harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I(f,g,h)=\iint_{\begin{subarray}{l}\xi=\xi_{1}+\xi_{2}\tau=\tau_{1}+\tau_{2}\end{subarray}}| 5\delta \xi^{4}+3\gamma \xi^{2}+2 \xi_1 |^{\frac{1}{2}}\hat{f}(\xi, \tau)\hat{g}(\xi_1, \tau_1)\hat{h}(-\xi_{2}, -\tau_{2}).
\|x\|_{L^p (\mathcal{L},\varphi)}^p =\|P_M x\|_{L^p (\mathcal{L},\varphi)}^p +\|Q_M x\|_{L^p (\mathcal{L},\varphi)}^p ,1\leq p<\infty ,\x\in L^p (\mathcal{L},\varphi) .
\Delta E_{CMC}^{*}=\sqrt{(\frac{L_{2}^{*}-L_{1}^{*}}{l\times S_{L}})^{2}+(\frac{C_{2}^{*}-C_{1}^{*}}{c\times S_{C}})^{2}+(\frac{\Delta H_{ab}^{*}}{S_{H}})^{2}}
\mathcal{M}\left\lbrace{}^{\Psi}B_{p}^{(\alpha,\beta)}(x,y)\right\rbrace =\int_{0}^{1}t^{x-1}(1-t)^{y-1}\int_{0}^{\infty}p^{s-1}{}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{t(1-t)}\right)dpdt.
{\hat\sigma}_{12}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})=\frac{i}{q}\;{\rm Tr}\;P_0^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})\left[P_{0,1}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}}),P_{0,2}^{(\Lambda,N)}(\tilde{\boldsymbol{\phi}})\right].
C \, T^{( \kappa \lambda ) \sigma}= - e \left( 2 m + n - 4 d \right) D^{\lambda \sigma \kappa}+ \frac{e}{2}\left( 2 a + c \right) \epsilon_{\mu \nu}{}^{\kappa \lambda}D^{\nu \sigma \mu}- \frac{e}{2}E \, \epsilon_{\mu}{}^{\sigma \kappa \lambda}D^{\mu}- \frac{e}{2}F \, G^{\sigma \kappa \lambda}\, .
q_k(y) = \frac{1}{\Delta_{k+1}(a)}\begin{vmatrix}1 1 \cdots 1 a_1 a_2 \cdots a_{k+1}\vdots \vdots \ddots \vdots \delta(y-a_1) \delta(y-a_2) \cdots \delta(y-a_{k+1}) \end{vmatrix}.
\Psi_{j}\left({\bf r}, t \right) = \frac{\mathrm{e}^{i{\bf k}_{j}\cdot{\bf r}- i E_{{\bf k}_{j}}t}}{\sqrt{2 \vert E_{{\bf k}_{j}}\vert \Omega}}, \quad \Phi \left({\bf r}, t \right) = \frac{g}{\mu_{0}^{2}}\frac{N}{2 \vert E_{{\bf k}_{j}}\vert \Omega}, \quad \Omega \rightarrow \infty ,
R_1 < \min\{I(X_1;Y_1),\, I(X_1;Y_2)\}+ \min\{I(X;Y_1|X_1,W_1),\, I(X;Y_2|X_1)\}:=I_1, R_2 < \min\{I(W_1;Y_1|X_1),\,I(W_1;Y_2|X)\}+ I(W;Y_2|X,W_1) := I_2.
\left( \begin{matrix}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{matrix}\right) \, \partial_{0}\left( \begin{matrix}{{\psi}}\\{{\chi}}\\ \end{matrix}\right) + \left( \begin{matrix}{{- 1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) \, \partial_{x}\left( \begin{matrix}{{\psi}}\\{{\chi}}\\ \end{matrix}\right) = \left( \begin{matrix}{{0}}\\{{0}}\\ \end{matrix}\right) \, .
(P_n(\xi_1\otimes \cdots \otimes \xi_n))(\theta_1,\cdots,\theta_n) = \frac{1}{n!}\left(\sum_{\sigma\in\mathfrak{S}n}\prod_{\substack{j<k \sigma(j)>\sigma(k)}}S(\theta_{\sigma(j)}-\theta_{\sigma(k)}) \cdot \xi_{\sigma(1)}(\theta_1)\cdots\xi_{\sigma(n)}(\theta_n)\right)
\frac{s_{0}}{x}+ \frac{s_{1}}{x^{2}}+ ... + \frac{s_{2r-1}}{x^{2r}}+ \sum_{k \geq 2r}\frac{s_{k}^{(r)}}{x^{k+1}}= \frac{Q_{r}(x)}{P_{r}(x)}\ ,
B(s) = 2^{s-2}\left( e^{i \frac{\pi}{2}(s-1)}+ e^{-i \frac{\pi}{2}(s-1)}\right) \frac{\Gamma \left(\frac{s+1}{2}\right) \Gamma \left(1 - \frac{s}{2}\right) \Gamma \left(s-\frac{1}{2}\right)}{{{\pi}}\Gamma(s+2)},
[\begin{matrix}1&1\\F_{1}&F_{2}\\P_{1}&P_{2}\end{matrix}][\begin{matrix}x_{1}\\x_{2}\end{matrix}]\le[\begin{matrix}L\\F\\P\end{matrix}],[\begin{matrix}x_{1}\\x_{2}\end{matrix}]\ge[\begin{matrix}0\\ 0\end{matrix}].
f_0= X_1^2X_2^4-X_1^4X_2^2, f_2=X_0^2X_1^2X_2^2-X_0^2X_1^4, f_1=X_0^4X_2^2-X_2^6,\quad\; f_3=X_0^4X_1^2-X_1^2X_2^4.
