harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nabla\times\mathbf{A}=\frac{1}{\rho}\begin{vmatrix}\begin{array}{ccc}\mathbf{e}_{\rho}\rho\mathbf{e}_{\phi}\mathbf{e}_{z}\frac{\partial}{\partial\rho}\frac{\partial}{\partial\phi}\frac{\partial}{\partial z}A_{\rho}\rho A_{\phi}A_{z}\end{array}\end{vmatrix}
\begin{aligned}- \iint \left ( \frac{1}{2}\frac{|m |^2}{\rho}+ F(\rho,\nabla\rho) \right ) \dot\theta(t) \le \int \left ( \frac{1}{2}\frac{|m |^2}{\rho}+ F(\rho,\nabla\rho) \right ) \Big |_{t=0}\theta(0)dx \, , \ \theta \in W^{1, \infty}[0, \infty) \ \mbox{compactly supported on $[0, \infty)$}, \end{aligned}
\mathbf{R_e}= \mathbf{R_e}^{mmse}+ \left(\mathbf{W}^{(g)}-\mathbf{W}^{(g)}_{mmse}\right)^H\mathbf{R}_{\mathbf{y}}\left(\mathbf{W}^{(g)}-\mathbf{W}^{(g)}_{mmse}\right)
{\cal P}_{0,n}^{A.J}(\kappa)\triangleq\Big\{\pi_t(x_t|y_{t-J}^{t-1}), ~t=0, 1, \ldots, n: \frac{1}{n+1}{\bf E}^{\pi}\Big( c^{A.N}_{0,n}(X^n, Y^{n-1}) \Big)\leq \kappa\Big\},~ \kappa \in [0,\infty)
\left( \begin{array}{c}{( d_{+})^{m}\chi_{1}^{( n )}( z , \bar{z})}\\{0}\\ \end{array}\right) \makebox{a n d}\left( \begin{array}{c}{0}\\{( d_{+})^{m - 1}\chi_{1}^{( n )}( z , \bar{z})}\\ \end{array}\right) ,
\mathcal{I}^{\mathcal{J}}_+ ((\phi_-,j_-'),(\phi_+,j_+'),T) = -\tau^{-1}_{\pm(T+T_0)}(\Pi_+^{\psi_+ + \phi_+,j_+', T}D_{+,T}^{j_+'})^{-1}\Pi_+^{\psi_+ + \phi_+,j_+',T}[D_{+,T}^{j_+'}(\psi_{+,T}+ \phi_{+,T}) + e_{+,T}& +\mathcal L_+(\psi_{-,T}+\phi_{-,T},\psi_{+,T}+\phi_{+,T}) +\mathcal{R}_+(\psi_{+,T}+\phi_{+,T})],
T(w,\bar{x},\bar{y}) = \left(\frac{f_{1}'(\bar{x}) - g_{1}'(\bar{y})}{f_{1}''(\bar{x})(\bar{y}-\bar{x})}\right)\left(\frac{g_{2}'(w) - f_{2}'(\bar{y})}{f_{2}''(\bar{y})(\bar{y}-w)}\right).
f(x_+) - f(x_-) = \frac{z-m}{z-y}\left(\varphi(x_+-y) -\varphi(x_--y) - \varphi(m-x_+)+\varphi(m-x_-)\right) +\frac{m-y}{z-y}\left(\varphi(z-x_+) -\varphi(z-x_-) - \varphi(m-x_+)+\varphi(m-x_-)\right).
\renewcommand{\arraystretch}{1.2}\begin{array}{rcl}\chi_{\mathcal{H}_{\frac12,0}}(\tau) = |\Lambda_{0,2}(\tau)|^2 = \chi_{\mathcal{V}_{-1/8,-1/8}}(\tau)\,, \chi_{\mathcal{H}_{\frac12,\frac12}}(\tau) = |\Lambda_{2,2}(\tau)|^2 = \chi_{\mathcal{V}_{3/8,3/8}}(\tau)\,, \end{array}
[z(u)]_{\alpha;\bar{\Omega}}=\sup_{x\neq y}\frac{|z(u)(x)-z(u)(y)|}{|x-y|^\alpha}&=\sup_{x\neq y}\left|\int_{\Omega}\frac{|u(x-z)|^p-|u(y-z)|^p}{(|x-y|^{\alpha})|z|^{n-2}}\,dz\right| &\leq [|u|^p]_{\alpha;\bar{\Omega}}\int_{\Omega}\frac{1}{|z|^{n-2}}\,dz &=C(\Omega) [|u|^p]_{\alpha;\bar{\Omega}}\leq C.
\Omega_1:=\left[1+\sqrt{\frac{\gamma_n}{4n}}-\frac{C_n}{\sqrt{4 n \gamma_n}},~1+\sqrt{\frac{\gamma_n}{4n}}+\frac{C_n}{\sqrt{4 n \gamma_n}}\right]\times \left[-\frac{n^{\tau/2}}{n^{1/4}},~\frac{n^{\tau/2}}{n^{1/4}}\right],
Z [ \eta_{l}] \approx \int{\cal D}[ \chi ] \exp \left[ -{\frac{e^{2}}{4 \pi^{2}m_{W}}}\int d^{3}x \left({\frac{1}{2}}( \nabla ( \chi_{\mathrm{c l}}- \eta_{l}) )^{2}- \omega^{2}\cos \chi_{\mathrm{c l}}\right) \right] ,
\bigcup\limits_{\mathcal{H}\times J\in NH}\mathcal{H}\times J=\bigcup_{\mathcal{K}\times I\in P^{1}ND_{\lambda}}\mathcal{K}\times I=\bigcup_{\mathcal{K}\times I\in P^{2}ND_{\lambda}}\mathcal{K}\times I.
|\tilde b(t,x)| \le\, |\tilde{b}(t,x)-\tilde{b}(t,x_0)|+\,|\tilde{b}(t,x_0)|\le\,\varphi(t)\,\omega(|x-x_0|)+\,|\tilde{b}(t,x_0)| &\le\,\omega(2R)\,\varphi(t)+|\tilde{b}(t,x_0)|t\in Ix,\,x_0\in B(o,R)\,.
