harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0=\Delta(s^\Lambda)^E=\Delta(s^\Lambda)^{E\cap \Lambda^i}+\sum_{\substack{\emptyset\neq G\subseteq E G\cap \Lambda^{e_i}\neq \emptyset \lambda\in \mathrm{MCE}(G)}}(-1)^{|G|}s_\lambda^\Lambda{s_\lambda^\Lambda}^*.
X_s=2ig_s\sum_{\lambda\in\Delta_s}x_{d}(\lambda\cdot q, \xi)E_{d}(\lambda),\quad Y_s=ig_s\sum_{\lambda\in\Delta_s}y_{d}(\lambda\cdot q, \xi)E_{d}(\lambda),\quad E_{d}(\lambda)_{\beta \gamma}=\delta_{\beta-\gamma,2\lambda}.
{\cal A}_{z^{\prime}=z}F =\frac{1}{2}\sum_{i=1}^{\infty}\frac{1}{k!}\partial_{z}^{i-1}{\rm Res}_{z^{\prime}=z}(z^{\prime}-z)^i F,\hspace{1em}{\cal B}_{z^{\prime}=z}F=0.
\begin{cases}u_t - \alpha^2u_{txx}+ c_0u_x + 3uu_x +\gamma u_{xxx}= \alpha^2(2u_xu_{xx}+uu_{xxx}) - \tilde{\rho}\tilde{\rho_x}-\tilde{\rho_x}, t > 0, x\in \mathbb{R}, \tilde{\rho_t}+(u\tilde{\rho})_x+u_x = 0 t > 0, x\in \mathbb{R}, \tilde{\rho}(0,x) = \tilde{\rho_0}, u(0,x) = u_0.\end{cases}
[\begin{matrix}ct^{\prime}\\x^{\prime}\\y^{\prime}\\z^{\prime}\end{matrix}]=[\begin{matrix}cosh\phi&-sinh\phi&0&0\\ -sinh\phi&cosh\phi&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{matrix}][\begin{matrix}ct\\x\\y\\z\end{matrix}],
- 6 b^{I}\nabla_{\alpha}\nabla^{\gamma}\nabla_{\gamma}Y^{I}- 6 \left[ \nabla^{\mu}\nabla_{\mu}b^{I}- \frac{1}{2 \sqrt{2}}h^{\prime I}+ \frac{3 \sqrt{2}}{5}h_{2}^{I}\right] \nabla_{\alpha}Y^{I}= 0 .
\begin{array}{ll}v_{0,0}=-(1-q)(1-q^2)ac, begin{align*}2pt] v_{1,0}=(1-q)\bigl(a(b+c)-q(a+c)\bigr), v_{0,1}=(1-q)\bigl(a+c-qa(b+c)\bigr), begin{align*}2pt] v_{2,0}=(1-q)(1-ab)+v_{0,2}, v_{1,1}=(1-q^{-1})(1-abq^2)-(q+q^{-1})v_{0,2}, \end{array}
\frac{1}{2}{(n-2)^2}\omega_{n-1}\Lambda \mathcal{H}(0)=-C_n \left(\frac{n-2}{4(n-1)}S_g(\bar{x}_0)-m_0^2 \right)\lim_{\alpha\to +\infty}\left(\mu_\alpha^{4-n}r_\alpha^{n-2}\right),
\Omega_{n, d}= \bigtimes_{i = 1}^n \Omega, \mathcal{A}_{n,d}= \bigotimes_{i = 1}^n \mathcal{A}, \mathbb{P}_{n, d, \theta}= \bigotimes_{i = 1}^n \mathbb{P}_{d, \theta}\theta \in \Theta_d,
\phi_{i}( x , y ) = \frac{\phi_{0}( x - y )}{v_{1}}\left[ v_{1}+ \frac{\sigma ( X ) + i \varphi^{0}( X )}{\sqrt{2}}\right] \delta_{i}{}^{5}+ \sqrt{2}i \frac{\phi_{0}( x - y )}{v_{2}}\varphi^{\rho}( X ) \left( S^{\rho}\right)_{i}{}^{5}+ \ldots ,
j_{s^{-1}_\chi(x)}= j_{s^{-1}_\chi(y) + 1}j_{s^{-1}_\chi(x)+1}= j_{s^{-1}_\chi(y)}\{g(x), g(y)\}= \{\ast, \cdot\}\{t\ast, t\}j_{s^{-1}_\chi(x)}= j_{s^{-1}_\chi(y)}j_{s^{-1}_\chi(x)+1}= j_{s^{-1}_\chi(y) + 1}.
\partial_x M^{n-1}(t) = \frac{\langle w \rangle_{f^{n-1}_t,\varphi'}\langle 1 \rangle_{f^{n-1}_t,\varphi}- \langle w \rangle_{f^{n-1}_t,\varphi}\langle 1 \rangle_{f^{n-1}_t,\varphi'}}{\langle 1 \rangle_{f^{n-1}_t,\varphi}^2}\,,
\delta (x-y)\, \left( \frac{C^{2}\, n^{4}\, \Theta (x_{0}-y_{0})}{\left( \left( x_{0}-y_{0}-i\, \epsilon \right)^{2}-\left(x_{s}-y_{s}\right)^{2}\right)^{2}}+\frac{C^{2}\, e^{2\, \alpha +2\, \alpha^{*}}\, n^{4}\, \Theta (x_{0}-y_{0})}{\left( \left( y_{0}-x_{0}-i\,\epsilon \right)^{2}-\left( x_{s}-y_{s}\right)^{2}\right)^{2}}\right.
