harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|\Pi_{\mathrm{ker}A^\perp}X_t| = \left| \Pi_{\mathrm{ker}A^\perp}X_0 + t \Pi_{\mathrm{ker}A^\perp}B(X_0,X_0)+ \Pi_{\mathrm{ker}A^\perp}\int_0^t (t-s) \frac{d}{ds}B(X_s,X_s)ds\right| & \ge c t|\Pi_{V_1}X_0||\Pi_{V_2}X_0| - C\sqrt{\delta}K^{r}- C K \tilde{\eta}\int_0^t |B(X_s,X_s)| ds,
\tilde{P}( (-\Delta)^{\tilde{s}})u := \sum_{i=1}^{M'}\tilde{\alpha}_i(-\Delta)^{\tilde{s}_i}u \tilde{s}=(\tilde{s}_1,\dots,\tilde{s}_{M'}), 0<\tilde{s}_1<\cdots<\tilde{s}_{M'}\leq s_M, \tilde{\alpha}_i\in L^\infty(\mathbb{R}^n)
\partial_t \phi(x,0) = -\frac{i}{\sqrt{2L}}v^{\rm top}(x)(a- b) - i \sum_{n=1}^\infty \sqrt{\frac{k_n}{2}}\left[ v^{\rm odd}(n,x)(a_n - b_n) + v^{\rm even}(n,x) (c_n-d_n)\right].
\left| f - \frac{P_{k+1}- BP_k}{Q_{k+1}- BQ_k}\right| = \frac{| (Q_{k+1}- BQ_k)f - (P_{k+1}- BP_k)|}{| Q_{k+1}- BQ_k|}&=\frac{| (Q_{k+1}f - P_{k+1}) +B(P_k - Q_kf)|}{|Q_{k+1}|}&= \frac{|B||Q_k f-P_k|}{|Q_{k+1}|}= \frac{|B|}{|Q_{k+1}|^2}.
^{G}\mathcal{D}^{\beta}f(x_{0})= &F-\lim_{n \rightarrow \infty}\frac{1}{\Gamma_{F}^{\alpha}(-\beta)}\left(\frac{S_{F}^{\alpha}(x_{0})}{n}\right)^{-\beta}\sum_{k=0}^{n-1}\frac{\Gamma_{F}^{\alpha}(k-\beta)}{\Gamma_{F}^{\alpha}(k+1)}&f\left(S_{F}^{\alpha}(x_{0})-k\frac{S_{F}^{\alpha}(x_{0})}{n}\right).
I(i,j)=\{\underbrace{n-m-j+1,\dots, n-m-j+i}_i, \underbrace{n-m+i+1,\dots, n}_{m-i}\}, &I^\circ(m-i,n-m-j)=\{\underbrace{1,\dots, i}_i,\underbrace{n-m-j+i+1,\dots, n-j}_{m-i}\}.
\lim_{k \to + \infty}\eta_{3k+1+K}= \frac{\sqrt{10}-2}{3}, \lim_{k \to + \infty}\eta_{3k+2+K}= \frac{\sqrt{10}-1}{3}. \lim_{k \to + \infty}t_{3k+1+K}= \frac{8-\sqrt{10}}{9}, \lim_{k \to + \infty}t_{3k+2+K}= \frac 23.
\begin{pmatrix}A_0 \cr B_0\end{pmatrix}=\begin{pmatrix}-k^{-1}\sin k \cr\cos k -e^{i\theta}\end{pmatrix},\begin{pmatrix}A_j \cr B_j\end{pmatrix}=e^{ij\theta}\begin{pmatrix}A_0 \cr B_0\end{pmatrix}(j\in \mathbf{Z}).
[z^n x_1^{\ell_1}\cdots x_k^{\ell_k}] (P_1+P_2+\cdots+P_k) = \frac{1}{n-1}[t^{n-2}x_1^{\ell_1}\cdots x_k^{\ell_k}] t^{-2}\Phi(t)^{n-1}&= \frac{1}{n-1}[t^{n}x_1^{\ell_1}\cdots x_k^{\ell_k}] \Phi(t)^{n-1},
=\frac{3}{2}1_{2n}- \frac{3}{4}\tilde{I}_{2n}\left(P_{2n|0}\circ\log_2 P_{2n|0}\right)1_{2n}+ \frac{1}{4}h_{2n|0}- \frac{3}{4}\tilde{I}_{2n}\left(P_{2n|0}\circ\log_2 P_{2n|0}\right)1_{2n}&= \frac{1}{4}h_{2n|0}+ \frac{3}{4}\tilde{I}_{2n}h_{2n|0}+ 1_{2n}
X^{n+1}(\alpha^{n+1}) = X^{n+1}(\alpha^{n}) + X^{n+1}(\beta^{n}) + \sum_{j=1}^{n}X^{j}(\alpha^{n}) \otimes X^{n+1-j}(\beta^{n}) &= X^{n+1}(\beta^{n+1}),
Q(x)=Q(x_1,\dotsc,x_{2s}) :=\sum_{r=1}^s x_{2r-1}x_{2r}, B(x,y)=B((x_1,\dotsc,x_{2s}),(y_1,\dotsc,y_{2s})) :=\sum_{r=1}^s (x_{2r-1}y_{2r}+x_{2r}y_{2r-1}).
\left| \frac{E_{n_{r}0}}{E_{_\mathrm{( g s )}}}\right| = \exp \left[ - 2^{4 / 3}\left( C_{n_{r}}- C_{1}\right) \left( \lambda_{_\mathrm{( g s )}}^{( \ast )}\right)^{- 1 / 3}\epsilon^{- 1 / 3}\right] \stackrel{( \epsilon \rightarrow 0 , n_{r}> 1 )}{\longrightarrow}0 .
