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Multi-Parameter Hardy Spaces Theory and Endpoint Estimates for Multi-Parameter Singular Integrals
The main purpose of this paper is to establish the theory of the multi-parameter Hardy spaces
More precisely, Street (2014) studied the
eBook
Eléments d'analyse Réelle
Ouvrage de synthèse avec des chapitres rédigés de manière relativement indépendante. Lignes directrices : approfondir les notions de base ; privilégier les applications à d'autres domaines des mathématiques ; bien analyser les hypothèses des théorèmes en proposant de nombreux contre-exemples ; élargir le champs des connaissances du lecteur.
eBook
Maximal Functions, Littlewood–Paley Theory, Riesz Transforms and Atomic Decomposition in the Multi-parameter Flag Setting
In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag
setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood–Paley square function
and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this
paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the
Plancherel–Pólya inequalities for flag quadratic forms; (2) introducing the maximal function and area function via flag Poisson kernels
and flag version of harmonic functions; (3) developing an atomic decomposition via the finite speed propagation and area function in
terms of flag heat semigroups. As a consequence of these real variable methods, we obtain the full characterisations of the
multi-parameter Hardy space with the flag structure.
eBook
Fourier Series in Several Variables with Applications to Partial Differential Equations
Discussing many results and studies from the literature, this work illustrates the value of Fourier series methods in solving difficult nonlinear PDEs. Using these methods, the author presents results for stationary Navier-Stokes equations, nonlinear reaction-diffusion systems, and quasilinear elliptic PDEs and resonance theory. He also establishes the connection between multiple Fourier series and number theory, presents the periodic Ca-theory of Calderon and Zygmund, and explores the extension of Fatou's famous work on antiderivatives and nontangential limits to higher dimensions. The importance of surface spherical harmonic functions is emphasized throughout.
eBook
Advanced Calculus of Several Variables
ADVANCED CALCULUS OF SEVERAL VARIABLES covers important topics of Transformations and topology on Euclidean in n-space Rn Functions of several variables, Differentiation in Rn, Multiple integrals and Integration in Rn. The topics have been presented in a simple clear and coherent style with a number of examples and exercises. Proofs have been made direct and simple. Unsolved problems just after relevant articles in the form of exercises and typical problems followed by suggestions have been given. This book will help the reader work on the problems of Numerical Analysis, Operations Research, Differential Equations and Engineering applications.
eBook
Paintings predict the distribution of species, or the challenge of selecting environmental predictors and evaluation statistics
Aim: Species distribution modelling, a family of statistical methods that predicts species distributions from a set of occurrences and environmental predictors, is now routinely applied in many macroecological studies. However, the reliability of evaluation metrics usually employed to validate these models remains questioned. Moreover, the emergence of online databases of environmental variables with global coverage, especially climatic, has favoured the use of the same set of standard predictors. Unfortunately, the selection of variables is too rarely based on a careful examination of the species' ecology. In this context, our aim was to highlight the importance of selecting ad hoc variables in species distribution models, and to assess the ability of classical evaluation statistics to identify models with no biological realism. Innovation: First, we reviewed the current practices in the field of species distribution modelling in terms of variable selection and model evaluation. Then, we computed distribution models of 509 European species using pseudo-predictors derived from paintings or using a real set of climatic and topographic predictors. We calculated model performance based on the area under the receiver operating curve (AUC) and true skill statistics (TSS), partitioning occurrences into training and test data with different levels of spatial independence. Most models computed from pseudo-predictors were classified as good and sometimes were even better evaluated than models computed using real environmental variables. However, on average they were better discriminated when the partitioning of occurrences allowed testing for model transferability. Main conclusions: These findings confirm the crucial importance of variable selection and the inability of current evaluation metrics to assess the biological significance of distribution models. We recommend that researchers carefully select variables according to the species' ecology and evaluate models only according to their capacity to be transfered in distant areas. Nevertheless, statistics of model evaluations must still be interpreted with great caution.
Journal Article
