harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|F_s^{i,r}(0)|= \ |f^i(V_s^{r,v},q(Z_s^{i,r,v}(m)))-f^i(V_s^{r,\bar{v}},q(Z_s^{i,r, v}(m)))| \leq \ \frac{C_vC_{\eta}}{C_{\eta}-C_v}|V^{r, v}_s-V^{r, \bar{v}}_s| \leq\frac{C_vC_{\eta}}{C_{\eta}-C_v}e^{-C_{\eta}(s-r)}|v-\bar{v}|,s\in [r,T],
P_{BS}^{\textrm{opt}}\!=\! \left[\frac{-(2 \theta A \!+\! d B \theta) \!+\! \sqrt{(2 \theta A \!+\! d B \theta)^{2}\!-\! 8 \theta A d (B \theta \!-\! a d)}}{4 \theta A d}\right]^{+}\!\!.
e^{{\cal{G}}_{\mathrm{c}}\{J \}}= \frac{{\cal{Z}}\{J \}}{{\cal{Z}}_{0}}= \frac{\int{\cal{D}}\{\phi \}e^{- S \{\phi \}+ ( J , \phi )}}{\int{\cal{D}}\{\phi \}e^{- S_{0}\{\phi \}}}.
M ( \Phi , \Phi^{\prime}, T ) = \left( \begin{array}{c c}{{- i \Phi_{I}({\bf X}^{0}) \tilde{D}{\bf X}^{I}}}&{{\overline{{{T}}}({\bf X}^{0})}}\\{{T ({\bf X}^{0})}}&{{- i \Phi_{I}^{\prime}({\bf X}^{0}) \tilde{D}{\bf X}^{I}}}\\ \end{array}\right) , \I = 1 , \dots , 9 \ .
\lim_{N\to \infty}\frac{1}{N}\#\{1\le n \le N \mid T^n x \in C_s \}=\lim_{N\to \infty}\frac{1}{N}\#\{1\le n \le N \mid \widetilde{T}^n (x;a) \in C_s \times A\}&= \mu (C_s)
\begin{array}{l}{{w_{2 i + 1}= x_{i}^{- 1}x_{i + 1}, ~ ~ i = 0 , 1 , 2 , \cdots , N - 1 ,}}\\{{w_{2 j}= y_{j}, ~ ~ j = 1 , 2 , \cdots , N - 1 ,}}\\{{w_{2 N}= \prod_{l = 1}^{N - 1}y_{l}^{- 1}.}}\\ \end{array}
S_{D}= - \int d^{4}\sigma ( \sqrt{- \mathrm{d e t}( G_{\mu \nu}+ \tilde{\cal F}_{\mu \nu}^{\prime})}+ \int ( C_{4}^{\prime}+ C_{2}^{\prime}\wedge \tilde{\cal F}^{\prime}- \frac{1}{2}C_{0}\tilde{F}^{\prime}\wedge \tilde{F}^{\prime}) .
\left(v^2-\left(\frac{2k}{n-2k}\right)^2\dot{v}^2\right)^{k-1}\left(v-\left( \frac{2k}{n-2k}\right)^2\ddot{v}\right)=\frac{n}{n-2k}v^{\frac{2kn}{n-2k}-1}.
\lambda_n^\beta =\frac12\psi\bigg(\frac{\beta n}2\bigg)-\frac12\psi\bigg(\frac{\beta (\kappa+n)}2\bigg), (\sigma_n^\beta)^2 =\frac14\psi'\bigg(\frac{\beta n}2\bigg)-\frac14\psi'\bigg(\frac{\beta (\kappa+n)}2\bigg).
\kappa(x,y) =\left(\frac{nb}{c-a}+n,\frac{nb}{c-a}\right)=\left(n\frac{B+C-A}{C-A},n\frac{B}{C-A}\right) =\left(PQ(P^{2}-Q^{2})\frac{2(P+Q)Q}{2Q^{2}},PQ(P^{2}-Q^{2})\frac{2PQ}{2Q^{2}}\right) =(P(P-Q),(P-Q)(P+Q)).
{\cal H}=-\frac{1}{4}\int_{-\infty}^\infty d\tau:[P^2+{X'}^2 -\frac{\sqrt{\pi}}{\beta v^2}(PX'P+\frac{1}{3}{X'}^3) -\frac{1}{2\beta\sqrt{\pi}}(\frac{v''}{3v^3}-\frac{(v')^2}{2v^4})X']:,
\frac{1}{2}\{(\begin{matrix}1\\ 1\\ 1\\ 1\end{matrix}),(\begin{matrix}1\\i\\ -1\\ -i\end{matrix}),(\begin{matrix}1\\ -1\\ 1\\ -1\end{matrix}),(\begin{matrix}1\\ -i\\ -1\\i\end{matrix})\}
p'_{k,j}+ p_{k,j-1}+ (\tilde f_{k,j}- \tilde f_{k,m+1}\tilde f_{m,j}) = p_{k+1,j}+ (f_{k,j}- f_{k,m+1}f_{m,j}), k = \overline{m+1, 2m}, \: j = \overline{1, m}.
