harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\times \int_0^\infty \frac{d\tau}{\tau}~ e^{2\pi \tau}~ e^{-2\pi \tau\alpha' k^2}~e^{- \frac{y^2 \tau}{2\pi\alpha'}}~{\rm e}^{2 \pi \tau}~ Tr \left(\prod_{n=1}^\infty e^{-2\pi \tau \alpha_{-n}\cdot \alpha_{n}}\right)=
|\mathbb{E}[g(\mathbf{V}_T)]-\mathbb{E}[g(\mathbf{Z})]| &=|\mathbb{E}[g(\mathbf{V}_T)h_R(\mathbf{V}_T)]+\mathbb{E}[g(\mathbf{V}_T)h_R^c(\mathbf{V}_T)]-\mathbb{E}[g(\mathbf{Z})h_R(\mathbf{Z})]-\mathbb{E}[g(\mathbf{Z})h_R^c(\mathbf{Z})]| &\ll|\mathbb{E}[g(\mathbf{V}_T)h_R(\mathbf{V}_T)]-\mathbb{E}[g(\mathbf{Z})h_R(\mathbf{Z})]|+\epsilon.
\frac{\partial\varphi(x,t)}{\partial t}=\nabla\cdot(D(x)\nabla\varphi(x,t))=D(x)\Delta\varphi(x,t)+\sum_{i=1}^{3}\frac{\partial D(x)}{\partial x_{i}}\frac{\partial\varphi(x,t)}{\partial x_{i}}
w\left({x,N}\right) = \binom{N}{x}p^x (1-p)^x (x = 0,1, \ldots ,N ), h\left({n,N}\right) = \frac{{\left({- 1}\right)^n n!}}{{\left({- N}\right)_n}}\left({\frac{{1 - p}}{p}}\right)^n .
\underset{T \rightarrow \infty}{lim}\underset{\kappa \in (0,1)}{sup}\mathbb{E}\left|\langle :X_T\otimes X_T:,\Psi_{\epsilon,\delta,\phi}^f \rangle - \eta_{f,\phi,\epsilon,\delta}^T\right|^2 = 0,\epsilon,\delta>0,\phi \in \mathcal{S},
\mu_{tf_k}([i]) \leq \exp(4tV(f) + tf_k(x) - P_k(t)) &\leq \exp(4tV(f) + t\sup f |_{[i]}- P_k(t)) = \exp(t(4V(f) + \sup f |_{[i]}- S_k)) \exp(tS_k- P_k(t)) \leq \exp(t(4V(f) + \sup f |_{[i]}- S)) \,.
A_{1}^{a}\big [ a_{1}^{a}, \varphi^{a}( x ) \big ] = a_{1}^{b}\, \left( U^{\dagger}[ \, \vec{\varphi}( x ) ] \right)^{b a}- u_{1}^{a}\left[ \, \vec{\varphi}( x ) \right] \; .
= -\left[ \frac{\sigma'(u) e^{uz}}{z^2}\right]_{1-\delta}^1 + \frac 1{z^2}\int_{1-\delta}^1 \sigma''(u) e^{uz}du= \frac 1{z^2}\int_{1-\delta}^1 \sigma''(u) e^{uz}du - \frac{\sigma'(1) e^{z}}{z^2}+O\left( \frac{e^{x(1-\delta)}}{|z|}\right) .
0\preceq{}\begin{bmatrix}\Gamma_1 0 0 \Gamma_2\end{bmatrix}^{-\frac{1}{2}}\begin{bmatrix}L_{B,11}L_{B,11}L_{B,21}L_{B,22}\end{bmatrix}\begin{bmatrix}\Gamma_1 0 0 \Gamma_2\end{bmatrix}^{-\frac{1}{2}}\preceq{}I.
S_{\mathrm{b o s o n}}= \frac{8 \pi}{e^{2}}\int d t d r \sqrt{A ( r ) B ( r )}\; \: r^{2}\: \: \left\{\frac{1}{\sqrt{B ( r )}}\frac{\partial a_{1}( r , t )}{\partial t}- \frac{1}{\sqrt{A ( r )}}\frac{\partial a_{0}( r , t )}{\partial r}\right\}^{2}
Z_{\kappa}exp_{\kappa}(-\beta x^{2});Z_{\kappa}=\sqrt{\frac{2\beta\kappa}{\pi}}(1+\frac{1}{2}\kappa)\frac{\Gamma(\frac{1}{2\kappa}+\frac{1}{4})}{\Gamma(\frac{1}{2\kappa}-\frac{1}{4})}
\frac{K}{2}\int_R \sqrt{-g}g^{rr}\phi(-k, r) \partial_r \phi(k, r) \frac{d^4k}{(2\pi)^4}- \frac{K}{2}\int_L \sqrt{-g}g^{rr}\phi(-k, r) \partial_r \phi(k, r) \frac{d^4k}{(2\pi)^4}\ ,
\begin{array}{cc}\left\vert \left\vert f\right\vert \right\vert_{X_{h}^{p}}=\left(\int\limits_{0}^{\infty}\left\vert f(t)\right\vert^{p}h^{^{\prime}}\left(x\right) dt\right)^{\frac{1}{p}}<\infty ,\ \ \ \ 1\leq p\leq \infty\end{array}
- \Omega_{u}( - u ) \, = \mathrm{i}\, \left( \begin{matrix}{{- 1}}&{{2}}\\{{- 1}}&{{1}}\\ \end{matrix}\right) \, \left( \begin{matrix}{{f^{( 1 )}( u )}}\\{{f^{( 2 )}( u )}}\\ \end{matrix}\right) = \mathrm{i}\, R \, \Omega_{u}( u )
(K^{-1})^S_S = \frac{1-A|z|^2}{P''(y)},\qquad (K^{-1})^T_T = \frac{(T+T^*)^2(1-A|z|^2)}{3(1-\frac{\bar{n}}{3}B(1-A|z|^2)\|\Pi\|)}
(u_s^n-H_s^n)d\gamma_s^{-\frac12}= (u_s^n-H_s^n)\gamma_s^{-\frac12}\left( -\frac12 \mu_s + \frac38 \sigma_s^2 \right) ds - (u_s^n - H_s^n) \gamma_s^{-\frac12}\frac12\sigma_s dW_s^1.
