harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{lcl}A^{(n,0)}(z) = \frac 12(1+z)= A^{(0)}(z)\,, A^{(n,m)}(z) = \displaystyle{\frac{1}{2^{n+1+m^{-\mu}}}(1+z)^{n-1}\bigl (z^2+2(2^{1+m^{-\mu}}-1)\z+1\bigr )}= = \frac{1+z}2 \, A^{(n-1,m)}(z) = A^{(0)}(z) \, A^{(n-1,m)}(z) \,, m> 0\,.\end{array}
\frac{A^{2n+2}}{V^{2n+1}}=\frac{2\pi}{(n+1)}{\mathrm{Vol{(M)}}}\,2^{2n+1}\,(n+1)^{2n+1}\frac{1}{\left(1-\frac{a^{2n+2}}{r^{2n+2}}\right)^n}
\prod_{\alpha=1}^{s - 1}(-1)^{s - \alpha}x_{i_{\alpha + 1}}x_{i_{\alpha + 2}}\cdots x_{i_s}= (-1)^{\frac{(s - 1) s}{2}}x_{i_2}x_{i_3}^2 x_{i_4}^3 \cdots x_{i_s}^{s - 1}.
I ( k , j , N ) \, = \, \delta_{j k}\, \frac{k ( k - 1 ) \cdots 1}{( N + 1 ) \cdots ( N + 2 - k )}\I ( 0 , 0 , N - k ) \, = \, \frac{k ! ( N - k ) !}{( N + 1 ) !}\, \pi \, \delta_{j k}\ \ .
f_{Cayley}={\rm det}\begin{pmatrix}q_1(s_1,s_2,\ldots,s_{d-1}) \ldots q_d(s_1,s_2,\ldots,s_{d-1}) begin{align*}3pt] q_1(t_1,s_2,\ldots,s_{d-1}) \ldots q_d(t_1,s_2,\ldots,s_{d-1}) begin{align*}3pt] \vdots \ddots \vdots begin{align*}3pt]q_1(t_1,t_2,\ldots,t_{d-1}) \ldots q_d(t_1,t_2,\ldots,t_{d-1}) begin{align*}3pt] \end{pmatrix}\Bigg/\prod_{i=1}^{d-1}(s_i - t_i).
\textsf{E}(\Xi^{2})^{p}\le (C\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2})^{p}\int_{0}^{\infty}\exp(-C_{1}z\log n\sum_{i\le N}\Vert B_{i}\Vert_{2}^{2})pz^{p-1}\, dz &=C_{2}^{p}p\int_{0}^{\infty}e^{-t}t^{p-1}\, dt\lesssim C_{2}^{p}p^{p+1}.
\forall k \geq 1, 0= - \int_0^T \int_{\mathbb{R}}\rho_{3}(s,u) \partial_s G_{k}(s,u) du ds- \int_0^t \int_{\mathbb{R}}\rho_{5}(s,u) [ - (- \Delta)^{\gamma/2}G_{k}](s,u) du ds.
{}_4F_3\bigg[\begin{matrix}-n \alpha \beta \gamma &1 \delta \epsilon\end{matrix}\bigg|\,1\bigg]=\frac{(\delta-\alpha)_n(\epsilon-\alpha)_n}{(\delta)_n(\epsilon)_n}\cdot{}_4F_3\bigg[\begin{matrix}-n \alpha 1-\beta 1-\gamma &1 1+\alpha-n-\delta 1+\alpha-n-\epsilon\end{matrix}\bigg|\,1\bigg],
\psi(h_1 \otimes x_1^l + \sum_{k=0}^{l-1}a_k(h_1 \otimes x_1^k)) =\psi(ad^l(\frac{c_1+c_0}{2})(h_1 \otimes 1)) =2^{-l}ad^l(w_{0,2})(w_{-1,0}) =(-1)^l w_{-1,l}+ \sum_{k=0}^{l-1}b_k(w_{-1,k}).
\Pi^{1}(t,r,u)(\Phi) = \frac{\mathbb{E}({\bf 1}_{\tau_{x}>u}\Phi(X_{u})f(r-u,x-X_{u})|\mathcal{G}_{u})}{\mathbb{E}({\bf 1}_{\tau_{x}>u}G(t-u,x-X_{u})|\mathcal{G}_{u})}, \Pi(t,u)(\Phi) = \frac{\mathbb{E}({\bf 1}_{\tau_{x}>u}\Phi(X_{u})G(t-u,x-X_{u})| \mathcal{G}_{u})}{[\mathbb{E}({\bf 1}_{\tau_{x}>u}G(t-u,x-X_{u})|\mathcal{G}_{u})]^{2}}
\int \prod_{i=1}^{2n}{dt_i\over 2\pi}d\theta_i ~ \Theta(t_1-t_2) \Theta(t_2-t_3) \cdots \Theta(t_{2n-1}-t_{2n})\prod_{i<j}| e^{it_i}-e^{it_j}- i e^{\frac{i}{2}(t_i+t_j)}\theta_i \theta_j | ={1\over 2^n}~,
\phi^{-2n}(z)\big(1 - \phi^{-2}\big)(z) = \left(1 + \sqrt{2}\frac{y^{1/2}}{n}+ O\left(\frac{y}{n^2}\right)\right)^{-2n}\left( 2\sqrt{2}\frac{y^{1/2}}{n}+ O\left(\frac{y}{n^2}\right)\right) \hphantom{\phi^{-2n}(z)\big(1 - \phi^{-2}\big)(z)}{}= \frac{2\sqrt{2}y^{1/2}}{n}e^{- 2\sqrt{2}y^{1/2}}\left(1 + O\left(\frac{y^{1/2}}{n}\right)\right).