t_1 := \min \max\{d(v,v') - \displaystyle c_{i_0}\sum_{e \in P(v,v'), \bar{x}_e > 0}\frac{t}{c_e},d(v,a) + \displaystyle c_{i_0}\sum_{e \in P(v,a), \bar{x}_e > 0}\frac{t}{c_e}\},
\frac{\partial \left(MA\right)}{\partial q_{j}}=\frac{\partial \left(MA_{j}\right)}{\partial q_{j}}=\left(1-\alpha\right)>0 &\frac{\partial \left(MH\right)}{\partial q_{j}}=\frac{\partial \left(MH_{j}\right)}{\partial q_{j}}=\frac{\left(1-\alpha\right)p^{2}_{j}}{\left[\left(1-\alpha\right)p_{j}+\alpha q_{j}\right]^{2}}>0
d s_{5}^{2}= - f^{- 2 / 3}( r ) \left( 1 -{\frac{r_{0}^{2}}{r^{2}}}\right) d t^{2}+ f^{1 / 3}( r ) \left[ \left( 1 -{\frac{r_{0}^{2}}{r^{2}}}\right)^{- 1}d r^{2}+ r^{2}d \Omega_{3}^{2}\right] ,
\frac{d^3\lambda_\pm}{d\theta^3}=2\sin \tau\cdot\left(\mp\frac{3}{2}\lambda_{2,-3}h^{-5/2}\mp\frac{1}{2}\lambda_{2,-1}h^{-3/2}\pm\frac{1}{2}\lambda_{2,1}h^{-1/2}+ O(1)\right).
\int_{-2/(3\pi)}^{2/(3\pi)}\frac{t^2\sin^2(t^{-1})dt}{(t-\frac{1}{\pi n})^2}=\int_{-2/(3\pi)}^{1/(\pi n+\pi/2)}\frac{t^2\sin^2(t^{-1})dt}{(t-\frac{1}{\pi n})^2}+\int_{1/(\pi n+\pi/2)}^{1/(\pi n-\pi/2)}\frac{t^2\sin^2(t^{-1})dt}{(t-\frac{1}{\pi n})^2}+\int_{1/(\pi n-\pi/2)}^{2/(3\pi)}\frac{t^2\sin^2(t^{-1})dt}{(t-\frac{1}{\pi n})^2}&:=I_1^{(n)}+I_2^{(n)}+I_3^{(n)}.
\sum_{n=0}^\infty\sum_{i=0}^\infty \sum_{j=0}^\infty A_{n,i,j}z^ix^jy^n(1-x)^{-(n+1)}(1-z)^{-(n+1)}= \sum_{k=0}^\infty \frac{z^k}{1-x(1-y)^{-k}}\,.
S_{d}= \int d^{d}x [ \sqrt{\mathrm{d e t}( \delta_{\mu \nu}+ 2 \pi \alpha^{\prime}\tilde{F}_{\mu \nu})}+ \frac{1}{2}\mid D_{\mu}\varphi \mid^{2}+ \lambda ( \mid \varphi \mid^{2})^{2}+ \frac{1}{2}i \Lambda^{\mu \nu}( \tilde{F}_{\mu \nu}- 2 \partial_{\mu}A_{\nu}) ] ,
sn[\frac{1}{5}K(k);k]=\{\frac{\sqrt{5}\vartheta_{01}[q(k)^{5}]}{\vartheta_{01}[q(k)]}-1\}\{\frac{5\vartheta_{01}[q(k)^{10}]^{2}}{\vartheta_{01}[q(k)^{2}]^{2}}-1\}^{-1}
\begin{cases}\nabla G({\bf u})-\sum\limits_{1\leq a\leq p}\sigma_{aa}({\bf u})\nabla F_{aa}({\bf u})-\sum\limits_{1\leq b< c\leq p}\sigma_{bc}({\bf u})\nabla F_{bc}({\bf u})={\bf 0}U^TU=\mathbb{I}_p.\end{cases}
\max\limits_{1\leq k\leq n-1}\left|W_{k, n}\right|=\max\limits_{1\leq k\leq n-1}\left|\sum\limits_{i=1}^k\sum\limits_{j=k+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right)\right|.
\sum_{\mu \in R(M_n(\lambda))}\prod_{\{d \mid \exists m \colon (m,d) \in B(\mu) \}}\binom{S(d)(q)}{[E(\mu,(m,d)) \colon (m,d) \in B(\mu)]}= \prod_{d \in B(\lambda)}\binom{-S_d(q)}{-E(\lambda,d)}
\bar{u}_i (x) := \bar{u}_{i-1}(x) + \frac{1}{\mu_i}\Gamma^t \big(\varphi_i|\tau_i|, \mu_ie_i\cdot x\big) t_i + \frac{1}{\mu_i}\Gamma^n \big(\varphi_i|\tau_i|, \mu_ie_i\cdot x\big) n_i\, .
E_1^{st}=H^{s+t}(X_{\Gamma},({\mathcal F}^{s,\bullet}/{\mathcal F}^{s+1,\bullet})^{\Gamma})\Rightarrow H^{s+t}(X_{\Gamma},{\mathcal{M}}^{\Gamma}\otimes_K\Omega^{\bullet}_{{X}_{\Gamma}})
log(\frac{\sum_{i=1}^{n}x_{i}}{n})\ge\frac{\sum_{i=1}^{n}log(x_{i})}{n}or\frac{x_{1}+x_{2}+\cdot\cdot\cdot+x_{n}}{n}\ge\sqrt[n]{x_{1}\cdot x_{2}\cdot\cdot\cdot x_{n}}
| \Psi_{2}^{1 1}( p ) \rangle = \int [ 1 2 3 ] \frac{\tilde{\delta}( p - p_{1 2 3}) f_{1 2}f_{( 1 2 ) 3}}{2 p_{1 2}^{+}( p^{-}- p_{1 2 3}^{-}) ( p^{-}- p_{( 1 2 ) 3}^{-})}| 1 2 3 \rangle .
