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 (pdf) of the random shear tensor due to point masses in the limit of an infinite number of stars. Up to this order, the pdf 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 star's mass. As a consequence, the pdf'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 pdf of the shear magnitude in the limit of an infinite number of stars is also presented. All the results on the random microlensing shear are given for a general point in the lens plane. Extending to the general random distributions (not necessarily uniform) 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 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. © 2009 American Institute of Physics.

Accelerating self-consistent field convergence with the augmented Roothaan-Hall energy function.

Based on Pulay's direct inversion iterative subspace (DIIS) approach, we present a method to accelerate self-consistent field (SCF) convergence. In this method, the quadratic augmented Roothaan-Hall (ARH) energy function, proposed recently by Høst and co-workers [J. Chem. Phys. 129, 124106 (2008)], is used as the object of minimization for obtaining the linear coefficients of Fock matrices within DIIS. This differs from the traditional DIIS of Pulay, which uses an object function derived from the commutator of the density and Fock matrices. Our results show that the present algorithm, abbreviated ADIIS, is more robust and efficient than the energy-DIIS (EDIIS) approach. In particular, several examples demonstrate that the combination of ADIIS and DIIS ("ADIIS+DIIS") is highly reliable and efficient in accelerating SCF convergence.

Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory

© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.A key component in calculations of exchange and correlation energies is the Coulomb operator, which requires the evaluation of two-electron integrals. For localized basis sets, these four-center integrals are most efficiently evaluated with the resolution of identity (RI) technique, which expands basis-function products in an auxiliary basis. In this work we show the practical applicability of a localized RI-variant ('RI-LVL'), which expands products of basis functions only in the subset of those auxiliary basis functions which are located at the same atoms as the basis functions. We demonstrate the accuracy of RI-LVL for Hartree-Fock calculations, for the PBE0 hybrid density functional, as well as for RPA and MP2 perturbation theory. Molecular test sets used include the S22 set of weakly interacting molecules, the G3 test set, as well as the G2-1 and BH76 test sets, and heavy elements including titanium dioxide, copper and gold clusters. Our RI-LVL implementation paves the way for linear-scaling RI-based hybrid functional calculations for large systems and for all-electron many-body perturbation theory with significantly reduced computational and memory cost.

A Measurement of the Proton Structure Function g2p at Low Q2

Experiments at Jefferson Lab have been conducted to extract the nucleon spin-dependent structure functions over a wide kinematic range. Higher moments of these quantities provide tests of QCD sum rules and predictions of chiral perturbation theory ($\chi$PT). While precise measurements of $g_{1}^n$, $g_{2}^n$, and $g_1^p$ have been extensively performed, the data of $g_2^p$ remain scarce. Discrepancies were found between existing data related to $g_2$ and theoretical predictions. Results on the proton at large $Q^2$ show a significant deviation from the Burkhardt-Cottingham sum rule, while results for the neutron generally follow this sum rule. The next-to-leading order $\chi$PT calculations exhibit discrepancy with data on the longitudinal-transverse polarizability $\delta_{LT}^n$. Further measurements of the proton spin structure function $g_2^p$ are desired to understand these discrepancies.

Experiment E08-027 (g2p) was conducted at Jefferson Lab in experimental Hall A in 2012. Inclusive measurements were performed with polarized electron beam and a polarized ammonia target to obtain the proton spin-dependent structure function $g_2^p$ at low Q$^2$ region (0.02$<$Q$^2$$<$0.2 GeV$^2$) for the first time. The results can be used to test the Burkhardt-Cottingham sum rule, and also allow us to extract the longitudinal-transverse spin polarizability of the proton, which will provide a benchmark test of $\chi$PT calculations. This thesis will present and discuss the very preliminary results of the transverse asymmetry and the spin-dependent structure functions $g_1^p$ and $g_2^p$ from the data analysis of the g2p experiment .

The Market for Apples: A Theory of Identity and Consumption

This paper presents an economic model of the effects of identity and social norms on consumption patterns. By incorporating qualitative studies in psychology and sociology, I propose a utility function that features two components – economic (functional) and identity elements. This setup is extended to analyze a market comprising a continuum of consumers, whose identity distribution along a spectrum of binary identities is described by a Beta distribution. I also introduce the notion of salience in the context of identity and consumption decisions. The key result of the model suggests that fundamental economic parameters, such as price elasticity and market demand, can be altered by identity elements. In addition, it predicts that firms in perfectly competitive markets may associate their products with certain types of identities, in order to reduce product substitutability and attain price-setting power.

