harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_0^1 d x_3\int_0^{x_3}d x_2 x_3^{2 \alpha' \alpha_{13}}(1-x_3)^{2 \alpha' \alpha_{34}}{x_2}^{2 \alpha' \alpha_{12}}(1-x_2)^{2 \alpha' \alpha_{24}}(x_3-x_2)^{2 \alpha' \alpha_{23}}\cdot\left\{{\kappa(x_2,x_3) \atop \lambda(x_2,x_3)}\right\},
Y_{3}^{1}=-\frac{1}{r^{3}}[\frac{7}{4\pi}\cdot\frac{3}{16}]^{1/2}(5z^{2}-r^{2})(x+iy)=-[\frac{7}{4\pi}\cdot\frac{3}{16}]^{1/2}(5cos^{2}\theta-1)(sin\theta e^{i\varphi})
\frac{\partial l}{\partial\mu}=\frac{\sum_{i=1}^{n}(x_{i}-\mu)}{\sigma^{2}}-\frac{2}{\sigma^{2}}\sum_{i=1}^{n}\frac{x_{i}e^{\frac{-2\mu x_{i}}{\sigma^{2}}}}{1+e^{\frac{-2\mu x_{i}}{\sigma^{2}}}}
\mathbf{y}=\mathcal{T}_{\mathbf{A}^{(1)},\mathbf{A}^{(2)}}\left(\mathbf{x}\right)\Leftrightarrow\mbox{Prod}\left(\mathbf{y}^{\top},\mathbf{y}\right)=\mbox{Prod}\left(\mathbf{x}^{\top},\mathbf{x}\right)
\forall j \in \left[ k+1 \right], T_j = \left\{x \in H : \sum_{i=1}^k \pi_j x_i = 0 \land \forall y \in T_1 \cup \ldots \cup T_{j-1}, \left| x \cap y \right| \leq k-2 \right\}& T = T_1 \cup T_2 \cup \ldots \cup T_{k+1}
\mathcal{Q}^\delta_s(h) =\iint_{(\mathbb{R}^+)^2}\delta^{2s}(x,y)h(x)h(y) \frac{dx dy}{\delta(x,y)}&=\int_{I(h)}\int_{I(h)}\delta^{2s}(x,y) h(x)h(y) dx dy &=\int_{I_-(h)}\int_{I_-(h)}+\int_{I_+(h)}\int_{I_+(h)}+\int_{I_-(h)}\int_{I_+(h)}+\int_{I_+(h)}\int_{I_-(h)}&= I_1+I_2+I_3+I_4.
\gamma_{{5}}=\left (\begin{array}{cc}\sigma_{3}0 \noalign{\medskip}0 \sigma_{3}\end{array}\right )\ ,\quad \Lambda^{\pm}=\frac{1}{2}(1\pm \gamma^{0}\gamma^{3})\ ,\quad \gamma^{0}\gamma^{3}=\left (\begin{array}{rr}{\bf 1}0 \noalign{\medskip}0{\bf -1}\end{array}\right )\ .
\|\sigma(u)\|_{\gamma(H,H^{a,r})}\leq \begin{cases}C\big(1+\|u\|_{H^{a,r}}\big), a\in[0,1], C\big(1+\|u\|_{H^{a,r}}+\|u\|_{H^{1,ar}}^a\big), a>1.\end{cases}
\displaystyle\sum\limits_{e=1}^{l_m}\sum\limits_{f=1}^{l_n}b^{st}_{ef}p^{m}_{\tilde{e}e}p^{n}_{\tilde{f}f}=\sum\limits_{e=1}^{l_m}\sum\limits_{f=1}^{l_n}b^{ef}_{\tilde{e}\tilde{f}}p^{m}_{es}p^{n}_{ft},
\mu^\ast(Z|C) =\mu^\ast(Z|C\sqcap V|A\sqcap W|B )+\mu^\ast(Z|C\sqcap V|A\sqcap (W|B)^\sqsubset) +\mu^\ast(Z|C\sqcap (V|A)^\sqsubset\sqcap W|B )+\mu^\ast(Z|C\sqcap (V|A)^\sqsubset\sqcap (W|B)^\sqsubset).
2(x-y)\cdot(\widetilde\mu(x)-\widetilde\mu(y))+(p_1-1)\cdot (\widetilde\sigma(x)-\widetilde\sigma(y))^2 &\qquad=2(y-x)\cdot(\widetilde\mu(y)-\widetilde\mu(x))+(p_1-1)\cdot (\widetilde\sigma(y)-\widetilde\sigma(x))^2,
\widehat \mu( \alpha) \propto \prod_{e \in V(\widehat G)}e^{w_e \alpha_e}\prod_{x\in V(G)}\left(\mathbf{1}_{\{\sum_{e: e\ni x}\alpha_e = 1\}}+ e^{\nu_x}\mathbf{1}_{\{\sum_{e: e\ni x}\alpha_e = 0\}}\right), \alpha \in \{0, 1\}^{V(\widehat G)}.