R^{(\lambda)}(0,1) = R^{(\lambda)}(1^\infty,2^\infty) = \lim_{n \to \infty}R^{(\lambda)}_n (1^n,2^n) &= \lim_{n \to \infty}\left(\sum_{k=1}^n c(1^k,1^{k-1})^{-1}+ \sum_{k=1}^n c(2^k,2^{k-1})^{-1}\right) & = 2 \lim_{n \to \infty}\sum_{k=1}^n (2\lambda)^{-k}= \frac{4\lambda}{1-2\lambda},
{\cal W}~ \equiv ~{\frac{1}{4 \rho^{2}}}~ \Big [ \cosh ( 2 \varphi_{2}) ~ ( \rho^{6}~ - ~ 2 ) ~ - ( 3 \rho^{6}~ + ~ 2 ) \Big ] \, , \qquad \rho ~ \equiv ~ e^{{\frac{1}{\sqrt{6}}}\varphi_{1}}\, .
\psi_{k}^{+}( x ) = \left( \begin{array}{l}{\eta_{k}^{S}}{\tilde{\eta}_{k}^{A}}\end{array}\right) \mathrm{~ ~ ~ ~ a n d ~ ~ ~ ~}\psi_{k}^{-}( x ) = \left( \begin{array}{l}{\eta_{k}^{A}}{\tilde{\eta}_{k}^{S}}\end{array}\right)
F_{L}^{( \alpha )}( \bar{z}) = - \overline{{{\Theta}}}_{\Lambda_{2 4}}\partial_{\bar{z}}\overline{{{\rho}}}- \overline{{{\Theta}}}_{\Lambda_{2 4}}\overline{{{\hat{G}}}}_{2}+ \frac{\pi^{2}}{6 \zeta ( 1 4 )}\overline{{{G}}}_{1 4}.
\mathcal{P}_1(r)=\omega_n^{-1}\int_{B(0,r)^c}\Big( 1 \land \frac{|x|^2}{r^2}\Big) J(|x|)dx \le \omega_n^{-1}\mathcal{P}(r) \le c_4 V(r)^{-2}, r>0.
\aligned R^{\alpha, \Omega}_\mu (x) = \int_\Omega \frac 1{|x - y|^{n-\alpha}}d\mu & = \int_{\Omega\setminus B_{2^{-i_0+2}\delta}}\frac 1{|x - y|^{n-\alpha}}d\mu + \int_{B_{2^{-i_0+2}\delta}\setminus \Omega^\delta_i}\frac 1{|x - y|^{n-\alpha}}d\mu + \int_{\Omega^\delta_i}\frac 1{|x - y|^{n-\alpha}}d\mu,\endaligned
\varepsilon_{k+1}= \frac{\tilde p - 1}{\tilde p}\sigma^{-\frac{1}{\tilde p-1}}\frac{(A_{k+1}- A_k)^{\frac{\tilde p}{\tilde p-1}}}{\delta}\|\nabla f(y_{k+1})\|^{\frac{\tilde p}{\tilde p-1}}&- \frac{A_{k+1}}{\delta}\frac{(N^2 -1)^{\frac{\tilde p-1}{2\tilde p-2}}}{2N}\epsilon^{\frac{1}{\tilde p-1}}\|\nabla f(y_{k+1})\|^{\frac{\tilde p}{\tilde p-1}},
\frac{F^2}{w_+}\left(f_1^{-1}\left(\frac{y}{n^2}\right)\right) + \frac{F^2}{w_-}\left(f_1^{-1}\left(\frac{y}{n^2}\right)\right) - \frac{F^2}{w_+}\left(f_1^{-1}\left(\frac{y}{(n+1)^2}\right)\right) \qquad{}+ \frac{F^2}{w_-}\left(f_1^{-1}\left(\frac{y}{(n+1)^2}\right)\right)= O\left(\frac{1}{n\log^3n}\right).
\left(\operatorname{Mod}_A\otimes_{\mathcal{C}}\operatorname{Mod}_A\right)^{\otimes}= QC(\operatorname{Spec}\,A\times_{\mathsf{M}_{\mathcal{C}^{\otimes}}}\operatorname{Spec}\,A) &= QC(\operatorname{Aut}^{\otimes}_{\omega})^{\otimes}&\simeq \operatorname{Mod}_{\mathcal{H}}^{\otimes}
T(M^1; x^0)=M^1, T(M^2; x^0)=\{(u_1, u_2) : u_1\leq 0\}, &T(M^3; x^0)= \{(u_1, u_2)\,:\, u_1\geq 0\}, \bigcap_{i=1}^3 T(M^i; x^0)=\{(u_1, u_2)\in\mathbb{R}^2:u_1=0, u_2\leq 0\}.
\chi_{n}( \xi_{1}) \equiv \langle \xi_{1}| n \rangle = \frac{( - i )^{n}}{\pi^{1 \! / \! 4}}\frac{1}{\sqrt{2^{n}n !}}H_{n}( \xi_{1}) e^{- \frac{1}{2}\xi_{1}^{2}},
M =\left(\begin{array}{cc|c|cc}m_{1,1}m_{1,2}m_{1,3}0 0 0 m_{2,2}m_{2,3}0 0 \hline 0 0 m_{3,3}0 0 \hline 0 0 0 1 0 0 0 0 0 1\end{array}\right),
\|\tilde{f}(j^-)- \tilde{f}(l)\|_2^2 =\sum_{r =j^-}^{l-1}\sum_{k=0}^{2^{r}-1}\Big( f_{0,r k}+Z_{r k}+W_{r k}\Big)^2 &\leq 3\sum_{r =j^-}^{l-1}\sum_{k=0}^{2^{r}-1}\Big( f_{0,rk}^2+ Z_{r k}^2+ W_{r k}^2\Big).
