Nonnegative matrix factorization (NMF) is the problem of approximating an input nonnegative matrix, $V$, as product two smaller matrices, $W$ and $H$. In this paper, we introduce a general framework to design multiplicative updates (MU) for NMF based on $\beta$-divergences ($\beta$-NMF) with disjoint equality constraints, penalty terms in objective function. By disjoint, mean that each variable...

Let $f$ be a locally univalent function on the unit disk $U$. We consider the normalized extensions of $f$ to the Euclidean unit ball $B^nsubseteqmathbb{C}^n$ given by $$Phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$  where $gammain[0,1/2]$, $z=(z_1,hat{z})in B^n$ and $$Psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$ in which $betain[0,1]$, $f(z_1)neq 0$ a...

For some $$\beta \ge 1/2$$ , a $$\varDelta _{\beta }$$ -metric graph $$G=(V,E,w)$$ is complete edge-weighted such that $$w(v,v)=0$$ $$w(u,v)=w(v,u)$$ and $$w(u,v) \le \beta \cdot (w(u,x)+w(x,v))$$ for all vertices $$u,v,x\in V$$ . A $$H=(V', E')$$ called spanning subgraph of $$G=(V, E)$$ if $$V'=V$$ $$E'\subseteq E$$ Given positive integer p, let H be G satisfying the three conditions: (i) ther...