harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Theta_a(\rho_{j+1}) =\frac{D_a(\rho_{j+1})}{\rho_{j+1}^{m-2}}= N_a(\rho_{j+1}) \frac{H_a(\rho_{j+1})}{\rho_{j+1}^{m-1}}\leq C N_a(\rho_{j+1}),
\begin{aligned}\circ: \Omega^k_{sk,g}(M) \times \Omega^l_{sk,g}(M) \rightarrow \Omega^*_{sk,g}(M), (\alpha, \beta) \mapsto \alpha \circ \beta = \frac{1}{l+1}\cdot \alpha \cdot d\beta + (-1)^{k+1}\frac{1}{k+1}d\alpha \cdot \beta,\end{aligned}
\mathfrak{l}_n(x_1, \ldots, x_n)=l_n(x_1, \ldots, x_n), x_i \in E & \mathfrak{l}_n(v_1, \ldots, v_n)=m_n(v_1, \ldots, v_n), v_i \in V & \mathfrak{l}_{k+n}(x_1, \ldots, x_k,v_1, \ldots, v_n )= \Phi_{k,n}(x_1, \ldots, x_k,v_1, \ldots, v_n ),
\begin{multlined}x^{k+4}\cdot x^{k+1}\cdot 17 x^4+x^{k+4}\cdot (2^k+k)x^k \cdot x^5+(2^{k+3}+k+3)x^{k+3}\cdot x^{k+1}\cdot x^5 = (2 k + 9\cdot2^k + 20) x^{2 k + 9},\end{multlined}
H_{R}= \sum_{i}\sum_{n}\frac{1}{2}\left[ e^{- 3 H t}L^{3}\pi_{\eta_{i}}( k_{n}, t ) \pi_{\eta_{i}}( -{\bf k}_{n}, t ) + \frac{e^{3 H t}}{L^{3}}( e^{- 2 H t}{\bf k}_{n}^{2}+ \mu_{i}^{2}) \eta_{i}({\bf k}_{n}, t ) \eta_{i}( -{\bf k}_{n}, t ) \right] ,
: \!{\cal F}[ X ] \! : \ = \exp \biggl \{\frac{1}{2}\int d^{2}z d^{2}z^{\prime}\ln | z - z^{\prime}|^{2}\frac{\delta}{\delta X ( z , \bar{z})}\frac{\delta}{\delta X ( z^{\prime}, \bar{z}^{\prime})}\biggr \}{\cal F}[ X ] .
f_1\left( e^{-\frac{\pi}{s}}\right) = \sqrt{s}f_1\left( e^{-\pi s}\right), \;\; f_3\left( e^{-\frac{\pi}{s}}\right) = f_3\left( e^{-\pi s}\right),\;\; f_2\left( e^{-\frac{\pi}{s}}\right) = f_4 \left( e^{-\pi s}\right) .
E_6=\sum_{j=2}^3\big(\|\partial_x^2u_{\neq}^j\|_{X_3}^2+\|\partial_x(\partial_z-\kappa\partial_y)u_{\neq}^j\|_{X_3}^2\big)+\|\partial_x\nabla W^2\|_{X_3}^2, &\|f\|_{Y_0^k}^2=\|f\|_{L^{\infty}H^k}^2+\nu\|\nabla f\|_{L^{2}H^k}^2.
(\alpha_{1}-\alpha_{2})\cdot\Delta T\cdot L_{D}=F\cdot(\frac{L_{D}}{E_{1}A_{1}}+\frac{L_{D}}{E_{2}A_{2}}+\frac{h_{S}}{A_{s}G_{s}}+\frac{h_{c}}{A_{c}G_{c}}+(\frac{2-\nu}{9G_{b}a}))
S_{gauged}= S_{LR}+ \frac{1}{2\pi}\int d^2 z \; \left( \frac{1}{2}A \bar{A}( 1+ B(x) \sin 2\alpha ) + \frac{1}{\sqrt{2}}A \bar{L}_R + \frac{1}{\sqrt{2}}L_L \bar{A}\right),
C(L_q^{2n+1}(r;\underline{m}))\cong C(L_{2n+1})^{\varrho_{\underline{m}}^r}\cong \left(\sum_{i=0}^n p_{(v_i,0)}\right)C^*(L_{2n+1}\times_{c}\mathbb{Z}_r)\left(\sum_{i=0}^n p_{(v_i,0)}\right).
\left. c_{2}( \beta ) \left( \frac 1 2 R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}- \frac 1 4 R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}\right) \right] ~ ~ ~ .
m(r,kY)= m\left(r,-\frac{R_{k-2}}{MY^{k-2}}+ \frac{h}{MY^{k-2}}\right) m(r,Y)\leq m\left(r,-\frac{R_{k-2}}{MY^{k-2}}\right) +m\left( r,\frac{h}{MY^{k-2}}\right) +\log 2. &
I_{3}:=\iint_{D_t}\varphi(u)|L\tilde{\Lambda}||Q_0(\ln \alpha,\tilde{\Lambda})|\lesssim \gamma \varepsilon^2, & I_4:= \iint_{D_t}\varphi(u)|L\tilde{\Lambda}||Q_0(\partial_x(\ln \alpha), \tilde{\Lambda})+Q_0(\ln \alpha, \partial_x \tilde{\Lambda})|\lesssim \gamma \varepsilon^2.
P_T \,\, = \,\,\begin{bmatrix}c_m c_{m-1}\cdots c_{m-e}c_{m-1}c_{m-2}\cdots c_{m-e-1}\vdots \vdots \ddots \vdots c_{d+2}c_{d+1}\cdots c_{d-e+2}c_{d+1}c_{d}\cdots c_{d-e+1}\end{bmatrix}.
