Minimum Principles for Ill-Posed Problems

Ill-posed problems Ax = h are discussed in which A is Hermitian,and postive definite; a bound ║Bx║ ≤ β is prescribed. A minimum principle is given for an approximate solution x^. Comparisons are made with the least-squares solutions of K. Miller, A. Tikhonov, et al. Applications are made to deconvolution, the backward heat equation, and the inversion of ill-conditioned matrices. If A and B are positive-definite, commuting matrices, the approximation x^ is shown to be about as accurate as the least-squares solution and to be more quickly and accurately computable

Numerical Simulation of Stationary and Non-Stationary Gaussian Random Processes

The purpose of this paper is to present numerical methods for the computation of samples of a Gaussian random process x(t) with prescribed autocorrelation or power spectral density

Confidence Intervals for Stereological Estimators with Infinite Variance

A statistical estimator is discussed for using two-dimensional electron-microscope data to estimate NV, the number of organelles per unit volume. Under general assumptions, the estimator is shown to be the unique unbiased estimator of NV. Though the estimator has infinite variance, large samples are shown to yield an approximately normally distributed statistic from which confidence intervals for NV can be obtained

The Effect of a Refractory Period on the Power Spectrum of Neuronal Discharge

The interspike intervals in steady-state neuron firing are assumed to be independently and identically distributed random variables. In the simplest model discussed, each interval is assumed to be the sum of a random neuron refractory period and a statistically independent interval due to a stationary external process, whose statistics are assumed known. The power spectral density (hence the autocorrelation) of the composite neuron-firing renewal process is derived from the known spectrum of the external process and from the unknown spectrum of the neuron-refraction process. The results are applied to spike trains recorded in a previous study [2] of single neurons in the visual cortex of an awake monkey. Two models are demonstrated that may produce peaks in the power spectrum near 40 Hz

Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation

G. Fichera [1] and other authors have investigated partial differential equations of the form [Eq. 1.1] in which the matrix (aij(x)) is required to be semidefinite. Equations of this type occur in the theory of random processes. A numerical analysis of some equations of this type has been by Cannon and Hill [9]. In this paper we consider a particular boundary value problem [Eq. 1.2] where we require [Eq. 1.3] and [Eq. 1.4]. A problem of this sort was discussed analytically by W. Fleming [2], but he did not obtain an explicit solution for T(x,0). The solution T(x,y) is related to a randomly-accelerated particle whose position ξ(t) satisfies the stochastic differential equation [Eq. 1.5] where w(t) is white Gaussian noise. If the initial position and velocity are ξ(0) = x and ξ'(0) = y, where |x| < 1, then T(x,y) is the expected value of the first time at which the position ξ(t) equals ±1. We obtain an analytic solution for T(x,y) in terms of hypergeometric functions and confluent hypergeometric functions. We use this analytic solution to test the validity of numerical methods which are applicable to general elliptic-parabolic equations (1.1). We show that, even though the truncation error for the difference equations does not tend to zero, nevertheless the difference methods give convergence of the difference methods. Each difference method requires the solution of a large number of simultaneous linear difference equations. We give iterative methods for solving these equations, and we prove that the iterations converge

The Justified Lawman: Cowboy Killings in the Modern Era

Stereotypes advanced by the popular media, do not necessarily have the power to directly determine an individual\u27s ways of thinking, but do help frame and reinforce already existing cultural beliefs, particularly within the context of the seemingly innocuous narrative fictions of cable television. These narratives not only simplify complex ideas, but also can further entrench or justify harmful social relations. My contention here, is that the popular television program Justified does precisely this by normalizing police violence and the ways that the police underpin and reproduce profoundly disparate class and racialized social order. While in many ways a typical police procedural, Justified is a particularly unique and hence useful case for analysis because its setting and subject matter focuses almost wholly upon the transgressions of the mostly white, rural poor as opposed to other popular examples from the genre such as The Wire which are routinely set in so-called urban ghettos. The justified violence of police in this particular context then, offers insight into a unique cultural script depicting how largely white rural poor are swept up in a moral and spatial purge deemed altogether necessary because of their cultural and perhaps biological failings