Authors:Jean-Luc Marichal; Pierre Mathonet Pages: 601 - 618 Abstract: The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0477-8 Issue No:Vol. 91, No. 4 (2017)
Authors:Gergely Kiss; Csaba Vincze Pages: 663 - 690 Abstract: The aim of the paper is to investigate the solutions of special inhomogeneous linear functional equations using spectral analysis in a translation invariant closed linear subspace of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The application of spectral analysis in some related varieties is a new and important trend in the theory of functional equations; especially they have successful applications in the case of homogeneous linear functional equations. The foundations of the theory can be found in Kiss and Varga (Aequat Math 88(1):151–162, 2014) and Kiss and Laczkovich (Aequat Math 89(2):301–328, 2015). We are going to adopt the main theoretical tools to solve some inhomogeneous problems due to Koclȩga-Kulpa and Szostok (Ann Math Sylesianae 22:27–40, 2008), see also Koclȩga-Kulpa and Szostok (Georgian Math J 16:725–736, 2009; Acta Math Hung 130(4):340–348, 2011). They are motivated by quadrature rules of approximate integration. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0490-y Issue No:Vol. 91, No. 4 (2017)
Authors:Gergely Kiss; Csaba Vincze Pages: 691 - 723 Abstract: As a continuation of our previous work [2] the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces of additive/multiadditive functions containing the restrictions of the solutions to finitely generated fields. The idea is based on the fundamental work of [5]. Using spectral analysis in some related varieties we can prove the existence of special solutions (automorphisms) of the functional equation but spectral synthesis allows us to describe the entire space of solutions on a large class of finitely generated fields. It is spanned by the so-called exponential monomials which can be given in terms of automorphisms of \({\mathbb C}\) and differential operators. We apply the general theory to some inhomogeneous problems motivated by quadrature rules of approximate integration [8], see also [7, 9]. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0482-y Issue No:Vol. 91, No. 4 (2017)
Authors:H. Ghahramani; M. N. Ghosseiri; S. Safari Pages: 725 - 738 Abstract: In this paper we pose some questions about superderivations on \({\mathbb {Z}}_{2}\) -graded rings. Then we consider the quaternion rings and upper triangular matrix rings with special \({\mathbb {Z}}_{2}\) -gradings and we check the answer to these questions about them. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0480-0 Issue No:Vol. 91, No. 4 (2017)
Authors:Maysam Maysami Sadr Pages: 739 - 743 Abstract: Let E, F be Banach spaces. In the case that F is reflexive we give a description for the solutions (f, g) of the Banach-orthogonality equation $$\begin{aligned} \langle f(x),g(\alpha )\rangle =\langle x,\alpha \rangle \qquad \forall x\in E,\forall \alpha \in E^*, \end{aligned}$$ where \(f:E\rightarrow F,g:E^*\rightarrow F^*\) are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that E and F are Hilbert spaces. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0466-y Issue No:Vol. 91, No. 4 (2017)
Authors:Piotr Maćkowiak Pages: 759 - 777 Abstract: In the paper we present results on the continuity of nonlinear superposition operators acting in the space of functions of bounded variation in the sense of Jordan. It is shown that the continuity of an autonomous superposition operator is automatically guaranteed if the acting condition is met. We also give a simple proof of the fact that a nonautonomous superposition operator generated by a continuously differentiable function is uniformly continuous on bounded sets. Moreover, we present necessary and sufficient conditions for the continuity of a superposition operator (autonomous or nonautonomous) in a general setting. Thus, we give the answers to two basic open problems mentioned in the monograph (Appell et al. in Bounded variation and around, series in nonlinear analysis and application, De Gruyter, Berlin, 2014). PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0491-x Issue No:Vol. 91, No. 4 (2017)