M_{a}^{\pm}=\mathop{\overleftarrow\prod}\limits_{i=1}^{t}\psi_{a}^{\pm}(z_{i})= \psi_{a}^{\pm}(z_{t})\psi_{a}^{\pm}(z_{t-1})\cdots\psi^{\pm}_{a}(z_{2})\psi_{a}^{\pm}(z_{1}).
(I-\lambda C_pG^{\omega,\lambda}_{p,p,p}(z))(I+\lambda C_p G^\omega_{p,p}(z))=I, &G^{\omega,\lambda}_{p,n,m}(z)=G^\omega_{n,m}(z)-\lambda G^\omega_{n,p}(z)C_pG^\omega_{p,m}(z) &\qquad\qquad\qquad\qquad+\lambda^2G^\omega_{n,p}(z)C_pG^{\omega,\lambda}_{p,p,p}(z)C_pG^\omega_{p,m}(z).
\min_{w \in \Sigma_{+}^{2}\cup \Sigma_{+}^{3}}\Re{f(w)}- \Re{f(z_{0})}= \Re{f(w_{3})}- \Re{f(z_{0})}\geq \Re{f(w_{2})}- \Re{f(z_{0})}\geq 2N^{-\frac{1}{6}},
\partial_i(\omega_if) \overset{(\ref{eq:extleibniz})}{=}-\omega_{i+1}f + \omega_{i}\partial_i(f) + (x_i - x_{i+1})\omega_{i+1}\partial_i(f) &\overset{(\ref{eq:relomega1}),(\ref{eq:relnh1})}{=}\omega_i \partial_i(f) + \partial_i(x_{i+1}\omega_{i+1}f) - \omega_{i+1}x_{i+1}\partial_i(f)
e_{a'}v_{a'+1,j}= f_{a'+1}^{(j)}e_{a'}f_{a'}^{(j)}v_{a'-1,j}& = f_{a'+1}^{(j)}f_{a'}^{(j)}e_{a'}v_{a'-1,j}+ f_{a'+1}^{(j)}f_{a'}^{(j-1)}[q^{h_{a'}};1-j]_q v_{a'-1,j}.
\tilde k_{y}^2 = 3 m (-u_{(1,3,2)}+ u_{(1,2,3)}- u_{(1,4,3)}+u_{(1,3,4)}-u_{(1,4,2)}+2u_\mathrm{Id}-u_{(2,4,2)}+ u_{(1,2,3)}-u_{(2,3,4)})
\frac{1}{2\pi}\int_{-\pi}^{\pi}(1-\cos(nx))g(x)dx=\frac{1}{\pi}\int_{0}^{\pi}(1-\cos(nx))g(x)dx=\int_{0}^{\pi}(1-\cos(y))\frac{1}{n\pi}\sum_{k=0}^{n-1}g\left(\frac{k\pi+y}{n}\right)dy
\begin{cases}u_\varepsilon(x)=\left(\frac{\varepsilon}{\varepsilon^2+r^2}\right)^{\frac{n-2}{2}}-\left(\frac{\varepsilon}{\varepsilon^2+\rho_0^2}\right)^{\frac{n-2}{2}}r\leq \rho_0, u_\varepsilon(x) = 0 r\geq \rho_0, \end{cases}
|\langle A\hat{k}_{\lambda}, \hat{k}_{\lambda}\rangle|^p{(1+ 2^{p-1}\displaystyle\sum_{n=3}^{\infty}\frac{1}{n^p})}<\frac{1}{2}(\frac{p}{p-1})^{p}\langle (|A|^{2p\alpha}+|A^*|^{2p(1-\alpha)}) \hat{k}_{\lambda}, \hat{k}_{\lambda}\rangle
\mathbb{P}_{((W_1,v_1),\dots, (W_n,v_n))=(x_1,\dots,x_n)}\left(\aleph_N-\aleph_n\geq 2^{-\frac{5}{2}}\lambda^{-1}\right)=\mathbb{P}_{(W_0,v_0)=x_t}\left(\aleph_{N-n}-\aleph_0\geq 2^{-\frac{5}{2}}\lambda^{-1}\right)
\int \frac{d^{d}k_{1}}{[ i \pi^{d / 2}]}~ ( q k_{1}) ~ P_{k_{1}, m_{1}}^{\nu_{1}+ 1}P_{k_{1}- q , m_{2}}^{\nu_{2}}= \nu_{2}q^{2}I_{\nu_{1}+ 1 ~ \nu_{2}+ 1}^{( d + 2 )}.
[a\bullet df] =(-1)^{\vert a\vert}\left( \delta(a\wedge df)-\delta a\wedge df -(-1)^{\vert a\vert}a\wedge\delta df\right) =\delta( df\wedge a)-(-1)^{\vert a\vert}\delta a \wedge df +a\wedge d\delta f =-df\wedge \delta a+\mathcal{L}(X)a+df\wedge \delta a =\mathcal{L}(X)a
B\begin{pmatrix}0 2 0\end{pmatrix}=B\begin{pmatrix}1 0 0 2 0 0\end{pmatrix}\begin{pmatrix}0 1\end{pmatrix}=\begin{pmatrix}1 0 0 2 0 0\end{pmatrix}\begin{pmatrix}1 1 0 1\end{pmatrix}\begin{pmatrix}0 1\end{pmatrix}=\begin{pmatrix}1 2 0\end{pmatrix},
\|f(T_L+\cdot)\|_{\cal{E}_{T_*-T_L}^{\frac{\beta-\gamma}2}\cal{C}^{\beta+1}}=\sup_{T_L<t\le T_*}(t-T_L)^{\frac{\beta-\gamma}2}\|f(t)\|_{\cal{C}^{\beta+1}}&\le\sup_{T_L<t\le T_*}t^{\frac{\beta-\gamma}2}\|f(t)\|_{\cal{C}^{\beta+1}}\le\|f\|_{\cal{E}^{\frac{\beta-\gamma}2}_{T_*}\cal{C}^{\beta+1}}.