{\cal C}_{kk}=\Big( \displaystyle\sum_{\sigma \subset U \atop |\sigma|=k}\prod_{u_i \in \sigma}(d_i)(d_i-1) \Big)\Big( \displaystyle\sum_{\sigma \subset W \atop |\sigma|=k}\prod_{w_i\in \sigma}(d_i')(d_i'-1) \Big)\Big( \dfrac{k!k!}{k^2}\Big)\:.
a \rhd b = \lambda_a(b) = a^{-1}\cdot (a\circ b),a\lhd b = \overline{\rho_{\bar{b}}(\bar{a})}= \overline{(\bar{b}\circ \bar{a})\cdot \bar{b}^{-1}}= \overline{\bar{a}\cdot b}\circ b.
\begin{array}{l}{{\partial_{0}x^{i}= - p^{i}, \qquad \partial_{0}p^{i}= - \partial_{1}\partial_{1}x^{i},}}\\{{( \partial_{0}+ \partial_{1}) \theta_{a}= 0 , \qquad ( \partial_{0}- \partial_{1}) \bar{\theta}_{\dot{a}}= 0 ,}}\\ \end{array}
\begin{cases}\Phi_S(t_{e_2}(k)) = \tau_{e_2}\Phi_S(k) k \in E_{2,-}\Phi_S(t_{e_2}^{-1}(k)) = \tau^{-1}_{e_2}\Phi_S(k) k \in E_{2,+}\Phi_S(c(k)) = \Theta \, \Phi_S(k) k \in E_{1,0}.\end{cases}
U_i = \{r(n) : \ell_i < n < 2^{k_i}\textup{and}T_f^n(x,t) \in X_{B-b}\}U_i' = \{r_i(n) : \ell_i < n < 2^{k_i}\textup{and}T_f^n(x,t) \in X_{B-b}\}
\mathbf E[H_q(t+dt)] = \mathbf E\Big[H_q(t)(1-(|V_t|+1)dt) + dt \cdot (|V_t|+1) \Big( H_q(t) - pq(1-q)\frac{d}{ds}H^\circ_s(t)\Big|_{s=q}+ H^\circ_{1-q+pq}(t)\Big)\Big].
[y_{(1\,p+1)},y_{(1\,l)}]_c=[y_{(1\,p)},[y_{p+1},y_{(1\,l)}]_c]_c+\chi_{(1\,l)}(g_{p+1})[y_{(1\,p)},y_{(1\,l)}]_cy_{p+1}&-\chi_{p+1}(g_{(1\,p)})y_{p+1}[y_{(1\,p)},y_{(1\,l)}]_c=\chi_{(1\,l)}(g_{p+1})[[y_{(1\,p)},y_{(1\,l)}]_c, y_{p+1}]_c,
\tilde \psi_{ni}(x) = \psi^K_{ni}(x) + \sum_{k=1}^{K}(\tilde H_{ni, k0}(x)\psi^K_{k0}(x) + \tilde H_{ni, k1}(x)\psi^K_{k1}(x)), n = \overline{1, K}, i=0,1.
\begin{array}{l}Q'=\frac{1}{2}(\tau+1)E^{-1}, q'=\frac{1}{2}(c+w), \theta'=\displaystyle{\frac{\beta}{\tau+1}}+\left\langle \displaystyle{\frac{1-\tau}{4}E^{-1}c}-\frac{1}{2}E^{-1}w,c\right\rangle+\displaystyle{\frac{\left\langle E^{-1}(w+\tau c),w+\tau c\right\rangle}{4(\tau+1)}}. \end{array}
m=m_0\left[1+\frac{\alpha}{4\pi}\log^2\frac{|eB|}{m_0^2}+\frac{5}{24}\left(\frac{\alpha}{2\pi}\log^2\frac{|eB|}{m_0^2}\right)^2+\frac{61}{720}\left(\frac{\alpha}{2\pi}\log^2\frac{|eB|}{m_0^2}\right)^3+\cdots. \right]
\displaystyle{\operatornamewithlimits{\mbox{minimize}}_{w; \, \alpha}}\ \displaystyle{\frac{1}{N}}\, \displaystyle{\sum_{i=1}^N}\, \ell\left( y_i - \sigma(w^Tx^i + \alpha_i) \, \right) + \gamma \, \displaystyle{\sum_{i=1}^m}\, \left[ \, \underbrace{\alpha_i \, | \, w_i \, | - h_i(w_i)}_{\mbox{surrogate sparsity function}}\, \right],
\| e^{(t-s) \Delta}|u(s)|^\alpha u(s) \|_{L^r}\le \| |u(s)|^\alpha u(s) \|_{L^r}= \| u(s)\|_{L^{( \alpha +1) r}}^{\alpha +1}= s^{- \frac{\alpha +1}{\alpha}+ \frac{N}{2 r}}\| f \|_{L^{(\alpha +1) r}}^{\alpha +1}.
