harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{q(q^2;q^2)_\infty^3}{(-q;q^2)_\infty^2}\omega(q) = \sum_{n=-\infty}^\infty \frac{(3n+2)q^{3n^2+14n+8}}{1-q^{12n+8}}-3\sum_{\substack{n=-\infty n\neq 0}}^\infty \frac{nq^{3n^2+2n}}{1-q^{12n}}&-\sum_{n=-\infty}^\infty \frac{(3n+2)q^{3n^2+8n+4}}{1-q^{12n+8}}+3\sum_{\substack{n=-\infty n\neq 0}}^\infty \frac{nq^{3n^2+8n}}{1-q^{12n}}
F_{m,r}(s,\beta) = \sum\limits_{n=0}^{\infty}\partial_s^n\left.F_{m,r}(s,\beta)\right|_{s=0}\frac{s^{n}}{n!}& =\sum\limits_{n=0}^{\infty}\left((-1)^n 2^{-(n+1)}\cdot C_{m,r}(\beta,n+1)\right)s^{n}
\begin{aligned}\sup_{n}\|\nabla R_\varepsilon u^{n}\|_{L^4(0,T;L^4)}, \sup_n \|R_\varepsilon u^n\cdot\nabla Q^{(n)}\|_{L^3(0,T;L^3)},dx\le C(\varepsilon) \sup_n \|Q^{(n)}\|_{L^2(0,T;H^2)\cap L^\infty(0,T;H^1)}<\infty, \sup_n \|u^n\|_{L^\infty(0,T;L^2)\cap L^2(0,T;H^1)}<\infty,\end{aligned}
\sum_{i}| 1_{X^{k}}\otimes b_{i}\otimes 1_{X^{k}}\rangle_{A_{[a-k,b+k]}}\ \langle 1_{X^{k}}\otimes b_{i}\otimes 1_{X^{k}}| &= \sum_{i}(1_{X^{k}}\otimes b_{i}\otimes 1_{X^{k}})\circ (1_{X^{k}}\otimes b^{*}_{i}\otimes 1_{X^{k}}) &=1_{X^{2k+b-a+1}\otimes Z}=id_{F^{k}_{[a,b]}(Z,c)}
\lambda(x, y) = b_1(m+\eta_1) \delta_{x^{1+},y}+d_1 \eta_1 \delta_{x^{1-},y}+ b_N(m+\eta_N) \delta_{x^{N+},y}+d_N \eta_N \delta_{x^{N-},y}+ \sum_i \eta_i (m+\eta_{i+1}) \delta_{x^{i,i+1},y}+\eta_{i+1}(m+\eta_{i}) \delta_{x^{i+1,i},y}.
\langle \mathrm{d}\sigma |_{\omega_{\varphi}}(u) , \mathrm{d}\sigma |_{\omega_{\varphi}}(v) \rangle_{\sigma^* \omega_{\varphi}}= \int (u\circ \sigma) (v\circ \sigma) (\sigma^*\omega_{\varphi})^n = \int uv \omega_{\varphi}^n = \langle u,v \rangle_{\omega_{\varphi}}.
||\mathbb{P}\xi||_{L^{\infty}(\mathbb{R}^{3})}\lesssim ||\mathbb{P}\xi||_{W^{1,4}(\mathbb{R}^{3})}\lesssim || \xi||_{W^{1,4}(\mathbb{R}^{3})}=|| \xi||_{W^{1,4}(B(0,R) )}.
\psi (\omega^{s_1}) \circ \psi (\omega^{s_2}) = \sum_{i=1}^{s_1n+1}(-1)^{(i-1)(s_2n)}\psi (\omega^{s_1}) \circ_i \psi (\omega^{s_2})= \sum_{l=0}^{s_1\frac{n}{r}}(-1)^{(lr)(s_2n)}\psi (\omega^{s_1+ s_2}) & = (s_1\frac{n}{r}+ 1) \psi (\omega^{s_1 + s_2})
G:= G(y,h_1,h_2,h_1',h_2')= &=F_1(h_1(y),y)h_1'(y)+F_2(x_2-h_2(y)+h_1(y),y)(h_2'(y)-h_1'(y)) &+F_3(x_2-h_2(y)+y,y)(1-h_2'(y)).
\!\!\!\!\!\!\bar{v}_0 = \bar{\textbf{w}}_{0}^{\ast}\textbf{v}_{0}\!=\sum\limits_{k\neq 0}\bar{\textbf{w}}_{0}^{\ast}{\mathbf{{H}}_{k}^{}}\textbf{w}_{k}x_{k}= \sum\limits_{k\neq 0}\sqrt{G_k P_k \mathit{l}\left({\Vert \bold{z}_k \Vert}_2\right)}a_k U_k,
yz =(\psi\otimes\operatorname{id})\bigl((a^*\otimes y)(\Delta b)\bigr)=(\psi\otimes\operatorname{id})\bigl((a^*\otimes 1)(\Delta y)(\Delta b)\bigr) &=(\psi\otimes\operatorname{id})\bigl((a^*\otimes 1)(\Delta(yb))\bigr),
\frac{(-q;q^3)_\infty(-q^2;q^3)_\infty}{(q;q^3)_\infty(q^2;q^3)_\infty}-1=\sum_{n\geq 1}\frac{(-q;q)_{n-1}}{(q;q)_{n-1}}\frac{2q^n}{1-q^n}\frac{q^{n^2-n}}{(q;q^2)_n}.
