harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g'(x) = \frac{1}{3}\left( -\frac{2}{(2x)^{4/3}}+ \frac{1}{\left(2x - \frac{3}{8}\right)^{4/3}}+ \frac{1}{\left(2x + \frac{5}{8}\right)^{4/3}}\right) = \frac{1}{3}\left(\left( \frac{1}{\left(2x -\frac{3}{8}\right)^{4/3}}- \frac{1}{(2x)^{4/3}}\right) - \left( \frac{1}{(2x)^{4/3}}- \frac{1}{\left(2x + \frac{5}{8}\right)^{4/3}}\right)\right) .
D_{\mu}f = \partial_{\mu}f -{\frac{i g}{2}}d^{B C C}( A_{\mu}^{B}\star f - f \star A_{\mu}^{B}) = \partial_{\mu}f -{\frac{i g}{2}}d^{B C C}\, [ A_{\mu}^{B}, f ]_{\mathrm{M}}
|D_{x}^{m}\left(|x|^{-d}\textnormal{H}_{k,0}^{q}(R)\right)|\leq N|x|^{-d-m}\sum_{l=1}^{m}\left|d\textnormal{H}_{k,l-1}^{q}(R)+2\textnormal{H}_{k,l}^{q}(R)\right|
[f]^{m+1}(x,y,z_{1}, \dots, z_{m-1}) &= [f]^{m}(x+y,z_{1}, \dots, z_{m-1}) - [f]^{m}(x,z_{1}, \dots, z_{m-1}) - [f]^{m}(y,z_{1},\dots, z_{m-1})
A_{\mu \nu \gamma}\, \Pi^{\mu \mu^{\prime}}\, \Pi^{\nu \nu^{\prime}}\, \Pi^{\gamma \gamma^{\prime}}\, \left( i \epsilon_{\mu^{\prime}\nu^{\prime}\gamma^{\prime}\alpha}\, ( X \partial^{\alpha}X^{\dagger}- X^{\dagger}\partial^{\alpha}X ) \, \phi^{N}\right)
=\hat{a}_{j}^{\dagger}\hat{a}_{i}\hat{a}_{l}^{\dagger}\hat{a}_{k}+\delta_{kl}\hat{a}_{j}^{\dagger}\hat{a}_{i}+\delta_{ij}\hat{a}_{l}^{\dagger}\hat{a}_{k}+\delta_{ij}\delta_{kl}
a=\frac{2}{3}\sqrt{-m_{a}^{2}+2m_{b}^{2}+2m_{c}^{2}}=\sqrt{2(b^{2}+c^{2})-4m_{a}^{2}}=\sqrt{\frac{b^{2}}{2}-c^{2}+2m_{b}^{2}}=\sqrt{\frac{c^{2}}{2}-b^{2}+2m_{c}^{2}}
A(1^{+}\cdot\cdot\cdot i^{-}\cdot\cdot\cdot j^{-}\cdot\cdot\cdot n^{+})=i(-g)^{n-2}\frac{\langle ij\rangle^{4}}{\langle12\rangle\langle23\rangle\cdot\cdot\cdot\langle(n-1)n\rangle\langle n1\rangle}
H^1_{0,L}(\mathcal{C}_\Omega) = \{v\in L^2(\mathcal{C}_\Omega): v =0 \,\,{\rm on}\,\, \partial_L\mathcal{C}_\Omega, int_{\mathcal{C}_\Omega}y^{1-2\alpha}|\nabla v|^2\,dxdy<\infty\}.
a^{3421}=a^{4321},a_{1}^{3421}=a_{1}^{4321},a_{2}^{3421}=a_{2}^{4321},a_{3}^{3421}=a_{3}^{4321},a_{4}^{3421}=a_{4}^{4321}, a_{12}^{3421}=a_{12}^{4321},a_{13}^{3421}=a_{13}^{4321},a_{14}^{3421}=a_{14}^{4321},a_{23}^{3421}=a_{23}^{4321},a_{24}^{3421}=a_{24}^{4321}, a_{123}^{3421}=a_{123}^{4321},a_{124}^{3421}=a_{124}^{4321}
G_r(s,x;\varphi)= \frac{1}{2\pi i}\int_{(\frac{3}{2}-\sigma)}\frac{\chi_f^{(m-r)}}{\chi_f}(1-s-w)L_f^{(r)}(s+w)\frac{K_\varphi(w)}{w}\times &\times\frac{\Gamma(s+w+\frac{k-1}{2})}{\Gamma(s+\frac{k-1}{2})}\left(\frac{x}{2\pi}e^{-i\frac{\pi}{2}{\rm sgn\;}t}\right)^wdw.
\int_{\Omega}d^{\prime}(t) \cdot \overline{\vartheta}\; \d x + \int_{\Omega}\nabla d(t) \cdot \overline{\nabla \vartheta}\; \d x = - \int_{\Omega}(u (t) \cdot \nabla) d (t) \cdot \overline{\vartheta}\; \d x + \int_{\Omega}\lvert \nabla d (t) \rvert^2 d (t) \cdot \overline{\vartheta}\; \d x.
I=\alpha =R_{\phi_o(x,u,v)}\gamma_{01}^\circ R_{[u\backslash (xv)]}^{-1},~\beta=L_u\gamma_{01}^\circ L_u^{-1}~\textrm{and}~\gamma=\gamma_{01}^\circ \gamma_{01}^\circ =\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]},~\beta=L_u\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}\mathbb{L}_u^{-1}~\textrm{and}~\gamma=\mathbb{R}_{\phi_o(x,u,v)}R_{[u\backslash (xv)]}
\mathbb{E}[\|w^0\|_{H^1(\mathbb{S}^2)}^2]=\mathbb{E}[\|w\|_{H^1(\mathbb{S}^2)}^2]\leq (\|\partial_x h\|^2_{L^2}+\|g\|_{L^\infty}^2C(\lambda_1,\lambda_2))=:K\, ,
E_xI_{T}(h)-E_xI_{T,n}(h){}= \int_{0}^{T/n}\int_{\Re^d}p_t(x,y)h(y)\,dydt-\frac{T}{n}h(x) &\quad{}+\sum_{k=2}^n\int_{(k-1)T/n}^{kT/n}\int_{\Re^d}\bigl(p_t(x,y)-p_{(k-1)T/n}(x,y)\bigr)h(y)\,dydt.