\Gamma^{( 2 )}[ \varphi_{1}] = \frac{\sigma^{2}}{4 ( 2 \pi )^{d}}\int d^{d}x \int d^{d}y \, \varphi_{1}^{2}( x ) \varphi_{1}^{2}( y ) \int d^{d}p \, e^{- i p ( y - x )}m_{2}^{d - 4}I ( p^{2}, m_{2}^{2}) .
\begin{array}{lcl}\{r,-r\}= \{r,(1-r)(1-r^{-1})^{-1}\}=\{r,1-r\}\{r,(1-r^{-1})^{-1}\}&= \{r,(1-r^{-1})^{-1}\}=\{r^{-1},1-r^{-1}\}&= 1,\end{array}
\frac{1}{(2\pi)^2}\int\limits_0^{\infty}\!d\vec{B}\exp \left[i\vec{B}(\vec{a}+ \vec{b}+\vec{c}+\vec{d}) \right] = \delta^{(2)}(\vec{a}+ \vec{b}+\vec{c}+\vec{d}) &= \delta (a\sin\varphi_1 + b\sin\varphi_2 + c\sin\varphi_3 +d\sin\varphi_4) &\times \delta (a\cos\varphi_1 + b\cos\varphi_2 + c\cos\varphi_3 +d\cos\varphi_4) \;,
|\frac{-32 \lambda'(t)}{\lambda(t)^{3}}\int_{t}^{\infty}dx \lambda''(x) K(x-t,\lambda(t))| \leq \frac{C \log^{3b}(t)}{t^{3}\log^{2b+2}(t)}\int_{t}^{\infty}|K(x-t,\lambda(t))| dx &\leq \frac{C}{t^{3}\log^{b+2}(t)}
\left(\begin{array}{c}<W(L_\mp)> <W(L_{W\mp})> \end{array}\right) =\left(\begin{array}{cc}q^{\pm{1\over 4}}(q^{\mp{1\over 4}}-q^{\pm{1\over 4}}) 0 q^{\mp{1\over 4}}\end{array}\right)\left(\begin{array}{c}<W(L_I)> <W(L_W)> \end{array}\right).
- \left(\frac{(D{v^\varepsilon},-\varepsilon)}{\sqrt{\varepsilon^2+|D{v^\varepsilon}|^2}}\right) - \frac{1}{2}{\left\langle \left(\mathbf{x},x_{n+2}\right), \frac{(D{v^\varepsilon},-\varepsilon)}{\sqrt{\varepsilon^2+|D{v^\varepsilon}|^2}}\right\rangle}= \frac{1}{\sqrt{\varepsilon^2+|D{v^\varepsilon}|^2}}\,,
\psi = u_{1,\alpha^+}\; u_{-1,\alpha^-}\prod_{r=1}^R u_{\tau_r,\alpha^+_r}\; u_{\bar{\tau}_r,\alpha^-_r}\;, \phi = u_{1,\gamma^+}\; u_{-1,\gamma^-}\;\prod_{r=1}^R u_{\tau_r,\gamma_r}\; u_{\bar{\tau}_r,\gamma_r}\;,
\hat{y}=\frac{1}{m}\sum_{j=1}^{m}\sum_{i=1}^{n}W_{j}(x_{i},x^{\prime})y_{i}=\sum_{i=1}^{n}(\frac{1}{m}\sum_{j=1}^{m}W_{j}(x_{i},x^{\prime}))y_{i}.
{\bf M}= \begin{bmatrix}1 0 0 0 0 0 1 1 0 1 0 1 0 0 0 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 0 1 0 0 0 1 1 1 0 0 0 0 1 1 1 0 1 0 \end{bmatrix}.
\mu(|T_{n}-1|>1) \le \mathbf{E}|T_{n}-1|^{\frac{p}{2}}\le 2^{\frac{p}{2}-1}\left\{\mathbf{E}|T_{n}-V_{n}|^{\frac{p}{2}}+\mathbf{E}|V_{n}-1|^{\frac{p}{2}}\right\}&\le Cn^{-\frac{p}{4}}(\|m\|_{L^p}^{p}+\|v\|_{\eta}^{p}).
\varphi(x,z) =\frac 1{z}\left\{\left(\frac{1-zx}{x-1}\right)\dfrac{\sin (x\sqrt{z})}{\sqrt{z}}+\left(-\frac{x}{x-1}+z\right)\cos(x\sqrt{z})\right\}, \psi(x,z) =\frac 1{z}\left\{\left(z-\frac{1}{x-1}\right)\dfrac{\sin (x\sqrt z)}{\sqrt{z}}+\left(\frac{x}{x-1}\right)\cos(x\sqrt{z})\right\};
{}~ [ \kern - 0 . 4 5 e m \llap ~ [ ~ J_{0}( x ) , J_{1}^{a b}( y ) ~ ] \kern - 0 . 4 5 e m \llap ~ ] ~ = M^{a b}~ \delta^{\prime}( x - y ) = ~ [ \kern - 0 . 4 5 e m \llap ~ [ ~ J_{1}( x ) , J_{0}^{a b}( y ) ~ ] \kern - 0 . 4 5 e m \llap ~
\partial_tQ =\sum w_kw_{kt}=-\det (a_{ij})\sum w_k\varrho_k-\varrho\sum w_k\nabla_k\det (a_{ij}) &=-\varrho\det(a_{ij})(p-n-1+e^{2w})|\nabla w|^2+\varrho\det(a_{ij})\sum a^{ij}\nabla_kw_{ij}w_k
B_{\Delta 1}^{j}= \frac{i}{2}\int [ 1 2 3 ] \left[ \sum_{l = 1}^{2}\left( p_{l}^{-}\frac{\partial}{\partial p_{l}^{j}}+ 2 p_{l}^{j}\frac{\partial}{\partial p_{l}^{+}}\right) r_{\Delta}( 1 2 ) \right] \tilde{\delta}\; a_{1}^{\dagger}a_{2}^{\dagger}a_{3}- h . c . \; ,
G_\mathrm{\tilde{\rho}}(\tilde{\rho})=1-\exp\left(-P_T^{-1}d^\alpha\tilde{\rho}\sigma^2_N \right)\exp\left(-\frac{\lambda}{N_{\mathrm{F}}}d^2\tilde{\rho}^{\frac{2}{\alpha}}\frac{2\pi^2}{\alpha\sin(2\pi/\alpha)}\right).
g_{( r , s )}^{[ k , l ]}= \mathrm{c o n s t .}{\frac{\operatorname{s i n}{\frac{\pi r}{l + 2}}\operatorname{s i n}{\frac{\pi s}{k + l + 2}}}{\sqrt{\operatorname{s i n}{\frac{\pi}{l + 2}}\operatorname{s i n}{\frac{\pi}{k + l + 2}}}}}.
