harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{\substack{p\leq n p\notin\mathcal{S}}}\beta_p\log p \leq \log(2n^2+l)\sum_{\substack{p \leq n p\in\mathcal{S}}}1+\dfrac{1}{\lambda}\sum_{i=1}^{r}e_i\log p_i+n\log 4.
s_{1,2}^{x_{1,2}}s_{2,3}^{x_{2,3}}\cdots{}s_{m-1,m}^{x_{m-1,m}}s_{1,3}^{x_{1,3}}\cdots s_{1,m-1}^{x_{1,m-1}}s_{2,m}^{x_{2,m}}s_{1,m}^{x_{1,m}}\cdot s_{1,2}^{-1}= s_{1,2}^{x_{1,2}}s_{2,3}^{x_{2,3}}\cdots s_{m-1,m}^{x_{m-1,m}}s_{1,3}^{x_{1,3}}\cdots s_{1,m-1}^{x_{1,m-1}}\cdot s_{1,2}^{-1}\cdot s_{2,m}^{x_{2,m}}\, s_{1,m}^{x_{1,m}+ x_{2,m}}.
\begin{cases}\nabla\times E_{\eta,\ell}-ik\mu H_{\eta,\ell}=0 \mbox{in}\Omega, \nabla\times H_{\eta,\ell}+ik\epsilon E_{\eta,\ell}=0 \mbox{in}\Omega, \nu\times E_{\eta,\ell}:=f_{\eta,\ell}\in TH^{-1/2}(\partial\Omega) \mbox{on}\partial\Omega,\end{cases}
K_{\bf a}^{( 1 )}( x , x^{\prime}, t ) \simeq -{\frac{1}{4 \pi t}}e^{-{\frac{( x^{2}+ x^{2})}{4 t}}}K_{0}({\frac{| x | | x^{\prime}|}{2 t}}) f ( p ( t ) ) ~ ~ ,
0= -\langle \tau_{H}, \alpha_{2}\rangle[(m_{1}+m_{2})(\cot\langle \alpha_{2}, H \rangle -\tan\langle \alpha_{2}, H \rangle )^{2}&+2m_{2}(\cot\langle 2\alpha_{1}, H \rangle -\tan\langle 2\alpha_{1}, H \rangle )^{2}](2\alpha_{1}+\alpha_{2}).
D((p_1,q_1)(r_1,s_1),(p,q)(r,s))\leq d_C(p_1,p)+d_C(q_1r_1,q_1r)+d_C(q_1r,qr)+ +d_R(q_1s_1, qs)=d_C(p_1,p)+d_C(r_1,r)+d_C((q_1q^{-1})qr,qr)+d_R(q_1s_1, qs)
\ell^{\mathrm{(I)}}(d_{\mathrm{I},j}) = \mathbf{1}_{j,\mathrm{L}}d_{\mathrm{I},j}^{-\alpha_\mathrm{L}}+ (1-\mathbf{1}_{j,\mathrm{L}}) d_{\mathrm{I},j}^{-\alpha_\mathrm{N}}
\frac{1}{n}\log\left(\mathbb{E}[\exp(n\nu_n(f))]\right) = \frac{1}{n}\sum \limits_{j = 1}^n \left(\log\left(\mathbb{E}\left[q_{2n, 2j - 1}^{x_{n, j}}p_{2n, 2j - 1}^{x_{n, j}}\right]\right) + \log\left(\mathbb{E}\left[q_{2n, 2j}^{y_{n, j}}p_{2n, 2j}^{x_{n, j}}\right]\right) + (3x_{n, j}+ y_{n, j})\log(2)\right) .
P_{C,2p}= 1+\ell(2p)t+\ell(4p)t^2+\ldots = \frac{1-t^4}{(1-t)(1-t)(1-t^2)}P_{C,3p}= 1+\ell(3p)t+\ell(6p)t^2+\ldots = \frac{1-t^3}{(1-t)(1-t)(1-t)}
T_{0,3}\lambda(\iota(t,\ell)) =\lambda(\iota(t/p,\ell)) + p^3\lambda(\iota(pt,p\ell)) + p(T_p\lambda)(\iota(t,\ell)) + p^4(T_p\lambda)(\iota(p^2t,p\ell))+ p^3\lambda(\iota(t/p,\ell/p)) + p^6\lambda(\iota(pt,\ell)).
\left\lbrace \begin{aligned}xy \equiv 0 \mod t^2 x^2-4txy \equiv 1 \mod t^2 y^2 \equiv -(n'+4k+4)t \mod t^2 \end{aligned}\Longrightarrow \left\lbrace \begin{aligned}x \equiv 1 \mod t^2 y \equiv 0 \mod t^2 \end{aligned}\right. \right.
\int_{\Omega}\int_{\Omega}\sum_{{i,j}_{j \neq i}}\overline{u}_{i}(x) \underline{u}_{j}(y) K(x,y) \, dy dx = \int_{\Omega}\int_{\Omega}\sum_{{i,j}_{j \neq i}}\underline{u}_{j}(x) \overline{u}_{i}(y) K(x,y) \, dy \, dx.
p_1(x_0)=\frac{-R_2 \left( (a_5 + 4 a_1 a_6)^2 + (a_3 - a_1 a_5 - 3 a_1^2 a_6)^2 \right)}{a_6^2 \left(a_5 + a_1 \left(-a_3 + a_1 a_5 + 4 a_6 + 3 a_1^2 a_6\right)\right)^3}.
\prod_{i=1}^{s}\frac{1}{1- (\sum_{j=1}^N z_{i,j}t_j)}=\prod_{i=1}^{s}\left(1+\left(\sum_{j=1}^N z_{i,j}t_j\right)+\left(\sum_{j=1}^N z_{i,j}t_j\right)^2+\cdots\right).
