harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{\Lambda}|f (x - j) | ~dx \leq C \int_{\Lambda}\frac{1}{|x - j|^{\alpha}}~dx \leq C L^d \frac{1}{\underset{x \in \Lambda}{\inf}~ | x - j|^{\alpha}}\leq C^{'}\frac{L^{d}}{|j|^{\alpha}}=: \sigma_j\,.
\lim_{J_2\ni m\to\infty}\int_{\varphi_n(\mathsf{U}_n)}\left|w_{ij}^{(m,n)}\right|^p\,d\mu(x)=\int_{\varphi_n(\mathsf{U}_n)}\left|u_{ij}^{(n)}(x)\right|^p \,d\mu(x) \ .
\begin{array}{r l l}{{F ( f_{0}^{- 1 / 2}f_{1}, f_{0}^{- 1 / 2}f_{2})}}&{{=}}&{{F ( f_{0}^{- 1 / 2}f_{2}, f_{0}^{- 1 / 2}f_{1}) \; ,}}\\{{\mathrm{~ a n d ~}F ( f_{0}^{- 1 / 2}f_{1}, f_{0}^{- 1 / 2}f_{2})}}&{{=}}&{{0 \mathrm{~ a t ~}\mathrm{P}_{\mathrm{J B}}\; .}}\\ \end{array}
\rightarrow[\begin{matrix}cos(\theta)&sin(\theta)&0\\ -sin(\theta)&cos(\theta)&0\\ 0&0&1\end{matrix}]\cdot\sqrt{\frac{2}{3}}[\begin{matrix}1&\frac{-1}{2}&\frac{-1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{matrix}]
V^{(n+1)}(t) = f(0) V^{(n)}(t) + \int_0^t f'(t-s) V^{(n)}(s) \, ds - g(0) V^{(n)}(t) - \int_t^T g'(s-t) V^{(n)}(s) \, ds begin{align*}6pt] + \sum_{k=0}^{n-1}(-1)^{n-k+1}g^{(n-k)}(T-t) V^{(k)}(T)
E_3^q=I_3-\sum_{j_3,j_2,j_1=0}^q C_{j_3j_2j_1}^2-\sum_{j_3,j_2,j_1=0}^q C_{j_2j_3j_1}C_{j_3j_2j_1}\ \ \ (i_1\ne i_2=i_3),
\begin{array}{r c l}{{\psi_{2 , 1}}}&{{\cong}}&{{\rho_{1 , 1}= L_{- 1}\psi_{1 , 1}}}\\{{\phi_{2 , 1}}}&{{\cong}}&{{\phi_{1 , 1}^{\prime}= L_{- 1}\phi_{1 , 1}}}\\{{\xi_{2 , 1}}}&{{\cong}}&{{\phi_{1 , 1}= \frac{1}{2}L_{1}\rho_{1 , 1}\, .}}\\ \end{array}
u_i < \{v \in \bar{v}: u_i < v < u_{i+1}\}< \{v \in \bar{v}_2 : u_i < v < u_{i+1}\}\cup \{w \in \bar{w}_2 : u_i < w < u_{i+1}\}< ~ \{w \in \bar{w}: u_i < w < u_{i+1}\}< u_{i+1}
\sum_{i=1}^{K}i(Q^N_i(t)-Q^N_{i+1}(t)) =\sum_{i=1}^{K}Q^N_i(t)-KQ^N_{K+1}(t) &=KN - \sum_{i=1}^{K}(N-Q^N_i(t))-KQ^N_{K+1}(t) &= KN-Z_1^N(t)-KZ_2^N(t),
-\begin{pmatrix}1.0000 -1.0000\mathbf{i}-1.0000 1.0000\mathbf{i}+1.0000 1.0000\mathbf{i}1.0000\end{pmatrix}W_{11}\begin{pmatrix}2.0000 -1.0000\mathbf{k}1.0000\mathbf{i}\end{pmatrix},
\Vert \varphi_{\varepsilon}\Vert_{L^{2}(\mathbb{P})}\leq1+\Vert \chi_{\varepsilon}\circ \tau_{0,1}\Vert_{L^{2}(\mathbb{P})}+\Vert \chi_{\varepsilon}\Vert_{L^{2}(\mathbb{P})}= 1+2\Vert \chi_{\varepsilon}\Vert_{L^{2}(\mathbb{P})}.
\begin{array}{l}\,\,\,\,\,\,(-1)^{\varepsilon(f_1)\varepsilon(f_3)}d_Sc_2(\xi|f_1,f_2,f_3)= =2c_{1n}\sigma_1\biggl((-1)^{\varepsilon(f_1)\varepsilon(f_3)}f_1(\xi)[\overleftarrow\partial\!\!_\alpha]^n\varepsilon^{[\alpha]_n}f_2(\xi)(\overleftarrow\partial\!\!_\beta\lambda_\beta\partial_\beta)^3f_3(\xi)+\hbox{cycle}(1,2,3)\biggr).\end{array}
\alpha B+\beta B^{1/2}WB^{1/2}+B^{1/2}W^2B^{1/2}&\qquad\quad+\int_{(0,\infty)}\biggl({1\over1+s}\,B^{1/2}WB^{1/2}-B^{1/2}h_s(W)B^{1/2}\biggr)\,d\mu(s) &\quad=B^{1/2}\biggl[\alpha I+\beta W+\gamma W^2+\int_{(0,\infty)}\phi_s(W)\,d\mu(s)\biggr]B^{1/2}=B^{1/2}f(W)B^{1/2}.
