harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
J_{-n}^{\rho}\Omega=v_{n}^{\rho}J_{-n}^{\rho}v_{n_{1}\cdot\cdot\cdot n_{m}}^{\rho_{1}\cdot\cdot\cdot\rho_{m}}=v_{nn_{1}\cdot\cdot\cdot n_{m}}^{\rho\rho_{1}\cdot\cdot\cdot\rho_{m}}.
\Omega'=\begin{pmatrix}0 \frac{1}{\prod_i(x-t_i)}\frac{\tilde{c}(x)}{\prod_i(x-t_i)\prod_j(x-q_j)}\hskip0.5cm\frac{-x^{n-1}+\tilde{d}(x)}{\prod_i(x-t_i)}-\sum_{j=1}^{n-3}\frac{1}{x-q_j}\end{pmatrix}dx
{\Delta_m}(L) = \Delta p - \frac{{u\Gamma{N_0}W}}{{- D\ln (1 - \varepsilon ) r_0^\alpha}}\times \frac{{{e^{({2^{\frac{b}{W}}}- 1)({\underline L}+ L)}}- 1}}{{{\underline L}+ L}}{\lambda_m}\pi R_s^2d_m^\alpha.
h_{k}(X_{1},X_{2},...,X_{n})=\sum_{1\le i_{1},i_{2},\cdot\cdot\cdot,i_{k}\le n}\frac{m_{1}!m_{2}!\cdot\cdot\cdot m_{n}!}{k!}X_{i_{1}}X_{i_{2}}\cdot\cdot\cdot X_{i_{k}},
\begin{cases}\partial_t\tilde{u}+ (u^s + \tilde{u}) \partial_x\tilde{u}+ \tilde v (u^s_y +\partial_y \tilde{u})= \partial^2_y\tilde{u}, \partial_x\tilde{u}+\partial_y\tilde{v}=0, \tilde{u}|_{y=0}= \tilde{v}|_{y=0}=0 , \ \lim\limits_{y\to+\infty}\tilde{u}= 0, \tilde{u}|_{t=0}=\tilde{u}_0 (x,y)\, .\end{cases}
{}Q(q, m+1) - Q(q, m) = \frac{1}{1-q^N}\left\{\frac{(-1)_{m+1}}{(-q^N)_{m+1}}- \frac{(-1)_{m}}{(-q^N)_{m}}\right\}{}= \frac{1}{(1-q^N)}\frac{(-1)_{m}}{(-q^N)_{m+1}}\left\{(1+q^m) - (1+q^{N+m})\right\}{}= \frac{(-1)_m q^m}{(-q^N)_{m+1}}= P(q, m+1)-P(q, m).
({\,\lower0.9pt\vbox{\hrule\hbox{\vrule height 0.2 cm \hskip 0.2 cm \vrule height 0.2cm}\hrule}\,}-m^2) F_{\alpha\beta}-2(\partial_{\beta}F_{\alpha\mu,\mu}-\partial_{\alpha}F_{\beta\mu,\mu})=0 \quad,
\begin{aligned}\biggl(\frac{1}{m}U_m\biggr)\hat{\Delta}= - \frac{1}{\sqrt{m}}y_m + \mathit{rest}_1, &\|\mathit{rest}_1\|\leq\|\hat{\Delta}\|\cdot\|\hat{X}_{\mathit{tls}}-X_0\|\cdot O_p(1). \end{aligned}
\widetilde{\mathbf{A}}_\nu=\begin{pmatrix}-\alpha\nu \nu 0 \cdots 0 \nu -\alpha\nu \nu \ddots \vdots 0 \nu \ddots \ddots 0 \vdots \ddots \ddots -\alpha\nu \nu 0 \cdots 0 \nu -\alpha\nu\end{pmatrix}.
\bigl \langle u_\Gamma '(t), z_\Gamma\bigr \rangle_{V_\Gamma^*, V_\Gamma}+\int_{\Omega}^{}\nabla \mu(t) \cdot \nabla z dx +\int_{\Gamma}^{}\nabla_\Gamma \mu_\Gamma(t) \cdot \nabla_\Gamma z_\Gamma d\Gamma =0
\frac{D(\phi)g(\phi)}{\phi-\alpha}\le \frac 34 (D_i-D_g)(r_i+\lambda_g)(\phi-\gamma)(\phi-\beta)\le \frac 34 (D_i-D_g)(r_i+\lambda_g)\gamma\beta &=\frac 14 r_i(D_i-D_g)(2+\omega).
\hat{\mathbf{d}}_{i,\rm MF}=\mathbf{A}^{\rm H}\mathbf{H}^{-1}_i \mathbf{y}_i, &\hat{\mathbf{d}}_{i,\rm ZF}=\mathbf{A}^{-1}\mathbf{H}^{-1}_i \mathbf{y}_i, &\hat{\mathbf{d}}_{i,\rm MMSE}=\left(\mathbf{R}_{i,\rm w}+ \mathbf{A}^{\rm H}\mathbf{H}^{\rm H}_i \mathbf{H}_i\mathbf{A}\right)^{-1}\mathbf{A}^{\rm H}\mathbf{H}^{\rm H}_i \mathbf{y}_i,
\lim\limits_{y \to \infty}{y^{\alpha+1}g(1-1/y) \over \int\limits_a^y s^\alpha g(1-1/s)\,ds}=\lim\limits_{y \to \infty}{(\alpha+1)y^\alpha g(1-1/y)+ y^{\alpha-1}g'(1-1/y) \over y^\alpha g(1-1/y)}&=\alpha+1+\lim\limits_{y \to \infty}{y^{\alpha-1}g'(1-1/y) \over y^\alpha g(1-1/y)}&=\alpha+1+\lim\limits_{x \rightarrow 1-}{(1-x)g'(x) \over g(x)}~=\alpha+1+L.
