harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{\nabla}_{X}\tilde{\nabla}_{Y}\tilde{\xi}=\frac{X(\beta )}{\beta^{2}}\mathcal{A}Y+\frac{1}{\beta}\nabla_{X}\nabla_{Y}\xi +\frac{(\beta^{2}-\gamma )}{\beta^{3}}g(\mathcal{A}X,\mathcal{A}Y)\xi .
\phi(0) \ge \lim_{t\to\infty}\phi(t) &= \lim_{t\to\infty}\left(S(C|M)_{\hat{\rho}_{CM}(t)}- \lambda\,S(A|M)_{\hat{\rho}_{AM}(t)}- (1-\lambda)\,S(B|M)_{\hat{\rho}_{BM}(t)}\right) &=n\left(\lambda\ln\frac{\eta}{\lambda}+ \left(1-\lambda\right)\ln\frac{|1-\eta|}{1-\lambda}\right)\;,
\sqrt{N}(\widehat{T}_N(f)-Err(f))=\dfrac{\sqrt{N}}{K}\sum_{k=1}^K\sum_{i=0}^{2m-1}\sum_{i-m<|y|\le m}\dfrac{1}{\sharp S_k(N)}\sum_{j\in S_k(N)}h_N(y,i,k,j),
\zeta(\sigma_{0}) = \dfrac{1}{4(-4+6\alpha+3\beta)^2}(2(8-6\alpha-3\beta)(6\alpha-6-3\beta)(6\alpha-6-3\beta &\quad{}+ \sqrt{(6-6\alpha+3\beta)^2 +8(6\alpha+3\beta-4)}) &\quad+48(6\alpha+3\beta-4)(2-2\alpha-\beta)),
\lim_{n \to \infty}\frac{1}{n}\inf_{y \in Z}\sum_{i=0}^{n-1}\mathfrak{d}(\sigma^ix,\sigma^{i}y) =\inf\left\{\int \mathfrak{d}\,dm \colon m \in \bigcup_{\nu \in \mathcal{M}_\sigma \colon \nu(Z)=1}J(\mu,\nu)\right\}&=\inf\{\overline{d}(\mu,\nu) \colon \nu \in\mathcal{M}_\sigma \nu(Z)=1\}.
\beta_{j,j'}= \beta \hat{\otimes}1: S{\frak C}(D_{r_i}^j) \to S C_0(V_{j,j'},Cl(V_{j,j'})) \hat{\otimes}{\frak C}(D_{r_i}^j) & \cong S{\frak C}(E_{r_i}^{j.j'}) & \hookrightarrow S{\frak C}(V_{j'})
T(\hat{\pi}^{(n)}_n) = \hat{T}^{(n)}(\hat{\pi}^{(n)}_n) = \hat{T}^{(n)}_n \leq \hat{T}^{(n)}(\pi) =\sum_{i=1}^{r}t^{(n)}(g_i) \leq \sum_{i=1}^{r}t(g_i) = T(\pi).
\sum_{k=0}^{n-1}\frac{(q;q^2)_k^2 (q;q^2)_{2k}}{(q^2;q^2)_{2k}(q^6;q^6)_k^2}[8k+1]q^{2k^2}\equiv q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr) \pmod{[n]\Phi_n(q)^2}, \sum_{k=0}^{(n-1)/2}\frac{(q;q^2)_k^2 (q;q^2)_{2k}}{(q^2;q^2)_{2k}(q^6;q^6)_k^2}[8k+1]q^{2k^2}\equiv q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr) \pmod{[n]\Phi_n(q)^2}.
\sum_{i\geq 0}G_i'(x,y)t^i-\sum_{i\geq 0}f_n\circ G_i'(x,y)t^{i+n}= x\ast y-(f_n(x)\ast y+x\ast f_n(y))t^n+f_n(x)\ast f_n(y)t^{2n}+\sum_{i\geq n}G_i(x,y)t^i-\sum_{i\geq n}(G_i(f_n(x),y)+G_i(x,f_n(y)))t^{i+n}+\sum_{i\geq n}G_i(f_n(x),f_n(y))t^{i+2n}.
(\nabla^2+k^2)u=0 D':=\mathbf{R}^3\setminus D, u_N=\zeta u S=\partial D, u=u_0+v, u_0=e^{ik\alpha\cdot x}, \alpha \in S^2, \frac{\partial v}{\partial r}-ikv=o\left(\frac{1}{r}\right),r=|x|\to\infty.
\|u_{\epsilon}\|_{q}^{q}=R^{q}\omega\epsilon^{\frac{(N-4)q}{2}}\int_{0}^{2}\frac{(\varphi^{q}(r)-1)r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{(N-4)q}{2}}}dr+R^{q}\omega\epsilon^{\frac{(N-4)q}{2}}\int_{0}^{2}\frac{r^{N-1}}{(\epsilon^{2}+r^{2})^{\frac{(N-4)q}{2}}}dr =I_{1}(\epsilon)+I_{2}(\epsilon).
\frac{P_{p,n,m}^{rc}(t)\cdot t^{-(np+mp)}\cdot \frac{t^{-n^2-nm}\cdot P_{n,m}(t)}{|GL_n(\mathbb F)|}\cdot|GL_n(\mathbb F)|}{|\mathbb F^{n\times n}\times\mathbb F^{n\times m}\times\mathbb F^{p\times n}\times\mathbb F^{p\times m}|}=P_{p,n,m}^{rc}(t)\cdot P_{n,m}(t).
\frac{\partial F}{\partial U}(1,\, U) = \frac{r}{2}- \frac{1}{2}\sum_{i=1}^r\frac{m-m_i+\lambda-mU}{[(m-m_i+\lambda-mU)^2+4\lambda m_i]^{1/2}}> 0, \frac{\partial^2F}{\partial U^2}(1,\, U) = \frac{m}{2}\sum_{i=1}^r \frac{4\lambda m_i}{\{(m-m_i+\lambda-mU)^2+4\lambda m_i\}^{3/2}}> 0 .
