harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-\left\langle \phi,f'(\phi)\phi_{k}+ \mathcal{K}*\phi_{k}- k\mathcal{K}_{1}*\phi_{k}' - \mathcal{K}_{1}*\phi' + \frac{1}{2}k\mathcal{K}_{2}*\phi'' \right\rangle = \langle\Psi_{3}^{0},\mathcal{A}_{1}\phi_{k}+ \mathcal{A}_{2}\Phi_{1}^{0}\rangle,
B_{k}^{-T}w_{k}=\left[\begin{array}{cc}B_{k-1}^{-T}-w_{k-1}^{T}\frac{\beta_{k-1}}{\alpha_{k}}\frac{1}{\alpha_{k}}\end{array}\right]\left[\begin{array}{c}-w_{k-1}\frac{\beta_{k-1}}{\alpha_{k}}\frac{1}{\alpha_{k}}\end{array}\right]=\left[\begin{array}{c}-\frac{\beta_{k-1}}{\alpha_{k}}\left(B_{k-1}^{-T}w_{k-1}\right) \|w_{k}\|^{2}\end{array}\right].
\beta_{i}=\alpha_{i}+\frac{1}{2}-u+r+\frac{u_{i}-r_{i}}{2}+\sum_{1\leqslant j<i}(u_{j}-r_{j}), \gamma_{i}=\alpha_{i}+\frac{1}{2}+u-r-\frac{u_{i}-r_{i}}{2}-\sum_{1\leqslant j<i}(u_{j}-r_{j}).
[\frac{(1-4k^{2}k^{\prime2})^{2}}{k^{2}k^{\prime2}}]^{\frac{5}{4}}\theta^{-5}+5[\frac{(1-4k^{2}k^{\prime2})^{2}}{k^{2}k^{\prime2}}]^{\frac{1}{4}}\theta^{-1}=2[\frac{(1-4k^{2}k^{\prime2})^{2}}{k^{2}k^{\prime2}}]^{\frac{1}{4}}
{\rm E}_1 \left( \eta_1, \cdots, \eta_p \right) =\eta^{a_1}_1 \cdots \eta^{a_p}_p \, \, h^a( \phi ) \partial_a \, \,\partial^{(1)}_{a_1}\cdots \partial^{(p)}_{a_p}F_p ( \phi, \cdots, \phi)
\mathbf{a}(\theta_{0}) =[1, e^{-j2\pi\frac{d}{\lambda}{sin}\theta_0}, \cdots, e^{-j2\pi(N-1)\frac{d}{\lambda}{sin}\theta_0}]^{T}&=[1, e^{-j\phi}, e^{-j2\phi}, \cdots, e^{-j(N-1)\phi}]^{T},
S^{\pm}[x]\simeq S_{\rm qc}[x_{\rm qc}]\mp\hbar\varphi [x_{\rm qc}]+\frac{1}{2}\int\limits_{0}^{t}{\rm d}\tau\left[\dot{\eta}^{2}-\left(\Phi^{2}\right)''\Bigl(x_{\rm qc}(\tau )\Bigr)\eta^{2}\right].
\mathbf{D}_{\mathrm{r}}=\left[\frac{\partial \mathbf{a}_{\mathrm{r}}(\mu^{\mathrm{r}}_1)}{\partial \mu^{\mathrm{r}}_1}\quad\cdots\frac{\partial \mathbf{a}_{\mathrm{r}}(\mu^{\mathrm{r}}_Z)}{\partial \mu^{\mathrm{r}}_Z}\right] =-j\mathbf{Z}_N\mathbf{A}_{\mathrm{r}}
=\sum\limits_{v\in e}\frac{\delta_E(e)}{|e|}[(avg(x))(e)-x(v))]x(v) =\sum\limits_{i= 1}^k\frac{\delta_E(e)}{|e|^2}\left[\left\{\sum\limits_{j=1}^k x(v_j)\right\}-|e|x(v_i))\right]x(v_i) =-\frac{1}{2}\frac{\delta_E(e)}{|e|^2}\sum_{i,j=1}^k(x(v_i)-x(v_j))^2\le 0.
\Phi_{(1- D)f}= -\frac{(f- f')'}{f- f'}=-\frac{(f(1+\Phi_f))'}{f(1+\Phi_f)}= -\frac{f'}{f}-\frac{\Phi_f'}{1+\Phi_f}=\Phi_f-\frac{\Phi_f'}{1+\Phi_f}.
2A = \{2a_0 +id:i=0,1,\ldots, 2k-2\}& = \{2a_0 +id:i=0,1,\ldots, k-1 \}\cup \{2a_0 + id:i= k,\ldots, 2k-1 \}& \subseteq \left( \{a_0 \}+ A \right) \cup \left( \{a_0 +kd\}+ A \right) & = X+A
|(\log_{e}|F(z)|^2)| \leq |(\log_{e}|(z-z_0)|^{24\nu_\tau^{(N)}(f)})|+|(\log_{e}|F_0(z)|^2)| &\leq |(\log_{e}|(z-z_0)|^{24\nu_\tau^{(N)}(f)})|+ M_2 \;\; ( M_2) &= |24\nu_\tau^{(N)}(f)\log_{e}\varepsilon|+M_2.
B^{A_{1}. . . A_{p}}= \epsilon^{A_{1}. . . A_{p}r B_{1}. . . B_{p}}\ \partial_{r}A_{B_{1}. . . B_{p}}+ p \epsilon^{A_{1}. . . A_{p}B_{1}r B_{2}. . . B_{p}}\A_{r B_{2}. . . B_{p}| | B_{1}}\
2\lambda K+\frac{4(\lambda -1)}{n}(2\left\vert b \right\vert-b)+(2(2-\lambda)\left\vert b\right\vert -\lambda b)p\omega^{-1}-2\lambda \left\vert b \right\vert p(p-1)\omega^{-2}\leq \,\, 2\lambda K+\frac{12(\lambda-1)\left\vert b \right\vert}{n}+(4-\lambda)p\left\vert b\right\vert \left\vert J\right\vert.