\left( \frac{\gamma_{m}}{w(Q)}\int_{Q}|f(y)|w(y)\,dy\right)^{p(x)}& \leq c\max \left( 1,\left( w(Q)\right)^{1-\frac{p\left( x\right)}{p_{Q}^{-}}}\right) \frac{1}{w(Q)}\int_{Q}\left\vert f(y)\right\vert^{p(y)}w(y)\,dy & +\frac{c\min (|Q|^{m},1)}{w(Q)}\int_{Q}\left\{(e+|x|)^{-m}+(e+|y|)^{-m}\right\}w(y)\,dy
M_{(n), s}(t) = P_n\Big[ \sigma(\cdot, X(\cdot)) \Big](s) + \int_s^t P_n\Big[ U(\cdot, r) \Big](s) dr + \int_s^t P_n\Big[ V(\cdot, r) \Big](s) dW(r) &+\int_s^t \nabla_x b(r, X(r)) M_{(n), s}(r) dr + \int_s^t \nabla_x \sigma(r, X(r)) M_{(n), s}(r) dW(r).
\left(Z(X_{A,\psi},T)\prod_{i=0}^{n-1}\big(1-q^iT\big) \right)^{(-1)^{n}}\mbox{and}\left(Z(X_{A',\psi},T)\prod_{i=0}^{n-1}\big(1-q^iT\big) \right)^{(-1)^{n}}
\|\theta_{x}\|_{L^{2}}\leq \left\|\Lambda^{\gamma/2}\theta\right\|_{L^{2}}, \left\|\theta_{i}\right\|_{L^{\infty}}+ \left\|\mathcal{H}\theta_{i}\right\|_{L^{\infty}}\leq C\left\|\theta_{i}\right\|_{H^{1}}, \ ,
\widehat{\omega}'((y_1, \zeta_{y_1}(\rho_1)), (y_2,\zeta_{y_2}(\rho_2)),(y_3 ,\zeta_{y_3}(\rho_3))) = \tilde{\nu}(y_1,y_2)(1;u(y_3)) \ \rho_1(\kappa_{y_2,u(y_3)}).
\eta_{jk}(t)=\sum_{\alpha=1}^n \frac{\partial e_{jk}^\alpha(w_k,t)}{\partial t}\frac{\partial}{\partial w_j^{\alpha}},\,\,\,w_k=e_{kj}(w_j,t),\,\,\,\,\,\,\,\,\pi_j(t)=\sum_{\alpha,\beta=1}^n \frac{\partial \Pi_{\alpha \beta}^{j}(w_j,t)}{\partial t}\frac{\partial}{\partial w_{j}^{\alpha}}\wedge \frac{\partial}{\partial w_{j}^{\beta}}
\widehat{T}(z_{0})\widehat{T}(z_{1}) =e^{\frac{i}{\hbar}\sigma(z_{0},z_{1})}\widehat{T}(z_{1})\widehat{T}(z_{0}) \widehat{T}(z_{0}+z_{1}) =e^{-\frac{i}{2\hbar}\sigma(z_{0},z_{1})}\widehat{T}(z_{0})\widehat{T}(z_{1})
B^{'}\left(\frac{2}{\alpha},1-\frac{2}{\alpha},\frac{1}{1+c\beta}\right) = \frac{\left(c\beta\right)^{1-\frac{2}{\alpha}}}{1-\frac{2}{\alpha}}+o\left(\beta^{1-\frac{2}{\alpha}}\right)\;, B^{'}\left(1+\frac{2}{\alpha},a-\frac{2}{\alpha},\frac{1}{1+c\beta}\right) = \frac{(c\beta)^{a-\frac{2}{\alpha}}}{a-\frac{2}{\alpha}}+o\left(\beta^{a-\frac{2}{\alpha}}\right)\;,
\mathbf{y}_j = \begin{pmatrix}h^H_{j,1}h_{j,1}d_1+h^H_{j,1}h_{j,2}d_2+\cdots+h^H_{j,1}h_{j,k}d_k h^H_{j,2}h_{j,1}d_1+h^H_{j,2}h_{j,2}d_2+\cdots+h^H_{j,2}h_{j,k}d_k \cdots h^H_{j,k}h_{j,1}d_1+h^H_{j,k}h_{j,2}d_2+\cdots+h^H_{j,k}h_{j,k}d_k\end{pmatrix}.
x \left(1-\frac{x}{4}\right)+y\left(1-\frac{x}{2}-\frac{y}{4}\right)+\left(1-\frac{x+y}{2}\right)^2 &=x+y-\frac{x^2+y^2}{4}-\frac{xy}{2}+1-x-y+\frac{x^2+2xy+y^2}{4}=1.
\delta_{v}j^{i_{1}\ldots i_{s}}= [ D - 2 + s ] ( v \cdot x ) j^{i_{1}\ldots i_{s}}+ \sum_{p = 1}^{s}( v^{i_{p}}x_{k}- x^{i_{p}}v_{k}) j^{i_{1}\ldots i_{p - 1}k \; i_{p + 1}\ldots i_{s}}
\bigg[ \bigsqcup_{i=1}^n A_i \times \{i\}\bigg] + \bigg[ \bigsqcup_{i=1}^m B_i \times \{i\}\bigg] = \bigg[ \bigg(\bigsqcup_{i=1}^n A_i \times \{i\}\bigg) \sqcup \bigg( \bigsqcup_{i=n+1}^{n+m}B_i \times \{i\}\bigg) \bigg] .
F_{\ast}\left( j_{F}^{p}(x) - j_{F}^{p}(y) \right) \le \left( F(x) + F(y) \right)^{p-\tau}\left( \sum_{i}\rho(x_{i}- y_{i})^{p \left(\frac{\tau-1}{p-1}\right)}\right)^{\frac{p-1}{p}}& \lesssim \left( F(x) + F(y) \right)^{p-\tau}\left( \sum_{i}\rho(x_{i}-y_{i})^{p}\right)^{\frac{\tau-1}{p}}& = (F(x) + F(y))^{p-\tau}F(x-y)^{\tau-1}.
