European Mathematical Society - a. vignoli
https://euro-math-soc.eu/author/vignoli
enEquivariant Degree Theory
https://euro-math-soc.eu/review/equivariant-degree-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>The aim of the book is the development and applications of the degree theory in the context of equivariant maps. (Equivariant simply means that the mapping has certain symmetries, e.g., being even/odd, periodic, rotational invariant, etc.). The theory is developed both in finite and infinite dimension. The first chapter gives necessary preliminaries. The second chapter brings the definition of the degree and studies its basic properties. As the definition is somewhat abstract (the degree is defined as an element of the group of equivariant homotopy classes of maps between two spheres), it is useful to compute the degree in various particular cases. This is accomplished in Chapter 3. The last and also the longest chapter, deals with applications to particular ODE’s and to bifurcation theory. The aim of the authors was to write a book that would be easily accessible even to non-specialists, thus the exposition is accompanied by a number of examples and the use of abstract special tools is limited. It is also worth noting that each chapter is accompanied by detailed bibliographical remarks.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">dpr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-ize" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. ize</a></li><li class="vocabulary-links field-item odd"><a href="/author/vignoli" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. vignoli</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/walter-de-gruyter" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">walter de gruyter</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2003</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-11-017550-9</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">€98</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Mon, 12 Sep 2011 13:49:22 +0000Anonymous39675 at https://euro-math-soc.euNonlinear Spectral Theory
https://euro-math-soc.eu/review/nonlinear-spectral-theory
<div class="field field-name-field-review-review field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><div class="tex2jax"><p>Nonlinear spectral theory is a relatively new field of mathematics, which is far from being complete, and many fundamental questions still remain open. The main focus of this book is therefore formulated by the authors as the following question: How should we define a spectrum for nonlinear operators in such a way that it preserves useful properties of the linear case but admits applications to a possibly large variety of nonlinear problems? Contrary to the linear case, the spectrum of a nonlinear operator contains practically no information on this operator. The authors convince the reader that it is not the intrinsic structure of the spectrum itself, which leads to interesting applications, but its property of being a useful tool for solving nonlinear equations. The book is an excellent presentation of the “state-of-the-art” of contemporary nonlinear spectral theory as well as a glimpse of the diversity of directions in which current research is moving.<br />
The whole text consists of 12 chapters. The authors recall basic facts on the spectrum of a bounded linear operator in the first chapter. In Chapter 2, some numerical characteristics providing quantitative descriptions of certain mapping properties of nonlinear operators are studied. The classical Kuratowski measure of noncompactness plays a key role here. Chapter 3 is devoted to general invertibility results. In particular, conditions that guarantee that the local invertibility of a nonlinear operator implies its global invertibility are of interest. The Rhodins and the Neuberger spectra are studied in Chapter 4. In Chapter 5, the authors study a spectrum for Lipschitz continuous operators, first proposed by Kachurovskij in 1969, and a spectrum for linearly bounded operators, introduced recently by Dörfner. Chapter 6 discusses the spectrum for certain special continuous operators introduced by Furi, Martelli and Vignoli in 1978, and its modification introduced recently by Appell, Giorgieri and Väth. The Feng spectrum is discussed in detail in Chapter 7. Chapter 8 is devoted to the study of “local spectrum” due to Väth, which in the literature is called “phantom”. In Chapter 9, the authors investigate a modification of the Feng spectrum proposed by Feng and Webb and another spectrum introduced by Singhof-Weyer and Weberand Infante-Webb. Chapter 10 is devoted to the study of nonlinear eigenvalue problems. The authors concentrate on the notion of a “nonlinear eigenvalue”, nonlinear analogue of the Krein-Rutman theorem, connected eigenvalues, etc. Chapter 11 contains a description on how numerical ranges may be used to localize the spectrum of a nonlinear operator on the real line or in the complex plane. Selected applications are presented in the last Chapter.<br />
The exposition of nonlinear spectral theory in this book is self-contained. All major statements are proved; each definition and notion is carefully illustrated by examples. To understand this text does not require any special knowledge and only modest background of nonlinear analysis and operator theory is required. The book is addressed not only to mathematicians working in analysis but also to non-specialists wanting to understand the development of spectral theory for nonlinear operators in the last 30 years. The bibliography is rather exhaustive and so this text will certainly serve as an excellent reference book for many years. I am convinced that at least one copy of this book should be in any mathematical library.</p>
</div></div></div></div><div class="field field-name-field-review-reviewer field-type-text field-label-inline clearfix"><div class="field-label">Reviewer: </div><div class="field-items"><div class="field-item even">pdr</div></div></div><span class="vocabulary field field-name-field-review-author field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Author: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/author/j-appell" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">j. appell</a></li><li class="vocabulary-links field-item odd"><a href="/author/e-de-pascale" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">e. de pascale</a></li><li class="vocabulary-links field-item even"><a href="/author/vignoli" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">a. vignoli</a></li></ul></span><span class="vocabulary field field-name-field-review-publisher field-type-taxonomy-term-reference field-label-inline clearfix"><h2 class="field-label">Publisher: </h2><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/publisher/walter-de-gruyter" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">walter de gruyter</a></li></ul></span><div class="field field-name-field-review-pub field-type-number-integer field-label-inline clearfix"><div class="field-label">Published: </div><div class="field-items"><div class="field-item even">2004</div></div></div><div class="field field-name-field-review-isbn field-type-text field-label-inline clearfix"><div class="field-label">ISBN: </div><div class="field-items"><div class="field-item even">3-11-018143-6</div></div></div><div class="field field-name-field-review-price field-type-text field-label-inline clearfix"><div class="field-label">Price: </div><div class="field-items"><div class="field-item even">€148</div></div></div><span class="vocabulary field field-name-field-review-msc field-type-taxonomy-term-reference field-label-hidden"><ul class="vocabulary-list"><li class="vocabulary-links field-item even"><a href="/msc/47-operator-theory" typeof="skos:Concept" property="rdfs:label skos:prefLabel" datatype="">47 Operator theory</a></li></ul></span>Mon, 23 May 2011 19:46:56 +0000Anonymous39131 at https://euro-math-soc.eu