G(x,v):=\left\lbrace\begin{array}{l}(1-(v_0\cdot v)^2)(\tilde E_1k_1-\tilde E_2 k_2)(x-{(x-x_0)\cdot (v-(v\cdot v_0) v_0)\over 1-(v\cdot v_0)^2}v,v_0,v)\textrm{when}v\not=\pm v_0, 0 \textrm{otherwise},\end{array}\right.
(\psi;g_1,\dots,g_l;[v],[y])\circ(\varphi;f_1,\dots,f_n;[u],[x]) &= (\psi\varphi^y;\gamma(g_1;((vy)_\ast\vec f)^\psi_1),\dots,\gamma(g_l;(vy)_\ast\vec f)^\psi_l;[\varphi^\ast(vy)u\varphi^\ast(y)^{-1}],[\varphi^\ast(y)x])\ ,
f(gdx^{i})=f(dx^{j}r_{j}^{i}(g))=(fdx^{j})r_{j}^{i}(g)=(dx^{k}r_{k}^{j}(f))r^{i}_{j}(g)=dx^{k}(r_{k}^{j}(f)r^{i}_{j}(g)).
M_\infty(\alpha) = M_\infty(\zeta\alpha^L) & \leq \max_{1\leq n\leq N}\{M(\zeta_n\alpha_n^{L_n})\}& \leq \max_{1\leq n\leq N}\{M(\zeta_n\alpha_n^{L_n})^{S_n}\}& = \max_{1\leq n\leq N}\{M(\alpha_n)\}& = M_\infty(\alpha).
{\alpha}_{1}= \displaystyle \frac{g^{2}C_{V}}{4 \pi}, \quad{\alpha}_{2}= - \displaystyle \frac{g^{2}C_{V}}{3 2 \pi}, \quad{\beta}_{1}= - \displaystyle \frac{g^{2}C_{V}}{1 6 \pi}, \quad{\beta}_{2}= 0 \, .
-\frac{1-r_s^2}{4}\left(\nabla_sv\cdot\nabla_sv\right)=z_j^2\frac{(1-|w_j|^2)^2}{(1+\overline{w}_jv^\prime)^4}\frac{-(1-r_s^2)}{4}\left(\nabla_sv^\prime\cdot\nabla_sv^\prime\right),
|\frac{U_p^{sp}}{a_p}f_m |_{S_{I,m}(\delta)\setminus S_{I,{m+1}}(\delta')}\leq |f_{m}|_{S_{I,m-1}(\delta)\setminus S_{I,{m}}(\delta')}&= |f_{m-1}|_{S_{I,m-1}(\delta)\setminus S_{I,{m}}(\delta')}
v_x=-\frac{1}{\kappa}[s_2X_x\tilde{u}_{\tilde{x}}-r_1X_x\tilde{v}_{\tilde{x}}+r_1(s_{1x}v+s_{2x}u+s_{3x})-s_2(r_{1x}u+r_{2x}v+r_{3x})],
\begin{aligned}\mathfrak{S}_{\delta, 1, \flat}^{\Lambda*\mathbb{1}}\ll \big((x^{171}D^{624})^{1/1026}+ D^{9/10}+ (x^{-1}D^3)^{1/2}& \hskip 3,5mm+ (x^{19}D^{68}H^{19})^{1/114}+ (x^5D^{17}H^5)^{1/30}\big)x^{\varepsilon}\end{aligned}
\frac{\rho_*}{\hat\rho}=\frac{\left(\frac{p_*}{\hat p}\right)+\left(\frac{\gamma-1}{\gamma+1}\right) +\frac{2\gamma\pi}{\hat p(\gamma+1)}}{\left( \frac{\gamma-1}{\gamma+1}\right)\left(\frac{p_*}{\hat p}\right)+1+\frac{2\gamma\pi}{\hat p(\gamma+1)}}\,.
[e_1, e_2] = -[e_2, e_1] = -3e_3, [e_1, e_3^*] = -[e_3^*, e_1] = e_1^* + 2 e_2^*, [e_2, e_3^*] = -[e_3^*, e_2] = -e_1^* + e_2^*; e_1 \cdot e_1 = e_2 \cdot e_2 = 2 e_3, e_1 \cdot e_2 = e_2 \cdot e_1 = e_3, e_1 \cdot e_3^* = e_3^* \cdot e_1 = e_1^*, e_2 \cdot e_3^* = e_3^* \cdot e_2 = e_1^* + e_2^*.
