harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}D^g Jd(hu^{-1})^\sharp= 2u^{-3}h du\otimes Jdu^\sharp - u^{-2}du\otimes Jdh^\sharp - u^{-2}dh\otimes Jdu^\sharp & + u^{-1}D^g(Jdh^\sharp) - u^{-2}h D^gJdu^\sharp du\otimes Jd(hu^{-2})^\sharp = u^{-2}du\otimes Jdh^\sharp - 2u^{-3}h du\otimes Jdu^\sharp\end{aligned}
S_{bos}\,=\, \int d^3 x \, \left( \pm \frac{i}{2}A_{\mu}\epsilon_{\mu \nu \lambda}\partial_{\nu}A_\lambda\,-\, \frac{i}{\sqrt{4 \pi}}s_{\mu}\epsilon_{\mu \nu \lambda}\partial_{\nu}A_{\lambda}\right) ~~~,
\|z-z'\|_{L^3_TL^r_x}\leq \eta_k^{1-\alpha}, \,\,\|z\|_{L^3_TL^r_x}\leq L(\eta_k), \, \,\|z'\|_{L^3_TL^r_x}\leq L(\eta_k) \, \, r=6,\infty, &\|z-z'\|_{L^2_TL^8_x}\leq \eta_k^{1-\alpha}, \|z\|_{L^2_TL^8_x}\leq L(\eta_k), \|z'\|_{L^2_TL^8_x}\leq L(\eta_k),
( \vec{x}_{a}- \vec{x}_{b}) A_{0 a b}( A_{0 a a}- A_{0 b b}) = \displaystyle{\sum_{c \neq a , b}\left( ( \vec{x}_{a}- \vec{x}_{c}) + ( \vec{x}_{b}- \vec{x}_{c}) \right) A_{0 a c}A_{0 b c}}\, .
i=1: -2(H_{q_1}-E_{1,5})-2(H_{p_1}-E_{1,7})-3E_{1-2}-2 E_{2-3}-E_{3-4}& -E_{5-6}-E_{7-8}=-2H_{q_1}-2H_{p_1}+ E_{1,\dots,8}, i=2: q_1\leftrightarrow q_2, p_1\leftrightarrow p_2, E_j \leftrightarrow E_{j+8}\ (j=1,\dots,8) \mbox{in the above},
\|\mathcal G^n(w_1)-\mathcal G^n(w_2)\|_{\mathcal C([0,T];D(A))}=\|\mathcal H^n(w_1-w_2)\|_{\mathcal C([0,T]; D(A))}&\le\frac{C\left(\Gamma(\alpha)T^{\alpha}\right)^n}{\Gamma(n\alpha+1)}\|w_1-w_2\|_{\mathcal C([0,T]; D(A))}
\frac{\partial\Psi}{\partial z}=0\quad\mbox{in}\,\Omega,\frac{\partial\widetilde \Psi}{\partial\overline z}=0\quad\mbox{in}\,\Omega,((\nu_1-i\nu_2)\Psi+(\nu_1+i\nu_2)\widetilde \Psi)\vert_{\Gamma_0}=0.
\sum_{j=1}^m \int_{{2^{k_{j-1}}}}^{2^{k_j}}\int_{\mathbb{R}^3}\Big(\int_{\mathbb{R}}F(y,x')F(x,x')h_{{\alpha}t}(x'-p) dx'\Big)^2 \rho_{2^{k_{j-1}}}(x-y) \,dx dy dp \frac{dt}{t}.
T_{2}-T_{1}=W_{byforce}=\int_{+\infty}^{r_{12}}-\frac{Gm_{1}m_{2}}{r^{2}}dr=+Gm_{1}m_{2}(\frac{1}{r_{12}}-\frac{1}{\infty})=\frac{Gm_{1}m_{2}}{r_{12}};
J_1 = \begin{pmatrix}0 0 0 0 0 -1 0 1 0 \end{pmatrix}, J_2 = \begin{pmatrix}0 0 1 0 0 0 -1 0 0 \end{pmatrix}, J_3 = \begin{pmatrix}0 -1 0 1 0 0 0 0 0 \end{pmatrix}
T_{\mu \nu}=g_{\mu \nu}T-F_{\mu \rho}\mbox{}^{a}F_{\nu}\mbox{}^{\rho a}+D_{\mu}\phi^aD_{\nu}\phi^a-\frac{1}{2}\ \phi^2\left(R_{\mu \nu}-\frac{1}{2}\g_{\mu \nu}R\right)
I_{\psi ,{\widetilde \psi}}\equiv{\frac{1}{2}}\int_{M}\epsilon^{\mu \nu \rho \sigma}\left[{\widetilde \psi}_{\mu}^{A^{\prime}}e_{A A^{\prime}\nu}D_{\rho}\psi_{\sigma}^{A}- \psi_{\mu}^{A}e_{A A^{\prime}\nu}D_{\rho}{\widetilde \psi}_{\sigma}^{A^{\prime}}\right] d^{4}x ,
f_{\mathbf{X}}(\mathbf{x})=\left(2\pi\right)^{-\frac{N}{2}}\left(\det\mathbf{\Sigma}_{\mathbf{X}}\right)^{-\frac{1}{2}}\exp\left[-\frac{1}{2}\mathbf{x}{}^{T}\mathbf{\Sigma}_{\mathbf{X}}^{-1}\mathbf{x}\right]=:\phi_{\mathbf{X}}(\mathbf{x};\mathbf{\Sigma}_{\mathbf{X}}).
