagger}( t )}\\ \end{array}\right)
B_{\mu \nu}^{c}= i e^{2}\{\epsilon_{\mu \nu \lambda}\partial_{\lambda}\phi - 2 \pi \int_{\Sigma_{C}}\delta ( x - y ( \sigma ) ) d \sigma_{\mu \nu}+ 2 \pi \oint \psi_{\mu}( \tau ) \psi_{\nu}( \tau ) \delta ( x - y ( \tau ) ) d \tau \}.
\mbox{There exists some $J$ such that \ $j>J\Rightarrow W_{i,j}=W_{i,j+1}$ for all $i$}. \mbox{There exists some $I$ such that\ $i>I\Rightarrow W_{i,j}=0$ for all $j$}. \textstyle \sum_{p\geq 0}W_{i+p,j+p}\geq \sum_{p\geq 0}W_{i+1+p,j+p}(\Leftrightarrow \lambda_i^{(j)}\geq \lambda_{i+1}^{(j)}).
d s^{2}= \left( 1 + \frac{\tilde{k}_{3}}{\hat{r}^{5}}\right)^{- 2 / 3}d x^{\mu}d x^{\nu}\eta_{\mu \nu}+ \left( 1 + \frac{\tilde{k}_{3}}{\hat{r}^{5}}\right)^{1 / 3}( d z^{2}+ d y^{\tilde{m}}d y^{\tilde{m}}) .
{\cal O}_{1}{}^{i}{}_{j}\left[ \begin{matrix}{\bar{\psi}_{n}^{j \dot{1}}}\\{\psi_{n 2}^{j}}\\ \end{matrix}\right] = - i m_{n}^{( 1 )}\left[ \begin{matrix}{\psi_{n 1}^{i}}\\{\bar{\psi}_{n}^{i \dot{2}}}\\ \end{matrix}\right]
f_{3}= - \frac{i e^{2}}{2 \pi^{2}}\ln \frac{\Lambda}{\mu}\, \sum_{i}c_{i}\int \frac{d^{4}k}{( 2 \pi )^{4}}\, \frac{M_{i}^{2}}{( k^{2}- M_{i}^{2})^{2}\Big ( ( k + p )^{2}- M_{i}^{2}\Big )}\qquad \quad
|\langle f,\phi_i \rangle| = | \langle \Phi a + w, \phi_i \rangle| \le |\langle \Phi a,\phi_i \rangle| + |\langle w,\phi_i \rangle| \le \nu_k \|a\|_2 + \|w\|_2 \|\phi_i\|_2 \le \nu_k \|a\|_2 + \epsilon.
\frac{(\sqrt{\mu^2+\alpha^2+\beta^2}-\mu )^k}{\sqrt{\mu^2+\alpha^2+\beta^2}}= \bigl(\sqrt{\mu^2+\alpha^2+\beta^2}\bigr)^{k-1}\biggl(1-\frac{\mu}{\sqrt{\mu^2+\alpha^2+\beta^2}}\biggr)^k,
\widehat{T}^{(p)}_{s_0,s_1,\ldots,s_k}(z)=\begin{vmatrix}\widehat{e}^{(p)}_{s_0}(z) \widehat{e}^{(p)}_{s_1}(z) \cdots \widehat{e}^{(p)}_{s_k}(z) \alpha_{1,s_0}\alpha_{1,s_1}\cdots \alpha_{1,s_k}\alpha_{2,s_0}\alpha_{2,s_1}\cdots \alpha_{2,s_k}\vdots \vdots \vdots \alpha_{k,s_0}\alpha_{k,s_1}\cdots \alpha_{k,s_k}\end{vmatrix}.
\left(z'(t_{+})\right)^{2}=\frac{2\gamma}{\frac{\partial^{2}h}{\partial z^{2}}\left(z_{+},\alpha\right)},\qquad\left(z'(t_{-})\right)^{2}=-\frac{2\gamma}{\frac{\partial^{2}h}{\partial z^{2}}\left(z_{-},\alpha\right)}.
