In this paper, we show that of all graphs of order n with matching number β, the graphs with maximal spectral radius are Kn if n = 2β or 2β + 1; if 2β + 2 ? n < 3β + 2; or if n = 3β + 2; if n > 3β + 2, where is the empty graph on t vertices. 相似文献
We improve – roughly by a factor 2 – the known bound on the multiplicity of the second eigenvalue of Schr?dinger operators
(i.e. Laplace plus potential) on closed surfaces. This gives four new topological types of surfaces for which Colin de Verdière's
conjecture relating the maximal multiplicity to the chromatic number of the surface is verified. The proof goes by defining
a space of "nodal splittings” of the surface, equipped with a double covering to which a Borsuk-Ulam type theorem is applied.
Received: 19 June 2001; revised version: 18 March 2002 /Published online: 17 June 2002 相似文献
The -th local cohomology module of a finitely generated graded module over a standard positively graded commutative Noetherian ring , with respect to the irrelevant ideal , is itself graded; all its graded components are finitely generated modules over , the component of of degree . It is known that the -th component of this local cohomology module is zero for all > 0$">. This paper is concerned with the asymptotic behaviour of as .
The smallest for which such study is interesting is the finiteness dimension of relative to , defined as the least integer for which is not finitely generated. Brodmann and Hellus have shown that is constant for all (that is, in their terminology, is asymptotically stable for ). The first main aim of this paper is to identify the ultimate constant value (under the mild assumption that is a homomorphic image of a regular ring): our answer is precisely the set of contractions to of certain relevant primes of whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology.
Brodmann and Hellus raised various questions about such asymptotic behaviour when f$">. They noted that Singh's study of a particular example (in which ) shows that need not be asymptotically stable for . The second main aim of this paper is to determine, for Singh's example, quite precisely for every integer , and, thereby, answer one of the questions raised by Brodmann and Hellus.
In a category with injective hulls and a cogenerator, the embeddings into injective hulls can never form a natural transformation,
unless all objects are injective. In particular, assigning to a field its algebraic closure, to a poset or Boolean algebra
its Mac-Neille completion, and to an R-module its injective envelope is not functorial, if one wants the respective embeddings to form a natural transformation.
Received January 21, 2000; accepted in final form August 10, 2001.
RID="h1"
RID="h2"
RID="h3"
ID="h1"The hospitality of York University is gratefully acknowledged by the first author.
ID="h2"Third author partially supported by the Grant Agency of the Czech Republic under Grant no. 201/99/0310, and the hospitality
of York University is also acknowledged.
ID="h3"Partial financial assistance by the Natural Sciences and Engineering Councel of Canada is acknowledged by the fourth
author. 相似文献
In the paper Geometric K-Theory for Lie Groups and Foliations, Baum and Connes conjecture in a remark following Corollary 2 of their famous Isomorphism conjecture that for a finitely generated group with torsion, the trace map tr: K0(C*)R maps K0(C*) onto the additive subgroup of Q generated by all rational numbers of the form 1/n where n is the order of a finite subgroup of . We construct a counterexample to this conjecture. 相似文献
Based on the Ben Artzi-Gohberg result concerning the equivalence between the invertibility of theL2-operator
and the exponential dichotomic evolution defined byA(t), the time-varying counterpart of the Redheffer theorem is considered under relaxed conditions. 相似文献