harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
D_{\alpha\beta}F_{{\mathbf i}{\mathbf j}}= i (D_{\alpha\beta}D_{{\mathbf i}{\mathbf j}}\Phi -D_{\alpha\beta}{\bar D}_{{\mathbf i}{\mathbf j}}{\bar\Phi}) =8\partial_{(\alpha\dot\alpha}D_{\beta)({\mathbf i}}{\bar D}^{\dot\alpha}_{{\mathbf j})}{\bar \Phi}=0
\sum_{\ell=0}^{L_j^+}N_{j,\ell,K}(\varepsilon) \lesssim \sum_{\ell=1}^{L_j^++1}\min\Big\{\ell , \varepsilon^{-2}L_K^{-2d}2^{-js(1+\alpha)}\ell^{-2\beta-1}\Big\}\le \sum_{\ell=1}^{\ell^-_*}\ell + \sum_{\ell=\ell^+_*}^{L^+_j+1}\varepsilon^{-2}L^{-2d}_K 2^{-js(1+\alpha)}\ell^{-2\beta-1}.
\lim\limits_{\varepsilon \longrightarrow 0}\phi_0 =\lim\limits_{\varepsilon \longrightarrow 0}\frac{\theta(x)((1+\beta v \sqrt{c})\sinh(\gamma)\sinh(\beta\gamma)+(\beta+v\sqrt{c})\cosh(\gamma)\cosh(\beta\gamma)}{2(1+(\cosh(\gamma)\sinh(\beta \gamma)+\beta \sqrt{c}\sinh(\gamma)\cosh(\beta\gamma))^2)}& = C_1\theta(x).
e^{u_0}= \alpha^d, e^\rho = \left(\frac{\mu}{\alpha}\right)^d, e^{-\rho}= \left(\frac{\mu}{\alpha}\right)^{-d}, \cosh \rho = \frac12\left( \left(\frac{\mu}{\alpha}\right)^d + \left(\frac{\mu}{\alpha}\right)^{-d}\right),
\frac{\partial w_{\beta}^{\varepsilon}}{\partial t}+A\Delta w_{\beta}^{\varepsilon}-\rho_{\beta}w_{\beta}^{\varepsilon}& =\mathbf{P}_{\varepsilon}^{\beta}w_{\beta}^{\varepsilon}+e^{\rho_{\beta}\left(t-T\right)}\mathbf{Q}_{\varepsilon}^{\beta}u +e^{\rho_{\beta}\left(t-T\right)}\left[F\left(x,t;u_{\beta}^{\varepsilon}\right)-F\left(x,t;u\right)\right].
A_{f,r}=\{(u,v_{1},v_{2},...,v_{d})\in\mathbb{R}^{d+1}:0\le u\le f(\frac{v_{1}}{u^{r}},\frac{v_{2}}{u^{r}},...,\frac{v_{d}}{u^{r}})^{\frac{1}{1+rd}}\}
\widehat f_{4n}= \sum_{i=1}^{\widetilde m}\Big(\frac{Y_{2, \widetilde x_i}-Y_{2, \widetilde x_{i-1}}}{\widetilde \Delta_i}\Big)^2 \mathbf{1}\big( \cdot \in [\widetilde x_{i-1}, \widetilde x_i)\big)
S = 2 \pi ( \det )^{1 / 4}\left\{r_{0}\delta_{i k}+ \sqrt{- 2 \Upsilon_{i j}\Upsilon_{j k}^{*}}\right\}= 2 \pi ( \det )^{1 / 4}\left\{r_{0}\delta_{i k}+ \sqrt{- 2 \tilde{\Upsilon}_{i j}\tilde{\Upsilon}_{j k}^{*}}\right\}\ .
a_m=\frac{1}{(q)_\infty}\sum_{j\in\mathbb{Z}}(-1)^jq^{rj^2-ij+m(r-1)j}\frac{1-q^{(m+2j)(i+1)}}{1-q^{m+2j}}\frac{(-1)^mq^{(m+2j)(m+1)}}{1+q^{m+2j}}.
\mbox{RHS}=\displaystyle\frac{(1-\tau^2)\zeta_{N-1}}{\prod_{j=1}^n(z_{N-1}-z_j\tau^2)}\sum_{k=1}^n \zeta_k\displaystyle \prod_{j=1 \atop j\neq k}^{n}\frac{(z_{N-1}-z_j)(z_k-z_j \tau^2)}{z_k -z_j}\displaystyle\overline{G}_{\varepsilon ,k}^{(n-1,l-1)}(\zeta').
\lambda_{1}^{2}= \dot{\lambda}+ \lambda \frac{q_{2}q_{3}}{q_{2}^{2}}, \lambda_{1}^{1}= \ddot{\lambda}+ \frac{\lambda}{q_{2}^{2}}\left[ q_{2}{\dot{q}}_{3}- \frac{q_{2}q_{3}}{q_{2}^{2}}( 2 q_{2}{\dot{q}}_{2}+ q_{2}q_{3}) \right] .
