harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_y m(s,x,t,0)=-\partial_y g(s,x,t,0)=\frac{-2x}{\sqrt{2\pi (t-s)^3}}e^{-\frac{x^2}{2(t-s)}}\times \mathbb{E}\left[ \left (-\int_{s}^{t}\phi''(u)B(u)du \right ) \vert B(s)=-x,B(t)=0 \right] =2\Phi(s,x,t) \left (-\phi'(s)x + \frac{1}{2}\int_{s}^{t}(\phi'(u))^2 du \right )=:-2\Upsilon(t)
\Big\|{u\choose \dot{u}}-V_0(t){u_0^+\choose u_1^+}\Big\|_{\dot H^1\times L^2}=\Big\|\int_t^\infty V_0(t-s){0\choose F(u(s))}ds\Big\|_{\dot H^1\times L^2}&\lesssim\|(|u|^{\frac{4}{n-2}})u\|_{L_t^1L_z^2((t,\infty)\times X)}&\lesssim\|u\|_{L_t^{(n+2)/(n-2)}L_z^{2(n+2)/(n-2)}((t,\infty)\times X)}^{\frac{4}{n-2}}&\rightarrow 0,t\rightarrow\infty.
\dot{x}^{2}( \tau , 0 ) = A^{2}\frac{[ f - g ]^{2}}{4 \acute{f}\acute{g}}, \dot{x}^{2}( \tau , l ) = A^{2}\frac{[ f_{l}- g_{l}]^{2}}{4 \acute{f}_{l}\acute{g}_{l}}.
G(x,y) = \frac{e^{ik_{+}\vert x \vert}}{\sqrt{\vert x \vert}}G^{\infty}(\hat x, y) + G_{Res}(x,y), & \nabla_y G(x,y) = \frac{e^{ik_{+}\vert x \vert}}{\sqrt{\vert x \vert}}H^{\infty}(\hat x,y ) + H_{Res}(x,y),
K_{\Sigma_m}=-2H+(m-2)F=-\frac{m+2}{m}H-\frac{m-2}{m}E, K_{\P(1,1,m)}=-\frac{m+2}{m}H, e_{c_m}=\frac{(m-2)^2}{m}, K_{\Sigma_m}^2=8, dH K_{\Sigma_m}=d(m+2), \chi(\Sigma_m,dH)=(md+2)(d+1)/2.
V=\{v_{0}=(\begin{matrix}e_{0}^{0}\\ \vdots\\e_{0}^{m-1}\end{matrix}),v_{1}=(\begin{matrix}e_{1}^{0}\\ \vdots\\e_{1}^{m-1}\end{matrix}),...,v_{p-1}=(\begin{matrix}e_{p-1}^{0}\\ \vdots\\e_{p-1}^{m-1}\end{matrix})\}
e_{r}^{n}:= \left\|G( \phi^{n})-\mu^{n}\phi^{n}\right\|_{\infty}<\varepsilon_{r}, &d_{v}^{n}:= \frac{\left\|\phi^{n}-\phi^{n-1}\right\|_{\infty}}{\tau}<\varepsilon_{v}, &e_{c}^{n}:=|\|\phi^n\|^2-1|<\varepsilon_{c},
\Pi_{k\,\boldsymbol{\nu}}(x,x)_{Z_x}:=\sum_{g_j\in Z_x}\Pi_{k\,\boldsymbol{\nu}}(x,x)_j,\Pi_{k\,\boldsymbol{\nu}}(x,x)_{G_x\setminus Z_x}:=\sum_{g_j\not\in Z_x}\Pi_{k\,\boldsymbol{\nu}}(x,x)_j.
E = - \frac{1}{4 \pi \alpha '}\int^{2\pi}_0 d \sigma [ \partial_0 t + i f (z^i \partial_- \bar{z}^i - \bar{z}^i \partial_- z^i + 2 i \bar{\psi}^i_R \psi^i_R ) ] = - \frac{1}{2}\frac{l}{\alpha '}+ 2 f J_R ~.
\frac{|(\tilde a_i(t,y)- \tilde a_j(t,y)) - (\tilde a_i(0,y)- \tilde a_j(0,y))|}{|(\tilde a_i(0,y)- \tilde a_j(0,y)) |}= \frac{|u_{ij}(t,y) - u_{ij(}0,y)|}{|u_{ij}(0,y) |}\le C_{ij}\|t\|^{r_{ij}}
\begin{array}{c}{{( \dot{x}, \dot{y}) = i ( a \alpha \zeta^{\alpha - 1}, b \beta \zeta^{\beta - 1}) = ( A \zeta^{\alpha - 1}, B \zeta^{\beta - 1})}}\\{{( i \dot{y}^{*}, - i \dot{x}^{*}) = ( i B^{*}( \zeta^{*})^{\beta - 1}, - i A^{*}( \zeta^{*})^{\alpha - 1})}}\\ \end{array}
g_{1,2,3}=(1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)\,, g_{2,3}=(1,4,7)(2,5,8)(3,6,9)(10,13,16)(11,14,17)(12,15,18)(19,22,25)(20,23,26)(21,24,27)\,, g_3 =(1,10,19)(2,11,20)(3,12,21)(4,13,22)(5,14,23)(6,15,24)(7,16,25)(8,17,26)(9,18,27)\,.
\sum_{\substack{X^{\theta_2-\theta_1}\le p_1\le \dots\le p_\ell p_1\cdots p_\ell\le X^{1-\theta_1}}}^* S_{p_1\cdots p_\ell}(X^{\theta_2-\theta_1})=o_{\mathcal{L}}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr),
\widetilde{\Psi}\big(\delta,\mathcal{X},\lambda,\mathcal{Y}\big) \!:=\!\left(\begin{matrix}L_{\mathcal{X}}'\big(\mathcal{X},\lambda,\mathcal{Y}\big)-\delta_{\!f}h(\mathcal{X})-\delta_h G(\mathcal{X})-\delta_G-\Pi_{K}\big(G(\mathcal{X})-\delta_{G}+\mathcal{Y}\big) \end{matrix}\right).