{\cal L}= \mu \int_{S^1\times R^{3+1}}\, tr \left\{- \frac{1}{2}{\cal F}_{MN}{\cal F}^{MN}+ \sum_P{\cal D}_M\Phi_P{\cal D}^M\Phi_P + \frac{1}{2}\sum_{P,Q}[\Phi_P,\Phi_Q]^2 \right\}
\lim_{n\rightarrow\infty}\frac{n_2'}{n}=\frac{R_1}{1-\delta_1}+\frac{R_2}{1-\delta_1\delta_2}\lim_{n\rightarrow\infty}\frac{n_2''}{n}=\frac{R_1}{1-\delta_1\delta_2}+\frac{R_2}{1-\delta_2}
| T_s\mu (\{-s+x \}) -T_t\mu\{-s+x\}| \geq | T_s\mu (\{-s+x \})| -| T_t\mu\{-s+x\}| =|\mu (\{x \})| -| \mu\{t-s+x\}| \geq a-\frac{a}{2}= \frac{a}{2}\,,
\sqrt{\frac{(2n+2)}{1-([\eta]_{m}-2^{-j})^2}}([\eta]_{m}-2^{-j}) =\sqrt{\frac{n+1}{m+1}}\sqrt{\eta^2+2m+2}\left(\frac{\eta}{\sqrt{\eta^2+2m+2}}-2^{-j}\frac{\eta^2}{2m+2}+r'\right) &=\sqrt{\frac{n+1}{m+1}}\eta-\sqrt{\frac{n+1}{m+1}}\sqrt{\eta^2+2m+2}\frac{2^{-j}\eta^2}{2m+2}+\sqrt{\frac{n+1}{m+1}}\sqrt{\eta^2+2m+2}r'.
G_{M N}= \Phi_{\cal B D}^{- 1}\left( \begin{array}{c c}{{g_{\mu \nu}}}&{{\, 0}}\\{{0}}&{{\, \Phi_{\cal B D}^{\frac{D - 2}{D - 4}}h_{m n}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\tilde{g}_{\mu \nu}}}&{{\, 0}}\\{{0}}&{{\, \Phi_{\cal B D}^{\frac{2}{D - 4}}h_{m n}}}\\ \end{array}\right) ,
S_{EH}+S_{GHY,0}=\frac{1}{16\pi}\int_{M}d^{4}x\sqrt{-g}R+\frac{1}{8\pi}\int_{\partial M}d^{3}y\epsilon\sqrt{h}K-\frac{1}{8\pi}\int_{\partial M}d^{3}y\epsilon\sqrt{h}K_{0},
G^{\rm (TM)}_{(3)}(\tau)= 360\tau^5\int_0^{\Omega/2}d\Omega' \int_0^{\Omega'}dK\,\frac{K\,\left[K^2+\Omega'\left(\Omega-\Omega'\right)\right]^2}{\sqrt{(\Omega'^2-K^2)\left[(\Omega-\Omega')^2-K^2\right]}}\,\left(\frac{1}{e^{\Omega'}-1}+\frac{1}{e^{\Omega-\Omega'}-1}\right).
T_g(h\circ f,\omega^W,h^E)-T_g(h,\omega^V,h^{f_*E}) -\int_{V_{\mu_n}/S}{\rm Td}_g(\overline{Th})T_g(f,\omega^W,h^E) = \widetilde{{\rm ch}}_g(\overline{E}(f,h,h\circ f))-\int_{W_{\mu_n}/S}\widetilde{{\rm Td}}_g(\overline{T}(f,h,h\circ f)){\rm ch}_g(\overline{E}).
\left[\left(\frac{H_0}{f}\right)^2-\left(\frac{f'}{f}\right)^2\frac{1}{b_{0}\,^2}\right]+\frac{2\alpha}{M^2}\left[\left(\frac{H_0}{f}\right)^2-\left(\frac{f'}{f}\right)^2\frac{1}{b_{0}\,^2}\right]^2=\frac{\Lambda_b}{6M^3}~~~.
r^1(x,y,t)=-2\frac{\partial p}{\partial \nu_x}(x,y,t),\;\; (x,t)\in \Sigma \times (0,+\infty ),\; y\in \overline{\Omega}, &r^{j+1}(x,y,t)=-2\int_0^t\int_\Sigma r^1(x,z,t-s)r^j(z,y,s)dA(z)ds,\;\; j\geq 1, &\hskip 7cm (x,t)\in \Sigma \times (0,+\infty ),\; y\in \overline{\Omega}.
\tilde{F}( A ) = \frac{1}{N^{2}}\log [ \frac{Z ( A )}{Z_{\mathrm{w e a k}}( A )}] \to \frac{1}{N^{2}}\log [ \oint \frac{d z}{2 \pi i}\frac{1}{z^{N + 1}}\frac{1}{1 - z}\exp ( - \sum_{n = 1}^{[ N / 2 ]}\frac{1}{2 n}z^{2 n}\alpha_{2 n}( A ) ) ]
\begin{array}{c}{\{\quad , \quad \}= i{\tilde{g}}^{A \bar{B}}\nabla_{A}\wedge{\bar{\nabla}}_{\bar{B}}+ g^{A \bar{B}}\frac{\partial}{\partial \theta^{A}}\wedge \frac{\partial}{\partial{\bar{\theta}}^{\bar{B}}}}\end{array}
\beta\left(u\alpha_{1}\right)^{b_{1}b}\left(v\alpha_{3}\right)^{b_{2}b}\beta^{-1}=\rho^{\kappa_{1}}\left(b\beta\beta_{0}\beta^{T}\right)\cdot \sigma \alpha_{1}, \beta\left(u\alpha_{1}\right)^{b_{3}b}\left(v\alpha_{3}\right)^{b_{4}b}\beta^{-1}=\rho^{\kappa_{2}}\left(b\beta\beta_{0}\beta^{T}\right)\cdot \tau \alpha_{3}
{\cal C}^{s,2d,c}_{pol,m}={\Big \{}f:{\mathbb R}^{2d}\rightarrow{\mathbb R}: \forall |x|\geq 1~\forall 0\leq |\gamma|\leq m~{\big |}D^{\gamma}_xf(z){\big |}\leq \frac{c}{1+|z|^s}{\Big \}}.
g([{f},{f}](\xi_i,Z),{f}X) = g({f}^2\,[\xi_i,Z] -{f}[\xi_i,{f}Z],{f}X) = - g({f}(\pounds_{\xi_i}{f})Z,{f}X) = g((\pounds_{\xi_i}{f})Z,QX) -\sum\nolimits_{j}\eta^j(X)\,\eta^j((\pounds_{\xi_i}{f})Z) .
