harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{}^{M\!L\!}\Gamma_{p}(x){}^{M\!L\!}\Gamma_{p}(y)=4\int_{0}^{\frac{\pi}{2}}\int_{0}^{\infty}r^{2(x+y)-1}\cos^{2x-1}\theta \sin^{2y-1}\theta \times E_{\alpha, \beta}^{\gamma}\left(-r^2\cos^2\theta -\frac{p}{r^2\cos^2\theta}\right) E_{\alpha, \beta}^{\gamma}\left(-r^2\sin^2\theta -\frac{p}{r^2\sin^2\theta}\right)drd\theta.
\frac{1}{N}\sum_{i=1}^N\sum_{j=1}^N\widehat G_{2}\widehat\Gamma_i\widehat\Sigma_{ii}^{-1}\bigl(\widehat BM_{z z}^{i j}\widehat B{}'-\widehat\Sigma_{z z}^{i j}\bigr)\widehat\Sigma_{j j}^{-1}I_{K+1}^{1}\hat\lambda_j'=0,
f_2(z)^2 A(z)(f_1(z)f_1(z+1)-B(z)) &=f_2(z)^2(\Delta f_1(z))^2=(f_1(z)\Delta f_2(z)-H(z))^2 &=f_1(z)^2(\Delta f_2(z))^2-2f_1(z)H(z)\Delta f_2(z)+H(z)^2 &=f_1(z)^2A(z)(f_2(z)f_2(z+1)-B(z))-2f_1(z)H(z)\Delta f_2(z)+H(z)^2.
= \; \left[ \prod_{a < b}^{N}\frac{( 2 n_{a}+ 2 n_{b}) !}{( 2 n_{a}) ! ( 2 n_{b}) !}\right]^{\frac{1}{q ( N - 1 )}}\; = \; \left[ \prod_{a < b}^{N}{\binom{2 n_{a}+ 2 n_{b}}{2 n_{a}}}\right]^{\frac{1}{q ( N - 1 )}}
I_3: =E\left[\int_t^T\langle 2(u(s)-\hat{\xi}(s)),f(s,u(s),\nabla u(s),v(s))-\hat{f}(s)\rangle \,ds|\mathcal{F}_t\right] &\leq E\left[\int_t^T\langle 2|u(s)-\hat{\xi}(s)|,|f_0|+L|u(s)|+L|\nabla u(s)|+L|v(s)|+|\hat{f}(s)|\rangle \,ds|\mathcal{F}_t\right].
P_G\left(x\right) = x^{n-t-1}\left(x+t\mu\right) \left(x-\mu\right)^t = x^{n-t-1}\left(x-\mu\right)^{t-2}\left[x^3+\left(t-2\right)\mu x^2+\left(1-2t\right)\mu^2 x+t\mu^3\right].
0=\int\limits_X \partial\xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \omega_{n-2}-\int\limits_X \xi \wedge \partial\overline{\rho_\omega^{2,\,0}}\wedge \omega_{n-2}- \int\limits_X \xi \wedge \overline{\rho_\omega^{2,\,0}}\wedge \partial \omega_{n-2}.
q_{\lambda}^{1/2}q_s^{-1}q_\mu^{1/2}\theta(\lambda n_s^{-1}(\mu)\mu_{n_s^{-1}}(-k))c_{s,-k}&=q_{\lambda}^{1/2}q_s^{-1}q_\mu^{1/2}q_{\lambda n_s^{-1}(\mu)\mu_{n_s^{-1}}(-k)}^{-1/2}E(\lambda n_s^{-1}(\mu)\mu_{n_s^{-1}}(-k))c_{s,-k}
S^{sumo}\le S(3;0,0)^9 S(3;2,2)^4 \le S(3;0,0)^9 S_U(3;2,2)^4 =: S_U^{sumo}\approx (6.6709 \times 10^{21})^9 \times (1.5976 \times 10^{11})^4 \approx 1.7045 \times 10^{241}.
|H_{C^0}(2c_{\mathbb{H}}(C) dH_C(y_n) + \eta_n) - H_{C^0}(2c_{\mathbb{H}}(C) dH_C(y_n))| \leq \|\eta_n\| \sup_{\theta\in (0,1)}\|dH_{C^0}(2c_{\mathbb{H}}(C) dH_C(y_n) + \theta \eta_n)\| \leq M \bigl( 2c_{\mathbb{H}}(C) \|dH_C(y_n)\| + \|\eta_n\|\bigr) \|\eta_n\|,
R(x_1, x_2)\, x_2 = - (3/4) \lambda^2 x_1 , R(x_1, x_i)\, x_i = - x_1 (\mbox{for $i = 3, \ldots, n-1$}) , R(x_1, x_n)\, x_n = - (1 - (1/4) \lambda^2) x_1 .
VI_2\leq \sum\limits_{k=1}^{\infty}\frac{e^{-C2^{2(k-1)}}2^{(k+1)n}}{(2^{k+1})^{n-\alpha}}\Big(\|\vec{b}\|_{*}M_{\alpha,r,w}f(x)+\|\vec{b}\|_{*}M_{\alpha,1}f(x)\Big) &\leq C\|\vec{b}\|_{*}M_{\alpha,r,w}f(x)+C\|\vec{b}\|_{*}M_{\alpha,1}f(x)
\bigl( 1-a_2 \bigr)^{-1/2}\iint f\left(x-\tfrac{a_1}{1-a_2}y\right)g(y) \exp\left( \tfrac{-y^2}{4\lambda(1-a_2)}\right) k_{\lambda\frac{a_1a_3}{1-a_2}}(x) k_{\lambda a_2}(y) &= \iint f\left(x-\tfrac{a_1}{1-a_2}y\right)g(y) k_{\lambda\frac{a_1a_3}{1-a_2}}(x) k_{\lambda a_2(1-a_2)}(y).