\frac{\eta^{i'i''}-\sum_{m',m''=1}^\infty (R^{-1}_{m'})_{j'}^{i'}\eta^{j'j''}(R^{-1}_{m''})_{j''}^{i''}(\psi')^{m'}(\psi'')^{m''}}{\psi'+\psi''}e_{i'}\otimes e_{i''}
\int_\Omega \xi |u|^2 d^{-4j}\rho \, d\mu_g \le \left( \frac{4}{N}\right)^j \int_\Omega \xi | \nabla^j u|^2 \rho\, d\mu_g & + \sum_{i=1}^j \left(\frac{4}{N}\right)^{j+1-i}\sup_\Omega (|\nabla \xi| d^{-4i+2}) \| \nabla^{j-i}u \|_{L_\rho^2(\Omega)}^2.
\begin{aligned}\frac{k}{12(k+2)}n^2 \geq\frac{k}{12(k+2)}[6(k+1)(k+2) n] &= \frac{k(k+1)}{2}n = \binom{k+1}{2}n &\geq \binom{k+1}{2}(n-k - 1) + \binom{k+1}{3}\end{aligned}
f_{1}^{*}(x_{e})+f_{2}^{*}(x_{b}) \,=\, \zeta \,\rho(x_{i}) + \zeta \, \Delta x^{2}\left( \sigma_{1}\sigma_{2}-\frac{1}{8}\right) \, \frac{\partial^{2}\rho}{\partial x^{2}}(x_{i})+{\rm{O}}(\Delta x^{3}).
c_2^{12}(A) = \int_0^1 \left(\frac{1}{2\pi}-\frac{\sqrt{6}}{\pi}\frac{1+ \cos(2\pi t)}{5+\sin(2\pi t)}\right)\ln(5+\sin(2\pi t))\,\mathrm{d}t = 0.003\dots\neq 0,
{\cal L}^*f(x) = \frac{1}{2V}\sum_{i,j = 1}^d \frac{\partial^2}{\partial x_i \partial x_j}\left(\Gamma_{i,j}(x) f(x)\right) -\sum_{i = 1}^d \frac{\partial}{\partial x_i}\left(b_i (x) f(x)\right).
P_{\textrm{out,F}}^{\textrm{ISAOC}}= \textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{NHF=0,FDF=0,NDF=1,NDN=1}\right\rbrace &+\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{FDF=0}\right\rbrace \times \left(1- \textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{NDF=1,NDN=1}\right\rbrace\right).
(1-\alpha)\Big\langle \nabla \mathrm{H};\mathrm{e}^{(2)}_{\mathrm{n}}\Big\rangle_{\mathbb{L}^{2}(\Omega)}\ = \ \Big\langle \nabla \mathrm{H}^{\textbf{in}};\mathrm{e}^{(2)}_{\mathrm{n}}\Big\rangle_{\mathbb{L}^{2}\Big(\Omega \Big)}+ \Big\langle \mathbb{T}\Big[ \nabla \mathrm{H}\Big];\ \mathrm{e}^{(2)}_\mathrm{n}\Big\rangle_{\mathbb{L}^{2}(\Omega)}.
\begin{aligned}\mathcal{II}\leq \frac{|d_1-d_3|}{2}\|\Delta \psi_{\theta}\|_{L^{p_0'}(\Omega_{\tau,\tau+2})}\|\varphi_\tau(u_1 + u_3)\|_{L^{p_0}(\Omega_{\tau,\tau+2})}&\leq \frac{|d_1-d_3|}{d_1+d_3}C_{\rm{mr},p_0'}\| \theta\|_{L^{p_0'}(\Omega_{\tau,\tau+2})}\|\varphi_\tau(u_1 + u_3)\|_{L^{p_0}(\Omega_{\tau,\tau+2})}\end{aligned}
\frac{{n-u-1\choose n-k-1}}{{n-u-1\choose k-1}}\le \prod_{i=k}^{n-k-1}\frac{n-3-i}{n-1-i}\le \frac{(k-1)(k-2)}{(n-k-1)(n-k-2)}\ \ \ \ \ \ 3\le u\le k.
j(E) = \frac{2^4 \cdot 5 \cdot 13^4 \cdot 17^3}{3^{13}}, j(E) = -\frac{2^{12}\cdot 5^3 \cdot 11 \cdot 13^4}{3^{13}}, j(E) = \frac{2^{18}\cdot 3^3 \cdot 13^4 \cdot 127^3 \cdot 139^3 \cdot 157^3 \cdot 283^3 \cdot 929}{5^{13}\cdot 61^{31}}.
\bar{A}_{\beta , 1}^{( 2 )}= \left. A_{\beta , 1}^{( 0 )}\right|_{{\cal C}_{\beta}}N^{( 2 )}( d - 2 ) \int_{\Sigma}+ \left. A_{\beta , 1}^{( 2 )}\right|_{{\cal C}_{\beta}}\int_{\Sigma}+ \left. A_{\beta , 1}^{( 1 )}\right|_{{\cal C}_{\beta}}( d - 2 ) \int_{\Sigma}.
\int |\chi_R(x-z) \nabla (u^{\xi}(t,x))|^2- \frac{3(p-1)}{2(p+1)}|\chi_R(x-z) u(t,x)|^{p+1}&\quad\quad\quad\quad\quad\quad\,+ \frac{3(q-1)}{2(q+1)}|\chi_R(x-z) u(t,x)|^{q+1}dx &= G(\chi_R(\cdot-z) u^\xi(t)) + O(R^{-2}) &\geq \delta \|\nabla(\chi_R(\cdot-z) u^{\xi}(t))\|^2_{L^2}+ O(R^{-2}).