A_{1,\lambda_{1}}=\left\{\mathcal{Y}_1(z)\psi_{1}(zq^{3/2})^{l}: l\in\mathbb{Z}_{\geq 0}\right\}; &A_{1,\lambda_{1}-\alpha_1}=\left\{x_{1}^{-}(z)_\bullet \mathcal{Y}_1(z)\psi_{1}(zq^{3/2})^{l}: l\in\mathbb{Z}_{\geq 0}\right\}.
s_{lj}=\frac 1{\gamma_{l-1}}\int_{-1}^1 h_j'(x)P_{l-1}(x)\,dx= \frac 1{\gamma_{l-1}}\bigg\{h_j(x)P_{l-1}(x)\big|_{-1}^1-\sum_{i=0}^N h_j(x_i)P_{l-1}'(x_i) \omega_i\bigg\}&=\frac 1{\gamma_{l-1}}\big\{\delta_{jN}+(-1)^l \delta_{j0}- \omega_j P_{l-1}'(x_j)\big\},
\frac{\overline{PA}\cdot\overline{QA}}{\overline{CA}\cdot\overline{AB}}+\frac{\overline{PB}\cdot\overline{QB}}{\overline{AB}\cdot\overline{BC}}+\frac{\overline{PC}\cdot\overline{QC}}{\overline{BC}\cdot\overline{CA}}=1.
\frac{2}{|\rho|}\bigg| \sum_{j=1}^{r}t_j \overline\chi(a_j) \bigg| \le 4\bigg| \sum_{j=1}^{r}t_j \overline\chi(a_j) \bigg| \le 4r\max_{1\le j\le r}|t_j| < \frac{12}5
\underline{\mathbf{n}}\left(e^{-q\tau_b^+};\tau_b^+<\zeta\right)=\lim_{x \downarrow 0}\frac{\mathbb{E}_x\left(e^{-q\tau_b^+};\tau_b^+<\tau_0^-\right)}{W(x)}=\lim_{x \downarrow 0}\frac{W_q(x)}{W(x)W_q(b)}=\frac{1}{W_q(b)}.
\sigma_{W_{\rm s}}^2 =\mathbb{E}\{|W_{\rm s}[k]|^2\}begin{align*}2pt] =\frac{1}{K}\sum_{n=0}^{K-1}\sum_{m=0}^{K-1}\mathbb{E}\{w_s[n]w_s^*[m]\}e^{-\frac{2\pi nkj}{K}+\frac{2\pi mkj}{K}}. begin{align*}2pt] =\frac{1}{K}\sum_{n=0}^{K-1}\mathbb{E}\{|w_s[n]|^2\}-\mathbb{E}^2\{w_s[n]\}.
\Psi_{s}( P ) = \frac{2^{- 1 / 4}\Gamma ( 1 + 2 i b P ) \Gamma \left( 1 + \frac{2 i P}{b}\right) \cos ( 2 \pi s P )}{- 2 i \pi P}( \pi \mu \gamma ( b^{2}) )^{- i P / b}~ .
\alpha_{1,2}\cdots \alpha_{1,k}=\beta (r+b_1) \alpha_{2,1}\cdots \alpha_{2,k}=\beta (r+b_2) \alpha_{3,1}\cdots \alpha_{3,k}=\beta (r+b_3) \alpha_{4,1}\cdots \alpha_{4,k}=\beta (r+b_4).
|k_bj_b+\sum_{j\in\mathbb{Z}_*}\frac{l_jj}{n-\frac12}|\geq \frac{1}{100n\sum_{b=1}^n|j_b|}\max\{|k|,\sum_{j\in\mathbb{Z}_*}|jl_j|\},b=1,\cdots,n.
E_{q \bar{q}}= \lim_{{\tilde{\Lambda}}\rightarrow \infty}\frac{1}{2 \pi}\int_{U_{0}}^{{\tilde{\Lambda}}}d x \sqrt{{U^{\prime}}^{2}+ \frac{U^{4}- U_{T}^{4}}{R^{4}}}= \frac{U_{T}}{\pi}\lim_{\Lambda \rightarrow \infty}\int_{a}^{\Lambda}d y \sqrt{\frac{y^{4}- 1}{y^{4}- a^{4}}}
\hat{y}^\mu_k|x^\mu,...,y^1_1,...,y^\mu_{l}+\,d s \,\delta^\mu_\rho \delta_{kl},...,y^4_{4N}\rangle =\,(y^\mu_{\rho}+\,d s \,\delta^\mu_\rho \delta_{kl})\cdot & \cdot |x^\mu,...,y^1_1,...,y^1_{l}+\,d s \,\delta^\mu_\rho \delta_{kl},...,y^4_{4N}\rangle .
\mathrm{L i}_{2}( e^{2 \pi i p / q}) = \frac{1}{q^{2}}\sum_{k = 1}^{q - 1}e^{2 \pi i k p / q}~ \psi^{( 1 )}\left( \frac{k}{q}\right) + \frac{\pi^{2}}{6 q^{2}},
\tilde{M}_{n}(t)=\sup_{s\in [0, t]}M_{n}(s)+ (1+t)^{n^3}\lesssim \big( \tilde{M}_{n}( t) \big)^{1- \epsilon}+ (1+t)^{n^3}, \Longrightarrow \tilde{M}_{n}(t)\lesssim (1+t)^{n^3}.
\begin{array}{rcl}i\Sigma_{ij}(p) = \displaystyle i\lambda^2\delta_{ij}\int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}[U(k_i,p)-T(k_i)], i\Sigma_{00}(p) = \displaystyle i\lambda^2\int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}[V(k_i,p)-T(k_i)], \end{array}
\widehat{S}_{( \pm 1 )}p^{( 0 )}= \widehat{S}_{( \pm 1 )}p^{( \widetilde{0})}= 0 , ~ ~ S_{( 1 )}^{2}p^{( 0 )}= S_{( 1 )}^{2}p^{( \widetilde{0})}= 0 , ~ ~ S_{( 0 )}^{2}p^{( 1 )}= S_{( 0 )}^{2}p^{( \widetilde{1})}= 0 ,
\frac{1}{2}\sum_{j}\log(1-2t\sigma^{2}_{j})+t\sum_{j}\sigma_{j}^{2}+nt\alpha \geq nt\alpha-2t^{2}\sum_{j}\sigma_{j}^{4}=n\alpha^{2}(\kappa-2\kappa^{2}\sum_{j}\sigma_{j}^{4}/n).
