harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
v_P(\det(A)) =v_P\left(\sum_{\sigma\in S_n}(\sigma)\prod^{n}_{i=1}A_{i,\sigma(i)}\right) \geq \min_{\substack{\sigma\in S_n \prod^{n}_{i=1}A_{i,\sigma(i)}\neq 0}}v_P\left(\prod^{n}_{i=1}A_{i,\sigma(i)}\right) \geq -n v_P(D).
(c\frac{d\tau}{dq})^{2}=-g_{\mu\nu}\frac{dx^{\mu}}{dq}\frac{dx^{\nu}}{dq}=(1-\frac{r_{s}}{r})c^{2}(\frac{dt}{dq})^{2}-\frac{1}{1-\frac{r_{s}}{r}}(\frac{dr}{dq})^{2}-r^{2}(\frac{d\varphi}{dq})^{2}.
[\hat n^\rho, v_1^\rho,\ldots, v_{r-1}^\rho]^{-1}= \begin{pmatrix}1 0 \cdots 0 -a_1 1 \cdots 0 \vdots 0 \ddots \vdots -a_{r-1}\cdot \cdots 1\end{pmatrix}[ n^\rho , v_1^\rho ,\ldots, v_{r-1}^\rho]^{-1},
\omega_{4}^{3}= i \{\frac{1}{2}d \operatorname{l n}( \frac{d \overline{{g}}_{2}}{d \overline{{z}}_{2}}\frac{d z_{2}}{d g_{2}}) + \frac{( g_{2}d \overline{{g}}_{2}- \overline{{g}}_{2}d g_{2})}{( 1 + g_{2}\overline{{g}}_{2})}\},
D (\Phi'' +{2r-r_0 \over r^2 K}\Phi' +{Z_2 Z_6 \over K^2}\omega^2\Phi) + P^1_2 (\Psi_2' + 2{\cal Z}_6') + Q^1_1 \Phi + Q^1_2 \Psi_2 +Q^1_3{\cal Z}_6 =0,
\widetilde{Y}_t =\widetilde{\xi}+\int^T_t\widetilde{f}(s,\widetilde{Y}_s,\widetilde{Z}_s,\widetilde{V}_s) ds+\int_t^Td\widetilde{R}_s -\int^T_t\widetilde{Z}_s dW_s begin{align*}-1pt] -\int_{t}^{T}\int_U\widetilde{V}_{s}(e)\widehat\pi(de,ds)-\int_t^Td\widetilde{M}_s,t\in[0,T],
S ({\cal M}) = \left( \begin{array}{c c}{{1}}&{{0}}\\{{b}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c c}{{\det G^{- 1 / 2}}}&{{0}}\\{{0}}&{{\det G^{1 / 2}G^{- 1}}}\\ \end{array}\right) \left( \begin{array}{c c}{{1}}&{{b^{T}}}\\{{0}}&{{1}}\\ \end{array}\right) .
v=\frac{d\xi}{dt}=\frac{1}{\nu_{i}}\frac{dn_{i}}{dt}=\frac{1}{\nu_{i}}\frac{d(C_{i}V)}{dt}=\frac{1}{\nu_{i}}(V\frac{dC_{i}}{dt}+C_{i}\frac{dV}{dt})
\frac{\log q_{k_{n+1}(x)}(x)- bk_n(x)}{\sigma_1\sqrt{2k_{n}(x)\log\log k_{n}(x)}}= \frac{\log q_{k_{n+1}(x)}(x)- bk_{n+1}(x)}{\sigma_1\sqrt{2k_{n}(x)\log\log k_{n}(x)}}+ \frac{b(k_{n+1}(x)-k_n(x))}{\sigma_1\sqrt{2k_{n}(x)\log\log k_{n}(x)}}
\begin{cases}c_{p,j}^j=c_{p,k}^k, \forall p\in \{1,2,\cdots, j\}c_{j,r}^j=c_{r,k}^k, \forall r\in \{j+1,j+2,\cdots, k-1\}c_{j,q}^j=c_{k,q}^k, \forall q\in \{k,k+2,\cdots, n\}.\end{cases}
D_{dR,K}(V) := (B_{dR}\otimes_{\mathbb{Q}_p}V)^{G_K}\supseteq D_{dR,K}^0(V):= (B_{dR}^+ \otimes_{\mathbb{Q}_p}V)^{G_K}\quad D_{cris,K}(V) := (B_{max,\mathbb{Q}_p}\otimes_{\mathbb{Q}_p}V)^{G_K}\ .