H_{\mu\nu\rho}= \epsilon_{\mu\nu\rho}\frac{r~r_1^2}{\left(r^2 + r_1^2\right)^2}\left(f_1f_5\right)^{1/4};\;\;\;\; H_{abc}= \epsilon_{abc}\frac{r_5^2}{r^3}\left(f_1f_5\right)^{-3/4}
\begin{cases}\rho_1{{\ddot \varphi}}+ a A^{\frac12}(A^{\frac12}\varphi + \psi) =0, \rho_2{{\ddot \psi}}+ b A \psi + a (A^{\frac12}\varphi + \psi) - \delta A^\gamma{\theta}= 0, \rho_3{{\dot \theta}}+ c A\theta + \delta A^\gamma{{\dot \psi}}=0,\end{cases}
\vert x\rangle=\phi^{+}(x)\vert0\rangle = \int d^n\mu(\mathbf{p}) \,e^{ipx}a_c^{*}(\mathbf{p})\vert0\rangle &= (2\pi)^{-n/2}\int d^n \mathbf{y}\left( \int d^n \mu(\mathbf{p}) \,\sqrt{2\omega_{\mathbf{p}}}\,e^{i\left(\omega_{\mathbf{p}}x^0+p_k(x-y)^{k}\right)}\right) \tilde{a}^*(\mathbf{y})\vert0\rangle,
m_{1}= m_{2}= m_{3}= m_{4}={\frac{C}{\sqrt{1 - 4 C^{2}}}}\, \qquad m_{5}= 0 \qquad Z_{\infty}^{5}= - Z_{\infty}^{6}= Z_{\infty}^{7}= - Z_{\infty}^{8}= C \ ,
f(t):=t^{p-1}\int_{t}^\infty e^{(t^2-s^2)/2}ds=t^{p-1}\int_{0}^\infty e^{-ts-s^2/2}ds\le t^{p-1}\int_0^\infty e^{-s^2/2}ds=\sqrt{\tfrac{\pi}{2}}t^{p-1}.
\bullet \; x\longmapsto 1/(x|\log(x)|^{8}) \textrm{is decreasing on}(0, \mu+\eta). &\bullet \; x\longmapsto \left|\frac{x}{\log(x)^6}\log\left(\frac{x}{\log(x)^6}\right)\right| = -\frac{x}{\log(x)^6}\log\left(\frac{x}{\log(x)^6}\right) \textrm{is increasing on}(0,\eta]. &\bullet \; x\longmapsto x(\log(x))^{-6}\textrm{is increasing on}(0,\eta].
\begin{aligned}\int_{B_r}u^{q-1}\left(u^m-k\right)_\pm\eta\nabla\eta\cdot\nabla v\,dx &\leq \frac{1}{2}\int_{B_r}\left(u^m-k\right)_\pm\left|\nabla\eta\right|^2\,dx+\frac{1}{2}\int_{B_r\cap\{(u^m-k)_\pm>0\}}u^{2(q-1)}\eta^2\left|\nabla v\right|^2\,dx.\end{aligned}
\psi_0(n,k)=(\epsilon_1\epsilon_2)^mm^{m (\epsilon\epsilon_1-\epsilon_2)}\frac{(n+\lambda+\epsilon_2km-\delta_{\epsilon_2,1}m)_{m}^{\epsilon_2}}{(n+\mu+\epsilon_1 km-\delta_{\epsilon_1,1}m)_{m}^{\epsilon\epsilon_1}},
=\sum_{j=1}^{r_i}j^2- \sum_{j=1}^{r_i}j-r_i \sum_{j=1}^{r_i}j+\frac{r_i(r_i+1)}{2}\sum_{j=1}^{r_i}1 =\frac{r_i(r_i+1)(2r_i+1)}{6}-\frac{r_i(r_i+1)}{2}-\frac{r^2_i(r_i+1)}{2}+\frac{r^2_i(r_i+1)}{2}=r_i(r_i+1)\left[ \frac{1}{6}(2r_i+1)-\frac{1}{2}\right] =r_i(r_i+1)\left[ \frac{r_i}{3}-\frac{1}{3}\right] =\frac{1}{3}r_i(r^2_i-1)
{\chi}_U(\mathcal{F})=\frac{[H^0_{c}(U,\mathcal{F})][H^2_c(U,\mathcal{F})]}{[H^1_c(U,\mathcal{F})][H^3_c(U,\mathcal{F})]\prod_{v\in S_{\infty}}[H^0(K_v,\mathcal{F}_v)]}=1
\varepsilon_p^D(L)=\frac{L^{p-D}}{2^{D+1}}\sum^{p-1}_{q=0}(-1)^{q+1}C_p^q\left(\sqrt{\pi}\right)^{q-D}\Gamma\left(\frac{D-q}{2}\right)A\left(1,\cdots,1;\frac{D-q}{2}\right)
0= \langle \eta_jy_1^*y_2^*, y_1^*\rangle = \langle \mu_{B^!}(y_1^*), \eta_jy_1^*y_2^*\rangle = \sum\limits_{i=1}^{n}k_i\langle e_i, \eta_jy_1^*y_2^*\rangle +k_{n+1}\langle y_1^*, \eta_jy_1^*y_2^*\rangle+ k_{n+2}\langle y_2^*, \eta_jy_1^*y_2^*\rangle = \sum\limits_{i=1}^{n}k_i\delta_{ij}=k_j.
\sum_{n=0}^\infty \bigg(\frac{r_1}s\bigg)^n \sum_{i=0}^n{n\choose i}_q^\alpha=\sum_{n=0}^\infty\sum_{i=0}^n\left[z^n{n\choose i}_q\right]^\alpha \le\left[\sum_{n=0}^\infty\sum_{i=0}^n z^n{n\choose i}_q\right]^\alpha.
\partial_t(|A|^2H^{-2}) = \alpha H^{-\alpha-1}\Delta(|A|^2H^{-2})+2(\alpha+1)H^{1-\alpha}(|A|^2H^{-2})^2 +4(\alpha-1)H^{-\alpha-1}(|A|^2H^{-2}) +2(1-\alpha)H^{-\alpha-1}-2(\alpha+1)H^{-\alpha-2}h_i^kh_k^jh_j^i+Q,
\mathcal{G}_{+,l}(me^{i\pi/2}, r) - \mathcal{G}_{-,l}(me^{i\pi/2},r) = \frac{ir}{\pi}\, e^{-i\pi|l|}(e^{2i\delta^{+}_l}- e^{2i\delta^{-}_l}) \, K^2_W(mr),
R[\bar{g}]e^{2u}\le \left( R[\bar{g}] - 2 \Delta u \right) \, \, n=2 \, R[\bar{g}] u^{\frac{n+2}{n-2}}\le \left( R[\bar{g}] u - \frac{4(n-1)}{n-2}\Delta u \right) \, \, n\ge3.
