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Showing papers in "Contemporary mathematics in 2006"


Journal Article
TL;DR: In this article, Wu et al. present a survey of Dirichlet-integral spaces of functions with finite Dirichlets integral functions and two important operators that act on this space of functions.
Abstract: D |f ′|2 dx dy. In particular, we will cover the basic structure of these functions their boundary values and their zeros along with two important operators that act on this space of functions the forward and backward shifts. This survey is by no means complete. For example, we will not cover the Toeplitz or Hankel operators on these functions, nor will we cover the important topic of interpolation. These topics are surveyed in a nice paper of Wu [64]. In order to make this survey more manageable, we will also restrict ourselves to this space of functions with finite Dirichlet integral and will not try to cover the many related Dirichlet-type spaces. We refer the reader to the papers [11, 41, 47, 54] for more on this.

76 citations


Journal Article
TL;DR: The following main results are proved: 1) Classification of p-groups all of whose subgroups of index p (square power) are abelian; 2) classification of p groups with 7 involutions; and 3) Classification of minimal nonmetacyclic p-group, classification of groups of odd order without normal elementary abelians of order p (power 3); and 4) Proofs of some basic counting theorems based on the new enumeration principle as discussed by the authors.
Abstract: The following main results are proved: 1) Classification of p-groups all of whose subgroups of index p (square power)are abelian. 2) Classification of p-groups with 7 involutions. 3) New proofs of some Blackburn's results (classification of minimal nonmetacyclic p-groups, classification of p-groups of odd order without normal elementary abelian subgroup of order p (power 3)). 4) Proofs of some basic counting theorems based on the new enumeration principle. 5) New proof of Ward's theorem on quaternion-free 2-groups. 6) Corrected proof of Iwasawa's theorem on the structure of modular p-groups. Our proofs are completely elementary.

46 citations


Journal Article
TL;DR: In this paper, the authors provide a comparative treatment of spectral theory for self-adjoint and non-selfadjoint operators connected with the defocusing and focusing nonlinear Schrodinger equation, of relevance to nonlinear optics.
Abstract: We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrodinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl–Titchmarsh half-line m-coefficients (and 2 × 2 matrix-valued M -matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoint context imply the blowup of the norm of spectral projections in the limit where the crossing point is approached.

42 citations


Journal Article
TL;DR: In this article, the authors describe new ways of constructing abstract saturated fusion systems, rst as fusion systems of spaces with certain properties, and then via certain graphs, and construct more concrete examples using that theorem, and show in many cases that they are ''exotic\" in the sense of not coming from any nite group.
Abstract: A saturated fusion system consists of a nite p-group S, together with a category which encodes \\conjugacy\" relations among subgroups of S, and which satis es certain axioms which are motivated by properties of the fusion in a Sylow p-subgroup of a nite group. We describe here new ways of constructing abstract saturated fusion systems, rst as fusion systems of spaces with certain properties, and then via certain graphs. A saturated fusion system consists of a nite p-group S, together with a category F whose objects are the subgroups of S, whose morphisms are group monomorphisms between those subgroups, and which satis es certain axioms modelled on the fusion category for the p-subgroups of a nite group. The precise de nition of a saturated fusion system is due to Puig [Pu], and our version of that de nition is given in Section 1. Saturated fusion systems mimic in several ways the structure of nite groups and their classifying spaces. Examples have been known for some time of \\exotic\" saturated fusion systems | systems which do not arise from the fusion in any nite group | but the construction of such examples is very complicated, and we are looking for simpler and more systematic ways to construct them. One consequence of the main result in this paper is a way of constructing a variety of examples of saturated fusion systems. Of the examples constructed using this technique, some are then shown by other means to be exotic. The de nition of a fusion system over a p-group S is simple, and in most cases it is clear whether or not a given category satis es it. In contrast, it is much harder to check whether a given fusion system is saturated. For example, for any map f : BS ! X, where S is a nite p-group and X is a topological space, the fusion system of X over (S; f) is a category FS;f(X) whose objects are the subgroups of S, and where MorFS;f (X)(P;Q) is the set of all monomorphisms ' 2 Hom(P;Q) such that (f jBP ) Æ B' ' (f jBQ). This is always a fusion system in the sense of De nition 1.1, but is not in general saturated. The central result in this paper is Theorem 2.1, where we list some conditions on the map f which ensure that the fusion system FS;f(X) is saturated. These conditions also ensure that FS;f(X) has an associated linking system (see De nition 1.3), and hence that X, S, and f de ne a p-local nite group. Afterwards, we construct more concrete examples using that theorem, and show in many cases that they are \\exotic\" in the sense of not coming from any nite group. For example, in Theorem 4.2, in certain cases when G is an amalgamated free product of nite groups, we apply our theorem to BGp to show that the fusion system of G (taken over a maximal p-subgroup of G) is 1991 Mathematics Subject Classi cation. Primary 55R35. Secondary 55R40, 20D20.