M =u(0) = \int_{\Omega}G(y) u^p(y) ~ dy = \frac{1}{2\pi}\int_{\Omega}\log(1/|y|)\, u^p ~ dy - \int_{\Omega}g u^p dy \geq \frac{1}{2\pi}\int_{\Omega}\log(1/|y|)\, u^p ~dy - C\int_{\Omega}u^p ~dy \geq \frac{1}{2\pi}\int_{\Omega}\log(1/|y|)\, u^p ~dy - C',
\Pr\{N^{(n)}>t\}=\exp\left(-\int_0^tn^{1/4}g(r_n)v(r_n)\,ds\right) \leq \exp\left(-\int_0^{\left(\frac{1}{1+c}\right)t}n^{1/4}\inf_S g(r) \inf_S v(r)\,ds\right) =\exp\left(-\frac{1}{1+c}t n^{1/4}\inf_S g(r) \inf_S v(r) \right)
1\geq \omega(\alpha,\beta) = \sum_{i=1}^d \frac{\alpha_i+\beta_i}{m_i}= \sum_{i=1}^{d_{\ell}}\frac{\alpha_i+\beta_i}{m_i}\geq \frac{1}{m_{d_{\ell}}}\sum_{i=1}^{d_{\ell}}\alpha_i+\beta_i = 1
\frac{d^m}{dx^m}\left. \frac{1}{h_\rho(r,x)}\right|_{x=1}=\frac{m!}{h_\rho(r,1)}\sum_{d=0}^m \sum_{\substack{i_1,i_2, \dots, i_d \geqslant 1 i_1+i_2+ \cdots +i_d=m}}G_\rho(r,i_1) \cdots G_\rho(r,i_d)
Z = \int{\cal D}A_\mu{\cal D}\Pi^\mu{\cal D}\phi{\cal D}\Pi_\phi{\cal D}\theta{\cal D}\Pi_\theta \prod_{\alpha,\beta = 1}^{2}\delta(\tilde{\Omega}_\alpha) \delta(\Gamma_\beta) \det \mid \{\tilde{\Omega}_\alpha,\Gamma_\beta\}\mid e^{iS'},
\|\Delta u_{\epsilon}\|_{2}^{2}=R^{2}(4-N)^{2}\omega\int_{0}^{\frac{2}{\epsilon}}\big[\frac{4r^{N-1}}{(1+r^{2})^{N-2}}+\frac{(2-N)^{2}r^{N+3}}{(1+r^{2})^{N}}+\frac{4(2-N)r^{N+1}}{(1+r^{2})^{N}}\big]dr+O(\epsilon^{N-4}) &=\|\Delta U_{\epsilon}\|_{2}^{2}+O(\epsilon^{N-4}).
\check{Q}_{ib}(\tau )\check{\vec{A}}_{a\perp}(\tau ,{\vec \eta}_i(\tau ))=Q_{ia}(\tau )[P\, e^{\Omega^{(\hat \gamma )}_{sa}(\eta^{(A)}(\tau ,-{\vec \eta}_i(\tau ))){\hat T}^a}]_{ab}\check{\vec{A}}_{b\perp}(\tau ,{\vec \eta}_i(\tau )),
\frac{1}{R(t)^{2\sigma_c}}\int\chi\bigg{(}\frac{x-x(T)}{R(t)}\bigg{)}|u^*|^2 =\frac{1}{A^{2\sigma_c}}\int|\mathcal{Q}_p|^2\big{(}1+\delta(p)\big{)}+O(b_c^{\frac{1}{40}}) &=\big{(}1+\delta(p)\big{)}\int|\mathcal{Q}_p|^2.
h_{\mu}\left(M,\boldsymbol{\alpha}_{p}\right) :=h_{\mu}(Y,\nu)+\int_{G}\int_{X}D\left(\alpha_{x,g}\parallel\alpha_{x,e}\right)\varphi_{g}(\pi(x))d\eta_{p}(x)d\mu(g) =h_{\mu}(Y,\nu)+\int_{G}\int_{Y}\int_{S_{y}}D\left(\alpha_{x,g}\parallel\alpha_{x,e}\right)d\eta_{p}^{y}dg\nu(y)d\mu(g)
\left|\frac{b_{t,s}^{2-\epsilon}}{a_{t,s}}\right| \leq\frac{D^{(t-s)(2-\epsilon)}}{C^{t-s}}\leq\frac{D^{(t-s)(2-\epsilon)}}{D^{k(t-s)}}=D^{-(k-2+\epsilon)(t-s)}.
\begin{array}{c}<j_{1}-m_{1}\j_{2}-m_{2}|j_{1}+j_{2}\ -m_{1}-m_{2}>_{q}= =<j_{2}m_{2}\j_{1}m_{1}|j_{1}+j_{2}\m_{1}+m_{2}>_{q}= =<j_{1}m_{1}\j_{2}m_{2}|j_{1}+j_{2}\m_{1}+m_{2}>_{q^{-1}}\end{array}
(e^{\tau x_1}(-\Delta_g)e^{-\tau x_1}w| T^*_\tau e^{-\tau x_1}(-\Delta_g)e^{\tau x_1}v)_{L^2(M)}&=(e^{\tau x_1}(-\Delta_g)e^{-\tau x_1}H_\tau e^{\tau x_1}(-\Delta_g)e^{-\tau x_1}w|v)_{L^2(M)}
\begin{cases}\frac{\partial^2 u}{\partial r^2}_{|_{r=1}}=\lambda\frac{\partial u}{\partial r}_{|_{r=1}}, -\frac{1}{r^2}{\Delta_S}\Big(\frac{\partial u}{\partial r}-\frac{u}{r}\Big)-\frac{\partial\Delta u}{\partial r}_{|_{r=1}}=\mu u_{|_{r=1}},\end{cases}
{\mathcal R}( x ) \frac{\delta}{\delta \varphi ( y )}S = - \frac{\delta F}{\delta \varphi ( x )}\int d z \, \Delta_{S}^{\mathrm r e t}( z , x ) \frac{\delta^{2}S}{\delta \varphi ( x ) \delta \varphi ( z )}= - \frac{\delta F}{\delta \varphi ( x )}\delta ( x - y ) \ .
\Phi(r)=-\frac{(4GM_{\bullet}/c)^{2}}{2r^{2}}[1+\frac{3(6GM_{\bullet}/c^{2})^{2}}{8r^{2}}]-\frac{GM_{\bullet}}{r}[1-\frac{(6GM_{\bullet}/c^{2})^{2}}{r^{2}}].