\Psi\left(\lambda\nu + (1-\lambda)\tilde{\nu}; \bar{\nu}\right) = \lambda\Psi(\nu;\bar{\nu}) + (1-\lambda)\Psi(\tilde{\nu};\bar{\nu}), J\left(\lambda\nu + (1-\lambda)\tilde{\nu}; \bar{\nu}\right) = \lambda J(\nu;\bar{\nu}) + (1-\lambda) J(\tilde{\nu};\bar{\nu}).
B= \ -\Big[(\sinh2t)^{-\frac{1}{2}}\frac{(x-y)^{2}}{(\sinh2t)^{2}}e^{-\varphi(t,x,y)}-(\sinh2t)^{-\frac{1}{2}}\frac{xy}{(\cosh t)^{2}}e^{-\varphi(t,x,y)}\Big] =: \ -[B_{1}-B_{2}].
\nu [ A ] \; = \; \frac{e}{4 \pi \hbar}\int d^{2}x \epsilon_{\mu \nu}F^{\mu \nu}\; = \; \frac{e}{2 \pi \hbar}\int_{- \frac{1}{2}T}^{\frac{1}{2}T}d t \int_{- \frac{1}{2}{R}}^{\frac{1}{2}{R}}d x ( \dot{A}_{1}- \partial A_{0})
k_{N}=\frac{1}{w}\cdot \frac{-\left[ kv^{2}+\delta \,\acute{}\,v+\delta^{2}\left( k-\lambda \right) \right] \,du^{2}+2\, \delta\,du\,dv}{\left[ v^{2}+\delta^{2}\left( \lambda^{2}+1\right) \right]\,du^{2}+2\, \delta \, \lambda \,du\,dv+dv^{2}}
S_{e f f}= \int d t [ P_{A}\dot{X^{A}}+ \dot{C}^{\mu}\bar{C}_{\mu}+ \dot{\eta}^{\mu}\bar{\eta}_{\mu}+ \Pi_{\mu}\dot{N}^{\mu}+ \{Q_{B}, \bar{C}_{\mu}N^{\mu}+ \bar{\eta}_{\mu}\chi^{\mu}\}] ,
\vartheta_{1}( \tau , \nu ) = \sqrt{- 1}( y^{- 1 / 2}- y^{1 / 2}) q^{1 / 8}\prod_{n = 1}^{\infty}( 1 - q^{n}) ( 1 - q^{n}y ) ( 1 - q^{n}y^{- 1}) \, ,
\partial{\mathcal M}_{k+1}(\beta)=\bigcup_{k_1+k_2=k}\bigcup_{i=1,\dots,k_2}\bigcup_{\beta_1+\beta_2=\beta}{\mathcal M}_{k_1+1}(\beta_1) \,\,{}_{{\rm ev}_0}\times_{{\rm ev}_i}{\mathcal M}_{k_2+1}(\beta_2),
\mathbb{E}e^{-T_{\infty}}\, = \, \mathbb{E}e^{- \sum_{j=1}^{\infty}Y_j}= \lim_{n \to \infty}\prod_{j=1}^{n}\mathbb{E}e^{-Y_j}= \lim_{n \to \infty}\prod_{j=1}^n \frac{\lambda_j}{1+\lambda_j}= \, \prod_{j=1}^{\infty}\frac{1}{1+\frac{1}{\lambda_j}}= \frac{1}{1+ \sum_{j=1}^{\infty}\frac{1}{\lambda_j}+ \cdots}.
I(C:Z|M)_{\hat{\sigma}_{CMZ}}\le I(AB:Z|M)_{\hat{\sigma}_{ABMZ}}&= I(A:Z|M)_{\hat{\sigma}_{AMZ}}+ I(B:Z|M)_{\hat{\sigma}_{BMZ}}&\phantom{=}+ I(A:B|MZ)_{\hat{\sigma}_{ABMZ}}- I(A:B|M)_{\hat{\sigma}_{ABM}}&\le I(A:Z|M)_{\hat{\sigma}_{AMZ}}+ I(B:Z|M)_{\hat{\sigma}_{BMZ}}\;.
\left(\sum_{l=1}^n \alpha_l v_k^l\right) \gamma_1(r\alpha) =r^{2n-1}\left(w_1\eta_k+\sum_{l=2}^{n-1}w_l v_k^l +\sqrt{1-|w|^2}e^{i\psi}v_k^n\right)\bar w_1 \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w) =r^{2n-1}\left(|w_1|^2 \eta_k +\sum_{l=2}^{n-1}\bar w_1 w_l v_k^l + \bar w_1 \sqrt{1-|w|^2}e^{i\psi}v_k^n\right) \frac{d\psi}{2\pi}\wedge d\beta_{n-1}(w)\ .
T(n_i)+T(n_{i+1}) <\left( \frac{5}{3}+ 3 \cdot \frac{9}{8}\right)\cdot \frac{1}{X_0}= \frac{121}{24}\cdot \frac{1}{X_0}&<7 \cdot \frac{3}{4}\cdot \frac{1}{X_0}\leq (k_i+k_{i+1}) \cdot \frac{3}{4}\cdot \frac{1}{X_0}.