\dot{b}\langle \eta_*, \psi_* \rangle_{L^2}=-12 |b|^2 b \langle \psi_*^2, \tilde{w}_2 \rangle_{L^2}+\frac{64}{\sqrt{c_*}}|b|^2 b + \frac{16}{3 \sqrt{c_*}}b \delta + \mathcal{O}(\delta^2 |b| + |b|^5),
A_0= \begin{pmatrix}0 1 0 0 0 0 0 0 0\end{pmatrix},A_1=\begin{pmatrix}p_2q_2 -p_2 p_1p_2 0 p_1q_1-p_2q_2-\theta^0_2 1 -t q_1 -p_1q_1-\theta^0_1\end{pmatrix}, A_2=\begin{pmatrix}0 0 0 q_2 -1 p_1 0 0 0\end{pmatrix}.
(\begin{matrix}n\\k\end{matrix})=\frac{n(n-1)\cdot\cdot\cdot(n-k+1)}{k(k-1)\cdot\cdot\cdot1}=\prod_{l=1}^{k}\frac{n-l+1}{l}=\prod_{l=0}^{k-1}\frac{n-l}{k-l}
\phi_a(x_0,\psi_0,\bar\psi_0,\tau_a)=\int_{x_0,\psi_0,\bar \psi_0}{\cal D}x^I_d{\cal D}\psi_{\alpha}{\cal D}\bar \psi_{\dot\alpha}V_a e^{-S(x^I_d,\psi_{\alpha},\bar\psi_{\dot\alpha})}
\Big( T \, \mathfrak{J}^L \, T^{-1}\Big) \Big( C \mapsto f^C \Big) = \big(T \, \mathfrak{J}^L \big) \big(f\big) = T \big( \mathfrak{J}^L f \big) = \Big( \, C \mapsto \big(\mathfrak{J}^L f\big)^C \, \Big).
J = \det \left[ \begin{array}{c c}{{\partial w_{i}/ \partial z_{j}}}&{{\partial w_{i}/ \partial \overline{{{z}}}_{j}}}\\{{\partial \overline{{{w}}}_{i}/ \partial z_{j}}}&{{\partial \overline{{{w}}}_{i}/ \partial \overline{{{z}}}_{j}}}\\ \end{array}\right] ,
\begin{cases}\dot{z}_i=A_i^0z_i+f_i(x_i), \dot{x}_{i1}=x_{i2}, \vdots \dot{x}_{in_i^x}=b_i^{\infty}u_i+g_{i}(z_i,x_i), y_i=x_{i1}, i=1,{\dots},N\end{cases}
\left|\det(I_n - i M_t) \right| = \prod_{k=1}^n \left|1-i \lambda_k(t) \right|=\prod_{k=1}^n \sqrt{1+\lambda_k^2(t)}&\geq \max_{k = 1, \ldots, n}\left|\lambda_k(t) \right| = \max_{x \in \mathbb{C}^n \setminus \{0\}}\frac{\left|\langle M_t x, x \rangle \right|}{\langle x, x\rangle}
\begin{array}{c}{{\bf J}={\bf p}\times{\bf x}+{\bf p}_{+}\times{\bf e}_{+}+{\bf p}_{1}\times{\bf e}_{1}=}\\{={\bf p}\times{\bf x}+ c ( 2 \epsilon - q ){\bf e}_{+}- 2 c{\bf e}_{-}.}\\ \end{array}
\begin{array}{ll}u_{j}(0,t)= \displaystyle\sum_{k=1}^{n}\int_{0}^{1}r_{jk}(x,t)u_{k}(x,t)\, dx, \;\;\; 1\le j\le m, [2mm] u_{j}(1,t)= \displaystyle\sum_{k=1}^{n}\int_{0}^{1}r_{jk}(x,t)u_{k}(x,t)\, dx, \;\;\; m<j\le n, \end{array}
\mathcal{E}(x,t) = \widetilde{\mathcal E}(x,t)|_{x_1>0}, \mathcal B(x,t) = \widetilde{\mathcal B}(x,t)|_{x_1>0}, \mathcal A(x,t) = \widetilde{\mathcal A}(x,t)|_{x_1>0},
q\dim_{\mathcal{U}}W=\frac{q\dim_{V_{\mathbb{Z}\beta}\otimes V_{\mathbb{Z}\beta}^{+}}W}{q\dim_{V_{\mathbb{Z}\beta}\otimes V_{\mathbb{Z}\beta}^{+}}\mathcal{U}}=\frac{1}{2}q\dim_{V_{\mathbb{Z}\beta}\otimes V_{\mathbb{Z}\beta}^{+}}W.
A_{1,1}: = \sup_{t > 0}\bigg( \int_0^t [\Phi_1]^{\frac{q}{p}}(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}\Phi[v;s]^{- \frac{1}{s}}(t), A_{1,2}: = \sup_{t > 0}W_*^{\frac{1}{q}}(t) \bigg( \int_0^t \bigg( \frac{\Phi_1(\tau)}{\Phi[v;s](\tau)}\bigg)^{\frac{s}{s-p}}\phi[v;s](\tau)\,d\tau \bigg)^{\frac{s - p}{p s}};
u^p_{\mu_t, p}(x) \cdot 2^p e^{-pt}= \int_X \cosh^p(d(x,y)) \cdot 2^p e^{-pt}d\mu_t(y) = \int_{\mathcal{G}X}\cosh^p(d(x, \pi \circ \phi_t (\gamma))) \cdot 2^p e^{-pt}d\nu(\gamma) = \int_{\mathcal{G}X}\cosh^p(d(x, \gamma(t)))) \cdot 2^p e^{-pt}d\nu(\gamma)
\begin{aligned}\frac{h\big[\kappa h^2 - 2h(1+\kappa) + 3\big]}{2L \big[2 - h(1+\kappa)\big]}\big\|g_{0}\big\|^2 + \frac{h}{2L \big[2 - h(1+\kappa)\big]}\|g_1\|^2{}\leq{}f_{0}-f_{1}\end{aligned}
\frac{p_1}{a_1^{d_1}}\, \frac{1}{\Gamma \left(\frac{d_1}{p_1}\right)}\, (d_1 - d_2) \, \frac{a_1^{d_1}}{p_1}\, \ln a_1 \, \Gamma \left(\frac{d_1}{p_1}\right)= (d_1 - d_2) \, \ln a_1
\|\varphi^{+}\|_{L^{m_{k}}(\partial B_1)}\leq(C m_0)^{\frac{1}{m_{0}}\sum_{j = 0}^{k-1}\left(\frac{2}{2^{\star}}\right)^{j}}\left(\frac{2^{\star}}{2}\right)^{\frac{1}{m_{0}}\sum_{j = 0}^{k-1}j\left(\frac{2}{2^{\star}}\right)^{j}}\|\varphi^{+}\|_{L^{m_{0}}(\partial B_1)}.
