Related papers: Synchronising primitive groups of diagonal type ex…
In a graph, we say that two nodes are topologically equivalent if their sets of first neighbors, excluding the two nodes, coincide. We prove that nonlinearly coupled heterogeneous oscillators located on a group of topologically equivalent…
Although the conjugacy classes of the general linear group are known, it is not obvious (from the canonic form of matrices) that two permutation matrices are similar if and only if they are conjugate as permutations in the symmetric group,…
Several constructions have been given for families of simple braces, but few examples are known of simple skew braces which are not braces. In this paper, we exhibit the first example of an infinite family of simple skew braces which are…
A transitive permutation group $G$ on a finite set $\Omega$ is said to be pre-primitive if every $G$-invariant partition of $\Omega$ is the orbit partition of a subgroup of $G$. It follows that pre-primitivity and quasiprimitivity are…
We prove that the total number $S_{n,132}(q)$ of copies of the pattern $q$ in all 132-avoiding permutations of length $n$ is the same for $q=231$, $q=312$, or $q=213$. We provide a combinatorial proof for this unexpected threefold symmetry.…
Problem 8.75 of the Kourovka Notebook [10], attributed to John G. Thompson, asks the following: Suppose $G$ is a finite primitive permutation group on $\Omega$, and $\alpha$, $\beta$ are distinct points of $\Omega$. Does there exist an…
In this article, we study symmetric $(v, k, \lambda)$ designs admitting a flag-transitive and point-primitive automorphism group $G$ whose socle is $PSU_{4}(q)$. We prove that there exist eight non-isomorphic such designs for which…
This paper is on the inverse parameterized differential Galois problem. We show that surprisingly many groups do not occur as parameterized differential Galois groups over K(x) even when K is algebraically closed. We then combine the method…
Sufficient conditions for synchronization of coupled Lienard-type oscillators are investigated via averaging technique. Coupling considered here is pairwise, unidirectional, and described by a nonlinear function (whose graph resides in the…
We provide abelianizations of differentiable actions of finite groups on smooth real manifolds. De Concini-Procesi wonderful models for (local) subspace arrangements and a careful analysis of linear actions on real vector spaces are at the…
Planar ornaments, a.k.a. wallpapers, are regular repetitive patterns which exhibit translational symmetry in two independent directions. There are exactly $17$ distinct planar symmetry groups. We present a fully automatic method for…
Sidon sequences and their generalizations have found during the years and especially recently various applications in coding theory. One of the most important applications of these sequences is in the connection of synchronization patterns.…
A permutation $\tau$ in the symmetric group $S_j$ is minimally overlapping if any two consecutive occurrences of $\tau$ in a permutation $\sigma$ can share at most one element. B\'ona \cite{B} showed that the proportion of minimal…
Generalising the work of Dey, we define the notion of ultra-synchronicity of sequences of real numbers. Let $B_{n,k},C_{n,k},P_{n,k},Q_{n,k}$ be the number of even permutations with $k$ descents, odd permutations with $k$ descents, even…
Using the properties of the ideal of the coordinate Hermite interpolation on n-dimensional grid [4], we prove that the extension k in k[x1, x2, ..., xn] / (f1(x1), ..., fn(xn)) has a primitive element if and only if at most one of the…
A {\it symmetric companion matrix} is a matrix of the form $A +A^T$ where $A$ is a companion matrix all of whose entries are in $\{0,1\}$ and $A^T$ is the transpose of $A.$ In this paper, we find the total number of primitive and the total…
We show that every element of PSL(2,q) is a commutator of elements of coprime orders. This is proved by showing first that in PSL(2,q) any two involutions are conjugate by an element of odd order.
We prove that the groups PSL_n(q) are (2,3)-generated for n=9,10 or 11 and all q. Actually, we find out explicit generators x_n and y_n of respective orders 2 and 3, for the groups SL_n(q).
We study a commutation pattern in which two affine families commute completely across the two families while each family retains internal noncommutativity. For one-dimensional affine groups over finite commutative rings, we prove a…
We show that given generators for subgroups $G$ and $H$ of $\mathrm{S}_n$, if $G$ is primitive then generators for $\mathrm{N}_H(G)$ may be computed in quasipolynomial time, namely $2^{O(\log^3 n)}$. The previous best known bound was simply…