Abstract. This paper will introduce noncommutative analogs of monomial symmetric functions and fundamental noncommutative symmetric functions. The expansion of ribbon Schur functions in both of these basis is nonnegative. With these functions at hand, one can derive a noncommutative Cauchy identity as well as study a noncommutative scalar product implied by Cauchy identity. This scalar product ...

In this paper we present the use of the ACL2 theorem prover to formalize and mechanically check a new proof of Dickson’s lemma about monomial sequences. Dickson’s lemma can be used to establish the termination of Büchberger’s algorithm to find the Gröbner basis of a polynomial ideal. This effort is related to a larger project which aims to develop a mechanically verified computer algebra system.

We show that for spaces with rational cohomology an exterior algebra on odd dimensional generators, one can define a torsion invariant which is a rational number. This may be interpreted as an absolute version of the multiplicative Euler characteristic associated to a rational homotopy equivalence. 0. Introduction. Reidemeister torsion of a finite complex X, with trivial action of the fundament...

Nowadays polynomial system solvers are involved in sophisticated computations in algebraic geometry as well as in practical engineering. The most popular algorithms are based on Gröbner bases, resultants, Macaulay matrices, or triangular decompositions. In all these algorithms, multivariate polynomials are expanded in a monomial basis, and the computations mainly reduce to linear algebra. The m...

The hull H(C) of a linear code C is defined by H(C) = C ∩ C⊥. A linear code with a complementary dual (LCD) is a linear code with H(C) = {0}. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enu...

This formula has been central in the development of path integral approaches to quantum theory, stochastic theory, and in quantum statistical mechanics [1, 2]. In particular, Suzuki has employed this to develop quantum statistical Monte Carlo methods [3, 4], general theory of path integrals with application to many-body theories and statistical physics [5], and more recently to mathematical phy...

Schur-Szegö composition of two polynomials of degree less or equal than a given positive integer n introduces an interesting semigroup structure on polynomial spaces and is one of the basic tools in the analytic theory of polynomials, see [4]. In the present paper we add several (apparently) new aspects to the previously known properties of this operation. Namely, we show how it interacts with ...

An efficient VLSI architecture implementation for barrel distortion correction in surveillance camera images is presented. The distortion correction model is based on least squares estimation method. To reduce the computing complexity, an odd-order polynomial to approximate the back-mapping expansion polynomial is used. By algebraic transformation, the approximated polynomial becomes a monomial...

We construct an explicit basis for the coordinate ring of the Bott-Samelson variety Zi associated to G = GL(n) and an arbitrary sequence of simple reflections i. Our basis is parametrized by certain standard tableaux and generalizes the Standard Monomial basis for Schubert varieties. In this paper, we prove the results announced in [LkMg] for the case of Type An−1 (the groups GL(n) and SL(n)). ...

This paper presents a new fast approach to improving stability in polynomial equation solving. Gröbner basis techniques for equation solving have been applied successfully to several geometric computer vision problems. However, in many cases these methods are plagued by numerical problems. An interesting approach to stabilising the computations is to study basis selection for the quotient space...