v(\mathcal F)\circ v(\mathcal E) = v(\mathcal O_{\Delta_X}) = \Delta_X \in \operatorname{CH}(X\times_K X) \otimes \mathbb Q, v(\mathcal E)\circ v(\mathcal F) = v(\mathcal O_{\Delta_Y}) = \Delta_Y \in \operatorname{CH}(Y\times_K Y) \otimes \mathbb Q.
\{Y^{D}_{\tau_{i}}\in D_0 ( 2 r , 2r) \setminus\cup_{k=1}^{i-1}J_k,\ \ \tau_{i}\leq \sigma_{i, li}\}\subset \cup_{k=1}^{li}\{|Y^D_{\sigma_{i, k}}- Y^D_{\sigma_{i, k-1}}|\geq r/(200li^3),Y^D_{\sigma_{i, k-1}}\in \widetilde{J}_{i}\},\ \ \i\geq i_0\, .
\delta_{G_{3}}( x ; 0 ) = -{\frac{1}{1 2}}{x^{- 6}}\left({\frac{\Gamma ( 5 / 2 )}{\Gamma ( 5 / 2 )}}- 2{\frac{\Gamma ( 4 )}{\Gamma ( 4 )}}+{\frac{\Gamma ( 5 )}{\Gamma ( 5 )}}\right) = 0 ,
\Omega_{M , 0}( \mathrm{m a t t e r}) \simeq 0 . 3 ~ , \qquad \mathrm{a n d}\qquad \Omega_{X , 0}( \mathrm{d a r k ~ e n e r g y}) \simeq 1 - \Omega_{M}\simeq 0 . 7 ~ ,
\int_{\partial D_R}e^{-\frac{i}{2}\theta_{\mathbf e}}(i\nabla+A_{\mathbf e})w_R\cdot\nu\, \psi_k\, ds =\frac{i k\sqrt\pi}{2}\frac{R^k}{R^k-1}\Big((R^k+1)\sqrt\pi-2\upsilon_R(1)\Big).
\displaystyle \int_{\Omega}\textbf{\textit{v}}\cdot \nabla{\varphi}\,d\textbf{x}+ \displaystyle \int_{\Omega}{\varphi}\,\mathrm{div}\, \textbf{\textit{v}}\, d\textbf{x}=\left\langle \textbf{\textit{v}}\cdot \textbf{\textit{n}},\varphi \right\rangle_{{W}^{-1/p,p}(\Gamma) \times{W}^{1/p,p'}(\Gamma)}.
e^{- 2 \left( \Phi + U \right)}( \partial_{0}\Psi )^{2}= \partial_{0}\left( e^{- 2 \left( \Phi + U \right)}\Psi \partial_{0}\Psi \right) - e^{- 2 \left( \Phi + U \right)}\partial_{0}\partial_{0}\Psi \ .
c_{-}= \left( \begin{array}{c c c}{0}&{0}&{0}\\{C_{-}}&{0}&{0}\\{0}&{- C_{-}^{T}}&{0}\\ \end{array}\right) , \qquad c_{+}= \left( \begin{array}{c c c}{0}&{C_{+}}&{0}\\{0}&{0}&{- C_{+}^{T}}\\{0}&{0}&{0}\\ \end{array}\right) .
V^{r}_{h}= Y^{r}_{h}\cap H_{0}^{1}(\Omega), \mathcal{V}^{r}_{h}= V^{r}_{h}\oplus{\rm i}V^{r}_{h},\mathbf{V}^{r}_{h}= \big( Y^{r}_{h}\big)^{3}\cap \mathbf{H}^{1}_{\rm t}(\Omega).
\sum_{1 \leq n \leq X}1_{\mathcal S}g_1(a_1 n + b_1) \prod_{j=2}^k 1_{\mathcal S}g_j(a_j n + b_j + h_j) &= \sum_{1 \leq n \leq X}1_{\mathcal S}g_1(a_1 n + b_1 + a_1 h_1) \prod_{j=2}^k 1_{\mathcal S}g_j(a_j n + b_j + h_j + a_j h_1) + O( H' )
f_0(d) ~\dot =~ \sum_{\ell=1}^{\infty}\frac{d^{2\ell +1}}{(\ell!)^2}= d \left[ I_0(2d) -1 \right] f_1(d) ~\dot =~ \sum_{h=1}^{\infty}\frac{d^{2h}}{h! (h-1)!}= d I_1(2d) \,,
(A-2I)y_{2}=(\begin{matrix}-1&0&0&0&0\\ 3&-1&0&0&0\\ 6&3&0&0&0\\ 10&6&3&0&0\\ 15&10&6&3&0\end{matrix})(\begin{matrix}0\\ 0\\ 0\\ 3\\ 0\end{matrix})=(\begin{matrix}0\\ 0\\ 0\\ 0\\ 9\end{matrix})=y_{1},
\frac{d \hat{J}(u_{0})}{d u_{0}}= \frac{d}{du_{0}}\left( \frac{1}{2}\int_{-1}^{1}J(u_{0}+ \epsilon v_0)dv_0 \right) = \frac{1}{2}\int_{-1}^{1}\frac{d J(u_{0}+ \epsilon v_0)}{\epsilon dv_0}dv_0 = \frac{J(u_{0}+\epsilon) - J(u_{0}-\epsilon)}{2\epsilon}.
