harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{v}{c}\equiv\frac{1}{c}\frac{dx}{dt}=\frac{w}{c}\frac{1}{\sqrt{1+(\frac{w}{c})^{2}}}=tanh(\eta)\equiv\frac{e^{2\eta}-1}{e^{2\eta}+1}=\pm\sqrt{1-(\frac{1}{\gamma})^{2}}
\prod_{p \leq X}\left(1-\frac{1}{p}\right)\left(1+\frac{\Delta_K(p)\iota_{K,f}(p)}{p}\right) \asymp \exp\left(\sum_{p \leq X}\frac{\Delta_K(p)\iota_{K,f}(p)-1}{p}\right) = \exp\left(\sum_{p \leq X}\frac{\Delta_K(p)-1}{p}+ O_{K,f}(1)\right) &\asymp_{K,f}\prod_{p \leq X}\left(1+\frac{\Delta_K(p)-1}{p}\right),
{}^{({\cal C}_{H}^{\cal R})}\Psi_{Y , X}\circ ( \triangleleft u ) \circ{}^{({\cal C}_{H}^{\cal R})}\Psi_{X , Y}={}^{\cal C}\Psi_{Y , X}\circ ( ( \triangleleft u ) \otimes ( \triangleleft u ) ) \circ{}^{\cal C}\Psi_{X , Y}
{\cal P}({\cal E}) =\sum_{\vert \Psi\rangle}\vert{\cal A}({\cal E})\vert^2= \int\limits_{-\infty}^\infty d\tau\, \int\limits_{-\infty}^\infty d\tau'\, e^{-i{\cal E}(\tau-\tau')}\,{\tilde G}\left[{\tilde x}(\tau),{\tilde x}(\tau')\right],
\begin{aligned}\nabla_{e_1}A (e_2) -\nabla_{e_2}A(e_1)-A[e_1,e_2] = \big[( 4 r^2-G)\,\cos\phi\,\sin\phi -\cos(2 \phi)\,e_2(r) \big]\,e_2 &-e_1(r)\, e_1.\end{aligned}
h^r \cdot\prod_{s=1}^{r}\bigl(1-\tfrac{s-1}{r^2}\bigl) \ge h^r\cdot\left(1-\sum_{s=1}^{r}\tfrac{s-1}{r^2}\right)=\tfrac{r+1}{2r}\cdot h^r >\tfrac{1}{2}h^r.
S^{(2)}_{3,0}(n)=\sum_{\substack{b_0-b_1+\cdots+(-1)^{k-1}b_{k-1}\equiv 0\mod 3 b_{k-1}b_{k-2}\cdots b_0<n_{k-1}n_{k-2}\cdots n_0}}(-1)^{b_0+\cdots+b_{k-1}},
u(t) =e^{t\Delta}[u_0+t\P\theta_0e_3]-B_1(u,u)-B_2(u,\theta) =\biggl(\int\theta_0\Bigr)t\,K(t)+t\,F(t)*V+e^{t\Delta}u_0-[B_1(u,u)+B_2(u,\theta)],
\mu'\big(B_s(y_j)\big) = \sum_{z_k\in B_s(y_j)}\omega_k 10^{-k}\bar r_k^k\leq C(n)\sum_{z_k\in B_s(y_j)}\mu^a(B_{\bar r_k/8}(z_k))\leq C(n)\mu^a(B_{2s}(y_j))\leq C(n) s^k\, ,
B^{(n)}_{(v,h)}=B^{(n)}_v+B_h, B^{(n)}_v(\eta) =\sum_{x\in\tau(\eta):\, s_x\ge1/n}\operatorname{div}^X v(x), B_h(\eta) =\int_X \frac{l'(s_x)}{l(s_x)}\, h(x)\,d\eta(x)+l(0)\int_X h(x)\,dx .
\|\nabla f_h(x)-\nabla f_h(y)\|_H^2 =\langle \nabla f_h(x)-\nabla f_h(y),\nabla f_h(x)-\nabla f_h(y)\rangle_H &=\langle L^{-1}(\nabla f(x)-\nabla f(y)),L^{-1}(\nabla f(x)-\nabla f(y))\rangle_L &=\langle \nabla f(x)-\nabla f(y),\nabla f(x)-\nabla f(y)\rangle_{L^{-1}}&=\|\nabla f(x)-\nabla f(y)\|_{L^{-1}}^2.
\mathcal A(t):= \left(\begin{array}{ccc}0 -1 -\div (A^{(4)}(t) \nabla (\cdot)) + p(t)\cdot \nabla(\cdot) - 2 q(t) \cdot \nabla (\cdot)\end{array}\right) \,, & X:=H^1_D(\Omega^{(4)}\setminus \Gamma^{(4)}(t_0)) \times L^2(\Omega^{(4)})\,, &Y:= \mathcal H \times H^1_D(\Omega^{(4)}\setminus \Gamma^{(4)}(t_0))\,,
MASE=mean(\frac{|e_{j}|}{\frac{1}{T-m}\sum_{t=m+1}^{T}|Y_{t}-Y_{t-m}|})=\frac{\frac{1}{J}\sum_{j}|e_{j}|}{\frac{1}{T-m}\sum_{t=m+1}^{T}|Y_{t}-Y_{t-m}|}
\sqrt{|\hat{\hat{g}}|}{\textstyle\frac{-1}{2\cdot 4!}}\hat{\hat{G}}{}^{2}=\sqrt{|g_{E}|}\left\{{\textstyle\frac{-1}{2\cdot 4!}}K^{6/7}F_{(4)}^{2}+{\textstyle\frac{1}{2\cdot 3!}}K^{-3/7}\vec{F}_{(3)}^{\T}{\cal M}^{-1}\vec{F}_{(3)}-{\textstyle\frac{1}{4}}K^{-12/7}F_{(2)}^{2}\right\}\, .