E = \int \sqrt{-g^{(3)}}T_{00}~d^3x = 4\pi \int_{r_h}^{+ \infty}e^{\frac{1}{2}B + A+C}\left( e^{-B}g_{i\bar{j}}\phi^i{'}{\bar{\phi}^{\bar{j}}}{'}+ e^{-2C}V_{\mathrm{BH}}+ V \right)dr \ .
G_{i j}={\frac{q^{2}}{Q^{2}}}[ ( D - 3 ) +{\frac{1}{c_{0}^{2}}}] M_{i j}^{e l}+{\frac{1}{2 q^{2}}}[{\frac{1}{c_{0}^{2}}}- ( D - 3 )^{2}] g_{i j}
\sum_{0\leq m<n<\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n})}{p^{m\beta}p^{n(1-\beta)}}= \sum_{m=0}^{\infty}\frac{\tau_{A\cup\{-\beta\}}(p^m) \tau_B(p^m)}{p^{m}}- \sum_{m=0}^{\infty}\frac{\tau_{A}(p^m) \tau_B(p^m)}{p^{m}}.
N_{m-(a-2)}=N_{m-(a-1)}+N_{m-(a+1)}N_{m-2(a-2)}=N_{m-2(a-1)}+2N_{m-2a}+N_{m-2(a+1)}N_{m-3(a-2)}=4N_{m-3(a-1)}-3N_{m-3a}+N_{m-3(a+1)}.
Z \mid_{\mathrm{o n - s h e l l}}= \int d ( \mathrm{z e r o ~ m o d e s}) \big [{\frac{\Pi_{j = 0}^{2}\mathrm{d e t}\Delta_{j F}^{( - 1 )^{j}}}{\Pi_{i = 1}^{4}\mathrm{d e t}\Delta_{i B}^{( - 1 )^{i}}}}\big ]^{\frac{1}{4}}.
| \mu^{k}_{i,\pm}| = \left| 1- \frac{\Delta t_k}{2}\left( \Delta t_k b^{k}_i \lambda^{k}_i + \eta \right) \pm i \frac{\Delta t_k}{2}\sqrt{4 b^{k}_i \lambda^{k}_i -\left( \Delta t_k b^{k}_i \lambda^{k}_i + \eta \right)^2}\right| = \sqrt{1- \eta \Delta t_k}.
\lim_{y\to\infty}{\rm e}^{\theta y}\mathbb{P}_i(T_y^+<\infty) = v_i(\theta)\sum_{j = \pm1}\frac{\pi^{\theta}_j[q_j - (\boldsymbol{\kappa}(-\theta)\boldsymbol{1})_j ]}{\theta v_j(\theta)\mu^+_\theta},
\hat\sigma\colon \xi_j'=\hat M_j(\xi\eta)\xi_j, \eta_j'=\hat M_j^{-1}(\xi\eta)\eta_j, 1\leq j\leq p, \hat\tau_i=\hat\Lambda_{ij}(\xi\eta)\eta_j, \eta_j'=\hat\Lambda_{ij}^{-1}(\xi\eta)\xi_j, \quad\hat\Lambda_{2j}=\hat\Lambda_{1j}^{-1}.
\widehat D_n(x|a_1,\dots,a_r) =\sum_{m=0}^n S_1(n,m)B_m(x+a_1+\cdots+a_r|a_1,\dots,a_r) &=\sum_{j=0}^n\left(\sum_{l=j}^n\binom{n}{l}S_1(l,j)\widehat D_{n-l}(a_1,\dots,a_r)\right)x^j &=\sum_{m=0}^n\binom{n}{m}\widehat D_{n-m}(a_1,\dots,a_r)(x)_m\,.
y_{1,k}(\theta) =-2 i t_{0,k-1}Q^-(-E) Q^+(-E), y_{1,t}(\theta) =-i\Bigl({\rm e}^{\frac{\pi}{k+2}i}Q^-(-E \Omega) Q^+(-E \Omega^{-1}) +{\rm e}^{-\frac{\pi}{k+2}i}Q^-(-E \Omega^{-1}) Q^+(-E \Omega)\Bigr).
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}\, ,
\mathcal{F}_\sigma: =F_\varepsilon(t_\sigma,x_\sigma,u^\varepsilon(t_\sigma,x_\sigma),\nabla_x\varphi(t_\sigma,x_\sigma,y_\sigma),X_\sigma) &-F^\varepsilon(t_\sigma,y_\sigma,v_\varepsilon(t_\sigma,y_\sigma),-\nabla_y\varphi(t_\sigma,x_\sigma,y_\sigma),-Y_\sigma).
f = - \sqrt{2 \pi}\left[ f_0(r) \delta(\omega) \tilde{w_1}(k) +\frac{1}{2}f_+(r) \delta(\omega - 2 \omega_0) \tilde{w_2}(k) + \frac{1}{2}f_-(r) \delta(\omega + 2 \omega_0) \tilde{w_2}(k) \right].
