harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
w(r,t) =w(r_{0}(t),t)-\int_{r}^{r_{0}(t)}w_{r}(\rho,t) d\rho \leqslant K +\int_{r}^{r_{0}(t)}\vert w_{r}(\rho,t)\vert d\rho \leqslant K+r^{1-N}\int_{r}^{r_{0}(t)}\rho^{N-1}\vert w_{r}(\rho,t)\vert d \rho \leqslant K+CKr^{1-N}.
\langle \Phi_{1}\rangle = \left( \frac{1}{\sqrt{2}}, 0 , 0 \right)^{T}, \quad \langle \Phi_{2}\rangle = \left( 0 , 0 , 0 \right)^{T}, \quad \langle \Phi_{3}\rangle = \left( 0 , 0 , 0 \right)^{T}.
h^{T}(t, \overline{\eta}) = \overline{S}^{V}_{T-t}g(\overline{\eta})\;\;\mbox{and}\; \overline{S}^{h^{T}}_{t,s}f(\overline{\eta}) = \frac{\overline{S}^{V}_{s-t}[f\overline{S}^{V}_{T-s}g](\overline{\eta})}{\overline{S}^{V}_{T-t}g(\overline{\eta})}\;.
\overline{v}=\frac{s_{1}+s_{2}+s_{3}+...+s_{n}}{t_{1}+t_{2}+t_{3}+...+t_{n}}=\frac{s_{1}+s_{2}+s_{3}+...+s_{n}}{\frac{s_{1}}{v_{1}}+\frac{s_{2}}{v_{2}}+\frac{s_{3}}{v_{3}}+...+\frac{s_{n}}{v_{n}}}
\mathcal{F}(\boldsymbol{D}^{-\sigma}u ) = (2\pi i\xi)^{-\sigma}\mathcal{F}(u) \mathcal{F}(\boldsymbol{D}^{-\sigma*}u )= (-2\pi i\xi)^{-\sigma}\mathcal{F}(u),\xi \ne 0,
\zeta(\mathcal{L}_n, s) = \pi^{2s-1/2}\frac{\Gamma(1/2-s)}{\Gamma(1-s)}n^{1-2s}+2\zeta_R(2s)+\frac{2s}{3}\pi^2 \zeta_R(2s-2)n^{-2}+o(n^{-2}) &= \pi^{2s-1/2}\frac{\Gamma(1/2-s)}{\Gamma(1-s)}n^{1-2s}+\zeta(\Delta_{\mathbb{S}^1},s)+\frac{s}{3}\pi^2 \zeta (\Delta_{\mathbb{S}^1},s-1)n^{-2}+o(n^{-2}).
\left( \begin{array}{c}{{x^{\prime}}}\\{{t^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\frac{1}{\sqrt{1 - \beta^{2}}}}}&{{\frac{\beta}{\sqrt{1 - \beta^{2}}}}}\\{{\frac{\beta}{\sqrt{1 - \beta^{2}}}}}&{{\frac{1}{\sqrt{1 - \beta^{2}}}}}\\ \end{array}\right) \left( \begin{array}{c}{{x}}\\{{t}}\\ \end{array}\right)
\deg (\sigma'^{m+1}(z_2)) = \deg((\tau_1\pi\tau_2\cdots\pi\tau_s\pi)^{m+1}(z_2)) =(d_1\cdots d_s)^{m+1}\geq (d_1\cdots d_s)^{m}=2\deg (\sigma'^{m}(z_2)).
\mathrm{det}\left(\begin{array}{llll}\Lambda f_1 \Lambda f_{2}\Lambda g \Lambda^2 f_1 \Lambda^2 f_{2}\Lambda^2 g \Lambda^3 f_1 \Lambda^3 f_{2}\Lambda^3 g \end{array}\right)=h'''P(z,w, h') +R(z,w, h',h'')=0~~\Gamma.
P(B) = P(N(t) - N(t_{n}+h_{n}) = 0) P(N(t_{1}) - N(0) = 0)\prod_{i=1}^{n-1}P(N(t_{i+1}) - N(t_{i}+ h_{i}) = 0) = e^{-m(t_{1})}e^{-(m(t) - m(t_{n}+ h_{n}))}\prod_{i=1}^{n-1}e^{-(m(t_{i+1})- m(t_{i}+ h_{i}))}= exp\left(-\left[m(t) + \sum_{i=1}^{n}m(t_{i}) - \sum_{i=1}^{n}m(t_{i}+ h_{i})\right]\right)
\mu < \omega<{\frac{\Big(\kappa m-{\frac{3}{2}}\sigma^2\Big)\mu+{\frac{4}{\overline t}}\mu\alpha_2-\beta}{\kappa m-{\frac{\sigma^2}{2}}}}=\mu+{\frac{\gamma\sigma^2\mu-\beta}{\kappa m-{\frac{\sigma^2}{2}}}},
\frac{\omega}{k}=(\frac{1}{2}(v_{A}^{2}+v_{s}^{2})\pm\frac{1}{2}\sqrt{(v_{A}^{2}+v_{s}^{2})^{2}-4v_{s}^{2}v_{A}^{2}cos^{2}\theta})^{\frac{1}{2}}
\cot^3(\eta ) \cot (2 \eta ) = \frac{\cot^4(\eta )}{8}-\frac{3 \cot^2(\eta )}{4}+\frac{\csc^4(\eta )}{8}-\frac{\csc^2(\eta )}{4}+\frac{1}{4}\cot^2(\eta ) \csc^2(\eta )+\frac{1}{8},
|N_t(e)| \le (d-2)\cdot\iota_{s,d}-1+2(t-2s+1)((d-2)\cdot\iota_{s,d}+1)+2\iota_{s,d}& = (2t-4s+3)(d-1)^s-2+2\frac{(d-1)^s-1}{d-2}.
