harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W_{\pi}(g_{t,a(\chi),v}) = q^{-\frac{t}{4}-\frac{n}{4}}\sum_{\pm}\gamma_F\left(-\frac{b}{2},\rho\right)\gamma_F\left(\pm Y,2\{-\frac{t}{4}\}\right) \chi\left(-\frac{b}{v}\right)\psi((\pm Y\varpi^{l+\frac{t}{2}}-b)\varpi^{-l})
\begin{aligned}A_{0}^{(k)}(z) =\frac{1}{z-a_{k}^{(0)}-\sum_{j=1}^{p}a_{k}^{(j)}\,A_{j-1}^{(k+1)}(z)}A_{j}^{(k)}(z) =A_{0}^{(k)}(z)\,A^{(k+1)}_{j-1}(z)1\leq j\leq p.\end{aligned}
G^{(0)}_{ij,kl}={\bf a}^2\int_0^{2\pi}\frac{dp_1 dp_2}{(2\pi)^2}\;\frac{e^{i(p_1(i-k)+p_2(j-l))}}{4-2(h_1\,\cos(p_1)+h_2\,\cos(p_2))}\; .
1:=[\begin{matrix}1\\ 0\\ 0\\ 0\end{matrix}];x:=[\begin{matrix}0\\ 1\\ 0\\ 0\end{matrix}];x^{2}:=[\begin{matrix}0\\ 0\\ 1\\ 0\end{matrix}];x^{3}:=[\begin{matrix}0\\ 0\\ 0\\ 1\end{matrix}]
\Pi(\tau_1,\tau_2) \!=\! \begin{bmatrix}\frac{U(0)}{h^2}\frac{1}{2h}U^\top\!(h+\tau_1)A_d \frac{1}{2h}U^\top\!(h+\tau_2)A_d \ast \frac{1}{2h}I_m \frac{1}{2}A_d^{\top}U(\tau_1-\tau_2)A_d \ast \ast \frac{1}{2h}I_m \end{bmatrix}\!,
\left\Vert xP_{u}\left(\frac{y_{0}}{x_{0}}\right)\right\Vert = \left\Vert \frac{xS_{u}(x_{0},y_{0})}{x_{0}^{d_{u}}}\right\Vert\geq \left\Vert \frac{x}{x_{0}^{d_{u}}}\right\Vert\geq \frac{1}{\Delta^{d_{u}}x_{1}^{d_{u}-d_{u-1}}}\geq \frac{1}{D}x_{1}^{-t}\geq \frac{1}{D}x_{0}^{-t}.
=\int{\cal D}j_\mu{\cal D}\lambda_\mu\exp\left[-\frac{1}{8\pi^2}\int d^4x d^4x' j_\mu(x)\frac{1}{(x-x')^2}j_\mu(x')-i\int d^4x \lambda_\mu j_\mu+2\zeta\int d^4x\cos\left(\frac{\left|\lambda_\mu\right|}{\Lambda}\right)\right],
\int_{\Omega}(1-\sigma)D^2 \tilde v_j:D^2\varphi +\sigma\Delta \tilde v_j\Delta\varphi dx=\eta\int_{\partial\Omega}\frac{\partial \tilde v_j}{\partial\nu}\frac{\partial\varphi}{\partial\nu}d\sigma\,,\ \ \ \forall\varphi\in \mathcal{H}^2_{0,D}(\Omega),
{\Omega^{- 1}}({X^a}{Y^b}{Z^c}{W^d}) ={q^{a - b}}{X^a}{Y^b}{Z^{c - 1}}{W^d},{\Omega^{' - 1}}({X^a}{Y^b}{Z^c}{W^d}) ={q^{b - a}}{X^a}{Y^b}{Z^c}{W^{d - 1}}.
\lim_{h\rightarrow0}\frac{1}{\|h\|}_{\!H^1}\! \bigg|K(u+h)-K(u)-\int_{[0,1]}u h \bigg| &= \lim_{h\rightarrow0}\frac{1}{2\|h\|}_{\!H^1}\!\bigg|\int_{[0,1]}h^2\bigg|\leq \lim_{h\rightarrow0}\frac{1}{2}\frac{\|h\|_{H^1}^2}{\|h\|_{H^1}}=0\,,
\frac{(r^6-1)(r^3-r)(r^6-r^3)}{(r^2-1)(r^2-r)(r^3-r^2)}=\frac{r(r^6-1)(r^2-1)(r^3-1)}{(r^2-1)(r-1)^2}=\frac{r(r^6-1)(r^2+r+1)}{(r-1)}.
\int_{\mathbb{R}^3}\Phi^+(f)|v|^2dv =\int_{v_1>0}e^{-\frac{1}{\tau|v_1|}\int^x_0\rho_f(y)dy}f_{L}(v)|v|^2dv\cr +\int_{v_1>0}\int_{0}^{x}\frac{1}{\tau|v_1|}e^{-\frac{\int^x_y\rho_f(z)dz}{\tau|v_1|}}\rho_f(y)\mathcal{M}_{\nu}(f)|v|^2dydv.
p_{c}\left(U,T_{1},T_{2}\right)={\rm E}\left[\min\left\{\frac{U}{K_{0}+1},1\right\}\right]=\sum_{k=0}^{U-1}{\rm Pr}\left(K_{0}=k\right)+\sum_{k=U}^{\infty}\frac{U}{k+1}{\rm Pr}\left(K_{0}=k\right)\;.
