harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\{j_{\alpha}^{0}, j_{\beta}^{0}\right\}\approx \left[ 2 \Gamma_{\alpha \beta}^{m}\cdot j_{m}^{0}+ \left( e m b^{*}S_{\alpha \beta}\left( \Delta \right) \right)_{r}\cdot \Lambda^{r}\right] \cdot \delta \left( \sigma - \sigma^{\prime}\right) \quad ,
= - 4 i \left({\epsilon}_{1}\cdot p_{1}{\epsilon}_{2}\cdot p_{2}\frac{1}{p^{2}+ m^{2}}+{\epsilon}_{2}\cdot p_{1}{\epsilon}_{1}\cdot p_{2}\frac{1}{q^{2}+ m^{2}}\right) - 2 i{\epsilon}_{1}\cdot{\epsilon}_{2}
2H_{0}\left[ \left( fg^{\prime}\right)^{2}-\left( f^{\prime}g\right)^{2}\right]^{\frac{3}{2}}=\left( fg^{\prime}\right)^{2}f^{\prime \prime}g-2fg\left( f^{\prime}g^{\prime}\right)^{2}+\left(f^{\prime}g\right)^{2}fg^{\prime \prime}.
\mu_i(x)-\mu_i(y)=r_i x_i \left(1- \sum_{j=1}^{d}a_{ij}x_j^+ \right)-r_i y_i \left(1- \sum_{j=1}^{d}a_{ij}y_j^+ \right) &= r_i(x_i-y_i)-r_i\left[\left(x_i\sum_{j=1}^{d}a_{ij}x_j^+\right)-\left(y_i\sum_{j=1}^{d}a_{ij}y_j^+\right)\right].
X = \left( \begin{array}{c c c c}{{0}}&{{- p \! \! /^{T}}}&{{0}}&{{0}}\\{{p \! \! /}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- 1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- 1}}\\ \end{array}\right)
\sum_{k: M_k\in B_{j+1}^{j+1}}\lambda_{k}\frac{\det(M_k)}{\det(A)}|A^{-1}-M_{k}^{-1}|\lesssim \sum_{k: M_k\in B_{j+1}^{j+1}}\lambda_{k}j^{2n'-2m_2'-m_1'+1}i^{-m_1'}&=\nu(B_{j+1}^{j+1})j^{2n'-2m_2'-m_1'+1}i^{-m_1'}\lesssim j^{2m_1'+3m_2'-3n'}i^{m_1'}j^{2n'-2m_2'-m_1'+1}i^{-m_1'}= j^{m_1'+m_2'-n'+1}\leq 1.
\Psi^{Ai}(\zeta) \sim \zeta^{-\frac{\sigma_3}{4}}\Psi^{Ai}_{0}\left(I+\sum_{j=1}^{\infty}\frac{\left(\Psi^{Ai}_{0}\right)^{-1}\Psi^{Ai}_{j}\left(\frac{2}{3}\right)^{-j}}{\zeta^{3j/2}}\right), \zeta\rightarrow\infty.
t^{\frac{N}{2}}u(x, t) =q_*[M_{0, 1}+ o(1)]U(|x|) -q_*\left[ \frac{N}{2}M_{0, 1}+o(1) \right] t^{-1}(UF_0)(|x|) & + t^{-2}O(|x|^4U(|x|))+ \sum^N_{i=1}[M_{1, i}+o(1)] t^{-1}U_1(|x|) Q_{1, i}\left( \frac{x}{|x|}\right) & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+t^{-2}U_1(|x|) O(|x|^2) +o(t^{-1})
\begin{aligned}\|U_0(t)u\|_{L^{p,2}_{i,j}}=\|\mathcal{R}_{i,j}U_0(t)u\|_{L^{p,2}_{d_{i,j}}}&=\|\widetilde{U}_0(t)\mathcal{R}_{i,j}u\|_{L^{p,2}_{d_{i,j}}}&\lesssim_p |t|^{-3(1/2-1/p)}\|\mathcal{R}_{i,j}u\|_{L^{p,2}_{d_{i,j}}}&\lesssim_p|t|^{-3(1/2-1/p)}\|u\|_{L^{p,2}_{i,j}}.\end{aligned}
\Gamma(F) := \sum_{j=1}^m \sum_{l=k_{j-1}}^{k_j-1}\int_{\mathbb{R}^4}\Big( \int_{\mathbb{R}}F(y,x') F(y,y') \omega_{2^l}(y-q) dy \Big)^2 |\vartheta|_{2^{k_j}}(x'-p) |\psi|_{2^l}(y'-p) \,dx' dy' dp dq .
e^{i\theta(t)}\zeta(1/2+it) = 2\, \Re \left(e^{-i\theta(t)}\sum_{n=1}^{n_1}\frac{e^{it \log n}}{\sqrt{n}}\right) - \frac{(-1)^{n_1}}{\sqrt{a}}\sum_{r=0}^m \frac{C_r(z)}{a^r}+ R_m(t).
\lambda\left( A_{\alpha}\right) =\max\left\{\lambda\left( A_{\alpha}\left( H\right) \right) :HG\right\}, \lambda_{\min}\left( A_{\alpha}\right) =\min\left\{\lambda_{\min}\left( A_{\alpha}\left( H\right) \right) :H\text{is a component of}G\right\}.
\frac{u_{0}'}{u_{0}}= \frac{\bar{v}_{0}'}{\bar{v}_{0}}, \frac{u_{0}''}{u_{0}}= \frac{\bar{v}_{0}''}{\bar{v}_{0}}, \frac{u_{0}'''}{u_{0}}= \frac{\bar{v}_{0}'''}{\bar{v}_{0}},
C_{H}^{\circ}\left(a,v,\varphi\right)\overset{{\rm def}}{=}\left\{a\right\}\cup\left\{u\in H\,:\,\left\langle u-a,v\right\rangle >\cos\varphi\,\left\Vert u-a\right\Vert \left\Vert v\right\Vert \right\}\mbox{.}
f_n= A_{2-n}A_{4-n}A_{6-n}\Big(\sum_{k=1}^n\psi_k\Big)+K_{6-n}\Big(\sum_{k=1}^n\psi_k\Big)+f = O(r^{n+1})(\log^3 r+\log^2 r+\log r+1)+O(r^{n+2})\log^4 r.