\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3n(n+1)+2n+1}}{1-q^{6n+3}}=\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3n(n+1)+4n+2}}{1-q^{6n+3}}\intertext{and}\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3n(n+1)}}{1-q^{2n+1}}=\sum_{n=-\infty}^\infty \frac{(-1)^nq^{3n(n+1)}(1+q^{2n+1}+q^{4n+2})}{1-q^{6n+3}}.
\frac{1}{n!}\int_{-R}^{R}\int_{-R}^R \widehat{\gamma}_n(t,s,x-y)dxdy=4\pi C_H^n R \int_{T_n(s)}\big(\ell_R*\widehat{g}_{\pmb{t}_n}^{(n)}\big)(0) d\pmb{t}_n,
\phi_1 = -h_b\frac{\sinh(\frac{z}{h_b}\sqrt{\mu}|\mathrm{D}|)}{\cosh(\sqrt{\mu}|\mathrm{D}|)}\frac{1}{\sqrt{\mu}|\mathrm{D}|}\nabla_X \cdot \big(\mathcal{L}_{1}^{\mu}[\beta b]\nabla_X\psi\big{)},
M_{I{\bar{J}}}^{2}= - \frac{4 g^{2}}{1 6 \pi^{2}}R_{I}R_{J}\sum_{p_{5}, \, p_{6}}\int_{0}^{\infty}d l \, \, \, \, \left( G^{I J}- \frac{2 \, G^{I K}G^{J L}{p}_{K}{p}_{L}}{l}\right) \, \, e^{- ( p_{M}G^{M N}p_{N}) / 2 l}
\frac{\partial\eta_k(\xi)}{\partial \xi}= \frac{\sigma_k}{\xi^2}+\frac{2\log(C_k)}{\xi^3}+\frac{k}{\xi^3}+\frac{k}{\xi^2\sigma_k}&-\left(\frac{1}{2\sigma_k}+\frac{2k\log(\sigma_k+\xi)}{\xi^3}+\frac{\log\left(\frac{1}{2}\left(1+\frac{\xi}{\sigma_k}\right)\right)}{\xi^3}+\frac{\xi^2-\sigma_k^2}{2\xi^2\sigma_k^2(\sigma_k+\xi)}\right).
\frac{\langle \Gamma \beta, \beta \rangle_{L_2}}{\langle \Gamma_e \beta, \beta \rangle_{L_2}}=\frac{\langle \Gamma_e \beta, \beta \rangle_{L_2}+\langle \Gamma_w \beta, \beta \rangle_{L_2}}{\langle \Gamma_e \beta, \beta \rangle_{L_2}}=1+\frac{\langle \Gamma_w \beta, \beta \rangle_{L_2}}{\langle \Gamma_e \beta, \beta \rangle_{L_2}}.
\xi_{CMH}=\frac{[\sum_{i=1}^{K}(A_{i}-\frac{N_{1i}M_{1i}}{T_{i}})]^{2}}{\sum_{i=1}^{K}\frac{N_{1i}N_{2i}M_{1i}M_{2i}}{T_{i}^{2}(T_{i}-1)}}.
\mathcal{E}_L(f+g)+\mathcal{E}_L(f-g) \leq L_{f+g}(\varphi_1)+L_{f-g}(\varphi_2)+\varepsilon &\leq L_{f+g}(\varphi_1\vee\varphi_2)+L_{f-g}(\varphi_1\vee \varphi_2)+\varepsilon &=2\left(L_f(\varphi_1\vee\varphi_2)+L_g(\varphi_1\vee\varphi_2)\right)+\varepsilon &\leq 2\left(\mathcal{E}_L(f)+\mathcal{E}_L(g)\right)+\varepsilon,
C^{-1}=\frac{1}{4}\begin{pmatrix}3 2 1 2 4 2 1 2 3\end{pmatrix}C(z) =\begin{pmatrix}[2]_z -1 0 -1 [2]_z -1 0 -1 [2]_z\end{pmatrix}\widetilde{C}(z) =\frac{1}{[4]_z}\begin{pmatrix}[3]_z [2]_z 1 [2]_z ([2]_z)^2 [2]_z 1 [2]_z [3]_z\end{pmatrix}.
\frac{df}{dE}=\frac{1}{m_{w}^{*}}\frac{dk}{dE}cot(\frac{kl_{w}}{2})-\frac{k}{m_{w}^{*}}csc^{2}(\frac{kl_{w}}{2})\times\frac{l_{w}}{2}\frac{dk}{dE}+\frac{1}{m_{b}^{*}}\frac{d\kappa}{dE}(9-2)
S_{j i}^{({\widehat G}/ H )}( \theta , q ) ~ = ~ \left( x^{- \rho_{H}\cdot h}\otimes x^{- \rho_{H}\cdot h}\right) ~ S_{j i}^{({\widehat G})}~ \left( x^{\rho_{H}\cdot h}\otimes x^{\rho_{H}\cdot h}\right) \ ,
J ( Z_{u}) \bullet_{\hbar}J ( Z_{v}) = Z_{u}\bullet_{\hbar}^{z}Z_{v}+ \sum_{i}y^{i}( Z_{\partial_{i}u}\bullet_{\hbar}^{z}Z_{v}+ Z_{u}\bullet_{\hbar}^{z}Z_{\partial_{i}v}) + \cdots \; , \quad \forall u , v \in N_{1}.
