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Several results about the union-closed sets conjecture are presented.

There has been growing interest in developing accurate models that can also be explained to humans. Unfortunately, if there exist multiple distinct but accurate models for some dataset, current machine learning methods are unlikely to find…

We obtain large deviations theorems for nonconventional sums with underlying process being a Markov process satisfying the Doeblin condition or a dynamical system such as subshift of finite type or hyperbolic or expanding transformation.

This paper provides a complete suite of axioms for a version of set theory that I call Explication. Explication borrows from the two most prominent existing systems of set theory. Explication starts with class variables. After several…

We present a new fragment of axiomatic set theory for pure sets and for the iteration of power sets within given transitive sets. It turns out that this formal system admits an interesting hierarchy of models with true membership relation…

In the present paper and as an application of Roth's theorem concerning the rational approximation of algebraic numbers, we give a sufficient condition that will assure us that a series of positive rational terms is a transcendental number.…

The standard model of particle physics is marvelously successful. However, it is obviously not a complete or final theory. I shall argue here that the structure of the standard model gives some quite concrete, compelling hints regarding…

The notion of clause set cycle abstracts a family of methods for automated inductive theorem proving based on the detection of cyclic dependencies between clause sets. By discerning the underlying logical features of clause set cycles, we…

Two types of approximation to the paradoxical Russell Set are presented, one approximating it from below, one from above. It is shown that any lower approximation gives rise to a better approximation containing it, and that any upper…

We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph…

We summarize some facts on chains (totally ordered sets), from an order-theoretic and from a topological point of view. We highlight the fact that many classical theorems that are true for partially ordered sets under some completeness…

Transfinite set theory including the axiom of choice supplies the following basic theorems: (1) Mappings between infinite sets can always be completed, such that at least one of the sets is exhausted. (2) The real numbers can be well…

We study the higher (or sequential) topological complexity $\mathrm{TC}_s$ of manifolds with abelian fundamental group. We give sufficient conditions for $\mathrm{TC}_s$ to be non-maximal in both the orientable and non-orientable cases. In…

Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which…

Using the technique of quasi difference sets we characterize geometry and automorphisms of configurations which can be presented as a join of some others, in particular - which can be presented as series of cyclically inscribed copies of…

Linear matrix Inequalities (LMIs) have had a major impact on control but formulating a problem as an LMI is an art. Recently there is the beginnings of a theory of which problems are in fact expressible as LMIs. For optimization purposes it…

In a recent paper (Appl. Math. Comput. 215, 1622--1645, 2009), the authors proposed a method of summation of some slowly convergent series. The purpose of this note is to give more theoretical analysis for this transformation, including the…

Evenly convex sets in a topological vector space are defined as the intersection of a family of open half spaces. We introduce a generalization of this concept in the conditional framework and provide a generalized version of the bipolar…

When supersymmetry is spontaneously broken it will be generically non-linearly realized. A method to describe the non-linear realization of supersymmetry is with constrained superfields. We discuss the basic features of this description and…

For any finite totally ordered set, the multisets of intervals form an abelian category. Various classes of subcategories admit natural combinatorial descriptions, and counting them yields familiar integer sequences. Surprisingly, in some…