Papers by Anatolij Prykarpatski
Proceedings of the International Geometry Center
We review main differential-algebraic structures \ lying in background of \ analytical constructi... more We review main differential-algebraic structures \ lying in background of \ analytical constructing multi-component Hamiltonian operators as derivatives on suitably constructed loop Lie algebras, generated by nonassociative noncommutative algebras. The related Balinsky-Novikov and \ Leibniz type algebraic structures are derived, a new nonassociative "Riemann" algebra is constructed, deeply related with infinite multi-component Riemann type integrable hierarchies. An approach, based on the classical Lie-Poisson structure on coadjoint orbits, closely related with those, analyzed in the present work and allowing effectively enough construction of Hamiltonian operators, is also briefly revisited. \ As the compatible Hamiltonian operators are constructed by means of suitable central extentions of the adjacent weak Lie algebras, generated by the right Leibniz and Riemann type nonassociative and noncommutative algebras, the problem of their description requires a detailed inves...
who had done so much for quantum statistical physics to become mature and so attractive.
Miskolc Mathematical Notes, 2006
A. The invariant ergodic measures for generalized Boole type transformatins are studied ma... more A. The invariant ergodic measures for generalized Boole type transformatins are studied making use of the invariant quasi-measure approach, based on some special solutions to the Frobenius-Perron operator.
The vortex structure of Langmuir turbulence in the interrupted magnetic z-pinch. Part 1 (in English)
Condensed Matter Physics, 2004
We analyze dynamical systems of general form possessing gradient (symmetric) and Hamiltonian (ant... more We analyze dynamical systems of general form possessing gradient (symmetric) and Hamiltonian (antisymmetric) flow parts. The relevance of such systems to self-organizing processes is discussed. Coherent structure formation and related gradient flows on matrix Grassmann type manifolds are considered. The corresponding graph model associated with the partition swap neighborhood problem is studied. The criterion for emerging gradient and Hamiltonian flows is established. As an example we consider nonlinear dynamics in a neuron network system described by a simulative vector field. A simple criterion was written in order to establish conditions for the formation of an oscillatory pattern in a model neural system under consideration.
Projection-Algebraic Scheme of Discrete Approximations for Linear and Nonlinear Differential Operator Equations in Banach Spaces
The solution set, realizability and convergence of the projection-algebraic method of discrete ap... more The solution set, realizability and convergence of the projection-algebraic method of discrete approximations for linear and nonlinear differential operator equations are studied. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)
We present new mathematical foundations of classical Maxwell–Lorentz electrodynamic models and re... more We present new mathematical foundations of classical Maxwell–Lorentz electrodynamic models and related charged particles interaction-radiation problems, and analyze the fundamental least action principles via canonical Lagrangian and Hamiltonian formalisms. The corresponding electrodynamic vacuum field theory aspects of the classical Maxwell–Lorentz theory are analyzed in detail. Electrodynamic models of charged point particle dynamics based on a Maxwell type vacuum field medium description are described, and new field theory concepts related to the mass particle paradigms are discussed. We also revisit and reanalyze the mathematical structure of the classical Lorentz force expression with respect to arbitrary inertial reference frames and present new interpretations of some classical special relativity theory relationships.
Arxiv preprint arXiv:1012.1024, 2010
A novel approach-based upon vertex operator representation-is devised to study the AKNS hierarchy... more A novel approach-based upon vertex operator representation-is devised to study the AKNS hierarchy. It is shown that this method reveals the remarkable properties of the AKNS hierarchy in relatively simple, rather natural and particularly effective ways. In addition, the connection of this vertex operator based approach with Lie-algebraic integrability schemes is analyzed and its relationship with τ-function representations is briefly discussed.
Journal of Nonlinear Mathematical Physics, 2004
We consider the general properties of the replicator dynamical system from the standpoint of its ... more We consider the general properties of the replicator dynamical system from the standpoint of its evolution and stability. Vector field analysis as well as spectral properties of such system has been studied. A Lyaponuv function for the investigation of the evolution of the system has been proposed. The generalization of replicator dynamics to the case of multi-agent systems is introduced. We propose a new mathematical model to describe the multi-agent interaction in complex system.
Central European Journal of Mathematics, 2007
The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie a... more The Hamiltonian representation for a hierarchy of Lax type equations on a dual space to the Lie algebra of integro-differential operators with matrix coefficients, extended by evolutions for eigenfunctions and adjoint eigenfunctions of the corresponding spectral problems, is obtained via some special Bäcklund transformation. The connection of this hierarchy with integrable by Lax twodimensional Davey-Stewartson type systems is studied.
