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In this paper we study arithmetical and structural features of a finite group that possesses exactly two conjugacy class sizes that are composite numbers.
The theory of finite automata applies to the study on relations of multiple zeta values.
A new sequential approach to investigations of structure of metric spaces at infinity is proposed. Criteria for finiteness and boundedness of metric spaces at infinity are found.
A well-known conjecture asserts that there are infinitely many primes $p$ for which $p - 1$ is a perfect square. We obtain upper and lower bounds of matching order on the number of pairs of distinct primes $p,q \le x$ for which $(p - 1)(q -…
Motivated by recent interests in predictive inference under distribution shift, we study the problem of approximating finite weighted exchangeable sequences by a mixture of finite sequences with independent terms. Various bounds are derived…
We describe the structure of finite Boolean inverse monoids and apply our results to the representation theory of finite inverse semigroups. We then generalize to semisimple Boolean inverse semigroups.
This paper offers a solution method that allows one to find exact values for a large class of convergent series of rational terms. Sums of this form arise often in problems dealing with Quantum Field Theory.
We give a broad survey of inequalities for the number of linear extensions of finite posets. We review many examples, discuss open problems, and present recent results on the subject. We emphasize the bounds, the equality conditions of the…
Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…
An iterative square root of a function $f$ is a function $g$ such that $g(g(\cdot))=f(\cdot)$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. This is used to prove that…
We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite dimensional basic classical simple Lie…
Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar…
This paper is an enhanced version of a more than decade-older paper with a similar title. Many formulae involving both finite and infinite sums of digamma and polygamma functions up to quadratic order, few of which appear in standard…
We introduce the notion of locally finite root supersystems as a generalization of both locally finite root systems and generalized root systems. We classify irreducible locally finite root supersystems.
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We…
For each infinite word over a given finite alphabet, we define an increasing sequence of rooted finite graphs, that can be thought as approximations of the famous Sierpinski carpet. These sequences naturally converge to an infinite rooted…
These notes are an introduction to the theory of quantum symmetries of finite and infinite sets, graphs, and locally compact spaces.
In this paper we study the number of finite topologies on an $n$-element set subject to various restrictions.
We study the asymptotics of the average number of squares (or quadratic residues) in Z_n and Z_n^*. Similar analyses are performed for cubes, square roots of 0 and 1, and cube roots of 0 and 1.
We obtain new uniform upper bounds for the (non necessarily symmetric) tensor rank of the multiplication in the extensions of the finite fields $\F_q$ for any prime or prime power $q\geq2$; moreover these uniform bounds lead to new…