\| \partial^{p}C_{2 m}^{( t , 1 )}\| \leq \left({\frac{1}{\Lambda^{2 m - 6}}}\right) P \left( l n{\frac{\Lambda}{\Lambda_{R}}}\right){\frac{1}{\Lambda^{p}}}\left({\frac{M}{\Lambda}}\right)^{\epsilon}
\nu^{1}_\epsilon \star \nu^{1}_\iota = \begin{cases}1,0,\lambda, \lambda-\frac{1}{2}, \nu^{0}_+, \nu^{0}_- \mbox{if}n \geq 5 1,0,\lambda, \lambda-\frac{1}{2}\mbox{if}n=m =3 ,4\end{cases}
\begin{cases}\dot{y}^n_{N_n-1}= -\dfrac{(y^n_{N_n-1})^2}{\kappa_n}\, p(y^n_{N_n-1}), \dot{y}^n_i = -\dfrac{(y^n_i)^2}{\kappa_n}\left[v_{i+1}^n(y^n_{i+1})-v_i^n(y^n_i)\right], i=0, \ldots, N_n-2 , y^n_i(0) = \bar{y}^n_i \doteq \dfrac{\kappa_n}{\bar{x}_{i+1}^n - \bar{x}_i^n}, i=0,\ldots,N_n-1.\end{cases}
\begin{cases}C^\varphi_{ijk}= - C^\varphi_{ikj}\, C^\varphi_{ikk}= 0 C^\varphi_{kki}= \alpha \varphi^a_{kk}\varphi^a_i C^\varphi_{ijk}+ C^\varphi_{jki}+ C^\varphi_{kij}= 0 \, .\end{cases}
\psi = \psi \left( \varepsilon_{0}^{a b}\right) = \left( 1 , 0 , \sqrt{{\frac{1}{2}}}\varepsilon_{0}^{a b}, 0 , \sqrt{{\frac{3 . 1}{4 . 2}}}\varepsilon_{0}^{\left( a b \right.}\varepsilon_{0}^{\left. c d \right)}, 0 , . . . \right) .
\widehat T^{\mu \nu}( q ) = -{\frac{e^{2}}{q^{2}}}\left( \eta^{\mu \nu}q^{2}- q^{\mu}q^{\nu}\right) \left[ \left. \hat{h}\right|_{q^{2}= 0}+ \left( \hat{h}( q^{2}) - \left. \hat{h}\right|_{q^{2}= 0}\right) \right] .
|\partial\varphi|_{\tilde{g}}^{2}+\sum_{i}\tilde{g}^{i\overline{i}}\geq C^{-1}\sum_{i}\left(\frac{|\varphi_{i}|^{2}}{\tilde{g}_{i\overline{i}}}+\tilde{g}_{i\overline{i}}^{\frac{1}{n-1}}\right) & \geq C^{-1}\sum_{i}|\varphi_{i}|^{\frac{2}{n}}& \geq C^{-1}|\partial\varphi|_{g}^{\frac{2}{n}},
y^2 - x^2y =(40960000000000 - 14580000000 j^5 + 531441 j^{10})\; x^6 - (86400000000 j^2 - 39366000 j^7 )\;x^5 &- (1/4 + 48600000 j^4 + 177147 j^9 /4)\;x^4 + (320000000 j + 291600 j^6 )\;x^3 - 1350000 j^3 \;x^2 &+ (1600000 + 243 j^5 )\;x-500 j^2
\begin{aligned}\left\|\partial_x^l \partial_t^j s_{p,N}(t) \partial^k_x v\right\|_{L_N^2}\leq C_{j,l,k}(1+t)^{-\frac{1}{4}-\frac{l+j}{2}}\|v\|_{L_N^1}, v\in H^k_N, \left\|\partial_x^l \partial_t^j s_{p,N}(t) \partial^k_x v\right\|_{L_N^2}\leq C_{j,l,k}(1+t)^{-\frac{l+j}{2}}\|v\|_{L_N^2}, v\in H^k_N,\end{aligned}
A(x)=\begin{pmatrix}0 x_1 x_1 -x_1 0 x_1x_2 -x_1 -x_1x_2 0 \end{pmatrix},S(y)=\begin{pmatrix}y_1^2 0 0 0 y_2^2 0 0 0 y_3 \end{pmatrix},
\begin{array}{l}{{\partial_{\tau}X^{2}( 0 ) = \partial_{\sigma}X^{1}( 0 ) = 0 ,}}\\{{\partial_{\tau}X^{1}( \pi ) + \partial_{\tau}X^{2}( \pi ) \cot ( \pi \vartheta ) = 0 ,}}\\{{\partial_{\sigma}X^{2}( \pi ) - \partial_{\sigma}X^{1}( \pi ) \cot ( \pi \vartheta ) = 0 ,}}\\ \end{array}
d_{\ell}= \sum_{j=1}^{m}\frac{(-1)^{\ell - 1}\log s_{j}}{2\pi}\int_{\lambda_{j}}^{\lambda_{j-1}}(-u)^{\ell-\frac{3}{2}}du = \sum_{j=1}^{m}\frac{(-1)^{\ell - 1}\log s_{j}}{\pi (2\ell -1)}\Big( |\lambda_{j}|^{\ell-\frac{1}{2}}-|\lambda_{j-1}|^{\ell-\frac{1}{2}}\Big).
