Generalized Appell bases

In the context of Köthe spaces we study the bases related with the backward unilateral weighted shift operator, the so-called generalized derivation operator, extending known results for spaces of analytic functions. These bases are a subclass of Sheffer sequences called generalized Appell sequences and they are closely connected with the isomorphisms invariant by the weighted shift. We use methods of the non classical umbral calculi to give conditions for a generalized Appell sequence to be a basis.     

Generalized bases of finite groups

Archiv der Mathematik - Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset...     

Generalized Jones traces and Kazhdan-Lusztig bases

We develop some applications of certain algebraic and combinatorial conditions on the elements of Coxeter groups, such as elementary proofs of the positivity of certain structure constants for the associated Kazhdan-Lusztig basis. We also explore some consequences of the existence of a Jones-type trace on the Hecke algebra of a Coxeter group, such as simple procedures for computing leading terms of certain Kazhdan-Lusztig polynomials.     

Generalized biorthogonal bases and tridiagonalisation of matrices

Summary. In connection with the breakdown problem of the Lanczos algorithm a theory of generalized biorthogonal bases is developed. The connection between the generalized biorthogonal bases of Krylov chains and look-ahead Lanczos recursions is worked out in detail. It is shown how generalized biorthogonal bases with “antidiagonal blocks” can be constructed with small computational effort. Finally a special look-ahead Lanczos algorithm is derived which requires minimal computational effort and storage. Received September 21, 1995 / Revised version received August 12, 1996     

On balanced bases

It is proved that either a given balanced basis of the algebra (n + 1)M₁ M_n or the corresponding complementary basis is of rank n + 1. This result enables us to claim that the algebra (n + 1)M₁ M_n is balanced if and only if the matrix algebra M_n admits a WP-decomposition, i.e., a family of n + 1 subalgebras conjugate to the diagonal algebra and such that any two algebras in this family intersect orthogonally (with respect to the form tr XY) and their intersection is the trivial subalgebra. Thus, the problem of whether or not the algebra (n + 1)M₁ M_n is balanced is equivalent to the well-known Winnie-the-Pooh problem on the existence of an orthogonal decomposition of a simple Lie algebra of type A_n–1 into the sum of Cartan subalgebras.     

On sum-product bases

We investigate additive-multiplicative bases in . Let , s>2, and . It is proved that , provided min {|B| ^s/2|A|^(s−2)/2,|A| ^s/2|B|^(s−2)/2}>p ^s/2. This note is supported by “Balaton Program Project” and OTKA grants K 61908, K 67676.     

Self-dual normal bases

Every Galois extension of odd degree has a self-dual normal basis.     

Computing border bases

This paper presents several algorithms that compute border bases of a zero-dimensional ideal. The first relates to the FGLM algorithm as it uses a linear basis transformation. In particular, it is able to compute border bases that do not contain a reduced Gröbner basis. The second algorithm is based on a generic algorithm by Bernard Mourrain originally designed for computing an ideal basis that need not be a border basis. Our fully detailed algorithm computes a border basis of a zero-dimensional ideal from a given set of generators. To obtain concrete instructions we appeal to a degree-compatible term ordering σ and hence compute a border basis that contains the reduced σ-Gröbner basis. We show an example in which this computation actually has advantages over Buchberger's algorithm. Moreover, we formulate and prove two optimizations of the Border Basis Algorithm which reduce the dimensions of the linear algebra subproblems.     

On weak bases

In this paper, we give an affirmative answer to Tanaka's question: Is a space X with a σ-hereditarily closure-preserving weak base g-metrizable? [Proc. Aroc. Amer. Math. Soc. 112 (1991) 283] and a negative answer to S. Lin's question: Is every weak base of a topological space a k-network? [S. Lin, Generalized Metric Spaces and Maps, Science Press, 1995, Problem 1.6.20]. We also discuss mapping theorems on weak bases and the product of weak bases.     

Optimal normal bases

Let K L be a finite Galois extension of fields, of degree n. Let G be the Galois group, and let (<) _{be a normal basis for L over K. An argument due to Mullin, Onyszchuk, Vanstone and Wilson (Discrete Appl. Math. 22 (1988/89), 149–161) shows that the matrix that describes the map x x on this basis has at least 2n - 1 nonzero entries. If it contains exactly 2n - 1 nonzero entries, then the normal basis is said to be optimal. In the present paper we determine all optimal normal bases. In the case that K is finite our result confirms a conjecture that was made by Mullin et al. on the basis of a computer search.}