\Theta^1 = \frac{\sqrt{\rho(e^{-4\psi}\Omega_t^2 + 4 \psi_t^2)}}{\sqrt{2}}\bigg( dt + \frac{e^{-4\psi}\Omega_t \Omega_\rho + 4\psi_t \psi_\rho}{e^{-4\psi}\Omega_t^2 + 4 \psi_t^2}\, d\rho \bigg), \Theta^2 = \frac{\sqrt{2\rho}\, e^{-2\psi}(\Omega_t \psi_\rho - \psi_t \Omega_\rho)}{\sqrt{e^{-4\psi}\Omega_t^2 + 4 \psi_t^2}}\, d\rho.
|K| \mathtt{f}^n_K\bar{\mathtt{f}}^{n+1/2}_K(\xi)+\Delta t_n \sum_{L\in\mathcal{N}(K)}\mathtt{f}^n_K\bar a_{K\to L}^n(\xi)=-|K|\Delta t_n\,\mathtt{f}^n_K \partial_\xi m^n_K(\xi),
{\cal D}^{(n)}_{q,t:**}(q^{-1/2})=\sum_{I\subset \{1,\dots,n\}}(-1)^{|I|}t^{|I|(|I|-1)/2}\prod_{1\le i\le n}z_i^{-1}\prod_{i\in I,j\notin I}\frac{z_j-t z_i}{z_j-z_i}\prod_{i\in I}T_i^{1/2}\prod_{i\notin I}T_i^{-1/2},
\begin{array}{lll}E^+_{\pm}(z)E^-_{\pm}(z)=E^-_{\pm}(z)E^+_{\pm}(z)=1 begin{align*}0.2cm] E^+_+(z)E^{\mp}_+(z)=E^{\mp}_+(z)E^+_+(z) begin{align*}0.2cm] E^-_+(z)E^{\mp}_{+}(z)=E^{\mp}_{+}(z)E^-_+(z) begin{align*}0.2cm] \partial_z(E_+^{\pm}(z)E_-^{\pm}(z))=\mp E_-^{\pm}(z)E_+^{\pm}(z)\sum\limits_{n\neq0}\frac{H(n)}{k}z^{-n-1}\end{array}
\sigma(\dot{q}_{i}\odot_{k}\dot{q}_{i})Z=\sum\limits_{j=1,\textrm{}j\neq i}^{n}\frac{m_{j}\left(Z-\sigma(q_{i}\odot_{k}q_{j})Z\right)}{(\sigma-\sigma(q_{i}\odot_{k}q_{j})^{2})^{\frac{3}{2}}},\textrm{}i\in\{1,...,n\}.
a_{m}^{\prime}\in a_{m}+I^{d}a_{j}^{\prime}=0\in I^{d+1}a_{i}^{\prime}\in I^{d}+I^{d}=I^{d}m<i\leq j a_{i}^{\prime}\in I^{d+1}+I^{d+d}=I^{d+1}i>j
c_1(n,k)=\sum_{i=0}^{k-1}{k\choose i}\sum_{j_1+j_2+\cdots+j_{k-i}=n-i}f(j_1)f(j_2)\cdots f(j_{k-i}) =\sum_{i=0}^{k-1}{k\choose i}a^{k-i}(a-1)^{n-2j+i}\sum_{j_1+j_2+\cdots+j_{k-i}=n-i}1 =\sum_{i=0}^{k-1}a^{k-i}(a-1)^{n-2k+i}{k\choose i}{n-k-1\choose k-i-1}.
\frac{\partial\nu_{s,\tau}}{\partial s}=\sum_{j=1}^{\dim\mathfrak{k}}\tilde{X}_{j}^{2}\nu_{s,\tau};\quad\frac{\partial\nu_{s,\tau}}{\partial u}=\sum_{j=1}^{\dim\mathfrak{k}}\tilde{X}_{j}\tilde{Y}_{j}\nu_{s,\tau}.
H_{1}= \exp ( - 4 ( 1 - y ) G ) [ ( - 4 ( 1 - y ) \partial_{\mu}\partial_{\mu}G ) [ - \frac{1}{1 2}- \frac{1}{8}( r - s ) ] - \frac{i}{2}( 1 - y ) \partial_{\mu}\partial_{\mu}F ( r + s ) ] ,
\frac{r\alpha^{\frac{4}{n-2}}K(a)}{4n(n-1)k}= \frac{\alpha^{\frac{4}{n-2}}J(u)}{4n(n-1)}+O(\Vert a \Vert) = \alpha^{\frac{4}{n-2}}c^{\frac{2}{n}}_{0}+O(\Vert a \Vert +\Vert v \Vert)+o_{\frac{1}{\lambda}}(1) = 1+O(\Vert a \Vert +\Vert v \Vert)+o_{\frac{1}{\lambda}}(1),
C=\textsf{a}dT^2+2\textsf{b}dTdX+(2\textsf{b}-\textsf{a})dX^2+(\textsf{b}-\textsf{a})dY^2+(\textsf{b}-\textsf{a})dZ^2+(\textsf{b}-\textsf{a})dW^2.
\Gamma_{N_v}(a,b)=[v^*, [v,a^*]]b-[v^*, [v,a^*b]]+a^*[v^*, [v,b]] \phantom{\Gamma_{N_v}(a,b)}=[v^*, [v,a^*]]b-[v^*, [v,a^*]b+a^*[v,b]]+a^*[v^*, [v,b]] \phantom{\Gamma_{N_v}(a,b)}=[v^*, a]^*[v^*, b]+[v,a]^*[v, b] \phantom{\Gamma_{N_v}(a,b)}=\Gamma_{v^*}(a, b)+\Gamma_v(a, b),
\delta \left( p \right) \star \delta \left( x \right) = \frac{1}{\pi \left| \theta \right|}\exp \left( - \frac{2 i x p}{\theta}\right) , \quad \delta \left( x \right) \star \delta \left( p \right) = \frac{1}{\pi \left| \theta \right|}\exp \left( \frac{2 i x p}{\theta}\right) .