1 = \frac{\partial q_{j}}{\partial q_{j}}= - \frac{\partial^{2}F}{\partial p_{j}\, \partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{j}}- \frac{1}{2}\frac{\partial^{3}F}{\partial{p_{j}}^{2}\, \partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{j}}\Delta p_{j}\; .
0 =\mathcal{L}_0(E_k)+\left(\frac{s_k^2}{2\lambda s_k^2+2u_k}-\frac{1}{2\lambda}\right)\mathcal{S}_{0,2}(U_k,U_k)+O(s_k) &=\mathcal{L}_0(E_k)-\frac{u_k}{2\lambda^2 s_k^2}\mathcal{S}_{0,2}(U_k,U_k)+O\left(s_k,\frac{u_k^2}{s_k^4}\right).
\Vert M-\hat M\Vert^{2}_{2}\leq \frac{a+1}{a-1}\Vert M\Vert^{2}_{2}+\dfrac{2a^{2}}{a-1}\Delta &\leq \frac{a+1}{a-1}n_1\,s^{2/q}+\dfrac{2a^{2}}{a-1}\Delta
\boxed{S_k(B(0),B_1) = \sum_{j=\frac{k-1}2}^{2N-\frac{k+1}2}\begin{pmatrix}j \frac{k-1}2 \end{pmatrix}\begin{pmatrix}2N-j-1 \frac{k-1}2 \end{pmatrix}B(0)^{2j}B_2(0)k=1,3,\dots, 2N-1\,.}
\sum_{k=0}^{\infty}\frac{1}{\Gamma \left( k+1\right)}I_{v-k}(q+2k;x)t^{k}=\sum_{n=0}^{\infty}\frac{2^{v+q+n-\frac{1}{2}}x^{n+v}}{\ \sqrt{\pi}\Gamma(2\upsilon +q+n+\frac{1}{2})\n!}\cdot \sum_{k=0}^{\infty}\frac{\Gamma(\upsilon +q+n+k)2^{k}t^{k}}{\ \k!x^{k}}.
\bigl \{A_{\mu}( \vec{x}, t ) , \dot{A}^{\nu}( \vec{y}, t ) \bigr \}= - \Bigl ( \delta_{\mu}^{\nu}+ ( \alpha - 1 ) \eta_{0 \mu}\delta_{0}^{\nu}\Bigr ) \delta ( \vec{x}- \vec{y}) .
-\theta Te^{-\theta T\mathcal{I}(\widetilde{\boldsymbol{\mathrm{x}}}_t;\widetilde{\boldsymbol{\mathrm{y}}}_t)}\frac{\partial\mathcal{I}(\widetilde{x}_t(i);\widetilde{y}_t(i))}{\partial\sigma_{t}(i)}+\lambda-\phi(i,i)=0,
\dfrac{M_{2}(\mathcal{NC}(G))}{|e( \mathcal{NC}(G) )|}- \dfrac{M_{1}(\mathcal{NC}(G))}{|v( \mathcal{NC}(G) )|}~~~~= \dfrac{2^{7k}(2^{5k}-8\cdot2^{k}-4)+2^{4k}(17\cdot2^{2k}-14)+2^{3k}(14\cdot2^{2k}-18)+2\cdot2^{2k}+8\cdot2^{k}}{2^{6k}(2^{3k}-4\cdot2^{k}-2)+2\cdot2^{3k}(3\cdot2^{2k}-1)+2^{2k}(4\cdot2^{2k}-3)}:= \dfrac{f(k)}{g(k)}.
\widetilde{\psi}_{k}^{(0^{2})}(x;1,2,\lambda_{1})=( 1-z)^{3/4}(1+z)^{5/4}\frac{\widetilde{P}_{1,2,k}^{( 0^{2})}(z; \lambda_1)}{\lambda_{1}-\frac{1}{3}(z+1)^{3}+\frac{1}{8}(z+1)^{4}},
\left(\sum_v \sum_j w_{k_j}(v)\right)^2 \le \left[\sum_v 1^2\right]\cdot\left[\sum_v \left(\sum_j w_{k_j}(v)\right)^2\right] = |V|\left[\sum_j \sum_v w_{k_j}(v)^2+2\sum_{j < j'}\sum_v w_{k_j}(v)w_{k_{j'}}(v)\right].
\begin{pmatrix}\chi_0(R) \chi_1(R) \dots \chi_{p}(R) \chi_1(R) \chi_2(R) \dots \chi_{p+1}(R) \vdots \vdots \vdots \chi_{p}(R) \chi_{p+1}(R) \dots \chi_{2p}(R) \end{pmatrix}\begin{pmatrix}\tilde\alpha_0 \tilde\alpha_1 \vdots \tilde \alpha_{p}\end{pmatrix}=\begin{pmatrix}1 0 \vdots 0\end{pmatrix}.
\Big|\partial_{\mathrm{x}_{i}}\Phi(\mathrm{x},t;\mathrm{y},\tau)\Big| \lesssim \frac{\alpha^\mathrm{r}}{(t-\tau)^\mathrm{r}}\frac{1}{|\mathrm{x}-\mathrm{y}|^{3-2\mathrm{r}}}, \ \ \ \mathrm{r}<2,\i=1,2,
R(l)=\frac{1}{2l}\sum_{n=1}^\infty \frac{A(n,1)}{n^\frac{3}{2}}(\frac{16}{ln})\sum_{\substack{\alpha \leq Y (\alpha,2ln)=1}}\frac{\mu(\alpha)}{\alpha^2}\sum_{\substack{k=-\infty k \neq 0}}^{\infty}(-1)^kG_k(ln)\widetilde \Phi_{n}(\frac{kX}{2\alpha^2 ln}).
