Recent zbMATH articles in MSC 39B https://www.zbmath.org/atom/cc/39B 2021-04-16T16:22:00+00:00 Werkzeug On quasi-convexity of the zero utility principle. https://www.zbmath.org/1456.39007 2021-04-16T16:22:00+00:00 "Chudziak, Jacek" https://www.zbmath.org/authors/?q=ai:chudziak.jacek The author recalls the zero utility principle in terms of a functional equation on the set of all non-negative bounded random variables provided with a preference relation. The bounded random variables represent the risks to be insured by an insurance company. The author investigates the quasi-convexity of the zero utility principle. The quasi-convexity ensures the economic requirement that diversification should not increase the risk. The main result of the paper consists in the proof that the quasi-convexity of the zero utility principle is equivalent to its convexity. Reviewer: Karel Zimmermann (Praha) Vector valued polynomials, exponential polynomials and vector valued harmonic analysis. https://www.zbmath.org/1456.43003 2021-04-16T16:22:00+00:00 "Laczkovich, M." https://www.zbmath.org/authors/?q=ai:laczkovich.miklos Let $$G$$ be a topological abelian semigroup with unit, $$E$$ be a Banach space, and let $$C(G,E)$$ stand for the set of all continuous functions from $$G$$ into $$E$$. A function $$f\in C(G,E)$$ is called a generalized polynomial if there is an $$n\ge 0$$ such that $$\Delta_{h_1}\dots\Delta_{h_{n+1}} f=0$$ for every $$h_1, \dots,h_{n+1}\in G$$ in which $$\Delta_h$$ denotes the difference operator. A function $$f\in C(G,E)$$ is said to be a polynomial if it is a generalized polynomial and the linear span of its translates is of finite dimension; $$f$$ is a $$w$$-polynomial if $$u\circ f$$ is a polynomial for every $$u$$ in the dual space of $$E$$, and $$f$$ is a local polynomial if it is a polynomial on every finitely generated subsemigroup. The author proves that each of the classes of polynomials, $$w$$-polynomials, generalized polynomials, local polynomials is contained in the next class. If $$G$$ is an abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. He also introduces the classes of exponential polynomials and $$w$$-exponential polynomials and investigates their representations and connection with polynomials and $$w$$-polynomials. He establishes spectral synthesis and analysis in the class $$C(G,E)$$. He shows that, if $$G$$ is an infinite and discrete abelian group and $$E$$ is a Banach space of infinite dimension, then spectral analysis fails in $$C(G,E)$$. If $$G$$ is discrete with finite torsion free rank and $$E$$ is a Banach space of finite dimension, then he proves that spectral synthesis holds in $$C(G,E)$$. Reviewer: Mohammad Sal Moslehian (Mashhad) Stability of pexiderized quadratic functional equation in non-Archimedean fuzzy normed spaces. https://www.zbmath.org/1456.39005 2021-04-16T16:22:00+00:00 "Eghbali, Nasrin" https://www.zbmath.org/authors/?q=ai:eghbali.nasrin Summary: We determine some stability results concerning the pexiderized quadratic functional equation in non-Archimedean fuzzy normed spaces. Our result can be regarded as a generalization of the stability phenomenon in the framework of $$\mathcal{L}$$-fuzzy normed spaces. A functional equation of tail-balance for continuous signals in the Condorcet jury theorem. https://www.zbmath.org/1456.91044 2021-04-16T16:22:00+00:00 "Alpern, Steve" https://www.zbmath.org/authors/?q=ai:alpern.steve "Chen, Bo" https://www.zbmath.org/authors/?q=ai:chen.bo.1|chen.bo.2|chen.bo.4|chen.bo.3 "Ostaszewski, Adam J." https://www.zbmath.org/authors/?q=ai:ostaszewski.adam-j Summary: Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability $$p$$, then the probability of a correct verdict tends to one as the jury size tends to infinity [\textit{Marquis de Condorcet}, Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix'', Paris: Imprim. Royale (1785)]. Recently, the first two authors [Eur. J. Oper. Res. 258, No. 3, 1072--1081 (2017; Zbl 1394.91113); Theory Decis. 83, No. 2, 259--282 (2017; Zbl 1395.91151)] developed a model where jurors sequentially receive independent signals from an interval according to a distribution which depends on the state of Nature and on the juror's ability'', and vote sequentially. This paper shows that, to mimic Condorcet's binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio $$\alpha(t)$$ of the probability that a mean-zero random variable satisfies $$X > t$$ given that $$|X| > t$$. In particular, we show that under natural symmetry assumptions the tail-balances $$\alpha(t)$$ uniquely determine the signal distribution and so the distributions assumed in [Zbl 1394.91113; Zbl 1395.91151] are uniquely determined for $$\alpha(t)$$ linear. Factorization of $C$-finite sequences. https://www.zbmath.org/1456.68231 2021-04-16T16:22:00+00:00 "Kauers, Manuel" https://www.zbmath.org/authors/?q=ai:kauers.manuel "Zeilberger, Doron" https://www.zbmath.org/authors/?q=ai:zeilberger.doron Summary: We discuss how to decide whether a given $C$-finite sequence can be written nontrivially as a product of two other $C$-finite sequences. For the entire collection see [Zbl 1391.33001]. On a generalization