harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\cos \alpha (\sin^2\theta - \cos^2\theta) \cos\alpha \sin\alpha \cos\theta + \frac{\cos^2\alpha \sin\theta}{\sin \alpha}\cos^2\alpha \cos\theta \sin \theta = \cos^2\alpha \sin\alpha \cos\theta(\sin^2\theta(1+\cot^2 \alpha) - \cos^2\theta) = \cos^2\alpha \sin\alpha \cos\theta (\sin^2\theta \csc^2 \alpha - \cos^2\theta).
\alpha=(\tilde{z}'_1,\tilde{z}'_2)+(\tilde{z}_1,\tilde{z}'_3)+ (\tilde{z}_2,\tilde{z}_3)+(\pi, \tilde{u})+ (\lambda_1,\tilde{z}'_2)+ (\lambda_2,\tilde{z}_1)+(\lambda_3,\tilde{z}_3)+(\pi, \lambda)
d s^{2}= g_{i j}d x^{i}d x^{j}= e^{- 2 \sigma ( \eta )}\eta_{\mu \nu}d x^{\mu}d x^{\nu}- d \eta^{2}, ~ ( \mu , \nu = 0 , 1 , 2 , 3 ; ~ d x^{5}= d \eta )
\frac{s^{\alpha\gamma-\beta}}{\Gamma(\gamma+\theta/\alpha)}\int_0^\infty \, t^{\gamma+\theta/\alpha-1}e^{-(\lambda+s^\alpha) t}\,dt = \frac{s^{\alpha\gamma-\beta}}{(\lambda+s^\alpha)^{\gamma+\theta/\alpha}}= \frac{s^{\alpha(\gamma+\theta/\alpha)-(\beta+\theta)}}{(\lambda+s^\alpha)^{\gamma+\theta/\alpha}}
d [ \sigma ] = \prod_{j = 1}^{2}\prod_{p , q}d ( \mathrm{R e}\, \sigma_{p q}^{j j}) d ( \mathrm{I m}\, \sigma_{p q}^{j j}) \prod_{j \neq l}\prod_{p , q}d \sigma_{p q}^{j l *}d \sigma_{p q}^{j l}
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-1;q^2)_k}{(q^2;q^2)_k}q^{(n-1)^2-k^2}begin{align*}7pt] =\sum_{k=0}^{n-1}\frac{(q;q^2)_n q^{k^2-k}}{(q^2;q^2)_k (q^2;q^2)_{n-k-1}(1-q^{2n-2k-1})}begin{align*}7pt] =\sum_{k=0}^{n-1}\frac{(q;q^2)_n q^{k^2-k}}{(q^2;q^2)_{n-1}(1-q^{2n-2k-1})}{n-1\brack k}_{q^2}.
\Delta_{\infty}u(x)=\frac{\langle Du,D^{2}uDu\rangle}{|Du|^{2}}=\frac{1}{|Du|^{2}}\sum_{i,j}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\frac{\partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}.
\lvert \omega \rvert =(t+1)\sum_{i=0}^{2t+1}n_i^2+\sum_{i=0}^{2t+1}in_i=2\left((t+1)\sum_{i=0}^{t-1}n_i^2+\sum_{i=0}^{t-1}(i-t-1)n_i)\right)
S ={\frac{1}{\sqrt{M^{2}}}}\varepsilon_{i j}L^{i}\tilde{D}L^{j}={\frac{L^{*}{}^{2}}{\sqrt{M^{2}}}}\varepsilon_{i j}\hat{k}^{i}\tilde{D}\hat{k}^{j}={\frac{L^{*}{}^{2}}{\sqrt{M^{2}}}}\overline{{\kappa}}_{2}\, .
(z_0+z_2)^{k+\frac{j_2}{T}}Y_{M^3}(u,z_0+z_2)I(w_1,z_2)w_2 = (z_2+z_0)^{k+\frac{j_2}{T}}I(Y_{M^1}(u,z_0)w_1,z_2)w_2,
\begin{aligned}\sup_{(x,t)\in \mathbb{R}^3\times \mathbb{R}_+}e^{\alpha \left(\delta (\left<\left<x\right>\right>-Mt)\right)^{\frac{2}{1-\gamma}}}\left| f(t,x) \right|_{L^2_\xi}\lesssim (1+t)\left\|f_0\right\|_{L^{2}(e^{3\alpha\left|\xi\right|^2}\mathcal{M})}.\end{aligned}
\mathcal P_{(\ker L)^\perp}\biggl(\begin{array}{c}q \int_0^1 u \,dz\end{array}\biggr) = \sum_{\mathbf k \in 2\pi \mathbb T^2 \setminus \lbrace (0,0)\rbrace}\bigl( b_{\mathbf k}^+ V_{\mathbf k}^+ + b_{\mathbf k}^- V_{\mathbf k}^- \bigr),
\sum_{m=1}^{N}\left\{h_{m}~s_{m}\left[ \theta_{n}\left( t\right) \right]\right\}=\rho_{n}\left[ \underline{\theta}\left( t\right) \right] ~\dot{\theta}_{n}\left( t\right) +\gamma_{n}\left[ \underline{\theta}\left(t\right) \right] ~.