\varphi(r)=S\cdot\frac{(Wr)\cdot tan^{-1}(Wr)-\frac{1}{2}log(1+(Wr)^{2})}{(Wr_{max})\cdot tan^{-1}(Wr_{max})-\frac{1}{2}log(1+(Wr_{max})^{2})}
\theta_t \tau_t \le \theta_{t-1}(1+ \tau_{t-1}), t = 2, \ldots, k, \theta_t \eta_t \le \theta_{t-1}(\mu + \eta_{t-1}), t = 2, \ldots, k, \eta_{t-1}\tau_{t}\ge 2 L_f \alpha_t, t = 2, \ldots, k, \eta_k (1+\tau_k) \ge 2 L_f, \alpha_t = \theta_{t-1}/ \theta_t, t = 2, \ldots, k,
A(x) =\sum_{l=0}^n\frac{1}{l!}\left(\frac{d}{dy}\right)^lA(y)\Big|_{y=y_0}(x-x_0)^l & +\frac{1}{n!}\int_{x_0}^x(x-y)^n\left(\frac{d}{dy}\right)^{n+1}A(y)dy.
\int_{\mathbb{R}^n}{}\frac{u(x)^p}{|x|^{\sigma}}(x\cdot \nabla u(x))\,dx ={}-(n-\alpha)\int_{\mathbb{R}^n}\frac{u(x)^{p+1}}{|x|^{\sigma}}\,dx - (n-\alpha)\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(z\cdot (x-z))u(x)^{p}u(z)^{p}}{|x-z|^{n-\alpha + 2}|x|^{\sigma}|z|^{\sigma}}\,dz dx.
\partial_\tau w_B=0,\quad\displaystyle\lim_{\eta\to 1}w_B\partial_\eta^2 w_B=-\frac{1}{2}\frac{1}{x+x_0},\quad\partial_\xi w_B|_{\eta=0}\leq -a_0,\partial_{\eta}^2 w_B|_{\eta=0}=0, &c(1-\eta)\leq w_B\leq c^{-1}(1-\eta)\sqrt{-\ln(\mu(1-\eta))},\partial_\xi w_B<0 in[0,X]\times[0,1),
|{\rm com}_1| = \left|M(k,\xi)\frac{N}{2}(1+k^2+(\xi-\theta\eta)^2)^{N/2-1})2(\xi-\theta\eta)\eta\right| & \lesssim \left((1+k^2+(\xi-\eta)^2)^{(N-1)/2}+(1+k^2+\xi^2)^{(N-1)/2}\right)|\eta|.
\quad\, \sum_{n = U}^{U+V}e\left(\overline{\xi}_{x + nw}\right) = \sum_{n = U}^{U+V}e\left(\xi_{x + nw}^e + \xi_{x + nw}^i\right) &= \sum_{n = U}^{U+V}e\left(\xi_{x}^e + \xi_{x + Uw}^i + n\left(\xi_y^e - \xi_x^e \right) + O\left(\frac{1}{R_1}\right) + O\left(\frac{V}{U}\right)\right).
P^0 = E = \frac{1}{2 e^2{\phi_0}^2}\int d^2x \left[ \frac{1}{2}F^2_{ij}+ \vert D_i\Phi\vert^2+ \frac{e^2}{4}(\vert\Phi\vert^2 -{\Phi_0}^2)^2 \right]
K(t,x,y) =\nabla_xm_{u_\mu}(t,x)\cdot\nabla_y m_{u_\iota}(t,y)+\kappa m_{u_\mu}(t,x) m_{u_\iota}(t,y)+2m_{\nabla u_\mu}(t,x) m_{u_\iota}(t,y).
Q(z)=(-1)^kT_{1,\ldots,k}V(z,z_1,\ldots,z_k)\bigg[\prod^k_{i=1}\Psi_p(z_i)\bigg]^{-1}\bigg[1+O\bigg(\frac{\Psi_p(z_k)}{\widetilde{\Psi}_{p,k}}\bigg)\bigg],
c_0^2 = - \frac 1{4\pi^2}\int \int |v (|x|)|^\frac 12 |v(|y|)|^\frac 12 |x-y|^2 \eta_+^* (x) \eta_+ (y) d^2 x d^2 y = \left[\int_0^\infty s^2 \eta_0 (s) |v (s)|^\frac 12 \right]^2
R_\mu T_{(\mu)}=\sum_{j_m \in [n_{\mu_m}]}\cdots \sum_{j_1 \in [n_{\mu_1}]}R_{\mu}\rho(u_{\mu_m, j_m}\cdots u_{\mu_1, j_1}) \rho(y_{\mu, j_1, \dots, j_m}) & =\sum_{j_m \in [n_{\mu_m}]}\cdots \sum_{j_1 \in [n_{\mu_1}]}W_{\mu_m, j_m}\cdots W_{\mu_1, j_1}\rho(y_{\mu, j_1, \dots, j_m}).
<V,\tilde{V}>_{H}=\sum_{j=0}^{N-1}\left(\int_j^{j+1}\rho_j \, u_{j}(x) \, \overline{\tilde{u}_{j}(x)}+ v_{j}(x) \, \overline{\tilde{v}_{j}(x)}dx\right), \, V = \left(\begin{array}{c}\underline{u}\underline{v}\end{array}\right), \tilde{V}= \left(\begin{array}{c}\underline{\tilde{u}}\underline{\tilde{v}}\end{array}\right),
\hat{\overline{Y}}_{knownN}=\frac{\hat{Y}_{pwr}}{N}=\frac{\frac{1}{n}\sum_{i=1}^{n}\frac{y_{i}^{\prime}}{p_{i}}}{N}\approx\frac{\sum_{i=1}^{n}\frac{y_{i}^{\prime}}{\pi_{i}}}{N}=\frac{\sum_{i=1}^{n}w_{i}y_{i}^{\prime}}{N}.
\mathbb{S}_{A^{-1}M N_1}^{\ell_1}\cap{\mathfrak{D}}_{j_1}^A\cap{\mathcal{T}}_{k_1}^{A'}\not= \emptyset, \mathbb{S}_{A^{-1}M N_1}^{\ell_1}\cap{\mathfrak{D}}_{j_1}^A \subset \bigcup_{\# k_1 \sim M}{\mathcal{T}}_{k_1}^{A'}.