\lim_{j\rightarrow\infty}[(\mathcal{K}^h)^{2k_j}(\xi_\tau^h)](\omega)=u(\omega)=\lim_{j\rightarrow\infty}[(\mathcal{K}^h)^{2k_j}(\eta_\tau^h)](\omega)\mathbb{P}\mbox{-a.s.}
\frac{1 1}{1 5}= \frac{1 1 \times 1 0}{1 5 \times 1 0}= \frac{1 1 0}{1 5 0}\frac{1 1}{1 8}= \frac{1 1 \times 1 0}{1 8 \times 1 0}= \frac{1 1 0}{1 8 0}
\psi = \left( \begin{array}{c}\psi_1 \psi_2 \end{array}\right) \propto \left( \begin{array}{c}\exp -2\pi\beta^{-1}\int^{x}_{-\infty}dy \left( \dot{\phi}+ \frac{\beta}{4\pi}\phi' \right) \exp -2\pi\beta^{-1}\int^{x}_{-\infty}dy \left( \dot{\phi}- \frac{\beta}{4\pi}\phi' \right) \end{array}\right) ,
R_1(x) = \frac{\alpha(\alpha-1)(\alpha-2)(\alpha-3) x^{\alpha-3}(1-x)}{24}& \times \Bigg( 1+2(1-x)((5-2\alpha)x + \alpha-4) \Bigg ), R_2(x) = \frac{\alpha(\alpha-1)(\alpha-2)(\alpha-3)(\alpha-4)}{120}& \times x^{\alpha-4}(1-x)(1-2x)(1-12x(1-x)).
\Vert w_{0j}-w_{0}\Vert_{W^{1,1}(\mathbb{R}^{N})}=\Vert w_{0j}-w_{0}\Vert_{W^{1,1}(B_{r_{j}})}\leqslant \Vert w_{0j}\Vert_{W^{1,1}(B_{r_{j}})}+C\Vert w_{0}\Vert_{W^{1,\infty}(\mathbb{R}^{N})}\vert B_{r_{j}}\vert,
{{\lim}_{v\to 0}}{{\mathcal{E}}_{v}}\left( A|B \right)={{A}^{\frac{1}{2}}}{{\exp}({{{A}^{-\frac{1}{2}}}B{{A}^{-\frac{1}{2}}}}}){{A}^{\frac{1}{2}}}=:{{\mathcal{E}}}\left( A|B \right)
|u|_{D^{3,q}}\leq C\big(|\rho u_t+\rho u\cdot\nabla u|_{D^{1,q}}+|H\times H|_{D^{1,q}}+|P|_{D^{2,q}}\big) \leq C(|u_t|_\infty+|\nabla u_t|_q+|u|_{D^{2,q}}+|H|_{D^{2,q}}+|P|_{D^{2,q}}).
\langle a_+(w) c{\bar c}V_{n}^0(0) c{\bar c}V_{m}^1(1)c{\bar c}V_{r}^\alpha(\infty)\prod_{i=1}^{L}\int V_{n_i}^0\prod_{j=1}^{M}\int V_{m_i}^1{1\over{R!}}(Q_-)^R\rangle
Df_k(B)E_{ij}=-\frac{1}{kf_k(B)^{k-1}}\sum_{\mbox{\tiny$\begin{array}{c}i\leq j_1<j_2<...<j_{k-2}\leq n, j_1,...,j_{k-2}\neq i,j \end{array}$}}\det{B_{(j,j_1,...,j_{k-2})(i,j_1,...,j_{k-2})}},
\begin{array}{l l l}{{0 = T_{\sigma^{+}\sigma^{+}}={\frac{1}{1 2}}\, t_{\sigma^{+}}\left( \sigma^{+}\right) = 0}}&{{,}}&{{x^{+}< x_{0}^{+}}}\\{{0 = T_{\sigma^{+}\sigma^{-}}= 0}}&{{,}}&{{x^{+}< x_{0}^{+}\; ,}}\\ \end{array}
\varphi(x)_{x^+=0}= \int_0^\infty dk^+ \int_{-\infty}^\infty \frac{dk_\perp}{\sqrt{2k^+ (2\pi)^2}}\left[ a({\bf k}) e^{-ik^+ x^- + ik_\perp x_\perp}+ a^\dagger({\bf k}) e^{ik^+ x^- - ik_\perp x_\perp}\right],
\int_{B_{\sigma}(x)}|D_{e}u|^{2}\leq \rho^{-2}\int_{B_{\sigma}(x)}|D u(z) \cdot v(x)|^{2}\leq 2\rho^{-2}\left( \int_{B_{\sigma}(x)}|D u(z) \cdot v(z)|^{2}+ \int_{B_{\sigma}(x)}|D u(z) \cdot (v(z) - v(x))|^{2}\right) \leq C \int_{B_{1}(0)}|D_{v}u|^{2}+ C \sigma^{2}\int_{B_{\sigma}(x)}|D u|^{2}\leq C \delta + C \Lambda \sigma^{m},
|E(G)|!p_G(x) =\sum_kF(G,k)x^k &=\sum_{e\in E(G)}\bigg(\sum_kF(G-e,k)x^k\bigg) &=\sum_{e\in E(G)}\bigg((|E(G)|-1)!\sum_kP(G-e,k)x^k\bigg) &=(|E(G)|-1)!\sum_{e\in E(G)}p_{G-e}(x).
S_p^*S_pS_q^*S_q= S_p^*f_{\Delta}S_pS_q^*f_{\Delta}S_q = f_{\Delta^p}f_{\Delta^q}= f_{\Delta^p\cap\Delta^q}= \begin{cases}f_{\Delta^r}Pp\cap Pq = Pr 0 Pp\cap Pq = \emptyset\end{cases}.
\Phi(A_n,D_n)(t)= X_n(t) + \frac{t-G_n(t)}{H_n(t) - G_n(t)}(Y_n(t) - X_n(t)) &= B(n) S(N_{t / b(n)}) + \frac{t - b(n)T(N_{t / b(n)})}{b(n)J_{N(t/b(n))+1}}B(n) J_{N(t/b(n))+1}\Lambda_{N(t/b(n))+1}&= B(n) \left( S(N_{t/b(n)})+ [t/b(n) - T(N_{t/b(n)})] \Lambda_{N(t/b(n))+1}\right) &= W_n(t).