W[\phi,\pi] =\mathcal{N}\exp\left( -\frac{1}{\hbar}\int dx\,\left( \left( \phi\left( x\right) \,\sqrt{m^{2}-\nabla_{x}^{2}}\,\phi\left( x\right)\right) + \left({\pi}\left( x\right) \,\left(\sqrt{m^{2}-\nabla_{x}^{2}}\right)^{-1}{\pi}\left( x\right) \right) \right) \right) ,
I(x)=\int_{-\pi}^{\pi}s\left( \frac{i(1-e^{i\theta})}{2(1+e^{i\theta})}\right) \exp\left( \frac{e^{i\theta}-1}{e^{i\theta}+1}x\right)e^{i\theta(m+1)}\left( e^{i\theta}+1\right)^{n-m-2}d\theta.
^{(3)}R = - 2\, \frac{\left( \theta ' \left( 1 - \frac{3M}{2r}\right)- \frac{1}{r \tan \theta}\right)^2+ \left( \frac{\sqrt{3}M}{2r}\theta ' \right)^2}{1 + r^2 (\theta ')^2\left( 1 - \frac{2M}{r}\right)}
\hat{\Psi}_{11}^{(\infty)}= B^{1/2}(c_1\tilde{\phi}_{\nu}+c_2\tilde{\phi}_{-\nu-1}), \hat{\Psi}_{12}^{(\infty)}= B^{1/2}(c_3\tilde{\phi}_{\nu}+c_4\tilde{\phi}_{-\nu-1}),
||f||_n^2 = \frac{1}{J^2}\Delta(a)' \Sigma_{n}^{q-1}\Delta(a) &\lesssim \frac{1}{J^2}\frac{1}{J-1}||\Delta(a)||_{J-1}^2 \lesssim \frac{1}{J^2}\frac{1}{J-1}||a||_J^2 &\lesssim \frac{1}{J^2}||Kf||_n^2
\begin{array}{ll}\phi_1 = 0 \phi_2 = \phi = \varphi(r )e^{i\theta}H_{\mu}= \frac{\sqrt{\beta}}{q}\frac{H(r)}{r}\delta^{\theta}_{\mu}\tilde D =\frac{1}{2\beta}\left(q(\varphi^2 - \eta^2)+ \alpha e \sigma^2 \right) G=0,\end{array}
\big\{W^{(\alpha,\beta)}_{j,m,n}\big\}_{j=1}^n=\big\{z_{j,m}^{(-\alpha-1,\beta-1)}\big\}_{j=1}^m \cup\big\{z_{j-m,n-m}^{(\alpha+1,\beta-1)}\big\}_{j=m+1}^{n}.
1 - \widehat F(s) \sim s^\alpha \sum_{m=- \infty}^\infty \left( 1- \exp \left[ 2^{(m - \{\alpha \log_2 s^{-1}\})/\alpha}\right] \right) 2^{-m + \{\alpha \log_2 s^{-1}\}}& =: s^\alpha q_0(s)
\omega_{\beta,\mu}(\prod_{i=1}^{m}\eta(A_{i}) B) = \lim_{V \to \infty}\omega_{\beta,\mu,\Lambda}(\prod_{i=1}^{m}\eta_{\Lambda}(A_{i}) B)=\lim_{V \to \infty}\omega_{\beta,\mu}(\prod_{i=1}^{m}\eta_{\Lambda})(A_{i}) B)\ .
\begin{bmatrix}S_1 0 0 T_1^{-1}\end{bmatrix}\begin{bmatrix}I_\ell B_{12}B_{21}B_{22}\end{bmatrix}\begin{bmatrix}S_1^{-1}-B_{12}0 I_{n-\ell}\end{bmatrix}= \begin{bmatrix}I_\ell 0 D C_{22}\end{bmatrix}
\big| R_{r,n}(w)\big| \leq2 \sum_{m=1}^\infty\, \frac{w^{r+1}}{(w^2 + 4\pi^2 m^2)^{(r+1)/2}}\leq2 \zeta\, \left( \frac{r+1}{2}\right) \left( \frac{w}{4\pi}\right)^{(r+1)/2}\, ,
[h_{i,r+2}, x_{i+1,s}^{\pm}] - 2[h_{i,r+1}, x_{i+1,s+1}^{\pm}] + [h_{i,r}, x_{i+1,s+2}^{\pm}] \pm (\varepsilon_1 + \varepsilon_2) ( h_{i,r+1}x_{i+1,s}^{\pm}+ x_{i+1,s}^{\pm}h_{i,r+1}) &\mp (\varepsilon_1 + \varepsilon_2) ( h_{i,r}x_{i+1,s+1}^{\pm}+ x_{i+1,s+1}^{\pm}h_{i,r}) + \varepsilon_1 \varepsilon_2 [h_{i,r}, x_{i+1,s}^{\pm}] = 0 ,
\varepsilon_{14}^N= \int_0^T\eta_{s,3}^N(\lambda_2)\Bigg(\int_\mathbb{T}\theta_s^S(v)\lambda_1(v)\left(G_s(v)-F_s(v)\right)dv &-\frac{1}{N}\sum_{j=0}^{N-1}\mathbb{E}S_s^N(j)\lambda_1\left(\frac{j}{N}\right)\left(G_s\left(\frac{j}{N}\right)-F_s\left(\frac{j}{N}\right)\right)\Bigg)ds,
A+B\ =\ [a_1+b_1, a_2+b_2]_\mathbb{Z}, A^* + B^*\ =\ \left[a_1 + b_1 - \frac{1}{2}, a_2 + b_2 + \frac{1}{2}\right], A - B\ =\ [a_1 - b_2, a_2 - b_1]_\mathbb{Z},
W_l(x;\tau_1,\dots,\tau_n) :=\sum_{0\le k \le n}\sum_{1\le i_1<\dots<i_k\le n}(-1)^k\tilde{M}(x-\tau_{i_1}-\dots-\tau_{i_k})^l &=\sum_{S}(-1)^{|S|}\tilde{M}(x-\sum_{i\in S}\tau_{i})^l,
\|\nabla v\|_{L^2(L^2)}=\|\tilde{\mathcal{M}}_{-\Delta}^*g\|_{L^2(L^2)}\lesssim\|g\|_{L^2(L^2)}, begin{align*}4pt]\mbox{hence}\qquad \|\Delta v\|_{L^2(\dot H^{-1})}\le\|\nabla v\|_{L^2(L^2)}\lesssim\|g\|_{L^2(L^2)}.