C l_{3 , 0}~ [ \sim M_{2}({\cal C}) ] ~ , C l_{1 , 2}~ [ \sim M_{2}({\cal C}) ] ~ , C l_{0 , 3}~ [ \sim{\cal H}\oplus{\cal H}] ~ , C l_{2 , 1}~ [ \sim M_{2}({\cal R}) \oplus M_{2}({\cal R}) ] ~ .
\int_{\mathcal L_{2j}}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh = -p \int_{\mathcal L_{2j}}1dh = -p \mathrm{vol}(\mathcal L_{2j}) = -p^{1-2j}(1-p^{-1})^2,
C_{\b c d}^{a}\equiv R_{\b c d}^{a}+{\frac{2}{d - 2}}\left( \delta_{\ [ d}^{a}R_{c ] b}+ g_{b [ c}R_{d ]}^{a}\right) +{\frac{2}{( d - 1 ) ( d - 2 )}}\delta_{\ [ c}^{a}g_{d ] b}R \, ,
\Phi_0=\sum_{N,M,P_i\geq0}\frac{\Gamma(3P_1+1)\;(-z_1)^{P_1}z_2^{P_2}z_3^{P_3}(-x)^My^N}{\left\{\begin{array}{c}\Gamma(P_1+M-N+1)\Gamma(P_2-2P_3+N+1)\Gamma(P_1-2P_2+P_3+1) \times\Gamma(N-3M+1)\Gamma^2(P_1+1)\Gamma(M+1) \times\Gamma(P_3+M-N+1)\Gamma(P_2-P_1+1)\end{array}\right\}}.
\hat{\bf G}= - \hat{\tau}_3 \left[{\bf I}\, \partial^2 + \hat{\bf P}_\pi \hat{\cal M}_\pi^2 + \hat{\bf P}_\sigma \hat{\cal M}_\sigma^2 \right]^{- 1}\, \rule[-3mm]{.14mm}{8.5mm}\raisebox{-2.85mm}{\scriptsize{$\; \vec{\varphi}_1 = \vec{\varphi}_2 = \vec{\varphi}$}}.
S = \{s\}\cup \{3s,4s,\ldots,(l_1 + 2)s \}&\cup \{(l_1 + 5)s,(l_1 + 6)s, \ldots ,(n+3)s\}&= [1]_s^{n+3}\setminus \{2s, (l_1+3)s, (l_1+4)s \}
[{\mit \Psi}( u , x^{\prime}, y^{\prime}, p^{\prime}) ,{\mit \Psi}^{\dagger}( u , x^{\prime \prime}, y^{\prime \prime}, p^{\prime \prime}) ] = \delta ( x^{\prime}- x^{\prime \prime}) \delta ( y^{\prime}- y^{\prime \prime}) \delta ( p^{\prime}- p^{\prime \prime}) ,
p_{|h_{1}|,\cdots,|h_{N}|}(x_{1},\dots,x_{N}) &=\prod_{\substack{k=1 (\mu_{1}\triangleq 0)}}^{N}\!\!\frac{2x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}e^{-\frac{x_{k}^{2}+\mu_{k}^{2}x_{1}^{2}}{\sigma^{2}(1-\mu_{k}^{2})}}I_{0}\left ({\frac{2~\mu_{k}x_{1}x_{k}}{\sigma^{2}(1-\mu_{k}^{2})}}\right)\!,
A = [ B_{R_{X_2}}(X_1)\setminus B_R(X_1) ] \cap [B_{R_{X_2}}(X_2) \setminus B_R(X_2)] R_1 = B_{R_{X_2}}(X_1)\setminus B_{R_{X_2}}(X_2) R_2 = B_{R_{X_2}}(X_2)\setminus B_{R_{X_2}}(X_1) R_3 = B_R(X_2)\setminus B_R(X_1) R_4 = B_R(X_1)\setminus B_R(X_2).
C^a(\xi,0) = \frac{i\xi}{\Lambda(\xi)(e^{2\xi}- 1)}\left [2J(\xi) - \xi J'(\xi) + J(\xi) J'(\xi) \right], C^a(0,-\eta) = -\frac{i\eta}{\Lambda(\eta)(e^{2\eta}- 1)}\left[ 2J(\eta) - \eta J'(\eta) - J(\eta)J'(\eta) \right].
\int_{F^\times \times K}|b|^{s}\chi(b)|a|^{-1}\left( r_i(k)f\right)(0,0,a^{-1}b)dk d^\times b = |a|^{s-1}\chi(a)\int_{F^\times \times K}\chi(b)|b|^{s}\left( r_i(k)f\right)(0,0, b)d^\times b.
{\displaystyle \partial_{t}\theta^{0}+\nabla\cdot\left(-\mathbb{K}\nabla\theta^{0}\right)+g_{0}\frac{\left|\Gamma_{R}\right|}{\left|Y_{1}\right|}\theta^{0}=\sum_{i=1}^{N}\left(\mathbb{T}^{i}\nabla^{\delta}u_{i}^{0}\right)\cdot\nabla\theta^{0}}\quad\mbox{in}\;Q_{T},
C^{sm}(G, L) = \prod_{g\in G/G_0}C^{sm}(gG_0, L) = \prod_{g\in G/G_0}\varinjlim_{H \subset G_0}C^{sm}(g G_0 / H, L) = \varinjlim_{\substack{g \in G / G_0 H_g \subset G_0}}\prod_{g \in G / G_0}C^{sm}(g G_0 / H_g, L).
