Let π be a set of primes. Recall that a group G is said to be a residually finite π-group if for every nonidentity element a of G there exists a homomorphism of the group G onto some finite π-group such that the image of the element a differs from 1. A group G will be said to be a virtually residually finite π-group if it contains a finite index subgroup which is a residually finite π-group. Recall that an element g in G is said to be π-radicable if g is an m-th power of an element of G for every positive π-number m. Let N be a nilpotent group and let all power subgroups in N are finitely separable. It is proved that N is a residually finite π-group if and only if N has no nonidentity π-radicable elements. Suppose now that π does not coincide with the set Π of all primes. Let π 0 be the complement of π in the set Π. And let T be a π 0 component of N i.e. T be a set of all elements of N whose orders are finite π 0 -numbers. We prove that the following three statements are equivalent: (1) the group N is a virtually residually finite π-group; (2) the subgroup T is finite and quotient group N/T is a residually finite π-group; (3) the subgroup T is finite and T coincides with the set of all π-radicable elements of N.

In the article is provided the review of tools used in new type object DBMS for increasing the efficiency of access to data. Described object DIM DBMS features based on the use of the classes of objects relations as objects sets (inheritance, inclusion, interaction and history) and objects relations (inheritance, internal inheritance, inclusion, internal inclusion, interaction and history). The description of subject domain is entered by means of an object and dynamic data model (OD-model), and DIM DBMS completeness for any OD-model is justified. ODQL object query language allowing to combine the exact description complexity with the simplicity of use due to two query levels introduction is described. For the elucidation of the most effective way of the appeal to DIM DBMS the study of various query technologies for this environment is conducted, and mechanisms for users work with it are developed and realized. For this purpose a software complex necessary for work with DIM DBMS is developed. The description of the "DIM Navigator" main software development concepts, necessary for data manipulation opportunity in the available DB by means of the graphic interface is provided. Software development "The Generator of ODQL-queries" is considered which is necessary for simplification of query creation to DIM DBMS, needlessly for the user to know the syntax of a modern query language. Problems of converting data from the existing DBMS in DIM DBMS are considered.

The article is published in the author’s wording

Push-down automata with independent counters (PDACs) combine the power of PDAs and Petri Nets. They were developed in [21, 15], as a tool of recognition of languages generated by Categorial Dependency Grammars (CDGs). CDGs are classical categorial grammars extended by oriented polarized valencies. They express both projective and non-projective dependencies between the words of a sentence. PDAC is a usual PDA equipped with a finite number of counters. The independence of counters means that their state has no effect on the choice of an automaton move. In the first part of the paper we compare some variants of PDACs and prove the equivalence of two variants of PDAs with independent counters: without syntactic and without semantic ε-loops. Some connections between PDAC-languages and Petri Net languages are noticed. Then we show that PDACs are equivalent to stack+bag push-down automata (SBPA) independently introduced by Søgaard and that ε-acyclic SBPAs recognize exactly CDG-languages. Multimodal Categorial Dependency Grammars (mmCDGs) were introduced in [4] as an extension of GDGs that allows control of some intersections of dependencies. The class of mmCDG-languages is rich enough and has good closure properties, that it forms AFL. In the second part of the paper we extend PDACs and introduce push-down automata with stacks of independent counters (PDASC). PDASCs extend PDACs twofold: (i) each counter is a stack of integers and (ii) there is a restriction function which allows to diminish a head of a counter only if the heads of all dependent counters are zeros. Our main result says that these PDASCs accept exactly the class of mmCDG- languages. The article is published in the author’s wording.

The solution stability of an initial boundary problem for a linear hybrid system of differential equations, which models the rotation of a rigid body with two elastic rods located in the same plane is studied in the paper. To an axis passing through the mass center of the rigid body perpendicularly to the rods location plane is applied the stabilizing moment proportional to the angle of the system rotation, derivative of the angle, integral of the angle. The external moment provides a feedback. A method of studying the behavior of solutions of the initial boundary problem is proposed. This method allows to exclude from the hybrid system of differential equations partial differential equations, which describe the dynamics of distributed elements of a mechanical system. It allows us to build one equation for an angle of the system rotation. Its characteristic equation defines the stability of solutions of all the system. In the space of feedback-coefficients the areas that provide the asymptotic stability of solutions of the initial boundary problem are built up.