\hat{w}_\alpha(x)=w_\alpha\left(\exp_{x_\alpha}(\mu_\alpha x)\right), &\hat{u}_\alpha(x)=u_\alpha\left(\exp_{x_\alpha}(\mu_\alpha x)\right), &\hat{g}_\alpha(x)=\left(\exp^\star_{x_\alpha}g\right)(\mu_\alpha x),
\langle\xi_0, \xi_0\rangle + \langle A_r\xi_0, \xi \rangle + \langle A_0\xi, \xi \rangle = (\xi_1^0)^2 + \dfrac{N-1}{r^2}(\xi_1)^2+\alpha \lambda |W|^{\alpha-1}We^{a(\alpha-1)t}\xi_1^0\xi_1+ & + (\xi_2^0)^2 + \dfrac{N-1}{r^2}(\xi_2)^2+\alpha \lambda |Z|^{\alpha-1}Ze^{a(\alpha-1)t}\xi_2^0\xi_2 + a(\xi_1)^{2}+ a(\xi_2)^2 - p(u^{p-1}+v^{p-1})\xi_1\xi_2.
a_1g_1a_1^{-1}=a_0(a_1'g_1a_1'^{-1})a_0^{-1}=a_0(a_2'g_2a_2'^{-1})a_0^{-1}=a_2g_2a_2^{-1}, &b_1^{-1}g_1b_1= b_0^{-1}(b_1'^{-1}g_1b_1')b_0= b_0^{-1}(b_2'^{-1}g_2b_2')b_0=b_2^{-1}g_2b_2.
\lim_{n \to \infty}n^{2m^2 - m -2ms-s}D^{(n)}_{\kappa_n}(x_i^{(n)}) = \frac{2^{2ms +s}}{(w_1 \cdots w_m)^{1+s}}z_i^{-\frac{s}{2}}\cdot A_I^{(s, w)}(\kappa, z_i),
L ( x , \dot{x}, \{q_{n}, \dot{q}_{n}\}, t ) = \frac{M}{2}\dot{x}^{2}- V ( x , t ) + \sum_{n}\left\{\frac{1}{2}m \dot{q}_{n}^{2}- \frac{1}{2}m \omega_{n}^{2}\left[ q_{n}- Q_{n}( x , t ) \right]^{2}\right\}.
\begin{aligned}a_h(\sigma_h,\tau_h) + c_h(\sigma_h,q_h) + c_h(\tau_h,p_h) + b(\sigma_h,v_h) + b(\tau_h,u_h) = (g,v_h), (\tau_h,v_h,q_h) \in \Sigma_h \times V_h \times \tilde Q_h. \end{aligned}
f_{\pm}^{1 / 2}( \theta , \phi ) = v \biggl [ - e^{- i ( \phi - \gamma ) / 2}\sin ( \beta / 2 ) \cos ( \theta / 2 ) + e^{i ( \phi - \gamma ) / 2}\cos ( \beta / 2 ) \sin ( \theta / 2 ) \biggr ] e^{\pm i \phi / 2}.
I_k=\|e^{3\nu^{1/3}t}\nabla G_{2,k}\|_{L^2L^2}^2+\|e^{3\nu^{1/3}t}\partial_xg_{3,k}\|_{L^2L^2}^2+\nu^{\frac{2}{3}}\|e^{3\nu^{1/3}t}\nabla(g_{3,k})_{\neq}\|_{L^2L^2}^2
\wedge_{i}\phi_B(b_1^i) = \pm \nu(\mathcal{E}_1)^{-1}(\wedge_{i}\phi_B\theta_1(a_1^i))\wedge (\wedge_{i}\phi_B\tau_1^{-1}(c_1^i)) = \pm \nu(\mathcal{E}_1)^{-1}M \wedge N .
q_0=\begin{pmatrix}1 0 0 1 \end{pmatrix}, q_1=\begin{pmatrix}i 0 0 -i \end{pmatrix}, q_2= \begin{pmatrix}0 i i 0 \end{pmatrix}, q_3= \begin{pmatrix}0 -1 1 0 \end{pmatrix}.
f_X(x)=\frac{1}{\lambda^{\theta}\Gamma(\theta)}x^{\theta-1}e^{-\frac{x}{\lambda}}=\frac{1}{\lambda\Gamma(\theta)}I^{1,0}_{0,1}\Bigg[\frac{x}{\lambda}\left|\begin{matrix}(\theta-1,1,1) \end{matrix}\right.\Bigg],\ \x>0.
K(w-z) = \frac{1}{2\pi i}\frac{1}{w-z}= \frac{1}{2\pi i}\frac{1}{w-z}= \frac{1}{2\pi i (w-a)}\frac{1}{1-\frac{z-a}{w-a}}= \frac{1}{2\pi i}\sum_{n=0}^{\infty}(z-a)^n(w-a)^{-(n+1)}= \frac{1}{2\pi i}\sum_{n=0}^{\infty}u_n(z)v_n(w),
|w(re^{i\theta})|^2{\rm Re}\Big(\frac{re^{i\theta}w'(re^{i\theta})}{w(re^{i\theta})}\Big)= r \int_0^r |w'(\rho e^{\theta})|^2 d\rho -r \int_0^r{\rm Re}(\rho^2e^{2i\theta}p(\rho e^{i\theta})) \cfrac{|w(\rho e^{i\theta})|^2}{\rho^2}d\rho.
