harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
x\mathcal{J}-\mathcal{K}= \begin{pmatrix}\zeta_{0}(x) -\phi_{1}^{R}(x) 0 0 \ldots -\phi_{1}^{L}(x) \zeta_{1}(x) -\phi_{2}^{R}(x) 0 \ldots 0 -\phi_{2}^{L}(x) \zeta_{2}(x) -\phi_{3}^{R}(x) \ldots \vdots \vdots \vdots \ddots \vdots\end{pmatrix},
\aligned\nabla_{j}\nabla_{i}|X|^{2}= 2\langle e_{i},e_{j}\rangle+2\langle X,X_{ij}\rangle = 2\delta_{ij}+2\langle X,\sum_{p}h_{ij}^{p^{\ast}}e_{p^{\ast}}\rangle = 2\delta_{ij}-2 \sum_{p}h_{ij}^{p^{\ast}}H^{p^{\ast}},\endaligned
={\cal E}+\left(\begin{array}{cccc}((p^2 - q^2)a + 2pqb) (pa + qb) (pb -qa) ((p^2 - q^2)a + 2pqb) (pa + qb) 0 0 (pa + qb) (pb -qa) 0 0 (pb -qa) ((p^2 - q^2)a + 2pqb) ( pa + qb) ( pb -qa) ((p^2 - q^2)a + 2pqb)\end{array}\right).
\Psi(z) =\gamma z\exp\left[\beta\frac{\eta_{\mu_{1}}(z)+1}{\eta_{\mu_{1}}(z)-1}\right] =\gamma z\exp\beta\left[1-\frac{2}{1-\eta_{\mu_{1}}(z)}\right] =e^{-\beta}\gamma z\exp\left[-2\beta\psi_{\mu_{1}}(z)\right].
\begin{aligned}L_{2}(q) = \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{8n^2 + 3n - 4j^2 - j}+ q^{8n^2 + 13n + 5- 4j^2-j}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{8n^2 + 11n + 3 - 4j^2 -3j}+ q^{8n^2 + 21n + 13 - 4j^2 -3j}\end{aligned}
p(\rho)-p(r)-p'(r)(\rho-r)\leq\begin{cases}c(\rho-r)^2, \rho\in(\alpha_1,\bar\rho-\alpha_1), 1+p'(r)r-p(r), \rho\in[0,\alpha_1], 2p(\rho), \rho\in[\bar\rho-\alpha_1,\bar\rho).\end{cases}
R =XE_{N}X^{-1}(E_{N}-I)=E_{N}(X^{-1}E_{N}-X^{-1})=E_{N}(E_{N}-X^{-1})=E_{N}(I-X^{-1})=R_{0}, L =(F_{N}-I)Y^{-1}F_{N}Y=(F_{N}Y^{-1}-Y^{-1})F_{N}=(F_{N}-Y^{-1})F_{N}=(I-Y^{-1})F_{N}=L_{0},
\begin{aligned}\mathbb{E}_{\Theta^{n}}\Big[\Psi \cdot \big( M^{\Theta,B}_f(t_1 )-M^{\Theta,B}_f(t_0) \big) \Big] \stackrel{n\rightarrow\infty}{\longrightarrow}\mathbb{E}_{\Theta}\Big[\Psi \cdot \big( M^{\Theta,B}_f(t_1 )-M^{\Theta,B}_f(t_0) \big) \Big].\end{aligned}
\sum_{\ell = 0}^L (\mathcal C_\ell^{\mu_2}\varphi(\triangle u_\ell))^\frac{1}{\mu_2+1}\leq(c_2^{\mu_2}c_4)^{\frac{1}{\mu_2 + 1}}\sum_{\ell = 0}^L h_\ell^{\frac{\beta-\mu_2\gamma}{\mu_2+1}}&= (c_2^{\mu_2}c_4 h_0^{\beta-\mu_2\gamma})^{\frac{1}{\mu_2 + 1}}\sum_{\ell = 0}^{L}s^{-\frac{(\beta - \mu_2\gamma)}{\mu_2+1}\ell}.
\mathbf{\Theta}^{-1}(\gamma)-\mathbf{\Lambda}-\mathbf{M}=\mathbf{0}, &\mathbf{\Lambda}_1(\mathbf{D}_1-\mathbf{\Theta}_{1}(\gamma))=\mathbf{0}, &\lambda_2(\gamma-\theta_{2}(\gamma))=0, &\mathbf{M}(\mathbf{\Sigma}_{\mathbf{S}_1,S_2}-\mathbf{\Theta}(\gamma))=\mathbf{0},
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{\left(\tau,v_{s+k+1},\omega_{k+1}\right) \in \mathcal{B}^+_{VI}}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,\alpha}\left( s+k-1 \right) T R^d \theta^{(d-1)/2}\end{aligned}
P_\mathrm{c}(T) =A_\mathrm{L}P_{\mathrm{c},\mathrm{L}}(T)=A_\mathrm{L}\mathbb P(\mbox{SIR}>T) &=A_\mathrm{L}\int_{0}^{R_\mathrm{B}}\mathbb{P}(C_\mathrm{L}M_rM_tr^{-\alpha_\mathrm{L}}>TI_r)\frac{2\pi\lambda r}{A_\mathrm{L}}\mathrm{e}^{-\lambda\pi r^2}\mathrm{d}r,
\prod\limits_{p|L}B_p^{S'}\Big(h\Big(v_p(L), v_p\Big(\frac{N_1}{{}L}\Big)\Big)\Big)=\sum_{r|L}\frac{1}{r}\;\prod\limits_{p | L}B_p^{S'}\Big(h\Big(0, v_p\Big(\frac{N_1}{r}\Big)\Big)\Big).
