Decay spectrum and decay subspace of normal operators

Let A be a self-adjoint operator on a Hilbert space. It is well known that A admits a unique decomposition into a direct sum of three self-adjoint operators A(p), A(ac) and A(sc) such that there exists an orthonormal basis of eigenvectors for the operator A(p) the operator A(ac) has purely absolutely continuous spectrum and the operator A(sc) has purely singular continuous spectrum. We show the existence of a natural further decomposition of the singular continuous component A c into a direct sum of two self-adjoint operators A(sc)(D) and A(sc)(ND). The corresponding subspaces and spectra are called decaying and purely non-decaying singular subspaces and spectra. Similar decompositions are also shown for unitary operators and for general normal operators.

Decay measures on locally compact abelian topological groups

Source: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS Volume: 131 Pages: 1257-1273 Part: Part 6 Published: 2001 Times Cited: 5 References: 23 Citation MapCitation Map beta Abstract: We show that the Banach space M of regular sigma-additive finite Borel complex-valued measures on a non-discrete locally compact Hausdorff topological Abelian group is the direct sum of two linear closed subspaces M-D and M-ND, where M-D is the set of measures mu is an element of M whose Fourier transform vanishes at infinity and M-ND is the set of measures mu is an element of M such that nu is not an element of MD for any nu is an element of M \ {0} absolutely continuous with respect to the variation \mu\. For any corresponding decomposition mu = mu(D) + mu(ND) (mu(D) is an element of M-D and mu(ND) is an element of M-ND) there exist a Borel set A = A(mu) such that mu(D) is the restriction of mu to A, therefore the measures mu(D) and mu(ND) are singular with respect to each other. The measures mu(D) and mu(ND) are real if mu is real and positive if mu is positive. In the case of singular continuous measures we have a refinement of Jordan's decomposition theorem. We provide series of examples of different behaviour of convolutions of measures from M-D and M-ND.

Prediction- and simulation-error based perceptron training: Solution space analysis and a novel combined training scheme

Previous papers have noted the difficulty in obtaining neural models which are stable under simulation when trained using prediction-error-based methods. Here the differences between series-parallel and parallel identification structures for training neural models are investigated. The effect of the error surface shape on training convergence and simulation performance is analysed using a standard algorithm operating in both training modes. A combined series-parallel/parallel training scheme is proposed, aiming to provide a more effective means of obtaining accurate neural simulation models. Simulation examples show the combined scheme is advantageous in circumstances where the solution space is known or suspected to be complex. (c) 2006 Elsevier B.V. All rights reserved.

Integrating Ballymun? Flawed progress in Ireland's largest estate regeneration scheme

Urban regeneration in the Republic of Ireland takes place in the context of the rapid, ‘Celtic Tiger’ economic growth of the 1990s. The boom transformed Irish society and led to greater affluence for many people, along with continuing and arguably worsening inequality for those excluded from its opportunities. In particular, Ireland’s small social rented sector has become the focus of the country’s most concentrated poverty and social exclusion. The Ballymun regeneration programme in North Dublin aims to facilitate physical, social and economic change in order to integrate the area more closely with the
more affluent surrounding suburbs. This article reviews the issues involved in restructuring such a large area of social exclusion within a rapidly changing European capital city, using a framework that disaggregates the concept of integration into three elements: market, citizenship and reciprocity. With just over half the physical refurbishment complete, progress has been made but some fundamental issues remain. The article concludes that although substantial advancement has been made with physical regeneration, progress with wider economic and social integration has been uneven and in some cases flawed.

UK Power Sector Challenge to National Allocation Plan Under the EU Emissions Trading Scheme Refused by the European Court of Justice

In Case T-130/06 Drax Power and others v European Commission, the Court of First Instance held that an application by Drax Power and others for annulment of Commission Decision (C(2006)426 final of 22 February 2006 concerning a proposed amendment to the National Allocation Plan notified by the UK in accordance with the EU Emissions Trading Directive was inadmissable. The Court ruled that the applicants could not be considered to be 'directly concerned' by the contested decision within the meaning of the fourth paragraph of Article 230 of the European Treaty, on legal standing: 'Any natural or legal person may, under the same conditions, institute proceedings against a decision addressed to that person or against a decision, which, although in the form of a regulation or a decision addressed to another persion, is of direct and individual concern to the former...'

Dissipative scheme to approach the boundary of two-qubit entangled mixed states

We discuss the generation of states close to the boundary family of maximally entangled mixed states as defined by the use of concurrence and linear entropy. The coupling of two qubits to a dissipation-affected bosonic mode is able to produce a bipartite state having, for all practical purposes, the entanglement and mixedness properties of one of such boundary states. We thoroughly study the effects that thermal and squeezed characters of the bosonic mode have in such a process and we discuss tolerance to qubit phase-damping mechanisms. The nondemanding nature of the scheme makes it realizable in a matter-light-based physical setup, which we address in some details.