Authors:Michael Schwarzenberger Pages: 779 - 783 Abstract: In this note we determine the unique solution to the functional equation \(f(x + y) ( x- y) = \left( f(x) -f (y) \right) (x + y)\) . We require no additional assumptions on the function \(f\) . Moreover we solve this functional equation if \(f\) is only defined on a finite interval. The interest in this type of functional equation is motivated by the study of symmetrizing measures for (the generator of) a Lévy-driven Ornstein–Uhlenbeck process. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0484-9 Issue No:Vol. 91, No. 4 (2017)
Authors:Marek Cezary Zdun Pages: 785 - 800 Abstract: Let I be an interval. We consider the non-monotonic convex self-mappings \(f:I\rightarrow I\) such that \(f^2\) is convex. They have the property that all iterates \(f^n\) are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every \(n\ge 2\) there exists a convex iterative root \(f^{1/2^n}\) of order \(2^n\) and the sequence \(\{f^{1/2^n}\}\) satisfies the condition of compatibility, that is \( f^{1/2^n}\circ f^{1/2^n}= f^{1/2^{n-1}}.\) A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions \(\{f^t,t>0\}\) such that \(f^t\circ f^s=f^{t+s}, \ \ t,s >0\) and \(f^1=f\) . We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions. PubDate: 2017-08-01 DOI: 10.1007/s00010-017-0483-x Issue No:Vol. 91, No. 4 (2017)
Authors:Ákos G. Horváth Pages: 401 - 418 Abstract: In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\) -dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\) . PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0465-4 Issue No:Vol. 91, No. 3 (2017)
Authors:Andrzej Olbryś Pages: 429 - 444 Abstract: In the present paper we introduce a notion of the \(\mathbb {K}\) -Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the \(\mathbb {K}\) -Riemann integral and the convexity notion is replaced by \(\mathbb {K}\) -convexity. PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0472-0 Issue No:Vol. 91, No. 3 (2017)
Authors:Mohamed Jleli; Donal O’Regan; Bessem Samet Pages: 479 - 490 Abstract: In this paper, some fractional integral inequalities involving m-convex functions are established. The presented results are generalizations of the obtained inequalities in Dragomir and Toader (Babeş-Bolyai Math 38:21–28, 1993). PubDate: 2017-06-01 DOI: 10.1007/s00010-017-0470-2 Issue No:Vol. 91, No. 3 (2017)
Authors:Daniel Eremita Pages: 563 - 578 Abstract: Let \(T_n(R)\) be the upper triangular matrix ring over a unital ring R. Suppose that \(B:T_n(R)\times T_n(R) \rightarrow T_n(R)\) is a biadditive map such that \(B(X,X)X = XB(X,X)\) for all \(X \in T_n(R)\) . We consider the problem of describing the form of the map \(X\mapsto B(X,X)\) . PubDate: 2017-06-01 DOI: 10.1007/s00010-016-0462-7 Issue No:Vol. 91, No. 3 (2017)
Authors:Dirk Keppens Abstract: In Keppens (Innov. Incidence Geom. 15: 119–139, 2017) we gave a state of the art concerning “projective planes” over finite rings. The current paper gives a complementary overview for “affine planes” over rings (including the important subclass of desarguesian affine Klingenberg and Hjelmslev planes). No essentially new material is presented here but we give a summary of known results with special attention to the finite case, filling a gap in the literature. PubDate: 2017-07-06 DOI: 10.1007/s00010-017-0497-4
Authors:Hamid Reza Moradi; Mohsen Erfanian Omidvar; Muhammad Adil Khan; Kazimierz Nikodem Abstract: In this paper we use basic properties of strongly convex functions to obtain new inequalities including Jensen type and Jensen–Mercer type inequalities. Applications for special means are pointed out as well. We also give a Jensen’s operator inequality for strongly convex functions. As a corollary, we improve the Hölder-McCarthy inequality under suitable conditions. More precisely we show that if \(Sp\left( A \right) \subset \left( 1,\infty \right) \) , then $$\begin{aligned} {{\left\langle Ax,x \right\rangle }^{r}}\le \left\langle {{A}^{r}}x,x \right\rangle -\frac{{{r}^{2}}-r}{2}\left( \left\langle {{A}^{2}}x,x \right\rangle -{{\left\langle Ax,x \right\rangle }^{2}} \right) ,\quad r\ge 2 \end{aligned}$$ and if \(Sp\left( A \right) \subset \left( 0,1 \right) \) , then $$\begin{aligned} \left\langle {{A}^{r}}x,x \right\rangle \le {{\left\langle Ax,x \right\rangle }^{r}}+\frac{r-{{r}^{2}}}{2}\left( {{\left\langle Ax,x \right\rangle }^{2}}-\left\langle {{A}^{2}}x,x \right\rangle \right) ,\quad 0<r<1 \end{aligned}$$ for each positive operator A and \(x\in \mathcal {H}\) with \(\left\ x \right\ =1\) . PubDate: 2017-07-04 DOI: 10.1007/s00010-017-0496-5