\tilde{V}_{\mu\nu'\rho'}^{(1)}=-2\pi^2\int d^4u'\,\left(\frac{(u'-y')_\sigma\,(u'-y')_{\rho'}}{(u'-y')^6}-\frac{1}{2}\frac{\delta_{\sigma\rho'}}{(u'-y')^4}\right)\frac{(u'-x')_\lambda}{(u'-x')^4}\,{\rm Tr}\,\gamma_5\gamma_\sigma\gamma_\lambda\gamma_{\nu'}\gamma_\mu \, .
G_{1, \varphi}(x) =\left(\begin{array}{cc}\frac{x - 9 i + 4 i e^{- i \varphi x}+ 4 i e^{i \varphi x}}{x + i}\frac{12 i - 4 i e^{- i \varphi x}- 8 i e^{i \varphi x}}{x + i}& \frac{- 6 i + 4 i e^{- i \varphi x}+ 2 i e^{i \varphi x}}{x + i}\frac{x + 9 i - 4 i e^{- i \varphi x}- 4 i e^{i \varphi x}}{x + i}\end{array}\right).
\tilde{g}=- \frac{T_{m}^{-1}g_{m}}{(T_{m}^{-1}g_{m},x)}=- \frac{\Psi_{m}g_{m}\Psi_{m}^{*}}{\left(\Psi_{m}g_{m}\Psi_{m}^{*}, \Psi_{m}\begin{pmatrix}1 0 0 -\mu\end{pmatrix}\Psi^{*}_{m}\right)}= \frac{\Psi_{m}g_{m}\Psi_{m}^{*}}{1-\mu|\psi|^{2}}.
\Psi_A=\,\sum^{N_A}_{k=1}\Big(n_{1k}(\varphi^{-1}_k(x),z_k)\,\exp(\imath \,\vartheta_{1k}(\varphi^{-1}_k(x),z_k))\break +n_{2k}(\varphi^{-1}_{k}(x),z_k)\,\exp(\imath \,\vartheta_{2k}(\varphi^{-1}_k(x),z_k))\Big)|\varphi^{-1}_{k}(x),z_k\rangle,
\left| K_{\theta, \tilde{\theta}_{(z,\eta), (z_0,\eta_0)}, \Phi}( (y,\omega), (z,\eta)) \right| = \sqrt{\frac{w(\Phi(\eta))}{w(\Phi(\omega))}}\cdot L_{\Phi(\eta)}[\theta, \tilde{\theta}_{(z,\eta), (z_0, \eta_0)}](A^{T}(\Phi(\eta)) \langle z-y \rangle,\Phi(\omega)-\Phi(\eta)),
S_{\mathrm{n o b o u n d}}={\frac{3 \pi M_{\mathrm{P}}^{2}}{2}}\int_{0}^{\infty}d r R ( r ) ( 1 - [ R^{\prime}( r ) ]^{2}) ={\frac{3 \pi^{2}}{8}}M_{\mathrm{P}}^{2}R^{2}( 0 ) \ .
{\textbf{W}}{\rm{=}}\left[{\underbrace{\frac{{\sqrt{\textbf{R}}{{\left[{1, - k}\right]}^T}}}{{\left\|{\left[{1, - k}\right]\sqrt{\textbf{R}}}\right\|}}}_{{{\textbf{w}}_1}},{{\textbf{w}}_2}}\right]
\begin{aligned}|x (t)-z_1(t)| \le \,|x(0)-z_1(0)| + t\gamma^2 \bar{k}_3(L_r,\delta) &+ \gamma \int_0^tL_r|x(\tau)-z_1(\tau)| + |\bar{y}(\tau)-z_2(\tau)|d\tau\end{aligned}
\Pi_{L}^{\mu \nu}= - \frac{4}{( 2 \pi )^{2}}\int_{0}^{1}\frac{d x}{( 1 + x )^{2}}e x p [ \frac{x}{1 + x}\frac{p^{2}}{\Lambda^{2}}] ( g^{\mu \nu}- \varepsilon^{\mu \nu}) ,
r_0:=2 \cdot (40 C_0\rho_0^2)^{4/5}, begin{align*}3pt]\delta_0=\min\bigg(L_0;\frac{\rho_0}{2};\frac{r_0^{5/4}}{2\rho_0^{1/4}}\bigg)=\min\bigg(L_0;{\rho_0}/{2};20\cdot 2^{5/4}\rho_0^{7/4}C_0\bigg),
\left|\langle(\lambda_0I-A)a,v\rangle_{X,X^\ast}\right| \ge\mathrm{Re}\left\langle(\lambda_0I-A)a,v\right\rangle_{X,X^\ast}&=\lambda_0\|a\|_{X}^2-\mathrm{Re}\langle Aa,v\rangle_{X,X^\ast}&\ge(\lambda_0-\omega)\|a\|_{X}^2.
B_{n}( F , G ) : = \sum \Theta^{i_{1}j_{1}}\Theta^{i_{2}j_{2}}\dots \Theta^{i_{n}j_{n}}\partial_{i_{1}}\partial_{i_{2}}\dots \partial_{i_{n}}F \ \partial_{j_{1}}\partial_{j_{2}}\dots \partial_{j_{n}}G ,
L_a f(\xi) =F J_a^* J_a F^* f(\xi) &=(2\pi)^{-d}\sum_{x\in\mathbb{Z}^d}\int_{\mathbb{T}^d}e^{i(\varphi_a(x,\xi)-\varphi_a(x,\eta))}f(\eta) d\eta, \tilde L_a f(\xi) = (2\pi)^{-d}\int_{\mathbb{R}^d}\int_{\mathbb{T}^d}e^{i(\varphi_a(x,\xi)-\varphi_a(x,\eta))}f(\eta) d\eta dx .