+2\sum_{j=1}^m\alpha_{k,j}\langle x_j-x_j^k,\varsigma_j^k\rangle+H_{k,\tau}(\Vert F(v^k,x)\Vert^2+\Vert x\Vert^2\epsilon_{k,\max}^2)\alpha_{k,\max}^2-CA_{k,\tau}\sum_{j=1}^m\left(g^+_{\omega_{k,j}}(j|x^k_j)\right)^2,
\left\Vert f \right\Vert_{S_p(G)}=\left\Vert f \right\Vert_1 +\left(\sum_{[\pi]\in\widehat{G}}d_\pi\bigl\Vert \widehat{f}(\pi)\bigr\Vert_{S^p(H_\pi)}^p\right)^{1/p}
T_{m,n}=f(m_1+n_1)+g(m_2+n_2)-f(m_1+n_1+2)-g(m_2+n_2) &\quad-f(m_1+n_1)-g(m_2+n_2+2)+f(m_1+n_1+2)+g(m_2+n_2+2) &=0,
\Phi^{\Delta}= \left( A_{\alpha \beta}^{( \sigma )}, \eta_{\alpha}^{\; ( \sigma )}, C^{( \sigma )}\right) ; \; \Phi_{\Delta}^{*}= \left( A_{\; ( \sigma )}^{* \alpha \beta}, \eta_{( \sigma )}^{* \alpha}, C_{( \sigma )}^{*}\right) ,
\mathcal{C}_{1,j}^{J}=\frac{1+2x_{1}}{2+2x_{1}}{\mathcal{A}}_{1}^{J\setminus j}\tilde{\mathcal{D}}_{1,j}^{J}+\frac{1+2x_{1}}{2x_{1}}\tilde{\mathcal{D}}_{1}^{J\setminus j}{\mathcal{A}}_{1,j}^{J}\,,
p(h_{p}^{3)}=1|h^{(2)})=g(\sum_{l=1}^{F_{2}^{m}}W_{lp}^{(3m)}h_{l}^{(2m)}+\sum_{l=1}^{F_{2}^{t}}W_{lp}^{(3t)}h_{l}^{(2t)}+b_{p}^{(3)})
\langle \, : a_{n}^{i}\, ( a_{m}^{j})^{\dagger}: \, \rangle_{\beta , \gamma}= \langle \, : ( a_{m}^{j})^{\dagger}\, a_{n}^{i}: \, \rangle_{\beta , \gamma}= \frac{\delta^{i j}\, \delta_{n m}}{e^{\epsilon_{n}}- 1}\, ,
\sum_{b\in \{0,1\}}\nu_{\pmb{x}}[\pmb{a}b]= \frac{\alpha^{q(\pmb{a})-1}}{(1+\alpha)^{q(\pmb{a})}}\frac{2}{2^{n+1-q(\pmb{a})}}=\nu_{\pmb{x}}[\pmb{a}].
h(0) =\mu (\omega,\mathbf{c})=S_{\omega,\mathbf{c}}(\Phi), h'(0) =-\frac{1}{\sqrt{\omega}}(2\omega Q(\Phi)+\mathbf{c}\cdot \mathbf{P}(\Phi)), h''(0) =\frac{3-d}{\omega}(2\omega Q(\Phi)+\mathbf{c}\cdot \mathbf{P}(\Phi)).
H^0(Z,\Omega_Z^1)\cong \bigoplus_{k\in G}H^1\left(\mathbb P^2,\mathcal J\left(\mathbb P^2,\sum_{j=1}^r\left\{\frac{km_j}{d}\right\}C_j\right)\left(\sum_{j=1}^r\left\{\frac{km_j}{d}\right\}d_j-3\right)\right).
\mathrel{\phantom{=}}x^k \left( \frac{l}{l+2}g'(x) y^2 + h'(x) \right) y^l dx - kx^{k-1}\left( \frac{4}{l+4}y^4 + \frac{2}{l+2}g(x) y^2 \right) y^l dx &= d(S_{l,k}), \ \mathrm{where}\S_{l,k}:= - x^k \left( \frac{4}{l+4}y^{l+4}+\frac{2}{l+2}g(x) y^{l+2}\right)
\mu_{c}=-\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}\left(\lambda_{c}a_{c}+\hat{A}_{1}\hat{Q}_{1}^{-1}\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]\right).
E(z,w) =y^w + \phi(w)y^{1-w}+ \frac{2\pi^w \sqrt{y}}{\Gamma(w)\zeta(2w)}\sum_{m \neq 0}\vert m \vert^{w-\frac{1}{2}}\sigma_{1-2w}(\vert m \vert) K_{w-\frac{1}{2}}(2\pi \vert m \vert y)e^{2\pi i m x},
v_m^{(i)}= \frac{1}{2\sqrt{a}}\left( (z_0^{(i)}\sqrt{a}+x_0^{(i)}\sqrt{c})(s+\sqrt{ac})^m + (z_0^{(i)}\sqrt{a}-x_0^{(i)}\sqrt{c})(s-\sqrt{ac})^m \right) w_n^{(j)}= \frac{1}{2\sqrt{b}}\left( (z_1^{(j)}\sqrt{b}+y_1^{(j)}\sqrt{c})(t+\sqrt{bc})^n + (z_1^{(j)}\sqrt{b}-y_1^{(j)}\sqrt{c})(t-\sqrt{bc})^n \right).
\frac{\delta}{\delta\{D_{\alpha}\}_nB_{\mu\nu}^c}\Delta_{0 b}^{\rho}(x)=\{D_{\alpha}\}_n gF_{\mu\nu}^c(x)X_{\rho}( A,gF)^{-}+ \{D_{\alpha}\}_n\epsilon^{\mu\nu\lambda\eta}gF_{\lambda\eta}^c(x)X_{\rho}( A,gF)^{+}
\Delta_0\frac{d^2}{d\epsilon_1d\bar\epsilon_2}\vert_{\epsilon=0}((\overline{f^{\epsilon\nu^{\epsilon\mu}}})^Tf^{\epsilon\nu^{\epsilon\mu}})\circ\Phi_1^{\epsilon\mu}=y^2(-\partial\mu_1\bar\nu_2^T-\bar\partial\bar\mu_2\nu_1+[\nu_1,\bar\nu_2^T]),
( G_{o}( k )^{a b}) = \left( \begin{array}{c c}{G_{0}( k )}&{0}\\{0}&{\widetilde{G}_{0}( k )}\\ \end{array}\right) = \left( \begin{array}{c c}{\frac{1}{k^{2}- m^{2}+ i \epsilon}}&{0}\\{0}&{\frac{- 1}{k^{2}- m^{2}- i \epsilon}}\\ \end{array}\right) .
