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The book is about differentiability of six operators on functions or pairs of functions: composition (f of g), integration (of f dg), multiplication and convolution of two functions, both varying, and the product integral and inverse operators for one function. The operators are differentiable with respect to p-variation norms with optimal remainder bounds. Thus the functions as arguments of the operators can be nonsmooth, possibly discontinuous, but four of the six operators turn out to be analytic (holomorphic) for some p-variation norms. The reader will need to know basic real analysis, including Riemann and Lebesgue integration. The book is intended for analysts, statisticians and probabilists. Analysts and statisticians have each studied the differentiability of some of the operators from different viewpoints, and this volume seeks to unify and expand their results.
Darwin's theory of evolution by natural selection was based on the observation that there is variation between individuals within the same species. This fundamental observation is a central concept in evolutionary biology. However, variation is only rarely treated directly. It has remained peripheral to the study of mechanisms of evolutionary change. The explosion of knowledge in genetics, developmental biology, and the ongoing synthesis of evolutionary and developmental biology has made it possible for us to study the factors that limit, enhance, or structure variation at the level of an animals' physical appearance and behavior. Knowledge of the significance of variability is crucial to this emerging synthesis. Variation situates the role of variability within this broad framework, bringing variation back to the center of the evolutionary stage. - Provides an overview of current thinking on variation in evolutionary biology, functional morphology, and evolutionary developmental biology - Written by a team of leading scholars specializing on the study of variation - Reviews of statistical analysis of variation by leading authorities - Key chapters focus on the role of the study of phenotypic variation for evolutionary, developmental, and post-genomic biology
Wood is the usual end product of a forestry operation. Because of its importance, numerous studies have been made relative to wood prop erties, the causes of wood variation, and how best to develop wood for desired products. There is voluminous literature related to these subjects, but it is neither well known nor appreciated by foresters because the publications are often not available or are not well understood by the forester or by those who use the wood. Frequently, the literature is confusing and contradictory, making it difficult for the nonspecialist to use what information is available. In order to produce and use wood efficiently, the variation pat terns within trees, among trees within species, and among species must be understood. This also requires some knowledge of the causes of variation and the effects of different wood properties upon utiliza tion. The information about variation patterns, their causes, and con trol and effect upon the product must be known by the tree grower, the tree breeder, and the tree harvester as well as by those who ultimately convert wood into a final, salable product.
A comprehensive account of the theory and applications of regular variation.
This volume is an analytical study of 18 works by Brahms, making skillful use of Schoenberg's provocative concept of developing variation. It traces a genuine evolution through Brahm's compositions, considering their relationship to each other.
Parametric variation in linguistic theory refers to the systematic grammatical variation permitted by the human language faculty. This book is a defence of the parametric approach to linguistic variation, set within the framework of the Minimalist Program.
Studies on Variation in Portuguese offers a collection of studies on a range of variable phenomena attested within and across varieties of Portuguese. The volume starts out with an overview of current issues in the study of intralinguistic variation and is divided in two parts. Part 1 is dedicated to research on variation within national varieties (Brazilian and European). Here, a multidimensional analysis that combines both the geographic and the social dimensions of variation emerges as a way to identify possible regional specificities and the directionality of some of the variants. Part 2 collects studies that compare the behavior of a particular linguistic variable across different varieties. The variable phenomena discussed concern several levels of grammar and are framed within different conceptions of variation, thus promoting confrontation of theoretical and methodological alternatives. Overall, the volume constitutes a significant contribution to the essential question of how to model variation at different levels.