Related papers: Constructions of difference sets in nonabelian 2-g…
We classify the category of finite-dimensional real division composition algebras having a non-abelian Lie algebra of derivations. Our complete and explicit classification is largely achieved by introducing the concept of a…
A (2,*)-group is a group that can be generated by two elements, one of which is an involution. We describe the method we have used to produce a census of all (2,*)-groups of order at most 6 000. Various well-known combinatorial structures…
This paper concerns finite groups of class (at most) two and of odd prime exponent $p$. Such a group is called special if the center lies within its derived group. Every group of class 2 and exponent $p$ can be uniquely expressed as the…
A $(v,k,\lambda, \mu)$-partial difference set (PDS) is a subset $D$ of size $k$ of a group $G$ of order $v$ such that every nonidentity element $g$ of $G$ can be expressed in either $\lambda$ or $\mu$ different ways as a product $xy^{-1}$,…
One of the very first results about designs over finite fields, by S. Thomas, is the existence of a cyclic 2-$(n,3,7)$ design over $\mathbb{F}_{2}$ for every integer $n$ coprime with 6. Here, by means of difference methods, we reprove and…
A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for homogeneous polynomials, has since been extended to the setting of field extensions and to that of…
We revisit the old idea of constructing difference sets from cyclotomic classes. Two constructions of skew Hadamard difference sets are given in the additive groups of finite fields using unions of cyclotomic classes of order $N=2p_1^m$,…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
For a system of N qubits, spanning a Hilbert space of dimension d=2^N, it is known that there exists d+1 mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases…
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A corresponding framework has been developed by Woronowicz, more…
Full set of autonomous completely solvable differential systems of equations in total differentials is built by basis of infinitesimal operators, universal invariant, and structure constants of admited multiparametric Lie group (abelian and…
Strongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest…
Twin-width is a recently introduced graph parameter with applications in algorithmics, combinatorics, and finite model theory. For graphs of bounded degree, finiteness of twin-width is preserved by quasi-isometry. Thus, through Cayley…
This paper presents new research in infinitesimal algebra by introducing the concept of an infinitesimal group and exploring its properties and ramifications. The author investigates first- and second-order subgroups of Lie groups and…
For $N \geq 2$, we study the structure of definable abelian group extensions of the additive group $(\mathbb{R}^N,+)$ by countable abelian (Borel) groups $G$. Given an extension $H$ of $(\mathbb{R}^N,+)$ by $G$, we measure the definability…
We give an algorithm for enumerating the regular nontrivial partial difference sets (PDS) in the group $G_n = C_{2^n}\times C_{2^n}$. We use our algorithm to obtain all of these PDS in $G_n$ for $2\leq n\leq 9$, and we obtain partial…
Higher gauge theory for non-abelian structure 2-groups faces significant challenges when extending beyond the fake-flat sector, which suffers from limited applicability in physical models. A promising resolution involves equipping 2-groups…
The DifSets package for GAP implements an algorithm for enumerating all difference sets in a group up to equivalence and provides access to a library of results. The algorithm functions by finding difference sums, which are potential images…
A set A of positive integers is called a perfect difference set if every nonzero integer has an unique representation as the difference of two elements of A. We construct dense perfect difference sets from dense Sidon sets. As a consequence…
In this paper we extensively study the notion of Hamiltonian structure for nonabelian differential-difference systems, exploring the link between the different algebraic (in terms of double Poisson algebras and vertex algebras) and…