\sum_{k=1}^\infty \|f_k^{(\ell)}\|_{L^\infty [0,1]}^2 \lesssim\sum_{m=1}^\infty \frac{r_{m+1}-r_m}{p_{r_m}}\left(\frac{p_{r_{m+1}}-p_{r_m}}{p_{r_m}}\right)^2\log^{2\ell}(p_{r_m}) &\lesssim\sum_{m=1}^\infty (r_{m+1}-r_m)\frac{(p_{r_{m+1}}-p_{r_m})^2}{p^3_{r_m}}\log^{2\ell}(p_{r_m})<\infty
\begin{aligned}[S^z, \widetilde{Q}^+] = \hphantom{+}\widetilde{Q}^+ \, , [S^+, \widetilde{Q}^-] = \hphantom{+}2\, \widetilde{Q}^z \, , [S^+, \widetilde{Q}^z] = -\widetilde{Q}^+ \, , & [S^z, \widetilde{Q}^-] = - \widetilde{Q}^- \, , [S^-, \widetilde{Q}^+] = -2\, \widetilde{Q}^z \, , [S^-, \widetilde{Q}^z] = \hphantom{+}\widetilde{Q}^- \, ,\end{aligned}
a(p)=\int_{-\infty}^\infty \!\int_{-\infty}^\infty\!\int_{-\infty}^\infty |xy|^p\Biggl\{\frac{1}{2\pi (1-\exp(-2|u|))^{1/2}}\exp\biggl( -\frac{1}{2(1-\exp(-2|u|))}(x^2+y^2 -2\exp(-|u|)|xy|)\biggl)-\phi(x)\phi(y)\Biggl\}dxdydu,
{\cal A}_{4}={}_{4 1 i}\langle V_{3}| ~_{j 2 3}\langle V_{3}| Y_{- 2}^{( i )}Y_{- 2}^{( j )}\frac{b_{o}^{( i )}}{L^{( i )}}W Q \frac{b_{o}^{( i )}}{L^{( i )}}| V_{2}\rangle_{i j}| \tilde{A}_{4}\rangle_{4}| \tilde{A}_{1}\rangle_{1}| \tilde{A}_{2}\rangle_{2}| \tilde{A}_{3}\rangle_{3}.
\frac{1}{|B|}\int_B|e^{-t_BL}((b-b_B)_{\sigma}f)(y)|dy &\leq \frac{1}{|B|}\int_B\int_{\mathbb{R}^n}|p_{t_B}(y,z)||(b(z)-b_B)_{\sigma}f(z)|dzdy &\leq \frac{1}{|B|}\int_B\int_{2B}|p_{t_B}(y,z)||(b(z)-b_B)_{\sigma}f(z)|dzdy &\quad+ \frac{1}{|B|}\int_B\sum\limits_{k=1}^{\infty}\int_{2^{k+1}B\setminus 2^kB}|p_{t_B}(y,z)||(b(z)-b_B)_{\sigma}f(z)|dzdy:=M+N.
(\widehat{G(z)\mspace{1.5mu}\xi})(\vec{k},\vec{p})=\sqrt{\frac{2}{\pi}}\;\frac{\hat{\xi}(\vec{p})+\hat{\xi}\!\left(\vec{k}\!-\frac{1}{2}\vec{p}\right)\mspace{-1.5mu}+\hat{\xi}\!\left(-\vec{k}\!-\frac{1}{2}\vec{p}\right)\!}{k^2+\frac{3}{4}p^2-z},z\in\rho(\mathcal{H}_0).
w_3 = s_1 s_{\alpha_2 + \alpha_4}s_{\alpha_3 + \alpha_4 + \alpha_5}s_{\alpha_2 + \alpha_3 + 2\alpha_4 + \alpha_5 + \alpha_6}s_{\alpha_2 + \alpha_3 + \alpha_4 + \alpha_5 +\alpha_6 + \alpha_7}s_{\alpha_7 + \alpha_8}s_6 s_{\alpha_2 + \alpha_4 + \alpha_5}.
\chi_{2,0}^{n,m}(x,y|\rho)\allowbreak=\allowbreak\sum_{k\geq0}\rho^{k}T_{k+n}(x)T_{k+m}(y)\allowbreak=\allowbreak (T_{n}(x)T_{m}(y)(w_{2}(x,y|\rho)-\rho^{4}) +\rho T_{n+1}(x)T_{m+1}(y)(1-2\rho^{2}+4\rho^{2}(x^{2}+y^{2})-4\rho xy) +\rho^{2}T_{n+2}(x)T_{m+2}(y)(1-4\rho xy)+\rho^{3}T_{n+3}(x)T_{m+3}(y))/w_{2}(x,y|\rho),
\frac{(y^{(m)}_{max})^2X}{\varphi(V)(\log X)^2}\Bigl(\sum_{\substack{u<R (u,VP_f)=1}}\frac{\mu(u)^2}{\varphi(u)}\Bigl)^{k-1}\ll\frac{(y^{(m)}_{max})^2\varphi(VP_f)^{k-1}X(\log X)^{k-3}}{\varphi(V)(VP_f)^{k-1}},
\{\chi_{\mu \nu}\}= \left( \begin{array}{c c c c}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\chi_{1 1}}}&{{\chi_{1 2}}}&{{\chi_{1 3}}}\\{{0}}&{{\chi_{1 2}}}&{{\chi_{2 2}}}&{{\chi_{2 3}}}\\{{0}}&{{\chi_{1 3}}}&{{\chi_{2 3}}}&{{\chi_{3 3}}}\\ \end{array}\right) ; \chi_{1 1}+ \chi_{2 2}+ \chi_{3 3}= 0 .
