On Cross-intersecting Families of Sets

A family of -element subsets and a family of k-element subsets of an n-element set are cross-intersecting if every set from has a nonempty intersection with every set from . We compare two previously established inequalities each related to the maximization of the product , and give a new and short proof for one of them. We also determine the maximum of for arbitrary positive weights ,_k.     

The Maximum Size of 3-Wise Intersecting and 3-Wise Union Families

Let be an n-uniform hypergraph on 2n vertices. Suppose that and holds for all F₁,F₂,F₃ ∈ . We prove that the size of is at most . The second author was supported by MEXT Grant-in-Aid for Scientific Research (B) 16340027     

Minimal Connected τ-Critical Hypergraphs

A hypergraph is τ-critical if τ(−{E})<τ() for every edge E ∈ , where τ() denotes the transversal number of . We show that if is a connected τ-critical hypergraph, then −{E} can be partitioned into τ()−1 stars of size at least two, for every edge E ∈ . An immediate corollary is that a connected τ-critical hypergraph has at least 2τ()−1 edges. This extends, in a very natural way, a classical theorem of Gallai on colour-critical graphs, and is equivalent to a theorem of Füredi on t-stable hypergraphs. We deduce a lower bound on the size of τ-critical hypergraphs of minimum degree at least two.     

Bipartite Subgraphs and Quasi-Randomness

We say that a family of graphs is p-quasi-random, 0<p<1, if it shares typical properties of the random graph G(n,p); for a definition, see below. We denote by the class of all graphs H for which and the number of not necessarily induced labeled copies of H in G_n is at most (1+o(1))p^e(H)n^v(H) imply that is p-quasi-random. In this note, we show that all complete bipartite graphs K_a,b, a,b2, belong to for all 0<p<1.Acknowledgments We would like to thank Andrew Thomason for fruitful discussions and Yoshi Kohayakawa for organizing Extended Workshop on Combinatorics in eq5 Paulo, Ubatuba, and Rio de Janeiro, where a part of this work was done. We also thank the referees for their careful work.The first author was partially supported by NSF grant INT-0072064The second author was partially supported by NSF grants DMS-9970622, DMS-0301228 and INT-0072064Final version received: October 24, 2003     

On Resolvable Multipartite G-Designs II

In this paper, it has been proved that and are factorable, where ⊗ denotes wreath product of graphs. As a consequence, a resolvable (k,n,k,2λ) multipartite –design exists for even k. These results generalize the results of Ushio on –factorizations of complete tripartite graphs.     

Topologically transitive extensions of bounded operators

Let X be any Banach space and T a bounded operator on X. An extension of the pair (X,T) consists of a Banach space in which X embeds isometrically through an isometry i and a bounded operator on such that When X is separable, it is additionally required that be separable. We say that is a topologically transitive extension of (X, T) when is topologically transitive on , i.e. for every pair of non-empty open subsets of there exists an integer n such that is non-empty. We show that any such pair (X,T) admits a topologically transitive extension , and that when H is a Hilbert space, (H,T) admits a topologically transitive extension where is also a Hilbert space. We show that these extensions are indeed chaotic.Mathematics Subject Classification (2000): 47 A 16     

Spanning Trails Connecting Given Edges

Suppose that is the set of connected graphs such that a graph G if and only if G satisfies both (F1) if X is an edge cut of G with |X|3, then there exists a vertex v of degree |X| such that X consists of all the edges incident with v in G, and (F2) for every v of degree 3, v lies in a k-cycle of G, where 2k3.In this paper, we show that if G and (G)3, then for every pair of edges e,fE(G), G has a trail with initial edge e and final edge f which contains all vertices of G. This result extends several former results.     

Integral varieties of the canonical cone structure on <Emphasis Type="Italic">G</Emphasis>/<Emphasis Type="Italic">P</Emphasis>

The canonical cone structure on a compact Hermitian symmetric space G/P is the fiber bundle where is the cone of the highest weight vectors under the action of the reductive part of P. It is known that the cone coincides with the cone of the vectors tangent to the lines in G/P passing through x, when we consider G/P as a projective variety under its homogeneous embedding into the projective space of the irreducible representation space V of G with highest weight associated to P. A subvariety X of G/P is said to be an integral variety of at all smooth points xG/P. Equivalently, an integral variety of is a subvariety of G/P whose embedded projective tangent space at each smooth point is a linear space We prove a kind of rigidity of the integral varieties under some dimension condition. After making a uniform setting to study the problem, we apply the theory of Lie algebra cohomology as a main tool. Finally we show that the dimension condition is necessary by constructing counterexamples.     

