harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D^{\mu \nu}( q ; \phi ) \equiv{\frac{- i}{q^{2}}}\Bigl \{g^{\mu \nu}+ i q^{\mu}\widetilde \phi^{\nu}( q ) - i{\widetilde \phi^{\mu *}}( q ) q^{\nu}+ q^{\mu}q^{\nu}\vert \widetilde \phi ( q ) \vert^{2}\Bigr \}\ ,
M_{\mu,\nu}(y) = y^{\nu+\frac{1}{2}}e^{\frac{y}{2}}\frac{\Gamma(1+2\nu)}{\Gamma(\nu+\mu+\frac{1}{2})\Gamma(\nu-\mu+\frac{1}{2})}\int_0^1 t^{\nu+\mu-\frac{1}{2}}(1-t)^{\nu-\mu-\frac{1}{2}}e^{-yt}dt, W_{\mu,\nu}(y) = y^{\nu+\frac{1}{2}}e^{\frac{y}{2}}\frac{1}{\Gamma(\nu-\mu+\frac{1}{2})}\int_1^{\infty}t^{\nu+\mu-\frac{1}{2}}(t-1)^{\nu-\mu-\frac{1}{2}}e^{-yt}dt.
\hbox{solv}\bigl({\hbox{O}(5,Q)\over \hbox{O}(5)\times\hbox{O}(Q)}\bigr)=\hbox{solv}\bigl({\hbox{O}(1,Q)\over \hbox{O}(Q)}\bigr)+\hbox{solv}\bigl({\hbox{O}(4,Q)\over O(4)\times\hbox{O}(Q)}\bigr)
L_{\phi}= - \frac{3}{4}c \int_{-R}^{R}e^{-2\sigma (y)}\sqrt{-\gamma}\partial_{\mu}\phi \partial^{\mu}\phi \, dy = -\frac{3}{2}\frac{kR^2}{e^{2kR}-1}\eta^{\mu \nu}\partial_\mu \phi \partial^\nu \phi.
\mathrm{(I)}~\mathrm{MDTE}_{(\boldsymbol{a},\boldsymbol{b})}(\mathbf{Y}) =\boldsymbol{\mu}+\frac{\mathbf{\Sigma}^{\frac{1}{2}}\boldsymbol{\delta}}{F_{\mathbf{Z}}(\boldsymbol{\xi_{a}},\boldsymbol{\xi_{b}})},
\tilde{G}(x,y)=\log|x-y|+\sum_{k=1}^\infty\big((\alpha\beta)^k(\log|\phi_{2k}(x)-y|+\log|\psi_{2k}(x)-y|) &-(\alpha\beta)^{k-1}(\alpha\log|\phi_{2k-1}(x)-\overline{y}|+\beta\log|\psi_{2k-1}(x)-\overline{y}|) \big),
\begin{aligned}a((u_1,v_1),(u_2,v_2))={}\int_{\Omega}\rho u_1u_2+ (\nabla u_1)^{\top}K(x) \nabla u_2 +(\nabla u_1)^{\top}a(x) \nabla u_2 \, dx &{}+{}\int_{\Omega}\rho v_1v_2+ (\nabla v_1)^{\top}K(x) \nabla v_2 +(\nabla v_1)^{\top}b(x) \nabla v_2 \, dx. \end{aligned}
{\frac{1}{A}}{\frac{\partial{\cal E}}{\partial m^{2}}}={\frac{1}{2}}\sum_{n = 1}^{\infty}\int \int{\frac{d k_{1}d k_{2}}{( 2 \pi )^{2}}}{\frac{1}{\sqrt{k_{1}^{2}+ k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+ m^{2}}}}\, .
|\tilde v(t)|^2 = 2\sum_{i=1}^m \int_{(0,t]}\langle v_i^{\ast}(s), v^{(2)}_n(s) \rangle\, dA(s) & + 2 \int_{(0,t]}(\tilde v_n(s)-h_n(s), dh(s)) + |h(t)|^2-K_n(t)
\left(\sqrt{1+s}+\sqrt{s}\right)^{-2\mu}= \left(\sqrt{1+s}-\sqrt{s}\right)^{2\mu}= \left(1+2s+2\sqrt{s(1+s)}\right)^{-\mu}= \left(1+2s-2\sqrt{s(1+s)}\right)^{\mu}
Q(E)=P^{(0)}(E)P^{(1)}(E)P^{(2)}(E)P^{(3)}(E),\;P^{(k)}(E)=E-15e_{k},\;(k=1,2,3), & P^{(0)}(E)=E^{4}-54g_{2}E^{2}-864g_{3}E-135g_{2}^{2}.
\left. \frac{{\cal V}_{\rm eff}}{N}\right|_{\rm spir}= \frac{m^2}{4\pi}\ln \left( \frac{m^2}{m_0^2}\right)+ \frac{m_0^2-m^2}{4\pi}- \frac{2}{\beta}\int \frac{{\rm d}k}{2\pi}\ln \left( 1 +{\rm e}^{-\beta \epsilon(k)}\right) -\frac{\mu^2}{2\pi}\ ,
\partial\cdot A=\partial^{\mu}\eta_{\mu\nu}A^{\nu}=\partial_{\nu}A^{\nu}=(\frac{\partial_{t}}{c},-\vec{\nabla})\cdot(\frac{\phi}{c},\vec{a})=\frac{\partial_{t}}{c}(\frac{\phi}{c})+\vec{\nabla}\cdot\vec{a}=\frac{\partial_{t}\phi}{c^{2}}+\vec{\nabla}\cdot\vec{a}=0
H(x_{1}, x_{2}) = \left( \begin{array}{cc}\bar{q}_{D}(x_{1}) q_{C}(x_{2}) D_{AB}(x_{1}) q_{C}(x_{2}) \bar{q}_{D}(x_{1}) \bar{D}_{EF}(x_{2}) D_{AB}(x_{1}) \bar{D}_{EF}(x_{2}) \end{array}\right)
(1-\tau)^{\beta_k}\sum_{j=1}^k (R-\rho)^{\beta_j}f_j(\rho) \leq P_0 \leq \liminf_{i\to\infty}(\delta^{i+1}P_{i+1}+ Q_0 + \delta Q_1 + \cdots + \delta^{i}Q_{i}).