\mathbb{E}[T_{a,b}^{n,n-1}(P_{N,n}^\beta)] = C_{N,n}^{\beta,b}\int_{-1}^1|h|^a(1-h^2)^{n\beta-{n-1\over 2}(n+b+1)}F_{1,\beta}(h)^{N-n}\,dh,
\langle m; d^p\rangle = \sum_{\substack{j\in J \mathcal{L}(e_j) < +\infty}}\langle m^{j}; d^p\rangle + \sum_{\substack{j\in J \mathcal{L}(e_j) = +\infty}}\langle m^{j}; d^p\rangle.
\frac{1}{\Gamma(\nu+1)}\frac{d}{ds}\int_0^s (s-u)^\nu \sqrt{Y_u}dB^1_u = \frac{1}{\Gamma(\nu)}\int_0^s\int_0^v \frac{1}{(s-v)^{1-\nu}}\sqrt{Y_u}dB^1_udv &=\frac{1}{\Gamma(\nu)}\frac{d}{ds}\int_0^s \frac{1}{(s-v)^{1-\nu}}\big(\int_0^v\sqrt{Y_u}dB^1_u\big)dv &= \frac{1}{\Gamma(\nu)}\frac{d}{ds}\int_0^s \frac{Z_v}{(s-v)^{1-\nu}}dv &=D^{1-\nu}Z_s.
\begin{aligned}d(x_1,x_2) \leq d(x_1, \bar x_2)+ d(\bar x_2, x_2) & \leq d(x_1,\bar x_2)+ \lambda d(f(\bar x_2),f(x_2)) & = d(x_1,\bar x_2)+ \lambda d(f(\bar x_2), f(x_1)) & \leq d(x_1, \pi^{-1}( y_2)) + \lambda^2 d(x_1, \pi^{-1}( y_2))^{\beta},\end{aligned}
\delta (\partial_x z )^{\natural,2,i,i}_{s,u,t}=- 2\sum_{l=1}^3 \partial_x W_{u,t}^{i,l}\delta(\partial_x z^l z^i)_{s,u}+2\sum_{l=1}^3\delta \partial_x\mathbb{W}^{i,l}_{s,u,t}z_s^lz_s^i-2\sum_{l=1}^3 \partial_x\mathbb{W}^{i,l}_{u,t}\delta(z^lz^i)_{s,u}&\quad+
g_{2/3, 3/4}^1 (w_1) = \frac{(1+\epsilon) \phi_{\ell}}{4}\log \left( 1 + \frac{w_1 P'}{4}\right) - \frac{|A^{\ast}|}{k_{\ell}}H_2 \left( \frac{w_1}{|A^{\ast}|}\right).
c_k'(0^+) = -C_s\left(\int_{0}^{+\infty}\ln |1 - \tau^4| \tau^{-(1 + 2s)}d\tau\right. & \qquad\quad\left.+\, (k - 2) \int_{0}^{\infty}\ln (1 + \tau^2) \tau^{-(1 + 2s)}d\tau\right) \mbox{for}\k \geq 2.
I(U_*+\varphi)= I(U_*) +\frac12\int_\Omega |\nabla \varphi|^2 + \int_\Omega \bigl( u_0^{2*-1}-\sum_{j=1}^k U_j^{2^*-1}\bigr)\varphi &-\frac1{2^*}\int_\Omega \Bigl( | U_*+\varphi|^{2^*}- | U_*|^{2^*}\Bigr).
{\left< T_{\sigma^{-}\sigma^{-}}^{\left( i n \right)}\left( \sigma^{-}\right) \right>}_{i n}\rightarrow -{\frac{1}{4 \delta^{2}}}+{\frac{\lambda^{2}}{4 8}}\left( 1 -{\frac{1}{{\left( 1 + \lambda \Delta e^{\lambda \sigma^{-}}\right)}^{2}}}\right) .
\begin{array}{l}A'_{\mu}=UA_{\mu}U^{-1}+\frac ig U\partial_{\mu}U^{-1}, Q'_{\mu}=UQ_{\mu}U^{-1},\,\,\, J'_{\mu}=UJ_{\mu}U^{-1}, C'_{\mu}=UCU^{-1},\,\,\, \overline{\xi}'=U\overline{\xi}U^{-1}, \overline{C}'=U\overline{C}U^{-1},\,\,\, \xi '=U\xi U^{-1}.\end{array}
\bar{E}_{2}= \frac{v^{2}}{2}\int \frac{d^{3}q}{( 2 \pi )^{2}}\Delta_{p}^{- 1}( - \vec{q}^{2}) \left\{h ( \vec{q}) h ( - \vec{q}) + \vec{p}( \vec{q}) \cdot \vec{p}( - \vec{q}) \right\}
\rho^{\omega}_{K,\mathbf{a},f}(X_{\mathbf{u}})=\rho^{\omega}_{K,\mathbf{a},f}(X_0+\sum_{i=1}^{m}u_iX_i)\le (m+1)\max\{\rho^{\omega}_{K,\mathbf{a},f}(X_i)\mid i\in \{0,1,\ldots,m\}\}.
p^*(Y_{j,k})=1+ p^* (X_{j,k}Y_{j-1,k})= 1+\frac{\Delta_{I(j+1,k)}\Delta_{J(j,k)}}{\Delta_{I(j,k-1)}\Delta_{I(j+1,k+1)}}=\frac{\Delta_{I(j,k)}\Delta_{J(j+1,k)}}{\Delta_{I(j,k-1)}\Delta_{I(j+1,k+1)}},
\dot{u}_i = v_i, \dot{v}_i = \frac{1}{\rho}\big(A_{iJKj}u_{j,K}- \beta_{Ji}\theta - (C_{iJKLIj}u_{j,IL}+ M_{iJKL}\tau_{,L}- B_{iJKj}v_{j})_{,K}\big)_{,J}, \dot{\tau}= \theta, \dot{\theta}= \frac{1}{a}\big(-\beta_{Ki}v_{i,K}+(m_{IJ}\theta_{,J}+ M_{jLKI}u_{j,LK}+ K_{IJ}\tau_{,J})_{,I}\big).