\Theta \left[ \begin{array}{lll}1/2 1/2 1/2 1/2 1/2 0\end{array}\right ]\stackrel{D_{\alpha_2^{-1}\alpha_3}}{\longrightarrow}\Theta \left[\begin{array}{lll}1/2 1/2 1/2 1/2 0 1/2\end{array}\right ],
D\big(p^{-1},{\rm sym}^m[g(\theta)]\big) = D(p^{-1},{\rm sym}^m[g(\theta_{m,p}^-)]) & + \tfrac{1}{2}D''(p^{-1},{\rm sym}^m[g(\theta_{m,p}^-)])(\theta-\theta_{m,p}^-)^2+ O\big((\theta-\theta_{m,p}^-)^3\big),
M(u(t),v(t)) = \|u(t)\|^2_{L^2}+ 2 \|v(t)\|^2_{L^2}, E(u(t),v(t)) = \frac{1}{2}(\|\nabla u(t)\|^2_{L^2}+ \kappa \|\nabla v(t)\|^2_{L^2}) - ( \langle v(t), u^2(t)\rangle).
b_{11}^{(k)}-b_{22}^{(k)}= b_{11}^{(k-1)}-b_{33}^{(k-1)}+2b_{13}^{(k-1)}\tan\phi_{k}, b_{44}^{(k)}-b_{33}^{(k)}= b_{22}^{(k-1)}-b_{44}^{(k-1)}+2b_{24}^{(k-1)}\tan\psi_{k}.
\rho(u(\cdot,t_2;s,u_0),u(\cdot,t_2;s,v_0)) =\rho(u(\cdot,t_2;t_1,u(\cdot,t_1;s,u_0)),u(\cdot,t_2;t_1,u(\cdot,t_1;s,v_0))) &<\rho(u(\cdot,t_1;s,u_0),u(\cdot,t_1;s,v_0)).
\aligned \psi_j =\frac1{N+1}\Big(\overline L^\alpha_{\underline{j\alpha}}+\frac12\overline F^i_k\overline\sigma_j+ \frac12\overline F^i_j\overline\sigma_k\Big)- \frac1{N+1}\Big(L^\alpha_{\underline{j\alpha}}+\frac12F^i_k\sigma_j+ \frac12F^i_j\sigma_k\Big). \endaligned
T_{A}= \sum_{s = 0}^{\infty}T_{A}^{s}\, , \qquad T_{A}^{0}= \Theta_{A}\, , \quad T_{A}^{s}= T_{A \, B_{1}, \ldots , B_{s}}^{s}( x ) Y^{B_{1}}, \ldots , Y^{B_{s}}\, ,
R_{\nu}(r) =\frac{(2\pi)^{-1/2}r^{\nu}}{\Gamma(\nu+1/2)}e^{i(r-\pi\nu/2-\pi/4)}\int_0^\infty e^{-rt}t^{\nu+1/2}[(1+it/2)^{\nu-1/2}-1] \, \frac{dt}{t}&+ \frac{(2\pi)^{-1/2}r^{\nu}}{\Gamma(\nu+1/2)}e^{-i(r-\pi\nu/2-\pi/4)}\int_0^\infty e^{-rt}t^{\nu+1/2}[(1-it/2)^{\nu-1/2}-1] \, \frac{dt}{t}.
s_{W_{\mathcal{I}}, X_{T^c}^k, Y_{T^c}^k}\stackrel{\eqref{qSimulatingConstruction}}{=}r_{W_{\mathcal{I}},X_{T^c}^k, Y_{T^c}^k}* \stackrel{}{=}p_{W_{\mathcal{I}}}\prod_{m=1}^k \left(p_{X_{T^c,m}|W_{T^c}, X_{T^c}^{m-1}, Y_{T^c}^{m-1}}s_{Y_{T^c,m}|X_{T^c,m}}^{(\lambda, T)}\right),
{\cal M}^{odd}= \frac 18 b_{ij}\left\{\epsilon^{ij}<p \cdot \psi p \cdot \bar \psi>^{odd}+ 2 i p_k \epsilon^{kj}\left (k^i + i <\psi^i p \cdot \bar \psi>^{odd}_o- <\psi^i p \cdot \psi >^{odd}_o \right) \right\}\;,
(1-t)P_{\mu}P_{N}=\sum_{\lambda: \lambda/\mu \in \mathfrak{h}_N}\varphi_{\lambda/\mu}(t)P_{\lambda},\varphi_{\lambda/\mu}(t)=\frac{b_{\lambda}(t)}{b_{\mu}(t)}\psi_{\lambda/\mu}(t),
\log Z[{v},{v}^{\dagger}]=\int d^{d}{x}d^{d}{y}{v}^{\dagger}({\mathbf{x}})\left( K\gamma^{0}\frac{\gamma \cdot ({\mathbf{x}}-{\mathbf{y}})}{|{\mathbf{x}}-{\mathbf{y}}|^{d+2m+1}}\right){v}({\mathbf{y}}),
K_{M N}^{( 0 )}\equiv \left( \begin{array}{c c c}{{g_{\mu}^{\; \rho}g_{\nu}^{\; \sigma}}}&{{0}}&{{- g_{\mu \nu}}}\\{{0}}&{{- g_{\mu}^{\; \rho}g_{\nu}^{\; \sigma}}}&{{0}}\\{{0}}&{{0}}&{{8}}\\ \end{array}\right)
[(b_1-b_2)^2-2]R_{1212}+[(b_2-b_3)^2-2]R_{2323}+[(b_1-b_3)^2-2]R_{1313}&=\frac{3}{2}|\tilde{A}|^2-\frac{3}{2}[|Ric|^2-\frac{1}{3}R^2]-\frac{1}{9}.
X^{m'}_{B_m}[T] = \lim_{\ell \to \infty}(X^{B_{\ell}}_{B_{m'}})_{B_m}[T] = \lim_{\ell \to \infty}X^{B_{\ell}}_{B_m}[T] =X^m[T] = X^m_{B_m}[T],
\widetilde{f}^{\gamma_{\pm,i}}_j(s,t):=\frac{\partial \beta^\pm_{s_{\pm,i}}}{\partial s}(s)f^{\gamma_{\pm,i}}_j(t)\otimes \pi_j (ds-idt)\in \Gamma (\dot F ,\wedge^{0,1}u^{*}T\widehat{W}).