\frac{\lambda}{\alpha}\frac{d}{dt}\int_U u^{\alpha}\xi^2 dx-\frac{\lambda}{\alpha}\int_U 2u^{\alpha}\xi \xi' dx &= \int_U \nabla\cdot \Big(K(|\nabla u+Z(u)|)(\nabla u+Z(u)) \Big) u^{\alpha-\lambda}\xi^2 dx +\int_U f u^{\alpha-\lambda}\xi^2 dx.
z(t) =C(t,0)(x_0-g(x))+S(t,0)(y_0-h(x))+\int_0^t S(t,s)f(s)ds+\int_0^t S(t,s)Bu(s,x)ds, v(t) =B^*S^*(b,t)R(a,\Upsilon_0^b)p(x(\cdot)), p(x(\cdot)) =x_b-C(t,0)(x_0-g(x))-S(t,0)(y_0-g(x))-\int_0^t S(t,s)f(s)ds
\omega = \sum_{i = 1}^{n}\frac{d u_{i}^{-}}{u_{i}^{-}}\wedge d \beta_{i}^{-}- \sum_{i < j}\frac{d \beta_{i}^{-}\wedge d \beta_{j}^{-}}{\beta_{i}^{--}\beta_{j}^{-}}+ \sum_{i = 1}^{n}\frac{d u_{i}^{+}}{u_{i}^{+}}\wedge d \beta_{i}^{+}- \sum_{i < j}\frac{d \beta_{i}^{+}\wedge d \beta_{j}^{+}}{\beta_{i}^{+}- \beta_{j}^{+}}.
P_{l}(z)=\displaystyle{\sum_{k=0}^{l}}(-1)^{l+k}\binom{l+k}{k}\binom{l}{k}\left( \frac{1+z}{2}\right)^{k}=\displaystyle{\sum_{k=0}^{l}}\frac{(-1)^{l+k}}{(k!)^{2}}\frac{(l+k)!}{(l-k)!}\left( \frac{1+z}{2}\right)^{k}
J = \mu\begin{bmatrix}\sum_{i=1}^3a_i c_i u_i \sum_{i=1}^3b_i c_i u_i \sum_{i=1}^3a_i d_i u_i \sum_{i=1}^3b_i d_i u_i\end{bmatrix}\begin{bmatrix}1/\bar{x}0 0 1/\bar{y}\end{bmatrix}.
P_{ME}=\frac{\sum[p_{t}\cdot\frac{1}{2}(q_{0}+q_{t})]}{\sum[p_{0}\cdot\frac{1}{2}(q_{0}+q_{t})]}=\frac{\sum[p_{t}\cdot(q_{0}+q_{t})]}{\sum[p_{0}\cdot(q_{0}+q_{t})]}
P_2= ~\{(\tau,1,-e^{-1/\tau^2})~|~\tau<0\}\cup\{(0,\pm1,0)\}\cup\{(\tau,1,-e^{-1/\tau^2})~|~\tau>0\}&~\cup\{(\tau,-1,e^{-1/\tau^2})~|~\tau<0\}\cup\{(\tau,-1,e^{-1/\tau^2})~|~\tau>0\}.
\begin{cases}-\frac{d^{2}\psi}{d\theta^{2}}+a(\theta )\psi=\mu_{k}\psi \psi(0) =e^{-2i\pi \widetilde{A}}\psi (2\pi ) \psi^{\prime}(0) =e^{-2i\pi \widetilde{A}}\psi^{\prime}(2\pi ),\end{cases}
2 i{\frac{( 1 2 ) ( 1 3 ) ( 2 3 )}{x_{1 2}^{4}x_{3 4}^{4}}}\; \left[{\frac{c}{s}}+ \beta_{0}{\frac{1 +{\frac{\beta_{1}}{\beta_{0}}}U +{\frac{\beta_{2}}{\beta_{0}}}U^{2}}{1 +{\frac{1 + s - t}{s}}U +{\frac{1}{s}}U^{2}}}\right] \; X_{2}^{\mu}
\Phi_m(\nu)=\gcd \left([m]_\nu/[m/p_1]_\nu,\ldots,[m]_\nu/[m/p_t]_\nu\right), \Phi_{p_k}(\nu^{m/p_k})=[m]_{\nu}/[m/p_k]_{\nu},k=1,\ldots, t, \Phi_n(\nu)=\Phi_{m}(\nu^{n/m}).
\mathbf{p}B_j^+ = \sum_{i=1}^{j}\sum_{s=1}^{j-i+1}\sum_{I\in \mathcal{I}_{ij}(s)}(-1)^{s-1}E^{e_I}B_i^+\frac{\prod_{\ell\in I^\complement}(h_\ell-h_j-1)}{\prod_{\ell=i}^{j-1}(h_\ell-h_j)},
\left( q \tfrac{d}{dq}\right)^{v}F(q)\big|_{q=\zeta}=\left( q \tfrac{d}{dq}\right)^{v}F(q; (\nu+1) n-1)\big|_{q=\zeta}=\left( q \tfrac{d}{dq}\right)^{v}F(q; N)\big|_{q=\zeta}.
Q_\sigma[u,u] - C\int_M \frac{|u(x)|^2}{|x|^2 \left( \log \frac{|x|}{\rho}+\frac{1}{2 \rho\, \sigma_0}\right)^{2}}\dx =2\pi \int_1^\infty (f'(r))^2\r\, \left( \log r +\frac{1}{2 \rho\, \sigma_0}\right)\, dr &\qquad\qquad-2\pi \left(C-\frac 14\right) \int_1^\infty f^2(r)\, r^{-1}\, \left( \log r +\frac{1}{2 \rho\, \sigma_0}\right)^{-1}\, dr.