F(y)=\begin{cases}F_{1},\ y\in\Pi_{1}=\{y=(y^{1},\ldots,y^{d})\in\mathbb{R}^{d}:\y^{d}<A\}F_{2},\ y\in\Pi_{2}=\{y=(y^{1},\ldots,y^{d})\in\mathbb{R}^{d}:\y^{d}\geqslant A\}\end{cases},
H_3=2\mu \delta_{J_1,J_2+J_3}{\sqrt{J_1 J_2 J_3}\over N}{J_3\over J_1}{\sin^2(\pi ny)\over\pi^2 (ny-m)^2}\Pi^{\prime J_1}_{2}(q_n)\bar{\Pi}^{\prime J_2}O^{\prime J_3}_{2}(s_m).
(F_{A})^{o}(\xi)=\sup_{x\in \mathbb R^{n}\setminus\{0\}}\frac{\langle x,\xi\rangle}{F_{A}(x)}=\sup_{y\in \mathbb R^{n}\setminus\{0\}}\frac{\langle A^{T}y,\xi\rangle}{F(y)}=\sup_{y\in \mathbb R^{n}\setminus\{0\}}\frac{\langle y, A\xi\rangle}{F(y)}= (F^{o})_{A}(\xi).
\hat{T}(x)=exp(-\frac{ix\cdot\hat{p}}{\hbar})=1-\frac{ix\cdot\hat{p}}{\hbar}-\frac{(x\cdot\hat{p})^{2}}{2\hbar^{2}}+\frac{i(x\cdot\hat{p})^{3}}{6\hbar^{3}}+\cdot\cdot\cdot
\zeta^{(d)}_{M}(s) =\Vert M\Vert\sum^{\infty}_{h=0}\sum_{z\in R^{(d)}_{M}(h)}m^{(d)}_{M}(z)S^{(d)}_{z}(s+2d) &=\Vert M\Vert\sum^{\infty}_{h=0}\sum_{z\in R^{(d)}_{M}(h)}m^{(d)}_{M}(z)T^{(d)}_{z}(u_s),
R_{M N}^{a b}= E_{M}^{\Lambda}E_{N}^{\Pi}\left\{\partial_{\Lambda}\Omega_{\Pi}^{\a b}+ \Omega_{\Lambda}^{\a c}\Omega_{\Pi c}^{\b}- \left( - \right)^{\Lambda \Pi}\left( \Lambda \leftrightarrow \Pi \right) \right\}
\zeta_{AV,2}(s_1,s_2,s_3) = \sum_{k=3}^\infty \left( \sum_{k/2 <m \leq k-1}\frac{1}{m^{s_1}(k-m)^{s_2}k^{\sigma_3}}\right) \frac{1}{k^{it_3}}&= \overline{\sum_{k=3}^\infty \overline{a_k}k^{it_3}}
I_{\pm}= 4\pi \int_0^{\vert a\vert^{3/2}}\frac{1+\vert\beta\vert^2\vert a \vert^2}{(1+\vert\beta\vert^2\vert a\vert^2 + 4\vert\lambda\vert^2 \vert a\vert^2 \vert b\vert^2)^2}d\vert b\vert^2.
H_{3}^{( 1 )}= x_{+}S_{0}^{2}- x_{-}S_{-}^{2}- 2 x_{0}S_{0}S_{-}+ 2 x_{+}k S_{0}+ 2 x_{0}( s - k ) S_{-}+ x_{+}( k^{2}- \frac{1}{1 2}) ,
P_{\tilde{p}}={\Omega_{p+2}\over{\sqrt{2}\kappa_D}}(p+1)(r_+r_-)^{{p+1}\over 2},\ \ \ \{\cal M}_{\tilde{p}}={{\Omega_{p+2}}\over{2\kappa^2_D}}[(p+2)r^{p+1}_+-r^{p+1}_-].
\|\mu\|_0 = \frac{1}{2}\bigl( \bigl|\|\mu^+\|_1+\| \mu^-\|_1\bigr| +\bigl| \|\mu^+\|_1 - \|\mu^-\|_1 \bigr|\bigr) = \frac{1}{2}\bigl( \|\mu\|_1 + \bigl| \mu^+(K)-\mu^-(K)\bigr| \bigr) = \frac{1}{2}\bigl( \|\mu\|_1 + \bigl| \mu(K)\bigl| \bigr)
\varphi ( x ) = \sum_{{\vec{k}}}{\frac{1}{\sqrt{( 2 \pi^{3})^{3}2 \omega ({\vec{k}})}}}( a_{{\vec{k}}}e^{- i \omega ({\vec{k}}) x^{0}+ i{\vec{k}}\cdot{\vec{x}}}+ a_{{\vec{k}}}^{\dag}e^{+ i \omega ({\vec{k}}) x^{0}- i{\vec{k}}\cdot{\vec{x}}})
R_{-}^{+}f_{-}^{+}= e^{+i\theta/2}f_{-}^{+}\hspace{0.5cm}R_{+}^{-}f_{+}^{-}= e^{+i\theta/2}f_{+}^{-}\hspace{0.5cm}R_{-}^{+}g_{-}^{+}= e^{+i\theta/2}g_{-}^{+}\hspace{0.5cm}R_{+}^{-}g_{+}^{-}= e^{+i\theta/2}g_{+}^{-},
Q ( \eta ) = \left({\frac{1 + \eta^{2}}{\eta^{2}}}\right)^{\frac{1}{2 - \gamma}}\left[ c_{1}\sin \left({\frac{2}{2 - \gamma}}\mathrm{{a r c t a n g}}\eta \right) + c_{2}\cos \left({\frac{2}{2 - \gamma}}{\mathrm{a r c t a n g}}\eta \right) \right]
M_{1}= \left( \begin{matrix}{{0}}\\{{m G_{1}e_{R}}}\\{{n G_{2}u_{R}}}\\{{n G_{3}d_{R}}}\\ \end{matrix}\right) ~ , ~ ~ M_{2}= \left( \begin{matrix}{{0}}\\{{m G_{4}\mu_{R}}}\\{{n G_{5}c_{R}}}\\{{n G_{6}s_{R}}}\\ \end{matrix}\right) \, .
