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

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

A Mathematical Theory of Stochastic Microlensing I. Random Time-Delay Functions and Lensing Maps

Stochastic microlensing is a central tool in probing dark matter on galactic scales. From first principles, we initiate the development of a mathematical theory of stochastic microlensing. Beginning with the random time delay function and associated lensing map, we determine exact expressions for the mean and variance of these transformations. We characterize the exact p.d.f. of a normalized random time delay function at the origin, showing that it is a shifted gamma distribution, which also holds at leading order in the limit of a large number of point masses at a general point of the lens plane. For the large number of point masses limit, we also prove that the asymptotic p.d.f. of the random lensing map under a specified scaling converges to a bivariate normal distribution. We show analytically that the p.d.f. of the random scaled lensing map at leading order depends on the magnitude of the scaled bending angle due purely to point masses as well as demonstrate explicitly how this radial symmetry is broken at the next order. Interestingly, we found at leading order a formula linking the expectation and variance of the normalized random time delay function to the first Betti number of its domain. We also determine an asymptotic p.d.f. for the random bending angle vector and find an integral expression for the probability of a lens plane point being near a fixed point. Lastly, we show explicitly how the results are affected by location in the lens plane. The results of this paper are relevant to the theory of random fields and provide a platform for further generalizations as well as analytical limits for checking astrophysical studies of stochastic microlensing.Comment: New layout, more details and discussion. To appear, Journal of Mathematical Physic

Experience and entrepreneurship: a career transition perspective

We cast entrepreneurship as one of three career choices – remaining with one’s employer, changing employers, or engaging in entrepreneurship – and theorize how the likelihood of entrepreneurship evolves over one’s career. We empirically demonstrate an inverted U-shaped relationship between accumulated experience and entrepreneurship across various industries and jobs. Despite detailed career history data and job displacement shocks that eliminate the current employer choice, we highlight the difficulty of inferring the mechanism underlying the observed relationship. These analyses motivate a formal career transitions model in which employer-specific and general skills accumulate with experience but potential employers observe only total skill. The upshot of our model is that entrepreneurial career transitions vary with two relative costs: (1) to an individual of forming a business and (2) to a potential employer of utilizing the individual’s employer-specific skills. We discuss how this model contributes new insights into entrepreneurial careers

The effect of bariatric surgery type on cardiac reverse remodelling

Introduction: Bariatric surgery is effective in reversing adverse cardiac remodelling in obesity. However, it is unclear whether the three commonly performed operations; Roux-en-Y Gastric Bypass (RYGB), Laparoscopic Sleeve Gastrectomy (LSG) and Laparoscopic Adjustable Gastric Band (LAGB) are equal in their ability to reverse remodelling. Methods: Fifty-eight patients underwent CMR to assess left ventricular mass (LVM), LV mass:volume ratio (LVMVR) and LV eccentricity index (LVei) before and after bariatric surgery (26 RYGB, 22 LSG and 10 LAGB), including 46 with short-term (median 251–273 days) and 43 with longer-term (median 983–1027 days) follow-up. Abdominal visceral adipose tissue (VAT) and epicardial adipose tissue (EAT) were also assessed. Results: All three procedures resulted in significant decreases in excess body weight (48–70%). Percentage change in VAT and EAT was significantly greater following RYGB and LSG compared to LAGB at both timepoints (VAT:RYGB −47% and −57%, LSG −47% and −54%, LAGB −31% and −25%; EAT:RYGB −13% and −14%, LSG –16% and −19%, LAGB −5% and −5%). Patients undergoing LAGB, whilst having reduced LVM (−1% and −4%), had a smaller decrease at both short (RYGB: −8%, p < 0.005; LSG: −11%, p < 0.0001) and long (RYGB: −12%, p = 0.009; LSG: −13%, p < 0.0001) term timepoints. There was a significant decrease in LVMVR at the long-term timepoint following both RYGB (−7%, p = 0.006) and LSG (−7%, p = 0.021), but not LAGB (−2%, p = 0.912). LVei appeared to decrease at the long-term timepoint in those undergoing RYGB (−3%, p = 0.063) and LSG (−4%, p = 0.015), but not in those undergoing LAGB (1%, p = 0.857). In all patients, the change in LVM correlated with change in VAT (r = 0.338, p = 0.0134), while the change in LVei correlated with change in EAT (r = 0.437, p = 0.001). Conclusions: RYGB and LSG appear to result in greater decreases in visceral adiposity, and greater reverse LV remodelling with larger reductions in LVM, concentric remodelling and pericardial restraint than LAGB