L_{n}(\alpha) := \sum_{x \in \mathbb{Z}^d}l(n,x)^{\alpha}, \alpha >0 L_n(0) :=\lim_{\alpha \downarrow 0}L_n(\alpha)= \sum_{x \in \mathbb{Z}^d}\mathbf{I}\{l(n,x)>0\}=\# R(n).
(R/\mathfrak{p}\otimes_R A, R/\mathfrak{p}\otimes_R P)=(R/\mathfrak{p}_{n-1}/\mathfrak{p}/\mathfrak{p}_{n-1}\otimes_{R/\mathfrak{p}_{n-1}}R/\mathfrak{p}_{n-1}\otimes_R A, R/\mathfrak{p}_{n-1}/\mathfrak{p}/\mathfrak{p}_{n-1}\otimes_{R/\mathfrak{p}_{n-1}}R/\mathfrak{p}_{n-1}\otimes_R P)
\lim_{t\rightarrow\infty}{\rm{dist}}_{\mathcal{H}_0}(S_0(t)B,\mathcal{A}_0):=\lim_{t\rightarrow\infty}\sup_{\varphi\in B}\inf_{\theta\in\mathcal{A}_0}\|S_0(t)\varphi-\theta\|_{\mathcal{H}_0}=0.
\begin{matrix}S&\Rightarrow^{L}&w_{1}A\chi&\Rightarrow&w_{1}\nu\chi&\Rightarrow^{*}&w_{1}w_{2}w_{3}\\S&\Rightarrow^{L}&w_{1}A\chi&\Rightarrow&w_{1}\omega\chi&\Rightarrow^{*}&w_{1}w_{2}^{\prime}w_{3}^{\prime},\\ \end{matrix}
Q_i=Q_{i-1}^{\alpha_i}+\sum\limits_{j=0}^{\alpha_i-1}\left(\sum\limits_{{\bar\gamma}_{i-1}}c_{j,i,{\bar\gamma}_{i-1}}\mathbf{Q}_{i-1}^{{\bar\gamma}_{i-1}}\right)Q_{i-1}^j,
\frac 12 \sum_{r=1}^{R}k_ru_{\infty}^{y_r}(2y_{r,i}y_{r,i}' - y_{r,i}^2 - y_{r,i}'^2) = \sum_{r=1}^{R}k_ru_{\infty}^{y_r}(y_{r,i}'y_{r,i}- \frac{1}{2}(y_{r,i}^2 + y_{r,i}'^2)).
u\left( x,t,x_{0}\right) =g_{0}\left( x,t,x_{0}\right) ,\partial_{n}u\left(x,t,x_{0}\right) =g_{1}\left( x,t,x_{0}\right) ,\forall \left(x,t,x_{0}\right) \in \Gamma \times \left[ 0,1\right] .
\lVert AB-A_0 B_0 \rVert^2 \sim \sum_{i=1}^{M-1}x_i^2 + \sum_{j=2}^{N}\sum_{i=1}^{M-1}\{x_i-(a_i b_1 -a^0_i b^0_1-a_i b_j +a^0_i b^0_j)\}^2 &\sim \sum_{i=1}^{M-1}\left\{x_i^2 + \sum_{j=2}^{N}(a_i b_1 -a^0_i b^0_1-a_i b_j +a^0_i b^0_j)^2 \right\}&= \sum_{i=1}^{M-1}\left[ x_i^2 + \sum_{j=2}^{N}\{a_i(b_j-b_1)-a^0_i(b^0_j-b^0_1)\}^2 \right].
\Phi^0(t,y,0) = \phi_{ref}+ \phi_c(t,y) - \phi^0(t,y,0) = \phi_c(t,y) + \ln\left(\frac{n^0(t,y,0)}{n_{ref}}\right), \lim_{z\rightarrow +\infty}\Phi^0(t,y,z) = 0.
\tau^{\varphi}(M,\mathcal{P}_1\cup \mathcal{P}_2;\mathfrak{e}_1\cup\mathfrak{e}_2,\mathfrak{h})= &=\mathfrak{T}(\mathfrak{h},\mathfrak{h}_1,\mathfrak{h}_2)\cdot\tau^{\varphi}(M_1,\mathcal{P}\cup\mathcal{P}_1;\mathfrak{e}_1,\mathfrak{h}_1)\cdot\tau^{\varphi}(M_2,\mathcal{P}'\cup\mathcal{P}_2;\mathfrak{e}_2,\mathfrak{h}_2).
\sum_{D \in \mathcal{D}_{v,d}(231,312)}y^dx^{des(\sigma(D))}= 1 + \sum\limits_{d=1}^\infty y^d(1+x)^{(v-2)d-1}&= 1 + \frac{1}{(1+x)}\sum\limits_{d=1}^\infty (y(1+x))^{(v-2)d}&= 1 - \frac{1}{(1+x)}+ \frac{1}{(1+x)(1-y(1+x)^{v-2})}&= \frac{x+yx(1+x)^{v-2}+1}{(1+x)(1-y(1+x)^{v-2}).}
\|j_{q}u\|^{2}_{L^{2}_{(0,q)}(M)}= \|u\|_{L^{2}_{(0,q)}(M)}^{2}\leq \varepsilon\|u\|_{graph}^{2}+ C_{\varepsilon}\|u\|_{W^{-1}_{(0,q)}(M)}^{2}\;.
e_{9b}=e_9|_{b=1}=16p^5 s^3-24p^4 s^3-72p^4s^2+12p^3 s^3+468 p^3 s^2-2s^3 p^2-24p^3s-426p^2 s^2 +24s p^2+111s^2 p+2p^2-18ps-3s^2+6s
\int k\psi^{k-1}f(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\nabla f\nabla\psi +n(k-2)\int\psi^{k}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}&\qquad+\int\psi^{k}(1+ |\nabla f|^{2})^{\frac{k-2}{2}}|\nabla f|^2+(k-1)\int\psi^{k}f(1+ |\nabla f|^{2})^{\frac{k-2}{2}}\Delta f=0.
