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Very recently, I. Altun, M. Aslantas and H. Sahin [1] introduced the notion of p -proximal contractions and surveyed the existence of best proximity points for such class of non-self mappings in metric spaces. In this note we show that this existence result is a straightforward consequence of the same conclusion in fixed point theory. [ABSTRACT FROM AUTHOR]

A decomposition of a non-empty simple graph G is a pair [G,P] such that P is a set of non-empty induced subgraphs of G, and every edge of G belongs to exactly one subgraph in P. The chromatic index χ ′ ([ G , P ]) of a decomposition [G,P] is the smallest number k for which there exists a k-coloring of the elements of P in such a way that for every element of P all of its edges have the same color, and if two members of P share at least one vertex, then they have different colors. A long standing conjecture of Erdős–Faber–Lovász states that every decomposition [ K n , P ] of the complete graph K n satisfies χ ′ ([ K n , P ]) ≤ n . In this paper we work with geometric graphs, and inspired by this formulation of the conjecture, we introduce the concept of chromatic index of a decomposition of the complete geometric graph. We present bounds for the chromatic index of several types of decompositions when the vertices of the graph are in general position. We also consider the particular case when the vertices are in convex position and present bounds for the chromatic index of a few types of decompositions. [ABSTRACT FROM AUTHOR]

Consider a multidimensional risk model, in which an insurer is exposed to more than one type of claims sharing a common accident-number process, and each type of the individual claims and the accident inter-arrival times are, respectively, extended negatively dependent and identically distributed nonnegative random variables. In various insurance contexts, it is hard to assume some specific dependence between the claims and the inter-arrival times due to their complex interplay, and hence arbitrary dependence is allowed in this paper. Under the assumption of the individual claims possessing dominatedly varying tails and some moment condition on the inter-arrival times, this paper establishes a precise large deviation result for the aggregate claim vector. [ABSTRACT FROM AUTHOR]

The paper establishes, with the help of loosenesses [3], results contained in [17]. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. We say that H is pronormal in G if for every g ∈ G , the subgroups H and H g are conjugate in ⟨ H , H g ⟩ ; H is called weakly pronormal in G if there exists a subgroup K of G such that G = HK and H ∩ K is pronormal in G. In this paper, we investigate the structure of G under the assumption that certain subgroups of G are weakly pronormal in G. Our results improve and generalize new recent results in the literature. [ABSTRACT FROM AUTHOR]

Let (t (m)) m ≥ 0 be Thue–Morse sequence and β > 2 be a Pisot or Salem number. In this paper, we prove that the real number ∑ m = 0 ∞ t (m 2) β m + 1 does not belong to Q (β). [ABSTRACT FROM AUTHOR]

In this paper we point out an interesting geometric structure of nonnegative metric curvature emerging from the hyperspaces of decomposable, non-locally connected homogeneous continua, where "smooth" and "non-smooth" partitions live together, similarly to the macroscopic and the quantum realms of the Universe. [ABSTRACT FROM AUTHOR]

This paper is devoted to the problem of where the critical points of a polynomial are relative to their zeros. Classical and new developments are surveyed along with illustrative examples. The paper finishes with a short proof of the sector theorem of Sendov and Sendov. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a new atomic decomposition for a dense class of weighted Hardy spaces H w p (R n) via the discrete Calderón-type reproducing formula and the weighted Littlewood–Paley–Stein theory, where w ∈ A ∞ is a Muckenhoupt's weight and 0 < p < ∞ . Our results can recover and improve the known ones in the literature by avoiding using the maximal function characterization and the Calderón–Zygmund decomposition. Moreover, we give a new proof of the weighted Hardy spaces estimates for generalized Calderón–Zygmund operators in terms of the atomic decomposition and the vector-valued singular integral operator theory. Although the theory of H w p (R n) is well known, we give new and simpler proofs, which in turn is amenable to utilization in general and nonclassical settings. [ABSTRACT FROM AUTHOR]

The Erdős–Lax Theorem states that, if P(z) is a polynomial of degree n having no zeros in | z | < 1 , then max | z | = 1 | P ′ (z) | ≤ n 2 max | z | = 1 | P (z) |. The problem of generalizing the Erdős–Lax Theorem to the class of polynomials having no zeros in | z | < K , K ≤ 1 is still not settled. Motivated by the above open problem, in this paper we prove some inequalities for a class of polynomials having no zeros in | z | < K , K ≤ 1 , and all these zeros lie either on a ray L emanating from the origin or zeros are symmetrically placed along this line L. Several important consequences are also discussed. Our results are the best possible and examples for the equality cases have been presented. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain some characterizations (or representations) of set-valued solutions defined on an Abelian monoid with values in a Hausdorff topological vector space of a new class of functional equations. [ABSTRACT FROM AUTHOR]

Given an integer μ , we study the numbers n that satisfy the condition μ n + ∑ p ∣ n 1 p ∈ N . This condition, which is reminiscent of the one satisfied by Giuga numbers ( μ = - 1 ), also includes the so-called weak primary pseudoperfect numbers ( μ = 1 ), see [9]. As a tribute to our late colleague Jonathan Sondow (1943–2020), we have named these numbers μ -Sondow numbers. In this paper, we give several different characterizations of these numbers, all of them suggested by well-known characterizations of the Giuga numbers. We also relate these numbers to the well-known Erdős–Moser equation and we present some conjectures about them. [ABSTRACT FROM AUTHOR]

A D(n)-m-tuple, where n is a non-zero integer, is a set of m distinct elements in a commutative ring R such that the product of any two distinct elements plus n is a perfect square in R. In this paper, we prove that there does not exist a D(−1)-quadruple { a , b , c , d } in the ring Z [ - k ] , k ≥ 2 with positive integers a < b < 16 a 2 - a - 2 + 2 k (8 a 2 + 3 a + 1) and integers c and d satisfying d < 0 < c . By combining that result with [14, Theorem 1.1] we were able to obtain a general result on the non-existence of a D(−1)-quadruple { a , b , c , d } in Z [ - k ] with integers a, b, c, d satisfying a < b ≤ 8 a - 3 . Furthermore, for a non-negative integer i and a positive integer j, we apply the obtained results in proving of the non-existence of D(−1)-quadruples containing powers of primes p i , q j with an arbitrary different primes p and q. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the following invariance equation involving two 2-variable generalized Bajraktarević means, i.e., we aim to solve the functional equation f - 1 ( p 1 (x) f (x) + p 2 (y) f (y) p 1 (x) + p 2 (y) ) + g - 1 ( q 1 (x) g (x) + q 2 (y) g (y) q 1 (x) + q 2 (y) ) = x + y (x , y ∈ I) , where I is a nonempty open real interval and f , g : I → R are continuous, strictly monotone and p 1 , p 2 , q 1 , q 2 : I → R + are unknown functions. The main result of the paper shows that, assuming four times continuous differentiability of f, g, twice continuous differentiability of p1 and p2 and assuming that p1 differs from p2 on a dense subset of I, a necessary and sufficient condition for the equality above is that the unknown functions are of the form f = u v , g = w z , and p 1 q 1 = p 2 q 2 = v z , where u , v , w , z : I → R are arbitrary solutions of the second-order linear differential equation F ′ ′ = γ F (γ ∈ R is arbitrarily fixed) such that v > 0 and z > 0 holds on I and { u , v } and { w , z } are linearly independent. [ABSTRACT FROM AUTHOR]