# Koshmanenko V. D.

### Limit distributions of conflict dynamical system with point spectra

Koshmanenko V. D., Voloshyna V. O.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 12. - pp. 1615-1624

We construct a model of conflict dynamical system whose limit states are associated with singular distributions. We prove that a criterion for appearance of a point spectrum in the limit distribution is the strategy of fixed priority. In all other cases, the limit distributions are pure singular continuous.

### Problem of optimal strategy in the models of conflict redistribution of the resource space

Koshmanenko V. D., Verigina I. V.

Ukr. Mat. Zh. - 2017. - 69, № 7. - pp. 905-911

The theory of conflict dynamical systems is applied to finding of the optimal strategy in the problem of redistribution of the resource space between two opponents. In the case of infinite fractal division of the space, we deduce an explicit formula for finding the Lebesgue measure of the occupied territory in terms of probability distributions. In particular, this formula gives the optimal strategy for the occupation of the whole territory. The necessary and sufficient condition for the parity distribution of the territory are presented.

### Hahn-Jordan decomposition as an equilibrium state of the conflict system

Koshmanenko V. D., Petrenko S. M.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 64-77

The notion of conflict system is introduced in terms of couples of probability measures. We construct several models of conflict systems and show that every trajectory with initial state given by a couple of measures $\mu, \nu$ converges to an equilibrium state specified by the normalized components $\mu_+, \nu_+$ of the classical Hahn – Jordan decomposition of the signed measure $\omega = \mu - \nu$.

### Quasipoint spectral measures in the theory of dynamical systems of conflict

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 187-199

In the framework of dynamical picture of interacting physical systems, the notion of a spectral measure with quasipoint spectrum is introduced. It is shown that, under conflict interaction with point measures, only quasipoint singularly continuous measures are admitted for the transformation into measures with purely point spectrum.

### Full measure of a set of singular continuous measures

Ukr. Mat. Zh. - 2009. - 61, № 1. - pp. 83-91

On the space of structurally similar measures, we construct a nontrivial measure **m** such that the subclass of all purely singular continuous measures is a set of full **m**-measure.

### Reconstruction of the spectral type of limiting distributions in dynamical conflict systems

Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 771–784

We establish the conditions of reconstruction of pure spectral types (pure point, pure absolutely continuous, or pure singularly continuous spectra) in the limiting distributions of dynamical systems with compositions of alternative conflict. In particular, it is shown that the point spectrum can be reconstructed starting from the states with pure singularly continuous spectra.

### Jacobi matrices associated with the inverse eigenvalue problem in the theory of singular perturbations of self-adjoint operators

Koshmanenko V. D., Tuhai H. V.

Ukr. Mat. Zh. - 2006. - 58, № 12. - pp. 1651–1662

We establish the relationship between the inverse eigenvalue problem and Jacobi matrices within the framework of the theory of singular perturbations of unbounded self-adjoint operators.

### Singular Perturbations of Self-Adjoint Operators Associated with Rigged Hilbert Spaces

Bozhok R. V., Koshmanenko V. D.

Ukr. Mat. Zh. - 2005. - 57, № 5. - pp. 622–632

Let *A* be an unbounded self-adjoint operator in a Hilbert separable space \(H_0\) with rigging \(H_ - \sqsupset H_0 \sqsupset H_ +\) such that \(D(A) = H_ +\) in the graph norm (here, \(D(A)\) is the domain of definition of *A*). Assume that \(H_ +\) is decomposed into the orthogonal sum \(H_ + = M \oplus N_ +\) so that the subspace \(M_ +\) is dense in \(H_0\). We construct and study a singularly perturbed operator *A* associated with a new rigging \(H_ - \sqsupset H_0 \sqsupset \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ +\), where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{H} _ + = M_ + = D(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{A} )\), and establish the relationship between the operators *A* and *A*.

### On the structure of resolvent of singularly perturbed operator solving the problem of eigenvalues

Koshmanenko V. D., Tuhai H. V.

Ukr. Mat. Zh. - 2004. - 56, № 9. - pp. 1292-1297

We investigate the structure of the resolvent of a singularly perturbed operator of finite rank that solves an eigenvalue problem.

### Invariant points of a dynamical system of conflict in the space of piecewise-uniformly distributed measures

Kharchenko N. V., Koshmanenko V. D.

Ukr. Mat. Zh. - 2004. - 56, № 7. - pp. 927–938

We prove a theorem on the existence and structure of invariant points of a dynamical system of conflict in the space of piecewise-uniformly distributed measures.

