Numerical Schemes for Rough Parabolic Equations

This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0, 1) perturbed by a non-linear rough signal. It is the continuation of [8, 7], where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H \textgreater{} 1/3.Comment: Applied Mathematics and Optimization, 201

An integrative environmental pollen diversity assessment and its importance for the Sustainable Development Goals

Pollen is at once intimately part of the reproductive cycle of seed plants and simultaneously highly relevant for the environment (pollinators, vector for nutrients, or organisms), people (food safety and health), and climate (cloud condensation nuclei and climate reconstruction). We provide an interdisciplinary perspective on the many and connected roles of pollen to foster a better integration of the currently disparate fields of pollen research, which would benefit from the sharing of general knowledge, technical advancements, or data processing solutions. We propose a more interdisciplinary and holistic research approach that encompasses total environmental pollen diversity (ePD) (wind and animal and occasionally water distributed pollen) at multiple levels of diversity (genotypic, phenotypic, physiological, chemical, and functional) across space and time. This interdisciplinary approach holds the potential to contribute to pressing human issues, including addressing United Nations Sustainable Development Goals, fostering social and political awareness of these tiny yet important and fascinating particles

Variable exponent Besov-Morrey spaces

In this paper we introduce Besov-Morrey spaces with all indices variable and study some fundamental properties. This includes a description in terms of Peetre maximal functions and atomic and molecular decompositions. This new scale of non-standard function spaces requires the introduction of variable exponent mixed Morrey-sequence spaces, which in turn are defined within the framework of semimodular spaces. In particular, we obtain a convolution inequality involving special radial kernels, which proves to be a key tool in this work.publishe

Global compactness results for nonlocal problems

International audienceWe obtain a Struwe type global compactness result for a class of nonlinear nonlocal problems involving the fractional $p-$Laplacian operator and nonlinearities at critical growth

Sobolev spaces on non-Lipschitz subsets of Rn with application to boundary integral equations on fractal screens

We study properties of the classical fractional Sobolev spaces on non-Lipschitz subsets of Rn. We investigate the extent to which the properties of these spaces, and the relations between them, that hold in the well-studied case of a Lipschitz open set, generalise to non-Lipschitz cases. Our motivation is to develop the functional analytic framework in which to formulate and analyse integral equations on non-Lipschitz sets. In particular we consider an application to boundary integral equations for wave scattering by planar screens that are non-Lipschitz, including cases where the screen is fractal or has fractal boundary

A randomised controlled trial investigating the effect of nutritional supplementation on visual function in normal, and age-related macular disease affected eyes: design and methodology [ISRCTN78467674]

BACKGROUND: Age-related macular disease is the leading cause of blind registration in the developed world. One aetiological hypothesis involves oxidation, and the intrinsic vulnerability of the retina to damage via this process. This has prompted interest in the role of antioxidants, particularly the carotenoids lutein and zeaxanthin, in the prevention and treatment of this eye disease. METHODS: The aim of this randomised controlled trial is to determine the effect of a nutritional supplement containing lutein, vitamins A, C and E, zinc, and copper on measures of visual function in people with and without age-related macular disease. Outcome measures are distance and near visual acuity, contrast sensitivity, colour vision, macular visual field, glare recovery, and fundus photography. Randomisation is achieved via a random number generator, and masking achieved by third party coding of the active and placebo containers. Data collection will take place at nine and 18 months, and statistical analysis will employ Student's t test. DISCUSSION: A paucity of treatment modalities for age-related macular disease has prompted research into the development of prevention strategies. A positive effect on normals may be indicative of a role of nutritional supplementation in preventing or delaying onset of the condition. An observed benefit in the age-related macular disease group may indicate a potential role of supplementation in prevention of progression, or even a degree reversal of the visual effects caused by this condition