\mathcal{M}_0 := \mathbb{N}\mathcal{M}_X := X \mathcal{M}_{\tau(\rho)}:= \{x \in \mathcal{M}_\tau^{\mathcal{M}_\rho}\mid x^* \in \mathcal{M}_{\widehat{\tau}}^{\mathcal{M}_{\widehat{\rho}}}x^* \gtrsim_{\tau(\rho)}x \}
c^{\rm eff}_g = \frac{6}{\pi^2}\sum_{a=1}^l L\Bigl( \frac{1}{1+y_a}\Bigr) \,, \qquad{\rm where}\quad \ln(1+y_a) = \sum_{b=1}^l ( N_{ab}+ g \delta_{ab}) \ln (1+y_a^{-1}) \,.
2^{N-1}=2b_0+b_{2^{n-1}}+(b_{2^{n-2}}+b_{3\cdot 2^{n-2}})+\cdots +\sum_{\substack{v_2(k)=1 b_k\in B_1}}b_k +\sum_{\substack{v_2(k)=0 b_k\in B_1}}b_k 2^{N-1}= \sum_{\substack{v_2(k)=1 b_k\in B_2}}b_k +\sum_{\substack{v_2(k)=0 b_k\in B_2}}b_k .
\overset{I}{A}_1\overset{J}{B}_2 = \overset{I}{S^{-1}(a_i)}_2\overset{IJ}{(R')}_{12}\overset{J}{B}_2\overset{IJ}{R}_{12}\overset{I}{A}_1\overset{IJ}{(R')}_{12}\overset{J}{(b_i)}_1,
\hat{x}_{k,j}= \left(\frac{m_{x_{k,j}}^{post}}{v_{x_{k,j}}^{post}}- \frac{m_{x_{k,j}}^{prio}}{v_{x_{k,j}}^{prio}}\right) / \left(1 + \frac{1}{v_{x_{k,j}}^{post}}- \frac{1}{v_{x_{k,j}}^{prio}}\right), \frac{\mu_{k,j}}{\sigma^2_{k,j}}= \left(1 + \frac{1}{v_{x_{k,j}}^{post}}- \frac{1}{v_{x_{k,j}}^{prio}}\right).
0 \geq \langle \nabla^{+}u_j(\hat{z}), \hat{z}-\hat{x}^* \rangle =\langle \nabla^{+}u_j(\hat{z}), \gamma_1 e_1 + \sum_{i=2}^{n-1}\gamma_i e_i \rangle &\geq \frac{\gamma_1}{|\nabla^{+}u_j(\hat{x}^*)|}\langle\nabla^{+}u_j(\hat{z}),\nabla^{+}u_j(\hat{x}^*) \rangle - C_0\epsilon.
\theta^{-1}= -\theta^2 +(t^4+4t)\theta -(t^5+3t) \Longrightarrow \theta^{-1}\equiv -\theta^2 \mod t \Longrightarrow (\theta^{-1})^3 \equiv -\theta^6 \equiv -(\theta^3)^2 \mod t \Longrightarrow (\theta^{-1})^3 \equiv -1 \mod t
2N\Big(1-\sqrt{\frac{5}{3}c}\Big) \leq2\big(N-\sqrt{N[1+(N-1)c]}\big) begin{align*}2 pt] \leq\sum\limits_{i=1}^{N}\|x-\varphi_{i}\|^{2}begin{align*}2 pt] \leq2\big(N+\sqrt{N[1+(N-1)c]}\big) begin{align*}2 pt] \leq2N\left(1+\sqrt{\frac{5}{3}c}\right).
\vec A_\theta = -\vec e_\phi \frac{1- K(r)}{e r}\ , \ \ vec A_\phi = \vec e_\theta \frac{1- K(r)}{e r}\sin \theta\ , \ \ \ \phi = \upsilon(H_1(r) \vec \tau \vec e_r + H_2(r) \lambda_8) \ ,
\begin{aligned}\sqrt{{\varepsilon}}> 2\sqrt{2}d_2\omega_{|a|}^{-\frac{1}{2}}&=2\sqrt{2}d_2(\theta_{|a|}+\theta_N+\sigma)^{\frac{1}{2}}(1-\theta_{|a|}-\theta_N-\sigma)^{-{\frac{1}{2}}}.\end{aligned}
\zeta ( s , z ) = \frac{z^{1 - s}}{s - 1}+ \frac{z^{- s}}{2}+ \frac{s}{1 2}z^{- s - 1}- \frac{s ( s + 1 )}{2 !}\int_{0}^{\infty}\frac{\overline{{{B}}}_{2}( t )}{( z + t )^{s + 2}}d t .
\begin{aligned}D_{f_q}(J_p (x_k), J_p (x_{k+1})) \leq \tilde{c}_q (\alpha) ~ \|J_{p}(x_{k+1})-J_{p}(x_{k})\|^{2}= \tilde c_q (\alpha) ~ \beta_{k}^{2}~ \|\nabla\Phi_{\alpha}(x_{k})\|^{2}. \end{aligned}
\lim_{s\rightarrow0}\Gamma(s)\left( \frac{1}{z^{s}}G(-1/z,s;B_{-b},A_{-c})-G(z,s;A_{-c},B_{b})\right) & =-2if(0)\lim_{s\rightarrow0}\Gamma(s)\sin(\pi s)L(s;B_{-b};0)+\sum\limits_{\mu=0}^{k-1}\sum\limits_{v=0}^{k-1}f(c(\mu+1))f^{\ast}(bv))I(z,0,1,0,0,0)
V_{(1,0)}= \left( (p^{-1}\mathbb{Z}_p/\mathbb{Z}_p)\times (p^{-1}\mathbb{Z}_p / \mathbb{Z}_p) \times (\mathbb{Z}_p / \mathbb{Z}_p)\right)(e_1) V_{(1,1)}= \left( (p^{-1}\mathbb{Z}_p/\mathbb{Z}_p)\times (p^{-2}\mathbb{Z}_p / \mathbb{Z}_p) \times (p^{-1}\mathbb{Z}_p / \mathbb{Z}_p)\right)(e_1e_2)
\mathcal D':= \Bigl \{\{(x,y) | x <0, y>0\}, \{(0,y) | y>0\}, \{(x,y) | x >0, y>0\}& \, \, \, \, \{(x,0) | x <0\}, \, \, \, \, \, \{(0,0)\}, \, \, \, \, \{(x,0) | x >0\}, & \, \, \, \, \{(x,y) | x <0, y<0\}, \{(0,y) | y<0\}, \{(x,y) | x >0, y<0\}\Bigr \}
\sum_{l\geq 2}^{L-1}\frac{(-1)^l}{l}\left(l(\sum_{\cup_{i=1}^{l-1}S_{i}=\{1,..,L-2\}}1)+l(\sum_{\cup_{i=1}^{l}S_{i}=\{1,..,L-2\}}1)\right) &=\sum_{l\geq 2}^{L-1}(-1)^l\left(\sum_{\cup_{i=1}^{l-1}S_{i}=\{1,..,L-2\}}1+\sum_{\cup_{i=1}^{l}S_{i}=\{1,..,L-2\}}1\right) =1.