F(z)^{m}=1+mf_{1}z+m((m-1)f_{1}^{2}+2f_{2})\frac{z^{2}}{2}+(m(m-1)(m-2)f_{1}^{3}+6m(m-1)f_{2}+6mf_{3})\frac{z^{3}}{6}+\cdot\cdot\cdot,
\pi_n^*\left( \sum_{T \in G_{0,n}}{\xi_{T}}_* \left(\prod_{v\in V(T)}\frac{1}{\psi_{h(v)}-1}\right) \right)=\sum_{T \in G_{0,n+1}^{ne}}{\xi_{T}}_* \left(\prod_{v\in V(T)}\frac{1}{\psi_{h(v)}-1}\right).
\begin{aligned}(3b+4)(w^2+2x^2+(b+1)y^2+2bz^2) = (w+2x+(b+1)y+2bz)^2+2(w-(b+1)y+bz)^2 &+(b+1)(w-2x+y)^2+2b(w+x-2z)^2.\end{aligned}
x^{k}=\sum_{i=0}^{n-k}(\begin{matrix}n-k\\i\end{matrix})\frac{1}{(\begin{matrix}n\\i\end{matrix})}b_{n-i,n}(x)=\frac{1}{(\begin{matrix}n\\k\end{matrix})}\sum_{j=k}^{n}(\begin{matrix}j\\k\end{matrix})b_{j,n}(x).
\overline{\zeta}_q'(0)= 4\xi_q'(-1)+2(n+1-q)(\xi_q(0)-\xi_q'(0))+\sum_{j=2}^{2n}(n+1-q)^j w_j\gamma_{n,q}(j-1) &+(n+1-q)\eta_q'(0)-2 \eta_q'(-1)-(n+1-q)\xi_q'(0)+2\xi_q'(-1) = 6 \xi'_q(-1)-3(n+1-q)\xi_q'(0)-2\eta_q'(-1)+(n+1-q)\eta'_q(0)+2(n+1-q)\xi_q(0) &+\sum_{j=2}^{2n}(n+1-q)^j w_j\gamma_{n,q}(j-1).
\begin{aligned}E=2\omega \sum_{i=1}^{N}k_i+\omega \sum_{i=1}^{N}\gamma_i+\frac{\omega N}{2},~~~~~k_i=0,1,2,\cdots \gamma_i=\frac{1}{2}(1+\sqrt{1+4\lambda_i+(d_i-1)(d_i-3)})~~~~~i=1,2,\cdots,N.\end{aligned}
(Lv,v) =(Lw,w)+2(v,\phi) (L\phi, w)+(v,\phi)^2 (L\phi,\phi) & = (Lw,w) \ge b \| w \|_{H^1}^2 = b \| v - (v, \phi) \phi \|_{H^1}^2 & \ge b (\| v \|_{H^1}- |(v,\phi)|\| \phi \|_{H^1})^2 \ge b \left( \frac{1}{2}\| v \|_{H^1}^2 - |(v,\phi)|^2\| \phi \|_{H^1}^2 \right).
{\cal G}=\frac{1}{2}M_{ij}d{\bf x}_i\cdot d{\bf x}_j+\frac{g^4}{2(4\pi)^2}(M^{-1})_{ij}(d\xi_i+{\bf W}_{ik}\cdot d{\bf x}_k)(d\xi_j+{\bf W}_{jl}\cdot d{\bf x}_l) .
\nabla_1 v^1 = \partial_1 v^1 + (\partial_2 \phi) v^2, \nabla_1 v^2 = \partial_1 v^2 - (\partial_2 \phi) v^1, \nabla_2 v^1 = \partial_2 v^1 - (\partial_1 \phi) v^2, \nabla_2 v^2 = \partial_2 v^2 + (\partial_1 \phi) v^1.
\mathrm{var}(Y_{n,i})=\frac{1}{b_{n}}\sum_{j=1}^{m}\frac{\lambda_{j}^{2}}{f(x_{j})\int K^{2}(u)du}\mathrm{var}(K(\frac{x_{j}-X_{0}}{b_{n}})) +\frac{2}{b_{n}}\sum_{p=1}^{m-1}\sum_{j=p+1}^{m}\frac{\lambda_{j}\lambda_{p}}{(f(x_{j})f(x_{p}))^{1/2}\int K^{2}(u)du}\mathrm{cov}(K(\frac{x_{j}-X_{0}}{b_{n}}),K(\frac{x_{p}-X_{0}}{b_{n}}))=I_{n}+II_{n}.
\|x_{n+1}^{\frac{1-2s}{2}}u_{1}\|_{L^{2}(B_{10}^{+})}^{2}\le\int_{0}^{10}\int_{\mathbb{R}^{n}}x_{n+1}^{1-2s}|u_{1}(x',x_{n+1})|^{2}\,dx'\,dx_{n+1}\le\bigg(\int_{0}^{10}x_{n+1}^{1-2s}\,dx_{n+1}\bigg)\|u\|_{L^{2}(B_{16}')}^{2}=C\|u\|_{L^{2}(B_{16}')}^{2}.
\frac{d^{2}x_{A}}{dt^{2}}=a_{AB}+\frac{dv_{B}}{dt}+\sum_{j=1}^{3}\frac{dx_{j}}{dt}\frac{du_{j}}{dt}+\sum_{j=1}^{3}x_{j}\frac{d^{2}u_{j}}{dt^{2}}.
{H}_{i n t}= \bar{\psi}* e \gamma^{\mu}\bar{B}_{\mu}\left[ 1 + \frac{e}{2}\gamma^{+}\frac{1}{i \partial_{-}- e \bar{B}_{-}}e \gamma^{\nu}\bar{B}_{\nu}\right] \ast \psi = \bar{\psi}\ast{\cal V}[ \bar{B}_{\mu}] \ast \psi .