T_{7}= ( \theta , \theta^{2}, \theta^{3}\vert \vert \theta , \theta^{2}, \theta^{3}\vert ( \textstyle{\frac{1}{7}})^{4}( \textstyle{\frac{2}{7}})^{4}( \textstyle{\frac{3}{7}})^{4}0^{4}) .
\theta(n_s^{-1}(\lambda))T_{n_s^{-1}}- T_{n_s^{-1}}\theta(\lambda) & = \sum_{k = 1}^n((n_s^{-2}(\lambda_s)^{k}\cdot (c'\lambda_s^{-1}n_s\lambda_s^{-1}n_s))\theta(n_s^{-2}(\lambda_s)^{k}n_s^{-2}n_s^{-1}(\lambda_s)^{-k + 1}) &+ (n_s^{-2}(\lambda_s)^{k}\cdot (n_s^{-2}c_{n_s}))\theta(n_s^{-2}(\lambda_s)^{k}n_s^{-1}(\lambda_s)^{-k}))\theta(\lambda).
(\nabla_{j}f)(y,m_{1},m_{2})=\frac{2\pi i}{\det\widetilde{T}}(C_{j1}y+C_{j2}m_{1}+C_{j3}m_{2})f(y,m_{1},m_{2})+\frac{C_{j4}}{\det\widetilde{T}}\frac{\partial f}{\partial y}(y,m_{1},m_{2}),
(\Phi_{ij}(x)_{i,j=1}^{n})^{*}(\Phi_{ij}(x))_{i,j=1}^{n}=(\varphi_{ij}\langle x,x\rangle)_{i,j=1}^{n}=(\Psi_{ij}(x)_{i,j=1}^{n})^{*}(\Psi_{ij}(y))_{i,j=1}^{n}
\begin{aligned}\frac{1}{2}\int_{\mathbb{R}^{N}}(\alpha u +\beta v)w dx \leqslant 6CK^{2}r_{0}^{1-N}+6CK+6CK^{2}+CK^{2}+6CKr_{0}^{\frac{1-N}{2}}\Vert f \Vert_{2}+6Cr_{0}\Vert f \Vert_{2}^{2}+CK^{\frac{4}{N+4}}\Vert f \Vert_{2}^{\frac{2N+4}{N+4}}+ 6C\sqrt{K}\Vert g_{1}\Vert_{2}+6C\sqrt{K}\Vert g_{2}\Vert_{2}.\end{aligned}
S_{l o c}~ = ~ \int_{d}\sqrt{g}\left( \Phi_{1}( \phi ) R + U ( \phi ) + \Phi_{2}( \phi ) R^{2}+ \Phi_{3}( \phi ) R_{\mu \nu}R^{\mu \nu}+ \Phi_{4}( \phi ) R_{\mu \nu \alpha \beta}R^{\mu \nu \alpha \beta}\right)
\mathbf{A}_{b}^{a}=\alpha_{b}^{a}-(\digamma_{b}^{a}-\mathfrak{f}_{b}^{a})\mathbf{m,}\mathbf{F}_{b}^{a}=\mathfrak{f}_{b}^{a}+D\mathfrak{f}_{b}^{a}\mathbf{m,}
\left( \begin{matrix}{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\star}}&{{\star}}&{{0}}&{{\star}}\\{{0}}&{{\star}}&{{\star}}&{{0}}&{{\star}}\\{{0}}&{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{\star}}&{{\star}}&{{0}}&{{\star}}\\ \end{matrix}\right)
\lim_{v\rightarrow 0+}u_{2i}(v) = \lim_{v\rightarrow 0+}\frac{\frac{1}{\theta}-\frac{1}{\sqrt{v^2+\theta^2}}}{v^{2}}= \lim_{v\rightarrow 0+}\frac{-v^2\left[\frac{g''(0)}{2!}+\frac{g^{(3)}(0)}{3!}v+\ldots\right]}{v^2}= -\frac{g''(0)}{2},
\tilde{f}_{1}^{e_{1}}\tilde{f}_{2}^{e_{2}}\tilde{f}_{3}^{e_{3}}= f_{1}^{e_{1}}f_{2}^{e_{2}}f_{3}^{e_{3}}f_{12}^{e_{1}+ e_{2}}f_{13}^{e_{1}+ e_{3}}f_{23}^{e_{2}+ e_{3}}f_{123}^{e_{1}+ e_{2}+ e_{3}}.
M_{n,\epsilon}= \big\{\max_{u\in[\epsilon,1]}X_{\hat{g},u}(z_1) \in [n-1,n]\big\},\,\,\,\,n\geq 1, M_{0,\epsilon}= \big\{\max_{u\in[\epsilon,1]}X_{\hat{g},u}(z_1) \leq 0\big\}.,
J_5 \geq -Cs^2 \lambda^4 \iint_Q \xi\left|A \nabla \eta \cdot \nabla \eta\right|^2 |u|^2dx dy d t -C\lambda^2\iint_{Q}\xi \left| A\nabla u\cdot \nabla \eta \right|^2 dx dy dt &-C s^2 \lambda^3 \int_0^T \int_{\omega_0}\xi|u|^2dx dy d t- C\lambda \iint_Q \xi A \nabla u \cdot \nabla u dx dy d t.