\mathbf{y}_{\scriptscriptstyle{A}}= \mathbf{H}_1 \mathbf{P}^{1/2}_{\scriptscriptstyle{A}}\mathbf{s}_{\scriptscriptstyle{A}}+ \mathbf{H}_2 \mathbf{P}^{1/2}_{\scriptscriptstyle{B}}\mathbf{s}_{\scriptscriptstyle{B}}+ \mathbf{H}_3 \,\mathbf{s}^p_{\scriptscriptstyle{A}}+ \mathbf{H}_4 \,\mathbf{s}^p_{\scriptscriptstyle{B}}+ \mathbf{n}_{\scriptscriptstyle{A}},
S ={\frac{1}{2 \pi}}\int d^{2}x \, \sqrt{- g}\left\{e^{- 2 \Phi}\left( R + 4{\left( \nabla \Phi \right)}^{2}+ 4 \lambda^{2}\right) -{\frac{1}{2}}\sum_{i = 1}^{N}{\left( \nabla f_{i}\right)}^{2}\right\}\; ,
0=\lim_{n\to\infty}\lim_{\varepsilon\to0}\int_X\left(\int\chi_n^\varepsilon d\beta_x\right)d\mu(x)=\lim_{n\to\infty}\int_X\beta_x(A_n)d\mu(x)=\int_X\beta_x({\cap}_nA_n)d\mu(x).
S_{s t r}={\frac{1}{2}}\int_{{\cal M}^{2}}d^{2}\xi \sqrt{- g}g^{m n}( \xi ) \partial_{m}\hat{X}^{\underline{{{m}}}}( \xi ) \partial_{n}\hat{X}^{\underline{{{n}}}}( \xi ) \eta_{\underline{{{m}}}\underline{{{n}}}}.
h= \sum \limits_{j\in \mathbb{J}}\Psi_j^*A_jh \iff \sum\limits_{j\in\mathbb{J}}(2-c_j)\langle A_jh, \Psi_jh\rangle = \|h\|^2 \iff \sum\limits_{j\in\mathbb{J}}c_j^2\| A_jh\|^2 = \|h\|^2.
\rho(L_{-1}^{\gamma})\, =\, h(z)^{-1}(L-c) \, \overset{\gamma}\cdots\, h(z)^{-1}(L-c) \, =\, h(z)^{-\gamma}P(L-c,\gamma), &\rho(L_{1}^{\alpha})\, =\, h(z)(L+c) \, \overset{\alpha}\cdots\, h(z)(L+c)\, =\, h(z)^{\alpha}P(L+c - \alpha+1, \alpha),
\int_0^t \xi \frac{\delta_{\zeta}\mathcal{N}(t,\zeta,\xi,v,\overline v)}{t}\vert_{\zeta = t}d\xi =\frac{1}{t}\int_0^t \xi \Big(\mathcal{N}(t,t,\xi,v,\overline v) - \mathcal{N}(t,0,\xi,v,\overline v) \Big)d\xi .
T_{B}^*f=-\frac 12 r(\overline z) (\mathcal C^{-1})^*\partial^{-1}_{z}(\mathcal C^*f)-\frac 12(1-r(\overline z)) (\widetilde{\mathcal C}^{-1})^*\partial^{-1}_{z}(\widetilde{\mathcal C}^*f).
E\big(u^P\bar{u}^Q\big)=E\left((N_1\otimes1)\big(u^P\overline{R}_1^{-1}\big(\bar{u}^Q\big)\big)\right)=E\left(N_1\big(u^P\big)\overline{R}_1^{-1}\big(\bar{u}^Q\big)\right),
I_{(Z_+,D_{Z,+})}(y,z)=ze^{t_+/z}\sum_{d\in\mathbb K_{+}}y^{d}\left(\prod_{i\in M_0}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right) \left(\prod_{0<a\leq v_1\cdot d}(v_1+az)\right)\textbf{1}_{[-d]}I_{D_{Z,+},d},
M^{-(a_1+a_2+a_4)l+a_3l}\sum_{m=2}^{N(\beta_2)}c_0^{\frac{m}{2}}M^{\frac{a_1l}{2}m}\alpha^m\sum_{n=2}^{\infty}|T_m^{l,(n)}(\beta_1)-T_m^{l,(n)}(\beta_2)|_l &\le c \beta_1^{-\frac{1}{2}}M^{-2a_1+a_2-a_3+2a_4+a_4l}\alpha^{-2}\le c \beta_1^{-\frac{1}{2}}M^{-a_1+a_2+a_4}.
f(x) = f(x) - P_\infty^W f(x) + P_\infty^W f(w) - f(w) &\leq \overline{\int_0^\infty}|\partial_t^+ P_t^W f(x)| dt + \overline{\int_0^\infty}|\partial_t^+ P_t^W f(w)| dt &\leq 2 \int_0^\infty e^{-Kt}D &= \frac{2D}K
v_N' =-Av_N +\frac1{t^{\mu_{N+1}}}\Big(- \sum_{\stackrel{1\le m,j\le N,}{\mu_m+\mu_j=\mu_{N+1}}}B(\xi_m,\xi_j)+\phi_{N+1}+\chi_{N+1}\Big) &- \sum_{n=1}^N \frac1{t^{\mu_n}}\Big( A\xi_n+\sum_{\stackrel{1\le m,j\le N,}{\mu_m+\mu_j=\mu_n}}B(\xi_m,\xi_j)-\phi_n - \chi_n\Big)+h_{N+1,4},
b(w) = (i_n^\ast \wedge j_n^\ast-1_n^\ast \wedge k_n^\ast)\otimes(i_n^\ast \otimes 1_1-1_n^\ast \otimes j_1+j_n^\ast \otimes k_1-k_n^\ast \otimes j_1) - (1_n^\ast \wedge j_n^\ast+i_n^\ast \wedge k_n^\ast)\otimes(1_n^\ast \otimes 1_1+i_n^\ast \otimes i_1+j_n^\ast \otimes j_1+k_n^\ast \otimes k_1).