\|(\nabla_{X}J)Y\|^{2}= -<R_{X,Y}X,Y>+<R_{X,Y}JX,JY> &=<\alpha(X,Y),\alpha((Y,X)>-<\alpha(X,X),\alpha(Y,Y)> &-<\alpha(X,JY),\alpha(Y,JX)>+<\alpha(X,JX),\alpha(Y,JY) &=\|\eta\|^{2}(-<X,Y>^{2}+<X,X><Y,Y> &+<X,JY><Y,JX>) &=\|\eta\|^{2}(\|X\|^{2}\|Y\|^{2}-<X,Y>^{2}-<X,Y>^{2}),
X_{us}(\lambda) = C\cup Sx_{s_2}\cup Sx_{s_1}\;,\; \lambda\in\Lambda^{++}\;; & X_{us}(k\omega_1) = C\cup Sx_{s_2}\;,\; k\geq 1 \;; & X_{us}(k\omega_2) = C\cup Sx_{s_1}\;,\; k\geq 1 \;.
\int_{-\infty}^\infty d \tau{\cal L}=\frac{16 C_4}{15}\frac{v^3}{b^6}+ \frac{256 C_6}{315}\frac{v^5}{b^{10}}+ \frac{2048 C_8}{3003}\frac{v^7}{b^{14}}+ O(v^9).
U\left({-{\tfrac{1}{2}}\gamma\alpha^{2},0}\right) =\pi^{-1/2}2^{\left({\gamma\alpha^{2}-1}\right) /4}\Gamma\left({{\frac{1}{4}}\gamma\alpha^{2}+{\frac{1}{4}}}\right) \sin\left({{\frac{1}{4}}\gamma\alpha^{2}\pi+{\frac{1}{4}}\pi}\right) ,
\frac{1}{4 \xi}( \partial_{\mu}A_{\mu}^{-})^{2}\ + \{\bf u}D_{\mu}^{-}\partial_{\mu}\bar{\bf v}\ + \, \frac{1}{4 \alpha}( D_{\mu}^{-}\Phi_{\mu})^{2}\ + \{\bf c}D_{\mu}^{+}D_{\mu}^{-}\bar{\bf b}\ + \, b a r{\bf b}D_{\mu}^{+}D_{\mu}^{-}{\bf u}
\frac{\partial}{\partial t}T\left( x,t\right) =-\frac{\partial}{\partial x}q\left( x,t\right) ,\;\;x\in \mathbb{R},\;t>0, \int_{0}^{1}\phi \left( \gamma \right) \,{}_{0}^{c}\mathrm{D}_{t}^{\gamma}q\left( x,t\right) \mathrm{d}\gamma =-\frac{\partial}{\partial x}T\left(x,t\right) ,\;\;x\in \mathbb{R},\;t>0,
\displaystyle\int_{\Omega}u_{\alpha}^{2}\left\{-div\left[\displaystyle\frac{\nabla c}{(c+\alpha)^{2}}\right]\right\}dx=\frac{1}{\lambda_{\alpha}}\displaystyle\int_{\Omega}\displaystyle\frac{|\nabla u_{\alpha}|^{2}}{c+\alpha}dx, \ \forall \ \alpha\in \mathcal{A}.
LP_{t}=-S\Phi(-a_{1}(S,M))+Me^{-r\tau}\Phi(-a_{2}(S,M))+\frac{S\sigma^{2}}{2r}(\Phi(a_{1}(S,M))-e^{-r\tau}(M/S)^{\frac{2r}{\sigma^{2}}}\Phi(a_{3}(S,M))).
E_1 \equiv-\frac{1}{N}\sum_{n=1}^{N-1}\sum_{l'=2}^{l-1}\sum_{l''=1}^{l'-1}e^{2il''\phi_n}&=-\frac{1}{N}\sum_{l'=2}^{l-1}\sum_{l''=1}^{l'-1}\left(\frac{1-e^{i2\pi l''}}{1-e^{i \pi y'' /N}}-1\right) &=\frac{1}{N}\sum_{l'=2}^{l-1}\sum_{l''=1}^{l'-1}1,
2\pi i\cdot f(z)=\int_{e^{-i\alpha}[0,+\infty)}\frac{f(\zeta) e^{p(\zeta-z)}}{\zeta-z}d\zeta-\int_{e^{i\alpha}[0,+\infty)}\frac{f(\zeta) e^{p(\zeta-z)}}{\zeta-z}d\zeta, z\in \Delta.
\mathrm{E}_{\theta,\varepsilon^{2}}\sum_{i=1}^{D}(\theta_{i}- \hat{\theta}_{\mathrm{MA},\beta, i})^{2}\leq \mathrm{E}_{\theta,\varepsilon^{2}}\sum_{d=1}^{D}\tilde{w}_{d}^{(D)}\hat{r}_{d}^{(D)}+ D\varepsilon^{2},
a\left( t\right) =W_{0}^{\ast}\left( t\right) aW_{0}\left( t\right)=a\left( 0:t\right) a^{\dag}\left( t\right) =W_{0}^{\ast}\left( t\right) a^{\dag}W_{0}\left( t\right) =a^{\dag}\left( 0:t\right)
Y_{m_1,m_2}(A) := \frac{1}{2}\sum_{\theta \in \{0,1\}^2}\prod_{j \in \mathbb{T}_{m_1,m_2}}\left| \alpha + \beta \exp \left\{2 \pi i \frac{j_1 + \theta_1 /2}{m_1}\right\}+ \gamma \exp \left\{2 \pi i \frac{j_2 + \theta_2 /2}{m_2}\right\}\right|.
V_{p}:= \Big\{u \in L^{2p}([0,T];L_{e}^{2}(\Omega)) \cap L^{2}([0,T];H_{e}^{\beta}(\Omega)) & g_{1-\alpha}*(u-u_{0}) \in C([0,T];L_{e}^{2}(\Omega)), (g_{1-\alpha}*(u-u_{0}))|_{t=0}= 0 \Big\},
\Lambda_1f'= g_1^{\frac{1}{4}}(g^{+,-}\big)^{-\frac{1}{2}}\Lambda' f +\Big(\frac{\partial g_1^{\frac{1}{4}}}{\partial y_n}+\sum_{j=1}^{n-1}g_0^{+,j}\frac{\partial g_1^{\frac{1}{4}}}{\partial y_j}\Big)g_1^{-\frac{1}{4}}f\Big|_{y_n=0}.
