Authors:Xujian Huang; Dongni Tan Pages: 401 - 413 Abstract: We give a positive answer to the Aleksandrov problem in n-normed spaces under the surjectivity assumption. Namely, we show that every surjective mapping preserving n-distance one is affine, and thus is an n-isometry. This is the first time the Aleksandrov problem is solved in n-normed spaces with only the surjectivity assumption even in the usual case \(n=2\) . Finally, when the target space is n-strictly convex, we prove that every mapping preserving two n-distances with an integer ratio is an affine n-isometry. PubDate: 2018-06-01 DOI: 10.1007/s00010-018-0539-6 Issue No:Vol. 92, No. 3 (2018)
Authors:Helmut Karzel; Sayed-Ghahreman Taherian Pages: 415 - 423 Abstract: We consider in a group \((G,\cdot )\) the ternary relation $$\begin{aligned} \kappa := \{(\alpha , \beta , \gamma ) \in G^3 \ \ \alpha \cdot \beta ^{-1} \cdot \gamma = \gamma \cdot \beta ^{-1} \cdot \alpha \} \end{aligned}$$ and show that \(\kappa \) is a ternary equivalence relation if and only if the set \( \mathfrak Z \) of centralizers of the group G forms a fibration of G (cf. Theorems 2, 3). Therefore G can be provided with an incidence structure $$\begin{aligned} \mathfrak G:= \{\gamma \cdot Z \ \ \gamma \in G , Z \in \mathfrak Z(G) \}. \end{aligned}$$ We study the automorphism group of \((G,\kappa )\) , i.e. all permutations \(\varphi \) of the set G such that \( (\alpha , \beta , \gamma ) \in \kappa \) implies \((\varphi (\alpha ),\varphi (\beta ),\varphi (\gamma ))\in \kappa \) . We show \(\mathrm{Aut}(G,\kappa )=\mathrm{Aut}(G,\mathfrak G)\) , \(\mathrm{Aut} (G,\cdot ) \subseteq \mathrm{Aut}(G,\kappa )\) and if \( \varphi \in \mathrm{Aut}(G,\kappa )\) with \(\varphi (1)=1\) and \(\varphi (\xi ^{-1})= (\varphi (\xi ))^{-1}\) for all \(\xi \in G\) then \(\varphi \) is an automorphism of \((G,\cdot )\) . This allows us to prove a representation theorem of \(\mathrm{Aut}(G,\kappa )\) (cf. Theorem 6) and that for \(\alpha \in G \) the maps $$\begin{aligned} \tilde{\alpha }\ : \ G \rightarrow G;~ \xi \mapsto \alpha \cdot \xi ^{-1} \cdot \alpha \end{aligned}$$ of the corresponding reflection structure \((G, \widetilde{G})\) (with \( \tilde{G} := \{\tilde{\gamma }\ \ \gamma \in G \}\) ) are point reflections. If \((G ,\cdot )\) is uniquely 2-divisible and if for \(\alpha \in G\) , \(\alpha ^{1\over 2}\) denotes the unique solution of \(\xi ^2=\alpha \) then with \(\alpha \odot \beta := \alpha ^{1\over 2} \cdot \beta \cdot \alpha ^{1\over 2}\) , the pair PubDate: 2018-06-01 DOI: 10.1007/s00010-018-0543-x Issue No:Vol. 92, No. 3 (2018)
Authors:Árpád Baricz; Khaled Mehrez Pages: 425 - 439 Abstract: In this paper our aim is to show some new inequalities of the Redheffer type for Bessel and modified Bessel functions of the first kind. The key tools in our proofs are some classical results on the monotonicity of quotients of differentiable functions as well as on the monotonicity of quotients of two power series. We also use some known results on the quotients of Bessel and modified Bessel functions of the first kind, and by using the monotonicity of the Dirichlet eta function we prove a sharp inequality for the tangent function. At the end of the paper a conjecture is stated, which may be of interest for further research. PubDate: 2018-06-01 DOI: 10.1007/s00010-018-0545-8 Issue No:Vol. 92, No. 3 (2018)
Authors:Oksana Bihun; Damiano Fulghesu Pages: 453 - 470 Abstract: In this paper we consider monic polynomials such that their coefficients coincide with their zeros. These polynomials were first introduced by S. Ulam. We combine methods of algebraic geometry and dynamical systems to prove several results. We obtain estimates on the number of Ulam polynomials of degree N. We provide additional methods to obtain algebraic identities satisfied by the zeros of Ulam polynomials, beyond the straightforward comparison of their zeros and coefficients. To address the question about the existence of orthogonal Ulam polynomial sequences, we show that the only Ulam polynomial eigenfunctions of hypergeometric type differential operators are the trivial Ulam polynomials \(\{x^N\}_{N=0}^\infty \) . We propose a family of solvable N-body problems such that their stable equilibria are the zeros of certain Ulam polynomials. PubDate: 2018-06-01 DOI: 10.1007/s00010-018-0546-7 Issue No:Vol. 92, No. 3 (2018)