U_{p,q}=\begin{bmatrix}1 0 0 0 0 0 0 0 1 0 0 a 0 0 0 0 1 0 b c 0 0 0 0 1 b 0 c 0 0 0 0 d 0 0 0 0 0 0 e f 0 0 0 0 0 e 0 f\end{bmatrix}
\begin{cases}G(t) = e^{tA}+ \displaystyle\int^t_0 e^{(t-s)A}\alpha A_1G(t-r)ds + \displaystyle\int^t_0\int^0_{-r}\beta(\theta)e^{(t-s)A}A_2G(t+\theta)d\theta ds,\,\,\,t\in [0, T], G(t) ={\rm O},\hskip 10pt t<0.\end{cases}
G_{l + \nu}\left( \left. r^{\prime \prime}, r^{\prime}; E \right| \lambda ; a ; \sigma \right) = G_{l + \nu}( r^{\prime \prime}, r^{\prime}; E | \lambda ) - \frac{G_{l + \nu}( r^{\prime \prime}, a ; E | \lambda ) \, G_{l + \nu}( a , r^{\prime}; E | \lambda )}{G_{l + \nu}( a , a ; E | \lambda ) - 1 / \sigma}\; .
\vec\mu=R_0\vec y,R_0=\begin{pmatrix}(-t)^{-1/4}0 0 0 (-t)^{-1/4}0 0 0 (-t)^{1/4}\end{pmatrix}\begin{pmatrix}1 1 1 1 -1 -1 0 \sqrt2 -\sqrt2\end{pmatrix},
ds^{2}=-(\frac{1-\frac{M}{2a(t)r}}{1+\frac{M}{2a(t)r}})^{2}c^{2}dt^{2}+(1+\frac{M}{2a(t)r})^{4}a^{2}(t)(dr^{2}+r^{2}d\Omega^{2}),
L^{i j}={\bf R}^{i}{\bf P}^{j}-{\bf R}^{j}{\bf P}^{i}, \, \, \, \quad L^{1^{\prime}i}= \frac{\alpha \left( H \right)}{\sqrt{- 2 H}}\left( \frac 1 2 L^{i j}{\bf P}_{j}+ \frac 1 2{\bf P}_{j}L^{i j}- \frac{\alpha \left( H \right){\bf R}^{i}}R \right) .
c_i =n_1+x_iN_1^{1/3}\,\,,\,\, c'_i=n_1+x'_iN_1^{1/3}d_i =n_1+1+y_iN_1^{1/3}\,\,,\,\, d'_i=n_1+1+y'_iN_1^{1/3}
LU \leq \| \nabla u_0^1 \|_{L^2 H^{N-1}}\| \nabla u_0^2 \|_{L^2 H^{N-1}}\lesssim \frac{C_0 \epsilon}{\nu^{3/2}}\frac{\epsilon}{\sqrt{\nu}}= \frac{1}{C_0}\left( \frac{C_0 \epsilon}{\nu}\right)^2,
[ \delta_{{\tilde{\epsilon}}{'}}, \delta_{\tilde{\epsilon}}] \psi_{i}= - 2 i{\tilde{\epsilon}}{\tilde{\epsilon}}{'}f_{i j k}J_{j}\psi_{k}- 2 i{\tilde{\epsilon}}{\tilde{\epsilon}}{'}f_{j l m}f_{j i k}x_{m}x_{k}{\dot{\psi}}_{l},
\left( \begin{array}{c}{A_{k}^{\mu}\left( \theta \right)}\\{\tilde{A}_{k}^{\mu \dagger}\left( \theta \right)}\\ \end{array}\right) ={\cal B}_{k}\left( \begin{array}{c}{A_{k}^{\mu}}\\{\tilde{A}_{k}^{\mu \dagger}}\\ \end{array}\right) ,
u_{n}\left( t,x\right) =\int_{0}^{t}\left[ L^{\nu}u_{n}\left( r,x\right)-\lambda u_{n}\left( r,x\right) +f_{n}\left( r,x\right) \right] dr,\left(t,x\right) \in \left[ 0,T\right] \times \mathbf{R}^{d}.
{\cal F}= \left( \begin{array}{c c c c c}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{F_{1}}&{0}&{0}\\{0}&{- F_{1}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{F_{2}}\\{0}&{0}&{0}&{- F_{2}}&{0}\\ \end{array}\right) .
\lim_{x \to \infty}\frac{1}{x}\#\bigg\{n \leq x \colon \frac{P_n(x) - D(x)}{\sqrt C(\log\log x)^{3/2}}< u \bigg\}= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^u e^{-t^2/2}\, dt.
\Delta_I\left(\overline{w}_0,\left(m_{ij}\right)_{m\geq i\geq 1}^{m<j\leq n}\right)=\sum_{\{\gamma\}:I^s\rightarrow I^t}\prod_{\gamma\in \{\gamma\}}\prod_{f\in \left\{\hat{\gamma}\right\}}X_f,
\displaystyle\int_{\mathbb{R}^N}b(x)|\Psi_n(x,0)|^{2_\alpha^*}dx=\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_n(x,0)|^{2_\alpha^*}dx-\displaystyle\int_{\mathbb{R}^N}b(x)|\omega_0|^{2_\alpha^*}dx+o_n(1).
T^{\,m'}\,e_{b_{2^{k-1}+l}+j-b_{l}}=\Biggl(\;\,\prod_{i=j-b_l+1}^{j-b_{l}+m'}w_{i}^{(k)}\Biggr)\,e_{b_{2^{k-1}+l}+j-b_{l}+m'},
\langle \bar{\psi}( x ) \psi ( x ) \, \bar{\psi}( \tilde{x}) \psi ( \tilde{x}) \rangle_{w}\cong -{\frac{\langle 1 \rangle_{w}}{( 4 \pi )^{2}}}{\frac{\vert \tilde{x}- x_{o}\vert^{4}}{r_{o}^{4}}}{\frac{2}{\vert x - x_{o}\vert^{4}}}
y_n(x) = Q_n(x;\alpha,\beta,N),\Lambda(x) = -x, A(n) = \frac{(n+\alpha+1)(n+\alpha+\beta+1)(N-n)}{(2n+\alpha+\beta+1)(2n+\alpha+\beta+2)},C(n) = \frac{n(n+\alpha+\beta+N+1)(n+\beta)}{(2n+\alpha+\beta)(2n+\alpha+\beta+1)}.
