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.

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.

Investigation of a ship's propeller jet using momentum decay and energy decay

Previous researchers use the velocity decay as an input to investigate the ship’s propeller jet induced scour. A researcher indicated that most of the equations used to predict the stability of various protection systems are often missing a physical background. The momentum decay and energy decay are currently proposed as an initial input for seabed scouring investigation, which are more sensible in physics. Computational fluid dynamics (CFD) and laser Doppler anemometry (LDA) experiments are used to obtain the velocity data and then transforming into momentum and energy decays. The findings proposed several exponential equations of velocity, momentum and energy decays to estimate the region exposed to the seabed scouring.

Decay of Cystalline Order and Equilibration during the Solid-to-Plasma Transition Induced by 20-fs Microfocused 92-eV Free-Electron-Laser Pulses

We have studied a solid-to-plasma transition by irradiating Al foils with the FLASH free electron laser at intensities up to 10(16) W/cm(2). Intense XUV self-emission shows spectral features that are consistent with emission from regions of high density, which go beyond single inner-shell photoionization of solids. Characteristic features of intrashell transitions allowed us to identify Auger heating of the electrons in the conduction band occurring immediately after the absorption of the XUV laser energy as the dominant mechanism. A simple model of a multicharge state inverse Auger effect is proposed to explain the target emission when the conduction band at solid density becomes more atomiclike as energy is transferred from the electrons to the ions. This allows one to determine, independent of plasma simulations, the electron temperature and density just after the decay of crystalline order and to characterize the early time evolution.

Temperature and water potential relations of tropical Trametes and other wood-decay fungi from the indigenous forests of Zimbabwe

Three isolates each, of nine different Trametes and five other wood inhabiting basidiomycetes, were collected from the indigenous forests of Zimbabwe, and the impact of temperature (20-60 degrees C), osmotic and matric potential (-0.5 to - 8.0 MPa), and their interactions on in vitro growth compared. Generally, there was no significant difference between growth of isolates of the same species in relation to temperature. Temperature relationships of the species studied correlated well with their geographic distributions. Species occurring in hot, dry regions tolerated a wide temperature range, with some showing unusually high thermotolerance (55 degrees, T. socotrana, T. cingulata and T. cervina). There were significant intra-strain differences for individual species in relation to solute potential on glycerol-modified media. Generally, growth of ail species was better on glycerol- and KCl-modified osmotic media than on a metrically-modified medium (PEG 8000) at 25, 30 and 37 degrees. The limits for growth on the osmotic media were significantly wider than matric medium, being - 4.5 to - 5.0 and - 2.5 to - 4.5 MPa, respectively. An Irpex sp. grew at lower water potentials than all other species, with good growth at - 7.0 MPa. This study suggests that the capacity of these fungi for effective growth over a range of temperatures, osmotic and matric potentials contributes to their rapid wood decay capacities in tropical climates.

The use and meanings of 'time of wetness' in understanding stone decay

Moisture is a well documented, and crucial, control on the nature of stone decay. The term time of wetness has frequently been adopted to describe how long a stone block is wet, with a view to understanding the impact of this on decay processes. Although this term has proved conceptually useful, it has been used in different ways, by different groups to mean mean quite different things. For example, the time of wetness for a stone block surface (the traditional understanding) may be quite different from that of a block interior, controlled by the different dynamics of wetting and drying in those zones. Thus, surface wetting will occur regularly (sometimes swiftly followed by drying, depending on the time of year), with block interior wetting requiring the accumulation of surface moisture to penetrate to depth (more likely in autumn and winter months), and drying out much more slowly. This relatively new but important perspective, framed in the context of climate change, is crucial to understanding the length of time stone may remain damp at depth following a period of prolonged precipitation. The nature and speed of drying is also relevant in quantifying time of wetness of both surfaces and the interior of building stones.
These ideas related to time of wetness have implications for decay processes, specifically how a prolonged time of deep wetness may re-focus the emphasis of salt weathering in natural building stones toward chemical action. Literature on chemical change is discussed, suggesting that chemical change occurring during periods of prolonged wetness is likely to be significant in itself, with implications for weakening the stone (in terms of, for example, cement dissolution or grain boundary weakening) and exacerbating physical damage from salt crystallisation when blocks finally dry out.