harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi (x_{1},x_{2})=-\dfrac{1}{2\pi}\sum_{k_{1}=1}^{+\infty}\dfrac{\cosh(\pi k_{1}(a-x_{2})/a)\cos (\pi k_{1}x_{1}/a)}{k_{1}\sinh (\pi k_{1})}-\dfrac{1}{12}+\dfrac{x_{2}}{4a}-\dfrac{x_{2}^{2}}{8a^{2}}
B_{II}^{(s, v)}(\kappa, z) = \frac{\partial}{\partial \kappa}A_{II}^{(s, v)}(\kappa, z) = \left| \boldsymbol{\phi}_{s, z}(\kappa) , \boldsymbol{\phi}_{s, z}'(\kappa), \cdots, \boldsymbol{\phi}_{s, z}^{(m-1)}(\kappa), \boldsymbol{\phi}_{s, z}^{(m+1)}(\kappa) \right|,
\sum_{{\bf q}_{k}}\frac{1}{\sqrt{2 E_{q_{k}}V}}e^{- i \overline{{Q}}_{k}\cdot z_{2}}\sum_{p_{j}}\frac{1}{\sqrt{2 E_{p_{j}}V}}e^{i \overline{{P}}_{j}\cdot z_{1}}\langle b_{-{\bf p}_{j}}^{\dagger}b_{-{\bf q}_{k}}\rangle
\Bigl ( [ \bar{\alpha}D , \alpha D ] - \alpha D \bar{\alpha}D + \frac{1}{2}( \alpha D )^{2}\bar{\alpha}^{2}- 2 \alpha^{2}\bar{\alpha}^{2}+ 2 ( 2 s + d - 3 ) \Bigr ) | \Phi \rangle = 0 ,
\Delta \varphi_{\nu+\mu}(z) = \frac{1}{2^{\nu+\mu}}\sum_{j=1}^{2^\nu}\sum_{k=1}^{2^\mu}\Delta_z\big[ \Phi(z,w_k^{(\mu)}(\nu,j;z))\big] \ge \frac{1}{2^{\nu+\mu}}\sum_{k=1}^{2^\mu}\Delta_z \big[\Phi(z,w_k^{(\mu)}(\nu,j_0;z))\big] \ge \frac{\tilde{L}}{2^\nu}\eqqcolon L
\|\tilde{g}_1\|_{C^{0,\gamma}(\mathcal{N}_{0}\cap (B_{1}\setminus B_{1/4}))}\leq C\lambda^{\frac{1}{2}-\alpha}\lambda^{1-\frac{n+1}{p}}\lambda^{-\frac{1}{2}}=C\lambda^{1-\alpha-\frac{n+1}{p}}.
{f}_1(\eta_1,u,x) = \kappa\eta_1^{\frac{1}{p}}+ \varphi(u,x),{f}_2(\eta_2,u,x) = \big\{\big[\max\{0,\varphi(u,x)- \eta_2\}\big]^p \big\},{f}_3(u,x) = \varphi(u,x),
H_{2,1}(F_{f}/2) =\dfrac{1}{48}\left(a^4_2 - 12 a^2_3 + 12 a_2 a_4\right) &=\dfrac{1}{36864}\left(-7c^4_{1}-8c^2_{1}c_2 -64c^2_{2}+96c_1 c_3\right).
{\tilde J}(\eta', p_\zeta):= \int_{|\eta| > R}d^3 \eta \int d^3 \zeta \; \frac{e^{-\sqrt{p_\zeta^2 + k_n^2}|\eta-\eta'|}}{|\eta-\eta'|}e^{-\tilde b_2 |m_{x\eta}\eta +m_{x\zeta}\zeta|}.
g'_1f'_1 = \begin{pmatrix}4 2 1 \end{pmatrix}, g'_1 f'_2 = \begin{pmatrix}12 6 1 \end{pmatrix}, g'_2f'_1 = \begin{pmatrix}12 2 1 \end{pmatrix}, g'_2f'_2 = \begin{pmatrix}4 6 1 \end{pmatrix}.
\mu_{\mathbb{C}}(z_1,z_2) = [(e^*_1(z_1) e^*_1(z_2) -e^*_2(z_1) e^*_2(z_2)] e_1 + [e^*_1(z_1) e^*_2(z_2)+e^*_2(z_1) e^*_1(z_2)] e_2 &= [(e^*_1(ae_1+be_2) e^*_1(ce_1+de_2) -e^*_2(ae_1+be_2) e^*_2(ce_1+de_2)] e_1 &\qquad+ [e^*_1(ae_1+be_2) e^*_2(ce_1+de_2)+e^*_2(ae_1+be_2) e^*_1(ce_1+de_2)] e_2 &=(ac-bd)e_1 + (ad+bc) e_2=(ac-bd,ad+bc) &=(ac-bd)+(ad+bc)i.
((1-q^{2d-2}) \chi_{\{P\}}+ \chi_{P^\sim}) ((1-q^{2d-2-s}) \chi_{H_s}+ (1-q^{s}) \chi_{H_s^\perp}+ \chi_{C(H_s)})^\top = (q^s-1)(q^{2d-2-s}-1)\frac{q^{2d-2}-1}{q^2-1}.
(A,b) \iff \exists \lambda, v v \neq 0, A v = \lambda v, b^\mathrm{T}v = 0 &\iff \exists \lambda, v v \neq 0, (A + \gamma b b^\mathrm{T})v = \lambda v, b^\mathrm{T}v = 0 &\iff (A + \gamma b b^\mathrm{T}, b) ,
E_1 := \{\mathbf{e}_q<\zeta\}\cup\{\tau_b^+<\zeta\}, E_2 := \{\mathbf{e}_q>\zeta,\tau_a^-<\zeta < \tau_b^+ \}, E_3 := \{\mathbf{e}_q>\zeta, \tau_0^- >\mathbf{e}_{r},\tau_a^- \wedge \tau_b^+>\zeta \}.