\mathbf\zeta(\Gamma[\mathbb{Z}_{p^{3}}]) = \sum\limits_{v \in V}{d_{v}e(v)}=\sum\limits_{v \in A_1}{d_{v}e(v)}+\sum\limits_{v \in A_2}{d_{v}e(v)}=(p-1)2+(p-1)2+.....+(p-1)2\quad{[p(p-1) times]}+ (p^{2}-2)1+(p^{2}-2)1+...+(p^{2}-2)1\quad{[(p-1)]times}. =2p(p-1)^{2}+(p^{2}-2)(p-1) =(p-1)[3p^{2}-2p-2]
var[Y_k] = k\sum_{j=1}^k \frac{(k-j+1)(k-j)}{j^2}& =k \sum_{j=1}^k \frac{k^2-2kj+j^2+k-j}{j^2}& = (k^3+k^2)\sum_{j=1}^k \frac{1}{j^2}-(2k^2 +k)\sum_{j=1}^k \frac{1}{j}+k^2 & \sim \frac{\pi^2}{6}k^3.
h(G)=\frac{1}{a_{3}}\int_{0}^{a_{3}}dx_{3}\frac{1}{a_{2}}\int_{0}^{a_{2}}dx_{2}\frac{1}{a_{1}}\int_{0}^{a_{1}}dx_{1}f(x_{1}\frac{a_{1}}{a_{1}}+x_{2}\frac{a_{2}}{a_{2}}+x_{3}\frac{a_{3}}{a_{3}})\cdot e^{-iG\cdot r}.
\frac{1}{\sqrt{2\pi}}\cdot\frac{t}{rs}\cdot 2^{-(\zeta-\frac12)}\,\Gamma(\zeta+1) \cdot \left(\frac{2rs}{r^2+s^2+t^2}\right)^{\zeta+1}\times\,_2\tilde{F}_1\left(\frac{\zeta+1}{2},\frac{\zeta+2}{2};\zeta+\frac12;\frac{4r^2s^2}{\left(r^2+s^2+t^2\right)^2}\right).
\begin{array}{lll}Y_0=2 J_1, Y_1=-J_0+J_2 , Y_2=J_0+J_2 , begin{align*}2pt]y^0=\frac12 P_1, y^1=\frac12 (P_0 +P_2), y^2=\frac12 (-P_0 +P_2) , \end{array}
\min_{0{}\leq{}i{}\leq{}N}\|\nabla f(x_i)\|^2{}\leq{}\frac{2L\,\big(f(x_0)-f_*\big)}{1 + \sum\limits_{i=0}^{N-1}\left[2 h_i - h_i^2 \frac{-\kappa}{2 \min\left(1, \frac{1}{h_i}\right) - (1+\kappa)}\right]}
\left.\Gamma^{(\infty)}(p_{1j}^{2}; m_{0}, L_{1},L_{2}, \lambda_{1B},\lambda_{2B}, \epsilon) \right|_{s.p.}= \left. \Gamma^{(0)}(p_{1j}^{2}; m_{0}, \lambda_{1B}', \epsilon) \right|_{s.p.}= g\kappa^{2\epsilon},
\mathcal L/q\mathcal L =\left(\oplus_{i\in I}\mathcal L(\lambda_i)\right)/q\left(\oplus_{i\in I}\mathcal L(\lambda_i)\right) &\cong \oplus_{i\in I}\left(\mathcal L(\lambda_i)/q\mathcal L(\lambda_i)\right) \cong \oplus_{i\in I}\left(\mathbb F\otimes_{\mathcal A_0}\mathcal L(\lambda_i)\right) cong \mathbb F\otimes_{\mathcal A_0}\mathcal L
\int_{0}^{\infty}u^{c-1}\exp \big ( -a u^2 - b u \big ) v = (2 a)^{\tfrac{-c}{2}}\Gamma (c) \exp \big ( \tfrac{b^{2}}{8 a}\big ) D_{-c}\big ( \tfrac{b}{\sqrt{2 a}}\big ) ,
C_j\otimes \vartheta D_j\vartheta^{-1}+C_j^*\otimes \vartheta D_j^* \vartheta^{-1}= 2(\Re C_j\otimes \vartheta \Re D_j\vartheta^{-1}+\Im C_j\otimes \vartheta \Im D_j \vartheta^{-1}),
\hat{\Phi}( \vec{x}, t ) = \sum_{\vec{p}}\frac{1}{\sqrt{2 \omega_{\vec{p}}V}}[ \hat{a}_{\vec{p}}e^{\frac{i}{\hbar}( \vec{p}\cdot \vec{x}- E t )}+{\hat{b}}_{\vec{p}}^{\dag}e^{- \frac{i}{\hbar}( \vec{p}\cdot \vec{x}- E t )}]
\begin{aligned}\mathbf{\Phi}_{\mathrm{r}}=\mathsf{diag}(\phi_{\mathrm{r},1},\ldots,\phi_{\mathrm{r},M}),~~ \mathbf{\Phi}_{\mathrm{t}}=\mathsf{diag}(\phi_{\mathrm{t},1},\ldots,\phi_{\mathrm{t},M}), \end{aligned}
\frac{E_{V}}{A}= \frac{1}{2}\int_{- \infty}^{\infty}\frac{d k_{y}d k_{z}}{( 2 \pi )^{2}}\sum_{n = 1}^{\infty}( ( \frac{n \pi}{a \mu})^{2}+ ( \frac{k_{y}}{\mu})^{2}+ ( \frac{k_{z}}{\mu})^{2})^{\frac 1 2}e^{- ( ( \frac{n \pi}{a \mu})^{2}+ ( \frac{k_{y}}{\mu})^{2}+ ( \frac{k_{z}}{\mu})^{2})^{\frac 1 2}t}\rfloor_{t = 0}\, .
q\big(o(C)-1\big)-u_{\overline{\chi},C}=\alpha_{C,\overline{\chi}}^{q}o(C)+\beta_{C,\overline{\chi}}^{q}\quad\mathrm{with}\quad\alpha_{C,\overline{\chi}}^{q}\in\mathbb{Z}\ \mathrm{and}\ 0\leq\beta_{C,\overline{\chi}}^{q}<o(C).