A\omega_z(y)=\frac1{b_0-a_0}\int_{a_0}^{b_0}\omega_z(t)\,dt\le\frac1{b_0}\int_0^{b_0}\omega_z(t)\,dt\le\frac1{b_0}\int_0^\infty\omega_z(t)\,dt=\frac2{b_0z}.
{\varphi_2}_t = d\Delta \varphi_2+\theta x\in \Omega, 0<t<T d\frac{\partial \varphi_2}{\partial \eta}=d\frac{\partial\varphi_0}{\partial\eta}x\in M, 0<t< T \varphi_2 =\varphi_0 x\in\Omega , t=0
P_N(a)\sim \frac{1}{1-{\rm e}^{\vartheta^\star_Z}}\frac{1}{\sqrt{2\pi N \Lambda_Z''(\vartheta_Z^\star)}}{\rm e}^{-NI_Z(a)}= \frac{1}{1-a\,\frac{1+\lambda}{1+a}}\frac{1}{\sqrt{2\pi N a(a+1)}}{\rm e}^{-NI_Z(a)}\,.
B = m + \frac{e^{2}}{2 \pi}\frac{\lambda + 1}{\lambda}\int_{0}^{\infty}d k \frac{k B}{A^{2}k^{2}+ B^{2}}\frac{1}{\sqrt{( p^{2}+ k^{2}+ \mu^{2})^{2}- 4 p^{2}k^{2}}}
E_{T}^{\prime}:y^{2}+\frac{a^{3}-2a^{2}b-2ab^{2}+2b^{3}}{2}xy-\frac{a^{2}b^{3}\left( a-2b\right) \left( a-b\right) \left( a^{2}-3ab+b^{2}\right)}{8}& =x^{3}-\frac{a\left( a-2b\right) \left( a-b\right) b^{3}}{4}x^{2}.
x_c\mathcal{O}(x_A)= x_cx_{A(\pi(k))}\cdots x_{A(\pi(j))}x_{A(\pi(j-1))}\cdots x_{(A(\pi(1))}\mathcal{O}(x_cx_A)= x_{A(\pi(k))}\cdots x_{A(\pi(j))}x_cx_{A(\pi(j-1))}\cdots x_{(A(\pi(1))}x_c\mathcal{O}(x_A) \mathcal{O}(x_cx_A)= [x_c,x_{A(\pi(k))}\cdots x_{A(\pi(j))}]x_{A(\pi(j 1))}\cdots x_{(A(\pi(1))}
\sum_{k=0}^{n-1}\frac{(q^{d+r};q^d)_k^{d-r-1}(q^r;q^d)_k^{r+1}q^{dk}}{(q^d;q^d)_k^d}\equiv\frac{(1-q^r)^{r+1}(q^d;q^d)_{n-1-(n+r)/d}}{(-1)^{n-1-(n+r)/d}(q^d;q^d)_{(n+r)/d}^{d-1}}q^{A(d,n,r)-r},
Z = \int D A D B D a e^{- S^{\prime}[ A , B ]}d e t ( i \raise . 1 5 e x \mathrm{/}\kern - . 5 7 e m \mathrm{\partial}+ \not \! \! A ) d e t ( i \gamma_{0}\partial_{t}+ a \gamma_{0}+ \not \! \! B ) .
\overline{c_1}(t) + \overline{c_3}(t) = \overline{c_{1,0}}+ \overline{c_{3,0}}=: M_{13}\overline{c_2}(t) + \overline{c_3}(t) + \overline{c_4}(t) = \overline{c_{2,0}}+ \overline{c_{3,0}}+ \overline{c_{4,0}}=: M_{234}.
\delta \Gamma^\sigma_{\sf{anom.}}= \frac{- 36 g^2}{(4 \pi)^2 M^2}\int_0^1 dx dy \; \delta(x+y-1) \Biggl\{\epsilon m Q^\sigma \left[\frac{2}{\epsilon}- \gamma + \ln \left(\frac{4 \pi \mu^2}{\Delta}\right) \right]\Biggr\}.
Z_{n+1,x}= \begin{cases}\displaystyle \frac{X_{n+1,x}}{W_{n,x}}, \mbox{if}W_{n,x}\geq t^{\ast}, \displaystyle \frac{X_{n+1,x}- t^{\ast}}{W_{n,x}+t^{\ast}}+ \frac{t^{\ast}}{\Tilde{t}}, \mbox{if}W_{n,x}< t^{\ast}.\end{cases}
\left \{K(\Lambda),K(\bar{\Lambda}) \right \}~=~ \sum_{I=1}^{n_1}k^1_I\int_{\partial D_1}\Lambda^1_Id\bar{\Lambda}^1_I~+\sum_{J=1}^{n_2}k^2_J\int_{\partial D_2}\Lambda^2_Jd \bar{\Lambda}^2_J~,
T_{z z}^{\Psi}= T_{z z}, \hspace{1 e m}T_{z \bar{z}}^{\Psi}={\cal O}_{0}\Psi \hspace{1 e m}T_{\bar{z}\bar{z}}^{\Psi}= T_{\bar{z}\bar{z}}, \hspace{1 e m}T_{\bar{z}z}^{\Psi}= \overline{{{\cal O}}}_{0}\Psi .
\int_{\Omega}\ln|x-y|dy \leq\int_{B_{1}(x)}\ln|x-y|dy+\int_{\Omega\smallsetminus B_{1}(x)}\ln|x-y|dy & \leq C+|\Omega|\ln(\max\{diam(\Omega),1\})d=2 \int_{\Omega}|x-y|^{2-d}dy \leq\int_{B_{1}(x)}|x-y|^{2-d}dy+\int_{\Omega\smallsetminus B_{1}(x)}|x-y|^{2-d}dy & \leq C+|\Omega|d>2
[y_1, y_1]=y_4, [y_2, y_1]=\frac{\alpha_5}{\alpha_1}y_4+\theta y_5, [y_1, y_3]=\frac{\beta_4}{\gamma_4}y_5, [y_2, y_3]=\frac{\gamma_2}{\gamma_4}y_5, [y_3, y_2]=y_5.