\mathcal{Z}_p:=\left\{f:\t^{1-\frac{3}{2p}}f\in C_b([0,\infty);L^p(\mathbb{R}^3)),\t^{\frac{3}{2}\left(1-\frac{1}{p}\right)}\partial_j f\in C_b([0,\infty);L^p(\mathbb{R}^3)),\j=1,2,3\right\}.
U_{n}=\bigcup_{q=2}^{\infty}\bigcup_{p=-\infty}^{\infty}\{x\in\mathbb{R}:0<|x-\frac{p}{q}|<\frac{1}{q^{n}}\}=\bigcup_{q=2}^{\infty}\bigcup_{p=-\infty}^{\infty}(\frac{p}{q}-\frac{1}{q^{n}},\frac{p}{q}+\frac{1}{q^{n}})\backslash\{\frac{p}{q}\}
\vert f_0\vert \leqslant \Vert f\Vert_{L_\infty}=\Vert Q_0^{-1}\Vert_{L_\infty}^{1/2}, \vert f_0^{-1}\vert \leqslant\Vert f^{-1}\Vert_{L_\infty}=\Vert Q_0\Vert^{1/2}_{L_\infty}.
= \frac{1}{3}+ \frac{1}{3}\times \frac{1}{4}+ \frac{1}{3}\times \frac{3}{4}\times \frac{1}{5}+ \frac{1}{3}\times \frac{3}{4}\times \frac{4}{5}\times \frac{1}{6}+ \frac{1}{3}\times \frac{3}{4}\times \frac{4}{5}\times \frac{5}{6}
\min_{\mathbf{W}_{k}\in\mathbb{H}^{N},k=1,...,K}~\sum_{k=1}^{K}\textrm{tr}(\mathbf{A}_{k}\mathbf{W}_{k}) &s.t.~\sum_{k=1}^{K}\textrm{tr}(\mathbf{B}_{m,k}\mathbf{W}_{k})\unrhd_{m}b_{m},m\!=\!1,...,M,\mathbf{W}_{k}\succeq0, k\!=\!1,...,K,
\sum\limits_{n = 0}^{\infty}\frac{q^{2n^2}}{(q;q)_{2n}}= \prod\limits_{n = 1}^{\infty}\frac{(1 + q^{8n - 3})(1 + q^{8n - 5})(1 - q^{8n})}{1 - q^{2n}},
\left|\|\nabla u_1\|^2_{L^2(\Omega)}-\|\nabla u_2\|^2_{L^2(\Omega)}\right| =\Big(\|\nabla u_1\|_{L^2(\Omega)}+\|\nabla u_2\|_{L^2(\Omega)}\Big)\,\left|\|\nabla u_1\|_{L^2(\Omega)}-\|\nabla u_2\|_{L^2(\Omega)}\right| &\le\Big(\|\nabla u_1\|_{L^2(\Omega)}+\|\nabla u_2\|_{L^2(\Omega)}\Big)\,\|u_1-u_2\|_{W^{1,2}_0(\Omega)},
H\alpha' P=\frac{[H\dot{r}\cos Ht+H^2\sqrt{b}\alpha' r\sin Ht]^2-H^2r^2(1+H^2r^2)[(1+H^2r^2)\cos^2Ht-H^2r^2]}{2[H^2r\dot{r}\sin Ht+H\sqrt{b}\alpha'\cos Ht]\sqrt{1+H^2r^2}\sqrt{(1+H^2r^2)\cos^2Ht-H^2r^2}}.
1 + d\bigg(\frac{-(1 - 3 d D) - \sqrt{(1 - d D )^{2}+ 2 a d^{2}C}}{2 d}- 2 D \bigg) & = 1 + d\bigg(\frac{-(1 - d D) - \sqrt{(1 - d D )^{2}+ 2 a d^{2}C}}{2 d}\bigg) & < 1 + d\bigg(\frac{-(1 - d D) - | 1 - d D |}{2 d}\bigg) \leq 1.