\Pi_{i}(p^{2}_{i}-m^{2}_{i})FT\frac{\delta^{4}}{\delta J^{4}}\begin{array}{c}lim \varepsilon \rightarrow 0\end{array}<<[i\delta \alpha \int d^{4}x\phi \partial^{2}_{0}\phi ]>>\mid_{_{J=0;\alpha \rightarrow \alpha^{-}_{0}}}
m(f(C_{f,\delta}\cap I)) \leq \sum_{n_1=1}^{N_1}\sum_{n_2 = 1}^{N_{2}(n_1)}m (f(I_{n_1,n_2})) \leq \sum_{n_1=1}^{N_1}(1-K) m(f(I_{n_1})) \le (1-K)^2 m(f(I))
(\overrightarrow{W}_{j+1}^\ell)^2 (\overline{W}_{j-1}- \overrightarrow{W}^\ell_j) = (\overrightarrow{W}_{j+1}^\ell)^2 \sum_{i=0}^{\ell-1}\psi_i (W_{j+i-1}- W_{j+i})
M_k:=\max\Big\{\tilde{f}\big(-\|u_k\|_{\infty}\big),\tilde{f}\big(\|u_k\|_{\infty}\big)\Big\}\ge (c_0/2)^{-1}\,\max_{[-\|u_k\|_{\infty},\|u_k\|_{\infty}]}\tilde{f},
(u_{\mu^*},v_{\mu^*}):=\Big(\Big(\frac{\mu^*}{\mu}\Big)^{\frac 12}u\Big(\Big(\frac{\mu^*}{\mu}\Big)^{\frac 12}x\,,\,\frac{\mu^*}{\mu}t\Big),\frac{\mu^*}{\mu}v\Big(\Big(\frac{\mu^*}{\mu}\Big)^{\frac 12}x,\frac{\mu^*}{\mu}t\Big)\Big)
H(r)=\sum_{i=1}^{m}\sum_{k=0}^{\infty}\textnormal{Res}_{z=\hat{d}_{ik}}\left[\frac{\prod_{j=1}^{m}\Gamma(d_{j}+\delta_{j}z)\prod_{j=1}^{n}\Gamma(1-c_{j}-\gamma_{j}z)}{\prod_{j=m+1}^{\mu}\Gamma(1-d_{j}-\delta_{j}z)\prod_{j=n+1}^{\nu}\Gamma(c_{j}+\gamma_{j}z)}r^{-z}\right],
\left\{\mu \lambda_1, \mu \lambda_2 \right\}\cdot \left\{\lambda_1^2, \lambda_1 \lambda_2, \lambda_2^2 \right\}=\left\{\mu \lambda_1^3, \mu \lambda_1^2\lambda_2, \mu \lambda_1\lambda_2^2, \mu \lambda_1^2\lambda_2, \mu \lambda_1\lambda_2^2, \mu \lambda_2^3 \right\}
\hat{W}= i \hat{\zeta}m \sum_{a = 1}^{N}[ \hat{\theta}_{a}^{2}, \hat{\theta}_{a}^{1}] = \hat{\zeta}m \left( \frac{N}{2}- \hat{n}\right) , \; \; \hat{n}= \sum_{a = 1}^{N}\hat{n}_{a}.
E\big[2^{\rho_0(d)}1_{\{\tau_1(d)=+\infty\}}\big]\leq \sum_{l=1}^{+\infty}2^lP(2\leq \theta(d)<+\infty)^{l-1}\leq 2\sum_{l=0}^{+\infty}(\frac{2C}{d^2})^l=\frac{2d^2}{d^2-2C}<4.
G(n_1,\ldots,n_r)=\sum_{d_1+\cdots +d_r=d}\frac{1}{d_1!\cdots d_r!}e_R(\mathcal I(1)^{[d_1]},\ldots, \mathcal I(r)^{[d_r]};M)n_1^{d_1}\cdots n_r^{d_r}.
\frac{\#\{P_j\in\mathfrak{P}_{\Gamma(q)}\colon \gamma_jP_j\gamma_j^{-1}=P_{0,l}\}}{[\Gamma_0:\Gamma(q)]}=\frac{\#(\Gamma(q)\backslash\Gamma_0/(\Gamma_0\cap P_{0,l}))}{[\Gamma_0:\Gamma(q)]}\leq \frac{1}{[\Gamma_0\cap N_{P_{0,l}}:\Gamma(q)\cap N_{P_{0,l}}]}.
\langle\langle\delta X\rangle\rangle =\delta X\bigg(\varphi^i+ i\hbar(\Gamma^{''-1})^{ij}\frac{\delta_l}{\delta\varphi^j},\;\psi^\alpha,\; \phi^*_A;\;\frac{i}{\hbar}\frac{\delta_l\Gamma}{\delta\varphi_i}, \rule{4cm}{0cm}
\langle \, a_{\omega}^{R \dag}a_{\omega^{\prime}}^{R}\, \rangle \,_{\mathrm{{i n}}}= \frac{1}{2 \pi}\, \int_{0}^{\infty}\, d \omega^{\prime \prime}\, \tilde{\beta}_{\omega^{\prime \prime}\omega}^{R *}\, \tilde{\beta}_{\omega^{\prime \prime}\omega^{\prime}}^{R}\, .
|F_m^l(\beta_1)-F_m^l(\beta_2)|_0 \le c\sum_{n=m}^{N(\beta_2)}2^{2n}c_0^{\frac{n-m}{2}}\Bigg(|J_n^{l+1}(\beta_1)-J_n^{l+1}(\beta_2)|_0 &\quad+\beta_1^{-\frac{1}{2}}\sum_{a=1}^2\|J_n^{l+1}(\beta_a)\|_{0,0}+\beta_1^{-1}\sum_{a=1}^2\|J_n^{l+1}(\beta_a)\|_{0,1}\Bigg).