\vartheta_1\bigl(\frac{w_1 +\bar{w}_2}{2\pi i t}| \frac{i}{t}\bigr) = \vartheta_1\bigl(\frac{\tau_{12}- i \delta}{2\pi t}- \frac{i}{2t}|\frac{i}{t}\bigr)= - i e^{\frac{\pi}{4t}+ i \frac{\tau_{12}}{2t}}\vartheta_4\bigl(\frac{\tau_{12}- i \delta}{2\pi t}|\frac{i}{t}\bigr)
[s, y_m] \bigl( [y_i, x_1] [y_{i'}, x_2] + [y_i, x_2] [y_{i'}, x_1] \bigr) = - [s, y_m] \bigl( [y_i, x_1] [x_2, y_{i'}] + [y_i, x_2] [x_1, y_{i'}] \bigr) \equiv 0 \pmod{I}.
\|\Delta f\|_{2}^2 = \|\Delta_r f \|_{2}^2 + \bigg\| \sum_{j=1}^n L_j^2f \bigg\|_{2}^2 + 2\mathbf{R}_{n}\sum_{j=1}^n \bigg\| \frac{L_jf}{|x|}\bigg\|_{2}^2 + 2\bigg\langle -\sum_{j=1}^n L_j^2 f_*, f_* \bigg\rangle,
\sum_{j=2}^{p}\sum_{s=1}^{j-1}\sum_{k\in\mathbb{Z}}\sum_{\ell\in\mathbb{Z}}a_{-s}^{(j)}\,A_{[k-1,j-s-1]}^{(1)}B_{[\ell-k-1,s-1]}^{(1)}W_{[n-\ell-1,0]}= \sum_{j=2}^{p}\sum_{s=1}^{j-1}\sum_{\ell\in\mathbb{Z}}\sum_{k\in\mathbb{Z}}a_{-s}^{(j)}\, A_{[\ell-1,j-s-1]}^{(1)}B_{[k-\ell-1,s-1]}^{(1)}W_{[n-k-1,0]}
\int_{\mathbb{R}^N}\nabla v_n(x) \cdot \nabla \varphi(x) \;dx + \int_{\mathbb{R}^N}V_0(x)v_n(x)\varphi(x)\;dx = \int_{\mathbb{R}^N}\dfrac{f_0(u_n(x))}{u_n(x)}v_n(x)\varphi(x)\;dx + o_n(1).
h_2(t) := t(t-1)(t^2+t+1), h_3(t) := \frac{6(4t^3-1)(4t^3+6t-1)(2t^2+2t-1)f(t)g(t)}{(4t^4-4t^3+6t^2+2t+1)(8t^4-8t^3-8t-1)},
[ \hat{x}_{1}^{\prime}, \hat{x}_{2}^{\prime}] = i \alpha^{2}, [ \hat{p}_{1}^{\prime}, \hat{p}_{2}^{\prime}] = i \alpha^{- 2}, [ \hat{x}_{i}^{\prime}, \hat{p}_{j}^{\prime}] = i \delta_{i j}
\lim_{\varepsilon \downarrow 0}P\left( \| s(v_t^{\delta_\varepsilon}) \|_{H^1}\leq \varepsilon^{\kappa '}\for \all \t \in [\widetilde{C}(\omega ) \varepsilon^{2\gamma +\frac{3}{2}}|\log \varepsilon |,T] \right) =1.
y_2:=\frac{\sum_{j=1}^{n-1}w_{j,n-2}b_{j,n-2}}{\sum_{j=1}^{n-1}b_{j,n-2}^2}\,, z_2:=\frac{\sum_{j=1}^{n-1}w_{j,n-2}b_{j,n-1}}{\sum_{j=1}^{n-1}b_{j,n-1}^2}
\lim_{\delta\to 0}\limsup_{n\to\infty}\,\P\,\bigg( \sup_{\substack{t_1-t_2<\delta 0\leq t_2\leq t_1\leq T}}\big\|u^n_{t_1}-u^n_{t_2}\big\|\,>\epsilon \bigg) = 0.
{{d^2t}\over{d\tilde{\lambda}^2}}+{\dot{n}\over n}\left({{dt}\over{d\tilde{\lambda}}}\right)^2+{{a\dot{a}}\over n^2}\sum_j\left({{dx^j}\over{d\tilde{\lambda}}}\right)^2=\left[nn^{\prime}\left({{dt}\over{d\tilde{\lambda}}}\right)^2-aa^{\prime}\sum_j\left({{dx^j}\over{d\tilde{\lambda}}}\right)^2-2{n^{\prime}\over n}\right]{{dt}\over{d\tilde{\lambda}}},
S(h, A)=S_0+\frac{ik}{2\pi}\int_{D\times R}dxdtd\tau \mbox{Tr}\partial_{\tau}[A_{+}*\partial_xh*h^{-1}]+\frac{ik}{4\pi}\int_{D\times R}dxdtd\tau\mbox{Tr}\partial_{\tau}(A_{+}*A_x),
\sum_{\{\delta\}}\int_{{\cal C}/\Gamma}dx ~ k(x,\gamma [x]) = \sum_{\delta \neq{\bf 1}}\sum_{n=1}^{\infty}\sum_{\beta \in \Gamma/\Gamma^{(\delta)}}\int_{\beta[{\cal C}/\Gamma]}dx ~ k_0(x, \delta^n [x] ) ~+~ \int_{{\cal C}/\Gamma}dx ~ k_0(x, x ),
0=\sum_{u=1}^mh_u\Lambda_{uj}'(z_j)-[\sum_{r,s=1}^n \Lambda_{M_{0j}}^{r,s}(z_j)\frac{\partial}{\partial z_j^r}\wedge \frac{\partial}{\partial z_j^s},\sum_{\alpha=1}^n g_j^{\alpha}(z_j)\frac{\partial}{\partial z_j^{\alpha}}],\,\,\,\,\,
x_{i_k}- x_* = (I - \gamma a_{i_k}a_{i_k}^*) x_{i_{k-1}}+ \gamma a_{i_k}y_{i_k}-x_{*}&=(I - \gamma a_{i_k}a_{i_k}^*) x_{i_{k-1}}+\gamma a_{i_k}(a_{i_{k}}^{*}x_{*}+w_{i_{k}})-x_{*}&=(I- \gamma a_{i_k}a_{i_k}^*)(x_{i_{k-1}}- x_*) + \gamma a_{i_k}w_{i_k}
\left|T(f,g;x)\right| \leq\frac{1}{2}\left(1-\sum^{n}_{k=0}{u^2_{n,k}(x)}\right)\cdot osc_{S_{\Delta_n}}(f)\cdot osc_{S_{\Delta_n}}(g) &\leq \frac{1}{2}\left(1-\frac{1}{2}\right)\cdot osc_{S_{\Delta_n}}(f)\cdot osc_{S_{\Delta_n}}(g) &\leq \frac{1}{4}\cdot osc_{S_{\Delta_n}}(f)\cdot osc_{S_{\Delta_n}}(g),
\ln (D_n(w\alpha_n)/D_n(\alpha_n))=\ln D_n(w\alpha_n)-\ln D_n(\alpha_n) \leq n^2\ln\cos\frac{w\alpha_n}{2}-\frac{1}{4}\ln\left(n\sin\frac{w\alpha_n}{2}\right) -n^2\ln\cos\frac{\alpha_n}{2}+\frac{1}{4}\ln\left(n\sin\frac{\alpha_n}{2}\right)+1 .