\langle\phi_{i}|e^{-i\hat{H}t/\hbar}\psi\rangle=\langle\phi_{i}(e^{-i\hat{H}t/\hbar})^{\dagger}|\psi\rangle=\langle\phi_{i}e^{-i\hat{H}^{\dagger}(-t)/\hbar}|\psi\rangle
\int_\Omega (\Phi'', \widehat \Phi)dx + \int_\Omega \langle \nabla\Phi, \nabla \widehat \Phi\rangle dx+\int_{\Gamma_1}(B\Phi, \widehat \Phi)d\Gamma +\int_\Omega (A\Phi, \widehat \Phi)dx =0
b_2 = -\frac 12 \frac{1}{X_{1+2}}\sum_{j=1}^{2}a_{2-j}a_{j-1}(2+1-j)! j! \sum_{Q\in Q^{(3,j)}}X_{Q}\Big|_{X_1=0=X_2}&= -\frac{1}{2X_{1+2}}\left( a_1 a_0 2! 1! \left( X_1+X_2 + X_{1+2}\right) + a_0 a_1 1! 2! \left( X_{1+2}+ X_{1}+ X_2\right) \right) \Big|_{X_1=0=X_2}&= -2 a_1.
u ( s^{\prime}) - 2 \frac{s^{\prime}}{s}\int_{0}^{s}d s^{\prime \prime}e A_{-}\left( x^{+}- u ( s^{\prime \prime}) \right) + 2 \int_{0}^{s^{\prime}}d s^{\prime \prime}e A_{-}\left( x^{+}- u ( s^{\prime \prime}) \right) = 0 \; .
\Phi^1((x_{0},{x}_1),\cdots,(x_{0},\mathbf{x}_k)) = (k!)^{-1/2}\langle 0\vert \phi_1(x_{0},\mathbf{x}_{1})\cdots\phi_1(x_{0},\mathbf{x}_{k})\vert\Phi^1 \rangle.
\psi(T_pT_q^*T_xT_y^*) = \psi(T_{pq^{-1}(q \vee x)}T_{yx^{-1}(q \vee x)}^*)=\begin{cases}e^{-\beta \theta(pq^{-1}(x\vee q))}\psi(T_{b^n}) e^{-\beta \theta(pq^{-1}(x\vee q))}\psi(T_{b^{-n}}^*) \end{cases}
=\frac{\phi(a)x}{an}\left(\sum^\infty_{\substack{m=1 (m,an)=1}}\frac{\mu(m)}{m^2}\right)+O(2^{\omega(a)}\sqrt{x}) &=\frac{\phi(a)x}{an}\cdot\frac{1}{\zeta^{(an)}(2)}+O(2^{\omega(a)}\sqrt{x}) &=\frac{\phi(a)x}{an}\cdot\frac{1}{\zeta(2)}\cdot\prod_{p\mid an}\frac{p^2}{p^2-1}+O(2^{\omega(a)}\sqrt{x}).
S(H_6,6f) =\frac{10f+c_a'''}{18}\hbox{where}c_a''':=\begin{cases}-19a-10 \hbox{if $a\equiv 0\pmod 6$}, a-18 \hbox{if $a\equiv 2\pmod 6$}, -a-19 \hbox{if $a\equiv 3\pmod 6$}, 19a+9 \hbox{if $a\equiv 5\pmod 6$},\end{cases}
\left ( \! \!\begin{array}{c}\partial_{\tau}\psi_1 \partial_{\tau}\psi_2 \end{array}\! \! \right ) =\left ( \! \! \begin{array}{c}\psi_2 - \rho \partial_{\rho}\psi_1 - \frac{2}{p-1}\psi_1 \Delta_{\rho}\psi_1 - \rho \partial_{\rho}\psi_2 - \frac{p+1}{p-1}\psi_2 + |\psi_1|^{p-1}\psi_1 \end{array}\! \! \right)
f_{k,\ell}(p^{\alpha}) :=\frac{1 + \sum_{i=1}^{\alpha}(\tau_k(p^i)\tau_{\ell}(p^i)-\tau_k(p^{i-1})\tau_{\ell}(p^{i-1}))X^i + \sum_{i=\alpha+1}^{\infty}( \tau_k(p^{\alpha}) \tau_{\ell-1}(p^i) + \tau_{\ell}(p^{\alpha}) \tau_{k-1}(p^i))X^i}{\Big( 1-\frac{1}{p}\Big)^{-(k-1)}+\Big( 1-\frac{1}{p}\Big)^{-(\ell-1)}-1}.
d_{G_{2}}(u) \ge \sigma_{2}(G) - d_{G_{1}}(u) - d_{G}(v) &\ge \big( 2(s_{1}+ s_{2}+ \varepsilon) - 1 \big) - (s_{1}- 2 + l + \varepsilon) - (s_{1}+ s_{2}- 2l + 2\varepsilon) &= s_{2}+ l + 1 - \varepsilon = (s_{2}- 2 + l +\varepsilon) + 3 - 2 \varepsilon \ge (s_{2}- 2 + l + \varepsilon) + 1.
\sum_{n = 0}^{\infty}A(n,r)q^{n}= \frac{1}{\prod_{j = 0}^{\infty}(1 - q^{2j + r})}+ \sum_{n = 1}^{\infty}\frac{q^{n(n+ 1)}}{(q^{2};q^{2})_{n}}\frac{1}{\prod_{j = n}^{\infty}(1 - q^{2j + r})}
E\Big[ \prod_{i=1}^p |\tilde{S}_{t_i}-\tilde{S}_{t_{i-1}}|_+^{-1}\Big] \le C |t_{\sigma(p)}-t_{\sigma(p-1)}|_+^{-1/2}E \Big[ \prod_{i=1}^{\ell-1}|\tilde{S}_{t_i}-\tilde{S}_{t_{i-1}}|_+^{-1}|\tilde{S}_{t_{\ell+1}}-\tilde{S}_{t_{\ell-1}}|_+^{-1}\prod_{i=\ell+2}^p |\tilde{S}_{t_i}-\tilde{S}_{t_{i-1}}|_+^{-1}\Big].
