harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal I(\rho^0,\rho^T)=\inf\{\mathcal S(\rho_t,\Phi_t)| (-\Delta_{\rho_t})^{\dagger}\Phi_t \in \mathcal T_{\rho_t}\mathcal P_{o}(G),\rho(0)=\rho^0, \rho(T)=\rho^T\}
A = \frac{\pi}{4}\left( L^{2}- \frac{1}{\mu^{2}}\right) + \frac{\pi \left( y^{2}+ z^{2}\right)}{\operatorname{l n}\left( \mu L \left( 1 - \frac{4 \left( w^{2}+ x^{2}\right)}{L^{2}}\right) \right)}
\sum_{n = 1}^N\frac{p_n^2}{2}+ \int_{k \in{\mathbb R}^3}\omega(k)a_k^{\dagger}a_k\,dk + \sum_{n = 1}^N\sqrt{\alpha}\int_{k \in{\mathbb R}^3}\frac{\chi_{\Lambda}}{\sqrt{\omega(k)}}(e^{ikx_n}a_k + e^{-ikx_n}a_k^{\dagger})\,dk.
K^{(2)}_{ij}:=\frac{\partial^{2}L^{(2)}}{\partial\ddot{q}^{i}\partial\ddot{q}^{j}},E^{(2)}_{ij}:=\frac{\partial^{2}L^{(2)}}{\partial\ddot{q}^{i}\partial\dot{q}^{j}}-\frac{\partial^{2}L^{(2)}}{\partial\ddot{q}^{j}\partial\dot{q}^{i}}.
\begin{array}{| c | | c | c | c | c |}\hline{\mathrm{c u r r e n t s}}&{J^{+ i j}}&{J^{+ - i}}&{J^{i j k}}&{J^{- i j}}\\ \hline{\hline \mathrm{d i m e n s i o n s}}&{- 1}&{1}&{1}&{3}\\ \hline{}\\ \end{array}
\langle \Psi_{f}| V | \Psi_{i}\rangle = \frac{2 \pi}{R}\lambda \frac{R^{y}}{( 2 \pi )^{y - 1}}\langle \Psi_{f}| \Phi ( 1 ) | \Psi_{i}\rangle_{\mathrm{p l a n e}}\ ,
\lvert u_n\rvert_S =\lvert(\phi v_0)_n\rvert_S-(n-j) &=\lvert(\phi v_0)_n\rvert_S-n+j &\geq \lvert(\phi v_0)_n\rvert_S-n+(n-2\lvert I\rvert+1) &=\lvert(\phi v_0)_n\rvert_S-2\lvert I\rvert+1
P_\lambda(x_1,\ldots,x_n;q,t) = \frac{\prod_{b\in\mu}1-q^{l(b)+1}t^{a(b)+1}}{\prod_{b\in\delta}1-q^{l(b)}t^{a(b)+1}}E_{0^n\mu}(x_1,\ldots,x_n;q,t),
\mu_n= n\pi-i \sum_{k=1}^{N+1}\frac{1}{(2i\mu_n)^k}\int_0^1 \varphi_1r_k\,dt + \frac{\hat\varphi_1(-n)+\hat\varphi_2(n)}{2}+ \frac{\ell^2_n}{n^{N+1}}.
\begin{array}{lcl}F_1\,=\,dD_0, \quad F_2\,=\,dD_1, F_3\,=\,dD_2- B_2\wedge dD_0, F_4\,=\,dD_3- B_2\wedge dD_1, F_5\,=\,dD_4- B_2\wedge dD_2 +{1\over 2}B_2\wedge B_2\wedge dD_0. ~\end{array}
L= \frac{1}{2}\left( \partial_\mu \varphi \right)^2 +\frac{1}{2}m^2 \varphi^2 + \frac{\lambda}{4!}\varphi^4 +\frac{1}{2}\xi R \varphi^2 + \Lambda_0 + \kappa R +\widetilde{a}_1R^2.
\Xi^{\lambda}(K_{a}F_{a}\otimes1\otimes E_{a}K_{a}^{-1}\otimes K_{a})=\sum_{i,j,m,n,o,p}(2c_{j}^{i}-\delta_{j}^{i})\pi(K_{a}F_{a})_{m}^{j}c_{n}^{m}\pi(E_{a}K_{a}^{-1})_{o}^{n}c_{p}^{o}\pi(K_{a}K_{\lambda})_{i}^{p}.
\int_{B_R}\int_{\mathbb{R}^N\setminus B_R}\frac{|\psi(x)|^p}{|x-y|^{N+\alpha\,p}}\,dx\,dy =\int_{B_r}\int_{\mathbb{R}^N\setminus B_R}\frac{|\psi(x)|^p}{|x-y|^{N+\alpha\,p}}\,dx\,dy &\le \frac{C}{\alpha\,(R-r)^{\alpha\,p}}\,\int_{B_r}|\psi|^p\,dx.
\hat{V}^{\epsilon}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{- q^{2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- q^{2}}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) \quad , \quad
\epsilon_{\rm Cas}(x)=\sum_j \frac{1}{2}\left( 1 - \frac{2}{\pi}\arctan\frac{\omega_j}{\sqrt{m^2- \omega_j^2}}\right)\omega_j\phi^2_{j}(x)-\sum_j\frac{1}{\pi}\sqrt{m^2-\omega^2_{j}}\phi_{j}^2(x) \ \ ,
B(exf,gyh) =B(exf,g)gyh+gB(exf,gyh)=-eB(g,x)fgyh+gB(exf,gyh) &=gB(exf,gyh)=gexfB(f,gyh)+gB(exf,gyh)f &=gexfgB(f,y)h+gB(exf,gyh)f=gB(exf,gyh)f.