A_n^{(1)}(z_1) \cdot \cdots \cdot A_n^{(1)}(z_k) = \frac{1}{f'(0)^k}E_n(z_1) \left(\prod_{j=1}^{k-1}T_\alpha^{-1}A_\alpha T_\alpha E_n^{-1}(z_j) E_n(z_{j+1})\right) T_\alpha^{-1}A_\alpha T_\alpha E_n^{-1}(z_k) &= E_n(z_1) \mathcal{O}(n^{k-1}) E_n^{-1}(z_k).
[U_u-\tilde{u}_{\tilde{\sigma}}X_{\sigma u}]u_{\mu\nu}=\tilde{u}_{\tilde{\sigma}\tilde{\rho}}(D_{\mu}X_{\rho})(D_{\nu}X_{\sigma}) + \tilde{u}_{\tilde{\sigma}}V_{\mu\nu}\cdot X_{\sigma}-V_{\mu\nu}\cdot U,
\sum_{i}(\phi_{1}(x_{i})-y_{i})\cdot D\phi_{1}|_{\phi_{1}^{-1}(\phi_{1}(x_{i}))}\int_{0}^{1}(D\phi_{t})_{|\phi_{1}^{-1}(\phi_{1}(x_{i}))}^{-1}\delta v_{t}|_{\phi_{t}\circ\phi_{1}^{-1}(\phi_{1}(x_{i}))}dt
t_1 =13-5-3-\max\big(\{-5,13-15-3\}\cup\{4-6,8-9,10-13\}\big)=6, t_2 =13-4-1-\max\big(\{-5+2,13-15-3+2,-4,13-15-1\}&\kern5cm\cup\{4-6+2,8-9+2,10-13+2\}\big)=7, t_3 =13-2-0-\max\big(\{-5+4,13-15-3+4,-4+2,13-15-1+2, &\kern4cm-2,13-15-0\}\cup\{4-6+4,8-9+4,10-13+4\}\big)=8.
\kappa_\rho:=\sup_{\rho <t<R}\frac{-(f(t)+2\rho)}{|f'(\rho)|\,(t-\rho)}, \lambda_\rho:=\sup \{t\in [\rho,R): \kappa_\rho+f'(t)<0\}, \tilde \theta_\rho:=\frac{\kappa_\rho}{2-\kappa_\rho}.
|(u-v)(x)| \le\sum_{i\in I_x}|u(x)-m_u^\gamma(2B_i)| &\le\sum_{i\in I_x}\Big(|u(x)-m_u^\gamma(B(x,R))|+|m_u^\gamma(B(x,R))-m_u^\gamma(2B_i)|\Big) &\le C2^{-ls}\sum_{j=l-4}^l M_{\gamma/C}\,g_j(x) &\le Cd(x,y)^s\tilde g_k(x).
R^\mu_{\ \nu}-{1\over 2}\delta^\mu_\nu R +2 l^2 \left[ \tau^\mu_{\ \nu}+\left(\alpha-{1\over 4}\right){\cal S}^\mu_{\ \nu}+{\beta\over 3}{\cal K}^\mu_{\ \nu}\right] ={\kappa^2 \over l}T^\mu_{\ \nu}\ .
R_{j,-p_j}\left(f_s\big|_{\{-p;s\},\gamma}(z)\right)= (R_{j,-p_j}f_s)\Big|_{\{-p+2e_j;s\},\gamma}(z)=\left(s-\frac{p_j}{2}\right)\cdot f_s\big|_{\{-p+2e_j;s\},\gamma}(z),
\left( \frac{1}{\alpha}{\bf I}- \Delta t{\bf D}_{xx}- \Delta t{\bf D}_{yy}\right){\bf U}^{k+1}= \left( \frac{1}{\alpha}{\bf I}+ \Delta t{\bf \bar{B}}+ \Delta t{\bf \hat{B}}\right){\bf U}^{k}+ \Delta t{\bf \bar{\Phi}}^k + \Delta t{\bf \hat{\Phi}}^k + \frac{\Delta t}{\alpha}{\bf F}^{k+1},
\begin{array}{r c l}{{u_{2 , t_{3}}}}&{{=}}&{{\kappa^{2}u_{2 , x x x}+ 3 u_{2}u_{2 , x}\, ,}}\\{{u_{2 , t_{5}}}}&{{=}}&{{1 0 \kappa^{4}u_{2 , x x x x x}+ 5 \kappa^{2}u_{2}u_{2 , x x x}+ 1 0 \kappa^{2}u_{2 , x}u_{2 , x x}+ \frac{1 5}{2}u_{2}^{2}u_{2 , x}\, .}}\\ \end{array}
\frac{\Gamma'}{\Gamma}(\frac{\frac12+ir+a}{2})+\frac{\Gamma'}{\Gamma}(\frac{\frac12-ir+a}{2})=-pv\int_0^\infty \frac{f_0(\nu)^{-\frac12+a}}{f_1(\nu)}\nu^{ir/2}d^\times\nu.