\left\lceil\frac{(m+n_1)n_2+q}{n_2(mn_1n_2+q)}\right\rceil=1, \left\lfloor\frac{(m+n_1)n_2+q+n_2(mn_1n_2+q)-1}{n_2(mn_1n_2+q)}\right\rfloor=1;
\int_{\mathcal{Y}^{k+1}}g(y_{[n,n+k]})P^{\mu}(dy_{[n,n+k]}|Y_{[0,n-1]}) &=\int_{\mathcal{Y}^{k+1}\times \mathcal{X}}g(y_{[n,n+k]})P^{\mu}(d(y_{[n,n+k]},x_{n})|Y_{[0,n-1]})
\frac{x_j^2 d(x_i/x_j)}{[a_1,\ldots,a_{n-2}]}=\pi^{*}\left( \pm \frac{x_j^2 d(x_i/x_j)}{[a_1,a_2,\ldots,\widehat{n},\ldots,a_{n-2}]}\right)
- \int\limits_{0}^{\infty}f_1(x \mid a_1, d_1, p_1) \, \left( \frac{x}{a_1}\right)^{p_1}\, dx = - \frac{\Gamma \left(\frac{d_1}{p_1}+ 1\right)}{\Gamma \left(\frac{d_1}{p_1}\right)}= - \frac{d_1}{p_1}.
X := E^\infty \cup \{\mu \in{\rm Path}(E) \mid r(\mu)\}\cup \{\mu \in{\rm Path}(E) \ \vert \r(\mu) \in \operatorname{Inf}(E)\}G_E := \{(\alpha x, |\alpha| - |\beta|, \beta x) \ \vert \ \alpha,\beta \in{\rm Path}(E), x \in X, r(\alpha) = r(\beta) = s(x)\}.
L^{K r}= \sum_{i}p_{i}E_{i i}- \frac{\nu}{2 \pi i}\sum_{i \neq j}\frac{\sigma ( z + \frac{\alpha_{i j}( D )}{\pi k}{\mathrm{I}m \tau})}{\sigma ( z ) \sigma ( \frac{\alpha_{i j}( D )}{\pi k}{\mathrm{I}m \tau})}E_{i j}
(1-\gamma_{n+3})\left(\displaystyle\frac{\gamma_{n+3}}{b}-1\right)=\displaystyle\frac{m\gamma_{n+2}}{\gamma_{n+1}+a\gamma_{n+2}}>\displaystyle\frac{m}{1+a}=(1-u_2^*)\left(\displaystyle\frac{u_2^*}{b}-1\right),
|\mathcal{A}| = 2^{\frac{8(8+1)}{2}}\cdot \prod_{i=1}^5 \left( \prod_{j=2}^{s_i}\left( 2^j-1 \right) \right) = 2^{36}\cdot (3) \cdot (3 \cdot 7) = 63 \cdot 2^{36}
r_{T_0+1}\leq r_1 + \sum_{t=1}^{T_0}\left(\frac{16}{\alpha^2T_0}\left(bLh_t^{\beta-1}\right)^2 + \frac{64}{\alpha^2 T_0^2}\left({\sf V}_2\bar{L}^2h_t^{2}+{\sf V}_3\sigma^2h_t^{-2}\right)\right)\,.
\sum_p \frac{1 - \operatorname{Re}( g(p) \overline{\chi(p)}p^{-it})}{p}\geq \sum_{t_m < p \leq t_{m+1}}\frac{1 - \operatorname{Re}( \overline{\chi(p)}p^{i(s_{m+1}-t)})}{p}\geq \sum_{\exp((\log t_{m+1})^{5/6}) < p \leq t_{m+1}}\frac{1 - \operatorname{Re}( \overline{\chi(p)}p^{i(s_{m+1}-t)})}{p}
\int_S\d\acute{\rho}:= \sum_{0\leq a < b \leq 3}\int_{I^2}\acute{\rho}_\sigma^{ab}(s,t)[\det \acute{J}_{ab}^\sigma(\hat{s})]\d\hat{s}, \int_S d|\acute{\rho}| := \int_{I^2}\left| \sum_{0\leq a < b \leq 3}\acute{\rho}_\sigma^{ab}(\hat{s})[\det \acute{J}_{ab}^\sigma](\hat{s})\right|\d\hat{s}.
\left| \int f d\mu - \int f d m\right| \le \sum_{n=0}^\infty \|f_n\|_\theta \left( \theta^{N+1}+\sqrt{2}\left(h_{m}(\sigma)-H_\mu(\xi|\xi_1^N)\right)^{\frac12}\right) & \le \frac{C}{1-\rho}\|f\|_\theta \left( \theta^{N+1}+\sqrt{2}\left(h_{m}(\sigma)-H_\mu(\xi|\xi_1^N)\right)^{\frac12}\right).
\int\limits_{\mathbb R^n_+}p^{z-I}h(q_p(x)) \, dp \, \Gamma(s) = x^{-z}\int\limits_{\mathbb R^n_+}p^{z-I}e^{-q(p)}\,dp \int\limits_0^\infty t^{s-1}h(s) \, ds.
\nabla_{\mu}\phi_{\mu_{1}\dots \mu_{k}}= ( \partial_{\mu}+ \omega_{\mu}) \phi_{\mu_{1}\dots \mu_{k}}- \sum_{i = 1}^{k}\Gamma_{\mu_{i}\mu}^{\lambda}\phi_{\mu_{1}\dots \mu_{i - 1}\lambda \mu_{i + 1}\dots \mu_{k}}
UW(f_1\oplus f_2,V_1\oplus V_2)U^* =W(f_1,V_1)\otimes W(f_2,V_2) Ud\Gamma (\omega_1\oplus \omega_2)U^* =d\Gamma (\omega_1)\otimes 1 +1\otimes d\Gamma ( \omega_2) U\varphi(f_1,f_2)U^* =\varphi(f_1)\otimes 1+1\otimes \varphi(f_2) Ua(f_1,f_2)U^* =a(f_1)\otimes 1+1\otimes a(f_2) Ua^{\dagger}(f_1,f_2)U^* =a^{\dagger}(f_1)\otimes 1+1\otimes a^{\dagger}(f_2).