\int_{B_R(0) \cap \Omega}e_n \cdot (\omega \times (c-u))\, dx = g\int_{B_R(0) \cap S}\eta\, dx' + \sigma \int_{\partial B_R \cap S}N \cdot \nu\, ds + \int_{\partial B_R(0) \cap \Omega}A \cdot N\, dS.
(D_0+i\lambda)(D_g+i\lambda)^{-1}= (D_0-D_g+D_g+i\lambda)(D_g+i\lambda)^{-1}= 1 + (D_0-D_g)(D_g+i\lambda)^{-1}= 1+D_0(D_g+i)^{-1}\frac{D_g+i}{D_g+i\lambda}- \frac{D_g}{D_g+i\lambda}.
\begin{array}{ll}\mathsf E K'_{\rho_1(x),b}(X_1)=(1/b)\mathsf EK_{\rho_1(x),b}(X_1)L_1(X_1;x) &= f'(x) + b\left(\frac{1}{12x^2}f(x) +\frac{1}{4}f''(x)\right)+o(b), &\mathsf E K'_{\rho_2(x),b}(X_1)=(x/2b^2)\mathsf E K_{\rho_2(x),b}(X_1)L_2(X_1;x) &=f'(x)\left(\frac{x}{2b}-\frac{b}{6x}\right)+f''(x)\left(\frac{7x}{48}+\frac{x^2}{2b}\right)+o(b).\end{array}
\begin{array}{l}\displaystyle \# \{\sum_{\substack{j\in J\setminus J_{k}}}X(u_{j})(t_{j+1}-t_{j})|\forall J_{k}=\{i_1,\cdots,i_k\}\subset J, 1\leq k\leq K\}=C_{n}^{K}(2^{K}-1).\end{array}
u(x,t) = \pm{\sf U}_\infty(x) + K{\sf D}_J (T-t)^{\frac{\gamma}{2}+J}|z|^\gamma (1+o(1)) = \pm{\sf U}_\infty(x) + K{\sf D}_J (T-t)^J \eta^\gamma |\xi|^\gamma (1+o(1)) |z|\to0.
< \tilde{\Sigma}( k_{1}, \theta_{1}) \tilde{\Sigma}( k_{2}, \theta_{2}) > = - \frac{1}{2 N}I ( k_{1}^{2}) \frac{D^{2}- 2 m}{k_{1}^{2}- 4 m^{2}}\delta_{1 2}^{5}\simeq - \frac{1 6 i}{N}\frac{D^{2}- 2 m}{\sqrt{- k_{1}^{2}+ 4 m^{2}}}\delta^{3}( k_{1}- k_{2}) \bar{\delta}_{1 2}.
\Lambda^p_{tt}=(D_\gamma^{\dot\gamma}\dot\gamma)^p-\dot\gamma^i\dot\gamma^j \Gamma^p_{\,\,ij}(\dot\gamma), \Lambda^p_{ts}=(D_\gamma^{\dot\gamma}U)^p-\dot\gamma^iU^j \Gamma^p_{\,\,ij}(\dot\gamma),
\sum_{k + l \geq 0}a_{kl}^{1}(\varphi(x)) (\lambda z_{1}+ z_{2})^{k}(\lambda z_{2})^{l}= \sum_{k + l \geq 0}\lambda a_{kl}^{1}(x) z_{1}^{k}z_{2}^{l}+ (z_{2}+ a_{00}^{2}(x)).
\int_0^1\int_{\|\varphi\|=r}\begin{vmatrix}\varphi(z) \end{vmatrix}^{p-1}\mathrm{d}s\mathrm{d}r=\int_0^1 2\pi r\int_{\|\varphi\|=r}\begin{vmatrix}\varphi(z) \end{vmatrix}^{p-2}\mathrm{d}\omega_r\mathrm{d}r
\left\| \sum_{i=M}^N \left( A^{\mu^i}_\pi h_i - M{h_i}\right) \right\|_p \le \sum_{i=M}^N \lambda^i \left\| h_i - M{h_i}\right\|_p &\le \left(\sum_{i=M}^N \lambda^{qi}\right)^{1/q}\, \left( \sum_{i=M}^N \left\| h_i - M{h_i}\right\|_p^p \right)^{1/p}\, .
\int_{\{2\|\mathbf{x}\|>{\eta}_{n}\}}\bigl|{g}_{\infty}(\mathbf{x})-g_{{\lambda}_{n}}(\mathbf{x}) \bigr|^p \,\mathrm{d}\mathbf{x}= \mathrm{o}(1)\qquad\mbox{as}n\to\infty,
\sup_{x\in B_r(y)}|u(x)-\phi(x)| &=\sup_{x\in B_r(y)}|u(x)-u(y)-(x-y)\cdot \nabla u(y)+\phi(y)+(x-y)\cdot \nabla \phi(y)-\phi(x)| &\leq (C+1) r^{1+\alpha}.
V_{\theta}^{D}= V_{0}^{D}\equiv V^{D}\equiv \sum_{n}\mu^{\epsilon}{\frac{1}{2}}\int{\frac{d^{D - 1}k}{( 2 \pi )^{D - 1}}}\log \left({\frac{k^{2}+ m_{n}^{2}( \varphi_{i}, D )}{\mu^{2}}}\right) , \quad \quad ( D \neq \mathrm{i n t e g e r}) .
