Equivalence Verification of Multipliers Based on Gr?bner Basis Method

2021 ◽  
Vol 10 (01) ◽  
pp. 343-350
Author(s):  
璇思 张
Keyword(s):  
Equivalence Verification ◽  
Bner Basis ◽  
Basis Method

Object Extent Determination for Algebraic Solid Models

1995 ◽  
Vol 117 (1) ◽  
pp. 20-26 ◽  
Author(s):  
M. A. Ganter ◽  
D. W. Storti
Keyword(s):  
Spatial Extent ◽  
Polynomial Functions ◽  
Profile Function ◽  
Variable Elimination ◽  
Property Evaluation ◽  
Solid Models ◽  
Bner Basis ◽  
Basis Method ◽  
Implicit Solid

This paper presents methods for determination of spatial extent of algebraic solid models. Algebraic solid models are a variation of implicit solid models defined by implicit polynomial functions with rational coefficients. Spatial extent information, which can be used to enhance the performance of visualization and property evaluation, includes silhouettes, outlines and profiles. Silhouettes are curves on the surface of the solid which separate portions of the surface which face towards or away from a given viewpoint. The projection of the silhouette onto the viewing plane gives the outline of the solid, and the bivariate implicit function which defines the area enclosed by the outline is called the profile. A method for outline determination is demonstrated using concepts from algebraic geometry including polar surfaces and variable elimination via the Gro¨bner basis method and/or resultants. Examples of outline generation are presented and a sample profile function is constructed.

scholarly journals The Investigation of Nonlinear Polynomial Control Systems

2021 ◽  
Vol 28 (3) ◽  
pp. 238-249
Author(s):  
Sergei Nikolaevich Chukanov ◽  
Ilya Stanislavovich Chukanov
Keyword(s):  
Nonlinear System ◽  
Control Systems ◽  
Polynomial System ◽  
Scalar Potential ◽  
Polynomial Equations ◽  
Wolfram Mathematica ◽  
Bner Bases ◽  
System Of Polynomial Equations ◽  
Bner Basis ◽  
Basis Method

The paper considers methods for estimating stability using Lyapunov functions, which are used for nonlinear polynomial control systems. The apparatus of the Gro¨bner basis method is used to assess the stability of a dynamical system. A description of the Gro¨bner basis method is given. To apply the method, the canonical relations of the nonlinear system are approximated by polynomials of the components of the state and control vectors. To calculate the Gro¨bner basis, the Buchberger algorithm is used, which is implemented in symbolic computation programs for solving systems of nonlinear polynomial equations. The use of the Gro¨bner basis for finding solutions of a nonlinear system of polynomial equations is considered, similar to the application of the Gauss method for solving a system of linear equations. The equilibrium states of a nonlinear polynomial system are determined as solutions of a nonlinear system of polynomial equations. An example of determining the equilibrium states of a nonlinear polynomial system using the Gro¨bner basis method is given. An example of finding the critical points of a nonlinear polynomial system using the Gro¨bner basis method and the Wolfram Mathematica application software is given. The Wolfram Mathematica program uses the function of determining the reduced Gro¨bner basis. The application of the Gro¨bner basis method for estimating the attraction domain of a nonlinear dynamic system with respect to the equilibrium point is considered. To determine the scalar potential, the vector field of the dynamic system is decomposed into gradient and vortex components. For the gradient component, the scalar potential and the Lyapunov function in polynomial form are determined by applying the homotopy operator. The use of Gro¨bner bases in the gradient method for finding the Lyapunov function of a nonlinear dynamical system is considered. The coordination of input-output signals of the system based on the construction of Gro¨bner bases is considered.

Institutionalism: Theory, Decision-making Basis, Method of Criticism

2009 ◽  
pp. 110-124 ◽  
Author(s):  
A. Moskovsky
Keyword(s):  
Decision Making ◽  
Institutional Economics ◽  
Economic Reforms ◽  
The State ◽  
Economic Approach ◽  
Main Attention ◽  
Technological Imperative ◽  
Logical Inconsistency ◽  
Basis Method

The author analyzes the state of institutional economics in contemporary Russia. It is characterized by arbitrary confusion of the ideas of «old», «new» and «mathematical» versions of institutionalism which results in logical inconsistency and even eclectics to be observed in the literature. The new and mathematical versions of institutionalism are shown to be based on legal, political and mathematical determinism tightly connected with the so-called «economic approach» (G. Becker). The main attention is paid to the discussion of theoretical and practical potential of the contemporary classical («old») institutionalism. The author focuses on its philosophical grounds and its technological imperative, the institution of science, the method of criticism, the opportunity of using classical institutionalist ideas as the ideology of economic reforms in Russia.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

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