D_{a\alpha}{\rm v}^{\beta}{}_{\gamma}(\theta)=\textstyle{\frac{2}{3}}\delta_{\alpha}{}^{\beta}D_{a\delta}{\rm v}^{\delta}{}_{\gamma}(\theta)-\textstyle{\frac{1}{3}}\delta^{\beta}{}_{\gamma}D_{a\delta}{\rm v}^{\delta}{}_{\alpha}(\theta)\,,
\left[ d m \right]_{2}^{4}= \prod_{i = 1}^{3}\frac{d z_{i}}{V_{i}^{'}( 0 )}\prod_{\mu = 1}^{2}\frac{d k_{\mu}}{k_{\mu}^{2}}\frac{d \xi_{1}d \eta_{1}}{( \xi_{1}- \eta_{1})^{2}}
\Phi_n(t)= \frac{-1}{4n^2}\sum_{i=1}^{\lfloor 4tn^2 \rfloor}\frac{a^2}{4x(i)}+\frac{a\sqrt{n}G_{n x(i)}}{\sqrt{\beta x(i)}}+\frac{a G^{(2)}_{nx(i)}}{\beta x(i)}.
\frac{1}{2}\tilde{g}_{a b}( \hat{\bf v}^{i}\cdot{\bf K}^{a}) ( \hat{\bf v}^{i}\cdot{\bf K}^{b}) = \frac{2 \pi^{2}}{3 e^{2}}\hat{\bf v}^{i}\cdot \hat{\bf v}^{i}\frac{r}{1 + 2 \mu r}
\begin{gathered}f^{2K}(\cdot, t) \xrightarrow{K\to\infty}f_0 \circ \big(\tilde{y}^\mathrm{even}_t\big)^{-1}, f^{2K+1}(\cdot, t) \xrightarrow{K\to\infty}f_0 \circ \big(\tilde{y}^\mathrm{odd}_t\big)^{-1},\end{gathered}
\exp{\left( - i \xi F_{1}- i \eta F_{2}\right)}= \left( \begin{matrix}{1}&{0}&{0}&{\xi}\\{0}&{1}&{0}&{\eta}\\{\xi}&{\eta}&{1}&{( \xi^{2}+ \eta^{2}) / 2}\\{0}&{0}&{0}&{1}\\ \end{matrix}\right) .
\eta_{2,2}(q\,|\,\tau, b) =\frac{\Gamma_2(q+b_0\,|\,\tau)}{\Gamma_2(b_0\,|\,\tau)}\frac{\Gamma_2(b_0+b_1\,|\,\tau)}{\Gamma_2(q+b_0+b_1\,|\,\tau)}\frac{\Gamma_2(b_0+b_2\,|\,\tau)}{\Gamma_2(q+b_0+b_2\,|\,\tau)}\frac{\Gamma_2(q+b_0+b_1+b_2\,|\,\tau)}{\Gamma_2(b_0+b_1+b_2\,|\,\tau)}.
\frac{1}{q(x)}\int_{x}^{\eta}\frac{2}{a(y)}q(y) dy = \e^{-\int_{0}^{x}r(u) du}\int_{x}^{\eta}\frac{2}{a(y)}e^{\int_{0}^{y}r(u) du}dy \leq \e^{\eta^2/(c_A-1)}\int_{x}^{\eta}\frac{2}{a(y)}dy \leq \e^{\eta^2/(c_A-1)}\frac{2\eta}{(c_A - 1)\mu}.
S_{Q, \zeta}(\varepsilon) = \biggl(\prod_{i=1}^{N}D_{\zeta_i}(\zeta_i \theta_i)^{- \frac{1}{m}}\biggr) \sum_{k \in (\mathbb{Z}/m\mathbb{Z})^N}\zeta^{\overline{Q(k)}}\biggl(\prod_{i=1}^{N}\frac{\theta_i^{(k^\mathsf{T}A)_i}}{(\zeta_i \theta_i ; \zeta_i)_{k_i}}\biggr) I_{Q, \zeta}(k, \varepsilon),
\mbox{wt}\left\{x^{N}z^{a}\overline{z}^{b}\right\}=\mbox{Min}\left(\mbox{wt}_{N,a,b}\left\{x^{N}z^{a}\overline{z}^{b}\right\}\right), \mbox{where $N,a,b\in\mathbb{N}$ and $a,b\neq 0$.}
\mathsf{A}(x,y) =\dfrac{x+y}{2}, \mathsf{G}(x,y) =\sqrt{xy}, \mathsf{H}(x,y) =\dfrac{2xy}{x+y}, \mathsf{L}(x,y) =\dfrac{x-y}{\log x-\log y}, \mathsf{P}_1(x,y) =x, \mathsf{P}_2(x,y) =y
r^{-3}\||F_A\|^2_{L^2(B_r(x))}\leq (r+1/2)^{-3}\| F_A\|^2_{L^2(B_{r+1/2}(x))}+ const. \|\pi_7(F_A)\|^2_{L^{7/2}(B_{r+1/2}(x))}& \leq const. \left( \| F_A\|_{L^2(\mathbb{B})}+ \|\pi_7(F_A)\|_{L^{7/2}(\mathbb{B})}\right)^2,
\mathcal{J}(\phi(\theta))-\theta \mathcal{J}(\phi^0)-(1-\theta)\mathcal{J}(\phi^1)= -\frac{\theta(1-\theta) \beta_a}{8h^2}(\xi_0-\xi_1)^2 -\frac{\theta(1-\theta)}{8h}\sum_{j=1}^N\epsilon_j(\xi_{j-1}-\xi_{j+1})^2 - \frac{\theta(1-\theta)}{8\beta_b}(\xi_N+\xi_{N+1})^2= 0.
