An Optimal Dimensionality Sampling Scheme on the Sphere for Antipodal Signals In Diffusion Magnetic Resonance Imaging

We propose a sampling scheme on the sphere and develop a corresponding spherical harmonic transform (SHT) for the accurate reconstruction of the diffusion signal in diffusion magnetic resonance imaging (dMRI). By exploiting the antipodal symmetry, we design a sampling scheme that requires the optimal number of samples on the sphere, equal to the degrees of freedom required to represent the antipodally symmetric band-limited diffusion signal in the spectral (spherical harmonic) domain. Compared with existing sampling schemes on the sphere that allow for the accurate reconstruction of the diffusion signal, the proposed sampling scheme reduces the number of samples required by a factor of two or more. We analyse the numerical accuracy of the proposed SHT and show through experiments that the proposed sampling allows for the accurate and rotationally invariant computation of the SHT to near machine precision accuracy.Comment: Will be published in the proceedings of the International Conference Acoustics, Speech and Signal Processing 2015 (ICASSP'2015

Associated primes of graded components of local cohomology modules

The i-th local cohomology module of a finitely generated graded module M over a standard positively graded commutative Noetherian ring R with respect to the irrelevant ideal R+, is itself graded; all its graded components are finitely generated modules over R-0, the component of R of degree 0. It is known that the n-th component H-R+(i) (M)(n) of this local cohomology module H-R+(i) (M) is zero for all nmuch greater than0. This paper is concerned with the asymptotic behaviour of Ass(R0)(H-R+(i) (M)(n)) as n--> -infinity. The smallest i for which such study is interesting is the finiteness dimension f of M relative to R+, defined as the least integer j for which H-R+(j) (M) is not finitely generated. Brodmann and Hellus have shown that AssR(0)(H-R+(f) (M)(n)) is constant for all nmuch less than0 ( that is in their terminology AssR(0)(H-R+(f) (M)(n)) is asymptotically stable for n--> -infinity). The first main aim of this paper is to identify the ultimate constant value ( under the mild assumption that R is a homomorphic image of a regular ring) : our answer is precisely the set of contractions to R-0 of certain relevant primes of R whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when i>f. They noted that Singh's study of a particular example ( in which f=2) shows that AssR(0)(H-R+(3) (R)(n)) need not be asymptotically stable for n--> -infinity. The second main aim of this paper is to determine, for Singh's example, AssR(0)(H-R+(3) (R)(n)) quite precisely for every integer n and, thereby answer one of the questions raised by Brodmann and Hellus

Luminosity Distributions within Rich Clusters - II: Demonstration and Verification via Simulation

We present detailed simulations of long exposure CCD images. The simulations are used to explore the validity of the statistical method for reconstructing the luminosity distribution of galaxies within a rich cluster i.e. by the subtraction of field number-counts from those of a sight-line through the cluster. In particular we use the simulations to establish the reliability of our observational data presented in Paper 3. Based on our intended CCD field-of-view (6.5 by 6.5 arcmins) and a 1-sigma detection limit of 26 mags per sq arcsecond, we conclude that the luminosity distribution can be robustly determined over a wide range of absolute magnitude (-23 < M_{R} < -16) provided: (a) the cluster has an Abell richness 1.5 or greater, (b) the cluster's redshift lies in the range 0.1 < z < 0.3, (c) the seeing is better than FWHM 1.25'' and (d) the photometric zero points are accurate to within Delta m = \pm 0.12. If these conditions are not met then the recovered luminosity distribution is unreliable and potentially grossly miss-leading. Finally although the method clearly has limitations, within these limitations the technique represents an extremely promising probe of galaxy evolution and environmental dependencies.Comment: 24 pages, 8 figures accepted for publication in MNRAS also available from http://star-www.st-and.ac.uk/~spd3/bib.htm

Dynamics of scalar dissipation in isotropic turbulence: a numerical and modelling study

The physical mechanisms underlying the dynamics of the dissipation of passive scalar fluctuations with a uniform mean gradient in stationary isotropic turbulence are studied using data from direct numerical simulations (DNS), at grid resolutions up to 5123. The ensemble-averaged Taylor-scale Reynolds number is up to about 240 and the Schmidt number is from ⅛ to 1. Special attention is given to statistics conditioned upon the energy dissipation rate because of their important role in the Lagrangian spectral relaxation (LSR) model of turbulent mixing. In general, the dominant physical processes are those of nonlinear amplification by strain rate fluctuations, and destruction by molecular diffusivity. Scalar dissipation tends to form elongated structures in space, with only a limited overlap with zones of intense energy dissipation. Scalar gradient fluctuations are preferentially aligned with the direction of most compressive strain rate, especially in regions of high energy dissipation. Both the nature of this alignment and the timescale of the resulting scalar gradient amplification appear to be nearly universal in regard to Reynolds and Schmidt numbers. Most of the terms appearing in the budget equation for conditional scalar dissipation show neutral behaviour at low energy dissipation but increased magnitudes at high energy dissipation. Although homogeneity requires that transport terms have a zero unconditional average, conditional molecular transport is found to be significant, especially at lower Reynolds or Schmidt numbers within the simulation data range. The physical insights obtained from DNS are used for a priori testing and development of the LSR model. In particular, based on the DNS data, improved functional forms are introduced for several model coefficients which were previously taken as constants. Similar improvements including new closure schemes for specific terms are also achieved for the modelling of conditional scalar variance

Creating Online Worlds: The Future Of Student Work

Traditional schoolwork is out of step with twenty-first century life and work. In the future, student work will focus on creating interactive online worlds instead of on writing paper-print essays

Supplying the Nuclear Arsenal

Originally published in 1996. Although the history of commercial-power nuclear reactors is well known, the story of the government reactors that produce weapons-grade plutonium and tritium has been shrouded in secrecy. Supplying the Nuclear Arsenal looks at the origin and development of these production reactors, Rodney Carlisle and Joan Zenzen describe a fifty-year government effort no less complex, expensive, and technologically demanding than the Polaris or Apollo programs—yet one about which most Americans know virtually nothing. Carlisle and Zenzen describe the evolution of the early reactors, the atomic weapons establishment that surrounded them, and the sometimes bitter struggles between business and political constituencies for their share of "nuclear pork." They show how, since the 1980s, aging production reactors have increased the risk of radioactive contamination of the atmosphere and water table. And they describe how the Department of Energy mounted a massive effort to find the right design for a new generation of reactors, only to abandon that effort with the end of the Cold War. Today, all American production reactors remain closed.Due to short half-life, the nation's supply of tritium, crucial to modern weapons, is rapidly dwindling. As countries like Iraq and North Korea threaten to join the nuclear club, the authors contend, the United States needs to revitalize tritium production capacity in order to maintain a viable nuclear deterrent. Meanwhile, as slowly decaying artifacts of the Cold War, the closed production reactors at Hanford, Washington, and Savannah River, South Carolina, loom ominously over the landscape

The Future Of Online Courses: How To Win The Marketing War

Many colleges are looking to online education to improve their revenue stream. Competition for online students is escalating. Designing online courses like video games can help win the marketing war