{\cal F}\left(L,\beta,\nu\right)= \frac{8 \pi^4}{\sqrt{2 \pi \alpha'}}\int_0^\infty \frac{ds}{s^{9/2}}\;e^{s-L^2/2s\alpha'}\Theta_2 \left(\nu \left \vert \frac{i\beta^2 s}{2\pi^3 \alpha'}\right. \right)\prod_{n=1}^\infty\left( \frac{1-e^{-(2n+1)s}}{1-e^{-2ns}}\right)^8,
\Pi_{\mu \nu}^{- +}= U_{\nu}^{+}( T_{\nu}^{+}U_{\mu}^{-}) ( T_{\mu}^{-}T_{\nu}^{+}U_{\mu}^{+}) ( T_{\nu}^{+}U_{\nu}^{-}) , \Pi_{\mu \nu}^{+ -}= \Pi_{\mu \nu}^{- +}( + \leftrightarrow - ) , \mu \neq \nu
\hphantom{{}={}}((uY)^{r-1})^i{}_{\alpha_1,\ldots,\alpha_{r-1}}* =u^i{}_{\alpha_1,\ldots,\alpha_{r-1},l}Y^l * \hphantom{{}={}}+(\textit{terms which do not involve $u^i{}_{j_1,\ldots,j_r}$ or $Y^i{}_{j_1,\ldots,j_{r-1}}$}) * \hphantom{{}={}}+u^i{}_lY^l{}_{\alpha_1,\ldots,\alpha_{r-1}},
S=\int d^{4}x(-2i\tilde{E}_{i}^{b}L_{t}A_{b}^{i}-2i(t^{a}A_{a}^{i})D_{b}\tilde{E}_{i}^{b}+2iN^{a}\tilde{E}_{i}^{b}F_{ab}^{i}+(\frac{N}{\sqrt{q}})\epsilon_{ijk}\tilde{E}_{i}^{a}\tilde{E}_{j}^{b}F_{ab}^{k})
p(t, y|x) = \frac{1}{2\pi}\int_{-\pi}^{\pi}dk e^{ik(y-x)}e^{-(1-\cos k)t}&= \frac{1}{2\pi a}\int_{-a\pi}^{a\pi}d\theta e^{i\theta(y-x)/a}\exp\left\{-\left( 1-\cos \left( \frac{\theta}{a}\right) \right) t \right\}&= \int_0^1 du \cos(u\pi(y-x)) \exp\left\{-\left( 1-\cos(u\pi) \right) t \right\}
\sigma_{\varepsilon}(2(TS-ST^{\ast})) =\sigma_{\varepsilon}([I\bullet T,S]_{\ast})=\sigma_{\varepsilon}([\psi(I)\bullet \psi(T),\psi(S)]_{\ast}) &=\sigma_{\varepsilon}(2(\psi(T)\psi(S)-\psi(S)\psi(T)^{\ast})).
\widetilde Z ( g ) ={\frac{1}{\sqrt{\pi}}}\int_{- \infty}^{\infty}d q e^{q^{2}- g q^{4}}={\frac{1}{( 2 g )^{1 / 4}}}e^{1 / ( 8 g )}D_{- 1 / 2}( - 1 / \sqrt{2 g}) ,
F_n(\alpha,\beta)=\frac{1}{\alpha^n}\prod_{k=0}^{n-1}\frac{\Gamma(\beta/\alpha + k)(k!)^2}{\Gamma(\beta/\alpha +n+k)}= \frac{1}{\alpha^n}\prod_{k=0}^{n-1}(k!)^2\prod_{j=0}^{n-1}\frac{\alpha}{\alpha(k+j)+\beta},
\begin{pmatrix}\tilde{V}_n \tilde{V}_{n-1}\dots \tilde{V}_1 \end{pmatrix}^t =M_{\tilde{\mathcal{V}}}\begin{pmatrix}\partial_{u_1}\partial_{u_2}\dots \partial_{u_n}\end{pmatrix}^t,\qquad\! \quad\! M_{\tilde{\mathcal{V}}}=M_{\mathcal{V}}\left({\partial x\over \partial u}\right)^{-1}\!.\!\!\!
\frac{\partial \alpha_j^{rv-1}(z_j,s)}{\partial z_j^p}=\frac{1}{2\pi i}\int_{|\xi-z_j^p|=\delta}\frac{\alpha_j^{rv-1}(z_j^1,...,\overset{}{\xi},...,z_j^n,s)}{(\xi-z_j^p)^2}d\xi
F_1(\Omega_n)= \frac{1}{2h}\int_{U_{h-t}(\Gamma_\infty)}j_1(p_n(y),\nabla b_{\Omega_n}(p_n(y)),H_{\Gamma_n}(p_n(y))) \, \det(dT_n) \, dy +\frac{1}{2h}\int_{U_{h}(\Gamma_n)\setminus{U_{h-t}(\Gamma_\infty)}}j_1(p_n(y),\nabla b_{\Omega_n}(p_n(y)),H_{\Gamma_n}(p_n(y))) \, \det(dT_n) \, dy.
C = - s A^{L^{\prime}}\; \; , \; \; \hspace{1 e m}\bar{{\cal P}}= - i s A_{0}\; \; , \; \; \hspace{1 e m}\mathrm{\boldmath ~ \p i ~}^{L}= \frac{\nabla}{\nabla^{2}}( s{\cal P}+ \rho^{( 2 )}+ \rho^{( 3 )}) \; \; , \; \; \hspace{1 e m}\pi_{0}= i s \bar{C}\; \; .
