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Root systems are sets with remarkable symmetries and therefore they appear in many situations in mathematics. Among others, denominator formulae of root systems are very beautiful and mysterious equations which have several meanings from a…
We revisit finite racks and quandles using a perspective based on permutations which can aid in the understanding of the structure. As a consequence we recover old results and prove new ones. We also present and analyze several examples.
By some hypergeometric summation theorems, the authors establish a series of new infinite summation formulas involving generalized harmonic numbers related to Riemann-Zeta function, with three different patterns.
A $\lambda$-quiddity of size $n$ is an $n$-tuple of elements from a fixed set, which is a solution to a matrix equation that arises in the study of Coxeter's friezes. The study of these solutions involves in particular the use of a notion…
Using a straightforward elementary approach, we derive numerous infinite arctangent summation formulas involving Fibonacci and Lucas numbers. While most of the results obtained are new, a couple of celebrated results appear as particular…
This paper aims to introduce high school students to the intriguing world of continued fractions, a mathematical concept that provides a unique representation of numbers. The study focuses on the exploration and development of the…
The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate…
A recent preprint (arXiv:1402.2632) introduced a growth procedure for planar maps, whose almost sure limit is "the uniform infinite 3-connected planar map". A classical construction of Brooks, Smith, Stone and Tutte (1940) associates a…
We summarize several results about non-simplicity, solvability and normal structure of finite groups related to the number of conjugacy classes appearing in the product or the power of conjugacy classes. We also collect some problems that…
Finite Unified Theories (FUTs) are N=1 supersymmetric Grand Unified Theories (GUTs) which can be made finite to all-loop orders, leading to a drastic reduction in the number of free parameters. By confronting the predictions of SU(5) FUTs…
The purpose of these notes is to collect in one place some facts on the category of finite totally ordered sets and some related categories. More specifically, we collect some results on them which will be useful for the study of iteratedly…
We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.
We introduce and study a combinatorially defined notion of root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root systems are…
We define and study the interpolated finite multiple harmonic $q$-series. A generating function of the sums of the interpolated finite multiple harmonic $q$-series with fixed weight, depth and $i$-height is computed. Some Ohno-Zagier type…
In this note unbounded hyperexpansive weighted composition operators are investigated. AS a consequence unbounded hyperexpansive multiplication and composition operators are characterized.
The structures of the ideals of Clifford algebras which can be both infinite dimensional and degenerate over the real numbers are investigated.
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
Let $\F$ be the finite field of odd prime power order $q$, We find explicit expressions for the number of triples $\{\al-1,\al,\al+1 \}$ of consecutive non-zero squares in $\F$ and similarly for the number of triples of consecutive…
Every semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
In this paper we look at the automorphisms of the multiplicative group of finite nearfields. We find partial results for the actual automorphism groups. We find counting techniques for the size of all finite nearfields. We then show that…