arXiv subjects

L. Stefanello

Publications and source records attributed to L. Stefanello.

7 recordsLinked to original sources

arXiv2025

Long-term behavior of casino games

We study the asymptotic behavior of the ratio of total return (or total profit) to total amount bet in a casino game. While the limit is well understood when the sequence of wagers is independent and identically distributed, here we consider the case in which bet sizes vary over time and may depend on past outcomes. We propose a general framework that yields such results under mild conditions on the conditional expectations of bets, returns, and profits. The set-up applies to many casino games (including compound games and those in which wagers are not immediately resolved), expressing the long-term behavior in terms of intrinsic parameters, namely return to player (RTP) and house advantage (HA). As an application, we examine the roulette win documented in Leigh's (1976) Thirteen against the Bank and attempt to quantify the likelihood that the story is true.

math.PR↗

arXiv2024

On Fuchs' problem for finitely generated abelian groups: The small torsion case

A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such group with the additional assumption that the torsion subgroups are small, for a suitable notion of small related to the Prüfer rank. As a concrete instance, we classify for each $n\ge2$ the realisable groups of the form $\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}^r$. Our tools require an investigation of the adjoint group of suitable radical rings of odd prime power order appearing in the picture, giving conditions under which the additive and adjoint groups are isomorphic. In the last section, we also deal with some groups of order a power of $2$, proving that the groups of the form $\mathbb{Z}/4\mathbb{Z}\times \mathbb{Z}/2^{u}\mathbb{Z}$ are realisable if and only if $0\le u\le 3$ or $2^u+1$ is a Fermat's prime.

math.AC↗

arXiv2023

On the connection between Hopf--Galois structures and skew braces

We present a different version of the well-known connection between Hopf--Galois structures and skew braces, building on a recent paper of A. Koch and P. J. Truman. We show that the known results that involve this connection easily carry over to this new perspective, and that new ones naturally appear. As an application, we present new insights on the study of the surjectivity of the Hopf--Galois correspondence, explaining in more detail the role of bi-skew braces in Hopf--Galois theory.

math.NT↗

arXiv2022

On bi-skew braces and brace blocks

L. N. Childs defined a bi-skew brace to be a skew brace such that if we swap the role of the two operations, then we find again a skew brace. In this paper, we give a systematic analysis of bi-skew braces. We study nilpotency and solubility, and connections between bi-skew braces and set-theoretic solutions of the Yang--Baxter equation. Further, we deal with Byott's conjecture in the case of bi-skew braces, and we use bi-skew braces as a tool to solve a classification problem proposed by L. Vendramin. In the final part, we investigate brace blocks, defined by A. Koch to be families of group operations on a given set such that any two of them yield a bi-skew brace. We provide a characterisation of brace blocks, illustrate how all known constructions in literature follow in a natural way from our characterisation, and give several new examples.

math.GR↗

arXiv2022

Brace blocks from bilinear maps and liftings of endomorphisms

We extend two constructions of Alan Koch, exhibiting methods to construct brace blocks, that is, families of group operations on a set $G$ such that any two of them induce a skew brace structure on $G$. We construct these operations by using bilinear maps and liftings of endomorphisms of quotient groups with respect to a central subgroup. We provide several examples of the construction, showing that there are brace blocks which consist of distinct operations of any given cardinality. One of the examples we give yields an answer to a question of Cornelius Greither. This example exhibits a sequence of distinct operations on the $p$-adic Heisenberg group $(G, \cdot)$ such that any two operations give a skew brace structure on $G$ and the sequence of operations converges to the original operation "$\cdot$".

math.GR↗

arXiv2022

Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples

Rota-Baxter operators for groups were recently introduced by L. Guo, H. Lang, and Y. Sheng. V. G. Bardakov and V. Gubarev showed that with each Rota-Baxter operator one can associate a skew brace. Skew braces on a group $G$ can be characterised in terms of certain gamma functions from $G$ to its automorphism group $\operatorname{Aut}(G)$, that are defined by a functional equation. For the skew braces obtained from a Rota-Baxter operator the corresponding gamma functions take values in the inner automorphism group $\operatorname{Inn}(G)$ of $G$. In this paper, we give a characterisation of the gamma functions on a group $G$, with values in $\operatorname{Inn}(G)$, that come from a Rota-Baxter operator, in terms of the vanishing of a certain element in a suitable second cohomology group. Exploiting this characterisation, we are able to exhibit examples of skew braces whose corresponding gamma functions take values in the inner automorphism group, but cannot be obtained from a Rota--Baxter operator. For gamma functions that can be obtained from a Rota-Baxter operators, we show how to get the latter from the former, exploiting the knowledge that a suitable central group extension splits.

math.GR↗

arXiv2021

From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures

The interplay between set-theoretic solutions of the Yang--Baxter equation of Mathematical Physics, skew braces, regular subgroups, and Hopf--Galois structures has spawned a considerable body of literature in recent years. In a recent paper, Alan Koch generalised a construction of Lindsay N.~Childs, showing how one can obtain bi-skew braces $(G, \cdot, \circ)$ from an endomorphism of a group $(G, \cdot)$ whose image is abelian. In this paper, we characterise the endomorphisms of a group $(G, \cdot)$ for which Koch's construction, and a variation on it, yield (bi-)skew braces. We show how the set-theoretic solutions of the Yang--Baxter equation derived by Koch's construction carry over to our more general situation, and discuss the related Hopf--Galois structures.

math.GR↗