p \sum_{k=1}^m|\tilde X_{k,R}(x+y)|^2+ \langle x, \tilde X_{0,R}(x+y)\rangle &=p\sum_{k=1}^m \left| X_{k}\Big(\pi_R(x+y)\Big)\right|^2+{|x|\over R}\Big \langle \pi_R(x), X_0(\pi_R(x+y))\Big \rangle.
\mu\bigl(\bigl\{I_\infty^k\bigr\}\bigr)=0 \mbox{and}\quad\mu_n\bigl({\mathcal{M}_{[n]:k}}\setminus \bigl\{I_n^k\bigr\}\bigr)<\infty\qquad\mbox{for all}n\in\mathbb{N},
I_{R_1}=\int_{\Gamma^-\setminus \Gamma^-_h}e^{-\tau (\mathbf d+\mathrm i\mathbf d^\perp)}\mathrm d\sigma,I_{R_2}=\int_{\Gamma^+\setminus \Gamma^+_h}e^{-\tau (\mathbf d+\mathrm i\mathbf d^\perp)}\mathrm d\sigma.
\Delta^{\mathrm{F}}( x , y \vert A ) ={\frac{- i}{2}}\int_{0}^{\infty}d \tau \int_{z ( 0 ) = y}^{z ( \tau ) = x}{\cal D}[ z ] \mathrm{P}\mathrm{e x p}\, i \int_{0}^{\tau}\mathrm{d}\tau \prime \{-{\frac{1}{2}}[ ({\frac{d z}{d \tau \prime}})^{2}+ m^{2}] - g z^{\mu}\prime A_{\mu}( z ) \}
\omega_{z,y}(s_J)=\omega_{z,y}((s(z)^{\alpha})^*\, s_J\,s(y)^{\beta})=\sum_{|K|=d-a}\sum_{|L|=a}\overline{(z^{\otimes \alpha})_{JK}}\, (y^{\otimes \beta})_{KL}\omega_{z,y}(s_L).
\frac{\partial f}{\partial t}+r\cdot\nabla_{r}f-\frac{1}{\hbar}\nabla_{r}V\cdot\nabla_{k}f+\sum_{\alpha=1}^{\infty}\frac{(-1)^{\alpha+1}}{\hbar4^{\alpha}(2\alpha+1)!}\times(\nabla_{r}\nabla_{k})^{2\alpha+1}Vf=(\frac{\partial f}{\partial t})_{c}
3\eta^2 + 1 +2\gamma (\sqrt{3}\eta - 1) > 3\eta^2 + 1 + 2(1+\sqrt{3}) \cdot (\sqrt{3}\eta - 1) = 3\eta^2 + ( 6 + 2\sqrt{3}) \eta - (1+2\sqrt{3}) = 3 \Big( \eta - \frac{\sqrt{5+4\sqrt{3}}- (1+\sqrt{3})}{\sqrt{3}}\Big) \cdot \Big( \eta + \frac{\sqrt{5+4\sqrt{3}}+ (1+\sqrt{3})}{\sqrt{3}}\Big) .
c^{m, E}_{j,l,r;err}(t-s,v,\omega):= \big(\frac{\hat{v}+\omega}{(1+\hat{v}\cdot \omega)^2}\big(1 -|\hat{v}^2| \big) + \omega - \frac{(\omega +\hat{v})\hat{v}\cdot \omega}{1+\hat{v}\cdot \omega}\big) \varphi_{l; r}(\tilde{v}+\omega ) \varphi_j(v) \varphi_{m;-10 M_t}( t-s )
s_n = \left[ \left(s^{(1)}\cdots s^{(d)}\right)^{-1/2}\pi^{-d/2}|\mathcal{S}|^{d/2}\right] \cdot n^{-d/2}\cdot |\mathcal{S}|^n + O\left( n^{-(d+1)/2}\cdot |\mathcal{S}|^n \right),
H_{\mu\nu}\rightarrow H_{\mu\nu}+ \frac{1}{m^2}\bigl( \lambda_1 R_\mu{}^\alpha{}_\nu{}^\beta H_{\alpha\beta}+ \lambda_2 R^\alpha{}_{(\mu}H_{\nu)\alpha}+ \lambda_3 R H_{\mu\nu}+ \lambda_4 G_{\mu\nu}R^{\alpha\beta}H_{\alpha\beta}\bigr)+ O \left(\frac{1}{m^4}\right)
U_0= \sup_{0\leq t \leq t_0}\|\theta_+(t)\|_2^2+2\nu\int_{0}^{t_0}\|\Lambda^{\frac{\alpha}{2}}\theta_+(t)\|_2^2 \,dt \lesssim \|\theta(0)\|_2^2 + \frac{t_0\lambda_0^{\frac{4}{p}-2-\alpha}}{\nu}\|f\|_p^2.
