On the Eventually Periodic Continued β-Fractions and Their Lévy Constants

In this paper, we consider continued β-fractions with golden ratio base β. We show that if the continued β-fraction expansion of a non-negative real number is eventually periodic, then it is the root of a quadratic irreducible polynomial with the coefficients in Z[β] and we conjecture the converse is false, which is different from Lagrange’s theorem for the regular continued fractions. We prove that the set of Lévy constants of the points with eventually periodic continued β-fraction expansion is dense in [c, +∞), where c=12logβ+2−5β+12.

On Rényi Permutation Entropy

Among various modifications of the permutation entropy defined as the Shannon entropy of the ordinal pattern distribution underlying a system, a variant based on Rényi entropies was considered in a few papers. This paper discusses the relatively new concept of Rényi permutation entropies in dependence of non-negative real number q parameterizing the family of Rényi entropies and providing the Shannon entropy for q=1. Its relationship to Kolmogorov–Sinai entropy and, for q=2, to the recently introduced symbolic correlation integral are touched.

On the Minimum Variable Connectivity Index of Unicyclic Graphs with a Given Order

The variable connectivity index, introduced by the chemist Milan Randić in the first quarter of 1990s, for a graph G is defined as ∑vw∈EGdv+γdw+γ−1/2, where γ is a non-negative real number and dw is the degree of a vertex w in G. We call this index as the variable Randić index and denote it by Rvγ. In this paper, we show that the graph created from the star graph of order n by adding an edge has the minimum Rvγ value among all unicyclic graphs of a fixed order n, for every n≥4 and γ≥0.

On comparison of the principles of equivalent utility and its applications

An insurance premium principle is a way of assigning to every risk, represented by a non-negative bounded random variable on a given probability space, a non-negative real number. Such a number is interpreted as a premium for the insuring risk. In this paper the implicitly defined principle of equivalent utility is investigated. Using the properties of the quasideviation means, we characterize a comparison in the class of principles of equivalent utility under Rank-Dependent Utility, one of the important behavioral models of decision making under risk. Then we apply this result to establish characterizations of equality and positive homogeneity of the principle. Some further applications are discussed as well.

LINEAR CENTERS WITH PERTURBATIONS OF DEGREE 2d + 5

We describe a method for studying the center and isochronicity problems for a class of differential systems in the form of linear center perturbed by homogeneous series of degree 2d + m, where d is a non-negative real number and m is a positive integer. As an application, we classify centers and isochronous centers for a particular case when m = 5.

Numerical criteria for integral dependence

We study multiplicity based criteria for integral dependence of modules or of standard graded algebras, known as ‘Rees criteria’. Rather than using the known numerical invariants, we achieve this goal with a more direct approach by introducing a multiplicity defined as a limit superior of a sequence of normalized lengths; this multiplicity is a non-negative real number that can be irrational.

ROTARU ALPHA - CONVEX FUNCTIONS

Let $S^*(a,b)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0) =f'(0) - 1 =0$, satisfying the condition $|(zf'(z)/f(z))- a|<b$, $a\in C$, $|a- 1|<b\le Re(a)$, $z\in E$. In this paper the class $S^*(\alpha, a, b)$ of functions $f$ analytic in $E$, with $f(0) = f'(0)- 1 =0$, $f(z)f'(z)/z\neq 0$ for $z$ in $E$ and satisfying in $E$ the condition $|J(\alpha,f)- a|<b$, $a \in C$, $|a-1|<b\le Re(a)$, where $J(\alpha, f) =(1- \alpha)(zf'(z)/f(z)) +\alpha((zf'(z))'/f'(z))$, $\alpha$ a non-negative real number is introduced. It is proved that $S^*(\alpha, a,b)\subset S^*(a,b)$, if $a> (4b/c)|Im(a)|$, $c=(b^2- |a- 1|^2)/b$. Further a representation formula for $f \in S^*(\alpha, a, b)$ and an inequality relating the coefficients of functions in $S^*(\alpha, a, b)$ are obtained.

Inclusion Relations for General Riesz Typical Means

Let α be a non-negative real number, λ≡{λ,n}(n≥0) a strictly increasing unbounded sequence with λ0≥0 and let be an arbitrary series with partial sums s≡{sn}. Writewhere s(t)=sn for λn<t≤λn+1, s(t)=0 for 0≤t≤λ0. The series ∑ an or the sequence of partial sums s={sn} is summable to ṡ by the Riesz method (R, λ, α) ifas ω→∞.

On non-negative spectrum in Banach algebras

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

On Orientations, Connectivity and Odd-Vertex-Pairings in Finite Graphs

The integer part of a non-negative real number p will be denoted by [p]. For any integer n, n* will denote the greatest even integer less than or equal to n, that is, n* = n or n — 1 according as n is even or odd respectively.The order of a set A, denoted by |A|, is the number of elements in A. The set whose elements are a1, a2, … , an will be denoted by {a1, a2 … , an. The empty set will be denoted by Λ. A set will be said to include each of its elements. A set separates two elements if it includes one but not both of them.An unoriented graph U consists of two disjoint sets V(U), E(U), the elements of V(U) being called vertices of U and the elements of V(U) being called edges of U, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices which the edge is said to join.