A linear projective ind-variety X is called 1-connected if any two points on it can be connected by a chain of lines l1, l2, ..., lk in X, such that li intersects li+1. A linear projective ind-variety X is called 2-connected if any point of X lies on a projective line in X and for any two lines l and l 0 in X there is a chain of lines l = l1, l2, ..., lk = l 0 , such that any pair (li , li+1) is contained in a projective plane P 2 in X. In this work we study an ind-variety X that is a complete intersection in the linear ind-Grassmannian G = lim−→G(km, nm). By definition, X is an intersection of G with a finite number of ind-hypersufaces Yi = lim−→Yi,m, m ≥ 1, of fixed degrees di , i = 1, ..., l, in the space P∞, in which the ind-Grassmannian G is embedded by Pl¨ucker. One can deduce from work [17] that X is 1-connected. Generalising this result we prove that X is 2-connected. We deduce from this property that any vector bundle E of finite rank on X is uniform, i. e. the restriction of E to all projective lines in X has the same splitting type. The motiavtion of this work is to extend theorems of Barth - Van de Ven - Tjurin - Sato type to complete intersections of finite codimension in ind-Grassmannians.

A finite group G with proper subgroups A and B has triple factorization G = ABA if every element g of G can be represented as g = aba0 , where a and a 0 are from A and b is from B. Such a triple factorization may be sometimes degenerate to AB-factorization. The task of finding triple factorizations for a group is fundamental and can be used for understanding the group structure. For instance, every simple finite group of Lie type has a natural factorization of such a type. Besides, the triple factorization is widely used in the study of graphs, geometries and varieties. The goal of this article is to find triple factorizations for sporadic groups of rank 3. We have proved the existence theorem of ABA-factorization for sporadic simple groups McL and F i22. There exist two rank 3 permutation representations of F i22. We have proved that ABA-factorizations exist in both cases.

In this paper, a principal architecture of common purpose CPU and its main components are discussed, CPUs evolution is considered and drawbacks that prevent future CPU development are mentioned. Further, solutions proposed so far are addressed and a new CPU architecture is introduced. The proposed architecture is based on wireless cache access that enables a reliable interaction between cores in multicore CPUs using terahertz band, 0.1-10THz. The presented architecture addresses the scalability problem of existing processors and may potentially allow to scale them to tens of cores. As in-depth analysis of the applicability of the suggested architecture requires accurate prediction of traffic in current and next generations of processors, we consider a set of approaches for traffic estimation in modern CPUs discussing their benefits and drawbacks. The authors identify traffic measurements by using existing software tools as the most promising approach for traffic estimation, and they use Intel Performance Counter Monitor for this purpose. Three types of CPU loads are considered including two artificial tests and background system load. For each load type the amount of data transmitted through the L2-L3 interface is reported for various input parameters including the number of active cores and their dependences on the number of cores and operational frequency.

We propose a new method of client-side data caching for relational databases with a central server and distant clients. Data are loaded into the client cache based on queries executed on the server. Every query has the corresponding DB table – the result of the query execution. These queries have a special form called "universal relational query" based on three fundamental Relational Algebra operations: selection, projection and natural join. We have to mention that such a form is the closest one to the natural language and the majority of database search queries can be expressed in this way. Besides, this form allows us to analyze query correctness by checking lossless join property. A subsequent query may be executed in a client’s local cache if we can determine that the query result is entirely contained in the cache. For this we compare truth spaces of the logical restrictions in a new user’s query and the results of the queries execution in the cache. Such a comparison can be performed analytically , without need in additional Database queries. This method may be used to define lacking data in the cache and execute the query on the server only for these data. To do this the analytical approach is also used, what distinguishes our paper from the existing technologies. We propose four theorems for testing the required conditions. The first and the third theorems conditions allow us to define the existence of required data in cache. The second and the fourth theorems state conditions to execute queries with cache only. The problem of cache data actualizations is not discussed in this paper. However, it can be solved by cataloging queries on the server and their serving by triggers in background mode. The article is published in the author’s wording.

