arXiv subjects

C. T. J. Dodson

Publications and source records attributed to C. T. J. Dodson.

13 recordsLinked to original sources

arXiv2012

A review of some recent work on hypercyclicity

Even linear operators on infinite-dimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinite-dimensional property, when iterations of the operator generate a dense subspace. A Frechet space admits a hypercyclic operator if and only if it is separable and infinite-dimensional. However, by considering the semigroups generated by multiples of operators, it is possible to obtain hypercyclic behaviour on finite dimensional spaces. This article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Frechet spaces.

math.FA↗

arXiv2011

Some recent work in Frechet geometry

Some recent work in Frechet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Frechet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Frechet structure needed for applications. A summary is given of recent results on hypercyclicity of operators on Frechet spaces.

math.DG↗

arXiv2010

Information geometry and entropy in a stochastic epidemic rate process

Epidemic models with inhomogeneous populations have been used to study major outbreaks and recently Britton and Lindenstrand \cite{BL} described the case when latency and infectivity have independent gamma distributions. They found that variability in these random variables had opposite effects on the epidemic growth rate. That rate increased with greater variability in latency but decreased with greater variability in infectivity. Here we extend their result by using the McKay bivariate gamma distribution for the joint distribution of latency and infectivity, recovering the above effects of variability but allowing possible correlation. We use methods of stochastic rate processes to obtain explicit solutions for the growth of the epidemic and the evolution of the inhomogeneity and information entropy. We obtain a closed analytic solution to the evolution of the distribution of the number of uninfected individuals as the epidemic proceeds, and a concomitant expression for the decay of entropy. The family of McKay bivariate gamma distributions has a tractable information geometry which provides a framework in which the evolution of distributions can be studied as the outbreak grows, with a natural distance structure for quantitative tracking of progress.

q-bio.PE↗

arXiv2010

An inhomogeneous stochastic rate process for evolution from states in an information geometric neighbourhood of uniform fitness

This study elaborates some examples of a simple evolutionary stochastic rate process where the population rate of change depends on the distribution of properties--so different cohorts change at different rates. We investigate the effect on the evolution arising from parametrized perturbations of uniformity for the initial inhomogeneity. The information geometric neighbourhood system yields also solutions for a wide range of other initial inhomogeneity distributions, including approximations to truncated Gaussians of arbitrarily small variance and distributions with pronounced extreme values. It is found that, under quite considerable alterations in the shape and variance of the initial distribution of inhomogeneity in unfitness, the decline of the mean does change markedly with the variation in starting conditions, but the net population evolution seems surprisingly stable.

q-bio.PE↗

arXiv2009

Information geometry for testing pseudorandom number generators

The information geometry of the 2-manifold of gamma probability density functions provides a framework in which pseudorandom number generators may be evaluated using a neighbourhood of the curve of exponential density functions. The process is illustrated using the pseudorandom number generator in Mathematica. This methodology may be useful to add to the current family of test procedures in real applications to finite sampling data.

stat.CO↗

arXiv2009

On the entropy flows to disorder

Gamma distributions, which contain the exponential as a special case, have a distinguished place in the representation of near-Poisson randomness for statistical processes; typically, they represent distributions of spacings between events or voids among objects. Here we look at the properties of the Shannon entropy function and calculate its corresponding flow curves. We consider univariate and bivariate gamma, as well as Weibull distributions which also include exponential distributions.

math-ph↗

arXiv2008

Quantifying galactic clustering and departures from randomness of the inter-galactic void probability function using information geometry

A number of recent studies have estimated the inter-galactic void probability function and investigated its departure from various random models. We study a family of parametric statistical models based on gamma distributions, which do give realistic descriptions for other stochastic porous media. Gamma distributions contain as a special case the exponential distributions, which correspond to the `random' void size probability arising from Poisson processes. The random case corresponds to the information-theoretic maximum entropy or maximum uncertainty model. Lower entropy models correspond on the one hand to more `clustered' structures or `more dispersed' structures than expected at random. The space of parameters is a surface with a natural Riemannian structure, the Fisher information metric. This surface contains the Poisson processes as an isometric embedding and provides the geometric setting for quantifying departures from randomness and perhaps on which may be written evolutionary dynamics for the void size distribution. Estimates are obtained for the two parameters of the void diameter distribution for an illustrative example of data published by Fairall.