I_0^{[k]}:I^{[\ell]}=(I_0^{[k]}:I_0^{[\ell]})\cap\big(I_0^{[k]}:\!\!\!\!\!\!\sum_{\substack{q\in[s] \nu(I_q)\ge\ell-1}}\!\!\!\!u_qI_q^{[\ell-1]}\big)=I_0^{[k]},
\forall n^0,x^X\preceq_X L1_X,y^X\exists z^X\preceq_X L1_X &\qquad\qquad\qquad\exists w^X\preceq_XL2^{\alpha_n+1}1_X\left(y\in Ax\rightarrow (w\in Az\land x=_Xz+_X\gamma_n w)\right).
N_{k}^{(p , 0)}= \int_{0}^{1}dt_{1}\int_{0}^{t_{1}}dt_{2}\ldots\int_{0}^{t_{p - k -1}}dt_{p - k}\int_{0}^{1}ds_{1}\int_{0}^{s_{1}}ds_{2}\ldots \int_{0}^{s_{k - 1}}ds_{k}\times
|u_1(j)\, u_2(j)\,\cdots\, u_n(j)|=\left |\begin{array}{cccc}u_1(j) u_2(j) \cdots u_n(j) u_1(j+1) u_2(j+1) \cdots u_n(j+1) \vdots \vdots \vdots u_1(j+n-1) u_2(j+n-1) \cdots u_n(j+n-1) \end{array}\right|,
\psi([v\otimes x]a) = \psi(v\otimes xa) = \psi \left(v\otimes \left(\sum_{j=1}^mx_ja_{j,1},\sum_{j=1}^mx_ja_{j,2},\ldots ,\sum_{j=1}^mx_ja_{j,m}\right)\right) = \left(v\otimes \sum_{j=1}^mx_ja_{j,1},v\otimes \sum_{j=1}^mx_ja_{j,2},\ldots ,v\otimes \sum_{j=1}^mx_ja_{j,m}\right) = (v\otimes x_1,v\otimes x_2,\ldots ,v\otimes x_m)a = \psi(v\otimes x)a.
F ( g , N ) = \sum_{k = 0}^{\infty}a_{k}( g - g_{*})^{k}+ \sum_{k = 0}^{\infty}b_{k}( g - g_{*})^{k + \gamma}+ O ( N^{- 2}) \ \ ( \gamma \in{\bf R}_{+}\setminus{\bf Z}, b_{0}\ne 0 ) .
\|\widehat{g}- g_0\|_{2,n}^2 := |\mathcal{D}_2|^{-1}\sum_{i\in\mathcal{D}_2}\int (\widehat{g}(x) - g_0(x))^2dP_{X_i}(x), \|\widehat{m}- m_0\|_{2,n}^2 := |\mathcal{D}_2|^{-1}\sum_{i\in\mathcal{D}_2}\int (\widehat{m}(x) - m_0(x))^2dP_{X_i}(x).
\sp M t^{n-1}e^{\omega t}\left|\frac{e^{-t(r+h+s\j)}- e^{-t(r+s\j)}}{h}- \partial_r e^{-t(r+s\j)}\right| = M t^{n-1}e^{\omega t}|e^{-t(r+s\j)}| \left|\frac{e^{-th}- 1}{h}+ t\right| \le M t^{n-1}e^{t\omega}e^{-tr}2t = 2 M t^n e^{t(r - \omega)}.
U\Pi\Gamma_2u=\dfrac{1}{\sqrt 2}\Big[\gamma^{-}_{1}(u_{-}-P_0^{-}\gamma_{0}^{-}u_{-})-\mu\gamma_{1}^{+}\big(u_{+}-P_0^{+}(\gamma_{0}^{+}u_{+},\gamma_{0}^{\partial}u_{+})\big)\Big],
\begin{aligned}\mathcal{L}_{DirA}(n,k)=\log \frac{P(y_r(n)|b_{a,k}(n)=1)}{P(y_r(n)|b_{a,k}(n)=0)}, \mathcal{L}_{DirB}(n,k)=\log \frac{P(y_r(n)|b_{b,k}(n)=1)}{P(y_r(n)|b_{b,k}(n)=0)},\end{aligned}
\left(\frac{1}{a_1}+ \frac{1}{a_1^2 - a_1+1}, \frac{1}{a_1}+ \frac{1}{a_1^2 - a_1}\right] = \left(\frac{1}{a_1}+ \frac{1}{a_1^2 - a_1+1}, \frac{1}{a_1}+ \frac{1}{a_1-1}\right]
\delta \hat\omega_{mab}=-i\kappa(2\bar{\varepsilon}\gamma_{[a}\hat{\cal R}_{b]m}-\bar\varepsilon\gamma_m\hat{\cal R}_{ab}) +2i\kappa^2(2i\bar{\varepsilon}\vec{\tau}\gamma_{ab}\psi_m\vec{t}+\frac{1}{\sqrt{3}}\bar{\varepsilon}\psi_m\hat{F}_{ab}+\frac{1}{2}\bar{\varepsilon}\gamma_{abcd}\psi_mv^{cd}).
\frac{\mathrm{d}}{\mathrm{d}z}\left(\det\left(\mathrm{id}-z\cdot M\right)\right)=\frac{\mathrm{d}}{\mathrm{d}z}\left(\left(1-z\right)\cdot G\left(z\right)\right)=-G\left(z\right)+\left(1-z\right)\cdot G'\left(z\right).
r_0dr_k\wedge d\bar{r}'=\sum_lr_ldi_k\wedge d\bar{r}_l+\sum_\alpha r_\alpha du_k\wedge d\bar{r}_\alpha \wedge d\bar{r}_\alpha '+\sum_\alpha r_\alpha dv_k\wedge d\bar{r}_\alpha \wedge d\bar{r}_\alpha ',
B = - \frac{1}{8}\bar{D}( 1 - \gamma_{5}) D ( W_{a}^{*})^{2}= ( D^{a})^{2}- \frac{1}{2}( F_{\mu \nu}^{a})^{2}- \frac{i}{2}F_{\mu \nu}^{a}\tilde{F}_{\mu \nu}^{a}+ O ( \theta )
\ddot{x}= - \Phi ( x ) \Phi^{\prime}( x ) - \Phi^{\prime \prime}( x ) \overline{{{\psi}}}\, \psi , \qquad \dot{\psi}= - \mathrm{i}\Phi^{\prime}( x ) \psi , \qquad \dot{\overline{{{\psi}}}}= \mathrm{i}\Phi^{\prime}( x ) \overline{{{\psi}}}.
