Minimal Involutive Bases

In this paper we present an algorithm for construction of minimal involutive polynomial bases which are Groebner bases of the special form. The most general involutive algorithms are based on the concept of involutive monomial division which leads to partition of variables into multiplicative and non-multiplicative. This partition gives thereby the self-consistent computational procedure for constructing an involutive basis by performing non-multiplicative prolongations and multiplicative reductions. Every specific involutive division generates a particular form of involutive computational procedure. In addition to three involutive divisions used by Thomas, Janet and Pommaret for analysis of partial differential equations we define two new ones. These two divisions, as well as Thomas division, do not depend on the order of variables. We prove noetherity, continuity and constructivity of the new divisions that provides correctness and termination of involutive algorithms for any finite set of input polynomials and any admissible monomial ordering. We show that, given an admissible monomial ordering, a monic minimal involutive basis is uniquely defined and thereby can be considered as canonical much like the reduced Groebner basis.Comment: 22 page

Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations

In this paper we present an algorithmic approach to the generation of fully conservative difference schemes for linear partial differential equations. The approach is based on enlargement of the equations in their integral conservation law form by extra integral relations between unknown functions and their derivatives, and on discretization of the obtained system. The structure of the discrete system depends on numerical approximation methods for the integrals occurring in the enlarged system. As a result of the discretization, a system of linear polynomial difference equations is derived for the unknown functions and their partial derivatives. A difference scheme is constructed by elimination of all the partial derivatives. The elimination can be achieved by selecting a proper elimination ranking and by computing a Gr\"obner basis of the linear difference ideal generated by the polynomials in the discrete system. For these purposes we use the difference form of Janet-like Gr\"obner bases and their implementation in Maple. As illustration of the described methods and algorithms, we construct a number of difference schemes for Burgers and Falkowich-Karman equations and discuss their numerical properties.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

On Computation of Groebner Bases for Linear Difference Systems

In this paper we present an algorithm for computing Groebner bases of linear ideals in a difference polynomial ring over a ground difference field. The input difference polynomials generating the ideal are also assumed to be linear. The algorithm is an adaptation to difference ideals of our polynomial algorithm based on Janet-like reductions.Comment: 5 pages, presented at ACAT-200

Church health and church growth in congregations of the Russian Church of Evangelical Christians

https://place.asburyseminary.edu/ecommonsatsdissertations/1280/thumbnail.jp

Effectiveness of introducing innovative solutions in machine-building as a factor of competitive immunity of the enterprise

The implementation of an innovation process at a machine-building enterprise provides a systematic approach to the formation of competitive immunity. The article deals with an assessment methodology which enables the necessary control over the innovation process implementation, minimizes costs for innovative activity, and helps to adjust the innovation process direction in the optimal way. The significance and attractiveness of this methodology is due to the fact that it can be applied for industrial enterprises of any economic branches regardless of the lifecycle phase of a product, enterprise and/or innovation process under study, because the list and number of specific indicators can be adjusted by the working group both before conducting the study and during the process of implementing the enterprise's innovative activity (at any stage of its implementation). The application of the competitive immunity formation strategy during the innovation processes implementation will help to create a model of industrial partnership development, which will ensure the competitive coexistence of machine-building enterprises. © 2020 Published under licence by IOP Publishing Ltd

Discrete Modified Nanostructural Wearproof Coatings TiN-Cu

Discrete modified nanostructural wearproof coatings Ti-N-Cu with crystallites size from 100 to 20 nanometers formed with ion-plasma vacuum-arc method. Copper amount in the received coatings made from 0 to 20 at. %, their hardness have considerably increased up to about 40-45 hPa in comparison with 20-22 hPa for Ti-N coverings. In this work, processes of structure and phase formation of Ti-N-Cu system coatings in a wide interval of copper concentration are investigated. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3531