\nu_n=\sum_{I\in \Sigma_{n}}\;\;\sum_{Q \in \mathcal{D}_{r/\sqrt{d}}(E):\; Q\cap \pi^{\mathbf{a}}([I]\cap E)\neq \emptyset}\frac{\mu_0([I])}{N_I}\cdot \delta_{z(Q)}, n=0,\ldots, \ell,
\frac{\,1\,}{\,n\,}\sum_{i=1}^{n}\mathbb E \big[ 2 b(s, X_{s,i}^{(u)}, \widehat{F}_{t,i}^{(u)}) (X_{s, i}- X_{t, i}) \vert \mathcal F_{t}] \le c_{5}\int_{\mathbb R}\lvert x\rvert^{2}{\mathrm d}m_{t, n}(x) + c_{6}\, .
\mathrm{d e t}M - ( \tilde{B}^{A_{1}A_{2}\ldots A_{N_{f}- N_{c}}}M_{A_{1}}{}^{B_{1}}M_{A_{2}}{}^{B_{2}}\ldots M_{A_{N_{f}- N_{c}}}{}^{B_{N_{f}- N_{c}}}B_{B_{1}B_{2}\ldots B_{N_{f}- N_{c}}}) ,
\underline x^t = (1-\lambda_t) [(1-\lambda_{t-1}) \bar x^{t-2}+ \lambda_{t-1}x^{t-1}] + \lambda_t x^{t-1}&= (1 - \lambda_t) [\underline x^{t-1}- \lambda_{t-1}x^{t-2}+ \lambda_{t-1}x^{t-1}]+ \lambda_t x^{t-1}&= (1-\lambda_t) \underline x^{t-1}+ (1-\lambda_t) \lambda_{t-1}(x^{t-1}- x^{t-2}) + \lambda_t x^{t-1}.
\eta_{1}( z ) = - 4 \sum_{n = 1}^{\infty}\left[ \int_{0}^{\infty}{\mathrm{d}}t{\frac{\sin ( t )}{( t^{2}+ n^{2}z^{2}/ 4 )^{2}}}- 2{\frac{1}{n^{2}z^{2}}}\right] .
Str \left(D_{a_1}\phi^{j_1}\ldots D_{a_l}\phi^{j_l}\phi^{i_1}\ldots \phi^{i_n}\underbrace{\phi^{i_{2(k+r)}'}\phi^{i_{2(k+r)-1}'}}\dots \underbrace{\phi^{i_2'}\phi^{i_1'}}\ldots F_{a_pa_{p-1}]}\right),
\sum_{x=y}^{z}\nu(x \in B, \: b(x)-x \leq k) =\sum_{x=y}^z \sum_{i=1}^k\nu(E_{x,x+i}) &\leq \sum_{x=y}^z \sum_{i=1}^k\sum_{j=0}^i\frac{a_{x+j}}{c(i)}\leq \sum_{x=y}^{z+k}\frac{k^2}{d(k)}a_x\le C(k).
\begin{aligned}e^{-W}\bigg(\sum_{i=1}^{l}\hat{H}_i\bigg)e^W= -\frac{\partial^2}{\partial r'^2}-\frac{\partial^2}{\partial r_l^2}+\omega^2(r'^2+r_l^2)+\frac{1}{r'^2}\big[-\hat{Z}_{l-1}+\frac{1}{4}(D_{l-1}-1)(D_{l-1}-3) \big] +\frac{1}{r_l^2}\big[-\hat{T}_l+\frac{1}{4}(d_l-1)(d_l-3) \big],\end{aligned}
x(t_{1},u_{1},v_{1})x(t_{2},u_{2},v_{2})=x(t_{1}+t_{2}, u_{1}+u_{2}-t_{1}t_{2}^{3\theta}, v_{1}+v_{2}-t_{2}u_{1}+t_{1}t_{2}^{3\theta+1}-t_{1}^{2}t_{2}^{3\theta}).
C^2 \geq -\frac{1}{2}\big( \delta(X) + K_X.\bar{C}\big) - \sum_{i=1}^s \frac{3}{2}m_{q_i}(\bar{C}) - n \geq -\frac{1}{2}\big( \delta(X) + K_X.\bar{C}\big) - \sum_{i=1}^s \frac{3}{2}(\bar{C}.A) - n \quad( n\geq s) \geq -\frac{1}{2}\big( \delta(X) + (K_X+3nA).\bar{C}\big) -n.
\psi = \left[ \begin{array}{c}{\psi_{1}}{\psi_{2}}\end{array}\right] , \; \; \; \; \; \psi^{+}= \left[ \begin{array}{c c}{\psi_{1}^{*}}{\psi_{2}^{*}}\end{array}\right] , \; \; \; \; \;
\mathcal{E}_m(r_0^{d}T_{r_0}\#\mu_\infty)=r_0^{md}\mathcal{E}_m(T_{r_0}\#\mu_\infty)=r_0^{md}r_0^{(1-m)d}\mathcal{E}_m(\mu_\infty)=r_0^{d}\mathcal{E}_m(\mu_\infty).
