Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group

We consider the asymptotic behavior as $n\to\infty$ of the spectra of random matrices of the form $$\frac{1}{\sqrt{n-1}}\sum_{k=1}^{n-1}Z_{nk}\rho_n ((k,k+1)),$$ where for each $n$ the random variables $Z_{nk}$ are i.i.d. standard Gaussian and the matrices $\rho_n((k,k+1))$ are obtained by applying an irreducible unitary representation $\rho_n$ of the symmetric group on $\{1,2,...,n\}$ to the transposition $(k,k+1)$ that interchanges $k$ and $k+1$ [thus, $\rho_n((k,k+1))$ is both unitary and self-adjoint, with all eigenvalues either +1 or -1]. Irreducible representations of the symmetric group on $\{1,2,...,n\}$ are indexed by partitions $\lambda_n$ of $n$. A consequence of the results we establish is that if $\lambda_{n,1}\ge\lambda_{n,2}\ge...\ge0$ is the partition of $n$ corresponding to $\rho_n$, $\mu_{n,1}\ge\mu_{n,2}\ge >...\ge0$ is the corresponding conjugate partition of $n$ (i.e., the Young diagram of $\mu_n$ is the transpose of the Young diagram of $\lambda_n$), $\lim_{n\to\infty}\frac{\lambda_{n,i}}{n}=p_i$ for each $i\ge1$, and $\lim_{n\to\infty}\frac{\mu_{n,j}}{n}=q_j$ for each $j\ge1$, then the spectral measure of the resulting random matrix converges in distribution to a random probability measure that is Gaussian with random mean $\theta Z$ and variance $1-\theta^2$, where $\theta$ is the constant $\sum_ip_i^2-\sum_jq_j^2$ and $Z$ is a standard Gaussian random variable.Comment: Published in at http://dx.doi.org/10.1214/08-AOP418 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Lensing Properties of Cored Galaxy Models

A method is developed to evaluate the magnifications of the images of galaxies with lensing potentials stratified on similar concentric ellipses. A simple contour integral is provided which enables the sums of the magnifications of even parity or odd parity or the central image to be easily calculated. The sums for pairs of images vary considerably with source position, while the signed sums can be remarkably uniform inside the tangential caustic in the absence of naked cusps. For a family of models in which the potential is a power-law of the elliptic radius, the number of visible images is found as a function of flattening, external shear and core radius. The magnification of the central image depends on the core radius and the slope of the potential. For typical source and lens redshifts, the missing central image leads to strong constraints; the mass distribution in the lensing galaxy must be nearly cusped, and the cusp must be isothermal or stronger. This is in accord with the cuspy cores seen in high resolution photometry of nearby, massive, early-type galaxies, which typically have the surface density falling like distance^{-1.3} outside a break radius of a few hundred parsecs. Cuspy cores by themselves can provide an explanation of the missing central images. Dark matter at large radii may alter the slope of the projected density; provided the slope remains isothermal or steeper and the break radius remains small, then the central image remains unobservable. The sensitivity of the radio maps must be increased fifty-fold to find the central images in abundance.Comment: 42 pages, 11 figures, ApJ in pres

No Need for Dark Matter in Galaxies?

Unhappily, there has been a maelstrom of problems for dark matter theories over the last few years and many serious difficulties still have no resolution in sight. This article reviews the evidence for dark matter in galaxies. The haloes built up by hierarchical merging in dark matter cosmogonies are cusped and dominated by dark matter at the center. Evidence from the microlensing optical depth towards Baade's Window and from dynamical modelling of the Galactic bar already suggests that the Galactic halo is not cusped. Similarly, evidence from the stability of unbarred disk galaxies, as well as the survival of fast bars in barred galaxies, suggests that the this result holds good more generally. Judged on the data from galactic scales alone, the case for dark matter is weak and non-standard theories of gravity provide a better description. Of course, non-standard theories of gravity have their own problems, but not on galactic scales.Comment: 8 pages, invited review for "IDM 2000: Third International Workshop on the Identification of Dark Matter", ed. N. Spooner (World Scientific

Microlensing Maps for the Galactic Bulge

Microlensing maps -- that is, contours of equal numbers of events per $10^6$ source stars -- are provided for the inner Galaxy under two alternative hypotheses : (1) the bulge is an oblate axisymmetric spheroid or (2) the bulge is a prolate bar. Oblate spheroids yield a total of $\sim 12$ events per year per $10^6$ stars at Baade's Window ($\sim 15$ events if the disk is maximal). The event rate is slightly lower for prolate bars viewed at $\sim 45^\circ$ and the maps have a characteristic asymmetry between positive and negative longitudes. Prolate bars can yield mild amplifications of the event rate if viewed almost down the long axis. The disk provides the dominant lensing population on the bulge major axis for |\ell | \gta 6^\circ. Measurements of the rate at major axis windows can test for disk dark matter or maximal disk models.Comment: 12 pages, Late

Sensitivity of the Lidar ratio to changes in size distribution and index of refraction

In order to invert lidar signals to obtain reliable extinction coefficients, sigma, a relationship between sigma and the backscatter coefficient, beta, must be given. These two coefficients are linearly related if the complex index of refraction, m, particle shape size distribution, N, does not change along the path illuminated by the laser beam. This, however, is generally not the case. An extensive Mie computation of the lidar ratio R = beta/sigma and the sensitivity of R to the changes in a parametric space defined by N and m were examined

Diagnosis and the management constituency of small-scale fisheries

Diagnosis and adaptive management can help improve the ability of small-scale fisheries (SSF) in the developing world to better cope with and adapt to both external drivers and internal sources of uncertainty. This paper presents a framework for diagnosis and adaptive management and discusses ways of implementing the first two phases of learning: diagnosis and mobilising an appropriate management constituency. The discussion addresses key issues and suggests suitable approaches and tools as well as numerous sources of further information. Diagnosis of a SSF defines the system to be managed, outlines the scope of the management problem in terms of threats and opportunities, and aims to construct realistic and desired future projections for the fishery. These steps can clarify objectives and lead to development of indicators necessary for adaptive management. Before management, however, it is important to mobilize a management constituency to enact change. Ways of identifying stakeholders and understanding both enabling and obstructive interactions and management structures are outlined. These preliminary learning phases for adaptive SSF management are expected to work best if legitimised by collaborative discussion among fishery stakeholders drawing on multiple knowledge systems and participatory approaches to assessment. (PDF contains 33 pages

Approaches and frameworks for management and research in small-scale fisheries in the developing world

Commonly adopted approaches to managing small-scale fisheries (SSFs) in developing countries do not ensure sustainability. Progress is impeded by a gap between innovative SSF research and slower-moving SSF management. The paper aims to bridge the gap by showing that the three primary bases of SSF management--ecosystem, stakeholders’ rights and resilience--are mutually consistent and complementary. It nominates the ecosystem approach as an appropriate starting point because it is established in national and international law and policy. Within this approach, the emerging resilience perspective and associated concepts of adaptive management and institutional learning can move management beyond traditional control and resource-use optimization, which largely ignore the different expectations of stakeholders; the complexity of ecosystem dynamics; and how ecological, social, political and economic subsystems are linked. Integrating a rights-based perspective helps balance the ecological bias of ecosystem-based and resilience approaches. The paper introduces three management implementation frameworks that can lend structure and order to research and management regardless of the management approach chosen. Finally, it outlines possible research approaches to overcome the heretofore limited capacity of fishery research to integrate across ecological, social and economic dimensions and so better serve the management objective of avoiding fishery failure by nurturing and preserving the ecological, social and institutional attributes that enable it to renew and reorganize itself. (PDF contains 29 pages