\alpha_\infty = \frac{1}{y^2}\alpha^\prime = \frac{z^3 - 4 z^2 + 6 z - 3}{z^3 - 2 z + 1}\beta_\infty = \frac{z^2}{y^2}\beta = \frac{z^3 - 2 z^2}{z^3 - 2 z + 1}
\Psi_{\nu}^{(\nu)}(x;\lambda_{1})=\psi_{\nu}^{( \nu)}(x)\left( \lambda_{1}+\int_{x_{0}}^{x}dt\frac{1}{( \psi_{\nu}^{(\nu)}(t))^{2}}\right) =\frac{\lambda_{1}+\int_{x_{0}}^{x}dt\psi_{\nu}^{2}(t)}{\psi_{\nu}(x)},\lambda_{1}\in \mathbb{R},
M(\ell, r)\otimes M(\ell', r')\cong \left(\bigoplus_{i=1}^{n-\ell_1}M(\ell_1-\ell_0-1+2i,r+r'+i-\ell_0)\right) \ \oplus \left(\bigoplus_{j=1}^{\ell+\ell'-n}M(n,r+r'+1-j)\right).
\widetilde{\mathsf{V}}\cdot \big( \mathsf{D}_{t}^{q}\cdot P \big) \big\vert_{U_{x,p}}= H_{0}^{q}\,P + H_{1}^{q}\, \big(\mathsf{D}_{t}^{1}\cdot P\big) + \dotsb + H_{q}^{q}\, \big(\mathsf{D}_{t}^{q}\cdot P\big);
\Big ( \sum_{m \ge 1}\sup_{|a|_U =1}| \nabla_k \nabla_{a}^G u_m (t,x)|^2 \Big)^{1/2}\le C |k|_K \, \sup_{t \in [0,T]}\| B(t, \cdot )\|_{C^{\alpha}_b(H, U)},\;\; \;\;\; k \in K,
{\cal D}_{D W}^{G}= i \gamma^{\mu}\left[ \partial_{\mu}+ \frac{2 \pi}{L}\left( \frac{a_{\mu}}{2}- g b_{\mu}\right) + \frac{2 \pi}{L}\hat{\eta}_{\mu}^{~ \nu}( \frac{r}{2}a_{\nu}) \right] ,
\langle A^T x , y \rangle = \langle x , B y \rangle = \langle B x_\perp , y \rangle , \forall y \in u^\perp \langle A^T x , u \rangle = - \frac 1 a \langle x , B x_\perp \rangle = - \frac 1 a \langle x_\perp , B x_\perp \rangle .
\zeta_{\mathrm{r e n}}^{D}( \alpha , R ) = \frac{m^{{\scriptstyle \frac{D}{2}}- 1}}{2^{D - 2}\pi^{D - 1}R^{{\scriptstyle \frac{D}{2}}- 1}}\sum_{n = 1}^{\infty}\frac{\cos ( 2 n \pi \alpha ) K_{{\scriptstyle \frac{D}{2}}- 1}( 2 n \pi R m )}{n^{{\scriptstyle \frac{D}{2}}- 1}}.
\|u_t\|_{L^2}^2 + 2 \int_0^t \|\nabla u_r\|^2_{L^2}dr = \|u_0\|_{L^2}^2 - 2 \int_0^t (F(u_r),\nabla u_r) - \frac{1}{2}( u_r^2, \mathrm{div}(\beta_j)) \dot{Z}_r^j dr \leq \|u_0\|_{L^2}^2 + \int_0^t \| \nabla u_r\|^2_{L^2}dr + \int_0^t \|F(u_r)\|_{L^2}^2 + \|\mathrm{div}( \beta_j )\|_{\infty}\|u_r\|_{L^2}^2 |\dot{Z}_r^j| dr .
\nabla^g = d - (dg) g^{-1}+ g p_{-1}g^{-1}dz + g u g^{-1}dz &= d + \Big( p_{-1}-{h^\vee}^{-1}\varphi\, \rho + r(z) + \frac{\langle r(z), \check \alpha_i \rangle -{h^\vee}^{-1}\varphi(z)}{z-x}\check e_i \Big) dz.
g(t_1,t_{k-1},t_k) =1-t_{k-1}-t_k+(k-1)t_{k-1}+kt_k &=1+(k-2)t_{k-1}+(k-1)t_k &\leq 1+(k-2)\frac{1-kt_k}{k-1}+(k-1)t_k &=\frac{2k-3}{k-1}+\frac{1}{k-1}t_k &\leq\frac{2k-3}{k-1}+\frac{1}{k-1}\frac{1}{k}&=\frac{2k-1}{k}.
