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A quasi-continuous dynamical system is a pair $(X,f)$ consisting of a topological space $X$ and a mapping $f: X\to X$ such that $f^n$ is quasi-continuous for all $n \in \mathbb N$, where $\mathbb N$ is the set of non-negative integers. In…
Topological phase transitions track changes in topological properties of a system and occur in real materials as well as quantum engineered systems, all of which differ greatly in terms of dimensionality, symmetries, interactions, and…
Mathematical descriptions of dynamical systems are deeply rooted in topological spaces defined by non-Euclidean geometry. This paper proposes leveraging structure-rich geometric spaces for machine learning to achieve structural…
We investigate several boundedness properties of function spaces considered as uniform spaces.
It has been recently shown that, under appropriate hypotheses, the $\omega$-limit sets of a dynamical system are characterized by internal chain transitivity. In this paper, we examine generalizations of these ideas in the context of the…
The purpose of this article is twofold. On one hand, we reveal the equivalence of shift of finite type between a one-sided shift $X$ and its associated hom tree-shift $\mathcal{T}_{X}$, as well as the equivalence in the sofic shift. On the…
The study of local function in topological spaces is remarkable. Various branches have been developed through this study. In this paper, we further consider the local function and exploring the various properties of the same by considering…
A topological dynamical system induces two natural systems, one is on the hyperspace and the other one is on the probability space. The connection among some dynamical properties on the original space and on the induced spaces are…
The purpose of this paper is to give, on one hand, a mathematical exposition of the main topological and geometrical properties of geometric transitions, on the other hand, a quick outline of their principal applications, both in…
Each complex network (or class of networks) presents specific topological features which characterize its connectivity and highly influence the dynamics of processes executed on the network. The analysis, discrimination, and synthesis of…
Adaptivity is a dynamical feature that is omnipresent in nature, socio-economics, and technology. For example, adaptive couplings appear in various real-world systems like the power grid, social, and neural networks, and they form the…
We investigate systems of transitive models of ZFC which are elementarily embeddable into each other and the influence of definability properties on such systems.
We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric…
This is a lightning introduction to some modern techniques used in the study of the statistical properties of hyperbolic dynamical systems. The emphasis is not in presenting a comprehensive theory but rather in fleshing out the main ideas…
We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and…
A salient feature of topological phases are surface states and many of the widely studied physical properties are directly tied to their existence. Although less explored, a variety of topological phases can however similarly be…
The work presented in this thesis concerns different aspects of dynamical processes on networks. The first subject considered is the theoretical modeling of exploration processes of complex networks, such as the ``traceroute'' process used…
We study different types of transitive points in CR-dynamical systems (X,G) with closed relations G on compact metric spaces X. We also introduce transitive and dense orbit transitive CR-dynamical systems and discuss their properties and…
The topology transition problem of transmission networks is becoming increasingly crucial with topological flexibility more widely leveraged to promote high renewable penetration. This paper proposes a novel methodology to address this…
The collective dynamics of interacting dynamical units on a network crucially depends on the properties of the network structure. Rather than considering large but finite graphs to capture the network, one often resorts to graph limits and…