Blackmore D. L.
Ukr. Mat. Zh. - 2018. - 70, № 12. - pp. 1660-1695
We present a brief review of the original results obtained by the authors in the theory of Delsarte –Lions transmutations of multidimensional spectral differential ope rators based on the classical works by Yu. M. Berezansky, V. A. Marchenko, B. M. Levitan, and R. G. Newton, on the well-known L. D. Faddeev’s survey, the book by L. P. Nyzhnyk, and the generalized De-Rham – Hodge theory suggested by I. V. Skrypnik and developed by the authors for the differential-operator complexes. The operator structure of Delsarte – Lions transformations and the properties of their Volterra factorizations are analyzed in detail. In particular, we study the differential-geometric and topological structures of the spectral properties of the Delsarte – Lions transmutations within the framework of the generalized De-Rham – Hodge theory.
Ukr. Mat. Zh. - 2015. - 67, № 2. - pp. 147-162
The Lax integrability of a two-component polynomial Burgers-type dynamical system is analyzed by using a differential-algebraic approach. Its linear adjoint matrix Lax representation is constructed. A related recursive operator and an infinite hierarchy of nonlinear Lax integrable dynamical systems of the Burgers–Korteweg–de-Vries type are obtained by the gradient-holonomic technique. The corresponding Lax representations are presented.
Invariant measures for discrete dynamical systems and ergodic properties of generalized Boole-type transformations
Ukr. Mat. Zh. - 2013. - 65, № 1. - pp. 44-57
Invariant ergodic measures for generalized Boole-type transformations are studied using an invariant quasimeasure generating function approach based on special solutions for the Frobenius - Perron operator. New two-dimensional Boole-type transformations are introduced, and their invariant measures and ergodicity properties are analyzed.
Ukr. Mat. Zh. - 2004. - 56, № 1. - pp. 88-96
On the basis of the structure of Casimir elements associated with general Hopf algebras, we construct Liouville–Arnold integrable flows related to naturally induced Poisson structures on an arbitrary coalgebra and their deformations. Some interesting special cases, including coalgebra structures related to the oscillatory Heisenberg–Weil algebra and integrable Hamiltonian systems adjoint to them, are considered.