1-s_1=s_4 1-s_2=s_3 s_1.\left(1-s_1\right)=s_1.\overline{s_2}s_3.\left(1-s_3\right)=s_2.\overline{s_1}s_2.\left(1-s_2\right)=s_2.\overline{s_1}=\overline{s_1.\left(1-s_1\right)}s_4.\left(1-s_4\right)=s_1.\overline{s_2}=\overline{s_3.\left(1-s_3\right)}
\begin{array}{lll}e(G-X) = e(G)-e(X,B) \geq \left(n-\left\lfloor k_{\ell}/2\right\rfloor+1\right)(s_\ell-1)+\left(m-s_\ell+1\right)\left(\left\lfloor k_{\ell}/2\right\rfloor-1\right) -s_{\ell-1}n &= \left(n+m-2s_{\ell}+2\right)(\lfloor k_{\ell}/2\rfloor-1). \end{array}
\tilde{Y}_{\mathbf{d}, \mathcal{A}}(abc) = \frac{(n - 1)^2 d_a d_b (d_b - 1) d_c}{d^2 n_a n_b (n_b - 1) n_c}\left(1 + \mathcal{O}\left( \frac{t^2}{n d}\right) \right) ,
\times \exp \bigg ( - \frac{i}{2}l^{2}e^{2}2 \mathrm{R e}\Big ( \varepsilon_{\mu \nu}\partial_{\nu}j_{\mu}^{( L )}, \tilde{Q}[ \delta ( - L , 0 ) - \delta ( L , 0 ) ] \Big ) \bigg ) .
K_{1}(x,\lambda(t)) = \int_{0}^{\infty}\frac{r dr}{\lambda(t)^{2}(1+\frac{r^{2}}{\lambda(t)^{2}})^{3}}\int_{0}^{x}\frac{\rho d\rho}{x}\left(1+\frac{r^{2}-1-\rho^{2}}{\sqrt{(r^{2}-1-\rho^{2})^{2}+4r^{2}}}\right)
P\Big(U_1 < U_5 , U_6 | \{U_3 < U_5 \}\cap \big\{\{U_3 < U_7\}\cup \{U_4 < U_8\}\big\}\Big) = \frac{861}{2100}= 0.41 > P\Big(U_1 < U_5 , U_6 | \{U_3 < U_5 \}\cap \{U_3 < U_7 \}\cap \{U_4 < U_8\}\Big) = \frac{336}{840}= 0.4.
k^+ = k_{\mathcal{W},g}^+ = \min \{k \mid 0 \le k \le l_S(g) \sigma(k) = \sigma^+ \}, k^- = k_{\mathcal{W},g}^- = \min \{k \mid 0 \le k \le l_S(g) \sigma(k) = \sigma^- \}
\|{F}^{(\delta)}_{2,N}v_0\|_p^p\lesssim\sum_{|k|\le N}\int_{\eta_{k-1}}^{\eta_k}v_1^{-p'}\bigl|G_{2,k}^{(\delta)}\bigr|^{p'}=:\bigl[\mathbf{G}^{(\delta)}_{2,N}(g)\bigr]^{p'}.
BB2=[\begin{matrix}(1-2\lambda-2\beta)&(\lambda-\alpha)&0&0\\ (\lambda+\alpha)&(1-2\lambda-2\beta)&(\lambda-\alpha)&0\\ 0&(\lambda+\alpha)&(1-2\lambda-2\beta)&(\lambda-\alpha)\\ 0&0&2\lambda&(1-2\lambda-2\beta)\end{matrix}].
H_{\mu \nu}= \mu \delta_{\mu , \nu}, \quad D_{\mu \nu}= \delta_{\mu , \nu}D_{\mu}, \quad D_{\mu}= - i g_{s}\left( \, z^{( 1 / 2 )}( \mu \cdot q ) + \sum_{\lambda \in \Delta_{s}, \ \lambda \cdot \mu = 1 / 2}z^{( 1 / 2 )}( \lambda \cdot q ) \right) ,
N(A^{C},\delta) \geq N(A\cap B_{\delta_1},\delta) N(A\cap B_{\delta_2},\delta_1) \dots N(A\cap B_{\delta_K},\delta_{K-1}) & \geq \delta^{O(\epsilon)}(\delta\delta_1\dots\delta_{K-1})^{-d\tau}& = \delta^{-d\tau(1+(1-\tau)+\dots+(1-\tau)^{K-1})+O(\epsilon)}& = \delta^{-d (1-(1-\tau)^K)+O(\epsilon)}\geq \delta^{-\frac{\sigma+d}{2}+O(\epsilon)}
\sum_{n=0}^\infty\frac{a_{n+1}}{n!}X^n =\alpha'=\exp(X)\exp(\exp(X)-1) &=\Bigl(\sum_{k=0}^\infty\frac{1}{k!}X^k\Bigr)\Bigl(\sum_{k=0}^\infty\frac{a_k}{k!}X^k\Bigr)=\sum_{n=0}^\infty\sum_{k=0}^n\frac{a_k}{k!(n-k)!}X^n.
\sum_{j = 1}^{n}q^{\rho_{j}- 2 ( n - j )}\Lambda_{j}^{1 / 2}\mu_{j}^{- 1 / 2}{\cal X}^{j}{\cal X}^{- j}+ \frac{q^{- 2 n + 1}}{1 + q}{\cal X}^{0}{\cal X}^{0}= - \frac{1}{a^{2}}.
X_{l,m}^{(L)}\equiv (\partial_{A_1}\cdots \partial_{A_{l+2m}}\omega^L ) \, K_{[l,m]}^{A_1\dots A_{l+2m},B_1\dots B_{l+2m}}\, (\partial_{B_1}\cdots \partial_{B_{l+2m}}\omega^{\prime L}) \;.