of the class of Jensen convex functions. https://www.zbmath.org/1456.26012 2021-04-16T16:22:00+00:00 "Lara, Teodoro" https://www.zbmath.org/authors/?q=ai:lara.teodoro "Quintero, Roy" https://www.zbmath.org/authors/?q=ai:quintero.roy "Rosales, Edgar" https://www.zbmath.org/authors/?q=ai:rosales.edgar "Sánchez, José Luis" https://www.zbmath.org/authors/?q=ai:sanchez.jose-luis Summary: The main objective of this article is to introduce a new class of real valued functions that include the well-known class of $${m}$$-convex functions introduced by Toader (1984). The members of this collection are called Jensen $${m}$$-convex and are defined, for $${m \in (0,1]}$$, via the functional inequality $f\left(\frac{x+y}{c_m} \right)\leq \frac{f(x)+f(y)}{c_m} \quad (x,y \in [0,b]),$ where $${c_{m} := \frac{m+1}{m}}$$. These functions generate a new kind of functional convexity that is studied in terms of its behavior with respect to basic algebraic operations such as sums, products, compositions, etc. in this paper. In particular, it is proved that any starshaped Jensen convex function is Jensen $${m}$$-convex. At the same time an interesting example (Example 3) shows how the classes of Jensen $${m}$$-convex functions depend on $${m}$$. All the techniques employed come from traditional basic calculus and most of the calculations have been done with Mathematica 8.0.0 and validated with Maple 15 as well as all the figures included. Some hyperstability results for a Cauchy-Jensen type functional equation in 2-Banach spaces. https://www.zbmath.org/1456.39008 2021-04-16T16:22:00+00:00 "Sayar, Khaled Yahya Naif" https://www.zbmath.org/authors/?q=ai:sayar.khaled-yahya-naif "Bergam, Amal" https://www.zbmath.org/authors/?q=ai:bergam.amal Let $$E$$ be a normed space and $$(Y, \|\cdot, \cdot\|)$$ be a real $$2$$- Banach space. The authors prove stability and hyperstability results for the Cauchy-Jensen equation $f\left(\frac{x+y}{2}+z\right)+f\left(\frac{x-y}{2}+z\right)=f(x)+2f(z)$ for all $$x, y, z \in E-\{0\}$$ such that $$\frac{x+y}{2}+z\neq 0$$ and $$\frac{x+y}{2}+\neq 0$$. Reviewer: Maryam Amyari (Mashhad) Local smooth conjugations of Frobenius endomorphisms. https://www.zbmath.org/1456.39004 2021-04-16T16:22:00+00:00 "Kalnitsky, V. S." https://www.zbmath.org/authors/?q=ai:kalnitskij.v-s "Petrov, A. N." https://www.zbmath.org/authors/?q=ai:petrov.andrei-n|petrov.a-n|petrov.alexander-n Summary: A generalization of the Böttcher equation is considered. It turned out that the parametrized Poisson integral, as a function of its parameters, satisfies an equation of the type described. The structure theorem for splitting maps of Frobenius endomorphisms in a ring and in an algebra over it is proved. The real field case is considered. The generalized Böttcher equation is solved for classical two-dimensional algebras and for the Poisson algebra. Some applications of Laplace transforms in analytic number theory. https://www.zbmath.org/1456.11154 2021-04-16T16:22:00+00:00 "Ivić, Aleksandar" https://www.zbmath.org/authors/?q=ai:ivic.aleksandar Summary: In this overview paper, presented at the meeting DANS14, Novi Sad, July 3--7, 2014, we give some applications of Laplace transforms to analytic number theory. These include the classical circle and divisor problem, moments of $$\vert\zeta(\tfrac12 + it)\vert$$, and a discussion of two functional equations connected to a work of Prof. \textit{B. Stankovic} [in: Math. Struct., comput. Math., math. Modelling (1974; Zbl 0269.45010)]. A fixed point approach to stability of $$k$$-th radical functional equation in non-Archimedean $$(n,\beta)$$-Banach spaces. https://www.zbmath.org/1456.39003 2021-04-16T16:22:00+00:00 "EL-Fassi, Iz-iddine" https://www.zbmath.org/authors/?q=ai:el-fassi.iz-iddine "Elqorachi, Elhoucien" https://www.zbmath.org/authors/?q=ai:elqorachi.elhoucien "Khodaei, Hamid" https://www.zbmath.org/authors/?q=ai:khodaei.hamid Summary: In this work, we prove a simple fixed point theorem in non-Archimedean $$(n,\beta)$$-Banach spaces, by applying this fixed point theorem, we will study the stability and the hyperstability of the $$k$$th radical-type functional equation: $f\left( \sqrt[k]{x^k+y^k}\right) = f(x)+f(y),$ where $$f$$ is a mapping on the set of real numbers and $$k$$ is a fixed positive integer. Furthermore, we give some important consequences from our main results. Solution and full classification of generalized binary functional equations of the type $$(3;3;0)$$. https://www.zbmath.org/1456.39006 2021-04-16T16:22:00+00:00 "Krainichuk, Halyna" https://www.zbmath.org/authors/?q=ai:krainichuk.halyna-v "Sokhatsky, Fedir" https://www.zbmath.org/authors/?q=ai:sokhatsky.fedir-m Summary: Generalized binary functional quasigroup equations in two individual variables with three appearances are under consideration. There exist five classes of the equations (two equations belong to the same class if there exists a relation between sets of their solutions). The quasigroup solution sets of equations from every class are given. In addition, it is proved that every parastrophe of a quasigroup has an orthogonal mate if the quasigroup has an orthogonal mate.