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

Detecting specific oscillatory regimes in the dynamics of erbium-doped fiber laser

A method for determining the oscillatory mode occurring in an erbiumdoped fiber laser with a modulated parameter is proposed. The method is based on using a continuous wavelet transform with a mother Morlet wavelet and analyzing the energy of the wavelet spectrum that corresponds to the relevant range of time scales

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 Grizzly, October 13, 1978

Frat Takes Charge: ZX To Clean New Men\u27s Dorm • Seniors Attack Teaching • News in Brief: Frosh Elections; Bause Gets Alumni Award; Espadas Presents Paper • Open Board Meeting • Portrait Of The Professor: Blanche Schultz • Yes and ELO: New Looks on Stage • Exhibit Coming • The Blue Oyster Cult -- Highly Underrated • The T.G. Party: A New Option At Ursinus • New Music Officers • Pancoast Honored by PACU • Byerly Speaks on Computer Innovations • Homecoming Excitement Builds • Harriers Overcome Injuries, Opposition • Hockey Gets A Lift • Football - Heartbreaker on Parent\u27s Day • Volleyball Rounduphttps://digitalcommons.ursinus.edu/grizzlynews/1002/thumbnail.jp

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

The Grizzly, October 20, 1978

Homecoming \u2778 Promises Color, Excitement • Judiciary Board Convicts Two • Shopping Center to Expand • On Personal Expression • Is Pledging All Fun and Games? • Ursinus\u27 Financial Aid Structure • SFARC Repairs Damage Policy • Gallagher Explores Amish • Springsteen & Dylan: Poet Laureates or Veritable Zeros? • The World\u27s Largest Hamburger • Paradise Lost: College Woods Gone Junkyard? • X-C: Dual Wins • Bears Fall Prey Again • Soccer Wins Five • News in Brief: Our New Look; Remember to Vote; Yom Kippur Celebration; Ursinus Announces Business Workshop; Library News Shortshttps://digitalcommons.ursinus.edu/grizzlynews/1003/thumbnail.jp