L_{\rm G}(Q^r,\dot{Q^r},V,M^r) \stackrel{\rm def}{\equiv}L_{\rm Q}(Q^r,V) +M_a^{s_a}(\dot{Q}_a^{s_a}- \bar{Q}_a^{s_a}) + M_a^{n_a}(\dot{Q}_a^{n_a}-V_a),
\begin{aligned}u(t)=\gamma_{1}(t)H_{1}(\alpha_{1}[u])+\delta_1(t)L_{1}(\beta_{1}[v])+\int_{0}^{1}k_1(t,s)g_1(s)f_1(s,u(s),v(s))\,ds+ G_1(u)(t), v(t)= \gamma_{2}(t)H_{2}(\alpha_{2}[v])+\delta_2(t)L_{2}(\beta_{2}[u])+\int_{0}^{1}k_2(t,s)g_2(s)f_2(s,u(s),v(s))\,ds+ G_2(v)(t),\end{aligned}
\frac{1}{p_1'}+ \frac{1}{p_2'}+ \frac{1}{p}= 1 , \phi_1 (g) \phi_2 (g^{-1}g') = \phi_1 (g)^{p_1/p_2'}\phi_2 (g^{-1}g')^{p_2/p_1'}( \phi_1 (g)^{p_1}\phi_2 (g^{-1}g')^{p_2})^{1/p}
f(x, y, z)= x^p+\sum_{i=0}^k(-1)^i\dbinom{k+r}{k-i}\dbinom{i+r-1}{i}t^{k-i}y^{r+i}z^{k+l-i}= x^p+\sum_{i=0}^k(-1)^ia_{k, r}(i)t^{k-i}y^{r+i}z^{k+l-i},
Q^TL_9(\lambda)Q=\left[\begin{array}{ccc|c}-P_4 \lambda P_4 0 0 \lambda P_4 \lambda P_3-P_2 \lambda P_2 -I_n 0 \lambda P_2 \lambda P_1+P_0 \lambda I_n \hline0 -I_n \lambda I_n 0\end{array}\right].
(1-u/p) \prod_{i \geq 3 \atop i \odd}(1-u^2/p^i) \frac{u^{|\lambda|}}{p^{n(\lambda)+|\lambda|}\prod_{i \geq 1}\prod_{j=1}^{\lfloor m_i(\lambda)/2 \rfloor}(1-1/p^{2j})}
\pi_{|U}=\sum_{i=1}^{k}\frac{\partial}{\partial q^{i}}\frac{\partial}{\partial p_{i}}+\frac{1}{2}\sum_{i,j=1}^{n-2k}\phi^{ij}(x)\frac{\partial}{\partial x^{i}}\frac{\partial}{\partial x^{j}},
\mathbb{E}_{\mathbf{H}, \mathbb{W}}\left [- \log_2 \left( \|\tilde{\mathbf{h}}_k \hat{\mathbf{Q}}^H \|^2 \right) \right ] =\log_2 e \sum_{m = K-1}^{n_T-2}\sum_{l=0}^{n_T -m-2}\frac{(n_T-2)!}{m!~ l!~ (n_T -m-2 - l) !}(-1)^{l}\frac{1}{m+l}
\begin{pmatrix}1 0 -\frac{1}{2}0 1 -\frac{1}{2}0 0 -\frac{1}{2}-1 0 0 0 -1 0 \end{pmatrix}\begin{pmatrix}x_1 x_2 x_3 \end{pmatrix}\geq \begin{pmatrix}-1 -1 -1 -1 -1 \end{pmatrix}.
\Delta\begin{pmatrix}\alpha \beta \gamma \delta\end{pmatrix}=\begin{pmatrix}\alpha \beta \gamma \delta\end{pmatrix}\otimes\begin{pmatrix}\alpha \beta \gamma \delta\end{pmatrix},\varepsilon\begin{pmatrix}\alpha \beta \gamma \delta\end{pmatrix}=\begin{pmatrix}1 0 0 1\end{pmatrix},
\gamma A_{0}^{a}= \eta_{1}^{a}, \; \gamma A_{i}^{a}= \partial_{i}\eta_{2}^{a}+ f_{\; \; b c}^{a}\eta_{2}^{b}A_{i}^{c}, \; \gamma B^{0 i}= \eta_{1}^{i}, \; \gamma B^{i j}= \partial^{\left[ i \right.}\eta_{2}^{\left. j \right]},
u^{\alpha^{n,\varepsilon,k}(^{t_k}\omega,\cdot)}(t_k, X_{t_k}^{\alpha,0,x}(^{t_k}\omega))=\,Y_{t_k,T}^{\alpha^{n,\varepsilon,k}(^{t_k}\omega,\cdot),t_k,X_{t_k}^{\alpha,0,x}(^{t_k}\omega)}=\,Y_{t_k,T}^{\alpha^{n,\varepsilon,k},t_k,X_{t_k}^{\alpha,0,x}}(^{t_k}\omega).
k_i = k_{i,0}x_0 + k_{i,1}x_1 + \cdots + k_{i,n}x_n, \ell_i = \ell_{i,0}x_0 + \ell_{i,1}x_1 + \cdots + \ell_{i,n}x_n, \mbox{and}m_i = m_{i,0}x_0 + m_{i,1}x_1 + \cdots + m_{i,n}x_n,
_{[a_1,\ldots,a_i]}F(x)_{[b_1,\ldots,b_j]}= \sum_{k=0}^{\infty}\frac{\left ( \prod_{m=1}^i \frac{\Gamma(a_m+k)}{\Gamma(a_m)}\right )}{\left ( \prod_{n=1}^j \frac{\Gamma(b_n +k)}{\Gamma(b_n)}\right )}x^k
\gamma_{r}^{\varepsilon}=\gamma_{\overline{\Omega}}^{B_{r}(y_{1})^{c}}+(1-\varepsilon)\gamma_{\overline{\Omega}}^{B_{r}(y_{1})}+\varepsilon(\pi_{1},\mathcal{T}_{y_{1}}^{y_{2}})_{\#}\gamma_{\overline{\Omega}}^{B_{r}(y_{1})},
\tilde{{\bf E}}{}={}\frac{1}{2}\left({\bf u}\mathrm{e}^{\mathrm{i}\Theta}{}+{}{\bf u}^{*}\mathrm{e}^{- \mathrm{i}\Theta}\right) \quad , \qquad \tilde{{\bf B}}{}={}\frac{1}{2}\bar{\bf k}{}\times{}\left({\bf u}\mathrm{e}^{\mathrm{i}\Theta}{}+{}{\bf u}^{*}\mathrm{e}^{- \mathrm{i}\Theta}\right) \quad ,
\tilde g_{f}=\ -(\xi^1)^2 - (\xi^2)^2 - (\xi^3)^2 - (\xi^4 )^2 + (\xi^5)^2 + (\xi^6)^2 +(\xi^7)^2 +(\xi^8 )^2,\text{and}\hat{\phi}=\ \xi^{1234}- \xi^{1256}+ \xi^{1278}- \xi^{1357}-\xi^{1368}- \xi^{1458}+ \xi^{1467}&+ \xi^{2358}- \xi^{2367}-\xi^{2457}-\xi^{2468}+ \xi^{3456}- \xi^{3478}+ \xi^{5678}.