\sum_{k=0}^{M}[4k-1]\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,cq^{-1},dq^{-1},q^{-1};q^2)_k}{(q^{2}/a,aq^2,bq^2,q^2/c,q^{2}/d,q^2;q^2)_k}\bigg(\frac{bq^7}{cd}\bigg)^k begin{align*}5pt]\: \:\equiv[n]\Theta_q(a,b,n)\sum_{k=0}^{(n+1)/2}\frac{(aq^{-1},q^{-1}/a,q^{-1}/b,q^3/cd;q^2)_k}{(q^2,q^{-2}/b,q^2/c,q^2/d;q^2)_k}q^{2k},
-\frac{\varphi_n(z_n) - \varphi_n(\overline z_m)}{z_n-\overline z_m}=\int_{U_n}\frac{t - \overline z_m - t + z_n}{(z_n-\overline z_m)(t-z_n)(t-\overline z_m)}d\widetilde\sigma(t) =\int_{U_n}\frac{d\widetilde\sigma(t)}{(t-z_n)(t-\overline z_m)},
\left\Vert(\Delta+z^2)^{-d}(p)-(4\pi)^{-1}\sum_{l=0}^{k}z^{-2d+2-2l}u_l(p)\frac{\Gamma(d+l-1)}{(d-1)!}\right\Vert\leq \tilde{C}_k(K)z^{-2d-2k}.
\sum_{i=1}^{n}\frac{\partial f}{\partial \kappa_{i}}=\frac{1}{k}H_{k}^{\frac{1-k}{k}}\sum_{i=1}^{n}\frac{\partial H_{k}}{\partial\kappa_{i}}=H_{k}^{\frac{1-k}{k}}H_{k-1}\geq 1,
\bigl(E^{(n,k)}_{1,1},\ldots,E^{(n,k)}_{1,k},E^{(n,k)}_{2,1},\ldots,E^{(n,k)}_{2,k},\ldots, E^{(n,k)}_{n,1},\ldots,E^{(n,k)}_{n,k}\bigr) =\bigl(\mathcal{E}_1,\mathcal{E}_2,....,\mathcal{E}_{nk}\bigr)
S_I(t_{n_j})(u_{n_j},\phi_{n_j})=e_1(L_{t_{n_j}-\ell}\chi_{n_j})\rightarrow e_1(\chi)\in\mathcal{A}\,\,\,\textit{in}\,\,\, H\times V_I\,\,\,\textit{as}\,\,\,j\rightarrow+\infty,
\left\{\begin{array}{c}(c_{2}-c_{3}+1-\kappa_{1})\mu_{1}=d(\kappa_{1}+c_{2}-c_{3}+1)+2(c_{2}-c_{3}-\sigma_{1})+3 (c_{2}-c_{3}+1-\kappa_{2})\mu_{2}=d(\kappa_{2}-c_{2}+c_{3}-1)+2(c_{2}-\sigma_{2}) (c_{2}-c_{3}+1-\kappa_{3})\mu_{3}=d(\kappa_{3}-c_{2}+c_{3}-1)+2(c_{3}-\sigma_{3}) \end{array}\right\}.
L_{v^{z}}\Psi + \partial_{z}v^{z}\Psi = \bar{L}_{v^{\bar{z}}}\Psi + \partial_{\bar{z}}v^{\bar{z}}\Psi = 0 \hspace{1 e m}( \partial_{\bar{z}}v^{z}= \partial_{z}v^{\bar{z}}= 0 ) .
E^{0,n}[(Y_{k}-E(Y_{k}|\xi_{km},\xi_{(k+1)m}))E(Y_{j}|\xi_{jm},\xi_{(j+1)m})] & =E^{0,n}[E^{0,jm}(Y_{k}-E(Y_{k}|\xi_{km},\xi_{(k+1)m}))E(Y_{j}|\xi_{jm},\xi_{(j+1)m})] & =E^{0,n}[E^{0,jm}(Y_{k}-E(Y_{k}|\mathcal{F}_{km}\vee\mathcal{F}^{(k+1)m}))E(Y_{j}|\xi_{jm},\xi_{(j+1)m})]=0
Z={V_{p+1}}\Gamma(0){{(2\pi{\alpha^2}\mbox{T})}^{-1/2}}\int{{d\tau}\over{\tau}}{{\left({{\mbox{T}}\over{4\pi\tau}}\right)}^{(p+2)/2}}\exp\left(-\mbox{T}{\mathcal{Y}^2}\tau\right){{|\eta(i\tau)|}^{1-d}},
\sigma_{11}^*(r^\prime)(r) =\sum\limits_{i, j}c_{ij}(\sigma_{11}^*(e_i^*)\sigma_{11}^*(e_j^*) +\sigma_{12}^*(e_i^*) \sigma_{21}^*(e_j^*))(\sum\limits_{k, l}d_{kl}e_ke_l) &=\sum\limits_{i, j}c_{ij}e_i^*e_j^*\big((\sigma_{11}\sigma_{11}+\sigma_{12}\sigma_{21})(\sum\limits_{k, l}d_{kl}e_ke_l)\big) &=r^\prime(\sigma_{11}(r)).
K_{r,a}:= \frac{\mathbb{C}\langle z_1 \dots, z_l \rangle}{\left\langle{\begin{array}{c}z_i \left( z_i^{\beta_i-2}\right) \left( z_{i+1}^{\beta_{i+1}-2}\right) \dots{\left( z_{j-1}^{\beta_{j-1}-2}\right)}\left( z_j^{\beta_j-2}\right) z_j=0 z_i z_j=0 \end{array}}\right\rangle}
\tau ={\frac{2 i}{\pi}}\ln{\frac{a}{\Lambda}}+{\frac{3 i}{\pi}}+ \sum_{k = 0}^{\infty}{\cal F}_{k}( 1 - 4 k ) ( 2 - 4 k ) \left({\frac{a}{\Lambda}}\right)^{- 4 k}.