R_{x}^{\alpha}f(x)\ =-\frac{1}{2\Gamma(2-\alpha)\cos\alpha\pi/2}\times\left[ \int_{-\infty}^{x}(x-x^{\prime})^{-\alpha+1}f^{(2)}(x^{\prime})dx^{\prime}+\int_{x}^{\infty}(x^{\prime}-x)^{-\alpha+1}f^{(2)}(x^{\prime})dx^{\prime}\right] ,1<\alpha<2,
S_{s}( m_{s , i}) ={\frac{b ( m_{s , i})}{4 8 \pi}}{\cal A}~ ~ , ~ ~ S_{d}( m_{d , i}) ={\frac{2 b ( m_{d , i})}{4 8 \pi}}{\cal A}~ ~ , ~ ~ S_{v}( m_{v , i}) ={\frac{3 b ( m_{v , i})}{4 8 \pi}}{\cal A}~ ~ .
\begin{aligned}\leq \left ( 2 \chi^{2}(1 + C_{p}^{2}) + 2 C_{p}^{2}h_{\infty}^{2}\lambda_{p}^{2}(4C_{p}^{2}+ 1) \right ) \|\nabla \omega_{k}\|_{L^{2}(0,s;L^{2})}^{2}& + \frac{1}{2}\|\nabla \mu_{k}\|_{L^{2}(0,s;L^{2})}^{2}+ C,\end{aligned}
\sqrt{2}=\sum_{k=0}^{\infty}\frac{(2k+1)!}{2^{3k+1}(k!)^{2}}=\frac{1}{2}+\frac{3}{8}+\frac{15}{64}+\frac{35}{256}+\frac{315}{4096}+\frac{693}{16384}+\cdot\cdot\cdot.
Z(g_0):=\min_{{\small \begin{array}{c}(u^1,g_0),(u^2,g_0)\in M\cap p^{-1}(g_0) (u^1,g_0)\ne (u^2,g_0)\end{array}}}\left\{\inf_{t>0}z(\varphi(t,\cdot;u^1,g_0)-\varphi(t,\cdot;u^2,g_0))\right\}.
\int d^{4}x^{\prime}\, d^{4}y^{\prime}\, \Gamma_{i k , i^{\prime}k^{\prime}}^{\left( 2 \right)}\left( x , y ; x^{\prime}, y^{\prime}\right) \psi^{i^{\prime}k^{\prime}}\left( x^{\prime}, y^{\prime}\right) = 0
{\hat{\lambda}_N}={1\over{8\pi}}\oint{d}\hat{s}(\xi){\hat{V}_N}(\xi){\partial_{\hat{n}}}{\hat{V}_N}(\xi)={1\over{8\pi}}\int{d^2}\sqrt{\hat{g}}{\hat{g}^{ab}}{\partial_a}{\hat{V}_N}{\partial_b}{\hat{V}_N}.
\begin{array}{rl}|N(e,le)- \theta(l)N(e,e)| \leq \theta(l)\theta(e)^2\left((W(le)-W(e))(W(e) -1)2 \sqrt{q}+( W(le)-W(e))(\sqrt{q}+1)\right) = (1-\frac{1}{l})\theta(e)^2\left(2W(e)(W(e)-1)\sqrt{q}+W(e)(\sqrt{q}+1)\right), \end{array}
F = V_{p}(8\pi^{2}\alpha ')^{-p/2}\int_{0}^{\infty}d\tau~\tau^{-4 +p/2}e^{-b^{2}/2\pi\alpha '{\tau}}\left[ e^{-\pi\tau /12}\prod_{n=1}^{\infty}(1-e^{-2\pi\tau n})\right]^{-9}\frac{(\theta_{1}(-i\epsilon /\pi\mid i\tau))^4}{\theta_{1}(-i2\epsilon/\pi \mid i\tau)}.
D_{\Phi(y)}G(\Phi(x),\Phi(y))D_{y_i}\Phi(y) = D_{\Phi(y)}\log|\Phi(x)-\Phi(y)|D_{y_i}\Phi(y) +\sum_{k=1}^\infty\alpha^kD_{\Phi(y)}[\log|{\bf k}+\Phi(x)-\Phi(y)|-\log|{\bf k}-\Phi(x)-\overline{\Phi(y)}|]\cdot D_{y_i}\Phi(y).
\mathbb{A}_{v_0}(\boldsymbol{\eta},\boldsymbol{\zeta},\boldsymbol{\alpha},\beta)-\lim_{n\to \infty}\dfrac{1}{n}\sum_{v\neq v_0}{{F_v}}(\boldsymbol{\alpha},\beta)(n)=V(\boldsymbol{\eta},\boldsymbol{\zeta},\boldsymbol{\alpha},\beta)\enspace.
\begin{cases}\frac{1}{2}\frac{d}{dt}\|a_{1}\|_{L^{2}}^{2}+\|\nabla a_{1}\|_{L^{2}}^{2}=0, p=2, \frac{1}{p}\frac{d}{dt}\|a_{1}\|_{L^{p}}^{p}+\frac{4(p-2)}{p^{2}}\|\nabla|a_{1}|^{\frac{p}{2}}\|_{L^{2}}^{2}+\||\nabla a_{1}||a_{1}|^{\frac{p-2}{2}}\|_{L^{2}}^{2}=0, p>2,\end{cases}
\mathcal I^\theta_\beta = \overline{\bigcup_{\alpha<\beta}\mathcal I^\theta_\alpha}= \overline{\bigcup_{\alpha<\beta}\mathcal I^\eta_\alpha \otimes \mathcal K(\ell_2(\kappa))}= \overline{\bigcup_{\alpha<\beta}\mathcal I^\eta_\alpha}\otimes \mathcal K(\ell_2(\kappa)) = \mathcal I_\beta^\eta \otimes \mathcal K(\ell_2(\kappa)).
