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.

Measurement of the duration of X-ray lasing pumped by an optical laser pulse of picosecond duration

Measurements of the duration of X-ray lasing pumped with picosecond pulses from the VULCAN optical laser are obtained using a streak camera with 700 fs temporal resolution. Combined with a temporal smearing due to the spectrometer employed, we have measured X-ray laser pulse durations for Ni-like silver at 13.9 nm with a total time resolution of 1.1 ps. For Ni-like silver, the X-ray laser output has a steep rise followed by an approximately exponential temporal decay with measured full-width at half-maximum (FWHM) of 3.7 (+/-0.5) ps. For Ne-like nickel lasing at 23.1 nm, the measured duration of lasing is approximate to10.7 (+/-1) ps (FWHM). An estimate of the duration of the X-ray laser gain has been obtained by temporally resolving spectrally integrated continuum and resonance line emission. For Ni-like silver, this time of emission is approximate to22 (+/-2) ps (FWHM), while for Ne-like nickel we measure approximate to35 (+/-2) ps (FWHM). Assuming that these times of emission correspond to the gain duration, we show that a simple model consistently relates the gain durations to the measured durations of X-ray lasing. (C) 2002 Elsevier Science B.V. All rights reserved.

A commentary on climate change, stone decay dynamics and the ‘greening’ of natural stone buildings: new perspectives on ‘deep wetting’

Environmental controls on stone decay processes are rapidly changing as a result of changing climate. UKCP09 projections for the 2020s (2010–2039) indicate that over much of the UK seasonality of precipitation will increase. Summer dryness and winter wetness are both set to increase, the latter linked to projected precipitation increases in autumn and spring months. If so, this could increase the time that stone structures remain wet and possibly the depth of moisture penetration, and it appears that building stone in Northern Ireland has already responded through an increased incidence of algal ‘greening’.This paper highlights the need for understanding the effects of climate change through a series of studies of largely sandstone structures. Current and projected climatic trends are therefore considered to have aesthetic, physical and chemical implications that are not currently built into our models of sandstone decay, especially with respect to the role played by deep-seated wetness on sandstone deterioration and decay progression and the feedbacks associated with, for example surface algal growth. In particular,it is proposed that algal biofilms will aid moisture retention and further facilitate moisture and dissolved salt penetration to depth. Thus, whilst the outer surface of stone may continue to experience frequent wetting and drying associated with individual precipitation events, the latter is less likely to be complete, and the interiors of building blocks may only experience wetting/drying in response to seasonal cycling. A possible consequence of deeper salt penetration could be a delay in the onset of surface deterioration,but more rapid and effective retreat once it commences as decay mechanisms ‘tap into a reservoir of deep salt’.

Understanding the decay of stone-built heritage

The problem of the decay and conservation of stone-built heritage is a complex one, requiring input across many disciplines to identify appropriate remedial steps and management strategies. Over the past few decades, earth scientists have brought a unique perspective to this challenging area, drawing on traditions and knowledge obtained from research into landscape development and the natural environment. This paper reviews the crucial themes that have arisen particularly, although not exclusively, from the work of physical geographers — themes that have sought to correct common misconceptions held by the public, as well as those directly engaged in construction and conservation, regarding the nature, causes and controls of building stone decay. It also looks to the future, suggesting how the behaviour of building stones (and hence the work of stone decay scientists) might alter in response to the looming challenge of climate change.

Micro-environmental change as a trigger for granite decay in offshore Irish lighthouses: implications for the long-term preservation of operational historic buildings.

Following automation of lighthouses around the coastline of Ireland, reports of accelerated deterioration of interior granite stonework have increased significantly with an associated deterioration in the historic structure and rise in related maintenance costs. Decay of granite stone- work primarily occurs through granular disintegration with the effective grusification of granite surfaces. A decay gradient exists within the towers whereby the condition of granite in the lower levels is much worse than elsewhere. The lower tower levels are also regions with highest rela- tive humidity values and greatest salt concentrations. Data indicate that post-automation decay may have been trig- gered by a change in micro-environmental conditions within the towers associated with increased episodes of condensation on stone surfaces. This in turn appears to have facilitated deposition and accumulation of hygro- scopic salts (e.g. NaCl) giving rise to widespread evidence of deliquescence in the lower tower levels. Evidence indicates that the main factors contributing to accelerated deterioration of interior granite stonework are changes in micro-environmental conditions, salt weathering, chemical weathering through the corrosive effect of strongly alkaline conditions on alumino-silicate minerals within the granite and finally, the mica-rich characteristics of the granite itself which increases its structural and chemical susceptibility to subaerial weathering processes by creating points of weakness within the granite. This case study demonstrates how seemingly minor changes in micro-environmental conditions can unintentionally trigger the rapid and extensive deterioration of a previously stable rock type and threaten the long-term future of nationally iconic opera- tional historic structures.