Supersaturation For Ramsey-Turán Problems

For an l-graph , the Turán number is the maximum number of edges in an n-vertex l-graph containing no copy of . The limit is known to exist [8]. The Ramsey–Turán density is defined similarly to except that we restrict to only those with independence number o(n). A result of Erdős and Sós [3] states that as long as for every edge E of there is another edge E′of for which |E∩E′|≥2. Therefore a natural question is whether there exists for which . Another variant proposed in [3] requires the stronger condition that every set of vertices of of size at least εn (0<ε<1) has density bounded below by some threshold. By definition, for every . However, even is not known for very many l-graphs when l>2. We prove the existence of a phenomenon similar to supersaturation for Turán problems for hypergraphs. As a consequence, we construct, for each l≥3, infinitely many l-graphs for which . We also prove that the 3-graph with triples 12a, 12b, 12c, 13a, 13b, 13c, 23a, 23b, 23c, abc, satisfies . The existence of a hypergraph satisfying was conjectured by Erdős and Sós [3], proved by Frankl and R?dl [6], and later by Sidorenko [14]. Our short proof is based on different ideas and is simpler than these earlier proofs. * Research supported in part by the National Science Foundation under grants DMS-9970325 and DMS-0400812, and an Alfred P. Sloan Research Fellowship. † Research supported in part by the National Science Foundation under grants DMS-0071261 and DMS-0300529.     

Discrepancy After Adding A Single Set

We show that the hereditary discrepancy of a hypergraph on n points increases by a factor of at most O(log n) when one adds a new edge to .     

More on Bounding Introspection in Modal Nonmonotonic Logics

Abstract By we denote the set of all propositional formulas. Let be the set of all clauses. Define . In Sec. 2 of this paper we prove that for normal modal logics , the notions of -expansions and -expansions coincide. In Sec. 3, we prove that if I consists of default clauses then the notions of -expansions for I and -expansions for I coincide. To this end, we first show, in Sec. 3, that the notion of -expansions for I is the same as that of -expansions for I. The project is supported by NSFC     

Proof Of A Conjecture Of Erdős On Triangles In Set-Systems

A triangle is a family of three sets A,B,C such that A∩B, B∩C, C∩A are each nonempty, and . Let be a family of r-element subsets of an n-element set, containing no triangle. Our main result implies that for r ≥ 3 and n ≥ 3r/2, we have This settles a longstanding conjecture of Erdős [7], by improving on earlier results of Bermond, Chvátal, Frankl, and Füredi. We also show that equality holds if and only if consists of all r-element subsets containing a fixed element. Analogous results are obtained for nonuniform families.     

Integral Equations on Function Spaces and Dichotomy on the Real Line

The purpose of this paper is to give new and general characterizations for uniform dichotomy and uniform exponential dichotomy of evolution families on the real line. We consider two general classes denoted and and we prove that if V,W are Banach function spaces with and , then the admissibility of the pair for an evolution family implies the uniform dichotomy of . In addition, we consider a subclass and we prove that if , then the admissibility of the pair implies the uniform exponential dichotomy of the family . This condition becomes necessary if . Finally, we present some applications of the main results.     

Translation Generalized Quadrangles In Even Characteristic

In this paper, we first introduce new objects called “translation generalized ovals” and “translation generalized ovoids”, and make a thorough study of these objects. We then obtain numerous new characterizations of the of Tits and the classical generalized quadrangle in even characteristic, including the complete classification of 2-transitive generalized ovals for the even case. Next, we prove a new strong characterization theorem for the of Tits. As a corollary, we obtain a purely geometric proof of a theorem of Johnson on semifield flocks. * The second author is a Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium).     

Weighted Non-Trivial Multiply Intersecting Families

Let n and r be positive integers. Suppose that a family satisfies F₁∩···∩F_r ≠∅ for all F₁, . . .,F_r ∈ and . We prove that there exists ε=ε(r) >0 such that holds for 1/2≤w≤1/2+ε if r≥13.     

