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Let $G$ be a finite abelian group and $A$ a subset of $G$. The spectrum of $A$ is the set of its large Fourier coefficients. Known combinatorial results on the structure of spectrum, such as Chang's theorem, become trivial in the regime…
The main goal of this paper is to extend [J. Algebra Appl. 20 (2021), 2150074] to generalized quaternion algebras, even when these algebras are not necessarily division rings. More precisely, in such cases, the image of a multilinear…
A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$; Lekkerkerker proved that the average number of summands for integers in $[F_n,…
We study some properties of a family of rings $R(I)_{a,b}$ that are obtained as quotients of the Rees algebra associated with a ring $R$ and an ideal $I$. In particular, we give a complete description of the spectrum of every member of the…
In this short note, we find an equivalent combinatorial condition only involving finite sums under which a centered Gaussian random vector with multinomial covariance matrix satisfies the Gaussian product inequality (GPI) conjecture. These…
The goal in thinning is to summarize a dataset using a small set of representative points. Remarkably, sub-Gaussian thinning algorithms like Kernel Halving and Compress can match the quality of uniform subsampling while substantially…
It has previously been demonstrated that the mesonic fields in chiral Lagrangians can be related to the quark-level operators of QCD sum-rules via energy-independent (constant) scale factor matrices constrained by chiral symmetry. This…
Recently, Sasada and Uozumi (2024) investigated connections between classical (deterministic) and random integrable models, discovering a hierarchy of quadrirational Yang-Baxter independence preserving (IP) maps together with related…
We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…
We study quantum fluctuations in quartet superfluid (QSF) of two-dimensional (2D) fermion mixtures with mass imbalance. Here QSF is a high-order superfluid that corresponds to the condensation of ($1+3$) clusters, each consisting of a light…
Starting with a quaternion difference equation with boundary conditions, a parameterized sequence which is complete in finite dimensional quaternion Hilbert space is derived. By employing the parameterized sequence as the kernel of discrete…
In the paper [25], written in collaboration with Gesine Reinert, we proved a universality principle for the Gaussian Wiener chaos. In the present work, we aim at providing an original example of application of this principle in the…
This is an introduction to rings and fields, written for a quarter-long undergraduate course. It includes the basic properties of ideals, modules, algebras and polynomials, the constructions of ring extensions and finite fields, some…
Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and…
This paper establishes an extension of Wolstenholme's theorem to the ring of Gaussian integers $\mathbb{Z}[i]$. For a prime $p > 7$, we prove that the sum $S_p$ of inverses of Gaussian integers in the set $\{n+mi \mid 1 \leq n, m \leq p-1,…
In this paper we investigate the structure of $\left(\frac{a,b}{\Z{n}}\right)$, the quaternion rings over $\Z{n}$. It is proved that these rings are isomorphic to $\left(\frac{-1,-1}{\Z{n}}\right)$ if $ a \equiv b\equiv -1 \pmod{4}$ or to…
We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra.
For a very general class of probability distributions in disordered Ising spin systems, in the thermodynamical limit, we prove the following property for overlaps among real replicas. Consider the overlaps among s replicas. Add one replica…
Gaussian processes are rich distributions over functions, which provide a Bayesian nonparametric approach to smoothing and interpolation. We introduce simple closed form kernels that can be used with Gaussian processes to discover patterns…
Let $f$ be a Rademacher or Steinhaus random multiplicative function. For various arithmetically interesting subsets $\mathcal A\subseteq [1, N]\cap\mathbb N$ such that the distribution of $\sum_{n\in \mathcal A} f(n)$ is approximately…