\hat{C}^{l\,l_1\,l_2}_{l\,j\,l-j}= (-1)^{l_1-j}{(l_1+l_2-l)!\over(l_1+l_2+l+1)! \Delta(l_1,l_2,l)}\left[{(l_1+j)!(l_2+l-j)!\over (l_1-j)!(l_2-l+j)!}\right]^{1/2},
\mathcal{E}(h,k) = \mathcal{L}^r(h,k) + \mathcal{U}^r(h,k) + O \bigg( \bigg( Q + \frac{Q^2}{C}\bigg) \frac{(XCQhk)^{\varepsilon}(h,k)}{\sqrt{hk}}\bigg) +O\Big((XCQhk)^{\varepsilon}\big( XC+ X^{-\frac{1}{2}}Q^{\frac{5}{2}}+ Q^{\frac{3}{2}}+X^2hk Q^{-96}\big) \Big).
J^{(1,2)}_2\leq 4 T_*L^{-2}_*\left(\sum\limits_{n=2}^{\infty}p_n(T_*)\left|\frac{2\widetilde{n}}{T_*}-1\right|^2\right)^{1/2}\leq 4\sqrt{35}T_*L^{-2}_* T^{-1/2}_*.
\int\frac{dx}{s^{9}}=\frac{1}{a^{8}}[\frac{x}{s}-\frac{3}{3}\frac{x^{3}}{s^{3}}+\frac{3}{5}\frac{x^{5}}{s^{5}}-\frac{1}{7}\frac{x^{7}}{s^{7}}]
\psi_{n_r,l}(r) \propto r^l (1+\lambda r^2)^{-\beta/(2\lambda)}P_{n_r}^{\left(l+\frac{d-2}{2}, -\frac{\beta}{\lambda}- \frac{1}{2}\right)}(1+2\lambda r^2), n_r=0, 1, 2, \ldots,
d_{k}\big((A,r),(B,s)\big) = d_{k}\big((A,r),(\{j\},{\scriptstyle \frac12})\big)+ d_{k}\big((\{j\},{\scriptstyle \frac12}),(B,s)\big) &= (r-{\scriptstyle \frac12})2^{k}+ ({\scriptstyle \frac12}- s)2^{k}=(r-s)2^{k}
\int_{T(\epsilon)}^{t}Mf'(x(u))\,du = \int_{T(\epsilon)}^{t}\frac{f'(x(u))}{f(x(u))}x'(u)\frac{Mf(x(u))}{x'(u)}\,du &\leq \frac{1}{1-\epsilon}\int_{T(\epsilon)}^{t}\frac{f'(x(u))}{f(x(u))}x'(u)\,du &= \frac{1}{1-\epsilon}\left(\log f(x(t))-\log f(x(T(\epsilon)))\right).
\#\{1\leq i\leq 2^n: \lambda_i=0\}=\#\{1\leq i\leq 2^{n-1}: \lambda_i=0\}+\#\{2^{n-1}+1\leq i\leq 2^n: \lambda_i=0\}& = \#\{1\leq i\leq 2^{n-1}: \lambda_i=0\}+ \#\{1\leq i\leq 2^{n-1}: \lambda_i=0\}-1 &= 2\#\{1\leq i\leq 2^{n-1}: \lambda_i=0\}-1.
( D_{3})_{\alpha}= \theta_{\alpha C}( \vec{X}_{B}\cdot \vec{P}_{A}) X_{D}^{a}{d^{C D}}_{F}f^{B A F}+ \frac{1}{2}( \gamma^{a b}\theta_{A})_{\alpha}X_{d B}( \vec{X}_{C}\cdot \vec{P}_{D}) ({c_{E}}^{A B}f^{E C D}+{c_{\lambda}}^{A B}f^{\lambda C D}) .
\begin{aligned}w(x_n,x^*) f[x_0,\cdots, x_n,x^*]=-\frac{f^{(n+1)}(\xi_1)}{(n+1)!}h^{n+2}\int^{\frac{x_n-x_0}{h}}_{\frac{x^*-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)\d\mu,\end{aligned}
\widehat{D}_{3}(q^2) = 2H_3, \widehat{D}_{5}^2(q^2) = 4H_5^2(1+4H_5), \widehat{D}_{7}(q^2) =2H_7\left(1+3H_7\right), \widehat{D}_{11}(q^2) =2H_{11}\left(1+4H_{11}+5H_{11}^2\right), \widehat{D}_{23}(q^2) = 2H_{23}\left(1+5H_{23}+13H_{23}^2+20H_{23}^3+20H_{23}^4+11H_{23}^5\right).
I(\alpha,\alpha '\left|\tau \right.)={\rm const}\cdot\frac{\det_{kj}\vartheta\left(\frac{\alpha_k-\alpha'_j}{2\pi}\left|\frac{i\tau}{2\pi N}\right.\right)}{J(\alpha )J(\alpha ')},~~~~J(\alpha )=\prod_{i<j}\sin\frac{\alpha_i-\alpha_j}{2},
\mathbf{d}_{3}=(0,\ldots,0,-a_{0}^{(2)},-a_{1}^{(1)},z-a_{2}^{(0)}), \mathbf{v}_{3}=(A_{0}^{(3)},\ldots,A_{p-4}^{(3)},-\sum_{j=3}^{p}a_{0}^{(j)}A_{j-3}^{(3)},-\sum_{j=2}^{p}a_{1}^{(j)}A_{j-2}^{(3)},-\sum_{j=1}^{p}a_{2}^{(j)}A_{j-1}^{(3)}).