\vert\phi\rangle={1\over\sqrt{2\pi}}\sum^\infty_{m=-\infty}e^{im\phi}\vert m;\alpha\rangle\left(\equiv\lim_{\varepsilon\to0}{1\over\sqrt{2\pi}}\sum^\infty_{m=-\infty}e^{im\phi-\vert m\vert\varepsilon}\vert m;\alpha\rangle\right)\ ,
F_{\pm N}= Z_{F}^{- 1}\int D \bar{\chi}D \chi p r o d_{i = 1}^{N}\bar{\chi}_{\pm}\chi_{\pm}( w_{i}) \exp ( - \int d^{2}x \bar{\chi}( i \! \! \not \! \partial +{\not \! \! A}^{( \pm N )}) \chi )
\sum_{\substack{\rho' = \beta' + i\gamma', |\gamma'| \leq T}}\left( \beta' - \frac{1}{2}\right) = \frac{T}{\pi}\log{\log{\frac{qT}{2\pi}}}+ \frac{T}{\pi}\left( \frac{1}{2}\log{m}- \log{\log{m}}\right) - \frac{2}{q}\operatorname{Li}\left( \frac{qT}{2\pi}\right) &\quad\quad+ O\left( m^{1/2}(\log{\log{(qT)}})^2 + m\log{\log{(qT)}}+ m^{1/2}\log{q}\right),
d(\beta) =\left(a_1T_1+a_2T_2 \right)^2 + \left(b_1T_1+b_2T_2 \right)^2+\left(a_1T_1+a_2T_2 \right) \left(b_1T_1+b_2T_2 \right) + (a_3^2+a_3b_3+b_3^2)T_3^2, d(\gamma) =\left(a_1T_1+a_2T_2 \right) \left( b_1T_1+b_2T_2 \right)^2 +\left(a_1T_1+a_2T_2 \right)^2 \left( b_1T_1+b_2T_2 \right) + (a_3^2b_3+a_3b_3^2)T_3^3.
\begin{array}{l}{H \equiv P_{+}= \int d x^{-}\big [ ( 1 / 8 ) \mathrm{t r}E^{2}- ( m / 2 ) \big ( \psi_{-}^{\dagger}( x ) \psi_{+}( x ) + \psi_{+}^{\dagger}( x ) \psi_{-}( x ) \big ) \big ]}\\{\left[ A_{-}( x^{-}, x^{+}) , E ( y^{-}, x^{+}) \right] = i \delta ( x^{-}- y^{-})}\\ \end{array}
\frac{\partial F}{\partial\theta}= F_{\ast}\left(\frac{\partial}{\partial\theta}\right)=\left(\frac{\cos\theta}{\sqrt{2}}\cos\phi,\frac{\cos\theta}{\sqrt{2}}\sin\phi,\frac{\cos\theta}{\sqrt{2}}\cos\psi,\frac{\cos\theta}{\sqrt{2}}\sin\psi, -\sin\theta\right)
\mathcal{E}_{\tau}(\rho):=\int_{\mathbb{R}^3}j_{m}(\rho(x)){\rm d}x - \frac{\tau}{2}\iint_{\mathbb{R}^{3}\times\mathbb{R}^{3}}\frac{\rho(x)\rho(y)}{\left|x-y\right|}{\rm d}x{\rm d}y + \int_{\mathbb{R}^3}V(x)\rho(x){\rm d}x.
{\widetilde Q}^{( n )}= \frac{1}{2}\log ( u v ) \sum_{r + s = n}\left( \begin{matrix}{n}\\{r}\\ \end{matrix}\right) \left( \begin{matrix}{n}\\{s}\\ \end{matrix}\right) u^{r}v^{s}- \sum_{r + 2 = n}\left( \begin{matrix}{n}\\{r}\\ \end{matrix}\right) \left( \begin{matrix}{n}\\{s}\\ \end{matrix}\right) c ( r , s ) u^{r}v^{s}
\mathcal{A}^{(\hat{\varphi},\nu)}_{(\alpha_0,\alpha_1)}(x,r,\eta):=\int_{S^1}(x,r)^*\lambda-\eta\int_{S^1}\kappa(t)H(r)dt-\int_{S^1}\beta_\nu(r)L_t(x,r)\,dt.
Z^{I}= \sum_{a = 1}^{a = 5}{\frac{q^{I}{}_{a}}{| \vec{x}- \vec{x}_{a}|}}\ , \qquad \sum_{a = 1}^{a = 5}q^{I}{}_{a}= 0 \ , \qquad I = 1 , 2 , 3 , 4 .
\hat{R}_{1 2}( \sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{1 2}^{- 1}( \sigma^{1}K^{\prime}\sigma^{1})_{1}= ( \sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{1 2}^{- 1}( \sigma^{1}K^{\prime}\sigma^{1})_{1}\hat{R}_{1 2}.
\Delta(x)= \sum_r x'_rk_{\delta_r}\otimes x_r(\delta_r\in Q^+, x_r\in U^+_{\delta_r}, x'_r\in U^+_{\gamma-\delta_r}), \Delta(y)= \sum_s y_s\otimes k_{-\gamma_s}y'_s(\gamma_s\in Q^+, y_s\in U^-_{-\gamma_s}, y'_s\in U^-_{-(\gamma-\gamma_s)})
\beta_{3}^{\prime}(k)=(\begin{matrix}2k\\k\end{matrix})\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})^{-1}(\begin{matrix}k\\j\end{matrix})(4)^{k-j}\sum_{m=0}^{j}(\begin{matrix}j\\m\end{matrix})^{4}=1,10,162,3124,66994,...