\ll\sum_{\ell\geq 1}\left|\sum_{(f,g)\in \mathcal{F}_k}b_{f,g}\lambda_{f\otimes g}(\ell)\right|^2e^{-\ell/L}=\sum_{\substack{(f_1,g_1)\in \mathcal{F}_k (f_2,g_2)\in \mathcal{F}_k}}b_{f_1,g_1}\overline{b_{f_2,g_2}}\sum_{\ell\geq 1}\lambda_{f_1\otimes g_1}(\ell)\lambda_{f_2\otimes g_2}(\ell)e^{-\ell/L}.
R_{(a_1,a_2, \dots , a_k)^{\uparrow}}^{n}=\binom{n+k-1}{k-1}- \sum_{m=1}^{k}(-1)^{m+1}\sum_{1\leq i_1 < i_2 < \dots <i_j \leq k}\binom{n-\sum_{l=1}^{m}a_{i_{l}}- m + k - 1}{k-1}\,.
\liminf_{n,m\to\infty}d(x_m,y_n)^{\frac{1}{p}}= \liminf_{n,m\to \infty}\|x_m-y_m\|_X \geq \liminf_{n,m\to \infty}\left( \|y_n-x\|_X -\|x_m-x\|_X \right) = \liminf_{n\to \infty}\|y_n-x\|_X x_m\to x \geq \|y-x\|_X
\sum_{i=1}^{d+1}\binom{d+1}{i}\sf(n-i) < \sum_{i=1}^{d+1}\binom{d+1}{i}(2d+2)^{n-i}= (2d+2)^n \sum_{i=1}^{d+1}\binom{d+1}{i}\Big(\frac{1}{2d+2}\Big)^i = (2d+2)^n \bigg(\Big(1 + \frac{1}{2d+2}\Big)^{d+1}- 1\bigg).
\Big{(}\mathcal{D}_tu_h^{n-\theta},v_h\Big{)}-\gamma(\nabla \sigma_h^{n-\theta},\nabla v_h)+(\nabla u_h^{n-\theta},\nabla v_h)+(f^{n-\theta}(u_h),v_h)= (g^{n-\theta},v_h),~\forall v_h\in L_h,
q_{l_1l_2l_3l_4}=\frac{\zeta^{1+|l_0|\delta_{l_4,0}}(\zeta+2)^{1+\chi(l_1+l_3)}}{(2\zeta+1)^{1+\chi(l_3+l_4)}}\,f(\zeta) &=1+\frac{(\zeta+1)^{1+|l_0|\delta_{l_3,0}}(\zeta-1)^{1+\chi(l_1+l_4)}}{(2\zeta+1)^{1+\chi(l_3+l_4)}}\,g(\zeta),
\mathbb{P}_{\mathbb{C}}^n\setminus \Lambda_0= \{[1\!:\! z_2\!:\!\cdots\!:\! z_{n+1}]: z_2,\!\cdots\!,\!z_{n+1}\!\in\!\mathbb{C}\}\cup \{[0\!:\! 1\!:\! z_3\!:\!\cdots\!:\! z_{n+1}]: z_3,\!\cdots\!,\! z_{n+1}\!\in\!\mathbb{C}\}\cup \{[0\!:\! 0\!:\! z_3\!:\!\cdots\!:\! z_{n}\!:\!1]: z_3,\!\cdots\!,\! z_{n}\!\in\!\mathbb{C}\}.
G_{\mu^{1/\alpha}\circledast\mathbf{b}_{\alpha,1}}(z) = (-z)^{\alpha-1}G_\mu(-(-z)^\alpha), z<0, \eta_{\mu^{1/\alpha}\circledast\mathbf{b}_{\alpha,1}}(z) = \eta_\mu(-(-z)^\alpha), z<0.
\frac{\partial W}{\partial\lambda_{i}}=\frac{\partial W}{\partial I_{1}}\frac{\partial I_{1}}{\partial\lambda_{i}}=2\lambda_{i}\frac{\partial W}{\partial I_{1}}and\frac{\partial^{2}W}{\partial\lambda_{i}\partial\lambda_{j}}=2\delta_{ij}\frac{\partial W}{\partial I_{1}}+4\lambda_{i}\lambda_{j}\frac{\partial^{2}W}{\partial I_{1}^{2}}.
\langle 0 | ( X^{1}( \sigma ) - \overline{{{X^{1}}}})^{2}| 0 \rangle \ = \ \sum_{m = 1}^{\infty}\frac{1}{m^{2}}\langle 0 | ( \alpha_{m}\alpha_{- m}+ \tilde{\alpha}_{m}\tilde{\alpha}_{- m}) | 0 \rangle \ = 2 \sum_{m = 1}^{\infty}\frac{1}{m}.
\sqrt{- g}L \equiv{\frac{1}{4 \pi{\cal A}_{P}}}\left( \sqrt{- g}{\cal G}\right) ={\frac{1}{4 \pi{\cal A}_{P}}}\left( \sqrt{- g}\, g^{i k}\left( \Gamma_{i \ell}^{m}\Gamma_{k m}^{\ell}- \Gamma_{i k}^{\ell}\Gamma_{\ell m}^{m}\right) \right) .
\frac{f (- s \triangle )- 1}{s \triangle}=\int_0^1\!\!{\mathrm{d}}\alpha_1 \alpha_1 (1-\alpha_1) \!\int_0^1\!\!{\mathrm{d}}\alpha_2 \,{\mathrm e}^{\alpha_1(1-\alpha_1)\alpha_2 s \triangle},
A=\mathrm{cov}(K(\frac{x_{j}-X_{0}}{b_{n}}),K(\frac{x_{p}-X_{k}}{b_{n}}))\leq\mathrm{cov}(K(\frac{x_{j}-X_{0}}{b_{n}}),K(\frac{x_{p}-X_{2}}{b_{n}})) =\frac{1}{b_{n}}\int\int K(\frac{x_{j}-u}{b_{n}})K(\frac{x_{p}-v}{b_{n}})(f_{2}(u,v)-f(u)f(v))dudv.