(1-|y|^2) \Big[|y|^2-2x\cdot y+1-\frac{(1-|y|^2)}{\rho}\Big] -(\rho+1)|x-y|^2 &=-|y|(|y|^2-2x\cdot y +|x|^2)-\frac{1-|y|^2}{1-|x|^2}|x-y|^2 = -|x-y|^2\Big(|y|^2+\frac{1-|y|^2}{1-|x|^2}\Big).
\{{\cal M}{}_{,}^{\otimes}v ( y ) \}= \Bigl ({\cal M}T_{\rho}\;{\cal G}_{\rho \sigma}( \ell , y ) \; - \; T_{\rho}{\cal M}\;{\cal G}_{\rho \sigma}( 0 , y ) \Bigr ) \otimes T_{\sigma}v ( y ) \quad ,
(Y\mathbf{d})Z =(\sup_{y\in Y}y\mathbf{d})Z=\inf_{z\in Z}\sup_{y\in Y}y\mathbf{d}z. Y(\mathbf{d}Z) =Y(\inf_{z\in Z}\mathbf{d}z)=\sup_{y\in Y}\inf_{z\in Z}y\mathbf{d}z.
\#\left(P_{g\oplus h}^{-1}\alpha\right) =\#\left(P_{g}^{-1}\gamma_{\, r}\oplus P_{h}^{-1}\beta\right)-1 =\#\left(P_{g}^{-1}\gamma_{\, r}\right)+\#\left( P_{h}^{-1}\beta\right)-1,
J_{r}= \oint p_{r}d r = 2 \int_{r_{1}}^{r_{2}}( 2 m E +{\frac{m Q^{2}}{2 \pi \epsilon_{0}r}}-{\frac{( J_{\theta}+ J_{\phi})^{2}}{4 \pi^{2}r^{2}}})^{\frac{1}{2}}d r ,
0 \to \left (\mathcal O_{H_{i_1 \cdots i_{l-1}}}/\mathcal J(\varphi)|_{H_{i_1 \cdots i_{l-1}}}\right) \otimes \mathcal O_{H_{i_1 \cdots i_{l-1}}}(-H_{i_l}) &\to \mathcal O_{H_{i_1 \cdots i_{l-1}}}/\mathcal J(\varphi)|_{H_{i_1 \cdots i_{l-1}}}\to \mathcal O_{H_{i_1 \cdots i_l}}/\mathcal J(\varphi)|_{H_{i_1 \cdots i_l}}\to 0
I_{m}= \int d^{4}x \sqrt{{\tilde{g}}}\left[ - \frac{1}{2}D_{\mu}\Phi ( D^{\mu}\Phi )^{*}- \frac{1}{2}D_{\mu}\Sigma ( D^{\mu}\Sigma )^{*}- \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{4}H_{\mu \nu}H^{\mu \nu}- V ( | \Phi | , | \Sigma | ) \right] ,
\lambda \int_{\mathbb R}\operatorname{sgn}(t)|f(t)|^2\,\mathrm dt = \lim_{x\to -\infty}\lambda U(x) = \lim_{x\to -\infty}V(x) = \int_{\mathbb R}|f'(t)|^2+ q(t) |f(t)|^2\,\mathrm dt.
\tilde{\vartheta}_{2}^{2}= \tilde{\vartheta}_{3}^{2}{\frac{\vartheta_{4}^{2}}{\vartheta_{3}^{2}}}\left( 1 - 4{\frac{\vartheta_{2}^{4}}{\vartheta_{3}^{4}}}\right) , \; \tilde{\vartheta}_{4}^{2}= \tilde{\vartheta}_{3}^{2}{\frac{\vartheta_{2}^{2}}{\vartheta_{3}^{2}}}\left( 1 - 4{\frac{\vartheta_{4}^{4}}{\vartheta_{3}^{4}}}\right) .
\begin{aligned}\{Pos(n)=k\}=\bigcup_{1\leq k_1<...<k_n=k}\bigcap_{i=1}^k \left( \{f_{k_i}>{\tfrac12}\}\cap\bigcap_{j\neq k_i,j<k}\{f_j\leq{\tfrac12}\}\right), \{pos(n)=k\}=\bigcup_{1\leq k_1<...<k_n=k}\bigcap_{i=1}^k \left( \{f_{k_i}\leq{\tfrac12}\}\cap\bigcap_{j\neq k_i,j<k}\{f_j>{\tfrac 12}\}\right).\end{aligned}
r = i\omega G_{\zeta, \mu}(Wr) + G_{\zeta, \mu}(V_1 r) - G_{\zeta, \mu}F, Q = G_{\zeta, \sigma}(V_2 Q) + G_{\zeta, \sigma}\big\{i\omega\nabla(\sigma\mu)\times r - \nabla_\zeta \times f \big\}.
G(\hat{\vec z}) = \frac{r -(\hat{\vec z},|\vec v|)}{\|\hat{\vec z}\|_p}\le \frac{r -(\vec z^*,|\vec v|)}{\|\hat{\vec z}\|_p}<\frac{r -(\vec z^*,|\vec v|)}{\|\vec z^*\|_p}= G(\vec z^*),
X_{a}( R^{c}{}_{b \mu \nu}) - X_{b}( R^{c}{}_{a \mu \nu}) + f^{c}{}_{a d}R^{d}{}_{b \mu \nu}+ f^{d}{}_{b a}R^{c}{}_{d \mu \nu}- f^{c}{}_{b d}R^{d}{}_{a \mu \nu}= 0 .