Q^{k_{1}}Q^{k_{2}}Q^{k_{3}}= \varrho_{k_{1}k_{2}k_{3}}( \sigma ) Q^{k_{\sigma ( 1 )}}Q^{k_{\sigma ( 2 )}}Q^{k_{\sigma ( 3 )}}+ 3{\eta}^{k_{1}k_{2}k_{3}}1 ,
S_{k, r}(j)= \sum\limits_{i=k-j}^k(-1)^{k-i}\dbinom{k+r}{i+r}\dbinom{i+r}{i+j-k}= \sum\limits_{i=k-j}^k(-1)^{k-i}\dbinom{k+r}{k-i}\dbinom{i+r}{j-(k-i)}= \sum\limits_{i=0}^j(-1)^i\dbinom{k+r}{i}\dbinom{k-i+r}{j-i}.
\delta(\d f_1\wedge\ldots\d f_k)= \sum_i(-1)^i\Delta f_i\,\d f_1\wedge\ldots \wedge\widehat{\d f_i}\wedge\ldots \wedge\d f_k &+\sum_{i<j}(-1)^{i+j}[\nabla f_i,\nabla f_j]^\flat\wedge\ldots\wedge \widehat{\d f_i}\wedge\ldots\wedge\widehat{\d f_j}\wedge\ldots\wedge\d f_k
(\delta_T f)(u,v,w)= -f([u,v,w]_T)+l_T(u,v,f(w))+m_T(u,f(v),w)+r_T(f(u),v,w) = -f(\rho(Tu,Tv)w)+[Tu,Tv,f(w)]+[Tu,f(v),Tw]-T\rho(Tu,f(v))w &+[f(u),Tv,Tw]-T\rho(f(u),Tv)w
K_{h_1}\circ K_{h_2}((x_1,x_2), A_1\times A_2)= \delta_{x_1}(A_1)\cdot K_{h_1}^2\circ K_{h_2}^2(x_2, A_2) &=\delta_{x_1}(A_1)\cdot\int_{D_2}K_{h}^2(x_2, A_2)\> d(\delta_{h_1}*_2\delta_{h_2})(h) &=\int_{D_2}K_{h}((x_1,x_2), A_1\times A_2)\> d(\delta_{h_1}*_2\delta_{h_2})(h)
\mathcal{X}^{t+1}=\mathop{\arg\min}_{\mathcal{X}\in{\mathbb{K}^{n\times p}_{l}}}\|\mathcal{X}-\mathcal{X}^{t}\|^2_{F(\mathcal{Q})}~~~ \mathcal{S}^{T}*\mathcal{A}*\mathcal{X}=\mathcal{S}^{T}* \mathcal{B},
\phi_+\left(\frac1{\gamma^*}\right)(z)=\frac1{\phi_-(\gamma)\big(\frac1{z^*}\big)^*},\vert z\vert <1, \phi_-\left(\frac1{\gamma^*}\right)(z)=\frac1{\phi_+(\gamma)\big(\frac1{z^*}\big)^*},\vert z \vert>1,
h_4(x^4+3xy+y^3) =x^4+3s^2xy+sy^3, h_2(x^4+3xy+y^3) =s^{-2}x^4+3xy+s^{-1}y^3, h_1(dx) =s\,dx-x\,ds, h_0(dx) =dx-\tfrac xs\,ds, h_1(y\,dx) =y\,dx-\tfrac{xy}s\,ds, h_1(y\,dx-x\,dy) =y\,dx-x\,dy, h_1(dx\wedge dy) =z\,dx\wedge dy+x\,dy\wedge dz+y\,dz\wedge dx .
L_\xi = \ \frac14 (\xi^{2n+1}A^h(\zeta,\eta) + \eta^{2n+1}D^a(-\xi,\zeta)) = \ \frac14 \left[- i \xi^{2n+1}\left( \eta +\frac{J(\zeta)^2 -\zeta^2}{4J(\zeta)}\right) + i \eta^{2n+1}\xi \right] = \i \eta \xi^{2n}\left( \frac{n}2 \zeta + \frac{J(\zeta)^2 -\zeta^2}{16 J(\zeta)}\right),
\left\vert j_{1}m_{1}j_{2}m_{2}\right\rangle =\sum_{j=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\sum_{m=-j}^{j}C_{m_{1},m_{2},m}^{j_{1},j_{2},j}\left\vert (j_{1}j_{2})jm\right\rangle ,
\begin{array}{c|cccc}\lambda \eta_1 \eta_2 \eta_3 \hline \lambda 0 0 0 \zeta_3^{2}\eta_1 0 0 \zeta_3 0 \eta_2 0 \zeta_3^{2}0 0 \eta_3 \zeta_3 0 0 0\end{array}
\begin{matrix}\overline{N}_{\Delta f}(f)&=&\int_{f-\Delta f/2}^{f+\Delta f/2}\overline{n}(f)df\\ &=&\overline{N}(f+\Delta f/2)-\overline{N}(f-\Delta f/2)\\ &\simeq&\frac{8\pi lwh}{c^{3}}\cdot f^{2}\cdot\Delta f\end{matrix}
B_2 = \left\{\begin{array}{ll}e_{(1,\ell_1-1,\ell_2),\ell_3+1,\ell_4}e_{(2,\ell_1-1,\ell_4),\ell_2+1,\ell_3}e_{(3,\ell_3,\ell_4),\ell_1,\ell_2}\end{array}: \; 1 \leq \ell_1 \leq s,\; 0 \leq \ell_2,\ell_3,\ell_4 \leq s-1\right\}
z(1-z)\frac{\partial^2}{\partial z^2}U_\epsilon(z,\bar{z})-\left[ c_\epsilon -(a_\epsilon +b_\epsilon+1)z\right]\frac{\partial}{\partial z}U_\epsilon(z,\bar{z})-a_\epsilon b_\epsilon U_\epsilon(z,\bar{z})=0,
{\cal L}_{\mathrm{e f f}}^{\beta , \mu}( E ) ={\cal L}_{\mathrm{e f f}}^{\beta , \mu}( 0 ) - \frac{( e E )^{2}}{2 4 \pi^{2}}\frac{| \mu - e A_{0}|}{\sqrt{( \mu - e A_{0})^{2}- m^{2}}}\theta \left( | \mu - e A_{0}| - m \right) \ ,
A_{k}-A_{k-2}&=3^{k-1}\times8k[\frac{-k^{2}+4k+3}{(k-1)k(k+1)(k+2)}]-2\times3^{k-1}[\frac{10}{(k-1)(k+1)}+\frac{3}{k(k+2)}]+2[\frac{k^{2}+4k+1}{(k-1)k(k+1)(k+2)}] &<0
U=|\phi|^2\left\{\sigma_1\beta_{11}(|\phi|^2-c^2) +\sigma_2\beta_{12}(|\chi|^2-{c'}^2)\right\}^2 +|\chi|^2\left\{\sigma_1\beta_{21}(|\phi|^2-c^2) +\sigma_2\beta_{22}(|\chi|^2-{c'}^2)\right\}^2,
Z ( t ) ={\frac{1}{\sqrt{2 \pi}}}{\frac{1}{g \hbar^{2}t^{3 / 2}}}\, \exp \! \left({\frac{t v^{4}}{4 g^{2}}}\right) K_{0}\! \left({\frac{t v^{4}}{4 g^{2}}}\right) .