\frac{\mathrm{e}^{- x^{2}/ 2 t}}{\left( 2 \pi t \right)^{D / 2}}= \int \frac{\mathrm{d}^{D}p}{\left( 2 \pi \right)^{D}}\; \mathrm{e}^{- i p \cdot x - \frac 1 2 p^{2}t},
\begin{cases}\begin{aligned}\lim_{m\rightarrow\infty}H^2(p_m)=\inf H^2=\bar H^2,\lim_{m\rightarrow\infty}|\nabla H^2(p_m)|=0, &\lim_{m\rightarrow\infty}|\nabla^{\perp}\vec H|^2(p_m)-\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2+\dfrac12(\bar H^2-S)(\bar H^2-3S+2)\geq 0.\end{aligned}\end{cases}
\overline{\alpha}_C(z)= \Big(1-\frac{2z}{2\alpha+n}\Big)\Big(1-\frac{2z}{n}\Big)\Big(1-\frac{2z}{-2\alpha+n}\Big) = \frac{3n-4\alpha^2-2}{n(n^2-4\alpha^2)}K_1(z)+\frac{6}{n(n^2-4\alpha^2)}K_3(z).
s^{(\pi)}_\lambda(X) = [Z^\lambda]\V^*_\pi(z_1;X) V^*_\pi(z_2;X) \cdots V^*_\pi(z_m;X)\cdot 1 \cr = [Z^{\lambda+\delta}]\ \prod_{1\le i<j\le m}(z_i-z_j)\ \prod_{\ell=1}^m\, L(z_\ell;X)\L_{\pi'}(Z) \cr = [s_\lambda(Z)] \L(XZ)\,L_{\pi'}(Z) \,.
\forall i_{0}\neq \ell , \breve{q}^{\ell}_{i_{0}, i_{0}}( x , \xi ) = \frac{\gamma w_{\ell}( x , \xi )}{1 + \vert \xi_{i_{0}}\vert^{2}+ \vert \partial_{i_{0}}\phi (x) \vert^{2}}.
( \tilde{X}^{I}; \tilde{F}_{I}) |_{\mathrm{1 - l o o p}}= ( 1 , T^{a}\eta_{a b}T^{b}, i T^{a}; i S , i S T^{a}\eta_{a b}T^{b}+ 2 i h ( T^{a}) - i T^{a}{\frac{\partial h}{\partial T^{a}}}, - S T^{a}+{\frac{\partial h}{\partial T^{a}}})
{R}_{1 2}( u - v ) Z_{2 1}^{- 1}T_{1}( u ) Z_{1 2}^{- 1}T_{2}( v ) = Z_{1 2}^{- 1}T_{2}( v ) Z_{2 1}^{- 1}T_{1}( u ){R}_{1 2}( u - v ) \ll{b q y b e t}
E_\lambda(x)= \begin{pmatrix}e^{-i\lambda x}0 0 e^{i\lambda x}\end{pmatrix},R= \begin{pmatrix}i 0 0 -i\end{pmatrix},\Phi (x) = \begin{pmatrix}0 \varphi_1(x) \varphi_2(x) 0 \end{pmatrix}\, .
t_{2}(N)=-t_{2}(N') \Longleftrightarrow t_{2}(N_{1})+t_{2}(N_{1}-2)=-2t_{2}(N_{1}-1) \Longleftrightarrow -2t_{2}\left(\frac{N_{1}-1}{2}\right)-2t_{2}\left(\frac{N_{1}-3}{2}\right)=-2t_{2}(N_{1}-1) \Longleftrightarrow t_{2}(N_{1}-1)=t_{2}\left(\frac{N_{1}-1}{2}\right)+t_{2}\left(\frac{N_{1}-3}{2}\right).
\|\mathcal{L}_n^c\|=\sup_{f\in C(K),\,f\neq 0}\frac{\|\mathcal{L}_{n}^cf\|_{L^\infty(K)}}{\|f\|_{L^\infty(K)}}\leq C_n\,\sqrt{M_n}\,\beta_{M_n}(\varepsilon)\;,
\int_{0}^{m}v^{\kappa}\left(t_{1},z\right)\;\d{z}-\int_{0}^{m}v^{\kappa}\left(t_{2},z\right)\;\d{z}+\int_{t_{1}}^{t_{2}}p^{\kappa}\left(t,0\right)\;\d{t}-\int_{t_{1}}^{t_{2}}p^{\kappa}\left(t,m\right)\;\d{t}=0.
A_1=\left\{\frac{1}{n_2}\left(\frac{i}{n_1}+\frac{j}{m}\right)+\frac{r}{n_2}\ \Big|\ 0< i<n_1,0<j<m,\frac{i}{n_1}+\frac{j}{m}<1,0\leq r<n_2\right\}, A_2=\left\{\frac{i}{n_2}+\frac{j}{n_1n_2m+q}\ \Big|\ 0<i<n_2, 0<j<n_1n_2m+q, \frac{i}{n_2}+\frac{j}{n_1n_2m+q}<1\right\}.
M_n =\sum_{j=0}^n (n-j+4) \frac{4}{\pi}2^{2j}\frac{1}{(n-j)!}\left(\frac{\pi}{2}\right)^{n-j}<\Bigl(\frac{16}{\pi}+\frac12\Bigr)e^{\pi/8}2^{2n}<8.3\cdot 2^{2n}, N_n =\sum_{j=0}^n \frac{4}{\pi}2^{2j}\frac{1}{(n-j)!}\left(\frac{\pi}{2}\right)^{n-j}<\frac{4}{\pi}e^{\pi/8}2^{2n}<1.9\cdot 2^{2n}.
A(B,\gamma) := \left[ \frac{\gamma^p\left| \left\{x \in B\,:\,\prod\limits_{i=1}^m \bigl|\frac{f_i(x)}{\| f_i \|_{w\mathcal{M}^{p_i}_{\phi_i}}}\bigr|> \gamma \right\}\right|}{\phi^{p}(R)|B|}\right]^{\frac{1}{p}}.