\prod_{i=1}^j a_f(p_i^{2m_i})\prod_{i=j+1}^r a_f(p_i^{l_i})\prod_{i=1}^j a_f(q_i^{m_i'})\prod_{i=j+1}^r a_f(q_i^{l'_i}) = \prod_{u=1}^t \left(\prod_{i \in \mathcal I(s_u)}a_f(s_u^{2i})\right),
\lim_{r \to 0}\frac{- \log \mu(\tau_r > s \mu(U_r)^{-\alpha})}{s\mu(U_r)^{1-\alpha}}= \lim_{r \to 0}\frac{\mu(\tau_r \le s\mu(U_r)^{-\alpha})}{s\mu(U_r)^{1-\alpha}}.
[ \: M_{\mu \nu}\: , \: M_{\rho \sigma}\: ] = - 4 \: \eta_{\begin{array}{c c}{{[ \mu}}&{{}}\\{{}}&{{[ \rho}}\\ \end{array}}M_{\begin{array}{c c}{{\nu ]}}&{{}}\\{{\bullet}}&{{\sigma ]}}\\ \end{array}}
\inf_{B_{\delta R_k}(x_k)}u+C \delta^{\gamma}\rho \leq C(\delta R_k)^{-\lambda}\inf_{B_1}u+C\rho+C \delta^{\gamma}\rho &< 2C(\delta R_k)^{-2\lambda}\inf_{B_1}u+\frac{M_{k-1}}{2}\rho &= \frac{M_{k-1}}{2}\left(\inf_{B_1}u+\rho\right) &< u(x_k)/2,
{}\widehat{w}_{i}^{(k)}(t)\leq\rho_{\epsilon}(t)=\frac{\tau -\epsilon}{\sigma}\cdot\exp\left [\frac{\nu\cdot (T-t)}{\sigma}\right ]\leq\frac{\tau}{\sigma}\cdot\exp\left (\frac{\nu\cdot T}{\sigma}\right )\mbox{for all $t\in [0,T]$,}
\sum_{j=1}^{n}\theta_\alpha \left(x - \frac{j}{n}+ \varepsilon_j \right) = \sum_{k \in \mathbb{Z}}e^{-\pi \alpha k^2}\left( \sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}\right)e^{2 \pi i k x}.
\begin{gathered}\Phi_{+\sqcup}^{a,b}: \{0,\ldots,a-1\}\sqcup \{0,\ldots,b-1\}\rightarrow \{0,\ldots,a+b-1\}\;, \Phi_{+\sqcup}^{a,b}((\chi,i))= \Phi_{+\chi}^{a,b}(i)\;.\end{gathered}
\begin{array}{ll}\begin{pmatrix}\lambda 0 0 \lambda^{-1}\end{pmatrix}, \lambda>1, \begin{pmatrix}a_1 b_1 c_1 d_1\end{pmatrix}, a_1d_1-b_1c_1=1, a_1+b_1=c_1+d_1>0\end{array}
\left( \begin{array}{c c c c}{{0}}&{{1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{0}}&{{1}}&{{0}}\\ \end{array}\right)^{2 n}= \left( \begin{array}{c c c c}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right)
q\varphi(S^{(n)}_{\xi}) \varphi(S^{(m)}_{\eta}) q = q \sum^{\infty}_{k=0}S^{(k)}_{\sum^{k}_{\ell = 0}U_{\ell,k-\ell}(\xi_{\ell}\otimes \eta_{k-\ell})}q &= q \sum^{\infty}_{k=0}S^{(k)}_{\sum^{k}_{\ell = 0}U_{\ell,k-\ell}( q \xi_{\ell}q \otimes q \eta_{k-\ell}q)}q = q\varphi(S^{(n)}_{\xi}) q \varphi(S^{(m)}_{\eta}) q
{\cal O}_{\pi^*{\cal S}}(A \sigma|_{\pi^{*}{\cal S}}+fF)={\cal O}_{\pi^*{\cal S}}(A \sigma|_{\pi^{*}{\cal S}}) \otimes{\cal O}_{\pi^*{\cal S}}(fF)={\cal O}_{\pi^*{\cal S}}(A \sigma|_{\pi^{*}{\cal S}}) \otimes\pi^*{\cal O}_{{\cal S}}(f).
\bigcup_{\xi\in Q_\ell^*}\sigma(D_\xi)=\adjustlimits\bigcup_{\xi\in Q_\ell^*}\bigcup_{n\geq 1}\left\{d^-_n(\xi),d^+_n(\xi)\right\}=\sigma(D^0)=(-\infty,-1]\cup[+1,+\infty).
|v_{4}(t,r)| \leq \begin{cases}\frac{C r}{t^{2}\log^{3b+2N-1}(t)}, r \leq \frac{t}{2}\frac{C}{\sqrt{r}t \log^{\frac{3N}{2}+3b-1}(t)}, r > \frac{t}{2}\end{cases}
[(A+B\xi+C\xi^2)\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u}, t_0\frac{\partial}{\partial u}+(t_1+t_2\xi+t_3\xi^2)\frac{\partial}{\partial \xi}] &=\left((-Bt_1+At_2)+2(-Ct_1+At_3)\xi+(-Ct_2+Bt_3)\xi^2 \right)\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u}
\lvert (i_1,j_1)\rvert_{W_n(w)}-\lvert (i_2,j_2)\rvert_{W_n(w)}= \lvert (i_1,j_1)\rvert_{W_n(w)}-\lvert (i_2,j_1)\rvert_{W_n(w)}&+\lvert (i_2,j_1)\rvert_{W_n(w)}-\lvert (i_2,j_2)\rvert_{W_n(w)}= i_1-i_2+ j_1-j_2
|q_n|_p =|a_1 a_2 \cdots a_n|_p,\n\geq 1, \begin{cases}|r_n|_p = |a_0 a_1 \cdots a_n|_p ,\n\geq 1, (a_0 \not = 0) |r_1|_p =1,\ |r_n|_p = |a_2 a_3 \cdots a_n|_p ,\n\geq 2, (\mbox{otherwise})\end{cases}\left|\alpha -\frac{r_n}{q_n}\right|_p = \frac{1}{|a_{n+1}|_p |q_n|_p^2},\n\geq 0.