P_{ED}^{Dispatch}:= \sum_{t\in\mathcal{T}}\left[ \sum_{i\in\mathcal{G}}p_{it}+ \sum_{i\in\mathcal{I}}( M_{it}- m_{it}) \right. \left. + \sum_{i\in\mathcal{W}}( W_{it}- w_{it}) + \sum_{i\in\mathcal{R}}( R_{it}- r_{it}) \right].
\det{\mathbf{A}}_{Q}=\prod_{1\le k\le n-1}^{\gets}\, \prod_{2\le m\le n-k+1}\, \prod_{T\in{Q \choose m}}(1-\sigma_T)^{(m-2)!\cdot (n-m)!}&= \prod_{2 \le m\le n}\, \prod_{T\in{Q \choose m}}(1-\sigma_T)^{(m-2)!\cdot (n-m)!\cdot (n-m+1)}
\frac{\tau(U^{\sigma}\times U^{\sigma})}{d(U_{\gamma_0},U^{\sigma}\times U^{\sigma})}SRO_{\gamma_0}(\Phi)=2\frac{\tau(G^{\sigma}\times G^{\theta})}{d(G_{\delta_0},G^{\sigma}\times G^{\theta})}STRO_{\delta_0}(f)
\begin{aligned}\Vert E_1-\nu\epsilon\partial_{YY}u^0_e+\sqrt{\epsilon}(R^{u,1}+R^{u,1}_p)+\epsilon\partial_xp^2_p+Er_1\Vert_{L^2}\leq C(L,\zeta)\epsilon^{3/4-\zeta}, \Vert E_3-\kappa\epsilon\partial_{YY}h^0_e+\sqrt{\epsilon}(R^{h,1}+R^{h,1}_p)+Er_2\Vert_{L^2}\leq C(L,\zeta)\epsilon^{3/4-\zeta}. \end{aligned}
\begin{aligned}\theta(\bar{u}) + I_K(\bar{\zeta}) = \lim\limits_{k\rightarrow+\infty}\theta(u^k) + I_K(\zeta^k) \leq \theta(\hat{u}) + I_K(\hat{\zeta}) + \beta\sum\limits_{i = n}^{+\infty}\|r^{i}\|^2.\end{aligned}
a(n;i,n) =a(n;i,n-1),\qquad\mbox{if $1\leq i\leq n-2$}, a(n;i,j) =a(n-1;i,j)+b(n;i-1),\qquad\mbox{if $2\leq i<j\leq n-1$}, a(n;i,j) =\sum_{k=1}^ja(n-1;i-1,k),\qquad\mbox{if $1\leq j<i-1\leq n-2$}, a(n;i,i-1) a(n-1,i-1),\qquad\mbox{if $2\leq i\leq n$}, b(n;i) =\sum_{k=1}^ib(n-1;k),\qquad\mbox{$1\leq i\leq n-1$}, a(n;n-1) =a(n-1), a(n;n) =a(n-1), b(n;n) =0.
\left( \xi_m(t,z), \xi_m(t,z') \Big| \mathcal B\right) &=\sum_{j, k=1}^m \int_0^t \gamma\left( B_{0,t}^j(t-s) - B_{0,t}^k(t-s) + \frac{s}{t}(z-z')\right) ds &\le \rho \sum_{j, k=1}^m \int_0^t \gamma\left( B_{0,t}^j(t-s) - B_{0,t}^k(t-s) \right) ds\,,
[S_{a,b}G - G S_{a,b}]_{x,y}= \frac{i}{\sqrt 2}[ \Delta_{a,b}G + \Delta_{b,a}G - G \Delta_{a,b}- G \Delta_{b,a}]_{x,y}= \frac{i}{\sqrt{2}}[ \delta_{a,x}G_{b,y}+ \delta_{b,x}G_{a,y}- G_{x,a}\delta_{b,y}- G_{x,b}\delta_{y,a}] \, .
v_it^m = m \partial_{\xi_{2\ell-i+1,m}}+(1-2\delta_{i,\ell})\partial_{\xi_{2\ell-i,m}}, v_{\ell + i}t^m = -m \partial_{\xi_{\ell-i+1,m}}+(1-\delta_{i,\ell})\partial_{\xi_{\ell-i,m}},
( S f ) ( y , x_{0}, \dots , x_{d - 1}) = \int \frac{\prod_{i = 0}^{d - 1}d p_{i}}{( 2 \pi y )^{d / 2}}\, f ( y , p_{0}, \dots , p_{d}) \, e^{\frac{i}{y}\sum_{i = 0}^{d - 1}p_{i}x_{i}}
DS_{N}(x, y) = \sum_{j, k = 1}^{N}\phi_{j}(x) c_{kj}\phi_{k}(y), S_{N}(x, y) = \sum_{j=1}^{N}\phi_{j}(x) \sum_{k=1}^{N+n}c_{kj}\epsilon\phi_{k}(y), IS_{N}(x, y) = -\epsilon(x, y) + \sum_{j,k=1}^{N+n}\epsilon\phi_{j}(x) c_{kj}\epsilon\phi_{k}(y).
\boldsymbol{S}(\boldsymbol{q})=\boldsymbol{V}_R=\begin{bmatrix}\lambda_1^{n-1}(q_1) \lambda_1^{n-2}(q_1) \ldots 1 \lambda_2^{n-1}(q_2) \lambda_2^{n-2}(q_2) \ldots 1 \ldots \ldots \ldots \ldots \lambda_n^{n-1}(q_{n}) \lambda_n^{n-2}(q_{n}) \ldots 1\end{bmatrix}.