[\begin{matrix}0&A^{*}\\A&0\end{matrix}]=(\frac{1}{\sqrt{2}}[\begin{matrix}V&V\\U&-U\end{matrix}])\cdot[\begin{matrix}\Sigma&0\\ 0&-\Sigma\end{matrix}]\cdot(\frac{1}{\sqrt{2}}[\begin{matrix}V^{*}&U^{*}\\V^{*}&-U^{*}\end{matrix}])
\begin{array}{rcl}\partial_t U(t,x) = \displaystyle\frac{1}{t^2}Z_* (t,-tx) -\frac{1}{t}\partial_t Z_* (t,-tx)+\frac{x}{t}\partial_x Z_* (t,-tx) & &\leq \displaystyle\frac{-1}{t}U(t,x) -\frac{1}{t}\underline{H}\big(\partial_x Z_* (t,-tx)\big)+\frac{x}{t}\partial_x Z_* (t,-tx) \end{array}
r^n \lambda^{-[\frac{n-k \log r}{\log \lambda}]-1}\cdot O_n^{-1}T R^n T^{-1}= \lambda^{-1}\cdot r^k \lambda^{\lbrace \frac{n-k \log r}{\log \lambda}\rbrace}\cdot O_n^{-1}T R^n T^{-1}
[u,e_2]+\lambda J[u,e_2] + J[u,Au+Be_1+Ce_2]-\lambda [u,Au+Be_1+Ce_2] = -\theta e_2-\lambda\theta Je_2 -BJe_1-C\theta Je_2 + \lambda Be_1+\lambda C\theta e_2 = (-\lambda\theta B-BY-C\theta B+\lambda B)e_1+(-\theta-\lambda\theta C -BZ-C^2\theta+\lambda C\theta)e_2+ &\qquad+(-\lambda \theta A-XB-C\theta A)u
(\varphi(\widehat{F}) \delta_G \beta) (k_1,k_2)(x_1,(a_2,x_2)) = \widehat{F}(k_1,k_2)({}^{x_1}a_2) \ \widehat{F}(k_1,k_2)(F(x_1,x_2)) \epsilon(k_1,k_2,x_1x_2)\ \epsilon(k_1,k_2,x_1)^{-1}
G_{j}(-rx_{j};r\vec{x},\vec{s}) = r^{-\frac{\sigma_{3}}{4}}\begin{pmatrix}1 0 - i\frac{x_{1}}{2}\sqrt{r}1\end{pmatrix}R(-x_{j})E_{-x_{j}}(-x_{j})\begin{pmatrix}\Psi_{j,11}\Psi_{j,12}\Psi_{j,21}\Psi_{j,22}\end{pmatrix}.
B_1({\bf z}) = \frac{z_1}{2z_0^{3/2}}, B_2({\bf z})=\frac{z_1^2}{2z_0^3}-\frac{z_2}{4z_0^2}, B_3({\bf z}) = \frac18 \frac{z_3}{z_0^{5/2}}-\frac{15}{16}\frac{z_1 z_2}{z_0^{7/2}}+ \frac{35}{32}\frac{z_1^3}{z_0^{9/2}}
A_{\mu}dx^{\mu}=\frac{1}{2gr}\left\{\begin{array}{l}\displaystyle{\tau_{\phi}^{n}\left[ H_{1}dr+(1-H_{2})rd\theta \right]}\displaystyle{-n\left[ \tau_{r}^{n}H_{3}+\tau_{\theta}^{n}\left( 1-H_{4}\right) \right]r\sin \theta d\phi}\end{array}\right\}
\lim_{\tau \rightarrow +\infty}\left\Vert \exp \left[ \tau A\right] \mathsf{p}^{\ast}-\left( \mathsf{p}^{\ast}\mathsf{,\hat{q}}_{\mathsf{1}}\right)_{2}\mathsf{\hat{p}}_{\mathsf{1}}\right\Vert_{2}=0
v_{1}(t,r) = \int_{t}^{\infty}ds\frac{\lambda''(s)}{r}\int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(1+r^{2}+\rho^{2})^{2}-4 r^{2}\rho^{2}}}\right)
H_{1,2}^2=\int_{\frac{3\pi}{2}}^{2\pi}\int_{\frac{1}{2}}^{\gamma}\frac{dr d\theta}{|\gamma-r|^\tau|re^{i\theta}+1|^q}\leq\int_{\frac{3\pi}{2}}^{2\pi}\int_{\frac{1}{2}}^{\gamma}\frac{dr d\theta}{|\gamma-r|^\tau}\leq\frac{\pi}{2(1-\tau)},
[\Gamma_\mu , \Gamma_\nu{]_\lambda}^\kappa \equiv(\Gamma_\mu \Gamma_\nu - \Gamma_\nu \Gamma_\mu{)_\lambda}^\kappa={\Gamma_{\mu\lambda}}^\sigma{\Gamma_{\nu\sigma}}^\kappa-{\Gamma_{\nu\lambda}}^\sigma{\Gamma_{\mu\sigma}}^\kappa.