\big(a\psi_x\big)_x - (b\psi)_x + c\psi + \psi_t = 0,\quad~\Omega \psi(x, T) = 2\beta_0(u(x, T) - w(x)),~0 \leq x \leq s(T) a(0, t)\psi_x(0, t) - b(0, t)\psi(0, t)=0,~0 \leq t \leq T \Big[a\psi_x - (b + s'(t))\psi\Big]_{x=s(t)}= 2\beta_1(u(s(t), t) - \mu(t)), ~0 \leq t \leq T
\begin{array}{r@{}c@{}ll}\partial_t u -{\rm div}\, (a(x)\nabla u ){}={}0 \mbox{in}(0,\infty ) \times \Omega , begin{align*}5pt] u = 0 \mbox{on}(0,\infty ) \times \partial\Omega , begin{align*}5pt] u (0,\cdot ) = u_{0}\mbox{in}\Omega ,\end{array}
D ( p ) = \left\{\frac{p^{2}}{4 \pi n \kappa}[ \zeta F ( p ) - 4 \kappa^{2}z ] - ( z \zeta - \frac{p^{4}}{4 \pi^{2}n^{2}}) \sqrt{F ( p )}\right\}\left[ \zeta^{2}F ( p ) - \frac{\kappa^{2}p^{4}}{\pi^{2}n^{2}}\right]^{- 1}
\Gamma^{\mu \nu}= ( p^{2}+ A ) \left( \eta^{\mu \nu}-{\frac{p^{\mu}p^{\nu}}{p^{2}}}\right) + B \,{\frac{\bar{\theta}^{\mu}\bar{\theta}^{\nu}}{\bar{\theta}^{2}}}+{\frac{p^{\mu}p^{\nu}}{\xi}}
f = \sum_{i=1}^m c_i \phi_i+ \sum_{i=1}^m \sum_{j=0}^\infty \sum_{k=0}^{2^{j-1}}d_{i,j,k}\ \psi_{i,j,k}= \sum_{(i,j,k) \in \Lambda}d_{i,j,k}\ \psi_{i,j,k}
H(z)=\frac{B(z)}{A(z)}=\frac{b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdot\cdot\cdot+b_{N}z^{-N}}{1+a_{1}z^{-1}+a_{2}z^{-2}+\cdot\cdot\cdot+a_{M}z^{-M}}
\Vert x^{n+1}-x^{n}\Vert_{2}\geq|x_{i^{\ast}}^{n+1}-x_{i^{\ast}}^{n}|=|x_{i^{\ast}}^{n+1}|+|x_{i^{\ast}}^{n}| &\geq\min_{i\in I}\{|x_{i}^{n+1}|+|x_{i}^{n}|\}\geq 2\eta_{\mu},
\phi_{m}= N_{m}e^{{\frac{\nu}{2}}k \rho}\left[ J_{\nu - 1}\left({\frac{2 m}{k}}\right) Y_{\nu}\left({\frac{2 m}{k}}e^{{\frac{k}{2}}\rho}\right) - Y_{\nu - 1}\left({\frac{2 m}{k}}\right) J_{\nu}\left({\frac{2 m}{k}}e^{{\frac{k}{2}}\rho}\right) \right]
\mathrm{Mant}_{G,b, \mu}(\rho)= \sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}\mathrm{Ind}^G_{P_b}(\boxtimes^k_{i=1}\mathrm{Mant}_{M_b, b'_i, \mu_{b,i}}(\rho_i)) \otimes [1][|\cdot|^{\langle \rho_G, \mu_b \rangle - \langle \rho_G, \mu \rangle}].
S_{5}( \xi ) = \sum_{n = 1}^{\infty}\frac{( - 1 )^{n}}{n^{2}}\frac 1{\xi^{2}}\int_{1}^{\infty}\frac{d z}{\sqrt{z^{2}- 1}}\frac 1{\left[ \ln \left( \alpha z n \xi \right) \right]^{3}}\ .