This paper is devoted to justifying the possibility of DBMS DIM usage and its interaction mechanism as an algorithmically complete implementation of an objectdynamic model. An extension for a static OD-model by including sets of algorithmic procedures which modify values of object attributes and also create, remove and modify objects themselves is considered. To ensure the possibility of modifying DIM DB data in a way equivalent to OD-model modifications, interaction and history relations between DIM objects are considered. To minimize the dependence from concrete language constructions, which describe OD-model algorithmic procedures, the reduction to the universal form – Turing machine is performed. A way to create a Turing machine equivalent to OD.MT in terms of DIM, where a special set of PL/ODQL procedures is used as a control unit and a functional table is proposed. Later, a mechanism to form a memory tape of such DIM.MT by encoding information about DIM object, and their subsequent decoding back to DIM objects is described. The process of work of such a machine is modelled by using an endless cycle of executing some PL/ODQL procedures of reading and writing objects from / to the memory tape. Basing on the earlier proved theorem about the static completeness of data representation in DIM, at the end of the paper the proof on the completeness of representation of the Objects attributes values dynamics is considered. The article is published in the author’s wording.

For the routing problem of tool permutations under the thermal cutting of parts from sheet material realized on CNC machines, questions connected with constructing precise (optimal) and heuristic algorithms used on the stage of mathematical simulation of route elements under sequential megalopolises circuit are investigated. Cutting points and points of tool cut-off are items (cities) of the above-mentioned megalopolises. In each megalopolis, interior works are provided. These works are connected with motion to the equidistant curve of the cut contour of a part from the cutting point and (with cutting completed) with motion from the equidistant curve to the tool cut-off (we keep in mind a working run). The problem about the time-optimal process of cutting which is a special variant of the generalized courier problem is investigated (the problem of the routing on the megalopolises with precedence conditions). An optimal procedure based on the dynamic programming and an effective heuristic algorithm realized on a multicore computer are proposed. A dynamic programming based procedure uses a special extension of the main problem. This extension provides the replacement of admissibility by precedence with the admissibility by deletion (from the list of tasks). Precedence conditions are used for decreasing computational complexity: it excludes the building of the whole array of the Bellman function values (this function is replaced by the layers system).

We study a problem about the number of lattice plane tilings by the given area centrosymmetrical polyominoes. A polyomino is a connected plane geomatric figure formed by joiining a finite number of unit squares edge to edge. At present, various combinatorial enumeration problems connected to the polyomino are actively studied. There are some interesting problems on enuneration of various classes of polyominoes and enumeration of tilings of finite regions or a plane by polyominoes. In particular, the tiling is a lattice tiling if each tile can be mapped to any other tile by a translation which maps the whole tiling to itself. Earlier we proved that, for the number T(n) of a lattice plane tilings by polyominoes of an area n, holds the inequalities 2n−3 + 2[ n−3 2 ] ≤ T(n) ≤ C(n + 1)3 (2, 7)n+1 . In the present work we prove a similar estimate for the number of lattice tilings with an additional central symmetry. Let Tc(n) be a number of lattice plane tilings by a given area centrosymmetrical polyominoes such that its translation lattice is a sublattice of Z 2 . It is proved that C1( √ 2)n ≤ Tc(n) ≤ C2n 2 ( √ 2.68)n . In the proof of a lower bound we give an explicit construction of required lattice plane tilings. The proof of an upper bound is based on a criterion of the existence of lattice plane tiling by polyominoes, and on the theory of self-avoiding walks on a square lattice.

We considered the problem of density wave propagation in a logistic equation with delay and diffusion (Fisher–Kolmogorov equation with delay). It was constructed a Ginzburg–Landau equation in order to study the qualitative behavior of the solution near the equilibrium state. The numerical analysis of wave propagation shows that for a sufficiently small delay this equation has a solution similar to the solution of a classical Fisher–Kolmogorov equation. The delay increasing leads to existence of the oscillatory component in spatial distribution of solutions. A further increase of delay leads to the destruction of the traveling wave. That is expressed in the fact that undamped spatio-temporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the delay is sufficiently large we observe intensive spatio-temporal fluctuations in the whole area of wave propagation.