harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{B_{2s}(x_0)}b^{ij,kl}v_{ij}\tau^{4}v_{kl}dx = \int_{B_{2s}(x_0)}b^{ij,kl}w_{ij}(\tau^4v)_{kl}dx &\qquad- \int_{B_{2s}(x_0)}b^{ij,kl}v_{ij}[ (\tau^{4})_{kl}v + (\tau^{4})_{l}v_{k}+ (\tau^{4})_{k}v_{l}] dx.
X_{n}(t,x) =\int_{0}^{t}b_{f_{n}}(s,x)\, ds+\sqrt{\sigma_{n}}\int_{0}^{t}M(ds,x)\,, X_{0}(t,x) =\int_{0}^{t}b_{f}(s,x)\, ds\,, \phi_{t}^{n}(x) =x+\int_{0}^{t}b_{f_{n}}(s,\phi_{s}^{n}(x))ds+\sqrt{\sigma_{n}}\int_{0}^{t}M(ds,\phi_{s}^{n}(x))\,, \phi_{t}^{0}(x) =x+\int_{0}^{t}b_{f}(s,\phi_{s}^{0}(x))ds\,
h^{-1}(I(a_1, a_2, \cdots, a_n))= \{(x_1, \cdots, x_n, \cdots): h(x_1, \cdots, x_n, \cdots) \in I(a_1, \cdots, a_n)\}= \{(x_1, \cdots, x_n, \cdots): x_1= a_1, \cdots, x_n= a_n\}= \{a_1\}\times\cdots\times\{a_n\}\times\mathbb{N}\times \cdots.
J_{B \mu}^{\prime}~ = ~ F_{\mu \nu}\partial^{\nu}\frac{1}{\partial_{i}\partial^{i}}{\overline{{{\cal P}}}}+ m^{2}( A_{\mu}+ \partial_{\mu}\rho ) \frac{1}{\partial_{i}\partial^{i}}{\overline{{{\cal P}}}}+ B \partial_{\mu}\frac{1}{\partial_{i}\partial^{i}}{\overline{{{\cal P}}}}.
\begin{array}{lcl}\{p^{-1}, f\}_{\theta}= -f'p^{-2}-{\theta}^{2}f'''p^{-4}-...-{\theta}^{2k}f^{(2k+1)}p^{-2k-2}-... \{p^{-2}, f\}_{\theta}= -2f'p^{-3}-4{\theta}^{2}f'''p^{-5}-...- (2k+2){\theta}^{2k}f^{(2k+1)}p^{-2k-3}-... \{p^{-3}, f\}_{\theta}= -3f'p^{-4}-10{\theta}^{2}f'''p^{-6}-...- \frac{(2k+3)(2k+2)}{2}{\theta}^{2k}f^{(2k+1)}p^{-2k-4}-...\end{array}
\phi(Q(j),Q(i)) = (l_j+\Delta(Q(j),j))-2(l_{i+1}-\Delta(i+1,Q(i)+1))-\Delta(Q(j),Q(i)+1) &\quad-\Delta(Q(i)+1-\alpha(Q(j),Q(i)),Q(i)+1) &=l_j-2l_{i+1}-[\Delta(j,Q(j)) + \Delta(Q(i)+1,i+1)+\Delta(Q(j),Q(i)+1)] &\quad- [\Delta(Q(i)+1,i+1)+\Delta(Q(i)+1-\alpha(Q(j),Q(i)),Q(i)+1)] & = l_j-2l_{i+1}-\Delta(j,i+1) -\Delta(Q(i)+1-\alpha(Q(j),Q(i)),i+1)\,\cdot
\Gamma_{n}=\ker \big(H_{n}(\L (W_{\leq n}))\overset{j_n}{\longrightarrow}W_n\big),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\forall n\geq 2.
U ( \lambda \mu \rho ) ={\frac{1}{d ( \rho )}}U_{k_{1}k_{2}}^{l_{1}l_{2}}\left( \begin{array}{c c c}{\lambda}&{\mu}&{\rho}\\{k_{1}}&{k_{2}}&{m}\\ \end{array}\right) \left( \begin{array}{c c c}{\lambda}&{\mu}&{\rho}\\{l_{1}}&{l_{2}}&{m}\\ \end{array}\right)^{\ast}.
F_2(n,k;\Delta^2)=-2\pi^2i\{\frac{1}{16k\mu^2}\frac{[\sqrt{k^2-\Delta^2}+k-i\mu]}{[k+i\frac{(\Delta^2-\mu^2)}{2\mu}]}+ \frac{1}{16 k \mu^2}\frac{[\sqrt{k^2 - \Delta^2}- k - i \mu]}{[k - i \frac{(\Delta^2 - \mu^2)}{2\mu}]}\},
R(\pi_\lambda(h_1)f)(h) =j_\lambda (h,z_0) (\pi_\lambda(h_1) f)(h\cdot z_0) =j_\lambda(h,z_0)j_\lambda (h_1^{-1},h\cdot z_0)f(h_1^{-1}h\cdot z_0) =j_\lambda(h_1^{-1}h,z_0)f(h_1^{-1}h\cdot z_0) =(\rho_\lambda (h_1)Rf)(h)\, .
\hat{T}_{0 0}= \frac{1}{2}\left[ \dot{\phi}^{2}+ \left( \vec{\nabla}\phi \right)^{2}+ m^{2}\phi^{2}+ \sigma \phi^{2}\right] = \frac{1}{2}\int d^{d}y \phi ( y ) \left( \hat{T}_{x}^{( 0 )}+ \hat{T}_{x}^{( 1 )}\right) \phi ( y )
\sum_{\substack{r=1 (r,n)=1}}^{\lfloor d/6\rfloor}\frac{1}{n-6r}\equiv \frac{1}{3}q_2(n)+\frac{1}{4}q_3(n)-\frac{1}{6}nq_2^2(n)-\frac{1}{8}nq_3^2(n) \pmod{n^2}.
