Differential equations play a relevant role in many disciplines and provide powerful tools for analysis and modeling in applied sciences. The book contains several classical and modern methods for the study of ordinary and partial differential equations. A broad space is reserved to Fourier and Laplace transforms together with their applications to the solution of boundary value and/or initial value problems for differential equations. Basic prerequisites concerning analytic functions of complex variable and Lp spaces are synthetically presented in the first two chapters. Techniques based on integral transforms and Fourier series are presented in specific chapters, first in the easier framework of integrable functions and later in the general framework of distributions. The less elementary distributional context allows to deal also with differential equations with highly irregular data and pulse signals. The theory is introduced concisely, while learning of miscellaneous methods is achieved step-by-step through the proposal of many exercises of increasing difficulty. Additional recap exercises are collected in dedicated sections. Several tables for easy reference of main formulas are available at the end of the book. The presentation is oriented mainly to students of Schools in Engineering, Sciences and Economy. The partition of various topics in several self-contained and independent sections allows an easy splitting in at least two didactic modules: one at undergraduate level, the other at graduate level. This text is the English translation of the Third Edition of the Italian book “Analisi Complessa, Trasformate, Equazioni Differenziali" published by Esculapio in 2015.
Deep comprehension of applied sciences requires a solid knowledge of Mathematical Analysis. For most of high level scientific research, the good understanding of Functional Analysis and weak solutions to differential equations is essential. This book aims to deal with the main topics that are necessary to achieve such a knowledge. Still, this is the goal of many other texts in advanced analysis; and then, what would be a good reason to read or to consult this book? In order to answer this question, let us introduce the three Authors. Alberto Ferrero got his degree in Mathematics in 2000 and presently he is researcher in Mathematical Analysis at the Università del Piemonte Orientale. Filippo Gazzola got his degree in Mathematics in 1987 and he is now full professor in Mathematical Analysis at the Politecnico di Milano. Maurizio Zanotti got his degree in Mechanical Engineering in 2004 and presently he is structural and machine designer and lecturer professor in Mathematical Analysis at the Politecnico di Milano. The three Authors, for the variety of their skills, decided to join their expertises to write this book. One of the reasons that should encourage its reading is that the presentation turns out to be a reasonable compromise among the essential mathematical rigor, the importance of the applications and the clearness, which is necessary to make the reference work pleasant to the readers, even to the inexperienced ones. The range of treated topics is quite wide and covers the main basic notions of the scientific research which is based upon mathematical models. We start from vector spaces and Lebesgue integral to reach the frontier of theoretical research such as the study of critical exponents for semilinear elliptic equations and recent problems in fluid dynamics. This long route passes through the theory of Banach and Hilbert spaces, Sobolev spaces, differential equations, Fourier and Laplace transforms, before which we recall some appropriate tools of Complex Analysis. We give all the proofs that have some didactic or applicative interest, while we omit the ones which are too technical or require too high level knowledge. This book has the ambitious purpose to be useful to a broad variety of readers. The first possible beneficiaries are of course the second or third year students of a scientific course of degree: in what follows they will find the topics that are necessary to approach more advanced studies in Mathematics and in other fields, especially Physics and Engineering. This text could be also useful to graduate students who want to start a Ph.D. course: indeed it contains the matter of a multidisciplinary Ph.D. course given by Filippo Gazzola for several years at Politecnico di Milano. Finally, this book could be addressed also to the ones who have already left education far-back but occasionally need to use mathematical tools: we refer both to university professors and their research, and to professionals and designers who want to model a certain phenomenon, but also to the nostalgics of the good old days when they were students. It is precisely for this last type of reader that we have also reported some elementary topics, such as the properties of numerical sets and of the integrals; moreover, every chapter is provided with examples and specific exercises aimed at the involvement of the reader.
Differential equations play a relevant role in many disciplines and provide powerful tools for analysis and modeling in applied sciences. The book contains several classical and modern methods for the study of ordinary and partial differential equations. A broad space is reserved to Fourier and Laplace transforms together with their applications to the solution of boundary value and/or initial value problems for differential equations. Basic prerequisites concerning analytic functions of complex variable and Lp spaces are synthetically presented in the first two chapters. Techniques based on integral transforms and Fourier series are presented in specific chapters, first in the easier framework of integrable functions and later in the general framework of distributions. The less elementary distributional context allows to deal also with differential equations with highly irregular data and pulse signals. The theory is introduced offhandedly and learning of miscellaneous methods is achieved step-by-step through the proposal of many exercises of increasing difficulty. Additional recap exercises are collected in dedicated sections. Several tables for easy reference of main formulas are available at the end of the book. The presentation is oriented mainly to students of Schools in Engineering, Sciences and Economy. The partition of various topics in several self-contained and independent sections allows an easy splitting in at least two didactic modules: one at undergraduate level, the other at graduate level. This text is the English translation of the Second Edition of the Italian book “Analisi Complessa, Trasformate, Equazioni Differenziali" published by Esculapio in 2013.
