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Nikolai A. Krylov

Publications and source records attributed to Nikolai A. Krylov.

At least 19 recordsLinked to original sources
arXiv2026

Periodicities in the Riordan arrays of polynomials over finite fields

We study periodicity properties of the 2-D $\bigl(p_1(t)/p_2(t),\, tp_3(t)\bigr)$ and 3-D $\bigl(p_1(t)/p_2(t),\, tp_3(t),\, p_4(t)\bigr)$ Riordan arrays over a finite field ${\mathbb F}_q$, where each $p_i(t)$ is a polynomial with $p_i(0)\neq 0$. We show that the columns of the 2-D Riordan array are eventually periodic sequences, where a circulant matrix generated by the coefficients of $p_3(t)$ determines the behavior of this periodicity as the column index grows indefinitely. Furthermore, we prove that the preperiodic column partial sums of the 2-D array are periodic, and present a family of the Riordan arrays for which such sequences of partial sums are identically zero. We also show that the layers of the 3-D Riordan array contain periodic orbits related to each other via powers of a circulant matrix generated by the coefficients of $p_4(t)$.

math.CO↗
arXiv2026

The symmetric groups $S_n, n\geq 4$, and finite non-abelian simple groups are not embeddable in any Riordan group

We prove that the symmetric group of degree greater than three cannot be embedded into the Riordan group with coefficients in any commutative ring. We also prove the impossibility to embed finite non-abelian simple groups. As a closely related topic, we show why all truncated Riordan groups are solvable, in stark contrast to the unsolvability of the infinite-sized Riordan groups. Finally, we give an explicit embedding of the alternating group $A_4$ into the Lagrange subgroup with coefficients in a certain commutative ring, and prove that $A_4$ cannot be embedded into a substitution group.

math.GR↗
arXiv2025

Lower central series of the Riordan group over the field with two elements

The Riordan group ${\cal R}$ over the field ${\mathbb F}_2$ is a split extension of the Appell subgroup by the Nottingham group ${\cal N}({\mathbb F}_2)$. Using the lower central series of the Nottingham group obtained by C. Leedham-Green and S. McKay, the lower central series of ${\cal R}({\mathbb F}_2)$ is calculated. Considering the Riordan group over an arbitrary commutative ring with identity, where all Riordan arrays have only 1s on the main diagonal, it is also proved that the abelianization of this group is isomorphic to the direct product of the abelianization of the corresponding Lagrange subgroup and the additive group of the ground ring.

math.GR↗
arXiv2025

On embeddability of Coxeter groups into the Riordan group

We discuss examples of linear representations of finite groups as subgroups of the Riordan group. In particular, we show that the symmetric group of degree three has no faithful representation as a subgroup of the Riordan group over the complex numbers, but can be embedded as a subgroup of the Riordan group over a field of characteristic three.

math.GR↗
arXiv2024

Periodic Column Partial Sums in the Riordan Array of a Polynomial

When $p(t)$ is a polynomial of degree $d$, $k$-th column of the Riordan array $\bigl(1/(1 - t^{d+1}), tp(t)\bigr)$ is an eventually periodic sequence with the repeating part beginning at the $1 + (k-1)(d+1)$-st term. The pre-periodic terms add up to the $(k-1)(d+1)$-st partial sum of the corresponding formal power series, and thus the Riordan array of $p(t)$ generates a sequence of column partial sums. We classify linear and quadratic polynomials, and present a particular family of polynomials of higher degrees, for which such sequences of column partial sums are eventually periodic.

math.CO↗
arXiv2023

Periodicity and Circulant Matrices in the Riordan Array of a Polynomial

We consider Riordan arrays $\bigl(1/(1-t^{d+1}), ~ tp(t)\bigr)$. These are infinite lower triangular matrices determined by the formal power series $1/(1-t^{d+1})$ and a polynomial $p(t)$ of degree $d$. Columns of such matrix are eventually periodic sequences with a period of $d + 1$, and circulant matrices are used to describe the long term behavior of such periodicity when the column's index grows indefinitely. We also discuss some combinatorially interesting sequences that appear through the corresponding A - and Z - sequences of such Riordan arrays.

