Methods to compute ring invariants and applications: a new class of exotic threefolds

We develop some methods to compute the Makar-Limanov and Derksen invariants, isomorphism classes and automorphism groups for k-domains B, which are constructed from certain Russell k-domains. We propose tools and techniques to distinguish between k-domains with the same Makar-Limanov and Derksen invariants. In particular, we introduce the exponential chain associated to certain modifications. We extract C-domains from the class B that have smooth contractible factorial Spec(B), which are diffeomorphic to R 6 but not isomorphic to C 3, that is, exotic C 3. We examine associated exponential chains to prove that exotic threefolds Spec(B) are not isomorphic to Spec(R), for any Russell C-domain R

LND-FILTRATIONS AND SEMI-RIGID DOMAINS

We investigate the filtration corresponding to the degree function induced by a non-zero locally nilpotent derivation and its associated graded algebra. We show that this kind of filtration, referred to as the LND-filtration, is the ideal candidate to study the structure of semi-rigid k-domains, that is, k-domains for which every non-zero locally nilpotent derivation gives rise to the same filtration. Indeed, the LND-filtration gives a very precise understanding of these structure, it is impeccable for the computation of the Makar-Limanov invariant, and it is an efficient tool to determine their isomorphism types and automorphism groups. Then, we construct a new interesting class of semi-rigid k-domains in which we elaborate the fundamental requirement of LND-filtrations. The importance of these new examples is due to the fact that they possess a relatively big set of invariant sub-algebras, which can not be recoverd by known invariants such as the Makar-Limanov and the Derksen invariants. Also, we define a new family of invariant sub-algebras as a generalization of the Derksen invariant. Finally, we introduce an algorithm to establish explicit isomorphisms between cylinders over non-isomorphic members of the new class, providing by that new counter-examples to the cancellation problem

Microarray image processing: A novel neural network framework

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.Due to the vast success of bioengineering techniques, a series of large-scale analysis tools has been developed to discover the functional organization of cells. Among them, cDNA microarray has emerged as a powerful technology that enables biologists to cDNA microarray technology has enabled biologists to study thousands of genes simultaneously within an entire organism, and thus obtain a better understanding of the gene interaction and regulation mechanisms involved. Although microarray technology has been developed so as to offer high tolerances, there exists high signal irregularity through the surface of the microarray image. The imperfection in the microarray image generation process causes noises of many types, which contaminate the resulting image. These errors and noises will propagate down through, and can significantly affect, all subsequent processing and analysis. Therefore, to realize the potential of such technology it is crucial to obtain high quality image data that would indeed reflect the underlying biology in the samples. One of the key steps in extracting information from a microarray image is segmentation: identifying which pixels within an image represent which gene. This area of spotted microarray image analysis has received relatively little attention relative to the advances in proceeding analysis stages. But, the lack of advanced image analysis, including the segmentation, results in sub-optimal data being used in all downstream analysis methods. Although there is recently much research on microarray image analysis with many methods have been proposed, some methods produce better results than others. In general, the most effective approaches require considerable run time (processing) power to process an entire image. Furthermore, there has been little progress on developing sufficiently fast yet efficient and effective algorithms the segmentation of the microarray image by using a highly sophisticated framework such as Cellular Neural Networks (CNNs). It is, therefore, the aim of this thesis to investigate and develop novel methods processing microarray images. The goal is to produce results that outperform the currently available approaches in terms of PSNR, k-means and ICC measurements.Aleppo University, Syri

On exit laws for semigroups in weak duality

summary:Let $\Bbb P:=(P_{t})_{t>0}$ be a measurable semigroup and $m$ a $\sigma$-finite positive measure on a Lusin space $X$. An $m$-exit law for $\Bbb P$ is a family $(f_{t})_{t>0}$ of nonnegative measurable functions on $X$ which are finite $m$-a.e. and satisfy for each $s,t >0$ $P_{s}f_{t}=f_{s+t}$ $m$-a.e. An excessive function $u$ is said to be in \Cal R if there exits an $m$-exit law $(f_{t})_{t>0}$ for $\Bbb P$ such that $u=\int_{0}^{\infty }f_{t}\,dt$, $m$-a.e. Let \Cal P be the cone of $m$-purely excessive functions with respect to $\Bbb P$ and \Cal I mV be the cone of $m$-potential functions. It is clear that \Cal I mV\subseteq \Cal R\subseteq \Cal P. In this paper we are interested in the converse inclusion. We extend some results already obtained under the assumption of the existence of a reference measure. Also, we give an integral representation of the mutual energy function