In this paper, we propose the concept of involutive direction ss a vector representation for the concept of involutive division proposed by Gerdt and hi co-workers. With this representation, most of the properties of involutive divisions such as Noetherity, Artinity, and constructivity, can be greatly simplified. A new algorithm to compute the involutive completion is also given. Based on the v...

In this paper we describe an efficient involutive algorithm for constructing Gröbner bases of polynomial ideals. The algorithm is based on the concept of involutive monomial division which restricts the conventional division in a certain way. In the presented algorithm a reduced Gröbner basis is the internally fixed subset of an involutive basis, and having computed the later, the former can be...

In this paper we consider parametric ideals and introduce a notion of comprehensive involutive system. This notion plays the same role in theory of involutive bases as the notion of comprehensive Gröbner system in theory of Gröbner bases. Given a parametric ideal, the space of parameters is decomposed into a finite set of cells. Each cell yields the corresponding involutive basis of the ideal f...

A selfadjoint involutive matrix J endows C with an indefinite inner product [·, ·] given by [x, y] := 〈Jx, y〉, x, y ∈ C. We present some inequalities of indefinite type involving the α−power mean and the chaotic order. These results are in the vein of those obtained by E. Kamei [6, 7].

In this paper we consider an algorithmic technique more general than that proposed by Zharkov and Blinkov for the involutive analysis of polynomial ideals. It is based on a new concept of involutive monomial division which is defined for a monomial set. Such a division provides for each monomial the self-consistent separation of the whole set of variables into two disjoint subsets. They are cal...

Binary symmetry constraints of the N -wave interaction equations in 1 + 1 and 2 + 1 dimensions are proposed to reduce the N -wave interaction equations into finite-dimensional Liouville integrable systems. A new involutive and functionally independent system of polynomial functions is generated from an arbitrary order square matrix Lax operator and used to show the Liouville integrability of th...

Recently the theory of Morita equivalence for involutive quantales and the notion of the interior tensor products of Hilbert modules over involutive quantales evolved considerably (see e.g. Paseka, 2002 and Paseka, 2001). The present paper is an attempt to put a part of this theory in a broader context of the bicategory of m-regular involutive quantales. For facts concerning quantales in genera...

In this paper we generalize the involutive methods and algorithms devised for polynomial ideals to differential ones generated by a finite set of linear differential polynomials in the differential polynomial ring over a zero characteristic differential field. Given a ranking of derivative terms and an involutive division, we formulate the involutivity conditions which form a basis of involutiv...

Gröbner Basis theory originated in the work of Buchberger [4] and is now considered to be one of the most important and useful areas of computer algebra. In 1993, Zharkov and Blinkov [13] proposed an alternative method of computing a commutative Gröbner Basis, namely the computation of an Involutive Basis. In the mid 1980’s, Mora showed [11] that Buchberger’s work could be generalised for nonco...