Fractional Bessel integrals and derivatives on semi-axes

In this paper we study fractional powers of the Bessel differential operator. The fractional powers are defined explicitly in the integral form without use of integral transforms in its definitions. Some general properties of the fractional powers of the Bessel differential operator are proved and some are listed. Among them are different variations of definitions, relations with the Mellin and Hankel transforms, group property, evaluation of resolvent integral operator in terms of the Wright or generalized Mittag-Leffler function

On some generalizations of the properties of the multidimensional generalized Erdélyi-Kober operators and their applications

In this paper we investigate the composition of a multidimensional generalized Erdélyi-Kober operator with differential operators of high order. In particular, with powers of the differential Bessel operator. Applications of proved properties to solving the Cauchy problem for a multidimensional polycaloric equation with a Bessel operator are show

ATHENA: A knowledge-based hybrid backpropagation-grammatical evolution neural network algorithm for discovering epistasis among quantitative trait Loci

<p>Abstract</p> <p>Background</p> <p>Growing interest and burgeoning technology for discovering genetic mechanisms that influence disease processes have ushered in a flood of genetic association studies over the last decade, yet little heritability in highly studied complex traits has been explained by genetic variation. Non-additive gene-gene interactions, which are not often explored, are thought to be one source of this "missing" heritability.</p> <p>Methods</p> <p>Stochastic methods employing evolutionary algorithms have demonstrated promise in being able to detect and model gene-gene and gene-environment interactions that influence human traits. Here we demonstrate modifications to a neural network algorithm in ATHENA (the Analysis Tool for Heritable and Environmental Network Associations) resulting in clear performance improvements for discovering gene-gene interactions that influence human traits. We employed an alternative tree-based crossover, backpropagation for locally fitting neural network weights, and incorporation of domain knowledge obtainable from publicly accessible biological databases for initializing the search for gene-gene interactions. We tested these modifications <it>in silico </it>using simulated datasets.</p> <p>Results</p> <p>We show that the alternative tree-based crossover modification resulted in a modest increase in the sensitivity of the ATHENA algorithm for discovering gene-gene interactions. The performance increase was highly statistically significant when backpropagation was used to locally fit NN weights. We also demonstrate that using domain knowledge to initialize the search for gene-gene interactions results in a large performance increase, especially when the search space is larger than the search coverage.</p> <p>Conclusions</p> <p>We show that a hybrid optimization procedure, alternative crossover strategies, and incorporation of domain knowledge from publicly available biological databases can result in marked increases in sensitivity and performance of the ATHENA algorithm for detecting and modelling gene-gene interactions that influence a complex human trait.</p

One-sided instance-based boundary sets

Instance retraction is a difficult problem for concept learning by version spaces. This chapter introduces a family of version-space representations called one-sided instance-based boundary sets. They are correct and efficiently computable representations for admissible concept languages. Compared to other representations, they are the most efficient useful(1) version-space representations for instance retraction

Analysing neurobiological models using communicating automata

Two important issues in computational modelling in cognitive neuroscience are: first, how to formally describe neuronal networks (i.e. biologically plausible models of the central nervous system), and second, how to analyse complex models, in particular, their dynamics and capacity to learn.We make progress towards these goals by presenting a communicating automata perspective on neuronal networks. Specifically, we describe neuronal networks and their biological mechanisms using Data-rich Communicating Automata, which extend classic automata theory with rich data types and communication.We use two case studies to illustrate our approach. In the first case study, we model a number of learning frameworks, which vary in respect of their biological detail, for instance the Backpropagation (BP) and the Generalized Recirculation (GeneRec) learning algorithms. We then used the SPIN model checker to investigate a number of behavioral properties of the neural learning algorithms. SPIN is a well-known model checker for reactive distributed systems, which has been successfully applied to many non-trivial problems. The verification results show that the biologically plausible GeneRec learning is less stable than BP learning. In the second case study, we presented a large scale (cognitive-level) neuronal network, which models an attentional spotlight mechanism in the visual system. A set of properties of this model was verified using Uppaal, a popular real-time model checker. The results show that the asynchronous processing supported by concurrency theory is not only a more biologically plausible way to model neural systems, but also provides a better performance in cognitive modelling of the brain than conventional artificial neural networks that use synchronous updates. Finally, we compared our approach with several other related theories that apply formal methods to cognitive modelling. In addition, the practical implications of the approach are discussed in the context of neuronal network based controllers