\langle w, [\widetilde{\mathcal{Q}}+\,(\Sigma_f + \mathcal{S}+\sigma\mathcal{A}\mathcal{A}^*, \Sigma_g + \mathcal{T}+ \sigma\mathcal{B}\mathcal{B}^*)]w\rangle = 0 \Rightarrow \|u\|\|v\| = 0.
\left( \begin{array}{c c}{{A_{1 2}}}&{{A_{2 2}}}\\{{- A_{1 1}}}&{{- A_{1 2}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{\partial_{\tau}a}}&{{\partial_{\tau}b}}\\{{\partial_{\tau}c}}&{{\partial_{\tau}d}}\\ \end{array}\right) \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right)^{- 1},
\begin{array}{lll}{}[J_{1},J_{2}]=iJ_{3}\,,~~~ ~~~{}[J_{2},J_{3}]=iJ_{1}\,,~~~ ~~~{}[J_{3},J_{1}]=iJ_{2}\,,{}{}{}{}[K_{1},K_{2}]=-iK_{3}\,,~~~ ~~~{}[K_{2},K_{3}]=iK_{1}\,,~~~ ~~~{}[K_{3},K_{1}]=iK_{2}\,,\end{array}
L = \frac{1}{4}\int_X \log \frac{\sum_{i_1 < \ldots <i_{r_E}}|e^{i\zeta}s_{i_1}\wedge \ldots \wedge e^{i\zeta}s_{i_{r_E}}|^2_{h_0}}{\sum_{i_1 < \ldots <i_{r_E}}|s_{i_1}\wedge \ldots \wedge s_{i_{r_E}}|^2_{h_0}}\omega^n + \frac{\delta}{\chi}\int_0^1 \langle\mu_{\overline{Z}}(e^{is\zeta}\phi_*),\zeta\rangle ds = L'
\omega_{_{\nabla}}=\frac12 r \, \eta_{1}\wedge d\eta_{1}+ \mathrm{Re}(q) \left( \eta_{2}\wedge d\eta_{2}- \eta_{3}\wedge d\eta_{3}\right)- \mathrm{Im}(q) \left(\eta_{2}\wedge d\eta_{3}+\eta_{3}\wedge d\eta_{2}\right),
\begin{aligned}s_b(N_k^{n}) =\frac{n+1}{2}(b-1)(2^{k}-1)+\frac{n+1}{2}b-2^{n-1}+2^{n-1}-\frac{n+1}{2}&=\frac{n+1}{2}(b-1)2^{k}.\end{aligned}
(\tau_i-\tau_{i-1})|_{w_{\ell_1,\ell_2,\dots,\ell_k}}=(w_{\ell_1,\ell_2,\dots,\ell_k}(i)-w_{\ell_1,\ell_2,\dots,\ell_k}(i-1))t, (\tau_i+\tau_{i-1})|_{w_{\ell_1,\ell_2,\dots,\ell_k}}=(w_{\ell_1,\ell_2,\dots,\ell_k}(i)+w_{\ell_1,\ell_2,\dots,\ell_k}(i-1))t.
T(z) = R(z)E_{0}(z)\Phi_{\mathrm{Be},0}(rf_{0}(z);\alpha)2^{-\alpha \sigma_{3}}s_{1}^{-\frac{\sigma_{3}}{2}}(rz)^{\frac{\alpha}{2}\sigma_{3}}\begin{pmatrix}1 s_{1}h(\frac{rz}{4}) 0 1\end{pmatrix}e^{-\sqrt{r}g(z)\sigma_{3}}.
\Lambda'\coloneqq \left\{((x;\tau_1\xi_1-\tau_2\xi_2),(t;\tau_1,\tau_2))\; \middle| \;\begin{aligned}\tau_1, \tau_2>0, (x;\xi_1) \in L_1, (x;\xi_2) \in L_2, & t=f_2(x;\xi_2)-f_1(x;\xi_1)\end{aligned}\right\}.
\prod_{t=1}^{\eta}\left(1-\frac{(1-2\varepsilon (-\varphi)^{-n})^2}{u_t^2}\right) \geq \prod_{t=1}^{\infty}\left(1-\frac{(1-2\varepsilon (-\varphi)^{-n})^2}{u_t^2}\right) \big(1-\mathcal{O}(\eta^{-1})\big).
