arXiv subjects

Alireza Mofidi

Publications and source records attributed to Alireza Mofidi.

4 recordsLinked to original sources

arXiv2022

On partial cubes, well-graded families and their duals with some applications in graphs

Well-graded families, extremal systems and maximum systems (the last two in the sense of VC-theory and Sauer-Shelah lemma on VC-dimension) are three important classes of set systems. This paper aims to study the notion of duality in the context of these classes of set systems and then use the obtained results for studying graphs. More specifically, we are concerned with the characterization of the finite set systems which themselves and their dual systems are both well-graded, extremal or maximum. On the way to this goal, and maybe also of independent interest, we study the structure of the well-graded families with the property that the size of the system is not much bigger than the size of its essential domain, that is, the set of elements of the domain which are shattered by the system as single element subsets. As another target of the paper, we use the above results to characterize graphs whose set systems of open or closed neighbourhoods, cliques or independent sets are well-graded, extremal or maximum. We clarify the relation of such graphs to the celebrated half-graphs. Through the paper, we frequently relate our investigations to the VC-dimension of the systems. Also we use one-inclusion graphs associated to set systems as an important technical tool.

math.CO↗

arXiv2022

On dominating graph of graphs, median graphs and partial cubes, and graphs in which complement of every minimal dominating set is minimal dominating

The dominating graph of a graph G is a graph whose vertices correspond to the dominating sets of G and two vertices are adjacent whenever their corresponding dominating sets differ in exactly one vertex. Studying properties of dominating graph has become an increasingly interesting subject in domination theory. On the other hand, median graphs and partial cubes are two fundamental graph classes in graph theory. In this paper, we make some new connections between domination theory and the theory of median graphs and partial cubes. As the main result, we show that the following conditions are equivalent for every graph $G \not \simeq C_4$ with no isolated vertex, and in particular, that the simple third condition completely characterizes first two ones in which three concepts of dominating graphs, median graphs and complement of minimal dominating sets get related: - The dominating graph of G is a median graph, - The complement of every minimal dominating set of G is a minimal dominating set, - Every vertex of G is either of degree 1 or adjacent to a vertex of degree 1. As another result, we prove that the dominating graph of every graph is a partial cube and also give some examples to show that not all partial cubes or median graphs are isomorphic to the dominating graph of a graph. The above-mentioned results, as another highlight of the paper, provide novel infinite sources of examples of median graphs and partial cubes.

math.CO↗

arXiv2019

Uniform logical proofs for Riesz representation theorem, Daniell-Stone theorem and Stone's representation theorem for probability algebras

Riesz representation theorem, Daniell-Stone theorem for Daniell integrals and Stone's representation theorem for probability and measure algebras are three important classical results in analysis concerning existence of measures with certain properties. Many proofs of these theorems can be found in the literature of analysis, from elementary ones which use ordinary techniques from measure theory, to more sophisticated ones, such as those employing techniques from nonstandard analysis, in particular for Riesz representation theorem. In this paper, as the first goal, we give new proofs for all these three theorems. Our proofs have a mild logical flavor and are uniform in the sense that they are all based on the same general idea and rely on the application of the same technical tool from logic to measure theory, namely logical compactness theorem. In fact, as the second goal of the paper, we try to reveal more the power of logical methods in analysis in particular measure theory, and make stronger connections between analysis and logic. We use the setting of "integration logic" which is a logical framework (and one of the forms of probability logics) for studying measure and probability structures by logical means. Indeed, we elaborate this setting and use its expressive power and a version of compactness theorem holding in it to show its application in measure theory by giving new proofs for the above-mentioned measure existence theorems. As mentioned, an advantage of these proofs is that they are all given in a uniform way since they are all based on the logical compactness theorem. The paper is mostly written for general mathematicians, in particular the people active in analysis or logic as the main audience. So it is self-contained and the reader does not need to have any advanced prerequisite knowledge from logic or measure theory.

math.CA↗

arXiv2015

On some dynamical aspects of NIP theories

We study some dynamical aspects of the action of automorphisms in model theory in particular in the presence of invariant measures. We give some characterizations for NIP theories in terms of dynamics of automorphisms and invariant measures for example in terms of compact systems, entropy and measure algebras. Moreover, we study the concept of symbolic representation for models. Amongst the results, we give some characterizations for dividing lines and combinatorial configurations such as independence property, order property and strictly order property in terms of symbolic representations.

math.LO↗