\|u\|_{\gamma}\leq \|u\|^{1-\theta_4}_{1}\|u\|^{\theta_4}_{\gamma+1}\,,\theta_4=\frac{\gamma^{2}-1}{\gamma^{2}}\,, begin{align*}.3cm]\|w\|_{\alpha}\leq \|w\|^{1-\theta_5}_{1}\|u\|^{\theta_5}_{\alpha+1}\,,\theta_5=\frac{\alpha^{2}-1}{\alpha^{2}}\,,
\langle x \mid K_{0}( T ) \mid y \rangle = \langle x \mid \exp \Bigl [ - T \bigl ( -{\partial}^{2}+ m^{2}\bigr ) \Bigr ] \mid y \rangle ={( 4 \pi T )}^{- d / 2}\exp ( -{\frac{( x - y )^{2}}{4 T}}) \, ,
\begin{array}{c | c}{{\mathrm{P h y s i c a l ~ s p a c e t i m e : ~}z}}&{{\mathrm{C o v e r i n g : ~}u = z^{1 / q}}}\\ \hline{{{\displaystyle{\frac{c}{2 4 \pi}}\int_{z}( R + 2 )}}}&{{{\displaystyle{\frac{c / q}{2 4 \pi}}\int_{u}( R + 2 )}}}\\{{\Downarrow}}&{{\Downarrow}}\\{{\{L_{n + r / q}, \c \}}}&{{\{Q_{n}, \ \{\displaystyle{\frac{c}{q}}\}}}}\\ \end{array}
y_1^{35}=u_1^{56}=:u_1, -y_1^{25}=u_2^{56}=:u_2, &y_2^{35}=u_1^{57}=:u_3, -y_2^{25}=u_2^{57}=:u_4, &v^{67}=u_1^{57}-u_2^{56}
-\frac{\underline{P}(0)}{\underline{Q}(0)}= \frac{\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (-as_{k-1})_{k=0}^{m}\right)}{a\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (1,0,\ldots,0)\right)}+ \frac{\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (s_k)_{k=0}^{m}\right)}{a\mathcal{D}\left((s^{(1)}_{k})_{k=0}^{2m-1}, (1,0,\ldots,0)\right)}
Tr_{(s+1)}(e^{-a^{\dagger}_{s}a_{s}/M^{2}})-Tr_{(s)}(e^{- a_{s}a^{\dagger}_{s}/M^{2}}) =Tr_{(s+1)}(e^{-a_{s}a^{\dagger}_{s}/M^{2}})-Tr_{(s)}(e^{- a_{s}a^{\dagger}_{s}/M^{2}}) = 1
p_{\mu}\; = \; q^{-3/2}\Bigl ( \frac{\partial L_{0}^m}{\partial\dot x^{\mu}}\Bigr ) \equiv (\dot x_{\mu}+q^{1/2}i \chi \psi_{\mu})[(\dot x +q^{1/2}i \chi \psi)^2 ]^{-1/2}\;m,
\begin{bmatrix}\phi(\Delta_n)^*\phi(\Delta_n) \phi(\Delta_n)^*\phi(a)^* \phi(a)\phi(\Delta_n) \phi(a)\phi(a)^*\end{bmatrix}\leq \begin{bmatrix}\phi(\Delta_n^*\Delta_n) \phi(\Delta_n^* a^*) \phi(a\Delta_n) \phi(aa^*)\end{bmatrix},
u^{c,\gamma}= u_{r}^{c,\gamma}e_{r}+u_{\theta}^{c,\gamma}e_{\theta}=\frac{(U_{\theta}^{c,\gamma})'}{r}e_{r}+\frac{U_{\theta}^{c,\gamma}}{r\sin\theta}e_{\theta}, p^{c,\gamma}= u_{r}^{c,\gamma}-\frac{1}{2}(u_{\theta}^{c,\gamma})^{2}=\frac{1}{r}\left((U_{\theta}^{c,\gamma})'-\frac{(U_{\theta}^{c,\gamma})^{2}}{2r\sin^{2}\theta}\right).
\times \exp \left( \sum_{b , b^{\prime}= 1}^{N}\sum_{j = 1}^{n_{b}}\sum_{j^{\prime}= 1}^{m_{b^{\prime}}}2 \frac{\pi}{\pi + g N}\frac{1}{N}\; \mathrm{K}_{0}\Big ( e \sqrt{\frac{N}{\pi + g N}}| x_{j}^{( b )}- y_{j^{\prime}}^{( b^{\prime})}| \Big ) \right)
ds^{2}=\frac{Re(dz\otimes d\overline{z})}{(1+|z|^{2})^{2}}=\frac{dx^{2}+dy^{2}}{(1+r^{2})^{2}}=\frac{1}{4}(d\varphi^{2}+sin^{2}\varphi d\theta^{2})=\frac{1}{4}ds_{us}^{2}
d s^{2}= \frac{1 - \left( \frac{t_{-}}{t}\right)^{2}}{- k + \left( \frac{t_{+}}{t}\right)^{2}}d y^{2}- \frac{d t^{2}}{\left( - k + \left( \frac{t_{+}}{t}\right)^{2}\right) \left( 1 - \left( \frac{t_{-}}{t}\right)^{2}\right)}+ t^{2}d \Omega_{k}^{2}
{\rm Var}({\cal N}^e(f_{\ell})) = \frac{1}{2^2 \cdot 3^3 \pi^2}\ell^2 \log \ell+O(\ell^2),{\rm Var}({\cal N}^s(f_{\ell})) = \frac{1}{2^2 \cdot 3^3 \pi^2}\ell^2 \log \ell+O(\ell^2).
\left\lfloor \frac{bq_{i}}{q_j}\right\rfloor - \left\lfloor \frac{cq_{i}}{q_j}\right\rfloor = \left\lfloor \frac{(b-c)q_{i}}{q_j}\right\rfloor \, , \, \, i\neq j, \mbox{and}c \left(\frac{1+\sum_{i\neq j}q_i}{q_j}\right)-\sum_{i\neq j}\left\lfloor \frac{cq_i}{q_j}\right\rfloor = 1.
= c_{k-1}+ \tau\frac{(k+1)q_{k+1}-kq_k}{k!}&\qquad+ (\tau-1)\frac{k^2 q_k-(k+1)q_{k+1}}{k!}&> c_{k-1}+ \tau\frac{(k+1)q_{k+1}-kq_k}{k!}
Z ( T ) = \lim_{N \rightarrow \infty}e^{i h J T}\prod_{j = 1}^{N}\int_{\mathrm{P B C}}{{d \mu \left({\xi_{j}^{*}, \xi_{j}}\right)}\exp \left\{{2 J{\ln \left({{\frac{1 + e^{- i \epsilon h}\xi_{j}^{*}\xi_{j - 1}}{1 + \xi_{j}^{*}\xi_{j}}}}\right)}}\right\}}\quad ,
\{|\nabla u_n|> t, u_n>1\}= \{|\nabla u_n|> t,1<u_n\leq k+1\}\cup \{|\nabla u_n| > t,u_n > k+1\}& \subset \{|\nabla u_n|> t,1<u_n\leq k+1\}\cup \{u_n > k+1\}\subset \Omega.