W^-(y_1,y_2|u_1) = \frac{1}{2}\sum_{u_2}W(y_1|u_1\oplus u_2)W(y_2|u_2), W^+(y_1,y_2,u_1|u_2) = \frac{1}{2}W(y_1|u_1\oplus u_2)W(y_2|u_2).
\xi'_i= \begin{cases}\xi_i i\not\in V_{\xi}, \xi_i-\frac{1}{2}i\in V_{\xi}\frac{1}{4}<\xi_i\le\frac{1}{2}, \xi_i+\frac{1}{2}i\in V_{\xi}-\frac{1}{2}\le\xi_i<-\frac{1}{4}, \end{cases}
\begin{aligned}(\mathcal{A}-\mathrm{I}) u(\varphi, x) =(1+\beta_x) u(\varphi, x+\beta)-u (\varphi, x)=u(\varphi, x+\beta)-u (\varphi, x)+\beta_x u(\varphi, x+\beta)= &=u_x(\varphi, x) \beta(\varphi, x)+ \beta_x(\varphi, x) u(\varphi, x)+O(\beta^2)= &=\varepsilon\partial_x (\beta_1 (\varphi, x)\,u(\varphi, x))+O(\varepsilon^2).\end{aligned}
h_1(s_1;x)=(\ell-1)\log(1-d_2N_1^{-1/3})+\frac 12\mu_1(1-d_2N_1^{-1/3})^2\cos 2N_1^{-1/3}s_1 - \xi_1(1-d_2N_1^{-1/3})\cos N_1^{-1/3}s_1
\left(\mathrm{curl}\, \sigma(y_\alpha),\phi\right)-\tfrac{\alpha}{\nu}\, b(y_\alpha,\phi,\mathrm{curl}\, \sigma(y_\alpha))=\left(\tfrac{\alpha}{\nu}\,\mathrm{curl}\, u_\alpha+\mathrm{curl}\, y_\alpha,\phi\right)\mbox{for all}\ \phi\in{\cal D}(\Omega).
\frac{1}{2}i\psi^\dagger_\pm\left(\partial_1-ieA^1\right)\psi_\pm-\frac{1}{2}i\left(\partial_1+ieA^1\right)\psi^\dagger_\pm\psi_\pm=\pm\frac{1}{4\pi}\left(\partial_1\phi_\pm\mp e\lambda A^1\right)^2\mp\frac{\pi}{12L^2}\ .
S_{p}= \left\{(u,v) \in X_{s,t,p}(\Omega)\,:\, \vert u(x_1)\vert^{\alpha(p)}\vert v(x_2) \vert^{\beta(p)}= 1\right\}\intertext{and}S_{\infty}= \left\{(u,v) \in X_{s,t,p}\,:\, \vert u(x_1)\vert^{\theta}\vert v(x_2) \vert^{1-\theta}= 1\right\}
\sum_{n\geq 0}a_nz^n + \sum_{n\geq 0}b_{n-1}\sqrt{p}z^n = \sum_{n\geq 0}(a_1+b_0\sqrt{p})^n z^n = \dfrac{1-a_1z+b_0\sqrt{p}z}{1-2a_1 z+ (a_1^2-b_0^2 p)z^2}.
\mathrm{I}\left(\xi_{1}^{x},\xi_{n}^{x}\right)=\sum_{\xi_{1}^{x}}\sum_{\xi_{n}^{x}}\log\left(\frac{P_{\mu,x}^{n-1}(\xi_{1}^{x},\xi_{n}^{x})}{P_{\mu,x}^{n}(L_{x},\xi_{n}^{x})}\right)P_{\mu,x}^{1}(L_{x},\xi_{1}^{x})P_{\mu,x}^{n-1}(\xi_{1}^{x},\xi_{n}^{x})
|v_{s_0}^R(x)|= c_{s_0}^R|w_{s_0}^R(x)|\geq \frac{1}{2n}cR;\sum\limits_{s=1}^n|v_s(x)|^2 = \sum\limits_{s=1}^n|v_s^R(x)|^2\geq c(n)R^2.
\begin{aligned}\frac{\left|\tilde{\mathcal{L}}(t)\right|}{D_1}=\left|\int_{D_{X,[l]}(0)}^{D_{X,[l]}(t)}\phi(s) ds\right|= \int^{D_{X,[l]}(0)}_{D_{X,[l]}(t)}\phi(s) ds &\geq \int^{D_{X,[l]}(0)}_{Be^{-tC}}\phi(s) ds= \frac{B^{1-\alpha}e^{tC(\alpha-1)}-\left(D_{X,[l]}(0)\right)^{1-\alpha}}{\alpha-1}.\end{aligned}
\mathcal{H}^{\frac{1}{\delta}}_{\infty}(U_{p,q, \delta})\geq (\mathcal{H}^{\frac{\log 2}{\log 3}}_{\infty}(U_{p,q, \delta}))^{\frac{\log 3}{\delta \log 2}}\geq \widetilde{C}(q^{-2\delta \frac{\log 2}{\log 3}})^{\frac{\log 3}{\delta\log 2}}=\widetilde{C}\mathcal{L}\Big(B\Big(\frac{p}{q},q^{-2}\Big)\Big).