+i\gamma_{tx}\frac{\partial^{2}A}{\partial t\partial x}+i\gamma_{ty}\frac{\partial^{2}A}{\partial t\partial y}-\frac{i}{2}\gamma_{xx}\frac{\partial^{2}A}{\partial x^{2}}-\frac{i}{2}\gamma_{yy}\frac{\partial^{2}A}{\partial y^{2}}+i\gamma_{xy}\frac{\partial^{2}A}{\partial x\partial y}+\cdot\cdot\cdot
|I_{3,c}|\leq \varrho\overline P\int_0^\tau\int_{\Omega_R}\left|\frac{\varrho-\rho}{\varepsilon}\right||s||\Delta\Psi| \leq c\left\|\frac{\varrho-\rho}{\varepsilon}\right\|_{L^2(0,T;L^2(\Omega))}\|s\|_{L^{q_1}(0,T;L^{q_1}(\Omega))}\|\Delta\Psi\|_{L^{q_2}(0,T;L^{q_2}(\Omega))}\leq c(D,T)\varepsilon^{2(1-\frac{1}{q_1}-\frac{1}{q_2})}
(\begin{matrix}t_{i}\\E_{i}\\ \end{matrix})\leftarrow(\begin{matrix}t_{i}\\E_{i}\\ \end{matrix})-(\begin{matrix}1&&0\\ \frac{\dot{M}_{i}(t_{i})}{1-e_{i}cosE_{i}}&&-\frac{1}{1-e_{i}cosE_{i}}\\ \end{matrix})(\begin{matrix}\Delta t_{i}\\ \Delta M_{i}\\ \end{matrix})
I =(x_4x_2x_1, x_4x_3x_1, x_4x_3x_2,x_5x_2x_1, x_5x_3x_1, x_5x_3x_2, x_5x_4x_1, x_5x_4x_2,x_6x_2x_1,\cr \hskip 2emx_6x_3x_1, x_6x_3x_2, x_6x_4x_1, x_6x_4x_3, x_6x_5x_2, x_6x_5x_3,x_6x_5x_4,x_7x_2x_1, x_7x_3x_1,\cr \hskip 2emx_7x_3x_2, x_7x_4x_2, x_7x_4x_3, x_7x_5x_1, x_7x_5x_3,x_7x_5x_4, x_7x_6x_1, x_7x_6x_2, x_7x_6x_4, x_7x_6x_5)
\widetilde{\psi}_{0}^{(n^{2})}\big(x;\omega ,N,-\Vert \psi_{n}\Vert^{2}\big)=E_{n}(\omega) \psi_{0}( x;\omega ,N) -\frac{W(\psi_{n},\psi_{0}\,|\, x)}{\Psi_{n}^{(n)}\big(x;\omega ,N,- \Vert \psi_{n}\Vert^{2}\big)}.
\eta_{y,w}=\sum_{(x,\kappa ,u)\in\eta :x\in D_{y,w}}\delta_{(x,\kappa ,u)}\mbox{\rm \and}\eta_{y,w}^g=\sum_{(x,\kappa ,u)\in\eta :x\in D_{y,w},u\leq u_g}\delta_{(x,\kappa ,u)}.
\Lambda \partial_{\Lambda}{\cal{G}}_{\mathrm{i r}}^{\Lambda , \Lambda_{0}}\{\varphi_{\bf{k}}\}= - \partial_{t}{\cal{G}}_{t}\{\tilde{\varphi}_{\bf{q}}\}+ \hat{\cal{R}}_{s}{\cal{G}}_{t}\{\tilde{\varphi}_{\bf{q}}\}+ \frac{N_{t}}{2}\left[ \eta_{t}+ D \ln Z_{t}\right] \; ,
\Big| \bigotimes_{i=1}^n x_i - \bigotimes_{i=1}^n \hat{x}_i \Big| \leq \sum_{k=1}^n \bigg( \prod_{i=1}^{k-1}|x_i| \bigg) \cdot \big| x_k - \hat{x}_k \big| \cdot \bigg( \prod_{i=k+1}^n \big| \hat{x}_i \big| \bigg)
K^{(1)}_{ij}:=\partial^{2}L/\partial\dot{q}^{i}\partial\dot{q}^{j}, &S_{i}:=-v^{j}\frac{\partial^{2}L}{\partial q^{i}\partial v^{j}}-\frac{\partial^{2}L}{\partial t\partial v^{i}}+\frac{\partial L}{\partial q^{i}}.
\sum_{k,l=1}^n \overline{g(x_k)}\> g(x_l) \> \tilde K_{\pi(x_k,x_l)}(x_k,\{x_l\})\> \tilde\omega_X(\{x_k\})\cdot &\cdot\tilde\alpha(\pi(x_k,x_l))\beta(\pi(x_k,x_l))\tilde\omega_D(\{\pi(x_k,x_l)\})\ge0.
\tilde{S}^{c}( x , y ) = S^{c}( x , y ) \gamma^{D + 1}, \; \; \left( \hat{{\cal P}}_{\mu}\tilde{\gamma}^{\mu}- m \gamma^{D + 1}\right) \tilde{S}^{c}( x , y ) = \delta^{D}( x - y ) ,
F^{(min)}_{ab}(\beta)=\prod_{\scriptstyle x=|a-b|+1 \atop\scriptstyle{\rm step}2}^{a+b-1}F_x^{(min)}(\beta)=\prod_{\scriptstyle x=|a-b|+1 \atop\scriptstyle{\rm step}2}^{h-|a+b-h|-1}F_x^{(min)}(\beta).
