Author

# Dave Applebaum

Bio: Dave Applebaum is an academic researcher from Nottingham Trent University. The author has an hindex of 1, co-authored 2 publication(s) receiving 2309 citation(s).

##### Papers

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TL;DR: The unsolvability of the Word Problem, originally given by Markov and Post, is proved in full as is the uncomputable of the busy beaver function.

Abstract: problem, Hilbert's decision problem and Turing's halting problem as examples of the former. Definitions are given, each in terms of an algorithm, of the informal concepts of computable function, decidable relation and semi-decidable relation. The next two substantial chapters introduce, in turn, a machine model of computability and a mathematical model of computability. For the machine model Hodel presents the concept of Unlimited Register Machine (URM) rather than a Turing machine, which dates back to the paper Computability of Recursive Functions, published in 1963 by J. C. Sheperdson and H. E. Sturgis. Turing's argument for the unsolvability of the Halting Problem is given and how he exploited this to resolve the Decision Problem is sketched. The unsolvability of the Word Problem, originally given by Markov and Post, is proved in full as is the uncomputability of the busy beaver function.

##### Cited by

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TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.

Abstract: Fractional kinetic equations of the diffusion, diffusion–advection, and Fokker–Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns. These fractional equations are derived asymptotically from basic random walk models, and from a generalised master equation. Several physical consequences are discussed which are relevant to dynamical processes in complex systems. Methods of solution are introduced and for some special cases exact solutions are calculated. This report demonstrates that fractional equations have come of age as a complementary tool in the description of anomalous transport processes.

6,586 citations

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TL;DR: The per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery, can be increased dramatically under this assumption, and a form of multiuser diversity via packet relaying is exploited.

Abstract: The capacity of ad hoc wireless networks is constrained by the mutual interference of concurrent transmissions between nodes. We study a model of an ad hoc network where n nodes communicate in random source-destination pairs. These nodes are assumed to be mobile. We examine the per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery. Under this assumption, the per-user throughput can increase dramatically when nodes are mobile rather than fixed. This improvement can be achieved by exploiting a form of multiuser diversity via packet relaying.

2,698 citations

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22 Apr 2001

TL;DR: The per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery, can be increased dramatically when the nodes are mobile rather than fixed, by exploiting node mobility as a type of multiuser diversity.

Abstract: The capacity of ad-hoc wireless networks is constrained by the mutual interference of concurrent transmissions between nodes. We study a model of an ad-hoc network where n nodes communicate in random source-destination pairs. These nodes are assumed to be mobile. We examine the per-session throughput for applications with loose delay constraints, such that the topology changes over the time-scale of packet delivery. Under this assumption, the per-user throughput can increase dramatically when the nodes are mobile rather than fixed. This improvement can be achieved by exploiting node mobility as a type of multiuser diversity.

1,376 citations

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TL;DR: Random Fields and Geometry as discussed by the authors is a comprehensive survey of the general theory of Gaussian random fields with a focus on geometric problems arising in the study of random fields, including continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities.

Abstract: * Recasts topics in random fields by following a completely new way of handling both geometry and probability
* Significant exposition of the work of others in the field
* Presentation is clear and pedagogical
* Excellent reference work as well as excellent work for self study
This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined.
The three parts to the monograph are quite distinct. Part I presents a user-friendly yet comprehensive background to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness, entropy and majorizing measures, Borell and Slepian inequalities. Part II gives a quick review of geometry, both integral and Riemannian, to provide the reader with the material needed for Part III, and to give some new results and new proofs of known results along the way. Topics such as Crofton formulae, curvature measures for stratified manifolds, critical point theory, and tube formulae are covered. In fact, this is the only concise, self-contained treatment of all of the above topics, which are necessary for the study of random fields. The new approach in Part III is devoted to the geometry of excursion sets of random fields and the related Euler characteristic approach to extremal probabilities.
"Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory. These applications, to appear in a forthcoming volume, will cover areas as widespread as brain imaging, physical oceanography, and astrophysics.

1,360 citations

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TL;DR: A number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed, including Ornstein-Uhlenbeck type processes, superpositions of such processes and Stochastic volatility models in one and more dimensions.

Abstract: With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse Gaussian marginals and various types of dependence structures are discussed. Ornstein-Uhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some discussion is given of the feasibility of making likelihood inference for these models.

1,267 citations