Visualizing elements of Sha[3] in genus 2 jacobians

Mazur proved that any element xi of order three in the Shafarevich-Tate group of an elliptic curve E over a number field k can be made visible in an abelian surface A in the sense that xi lies in the kernel of the natural homomorphism between the cohomology groups H^1(k,E) -> H^1(k,A). However, the abelian surface in Mazur's construction is almost never a jacobian of a genus 2 curve. In this paper we show that any element of order three in the Shafarevich-Tate group of an elliptic curve over a number field can be visualized in the jacobians of a genus 2 curve. Moreover, we describe how to get explicit models of the genus 2 curves involved.Comment: 12 page

Bounds on the diameter of Cayley graphs of the symmetric group

In this paper we are concerned with the conjecture that, for any set of generators S of the symmetric group of degree n, the word length in terms of S of every permutation is bounded above by a polynomial of n. We prove this conjecture for sets of generators containing a permutation fixing at least 37% of the points.Comment: 17 pages, 6 table

Computing automorphic forms on Shimura curves over fields with arbitrary class number

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms.Comment: 15 pages; final submission to ANTS I

Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ can be isometrically embedded into or even be isometrically equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. We provide several examples, such as Mat\'ern kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are isometrically equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator $\mathbf{P}$. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the "best" kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D. thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}

Pathprinting: An integrative approach to understand the functional basis of disease

New strategies to combat complex human disease require systems approaches to biology that integrate experiments from cell lines, primary tissues and model organisms. We have developed Pathprint, a functional approach that compares gene expression profiles in a set of pathways, networks and transcriptionally regulated targets. It can be applied universally to gene expression profiles across species. Integration of large-scale profiling methods and curation of the public repository overcomes platform, species and batch effects to yield a standard measure of functional distance between experiments. We show that pathprints combine mouse and human blood developmental lineage, and can be used to identify new prognostic indicators in acute myeloid leukemia. The code and resources are available at http://​compbio.​sph.​harvard.​edu/​hidelab/​pathprin

Elliptic curves of large rank and small conductor

For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.Comment: 16 pages, including tables and one .eps figure; to appear in the Proceedings of ANTS-6 (June 2004, Burlington, VT). Revised somewhat after comments by J.Silverman on the previous draft, and again to get the correct page break

Disparate MgII Absorption Statistics towards Quasars and Gamma-Ray Bursts : A Possible Explanation

We examine the recent report by Prochter et al. (2006) that gamma-ray burst (GRB) sight lines have a much higher incidence of strong MgII absorption than quasar sight lines. We propose that the discrepancy is due to the different beam sizes of GRBs and quasars, and that the intervening MgII systems are clumpy with the dense part of each cloudlet of a similar size as the quasars, i.e. < 10^16 cm, but bigger than GRBs. We also discuss observational predictions of our proposed model. Most notably, in some cases the intervening MgII absorbers in GRB spectra should be seen varying, and quasars with smaller sizes should show an increased rate of strong MgII absorbers. In fact, our prediction of variable MgII lines in the GRB spectra has been now confirmed by Hao et al. (2007), who observed intervening FeII and MgII lines at z=1.48 to be strongly variable in the multi-epoch spectra of z=4.05 GRB060206.Comment: 12 pages, 2 figures; substantially revised model calculation; accepted for publication in Astrophysics & Space Science as a Lette

Indications for cardiovascular magnetic resonance in children with congenital and acquired heart disease: an expert consensus paper of the Imaging Working Group of the AEPC and the Cardiovascular Magnetic Resonance Section of the EACVI

This article provides expert opinion on the use of cardiovascular magnetic resonance (CMR) in young patients with congenital heart disease (CHD) and in specific clinical situations. As peculiar challenges apply to imaging children, paediatric aspects are repeatedly discussed. The first section of the paper addresses settings and techniques, including the basic sequences used in paediatric CMR, safety, and sedation. In the second section, the indication, application, and clinical relevance of CMR in the most frequent CHD are discussed in detail. In the current era of multimodality imaging, the strengths of CMR are compared with other imaging modalities. At the end of each chapter, a brief summary with expert consensus key points is provided. The recommendations provided are strongly clinically oriented. The paper addresses not only imagers performing CMR, but also clinical cardiologists who want to know which information can be obtained by CMR and how to integrate it in clinical decision-makin