\lim_{r \rightarrow 0}\frac{\log \mu ( B^T_{\xi, r}( q_\xi (x) ) )}{\log r}= \lim_{r \rightarrow 0}\frac{\log \mu ( \Sigma_r ( \xi , x ) )}{\log r}= \frac{h_\nu - h_{\mu}^{ss}}{- \int \log \lambda \, d \nu}
\frac{1}{N \theta}\log \left( \int_{\mathbb{R}^{d \times N}}\exp \left( - \beta \mathcal{H}^{*}_{N}\right) d \pi^{\otimes N}\right) = \frac{1}{N \theta}\log \left( \int_{\mathbb{R}^{d \times N}}\exp \left( - \beta \mathcal{H}_{N}\right) d X_{N}\right) - \frac{\log z^{*}}{\theta}.
\begin{array}{rcl}Z[J_\Phi,J_\Sigma,K_\Phi,K_\Sigma] \!\! = \!\! \int{\cal D}\Phi \,{\cal D}\Sigma \, \exp [ i \int_x L(\Phi,\Sigma)+i \int_x (J_\Phi \Phi + J_\Sigma \Sigma) &{}~~~~~~~~~~~~~~~~~+i \int_{x,y}(\Phi K_\Phi \Phi/2 + \Sigma K_\Sigma \Sigma/2) ]~,\end{array}
\|\hat{F}_2^l\|_{0,r}\le \|\hat{F}_2^{l+1}\|_{0,r}+c c_0^{-1}M^{-l-1}\alpha^{-4}\le c c_0^{-1}\sum_{j=l}^{N_+-1}M^{-j-1}\alpha^{-4}&\le c c_0^{-1}M^{-l-1}\alpha^{-4},\ (\forall r\in\{0,1\}).
\gamma_{p}:=C_{p}^{(2)}+\frac{C_{p}^{(1)}}{1+\frac{1}{\pi p}}-\gamma-\frac{2e^{2\pi p}\,Q_{2\pi p}(0)}{1+\frac{1}{\pi p}}-\log(2\pi)\left(1-\frac{1}{1+\frac{1}{\pi p}}\right),
\nu_3( k_i \ell_i m_i ) =\begin{bmatrix}k_{i,0}\ell_{i,0}m_{i,0}k_{i,0}\ell_{i,0}m_{i,1}+ k_{i,0}\ell_{i,1}m_{i,0}+ k_{i,1}\ell_{i,0}m_{i,0}\vdots k_{i,n-1}\ell_{i,n-1}m_{i,n}+ k_{i,n-1}\ell_{i,n}m_{i,n-1}+ k_{i,n}\ell_{i,n-1}m_{i,n-1}k_{i,n}\ell_{i,n}m_{i,n}\end{bmatrix}
D_{\vec f}(\vec{\textbf{P}}, \vec{\textbf{Q}};i )= \int_{X}\left[f_1 \left(\frac{p_1}{q_1}\right) q_1 \right]^{\frac{i}{n}}\left[f_2 \left(\frac{p_2}{q_2}\right) q_2 \right]^{\frac{n-i}{n}}d\mu.
0=\frac{f(x_{0}+h)-f(x_{0})}{h}+\frac{f(x_{0}-h)-f(x_{0})}{h}-2\frac{f^{(2)}(x_{0})}{2!}h-2\frac{f^{(4)}(x_{0})}{4!}h^{3}+\cdot\cdot\cdot
\Phi((a,i'))=(a-\sum_{k=1}^{n+i}\alpha^{-kh}(g),-(n+i)\cdot h)=(a-\sum_{k=1}^{n+i}\alpha^{-kh}(g),-(n+i)\cdot h)= (a-\sum_{k=1}^{n}\alpha^{-kh}(g)-\sum_{k=n+1}^{n+i}\alpha^{-kh}(g),-i\cdot h)=(a-\sum_{k=n+1}^{n+i}\alpha^{-kh}(g),-i\cdot h)= (a-\sum_{k=1}^{i}\alpha^{-(k+n)h}(g),-i\cdot h)= (a-\sum_{k=1}^{i}\alpha^{-kh}(g),-i\cdot h)=\Phi((a,i)),
l_1,\ldots,l_r\in\left\{1,2,\ldots,n\right\},\,\,\, 1\leq j_1\leq\ldots\leq j_s\leq n,\,\,\, u_{1},\ldots,u_s\in\mathbb{Z},\,\,\, r,s\geq 0, & u_{l}\geq u_{m}l\leq m j_l =j_m.
\left|e^{-\frac{1}{2}\int_0^tB_i^2(s)ds}f\right|_q\leq \mathcal{B}|f|_q,\ \forall f\in e^{\int_0^tB_i^2(s)ds}[L^q(\mathbb{R}^d)],\t\geq0,\i=1,2,...,N,
c^{(1)}(1,1):=1, &c^{(n)}(l,m_1,m_2,\cdots,m_l) &:=\sum_{i=1}^l\big(1_{l\ge 2}1_{m_i=1}c^{(n-1)}(l-1,m_1,\cdots,m_{i-1},\overbrace{m_i},m_{i+1},\cdots,m_l) &\qquad+1_{l\le n-1}1_{m_i\neq 1}c^{(n-1)}(l,m_1,\cdots,m_{i-1},m_i-1,m_{i+1},\cdots,m_l)\big).
