Theory and Applications of Mathematics and Computer Science

On Some Nonuniform Dichotomic Behaviors of Discrete Skew-product Semiflows

2025-02-17T16:05:57+02:00

In this paper we approach concepts of nonuniform dichotomy for the case of discrete skew-product semiflows. Di erent
characterizations of this properties are given from the point of view of invariant and strongly invariant projector families.

Polynomial Stability in Average for Cocycles of Linear Operators

2025-02-17T16:06:59+02:00

In the present paper we deal with the concept of polynomial stability in average. We obtain two characterization theorems
that describe the concept mentioned above. In fact, we give a logarithmic criterion and a Datko type theorem for cocycles of linear
operators.

Ghrist Barcoded Video Frames. Application in Detecting Persistent Visual Scene Surface Shapes captured in Videos

2025-02-17T16:08:59+02:00

This article introduces an application of Ghrist barcodes in the study of persistent Betti numbers derived from vortex nerve
complexes found in triangulations of video frames. A Ghrist barcode (also called a persistence barcode) is a topology of data pictograph
useful in representing the persistence of the features of changing shapes. The basic approach is to introduce a free Abelian
group representation of intersecting filled polygons on the barycenters of the triangles of Alexandroff nerves. An Alexandroff nerve
is a maximal collection of triangles of a common vertex in the triangulation of a finite, bounded planar region. In our case, the
planar region is a video frame. A Betti number is a count of the number of generators is a finite Abelian group. The focus here
is on the persistent Betti numbers across sequences of triangulated video frames. Each Betti number is mapped to an entry in a
Ghrist barcode. Two main results are given, namely, vortex nerves are Edelsbrunner-Harer nerve complexes and the Betti number
of a vortex nerve equals k + 2 for a vortex nerve containing k edges attached between a pair of vortex cycles in the nerve.