Quasivarieties of Metric Algebras

We introduce concepts of a continuous family of quasi-identities and of a continuous quasivariety. For continuous quasivarieties, a characterization theorem and an analog of the Birkhoff theorem on subdirect decomposition are proved. Also we point out the way of constructing examples of continuous quasivarieties and furnish the characterization of a relative congruence lattice of systems in the quasivarieties in question. Lastly, we re-prove the Hahn–Banach theorem on extension of a linear functional.     

Tri-Hamiltonian Structures of Egorov Systems of Hydrodynamic Type

We prove a simple condition under which the metric corresponding to a diagonalizable semi-Hamiltonian hydrodynamic type system belongs to the class of Egorov (potential) metrics. For Egorov diagonal hydrodynamic type systems satisfying natural semisimplicity and homogeneity conditions, we prove necessary and sufficient conditions under which the third structure is local or corresponds to a metric of constant curvature. The results are illustrated by some well-known physical examples of such systems.     

The metric of anti-Thorpe spaces, n = 4k, k > 1

We show that in dimension 8, a semiflat metric is flat and that in dimension (8k+4) higher than 8, a semiflat metric does not necessarily imply flat. This revised version was published online in June 2006 with corrections to the Cover Date.     

For cones in Hilbert spaces, regularity is equivalent to self-duality

The theorem stating that a cone in a Hilbert space is regular if and only if it is self-dual is proved and applied to obtain new proofs of earlier results.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 616–621, October, 1998.     

The Scalar Fields with Negative Kinetic Energy, Dark Matter and Dark Energy

The inhomogeneous cosmological model with generalized nonstatic Majumdar-Papapetrou metric is considered. The scalar field with negative kinetic energy and some usual matter sources of the gravitational field such as two-component nonlinear sigma model and perfect fluid are presented. Some exact solutions in these models are obtained and analyzed. In particular it is shown that the latent mass effect and effect of accelerating expansion (quintessence) of the Universe exist in these models. The 5-dimensional generalization of the model is presented, too.     

The rigidity theorem for harmonic maps from Berwald manifolds

In this paper, we can prove that any non‐degenerate strongly harmonic map ? from a compact Berwald manifold with nonnegative general Ricci curvature to a Landsberg manifold with non‐positive flag curvature must be totally geodesic, which generalizes the result of Eells and Sampson ([2]).     

Evolution equation for thermodynamic systems with correlated states

We have derived time evolution equations for thermodynamic systems with correlated states, considering the change in a probability density, entropy, and phase space metric at fixed phase‐space locations and obtained simple relations between these quantities for the separate systems. © 2014 Wiley Periodicals, Inc. Complexity 21: 162–165, 2015     

Halpern- and Mann-Type Algorithms for Fixed Points and Inclusion Problems on Hadamard Manifolds

In this article, we consider an inclusion problem which is defined by means of a sum of a single-valued vector field and a set-valued vector field defined on a Hadamard manifold. We propose Halpern-type and Mann-type algorithms for finding a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. Some particular cases of our problem and algorithm are also discussed. We study the convergence of the proposed algorithm to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of the inclusion problem defined on a Hadamard manifold. As applications of our results and algorithms, we derive the solution methods and their convergence results for the optimization problems, variational inequality problems and equilibrium problems in the setting of Hadamard manifolds.