0=\sum_{\substack{d_1+\cdots+d_n+e=g-1 d_i\geq 0, e\geq 1}}\prod_{i=1}^{n}Q_{d_i}(a_i) Q_{e+1}(a_{n+1}) \prod_{i=1}^{n}\psi_i^{d_i}\kappa_e
\sum_{\chi \in S_{n,q}}\left( q^{-\frac{n+1}{2}}\sum_{a_1,\dots,a_n \in \mathbb F_q}\chi( 1+ \sum_i a_i x^i) \psi(a_n) \right)^m = \sum_{\substack{\chi \in S_{n,q}\psi_\chi=\psi}}\epsilon_\chi^m = \sum_{\substack{\chi \in S_{n,q}\psi_\chi=\psi}}\mu = q^{n-1}\mu
||\frac{\lambda(t) |v_{5}(t,r) \partial_{r}v_{5}(t,r)|}{r(r^{2}+\lambda(t)^{2})}\vert_{r = R\lambda(t)}||_{L^{2}( R dR)}\leq C \lambda(t) ||\frac{v_{5}(t,r)}{r(r^{2}+\lambda(t)^{2})}||_{L^{\infty}}\cdot ||(\partial_{2}v_{5})(t,\cdot \lambda(t))||_{L^{2}(R dR)}&\leq \frac{C}{t^{21/4}\log^{b-6+\frac{5N}{2}}(t)}
|y_\ell(p, q)| = \bigg| \sum_{s=0}^{M-1}\sum_{t=0}^{N-1}\bar{d}_\ell(s, t) \rho(s, t) e^{-2\pi i ps/M}e^{-2\pi i qt/N}\bigg|,\ell = 1, \dots, L,
w_4([z,i,iz]) = 2\frac{Y(z)-Y(-z)}{x(z)-x(-z)}\cdot\frac{Y(i)-Y(-iz^2)}{x(i)-x(-iz^2)}\cdot\frac{Y((iz)^{-1})-Y(-iz)}{x((iz)^{-1})-x(-iz)}\equiv 2\frac{g(z)-g(-z)}{f(z)-f(-z)}\cdot\frac{g(i)-g(-iz^2)}{f(i)-f(-iz^2)}\cdot\frac{g((iz)^{-1})-g(-iz)}{f((iz)^{-1})-f(-iz)}(\mathrm{mod}\, z^2).
2^{J_{n,l}(\boldsymbol{\alpha})}=\left(\frac{R}{M}\right)^{\left[\sum_{l=1}^d\alpha_l\left(\frac{1}{d}+\frac{1}{2\alpha{*}}-\frac{1}{2\alpha_l}\right)\right]^{-1}}\left(\frac{n}{\log{n}}\right)^{\frac{\alpha^{*}}{\alpha_l(2\alpha^{*}+d)}}.
\begin{array}{cccc}\mathbf{A}[1] =\left[ \begin{array}{c}\mathbf{a}_{11}^T \vdots \mathbf{a}_{M1}^T \end{array}\right] \cdots \mathbf{A}[n_2] =\left[ \begin{array}{c}\mathbf{a}_{1n_2}^T \vdots \mathbf{a}_{Mn_2}^T \end{array}\right]\end{array}
f^{-1}(f(x)+b_i) =g^{-1}(p_ig(x)+q_i), g\circ f^{-1}(y+b_i) =p_ig\circ f^{-1}(y)+q_i, m(y+b_i) =p_im(y)+q_i \textrm{, where}m(y)=g\circ f^{-1}(y).
k_{\delta}(\cdot) =\frac{1}{\left\Vert k_{x}\right\Vert \delta(x,y)}\left( k_{x}\left( \cdot\right) -\frac{k_{xy}k_{y}\left(\cdot\right)}{k_{yy}}\right) m_{xy}(\cdot) =m_{\delta}(\cdot)=\frac{1}{\delta(x,y)}\left(1-\frac{k_{xy}k_{y}\left( \cdot\right)}{k_{yy}k_{x}\left( \cdot\right)}\right) .
{\cal R}\left( \lambda ,\mu \right) =\left(\begin{array}{cccc}\frac{\lambda q-\mu q^{-1}}{\lambda -\mu}0 0 0 0 1 \frac \mu{\lambda -\mu}\left( q-q^{-1}\right) 0 0 \frac \lambda{\lambda -\mu}\left( q-q^{-1}\right) 1 0 0 0 0 \frac{\lambda q-\mu q^{-1}}{\lambda -\mu}\end{array}\right)
I_3\leq \int_{|r|>2|t|}\frac{|F_s(r)|}{|r-t|^{2-\frac1{2s}}}dr+\int_{|r|>2|t|}\frac{|F_s(t)|}{|r-t|^{2-\frac1{2s}}}dr=I_{31}+I_{32}.
\left( \begin{array}{c}{\psi_{1}}\\{\psi_{2}}\\ \end{array}\right) \equiv \left( \begin{array}{c c c c}{1}&{\psi_{1}}&{}&{}\\{0}&{0}&{}&{}\\{}&{}&{1}&{0}\\{}&{}&{\psi_{2}}&{0}\\ \end{array}\right) .
E\{C_{i_1}\cdots C_{i_k}\}= \frac{(\phi \theta)^k}{(1-m/n)^{1-\phi \theta}}\frac{1}{i_1 i_2\cdots i_k}\left(1+O(\lfloor n/i_1 \rfloor\cdots\lfloor n/i_k \rfloor \rho^{\min(i_1,\dots,i_k)})\right),
\lambda_{k}= \ell^2 k^2 + C_{1}+ c_{k}, &\phi_{k}(x) = C_{2}\, a(x)^{-1/4}\sin(k\xi(x)) + O(k^{-1}), &\phi_{k}'(x) = C_{2}\,a(x)^{-3/4}\, k\, \cos(k\xi(x)) + O(1),
\operatorname{Tr}\left\{\mathbf{nMSE}^{(g)}\right\}\triangleq\operatorname{Tr}\left\{\left(\sum_{l=0}^{L_g-1}\mathbf{R}^{(g)}_{code}(l) \otimes\mathbf{SNR}^{(g)}_{mimo}(l)+\mathbf{I}_{TD}\right)^{-1}\right\}+\left(K_g\left(\sum_{l=0}^{L_g-1}r_{g,l}\right)-TD\right)
\langle \varphi,\psi \rangle := \int_{\mathbb{R}^n}\varphi \psi \, dx, \varphi,\psi \in L^2(\mathbb{R}^n), (\varphi,\psi)_{\Omega}:= \int_{\Omega}\varphi \psi \, dx, \varphi ,\psi \in L^2(\Omega).