\begin{array}{rclcrcl}y^{1,+}= y_{[1]}^{1}\cdots y^{m,+}= y_{[1]}^{m}\vdots \vdots y_{[r^{1}-1]}^{1,+}= y_{[r^{1}]}^{1}\cdots y_{[r^{m}-1]}^{m,+}= y_{[r^{m}]}^{m}\,.\end{array}
\sup_{t\in[0,T]}\frac{1}{\sqrt{N}}\left|Q_{K+1}^N(t)-f(N)-(Y^N_\infty(t)-\lambda (N))\right| = \sup_{t\in[0,T]}\frac{1}{\sqrt{N}}\left|\sum_{i=1}^B Q_{i}^N(t)+\sum_{i=1}^K(N- Q_{i}^N(t))-\sum_{i=K+2}^BQ^N_i(t)-KN-f(N)-(Y^N_\infty(t)-\lambda (N))\right| = \sup_{t\in[0,T]}\frac{1}{\sqrt{N}}\left[Y^N(t)-Y^N_\infty(t)+D_N^+(t)-D^N_-(t)\right]\to 0,
\sup_{\substack{\eta,{\zeta}\in V, \boldsymbol{\tau},\boldsymbol{\rho}\in H(\emph{div},\Omega), v_{\zeta}= \mathbb{P}_{[u_a,u_b]}\{u_d - \frac{1}{\lambda}{\zeta}\}}}\mathcal{J}^\ominus(\eta,{\zeta},\boldsymbol{\tau},\boldsymbol{\rho},v_{\zeta}) = \mathcal{J}(y(u),u).
\nabla_x q(s,t;y(s,t),\xi) \partial_x^\alpha\nabla_x y(s,t) &=-\sum_{0\lneq\alpha^\prime\leq\alpha}\binom{\alpha}{\alpha^\prime}\partial_x^{\alpha^\prime}\left[\nabla_x q(s,t;y(s,t),\xi)\right] \partial_x^{\alpha-\alpha^\prime}\nabla_x y(s,t).
I_{m,n}=\frac{\partial\mu^{T}}{\partial\theta_{m}}\Sigma^{-1}\frac{\partial\mu}{\partial\theta_{n}}+\frac{1}{2}tr(\Sigma^{-1}\frac{\partial\Sigma}{\partial\theta_{m}}\Sigma^{-1}\frac{\partial\Sigma}{\partial\theta_{n}}),
{\bf \Omega}^{\prime}( x_{1}, \ldots , x_{N}) = \prod_{i = 1}^{N / 4}f ( x_{4 i - 1}- x_{4 i}) \bar{f}( x_{4 i - 3}- x_{4 i - 2}) \, \, \delta \left( \sum_{i = 1}^{N / 4}( x_{4 i - 3}- x_{4 i - 1}) \right) \, . \nonumber
{\tilde{Z}}_{m}^{T}~ = ~ \int \prod_{\mu ; <{i j}>}d P_{i j}^{\mu}e^{\{- \sum_{\mu ; <{i j}>}( P_{i j}^{\mu})^{2}- l n ( \sum_{\mu}( P^{\mu})^{2}) \}}\cdot{c o n s t r a i n t s}~ ~ ,
\dot{\tilde x}_2 =-\vert \beta \vert^{-1}\lambda_{\pm}+(\beta)\tilde w, \dot{\tilde w}=f(w)(-\vert b\vert \tilde z_2+\vert \beta \vert \tilde x_2 \tilde w), \dot{\tilde z}_2 =-\vert b\vert^{-1}\lambda_{\mp}\tilde w,
\sum_{k=1}^N \Delta t_k \left| P_{T-t_{k}}\bar \L f (y_{t_{k-1\over \epsilon}}^\epsilon)-P_{T-t_{k-1}}\bar \L f(y_{t_{k-1\over \epsilon}}^\epsilon)\right|\le \epsilon c_1(T)\left( 1+\lambda_{q_1}(V(y_0))\right).
\mathbf{E}^\prime(\mathbf{x}) =\frac{1}{4\pi}\int\rho_e^\prime(\mathbf{x}')\frac{\mathbf{r}}{r^3}\,d^3x' -\frac{1}{4\pi}\int\mathbf{J}_m^\prime (\mathbf{x}')\,{\bf\times}\,\frac{\mathbf{r}}{r^3}\,d^3x',
V = q \left({{\cal A}_{0}\left( y \right) -{\cal A}_{0}\left({y^{\prime}}\right)}\right) = \frac{{q^{2}\sqrt \pi}}{{2 g}}\sqrt{\frac{a - 1}{{a}}}\left({1 - e^{- \frac{g}{{\sqrt \pi}}\sqrt{\frac{a}{{a - 1}}}| y - y^{\prime}|}}\right) .
\tilde \beta T_t=\beta(X^1)^5,U^1_x=0,U^1T_t=X^1,U^0T_t=X^1_tx+X^0_t, U^0_t=-\tilde\alpha T_tU^0,U^1_tX^1-\alpha U^1 X^1+U^1U^0_x T_t + \tilde\alpha U^1 T_t X^1=0.
[(T-L_{E_{i}(T)})b] \otimes [(S-R_{E_j(S)}) 1_B] = [(T-L_{E_{i}(T)})R_b 1_B] \otimes [(S-R_{E_j(S)}) 1_B] & = [R_b(T-L_{E_{i}(T)}) 1_B] \otimes [(S-R_{E_j(S)}) 1_B] & = [(T-L_{E_{i}(T)}) 1_B] \otimes [L_b(S-R_{E_j(S)}) 1_B] & = [(T-L_{E_{i}(T)}) 1_B] \otimes [(S-R_{E_j(S)}) L_b1_B] & = [(T-L_{E_{i}(T)})1_B] \otimes [(S-R_{E_j(S)}) b] .