\pi_i(\boldsymbol{d}_i;\boldsymbol{d}_{-i}) = \sum_{t \in \mathcal{T}}{\left\{-p^{[t]}\!\!\left(\boldsymbol{d}^{[t]}\right) \cdot d_i^{[t]}{- c_{i}\left(d_i^{[t]}\right)}\right\}}
\frac{F_{\pi}^{2}}{1 6}\ \frac{1}{r}\, \, \partial_{r}\left( \, r F_{d}^{\prime}\right) = \frac{\partial \omega_{e}}{\partial F_{d}}\, \frac{\partial \rho_{m}}{\partial \omega_{e}}\quad \mathrm{f o r ~ d > 1 ~ .}
\sum_{l=0}^k a_l z^l \sum_{m=0}^{n-2}(-1)^{n-m-2}(m+2)(m+1) e_{n-m-2}z^m + \sum_{l=0}^{k-1}b_l z^l \sum_{m=0}^{n-1}(-1)^{n-m-1}(m+1) e_{n-m-1}z^m + \sum_{l=0}^{k-2}c_l z^l \sum_{m=0}^n (-1)^{n-m}e_{n-m}z^m = 0.
\Lambda^i_{d\leq r_0}(r_0) =\int^{r_0+d}_{r_0-d}\int^{\arccos \frac{r^2+r_0^2-d^2}{2r_0r}}_{-\arccos \frac{r^2+r_0^2-d^2}{2r_0r}}\mathrm{P}^i (r) \lambda_u d\theta rdr =2\lambda_u\int^{r_0+d}_{r_0-d}\mathrm{P}^i(r) r\arccos \frac{r^2+r_0^2-d^2}{2r_0r}dr.
\int_{\mathcal{O}_\delta^c}\frac{J_p(|x|^{-\beta}-|y|^{-\beta})}{|x-y|^{N+s\,p}}\,(\varphi(x)-\varphi(y))\,dy\,dx=2\,\int_{\mathbb{R}^N}\left(\int_{\mathcal{K}_\delta(x)^c}\frac{J_p(|x|^{-\beta}-|y|^{-\beta})}{|x-y|^{N+s\,p}}\,dy\right)\,\varphi(x)\,dx,
\left| \frac{v_1^{\sigma}(\phi_{\sigma}(x_n+x)-\phi_{\sigma}(x_n))e^{2t_n}+v_2^{\sigma}}{\|v^{\sigma}\|}\right| = \left|(\phi_{\sigma}(x_n+x)-\phi_{\sigma}(x_n))(\rho(\alpha\beta)^n)^{-1}\frac{v_1^{\sigma}}{\|v^{\sigma}\|}+ \frac{v_2^{\sigma}}{\|v^{\sigma}\|}\right|,
U_{1 .}= \left( \begin{array}{c c c c c c c}{{\sqrt{Q}}}\\{{2}}&{{0}}\\{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{\sqrt{Q}}}\\{{Q - \alpha}}&{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\ \end{array}\right)
V(z h_K ; 1) = \frac{1}{n}\int_{S^{n-1}}z h_K dS_K = \int_{S^{n-1}}z dV_K V(z^2 h_K ; 1) = \frac{1}{n}\int_{S^{n-1}}z^2 h_K dS_K = \int_{S^{n-1}}z^2 dV_K V(z h_K ; 2) = \frac{1}{n}\int_{S^{n-1}}z h_K \SS(z h_K, h_K,\ldots,h_K) d\theta = \int_{S^{n-1}}(\tilde{L}_K z) z dV_K .
B_{p}^{m}=\frac{\Gamma( \frac{n}{2})\Gamma(m+1)\Gamma(m+\frac{n}{2}+\alpha)\Gamma(\frac{n}{p}+\alpha)}{2\pi^{n/2}\Gamma(m+\frac{n}{p}+\alpha+\frac{n}{2})}\times \left(\frac{n(n+m-1)\Gamma(\frac{n}{2})\Gamma(pm-n+1)}{2\Gamma(pm-\frac{n}{2}+1)}\right)^{1/p}.
{\cal A}( p_{1}, p_{2}, p_{3}, p_{4}) ~ = ~ V_{C K G}^{- 1}\int \prod_{i = 1}^{4}d z_{i}~ <{V_{F}( 1 ) ~ V_{B}( 2 ) ~ [ V_{B}( 3 ) ~ V_{F}( 4 ) ~ + ~ V_{B}( 4 ) ~ V_{F}( 3 ) ]}> ~ ,
^{+}S_{2}= \frac{i}{2}{^+ \tau}\int_{M}e F^{\mu \nu i}F^{\alpha \beta j}g_{a b}\,{\cal R}_{\mu \nu}^{a b}\,{\cal R}_{\alpha \beta}^{c d}\, \epsilon_{a b c d}- \frac{1}{2}{^+ \tau}\int_{M}e F^{\mu \nu i}F^{\alpha \beta j}g_{i j}\,{\cal R}_{\mu \nu}^{a b}\,{\cal R}_{\alpha \beta}^{c d}\delta_{a b , c d}\, .