\left\| u- p \right\|_{L^2(\mathcal{B}_r^+)}^2 = \sum\limits_{k=5}^{\infty}|a_k|^2 \| p_k\|_{L^2(\mathcal{B}_r^+)}^2 \leq \sum\limits_{k=5}^{\infty}|a_k|^2 r^{10}|\mathcal{B}_r^+|\| p_k\|_{L^2(\mathcal{B}_1^+)}^2 &\leq r^{10+ 2n}C(\bar{c}),
|u(x)-u(x+y)| =|u(x)-u(x+t_1y)+u(x+t_2y)-u(x+y)| &\le |u(x)-u(x+t_1y)|+|u(x+t_2y)-u(x+y)| &=\left|\int_0^{t_1}(\nabla u(x+ty),y)dt\right|+\left|\int_{t_2}^1(\nabla u(x+ty),y)dt\right| \mbox{(by \eqref{u'-bound})}\le C_4|y|\left(\int_0^{t_1}+\int_{t_2}^1\right) |x+ty|^{p-1}dt\le C|y|\big(|x|^{p-1}+|y|^{p-1}\big),
\Phi_{SHO}(x,p) &\ \ =\prod_{i=1}^{n}(2 mn x +( i-k_{1})2m) (2mn x +(i-k_{1})2m +m(-\epsilon_{1}s_{x}+2\epsilon_{1}\nu_{x,s_{x}})) &\ \ \times \prod_{j=1}^{m}( 2 mn (p+1) - 2 mn x + 2 n(j-k_{2}))(2mn (p+1) -2mnx +2n (j-k_{2})+n (-\epsilon_{2}s_{y}+2\epsilon_{2}\nu_{y,s_{y}})).
\sum_{n=0}^\infty\overline{\textrm{spt}}_\omega{\left(2\times5^{2m}n+5^{2m}\right)}q^n \equiv x_{2m,1}E(q^5)^2E(q^{50}) \quad\times\left(\dfrac{1}{R(q^{10})}-q^2-q^4R(q^{10})\right)\pmod{5^{2m}},
\omega_{h,k,n}(z) I_{h,k,n}(z) = \int_{-\frac{1}{k(k+k')}}^{\frac{1}{k(k+k'')}}\exp(A(z,v;h,k,n)+g_{h,k}(z,\alpha_n-iv)+ \ln \omega_{h,k,n}(z)) dv.
\alpha = \frac{c_1 h_{3/2}h_0^{1/2}+ c_{3/2}h_1^{3/2}}{h_0^{3/2}}, \beta=\frac{h_1^{5/3}}{(3 h_{3/2})^{2/3}h_0}, \gamma = \beta \cdot \frac{c_{3/2}h_1^{3/2}}{\alpha \cdot h_0^{3/2}}\, .
A_\pm=\frac{1}{p}\int_{\Omega_\mp^c\times \Omega_\mp^c}\frac{|u^+(x)-u^+(y)|^p}{|x-y|^{N+ps}}\, dx\, dyB_\pm=\frac{\lambda}{r}\int_{\Omega_\pm}|u|^r\, dxC_\pm=\frac{\mu}{q}\int_{\Omega_\pm}\frac{|u|^q}{|x|^\alpha}\, dx.
\left\vert \mathrm{STr\,}e^{-t\Box_{b,m}}(x,x)-\sum\limits^{p_r}_{s=1}e^{\frac{2\pi(s-1)}{p_r}mi}\sum^{N_0}_{j=0}t^{-n+j}\mathrm{STr\,}\alpha_{n-j}(x)\Bigr)\right\vert &\leq C_{N_0}\Bigr(t^{-n+N_0+1}+e^{-\frac{\varepsilon_0\hat d(x,X^r_{\mathrm{sing\,}})^2}{t}}\Bigr),\ \ \forall \,0<t<\delta,\ \ \forall x\in X_{p_r},
I_m := \frac{1}{m}\sum_{j = -m/2+1}^{m/2}\frac{e^{-2\pi i \delta \frac{j+\theta_1/2}{m}}}{1 - \exp \left\{- 2 \pi i \frac{j + \theta_1/2}{m}\right\}+ \frac{z}{m}}= G_\delta(z/m) + B_{\delta,m},
\sum_{k=0}^{n-1}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^2/a,aq^2,bq^2,q^2/c,q^2/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k\equiv0\pmod{\Phi_n(q)}.
|(Y^*\tilde{N}_pW^{\dagger})_{2u,k}|<8(r-1)!\;2p\sum_{v=0}^u\sum_{t=0}^{2p-1}\frac{(u!)^2(r-1)!(2t+1)}{(u-v)!(u+v)!(r-t-1)!(r+t)!}\tau(k) &<16p\cdot 4p\cdot(u+1)\sum_{t=0}^{2p-1}\frac{((r-1)!)^2}{(r-t-1)!(r+t)!}\tau(k)<16p\cdot 4p\cdot p\cdot 2p\frac{1}{r}\tau(k)=\frac{128p^4\tau(k)}{r}.
\big( \tfrac{\partial}{\partial x}\big[ \mathbf{Y}^{0,x,\varpi,\mathbf{u}}_t \big]_{\P, \mathcal{B}(H^{\#_\varpi+1})}\big) h = \big[ \mathbf{Y}^{1,(x,h),\varpi,\mathbf{u}}_t \big]_{\P, \mathcal{B}(H^{\#_\varpi+1})}.