\psi_{\kappa}(\theta, x) \ = \ \inf_{v_2 \in{\mathcal M}_2}E^{v_1, v_2}_x \Big[e^{\frac{\kappa \|r_2\|_{\infty}}{\alpha}}e^{\theta \int^{T_\kappa}_0e^{-\alpha t}r_2(X(t), \hat v_1(\theta e^{-\alpha t}, X(t)), v_2(t, X(t)))dt}\Big].
\frac{\tilde\lambda_j}{\tilde\lambda_j-\tilde\lambda_{j+1}}=1+ \frac{\tilde\lambda_{j+1}}{\tilde\lambda_j-\tilde\lambda_{j+1}}\leq 1+2\frac{\lambda_j}{\lambda_{j-1}-\lambda_{j}}\leq 2\frac{\lambda_{j-1}}{\lambda_{j-1}-\lambda_{j}},
B_t^\sigma :=\int_0^{\sigma \wedge t}\ell(X_s) \ds C_t^\sigma :=\int_0^{\sigma \wedge t}Q(X_s) \ds \nu^\sigma(dt,dy) :=\Big(N(X_t, dy) + a(X_t) \varepsilon_{\widehat{\partial}-X_t}(dy) \Big) \dt
\sum_{k=1}^{\infty}\left(\,_{3}\phi_{3}\left[\begin{matrix}0, q, q q^{3/2}, -q^{3/2}, -q \end{matrix}\ ;q, -\lambda_{k}^{2}(u) \right]\right)^{\!-1}\!P_{n}(\lambda_{k}(u);q) P_{m}(\lambda_{k}(u);q) =\frac{1}{1-q}\delta_{mn}
\phi_{m_{1}, n_{1}}\times \phi_{m_{2}, n_{2}}= \sum_{m_{3}= | m_{1}- m_{2}| + 1}^{m_{1}+ m_{2}- 1}\sum_{n_{3}= | n_{1}- n_{2}| + 1}^{n_{1}+ n_{2}- 1}\phi_{m_{3}, n_{3}}\, ,
T^A_B=\frac{\delta(y)}{b_0}{\rm diag}(V_\star+\rho_\star,V_\star-p_\star,V_\star-p_\star,V_\star-p_\star,0)+\frac{\delta(1-y)}{b_0}{\rm diag}(-V+\rho,-V-p,V-p,V-p,0),
\left(z^{\prime\prime}\right)^{-1}\rho\left(g^{\prime\prime}\right)^{-1}-z\rho\left(g\right)=\left(z^{\prime\prime}\right)^{-1}\cdot\left(\rho\left(g^{\prime\prime}\right)^{-1}-z^{\prime\prime}z\rho\left(g\right)\right).
Q_{h}( F,t)=exp\left(\frac{-pt^{p}\theta}{e^{px}-t^{p}}\right)\cdot\frac{(1-t^{p}e^{-px})^{k+g_{Y}-1}}{(1-t^{p}e^{-2px})^{2g_{Y}-2}}\cdot \prod_{j=0}^{p-1}\left\{\frac{(1-\nu^{j}te^{-2x})^{m_{j,2}+m'_{p-j,-2}}}{(1-\nu^{j}te^{-x})^{m_{j,1}}}\right\}.
I(s) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\sigma_{2it}(m) \sigma_{2it}(n)}{(mn)^{it}|\zeta(1 + 2it)|^2}\cosh(\pi t) \Gamma(s - \tfrac12 + it)\Gamma(s - \tfrac12 - it)\d t
\sum_{f\in\mathfrak{F}_{\eta;\gamma}}\prod_{(x', y')\in E(f)}\nu(x'-y')=\sum_{\xi\subseteq\gamma}\prod_{(y\in\xi)}\nu(x-y)\sum_{f\in\mathfrak{F}_{\eta^{\hat{x}}\cup\xi;\gamma\setminus\xi}}\prod_{(x', y')\in E(f)}\nu(x'-y').
E_{P}\left\{E_{P}\left[ Y\mid A=a,\mathbf{O}_{min}\right] \right\}=E_{P}\left[ \frac{I_{a}(A)Y}{\pi_{a}(\mathbf{O}_{min},P)}\right] =E_{P}\left[ \frac{I_{a}(A)Y}{\pi_{a}(\mathbf{O},P)}\right] =\chi_{a}(P;\mathcal{G})
=\rho\left[ \frac{\partial u_{1}}{\partial t}+u_{1}\frac{\partial u_{1}}{\partial x}+u_{2}\frac{\partial u_{1}}{\partial y}\right] +\frac{Ke^{f(s)}}{a(t)^{2}}\frac{x}{r}\frac{\partial}{a(t)\partial\left( \frac{r}{a(t)}\right)}f(s)+\frac{x}{r}\rho\frac{\partial}{\partial r}\Phi
\widehat{T}_{f}\widehat{T}_{g}=\begin{bmatrix}T_{f}T_{g}+{\Gamma^{*}_{\bar{f}}}{\Gamma_{g}}T_{f}{\Gamma^{*}_{\bar{g}}}+{\Gamma^{*}_{\bar{f}}}{S_{g}}\Gamma_{f}T_{g}+S_{f}{\Gamma_{g}}\Gamma_{f}{\Gamma^{*}_{\bar{g}}}+S_{f}S_{g}\end{bmatrix}=\begin{bmatrix}T_{fg}{\Gamma^{*}_{\overline{fg}}}\Gamma_{fg}S_{fg}\end{bmatrix}=\widehat{T}_{fg},
\alpha_{r-1}\leq \frac{2^{r+1}r \alpha_r}{(\lambda(1-\gamma)(1- \beta')^{1+\gamma})^{r-1}}+\frac{2(r-1)\gamma}{\lambda^{r-2}}&\leq \frac{k2^{4k}}{\lambda^{r-1}}\alpha_r+\frac{2k\gamma}{\lambda^{r-2}}&\leq \lambda^{-r}\alpha_r+\lambda^{k^2-r}&\leq \lambda^{-k}((k-r+1) \lambda^{kr})+\lambda^{kr-k}&=(k-(r-1)+1) \lambda^{k(r-1)},
\left| \frac{B \left( x^{i}, v \right)}{a \left( \lambda \right)}\right| = h_{[ 0 ]}^{2}h_{5 ( 0 )}( x^{i}) \left[ \left( \sqrt{| \eta_{5}\left( x^{i}, v \right) |}\right)^{\ast}\right]^{2}.