R_n=\frac{1}{n}\sup_i \left| \left| \Big( (\mathbf{y}^M_i)^* \mathbf{y}_i^M-\Sigma \Big) \right| \right| =\frac{1}{n}\sup_i \left| \left| \Big( (\mathbf{x}_i^* \mathbf{x}_i)\Big(\mathbf{b}^*(\frac{i}{n})\mathbf{b}(\frac{i}{n})\Big)-\Sigma \Big) \right| \right| & \leq C \frac{\zeta_c^2}{n},
| \varphi_{k}\dot{\varphi}_{k^{\prime}}- \dot{\varphi}_{k}\varphi_{k^{\prime}}| = | \dot{\varphi}_{k}|^{2}| \varphi_{k^{\prime}}|^{2}+ | \dot{\varphi}_{k}|^{2}| \varphi_{k^{\prime}}|^{2}+ \frac 1 2 ( \dot{\varphi}_{k}\varphi_{k}^{*}+ \varphi_{k}\dot{\varphi}_{k}^{*}) ( \dot{\varphi}_{k^{\prime}}\varphi_{k^{\prime}}^{*}+ \varphi_{k^{\prime}}\dot{\varphi}_{k^{\prime}}^{*}) + 2
\sum_{M\geq1}g_r(M)\,z^M =\sum_{n\geq0}\sum_{d=1}^{r-1}\frac{z^{nr+d}}{(1-z)^{n(r-1)+d}}=\sum_{d=1}^{r-1}\left(\frac{z}{1-z}\right)^d\sum_{n\geq0}\left(\frac{z^r}{(1-z)^{r-1}}\right)^n &=\frac{z}{1-z}\frac{1-\left(\frac{z}{1-z}\right)^{r-1}}{1-\frac{z}{1-z}}\frac1{1-\frac{z^r}{(1-z)^{r-1}}}.
\|\mathbf{u}^{[k+1]}-\mathbf{u}^{\star [k+1]}\|_{\mathbf{G}}\leq \left( \frac{1}{\sqrt{1+\delta_{}}}\right)^{k+1}\|\mathbf{u}^{[0]}-\mathbf{u}^{\star [0]}\| + \sum_{i=0}^{k}\left(\frac{1}{\sqrt{1+\delta_{}}}\right)^{k-i+1}\|\mathbf{u}^{\star [i]}- \mathbf{u}^{\star [i+1]}\|_{\mathbf{G}},
\Sigma=[\begin{matrix}\Sigma_{XX}&\Sigma_{XY}&\Sigma_{XZ}\\ \Sigma_{YX}&\Sigma_{YY}&\Sigma_{YZ}\\ \Sigma_{ZX}&\Sigma_{ZY}&\Sigma_{ZZ}\end{matrix}]=[\begin{matrix}C_{11}&C_{12}\\C_{21}&C_{22}\\ \end{matrix}]
\forall \beta \in \mathcal{C}^{\infty}(M): \bar D_{\bar X}( \beta s \circ f) = \bar X(\beta \circ f) (s \circ f) + (\beta \circ f) \bar D_{\bar X}(s \circ f), \forall \alpha \in \mathcal{C}^{\infty}(N): \bar D_{\alpha \bar X}(s \circ f) = \alpha \bar D_{\bar X}(s \circ f).
\limsup_{\varepsilon \rightarrow 0^{+}}\limsup_{n \rightarrow \infty}\mathbb{E}_{\mu_n}\Big[ \Big| \int_{0}^{t}\frac{1}{n}\sum_{x}\Phi_{n}(s, \tfrac{x}{n}) [\eta_{s}(x)- \overleftarrow{\eta}_{s}^{\ell}(x) ]\eta_{s}(x+1) ds \Big| \Big] =0.
x_{k+2}=\sum_{l=2}^kA_{k-l}x_{l+1}+ a x_0 A_k + [2(a+kd)-(a+(k-1)d)]r^{k+1}x_0 &\quad\quad+ a^2x_0 A_{k-1}+ 2arA_{k-1}x_0 -arA_{k-1}x_0+drA_{k-1}x_0 &=\sum_{l=2}^kA_{k-l}x_{l+1}+A_{k+1}x_0 + A_k x_1 + A_{k-1}x_2=\sum_{l=0}^{k+1}A_{k+1-l}x_l.
\boldsymbol{\varpi}_j :=\frac{1}{\det^{M}\big(1+M^{-1}V_j^*\hat{X}_j^{-1}V_j \Omega_jT_j^{-1}\hat{B}_j^{-1}T_j\Xi_j\big)}&=\exp\Big{\{}-M\log\det\big(1+M^{-1}V_j^*\hat{X}_j^{-1}V_j \Omega_jT_j^{-1}\hat{B}_j^{-1}T_j\Xi_j\big)\Big{\}}&= 1+\sum_{\ell=1}^4 \frac{1}{M^{\ell-1}}\mathfrak{p}_\ell(\hat{X}_j, \hat{B}_j, V_j,T_j, \Omega_j, \Xi_j).