\varphi_+\mathfrak{p}= \{\mathfrak{p}_+ \setminus\{n+1\},\;(\mathfrak{p}_i)_{i\in I},\; \mathfrak{p}_- \}\varphi_+^{-1}\mathfrak{q}= \{\mathfrak{q}_+ + \{n+1\},(\mathfrak{q}_j)_{j \in J},\mathfrak{q}_- + \}a_{c_{n+1}}^+ \lfloor V \rfloor_\mathfrak{q}= \lfloor W \rfloor_{\varphi_+^{-1}\mathfrak{q}}.
\|r^{k_1}\widehat{F_k}\|_{\mathcal L^2(\mathbb R^n)}^2 =\int_0^\infty|\widehat{F_k}|^2\,r^{n+2k_1-1}\,dr =\|\widehat{F_k}\|_{\mathcal L^2(\mathbb R^{n+2k_1})}&=C\,\|F_k\|_{\mathcal L^2(\mathbb R^{n+2k_1})}=C\int_0^\infty\left|\frac{\psi_k(r)}{r^{k_1}}\right|^2\,r^{n+2k_1-1}\,dr &=C\,\int_0^\infty|\psi_k(r)|^2\,r^{n-1}\,dr,
\begin{cases}K_{1}(z)\sim (\frac{\pi}{2})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{-z}, \arg z\in(-\frac{3\pi}{2},\frac{3\pi}{2}); begin{align*}0.2cm]I_{1}(z)\sim (\frac{1}{2\pi})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{z}-i(\frac{1}{2\pi})^{\frac{1}{2}}z^{-\frac{1}{2}}e^{-z}, \arg z\in(-\frac{\pi}{2},\frac{3\pi}{2}).\end{cases}
F\left(\frac1i \partial_x\widetilde{G}_{\lambda}*(um_1(\lambda+))\right) = \xi\frac{\chi_{\real^-}}{\xi-\lambda}F(um_1(\lambda+)) &= \chi_{\real^-}F(um_1(\lambda+)) + \lambda \frac{\chi_{\real^-}}{\xi-\lambda}F(um_1(\lambda+)) &= F[C_-(um_1(\lambda+))+\lambda \widetilde{G}_{\lambda}*(um_1(\lambda+))].
\left\|\left\{\mathbf{1}_P2^{-jn(\tau-\frac{1}{p})}\right\}_{j\geq j_P}\right\|_{L\dot A_{p,q}}=\left\|\mathbf{1}_P\right\|_{L^p}\left\|\left\{2^{-jn(\tau-\frac{1}{p})}\right\}_{j\geq j_P}\right\|_{\ell^q}&\sim|P|^{\frac{1}{p}}2^{-j_Pn(\tau-\frac{1}{p})}=|P|^{\frac{1}{p}}|P|^{\tau-\frac{1}{p}}=|P|^{\tau}
\left( \left| \frac{e^{\lambda y}\sin(\theta-\lambda\xi) - \sin\theta}{\lambda}\right|^2 + \left| \frac{e^{\lambda y}\cos(\theta-\lambda\xi)-\cos\theta}{\lambda}\right|^2 \right)^2 & + 4 \left( \frac{e^{2\lambda y}- 1}{2\lambda^2 y}(-\xi + \lambda z) + \frac1{\lambda^2}e^{\lambda y}\sin(\lambda \xi) \right)^2
\langle \mathrm{T r}Z^{J}~ \mathrm{T r}\bar{Z}^{J}\rangle = J N^{J}\Bigg \{1 + \sum_{h = 1}^{\infty}{\frac{a_{h , 4 h}}{( 4 h ) !}}{\frac{J^{4 h}}{N^{2 h}}}+ \ldots \Bigg \}
\frac{\partial V_1}{\partial t_{1,j,a}}= B_{1,j,a}V_1, \frac{\partial V_1}{\partial t_{2,j,a}}= B_{2,j,a}V_1, \frac{\partial \tilde V_2}{\partial t_{1,j,a}}=B_{1,j,a}\tilde V_2, \frac{\partial \tilde V_2}{\partial t_{2,j,a}}= B_{2,j,a}\tilde V_2,
d_0\tilde\varphi(x,x+n) =\sum_{i=0}^{x+n-1}\varphi(i,i+1)-\sum_{i=0}^{x-1}\varphi(i,i+1) =\sum_{i=x}^{x+n-1}\varphi(i,i+1) =\begin{cases}\varphi(x,x+n)-\varphi_n(x) x=x_1,\ldots,x_{r_n}\varphi(x,x+n) \mbox{otherwise}. \end{cases}
\mathcal{C}=-\frac{1}{\,\Delta_p(\{\hat{v}^{-1}\})}\sum_{\mu=1}^m(-1)^{\mu}\int_0^\infty d\lambda\, \left(\det\begin{bmatrix}\left[\frac{\lambda^{m-\mu}}{\Gamma(m-\mu+1)}\frac{\exp(-\lambda/\hat{v}_j)}{\hat{v}_j^m}\right]_{j=1,...,p}\left[\hat{v}_j^{-i+1}\right]_{\substack{i,j=1,...,p (i\neq \mu)}}\end{bmatrix}\right)\, \log_2 (1+\lambda).
S_7 =m_K \int_{Y}^R \alpha(u)\left( V^{(m)}\left(\frac{\log u}{\log R}\right)\right)^2 \frac{du}{u\log u}+ \int_{Y}^R \alpha(u)\left( V^{(m)}\left(\frac{\log u}{\log R}\right)\right)^2 \frac{dE(u)}{u\log u}&:=S_8+S_9.
