Chern character in equivariant entire cyclic cohomology

We construct the Chern character in the equivariant entire cyclic cohomology. We prove a general index theorem for theG-invariant Dirac operator.Supported in part by the Department of Energy under Grant DE-FG02-88ER25065.     

Factorization of the Non-Stationary Schrödinger Operator

We consider a factorization of the non-stationary Schr?dinger operator based on the parabolic Dirac operator introduced by Cerejeiras, K?hler and Sommen. Based on the fundamental solution for the parabolic Dirac operators, we shall construct appropriated Teodurescu and Cauchy-Bitsadze operators. Afterwards we will describe how to solve the nonlinear Schr?dinger equation using Banach fixed point theorem.     

The Howe Duality for the Dunkl Version of the Dirac Operator

The classical Fischer decomposition of polynomials on Euclidean space makes it possible to express any polynomial as a sum of harmonic polynomials multiplied by powers of |x|². A deformation of the Laplace operator was recently introduced by Ch.F. Dunkl. It has the property that the symmetry with respect to the orthogonal group is broken to a finite subgroup generated by reflections (a Coxeter group). It was shown by B. ?rsted and S. Ben Said that there is a deformation of the Fischer decomposition for polynomials with respect to the Dunkl harmonic functions. In Clifford analysis, a Dunkl version of the Dirac operator was introduced and studied by P. Cerejeiras, U. K?hler and G. Ren. The aim of the article is to describe an analogue of the Fischer decomposition for solutions of the Dunkl Dirac operator. The main methods used are coming from representation theory, in particular, from ideas connected with Howe dual pairs. This paper is dedicated to the memory of our friend and colleague Jarolim Bureš to recall our long lasting cooperation on many topics in mathematics The work presented here is a part of the research project MSM 0021620839 and was supported also by the grant GA ČR 201/05/2117. Received: October, 2007. Accepted: February, 2008.     

Opial-type inequalities for differential operators

Some continuous and discrete versions of Opial-type inequalities which are readily applicable to differential and difference operators are established. These generalize earlier results of Anastassiou and Pe?ari?, and of Koliha and Pe?ari?.     

Decomposing simple permutations,with enumerative consequences

We prove that every sufficiently long simple permutation contains two long almost disjoint simple subsequences, and then we show how this result has enumerative consequences. For example, it implies that, for any r, the number of permutations with at most r copies of 132 has an algebraic generating function (this was previously proved, constructively, by Bóna and (independently) Mansour and Vainshtein). Supported by a Royal Society Dorothy Hodgkin Research Fellowship. Supported by EPSRC grant GR/S53503/01.     

Quantitative Bessaga–Pełczyński property and quantitative Rosenthal property

We prove that c₀ and , where K is a dispersed compact Hausdorff space, enjoy a quantitative version of the Bessaga–Pe?czyński property. We also prove that l₁ possesses a quantitative version of the Pe?czyński property. Finally, we show that has a quantitative version of the Rosenthal property for any finite measure μ.     

Joint seminormality and Dirac operators

In this article, an approach to joint seminormality based on the theory of Dirac and Laplace operators on Dirac vector bundles is presented. To eachn-tuple of bounded linear operators on a complex Hilbert space we first associate a Dirac bundle furnished with a metric-preserving linear connection defined in terms of thatn-tuple. Employing standard spin geometry techniques we next get a Bochner type and two Bochner-Kodaira type identities in multivariable operator theory. Further, four different classes of jointly seminormal tuples are introduced by imposing semidefiniteness conditions on the remainders in the corresponding Bochner-Kodaira identities. Thus we create a setting in which the classical Bochner's method can be put into action. In effect, we derive some vanishing theorems regarding various spectral sets associated with commuting tuples. In the last part of this article we investigate a rather general concept of seminormality for self-adjoint tuples with an even or odd number of entries.     

A Krasnosel’skii-type theorem for paths of bounded length

Let S be a nonempty closed, simply connected set in the plane, and let α τ; 0. If every three points of 5 see a common point of S via paths of length at most α, then for some point s0 of S, s0 sees each point of S via such a path. That is, S is starshaped via paths of length at most α. Supported in part by NSF grant DMS-9207019     

Ordered sets,pagenumbers and planarity

General bounds for the pagenumber of an ordered set are developed. One bound is derived by first showing that the maximum number of edges in the diagram of a planar ordered setP is 2v(P)-2-ht(P). A construction is given to show that the pagenumber of the product ofn chains is no more than 2n-2. Lastly, some open questions are discussed.Supported in part by NSERC of Canada.Supported by an NSERC Summer Scholarship.     

On the mutually non isomorphic ℓp(ℓq) spaces

We extend a result of Pe?czyński showing that {?_p(?_q): 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of ?_p(?_q) are also given.     

Enumeration of order preserving maps

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 2 ^2n/3 order preserving maps (and 2 ² in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.Supported by ONR Contract N00014-85-K-0769.Supported by NSF Grant DMS-9011850.Supported by NSERC Grants 69-3378 and 69-0259.     