R e \, \tilde{\Pi}_{F}^{( q ) \, \nu \mu}( Q ) = - \frac{g^{2}}{2}\, N_{f}\, R e \int \frac{d^{\, 4}K}{( 2 \pi )^{4}}\, t r \left[ \gamma^{\mu}\, S^{( 1 1 )}( K - Q ) \, \gamma^{\nu}\, S^{( 1 1 )}( K ) \right] \, .
F\supset \bigcap_{N=1}^{\infty}\bigcup_{n=N}^{\infty}\bigcup_{w\in \Sigma_{\beta_1}^n, v\in \Sigma_{\beta_2}^n}\tilde{J}_{n,\beta_1}(w)\times \tilde{J}_{n,\beta_2}(v):=\widetilde{F}.
f \circ (g_1,\ldots,g_n) \left(x^1_1,\ldots,x^1_m,x^2_1,\ldots,x^2_m,\ldots,x^n_1,\ldots,x^n_m \right)= &= f \left(g_1(x^1_1,\ldots,x^1_m),g_2(x^2_1,\ldots,x^2_m),\ldots,g_n(x^n_1,\ldots,x^n_m) \right).
E \Biggl[\check{\zeta}f_1\bigl(\check{\mathbf{L}}^n\bigr)\prod_{j=1}^{J-1}Y^n_{j-}Y^n_{j+}\Biggr] \to E \Biggl[\zeta f_1\bigl(\mathcal{W}^{S_{r_J}}\bigr)f_2\bigl(\widetilde{\mathcal{W}}(\beta)\bigr)\prod_{j=1}^JY_{j-}Y_{j+}\Biggr].
\psi (t)=\begin{cases}0, \mbox{if $b_{2n-2}\leq t <a_{2n-1}$}, \dfrac{t-a_{2n-1}}{b_{2n-1}-a_{2n-1}}, \mbox{if $a_{2n-1}\leq t <b_{2n-1}$}, 1, \mbox{if $b_{2n-1}\leq t <a_{2n}$,}\dfrac{b_{2n}-t}{b_{2n}-a_{2n}}, \mbox{if $a_{2n}\leq t <b_{2n}$}.\end{cases}
\begin{aligned}([1, g],1) ([\epsilon, 1], 1) =([\epsilon, g^{s(\epsilon)}], \widetilde{C}_{X^{\ast}}([1, g], [\epsilon, 1])) &=([\epsilon, g^{s(\epsilon)}], \nu( \epsilon,g)\widetilde{c}_{X^{\ast}}(g^{s(\epsilon)}, 1)) &=([\epsilon, g^{s(\epsilon)}], \nu( \epsilon,g)) &=([\epsilon, 1], 1)([1, g^{s(\epsilon)}],\nu( \epsilon,g)).\end{aligned}
\beta_{s,a^2}^2=\frac{r^2(s,a_s^1,a^2)-r^2(s,a_s^1,a_s^2)}{r^2(s,a_s^1,a^2)-\sum_{s'\in S}\left(\frac{\mu(s',s,a_s^1,a^2)}{||\mu||}+\delta(s,s')\right)r^2(s',a_{s'}^1,a_{s'}^2)},
\phi'(s)= \beta_- Ms^{\beta_--1}(\rho^{\epsilon}- \frac{\beta_- + \epsilon}{\beta_-}s^{\epsilon}), \phi''(s):= \beta_-(\beta_--1) \frac{\phi(s)}{s^2}- M \epsilon (2 \beta_- + \epsilon -1)s^{\beta_- +\epsilon-2}.
\frac{1}{\left(x^{2}+t^{2}\right)^{s}}=\frac{\sqrt{\pi}\,2^{\frac{1}{2}-s}}{\Gamma(s)t^{s-\frac{1}{2}}}\intop_{0}^{\infty}y^{s-\frac{1}{2}}J_{s-\frac{1}{2}}(ty)\,e^{-xy}\,dy,
\frac{\partial F_{\alpha}^{m}}{\partial X^{\beta}}=F_{\mu}^{m}\frac{C^{\mu\gamma}}{2}(\frac{\partial C_{\alpha\gamma}}{\partial X^{\beta}}+\frac{\partial C_{\beta\gamma}}{\partial X^{\alpha}}-\frac{\partial C_{\alpha\beta}}{\partial X^{\gamma}})
\begin{aligned}D_1(x,x,y) = x, D_n(x,y,y) = y, \begin{aligned}D_i(x,y,x) = x 1\leq i\leq n, D_i(x,y,y) = D_{i+1}(x,x,y) 1\leq i<n. \end{aligned}\end{aligned}
H_{k}\left( \theta,\theta^{\ast}\right)^{\ast}:=\sum_{\mathbf{b}=\left(b_{1},\dots,b_{k}\right) \in\left\{\theta,\theta^{\ast}\right\}^{k}}C_{k}\left( \mathbf{b}\right) b_{k}^{\ast}\dots b_{1}^{\ast}
z_{tt}(t, x) - \bar{a}_{ij}(t, x) \triangle z(t, x) = \bar{f}(t, x)\phantom{0}(t, x) \in (0, T) \times \Omega, z(t, x) = 0\phantom{\bar{f}(t, x)}(t, x) \in (0, T) \times \partial \Omega, z(0, x) = z^{0}(x), z_{t}(0, x) = z^{1}(x)\phantom{00}x \in \Omega
{\bf S}= \int d^8zE^{-1}\left[\frac{1}{2}{\bf D}_{\alpha\dot{\alpha}}L{\bf D}^{\alpha\dot{\alpha}}L -\frac{\imath}{12}\left(\nabla_{\alpha}T_{\dot{\alpha}}+ \nabla_{\dot{\alpha}}T_{\alpha}\right){\bf D}^{\alpha\dot{\alpha}}L\right]
E_{-x_{1}}(z) = P^{(\infty)}(z)e^{-\frac{\pi i \alpha}{2}\theta(z)\sigma_{3}}s_{2}^{\frac{\sigma_{3}}{2}}N^{-1}\left( 2\pi \sqrt{r}f_{-x_{1}}(z)^{1/2}\right)^{\frac{\sigma_{3}}{2}}.