G_{\mu \nu}- 3 \mu^{2}g_{\mu \nu}- \frac{3}{2}\lambda \sigma \phi^{2}g_{\mu \nu}= 6 \phi^{- 2}( \Sigma_{\mu \nu}( e^{\sigma}g_{\mu \nu}) + \tau_{\mu \nu}) + 6 t_{\mu \nu}~ ,
(F^{f(x)})^{g(x)}=F^{b_n x^n f(x)}F^{b_{n+1}x^{n+1}f(x)}\cdots F^{b_{n+m}x^{n+m}f(x)}&=F^{b_n x^n f(x)+b_{n+1}x^{n+1}+\cdots+ b_{n+m}x^{n+m}f(x)}&=F^{f(x)g(x)}.
\begin{aligned}\|c\varphi \|_{W^{1,6}(\Omega)}&\le C( \|\Delta(c \varphi) \|_{L^{\frac{3}{2}}(\Omega)}+\| c \varphi\|_{L^{\frac{3}{2}}(\Omega)}+ \| (\gamma-c) \varphi \|_{W^{\frac{1}{3},\frac{3}{2}}(\partial\Omega)}+\| c\nabla \varphi \cdot \nu \|_{W^{\frac{1}{3},\frac{3}{2}}(\partial\Omega)}).\end{aligned}
S^{c}= \frac{1}{3 2 \pi^{2}}\int_{0}^{\infty}d e_{0}\left( \det \frac{\sinh \frac{g e_{0}F}{2}}{g F}\right)^{- \frac{1}{2}}\Phi [ x_{c o m b}, e_{0}] e^{I [ x_{c o m b}, e_{0}]},
2 N^{2}{\frac{g^{2}( M )}{M}}\int_{0}^{M}{\frac{d^{3}k}{( 2 \pi )^{3}}}{\frac{1}{k^{2}+ \sigma^{2}}}={\frac{N^{2}g^{2}( M )}{\pi^{2}}}\left( 1 -{\frac{\sigma}{M}}\operatorname{a r c t a n}{\frac{M}{\sigma}}\right) = N
Lf(x,t) =\frac{1}{2\pi}\int_{\mathbb{R}^{d_2}}H(|\lambda|)f^\lambda(x)e^{-i\lambda\cdot t}\,d\lambda =\frac{1}{2\pi}\int_0^\infty a^{d_2-1}\left(\int_{S^{d_2-1}}H(a)f^{a\epsilon}(x)e^{-ia t\cdot\epsilon}\,d\sigma(\epsilon)\right)\,da
\frac{\partial W}{\partial z}(r,z)=\frac{\partial}{\partial z}\int_{\{x:\,w(x,z)>w^{\circledast}(r,z)\}}w(x,z)\,d\gamma(x)=\int_{\{x:\,w(x,z)>w^{\circledast}(r,z)\}}\frac{\partial}{\partial z}w(x,z)\,d\gamma(x).
\sum_{j\le j_0}\psi_j2^{jr}\Vert\dot{\Delta}_j(T_{f}g)\Vert_{L^2}\le C \Big(\sum_{j'\le j_0+4}2^{j'r_1}\Vert \dot \Delta_{j'}f\Vert_{L^\infty}\Big) \sum_{j\le j_0}\sum_{\vert j-j'\vert \le 4}2^{j'(r-r_1)}\Vert \dot \Delta_{j'}g \Vert_{L^2}\psi_j.
1 =\frac{e^{\frac{h^2_1+|\nabla h_1|^2}{2}}\det (\nabla^2 h_2+h_2I)h_2^{1-p}}{e^{\frac{h_2^2+|\nabla h_2|^2}{2}}\det (\nabla^2 h_1+h_1I)h_1^{1-p}}& \geq \frac{e^{\frac{h^2_1+|\nabla h_1|^2}{2}}}{e^{\frac{h_2^2+|\nabla h_2|^2}{2}}}F(x_0)^{p-(n+1)}.
Q_{\mathsf{M}}(\lambda )=\sum_{a=1}^{\mathsf{N}}\prod_{\substack{b=1 b\neq a}}^{\mathsf{N}}\frac{\cosh 2\lambda -\cosh 2\zeta_{b}^{(0)}}{\cosh2\zeta_{a}^{(0)}-\cosh 2\zeta_{b}^{(0)}}Q(\zeta_{a}^{(0)})=\lambda_{\mathsf{M}+1}^{(\mathsf{M})}\prod_{b=1}^{\mathsf{M}}\left( \cosh 2\lambda-\cosh 2\lambda_{b}^{(\mathsf{M})}\right) .
f_{E_{6}}= \frac{e^{- S_{1}- i \frac{x_{0}I_{3}}{y ( x_{0}^{2}+ y^{2})}}}{( y^{2}+ x_{0}^{2}) S_{1}}\propto \frac{K_{1 / 2}( S_{1}) e^{- i \frac{x_{0}I_{3}}{y ( x_{0}^{2}+ y^{2})}}}{( y^{2}+ x_{0}^{2}) \, \sqrt{S_{1}}}\, .
\bigl(R_{k,j}^\mathbf{a}\bigr)_{0 \le j \le q_k, k \ge K_0}\stackrel{\cal D}{=}\bigl(Y_{k,j}(\mathbf{a}_{k, 3j}, \mathbf{a}_{k, 3j+3})\bigr)_{0 \le j \le q_k, k \ge K_0}
\begin{array}{ll}T_{N}^{\mu\nu}=\displaystyle{\frac{\partial{\cal L}}{\partial A^{a}_{\lambda,\mu}}A^{a,\nu}_{\lambda}-\eta^{\mu\nu}{\cal L}}{}{}{}=-\Pi^{a\mu\lambda}A^{a,\nu}_{\lambda}-\eta^{\mu\nu}{\cal L}\end{array}
c_1(\varepsilon)=\frac12 (4\pi)^{-4+\varepsilon}\Gamma^2\Big(1-\frac\varepsilon2\Big)\Big[\zeta^2(2-\varepsilon)+(1-\cos\pi\varepsilon)B(3-2\varepsilon,-1+\varepsilon)\zeta(3-2\varepsilon)\Big]
x_{(n,j)}= - \sum_{k=j}^{n-1}x_{(k,n+j-k)}, & x_{(n-2,j)}= - \sum_{k=1}^{j-1}x_{(n-2-j+k,k)}- x_{(n-1,j+1)}- x_{(n,j+2)}, & x_{(n-1,j)}= - \sum_{i=1}^{n-2}x_{(i,j)}- x_{(n,j)},
Q_{-\sigma,\Omega}^N[u]=-\frac{2\sigma}{C_\sigma}\cdot \widetilde{\cal E}_{-\sigma}^N(w_{-\sigma}^N);(-\Delta_{\Omega})^{-\sigma}_Nu(x)=\frac{2\sigma}{C_\sigma}\,w_{-\sigma}^N(x,0).
g(x,v) = \int_0^1 \frac{e^{-(1-y)/ \kappa |v|}}{\kappa |v| (1-e^{-1/\kappa |v|})}\rho_g (x+y) \left( \alpha\mathcal{M}_{T(x+y)}(v) + (1-\alpha) \mathcal{M}_{\tau(x+y)}(v) \right) \mathrm{d}y.