p\left({{d_{pF}}\leq \delta}\right) = \frac{M}{{k!}}\int_{{{\mathbf{\Omega}}_1}}\prod\limits_{i = 1}^k{{{\left({{{\sin}^2}{\theta_i}}\right)}^{n - 2k}}}\,\,\, \times \prod\limits_{1 \leq i < j}^k{{{\left({{{\sin}^2}{\theta_i}-{{\sin}^2}{\theta_j}}\right)}^2}}d{{\sin}^2}{\theta_1}\cdots d{{\sin}^2}{\theta_k}
\widehat{q}_{y}(\textbf{x}) = \sum_{j:Y_j=y}\delta_{j}K\left( \dfrac{\textbf{X}_j - \textbf{x}}{h_{n}}\right) \Big/ \sum_{j:Y_j=y}K\left( \dfrac{\textbf{X}_j - \textbf{x}}{h_{n}}\right),
\Sigma_{n,i}=2q^{2i+2}\left[\mu_{n-i-1}(q^{2})+2\frac{q^{4}\mu_{n-i-3}(q^{2})+q^{2}-1}{q^{2}-1}\right]\;, \overline{\alpha_{n,i}}=\lim_{q\to\infty}\frac{B''-\sqrt{B''^{2}-4\delta q^{2n-1}(q^{2}-1)C''}}{q^{2}-1}\; ,
\mathbf{f}:=\left( \mathbf{x};\mathbf{y};\mathbf{z}\right):=\left( x_{1},\thinspace \mathellipsis \thinspace ,\thinspace x_{m};y_{1},\mathellipsis \thinspace ,\thinspace y_{n};\thinspace z_{1},\thinspace \mathellipsis \thinspace ,\thinspace z_{p}\right)\in R^{m+n+p}.
{\mathbb E}\,{\rm e}^{-\alpha S_{N_{\beta,\omega}}-\gamma U_{N_{\beta,\omega}}}&= \exp\left[-\int_0^\infty \int_{\mathbb R}\frac{1}{t}{\rm e}^{-\beta t}(1-{\rm e}^{-\omega t})(1-{\rm e}^{-\gamma t-\alpha x})\,{\mathbb P}(Y(t)\in{\rm d}x)\,{\rm d}t\right].
d^2\!f(\overline{x}|\overline{v})(w)\ge\sup_{(\xi,u)\in\Gamma_{\overline{x},\overline{v}}}\Big\{d^2\vartheta(F(\overline{x})|\xi)(dF(\overline{x})(w))+{\textstyle\sum_{i=1}^m}\xi_id^2F_i(\overline{x}|u_i)(w)\Big\}
V^{(2)}_2(r) = \lambda A - \frac{\lambda A}{1+\lambda r^2}+ \frac{\lambda B_1}{(1+\lambda r^2)^2}+ \frac{\lambda B_2}{(1+\lambda r^2)^3}+ \frac{\lambda B_3}{(1+\lambda r^2)^4}+ \frac{\lambda B_4}{(1+\lambda r^2)^5}
\mathcal{O}^-_i := \left\{\left((A^+,\underline{A}^+),(B^+,\underline{B}^-)\right), \left((A^+,\underline{A}^+),(B^-,\underline{B}^+)\right): A,B\in \binom{[n]}{k}\mbox{and}|A\cap B| = k-i \right\}.
\left(\sum_{a \in \mathcal{A}}P_A(a) P_{V_a}\right)\left(\sum_{b \in \mathcal{B}}P_B(b) P_{V_b}\right) = \sum_{(a,b) \in \mathcal{A}\times \mathcal{B}}P_A(a) P_B(b) P_{V_a}P_{V_b}.
{Q_2}(M,P_i,\tau)= \left\{e^{\frac{1}{24}E_2(\tau)A_3}\widehat{A}(TM,\nabla^{TM}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^{m-\frac{1}{2}}}(\widetilde{T_CM})\right]\right. &\left.\cdot\varphi(\tau)^{8}{\rm ch}(\mathcal{V}_i)\right\}^{(12)}.
\int_0^t\int_{\Omega}u_{ss}\psi_1\,dx\,ds-\int_0^t\int_{\Omega}\Delta u\psi_1\,dx\,ds-\int_0^t\int_{\Omega}\Delta u_s\psi_1\,dx\,ds=\int_0^t\int_{\Omega}|u|^p\psi_1\,dx\,ds.
O_{\texttt{b}}(\boldsymbol{\xi};q,q_0):=\prod_{1\leq j <k\leq n}(1-2q\cos(\xi_j-\xi_k) +q^2) (1-2q\cos(\xi_j+\xi_k) +q^2) &\times \prod_{1\leq j\leq n}(1-2q_0\cos(\xi_j) +q_0^2) .
\omega \mid U_{\pm}= \int_{N}\frac{- 2 i S_{0}}{\left( 1 + z_{\pm}( \vec{x}) \overline{{z}}_{\pm}( \vec{x}) \right)^{2}}\delta \overline{{z}}_{\pm}( \vec{x}) \wedge \delta z_{\pm}( \vec{x}) d \mu ( \vec{x}) .
T r \gamma_{1 , 3 , ( 0 , \pm 1 , 0 )}- \frac{1}{3}T r [ \gamma_{1 ,{\tilde{7}}_{3}, ( 0 )}\gamma_{W 4 ,{\tilde{7}}_{3}}^{\pm 1}] - \frac{1}{3}T r \gamma_{1 , 7_{4}, ( \pm 1 )}+ \frac{1}{3}T r [ \gamma_{1 , 7_{5}, ( 0 )}\gamma_{W 4 , 7_{5}}^{\pm 1}] = 0
\phi = \left[ \frac{( \lambda / 6 \kappa^{- 2}l^{- 1})^{2}- \Lambda_{4}l^{2}/ 3 - 1}{( \bar{\lambda}/ 6 \kappa^{- 2}l^{- 1})^{2}- \phi^{- 2}\Lambda_{4}l^{2}/ 3 - 1}\right]^{1 / 5}.