\tilde{m_{0}}^{2}\stackrel{p \to \infty}{\rightarrow}\frac{m_{0}^{2}}{2 \operatorname{c o s h}^{2}p \sqrt{\alpha^{\prime}}}\simeq \frac{1}{2}m_{0}^{2}e^{- 2 \sqrt{\alpha^{\prime}}p}\, .
\Phi ( x , \theta ) = A ( x ) + \theta^{\mu}V_{\mu}( x ) +{\textstyle \frac 1 2}\theta^{\mu}\theta^{\nu}F_{\mu \nu}( x ) +{\textstyle \frac 1 6}\theta^{\mu}\theta^{\nu}\theta^{\rho}A_{\mu \nu \rho}( x ) +{\textstyle \frac{1}{2 4}}\theta^{\mu}\theta^{\nu}\theta^{\rho}\theta^{\sigma}B_{\mu \nu \rho \sigma}( x )
q_{C_n}= \sum_{C_{n+1}\!\hbox{\tiny{child of}}\! C_n}\Big(\frac{\mu(C_{n+1})}{\mu(C_n)}-\frac{\nu(C_{n+1})}{\nu(C_n)}\Big)_+=1 - \sum_{C_{n+1}\!\hbox{\tiny{child of}}\! C_n}\rho_{C_{n+1}}.
\dot{V}=\dot{x}\Big(-\nabla^2 f(x)\dot{x}- \Big[\frac{\partial^2 g}{\partial^2 x}\Big]^Tg(x)\dot{x}- \Big[\frac{\partial g}{\partial x}\Big]^T\Big[\frac{\partial g}{\partial x}\Big]\dot{x}\Big) \leq \dot{x}^T[-\nabla^2f(x)-k]\dot{x}\leq -\gamma V(x)
|A| = |\mathcal{L}(G - \sum_{i = 1}^{N}P_i)| - \sum_{j = 1}^{N}|A_j| + \dots + (-1)^{K - g + 1}\sum_{j_1, \dots, j_{K - g + 1}\in [N], j_1 \neq \dots \neq j_{K - g + 1}}|\cap_{\alpha = 1}^{K - g + 1}A_{j_\alpha}|.
( \mathrm{M a s s})^{2}={\frac{1}{1 6 \lambda_{2}^{( 0 )}}}\hat{\alpha}^{a}( I + L )_{a b}\hat{\alpha}^{b}={\frac{1}{8 \lambda_{2}^{( 0 )}}}( \hat{\alpha}_{R})^{2}~ ,
I_5 = \Big[-\frac{n^2-8n+8}{3(n-1)(n-2)}|W(p)|^2r^{4-n}-\frac{32}{9(n-2)}\sum_{k,l,s}\big((W_{ikls}(p)+W_{ilks}(p))x^i\big)^2 r^{2-n}~~ -\frac{16(7n-8)}{n-2}\sigma_1(A)_{,ij}(p)x^ix^j r^{2-n}\Big](n-6)+O(r^{5-n}).
\mathfrak{F}_{t,k,q}^{(\chi_{x}^{(\Omega )}(\omega ))}\left( \Lambda\right) =\mathfrak{F}_{t,k,q}^{(\omega )}\left( \Lambda +x\right) \ ,\Lambda \in \mathcal{P}_{f}(\mathfrak{L}),\x\in \mathbb{Z}^{d}\ .
\mathbb{E}^{0}({L_{b}\bf 1}_{a<\tau_{x}<b}|\mathcal{F}_{t}^{Q}) = \mathbb{P}^{0}(a<\tau_{x}<b) + \displaystyle \int_{0}^{t}\mathbb{E}^{0}({\bf 1}_{\tau_{x}>u}L_{u}h(X_{u})[G(a-u,x-\!X_{u})-G(b-u,x-\!X_{u})]| \mathcal{F}_{u}^{Q})dQ_{u}.
(\lambda t_\alpha + \mu e^{\alpha}) \cdot (x e^\beta + e^{\alpha + \beta}) = \big(\lambda x a \vert \alpha \cap \beta \vert + \mu b_{\alpha,\beta}\big)e^\beta & + \big(\lambda a \vert \alpha \cap (\alpha + \beta) \vert + \mu x b_{\alpha, \beta}\big) e^{\alpha + \beta}
z=x+iy ,\bar{z}=x-iy \mbox{avec}i=\sqrt{-1}s=u_1+u_2,t=v_1+v_2,p=u_1^2+u_2^2,q=v_1^2+v_2^2 a=a_1+a_2,r=u_1v_1+u_2v_2,d=a_1u_1+a_2u_2,f=a_1v_1+a_2v_2
\|\hat{p}^{(k)}(\cdot|Z^n)-\mu_k^{(b)}\|_1 \leq{t\over n-k}\sum_{i=1}^{k+g}\|\hat{p}^{(1)}(\cdot|S^{(i),t})-\mu_k^{(b)}(\cdot)\|_1+|{t(k+g)\over n-k}-1|+{k+g\over n-k}.
(\Lambda_1)^*\cap(\Lambda_2)^*\cap Z_* = (M_1)_*\cap(M_2)_*\cup(Z^*\cap Z_*) (\Lambda_1)_*\cap(\Lambda_2)_*\cap Z^* = (M_1)^*\cap(M_2)^*\cup(Z^*\cap Z_*) \emptyset = (M_1)_*\cap(M_2)^* =(M_1)^*\cap(M_2)_*.