\| u \|_{C^{\gamma}(U)}:= \| u\|_{L^{\infty}(U)}+ \| u\|_{\dot C^{\gamma}(U)}, &\| u\|_{C^{k,\gamma}(U)}:=\| u\|_{C^{k}(U)}+ \sum_{|\beta|=k}\| D^{\beta}u \|_{\dot C^{\gamma}(U)}.
[\mathcal S(N)](z_1,\dots,z_N;\tau) = \prod_{n=1}^N q^{\frac{1}{24}}[\mathcal S(1)](z_n;\tau)= q^{\frac{N}{24}}\prod_{n=1}^N\frac{1}{\left(\zeta_n^{-1}q^{\frac{1}{2}}\right)_{\infty}\left(\zeta_nq^{\frac{1}{2}}\right)_{\infty}}.
\theta\, f_{1,\ell}= \frac{\theta}{4}\, \sqrt{\frac{i \pi}{2}}\,e^{i \frac{\Phi}{2}}\,\, k^{3/2}\,\, \left( 1 + \frac{\Phi}{2 \pi \ell}+{\cal O}(\ell^{-2})\right).
\Big(\bar{x}^{(1)}_\nu(\theta_t\omega)e^{O_t^{(1)}(\omega)},\bar{x}^{(2)}_\nu(\theta_t\omega)e^{O_t^{(2)}(\omega)},\dots,\bar{x}^{(N)}_\nu(\theta_t\omega)e^{O_t^{(N)}(\omega)}\Big)^{\top}
\overline{\pi_1( \mathrm{\mathsf{x}}+1)}-\overline{\pi_1( \mathrm{\mathsf{x}})}=-\sum_{n=1}^\infty\frac{1}{n!}\int_{\lambda( \mathrm{\mathsf{x}})}^{\lambda( \mathrm{\mathsf{x}}+1)}e^{-t}\,t^{n-1}\;dt\sim\frac{-1}{\lambda( \mathrm{\mathsf{x}})}
868t^6 + 2569t^5 + 616t^4 + 1646t^3 + 92 < 868t^6 + 2569t^5 + 616t^4 + 1646t^3 + 92\biggr\rvert_{t=0.6263716633 + 10^{-10}}& < 891.450148292474 & < 974.25358710372530451456 & = 1744t^2 + 463t\biggr\rvert_{t=0.6263716633 - 10^{-10}}< 1744t^2 + 463t,
A = \sum_{\mu=1}^{N_d}\sigma^A_\mu \left\vert f^A_\mu \right\rangle \left\langle g^A_\mu \right\vert \ , \ \B = \sum_{\mu=1}^{N_s}\sigma^B_\mu \left\vert f^B_\mu \right\rangle \left\langle g^B_\mu \right\vert \ .
\Pi_{a b}( k ) = i g \, D_{a a^{\prime}}^{- 1}( k ) \, \tau_{b x}^{3}\left( \int \frac{d^{6}p}{( 2 \pi )^{6}}\Gamma_{x a^{\prime}b}( p - k / 2 , k , - ( p + k / 2 ) ) \right) \, .
\mathbb{R}^{n}\rightarrow^{c}\mathbb{C}^{n}=\mathbb{R}^{n}+i\mathbb{R}^{n}\rightarrow^{r}\mathbb{R}^{n}\oplus\mathbb{R}^{n}, \mathbb{C}^{n}\rightarrow^{q}\mathbb{H}^{n}=\mathbb{C}^{n}+j\mathbb{C}^{n}\rightarrow^{c^{\prime}}\mathbb{C}^{n}\oplus\mathbb{C}^{n}
(X, \dot{\delta_X}Df \wedge \delta_X Dg) = -(X, \delta_X Dg \wedge \dot{\delta_X}Df) = -(\rho_X \delta_X D g, \dot \delta_X f) &=- (\dot{d_X}\rho_X \delta_X Dg, Df) = -([\dot X, \rho_X \delta_X Dg], Df) & = (\dot X, -O_{X, Dg}Df).
\frac{1}{T}\sum_{t=1}^{\lfloor sT \rfloor-1}\Bar{\Theta}^{d,T}(\frac{t}{T},x)\left( f'(\frac{t}{T})+O(T^{-1}) \right)=\int_0^{\frac{\lfloor sT \rfloor}{T}}\Theta( u ,x) f'(u) \mathrm{d}u +o(1).
A = \begin{pmatrix}19 -1 -1 18 -1 \quad \ddots \ddots \ddots \quad \quad -1 1 -1 \quad \quad \quad -1 1 -1 \quad \quad \quad \quad \ddots \ddots \ddots \quad \quad \quad \quad \quad -1 19 -1 \quad \quad \quad \quad \quad \quad -1 20\end{pmatrix}
\prod_{d\mid n}\binom{kd-1}{\lfloor d/4\rfloor}^{\mu(n/d)}\equiv (-1)^{\phi_4(n)}\left\{8^{k\phi(n)}+(-1)^{\frac{n+1}{2}}k(2k-1)n^{\phi(n)}\phi_{J_{4}}^{(2-\phi(n))}(n)E_{\phi(n)-2}\right\}\pmod{n^3};
P_n^{(m)}(x_i ) = \frac{2}{n}\sum\limits_{j = 0}^n \sum\limits_{k = 0}^n \theta_j \theta_k f_j T_k (x_j )T_k^{(m)}(x_i ) = \sum\limits_{j = 0}^n{d_{ij}^{(m)}f_j}, m \ge 0,
\widehat{\pmb{x}}_k = \pmb{W}_k^H\pmb{y}_k = \underbrace{\pmb{W}_k^H\pmb{H}_k\pmb{F}_k\pmb{x}_k}_{}+ \underbrace{\sum_{\ell = 1,\ell \ne k}^K \pmb{W}_k^H\pmb{H}_k\pmb{F}_\ell\pmb{x}_\ell}_{}+ \underbrace{\pmb{W}_k^H\pmb{n}_k}_{}.