\sum_{j=1}^k \frac{|Q_j|^{-2/d}f(\gamma(|Q_j|))}{|Q_B|^{-2/d}f(\gamma(|Q_B|))}\leq \sum_{j=1}^k \left(2^{d(k-j)}\right)^{-2/d}2^{(2-\alpha)(k-j)}&= \sum_{j=1}^k 2^{-\alpha(k-j)}= \frac{1-2^{-\alpha k}}{1-2^{-\alpha}}\leq \frac{1}{1-{2^{-\alpha}}}.
\omega_\ell = \begin{cases}N \ell=0, \frac{N\left(1-\frac{1}{N}\right)^\ell}{N+(\ell+1-N)\left(1-\frac{1}{N}\right)^\ell}\ell\in[\overline{L}-1], 0 \ell=\overline{L}.\end{cases}
G(Z) =(\ell-1)\sum_{j=0}^{m-2}\Big(\frac y\Delta(m-j)(m-j-1)F_j(Z) \:+2\frac v\Delta(m-j-1)(j+1)F_{j+1}(Z)+\frac{y'}\Delta(j+2)(j+1)F_{j+2}(Z)\Big)S^{m-2-j}T^j +2i\sum_{j=0}^{m-2}\Big((j+2)(j+1)\frac{\partial F_{j+2}}{\partial\tau}(Z) \:-(m-j-1)(j+1)\frac{\partial F_{j+1}}{\partial z}(Z)+(m-j)(m-j-1)\frac{\partial F_j}{\partial\tau'}(Z)\Big)S^{m-2-j}T^j.
p\mapsto H(\xi_{n}|X,\eta_{p})=\int_{X}H(\xi_{n}^{x})d\eta_{p}(x)=\int_{Y}\int_{\pi^{-1}(y)}H(\theta_{L_{x}\ast}\mathbb{P}_{\mu,n}^{y})d\eta_{p}^{y}(x)d\nu(y)
\left|\left(\delta_{a_1^{-1}e_1}-\delta_0\right)*\sigma_j\right|([-r,r]^s) =\left|\left(\delta_{N_j a_1^{-1}e_1}-\delta_0\right)*\rho_j\right|([-r,r]^s) &\le \left(\delta_{N_j a_1^{-1}e_1}+\delta_0\right)*\rho_j([-r,r]^s)\to 0,
\zeta^{\wedge}_\Gamma(s) = \frac{\zeta(5s-15) \zeta(9s-29) \zeta(10s-30) \zeta(11s-30) \zeta(15s-45) \zeta(21s-61)}{\zeta(10s-29) \zeta(30s-90)}\; \widetilde{\psi}(s),
S = \sum_{\substack{0\leq k\leq d-1 (y_0+kN,d)=1}}1 &= \sum_{0\leq k\leq d-1}\sum_{\delta \, \mid\, (y_0+kN,d)}\mu(\delta) &= \sum_{\delta \, \mid\, d}\mu(\delta) \sum_{\substack{0\leq k\leq d-1 \delta \, \mid \, y_0+kN}}1 = \sum_{\delta \, \mid\, d}\mu(\delta) \sum_{\substack{0\leq k\leq d-1 kN\equiv -y_0 \pmod{\delta}}}1.
\pi^{-1}(\underline{j}) = \{w_0\subsetneq \cdots \subsetneq w_m\,|\,c\in w_0c\leq j \}\pi^{-1}(\underline{i}) \cap V_+\pi^{-1}(\underline{j}) = \{w_0\subsetneq \cdots \subsetneq w_m\,|\, a\in w_0a\leq ib\in w_m b\leq j \}
\alpha^{(t)}_k(\theta_{t,j})_{j=1}^{g}= -\sum_{j=1}^g 2\Re\left(\sum_{\ell=0}^{k-1}\frac{Z(\gamma_{j,\ell}^{-1})}{k\gamma_{j,\ell}^{1-k}Z'(\gamma_j^{-1})}\frac{\gamma_{j,\ell}}{\gamma_{j,\ell}-1}e^{i(\theta_{t,j}+ 2\pi[\ell t]/k)}\right)
\lim_{n\rightarrow\infty}\mathbb{\int}_{0}^{\infty}h\left(y\right)n\mu_{1}\left(a_{n}^{-1}dy\right) =\lim_{n\rightarrow\infty}\mathbb{\int}_{a}^{b}\frac{\partial h\left(y\right)}{\partial y}n\bar{\mu}_{1}\left(a_{n}^{-1}y\right)dy =\mathbb{\int}_{a}^{b}\frac{\partial h\left(y\right)}{\partial y}\frac{y^{-\alpha}}{\Gamma\left(1-\alpha\right)}dy,
N_3(f,H)=-f-3S(H,f)+\frac{9}{2}S(H_3,3f)=\frac{c_a''+1}{4}=\begin{cases}-2a-1 \hbox{if $a\equiv 0\pmod 3$}, 2a+1 \hbox{if $a\equiv 2 \pmod 3$},\end{cases}
\bigcup\limits_{\left(F,s\right)\in \mathcal{F}^{p}_{j}}I_T\left(F,s,\delta_{p-1}\right)=\bigcup\limits_{\left(F,s\right)\in \mathcal{T}^{p}_{j}}I_T\left(F,s,\delta_{p-1}\right).