{C}_n(\kappa,{\bf P}_{Y_1})= \sup_{\frac{1}{n}{\bf E}\big\{\sum_{t=1}^n ||X_t||_{{\mathbb R}^{n_x}}^2\big\}\leq \kappa}\frac{1}{2}\sum_{t=1}^n \ln \Big\{\frac{\det( K_{I_t})}{\det(K_{\hat{I}_t})}\Big\}^+
\begin{aligned}\theta(u^k) + I_K(\zeta^k) \leq \theta(\hat{u}) + I_K(\hat{\zeta}) - [ \langle \lambda^k, \hat{\zeta}- \zeta^k\rangle + \langle D_u\theta(u^k), \hat{u}- u^k \rangle_\mathcal{U}] & \leq \theta(\hat{u}) + I_K(\hat{\zeta}) + \beta\sum\limits_{i = n}^{k}\|r^{i}\|^2 + |\langle \lambda^n, r^k\rangle|.\end{aligned}
\int_{\mathbb{R}_+}e^{uy}\nu_t(dy)= \Big (\frac{d-\frac{\sigma^2du}{2a}+\big(\frac{\sigma^2d}{2a}-1\big)ue^{-at}}{d-u}\Big )^{\frac{c}{a-\frac{\sigma^2d}{2}}}= \Big (\frac{d-uL_1(t)}{d-u}\Big )^{\frac{c}{a-\frac{\sigma^2d}{2}}}= \Big( L_1(t)+\big(1-L_1(t)\big) \frac{1}{1-\frac{u}{d}}\Big)^{\frac{c}{a-\frac{\sigma^2d}{2}}},
j^n_i(t_0) = j^{n - 1}_i(t_0) j^{k}_j(t_0) = j^{k - 1}_j(t_0) j^n_i(t_0) = j^{n - 1}_i(t_0) j^{k}_{j + 1}(t_0) = j^{k - 1}_j(t_0) j^n_{i+1}(t_0) = j^{n - 1}_i(t_0) j^{k}_j(t_0) = j^{k - 1}_j(t_0) j^n_{i+1}(t_0) = j^{n - 1}_i(t_0) j^{k}_{j+1}(t_0) = j^{k - 1}_j(t_0).
g(\beta,\mu) = \sum_j \frac{\exp\{\beta(\varepsilon_j - \mu)\}}{(\exp\{\beta(\varepsilon_j - \mu)\}- 1)^2}\left( \begin{array}{c c}(\varepsilon_j - \mu)^2 - \beta(\varepsilon_j - \mu) -\beta(\varepsilon_j - \mu) \beta^2\end{array}\right).
\varepsilon \leq \varepsilon_0:=\left[(4|g|_\alpha+|f|_{\infty}){\sf adiam}(D)\vee 1\right]^{-\frac{2}{\alpha}}\gamma^{\frac{2}{\alpha}}d^{-1}. \intertext{Also, choose}K := \left\lceil{\sf diam}(D)\left(\frac{8\left(|g|_\alpha+\frac{{\rm diam}(D)^2|f|_\alpha+2{\rm diam}(D)|f|_\infty}{d}\right)+1}{\gamma}\right)^{1/\alpha}\right\rceil.
\frac 1{2 \Gamma \left( - \frac 1 2 \right)}\sum_{j = 0}^{d}\frac{2 a_{j}}{j - d - 1}+ \frac 1{2 \Gamma \left( - \frac 1 2 \right)}r_{d + 1}\left( - \frac 1 2 \right) + \left. \frac{\mu^{s + 1}}{\Gamma \left( \frac s 2 \right)}\frac{a_{d + 1}}{s + 1}\right\rfloor_{s = - 1}=
\delta_\sigma= \begin{cases}\delta, \sigma_1\sigma_2\sigma_3 \neq 0 \delta^*, (\sigma_1,\sigma_2,\sigma_3)=(\pm,\mp,0),(\pm,0,\mp) 2 \max \{\eta, 1\}, \sigma_1=0,\sigma_2 \sigma_3 \neq 0 \min\{\eta, 1\}-\epsilon, \vec \sigma \neq \vec 0\end{cases}
f_9f_{12}f_{15}t + f_6f_{11}f_{15}t = -f_{14}t f_9f_{26}\widehat{c}+ f_6f_{29}\widetilde{c}= -f_{14}t -f_Df_{25}\widehat{c}+ -f_Df_{28}\widetilde{c}= -f_{14}t.
\sum_{k=n+1}^{m}f(k)=\int_n^m f(u) du +\sum_{l=1}^{p}(-1)^l \frac{B_l(0)}{l!}\{f^{(l-1)}(m)-f^{(l-1)}(n)\}+ \frac{(-1)^{p-1}}{p!}\int_n^m B_p(u-[u]) f^{(p)}(u) du,
\begin{aligned}f_1(z) = \frac{p(z)+\tilde{p}(z)}{p(z)-\tilde{p}(z)}+ \frac{v(z)}{p(z)-\tilde{p}(z)}&= -\overline{f_1(1/\bar{z})}= - \frac{\overline{p(1/\bar{z})}+\overline{\tilde{p}(1/\bar{z})}}{\overline{p(1/\bar{z})}-\overline{\tilde{p}(1/\bar{z})}}-\frac{\overline{v(1/\bar{z})}}{\overline{p(1/\bar{z})}-\overline{\tilde{p}(1/\bar{z})}}\end{aligned}
\frac{1}{2}{\rm dim}\;{\rm ker}\; (A-1) &=\frac{1}{2}\left[{\rm dim}\;{\rm ker}\; (P_{\rm F}-\mathfrak{D}_a^\ast P_{\rm F}\mathfrak{D}_a-1)+{\rm dim}\;{\rm ker}\; (P_{\rm F}-\mathfrak{D}_aP_{\rm F}\mathfrak{D}_a^\ast +1)\right] &={\rm dim}\;{\rm ker}\; (P_{\rm F}-\mathfrak{D}_a^\ast P_{\rm F}\mathfrak{D}_a-1),
\left[\Upsilon \Theta\right]^I_J = \sum_{K\in \{1,\ldots,d\}^n}\Upsilon^I_K \Theta^K_J = \sum_{k_1=1}^d \cdots \sum_{k_n=1}^d \Upsilon^{i_1, \ldots, i_n}_{k_1, \ldots, k_n}\Upsilon^{k_1, \ldots, k_n}_{j_1, \ldots, j_n}.