\log |a(t, s)| \lesssim \big( \log (|a(t,s)| + |b(t,s)| ) \big)^2 = \big( \log \|G (t,s) \|_{op}\big)^2 = \big( \log \|G_{-}(t,s) \|_{op}+ \log \|G_{+}(t,s) \|_{op}\big)^2 \lesssim \big( \log \|G_- (+\infty, s) \|_{op}\big)^2 + \big( \log \|G_+(t,s) \|_{op}\big)^2 \lesssim \log |a_{-}(+\infty, s)| + \log |a_{+}(t, s)|.
\mathbb{P}(\bigcup_{i=1}^{n}A_{i})=\sum_{j=1}^{n}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n-1}(-1)^{j-1}S_{j}\le\sum_{j=1}^{n-3}(-1)^{j-1}S_{j}\le...\le\sum_{j=1}^{1}(-1)^{j-1}S_{j}
\overline{{P}}= \varepsilon^{\mu , \mu}+ \varepsilon^{\widetilde{\mu}, \widetilde{\mu}}P = \varepsilon^{0 , 0}+ \frac 1 2 \varepsilon^{[ \mu \nu ] , [ \mu \nu ]}+ \varepsilon^{\widetilde{0}, \widetilde{0}}
\dfrac{M_{2}(\mathcal{NC}(G))}{|e( \mathcal{NC}(G))|}- \dfrac{M_{1}(\mathcal{NC}(G))}{|v( \mathcal{NC}(G))|}= \dfrac{64n^{3}(n-6)+ 64n(13n - 12) + 256}{24(n-1)(2n-1)}:= \dfrac{f(n)}{g(n)}.
\mathbf{r}^{\prime \prime \prime}(s)=\left(\begin{array}{c}0,-\kappa^{\prime}\sinh \left( \int \tau (s)\,ds\right) -\cosh \left( \int\tau (s)\,ds\right) \kappa (s)\tau (s), \cosh \left( \int \tau (s)\,ds\right) \kappa^{\prime}+\sinh \left( \int\tau (s)\,ds\right) \kappa (s)\tau (s)\end{array}\right).
I^{(1)}\cdot J^{(1)}=\sum_{n=-1}^{+1}(-1)^{n}I_{n}^{(1)}J_{-n}^{(1)}=I_{0}^{(1)}J_{0}^{(1)}-I_{-1}^{(1)}J_{+1}^{(1)}-I_{+1}^{(1)}J_{-1}^{(1)},
\beta_i:=\alpha_{7-i}\ (1\leq i\leq 4),\beta_5:=\alpha_1,\beta_6:=\alpha_2+\alpha_4+\alpha_5+\alpha_6+\alpha_7, &\gamma_1:=\alpha_1+2\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+\alpha_6.
M\sum\limits_{|\alpha|=0}^{3}\left(\|\sqrt{B_0}\partial_\tau^\alpha u_0\|_{L^2}+\|\sqrt{A_0}\partial_\tau^\alpha\tilde{h}_0\|_{L^2}\right)+\sum\limits_{|\alpha|=0}^{2}\left(\|\sqrt{B_0}\partial_\tau^\alpha\partial_y u_0\|_{L^2}+\|\sqrt{A_0}\partial_\tau^\alpha\partial_y\tilde{h}_0\|_{L^2}\right)\le\varepsilon,
G_x(u)- G_x(u')=\int_{F_\nu(a)-u}^{F_\nu(a)-u'}F_\nu^{-1}(v)\,dv - \int_{1+u'-F_\mu(x)}^{1+u-F_\mu(x)}F_\nu^{-1}(v)\,dv \leq (u-u')a - (u-u')b.
n_p\bigl(a,a+b,a J_{k-1}(v)+b J_k(v)\bigr) &=\frac{1}{2}\biggl(-v(a-r^2)+(a-1)(b-1)+\frac{v(a+r)(a-r)J_{k-1}(v)}{J_k(v)}&+v(a-r)\bigl(J_k(v)-J_{k-1}(v)\bigr)\biggr) &+\frac{p}{2}a J_k(v)\bigl(2(v a+b)-v(J_k(v)-J_{k-1}(v)\bigr)-\frac{p^2}{2}a v J_k(v)\bigl(J_k(v)-J_{k-1}(v)\bigr)\,,
Z\left( \vartheta_{2}+i\rho_{k}\right)_{II}=M\sinh \left( \vartheta_{2}+i\rho_{k}\right)_{II}+\chi (\vartheta_{2}-\vartheta_{1}+i\rho_{k})_{II}+\ldots =2\pi I^{(k)}_{2}\, \, ,k=0,\ldots ,s
|\partial^\alpha \Psi_1| \le C\sum_{|{\alpha_1}|\le|\alpha|}\|\partial^{\alpha_1}f\|_{L^2_{p,\mathcal{I}}}\|\partial^{\alpha-{\alpha_1}}f\|_{L^2_{p,\mathcal{I}}}\cr \le C\sqrt{E(f)(t)}\sum_{|{\alpha_1}|\le |\alpha|}\|\partial^{\alpha_1}f\|_{L^2_{p,\mathcal{I}}}.
