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.     

Invariants and spaces of zero solutions of linear partial differential operators

It is shown that for open convex , d > 1 and a nontrivial polynomial P the space does not have property . If P is elliptic or homogeneous, then this holds for every open Ω. For even cannot occur and if it occurs for some Ω, then P must be hypoelliptic. Received: 18 July 2005     

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.     

Coincidences of Centers of Edge-Incentric, or Balloon, Simplices

An edge-incentric d-simplex is defined to be a d-simplex S which admits a (d − 1)-sphere that touches all the edges of S internally. The center of such a sphere is called the edge-incenter of S and is denoted by . Equivalently, S is edge-incentric if and only if its vertices are the centers of d + 1 (d − 1)-spheres in mutual external touch, and for this reason one may call such an S a balloon d-simplex. An orthocentric d-simplex is a d-simplex in which the altitudes are concurrent. The point of concurrence is called the orthocenter and is denoted by . The spaces of edge-incentric and of orthocentric d-simplices have the same dimension d in the sense that a d-simplex in either space can be parametrized, up to shape, by d numbers. Edge-incentric and orthocentric tetrahedra are the first two of the four special classes of tetrahedra studied in [1, Chapter IX.B, pp. 294–333]. The degree of regularity implied by the coincidence of two or more centers of a general d-simplex is investigated in [8], where it is shown that the coincidence of the centroid , the circumcenter , and the incenter does not imply much regularity. For an orthocentric d-simplex S, however, it is proved in [9] that if any two of the centers , and coincide, then S is regular. In this paper, the same question is addressed for edge-incentric d-simplices. Among other things, it is proved that if any three of the centers , and of an edge-incentric d-simplex S coincide, then S is regular, and it is also shown that none of the coincidences , and implies regularity (except when d ≤ 3, d ≤ 4, and d ≤ 6, respectively). In contrast with the afore-mentioned results for orthocentric d-simplices, this emphasizes once more the feeling that, regarding many important properties, orthocentric d-simplices are the true generalizations of triangles. Several open questions are posed. Received: June 19, 2006.     

A class number formula for the p-cyclotomic field

Let p be an odd prime number and . Let be the classical Stickelberger ideal of the group ring . Iwasawa [6] proved that the index equals the relative class number of . In [2], [4] we defined for each subgroup H of G a Stickelberger ideal of , and studied some of its properties. In this note, we prove that when mod 4 and [G : H] = 2, the index equals the quotient . Received: 13 January 2006     

Remarks on balance in generalized Tate cohomology

We consider two pairs of complete hereditary cotorsion theories on the category of left R-modules, such that We prove that for any left R-modules M, N and for any n ≧ 1, the generalized Tate cohomology modules can be computed either using a left of M and a left of M or using a right a right of N. Received: 17 December 2004     

Elliptic operators with unbounded diffusion coefficients in L 2 spaces with respect to invariant measures

We study the self-adjoint and dissipative realization A of a second order elliptic differential operator with unbounded regular coefficients in , where μ(dx) = ρ (x)dx is the associated invariant measure. We prove a maximal regularity result under suitable assumptions, that generalize the well known conditions in the case of constant diffusion part. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

{\left( {\mathfrak{V},\mathfrak{W}} \right)}{\text{-cotorsion pairs}}

Let R be a unital associative ring and two classes of left R-modules. In this paper we introduce the notion of a In analogy to classical cotorsion pairs as defined by Salce [10], a pair of subclasses and is called a if it is maximal with respect to the classes and the condition for all and Basic properties of are stated and several examples in the category of abelian groups are studied. Received: 17 March 2005     

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     

Strongly connective degree sets II: perfect derived subgroups

When G is a finite nonabelian group, we associate the common-divisor graph with G by letting nontrivial degrees in cd(G) = {χ(1) | χ∈Irr(G)} be the vertices and making distinct vertices adjacent if they have a common nontrivial divisor. A set of vertices for this graph is said to be strongly connective for cd(G) if there is some prime which divides every member of , and every vertex outside of is adjacent to some member of . When G is nonsolvable, we provide sufficiency conditions for cd(G) to have a strongly connective subset. We also extend a previously known result about groups with nonabelian solvable quotients, and prove for arbitrary groups G that if the associated graph is connected and has a diameter bounded by 2, then indeed cd(G) has a strongly connective subset. The major focus is on when the derived subgroup G′ is perfect. Received: 23 July 2005     

A singular Moser-Trudinger embedding and its applications

Let Ω be a bounded domain in , we prove the singular Moser-Trudinger embedding: if and only if where and . We will also study the corresponding critical exponent problem.     

Explicit Bounded-Degree Unique-Neighbor Concentrators

Here we solve an open problem considered by various researchers by presenting the first explicit constructions of an infinite family of bounded-degree ‘unique-neighbor’ concentrators Γ; i.e., there are strictly positive constants α and ε, such that all Γ = (X,Y,E(Γ)) ∈ satisfy the following properties. The output-set Y has cardinality times that of the input-set X, and for each subset S of X with no more than α|X| vertices, there are at least ε|S| vertices in Y that are adjacent in Γ to exactly one vertex in S. Also, the construction of is simple to specify, and each has fewer than edges. We then modify to obtain explicit unique-neighbor concentrators of maximum degree 3. * Supported by NSF grant CCR98210-58 and ARO grant DAAH04-96-1-0013.     

On convex perturbations with a bounded isotropic constant

Let be a convex body and ɛ > 0. We prove the existence of another convex body , whose Banach–Mazur distance from K is bounded by 1 + ɛ, such that the isotropic constant of K’ is smaller than , where c > 0 is a universal constant. As an application of our result, we present a slight improvement on the best general upper bound for the isotropic constant, due to Bourgain. The author is a Clay Research Fellow, and was also supported by NSF grant #DMS-0456590. Received: November 2005; Accepted: February 2006     

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 .     

Lefschetz decompositions and categorical resolutions of singularities

Let Y be a singular algebraic variety and let be a resolution of singularities of Y. Assume that the exceptional locus of over Y is an irreducible divisor in . For every Lefschetz decomposition of the bounded derived category of coherent sheaves on we construct a triangulated subcategory ) which gives a desingularization of . If the Lefschetz decomposition is generated by a vector bundle tilting over Y then is a noncommutative resolution, and if the Lefschetz decomposition is rectangular, then is a crepant resolution.     

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

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