\Phi(x_{1}, \ldots, x_{l})= \sum_{k=1}^{n}\dfrac{1}{\binom{l}{p_{k}}}\sum_{\mathrm{card}(I)=p_{k}}\left(\prod_{j\in \left\{1, \ldots, l\right\}\setminus I}x_{j}\right)\cdot f_{k}\left(\prod_{i\in I}x_{i}\right) \left(x_{1}, \ldots, x_l\in R\right),
I = \int{\rm d}^4 x{\rm d}^4 y h_{\mu\nu}(x) S_{sub}^{\mu\nu\,\alpha\beta}(x-y) h_{\alpha\beta}(y) + \int{\rm d}^4 x{\rm d}^4 y J^{\mu\nu}(x) X_{\mu\nu\, \lambda}(x-y) \eta^{\lambda}(y) + \cdots.
x^{-\alpha}G_{p,q+1}^{m,n}(\begin{matrix}a_{p}\\b_{q},\alpha\end{matrix}|\eta x)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}e^{\omega x}\omega^{\alpha-1}G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|\frac{\eta}{\omega})d\omega,
\frac{L}{N}\sum_{n_1,n_2\neq0}\widehat{\Delta}\left(\frac{Ln_1}{N}\right)\widehat{\Delta}\left(\frac{Ln_2}{N}\right)\delta(n_1w_r-n_2w_s)=\frac{L}{N}\sum_{n_0\neq0}\widehat{\Delta}\left(\frac{Lw_rn_0}{Nd}\right)\widehat{\Delta}\left(\frac{Lw_sn_0}{Nd}\right).
H\Big(\frac{1}{|\omega_d^t|}\int_{\omega_d^t}|u'(t,x)|^2 dx\Big) \leq \frac{1}{|\omega_d^t|}\int_{\omega_d^t}H (|u'(t,x)|^2)dx &=\frac{1}{|\omega_d^t|}\int_{\omega_d^t}|u'(t,x)|g(|u'(t,x)|)dx &\leq \frac{1}{|\omega_d^t|k}\int_{\omega_d^t}u'(t,x) \rho(x,u'(t,x))dx \,.
S\partial_\eta F(h)= \left.\frac{d}{dw}\right|_{w=0}SF(h+w\,\eta) = \,\frac{1}{2\pi i}\int_{|w|=\frac{1}{2}|\eta|_{-p}^{-1}}\frac{SF(h+w\,\eta)}{w^2}\,dw,
\sum^{+\infty}_{l=1}\,\beta_l \mathcal{G}_l =\, \beta_{\mathcal{M}}\mathcal{M}+\,\tilde{\beta}_{\tilde{\mathcal{M}}}\tilde{\mathcal{M}}+\,\beta_A \mathcal{F}_A+\,\tilde{\beta}_{\tilde{\mathcal{F}}}\tilde{\mathcal{F}}+ +\beta_{\mathcal{M}\,\mathcal{F}}M\cdot \mathcal{F}_A\,+\beta_{AB}\mathcal{F}_{AB}\,+...
\theta_K^n + \sum_{r = 1}^{n-1}\left(\sum_{i = 1}^{n-r}\sum_{j = 0}^{i}\frac{a_{ij}p_{(n-i)r}(K)}{(1-K)^{j+\rho(n-i,r)}}\right)\theta_K^r + \sum_{j = 0}^n\frac{a_{nj}}{(1-K)^j},
I_{1}= \int_{0}^{+ \infty}d t_{1}\, d t_{2}\, \exp \Bigl [ - \frac{t_{1}^{2}}{2 \alpha}- \frac{t_{2}^{2}}{2 \alpha}- \frac{t_{1}t_{2}}{2 \alpha}\Bigr ] \sin ( t_{1}\frac{\alpha_{1}}{\alpha}) \sin ( t_{2}\frac{\alpha_{1}}{\alpha}) .