\Phi'' - \partial_{\alpha}\partial^{\alpha}\Phi + \Phi\biggl[ \frac{5}{2}{\cal H}'+ \frac{\varphi'''}{\varphi'}- \frac{9}{4}{\cal H}^2-{\cal H}\frac{\varphi''}{\varphi'}- 2 \biggl( \frac{\varphi''}{\varphi'}\biggr)^2 \biggr]=0.
u=1-\frac{\vartheta_{[0,1]}^3(p)}{\vartheta_{[0,2]}^3(p)}=1-\frac{\vartheta_{(-1,0,0,1)/3}^3(\tau(\eta),\zeta(p))}{\vartheta_{(-2,0,0,2)/3}^3(\tau(\eta),\zeta(p))}.
\sigma_{\vec{\alpha}}(z)=\vartheta\left[\begin{array}{c}\alpha_2 /n +1/2 \alpha_1 /n+1/2\end{array}\right](z,\tau ), ~~~~~~\theta^{(j)}(z)=\vartheta\left[\begin{array}{c}1/2 - j/n 1/2 \end{array}\right](z,n\tau ),
XY(r+\mathbf{m}_i)=X(s+\mathbf{m}_{i-1})=\sigma(ts)+\mathbf{m}_i=\sigma(t)r+\mathbf{m}_i, YX(r+\mathbf{m}_i)=Y(\sigma(tr)+\mathbf{m}_{i+1})=tr+\mathbf{m}_i.
\begin{array}{l}1^0 \quad E_{j}(s){\neq}E_{k}(s) \quad \forall s \in [0,1],\quad j{\neq}k , 2^0 \quad \forall j \P_j (s) mbox{is double continuously}\qquad \qquad \quad\mbox{differentiable function of}\s \in [0,1] ,\end{array}
u\times v=\epsilon_{ijk}\hat{u}_{j}\hat{v}_{k}e_{i}=\epsilon_{ijk}\frac{\partial x_{j}}{\partial q^{m}}\frac{\partial x_{k}}{\partial q^{n}}\frac{\partial x_{i}}{\partial q^{s}}u^{m}v^{n}b^{s}=[(b_{m}\times b_{n})\cdot b_{s}]u^{m}v^{n}b^{s}=E_{smn}u^{m}v^{n}b^{s}
P_N (y_i, x_2, \dots, x_N){}= S_{N-1,i}(x_2, \dots, x_N) &{}\times (q-q^{-1})\,\prod_{k = 2}^N (qx_k - q^{-1}y_i)\,\prod_{\substack{j=1 j\neq i}}^N (qy_i - q^{-1}y_j),
U=[\begin{matrix}0&1\\ 1&0\\ 0&0\\ 0&0\\ 0&0\end{matrix}],\Sigma=[\begin{matrix}2&0\\ 0&1\end{matrix}],V_{h}=[\begin{matrix}1/\sqrt{2}&-1/\sqrt{2}\\ 1/\sqrt{2}&1/\sqrt{2}\end{matrix}].
p(x_{1},...,x_{n})=(\prod_{i=1}^{n-1}\frac{1}{\sqrt{2\pi\sigma(x_{i})^{2}\Delta t_{i}}})exp(-\sum_{i=1}^{n-1}L(x_{i},\frac{x_{i+1}-x_{i}}{\Delta t_{i}})\Delta t_{i})
u_{3}=b+1 u_{4}=ab+a u_{5}=ba u_{6}=ba+t^{n+1}+nt^{n-1}+(n-1)t^{n-2}u_{7}=ba+t^{n+1}+nt^{n-1}+nt^{n-2}u_{8}=ab+a+t^{n+1}=ba+a+t^{n+1}+nt^{n-1}+nt^{n-2}
\begin{pmatrix}a^4\zeta^4 & a^2\zeta^2 1 a^{-2}\zeta^{-2}& a^{-4}\zeta^{-4}\end{pmatrix}\textrm{and}\begin{pmatrix}a^4 & a^2 1 a^{-2}& a^{-4}\end{pmatrix}
m\left[\begin{array}{cc}0 0 0 d_{22}(B) \end{array}\right]\left[\begin{array}{cc}I_{\mathcal{A}}0 N 0 \end{array}\right]+n\left[\begin{array}{cc}I_{\mathcal{A}}0 N 0 \end{array}\right]\left[\begin{array}{cc}0 0 0 d_{22}(B) \end{array}\right]=0.
\sum_{p_{n+1}\le X}\frac{\log^{\alpha}d_n}{d_n}\sim\begin{cases}\frac{X}{\log^2 X}\frac{(\log\log X)^{1+\alpha}}{1+\alpha}\alpha>-1, \quad\frac{X\log\log\log X}{\log^2 X}\alpha=-1.\end{cases}
\lim_{N\to\infty}\frac{\#\mathcal{A}_N\cap \Gamma_\alpha^{m,\varepsilon}}{\mathcal{S}_{\mathcal{A}_N,\alpha}}= \mathcal{R}(\Gamma_\alpha^{m,\varepsilon}) , 1\le m\le M_\alpha^{(\varepsilon)}
\sum_{1 \leq n \leq X}g_1(a_1 n + b_1) \prod_{j=2}^k g_j(a_j n + b_j + h_j) & = \sum_{1 \leq n \leq X}1_{\mathcal S}g_1(a_1 n + b_1) \prod_{j=2}^k 1_{\mathcal S}g_j(a_j n + b_j + h_j) + O \Big ( k A X \frac{\log W}{\log H}\Big ).