- u Q(X)^{-1}v = - \begin{pmatrix}u_1 u_2\end{pmatrix}\begin{pmatrix}Q_1(X)^{-1}0 0 Q_2(X)^{-1}\end{pmatrix}\begin{pmatrix}v_1 v_2\end{pmatrix}&= - u_1 Q_1(X)^{-1}v_1 - u_2 Q_2(X)^{-1}v_2 &= r_1(X) + r_2(X).
m^{(t)}_{v\to w}= \frac1{Z^{(t)}_{v\to w}}\left\langle f_v,\, \bigotimes_{w'\in\partial v\setminus\{w\}}m^{(t)}_{w'\to v}\right\rangle_{\mathcal{V}_{\partial v\setminus w}}m^{(t)}_{w\to v}= \frac1{Z^{(t)}_{w\to v}}\left\langle \bigotimes_{v'\in\partial w\setminus\{v\}}m^{(t-1)}_{v'\to w},\, g_w\right\rangle_{\mathcal{V}_{\partial w\setminus v}}
\lim_{h\to0}\frac{1}{h}(p_{ii}(t,t+h,u)-1) =-\lim_{h\to0}\sum_{j\neq i}\frac{1}{h}p_{ij}(t,t+h,u) &=-\sum_{j\neq i}\lim_{h\to0}\frac{1}{h}p_{ij}(t,t+h,u) &=-\sum_{j\neq i}q_{ij}(t,u) = q_{ii}(t,u),
\alpha_j \left. \Big( \frac{T^2c}{y^{\frac{3}{2}}}\Big)^{2j}g\Big(\frac{m^2 y}{N}\Big)\widehat{k^*}\Big(\frac{MTc}{2 \pi^2 \sqrt{y}}\Big)e\Big(\frac{-T^2c}{4 \pi^2 \sqrt{y}}\Big)y^{-{13}/{12}}\right|_{y_0}=: a_{2j}(y_0).
\sum_{n\in\mathbf{B}_N}e^{2\pi i \xi n}=\sum_{n=1}^N\psi(n)e^{2\pi i \xi n}+\sum_{n=1}^N\big(\Phi\big(\varphi_1(n)-\psi(n) \big)-\Phi\big(\varphi_1(n)\big)\big)e^{2\pi i \xi n}.
\texttt{wpterm[k]}= \max\Bigl\{\texttt{mag}(40(L+2)),\texttt{mag}\Bigl(68\frac{(L+2)^2\, A}{\varepsilon_3}\frac{\Gamma(k+\frac12)^{1/2}}{(B_1 a\sqrt{\pi})^k}\Bigr)\Bigr\}.
\mathbf{Pr}\left( \bigg\|\frac{d}{n\epsilon}\sum_{i=1}^{n}J(u_0 + \epsilon v_{0,i})v_{0,i}- \nabla \hat{J}(u_0) \bigg\| \geq t \right) \leq 2\exp\left( - \frac{cnt^2}{K^2}\right).
\delta_B ( \left[ v^3 \theta^2 \right] ) =-i v^2 \left( \left( \frac{d}{\phi d \phi}\right) f^{(2)}_1 v^i \phi^j \phi^k + f^{(2)}_1 v^i \delta^{jk}\right) (\epsilon \gamma^k \theta ) (\theta \gamma^{ij}\theta ) .
\frac{- i}{4 r}\, ( \delta_{A B}- e_{A}e_{B}) \, ( \delta_{s t}- E_{s}E_{t}) \, \left( \vec{\Theta}_{\hat{\alpha}}\times \vec{\Theta}_{\hat{\alpha^{\prime}}}\right)_{B}\gamma_{\hat{\alpha}\hat{\alpha}^{\prime}}^{s}\, ,
[\eta^H_{gh}\cdot (\eta^H_h)^{-1}] \circ (\lambda^H_h)^{-1}= \eta^H_g - [\eta^H_g\lambda^H_h - \eta^H_{gh}\cdot (\eta^H_h)^{-1}] \circ (\lambda^H_h)^{-1}= \eta^H_g + \omega_g^{-1}\circ \varDelta^H(g,h) \circ \lambda^H_g
1 \rightarrow [1,2]\Leftrightarrow (1,3)/[2,4]; -2 \rightarrow (1,3)/[2,4]\Leftrightarrow [1,6]/(3,5)/[2,4]=[1,6]/[2,4]/(3,5); 4 \rightarrow [1,6]/[2,4]/(3,5)\Leftrightarrow [1,6]/[2,8]/(4,7)/(3,5)=[1,6]/[2,8]/(3,5)/(4,7); -1 \rightarrow [1,6]/[2,8]/(3,5)/(4,7)\Leftrightarrow [1,6]/[2,8]/(3,9)/[5,10]/(4,7)=[1,6]/[2,8]/(3,9)/(4,7)/[5,10].
A^{T B A}= \frac{1}{4 \left( \sin \frac{\pi}{1 5}+ \sin \frac{\pi}{3}+ \sin \frac{2 \pi}{5}+ \sin \frac{3 \pi}{5}+ \sin \frac{2 \pi}{3}+ \sin \frac{1 4 \pi}{1 5}\right)}= . 0 6 1 7 3 \ldots
|\gamma_{\theta}|_{g(t_2)}\leq e^{Ct_2 \xi^{-1}\sqrt{S}}|\gamma_{\theta}|_{g(0)}\leq e^{CT_0 \xi^{-1}\sqrt{S}}|\gamma_{\theta}|_{g(0)}=e^{CT_0 \xi^{-1}\sqrt{S}}L.
\Omega_2\mid_{B_0}=\frac{dw_1\wedge dw_3}{f_{w_0}},~\Omega_2\mid_{B_1}=\frac{dw_3\wedge dw_0}{f_{w_1}},~\Omega_2\mid_{B_3}=\frac{dw_0\wedge dw_1}{f_{w_3}},
H'(r_0) =(n+1-2s)r_0^{n-2s}\int_{\partial B_1}|y|^{1-2s}\mu(r_0z)U^2(r_0z)dH^n(z) &+r_0^{n+1-2s}\int_{\partial B_1}|y|^{1-2s}\frac{\partial}{\partial r}|_{r=r_0}\mu(rz)U^2(r_0z)dH^n(z) &+2r_0^{n+1-2s}\int_{\partial B_1}|y|^{1-2s}\mu(r_0z)U(r_0z)U_{\nu}(r_0z)dH^n(z) &=(\frac{n+1-2s}{r_0}+O(1))H(r_0)+2\int_{\partial B_{r_0}}|y|^{1-2s}\mu(z)U(z)U_{\nu}(z)dH^n(z).