\mu^{l}=f_{\;bc}^{a}f_{\;de}^{c}\left( \epsilon^{0ijl}A_{j}^{e}\eta_{i}^{\left( 2\right) d}-{\cal P}^{\left( 2\right) le}\eta^{d}\right)\left( B_{a}^{0k}A_{k}^{b}+\eta_{a}^{\left( 2\right) k}{\cal P}_{k}^{\left(2\right) b}+\eta_{a}{\cal P}^{b}\right) .
\ln_{q}b_{q}\left( k;n,r\right) =\ln_{q}\left[\begin{array}[c]{l}n k\end{array}\right]_{q}& \quad+\frac{1}{2-q}\left( k^{2-q}\ln_{2-q}r+\left( n-k\right)^{2-q}\ln_{2-q}\left( 1-r\right) \right) +C_{q}
\int_{-\infty}^{\xi_1}e^{x_j(\zeta_j-z_j)}dx_j=\frac{e^{\xi_1(\zeta_j-z_j)}}{\zeta_j-z_j}\,\,,\,\,\int_{-\infty}^{\Delta\xi}e^{y_j(\omega_j-w_j)}dy_j=\frac{e^{\Delta\xi(\omega_j-w_j)}}{\omega_j-w_j},
\|f_2f_3\|_{L^{p_2}_tL^3_x}^2 = \|f_2^2f_3^2\|_{L^{\frac{p_2}{2}}_t L^{\frac32}_x}\leq \|f_2f_3^2\|_{L^r_tL^2_x}\|f_2\|_{L^s_tL^6_x},
\Delta(p_{\gamma'},\widetilde{p})=\min\{|\gamma'-\gamma_i| \mid i=1,\ldots,t\}= =\min\{|\alpha'^{l^m}-\alpha_i^{l^m}| \mid i=1,\ldots,t\}=\min\{|\alpha'-\alpha_i|^{l^m}\mid i=1,\ldots,t\}=\Delta(p_{\alpha'},p)^{l^m}
{\rm Ric}_{ca}=R^b_{c,ab}\,,\,{\rm Ric}_{c\overline{a}}=R^b_{c,\overline{a}b}\,,\,{\rm Ric}_{\overline{c}\,\overline{a}}=\overline{{\rm Ric}_{ca}}\,,\,{\rm Ric}_{\overline{c}a}=\overline{{\rm Ric}_{c\overline{a}}}.
SP_{u}\:=\: \begin{pmatrix}S_{1}\vec{u}\vec{u}^{*}S_{1}^{*}\overline{u}_{n}S_{1}\vec{u}u_{n}( S_{1}\vec{u})^{*}|u_{n}|^{2}\end{pmatrix}= \begin{pmatrix}S_{1}0 0 1 \end{pmatrix}\, P_{u}\, \begin{pmatrix}S_{1}0 0 1 \end{pmatrix}^{*}.
\frac{\langle 0|0_V\rangle_1 (\lambda_2,\lambda_0)}{\langle 0|0_V\rangle_1({\lambda}^{\prime}_2,{\lambda}^{\prime}_0)}=e^{-[S_0(\lambda_2,\lambda_0)-S_0(\lambda_2',\lambda_0')]}\left[\frac{\det' \hat{{\cal M}}(\lambda_2,\lambda_0)}{\det' \hat{{\cal M}}(\lambda_2',\lambda_0')}\right]^{-\frac12}{\cal Z}
\theta_s^1(a_0\otimes a_1\otimes\cdots\otimes a_{k-1}\otimes a_{k}a_{k+1}\otimes a_{k+2}\otimes\cdots\otimes a_p) ={\rm Tr}\left(\Gamma a_0\left(\prod_{l=1}^{k-1}b_l(a_l)\right)b_k(a_ka_{k+1})\left(\prod_{l=k+1}^{m-2}b_l(a_{l+1})\right)\delta^2(a_m)\left(\prod_{l=m}^{p-1}b_{l+1}(a_{l+1})\right)\cdot T\right).
|\varphi(z)|\leq\sum_{n=0}^{\infty}|\varphi_{n}C_{n}^{\lambda,\beta}(z)| \leq\sum_{n=0}^{\infty}|\varphi_{n}||C_{n}^{\lambda,\beta}(z)|\leq\mathrm{e}^{\varepsilon|z|}C_{\varepsilon}\sum_{n=0}^{\infty}n!|\varphi_{n}|\sigma_{\varepsilon}^{-n}.