\begin{array}{rcl}\frac{A^{2n+2}}{V^{2n+1}}= 4\pi(n+1)\,\lambda^n\left(\frac{n+1}{2^n}\right)^{n}\mathrm{Vol(M)}\,\rho^{n+1}(\rho+1)^{n(n+1)}\times \frac{\left({\rm{Appell}}F_{1}[n+1,2,-n,n+2,1-\rho,\frac{1-\rho}{2}]\right)^{n+1}}{\left(_2F_{1}[1+n,-n,n+2,\frac{1-\rho}{2}]\right)^{2n+1}}.\end{array}
\int^s_0\int^\eta_0g_6(r^2-\xi^2)d\xi d\eta+\int^r_0 \rho g_6(\rho^2)d\rho= \frac{1}{2}\int_0^{r^2-s^2}g_6(\xi)\,d\xi + s\int_0^sg_6(r^2-\xi^2)\,d\xi,
J^{i^{\prime}j^{\prime}}= \widehat{J}^{i^{\prime}j^{\prime}}- \frac{i}{4}\int d \sigma ( \psi_{+}^{1}\gamma^{i^{\prime}j^{\prime}}\psi_{+}^{1}+ \psi_{-}^{2}\gamma^{i^{\prime}j^{\prime}}\psi_{-}^{2}) ~ .
S_{\rho}=\left(\frac{n - 2}{n+1}\right)^2 S_{\hat{\rho}}+ \frac{9}{(n+1)^2}S_{\tau}+ \frac{6 (n -2)}{(n+1)^2}\left[ \sum_{1 \leq p < q \leq m}\hat{\rho}^{(pq)}\tau^{(pq)}-{m \choose 2}\mu_{\hat{\rho}\tau}\right] .
\lbrace{\cal F}_{i},{\overline{{{Q}}}}_{a}\rbrace^{\ast}= \frac{i}{2}{( \sigma_{i})}_{a}^{c}{\overline{{{Q}}}}_{c}, \quad \lbrace{\cal F}_{i}, Q^{a}\rbrace^{\ast}= - \frac{i}{2}{( \sigma_{i})}_{c}^{a}Q^{c}.
\frac{zD_{q}\mathcal{F}(z)}{\mathcal{F}(z)}= 1 +\frac{1-[2_{q}]}{3}a_{2}z + \left(\frac{[3]_{q}-1}{10}a_{3}+\frac{1-[2_{q}]}{9}a_{2}^{2}\right) z^2 + \dots
\left\{(x,y)\in I_{n,\beta}(w)\times J_{n,\beta}(k): |T_{\beta}^nx-f(x,y)|<\varphi(n)\right\}&\subset \left\{x\in I_{n,\beta}(w): |T_{\beta}^nx-f(x,k\varphi(n)/\beta^n)|<2\varphi(n)\right\}\times J_{n,\beta}(k) &:=\tilde{I}_{n,\beta}(w,k)\times J_{n,\beta}(k),
S^{-1}\left(\begin{bmatrix}\sqrt{|u_n|}0 0 \sqrt{|u_n|}\end{bmatrix}\begin{bmatrix}u_n+p/u_n u_{\tau}\end{bmatrix}-\begin{bmatrix}R_1 R_2\end{bmatrix}p/ \sqrt{|u_n|}\right) = G
\biggl ( f_{1}*_{1}( f_{2}f_{3}) \biggr ) ( \xi ) + f_{1}( \xi ) \biggl ( f_{2}*_{1}f_{3}\biggr ) ( \xi ) - \biggl ( f_{1}*_{1}f_{2}\biggr ) ( \xi ) f_{3}( \xi ) - \biggl ( ( f_{1}f_{2}) *_{1}f_{3}\biggr ) ( \xi ) = 0 .
\begin{cases}\partial_tw^n + (u^s+{u}^{n-1})\partial_x w^{n}+ (u^s_{yy}+\partial_y w^{n-1}){v}^{n}= \partial^2_{y}w^n + \epsilon \partial^2_{x}w^n , & \partial_y w^n |_{y=0}=0 &w^n|_{t=0}=\tilde{w}_{0, \epsilon},\end{cases}
\mathcal{J}= a^2 + a \sqrt{\rho_g}\langle h^\perp, \mathcal{M}_\tau \rangle - a^2 - a \frac{1}{\sqrt{\rho_g}}\int \left(1+ \rho_g \frac{\mathcal{M}_\tau}{g}\right) h^\perp - \frac{1}{2}\int \left(1+ \rho_g \frac{\mathcal{M}_\tau}{g}\right) \frac{(h^\perp)^2}{g}&= - \frac{1}{2}\int \left( 1+ \rho_g \frac{\mathcal{M}_\tau}{g}\right) \frac{(h^\perp)^2}{g}.
A_1 = \begin{bmatrix}-I/2 -I/2 O O \end{bmatrix}, A_2 = \begin{bmatrix}O O I/2 -I \end{bmatrix}, A_3 = \begin{bmatrix}O O I -I \end{bmatrix}, A_4 = \begin{bmatrix}-I/2 -I O O \end{bmatrix}.