a_{0,2}(\ell,k)=\frac 1{(2\pi i)^4}\int_{\gamma_{r_2}}dz\int_{\Gamma_{D_2}}dw\int_{\gamma_{\tau_1}}d\zeta\int_{\gamma_{\tau_2}}d\omega\frac{G_{n_1,\mu_1,\xi_1}(z)G_{\Delta n,\Delta\mu,\Delta\xi}(w)}{G_{k,\mu_1,\xi_1}(\zeta)G_{n_2+1-\ell,\Delta\mu,\Delta\xi}(\omega)(z-w)(z-\zeta)(w-\omega)}.
\lim_{\vec{p}\rightarrow 0}\frac{\left( C_{1}\left( \left| \vec{p}\right| \right) + C_{1}^{*}\left( \left| \vec{p}\right| \right) + 2 \left| C_{1}\left( \left| \vec{p}\right| \right) \right|^{2}\right) \left| \vec{p}\right|^{3}\frac 1{\left( 1 - \left( 1 - \kappa \left| \vec{p}\right|^{2}\right)^{2}\right)}}{4 p_{0}}= \frac C{2 \left( 2 \pi \right)^{4}}
\int_{\Omega}(|x|^{-\mu}*F(u_k)) f(u_k) u_k \phi =\lim_{k \to \infty}m(\|u_k\|^n) \int_{\Omega}|\nabla u_k|^n - \nabla u_k|^{n-2}\nabla u_k. \nabla u \phi ~dx &+ \int_{\Omega}(|x|^{-\mu}*F(u)) f(u) u \phi ~dx +o_k(1).
A= \widetilde{C}\sup_{s\in[1,\sigma]}\left\|X_{H}\right\|_{J_s}\left(2C_1\widetilde{C}+\frac{2}{C_0}+C_1\right)\leq \widetilde{C}\underbrace{\left\|X^{\lambda_s}_H\right\|_{\infty}(3C_1 + \frac{2}{C_0})}_{=:K}< \frac{1}{2},
\begin{array}{c c c c c}{{\delta \sqrt{- g}}}&{{=}}&{{-{\frac{1}{2}}\sqrt{- g}\, g_{a b}\, \delta g^{a b}}}&{{\}}&{{}}\\{{\delta R}}&{{=}}&{{\delta \left( R_{a b}\, g^{a b}\right)}}&{{=}}&{{\delta R_{a b}\, g^{a b}+ R_{a b}\, \delta g^{a b}\; ,}}\\ \end{array}
\phi_t^*\left(\sum u_{i}\sigma_i\right) = e^{\lambda_1(t-r)}\cos(\varphi)\,\sigma_1 + e^{\lambda_2(t-r)}\sin(\varphi)\,\sigma_2 &= e^{\lambda_1(t')}\cos(\varphi)\,\sigma_1 + e^{\lambda_2(t')}\sin(\varphi)\,\sigma_2\stackrel{t=r}{=}\left( \cos(\varphi)\,\sigma_1 +\sin(\varphi)\,\sigma_2 \right)
\bar{g}_{x y}= G_{I J}\frac{\partial{X^{I}}}{\partial{{\phi}^{x}}}\frac{\partial{X^{J}}}{\partial{{\phi}^{y}}}= G_{x y}+ G_{y 0}\frac{\partial{X^{0}}}{\partial{X^{x}}}+ G_{x 0}\frac{\partial{X^{0}}}{\partial{X^{y}}}+ G_{0 0}\frac{\partial{X^{0}}}{\partial{X^{x}}}\frac{\partial{X^{0}}}{\partial{X^{y}}}.
[s (E), k_0 (E) ] = \left( \begin{matrix}s_{ll}(E) \beta_l E s_{lr}(E) \beta_l E s_{rl}(E) \beta_r E s_{rr}(E) \beta_r E \end{matrix}\right) - \left( \begin{matrix}s_{ll}(E) \beta_l E s_{lr}(E) \beta_r E s_{rl}(E) \beta_l E s_{rr}(E) \beta_r E \end{matrix}\right) ,
\|\mathfrak{e}'_{k+1}-\mathfrak{e}'_{k}\|_{L^2_1}=\Big\|s^{k+1}\chi_0^{k+2}J^k\Big(\frac{\mathfrak{e}'_0}{\chi_0}\Big)\Big\|_{L^2_1}\leq \frac{1}{2^{k}}\kappa_1
\left| V^{\star}_{k_\ell}(x_\ell)+V^{\star}_{k_\ell}(-x_\ell)-2\,V^{\star}_{k_\ell}(0)\right|\leq \max_{|x|\leq x_\ell}\left| V''_{k_{\ell}}(x)\right|\cdot x^2_{\ell}\leq &2C'_4 \frac{\omega (\delta_{k_\ell})}{\lambda{k_\ell})\delta^{2}_{k_\ell}}\cdot A^2_{\ell}\delta^2_{k_\ell}=C'_4\omega(\delta_{k_\ell}).