\Big ( \begin{array}{c c}{{\mathrm{e}^{V}}}\\ \end{array}\Big ) = \frac 1{\sqrt{( \bar{z}_{+}\cdot z_{+}) ( \bar{z}_{-}\cdot z_{-}) - ( \bar{z}_{+}\cdot z_{-}) ( \bar{z}_{-}\cdot z_{+})}}\left( \begin{array}{c c}{{\bar{z}_{-}\cdot z_{-}}}&{{- \bar{z}_{-}\cdot z_{+}}}\\{{- \bar{z}_{+}\cdot z_{-}}}&{{\bar{z}_{+}\cdot z_{+}}}\\ \end{array}\right) \ ,
(\vec{X}(x_1), \vec{X}(x_2) )= \sum_{\vec{k}\in K}\left\{R(k, |\tilde{t}_k(x_1)- \tilde{t}_k(x_2)| ) + b(k)^2 \right\}\mbox{e}^{i k_\mu(x_1^\mu-x_2^\mu)}\;, \hspace{1cm}x_1,x_2 \in{\sf M}^4\;.
G^{(4)}(u_1-{{u}},u_2,u_3,u_4)= = \frac{1}{a}\exp \left( \frac{3{u_3}^4}{256 a^{12}}+\frac{{u_2}{u_3}^2}{16 a^8}+\frac{({u_1-u}){u_3}}{4 a^4}\right) \times \Lambda \left( \frac{3{u_3}^2}{8 a^6}+\frac{{u_2}}{a^2},\frac{{u_3}^3}{8 a^9}+\frac{{u_2}{u_3}}{2 a^5}+\frac{{u_1-u}}{a}\right)
F_{BEMWE}(x_{1},x_{2})=\left[ 1-e^{-\lambda \alpha (e^{(x_{1}/\alpha)^{\beta}}-1)}\right]^{\gamma_{1}}\left[ 1-e^{-\lambda \alpha(e^{(x_{2}/\alpha )^{\beta}}-1)}\right]^{\gamma_{2}}\left[ 1-e^{-\lambda\alpha (e^{(z/\alpha )^{\beta}}-1)}\right]^{\gamma_{3}},
\widetilde X_1 = \left( \begin{array}{ccccc}0.4349 0.1406 0.0652 0.4912 0.9345 0 0.3751 0.0682 0.7668 0.0518 0 0.3383 0.6881 0.9466 0.1009 0 0 0 0.5917 0 0.2989 0.0878 0.8343 0.2834 0.0762\end{array}\right),
G_{\,+}^{\ell k}(q^0) = i \int^\infty_{-\infty}dx^0 e^{iq^0 x^0}\int d\vec{x}\left( \Phi_p, \left( j^\ell (\vec{x}, x^0) j^k (\vec{y}, 0)\right)_+ \Phi_p \right)
\Psi \left( x\rightarrow -\infty \right) =e^{ipx}\left(\begin{array}{c}0 1\end{array}\right) +e^{-ipx}\left[ B\left(\begin{array}{c}0 1\end{array}\right) -\frac{im}p\left(\begin{array}{c}1 0\end{array}\right) \int V\left( x^{\prime}\right){\psi^{+}}^2\left( x^{\prime}\right) dx^{\prime}\right] .
T_{s_1}^{(1)}P_{s_1}^{\perp}\otimes \ldots \otimes T_{s_k}^{(1)}P_{s_k}^{\perp}\otimes T_{s_{k+1}}^{(1)}P_{s_{k+1}}\otimes \ldots \otimes T_{s_{2d}}^{(1)}P_{s_{2d}}\otimes \overline{x}\mapsto \mid e_{s_1}\otimes \ldots e_{s_k}\rangle \langle e_{s_{k+1}}\otimes \ldots \otimes e_{s_{2d}}\mid \otimes \overline{x},
u(\cdot,t) = \cos (t D) u_0 + \frac{\sin (t D)}{D}u_1 + \int_0^t \frac{\sin (t-s) D}{D}F(\cdot, s)\, ds; \partial_t u (\cdot,t) = -D \sin (t D) u_0 + \cos (t D) u_1 + \int_0^t \left[\cos (t-s) D\right] F(\cdot, s)\, ds;
\sum_{|z_k|<1}\int_{D(z_k, 3r)}|f(z)|^pe^{-\frac{\alpha p}{2}|z|^2}dV(z)\leq M \sum_{|z_k|<1}\int_{D(z_k, 3r)}\frac{|f(z)|^p |z|^{mp}}{e^{\frac{\alpha p}{2}|z|^2}}dV(z) \lesssim \sum_{k=1}^\infty\int_{D(z_k, 3r)}|f(z)|z|^m|^pe^{-\frac{\alpha p}{2}|z|^2}dV(z).
f(G_K) \geq \frac{1}{K}f(O_K)+\left(1-\frac{1}{K}\right)f(G_{K-1}) &\quad\vdots &\geq \frac{1}{K}f(O_K)\sum\limits_{i=0}^{K-1}\left(1-\frac{1}{K}\right)^i &=\left(1-\left(1-\frac{1}{K}\right)^K\right)f(O_K).