\hat{a}_{m,*}\omega_{\hat{X}_m}= R\hat{a}_{m,*}\omega_{\hat{X}_m}=R\hat{a}_{m,*}R\mathcal{H}om(\mathcal{O}_{\hat{X}_m},\omega_{\hat{X}_m}) = R\mathcal{H}om(R\hat{a}_{m,*}\mathcal{O}_{\hat{X}_m},\omega_{\hat{A}}).
g_I := \left[-\tfrac{3}{2}\big(I^2 + 1\big) y^2 + 2 I p^2 - \tfrac{1}{2}q^2\right] \! dx^2 - 4 I p \,dx \,dy + q \,dx \,dp \hphantom{g_I :=}{}- 3 p \,dx \,dq - 3 \,dx \,dz - 3 I \,dy^2 + 3 \,dy \,dq - 2 \,dp^2 .
i\iint_{\Omega(t)}ydxdy=i\iint_{\Omega(t)}\textrm{div}\left(0,\frac{y^{2}}{2}\right)dxdy= i\int_{\partial\Omega(t)}\left(0,\frac{y^{2}}{2}\right)\cdot \left(\frac{y_{\alpha}}{|z_{\alpha}|},-\frac{x_{\alpha}}{|z_{\alpha}|}\right)ds = -\frac{i}{2}\int_{0}^{2\pi}y^{2}x_{\alpha}d\alpha=-\frac{i}{2}\int_{0}^{2\pi}y^{2}(x_{\alpha}+iy_{\alpha})d\alpha.
\langle A\rangle=\int_{PS}p^{-1}(\vec{r})\frac{A_{\vec{r}}}{p^{-1}(\vec{r})}e^{-\beta E_{\vec{r}}}/Zd\vec{r}=\int_{PS}p^{-1}(\vec{r})A_{\vec{r}}^{*}e^{-\beta E_{\vec{r}}}/Zd\vec{r}
\Big(\frac{N}{2 \pi \hbar T}\Big)^{n/2}\prod_{i=1}^{n}\prod_{k=1}^{N-1}( \frac{N^2}{4 \pi \hbar T})^{1/2}\Big( \frac{k^2 \pi^2}{2 N^2 (1-\cos\frac{k \pi}{N})}\Big)^{1/2}dv^k_i.
\frac{-1}{m}P_k = P_k[B,C]P_k &= P_kBP_{k-1}P_{k-1}CP_k-P_kCP_{k-1}P_{k-1}BP_k &+P_kBP_kP_kCP_k-P_kCP_kP_kBP_k &+P_kBP_{k+1}P_{k+1}CP_k-P_kCP_{k+1}P_{k+1}BP_k.
H_{m, n}^-(x) = 2 \int_{-\infty}^\infty \frac{I_{-2 i t}(x) - I_{2 i t}(x)}{\sin(2 i t \pi)}\sinh(\pi t) k(t) V(m^2 n, t) t \, d t &= -4 \int_{-\infty}^\infty \frac{I_{2 i t}(x)}{\sin(2 i t \pi)}\sinh(\pi t) k(t) V(m^2 n, t) t \, d t
\int_D fg = \int_0^\infty \int_0^\infty |\{f > s\}\cap \{g > t\}|_M ds dt &\leq \int_0^\infty \int_0^\infty \min\{|f > s|_M, |g > t|_M \}ds dt &= \int_0^\infty \int_0^\infty |\{S^{D,N}f > s\}\cap \{S^{D,N}g > t\}|_N ds dt &= \int_{S^N D}S^{D,N}f S^{D,N}g .
\int_{Q^{(4)}\backslash E(\lambda_1)}(|Du|+|F|+1)^{p(z)-1}\left[\lambda^{(2-p_0)/(2p_0)}(\rho_2-\rho_1)^{-1}|\widetilde{v}|+|D\widetilde{v}|\right]dz &\leq c\mu\lambda_1|Q^{(4)}\backslash E(\lambda_1)|+c\mu s^{-1}\int_{Q^{(4)}\backslash E(\lambda_1)}|v|^2dz.
b_s^{}= (-1)^{2n+1-m-s}q (j^* \sigma_{2n+1-m,\dots,p+2,p,\dots,2n+2-2m,1}^+)^\vee &+ (-1)^{2n+1-m-s}q (j^* \sigma_{2n+1-m,\dots,p+1,p-1,\dots,2n+2-2m}^+)^\vee = (-1)^{2n+1-m-s}q \sigma_{2n-m,\dots,p+1,p-1,\dots,2n+1-2m}^\vee &+ (-1)^{2n+1-m-s}q (\tau_{2n+1-m,\dots,p+1,p-1,\dots,2n+2-2m}')^\vee.
B(r,t)=-\frac{\mu_{0}}{4\pi}\int[\frac{r-r^{\prime}}{|r-r^{\prime}|^{3}}\times J(r^{\prime},t_{r})+\frac{r-r^{\prime}}{|r-r^{\prime}|^{2}}\times\frac{1}{c}\frac{\partial J(r^{\prime},t_{r})}{\partial t}]dV^{\prime},
\begin{array}{c}\frac{1}{\max_{i}\|y_{i}^{(k)}\|}\underset{i=1}{\overset{m+1}{\sum}}y_{i}^{(k)}=\frac{1}{\max_{i}\|y_{i}^{(k)}\|}[d-x_{m+1}^{(k)}],\end{array}
E =\frac{\lambda_1^2\lambda_2\lambda_4^2\lambda_5}{A^2BD^2}&\approx\frac{\lambda_1^2\lambda_2\lambda_4^2\lambda_5(Mt)^{6/7}}{m^2(\lambda_1\lambda_2\lambda_3)^{1/3}(Mt)^{2/7}}&=M_E(Mt)^{4/7}.