G_-(x) \le(4 \pi)^{-\frac{N}{2}}\int_0^\infty e^{-\frac{|x|^2}{4 \delta}}\delta^{-\frac{N}{2}}e^{-\delta}d\delta=\begin{cases}C |x|^{-N+2}+ o(|x|^{-N+2}), N \ge 3, C \log \frac{1}{|x|}+ o \left( \log \frac{1}{|x|}\right), N=2, C + o(1), N=1.\end{cases}
THD_{R}=\frac{\sqrt{V_{2}^{2}+V_{3}^{2}+V_{4}^{2}+\cdot\cdot\cdot}}{\sqrt{V_{1}^{2}+V_{2}^{2}+V_{3}^{2}+\cdot\cdot\cdot}}=\frac{THD_{F}}{\sqrt{1+THD_{F}^{2}}}
\int_{0}^{+\infty}F(x)\cosh\Big(\frac{x}{2}\Big)\d x = 2\Big(\int_{0}^{+\infty}\Phi(x)\cosh\Big(\frac{x}{2}\Big)\d x\Big)^2, \int_{0}^{+\infty}F(x)\cos(xt)\d x = 2\Big(\int_{0}^{+\infty}\Phi(x)\cos(xt)\d x\Big)^2,
\langle{f'_{{l'}^{(0)}}}^{\mspace{-25mu}(1)}, F_i^{(t)}\varphi(G_{\varphi_t})E_i^{(N)}.v_{-\lambda_1}\rangle =(G_{\lambda_1}^{\mathrm{up}}((\pi_{\lambda_1}(\ast b_-)),\omega(F_i^{(t)}\varphi(G_{\varphi_t})E_i^{(N)}).v_{\lambda_1})_{\lambda_1}&=(G_{\lambda_1}^{\mathrm{up}}((\pi_{\lambda_1}(\ast b_-)), E_i^{(t)}\ast(G_{\varphi_t})F_i^{(N)}.v_{\lambda_1})_{\lambda_1}.
\frac{\partial y}{\partial t}-J\Big(x, y, x\frac{\partial y}{\partial x}\Big)v=0, &\frac{\partial v}{\partial t}+H_1\Big(x, y, x\frac{\partial y}{\partial x}, v\Big)\mathcal{L}\Big(\frac{\partial}{\partial x}\Big)y+H_2\Big(x,y, x\frac{\partial y}{\partial x}, v, x\frac{\partial v}{\partial x}\Big)=0
\left(\begin{array}{c}\dot \rho \dot\phi\end{array}\right)=\left(\begin{array}{c}A^+ \rho \omega^+\end{array}\right)+P\left(\begin{array}{c}u_0^+(\phi)+u_1^+(\phi)\rho+H_1^+ w^+(\phi)+H_2^+\end{array}\right)+P\mathfrak{D}_+ (G-G^{\nu+2})\circ T_+(\rho,\phi),
x\succeq v =L_\succ^\alpha(x)v=L_\succ\circ(\alpha^2\otimes Id)(x\otimes v)=\alpha^2(x)\succ v, v\preceq x =R_\prec^\alpha(x)v=R_\prec\circ(id\otimes\alpha^2)(v\otimes x)=v\prec\alpha^2(x),
\mu(H) = \mu(H') := (0,4)\setminus \{E_{\pm}(k)\}\ \ \ \d=1, \mu(H) := [0,E(k)) \cup (4d-E(k),4d] \ \ \ \d\geqslant 2, \mu(H') := [0,E'(k))\cup(4d-E'(k),4d] \ \ \ \d\geqslant 2.
J_{2,n}(z,\tau) = \delta_{n,1}\frac{y}{y - 1}+ B_n - n \sum_{k, r \geq 1}r^{n - 1}(y^k + (-1)^n y^{-k}) q^{k r}J_{3,n}(z, \tau) = -B_n \Big( 1 - \frac{1}{2^{n - 1}}\Big) - n \sum_{k , r \geq 1}\left( r - 1/2 \right)^{n - 1}(y^k + (-1)^n y^{-k}) q^{k (r - \frac{1}{2})},
\real\left(\frac{z\mathtt{g}_{a, \nu}'(z)}{\mathtt{g}_{a, \nu}(z)}\right)\geq \frac{r\mathtt{g}_{a, \nu}'(r)}{\mathtt{g}_{a, \nu}(r)}> \frac{\mathtt{g}_{a, \nu}'(1)}{\mathtt{g}_{a, \nu}(1)}= a (1-\nu)+ a^{a/2}\left(a\nu-a+1-\sum_{n=1}^\infty \frac{2}{\mathtt{j}_{\nu, n}^2-1}\right).
( - \tilde{g})^{- 1 / 2}\partial_{\mu}\left( \sqrt{- \tilde{g}}\tilde{g}^{\mu \nu}\partial_{\nu}\phi \right) - \frac{\alpha M^{4}}{M_{p}( \zeta + b )}e^{- 2 \alpha \phi / M_{p}}+ \frac{\alpha m}{M_{p}}F ( \zeta ) \overline{{{\Psi}}}^{\prime}\Psi^{\prime}= 0 .
