Related papers: Integer group determinants for ${\rm C}_{2}^{2} \r…

We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C \log \log n$, where $C=(1-e^{-\gamma})^{-1} = 2.280...$ and $\gamma$ is…

Under binary matrices we mean matrices whose entries take one of two values. In this paper, explicit formulae for calculating the determinant of some type of binary Toeplitz matrices are obtained. Examples of the application of the…

We enumerate total cyclic orders on $\left\{1,\ldots,n\right\}$ where we prescribe the relative cyclic order of consecutive triples $(i,{i+1},{i+2})$, these integers being taken modulo $n$. In some cases, the problem reduces to the…

Let $1=d_{1}<d_{2}< \cdots < d_{\tau(n)}=n$ denote the ordered sequence of the positive divisors of an integer $n$. We are interested in estimating the arithmetic function $$ V(n) := \prod_{1 \le i < j \le \tau(n)}(d_{j}-d_{i}) \quad (n \ge…

We discuss the application of the determinantal method to the proof of the Riemann hypothesis. We start from the fact that, if a certain doubly infinite set of determinants are all positive, then the hypothesis is true. This approach…

Let $p$ be a an odd prime and let $G$ be a finite $p$-group with cyclic commutator subgroup $G'$. We prove that the exponent and the abelianization of the centralizer of $G'$ in $G$ are determined by the group algebra of $G$ over any field…

Let $n \in \mathbb{Z}_{\geqslant 2}$. By $P(n)$ we denote the set of all prime divisors of the integers in the sequence $n, n^2-1, (n^2-1)^2-1, \dots$. We ask whether the set $P(n)$ determines $n$ uniquely under the assumption that $n \neq…

We study the probability of a given element, in the commutator subgroup of a group, to be equal to a commutator of two randomly chosen group elements, and compute explicit formulas for calculating this probability for some interesting…

This paper is devoted to the units of integral group rings of cyclic $2$-groups of small orders, namely, the orders of $2^n$ for $n<8$. Immediately we should note the issues our consideration describe in the introduction in more detail.…

We prove that there is an absolute constant $c>0$ with the following property: if $Z/pZ$ denotes the group of prime order $p$, and a subset $A\subset Z/pZ$ satisfies $1<|A|<p/2$, then for any positive integer…

We solve the problem of effectively computing the $a$-invariant of ladder determinantal rings. In the case of a one-sided ladder, we provide a compact formula, while, for a large family of two-sided ladders, we provide an algorithmic…

We fill the gaps in A. Gica's determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all the positive and odd integers $x\leq\sqrt{d}$. We also…

For a group $G$, $\mathcal{F}_G$ denotes the set of all non-empty finite subsets of $G$. We extend the finitary coarse structure of $G$ from $G\times G$ to $\mathcal{F}_G\times \mathcal{F}_G$ and say that a macro-uniform mapping $f:…

The paper is devoted to finding a homomorphic image for the $c$-nilpotent multiplier of the verbal product of a family of groups with respect to a variety ${\mathcal V}$ when ${\mathcal V} \subseteq {\mathcal N}_{c}$ or ${\mathcal…

We study continuous 2-valued dynamical systems with discrete time (dynamics) on $\mathbb{C}$. The main question addressed is whether a 2-valued dynamics can be defined by the action of a 2-valued group. We construct a class of strongly…

We provide a family of group measure space II_1 factors for which all finite index subfactors can be explicitly listed. In particular, the set of all indices of irreducible subfactors can be computed. Concrete examples show that this index…

A method for finding an optimum $n$-dimensional commutative group code of a given order $M$ is presented. The approach explores the structure of lattices related to these codes and provides a significant reduction in the number of…

We compute the determinant of $\sum_{n=1}^{N} \vec{A}^{(n)} \otimes \vec{B}^{(n)}$, where $\vec{A}^{(n)}$ is square and ${\vec{B}^{(n)}=\vec{x}^{(n)}{\vec{y}^{(n)}}^T}$ where $\vec{x}^{(n)}$ and $\vec{y}^{(n)}$ have length $N$.

This paper considers solutions (x_1, x_2, ..., x_n) to the cyclic system of n simultaneous congruences r (x_1x_2 ...x_n)/x_i = s (mod |x_i|), for fixed nonzero integers r,s with r>0 and gcd(r,s)=1. It shows this system has a finite number…

A ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive code ${\cal C}\subseteq{\mathbb{Z}}_2^\alpha\times{\mathbb{Z}}_4^\beta$ is called cyclic if the set of coordinates can be partitioned into two subsets, the set of ${\mathbb{Z}}_2$ and the set of…