\Pi_{\mu_{1}\mu_{2}}^{1}( k , u ) = \frac{1}{2 \pi i}\int \frac{\mathrm{d}^{3}Q}{\left( 2 \pi \right)^{3}}\int_{\cal{C}}\mathrm{d Q_{0}}N \left( Q_{0}\right) t_{\mu_{1}\mu_{2}}^{\alpha \beta \rho \sigma}\left( Q_{0},{\bf Q}, Q_{0}+ k_{0},{\bf Q}+{\bf k}\right) D_{\alpha \beta}( Q ) D_{\rho \sigma}( Q + k ) ,
\lvert (i,j)\rvert_{W_{n_1}(w)}-\lvert (i,j)\rvert_{W_{n_2}(w)}= (\lvert l_i \rvert_S+\lvert m_{n_1}\rvert_S+\lvert r_j \rvert_S) &-(\lvert l_i \rvert_S+\lvert m_{n_2}\rvert_S+\lvert r_j \rvert_S) = \lvert m_{n_1}\rvert_S-\lvert m_{n_2}\rvert_S.
\varepsilon_{D}\left( L \right) = \frac{c ( D ) \pi^{( D - 1 ) / 2}}{2^{D - 1}L^{D - 1}}\, \Gamma \left( \frac{1 - D}{2}\right) \sum_{n = 0}^{\infty}( n + \frac 1{2})^{D - 1}.
-l_1^{(1)}\hat{\bf a}_1^{(1)}- \sum_{i=2}^M l_0^{(i)}\hat{\bf a}_0^{(i)}= l_0^{(1)}\hat{\bf a}_0^{(1)}+ \sum_{j=2}^{N_1}l_j^{(1)}\hat{\bf a}_j^{(1)}+ \sum_{i=2}^M\sum_{j=1}^{N_i}l_j^{(i)}\hat{\bf a}_j^{(i)}.
\mathrm{SG}_m:=\left\{{\left.\begin{pmatrix}\cos(2mq^4) -\sin(2mq^4) 0 0 q^1 \sin(2mq^4) \cos(2mq^4) 0 0 q^2 0 0 1 0 q^3 0 0 0 1 q^4 0 0 0 0 1\end{pmatrix}\right|q\in\mathbb{R}^4}\right\}.
\chi (t)=\frac{\sum\nolimits_{k=0}^{t}k3^{k}}{t\sum\nolimits_{k=0}^{t}3^{k}}=\frac{(3t-\frac{3}{2})+\frac{3}{2}\frac{1}{3^{t}}}{(3t)(1-\frac{1}{3^{t+1}})}\leq 1,
\overline{V}(2,f_{(1,1)},\pi_n)\overset{\mathbb{P}}{\longrightarrow}\int_0^T \rho_s \sigma^{(1)}_s \sigma^{(2)}_s d{H_{0,0,2}}(s)=\int_0^T \rho_s \sigma^{(1)}_s \sigma^{(2)}_s ds
\rho_k-\rho_{k+1}\geq \frac{(X_{i_{k+1}^{**}}^\top M_{\mathcal{S}_{k}}Y)^2}{X_{i_{k+1}^{**}}^\top X_{i_{k+1}^{**}}}-n\lambda_n\sum_{j=1}^{s_{k}+1}|\hat{\beta}_j^{**}|.
f(x)=\bigoplus_{n=0}^{\infty}(a_{n}\otimes x^{\otimes s_{n}})=a_{0}\otimes x^{\otimes s_{0}}\oplus a_{1}\otimes x^{\otimes s_{1}}\oplus\cdots\oplus a_{n-1}\otimes x^{\otimes s_{n-1}}\oplus a_{n}\otimes x^{\otimes s_{n}}\cdots ,
\begin{array}{r l}{{u_{1 0}}}&{{= u_{0}+ \frac{p^{+}}{\sqrt{2}}}}\\{{v_{1 0}}}&{{= \frac{1}{u_{0}+ \frac{p^{+}}{\sqrt{2}}}\left\{1 - \frac{[ 1 - ( u_{0}+ \frac{p^{+}}{\sqrt{2}}) ( v_{0}+ p^{-}\sigma_{p e r}^{+}) ] \times [ 1 - ( u_{0}+ \frac{p^{+}}{\sqrt{2}}) ( v_{0}+ p^{-}\sigma_{p e r}^{-}) ]}{[ 1 - ( u_{0}+ \frac{p^{+}}{\sqrt{2}}) ( v_{0}- \frac{p^{-}}{\sqrt{2}}) ]}\right\}}}\\ \end{array}
\begin{aligned}\|Af\|_{\infty}\leq 2\|f\|_{(2)}\sup_{|x| \leq R}\left( |b(x)|+|Q(x)| + \int_{y \neq 0}\min\{|y|^2,1\}\, \nu(x,dy) \right) + \|f\|_{\infty}\sup_{|x|>R}\nu(x,\overline{B(-x,R)}). \end{aligned}
\begin{array}[c]{l}\left.\frac{\partial V}{\partial u}(s,\bar{X}^{t,x;\bar{u}}(s),W(s,\bar{X}^{t,x;\bar{u}}(s)),W_x(s,\bar{X}^{t,x;\bar{u}}(s)),u)\right\vert_{u=\bar{u}(s)}=\left( 1-W_{x}(s,\bar{X}^{t,x;\bar{u}}(s))\sigma_{z}\left( s\right) \right)^{-1}W_{x}(s,\bar{X}^{t,x;\bar{u}}(s))\sigma_{u}(s).\end{array}
\langle \mathbb{A}_r \frac{\sigma_k^i-2\sigma_k^{i-1}+\sigma_k^{i-2}}{\delta_k^2}, \sigma_k^i-\sigma_k^{i-1}\rangle+\langle \frac{p_k^i-2p_k^{i-1}+p_k^{i-2}}{\delta_k^2}, \sigma_k^i-\sigma_k^{i-1}\rangle =\langle \frac{E u_k^i-2 E u_k^{i-1}+E u_k^{i-2}}{\delta_k^2}, \sigma_k^i-\sigma_k^{i-1}\rangle
\left( \begin{array}{cc}M_h -K_h -K_h -\lambda^{-1}M_{h,\mathcal{I}_{k}}\end{array}\right) \left( \begin{array}{c}\underline{y}_{h,k}\underline{p}_{h,k}\end{array}\right) = \left( \begin{array}{c}\underline{y}_d -(\underline{f}+\underline{u}_d) - \underline{u}_a - \underline{u}_b \end{array}\right),
\sum_{u\in U}d_s(u)\, \le\,\tau^{r-s}\mu(U)nd+\tau^{r-s}nd\,2^{{r\choose2}-1}\sum_{j=2}^{r-s+1}2^{-{s+j-2\choose2}-s+1}\delta_j &=\,\tau^{r-s}\mu(U)nd+\tau^{r-s}nd\,2^{{r\choose2}-1}\sum_{j=2}^{r-s+1}2^{-{s-1\choose2}-(s-1)j-{j-1\choose2}}\delta_j &\le\,\tau^{r-s}\mu(U)nd+4^{1-s}\tau^{r-s}nd\,\delta(G,\tau)
\sum_{m \leq Y}\frac{c_m}{m^l}= \int_1^Y \frac{d \left( \sum_{m \leq u}c_m \right)}{u^l}&= \frac{\sum_{m \leq u}c_m}{u^l}\bigg|_1^Y - \int_1^Y (-l) \frac{\sum_{m \leq u}c_m}{u^{l+1}}du.