\mathcal{F}^{dec}_{r,D}(C):=\left\{\theta: \ \forall \varepsilon \in (0,1), \ \sum_{k>D_\varepsilon}\theta_k^2\leq CD_\varepsilon^{-2s}\right\}=\left\{\theta:\sup_{K\in \mathbb{N}^*}K^{2s}\sum_{k>K}\theta_k^2\leq C\right\}
( D - 2 ) ({\frac{\alpha^{2}}{2}}+ q ) - d_{1}^{2}+ ( D - 2 ) \sum_{i = 1}^{N}r_{i}\Delta_{a i}^{( 1 )}\Delta_{a^{\prime}i}^{( 1 )}= 0 , ~ ~ a \not = a^{\prime}
\rho(m,\alpha,\beta,\gamma,\delta;q)= \frac{(\alpha q,\beta\delta q, \gamma q, \gamma\delta q;q)_m\big(1-\gamma\delta q^{2m+1}\big)}{\big(q,\alpha^{-1}\gamma\delta q, \beta^{-1}\gamma q, \delta q;q\big)_m(\alpha\beta q)^m(1-\gamma\delta q)},
f ( \langle \mathrm m_{t, n}, g \rangle) \, -\, f( \langle \mathrm m_{0, n}, g \rangle) - \int^{t}_{0}f^{\prime}( \langle \mathrm m_{s, n}, g \rangle) \Big(\frac{\,1\,}{\,n\,}\sum_{i=1}^{n}g^{\prime}( X_{s, i}^{(u)}) b ( s, X_{s, i}^{(u)}, \widehat{F}_{s, i}^{(u)}) + \frac{\,1\,}{\,2\,}\langle \mathrm m_{s, n}, g^{\prime\prime}\rangle\Big){\mathrm d}s
\frac{1}{2\pi}\int_{0}^{2\pi}e^{-100(x-1)^2}e^{-ikx}dx=\frac{e^{-\frac{k^2}{400}-ik}}{40\sqrt{\pi}}\left(\operatorname{erf}\left(\frac{ik+400\pi-200}{20}\right)-\operatorname{erf}\left(\frac{ik-200}{20}\right)\right)~(k\in\mathbb{Z}),
\limsup_{n\to \infty}D_n ~\le ~ \limsup_{n\to \infty}\Big\{\frac{||f||_\infty}{\delta^2 n}\sum \limits_{j = 1}^n \frac{1}{n - j + 1}+ \frac{\epsilon}{\delta^2}\sum \limits_{j = 1}^n \frac{1}{(n - j + 1)^2}\Big\}~=~ \frac{\epsilon}{\delta^2}\sum \limits_{j = 1}^\infty \frac{1}{j^2}~.
\left[ \left( \frac{\partial}{\partial Y}- \frac{2 \pi N_{A}}{\kappa L_{2}}\frac{\partial}{\partial \theta}\right) , \left( \frac{N_{A}}{L_{1}}\theta + \frac{2 \pi{N_{A}}^{2}}{\kappa L_{1}L_{2}}Y \right) \right] = 0
(x^-_{i_p,i_{p+1}}\otimes t^{s_{i_p, i_{p+1}}})v_{\nu,\lambda}=0,\ \ 1\le p\le k-1 \implies (x^-_{i,j}\otimes t^{s_{i,j}})v_{\nu,\lambda}=0,\ \ \ 1\le i\le j\le n.
\begin{aligned}E Q_{2 k}(\theta) =-P_{2 k+1}(\theta- \alpha)-P_{2 k-1}(\theta) E^{2}Q_{2 k-1}(\theta) =P_{2 k+1}(\theta- \alpha)+P_{2 k-1}(\theta)+P_{2 k-1}(\theta- \alpha)+P_{2 k-3}(\theta)\end{aligned}
\begin{array}{c}W = X_{3}X_{8}X_{13}- X_{8}X_{9}X_{11}- X_{5}X_{6}X_{13}- X_{1}X_{3}X_{4}X_{10}X_{12}+ X_{7}X_{9}X_{12}+ X_{1}X_{2}X_{5}X_{10}X_{11}+ X_{4}X_{6}X_{14}- X_{2}X_{7}X_{14}.\end{array}
\lambda_n (w_n)_{ij}- |\lambda_n| \delta^{kl}\frac{\partial^2 (w_n)_{ij}}{\partial y^k \partial y^l}- |\lambda_n| \widetilde{Z}(w_n)_{ij}= \frac{\left(f_n\left( \frac{1}{\sqrt{|\lambda_n|}}y \right)\right)_{ij}}{\|R_{\lambda_n}f_n\|_{C^{0,\alpha}}}.
\begin{array}{lll}\langle (\mathcal L_{\mu_2}-|\omega| J_b)\nu,\nu\rangle = \Big(\frac{1}{\gamma}-|\omega|-\frac{\sqrt{\mu}}{\gamma^2}\frac{1}{\sqrt{\mu_2}}-\eta \frac{\sqrt{\mu}}{\gamma^2}\Big)\|\nu\|^2+(\alpha-\mu b|\omega|- \frac{1}{4\eta}\frac{\sqrt{\mu}}{\gamma^2})\|\nu'\|^2 & = r\|\nu\|^2 + \theta \|\nu'\|^2,\end{array}
u_\varphi(t) = \left(\begin{array}{c c c}1 0 c_2 \, t^{p^{e_2}}0 1 c_1 \, t^{p^{e_1}}0 0 1\end{array}\right) (\, c_1, c_2 \in k , e_1, e_2 \geq 0 \,) ,
{\cal F}[ \rho ] = a^{N}\! \! \! \int \! \! d^{N}\! \varphi \G \left\{{\frac{1}{2}}\left({\frac{\partial \rho}{\partial \varphi_{a}}}\right)^{2}+{\frac{D}{4}}\rho^{2}\left( 1 - 2 \ln \rho \right) \right\}\quad .
Y = \left( \begin{array}{c c c c}{\langle 0 , 0 |}&{0}&{\cdots}&{0}\\{\langle 1 , 0 |}&{0}&{\cdots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{\langle k^{\prime}- 1 , 0 |}&{0}&{\cdots}&{0}\\ \end{array}\right) .
