Related papers: Number of $k$-normal elements over a finite field

We use the remarkable distance estimate of Ilya Kachkovskiy and Yuri Safarov, to show that if $H$ is a nonseparable Hilbert space and $K$ is any closed ideal in $B(H)$ that is not the ideal of compact operators, then any normal element of…

Let $\mathcal{O}$ be an order of index $m$ in the maximal order of a quadratic number field $k=\mathbb{Q}(\sqrt{d})$. Let $\underline{\mathbf{O}}_{d,m}$ be the orthogonal $\mathbb{Z}$-group of the associated norm form $q_{d,m}$. We describe…

In 1995, Isaacs, Kantor and Spaltenstein proved that for a finite simple classical group G defined over a field with q elements, and for a prime divisor p of |G| distinct from the characteristic, the proportion of p-singular elements in G…

We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In…

Let $\mathbb {F}_q$ be finite field with $q$ elements. Let $\alpha\leqslant n$ be positive integers. Consider the general linear group $\mathrm{GL}(\alpha+n, \mathbb {F}_q) $ and its subgroup $H(n)$, which fixes the first $\alpha$ basis…

In this paper we introduce the concept of inessential element of a standard basis of I, where I is any homogeneous ideal of a polynomial ring. An inessential element is, roughly speaking, a form of the basis whose omission produces an ideal…

Apart from the qualitative features described in \cite{chm}, the renormalization group equation derived for the rotation of the fermion mass matrices are amenable to quantitative study. The equation depends on a coupling and a fudge factor…

We will examine a particular mathematical derivation in a paper by P. Falkensteiner and H. Grosse (F&G) [1]. In [1] a quantity "delta(A)" is defined. This quantity is generated when the normal ordered generalized charge operator undergoes a…

Schwick, in [6], states that let $\mathcal{F}$ be a family of meromorphic functions on a domain $D$ and if for each $f\in\mathcal{F}$, $(f^n)^{(k)}\neq 1$, for $z\in D$, where $n, k$ are positive integers such that $n\geq k+3$, then…

In this paper, we extend the results obtained by Cortes-Ferrero-Juriaans (2009) for the quaternion over the ring Colombeau's simplified generalized numbers, denoted by $\overline{\mathbb{H}}_s$, to the quaternion over the ring of…

In 2011, Fici and Lipt\'ak introduced prefix normal words. A binary word is prefix normal if it has no factor (substring) that contains more occurrences of the letter 1 than the prefix of the same length. Among the open problems regarding…

We introduce the fundamental group ${\mathcal F}(A)$ of a unital simple $C^*$-algebra $A$ with a unique normalized trace. We compute fundamental groups ${\mathcal F}(A)$ of several nuclear or non-nuclear $C^*$-algebras $A$. K-theoretical…

In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(\alpha)}{(x)}=\sum_{j=1}^{k}w(k,j)^{\alpha}x^{j-1}, \end{equation*} where $k,\alpha$ are positive integers and $w(k,j)=\frac{1}{j}\binom{k-1}{j-1}\binom{k+j}{j-1}$. Let $(x)_{0}=1$ and…

The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called…

We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all…

We give an upper bound for the norm of the determinant of additively indecomposable, totally positive definite quadratic forms defined over the ring of integers of totally real number fields. We apply these results to find lower and upper…

Consider the average of the first n k-th powers. We pose and answer the following natural question: For which values of n and k is this average an integer? If k is odd the answer is easy; it is an integer as long as n is incongruent to 2…

A.A. Suslin proved a normality theorem for an elementary linear group, which says that an elementary linear group of size bigger than or equal to 3 over a commutative ring with unity is normal in the general linear group of same size.…

Let $\mathbf{F}_q$ be a finite field of $q$ elements. We show that the normalized Jacobi sum $J(\chi,\eta)/\sqrt{q}$, for each fixed non-trivial multiplicative character $\eta$, becomes equidistributed in the unit circle as…

We present a new class of binary words: the prefix normal words. They are defined by the property that for any given length $k$, no factor of length $k$ has more $a$'s than the prefix of the same length. These words arise in the context of…