\mathbb{T}^{(i)}( M_1\otimes\dots \otimes M_{i-1}\otimes M_i\otimes M_{i+1}\otimes M_{i+2}\otimes\dots \otimes M_n)\mathbb{T}^{(i)}= M_1 \otimes\dots\otimes M_{i-1}\otimes M_{i+1}\otimes M_{i}\otimes M_{i+2}\otimes \dots \otimes M_n
\|p_{j}(\varkappa)-p_{j}(y)\|_{\mathbb{X}}\le KL_{\nu_{j}}\|\varkappa-y\|_{\mathbb{X}}+KK_{1}b\|\tilde{\varkappa}_{\tau}-\tilde{y}_{\tau}\|_{D}\le KL_{\nu_{j}}\|\varkappa-y\|_{\mathbb{X}}+KK_{1}\gamma b\|\varkappa-y\|_{\mathbb{X}},\ \mbox{for}\j=1,\ldots,n.
\|\tilde{q}-q\|_{L^2(M)}^2 =\sum_{k\le \ell}(\tilde{q}-q,\phi_k)^2+\sum_{k>\ell}(\tilde{q}-q,\phi_k)^2 &\le \sum_{k\le \ell}(\tilde{q}-q,\phi_k)^2 +\frac{1}{\lambda_{\ell+1}}\sum_{k>\ell}\lambda_k(\tilde{q}-q,\phi_k)^2 &\le \sum_{k\le \ell}(\tilde{q}-q,\phi_k)^2 +\frac{N^2}{(\ell+1)^{2/n}}.
\tilde{E}_{\lambda \mu}= -\hbar^{-1}(-\gamma_{i,p}^{(s)}) \displaystyle\prod_{(s',p') \neq (s,p)}\dfrac{1}{\nu_{i,p}^{(s)}- \nu_{i,p'}^{(s')}}, \tilde{F}_{\lambda \mu}= -\hbar^{-1}\beta_{i,p}^{(s)}\displaystyle\prod_{(s',p') \neq (s,p)}\dfrac{1}{\nu_{i,p}^{(s)}- \nu_{i,p'}^{(s')}}.
\mathrm{I = \int \frac{d q_{0}}{2 \p i}\int \frac{d k_{0}}{2 \p i}\, \frac{k_{0}}{( k_{0}^{2}- \vec{k}^{2}+ i \e p s i l o n )}\, \frac{q_{0}}{( q_{0}^{2}- \vec{q}^{2}+ i \e p s i l o n )}\, , \qquad \e p s i l o n > 0 ,}
\left| Q_2(t,s_1,s_2) \right| \lesssim \varepsilon^{3/2}\left(\alpha\beta\right)^{-3/2}\left( \frac{1}{\alpha}+ \frac{1}{\beta}\right)^{-5/2}= \frac{\varepsilon^{3/2}\alpha\beta}{\left(\alpha+\beta\right)^{5/2}}.
ds_5^2 \approx e^{-2|y|/\ell}\left[- \rho^2\frac{\Delta}{\Sigma^2}dt^2+ \frac{\Sigma^2}{\rho^2}\left(d\varphi + \frac{2G_4Jr}{\Sigma^2}dt\right)^2\sin^2{\theta}+ \frac{\rho^2}{\Delta}dr^2 + \rho^2 d\theta^2\right ]+ dy^2 ,
\lim_{\left\|w\right\|_{X^{1/2}_0(\mathcal{C}_\Omega)}\to 0}\frac{\Psi(w)}{\left\|w\right\|^2_{X^{1/2}_0(\mathcal{C}_\Omega)}}=0\,\,\,\,\,\,{\rm and}\,\,\,\,\,\, \lim_{\left\|w\right\|_{X^{1/2}_0(\mathcal{C}_\Omega)}\to \infty}\frac{\Psi(w)}{\left\|w\right\|^2_{X^{1/2}_0(\mathcal{C}_\Omega)}}=0.
\int_G \int_X \big| s(g^{-1}x) - s(x) \big|^2 \, d\mu^{*k}(g) \, d\nu(x)= 2 \cdot \Big( \int_X |s(x)|^2 \, d\nu(x) - \int_X s(x) \, \Big( \int_G s(g^{-1}x) \, d\mu^{*k}(g) \Big) \, d\nu(x) \Big),
K_{\Psi,w}(t_{1},t_{2},\ldots , t_{l})=\prod_{j=1}^{l}\Gamma (-\beta_{j})^{-1}\times\{\exp (-t_{1}F_{i_{1}})\cdots\exp(-t_{l}F_{i_{l}})\prod_{j=1}^{l}t_{j}^{-\beta_{j}-1}\Psi\}.
\left\| \nabla^{k}_{x}\left(K_{3}(t)g - \mathcal{F}^{-1}\left[e^{-\frac{\nu t |\xi|^{2 \sigma}}{2}}\frac{\sin (t |\xi|)}{|\xi|}\right] \ast g\right)\right\|_{2}& \le C (1+t)^{-\frac{n}{4 \sigma}-1-\frac{k}{2 \sigma}+\frac{1}{\sigma}}\| g \|_{1}+C e^{-ct}\| \nabla^{(k-2 \sigma)_{+}}_{x}g\|_{2},
\sum_{i=3,4}\big|\mathbf{P}\big( T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x,\zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta)\big) \big|+ 2^{(\gamma_1-\gamma_2)M_{t^{\star}}}\big| T_{k,j;n}^{\mu,i}( \mathfrak{m}, E)(t,x,\zeta) + \hat{\zeta}\times T_{k,j;n}^{\mu,i}( \mathfrak{m}, B)(t,x, \zeta) \big|
\Big(\int_{0}^{1}{r_N}^{c_N}(1-r_N)^{\alpha\cdot(\mathbf a+\mathbf 1)+\sum_{j=1}^{N-1}(c_j+1)}dr_N\Big) &\times\Big(\int_{\mathbb B^{N-1}_+}(1-\sum_{j=1}^{N-1}t_j)^{\alpha\cdot(\mathbf a+\mathbf 1)}\prod_{j=1}^{N-1}t_j^{c_j}dt_1\dots dt_{N-1}\Big).