math.CO↗
arXiv2020

Combinatorial properties of irreducible Laguerre polynomials in two variables

Following our earlier work, where doubly indexed and irreducible over Q two-variable Laguerre polynomials were introduced, we prove for such polynomials some recurrence formulas and obtain a generating function. In addition, we show how certain sums of such polynomials with a fixed total degree relate to some standard polynomials.

math.CA↗
arXiv2019

Permutations with a distinct divisor property

A finite group of order $n$ is said to have the distinct divisor property (DDP) if there exists a permutation $g_1,\ldots, g_n$ of its elements such that $g_i^{-1}g_{i+1} \neq g_j^{-1}g_{j+1}$ for all $1\leq i<j<n$. We show that an abelian group is DDP if and only if it has a unique element of order 2. We also describe a construction of DDP groups via group extensions by abelian groups and show that there exist infinitely many non abelian DDP groups.

math.GR↗
arXiv2017

On the subgroup generated by solutions of Pell's equation

Equivalence classes of solutions of the Diophantine equation $a^2+mb^2=c^2$ form an infinitely generated abelian group $G_m$, where $m$ is a fixed square-free positive integer. Solutions of Pell's equation $x^2-my^2=1$ generate a subgroup $P_m$ of $G_m$. We prove that $P_m$ and $G_m/P_m$ have infinite rank for all $m>1$. We also give several examples of $m$ for which $G_m/P_m$ has nontrivial torsion.

math.NT↗
arXiv2014

A basis of the group of primitive almost pythagorean triples

Let $m$ be a fixed square-free positive integer, then equivalence classes of solutions of Diophantine equation $x^2+m\cdot y^2=z^2$ form an infinitely generated abelian group under the operation induced by the complex multiplication. A basis of this group is constructed here using prime ideals and the ideal class group of the field $\mathbb Q (\sqrt{-m})$.

math.NT↗
arXiv2012

The Group of Primitive Almost Pythagorean Triples

We consider the triples of integer numbers that are solutions of the equation $x^2+qy^2=z^2$, where $q$ is a fixed, square-free arbitrary positive integer. The set of equivalence classes of these triples forms an abelian group under the operation coming from complex multiplication. We investigate the algebraic structure of this group and give a complete analysis when $q\in\{2,3,5,6\}$.

math.NT↗
arXiv2010

Zeta(n) via hyperbolic functions

We present here an approach to a computation of $ζ(2)$ by changing variables in the double integral using hyperbolic trig functions. We also apply this approach to present $ζ(n)$, when $n>2$, as a definite improper integral of single variable.

math.CA↗
arXiv2009

Angle contraction between geodesics

We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of such systems are observed when the plane is replaced by a regular surface in ${\mathbb R}^3$ and lines are replaced by geodesics. An application of our results to the classification of points on the surface as elliptic, hyperbolic or parabolic is also presented.

math.DS↗
arXiv2006

Pseudo-isotopy classes of diffeomorphisms of the unknotted pairs $(S^{n+2},~S^n)$ and $(S^{2p+2},~S^p\times S^p)$

We consider two pairs: the standard unknotted $n$-sphere in $S^{n+2}$, and the product of two $p$-spheres trivially embedded in $S^{2p+2}$, and study orientation preserving diffeomorphisms of these pairs. Pseudo-isotopy classes of such diffeomorphisms form subgroups of the mapping class groups of $S^n$ and $S^p\times S^p$ respectively and we determine the algebraic structure of such subgroups when $n>4$ and $p>1$.

math.GT↗
arXiv2004

Kernel of the variation operator and periodicity of the open books

We consider a parallelizable $2n$-manifold $F$ which has the homotopy type of the wedge product of $n$-spheres and show that the group of pseudo-isotopy classes of orientation preserving diffeomorphisms that keep the boundary $\partial F$ pointwise fixed and induce the trivial variation operator is a central extension of the group of all homotopy $(2n+1)$-spheres by $H_n\bigl(F; Sπ_n(SO(n))\bigr)$. Then we apply this result to study the periodicity properties of branched cyclic covers of manifolds with simple open book decompositions and extend the previous results of Durfee, Kauffman and Stevens to dimensions 7 and 15.

math.GT↗
arXiv2004

Relative mapping class group of $S^p\times D^q$

Algebraic structure of the group of pseudo-isotopy classes of diffeomorphisms of the trivial disk bundle over the standard sphere which restrict to the identity map on the boundary is determined.

math.AT↗