\int_{S^{n-1}}H_m^n(\xi)\, \widetilde{W}_{l,\xi}^{(q)}(0) \, d\xi = \frac{(n-1-l) \lambda_m(q)}{( n-1-q-l )(n-1)}\int_{S^{n-1}}H_m^n(\xi)\, \rho_K^{n-1-q-l}(\xi)\, d\xi
A\langle K^{\ast}x,K^{\ast}x\rangle_{\mathcal{A}}=A\langle K^{\ast}Q^{-1}Qx,K^{\ast}Q^{-1}Qx\rangle_{\mathcal{A}}&=A\langle Q^{-1}K^{\ast}Qx,Q^{-1}K^{\ast}Qx\rangle_{\mathcal{A}}&\leq\|Q^{-1}\|^{2}\int_{\Omega}\langle T_{\omega}Qx,T_{\omega}Qx\rangle_{\mathcal{A}}d\mu(\omega).
X\triangleright(\mathsf{M}_{m}^{n})_{j}^{i}=(X_{(1)}\triangleright u_{i}^{m*})(X_{(2)}\triangleright u_{j}^{n}= \sum_{k,\ell}\pi(S(X_{(1)}))_{k}^{i}u_{k}^{m*}\pi(X_{(2)})_{j}^{\ell}u_{\ell}^{n}=\sum_{k,\ell}\pi(S(X_{(1)}))_{k}^{i}(\mathsf{M}_{m}^{n})_{\ell}^{k}\pi(X_{(2)})_{j}^{\ell}.
\begin{array}[c]{ll}\mathcal{H}(s,x,y,z,u,p,q,P) &= pb(s,x,y,z+\Delta(s),u)+q\sigma(s,x,y,z+\Delta(s),u) +g(s,x,y,z+\Delta(s),u) &\ \ +\frac{1}{2}P(\sigma(s,x,y,z+\Delta(s),u)-\sigma(s,\bar{x}(s),\bar{y}(s),\bar{z}(s),\bar{u}(s)))^{2}\,\end{array}
|(B(v)-B(u),\,\eta)_H| =|b(\eta,\,u,\,\eta)|\le|\eta|_{L^4(\Omega,\,T\Omega)}|\eta|_{L^4(\Omega,\,T\Omega)}|u|_{H^1(\Omega,\,T\Omega)}&\le C|\eta|_{H^{\frac{1}{2}}(\Omega,\,T\Omega)}^2|u|_{H^1(\Omega,\,T\Omega)}\le C_1|\eta|_H|A^{\frac{1}{2}}\eta|_H|A^{\frac{1}{2}}u|_H
\langle I'(u),v\rangle=\int_{{\mathbb{R}}^3}\Big((-\Delta)^{\frac{s}{2}}u \cdot (-\Delta)^{\frac{s}{2}}v+V(x)uv+\phi_u^{t}uv-f(x,u)v\Big)\,\mathrm{d}x,\forall v\in E.
\widetilde{\Psi}^{c}_{R}(x;k) = \left( \begin{array}{c}\frac{\Psi^1_R(x)}{\frac{\pi}{L}(k+\frac{1}{2})-gq}\frac{\Psi^2_R(x)}{\frac{\pi}{L}(k+\frac{1}{2})+gq}\end{array}\right)^c,
\omega = \left( \begin{array}{c c}{\gamma^{\mu}\omega_{1 \mu}}&{\begin{array}{c}{\omega_{1}}\\{\omega_{2}}\\ \end{array}}\\{\begin{array}{c c}{\tilde{\omega}_{1}}&{\tilde{\omega}_{2}}\\ \end{array}}&{\gamma^{\mu}\omega_{2 \mu}}\\ \end{array}\right)
S_{s t a t}\equiv \log d ( N_{L 0}, N_{R 0}) \approx 4 \pi \sqrt{N_{L 0}}= 4 \pi \left( N_{L}-{\frac{1}{2}}\sum_{i = 1}^{[{\frac{D - 1}{2}}]}J_{i}^{2}\right)^{\frac{1}{2}},
\rho^n \Psi_\rho(\theta,\theta_2,\dots,\theta_{n-1},\varphi) &=\rho^n\sum_{l=0}^\infty\sum_{k\in\mathcal M_{n-1}(l)}a_l^k(\Psi_\rho)\,[Y_l^k(\rho r,\theta_2,\dots,\theta_{n-1},\varphi)+\mathcal O(\rho^3)],\qquad\rho\to0.
\sup_{\Omega}w \leq \sup_{\partial \Omega}\psi +C \left\|\frac{f^-}{d^{(n-1)/n}}\right\|_{L^n(\Omega)}&\leq \sup_{\partial \Omega}\psi +C \left\| f^-\right \|_{L^n(\Omega)}\sup_{\Omega}(d^{(1-n)/n}),
{}_3F_2\bigg[\begin{matrix}\alpha \beta \gamma &\alpha-\beta+1 \alpha-\gamma+1\end{matrix}\bigg|\,z\bigg] = (1-z)^{-\alpha}{}_3F_2\bigg[\begin{matrix}\alpha-\beta-\gamma+1 \frac12\alpha \frac12\alpha+\frac12 &\alpha-\beta+1 \alpha-\gamma+1\end{matrix}\bigg|\,-\frac{4z}{(1-z)^2}\bigg].
v^{\mu}(x_{+},\theta)=a^{\mu}+2i\theta\sigma^{\mu}\bar{\xi}+w^{\mu}_{~\nu}x_{+}^{\nu}+\lambda x^{\mu}_{+}-2\theta\sigma^{\mu}(x_{+}\cdot\tilde{\sigma})\tilde{\zeta}+b^{\mu}x_{+}^{2}-2x_{+}^{\mu}b\cdot x_{+}
\lim_{n \rightarrow \infty}\left |{\mathbb E}\left ( f \left( \frac{S_{[ns_1]}}{\sqrt n}, \ldots , \frac{S_{[ns_d]}}{\sqrt n}\right) \right)-{\mathbb E}\left ( f \left( \frac{M_{[ns_1]}}{\sqrt n}, \ldots , \frac{M_{[ns_d]}}{\sqrt n}\right) \right)\right |=0 \, .