\widehat{\Phi}_e *\widehat{\Phi}_e=\lim_{\epsilon\to 0}a_+K^{(3)}|f^{(3)\prime}_1(i+\epsilon)|^{\frac{1}{4}}\left(e^{-\frac{1}{2}\rho(i)}e^{-\frac{1}{2}\bar{\rho}(i)}\cdot e^{\frac{1}{2}\rho(i-\epsilon)}e^{\frac{1}{2}\bar{\rho}(i-\epsilon)}\right)\widehat{\Phi}_e.
A_1=\sum^{2m}_{\stackrel{i=0}{i\even}}(\mathcal{R}^{m,\frac{2m-i}{2}}_{\frac{2m-i}{2},\frac{2m-i}{2}}+\mathcal{L}^{m,\frac{2m-i}{2}}_{\frac{2m-i}{2},\frac{2m-i}{2}})+\sum^{2m}_{\stackrel{i=0}{i\odd}}(\mathcal{R}^{m,\frac{2m-1-i}{2}}_{\frac{2m-1-i}{2},\frac{2m+1-i}{2}}+\mathcal{L}^{m,\frac{2m-1-i}{2}}_{\frac{2m+1-i}{2},\frac{2m-1-i}{2}})
L^2(J;V) =\{v:J\to V \mid \|v\|_{L^2(J;V)}<\infty\}, \|v\|_{L^2(J;V)}^2=\int_J\|v\|^2_V~dt, H^1(J;V') =\{v:J\to V \mid \|v\|_{H^1(J;V')}<\infty\}, \|v\|_{H^1(J;V')}^2=\int_J (\|v\|^2_{V'}+\|v'\|^2_{V'})~dt.
\begin{array}{l}{\inf}\; \mathbb{E}_{\xi_2,\ldots,\xi_{T_{\max}},D_2,\ldots,D_{T_{\max}}}\Big[ \displaystyle{\sum_{t=1}^{T_{\max}}}\; D_{t}f_t(x_t, x_{t-1}, \xi_t ) + ( D_{t-1}- D_t ){\overline{f}}_t(x_t, x_{t-1}, \xi_t )\Big] x_t \in X_t( x_{t-1}, \xi_t )\;\mbox{a.s.}, x_{t}\;{\overline{\mathcal{F}}}_t\mbox{-measurable,}t=1,\ldots,T_{\max},\end{array}
q^n-\left(4r^2\left\lceil\frac{\deg d}{2}\right\rceil + 2r\left\lceil\frac{\deg d}{2}\right\rceil + 2r + 5\right)q^{n/2}- \left(4r^2\left\lceil\frac{\deg d}{2}\right\rceil + 2r\left\lceil\frac{\deg d}{2}\right\rceil\right)=0,
\Phi_1^{(n)}(m;p_1, p_2\ldots p_{n-1};q;z_1,z_2,\ldots z_n) =\frac{\Gamma(q)}{\Gamma(m)\Gamma(q-m)}\int_0^1v^{m-1}(1-v)^{q-m-1}& \times\prod_{i=1}^{n-1}(1-v\,z_i)^{-p_i}\,\exp(v\,z_n)dv,
A\otimes B:= \begin{bmatrix}a_{11}B a_{12}B \cdots a_{1n}B a_{21}B a_{22}B \cdots a_{2n}B \vdots \vdots \vdots a_{m1}B a_{m2}B \cdots a_{mn}B \end{bmatrix}\in \mathbb R^{mp\times nq}.
|I_1| \leq(d-1) \, \|x\widetilde f_{R,\delta}\| \left\|\frac{u_{R,\delta}}{|x|}\right\| = (d-1) \, \|x\widetilde f_{R,\delta}\| \left\|\frac{u^-_{R,\delta}}{|x|}\right\| \leq\frac{2(d-1)}{d-2}\left(\Lambda\,\|\nabla u^-_{R,\delta}\|^2 +\epsilon^2(R) \, \|\nabla u^-_{R,\delta}\|\right) \,.
\ \ \ \ \ \ \ \ \ \ \ \int_{\textstyle\prod_{t=1}^m [b_{t,1}, b_{t,2}]}\ \frac{du_1du_2\ldots du_m}{\displaystyle \sum_{t=1}^m\left(u_t-c_t\right)^{2s}\left(u_t-d_t \right)^{2s}}\ \ \ \ \ \le M\Bigl(\max\Big\{b_{1,2}-b_{1,1},b_{2,2}-b_{2,1},\ldots,b_{m,2}-b_{m,1}\Big\}\Big)^{m-4s}.\hskip 2.6cm\square
\lim_{p\rightarrow\infty}\Lambda_{p}(B_{R})^{\frac{1}{p}}=\lim_{p\rightarrow\infty}\left( \frac{N\omega_{N}}{R^{p-N}}\right)^{\frac{1}{p}}\left(\frac{p-N}{p-1}\right)^{\frac{p-1}{p}}=\frac{1}{R}=\frac{1}{\left\Vert\rho\right\Vert_{\infty}}
S_2 = \sum_{s=0}^{p'}v^{2s(a+2s)}\begin{bmatrix}p' s\end{bmatrix}_{v^2}\prod_{k=1}^{p'-s}(1-v^{2a+4s+4k}) \cdot (1-v^{2a+ 4p}) =1-v^{2a+ 4p}.
\theta_n(\lambda)(1-\theta_n(\lambda))\leq \frac{a}{n}\sum_{i\geq 1}\sum_{k=1}^{n}\bigotimes_{j\geq 1}P_j\left(E(c_i,S_k,\eta) \right)\mbox{Inf}_{{C}_i}^{\eta}(1_{B_n^{{\eta}}}) +O(\eta).