\frac{u_i p_j}{p_\ast}\cdot \left(\frac{1}{u_i}+O(n^{-1/2}) \right) \cdot \left(\frac{1}{p_j}+O(n^{-1/2})\right) \cdot \left( p_\ast +O(\theta^n)\right) = 1 +O(n^{-1/2})
\frac14\left|(\log t-\log s)w\right| = \frac14\left|(\log s-\log t)\frac{w}{\theta(s,t)}\right|\theta(s,t) \leq \frac14\Psi\big(\frac{w}{\theta(s,t)}\big)\theta(s,t) +\frac14 \Psi^{*}\big(\log s-\log t\big)\theta(s,t) &= \frac14\Psi\big(\frac{w}{\theta(s,t)}\big)\theta(s,t) + G_{\Psi^{*}}(s,t)\;.
\#\Big\{(\mu_i,\mu_j) = (0,2), (\nu_i,\nu_j) = (1,1) \Big\}+\#\Big\{(\mu_i,\mu_j) = (1,2), (\nu_i,\nu_j) = (1,1) \Big\}+ &\#\Big\{(\mu_i,\mu_j) = (2,2), (\nu_i,\nu_j) = (1,1) \Big\}.
\Gamma \, [ A ] = - \, i \, N_{f}\, \mathrm{T r}\, \biggl [ - \, \frac{1}{p - m + i \epsilon}\, A - \frac{1}{2}\, \frac{1}{p - m + i \epsilon}\, A \, \frac{1}{p - m + i \epsilon}\, A + \cdots \biggr ]
+\xi^0\wedge\xi^3\wedge\eta^0\wedge\eta^3+\xi^0\wedge\xi^3\wedge\eta^1\wedge\eta^2+\xi^1\wedge\xi^2\wedge\eta^0\wedge\eta^3+\xi^1\wedge\xi^2\wedge\eta^1\wedge\eta^2,
\dim X_{\lambda,\mu}(t)= d_\lambda d_\mu \prod_{i=1}^r\frac{\binom{t+\lambda_i-i-s}{\lambda_i}}{\binom{\lambda_i+r-i}{\lambda_i}}\prod_{j=1}^s\frac{\binom{t+\mu_j-j-r}{\mu_j}}{\binom{\mu_j+s-j}{\mu_j}}\prod_{i=1}^r\prod_{j=1}^s\frac{t+1+\lambda_i+\mu_j-i-j}{t+1-i-j},
\mathbb{E}\big[ \mathfrak{m}^{(p,p)}\big]=\mathbb{E}\big[O_\prec(\hat{\Pi}^2)\mathfrak{m}^{(p-1,p)}\big]+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{m}^{(p-2,p)}\big]+\mathbb{E}\big[O_\prec(\Psi^2\hat{\Pi}^2) \mathfrak{m}^{(p-1,p-1)}\big].
\sum_{j = 0}^k{\sum_{s = 0}^j{\binom kj\binom js\left({\frac{{f_3}}{{f_1}}}\right)^j\left({\frac{{f_2}}{{f_3}}}\right)^s X_{m - ak + (a - c)j + (c - b)s}}}= \frac{{X_m}}{{f_1{}^k}}\,,
\begin{array}{r c l}{F_{0 1}}&{=}&{\left( Q ( r^{2}- N^{2}) - 2 N P r \right) / R^{4},}\\{}&{}&{}\\{F_{2 3}}&{=}&{- \left( P ( r^{2}- N^{2}) + 2 N Q r \right) / R^{4},}\\ \end{array}
\frac{D_{a + 1}( u )}{U_{a - 1}( u )}= \frac{\operatorname{s i n}( \xi_{a}+ u ) \operatorname{s i n}( \xi_{a}+ a \gamma - u )}{\operatorname{s i n}( \xi_{a}- u ) \operatorname{s i n}( \xi_{a}+ a \gamma + u )}
\frac{-2q}{1+q}{}_2\phi_1 \left(\genfrac{}{}{0pt}{}{-q,-q}{-q^2};q,-q\right) = \frac{-2q}{1+q}\frac{(q^2;q)_\infty}{(-q;q)_\infty}{}_2\phi_2 \left(\genfrac{}{}{0pt}{}{-q,q}{-q^2,q^2};q,q^2\right).
\Psi( E_{j+1}(x, d-1)) - \Psi(E_j(x, d-1)) = = 2 \sqrt{d-1}\left( \cot\left( \dfrac{\pi}{2j+4}\right) - 1 \right) - 2 \sqrt{d-1}\left( \csc\left( \dfrac{\pi}{2j+2}\right) - 1 \right) = 2 \sqrt{d-1}\left( \cot\left( \dfrac{\pi}{2j+4}\right) - \csc\left( \dfrac{\pi}{2j+2}\right) \right) = f_j \, \sqrt{d-1}\, .
s'=k_{-1}^*c_1-k^*_1s(e_0-c_1-c_2) c_1'=k_1^*s(e_0-c_1-c_2)-(k_{-1}^*+k_2^*)c_1 c_2'=\frac{-(i_0-c_2)[k_1^*s(e_0-c_1-c_2)-(k_{-1}^*+k_2^*)c_2]}{\kappa+e_0+i_0-c_1-2c_2}
m_j(R_{\theta_j/N^k}(U)) = m_j(R_{N^k\theta_{j+k}}(U))= m_{j+k}(p_N^{-k}(R_{N^k\theta_{j+k}}(U))) &= m_{j+k}(R_{\theta_{j+k}}(p_N^{-k}(U)))\le e^{\beta/N^{j+k}}m_{j+k}(p_N^{-k}(U))=e^{\beta/N^{j+k}}m_j(U),
E\left[(\sigma'\sigma)(X_{\eta_n(s_1)}) (\sigma'\sigma)(X_{\eta_n(s_2)}) \Xi^{n,2}_{s_1}\Xi^{n,2}_{s_2}\right] = E\left[(\sigma'\sigma)(X_{\eta_n(s_1)}) (\sigma'\sigma)(X_{\eta_n(s_2)}) \Xi^{n,2}_{s_1}E[ \Xi^{n,2}_{s_2}| \mathcal{F}_{\eta_n(s_2)}\right]]=0.
