On the analytic structure of the Henon-Heiles system

Solutions of the Henon-Heiles hamiltonian that are analytically continued into the complex time domain are found to possess a neutral boundary with a self-similar structure.     

On functions behaving like additive functions

Let 0 < 1. In the paper we consider the following inequality: |f(x + y) – f(x) – f(y)| min{|f(x + y)|, |f(x) + f(y)|}, wheref: R R. Solutions and continuous solutions of this inequality are investigated. They have similar properties as additive functions, e.g. if the solution is bounded above (below) on a set of positive inner Lebesgue measure then it is continuous. Some sufficient condition for this inequality is also given.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday     

Lipschitz Stability of the Cauchy and Jensen Equations

Let G be an amenable metric semigroup with nonempty center, let E be a reflexive Banach space, and let ?: G → E be a given function. By C?: G × G → E we understand the Cauchy difference of the function /, i.e.: $$ {\cal C}f(x,y):=f(x+y)- f(x)- f(y)\ {\rm for}\ x,y\in G. $$ We prove that if the function C(f) is Lipschitz then there exists an additive function A: G → E such that f ? A is Lipschitz with the same constant. Analogous result for Jensen equation is also proved. As a corollary we obtain the stability of the Cauchy and Jensen equations in the Lipschitz norms.     

Quasi-additive functions

Summary Let 0 < 1 and letX, Y be real normed spaces. In this paper we consider the following functional inequality:f(x + y) – f(x) – f(y) min{f(x + y), f(x) + f(y)} forx, y R, wheref: X Y. Mainly continuous solutions are investigated. In the case whereY = R some necessary and some sufficient conditions for this inequality are given.Let 0 <1. The following functional inequality has been considered in [5]:f(x + y) – f(x) – f(y) min{f(x + y), f(x) + f(y)} forx, y R, wheref: R R. It appeared that the solutions of this inequality have properties very similar to those of additive functions (cf. [1], [2], [3]). The inequality under consideration seems to be interesting also because of its physical interpretation (cf. [5]). In this paper we shall consider this inequality in a more general case, wheref is defined on a real normed space and takes its values in another real normed space.The first part of the paper concerns the general case; in the second part we assume that the range off is inR.     

Morphology and optical properties of thin silica films containing bimetallic Ag/Au nanoparticles

We have studied the optical spectra in the UV and visible regions, the morphology by scanning electron microscopy (SEM), and the X-ray photoelectron spectra (XPS) of bimetallic Ag/Au nanoparticles incorporated into transparent silicate films in the sol-gel transition stage. The bimetallic nanoparticles, obtained by a combination of photoreduction and thermal reduction, form structures of the alloy or core-shell type. Translated from Teoreticheskaya i éksperimental'naya Khimiya, Vol. 44, No. 6, pp. 348–353, November–December, 2008.     

General stability of functional equations of linear type

Making use of a dynamical systems notion called shadowing, we prove a stability result for linear functional equations in metric groups. As a corollary we obtain stability of the quadratic functional equation in the case when the target space is a metric group satisfying some local 2-divisibility condition.