Aequationes Math Research Articles

A Mazur–Ulam theorem for probabilistic normed spaces

A classical theorem of S. Mazur and S. Ulam asserts that any surjective isometry between two normed spaces is an affine mapping. D. Mushtari proved in 1968 the same result in the case of random normed spaces in the sense of A. Sherstnev. The aim of the present paper is to show that the result holds also for the probabilistic normed spaces as defined by C. Alsina, B. Schweizer and A. Sklar, Aequationes Math. 46 (1993), 91–98.

On a functional equation involving group actions

During the forty-first ISFE in Noszvaj, Hungary, G. Guzik posed a problem on a functional equation involving group actions which arose in a generalization of Bargman theory occurring in Quantum Mechanics. (Cf. 18. Problem and Remark in “Report of Meeting", Aequationes Math., 67 (2004), 312–313.) Let (G, ·) be a group which is acting on a set X and let (K, +) be an abelian group. Describe all functions $$f : G \times G \times X \rightarrow K$$ satisfying $$f(g1, g2, x) + f(g1g2, g3, x) = f(g2, g3, g^{-1}_{1} x) + f(g1, g2g3, x)$$ for all $$g1, g2, g3 \in G$$ and $$x \in X$$ . This problem was solved in a particular case by B. Ebanks. (Cf. 19. Remark in “Report of Meeting", Aequationes Math. 67 (2004), p. 313.) We present the general solution of this problem.

Hausdorff and packing dimensions of non-normal tuples of numbers: Non-linearity and divergence points

We apply the results in [L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages, J. Math. Pures Appl. 82 (2003) 1591–1649; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. III, Aequationes Math. 71 (2006) 29–53; L. Olsen, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension, Bull. Sci. Math. 132 (2008) 650–678; L. Olsen, S. Winter, Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. II: Non-linearity, divergence points and Banach space valued spectra, Bull. Sci. Math. 131 (2007) 518–558] to give a systematic and detailed account of the Hausdorff and packing dimensions of sets of d-tuples of numbers defined in terms of the asymptotic behaviour of the frequencies of strings of digits in their N-adic expansion.

Convex combinations of nilpotent triangular norms

In this paper we deal with the open problem of convex combinations of continuous triangular norms stated by Alsina, Frank, and Schweizer [C. Alsina, M.J. Frank, B. Schweizer, Problems on associative functions, Aequationes Math. 66 (2003) 128–140, Problems 5 and 6]. They pose a question whether a non-trivial convex combination of triangular norms can ever be a triangular norm. The main result of this paper gives a negative answer to the question for any pair of continuous Archimedean triangular norms with different supports. With the help of this result we show that a non-trivial convex combination of nilpotent t-norms is never a t-norm. The main result also gives an alternative proof to the result presented by Ouyang and Fang [Y. Ouyang, J. Fang, Some observations about the convex combination of continuous triangular norms, Nonlinear Anal., 68 (11) (2008) 3382–3387, Theorem 3.1]. In proof of the main theorem we utilize the Reidmeister condition known from the web geometry.

Functional Equations and μ-Spherical Functions

Abstract We will study the properties of solutions 𝑓, {𝑔𝑖}, {ℎ𝑖} ∈ 𝐶𝑏(𝐺) of the functional equation where 𝐺 is a Hausdorff locally compact topological group, 𝐾 a compact subgroup of morphisms of 𝐺, χ a character on 𝐾, and μ a 𝐾-invariant measure on 𝐺. This equation provides a common generalization of many functional equations (D'Alembert's, Badora's, Cauchy's, Gajda's, Stetkaer's, Wilson's equations) on groups. First we obtain the solutions of Badora's equation [Aequationes Math. 43: 72–89, 1992] under the condition that (𝐺,𝐾) is a Gelfand pair. This result completes the one obtained in [Badora, Aequationes Math. 43: 72–89, 1992] and [Elqorachi, Akkouchi, Bakali and Bouikhalene, Georgian Math. J. 11: 449–466, 2004]. Then we point out some of the relations of the general equation to the matrix Badora functional equation and obtain explicit solution formulas of the equation in question for some particular cases. The results presented in this paper may be viewed as a continuation and a generalization of Stetkær's, Badora's, and the authors' works.