{[\bar{x}, \bar{y}]}= - \bar{T}(\bar{x}, \bar{y}) + \bar{R}(\bar{x},\bar{y}) = -\overline{T(x,y)}+ \overline{R(x,y)},{[\bar{E}_1, \bar{y}]}= \bar{E}\bar{y}= \overline{Ey},{[\bar{E}_1, \bar{E}_2]}= - (\bar{E_1}\bar{E_2}- \bar{E_2}\bar{E_1}) = - (\overline{E_1 E_2}- \overline{E_2 E_1}),
I_{S t m o d}= - \frac{1}{4}F_{m n [ A ]}F_{[ A ]}^{m n}- \frac{1}{2}\mu^{2}[ A_{m}+ \frac{1}{2 \mu}l_{m}] [ A^{m}+ \frac{1}{2 \mu}l_{m}] + \frac{1}{4}\epsilon^{m n p q}B_{m n}\partial_{p}l_{q}.
\mathbb{E}\left(e^{-s S_{\alpha_1,\lambda_1,\alpha_2,\lambda_2}(t)}\right) = e^{-t\left(c_1\left((s+\lambda_1)^{\alpha_1}-\lambda_1^{\alpha_1}\right) + c_2\left((s+\lambda_2)^{\alpha_2}-\lambda_2^{\alpha_2}\right)\right)},\;s> 0,
v_{q^{\prime}m^{\prime}k^{\prime}}( t^{\prime}, r^{\prime}, \theta^{\prime}, z^{\prime}) = N_{1}\, e^{i k^{\prime}z^{\prime}+ i m^{\prime}\theta^{\prime}}e^{- i \omega^{\prime}t^{\prime}}J_{m^{\prime}}( q^{\prime}r^{\prime}) ,
{\frac{1}{2}}\bar{\theta}( 1 - \Gamma^{5}){\slash{\Pi_{\bar{z}}}}( 1 - \Gamma^{5}) \partial_{z}\theta ={\frac{1}{2}}\bar{\theta}( 1 - \Gamma^{5}) \left( \gamma_{0}\Pi_{\bar{z}}{}^{0}+ \sum_{a = 5}^{a = 9}\gamma_{a}\Pi_{\bar{z}}{}^{a}\right) \partial_{z}\theta \ .
\phi_2(2cq_1+a_1+2cq_2+a_2) =6q_1+6q_2+a'_1+a'_2 &\leq \phi_2(6q_1+6q_2+s_1+s_2) &\leq 6q_1+6q_2+b'_1+b'_2 &= \phi_2(6q_1+b_1+6q_2+b_2)
\begin{aligned}\Theta_{k_1k_2}(M) = \Big\{\theta = (\theta_{ij})\in\mathbb{R}^{n_1\times n_2}: \theta_{ij}=Q_{z_1(i)z_2(j)}, z_1\in[k_1]^{n_1},z_2\in[k_2]^{n_2}, & ~~~ Q\in[-M,M]^{k_1\times k_2}\Big\}.\end{aligned}
\frac{a^{2}}{\alpha}\underset{i=1}{\overset{m}{\sum}}\Big\{2R(e_{i},S(U))\nabla_{e_{i}}U+(\nabla_{e_{i}}R)(e_{i},S(U))U+R(e_{i},\nabla_{e_{i}}S(U))U\Big\}\:\: and \:\: U \:\: are \:\: collinear.
\left(\begin{array}{c}z_1(x,t) z_2(x,t)\end{array}\right)= \left(\begin{array}{cc}\cos(\Omega t) \sin(\Omega t) -\sin(\Omega t) \cos(\Omega t)\end{array}\right)\left(\begin{array}{c}(1 + R(x))\cos(x) (r + R(x))\sin(x)\end{array}\right)
A_{0}= 0 , \; A_{D}= E x^{0}, \; A_{i}= A_{i}^{\perp}= - H_{j}\, x_{i + 1}\, \delta_{i , 2 j - 1}, \; j = 1 , \ldots , [ d / 2 ] - 1 , \; i = 1 , \ldots , D - 1 .
B =4h^{ij}h_{jl}R_{\mi\}^{l\ \ \m}-4 h^{ij}h^{lm}R_{iljm}-2h^{ij}(\nabla_j R_{\nu li}^{\ \ \l}+\nabla_l R_{\nu ij}^{\ \ \l}),\quad\mbox{and}D =h^{ij}\left(H\lambda_{,ij}+2H_{,i}\lambda_{,j}\right)-\frac{1}{2}H(H\Delta\lambda+2\langle\nabla H,\nabla\lambda\rangle).
\frac{BC_{2,18}}{\Pi(18)}=\sum_{k=1}^{18}\binom{19}{k+1}(-D_2)^k\frac{k}{D_2^{k-1}D_3}&=\frac{D_2}{D_3}\sum_{k=1}^{18}(-1)^k\binom{19}{k+1}k=-\frac{D_2}{D_3}\,.
(X_{a}^{n+1})_{H^{n+1}}=(\widetilde{\widehat{X}^{n}_{a}})_{H^{n+1}}+(\widetilde{(\xi^{c})^{\bar{P}^{(n)}}})_{H^{n+1}}+(\xi^{A})^{P^{n+1}}_{H^{n+1}}, &(X_{b}^{n+1})_{H^{n+1}}=(\widetilde{\widehat{X}^{n}_{b}})_{H^{n+1}}+(\widetilde{(\xi^{d})^{\bar{P}^{(n)}}})_{H^{n+1}}+(\xi^{B})^{P^{n+1}}_{H^{n+1}}
\Phi_{b}( \vec{y}_{1}, \eta ) . . . \Phi_{b}( \vec{y}_{n / 2}, \eta ) \Phi_{b}( \vec{y}_{n / 2 + 1}, - \eta ) . . . \Phi_{b}( \vec{y}_{n}, \eta ) = \Psi_{1}+ \bar{\eta}\Psi + \bar{\Psi}\eta + \bar{\eta}\eta \Psi
\!\!\!\! - \frac{1}{2}\sum_{n_1,n_2}\, \log \left(\frac{\mid \tau \mid^2 n_1^2 - 2 \tau_1 n_1 n_2+ n_2^2 + \lambda^2}{\mid \tau \mid^2 n_1^2 -2 \tau_1 n_1 n_2 + n_2^2}\right) \;.
