Andrew Vogt

Mathematical analysis of the asymptotic behavior of the leslie population matrix model

Let L be a Leslie population matrix. Leslie (1945) and others have shown that the matrix L has a leading positive eigenvalue λ0 and that in general: lim ⁡ t → ∞ L t X λ 0 t = γ X λ 0 , ( 1 ) where X λ 0 is an eigenvector corresponding to λ0, X is any initial population vector, and γ is a scalar quantity determined by X. In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms of λ0 and the entries of the matrix L the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is non-zero. To stabilize a population, the matrix L must be adjusted so that λ0 = 1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stabilization which yields as a byproduct an easily accessible scalar measure of Ls tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.

Approximation by neural networks is not continuous

Abstract It is shown that in a Banach space X satisfying mild conditions, for its infinite, linearly independent subset G, there is no continuous best approximation map from X to the n-span, span n G . The hypotheses are satisfied when X is an L p -space, 1 span n G is not a subspace of X, it is also shown that there is no continuous map from X to span n G within any positive constant of a best approximation.

POSITION AND MOMENTUM DISTRIBUTIONS DO NOT DETERMINE THE QUANTUM MECHANICAL STATE

Publisher Summary This chapter presents examples that are constructed to show that the state of an elementary quantum mechanical system need not be uniquely determined by the associated position distribution and the associated momentum distribution. It explains that pure states are not distinguishable by their position statistics or by their momentum statistics in the quantum mechanical state. The position statistics and the momentum statistics play very different roles in quantum mechanics from their roles in classical mechanics. The chapter explains that distinct pure states exist that not only have the same position and momentum distributions but also the same distributions with respect to other significant observables, including, under certain circumstances, the energy.

Continuity of Approximation by Neural Networks in Lp Spaces

Devices such as neural networks typically approximate the elements of some function space X by elements of a nontrivial finite union M of finite-dimensional spaces. It is shown that if X=Lp(Ω) (1<p<∞ and Ω⊂Rd), then for any positive constant Γ and any continuous function φ from X to M, ‖f−φ(f)‖>‖f−M‖+Γ for some f in X. Thus, no continuous finite neural network approximation can be within any positive constant of a best approximation in the Lp-norm.

On the electrochemical response of rough electrodes: Part II. The transient response in the presence of slow faradaic processes

Abstract For an electrode with self-similar roughness, explicit expressions are given for the voltage transient following a current step, and for the current transient following a voltage step.

On the electrochemical response of rough electrodes: Part I. The current transient following a voltage step in the absence of faradaic processes

Abstract For a rough solid electrode the current response to a voltage step in the absence of a faradaic process is described analytically. At short times t , the transient response is expressed by a power series in t α , where α is the interfacial roughness parameter. At long times, the response is expressed by an asymptotic series in t −α .

Best approximation by linear combinations of characteristic functions of half-spaces

It is shown that for any positive integer n and any function f in Lp([0,1]d) with p ∈ [1,∞) there exist n half-spaces such that f has a best approximation by a linear combination of their characteristic functions. Further, any sequence of linear combinations of n half-space characteristic functions converging in distance to the best approximation distance has a subsequence converging to a best approximation, i.e., the set of such n-fold linear combinations is an approximatively compact set.

Best approximation by Heaviside perceptron networks

In Lp-spaces with p an integer from [1, infinity) there exists a best approximation mapping to the set of functions computable by Heaviside perceptron networks with n hidden units; however for p an integer from (1, infinity) such best approximation is not unique and cannot be continuous.

Unique continuation of some dispersive waves

We demonstrate that a complex-valued wave on space-time Rn+1 , obtained from square-integrable but not necessarily smooth nonzero initial data and having a suitable Hamiltonian generator or dispersion relation, cannot vanish on a measurable rectangle in Rn+1 . Nor can the product of two such waves vanish in such a rectangle - even waves arising from distinct Hamiltonians or dispersion relations. Examples include the solutions of the free particle Schrodinger equation, the positive energy solutions of the free particle Klein-Gordon and Dirac equations, and the positive frequency solutions of the wave equation. Nonlocalization results of this type were obtained by the physicist G. C. Hegerfeldt.

A simple model for desorption transients

Abstract On the basis of a linear adsorption isotherm, the time dependence of the interfacial excess is calculated for semi-infinite planar diffusion and desorption—adsorption kinetics.