Generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation

Using the fixed point method, we prove the generalized Ulam-Hyers stability of random homomorphisms in random normed algebras associated with the Cauchy functional equation.     

On the stability of the additive Cauchy functional equation in random normed spaces

In this paper, we prove a stability result for the additive Cauchy functional equation in random normed spaces, related to the main theorem from the paper [D. Mihe?, V. Radu, On the stability of the additive Cauchy functional equation in random normed spaces, J. Math. Anal. Appl. 343 (2008) 567–572].     

On the stability of the additive Cauchy functional equation in random normed spaces

Some stability results for the functional equations of Cauchy and Jensen in probabilistic setting are proved by using the fixed point method.     

Symmetric block bases in finite-dimensional normed spaces

It is shown that for 1 ≦p < ∞, any basisC-equivalent to the unit vector basis ofl _p ⁿ has a (1 + ε)-symmetric block basis of cardinality proportional ton/logn. When 1 <p < ∞, an upper bound proportional ton log logn/logn is also obtained. These results extend results of Amir and Milman in [2].     

Superstability of the Cauchy, Jensen and Isometry Equations

In the first part of our paper we generalize the results obtained by Józef Tabor in [12] concerning the superstability of the Cauchy and Jensen functional equation almost everywhere. In the second part we prove a general theorem on the superstablity of the Isometry equation in inner product spaces. As a corollary we determine when the Isometry Equation is superstable in the integral norm (this is a partial answer to [14]).     

A finite-dimensional normed space with two non-equivalent symmetric bases

A sequence of finite-dimensional normed spaces is constructed, each with two symmetric bases, such that the sequence of equivalence constants between these bases is unbounded. An essential tool in the proof is the edge-isoperimetric inequality in the discrete cube.     

Lie algebras with finite-dimensional polynomial centralizer

We give criteria for finite or infinite dimensionality of the polynomial centralizer of the Lie algebra of a linear Lie group, in terms of invariants and relative invariants of the group. In the finite-dimensional scenario some applications to normal forms and to certain equations with fundamental solutions are presented.     

On multi-orthogonal bases in finite-dimensional non-Archimedean normed spaces

The paper deals with the problem of the existence multi-orthogonal bases in finite-dimensional normed spaces over K, where K is a non-Archimedean complete valued field.     

Identities of unitary finite-dimensional algebras

We deal with growth functions of sequences of codimensions of identities in finite-dimensional algebras with unity over a field of characteristic zero. For three-dimensional algebras, it is proved that the codimension sequence grows asymptotically as a ⁿ , where a is 1, 2, or 3. For arbitrary finite-dimensional algebras, it is shown that the codimension growth either is polynomial or is not slower than 2 ⁿ . We give an example of a finite-dimensional algebra with growth rate an with fractional exponent a = \frac3³?{4} + 1 a = \frac{3}{{\sqrt[3]{4}}} + 1 .     

Inner and outerj-radii of convex bodies in finite-dimensional normed spaces

This paper is concerned with the various inner and outer radii of a convex bodyC in ad-dimensional normed space. The innerj-radiusr _j (C) is the radius of a largestj-ball contained inC, and the outerj-radiusR _j (C) measures how wellC can be approximated, in a minimax sense, by a (d —j)-flat. In particular,r _d (C) andR _d (C) are the usual inradius and circumradius ofC, while 2r ₁(C) and 2R ₁(C) areC's diameter and width.Motivation for the computation of polytope radii has arisen from problems in computer science and mathematical programming. The radii of polytopes are studied in [GK1] and [GK2] from the viewpoint of the theory of computational complexity. This present paper establishes the basic geometric and algebraic properties of radii that are needed in that study.Much of this paper was written when both authors were visiting the Institute for Mathematics and Its Applications, 206 Church Street S.E., Minneapolis, MN 55455, USA. The research of P. Gritzmann was supported in part by the Alexander-von-Humboldt Stiftung and the Deutsche Forschungsgemeinschaft. V. Klee's research was supported in part by the National Science Foundation.     

Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation

In this paper, we prove the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras associated to the Pexiderized Cauchy functional equation. This is applied to investigate homomorphisms between quasi-Banach algebras. The concept of Hyers-Ulam-Rassias stability originated from Th.M. Rassias' stability theorem that appeared in his paper [Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978) 297-300].     

Superstability of the generalized orthogonality equation on restricted domains

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation |〈f(x), f(y)〉| = |〈x,y〉|. In this paper, we will extend the result of Chmielinski by proving a theorem: LetD _n be a suitable subset of ℝⁿ. If a function f:D _n → ℝⁿ satisfies the inequality ∥〈f(x), f(y)〉| |〈x,y〉∥ ≤ φ(x,y) for an appropriate control function φ(x, y) and for allx, y ∈ D _n, thenf satisfies the generalized orthogonality equation for anyx, y ∈ D _n.     

Some New Results on the Superstability of the Cauchy Equation on Semigroups

In this paper the superstability of the Cauchy equation can be proved for complex valued functions on commutative semigroup under some suitable conditions. Furthermore, in this note, we give a partial affirmative answer to problem 18, in the 31st ISFE.     

On functional representation of normed algebras

Let A be a commutative algebra over complex numbers with a space norm ‖⋅‖ making the multiplication on A separately continuous. We will study the Gelfand representation of this type of normed algebra. In particular, we look at the cases where the standard Gelfand representation (i.e., the use of supremum-norm on the Gelfand transform algebra ) gives different properties from the original algebra (A,‖⋅‖). We show that there are even Banach algebras for which this type of difficulty may happen. We will provide with some weighted supremum-norm and by using these weights we can avoid the difficulties mentioned above. For the definition of these weights we adopt the ideas of Cochran represented in [A.C. Cochran, Representation of A-convex algebras, Proc. Amer. Math. Soc. 30 (1973) 473-479].     

On the sum of two Lie algebras with finite-dimensional commutants

We prove that an infinite-dimensional Lie algebra over an arbitrary field which is decomposable into the sum of two of its subalgebras with finite-dimensional commutants is almost solvable.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1089–1096, August, 1995.