Regular rapidly decreasing nonlinear generalized functions. Application to microlocal regularity

We present new types of regularity for nonlinear generalized functions, based on the notion of regular growth with respect to the regularizing parameter of Colombeau's simplified model. This generalizes the notion of G^{\infty }-regularity introduced by M. Oberguggenberger. A key point is that these regularities can be characterized, for compactly supported generalized functions, by a property of their Fourier transform. This opens the door to microanalysis of singularities of generalized functions, with respect to these regularities. We present a complete study of this topic, including properties of the Fourier transform (exchange and regularity theorems) and relationship with classical theory, via suitable results of embeddings.Comment: Submitted to the Journal of Mathematical Analysis and Application

Remarks on the embedding of spaces of distributions into spaces of Colombeau generalized functions

We present some remarks about the embedding of spaces of Schwartz distributions into spaces of Colombeau generalized functions. We show that the various constructions of such embeddings existing in the literature lead in fact to the same one.Comment: 13 page

Alpha-invariant of Toric Line Bundles

We generalize the work of Jian Song to compute the alpha invariant of any (nef and big) toric line bundle in terms of the associated polytope. We use the analytic version of the computation of the log canonical threshold of monomial ideals to give the log canonical threshold of any non-negatively curved singular hermitian metric on the line bundle, and deduce the alpha invariant from this.Comment: 13 pages. Minor corrections, and references added. Published versio

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

Semi-pointed partition posets and Species

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Fa{\`a} di Bruno Hopf algebra.Comment: 27 page

Pulmonary arterial hypertension: the burden of disease and impact on quality of life.

Pulmonary arterial hypertension (PAH) is a debilitating disease that pervades all aspects of a patients daily life. It is also increasingly acknowledged that the burden of PAH extends to older patients and carers. Until recently, the adverse effect of disease symptoms on the physical, emotional and social factors governing patient health-related quality of life (HRQoL) remained largely unrecognised. With a shift in therapeutic objectives to longer term improvements and HRQoL benefits, clinical trials now frequently include HRQoL measures as study end-points. Most HRQoL instruments used in patients with PAH are generic or non-disease-specific questionnaires and therefore may not accurately capture PAH disease burden. New PAH-specific HRQoL instruments currently undergoing validation include emPHasis-10 and Pulmonary Arterial Hypertension-Symptoms and Impact (PAH-SYMPACT; Actelion Pharmaceuticals Ltd, Allschwil, Switzerland). Using various HRQoL measures, pharmacological therapies have been shown to improve HRQoL in patients with PAH. Patients also derive HRQoL benefits from nonpharmacological strategies, which include the emotional support provided by multidisciplinary care and support groups that is fundamental to patient wellbeing. Looking to the future, validated PAH-specific HRQoL instruments together with dedicated guidelines and procedures are essential to support the translation of HRQoL scores to the clinic, thus enabling a holistic treatment approach to the management of patients with PAH