\frac{\partial u}{\partial t}-\Delta u + \xi + \pi(u) + \lambda w = f\mbox{a.e.\in}Q, \partial_\nu u+ \frac{\partial u_{\Gamma}}{\partial t}-\Delta_{\Gamma}u_{\Gamma}+ \xi_{\Gamma}+\pi_{\Gamma}(u_{\Gamma}) + \lambda w_{\Gamma}= f_\Gamma \mbox{a.e.\on}\Sigma.
\left\langle \alpha_{1},\alpha_{2},...,\alpha_{2N},t\right| U_{L}\left| \alpha_{1}',\alpha_{2}',...,\alpha_{2N}',t\right\rangle =R^{\alpha_{2}'\alpha_{3}'}_{\alpha_{1}\alpha_{2}}\: R^{\alpha_{4}'\alpha_{5}'}_{\alpha_{3}\alpha_{4}}\: ...\: R^{\alpha_{2N}'\alpha_{1}'}_{\alpha_{2N-1}\alpha_{2N}}
\begin{aligned}(u_0\otimes u_0)^r\Delta(u_n)={}P_0(u_0\otimes u_0)+\sum_{i=1}^{n-1}P_i(u_0\otimes u_0)\Delta(u_i) \ + & \sum_{i\leq j}P_{i,j}(u_0\otimes u_0)\Delta(u_iu_j)+\Delta(H).\end{aligned}
J_i^N(\mathfrak{m}^N, \gamma^N, \widetilde\gamma ):= \mathbb{E}\left[ \sum_{t=0}^{T-1}f \left(t,X^{i,N}_t,\mu^{i,N}_t,\widetilde{\Phi}_i\left(t,X^{i,N},\mu^{i,N}\right)\right) +F\left(X^{i,N}_T,\mu^{i,N}_T\right) \right].
\mathbb{C}T^*X = T^{*1,0}X \oplus T^{*0,1}X \oplus \{\lambda \omega_0; \lambda \in \mathbb{C}\, \}, &\mathbb{C}TX =T^{1,0}X \oplus T^{0,1}X \oplus \{\lambda T; \lambda \in \mathbb{C}\, \}.
L^2 \int_{\Sigma \cap \{|x_{n+1}| \leq L\}}e^{-\frac{|x|^2}{4}}u^2 + L \int_{\Sigma \cap \{|x_{n+1}|=L\}}e^{-\frac{|x|^2}{4}}\, u^2 \leq C \int_{\Sigma \cap \{|x_{n+1}| \leq L\}}e^{-\frac{|x|^2}{4}}\, |\nabla^\Sigma u|^2 + C L^2 \int_{\Sigma \cap \{|x_{n+1}| \leq \frac{L}{2}\}}e^{-\frac{|x|^2}{4}}\, u^2.
f_{Q_j}(y;\rho_1,\rho_2) =\frac{\sigma_j^2}{\rho_1^2 - \rho_2^2}\left(F_{\chi^2_{n}(\frac{\rho_2^2}{\sigma_j^2})}(y)-F_{\chi^2_{n}(\frac{\rho_1^2}{\sigma_j^2})}(y)\right),
\overline{M_{i+1}}=(\mathop{q-2}\limits_{m-1},\underbrace{q-1, \ldots, q-1,}_{m-2-j_1}\mathop{q-2}\limits_{j_1},\ldots,\mathop{q-2}_{j_{i-1}},\underbrace{q-1, \ldots, q-1}_{j_{i-1}-j_i-1},\mathop{q-2}_{j_i},\underbrace{q-1, \ldots, q-1}_{j_i}),
U(\vec{a},\Lambda) \phi^{\alpha,n}(f) U(\vec{a},\Lambda)^{-1}\left( S, g_\beta \otimes \rho_n(E^\beta)\right) &=\left( S, f(\Lambda^{-1}(\cdot - \vec{a}) )^{S^\flat}(\cdot)(\bar{\Lambda}[\hat{H}(\rho) \hat{f}_0 \pm \hat{P}(\rho)\hat{f}_1] )\cdot g_\beta \otimes \rho_n(E^\beta)\right),
\rho ( \omega ) ={\frac{1}{Z}}\sum_{m}e^{- \beta E_{m}}\sum_{n > m}\vert \langle n \vert \phi \vert m \rangle \vert^{2}2 \omega \left( 1 - e^{- \beta \omega}\right) \delta ( \omega - E_{n}+ E_{m}) .
\phi(h)^{\phi(\sigma, g)}=(\sum^m_{k=1}(a_k e_k +ib_ke^{\ast}_k),t)^{(s(-1),g)}=(\sum^m_{k=1}(a_k e_k -ib_ke^{\ast}_k),-t)^{g}&=([\sum^m_{k=1}(a_k e_k -ib_ke^{\ast}_k)]g,-t)=\phi(h^{(\sigma, g)}).
\int \Big\{\sum_{j=1}^{m-1}\big[(\partial_{z_{j-1}}- \partial_{z_{j}}) \sqrt{g_t}\big]^2 \Big\}\prod_{j=1}^{m-1}2 e^{-2 z_j}d z_j \le \frac{1}{2t}H(\nu_0^{(m)}| \mu^{(m,2)}_\star) + \frac 1m\,.