{\frac{{d^{2}H_C}}{{dz^{2}}}}+\left( \alpha +{\frac{{\gamma+1}}{{z-1}}}+{\frac{{\beta+1}}{{z}}}\right){\frac{{dH_C}}{{dz}}}+\left({\frac{{\mu}}{{z}}}+{\frac{{\nu}}{{z-1}}}\right)H_C =0,
L^{( - )}= \left( \begin{array}{c c}{{( - 1 )^{F}q^{- \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right)}}}&{{0}}\\{{( q^{- 1}- q ) ( - 1 )^{F}Y^{-}}}&{{( - 1 )^{F}q^{- \left( \begin{array}{c c}{{0}}&{{0}}\\{{0}}&{{1}}\\ \end{array}\right)}}}\\ \end{array}\right)
D_{\nu}( z ) = \sqrt{{\frac{2}{\pi}}}2^{\nu}e^{-{\frac{\pi}{2}}\nu i}e^{{\frac{z^{2}}{4}}}\int_{- \infty}^{\infty}x^{\nu}e^{- 2 x^{2}+ 2 i x z}d x , \mathrm{R e}\nu > - 1 .
\tilde{p}(t,z)=\tilde{p}(0,z) \exp(-\int_0^t \frac{a_0(s,z)}{b_0(s,z)}dB(s) - \frac{1}{2}\int_0^t (\frac{a_0(s,z)}{b_0(s,z)})^{2}ds),
\left[ \mu \frac{\partial}{\partial \mu}~ + ~ \sum_{\kappa = 0}^{\infty}\tilde{\beta}_{( \kappa )}( g_{\alpha}) \frac{\partial}{\partial g_{\kappa}}~ + ~ \tilde{\gamma}_{m}( g_{\alpha}) m \frac{\partial}{\partial m}~ + ~ \frac{n}{2}\tilde{\gamma}( g_{\alpha}) \right] G^{( n )}( p , m , \mu , g_{\beta}) ~ = ~ 0
\sum_{j=2}^{m}\frac{s_{j+1}-s_{j}}{2\pi i}\big( \Psi_{j,11}\partial_{s_{k}}\Psi_{j,21}-\Psi_{j,21}\partial_{s_{k}}\Psi_{j,11}\big) = \sum_{j=2}^{m}\beta_{j}\partial_{s_{k}}\log \frac{\Gamma(1+\beta_{j})}{\Gamma(1-\beta_{j})}.
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) E_{5}\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{3}\log(t)}+\frac{C \sup_{x \geq t}\left(\frac{x|e'''(x)|}{\lambda(x)^{3-2\alpha}}\right)}{\log^{(2-2\alpha)b}(t)t}
\phi\left[ \begin{array}{cc}A M 0 B \end{array}\right]=\left[ \begin{array}{cc}f_{11}(A)+g_{11}(M)+h_{11}(B) f_{12}(A)+g_{12}(M)+h_{12}(B) 0 f_{22}(A)+g_{22}(M)+h_{22}(B) \end{array}\right].
\mbox{LHS}=\sum_{a=0}^{p-1}\sum_{b=0}^a\sum_{c=0}^b\binom{b}c^2\binom{a}b^2=\sum_{a=0}^{p-1}\sum_{b=0}^a\binom{2b}b\binom{a}b^2=\sum_{b=0}^{p-1}\binom{2b}b\sum_{a=b}^{p-1}\binom{a}b^2.
\det(A_{\ell\rightarrow-b})=\sum_{k=1}^d (-b_{j_k})(-1)^{j_k+\ell}A_{(\{j_k\},\{\ell\})}+\!\!\!\sum_{k=m-m_0+1}^m(-b_k) (-1)^{k+\ell}A_{(\{k\},\{\ell\})}.
G^{(1)}_{\mathcal R}(x,y) = &\frac{e^{ik_{+}\vert x \vert}}{\sqrt{\vert x \vert}}\frac{e^{i\frac{\pi}4}}{\sqrt{8\pi k_{+}}}\left(\frac{2\cos^2\theta_{\hat x}-1-n^2}{n^2-1}\right)e^{-i k_{+}\vert y \vert\cos(\theta_{\hat x}{+}\theta_{\hat y})}+ G^{(1)}_{\mathcal R,Res}(x,y)
Q(z,u,v)=exp(\frac{v}{u}(\frac{zu}{1}+\frac{z^{2}u^{2}}{2}+\frac{z^{3}u^{3}}{3}+\frac{z^{4}u^{4}}{4}+\frac{z^{5}u^{5}}{5}+\cdot\cdot\cdot))
\frac{G'(s)}{G(s)}=5\log(p)-8\log(\pi)+\frac{3}{2}\psi\left(\frac{s}{2}\right)+\frac{3}{2}\psi\left(\frac{s+1}{2}\right)+2\psi\left(\frac{s+k-1}{2}\right) &+ 2\psi\left(\frac{s+k-1}{2}\right) + \frac{1}{2}\psi\left(\frac{s+2k-2}{2}\right)+\frac{1}{2}\psi\left(\frac{s+2k-1}{2}\right).
\tau_{S}^\star:=\inf \{t \geq S:\,\, V_t=h_1(t,V_t,\mathbb{P}_{V_t})\}\,\,\, \,\,\,\sigma_{S}^\star:=\inf \{t \geq S:\,\, V_t=h_2(t,V_t,\mathbb{P}_{V_t})\}.
< \tilde{T}_{J , J}\tilde{W}_{J_{1}, m_{1}}\prod_{i = 2}^{N}\tilde{T}_{J_{i}, J_{i}}> = \frac{( J_{1}- m_{1}) ( J + J_{1}- 1 )}{J_{1}}< \tilde{W}_{J + J_{1}- 1 , J + m_{1}}\prod_{i = 2}^{N}\tilde{T}_{J_{i}, J_{i}}>
{\Delta}S = S_F - S_I = 4{\pi}(R^2 - C^2){\mu}-{\pi}^{2}R^{3}\Big(1 - \frac{2}{\pi}(\frac{-C}{2R}{\sqrt{1 - C^{2}/R^2}}+ \frac{cos^{-1}(C/R)}{2}) \Big){\sigma}
\mathbf{m}^{1}=\overline{\mathbf{m}}^{1},\mathbf{m}^{2}=\overline{\mathbf{m}}^{2}\mathbf{m=\mathbf{m}^{1}+i\mathbf{m}^{2},}d\mathbf{m}^{1}=1,d\mathbf{m}^{2}=0.
