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     

Properties of Auto- and Antiautomorphisms of Maximal Chain Structures and their Relations to i-Perspectivities

Let E be a non empty set, let P : = E × E, := {x × E|x ∈ E}, := {E × x|x ∈ E}, and := {C ∈ 2 ^P |∀X ∈ : |C ∩ X| = 1} and let . Then the quadruple resp. is called chain structure resp. maximal chain structure. We consider the maximal chain structure as an envelope of the chain structure . Particular chain structures are webs, 2-structures, (coordinatized) affine planes, hyperbola structures or Minkowski planes. Here we study in detail the groups of automorphisms , , , related to a maximal chain structure . The set of all chains can be turned in a group such that the subgroup of generated by the left-, by the right-translations and by ι the inverse map of is isomorphic to (cf. (2.14)).     

Minimal dynamical systems on the product of the Cantor set and the circle II

Let X be the Cantor set and φ be a minimal homeomorphism on . We show that the crossed product C^*-algebra is a simple A -algebra provided that the associated cocycle takes its values in rotations on . Given two minimal systems and such that φ and ψ arise from cocycles with values in isometric homeomorphisms on , we show that two systems are approximately K-conjugate when they have the same K-theoretical information.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

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.     

Fong characters and chains of normal subgroups

In this note, we show that if is a π-partial character of the π-separable group is a chain of normal subgroups of G, and H is a Hall π-subgroup of G, then has a Fong character α Irr(H) such that for every subgroup , every irreducible constituent of α _H∩N is Fong for N. We also show that if is quasi-primitive, then for every normal subgroup M of G the irreducible constituents of are Fong for M. Received: 21 July 2006 Revised: 17 January 2007     

Equidistribution of Rational Matrices in their Conjugacy Classes

Let G be a connected simply connected almost -simple algebraic group with non-compact and a cocompact congruence subgroup. For any homogeneous manifold of finite volume, and a , we show that the Hecke orbit T _a (x ₀ H) is equidistributed on as , provided H is a non-compact commutative reductive subgroup of G. As a corollary, we generalize the equidistribution result of Hecke points ([COU], [EO1]) to homogeneous spaces G/H. As a concrete application, we describe the equidistribution result in the rational matrices with a given characteristic polynomial. The second author partially supported by DMS 0333397. Received: May 2005 Revision: March 2006 Accepted: June 2006     

Polar Graphs and Corresponding Involution Sets, Loops and Steiner Triple Systems

A 1-factorization (or parallelism) of the complete graph with loops is called polar if each 1-factor (parallel class) contains exactly one loop and for any three distinct vertices x₁, x₂, x₃, if {x₁} and {x₂, x₃} belong to a 1-factor then the same holds for any permutation of the set {1, 2, 3}. To a polar graph there corresponds a polar involution set , an idempotent totally symmetric quasigroup (P, *), a commutative, weak inverse property loop (P, + ) of exponent 3 and a Steiner triple system . We have: satisfies the trapezium axiom is self-distributive ⇔ (P, + ) is a Moufang loop is an affine triple system; and: satisfies the quadrangle axiom is a group is an affine space.     

<Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras of Bergman Type Operators with Piecewise Continuous Coefficients

The C ^*-algebra generated by the n poly-Bergman and m antipoly-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L ²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional convolution operators with symbols admitting homogeneous discontinuities we reduce the study to simpler C ^*-algebras associated with points and pairs . Applying a symbol calculus for the abstract unital C ^*-algebras generated by N orthogonal projections sum of which equals the unit and by M = n + m one-dimensional orthogonal projections and using relations for the Gauss hypergeometric function, we study local algebras at points being the discontinuity points of coefficients. A symbol calculus for the C ^*-algebra is constructed and a Fredholm criterion for the operators is obtained.     

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.     

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.     

On the Matlis duals of local cohomology modules

Let ( ) be a commutative Noetherian local ring with non-zero identity, an ideal of R and M a finitely generated R-module with . Let D(–) := Hom _R (–, E) be the Matlis dual functor, where is the injective hull of the residue field . We show that, for a positive integer n, if there exists a regular sequence and the i-th local cohomology module H ⁱ _a (M) of M with respect to is zero for all i with i > n then The author was partially supported by a grant from Institute for Studies in Theoretical Physics and Mathematics (IPM) Iran (No. 85130023). Received: 9 August 2006     

On the Schwarz reflection principle for monogenic functions

Let ∑ be either an oriented hyperplane or the unit sphere in , let be open and connected and let be an open and connected domain in such that . If in is a null solution of the Dirac operator (also called a monogenic function in ) which is continuously extendable to , then conditions upon are given enabling the monogenic extension of across . In such a way Schwarz reflection type principles for monogenic functions are established in the Spin (1) and Spin cases. The Spin (1) case includes the classical Schwarz reflection principle for holomorphic functions in the plane. The Spin case deals with so-called “half boundary value problems” for the Dirac operator. Received: 2 February 2006     

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

An Operator Corona Theorem for Some Subspaces of <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript>

Let E, E_* be separable Hilbert spaces. If S is an open subset of , then denotes the space of all functions that are holomorphic in , and bounded and continuous on . In this article we prove the following results:

Parametrization of Contractive Block Operator Matrices and Passive Discrete-Time Systems

Passive linear systems τ = have their transfer function in the Schur class S . Using a parametrization of contractive block operators the transfer function is connected to the Sz.-Nagy–Foiaş characteristic function of the contraction A. This gives a new aspect and some explicit formulas for studying the interplay between the system τ and the functions and . The method leads to some new results for linear passive discrete-time systems. Also new proofs for some known facts in the theory of these systems are obtained. Dedicated to Eduard Tsekanovskiĭ on the occasion of his seventieth birthday This work was supported by the Research Institute for Technology at the University of Vaasa. The first author was also supported by the Academy of Finland (projects 212146, 117617) and the Dutch Organization for Scientific Research N.W.O. (B 61-553). Received: December 22, 2006. Revised: February 6, 2007.     

Distances entre exponentielles et puissances d’éléments de certaines algèbres de Banach

Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let . We show that if then . We also show, assuming a suitable spectral condition on x, that if , then Received: 12 July 2006 Revised: 31 January 2007     

Furstenberg family and chaos

A Furstenberg family F is a family,consisting of some subsets of the set of positive integers,which is hereditary upwards,i.e.A■B and A∈F imply B∈F.For a given system (i.e.,a pair of a complete metric space and a continuous self-map of the space) and for a Furstenberg family F, the definition of F-scrambled pairs of points in the space has been given,which brings the well-known scrambled pairs in Li-Yorke sense and the scrambled pairs in distribution sense to be F-scrambled pairs corresponding respectively to suitable Furstenberg family F.In the present paper we explore the basic properties of the set of F-scrambled pairs of a system.The generically F-chaotic system and the generically strongly F-chaotic system are defined.A criterion for a generically strongly F-chaotic system is showed.