Law of large numbers and central limit theorem for Donsker's delta function of diffusions I

Limit theorems in the space of Hida distributions, similar to the law of large numbers and the central limit theorem, are shown for composites of the Dirac distribution with solutions of one-dimensional, non-degenerate Itô equations.Supported by National Science Foundation under grant DMS-9001859.Supported by the Louisiana Education Quality Support Fund under grant (91–93) RD-A-08.Supported by the Council on Research of Louisiana State University.     

Bounded Berezin-Toeplitz Operators on the Segal-Bargmann Space

We discuss the boundedness of Berezin-Toeplitz operators on a generalized Segal-Bargmann space (Fock space) over the complex n-space. This space is characterized by the image of a global Bargmann-type transform introduced by Sj?strand. We also obtain the deformation estimates of the composition of Berezin-Toeplitz operators whose symbols and their derivatives up to order three are in the Wiener algebra of Sj?strand. Our method of proofs is based on the pseudodifferential calculus and the heat flow determined by the phase function of the Bargmann transform. Supported by the JSPS Grant-in-Aid for Scientific Research #20540151.     

Every triangle-free planar graph has a planar upward drawing

A planar ordered set has a triangle-free, planar covering graph; on the other hand, there are nonplanar ordered sets whose covering graphs are planar. We show thatevery triangle-free planar graph has a planar upward drawing. This planar upward drawing can be constructed in time, polynomial in the number of vertices.Our results shed light on the apparently difficult problem, of long-standing, whether there is aneffective planarity-testing procedure for an ordered set.Supported in part by the Alexander von Humboldt Stiftung.Supported in part by the Deutsche Forschungsgemeinschaft.     

Dirac eigenvalue estimates on surfaces

We prove lower Dirac eigenvalue bounds for closed surfaces with a spin structure whose Arf invariant equals 1. Besides the area only one geometric quantity enters in these estimates, the spin-cut-diameter which depends on the choice of spin structure. It can be expressed in terms of various distances on the surfaces or, alternatively, by stable norms of certain cohomology classes. In case of the 2-torus we obtain a positive lower bound for all Riemannian metrics and all nontrivial spin structures. For higher genus g the estimate is given by The corresponding estimate also holds for the -spectrum of the Dirac operator on a noncompact complete surface of finite area. As a corollary we get positive lower bounds on the Willmore integral for all 2-tori embedded in . Received: 15 May 2001; in final form: 11 September 2001 / Published online: 1 February 2002     

Volume approximation of smooth convex bodies by three-polytopes of restricted number of edges

For a given convex body K in with C ² boundary, let P ^c _n be the circumscribed polytope of minimal volume with at most n edges, and let P ⁱ _n be the inscribed polytope of maximal volume with at most n edges. Besides presenting an asymptotic formula for the volume difference as n tends to infinity in both cases, we prove that the typical faces of P ^c _n and P ⁱ _n are asymptotically regular triangles and squares, respectively, in a suitable sense. Supported by OTKA grants 043520 and 049301, and by the EU Marie Curie grants Discconvgeo, Budalggeo and PHD. Authors’ addresses: Károly J. B?r?czky, Alfréd Rényi Institute of Mathematics, P.O. Box 127, Budapest H–1364, Hungary, and Department of Geometry, Roland E?tv?s University, Pázmány Péter sétány 1/C, Budapest 1117, Hungary; Salvador S. Gomis, Department of Mathematical Analysis, University of Alicante, 03080 Alicante, Spain; Péter Tick, Gyűrű utca 24, Budapest H–1039, Hungary     

Some new rich subspaces of C. Applications

In this paper, we are interested in a class of subspaces of C, introduced by Bourgain [Studia Math. 77 (1984) 245-253]. Wojtaszczyk called them rich in his monograph [Banach Spaces for Analysts, Cambridge Univ. Press, 1991]. We give some new examples of such spaces: this allows us to recover previous results of Godefroy-Saab and Kysliakov on spaces with reflexive annihilator in a very simple way. We construct some other examples of rich spaces, hence having property (V) of Pe?czyński and Dunford-Pettis property. We also recover the results due to Bourgain and Saccone saying that spaces of uniformly convergent Fourier series share these properties, by only using the main result of [Studia Math. 77 (1984) 245-253] and some very elementary arguments. We generalize too these results.     

On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

Generalizations of Bohr's inequality in Hilbert C*-modules

We present a new operator equality in the framework of Hilbert C*-modules. As a consequence, we get an extension of the Euler–Lagrange type identity in the setting of Hilbert bundles as well as several generalized operator Bohr's inequalities due to O. Hirzallah, W.-S. Cheung, J.E. Pe?ari? and F. Zhang.     

Cycles through a prescribed vertex set in N-connected graphs

A well-known result of Dirac (Math. Nachr. 22 (1960) 61) says that given n vertices in an n-connected G, G has a cycle through all of them. In this paper, we generalize Dirac's result as follows:Given at most vertices in an n-connected graph G when n3 and , then G has a cycle through exactly n vertices of them.This improves the previous known bound given by Kaneko and Saito (J. Graph Theory 15(6) (1991) 655).