In this paper we give a short proof of the main results of Kumam, Dung and
Sitthithakerngkiet (P. Kumam, N.V. Dung, K. Sitthithakerngkiet, A
Generalization of Ciric Fixed Point Theorems, FILOMAT 29:7 (2015),
1549-1556).
AbstractWe give a short proof of Mirsky’s result regarding the extreme points of the convex polytope of doubly substochastic matrices via Birkhoff’s Theorem and the doubly stochastic completion of doubly sub-stochastic matrices. In addition, we give an alternative proof of the extreme points of the convex polytopes of symmetric doubly substochastic matrices via its corresponding loopy graphs.
AbstractA method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented.
The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies.
Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.
AbstractA remarkable theorem of Joris states that a function f is $$C^\infty $$
C
∞
if two relatively prime powers of f are $$C^\infty $$
C
∞
. Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.