\begin{cases}(\hat{\nabla}-\imath\hat{A}_n)^{2}u_n+\frac{1}{\epsilon^2}(1-|u_n|^2)u_n+P_n=0 \Omega, \nabla\times(\nabla\times\vec{A})=(j_1,j_2,j_3) \mathbb{R}^3, (\hat{\nabla}-\imath\hat{A}_n)u_n\cdot\vec{n}=0 \partial\Omega, \nabla\times\vec{A}-h_{ex}\vec{e}_3\in L^2(\mathbb{R}^3;\mathbb{R}^3)\end{cases}
G_{0}( \kappa , \kappa^{\prime}) = \left[ \frac{\theta ( \kappa )}{\kappa}\right]^{1 / 2}\left[ \frac{\theta ( \kappa^{\prime})}{\kappa^{\prime}}\right]^{1 / 2}\, \frac{1}{4 \cosh \left[ \frac{\pi}{4}( \kappa - \kappa^{\prime}) \right]}.
\sum_{\stackrel{k'+k''=k}{k',k''\not=0}}{\rm min}^2\{1,\ell(d(k',0)+d(k'',0))\}d^{-4}(k',0)d^{-1}(k'',0)\lesssim\min\{d^{-1}(k,0),\ell^2d(k,0)\}.
\frac{d^2 \psi(\zeta)}{d \zeta^2}+ \frac{2 \zeta -1}{\zeta(\zeta-1)}\frac{d \psi(\zeta)}{d \zeta}-\frac{q + r(1 - 2 \zeta)^2}{\zeta^2(\zeta-1)^2}\psi(\zeta) = 0
\gamma_s^{(N)}(K_{4_1})=\sum_{i=0}^{\underline s-1}\{s+i, 2i+1\}\sum_{k=s-i}^{s+i}\dfrac{\{k\}_+}{\{k\}}+ 2\sum_{i={\underline s}}^{\overline s-1}\widetilde{\{s+i, 2i+1\}},
\frac{d}{d\epsilon}E(\epsilon)K(\sqrt{1-\epsilon^{2}})=\frac{1}{\epsilon(1-\epsilon^{2})}[-E(\epsilon)E(\sqrt{1-\epsilon^{2}})+E(\epsilon)K(\sqrt{1-\epsilon^{2}})-(1-\epsilon^{2})K(\epsilon)K(\sqrt{1-\epsilon^{2}})]
\lim_{n\to\infty}\mu\left(\left\{x\in X \mid \frac{S_nf(x)}{\sqrt{n}}\le z \right\}\right) = \frac{1}{\sigma_f\sqrt{2\pi}}\int_{-\infty}^z\exp\left(-\frac{t^2}{2\sigma_f^2}\right)dt.
\tau_{j l}= \pi + \mathrm{i}\; \mathrm{A r c h}( - c_{j l}) = \pi + \frac{\mathrm{i}}{2}\ln \left( \frac{k_{j l}^{2}- m_{j}^{2}- m_{l}^{2}+ \sqrt{\lambda ( m_{j}^{2}, m_{l}^{2}, k_{j l}^{2})}}{k_{j l}^{2}- m_{j}^{2}- m_{l}^{2}- \sqrt{\lambda ( m_{j}^{2}, m_{l}^{2}, k_{j l}^{2})}}\right) .
\underset{X\in\mathbb{R}^{2\times2}}{\mbox{minimize}}f(X)+\gamma\mathcal{T}_1(X)=\underbrace{\begin{psmallmatrix}-\gamma 0 0 0\end{psmallmatrix}\bullet X}_{f(X)}+\gamma\underbrace{\sigma_2(X)}_{\mathcal{T}_1(X)}.
\mu_k =\inf\limits_{\substack{\mathfrak{M}\subset P^{+}_n\mathsf{H}^{1/2}(\mathbb{R}^n;\mathbb{C}^{2(n-1)}) \dim \mathfrak{M}=k}}\sup\limits_{\psi \in (\mathfrak{M}\oplus P^{-}_n\mathsf{H}^{1/2}(\mathbb{R}^n;\mathbb{C}^{2(n-1)}))\setminus\{0\}}\frac{\mathtt{d}_{n}[\psi]+\mathtt{v}[\psi]}{\|\psi\|^2}.
\begin{array}{ccccccccc}\theta (q,x) = \theta (-v,x) = \psi_1+\psi_2 ,{\rm where}\psi_1(v,x) := \theta (v^4,-x^2/v){\rm and}\psi_2(v,x) := -vx\theta (v^4,-vx^2)~;\end{array}
{\cal L}^{D}= \sqrt{- g^{D}}\left( R^{D}- \frac{1}{2}\partial_{\mu}\Phi \partial_{\nu}\Phi -{\frac{1}{2 . 3 !}}e^{- l \Phi}F_{\mu \nu \sigma}^{( 3 )}F^{( 3 ) \mu \nu \sigma}\right) ,
K_{\omega,\mathbf{c}}(V^{\lambda}) =2L(V^{\lambda})+3N(V^{\lambda})+2\omega M(V^{\lambda})+2\mathbf{c}\cdot \mathbf{P}(V^{\lambda}) &=2\lambda^2L(V)+3\lambda^{\frac{d}{2}+1}N(V)+2\omega M(V)+2\lambda \mathbf{c}\cdot \mathbf{P}(V),
\left( \begin{array}{c}M_{\pi , \rho}\mbox{is a meander}\mbox{(in the sense of Def. \ref{def:31}.2)}\end{array}\right) \ \Leftrightarrow \ \Bigl( \, A ( \pi ) \, \widetilde{\vee}\, A( \rho ) = 1_{2n}\Bigr),
\frac{d^r}{dt^r}\bigl[e^{-b_0 t}\prod\limits_{j=1}^N (1-e^{-b_jt})\bigr] = (-1)^r \sum\limits_{p=0}^N (-1)^p\sum\limits_{k_1<\cdots<k_p=1}^N \bigl(b_0+\sum b_{k_j}\bigr)^r\exp\bigl(-(b_0+\sum b_{k_j})t\bigr).