\big[\partial_{ij}y_0, ~ \partial_i\vec b_0, ~ \tilde d_0\big](x') =\big[\partial_1y_0, ~ \partial_2\vec y_0, ~ \vec b_0\big](x')\cdot\left[\begin{array}{ccc}\Gamma_{ij}^1 \Gamma_{i3}^1 \Gamma_{33}^1 \Gamma_{ij}^2 \Gamma_{i3}^2 \Gamma_{33}^2 \Gamma_{ij}^3 \Gamma_{i3}^3 \Gamma_{33}^3 \end{array}\right](x',0) \qquad\mbox{for}\; i,j=1,2,
\sum_{X_0 \leq n < X_0 + X}\nu(n) = \sum_{X_0 < n \leq X_0 + X}\!\! \big(\sum_{\substack{d | n d \leq D}}\lambda_d\big)^2 = \sum_{d,d' \leq D}\lambda_d \lambda_{d'}\!\!\!\! \sum_{\substack{X_0 \leq n < X_0 + X d, d' | n}}1.
\varphi(\alpha_h(g))(s) = \dot{\alpha}_{s^{-1}}(\alpha_h(g)) = \dot{\alpha}_{s^{-1}\pi(h)}(g) = \dot{\alpha}_{(\pi(h)^{-1}s)^{-1}}(g) &= \varphi(g)(\pi(h)^{-1}s) = \beta_h(\varphi(g))(s)\ ,
\|g_1 - g_2\|_{L_{2}(P)}= \sqrt{P((f_1(X) - Y)^2 - (f_2(X) - Y)^2)^2}&\le 2\sqrt{P((f_1(X) - f_2(X))^2}&= 2\|f_1 - f_2\|_{L_{2}(P)}.
{\cal F}^{\prime \prime}D^{\alpha \beta}{\cal A}| = D^{\alpha \beta}{\cal F}^{\prime}| -{\cal F}^{\prime \prime \prime}D^{\alpha{\mathbf i}}{\cal A}D_{\mathbf i}^{\beta}{\cal A}| \stackrel{| x | \to \infty}{\longrightarrow}D^{\alpha \beta}{\cal F}^{\prime}| .
[p_{r(0^{(n)}1^{(n)}0^{(n)})}]_0 = [p_{r(0^{(n)}1^{(n)}0^{(n)}0^{(n)})}]_0 + [p_{r(0^{(n)}1^{(n)}0^{(n)}1^{(n)})}]_0 & = [p_{r(1^{(n)}0^{(n)}1^{(n)}0^{(n)}0^{(n)}1^{(n)})}]_0 + [p_{r(0^{(n+1)}0^{(n+1)})}]_0 & = [p_{r(1^{(n+1)}1^{(n+1)}0^{(n+1)})}]_0 + [p_{r(0^{(n+1)}0^{(n+1)})}]_0 & = 2 [p_{r(0^{(n+1)}0^{(n+1)})}]_0,
V(\bar{\vec{x}})=L(\vec{z}^{*}(\bar{\vec{x}}),\boldsymbol{\lambda}^{*}(\bar{\vec{x}}),\bar{\vec{x}})=f(\vec{z}^{*}(\bar{\vec{x}}),\bar{\vec{x}}) + [\boldsymbol{\lambda}^{*}(\bar{\vec{x}})]^{\top}\vec{g}(\vec{z}^{*}(\bar{\vec{x}}),\bar{\vec{x}}).
|f_3| \lesssim b_c^{\frac{5}{2}}\bigg{(}\int_{y<\kappa B}\psi_B(|\varepsilon|^p+\varepsilon^2)+e^{-\frac{\kappa B}{20}}\|\varepsilon\|_{L^{\infty}}^2\bigg{)}&\leq \frac{\mu_0}{100}\int(\varepsilon_y^2+\varepsilon^2)\varphi'_B+Cb_c^{\frac{7}{2}}.
\frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{h-h'}}{1 - e^{- (\lambda+\theta_1 \pi i +Tw)}}= \mathrm{1}_{h'=h}+ \frac{1}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{h-h'}e^{- (\lambda+\theta_1 \pi i+Tw)}}{1 - e^{- (\lambda+\theta_1 \pi i +Tw)}}.
A_{4}= \int_{- L / 2}^{L / 2}\! \! \! \! d x \, \left[ \left. \mathrm{I m}\left( \phi \partial_{x}\bar{\phi}\right) \right|_{t = - T / 2}- \left. \mathrm{I m}\left( \phi \partial_{x}\bar{\phi}\right) \right|_{t = T / 2}\right] .
Q_2 =\mathrm{P}\left( z_m>x_m, x_m< \frac{\epsilon_{1,m}}{\rho}, \frac{\epsilon_{1,m}}{\rho}> x_m \right) \leq \mathrm{P}\left( x_m< \frac{\epsilon_{1,m}}{\rho}\right) \sim \frac{1}{\rho^{M-m+1}},
\int \frac{n^2}{u^{n+1}}\int_0^{u}r^{n-1}K_A(r) \mathrm{d}r \mathrm{d}u = \frac{-n}{u^n}\int_0^{u}r^{n-1}K_A(r) \mathrm{d}r + n \int \frac{K_A(u)}{u}\mathrm{d}u.
a_{{h}}:=H_j^\pi(\sigma_{{h}})-H_j^\pi(\sigma_{{h}-1}) & = \left( H_j^\pi(\sigma_{{h-1}}) +\langle x_{m+v-h}\lambda, \alpha_j^\vee \rangle(\sigma_{{h}}-\sigma_{{h-1}}) \right)-H_j^\pi(\sigma_{{h-1}}) & = \langle x_{m+v-h}\lambda, \alpha_j^\vee \rangle(\sigma_{{h}}- \sigma_{{h-1}}).
\sum_{i}\frac{1}{p_{i}}-\sum_{i<j}\frac{1}{p_{i}p_{j}}+\sum_{i<j<k}\frac{1}{p_{i}p_{j}p_{k}}-\cdot\cdot\cdot\pm(-1)^{N+1}\frac{1}{p_{1}\cdot\cdot\cdot p_{N}}.(2)
\phi^1=du_{xx}-a_1dx-a_2dy-a_3dz,\phi^2=du_{xy}-a_2dx-a_5dy-a_6dz, \phi^3=du_{yy}-a_5dx-a_8dy-a_9dz\phi^4:=du_{xz}-a_3dx-a_6dy-(a_1e^{u_{xx}}-a_5)dz, \phi^5:=du_{yz}-a_6dx-a_9dy-(a_2e^{u_{xx}}-a_8)dz.