\tilde{sec}(U, V)= \frac{\tilde{g}(\tilde{R}(U,V)V, U)}{\tilde{g}(U, U) \tilde{g}(V, V) - \tilde{g}(U, V)^2}= \frac{1}{\lambda^2}g(\tilde{R}(U,V)V, U).
\mathcal{E}_{n+2m}:=\left\{P_{m,j}^{(\alpha, \beta)}:j=m, m+1, \hdots, n+2m \right\}\mathcal{F}_{n+2m}:=\left\{q\in \mathcal{P}_{n+2m}:(1+x_i)q'(x_i)+\beta q(x_i) = 0\right\},
\sum_{n = - \infty}^{\infty}\; e^{- c^{2}n^{2}}= \theta_{3}( 0 , e^{- c^{2}}) = \sqrt{\frac{\pi}{c^{2}}}\, \theta_{3}( 0 , e^{- \pi^{2}/ c^{2}}) = \sqrt{\frac{\pi}{c^{2}}}\sum_{n = - \infty}^{\infty}\, \exp [ - \pi^{2}n^{2}/ c^{2}] .
\sum_{(k,m) \in R_2}\frac{(k+m)^2 |\widehat f(k+m) |^2}{k^2+m^2}\big| \widehat \iota_\varepsilon(k,m) - \widehat \iota_\delta(k,m)\big|^2 \leq C^2(\varepsilon -\delta)^2 \sum_{(\ell,n) \in R_2}\ell^2 |\widehat f(\ell)|^2 \leq 2 C^3 (\varepsilon-\delta) \|f'\|^2.
V^{\omega}_{0,b}(x):=\mathrm{E}_{x}\bigg[\int_0^{\infty}e^{-q t}\mathrm{d}D^{(0,b)}_t-\phi\int_0^{\infty}e^{-q t}\mathrm{d}R^{(0,b)}_t+\lambda\int_{0}^{\infty}e^{-q t}\omega(U^{(0,b)}_{t})\mathrm{d}t\bigg].
S= -{T \over 4}\int d^{p+1}\xi \exp(-\phi) (-g)^{1/4}(-\gamma)^{1/4}\left[ \gamma^{ij}(g_{ij}- g^{kl}{\cal F}_{ik}{\cal F}_{lj})\varphi^{p-3 \over p+1}- (p-3) \varphi \right].
G(v(t)) = \frac{C_0}{2\lambda}Z_0(v(t)) + \frac{b_0 (2d-1)}{4}\sum_{\mu=0}^{L-1}\langle \partial_t^{\mu+1}v(t), \partial_t^\mu v(t) \rangle + \frac{b_0(2d-1)}{8}\sum_{\mu=0}^{L-1}\langle \partial_t^\mu v(t), B_\lambda(t) \partial_t^\mu v(t) \rangle + \sum_{\mu=0}^{L-1}\langle \partial_t^{\mu+1}v(t), [h ; \nabla \partial_t^\mu v(t)] \rangle,
\begin{aligned}S_{\lambda}(T_{\alpha,q}v)(x) =v\left(\left( -\lambda (q-1) \log_{q}\frac{|x|}{R}\right)^{-\alpha}\right) &=(D_{\lambda^{-\alpha}}v)\left(\left( -(q-1)\log_{q}\frac{|x|}{R}\right)^{-\alpha}\right)=T_{\alpha,q}(D_{\lambda^{-\alpha}}v)(x)\end{aligned}
\frac{d}{d\lambda}S_{\omega,\mathbf{c}}(\Phi^{\lambda})\big|_{\lambda=1}=\sum_{j=1}^3\left.\left\langle (D_jS_{\omega,\mathbf{c}})(\Phi^{\lambda}), \frac{d}{d\lambda}\Phi^{\lambda}\right\rangle\right|_{\lambda =1}=\sum_{j=1}^3\left\langle D_jS_{\omega,\mathbf{c}}(\Phi), \frac{d}{2}\Phi+\nabla \cdot \Phi\right\rangle.
\root n\of{a_1 \cdot a_n}= \overset{``f(average)" \swarrow}{\exp (n^{-1}\sum_{j=1}^nA_j)}\overset{(\, )}\le \overset{``average(f)"\swarrow}{n^{-1}\sum_{j=1}^n\exp A_j}=n^{-1}\sum_{j=1}^na_j\, .
\|u_H^{n}\|^2+\Delta t\sum_{k=2}^n\|\nabla u_H^{k-\theta}\|^2+\gamma\Delta t\sum_{k=2}^n\|\sigma_H^{k-\theta}\|^2\leq C(\|u_H^{0}\|^2+\Delta t\sum_{k=1}^n\|g^{k}\|^2).
P(x) = \begin{cases}-(x+1)^2 x < -1 0 -1 \leq x \leq 1 (x-1)^2 x > 1, \end{cases}P^{-1}(y) = \begin{cases}- \sqrt{|y|}- 1 y < 0 \sqrt y + 1 y > 0. \end{cases}
\prod_{i=1}^{2k-\ell-1}\frac{f\Bigl( \|(R_{n}+a(R_{n})\rho)\theta + y_i\|e_1 \Bigr)}{f(R_{n}e_1)}\leq \prod_{i=1}^{2k-\ell-1}C \left( 1 + \frac{a(R_{n})}{R_{n}}\bigl( \rho + \xi_n(\rho,\theta,y_i) \bigr) \right)^{\gamma}&\leq C^* \bigl( \max \{\rho,1 \}\bigr)^{\gamma (2k-\ell-1)}.
b_1 = 1\, , \ \b_2 = \frac{1}{2m}\, , \ \b_3 = \Delta\, , &B_1 = \frac{9+8m}{5+8m}\, , \ \B_2 = \frac{1}{5+8m}\, , \ \B_3 = \Delta - \frac{16m}{k(5+8m)}\, .