f =(I-(I-e^{-r^2L})^{2M})f + (I-e^{-r^2L})^{2M}f &=\sum_{k=1}^{2M}c_k e^{-kr^2L}f + (r^2L)^{-M}(I-e^{-r^2L})^{2M}(r^2L)^Mf,
b_{j-1}=\frac{2}{3}\Big(2+c_1+(2j+1)a_{j}+b_j^2+\frac{2\big(a_j-(2j+2)b_j\big)^2}{b_j}\Big)>0, a_{j-1}=\frac{8b_{j-1}^2}{a_{j}}>0,
f(z)=\bigg(\overset{\infty}{\underset{n=0}{\sum}}a_{n}z^n\bigg) \bigg(\overset{\infty}{\underset{n=0}{\sum}}b_{n}z^{-n}\bigg) \; \; f(T)=\bigg(\overset{\infty}{\underset{n=0}{\sum}}a_{n}T^n\bigg) \bigg(\overset{\infty}{\underset{n=0}{\sum}}b_{n}T^{-n}\bigg),
a^+(\varphi)f^{(n)}=\varphi\otimes f^{(n)},f^{(n)}\in H_0^{\otimes n}, a^-(\varphi)h_1\otimes h_2\otimes\dots\otimes h_n =(\varphi,h_1)_{H_0}\,h_2\otimes\dots\otimes h_n, h_1,h_2,\dots,h_n\in H_0.
\sum_{\sigma=0}^{p^k-1}\mathcal{C}(\mathbf{a}_\sigma^{t_1},\mathbf{a}_\sigma^{t_2})(0)=\sum_{\sigma=0}^{p^{k}-1}\sum_{r=0}^{p^{m}-1}\omega_q^{\left(a^{t_1}_{\sigma,r}-a^{t_2}_{\sigma,r}\right)}=0.
\langle y,(-u_i,1,0,0)\rangle\leq0,\i=1,\ldots,m,\, \langle y,-e_{s+1}+e_{s+3}\rangle\leq 0, &\langle y,-e_i \rangle\leq 0,\, i=1,\ldots,s,\, \langle y,-e_{s+3}\rangle\leq 0, &\langle y, (\gamma_j,0,0,0)-d_je_{s+2}+e_{s+3}\rangle\leq 0,\,\langle y, e_{s+2}\rangle\leq 1,\, \langle y, -e_{s+2}\rangle\leq -1.
\beta F = - \left({\frac{2^{2 1}3^{2}5^{7}\pi^{1 4}}{7^{1 9}}}\right)^{1 / 5}\! \! N^{2}\left({\frac{T}{( g_{Y M}^{2}N )^{1 / 3}}}\right)^{9 / 5}= - 4 . 1 1 5 ~ N^{2}\left({\frac{T}{( g_{Y M}^{2}N )^{1 / 3}}}\right)^{1 . 8}
A_{0}^{\prime}( y ) ( 4 \eta ( y , t ) + \sigma ( y , t ) ) + \frac{\kappa^{2}}{3}\left( \xi_{0}^{\prime \prime}( y ) \delta \xi ( y , t ) - \xi_{0}^{\prime}( y ) \delta \xi^{\prime}( y , t ) + ( \xi_{0}^{\prime}( y ) )^{2}\frac{\delta b ( y , t )}{b_{0}}\right) = 0 + O ( m_{B}^{4}/ k^{4}) .
(\ref{mainlower}) \gtrsim_M \Big[ \sum_{j=M}^{L-M}{\Big(\sum_{n=1}^{L-j}{2^{-{t_n}({s}+d-{d}/{p})}}\Big)^p}\Big]^{{1}/{p}}\geq \Big[ \sum_{j=\lfloor L/3\rfloor}^{\lfloor L/2\rfloor}{\Big(\sum_{n=1}^{\lfloor L/3\rfloor}{2^{-{t_n}({s}+d-{d}/{p})}}\Big)^p}\Big]^{{1}/{p}}\gtrsim L^{{1}/{p}}L^{-({s}+d-{d}/{p})}\log{L}
S_{11}(s,t) = (-4\pi) \left[\frac{c_0^2}6 + \omega(s) + \omega(t)\right]g_z (s)g_z (t) S_{12}(s,t) = (-4\pi)\omega(t) g_z (s) g_z (t) , S_{21}(s,t) = (-4\pi)\omega(s) g_z (s) g_z (t) ,
b_i(v_1,\dots,v_N) :=b \Bigl(\frac1N \sum_{j=1}^N \delta_{v_j}\star \phi_{\eta_N}, v_i \Bigr) = \frac1N \sum_{j=1}^N b_{\eta_N}(v_i - v_j), \sigma_i(v_1,\dots,v_N) := \biggl[a \Bigl( \frac1N \sum_{j=1}^N \delta_{v_j}\star \phi_{\eta_N},v_i \Bigr) \biggr]^{\frac12}= \biggl[ \frac1N \sum_{j=1}^N a_{\eta_N}(v_i - v_j) \biggr]^{\frac12}.
I(\xi u) =\frac{1}{bp^2}M_{\xi u}^p-\int_0^TF(t,\xi u(t))dt-\frac{a^p}{bp^2}&\leq\frac{1}{bp^2}M_{\xi u}^p-c_1\int_0^T|\xi u(t)|^\mu dt+c_2T-\frac{a^p}{bp^2}&=\frac{1}{bp^2}(a+b\xi^p\|u\|_{E^{\alpha,p}}^p)^p-c_1\xi^\mu\|u\|_{L^\mu}^\mu+c_2T-\frac{a^p}{bp^2}&\rightarrow-\infty\ \ \mbox{as}\ \xi\rightarrow\infty.
D_{n}(\mathbf{g}) = \left( \partial_{z}- \frac{g_{n{-}1}{-}(n{-}1)}{z}\right) \left( \partial_{z}- \frac{g_{n{-}2}-(n{-}2)}{z}\right) \cdots \left( \partial_{z}- \frac{g_{0}}{z}\right)\,.