\frac{4^{c}+x^{c}}{T_0}\sum_{n=1}^{\infty}\frac{\sigma_{a_1,\beta}(n)\sigma_{a_2,\beta}(n)}{n^{c}}\ll \frac{x}{T_0}f(a_1,a_2,c;\b)\ll\frac{x}{T_0}\log^4x ,
\frac{i}{4\pi}\int_{-\infty}^{\infty}\left(\frac{\sinh\left(\frac{\alpha}{2}\right)}{\cosh\left(\frac{\alpha}{2}\right)}\right)e^{-i\alpha x}d\alpha=\int_{-\infty}^{\infty}\frac{\cosh(\pi u)}{\sinh(2\pi u)}\delta(x-u)du
\liminf_{k\to+\infty}\Vert u(t_{n_k},\cdot+x(t_{n_k}))\Vert_{H^1(K)}^2\leq2\Vert \varphi_{c^\star}\Vert_{L^2(K)}^2=\Vert \varphi_{c^\star}\Vert_{H^1(K)}^2,
I(G)= \ (x_1x_2,x_2x_3,x_2x_4,x_2x_5,x_2x_6,x_2x_{12},x_1x_4,x_1x_6,x_1x_7,x_1x_8,x_2x_{12},x_3x_5, &\ \x_3x_8, x_3x_{11}, x_3x_{12}, x_4x_5, x_4x_9, x_4x_{10}, x_5x_7, x_5x_9, x_6x_7, x_6x_{10}, x_6x_{11}, x_7x_8, &\ \x_7x_9, x_7x_{12}, x_8x_{11}, x_9x_{10}, x_9x_{12}, x_{10}x_{11}, x_{10}x_{12}, x_{11}x_{12}).
\mu(X(s_1)\in B_1,\cdots, X(s_r)\in B_r)=\frac{1}{(2\pi)^{\frac{r}{2}}.|C|}\int_{B_1\times \cdots \times B_r}\exp\big(-\frac{1}{2}.{}^{t}{x}.C^{-1}.x\big) dx,
\begin{aligned}0 \equiv\xi^2_1\wedge\eta^1\wedge\eta^0+\xi^\varsigma_1\wedge\eta^1\wedge\eta^2 \mod\{\eta^{\overline{2}}\}\Rightarrow 0 \equiv \xi^2_1,\xi^\varsigma_1 \mod\{\eta^0,\eta^1,\eta^2,\eta^{\overline{2}}\}.\end{aligned}
divF=\nabla\cdot F=\frac{1}{r^{2}}\frac{\partial}{\partial r}(r^{2}F_{r})+\frac{1}{rsin\theta}\frac{\partial}{\partial\theta}(sin\theta F_{\theta})+\frac{1}{rsin\theta}\frac{\partial F_{\varphi}}{\partial\varphi}.
e^{\Omega_{12,3}/2}=\Phi_{3,1,2}e^{\Omega_{13}/2}\Phi_{1,3,2}^{-1}e^{\Omega_{23}/2}\Phi_{1,2,3}&e^{\Omega_{1,23}/2}=\Phi_{2,3,1}^{-1}e^{\Omega_{13}/2}\Phi_{2,1,3}e^{\Omega_{12}/2}\Phi_{1,2,3}^{-1}
\int_{\{|t-\xi|>1-r\}}\frac{1-r}{|1-\bar{t}r\xi||1-\bar{t}\xi|}d \mu(t) \leq \sum_{n=0}^\infty \int_{I(n+1) \setminus I(n)}\frac{1-r}{2^{2n}(1-r)^2}d\mu(t) = \sum_{n=0}^{\infty}\frac{\mu (I(n+1) \setminus I(n))}{2^{2n}(1-r)}\leq C_2\ ,
\hat{V}( y ) ={\frac{1}{8}}( \alpha^{2}- \beta^{2}) y^{2}+ \sigma_{2}[ - ( n +{\frac{1}{4}}) \beta +{\frac{\alpha \beta}{4}}y^{2}-{\frac{\alpha}{4}}\tan{\frac{\beta y^{2}}{2}}] \cos{\frac{\beta y^{2}}{2}}+
\|\xi_i\|^p =\sum_{t\in G}\|v_{t^{-1}}\pi(g_i(t))\xi\|^p=\sum_{t\in G}\int |\pi_0(g_i(t))\xi|^p \, d\mu=\int \pi_0\left(\sum_{t\in G}g_i(t)^p\right)| \xi|^p \, d\mu &\longrightarrow \int |\xi|^p \, d\mu=\|\xi\|^p.
U_{L}(\theta,P_{BS},p_{s},\rho_{e}) = \mu T_{s}- E &~~= \mu (1-\theta) \bigg[ \log \bigg( 1 + \frac{p_{s}|h_{s}|^{2}}{\sigma_{s}^{2}}\bigg) - \log(1+\rho_{e}) \bigg] ~~~~~~~~~~ - \lambda \theta P_{BS}\|\mathbf{h}\|^{2}.