\operatorname*{Re}\sum_{i=1}^d z_i^2 = \operatorname*{Re}\sum_{i=1}^d (x_i+\zeta_i)^2 \ge \sum_{i=1}^d \left(x_i^2 - 2 |x_i| |\zeta_i| - |\zeta_i|^2\right)\ge \|x\|^2 - \delta \|x\|^2 - \delta^{-1}\|\zeta\|^2 - \|\zeta\|^2 &> (1 - \delta - D^2/\delta - D^2)|x|^2 = (1 - 2 D - D^2)\|x\|^2 \stackrel{D = \sqrt{2}-1}{=}0.
\left\|x-\sum_{n=1}^{\lceil\lambda m\rceil}e^*_{\rho_x(n)}(x)e_{\rho_x(n)}\right\| \ \leqslant\ \mathbf C_\ell\left\|x-\sum_{n=1}^{m}e^*_{\rho_x(n)}(x)e_{\rho_x(n)}\right\| \ \leqslant\ \mathbf C_\ell\mathbf C_{1, rp}\sigma_m^{\mathcal{U}_{X}, R}(x).
[b_k(s), a_p^* (t) ] \ = \ \chi (p - k )\, e^{i(t-s) E_p}a_{p - k}(s) \ , [ b_k (s) , a_h^* (t) ] \ = \ - \chi^\perp ( h + k )\, e^{i(t-s) E_h}a_{h + k}(s) \ .
R^{n+1/2}a(z^{-n-1})^* T_R(\xi \otimes \eta) = T_R(a(s^{1/2}z^{-1}(sz^{-1}+ \overline{w})^n)^*\xi \otimes \eta) + (-1)^{p(\xi)}T_R(\xi \otimes \eta^{\prime\prime}).
P ( \mathrm{\boldmath ~ p ~}) = l \left( | f_{0}|^{2}\left[ | g_{p 0}|^{2}- ( - \eta )^{3}\frac{H^{2}}{2 \sqrt{( - p \eta )^{2}}}\right] + \int_{3 / 2}^{\infty}\, d m \ | f_{m}|^{2}\left[ | g_{p m}|^{2}- ( - \eta )^{3}\frac{H^{2}}{2 \omega}\right] \right) \ .
\frac{\mathrm{d}P_{\boldsymbol{\theta}}}{\mathrm{d}P_{\hat{\boldsymbol{\theta}}\left(\boldsymbol{x}\right)}}\left(\boldsymbol{x}\right)=\exp\left(-D_{A}\left(\boldsymbol{\theta},\hat{\boldsymbol{\theta}}\left(\boldsymbol{x}\right)\right)\right).
|F(x,t)-F(y,s)| =\big(|f_t(x)-f_s(y)|^2+|t-s|^2\big)^{1/2}&=\big(|f_t(x)-f_t(y)+f_t(y)-f_s(y)|^2+|t-s|^2\big)^{1/2}&=\big(|f_t(x)-f_t(y)|^2+|f_t(y)-f_s(y)|^2 &\qquad\qquad\qquad+2\,|f_t(x)-f_t(y)||f_t(y)-f_s(y)|+|t-s|^2\big)^{1/2}
y_{k,i}=r_{k,i}+{\textbf{w}}_k^{\mathsf{H}}\left(\sqrt{\alpha_k}{\textbf{h}}_kx_{k,i}+\sqrt{\alpha_{k'}}{\textbf{h}}_{k'}x_{{k'},i}\right) &+\sum\nolimits_{j\neq k}{\textbf{w}}_k^{\mathsf{H}}\left(\sqrt{\alpha_j}{\textbf{h}}_jx_{j,i}+\sqrt{\alpha_{j'}}{\textbf{h}}_{j'}x_{{j'},i}\right)+n_{k,i},
[n_i]_q := \frac{1-q^{2n_i}}{1-q^2}, [n_i]_q! := \Pi_{k=1}^{n_i}[k]_q, \tilde A_i\vert 0,0\rangle = 0, \langle 0,0\vert 0,0\rangle =1,i=1,2.
\sharp S(\varepsilon,K,\Lambda,B)=\frac{\Theta(\Lambda)\varepsilon K^2}{2}B^{2-\frac{1}{r}}+O(K^\frac{3}{2}b^{\frac{1}{4}}\det(\Lambda)^\frac{1}{2}B^{\frac{3}{4}(2-\frac{1}{r})}\log B+Kb^{\frac{3}{2}}B^{1-\frac{1}{2r}}\log B).