26 citations


Journal Article
TL;DR: In this article, a simple heuristic approach to the root problem in braid groups was used to show that cryptographic parameters proposed in this context must be considered as insecure, and the secret key of an authentication system based on this root problem was extracted within seconds.
Abstract: Using a simple heuristic approach to the root problem in braid groups, we show that cryptographic parameters proposed in this context must be considered as insecure. In our experiments we can, often within seconds, extract the secret key of an authentication system based on the root problem in braid groups.

17 citations


Journal Article
TL;DR: For Jacobi polynomials associated with root system BC2, a new pair of lowering and raising operators is presented in this paper. But the authors do not discuss the effect of raising or lowering the degree on the parameters of the polynomial.
Abstract: This paper discusses operators lowering or raising the degree but preserving the parameters of special orthogonal polynomials. Results for one-variable classical (q-)orthogonal polynomials are surveyed. For Jacobi polynomials associated with root system BC2 a new pair of lowering and raising operators is obtained.

12 citations



Journal ArticleDOI
TL;DR: The proceedings of a conference held at the Courant Institute in 2006 to celebrate the 60th birthday of Percy A. Deift as mentioned in this paper reflected the wide-ranging contributions of Professor Deift to analysis with emphasis on recent developments in Random Matrix Theory and integrable systems.
Abstract: This volume contains the proceedings of a conference held at the Courant Institute in 2006 to celebrate the 60th birthday of Percy A. Deift. The program reflected the wide-ranging contributions of Professor Deift to analysis with emphasis on recent developments in Random Matrix Theory and integrable systems. The articles in this volume present a broad view on the state of the art in these fields. Topics on random matrices include the distributions and stochastic processes associated with local eigenvalue statistics, as well as their appearance in combinatorial models such as TASEP, last passage percolation and tilings. The contributions in integrable systems mostly deal with focusing NLS, the Camassa-Holm equation and the Toda lattice. A number of papers are devoted to techniques that are used in both fields. These techniques are related to orthogonal polynomials, operator determinants, special functions, Riemann-Hilbert problems, direct and inverse spectral theory. Of special interest is the article of Percy Deift in which he discusses some open problems of Random Matrix Theory and the theory of integrable systems.

8 citations





Journal Article
TL;DR: Under certain regularity conditions on the norm, the functions in a nontrivial invariant subspace of the backward shift operator have meromorphic pseudocontinuations in the Nevanlinna class of the exterior of the unit disk as discussed by the authors.
Abstract: It is shown that under certain regularity conditions on the norm, the functions in a nontrivial invariant subspace of the backward shift operator have meromorphic pseudocontinuations in the Nevanlinna class of the exterior of the unit disk. In addition, we provide a regularity condition which implies that the subspace itself is contained in the Nevanlinna class of the disc.



Journal Article
TL;DR: Experimental evidence shows that this attack on a public key encryption scheme suggested by Shpilrain and Zapata is practical and works for the proposed parameters and a way to repair the encryption scheme is given.
Abstract: We give an attack on a public key encryption scheme suggested by Shpilrain and Zapata. Experimental evidence shows that this attack is practical and works for the proposed parameters. We give a way to repair the encryption scheme so that our attack does not work anymore. However, we also expose weak points of the scheme that do not seem to be repairable in an obvious manner.


Journal Article
TL;DR: For a group satisfying the Baum-Connes conjecture, finitely generated projectives over the reduced group C*-algebra satisfy a condition reminiscent of the Bass conjecture as discussed by the authors.
Abstract: We study the universal (Hattori-Stallings) trace on the K-theory of Banach algebras containing the complex group ring. As a result, we prove that for a group satisfying the Baum-Connes conjecture, finitely generated projectives over the reduced group C*-algebra satisfy a condition reminiscent of the Bass conjecture. An immediate consequence is that in the torsion-free case, the change of ring map from the reduced group C*-algebra to the von Neumann algebra of the group induces the zero map at the level of reduced K-theory.