2\lambda mT_{m,0}+3m\frac{\partial T_{1,0}}{\partial M_0}\lambda^mM_0+3m\frac{\partial T_{-1,0}}{\partial M_0}\lambda^{m+2}M_0+\lambda^mT_{1,0}-\lambda^{m+2}T_{-1,0}=(m+1)T_{m+1,0}+(m-1)T_{m-1,0}\lambda^2.
\inf_{\substack{\eta \colon\, |\eta| \le \delta \log L, \eta(z) = 0 \,\forall\, z \in [-\ell_L, \ell_L]}}\, \P_\eta \left( \bigcup_{i \in [0, M_L -1]}G^{(0,T_i)}_{T_1}\cap \Lambda^{(0,T_i)}_{T_1}\right) \ge 1 - c e^{-c^{-1}L^{\varepsilon}}.
\begin{alignedat}{12}C_{\alpha,\frac{\theta}{\alpha}}:= C_{\alpha}\oplus \alpha C_{\frac{\theta}{\alpha}}\, \bar{\alpha}, C_{\alpha,\frac{\theta}{\alpha}}(g_1+\alpha g_2)= C_{\alpha}g_1+\alpha C_{\frac{\theta}{\alpha}}g_2=\alpha \bar{z}\,\bar{g_1}+\theta \bar{z}\,\bar{g_2}\end{alignedat}
s_n = n \sum \limits_{\substack{m_1,...,m_n \geq 0 m_1 + 2m_2 + ... + nm_n = i}}(-1)^{m_2 + m_4 + ...}\cdot \frac{(m_1 + m_2 + ... + m_n - 1)!}{m_1!m_2! \cdots m_n!}\cdot \sigma_1^{m_1}\cdots \sigma_n^{m_n}.
{\cal Z}= \int D{\bf x}( \tau ) D{\bf h}\exp \left\{- \int d^{3}x \left[ \frac{1}{4 \eta^{2}}H_{\mu \nu}^{2}+ 2 g_{m}^{2}{\bf h}^{2}- 2 \pi i h_{\mu}\delta_{\mu}\right] \right\},
\underline \tau = (\tau_0,\ldots,\tau_r), \underline \tau^{-}(j) = (\tau_0, \ldots, \tau_{j-1},\tau_{j+1},\ldots,\tau_r), \underline \tau [j] = (\tau_0,\ldots,-\tau_j,\ldots,\tau_r ) , | \underline \tau | = \tau_0+ \tau_1+\ldots+\tau_r,
\begin{array}{cccc}\bar{\varepsilon}_{i}\gamma^{A}\varepsilon^{\prime i}=0~~~ ~~~\bar{\rho}_{i}\tilde{\gamma}^{A}\rho^{\prime i}=0~~~ ~~~\bar{\varepsilon}_{i}\gamma^{[A}\tilde{\gamma}^{\mu]}\rho^{i}=0~~~ ~~~A=4,5\end{array}
n \leq \frac{q}{r}-(1+2\beta) \Longleftrightarrow q - (rn+r+\beta r) \geq \beta r & \Longrightarrow (q-(rn+r+\beta r))(q-1) - \beta r \geq 0 & \Longleftrightarrow q^2-q(rn+r+\beta r +1) + r(n+1) \geq 0 & \Longleftrightarrow -1 + \frac{n+1}{q}+\frac{q-1}{r}\geq \beta + n,
U=\mu_{1}\ell_{\bf i}+\mu_{2}\ell_{\bf j}+\mu_{3}\ell_{\bf k}+c_{0}u_{0}+c_{1}u_{\bf i}+c_{2}u_{\bf j}+c_{3}u_{\bf k}&\qquad\qquad\qquad\qquad\qquad\qquad\qquad+d_{1}F_{\bf i}+d_{2}F_{\bf j}+d_{3}F_{\bf k}=0.
\mathcal{F}\left( f\right) (\xi)=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{-i\xi\cdot x}f(x)dx,\quad\mathcal{F}^{-1}( g) (x):=\frac{1}{(2\pi)^{d/2}}\int_{\mathbb{R}^{d}}e^{i\xi\cdot x}g(\xi)d\xi,
\hat{\sigma}[H(u,u)g_{ij}](x,y)(\xi ,h_{ij}) =(h^i_l\xi_i\xi_su^su^l-\frac{1}{2}h(u,u)-u^s\xi_su^l\xi_ltr(h))\tilde{g}_{ij}, &=-\frac{1}{2}h(u,u)\tilde{g}_{ij}=-\partial_tlog \tilde{F}\tilde{g}_{ij}.
\theta(z,.)*E_{\mathfrak{a}}(.,1/2+it)= \sum_{\mathfrak{a}^{\prime}\in \mathcal{E}^{\prime}_{4k}}\alpha_{\mathfrak{a}\mathfrak{a}^{\prime}}(t) E_{\mathfrak{a}^{\prime}}(z,1/2+it/2),
\int_{\infty}d \Omega_{( 1 )}{}^{*}F_{0 \Omega_{( 1 )}}^{I}+ \int_{\infty}d \Omega_{( 2 )}{}^{*}F_{0 \Omega_{( 2 )}}^{I}= \int_{\infty}d \Omega_{( 3 )}{}^{*}F_{0 \Omega_{( 3 )}}^{I}.