\frac{c_n^2-1}{c_n^2+1}\bigg\|\binom{\varphi}{L_n \varphi}\bigg\|^2 \leq \mathtt{d}_n\bigg[\binom{\varphi}{L_n \varphi}\bigg]- \frac{1}{4-n}\int_{\mathbb{R}^n}\frac{1}{|\mathbf{x}|}\bigg|\binom{\varphi(\mathbf{x})}{\big(L_n \varphi\big)(\mathbf{x})}\bigg|^2\mathrm{d}\mathbf{x}
E_{A}^{I - L - M}\biggl ( E_{C}^{L}( f ) D_{N}E_{D}^{M}( g ) \biggr ) = \sum_{P}( - 1 )^{\vert P \vert + \vert I \vert + \vert L \vert + \vert M \vert}{\binom{P}{I - L - M}}
\begin{pmatrix}{\dot{f}}_1^0 \vdots{\dot{f}}_N^0\end{pmatrix}=\begin{pmatrix}\lambda_{11}^0 \cdots \lambda_{1N}^0 \vdots \ddots \vdots \lambda_{N1}^0 \cdots \lambda_{NN}^0\end{pmatrix}\begin{pmatrix}f_1^0 \vdots f_N^0\end{pmatrix},
^{(n)}_{\texttt{b}}:=\left\{\bigl(X_1(\boldsymbol{\xi}),\ldots ,X_n(\boldsymbol{\xi})\bigr) \mid \boldsymbol{\xi}\in\mathbb{A}^{(n)}_{\texttt{b}}\right\}, \boldsymbol{X}^{(m,n)}_{\texttt{b};\lambda}:= \bigl(X_1( \boldsymbol{\xi}^{(m,n)}_{\texttt{b};\lambda}),\ldots ,X_n( \boldsymbol{\xi}^{(m,n)}_{\texttt{b};\lambda})\bigr) .
\phi([\sum^m_{k=1}(a_k +ib_k ) f_k, t]^{\sigma}) =\phi([\sum^m_{k=1}(a_k -ib_k ) f_k, -t])=[\sum^m_{k=1}a_k e_k -b_k e_k^{\ast},-t] &=[\sum^m_{k=1}a_k e_k +b_k e_k^{\ast},t]^{s(-1)}=\phi([\sum^m_{k=1}(a_k +ib_k ) f_k, t])^{\phi(\sigma)}.
A_{3}= \frac{1}{\sqrt{2}}f_{L}^{\bar{A}\bar{B}\bar{C}}f_{R}^{A B C}\Phi^{\bar{A}A}\Phi^{\bar{B}B}\Phi^{\bar{C}C}= \frac{1}{\sqrt{2}}\mathrm{T r \,}\{[ \Phi , \Phi ] \Phi \},
U_{\delta,\xi}(x)=\alpha_n \frac{\delta^{\frac{n-2}{2}}}{(\delta^2+|x-\xi|^2)^{\frac{n-2}{2}}}, V_{\delta,\xi}(x):=\frac{\delta^\frac{n-2}{2}}{(\delta+|x-\xi|)^{n-3}}.
f_1^{\prime}(0)= \frac{1}{2}\textbf{Tr}\left(\psi^{\prime}(v) \diamond (d_x+d_s)\right)= -\frac{1}{2}\textbf{Tr}\left(\psi^{\prime}(v) \diamond \psi^{\prime}(v)\right)= - 2 \delta(v)^2
\begin{array}{c}d\tau =\frac{d^4k_1}{(2\pi )^4}\frac{d^4k_2}{(2\pi )^4}(2\pi )^4\delta^4(p_1+p_2-k_1-k_2)\pi \delta (k_1^2-M^2)\theta (k_1^0) \times \pi \delta (k_2^2-M^2)\theta (k_2^0)\end{array}
S=\int_M E^{\mu \nu \alpha \beta}\left( c_1\,{^+G}_{\mu \nu}^{ab}\,{^+G}_{\alpha \beta}^{cd}+ c_2\,{^-G}_{\mu \nu}^{ab}\,{^-G}_{\alpha \beta}^{cd}+ c_3\,{^+{\cal R}}_{\mu \nu}^{ab}\,{^+G}_{\alpha \beta}^{cd}+ c_4\,{^-{\cal R}}_{\mu \nu}^{ab}\,{^-G}_{\alpha \beta}^{cd}\right)\epsilon_{abcd}
\kappa_4^2 \equiv \frac{\sqrt{2}(\Delta +4)}{8}\kappa^2 \sigma,\quad \tilde{\kappa}_4^2 \equiv \kappa_4^2(1-\varphi)^{\frac{8}{3(\Delta+4)}}=\kappa_4^2 (1-\varphi)^{\frac{2}{3 b^2+1}}.
E_{a,b}^{(M)}E_{c,d}^{(N)}=\begin{cases}(-1)^{\overline{E}_{a,b}\overline{E}_{c,d}}E_{c,d}E_{a,b}+(q_b-q_b^{-1})E_{a,d}E_{c,b}, M=N=1; \sum_{t=0}^{\min(M,N)}q_{b}^{\frac{t(t-1)}{2}}(q_b-q_b^{-1})^t[t]_q!E_{c,b}^{(t)}E_{c,d}^{(N-t)}E_{a,b}^{(M-t)}E_{a,d}^{(t)}, .\end{cases}
\bar{d}_0 = \check{d}_{\xi}\vee \check{\eta}\check{\eta}\geq \eta = \frac{\log\left(\psi_{\bar{A}}|\bar{C}|_{S_\infty}^2/s_{d_0}(Hg_{0,d_0})\right)}{2\log\left( 1/\rho(\bar{A}) \right)},
{\cal H}= \pi^{\mu}{\dot{A}}_{\mu}+ \pi^{<\mu \nu>}{\dot{B}}_{< \mu \nu >}+ \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{12}H_{\mu \nu \lambda}H^{\mu \nu \lambda}+ \frac{m}{6}\epsilon^{\mu \nu \lambda \rho}H_{\mu \nu \lambda}A_{\rho}
\sum_{1\leq j,k \leq n}|\langle \tau_j, \tau_k\rangle |^{2m}=\sum_{j=1}^n\sum_{k=1}^n|\langle \tau_j, \tau_k\rangle |^{2m}\geq \frac{n^2}{{d+m-1\choose m}}, \forall m \in \mathbb{N}.
|y_0-y_{k+1}|>\max(|y_0-y_i|,|y_i-y_{k+1}|)\geq \min(|y_0-y_i|,|y_i-y_{k+1}|)> \min\left(\left|y_0^2-y_i^2\right|/4,\left|y_i^2-y_{k+1}^2\right|/4\right)=\left|(1+|u_i|/\ln n)^2-1\right|S(I)^2/4 \geq\varepsilon_0 (\ln n)^{-1}.