S_{+}( \eta ) = \left( \begin{matrix}{{\cosh \eta}}&{{\sinh \eta}}&{{0}}&{{0}}\\{{\sinh \eta}}&{{\cosh \eta}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{\cosh \eta}}&{{\sinh \eta}}\\{{0}}&{{0}}&{{\sinh \eta}}&{{\cosh \eta}}\\ \end{matrix}\right) .
F(p,s) \leq F(p, 1.7)^{\frac{10s-13}{4}}F(p, 1.3)^{\frac{17-10s}{4}}&\leq \Big(G(p, 1.7)\Big)^{\frac{10s-13}{4}}\Big(e^{2p/17}G(p, 1.7)\Big)^{\frac{17-10s}{4}}&= e^{\frac{p}{2}\frac{17-10s}{17}}1.7^{-p/2}2^{3p/2-1}\Gamma(p/2).
\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{n(-1)^{n-1}q^{\frac{n(n+1)}{2}}}{(q)_n(1-q^n)}=\frac{1}{(q)_{\infty}}\sum_{n=1}^{\infty}\frac{q^n}{(1-q^n)^2}-\sum_{j=1}^{\infty}\frac{q^{j^2}}{(q)_{j}^{2}}\sum_{n=1}^{j}\frac{q^n}{(1-q^n)^2}.
{}_2F_1(a\; ,\; b\; ;\; c\; ;\; z) = 1 +{ab\over c}\; z +{a(a+1)b(b+1)\over c(c+1)}\;{z^2\over 2!}+{a(a+1)(a+2)b(b+1)(b+2)\over c(c+1)(c+2)}\;{z^3\over 3!}+ \cdots
{\cal M}= e^{2 \phi}\left( \begin{array}{c c}{| \lambda |^{2}}&{a}\\{}&{}\\{a}&{1}\\ \end{array}\right) ,{\cal M}^{- 1}= e^{2 \phi}\left( \begin{array}{c c}{1}&{- a}\\{}&{}\\{- a}&{| \lambda |^{2}}\\ \end{array}\right) ,
N_q(\tilde{\mathcal{X}}) = N_q(\mathcal{X})q +1 = \theta_q(\frac{m}{2})q^{\frac{m}{2}}(d-1) + \theta_q(\frac{m}{2}-1)q +1 = \theta_q(\frac{m}{2}) \Bigl( (d-1) q^{\frac{m}{2}}+1 \Bigr).
\left( \omega_{\alpha}^{g_{1}g_{2}, g_{3}}\right) \, \left( g_{3}^{*}\omega_{\alpha}^{g_{1}, g_{2}}\right) \: = \: \left( \omega_{\alpha}^{g_{1}, g_{2}g_{3}}\right) \, \left( \omega_{\alpha}^{g_{2}, g_{3}}\right)
E^{X^{(n)}}\left[G_{\delta,M}^2\wedge K\right]-E\left[\left(\int_0^1 Z^{(0,0)}(x)\,dx\right)^2\right]\overset{Lem.\,\ref{calc}}{=}E^{X^{(n)}}\left[G_{\delta,M}^2\wedge K\right]-\frac{5}{3}\ge -2\epsilon.
\left\{\begin{array}{c c c}{{a}}&{{b}}&{{i}}\\{{c}}&{{d}}&{{j}}\\ \end{array}\right\}= \frac{T e t \left[ \begin{array}{c c c}{{a}}&{{b}}&{{i}}\\{{c}}&{{d}}&{{j}}\\ \end{array}\right] \Delta_{i}}{\theta ( a , d , i ) \theta ( b , c , i )}.
\begin{aligned}M(x):=\vartheta_0^{-1}\left(C(u_e,h_e,u_b)+\Vert \langle y\rangle^{l+1}\partial_y (u,h)\Vert_{L^\infty}+\Vert \langle y\rangle^{l+1}\partial_y^2 (u,h)\Vert_{L^\infty}\right).\end{aligned}
\begin{aligned}\big(\sum_{-\delta N\le j\le 0}2^{-\frac{1}{2}jq}I^q_j\big)^{\frac{1}{q}}\ge \frac{C}{(\ln N)^3N^{\frac{1}{2}}}(\sum_{-\delta N\le j\le 0}1)^{\frac{1}{q}}=\frac{CN^{\frac{1}{q}-\frac{1}{2}}}{(\ln N)^3}.\end{aligned}
L(s, f\times\bar{g}) = \zeta^{(N)}(2s) \sum_{m=1}^\infty \frac{f_m\overline{\rho_g(m)}}{m^s}\quad\quad\zeta^{(N)}(s)= \zeta(s)\prod_{p\mid N}\bigl(1 - p^{-s}\bigr).
\langle w,\nabla^2 h_{k}(y)w\rangle+\langle\nabla h_{k}(y), z\rangle =\lim_{\tau\downarrow 0}\frac{2[h_k(y+\tau w+\frac{1}{2}\tau^2z)-h_k(y)-\tau dh_k(y)(w)]}{\tau^2}=\lim_{\tau\downarrow 0}\frac{2[h(y+\tau w+\frac{1}{2}\tau^2z)-h(y)-\tau dh(y)(w)]}{\tau^2}.
\tilde{s}^i_x=s^i_x-\frac{s^i_x(q_i)}{s^i_y(q_i)}s^i_y~{\rm and}~ \tilde{s}^i_y=s^i_y-\frac{s^i_y(p_i)}{s^i_x(p_i)}s^i_x.