Large deviations of the maximal eigenvalue of random matrices

We present detailed computations of the 'at least finite' terms (three dominant orders) of the free energy in a one-cut matrix model with a hard edge a, in beta-ensembles, with any polynomial potential. beta is a positive number, so not restricted to the standard values beta = 1 (hermitian matrices), beta = 1/2 (symmetric matrices), beta = 2 (quaternionic self-dual matrices). This model allows to study the statistic of the maximum eigenvalue of random matrices. We compute the large deviation function to the left of the expected maximum. We specialize our results to the gaussian beta-ensembles and check them numerically. Our method is based on general results and procedures already developed in the literature to solve the Pastur equations (also called "loop equations"). It allows to compute the left tail of the analog of Tracy-Widom laws for any beta, including the constant term.Comment: 62 pages, 4 figures, pdflatex ; v2 bibliography corrected ; v3 typos corrected and preprint added ; v4 few more numbers adde

Lily Day, Wednesday, March 4, 1964

Comments on past research in Ohio on Lilies / D. C. Kiplinger -- Lily investigations / Abdel-Alim M. Shoushan -- Further studies on causes and control of leaf scorch in Croft Easter lily / N. W. Stuart, William Skou, and D. C. Kiplinger -- Fertilizer and lime affect amount of leaf scorch in Croft Easter lilies / Neil W. Stuart, K. S. Nelson and D. C. Kiplinger -- Comparative development of Ace and Nellie White lilies / Dennis Rider and D. C. Kiplinger -- Experiments with potted lilies, 1961-1962 / D. C. Kiplinger, Robert O. Miller, Howard Jones, and Dennis Rider -- Lily culture and timing for Easter, 1964 / D. C. Kiplinger and Robert O. Miller -- High temperature treatment of Easter lily bulbs / Robert O. Miller and D. C. Kiplinger -- An investigation of causes of variation in the growth of commercial and experimental lilies / Robert O. Miller and D. C. Kiplinger -- Applying terraclor with or without dexon / D. C. Kiplinger, Robert O. Miller and Leonard J. Her

One-component plasma on a spherical annulus and a random matrix ensemble

The two-dimensional one-component plasma at the special coupling \beta = 2 is known to be exactly solvable, for its free energy and all of its correlations, on a variety of surfaces and with various boundary conditions. Here we study this system confined to a spherical annulus with soft wall boundary conditions, paying special attention to the resulting asymptotic forms from the viewpoint of expected general properties of the two-dimensional plasma. Our study is motivated by the realization of the Boltzmann factor for the plasma system with \beta = 2, after stereographic projection from the sphere to the complex plane, by a certain random matrix ensemble constructed out of complex Gaussian and Haar distributed unitary matrices.Comment: v2, typos and references corrected, 24 pages, 1 figur

The Grizzly, November 10, 1978

Physical Education Program To Change • Task Force Continues Recommendations • No Funds For Bomberger • Forum: High Strung • Hockey Not Safe • Staffer Clears Misinterpretation • Dining Service Transitions • Letters to the Editor • Portrait of the Professor: Gayle A. Byerly • For Whom The Walls Toil • Egdon Heath - A New Look For Monday Night • The Good Doctor Makes House Call To Protheatre • Eighteen Named to Who\u27s Who • Free V. D. Clinic • GM: Looking Good For \u2779 • Sports Profile: Keith Kemper • Thinclads Nab Third At MAC\u27s • Soccer Kicks Moravian • Bears Blast Dickinson • Gymnastics Get New Coach • Hockey Ends • Women\u27s B-Ball Preview • News in Brief: Senior Symposium Cancelled; Deans Attend State Conventionhttps://digitalcommons.ursinus.edu/grizzlynews/1006/thumbnail.jp

Capillary filling with wall corrugations] Capillary filling in microchannels with wall corrugations: A comparative study of the Concus-Finn criterion by continuum, kinetic and atomistic approaches

We study the impact of wall corrugations in microchannels on the process of capillary filling by means of three broadly used methods - Computational Fluid Dynamics (CFD), Lattice-Boltzmann Equations (LBE) and Molecular Dynamics (MD). The numerical results of these approaches are compared and tested against the Concus-Finn (CF) criterion, which predicts pinning of the contact line at rectangular ridges perpendicular to flow for contact angles theta > 45. While for theta = 30, theta = 40 (no flow) and theta = 60 (flow) all methods are found to produce data consistent with the CF criterion, at theta = 50 the numerical experiments provide different results. Whilst pinning of the liquid front is observed both in the LB and CFD simulations, MD simulations show that molecular fluctuations allow front propagation even above the critical value predicted by the deterministic CF criterion, thereby introducing a sensitivity to the obstacle heigth.Comment: 25 pages, 8 figures, Langmuir in pres