\|f_1f_2\|_{L^2}^2= \sum\limits_{k\in\mathbb{Z}}\|f_1f_{2,k}\|_{L^2}^2\leq \sum\limits_{k\in\mathbb{Z}}C\|f_1\|_{H^{1}}^2\big(\|f_{2,k}\|_{L^2}^2+\|\partial_zf_{2,k}\|_{L^2}^2\big) = C\|f_1\|_{H^{1}}^2\big(\|f_{2}\|_{L^2}^2+\|\partial_zf_{2}\|_{L^2}^2\big),
\sum_{\mathbf{v}\in P_{\mathbf{c}}}\exp{\Big(2\pi i \frac{\langle \pmb{\xi}, \mathbf{v}-\mathbf{c}\rangle}{p^{2k}}\Big)}=\exp{\Big(2\pi i \frac{\langle \pmb{\xi}, \mathbf{u}-\mathbf{c}\rangle}{p^{2k}}\Big)}\sum_{\mathbf{v}\in P_{\mathbf{c}}}\exp{\Big(2\pi i \frac{\langle \pmb{\xi}, \mathbf{v}-\mathbf{c}\rangle}{p^{2k}}\Big)}
U_{2m}^{(2)}=U_{m+1}U_{m-1}+q^{m-1}&=U_{m-1}\left( pU_{m}-qU_{m-1}\right) +q^{m-1}&=pU_{m-1}U_{m}-qU_{m-1}^{2}+q^{m-1}&=pU_{2m-1}^{(2)}-qU_{2m-2}^{(2)}+q^{m-1}.
\mathcal{G}^{\neq}\left( \delta_{x}- \delta_{x}^{\epsilon},{\rm emp}_{N}\right) = \int_{y:0<|y-x| \leq r}h^{\delta_{x}- \delta_{x}^{\epsilon}}d{\rm emp}_{N}(y) + \int_{y:|y-x| \geq r}h^{\delta_{x}- \delta_{x}^{\epsilon}}d{\rm emp}_{N}(y),
V^{\dagger}= V , \quad A_{\mu}^{\dagger}= A_{\mu}, \quad F_{\mu \nu}^{\dagger}= F_{\mu \nu}, \quad \left( D_{\mu}X \right)^{\dagger}= D_{\mu}X \quad \mathrm{i f}\quad X^{\dagger}= X \vspace *{- 0 . 1 5 c m}
\begin{aligned}\rho^0= \rho^{\rm in}, (\rho^{n+1}, \phi^{n+1}) = \arg \min_{(\rho, \phi)\in \mathcal{A}}\left\{\frac{1}{2\tau}d^2(\rho^n, \rho )+E(\rho, \phi) \right\}.\end{aligned}
\left[\cos(\kappa_j x) + \frac{\xi}{\kappa_j}\sin(\kappa_j |x|)\right]\cos(\kappa_j t) = \frac{\xi L}{2}\left[ \frac{1}{Z_j^2}+ 2\sum_{n=1}^\infty \frac{\cos(k_n x)\cos(k_n t)}{Z_j^2-(\pi n)^2}\right]
B_{0}( M_{*}^{2}) = \frac{\Gamma ( \frac{d}{2}) \Gamma ( d - 2 + 2 k ) \Gamma ( k + \frac{1}{2})}{\sqrt{\pi}\Gamma ( d - 2 ) \Gamma ( 2 k + 1 ) \Gamma ( k + \frac{d}{2})}M_{\ast}^{2 k}\ \cdot
R^n_{\frac{it}{\sigma \sqrt{n}}}1(v) - e^{- \frac{t^2}{2}}= \lambda^n_{\frac{it}{\sigma \sqrt{n}}}\Pi_{\frac{it}{\sigma \sqrt{n}}}1(v) + N_{\frac{it}{\sigma \sqrt{n}}}^n 1(v) - e^{- \frac{t^2}{2}}=: \sum_{j=1}^3 I_j(t),
z^{-\nu}J_{\nu}(z)=\frac{2^{1-\nu}}{\sqrt{\pi}\Gamma(\nu+\frac{1}{2})}\,\intop_{0}^{1}\left(1-u^{2}\right)^{\nu-\frac{1}{2}}\,\cos(zu)\,\,du,\,\,\,\,\,\,(\nu)>-\frac{1}{2},\,\,\,z\in\mathbb{C},
\mathrm{d}s_{*}^{2}= r_{0}^{2}\left[ \mathcal{R}^{2}\left\{u^{2}\mathrm{d}x_{d}^{2}+ \left({\frac{\mathrm{d}u}{u}}\right)^{2}\right\}+ \mathrm{d}\Omega_{\tilde{d}+ 1}^{2}\right] .
\frac{1}{k^2k\cdot n}\rightarrow \frac{1}{k^2+i\varepsilon}\left\{\frac{\Theta(-k^0)}{k\cdot n-i\xi}+\frac{\Theta(k^0)}{k\cdot n+i\xi}\right\},\hspace{.5cm}\begin{array}{c}\varepsilon\equiv 2\epsilon k_0^2 \rightarrow 0^{+}\xi \equiv \epsilon \mid k^0 \mid n^0\rightarrow 0^{+}\end{array},
h(x) &\le \Bigl(g_{9-}(x)+\frac{2}{6^{10}}(2^{10}x - 426)(24596 -3^{10}x)\Bigr)\Bigr|_{x=\frac{24169}{58025}}+ \frac1{10\cdot 6^{10}}- \eta &=-\frac{11899418967263802834461455793906308311731}{4952561012067063864753471157095621144079791360000}<0,
P\left(\sum_{0\leq t\leq T}\left(\mu^N_{t,k}(f)-\mu^N_{t-,k}(f)\right)^2\geq \epsilon\right) \leq e^{-N^2\epsilon}\mathbb{E}\exp\left(\|f\|^2_\infty Y(NK_2 T)\right) &=e^{-N^2\epsilon}e^{NK_2T(e^{\|f\|^2_\infty}-1)}
\begin{aligned}\Delta(t) \geq K\Big(\varepsilon_0-\eta(1+\sin\gamma_0)(1+\cos\gamma_0)\Big)-M\Big(1+\frac{1}{\varepsilon_0-\eta(1+\sin\gamma_0)}\Big) & \mbox{and}R(t) \geq \varepsilon_0.\end{aligned}
C_{2}=\frac{1-\frac{1}{2}}{\left(1-\frac{1}{2}\right)^2}\cdot\frac{1-\frac{2}{3}}{\left(1-\frac{1}{3}\right)^2}\cdot\frac{1-\frac{1}{5}}{\left(1-\frac{1}{5}\right)^2}\cdot\frac{1-\frac{2}{7}}{\left(1-\frac{1}{7}\right)^2}\cdot\prod_{7<p}\frac{1-\frac{2}{7}}{\left(1-\frac{1}{7}\right)^2}\approx 1.76\;.