M_{i}= \left( \begin{array}{c c}{a_{r}}&{- 1}\\{1}&{0}\\ \end{array}\right) \left( \begin{array}{c c}{a_{r - 1}}&{- 1}\\{1}&{0}\\ \end{array}\right) \left( \begin{array}{c c}{a_{1}}&{- 1}\\{1}&{0}\\ \end{array}\right)
s_{-r}(N)=s_{-r}\left(\prod_{j=1}^v q_j^{\beta_j}\right)=\prod_{j=1}^v s_{-r}(q_j^{\beta_j})\geq\prod_{j=1}^v s_{-r}(q_j)>\prod_{i=m+1}^{\infty}s_{-r}(p_i).
\partial_z W(z) \vert_{z=z_0}=-e^{W(z_0)-\sqrt{2}b \kappa \Xi(z_0)}\frac{\sqrt{2}}{6}\lambda_0,\quad \partial_z \Xi(z) \vert_{z=z_0}=- e^{W(z_0)-\sqrt{2}b \kappa \Xi(z_0)}b \kappa^{-1}\lambda_0.
ds_{T^{1,1}}^2={1\over 9}\bigg(d\psi + \sum_{i=1}^2 \cos \theta_i d\phi_i\bigg)^2+{1\over 6}\sum_{i=1}^2 \left(d\theta_i^2 +{\rm sin}^2\theta_i d\phi_i^2 \right)\ .
g_{1}=x_{0}^{n+1}-x_{3}^{n}=\sum_{\mu=1}^{n-1}\alpha_{\mu}\left[x_{2}^{(n+1)-\mu}x_{3}^{\mu}-x_{0}^{(n+1)-\mu}x_{1}^{\mu+1}\right] &+\sum_{t=0}^{n}\beta_{t}\left[x_{1}^{(n+1)-t}x_{3}^{t}-x_{0}^{n-t}x_{2}^{t+1}\right]+\gamma_{2}(x_{1}x_{2}-x_{0}x_{3}),
\lambda |\widetilde{Z}|^2 + O(|\widetilde{Z}||z_{n}| + |z_{n}|^2)+o(|Z|^2)=\widetilde{z}AUA^{*}\overline{\widetilde{z}}^t + O(|\widetilde{Z}||z_{n}|+|z_{n}|^2)+o(|Z|^2),
0 <m(B_{\eta}(\tilde{y}))=\int_{l-r_0}^{l}m_r(B_{\eta}(\tilde{y}))\mathcal{L}^1(dr) =\int_{I'}m_r(B_{\eta}(\tilde{y}))\,\mathcal{L}^1(dr) \leq \int_{I'}m_r(A)\,\mathcal{L}^1(dr)=0.
<\psi_{qG}|\psi_{qG}>= \frac{\hbar\sqrt{2(q+1)}}{2(q-1)|p|}\int\limits_{\frac{|\Im(p)|^2}{|p|^2}}^{\infty}s^{-\frac{1}{2}}\left\{s+\frac{[\Re(p)]^2}{|p|^2}\right\}^{\frac{2}{1-q}}\;ds.
z_1 = \frac{1}{4a+3}\left(2a - 2b_1 - l - \frac{d-3}{2}\pm \Delta\right), \Delta = \left[4(a+b_1)^2 + 4(a+2b_1) + 1 - \left(l + \frac{d-1}{2}\right)\left(4a - 4b_1 - l - \frac{d-5}{2}\right) \right]^{1/2},
=\frac{-f_1}{2e_1}+\sqrt{\frac{\Delta_1^2}{4e_1^2}-\frac{1}{e_1}\sqrt[1/6]{\frac{-f_1}{2e_1}+\sqrt{\frac{\Delta_1^2}{4e_1^2}-\frac{1}{e_1}\sqrt[1/6]{\frac{-f_1}{2e_1}+\sqrt{\frac{\Delta_1^2}{4e_1^2}+...}}}}}
C^{\textrm{main}}_{(cp+a,dq+b)}= \frac{(p c^2 + 2ac - p c)C_\tau}{2}+ \left(c+\frac{a}{p}\right) dC_\sigma + \frac{(q d^2 + 2 bd - q d)C_\rho}{2}&+ d \sum_{l=0}^{q-1}l C_{\rho,(0,-l-1)}+ c \sum_{k=0}^{p-1}k C_{\tau,(-k-1,b)}+ cp \sum_{l=0}^{b-1}C_{\sigma,(0,l)}.
\begin{array}{l l}{d s_{6}^{2}=}&{e^{2 \lambda}\left( - d t^{2}+ d z^{2}\right) + e^{2 \varphi}d x^{\mu}d x^{\mu},}\\{H_{e}=}&{d \left( e^{2 \lambda}d t \wedge d z \right) ,}\\ \end{array}
v^*_{T,\ell}= y^*_\ell \cdot q^*_T + (1-y^*_\ell) \cdot 0 = y^*_\ell \cdot v^1_{T, \ell}+ (1-y^*_\ell) \cdot v^0_{T, \ell}, & \bar v^*_{T,\ell}= y^*_\ell \cdot \bar q^*_T + (1-y^*_\ell) \cdot 0 = y^*_\ell \cdot \bar v^1_{T, \ell}+ (1-y^*_\ell) \cdot \bar v^0_{T, \ell},
\simeq \frac{- \lambda_{1}( a^{+}\Gamma_{1}\Lambda_{3}) + i \lambda_{2}( \Lambda_{3}\Gamma_{2}\Gamma_{3}a^{+})}{\sqrt{2}}u_{R}( \lambda_{1}, \lambda_{2}) \mid 0 \rangle \mid \psi ( R , \Lambda_{3}) \rangle .
