Entire Real Line Research Articles

Phase operator and phase fluctuations of spins

It is shown that a Hermitian phase operator exists for quantum spins. Its spectrum is not continuous but has the values (2 pi /(2S+1))n, 0<or=n<or=2S for spin S. The precession of a high-S spin in a magnetic field is shown to consist essentially of a set of jumps from one value of phase to the next one, at equal time intervals. At intermediate times the value of the phase is uncertain. By going to a larger Hilbert space than the spin space, a Hermitian phase operator whose eigenvalue spectrum covers the entire real line is defined, and its relation to the formerly defined operator is clarified. Possible applications to spin and other systems are outlined.

On the Behavior of Characteristic Functions on the Real Line

This paper is concerned with the following question: if a characteristic function satisfies a certain property at the origin, what can be said about its behavior on the entire real line? If $k$ is an even integer and $f(u)$ is a characteristic function, then the existence of $f^{(k)}(0)$ implies the existence of $f^{(k)}(u)$ for all $u$. If $k$ is an odd integer, then it is possible to construct a characteristic function $f(u)$ such that $f^{(k)}(0)$ exists but $f^{(k)}(u)$ fails to exist for almost all $u$. However the existence of $f^{(k)}(0)$, when $k$ is odd, implies that $f(u)$ satisfies a $k$th order smoothness condition uniformly on the real line and thus $f(u)$ has many of the properties of a characteristic function with a continuous $k$th derivative. Several other results are obtained that show that if a characteristic function has a property $P$ at 0 then it either has the same property everywhere on the real line or comes close to having the property everywhere.

Absolute error rate-distortion functions for sources with constrained magnitudes (Corresp.)

The explicit evaluation of the absolute error rate-distortion function has been obtained previously for a class of i.i.d, sources with probability densities supported on the entire real line. The technique is extended to various i.i.d, sources with densities supported on a proper subset of the real line, such as uniform and one-sided exponential densities. Comparisons of these rate-distortion functions with their Shannon lower bounds are also given.

Reconstruction of a stationary Gaussian process from its sign-changes

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.

Reconstruction of a stationary Gaussian process from its sign-changes

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.

A note on Lévy’s Brownian process on the Hilbert sphere

Let X ( t ) X(t) , t in l 2 {l_2} , be Lévy’s separable Brownian process. Let S be the unit Hilbert sphere. It is shown that with probability 1, the image of X ( t ) , t ∈ S X(t),t \in S , is the entire real line.

Expansion of multivariate weakly stationary stochastic processes

Two distinct series representations are derived for the class of second order, mean-square continuous, weakly stationary multivariate stochastic processes. The representations, converging in the stochastic mean, have orthogonal coefficients and are valid over the entire real line. A constructive procedure for obtaining explicit representation for any given spectral density matrix is presented.

Isomorphism and approximation of general state Markov processes

1. Summary. Given below is a brief description of the main results of this paper: ?3. It is shown that the study of a Markov process X,(Q , v) where Q is a general state space, E a separable a-field of subsets of Q, and v a a-finite stationary measure for the process, can essentially be reduced to the study of a real-valued Markov process Yn(K, A, A) where K is a bounded interval or the entire real line, A the Borel sets of K, and A Lebesgue measure, stationary for the process. To show this a notion of isomorphism of processes is introduced. The main tool is the geometric-isomorphism theorem of Halmos and von Neumann. ?4. Processes Xn(R, A, A) are discussed where R is the real line, and A and A are as defined in ? 3. The selection of R rather than a bounded interval K is made because we are particularly interested in infinite stationary measures and the infinite case introduces a few technical difficulties. Treatment of the finite case is practically identical to that of the infinite case except that it is simpler. The basic result is that the process Xn(R, A, A) may be approximated in a certain way by processes (called k-processes) which are essentially Markov chains on a countable state space. The nature of the approximation is weak convergence of measures on function space. Such an approximation is useful because certain results difficult to prove directly for processes on a continuous state space may be easy to prove for Markov chains, and then carried over to the general process by using the k-processes. (Similar approaches have been used to approximate continuous time by discrete time processes; in this paper it is the state space rather than the time parameter which is discretized-time is discrete throughout.) Next, if the process is conservative and ergodic (that is, the shift T is; see [9] and Halmos' Lectures on ergodic theory) we derive a few conclusions about the nature of the k-processes. ?5. As an application of ?4, we give a probabilistic proof of Birkhoff's ergodic theorem. ?6. Using the isomorphism of ?3, we extend the results of the paper to processes Xn(Q, , v) for E separable.

Bayesian Approach to Life Testing and Reliability Estimation

Abstract Bayesian analysis of the exponential model, based on life tests that are terminated at preassigned time points or after preassigned number of failures, has been developed. For the prior distribution of the parameter involved, uniform, inverted gamma and exponential densities have been examined. The estimation of the reliability function has also been carried out by using Bayesian methods and the case of ‘attribute testing’ has been considered briefly. The role of prior quasi-densities when a life tester has no prior information has been illustrated and it has been observed that the reliability estimate for a diffuse prior which is uniform over the entire positive real line closely resembles the classical MVU estimate obtained by Pugh. It has also been noted that the Bayes estimate of the exponential parameter θ for a prior quasi-density of the form 1/θ2 coincides with the classical MVU estimate of Epstein and Sobel. Further, it has been proved that in a wide class of prior densities, proper or im...

The existence of steady shock waves in nonlinear materials with memory

In recent years there has been extensive work on wave propagation in nonlinear materials with fading memory. Shock waves in such materials have been studied by COLEMAN, GURTIN, & HERRERA [1], and COLEMAN & GURTIN [2], while acceleration waves and higher-order discontinuities have been investigated by COLEMAN t~ GURTIN [1, 3-7], VARLEY [8], COLEMAN, GREENBERG, d~ GURTIN [9], and WANG • BOWEN [10]. PIPKIN [11], using a special constitutive equation, has obtained exact solutions of the one-dimensional steady flow equations; these solutions possess both acceleration waves and shock waves. Here I show that for a large class of nonlinear viscoelastic materials it is possible to find one-dimensional steady solutions exhibiting shock waves; thus, this investigation may be regarded as a generalization of PIPKIN'S results. Although this article deals only with compressive motions*, it is possible, by an obvious modification of the underlying hypotheses**, to derive corresponding results for expansive motions***. In section 2 the relevant balance laws are stated, and a representation theorem for steady motions is established. The theorem asserts that if the body occupies the entire real line while in a fixed reference configuration ~ with constant density, then every steady motion (X, t ) -~x(X, t), with material points labeled by their positions X in ~, is of the form