I = \int_{R^{2}}d^{2}z \frac{( y_{2}\bar{y}_{2})^{\frac{1}{4}}}{\bar{y}_{2}}( y_{1}\bar{y}_{1})^{- \frac{1}{4}}e^{- | x_{\perp}| ( \sqrt{y_{1}\bar{y}_{1}}+ \sqrt{y_{2}\bar{y}_{2}})}
\mathcal{J}(x^{u,v}, u, v) &\geq\sum\limits_{i=1}^{m+1}\left[\Phi (x^{u,v}(t_i))+ \int_{t_{i-1}}^{t_i}\left\lbrace\varphi(t) + c_1\|x(t)\| + c_2\|u(t)\|_{Y}^{p}+ c_3\|v(t)\|_{Y}^{p}\right\rbrace dt\right] &\geq -\eta>-\infty,~ i=1,\ldots,m+1.
B(w,z)=\frac{1}{(q/z;q)_{\infty}}\left(1+\sum_{k=1}^{K(z)}\frac{(-1)^{k}q^{k(k-1)/2}w^{k}}{(q;q)_{k}(z;q)_{k}}\right)\!,\C(w,z)=\frac{\tilde{C}(w,z)}{B(w,z)},
W_{\varepsilon,\xi(y)}(x) =\gamma(\xi(y))U_{\varepsilon}(\sqrt{A(\xi(y))}\exp_{\xi(y)}^{-1}(x))\chi(\exp_{\xi(y)}^{-1}(x)) & =\tilde{\gamma}(y)U_{\varepsilon}(\sqrt{\tilde{A}(y)}\mathcal{E}(y,x))\chi(\mathcal{E}(y,x))
F(k) = \sum_{k=r\ell_1n_1\ell_2n_2}\mu^2(r\ell_1n_1)\mu^2(r\ell_2n_2) h(r) f_1(\ell_1)f_2(\ell_2) g_1(n_1)g_2(n_2) \prod_{p|(\ell_1,\ell_2)}t(p).
|I_3|/(-\log y) \le \int_1^T \dfrac{y^t\log t}{t}\,dt +\int_T^\infty \dfrac{y^t\log t}{t}\,dt &\le \dfrac{\log T}{1}\int_1^T y^t\,dt +\dfrac{\log T}{T}\int_T^\infty y^t\,dt &=\dfrac{\log T}{\log y}(y^T-y)-\dfrac{\log T}{T\log y}y^T,
f_1(z,\xi,w)=g_1(z,\bar z)+( a^-(z,\xi,w)-\hat a^-(z,\xi,w))\eta+wt(N+s), f_2(z,\xi,w)=g_2(z,\bar z,\xi)+wt(N+s), f_3(z,\xi,w)=(2\bar z+\xi)\bar g_1(\bar z,z)+\bar zg_2(z,\bar z,\xi)+wt(2N)+wt(N'+s).
\frac{s_{\lambda}(1^{\ell(\lambda)+1})}{\ell(\lambda)+1}\geq\frac{s_{\mu}(1^{\ell(\lambda)+1})}{\ell(\lambda)+1}&\geq\frac{s_{\mu}(1^{\ell(\lambda)})}{\ell(\lambda)}+1 &=\frac{s_{\lambda}(1^{\ell(\lambda)})}{\ell(\lambda)}+1.
{\bf E}\Biggl[\exp\Bigl(i\sum_{j=1}^n q_j P\bigl(\mathcal{A}_{\varepsilon}(t_j)\bigr)\Bigr)\Biggr] = \exp\Bigl(\mu\sum_{p=1}^n\sum_{k=1}^p \alpha_{p,k}\,\rho_{\varepsilon}(t_p-t_k)\Bigr),
\mathcal{U}=\frac{T_{\rm s}^2}{\sqrt{2\pi}}\,e^{-\frac{2\psi_2\tau_{\rm d}}{N_{\rm a}}+\frac{2\psi_1^2\tau_{\rm d}^2}{N_{\rm a}^2}}\left( \psi_{1}^2\,{\mathcal{U}_1}+2\psi_1\psi_2\,\mathcal{U}_2+\psi_2^2\,\mathcal{U}_3\right),
f(x^2) ={8i\left (m^2\right )^4\over \pi^3}\,\int_0^{\infty}d\kappa_1\,\int_0^{\infty}d\kappa_2\,\kappa_1\kappa_2\,e^{-i(\kappa_1 + \kappa_2)(m^2x^2 - i\delta )}\>e^{-i(1-i\epsilon )(\kappa_1 + \kappa_2)/(4\kappa_1\kappa_2)}.
