arXiv subjects
Publications and source records attributed to Diana Savin.
Let $\left(P_{n}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan numbers and let $\left(M_{p_{n}}\right)_{n\geq0}$ be the sequence of bi-periodic Padovan matrices. In this article we study when these matrices are diagonalizable and we obtain a certain connection with the Lucas number sequence. We also obtain some connections of these matrices with the generating matrix $Q$ for the Padovan numbers.
In this paper, we investigate bi-periodic Padovan and bi-periodic Perrin quaternions in the quaternion algebra Q_Zp. We introduce the bi-periodic Perrin sequence and clarify its structural relationship with the bi-periodic Padovan sequence. By extending these sequences to the quaternion setting, we analyze their norm properties in the modular framework. For suitable choices of twin prime coefficients, we derive explicit criteria characterizing zero divisors and invertible elements in Q_Zp.
We obtain inequalities involving the entropy of a positive integer and the divergence of two positive integers, respectively the entropy of an ideal and the divergence of two ideals in a ring of algebraic integers. Among the important results, we show that the minimal entropy arises for sharp localization, and the maximal entropy occurs for equidistribution. We also study other interesting estimates of entropy and divergence for numbers and for ideals. Finally, we determine the entropies of probability distributions on infinite trees of Schur {\sigma}-groups, which are realized by 3-class field tower groups of imaginary quadratic number fields.
Let $A$ be a unitary ring and let $(\mathbf{I(A),\subseteq })$ be the lattice of ideals of the ring $A.$ In this article we will study the property of the lattice $(\mathbf{I(A),\subseteq})$ to be Noetherian or not, for various types of rings $A$. In the last section of the article we study certain rings that are not Boolean rings, but all their ideals are idempotent.
It is known that the quaternion algebras are central simple algebras and also clifford algebras. In this paper, we introduce a new class of quaternions called Lucas-Leonardo p-quaternions and derive several fundamental properties of these numbers. Furthermore, we investigate some applications related to companion sequences associated with Leonardo quaternions. In particular, we determine Lucas-Leonardo quaternions and Francois quaternions, which are zero divisors and invertible elements in the quaternion algebra over certain finite fields.
Let $p_{1}, p_{2}$ be two distinct prime integers, let $n$ be a positive integer, $n$$\geq 3$ and let $ξ_{n} $ be a primitive root of order $n$ of the unity. In this paper we obtain a complete characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{3,4,6,7,8,9,11,12\right\}$ and also we obtain a characterization for a quaternion algebra $H\left(p_{1}, p_{2}\right)$ to be a division algebra over the $n$th cyclotomic field $\mathbb{Q}\left(ξ_{n}\right)$, when $n$$\in$$\left\{5,10\right\}$. In the 4th section we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{n}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $n=l^{k},$ with $l$ a prime integer, $l\equiv 3$ (mod $4$) and $k$ a positive integer. In the last section of this article we obtain a complete characterization for a quaternion algebra $H_{\mathbb{Q}\left(ξ_{l}\right)}\left(p_{1}, p_{2}\right)$ to be a division algebra, when $l$ is a Fermat prime number.
The aim of this paper is to study certain properties of the Kullback-Leibler distance between two positive integer numbers or between two ideals. We present some results related the entropy of a positive integer number and the divergence of two numbers. We also study the entropy of some types of ideals and the divergence of two ideals. Finally, we find some inequalities, involving the entropy H of an exponential divisor of a positive integer, respectively the entropy H of an exponential divisor of an ideal.
In this article we find some properties of certain types of entropies of a natural number. Also, regarding the entropy H of a natural number, introduced by Minculete and Pozna, we generalize this notion for ideals and we find some of its properties. In the last section we find some inequalities, involving the entropy H of an exponential divisor of a positive integer, respectively the entropy H of an exponential divisor of an ideal.
Let $p$ be an odd prime, let $K=\mathbb{Q}(ε)$ where $ε$ is a primitive cubic root of unity, and let $L$ be the Kummer field $\mathbb{Q}\left(ε, \sqrt[3]α\right)$. In this paper we obtain a characterization of the splitting behavior of the symbol algebras $\left( \frac{α,p}{K,ε}\right)$ and $\left( \frac{α,p^{h_{p}}}{K,ε}\right)$, where $h_{p}$ is the order in the class group $Cl\left(L\right)$ of a prime ideal of $\mathcal{O}_L$ which divides $p\mathcal{O}_L.$
Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an abelian $\ell$-extension unramified over $F$ assuming $\ell$ divides the class number of $F$. In this paper, we obtain a complete characterization of division quaternion algebras $H_{K}(p, q)$ over $K$.
In this paper we provide some applications of the norm form in some quaternion division algebras over rational field and we give some properties of Fibonacci sequence and Fibonacci sequence in connection with quaternion elements. We define a monoid structure over a fnite set on which we will prove that the defined Fibonacci sequence is stationary, we provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four-square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quaternions defined on some division quaternion algebras and we define and give properties of some special quaternions by using Fibonacci sequences.
In this paper, we provide properties and applications of some special integer sequences. We generalize and give some properties of Pisano period. Moreover, we provide a new application in Cryptography and applications of some quaternion elements.
In this paper, we provide new applications of Fibonacci and Lucas numbers. In some circumstances, we find algebraic structures on some sets defined with these numbers, we generalize Fibonacci and Lucas numbers by using an arbitrary binary relation over the real fields instead of addition of the real numbers and we give properties of the new obtained sequences. Moreover, by using some relations between Fibonacci and Lucas numbers, we provide a method to find new examples of split quaternion algebras and we give new properties of these elements.
Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $ H_K(α, m)$ a quaternion algebra over $K$, where $α\in \mathcal{O}_K$. In this paper we give necessary and sufficient conditions for $ H_K(α, m)$ to split over $K$ for some values of $α$, and we obtain a complete characterization of division quaternion algebras $ H_K(α, m)$ over $K$ whenever $α$ and $m$ are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers.
In this paper, we generalize the arithmetic functions $τ$ and $τ^{\left(e\right)}$ in algebraic number fields and we find some properties of these functions.
Horadam symbol elemnts are introduced. Certain properties of these elements are explored. Some well known identities such as Catalan identity, Cassini formula and d'Ocagne's identity are obtained for these elements.
In this paper, we define a special class of elements in the algebras obtained by the Cayley Dickson process, called l elements. We find conditions such that these elements to be invertible. These conditions can be very useful for finding new identities, identities which can help us in the study of the properties of these algebras.
Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of quaternion division algebras H_K(p,q) over some dihedral fields K.