\mathbf{g}_i = \Bigl( \underbrace{a_{i,1}, \ldots, a_{i,1}}_{x_1}, \underbrace{a_{i,2}, \ldots, a_{i,2}}_{x_2}, \cdots, \underbrace{a_{i,2^k-1}, \ldots, a_{i,2^k-1}}_{x_{2^k-1}}\Bigr)
\lambda(y^{n-1}_{n-J}) &=-\log\Big(\sum_{x_n}\exp{\Big\{\sum_{y_{n}}\log\big(r_{n}(x_n|y_n,y^{n-1}_{n-M}\big)q_n(y_n|y^{n-1}_{n-M},x_n)-1-s\gamma_n(x_n,y^{n-1}_{n-N})\Big\}}\Big).
\tau(X_{p_{\ell(\gamma_\ell)}})\equiv\tau(\ell(\gamma_\ell)):=({i_1(\gamma_{i_1})},{i_2(\gamma_{i_2})},\ldots, i_f(\gamma_{i_f}), i_{f+1}(\gamma_{i_{f+1}})),i_1=\gamma_{i_1}=s_{1}=1; i_{f+1}=\ell.
\begin{array}{lll}A = \tau E^*A^{-1}E, (\tau E^* A^{-1}+I)b = w+\tau E^{*}A^{-1}c, \gamma = \displaystyle{\frac{\beta+\langle\tau (c-b),\frac{1}{2}A^{-1}(c-b)\rangle}{\tau+1}}.\end{array}
\frac{\Gamma'}{\Gamma}\Big(\frac{1}{8}+i\frac{t}{2}\Big) -\frac{\Gamma'}{\Gamma}\Big(1+i\frac{t}{2}\Big)= - \frac{8}{1+4it}+ \frac{\Gamma'}{\Gamma}\Big(\frac{9}{8}+i\frac{t}{2}\Big) -\frac{\Gamma'}{\Gamma}\Big(1+i\frac{t}{2}\Big),
\Pi_{o d d}( p ) = - \frac{4 m g^{2}}{\beta}\sum_{n = - \infty}^{\infty}\int_{0}^{1}d x \int \frac{d^{2}l}{( 2 \pi )^{2}}[ \frac{1 - \frac{1}{2}\{e^{i l \tilde{p}}+ e^{- i l \tilde{p}}\}}{( M_{1}^{2}+ ( ( 2 n + 1 ) \frac{\pi}{\beta}+ i \mu )^{2}}] ,
\begin{cases}c'_{\ge \ell}(z,\zeta) \textrm{if}1 \leq \ell \leq n-k c'_{\ge \ell}(z,\zeta) + \bigl(E\cdot C^\zeta_{\ge n-k+2}\bigr)_\ell + (-1)^{n+k}q \binom{k-1}{n-\ell}c^\zeta \textrm{if}n-k+1 \le \ell \le n \/. \end{cases}
\begin{array}{rl}= \{\strut \displaystyle -2 a_{1}\over\displaystyle 2 a_{1}+a_{0}(s+1)-{\strut \displaystyle 3 a_{0}a_{2}(s+1) \over\displaystyle 3 a_{2}+a_{1}(s+2)-{\strut \displaystyle 4 a_{1}a_{3}(s+2) \over\displaystyle 4 a_{3}+a_{2}(s+3)}}}& \ddots\end{array}
c_2(E) = \omega_1^2 + \omega_2^2 +\omega_3^2 + \left(\omega_1+\omega_2+\omega_3\right)^2 = 2\left(\omega_1\omega_2+\omega_1\omega_3+\omega_2\omega_3\right) + 2\sum\omega_i^2 =2\left(h + \sum \omega_i^2\right)
\mathbb{P}\left( S_{n}\geq x_{n}\sigma_{n}\right)=(1+o(1))\bigg[x_n^{-t}\sum_{r,s\in\mathbb{Z}}b_{n,r,s}^t h\Big(\frac{x_n}{b_{n,r,s}}\Big)+1-\Phi(x_{n})\bigg].
\nu (\Delta \varphi_{\varepsilon}(s), \varphi_{\varepsilon}(s)-m_0)_H = - \nu \| \nabla \varphi_{\varepsilon}(s)\|^2_H - m_0 \int_{\Omega}\Delta \varphi_{\varepsilon}(s) = - \nu \| \nabla \varphi_{\varepsilon}(s)\|^2_H \leq 0.
2 9 g^{2}\int d^{3}k \frac{1}{k^{2}- 1 5 N g^{2}k / 3 2}\rightarrow g^{2}\int d^{3}k [ \frac{1}{k^{2}}+ \frac{1 5 N g^{2}}{3 2 k^{3}}+ \cdots ] .
m\rq_{1,1}(x,y)-m_{1,1}(x,y)=m_1(\phi_1(x),y)+m_1(x,\phi_1(y))-\phi_1(m_1(x,y))=\delta_1 \phi_1(x,y). & m\rq_{2,1}(x,y)-m_{2,1}(x,y)=m_2(\phi_1(x),y)+m_2(x,\phi_1(y))-\phi_1(m_2(x,y))=\delta_2 \phi_1(x,y).
f(T_{1k}T_{1k+1},T_{1k+1})=\sum_{r,s\in{\mathbb F}_p}a_{r,s}(T_{1k}T_{1k+1})^rT_{1k+1}^s=\sum_{r,s\in{\mathbb F}_p}a_{r,s}T_{1k}^rT_{1k+1}^{r+s}= \sum_{r,t\in{\mathbb F}_p}a_{r,t-r}T_{1k}^rT_{1k+1}^t=\sum_{r,t\in{\mathbb F}_p}a_{r,t}T_{1k}^rT_{1k+1}^t.