\mathcal{T}(f)(\xi)= \left(\widehat{f}\left(\xi+u(p)\right)\right)_{p \in \mathbb N_0}=r(\xi)\left(\widehat{\varphi}_i\left(\xi+u(p)\right)\right)_{p \in \mathbb N_0}=r(\xi)\mathcal{T}(\varphi_i)(\xi),
\begin{aligned}f_{1}(U,D_{x}v) =g_{1}(U,D_{x}v)-A^{i}_{12}(u,v)\partial_{i}v f_{2}(U,D_{x}U) =g_{2}(U,D_{x}U)-A^{i}_{21}(u,v)\partial_{i}u-A^{i}_{22}(u,v)\partial_{i}v.\end{aligned}
C_{\alpha,n}^{a,+}(\varphi) =(-1)^{n}C_{\alpha}^+(D^n\varphi):=(-1)^{n+1}\lim_{x\searrow l_\alpha}D^{n+1}\varphi(x)(x-l_\alpha)^{1+a}, C_{\alpha,n}^{a,-}(\varphi) =C_{\alpha}^-(D^n\varphi):=\lim_{x\nearrow r_\alpha}D^{n+1}\varphi(x)(r_\alpha-x)^{1+a}
\limsup_{n \to \infty}\frac{1}{nk}\log a_{nk}(v) =\limsup_{n \to \infty}\frac{1}{nk}\log a_{(n+1)k}(v) & \leq \limsup_{n \to \infty}\frac{1}{nk}\log a_{nk+\ell}(v) \leq \limsup_{n \to \infty}\frac{1}{nk}\log a_{nk}(v)
\sum_{n=0}^{\infty}B_{n, p}\frac{t^{n}}{n!}= \frac{(p+1)(t-H_{p})e^{pt}}{(e^{t}-1)^{p+1}}+(p+1)\sum_{k=1}^{p}\binom{p}{k}\frac{H_{k}}{(e^{t}-1)^{k+1}},
\vert \min{\bf I_l^t}(A_1)-\min{\bf I_l^t}(A_2)\vert>0 \hbox{and, if}\min{\bf \Sigma^t}({\bf I_l^t}(A_1))=\min{\bf \Sigma^t}({\bf I_l^t}(A_2)), \hbox{their respective multiplicities in the vectors}\quad{\bf \Sigma^t}({\bf I_l^t}(A_1))\,\,\hbox{and}\,\,\,{\bf \Sigma^t}({\bf I_l^t}(A_2)) \hbox{are different,}
\sum_{m_1}\lambda_{1}(m_1)e\left(\frac{dm_1}{c}\right)W_{\eta H}\left(\frac{m_1-n}{H}\right)=\frac{H}{c}\sum_{m_1}\lambda_{1}(m_1)e\left(-\frac{\bar{d}m_1}{c}\right)W^{\star}_{\eta H}\left(\frac{m_1n}{c^2},\frac{m_1H}{c^2}\right),
\langle f,g\rangle_{\mathfrak{F}^2_{\mathbb{C}}}:=\int\bar f\,g\,d\mu(z,\bar z) \ , \ \ \|f\|_{\mathfrak{F}^2_{\mathbb{C}}}^2:=\langle f,f\rangle_{\mathfrak{F}^2_{\mathbb{C}}}\ .
\hat{f}(\chi_i \otimes \varphi_i ) = \chi_i(\varphi_i \circ f ) &= \int_G \overline{\chi_i (x)}\varphi_i(1_G(x)a)dx &= \varphi_i (a) \int_G \overline{\chi_i(x)}dx &= \varphi_i (a) \chi_i ( 1_G ) &= \varphi_i (a) &= \hat{a}(\varphi_i) \;\;\;\;\;\;\;\;\;\;\;\; ( 1 \leq i \leq n ).
\times \frac{[ 1 + A{\Bigl ( \frac{q}{| m |}\Bigr )}^{- 2 ( 1 - F )}] K_{F}^{2}( q r ) - [ 1 + A^{- 1}{\Bigl ( \frac{q}{| m |}\Bigr )}^{- 2 F}] K_{1 - F}^{2}( q r )}{A{\Bigl ( \frac{q}{| m |}\Bigr )}^{2 F}+ 2 + A^{- 1}{\Bigl ( \frac{q}{| m |}\Bigr )}^{2 ( 1 - F )}}.
V\Big[ \begin{array}{c}\alpha + n b_3 +{m}b_4 \cr\beta + \tilde{n}b_3 + \tilde{m}b_4 \end{array}\Big] = V\Big[ \begin{array}{c}\alpha + n b_3 +{m}b_4 +n_1 b_1 + m_1 b_2 \cr\beta + \tilde{n}b_3 + \tilde{m}b_4 +n_2 b_1 + m_2 b_2\end{array}\Big],
\sin \gamma_{12}' = \frac{\sin \gamma_{12}- \frac{\cal V}{c^2}\dot \lambda_3}{1 - \frac{\cal V}c \cos \gamma_{12}}~, ~~~~~ \cos \gamma_{12}' = \frac{\cos \gamma_{12}- \frac{\cal V}{c}}{1 - \frac{\cal V}c \cos \gamma_{12}}~,
H^{1,\frac{1}{2}}\Big(\Omega \times \mathbb{R},\mathcal{L}\Big) := \Big\{\mathrm{u}\in H^{1,\frac{1}{2}}\Big(\Omega \times \mathbb{R}\Big) : \mathcal{L}u \in \mathrm{L}^2\Big(\Omega \times \mathbb{R}\Big)\Big\},
|\langle x\xi_i,\xi_i\rangle-\tau(x\theta_t(q))|\leq\delta x\in Xx=(\theta_t(y)-y)^\ast(\theta_t(y)-y)y\in Y \; , &\|\theta_t(y)\xi_{i}-\xi_{i}\theta_t(y)\|_2\leq\delta y\in Y \; .
\phi_1(u_s)=\frac{-\frac{a_1+a_2}{b_1-b_2}\pm\sqrt{1+\frac{4a_1a_2}{(b_1-b_2)^2}}}{1+\frac{a_1-a_2}{b_1-b_2}}\;,x_{2s}=\alpha(\phi_1(u_s)-1)^2\;,x_{3s}=-\alpha(\phi_1(u_s)+1)^2\;.