### On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

Dudkin M. Ye., Koshmanenko V. D.

Ukr. Mat. Zh. - 2003. - 55, № 9. - pp. 1269-1276

We discuss purely singular finite-rank perturbations of a self-adjoint operator *A* in a Hilbert space ℋ. The perturbed operators \(\tilde A\) are defined by the Krein resolvent formula \((\tilde A - z)^{ - 1} = (A - z)^{ - 1} + B_z \) , Im *z* ≠ 0, where *B* _{z} are finite-rank operators such that dom *B* _{z} ∩ dom *A* = |0}. For an arbitrary system of orthonormal vectors \(\{ \psi _i \} _{i = 1}^{n < \infty } \) satisfying the condition span |ψ_{ i }} ∩ dom *A* = |0} and an arbitrary collection of real numbers \({\lambda}_i \in {\mathbb{R}}^1\) , we construct an operator \(\tilde A\) that solves the eigenvalue problem \(\tilde A\psi _i = {\lambda}_i {\psi}_i , i = 1, \ldots ,n\) . We prove the uniqueness of \(\tilde A\) under the condition that rank *B* _{z} = *n*.

### Theorem on Conflict for a Pair of Stochastic Vectors

Ukr. Mat. Zh. - 2003. - 55, № 4. - pp. 555-560

We investigate a mathematical model of conflict with a discrete collection of positions.

### Regularized approximations of singular perturbations from the $H_2$-class

Ukr. Mat. Zh. - 2000. - 52, № 5. - pp. 626-637

For a sequence of singular perturbations belonging to the $H_1$-class and converging to a given singular perturbation from the $H_2$-class, we find a method of additive regularization that guarantees the strong resolvent convergence of perturbed operators.

### Singular perturbations of finite rank. Point spectrum

Koshmanenko V. D., Samoilenko O. V.

Ukr. Mat. Zh. - 1997. - 49, № 9. - pp. 1186–1194

We establish necessary and sufficient conditions for the appearance of an additional point spectrum under singular perturbations of finite rank.

### On characteristic properties of singular operators

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1484-1493

For a linear operator*S* in a Hilbert space ℋ, the relationship between the following properties is investigated: (i)*S* is singular (= nowhere closable), (ii) the set ker*S* is dense in ℋ, and (iii)D(*S*)∩ℛ(*S*)={0}.

### New aspects of Krein's extension theory

Brasche J. F., Koshmanenko V. D., Neidhardt H.

Ukr. Mat. Zh. - 1994. - 46, № 1-2. - pp. 37–54

The extension problem for closed symmetric operators with a gap is studied. A new kind of parametrization of extensions (the so-called Krein model) is developed. The notion of a singular operator plays the key role in our approach. We give the explicit description of extensions and establish the spectral properties of extended operators.

### On the definition of singular bilinear forms and singular linear operators

Karwowski W., Koshmanenko V. D.

Ukr. Mat. Zh. - 1993. - 45, № 8. - pp. 1084–1089

We analyze various definitions of the concepts of a singular operator and a singular bilinear form and propose the most suitable ones. We also study the simplest properties of these objects.

### Toward the rank-one singular perturbation theory of self-adjoint operators

Ukr. Mat. Zh. - 1991. - 43, № 11. - pp. 1559–1566

### Generalized asymptotic constants

Koshmanenko V. D., Wollenberg M.

Ukr. Mat. Zh. - 1986. - 38, № 4. - pp. 416–421

### Construction of wave operators with respect to a perturbed semigroup

Ukr. Mat. Zh. - 1985. - 37, № 5. - pp. 634–636

### Scattering problem in the theory of singular perturbations of self-adjoint operators

Koshmanenko V. D., Neidhardt H., Wollenberg M.

Ukr. Mat. Zh. - 1984. - 36, № 1. - pp. 7 - 12

### Structure of the general solution of the inverse scattering problem in an abstract formulation

Ukr. Mat. Zh. - 1980. - 32, № 4. - pp. 499–506

### The unitarity of the S-operator in Haag-Ruelle scattering theory

Ukr. Mat. Zh. - 1974. - 26, № 4. - pp. 552–557

### Example of a locally commutative operator field

Ukr. Mat. Zh. - 1973. - 25, № 3. - pp. 379—382

### Weakened axiom of local commutativity

Ukr. Mat. Zh. - 1970. - 22, № 2. - pp. 236–242

### On the normal form of operators

Ukr. Mat. Zh. - 1969. - 21, № 2. - pp. 210–219