{}_3F_2\bigg[\begin{matrix}\alpha 1-\alpha \beta &1 2\beta\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(\frac12)\Gamma(\frac12+\beta)\Gamma(\beta)}{\Gamma(\frac12+\frac12\alpha)\Gamma(1-\frac12\alpha)\Gamma(\frac12+\beta-\frac12\alpha)\Gamma(\beta+\frac12\alpha)}.
b_{+-}(\xi,\eta) = \frac{i}{4}\frac{|\xi| \widetilde{m}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|}- \frac{i}{4}\frac{|\xi|^{1/2}\widetilde{q}_2(\xi,\eta)}{|\xi-\eta|^{1/2}|\eta|^{1/2}},
E_{1,l}= \lambda[2a(2l+d+4) + 2b_1(4a-2l-d+7-4z_1) - 16b_2z_1(z_1-1){}- l(l+d-1) + 6], \psi_{1,l}(r) \propto r^l (1+\lambda r^2)^{-a-1}[1 - z_1(1+\lambda r^2)] \exp\left(- \frac{b_1}{1+\lambda r^2}- \frac{b_2}{(1+\lambda r^2)^2}\right),
\begin{aligned}\lvert E_{loc}(x_{j}(t),u(t))-E(x_{j,0})\rvert \leq \int_{0}^{t}\lvert\frac{d}{d\tau}E_{loc}(x_{j}(\tau), u(t))\rvert d\tau \leq \int_{0}^{T}\lvert\frac{d}{dt}E_{loc}(x_{j}(t),u(t))\rvert dt \leq E_{1}+E_{2},\end{aligned}
\begin{aligned}d\widehat X_i(t) = b_i(\widehat Z(t))\, dt+\sum_{k=1}^{n+m}\varsigma_{ik}(\widehat Z(t))\, d\widehat W_k(t), \quad\forall\, t>0, d\widehat Y_l(t) = e_l(\widehat Z(t))\, dt+\sum_{k=1}^{n+m}\varsigma_{n+l,k}(\widehat Z(t))\, d\widehat W_k(t),\quad\forall\, t>0,\end{aligned}
\left\vert \int(\psi\circ R_{k})\,\left( u-\frac{1}{M}{\textstyle\sum_{j=0}^{M-1}}\widehat{T}^{j}u\right) \,d\mu\right\vert \leq2\sup\left\vert\psi\right\vert \,\int_{\{\varphi_{E_{k}}\leq M\}}u\,d\mu & +\mathrm{Lip}(\psi)\,\gamma_{T}(\mu(E_{k}))\,M
D_xS = \sum_{(x_1,\hdots,x_k)\in(\eta_t+\delta_x)^k_{\neq}}f(x_1,\hdots,x_k) - \sum_{(x_1,\hdots,x_k)\in\eta^k_{t,\neq}}f(x_1,\hdots,x_k) & = \sum_{(x_1,\hdots,x_{k-1})\in\eta^{k-1}_{t,\neq}}\big(f(x,x_1,\hdots,x_{k-1}) + \hdots + f(x_1,\hdots,x_{k-1},x)\big) & = k\sum_{(x_1,\hdots,x_{k-1})\in\eta^{k-1}_{t,\neq}}f(x,x_1,\hdots,x_{k-1})
\int_{K}\rho_{t}(xy^{-1})f_{\pi,A}(x)~dx =\sum_{n=0}^{\infty}\frac{t^{n}}{2^{n}n!}(\Delta_{K})^{n}f_{\pi,A}(x) & =\mathrm{Tr}(\pi(x)e^{tC_{\pi}/2}A)
\prod_{j=1}^k\lambda_j((B^\frac{p}{2}A^pB^\frac{p}{2})^\frac{1}{p}) = \ \lambda_1\Big(\big((\wedge^{k}B)^\frac{p}{2}(\wedge^{k}A)^p(\wedge^{k}B)^\frac{p}{2}\big)^\frac{1}{p}\Big) \geq \ \lambda_1\Big((\wedge^{k}B)^\frac{1}{2}(\wedge^{k}A)(\wedge^{k}B)^\frac{1}{2}\Big) = \prod_{j=1}^k\lambda_j(B^\frac{1}{2}AB^\frac{1}{2}).
A_n(t) =: \sum_{i=1}^{2s+4}\sum_{k=0}^{n}\frac{a_{n,i,k}}{(t+k)^i}, B_n(t) =: \sum_{i=1}^{s+2}\sum_{k=0}^{n}\frac{b_{n,i,k}}{(t+k)^i}.
\begin{cases}\underline h(t):=(c_0-\delta) t+K, \ \ \t\geq 0, \underline U(t,x):=(1-\epsilon) \big[\Phi(x-\underline h(t))+\Phi(-x-\underline h(t))-\mathbf{u}^*\big], \ \ \t\geq 0, \x\in [-\underline h(t),\underline h(t)],\end{cases}
\left[ \mathbf{A}\operatorname{div}\mathbf{B}-\mathbf{B}\operatorname{div}\mathbf{A}+\mathbf{A}\times\operatorname{curl}\mathbf{B}-\mathbf{B}\times\operatorname{curl}\mathbf{A}-\operatorname{curl}\left( \mathbf{A}\times\mathbf{B}\right) \right]_{p}& \quad=A_{q}\frac{\partial B_{q}}{\partial x_{p}}-B_{q}\frac{\partial A_{q}}{\partial x_{p}},
\frac{d \phi_{1}}{d x}= \frac{\partial W}{\partial \phi_{1}}= \sqrt{2}( 4 \phi_{1}^{2}+ 2 \sigma \phi_{2}^{2}- 1 ) \hspace{1 c m}; \hspace{1 c m}\frac{d \phi_{2}}{d x}= \frac{\partial W}{\partial \phi_{2}}= 4 \sqrt{2}\sigma \phi_{1}\phi_{2}
\sum_{j=-\infty}^{\infty}(2^{j}\rho)^d\int_{S^{d-1}}\sup_{r\in [2^{j}\rho,2^{j+1}\rho]}|\mathbf{K}^{\varepsilon,n}_{e,\rho}(r\theta)|d\mathcal{H}^{d-1}(\theta)\lesssim|\log(\varepsilon)|.