(\partial_{t}-\underline{\Delta})(tF) =F+t(\partial_{t}-\underline{\Delta})(\varphi \phi) &= F-2tg'(\overline{\nabla}\varphi,\overline{\nabla}\phi)+t\varphi(\partial_{t}-\underline{\Delta})\phi+t\phi(\partial_{t}-\underline{\Delta})\varphi.
\mathcal{I}_1 \le |\mathcal{I}_2|+|\mathcal{I}_3|+\int_\Omega \left|\delta_h f\right|\, \left|\frac{\delta_h u}{|h|^{\gamma+t}}\right|\,\eta^p\,dx &\le |\mathcal{I}_2|+|\mathcal{I}_3|+(1-s)^\frac{1}{p-1}\,R^{s\,p'}\,\int_{B_R}\left|\frac{\delta_h f}{|h|^s}\right|^{p'}\,dx+\frac{1}{(1-s)\,R^{s\,p}}\int_{B_R}\left|\frac{\delta_h u}{|h|^{\gamma+t-s}}\right|^p\,dx.
U_{1,n}:= \sqrt{n}\left(S_{1,n}(\beta^\prime) - E(S_{1,n}(\beta^\prime)) \right)= \frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t,\beta^\prime}^2 - E e_{t,\beta'}^2; U_{2,n}:= \frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t,\beta^\prime}c_{nt}
\tilde{\gamma}_{1}(\gamma_1, Z_{a}h, V_a) =\tilde{\gamma}_{1}(\gamma_1, 0, V_a)+\tilde{\gamma}_{1}(\gamma_1, Z_{a}h, V_a)-\tilde{\gamma}_{1}(\gamma_1, 0, V_a)=\gamma_1+O(1)Z_{a}h.
d\,R_{g}(A|_{h})=a^{M}(h)\frac{\partial x^{N}(hg)}{\partial x^{M}(h)}\frac{\partial}{\partial x^{N}(hg)}=a^{M}(h) R^{N}_{M}(hg, h)\frac{\partial}{\partial x^{N}(hg)}\,,
{F_{\alpha_{1}\alpha_{2}}}{F_{\alpha_{2}\alpha_{3}}}{F_{\alpha_{3}\alpha_{4}}}{F_{\alpha_{4}\alpha_{1}}}\rightarrow{F_{1 \alpha_{1}\alpha_{2}}}{F_{2 \alpha_{2}\alpha_{3}}}{F_{3 \alpha_{3}\alpha_{4}}}{F_{4 \alpha_{4}\alpha_{1}}}
R_{h}(t,s)= \tfrac{c^2_{h_{t,s}}}{c_{(h(t))}c_{(h(s))}}\ \big[\tfrac{1}{2}\big({|t|}^{2h_{t,s}}+{|s|}^{2h_{t,s}}-{|t-s|}^{2h_{t,s}}\big)\big],
I_1^{(n)}\leq \int_{-2/(3\pi)}^{1/(\pi n+\pi/2)}\frac{t^2dt}{(t-\frac{1}{\pi n})^2}=\bigg[t-\frac{1}{(\pi n)^2}\frac{1}{(t-\frac{1}{\pi n})}+\frac{2}{\pi n}\log \bigg\vert t-\frac{1}{\pi n}\bigg\vert\bigg]_{-2/(3\pi)}^{1/(\pi n+\pi/2)}&=\frac{2}{3\pi}+O(n^{-1}\log n).
\begin{array}{c}{{u ( \sigma^{+}, \sigma^{-}) = u_{0}+ \frac{p^{+}}2 \left[ ( \sigma^{+}+ f ( \sigma^{+}) ) + ( \sigma^{-}- g ( \sigma^{-}) ) \right]}}\\{{v ( \sigma^{+}, \sigma^{-}) = v_{0}+ \frac{p^{-}}2 \left[ ( \sigma^{+}- f ( \sigma^{+}) ) + ( \sigma^{-}+ g ( \sigma^{-}) ) \right]}}\\ \end{array}
H_i(\sigma) := H_{f_i}(\sigma) = \left\lbrace\beta\in K_{p+1}\ |\ \beta \supset \sigma f_i(\beta)\leq f_i(\sigma)\right\rbrace, T_i(\sigma) := T_{f_i}(\sigma) = \left\lbrace\alpha\in K_{p-1}\ |\ \alpha \subset \sigma f_i(\alpha)\geq f_i(\sigma)\right\rbrace.