{\cal L}_F=\bar\psi i\gamma^\mu (\partial_\mu-ieA_\mu)\psi -m\bar\psi\psi+{g\over 2}\mid\bar\psi\gamma^\mu\psi\mid^2 -{{pe^2}\over 4}\mid A_{\mu\nu}\mid^2+{{qe^2}\over 2}\epsilon_{\mu\nu\lambda}A^\mu A^{\nu\lambda}.
\overline{{{\alpha}}}_{3}= \left( \begin{array}{c c c c}{{0}}&{{- i}}&{{0}}&{{0}}\\{{i}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- i}}\\{{0}}&{{0}}&{{i}}&{{0}}\\ \end{array}\right) , \hspace{0 . 3 i n}\overline{{{\alpha}}}_{4}= \left( \begin{array}{c c c c}{{- i}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- i}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- i}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- i}}\\ \end{array}\right) .
{E_{m}^{[\alpha,\beta],\rho,\varepsilon}=\left\{x\in\widetilde{E}_{m}^{[\alpha,\beta],\rho,\varepsilon}: \forall r\leq \rho, \ \frac{3}{4}m(B(x,r)) \leq m(B(x,r)\cap \widetilde{E}_{m}^{[\alpha,\beta],\rho,\varepsilon})\right\}}
\mathrm{N = 8}\, \mathrm{\bf ~ s p i n ~ 2}\, \stackrel{N = 2}{\longrightarrow}\, \mathrm{\bf ~ s p i n ~ 2}+ 6 \times \mathrm{\bf ~ s p i n ~ 3 / 2}+ 1 5 \times \mathrm{\bf ~ v e c t . ~ m u l t .}+ 1 0 \times \mathrm{\bf ~ h y p e r m u l t .}
\left| \frac t{e^t-1}-1+\frac 1 2 t-\frac 1{12}t^2-\sum_{k=2}^{N-1}\frac{B_{2k}}{(2k)!}t^{2k}\right|\leq \frac{|B_{2N}|}{(2N)!}t^{2N},t\geq 0.
i\hbar\gamma^{\mu}\partial_{\mu}+mc\equiv i\hbar(\gamma^{0}\partial_{0}+\gamma^{1}\partial_{1}+\gamma^{2}\partial_{2}+\gamma^{3}\partial_{3})+mc(\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix})
\frac{d^{m+1}G}{dv^{m+1}}=v^3\frac{d^{m+1}\hat{B}}{dv^{m+1}}+(c_{2,m}+3)v^2\frac{d^{m}\hat{B}}{dv^{m}}&\qquad+(c_{1,m}+2c_{2,m})v\frac{d^{m-1}\hat{B}}{dv^{m-1}}+(c_{0,m}+c_{1,m})\frac{d^{m-2}\hat{B}}{dv^{m-2}},
\epsilon \mu \sum_{k=1}^{\infty}a_{k1}\tilde{P}^{k-1}\tilde{P}^{(1)}e^{-\alpha k}\sum_{l=k}^{\infty}e^{-\alpha l}\tilde{P}^l r = \epsilon \mu \sum_{k=1}^{\infty}a_{k1}\tilde{P}^{k-1}\tilde{P}^{(1)}e^{-\alpha k}(I-e^{-\alpha}\tilde{P})^{-1}r,
\int_{-\frac{1}{2}}^{\frac{1}{2}}v(\sigma)\Bigl(\int_{{\mathbb{R}}^n}u(s-\varepsilon(1-2\sigma),x)\overline{\phi}(s-\varepsilon(1-2\sigma),x)\,{\rm d}x\Bigr)\,{\rm d}\sigma = \int_{-\frac{1}{2}}^{\frac{1}{2}}v(\sigma)\Bigl(\int_{{\mathbb{R}}^n}u(t-2\varepsilon(1-\sigma),x)\overline{\phi}(t-2\varepsilon(1-\sigma),x)\,{\rm d}x\Bigr)\,{\rm d}\sigma.
\theta^1 :=du-u_xdx-u_ydy-u_zdz,\theta^2:=du_x-F_1dx-F_2dy-F_3dz, \theta^3 :=du_y-F_2d x-u_{yy}dy-u_{yz}dz,\theta^4:=du_z-F_3d x-u_{yz}dy-u_{zz}dz,
J_{\mathcal{M},T}(x) = h(x) x \in X_T J_{\mathcal{M},t}(x) =\inf_{a \in \psi(x)}\{c(x,a) + \int_{X_{t+1}}J_{\mathcal{M},t+1}(y)Q_{t}(dy|x,a) \}x \in X_t, t \in [T-1]
g_{p,\mu,v}(t) =-t^{\frac{p-2}{p-1}}\left(t\|v'\|_{L^p_{p-1}}^p+\|v\|_{L^p_{\alpha_0}}^p\right)^{\frac{1}{p-1}}&\quad+\dfrac{\mu}{p(p-1)}\dfrac{\|v\|_{L^{pp'}_{\alpha_0}}^{pp'}}{\|v'\|^p_{L^p_{p-1}}\|v\|_{L^{p}_{\alpha_0}}^{p}}\left(-t(p-1)\|v'\|^p_{L^p_{p-1}}+\|v\|_{L^{p}_{\alpha_0}}^{p}\right).