\left| \varphi_s (\omega, x) - \varphi_s (\omega,y) \right| \leq \left| \varphi_{\hat{\tau}(\omega)}(\omega, x) - \varphi_{\hat{\tau}(\omega)}(\omega,y) \right| \; e^{-2 \left( s - \hat{\tau}(\omega) \right)}&= 4 \; e^{-2 \left( s - \hat{\tau}(\omega) \right)}\leq 4
\rho\dot{\eta}\ge-\frac{1}{T}\nabla\cdot q+\frac{1}{T^{2}}q\cdot\nabla T+\frac{\rho s}{T}or\rho\dot{\eta}\ge-\frac{1}{T}(\nabla\cdot q-\rho s)+\frac{1}{T^{2}}q\cdot\nabla T.
d s^{2}= Z^{- 1 / 2}\eta_{\alpha \beta}d x^{\alpha}d x^{\beta}+ Z^{1 / 2}e^{- \phi ( x^{i})}\delta_{i j}d x^{i}d x^{j}+ Z^{1 / 2}\delta_{m n}d x^{m}d x^{n}
f_{\vec k}(\Omega)~=~-\sum_{l=0}^{\infty}~(2l+1)~\int dr~r^2~U(r)~\frac{f(r)}{r}~j_l(kr)~i^l \int d\theta \sin \theta P_l(\cos\theta) \int d\phi ~Y_{lm}(\theta,\phi)~,
\Phi^{- 1}( p ) \sim \frac{1}{A}\left[ 1 - \frac{A^{\prime}}{( p^{2})^{\omega}}\right] , \sigma^{- 1}( p ) \sim \frac{1}{B}\left[ 1 - \frac{B^{\prime}}{( p^{2})^{\omega}}\right]
J(z)= \frac{\sqrt\theta}{B(a,b)}\sum_{n=0}^\infty\frac{(-1)^n}{\Gamma(b-n)n!}\int_{\theta/z^2}^{\infty}y^{-1/2}\exp\{-(a+n)y \}\mathrm{d}y = \sqrt{\pi\theta}\frac{\Gamma(b)}{B(a,b)}\sum_{n=0}^{\infty}\frac{(-1)^n}{\Gamma(b-n)n!}\frac{1}{\sqrt{a+n}}\operatorname{erfc}(\sqrt{\theta/z^2}\sqrt{a+n}),
g_0^{[2]}(x_0,0,x_0) =\frac{d}{dx}g_0^{[1]}(x,0)\Big|_{x=x_0}=\frac{d}{dx}\Big(\frac{xf_0(x)-0}{x-0}\Big)\Big|_{x=x_0}=f'_0(x_0)=f^{[1]}_0(x_0,x_0).
\psi^{\gamma}_{(2)i}({\bf{r}})=-f^{\alpha\beta b}\,f^{b\delta\gamma}\,{\cal{X}}^\alpha({\bf{r}})\,{\cal{Q}}_{(2)i}^{\beta}({\bf{r}})\,{\cal{X}}^\delta({\bf{r}})=-f^{\alpha\beta b}\,f^{b\delta\gamma}\,{\cal{X}}^\alpha({\bf{r}})\,[a_i^\beta ({\bf{r}})+{\textstyle\frac{2}{3}}x_i^\beta({\bf{r}})]\,{\cal{X}}^\delta({\bf{r}})\;,
\frac{\int_{0}^{a}\left|r j_{l}(Kr) \right|^{2}dr}{\left| \int_{0}^{a}j_{l}(kr) j_{l}(Kr) r^2 dr \right|^2}= f_{0}^{(2)}+f_{2}^{(2)}\, \chi^2+O[\chi^3]
\int_\epsilon^\infty (\partial_x^lu_0^\mu)^2(x) \; dx = \int_\epsilon^\infty (\rho_\mu\ast\partial_x^lu_01_{[0,\infty)})^2(x) \; dx &\leq \|\rho_\mu\|_1^2 \int_\epsilon^\infty (\partial_x^lu_0)^2(x) \; dx &\leq \|\partial_x^lu_0\|_{L^2((0,\infty))}^2.
\left(\begin{array}{c}\theta^1 \theta^2\end{array}\right)\Rightarrow\left(\begin{array}{c}\tilde\theta^1 \tilde\theta^2\end{array}\right)=\left(\begin{array}{cc}\cos\alpha \sin\alpha -\sin\alpha \cos\alpha\end{array}\right)\;\left(\begin{array}{c}\theta^1 \theta^2\end{array}\right)\;,
\ell := \left\lfloor \frac{\max \{0 , D - e_G(A',B') \}}2 \right\rfloor \ \lambda n := \lfloor \phi n /2 \rfloor = \begin{cases}(\phi n -1)/2 \textrm{if $D$ is odd,}\phi n/2 \textrm{if $D$ is even.}\end{cases}
\rho_{U,n}(t) = \frac{f_{U}(t)-g_{U}(t)}{u_{U}(t)-v_{U}(t)}\le \frac{(1+\eta)^2f_X((1+\eta)^2t)-(1-\eta)^2g_X((1-\eta)^2t)}{u_X((1-\eta)^2t)-v_X((1+\eta)^2t)}.
\left(4\sum_{e\in E}\frac{\delta_E(e)}{|e|^2}\sum\limits_{\{u,v\}\subset e\cap V_1}y(u)y(v)+\alpha\right) &=\left(2\sum_{e\in E}\frac{\delta_E(e)}{|e|^2}\sum\limits_{\{u,v\}\subset e\cap V_1}y(u)y(v)+\sum_{e\in \partial V_1}\frac{\delta_E(e)}{|e|^2}\sum\limits_{u\in e\cap V_2;v\in e\cap V_1}z_2(u)z_2(v)\right)\ge 0.
X = \left( \begin{array}{c c}{{K^{2}+ q^{- 1}( q - q^{- 1})^{2}F E}}&{{( q - q^{- 1}) F K^{- 1}}}\\{{( q - q^{- 1}) K^{- 1}E}}&{{K^{- 2}}}\\ \end{array}\right)
\lim_{\varepsilon \to 0^+}\frac{Du(x+\varepsilon Q)}{|Du|(x+\varepsilon Q)}=\lim_{\varepsilon \to 0^+}\frac{D^cu(x+\varepsilon Q)}{|D^cu|(x+\varepsilon Q)}=A(x),\ |D^c u|-a.e.\x\in\Omega
a:=\frac{\gamma}{m^2}+\frac{1}{m}\sum_{k=1}^N\lambda_k+\sum_{k>N}\lambda_k k^{-2s},a_1:=1-\frac{1}{\gamma m}\sum_{k>N}\frac{c_k}{k^{-2s}\lambda_k}-\frac{m}{\gamma}\sum_{k>N}\frac{k^{-2s}c_k}{\lambda_k}.