\delta R_{A B}= \partial_{C}\delta \Gamma_{A B}^{C}- \partial_{B}\delta \Gamma_{A C}^{C}+ \overline{{{\Gamma}}}_{A B}^{C}\delta \Gamma_{C D}^{D}+ \delta \Gamma_{A B}^{C}\overline{{{\Gamma}}}_{C D}^{D}- \delta \Gamma_{B C}^{D}\overline{{{\Gamma}}}_{A D}^{C}- \overline{{{\Gamma}}}_{B C}^{D}\delta \Gamma_{A D}^{C}.
\left\{\hat{P}^{\mu}\hat{P}_{\mu}-e\left( \hat{P}^{\mu}\hat{A}_{\mu}+\hat{A}^{\mu}\hat{P}_{\mu}\right) +e^{2}\hat{A}^{\mu}\hat{A}_{\mu}+e\sigma^{\mu \nu}\left[ \hat{P}^{\nu},\hat{A}_{\mu}\right]-m^{2}\right\}\psi =0.
\begin{aligned}\|(u, v, \theta)(t)\|^2 + 2 \int_0^t\Big(\nu \|\nabla u\|^2 + \eta \| \nabla v \|^2 + \mu \|\nabla \theta\|^2 + \sigma_1 \|u\|_{\alpha+1}^{\alpha+1}+ \sigma_2 \|v\|_{\beta+1}^{\beta+1}\Big) \,d\ell =\|(u_0, v_0, \theta_0)\|^2, \end{aligned}
S = \frac{1}{\beta^{2}}\int d^{4}x ~ d^{2}\theta ~ d^{2}\bar{\theta}~ t r \left[ - \frac{1}{2}( e^{- V}D^{\alpha}e^{V}) \bar{D}^{2}( e^{- V}D_{\alpha}e^{V}) - e^{- V}\bar{\Sigma}e^{V}\Sigma \right] \, .
\frac d{ds}\left[ \begin{array}{c}\gamma^0 \gamma^1 \pi^0 \pi^1\end{array}\right] =\left[ \begin{array}{cccc}2i{\cal H}0 -2i 0 0 2i{\cal H}0 -2i 0 -eE 0 0 -eE 0 0 0\end{array}\right] \left[ \begin{array}{c}\gamma^0 \gamma^1 \pi^0 \pi^1\end{array}\right] ,
(m-(k-\ell))x_{i_1}\ldots x_{i_\ell}e_{k-\ell}(B)= x_1\ldots x_{i_{\ell-1}}\sum_{b \in B}e_{k-\ell+1}(\{i_\ell\}\sqcup B\setminus \{b\})- &x_1\ldots x_{i_{\ell-1}}(m-(k-\ell+1))e_{k-\ell+1}(B).
\mathbf{p}_{h,i,j}= \mathbf{p}_{d}+ p_{h,i,j}\mathbf{R}_{\phi_{h,i,j}}\mathbf{r}_{i,j}\phi_{h,i,j}= \sin^{-1}\frac{r_c}{r_{i,j}}p_{h,i,j}= \sqrt{(r_{i,j})^2 - r_c^2}
\lambda_{\alpha}\lambda_{\beta}=K_{\alpha\beta}^{\gamma}\lambda_{\gamma}, \ \quad\lambda_{\alpha}\lambda_{\beta}\lambda_{\delta}=K_{\alpha\beta}^{\gamma}\lambda_{\gamma}\lambda_{\delta}, K_{\alpha\beta\delta}^{\varepsilon}\lambda_{\varepsilon}=K_{\alpha\beta}^{\gamma}K_{\gamma\delta}^{\varepsilon}\lambda_{\varepsilon},
\begin{aligned}f^-(x_j) =f^-_j=f^-(x^*)-f^-_x(x^*) \alpha+O(h^2), f^+(x_j) =f^+_j=f^+(x^*)-f^+_x(x^*) \alpha+O(h^2), f^-(x_{j+1}) =f^-_{j+1}=f^-(x^*)+f^-_x(x^*) (h-\alpha) +O(h^2), f^+(x_{j+1}) =f^+_{j+1}=f^+(x^*)+f^+_x(x^*) (h-\alpha) +O(h^2),\end{aligned}
\begin{cases}\dfrac{\partial u}{\partial t}= \nabla \cdot (d \nabla u) + (m(x) + a(x)|u|^{p-2})u \mbox{in $(0, \infty)\times \Omega$}, u(0,x) = u_0(x) \geq 0 \mbox{in $\Omega$}, (d\nabla u) \cdot \mathbf{n}= b(x)|u|^{q-2}u \mbox{on $(0, \infty)\times \partial \Omega$}, \end{cases}
\!\!\!\!\!\! \sigma_0^2 = \bar{q}_0 = \sum\limits_{k>0:\bold{z}_k\in \Phi_2}{G_k \mathit{l}\left({\Vert \bold{z}_k \Vert}_2\right)}g_k P_k = \bar{q}_0^{(1)}+ \bar{q}_0^{(2)}+ \bar{q}_0^{(3)},
\frac{\partial C^{r}(u,v)}{\partial r}=\frac{1}{2\pi\sqrt{1-r^2}}\exp\Big(-\frac{\Phi^{-1}(u)^2+\Phi^{-1}(v)^2-2r\Phi^{-1}(u)\Phi^{-1}(v)}{2(1-r^2)}\Big).