\int_{B_{R}(0)}{}\frac{z\cdot \nabla u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz = R\int_{\partial B_{R}(0)}\frac{u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,ds{}- (n-\sigma)\int_{B_{R}(0)}\frac{u(z)^{p}}{|x-z|^{n-\alpha}|z|^{\sigma}}\,dz - (n-\alpha)\int_{B_{R}(0)}\frac{(z\cdot (x-z))u(z)^{p}}{|x-z|^{n-\alpha + 2}|z|^{\sigma}}\,dz,
\bar W_{L a b \underline L}= \frac{1}{2}\tilde \sigma_{ab}\rho + \frac{1}{4}\epsilon_{ab}\sigma \bar W_{abcL}=-\epsilon_{ab}\epsilon_{cd}\beta^d \bar W_{abc\underline L}=\epsilon_{ab}\epsilon_{cd}\underline{\beta}^d \bar W_{ab\underline{L}L}=\frac{1}{2}\epsilon_{ab}\sigma.
\left\langle U_{\lambda}\left(w\right)\right\rangle_{w'}=\prod_{X'_{i'}\in{\it G}}\prod_{\left(n'^{*}_{i'}\right)^{k}_{i'=0}}\left\langle U_{\lambda}\left(w\right)\right\rangle^{\left({\it G}_{1}\right)_{X'_{i'}}}_{w'_{n'^{*}_{i'}}}
\big( a(D)u \big)(x) = \sum_{|\alpha|\leqslant m}a_{\alpha}\int_{\mathbb{R}^n}e^{2 \pi i x \cdot \xi}\xi^{\alpha}\hat{u}(\xi) \, d\xi = \int_{\mathbb{R}^n}e^{2 \pi i x \cdot \xi}a(\xi) \hat{u}(\xi) \, d\xi,
\px'+py'+\delta z'+\log{\lambda_{G\times G;\varphi}(x',y',z')}= \ 2p\bar{x}+\delta z'+\log{\lambda_{G\times G;\varphi}(x',y',z')}\ge \ 2p\bar{x}+\delta z'+ \log{\lambda_{G\times G;\varphi}(\bar{x},\bar{x},z')},
\begin{aligned}\int_X (D_A \Phi, D_A(|\Phi|^2 \Phi)) \,d\mu_g = \int_X \big[ |\Phi|^2 |\nabla_A \Phi|^2 + \frac{1}{2}|\nabla |\Phi|^2|^2 + \frac{1}{4}R_g |\Phi|^4 - 2(iF_A^+, |\Phi|^2\sigma(\Phi)) \big] \,d\mu_g\end{aligned}
\int_\Omega \gamma^{-\frac{p}{2-p}}(v)dx\leq \int_\Omega \left(b v^{k}+\gamma^{-1}(s_b)\right)^{\frac{p}{2-p}}dx \leq \int_\Omega \left(b\left(\frac{1}{1-\gamma(a)}(w+K)\right)^k+\gamma^{-1}(s_b)\right)^{\frac{p}{2-p}}dx \leq C\int_\Omega w^{\frac{pk}{2-p}}dx+C,
E[(1+\lambda t)^{\frac{U_1}{\lambda}}] = \int_0^1 (1+\lambda t)^{\frac{u_1}{\lambda}}f(u_1) du_1 = \int_0^1 e^{\frac{u_1}{\lambda}\log(1+\lambda t)}du_1 &= \frac{1}{\frac{1}{\lambda}\log(1+\lambda t)}\big( (1+\lambda t)^{\frac{1}{\lambda}}-1\big).
{\cal L}= \frac 1 2 \partial_{\mu}\phi \partial^{\mu}\phi - \frac{m^{2}}{2 \beta^{2}}\cosh \sqrt 2 \beta \phi - \bar{\psi}i \gamma^{\mu}\partial_{\mu}\psi + m \bar{\psi}\psi \cosh \frac{\beta \phi}{\sqrt 2},
\lambda_1 = \frac{1}{\beta-\eta}, \lambda_2= \frac{1}{\beta+\eta}, \lambda_3 ={\frac{\alpha+\mu}{\alpha\,\gamma-\alpha\,\tau-{\delta}^{2}-2\,\nu\, \delta+\gamma\,\mu-\mu\,\tau-{\nu}^{2}}}, \mbox{and}\lambda_4 ={\frac{\alpha-\mu}{\alpha\,\gamma+\alpha\,\tau-{\delta}^{2}+2\,\nu\,\delta-\gamma\,\mu-\mu\,\tau-{\nu}^{2}}}.
\mathbf{M=}(m_{ij})_{i,j=1}^{N}=\left(\begin{array}{ccccc}m_{11}m_{12}... ... m_{1N}0 m_{22}... ... m_{2N}0 0 m_{33}... ... ... ... ... ... ... ... ... ... ... ... 0 0 ... 0 m_{NN}\end{array}\right) .
\phi_t(t_{n-\theta})= \frac{(3-2\theta)\phi(t_{n})-(4-4\theta)\phi(t_{n-1})+(1-2\theta)\phi(t_{n-2})}{2\Delta t}+O(\Delta t^2) = \frac{(3-2\theta)\phi^{n}-(4-4\theta)\phi^{n-1}+(1-2\theta)\phi^{n-2}}{2\Delta t}\triangleq \mathcal{D}_t\phi^{n-\theta},~n\geq2.
\mathcal L\phi \equiv L\phi_{tt}-2M\phi_{ts}+N\phi_{s s}+\left(\frac{A_{1}}N+\frac{2Mh_t}{h}\right)\phi_s+\left(\frac{A_{2}}N-\frac{2Lh_t}h\right)\phi_t &=\frac L{Nh}(h_{t t}h-2h^2_t)+A_{2}h_t\frac1{N^2}+h\frac{A_0}{N^2}.
\vec{\sigma}\cdot \vec{p}\tilde{\xi}_{\lambda}( \vec{p}) = 2 p \lambda \tilde{\xi}_{\lambda}( \vec{p}) , \quad \vec{\sigma}\cdot \vec{p}\tilde{\eta}_{\lambda}( \vec{p}) = - 2 p \lambda \tilde{\eta}_{\lambda}( \vec{p}) .