\lim\limits_{s\uparrow1}\dfrac{g_{x}(s)-g_{x}(1)}{s-1}={\textstyle\sum\limits_{n=1}^{\infty}}\lim\limits_{s\uparrow1}\left( 1+s+...+s^{n-1}\right) \pi_{x}(n) & ={\textstyle\sum\limits_{n=1}^{\infty}}n\pi_{x}(n)=Tx.
\begin{aligned}\left(\Delta \circ S\right)(x_i) =\Delta \left(-x_1^{-1}x_ix_1^{-1}\right) &=-\left( x_1^{-1}\otimes x_1^{-1}\right)\left( x_i\otimes x_1+x_1\otimes x_i \right)\left( x_1^{-1}\otimes x_1^{-1}\right) &= -x_1^{-1}\otimes x_1^{-1}x_ix_1^{-1}-x_1^{-1}x_ix_1^{-1}\otimes x_1^{-1}.\end{aligned}
|\kappa|^2(p_j,t_j) \left(T-\tfrac{1}{j}- t_j\right) = \max\left\{\left(|\kappa|^2(p,t) \left(T-\tfrac{1}{j}- t\right)\right) : t\in[0,T-\tfrac{1}{j}], p\in[a,b]\right\}.
\int_{0}^{R_{0}}\mu\xi(\mu,t_{0})\d\mu=\int_{0}^{R_{0}}\mu\xi^{\mathrm{in}}(\mu)\d\mu-R_{0}\int_{0}^{t_{0}}\int_{0}^{R_{0}}\nu\xi(R_{0},s)\xi(\nu,s)\Lambda(R_{0},\nu)\d\nu ds.
L_{d}DGHI(p\|q)= \frac{\sum_{j}p_{j}}{a-b}\left[\left(\overline{MG}\right)^{a-1}-\left(\overline{MG}\right)^{b-1}\right] &-\frac{\sum_{j}p_{j}}{a-b}\left[\left(\overline{MH}\right)^{a-1}-\left(\overline{MH}\right)^{b-1}\right]
\begin{array}{l c r}{{u_{1}^{+}= \cos{\frac{\theta}{2}}\exp{\frac{i}{2}}( \psi + \phi )}}\\{{u_{2}^{+}= \sin{\frac{\theta}{2}}\exp{\frac{i}{2}}( \psi - \phi )}}\\{{u_{1}^{-}= \sin{\frac{\theta}{2}}\exp{\frac{- i}{2}}( \psi + \phi )}}\\{{u_{2}^{-}= \cos{\frac{\theta}{2}}\exp{\frac{- i}{2}}( \psi - \phi ) .}}\\ \end{array}
\sum_{\gamma\in\Gamma}\frac{|\gamma'(f({\bf y}))|^{\Delta}}{|f({\bf x})-\gamma f({\bf y})|^{2\Delta}}\;=\;|f'({\bf x})|^{-\Delta}|f'({\bf y})|^{-\Delta}\sum_{\gamma\in f^{-1}\Gamma f}\frac{|\gamma'({\bf y})|^{\Delta}}{|{\bf x}-\gamma{\bf y}|^{2\Delta}}.
\lambda\left( \bigcup_{i=1}^{\infty}(s_i, s_i + \alpha \sigma_i / 8) \cap \{r \in (0, s_0) : \|v'(r)\| \leq m\}\right) \leq \lambda \left( \bigcup_{i=1}^{\infty}(s_i, s_i + \alpha \sigma_i/8) \right) & = \alpha \left(\sum_{i=1}^{\infty}\sigma_i\right)/8 & \leq \alpha s_0 /4,
||\sqrt{\omega}\lambda(x) K(\partial_{1}y(x,\frac{\cdot}{\lambda(x)^{2}}))(\omega \lambda(x)^{2}) ||_{L^{2}(\rho(\omega \lambda(x)^{2})d\omega)}&\leq C\left(||\partial_{1}y(x)||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}+||\sqrt{\omega}\lambda(x)\partial_{1}y(x) ||_{L^{2}(\rho(\omega \lambda(x)^{2}) d\omega)}\right)
\frac{1}{\mathrm{L}(a,b)}=\int_0^\infty \frac{dt}{(t+a)(t+b)}, \frac{1}{\mathrm{AG}(a,b)}=\frac{2}{\pi}\int_0^\infty\frac{dt}{\sqrt{(t^2+a^2)(t^2+b^2)}}.
\left[\begin{array}{cc}p_{n-1}p_{n}q_{n-1}q_{n}\end{array}\right]=\left[\begin{array}{cc}0 1 1 a_1\end{array}\right]\left[\begin{array}{cc}0 1 1 a_2\end{array}\right]\cdots\left[\begin{array}{cc}0 1 1 a_{n-1}\end{array}\right]\left[\begin{array}{cc}0 1 1 a_{n}\end{array}\right]
\varphi\left(s(h_{y+n})^{m_{1}}s(h_0)^{k_1}\cdots s(h_{y+1})^{m_{n}}s(h_0)^{k_n}s(h_{y})^{m_{n+1}}\right) = \varphi\left(s(h_0)^{k_1 + \cdots + k_n}\right) \prod^{n+1}_{j=1}\varphi\left(s(h_{y+n+1-j})^{m_j}\right)
\vec{w}= \left( \begin{array}{c}{{w_{3}+ \frac{1}{6}}}\\{{6 w_{3}}}\\{{w_{3}}}\\ \end{array}\right) \; , \qquad \vec{k}= \left( \begin{array}{c}{{k_{3}+ \frac{1}{6}\ln c_{0}^{2}}}\\{{6 k_{3}}}\\{{k_{3}}}\\ \end{array}\right)
C^{\ast}\left( \nabla u\right) =C^{\ast}\left( u^{\prime}\left( C\left(x\right) \right) x^{\ast}\right) =\left\vert u^{\prime}\left( C\left(x\right) \right) \right\vert^{p}C^{\ast}\left( x^{\ast}\right)=\left\vert u^{\prime}\left( C\left( x\right) \right) \right\vert^{p}\frac{q}{p}C\left( x\right)
r_6 = R_{1313}R_{2323}S_{12232}-R_{1213}R_{2323}S_{12233}-R_{1213}R_{2323}S_{13232}+R_{1212}R_{2323}S_{13233}& = \{(b+c)(R_{1212}-R_{1313})-(a-d)R_{1213}\}R_{2323}{}^2 & = -(b-c) \{(a-d)^2+4(b+c)^2 \}\{(b+c)^2-ad \}^2.