R_p(x) - L_p(x) = -\frac2{p(1+p)}\, \log(1-x^p) - \frac1{p(1+2p)}\, \log(1+x^p) > \frac2{p(1+p)}\,x^p - \frac1{p(1+2p)}\,x^p = \frac{x^p}{(1+p)(1+2p)}>0\, .
\frac{1}{\pi}\iint_{\Omega,|z-w|>\epsilon}\frac{\overline{F(z)}}{(z-w)^{2}}dxdy=Cf(w)-\frac{1}{2\pi i}\int_{|z-w|=\epsilon}\frac{\overline{F(z)}}{z-w}d\overline{z}=Cf(w).
{I^*}=\|{\tilde \sigma^*_1}\|_{L^\infty (\mathbb{R}^2 \times\mathbb{R}_+)}+ (1/A_0)\left\|{g_0}\right\|_{L^\infty \left({\mathbb{R}_+}\right)}N_1 \left\| Q^* \right\|_{L^\infty \left({\Omega}\right)},
\mathrm{e}^{-vU_{\tau(t)}(\psi)}\, =\,1+\int_{0}^{t}\mathrm{d}s\ \mathrm{e}^{-vU_{\tau(s)}(\psi)}\,\frac{X_{\tau(s)}\bigl(v^{1+\beta}\psi^{1+\beta}(s,\cdot)\bigr)}{X_{\tau(s)}\bigl(\psi^{1+\beta}(s,\cdot)\bigr)}+ & =\,1+\int_{0}^{t}\mathrm{d}s\ \mathrm{e}^{-vU_{\tau(s)}(\psi)}\,v^{1+\beta}+t\leq t^{\ast}.
I_1 = \frac{1}{2\pi}\int_{-\infty+ic}^{\infty+ic}\frac{\zeta'}{\zeta}(s) \Bigl( \Phi_1(\phi;z)\overline{\Phi_1(\phi;\bar{z})}+ \Phi_1(\phi;-z)\overline{\Phi_1(\phi;-\bar{z})}\Bigr) \, dz,
\begin{array}{r l}{{\mathrm{g r a v i t y}:}}&{{( 3 , 3 ) + 2 ( 2 , 3 ) + ( 1 , 3 ) \nonumber}}\\{{\mathrm{t e n s o r}:}}&{{( 3 , 1 ) + 2 ( 2 , 1 ) + ( 1 , 1 ) \nonumber}}\\{{\mathrm{v e c t o r}:}}&{{( 2 , 2 ) + 2 ( 1 , 2 ) \nonumber}}\\{{\mathrm{h y p e r}:}}&{{2 ( 2 , 1 ) + 4 ( 1 , 1 ) .}}\\ \end{array}
j_{\nu, k}=\left(k+\frac{\nu}{2}-\frac{1}{4}\right) \pi-\frac{4 \nu^{2}-1}{8\left(k+\frac{\nu}{2}-\frac{1}{4}\right) \pi}+O\left(\frac{1}{k^{3}}\right), k \rightarrow\infty
X^2_+(t) =\sum_{i:\gamma^+_i>\gamma_*}(\xi_i^+(t))^2+\sum_{i:\gamma^-_i>\gamma_*}(\xi_i^-(t))^2, X^2_-(t) =\sum_{i:\gamma^+_i<\gamma_*}(\xi_i^+(t))^2+\sum_{i:\gamma^-_i<\gamma_*}(\xi_i^-(t))^2.
M_{G, H, I}= \left( \begin{array}{cccc}\frac{n_1 + 2}{2}0 -\frac{n_1 + 2}{2}0 0 \frac{n_2 + 1}{2}0 -\frac{n_2 + 1}{2}-\frac{n_1 + 2}{2}0 \frac{n_1 + 2}{2}0 0 -\frac{n_2 + 1}{2}0 \frac{n_2 + 1}{2}\end{array}\right),
|\int_{t}^{\infty}ds \frac{\lambda''(s)}{r}\int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\frac{4 r \left(p^2+r^2+1\right)}{\left(\left(p^2-r^2+1\right)^2+4 r^2\right)^{3/2}}| &\leq C_{x \geq t}\left(|\lambda''(x)| (1+(x-t)^{2})\right) \frac{1}{r^{2}}
\begin{array}{r c l}Q = \int_0^L dx\,\psi^\dagger\psi=-\lambda(p_+-p_-)\ , Q_5 = \int_0^L dx\psi^\dagger\gamma_5\psi=-\frac{i\lambda\sqrt{2\mu}}{2\sqrt{\pi}\sqrt{L}}\,\left[\varphi_0-\varphi^\dagger_0\right]\ .\end{array}
e_{p,q,n}\left(\tau\right) =\left(b_{1,n-1}\ln\left(2\tau\sqrt{3p\left(q+1\right)}\right)+b_{2,n-1}\right) \qquad\times\left(12\tau^{2}p\left(q+1\right)\right){}^{n},\, n\in[1,d'],
\dfrac{\langle x_{i,r}^+[\lambda], [\mu] \rangle_{H}}{\langle [\mu], [\mu] \rangle_{H}}=\delta((\mu, \lambda) \in \mathfrak{P}_i({\bf v}))(-1)^{\delta_{i,0}- (v_i(\lambda) - v_{i-1}(\lambda)) + 1}c_1(\mathcal{L}_i({\bf v}))^r \dfrac{e(N_{\mu \lambda})}{e(T_{\mu})}.