\delta ( b ) ~ = ~ \frac{1}{4 \pi}~ \int_{0}^{\infty}~ \frac{d t}{t}~ ( 8 \pi^{2}\alpha^{\prime}t )^{- N N / 2}~ e^{- b^{2}t / 2 \pi \alpha^{\prime}}\cdot \left( B \times J \right) ~ ~ ,
\Vert \mathfrak{R}_{\Phi}^{f,\gamma}\Vert_{\mathcal{U}}\leq \mathfrak{F}_{\gamma}\left( \Phi ,f\right) \lim_{\gamma\rightarrow 0^{+}}\Vert \mathfrak{R}_{\Phi}^{f,\gamma}\Vert_{\mathcal{U}}=0\ .
s x_j = s^{w_j}x_j, \j = 0, \ldots, n s y_j = s^{-w_j}y_j, \j = 0, \ldots, n s x_{n+1}= s^{-w_1-\cdots-w_n}x_{n+1}, s y_{n+1}= s^{w_1+\cdots+w_n}y_{n+1}.
g_{m,n}(s,t):=\begin{cases}\displaystyle \sum_{j=1}^{m}\left\langle e_j,g\left(s,\tfrac{t^n_k+t^n_{k+1}}{2}\right)\right\rangle e_j t\in (t_{k}^n,t_{k+1}^n), k=0, \ldots, N_n-1, \displaystyle\sum_{j=1}^{m}\langle e_j,g(s,t^n_k)\rangle e_j, t=t_{k}^n, k=0, \ldots, N_n, \end{cases}
V(x) = x^6 + 6a |x|^5 + (9a^2 - 4b) x^4 - (12ab - 2c) |x|^3 + (4b^2 + 6ac - 7) x^2 - \left(18a + 4bc - \frac{2}{c}\right) |x|, E_2^{(-)}= - 10b - c^2 - \frac{6a}{c}+ \frac{2}{c^2}, \psi_2^{(-)}(x) = e^{- \frac{1}{4}x^4 - a |x|^3 + b x^2 - c |x|}x \left(|x| + \frac{1}{c}\right),
\bigl( (g_1,g_2 ),g_3 \bigr) \cdot \bigl( (h_1,h_2),h_3 \bigr) =\bigl(Z^{d(g_3) \, d(h_1,h_2)}(g_1,g_2)\cdot(h_1,h_2), g_3 h_3 \bigr) & = \bigl( Z_1^{d(g_3) (d(h_1) + d(h_2))}Z_1^{d(g_2)d(h_1)}g_1 h_1, g_2 h_2,g_3 h_3 \bigr) & = \bigl( Z_1^{d(g_3) d(h_1) + d(g_3) d(h_2) + d(g_2)d(h_1)}g_1 h_1, g_2 h_2,g_3 h_3 \bigr).
\dot{z}_{n+1}= Q_{\overline{\eta}_{n+1}}( (\nabla_{\overline{u}_{n+1}})_{\overline{\eta}_{n+1}}(z_{n+1}) ) - P_{\overline{\eta}_{n+1}}( (\nabla_{z_{n+1}\circ \overline{\eta}_{n+1}^{-1}})_{\overline{\eta}_{n+1}}+(\nabla h_{n+1}\circ{\overline{\eta}_{n+1}}) ) + \nabla H_{n+1}\circ \overline{\eta}_{n+1},
N(\varphi_j^a, R_0,\lambda_j^a,A_a)= \frac{\int_{D_{R_0}}|(i\nabla + A_a)\varphi_j^a|^2\,dx - \lambda_j^a \int_{D_{R_0}}|\varphi_j^a|^2\,dx}{H(\varphi_j^a, R_0)}\leq \frac{2\Lambda}{C_0}.
g(x_1,\cdots,x_{d-1},x_d)=\sum_{k=-r}^rg_k(e^{\frac{2\pi i x_1}{T_1}},e^{-\frac{2\pi i x_1}{T_1}}, \cdots, e^{\frac{2\pi i x_{d-1}}{T_{d-1}}},e^{-\frac{2\pi i x_{d-1}}{T_{d-1}}}) e^{k\frac{2\pi i x_d}{T_d}}
\varprojlim_n H^i_{Iw}(L_\infty/L,V_n) = \varprojlim_n \varprojlim_K H^i(K,V_n) = \varprojlim_K \varprojlim_n H^i(K,V_n) = \varprojlim_K H^i(K,V) = H^i_{Iw}(L_\infty/L,V) \ .
\pi^{(i)}_{j,k}((u'^{(i)}_j\nabla^{(i)}_j-\nabla^{(i)}_j u'^{(i)}_j) u^{(i)}_j|_{V^{(i)}_{j,k}}) = \left( \gamma^{(i)}_{j,k}\big( \nu^{(i)}_j(w^{(i)}_j)+kdz_i/r^{(i)}_jz_i \big) -\big( \nu^{(i)}_j(w^{(i)}_j)+kdz_i/r^{(i)}_jz_i \big) \gamma^{(i)}_{j,k}\right) \beta^{(i)}_{j,k}\pi^{(i)}_{j,k}=0
\begin{array}{r c l}{{p^{\prime}}}&{{=}}&{{P ( p )}}\\{{}}&{{=}}&{{p^{8}+ 8 p^{7}( 1 - p ) + 2 8 p^{6}( 1 - p )^{2}+ 5 6 p^{5}( 1 - p )^{3}+ \frac{5 2 5}{8}p^{4}( 1 - p )^{4}+ \frac{3 5}{2}p^{3}( 1 - p )^{5}}}\\{{}}&{{=}}&{{\frac{1 0 5}{8}p^{8}- 5 5 p^{7}+ \frac{3 1 5}{4}p^{6}- \frac{6 3}{2}p^{5}- \frac{1 7 5}{8}p^{4}+ \frac{3 5}{2}p^{3}\, \, \, \, \, . \, \, \, \, \,}}\\ \end{array}
\mu =\frac{\Gamma \left( \beta^{2}\right)}{\pi \Gamma \left( 1-\beta^{2}\right)}\left[ \frac{M\sqrt{\pi}\Gamma \left( \frac{1+\xi}{2}\right)}{2\Gamma \left( \frac{\xi}{2}\right)}\right]^{2-2\beta^{2}},
d s^{2}={\frac{X^{1 / 2}}{\rho}}( e^{2 A}d x^{2}+ d r^{2}) +{\frac{X^{1 / 2}}{\rho^{3}}}( d \theta^{2}+{\frac{\rho^{6}\cos^{2}\theta}{X}}d \Omega_{3}^{2}+{\frac{\sin^{2}\theta}{X}}d \phi^{2})
\| A^s (fg) - f A^s g - g A^s f \|_p \lesssim \| f \|_{p_1}\| D^s g \|_{p_2}, 1<p_1,p_2<\infty,\; \frac 1{p_1}+\frac 1{p_2}=\frac 1p; &\| A^s (fg) - g A^s f \|_p \lesssim \| f \|_p \| D^s g \|_{\operatorname{BMO}}, 1<p<\infty; &\| A^s (fg) - fA^s g - g A^s f \|_p \lesssim \| f \|_{\operatorname{BMO}}\| D^s g \|_p, 1<p<\infty.