\int_{0}^{T}\langle A^{0}U^{\delta}_{t}+A^{i}\partial_{i}U^{\delta}+DU^{\delta}-B^{ij}\partial_{i}\partial_{j}U^{\delta}-\delta\Lambda\Delta U^{\delta},\phi\rangle dt=\int_{0}^{T}\langle f,\phi\rangle
\int \frac{xk(qx)y(x)}{( -x^2(1-q)^2;q^2)_\infty}\Big( D_q u(x)+ u(x) u(qx)+\frac{1-q^{1-\nu}(1-q)[\nu]^2}{x}u(qx)+\frac{qx^2-q^{1-\nu}[\nu]^2}{x^2}\Big) d_qx & = \frac{xk(x)}{( -x^2(1-q)^2;q^2)_\infty}\Big ( y(x)u(x) - D_{q^{-1}}y(x)\Big).
T_4 \ni t_x \xrightarrow{[\chi_p]\alpha}\big( [\chi_p]\alpha \big)(t_x)= \chi_p (\alpha^{-1}[t_x]) = \chi_p (\alpha^{-1}x{\alpha^{*}}^{-1}) = \chi_{\alpha p \alpha^{*}}(x)= \chi_{\alpha p\alpha^{*}}(t_x),
-\|f\|_{}\|u_N-u_M\|^2 \le -(u_N-u_M,f(u_N)-f(u_M)) = -(A^{-1}(f(u_N)-f(u_M)),f(u_N)-f(u_M))-(E_{M,N},f(u_N)-f(u_M)) \le -\gamma\|A^{-1}(f(u_N)-f(u_M))\|^2+\|E_{M,N}\|\cdot\|f(u_N)-f(u_M)\|.
q(t,s,x) :=1_{s<t}\mathcal{F}^{-1}\left\{\exp\left(\int_{s}^{t}(\psi(r,(t-s)^{-\frac{1}{\gamma}}\xi))dr\right)\right\}=(t-s)^{\frac{d}{\gamma}}e^{\lambda(t-s)}p_{\lambda}(t,s,(t-s)^{\frac{1}{\gamma}}x).
\frac{1}{\lfloor n/k_n\rfloor m_n}\sum_{j=1}^{\lfloor n/k_n\rfloor}\sum_{i=(j-1)k_n+1}^{(j-1)k_n+m_n}\mathbb{P}\bigl(\mathcal{A}_{i,j}^n \bigr) = o(1).
\pi_*(N^{(1)}.N^{(2)}.\pi^*\lambda_i) = d^2 \lambda_i \pi_*(N^{(1)}.N^{(2)}.D^{(i)}) =d^2 R^{(i)}\pi_*((N^{(1)})^2. N^{(2)}) = -d^2 \lambda_1 \pi_*(N^{(1)}. (N^{(2)})^2) = -d^2 \lambda_2
\rho^{D}_{m_0\rightarrow m_{1}+\cdots +m_{N}}= \left(\prod_{i=1}^N\int\frac{d^n\!\vec{\mu_i}}{(2\pi)^n}\right) \int d^n\vec{k}\,\,{\rm e}^{i\vec{k}.\sum_i\vec{\mu_i}}\, \rho^{D-n}_{m_0\rightarrow m_{1\mu_1}+\cdots +m_{N\mu_N}}
(z-\mu^{(t)})H_{k}^{(s,t)}H_{k-1}^{(s,t+1)}(z)=H_{k}^{(s,t+1)}H_{k-1}^{(s,t)}(z)+H_{k-1}^{(s,t+1)}H_{k}^{(s,t)}(z), &H_{k}^{(s,t+1)}H_k^{(s,t)}(z)=H_{k+1}^{(s,t)}H_{k-1}^{(s,t+1)}(z)+H_{k}^{(s,t)}H_{k}^{(s,t+1)}(z).
\| D^k \Delta^{-1}\nabla^{\perp}\omega -\sum_{j=1}^{\infty}(D^k \Delta^{-1}\nabla^{\perp}\tilde \omega^j) (t,x-x_j) \|_{\infty}\le \; \sum_{j=1}^{\infty}\| D^k \Delta^{-1}\nabla^{\perp}(\omega^j -\tilde \omega^j ) \|_{\infty}\lesssim_k \sum_{j=1}^{\infty}\| \omega^j -\tilde \omega^j \|_{H^{k_0+10}}\le C_k.
I_\infty(\varepsilon) = \frac{(z_1z_2)^{-\frac{s}{2}}}{2}\cdot \frac{A_I^{(s,w)}(\varepsilon, z_1) B_I^{(s,w)}(\varepsilon, z_2) - A_I^{(s,w)}(\varepsilon, z_2) B_I^{(s,w)}(\varepsilon, z_1)}{\big[C_I^{(s, w)}(\varepsilon) \big]^2 (z_1-z_2)}\cdot \varepsilon.
\begin{array}{ccl}r_{p'}= R(p') - \Big|\Big\{v' \in T_{p'}(\widetilde{\varphi}) \ | \d_{v'}(\widetilde{\varphi}) =0\Big\}\Big| &= 2 d_{p'}(\widetilde{\varphi})- 2 - \Big|\Big\{v' \in T_{p'}(\widetilde{\varphi}) \ | \d_{v'}(\widetilde{\varphi}) =0 \Big\}\Big| \end{array}
\left( - \frac{1+h_q^2}{2h_p^2}+ B - g \rho (h-d) \right)_p + \left( \frac{h_q}{h_p}\right)_q + g\rho h_p = 0,\textrm{in}\bigcup_i \mathcal{R}_i, -\frac{1+h_q^2}{2h_p^2}- g\rho h + \frac{Q}{2}= 0, \textrm{on}\{p = 0 \}, h = 0, \textrm{on}\{p = p_0\}.