I_{0,1}=\frac{\mu^\epsilon}{(2\pi)^{3-\epsilon}}\int\frac{d^{3-\epsilon}\vec q}{2|\vec q|}\left(\frac{1}{k^2+2q\cdot k}+ \frac{1}{k^2-2q\cdot k}\right)\left[\frac{\vec q^2}{k^2}\left(1-x^2\right)\right]^{0,1}\left[\frac 1 2 + N\left(\frac{|\vec q|}{T}\right)\right].
H(G^{\{v\}}_{-1/2}v_{\nu,\ell_0},G^{\{v\}}_{-1/2}v_{\nu,\ell_0})= -2(k+h^\vee)\langle \phi(v),v\rangle l_0+\langle \phi(v),v\rangle(\nu|\nu+2\rho^\natural) &-2(k+1)\langle \phi(v),v\rangle(\xi|\nu)+2(\xi|\nu)^2\langle \phi(v),v\rangle,
a^{n+m}_{k^m \cdot \bar s}(\omega) =F_{s}^{(n)}(\omega) \circ F_{(00\ldots 0)}^{(m)}(\theta^n\omega) (p(\theta^{n+m}\omega)) &= F_{s}^{(n)}(\omega) (p(\theta^n\omega))= a^n_{\bar s}(\omega).
\mu (X) = \kappa_1 ( \mu ), mu (X^2) = \kappa_2 ( \mu ) + \kappa_1 ( \mu )^2, mu (X^3) = \kappa_3 ( \mu ) + 3 \kappa_1 (\mu ) \kappa_2 ( \mu ) + \kappa_1 ( \mu )^3,
|g_{\lambda_i}(z)^{-\frac{1}{2}}-\lambda_i^{-\frac{1}{2}}| =\frac{|g_{\lambda_i}(z)-\lambda_i|}{|\lambda_ig_{\lambda_i}(z)|^{\frac{1}{2}}|g_{\lambda_i}(z)^{\frac{1}{2}}+\lambda_i^{\frac{1}{2}}|}&=\frac{|\lambda_iz^2-2z|}{|\lambda_ig_{\lambda_i}(z)|^{\frac{1}{2}}|g_{\lambda_i}(z)^{\frac{1}{2}}+\lambda_i^{\frac{1}{2}}|}.
\begin{array}{ll}\displaystyle\partial_{t}u_j +a_j(x,t)\partial_{x}u_j +\sum_{k=1}^{n}b_{jk}(x,t)u_k +\sum_{k=1}^{n}\int_{0}^{x}g_{jk}(y,t)u_k(y,t)dy [2mm]=\displaystyle \sum_{k=m+1}^{n}h_{jk}(x,t)u_k(0,t) +\sum_{k=1}^{m}h_{jk}(x,t)u_k(1,t) + f_j(x,t), \;\;\; x\in(0,1),\; j\le n, \end{array}
\rho_{min,GS}^{\left(J'_{D}\right)}\overset{}{\geq}\gamma_{min}^{\left(J'_{D}\right)}\overset{}{\geq}\gamma_{min}^{\left(J_{\varsigma}\right)}\overset{}{\geq}\zeta_{min}^{\left(J_{\varsigma}\right)}:=\rho_{min,GS}^{LB},
V_{G}^{T}( \Phi_{0}) = V_{G E P}^{2 + 1}( \Phi_{0}) + \frac{x - 1}{8 \pi \hat{\beta}}-{\frac{x}{8 \pi \hat{\beta}}}\ln ({\hat{\beta}}^{2}x ) +{\frac{1}{8 \pi \hat{\beta}}}\ln{\hat{\beta}}^{2}+ \frac{x^{\frac{3}{2}}- 1}{1 2 \pi}- \frac{\hat{\beta}( x^{2}- 1 )}{9 2 \pi}\; .
J_{D 0}( p ) = \frac{\mathrm{P f}( \theta^{- 1})}{2 \pi}\int d^{2}x ~ \left[ e^{i p x}P \exp \left( i \int_{0}^{1}\hat{A}_{i}( x + l \tau ) l^{i}d \tau \right) \right] \; ,
\max_{M=1,\dots, N}\left\|\sum_{n=1}^{M}(v(t_{n})-v_*(t_{n}))\right\|_\sigma^2 \leq\left(\sum_{n=1}^{N}\left\|(v(t_{n})-v_*(t_{n}))\right\|_\sigma\right)^2\lesssim \tau^2.
