Intersection theorems for systems of finite vector spaces

AbstractA theorem of Erdös, Ko and Rado states that if S is an n-element set and F is a family of k-element subsets of S, k⩽ 12n, such that no two members of F are disjoint, then …F… ⩽ (n - 1k - 1). In this paper we investigate the analogous problem for finite vector spaces.Let F be a family of k-dimensional subspaces of an n-dimensional vector space over a field of q elements such that members of F intersect pairwise non-trivially. Employing a method of Katona, we show that for n ⩾ 2k, …F… ⩽ (k/n) [nk]q. By a more detailed analysis, we obtain that for n ⩾ 2k + 1, …F… ⩽ [n - 1k - 1]q, which is a best possible bound. The argument employed is generalized to the problem of finding a bound on the size of F when its members have pairwise intersection dimension no smaller than r. Again best possible results are obtained for n ⩾ 2k + 2 and n ⩾ 2k + 1, q ⩾ 3. Application of these methods to the analogous subset problem leads to improvement on the Erdös-Ko-Rado bounds

Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of G\"artner et al.\ applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant).Comment: Minor edits, added two references; results unchange

A unified theory of cone metric spaces and its applications to the fixed point theory

In this paper we develop a unified theory for cone metric spaces over a solid vector space. As an application of the new theory we present full statements of the iterated contraction principle and the Banach contraction principle in cone metric spaces over a solid vector space.Comment: 51 page

Hodge cohomology of gravitational instantons

We study the space of L^2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincar\'e metrics on Q-rank 1 ends of locally symmetric spaces and on the complements of smooth divisors in K\"ahler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L^2 signature formula implied by our result is closely related to the one proved by Dai [dai] and more generally by Vaillant [Va], and identifies Dai's tau invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [Car] and the forthcoming paper of Cheeger and Dai [CD]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about L^2 harmonic forms in duality theories in string theory.Comment: 45 pages; corrected final version. To appear in Duke Math. Journa

Random fixed point theorems under mild continuity assumptions

In this paper, we study the existence of the random fixed points under mild continuity assumptions. The main theorems consider the almost lower semicontinuous operators defined on Frechet spaces and also operators having properties weaker than lower semicontinuity. Our results either extend or improve corresponding ones present in literature.Comment: 15 page