<A_2|n>=\left(\frac{k\tau_2}{2\pi^2|\tau|^2}\right)^{1/4}\,\frac{(-i)^n}{\sqrt{n!}}\,\left(\frac{i\bar{\tau}}{2\tau_2}\right)^{n/2}\,e^{\frac{k}{4i\pi\tau}A_2^2}\,P_n\left(\sqrt{\frac{k}{2i\pi\bar{\tau}}}\,A_2;\frac{i\bar{\tau}}{-i\tau}\right)\ \ \ .
B_{g_1^{M_1},g_2^{M_2}}(X_1,Y_1,X_2,Y_2) =B_{g_1,g_2}(X_1,Y_1,X_2,Y_2)^{(M_1,M_2)}&=B_{g_1,g_2}(X_1',Y_1',X_2',Y_2')
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) \chi_{\geq 1}(\frac{2r}{\log^{N}(t)})\left(v_{1}+v_{2}+v_{3}\right)\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{3}\log^{2b+2N+1}(t)}+ \frac{C}{t^{2}\log^{1+b\alpha +b+2N}(t)}
\mathrm{D}_\mathcal{F}^*(x,r)=\limsup_{n\to\infty}\frac{1}{|F_n|}\int_{F_n}1_{B(x,r)}(gx)dg,x\in X\textrm{and}r>0,\intertext{and}Q_{\textit{wap}}(G\curvearrowright X)=\left\{x\in X\,|\,\mathrm{D}_\mathcal{F}^*(x,\epsilon)>0\ \forall \epsilon>0\right\}.
\psi_{m}^{\alpha}( x^{i}, t ) = e_{m}^{( \mu )}\sigma_{( \mu )}^{\alpha \dot{\beta}}\bar{\lambda}_{\dot{\beta}}( x^{i}) \bar{\lambda}( t ) , \qquad \bar{\psi}_{m}^{\dot{\alpha}}( x^{i}, t ) = e_{m}^{( \mu )}\sigma_{( \mu )}^{\dot{\alpha}\beta}\lambda_{\beta}( x^{i}) \lambda ( t ) ,
\begin{array}{l}{3 ( 3 , 2 )_{\frac{1}{6}}+ 3 ( \overline{{3}}, 1 )_{\frac{1}{3}}+ 3 ( \overline{{3}}, 1 )_{- \frac{2}{3}}+ 1 2 ( 1 , 2 )_{- \frac{1}{2}}+ 9 ( 1 , 2 )_{\frac{1}{2}}}\\{\ + 1 2 ( 1 , 1 )_{1}+ 9 ( 1 , 1 )_{0}+ 9 ( 1 , 1 )_{- 1}.}\\ \end{array}
f_{\psi_n, \psi_d}(x, y) = \frac{e^{-\frac{1}{2(1-\rho)}\left(\frac{x}{s_n^2}+\frac{y-1}{s_d^2}\right)}}{4 s_n^2 s_d^2 (1-\rho)}I_0\left(\frac{\sqrt{\rho}}{1-\rho}\frac{\sqrt{x(y-1)}}{s_n s_d}\right).
x_3 x_1-x_1x_2 =0, x_4x_2+x_2x_4=0, x_3 x_2-x_2x_3 =0, x_3x_4+x_4x_3=0, x_4 x_1+x_1x_4 =0, x_1x_2+x_2x_1=t.
\mathcal{N}^-_x =\bigcup_{A \in \mathcal{A}_k: A \ni x}\mathcal{S}^-_{A,\infty}, \mathcal{N}^0_x =\bigcup_{A \in\mathcal{A}_k: A \ni x}\mathcal{S}^0_{A,\infty}, \mathcal{N}^+_x =\bigcup_{A \in \mathcal{A}_k: A \ni x}\mathcal{S}^+_{A,\infty}.
\frac{1}{n-1}\sum_{r=0}^{n-\ell}\binom{4(h-1)(n-1)+r-1}{r}\binom{3n-3-r}{\ell}\binom{2(k+1-2h)(n-1)}{n-r-\ell}&-\frac{1}{2n-1}\sum_{r=0}^{n-\ell}\binom{2(h-1)(2n-1)+r-1}{r}\binom{3n-2-r}{\ell}\binom{(k+1-2h)(2n-1)}{n-r-\ell}
g \left( q^{i}, p_{i}, z^{a}, \overline{{p}}_{a}\right) = \sum_{k = 1}^{\infty}\frac 1{k ! \lambda^{k}}\left[ C_{a_{1}}^{0}, \left[ C_{a_{2}}^{0}, \ldots , \left[ C_{a_{k}}^{0}, \tilde{H}\right] \ldots \right] \right] z^{a_{1}}\ldots z^{a_{k}}.
\langle \mathcal L_\varphi u, w \rangle_{\Omega}= \langle u ,\mathcal L^{*}_{\varphi}w\rangle_{\mathbb{R}^n}= \widetilde L(\mathcal L^{*}_{\varphi}w) = L(\mathcal L^{*}_{\varphi}w) = \langle v, w\rangle_{\Omega}.
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1-z^n}\right)^{\frac{m^1}{n^2}}= \sqrt{\frac{1}{1-z}}\; \exp\left\{\frac{1}{2}\frac{z}{1-z}\right\},
P_{out}(\gamma_{th}) = 1- \int_{0}^{\infty}(\mathcal{J}_{R,I}+\mathcal{J}_{R,II}) &\times(\mathcal{J}_{D,I}+\mathcal{J}_{D,II})\frac{z_{2}^{N-1}e^{-\frac{z_{2}}{P_{PU_{tx}}\nu_{2}}}}{\Gamma(N)(P_{PU_{tx}}\nu_{2})^N}dz_{2},
{\cal P}( z | \omega_{1}, \omega_{2}) = \frac{1}{z^{2}}+ \sum_{m , n}{}^{'}\left[ \frac{1}{( z - 2 m \omega_{1}- 2 n \omega_{2})^{2}}- \frac{1}{( 2 m \omega_{1}+ 2 n \omega_{2})^{2}}\right] .
