Distributivity,binary relations,and standard bases

In the author’s previous papers, the connection between generating syzygy modules by binary relations, the property of a commutative ring to be arithmetical (that is to have a distributive ideal lattice), and the use of the so-called S-polynomials in the standard basis theory were discussed. In this note, these connections are considered in a more general context. As an illustration of the usefulness of these considerations, a simple proof of some well-known fact from commutative algebra is given.     

Projective resolutions,Markushevich bases,and equivalent norms

Mathematical Notes -     

{\mathfrak {sl}}_3-Web bases,intermediate crystal bases and categorification

We give an explicit graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra \(K_S\) . In order to do this, we identify Kuperberg’s basis for the \({\mathfrak {sl}}_3\) -web space \(W_S\) with a version of Leclerc–Toffin’s intermediate crystal basis and we identify Brundan, Kleshchev and Wang’s degree of tableaux with the weight of flows on webs and the \(q\) -degree of foams. We use these observations to give a “foamy” version of Hu and Mathas graded cellular basis of the cyclotomic Hecke algebra which turns out to be a graded cellular basis of the \({\mathfrak {sl}}_3\) -web algebra. We restrict ourselves to the \({\mathfrak {sl}}_3\) case over \(\mathbb {C}\) here, but our approach should, up to the combinatorics of \({\mathfrak {sl}}_N\) -webs, work for all \(N>1\) or over \(\mathbb {Z}\) .     

Height bases, level bases, and the equality of the height and the level characteristics of an M-matrix

Let A be an M-matrix. We introduce the concepts of height basis, level basis, and height-level basis for the generalized nullspace of A. We explore the properties of such bases and of induced matrices. We use these results to prove some new conditions for the equality of the (spectral) height (Weyr) characteristic and the (graph theoretic) level characteristic of A, and to simplify proofs of known conditions. We also prove the existence of a Jordan basis for the generalized nullspace with all chains of maximal length nonnegative.     

Automatic continuity,bases, and radicals in metrizable algegbras

The automatic continuity of a linear multiplicative operator T: XY, where X and Y are real complete metrizable algebras and Y semi-simple, is proved. It is shown that a complex Frechét algebra with absolute orthogonal basis (x_i) (orthogonal in the sense that x_iX_j=0 if i j) is a commutative symmetric involution algebra. Hence, we are able to derive the well-known result that every multiplicative linear functional defined on such an algebra is continuous. The concept of an orthogonal Markushevich basis in a topological algebra is introduced and is applied to show that, given an arbitrary closed subspace Y of a separable Banach space X, a commutative multiplicative operation whose radical is Y may be introduced on X. A theorem demonstrating the automatic continuity of positive functionals is proved.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 8, pp. 1129–1132, August, 1992.     

Incidence matrices, geometrical bases, combinatorial prebases and matroids

For a relation A (C × D), where C,D are two finite sets, and an ordering σ of C we construct a matroid M(σ) on the set D. For the relation A with the incidence matrix  we also define a geometrical basis with respect to F, where F is a subset of the set of all circuits of the column matroid on Â. Geometrical bases are certain bases of this column matroid. We establish connections between the bases of matroids M(σ) and the geometrical bases of A with respect to F. These connections give a combinatorial way of constructing bases of the column matroid on  using a subset F of its circuits.

We also consider a matroid M and the incidence relation between what we call the extended circuits of M and the bases of M. Applying the technique above we obtain the matroids M(σ) on the set of bases of the matroid M. In case of the incidence relation between vertices and edges of a graph this technique yields a unique matroid, the usual matroid of the graph.

Some particular relations are considered: a class of relations with a certain property (the T-property) and the relation of inclusion of chambers in simplices in an affine point configuration.     

Weak compactness, transfinite Schauder bases and differentiability

We provide a characterization of the class of nonseparable Banach spaces that contain a nonseparable weakly compact set (respectively a relatively weakly compact transfinite basic sequence) in terms of differentiability properties of those spaces.     

Rotation invariant,Riesz bases of directional wavelets

This article addresses the issue of designing bases for $L^2({\mathbb{R}}^2)$ that are generated by translations, rotations and dilations of a single mother wavelet ψ. We show how this construction can be simplified by setting an odd number of directions and by choosing properly the phase of the Fourier transform of ψ. A large part of the article is devoted to the proof of theorems that give sufficient conditions for ψ to generate a Riesz sequence and a Riesz basis for $L^2({\mathbb{R}}^2)$. An example of Riesz sequence whose restriction to each scale is orthonormal is set. Theoretical results are confirmed by numerical experiments where a discrete directional wavelet transform is introduced.     

A note on fundamental,non-fundamental,and robust cycle bases

In many biological systems, robustness is achieved by redundant wiring, and reflected by the presence of cycles in the graphs connecting the systems’ components. When analyzing such graphs, cyclically robust cycle bases of are of interest since they can be used to generate all cycles of a given 2-connected graph by iteratively adding basis cycles. It is known that strictly fundamental (or Kirchhoff) bases, i.e., those that can be derived from a spanning tree, are not necessarily cyclically robust. Here we note that, conversely, cyclically robust bases (even of planar graphs) are not necessarily fundamental. Furthermore, we present a class of cubic graphs for which cyclically robust bases can be explicitly constructed.     

Subexpansions,superexpansions and uniqueness properties in non-integer bases

The β-expansions, i.e., greedy expansions with respect to non-integer bases q>1, were introduced by Réenyi and then investigated by many authors. Some years ago, Erdős, Horváth and Joó found the surprising fact that there exist infinitely many numbers 1<q<2 for which the β-expansion of 1 is the unique possible expansion with coefficients 0 or 1. Subsequently, the unique expansions were characterized in [9] and this characterization led to the determination (in [17]) of the smallest number q having this curious property. It is intimately related to the classical Thue-Morse sequence. Allouche and Cosnard recently proved that this q is transcendental. The purpose of this paper is to extend the previous results for expansions in arbitrary non-integer bases q>1. We also determine the smallest q having the corresponding uniqueness property in each case, and we prove that all of them are transcendental. We will also obtain some probably new properties of the Thue-Morse sequence. In the last section we answer a question concerning the existence of universal expansions, a notion introduced in [12]. This revised version was published online in June 2006 with corrections to the Cover Date.     

Gröbner bases, H-bases and interpolation

The paper is concerned with a construction for H-bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchberger's algorithm to construct these H-bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.

     

Vanishing theorems, boundedness and hyperbolicity over higher-dimensional bases

A vanishing theorem is proved for families over higher dimensional bases and applied to generalize some Shafarevich type statements to that setting.

     

Generalized markushevich bases

Conditions for the embedding of a Banach space as a dense subspace of a continuous function space are studied. Necessary and sufficient conditions for a Banach space to be isomorphic to aP ₁ space are found.     

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.