\tfrac{1}{4\pi\gamma^2}S^{{\rm cl},\mu \gamma^2}_{{\rm L}}( \hat g,g):=\frac{1}{4\pi}\int_M\big(|d\omega|^2_{g}+ \frac{2}{\gamma}K_{g}\omega+4\pi\mu e^{\gamma\omega}\big){\rm dv}_{g}
\deg\left(_\infty\left(\frac{\Delta}{\Delta_{N,12}}\right)\right)=2^{n_1-1}3^{n_2-1}7^{n_3-1}\cdot 3\cdot 4\cdot 8-1 &\deg\left(_\infty\left(\frac{\Delta_N}{\Delta_{N,12}}\right)\right)=2^{n_1-1}3^{n_2-1}7^{n_3-1}( 3\cdot 4\cdot 8-2\cdot 3\cdot 7)=2^{n_1-1}3^{n_2-1}7^{n_3-1}\cdot 54.
\tilde{B}( m ; i - k ) = \sum_{a = 0}^{P - 1}\sum_{l = 0}^{Q - 1}B ( a - P l ) \mathrm{e}^{- i{\frac{2 \pi}{K}}( a - P l ) ( i - k )}\mathrm{e}^{- i{\frac{2 \pi}{P}}m a}.
\phi_{{\rm L}}(M,[g],\tau) =\langle\Phi_{{\rm L}}(\nabla^{TM},g,d,\tau),[M]\rangle, \phi_{{\rm W}}(M,[g],\tau) =\langle\Phi_{{\rm W}}(\nabla^{TM},g,d,\tau),[M]\rangle, \phi'_{{\rm W}}(M,[g],\tau) =\langle\Phi'_{{\rm W}}(\nabla^{TM},g,d,\tau),[M]\rangle, \psi_{{\rm W},\,j}(M,[g],\tau) =\langle\Psi_{{\rm W},\,j}(\nabla^{TM},g,d,\tau),[M]\rangle, j=1,2,3.
- \frac{V_{J J}}{V}= \frac{1}{3}\Big [ \beta + \frac{1}{J}\Big ]^{2}\bigg ( 1 + \Big [ \beta + \frac{1}{J}\Big ] \Big [ J + \frac{1}{r_{O M}}\Big ( \frac{6 V^{4}}{d}\Big )^{1 / 3}\Big ] \bigg ) ,
\mathbf{C}_{\mathbf{y}_\mathcal{Q}}=\frac{2}{\pi}\left(^{-1}\left(\mathbf{K}\mathfrak{R}\{\mathbf{C}_{\mathbf{y}}\}\mathbf{K}\right)+j^{-1}\left(\mathbf{K}\mathfrak{I}\{\mathbf{C}_{\mathbf{y}}\}\mathbf{K}\right)\right)
G(z=1\,|\,\tau)= 1, G(z+1\,|\,\tau) = \Gamma\Big(\frac{z}{\tau}\Bigr)\,G(z\,|\,\tau), G(z+\tau\,|\,\tau) = (2\pi)^{\frac{\tau-1}{2}}\,\tau^{-z+\frac{1}{2}}\,\Gamma(z)\,G(z\,|\,\tau).
z_{1}= \frac{1}{\sqrt{2}}( x^{1}+ i x^{n + 1}) \, , \frac{1}{\sqrt{2}}( x^{2}+ i x^{n + 2}) \, , \dots \, , z_{n}= \frac{1}{\sqrt{2}}( x^{n}+ i x^{2 n}) \, .
\displaystyle\#(\{x,y,z\}:xy,zy,zx\notin E(G)) \geq \frac{m(4m-n^2)}{3n}&=\frac{(\binom{n}{2}- C'n^{3/2})(4\binom{n}{2}- 4C'n^{3/2}-n^2)}{3n}&=\frac{n^4-o(n^4)}{6n}&=\frac{n^{3}}{6}-o(n^3).
\langle{\hat{\sigma}}_{1}{\hat{\sigma}}_{n_{1}}. . .{\hat{\sigma}}_{n_{s}}\rangle_{g}= \sum_{i = 1}^{s}C ({\hat{\sigma}}_{1},{\hat{\sigma}}_{n_{i}}) \langle{\hat{\sigma}}_{n_{1}}. . .{\hat{\sigma}}_{n_{s}}\rangle_{g}= ( 2 g - 2 + n ) \langle{\hat{\sigma}}_{n_{1}}. . .{\hat{\sigma}}_{n_{s}}\rangle_{g}
\int_{-1}^{1}T_n(z)T_m(z)\frac{1}{\sqrt{1-z^2}}dz = \begin{cases}0 \mbox{if}n\not=m, \pi \mbox{if}n =m=0, \pi /2 \mbox{if}n =m\neq 0, \end{cases}\int_{-1}^{1}U_n(z)U_m(z)\sqrt{1-z^2}dz = \begin{cases}0 \mbox{if}n\not=m, \pi /2 \mbox{if}n =m. \end{cases}
\begin{aligned}0 \geq \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii}- \frac{\beta}{2n}F^{11}U_{11}+ \gamma_2 \sum_{U_{ii}< 0}F^{ii}U_{ii}&\geq \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii}+ \frac{\beta(A + M)}{8n(n-k+1)}\sum_i F^{ii}- \gamma_2 C_0 (1 + M) \sum_{i}F^{ii}& \geq \frac{\gamma_1 (A + M)}{2}\sum_i F^{ii},\end{aligned}
\int_{\left\vert_{\!_{\!_{\,}}}y-c\,(t-s)^{1/2}\right\vert >\frac{c}{2}(t-s)^{1/2}}\mathrm{d}y\p_{t-s}\bigl(c\,(t-s)^{1/2}-y\bigr)\mathsf{1}_{B_{2}(0)}(y)X_{s}(y) \leq\frac{2c}{(2\pi)^{1/2}}W_{B_{3}(0)}\sum_{j=0}^{\infty}e^{-c^2(1/2+j)^{2}/2}\leq CW_{B_{3}(0)}c^{-2}.