w(D)=-2\pi C\rho_{1}\int_{z=D}^{z=\infty}dz\int_{x=0}^{x=\infty}\frac{xdx}{(z^{2}+x^{2})^{3}}=\frac{2\pi C\rho_{1}}{4}\int_{D}^{\infty}\frac{dz}{z^{4}}=-\frac{\pi C\rho_{1}}{6D^{3}}
F_{m}^{\n}= \left[ \begin{array}{c c c}{0}&{\chi{\vec{\partial}}\phi^{9}}&{0}\\{\chi{\vec{\partial}}\phi^{9}}&{0}&{-{\vec{\partial}}\phi^{9}}\\{0}&{+{\vec{\partial}}\phi^{9}}&{0}\\ \end{array}\right] \ ,
e_{1}= \left| \begin{array}{c}{\overrightarrow{e_{1}}}\\{0}\\ \end{array}\right| , \quad e_{2}= \left| \begin{array}{c}{\overrightarrow{e_{2}}}\\{0}\\ \end{array}\right| , \quad e_{3}= \left| \begin{array}{c}{\overrightarrow{e_{3}}}\\{0}\\ \end{array}\right| , \quad e_{4}= \left| \begin{array}{c}{0}\\{1}\\ \end{array}\right| .
\partial_s^\ell H_0^\varphi (s,t)= \int_0^t\int_{-\infty}^{\infty}\partial_s K(s-\xi,t-\tau)\cdot \partial_\xi^{\ell-1}h(\varphi)(\xi,\tau)d\xi d\tau = \int_0^t\int_{-\infty}^{\infty}K(s-\xi,t-\tau)\cdot \partial_\xi^\ell h(\varphi)(\xi,\tau)d\xi d\tau
xh(x) =xf(p_\sigma( f^{-1}(x)))=x\phi(p_\sigma( f^{-1}(x)))=\phi( f^{-1}(x))\phi(p_\sigma( f^{-1}(x))) &=\phi(f^{-1}(x)p_\sigma( f^{-1}(x)))=\phi(q_\sigma( f^{-1}(x)))=\phi(q_{\sigma'}(x))=q_{\sigma'}(x).
(\begin{matrix}\rho_{x_{1}}\\ \rho_{x_{2}}\end{matrix})=(\begin{matrix}a_{1,x_{1}}&a_{1,x_{2}}\\a_{2,x_{1}}&a_{2,x_{2}}\end{matrix})^{-1}(\begin{matrix}r_{x_{1}}\\r_{x_{2}}\end{matrix})
(1+o(1))\exp(\tfrac{1}{4}(1-d^2)) \frac{(dn)!}{(dn/2)!2^{(dn)/2}(d!)^n}& = (1+o(1)) \sqrt{2}\exp(\tfrac{1}{4}(1-d^2)) \left(\frac{d^d n^d}{e^d(d!)^2}\right)^{n/2}& = \exp((1+o(1))\tfrac{1}{2}dn \log n),
K(S,S)(\eta) = \frac{1}{\sqrt{|\eta|}}\int e^{3i \mu^2/4}\tilde S \left(\eta, \frac{\mu}{\sqrt{| \eta|}}\right) d \mu \sqrt{|\eta|}K(S,S)(\eta) = \int_{|\mu| \le |\eta|^{3/2}/2}+ \int_{| \eta|^{3/2}/2 \le |\mu|}= T_{2,1}+T_{2,2}.
\boldsymbol{\phi}_n(f) = (-1)^n\int f_R(x_{n-1}) S(x_{n-1},x_{n-2}) S(x_{n-2},x_{n-3})\dots S(x_1,x_{0}) \; \left( \lambda(x_{n-1})\dots \lambda(x_0)\right) \; \phi_0(x_0)\; r(x_{n-1},\dots, x_{0}) \; dx_0\dots dx_n\;.
{\mathfrak p}_{m}(t;\delta)=\int_{-\infty}^{\infty}1_{[-1,1]}\biggl(\frac{u}{1+\frac{1}{2}\delta}\biggr) 1_{[-1,1]}^{(*m)}\biggr(\frac{t-u}{\frac{1}{2}\delta/m}\biggr) \frac{m\,du}{2^m \frac{1}{2}\delta}.
\mathbb P \bigg( (\xi_{1}\in dx_1, \cdots, \xi_{n}\in dx_n), A(T) = n \bigg) =~ \mathbb P \bigg(\xi_{\pi(1)}\in dx_1, \ldots \xi_{\pi(n)}\in dx_n,\sum_{l=1}^n \xi_{\pi(l)}\leq T, \sum_{l=1}^n\xi_{\pi(l)}+ \xi_{n+1}> T \bigg),
|I_1(x')|=\left\vert \int_{\Omega'}\big( |v(x', z)|^p - |v(y', z)|^p \big) \delta_{\varepsilon,x'_0}(y') \, dy'dz \right\vert \le C_{\theta,p}|\lambda|^{-1/2}\|f\|_{L^\infty_H L^p_z(\Omega)}\|v\|_{L^\infty_H L^p_z(\Omega)}^{p-1}.
\varepsilon_p^D\left(L_{1}, L_{2},\cdots,L_{p}\right)=-\frac{\pi^{(D-p)/2}}{2^{D-p+1}}\Gamma\left(-\frac{D-p}{2}\right)E_p\left(\frac{1}{L_1^2},\frac{1}{L_2^2},\cdots, \frac{1}{L_p^2};-\frac{D-p}{2}\right)\hspace{3cm}
\begin{aligned}U_{11}= \, - \frac{a}{\phi}|\nabla u|^2 + A^{11}+ \frac{1}{\nabla_1 u}\sum_{k \geq 2}\nabla_k u A^{1k}\leq \, - \frac{a}{\phi}|\nabla u|^2 + C (1 + |\nabla u| + |\nabla u|^{\gamma-2}) < 0\end{aligned}
\mathrm{e i t h e r}\quad{\cal S}^{a b}= \tilde{S}^{a b}= - \frac{i}{4}[ \tilde{a}^{a}, \tilde{a}^{b}] = - \frac{i}{4}[ \tilde{\gamma}^{a}, \tilde{\gamma}^{b}] , \quad \mathrm{o r}\qquad{\cal S}^{a b}= \tilde{\tilde{S}}{}^{a b}= - \frac{i}{4}[ \tilde{\tilde{a}}{}^{a}, \tilde{\tilde{a}}{}^{b}] .