\left( \sum_{n \leq X}\lvert S_f(n) \rvert \right)^2 = \left( \sum_{n \leq X}\lvert S_f(n) \rvert \cdot 1 \right)^2 \leq \sum_{n \leq X}\lvert S_f(n) \rvert^2 \sum_{m \leq X}1 = X \sum_{n \leq X}\lvert S_f(n) \rvert^2 \ll X^{k-1 + \frac{5}{2}}.
\omega'_{\log}( b_1,\ldots,b_{p-1},a_I\cdot \tau^{(r+k)I},b_{p+1},\ldots,b_k)=(p\circ{\pi}_{0,k})^*\tau^{rI}\cdot \omega'_{\log,p,I}(b_1,\ldots,b_{p-1},a_I,b_{p+1},\ldots,b_k).
\left( \xi_m(x_0,z), \xi_m(x_0,z')\Big| \mathcal B \right)=\sum_{j, k=1}^m \int_0^t \gamma_{\epsilon_n}\left( B_{0,t}^j(t-s) - B_{0,t}^k(t-s) + \frac{s}{t}(z-z')\right) ds\,.
\sum_{j = 1}^{N}\left( \frac{m}{2 \epsilon}\Delta{r_{j}}^{2}+ \frac{m}{2 \epsilon}{r_{j}}^{2}\left( 1 - \frac{1}{2}\frac{\Delta r_{j}}{r_{j}}\right)^{2}\Delta{\theta_{j}}^{2}\right) .
\sum_{k=0}^n \binom{p+c+k}{p}\binom{q+n-k}{q}\binom{c+k}{c}\binom{n-k}{d}H_{p+c+k}={}\binom{n+p+q+c+1}{n-d}\binom{p+c}{c}\binom{q+d}{d}(H_{n+p+q+c+1}-H_{p+q+c+d+1}+H_{p+c})
+A_{kp}\frac{f^{(p)}(z+c_{k})}{f(z+c_{k})}\frac{f(z+c_{k})}{f(z+c_{l})}+\sum_{j=1}^{m}A_{lj}\frac{f^{(j)}(z+c_{l})}{f(z+c_{l})}-\frac{F(z)}{f(z+c_{l})}.
x(t) = \frac{x_{(2)}(t)(x_{(3)}(t) - x_{(1)}(t)) + \rho x_{(3)}(t) (x_{(1)}(t) - x_{(2)}(t))}{(x_{(3)}(t) - x_{(1)}(t)) + \rho (x_{(1)}(t) - x_{(2)}(t))},
\begin{array}{lllll}\hat{f}(\xi) = \displaystyle \hat{u_*}+i\int_1^t\int e^{\frac 12i(\eta^2-2\xi\eta)s}e^{-s}{\hat{f}(\xi-\eta,s)\hat{v_*}(\eta,s)}d\eta ds & \displaystyle +i\int_1^t\int_1^s\iint e^{-i\eta\sigma s}F(s,\xi,\xi-\sigma,\eta)dsd\eta d\sigma.\end{array}
\tilde{q}_{\scriptscriptstyle L}(t) =-2\sigma_1 e^{-j\phi_L-2j\omega_1t}(2\sigma_1(t-t_L)), t<t_L \tilde{q}_{\scriptscriptstyle R}(t) = -2\sigma_1 e^{-j\phi_R-2j\omega_1t}(2\sigma_1(t-t_R)), t>t_R
{\cal G}_{0}( \omega ,{\bf k}; z , z^{\prime}) = \int_{- \infty}^{\infty}\frac{d q}{2 \pi}\, \frac{e^{i q ( z - z^{\prime})}}{\omega^{2}+{\bf k}^{2}+ q^{2}+ m^{2}}= \frac{e^{- \sigma | z - z^{\prime}|}}{2 \sigma}\, ,
\mathcal{K}_m' (x) v (s) = \tilde{h}\sum_{j=1}^m \sum_{i=1}^\rho w_i \; \frac{\partial \kappa}{\partial u}(s, \zeta_i^j, x (\zeta_i^j)) v (\zeta_i^j), \;\;\; s \in [a, b].
\left(\left(I_r\otimes \left[\begin{array}{cccc}C_1 \vdots C_n\end{array}\right]\right)A\left(I_r\otimes \left[\begin{array}{cccc}D_1 \vdots D_n\end{array}\right]\right)'\right) = \sum^n_{i=1}C_i(A)D_i'.
\begin{array}{c}a^{A*}=a^{A}~~~~b^{A*}=b^{A}~~~~\omega_{AB}{}^{*}=\omega_{AB}~~~~\lambda^{*}=\lambda~~~~T^{\dagger}=-T{}\bar{\rho}_{i}=\rho^{i\dagger}\gamma^{0}~~~~~~~\bar{\varepsilon}_{i}=\varepsilon^{i\dagger}\tilde{\gamma}^{0}\end{array}
\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty}\cdot\cdot\cdot\int_{-\infty}^{\infty}\prod_{i=1}^{n}e^{-t_{i}^{2}/2}\prod_{1\le i<j\le n}|t_{i}-t_{j}|^{2\gamma}dt_{1}\cdot\cdot\cdot dt_{n}=\prod_{j=1}^{n}\frac{\Gamma(1+j\gamma)}{\Gamma(1+\gamma)}.