S_{n}= \frac{2^{n}- 1}{2 - 1}= \frac{n ( 1 + 4 n - 3 )}{2}= 2^{n}- 1 = a_{1}^{2}+ a_{2}^{2}+ \cdots a_{n}^{2}= \frac{1 ( 1 - 4^{n})}{1 - 4}
g=\left( \begin{array}{cccc}\psi_{1}^{(n-1)}\psi_{2}^{(n-1)}\dots \psi_{n}^{(n-1)}\dots \dots \dots \dots \psi_{1}^{\prime}\psi_{2}^{\prime}\dots \psi_{n}^{\prime}\psi_{1}\psi_{2}\dots \psi_{n}\end{array}\right) \, \, ,
P\left(M(t)>\gamma t\right) = P\left(\exists u\in \mathcal{N}_t^*\ :\ \sup_{0\le s\le t}|X_u(s)|>\gamma t\right) & \le E(|\mathcal{N}_t^*|)\mathbf{P}_0\left(\sup_{0\le s\le t}|B(s)|>\gamma t\right),
\big| c^{\iota_1\iota_2\iota_3}(\xi,\eta,\sigma) \cdot \varphi'_{k_1}(\xi-\eta)\varphi'_{k_2}(\eta-\sigma)\varphi'_{k_3}(\sigma)\big| &\lesssim 2^{3\max(k_1,k_2,k_3)/2}2^{[\mathrm{med}(k_1,k_2,k_3)-\min(k_1,k_2,k_3)]/2}.
{\cal L}= -{1\over 4}E^{(11)}[{\cal R}^{(11)}+{1\over{12}}F^{(11)}_{MNPQ}F^{(11)\MNPQ}-{8\over{12^4}}\varepsilon^{M_1 \cdots M_{11}}F_{M_1 \cdots M_4}F_{M_5 \cdots M_8}A_{M_9 M_{10}M_{11}}],
c |\log(y-l_\alpha)-\log(x-l_\alpha)|=\int_x^{y}\frac{c}{s-l_\alpha}ds\leq \Big|\int_x^{y}D^{n+1}\varphi(s)ds\Big| &\leq |D^{n}\varphi(x)-D^{n}\varphi(y)|\leq \|D^{n}\varphi\|_{C^{\tau}}|(y-l_\alpha)-(x-l_\alpha)|^{\tau}.
(L^\infty(T_{12},\dots,T_{1n}))'= (T_{1k}\mid 2\leq k\leq n)'=\bigcap_{k=2}^n(T_{1k})'=\otimes_{k=2}^n(T_{1k})' = \otimes_{k=2}^n(L^\infty(T_{1k}))'=\otimes_{k=2}^nL^\infty(T_{1k})=L^\infty(T_{12},\dots,T_{1n}).
\mathbb{E}\Gamma'^{+}_{0}(\tau+i\eta)\overline{\Gamma'^{+}_{0}(\tau+i\eta)}= \frac{1}{2\pi}\int_{\mathbb{R}}\,dk |k|\widehat{r_{\tau_1,\eta_1}}(k)\overline{\widehat{r_{\tau_2,\eta_2}}(k)}
\begin{cases}\frac{\partial\varphi}{\partial t}\left(x,t\right)+H\left(t,\nabla_{x}\varphi\left(x,t\right)\right)=0 \,\mathbb{R}^{n}\times\left(0,+\infty\right), \varphi\left(x,0\right)=J\left(x\right) \forall x\in\mathbb{R}^{n},\end{cases}
(M(s,s+t])^2 = \big( \sum_{j=1}^{N(s)}L_j(s-T_j,s+t-T_j)\big)^2 + 2 \sum_{j=1}^{N(s)}L_j(s-T_j,s+t-T_j]\sum_{j=N(s)+1}^{N(s+t)}L_j(s+t-T_j) + \big( \sum_{j=N(s)+1}^{N(s+t)}L_j(s+t-T_j) \big)^2 =: \mathrm{I}+ 2\mathrm{II}+\mathrm{III}
\frac{nC-H|h|^2}{nH^{1-\sigma}}-|h|^2f_{\sigma}\geq \frac{nf_{\sigma}H |h|^2 + \frac{1}{2}f_{\sigma}H^3}{nH}- |h|^2 f_{\sigma}= f_{\sigma}\frac{H^2}{2n},
\cos d_{PS}= \cos x \cos y+ \sin x \sin y \cos \theta = \cos (x-y) + \sin x \sin y (\cos \theta -1) \cos d({P_{\eta'}}, S_{\eta}) \ge \cos ((1-\eta) x) \cos ((1-\eta')y)+ \sin ((1-\eta)x) \sin ((1-\eta')y )\cos \theta & = \cos ((1-\eta)x- (1-\eta')y) + \sin ((1-\eta)x) \sin ((1-\eta')y) (\cos \theta -1),
A=[\begin{matrix}a_{0}&a_{1}&a_{2}&...&...&a_{n-1}\\a_{1}&a_{2}&&&&\vdots\\a_{2}&&&&&\vdots\\ \vdots&&&&&a_{2n-4}\\ \vdots&&&&a_{2n-4}&a_{2n-3}\\a_{n-1}&...&...&a_{2n-4}&a_{2n-3}&a_{2n-2}\end{matrix}].