u ( t , x ; a , \nu ) = \frac{\nu}{a}\, \ln < 1 > \equiv \frac{\nu}{a}\, \ln \left\{\frac{1}{\sqrt{4 \pi \nu t}}\int_{- \infty}^{\infty}\, d y \, \exp \left( - \frac{( x - y )^{2}}{4 \nu t}+ \frac{a f ( y )}{\nu}\right) \right\}
G(q,z)=\frac{(q^2z)^{p-1}}{1-q^2z\left(\frac{(q^3z)^{p-1}}{1-q^3z\left(\frac{(q^4z)^{p-1}}{1-\dots}+\frac{1-(q^4z)^{p-1}}{1-q^4z}\right)}+\frac{1-(q^3z)^{p-1}}{1-q^3z}\right)}+\frac{1-(q^2z)^{p-1}}{1-q^2z}.
\mathrm{corank}_{\Lambda(G)}(\displaystyle\bigoplus_{v \in S}J_v^{\pm}(E_{p^{\infty}}/L_\infty))=\mathrm{corank}_{\Lambda(G)}(\displaystyle\bigoplus_{v \in S_p}J_v^{\pm}(E_{p^{\infty}}/L_\infty)).
{\bf E}_v^{\otimes M}\{\bf E}_y^{\otimes M}\sum_{\gamma_1,...,\gamma_n = 1}^M \prod_{j=1}^n v_{\gamma_j}\exp(2 \pi i q_j \cdot y_{L,\gamma_j}) = \sum_{A \in \mathcal{A}_n}1_{\{|A| \leq M \}}\frac{M!}{(M-|A|)!}\prod_{a \in A}\left\{m_{|a|}\delta_* \left( \sum_{l \in a}q_l \right) \right\}
B^s_{p,\infty}(\mathbb{R}^d) = \left\{f\in \mathcal{S}'(\mathbb{R}^d) \,\,|\,\, f \stackrel{\mathcal{S}'}{=}\sum_{j = 0}^{\infty}a_j(x);\,\, \mathrm{supp}F a_j \subset M_j;\right. \left. \| \{a_j \}\|_{l^s_\infty(L_p)}= \sup_{j \in \mathbb{Z}_+}2^{sj}\| a_j \|_{L_p(\mathbb{R}^d)}< \infty \right\},\,\,\,\,\,\, q = \infty
\mathbb{P}_{n}^{(1)}=\frac{1}{n}{\displaystyle\sum\limits_{j=1}^{m_{1}}}\frac{P(A_{j}^{(1)})}{\mathbb{P}_{n}(A_{j}^{(1)})}{\displaystyle\sum\limits_{X_{i}\in A_{j}^{(1)}}}\delta_{X_{i}},
{\cal F}( s ) = \frac{\sqrt{\pi}}{\Gamma ( s )}\sum_{\mu_{k}}\int_{0}^{\infty}d t ~ t^{s - 3 / 2}\sum_{n \in Z \! \! \! Z}\exp \left[ - \frac{\beta^{2}}{4 \pi^{2}}( m^{2}+ \mu_{k}^{2}) t - \frac{\pi^{2}n^{2}}{t}- 2 \pi i \left( a + \frac{1}{2}\right) n \right] .
\ddot \phi_k(0)= D^2_{\phi\phi}\psi(0,\mu^*_{k})[\phi^*_k,\phi^*_k]+2D^2_{\phi\mu}\psi(0,\mu^*_{k})[\phi^*_k,\dot \mu_{k}(0)] + D^2_{\mu\mu}\psi(0,\mu^*_{k})[\dot \mu_k(0),\dot \mu_{k}(0)] &+D_\mu\psi(0,\mu^*_{k})\dot \mu_{k}(0).
(\alpha,\beta)\otimes (\gamma,\delta) = (1,0)\begin{bmatrix}\alpha \beta 0 0\end{bmatrix}\otimes(1,0)\begin{bmatrix}\gamma \delta 0 0 \end{bmatrix}& = (1,0)\otimes \left((1,0)\begin{bmatrix}\gamma \delta 0 0 \end{bmatrix}\begin{bmatrix}0 -\beta 0 \alpha \end{bmatrix}\right) = (1,0)\otimes (\alpha\delta-\beta\gamma,0).
\quad\int_D |\nabla u|^2 + \int_D \frac{1}{2}\tilde{V}|u|^2 \geq \int_D |\nabla v|^2 + \int_D \frac{1}{2}\tilde{V}\left(2 |v|^2 + 2 |u_D|^2\right) = \int_D |\nabla v|^2 + \int_D \tilde{V}|v|^2 + \left|\frac{1}{|D|}\int_D u\right|^2 \int_D \tilde{V}\ge \frac{1}{|D|}\int_D \tilde{V}\int_D |u|^2 .
\left<Z_{k},\mathcal{B}_{kk}^{k}\left(Z_{k}\right)\right>\left<Z_{k}, Z_{k}\right>=\displaystyle\sum_{l'=1}^{p-q}\varphi_{kl'}^{kk}\left(Z_{k},Z_{k}\right)\overline{\varphi_{kl'}^{kk}\left(Z_{k},Z_{k}\right)}.
C(n,p,r) = \frac{(2\omega_n)^{-\lambda/n}}{2^{1+\lambda/n}}\, \frac1{(pr)^{1+\lambda/n}}\bigg(\frac\lambda n\bigg)^{-\frac \lambda n}\bigg(\max\bigg\{\frac{r}{1-r}, \frac{p}{1-p}\bigg\}\bigg)^{- \frac \lambda n}.