On A Hypergraph Turán Problem Of Frankl

Let be the 2k-uniform hypergraph obtained by letting P₁, . . .,P_r be pairwise disjoint sets of size k and taking as edges all sets P_i∪P_j with i ≠ j. This can be thought of as the ‘k-expansion’ of the complete graph K_r: each vertex has been replaced with a set of size k. An example of a hypergraph with vertex set V that does not contain can be obtained by partitioning V = V1 ∪V₂ and taking as edges all sets of size 2k that intersect each of V₁ and V₂ in an odd number of elements. Let denote a hypergraph on n vertices obtained by this construction that has as many edges as possible. For n sufficiently large we prove a conjecture of Frankl, which states that any hypergraph on n vertices that contains no has at most as many edges as . Sidorenko has given an upper bound of for the Tur′an density of for any r, and a construction establishing a matching lower bound when r is of the form 2^p+1. In this paper we also show that when r=2^p+1, any -free hypergraph of density looks approximately like Sidorenko’s construction. On the other hand, when r is not of this form, we show that corresponding constructions do not exist and improve the upper bound on the Turán density of to , where c(r) is a constant depending only on r. The backbone of our arguments is a strategy of first proving approximate structure theorems, and then showing that any imperfections in the structure must lead to a suboptimal configuration. The tools for its realisation draw on extremal graph theory, linear algebra, the Kruskal–Katona theorem and properties of Krawtchouck polynomials. * Research supported in part by NSF grants DMS-0355497, DMS-0106589, and by an Alfred P. Sloan fellowship.     

One-sided Grothendieck quotients

We introduce one-sided thick subcategories of an arbitrary preadditive category and define a quotient category . When is abelian, this concept specializes to Grothendieck’s quotient for two-sided thick . We determine the left noetherian rings for which the injective modules form a left thick subcategory. We exhibit a class of one-sided thick subcategories in categories of coherent functors which are ubiquitous in representation theory. Received: 14 November 2006 Revised: 12 March 2007     

The Reduced Minimum Modulus in <Emphasis>C</Emphasis><Superscript>*</Superscript>-Algebras

We define the reduced minimum modulus of a nonzero element a in a unital C ^*-algebra by . We prove that . Applying this result to and its closed two side ideal , we get that dist , and for any if RR = 0, where and is the quotient homomorphism and . These results generalize corresponding results in Hilbert spaces.     

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

Schur Complements in Krein Spaces

The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded) J-selfadjoint operator A (with the unique factorization property) acting on a Krein space and a suitable closed subspace of , the Schur complement of A to is defined. The basic properties of are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive operators on a Hilbert space. To the memory of Professor Mischa Cotlar     

Exponential Stability of Traveling Pulse Solutions of a Singularly Perturbed System of Integral Differential Equations Arising From Excitatory Neuronal Networks

We establish the exponential stability of fast traveling pulse solutions to nonlinear singularly per-turbed systems of integral differential equations arising from neuronal networks.It has been proved that expo-nential stability of these orbits is equivalent to linear stability.Let (?) be the linear differential operator obtainedby linearizing the nonlinear system about its fast pulse,and let σ((?)) be the spectrum of (?).The linearizedstability criterion says that if max{Reλ:λ∈σ((?)),λ≠0}(?)-D,for some positive constant D,and λ=0 is asimple eigenvalue of (?)(ε),then the stability follows immediately (see [13] and [37]).Therefore,to establish theexponential stability of the fast pulse,it suffices to investigate the spectrum of the operator (?).It is relativelyeasy to find the continuous spectrum,but it is very difficult to find the isolated spectrum.The real part ofthe continuous spectrum has a uniformly negative upper bound,hence it causes no threat to the stability.Itremains to see if the isolated spectrum is safe.Eigenvalue functions (see [14] and [35,36]) have been a powerful tool to study the isolated spectrum of the as-sociated linear differential operators because the zeros of the eigenvalue functions coincide with the eigenvaluesof the operators.There have been some known methods to define eigenvalue functions for nonlinear systems ofreaction diffusion equations and for nonlinear dispersive wave equations.But for integral differential equations,we have to use different ideas to construct eigenvalue functions.We will use the method of variation of param-eters to construct the eigenvalue functions in the complex plane C.By analyzing the eigenvalue functions,wefind that there are no nonzero eigenvalues of (?) in {λ∈C:Reλ(?)-D} for the fast traveling pulse.Moreoverλ=0 is simple.This implies that the exponential stability of the fast orbits is true.     