J(v_n)=\left(\frac{1}{p'}-\frac12\right)\|v_n\|_{p'}^{p'}+\frac12J'(v_n)v_n \geq\left(\frac{1}{p'}-\frac12\right)\|v_n\|_{p'}^{p'}-\frac12\|J'(v_n)\|_\ast\|v_n\|_{p'}
\begin{aligned}\frac{\partial^2q}{\partial u\partial v}+\frac{\mu_1(1-r^2)+2\mu_2qr}{-\mu_1q(1-r^2)+\mu_2(1-q^2)r}\frac{\partial q}{\partial u}\frac{\partial q}{\partial v}+\frac{(\mu_1^2-\mu_2^2)(1-q^2)(1+r^2)q}{[-\mu_1q(1-r^2)+\mu_2(1-q^2)r][-\mu_1(1-q^2)r+\mu_2q(1-r^2)]}\frac{\partial r}{\partial u}\frac{\partial q}{\partial v}=0\end{aligned}
f_A ={\pi(e_1+e_2)Et e^{-\pi t \epsilon^2 / 2}\over 2sin(\pi t \epsilon/2)}prod_{n=1}^{\infty}{(1-e^{-\pi t n})^2 \over (1-e^{-\pi t(n+i\epsilon)})(1-e^{-\pi t(n-i\epsilon)})}\ .
\textstyle(D_{-x_1}+D_{-x_2})^{k+1}\frac{dz_1dz_2}{(z_1-z_2)^{2(k+1)}}=\frac{(2k+1)!!dx_1dx_2}{(x_1x_2)^{2k+2}}=\frac{1}{(2k+1)!!}(D_{-x_1}D_{-x_2})^{k+1}dx_1dx_2.
|f|^2_{I_F\otimes h_E,\omega}=\sum_j|f_j|^2_{h_E,\omega}=\sum_{j,I}|f_{jI}|^2_{h_E}, ||f||^2_{I_F\otimes h_E,\omega}=\sum_{j,I}\int_U|f_{jI}|^2_{h_E}dV_\omega.
I_{k-1}= [a_n \!+\! b_{k-1}\!-\!a_{k-1}- f(a_n) \!+\! 1,a_n \!+\! b_{k-1}\!-\!a_{k-1}+ f(a_n)\!-\!1] I_k = [a_n \!+\! b_k\!-\!a_k - f(a_n) \!+\! 1,a_n \!+\! b_k\!-\!a_k + f(a_n)\!-\!1]
\sum\limits_{k=1}^{N^{(\ell)}}\sum\limits_{j=1}\alpha_k \binom{\ell+1}{j}\frac{k!}{(k-j)!}H^{k-j}P_{\ell+1-j}P_{\ell-1}^{N^{(\ell)}-k}\big|u_0^{(\ell)}\big\rangle = 0.
\xi_n(t) =\frac{n^2}{t^2}\left( \left(1-\cos\left(t \ln\left(1-\frac{1}{n}\right) \right) \right)^2 + \sin^2\left(t \ln\left(1-\frac{1}{n}\right)\right) \right) &\leq \frac{n^2}{t^2}\left( 2\left( t \ln\left(1-\frac{1}{n}\right) \right)^2\right)=2 n^2 \left(\ln\left(1-\frac{1}{n}\right) \right)^2 \leq 8.
f(X_t) = f(X_0) +\sum_j \int_0^t \partial_j f(X_{s^-}) dX_{j,s}+ \frac{1}{2}\sum_{i,j}\int_0^t \partial_{i}\partial_j f(X_{s^-}) d[X_i,X_j]_s &+ \sum_{s \leq t}\left( \Delta f(X_s) - \sum_j \partial_j f(X_{s^-}) \Delta X_{j,s}- \sum_{i,j}\frac{\partial_j \partial_i f(X_{s^-})}{2}\Delta X_{i,s}\Delta X_{j,s}\right)\;.
\prod_{c\sigma}m_{\sharp c}(\mu)=e^{n L\kappa_1(\mu)}\cdot\displaystyle\sum_{\substack{\Gamma \textup{simple chain in}[1_{\mathfrak{S}_n},\sigma] \Gamma=(\sigma_0,\ldots,\sigma_{|\Gamma|}),\sigma_{|\Gamma|}=\sigma}}\frac{1}{|\Gamma| !}\displaystyle\prod_{i=1}^{|\Gamma|}L\kappa_{d(\sigma_i,\sigma_{i-1})+1}\left(\mu \right).