\gamma_2=\sigma_2 (C^+_K)^{\alpha\beta}1_{\{C^+_K>0\}}+\sigma_1 (C^-_K)^{\alpha\beta}1_{\{C^-_K>0\}}\;\; \;\; \gamma_1=\sigma_2 |C^+_K|^{\alpha\beta}1_{\{C^+_K<0\}}+\sigma_1 |C^-_K|^{\alpha\beta}1_{\{C^-_K<0\}}.
\int_{\mathbb{R}_{\geqslant0}}f(y)\rho_{t}^{\ast n}(\mathrm{d}y) =\int_{\mathbb{R}_{\geqslant0}}\cdots\int_{\mathbb{R}_{\geqslant0}}f(y_{1}+\cdots+y_{n})\rho_{t}(\mathrm{d}y_{1})\cdots\rho_{t}(\mathrm{d}y_{n}) \leqslant c_{1}^{n}\left(\int_{\mathbb{R}_{\geqslant0}}f(y)\rho_{t}(\mathrm{d}y)\right)^{n}<\infty,
\begin{aligned}\|M(\widetilde{v}^{k}-v^*)\|^2_H+\frac{1-\tau^k}{\tau^k}\underbrace{\|M(\breve{v}^{k-1}-v^*)\|^2_H}_{:=D^{k-1}}= \frac{1-\tau^k}{(\tau^k)^2}\underbrace{\|M(\breve{v}^{k}-\breve{v}^{k-1})\|^2_H}_{:=E^k}+\frac{1}{\tau^k}\underbrace{\|M(\breve{v}^{k}-v^*)\|^2_H}_{D^k}.\end{aligned}
A= \widetilde{C}\sup_{s\in[s_0,s_1]}\left\|X_{H}\right\|_{J_s}\left(2C_1\widetilde{C}+\frac{2}{C_0}+C_1\right)\leq \widetilde{C}\underbrace{\left\|X^{\lambda_s}_H\right\|_{\infty}(3C_1 + \frac{2}{C_0})}_{=:K}< \frac{1}{2},
\int |\xi|^{2\ell}|\hat u|^2 =\int \bigl(|\xi|^{2\ell}|\hat u|^{2\ell/m}\bigr) |\hat u|^{2(m-\ell)/m}\leq \||\xi|^{2\ell}|\hat u|^{2\ell/m}\|_{L^{m/\ell}}\| |\hat u|^{2(m-\ell)/m}\|_{L^{(m-\ell)/\ell}}&=\biggl(\int |\xi|^{2m}|\hat u|^2\biggr)^{\ell/m}\biggl(\int |\hat u|^2\biggr)^{1-\ell/m}.
\Lambda(z,t):=\sum_{r,s=1}^n h_{rs}^j(z,t)\frac{\partial}{\partial z_r}\wedge \frac{\partial}{\partial z_s}=\sum_{r,s=1}^n h_{rs}(z,t)\frac{\partial}{\partial z_r}\wedge \frac{\partial}{\partial z_s}.
J \le \frac{2\ln(\hat{\zeta}/\sqrt{T})}{\ln(\tau)}=\frac{\ln(\hat{\zeta}^2/T)}{\ln\left(\frac{\sqrt{\kappa}}{\sqrt{\kappa}+1}\right)}= \frac{\ln(T/\hat{\zeta}^2)}{\ln(1+{1}/{\sqrt{\kappa}})}\le \left(\sqrt{\kappa}+\frac{1}{2}\right) \ln\left(\frac{T}{\hat{\zeta}^2}\right),
\begin{array}{rcr}\{\eta,c\}= -c \{\eta,c^*\}= c^* \{\eta,z\}= -2z \{\eta,z^*\}= 2z^* \{c,z^*\}= \frac{1}{\sqrt{2}}c^* \end{array}\;\;\;\; \begin{array}{rcr}\{c^*,c\}= 1 \{z^*,z\}= -\frac{1}{2}\eta \{c,z\}= 0 \{c^*,z^*\}= 0 \{c^*,z\}= \frac{1}{\sqrt{2}}c\end{array}
A_{Q(K)_{2,1}}(x_0 \otimes \eta_0) = -2A_K(\eta_0)-4 = 2\tau(K) -4 A_{Q(K)_{2,1}}(x_2 \otimes \xi_0) = -2A_K(\xi_0)-4 = -2\tau(K) -4 A_{Q(K)_{2,1}}(y_2 \otimes \xi_0) = -2A_K(\xi_0)-2 = -2\tau(K) -2.
\| F \|_{X_{R}(I \times \mathbf{R}^{d})}= R^{\frac{1}{q}- 1}\| \psi(Rx) F \|_{L_{t}^{q}L_{x}^{2}}+ R^{\frac{1}{q}- 1}\sum_{j \geq 0}2^{j(1 - \frac{1}{q})}\| \chi(2^{-j}Rx) F \|_{L_{t}^{q}L_{x}^{2}}.
C(X,Y,Z) = (\nabla_{Z})(\mathring{Ric})(X,Y) - (\nabla_{Y})(\mathring{Ric})(X,Z) = \frac{1}{V}\left(T(X,Y,Z) - (\mathring{Ric}(X,Y)dV(Z) - \mathring{Ric}(X,Z)dV(Y)) \right).
\sum_{A , B}\int_{x}^{\infty}d \theta \frac{L_{A}^{>}( \theta )}{d \theta}( \Phi_{A B}* L_{A}^{>}) ( \theta ) \simeq \sum_{A}\int_{\epsilon_{A}^{>}( x )}^{\epsilon_{A}^{>}( \infty )}d \epsilon_{A}^{>}\frac{\epsilon_{A}^{>}}{( 1 + e^{\epsilon_{A}^{>}})}- \sum_{A}\int_{x}^{\infty}d \theta M_{A}e^{\theta}L_{A}^{>}( \theta ) .