|\{(v,w) \in \operatorname{Inv}(\varphi'_{\mathcal{L}_{D'}T'_1}) \mid v w_0, \,w w_0\}| &= (|D'|-1) \cdot |\{(v,w) \in \operatorname{Inv}(\varphi_{\mathcal{L}_{D'}T_1}) \mid w=w_0 \}|
-{\rm Tr}\left(P_-\Gamma_7\frac{\mu^2+\hat{\phi}^i\hat{\phi}^i}{-(\Gamma\cdot\Delta)^2+\mu^2}\right)=-\int d^3x\{\rm tr}\left( P_-\Gamma_7\Gamma_\mu{\partial}_\mu\,\langle x|(\Gamma\cdot\Delta +\mu)^{-1}| x\rangle\right)
\mathcal{N_{\alpha}}\left(B_\rho(z_0,2^{i+1}\alpha)~\Big|~E\right)~\leq~\sum_{j=1}^{2^{{\bf d}(E)}}\mathcal{N_{\alpha}}\left(B_\rho(x_j,2^{i}\alpha)~\Big|~E\right)~\leq~2^{{\bf d}(E)}\cdot 2^{i{\bf d}(E)}~=~2^{(i+1){\bf d}(E)}.
\eta(\eta(\tau_i, \tau_j),\eta(\tau_i,\tau_\ell))=(\tau_n \tau_1)^{i-j}\tau_n^{q-1+D(n-j+\ell)}\tau_{n-j+\ell+1}=\eta(\eta(\tau_j, \tau_i),\eta(\tau_j,\tau_\ell))
\Phi_{-h}y_0= \Phi_{-h}E_h y_{-1}= \Phi_{-h}\left( \Phi_h y_{-1}+c_{p+1}(E_{-h}y_0)h^{p+1}+O(h^{p+2})\right) &=E_{-h}y_{0}+c_{p+1}(y_0)h^{p+1}+O(h^{p+2}),
\zeta W^{(n)}_\lambda(t;\zeta) =\mathbb{P}^{(n)}_{\lambda,j}(t;\partial)W^{(n)}_\varnothing(t;\zeta)\ ,j=1,2,\dots n\ , \partial_\zeta W^{(n)}_\lambda(t;\zeta) =\beta_{p,p'}\sum_{j=1}^n\mathbb{Q}^{(n)}_{\lambda,j}(t;\partial)W^{(n)}_\varnothing(t;\zeta)\ ,
||\Upsilon_{3i}||^2 \leq\phi_{1i}(\hat z_i)||\hat z_i||^2+\phi_{2i}(e_i)||e_i||^2 \sum_{i=1}^n[\hat \gamma_{3i}(e_i)+ \phi_{2i}(e_i)]{e_i}^2\leq\sum_{i=1}^n \tilde\phi_i(e_{vi})e_{vi}^2
\mu^{(r)}(F) = \mu^{(r)}(B_{r}) - \mu^{(r)}(B_{r}\setminus F) > \mu^{(r)}(B_{r}) - \delta &> \mu^{(r)}(B_{r}) + \varepsilon + \nu(B_{r}) -\mu(B_{r}) \geq \nu^{(r)}(F^{\varepsilon}) + \varepsilon.
0 \le \{\mathcal{F}[v](x_0) - B(x_0, v(x_0), Dv(x_0))\}-\{\mathcal{F}[u](x_0)-B(x_0, u(x_0), Du(x_0))\}= F(D^2v(x_0)-A(x_0, v(x_0), Dv(x_0)) - F(D^2u(x_0)-A(x_0, u(x_0), Du(x_0)) + B(x_0, u(x_0), Du(x_0)) - B(x_0, v(x_0), Dv(x_0)) < 0,
\int_{\Gamma_j}p_{t_j}(z_j-\xi_j)d\xi_j =\lim_{R\rightarrow\infty}\frac{1}{\pi}\int_{E_R}\frac{t_j}{(\xi_j-(z_j+it_j))(\xi_j-(z_j-it_j))}d\xi_j &=\lim_{R\rightarrow\infty}\frac{1}{\pi}\frac{2\pi i\,t_j}{(z_j+it_j)-(z_j-it_j)}=1.
{\cal L}_{p}={\cal L}_{p}^{reg}+ 4p \epsilon_{\phi \rho i_1...i_{2p-2}}\epsilon^{\phi \rho j_1... j_{2p-2}}R^{\phi \rho}_{(con) \ \phi \rho}R^{i_1 i_2}_{\ \j_1 j_2}...R^{i_{2p-3}i_{2p-2}}_{\ \j_{2p-3}j_{2p-2}}
\int_{D(z_0,\frac{r}{2}+\rho)}\|h-h_{(r)}\|_{L^1(D(z,\frac{r}{2}))}dz= \int_{D(z_0,\frac{r}{2})}\|h-h_{(r)}\|_{L^1(D(z,\frac{r}{2}+\rho))}dz \geq \int_{D(z_0,\frac{r}{2})}\|h-h_{(r)}\|_{L^1(D(z_0,\rho))}dz = \pi\left(\frac{r}{2}\right)^2\|h-h_{(r)}\|_{L^1(D(z_0,\rho))},
\sum_n\left\{\frac{(-1)^n}{2(n+1)!}(Ad_{\Phi})^n\partial^2\Phi^a+\frac{(-1)^{n+1}}{2(n+2)!}\sum_{k=0}^{n-1}(Ad_{\Phi})^kAd_{\partial_\mu\Phi}(Ad_{\Phi})^{n-k}\partial^\nu\Phi^a\right\}=0,
F_{ab}=\int_{a}^{b}dx\; e^{i\beta f(x)}=\frac{1}{i\beta}\left[ \frac{1}{f{'}}e^{i\beta z}|_{f(a)}^{f(b)}-\int_{f(a)}^{f(b)}dz\; e^{i\beta z}\frac{d}{dz}(f{'})^{-1}\right] .
f_t(x,v) = \mathcal{Z}_t^{-1}\exp\bigg(-\frac{(v-A(t,x))^2}{2 B(t,x)}\bigg)\,, \mathcal{Z}_t := \int_{\mathbb{T}}\! \mathrm{d}x\, \sqrt{2 \pi B(t,x)}\,, \mbox{and}\,\, B(t,x)>0\,,
(a\tfrac{1}{c\overline{c}}y^2+\tfrac{1}{d\overline{d}}z^2 - (a{+}1)x^2 -a^2-b^2-a)^2\ +\b^2(x^2-\tfrac{1}{c\overline{c}}y^2+1)^2\ -\ 4b^2x^2\ =\ 0\,.