\gamma_{+}({\bf p})=-\frac{e^{2}}{2}\int\limits_{-\infty}^{+\infty}dt~\left[\frac{p_{3}}{{\bf p}^{4}}p_{k}\epsilon_{ijk}+\pi\delta(p_{1}, p_{2})\Delta(p_{3})\epsilon_{ij3}\right]a_{j}\dot{a}_{i}.
\alpha_{ik}Z_k = Z_i \alpha_{ik}k \Rightarrow & \bigg[ K(Z_i, Z_j)\big(\sum_{k=1}^N \alpha_{ik}P_{kj}\big) \bigg]_{ij}=\bigg[ \sum_{k=1}^N \alpha_{ik}K(Z_k, Z_j)(P_{kj}) \bigg]_{ij}.
\frac{\partial u}{\partial t}+\frac{1}{2}\nabla(u^{2})+(\nabla\times u)\times u+\frac{\nabla p}{\rho}=g=\frac{\partial u}{\partial t}+\frac{1}{2}\nabla(u^{2})-u\times(\nabla\times u)+\frac{\nabla p}{\rho}
\big( \phi_t = \mathrm{id}_A + t (\mathrm{ad}^l_a - \mathrm{ad}^r_a ) + \sum_{i \geq 2}t^i \phi_i, ~ \psi_t = \mathrm{id}_A + t (l_a - r_a + H (a, T-) - H (T-, a)) + \sum_{i \geq 2}t^i \psi_i \big)
\xi^i=-{\rho dx_0^i\over D^{3/2}}-{(x-x_0)^i[2\rho^2+(x-x_0)^2]d\rho\over \rho^2 D^{3/2}},\quad\quad\omega^u_{(0,1)}=q^tI_u\xi=-{(x-x_0)^tI^udx_0\over D^2}.
= \frac 1{1 6}{\cal R}_{a b c d}\bar{\Psi}^{a}( 1 + \gamma_{p}) \Psi^{c}\bar{\Psi}^{b}( 1 + \gamma_{p}) \Psi^{d}= \frac 1{1 6}{\cal R}_{a b c d}\tilde{V}_{i}^{a}V_{j}^{c}\tilde{V}_{k}^{b}V_{l}^{d}\bar{X}^{i}( 1 + \gamma_{p}) X^{j}\bar{X}^{k}( 1 + \gamma_{p}) X^{l}~ .
\begin{array}{l l}{\delta_{( \xi )}^{S}M^{\mu \nu}={\cal L}_{\xi}( \sqrt{g}g^{\mu \nu}) ,}&{}\\{\delta_{( \xi )}^{S}L^{\mu \nu}={\cal L}_{\xi}\lbrack M^{\mu \nu}- s ( \sqrt{g}g^{\mu \nu}) \rbrack .}\\ \end{array}
g(k):= \int_0^b[\tilde{q}(x)-q(x)]y(x,\lambda)\tilde{y}(x,\lambda)dx+(\tilde{h}-h) &+a_1(\tilde{a}_2-a_2)y(\textstyle{\frac{1}{2}-0},\lambda)\tilde{y}(\textstyle{\frac{1}{2}-0},\lambda).
M_{\alpha_1}^{(\varphi'_{1},\varphi'_{2})}:=p_1\big( (p_2|_{C_{\alpha_1}})^{-1}((\varphi_1',\varphi_2')) \big) = \{x \in M_{\alpha_1}\mid \theta_{(\varphi'_{1},\varphi'_{2})}(f(x))=\theta_{(\varphi_{1},\varphi_{2})}(x)\}
N_{\mu \nu \rho}\left(0, x, y \right) = -\frac{i g^3 f^2}{512\pi^8}\, x'^{6}y'^{6}\,J_{\nu\nu'}(x')\,J_{\rho \rho'}\left({y'}\right)\, \frac{{\rm Tr}\, \gamma_{5}\gamma_{\mu}\gamma_{\rho'}\gamma_{\nu'}\, \left(/\hspace{-2mm}x' - /\hspace{-2mm}y' \right)}{\left( x' - y' \right)^4}\, .
P\left(^{(r,s)}S_t > \eta x,\ \Delta X_t^{(r+1)}\le \delta x b_t \right) = P\left(^{(r,s)}X_t-a_t > \eta xb_t,\ \Delta X_t^{(r+1)}\le \delta x b_t \right) \le P\left(^{(r,s)}X_t - t\nu(\delta x b_t) > \frac 1 2 \eta x b_t,\ \Delta X_t^{(r+1)}\le \delta x b_t\right).
\sum_{(d,2)=1}\left( \frac{d}{p}\right) W\left( \frac{d\alpha^2}{X}\right)=\sum_{d}\left( \frac{d}{p}\right) W\left(\frac{d\alpha^2}{X}\right)-\left( \frac{2}{p}\right)\sum_{d}\left( \frac{d}{p}\right) W\left( \frac{2 d\alpha^2}{X}\right).