\begin{aligned}p(t)(\partial_t\varphi(t,q)) =\frac{1}{2}g_{q(t)}(u(q(t),p(t)),u(q(t),p(t)))=h(q(t),p(t)) &=h(q(0),p(0))=h(q_0,n(q_0\bar p_0)=h(\bar q_0,\bar p_0)=c^2.\end{aligned}
e^{z L_{- 1}+ x J_{0}^{-}}( J_{0}^{+}+ D_{\jmath_{0}}^{+}) e^{- z L_{- 1}- x J_{0}^{-}}= J_{0}^{+}- 2 x J_{0}^{0}+ 2 \jmath_{0}x - x^{2}\partial_{x}= ( J_{0}^{+}- D_{\jmath_{1}}^{+}) - 2 x ( J_{0}^{0}- D_{\jmath_{1}}^{0}- \jmath_{0})
\|\nabla u_R\|^p_{L^p(B_{1/2})}\le \frac{Ci_0^{2p}}{(\gamma_{i_0}-1)^p}\,\left[(1-s)\,\mathcal{M}_{\gamma_{i_0-1}}+h_0^{-p\,\gamma_{i_0}}\,\|u_R\|^p_{L^p(B_1)}\right],
(\rho_1 \rho_n^q)^{n-j+1}\rho_{i}=\rho_1 \rho_n^q\rho_{{i+n-j}}\rho_n^{q(n-j)+B(i,j)}=\rho_{{n-j+i+1}}\rho_n^{q(n-j+1)+B(i,j)+D(n-j+i)}=\rho_{{n-j+i+1}}\rho_n^{q(n-j+1)+B(j,j)+D(n-j+i)}=\rho_{{n-j+i+1}}\rho_n^{q-1+D(n-j+i)}\rho_n\rho_n^{q(n-j)+B(j,j)}=\rho_{{n-j+i+1}}\rho_n^{q-1+D(n-j+i)}(\rho_1 \rho_n^q)^{n-j}\rho_j,
\eta_{t,1}^N(f)= \eta_{0,1}^N(f)+\widetilde{\mathfrak{M}}_{f,k}^N(t)-\int_0^tK_{15,s}^N(f)ds &-\int_0^t\mu_{s,1}^N\bigotimes\mu_{s,3}^N\left(\lambda(\cdot, \ast)(\tilde{G}(\cdot)-\tilde{F}(\cdot))\right)ds+o(1),
c_1(\theta_1)\gamma_1(\theta_1)\dfrac{\partial\theta_1}{\partial t}=\dfrac{1}{r^{\nu}}\dfrac{\partial}{\partial r}\bigg(\lambda_1(\theta_1)r^{\nu}\dfrac{\partial\theta_1}{\partial r}\bigg),\;\;\;\alpha(t)<r<\beta(t),\;\;\;0<\nu<1,
P(t,S)=p^{E}(t,S) + \int_{0}^{t}rK e^{-r(t-\xi)}\aleph (-d_{2}(S, t-\xi, \mathcal{B}(\xi)))\mathrm{d}\xi - \int_{0}^{t}\delta S e^{-\delta (t- \xi)}\aleph (-d_{1}(S, t- \xi, \mathcal{B}(\xi))) \mathrm{d}\xi,
s_{M_{t,n}^*}\,(\alpha,u)= \frac{1}{\tilde\sigma\sqrt n}s_{L_{nt}^*}(\alpha,u) = \frac{1}{\tilde\sigma\sqrt n}\left(s_{S_{\lfloor nt\rfloor}}(\alpha,u)+(nt-\lfloor nt\rfloor)s_{X_{\lfloor nt\rfloor+1}^*}(\alpha,u)\right).
P_{m}:=\sum_{i=0}^{m}p_{i}\to\infty\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Q_{n}:=\sum_{j=0}^{n}q_{j}\to\infty\,\,\,\,\,\,\,\,m, n\to\infty.
\Omega^{\alpha \beta}= v_{~ \gamma}^{\alpha}( y ) v_{~ \delta}^{\beta}( y ) \left[ ( \gamma_{a})^{\gamma \delta}e^{a}( x ) + R ( \gamma_{a b})^{\gamma \delta}\omega^{a b}( x ) + \theta^{( \gamma}{\cal D}\theta^{\delta )}+ ( d v v^{- 1})^{\gamma \delta}\right] ,
\lim_{d\rightarrow+\infty}P\big(\sup_{0\leq t\leq T}|\frac{|F_t^{d,p}|}{r^d}-pe^{-t}|>\epsilon/2\big) = \lim_{d\rightarrow+\infty}P\big(\sup_{0\leq t\leq T}|\frac{|G_t^{d,p}|}{r^d}-pe^{-t}|>\epsilon/2\big)\leq\lim_{d\rightarrow+\infty}\frac{4e^Tp}{\epsilon^2r^d}=0
B_{0}( x ) = \frac{1 - 4 \nu^{2}}{8 \mathrm{a r c t a n h}^{2}\sqrt{x}}- \frac{\sqrt{x}}{8 \mathrm{a r c t a n h}\sqrt{x}}\left( 1 + \frac{1 - 4 \nu^{2}}{x}\right) \, ,
\begin{aligned}x^{k+1}= \mathrm{prox}_g^{T_k}(x^k - T_k (A^T\Bar{y}^k + \nabla h(x^k))) &\hat{y}_i^{k+1}= \mathrm{prox}_{f_i^*}^{S_k^i}(y_i^k + S_k^i A_i x^{k+1}), \ \i = 1,...,n,\end{aligned}
P_p= \frac{1}{2}\mathbf{w}_1^H\mathbf{w}_1, P_r= \frac{1}{2}{\rm tr}\{\mathbf{F}\mathbf{H}_0\mathbf{w}_1\mathbf{w}_1^H \mathbf{H}_0^H\mathbf{F}^H+\sigma^2\mathbf{F}\mathbf{F}^H\}, P_s= \frac{1}{2}\mathbf{w}_2^H\mathbf{w}_2.