A new Riemann type hydrodynamical hierarchy and its integrability analysis
The work is devoted to studying the vacuum structure, special relativity, electrodynamics of inte... more The work is devoted to studying the vacuum structure, special relativity, electrodynamics of interacting charged point particles and quantum mechanics, and is a continuation of [6, 7]. Based on the vacuum field theory no-geometry approach, the Lagrangian and Hamiltonian reformulation of some alternative classical electrodynamics models is devised. The Dirac type quantization procedure, based on the canonical Hamiltonian formulation, is developed for some alternative electrodynamics models. Within an approach developed a possibility of the combined description both of electrodynamics and gravity is analyzed.
This paper is dedicated to the memory of 95-th birthday and 10-th death anniversaries of the math... more This paper is dedicated to the memory of 95-th birthday and 10-th death anniversaries of the mathematics and physics luminary of the former century academician Nikolay Nikolayevich Bogoliubov Abstract. The structure properties of multidimensional Delsarte transmutation operators in parametirc functional spaces are studied by means of differential-geometric tools. It is shown that kernels of the corresponding integral operator expressions depend on the topological structure of related homological cycles in the coordinate space. As a natural realization of the construction presented we build pairs of Lax type commutive differential operator expressions related via a Darboux-Backlund transformation having a lot of applications in solition theory. Some results are also sketched concerning theory of Delsarte transmutation operators for affine polynomial pencils of multidimensional differential operators.
Condensed Matter Physics, 2013
The non-relativistic current algebra approach is analyzed subject to its application to studying ... more The non-relativistic current algebra approach is analyzed subject to its application to studying the distribution functions of many-particle systems at the temperature equilibrium and their stability properties. We show that the classical Bogolubov generating functional method is a very effective tool for constructing the irreducible current algebra representations and the corresponding different generalized measure expansions including collective variables transform. The effective Hamiltonian operator construction and its spectrum peculiarities subject to the stability of equilibrium many-particle systems are discussed.
Mathematics, 2015
We review new electrodynamics models of interacting charged point particles and related fundament... more We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field theory approach to the Lagrangian and Hamiltonian, the formulations of alternative classical electrodynamics models are analyzed in detail and their Dirac type quantization is suggested. Problems closely related to the radiation reaction force and electron mass inertia are analyzed. The validity of the Abraham-Lorentz electromagnetic electron mass origin hypothesis is argued. The related electromagnetic Dirac-Fock-Podolsky problem and symplectic properties of the Maxwell and Yang-Mills type dynamical systems are analyzed. The crucial importance of the remaining reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized.
Reports on Mathematical Physics, 2011
A gradient-holonomic approach for the Lax type integrability analysis of differentialdiscrete dyn... more A gradient-holonomic approach for the Lax type integrability analysis of differentialdiscrete dynamical systems is described. The asymptotical solutions to the related Lax equation are studied, the related gradient identity subject to its relationship to a suitable Lax type spectral problem is analyzed in detail. The integrability of the discrete nonlinear Schrödinger, Ragnisco-Tu and Burgers-Riemann type dynamical systems is treated, in particular, their conservation laws, compatible Poissonian structures and discrete Lax type spectral problems are obtained within the gradient-holonomic approach.
Nonlinearity, 2006
Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky e... more Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm.The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The two-and four-dimensional invariant reductions are studied in detail. The well defined regularization of the model is constructed and its Lax type integrability is discussed.
Journal of Physics A: Mathematical and Theoretical, 2009
Symplectic structures associated to connection forms on certain types of principal fiber bundles ... more Symplectic structures associated to connection forms on certain types of principal fiber bundles are constructed via analysis of reduced geometric structures on fibered manifolds invariant under naturally related symmetry groups.This approach is then applied to nonstandard Hamiltonian analysis of of dynamical systems of Maxwell and Yang-Mills type. A symplectic reduction theory of the classical Maxwell equations is formulated so as to naturally include the Lorentz condition (ensuring the existence of electromagnetic waves), thereby solving the well known Dirac-Fock-Podolsky problem. Symplectically reduced Poissonian structures and the related classical minimal interaction principle for the Yang-Mills equations are also considered.
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Papers by Anatolij Prykarpatski