A solution to a problem raised in Luce and Marley (2005)

Luce and Marley [2005. Ranked additive utility representations of gambles: Old and new axiomatizations. Journal of Risk and Uncertainty, 30, 21–62] examined various relations between mathematical forms for the utility of joint receipt ⊕ of gambles and for the utility of uncertain gambles. Their assumptions lead to a bisymmetry functional equation which, when the gambles are ranked, is defined on a restricted domain. Maksa [1999. Solution of generalized bisymmetry type equations without surjectivity. Aequationes Mathematicae, 57, 50–74] solved the general case and Kocsis [2007. A bisymmetry equation on restricted domain. Aequationes Mathematicae, 73, 280–284] presents the solution for the ranked case. The latter solution allows us to solve open problem 5 in Luce and Marley (2005) by showing that the assumptions of their Theorem 19 for an order-preserving ranked additive utility (RAU) representation U imply that U is a ranked weighted utility (RWU) representation that is additive over ⊕ .

Partial cubes are distance graphs

Chatrand et al. [Graph similarity and distance in graphs, Aequationes Math. 55 (1998) 129–145] have recently conjectured that all bipartite graphs are distance graphs. Here we show that all graphs of a large subclass of bipartite graphs, i.e. partial cubes, are distance graphs.

A remark on a logarithmic functional equation

We revisit the logarithmic functional equation of Heuvers and Kannappan [K.J. Heuvers, Pl. Kannappan, A third logarithmic functional equation and Pexider generalizations, Aequationes Math. 70 (2005) 117–121] and give a simple proof of the result and discuss the locally integrable solutions of the equation.

Solutions of two functional equations using a result of M. Sablik

We rephrase Lemma 2.3 from (M. Sablik, Taylor’s theorem and functional equations, Aequationes Math. 60 (2000), 258–267) in order to solve two functional equations. First, the following “Taylor-like” functional equation $$ f(x)\, = \,{\sum\limits_{k = 0}^n {g_{k} (x + t(y - x))((t(x - y))^{k} ) + (\Phi (x) - \Phi (y))((t(x - y))^{n} )} } $$ and a second one stemming from Simpson’s rule: $$ f(x) - f(y) = [h(sx + ty) + \Phi (x) + \Psi (y)](x - y) $$ .

Erratum to: “On nonstrict means” [Aequationes Math. 54(3) (1997), 308–327

Erratum to: “On nonstrict means” [Aequationes Math. 54(3) (1997), 308–327

Erratum to: “Skew-commuting and commuting mappings in rings” [Aequationes Math. 64 (2002), 136–144

Erratum to: “Skew-commuting and commuting mappings in rings” [Aequationes Math. 64 (2002), 136–144

Stability of functional equations of Drygas in the space of Schwartz distributions

In this paper we reformulate and prove the stability theorems of S.M. Jung and P.K. Sahoo [S.M. Jung, P.K. Sahoo, Stability of a functional equation of Drygas, Aequationes Math. 64 (2002) 263–273] in the spaces of generalized functions such as the Schwartz distributions and tempered distributions.

Gelfand Pairs and Generalized d'Alembert's and Cauchy's Functional Equations

Abstract We show that Cauchy's, d'Alembert's functional equations and their generalizations are the functional equations for bounded spherical functions associated to some Gel'fand pairs. Our results are very close to the ones obtained by Stetkær in [Aequationes Math. 48: 220–227, 1994].

Badora's Equation on Non-Abelian Locally Compact Groups

Abstract This paper is mainly concerned with the following functional equation where 𝐺 is a locally compact group, 𝐾 a compact subgroup of its morphisms, and μ is a generalized Gelfand measure. It is shown that continuous and bounded solutions of this equation can be expressed in terms of μ-spherical functions. This extends the previous results obtained by Badora (Aequationes Math. 43: 72–89, 1992) on locally compact abelian groups. In the case where 𝐺 is a connected Lie group, we characterize solutions of the equation in question as joint eigenfunctions of certain operators associated to the left invariant differential operators.

On a functional equation related to power means

In the paper On the mutual noncompatibility of homogeneous analytic non-power means (Aequationes Math. 45 (1993)) M. E. Kuczma considered analytic solutions of the functional equation $ x + g(y + f(x)) = y + g(x + f(y)) $ on the real line. Solutions in the class of twice differentiable functions are given in the author’s paper Differentiable solutions of a functional equation related to the non-power means (Aequationes Math. 55 (1998)). During the 38th International Symposium on Functional Equations N. Brillouet-Belluot presented the proof that differentiable solutions of this equation have the same form as in the previous cases. We present solutions of the equation in the class of monotonic Jensen convex or Jensen concave functions on the real line. This time we get already some new families of solutions.