\Lambda:= \left(\begin{array}{ccc}\mbox{log}\vert \sigma_1(\gamma_1) \vert \dots \mbox{log}\vert \sigma_d(\gamma_1) \vert \vdots \dots \vdots \mbox{log}\vert \sigma_1(\gamma_{d-1}) \vert \dots \mbox{log}\vert \sigma_d(\gamma_{d-1}) \vert\end{array}\right)
\overline{{G}}_{\sigma}^{( n )}( \zeta^{\prime}, x^{2}\zeta_{2 n})^{\varepsilon^{\prime}\varepsilon_{2 n}}= \sigma \overline{{G}}_{\sigma}^{( n )}( \zeta_{2 n}, \zeta^{\prime})^{\varepsilon_{2 n}\varepsilon^{\prime}}\prod_{j = 1}^{2 n - 1}\frac{\zeta_{2 n}}{\zeta_{j}}.
\omega_{a,b}([x,y],z) =\omega_{a,b}(x,[y,z])=-(-1)^{|y||z|}\omega_{a,b}(x,[z,y]) &=-(-1)^{|y||z|}\omega_{a,b}([x,z],y)=(-1)^{|y||z|+|x||z|}\omega_{a,b}(z,[x,y]).
\widehat{f}_n(k) =2\int_0^1f_1(nx) \cos(k\pi x)\mathrm{d}x &=2\int_0^1\Big(\sum_{m=1}^\infty \widehat{f}_1(m) \cos(m\pi nx)\Big)\cos(k\pi x) \mathrm{d}x &=2\sum_{m=1}^\infty \widehat{f}_1(m)\int_0^1\cos(mn\pi x)\cos(k\pi x)\mathrm{d}x &=\begin{cases}b_m(p) mn=k,\m\equiv_2 1 0 . \end{cases}
\gamma_j = -(j-1) + \frac{(n-1)}{2}\sum_{\ell=1}^{j-1}\left(\frac{1}{t_\ell}- \frac{1}{r_\ell}\right) &= -(j-1) + \frac{(n-1)}{2}\sum_{\ell=1}^{j-2}\left(\frac{1}{t_\ell}- \frac{1}{r_{\ell+1}}\right).
df=\frac{(\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}})^{2}}{\frac{(\frac{s_{1}^{2}}{n_{1}})^{2}}{n_{1}-1}+\frac{(\frac{s_{2}^{2}}{n_{2}})^{2}}{n_{2}-1}}
\frac 38 I = 3(K_{13}-K_{12})(1-K_{12}-K_{13}) +3 (1-2K_{12}-K_{13}) K_{13}+ 2(K_{13}-K_{12})^2 - 2(K_{13}-K_{12})(1-K_{12}-2K_{13}) - (1-K_{12}-2K_{13})^2 = -1+K_{12}+8K_{13}+2K_{12}^2-4K_{13}^2-16K_{12}K_{13}= -4z^2 + 8(1+sm-2m)z + [-1+(1-8s)m+2(1+8s-2s^2)m^2]
\delta^2 \geq \int_0^x \{e^{\phi(u)/2}- f_0^{1/2}(u)\}^2 \, du = \int_0^x (1-e^{\psi(u)/2})^2 \frac{e^{-u}}{1 - e^{-a}}\, du &\geq (1-e^{\psi(x)/2})^2\frac{1 - e^{-x}}{1 - e^{-a}}\geq (1-e^{\psi(x)/2})^2\frac{e^{-x}(e - 1)}{1 - e^{-a}},
M_3=\sum_{\substack{0\leq m_2\leq j-i-1 \gcd(l+m_1+m_2,j)\{\rm{is}}\k-{\rm{free}}\gcd(l+m_1+m_2,j+1)\{\rm{is}}\k-{\rm{free}}}}C_{\alpha}(j-i-1,m_2)
\left( \frac{\dot{a}_{0}}{a_{0}}\right)^{2}+ \frac{K}{a_{0}^{2}}= 2 k^{2}\frac{( - 1 - \bar{\rho}_{0}\bar{\rho}_{\frac 1 2}) \sinh ( k b ) + ( \bar{\rho}_{0}+ \bar{\rho}_{\frac 1 2}) \cosh ( k b )}{\sinh ( k b ) - \bar{\rho}_{\frac 1 2}( \cosh ( k b ) - 1 )}.
p\sum_{k=0}^{n-1}\frac{\binom{2k}k^2}{(2k+1)16^k}\sum_{j=1}^k\frac1{2j-1}\equiv \sum_{k=0}^{n-1}\frac{\binom{2k}k^2}{(2k+1)16^k}-\sum_{k=0}^{n-1}\frac{\binom nk\binom{2k}k}{(2k+1)(-4)^k}\pmod{p^2}
\mu_{\alpha}^{\left( 1\right)}:=\Gamma\left( \alpha+1\right) ,\mu_{\alpha}^{\left( 2\right)}\left( \rho_{1}\right) :=\frac{\Gamma\left( \alpha\right) \left( 1-\left( 1-\rho_{1}\right)^{\alpha}\right)}{\rho_{1}\left( 1-\rho_{1}\right)^{\alpha}},
X(z)^{-1}= \overline{\tilde{X}(z)}^{T}= \left ( \begin{array}{cc}\overline{\tilde{\mathcal{A}}(z)}^{T}\overline{\tilde{\mathcal{B}}(z)}^{T}\overline{\tilde{\mathcal{A}}(-z)}^{T}\overline{\tilde{\mathcal{B}}(-z)}^{T}\end{array}\right ) .
\int_{B_{1/2}}r^{2-n}|\nabla u|^2 \d x = \sum_{j = 0}^{\infty}\int_{B_{r_{j+1}}\setminus B_{r_{j+2}}}r^{2-n}|\nabla u|^2 \d x & \leq C \sum_{j = 0}^{\infty}r_j^{2-n}\int_{B_{r_{j+1}}\setminus B_{r_{j+2}}}|\nabla u|^2 \d x.