\mathcal{U}_2(\underline{\tilde{g}}) - \min_{g_q\in \mathbb{Q}}\mathcal{U}_2(g_c^{-1}{\tilde{g}}g_q g_c) =\mathcal{U}_2(\underline{\tilde{g}}) - \min_{g_q\in \mathbb{Q}}\mathcal{U}_2(\underline{\tilde{g}}g_c^{-1}g_q g_c) & = \mathcal{U}_{1}(\tilde{g}) - \min_{g_q\in \mathbb{Q}}\mathcal{U}_{1}(\tilde{g}g_q)> \delta,
\phi_m(q_j) = \lambda_m(q_j) \int_0^1 G_0 (x,\xi)(\phi_m(q_j)(\xi)-\phi^*_m(\xi))d\xi &-\int_0^1 G_0(x,\xi) q_j(\xi) (\phi_m(q_j)(\xi)-\phi^*_m(\xi))d\xi+ + \lambda_m(q_j) \int_0^1 G_0(x,\xi)\phi^*_m(\xi)d\xi -\int_0^1 G_0(x,\xi) q_j(\xi) \phi^*_m(\xi)d\xi.
F(t,x,y)=\int_{\mathbb{R}}\exp\big[\int_0^t\int_{\mathbb{R}}ix\psi(s,\zeta)\tilde{N}(ds,d\zeta) +\int_0^t ix\beta(s)dB(s) & +\int_t^{T_0}\int_{\mathbb{R}}(e^{ix\psi(s,\zeta)}-1-ix\psi(s,\zeta))\nu(d\zeta)ds-\int_t^{T_0}\frac{1}{2}x^2\beta^2(s)ds-ixy\big]dx.
\Psi(\beta v)= \sum_{j=1}^{N}\Psi(\beta v^j) =\frac{1}{2}\sum_{j=1}^{4N}\psi(\beta z_j) & \leq \frac{4N}{2}\psi\left(\beta \varrho \left(\frac{1}{4N}\sum_{i=1}^{4N}\psi(z_i)\right)\right) & = 2N \psi\left(\beta \varrho \left(\frac{\Psi(v)}{4N}\right)\right).
\Vert \mathbf{\boldsymbol{\chi}}(P^{t+1})\Vert_1 = \sum_{i=1}^\lambda \mathbf{\boldsymbol{\chi}}(P^t) \boldsymbol{\chi}(P_{i})^T = \Vert \mathbf{\boldsymbol{\chi}}(P^t)\mathbf{D}_{\mathcal{C}}\Vert_1\,.
\frac{1}{1-\theta}\|u_h^{n}\|^2+4\Delta t\sum_{k=2}^n\|\nabla u_h^{k-\theta}\|^2+4\gamma\Delta t\sum_{k=2}^n\|\sigma_h^{k-\theta}\|^2 \leq C\Delta t\sum_{k=0}^n\|u_h^{k}\|^2+2\Delta t\sum_{k=1}^n\|g^{k}\|^2.
(F(x,u)-F(\tilde x,u))^T(x-\tilde x) = \sum_{l=1}^{\mathcal{L}}\sum_{i=1}^N(c_{il}'(g_{il})-c_{il}'(\tilde g_{il}))(g_{il}- \tilde g_{il}) - \sum_{l=1}^{\mathcal{L}}\sum_{i=1}^N p_{l}'(u_l) (s_{il}- \tilde s_{il})^2.
L(y_{i1}):= ba^{-10^{J+10i-4}}L(y_{i2}):= a^{10^{J+10l_{i2}}}ba^{-10^{J+10i-3}}L(z_{i3}):= a^{10^{J+10i-2}}ba^{-10^{J+10k_{i3}}}L(z_{i4}):= a^{10^{J+10i-1}}b .
M_{123123}(\mathbf{z}) = \begin{pmatrix}1 0 0 1 0 0 0 1 0 0 1 0 0 0 1 0 0 1 -1 0 0 1 0 0 z_{5,1}-1 0 -z_{5,1}1 0 z_{6,1}z_{6,2}-1 -z_{6,1}-z_{6,2}1 \end{pmatrix}.
\phi \mathcal{T}_{U}X+\mathcal{B}\nabla^{^{M_1}}_{U}X-\eta(X)U =\hat{\nabla}_{U}\mathcal{B}X+\mathcal{T}_{U}\mathcal{C}X, \omega \mathcal{T}_{U}X+\mathcal{C}\nabla^{^{M_1}}_{U}X =\mathcal{T}_{U}\mathcal{B}X+\mathcal{H}\nabla^{^{M_1}}_{U}\mathcal{C}X,
R_{T_{N}}\left( \rho\right) =\mathcal{L}^{-1}\left(\frac{e^{x}}{\sqrt{2x}}\mathcal{L}^{-1}\left( \left( 2\pi\right)^{3/2}e^{\frac{1}{2}t^{2}}m_{\Sigma}\left( t/\kappa\right) \right) \left(\sqrt{2x}\right)\right) \left( \rho^{2}\right)
\widehat{m}_1^{[s(x)]}(x)=m^{[s(x)]}_1(x)+\frac{U_p'(x)}{r_p(x)U_p(x)}\sum_{k=2}^\gamma\frac{m^{[s(x)]}_k(x)}{(k-1)!}r^{k-1}(x)+O(p(x))
| \kappa , \Omega \rangle ( z ) = \sqrt{A_{1}( \kappa )}\, \sum_{m = 1}^{\infty}V_{m - 1}^{( 1 )}( \kappa ) \, \frac{z^{m}}{m}\equiv \sum_{m = 1}^{\infty}\langle \kappa , 1 | m - 1 , 1 \rangle \, \frac{z^{m}}{\sqrt{m}}.
\beta_k(x)=\sup\Big\{\epsilon\geq 0\big|\exists_{\mathcal{C}\in C_{n,k}(\epsilon)}\exists_{W\in G_{m,k}}\exists_{T\in O(V_{\mathcal{C}},W)}\forall_{y\in (x+\mathcal{C})\cap B(x,\epsilon)}&u(x)-u(y)=T(x-y) \Big\},
\left(\frac{2}{e^{t}+1}\right)^{r}e^{xt}=\sum_{n=0}^{\infty}E_{n}^{\left(r\right)}\left(x\right)\frac{t^{n}}{n!},\quad\left(\mbox{see \cite{key-1,key-2,key-3,key-4,key-5,key-6,key-7,key-8,key-9,key-10,key-11,key-12}}\right).