\rho^{{y}}(x)=\begin{cases}\displaystyle-\frac{1}{\Gamma(2+\alpha)}x^{1+\alpha}+\frac{C}{\Gamma(1+\alpha)}x^\alpha\quad \displaystyle-\frac{1}{\Gamma(2+\alpha)}(x^{1+\alpha}-L^{1+\alpha})\quad \end{cases}
\begin{array}{c}{6 \beta^{\prime}v^{2}v^{\prime}+ 2 \beta^{\prime}( D - 1 ) v^{3}+ 2 \beta^{\prime}v^{\prime}= \alpha^{\prime}}\\{v^{\prime}= I_{1}( \alpha^{\prime}) / I_{0}( \alpha^{\prime})}\\ \end{array}
\mathbf{E}\left[\|\mathcal{X}^{t+1}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}|\mathcal{X}^{t}\right] \leq \left(1-\mathop{\min}_{k=1,2,\cdots,l}\lambda_{\min}(\mathbf{E}[\widehat{\mathcal{Z}}_{(k)}])\right)\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}.
(n-k) f_1 f^n_k = -\sum_{j=1}^{d-1}(n(j+1) - k + j) f_{j+1}f^n_{k-j}, (nd - k) f_d f^n_k = -\sum_{j=1}^{d-1}(n(d-j) - k - j) f_{d-j}f^n_{k+j},
\lim_{N \to \infty}\sum^M_{j = 0}\Big( \log(\nu_j + N) - \psi(\nu_j + N) + \frac{w}{\nu_j + N}\Big) = \gamma(w + \frac{1}{2}), \lim_{N \to \infty}\sum^M_{j = 0}\psi'(\nu_j + N) = \gamma,
G_{{\cal Q}^{4}}( x_{p}) = \frac{3^{3}\Gamma ( 1 1 )}{2^{7}\left( \pi^{3}\Gamma ( 4 ) \right)^{4}}\: g^{8}e^{- \frac{8 \pi^{2}}{g^{2}}+ i \theta}x_{1 2}^{4}x_{3 4}^{4}\prod_{p < q}\frac{\partial}{\partial x_{p q}^{2}}B ( x_{p q}) ,
\sum_{\vec{k}}\left\{\delta \Pi \ \dot{\varphi}( \vec{k}, t ) + \delta \dot{\varphi}\ \Pi ( \vec{k}, t ) - \left[ \delta \varphi \{\frac{\partial H}{\partial \varphi ( \vec{k}, t )}}+ \delta \Pi \{\frac{\partial H}{\partial \Pi ( \vec{k}, t )}}\right] \right\}= \dot{\Omega}\ .
S_{a_{1},a_{1}+b_{1}}^{a_{2},a_{2}+b_{2},p_{2}}\left( p_{1},k\right)=S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1},k\right) +\left( k+1\right)S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1},k+1\right) .
\sum_{n=0}^{2^{k_i}- 1}1_{[-B-b,-B+b] \cup [B-b,B+b]}\left( t + \sum_{j=0}^{n-1}f(T^j x) \right)\le2\eta \sum_{n=0}^{2^{k_i}- 1}1_{[-B-b,B+b]}\left( t + \sum_{j=0}^{n-1}f(T^j x) \right)
D_{\mu , x}^{a b}i G^{[ s ]}( x , x^{\prime})_{\nu \rho}^{b c}={\frac{1}{2}}\int_{0}^{\infty}d T \int_{x ( 0 ) = x^{\prime}, x ( T ) = x}D x ( t ) \dot{x}_{\mu}( T ) e x p \left( -{\frac{1}{4}}\int_{0}^{T}d t \, \dot{x}^{2}\right) \Phi_{\nu \rho}^{[ 1 ]}[ C^{x x^{\prime}}] U [ C^{x x^{\prime}}]^{a c}
\mathfrak{P}_{t}f(x)-\mathfrak{P}_{s}f(x) =\mathfrak{S}_{t}f(x)-\mathfrak{S}_{s}f(x)+\Pi_{x}\left[\int_{s}^{t}F_{r}(\xi_{t-r}){\rm d}r\right] &+\Pi_{x}\left[\int_{0}^{s}F_{r}(\xi_{t-r})-F_{r}(\xi_{s-r}){\rm d}r\right].
\sup_{u \in X}\left\| y - m^\mathcal G_N(u) \right\|^2 \leq \sup_{u \in X}\| y - \mathcal G(u) \|^2 + \sup_{u \in X}\left\| \mathcal G(u) - m^\mathcal G_N(u) \right\|^2 =: - \ln C_1,
S^{\mathrm{h e}}= \frac{\mu^{4}}{2 \kappa_{1 0}^{2}}\int d^{6}x \sqrt{- g^{\mathrm{h e}}}e^{- 2 \varphi^{\mathrm{h e}}}\Bigg [ R ( g^{\mathrm{h e}}) + 4 \partial_{\mu}\varphi^{\mathrm{h e}}\partial^{\mu}\varphi^{\mathrm{h e}}
\liminf_{j\rightarrow \infty}\left( \int_\Omega |u_j(x)|^{2^*}dx - \int_\Omega |u_\infty(x)|^{2^*}dx\right) = \liminf_{j\rightarrow \infty}\int_\Omega |u_j(x)-u_\infty(x)|^{2^*}dx
F_{m}( 0 ) [ l ] = \frac{1}{k}\sum_{n = 0}^{k - 1}A_{m}( n ) [ l ] = \frac{l^{2 \Delta_{a}}}{( 2 \pi )^{2 \Delta_{a}}}g ( a ) N ( l ) \, , \quad a = \frac{m \beta}{k}\, ,