A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula

Continuing our development of a mathematical theory of stochastic microlensing, we study the random shear and expected number of random lensed images of different types. In particular, we characterize the first three leading terms in the asymptotic expression of the joint probability density function (p.d.f.) of the random shear tensor at a general point in the lens plane due to point masses in the limit of an infinite number of stars. Up to this order, the p.d.f. depends on the magnitude of the shear tensor, the optical depth, and the mean number of stars through a combination of radial position and the stars' masses. As a consequence, the p.d.f.s of the shear components are seen to converge, in the limit of an infinite number of stars, to shifted Cauchy distributions, which shows that the shear components have heavy tails in that limit. The asymptotic p.d.f. of the shear magnitude in the limit of an infinite number of stars is also presented. Extending to general random distributions of the lenses, we employ the Kac-Rice formula and Morse theory to deduce general formulas for the expected total number of images and the expected number of saddle images. We further generalize these results by considering random sources defined on a countable compact covering of the light source plane. This is done to introduce the notion of {\it global} expected number of positive parity images due to a general lensing map. Applying the result to microlensing, we calculate the asymptotic global expected number of minimum images in the limit of an infinite number of stars, where the stars are uniformly distributed. This global expectation is bounded, while the global expected number of images and the global expected number of saddle images diverge as the order of the number of stars.Comment: To appear in JM

Accelerator Design for the CHESS-U Upgrade

During the summer and fall of 2018 the Cornell High Energy Synchrotron Source (CHESS) is undergoing an upgrade to increase high-energy flux for x-ray users. The upgrade requires replacing one-sixth of the Cornell Electron Storage Ring (CESR), inverting the polarity of half of the CHESS beam lines, and switching to single-beam on-axis operation. The new sextant is comprised of six double-bend achromats (DBAs) with combined-function dipole-quadrupoles. Although the DBA design is widely utilized and well understood, the constraints for the CESR modifications make the CHESS-U lattice unique. This paper describes the design objectives, constraints, and implementation for the CESR accelerator upgrade for CHESS-U

Large droplet impact on water layers

The impact of large droplets onto an otherwise undisturbed layer of water is considered. The work, which is motivated primarily with regard to aircraft icing, is to try and help understand the role of splashing on the formation of ice on a wing, in particular for large droplets where splash appears, to have a significant effect. Analytical and numerical approaches are used to investigate a single droplet impact onto a water layer. The flow for small times after impact is determined analytically, for both direct and oblique impacts. The impact is also examined numerically using the volume of fluid (VOF) method. At small times there are promising comparisons between the numerical results, the analytical solution and experimental work capturing the ejector sheet. At larger times there is qualitative agreement with experiments and related simulations. Various cases are considered, varying the droplet size to layer depth ratio, including surface roughness, droplet distortion and air effects. The amount of fluid splashed by such an impact is examined and is found to increase with droplet size and to be significantly influenced by surface roughness. The makeup of the splash is also considered, tracking the incoming fluid, and the splash is found to consist mostly of fluid originating in the layer

Measurement and Compensation of Horizontal Crabbing at the Cornell Electron Storage Ring Test Accelerator

In storage rings, horizontal dispersion in the rf cavities introduces horizontal-longitudinal (xz) coupling, contributing to beam tilt in the xz plane. This coupling can be characterized by a "crabbing" dispersion term {\zeta}a that appears in the normal mode decomposition of the 1-turn transfer matrix. {\zeta}a is proportional to the rf cavity voltage and the horizontal dispersion in the cavity. We report experiments at the Cornell Electron Storage Ring Test Accelerator (CesrTA) where xz coupling was explored using three lattices with distinct crabbing properties. We characterize the xz coupling for each case by measuring the horizontal projection of the beam with a beam size monitor. The three lattice configurations correspond to a) 16 mrad xz tilt at the beam size monitor source point, b) compensation of the {\zeta}a introduced by one of two pairs of RF cavities with the second, and c) zero dispersion in RF cavities, eliminating {\zeta}a entirely. Additionally, intrabeam scattering (IBS) is evident in our measurements of beam size vs. rf voltage.Comment: 5 figures, 10 page

Edge scaling limits for a family of non-Hermitian random matrix ensembles

A family of random matrix ensembles interpolating between the GUE and the Ginibre ensemble of $n\times n$ matrices with iid centered complex Gaussian entries is considered. The asymptotic spectral distribution in these models is uniform in an ellipse in the complex plane, which collapses to an interval of the real line as the degree of non-Hermiticity diminishes. Scaling limit theorems are proven for the eigenvalue point process at the rightmost edge of the spectrum, and it is shown that a non-trivial transition occurs between Poisson and Airy point process statistics when the ratio of the axes of the supporting ellipse is of order $n^{-1/3}$. In this regime, the family of limiting probability distributions of the maximum of the real parts of the eigenvalues interpolates between the Gumbel and Tracy-Widom distributions.Comment: 44 page

A relativistically covariant version of Bohm's quantum field theory for the scalar field