\left\langle \psi_{a , b}^{\left[ A , B \right]}( x ) , \psi_{a^{\prime}, b^{\prime}}^{\left[ A , B \right] *}( y ) \right\rangle_{0}= P_{a , b ; a^{\prime}, b^{\prime}}( i \partial_{x}) i \Delta^{( + )}( x - y ) \,
\mathbb P \left(|S_n|>(1+\varepsilon)\sigma_n\sqrt{2\ln\Psi (n)}\right) &=2 \left(1-\Phi(x_n)\right)\left(1+o(1)\right) &=2(2\pi)^{-1/2}\frac{1}{(1+\varepsilon)\sqrt{2\ln\Psi (n)}}\exp \left(-(1+\varepsilon)^2\ln\Psi (n)\right)(1+o(1)) &\propto \frac{1}{\sqrt{\ln\Psi (n)}}\big(\Psi (n) \big)^{-(1+\varepsilon)^2}.
\nabla u(x,t)= \big(\mathbb{I}_3+t\nabla u_0(x-tu(x,t))\big)^{-1}\nabla u_0(x-tu(x,t)) = \big(\mathbb{I}_3+t\nabla u_0(\xi_0)\big)^{-1}\nabla u_0(\xi_0), x\in V(t).
\mathcal{D}\left(\mathcal{B}(t)\right) = \left\lbrace \rho \in Z: \rho' \in Z~~~~\int_{-\infty}^{0}J(\theta) \rho'(\theta) \d\theta \ni \mathcal{A}\rho(0) + f(t), ~\rho(0) \in \mathcal{D}\left(\mathcal{A}\right) \right\rbrace.
B^0(\varepsilon ;\vartheta) =\mathcal{A}^0+\varepsilon \vartheta\biggl(-b(\mathbf{D})^*V-V^*b(\mathbf{D})+\sum_{j=1}^d(\overline{a_j+a_j^*})D_j\biggr) &+\varepsilon^2\vartheta^2(\overline{Q}-W+c_5\overline{Q_0}).
\phi_{k+1}(x) := f(x_{k+1}) + \frac{\mu}{2}\|x - z_{k+1}\|^2 \overset{\eqref{Eq:Est}}{=}(1 - \tau_k) \phi_k(x) + \tau_k f_k(x)\overset{\eqref{Eq:Under}}{\leq}(1 - \tau_k) \phi_k(x) + \tau_k f(x).
T_1(y) ={\sf A}_1 + O(|y|^{-(n-4)}) + O(|y|^{-2}) |y|\to\infty (n\geq5), |\nabla_yT_1(y)| = O(|y|^{-(n-3)}) + O(|y|^{-3}) |y|\to\infty (n\geq5).
A+A'-B'B-(p-2)(T_{B}+\Lambda)'(T_{B}+\Lambda) ={\rm diag}\Bigl\{2+\lambda^{2}-(p-2)\Big(c+\frac{\lambda+\mu}{2}\Big)^{\!2},\ 2+\mu^{2}-(p-2)\Big(c-\frac{\lambda+\mu}{2}\Big)^{\!2}\Bigr\}=:{\rm diag}\{g(c),\,h(c)\}.
\mathtt{q}^{\nu}_n[\varphi]:=\int\limits_{\mathbb{R}^n}\bigg( \frac{|K_n \varphi|^2}{1+\sqrt{1-\big((4-n)\nu\big)^2}+ \frac{\nu}{|\mathbf{x}|}}+ \bigg(1-\sqrt{1-\big((4-n)\nu\big)^2}-\frac{\nu}{|\mathbf{x}|}\bigg) |\varphi|^2\bigg)\mathrm{d}\mathbf{x}.
\mathbf{c}_n(\lambda) = \mathbf{c}_{-n}(\lambda) = \frac{1}{\sqrt{\pi}}\frac{\Gamma(\lambda)\Gamma(\lambda+\frac{1}{2})}{\Gamma(\lambda+\frac{1+n}{2})\Gamma(\lambda+\frac{1-n}{2})}, \;\;\; \lambda \in \mathfrak{a}_\mathbb{C}^*.
i^{t}=\mathop{\arg\max}_{i=1,\cdots,q}f_{i}(\mathcal{X}^{t})=\mathop{\arg\max}_{i=1,\cdots,q}\|\mathcal{A}*\mathcal{X}^{t}-\mathcal{B}\|_{F(\mathcal{G}_{i})}^{2},
\mu\left( \left(\Omega_\delta \cap \left(\overline{J_u}\setminus J_u \right)\setminus \Sigma_\delta \right) \right) = \mathcal{H}^1(\underbrace{J_u \cap (\Omega_\delta \cap \left(\overline{J_u}\setminus J_u \right) \setminus \Sigma_\delta}_{=\varnothing})) = 0,
\frac{dA}{dt}=\frac{B}{C}+\frac{C}{B}+2, \frac{dB}{dt}= \frac{C}{A}+\frac{D}{C}-\frac{B^{2}}{AC}, &\frac{dC}{dt}=\frac{B}{A}+\frac{D}{B}-\frac{C^{2}}{AB}, \frac{dD}{dt}= -\frac{D^{2}}{BC}.
\begin{aligned}\sup_{j \geqslant \, 0}\Bigl( \| v^j\|^2_{L^2_{dx}}+\int_{\mathbb{T}^3}G(\Xi^j) \,dx \Bigr) + \sum_{j=1}^{\infty}\|\Theta^j - \Theta^{j-1}\|_{L^2_{dx}}^2 \leqslant E_0.\end{aligned}
\lambda^{^{S , A}}= \left( \begin{matrix}{\zeta_{\lambda}\Theta_{[ j ]}\phi_{_L}^{\ast}( p^{\mu})}\\{\phi_{_L}( p^{\mu})}\\ \end{matrix}\right) , \quad \rho^{^{S , A}}= \left( \begin{matrix}{\phi_{_R}( p^{\mu})}\\{( \zeta_{\rho}\Theta_{[ j ]})^{\ast}\phi_{_R}^{\ast}( p^{\mu})}\\ \end{matrix}\right) .