K=\frac{|\gamma B_{effective}|}{|\frac{d\varphi}{dt}|}=\frac{\gamma\sqrt{A^{2}(t)+(B_{0}-\frac{\omega(t)}{\gamma})^{2}}}{\frac{d}{dt}(arctan(\frac{A(t)}{B_{0}-\frac{\omega(t)}{\gamma}}))}
v_k(f,x) = \sum_{a \subseteq x}(-1)^{\dim(a)-k}c(f,a) \mu_k(a) = (-1)^{\dim(x)-k}c(f,x) \mu_k(x) + \sum_{a \subseteq \partial x}(-1)^{\dim(a)-k}c(f,a) \mu_k(a) = c(f,x) \mu_k(\mathring{x}) + v_k(f,\partial x).
\begin{aligned}u_{t}=\Delta u -\chi \nabla \cdot (u\nabla w), x \in \Omega, t>0, v_{t}=\Delta v - \xi \nabla \cdot (v \nabla w), x \in \Omega, t>0, w_{t}=\Delta w - \lambda w + \alpha u +\beta v, x \in \Omega, t>0 ,\end{aligned}
\mathbf{\Sigma}(x_1)=\frac{(-x_1)^{-3/2}}{2\sqrt{2\pi}}\left[C(-x_1)^{i\epsilon}\bar{\mathbf{w}}+\bar{C}(-x_1)^{-i\epsilon}\mathbf{w}+ C_3(-x_1)^{-\kappa}\mathbf{w}_3 + C_4(-x_1)^{\kappa}\mathbf{w}_4\right],
\sum_{i \in S_e}B_i = \sum_{i \in S_e}\Bigg\lceil \frac{f_i}{\sum_{j \in S_e}f_j}\cdot B \Bigg\rceil \leq \sum_{i \in S_e}\Bigg( \frac{f_i}{\sum_{j \in S_e}f_j}\cdot B + 1\Bigg) = B + |S_e| \leq B + k.
\mathbf{f}:=\left( \mathbf{x};\mathbf{y};\mathbf{z}\right):=\left( x_{1},\thinspace \mathellipsis \thinspace ,\thinspace x_{m};y_{1},\mathellipsis \thinspace ,\thinspace y_{n};\thinspace z_{1},\thinspace \mathellipsis \thinspace ,\thinspace z_{p}\right)\in R^{m+n+p}
\int_{Z_{\bar{z}_a}}F(z)\mathcal{D}_\lambda\delta_{\bar{z}}(\mathrm{L}(z))=\int_{Z_{0}}F(z)\mathcal{D}_\lambda\delta(\mathrm{L}(z))=\sum_{z_0}\left(\mathrm{Det}_\lambda \mathrm{L}'_{z_0}\right)^{-1}F(z_0)
\Big|\partial_x \phi(x,y)-\begin{pmatrix}1 0\end{pmatrix}\Big|\leq C\tau + \left|\frac{-1+|M|^{1/2}}{x^2 |M|^{1/2}}y(1+x)(\partial_y \phi_2(\ldots)-1)\right| &+\frac{1}{|M|^{1/2}}\left|\phi_2(\ldots)-y\frac{-1+|M|^{1/2}}{x}\right| \leq C\tau+C\tau+\frac{1}{|M|^{1/2}}C|M|^{1/2}\tau = C\tau,
l(x):= \begin{cases}\frac{u_+(x)-u_-(x)}{v_+(\xi(x))-v_-(\xi(x))}, v_+(\xi(x))\neq v_-(\xi(x)) \frac{\partial_x(u_{+}(x)-u_{-}(x))}{\partial_x(v_{+}(\xi(x))-v_{-}(\xi(x)))}, v_+(\xi(x))= v_-(\xi(x)) \end{cases}
\frac{1}{|B_{r}|}\int_{B_{r}}|X_{i}X_{j}u\left( x\right) -(X_{i}X_{j}u)_{B_{r}}|dx & \leq\frac{c}{k}\sum_{i,j=1}^{q}\frac{1}{|B_{kr}|}\int_{B_{kr}}|X_{i}X_{j}u\left( x\right) |dx+ck^{2+Q/p}\left( \frac{1}{|B_{kr}|}\int_{B_{kr}}|\overline{L}u\left( x\right) |^{p}dx\right)^{1/p}.
\mathrm{unfold}_{i}(T) := \mathrm{unfold}_{\{i\}, [k] \setminus \{i\}}(T) = w^{(i)}(w^{(1)}\otimes \cdots \otimes w^{(i-1)}\otimes w^{(i+1)}\otimes \cdots \otimes w^{(k)})^\top.
