2011 Summary of Scientific Activities

Laboratory "Algebra and Topology"

The preradicals of special type (standard preradicals) in module categories have been studied and using them the four new operations has been defined in the lattice of submodules of an arbitrary module. The basic properties of these operations have been shown, as well as some relations between them and the lattice operations of the lattice of submodules (Dr. habil. A. Kashu).

The necessary and sufficient conditions of the functional completenes of the systems of formulas of the chain extensions of dual intuitionistic logic have been established (Dr. habil. M. Ratsiu).

It has been proved that the provability-intuitionistic logic is not finite approximable relative to model-completeness (PhD O. Izbash).

Six classes of quasigroups defined by the sets of their parastrophies have been established. These classes have been characterized by means of four identities with two variables. The quasigroups with six different parastrophies (DC-quasigroups) have been studied. A criterion when a quasigroup is a DC-quasigroup (a DC-T-quasigroup, a DC-IP-quasigroup) has been given. The existence of DC-T-quasigroups of any order n>6 has been proved (PhD G. Belyavskaya, T. Popovich).

Some free commutative radical rings have been constructed (Dr. habil. Iu. Ryabukhin).

The necessary and sufficient conditions for existence of an orthogonal complement for a groupoid have been determined and the number of orthogonal complements of the groupoids have been found. Pairs of orthogonal groupoids with some "good" properties necessary for combinatorial applications have been constructed (PhD V. Izbash).

It has been proved that a quasigroup is A-nuclear if and only if it is group isotope; a quasigroup is A-central if and only if it is abelian group isotope. Conditions of the coincidence and normality of nuclei in (α, β, γ)-inverse quasigroups have been given. Nuclei of loops that are inverse to a fixed loop have been researched (Dr. habil. V. Shcherbacov).

Properties of transformation of loop transversals, in a case when the operations obtained from these transversals are isomorphic (isotopic), have been described (PhD E. Kuznetsov).

Structure theorems on isotopes of operations constructed from loop transversals have been proved (S. Botnari).

The number of one-point extentions of a topology on a finite set has been estimated (Dr. habil. V. Arnautov).

The topological rings with no more that two nontrivial closed ideals have been described, in terms of ideal extentions of topological rings (PhD V. Popa). The structure of some types of locally compact abelian groups with the property that their rings of continuous endomorphisms are Zorn rings has been obtained (S. Cruglea, PhD V. Popa). Also, the structure of densely divisible, tortion-free, locally compact abelian groups which are + - complemented or ∩ - complemented has been described (Iu. Jardan, PhD V. Popa).

Inner geometry of cusps has been investigated for hyperbolic manyfolds of different dimensions. Methods of construction of hyperbolic manyfolds with cusps over different Euclidean space forms have been analyzed (PhD F. Damian). Monohedral tilings and the known dihedral tilings have been analyzed in terms of transitivity classes of cells (PhD E. Zamorzaeva).

Metric methods of construction of three-dimensional polyhedra have been obtained (PhD I. Gutsul).

Laboratory "Differential Equations"

Using affine invariant polynomials a complete classification of all finite weak singularities or the family of quadratic systems of differential equations has been done. Necessary and sufficient conditions for the coexistence of different types of weak singularities (focus, center, saddle) have been determined (Dr. habil. N. Vulpe).

The basic properties of cubic systems of differential equations with invariant lines have been established and in the case when the infinite line is filled up with singularities, the possible configurations of these lines have been constructed (Dr. habil. A. Suba).

The theory of normal functions on infinite dimensional complex Banach manifolds has been constructed. A generalization of the classical Lindelof-Ghering-Lohwater theorem for bounded domains in C^n with C^2-smooth boundary has been obtained (Dr. habil. P. Dovbush).

Properties of solutions of the Pfaff equations associated with the systems of first order differential equations have been studied. Geometric properties of the Euler and Navier-Stokes equations have been studied (PhD V. Dryuma).

Invariant conditions for the existence of a center at the origin of coordinates for the class of planar cubic systems in the case when the infinite line is filled up with singularities have been established (PhD Iu. Calin, PhD V. Baltag).

A minimal polynomial basis of the center affine comitants (up to degree 14) for the planar differential systems with homogeneous polynomials of degree four on the right-hand sides have been constructed (PhD Iu. Calin, S. Ciubotaru).

The center affine invariant conditions for the existence of an integrating factor Lie of the family of planar quadratic differential systems have been determined. For the classes of planar differential systems s2(1,3) and s2(0,1,2) integrating factors Lie have been constructed (Dr. habil. M. Popa, V. Orlov).

The Hilbert series of the algebra of unimodular comitants for planar differential systems with nonlinear homogeneities of degree five have been constructed. The connection between Hilbert series of these algebras for planar differential systems with nonlinear homogeneities of odd degree have been established (Dr. habil. M. Popa, V. Pricop).

The planar cubic differential systems possessing six invariant lines with six distinct slopes have been determined, and the qualitative study of these systems has been done (PhD V. Putuntica).

Laboratory "Mathematical Modeling"

The existence of optimal stationary strategies for discrete-time stochastic optimal control problems and Markov decision problems has been proved. Some polynomial algorithms for finding optimal stationary strategies for these problems have been developed and have been grounded (Dr. habil. D. Lozovanu).

The problem for the transition density of transport random process in Rm space of arbitrary dimension m≥2 has been solved completely. The result was published by Dr. habil. A. Kolesnik and M. Pinsky in international journal with impact factor: Journal of Statistical Physics, 2011, vol. 142, pp. 828-846.

A two-dimensional mathematical model has been developed and basic relations for the problem of interaction between soil and elasto-plastic shell filled with liquids under explosive loading have been established. The developed software allows to obtain a detailed picture of the dynamic loading process (Dr. habil. B. Rybakin, PhD G. Secrieru).

The problem of endogenous economic growth which includes research and development sector under the influence of a stochastic factor has been solved (PhD E. Naval).

Laboratory "Programming Systems"

Obligatory hybrid networks of evolutionary processors (OHNEP) with insertion/deletion operations (without substitution) were investigated. These networks are not computationally complete, but the completeness can be achieved by changing the notion of obligatory operation. For very simple evolutionary processors with a single insertion/deletion operation for each node, an original method of building a computationally complete OHNEP with 182 nodes was proposed (PhD A. Alhazov, Dr. habil. Iu. Rogojin).

Laboratory "Information Systems"

An approach for evaluation of medical staff based on expert precedents has been developed and tested (PhD O. Burlaca, O. Popcova, Iu. Secrieru).

Knowledge, decision rules and graphic contents concerning ultrasound examination of biliary tree have been acquired, formalized and stored (Dr. habil. C. Gaindric, Dr. habil. S. Cojocaru, PhD L. Burtseva, PhD G. Magariu).