Remarks on ample vector bundles of curve genus two

Let be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a -bundle over and that for any fiber F of the bundle projection . The pairs with = 2 are classified, where is the curve genus of . This allows us to improve some previous results. Received: 13 June 2006     

Stable Ranks of Banach Algebras of Operator-Valued Analytic Functions

Let E be a separable infinite-dimensional Hilbert space, and let denote the algebra of all functions that are holomorphic. If is a subalgebra of , then using an algebraic result of Corach and Larotonda, we derive that under some conditions, the Bass stable rank of is infinite. In particular, we deduce that the Bass (and hence topological stable ranks) of the Hardy algebra , the disk algebra and the Wiener algebra are all infinite. Submitted: October 10, 2007., Revised: January 11, 2008., Accepted: January 12, 2007.     

Minimal invariant varieties and first integrals for algebraic foliations

Let X be an irreducible algebraic variety over ℂ, endowed with an algebraic foliation . In this paper, we introduce the notion of minimal invariant variety V( , Y) with respect to ( , Y), where Y is a subvariety of X. If Y = {x} is a smooth point where the foliation is regular, its minimal invariant variety is simply the Zariski closure of the leaf passing through x. First we prove that for very generic x, the varieties V( , x) have the same dimension p. Second we generalize a result due to X. Gomez- Mont (see [G-M]). More precisely, we prove the existence of a dominant rational map F : X → Z, where Z has dimension (n − p), such that for very generic x, the Zariski closure of F⁻¹(F(x)) is one and only one minimal invariant variety of a point. We end up with an example illustrating both results.     

Bounded width problems and algebras

Let be finite relational structure of finite type, and let CSP denote the following decision problem: if is a given structure of the same type as , is there a homomorphism from to ? To each relational structure is associated naturally an algebra whose structure determines the complexity of the associated decision problem. We investigate those finite algebras arising from CSP’s of so-called bounded width, i.e., for which local consistency algorithms effectively decide the problem. We show that if a CSP has bounded width then the variety generated by the associated algebra omits the Hobby-McKenzie types 1 and 2. This provides a method to prove that certain CSP’s do not have bounded width. We give several applications, answering a question of Nešetřil and Zhu [26], by showing that various graph homomorphism problems do not have bounded width. Feder and Vardi [17] have shown that every CSP is polynomial-time equivalent to the retraction problem for a poset we call the Feder − Vardi poset of the structure. We show that, in the case where the structure has a single relation, if the retraction problem for the Feder-Vardi poset has bounded width then the CSP for the structure also has bounded width. This is used to exhibit a finite order-primal algebra whose variety admits type 2 but omits type 1 (provided P ≠ NP). Presented by M. Valeriote. Received January 8, 2005; accepted in final form April 3, 2006. The first author’s research is supported by a grant from NSERC and the Centre de Recherches Mathématiques. The second author’s research is supported by OTKA no. 034175 and 48809 and T 037877. Part of this research was conducted while the second author was visiting Concordia University in Montréal and also when the first author was visiting the Bolyai Institute in Szeged. The support of NSERC, OTKA and the Bolyai Institute is gratefully acknowledged.     

The general dévissage theorem for Witt groups of schemes

Let be a closed subscheme of the noetherian scheme X. We show that if X has a dualizing complex then there exists a dualizing complex of Z such that there is an isomorphism of coherent Witt groups for all . Received: 3 March 2006     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

On Shadows Of Intersecting Families

The shadow minimization problem for t-intersecting systems of finite sets is considered. Let be a family of k-subsets of . The -shadow of is the set of all (k-)-subsets contained in the members of . Let be a t-intersecting family (any two members have at least t elements in common) with . Given k,t,m the problem is to minimize (over all choices of ). In this paper we solve this problem when m is big enough.