\frac{a n^{2}}{2 a n - 5}- \frac{( a_{n - 1})^{2}}{2 a_{n - 1}- 5}= \frac{( a_{n - 1})^{2}}{2 a_{n - 1}- 5}- \frac{( a_{n - 2})^{2}}{2 a_{n - 2}- 5}
w(y_1,x')=\int \psi(t)e^{-\frac{i}{h}A(t,y_1,x',hD_{x'})}w|_{x_1=t}dt-\frac{i}{h}\int \psi(t) \int_{y_1}^{t}e^{-\frac{i}{h}A(s,y_1,x',hD_{x'})}f(s,x')dsdt
\left[-\frac{1}{\rho}\frac{d}{d\rho}\left(\rho\frac{d}{d\rho}\right)+\frac{k^{2}}{\rho^{2}}+\rho^{2}\right]f_{k}(\rho)=\left(\frac{2}{eB}\right)\left[E^{2}-m^{2}+eB(k+1)\right]f_{k}(\rho),
0 =\nabla f(\bar x)+\sum\limits_{i\in I^g(\bar x)}\lambda_i\nabla g_i(\bar x)+\sum\limits_{j\in\mathcal P}\rho_j\nabla h_j(\bar x) &+\sum\limits_{l\in I^{0+}(\bar x)}\xi_l\nabla G_l(\bar x)+\sum\limits_{l\in I^{+0}(\bar x)}\xi_l\nabla H_l(\bar x)+\sum\limits_{l\in I^{00}(\bar x)}\xi_l(\alpha_l\nabla G_l(\bar x)+\beta_l\nabla H_l(\bar x)).
\begin{aligned}\left[(\rho(r^2u)_x)_xr^2\right]_tu_t=\left[(\rho r^4u_x)_tu_t\right]_x+\left(2 u^2u_t\right)_x-\rho(r^2u_{xt})^2-2\rho^{-1}\frac{u_t^2}{r^2}+K.\end{aligned}
\Vert(v_{s},v_{u})\Vert \leq \Vert v_{s}\Vert+\Vert v_{u}\Vert \leq 2(\Vert v_{s}\Vert^{2}+\Vert v_{u}\Vert^{2})^{1/2}\leq 2(\Vert v_{s}\Vert^{2}_{1}+\Vert v_{u}\Vert^{2}_{1})^{1/2}=2\Vert (v_{s},v_{u})\Vert_{1}.
\Big\|\widetilde \Phi_{M_n,n}( \widetilde g)(\cdot,t)-\widetilde \Phi_{M_n,n}( g)(\cdot,t)\Big\|_{L^2(\Omega)}^2 = \frac{1}{n^2}\Big[ \sum_{k=1}^n \vartheta\xi_{k}(t)\Big]^2 &+\sum_{p=1}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}\left( \dfrac{\pi}{n}\sum_{k=1}^n \vartheta\xi_{k}(s) \phi_p(x_k) \right) ds\Bigg]^2
[ L_{m}, \mathbf{H}_{n}\bigl ({\cal O}^{\mu}\bigr ) ] = [ L_{m}, [ \mathbf{H}_{n},{\cal O}^{\mu}] ] = [ \mathbf{H}_{n}, [ L_{m},{\cal O}^{\mu}] ] - [{\cal O}^{\mu}, [ L_{m}, \mathbf{H}_{n}] ] = 0 \quad .
\lim_{\varepsilon\to 0}\frac{\gamma_\varepsilon^2-\tilde{N}_\varepsilon}{\sqrt{\tilde{N}_\varepsilon}}=+\infty\,, & \frac{\gamma_\varepsilon^2-\tilde{N}_\varepsilon}{\sqrt{\tilde{N}_\varepsilon}}=O(1)\,.
\frac{Z_{\mathrm{GJ}}\left(s,c_{v_{\pi_{\nu}}^\circ,\check{v}_{\pi_{\nu}}^{\circ}},\Phi_0\right)}{L\left(n\left(s+\frac{n-1}{2}\right),\omega_{\pi_{\nu}}\right)}=\int\limits_{Z_n(k_\nu)\backslash T^-}\mu(\underline{t})c_{v_{\pi_{\nu}}^\circ,\check{v}_{\pi_{\nu}}^{\circ}}(\underline{t})\left|\det \left( \underline{t}\right)\right|^{s+\frac{n-1}{2}}d\underline{t},
\mathcal{N}(d)=\{n\in\mathbb{N}\; : \; 1\leq n \leq d, \;\; n^2+n+1\equiv 0\,(d) \}\,, &\mathcal{N}'(d)=\{n\in\mathbb{N}\; : \; 1\leq n \leq d^2, \;\; n^2+n+1\equiv 0\,(d^2) \}\,.
[ X^{0,x}_t - e^{tA}x ]_{\P, \mathcal{B}(H)}= \int^t_0 e^{(t-s) A}F( X^{0,x}_s ) \,{\bf ds}+ \int^t_0 e^{(t-s) A}B( X^{0,x}_s ) \, dW_s .
B_m(\theta_1,x+iy) = \sum_{0 \leq j \leq m-1}\left\{\log \left| 1+ e^{2 \pi i \frac{j + \theta_1/2}{m}}+ \frac{x + iy}{m}\right| - \int_{-1/2}^{1/2}\log \left| 1+ e^{2 \pi i \frac{j + \theta_1/2 + \phi}{m}}+ \frac{x + iy}{m}\right| \mathrm{d}\phi \right\}
\Big| \frac{1}{2m \ell n}\sum_{x}\Phi_{n}(s,\tfrac{x}{n}) \sum_{j=1}^{m-1}\sum_{z = x-\ell}^{x-1}\int_{\Omega}[ \eta(z) - \eta(z - j \ell) ] \overrightarrow{\eta}^{\varepsilon n}(x+1) [f(\eta) - f(\eta^{z,z- j \ell}) ] d\nu_{b}\Big|.