\biggl ( \frac{S}{\Lambda^{3}}\biggr )^{\frac{2 N_{c}- N_{f}}{N_{f}}}- \biggl ( \frac{S}{\Lambda^{3}}\biggr )^{\frac{N_{c}- N_{f}}{N_{f}}}+ \frac{\Lambda^{3}}{m^{2}}= 0 .
=\sum\limits_{(M_b, \mu_b) \in \mathcal{I}^{G, \mu}_{M_b, b'}}(\mathrm{Ind}^G_{P_b}\circ \otimes^k_{i=1}[\mathcal{M}_{M_{b_i}, b'_i, \mu_{b i}}]\circ (\delta_{P_b}\otimes \mathrm{Jac}^G_{P^{op}_b}))\otimes [1][|\cdot|^{\langle \rho_G, \mu_b \rangle - \langle \rho_G, \mu \rangle}],
BB1=[\begin{matrix}(1-2\lambda-\beta)&(\lambda-\alpha)&0&0\\ (\lambda+\alpha)&(1-2\lambda-\beta)&(\lambda-\alpha)&0\\ 0&(\lambda+\alpha)&(1-2\lambda-\beta)&(\lambda-\alpha)\\ 0&0&2\lambda&(1-2\lambda-\beta)\end{matrix}],
\langle (0, x_i), L_{a,b}^d(p_i,0) \rangle = d ab^{d-1}\langle L x_i, L^{d-1}p_i \rangle + O(b^{d-2}) = O(b^{d-2}) \langle (Lx_i, 0), L_{a,b}^d(p_i,0) \rangle = d ab^{d-1}\langle Lx_i, L^{d-1}p_i \rangle + O(b^{d-2}) = O(b^{d-2}) \langle (p_i, 0), L_{a,b}^d(p_i,0) \rangle = d ab^{d-1}\langle p_i, L^{d-1}p_i \rangle + O(b^{d-2}).
(\omega \pi)(x') = - \int_{\mathbb R^2}(\nabla_{y'}\Psi)(x', y') \cdot \big[ \omega F + 2(\nabla_{y'}\omega)\pi \big](y') \, dy'- \int_{\mathbb R^2}\Psi(x', y') \big[(\nabla_H\omega)\cdot F + (\Delta_H\omega)\pi \big](y') \, dy'.
\int\limits_0^\infty{{x^{s - 1}}{e^{- ax}}\cos \left({nx}\right)dx}= \frac{{\Gamma \left( s \right)\cos \left\{{{{\tan}^{- 1}}\left({\frac{n}{a}}\right)}\right\}}}{{{{\left({{a^2}+{n^2}}\right)}^{\frac{s}{2}}}}}
\lim_{n \rightarrow \infty}\frac{h_{v}^{(n+1)}}{\rho_{\alpha}^{(n)}}=\sum_{\mathcal{E}_{\beta}\preceq \mathcal{E}_{\alpha}}\left(\sum_{w\in \mathcal{E}_{\beta}}a_{vw}h_w^{(1)}\right)
\alpha - 1 = \frac{3^3 - 3}{3}-1 = 7 \neq 0 \mod (3), \alpha - 1 = \frac{4^3 - 4}{3}- 1 = 19 \neq 0 \mod(3), \alpha - 1 = \frac{9^3 - 9}{3}-1 = 239 \neq 0 \mod (3),
N(u)(t,r)=\left(\frac{\sin(2u(t,r))-2u(t,r)}{2r^{2}}\right)\cos(2Q_{\frac{1}{\lambda(t)}}(r)) + \left(\frac{\cos(2u(t,r))-1}{2r^{2}}\right)\sin(2Q_{\frac{1}{\lambda(t)}}(r)+2v_{corr})
\psi = u_{1,\alpha^+}\; u_{-1,\alpha^-}\prod_{r=1}^R u_{\tau_r,\alpha^+_r}\; u_{\bar{\tau}_r,\alpha^-_r}\;, \phi = u_{1,\alpha^++\beta^+}\; u_{-1,\alpha^-+\beta^-}\;\prod_{r=1}^R u_{\tau_r,\alpha_r+\beta_r}\; u_{\bar{\tau}_r,\alpha_r+\beta_r}\;,
N ( L \to 0 , L^{\prime}, D ) ={\frac{1}{\sqrt{\pi L}}}e^{L^{\prime}\sqrt \tau}{\frac{\partial}{\partial D}}e^{-{\frac{3}{2}}L^{\prime}\sqrt \tau \coth^{2}\left( \sqrt{\frac{3}{2}}\tau^{1 / 4}D \right)}+ O ( L^{0})
\mathbf v=\sum_{i=1}^N\xi^i(x,u)\frac{\partial}{\partial x_i}+\phi(x,u)\frac{\partial}{\partial u}+\sum_{|\sigma|\geq 1}\phi^\sigma(x,u,\dots,\partial^{\kappa}_xu,\dots)\frac{\partial^{|\sigma|}}{\partial u^\sigma},u^\sigma\coloneqq\frac{\partial^{|\sigma|}u}{\partial x^\sigma},
\int_{\omega_-}\frac{\delta^{n}}{(\delta^2+|x-e_n|^2)^{\frac{n-2}{2}}(\delta^2+|x+e_n|^2)^{\frac{n+2}{2}}}\, dx\leq C2^{n-2}\delta^{n-2}\left(1+\int_1^{\frac{1}{2\delta}}\frac{r^{n-1}}{(1+r^2)^\frac{n+2}{2}}\, dr\right)=O(\delta^{n-2}),
\lambda_k =\zeta(\mathfrak n_1) \times \langle \mathfrak m_1 \rangle \times \ldots \times \zeta(\mathfrak n_{k})\times \langle \mathfrak m_{k}\rangle \times \zeta(\mathfrak n_{k+1}+\ldots +\mathfrak n_{r+1}+\mathfrak m_{k+1}+\ldots +\mathfrak m_{r})
Y(u_0) = \sum_{\substack{d_1, \ldots, d_{2t}: d_1 = 1, \,\, |d_{i+1}- d_i| \le 1}}\,\, \sum_{a_1, \ldots, a_{2t}\in [2t]}\,\, \sum_{\substack{u_1, \ldots, u_{2t}: u_i \in B(d_i, a_i)}}W(u_0, u_1, \ldots, u_{2t}, u_0) .