1+\sum_{d=0}^\infty \big((d+2)^a-(d+1)^a\big)F_{d+2}(n) =1+\sum_{i=0}^{a-1}\left(v_i \sum_{d=0}^\infty (d+2)^iF_{d+2}(n)\right) &\rightarrow 1+\sum_{i=0}^{a-1}\left(v_i \sum_{d=0}^\infty (d+2)^iF_{d+2}(\infty)\right) &=\sum_{d=0}^\infty (d+2)^a\Pr{\mathbf{Y}= d+2},
\frac{1}{2}\frac{d}{dt}(\|u\|_{L^{2}}^{2}+\|\nabla\Psi\|_{L^{2}}^{2})+\|\nabla u\|_{L^{2}}^{2}+\|\Delta\Psi\|_{L^{2}}^{2}+\int_{\mathbb{R}^{3}}(v+w)|\nabla\Psi|^{2}dx=0,
d_l^2(x_0 - x_l) + d_r^2(x_r - x_0) - \frac{\left( d_l (x_l - x_0) + d_r (x_r - x_0) \right)^2}{x_r - x_l}= \frac{(x_0 - x_l)(x_r - x_0)}{x_r - x_l}\left( d_r - d_l \right)^2,
\left\|r_2^n \right\| \leq \sum\limits_{q=1}^{2}\left( 2 \breve{\mu}_{n-1}^{(q)}\int_{0}^{k}\left\|\mathbf{B}_qu_{t}(\zeta)\right\|d\zeta + k\sum\limits_{j=2}^{n}\breve{\mu}_{n-j}^{(q)}\int_{t_{j-1}}^{t_{j}}\left\|\mathbf{B}_qu_{tt}(\zeta)\right\| d\zeta \right).
d_{(3)^*}(\varphi) := (\, \dim_k \, (V)^\varphi_{\ell_1}, \; \dim_k \, (V)^\varphi_{\ell_2}\,) , d'_{(3)^*}(\varphi) := (\, \dim_k \, (V')^\varphi_{\ell_1}, \; \dim_k \, (V')^\varphi_{\ell_2}\,) .
\{f_j,f_k\}=\sum_{\ell=1}^n\sum_{i=0}^{m-1}\sum_{\alpha=1}^d\left(\frac{\partial f_j}{\partial[\tilde{w}_i]_{\alpha,\ell}}\frac{\partial f_k}{\partial[\tilde{v}_i]_{\ell,\alpha}}-\frac{\partial f_j}{\partial[\tilde{v}_i]_{\ell,\alpha}}\frac{\partial f_k}{\partial[\tilde{w}_i]_{\alpha,\ell}}\right).
\Psi_{\rm{R}}= \begin{cases}\left( \psi_{\rm{R}}(1), \cdots, \psi_{\rm{R}}(|N(\tau_i) - N(\tau_{i+1})|) \right), ~ N(\tau_i) \neq N(\tau_{i+1}) \quad\quad\quad\quad\quad\,\,\,\,\varnothing, ~ N(\tau_i) = N(\tau_{i+1}).\end{cases}
I_{\rm one-loop}= (e^{i W})_{\rm one-loop}=\prod_{l=0}^{\infty}{}\left\{{(\det{\cal D}_{3/2})^2 \over \det{\cal D}_{2}\det{\cal D}_{1}}\right\}=\prod_{l=0}^{\infty}{}\left\{{\det{\cal D}_{3/2}\over \det{\cal D}_{1}}\right\}^2=1.
\|w(\cdot,t)\|_{L^2(\Omega)}= \int_{0}^t\|S(t-\tau)\|\cdot\|f_\tau(\cdot,\tau)\|_{L^2(\Omega)}d\tau \leq \int_{0}^tC\|f(\cdot,T)\|_{L^2(\Omega)}d\tau \leq C\|f(\cdot,T)\|_{L^2(\Omega)},
\Psi_0(z) = 1 - \frac{1}{z}+ \mathcal{O}(z^{-2}), \Psi_1(z) = \frac{1}{2z}+ \mathcal{O}(z^{-3/2}), \Psi_2(z) = \frac{1}{2z}+ \mathcal{O}(z^{-3/2}),
\lim_{N^\prime}\int_{T_1}^{T_2}\int \left| \omega \cdot (v_2 - v_1) \right|^2f_{N^\prime}^{(2)}(t,x_1,v_1,x_1+\varepsilon \omega,v_2)d\omega dx_1 dv_1 dv_2 dt = 0
f_j(u_{i+k})=\dfrac{[a\pm u_{i+k}]_j}{[b\pm u_{i+k}]_{j}}=\dfrac{[a\pm u_{i}]_{j}}{[b\pm u_{i}]_{j}}\dfrac{[u_i-a+\delta]_k[u_{i}+a+j\delta]_{k}[u_i+b]_k[u_i-b+(1-j)\delta]_k}{[u_i+a]_k[[u_i-a+(1-j)\delta]_k[u_i-b+\delta]_k[u_{i}+b+j\delta]_k}.
\Omega_{i}^{need}=M_{A,i}\left( S^{need}\right), \Omega_{i}^{reg}=M_{A,i}\left( S^{reg}\right) , M_{i}^{need}(\xi ,\eta ) =M_{A,i}\left( \lambda p\varepsilon \xi ,\eta\right), M_{i}^{reg}(\xi ,\eta ) =M_{A,i}\left( \lambda p\varepsilon+(1-\lambda p\varepsilon )\xi ,\eta \right) .