H = \int d x \left\{\frac{1}{2}\left[ \xi_{5}^{2}+ \xi_{6}^{2}+ ( \partial_{1}\xi_{3})^{2}- e^{2}\xi_{2}^{2}\right] + e \xi_{6}\xi_{2}+ \xi_{1}( \partial_{1}\xi_{5}+ e \partial_{1}\xi_{3}) + \lambda \xi_{4}\right\}.
\eta (1-\eta )\bar \varphi'' (\eta )+\left(\frac{n}{2}+\frac{3-\beta}{2}-\left(n+\frac{3-\beta}{2}\right)\eta\right)\bar \varphi' (\eta )-\left(\frac{n^2}{4}+\frac{n(1-\beta)}{4}\right)\bar \varphi(\eta )=0.
S_1=\begin{pmatrix}-0.5000+2.000\mathbf{i}-1.0000\mathbf{k}1.0000-3.0000\mathbf{i}+1.0000\mathbf{j}+1.0000\mathbf{k}-2.0000-0.5000\mathbf{i}+1.0000\mathbf{j}\end{pmatrix}
\sum_{0\leq n<m <\infty}\frac{\tau_A(p^{m}) \tau_B(p^{n}) \left( 1-\frac{p^w}{p^{2}}\right)}{p^{m(1-\alpha)}p^{n(2-\beta-w)}}= \frac{\tau_A(p )}{p^{1-\alpha}}- \frac{\tau_A(p )}{p^{3-\alpha-w}}+ O\left( \frac{1}{p^{1+\varepsilon}}\right).
R_{\nu \mu}^{\pm}= \delta_{\mu}^{\pm}\partial_{\nu}H^{\pm}- \delta_{\nu}^{\pm}\partial_{\mu}H^{\pm}+ \delta_{\mu}^{\pm}[ \omega_{\nu}, H^{\pm}] - \delta_{\nu}^{\pm}[ \omega_{\mu}, H^{\pm}] \, ,
\frac{1}{q}\frac{d|Y(t)|_{L^p}^p}{dt}=\langle AY(t),Y(t)|Y(t)|^{p-2}\rangle+\langle F(Y(t)+Z(t)),Y(t)|Y(t)|^{p-2}\rangle &\le -\lambda_1\frac{4(p-1)}{p^2}|Y(t)|_{L^p}^p+\epsilon|Y(t)|_{L^p}^p+C_\epsilon|F(Z(t))|_{L^p}^p+\lambda_F|Y(t)|_{L^p}^p,
(\partial_y\phi)(x,\rho(x,z))=\frac{1}{\partial_z\rho}\partial_zv(x,z)=:(\Lambda_1v)(x,z), &(\nabla_x\phi)(x,\rho(x,z))=\big(\nabla_x-\frac{\nabla_x\rho}{\partial_z\rho}\partial_z\big)v(x,z)=:(\Lambda_2v)(x,z).
\lim_{\xi \to -0}\frac{K_r(\xi-y)-K_r(\xi+y)}{|\xi|}&= \lim_{\xi \to -0}\left(\frac{K_r(y-\xi)-K_r(y)}{\xi}+\frac{K_r(y)-K_r(y+\xi)}{\xi}\right)\frac{\xi}{|\xi|}=2K_r^\prime(y)
V_{N;g}={\cal C}_{g}^{open}\prod_{i=1}^{N}\bigg[{}_i\!<\!p_{i};O_{a}|\bigg] \,\,\delta \left( \sum_{i=1}^{N}p_{i}\right) \int [{\mbox d}m]^{N}_{g}\,{\hat{V}}_{N;0}
d s^{2}= \frac{1}{1 - \xi^{0}\xi^{0}}d \xi^{0}d \xi^{0}- ( 1 - \xi^{0}\xi^{0}) \frac{d{\bf x}( I -{\bf x}^{\prime}{\bf x})^{- 1}d{\bf x}^{\prime}}{1 -{\bf x}{\bf x}^{\prime}}~ ,
| D_f| \ = \ \dim \left( \overline{S}/(x_m^{r+1}, x_{m+1}, \dots, x_n)\right)_d &=\ \dim \big ( \overline{S}/(x_{m+1}, \dots, x_{n})\big)_d - \dim \big( \overline{S}/(x_{m+1}, \dots, x_{n})\big)_{d-r-1}+ \dim \big( \overline{S}/(x_{m+1}, \dots, x_{n})\big)_0 &=\ \left( \binom{m+d}{d}- m-1 \right)- \binom{d-r+m-1}{m}+1.
P^\eta(X_T^{\vec v}-X_1^{\vec v}\leq (v-\varepsilon)T+1) \leq P^\eta\Big(\sum_{m=1}^{T-1}Y_m\leq (v-\varepsilon)T+1\Big) +\sum_{m=1}^{T-1}\P^\eta(\mathcal E^c_{m-1}).
M_{Y_{7}Y_{6}}^{2}= 2 \left( \begin{array}{c c}{{H}}&{{- 2 i c}}\\{{2 i c}}&{{H}}\\ \end{array}\right) . M_{Y_{5}Y_{4}}^{2}= 2 \left( \begin{array}{c c}{{H}}&{{- 2 i c}}\\{{2 i c}}&{{H}}\\ \end{array}\right) .