\sum_{j=1}^N\left(\max_{i \in \mathcal{A}^{\star}_j}\ell_{i,s}^{\prime}+\max_{i \in ([1:N]\backslash \{j\}) \backslash \mathcal{A}^{\star}_j}r_{i,s}^{\prime}\right)\geq \sum_{j=1}^{N-1}\left(\max_{i \in \mathcal{A}_{{\rm{F}}j}}\ell_{i,s}^{\prime}+ \max_{i \in [1:N] \backslash \mathcal{A}_{{\rm{F}}j}}r_{i,s}^{\prime}\right),
s_{e_\nu}= s_{e_\nu}p_{s_E(e_1)}= s_{e_{\nu}}s_{e_1}s_{e_1}^* = s_{e_{\nu}}s_{e_1}p_{s_E ( e_2)}s_{e_1}^* = s_{e_{\nu}}s_{e_1}s_{e_2}s_{e_2}^*s_{e_1}^* &\ \vdots &= s_{e_{\nu}}s_{\mu}s_{\mu}^* &= s_\nu s_\mu^* &= \Psi ( t_{\overline{\nu}}t_\mu) \in \psi ( C^*(F) ).
T_L\ =\ -3\, \omega\, e^{-b_L d}\coth \left[\omega (x_s\! -\! L)\right] \left( \frac{\tanh \left[\, \frac{\omega}{2}(x_s\! -\! L)\, \right]}{\tanh \left( \frac{\omega x_s}{2}\right)}\right)^{\cosh^{-1}\left[\omega (x_s\! -\! L)\right]}\ .
\begin{aligned}\left\| (\Phi^j_I)^Tr^j\right\| = \max_{i \in I}\left| \langle \phi^j_i,r^j\rangle\right| \geq \displaystyle\sqrt{\frac{1}{|\Omega|}\sum_{i \in \Omega}\left|\langle\phi^j_i,r^j\rangle\right|^2}&=\frac{1}{\sqrt{K}}\left\|(\Phi^j_{\Omega})^Tr^j\right\|.\end{aligned}
\|u_i\|_2 \leq\sqrt{k/(t-1)}\alpha \leq\frac{\|h_{S_0}\|_2}{\sqrt{t-1}}+\frac{2\sigma_k(x_0)_1+\rho}{\sqrt{k(t-1)}}\leq\frac{\|h_{S_0}+h^{(1)}\|_2}{\sqrt{t-1}}+\frac{2\sigma_k(x_0)_1+\rho}{\sqrt{k(t-1)}}=\frac{z+R}{\sqrt{t-1}},
f({\bf r})=\frac{1}{2 \pi}\,e^{i \theta}\,\left(\frac{\partial}{\partial r}+\frac{i}{r}\, \frac{\partial}{\partial \theta}\right)\,\int_{D}\,d^2 r^{\prime}\, \ln{|{\bf r - r^{\prime}}|}\,e^{2 \phi({\bf r^{\prime}})}\,b(r^{\prime}e^{i \theta^{\prime}})+ C\,,
\begin{array}{ll}\log(\rho(n,l))=\frac{2l}{n-k_0}\log\left({\frac{n-k_0}{2p}}\right)+\frac{2l}{n-k_0}\log({B}(\frac{n-k_0}{2},0.5)) +\frac{2l}{n-k_0}\log(\alpha)\end{array}
M_{11}=\left(\begin{array}{cc}1 0 0 0\end{array}\right),\,\,M_{12}=\left(\begin{array}{cc}1 1 0 0\end{array}\right),\,\,M_{21}=\left(\begin{array}{cc}0 0 1 0\end{array}\right),\,\,M_{22}=\left(\begin{array}{cc}0 0 1 1\end{array}\right).
I_{\operatorname{l o g}}= \frac{1 6 \times 8}{2 7 \times 7}\frac{\pi^{2}}{2}( \frac{6}{\pi^{2}})^{4}\frac{1}{( x_{1 2}^{2})^{4}( x_{3 4}^{2})^{4}}\operatorname{l n}s ,
x(r,s)_k = \sum_{(b_j)_{j\ge 0}\in \mathbf{S}(r,s)_k}(x\otimes 1)^{(b_0)}(x\otimes t)^{(b_1)}\cdots (x\otimes t^{k-1})^{(b_{k-1})},{}_kx(r,s) = \sum_{(b_j)_{j\ge0}\in{}_k\mathbf{S}(r,s)}(x\otimes t^k)^{(b_k)}(x\otimes t^{k+1})^{(b_{k+1})}\cdots (x\otimes t^{s})^{(b_s)}.
A^{ab}=\tilde{g}^{ap}\tilde{g}^{bq}\nabla_{p}l_{q}=z\overline{m}^{a}m^{b}+\overline{z}m^{a}\overline{m}^{b}+\overline{\sigma}m^{a}m^{b}+\sigma\overline{m}^{a}\overline{m}^{b}.
\sum_{N\geq n}\frac{|A(N,n)|^2\|P_{N-n}^{(2)}\|_{HS}^2}{\lambda_N(\nu)}\leq C\sum_{N\geq n}\frac{(N+1)^{2k}(N-n+1)^{d-1}}{(N+1)^{\rho-\nu}}= C \sum_\ell \frac{(\ell+1)^{d-1}}{(\ell+n+1)^{\rho-\nu-2k}}\leq \frac{C'}{(n+1)^{\rho-\nu-2k-d}}\leq \frac{C''}{\lambda^\flat_n(\nu+2k)}.