\Delta_{s}\frac{\lambda ( s )}{B_{k_{1}k_{2}}( s^{2})}= \frac{1}{B_{k_{1}k_{2}}( s^{2})}\sum_{j = 1}^{2}\sum_{p = 1}^{k_{j}}\left( \frac{\partial^{2}}{\partial s_{p j}^{2}}+ \frac{1}{s_{p j}}\frac{\partial}{\partial s_{p j}}- 4 D_{p j}\right) \lambda ( s ) \ ,
0=\nabla \cdot u =\sum_{k\in \mathbb{Z}^2 \times \mathbb{N}}\hat{u}_1(t;k)\frac{\partial}{\partial x_1}E(x;k)+\sum_{k\in \mathbb{Z}^2 \times \mathbb{N}}\hat{u}_2(t;k)\frac{\partial}{\partial x_2}E(x;k)+\frac{\partial u_3}{\partial x_3},
\mathbf{W}_{}= \begin{bmatrix}\mathbf{W}^1 \mathbf{0}\cdots \mathbf{0}\mathbf{0}\mathbf{W}^2 \cdots \mathbf{0}\vdots \vdots \ddots \vdots \mathbf{0}\mathbf{0}\cdots \mathbf{W}^G \end{bmatrix}\in \mathbb{C}^{N \times K}.
\int_{t_0}^{t_1}\int_{K_{R}}u_{t}2 h\left( \Psi^2 \right)\Psi \Psi' \zeta^{q}\,dx \,dt = \int_{t_0}^{t_1}\int_{K_{R}}\left[\frac{d}{dt}H\left( \Psi^2 \right)\right]\zeta^{q}\,dx \,dt = \int_{K_{R}\times \{t_1\}}H\left( \Psi^2\right)\zeta^{q}\,dx - \int_{K_{R}\times \{t_0\}}H\left( \Psi^2\right)\zeta^{q}\,dx.
\Psi_{\widetilde{\pi}_F}(\widetilde{\pi}_F(f)v) =\Psi_{\widetilde{\pi}_F}(\sigma_{\zeta_1}(f)v_{\zeta_1}\oplus\cdots\oplus \sigma_{\zeta_m}(f)v_{\zeta_m}) &=\sum_{j=1}^m\Vert \sigma_{\zeta_j}(f)v_{\zeta_j}\Vert^2\Psi_{\sigma_{\zeta_j}}(\sigma_{\zeta_j}(f)v_{\zeta_j})
\begin{aligned}{\mathcal I}_{311}(t) =- \omega \Big[ \big(B_{+, \omega}(t)\big)^2 - \big(B_{-, \omega}(t)\big)^2 \Big] + KR\cos\gamma_0 \Big[ \big(B_{+, \omega}(t)\big)^2 + \big(B_{-, \omega}(t)\big)^2 \Big], &{\mathcal I}_{32}(t) +{\mathcal I}_{312}(t) \geq KR \sin\gamma_0 \Gamma_{\gamma_0, \omega}^+(t).\end{aligned}
VI_2\leq \sum\limits_{k=1}^{\infty}\frac{e^{-C2^{2(k-1)}}2^{(k+1)n}}{|2^{k+1}B|}\int_{2^{k+1}B}|L^{-\alpha/2}(\prod\limits_{j=1}^m(b_j-(b_j)_B)f_1)(z)|dz
\gamma_{J}(t^{J}(a,b)) =t^{\mathcal{I}(\mathcal{C}(J))}(\gamma_{J}(a),\gamma_{J}b)=t^{\mathcal{I}(\mathcal{C}(J))}([\Delta^J a,\nabla^J a],[\Delta^J b,\nabla^J b]) &=t^{\mathcal{I}(\mathcal{C}(J))}([ a,a],[ b, b])=t^{\mathcal{I}(\mathcal{C}(J))}(\Delta^{\mathcal{I}(\mathcal{C}(J))}[ a,b],\nabla^{\mathcal{I}(\mathcal{C}(J))}[ a, b]) &=[a,b],
A_{k}(t,r,\theta,\varphi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\Bigr[c_{l}(r)\partial_{k}Y_{lm}(\theta,\varphi)+T_{l}(r)\varepsilon_{kp}\partial^{p}Y_{lm}(\theta,\varphi)\Bigr]e^{i\omega t}.
g_{\phi \phi}= \alpha \left[ ( C_{1 0 0}^{2}- C_{1 1 0}C_{0 0 0}) \phi^{2}- ( C_{1 1 1}C_{0 0 0}- C_{1 1 0}C_{0 0 1}) \phi + C_{1 1 0}^{2}- C_{0 0 0}C_{1 0 0}\right] \ ,
P_n(x) = \log 2 \cdot \omega_0(n) + \frac14 \sum_{2\le q\le X}\omega_{q}(n)^2 \Lambda(q) = \log 2 \cdot (\mu(\omega_0) + F_{\omega_0}(n)) + \frac14 \sum_{2\le q\le X}(\mu(\omega_q) + F_{\omega_q}(n))^2 \Lambda(q).
c^{0}= \left[ ( 1 - \mu^{2}k^{2}r^{4}) \left( 1 - \mu^{2}/ k^{2}\right) \right]^{\frac{1}{4}\left( 1 - \frac{\mu}{k}\right)}\sqrt{\left( 1 + \mu / k \right) ( 1 + \mu k r^{2})}\; .