Z_N(s,\alpha) = \sum_{n=0}^{\infty}(n+\alpha)^{-s}\frac{1}{2\pi i}\int_{c-i \infty}^{c+i \infty}\widehat{\phi}(w) \left(\frac{n+\alpha}{N}\right)^{-w}\,dw &= \frac{1}{2\pi i}\sum_{n=0}^{\infty}\int_{c-i \infty}^{c+i \infty}(n+\alpha)^{-(s+w)}\widehat{\phi}(w) N^w \,dw.
\eta(L^{\rm opt}) \approx \begin{cases}\frac{8}{9}\cdot \frac{M^2}{M^2-1}M ~{\rm mod}~3 = 0 \frac{8}{9}\cdot \frac{(M + \frac{1}{2})^2}{M (M+1)}M ~{\rm mod}~3 = 1 \frac{8}{9}\cdot \frac{(M - \frac{1}{2})^2}{M (M-1)}M ~{\rm mod}~3 = 2. \end{cases}
\hat{w}_{k}^{(i)}=w_{k-1}^{(i)}\frac{p(y_{k}|x_{k}^{(i)})p(x_{k}^{(i)}|x_{k-1}^{(i)})}{\pi(x_{k}^{(i)}|x_{0:k-1}^{(i)},y_{0:k})}.
\varphi \equiv \int d \Phi \left[{\frac{3}{4}}\left({\frac{B_{g}^{\prime}}{B_{g}}}\right)^{2}+ 2{\frac{B_{\Phi}^{\prime}}{B_{g}}}+ 2{\frac{B_{\Phi}}{B_{g}}}\right]^{1 / 2}
A_{112}=\frac{\left( 4\alpha-1\right) |\delta|^{-2\alpha}}{2w^{3-4\alpha}}\left[ \kappa v^{3}+2\delta \,\acute{}\,v^{2}+\delta^{2}\left( \kappa-\lambda \right) v-\delta^{2}\delta \,\acute{}\, \right] ,
C_{4}=\sum_{\substack{1\leq i\leq p-1 i\ne (4a)^{-1}c^{2}\bigl(\frac{i}{p}\bigr)=\bigl(\frac{a}{p}\bigr)}}(p^{e-2d-1}+p^{m-d-1})w_{c}^{p^{e-2}-\bigl(\frac{-1}{p}\bigr)p^{m+d-1}}\prod_{\substack{1\leq j\leq p-1}}w_{j+c}^{p^{e-2}-\bigl(\frac{j^{2}-4ai}{p}\bigr)p^{m+d-1}}.
\delta_\tau\bar\varphi_\Sigma((f\otimes \mu)\otimes \Phi)(z,t)= f(z,t)\cdot \left(\frac{\partial F}{\partial x}(0,0)+\frac{\partial F}{\partial y}(0,0)\right)=f(z,t)\cdot \left(\Phi( \underline{\delta s}(z^{-1}\mu))(z,t)\right) =\varphi_\Sigma((f\otimes \mu)\otimes \Phi)(z,t).
\int_z^\infty u^o(t)\,\frac{dt}{t^q}\le u^o(z)\int_z^\infty \,\frac{dt}{t^q}=u^o(z)\frac{z^{-q}}{q-1}\int_0^z\,dt\le \frac{z^{-q}}{q-1}\int_0^zu^o(t)\,dt.
H^{0}(\pi^*{\cal S},{\cal O}_{\pi^{*}{\cal S}}(A^{\prime}\sigma|_{\pi^*{\cal S}}- B^{\prime}F))=H^{0}({\cal S},{\cal O}_{{\cal S}}(-B^{\prime}) \oplus\bigoplus_{i=2}^{A^{\prime}}(-B^{\prime}+ i(r-2)).
\operatorname{Hyp}\big(S^1,G\big)=\lim_{r\downarrow 1}H^0\big(\{1<\vert z\vert <r\},G\big)\times_{H^0(S^1,G)}\lim_{r\uparrow 1}H^0\big(\{r<\vert z\vert <1\},G\big).
{\cal A}_\xi^\lambda = \left\{u \in \Theta_{2\delta}\,:\,\Phi_{\Gamma}(u)\ge 0,\, \left|\left|u\right|\right|_{\lambda,\mathbb{R}^3\setminus \Omega'_{\Gamma}}\leq \xi\, \mbox{and}\, | I_{\lambda}(u)-c_{\Gamma}| \leq \xi\right\}.
\sum_{t=1}^T\alpha_{t}=\sum_{t=1}^T\frac{1}{\mu t}=\sum_{t=2}^{T}\frac{1}{\mu t}+\frac{1}{\mu}\le\int_{1}^{T}\frac{1}{\mu t}dt+\frac{1}{\mu}\le\frac{1}{\mu}(\log(T)+1).
f_{D \cup \{u\},S}(v) \prod_{x \in S \setminus B}x_{3}= (-1)^{t+1}f_{D \cup \{u\},S}(w) \prod_{x \in S \setminus B}x_{2}f_{D \cup \{v\},S}(w) \prod_{x \in S \setminus B}x_{1}= (-1)^{t+1}f_{D \cup \{v\},S}(u) \prod_{x \in S \setminus B}x_{3}.
- \int_{Q_t}\partial_t \mu_{\varepsilon}\partial_t \varphi_{\varepsilon}= \frac{1}{2}\| \partial_t \varphi_{\varepsilon}(t) \|^2_{V^*}- \frac{1}{2}\| \partial_t \varphi_{\varepsilon}(0) \|^2_{V^*},
\Big\| \big|\frac1K{}^{\vee}{\mathfrak{m}_{m,m',l,l'}}(\frac{\cdot}{K}) \big| \ast_{y_1}\phi \Big\|_{L^1(w_{[0,K]})}= \int \phi(y_1)\int \Big|\frac1K{}^{\vee}{\mathfrak{m}_{m,m',l,l'}}(\frac{z-y_1}{K}) \Big|\, dw_{[0,K]}(z)\, dy_1.