E(t) =\frac{1}{2}|| u_t||^2_{L^2(\mathbb{H}^n)}+ \frac{m}{2}||u||^2_{L^2(\mathbb{H}^n)}+ \frac{1}{2}||\nabla_{H}u||^2_{L^2(\mathbb{H}^n)}- \int_{\mathbb{H}^n}F(u) dx,
E_{q}(z)=\sum_{n=0}^{\infty}\frac{q^{n(n-1)/2}}{(q;q)_{n}}z^{n}(z\in\mathbb{C}) \mbox{and}e_{q}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{(q;q)_{n}}(|z|<1),
{\bf L}={\bf K}^{-1/2}\left({\bf H}+ \left[\begin{array}{cc}{\bf 0}{\bf 0}{\bf 0}{\bf G}^{-1}\end{array}\right] \right)^{-1}\left[\begin{array}{cc}{\bf 0}{\bf 0}{\bf 0}{\bf G}^{-1}\end{array}\right]{\bf K}^{1/2}.
a_{2i-1,\tau}=\frac12 \left[n \atop i-1 \right]\sum_{j=0}^{i-l}(-1)^jq^{j(j-1)}\left[i \atop j\right]\left(\frac{|Y|}{q^{(2n+1)(n+1+j-i)}}-1\right), a_{2i,\tau}=\frac12(q^{2i}+\tau\eta(-1)^iq^i)\left[n \atop i\right]\sum_{j=0}^{i-l}(-1)^jq^{j(j-1)}\left[i \atop j\right]\left(\frac{|Y|}{q^{(2n+1)(n+1+j-i)}}-1\right),
G_{n,m}=(-1)^n\bigl\{f^{(n+2m)}f^2-f^{(n+m)}f^{(m)}f-f^{(n)}f^{(2m)}f+f^{(n)}\bigl[f^{(m)}\bigr]^2\bigr\}, H_{n,m}=(-1)^n\bigl\{f^{(n+2m)}f^2-2f^{(n+m)}f^{(m)}f+f^{(n)}\bigl[f^{(m)}\bigr]^2\bigr\}, I_{n,m}=(-1)^n\bigl\{f^{(n+2m)}f^2-2f^{(n)}f^{(2m)}f+f^{(n)}\bigl[f^{(m)}\bigr]^2\bigr\}
\left\langle [\gamma_1]_{\vec s^1}\bar{\psi}^{a_1},\ldots, [\gamma_m]_{\vec s^m}\bar{\psi}^{a_m}\right\rangle_{g,\{\vec s^j\}_{j=1}^m,\beta}^{X_{D,\infty}}:=\left[\left(\prod_{i=1}^n r_i^{s_{i,-}}\right)\left\langle \gamma_1\bar{\psi}^{a_1},\ldots, \gamma_m\bar{\psi}^{a_m}\right\rangle_{g,\{\vec s^j\}_{j=1}^m,\beta}^{X_{D,\vec r}}\right]_{\prod_{i=1}^n r_i^0}
R^{(d)}_T(0)\leq R^{(d)}_{\prod_{i=1}^dT_i}(0), \leq \mathcal{F}(d,T_1)r^{(1)}_{T_1}(0)\prod_{i=1}^{d}T_i \leq 2T\mathcal{F}(d,T_1)r^{(1)}_{T_1}(0) \leq 2T\mathcal{F}(d,\lfloor T^{\frac{1}{d}}\rfloor)r^{(1)}_{\lfloor T^{\frac{1}{d}}\rfloor}(0),
U_{2n,N-x}= U_{2n,N+x+1}= \frac{(-1)^n}{\sqrt{2}}\tilde R_n(\lambda(x);\alpha,\beta,\gamma,\delta+1), U_{2n+1,N-x}= -U_{2n+1,N+x+1}= -\frac{(-1)^n}{\sqrt{2}}\tilde R_n(\lambda(x);\alpha,\beta+1,\gamma+1,\delta),
\sum_{i}\left[ \Psi \right]_{i}\left( a_{\alpha i}\Delta p_{\alpha}+b_{\alpha i}^{\mu}\partial_{\mu}\left( \Delta p_{\alpha}\right)\right) \equiv -\sum_{i}\partial_{\mu}\left\{\frac{\partial L}{\partial\left( \partial_{\mu}\psi_{i}\right)}\left( a_{\alpha i}\Delta p_{\alpha}+b_{\alpha i}^{\mu}\partial_{\mu}\left( \Delta p_{\alpha}\right)\right) \right\},
V_{\tilde{\rho}}^*xV_{\tilde{\rho}}\Lambda (y) = \sqrt{d(\rho )}^{-1}V_{\tilde{\rho}}^*\Lambda (x\tilde{\rho}(y)) = \Lambda (\phi_{\tilde{\rho}}(x\tilde{\rho}(y))) =\phi_{\tilde{\rho}}(x) \Lambda (y).
\mu_z=|d_{\mathrm{min}}|^2 \left( \frac{\alpha_0b_0^{\mathtt{opt}}}{P_0+1}|y_0[k-1]|^2 \right.\left. +\sum \limits_{i=1}^{R}\frac{\alpha_i b_i^{\mathrm{opt}}}{\rho_i+1}|y_i[k-1]|^2\right)
D_6 := \{x_1 + x_2 + x_3 = 0,\q_1 + q_2 = 0 \}D_7 := \{x_1 + x_2 - x_3 = 0,\q_1 + q_2 = 0 \}D_8 := \{x_1 - x_2 + x_3 = 0,\q_1 + q_2 = 0 \}.