\textrm{Pic}^0_{C_{K}/K}(\overline{k})\cong \mu^{Z^K/Y^K}(\overline{k})=\frac{\mu^{Z^K}(\overline{k})}{\mu^{Y^K}(\overline{k})}=\dfrac{\left(\overline{k}\otimes_k \frac{\mathcal{O}_{\mathbb{P}^1_{K}, \infty}}{\mathfrak{C}}\right)^*}{\left(\overline{k}\otimes_k \frac{\mathcal{O}_{C_K, P_\infty}}{\mathfrak{C}}\right)^*}.
\mathcal{L}_5:=\Phi_2^{-1}\mathcal{L}_4 \Phi_2=\Pi_S^{\perp}(\mathcal{D}_{\omega}+m_3 \partial_{xxx}+(\tilde{d}_1+\varepsilon^2 c(\xi))\,\partial_x + R_5)\Pi_S^{\perp}, &R_5:=(\Phi_2^{-1}-\mathrm{I})\Pi_S^{\perp}(\tilde{d}_1+\varepsilon^2 c(\xi))\partial_x+\Phi_2^{-1}\Pi_S^{\perp}\tilde{R}_5.
A = \pm \prod_{p \in{\cal P}_3}p^{a_p-b_p+\sum_{i=1}^n (\alpha_i-\beta_i) z_{ip}^0}, B = \pm \prod_{p \in{\cal P}_2}p^{b_p-c_p+\sum_{i=1}^n (\beta_i-\gamma_i) z_{ip}^0}, C = \pm \prod_{p \in{\cal P}_1}p^{c_p-a_p+\sum_{i=1}^n (\gamma_i-\alpha_i) z_{ip}^0}.
{R}_{1 2}( u - v ) Z_{2 1}^{- 1}( u , v ) T_{1}( u ) Z_{1 2}^{- 1}( v , u ) T_{2}( v ) = Z_{1 2}^{- 1}( v , u ) T_{2}( v ) Z_{2 1}^{- 1}( u , v ) T_{1}( u ){R}_{1 2}( u - v ) . \ll{b g q y b e l}
C = 2\gamma-2\sum_{p}\frac{\log p}{p^2-p+1}\ \ + \sum_{k=1}^{\infty}\frac{\mu(k)C_1(k)}{k^2}\left(-2\sum_{p|k}\frac{(p-1)p\log p}{p^2-p+1}+\log\left(\frac{(k')^2}{k}\right)\right).
G_{\{\sigma \}}( t , t^{\prime}) = \sum_{\Gamma}\frac{\exp \pi i [ \Omega_{\Gamma}( \{\sigma_{s}\}) + \sum_{s}2 l_{1 s}\sigma_{s}( J_{s}( z ) - J_{s}( z^{\prime}) ) ]}{[ z - g_{\Gamma}( z^{\prime}) ] Q_{\Gamma}^{3}( z^{\prime})}
A_{L , N^{\prime}, N^{\prime \prime}}^{( c / 2 \mu )}= \left[ 2^{L + 1}\Gamma ( L + 1 ) \right]^{2}\sqrt{\frac{n^{\prime}! N^{\prime}}{2 \pi \Gamma ( N^{\prime}+ L + 1 )}}\sqrt{\frac{n^{\prime \prime}! N^{\prime \prime}}{2 \pi \Gamma ( N^{\prime \prime}+ L + 1 )}}B_{L , N^{\prime}, N^{\prime \prime}}^{( c / 2 \mu )}
T_{1}X_{1}T_{3}X_{1}T_{1}= - 4 q^{2}\, \partial_{\mu}\phi_{c l}\, \partial^{\nu}\phi_{c l}\, T_{1}, \qquad T_{3}X_{1}T_{1}X_{1}T_{3}= - 4 q^{2}\, \partial_{\mu}\phi_{c l}\, \partial^{\nu}\phi_{c l}\, T_{3}\quad \quad \quad
M(a_0, \ldots, a_n) :=\left[ \begin{array}{cc}a_0 1 1 0 \end{array}\right]\left[ \begin{array}{cc}a_1 1 1 0 \end{array}\right]\cdots\left[ \begin{array}{cc}a_n 1 1 0 \end{array}\right] =\left[ \begin{array}{cc}p_n p_{n-1}q_n q_{n-1}\end{array}\right] .
\hat{H}( t ) = \int_{0}^{\infty}{\frac{d p}{2 \pi}}\Big ( \hat{\Pi}^{\dagger}( p ) \hat{\Pi}( p ) + ( m^{2}( t ) + p^{2}) \; \hat{\Phi}^{\dagger}( p ) \hat{\Phi}( p ) \Big ) \; ,
\begin{array}{l}{{\phi_{k}( \chi ) = \displaystyle \frac{1}{\omega_{n - 1}}\int_{u u^{\prime}= 1}a_{k}( u ) P_{\alpha_{k}( m_{0})}( \chi , u ) \dot{u}~ ,}}\\{{\psi_{k}( \chi ) = \displaystyle \frac{1}{2 \pi}\int_{u u^{\prime}= 1}b_{k}( u ) P_{\alpha_{k}( m_{0})}( \chi , u ) \dot{u}~ .}}\\ \end{array}
\left|\mathbb{E}\left(\widehat{\alpha}_{t,\varepsilon}-\widehat{\alpha}_{t,\eta}\right)^n\right| \leq\frac{1}{(2\pi)^n}\int_{E^{n}}\int_{\mathbb{R}^n}\prod_{i=1}^n\left|e^{-\frac{\varepsilon}2|p_i|^2}-e^{-\frac{\eta}2|p_i|^2}\right|\prod_{j=1}^\infty(1+4\lambda_{j}^2)^{-\frac14}dpd\ell.