\mathcal{A}_{h}(u)=\left(\begin{array}{cc}{\rm e}^h u_{x_1x_1}+{\rm e}^{-h}u_{y_1y_1}+u_{y_1}+{\rm e}^h h_{x_1}u_{x_1}-{\rm e}^{-h}h_{y_1}u_{y_1}u_{x_1x_2}{\rm e}^{h}u_{x_1x_2}u_{x_2x_2}\end{array}\right)
cond_{2}(\mathbb{A}^{\varepsilon})=||\mathbb{A}^{\varepsilon}||_{2}||(\mathbb{A}^{\varepsilon})^{-1}||_{2}\leq \frac{M\max (\nu_u^2, \nu_{\ast}^2 + \nu_q^2)}{\beta' \min\{\mu_u^{2},\mu_*^2\}}\,,
V_{\rho_{-}}(\rho_{-},\rho_{+},t)=([H]_{\rho_{-},\rho_{+}}-H'(\rho_{-}))b(\rho_{-},t) V_{\rho_{+}}(\rho_{-},\rho_{+},t)=([H]_{\rho_{-},\rho_{+}}-H'(\rho_{+}))b(\rho_{+},t)
\left(\Pi_{\lambda_1,\lambda_2}g\right)^{-,-}_{i+\frac{1}{2},j+\frac{1}{2}}= g^{-,-}_{i+\frac{1}{2},j+\frac{1}{2}}, \mbox{if}\left(i,j\right) \in \left(\,\gamma_2,\gamma_1\,\right)
D_h^{(\theta)}= \Op(ad(G_h^{(\theta)})Ad(O)) - Op(ad(G_h^{(\theta)})) Op(Ad(O)) = \ \int_{h_\theta}^0 Op(ad(G_h^{(\theta)}) ad (\Psi_k) Ad(O_{<k})) dk & \ - \int_{h_\theta}^0 Op(ad(G_h^{(\theta)}) Op( ad(\Psi_k) Ad(O_{<k})) dk + \Op(ad(G_h^{(\theta)})Ad(O_{<h_\theta})) - Op(ad(G_h^{(\theta)})) Op(Ad(O_{<h_\theta}))
d_{n}^k(x_{r,k}) =0, d_{n}^k(s_{r,k}) = Y_{n-r+1,k}, \intertext{and}d_{n}^{k,k+1}(x_{r,k}) =0, d_{n}^{k,k+1}(\xi_{k+1}) =0, d_{n}^{k,k+1}(s_{r,k+1}) = Y_{n-r+1,k+1},
\begin{array}{l l}{\hat{q}}&{= \frac{2}{( k^{+}+ 1 )}\ell_{3}^{+}}\\{}&{}\\{\hat{h}}&{= \frac{1}{( k^{+}+ 1 )}\{\ell^{+}( \ell^{+}+ 1 ) - ( \ell_{3}^{+})^{2}\}}\\ \end{array}
\Delta_{BC}=R_{BC}-R_{0}=\frac{1}{2}\log\left(\frac{1+\sum_{i=1}^{m}\gamma_i}{1+\gamma_m}\right) & \leq \frac{1}{2}\log\left(\frac{1 + m\gamma_m}{1+\gamma_m}\right)<\frac{1}{2}\log(m),
\mu(\mathbf{\Phi}) = \max\limits_{\substack{1\leq i,j\leq L 0\leq c_{1},c_{2}\leq M-1}}\frac{\mid\langle \mathbf{s}_{\mathbf{A}_{i}}^{(c_{1})}, \mathbf{s}_{\mathbf{A}_{j}}^{(c_{2})}\rangle\mid}{M},
\mathcal{E}_{s}:C(X)\to\tilde{L}^{\infty}(X) \mathcal{E}_{s}f\left(u,v,x_{0}\right) =\lim_{N\to\infty}\frac{1}{N+1}\sum_{n=0}^{N}s^{n}.\left(f\circ\xi\right)\left(u,v,x_{0}\right)=\mathbb{E}_{\lambda}\left(\tilde{f}\left(u,\cdot\right)|\mathcal{F}^{s}\right)(x_{0}).
S_{W Z}^{N D 3}= \mu_{3}\int_{R^{3 + 1}}\mathrm{T r}\left( C^{( 4 )}+ i ( 2 \pi \alpha^{\prime}) i_{\Phi}i_{\Phi}B^{( 6 )}+ \frac 1 2 C^{( 2 )}\wedge B^{( 2 )}- ( 2 \pi \alpha^{\prime}) B^{( 2 )}\wedge{\tilde{\cal F}}+ \dots \right) \, ,
\bar{\rho}_{{}_{\rm REG}}(r) = - \frac{m\,r^{-1}}{(2\pi)^{\frac{3}{2}}}\int_{0}^{\infty}ds \,\exp (-s^{2}- \frac{m^{2}r^{2}}{2{s}^{2}})[I_{1 - F}(s^{2}) - I_{F}(s^{2})],
W_M(S;\varphi)=\mathbb{E}\sup_{x\in S}\bigg\langle x,\frac{1}{\sqrt{M}}\sum_{i=1}^M\epsilon_i\varphi_i\bigg\rangle=\mathbb{E}\sup_{x\in S}\langle x,\varphi\rangle\leq \rho^{-1}\mathbb{E}\|\varphi\|_\infty\leq3\rho^{-1}\sqrt{v\log N},
\lambda_b\ast g=\frac{1}{2}\left[e^{ib\left(\frac{\alpha-\beta}{2}\right)}+e^{-ib\left(\frac{\alpha-\beta}{2}\right)}\right]g- \frac{1}{2}\left[\e^{ib\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{b}+e^{-ib\left(\frac{\alpha+\beta}{2}\right)}\,\delta_{- b}\right]\ast g,
\Omega_1=\left\lbrace y\in\Omega:\delta(y)\geq\frac32\,\delta(x)\right\rbrace, & \Omega_2=\left\lbrace y\in\Omega:\frac12\,\delta(x)<\delta(y)<\frac32\,\delta(x)\right\rbrace, & \Omega_3=\left\lbrace y\in\Omega:\delta(y)\leq\frac12\,\delta(x)\right\rbrace.
\iint_{{Q}_{n}}G^{r}\left(\frac{{\zeta}_{n}u_{n}}{\rho_n}\right){u}_{n}^{s}{\zeta}_{n}^{q}\,dx\,dt &\leq \gamma\left(1+\frac 1\theta\right) 2^{n(g_1+2)}&\times k^{s-2/(N+1)}\rho^{-N/(N+1)}G^{r+1/(N+1)}\left(\frac k{\rho}\right)|A_n|^{(N+2)/(N+1)}.