-\sum_{n=-\infty}^\infty \frac{(6n-1)(-1)^nq^{3n^2+4n+1}}{1+q^{6n-1}}-\sum_{n=-\infty}^\infty \frac{(6n-3)(-1)^nq^{3n^2+2n}}{1+q^{6n-3}}&=-\frac{[q,q^2,q^2,-q^2,q^3;q^6]_\infty(q^6;q^6)^4_\infty}{[-q;q^6]_\infty}+4\frac{(q^2;q^2)_\infty^2}{(q;q^2)_\infty^2}\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+5)+2}}{1+q^{6n}}.
Z_{r_1}(d_2(f), s+1) =\int_{F^\times}\int_{F}f(0,0,0,y,0,0)\psi(ay)dy|a|^{s+1}d^\times a =\int_{|a|\le q^A}\int_{F}f(0,0,0,y,0,0)\psi(ay)dy|a|^{s+1}d^\times a =\int_{F}\int_{|a|\le q^{A}}f(0,0,0,y,0,0)\psi(ay)|a|^{s+1}d^\times a dy.
\gamma_\mu(2k)= \frac{2^{2k}k!\Gamma\left(k+\mu+\frac{1}{2}\right)}{\Gamma\left(\mu+\frac{1}{2}\right)},~~~\gamma_\mu(2k+1)= \frac{2^{2k+1}k!\Gamma\left(k+\mu+\frac{3}{2}\right)}{\Gamma\left(\mu+\frac{1}{2}\right)}.
\rightarrow\sqrt{\frac{2}{3}}[\begin{matrix}cos(\theta)&cos(\theta-\frac{2\pi}{3})&cos(\theta+\frac{2\pi}{3})\\ -sin(\theta)&-sin(\theta-\frac{2\pi}{3})&-sin(\theta+\frac{2\pi}{3})\\ \frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}&\frac{\sqrt{2}}{2}\end{matrix}]
V \left( \sigma , \pi \right) = \frac{1}{2 \lambda}\left( \sigma^{2}+ \pi^{2}\right) + i \, \mathrm{S p}\ \log < x \mid \left[ i \gamma^{\mu}\left( x \right) \nabla_{\mu}- \left( \sigma + i \gamma_{5}\pi \right) \right] \mid x >
\Gamma(k) =\begin{bmatrix}\Gamma(k)_{1,1}\Gamma(k)_{2,1}\vdots \Gamma(k)_{n,1}\end{bmatrix}\otimes\begin{bmatrix}\Gamma(k)_{1,1}\Gamma(k)_{1,2}\vdots \Gamma(k)_{1,n}\end{bmatrix}^{\top}
\mathcal{D}_0=\Big\{\big(\mathbf{U}_0(1),\mathbf{U}_1(1,1,1,I),\mathbf{U}_2(1,1),\mathbf{Y}_1,\mathbf{Y}_2\big)\in\mathcal{T}_\delta^{n}(Q_{U_0,U_1,U_2,Y_1,Y_2})\Big\}
\delta a_{a}^{+}= ( \epsilon_{b}^{\beta}\theta^{+ b}\partial_{\beta}^{-}+ \epsilon_{b}^{\beta}\theta^{- b}\partial_{\beta}^{+}- \epsilon^{+ b}\partial_{b}^{-}) a_{a}^{+}- \epsilon_{a}^{\alpha}A_{\alpha}^{+}\ .
0 = \ \biggl(t^{-1}d_0 + \sum_{i=1}^{n}(t^{-1}d_i'x_i - t^{-1}d_i''d_i)\biggr) \biggl(b_0' + \sum_{j=1}^{n}b_j'x_j\biggr) = \t^{-1}\biggl(d_0 + \sum_{i=1}^{n}(d_i'x_i - d_i''d_i)\biggr) \biggl(b_0' + \sum_{j=1}^{n}b_j'x_j\biggr)
C_J=\left (\frac{\partial E}{\partial T}\right)_J= 12 \pi^{11/4}\kappa^{-3/2}N^{1/2}m^{3/2}r_+^{-1}\frac{2r_+^2 +l^2}{2r_+^4 +5l^2r_+^2 -l^4},
f_1(x,y) =x^{12}(y_0+\pi^{j+1}y)^4((y_0-1+\pi^{j+1}y)^2+ \pi^{8n}x^8(y_0+\pi^{j+1}y)^4), f_2(x,y) =x^{12}(1+\pi^{j+1}y)^4(y^2+\pi^{8n-(2+2j)}x^8(1+\pi^{j+1}y)^4), f_3(x,y) =x^{12}(1+\pi^{j+1}y)^4(y^2+x^8(1+\pi^{j+1}y)^4), \intertext{and}f_4(x,y) =x^{12}(1+\pi^{j+1}y)^4(\pi^{2+2j-8n}y^2+x^8(1+\pi^{j+1}y)^4).