math-ph↗

arXiv2008

Second order structures for sprays and connections on Frechet manifolds

Ambrose, Palais and Singer \cite{Ambrose} introduced the concept of second order structures on finite dimensional manifolds. Kumar and Viswanath \cite{Kumar} extended these results to the category of Banach manifolds. In the present paper all of these results are generalized to a large class of Frechet manifolds. It is proved that the existence of Christoffel and Hessian structures, connections, sprays and dissections are equivalent on those Frechet manifolds which can be considered as projective limits of Banach manifolds. These concepts provide also an alternative way for the study of ordinary differential equations on non-Banach infinite dimensional manifolds. Concrete examples of the structures are provided using direct and flat connections.

math.DG↗

arXiv2008

A note on quantum chaology and gamma approximations to eigenvalue spacings for infinite random matrices

Quantum counterparts of certain simple classical systems can exhibit chaotic behaviour through the statistics of their energy levels and the irregular spectra of chaotic systems are modelled by eigenvalues of infinite random matrices. We use known bounds on the distribution function for eigenvalue spacings for the Gaussian orthogonal ensemble (GOE) of infinite random real symmetric matrices and show that gamma distributions, which have an important uniqueness property, can yield an approximation to the GOE distribution. That has the advantage that then both chaotic and non chaotic cases fit in the information geometric framework of the manifold of gamma distributions, which has been the subject of recent work on neighbourhoods of randomness for general stochastic systems. Additionally, gamma distributions give approximations, to eigenvalue spacings for the Gaussian unitary ensemble (GUE) of infinite random hermitian matrices and for the Gaussian symplectic ensemble (GSE) of infinite random hermitian matrices with real quaternionic elements, except near the origin. Gamma distributions do not precisely model the various analytic systems discussed here, but some features may be useful in studies of qualitative generic properties in applications to data from real systems which manifestly seem to exhibit behaviour reminiscent of near-random processes.

math-ph↗

arXiv2006

Quantifying galactic clustering and departures from randomness of the inter-galactic void probability function using information geometry

We study a family of parametric statistical models based on gamma distributions, which do give realistic descriptions for other stochastic porous media. Gamma distributions contain as a special case the exponential distributions, which correspond to the `random' void size probability arising from Poisson processes. The space of parameters is a surface with a natural Riemannian metric structure. This surface contains the Poisson processes as an isometric embedding and a recent theorem shows that it contains neighbourhoods of all departures from randomness. The method provides thereby a geometric setting for quantifying departures from randomness and on which may be formulated cosmological evolutionary dynamics for galactic clustering and for the concomitant development of the void size distribution. The 2dFGRS data offer the possibility of more detailed investigation of this approach than was possible when it was originally suggested and some parameter estimations are given.

astro-ph↗

arXiv2003

Neighbourhoods of independence for random processes

The Freund family of distributions becomes a Riemannian 4-manifold with Fisher information as metric; we derive the induced $α$-geometry, i.e., the $α$-curvature, $α$-Ricci curvature with its eigenvales and eigenvectors, the $α$-scalar curvature etc. We show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere. We consider special cases as submanifolds and discuss their geometrical structures; one submanifold yields examples of neighbourhoods of the independent case for bivariate distributions having identical exponential marginals. Thus, since exponential distributions complement Poisson point processes, we obtain a means to discuss the neighbourhood of independence for random processes.

math.DG↗

arXiv2003

Information geometric neighbourhoods of randomness and geometry of the McKay bivariate gamma 3-manifold

We show that gamma distributions provide models for departures from randomness since every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions, using an information theoretic metric topology. We derive also the information geometry of the 3-manifold of McKay bivariate gamma distributions, which can provide a metrization of departures from randomness and departures from independence for bivariate processes. The curvature objects are derived, including those on three submanifolds. As in the case of bivariate normal manifolds, we have negative scalar curvature but here it is not constant and we show how it depends on correlation. These results have applications, for example, in the characterization of stochastic materials.

math.DG↗

arXiv2003

Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. Here we extend this result to $M$ modeled on an arbitrarily chosen Banach space and more generally to those Fréchet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.

math.DG↗