\phi\big(s,t,\theta_h \omega,x_0\big) = \phi\big(\tau_{N-1},\tau_N,\theta_h \omega, \cdot\big) \circ \cdots \circ \phi\big(\tau_{1},\tau_0,\theta_h \omega, x_0\big) = \phi\big(\tau_{N-1}+ h,\tau_N + h, \omega, \cdot\big) \circ \cdots \circ \phi\big(\tau_{1}+ h,\tau_0 + h, \omega, x_0\big) = \phi\big(s + h,t + h,\omega,x_0\big).
\Vert (B_\varepsilon (\vartheta )+\lambda_0 Q_0^\varepsilon )^{-1}-(B^0(\vartheta )+\lambda_0 \overline{Q_0})^{-1}-\varepsilon \mathcal{K}(\varepsilon ;\vartheta)\Vert_{L_2(\mathbb{R}^d)\rightarrow H^1(\mathbb{R}^d)}\leqslant \widehat{C}_3\varepsilon , 0 < \vartheta \leqslant 1,0< \varepsilon \leqslant \vartheta^{-1},
S_{2}= \left( \begin{matrix}{{\cos ( \psi / 2 )}}&{{- \sin ( \psi / 2 )}}\\{{\sin ( \psi / 2 )}}&{{\cos ( \psi / 2 )}}\\ \end{matrix}\right) \left( \begin{matrix}{{e^{\lambda / 2}}}&{{0}}\\{{0}}&{{e^{- \lambda / 2}}}\\ \end{matrix}\right) \left( \begin{matrix}{{\cos ( \psi / 2 )}}&{{\sin ( \psi / 2 )}}\\{{- \sin ( \psi / 2 )}}&{{\cos ( \psi / 2 )}}\\ \end{matrix}\right) ,
\partial_{\alpha}\, ( \, \Gamma^{\, \alpha , \, \alpha_{0}}( \Phi_{c}) - \frac{1}{2}< \Phi_{c}, \{\tilde{C}_{\alpha}^{\, \alpha_{0}}\}^{\, - 1}\Phi_{c}> ) = [ \partial_{\alpha}\, \tilde{\Delta}( \alpha , \alpha_{0}) \, ] \L^{\, \alpha , \, \alpha_{0}}( \Phi ) \, \vert_{\, \Phi = \tilde{C}_{\alpha}^{\, \alpha_{0}}\! J ( \Phi_{c})}\quad .
f_{\theta_1,\theta_2}(x,y;t) ={\cal B}_{\theta_1,\theta_2}(y)\Big \{1 + \sum_{n=1}^\infty e^{-n(n+\theta -1)t/2}{\widehat{P}}^{(\theta_1,\theta_2)}_n(x){\widehat{P}}^{(\theta_1,\theta_2)}_n(y)\Big \},
Z_N(\zeta)=\zeta\left(\prod_{i=1}^mK_{N,L_i}\right)\mathrm{det}\left[\left[\zeta^2\alpha_{2j-1,2l}+\beta_{2j-1,2l}\right]\left[\mu_{2j-1}\right]\right]_{\substack{j=1,\ldots,(N+1)/2, l=1,\ldots,(N-1)/2}}.
\int_M\langle\nabla f_i, \nabla f\rangle d\mu=\lambda_1(M, \mu)\int_Mf_ifd\mu=\lambda_1(M, \mu)\int_Mf_i^2d\mu = \int_M\langle\nabla f_i, \nabla f_i\rangle d\mu, \,\,\,\, i=0,1.
\mathbb{Z}_2^{[\delta\times k]}\coloneqq\left\{A = (A_{i,j}) \in \mathbb{Z}_2^{N\times k}\left|\begin{array}{l}\sum\limits_{j = 1,\ldots, k}A_{i,j}=\delta_i\,(i\in[N]), \sum\limits_{i = 1,\ldots, N}A_{i,j}= 1 \,\, (j\in[k])\end{array}\right.\right\}.
\Omega_{s_{f}}:= \left\{\omega\in\Omega:\exists\gamma\sqsubset G\left(\omega\right) \left( \overline{\partial}\gamma\right)^{\ast}\supset s_{f}\ ;\ \omega_{e}=1,\right. & \left. \forall e\in\left\{e_{-}\left( s_{f}\right) ,e_{-}\left(s_{f}\right) +u\right\}\right\}\ ,
\left(1-\eta^2\right)^n\alpha^{t-\left(1-\eta^2\right)n}\geq \left(1-\eta^2\right)^n \alpha^{\eta^2 n}= \left(1-\eta^2\right)^n e^{-\log(1/\alpha)\eta^2 n}\geq \left(1-\eta^2\right)^n \left(1 - \log\big(\tfrac{1}{\alpha}\big) \eta^2\right)^n \geq \left(1-\eta\right)^n.
[\mathcal{O}_{X(i,j) \cup X(i-1,j-1)}] = [\mathcal{O}_{Y_{h,p}^h}\oplus \mathcal{O}_{Y_{h-1,p}^{h-1}}] - [\widetilde{S \otimes T}_{/<I_1, I_2}] &= [\mathcal{O}_{Y_{h,p}^h}] + [\mathcal{O}_{Y_{h-1,p}^{h-1}}] - [\mathcal{O}_{Y_{h+1,p}^{h-1}}].