I = \int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}}d \mu f^{- \tilde{\gamma}}\left[ ( K^{\prime})^{2}+ ( \tilde{\gamma}^{2}- 9 / 4 ) K^{2}\right] >{-{\frac{9}{4}}}\int_{-{\frac{\pi}{2}}}^{\frac{\pi}{2}}d \mu f^{- \tilde{\gamma}}K^{2}\ ,
\sum_{i=0}^p\lambda_if(V_i^+,V_i^-) =\sum_{i=0}^p \int_{t_{i-1}}^{t_i}f(V_t^+(\vec x),V_t^-(\vec x))dt &=\int_{0}^{t_p}f(V_t^+(\vec x),V_t^-(\vec x))dt=f^L(\vec x).
f ~ = ~ \left( 1 + \frac{a^{2}\sinh^{2}\alpha}{\bar{r}^{2}}\right) ~ \left( 1 + \frac{a^{2}\sinh^{2}\gamma}{\bar{r}^{2}}\right) ~ \left( 1 + \frac{a^{2}\sinh^{2}\sigma}{\bar{r}^{2}}\right)
n B_{2}+ l A_{2}= n U_{l}^{n}S t r \left( \ldots \phi^{i_{2 k}^{\prime}}\ldots \underbrace{\phi^{i_{2 k - 1}^{\prime}}\phi^{i_{n}}}\ldots \partial_{a_{l}}\phi^{j_{l}}\ldots \right) + l U_{l}^{n}S t r \left( \ldots \phi^{i_{2 k}^{\prime}}\ldots \underbrace{\phi^{i_{2 k - 1}^{\prime}}\partial_{a_{l}}\phi^{j_{l}}}\ldots \right)
\begin{array}{r c l}{{x^{i}x^{j}}}&{{=}}&{{q x^{j}x^{i}~ , ~ ~ \mathrm{f o r}~ i < j ~ ~ \mathrm{a n d}~ ~ i \not = - j ~ ,}}\\{{x^{i}x^{- i}}}&{{=}}&{{x^{- i}x^{i}+ \lambda q^{i - 2}L_{i - 1}}}\\{{}}&{{=}}&{{q^{- 2}x^{- i}x^{i}+ \lambda q^{i - 2}L_{i}~ , ~ ~ \mathrm{f o r}~ i > 0 ~ ,}}\\ \end{array}
\sum_{1\leq j,k \leq n}f_j(\tau_k)^mf_k(\tau_j)^m=\sum_{j=1}^n\sum_{k=1}^nf_j(\tau_k)^mf_k(\tau_j)^m\geq \frac{1}{(G_{f,\tau}^{\circ^m})}\left(\sum_{j=1}^nf_j(\tau_j)^m\right)^2.
\frac{2}{3}{u}_{j-\frac{1}{2}}^-+\frac{1}{3}{u}_{j+\frac{1}{2}}^- =\frac{1}{6}(\bar{u}_{j-1}+5\bar{u}_{j}), \frac{1}{3}{u}_{j-\frac{1}{2}}^-+\frac{2}{3}{u}_{j+\frac{1}{2}}^- =\frac{1}{6}(5\bar{u}_{j}+\bar{u}_{j+1}), \frac{2}{3}{u}_{j+\frac{1}{2}}^-+\frac{1}{3}{u}_{j+\frac{3}{2}}^- =\frac{1}{6}(\bar{u}_{j}+5\bar{u}_{j+1}).
\gamma_* \O(x,y)(ze) = \O(x,y)\otimes \rho_* \O_\rho(z) - \O(x-3,y-1)\otimes \rho_* \O_\rho(z-1) \sum_{0 \leq h \leq z}\O(x-h,y)-\sum_{0 \leq k \leq -z -2}\O(1+x+k,y)-\sum_{0 \leq j \leq z-1}\O(x-3-j,y-1)+\sum_{0 \leq l \leq -z-1}\O(x+l-2,y-1)
\sum_{k = 0}^{+ \infty}\alpha_k \epsilon_V[t_k] \leq 2 G^2 \sum_{k = 0}^{+ \infty}\alpha_k \sum_{h = 1}^{k - c(k)}\alpha_{k-h+1}& \leq 2 G^2 \sum_{k = 0}^{+ \infty}\alpha_k \sum_{h = 1}^{M-1}\alpha_{k-h+1}& \leq 2 G^2 \sum_{k = 0}^{+ \infty}\sum_{h = 1}^{M-1}\eta^2_{k-h+1}
\overline{\omega}^m\left(\frac{b^{d-1}d^d}{a^d(d-1)^{d-1}}\right)\prod_{i=0}^{r-1}\prod_{h=1}^{d-2}\frac{\Gamma_p(\langle(\frac{h}{d-1}-\frac{m}{q-1})p^i\rangle)}{\Gamma_p(\langle\frac{hp^i}{d-1}\rangle)}\prod_{\substack{h=1 h\neq\frac{d}{2}}}^{d-1}\frac{\Gamma_p(\langle(\frac{h}{d}+\frac{m}{q-1})p^i\rangle)}{\Gamma_p(\langle\frac{hp^i}{d}\rangle)}=\phi(b).
[z^{2n}][u]g(z,u)=[z^{2n}]\frac{1}{1-z}(\frac{z^{n+1}}{n+1}+\frac{z^{n+2}}{n+2}+\cdot\cdot\cdot)=\sum_{k=n+1}^{2n}\frac{1}{k}=H_{2n}-H_{n}.
U^{a_1b_1}U^{a_2b_2}U^{a_3b_3}f_{\tau \alpha,\tau_1 \alpha_1,\tau_2\alpha_2, \tau_3 \alpha_3,b_1,b_2,b_3}\Psi= f_{\tau \alpha,\tau_1 \alpha_1,\tau_2 \alpha_2,\tau_3 \alpha_3,a_1,a_2,a_3}\Psi,
E[(\sigma'\sigma)^2(X_{\eta_n(s_1)})\Lambda^n_{s_1}(\sigma'\sigma)^2(X_{\eta_n(s_2)})\Lambda^n_{s_2}] =E[(\sigma'\sigma)^2(X_{\eta_n(s_1)})\Lambda^n_{s_1}(\sigma'\sigma)^2(X_{\eta_n(s_2)})E[\Lambda^n_{s_2}|\mathcal{F}_{\eta_n(s_2)}] ]=0.