S_{\rm eff}[\chi] = \int\! d^2x \, \Bigl({1\over 2}\,\partial_{\alpha}\chi \partial_{\alpha}\chi -{\delta v\over 2}\,\chi^{2}+{\pi V\over N}\,\chi^4 +{\cal O}\bigl(\chi^{6}/N^{2}\bigr)\Bigr),
\begin{aligned}\tilde p= \tilde p[\Lambda,\phi,\alpha]:= ~{}\kappa \partial_x\alpha \left[ (\partial_t\Lambda)^2+(\partial_x\Lambda)^2 +4\sinh^2(\Lambda)\Big((\partial_t \phi)^2+(\partial_x\phi)^2\Big) \right] &~{}-2\kappa \partial_t\alpha\left(\partial_x\Lambda \partial_t\Lambda +4\partial_x \phi \partial_t \phi \sinh^2(\Lambda) \right).\end{aligned}
\frac{W_{1}^{\prime}}{2 \sqrt{W_{1}}}\left[ 1 + \frac{R^{2}}{\Pi^{2}}\left( 1 - \frac{1}{R_{x}}\right) \right] + \frac{\sqrt{W_{1}}}{2}W_{2}^{\prime}R_{x}( R_{x}- 1 ) = 0
\left\langle \left( \gamma;d_{1},...,d_{p}\right) ,\int_{s}^{t}\widetilde{\omega}\left( u\right) \mathbf{d}u\right\rangle & =\int_{s}^{t}\left\langle \left( \gamma;d_{1},...,d_{p}\right),\widetilde{\omega}\left( u\right) \right\rangle \mathbf{d}u
- \sum_{m=1}^{\infty}\frac{q^{N+m}}{1+q^{N+m}}- \sum_{m=1}^{\infty}\frac{q^{m}}{1-q^{m}}=-\sum_{m=1}^{N}\frac{q^m}{1-q^m}- \sum_{m=N+1}^{\infty}\frac{2q^m}{1-q^{2m}}.
\int_{M}\left[ 2u_{p}\cdot u_{p}^{\left( 2\right)}+\left( u_{p}^{\left(1\right)}\right)^{2}+u_{p}\cdot u_{p}^{\left( 1\right)}trh+\frac{1}{8}u_{p}^{2}\left( trh\right)^{2}-\frac{1}{4}u_{p}^{2}\left\vert h\right\vert^{2}\right] d\mu =0.
\frac{1}{2\pi i}\int_{-\frac{1}{2}-i\infty}^{-\frac{1}{2}+i\infty}\zeta_K(s+1)\frac{x^s}{s}\frac{n_K!}{\prod_{j=1}^{n_K}(s+j)}ds=\frac{n_K!}{2\pi i}\int_{-\frac{1}{2}-i\infty}^{-\frac{1}{2}+i\infty}\zeta_K(s+1)\frac{\Gamma(s)}{\Gamma(n_K+1+s)}x^s ds.
\{Q_{\alpha}, Q_{\beta}\}= ( \Gamma^{\mu}{\cal C})_{\alpha \beta}P_{\mu}+ ( \Gamma^{\mu_{1}\mu_{2}}{\cal C})_{\alpha \beta}Z_{\mu_{1}\mu_{2}}+ ( \Gamma^{\mu_{1}\cdots \mu_{5}}{\cal C})_{\alpha \beta}X_{\mu_{1}\cdots \mu_{5}},
{{f}_{{{I}_{x}}}}\left( I|{{I}_{y}}\right)=\frac{{{\gamma}_{1}}{{\left( I/{{I}_{y}}\right)}^{{{m}_{1}}{{\gamma}_{1}}-1}}}{{{I}_{y}}{{\left({{{\Omega}_{1}}}/{{{m}_{1}}}\; \right)}^{{{m}_{1}}}}\Gamma \left({{m}_{1}}\right)}\exp \left( -\frac{{{m}_{1}}}{{{\Omega}_{1}}}{{\left( \frac{I}{{{I}_{y}}}\right)}^{{{\gamma}_{1}}}}\right)
Inf_H^{H\times 1}a \circ Inf^{1\times K}_Kb =A(H\times \overleftarrow{1}\times K)(A(_{H\times 1}H_H)(a)\times A(_{1\times K}K_K)(b)) &=A(H\times \overleftarrow{1}\times K \circ \; H \times K_{H\times K})(a \times b)=a\times b
T_{\mu \nu}^{\mathrm{r e n}}= T_{\mu \nu}^{\mathrm{r e g}}+ g_{\mu \nu}\left( \delta \Lambda + \frac{1}{2}\delta m^{2}\phi^{2}( x ) + \frac{\delta \lambda}{4 !}\phi^{4}( x ) \right) + A ( g_{\mu \nu}\partial_{\alpha}\partial^{\alpha}- \partial_{\mu}\partial_{\nu}) \phi^{2}( x ) \; .
\left({{\dot{a}}\over a}\right)^2={{(C+E)^2g_Se^{2\phi}-|g_{00}|[g_Sg(g^2h^2+b^2)+\ell^2e^{2\phi}]}\over{4|g_{00}|g_{rr}g_Sg(g^2h^2+b^2)}}\left({{g^{\prime}}\over g}\right)^2.
\begin{array}{lc}(P_jQ_{jk}P_{k}w)(x,t) = c_j(x_j,x,t) [2mm]\displaystyle \quad\times\int_{0}^{1}r_{jk}(\xi,\omega_j(x_j,\xi,t))c_k(x_k,\xi,\omega_j(x_j,\xi,t))\int_{0}^{1}w(\eta,\omega_k(x_k,\xi,t))\, d\eta d\xi. \end{array}
\textrm{Pr}\left\{\sum_{i=1}^{E^{\textrm{all}}_{k'}}\rho \log_2\left(1+\bar{\gamma}_{k'}^{\textrm{T}}\frac{|h_i|^2}{1+|g_i|^2}\right)< N_{k'}\right\}\leq p_\textrm{o}&\bar{\gamma}_i\geq \bar{\gamma}_{k'}^{\textrm{T}}, \forall i=1,2,...,E^{\textrm{all}}_{k'}.