R_{u,v}^{J,q}(q)= R_{us_j,vs_j}^{J,q}(q) begin{align*}5pt]= (-1)^{\ell(vs_j)-\ell(us_j)}(1-q)^2 \prod_{k\in D(us_j,vs_j)}\left(1-q^{a_k(us_j,vs_j)}\right) begin{align*}5pt]= (-1)^{\ell(v)-\ell(u)}(1-q)^2\prod_{k\in D(u,v)}\left(1-q^{a_k(u,v)}\right),
\{x^\mu ,\theta_\alpha\,, \bar\theta_{\dot\alpha}\}:\qquad\delta\theta_\alpha=\varepsilon_\alpha\,,\quad \delta\bar\theta_{\dot\alpha}=\bar\varepsilon_{\dot\alpha}\,, \quad \delta x_{\alpha\dot\alpha}= -2i\,\theta_{\alpha}\bar\varepsilon_{\dot\alpha}-2i\,\bar\theta_{\dot\alpha}\varepsilon_{\alpha}
\frac{d}{dt}u_k^K=\frac{d}{dt}f(K_u^2)u =[B_u,f(K_u^2)]u+f(K_u^2)(B_uu-iJH_uu) &=B_uf(K_u^2)u-iJf(K_u^2)H_uu &=B_uu_k^K-iJw_k^K,
a_1 \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+a_2 \frac{d b_1}{b_1}\wedge \frac{d b_3}{b_3}+a_3 \frac{d b_2}{b_2}\wedge \frac{db_3}{b_3}= a_1'' \frac{d b_1}{b_1}\wedge \frac{d b_2}{b_2}+ \frac{d z_1}{z_1}\wedge \frac{d z_3'}{z_3'}+ \frac{d z_2}{z_2}\wedge \frac{dz_3}{z_3}
\langle H_1, H_1\rangle_G =\int_{P(E)/M}(\frac{H_1}{G})^2\Xi^{r-1}+r\left(\int_{P(E)/M}(\frac{H_1}{G})^2\Xi^{r-1}- (\int_{P(E)/M}\frac{H_1}{G}\Xi^{r-1})^2\right) \geq \int_{P(E)/M}(\frac{H_1}{G})^2\Xi^{r-1},
u=\frac{2C_0}{r^2}\left(x-y\tan\left(C_0\arctan\frac yx\right)\right), v=\frac{2C_0}{r^2}\left(y+x\tan\left(C_0\arctan\frac yx\right)\right), u=\frac y{r^2\arctan(y/x)},v=-\frac x{r^2\arctan(y/x)}.
\|P_{n}T^{\,j}P_{l}\,x\| \le C_{l}\,\cdot\,(b_{l+1}-b_{l})^{1-\frac1{p}}\,\cdot\,\Bigl\|\sum_{k=b_{l+1}-N}^{b_{l+1}-1}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}w_{s}\Bigr)\,x_{k}e_{k}\Bigr\| &\le C_{l}\,\cdot\,(b_{l+1}-b_{l})^{1-\frac1{p}}\,\cdot\,\sup_{b_{l+1}-N\le k<b_{l+1}}\Bigl(\prod_{s=k+1}^{b_{l+1}-1}|w_{s}\,|\Bigr)\,\cdot\,\|P_{l}\,x\|.
\beta:=c_{n,\sigma p}\left(\int_{B_{4}\setminus B_{3}}\frac{\phi^{p-1}(y)}{|y|^{n+\sigma p}}dy+\int_{B_{6}\setminus B_{4}}\frac{dy}{|y|^{n+\sigma p}}+\int_{B_{6}^{c}}\frac{(1-\psi(y))^{p-1}}{|y|^{n+\sigma p}}dy\right).
\mathcal{C}_{\mathcal{M}^{}}\left( \tau;\rho\right) =\frac{8}{\lambda_{\mathcal{M}}}\sqrt{\frac{1-\rho}{1+\rho}}\left[ -\frac{3}{4}\lambda_{\mathcal{M}}+\frac{1}{4}\frac{\sinh\left( \lambda_{\mathcal{M}}\tau\right)}{\tau}+\frac{\tanh\left( \frac{1}{2}\lambda_{\mathcal{M}}\tau\right)}{\tau}\right] .
\gamma_{E}=\alpha \mathbf{g}\mathbf{v}_{I}\left(\frac{1-\alpha}{N_t-1}\mathbf{g}\mathbf{V}_{N}\mathbf{\Omega}\mathbf{V}_{N}^{\dag}\mathbf{g}^{\dag}+\frac{1}{\mu_E}\right)^{-1}\mathbf{v}_{I}^{\dag}\mathbf{g}^{\dag},
\phi_{jk}=-\begin{vmatrix}z_{1j}z_{1k}z_{4j}z_{4k}\end{vmatrix}-\begin{vmatrix}z_{2j}z_{2k}z_{5j}z_{5k}\end{vmatrix}-\begin{vmatrix}z_{3j}z_{3k}z_{6j}z_{6k}\end{vmatrix}.
\begin{array}{l}{\bf{R}}_{t,k}= E_{{\bf{H}}_k}\left[{{\bf{H}}_k^H{\bf{H}}_k}\right] ={\bf{U}}_{{\rm{T}}_k}\boldsymbol{\Gamma}_{{\rm{T}}_k}{\bf{U}}_{{\rm{T}}_k}^H{\bf{R}}_{r,k}= E_{{\bf{H}}_k}\left[{{\bf{H}}_k{\bf{H}}_k^H}\right] ={\bf{U}}_{{\rm{R}}_k}\boldsymbol{\Gamma}_{{\rm{R}}_k}{\bf{U}}_{{\rm{R}}_k}^H \end{array}
\bar{F}(x) \leq \rho_{1}\left[ F(0) + f^{*}(\gamma) \right]\bar{B}^{r}(x) + \rho_{2}\bar{V}^{r}(x) + \rho_{1}f^{*}(\gamma) \bar{F}_{\gamma}(x-\Delta)B^{r}(\Delta)+\bar{F}_{\gamma}(0)\left[ \bar{B}^{r}(x-\Delta) - \bar{B}^{r}(x) \right] & + \rho_{1}f^{*}(\gamma)\int_{\Delta}^{x-\Delta}\bar{F}_{\gamma}(x-u)dB^{r}(u).