(E_0=M\oplus K_X^{n-1}\oplus K_X^{n-2}\oplus \cdots \oplus K_X^{2-n}\oplus K_X^{1-n}\oplus M^{-1}, \quad\theta_0=\begin{pmatrix}0 \mu 0 &1 0 & \ddots \ddots & 1 0 & \mu 0\end{pmatrix}).
\textnormal{Tr}(A)= \sum_{[\xi]\in \widehat{G}}d_\xi\smallint\limits_G\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=\lim_{\lambda\rightarrow \infty}\sum_{[\xi]\in \widehat{G}:|\xi|\leq \lambda}d_\xi\smallint\limits_G\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx
\left\{\begin{array}{c}\mathfrak q' \updownarrow \mathfrak q''\end{array}:u(\mathfrak q')=\mathfrak p', u(\mathfrak q'')=\mathfrak p'', \mathtt{In}(\mathfrak q'')=J,\mathtt{In}(\mathfrak q')=\mathtt{Out}(\mathfrak q''), \mathtt{Out}(\mathfrak q')=I\right\}
\iint_Q \hat{y}(L^*u - \alpha_1 z^1{1}_{\mathcal{O}_{1,d}}- \alpha_2 z^2{1}_{\mathcal{O}_{2,d}})dxdt + \sum_{i=1}^2 \iint_Q \hat{p}^i(Lz^i + \dfrac{1}{\mu_i}u{1}_{\mathcal{O}_i})dxdt = (y_0,u(0))+ \iint_{\mathcal{O}\times (0,T)}\hat{y}u\dxdt+ \iint_Q (G u{G}_1 z^1 +{G}_{2}z^2)dxdt
\frac{1}{2}\int_{0}^{\infty}\frac{d x}{x^{1 / 2}}\Big ( \operatorname{c o s}\big ( \frac{\mu_{E}^{2}}{2}x \big ) - \operatorname{s i n}\big ( \frac{\mu_{E}^{2}}{2}x \big ) \Big ) = 0 .
S_1 := \sum_{n\leq x \atop P^+(n)\leq \mathcal{L}}n^{-a}&= \psi(x,\mathcal{L}) x^{-a}+ a \int_1^x \psi(w,\mathcal{L}) w^{-1-a}\dw &\ll 1+\int_1^x w^{-a}\exp \bigg( -\bigg(\frac{\log w}{\log \mathcal{L}}\bigg)\bigg\{\log\bigg(\frac{\log w}{\log \mathcal{L}}\bigg) &\qquad+\log_2\bigg(\frac{\log w}{\log\mathcal{L}}\bigg)-1 \bigg\}\bigg) \dw.
{\rm Var}\Bigg[\sum_{\ell=2}^n \sigma^2_{n\ell}\Bigg] \leq\frac{4{\rm Var}[u_{n1}^2]}{n({\rm E}[u_{n1}^2])^2}+\frac{16}{p_n-1}\leq \frac{4{\rm E}[u_{n1}^4]}{n({\rm E}[u_{n1}^2])^2}+\frac{16}{p_n-1}\to 0,
\mathbb A_L^{\tilde V}\bigl(Q_L\bigr):= \mathcal H\bigl(Q_L|Q_L^{\tilde V}\bigr)=\mathcal H\bigl(P_L|P_L^{\tilde V}\bigr) = \frac 12 \int_0^T \Phi_L\bigl(\mu^L_t,\jmath^L_t,F^{\tilde V_t}(\mu^L_t)\bigr)\;\!\mathrm dt.
S_{\mathcal{H}}(x) = \sum_{n=0}^\infty S^n_{\mathcal{H}}(x), S^n_{\mathcal{H}}(x) := \int_{0<u_1<\ldots<u_n<T}d(u_1,x_{u_1}) \otimes \ldots \otimes d(u_n,x_{u_n}),
\nabla_y\left(\frac{i}{4}H^{(1)}_0(k_{+}\vert x-y\vert )\right) = \frac{e^{ik_{+}\vert x \vert}}{\sqrt{\vert x \vert}}e^{-i\frac{\pi}4}{\sqrt{\frac{k_+}{8\pi}}}\left(\hat x e^{-ik_{+}\hat x\cdot y}+O\left({\vert x \vert^{-1}}\right)\right)
\int_{|y| \le t^{\beta/\alpha}}\left|\nabla I_{B}\right| dy \le C \int_{|y| \le t^{\beta/\alpha}}|y|^{\alpha - 1 - 1 - d + d}t^{-\beta}dy = Ct^{-\beta}(t^{\beta/\alpha})^{\alpha - 1}= C t^{-\beta/\alpha}.
CKN\left( N,p,\mu,s\right) =\sup_{u\in D_{\mu}^{1,p}\left(\mathbb{R}^{N};\frac{dx}{\left\vert x\right\vert^{s}}\right)}\frac{\left(\int\limits_{\mathbb{R}^{N}}\left\vert u\right\vert^{r}\frac{dx}{\left\vert x\right\vert^{s}}\right)^{1/r}}{\left( \int\limits_{\mathbb{R}^{N}}\left\vert \nabla u\right\vert^{p}\frac{dx}{\left\vert x\right\vert^{\mu}}\right)^{1/p}}.
\chi_{1}(\omega)=\frac{1}{\pi}P\int_{-\infty}^{\infty}\frac{\omega^{\prime}\chi_{2}(\omega^{\prime})}{\omega^{\prime2}-\omega^{2}}d\omega^{\prime}+\frac{\omega}{\pi}P\int_{-\infty}^{\infty}\frac{\chi_{2}(\omega^{\prime})}{\omega^{\prime2}-\omega^{2}}d\omega^{\prime}.
\int_r^\infty y^{d+1}e^{-\frac{y^2}{4s}}\,dy =2sr^{d}e^{-\frac{r^2}{4s}}+2ds\int_r^\infty y^{d-1}e^{-\frac{y^2}{4s}}\,dy \geq 2sr^{d}e^{-\frac{r^2}{4s}}\,.