(1-q)\zeta_{2}(q)\phi\int_{\rho B_{1}}\int_{\rho B_{1}}(\psi(x)-\psi(y))^{2}(w^{2}(s,x) + w^{2}(s,y))k(x,y)dxdy & \leq c_{2}C(\delta,\Lambda)(\rho-\rho')^{-2\beta}\phi\int_{\rho B_{1}}w^{2}(s,x)dx= c_{3}(\rho-\rho')^{-2\beta}\phi\int_{\rho B_{1}}w^{2}(s,x)dx,
l_{0}^{e}=(l_{0}-l_{1}-l_{2}-l_{3})/2-1,l_{1}^{e}=(l_{0}-l_{1}+l_{2}+l_{3})/2, & l_{2}^{e}=(l_{0}+l_{1}-l_{2}+l_{3})/2,l_{3}^{e}=(l_{0}+l_{1}+l_{2}-l_{3})/2.
y_1 = \rho T_{d-1}\dotsm T_{2}T_1 , y_i = T_{i-1}^{-1}\dotsm T_2^{-1}T_1^{-1}\rho T_{d-1}\dotsm T_{i+1}T_i, i=2,\ldots, d-1, \intertext{resp.}y_1^* = \rho T_{d-1}^{-1}\dotsm T_{2}^{-1}T_1^{-1}, y_i^* = T_{i-1}\dotsm T_2T_1\rho T_{d-1}^{-1}\dotsm T_{i+1}^{-1}T_i^{-1}, i=2,\ldots,d-1.
\sqrt{|D|}L_D(1)\frac{1}{2\pi}\sum_{Q \in \mathcal{Q}_{d}/ \mathrm{SL}_2(\mathbb{Z})}\int_{\mathrm{SL}_2(\mathbb{Z})_Q \backslash S_Q}\frac{\sqrt{d}dz}{Q(z,1)}= \sqrt{|D|}L_D(1)\frac{h(d)\mathrm{log}\varepsilon_d}{\pi},
\varphi_{g_k}([g_i,h_j])={g_k}[g_i,h_j]{g_k}^{-1}=[g_k g_i,h_j][h_j,g_k] \varphi_{h_k}([g_i,h_j])={h_k}[g_i,h_j]{h_k}^{-1}=[h_k,g_i][g_i,h_k h_j],
h := \left( \prod\limits_{j = 1}^s \frac{1}{y_j}\right) \left( \sum\limits_{k,k' = 1}^s \frac{1}{y_k y_{k'}}d x_k \otimes d x_{k'}+ d y_k \otimes d y_{k'}\right) + d x_{s+1}^2 + d y_{s+1}^2,
{\cal K}^{[ j ]}= \mathrm{l o g}\| \Lambda^{[ j ]}\|^{2}\equiv \mathrm{l o g}[ < 0 |{\bar{X}}^{[ j ] 1}\cdots{\bar{X}}^{[ j ] j}{X}^{[ j ] j}\cdots{X}^{[ j ] 1}| 0 > ] .
\mathcal{K}_-(x,k) = b_-^+(k) e^{ikx}+ b_-^-(k) e^{-ikx}, & \mbox{with}b_-^+(k) = \mathbf{1}_{+}(k) + \mathbf{1}_{-}(k)T(-k), b^-_-(k) = \mathbf{1}_{+}(k)R_-(k).
\left( \dfrac{(f+W^{2})}{W^{2}}\left\langle \Pi_{\ast}\bar{\nabla}_{D_{0}}D_{0},Du \right\rangle \right)_{k}= \dfrac{(f+W^{2})}{2fW^{2}}[(\dfrac{f_{l}f_{k}}{f}-f_{k;l})WN^{l}+ f^{l}\hat{u}_{l;k}] + \left\langle \Pi_{\ast}\bar{\nabla}_{D_{0}}D_{0},Du \right\rangle\dfrac{1}{W^{2}}(f_{k}+2 \dfrac{f\eta_{k}}{\eta}),
\begin{aligned}\lim_{n\rightarrow\infty}\bigg[ \sup_{t\in\mathbb{R_+}}\mathbb{E}e^{-\beta t}|y_{(n,t)}-\hat{y}^n_t|^2+\mathbb{E}\int_0^\infty\mathbb{I}_{[0,n]}e^{-\beta t}\Big(|y_{(n,t)}-\hat{y}^n_t|^2+|z_{(n,t)}-\hat{z}^n_t|^2 &\quad+|\tilde{z}_{(n,t)}-\hat{\tilde{z}}^n_t|^2+||\gamma_{(n,t)}-\hat{\gamma}^n_t||^2\Big)dt\bigg]=0.\end{aligned}
(u_{h}=\mathrm{pr}_{\underline{S}}\colon \underline{S}\times_{h,\mathfrak{Z},u}\mathfrak{U}_{X}\to\underline{S},\Theta_{u,h}=\mathrm{id}_{\underline{S}_{\mathbb{C}}}\times_{\mathfrak{Z}_{\mathbb{C}}}\Theta_{u})
\widehat u_{\tau}(t):=\widehat u_{\tau,m}^S(t):=(\Phi_{j,t-t_m}^S\widehat{\Phi_{j,\tau}^D})\prod_{j=1}^{m-1}\big(\Phi_{j,\tau}^S\widehat{\Phi_{j,\tau}^D}\big)u_\tau(0),t\in T_m,
\int_{\lambda_2}^{\lambda}Q^{1/2}(\eta,x) d \eta=\left\{\int_{\lambda_2}^{\lambda_1}+\int_{\lambda_1}^{\lambda_*}+\int_{\lambda_*}^{\lambda}\right\}Q^{1/2}(\eta,x) d\eta:=I_0+I_1+I_2.