v_t+(-1)^m D_d^m(\tilde{A}^{\hat{\alpha}\hat{\alpha}}D^m_d v)+(-1)^{m}\sum_{^{|\alpha|=|\beta|=m}_{(\alpha,\beta)\neq (\hat{\alpha},\hat{\alpha})}}\mu^{\alpha_d+\beta_d-2m}D^\alpha(\tilde{A}^{\alpha\beta}D^\beta v) &+\lambda\mu^{{-2m}}v=\sum_{|\alpha|\le m}\mu^{\alpha_d-2m}D^\alpha f_\alpha
\begin{cases}P( (-\Delta)^s)_1 \tilde{u}+ q\tilde{u}= \left(\alpha_m^2-\alpha_m^1\right)(-\Delta)^{s_m}u_2 \Omega, \mathcal{L}\tilde{u}=0 \Omega^c, \tilde{u}=0 \Omega^c. \end{cases}
\begin{array}{l l l}{{X \equiv X^{i = 5}\quad}}&{{X^{i}\sim \epsilon^{i j k l m}\Omega_{j k}\Omega_{l m}}}&{{\qquad 1 0 \times 1 0 \rightarrow 5^{*}}}\\{{Y \equiv Y_{i = 5}\quad}}&{{Y_{i}\sim \Phi^{j}\Omega_{j i}}}&{{\qquad 5^{*}\times 1 0 \rightarrow 5}}\\{{S}}&{{S \sim \mathrm{t r}\left( W^{\dot{\alpha}}W_{\dot{\alpha}}\right)}}&{{\qquad 2 4 \times 2 4 \rightarrow 1}}\\ \end{array}
B(F_p(x_t), \pi(\phi(\overrightarrow{x_t\xi})), f(\xi)) = B(F_p(x_t), \pi(\phi(\overrightarrow{x\xi})), f(\xi)) + B(\pi(\phi(\overrightarrow{x\xi})), \pi(\phi(\overrightarrow{x_t\xi})), f(\xi)) = B(F_p(x_t), \pi(\phi(\overrightarrow{x\xi})), f(\xi)) + B(x, x_t, \xi)
R^{(1)}(\theta )=\frac{\left( \frac{1}{2}\right)_{\theta}\left(\frac{1}{2\lambda}+1\right)_{\theta}}{\left( \frac{1}{2\lambda}+\frac{3}{2}\right)_{\theta}}\frac{\left( \frac{\eta}{\pi \lambda}-\frac{1}{2}\right)_{\theta}}{\left( \frac{\eta}{\pi \lambda}+\frac{1}{2}\right)_{\theta}}\ .
h_{1-L}^{\mathcal{L}}(t)=h_{1-L}^{\mathcal{L}_+}(t)-(h_{0}^{(\mathcal{L}_{1,L+1})_+}(t)-h_{0}^{\mathcal{L}_{1,L+1}}(t))\prod_{\alpha=2}^Lh_{-1}^{\mathcal{K}_\alpha}(t)
H_{k+1}^{(s,t)}=\left|\begin{array}{cccc}f_{s}^{(t)}f_{s+1}^{(t)}\cdots f_{s+k}^{(t)}begin{align*}2pt]f_{s}^{(t+1)}f_{s+1}^{(t+1)}\cdots f_{s+k}^{(t+1)}\vdots \vdots \vdots \vdots f_{s+k-1}^{(t+1)}f_{s+k}^{(t+1)}\cdots f_{s+2k-1}^{(t+1)}\end{array}\right|.
J(x,y)=\frac{s}{\Gamma(1-s)}\int_0^\infty\frac{p_{\Omega}(t,x,y)}{t^{1+s}}\;dt\quad\hbox{and}\kappa(x)=\frac{s}{\Gamma(1-s)}\int_\Omega\left(1-\int_\Omega p_{\Omega}(t,x,y)\;dy\right)\frac{dt}{t^{1+s}}
\bar{E}_{2}= \frac{v^{2}}{2}\int \frac{d^{3}q}{( 2 \pi )^{2}}\Delta_{p}^{- 1}( - \vec{q}\,^{2}) \left\{h ( \vec{q}) h ( - \vec{q}) + \vec{p}( \vec{q}) \cdot \vec{p}( - \vec{q}) \right\}
\left( \Pi_{N}u\right)^{i}\left( \theta\right) =\sum_{k}u_{k}^{i}\varphi_{N,k}\left( \theta\right) ,\ 1\leq i<5 \left( M\left( x\left( \theta\right) \right) \Pi_{N}u\right)^{j}\left(\theta\right) =\sum_{k}\sum_{i}\varphi_{N,k}\left( \theta\right)\left( M_{i}^{j}\left( x\left( \theta\right) \right) u_{k}^{i}\right)
\int d^{4}v V_{\rho}( y , v , 0 ) = \frac{i g}{1 6 \pi^{4}}( f^{2}- \frac{1}{2}g^{2}) \gamma_{5}\gamma_{\sigma}\left( \frac{y_{\sigma}y_{\rho}}{y^{6}}- \frac{\delta_{\sigma \rho}}{4 y^{4}}\right) .
\varepsilon_{k+1}= - \frac{\sigma}{2}\|z_{k+1}- z_{k}\|^2 + \ \frac{A_{k+1}}{2\epsilon}\|\tau_k z_k + (1- \tau_k) y_k - y_{k+1}\|^2 + A_{k+1}\psi(y_{k+1}) - A_k \psi(y_k) - \alpha_k\psi(z_{k+1}) .