\biggl(\sum_{j=1}^dc_j\alpha_j(x)\biggr)^m =\sum_{l_1+\dots+l_r=m}\frac{m!}{l_1!\cdots l_r!}c_1^{l_1}\cdots c_r^{l_r}\alpha_1(x)^{l_1}\cdots \alpha_r(x)^{l_r}.
\left\langle P\left( a_{\hbar},a_{\hbar}^{\dag}\right) \right\rangle_{U_{\hbar}\left( \alpha\right) \psi}=P\left( \alpha,\bar{\alpha}\right) +O\left( \sqrt{\hbar}\right) \operatorname*{Var}\nolimits_{U_{\hbar}\left( \alpha\right) \psi}\left(P\left( a_{\hbar},a_{\hbar}^{\ast}\right) \right) =O\left( \sqrt{\hbar}\right) .
\hat{\pi}(t,z) =-\frac{\alpha(t)\int_{\mathbb{R}_+}y(t,x,z)p'(t,x,z)dx}{\beta^2(t)\int_{\mathbb{R}_+}y(t,x,z)p''(t,x,z)dx}&=-\frac{\alpha(t)\mathbb{E}[p'(t,X(t),z)|\mathcal{R}_t]}{\beta^2(t)\mathbb{E}[p''(t,X(t),z)|\mathcal{R}_t]}.
\frac{A_i^{(0)}}{A}e^{(ball)}(A^{(0)}_i) + \frac{A_j^{(0)}}{A}e^{(ball)}(A^{(0)}_j) = \left( \theta^{2/3}+ (1-\theta)^{2/3}\right) \left(A^{(0)}_i + A^{(0)}_j \right)^{2/3}\frac{P(B)}{A\, |B|^{2/3}}& + \left( \theta^{5/3}+ (1-\theta)^{5/3}\right) \left( A^{(0)}_i + A^{(0)}_j\right)^{5/3}\frac{D(B)}{A\, |B|^{5/3}}
(\mathcal{F}\otimes 1)(\Delta\otimes\mathrm{id})(\mathcal{F}) = (1\otimes\mathcal{F})(\mathrm{id}\otimes\Delta)(\mathcal{F}), (\epsilon\otimes\mathrm{id})(\mathcal{F}) = 1=(\mathrm{id}\otimes\epsilon)(\mathcal{F}).
{\alpha}( t^{0}\otimes h_{{\alpha}_{k}^{0}}^{0}) ={\alpha}^{0}( h_{{\alpha}_{k}^{0}}^{0}) \mathrm{( f o r k = 1 , 2 , \ldots , \ell )},{\alpha}( c ) = 0 ,{\alpha}( d ) = 0 .
\pi^{2+k}\sum_{\substack{\sum_{j=1}^kjm_j=k}}\frac{\left(\sum_{j=1}^km_j\right)!~2^k\prod_{j=1}^k\left[\cos\left(2\pi x+j\frac{\pi}{2}\right)\right]^{m_j}}{2^{\sum_{j=1}^km_j}\left(\prod_{j=1}^km_j!\right)\left[\sin^2(\pi x)\right]^{\sum_{j=1}^km_j+1}\prod_{j=1}^k(j!)^{m_j}}=\sum_{n=-\infty}^{\infty}\frac{(-1)^k(k+1)}{(x-n)^{k+2}},
\partial_\eta K(z,w) = -\frac{1}{2\eta}K(z,w) & +\frac{1}{2\eta}\int e^{3i\eta\nu^2/4}\left(\eta-\frac{\nu}{2}\right)\left(z'\left(\frac{\eta+\nu}{2}\right)w\left(\frac{\eta-\nu}{2}\right) + z\left(\frac{\eta+\nu}{2}\right)w'\left(\frac{\eta-\nu}{2}\right)\right)d\nu
\begin{aligned}p_1(x):=\frac{(x-a)^2}{(b-a)^2}, p_2(x):=\frac{(x-a)(2x-a-b)}{(b-a)^2}, &p_3(x):=\frac{(b-x)(a+b-2x)}{(b-a)^2}, p_4(x):=\frac{(b-x)^2}{(b-a)^2},\end{aligned}
H_c=\int dx\left[\theta(3{\sigma}^{\prime}{\pi}_{\sigma}+2\sigma{\pi}^{\prime}_{\sigma}+{\pi}_{\phi}{\phi}^{\prime})+{\sigma}^2{\pi}_{\sigma}{\pi}_{\phi}-{\frac 1 2}{\sigma}^3{\pi}^2_{\sigma}+{\frac 1 2}\sigma({\phi}^{\prime})^2-{\sigma}^{\prime}{\phi}^{\prime}\right]
{S_{i}}={S_{\max}- \frac{(S_{\max}- S_{\min}){(fit({{\textbf{p}}}^i) - \mathop{\min}\limits_{j = 1, \cdots ,W}fit({\textbf{p}}^j))}}{{\mathop{\max}\limits_{j = 1, \cdots ,W}fit({\textbf{p}}^j) - \mathop{\min}\limits_{j = 1, \cdots ,W}fit({\textbf{p}}^j)}}},
\left|\sum_{k=1}^N B_{0,k-1}t_k+B_{0,N}\tau_N-\tau_0+\frac{B_{0,N}A_N+A_0}{2}\left(y_N-y_0\right)\right| \leq \left(C_AC_{|A|}c^{\tau}_J +\frac{c_J^A}{\lambda}\sum_{j=1}^N \left|y_{j+1}-y_j\right|\right)\left|A_0\right|\sum_{j=1}^{N}\hat{b}_{j-1,j}\exp \left(\frac{C_A}{\lambda}\sum_{k=1}^N\hat{b}_{k-1,k}\right),
d\frac{\alpha-2}{2}\displaystyle\int_{\mathbb{R}^{d}}G(u)dx \leq d\displaystyle\int_{\mathbb{R}^{d}}\widetilde{G}(u)dx=s_1|\nabla_{s_{1}}u|_{2}^{2}+s_2|\nabla_{s_{2}}u|_{2}^{2}-\displaystyle\int_{\mathbb{R}^{d}}W(x)u^{2}dx &\leq(s_1+\sigma_{2})|\nabla_{s_{1}}u|_{2}^{2}+(s_2+\sigma_{2})|\nabla_{s_{2}}u|_{2}^{2}.