\big|P(\alpha_i)\big|_v \le p^2(N-1) \frac{1}{\delta}\prod_{k=1}^s \max\{|\alpha_i-\alpha_k|_v,\,\delta\}^{n_k}\le p^3 N \prod_{k=1}^s \max\{|\alpha_i-\alpha_k|_v,\,\delta\}^{n_k}.
\chi_{1,1}^{n,m}(x,y|\rho)\allowbreak=\allowbreak\sum_{j\geq0}\rho^{j}U_{m+j}(x)T_{n+j}(y)\allowbreak= (T_{n}(y)U_{m}(x)(w_{2}(x,y|\rho)-\rho^{4}) +\rho T_{n+1}(y)U_{m+1}(x)(1-2\rho^{2}+4\rho^{2}(x^{2}+y^{2})-4\rho xy) +\rho^{2}T_{n+2}(y)U_{m+2}(y)(1-4\rho xy)+\rho^{3}T_{n+3}(y)U_{m+3}(y))/w_{2}(x,y|\rho).
\vec{x}=\vec{a}G=(\begin{matrix}1&0&1&1\end{matrix})(\begin{matrix}1&0&0&0&1&1&0\\ 0&1&0&0&1&0&1\\ 0&0&1&0&0&1&1\\ 0&0&0&1&1&1&1\\ \end{matrix})=(\begin{matrix}1&0&1&1&2&3&2\end{matrix})=(\begin{matrix}1&0&1&1&0&1&0\end{matrix})
\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln|\omega(x,0)|dx = \displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2\ln\frac{|\omega(x,0)|}{\|\omega\|_X}dx+\ln\|\omega\|_X\displaystyle\int_{\mathbb{R}^N}a(x)|\omega(x,0)|^2dx = I_1+I_2,
\int_{\Psi_m(T)(t_0)\wedge \sigma_k(B)}^{T(t_0) \wedge \sigma_k(B)}\sup_{|x|<R}| A(x+B(s))|^2ds \begin{cases}\to 0\m \downarrow 0\ \nu^0, &\leq \displaystyle \sup_{|z|< R+ k}|A(z)|^2 \T(t_0)<\infty. \end{cases}
\int_{\Lambda}\left[\prod_{i=1}^{n_a}d^2x_{i}^{(a)}d^2y_{i}^{(a)}\prod_{j=1}^{n_b}d^2x_{j}^{(b)}d^2y_{j}^{(b)}\right]\Big| B(a,b) \Big|^{\frac{g}{\pi+gN}q(N-1)}\;
0 = \biggl | \begin{matrix}{A_{j m}( x )}&{A_{j m}( y )}\\{A_{m k}( x )}&{A_{m k}( y )}\\ \end{matrix}\biggr | - \biggl | \begin{matrix}{\psi_{j m}( x )}&{\psi_{j m}( y )}\\{\psi_{m k}( x )}&{\psi_{m k}( y )}\\ \end{matrix}\biggr | A_{j m k}( x + y ) .
{\cal Z}_{_I}= \int D\overline{\xi}^\prime D\xi^\prime D\overline{\chi}D\chi \exp \left\{i \int d^2 x \left[ -i \overline{\chi}\partial \!\!\! / \chi - \frac{1}{m}\overline{\chi}\partial \!\!\! / \partial \!\!\! / \chi + m \overline{\xi}^\prime \partial \!\!\! / \partial \!\!\! / \xi^\prime \right] \right\}.
{\cal L}= \frac{3}{4}\Omega^{\mu,\nu\alpha\beta}\Omega_{\nu,\mu\alpha\beta}- \frac{3}{4}\Omega^{\alpha\beta}\Omega_{\alpha\beta}- \frac{1}{2}\Omega^{\mu,\nu\alpha\beta}T_{\nu\alpha\beta,\mu}+ \frac{3}{2}\Omega^{\alpha\beta}T_{\alpha\beta}
\hat{A}_{\underline{m}}\cong \lim_{j\rightarrow\infty}A/\underline{m}^j \cong \lim_{j\rightarrow\infty}\left( \times_{i=1}^n A/\underline{m}^j_i\right) \cong \times_{i=1}^n\lim_{j\rightarrow\infty}A/\underline{m}^j_i \cong \times_{i=1}^n \hat{A}_{\underline{m}_i},
\sup_{Q'\in \mathcal{D}_\nu}\sum_{Q\in \mathcal{D}_\nu}(1+2^\nu{\rm dist}(Q,Q'))^{N}\|1_Q e^{-tL}1_{Q'}\|_{2\to 2}\leq Ce^{M_{0}t}, N=\lfloor n/2\rfloor+1.
S_{r}^{\alpha}={\frac{1}{\sqrt{2}}}\left( \begin{array}{l l}{f_{I}^{\Lambda}+ \mathrm{i}h_{\Lambda I}}{\bar{f}_{I}^{\Lambda}+ \mathrm{i}\bar{h}_{\Lambda I}}{f_{I}^{\Lambda}- \mathrm{i}h_{\Lambda I}}{\bar{f}_{I}^{\Lambda}- \mathrm{i}\bar{h}_{\Lambda I}}\end{array}\right)
S_{1}= \int d \tau \left( \begin{array}{c}{{p_{1}\cdot \dot{x}_{1}- p_{1}\cdot \lambda_{2}\, A_{1}}}\\{{+ \bar{\theta}_{1}\gamma^{\mu \nu}\dot{\theta}_{1}\, p_{1 \mu}\lambda_{2 \nu}- \frac 1 2 e_{1}p_{1}^{2}}}\\{{+ \bar{\theta}_{1}\gamma^{1 1 , \nu}\dot{\theta}_{1}\, m_{1}\lambda_{2 \nu}- \frac 1 2 e_{1}m_{1}^{2}}}\\ \end{array}\right) .