V(T) = \sum_{n=1}^{\infty}V(T | N=n) P(N=n) + \sum_{n=1}^{\infty}E(T|N=n)^2 (1-P(N=n)) P(N=n) - 2 \sum_{n=1}^{\infty}\sum_{m=1}^{n-1}E(T | N=n) P(N=n) E(T | N=m) P(N=m)
\epsilon_{a_{1}a_{2}\cdots a_{n}}=\prod_{i=1}^{n-1}\left[\frac{1}{i!}\prod_{j=i+1}^{n}\left(a_{j}-a_{i}\right)\right]=\frac{1}{S(n-1)}\prod_{1\le i<j\le n}\left(a_{j}-a_{i}\right)
_2\tilde F_1\left(a,b;c,z\right) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\cdot \,_2\tilde F_1\left(a,b;a+b-c+1,1-z\right) + \frac{\Gamma(c)\Gamma(a+b-c) (1-z)^{c-a-b}}{\Gamma(a)\Gamma(b)}\cdot \,_2\tilde F_1\left(c-a,c-b;c-a-b+1,1-z\right)
\eta^{(k-1)}(e_k, \dots, e_n) = (-1)^k \iota_{\rho(e_k)}\eta^{(k)}(e_{k+1}, \dots, e_n) + \operatorname{cycl}(e_k, \dots, e_n), \iota_{\rho(e_k)}\eta^{(k)}(e_{k+1}, \dots, e_{k+m}, \dots, e_n)+ \iota_{\rho(e_{k+m})}\eta^{(k)}(e_{k+1}, \dots, e_k, \dots, e_n) =0, k=1, \dots, n-1, m=1, \dots, n-k,
E_{M_i}(x\theta_t(a_n)y) = e_{M_i}(xJy^\ast JU_t(a_n\Omega)) & = \sum_{s=0}^k\sum_{r=0}^m e_{M_i}(\ell(\xi_1)\cdots\ell(\xi_s)\ell(J\xi_{s+1})^\ast\cdots\ell(J\xi_k)^\ast r(\eta_m)\cdots r(\eta_{r+1})r(J\eta_{r})^\ast\cdots r(J\eta_1)^\ast U_t(a_n\Omega)) \; ,
\vert p,\sigma,q,\lambda \colon f(p) \rangle = exp\biggl[-e\int\sum_{\lambda=1,2}{{dk^+}\over{\sqrt{2k^+}}}\int{{d^2k_\perp}\over{(2\pi)^{3 \over 2}}}\bigl[f(k,\lambda \colon p)a^\dagger(k,\lambda) - f^*(k,\lambda \colon p)a(k,\lambda)\bigr] \biggr]\vert p,\sigma,q,\lambda \rangle\;.
A_{\sc s}^{2}\simeq \frac{1}{7 5 \pi^{2}}\bigg ( \frac{1}{M_{P l}^{2}\sqrt{1 + 4 \alpha \Lambda_{b}/ 3 M^{5}}}\bigg )^{3}{\frac{V^{3}}{V^{\prime 2}}}\left. \left[ 1 + \frac{V}{2 \Lambda_{2}^{+}}\right]^{3}\right|_{k = a H}.
\begin{array}{cl}\null \Vert f(t, x^p(\cdot,S,a)_t, u(t,S,a)) - f(t,x^{p-1}(\cdot,S,a)_t, u(t,S,a)) \Vert \leq c_3 \cdot \Vert x^p(\cdot,S,a)_t -x^{p-1}(\cdot,S,a)_t \Vert \leq c_3 \frac{(c_3 t)^{p-1}}{(p-1)!}c_2 \cdot \Vert a \Vert = \frac{c_3^p t^{p-1}}{(p-1)!}c_2 \cdot \Vert a \Vert\end{array}
\begin{gathered}A_{n_1 n_2}=\sum_{p=0}^\infty(n_1-n_2+2p+1)(\varphi_{n_1+p+1}\varphi_{n_2-p}+\varphi_{n_1+p}\varphi_{n_2-p-1}) -(n_1-n_2)\varphi_{n_1}\varphi_{n_2}-2\sum_{p=1}^\infty(n_1-n_2+2p)\varphi_{n_1+p}\varphi_{n_2-p}\end{gathered}
B(\theta)={\sqrt{2}\over \sqrt{\cosh 2\lambda\theta}}\exp\left[-\int_{-\infty}^\infty{d\omega\over \omega}e^{i(2P-1)\omega\theta/\pi}~~{2\sinh [(P-3/2)\omega]+\sinh[(P-1/2)\omega]\over4\sinh[\omega]\cosh[(P-1/2)\omega)]}\right]
\tilde{S}^{\prime}\ [ X^{\mu}, b_{\xi}] = - | P_{y}| \int d \xi \e^{- \phi}\sqrt{| g_{\xi \xi}|}+ P_{y}\int d \xi \left( C^{( 1 )}{}_{\xi}-{\textstyle \frac{m}{2}}\ ( 2 \pi \alpha^{\prime}) \b_{\xi}\right) \, ,
\frac{\partial^2 \varphi_E (\tau)}{\partial \tau^2}+ 3\Bigl(\frac{\partial a(\tau)/ \partial \tau}{a(\tau)}\Bigr)\frac{\partial \varphi_E (\tau)}{\partial \tau}- \frac{\delta^2V(\hat{\phi})}{\delta \hat{\phi}^2}\varphi_E (\tau) = 0.
D_{x_j}(E_{ab}) = D_{x_j}\left[g_{a,b}-\frac{g_{a,z}g_{,b}}{g_z}\right] = g_{aj,b}-\frac{g_{aj,z}g_{,b}}{g_z}-\frac{g_{a,z}g_{j,b}}{g_z}+\frac{g_{j,z}g_{a,z}g_{,b}}{g_z^2}= -\frac{g_{a,z}}{g_z}E_{jb}+E_{l,b}D_{p_l}g_{aj}.