S_{E}= - \frac{1}{16\pi G_{d-1}}\int_{\Sigma}d^{d-1}x \sqrt{g}\left[ R -m e^{\frac{4\phi}{\sqrt{d-2}(d-3)}}- \frac{4}{d-3}(\partial\phi)^2 - \frac{1}{4}e^{\frac{-4\sqrt{d-2}}{d-3}\phi}F^2 \right],
f(x) =\int_{D^c}P_{B(x,\varepsilon)}(x,y)\lambda(dy)+\int_{D\setminus B(x,\varepsilon)}f(y)P_{B(x,\varepsilon)}(x,y)dy, f(x_n) =\int_{D^c}P_{B(x,\varepsilon)}(x_n,y)\lambda(dy)+\int_{D\setminus B(x,\varepsilon)}f(y)P_{B(x,\varepsilon)}(x_n,y)dy.
L_{BV}^{a,\mathrm{I}}\Big\vert_{A^\dag = c^\dag =0}= \mathbb{L}_{BRST}^{\mathrm{I}}L_{BV}^{b,\mathrm{I}}\Big\vert_{A^\dag = c^\dag =0}= \mathbb{L}_{BRST}^{\mathrm{I}}+ cF_A + (d-\mathcal{L}_Q)(cA) \approx \mathbb{L}_{BRST}^{\mathrm{II}}+ (d-\mathcal{L}_Q)(cA)
\frac1{\tau}-\frac1{\delta_{m+1}}=\frac1{\widetilde{\delta}_{m+1}}-\frac1{\delta_{m+1}}=\frac 1{p_{m+1}}-\frac 1{q_{m+1}}=\frac 1{q}-\frac 1{p}=\frac 1{s}-\frac 1{p},
\frac{\big(x q^{-L}/\xi, q^{-L}/(\beta_1 \xi ), q^{-L}/(\beta_2 \xi ) ;q\big)_{\infty}}{\big(x q^{-\lambda -1-L}/ \xi, q^{-L}/(\alpha_1 \xi ), q^{1-L}/(\alpha_2 \xi ) ;q\big)_{\infty}}=0
\tilde{V}^{ss}_r ( \xi , x , y ) := \left\{(\tilde{\xi}, \tilde{x}, \tilde{y}) : \tilde{x}\in I_{N_r (x)}(x) , \, | \xi - \tilde{\xi}| \leqslant r \mbox{and}\left| \ell^{ss}_{(\tilde{\xi}, \tilde{x}, \tilde{y})}( x ) - y \right| \leqslant r \right\},
|A|\leqslant O(1) \big(|u^a_x|+|u^a_t|+|u^a_x|^2+|u^a_t|^2+|u^a_{xx}|+|u^a_{tt}|\big)\leqslant O(1) \big(|u^a_x|+|u^p_{xx}|+|u^p_{tt}|\big),
|(q+\xi)^2 - E - i \eta |^2 = | q^2 - E + \xi^2 + \xi q + q \xi - i \eta |^2 = (q^2 - E )^2 + 2 (q^2 - E) ( \xi q + q \xi + \xi^2 ) + ( \xi q + q \xi + \xi^2 )^2 + \eta^2.
2 \left( \frac{\Theta g_{m}^{2}\eta}{4 \pi}\right)^{2}\left( s_{a}^{( c )}\right)^{2}\int d^{4}x d^{4}y \tilde{\cal F}_{\mu \nu}( x ){\cal D}_{m}^{( 4 )}( x - y ) \tilde{\cal F}_{\mu \nu}( y ) ,
\frac{p_{r}^{2}}{g_{r r}}+ \frac{p_{\theta}^{2}}{g_{\theta \theta}}= \frac{p_{\phi}- \left( \frac{g_{\phi \phi}E^{2}}{\cal D}+ V \right) / \left( \frac{2 g_{t \phi}}{\cal D}E \right)}{\frac{-{\cal D}}{2 g_{t \phi}E}},
T_f^{s_{kn}(\tilde{z}_0,\tilde{q}(w);t)}(\tilde{q}(w))=(x,0)\quad\mbox{but}T_f^{s_{(k+1)n}(\tilde{z}_0,\tilde{q}(w);t)}(\tilde{q}(w))\notin H_{k+1}\times\{0\}
\mathcal{A}_{1}(a) = \left(\bigcup_{k=2}^{+\infty}\bigcup_{q_l\in \mathbb{Q}^+}\bigcap_{r_j\in \mathbb{Q}^+}\mathcal{D}(k+r_j,a+q_l)\right)^c, \mathcal{A}_{2}(a) = \left(\bigcup_{k=2}^{+\infty}\bigcap_{r_j\in \mathbb{Q}^+}\mathcal{D}(k+r_j,a)\right)^c.