\int_{\mathbb{R}^4}F(x+u,y)G(x,y+u)F(x+v,y)G(x,y+v) \delta_{(0,0)}(u,v) \,dx dy du dv = \int_{\mathbb{R}^4}F(x,y)^2 G(x,y)^2 \,dx dy \leq \|F\|_{\textup{L}^4(\mathbb{R}^2)}^2 \|G\|_{\textup{L}^4(\mathbb{R}^2)}^2 = 1 ,
E_m(z)=\displaystyle\prod_{n\in\mathbb{Z}^*, n\neq m}\left|1+\frac{iz}{\lambda_n}\right| \ \ \ \ \Q_m=\displaystyle\prod_{n\in\mathbb{Z}^*, n\neq m}\left|\frac{\lambda_n}{\lambda_n-\lambda_m}\right|.
\int_{\Omega}\sum_{i,j=1}^N a_{ij}\frac{\partial u}{\partial x_j}\frac{\partial u}{\partial x_i}\leq \lim_{m\rightarrow \infty}\inf \int_{\Omega}\sum_{i,j=1}^N a_{ij}\frac{\partial v_m}{\partial x_j}\frac{\partial v_m}{\partial x_i}.
\mbox{(f)}+\mbox{(g)}= -\frac{d_{\Gamma}g^2}{2x_{ij}^4}\sum_{k}\frac{x_{jk}^{\mu}x_{ik}^{\nu}+x_{jk}^{\nu}x_{ik}^{\mu}-\delta^{\mu\nu}x_{jk}\cdot x_{ik}}{x_{jk}^2 \, x_{ik}^2},
(2\pi)^{-1}\int_{-\infty}^{\infty}e^{-ix}(A - ix)^a ( B - ix)^b dx = \frac{1}{\Gamma(a) \Gamma(b)}\int_0^1 (1-t)^{a-1}t^{b-1}e^{-(A - (A-B)t)}dt
a_j\cos\theta_{j+1}^\bot+b_j\sin\theta_{j+1}^\bot =a_{j+1}\cos\theta_{j+1}^\bot+b_{j+1}\sin\theta_{j+1}^\bot > \max_{k\neq j,j+1}\{a_k\cos\theta_{j+1}^\bot +b_k\sin\theta_{j+1}^\bot\},
| \Phi \rangle = \sum_{r , s}\, b_{\overline{\jmath}_{1}. . . \overline{\jmath}_{s}}^{i_{1}. . . i_{r}}\, \lambda_{i_{1}}\dots \lambda_{i_{r}}\, \psi^{\overline{\jmath}_{1}}\dots \psi^{\overline{\jmath}_{s}}| 0 \rangle ~ ,
S_{-}( \eta ) = \left( \begin{matrix}{{\cosh \eta}}&{{\sinh \eta}}&{{0}}&{{0}}\\{{\sinh \eta}}&{{\cosh \eta}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\cosh \eta}}&{{- \sinh \eta}}\\{{0}}&{{0}}&{{- \sinh \eta}}&{{\cosh \eta}}\\ \end{matrix}\right) ,
I^{N,aV}_{\beta}(\lambda^1,\ldots, \lambda^k) =\biggl(1+O\Bigl(\frac{1}{N}\Bigr)\biggr)e^{\sum_{l=0}^2 N^{2-l}F_l^{a}(L^N_1,\cdots, L^{N}_d,\tau^N_B)},
d(u,\tilde{u})= - \frac12 \sum_{n=1}^M \Big( \|u_{-}^n\|_{t_n}^2 - \|u_{+}^{n-1}\|_{t_{n-1}}^2 \Big) + \frac12 \|u_{-}^M\|_{t_M}^2 + \frac12 \sum_{n=1}^M \|[u]^{n-1}\|_{t_{n-1}}^2.
(n-4) \langle J_{ab}, F \rangle =\sum_{k\neq i,j}\langle [J_{ab},J_{ki}],\omega_{ki}\rangle + \langle [J_{ab},J_{kj}],\omega_{kj}\rangle &= \sum_{k\in\{a,b\}}\langle [J_{ab},J_{ki}],\omega_{ki}\rangle + \langle [J_{ab},J_{kj}],\omega_{kj}\rangle
I_{m,q}(\vec{c})=\left<\vec{v},e_d\right>\int_{\xi}p_{\frac{q}{Q}}(\xi)\int_{t}w_0((\tilde{\vec{u}},\beta)) e\big( \xi y-\frac{\left<\vec{t}, \vec{c}\right>}{mq}\big) d\tilde{u}d\beta d \xi,
G(z) = -\frac{1}{2}\int_{\partial \mathbb{D}}\frac{(b-a)^2}{4}\frac{1}{v}\left ( v - \frac{1}{v}\right )^2 \frac{\psi(J_{[a,b]}(v))}{z - J_{[a,b]}(v)}\mathrm{d}v,
\sum_{\substack{q\leq \frac{X^{1/2}}{M^{6/5}}(\log X)^{-B}(q,Mp_0)=1}}\max_{(a,qM)=1}\;\Big\lvert\psi(X;qM,a) - \frac{\psi(X)}{\varphi(qM)}\Big\rvert\ll\frac{X}{\varphi(M)}(\log X)^{-A},
\tilde{a}=\frac{a}{a_{0}},\rho_{c}=\frac{3H_{0}^{2}}{8\pi G},\Omega=\frac{\rho}{\rho_{c}},t=\frac{\tilde{t}}{H_{0}},\Omega_{k}=-\frac{kc^{2}}{H_{0}^{2}a_{0}^{2}},
\varphi(a_1\pi_{\chi}(B_1)+a_2\pi_{\chi}(B_2))=\dfrac{1}{3}(a_2,a_1), &\varphi'(a_1\pi_{\overline{\chi}}(B_1)+a_2\pi_{\overline{\chi}}(B_2))=\dfrac{1}{3}(a_2,a_1).