P^{( N_{f})}( \lambda_{\mathrm{m i n}}, m ) = 2 \lambda_{\mathrm{m i n}}P^{( N_{f})}( x_{\mathrm{m i n}}, m ) = - 2 \lambda_{\mathrm{m i n}}\left. \frac{d E ( 0 , s )}{d s}\right|_{s = \lambda_{\mathrm{m i n}}^{2}}\: .
(e'_i)^{\varepsilon_i(\hat{b})}(F_i^{(N)}G^{\mathrm{low}}(b)) &=\sum_{s=0}^Nq_i^{-2\varepsilon_i(\hat{b})N+(\varepsilon_i(\hat{b})+N)s-\frac{1}{2}s(s-1)}\left[ \begin{array}{c}\varepsilon_i(\hat{b}) s \end{array}\right]_iF_i^{(N-s)}(e'_i)^{\varepsilon_i(\hat{b})-s}G^{\mathrm{low}}(b).
\mathbf{T}_1(E) = 1-i\Bigl({\rm e}^{\frac{\pi}{3}i}Q^{-}(-E\Omega) Q^{+}(-E \Omega^{-1})+{\rm e}^{-\frac{\pi}{3}i}Q^{+}(-E\Omega) Q^{-}(-E\Omega^{-1})\Bigr)+O(h^2).
{\rm Coeff}\left[\prod_{1 \leq i<j \leq n}\left(\frac{1-z_j/z_i}{1- t z_j/z_i}\right)Q_{\lambda/\mu}(z_1,\dots,z_n;t),z_1^{\nu_1}\dots z_n^{\nu_n}\right]=f^{\lambda}_{\mu\nu}(t)b_{\nu}(t).
\tilde{U}(nx) = \lambda_{nx}^{-1}(nx) &= n\lambda_{nx}^{-1}(x) &= n((\sigma_{x}\cdots\sigma_{U^{n-1}x})^{-1}(x)) &= n(\sigma_{U_{n-1}}^{-1}\cdots\sigma_{x}^{-1}(x)) &= n(U^{n}(x)),
\psi_{0,2}( x, y )( v_1, v_2 ) = \big( \tfrac{\partial^2}{\partial y^2}\psi( x, y ) \big)( v_1, v_2 ) , \psi_{1, 1}( x, y )( v_1, v_2 ) = \big( \tfrac{\partial}{\partial y}\tfrac{\partial}{\partial x}\psi( x, y ) \big)( v_1, v_2 ) .
\nabla_{y_{n+1}}\mathcal{L}(y^*,\alpha^*,\beta^*,\gamma^*,(\delta^\pm)^*,\pi^*) = -y^*_n - \beta^*_{n+1}+ \gamma^* = 0, \nabla_{y_{n+2}}\mathcal{L}(y^*,\alpha^*,\beta^*,\gamma^*,(\delta^\pm)^*,\pi^*) = -y^*_n - (\delta^-)^* + (\delta^+)^* = 0
e^{b_1^{(1)}-ib_2^{(1)}}v_{1s}^g(s,\tau,\beta^{(1)}(s,\tau,\hat y'))J_1=e^{b_1^{(2)}-ib_2^{(2)}}v_{2s}^g(s,\tau,\beta^{(2)}(s,\tau,\hat y'))J_2.
\displaystyle\vert \nabla_xl_p(x_0,y_0)\vert =\frac{2(p-x_0)\cdot\nu_{x_0}}{\vert x_0-p\vert}\not=0,\displaystyle\nu_{x_0}=-\frac{1}{\vert\nabla_x l_p(x_0,y_0)\vert}\nabla_x l_p(x_0,y_0)=-\hat{\nu}_{x_0}, \displaystyle\nu_{y_0}=\hat{\nu}_{y_0}=\nabla_yl_p(x_0,y_0).
f(A) = \sum_p f(A_p) = \sum_{\substack{p< x p\notin A}}f(A_p) + \sum_{p\in A, p\ge x}f(p) & \le e^\gamma\sum_{\substack{p< x p\notin A}}{\rm d}({\rm L}_p) + \sum_{p\in A, p\ge x}f(p) & \le e^\gamma\sum_{p}{\rm d}({\rm L}_p) + \big(e^\gamma+o_x(1)\big)\sum_{p\ge x}{\rm d}({\rm L}_p) \ \le \e^\gamma+o_x(1)
(A \circ B)^{\#}(z) =|\det (A \circ B)' (z)|^{\frac{N-p}{Np}}f((A \circ B)(z)) &=|\det A' ( B (z) ) \cdot \det B'(z)|^{\frac{N-p}{Np}}f(A (B(z) ) ) &=|\det B'(z)|^{\frac{N-p}{Np}}(A^{\#}f )(B(z) ) &= [(B^{\#}\circ A^{\#}) f] (z),
S=\frac{1}{3}\begin{pmatrix}1-\zeta_3 1-\zeta_3^2 \zeta_3-\zeta_3^3 1-\zeta_3^2 1-\zeta_3 \zeta_3^2-\zeta_3 \zeta_3-\zeta_3^2 \zeta_3^2-\zeta_3 \zeta_3-\zeta_3^2\end{pmatrix}T=\begin{pmatrix}1 0 0 0 1 0 0 0 \zeta_3^2\end{pmatrix}
u_{k+1}=\sum_{l=1}^{k}C_{k-l+1}\frac{\partial_{\epsilon}^{l-1}u}{(l-1)!}+\sum_{l=1}^{k}D_{k-l+1}\frac{\partial_{\epsilon}^{l-1}v}{(l-1)!}+\frac{\partial_\epsilon^{k}u}{k!}.
