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.

A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.

Evolution PDEs and augmented eigenfunctions. half line.

The solution of an initial-boundary value problem for a linear evolution partial differential equation posed on the half-line can be represented in terms of an integral in the complex (spectral) plane. This representation is obtained by the unified transform introduced by Fokas in the 90's. On the other hand, it is known that many initial-boundary value problems can be solved via a classical transform pair, constructed via the spectral analysis of the associated spatial operator. For example, the Dirichlet problem for the heat equation can be solved by applying the Fourier sine transform pair. However, for many other initial-boundary value problems there is no suitable transform pair in the classical literature. Here we pose and answer two related questions: Given any well-posed initial-boundary value problem, does there exist a (non-classical) transform pair suitable for solving that problem? If so, can this transform pair be constructed via the spectral analysis of a differential operator? The answer to both of these questions is positive and given in terms of augmented eigenfunctions, a novel class of spectral functionals. These are eigenfunctions of a suitable differential operator in a certain generalised sense, they provide an effective spectral representation of the operator, and are associated with a transform pair suitable to solve the given initial-boundary value problem.

Relationships Among Murine CD11chigh Dendritic Cell Subsets as Revealed by Baseline Gene Expression Patterns

The functional relationships and properties of different subtypes of dendritic cells (DC) remain largely undefined. To better characterize these cells, we used global gene analysis to determine gene expression patterns among murine CD11c(high) DC subsets. CD4(+), CD8alpha(+), and CD8alpha(-) CD4(-) (double negative (DN)) DC were purified from spleens of normal C57/BL6 mice and analyzed using Affymetrix microarrays. The CD4(+) and CD8alpha(+) DC subsets showed distinct basal expression profiles differing by >200 individual genes. These included known DC subset markers as well as previously unrecognized, differentially expressed CD Ags such as CD1d, CD5, CD22, and CD72. Flow cytometric analysis confirmed differential expression in nine of nine cases, thereby validating the microarray analysis. Interestingly, the microarray expression profiles for DN cells strongly resembled those of CD4(+) DC, differing from them by <25 genes. This suggests that CD4(+) and DN DC are closely related phylogenetically, whereas CD8alpha(+) DC represent a more distant lineage, supporting the historical distinction between CD8alpha(+) and CD8alpha(-) DC. However, staining patterns revealed that in contrast to CD4(+) DC, the DN subset is heterogeneous and comprises at least two subpopulations. Gene Ontology and literature mining analyses of genes expressed differentially among DC subsets indicated strong associations with immune response parameters as well as cell differentiation and signaling. Such associations offer clues to possible unique functions of the CD11c(high) DC subsets that to date have been difficult to define as rigid distinctions.

Living the Revolution: urban communes & soviet socialism, 1917-1932

This book follows a revolutionary trend popular among young activists and would-be radicals after 1917, the formation of collective units of cohabitation and association known as 'urban communes'. In these spaces, activists tried to live what they understood as the 'socialist lifestyle', self-consciously putting Marxist and Bolshevik theories into practice. By telling the story of the urban communes, this book reveals how grand revolutionary ideals, such as collectivism, equality, proletarian ethics, and modern practice, were experienced, understood, and appropriated on a human level. This enables us to better understand the messy realities of the early Soviet state, showing how ideological beliefs and revolutionary contingencies actually came into being during this time.

On mathematical and physical principles of transformations of the coherent radar backscatter matrix

The congruential rule advanced by Graves for polarization basis transformation of the radar backscatter matrix is now often misinterpreted as an example of consimilarity transformation. However, consimilarity transformations imply a physically unrealistic antilinear time-reversal operation. This is just one of the approaches found in literature to the description of transformations where the role of conjugation has been misunderstood. In this paper, the different approaches are examined in particular in respect to the role of conjugation. In order to justify and correctly derive the congruential rule for polarization basis transformation and properly place the role of conjugation, the origin of the problem is traced back to the derivation of the antenna height from the transmitted field. In fact, careful consideration of the role played by the Green’s dyadic operator relating the antenna height to the transmitted field shows that, under general unitary basis transformation, it is not justified to assume a scalar relationship between them. Invariance of the voltage equation shows that antenna states and wave states must in fact lie in dual spaces, a distinction not captured in conventional Jones vector formalism. Introducing spinor formalism, and with the use of an alternate spin frame for the transmitted field a mathematically consistent implementation of the directional wave formalism is obtained. Examples are given comparing the wider generality of the congruential rule in both active and passive transformations with the consimilarity rule.

The evolution of green leases: towards inter-organizational environmental governance

Improving the environmental performance of non-domestic buildings is a complex and ‘wicked’ problem due to conflicting interests and incentives. This is particularly challenging in tenanted spaces, where landlord and tenant interactions are regulated through leases that traditionally ignore environmental considerations. ‘Green leasing’ is conceptualized as a form of ‘middle-out’ inter-organizational environmental governance that operates between organizations, alongside other drivers. This paper investigates how leases are evolving to become ‘greener’ in the UK and Australia, providing evidence from five varied sources on: (1) UK office and retail leases, (2) UK retail sector energy management, (3) a major UK retailer case study; (4) office leasing in Sydney, and (5) expert interviews on Australian retail leases. With some exceptions, the evidence reveals an increasing trend towards green leases in prime offices in both countries, but not in retail or sub-prime offices. Generally introduced by landlords, adopted green leases contain a variety of ambitions and levels of enforcement. As an evolving form of private–private environmental governance, green leases form a valuable framework for further tenant–landlord cooperation within properties and across portfolios. This increased cohesion could create new opportunities for polycentric governance, particularly at the interface of cities and the property industry.