I^{i}_{i,j}(t) =\int_{t_{0}}^{t}D_{t_{0}+}^{\alpha}(\lambda_{i}-\lambda_{j})_{t_{0}}^{-1}(s)D_{t-}^{1-\alpha}\lambda_{i,t-}(s)ds +(\lambda_{i}(t_{0})-\lambda_{j}(t_{0}))^{-1}\left( \lambda_{i}(t)-\lambda_{i}(t_{0}))\right) ,
\varPhi_{k,T}^{h}(\lambda_{1},\ldots,\lambda_{k-1}):=\frac{1}{(2\pi)^{k-1}H_{k,T}(0)}\prod_{j=1}^{k-1}H_{1,T}(\lambda_{j})H_{1,T}\Biggl(-\sum_{j=1}^{k-1}\lambda_{j}\Biggr)
\widetilde{IV}_{A,i}=\varepsilon\int_{0}^{t}\int_{\mathbb{T}^{d}}\sum_{i=1}^{d}\sum_{j=1}^{d}(\partial^{\alpha}\partial_{x_{i}}\mu^{n+1})(\mu^{n}+\bar{m})(\Theta_{p_{i}p_{j}})(\partial^{\alpha}\partial_{x_{j}}w^{n})\dxd\tau,
(\sigma^{(i)}y^{(k)})^{(k+1)}+ (\sigma^{(i)}y^{(k+1)})^{(k)}= \sum_{s = 0}^{i + 1}(-1)^s (C_{i+1}^s - 2 C_i^{s-1})(\sigma y^{(s + k)})^{(i + 1 - s + k)}, ( (\sigma^{(i)}y^{(k)})^{(k+1)}+ (\sigma^{(i)}y^{(k+1)})^{(k)}, z) = \sum_{s = 0}^{i + 1}(-1)^{i + k + 1}(C_{i + 1}^s - 2 C_i^s) ( \sigma y^{(s + k)}, z^{(i + 1 - s + k)}),
\|M(\chi_{B})^{1/s}\|_{\mathcal M^{p}(\varphi,w)'}\lesssim \sum_{k=1}^\infty 2^{k\left (\frac n{s'}+\frac{\lambda_1}p\right)}|B|^{1+\frac{\lambda_1}{np}}w(2^kB)^{\left (\frac{\lambda_2}n-1\right)\frac 1p}.
M ( \lambda , L , m , d ) = \lambda^{2}\int d^{d - 1}r \int d^{d - 1}r^{\prime}\int_{0}^{L}d z \int_{0}^{L}d z^{\prime}F ( \vec{r}, \vec{r}^{\prime}, z , z^{\prime}) ( G_{0}^{( 2 )}( \vec{r}- \vec{{r}^{\prime}}, z , z^{\prime}) )^{2},
F_{2}^{\prime\prime\prime}+F_{2}F_{2}^{\prime\prime}-F_{2}^{\prime2}+1=0,\frac{p_{o2}-p_{2}}{\rho_{2}}=\frac{1}{2}k_{2}^{2}x^{2}+k_{2}\nu_{2}F_{2}^{\prime}+\frac{1}{2}k_{2}\nu_{2}F_{2}^{2}.
e_1 = \begin{bmatrix}\frac{1}{2}\frac{1}{2}0 0\end{bmatrix}^T, e_2 = \begin{bmatrix}\frac{1}{2}-\frac{1}{2}0 0\end{bmatrix}^T, t_1 = \begin{bmatrix}0 0 1 0 \end{bmatrix}^T, t_2 = \begin{bmatrix}0 0 0 1 \end{bmatrix}^T.
\sum_{\sigma \in S_{b,c}}\prod_{j=1}^N \bigg( \prod_{h=1}^{\sigma(\vec{A})_j -1}\Big( \sum_{k=j+1}^N \sigma(\vec{A})_k + h \Big) \bigg) = \sum_{\sigma \in S_{b+1,c}}\prod_{j=1}^{N+1}\bigg( \prod_{h=1}^{\sigma(\vec{A}')_j -1}\Big( \sum_{k=j+1}^{N+1}\sigma(\vec{A}')_k + h \Big) \bigg).
\nu_{f,g}^\epsilon(\vec{t})=\int_E\int_{\Phi_y^{-1}(\vec{t})}\,\mu_f^\epsilon(x)\,\mu_g^\epsilon(z)\,\frac{1}{|J_2\Phi_y|}\,d\mathcal{H}^{2d-2}(x,z)\,d\mu(y).
M R_{P}M^{- 1}= \left( \begin{array}{c c}{- \cos 2 ( \phi_{P}+ \phi_{C})}&{- \sin 2 ( \phi_{P}+ \phi_{C})}\\{- \sin 2 ( \phi_{P}+ \phi_{C})}&{\cos 2 ( \phi_{P}+ \phi_{C})}\\ \end{array}\right) .
S=\pi \left\{(m^{2}-|{\cal Z}_{1}|^{2})+ (m^{2}-|{\cal Z}_{2}|^{2})+ 2\sqrt{(m^{2}-|{\cal Z}_{1}|^{2})(m^{2}-|{\cal Z}_{2}|^{2}) -J^{2}}\, .\right\}\, .