A={\rm diag}\left(A_1,\cdots, A_k\right),A_k=\begin{pmatrix}B_k 1 0 \cdots 0 0 B_k 1 \cdots 0 \vdots \vdots \vdots \ddots \vdots 0 0 0 \cdots B_k\end{pmatrix},
\begin{aligned}I_2=\iint\partial_Y\left(\frac{h^0_e}{u^0_e}\right)\partial_{YY}w_2\cdot\partial_Y w_1\lesssim \left\Vert \partial_Y\left(\frac{h^0_e}{u^0_e}\right)\right\Vert_{L^\infty}\Vert \nabla \partial_Yw_2\Vert_{L^2}\Vert \partial_Y w_1\Vert_{L^2}\lesssim \Vert \nabla \partial_Yw_2\Vert_{L^2}\Vert w_1\Vert_{H^1},\end{aligned}
A_{\ell}^{\pm}=(\Gamma_{\{\ell+1,\ell+2\}}\pm \Gamma_{[\ell+2]\setminus\{\ell+1\}})(\Gamma_{[\ell+1]}\mp 1/2), B_{\ell}^{\pm}=-(\Gamma_{\{\ell+2\}}\pm \Gamma_{[\ell+2]})(\Gamma_{[\ell]}\pm \Gamma_{\{\ell+1\}}).
a_h(u_h,\phi_h) := -\sum_{i = 1}^{N_x}\int_{I_i}\frac{u_h^2}{2}\partial_x\phi_h\,{\rm d}x - \sum_{i = 0}^{N_x-1}\left(\frac{\widehat{u_h^2}[\![\phi_h]\!]}{2}\right)_{i+1/2}
\frac{n^2\alpha_n^2}{8}=\frac{n^2F_n^2(x)}{32}=\frac{32\ln n}{32}+\frac{8x+3\ln(2\ln n)}{(2\ln n)^{\frac{1}{2}}}\frac{(32\ln n)^{\frac{1}{2}}}{32}+\frac{(8x+3\ln(2\ln n))^2}{32\cdot4\cdot(2\ln n)}&=\ln n+\frac{8x+3\ln(2\ln n)}{8}+o(1)
\frac{\sigma^2_{P_n,j+R_1}(\theta^\prime_n)}{\sigma^2_{P_n,j}(\theta^\prime_n)}=\frac{\sigma^2_{P_n,j}(\theta^\prime_n) + Var_{P_n}(t_j(X_i,\theta^\prime_n))+2Cov_{P_n}(m_j(X_i,\theta^\prime_n),t_j(X_i,\theta^\prime_n))}{\sigma^2_{P_n,j}(\theta^\prime_n)}\to 1,
\textsf{P}\{\Xi^{2}>Cz\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2}\}=\textsf{P}\{\Xi^{2}>C\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2}\}&\le \exp(-C_{1}z\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2}).
\left\|\sum_{j \in \mathbb{S}}c_jy_j\right\| \leq \left\|\sum_{j \in \mathbb{S}}c_j(x_j-y_j)\right\| +\left\|\sum_{j \in \mathbb{S}}c_jx_j\right\|=(1+\alpha)\left\|\sum_{j \in \mathbb{S}}c_jx_j\right\|+\beta\left\|\sum_{j \in \mathbb{S}}c_jy_j\right\|+\gamma \left(\sum\limits_{j\in \mathbb{S}}|c_j|^2\right)^\frac{1}{2}.
\begin{aligned}0\le I_1 \le |\Upsilon(x)|^{p-1}\int_{|y|>\frac{3}{2}|x|}\dfrac{dy}{|x-y|^{N+sp}}=C(N,s,p)\dfrac{|\Upsilon(x)|^{p-1}}{|x|^{ps}}\le \dfrac{C(N,s,p,C_1,\alpha)}{|x|^{\alpha(p-1)+ps}}. \end{aligned}
\Phi_{t}\left( z_{b},z_{d}\right) =\mathbf{E}\left( z_{b}^{N_{b}\left(t\right)}z_{d}^{N_{d}\left( t\right)}\right) \rightarrow \rho_{e}\mathbf{E}\left( \left( z_{b}^{{}}z_{d}\right)^{N_{d}\left( \infty \right)}\mid \right) ,
\tilde{\Pi}_{R{( 2 )}}^{( \beta )}( \lambda , p ) = - \frac{8 e^{4}}{p^{2}}m^{2}I ( T / \lambda ) \int d^{2}q n_{F}( | q^{0}| ) \left( q^{2}+ m^{2}\right) \delta^{\prime \prime}( q^{2}- m^{2}) .
{T}_{u}v=\sum_j{\dot{S}_{j-1}u}{\dot{\Delta}_j v}, \textrm{and}{R}(u,v)=\sum_j \dot{\Delta}_j u\widetilde{\dot{\Delta}}_j v, \widetilde{\dot{\Delta}}_j := \dot{\Delta}_{j-1}+\dot{\Delta}_j + \dot{\Delta}_{j+1}.
I_{p k}\left( M_{p}\right) = \int_{M_{p}}^{\infty}\, d \omega \, \omega \left( \omega + M_{p}\right)^{- 1 / 2}\left( \omega - M_{p}\right)^{- 1 / 2}\left( \omega + \mu \right)^{- k}\left( \omega - \mu \right)^{k}e^{- 2 k \omega}.
[a]_{\times}=[\begin{matrix}0&c_{2}d_{1}-c_{1}d_{2}&c_{3}d_{1}-c_{1}d_{3}\\c_{1}d_{2}-c_{2}d_{1}&0&c_{3}d_{2}-c_{2}d_{3}\\c_{1}d_{3}-c_{3}d_{1}&c_{2}d_{3}-c_{3}d_{2}&0\end{matrix}]
\nabla \mathcal{K}_{t/2}(x)=-\frac{x}{t}\,\mathcal{K}_{t/2}(x),\qquad\|\nabla \mathcal{K}_t\|_{L^1(\mathbb{R}^N)}\le \frac{C}{\sqrt{t}},\int_{\mathbb{R}^N}\frac{\partial}{\partial t}\mathcal{K}_{t}(y)\,dy=0
| \widehat f |^2 \le \sum_{j=1}^{\log R^{1/2}}\sum_{(k,k^\prime):\theta^j_k\approx \theta^j_{k^\prime}}|\widehat{f_k^j}\widehat{f_{k'}^j}| + \sum_{k}|\widehat{f_k^{j_\circ}}|^2.