H = \int d^{6}x \left( \frac{1}{4 8}F_{i j k l}^{2}- 3 \left( \pi_{i j k}- 4 \xi \varepsilon_{0 i j k l m n}A^{l m n}\right)^{2}+ A^{0 i j}G_{i j}^{( 2 )}\right) ,
= \frac{1}{16 R}\ \Big[ \ \frac{3}{r^5}+ \sum^{\infty}_{m=1}e^{-mr/R}(\frac{m^2}{R^2}\frac{1}{r^3}+ \frac{m}{R}\frac{3}{r^4}+ \frac{3}{r^5}) 2 \cos (mx_{11}/R)~ \Big]
{\widehat{A}(TX,\nabla^{TX})}{\rm ch}(2^k\Theta_3(T_{C}X))=\prod_{j=1}^{k}\frac{2 x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\frac{\theta_3(x_j,\tau)}{\theta_3(0,\tau)},
\begin{aligned}\mathcal{R}_{k-1}q_0(t)=\mathcal{R}_{k-1}\tilde{q}_0(t), \mathcal{R}_{k}q_0(t) =\mathcal{R}_{k}\tilde{q}_0(t) \mathcal{R}_{k-1}p(t)=\mathcal{R}_{k-1}\tilde{p}(t), \mathcal{R}_{k}p(t) =\mathcal{R}_{k}\tilde{p}(t).\end{aligned}
{\rm Ent}[\mathbf{P}_{1}|\mathbf{P}_{2}] = \begin{cases}\int \frac{d \mathbf{P}_{1}}{d \mathbf{P}_{2}}\log \left( \frac{d \mathbf{P}_{1}}{d \mathbf{P}_{2}}\right) d \mathbf{P}_{2}\{\rm if}\ \mathbf{P}_{1}\ll \mathbf{P}_{2}\infty \{\rm o.w,}\end{cases}
q=[v_{1}[\begin{matrix}c_{1}&0&\cdot\cdot\cdot&0\\ \end{matrix}]+v_{2}[\begin{matrix}0&c_{2}&\cdot\cdot\cdot&0\\ \end{matrix}]+...+v_{n}[\begin{matrix}0&0&\cdot\cdot\cdot&c_{n}\\ \end{matrix}]][\begin{matrix}1\\e^{\lambda_{2}t}\\ \vdots\\e^{\lambda_{n}t}\\ \end{matrix}]
{\phi}^{s e l f}( x^{+}) = \int_{0}^{L}d y^{-}\int_{0}^{L}d z^{-}\int_{0}^{L}d{\eta}^{-}{\Phi}^{\star , + +}( z^{-}, x^{+}| y^{-}, x^{+}) i{\epsilon}( y^{-}-{\eta}^{-}){\Phi}^{+ +}({\eta}^{-}, x^{+}| z^{-}, x^{+})
\hat{f}( \eta ) \sim \left( 1 - \kappa \eta^{2}/ 2 \right)^{- 1 / 4}\exp \left[ - \frac{\kappa}2 \int \frac{\eta d \eta}{( 1 - \eta ) \left( 1 - \kappa \eta^{2}/ 2 \right)}\right] .
\begin{aligned}F_{k+1}= F_k (I_m - H_k G_k)^{-1}F_k, \ \ \E_{k+1}= E_k (I_n - G_k H_k)^{-1}E_k, H_{k+1}= H_k + F_k (I_m-H_k G_k)^{-1}H_k E_k, \ \ \G_{k+1}= G_k + E_k (I_n-G_k H_k)^{-1}G_k F_k. \end{aligned}
\begin{aligned}{\rho}_{i}\leq C \rho_{i-1}\ \ \forall i \in \{\ 1,2,3,4,5\}, &{\rho}_0 \leq C \rho \leq C \rho_5^2 & |\rho_i \rho_{i,t}| \leq C |\rho_{i-1}|^2, \ \ \forall i \in \{\ 2,3,4,5\}\ .\end{aligned}
\delta S_{5}= \frac{\sqrt{2}}{8(2\pi)^3 \kappa^2 T_5}\frac{1}{6\sqrt{2}}\frac{\kappa^2}{\lambda^2}\int_{M^{11}_D}d( \hat{I}^1_2 \wedge X_8 )= - \frac{1}{48(2\pi)^3 \lambda^2 T_5}\int_{M^{10}}\hat{I}^1_2 \wedge X_8
G ( A{\mathbin \mathrm{\vphantom{X}\hspace{1 . 1 p t}\hspace{- 0 . 5 p t}\_\hspace{- 0 . 3 e m}\_\v r u l e w i d t h 0 . 5 p t \hspace{1 . 7 p t}}}B , C ) = G ( B , \tilde{A}\wedge C ) , \, \, \, \, \, \, \forall C \in \bigwedge ( V ) ,
\bar{H}_{\rm g}=H_{\rm g}(q^r,p^r) + \lambda_a^{(1)r_a}\pi_a^{r_a}+\lambda_a^{(2)r_a}\psi_a^{r_a}+ \lambda_a^{(3)}\theta_a +\lambda_a^{(4)}\gamma_a,
c_{n,k}=\lim_{\alpha \to \rho+1}c_{n,k}^\alpha & = (-1)^{\frac{|\lambda|}{2}}\frac{\Gamma_k(\rho)\Gamma_k\left(\frac{k+1}{2}\right)}{\Gamma_k\left(\frac{k}{2}\right)\Gamma_k\left(\frac{n+1+\lambda}{2}\right)}\lim_{\alpha \to \rho+1}\frac{\Gamma_k\left(\frac{-\alpha+\rho+\lambda}{2}\right)}{\Gamma_k\left(\frac{-\alpha+\rho}{2}\right)}.
\mathcal I_{\infty}^{\sharp}(\mathbf h,\breve{\mathbf g},\pmb{\phi}) = \frac{L(1,\pi_{\infty},\mathrm{Ad})L(1,\tau_{\infty}, \mathrm{Ad})}{L(1/2,\pi_{\infty}\times \mathrm{Ad}(g))}\alpha_{\infty}^{\sharp}(\mathbf h,\breve{\mathbf g},\pmb{\phi}) = 1.
