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

ON THE NUMBER OF CYCLES OF GRAPHS AND VC-DIMENSION

The number of the cycles in a graph is an important well-known parameter in graph theory and there are a lot of investigations carried out in the literature for finding suitable bounds for it. In this paper, we delve into studying this parameter and the cycle structure of graphs through the lens of the cycle hypergraphs and VC-dimension and find some new bounds for it, where the cycle hypergraph of a graph is a hypergraph with the edges of the graph as its vertices and the edge sets of the cycles as its hyperedges respectively. Note that VC-dimension is an important notion in extremal combinatorics, graph theory, statistics and machine learning. We investigate cycle hypergraph from the perspective of VC-theory, specially the celebrated Sauer-Shelah lemma, in order to give our upper and lower bounds for the number of the cycles in terms of the (dual) VC-dimension of the cycle hypergraph and nullity of graph. We compute VC-dimension and the mentioned bounds in some graph classes and also show that in certain classes, our bounds are sharper than many previous ones in the literature

On some aspects of measure and probability logics and a new logical proof for a theorem of Stone

One of the functions of mathematical logic is studying mathematical objects and notions by logical means. There are several important representation theorems in analysis. Amongst them, there is a well-known classical one which concerns probability algebras. There are quite a few proofs of this result in the literature. This paper pursue two main goals. One is to consider some aspects of measure and probability logics and expose a novel proof for the mentioned representation theorem using ideas from logic and by application of an important result from model theory. The second and even more important goal is to present more connections between two fields of analysis and logic and reveal more the strength of logical methods and tools in analysis. The paper is mostly written for general mathematicians, in particular the people who are active in analysis or logic as the main audience. It is self-contained and includes all prerequisites from logic and analysis