B^{0}_T =C\left(1+\|\tilde w_{0, \epsilon}\|_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}+(1+\frac{1}{\epsilon}) \|\tilde w_{0, \epsilon}\|^2_{H^{m+2}_{k+\ell}(\mathbb{R}^2_+)}\right) &\le C\left(1+\bar \zeta+(1+\frac{1}{\epsilon}) \bar \zeta^2 \right) ,
\Omega_1(E)=\frac{V\pi^{-51/2}}{\Gamma\left(\frac{25}{2}\right)2^{38}}\sum_{r=1}^{\infty}\sum_{k=0}^{\infty}r^{-26}a_k^2E\left(E^2-\frac{4 r^2 k}{\alpha'}\right)^{23/2}{\cal H}\left(E^2-\frac{4 r^2 k}{\alpha'}\right)\vspace{4mm}
\frac{3}{3-2\sqrt{5}}=\frac{3}{3-2\sqrt{5}}\cdot\frac{3+2\sqrt{5}}{3+2\sqrt{5}}=\frac{3(3+2\sqrt{5})}{3^{2}-(2\sqrt{5})^{2}}=\frac{3(3+2\sqrt{5})}{9-20}=-\frac{9+6\sqrt{5}}{11}
I(u_1,\dots,u_d) =\min_{(w_1,\dots,w_d)\in\mathcal{N}_d}I(w_1,\dots,w_d) \le \underset{w_i=0,\i\notin P}{\min_{(w_1,\dots,w_d)\in\mathcal{N}_d}}I(w_1,\dots,w_d) &=c((\lambda_i;\mu_i;b_{ij})_{(i,j)\in P^2}).
\frac{\overline{GB}}{\overline{GA}}\times\frac{\overline{HA}}{\overline{HF}}\times\frac{\overline{KF}}{\overline{KE}}\times\frac{\overline{GE}}{\overline{GD}}\times\frac{\overline{HD}}{\overline{HC}}\times\frac{\overline{KC}}{\overline{KB}}=1.
T(r, f_j) = \frac1{2\pi}\int_{|z|=r}\log^+|f_j| \,d\theta \leq \frac1{4\pi}\int_{|z|=r}\log( 1+ |f_j|^2) \,d\theta &= \frac1{2}\log \left(1+ \frac1{2\pi}\int_{|z|=r}|f_j|^2 \,d\theta \right)\leq \frac12 \log^+T_2(r)+O(1) &= O(\log^+T_{\exp_Af}(r)) +O(1).
\int_{H_n}\left((u')^2 - (w_n')^2 \right) = \int_{H_n}\left((u')^2 - (l_n')^2 \right)+ \int_{H_n}\left((l_n')^2 - (w_n')^2 \right) & \geq \int_{H_n}\left( (u')^2 - (l_n')^2 \right) - \int_{H_n}|(l_n')^2 - (w_n')^2| .
\Gamma_{I}^{\phi_{S}^{0}}( - i \Omega_{m}, \stackrel{\rightharpoonup}{p}) = - G / \sum_{Q}g_{Q Q}( - \Omega_{m}^{2}-{\stackrel{\rightharpoonup}{p}}^{2}- 4 m_{Q}^{2}) K_{Q}^{T}( - i \Omega_{m}, \stackrel{\rightharpoonup}{p}) .
e_{\underline{{\alpha}}}^{\alpha}( x , \phi , \eta ) = ( f ( \phi , \eta_{i}) )_{\beta}^{\alpha}\left[ 1 -{\frac{1}{2}}x^{m}\Gamma_{m}\Gamma_{r}( 1 - \tilde{\Gamma}) \right]_{\underline{{\alpha}}}^{\beta}\ .
1=F_{n,0}\geq \dots \geq F_{n,k_0(n)-1}\geq F_{n,k_0(n)}\geq q^{n/4(1\wedge \gamma)}>F_{n,k_0(n)+1}\geq \ldots, 1=\tilde{F}_{n,0}\geq \dots \geq \tilde{F}_{n,d_0(n)-1}\geq \tilde{F}_{n,d_0(n)}\geq q^{n/4(1\wedge\gamma)}> \tilde{F}_{n,d_0(n)+1}\geq \ldots.
(g_B\cdot f_1\otimes f_2)(u_1\otimes u_2) = g_B(f_1(u_1)\otimes f_2(u_2)) = f_2(u_2)\otimes f_1(u_1) = (f_2\otimes f_1)(g_A(u_1\otimes u_2)) = (f_2\otimes f_1 \cdot g_A)(u_1\otimes u_2)
d_M(se^{i\theta},\phi(se^{i\theta})) = |\frac{\frac{se^{i\theta}+ r}{1+rse^{i\theta}}- se^{i\theta}}{1 - se^{-i\theta}\frac{se^{i\theta}+ r}{1+rse^{i\theta}}}| & = |\frac{se^{i\theta}+ r - se^{i\theta}(1+rse^{i\theta})}{(1+rse^{i\theta}) - se^{-i\theta}(se^{i\theta}+ r)}| & = |\frac{r(1-s^2e^{2i\theta})}{1 - s^2 + rs(e^{i\theta}- e^{-i\theta})}|,
\underset{_{\substack{(\rho )=\frac{1}{2}Y<\left\vert \gamma \right\vert \leq eT}}}{\sum}\frac{(x+h)^{\rho +1}}{\rho\left( \rho +1\right)}=O\left( x^{\alpha}(\log x)^{\beta}(\log \log x)^{\beta}\right)
{}^{M\!L\!\!}\textrm{F}_{p}(a,b;c;z) = \frac{1}{2\pi i}\int_{-i\infty}^{-i\infty}\frac{{}^{M\!L\!}\Gamma(s)B(b+s,c+s-b)}{B(b,\; c-b)}{}_{2}\textrm{F}_{1}(a,b+s;c+2s;z)p^{-s}ds.