2 \, \Im m \, \int \frac{d x}{( 2 \pi )^{2}i}\varphi_{\pm}^{,}( x + i \eta ) \log \left( 1 + ( - )^{\delta}e^{i Z_{\pm}( x + i \eta )}\right) = \pm \left( \frac{1}{2 4}- \frac{{\cal Q}_{\pm}^{2}( \mp \infty )}{1 6 \pi^{2}}\frac{p + 1}{p}\right)
\frac{2}{p^{2^n}}= \frac{p^{2^n}- 1}{p^{2^{n+1}}}+ \frac{p^{2^n}+ 1}{p^{2^{n+1}}}\ \ \frac{p^{2^n}\pm 1}{p^{2^{n+1}}}= \frac{p^{2^n}\pm 1}{2}\frac{2}{p^{2^{n+1}}}.
J(x, y) =J(-x, -y) &=J\left(x,\, -\frac{2a_2}{1+a_1}x-y\right)=J\left(-x,\, \frac{2a_2}{1+a_1}x+y\right) &=J\left(-x-\frac{2a_1}{1+a_2}y,\,y\right)=J\left(x+\frac{2a_1}{1+a_2}y,\,-y\right).
{\mathcal A}^{( < )}= \frac{\hbar^{2}}{2 M}\frac{\hat{\lambda}B^{2}}{2}\, \xi \, I_{0}( \xi ) \, I_{0}^{\prime}( \xi ) \stackrel{( a \rightarrow 0 )}{=}E_{\mathrm{( g s )}}\, \times O \left( \xi^{2}\ln \xi \right) \; ;
*^{-1}\frac{\partial}{\partial \xi_i}*=\eta^{i\bar{j}}\bar{\xi}_j, *^{-1}\frac{\partial}{\partial \bar{\xi}_i}*=\eta^{\bar{i}j}\xi_j, *^{-1}\xi_i* =\eta_{\bar{j}i}\frac{\partial}{\partial \bar{\xi}_j}, *^{-1}\bar{\xi}_i* =\eta_{j\bar{i}}\frac{\partial}{\partial \xi_j},
\phi_{m a g}( \eta , z ) ~ \approx ~ \phi_{e l e c}( \eta , z ) ~ \approx ~ \frac{h_{r}}\eta ~ - ~ \frac{p^{2}}{2}\left( h_{r}\eta ~ - ~ \frac{h_{0}\eta^{2}}{3}~ + ~ h_{0}z^{2}\right)
\begin{array}{l l}{{< L_{1}^{\pm}, M_{2}> = R_{1 2}^{\pm}\; , \quad}}&{{\qquad < \tilde{L}_{1}^{\pm}, \tilde{M}_{2}> = R_{1 2}^{\pm}\; ,}}\\{{< L_{1}^{\pm}, \tilde{M}_{2}> = A_{1 2}\; , \quad}}&{{\qquad < \tilde{L}_{1}^{\pm}, M_{2}> = \tilde{A}_{1 2}\; .}}\\ \end{array}
\sum_{v \in V}\left( \prod_{i=1}^k x_i(v)-4 \delta \right) \geq (2^{-k}-4 \delta)N=(2^{-k}-4 \delta)(2^k+\varepsilon)n\geq(1+2^{-k}\varepsilon-2^{k+3}\delta)n\geq n,
\frac{\overline{{I}}_{6}- \overline{{I}}_{4}}{\overline{{I}}_{4}- \overline{{I}}_{2}}={\frac{{. 9 9 5 1 2 2 6 1 4 6 \ldots - . 9 9 5 2 2 0 3 9 4 5 \ldots}}{{. 9 9 5 2 2 0 3 9 4 5 \ldots - . 9 9 5 5 7 5 2 2 1 2 \ldots}}}= . 2 7 5 5 7 \ldots
L_{n-2p}^{(p)}=(n-2p)(\tfrac{2J}{n})^{\tfrac{n-2p}{2}}R_{\tfrac{n-2p}{2}}(y^{(+)};0) =(n-2p) \prod_{l=1}^{\tfrac{n-2p}{2}}\big[\delta^h\d+(\tfrac{\beta}{2}-l)(\tfrac{\beta}{2}+l-1)\tfrac{2J}{n}\big].
\frac{\partial P_p}{\partial t}= \frac{1}{2}\frac{\partial}{\partial\phi}\left( \frac{H^{3/2}(\phi)}{2\pi}\, \frac{\partial}{\partial\phi}\Bigl( \frac{H^{3/2}(\phi)}{2\pi}P_p \Bigr) +\frac{V'(\phi)}{3H(\phi)}\, P_p\right) + 3H(\phi) P_p\ .
\theta(c_s x,\tau_s):=\sum_{k\in\mathbb{Z}}\exp (\pi i k^2 \tau_s + 2c_s\pi i k x)= (\texttt{q}_s;\texttt{q}_s)_\infty\prod_{k=1}^\infty \left(1 + \texttt{q}_s^{k-{\frac 12}}\texttt{t}_s^{x}\right)\left(1 +\texttt{q}_s^{k-{\frac 12}}\texttt{t}_s^{- x}\right)
L_{n}^{\alpha}( x ) = \frac{1}{\pi}e^{\frac{x}{2}}x^{- ( \frac{2 \alpha - 1}{4})}n^{( \frac{2 \alpha - 1}{4})}\cos \big [ 2 ( n x )^{\frac{1}{2}}- \frac{\alpha \pi}{2}- \frac{\pi}{4}\big ] + \mathcal{O}( n^{\frac{2 \alpha - 3}{4}}) .