32,737 research outputs found
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
The Decomposition Theorem and the topology of algebraic maps
We give a motivated introduction to the theory of perverse sheaves,
culminating in the Decomposition Theorem of Beilinson, Bernstein, Deligne and
Gabber. A goal of this survey is to show how the theory develops naturally from
classical constructions used in the study of topological properties of
algebraic varieties. While most proofs are omitted, we discuss several
approaches to the Decomposition Theorem, indicate some important applications
and examples.Comment: 117 pages. New title. Major structure changes. Final version of a
survey to appear in the Bulletin of the AM
The Development of Intersection Homology Theory
This historical introduction is in two parts. The first is reprinted with
permission from ``A century of mathematics in America, Part II,'' Hist. Math.,
2, Amer. Math. Soc., 1989, pp.543-585. Virtually no change has been made to the
original text. In particular, Section 8 is followed by the original list of
references. However, the text has been supplemented by a series of endnotes,
collected in the new Section 9 and followed by a second list of references. If
a citation is made to the first list, then its reference number is simply
enclosed in brackets -- for example, [36]. However, if a citation is made to
the second list, then its number is followed by an `S' -- for example, [36S].
Further, if a subject in the reprint is elaborated on in an endnote, then the
subject is flagged in the margin by the number of the corresponding endnote,
and the endnote includes in its heading, between parentheses, the page number
or numbers on which the subject appears in the reprint below. Finally, all
cross-references appear as hypertext links in the dvi and pdf copies.Comment: 58 pages, hypertext links added; appeared in Part 3 of the special
issue of Pure and Applied Mathematics Quarterly in honor of Robert
MacPherson. However, the flags in the margin were unfortunately (and
inexplicably) omitted from the published versio
Observation and inverse problems in coupled cell networks
A coupled cell network is a model for many situations such as food webs in
ecosystems, cellular metabolism, economical networks... It consists in a
directed graph , each node (or cell) representing an agent of the network
and each directed arrow representing which agent acts on which one. It yields a
system of differential equations , where the component
of depends only on the cells for which the arrow
exists in . In this paper, we investigate the observation problems in
coupled cell networks: can one deduce the behaviour of the whole network
(oscillations, stabilisation etc.) by observing only one of the cells? We show
that the natural observation properties holds for almost all the interactions
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