(\log G_2)^{(i)}(s,x) =-(-2)^{i-1}\int_{\sqrt{x}}^{x}(\log t)^{i-1}t^{-2s}dt \left( 1+O_{\varepsilon,i}\left( L(x)^{-c}\right)\right) &\asymp_{\varepsilon,i}(-1)^i (\log x)^{i-1}\int_{\sqrt{x}}^{x}t^{-2s}dt.
\left[ g_{\mu \nu}( u ) \right] = \left( \begin{array}{c | c}{{1}}&{{u^{0}\vec{u}^{\mathrm{T}}}}\\ \hline{{u^{0}\vec{u}}}&{{- I + \vec{u}\otimes \vec{u}^{\mathrm{T}}( u^{0})^{2}}}\\ \end{array}\right) ,
H_{B}= 1 +{\frac{r_{B}^{7 - B}}{V_{a}}}{\frac{1}{r^{7 - B - a}}}= 1 +{\frac{g_{s}N_{B}}{V_{a}/ \ell_{s}^{a}}}\left({\frac{\ell_{s}}{| x_{\perp}|}}\right)^{7 - A - b}.
n^{-1}\sum_{j=0}^{[nt]-1}\{U_nL_n (m_nm_n^T)\}\circ f_n^j \, -\,t\Sigma = n^{-1}\sum_{j=0}^{[nt]-1}\breve{\phi}_n\circ f_n^j+ n^{-1}[nt]\Sigma_n-t\Sigma \to_{\mu_{\Delta_n}}0,
\mathbf{x}^{-}_{\alpha}(r,s)=\sum_{(b_p)\in\mathbf{S}(r,s)}(x^{-}_{\alpha}\otimes 1)^{(b_0)}(x^{-}_{\alpha}\otimes t)^{(b_1)}\cdots(x^{-}_{\alpha}\otimes t^s)^{(b_s)},
S : \left( \begin{array}{c}{{a_{D}}}\\{{a}}\\ \end{array}\right) \longrightarrow \left( \begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c}{{a_{D}}}\\{{a}}\\ \end{array}\right) = \left( \begin{array}{c}{{a}}\\{{- a_{D}}}\\ \end{array}\right) .
\|V^0_s\| =\sup_{(x,t,\xi)\in \{0,\cdots,L-1\}\times\{0,\cdots,n-1\}\times\{1,-1\}}\sum_{(y,u,\zeta)\in \{0,\cdots,L-1\}\times\{0,\cdots,n-1\}\times\{1,-1\}}&\quad\cdot \left|\frac{\gamma}{2L}\delta_{x,y}\delta_{t,u}(\delta_{\xi,1}\delta_{\zeta,-1}-\delta_{\xi,-1}\delta_{\zeta,1})\right| &=\frac{1}{2}|\gamma|L^{-1}.
T : \Upsilon_{l,\, m}= -i \sigma_y \Upsilon^*_{l,\, m}=\left(\begin{array}{r}-\beta^*_{l,\, m}\alpha^*_{l,\, m}\end{array}\right) = (-1)^{l-m}\,\left(\begin{array}{c}\alpha_{l,\,- m}\beta_{l,\,- m}\end{array}\right).
{\cal H}= \int d^{d}x \Bigl [{\frac{1}{2}}\sum_{i}( \partial_{\mu}\Phi_{i})^{2}+{\frac{1}{2}}\sum_{i}r_{i}\Phi_{i}^{2}+{\frac{1}{4 !}}\sum_{i j k l}u_{i j k l}\Phi_{i}\Phi_{j}\Phi_{k}\Phi_{l}\Bigr ] ,
\big[a^{\dagger}_k,K^+_{ij}\big] = [a_k,K^-_{ij}]=0, [a_i,K^+_{kj}] = \tfrac{1}{2}\delta_{ik}a^{\dagger}_j+\tfrac{1}{2}\delta_{ij}a^{\dagger}_k , \big[K^-_{kj},a^{\dagger}_i\big] = \tfrac{1}{2}\delta_{ik}a_j+\tfrac{1}{2}\delta_{ij}a_k , \big[K^0_{ij},a^{\dagger}_k\big] = \tfrac{1}{2}\delta_{jk}a^{\dagger}_i,\big[a_k,K^0_{ij}\big]= \tfrac{1}{2}\delta_{ik}a_{j}.
{s}^H_{g_{f,J}}=-u^{\frac{2}{n}}\sum_{i,j=1}^nH_{ij,ij}+2u^{\frac{2}{n}-1}\sum_{i,j=1}^na_iH_{ij,j}-2u^{\frac{2}{n}-2}\sum_{i,j=1}^na_ia_jH_{ij},
\partial \mathbb{W}_{0}\triangleq X^{+}\setminus \mathbb{W}_{0}= \{\psi = (\psi_{1}, \psi_{2}, \psi_{3}, \psi_{4}) \in X^{+}: \psi_{2}(\cdot) \equiv 0 \psi_{4}(\cdot) \equiv 0 \}.
\begin{aligned}\|\beta\|_{L^{\infty}(N\cap B_2)}\leq C\|\underline{\beta}\|_{L^{2}(N\cap B_4)}, \|y_i\|_{L^{\infty}(N\cap B_2)}\leq C\|\underline{y}\|_{L^{2}(N\cap B_4)}\end{aligned}
\Phi_n(t)= \frac{-1}{4n^2}\sum_{i=1}^{\lfloor 4tn^2 \rfloor}\frac{a^2}{4 x(i)}+\frac{a\sqrt{n}G_{n x(i)}}{\sqrt{\beta x(i)}}+\frac{a G^{(2)}_{nx(i)}}{\beta x(i)},x(i)=X^{n}(i\Delta t),
S ={\frac{1}{2 \kappa_{1 0}^{2}}}\int d x^{1 0}\sqrt{- \hat{G}}\, e^{- \hat{\Phi}}\left[{\cal R}_{\hat{G}}+ \partial_{M}\hat{\Phi}\partial^{M}\hat{\Phi}-{\frac{1}{2 \cdot 3 !}}\hat{H}_{M N P}\hat{H}^{M N P}\right] ,
\mathcal{R}_{1,0}(t,u)= \int_0^t e^{( t - \xi) \mathcal{L}}\Big( f \left(u(\xi), \overline u(\xi) \right) - f \left(e^{\xi\mathcal{L}}u_0, e^{\xi \overline{\mathcal{L}}}\overline u_0\right)\Big)d\xi.
