Characterization of Locally Finite Simple Groups of the Type <Superscript>3</Superscript><Emphasis Type="Italic">D</Emphasis><Subscript>4</Subscript> Over Fields of Odd Characteristic in the Class of Periodic Groups

We prove that a periodic group is locally finite, given that each finite subgroup of the group lies in a subgroup isomorphic to a finite simple group of Lie type ³D₄ over a field of odd characteristic.     

Interpretation of contradictory images by means of systems of linear inequalities

We consider the problem of interpretation of three-dimensional images from their flat projections up to the set of visible faces. For projections of convex polyhedra, we present an interpretation algorithm based on maximal feasible subsystems of a certain infeasible system of linear inequalities modeling the visibility requirement for faces. A number of model examples are given; in particular, the algorithm is applied to the interpretation of the Necker cube.     

Synchronization of relaxational self-oscillating systems: Synchronization in neural networks

A general problem of the synchronization and mutual synchronization of relaxational self-oscillating systems is formulated. A direct way of describing the synchronization of relaxational systems on the basis of Kronecker’s inequalities is proposed. The solution to the problem formulated by N. Wiener and A. Rosenbluth of forming a single rhythm in a system of coupled relaxational oscillators is described. Specific transient processes in the synchronization of relaxational systems are considered. Burst synchronization in neural networks and synchronization in distributed relaxational systems are also described.     

Groups of exponent 60 with prescribed orders of elements

We prove that a group which contains elements of orders 1, 2, 3, 4, 5 and does not contain elements of any other order is locally finite and isomorphic either to an alternating group of degree 6 or to an extension of a nontrivial elementary Abelian 2-group by an alternating group of degree 5. This article was written during my visit to the University of Manitoba, Canada, and supported by RFFR grant No. 99-01-00550. Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 329–346, May–June, 2000.     

Nonlinear concave spiral autowaves and their applications

Works devoted to concave spiral autowaves are reviewed. Concave spiral phase autowaves do not transfer energy in self-oscillating active media. Concave spiral autowaves that transfer energy are predicted and found in certain inhomogeneous active media. Ways of studying them mathematically are developed. Applications of spiral concave autowaves that transfer energy in different natural processes (in physics, chemistry, hydrodynamics, biology, and meteorology) are considered.     

Finite groups admitting an automorphism of prime order with central fixed points

In 1957, Higman showed that a Lie algebra admitting a fixed-point-free automorphism is nilpotent, and that an analogous result also holds for a finite soluble group. Two years later, Thompson proved that a finite group having a fixed-point-free automorphism of prime order is soluble, and consequently nilpotent. Generalizing that situation, a few years ago, Kharchenko set up a conjecture on the solubility of a Lie algebra L admitting an automorphism of prime order whose fixed points lie in the center of L. A similar conjecture applies also with finite groups. Here we affirm the latter for the case where the order p of an automorphism is equal to 2 and deny it for all p>3. Supported by RFFR grants Nos. 93-01-01501 and 96-01-01893. Translated fromAlgebra i Logika, Vol. 35, No. 6, pp. 699–708, November–December, 1996.     

A new characterization of A 5

Let G be a group and τ _e (G) the set of numbers of elements of G of the same order. In this paper, by τ _e (G), we give a new characterization of A ₅, where A ₅ is the alternating group of degree 5. We get the theorem following: Theorem. Let G be a group, ${G\cong A_5}$ if and only if τ _e (G) = τ _e (A ₅) = {1, 15, 20, 24}.