\tilde H_n:=\left(\begin{array}{cc}H_n 0 0 H_n\end{array}\right)=\left(\begin{array}{cccc}A_n+B_n 0 0 0 0 A_n 0 0 0 0 A_n+B_n 0 0 0 0 A_n \end{array}\right), \ \n\ge 1,
\Omega^{(\alpha,\theta)}(Q_{XYU}) =-\log\mathbb{E}_{Q_{XYU}}\Big[\exp\big(-\theta\omega_{Q_{XYU}}^{(\alpha)}(X,Y|U)\big)\Big] =-\log\sum_{x,y,u}Q_{XYU}^{1-\theta(1+\bar{\alpha})}(x,y,u)\left(Q_{U}(u)\pi_{XY}(x,y)\right)^{\theta}\qquad\times\left(Q_{U|XY}(u|x,y)Q_{X|U}(x|u)Q_{Y|U}(y|u)\right)^{\theta\bar{\alpha}}
h''(x)+ (p-m)\tanh x\frac{h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}h'(x) &+\frac{m-1}{2}\frac{(3-p)h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}\sin 2h(x)=0.
\begin{array}{c}bF_{v_yv_y}+c_{v_y}=0, z^1+z^2F_{u_1}+bF_{v_yu_1}+c_{u_1}=0, -z^1F_{v_y}+z^2F_{u_x}+bF_{v_yv_x}+c_{v_x}=0.\end{array}
\Bigl\|\sum_{\eta\in G_{u}}j(b)(\eta\gamma)^{*}\, j(b)(\eta\gamma) \Bigr\| = \Bigl\| \sum_{\eta\in G_{v}}j(b)(\eta)^{*}\, j(b)(\eta) \Bigr\| \le \|b\|_{r}^{2}.
C_{1}=[\begin{matrix}0&0&0\\ 0&0&1\\ 0&-1&0\end{matrix}],C_{2}=[\begin{matrix}0&0&-1\\ 0&0&0\\ 1&0&0\end{matrix}],C_{3}=[\begin{matrix}0&1&0\\ -1&0&0\\ 0&0&0\end{matrix}]
\int\nolimits_{\lbrack0,2\pi)}\left( \overline{\alpha_{k_{s}}(t)}\right)^{-1}(f_{t},\Psi_{k_{s},t}^{\ast})u_{k_{s}}(t)dt=\left( b(s)\right)^{-1}\int\nolimits_{[0,h]}\left\vert \left( \alpha_{k_{s}}(t)\right)^{-1}u_{k_{s}}(t)u_{k_{s}}^{\ast}(t)\right\vert dt=1.
\sqrt{\frac{16}{\pi}}\phi\left(\left[\frac{1}{2}-\sigma\right] + j\omega \right) = \frac{z\left(\left[\frac{1}{2}-\sigma\right] + j\omega \right)}{\left(\left[\frac{1}{2}-\sigma \right] + j\omega \right)}+ \frac{z\left(\left[\frac{1}{2}+\sigma\right] - j\omega\right)}{\left(\left[\frac{1}{2}+\sigma \right] - j\omega \right)}.
\delta_K\mu (y) ( x_1,(a_2,x_2),(a_3,x_3 )) = \mu(x_1, (a_2,x_2),(a_3,x_3)) \ \mu(yx_1, (a_2,x_2),(a_3,x_3))^{-1}= \frac{\widehat{F}(yx_1,x_2)(a_3) \ \epsilon(yx_1,x_2,x_3)}{\widehat{F}(x_1,x_2)(a_3) \ \epsilon(x_1,x_2,x_3)},
\max_{y\in \Omega_\varepsilon}|y-x_\varepsilon|^2 |\Delta u_\varepsilon(y)| u_\varepsilon(y)=|y_\varepsilon-x_\varepsilon|^2 |\Delta u_\varepsilon(y_\varepsilon)| u_\varepsilon(y_\varepsilon)\to +\infty
v_{3}(J(\ell-2,m) - v_{3}(J(\ell,m)) = \frac{1}{8}\left(a_{2}(J(\ell-2,m)) +a_{2}(J(\ell,m))+\frac{m^{2}}{4}\right) &= \frac{1}{8}\left( \frac{(\ell-2)m}{4}+ \frac{\ell m}{4}+\frac{m^{2}}{4}\right)= \frac{m}{32}(2\ell-2+m) .
f_{k}(z,\lambda):=\frac{1}{2\pi i}\intop_{\sigma-i\infty}^{\sigma+i\infty}\Gamma(s)\,(s,\lambda)_{k}\,z^{-s}ds=\left(\frac{\lambda-z}{\lambda+z}\right)^{k}\,e^{-z},\,\,\,\,\sigma>0,\,\,(z)>0.