\mu_A P\mu_A^{-1}=(\alpha^{-1}\tau\overline{\alpha})P(\overline{\alpha}^{-1}\tau\alpha)=\alpha^{-1}\tau(\overline{\alpha}\overline{P}\overline{\alpha}^{-1})\tau\alpha &=\alpha^{-1}(\tau \overline{A}\tau)\alpha =\alpha^{-1}A\alpha=P.
\frac{\overline{Z}_r^{p,q}}{\overline{B}_r^{p,q}}= \frac{Z_{r}^{p,q}/ (Z_{r}^{p,q}\cap Z_{r-2}^{p+1,q-1})}{B_{r}^{p,q}/ (B_{r}^{p,q}\cap Z_{r-2}^{p+1,q-1})}.
M_{f,a,\beta}(x) \leq \max\biggl\{\max_{r=1,\ldots,m}\frac{A_r'}{(1-x^2)^{2r}}\, , \, \sup_{y: 0 < |y-x| \leq r_a(x)}\frac{A_{m+1}'f(y)}{(1-y^2)^{2(m+1)}f(x)}\biggr\}&\leq \frac{B_{m+1}'}{(1-x^2)^{2(m+1)}}\leq a\bigl(f(x)\bigr),
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) v_{4}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{3}\log^{2b+2N}(t)}+ \frac{C}{t^{2}\log^{b +2N -1}(t)}
\sum\limits_{i=1}^{\lfloor nr\rfloor}\sum\limits_{j=\lfloor nr\rfloor+1}^n\left(1_{\{X_i\leq X_j\}}-\frac{1}{2}\right)-\frac{\lfloor nr\rfloor}{\lfloor n\lambda\rfloor}\sum\limits_{i=1}^{\lfloor n\lambda\rfloor}\sum\limits_{j=\lfloor n\lambda\rfloor+1}^n\left(1_{\{X_i\leq X_j\}}-\frac{1}{2}\right)=0
\Gamma_p\left(\frac{1}{2}\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle\right)\Gamma_p\left(\frac{1}{2}\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle+\frac{1}{2}\right) &=\Gamma_p\left(\left\langle\frac{1}{4}-\frac{j}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{3}{4}-\frac{j}{p-1}\right\rangle\right)
(\rho \otimes_0 \varphi)^{\sigma',s'}_{\sigma,s}(a'ga) = \rho_{a'ga\sigma}^{\sigma'}\cdot \varphi_s^{s'}(a'ga) = \rho_{g\sigma}^{(a')^{-1}\sigma'}\cdot \varphi_s^{s'}(g) = \rho_{g\sigma}^{\sigma'}\cdot \varphi_s^{s'}(g) = (\rho \otimes_0 \varphi)^{\sigma',s'}_{\sigma,s}(g),
A ( E , \nabla ) = \left( \mu_{2}( E ) - \frac{i}{2 \pi}\frac{\left( \mu_{1}( E )^{+}+ \left( \iota ( \theta ) \mu_{2}( E ) \right)^{+}\right)^{2}}{\mu_{0}( E ) + \iota ( \theta ) \mu_{1}( E ) + \frac{1}{2}\iota ( \theta )^{2}\mu_{2}( E )}\right) [ T ] ,
{\rm Prob}(S_B(x) \le s)\ =\ \lim_{k\to\infty}\lim_{\epsilon\to 0}\frac{\sum_{l=0}^{k-1}\mu(\exp(U_{\epsilon}([\log B^l,\log B^ls)X)))}{\mu(\exp(U_{\epsilon}([0,\log B^k)X)))}=\int_0^s\psi(t)\dt
\Lambda(T_{ij},\lambda_{i},\lambda_{j})=\frac{T!}{\prod_{ij}Tij!}+\sum_{i}\lambda_{i}(T_{i}-\sum_{j}T_{ij})+\sum_{j}\lambda_{j}(T_{j}-\sum_{i}T_{ij})+\beta(C-\sum_{i}\sum_{j}T_{ij}C_{ij})
\left<{\cal O}\left[ j_{\mu}^{M}\right] \right>_{j_{\mu}^{M}}= \prod_{i = 1}^{N}\int d^{4}y^{i}{\cal D}z^{i}\mu \left[ z^{i}\right] \sum_{q_{a}= \pm 1}^{}{\cal O}\left[ j_{\mu}^{M{}( N )}\right] .
u_1'(x) = - \int_{x_0}^{x}\left[ t\mu_1( 1 - u_1) u_1 + (1-t) \frac{\mu_1}{\alpha^2}\left(\alpha - (\alpha u_{1}- d u_2)^+\right)(\alpha u_{1}- d u_2)^+ \right] + k \int_{x_0}^{x}\omega u_1 u_2
|F_1 | = (V_1(z_{-N},x_{-N+1})-V_1(z_{-N},z_{-N+1}))^2 & \leq |V_{12}(z_{-N},c_1)||x_{-N+1}-z_{-N+1}| \cdot (|V_1(z_{-N},x_{-N+1})|+|V_1(z_{-N},z_{N+1})|) \leq \frac{\kappa_2}{4}|x_{-N+1}-z_{-N+1}|,
(u^-_1)^2(1,y) = \int_0^{1}\frac{\partial}{\partial x_1}(x_1 (u^-_1)^2(x_1,y) ) dx_1 = \int_0^1 (2 x_1 u^-_1(x_1,y) u^-_{11}(x_1,y) + (u^-_1)^2(x_1,y)) dx_1 .
ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}=(\begin{matrix}cdt&dx&dy&dz\end{matrix})(\begin{matrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})(\begin{matrix}cdt\\dx\\dy\\dz\end{matrix})
H_\xi \coloneqq\begin{bmatrix}A_\xi 0 0 -A_\xi^{\mathsf{H}}\end{bmatrix}- \begin{bmatrix}B C^{\mathsf{H}}\end{bmatrix}\begin{pmatrix}D_\xi^{\mathsf{H}}+D_\xi \end{pmatrix}^{-1}\begin{bmatrix}C -B^{\mathsf{H}}\end{bmatrix}.
F_{4}( a , b ; b , b ; x , y ) = ( 1 - x - y )^{- a}{}_{2}F_{1}({\textstyle{\frac{1}{2}}}a ,{\textstyle{\frac{1}{2}}}( a + 1 ) ; b ; \frac{4 x y}{( 1 - x - y )^{2}}) ,
\| u^{*}(\varphi(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\rho (p\otimes a)\oplus \cdots \oplus\bigoplus_{k=1}^m f(z_k)\rho (p\otimes a)}^m) u & - \psi(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\rho (p\otimes a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\rho (p\otimes a)}^m\| < \varepsilon
\begin{array}{r c l}{{\nu}}&{{=}}&{{\left( E_{\mathrm{e - v i b}}^{\prime}- E_{\mathrm{e - v i b}}^{\prime \prime}\right) / h c}}\\{{}}&{{=}}&{{( E_{0}^{\prime}- E_{0}^{\prime \prime}) / h c + ( G_{q^{\prime}}^{\prime}- G_{q^{\prime \prime}}^{\prime \prime}) = \nu_{e}+ \nu_{v}~ ,}}\\ \end{array}
\begin{cases}\frac{1}{p_{2}}\leq \frac{1}{p_{1}}s_{2}+R(\mathbf{p},\mathbf{q},\alpha_{1},\alpha_{2})+\frac{n(1-\alpha_{1}\vee \alpha_{2})}{q_{2}}<s_{1}+\frac{n(1-\alpha_{1}\vee \alpha_{2})}{q_{1}}\frac{1}{q_{2}}>\frac{1}{q_{1}}\end{cases}
q^TAq-\lambda_0q^Tq =(\sum_{i,j}k_ik_j\bar{q}_i^TA\bar{q}_j+2p\bar{\beta}^TA\sum_{i}k_i\bar{q}_i+p^2\hat{\beta}^T\hat{\beta})-\lambda_0q^Tq &\equiv 2p\hat{\beta}^TA\sum_{i}k_i\bar{q}_i-2p\lambda_0\hat{\beta}^Tq ~({\rm mod}~p^2),
r_+({\bf R}^*) = r_+({\bf R}^{*1/2}({\bf W}_1 -{\bf W}_2 +{\bf M}){\bf R}^{*1/2}) &= r_+({\bf R}^{*1/2}({\bf W}_1 -{\bf W}_2){\bf R}^{*1/2}) &\leq r_+({\bf W}_1 -{\bf W}_2) = rank({\bf W}_+)
\|P([z])\| = \frac{|P(z)|}{\left(\sum_{i=0}^n |z_i|^2\right)^{\frac{d}{2}}}\le \frac{\sum_{|I|=d}|a_I||z^I|}{\left(\sum_{i=0}^n |z_i|^2\right)^{\frac{d}{2}}}\le \frac{\max_{|I|=d}|z^I|}{\left(\sum_{i=0}^n |z_i|^2\right)^{\frac{d}{2}}}\|P\|_1
\frac12 \| (\nabla_xu, u_t)(t) \|_{L^2}^2 + \int_0^t \int_{\mathbb{R}^n}b(\tau) u_t(\tau, x)^2\,dxd\tau &= \frac12 \| (\nabla_x u_0, u_1) \|_{L^2}^2+ \int_0^t \int_{\mathbb{R}^n}F(\tau, x) u_t(\tau,x) \,dxd\tau
\dot{\bar{q}}^{c,k}(t) = \frac{d}{dt}|C_k|^{-1}\sum\limits_{j\in C_k}q^c_j(t) & = |C_k|^{-1}\sum\limits_{j\in C_k}\frac{d}{dt}q^c_j(t) & =|C_k|^{-1}\sum\limits_{j\in C_k}\dot q^c_j(t),
\mathcal{Z}^*_K (\underline{\Delta^{J-1}}, \underline{\partial \Delta^{J-1}}) (\underline{\Delta^{J-1}}, \underline{\partial \Delta^{J-1}}) =\{( \Delta^{j_i-1}, \partial \Delta^{j_i-1})\}_{i \in \{1, \ldots, m\}}.
\widetilde{\Lambda}^{\varepsilon}_{\vec{\mathbb{P}},I}(f_{1},f_{2},f_{3})\lesssim|E_{1}|^{\theta_{1}}|E_{2}|^{\theta_{2}}\sum_{k'\in\mathbb{Z}}\widetilde{C}^{\varepsilon}_{k'}\sum_{P'\in\mathbb{P}'_{k'}}\{\prod_{j=1}^{3}(\int_{\mathbb{R}}|\langle f_{j},\Phi^{j}_{P'_{j}}\rangle|^{2}dx_{2})^{\frac{1-\theta_{j}}{2}}\}.
( - m_{\pi}^{2}+ \partial_{\bot}^{2}) \omega_{\pi}= \frac{\lambda}{2 L}\int_{- L}^{L}d x^{-}( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) ,