A_{2}= - \frac{i \sqrt{2}G \hbar D_{1}\sin ( \frac{1}{\hbar}p_{3}^{( 1 )}L )}{( E + p_{3}^{( 2 )}) \left[ e^{- \frac{i}{\hbar}p_{3}^{( 2 )}L}- \frac{( G \hbar D_{1})^{2}}{( E + p_{3}^{( 2 )})^{2}}e^{\frac{i}{\hbar}p_{3}^{( 2 )}L}\right]}
\begin{aligned}\mathbb{B}: =(24 c_1^2-12 c_4) D_S^6\{\mathrm{I}-2 D_{-2}U D_S^2 \}+(\frac{14}{3}c_2^2-4 c_6) D_S^4 &+(4 c_6-\frac{16}{3}c_2^2) \{D_S^4 U+D_S^2 U D_S^2\}-6 c_3^2 \mathrm{I}+12 (c_2 c_3-c_7) D_S^2 &+(24 c_7-16 c_2 c_3) D_S^2 U,\end{aligned}
\frac{\cosh \left( \nu \pi \right)}{\pi}\left| K_{_{\frac{1}{2}+i\nu_{1}}}\left( x\right) \right|^{2}-\frac{e^{-\Lambda x}}{2x}\ ,\ \frac{\cosh\left( \nu \pi \right)}{\pi}\left| K_{_{\frac{1}{2}+i\nu_{2}}}\left(x\right) \right|^{2}-\frac{e^{-\Lambda x}}{2x}\ ,
\mathsf{d}(x,y) = \inf_{u \in G , w \in{\rm Pic}^0(X)}\widetilde{\mathsf{d}}(t+v_1 - u-w,s+v_2- u-w) &= \inf_{u \in G , w \in{\rm Pic}^0(X)}\left( \mathsf{d}_{t-u}(t+v_1 - u-w,t-u ) + \mathsf{d}_{s-u}(s-u, s+v_2- u-w) \right) + \delta &=\inf_{w \in{\rm Pic}^0(X)}\left( \mathsf{d}_{0}(v_1 , w ) + \mathsf{d}_{0}(v_2,w) \right) + \delta
\mathbb{E}_{\lambda}^d\xi_t(O) \leq e^{-t}\sum_{n=0}^{\infty}\frac{t^n\lambda^n}{n!}\big[\sum_{\overrightarrow{x}\in V_n}(E{\widetilde{\rho}}^2)^{|x_n|-1}\big]M^{2n}&=e^{-t}\sum_{n=0}^{\infty}\frac{t^n\lambda^nM^{2n}}{n!E{\widetilde{\rho}}^2}\big[\sum_{\overrightarrow{x}\in V_n}(E{\widetilde{\rho}}^2)^{|x_n|}\big]
l(t) = \frac 12\Bigl(\frac{b}{t^3}+\frac{2\lambda}{t^4}- \frac{\kappa^2u^*(0, 0)^2}{24}\Bigr)^2 + \frac 12\Bigl(\frac{b}{t^3}-\frac{3\lambda}{t^4}- \frac{\kappa^2u^*(0, 0)^2}{144}\Bigr)^2,
C_2 = C_2(p,m,k,j,u) = &\inf_{\substack{0 < b < m \text{$a$ satisfies all of \eqref{eq:4aconditions}}}}C_1(aq, (m-b)q, k)^{1/q}C_1(ap+(m-b)p+u-(p/q), bp, j)^{1/p}.
{\cal L}_{\mathrm{Y M}}= -{\frac{1}{4 g_{\mathrm{Y M}}^{2}}}G_{\mu \nu}^{a}G^{a \mu \nu}+{\frac{\theta}{3 2 \pi^{2}}}G_{\mu \nu}^{a}{\widetilde G}^{a \mu \nu}= -{\frac{1}{4}}\mathrm{R e}\left( \tau \left[ G_{\mu \nu}^{a}G^{a \mu \nu}+ i G_{\mu \nu}^{a}{\widetilde G}^{a \mu \nu}\right] \right) ~ .
[e_{\alpha_j+\alpha_4},e_{\alpha_i}] = e_{\alpha_i+\alpha_j+\alpha_4}= [e_{\alpha_i+\alpha_4},e_{\alpha_j}], [e_{\alpha_j+\alpha_4},f_{\alpha_i}] = f_{\alpha_i+\alpha_j+\alpha_4}= [e_{\alpha_i+\alpha_4},f_{\alpha_j}]
I_0[u_0,u_1] := \int_{\Omega}\left[ ( |u_1(x)|^2 + |\nabla u_0(x)|^2 + |u_0(x)|^{p+1}) \langle x \rangle^{\alpha}+ |u_0(x)|^2 \langle x \rangle^{-\alpha}\right] \langle x \rangle^{\lambda(2-\alpha)}\,dx < \infty
\rho \Lambda^{2}\left( \frac{1}{g}- 1 \right) = - \rho^{3}\ln \frac{( \Lambda l )^{2}}{2}+ \gamma \rho^{3}+ \frac{\rho}{l^{2}}\ln \frac{( \rho l )^{2}}{4 \pi}+ \frac{2 \rho}{l^{2}}\ln \Gamma \left( \frac{( \rho l )^{2}}{2}\right) + O \left( \frac{1}{\Lambda}\right) ,
\frac{i}{2}T r D^{- 1}G = \frac{1}{1 2 N}\int d^{D}x \left( \lambda_{0}+ \frac{\eta_{0}\phi^{2}}{1 0 N}\right) [ G_{a a}( x , x ) ]^{2}+ \frac{3 \eta_{0}}{6 ! N^{2}}\int d^{D}x [ G_{a a}( x , x ) ]^{3}+ c t e .
\frac{d X^\mu(u,p,t)}{d t}=\,\sum^{8N}_{k=1}\,\frac{\partial X^\mu}{\partial u^\rho_k}\,\frac{d u^\rho_k}{d t}+\,\sum^{8N}_{k=1}\,\frac{\partial X^\mu}{\partial p_{\rho k}}\,\frac{d p_{\rho k}}{d t}.
L=-\int_{-\infty}^{\infty}f(x)lnf(x)dx-\lambda_{0}(1-\int_{-\infty}^{\infty}f(x)dx)-\lambda_{1}(\mu-\int_{-\infty}^{\infty}f(x)xdx)-\lambda_{2}(\sigma^{2}-\int_{-\infty}^{\infty}f(x)(x-\mu)^{2}dx).