N_{\mathrm{c}m}=\sum_{n=1}^{m}\frac{n!}{2}\mathcal{C}_{n}^{m}\left[\frac{N_{\mathrm{d}\,n+1}}{(n+1)!}-N_{\mathrm{d}1}\frac{N_{\mathrm{d}n}}{n!}\right].
v_{t}+(-\triangle)^{\alpha/2}v =2Ce^{2Ct}u+e^{2Ct}(u_{t}+(-\triangle)^{\alpha/2}u) &=2Ce^{2Ct}u+e^{2Ct}2Cu^{p}&\geq 2C(v+(e^{2Ct})^{1-p}v^{p}) &\geq 2Cv+v^{p}>C(v+v^{p}),
d s_{1 1}^{2}= e^{\frac{1}{2 \sqrt{3}}\sigma - \frac{1}{6}\phi}( e^{- \frac{2}{\sqrt{3}}\sigma}g_{\mu \nu}d x^{\mu}d x^{\nu}+ g_{m n}^{c}d y^{m}d y^{n}) + e^{\frac{4}{3}\phi}d z^{2}\quad .
\frac{d}{dt}|H|^q_q= q \int_{\Omega}\big(H\cdot \nabla u-u\cdot \nabla H-H u\big) \cdot H|H|^{q-2}x = q \int_{\Omega}\big(H\cdot D( u)-u\cdot \nabla H-H u\big) \cdot H|H|^{q-2}x.
\begin{aligned}H(\rho_0 | M_{\Omega_{\rho_0}}) \ge H(\rho_0 | M_{\Omega_{\rho_0}})-H(\rho_t | M_{\Omega_{\rho_t}})=-\int_{0}^{t}\frac{d}{ds}H(\rho_s | M_{\Omega_{\rho_s}})ds &=\int_{0}^{t}I(\rho_s | M_{\Omega_{\rho_s}})ds \ge t\min_{s\in[0,t]}I(\rho_s | M_{\Omega_{\rho_s}}).\end{aligned}
T_f^n(z_i,t) =\big( T^n(z_i),t + f(z_i) + \cdots + f(T^{n-1}z_i) \big) &=\big( T^n(z_i),t + f(x) + \cdots + f(T^{n-1}x) + f(T^{\ell_i}z_i) - f(T^{\ell_i}x) \big) &=V^v \big( T^n(z_i),t + f(x) + \cdots + f(T^{n-1}x) \big)
\tilde{\gamma}^{a}p_{0 a}= 0 = \tilde{\gamma}^{a}f^{\mu}{}_{a}p_{0 \mu}= ( \tilde{\gamma}^{m}f^{\alpha}{}_{m}+ \tilde{\gamma}^{h}f^{\alpha}{}_{h}) p_{0 \alpha}+ ( \tilde{\gamma}^{m}f^{\sigma}{}_{m}+ \tilde{\gamma}^{h}f^{\sigma}{}_{h}) p_{0 \sigma},
\mathrm{d}\varepsilon = 16 \, \frac{1}{8}\, \frac{1}{15}\, \frac{1}{8\pi^2}|J| \, \left( \mathrm{d}^{(1/30,1/15)}+ \mathrm{d}^{(1/30,4/15)}+ \mathrm{d}^{(1/15,11/30)}+ \mathrm{d}^{(2/15,7/30)}\right) & + 36 \, \frac{1}{8}\, \frac{10}{3}\, \frac{1}{15}\, \frac{1}{8\pi^2}|J| \, \mathrm{d}_6 \times \mathrm{d}_6.
Q(h_t, \nabla h_t , \nabla^2 h_t) = (g_{eucl}+ h_t)^{-1}\ast(g_{eucl}+ h_t)^{-1}\ast\nabla h_t \ast \nabla h_t \qquad\qquad\qquad\qquad+ \nabla\ast \left( \left((g_{eucl}+h_t)^{-1}- g_{eucl}^{-1}\right) \ast \nabla h\right)
A_\epsilon =A_\epsilon + (n+1)H_0 V_\epsilon &=-(n+1)a_n \int_{z_1}^{z_2}\Big( H(z,\epsilon) - H_0 \Big) h^nh_\epsilon dz+a_n \left[\frac{h^n h_z h_\epsilon}{\sqrt{1+h_z^2}}\right]_{z_1}^{z_2}.
\psi_{N}^{-}= \frac{1}{[ 2 r ( r + r_{3}) ]^{\frac{1}{2}}}\left( \begin{array}{c}{{- r_{1}+ i r_{2}}}\\{{r + r_{3}}}\\ \end{array}\right) = \left( \begin{array}{c}{{- e^{- i \phi}\sin \frac{\theta}{2}}}\\{{\cos \frac{\theta}{2}}}\\ \end{array}\right)
\|F\otimes G\|^2_{\mathcal{G}_q(\mathcal{H}_+,r,\alpha)}\le C_2 \|F\|^2_{\mathcal{G}_q(\mathcal{H}_+,s,\alpha)}\|G\|^2_{\mathcal{G}_q(\mathcal{H}_+,s,\alpha)}&\qquad\times \left(\sum_{n=0}^\infty \bigg(\frac{r_1}s\bigg)^n \sum_{i=0}^n{n\choose i}_q^\alpha\right)\,.