E_{e f f ( 3 + 1 )}= - \frac{B_{m}}{2 \pi^{2}}\int_{0}^{+ \infty}y \ln \left( \frac{y^{2}+ m_{f}^{2}/ B_{m}}{m_{f}^{2}/ B_{m}}\right) \left( G ( y , B_{m}d^{2}) - c \right) d y
\int d^2x \left\{2\chi_{\ell}^{\dagger}\left(\partial_z\partial_{\bar{z}}+\partial_{\bar{z}}\partial_z\right)\phi_{\ell}-2\phi_{\ell}^{\dagger}\left(\partial_z\partial_{\bar{z}}+\partial_{\bar{z}}\partial_z\right)\chi_{\ell}\right\}=-4\sin\delta_{\ell}\, .
d_{\ell}= \sum_{j=2}^{m}\frac{(-1)^{\ell - 1}\log s_{j}}{2\pi}\int_{\lambda_{j}}^{\lambda_{j-1}}(-u)^{\ell-\frac{3}{2}}du = \sum_{j=2}^{m}\frac{(-1)^{\ell - 1}\log s_{j}}{\pi (2\ell -1)}\Big( |\lambda_{j}|^{\ell-\frac{1}{2}}-|\lambda_{j-1}|^{\ell-\frac{1}{2}}\Big),
2 \int_{0}^{\infty}a^{d}( w ) q ( w ) ~ d w = 2 \int_{0}^{\infty}a^{d}( w ) \biggl [ 1 - 2 \epsilon \frac{{\cal H}^{2}}{a^{2}}( d - 2 ) ( d - 1 ) \biggr ] ~ d w ,
\left[ \bar{\psi}_{n}^{( K )}, \Phi_{\chi_{k}\otimes \chi_{\bar{k}}}^{\beta}\right] \ = P h i_{\chi_{k}\otimes \bar{k}\left( \delta_{1}\left( \bar{\psi}^{( K )}\right)_{n}\right) \chi_{\bar{k}}}^{\beta}\ ,
S_{1}-S_{2}+S_{3}+S_{4}+S_{5}+S_{6}-S_{7}= \displaystyle\sum_{l=1}^{p-q}\left(\varphi_{il}^{\star\star}\left(Z\right)\right)^{(2)}\overline{\left(\varphi_{jl}^{\star\star}\left(Z\right)\right)^{(2)}}+2\mbox{Re}\left\{\displaystyle\sum_{l=1}^{p-q}\overline{z}_{il}b_{jl}\left(Z\right) \right\},\quad\mbox{for all $i,j=1,\dots,q$.}
\tilde \sigma(D) (f_j, \tilde h_j )(x) = \int \sigma(\xi,-\xi-\eta) \chi_1(\frac{\xi}{2^j}) \chi_2(\frac{\eta}{2^j}) \hat{f_j}(\xi) \widehat{\tilde h_j}(\eta) e^{i (\xi +\eta) \cdot x}d\xi d\eta,
B_1 =B_{(00)(01)}\cup B_{(00)(11)}, B_{-1}= B_{(00)(0-1)}\cup B_{(00)(-1-1)}, B_2 =B_{(01)(01)}\cup B_{(11)(11)}, B_{-2}=B_{(0-1)(0-1)}\cup B_{(-1-1)(-1-1)}, B_3 =B_{(01)(00)}, B_{-3}=B_{(0-1)(00)}, B_4 =B_{(01)(11)}, B_{-4}=B_{(0-1)(-1-1)},
\sigma=S\left(\phi_1^i \frac{u_1}{u_0},\ \phi_1^j (x-uu_1^{-1}),\ \phi_1^k (2t+u_1^{-2}),\i=1,\ 2,\ \ldots,\n_1,\j=1,\ 2,\ \ldots,\n_2,\k=1,\ 2,\ \ldots,\n_3\right)u_0,
{\mathcal{L}_{S D}}= \beta^{2}g \left( m^{2}A^{2}/ 2 \beta^{2}\right) - \frac{\beta^{2}}2 h \left( m^{2}B^{2}/ 2 \beta^{3}\right) - \frac{\chi \theta}{2}\varepsilon^{\mu \nu \rho \lambda}\partial_{\mu}B_{\nu \rho}A_{\lambda}.
V^{C}_{\sigma}(C_{1}, C_{2}) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}f_{XYZS}(x,y,z,s) G_{\sigma}(x-y) G_{\sigma}(z-s) \mathrm{d}x\mathrm{d}y \mathrm{d}z\mathrm{d}s
G_{\mu\rho}^{ac}(\theta)\, M^{\rho\nu}_{cb}(\theta) \;=\;\delta_\mu^\nu \delta^a_b \;\;,\;\;M^{\mu\nu}_{ab}\,=\, g^{\mu\nu}\delta_{ab}+ g^{-\frac{1}{2}}\epsilon^{\mu\lambda\nu}f^{acb}\theta_\lambda^c\;.
\bigl((\tilde{\Delta}\otimes\operatorname{id})(\Delta a)\bigr)(1\otimes1\otimes b)=(\Delta\otimes\operatorname{id})\bigl((\Delta a)(1\otimes b)\bigr), &(a\otimes1\otimes1)\bigl((\operatorname{id}\otimes\tilde{\Delta})(\Delta b)\bigr)=(\operatorname{id}\otimes\Delta)\bigl((a\otimes1)(\Delta b)\bigr).
\frac{g(3)}{g(s)}= n^{s-3}\cdot p^{\binom{s}{3}-1}= \Big[np^{\frac{s^2+2}{6}}\Big]^{s-3}\geq \Big[np^{\frac{s(s+1)}{6}}\Big]^{s-3}\geq \Big[np^{\frac{(t-1)(t-2)}{6}}\Big]^{s-3}\geq 1.
H_{D}= \int d^{3}\sigma \lambda^{\mu}( \tau , \vec{\sigma}){\cal H}_{\mu}( \tau , \vec{\sigma}) + \lambda_{\tau}( \tau , \vec{\sigma}) \pi^{\tau}( \tau , \vec{\sigma}) - A_{\tau}( \tau , \vec{\sigma}) \Gamma ( \tau , \vec{\sigma}) ] .