\begin{aligned}^2 (\ell_1, \ell_2) := \det \tilde{W}= - \det \Big{\{}\Big( (X_1 + i Y_1) M_1^2 (X_1^t + i Y_1^t) \Big)\cdot \Big((X_2 - iY_2) M_2^2 (X_2^t - i Y_2^t)\Big) \Big{\}}&= - \frac{\det^2 (X_1 + i Y_1)}{\det (X_1^t X_1 + Y_1^t Y_1)}\cdot\frac{\det^2 (X_2 - i Y_2)}{\det (X_2^t X_2 + Y_2^t Y_2)}.\end{aligned}
{f^{\phi}}_{0\frac{1}{2}0}={f^{\phi}}_{1\frac{1}{2}1}=2^{(\pi -2\rho )/4\pi}{f^{\phi}}_{\frac{1}{2}0\frac{1}{2}}=2^{(\pi -2\rho )/4\pi}{f^{\phi}}_{\frac{1}{2}1\frac{1}{2}}=\sqrt{K(i\rho )2^{(\pi -\rho )/2\pi}\cos \left( \frac{\rho -\pi}{4}\right)}
c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast\ast}= \frac{\Gamma\left(\frac{n-1}{2}\right)}{(2\pi)^{(n-1)/2}}\bigg\{\int_{0}^{\infty}t^{\frac{n-3}{2}}\left[(2+3\sqrt{2})\left(t+\frac{\xi_{\boldsymbol{s},k}^{2}}{2}\right)+3\right]\exp\left(-\sqrt{2t+\xi_{\boldsymbol{s},k}^{2}}\right)\mathrm{d}t\bigg\}^{-1}
\phi \nu_x v_i = \phi v_{i+1}= \gamma_{i+1}u_{i+1}= \gamma_{i}q^{-1}(1 - q^{2(i+1)}) u_{i+1}, \nu_x \phi v_i = \gamma_i \nu_x u_i = \gamma_i q^{-1}(1 - q^{2(i+1)}) u_{i+1}.
\operatorname{t a n}\hat{\theta}= - \frac{Q_{\mathrm{F 1}}}{Q_{\mathrm{D}p}}= \frac{V_{1}T_{\mathrm{F 1}}M}{V_{p}T_{\mathrm{D}p}N}= g_{s}\frac{( 2 \pi l_{s})^{p - 1}}{V_{p - 1}}\frac{M}{N}
< \bar{\psi}_{R}( x_{1}) \psi_{R}( y_{1}) . . . . . . \bar{\psi}_{R}( x_{p}) \psi_{R}( y_{p}) \bar{\psi}_{L}( w_{1}) \psi_{L}( z_{1}) . . . . . . \bar{\psi}_{L}( w_{q}) \psi_{L}( z_{q}) > ,
(\Delta_{I_{\lambda, v}\times J}e^{i\lambda\varphi_{\lambda,v}})^\wedge(u, P^\ast v) &=\frac{1}{(2\pi)^{k+m}}\int_{x\in I_{\lambda, v}}\int_{y\in J}\Delta_{I_{\lambda, v}}(x) \Delta_J(y)e^{i(v, Py+y_0)}e^{-i(u,x)}e^{-i(P^\ast v, y)}dx dy &=e^{i(v, y_0)}\widehat{\Delta_J}(0) \widehat{\Delta_{I_{\lambda,v}}}(u),
\textrm{\emph{Var}}_{\{Y<q(\alpha_1)\}}[X] =\textrm{\emph{Var}}_{\{q(\alpha_1)<Y<q(\alpha_2)\}}[X] & =\ldots & =\textrm{\emph{Var}}_{\{q(\alpha_{k-1})<Y<q(\alpha_k)\}}[X]=\textrm{\emph{Var}}_{\{q(\alpha_k)<Y\}}[X],
\tilde{S}_{2}[ \delta b_{i}] = \frac{1}{2 \pi^{2}g_{o}^{2}\alpha^{\, \prime \, \frac{p + 1}{2}}}\int \frac{d^{p + 1}k}{( 2 \pi )^{p + 1}}\; \frac{k_{\alpha}^{2}}{2}\delta b_{i}( - k ) \delta b^{i}( k )
\overline{C}_{X^{\ast}}([y_1,g_1], [y_2, g_2])=m_{X^{\ast}}(g_1^{y_2}g_2)^{-1}m_{X^{\ast}}(g_1)m_{X^{\ast}}(g_2)\widetilde{C}_{X^{\ast}}([y_1,g_1], [y_2, g_2]).
N_2 =q^{\max\{n_2,n_4\}-\min\{n_2,n_4\}}(q^{\min\{n_2,n_4\}-n_3}-(q-1)^{\min\{n_2,n_4\}-n_3})L_3 &=q^2(q-(q-1))H_3(q-1)^{n_3-s_3}\prod\limits_{j=1}^{s_3}\gcd(q-1,d_j^{(3)})=196.
\Bigg| \ln \frac{\mathbf{P}\big( (\hat{\theta}_n -\theta )\sqrt{\Sigma_{k=1}^n X_{k-1}^2}> x \sigma \big)}{1- \Phi(x)}\Bigg|\leq c \, \Bigg( \frac{x^3}{\sqrt{n}}+ (1+x) \frac{\ln n}{\sqrt{n}}\Bigg).
\frac{d\Pr\left[ Y_{j}=I\right]}{dt}=E\left[ -\delta1_{\left\{Y_{j}=I\right\}}+1_{\left\{Y_{j}=S\right\}}\beta\sum_{k=1}^{N}a_{kj}1_{\left\{Y_{k}=I\right\}}\right]
\psi_{1}= \frac{e^{2}Q^{2}}{1 6 \pi^{2}}[ \ln^{2}( q^{2}/ \mu^{2}) - \ln^{2}( p_{1}^{2}/ \mu^{2}) - \ln^{2}( p_{2}^{2}/ \mu^{2}) ] + \frac{g^{2}}{1 6 \pi^{2}}
\rho^{2}\frac{\partial^{2}W}{\partial r^{2}}+2\rho\sum\limits_{i=1}^{N}\gamma_{i}\frac{\partial^{2}W}{\partial r\partial\alpha_{i}}+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\gamma_{i}\gamma_{j}\frac{\partial^{2}W}{\partial\alpha_{i}\partial\alpha_{j}}\leq 0
V^{e}( r , \epsilon ) = 2 e^{- 2 \sqrt 2 r}f_{e}( r , \epsilon ) \{1 - 2 ({\frac{3 - \epsilon}{2 - \epsilon}}) ({\frac{2 - \epsilon}{4 - \epsilon}})^{( 4 - \epsilon ) / 2}e^{- ( 2 - \epsilon ) \sqrt 2 r}\}.
