Generalized Integral Operators and Schwartz Kernel Theorem

In connection with the classical Schwartz kernel theorem, we show that in the framework of Colombeau generalized functions a large class of linear mappings admit integral kernels. To do this, we need to introduce news spaces of generalized functions with slow growth and the corresponding adapted linear mappings. Finally, we show that in some sense Schwartz' result is contained in our main theorem.Comment: 18 page

Generalized Fourier Integral Operators on spaces of Colombeau type

Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distributional data

An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms

We offer an axiomatic definition of a differential algebra of generalized functions over an algebraically closed non-Archimedean field. This algebra is of Colombeau type in the sense that it contains a copy of the space of Schwartz distributions. We study the uniqueness of the objects we define and the consistency of our axioms. Next, we identify an inconsistency in the conventional Laplace transform theory. As an application we offer a free of contradictions alternative in the framework of our algebra of generalized functions. The article is aimed at mathematicians, physicists and engineers who are interested in the non-linear theory of generalized functions, but who are not necessarily familiar with the original Colombeau theory. We assume, however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page

Isomorphisms of algebras of Colombeau generalized functions

We show that for smooth manifolds X and Y, any isomorphism between the special algebra of Colombeau generalized functions on X, resp. Y is given by composition with a unique Colombeau generalized function from Y to X. We also identify the multiplicative linear functionals from the special algebra of Colombeau generalized functions on X to the ring of Colombeau generalized numbers. Up to multiplication with an idempotent generalized number, they are given by an evaluation map at a compactly supported generalized point on X.Comment: 10 page

On the Geroch-Traschen class of metrics

We compare two approaches to semi-Riemannian metrics of low regularity. The maximally 'reasonable' distributional setting of Geroch and Traschen is shown to be consistently contained in the more general setting of nonlinear distributional geometry in the sense of Colombea

Evaluation of elicitation methods to quantify Bayes linear models

The Bayes linear methodology allows decision makers to express their subjective beliefs and adjust these beliefs as observations are made. It is similar in spirit to probabilistic Bayesian approaches, but differs as it uses expectation as its primitive. While substantial work has been carried out in Bayes linear analysis, both in terms of theory development and application, there is little published material on the elicitation of structured expert judgement to quantify models. This paper investigates different methods that could be used by analysts when creating an elicitation process. The theoretical underpinnings of the elicitation methods developed are explored and an evaluation of their use is presented. This work was motivated by, and is a precursor to, an industrial application of Bayes linear modelling of the reliability of defence systems. An illustrative example demonstrates how the methods can be used in practice

Psychometric validation of the European Organisation for Research and Treatment of Cancer–Quality of Life Questionnaire Sexual Health (EORTC QLQ-SH22)

BACKGROUND: The European Organisation for Research and Treatment of Cancer (EORTC) Quality of Life Group developed a questionnaire to assess sexual health in patients with cancer and cancer survivors. This study evaluates the psychometric properties of the questionnaire. METHODS: The 22-item EORTC sexual health questionnaire (EORTC QLQ-SH22) was administered with the EORTC QLQ-C30 to 444 patients with cancer. The hypothesised scale structure, reliability and validity were evaluated through standardised psychometric procedures. RESULTS: The cross-cultural field study showed that the majority of patients (94.7%) were able to complete the QLQ-SH22 in less than 20 min; 89% of the study participants did not need any help to fill in the questionnaire. Multi-item multi-trait scaling analysis confirmed the hypothesised scale structure with two multi-item scales (sexual satisfaction, sexual pain) and 11 single items (including five conditional items and four gender-specific items). The internal consistency yielded acceptable Cronbach's alpha coefficients (.90 for the sexual satisfaction scale, .80 for the sexual pain scale). The test-retest correlations (Pearson's r) ranged from .70 to .93 except for the scale communication with professionals (.67) and male body image (.69). The QLQ-SH22 discriminates well between subgroups of patients differing in terms of their performance and treatment status. CONCLUSION: The study supports the reliability, the content and construct validity of the QLQ-SH22. The newly developed questionnaire is clinically applicable to assess sexual health of patients with cancer at different treatment stages and during survivorship for clinical trials and for clinical practice