Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions

We consider in this paper the solvability of linear integral equations on the real line, in operator form (λ−K)φ=ψ, where and K is an integral operator. We impose conditions on the kernel, k, of K which ensure that K is bounded as an operator on . Let Xa denote the weighted space as |s|→∞}. Our first result is that if, additionally, |k(s,t)|⩽κ(s−t), with and κ(s)=O(|s|−b) as |s|→∞, for some b>1, then the spectrum of K is the same on Xa as on X, for 0of operators, , which ensure that, if λ≠0 and λφ=Kkφ has only the trivial solution in X, for all k∈W, then, for 0⩽a⩽b, (λ−K)φ=ψ has exactly one solution φ∈Xa for every k∈W and ψ∈Xa. These conditions ensure further that is bounded uniformly in k∈W, for 0⩽a⩽b. As a particular application we consider the case when the kernel takes the form k(s,t)=κ(s−t)z(t), with , , and κ(s)=O(|s|−b) as |s|→∞, for some b>1. As an example where kernels of this latter form occur we discuss a boundary integral equation formulation of an impedance boundary value problem for the Helmholtz equation in a half-plane.

The full infinite dimensional moment problem on semi-algebraic sets of generalized functions

We consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon–Nikodym density w.r.t. the Lebesgue measure.

On asymptotic behavior at infinity and the finite section method for integral equations on the half-line

e consider integral equations on the half-line of the form and the finite section approximation to x obtained by replacing the infinite limit of integration by the finite limit β. We establish conditions under which, if the finite section method is stable for the original integral equation (i.e. exists and is uniformly bounded in the space of bounded continuous functions for all sufficiently large β), then it is stable also for a perturbed equation in which the kernel k is replaced by k + h. The class of perturbations allowed includes all compact and some non-compact perturbations of the integral operator. Using this result we study the stability and convergence of the finite section method in the space of continuous functions x for which ()()()=−∫∞dttxt,sk)s(x0()syβxβx()sxsp+1 is bounded. With the additional assumption that ()(tskt,sk−≤ where ()()(),qsomefor,sassOskandRLkq11>+∞→=∈− we show that the finite-section method is stable in the weighted space for ,qp≤≤0 provided it is stable on the space of bounded continuous functions. With these results we establish error bounds in weighted spaces for x - xβ and precise information on the asymptotic behaviour at infinity of x. We consider in particular the case when the integral operator is a perturbation of a Wiener-Hopf operator and illustrate this case with a Wiener-Hopf integral equation arising in acoustics.

Determination of the impact of continuous defoliation of Lolium perenne and Trifolium repens on bacterial and fungal community structure in rhizosphere soil

To determine the effects of defoliation on microbial community structure, rhizosphere soil samples were taken pre-, and post-defoliation from the root tip and mature root regions of Trifolium repens L. and Lolium perenne L. Microbial DNA isolated from samples was used to generate polymerase chain reaction-denaturing gradient gel electrophoresis molecular profiles of bacterial and fungal communities. Bacterial plate counts were also obtained. Neither plant species nor defoliation affected the bacterial and fungal community structures in both the root tip and mature root regions, but there were significant differences in the bacterial and fungal community profiles between the two root regions for each plant. Prior to defoliation, there was no difference between plants for bacterial plate counts of soils from the root tip regions; however, counts were greater in the mature root region of L. perenne than T. repens. Bacterial plate counts for T. repens were higher in the root tip than the mature root region. After defoliation, there was no effect of plant type, position along the root or defoliation status on bacterial plate counts, although there were significant increases in bacterial plate counts with time. The results indicate that a general effect existed during maturation in the root regions of each plant, which had a greater impact on microbial community structure than either plant type or the effect of defoliation. In addition there were no generic consequences with regard to microbial populations in the rhizosphere as a response to plant defoliation.

Comparing correlations of continuous observations from two independent populations using a sequential approach

A sequential study design generally makes more efficient use of available information than a fixed sample counterpart of equal power. This feature is gradually being exploited by researchers in genetic and epidemiological investigations that utilize banked biological resources and in studies where time, cost and ethics are prominent considerations. Recent work in this area has focussed on the sequential analysis of matched case-control studies with a dichotomous trait. In this paper, we extend the sequential approach to a comparison of the associations within two independent groups of paired continuous observations. Such a comparison is particularly relevant in familial studies of phenotypic correlation using twins. We develop a sequential twin method based on the intraclass correlation and show that use of sequential methodology can lead to a substantial reduction in the number of observations without compromising the study error rates. Additionally, our approach permits straightforward allowance for other explanatory factors in the analysis. We illustrate our method in a sequential heritability study of dysplasia that allows for the effect of body mass index and compares monozygotes with pairs of singleton sisters. Copyright (c) 2006 John Wiley & Sons, Ltd.