\widetilde \zeta_{j,N}= \gamma + \zeta_{j,N}\,, \zeta_{j,N}\dot = \sum_{\ell=1}^{\min\{j,2N-j-1\}}\gamma^{2\ell +1}\begin{pmatrix}j \ell\end{pmatrix}\begin{pmatrix}2N-j-1 \ell\end{pmatrix}\,.
T^{(2)}_{L,(1,0)}(Q_n)(1) = \Delta^{(2)}_{K_n}(1) = \exp\left (\dfrac{vol(K_n)}{6 \pi}\right ) \underset{n \to \infty}{\longrightarrow}\exp\left (\dfrac{vol(L)}{6 \pi}\right ) = T^{(2)}_{L,(1,0)}(1).
I_2 =\sum_{1\le|\alpha|\le L}\dfrac{\lambda^{-\frac{|\alpha|}{2}-\frac{n}{4}}}{\alpha !}\mathcal{F}[(-y)^\alpha \phi (y)](\lambda^{-\frac{1}{2}}\eta) \mathcal{F}[(\partial^\alpha \widetilde{\chi})\chi^2 u] (\eta +\lambda\xi).
\varphi=\bigl(\frac i{2\pi}\bigr)^{n-1}\sum_{j,k=1}^n\varphi_{jk}\Bigl(\frac{dz_1}{1-|z_1|^2}\wedge\stackrel{\stackrel{\hat \j}{\smile}}{\dots}\wedge\frac{dz_n}{1-|z_n|^2}\wedge \frac{d\bar z_1}{1-|z_1|^2}\wedge\stackrel{\stackrel{\hat k}{\smile}}{\dots}\wedge\frac{d\bar z_n}{1-|z_n|^2}\Bigr)
\sum_{l}\int_0^{\infty}\,dr r \,\psi_{\scriptscriptstyle{l,L}}^{\ast}(k, r)\,\psi_{\scriptscriptstyle{l,L^{\prime}}}(k^{\prime}, r)=\delta_{\scriptscriptstyle{L,L^{\prime}}}\,\delta(E - E^{\prime})\,.
I(0)=\sup_{x\in\Omega}\int_{\Omega\,\cap\,B_{H_0}(x,1)}|z(y,0)|\,dy=\sup_{x\in\Omega}\int_{\Omega\,\cap\,B_{H_0}(x,1)}e^{- H_0(y)^2}|\phi(y)|\,dy=1.
\begin{aligned}\d I ={}J\,\omega_1{}+ I_2\,\omega_2{}+ I_3\,\omega_3\,,{}\d J ={}-KI\,\omega_1{}+ J_2\,\omega_2{}+ J_3\,\omega_3\,,{}\d K ={}K_1\,\omega_1{}+ K_2\,\omega_2{}+ 0\,\omega_3\,.{}\end{aligned}
\tau^{\rm nonpl}_{\rm nc}(q)=\frac{-1}{32\pi}\left(\frac{|\theta q|^2}{4}+\frac{1}{\Lambda^2}\right)^{-\frac12}+ \frac{|q|}{16\pi}\int^1_0 \frac{d\alpha}{\sqrt{\alpha(1 - \alpha)}}\exp\left(- |\theta q| |q| \sqrt{\alpha(1 - \alpha)}\right)
\alpha_{n,i}(q^{2})+\alpha'_{n,i}(q^{2})=2q^{2i+2}\mu_{n-i-1}(q^{2})+4\frac{\mu_{n-i-2}(q^{2})(q^{n-i-1}-1)}{q^{n-3i-5}(q^{2}-1)}=\Sigma_{n,i}\;.
\Pi\circ \bar{Q}\psi (\omega_0,\hat{v}) =\int \bar{Q}\psi(\omega_1,\omega_0,\hat{v}) \; dK_{\omega_0}(\omega_1) &=\int \int\psi(\omega_2,\omega_1, \hat{A}(\omega_1,\omega_0)\hat{v}) \; dK_{\omega_1}(\omega_2) \, dK_{\omega_0}(\omega_1).
\theta_2(x) ={\sf c}_2 |x|^2{\sf U}_\infty^{p-1}|x|^\frac{2(p-q)}{1-q}\theta_0 +{\sf h}_2(x) ={a}_2 |x|^\frac{4(p-q)}{1-q}\theta_0 +{\sf h}_2(x) ({a}_2\not=0),{\sf h}_2(x) = |x|^\frac{4(p-q)}{1-q}\theta_0 \sum_{i=1}^{N'}b_i |x|^\frac{2(p-q)i}{1-q}.
\left( \begin{array}{c}\dot{y}_1 \dot{y}_2 \end{array}\right) = \left( \begin{array}{cc}m(l) 1 0 -[n-2-m(l)] \end{array}\right) \left( \begin{array}{c}y_1 y_2 \end{array}\right) +\left( \begin{array}{c}0 - g(y_1,s;l)\end{array}\right)
S_{f_n}(\rho\|\sigma) =\sup_{x(\cdot)}\biggl[f_n(0^+)\sigma(1)+f_n'(+\infty)\rho(1) &\qquad-\int_{[1/n,n]}\bigl\{\sigma((1-x(s))^*(1-x(s)))+s^{-1}\rho(x(s)x(s)^*)\bigr\}(1+s)\,d\nu_n(s)\biggr].