\begin{aligned}|\eta_j|\le 1,|D_v \eta_j|\le N2^{-j}R^{-1}, |D^2_v \eta_j|\le N2^{-2j}R^{-2}, & |D_x \eta_j|\le N2^{-3j}R^{-3}, |\partial_t \eta_j| \le N2^{-2j}R^{-2}.\end{aligned}
\iota(g',g') \cdot g_0^{-1}= \begin{pmatrix}\frac{a}{2}-\frac{a}{2}\frac{b}{2}\frac{b}{2}\frac{c}{2}-\frac{c}{2}\frac{d}{2}\frac{d}{2}c c d -d -a -a -b b \end{pmatrix},
LL\left(|\varrho_{ZZ}(L,L)|^2\right)=-4 \varrho_{ZZZ}(L,L,L) \overline{\varrho_{ZZ}(N,L)}- 2 \varrho_{ZZ}(L,L) (L \overline{\varrho_{ZZ}(N,L)}) + \overline{\varrho_{ZZ}(L,L)}\left(\varrho_{ZZZZ}(L,L,L,L) - 3 \varrho_{ZZZ}(L,L,\varrho_{ZZ}\cdot L)\right).
\delta_{\mathrm{S}}( k ) = \delta_{\mathrm{A}}( k ) + \operatorname{a r c t a n}( \frac{2 \: \mathrm{I m}\: \beta^{\prime}( k , 0 )}{k + 2 \: \mathrm{R e}\: \beta^{\prime}( k , 0 )}) .
\Big|\sum_{\boldsymbol{\sigma}\in S_n}\mathop{\mathrm{sgn}}(\boldsymbol{\sigma}) \prod_{i=1}^n\sum_{y\in\mathcal{S}_{\sigma_i}}a^\varepsilon_{\mathfrak{m}_i,y}(\lambda) -\sum_{\boldsymbol{\sigma}\in S_n}\mathop{\mathrm{sgn}}(\boldsymbol{\sigma})\prod_{i=1}^n a^{\mathbf{Q}}_{i,\sigma_i}(\lambda) \Big| \leq\tilde{C}(\tilde C+|\lambda|)^{n-1}\varepsilon^\alpha.
\Gamma(\frac{1}{2}+bi)\Gamma(1-(\frac{1}{2}+bi))=\Gamma(\frac{1}{2}+bi)\Gamma(\frac{1}{2}-bi)=\frac{\pi}{sin\pi(\frac{1}{2}+bi)}=\frac{\pi}{cos\pi bi}=\frac{\pi}{cosh\pi b}
{\cal V}^{( 0 )}( \Phi ) ={\frac{1}{2}}I_{0}( \mu^{2}) -{\frac{1}{2}}I_{0}({\cal M}^{2}) +{\frac{{\cal M}^{2}}{4}}I_{1}({\cal M}^{2}) +{\frac{m^{2}}{\beta^{2}}}.
G_{n}(x_{1},\dots ,x_{n})=\left< \Omega \right| T\phi_{H}(x_{1})\dots \phi_{H}(x_{n})\left| \Omega \right> =\frac{\left< 0\right| T\phi_{D}(x_{1})\dots \phi_{D}(x_{n})\;{\cal{S}}\left| 0\right>}{\left< 0\right|{\cal{S}}\left| 0\right>}.
\langle\,\Lambda \,\vert\Lambda\,\rangle\,=1\,,\,\,\langle\,\Phi \vert X\Psi\,\rangle\,=\,\langle\,{^tX}\Phi\vert\Psi \,\rangle\,,\,\,\langle\,\Phi\vert\Psi\,\rangle = \langle\,\Psi \vert\Phi\,\rangle\,.
d_1 = 1 + 0 = 0 + 1 = 1 (n = 1); d_2 = 0 + 1 = 1 + 0 = 1 (n = 2); d_3 = 3 + 0 = 2 + 1 = 3 (n = 3); d_4 = 8 + 3 = 9 + 2 = 11 (n = 4); d_5 = 45 + 8 = 44 + 9 = 53 (n = 5)\ldots
a\sharp b = ab + \frac{h}{2i}\{a,b\}+ \left(\frac{h}{2i}\right)^2 \left[\frac{1}{2}\partial^2_x a \partial^2_{\xi}b + \frac{1}{2}\partial^2_{\xi}a \partial^2_x b - \partial_x \partial_{\xi}a \partial_x \partial_{\xi}b\right] +r^{a\sharp b}_2 \,,
\langle \zeta ( t , \vec{x}) \zeta ( t , \vec{x}) \rangle = \frac{2 \langle \delta \bar{\psi}( t , \vec{x}) \psi ( t ) \cdot \bar{\psi}( t ) \delta \psi ( t , \vec{x}) \rangle}{( \bar{\psi}\psi )^{2}},
b(X,t+t_0(X)) & = \nu^{(i)}(x) \nu^{(j)}(x+ r_0 y(x, \frac{t-t_0(X)}{r_0})) -\nu^{(j)}(x) \nu^{(i)}(x+ r_0 y(x,\frac{t-t_0(X)}{r_0})) ,
\sum_{\alpha}''' w\left(\frac{N(\alpha)}{X}\right) \log N(\alpha) = \frac{1}{2\pi i}\int_{(2)}\sum_{\alpha}\frac{\nu(\alpha) \log N(\alpha)}{N(\alpha)^s}X^s \mathfrak{w}(s) ds= -\frac{1}{2\pi i}\int_{(2)}\frac{\partial I(s)}{\partial s}X^s \mathfrak{w}(s) ds.