On lacets and their manifolds

By topological lacet we mean the embedding, with self-intersections, of a closed curve in a 2-manifold, such that (1) all self-intersections are simple double points at which the curve crosses itself, and (2) the complement of the curve in the 2-manifold is a 2-colorable family of discs. Such embeddings are characterized, up to homotopy, by a combinatorial lacet, that is, by a double occurrence word Δ on an alphabet P and a bipartition (K,L) of the set P, P representing the set of self-intersections of the curve, Δ their sequence in one complete run along the curve, and K,L the two types of reentry possible at a crossing. Passing by the construction of a map associated to the lacet, we show that every combinatorial lacet is representable on a 2-manifold. Beginning with a problem on curves in the plane posed and partially solved by Gauss (1840), and introducing certain linear transformations of vector spaces over GF(2) , depending on Δ and (K,L), Rosenstiehl [C. R. Acad. Sci. Paris Ser. A 283 (1976) 551] and Lins et al. [Aequationes Math. 33 (1987) 81] characterized combinatorial lacets representable in the Euclidean plane and projective plane, respectively, by a simple property of the enlacement graph of pairs of letters in Δ. We prove here that a characterisation of the same sort extends naturally to the case of arbitrary 2-manifolds. In particular, we present procedures for determining if a combinatorial lacet is representable on the torus, or on the Klein bottle.

A conversation with Johannes H. B. Kemperman

Johannes H. B. Kemperman, born in 1924, received his Bachelor of Science in 1945 and Ph.D. in 1950, each in mathematics and physics, from the University of Amsterdam. From 1948 to 1951 he was a Research Associate at the Mathematical Centre in Amsterdam. During 1951–1953 he was a Visiting Professor on a Fulbright grant at Purdue University, and subsequently he joined the faculty and stayed at Purdue for 10 years, becoming a full Professor in 1959. In 1961 he went to the University of Rochester, becoming the Fayerweather Professor of Mathematics in 1970. He stayed at Rochester for 25 years. In 1985 he joined the faculty at Rutgers University as a Professor in the Statistics Department and also a voting member of the Mathematics Department. He retired to emeritus status at Rutgers in 1995. He served in editorial posts at the Annals of Mathematical Statistics, Annals of Probability, Annals of Statistics, Aequationes Mathematicae and Stochastic Processes and Applications. He is a Fellow of the Institute of Mathematical Statistics and the American Association for the Advancement of Sciences and is a Correspondent of the Royal Dutch Academy of Sciences (Amsterdam) . He has produced 23 Ph.D. students, 3 books and over 100 publications in analysis, number theory, group theory, probability, statistics, functional equations, mathematical biology and other areas.

Remarks on Bárány's theorem and affine selections

Using the ‘multiplied’ version of Helly's theorem given by Bárány (Discrete Math. 40 (1982) 141–152) we generalize some selection and separation results obtained in Behrends and Nikodem (Studia Math. 116 (1) (1995) 43–48), Nikodem and Wa̧sowicz (Aequationes Math. 49 (1995) 160–164) and Wa̧sowicz (J. Appl. Anal. 1 (2) (1995) 173–179). In particular, it is shown that if three set-valued functions Φ 1,Φ 2,Φ 3:I→ cc(R) satisfy the condition Φ i(tx+(1−t)y)∩[tΦ j(x)+(1−t)Φ k(y)]≠∅ for all x,y∈I,t∈[0,1] and every permutation (i,j,k) of the set {1,2,3} then at least one of them has an affine selection.

On the significance of the theory of convex polyhedra for formal algebra

This paper was originally published as Über die Bedeutung der Theorie der konvexen Polyeder für die formale Algebra in Jahresbericht d. Deutschen Math. Ver. 30 (1922): 98-99. It is based on a talk given in Jena to the German Mathematical Society on 23 September 1921. Quoted in Ostrowski's later paper On Multiplication and Factorization of Polynomials. I Lexicographic Orderings and Extreme Aggregates of Terms. Aequationes Math. 13 (1975): 201-228.

Corrigendum to "On trigonometric functional equations of rectangular type" in Aequationes Math. 53 (1997), pp. 36-53

Corrigendum to "On trigonometric functional equations of rectangular type" in Aequationes Math. 53 (1997), pp. 36-53