\int_{B_{xy}}\omega(B_{xu})^{-\frac{n-1}{n}}+\omega(B_{yu})^{-\frac{n-1}{n}}\Big) \omega(u)du \leq 2C\int_{2B}\omega(B_{xu})^{-\frac{n-1}{n}}\omega(u)du \leq C \omega(B)^{\frac{1}{n}}.
\frac{-1}{(-q)_{\infty}}\sum_{n=1}^{\infty}\frac{nq^{n(n+1)/2}}{(q)_n}+\sum_{n=1}^{\infty}\frac{(-1)^nq^{n(n+1)}}{(q^2;q^2)_n(1-q^n)}=\sum_{n=1}^{\infty}\frac{(-q)^n}{(q)_n(1-q^n)}.
\boldsymbol{\Sigma}_{\frac{n+1}{2}}:= \mathrm{diag}\left(\lambda_{3},\lambda_{5},\ldots,\lambda_{n}, \lambda_{n+2}\right) + \lambda_{1}\mathbf{u}\mathbf{u}^{\top}\boldsymbol{\Gamma}_{\frac{n-1}{2}}:= \mathrm{diag}\left(\lambda_{2},\lambda_{4},\ldots,\lambda_{n-1}\right) + \lambda_{n+1}\mathbf{v}\mathbf{v}^{\top}
\frac{1}{(\sigma_{3}^{y})^{2}}=\frac{R_{0}+R_{90}}{(1+R_{0})R_{90}}\frac{1}{(\sigma_{1}^{y})^{2}};\frac{1}{(\sigma_{2}^{y})^{2}}=\frac{R_{0}(1+R_{90})}{(1+R_{0})R_{90}}\frac{1}{(\sigma_{1}^{y})^{2}}
[ V_\alpha, V_\beta ] = \epsilon_{\alpha\beta}^{~~~\!\gamma}V_\gamma,\qquad[ V_\alpha, V_m^a ] = V_m^b (\sigma_\alpha)_b^{~a},\qquad\{V_m^a, V_m^b \}= - m^2 (\sigma^\alpha)^{ab}V_\alpha
\prod_{a = 1}^{N}\overline{{\psi}}_{\alpha}^{( a )}( x^{( \pi ( a ) )}) = \mathrm{s i g n}( \pi^{- 1}) \prod_{a = 1}^{N}\overline{{\psi}}_{\alpha}^{( \pi^{- 1}( a ) )}( x^{( a )}) ,
r = 4b^2 + 6ac - 2n - 3, q = -2 \sum_{i=1}^n x_i + 6a(n+1) + 4bc, E = 2 \sum_{i=1}^n x_i^2 - 6a \sum_{i=1}^n x_i - 2b(2n+1) - c^2,
\Big\|\sum_{k\in\Gamma_n}T_k\varphi \Big\|_{\textup{L}^2(G)}^2= \Big\|\sum_{k\in\Gamma_{n-1}}T_k\varphi + T_m\varphi \Big\|_{\textup{L}^2(G)}^2 & = \Big\|\sum_{k\in\Gamma_{n-1}}T_k\varphi\Big\|_{\textup{L}^2(G)}^2 + \|T_m\varphi\|_{\textup{L}^2(G)}^2= \Big\|\sum_{k\in\Gamma_{n-1}}T_k\varphi\Big\|_{\textup{L}^2(G)}^2 + \|\varphi\|_{\textup{L}^2(G)}^2.
\int_{4\pi}d\Omega^{\prime}\int_{0}^{\infty}dE^{\prime}\Sigma_{s}(r,E^{\prime}\rightarrow E,\hat{\Omega}^{\prime}\rightarrow\hat{\Omega},t)\psi(r,E^{\prime},\hat{\Omega}^{\prime},t)+s(r,E,\hat{\Omega},t)
[X+r+\xi,Y+t+\eta] ={}[X,Y] + L_{X}\eta - i_{Y}d\xi + i_{Y}i_{X}H_0 - [r,t] - F_0(X,Y) + d^{\theta_0}_Xt - d^{\theta_0}_Y r + 2c(d^{\theta_0}r,t) + 2c(F_0(X,\cdot),t) - 2c(F_0(Y,\cdot),r).
\mathcal{F}^{-1}\Delta_0(h_1 h_2)\mathcal{F}=\mathcal{F}^{-1}\Delta_0(h_1)\mathcal{F}\mathcal{F}^{-1}\Delta_0(h_2)\mathcal{F}=\Delta_H(h_1)\Delta_H(h_2) = \Delta_H(h_1 h_2),
{\rm Hilb}\big({\rm CYB}(B_2)^{!},t\big)=(1,4,3),{\rm Hilb}\big({\rm CYB}(B_3)^{!},t\big)=(1,9,13,2),{\rm Hilb}\big({\rm CYB}(B_4)^{!},t\big)=(1,16,46,28,5),{\rm Hilb}\big({\rm CYB}(B_5)^{!},t\big)=(1,25,130,200,101,12),{\rm Hilb}\big({\rm CYB}(D_4)^{!},t\big)=(1,12,34,24,4),{\rm Hilb}\big({\rm CYB}(D_5)^{!},t\big)=(1,20,110,190,96,11).
\textstyle P_{s,c}(T)=\int_{r>0}^{R}\frac{2r}{R^2}\exp\{-\pi r^2 (T )^{2/\alpha}C(L_p(r),\alpha)\lambda_p \}\mathbb{E}_q\left[\frac{P_p}{P_p+Tr^\alpha P_m{q}^{-\alpha}}e^{\{-\pi r^2 \left(\frac{\eta T P_m}{P_p}\right)^{2/\alpha}C(L_c(r),\alpha)\lambda_m}\}\right]\mathrm{d}r