Theory and Practice in Sustainability Science: Influence of Urban Form on the Urban Heat Island and Implications for Urban Systems

As the world population continues to grow past seven billion people and global challenges continue to persist including resource availability, biodiversity loss, climate change and human well-being, a new science is required that can address the integrated nature of these challenges and the multiple scales on which they are manifest. Sustainability science has emerged to fill this role. In the fifteen years since it was first called for in the pages of Science, it has rapidly matured, however its place in the history of science and the way it is practiced today must be continually evaluated. In Part I, two chapters address this theoretical and practical grounding. Part II transitions to the applied practice of sustainability science in addressing the urban heat island (UHI) challenge wherein the climate of urban areas are warmer than their surrounding rural environs. The UHI has become increasingly important within the study of earth sciences given the increased focus on climate change and as the balance of humans now live in urban areas.

In Chapter 2 a novel contribution to the historical context of sustainability is argued. Sustainability as a concept characterizing the relationship between humans and nature emerged in the mid to late 20th century as a response to findings used to also characterize the Anthropocene. Emerging from the human-nature relationships that came before it, evidence is provided that suggests Sustainability was enabled by technology and a reorientation of world-view and is unique in its global boundary, systematic approach and ambition for both well being and the continued availability of resources and Earth system function. Sustainability is further an ambition that has wide appeal, making it one of the first normative concepts of the Anthropocene.

Despite its widespread emergence and adoption, sustainability science continues to suffer from definitional ambiguity within the academe. In Chapter 3, a review of efforts to provide direction and structure to the science reveals a continuum of approaches anchored at either end by differing visions of how the science interfaces with practice (solutions). At one end, basic science of societally defined problems informs decisions about possible solutions and their application. At the other end, applied research directly affects the options available to decision makers. While clear from the literature, survey data further suggests that the dichotomy does not appear to be as apparent in the minds of practitioners.

In Chapter 4, the UHI is first addressed at the synoptic, mesoscale. Urban climate is the most immediate manifestation of the warming global climate for the majority of people on earth. Nearly half of those people live in small to medium sized cities, an understudied scale in urban climate research. Widespread characterization would be useful to decision makers in planning and design. Using a multi-method approach, the mesoscale UHI in the study region is characterized and the secular trend over the last sixty years evaluated. Under isolated ideal conditions the findings indicate a UHI of 5.3 ± 0.97 °C to be present in the study area, the magnitude of which is growing over time.

Although urban heat islands (UHI) are well studied, there remain no panaceas for local scale mitigation and adaptation methods, therefore continued attention to characterization of the phenomenon in urban centers of different scales around the globe is required. In Chapter 5, a local scale analysis of the canopy layer and surface UHI in a medium sized city in North Carolina, USA is conducted using multiple methods including stationary urban sensors, mobile transects and remote sensing. Focusing on the ideal conditions for UHI development during an anticyclonic summer heat event, the study observes a range of UHI intensity depending on the method of observation: 8.7 °C from the stationary urban sensors; 6.9 °C from mobile transects; and, 2.2 °C from remote sensing. Additional attention is paid to the diurnal dynamics of the UHI and its correlation with vegetation indices, dewpoint and albedo. Evapotranspiration is shown to drive dynamics in the study region.

Finally, recognizing that a bridge must be established between the physical science community studying the Urban Heat Island (UHI) effect, and the planning community and decision makers implementing urban form and development policies, Chapter 6 evaluates multiple urban form characterization methods. Methods evaluated include local climate zones (LCZ), national land cover database (NCLD) classes and urban cluster analysis (UCA) to determine their utility in describing the distribution of the UHI based on three standard observation types 1) fixed urban temperature sensors, 2) mobile transects and, 3) remote sensing. Bivariate, regression and ANOVA tests are used to conduct the analyses. Findings indicate that the NLCD classes are best correlated to the UHI intensity and distribution in the study area. Further, while the UCA method is not useful directly, the variables included in the method are predictive based on regression analysis so the potential for better model design exists. Land cover variables including albedo, impervious surface fraction and pervious surface fraction are found to dominate the distribution of the UHI in the study area regardless of observation method.

Chapter 7 provides a summary of findings, and offers a brief analysis of their implications for both the scientific discourse generally, and the study area specifically. In general, the work undertaken does not achieve the full ambition of sustainability science, additional work is required to translate findings to practice and more fully evaluate adoption. The implications for planning and development in the local region are addressed in the context of a major light-rail infrastructure project including several systems level considerations like human health and development. Finally, several avenues for future work are outlined. Within the theoretical development of sustainability science, these pathways include more robust evaluations of the theoretical and actual practice. Within the UHI context, these include development of an integrated urban form characterization model, application of study methodology in other geographic areas and at different scales, and use of novel experimental methods including distributed sensor networks and citizen science.