Authors:Poo-Sung Park Pages: 487 - 495 Abstract: P. V. Chung showed that there are many multiplicative functions f which satisfy \(f(m^2+n^2) = f(m^2)+f(n^2)\) for all positive integers m and n. In this article, we show that if more than 2 squares in the additive condition are involved, then such f is uniquely determined. That is, if a multiplicative function f satisfies $$\begin{aligned} f(a_1^2 + a_2^2 + \cdots + a_k^2) = f(a_1^2) + f(a_2^2) + \cdots + f(a_k^2) \end{aligned}$$ for arbitrary positive integers \(a_i\) , then f is the identity function. In this sense, we call the set of all positive squares a k-additive uniqueness set for multiplicative functions. PubDate: 2018-06-01 DOI: 10.1007/s00010-017-0517-4 Issue No:Vol. 92, No. 3 (2018)
Authors:Boštjan Brešar; Sandi Klavžar; Douglas F. Rall; Kirsti Wash Pages: 497 - 513 Abstract: The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\) , \(i\in [k]\) , where each \(V_i\) is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that \(\omega (G) = a\) , \(\chi (G) = b\) , and \(\chi _{\rho }(G) = c\) . If so, we say that (a, b, c) is realizable. It is proved that \(b=c\ge 3\) implies \(a=b\) , and that triples \((2,k,k+1)\) and \((2,k,k+2)\) are not realizable as soon as \(k\ge 4\) . Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on \(\chi _{\rho }(G)\) in terms of \(\Delta (G)\) and \(\alpha (G)\) is also proved. PubDate: 2018-06-01 DOI: 10.1007/s00010-017-0520-9 Issue No:Vol. 92, No. 3 (2018)
Authors:Eszter Gyimesi; Gábor Nyul Pages: 515 - 527 Abstract: After his extensive study of Whitney numbers, Benoumhani introduced Dowling numbers and polynomials as generalizations of the well-known Bell numbers and polynomials. Later, Cheon and Jung gave the r-generalization of these notions. Based on our recent combinatorial interpretation of r-Whitney numbers, in this paper we derive several new properties of r-Dowling polynomials and we present alternative proofs of some previously known ones. PubDate: 2018-06-01 DOI: 10.1007/s00010-017-0538-z Issue No:Vol. 92, No. 3 (2018)
Authors:Carlos Benítez; Pedro Martín; Diego Yáñez Pages: 529 - 541 Abstract: Let X be a real normed space with unit sphere S. We prove that X is an inner product space if and only if there exists a real number \(\rho =\sqrt{(1+\cos \frac{2k\pi }{2m+1})/2}, (k=1,2,\ldots ,m; m=1,2,\ldots )\) , such that every chord of S that supports \(\rho S\) touches \(\rho S\) at its middle point. If this condition holds, then every point \(u\in S\) is a vertex of a regular polygon that is inscribed in S and circumscribed about \(\rho S\) . PubDate: 2018-06-01 DOI: 10.1007/s00010-018-0540-0 Issue No:Vol. 92, No. 3 (2018)
Authors:Gyula Maksa Pages: 543 - 547 Abstract: In this paper, we give the solution of a problem formulated in Kominek and Sikorska (Aequationes Math 90:107–121, 2016) in connection with the functional equation $$\begin{aligned} f(xy)-f(x)-f(y)=g(x+y)-g(x)g(y). \end{aligned}$$ Our result can also be interpreted in the way that, under some additional condition, the logarithmic and the exponential Cauchy equations are strongly alien. PubDate: 2018-06-01 DOI: 10.1007/s00010-017-0535-2 Issue No:Vol. 92, No. 3 (2018)
Authors:Maliheh Hosseini; Juan J. Font Pages: 549 - 561 Abstract: Let G be a locally compact abelian group and B be a commutative Banach algebra. Let \(L^{1}(G, B)\) be the Banach algebra of B-valued Bochner integrable functions on G. In this paper we provide a complete description of continuous disjointness preserving maps on \(L^{1}(G, B)\) -algebras based on a scarcely used tool: the vector-valued Fourier transform. We also present necessary and sufficient conditions for these operators to be compact. PubDate: 2018-06-01 DOI: 10.1007/s00010-018-0547-6 Issue No:Vol. 92, No. 3 (2018)
Authors:Péter Kutas Pages: 563 - 575 Abstract: Let C be an affine plane curve. We consider additive functions \(f{:}\; K\rightarrow K\) for which \(f(x)f(y)=0\) , whenever \((x,y)\in C\) . We show that if \(K=\mathbb {R}\) and C is the hyperbola with defining equation \(xy=1\) , then there exist nonzero additive functions with this property. Moreover, we show that such a nonzero f exists for a field K if and only if K is transcendental over \(\mathbb Q\) or over \(\mathbb {F}_p\) , the finite field with p elements. We also consider the general question when K is a finite field. We show that if the degree of the curve C is large enough compared to the characteristic of K, then f must be identically zero. PubDate: 2018-06-01 DOI: 10.1007/s00010-017-0521-8 Issue No:Vol. 92, No. 3 (2018)