H_{(X,D)}(y)=e^{\frac{t}{2\pi i}}\sum_{\substack{d\in\mathbb K (k_1,\ldots,k_m)\in (\mathbb Z_{\geq 0})^m}}y^d\left(\frac{\Gamma (1+\frac{D}{2\pi i}+D\cdot d)}{\prod_{i=1}^m\Gamma (1+\frac{\bar D_i}{2\pi i}+D_i\cdot d)}\right)\textbf{1}_{[d]}[\textbf{1}]_{D\cdot d-\sum_{i=1}^mk_ia_i}\frac{\prod_{i=1}^m x_i^{k_i}}{\prod_{i=1}^m(k_i!)}.
(d^2+1)(t-s+2)^k \ge \sum_{v\in B}|X_v| = \sum_{v\in B}(t-s+2)^{k-d_v+d'_v}d^2+1 \ge \sum_{v\in B}(t-s+2)^{d'_v-d_v}\ge m\cdot (t-s+2)^{\frac{1}{m}(\sum_{v\in B}d'_v - \sum_{v\in B}d_v )}
(x \otimes y)^{*_\beta \, \otimes_\beta \, *_\beta}= \beta(a^{-1}b,a^{-1})\beta(b^{-1}c,b^{-1})\beta(b^{-1}c,a^{-1}b)\,x^* \otimes y^* &= \beta(a^{-1}c,a^{-1})\, x^* \otimes y^*.
t^\omega_{12}= \frac{1}{4 \pi}\:\int_0^{2 \pi}\frac{t_1 \:\overline{t_2}}{|t_2|^2}\:d\alpha= \frac{1}{4 \pi}\:\int_0^{2 \pi}\frac{t_1}{t_2}\:d\alpha &= \frac{1}{2 \pi}\:\int_0^{\pi}\frac{f_{\infty\: 2}^+ - f_{\infty\: 2}^- \:e^{2 i \alpha}}{-f_{\infty\:1}^+ + f_{\infty\:1}^- \:e^{2 i \alpha}}\:d\alpha \:.
S_{\mathrm{e f f}}= - \left({\frac{m}{2 \sqrt{\pi}}}\right)^{d - 1}{\frac{B}{2 \pi}}\int_{0}^{\infty}{\frac{d s}{s^{( d + 1 ) / 2}}}\, e^{- s}\, \left[ \coth \left({\frac{B s}{m^{2}}}\right) -{\frac{m^{2}}{B s}}\right] \theta_{4}{\left({\frac{i \beta \mu}{2}}\left|{\frac{i \beta^{2}m^{2}}{4 \pi s}}\right. \right)}
\langle\widetilde\Gamma_a(\cdot,y), \mbox{div $\varphi$}\rangle=\int_0^\infty \left\langle O(at)^T\left(K(O(at)x-y,t)-\frac{e^{-1/4t}}{8\pi t}\mathbb I\right),\,\mbox{div $\varphi$}\right\rangle\,dt
x_2 = 2^{1/(2k-1)}\left(\vert b\vert^{-1/(2k-1)}\vert \beta\vert^{-(k-1)/(2k-1)}+\mathcal O(z)\right) (-\phi^{[k]}z)^{-1/(2k-1)}\tilde x_2, y_2 = 2^{2/(2k-1)}\left(\vert b\vert^{-2/(2k-1)}\vert\beta \vert^{1/(2k-1)}+\mathcal O(z)\right) (-\phi^{[k]}z)^{-2/(2k-1)}\tilde y_2
S(x,t)-1=\dfrac{t}{x^{r-2}}\left[S(x,t)-1-\sum\limits_{i=1}^{r-2}x^i\,S_i(t)\right]+t\,x^2\,S^2(x,t),S_i(t):=\sum\limits_{j=0}^{\infty}s_{n,j}\,t^j.
\sum_{k=0}^{N-1}\cos(2 \pi k/N)^2 = \frac{1}{2}\sum_{k=0}^{N-1}\left(1 + \cos(2 \pi 2k/N)\right) = \frac{N}{2}+ \frac{1}{2}\sum_{k=0}^{N-1}\cos(2 \pi 2k/N).
S_{thermo}=2\pi \left[{4\over{(D-3)^2}}Q^{(1)}_1Q^{(2)}_1(2m)^{2\over{D-3}}-{{2}\over{(D-3)}}\sum_{i=1}^{[{{D-1}\over 2}]}J^2_i\right]^{1\over 2}.
\rho(i,i,t-s) =\int_{-\infty}^{t}\int_{-\infty}^{s}\gamma_{i}(t-u)\gamma_{i}(s-v)\rho(i-1,i-1,|u-v|)dudv &\qquad+\int_{-\infty}^{s}\gamma_{i}(t-u)\gamma_{i}(s-u)\overline{\gamma}_{0}m_{i-1}du,
D_x^2(\Delta^i)^{-1}(p,p) = D_x^2 \L_(p,p) - D_x^2 \L_{R_i}( p, p) - D_x^2 \L_{R_j}( p,p)+ D_x^2 \L_{R_{ij}}(p,p)+ O(\epsilon^2) D_x (\Delta^i)^{-1}(p,p) = -D_x \L_{R_i}( p, p) - D_x \L_{R_j}( p,p)+ D_x \L_{R_{ij}}(p,p)+ O(\epsilon)
\int_{B_\rho(x_0)}v^{\frac{p+2}{2}}\xi^2 dx = \int_{B_\rho(x_0)}v^{\frac{p}{2}}|\nabla f|^2 \xi^2 dx = \int_{B_\rho(x_0)}v^{\frac{p}{2}}f_{x_i}[f(x) - f(0)]_{x_i}\xi^2 dx.
