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.

A theory of buyer-seller networks

This paper introduces a new model of exchange: networks, rather than markets, of buyers and sellers. It begins with the empirically motivated premise that a buyer and seller must have a relationship, a "link," to exchange goods. Networks - buyers, sellers, and the pattern of links connecting them - are common exchange environments. This paper develops a methodology to study network structures and explains why agents may form networks. In a model that captures characteristics of a variety of industries, the paper shows that buyers and sellers, acting strategically in their own self-interests, can form the network structures that maximize overall welfare.

Hannah Arendt's Theory of Political Judgment

Hannah Arendt's theory of political judgment has been an ongoing perplexity among scholars who have written on her. As a result, her theory of judgment is often treated as a suggestive but unfinished aspect of her thought. Drawing on a wider array of sources than is commonly utilized, I argue that her theory of political judgment was in fact the heart of her work. Arendt's project, in other words, centered around reestablishing the possibility of political judgment in a modern world that historically has progressively undermined it. In the dissertation, I systematically develop an account of Arendt's fundamentally political and non-sovereign notion of judgment. We discover that individual judgment is not arbitrary, and that even in the complex circumstances of the modern world there are valid structures of judgment which can be developed and dependably relied upon. The result of this work articulates a theory of practical reason which is highly compelling: it provides orientation for human agency which does not rob it of its free and spontaneous character; shows how we can improve and cultivate our political judgment; and points the way toward the profoundly intersubjective form of political philosophy Arendt ultimately hoped to develop.

A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

Event memory: A theory of memory for laboratory, autobiographical, and fictional events.

An event memory is a mental construction of a scene recalled as a single occurrence. It therefore requires the hippocampus and ventral visual stream needed for all scene construction. The construction need not come with a sense of reliving or be made by a participant in the event, and it can be a summary of occurrences from more than one encoding. The mental construction, or physical rendering, of any scene must be done from a specific location and time; this introduces a "self" located in space and time, which is a necessary, but need not be a sufficient, condition for a sense of reliving. We base our theory on scene construction rather than reliving because this allows the integration of many literatures and because there is more accumulated knowledge about scene construction's phenomenology, behavior, and neural basis. Event memory differs from episodic memory in that it does not conflate the independent dimensions of whether or not a memory is relived, is about the self, is recalled voluntarily, or is based on a single encoding with whether it is recalled as a single occurrence of a scene. Thus, we argue that event memory provides a clearer contrast to semantic memory, which also can be about the self, be recalled voluntarily, and be from a unique encoding; allows for a more comprehensive dimensional account of the structure of explicit memory; and better accounts for laboratory and real-world behavioral and neural results, including those from neuropsychology and neuroimaging, than does episodic memory.