A_r:=\frac{f\left(-q^2\right)^6}{q^2f\left(-q^{10}\right)^6}=\frac{\eta_1\left(\sqrt{-r}\right)^6}{\eta_1\left(5\sqrt{-r}\right)^6}=R\left(q^2\right)^{-5}-11-R\left(q^2\right)^5,
\sup_k \left\|\Delta \left(\sum_{i \ge 0}^ke^{-\lambda_it}\phi_i^2\right)\right\|_{H^{1, 2}}= \sup_k\left\|2\sum_{i \ge 0}^ke^{-\lambda_it}\left(-\lambda_i\phi_i^2 +|\nabla \phi_i|^2\right)\right\|_{H^{1, 2}}<\infty.
Err(x^{k}):=\max_{i}|x^{k}(\hat\omega_{i})\!-\!\Pi_{C(\hat\o_{i})}(x^{k}(\hat\omega_{i}))\!-\!\hat F(x^{k})(\hat\omega_{i})\!+\!\lambda^{k}(\hat\omega_{i}))|\!+\!\sum_{i=1}^{m}| x^{k}(\hat\omega_{i})\!-\!y^{k}(\hat\omega_{i})|^2p_{i}<\varepsilon
|3\Phi(\xi_1,\eta_1,-\xi_2,-\eta_2) + \tau_2 -\xi_2^3 - \eta_2^3| \lesssim \max (L_0,L_1), |\partial_{\eta_1}\Phi(\xi_1,\eta_1,-\xi_2,-\eta_2)| =|(\eta_1 + (\eta_2-\eta_1) )\, (\eta_1- (\eta_2-\eta_1))| & \geq (|\eta_1| - |\eta_2-\eta_1| )^2 \gtrsim A^{-2}N_1^2.
\sum_{n<R(n)^{{\mathcal{S}}/{\mathcal{T}}}}\dfrac{R(n)^{t}}{n^s}\ln\dfrac{R(n)^{\mathcal{S}}}{n^{\mathcal{T}}}=\sum_{n>R(n)^{{\mathcal{S}}/{\mathcal{T}}}}\dfrac{R(n)^{t}}{n^s}\ln\dfrac{n^{\mathcal{T}}}{R(n)^{\mathcal{S}}}.
\lambda_{0}\int_{0}^{T}v(t)f(u(t))~\!dt = \int_{0}^{T}\bigl{(}u''(t)f'(u(t))v(t) - v''(t)f(u(t)) \bigr{)}~\!dt = \Bigl{[}-v'(t)f(u(t)) \Bigr{]}_{t=0}^{t=T}+ \int_{0}^{T}\bigl{(}u''(t)f'(u(t))v(t) + v'(t)f'(u(t))u'(t) \bigr{)}~\!dt = \int_{0}^{T}\bigl{(}u''(t)f'(u(t))v(t) + v'(t)f'(u(t))u'(t) \bigr{)}~\!dt.
\binom{c_nkn}l\frac{(k(n-s-1))_{c_nkn-l}}{(kn)_{c_nkn}}=\frac{\binom{c_nkn}l}{\prod_{i=1}^l(kn-c_nkn+i)}\prod_{t=0}^{c_nkn-l-1}\frac{k(n-s-1)-t}{kn-t}.
(\tau \sigma)^{-1}\|\bar x - \frac{1}{k}\displaystyle\sum_{i=1}^k x^{i+1}\|^2 + \|\bar u - \hat u^k \|^2_{\mathcal{D}_1 + \mathcal{S}}+\|\bar v - \hat v^k\|^2_{\mathcal{Q}_{22}+ \mathcal{D}_{2}+ \mathcal{T}+ \sigma\mathcal{B}\mathcal{B}^*}+ \frac{1}{2}\|\bar w - \hat w^k\|^2_{\mathcal{Q}}\leq \Phi_1(\bar{u}, \bar{v}, \bar{x}).
dU_t = \left[\left[\left(\mathbf{S}- \mathbf{I}\right)d\mathbf{\Lambda}^\top_t\right] + d\mathbf{A}^{\dagger}_t \mathbf{L}- \mathbf{L}^{\dagger}\mathbf{S}d\mathbf{A}_t - \left(\dfrac{1}{2}\mathbf{L}^{\dagger}\mathbf{L}+iH\right)dt\right]U_t \;, U_0 = I
F(x + x_1 \varepsilon_1 + x_2 \varepsilon_2 + \cdots + x_n \varepsilon_n) = \sum_{S \subseteq \{1,2,\ldots,n\}}F^{(|S|)}(x) \prod_{i \in S}x_i \varepsilon_i = \sum_{S \subseteq \{1,2,\ldots,n\}}\left( \sum_{T \subseteq S}f_{S-T}^{(|T|)}(x) \prod_{i \in T}x_i \right) \varepsilon_S
| \dot A_R(t) | = | x_{R}(t) \wedge \nabla W(x_{R}(t)) | \le \frac{C_{+}}{r_{R}^{\beta}(t)}\le C_{+}\left[ \frac{2\alpha}{(2-\alpha)m}\right]^{\beta/(\alpha+2)}\dfrac{1}{(t-t^{+}_{R})^{2\beta/(\alpha+2)}}
[x_\alpha^+\otimes t^{s}, (r+1)Y_\alpha[s]^{(r+1)}] =[x_\alpha^+\otimes t^{s}, Y_\alpha[s]Y_\alpha[s]^{(r)}] &=[x_\alpha^+\otimes t^{s},Y_\alpha[s]]Y_\alpha[s]^{(r)}+Y_\alpha[s][x_\alpha^+\otimes t^{s},Y_\alpha[s]^{(r)}]
A_{i}^{\gamma}\,^{T}({\bf r}) ={\delta}_{{i}\,{\gamma}}\,{\cal T}_{A}( r ) + \frac{r_{i}\, r_{\gamma}}{r^{2}}\,{\cal T}_{B}( r ) +{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\cal T}_{C}( r )