\Pi_{i}( p_{i}^{2}- m_{i}^{2}) F T \frac{\delta^{4}}{\delta J^{4}}\begin{array}{c}{{l i m}}\\{{\varepsilon \rightarrow 0}}\\ \end{array}< < [ i \delta \alpha \int d^{4}x \phi \partial_{0}^{2}\phi ] > > \mid_{_{J = 0 ; \alpha \rightarrow \alpha_{0}^{-}}}
p_j(x_{\lambda}- x_{\eta}) = \inf_{p_j(z) =0}\left \{p_j(x_{\lambda}- x_{\eta}) - p_j(z) \right \}& \leqslant \inf_{p_j(z) =0}p_j(x_{\lambda}- x_{\eta}- z) & = \| [x_{\lambda}]_j - [x_{\eta}]_j \|_j < \varepsilon,
a\cdot b =\sum_{h\in S}a_{h}t^{h}\cdot\sum_{h\in S'}b_{h}t^{h}=\sum_{l\in S+S'}\left(\sum_{\substack{h\in S h'\in S' h+h'=l}}a_{h}b_{h'}p^{-\beta(h,h')}\right)t^{l}= &=\sum_{l\in S+S'}\left(\sum_{\substack{h\in S h'\in S' h+h'=l}}b_{h'}a_{h}p^{-\beta(h',h)}\right)t^{l}= &=\sum_{h\in S'}b_{h}t^{h}\cdot\sum_{h\in S}a_{h}t^{h}=b\cdot a.
\gamma B^{* \mu \nu}= \gamma \eta^{* \mu}= \gamma \rho^{*}= 0 ; \ \, g a m m a A_{a}^{* \mu}= C_{a b c}A^{* b \mu}C^{c}; \ \ \gamma C_{a}^{*}= - C_{a b c}C^{* b}C^{c};
F_4(t,y_1,y_2,y_3,y_4):= -(1/2N)\Big\{v_N(y_1-y_2)u(y_3,y_1)u(y_2,y_4) &+\int \mathrm{d}x \{\bar p(y_2,x)v_N(y_1-x)u(x,y_4)\}u(y_3,y_1) &+\int \mathrm{d}x \{\bar p(y_1,x)v_N(x-y_2)u(y_3,x)\}u(y_2,y_4) &+\int\mathrm{d}x_1\mathrm{d}x_2 \{\bar p(y_1,x_1)p(x_2,y_2)v_N(x_1-x_2)u(y_3,x_1)u(x_2,y_4)\}\Big\}.
\sum_{\substack{n \geq 0 M = (m_1, \cdots, m_\mathcal{N})}}p(n,M) q^n u_{1}^{m_1}\cdots u_{\mathcal{N}}^{m_\mathcal{N}}= \prod_{n=0}^{\infty}\frac{1}{(1-q^{n+1})}\dfrac{\prod\limits_{t \in T_{}}(1+u_{\{t \}}q^n)}{\prod\limits_{t \in T_{}}(1-u_{\{t \}}q^n)},
J^{NLS}_2 = \left( \begin{array}{cc}D + \varepsilon^{2}\left\{\begin{array}{c}\eta^{-1}\, D^3 + D^3 \, \eta^{-1}-\frac{1}{2}\left[ (\eta^{-1})_{xx}\, D + D \, (\eta^{-1})_{xx}\right]\end{array}\right\}\frac{1}{2}\, D \, \upsilon \frac{1}{2}\, \upsilon D \frac{1}{2}( \eta \, D + D \, \eta ) \end{array}\right)
A\otimes B = \left(\begin{matrix}A_{1,1}B A_{1,2}B \cdots A_{1,q}B A_{2,1}B A_{2,2}B \cdots A_{2,q}B \vdots \vdots \ddots \vdots A_{p,1}B A_{p,2}B \cdots A_{p,q}B \end{matrix}\right).
U^{(l)}=\left\{\left(\begin{array}{ccc}z x_1 x_2 &I_{k-l}u & I_{k-l}\end{array}\right):z\in N_{2l}\right\}, &\psi^{(l)}(u)=\sum_{i=1}^{l}\psi((-1)^{k-i}z_{2i-1,2i})\psi(\mathrm{tr}(u)).
\varliminf_{i \to \infty}\frac{\log_p \lvert K P_i(G) \cap Z : P_i(G) \cap Z \rvert}{\log_p \lvert Z : P_i(G) \cap Z \rvert}\le \varliminf_{k \to \infty}\frac{\log_p \lvert K \varrho_k \cap Z \varrho_k \rvert}{\log_p \lvert Z \varrho_k \rvert}= \lim_{k \to \infty}\frac{(2m-1) p^{k-n}+1}{p^k+1}= (2m-1)/p^n.