\begin{aligned}\mathbf{F}_R^{(k+1)}\approx \frac{1}{\sqrt{N_t}}\{(1 + j\delta_{m,n}^{(k)}) e^{j\phi_{m,n}^{(k)}}\}= \mathbf{F}_R^{(k)}+ \frac{j}{\sqrt{N_t}}\{\delta_{m,n}^{(k)}\cdot e^{j\phi_{m,n}^{(k)}}\}= \mathbf{F}_R^{(k)}+ \{\delta_{m,n}^{(k)}\}\circ \frac{j}{\sqrt{N_t}}\{e^{j\phi_{m,n}^{(k)}}\},\end{aligned}
S = \int_{M}d^3 x \; \epsilon^{\sigma\mu\nu}\left [{B_\sigma}\!^a({F_{\mu\nu}}\!^a + \epsilon^{abc}{\phi_\nu}\!^b{\phi_\mu}\!^c) +{\eta_\sigma}\!^a D_{[\mu}{\phi_{\nu]}}^a \right ] \; .
h_t(x) = \sum_{p = 1}^{q}\sum_{s=1}^m \bigg( e^{-i \langle x, \xi_p \rangle}g_s(\xi_p) \rho_{st}(V_p ) + e^{i \langle x, \xi_p \rangle}g_s(-\xi_p) \rho_{st}(-V_p) \bigg),
\left\Vert f\right\Vert_{\widehat{w}_{\alpha,\beta}}:=\left( \int_{-1}^{1}\left\vert f(x)\right\vert^{2}\widehat{w}_{\alpha,\beta}(x)\,dx\right)^{1/2}\quad(f\in L^{2}((-1,1);\widehat{w}_{\alpha,\beta}))
D_{L}\oplus\delta_{R}=[\begin{matrix}D_{L}&0\\ 0&\delta_{R}\end{matrix}]=i[\begin{matrix}I&0\\ 0&I\end{matrix}]\partial_{t}+i[\begin{matrix}-\sigma^{k}&0\\ 0&\sigma^{k}\end{matrix}]\nabla_{k}+m[\begin{matrix}\eta\omega K&0\\ 0&-\eta\omega K\end{matrix}]
\Big|\sum_{\mathrm{N}\mathfrak{n}< x}\frac{1}{\mathrm{N}\mathfrak{n}}\Big(1-\frac{\mathrm{N}\mathfrak{n}}{x}\Big)^{n_K}-\kappa_K\Big(\log x-\sum_{j=1}^{n_K}\frac{1}{j}\Big)-\kappa_K\gamma_K\Big|\ll_{\epsilon}\big( n_K^{n_K}D_K \big)^{1/4+\epsilon}x^{-1/2}.
\frac{\mathrm{d}^{\beta}}{\mathrm{d}t^{\beta}}p_{\beta}(n,t)=-(\lambda_{1}+\lambda_{2}+\dots+\lambda_{k})p_{\beta}(n,t)+\sum_{j=1}^{\min\{n,k\}}\lambda_{j}p_{\beta}(n-j,t),\n\ge0,
\mathcal A :=V^2(1 -\eta)^2, \mathcal B :=V^2(1 -2\nu^2) -(1 +2V^2 +2\nu^2V^2 +\nu^4V^4)\eta + (1+ V^2 +\nu^4V^4)\eta^2, \mathcal C :=\nu^4V^2 -\nu^2(1 +V^2 +\nu^2V^4 +\nu^4V^4)\eta +\nu^4V^2(1 +V^2)\eta^2.
\frac{\det'(-\partial_\tau^2 + V^{\prime\prime})}{\det(-\partial_\tau^2 + \omega_+^2)}=\frac{e^{(\omega_- - \omega_+)(T/2 - \tau_I)}}{2\omega_-F_+F_-}\int_{-\infty}^{\infty}d\tau F^2(\tau).
\left( y^{( 1 ) a_{2 k + 1}}, \pi_{a_{2 k + 1}}^{( 1 )}\right) , \; \left( y^{( 2 ) a_{2 k + 1}}, \pi_{a_{2 k + 1}}^{( 2 )}\right) , \; k = 0 , \cdots , a ,
\gamma_{i}=(t_{\alpha_3}\circ t_{\alpha_2}\circ t_{\alpha_1})^i(\beta), \gamma_{i}^{(j)}=t_{\alpha_1}^j(\gamma_{i}), \beta^{(j)}=t_{\alpha_1}^j(\beta), \rho_{i}=(t_{a_3}\circ t_{a_2}\circ t_{a_1})^i(\beta), \rho_{i}^{(j)}=t_{a_1}^j(\gamma_{i}), b^{(j)}=t_{a_1}^j(\beta).
\Lambda(s, f_\chi)=\int_0^\infty f_\chi(iy) y^{s+\frac{k-1}{2}}\; \frac{dy}{y}=\frac{\Gamma\left(s+\frac{k-1}{2}\right)}{(2\pi)^{s+\frac{k-1}{2}}}L(s, f_\chi).
R_0 = \max\left\{|\varphi_p(\alpha +A\beta)|: \alpha \in \mathfrak D_0, \beta \in \mathfrak D, 0\leq p \leq q-1 \right\}, \and E_0 =\max\displaystyle \Big\{\frac{1}{N}\sum_{n \in{\mathbb{Z}}^2_N}|\varphi_p(n)|: 0\leq p \leq q-1\Big\}.