E\left[T_1(T_1 + 1)\right] = \left(\frac{\nu_1}{\alpha_1}\right)^2 M^2 \left(H_{\nu_1 M}^2 + \sum_{j=1}^{\nu_1 M}\frac{1}{j^2}\right)+ O\left(e^{-\varepsilon M}\right) &= (\nu_1 + \lambda \nu_2)^2 M^2 \left(H_{\nu_1 M}^2 + \sum_{j=1}^{\nu_1 M}\frac{1}{j^2}\right) + O\left(e^{-\varepsilon M}\right)
\log\biggl(\frac{\Gamma(1+x/p)^{1/x}}{x^{1/p}}\biggr) =\frac{1}{x}\log\Gamma(1+x/p)-\frac{1}{p}\log(x) &\ge \frac{1}{x}\log(c_p)+\frac{1}{x}\sum_{j=0}^{k-1}\log(x/p-j)-\frac{1}{p}\log(x) &\ge \frac{1}{x}\log(c_p)+\frac{1}{x}\int_{x/p-k}^{x/p}\log(t)dt-\frac{1}{p}\log(x) &=\frac{1}{x}\log(c_p)+\frac{1}{p}\log(1/p)-\frac{1}{p}-\frac{1}{x}f(x/p-k),
(\#S_1) \mu_{S_1}= \sum_{i \in S_1}(|x_i|^2 - \alpha_i) = \frac{\gamma}{2}- \alpha_{S_1}, (\#S_2) \mu_{S_2}= \sum_{i \in S_1}(|x_i|^2 - \alpha_i) = \frac{\gamma}{2}- \alpha_{S_2}, (\#S_3) \mu_{S_3}= \sum_{i \in S_1}(-|y_i|^2 - \alpha_i) = -\delta - \alpha_{S_3},
s(\check{x})=\biggl\{\left( \frac{1}{r}A\check{x}+\frac{1}{r}p+\frac{2\pi}{r}A\mathcal{B}M_s \right) \in \check{\pi}^{-1}(\check{x})\approx \mathbb{R}^n/2\pi \mathbb{Z}^n \ | &M_s=(m_1,\cdots, m_s, 0,\cdots, 0)^t\in \mathbb{Z}^n, \ 0\leq m_i\leq r_i'-1, \i=1,\cdots, s \biggr\},
\tilde{f}^{( 1 )}= \left( \begin{array}{c c c c}{{0}}&{{- \delta_{i j}}}&{{2 a_{i}}}&{{0}}\\{{\delta_{i j}}}&{{0}}&{{0}}&{{0}}\\{{- 2 a_{i}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right) ,
\lim_{q\to\zeta}\frac{(q^r;q^d)_{\ell e+k}}{(q^d;q^d)_{\ell e+k}}=\frac{v\,(v+d)\dotsb(v+(\ell-1)d)}{d\cdot2d\,\dotsb\,\ell d}\,\lim_{q\to\zeta}\frac{(q^r;q^d)_k}{(q^d;q^d)_k}&={v/d+\ell-1\choose\ell}\lim_{q\to\zeta}\frac{(q^r;q^d)_k}{(q^d;q^d)_k}
\Tilde{A}:= \begin{pmatrix}\textbf{1}\boldsymbol{0}\dots \boldsymbol{0}\boldsymbol{\beta_1^2}\textbf{1}\dots \boldsymbol{0}\vdots \vdots \dots \vdots \boldsymbol{\beta_1^m}\boldsymbol{\beta_2^m}\dots \textbf{1}\end{pmatrix}_{m \times m},
= \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{1}{N}\sum\limits_{i=1}^N \frac{1}{(4\pi^{2}\sigma^{4})}exp \left ( - \frac{(u_1-b)^{2}}{\sigma^{2}}\right) exp \left( - \frac{(u_2-b')^{2}}{\sigma^{2}}\right) exp \left( - \frac{c}{2\sigma^{2}}\right ) \mathrm{d}u_1\mathrm{d}u_2
\mathcal{E}=\left\{u\in C^0(\bar \Omega)\,|\, u\ge 0,\,\,{\int_\Omega u(x)dx\le M_0^*}, \int_\Omega u^{-p}(x)dx\le M_1^*,\,\, \int_\Omega u^{q}(x)dx\le M_2^*\right\}
\sqrt{\alpha}\left(\frac{\gamma}{4}-\frac{\log(4\beta)}{4}+\sum_{n=1}^{\infty}d(n)\,K_{0}(2n\alpha)\right)=\sqrt{\beta}\left(\frac{\gamma}{4}-\frac{\log(4\alpha)}{4}+\sum_{n=1}^{\infty}d(n)\,K_{0}(2n\beta)\right).
E \bigl( Z_t^H(a,b) - Z_s^H(a,b)\bigr)^2 = \bigl(a^2 + b^2\bigr) | t-s|^{2H}&- 2^{2H}ab \bigl( | t |^{2H}+ | s |^{2H}\bigr) + 2ab | t + s |^{2H};
0 = \int_a^t\langle L_{\lambda}[X], X \rangle ds = -\int_a^t \langle X'', X \rangle ds + \int_a^t \langle (R - \lambda I)X, X \rangle ds = \int_a^t \|X'\|^2 ds - \langle X', X \rangle|_a^t + \int_a^t \langle (R - \lambda I)X, X \rangle ds = \int_a^t \|X'\|^2 + \langle (R-\lambda I)X, X \rangle ds.
e_{0}( g_{4}) = \frac{1}{2}\ln a + \frac{1}{2 4}( a - 1 ) ( a - 9 ) = \sum_{k = 1}^{\infty}( 3 g_{4})^{k}\, \frac{( 2 k - 1 ) !}{k ! \, ( k + 2 ) !}
\sup_{z,w\in Y_{N,\varepsilon}}\bigg|g_{X_{N},\mathrm{hyp}}(z,w)-\sum_{\gamma\in S_{\Gamma_{X_{N}}}(\delta;z,w)}g_{\mathbb{H}}(z,\gamma w)\bigg|=O_{X_{q_{\mathcal{N}}},\varepsilon,\alpha,\delta}(1).
V_q = \frac{(g \! + \! N)(g \! + \! N \! + \! 1)}{2 \sin^2 \! x}+ \frac{(h \! + \! N \! - \! 1)(h \! + \! N \! - \! 2)}{2 \cos^2 x}- \frac{(2N \! + \! h \! - \! g \! - \! 1)^2 + (2n \! + \! g \! + \! h \! + \! 2N)^2}{4}.
\mathbb{E}\left\{\sup\limits_{t}\left\Vert \langle X-x \rangle^{bd}e^{-itH_\omega}P_\omega(I) \delta_x \right\Vert^s \right\}\leq C\sum\limits_k 2^{\frac{sbd}{2}+(k+1)sbd-k(pd-s\nu)}\leq C\sum\limits_k \left(2^{sbd-pd+s\nu}\right)^k<\infty.