\frac{f(\omega h_j,1)}{g(\omega h_j)}= e^{\omega h_j}- \frac{(-1)^{p+1}(\omega h_j)^{p+1}}{2g(\omega h_j)}\left[ R^-_{p+1}(1) + \sum_{k=1}^\infty \left(\frac{\omega h_j}{2}\right)^k R^{-,(-k)}_{p+1}(1) \right].
y_{\textrm{FN}}=\sqrt{\beta_{\textrm{F}}\eta P_\textrm{B}d_{\textrm{BF}}^{-\alpha}d_{\textrm{NF}}^{-\alpha}|h_{\textrm{BF}}|^2}h_{\textrm{NF}}x_{\textrm{N}}+n_{\textrm{N}},
\begin{aligned}A_{1}U_{0}=C_{1},\V_{0}B_{1}=D_{1}, &A_{2}X_{0}=C_{2},\X_{0}B_{2}=D_{2}, &A_{3}Y_{0}=C_{3},\Y_{0}B_{3}=D_{3}, &A_{4}Z_{0}=C_{4},\Z_{0}B_{4}=D_{4},\end{aligned}
\lvert g(t-s,x-y)-g(t-s,x'-y)\rvert^2 z^2 \leq C(t-s)^{-d}z^2\Bigl\lvert e^{-\frac{\lvert x-y\rvert^2}{2 (t-s)}}- e^{-\frac{\lvert x'-y\rvert^2}{2 (t-s)}}\Bigr\rvert^2 &\leq C(t-s)^{-\frac d2 q}z^q\Bigl\lvert e^{-\frac{\lvert x-y\rvert^2}{2 (t-s)}}- e^{-\frac{\lvert x'-y\rvert^2}{2 (t-s)}}\Bigr\rvert^q &=C\lvert g(t-s,x-y)-g(t-s,x'-y)\rvert^qz^q,
{\cal{G}}_{k_\perp}^{\pm(\eta)}(t,t^\prime) = -{\Gamma(-\nu) \over 2 \sqrt{\pi}}e^{-{i \pi \over 4}}{1 \over S^\prime}D_\nu \left({1+i \over \sqrt{\hbar}}S(t) \right) D_\nu \left( -{1+i \over \sqrt{\hbar}}S(t^\prime) \right)
\varphi_{Z_{n}}(t)=\varphi_{\sum_{i=1}^{n}\frac{1}{\sqrt{n}}Y_{i}}(t)=\varphi_{Y_{1}}(\frac{t}{\sqrt{n}})\varphi_{Y_{2}}(\frac{t}{\sqrt{n}})\cdot\cdot\cdot\varphi_{Y_{n}}(\frac{t}{\sqrt{n}})=[\varphi_{Y_{1}}(\frac{t}{\sqrt{n}})]^{n},
e^{{\frac{i \hbar}{2}}( \stackrel{\leftarrow}{\partial}_{x}\stackrel{\rightarrow}{\partial}_{p}- \stackrel{\leftarrow}{\partial}_{p}\stackrel{\rightarrow}{\partial}_{x}) + \hbar \stackrel{\leftarrow}{\partial}_{\theta}\stackrel{\rightarrow}{\partial}_{\theta}}.
\frac{\partial \overline{{{H}}}_{0}}{\partial q_{n , \alpha_{n}}}\left( q_{n , \alpha_{n}}, p_{n , m_{n}- 1}\right) = - \frac{\partial L_{0}}{\partial q_{n , \alpha_{n}}}\left( q_{n , \alpha_{n}}, \dot{q}_{n , m_{n}- 1}( q_{n , \alpha_{n}}, p_{n , m_{n}- 1}) \right) \ ,
\mathrm{\mathrm{}^* \!}F_{\mu \nu}( x ) = - \frac{2 \sqrt{6}}{\bar{N}}\int \delta \xi d s H_{\mu \nu \rho}[ \xi | s ] \dot{\xi}^{\rho}( s ) \dot{\xi}( s )^{- 2}\delta ( x - \xi ( s ) ) ,
v^{(0)}_0(x_1,-l_1,t) = v^{(1,m)}_0(x_1,-l_1,t), m=1, 2, h_0 \partial_{x_2}v^{(0)}_0(x_1,-l_1,t) = h_{1,1}\partial_{x_2}v^{(1,1)}_0(x_1,-l_1,t) + h_{1,2}\partial_{x_2}v^{(1,2)}_0(x_1,-l_1,t)
\varphi(e_{i+k-1})=\frac{1}{(n-k+1)b^{\frac{n}{n-k+1}}\zeta^{ni}}\frac{w^{n+1}-bw^k}{w-b^{\frac{1}{n-k+1}}\zeta^i},\ \mbox{for}1\leq i\leq n-k+1.
C_{I J K}\tilde{h}^{I}\tilde{h}^{J}\tilde{h}^{K}= a^{3}( y ) C_{I J K}h^{I}h^{J}h^{K}= a^{3}( y ) \qquad \mathrm{o r}\qquad a^{2}( y ) = h^{I}H_{I}.
\omega_2(t,r,\theta) = - \frac{F(r,\theta)}{\rho_0 C^{5/3}}\frac{\left( \frac{\rho}{\rho_0}\right)^{2/3}\ln\frac{\rho}{\rho_0}}{\left( \ln \frac{\rho}{\rho_0}+ 1\right)^{8/3}}.
x(t,s_0,x_0) =U(t,s_0)x_0 +\int_{s_0}^{t}U(t,s){\rm d}[g(s)-g(s_0)]-\sum_{s_0< \tau \leq t}\Delta^{-}U(t,\tau)\Delta^{-}g(\tau) &\ \ + \sum_{s_0\leq \tau < t}\Delta^{+}U(t,\tau)\Delta^{+}g(\tau).