\int_{\mathcal U(n)}\det \left(A-UBU^*\right)^{p}\theta(A-UBU^*) \, dU= c_{n,p}\frac{\det \left[\left( a_j - b_k\right)_+^{p+n-1}\right]_{j,k=1}^n}{\Delta(a) \Delta(b)}.
\nabla(df)_{ij}^{\alpha}=\frac{\partial^{2}f^{\alpha}}{\partial x^{i}\partial x^{j}}-\sum_{k=1}^{m}\Gamma(g)_{ij}^{k}\frac{\partial f^{\alpha}}{\partial x^{k}}+\sum_{\beta=1}^{n}\sum_{\gamma=1}^{n}\frac{\partial f^{\beta}}{\partial x^{i}}\frac{\partial f^{\gamma}}{\partial x^{j}}\Gamma(h)_{\beta\gamma}^{\alpha}\circ f.
s_1^{i+1}\, \triangleleft\, s_{p}^i \text{if $1 \le i \le d-q-1$ and $\alpha_{s_1^{i+1}}=1$,}\intertext{and}s_0^{j+1}\, \triangleleft\, s_{p}^{j}\text{if $d-q\le j \le d-1$ and $\alpha_{s_0^{j+1}}=1$,}
\mathcal K_{j}^{p,q}:=\mathcal K_{\lceil\frac j2+1\rceil,\lfloor\frac j2+1\rfloor}^{p,q}= \frac{\Gamma(\lceil\frac j2\rceil+\frac12)}{\Gamma(p)\Gamma(p+q+j+\frac12)}\sum_{\ell=1}^{q}\frac{\Gamma(j+\ell+\frac12)\Gamma(p+q-\ell)}{\Gamma(\lceil\frac j2\rceil+\ell+\frac12)\Gamma(q-\ell+1)}.
\overset{0}{H}(X,Y)Z = \overset{g}{R}(X,Y)Z + \frac{1}{4(n-1)}( \overset{g}{R}ic (X,Z)Y - \overset{g}{R}ic (Y,Z)X ) + \frac{1}{4}( 2\overset{g}{R}ic (AX,Y)AZ + \overset{g}{R}ic (AX,Z)AY - \overset{g}{R}ic (AY,Z)AX)
\sum_A M_A \int\limits_{x}^{\infty}d \theta \frac{d \epsilon_A^{>}}{d \theta}L_A^{>}(\theta)= \sum_A M_A \int\limits_{\epsilon_A^{>}(x)}^{\epsilon_A^{>}(\infty)}d \epsilon_A^{>}\ln ( 1 + e^{\epsilon_A^{>}(\theta)}),
= - n ( U_{0}^{n}|_{i_{n}\leftrightarrow i_{2 k}^{\prime}}+ U_{0}^{n}|_{i_{n}\leftrightarrow i_{2 k - 1}^{\prime}}) S t r \left( \ldots \underbrace{\phi^{i_{2 k}^{\prime}}\phi^{i_{2 k - 1}^{\prime}}}\ldots \right)
a_{nm}(1) \equiv f_0(1) = G_{g_0}(1) =\sum_{d\in D_n}\sum_{e \mid d}\mu(\frac{d}{e}) a_{me}(1) &=\sum_{e \mid n}a_{me}(1) \sum_{d: e\mid d \mid n, \, d \neq n}\mu(\frac{d}{e}) \bmod n^{1+\min \{p-2,r-1\}}.
\mathbb{P}(\bigcup_{i=1}^{n}A_{i})=\sum_{j=1}^{n}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n-2}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n-4}(-1)^{j-1}S_{j}\le...\le\sum_{j=1}^{1}(-1)^{j-1}S_{j}
\theta_{1 3}= \Gamma_{1}e^{- \sum \lambda^{n}t_{n}}\; \; ; \; \; \theta_{2 3}= \Gamma_{2}e^{- \sum \lambda^{n}t_{n}}\; \; ; \; \; \theta_{3 3}= \psi_{B A}e^{- \sum \lambda^{n}t_{n}}\, ,
f(w_1,w_2) =f(w_1+w_0,w_2+w_0) \rightarrow f(w_1+w_0,w_2+w_0), f(w_1,w_2) =f(w_1, (w_2-w_1)+w_1) \rightarrow f(0,w_2-w_1), f(w_1,w_2) =f((w_1-w_2)+w_2, w_2) \rightarrow f(w_1-w_2,0).
[L_{p_{\epsilon(1)}\left(T_1E(S_1)\right)}(1-p_{\epsilon(k_{2})}) S_2 1_B] \otimes \cdots \otimes [(1-p_{\epsilon(k_{m})}) S_m1_B] = E_{D_2}(\mu_1(T_1), \mu_2(T_2),\ldots, \mu_n(T_n))
A=\alpha_1 \alpha_3=\alpha_2\alpha_4, begin{align*}1.5pt] B_1=\alpha_2 \beta_4, B_2=\alpha_1 \beta_3,B_3=-\alpha_4\beta_2,B_4 = -\alpha_3\beta_1, begin{align*}1.5pt] \Gamma_2 = -\beta_2\beta_4\quad\Gamma_3= -\beta_1\beta_3,
\delta_0^k(a)(D_0+i\lambda)^{-p-1}= \delta_0^k(a)(D+i\lambda)^{-p-1}\left(\frac{D+i\lambda}{D_0+i\lambda}\right)^{p+1}= \left(\sum_{l=0}^k \binom{k}{l}\delta_1^{k-l}(\delta^l(a))(D+i\lambda)^{-p-1}\right)\left(\frac{D+i\lambda}{D_0+i\lambda}\right)^{p+1}.
\begin{aligned}u = (-16k)^{-1/3}r^{2/3}(t(1+(c_2+c_3)t))^{-2/3}&\times\exp\big(\i c_1 - \i r^2(1+2(c_2+c_3)t)/(8t(1+(c_2+c_3)t))\big), &p = 3,n=4/3,k<0 ;\end{aligned}