|F_4(q)| = q^{24}(q^{12}-1)(q^8-1)(q^6-1)(q^2-1) = q^{24}(q-1)^4 (q+1)^4 (q^2+1)^2 (q^2-q+1)^2 (q^2+q+1)^2 (q^4+1) (q^4-q^2+1)\,,
F_X(x) =\frac{(1-\beta^2/\alpha^2)^{\nu+1/2}}{2\sqrt{\pi}\Gamma(\nu+1/2)}\sum_{k=0}^\infty\frac{(-1)^k}{k!}\bigg(\frac{2\beta}{\alpha}\bigg)^k\Gamma\bigg(\frac{k+1}{2}\bigg)\Gamma\bigg(\nu+\frac{k+1}{2}\bigg)\tilde{G}_{\nu+k,\nu}(-\alpha (x-\mu)).
\begin{aligned}\mathcal{P}9:~ \min_{\bold{\Theta}}~ P(\bold{\Theta})=\Phi\left(\frac{R-\overline{C}_{B}(\bold{\Theta})+\overline{C}_{E}(\bold{\Theta})}{\sqrt{V_{S}(\bold{\Theta})}}\right), &s.t.~\mathcal{C}_3. \end{aligned}
\Big|\int_{\mathbb{R}}\phi|R_n|^2 \overline{R_n}dx\Big| \leq\sum_{j_1,j_2,j_3=2}^J\Big|\int_{\mathbb{R}}\phi(x) \overline{\phi^{j_1}(\cdot-x_n^{j_1})}\phi^{j_2}(x-x_n^{j_2})\overline{\phi^{j_3}(\cdot-x_n^{j_3})}dx\Big| &\quad+\Big|\int_{\mathbb{R}}\phi|R_n^J|^2\overline{R_n^J}dx\Big| &\leq o_n(1)+\|\phi\|_{L^4}\|R_n^J\|_{L^4}^3\to 0\textup{as}n, J\to\infty.
\left( \partial^{\gamma}\partial_{\alpha}T^{( \alpha \nu ) \beta}- \partial^{\beta}\partial_{\alpha}T^{( \alpha \nu ) \gamma}\right) = - \frac{D^{2}}{a e^{2}}\partial^{2}T^{( \beta \gamma ) \nu}+ \frac{D}{2 e^{2}}\partial^{2}\epsilon^{\beta \gamma}{}_{\kappa \lambda}T^{( \kappa \lambda ) \nu}.
\prod_{n=1}^{\infty}(1-q^{n})^{3}= \lim_{\epsilon \rightarrow 0}\sum_{-\infty}^{\infty}\frac{1}{\epsilon}\left( \frac{-1}{1-\epsilon}\right)^{n}q^{\frac{1}{2}n^{2}i\frac{1}{2}n}
\left(\begin{array}{c}\dot I \dot\varphi\end{array}\right)=\left(\begin{array}{c}A^\nu I \omega^\nu\end{array}\right)+P\left(\begin{array}{c}u_0^\nu(\varphi)+u_1^\nu(\varphi)I+H_1^\nu w^\nu(\varphi)+H_2^\nu\end{array}\right)+P\mathfrak{D}_\nu (G\circ T_\nu-G^{\nu+1}\circ T_\nu),
\int_{x(0)+r-1}^\infty y_n(0,x)dx =\int_{x(0)+r-1}^{+\infty}y_n\big(t,q_n(t,x)\big)q_{x,n}^{3/2}(t,x)dx \leq C_0^{1/2}\int_{x(0)+r-1}^{+\infty}y_n\big(t,q_n(t,x)\big)q_{x,n}(t,x)dx
\d B_{\mu \nu}= \eta_{1 1}\theta^{1}\partial_{[ \mu}V_{\nu ]}+ \eta_{A B}\theta^{A}\partial_{[ \mu}W_{\nu ]}^{B}\rightarrow F = \eta_{1 1}\frac{( X^{1})^{3}}{X^{0}}+ \eta_{A B}\frac{X^{1}}{X^{0}}X^{A}X^{B}
\int d^{d}x \left\{f + u^{\dagger}\Gamma ( \tau \gamma_{0}\gamma^{i}\partial_{i}) u + u^{\dagger}\Xi ( \tau \gamma_{0}\gamma^{i}\partial_{i}){v}+{v}^{\dagger}\Pi ( \tau \gamma_{0}\gamma^{i}\partial_{i}) u +{v}^{\dagger}\Upsilon ( \tau \gamma_{0}\gamma^{i}\partial_{i}){v}\right\}.
\sum_{i=0}^{n-3}\binom{n-2}{i}\frac{\partial^{i+1}t}{\partial v^{i+1}}\frac{\partial^{n-2-i}\tilde{A}}{\partial v^{n-2-i}}+\frac{\partial^{n-1}t}{\partial v^{n-1}}\tilde{A}.
N_{m}=N_{m-1}+N_{m-3}&=(10N_{m-7}-N_{m-13}+N_{m-19})+(10N_{m-9}-N_{m-15}+N_{m-21}) &=10(N_{m-7}+N_{m-9})-(N_{m-13}+N_{m-15})+(N_{m-19}+N_{m-21}) &=10N_{m-6}+5N_{m-12}+N_{m-18},