Q_{n,\alpha}(r) = \frac{1}{\widehat{Z}_{n,\alpha}}\left( \frac{n-1}{n}\right)^{(n-1)(n+1+\alpha)}\int_{0}^\infty y^\alpha e^{-n y}Z_{n-1,\alpha}\left(\frac{n}{n-1}y;r \right) dy,
|\left(\frac{\cos(2(v_{1}+v_{2}+v_{3}+v_{4}))-1}{2r^{2}}\right)\left(\sin(2Q_{\frac{1}{\lambda(t)}})\left(\cos(2v_{5})-1\right)+\cos(2Q_{\frac{1}{\lambda(t)}})\sin(2v_{5})\right)| &\leq \frac{C (v_{1}+v_{2}+v_{3}+v_{4})^{2}}{r^{2}}\left(\frac{r \lambda(t) v_{5}^{2}}{r^{2}+\lambda(t)^{2}}+ |v_{5}|\right)
\mathfrak{H}:=\left(\begin{array}{ccccccc}H_{2}H_1 H_0 & H_2 H_1 H_0 & H_2 H_1 H_0 & H_2 H_1 H_0 & H_2 H_1 H_0\end{array}\right) \in\mathbb F^{5\times 14}.
N_{T}=\left[\begin{array}{ccccc}A_0 T 0 \cdots 0 -A_2 \, T_2/ (\gamma T_0) A_1 A_0 \cdots 0 0 A_2 A_1 \ddots 0 0 \vdots \vdots \vdots \vdots \vdots 0 0 \cdots A_1 A_0 0 0 \cdots A_2 A_1 - A_2T_1/\gamma T_0 \end{array}\right]
Z(T) =\lim_{N \rightarrow \infty}e^{ihJT}\prod\limits_{j=1}^N\int\limits_{\rm PBC}{{d\mu \left({\xi_j^*,\xi_j}\right)}\exp\left\{{2J{\ln\left({{{1+e^{-i\epsilon h}\xi_j^*\xi_{j-1}}\over{1+\xi_j^*\xi_j}}}\right)}}\right\}}\quad,
(T_{s_k})^* = \big(-t^{-\frac12}+(1+s_k)t^{-\frac12}\Delta_{k+1,k})\big)^* = -t^{\frac12}+ t^{\frac12}\overline{\Delta_{k+1,k}}(1+s_k^*) &= -t^{\frac12}+ t^{-\frac12}\Delta_{k+1,k}\Big(1+\frac{\Delta_{k,k+1}}{\Delta_{k+1,k}}s_k\Big) = -t^{\frac12}+ (1+s_k)t^{-\frac12}\Delta_{k+1,k}= T^{-1}_{s_k}.
\big(A_{-\alpha c^2/2 , \alpha c^2/2}- (\lambda + c^2/2) \big)^{-1}= \left( A_{0}-(\lambda +c^2/2)\right)^{-1}&+ \Phi_{\lambda + c^2/2}\left( I - \alpha c^2 M_3 \mathcal{C}_{\lambda + c^2/2}\right)^{-1}\alpha c^2 M_3 \Phi_{\overline{\lambda}+ c^2/2}^{\ast}.
R^{[i,j]}\ge \log_2 \left( 1+ \frac{\frac{\left\|\mathbf{b}_{j}^{[i]}\right\|^2}{1+ \frac{(S-1)^4\alpha^2}{1-(S-1)\alpha^2}}}{\frac{S}{\mathsf{SNR}}+\sum_{k\neq i}^{K}\sum_{s=1}^{S}\left|{\mathbf{f}_{k}^{[i,j]}}^{\mathsf{H}}\mathbf{v}^{[k,s]}\right|^2}\right).
{\rho}_{(1), \infty}^{(\rm cJ)}(x;\beta,p,q) |_{\beta = 2 \atop q=0}={\pi^2 |x| \over 2}\Big ( (J_{p-1/2}(\pi x))^2 + (J_{p+1/2}(\pi x))^2 -{2 p \over \pi x}J_{p-1/2}(\pi x) J_{p+1/2}(\pi x) \Big ).
\rho \circ l = r \circ \rho \rho(D_X(a)) = D^r_X(\rho(a)) l([a,b]) = [a,l(b)] - D_{\rho(b)}(a) D_X([a,b]) = [a,D_X(b)]+[D_X(a),b] +D_{[\rho(b),X]}(a) - D_{[\rho(a),X]}(b),
D_{r-1}^{2}\sum_{k=0}^{r-1}\frac{P_{k}(x)P_{k}(y)}{D_{k}D_{k-1}}= \frac{P_{r}(x)P_{r-1}(y) - P_{r}(y)P_{r-1}(x)}{x-y}\ , \ \D_{-1}:= 1 \ , \x\neq y \ ,
\begin{aligned}\|J_2\|_{L_2 (Q_r)}\le N (d) \sum_{k = 2}^{\infty}2^{-k}\big(|(-\Delta_x)^{1/3}u_{\varepsilon}- ((-\Delta_x)^{1/3}u_{\varepsilon})_{Q_{1, 2^k}}|^2\big)^{1/2}_{Q_{1, 2^k}}.\end{aligned}
J(r) = \frac{1}{1+\alpha^2}\int_0^\pi\int_0^\pi\frac{(1+\alpha^2 -\alpha^2 \cos(\theta_1) - \cos(\varphi_1) ) \sin(\theta_1)^{m-2}\sin(\varphi_1)^{n-2}}{( r^2 + 1 - 2r\cos(\theta_1) + \alpha^2 ( r^2 + 1 - 2 r \cos(\varphi_1))^{\frac{N+s}{2}}}d\theta_1 d\varphi_1 .
\Bigl|\sum_{k=0}^{N_n-1}U_\omega\circ F_\omega^k\Bigr|_{L^p(\Delta)}\le |r|_a^{1/p}Biggl|\max_{j\le n}\Bigl|\sum_{k=0}^{j-1}U_\omega\circ F_\omega^k\Bigr|\Biggr|_{L^{bp}(Y\times S^1)}\ll n^{1/2}.