q[\frac{1}{2}\sqrt{2}(\Phi^{-1/2}+\Phi^{-1})]=exp\{-\pi K[\frac{1}{2}\sqrt{2}(\Phi^{-1/2}-\Phi^{-1})]\div K[\frac{1}{2}\sqrt{2}(\Phi^{-1/2}+\Phi^{-1})]\}=exp(-\frac{1}{5}\sqrt{5}\pi)
t_{s,p}=\frac{\beta'(b_s)({a_1}-{b_s})t_{s,1}+{\beta'(b_1)}({b_1}-{a_1})t_{1,1}+\beta'(b_1)({a_p}-{b_1})t_{1,p}}{{\beta'(b_s)}({a_p}-{b_s})}
\mathcal{B}_0(\hat{{\mathbf U}},\hat{\Psi})\partial_t{\mathbf V}+\mathcal{B}_1(\hat{{\mathbf U}},\hat{\Psi})\partial_1{\mathbf V}+\mathcal{B}_2(\hat{{\mathbf U}},\hat{\Psi})\partial_2{\mathbf V}+\mathcal{B}_3(\hat{{\mathbf U}},\hat{\Psi}){\mathbf V}=\tilde{\mathcal{F}}(\hat{{\mathbf U}},\hat{\Psi}),
\mathbb{P}\left(S_d \geq y\right) \leq \mathbb{E}~e^{t(S_d - y)}\leq e^{-ty}\mathbb{E}~e^{t S_d}= e^{-t y}\prod_{j=1}^{d}{\mathbb{E}e^{t X_j}}\leq e^{-t y}\left(1 + \sqrt{2\pi}\frac{t}{\sqrt{n}}\right)^d\exp\left(\frac{t^2d}{8n}\right).
\ \ \sum_{\alpha=1}^{\infty}(-v)^{\alpha-1}\sum_{n=1}^{\alpha}c_{n}(\alpha) \frac{x^n\sin(nx)}{\sin^nx}\ =\ \sum_{n=1}^{\infty}\frac{x^n\sin(nx)}{\sin^nx}\, \frac{v^n}{v(v+1) \cdots (v+n)}\,.
{\cal D}_{x^{2 \xi}}^{2}V_{1 , 0}( w ) ={\frac{1}{( x^{\xi}- x^{- \xi})^{2}}}\left( \oint_{w}z^{- 3}f_{1 , 0}\left({\frac{w}{z}}\right) T ( z ) V_{1 , 0}( w ) d z - ( x^{\xi}- x^{- \xi}) w^{- 2}V_{1 , 0}( w ) \right) .
\frac{SO(4,n_A n_B )}{SO(4) \times SO(n_An_B)}\to\frac{SO(2,n_1 n_3 + n_2 n_4)}{SO(2) \times SO(n_1 n_3 + n_2 n_4)}\times \frac{SO(2,n_1 n_4 + n_2 n_3)}{SO(2) \times SO(n_1 n_4 + n_2 n_3)}.
Q_3^k = E\bigg[\sup_{i\le 0}e^{-\beta \tau_i}Y_i-\max_{i\le n}e^{-\beta \tau_i}Y_i\bigg]^{\!k}&\le E\bigg[\sum_{i>n}e^{-k\beta\tau_i}Y_i^k\bigg] &=E(Y_1^k)\,E\bigg[\sum_{i>n}e^{-k\beta\tau_i}\bigg] &\le \sup_{t\ge 0}E\big(\zeta_i^k(t)\big)\,E\bigg[\sum_{i>n}e^{-k\beta \tau_i}\bigg],
M_J =\begin{pmatrix}\rho^{(J)}_1 1 0 \cdots 0 - \rho^{(J)}_2 0 1 \cdots 0 \vdots \vdots \vdots \cdots \vdots (-1)^{N-2}~ \rho^{(J)}_{N-1}0 0 \cdots 1 (-1)^{N-2}~ \rho^{(J)}_N 0 0 \cdots 0\end{pmatrix}~~~ : J \in \left\lbrace L,R \right\rbrace
A^T\tilde{U}_k = \begin{bmatrix}\tilde{V}_k q_{k+1}\end{bmatrix}\tilde{J}_k^T,\tilde{J}_k=\begin{bmatrix}\tilde{\sigma}_1 \rho_1 \tilde{\sigma}_2 \rho_2 \ddots \vdots \tilde{\sigma}_k \rho_k \end{bmatrix},
i D_{T_{\mu \nu}}^{\underline{{a}}\underline{{b}}}( k ) = \frac{i \delta_{\underline{{a}}\underline{{b}}}}{k^{2}- m_{G}^{2}}\left[ - g_{\mu \nu}+ \frac{k_{\mu}k_{\nu}}{k^{2}}\right] \; .
\theta^s_{a,\omega,n}(\kappa_{a,\omega,n}(w^u_q(\tau))) =\langle \gamma^s_{q,(-h,h)}(w^u_q(\tau)), e_y(w^u_q(\tau))\rangle &=\psi^s_{q,(-h,h)}(\tau)\cdot \langle{\gamma}^{\perp}_{q,(-h,h)}(w^u_q(\tau)), e_y(w^u_q(\tau))\rangle+\langle{\gamma}^0_{q,(-h,h)}(w^u_q(\tau)), e_y(w^u_q(\tau))\rangle
\sum_{j=1}^n|A_{ij}|^pe^{u_i+v_j}- n + (1/m)\gamma e^{u_i}= 0,i = 1,\ldots,m \sum_{i=1}^m|A_{ij}|^pe^{u_i+v_j}- m + (1/n)\gamma e^{v_j}= 0,j = 1,\ldots,n.
\begin{aligned}i=i_1, j=i_2, t_1=T\gamma_g(2\tau+1), t_2=T\gamma_g(2\tau+1)+2\tau+1, & \alpha_c= \frac{\psi(X(0)+\rho) e^{-n^{\infty}\kappa(3\tau+2)}}{1+ n^{\infty}\kappa}\mbox{and}\underline{\psi}= \psi(X(0)+\rho)\end{aligned}
S_{( d u s t )}^{'}[ T , Z^{r}, M , W_{r};{}^{4}g_{\mu \nu}] = -{\frac{1}{2}}\int d^{4}x \sqrt{{}^{4}g}( \mu n ) \Big ( U_{\mu}{}^{4}g^{\mu \nu}U_{\nu}- \epsilon \Big ) ,