Authors:N. H. Bingham; A. J. Ostaszewski Abstract: Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice, DC rather than the Axiom of Choice, AC. Berz used the Hahn–Banach theorem over \({\mathbb {Q}}\) to prove that the graph of a measurable sublinear function that is \({\mathbb {Q}}_{+}\) -homogeneous consists of two half-lines through the origin. We give a category form of the Berz theorem. Our proof is simpler than that of the classical measure-theoretic Berz theorem, our result contains Berz’s theorem rather than simply being an analogue of it, and we use only DC rather than AC. Furthermore, the category form easily generalizes: the graph of a Baire sublinear function defined on a Banach space is a cone. The results are seen to be of automatic-continuity type. We use Christensen Haar null sets to extend the category approach beyond the locally compact setting where Haar measure exists. We extend Berz’s result from Euclidean to Banach spaces, and beyond. Passing from sublinearity to convexity, we extend the Bernstein–Doetsch theorem and related continuity results, allowing our conditions to be ‘local’—holding off some exceptional set. PubDate: 2017-07-01 DOI: 10.1007/s00010-017-0486-7
Authors:Sokol Bush Kaliaj Abstract: In this paper, we utilize some fixed point theorems of contractive type to present a few existence and uniqueness theorems for a functional equation arising in dynamic programming of continuous multistage decision processes. PubDate: 2017-06-06 DOI: 10.1007/s00010-017-0495-6
Authors:Paweł Siedlecki Abstract: A generalized solution operator is a mapping abstractly describing a computational problem and its approximate solutions. It assigns a set of \(\varepsilon \) -approximations of a solution to the problem instance f and accuracy of approximation \(\varepsilon \) . In this paper we study generalized solution operators for which the accuracy of approximation is described by elements of a complete lattice equipped with a compatible monoid structure, namely, a quantale. We provide examples of computational problems for which the accuracy of approximation of a solution is measured by such objects. We show that the sets of \(\varepsilon \) -approximations are, roughly, closed balls with radii \(\varepsilon \) with respect to a certain family of quantale-valued generalized metrics induced by a generalized solution operator. PubDate: 2017-05-29 DOI: 10.1007/s00010-017-0485-8
Authors:M. Rajesh Kannan Abstract: In this article we introduce the notion of P-proper splitting for square matrices. For an inconsistent linear system of equations \(Ax =b\) , we associate an iterative method based on a P-proper splitting of A, which if convergent, converges to the best least squares solution of this system. We extend a result of Stein, using which we prove that if A is positive semidefinite, then the said iterative method converges. Also, we generalize Sylvester’s law of inertia and as an application of this generalization we establish some properties of P-proper splittings. Finally, we prove a comparison theorem for iterative methods associated with P-proper splittings of a positive semidefinite matrix. PubDate: 2017-05-27 DOI: 10.1007/s00010-017-0492-9
Authors:Dijana Mosić Abstract: We study the generalization of Jacobson’s lemma for the group inverse, Drazin inverse, generalized Drazin inverse and pseudo Drazin inverse of \(1-bd\) (or \(1-ac\) ) in a ring when \(1-ac\) (or \(1-bd\) ) has a corresponding inverse, \(acd=dbd\) and \(bdb=bac\) (or \(dba=aca\) ). Thus, we recover some recent results. PubDate: 2017-04-04 DOI: 10.1007/s00010-017-0476-9
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