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Function Spaces of Logarithmic Smoothness: Embeddings and Characterizations
In this paper we present a comprehensive treatment of function spaces with logarithmic smoothness (Besov, Sobolev, Triebel-Lizorkin).
We establish the following results:
The key tools behind our results
are limiting interpolation techniques and new characterizations of Besov and Sobolev norms in terms of the behavior of the Fourier
transforms for functions such that their Fourier transforms are of monotone type or lacunary series.
eBook
distance‐based framework for measuring functional diversity from multiple traits
A new framework for measuring functional diversity (FD) from multiple traits has recently been proposed. This framework was mostly limited to quantitative traits without missing values and to situations in which there are more species than traits, although the authors had suggested a way to extend their framework to other trait types. The main purpose of this note is to further develop this suggestion. We describe a highly flexible distance‐based framework to measure different facets of FD in multidimensional trait space from any distance or dissimilarity measure, any number of traits, and from different trait types (i.e., quantitative, semi‐quantitative, and qualitative). This new approach allows for missing trait values and the weighting of individual traits. We also present a new multidimensional FD index, called functional dispersion (FDis), which is closely related to Rao's quadratic entropy. FDis is the multivariate analogue of the weighted mean absolute deviation (MAD), in which the weights are species relative abundances. For unweighted presence–absence data, FDis can be used for a formal statistical test of differences in FD. We provide the “FD” R language package to easily implement our distance‐based FD framework.
Journal Article
The Oxford reference guide to lexical functional grammar
This volume is the most comprehensive reference work to date on Lexical Functional Grammar. The authors provide detailed and extensive coverage of the analysis of syntax, semantics, morphology, prosody, and information structure, and how these aspects of linguistic structure interact in the nontransformational framework of LFG. 0The book is divided into three parts. The first part examines the syntactic theory and formal architecture of LFG, with detailed explanations and comprehensive illustration, providing an unparalleled introduction to the fundamentals of the theory. Part two explores non-syntactic levels of linguistic structure, including the syntax-semantics interface and semantic representation, argument structure, information structure, prosodic structure, and morphological structure, and how these are related0in the projection architecture of LFG. Chapters in the third part illustrate the theory more explicitly by presenting explorations of the syntax and semantics of a range of representative linguistic phenomena: modification, anaphora, control, coordination, and long-distance dependencies. The final chapter discusses LFG-based work not covered elsewhere in the book, as well as new developments in the theory.0The volume will be an invaluable reference for graduate and advanced undergraduate students and researchers in a wide range of linguistic sub-fields, including syntax, morphology, semantics, information structure, and prosody, as well as those working in language documentation and description.
Book
Methods for Scalar-on-Function Regression
Recent years have seen an explosion of activity in the field of functional data analysis (FDA), in which curves, spectra, images and so on are considered as basic functional data units. A central problem in FDA is how to fit regression models with scalar responses and functional data points as predictors. We review some of the main approaches to this problem, categorising the basic model types as linear, non-linear and non-parametric. We discuss publicly available software packages and illustrate some of the procedures by application to a functional magnetic resonance imaging data set.
Journal Article
Introducing Elixir : getting started in functional programming
Smooth, powerful, and small, the Elixir programming language is an excellent place for newcomers to learn about functional programming. This book shows readers how Elixir combines the robust functional programming of Erlang with an approach that looks more like Ruby. Readers will learn how Elixir simplifies some of Erlang's odder corners and reaches toward metaprogramming with powerful macro features. Updated to cover Elixir 1.4.-- Source other than the Library of Congress.
Book
mFD: an R package to compute and illustrate the multiple facets of functional diversity
Functional diversity (FD), the diversity of organism attributes that relates to their interactions with the abiotic and biotic environment, has been increasingly used for the last two decades in ecology, biogeography and conservation. Yet, FD has many facets and their estimations are not standardized nor embedded in a single tool. mFD (multifaceted functional diversity) is an R package that uses matrices of species assemblages and species trait values as building blocks to compute most FD indices. mFD is firstly based on two functions allowing the user to summarize trait and assemblage data. Then it calculates trait-based distances between species pairs, informs the user whether species have to be clustered into functional entities and finally computes multidimensional functional space. To let the user choose the most appropriate functional space for computing multidimensional functional diversity indices, two mFD functions allow assessing and illustrating the quality of each functional space. Next, mFD provides 6 core functions to calculate 16 existing FD indices based on trait-based distances, functional entities or species position in a functional space. The mFD package also provides graphical functions based on the ggplot library to illustrate FD values through customizable and high-resolution plots of species distribution among functional entities or in a multidimensional space. All functions include internal validation processes to check for errors in data formatting which return detailed error messages. To facilitate the use of mFD framework, we built an associated website hosting five tutorials illustrating the use of all the functions step by step.
Journal Article
Embeddings of Decomposition Spaces
Many smoothness spaces in harmonic analysis are decomposition spaces. In this paper we ask: Given two such spaces, is there an
embedding between the two?
A decomposition space
We establish readily verifiable criteria which ensure the
existence of a continuous inclusion (“an embedding”)
In a nutshell, in order to apply the embedding results presented in this
article, no knowledge of Fourier analysis is required; instead, one only has to study the geometric properties of the involved
coverings, so that one can decide the finiteness of certain sequence space norms defined in terms of the coverings.
These
sufficient criteria are quite sharp: For almost arbitrary coverings and certain ranges of
We also prove a
The resulting embedding theory is illustrated by applications
to
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