79 research outputs found
Method of constructing exactly solvable chaos
We present a new systematic method of constructing rational mappings as
ergordic transformations with nonuniform invariant measures on the unit
interval [0,1]. As a result, we obtain a two-parameter family of rational
mappings that have a special property in that their invariant measures can be
explicitly written in terms of algebraic functions of parameters and a
dynamical variable. Furthermore, it is shown here that this family is the most
generalized class of rational mappings possessing the property of exactly
solvable chaos on the unit interval, including the Ulam=Neumann map y=4x(1-x).
Based on the present method, we can produce a series of rational mappings
resembling the asymmetric shape of the experimentally obtained first return
maps of the Beloussof-Zhabotinski chemical reaction, and we can match some
rational functions with other experimentally obtained first return maps in a
systematic manner.Comment: 12 pages, 2 figures, REVTEX. Title was changed. Generalized Chebyshev
maps including the precise form of two-parameter generalized cubic maps were
added. Accepted for publication in Phys. Rev. E(1997
Approximate Homomorphisms of Ternary Semigroups
A mapping between ternary semigroups will be
called a ternary homomorphism if . In this paper,
we prove the generalized Hyers--Ulam--Rassias stability of mappings of
commutative semigroups into Banach spaces. In addition, we establish the
superstability of ternary homomorphisms into Banach algebras endowed with
multiplicative norms.Comment: 10 page
Orthogonalities and functional equations
In this survey we show how various notions of orthogonality appear in the theory of functional equations. After introducing some orthogonality relations, we give examples of functional equations postulated for orthogonal vectors only. We show their solutions as well as some applications. Then we discuss the problem of stability of some of them considering various aspects of the problem. In the sequel, we mention the orthogonality equation and the problem of preserving orthogonality. Last, but not least, in addition to presenting results, we state some open problems concerning these topics. Taking into account the big amount of results concerning functional equations postulated for orthogonal vectors which have appeared in the literature during the last decades, we restrict ourselves to the most classical equations
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