arXiv subjects

Nicusor Minculete

Publications and source records attributed to Nicusor Minculete.

8 recordsLinked to original sources

arXiv2024

Some results about entropy and divergence in number theory

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.

math.NT↗

arXiv2023

Some properties of a type of the entropy of an ideal and the divergence of two ideals

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.

math.NT↗

arXiv2022

A type of the entropy 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.

math.NT↗

arXiv2020

Inequalities from Lorentz-Finsler norms

We show that Lorentz-Finsler geometry offers a powerful tool in obtaining inequalities. With this aim, we first point out that a series of famous inequalities such as: the (weighted) arithmetic-geometric mean inequality, Aczél's, Popoviciu's and Bellman's inequalities, are all particular cases of a reverse Cauchy-Schwarz, respectively, of a reverse triangle inequality holding in Lorentz-Finsler geometry. Then, we use the same method to prove some completely new inequalities, including two refinements of Aczél's inequality.

math.DG↗

arXiv2020

The weak n-inner product space

In this article we study a generalization of the n-inner product which we name weak n-inner product. As particular case we consider the n-iterated 2-inner product and we give its representation in terms of the standard k-inner products, k<= n, using the Dodgson's identity for determinants. Finally, we present several applications, including a brief characterization of a linear regression model for the random variables in discrete case and a generalization of the Chebyshev functional using the n-iterated 2-inner product.

math.CA↗

arXiv2020

Bounds for the $p$-angular distance and characterizations of inner product spaces

Based on a suitable improvement of a triangle inequality, we derive new mutual bounds for $p$-angular distance $α_p[x,y]=\big\Vert \Vert x\Vert^{p-1}x- \Vert y\Vert^{p-1}y\big\Vert$, in a normed linear space $X$. We show that our estimates are more accurate than the previously known upper bounds established by Dragomir, Hile and Maligranda. Next, we give several characterizations of inner product spaces with regard to the $p$-angular distance. In particular, we prove that if $|p|\geq |q|$, $p\neq q$, then $X$ is an inner product space if and only if for every $x,y\in X\setminus \{0\}$, $${α_p[x,y]}\geq \frac{{\|x\|^{p}+\|y\|^{p} }}{\|x\|^{q}+\|y\|^{q} }α_q[x,y].$$

math.FA↗

arXiv2019

Some generalizations of the functions $τ$ and $τ^{\left(e\right)}$ in algebraic number fields

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.

math.NT↗

arXiv2012

A Geometric Way to Generate Blundon Type Inequalities

We present a geometric way to generate Blundon type inequalities. Theorem 3.1 gives the formula for cosPOQ in terms of the barycentric coordinates of the points P and Q with respect to a given triangle. This formula implies Blundon type inequalities generated by the points P and Q (Theorem 3.2). Some applications are given in the last section by choosing special points P and Q.

math.GM↗