D_{\mu \nu}^{( 0 )}= \frac{1}{p^{2}+ \kappa^{2}}\left[ ( \delta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}) - \kappa \epsilon_{\mu \nu \lambda}{\frac{p_{\lambda}}{p^{2}}}\right] + \xi \frac{p_{\mu}p_{\nu}}{( p^{2})^{2}},
G_{1/2,1/2,n}(x,t)=\frac{\Gamma\left(\frac{n+1}{2}\right)}{2^n \pi^{n/2+1}t^{n/2}}e^{{|x|^2}/{4t}}\Gamma\left(\frac{1-n}{2},\frac{|x|^2}{4t}\right).
v^{\eta}(t,x) =v^{\eta}(t_{0},x)e^{-\int_{t_{0}}^{t}(1-a^{\eta}(s,x))ds}+\int_{t_{0}}^{t}b^{\eta}(\tau,x)e^{-\int_{\tau}^{t}(1-a^{\eta}(s,x))ds}d\tau &\quad\quad+\int_{t_{0}}^{t}\tilde{a}^{\eta}(\tau,x)e^{-\int_{\tau}^{t}(1-a^{\eta}(s,x))ds}d\tau.
(\tilde{u}(t),\varphi(t))_{\mathcal{H}}-(\tilde{u}_0,\varphi(0))_{\mathcal{H}}= \int_0^t \Big[ (\tilde{u},\partial_t \varphi)_{\mathcal{H}}+2\nu a(\tilde{u},\varphi)+2\beta \nu a(H,\varphi) -b(\tilde{u},\varphi,\tilde{u})-\beta b(H,\varphi,\tilde{u})-\beta b(\tilde{u},\varphi, H) \Big]dt.
dz(s) = \, \widetilde{b}(z(s))\, ds + \sqrt{2\beta^{-1}}\widetilde{\sigma}(z(s))\, d\widetilde{w}(s) = \, \widetilde{b}(z(s))\, ds + \sqrt{2\beta^{-1}}\widetilde{\sigma}(z(s))\Big[\big(\nabla\xi a \nabla\xi^T\big)^{-\frac{1}{2}}\nabla\xi\sigma\Big](x(s))\, dw(s)\,.
\mathcal{H}=J_1^2+J_2^2+J_3^2+(s_1^2+s_2^2+s_3^2)\left(\frac{\mu_1^2-\mu_1 R_1}{s_1^2}+\frac{\mu_2^2-\mu_2R_2}{s_2^2}+\frac{\mu_3^2-\mu_3R_3}{s_3^2}\right),
0 =(\tilde{B}(\tilde{u},v),w)+(\tilde{B}(\tilde{u},w),v) &=\sum_{h,j,k}(B^h_{jk}\eta_j v_k w_h)+\sum_{h,j,k}(B^h_{jk}\eta_j w_k v_h) &=\sum_{h,j,k}(B^h_{jk}\eta_j v_k w_h+B^k_{jh}\eta_j w_h v_k),
J = \frac{1}{i ( 2 \pi )^{2}}\left[ \int D v \; D p \; \delta^{4}\left( \int v d \tau \right) \exp \left\{i \int d \tau \left( - \frac{v^{2}}{2}+ \frac{p^{2}}{2}\right) \right\}\right]^{- 1}
\frac{\partial \zeta^{( \lambda \mu \alpha )}}{\partial \zeta^{( \tau , \nu , \beta )}}= \psi_{( \tau , \nu , \beta )}^{( \lambda \mu \alpha )}= \frac{1}{2}\left( D_{( \tau , \nu , \beta )}^{( \lambda \mu \alpha )}+ \omega_{( \tau , \nu , \beta )}^{( \rho , \omega , \gamma )}D_{( \rho , \omega , \gamma )}^{( \lambda \mu \alpha )}\right) .
p^{\kappa}(t,x,y)\ge c_5\begin{cases}\Phi^{-1}(t^{-1})^d |x-y| \le 3\Phi^{-1}(t^{-1})^{-1}, t j\left( |x-y|\right) |x-y| > 3\Phi^{-1}(t^{-1})^{-1}.\end{cases}
0 =\int_\Omega [c_1{\rm div}(a\nabla)\varphi + 2a \nabla c_1\cdot\nabla \varphi)]c_1\varphi dx= \int_\Omega{\rm div}[c_1^2a\nabla\varphi]\varphi dx = -\int_\Omega c_1^2 a|\nabla\varphi|^2dx.
\frac{d(p_{n+1},q_{n+1})}{1+d(p_{n+1},q_{n+1})}\min\bigl\{1,\dfrac{d(a_j,b_j)}{C_n\alpha_j'}\bigr\}\leq \frac{d(c_j,d_j)}{\alpha_{m+j}+\beta_k}&=\dfrac{1}{\frac{\alpha_j'}{d(a_j,b_j)}+\frac{\lambda_{n+1}}{d(p_{n+1},q_{n+1})}}.
\tilde{\Theta}_\pm D_a A\varphi =\tilde{\Theta}_\pm D_a\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=-\tilde{\Theta}_\pm\gamma^{(2n+1)}D_a(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\tilde{\Theta}_\pm\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)D_a\varphi=-A\tilde{\Theta}_\pm D_a\varphi.
\frac{2}{(1-z)^{N}(1+z)^{M}}\frac{d}{dz}B(z){}=\big( 1-z^{2}\big) \frac{N+M+n+1}{4n}\big( P_{n-1}^{( N+1,M+1)}(z) \big)^{2}-\big( P_{n}^{(N,M)}(z) \big)^{2}.
y_1=\frac{1}{\sqrt{\mu}}x_3, \, y_2=\frac{1}{\sqrt{2}}(x_1-x_2), \, y_3=\frac{1}{\sqrt{2}}(x_1+x_2); [y_1,y_2]=\frac{1}{2\sqrt{\mu}}(y_2-y_3), \, [y_2,y_3]=0, \, [y_3,y_1]=-\frac{1}{2\sqrt{\mu}}(y_2+3y_3).
b_{m+2}=\frac{-\lambda}{m+1}b_m\quad\mbox{and}b_{m+2j}=\frac{-\lambda}{j(m+j)}\prod_{\nu=1}^{j-1}\left(1-\frac{\lambda}{\nu(m+\nu)}\right)b_m\quad\quad\forall j\geq2.