N^{-2}\sum_{k=1}^{N}\sum_{i<k}\mathbb{E}_{k}(G^{(1)}_{k})_{ii}\mathbb{E}_{k}(G^{(2)}_{k})_{ii}\beta_{ik}= N^{-2}s(z_{1})s(z_{2})\sum_{k=1}^{N}\sum_{i<k}\beta_{ik}+O_{L^1}(\sqrt{d_{N}/N})
\prod_{i = 3}^{N - 1}( \int d z_{i}\oint d z_{i}^{\prime}) \langle{\cal O}^{1}( z_{1}){\cal O}^{2}( z_{2}) b ( z_{3}^{\prime}){\cal O}^{3}( z_{3}) \cdots b ( z_{N - 1}^{\prime}){\cal O}^{N - 1}( z_{n - 1}){\cal O}^{N}( z_{N}) \rangle ,
P_I( Y^I_t ) = e^{A t}P_I( \xi ) + \int^t_0 e^{A ( t - s )}P_I F\big( P_I( Y^I_s ) \big) \, ds + \int^t_0 e^{A (t - s)}P_I B\big( P_I( Y^I_s ) \big) \, dW_s .
\int_{\mathbb{R}^{2N}}\frac{J_p(U(x)-U(y))\, \big(g_{\alpha,M}(U(x))-g_{\alpha,M}(U(y))\big)}{|x-y|^{N+s\,p}}\,dx\,dy=\mathcal{S}_{p,s}\, \int_{\mathbb{R}^N}U^{p^*-p}\,U_{M}^{\alpha-1}\,U^p\,dx.
y_0 =f_0(x_0,\dots, x_n)= \sum_{|I|= m}a_{0,I}x^I y_1 =f_1(x_0,\dots, x_n)= \sum_{|I|= m}a_{1,I}x^I \cdots y_n =f_n(x_0,\dots, x_n)= \sum_{|I|= m}a_{n,I}x^I
g_{k k}= \left( \omega^{2}g^{E}+{\frac{k^{2}}{\epsilon \epsilon^{\prime}}}g^{H}+{\frac{1}{\epsilon}}{\frac{\partial}{\partial z}}{\frac{1}{\epsilon^{\prime}}}{\frac{\partial}{\partial z^{\prime}}}g^{H}\right) \bigg |_{z = z^{\prime}}.
\begin{array}{lll}x_i = \left(\prod_{j=1}^rx_j(1)^{a_{ij}(1)}\right)x_{s+1}(1)^{a_{i,r+1}(1)}\mbox{for}1\le i\le r\mbox{and}x_{s+1}= \left(\prod_{j=1}^rx_j(1)^{a_{r+1,j}(1)}\right)x_{s+1}(1)^{a_{r+1,r+1}(1)}\end{array}
\sum_{\ell =0}^L s^{-\frac{\beta-\gamma \mu_2}{\mu_2+1}\ell}= (L + 1)\leq \frac{1}{\alpha\log(s)}\log(2c_1(h_0s^2)^\alpha\varepsilon^{-1}) &\leq \frac{1+\log(2c_1(h_0 s^2)^{\alpha}}{\alpha\log(s)}\log(\varepsilon^{-1}),
i\vert i+1 \vert i+2=0=i\vert i-1\vert i-2, i=0,\ldots,d-1; i\vert i+1\vert i = i\vert i-1\vert i, i=1,\ldots, d-1; 0 \vert 1 \vert 0 = (-1)^d (0 \vert d-1 \vert 0).
\sqrt{u}\acute \mu_{k,j}^\ell(u)=\begin{cases}-\frac{2k}{2k+1}\nu^\ell_{k-1,1}+o(1), j=1 O(u), j\in\mathbb Z\cap[2,k] -\nu^{\ell}_{k-1,j-k-1}+o(1), j\in\mathbb Z\cap[k+1,2k-2] \end{cases}
(C_2)^{2}=0,\D^{2}=C_1,\E^{2}=0, C_2 D =C_2= - D C_2,\D E=E =-E D, C_2 E=\frac{1}{2}(C_{1}-D),\E C_2 = \frac{1}{2}(C_{1}+ D), & [C_{2}, D] = 2 C_{2},\ [C_{2}, E]= - D,\ [D, E]=2E.
\begin{aligned}\frac{2\omega_0+\omega_1}{3}\tilde{u}_{j-\frac{1}{2}}+\frac{\omega_0+2(\omega_1+\omega_2)}{3}\tilde{u}_{j+\frac{1}{2}}+\frac{1}{3}\omega_2\tilde{u}_{j+\frac{3}{2}}= \frac{1}{6}\omega_0\bar{u}_{j-1}+\frac{5(\omega_0+\omega_1)+\omega_2}{6}\bar{u}_j+\frac{\omega_1+5\omega_2}{6}\bar{u}_{j+1}, \end{aligned}
f_1^{\prime\prime}(\alpha) \leq \psi^{\prime\prime}\left(\lambda_2(v)-2 \sqrt{2}\alpha \delta(v)\right) (\|d_x\|^2+\|d_s\|^2)= &=2 \delta(v)^2\psi^{\prime\prime}\left(\lambda_2(v)-2 \sqrt{2}\alpha \delta(v)\right),
\min f(u^*) + \langle p_{u^*}, u - u^*\rangle_{\mathcal{U}}+ \frac{c}{2}\|u - u^*\|_{\mathcal{U}}^2 \mbox{s.t.}\ \S_{u^*}u \in K = \mathcal{K}- G(u^*) + S_{u^*}u^*
F^{(d)}_C\left(\begin{array}{c}a,b c_1,\ldots,c_d\end{array};x_1,\ldots,x_d\right) =\displaystyle{\sum_{n=(n_1,\ldots,n_d)\in(\mathbb{Z}_{\ge 0})^d}\frac{(a)_{|n|}(b)_{|n|}}{(c_1)_{n_1}\cdots (c_d)_{n_d}}\frac{x_1^{n_1}}{n_1!}\cdots \frac{x_d^{n_d}}{n_d!}}& \qquad\qquad\qquad\qquad\qquad\qquad(|x_1|^{\frac{1}{2}}+\cdots+|x_d|^{\frac{1}{2}}<1)
\sum_{f:\,(u,v)\in f}B((u,v),f) =\frac{1}{(r-1)!}\sum_{f:\,(u,v)\in f}B_1(u,\pi_1(f))B_2(v,\pi_2(f)) & =\sum_{e_1:\,u\in e_1}\sum_{e_2:\,v\in e_2}B_1(u,e_1)B_2(v,e_2) & =1.