[p_{r(0^{(n)}0^{(n)})}]_0 = [p_{r(1^{(n)}0^{(n)}0^{(n)}1^{(n)})}]_0 = [p_{r(1^{(n+1)}0^{(n+1)})}]_0 & = [p_{r(1^{(n+1)}0^{(n+1)}0^{(n+1)})}]_0 + [p_{r(1^{(n+1)}0^{(n+1)}1^{(n+1)})}]_0 & = [p_{r(0^{(n+1)}0^{(n+1)}1^{(n+1)})}]_0 + [p_{r(1^{(n+1)}0^{(n+1)}1^{(n+1)})}]_0 & = [p_{r(0^{(n+1)}1^{(n+1)}}]_0 = [p_{r(1^{(n)}1^{(n)})}]_0
\min|a(t-p)|^2+|c(t)|^2 =|\frac{1}{2}(-c(t)+a(t-p))|^2+|\frac{1}{2}(c(t)-a(t-p))|^2 =\frac{1}{2}|(c(t)-a(t-p))|^2 =\frac{1}{2}|f(t)-f(t-p)-\delta|^2,
\mathbf{r}^{\prime}=\mathbf{r}+\left[ \left( \gamma-1\right) \frac{\mathbf{v}\cdot\mathbf{r}}{v^{2}}-\gamma t\right] \mathbf{v},t^{\prime}=\gamma\left( t-\frac{\mathbf{v}\cdot\mathbf{r}}{c^{2}}\right) ,
\frac{e^{-jk_{0}r}}{r}=\frac{1}{j2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dk_{x}dk_{y}e^{-j(k_{x}x+k_{y}y)}\frac{e^{-jk_{z}|z|}}{k_{z}}
{d^{\dagger}}^{(i)}( \eta^n \cdot \tau^n{\cal{G}}) \;\; = \;\; \eta^n \cdot \tau^n \,{d^{\dagger}}^{(i)}{\cal{G}}+{\textsf{P}}_N (n) \; \eta^{n-1}\cdot \tau^{n-1}\,{d^{\dagger}}^{(i)}{\cal{G}}
E := \left\{x \in I_{(\omega_1 , \ldots , \omega_M)}: \inf_{r > 0}\frac{\mu \left(\Sigma_r ( \xi , x ) \cap R_{(\omega_1 , \ldots , \omega_M)}\right)}{\mu \left( \Sigma_r ( \xi , x ) \right)}< e^{-n}\right\}\mbox{, and}
\min_{t\in\mathbb{R}}d_H\bigl(Y_0^{N}(t),R_{-t}(Y)\bigr) =\min_{t\in\mathbb{R}}d_H\bigl(X_0^{N}(t-\eta),R_{-(t-\eta)}(X)\bigr) &=\min_{t\in\mathbb{R}}d_H\bigl(X_0^{N}(t),R_{-(t)}(X)\bigr) &=d_H\bigl(X_0^{N}(\tau),R_{-\tau}(X)\bigr).
G_d:=-2[\bar{q}, \mathcal{H}]\frac{\partial_{\alpha}\bar{\mathfrak{F}}}{\zeta_{\alpha}}-2[\bar{\mathfrak{F}},\mathcal{H}]\frac{\partial_{\alpha}\bar{q}}{\zeta_{\alpha}}-2[\bar{q},\mathcal{H}]\frac{\partial_{\alpha}\bar{q}}{\zeta_{\alpha}}-4D_t q.
\frac{1}{2}\log\frac{P(\gamma)+N_1}{N_1}&=I(\mathbf{S}_1,S_2;\mathbf{S}_1(\gamma),S_2(\gamma)) &=I(\mathbf{S}_1,S_2;\mathbf{S}_1(\gamma),S_2(\gamma),S_0(\gamma)) &=I(\mathbf{S}_1,S_2;\mathbf{S}_1(\gamma),S_0(\gamma))+I(S_2;S_2(\gamma)|S_0(\gamma)) &=\frac{1}{2}\log\frac{P_a+P_{d,1}+N_1}{N_1}+I(S_2;S_2(\gamma)|S_0(\gamma)),
\begin{array}{c}[s_{\mu\nu},s_{\lambda\rho}]=-\eta_{\mu\lambda}s_{\nu\rho}+\eta_{\mu\rho}s_{\nu\lambda}+\eta_{\nu\lambda}s_{\mu\rho}-\eta_{\nu\rho}s_{\mu\lambda}\,,{}{[}{s^{a}_{~b}},{}{s^{c}_{~d}}{]}=2(\delta^{a}_{~d}s^{c}_{~b}-\delta^{c}_{~b}s^{a}_{~d})\,,\end{array}
R = \left[ \begin{array}{l l l l}{{q}}&{{\,}}&{{\,}}&{{\,}}\\{{\,}}&{{1}}&{{0}}&{{\,}}\\{{\,}}&{{\lambda}}&{{1}}&{{\,}}\\{{\,}}&{{\,}}&{{\,}}&{{q}}\\ \end{array}\right] \quad , \quad{\cal P}= \left[ \begin{array}{l l l l}{{1}}&{{\,}}&{{\,}}&{{\,}}\\{{\,}}&{{0}}&{{1}}&{{\,}}\\{{\,}}&{{1}}&{{0}}&{{\,}}\\{{\,}}&{{\,}}&{{\,}}&{{1}}\\ \end{array}\right] \quad .