If you told a woman her sex had a shared, long-lived history with weasels, she might deck you. But those familiar with mythology know better: that the connection between women and weasels is an ancient and favorable one, based in the Greek myth of a midwife who tricked the gods to ease Heracles’s birth—and was turned into a weasel by Hera as punishment. Following this story as it is retold over centuries in literature and art, Women and Weasels takes us on a journey through mythology and ancient belief, revising our understanding of myth, heroism, and the status of women and animals in Western culture. Maurizio Bettini recounts and analyzes a variety of key literary and visual moments that highlight the weasel’s many attributes. We learn of its legendary sexual and childbearing habits and symbolic association with witchcraft and midwifery, its role as a domestic pet favored by women, and its ability to slip in and out of tight spaces. The weasel, Bettini reveals, is present at many unexpected moments in human history, assisting women in labor and thwarting enemies who might plot their ruin. With a parade of symbolic associations between weasels and women—witches, prostitutes, midwives, sisters-in-law, brides, mothers, and heroes—Bettini brings to life one of the most venerable and enduring myths of Western culture.
The main aim of the research presented in this thesis was to expand the substrate scope of the iridium-catalyzed asymmetric hydrogenation, which represents an extremely useful methodology for the enantioselective synthesis of chiral molecules.While this chemistry has been developed and investigated mainly for the hydrogenation of unfunctionalized olefins, so far only little attention has been given to functionalized olefins. Therefore, the different research projects presented in this thesis dealt with the application of iridium catalysts to the reduction of particularly valuable substrates that are difficult to hydrogenate enantioselectively with other methods. The first chapter of this thesis gives a general introduction on asymmetric hydrogenation and the role of iridium catalysts in this context. The following two chapters deal with the investigation of new substrates in the iridium-catalyzed asymmetric hydrogenation using various N,Pligands developed in the Pfaltz group,and give an account of the superior results that have been obtained with such catalysts compared to thoserepresenting the state-of-the-art. In particular, chapter twoconcerns the reduction of vinylsilanes, for which the judicious choice of the best catalyst for each specific substrate was required to achieve good results in term of both chemical and optical yield. On the contrary, a pyridinyl phosphitine bearing a 2,6-difluorophenyl group on the oxazoline ringwas best suited for a broad array of 2-alkyl- and aryl- substituted maleic acid dimethyldiester, as reported in chapter three. Such process turned out to be enantioconvergent, allowing the hydrogenation of mixtures of maleates and fumarates in high enantiomeric excesses. In chapter fourthe deployment of environmentally friendly solvents such as THF and 2-MeTHF in the iridium-catalyzed asymmetric hydrogenation of 3,3-disubstituted allylic alcohols is described. Finally, chapter five of this dissertation deals with the development of new NHC ligands for the iridium-catalyzed asymmetric hydrogenation of acid-labile substrates such as tert-butyloxycarbonyl protected allylic alcohols. Experimental details and characterization of the substances discussed in the main body of this manuscript is reported in the experimental section that constitutes chapter six.
Differential equations play a relevant role in many disciplines and provide powerful tools for analysis and modeling in applied sciences. The book contains several classical and modern methods for the study of ordinary and partial differential equations. A broad space is reserved to Fourier and Laplace transforms together with their applications to the solution of boundary value and/or initial value problems for differential equations. Basic prerequisites concerning analytic functions of complex variable and Lp spaces are synthetically presented in the first two chapters. Techniques based on integral transforms and Fourier series are presented in specific chapters, first in the easier framework of integrable functions and later in the general framework of distributions. The less elementary distributional context allows to deal also with differential equations with highly irregular data and pulse signals. The theory is introduced concisely, while learning of miscellaneous methods is achieved step-by-step through the proposal of many exercises of increasing difficulty. Additional recap exercises are collected in dedicated sections. Several tables for easy reference of main formulas are available at the end of the book. The presentation is oriented mainly to students of Schools in Engineering, Sciences and Economy. The partition of various topics in several self-contained and independent sections allows an easy splitting in at least two didactic modules: one at undergraduate level, the other at graduate level. This text is the English translation of the Third Edition of the Italian book “Analisi Complessa, Trasformate, Equazioni Differenziali" published by Esculapio in 2015.
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