\frac{1}{2\pi i}\,\intop_{\mu-i\infty}^{\mu+i\infty}\Gamma(z)\,\Gamma(s-z)\,\zeta_{p}(2z)\,x^{2z}dz=\frac{1}{2\pi i}\,\intop_{\frac{1}{2}-\mu-i\infty}^{\frac{1}{2}-\mu+i\infty}\Gamma(z)\,\Gamma(s-z)\,\zeta_{p}(2z)\,x^{2z}dz-\frac{1}{2}\,\frac{\Gamma(s)}{1+\frac{1}{\pi p}}+\frac{\sqrt{\pi}\,x}{2}\Gamma\left(s-\frac{1}{2}\right).
a_{1}=x+v,\,\,\,\,\,\,\,\,\,\,a_{2}=2\,x-4\,v,\,\,\,\,\,\,\,\,\,\,a_{3}=-\frac{1}{3}\,x+v+\frac{y}{9},\,\,\,\,\,\,\,\,\,\,a_{4}=z\zeta .
\int_{G}dg \, \overline{\langle l_1, m_1 | T_g | l_1, m_1' \rangle}\langle l_2, m_2 | T_g | l_2, m_2' \rangle ={\delta^{l_1 l_2}\delta_{m_1 m_2}\delta_{m_1' m_2'}\over{\rm dim}_{l_1}},
1 = uu^* = \left( \sum_{i=1}^n \lambda_i \alpha_i \beta_i^* \right) \left( \sum_{j=1}^n \overline{\lambda_i}\beta_j \alpha_j^* \right) = \sum_{i=1}^n |\lambda_i|^2 \alpha_i \alpha_i^* = \sum_{i=1}^n \alpha_i \alpha_i^*.
\Psi(s) = (-1)^nC_2\frac{\Gamma(2k_4+k_1+n)}{\Gamma(2k_4+k_1)}s^{k_4}(1-k_3s)^{k_5}\_2F_1\left(-n, n+2(k_4+k_5)+\frac{k_2}{k_3}-1; 2k_4+k_1, k_3s\right) &= N_ns^{k_4}(1-k_3s)^{k_5}\_2F_1\left(-n,\n+2(k_4+k_5)+\frac{k_2}{k_3}-1;\ 2k_4+k_1,\k_3s\right), k_{3}\ne 0,
\dfrac{M_{1}(\mathcal{NC}(D_{2m}))}{|v( \mathcal{NC}(D_{2m}) )|}= \dfrac{m(m-1)(5m-4)}{2m-1}\dfrac{M_{2}(\mathcal{NC}(D_{2m}))}{|e( \mathcal{NC}(D_{2m}) )|}= \dfrac{m(m-1)(8m^{2}-12m+4)}{3m(m-1)}.
\begin{aligned}2\rho_k\int_{B_R^c(0)}{\rm ln}\big(1+|x|^2\big)\big|\tilde u_{2,k}(x)+\tilde u_{1,k}(x)\big|{\rm d}x+2\int_{B_R^c(0)}\left|\tilde u_{2,k}(x)+\tilde u_{1,k}(x)\right|{\rm d}x \quad\cdot\Big(\|\tilde u_{2,k}\|_*^2+C\|\tilde u_{2,k}\|^2\Big) < \frac{\epsilon}{2}\ \ \mbox{as}\ \k\to\infty.\end{aligned}
\frac{d^k}{dy^k}\frac{(wy)^{j+1}}{(1-wy)^{j+1}}= \frac{d^k}{dy^k}\sum_{s=0}^{j+1}\binom{j+1}{s}(-1)^{s}\frac{1}{(1-wy)^s}& = \sum_{s=1}^{j+1}\binom{j+1}{s}(-1)^{s}\frac{\Gamma(s+k)}{\Gamma(s)}\frac{w^k}{(1-wy)^{s+k}}& = -\sum_{s=0}^{j}\binom{j+1}{s+1}(-1)^{s}\frac{\Gamma(s+k+1)}{\Gamma(s+1)}\frac{w^k}{(1-wy)^{s+k+1}}.
S = \oint d \tau \int d^{2}r_{\perp}\left({\frac{1}{2}}{\Omega}^{-}{\vec{\nabla}}^{2}{\dot{\Omega}}^{+}-{\frac{1}{2}}{\Omega}^{+}{\vec{\nabla}}^{2}{\dot{\Omega}}^{-}+{\dot{k}}^{+}{\Omega}^{-}-{\dot{k}}^{-}{\Omega}^{+}\right) .
2 \pi{\rho}_{a , j}( \beta ) = \sigma_{j}^{( \infty )}* \rho_{a , 0}- \sum_{b = 1}^{N - 1}\sum_{l = 1}^{\infty}A_{j l}^{( \infty )}* K_{a b}^{( N )}* \widetilde \rho_{b , l}( \beta )
\frac{dr}{ds}=T=\frac{-a}{\sqrt{a^{2}+b^{2}}}sin\frac{s}{\sqrt{a^{2}+b^{2}}}i+\frac{a}{\sqrt{a^{2}+b^{2}}}cos\frac{s}{\sqrt{a^{2}+b^{2}}}j+\frac{b}{\sqrt{a^{2}+b^{2}}}k
\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
\mu \left( \biggl\{\omega \in \Omega : \frac{1}{n}\sum_{i=1}^n h \circ Z_i \geq \varepsilon \biggr\}\right) \leq 2 \exp \left( - \frac{n \varepsilon^2}{8 (\log n)^{\frac{2}{\gamma}}(\sigma^2 + \varepsilon B/3)}\right),