\Pr_{{\cal G}}(-\tilde x,g^0, \tau) = \arg\min\limits_{g \in{\cal G}}\left\{-\langle \tilde x + \tau J_f'(g^0), g \rangle + (1 + \tau) J_f(g)\right\}= \arg\max\limits_{g \in{\cal G}}\left\{\langle z, g \rangle - J_f(g) \right\}= \nabla f(z).
\begin{array}{ll}L_c(x,Y, \mu,\lambda):= f(x)+ \theta_c (F(x)+Y/c)-\displaystyle\frac{\|Y\|^2}{2c}begin{align*}8pt] +\langle \mu, h(x) \rangle+\displaystyle \frac{c}{2}\|h(x)\|^2+ \frac{1}{2c}\left[ \|\Pi_{K^*}(\lambda-c g(x))\|^2-\|\lambda\|^2\right]\, ,\end{array}
\lim_{n \rightarrow +\infty}\left( \frac{\lambda_{j',n}}{\lambda_{j,n}}+ \frac{\lambda_{j,n}}{\lambda_{j',n}}+ \frac{|x_{j,n}- x_{j',n}|}{\lambda_{j,n}}+\frac{|t_{j,n}- t_{j',n}|}{\lambda_{j,n}}\right) = + \infty
a_{n,k+1}= \sum_{q=1}^k 4^q\sum_{j=1}^n a_{n-j,q}\, a_{j-1,q}<\sum_{q=1}^k 4^{nq}\sum_{j=1}^n C_{n-j}\, C_{j-1}< \sum_{q=1}^k 4^{nq}< C_n \,4^{n(k+1)},
\varphi_{i,e}^{{\rm even}}(x) :=\frac{\varphi_{i,e}(x',x_d)+\varphi_{i,e}(x',-x_d)}{2}, \varphi_{i,e}^{{\rm odd}}(x) :=\frac{\varphi_{i,e}(x',x_d)-\varphi_{i,e}(x',-x_d)}{2}.
\Gamma_{d i v}^{m a t}= \frac{1}{\varepsilon}\int \left( \gamma_{1}\frac{1}{2}g^{\mu \nu}\partial_{\mu}\varphi \partial_{\nu}\varphi - \gamma_{2}\frac{\varphi^{4}}{4 !}+ \gamma_{3}\frac{1}{2}R \varphi^{2}\right) \sqrt{- g}d^{4}x .
p < 4f(\kappa,\gamma, t)^2 & = 4 \left( \prod_{i=\kappa-t}^{\kappa}p_{\gamma i}\right)^2 & \leq 4 \prod_{i = \kappa-t}^\kappa \left( i\gamma (\ln{i\gamma}+ \ln\ln(i\gamma)) \right)^2 & \leq 4 \gamma^{2t}\left( \frac{\kappa!}{(\kappa-t-1)!}\right)^2 \left( \ln \kappa\gamma + \ln\ln \kappa\gamma \right)^{2t}
\begin{aligned}d_i[g_1,\ldots,g_n] = \begin{cases}[g_2,\ldots,g_n] i = 0, [g_1,\ldots, g_{i-1},g_i g_{i+1}, g_{i+2},\ldots,g_n] 0<i<n, [g_1,\ldots,g_{n-1}] i = n, \end{cases}s_i[g_1,\ldots,g_n] = [g_1,\ldots,g_i,e,g_{i+1},\ldots,g_n].\end{aligned}
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{( - 1)^{n(j + s) + j}\binom kj\binom{k-j}s\frac{{G_{n + r}^j G_n^s}}{{G_r^{j + s}}}H_{m - rk + nj + (n + r)s}}}= \left({\frac{{G_0}}{{G_r}}}\right)^k H_m,G_r\ne 0\,.
f_k(x,y) := k^2 \int_{\left[ \frac{i}{k}, \frac{i+1}{k}\right) \times \left[ \frac{j}{k}, \frac{j+1}{k}\right)}f \,d\lambda \text{if}(x,y) \in \left[ \frac{i}{k}, \frac{i+1}{k}\right) \times \left[ \frac{j}{k}, \frac{j+1}{k}\right)
b_n =(7-n)2^{n-3}-2+\sum_{a=1}^{n-2}2^{a-1}e_{n-1-a}+\sum_{a=2}^{n-3}\sum_{\ell=1}^{n-2-a}\sum_{i=0}^{a-1}2^{n-2-a-\ell}\binom{a-1}{i}\binom{\ell+i}{i}&\quad+\sum_{a=2}^{n-1}\left(2^{n-1-a}-1-\binom{n-a}{2}\right)(a-1)2^{a-2}+\sum_{a=3}^{n-1}\sum_{\ell=1}^{a-2}\sum_{j=0}^{\ell}\binom{n-1-a+j}{j}e_{a-1,\ell}, n \geq 3,
\int_{\Omega_{\epsilon}}\frac{Lw(x,y)}{Z(x,y)-z_{0}}\dxdy =-\int_{\partial \Omega_{\epsilon}}\frac{w(x,y)}{Z(x,y)-z_{0}}dZ(x,y) &=-\int_{\partial \Omega}\frac{w(x,y)}{Z(x,y)-z_{0}}dZ(x,y) + \int_{\partial K_{\epsilon}}\frac{w(x,y)}{Z(x,y)-z_{0}}dZ(x,y);
G_{\vert s- r\vert}(z-y)^2 + G_{s+ r}(z-y)^2= \frac{G_{\vert s- r\vert/2}(z-y)}{\sqrt{8\pi\vert s- r\vert}}+ \frac{G_{(s+ r)/2}(z-y)}{\sqrt{8\pi (s+ r)}}\,.
|\mu^s| = \sum_{j=1}^{s}\left(m(t-2j)+t\binom{m}{2}\right)-\binom{ms}{2}&=-s^2\cdot (m^2+2m)/{2}+s\cdot (m^2t+mt-m)/{2}&= - \frac{m^2+2m}{2}\cdot(s- \alpha_{m,t}(-1))^2+ \frac{m^2+2m}{2}\cdot \alpha^2_{m,t}(-1).