\|A_{j+1}\|' = \epsilon \|[\alpha, \sum_{i=0}^j A_i\beta B_{j-i}]\|'\le 2 \epsilon \sum_{i=0}^j \|\alpha\|'\| A_i\|' \|\beta\|'\|B_{j-i}\|' &= 2\epsilon \|\alpha\|'^2 \sum_{i=0}^j C_i C_{j-i}.
I(X_1^L=x_1^L ; Y_1^L ) = \sum_{y_1^L \in \mathcal{Y}^L}W^L(y_1^L | x_1^L) \log \frac{|\mathcal{A}| \W^L(y_1^L | x_1^L)}{\displaystyle \sum_{\tilde{x}_1^L \in \mathcal{A}}W^L(y_1^L | \tilde{x}_1^L)}.
\big|\bigcup\nolimits_{j}\partial T^l_{j}\big|_{\cal H}\le \big|\bigcup\nolimits_{j}\partial T^{l-1}_{j}\big|_{\cal H}+ \big|\partial X_l' \setminus \big(\bigcup\nolimits^{l-1}_{i=1}\partial X_i' \cup \bigcup\nolimits_{j}T_j^{0}\big)\big|_{\cal H}+ \tfrac{1}{2}\big|\partial X_l' \cap \bigcup\nolimits_{j}T^{0}_j\big|_{\cal H}.
|\Delta_y(z_1)-\Delta_y(z_2)| = |\Pi_{\theta(y,z_3)}(z_1-z_2)| &\le |\Pi_{\theta(y,x_Q)}(z_1-z_2)| + \|\Pi_{\theta(y,z_3)}-\Pi_{\theta(y,x_Q)}\||z_1-z_2| &\lesssim |\Pi_{\theta(y,x_Q)}(z_1-z_2)| + 2^{-m_j}2^{-m_j}&\le |\Pi_{\theta(y,x_Q)}(z_1)-\Pi_{\theta(y,x_Q)}(z_2)| + 2\cdot 2^{-m_{j+1}}.
[\begin{matrix}SV_{2C}&SV_{2Y}\\SV_{1C}&SV_{1Y}\end{matrix}][\begin{matrix}d\overline{C}\\d\overline{Y}\end{matrix}]=-[\begin{matrix}DV_{2C}+\lambda DV_{2Y}\\DV_{1C}+\lambda DV_{1C}\end{matrix}]dh,
|\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}ds \lambda''(s)K(s-t,\lambda(t))| \leq \frac{C}{t^{2}\log^{b+1}(t)}&|\frac{16}{\lambda(t)^{2}}\int_{t}^{\infty}dx \lambda''(x)\left(K_{1}(x-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+x-t)}\right)| \leq \frac{C}{t^{2}\log^{b+1}(t)}
\|\nabla h_3\|_{L^2}\leq \|h_3\|_{H^1}\leq C\|\Delta\kappa\|_{H^1}\| \partial_z h_2\|_{H^2}+C\|\nabla\kappa\|_{H^2}\| \nabla\partial_z h_2\|_{H^1}\leq C\|\kappa\|_{H^3}\| h_2\|_{H^3}&\leq Cc_{4}\|\Delta h_2\|_{H^1}=Cc_{4}\| \partial_yVf\|_{H^1}\leq Cc_{4}\|f\|_{H^1}\leq Cc_{4}\|\nabla f\|_{L^2},
( 2 L(q^{2}) - 7 L(q^{7}) )^{2}= \sum_{\delta|14}x_{\delta}M(q^{\delta}) + \sum_{j=1}^{4}y_{j}A_{j}(q) = \sum_{\delta|14}x_{\delta}+ \sum_{i=1}^{\infty}\biggl(\,240\,\sum_{\delta|14}x_{\delta}\sigma_{3}(\frac{n}{\delta}) + \sum_{j=1}^{4}y_{j}a_{j}(n)\,\biggr)\,q^{n}.
||fg||_{r^{\prime}}^{r^{\prime}}=||(fg)^{r^{\prime}}||_{1}\le||f^{r^{\prime}}||_{p/r^{\prime}}\cdot||g^{r^{\prime}}||_{q/r^{\prime}}=||f||_{p}^{r^{\prime}}\cdot||g||_{q}^{r^{\prime}}
F(\kappa_1) = P(E_{\kappa_1}) - \kappa_1 |E_{\kappa_1}| \le P(E_{\kappa_2}) - \kappa_1 |E_{\kappa_2}| &< P(E_{\kappa_2}) - \kappa_2 |E_{\kappa_2}| = F(\kappa_2).
\begin{aligned}z=x_{\lambda}= \left[ \lambda |x|^{-\frac{n-p}{p}\frac{\theta}{\theta-1}}+(1-\lambda)R^{-\frac{n-p}{p}\frac{\theta}{\theta-1}}\right]^{-\frac{p}{n-p}\frac{\theta-1}{\theta}}\frac{x}{|x|}&=R\exp_{q}\left[\lambda \log_{q}\frac{|x|}{R}\right]\frac{x}{|x|}.\end{aligned}
|\alpha_1 + \rho \beta_1|^2 = |\rho \beta_1 + \mathrm{Re}(\alpha_1)|^2 + \mathrm{Im}(\alpha_1)^2 &\ge \rho^2 \beta_1^2 + |\alpha_1|^2 - 2 \rho \beta_1 |\mathrm{Re}(\alpha_1)| &\ge \beta_1^2 (\rho^2 + c_1^2 \lambda^{10}- 2 \delta/c_1 \lambda^5). &\ge \beta_1^2 (\rho^2 + (c_1^2 - 2 c/c_1) \lambda^{10}),