\begin{bmatrix}\dot{x}\dot{y}\end{bmatrix}=\begin{cases}\begin{bmatrix}\lambda_L x + y -x + \lambda_L y \end{bmatrix}, x < 0, \begin{bmatrix}\lambda_R x + y + \mu -x + \lambda_R (y + \mu) \end{bmatrix}, x > 0.\end{cases}
-(\vec{\Theta}_\alpha\cdot\vec{e}\,)(\gamma^t\gamma^u)_{\alpha\beta}E_uE_sM_{st}^{\parallel}\psi_0=-\frac{\i}{2}[\vec{\Theta}_\beta\cdot\vec{e}\,,M_{ut}^{\parallel}M_{ut}^{\parallel}]\psi_0-\i\frac{d^2-d}{8}(\vec{\Theta}_\beta\cdot\vec{e}\,)\psi_0\;.
mnS = \sum_{i=1}^{m-1}\sum_{j=1}^{n}( x_{i,j}+ x_{i+1,j}+ x_{i,j+1}+ x_{i+1,j+1}) + \; \sum_{i=1}^{m}( x_{m,j}+ x_{m,j+1}+ x_{1,n-j}+ x_{1,n-j+1}) \; = \; 4\biggl( \sum_{k=1}^{mn}k \biggr) = (2mn)(mn+1),
\frac{p(p-1)}{14}\sum_{m=0}^{6}(16f^{2}-6fn_{j}+4f-n_{j}^{2}-2n_{j}) &=\frac{p(p-1)}{14}\sum_{m=0}^{6}(16f^{2}+4f-(6f+2)((m,m)_{7}+2(m,m+1)_{7}+(m+1,m+1)_{7})- &((m,m)_{7}+2(m,m+1)_{7}+(m+1,m+1)_{7})^{2}).
\displaystyle{\lim_{k \to +\infty}\int_{A^\ast}f(x, u_k, v_k,\nabla u_k)dx = \int_{J_u \cap A^\ast}K_p(x,0,c,d,e_N)d{\cal H}^{N-1}+ \int_{A^\ast}f(x,u(x), v(x),0)dx.}
\frac{\partial^2 \widetilde{R}_{1,k}}{\partial\tau^2}= -\frac{1}{ln2}\bigg(\underbrace{\frac{1-\alpha}{T}\cdot\frac{1}{\tau(1+h\tau)}}_{\geq 0}+ \underbrace{\frac{T-(1-\alpha)\tau}{T\tau}\cdot\frac{h}{(1+h\tau)^2}}_{\geq 0}+ \underbrace{\frac{1}{(1+h\tau)\tau^2}}_{> 0}\bigg).
\sum_{0\le |q|+2\kappa\le |m|+2\ell + k - 2}c^{m ,\ell}_{q,\kappa}x^q t^\kappa = \sum_{|m| + 2\ell \leq |q| + 2\kappa \leq k-1}c^{m ,\ell}_{q,\kappa}x^q t^\kappa + \sum_{k\le |q|+2\kappa\le |m|+2\ell + k - 2}c^{m ,\ell}_{q,\kappa}x^q t^\kappa.
\frac{1}{\lambda_{r}}\sigma_{0}- \frac{2}{\pi}\sigma_{0}+ \frac{1}{\pi r}\frac{\displaystyle \Gamma \left( \frac{3}{2}+ i m r \right) \Gamma \left( \frac{3}{2}- i m r \right)}{\displaystyle \Gamma \left( 1 + i m r \right) \Gamma \left( 1 - i m r \right)}= 0 \, ,
\int_0^{j_{\alpha,1}}t \Big( |J_{\alpha+1}(t)|^2 + |J_\alpha(t)|^2 \Big) \frac{(j_{\alpha,1}^2-t^2)^2}{2t^4}\,dt > \sum_{i=1}^m \Big( f(p_i)-f(p_{i-1}) \Big) g(p_i)
\nu(\mathcal{E}_k) \leq \inf_{\substack{A, S \s.t. \mathcal{E}_k = ASA^*}}||A||^2\nu(S) \leq ||\sigma||_p^2 \leq C_p^{-\frac{2}{p}}\inf_{\substack{A, S \s.t. \mathcal{E}_k = ASA^*}}||A||^2\nu(S)
\begin{aligned}B_1 [h]:=\alpha_{1, 2}\,h_x, B_2 [h]:=\alpha_{0, 2}\,h, B_3 [h]:=-(\alpha_{1,1})_x\,\beta_1\,h_x, begin{align*}2mm] B_4 [h]:=-(\alpha_{0, 1})_x\,\beta_1, B_5[h]:=-\partial_x \overline{\mathcal{R}}_2 [h].\end{aligned}
\overline{\zeta}_q(s)=\frac{1}{\Gamma(s)}\sum_{j=0}^\infty (n+1-q)^j \frac{\Gamma(s+j-1)}{\Gamma(j+1)\Gamma(2s-2+j)}\int_0^\infty t^{2s+j-2}S_{n,q}(e^{-t}) \exp(-(n+1-q)t)dt.
\widetilde{\textnormal{(I)}}= -\frac{4(p-1)}{p}\int (a\nabla (\eta u_k^{p/2}),\nabla (\eta u_k^{p/2}))\;dx +\frac{4(p-2)}{p}\int u_k^{p/2}(a\nabla (\eta u_k^{p/2}),\nabla \eta)\;dx+\frac{4}{p}\int u_k^{p}(a\nabla \eta,\nabla \eta)\;dx.
(\widehat{F}\varphi,\varphi)=\sigma\sum_{j=1}^{\beta_i^+}c_j^2\mu_j^{-1}(\lambda)+O(1), \varphi=\sum_{j=1}^{\beta_i^+}c_j\varphi_j\in \widehat{\Phi}^+, \sigma(\lambda-\lambda_0)\to + 0.