=(\frac{l^{\prime\prime}(l^{-1}(x))}{l^{\prime}(l^{-1}(x))^{3}},\frac{1}{l^{\prime}(l^{-1}(x))^{2}l^{-1}(x)^{2}}+\frac{l^{\prime\prime}(l^{-1}(x))}{l^{\prime}(l^{-1}(x))^{3}l^{-1}(x)})
\begin{matrix}{{\quad}}\\{{\vec{\varphi}( x )}}\\{{\quad}}\\ \end{matrix}= \left( \begin{matrix}{{c_{1}\: \cos ( r \, x ) + c_{2}\: \sin ( r \, x )}}\\{{- c_{1}\: \sin ( r \, x ) + c_{2}\: \cos ( r \, x )}}\\{{c_{3}\: x}}\\ \end{matrix}\right) \; ,
\textrm{rank}\left(\mathbf{V}\right) =\sum_{m}\frac{\log\left(1+\frac{\lambda_{m}}{\varepsilon}\right)}{\log\left(1+\frac{1}{\varepsilon}\right)}+o\left(\varepsilon\right) =\frac{M\log\left(\frac{1}{\varepsilon}\right)+\log\left(\left|\mathbf{V}+\varepsilon\mathbf{I}\right|\right)}{\log\left(1+\frac{1}{\varepsilon}\right)}+o\left(\varepsilon\right)
\overline{{x}}= \left[ \frac \alpha{1 - \alpha^{2}+ 6 \gamma ( \gamma - 1 )}\frac{1 2 \gamma}{\alpha + 2 \gamma}\left( \frac{g_{i}^{N}- \gamma / \alpha f_{i}^{N}}{g_{i}^{N}+ \gamma / \alpha f_{i}^{N}}\right) \right] \ .
(I\otimes \Delta^e)\Delta^em(a) =(I\otimes \Delta^e)(m\otimes m)\Delta(a)=m(a_{(1)})\otimes \Delta^em(a_{(2)}) &=m(a_{(1)})\otimes (m\otimes m)\Delta(a_{(2)})=m(a_{(1)})\otimes m(a_{(21)})\otimes m(a_{(22)}),
y_{\textrm{BF}}=\left( \sqrt{P_{\textrm{N}}}x_{\textrm{N}}+\sqrt{P_{\textrm{F}}}x_{\textrm{F}}\right)\sqrt{d_{\textrm{BF}}^{-\alpha}}h_{\textrm{BF}}+n_{\textrm{F}},
\sum_{j=1}^m \sum_{\overset{l_1+\ldots+l_j=m}{l_i \geq 1}}\frac{1}{l_1! \cdots l_j!}\leq \sum_{j=1}^m \sum_{l_1+\ldots+l_j=m}\frac{1}{l_1! \cdots l_j!}= \sum_{j=1}^m \frac{(1+1+ \ldots+1)^m}{m!}= \frac{m^{m}}{m!}\leq \frac{m^{1/2}}{\sqrt{2\pi}}{\rm e}^{m}.
\int^{\infty}_{0}f_{1}(\tau)e^{-\alpha_1 \tau}x'(t-\tau)d\tau +y'(t)=\int^{\infty}_{0}f_{1}(\tau)e^{-\alpha_1 \tau}n(x(t-\tau))d\tau-a\varphi_{1}(y)\leq M_{1}G_{1}-ak_{1}y,
\chi_{L}=\frac{<p_{\psi}|M|p_{\psi}>}{<p_{\psi}|p_{\psi}>}=\frac{<\psi|M|\psi>+\epsilon^{2}<p_{\psi}^{1}|M|p_{\psi}^{1}>+\epsilon^{4}<p_{\psi}^{2}|M|p_{\psi}^{2}>}{<\psi|\psi>+\epsilon^{2}<p_{\psi}^{1}|p_{\psi}^{1}>+\epsilon^{4}<p_{\psi}^{2}|p_{\psi}^{2}>}.
<{\pounds}_{H}^{( 2 )}> = \frac{\eta_{0}}{2 \pi^{2}}p_{0}\left( \left| \psi_{0}\right|^{2}+ \left| \varphi_{0}\right|^{2}\right) \int \left[ d^{2}k \right] \left( 3 \left| k \right|^{2}- 4 p_{0}^{2}\right) D_{P}^{- 1}
|G_{2}^{(n_{l})}[g,m_{l}]| = |G_{2}^{(n_{l}-n_{l-1})}|^{|\Omega_{n_{l-1}}|}\bigl(\tfrac{1}{2}+o(1)\bigr)^{|\Omega_{n_{l-1}}|}= |G_{2}^{(n_{l})}|\Bigl(\frac{1}{2^{|\Omega_{n_{l-1}}|}}+ o(1)\Bigr) = |G_{2}^{(n_{l})}|\Bigl(\frac{1}{|G_{2}^{(n_{l-1})}|}+ o(1)\Bigr),
U_{1,n}:= S_{1,n}(\beta^\prime) - E\left(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}:= \Omega_{1}^{-1}\frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t,\beta^\prime}x_{nt}= \frac{1}{\sqrt{n}}\sum_{t=1}^{n}e_{t,\beta^\prime}c_{nt}
A := \{(f_1,f_2) : f_i \in C[0,\infty) f_i(0) = 0 i=1,2\lambda_0(f_1,f_2), \lambda_1(f_1,f_2) < \infty \}A^n := \{(f_1,f_2) : f_i \in C[0,\infty) f_i(0) = 0 i=1,2, \lambda^n_0(f_1,f_2), \lambda^n_1(f_1,f_2)< \infty \}.
\int d^{4}x \sqrt{g_{{}_{E}\pm}}{\cal K}_{{}_{E}\pm}^{n}= \int d^{4}x \left({\frac{z_{\pm}}{z_{0}}}\right)^{( 4 - n ) \beta}{\frac{1}{z_{\pm}^{n}}}\propto \int d^{4}x r_{\pm}^{4 + ( 4 / 3 ) D_{2}- ( n / 3 ) D_{2}},
\begin{array}{r l}{{\displaystyle \Delta_{\pm}}}&{{= \displaystyle \frac{c}{2 4}\pm \left( I_{H}^{\pm}- 2 I_{S}^{\pm}- I_{C}^{\pm}- I_{W}^{\pm}\right) + \displaystyle \frac{1}{2 \pi}\Sigma_{\pm}}}\\{{}}&{{\mp \displaystyle \frac{1}{2 \pi}\displaystyle \int_{- \infty}^{\infty}\displaystyle \frac{d x}{2 \pi}\varphi_{\pm}^{,}( x ){\cal Q}_{\pm}( x ) \: ,}}\\ \end{array}
e^{-\lambda t}\hat{S}_{j,+}^k(t) = \nu e^{-\lambda t}\int_0^t Ye^{\lambda s}p_{j,+}^k(t-s) ds = \nu Y \int_0^t e^{-\lambda s}p_{j,+}^k(s) ds \leq \frac{\nu Y}{\lambda}\theta^k.
