# V. D. Koshmanenko

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### Dynamical systems of conflict in terms of structural measures

Volodymyr Koshmanenko, Inga Verygina

Methods Funct. Anal. Topology **22** (2016), no. 1, 81-93

We investigate the dynamical systems modeling conflict processes between a pair of opponents. We assume that opponents are given on a common space by distributions (probability measures) having the similar or self-similar structure. Our main result states the existence of the controlled conflict in which one of the opponents occupies almost whole conflicting space. Besides, we compare conflicting effects stipulated by the rough structural approximation under controlled redistributions of starting measures.

### Existence theorems of the ω-limit states for conflict dynamical systems

Methods Funct. Anal. Topology **20** (2014), no. 4, 379-390

We introduce a notion of the conflict dynamical system in terms of probability measures, study the behavior of trajectories of such systems, and prove the existence theorems of the ω-limit states.

### Nonzero capacity sets and dense subspaces in scales of Sobolev spaces

Mykola E. Dudkin, Volodymyr D. Koshmanenko

Methods Funct. Anal. Topology **20** (2014), no. 3, 213-218

We show that for a compact set $K\subset{\mathbb R}^n$ of nonzero $\alpha$-capacity, $C_\alpha(K)>0$, $\alpha\geq 1$, the subspace $\overset{\circ}{W}{^{\alpha,2}}(\Omega)$, $\Omega={\mathbb R}^n\setminus K$ in ${W}{^{\alpha,2}}({\mathbb R}^n)$ is dense in $W^{m,2}({\mathbb R}^n)$, $m\leq\alpha-1$, iff the $m$-capacity of $K$ is zero, $C_{m}(K)=0$.

### On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, G. Torbin

Methods Funct. Anal. Topology **17** (2011), no. 2, 97-111

We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

### The infinite direct products of probability measures and structural similarity

Methods Funct. Anal. Topology **17** (2011), no. 1, 20-28

We show that any similar structure measure on the segment $[0,1]$ is an image-measure of the appropriate constructed infinite direct product of discrete probability measures.

### Origination of the singular continuous spectrum in the conflict dynamical systems

Methods Funct. Anal. Topology **15** (2009), no. 1, 15-30

We study the spectral properties of the limiting measures in the conflict dynamical systems modeling the alternative interaction between opponents. It has been established that typical trajectories of such systems converge to the invariant mutually singular measures. We show that "almost always" the limiting measures are purely singular continuous. Besides we find the conditions under which the limiting measures belong to one of the spectral type: pure singular continuous, pure point, or pure absolutely continuous.

### Dynamics of discrete conflict interactions between non-annihilating opponents

Sergio Albeverio, Maksym Bodnarchuk, Volodymyr Koshmanenko

Methods Funct. Anal. Topology **11** (2005), no. 4, 309-319

### Dense subspaces in scales of Hilbert spaces

S. Albeverio, R. Bozhok, M. Dudkin, V. Koshmanenko

Methods Funct. Anal. Topology **11** (2005), no. 2, 156-169

### Fine structure of the singular continuous spectrum

Sergio Albeverio, Volodymyr Koshmanenko, Grygoriy Torbin

Methods Funct. Anal. Topology **9** (2003), no. 2, 101-119

### A variant of the inverse negative eigenvalues problem in singular perturbation theory

Methods Funct. Anal. Topology **8** (2002), no. 1, 49-69

### Generalized eigenfunctions under singular perturbation

S. Albeverio, V. Koshmanenko, K. A. Makarov

Methods Funct. Anal. Topology **5** (1999), no. 1, 13-28

### Lippmann-Schwinger equation in the singular perturbation theory

S. Albeverio, J. F. Brasche, V. Koshmanenko

Methods Funct. Anal. Topology **3** (1997), no. 1, 1-27