Erik P. Van Den BanMathematisch Instituut Universiteit Utrecht, University of UtrechtHenrik SchlichtkrullMatematisk Institut Københavns Universitet, University of Convenhagen
David DrasinDepartment of Mathematics, Purdue UniversityPekka PankkaDepartment of Mathematics and Statistics, P.O. Box 68, (Gustaf Hällströmin katu 2b), University of Helsinki, Finland
We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .
In this paper, a homotopy continuation method for the computation of nonnegative
Z-/H-eigenpairs of a nonnegative tensor is presented. We show that the homotopy
continuation method is guaranteed to compute a nonnegative eigenpair.
Additionally, using degree analysis, we show that the number of positive
Z-eigenpairs of an irreducible nonnegative tensor is odd. A novel homotopy
continuation method is proposed to compute an odd number of positive Z-eigenpairs, and some numerical results are presented.
We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that the Ricci curvature in the sense of Bakry and Emery is bounded below by 1 on locally finite graphs. The Ricci flat graph in the sense of Chung and Yau is proved to be a graph with Ricci curvature bounded below by zero. We also get an estimate for the eigenvalue of Laplace operator on finite graphs: 1 dD (exp (dD+ 1) 1)