< \hat{{\cal{J}}}_{\mu_{1}\cdots \mu_{s}}^{( s )}( x ) \hat{{\cal{J}}}_{\nu_{1}\cdots \nu_{s}}^{( s )}( 0 ) > ={\prod}_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}\left({\frac{c_{s}( t )}{x^{4}}}\right) .
\Delta(n\mid\{1,1\}) =\Delta(n-2\mid\{1,1\}) + \Delta(n-2\mid\{1\}) &=\Delta(n-2\mid\{1\}) + 1 &=\Delta(n-4\mid\{1,1\}) + \Delta(n-4\mid\{1\}) +1 &=\ldots &=\Delta(n-2k\mid\{1,1\}) + k=0 + k=\lfloor\frac{n-1}2\rfloor.
\pi_{i}= g \bar{\Gamma}\psi_{i}^{\prime}\sqrt{2 V_{e f f}- \rho_{f}- p_{f}}e^{3 T}, \ \ \ \pi_{f}= g \bar{\Gamma}({\frac{n_{f}^{2}\theta_{f}^{\prime}}{\rho_{f}+ p_{f}}}) \sqrt{2 V_{e f f}- \rho_{f}- p_{f})}e^{3 T},
i\widetilde{\mathcal{G}}_{\alpha \beta}(x,y)=\sum_{m=0}^{\infty}\left(-\frac{i}{\hbar}\right)^{m}\frac{1}{m!}\int_{-\infty}^{\infty}dt_{1}\cdots\int_{-\infty}^{\infty}dt_{m}\langle\phi_{0}|T[\hat{H}_{1}(t_{1})\cdots\hat{H}_{1}(t_{m})\hat{\psi}_{\mathrm{H}\alpha}(x)\hat{\psi}_{\mathrm{H}\beta}^{\dagger}(y)]|\phi_{0}\rangle,
\begin{array}{l@{{}{}}l}H_{L+1}= c_1H_{L}+\cdots+\left( c_{L-1}+1 \right) H_2+\left( c_{L}-1 \right) H_1 &= c_1\left( G_{L}+1 \right) +\cdots+\left( c_{L-1}+1 \right) G_2+\left( c_{L}-1 \right) G_1 = c_1+G_2-G_1+\sum_{i=1}^{L}c_{i}G_{L+1-i}= c_1+c_1 +G_{L+1}= 2c_1+G_{L+1}. \end{array}
H=h_1(x) v + h_2(x) v^{a+1}\bar{v}^{a},a\neq 0, &G= - \Psi(h_1(x) v + h_2(x) v^{a+1}\bar{v}^{a}) -\i kv^{1+p/2}\bar{v}^{p/2},a\neq 0.
\|a+x\|_\sharp=(1+c)\bigg\|\frac{1}{1+c}\cdot a+\frac{c}{1+c}\cdot v\bigg\|_\sharp=(1+c)\|a\|_\sharp=\|a\|_\sharp+c\|v\|_\sharp=\|a\|_\sharp+\|x\|_\sharp,
\lambda = \frac{\eta\,\exp\frac{S(A|M)_{\hat{\rho}_{AM}(t)}}{n}}{\eta\,\exp\frac{S(A|M)_{\hat{\rho}_{AM}(t)}}{n}+ \left|1-\eta\right|\exp\frac{S(B|M)_{\hat{\rho}_{BM}(t)}}{n}}\;,
\mathbb{E}[\eta^2_t]=\mathbb{E}[\mathbb{E}(\eta^2_t|R_{t-1},N_t)] \leq \mathbb{E}[\mathbb{E}(^2(n,pN_t)|N_t)] &\leq np\mathbb{E}[N_t]+(np)^2\mathbb{E}[N^2_t].
\widehat{M}:= \left( \begin{array}{c}\widehat{m}_1\times n\_\_\_\_\widehat{m}_2\times n\_\_\_\_\widehat{m}_3\times n\_\_\_\_\end{array}\right), \widehat{m}:= \left( \begin{array}{c}\widehat{m}_1 \_\_\_\_\widehat{m}_2\_\_\_\_\widehat{m}_3 \_\_\_\_\end{array}\right).
\Xi^{\nu \rho}( x ) = \int d^{3}x_{2}. . . . d^{3}x_{n}F_{\mu_{2}\nu_{2}}( x_{2}) . . . . F_{\mu_{n}\nu_{n}}( x_{n}) \Omega^{\nu \rho \mu_{2}\nu_{2}. . . \mu_{n}\nu_{n}}( x , x_{2}, . . . , x_{n}) \; .
\Omega_{s}= - \mathrm{d}x^{a}\wedge \mathrm{d}p_{a}+ \Omega_{\cal L}\qquad \Omega_{\cal L}= - 2 i s \frac{\mathrm{d}z \wedge \mathrm{d}\bar{z}}{( 1 - z \bar{z})^{2}},
\sum_{m=0}^N\alpha^mc_A^{\frac{m}{2}}\|S_m^{(1)}-S_m^{(2)}\|_{L^1}&\le \left(|e^{W_0^{(1)}}-e^{W_0^{(2)}}|+e^{|W_0^{(2)}|}\sum_{m=1}^N(\alpha+1)^mc_A^{\frac{m}{2}}\|W_m^{(1)}-W_m^{(2)}\|_{L^1}\right) &\quad\cdot e^{\sup_{j\in\{1,2\}}\sum_{m=1}^N(\alpha+1)^mc_A^{\frac{m}{2}}\|W_m^{(j)}\|_{L^1}}.
ln\gamma_{i}=\frac{\partial(\frac{G^{ex}}{W_{w}RT})}{\partial b_{i}}=\frac{z_{i}^{2}}{2}f^{\prime}+2\sum_{j}\lambda_{ij}b_{j}+\frac{z_{i}^{2}}{2}\sum_{j}\sum_{k}\lambda_{jk}^{\prime}b_{j}b_{k}+3\sum_{j}\sum_{k}\mu_{ijk}b_{j}b_{k}+\cdot\cdot\cdot
L_{1}^{\ast}=\frac{\Gamma(t-1)\left[\xi_{p}^{2}\left(1+\xi_{p}^{2}\right)^{-(t-1)}-\xi_{q}^{2}\left(1+\xi_{q}^{2}\right)^{-(t-1)}\right]}{2\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
S = \int_{0}^{L}d y \int d^{4}x \sqrt{- G}\left( - \frac{1}{2}G^{M N}\partial_{M}\Phi \partial_{N}\Phi + \frac{1}{\sqrt{- G}}\partial_{M}\left( \sqrt{- G}G^{M N}\Phi \partial_{N}\Phi \right) - \frac{1}{2}V ( \Phi ) \right) \ .