\frac{dQ_f^n}{dQ_0^n}= \exp\Big( 2\sqrt{n}\int_0^1 \sqrt{f(t)}dW_t - 2n \big\|\sqrt{f}\big\|_2^2\Big)= \frac{dQ_f^{\lfloor n/2\rfloor}}{dQ_0^{\lfloor n/2\rfloor}}\frac{dQ_f^{\lceil n/2\rceil}}{dQ_0^{\lceil n/2\rceil}}
\frac1{d_{\nu,n}^2 - 2 \nu d_{\nu,n}+ \nu^2 / t^2}\quad f^{-1/4}(\epsilon,t) \quad\frac{S_u^+ \left( \nu,t f^(-1/2) (\epsilon,t) \right)}{S_u^+ \left( \nu,t \right)}\times
\begin{aligned}c(M_n) \ge \frac{1}{627419520}\Big(3\Big(1-\frac{1}{d^2}\Big)n - \frac{15d-3}{2d}-d+1\Big) \ge \frac{1}{627419520}\Big(3\Big(1-\frac{1}{d^2}\Big)n - (d+7)\Big)\end{aligned}
\operatorname{Hor}^{(\varphi_t)_*\Gamma_0}_{\varphi_t(p)}=T\varphi_t\!\cdot\!\operatorname{Hor}^{\Gamma_0}_p=T\Phi_{\tau_t(p)}\!\cdot\!\left(\operatorname{Hor}^{\Gamma_0}_p+\left(TR_{\tau_t(p)^{-1}}T\tau_t\!\cdot\!\operatorname{Hor}^{\Gamma_0}_p\right)_P(p)\right).
\mathbf{r}^{(k)}=g_{\Phi,\rho^{(k)}}\left(\mathbf{x}^{(k-1)}, \mathbf{y}\right)=\mathbf{x}^{(k-1)}-\rho^{(k)}\Phi^{\top}\left(\Phi\mathbf{x}^{(k-1)}-\mathbf{y}\right).
E_\pi(T_1, \ldots, T_n) = E\left(T'_{q_1}T'_{q_2}\cdots T'_{q_{p_1}}R_{E_{\pi|_{V_0}}((T_1, \ldots, T_n)|_{V_0})}T'_{q_{p_1+1}}\cdots T'_{q_{p_2}}\right),
\frac{\pi}{M}{\displaystyle\sum_{k=0}^{K}}{\displaystyle\sum_{\ell=1}^{a_{k}}}{\displaystyle\sum_{j=1}^{N}}{\displaystyle\sum_{s=1}^{M}}\left\vert \lambda_{j,\left( k,\ell\right)}\cdot t_{j,\left(k,\ell\right)}^{-\frac{k}{2}}\cdot Y_{\left( k,\ell\right)}\left(\frac{2\pi s}{M}\right) \right\vert \leq\sqrt{\pi}\left\Vert w\right\Vert ,
\delta^{4}(\varphi_{rest_{1}}^{1})=\varphi_{rest_{1},[4]}^{1}(q^{1},q^{2},q^{3},\omega^{1},\omega^{2},\omega^{3},v_{1}^{1},v_{1,[1]}^{1},v_{1,[2]}^{1},u^{2})\,.
\lambda^{\mathrm{\footnotesize{G N}}}~ = ~ \mu - 2 - \frac{2 ( \mu - 2 ) ( \mu - 1 ) \Gamma ( 2 \mu )}{( 2 \mu - 3 ) \Gamma ( 2 - \mu ) \Gamma^{2}( \mu ) \Gamma ( \mu + 1 ) N}
u_{ij}u_{ik}u_{jk}-u_{jk}u_{ik}u_{ij}=[u_{ik},q_{ij}]=[q_{jk},u_{ik}], u_{ij}u_{jk}u_{ij}-u_{jk}u_{ij}u_{jk}= q_{ij}u_{ik}-u_{ik}q_{jk}.
\lim_{n\to\infty}\, \frac{\omega^{2n+1}}{(2n)!}\, \sum_{j\leq a_n + y\sqrt{b_n}}(2j)!{n\brace j}_1 =\frac{1}{\sqrt{2\pi}}\int\limits_{-\infty}^y e^{-t^2/2}dt.
|\bar L|^2 =|\mathbf{r}-\frac{r^4}{|L|^2-\mu^2}(i\xi, \mathbf{r}'\lrcorner F)|^2 \cdot | \mathbf{r}'-\frac{r'}{r}\mathbf{r}|^2-(\mathbf{r}-\frac{r^4}{|L|^2-\mu^2}(i\xi, \mathbf{r}'\lrcorner F),\mathbf{r}'-\frac{r'}{r}\mathbf{r})^2 &=|\mathbf{r}\wedge \mathbf{r}'|^2+\frac{\lambda^2}{k}=|L|^2-|\lambda|^2+\frac{\lambda^2}{k}>\frac{\lambda^2}{k}.
P_{\textrm{isout,opt}}^{\textrm{ISANC}}= \left(1-\exp \left(-\frac{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^{2}}{\lambda_{\textrm{BN}}P_{\textrm{B}}}\left(2^{2 R}-1\right)\right)\right) \times \left(1-\exp \left(-\frac{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^{2}}{\lambda_{\textrm{BF}}P_{\textrm{B}}}\left(2^{2 R}-1\right)\right)\right).