Significant redox insensitivity of the functions of the SARS-CoV spike glycoprotein - Comparison with HIV envelope

The capacity of the surface glycoproteins of enveloped viruses to mediate virus/cell binding and membrane fusion requires a proper thiol/disulfide balance. Chemical manipulation of their redox state using reducing agents or free sulfhydryl reagents affects virus/cell interaction. Conversely, natural thiol/disulfide rearrangements often occur during the cell interaction to trigger fusogenicity, hence the virus entry. We examined the relationship between the redox state of the 20 cysteine residues of the SARS-CoV (severe acute respiratory syndrome coronavirus) Spike glycoprotein S1 subdomain and its functional properties. Mature S1 exhibited similar to 4 unpaired cysteines, and chemically reduced S1 displaying up to similar to 6 additional unpaired cysteines still bound ACE2 and enabled fusion. In addition, virus/cell membrane fusion occurred in the presence of sulfhydryl-blocking reagents and oxidoreductase inhibitors. Thus, in contrast to various viruses including HIV (human immunodeficiency virus) examined in parallel, the functions of the SARS-CoV Spike glycoprotein exhibit a significant and surprising independence of redox state, which may contribute to the wide host range of the virus. These data suggest clues for molecularly engineering vaccine immunogens.

Optimization of apodization functions in terahertz transient spectrometry

A quadratic programming optimization procedure for designing asymmetric apodization windows tailored to the shape of time-domain sample waveforms recorded using a terahertz transient spectrometer is proposed. By artificially degrading the waveforms, the performance of the designed window in both the time and the frequency domains is compared with that of conventional rectangular, triangular (Mertz), and Hamming windows. Examples of window optimization assuming Gaussian functions as the building elements of the apodization window are provided. The formulation is sufficiently general to accommodate other basis functions. (C) 2007 Optical Society of America

Discrete time representation of continuous time ARMA processes

This paper derives exact discrete time representations for data generated by a continuous time autoregressive moving average (ARMA) system with mixed stock and flow data. The representations for systems comprised entirely of stocks or of flows are also given. In each case the discrete time representations are shown to be of ARMA form, the orders depending on those of the continuous time system. Three examples and applications are also provided, two of which concern the stationary ARMA(2, 1) model with stock variables (with applications to sunspot data and a short-term interest rate) and one concerning the nonstationary ARMA(2, 1) model with a flow variable (with an application to U.S. nondurable consumers’ expenditure). In all three examples the presence of an MA(1) component in the continuous time system has a dramatic impact on eradicating unaccounted-for serial correlation that is present in the discrete time version of the ARMA(2, 0) specification, even though the form of the discrete time model is ARMA(2, 1) for both models.

Asymptotic expansions for Taylor coefficients of the composition of two functions

We give an asymptotic expansion for the Taylor coe±cients of L(P(z)) where L(z) is analytic in the open unit disc whose Taylor coe±cients vary `smoothly' and P(z) is a probability generating function. We show how this result applies to a variety of problems, amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal sequences.

Root growth models: towards a new generation of continuous approaches

Models of root system growth emerged in the early 1970s, and were based on mathematical representations of root length distribution in soil. The last decade has seen the development of more complex architectural models and the use of computer-intensive approaches to study developmental and environmental processes in greater detail. There is a pressing need for predictive technologies that can integrate root system knowledge, scaling from molecular to ensembles of plants. This paper makes the case for more widespread use of simpler models of root systems based on continuous descriptions of their structure. A new theoretical framework is presented that describes the dynamics of root density distributions as a function of individual root developmental parameters such as rates of lateral root initiation, elongation, mortality, and gravitropsm. The simulations resulting from such equations can be performed most efficiently in discretized domains that deform as a result of growth, and that can be used to model the growth of many interacting root systems. The modelling principles described help to bridge the gap between continuum and architectural approaches, and enhance our understanding of the spatial development of root systems. Our simulations suggest that root systems develop in travelling wave patterns of meristems, revealing order in otherwise spatially complex and heterogeneous systems. Such knowledge should assist physiologists and geneticists to appreciate how meristem dynamics contribute to the pattern of growth and functioning of root systems in the field.