\lim_{n\to+\infty}\frac{C_{\beta,n-2,1}}{C_{\beta,n}n^{\beta}}=(2\pi)^{-1}\frac{(\beta /2)^{\beta}(\Gamma(\beta/2+1))^{3}}{\Gamma(3\beta/2 +1)\Gamma(\beta+1)}:=A_{\beta}.
xP_{n}=P_{n+1}+\left( l_{nn}+m_{n+1}\right) P_{n}+\sum\nolimits_{i=0}^{d-2}\left( l_{n,n-1-i}+l_{n,n-i}m_{n-i}\right)P_{n-i-1}&+l_{n,n-d+1}m_{n-d+1}P_{n-d}
\|f\|_{L^{p l_{j+1}}}^{2p}\leq C \|f\|_{W^{2,l_{j}}}^{2}\|f\|_{L^{6}}^{2p-2}\Leftrightarrow \frac{1}{l_{j+1}p}=\left ( \frac{1}{l_{j}}- \frac{2}{3}\right )\frac{1}{p}+ \frac{1}{6}- \frac{1}{6p}.
T_t= \left\langle e_I \: | \: d(I,I_0)\leq k;\:\!\!\!\!\sum_{J\in \Delta(I)^+_j}\!\!\!\!t^{d(I,J)}c_{(I,J)}e_J\: | \: d(I,I_0)\leq k,\j=1,\dots,n_1\right \rangle,
z_{1}\mathcal{S}_{1}(z_{1},z_{2})(k) = \mathcal{S}_{1,1}+\mathcal{S}_{1,2}+\mathcal{S}_{1,3}+\mathcal{S}_{1,4}+\mathcal{S}_{1,5}+\mathcal{S}_{1,6}
J_{am}{J^{m}}_{b}=\frac{\theta^{2}}{9}h_{ab}+\frac{2\theta}{3}(\sigma_{ab}+\omega_{ab})+(\sigma_{am}{\sigma^{m}}_{b}+\omega_{am}{\omega^{m}}_{b})+(\sigma_{am}{\omega^{m}}_{b}+\omega_{am}{\sigma^{m}}_{b})
\begin{array}{rl}{\displaystyle\left\{p_{\mu},x^{\nu}\right\}=-i\delta_{\mu}^{\nu},}{\displaystyle\left\{p_g,g\right\}=-i,}begin{align*}0.4cm]{\displaystyle\left\{\pi_{\alpha}^i,\theta_j^{\beta}\right\}=-i\delta_j^i\delta_{\alpha}^{\beta},}{\displaystyle\left\{\bar\pi_{\dot\alpha i},\bar\theta^{\dot\beta j}\right\}=-i\delta_i^j\delta^{\dot\beta}_{\dot\alpha}.}\end{array}
\mathfrak{k}_{0}= \{X \in \mathfrak{k}_{1}\cap \mathfrak{k}_{2}\mid [\mathfrak{a}, X] =\{0\}\}, V(\mathfrak{k}_{1}\cap \mathfrak{m}_{2}) = \{X \in \mathfrak{k}_{1}\cap \mathfrak{m}_{2}\mid [\mathfrak{a}, X] =\{0\}\}, V(\mathfrak{m}_{1}\cap \mathfrak{k}_{2}) = \{X \in \mathfrak{m}_{1}\cap \mathfrak{k}_{2}\mid [\mathfrak{a}, X] =\{0\}\}.
T^{L}= \left( \begin{matrix}{1}&{0}\\{0}&{0}\\ \end{matrix}\right) , T^{R}= \left( \begin{matrix}{0}&{0}\\{0}&{1}\\ \end{matrix}\right) , Q = \left( \begin{matrix}{0}&{1}\\{0}&{0}\\ \end{matrix}\right) , Q^{\dagger}= \left( \begin{matrix}{0}&{0}\\{1}&{0}\\ \end{matrix}\right) .
\partial_a \ln V_M \Gamma_{\hat{a}}\eta-{1\over V_M}\partial_a\tilde{E}_M\Gamma_{\hat{a}}\Gamma_M^\perp \eta -{1\over 2}\sum_N (\bar{q}_{NM}-1){1\over V_N}\partial_a E_N\Gamma_{\hat{a}}\Gamma_N \eta \qquad\qquad
c^* = \mathbb{E}\biggl[ \sup_{s \in [0,1]}(c^* \sqrt{s}+ W_{s}) \biggr] - C \sqrt{h}= \mathbb{E}\biggl[ \sup_{s \in [0,1]}(c^* \sqrt{s}+ W_{s}) - (c^*+W_{1}) \biggr] + c^* - C \sqrt{h}.
\left[ \begin{array}{c}\dot{z}_{1}\dot{z}_{2}\end{array}\right] =\left[ \begin{array}{cc}0 1 -1 0\end{array}\right] \left[ \begin{array}{cc}\left( \alpha +\beta p\left( t\right) \right) 0 0 1\end{array}\right] \left[ \begin{array}{c}z_{1}z_{2}\end{array}\right]
\int_0^t T^{x'}f(dr, \cdot) - \int_0^t T^{x}f(dr, \cdot) = \int_0^t T^{x + \theta' - \theta}f(dr, \cdot) - \int_0^t T^{x}f(dr, 0) = \int_0^t T^{x}f(dr, \theta'_r-\theta_r ) - \int_0^t T^{x}f(dr, 0)
X^{2}(1- 4X)(1-12X+16X^{2})\frac{d^{3}Z}{dX^{3}}+3X(1-24X+128X^{2}-160X^{3})\frac{d^{2}Z}{dX^{2}}&+(1-56X+464X^{2}-784X^{3})\frac{dZ}{dX}-4(1-20X+54X^{2})Z=0.