\left\|w_{n}-x^{*}\right\|^{2}=\left\|(1-\beta_{n})(z_{n}-x^{*})+\beta_{n}(Uz_{n}-x^{*})\right\|^{2}= (1-\beta_{n})\left\|z_{n}-x^{*}\right\|^{2}+\beta_{n}\left\|Uz_{n}-x^{*}\right\|^{2}-\beta_{n}(1-\beta_{n})\left\|Uz_{n}-z_{n}\right\|^{2}\leq \left\|z_{n}-x^{*}\right\|^{2}-\beta_{n}(1-\beta_{n})\left\|Uz_{n}-z_{n}\right\|^{2},
{\it L}= \sqrt{-g}\biggl[\phi\biggl(R -\omega\frac{\phi_{;\rho}\phi^{;\rho}}{\phi^2}-\frac{\Psi_{;\rho}\Psi^{;\rho}}{\phi^{2}}\biggr) -\frac{1}{2}\xi_{;\rho}\xi^{;\rho}\biggr] + L_{\rm r},
\langle \,0\,|\,\mathcal{O}(x)\,|\,p_{1},\dots ,p_{n}\,\rangle^{in}=e^{-ix(p_{1}+\dots +p_{n})}\,f_{n}^{\mathcal{O}}(\theta_{1},\dots,\theta_{n})~,~~~\mathrm{for}~~\theta_{1}>\dots >\theta_{n}
j_{+ +}^{n m}= a_{n}^{\dagger}a_{m}, \hspace{1 c m}j_{--}^{n m}= b_{n}^{\dagger}b_{m}, \hspace{1 c m}j_{+ -}^{n m}= a_{n}^{\dagger}b_{m}, \hspace{1 c m}j_{- +}^{n m}= b_{n}^{\dagger}a_{m},
12\pi^{1/2}|d|^{\frac 32}|b_{\psi}(d)|^2 = \frac{1}{\langle \varphi,\varphi \rangle}\sum_{Q\in \Gamma\backslash\mathcal Q_{d^2}}\chi_d(Q)\begin{dcases}\int_{C_Q}\varphi(z) y^{-1}\, |dz| d>0, \int_{C_Q}i\partial_z \varphi(z) \, dz d<0.\end{dcases}
G_{f;M}=\min_{i_1,i_2,i_3=0,\dots,M-1}\left[f_+\left(\frac{i_1}{M},\frac{i_2}{M},\frac{i_3}{M}\right)-f_-\left(\frac{i_1+1}{M},\frac{i_2+1}{M},\frac{i_3+1}{M}\right)\right].
\lim_{n\to+\infty}\frac{D_n((1+z/\ln n) G_n(x)/2)}{D_n(G_n(x)/2)}\frac{(1+z/\ln n)a_1}{\sqrt{4-(1+z/\ln n)^2a_1^2}}= \lim_{n\to+\infty}\frac{D_n((1+z/\ln n) G_n(x)/2)}{D_n(G_n(x)/2)}\frac{a_1}{\sqrt{4-a_1^2}}=e^{-2z}\frac{a_1}{\sqrt{4-a_1^2}}.
\aligned \lim\limits_{n\to\infty}(\|u_n\|^2_{E_a}-\|u_n-\tau_{k_{1,n}}v_1\|^2_{E_a}) = \lim\limits_{n\to\infty}(\|\tau_{-k_{1,n}}u_n\|^2_{E_a}-\|\tau_{-k_{1,n}}u_n-v_1\|^2_{E_a}) = \lim\limits_{n\to\infty}2(\tau_{-k_{1,n}}u_n, v_1)_{E_a}-\|v_1\|^2_{E_a}= \|v_1\|^2_{E_a}.\endaligned
{S_{L}}\left[ \phi , \hat{g}\right] ={\frac{1}{8 \pi}}\int{d^{2}}\xi \sqrt{\hat{g}}\left({\frac{1}{2}}\phi \hat{\Delta}\phi + Q \hat{R}\phi \right) +{\frac{Q}{4 \pi}}\oint{d}\hat{s}{k_{\hat{g}}}\phi + 2{\mu_{2}^{2}}\int{d^{2}}\xi \sqrt{\hat{g}}{e^{\alpha \phi}}+{\lambda_{2}}\oint{d}\hat{s}{e^{\alpha \phi / 2}}.
D_{2}( A ) \Psi_{_{l , j = l \pm \frac{1}{2}, m , q}}( \theta , \phi ) = \pm \sqrt{2 \lambda \alpha^{2}[ ( j + \frac{1}{2})^{2}- q^{2}]}\Psi_{_{l , j = l \pm \frac{1}{2}, m , q}}( \theta , \phi ) ,
D^{\mu \nu}( k ) = \left( g^{\mu \nu}- \frac{k^{\mu}k^{\nu}}{k^{2}}\right) \frac{1}{e^{2}/ \pi - k^{2}}- \frac{1}{\lambda}\frac{k^{\mu}k^{\nu}}{( k^{2})^{2}}\; ,
\Phi^{(3)}_{[0,1,2]}(E,{\bf g}) =\Phi^{(3)}_{[1,2,3]}(E,{\bf g})\begin{pmatrix}\tau^{(1)}_1(E,{\bf g}) 1 0 -\tau^{(2)}_1(E,{\bf g}) 0 1 1 0 0\end{pmatrix}
\frac{FP}{k_{B}T}=\frac{1}{4}(1-\frac{x}{L_{0}}+\frac{F}{K_{0}})^{-2}-\frac{1}{4}+\frac{x}{L_{0}}-\frac{F}{K_{0}}-0.8(\frac{x}{L_{0}}-\frac{F}{K_{0}})^{2.15}
S_{j k}= \frac{g^{2}( \mu_{j}, \lambda_{k})}{f ( \mu_{j}, \lambda_{k})}e^{\phi^{\kappa}( \mu_{j}) + \phi^{\nu}( \lambda_{k})}+ \frac{g^{2}( \lambda_{k}, \mu_{j})}{f ( \lambda_{k}, \mu_{j})}e^{\phi^{\kappa}( \lambda_{k}) + \phi^{\nu}( \mu_{j})}
V_i\lrcorner (V^*_{j_0}\wedge \cdots \wedge V^*_{j_{\ell}}) = \sum_{m=0}^{\ell}(-1)^{m}V^*_{j_0}\wedge \cdots (V_i \lrcorner V^*_{j_m}) \cdots \wedge V^*_{j_{\ell}},