\left\{\varphi=\big(\varphi_0(z_0,z_1,z_2),\psi(z_1,z_2)\big)\,\big\vert\,\varphi_0\in\mathrm{PGL}(2,\mathbb{C}[z_1,z_2]),\, \psi\in\mathrm{Bir}(\mathbb{P}^2_\mathbb{C})\right\}.
K_{X' / X}= \sum_{j=1}^u \left( \kappa(p_j)b_{y_j}+\kappa(p_j)-1-\vartheta_X (y,\kappa(p_j)p_j,\kappa(p_j)) \right) D_{(y_j,p_j)}' + \sum_{i=1}^e \left(-1-\vartheta_X (C,\rho_i,0) \right) D_{\rho_i}' ,
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( 1-z^n \right)^{\frac{m^4}{n^5}}= \sqrt[3]{1-z}\; \exp\left\{\frac{z(2z-7)}{10(1-z)^2}+ \frac{1}{30}Li_3(z) \right\}.
\begin{aligned}\mathbb{P}_0\left(\widetilde{\tau}^{(n,\delta)}_i>s\right) =\mathbb{P}_0\left(X_{N_r}^{(n)}<i,\,\forall 0\le r\le s\right) =\mathbb{P}_0(X_m^{(n)}<i,\,\forall m\le N_s)=\mathbb{P}_0\left(\tau^{(n)}_i>N_s\right).\end{aligned}
\log P_t (e^f) - P_t f \leq \log P_{t}\left( e^{\alpha\Gamma(f)}\right) \int_0^t \frac{2\rho}{2\rho \theta(s)-1+e^{-2\rho (t-s)}}ds & =\log P_{t}\left( e^{\alpha\Gamma(f)}\right) \log \left( \frac{2\rho \alpha e^{2\rho t}}{2\rho \alpha e^{2\rho t}+1}\times \frac{2\rho \alpha+e^{-2\rho t}}{2\rho \alpha - 1 +e^{-2\rho t}}\right)
\frac{d}{dt}|\overline{u}^{k+1}|^2_2+\alpha |\phi^{k+1}\nabla\overline{u}^{k+1}|^2_2 \leq B^k_\nu(t)|\overline{u}^{k+1}|^2_2+B^k(t)|\overline{\phi}^{k+1}|^2_{2}+\nu (|\phi^k \nabla \overline{u}^k|^2_2+ | \overline{u}^{k}|^2_2),
\int |g|^2e^{-\phi-\varphi}= \int |g|^2 e^{-\varphi -\phi_{\varepsilon}}e^{-\phi+\phi_{\varepsilon}}&\leq \Big( \int |g|^{2p}e^{-p(\varphi + \phi_{\varepsilon})}\Big)^{1/p}\cdot \Big( \int e^{-q(\phi- \phi_{\varepsilon})}\Big)^{1/q},
\frac{[Ext^1_U(j_{*}\hat{T},\mathbb{G}_m)]}{[Ext^1_U(j_{*}\hat{T}',\mathbb{G}_m)]}=\frac{\prod_{v\in S_{\infty}}[H^0(K_v,\hat{T}'')]}{[\ker(\hat{\lambda}(H^0_c))]q(\lambda(O_{K,S}))}\frac{q(\hat{\lambda}(H^1_c))}{q(\hat{\lambda}(H^0_c))}.
v_{0}=\epsilon,v_{1}=\epsilon(2c+3),\v_{m+2}=(2c+2)v_{m+1}-v_{m},\m\geq0, & u_{0}=\epsilon,u_{1}=\epsilon(2c+1),\u_{m+2}=(2c+2)u_{m+1}-u_{m},\m\geq0.
p_{UV}((x^n, y^n), z^n)=p_{X^n, Y^n, Z^n}(x^n, y^n, z^n|X^n\in\mathcal{A}, Y^n\in \mathcal{B}) &u_1=(\mathbf{x}_1,\mathbf{y}_1) &u_2=(\mathbf{x}_2,\mathbf{y}_2).
\sum_{k=0}^{\infty}\sum_{r_1=1}^{n}\frac{a_{r_1}b_{r_1}c_{r_1}}{(r_1+k+1)^3}= \sum_{r_1=1}^{n}a_{r_1}b_{r_1}c_{r_1}\left(\zeta(3)-H^{(3)}_{r_1}\right);
\frac{\pi}{2}=\prod_{n=1}^{\infty}\frac{(2n)(2n)}{(2n-1)(2n+1)}=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot\frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdot\cdot\cdot
\beta(r)\equiv1-\frac{0.5\langle{V_{t}}^{2}(|r|)\rangle}{\langle{V_{r}}^{2}\rangle(|r|)}=1-\frac{\langle{V_{\theta}}^{2}(|r|)\rangle}{\langle{V_{r}}^{2}\rangle(|r|)}=-\frac{r^{2}}{r_{0}^{2}-r^{2}}\le0;
F(x)=\frac{1}{2}+\frac{\sqrt{q-1}\Gamma(\frac{1}{q-1})x\sqrt{\beta}{}_{2}F_{1}(\frac{1}{2},\frac{1}{q-1};\frac{3}{2};-(q-1)\beta x^{2})}{\sqrt{\pi}\Gamma(\frac{3-q}{2(q-1)})},
C_{n}\sim\frac{B(r)}{r^{\alpha}\Gamma(\beta)}n^{\beta-1}(\frac{1}{r})^{n}=\frac{-\frac{1}{2}}{(\frac{1}{4})^{1}\Gamma(-\frac{1}{2})}n^{-\frac{1}{2}-1}(\frac{1}{\frac{1}{4}})^{n}=\frac{4^{n}}{n^{\frac{3}{2}}\sqrt{\pi}}.