\Lambda_2 (\lambda)(f, g) = \sum_{k = 1}^\infty \partial_\nu(\Delta\phi_k)\Big\vert_{\partial B_R}\frac{1}{\lambda_k - \lambda}\Big( \int_{\partial B_R}\partial_\nu\phi_k f{\rm d}s(y) + \int_{\partial B_R}\partial_\nu(\Delta\phi_k)g{\rm d}s(y)\Big).
\left\Vert \Phi \right\Vert_{1\rightarrow p}^{p}=\sup_{\rho \in \mathfrak{S}\left( \mathcal{H}_{A}\right)}\mathrm{Tr}\Phi [ \rho ]^{p}=\sup_{\psi \in\mathcal{H}_{A}}\mathrm{Tr}\Phi [P_{\psi}]^{p},
f_{Cayley}(s_1,\ldots,s_{d-1},t_1^{\ast},\ldots,t_{d-1}^{\ast}) = \sum_{i_1=0}^{\tau_1}\cdots\sum_{i_{d-1}=0}^{\tau_{d-1}}Y_{i_1,\ldots,i_{d-1}}\prod_{k=1}^{d-1}\phi_{i_k}(s_k).
\Delta S = - \frac{1}{8}\left({\frac{1}{4 \pi}}\right)^{3 / 2}\int d^{3}x{\frac{\left( \vec{\nabla}B \right)^{2}}{B^{3 / 2}}}\int_{0}^{\infty}{\frac{d s}{s^{1 / 2}}}\, e^{- m^{2}s / B}\left({\frac{d}{d s}}\right)^{3}\left[ s \, c o t h ( s ) \right]
S_{UV\psi}= N{\rm tr}_N\psi^{\dagger}_i\psi_i+N\frac{1}{2}{\rm tr}_N(U^2+{\rm e}^{2\beta}V^2) -Ng{\rm tr}_N(\sum_i \psi_iU\psi_i^{\dagger}+\sum_{i,j}L_{ij}\psi_jV\psi^{\dagger}_i).
\sum_{{\bf{q}}_{1}\neq{\bf{0}},{\bf{q}}}{\bar{n}}_{{\bf{k}}-{\bf{q}}/2 +{\bf{q}}_{1}}e^{i\mbox{}\theta({\bf{k}}+{\bf{q}}_{1}/2, -{\bf{q}}+{\bf{q}}_{1})}e^{-i\mbox{}\theta({\bf{k}}-{\bf{q}}/2 +{\bf{q}}_{1}/2,{\bf{q}}_{1})}= 0
i \frac{\partial}{\partial t}\langle \vec{q}\,^{\prime}, t \mid \vec{q}, 0 \rangle = - \frac{1}{2}\sum_{\nu = 1}^{3}\frac{\partial^{2}}{\partial q_{\nu}^{2}}\langle \vec{q}\,^{\prime}, t \mid \vec{q}, 0 \rangle + V ( \vec{q}\,^{\prime}) \langle \vec{q}\,^{\prime}, t \mid \vec{q}, 0 \rangle
e^{-2\phi}= \left(\Bigl(e^{-2\hat \phi_0}+{2\tilde m\over r}\Bigr)\Bigl(K_0+{2\hat m \over r}\Bigr) -\left[(A_n)_0 +{2 q_n \over r}\right ]^2\right)^{1\over2}\ .
y_{1L}'(y_3)=\frac{-1/2\alpha}{1+\phi_1(u_s)+y_{1L}(y_3)}\;,y_{2L}'(y_3)=\frac{1-\phi_1(u_s)-y_{1L}(y_3)}{1+\phi_1(u_s)+y_{1L}(y_3)}\;.
\int\limits_{0\leq x_n\leq \dots \leq x_1\leq x}\!\!\!\!\! E_\lambda\Big(x+2\sum_{k=1}^{n-1}(-1)^kx_k+ (-1)^nx_n\Big) \big( \prod_{j=1}^nR \Phi(x_j) \big) M_3(x_n,\lambda) dx_n\cdots dx_1.
H(m,Y^{\nu,\ell_0}(u,z)m') =\beta(Y^{\nu,\ell_0}(u,z)m')(m)=\check Y^{\nu,\ell_0}(u,z)\beta(m')(m) &=\beta(m')(Y^{\nu,\ell_0}(A(z)u,z^{-1})m),
\begin{cases}\dim V_1 \cap V_2 \geq k \vert S_1 \cap S_2 \vert +1-g \dim V_1 \cap V_2= k \vert S_1 \cap S_2 \vert+1-g\ \mathrm{for}\k \vert S_1 \cap S_2 \vert\geq 2g-1\end{cases}
S_E =\frac{1}{2\kappa_{11}^2}\int_M \mathrm{dvol}_g R S_M =-\frac{1}{4\kappa_{11}^2}\int_M \mathrm{dvol}_g |G|^2 S_{CS}=-\frac{1}{12\kappa_{11}^2}\int_M C\wedge G\wedge G.
