Vekua theory for the Helmholtz operator

Vekua operators map harmonic functions defined on domain in \mathbb R2R2 to solutions of elliptic partial differential equations on the same domain and vice versa. In this paper, following the original work of I. Vekua (Ilja Vekua (1907–1977), Soviet-Georgian mathematician), we define Vekua operators in the case of the Helmholtz equation in a completely explicit fashion, in any space dimension N ≥ 2. We prove (i) that they actually transform harmonic functions and Helmholtz solutions into each other; (ii) that they are inverse to each other; and (iii) that they are continuous in any Sobolev norm in star-shaped Lipschitz domains. Finally, we define and compute the generalized harmonic polynomials as the Vekua transforms of harmonic polynomials. These results are instrumental in proving approximation estimates for solutions of the Helmholtz equation in spaces of circular, spherical, and plane waves.

Operator space structure of JC*-triples and TROs, I

We embark upon a systematic investigation of operator space structure of JC*-triples via a study of the TROs (ternary rings of operators) they generate. Our approach is to introduce and develop a variety of universal objects, including universal TROs, by which means we are able to describe all possible operator space structures of a JC*-triple. Via the concept of reversibility we obtain characterisations of universal TROs over a wide range of examples. We apply our results to obtain explicit descriptions of operator space structures of Cartan factors regardless of dimension

On the operator space structure of Hilbert spaces

Operator spaces of Hilbertian JC∗ -triples E are considered in the light of the universal ternary ring of operators (TRO) introduced in recent work. For these operator spaces, it is shown that their triple envelope (in the sense of Hamana) is the TRO they generate, that a complete isometry between any two of them is always the restriction of a TRO isomorphism and that distinct operator space structures on a fixed E are never completely isometric. In the infinite-dimensional cases, operator space structure is shown to be characterized by severe and definite restrictions upon finite-dimensional subspaces. Injective envelopes are explicitly computed.

The paradox of nationalism: the common denominator of radical right and radical left euroscepticism

What can explain the strong euroscepticism of radical parties of both the right and the left? This article argues that the answer lies in the paradoxical role of nationalism as a central element in both party families, motivating opposition towards European integration. Conventionally, the link between nationalism and euroscepticism is understood solely as a prerogative of radical right-wing parties, whereas radical left-wing euroscepticism is associated with opposition to the neoliberal character of the European Union.This article contests this view. It argues that nationalism cuts across party lines and constitutes the common denominator of both radical right-wing and radical left-wing euroscepticism. It adopts a mixed-methods approach, combining intensive case study analysis with quantitative analysis of party manifestos. First, it traces the link between nationalism and euroscepticism in Greece and France in order to demonstrate the internal validity of the argument. It then undertakes a cross-country statistical estimation to assess the external validity of the argument and its generalisability across Europe.

The civic zeitgeist: nationalism and liberal values in the European radical right

This article examines the relationship between nationalism and liberal values, and more specifically the redefinition of boundaries between national communities and others in the rhetoric of radical right parties in Europe. The aim is to examine the tension between radical right party discourse and the increasing need to shape this discourse in liberal terms. We argue that the radical right parties that successfully operate within the democratic system tend to be those best able to tailor their discourse to the liberal and civic characteristics of national identity so as to present themselves and their ideologies as the true authentic defenders of the nation's unique reputation for democracy, diversity and tolerance. Comparing the success of a number of European radical right parties ranging from the most electorally successful SVP to the more mixed BNP, FN and NPD, we show that the parties that effectively deploy the symbolic resources of national identity through a predominantly voluntaristic prism tend to be the ones that fare better within their respective political systems. In doing so, we challenge the conventional view in the study of nationalism which expects civic values to shield countries from radicalism and extremism.

Drug control, human rights, and the right to the highest attainable standard of health: a reply to Saul Takahashi

A recent article in this journal challenged claims that a human rights framework should be applied to drug control. This article questions the author’s assertions and reframes them in the context of socio-legal drug scholarship, aiming to build on the discourse concerning human rights and drug use. It is submitted that a rights-based approach is a necessary, indeed obligatory, ethical and legal framework through which to address drug use and that international human rights law provides the proper scope for determining where interferences with individual human rights might be justified on certain, limited grounds.

Universally reversible JC*-triples and operator spaces

Abstract. We prove that the vast majority of JC∗-triples satisfy the condition of universal reversibility. Our characterisation is that a JC∗-triple is universally reversible if and only if it has no triple homomorphisms onto Hilbert spaces of dimension greater than two nor onto spin factors of dimension greater than four. We establish corresponding characterisations in the cases of JW∗-triples and of TROs (regarded as JC∗-triples). We show that the distinct natural operator space structures on a universally reversible JC∗-triple E are in bijective correspondence with a distinguished class of ideals in its universal TRO, identify the Shilov boundaries of these operator spaces and prove that E has a unique natural operator space structure precisely when E contains no ideal isometric to a nonabelian TRO. We deduce some decomposition and completely contractive properties of triple homomorphisms on TROs.

Valuation 'accuracy' within the Right to Buy process

Based on surveys undertaken with local authorities and valuers who provide the valuations on which purchase prices for local authority houses under the Right to Buy are based, this paper reports on research which aims to establish the reasons for the differences between the initial valuations provided by the local authority valuers and those provided by the District Valuer on appeal. The paper reports on the reasons why tenants appeal the initial valuation and discusses issues of valuation accuracy, uncertainty and the different and imperfect data available to valuers employed by the organisations involved, as well as the factors within the valuation process, including the absence of any requirement to agree a value, which contribute to the different outcomes.

'Quasi'-norm of an arithmetical convolution operator and the order of the Riemann zeta function

In this paper we study Dirichlet convolution with a given arithmetical function f as a linear mapping 'f that sends a sequence (an) to (bn) where bn = Pdjn f(d)an=d. We investigate when this is a bounded operator on l2 and ¯nd the operator norm. Of particular interest is the case f(n) = n¡® for its connection to the Riemann zeta function on the line 1, 'f is bounded with k'f k = ³(®). For the unbounded case, we show that 'f : M2 ! M2 where M2 is the subset of l2 of multiplicative sequences, for many f 2 M2. Consequently, we study the `quasi'-norm sup kak = T a 2M2 k'fak kak for large T, which measures the `size' of 'f on M2. For the f(n) = n¡® case, we show this quasi-norm has a striking resemblance to the conjectured maximal order of j³(® + iT )j for ® > 12 .

A generalized collectively compact operator theory with an application to integral equations on unbounded domains

In this paper a generalization of collectively compact operator theory in Banach spaces is developed. A feature of the new theory is that the operators involved are no longer required to be compact in the norm topology. Instead it is required that the image of a bounded set under the operator family is sequentially compact in a weaker topology. As an application, the theory developed is used to establish solvability results for a class of systems of second kind integral equations on unbounded domains, this class including in particular systems of Wiener-Hopf integral equations with L1 convolutions kernels

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.