Authors:Bruce Ebanks Pages: 581 - 597 Abstract: We continue the study of additive functions \(f_k:R\rightarrow F \;(1\le k\le n)\) linked by an equation of the form \(\sum _{k=1}^n p_k(x)f_k(q_k(x))=0\) , where the \(p_k\) and \(q_k\) are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case \(\sum _{k=1}^n x^{m_k}f_k(x^{j_k})=0\) in which the \(p_k\) and \(q_k\) are monomials. In this case we show that if there is no duplication, i.e. if \((m_k,j_k)\ne (m_p,j_p)\) for \(k\ne p\) , then each \(f_k\) is the sum of a linear function and a derivation of order at most \(n-1\) . Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way. PubDate: 2018-06-01 DOI: 10.1007/s00010-017-0537-0 Issue No:Vol. 92, No. 3 (2018)
Authors:Aureliano M. Robles-Pérez; José Carlos Rosales Abstract: A problem about how to transport profitably a group of cars leads us to studying the set T formed by the integers n such that the system of inequalities, with non-negative integer coefficients, $$\begin{aligned} a_1x_1 +\cdots + a_px_p + \alpha \le n \le b_1x_1 +\cdots + b_px_p - \beta \end{aligned}$$ has at least one solution in \({\mathbb N}^p\) . We prove that \(T\cup \{0\}\) is a submonoid of \(({\mathbb N},+)\) and, moreover, we give algorithmic processes to compute T. PubDate: 2018-06-05 DOI: 10.1007/s00010-018-0572-5
Abstract: The notion of \((m,M,\Psi )\) -Schur-convexity is introduced and functions generating \((m,M,\Psi )\) -Schur-convex sums are investigated. An extension of the Hardy–Littlewood–Pólya majorization theorem is obtained. A counterpart of the result of Ng stating that a function generates \((m,M,\Psi )\) -Schur-convex sums if and only if it is \((m,M,\psi )\) -Wright-convex is proved and a characterization of \((m,M,\psi )\) -Wright-convex functions is given. PubDate: 2018-05-30 DOI: 10.1007/s00010-018-0569-0
Abstract: In this paper, we study mappings, which approximately preserve angles between inner product spaces. We also introduce a notion of angle in normed spaces. The notion of angle, considered in this part, relates to the well-known Birkhoff–James orthogonality. Based on it, we express a characterization for approximate Birkhoff–James orthogonality, introduced in the literature, through this notion of angle. Then we return to the issue of mappings which approximately preserve angle stating some results in normed spaces. PubDate: 2018-05-30 DOI: 10.1007/s00010-018-0571-6
Authors:Dijana Mosić Abstract: We extend the notation of the CMP inverse for a square matrix to a rectangular matrix. Precisely, we define and characterize a new generalized inverse called the weighted CMP inverse. Also, we investigate properties of the weighted CMP inverse using a representation by block matrices. Some new characterizations and properties of the CMP inverse are obtained. PubDate: 2018-05-14 DOI: 10.1007/s00010-018-0570-7
Authors:S. S. Linchuk; Yu. S. Linchuk Abstract: In the original publication linking to CrossRef, MATH, MathSciNet were missed for the References [9–12]. The correct References are given below. PubDate: 2018-03-30 DOI: 10.1007/s00010-018-0551-x
Authors:Mihai Monea; Dan Ştefan Marinescu Abstract: We give another elementary proof for the majorization principle for Wright-convex functions. This inequality is due to Ng. PubDate: 2018-03-30 DOI: 10.1007/s00010-018-0549-4
Authors:Nikolai Kolev; Jayme Pinto Abstract: We introduce a new probability aging notion via a functional equation based on the tail invariance of Sibuya’s dependence function which is specified as the ratio between the joint survival function and the product of its marginal survival functions. Solutions of the functional equation are generated by Gumbel’s type I bivariate exponential distribution and independence law. In a particular setting, we construct a version of Gumbel’s law with a singular component. PubDate: 2018-03-28 DOI: 10.1007/s00010-018-0544-9
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