S_{\alpha}(\epsilon,R)=\frac{\epsilon q^{2}R^{2}}{p^{2}+q^{2}}+\frac{1}{q^{2}\epsilon}\left\{\frac{\epsilon q^2 R}{\sqrt{p^{2}+q^{2}}}\right\}\left(1-\left\{\frac{\epsilon q^2 R}{\sqrt{p^{2}+q^{2}}}\right\}\right)+O\left(1+(R\epsilon)^{2}\right),
W_2 = \int_{\mathcal{X}\times \mathcal{X}}f(x)f(y) \int_{l_x}^{v_x}\int_{l_y}^{v_y}h(u,v) \,d(G_{n,x,y}- F_{n,x}F_{n,y})(u,v) \, dx \, dy + \frac{1}{n}.
[ J_{0}^{k}, \Psi^{a b}( z ) ] | \Psi^{c d}\rangle + \Psi^{a b}( z ) J_{0}^{k}| \Psi^{c d}\rangle = z^{- 2 \Delta_{O}+ \Delta_{\phi}}C_{n}^{a b , c d}f_{e}^{k n}| \phi^{e}\rangle + . . .
\alpha:=\frac{\bar{\beta}_3}{2 \bar{\beta}_3 + d - 2}(\frac{1}{\beta_1}+ \frac{1}{\beta_2}) = \frac{\bar{\beta}_3}{2 \bar{\beta}_3 + d - 2}(\frac{d}{\bar{\beta}}- \frac{d-2}{\bar{\beta_3}}) = \frac{\bar{\beta}_3 d - (d-2) \bar{\beta}}{\bar{\beta}(2 \bar{\beta}_3 + d - 2)}.
\sum_{i=0}^N f^{-1}(i)-f^{-1}(1) =\sum_{i=3}^N f^{-1}(i)+f^{-1}(0)+f^{-1}(2) &=\sum_{i=3}^N f^{-1}(i)+ \int_0^2 f^{-1}(t)\mathrm{d}t &\leq \int_2^N f^{-1}(t)\mathrm{d}t+\int_0^2 f^{-1}(t)\mathrm{d}t = 1.
\alpha(\nu,t) := \left( \begin{array}{cc|cc}\nu+t 0 \sqrt{\nu^2-\frac{1}{4}}0 0 \nu+t 0 -\sqrt{\nu^2-\frac{1}{4}}\hline \sqrt{\nu^2-\frac{1}{4}}0 \nu 0 0 -\sqrt{\nu^2-\frac{1}{4}}0 \nu \end{array}\right)\;.
{\rm dim}\;{\rm ker}\; \mathfrak{T}_\chi-{\rm dim}\;{\rm ker}\; \mathfrak{T}_\chi^\ast\ \ \mbox{modulo}\ 2 &={\rm dim}\;{\rm ker}\; \mathfrak{T}_\chi+{\rm dim}\;{\rm ker}\; \mathfrak{T}_\chi^\ast\ \ \mbox{modulo}\ 2 &={\rm dim}\;{\rm ker}\;(\mathcal{P}_{\rm D}-U\mathcal{P}_{\rm D}U-1)\ \ \mbox{modulo}\ 2,
\sum_{\substack{U \pmod{D}U\equiv S \pmod{Q}}}\psi (U) = \prod_{i=1}^{m}\sum_{\substack{U_i \pmod{P_i^{\alpha_i}}U_i \equiv S \pmod{P_i^{\beta_i}}}}\psi_{P_i^{\alpha_i}}(U_i).
\sum_{b}\sum_{\alpha , \gamma}\; \sum_{\beta , \beta^{\prime}{\mathrm{\scriptsize{~ w i t h ~}}}\beta \neq \beta^{\prime}}P_{( b \beta )}^{( a \alpha )}\: P_{( c \gamma )}^{( b \beta )}\; = \; \delta^{2}\: Q_{c}^{a}( x , y ) \; .
T\sim\frac{\mu_{x}+\mathbb{N}(0,\sigma_{x}^{2})}{\mu_{y}+\mathbb{N}(0,\sigma_{y}^{2})}=\frac{\mu_{x}+X}{\mu_{y}+Y}=\frac{\mu_{x}}{\mu_{y}}\frac{1+\frac{X}{\mu_{x}}}{1+\frac{Y}{\mu_{y}}}
\begin{aligned}x_1(t)-x_2(t) = x_1(0)-x_2(0)+\int_{0}^t(v_1(s)-v_2(s)) ds &= x_1(0)-x_2(0)+\int_{0}^t(e^{i\theta_1(s)}-e^{i\theta_2(s)}) ds &= x_1(0)-x_2(0)-2\int_{0}^t(0,\sin(\theta_2)) ds, v_1(t)-v_2(t) = -2(0, \sin(\theta_2(t))).\end{aligned}
d^{2}(f)(x,y,z) =f\left(\alpha(x),[y,z] \right)+f\left(\alpha(y),[x,z] \right)+f\left(\alpha(z),[x,y] \right) + \rho(\alpha(x))f(y,z)+\rho(\alpha(y))f(x,z) +\rho(\alpha(z))f(x,y).
\varrho^\star([[x,y],\alpha(z)])(\beta^{-2})^* =\varrho^{\star}(\alpha^{2}(z))\varrho^{\star}(\alpha (y))\varrho^{\star}(x)- \varrho^{\star}(\alpha^{2}(y))\varrho^{\star}(\alpha (x))\varrho^{\star}(z) &\quad- \varrho^{\star}(\alpha([z, x]))\varrho^{\star}(\alpha(y))(\beta^{-1})^* + \varrho^{\star}(\alpha^{2}(x))\varrho^{\star}([y, z])(\beta^{-1})^*.
\mathrm{Var}\bigl( \mathbb{E}\bigl\{\hat{\mu}| J \bigr\}\bigr) \leq \mathbb{E}\Bigl[ \Bigl\{\mathbb{E}( \hat{\mu}| J ) - \mathbb{E}( \hat{\mu}| J = j_0-1) \Bigr\}^2 \Bigr] \leq (T+M)^2 \delta + 8(T+M)^2\exp(-n \alpha^2/512).