\mu =\frac{1}{P_{}}\sum_{l=1}^{L}w_{l}\left(\mathbf{\Omega}_{l}^{-1}-\left(\mathbf{\Omega}_{l}+\mathbf{H}_{l,l}\mathbf{\Sigma}_{l}\mathbf{H}_{l,l}^{\dagger}\right)^{-1}\right)
\sum_{j=1}^{k}\prod_{i=k}^{k+j-2}q_i^\star=\sum_{j=1}^{k}\prod_{i=k}^{k+j-2}q_i=\sum_{j=1}^{k}\left(\frac{k+1}{k+j}\right)^{\beta}=(k+1)^\beta\sum_{j=k+1}^{2k}\frac{1}{j^\beta}
e_{i}={\bf 1}\otimes \cdots \otimes \left(\begin{array}{cccc}0 0 0 0 0 q^{-1}-1 0 0 -1 q 0 0 0 0 0 \end{array}\right) \otimes \cdots \otimes{\bf 1}\;\;\; \in{\bf C}^{\otimes 2l}\;,
F ( a , b ; z ) = 1 +{\frac{a}{b}}\; z +{\frac{a ( a + 1 )}{b ( b + 1 )}}\;{\frac{z^{2}}{2 !}}+{\frac{a ( a + 1 ) ( a + 2 )}{b ( b + 1 ) ( b + 2 )}}\;{\frac{z^{3}}{3 !}}+ \cdots
J_{\varphi}(t,x^{(t)},m^{(t)},w)=\mathbb{E}_{\varphi}\left[\sum_{s=t}^{T-1}f(s,X_s,\mu_s,w\circ\varphi(s,X^{(s)},\mu^{(s)}))+ F(X_T,\mu_T)| X^{(t)}=x^{(t)}, \mu^{(t)}=m^{(t)}\right].
s_n = \frac{1}{\lambda_n}\big(\nabla u(x_n)-\nabla u(p_n) \big) + \nabla h(p_n)-\nabla h(x_n)+\frac{\alpha_n}{\lambda_n}(p_{n-1}-x_{n-1}) & + \frac{\alpha_n\lambda_{n-1}}{\lambda_n}\big(\nabla h(x_{n-1})-\nabla h(p_{n-1})\big).
\begin{array}{l c l}{{\mathcal L}_{3}^{( 0 , 3 )}}&{=}&{p^{3}+ u_{2}p + ( u_{3}- \theta u_{2}^{\prime})}\\{{\tilde{\mathcal L}}_{- 3}^{( - 3 , - 1 )}}&{=}&{{v_{0}\star p}^{- 3}+ v_{- 1}\star p^{- 2}+ v_{- 2}\star p^{- 1}}\\ \end{array}
(s_0\varphi)_1(x,y) = x_1, \mbox{if $x_2\sim 0$}, (s_\lambda\varphi)_{\lambda+1}(x,y) = \varphi_\lambda(s^\lambda x,y) + x_{\lambda+1}- \frac{1}{2}(x_\lambda +x_{\lambda +2}), \mbox{if $x_\lambda \sim x_{\lambda +2}$}, (s_k\varphi)_{k+1}(x,y) = x_{k+1}, \mbox{if $x_k \sim 1$}.
(y_1, g_1, \epsilon_1)\cdot (y_2, g_2, \epsilon_2) =(y_1y_2, [g_1, \epsilon_1]^{y_2}[g_2, \epsilon_2]) & =(y_1y_2, [g_1^{y_2}, \nu(y_2, g_1)\epsilon_1][g_2, \epsilon_2]) &=(y_1y_2, [g_1^{y_2}g_2, \nu( y_2,g_1)\widetilde{c}_{X^{\ast}}(g_1^{y_2}, g_2)\epsilon_1\epsilon_2]).
T_{\tau \tau}= T_{\tau \tau}^{( 0 )}+{\frac{1}{4 8 \pi}}({\frac{2}{\beta_{H}^{2}}}+ 2 g_{\rho}^{\prime \prime}-{\frac{3}{2}}{\frac{( g_{\rho}^{\prime})^{2}}{g}}) ,
\|u\|_{\widetilde{L}^q_T(\dot{B}^s_{p,r})}:= \big{\Vert}\big\{2^{js}\Vert{\dot{\Delta}_ju}\Vert_{L^q_T(L^p)}\big\}_{j\in\mathbb{Z}}\big{\Vert}_{\ell^r(\mathbb{Z})},
Q_6 = \Big(\partial_{u_3}+ a_2 \partial_{u_1}+ a_2^2 u_1 + (\gamma_4 + \tfrac{4}{3}a_2^2) a_2 u_3\Big)^2 -\big(\gamma_6 + \tfrac{5}{3}a_2 \gamma_4 + \tfrac{125}{27}a_2^3\big).
p_{-2}(a) =\sum_{\alpha_{1}+\alpha_{2}=a}p(\alpha_{1}) p(\alpha_{2}), p_{-2}(b) =\sum_{\beta_{1}+\beta_{2}=b}p(\beta_{1})p(\beta_{2}), p_{-2}(a+b) =\sum_{\gamma_{1}+\gamma_{2}=a+b}p(\gamma_{1})p(\gamma_{2}),
(-1)^{m+1}\sum_{j=0}^{k}(\begin{matrix}k\\j\end{matrix})\frac{B_{m+1+j}}{m+1+j}+(-1)^{k+1}\sum_{j=0}^{m}(\begin{matrix}m\\j\end{matrix})\frac{B_{k+1+j}}{k+1+j}=\frac{k!m!}{(k+m+1)!}
\left\{\begin{array}{c}\left(\begin{array}{clc}\frac{\delta - \sqrt{\delta^2 + \omega^{\ast}\omega}}{\omega}, 1\end{array}\right) \left(\begin{array}{clc}\frac{\delta + \sqrt{\delta^2 +\omega^{\ast}\omega}}{\omega}, 1\end{array}\right) \end{array}\right\}.
\|x^k-J(x^k,\alpha_k)\|^2 =\|x^k-x^{k+1}\|^2\le\frac{\alpha_k}{1-\delta}\left[(f+g)(x^k)-(f+g)(x^{k+1})\right] &\le\frac{\sigma}{1-\delta}\left[(f+g)(x^k)-(f+g)(x^{k+1})\right],