\frac{1}{|r_{j}-r_{i}|}=\frac{1}{|R_{AB}-(r_{Ai}-r_{Bj})|}=\sum_{L=0}^{\infty}\sum_{M=-L}^{L}(-1)^{M}I_{L}^{-M}(R_{AB})R_{L}^{M}(r_{Ai}-r_{Bj}),
\begin{aligned}\overline G_i (h, q) - G(0, 0) + \gamma \geq - \, \frac{C_2}{T_i (h)}\int_0^{T_i (h)}(1 + |q|)|DH(X(t, x))| \, dt = - (1 + |q|) \frac{C_2 L_i (h)}{T_i (h)}\geq - \, \frac{C_2 L_i (h)}{T_i (h)}- \gamma C_2 L_i (h).\end{aligned}
S\!\times\! (Z\!\times\!{\mathbf A}^1) \to Z\!\times\!{\mathbf A}^1, (s, (z, a)) \!\mapsto\!\begin{cases}\hskip -.5mm (z, a\!+\!f(z)s) \mbox{if ${\rm char}\,k\!=\!0$}, \hskip -.5mm (z, a\!+\!f_1(z)s^{p^{i_1}}\!+\!\cdots \!+\! f_d(z)s^{p^{i_d}}) \mbox{if ${\rm char}\,k\!=\!p\!>\!0$}. \end{cases}
|\frac{-32 \lambda'(t)}{\lambda(t)^{3}}\int_{t}^{\infty}ds \lambda''(s) \left(K_{1}(s-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+s-t)}\right)| &\leq \frac{C}{t^{3}\log^{2-b}(t)}\int_{t}^{\infty}|K_{1}(s-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+s-t)}| ds &\leq \frac{C}{t^{3}\log^{b+2}(t)}
(\Psi_l,\Psi_l) = |U_{\rho_l}(k_0)|^{-1}\int_{U_{\rho_l}(k_0)}dk \int_{F_k^n}\Big|\phi\Big(k;p_1,\ldots,p_n \Big)\Big|^2\,dp & \times \int_{F_{K-k}^{N-n}}\Big|\psi\Big(K-k;p_{n+1},\ldots,p_N\Big)\Big|^2\,dp=1.
\begin{aligned}\eta_2^\varepsilon(x,t) = \varepsilon d(x,t)\Bigl(\bar{g}_1\bar{\zeta}_0-\bar{g}_0\bar{\zeta}_1\Bigr)(x,t)+\frac{1}{2}d(x,t)^2\Bigl(\bar{\zeta}_0-\bar{\zeta}_1\Bigr)(x,t), \rho_\eta^\varepsilon(x,t) = \bar{\eta}(x,t)-k_d^{-1}d(x,t)\Bigl(\overline{V_\Gamma\eta}\Bigr)(x,t)+\eta_2^\varepsilon(x,t). \end{aligned}
\lim_{r \to r_{+}}F ( r ) ={\frac{1}{4 8 \pi}}\lim_{r \to r_{+}}{\frac{f^{\prime \prime \prime}}{f^{\prime}}}+{\frac{\lambda^{2}}{4 \pi}}\lim_{r \to r_{+}}( \ln f -{\frac{1}{f}}+ 1 ) .
(\pi_*(E\oplus F))^+ =(\pi_*E)^+\oplus(\pi_*F)^+=(K_E^+\oplus K_F^+)\oplus(L_E^+\oplus L_F^+), (\pi_*(E\oplus F))^- =(\pi_*E)^-\oplus(\pi_*F)^-=(K_E^-\oplus K_F^-)\oplus(L_E^-\oplus L_F^-).
\log \left( d^2_{R_i}(p_1,p_2) \right) = \log \left(g_x^{-1}(x_1 + x_2)^2 + g_y^{-1}(y_2-y_1)^2 \right) + \cdots \log \left( d^2_{R_{ij}}(p_1,p_2) \right) = \log \left( g_x^{-1}(x_1 + x_2)^2 + g_y^{-1}(y_1+y_2)^2 \right) + \cdots
\sigma_{1}=\begin{pmatrix}\-1 0 0 0 1 0 0 0 -1\end{pmatrix},\sigma_{2}= \begin{pmatrix}-\frac{1}{2}\alpha 0 -\frac{3}{4\alpha}-\frac{1}{2}0 0 0 1\end{pmatrix}\alpha \neq 0.
X_0 \times_{Y_0}Y_1 \to Y_0, X_1\times_{Y_1}B^Y_2 \to X_0\times X_0 \times_{Y_0\times Y_0}Y_1, B_2^X \cong B_2^Y\times_{Y_1\times Y_1}X_1 \times X_1\times_{Y_0\times Y_0}X_0 \times X_0
\tilde{A}_{|c|,E}(\theta)=\frac{1}{\sqrt{|c|(\theta)|c|(\theta-\alpha)}}D_{|c|,E}(\theta)=\frac{1}{\sqrt{|c|(\theta)|c|(\theta-\alpha)}}\left(\begin{matrix}E-v(\theta)\ \ -|c|(\theta-\alpha) |c|(\theta)\ \ 0\end{matrix}\right).
\frac{1}{2}\frac{d}{dt}\left\| \Lambda^{1/2}\theta \right\|^{2}_{L^{2}}+\nu\left\|\Lambda^{(1+\gamma)/2}\theta\right\|^{2}_{L^{2}}=\left(\frac{1}{2}-\delta\right)\int_{\mathbb{T}}\left[\theta\left(\Lambda \theta \right)^{2}\right]dx- \frac{1}{2}\int_{\mathbb{T}}\left[\theta \left(\theta_{x}\right)^{2}\right]dx,
P_{\textrm{isout}}^{\textrm{ISAOC}}= \left(1-\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FDF=1,FDN=1}\right\rbrace\right)\times\left(1-\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{NDF=1,NDN=1}\right\rbrace\right).