\frac{C}{t^{2}\log^{3b+3N}(t)}\geq ||\partial_{r}v_{4,c}(t,r) + \frac{v_{4,c}(t,r)}{r}||_{L^{2}(r dr)}= ||\xi \widehat{v_{4,c}}(t,\xi)||_{L^{2}(\xi d\xi)}
\begin{array}{r c l}{{}}&{{}}&{{T_{i}( f_{i}) = - t_{i}^{- 1}e_{i}~ ,}}\\{{}}&{{}}&{{T_{i}( f_{j}) = \displaystyle \sum_{m = 0}^{- a_{i j}}( - 1 )^{m - a_{i j}}q_{i}^{m}\frac{f_{i}^{m}}{[ m ]_{q_{i}}!}f_{j}\frac{f_{i}^{- a_{i j}- m}}{[ - a_{i j}- m ]_{q_{i}}!}~ , \mathrm{i f ~}i \not = j ~ ;}}\\ \end{array}
\| h_r(s,t)- I(s,t)\| = \left\| \frac{u}{\|u\|}- v \right\| \leq \left\| \frac{u}{\|u\|}- \frac{u}{r}\right\| + \left\| \frac{u}{r}- v \right\| \leq \frac{(r+2K)2K}{r(r-rK)}+ \frac{2K}{r},
\frac{1}{\Gamma(2-\alpha)}\int_0^x{(x-t)^{1-\alpha}}\int_0^t (t-z)^{\alpha - 2}E_{\alpha, \alpha-1}(\lambda (t-z)^{\alpha})f'(z)dz dt & = \frac{1}{\Gamma(2-\alpha)}\int_0^xf'(z)\int_z^x{(x-t)^{1-\alpha}}(t-z)^{\alpha - 2}E_{\alpha, \alpha-1}(\lambda (t-z)^{\alpha}) dtdz
S = \int \frac{1}{1 6 \pi G}R \sqrt{- g}d^{4}x - \int \left[ \frac{1}{4}F_{\mu \nu}^{a}F^{a \mu \nu}+ \frac{1}{2}D_{\mu}\phi^{a}D^{\mu}\phi^{a}+ \frac{1}{4}\lambda ( \phi^{a}\phi^{a}- v^{2})^{2}\right] \sqrt{- g}d^{4}x \
\;\|I_4+I_5+I_6\|_{L^2}\leq \;C\|A^{-\beta s}(Id - e^{-h\nu A^s})\|_{\mathcal{L}(L^2)}\|A^{\beta s}e^{-t\nu A^s}\|_{\mathcal{L}(L^2)}\|A^{1-s/2}f(w_1,w_2)(t)\|_{L^2}&\;+C\|(Id-e^{-2h\nu A^s})\|_{\mathcal{L}(L^2)}\|A^{1-s/2}(f(w_1,w_2)(t+h)-f(w_1,w_2)(t))\|_{L^2}\leq \;Ch^\beta t^{-(\beta+1/2)}R_1R_2,
C_{\left( A,i\right) \left( B,j\right)}{}^{\left( C,k\right)}=\left( K_{ij}{}^{k}-K_{ij}{}^{k+n}\right) C_{AB}{}^{C}& =\left( K_{ji}{}^{k}-K_{ji}{}^{k+n}\right) C_{AB}{}^{C}& =-\left( K_{ji}{}^{k}-K_{ji}{}^{k+n}\right) C_{BA}{}^{C}& =-C_{\left( B,j\right) \left( A,i\right)}{}^{\left( C,k\right)},
\begin{array}{r c l}{\nu}&{=}&{\left( E_{\mathrm{e - v i b}}^{\prime}- E_{\mathrm{e - v i b}}^{\prime \prime}\right) / h c}\\{}&{=}&{( E_{0}^{\prime}- E_{0}^{\prime \prime}) / h c + ( G_{q^{\prime}}^{\prime}- G_{q^{\prime \prime}}^{\prime \prime}) = \nu_{e}+ \nu_{v},}\\ \end{array}
\leq \frac{1}{(\log N)^{2}}\displaystyle\sum_{n=N+1}^{N^{2}}\sqrt{n(n-1)}(\log(\frac{n}{n-1}))^2 & \leq\frac{1}{(\log N)^{2}}\displaystyle\sum_{n=N+1}^{N^{2}}\frac{\sqrt{n(n-1)}}{(n-1)^{2}}& \leq \frac{1}{\log N}+ \frac{N}{2(N^2-1)(\log N)^{2}},
+\infty>\int_\Omega \frac{1}{d_\Omega^\alpha}\,dx\ge \frac{1}{2^\alpha\,r^\alpha}\,\int_{B_{r}(x_0)}\,dx=\frac{\omega_N}{2^\alpha}\,r^{N-\alpha}=\frac{\omega_N}{2^\alpha}\,d_\Omega(x_0)^{N-\alpha}.
E_{I_n}^{n}\simeq E_{I_{n-1}}^n \times_{s,\alpha_n}E^0_{I_{n}}\simeq (\cdots((E^n \times_{s,\alpha_1}E^0_{I_1}) \times_{s,\alpha_2}E^0_{I_2})\times_{s,\alpha_3}\cdots ) \times_{s,\alpha_n}E_{I_n}^0 &\simeq E^n \times_{s, \alpha_1 \circ \cdots \circ \alpha_n}E_{I_n}^0= E^n \times_{s, \alpha}E_{I_n}^0.
+\,\frac{D-2}{2\pi}\frac{m}{M_0^2}\,\frac{1}{R^2}\,-\,(D-2)\,\sqrt{\frac{2}{\pi\alpha r_s}}\,\frac{1}{\sqrt{R}}\,\exp[-2\sqrt{2}\,R/\sqrt{\alpha}r_s]\,{,}\quad R\to\infty\,{.}
T=\frac{S_{n}S_{n+3}}{S_{n+1}S_{n+2}}+\alpha(\frac{S_{n+1}^2}{S_{n}S_{n+2}}+\frac{S_{n+2}^2}{S_{n+1}S_{n+3}})+\beta\frac{S_{n+1}S_{n+2}}{S_{n}S_{n+3}}.
\psi^{(m)}(t) = (-1)^{m+1}m! \sum_{n=0}^{\infty}\frac{1}{(n+t)^{m+1}},t>0 \psi_{p,q}^{(m)}(t) = (\ln q)^{m+1}\sum_{n=1}^{p}\frac{n^{m}q^{nt}}{1-q^{n}},t>0.