\phi_{\alpha^\ast \mathcal{L}}(x) = t_x^\ast (\alpha^\ast \mathcal{L}) \otimes (\alpha^\ast \mathcal{L})^{-1}&= \alpha^\ast(t_{\alpha(x)}^\ast \mathcal{L}) \otimes \alpha^\ast (\mathcal{L})^{-1}&= \alpha^\ast ( t_{\alpha(x)}^{\ast}\mathcal{L}\otimes \mathcal{L}^{-1}) &= \alpha^\vee (\phi_\mathcal{L}(\alpha(x)))
f_{(0,\ldots,0,i_{2^{m-1}}=s,0,\ldots,0)}(t) = ~ \frac{(s+2)!}{2}\cdot (2^m-1)!^2 \cdot (2^{m-1}-1)!^{2s}\cdot g(t) \cdot \binom{t+2^m-1}{2^m-1}^2 \binom{t+2^{m-1}-1}{2^{m-1}-1}^{s}\binom{t+2^m-1}{2^{m-1}-1}^{s}.
\sum_{n=1}^N\mathbf{1}_{[0,y)}(x_n) \geq \sum_{n=1}^{N}\mathbf{1}_{[0,\beta_H)}(x_n) = \sum_{h=0}^{H-1}\sum_{n=1}^N\left(\mathbf{1}_{K_{h}}(x_n)-\lambda(K_h)\right).
{\cal Z}_c\simeq\exp\left\{-\frac{g^2}{24}\int d^4x\int d^4y\left[2\Sigma_{\mu\nu}(x)\Sigma_{\mu\nu}(y)\hat D\left((x-y)^2\right)+j_\mu(x)j_\mu(y)\int\limits_{(x-y)^2}^{+\infty}d\lambda\hat D_1(\lambda)\right]\right\},
A(\gamma)e^{i\left\langle b+t,x\right\rangle}=\sum_{\gamma_{1}\in\Gamma(k+)}\frac{q_{\gamma_{1}}e^{i\left\langle b+\gamma_{1}+t,x\right\rangle}}{\mid\gamma+t\mid^{2}-\mid b+\gamma_{1}+t\mid^{2}}.
\int d^{10}x sqrt{-\hat{g}}\ \left[{\textstyle\frac{3}{4}}\hat{G}^{2}\right] =\int d^{9}x\sqrt{g}\ \left[{\textstyle\frac{3}{4}}k G^{2}-{\textstyle\frac{3}{4}}k^{-1}\left( H^{(2)}-\ell H^{(1)}\right)^{2}\right]\, .
\Delta \Omega = -{\frac{1}{\beta}}\sum_{m = 1}^{\infty}\frac{( - 1 )^{m}}{m !}\int_{0}^{\beta}d \tau_{1}. . . d \tau_{m}\langle T_{\tau}( H_{I}( \tau_{1}) . . . H_{I}( \tau_{m}) ) \rangle_{c o n n}^{\beta}.
\begin{aligned}\left|\sqrt{x_ix_j}u_{x_ix_j}(P)\right| \leq \frac{\left|\sqrt{x_ix_j}u_{x_ix_j}(P) - \sqrt{x_ix'_j}u_{x_ix_j}(P')\right|}{\rho^{\alpha}(P,P')}\rho^{\alpha}(P,P') &+ \left|\left(\sqrt{x_ix_j}-\sqrt{x_ix'_j}\right) u_{x_ix_j}(P')\right| + 2 \eta^{-1}\|\sqrt{x_ix_j}u_{x_i}\|_{C([0,T]\times\bar S_{n,m})}.\end{aligned}
\int_\Omega |\nabla u_{n,k}|^2 + \int_\Omega \lambda V |u_{n,k}|^2 \leq b_\nabla 2^{- (d-1) m n}\cdot 2^{\gamma m n}- \lambda b_V 2^{- \gamma m n \frac{2}{d}(-1 + \epsilon)}b_2 2^{-(d-1) m n}2^{- \gamma m n}= b_\nabla 2^{-(d-1) m n}\cdot 2^{\gamma m n}\left(1 - 2\lambda 2^{- \gamma m n \frac{2}{d}(-1 + \epsilon)}\cdot 2^{- 2 \gamma m n}\right).
3 2 \lambda{\frac{( \gamma -{\frac{1}{1 6}}) ( \gamma -{\frac{9}{1 6}}) ( \lambda^{2}- 8 \gamma ) ( \lambda^{2}- 2 )}{( 2 \lambda^{2}- 1 )^{2}( 4 \lambda^{4}- 6 \gamma \lambda^{2}+ 2 2 5 )}}C_{\lambda},
p(t,x,y)=\mathbb{E}\left[\delta_{y}\left(\Phi_{1}(x;\varepsilon B)\right)\right] \ge C\varepsilon^{-N}\cdot\mathbb{E}\left[\delta_{0}\left(\frac{\Phi_{1}(x;\varepsilon B+h)-\Phi_{1}(x;h)}{\varepsilon}\right){\rm e}^{-I\left(\frac{h}{\varepsilon}\right)}\right].
na_n(\vec q\,',\vec q) + \sum_{\nu=1}^3 (q'_{\nu}-q_{\nu}) \frac{\partial a_n(\vec q\,',\vec q)}{\partial q'_{\nu}}= \frac{1}{2}\Delta_{\vec q\,'}a_{n-1}(\vec q\,',\vec q) - V(\vec q\,')a_{n-1}(\vec q\,',\vec q)
f_{a_1}\equiv \phi (f_{a_1}(x) = x_{a_1}\lor x_{a_k}) c_j \equiv a_{b_j}1 \le j \le l a_{b_j + 1}\not\equiv c_j 1 \le j \le l a_{i+1}\equiv a_i i \le k - 1, a_i \notin B.
K^d_{(h)}\simeq{1\over (d-1)}{1\over \Sigma_d}\left({1\over \rho^{d-1}}+{b_0^{(d)}h\over \rho^{d-2}}+...+{b_k^{(d)}h^k\over \rho^{d-k-1}}+...+b^{(d)}_{d-1}h^{d-1}\ln (\rho h) \right)~~,