E_{\rm abs}= \max_{t \in [0,T]}\left| \bar{y}(t) - y(t) \right| \mbox{and}E_{\rm rel}= \frac{1}{T}\int_0^T \frac{\left| \bar{y}(t) - y(t) \right|}{\left| y(t) \right|}\;{\rm d}t
(2i)^{N_+}\kappa (a)=\prod\limits_{\alpha >0}^{}\left( e^{i\pi (a,\alpha )/a_0}-e^{-i\pi (a,\alpha)/a_0}\right)=\sum\limits_{\hat{s}\in W}^{}\det \hat{s}\exp \left(\frac{2\pi i}{a_0}(\hat{s}\rho ,a)\right)
\sum\limits_{k = 1}^m{{a^{\frac{{2k - 1}}{{2m}}}}{b^{\frac{{2m - \left({2k - 1}\right)}}{{2m}}}}}\ge \sum\limits_{k = 1}^m{{a^{\frac{k}{{m + 1}}}}{b^{\frac{{m + 1 - k}}{{m + 1}}}}}, \,\,\,\,\left({a,b > 0,m \in \mathbb{N}}\right).
|\angle{i_0(w)i_0(x)i_0(y)}-\angle{i(w)i(x)m}| \leq |\angle{i_0(w)i_0(x)i_0(y)}-\angle{w'i_0(\bar{x})i_0(\bar{y})}|+|\angle{i(w)i(x)m}-\angle{w''i(\bar{x})m}|+|\angle{w'i_0(\bar{x})i_0(\bar{y})}-\angle{w''i(\bar{x})m}| \leq \frac{\delta}{100}+\frac{\delta}{100}+\frac{\delta}{100}<\frac{\delta}{10}.
\begin{array}{l}\int_{(-R,R)\times (0,T) \times C}|\partial_{z}\phi_{\epsilon}|dzdtdx =\int_{(-R,R)\times (0,T) \times C}\partial_{z}\phi_{\epsilon}dzdtdx = \int_{(0,T)\times C}\phi_{\epsilon}(R,t,x)dtdx -\int_{(0,T)\times C}\phi_{\epsilon}(-R,t,x)dtdx \rightarrow \int_{(0,T)\times C}p(t,x) dtdx \ \hbox{as}\R\rightarrow +\infty \end{array}
V(\alpha x+(1-\alpha) y) =\liminf_{t\to 1}\Phi^{-1}(Q_{1-t}f(\alpha x+(1-\alpha)y)) &\ge\liminf_{t\to 1}\{\alpha \Phi^{-1}(Q_{1-t}f(x)) +(1-\alpha)\Phi^{-1}(Q_{1-t}f(y))\}&\ge \alpha V(x) + (1-\alpha)V(y)
f_1 = Y^{p^s}+\tfrac{c_1}{c_0}\Delta^{p^{s-1}}X^{p^s-2p^{s-1}}+\tfrac{c_2}{c_0}Y^{p^{s-1}}X^{p^s-p^{s-1}}+\tfrac{c_3}{c_0}\Delta^{p^{s-2}}X^{p^{s}-2p^{s-2}}+ \ldots +\tfrac{c_r}{c_0}YX^{p^{s}-1}
d_{m}(a_i^*\otimes u^*) =(d_{m}(a_i^*\otimes 1))(1\otimes u^*)+ (-1)^{|a_i|}(a_i^*\otimes 1)d_{m}(1\otimes u^*) &=(-1)^{\vert a_{i}\vert}(a_i^*\otimes 1)(a_1^*\otimes 1) = (-1)^{\vert a_{i}\vert}c_{i1}(z^*\otimes 1).
S_{f i}= e^{- i [ ( u + w ) / 2 ]^{2}T / 2}\! \int \! d^{9}b e^{- i q_{\mu}b_{\mu}}\langle{\cal H}^{3}| \langle{\cal H}^{4}| e^{i \Gamma_{\mathrm{( 1 )}}( u_{\mu}+ w_{\mu}/ 2 , b_{\mu}, \theta^{3})}|{\cal H}^{1}\rangle |{\cal H}^{2}\rangle
{}^{c_i}({}^{c_1^{\delta_1}\ldots c_{i-1}^{\delta_{i-1}}c_{i+1}^{\delta_{i+1}}\ldots c_r^{\delta_r}}H_{[i]})={}^{c_1^{\delta_1}\ldots c_{i-1}^{\delta_{i-1}}c_ic_{i+1}^{\delta_{i+1}}\ldots c_r^{\delta_r}}H_{[i]}={}^{c_1^{\delta_1}\ldots c_{i-1}^{\delta_{i-1}}c_{i+1}^{\delta_{i+1}}\ldots c_r^{\delta_r}c_i}H_{[i]}.
\frac{E\Vert S_n(\theta)\Vert^2}{n}=\frac{2\pi}{n}\,\sum_{k=1}^n E\Vert Z^{(0)}_{n-k}(\theta)\Vert^2 + \frac{1}{n}\, E\big\|E[S_n(\theta)\vert \mathcal{G}_0]\big\|^2 &= 2\pi \, E\Vert Z^{(0)}(\theta)\Vert^2 \; + \; o(1),
1+\|x+y\|^p \le 1+(\|x\|+\|y\|)^p =1+\sum_{k=0}^p\binom{p}{k}\|x\|^{p-k}\|y\|^k =(1+\|x\|^p)\left(1+\sum_{k=1}^p\binom{p}{k}\tfrac{\|x\|^{p-k}}{1+\|x\|^p}\|y\|^k \right) \leq (1+\|x\|^p)\left(1+\sum_{k=1}^p\binom{p}{k}\|y\|^k \right) =(1+\|y\|)^p(1+\|x\|^p).