We give a relativistically covariant, wave-functional formulation of Bohm's quantum field theory for the scalar field based on a general foliation of space-time by space-like hypersurfaces. The wave functional, which guides the evolution of the field, is space-time-foliation independent but the field itself is not. Hence, in order to have a theory in which the field may be considered a beable, some extra rule must be given to determine the foliation. We suggest one such rule based on the eigen vectors of the energy-momentum tensor of the field itself.Comment: 1 figure. Submitted to J Phys A. 20/05/04 replacement has additional references and a few minor changes made for clarity. Accepted by J Phys

Capacity Market for Distribution System Operator – with Reliability Transactions – Considering Critical Loads and Microgrids

Conventional distribution system (DS) asset planning methods consider energy only from transmission systems (TS) and not from distributed energy resources (DER), leading to expensive plans. Newer transactive energy DS (TEDS) asset planning models, built on capacity market mechanisms, consider energy from both TS and DERs, leading to lower-cost plans and maximizing social welfare. However, in both methods the cost of higher reliability requirements for some users are socialized across all users, leading to lower social welfare. In this paper, a novel transactive energy capacity market (TECM) model is proposed for DS asset planning. It builds on TEDS incremental capacity auction models by provisioning for critical loads to bid and receive superior reliability as a service. The TECM model considers these reliability transactions, in addition, to selling energy transactions from TS and DERs, buying energy transactions from loads, and asset upgrade transactions from the network operator. The TECM model allows for islanded microgrids and network reconfiguration to maximize social welfare. The TECM model is assessed on several case studies, demonstrating that it achieves higher social welfare and a lower plan cost

Análise de endemismo de táxons neotropicais de Pentatomidae (Hemiptera: Heteroptera)

The definition of areas of endemism is central to studies of historical biogeography, and their interrelationships are fundamental questions. Consistent hypotheses for the evolution of Pentatomidae in the Neotropical region depend on the accuracy of the units employed in the analyses, which in the case of studies of historical biogeography, may be areas of endemism. In this study, the distribution patterns of 222 species, belonging to 14 Pentatomidae (Hemiptera) genera, predominantly neotropical, were studied with the Analysis of Endemicity (NDM) to identify possible areas of endemism and to correlate them to previously delimited areas. The search by areas of endemism was carried out using grid-cell units of 2.5° and 5° latitude-longitude. The analysis based on groupings of grid-cells of 2.5° of latitude-longitude allowed the identification of 51 areas of endemism, the consensus of these areas resulted in four clusters of grid-cells. The second analysis, with grid-cells units of 5° latitude-longitude, resulted in 109 areas of endemism. The flexible consensus employed resulted in 17 areas of endemism. The analyses were sensitive to the identification of areas of endemism in different scales in the Atlantic Forest. The Amazonian region was identified as a single area in the area of consensus, and its southeastern portion shares elements with the Chacoan and Paraná subregions. The distribution data of the taxa studied, with different units of analysis, did not allow the identification of individual areas of endemism for the Cerrado and Caatinga. The areas of endemism identified here should be seen as primary biogeographic hypotheses.A definição de áreas de endemismo é central aos estudos de Biogeografia Histórica e suas inter-relações são questões fundamentais. Hipóteses consistentes sobre a evolução de Pentatomidae (Hemiptera) na Região Neotropical dependem da acuidade das unidades empregadas nas análises, que no caso de estudos de biogeografia histórica, podem ser áreas endêmicas. Neste trabalho foram estudados os padrões de distribuição de 222 espécies, pertencentes a 14 gêneros de Pentatomidae, com ocorrência predominantemente neotropical, com base em uma Análise de Endemicidade (NDM) a fim de inferir possíveis áreas endêmicas e relacioná-las a áreas previamente delimitadas. A busca por áreas endêmicas foi realizada com quadrículas de 2,5° e 5° latitude-longitude. A análise com base em agrupamentos de 2,5° latitude-longitude permitiu identificar 51 áreas de endemismo, sendo que o consenso destas áreas resultou em quatro agrupamentos de quadrículas. A segunda análise, com quadrículas de 5° latitude-longitude, resultou em 109 áreas de endemismo. O consenso flexível empregado resultou em 17 áreas de endemismo. As análises foram sensíveis à identificação de áreas de endemismo na Mata Atlântica em diferentes escalas. A região Amazônica foi identificada como uma área única no consenso, sendo que a porção sudeste compartilha elementos com as sub-regiões do Chaco e Paraná. Os dados de distribuição dos táxons estudados, com diferentes unidades de análises, não permitiram a identificação de áreas endêmicas para o Cerrado e a Caatinga. As áreas de endemismo aqui identificadas devem ser tratadas como hipóteses biogeográficas primárias.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Universidade Federal do Rio Grande do Sul Laboratório de Entomologia Sistemática Departamento de ZoologiaUniversidade Federal do Paraná Departamento de Zoologia Programa de Pós-Graduação em EntomologiaUniversidade Federal de São Paulo (UNIFESP) Departamento de Ciências BiológicasUNIFESP, Depto. de Ciências BiológicasSciEL