Associated primes and cofiniteness of local cohomology modules

Let be an ideal of Noetherian ring R and let s be a non-negative integer. Let M be an R-module such that is finite R-module. If s is the first integer such that the local cohomology module is non -cofinite, then we show that is finite. In particular, the set of associated primes of is finite. Let be a local Noetherian ring and let M be a finite R-module. We study the last integer n such that the local cohomology module is not -cofinite and show that n just depends on the support of M.The research of the first author was supported in part by a grant from IPM (No. 83130114).The second author was supported by a grant from University of Tehran (No. 6103023/1/01).     

The cofinality of the saturated uncountable random graph

Assuming CH, let be the saturated random graph of cardinality ₁. In this paper we prove that it is consistent that and can be any two prescribed regular cardinals subject only to the requirement      

Modified Ariki-Koike algebras and cyclotomic <Emphasis Type="Italic">q</Emphasis>-Schur algebras

The modified Ariki-Koike algebra is a variation of the original Ariki-Koike algebra over an integral domain R. When R is a rational function field over the independent parameters, But for general R, is not isomorphic to , and has a simpler structure than . In this paper, we construct a cellular basis of which has a similar property as the cellular basis of introduced by Dipper-James-Mathas. By comparing these two cellular bases, we obtain some estimate on the decomposition numbers of in terms of the decomposition numbers of . We also prove the integral form of the Schur-Weyl reciprocity between a certain quantum algebra U_q and on the tensor space      

On the logarithmic plurigenera of complements of plane curves

Let B be a (not necessarily irreducible) plane curve in ². In the present article, we prove that if and only if Moreover, we determine the curve B when and Mathematics Subject Classification (2000): 14R05, 14H50, 14J26     

Analysis of nonsmooth vector-valued functions associated with second-order cones

Let be the Lorentz/second-order cone in . For any function f from to , one can define a corresponding function f^soc(x) on by applying f to the spectral values of the spectral decomposition of x with respect to . We show that this vector-valued function inherits from f the properties of continuity, (local) Lipschitz continuity, directional differentiability, Fréchet differentiability, continuous differentiability, as well as (-order) semismoothness. These results are useful for designing and analyzing smoothing methods and nonsmooth methods for solving second-order cone programs and complementarity problems.Mathematics Subject Classification (1991): 26A27, 26B05, 26B35, 49J52, 90C33, 65K05     

The feasibility pump

In this paper we consider the NP-hard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{c^Tx:Ax≥b,x_j integer ∀j ∈ }. Trivially, a feasible solution can be defined as a point x* ∈ P:={x:Ax≥b} that is equal to its rounding , where the rounded point is defined by := x*_j if j ∈ and := x*_j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function Δ(x*, ), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP.We start from any x* ∈ P, and define its rounding . At each FP iteration we look for a point x* ∈ P that is as close as possible to the current by solving the problem min {Δ(x, ): x ∈ P}. Assuming Δ(x, ) is chosen appropriately, this is an easily solvable LP problem. If Δ(x*, )=0, then x* is a feasible MIP solution and we are done. Otherwise, we replace by the rounding of x*, and repeat.We report computational results on a set of 83 difficult 0-1 MIPs, using the commercial software ILOG-Cplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOG-Cplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOG-Cplex could not find any feasible solution at the root node for 19 problems in our test-bed, whereas FP was unsuccessful in just 3 cases.     

A Lower Bound for the Rectilinear Crossing Number

We give a new lower bound for the rectilinear crossing number of the complete geometric graph K_n. We prove that and we extend the proof of the result to pseudolinear drawings of K_n. Dedicated to the memory of our good friend and mentor Víctor Neumann-Lara. Received: April, 2003 Final version received: March 18, 2005     

Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

On an eigenvalue problem related to the critical exponent

In this paper we study the eigenvalue problemwhere is a smooth bounded domain, and u is a positive solution of the problemsuch thatwhere S is the best Sobolev constant for the embedding of H¹₀() into L²*(), We prove several estimates for the eigenvalues _i, of (I), i=2,..,N+2 and some qualitative properties of the corresponding eigenfunctions.Supported by M.I.U.R., project Variational methods and nonlinear differential equations.     

On Cyclic Packing of a Tree

We prove that there exists a packing of copies of a tree of size into K_n. Moreover, the proof provides an easy algorithm.Acknowledgments. The research of the second author was partially supported by Deutscher Akademischer Austauschdienst.     

On certain categories of modules for affine Lie algebras

In this paper, we exploit basic formal variable techniques to study certain categories of modules for an (untwisted) affine Lie algebra , motivated by Chari-Pressleys work on certain integrable modules. We define and study two categories and of -modules using generating functions, where is proved to contain the well known evaluation modules and to unify highest weight modules, evaluation modules and their tensor product modules. We classify integrable irreducible -modules in categories and and we determine the isomorphism classes of those irreducible modules. Finally we prove a result that relates fusion rules in the context of vertex operator algebras with integrable irreducible modules of Chari-Pressley.in final form: 12 November 2003Partially supported by a NSA grant and a grant from Rutgers Research Council.