\int_{0}^{T}(\phi(\tilde{u}'(\xi)))' \tilde{u}(\xi) \,d\xi =\int_{0}^{T}\phi(\tilde{u}'(\xi)) \tilde{u}'(\xi) \,d\xi &\geq \int_{0}^{T}\bigl{(}b|\tilde{u}'(\xi)|-K_{b}\bigr{)}\,d\xi= b\|\tilde{u}'\|_{L^{1}}-K_{b}T.
\frac{1}{2}x|P_n\varphi(x)|^2\bigg|_{-\infty}^{\infty}-\frac{1}{2}\int_{\mathbb{R}}|P_n\varphi|^2~dx +\frac{1}{2\pi}\int_0^{\infty}(\psi_n \widehat{u\varphi})'\psi_n\bar{\hat\varphi}~d\xi=0.
s - a_{i^* n_{i^*}}+ (|M_P - M_0| - 1)(b-s) + a_{i^* n_{i^*}}\cdot \frac{a_{i^* j'}}{a_{i^* n_{i^*}}+ b - s}\cdot \frac{a_{i^* n_{i^*}}+ b - s}{a_{i^* j'}}& = b + (|M_P - M_0| - 2)(b-s).
\delta^g_R(\pi) = g(\pi) - g(\pi') = \sum_{i=1}^{k-t+2}\bigl( h_i(\pi_1 \dotsm \pi_k, n) - h'_i(\pi_1 \dotsm \pi_k) \bigr) - \sum_{i=1}^{k-t-|R|+2}\bigl( h_i(\pi'_1 \dotsm \pi'_{k-|R|}, n-|R|) - h'_i(\pi'_1 \dotsm \pi'_{k-|R|}) \bigr)
\hat{B}_{x ; \hat{k}, \hat{l}}+ \hat{B}_{x ; \hat{n}, \hat{k}}+ \hat{B}_{x ; \hat{l}, \hat{n}}+ \hat{B}_{x + \hat{n}; \hat{l}, \hat{k}}+ \hat{B}_{x + \hat{l}; \hat{k}, \hat{n}}+ \hat{B}_{x + \hat{k}; \hat{n}, \hat{l}}= 0 \ .
I ( \tau ) = \exp \left( \int \frac{d^{2}p}{( 2 \pi )^{2}}\; e^{- 2 \frac{| p |}{n}}\; \hat{K}( p ) \; 2 \sum_{i , j = 1}^{M}\Big ( e^{- i p w_{i}}- e^{- i p z_{i}}\Big ) \Big ( e^{+ i p w_{j}}- e^{+ i p z_{j}}\Big ) \right)
\# \mathcal{N}_A(y) = \frac{T}{\phi(Q)}\frac{e^{\frac{[T \log y]+1}{T}}e^{\frac{\xi}{T}}}{[T \log y]+1}\frac{1 - e^{-\frac{1}{9T}}}{e^{\frac{1}{T}}-1}+O\bigg( \frac{T}{\phi(Q)}\frac{e^{\frac{T \log y+1}{T}}}{(T \log y+1)^2/T^2}+ \frac{T^2}{\phi(Q)}\bigg).
\gamma_\Psi A'^{* \mu}= 0, \; \gamma_\Psi \phi'^* = 0, \;\gamma_\Psi C'^* = - \partial_\mu A'^{*\mu},\; \gamma_\Psi b'^* = - \overline{C}'^*, \; \gamma_\Psi\overline{C}'^* = 0.
\Psi_{N}( \beta ) = ( 2 \pi )^{- N}\int_{0}^{2 \pi}\cdots \int_{0}^{2 \pi}\prod_{j < k}\left\vert \exp ( i \theta_{j}) - \exp ( i \theta_{k}) \right\vert^{\beta}\, d \theta_{1}\cdots d \theta_{N}
\sup_{f \in \mathcal{F}_{d,\theta}}| \mathbb{E}_f(\hat{H}_n) - H| = O \biggl( \max \biggl\{\frac{k^{\frac{\alpha}{\alpha+d}- \epsilon}}{n^{\frac{\alpha}{\alpha+d}- \epsilon}}\, , \, \frac{k^{\frac{\beta}{d}}}{n^{\frac{\beta}{d}}}\biggr\}\biggr).
Q_0=P_{n,\lambda}^{(0)}=\{7,28,49\}, &Q_1=\{1,~4,~16\},~~~Q_2=\{10,40,34\},~~~Q_3=\{13,52,19\}, &Q_4=\{22,25,37\},~~~Q_5=\{31,61,55\},~~~Q_6=\{43,46,58\};
K_{1}(x,\lambda(t)) =\int_{0}^{\infty}\frac{r dr}{\lambda(t)^{2}(1+\frac{r^{2}}{\lambda(t)^{2}})^{3}}\int_{0}^{x}\frac{\rho d\rho}{x}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(r^{2}-1-\rho^{2})^{2}+4r^{2}}}\right)
g_c\simeq 1 + \frac{D-1}{2N_c}- \frac{(D-1)^2(4-D)}{2N_c(D-2)}\left[ 1-\frac{1}{g_{Vc}}\log(1+g_{Vc}) \right] + O\left( \frac{1}{N_c^2}\log N_c \right) \ .
\nu^{\mu}= P^{0,1}_{{\rm End}E}((\overline{\partial \chi_1^{\mu\oplus 0}})(\nu_1 d\bar z -\bar\mu\nu_2 d\bar z) )\circ(\chi_1^{\mu\oplus 0})^{-1}) +P^{1,0}_{{\rm End}E}((\partial \chi_1^{\mu\oplus 0}(\nu_2 dz-\mu\nu_1dz) )\circ(\chi_1^{\mu\oplus 0})^{-1})