\Phi(r) = \Phi(\varphi(r)) =\frac1{r^2 - \sqrt2 r + 1}\left(\sqrt2 r - 1, r^2 - \sqrt2 r\right) &= \frac{\left(\sqrt 2 xy - y^2, x^2 -\sqrt 2 xy \right)}{x^2 -\sqrt 2 xy + y^2}.
I_{321}= -r\cos(kx)\sin(x)\int_{0}^{2\pi}\log\left((1+r^2)+(r^2-1)\cos(y))\right)\cos(y-x)dy & = -r\cos(kx)\sin(x)\cos(x)(-2\pi)\left(\frac{1-r}{1+r}\right) I_{322}= \cos(kx)\cos(x)\int_{0}^{2\pi}\log\left((1+r^2)+(r^2-1)\cos(y))\right)\sin(y-x)dy & = -\sin(x)\cos(kx)\cos(x)(-2\pi)\left(\frac{1-r}{1+r}\right)
t_{\Omega \Sigma}^{( 1 )}{}^{\Pi \Gamma}{}_{\Delta \Lambda}= \delta_{[ \Delta}^{[ \Pi}\delta_{\Lambda ] [ \Omega}\delta_{\Sigma ]}^{\Gamma ]}\, , \qquad t_{\Omega \Sigma}^{( 2 )}{}^{\Pi \Gamma \Delta \Lambda}= 0 \, .
\frac{d}{d\rho}g_{(n)}(s, W_0^s, Y_{(n), 0}^s)=\left(\frac{d}{d\rho}g(s, W_0^s, Y_{0}^s)\right) \mathbf{1}_{\int_0^s g^2(t, W_0^t, Y_{0}^t) dt < n}\mbox{a.s.},
\lambda_n^{1/2}B_n^2L_{m(n)}^2(1) \le B_{m(n)}^2L_{m(n)}^2(1) =\sum_{k=1}^{m(n)}\mathbf{E}[X_k^2;|X_k|>B_{m(n)}] &\le\sum_{k=1}^n\mathbf{E}[X_k^2;|X_k|>B_{m(n)}]\le\sum_{k=1}^n\mathbf{E}[X_k^2;|X_k|>\lambda_nB_n] &=B_n^2L_n^2(\lambda_n)\le B_n^2\lambda_n^2.
\left[\begin{array}{l}H_{j,e}^{[s]}H_{j,o}^{[s]}\end{array}\right]=\left[\begin{array}{ll}I_{n_j}-U_j^{[s+1]}0 I_{n_j'}\end{array}\right]\left[\begin{array}{ll}I_{n_j}0 P_j^{[s+1]}I_{n_j'}\end{array}\right]\left[\begin{array}{l}D_j 0\end{array}\right],
R=4\langle S\overline{v},(II)\overline{v}\rangle-4\langle A\overline{v},(I)\overline{v}\rangle-4\langle A\overline{v},(III)\overline{v}\rangle+4\langle(I)\overline{v},(II)\overline{v}\rangle+4\langle(II)\overline{v},(III)\overline{v}\rangle.
p_n^+:=P\left(\ln\rho_0>0,\omega_0^0\leq\frac{1}{n}\right) p_n^-:=P\left(\ln\rho_0<0,\omega_0^0\leq\frac{1}{n}\right) p_n^0:=P\left(\ln\rho_0=0,\omega_0^0\leq\frac{1}{n}\right)
(Ds)_{\beta \gamma}= \delta_{\beta,\gamma}(Ds)_{\beta},\quad (Ds)_{\beta}=-i\sum_{\lambda\in\Delta_s,\ \beta\cdot\lambda=1}\left[ g_{s_1}z_{d}^{(1/2)}(\lambda\cdot q,\xi)+ g_{s_2}z_{d}^{(1/4)}(\lambda\cdot q, \xi)\right],
\Re\Big\{\sum_{n=1}^{\infty}z_n^{m_0}\Big\}\geq \tfrac{\epsilon}{50}|z_1|^{m_0}\geq \tfrac{\epsilon}{50}(\alpha+1-\beta')^{-2m_0}\geq \tfrac{\epsilon}{50}\alpha^{-2m_0}\exp(-\tfrac{2m_0}{\alpha}(1-\beta') ),
\big\| \big(\tfrac{\partial}{\partial x}\psi\big)( x, y ) \, v_1 \big\|_V= \left\| u_{0,2}( t, x ) \big( F( y ), v_1 \big) \right\|_V\leq \tfrac{c_{-\vartheta, 0}}{( T - t )^{\vartheta}}\, \| F( y ) \|_{H_{-\vartheta}},
\small\mathbf{v}_{i,j}= \mathbf{y}_{i,j}- \mathbf{H}_{i,j}\mathbf{x}_{i,j}= \begin{cases}\mathbf{n}_{i,j}, (i,j) \in \mathbb{S}_b, b \notin \mathbb{I}, \mathbf{H_I}_{i,j}\mathbf{x_I}_{i,j}+ \mathbf{n}_{i,j}, (i,j) \in \mathbb{S}_b, b\in \mathbb{I},\end{cases}
\begin{gathered}b\left(x\right)=\frac{8 x H^2}{(H^2+x^2)(9 H^2+ x^2)}+2H\frac{4x H}{((1-H)^2+x^2)((1+H)^2+x^2)}-\sum_{n=2}^{\infty}\left|\binom{2 H}{n}\right|\frac{4x n H}{((n-H)^2+x^2)((n+H)^2+x^2)}\end{gathered}