( i \gamma^{\mu}\partial_{\mu}- m_{1}) \Psi_{1}= 2 \pi \lambda \varepsilon_{\mu \nu}J_{2}^{\nu}\gamma^{\mu}\Psi_{1}, \quad ( i \gamma^{\mu}\partial_{\mu}- m_{2}) \Psi_{2}= 2 \pi \lambda \varepsilon_{\nu \mu}J_{1}^{\nu}\gamma^{\mu}\Psi_{2},
\left[\overline{\mathcal{DR}}_2(d)\right]= (d^2-1)\left(\frac{d^2-1}{4}\left(\psi_1+\psi_2 \right)\left(\delta_{1,2}-\delta_{1,1}\right)-(d^2+1)\delta_2\left(\delta_2+\frac{7}{10}\delta_{1,2}\right) \right).
\begin{cases}f(y_k,y)+\lambda_{k-1}\langle \overrightarrow{x_{k-1}y_k},\overrightarrow{y_ky}\rangle \geq 0 ,\ \ \ \ \ \forall y\in K, x_k=\alpha_k u\oplus(1-\alpha_k)y_k, \end{cases}
\tau_1(t) ={|\hat C|}^{1/2}\,\Big(1-\frac{2({|\hat C|}^{1/2}-\tau_1^0)}{({|\hat C|}^{1/2}\!+\tau_1^0)\,e^{-2\,t\,{|\hat C|}^{1/2}}\!+{|\hat C|}^{1/2}\!-\tau_1^0}\Big), \ \ \tau_1(0)=\tau_1^0\in [-{|\hat C|}^{\frac12},{|\hat C|}^{\frac12}],
\int_{Z}b(z)\,d(\nu^{r(h)}\cdot h)(z) =\int_{G}b((\gamma^{-1}\cdot z)\cdot h)\,d\lambda^{r(z)}(\gamma) =\int_{G}b(\gamma^{-1}\cdot (z\cdot h))\,d\lambda^{r(z\cdot h}(\gamma) = \int_{Z}b(z)\,d\nu^{s(h)}(z).
|\bar a_{ij}(\hat x) - \bar a_{ij}( \hat y ) | \leq \breve c \, | \hat x - \hat y | \max_{B ( x, |x|/2)}\left\{\left( |\nabla \bar u(z) | + | \nabla G(z) | \right)^{p-3}\sum_{i,j= 1}^n \, ( | u_{z_i z_j}(z) | + | G_{z_i z_j}(z) | ) \right\}& \leq \breve c^2 | \hat x - \hat y | \, |x|^{\frac{(2-p)n - 1}{p-1}}
\|\bullet\| =\|\bullet\|_{L^{2}(\mathbb{R}_{+}^{n+1})}, \|\bullet\|_{0}=\|\bullet\|_{L^{2}(\mathbb{R}^{n}\times\{0\})}\langle\bullet,\bullet\rangle =\langle\bullet,\bullet\rangle_{L^{2}(\mathbb{R}_{+}^{n+1})}, \langle\bullet,\bullet\rangle_{0}=\langle\bullet,\bullet\rangle_{L^{2}(\mathbb{R}^{n}\times\{0\})}
\left\Vert E^{l}u\right\Vert_{2,\widetilde{P}_{0}(h^{l-1})_{\delta}}=\left\Vert EE^{l-1}u\right\Vert_{2,\widetilde{P}_{0}(h^{l-1})_{\delta}}\leq C\left\Vert E^{l-1}u\right\Vert_{2,(h^{l-1})_{\delta}},
\left( \prod_{i = 1}^{N}G ( x , t_{i}) \right) = \sum_{m_{1}= 0}\sum_{m_{2}= 0}\cdots \sum_{m_{N - 1}= 0}\sum_{n_{N}}\det ( A_{n_{j}+ i - j}) \left( \det U \right)^{n_{N}}\chi_{( \ell_{1}, \ell_{2}, \cdots , \ell_{N})}( U ) .
\begin{array}{c}min\Big\{max\Big[rV^{ij}(x)-\mathcal{A}V^{ij}(x)-f(x,i,j);\qquad\qquad\qquad\qquad V^{ij}(x)-N^{ij}[V](x)\Big];V^{ij}(x)-M^{ij}[V](x)\Big\}=0.\end{array}
\left\langle\prod_{k=1}^M\det(x_k-X)\right\rangle_{n\times n}=\frac{a_\varnothing(\partial)}{a_\varnothing(\zeta)}\prod_{k=1}^M\psi^{(1)}(t;\zeta_k)\mathrm{as}\N\to\infty\ ,\ \varepsilon\to 0\ ,
\varepsilon_i = \alpha_n + \alpha_{n-1}+ \dots + \alpha_i, \;\; i < n, \varepsilon'_i = \alpha'_n + 2\alpha_{n-1}+ \dots + 2\alpha_i \;\; i < n, \varepsilon_n = \alpha_n, \;\; \varepsilon'_i = 2\varepsilon_n = 2\alpha_n = \alpha'_n.
\pi_{\mathcal{L}}(\mathsf{a}_s) = \begin{cases}-v\tilde{A}_{s}^{-1}1_{\mathcal{L}}s \in W_{\mathcal{L}}^\circ -\tilde{A}_s1_{\mathcal{L}}s \not\in W_{\mathcal{L}}^\circ \end{cases}
\hat{L}_{uu}(t) = L_{uu}(t,u_0(t), \dot{u}_0(t)), \hat{L}_{u\dot{u}}(t) = L_{u\dot{u}}(t,u_0(t), \dot{u}_0(t)), \hat{L}_{\dot{u}\dot{u}}(t)= L_{\dot{u}\dot{u}}(t,u_0(t), \dot{u}_0(t)),
G ( x^{\prime}, t^{\prime}; x^{\prime}, t^{\prime}) = i \int \int \frac{d \omega^{\prime \prime}}{2 \pi}\frac{d k^{\prime \prime}}{2 \pi}\frac{1}{\omega^{2}- k^{2}- 4 + i \rho}\left( 1 + K_{0}( k^{\prime \prime}) e^{- 2 i k^{\prime \prime}x^{\prime}}\right) .
\begin{array}{c}{U_{0}( z ) = \frac 1 2 ( u_{0}+ u_{- 1}) + \frac 1{\sqrt{2}}( u_{0}- u_{- 1}) \, z_{p e r}}{V_{0}( z ) = \frac 1 2 ( v_{0}+ v_{- 1}) + \frac 1{\sqrt{2}}( v_{0}- v_{- 1}) \, z_{p e r},}\end{array}