\tilde{F}(A)=\frac{1}{N^2}\log [\frac{Z(A)}{Z_{{\rm weak}}(A)}] \to\frac{1}{N^2}\log [ \oint \frac{dz}{2\pi i}\frac{1}{z^{N+1}}\frac{1}{1-z}\exp(-\sum_{n=1}^{[N/2]}\frac{1}{2n}z^{2n}\alpha_{2n}(A) ) ]
\lim_{N \rightarrow \infty}\langle \frac{1}{N}\mathrm{T r}\xi^{2 k_{1}}\cdots \frac{1}{N}\mathrm{T r}\xi^{2 k_{p}}\rangle = \frac{1}{N}\mathrm{T r}{\hat{\xi}}^{2 k_{1}}\cdots \frac{1}{N}\mathrm{T r}{\hat{\xi}}^{2 k_{p}}
\hat{C}_0(u_n)=2v_n\sum^{[t/(2v_n)]-1}_{j=0}\biggl(c_j^0(u_n)-\frac{1}{u_n^2(k_n-1)}\bigl(\sinh\bigl(u_n^2c_j^0(u_n)\bigr)\bigr)^2 \biggr),
K[t,p,q,x] = \frac{\pi^2}{2 t^2}\big(\mathrm{e}^{- t(p^2 x - q^2x^2)}- \mathrm{e}^{- t(p^2 x - q^2 x^2)- p\circ p/t}\cos2 x\theta(p,q) \big)~.
\mathrm{Sub}(D)(x_0,\xi)=\gamma^j(x_0)b_j(x_0)+\psi(x_0)-\frac{1}{2}\big(\frac{\partial\gamma^k}{\partial x^k}\big)(x_0)=\big(\frac{\gamma^jb_j+b_j\gamma^j}{2}+\psi\big)(x_0).
\texttt{LHS}\eqref{id:2ref2}\simeq\int_a^b x^{\xi + \lambda}\, dx\simeq\begin{cases}(b-a) b^{\xi + \lambda}\xi + \lambda > -1, \log \Big( 1 + \frac{b-a}{a}\Big) \xi + \lambda = -1, (b-a) b^{-1}a^{\xi + \lambda + 1}\xi + \lambda < -1\end{cases}& \simeq \texttt{RHS}\eqref{id:2ref2}.
\frac{d}{dt}A_1(t)+B_1(t)\leq K(t)\left(\log\sum_{i=1}^nA_i(t)\right)A_1(t), &\frac{d}{dt}A_i(t)+B_i(t)\leq K(t)\left(\log\sum_{i=1}^nA_i(t)\right)A_i(t)+\zeta A_{i-1}^\alpha(t)B_{i-1}(t),
\dim_{E}\mathrm{Hom}_{G}(I_{\sigma}, P(\chi)) = \dim_{E}\mathrm{Hom}_{G}(I_{\sigma}\otimes_{\mathfrak{Z}_{\Omega}}\kappa(\mathfrak{m}), P(\chi)) = \dim_{E}\mathrm{Hom}_{G}(P(\chi)^{\oplus n_{\sigma}}, P(\chi)) = n_{\sigma}
\begin{aligned}\left(C_h-e^{-\frac{1}{2}\zeta hj^s}\right)\vert \eta(\mathbf 0)v_j(\mathbf 0)\vert \lesssim j^{-\beta+s}+j^{\gamma-(\alpha+1)s}+j^{-\beta+\frac{1}{2}s}+j^{-\beta+(-\alpha +\frac{1}{2})s}+j^{\gamma-\alpha s}+j^{-\frac{13}{24}s}+j^{-\frac{17}{12}s}, \end{aligned}
Q = \frac{2\pi a^3 c}{\beta}\sum_{j=1}^{\infty}\frac{\sinh (\beta \mu j)}{j}K_2(\beta c^{1 \over 2}j)- 2a^3 \sum_{l=-\infty}^{\infty}\frac{\bigl[c - ( \mu + i \frac{2\pi l}{\beta})^2 \bigr]^{1 \over 2}}{e^{-a A}- 1}
\psi_1(n_1; \lambda_1) \psi_2(n_2; \lambda_2) \cdots \psi_g(n_g; \lambda_g) & = (-1)^g \sum_{k_g = 1}^{n_g}\cdots \sum_{k_1 = 1}^{n_1}(-1)^{k_1 + \cdots + k_g}\binom{n_1}{k_1}\cdots \binom{n_g}{k_g}\frac{(k_1 p_1) \cdots (k_g p_g)}{(\lambda_1 + k_1 p_1) \cdots (\lambda_g + k_g p_g)}.
\begin{aligned}F_{k,N}^{\pm}(t,{\rm sym}^m):= \frac{1}{|\mathcal{H}^*_k(N)|}\sum_{\substack{f\in \mathcal{H}^*_k(N) L(1,{\rm sym}^m f)\gtreqless (B_m^\pm t)^{\pm A_m^\pm}}}1.\end{aligned}
( \Delta S )^{0}= T r \left[ \left( \begin{matrix}{{K^{\mu}{}_{\nu}}}&{{K^{\mu}{}_{F}}}\\{{K^{F}{}_{\nu}}}&{{K^{F}{}_{F}}}\\ \end{matrix}\right) \exp \frac{1}{M^{2}}\tilde{\cal R}\right] \ ; \qquad \tilde{\cal R}= \left( \begin{matrix}{{S_{\mu \nu}}}&{{q_{\mu}}}\\{{- \tilde{\partial}q_{\nu}^{T}}}&{{\tilde{\partial}\tilde{\nabla}}}\\ \end{matrix}\right) \ .
=\frac{M_{i}+\eta_{i}}{\sqrt{2}}\left( \begin{array}{c}\psi_{lR}\psi_{l}+\psi_{u}\end{array}\right) =\frac{M_{i}+\eta_{i}}{\sqrt{2}}\left( \begin{array}{c}\psi_{1R}\sqrt{2}\psi_{1L}+\psi_{1R}\end{array}\right) .
f_{1}= 1 + \frac{c_{1}Q_{1}}{x^{2}}, \hspace{1 c m}f_{5}= 1 + \frac{c_{5}Q_{5}}{x^{2}}, \hspace{1 c m}k = \frac{c_{P}N}{x^{2}}\hspace{1 c m}x^{2}= x_{1}^{2}+ \cdots x_{4}^{2}