\left( \begin{array}{c}{{- p_{0}a u_{3}i + p_{x}j a u_{3}i}}\\{{p_{y}a u_{3}i + p_{z}j a u_{3}i}}\\{{p_{0}a u_{1}i - p_{x}j d u_{1}i + p_{y}a u_{2}i - p_{z}j d u_{1}i}}\\ \end{array}\right) = m \; \left( \begin{array}{c}{{u_{1}}}\\{{u_{2}}}\\{{u_{3}}}\\ \end{array}\right)
V_{i \bar{j}}\Big [ \begin{array}{c c}{{b_{3}}}&{{b_{4}}}\\{{\emptyset}}&{{\emptyset}}\\ \end{array}\Big ] = a_{2 3}^{i}\bar{a}_{4 5}^{j}( a_{1 3 4 5 2 6}\bar{a}_{1 6 4 3 2 5}) | ( 1 2 4 | 3 5 6 )^{2}( 1 3 5 | 2 4 6 ) | ,
\frac{\partial Y}{\partial x}=\left(\frac{A_2}{x^2}+\frac{A_1}{x}+A_0\right)Y ,\frac{\partial Y}{\partial t_1}=\left( \frac{B_{11}}{x}+B_{10}\right)Y ,\frac{\partial Y}{\partial t_2}=\left( B_{21}x+B_{20}\right)Y ,
{\rm reg}_{0}(e_{0}^{n}(u*e_{1})) =\sum_{i+j=n}{\rm reg}_{0}(e_{0}^{i}u)*{\rm reg}_{0}(e_{0}^{j}e_{1})=\sum_{i+j=n}(-1)^{j}{\rm reg}_{0}(e_{0}^{i}u)*e_{1}e_{0}^{j}.
P_{a}=aa^{T}=[\begin{matrix}a_{x}\\a_{y}\\a_{z}\end{matrix}][\begin{matrix}a_{x}&a_{y}&a_{z}\end{matrix}]=[\begin{matrix}a_{x}^{2}&a_{x}a_{y}&a_{x}a_{z}\\a_{x}a_{y}&a_{y}^{2}&a_{y}a_{z}\\a_{x}a_{z}&a_{y}a_{z}&a_{z}^{2}\\ \end{matrix}]
\int_{(0,+\infty)}f(t)\,d\|E_{\rho,\sigma}(t)\xi_\sigma\|^2 \ge\int_{(0,+\infty)}(a+bt)\,d\|E_{\rho,\sigma}(t)\xi_\sigma\|^2 &=a\|s_M(\rho)s_{M'}(\sigma)\xi_\sigma\|^2+b\|\Delta_{\rho,\sigma}^{1/2}\xi_\sigma\|^2 &=a\|s_M(\rho)\xi_\sigma\|^2+b\|s_M(\sigma)\xi_\rho\|^2 &=a\sigma(s_M(\rho))+b\rho(s_M(\sigma))>-\infty,
\mu_\omega (C_u) = \mu_\omega ( f^{(\omega_1)}_{u_1}\circ \circ \circ f^{(\omega_n)}_{u_n}([0,1]) ) = \eta^{(\omega)}([u_1,...,u_n]) = \prod_{i=1}^n \mathbf{p}^{(\omega_i)}_{u_i}.
\int_{R_{0}}^{R(T)}\frac{dR}{R}=\int_{T_{0}}^{T}\alpha dT\Rightarrow ln(R)|_{R_{0}}^{R(T)}=\alpha(T-T_{0})\Rightarrow ln(\frac{R(T)}{R_{0}})=\alpha(T-T_{0})\Rightarrow R(T)=R_{0}e^{\alpha(T-T_{0})}
\min_{t\in [0,T]}\min_{i=1}^N\rho_i(t)>0, a.s. & \max_{ij\in E}|S_i(t)-S_j(t)|\le \sqrt{\frac{C_{Kin}}{\min\limits_{t\in [0,T]}\min\limits_{i=1}^N\rho_i(t)}}<\infty, a.s.
\mu_{n+1}:=\Big(1-\frac{\gamma_{n+1}}{\lambda_{n+1}-\lambda_0}\Big)\prod_{1\leq p\ne n + 1}\frac{\displaystyle{\Big( 1-\frac{\gamma_p}{\lambda_p-\lambda_{n+1}}\Big)}}{\displaystyle{\Big(1-\frac{\gamma_p}{\lambda_p-\lambda_{n}-1}\Big)}}\ .
M_{i j}\equiv \left( \begin{array}{c c c c c}{2}&{1}&{\cdots}&{\cdots}&{1}\\{1}&{2}&{}&{}&{\vdots}\\{\vdots}&{}&{\ddots}&{}&{\vdots}\\{\vdots}&{}&{}&{\ddots}&{\vdots}\\{1}&{\cdots}&{\cdots}&{\cdots}&{2}\\ \end{array}\right) .
\bar{\cal K}= \left( \begin{array}{c c}{- \frac{d^{2}}{d x^{2}}+ 4 - \frac{6}{\cosh^{2}x}}&{0}\\{0}&{- \frac{d^{2}}{d x^{2}}+ \sigma^{2}- \frac{2}{\cosh^{2}x}}\\ \end{array}\right)
\left\| M_{n+1}\right\|^{2p}\leq \left( \left\| M_{n}\right\|^{2}+ M^{2}\right)^{p}+ 2 \left\langle M_{n}, \xi_{n+1}\right\rangle \left( \left\| M_{n}\right\|^{2}+ M^{2}\right)^{p-1}& + \sum_{k=2}^{p}\binom{p}{k}\left| 2 \left\langle M_{n}, \xi_{n+1}\right\rangle \right|^{k}\left( \left\| M_{n}\right\|^{2}+ M^{2}\right)^{p-k}.