\theta(x) \triangleq \max_i \{g_i(x)_+\}- \kappa(x) = \lambda \left(\max_i \{g_i(x)_+\}- \min_d \left\{\max_i \left\{\tilde g_i(d; x)_+ \right\}\, | \, \|d\|_\infty \le \rho, \, d \in K - x\right\}\right)
\phi(t):=-\frac{pe^{-qt}}{p+2q}, &b(s,\tau):=-pe^{-qs}e^{-(p+q)\tau}, &g(s,\tau ; 0,t):=b(t-s,\tau)\frac{1}{1-\phi^2(t)}+b(s,\tau)\frac{\phi(t)}{1-\phi^2(t)}.
\sum n^{m}=\frac{1}{m+1}(B_{0}n^{m+1}+(\begin{matrix}m+1\\ 1\end{matrix})B_{1}^{+}n^{m}+(\begin{matrix}m+1\\ 2\end{matrix})B_{2}n^{m-1}+\cdot\cdot\cdot+(\begin{matrix}m+1\\m\end{matrix})B_{m}n).
\left.\begin{tabular}[c]{l}$\mathcal{F}\left\{_{-\infty}D_{x}^{\alpha}f(x)\right\}=(i\omega)^{\alpha}g(\omega),$ $\mathcal{F}\left\{_{\infty}D_{x}^{\alpha}f(x)\right\}=(-i\omega)^{\alpha}g(\omega)$\end{tabular}\ \ \ \ \ \ \ \ \right.
\Psi = ( 1 -{\frac{\rho}{4}}x^{+}x^{-})^{\frac{1}{2}}\left( \begin{array}{c}{e^{-{\frac{\theta}{2}}}e^{\imath v}p ( x^{-})}{e^{{\frac{\theta}{2}}}e^{\imath u}f ( x^{+})}\end{array}\right)
\mathrm{Var}Y_N = \frac{1}{N^2}\sum_{i,j,k,l}\phi\left( \frac{i}{N},\frac{j}{N},k,l\right) g\left(\frac{i}{N},\frac{j}{N}\right)g\left(\frac{i+k}{N},\frac{j+l}{N}\right) \exp(\mathrm{i}2\pi(sk-tl)).
a_1(\xi,\eta) = |\xi-\eta|^{3/2}+ O \big( |\xi-\eta|^{5/2}|\eta|^{-1}\big), a_2(\xi,\eta) = |\xi-\eta|^{3/2}a_3(\xi,\eta) = -\frac{1}{2}|\xi-\eta|^{3/2}+ O \big( |\xi-\eta|^{5/2}|\eta|^{-1}\big),a_4(\xi,\eta) = |\xi-\eta|^{2}|\eta|^{-1/2}+ O \big( |\xi-\eta|^{3}|\eta|^{-3/2}\big).
M =\left(\begin{array}{ccc|ccc|c}0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \hline 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 m_{6,6}m_{6,7}\hline0 0 0 0 0 0 0\end{array}\right).
\begin{aligned}k(n,m,\alpha^\prime) \, \mbox{is constant for}\, \alpha^\prime \in \left(0, \frac 1 2\right), \, \mbox{and}\, \alpha^\prime \in \left(\frac 1 2, 1\right) \, \mbox{respectively}, &\mbox{except for half integers}\, \alpha \in \left\{0,\frac 1 2, 1\right\}.\end{aligned}
\begin{array}{c}{2 \pi{\cal E}= \int_{M_{3}}d ( \mathrm{t r}( \Phi_{A}H^{A}) ) + \int_{M_{3}}d ( \mathrm{t r}( \Phi_{B}K^{B}) )}\\{= \int_{S^{2}}\mathrm{t r}( \Phi_{A}H^{A}) + \int_{S^{2}}\mathrm{t r}( \Phi_{B}K^{B})}\\ \end{array}
y^k_{xx}=-\Gamma_{11}^k - (2\Gamma_{1i}^k-\delta^k_i\Gamma_{11}^1) y^i_x - (\Gamma_{ij}^k - 2\delta^k_i \Gamma_{1j}^1 )y^i_x y^j_x + \Gamma_{ij}^1 \, y^i_x y^j_x y^k_x\,,k=2,\dots,N\,,\,\,\, i,j\geq 2\,,
\begin{array}{ll}\left\lbrace \begin{array}{l}\frac{dQ_1}{dt}+ Q_1^2 - Q_2^2 + V''[x_c]=0\,, Q_1(0)=0\,, \end{array}\right. \left\lbrace \begin{array}{ll}\frac{dQ_2}{dt}+ Q_1 Q_2 + Q_2 Q_1 =0\,, Q_2(0)=b I_N\,. \end{array}\right.\end{array}
{{\bf{Y}}^{[1]}}(n) -{{\bf{Y}}^{[1]}}({t_2}) ={{\bf{H}}^{[11]}}(n){\bf{V}}_1^{[1]}(n){{\bf{u}}^{[1]}}+{{\bf{H}}^{[12]}}(n){\bf{V}}_1^{[2]}(n){{\bf{u}}^{[2]}}.