P_{}= P^R_r = M^2\cdot (\mu^R_h|g|\beta^R)^2 + M\cdot\big\{\sigma^R_h|g|^2(\beta^R)^2 + 2\mu^R_h|g|\beta^R \mathbb{E}[|h^R_d|] \big\}+\mathbb{E}[|h^R_d|^2],
\gamma_0^i = (\lambda z_i, d(\lambda) - d(\mu^i), \mu^i z_i) \gamma_1^i = (\mu^i z_i, -d(\lambda^i_n), \lambda_n \mu^i z_i) & \vdots \gamma_n^i = (\lambda^i_2 \cdots \lambda^i_n \mu^i z_i, -d(\lambda^i_1), \lambda_1^i \cdots \lambda_n^i \mu^i z_i).
\gamma^1_{44}= 1-\frac{k}{4\ell}e^{-y}+O(e^{-2y}), \gamma^2_{33}= -\cot(\phi) +O(e^{-2y}) =O(e^{-\frac{3y}{4}}), \gamma_{23}^4 = \frac{k}{4\ell}e^{-y}+O(e^{-2y}), \gamma_{32}^4 = -\frac{k}{4\ell}e^{-y}+O(e^{-2y}), \gamma_{42}^3 = -\frac{k}{4\ell}e^{-y}+O(e^{-2y}) .
B_{3}(y) = \int_{0}^{\infty}\frac{d\rho}{(1+2(\rho^{2}+y^{2})+(\rho^{2}-y^{2})^{2})^{1/2}}\leq C \int_{0}^{2y}\frac{d\rho}{y}+ \int_{2y}^{\infty}\frac{d\rho}{(1+\rho^{2})}\leq C
\mathbb{Q}(S,2) =\left\{b\in\mathbb{Q}^*/(\mathbb{Q}^*)^2 :_l(b)\equiv 0 ~(\bmod{2}) ~ ~ l\neq 2, p,q\right\}&=\left\{\pm 1,\; \pm 2, \;\pm p, \;\pm q, \;\pm 2p, \;\pm 2q, \;\pm pq, \;\pm 2pq \right\}.
\sum_{k=1}^{n}\varphi(k)=\frac{1}{2}(1+\sum_{k=1}^{n}\mu(k)\lfloor\frac{n}{k}\rfloor^{2})=\frac{3}{\pi^{2}}n^{2}+O(n(logn)^{\frac{2}{3}}(loglogn)^{\frac{4}{3}})
\frac{d}{dt}\left(1-\frac{1}{16\pi}\int_{\Sigma_t}(H^2-4)d\mu_t\right) =-\left(1-\frac{1}{16\pi}\int_{\Sigma_t}(H^2-4)d\mu_t\right)+\frac{1}{8\pi}\int_{\Sigma_t}\frac{|\nabla H|^2}{H^2}d\mu_t.
U_{1 , 1}( \alpha ) = U ( \alpha ) = \frac{\left[ \pi \mu \gamma ( b^{2}) \right]^{- \alpha / b}\Gamma ( 2 + b^{2}) \Gamma ( 1 + 1 / b^{2})}{\Gamma ( 2 + b^{2}- 2 b \alpha ) \Gamma ( 1 + b^{- 2}- 2 \alpha / b )}
\Vert T_\phi^{A_0,A_1}\colon \mathcal S^\infty_n \to \mathcal S^\infty_n\Vert = \bigl\Vert\{\phi(\lambda_i^{(0)},\lambda_k^{(1)})\}_{i,k=1}^n\colon \mathcal S^\infty_n \to \mathcal S^\infty_n\bigr\Vert.
R_{2,H,f}(V_k)-R_{2,H,f}(W) &=\int_0^1(T^{H_t,H_t}_{f^{[1]}}(V_k-W)-T^{H,H_t}_{f^{[1]}}(V_k-W)+T^{H,H_t}_{f^{[1]}}(V_k-W)-T^{H,H}_{f^{[1]}}(V_k-W))dt &=\int_0^1(T^{H_t,H,H_t}_{f^{[2]}}(W+t(V_k-W),V_k-W)+T^{H,H_t,H}_{f^{[2]}}(V_k-W,W+t(V_k-W)))dt.
Q_{\vec \ell}(t,x) = Q \left (\left ( -\frac{t}{\sqrt{1 - |\vec \ell|^2}}+ \frac{1}{|\vec \ell|^2}\left ( \frac{1}{\sqrt{1 - |\vec \ell|^2}}- 1 \right )\vec \ell \cdot x\right ) \vec \ell + x\right )= Q_{\vec \ell}(0, x - t\vec \ell),
\Delta \theta = 4 \pi G_{0}\left[ U - I^{2}\left( \frac{3}{2}- \ln \left( \frac{p}{r_{0}}\right) \right) \right] + 4 \pi \hspace{0 . 1 c m}\ln ( 2 ) I^{2}G_{0}+ O ( G_{0}^{2}) \, .
\int_{\mathbb{R}^{d}}\prod_{j=1}^{d}f_{j}(\pi_{j}(x))^{1/(d-1)}dx\le\prod_{j=1}^{d}(\int_{\mathbb{R}^{d-1}}f_{j}(\hat{x}_{j})d\hat{x}_{j})^{1/(d-1)}.
\mathcal{F}(f(x))''(\xi) = \int_{0}^{\frac{2}{\sqrt{\xi}}}f(x,r) \partial_{2}^{2}\phi(r,\xi) dr + \int_{\frac{2}{\sqrt{\xi}}}^{\infty}f(x,r) \partial_{2}^{2}\phi(r,\xi) dr