Generalized manifolds are a most fascinating subject to study. They were introduced in the 1930s, when topologists tried to detect topological manifolds among more general spaces (this is nowadays called the manifold recognition problem). As such, generalized manifolds have served to understand the nature of genuine manifolds. However, it soon became more important to study the category of generalized manifolds itself. A breakthrough was made in the 1990s, when several topologists discovered a systematic way of constructing higher-dimensional generalized manifolds, based on advanced surgery techniques. In fact, the development of controlled surgery theory and the study of generalized manifolds developed in parallel. In this process, earlier studies of geometric surgery turned out to be very helpful. Generalized manifolds will continue to be an attractive subject to study, for there remain several unsolved fundamental problems. Moreover, they hold promise for new research, e.g. for finding appropriate structures on these spaces which could bring to light geometric (or even analytic) aspects of higher-dimensional generalized manifolds. This is the first book to systematically collect the most important material on higherdimensional generalized manifolds and controlled surgery. It is self-contained and its extensive list of references reflects the historic development. The book is based on our graduate courses and seminars, as well as our talks given at various meetings, and is suitable for advanced graduate students and researchers in algebraic and geometric topology.
This is the first monograph dealing with the analysis of nonlinear problems described by partial differential equations with variable exponent and nonhomogeneous differential operators. This analysis is developed by using variational and topological methods. The authors are interested in many problems of huge interest both for mathematicians and for physicists. We refer only to the spectral analysis of the associated differential operators. Due to the presence of nonstandard nonlinear terms, the authors point out several striking phenomena like concentration properties of the spectrum. It is showed that the spectrum is no longer discrete (as for the standard Laplace operator) and a continuous spectrum can appear either in a neighborhood of the origin or at infinity. The authors of this monograph also elaborate general methods for the bifurcation analysis of solutions, the study of discrete anisotropic problems with variable exponents, etc. The monograph is mainly based on the authors’ original results in the field but some important recent contributions of other authors have been also introduced in this volume. The study of this field has become extremely active in the last decade due to several powerful applications of models involving variable exponents to concrete problems arising in electrorheological and thermorheological fluids, robotics, and image reconstruction.
This is the first monograph devoted to the variational analysis of nonlocal problems. The volume provides researchers and graduate students with a thorough introduction to variational and topological methods in the framework of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory. The key features of this new monograph are the following: (i) it presents a modern, unified approach to analyzing nonlocal equations; (ii) it examines a broad range of problems described by nonlocal operators that can be extended to other classes of related problems; and (iii) it reveals a number of surprising interactions among various topics.
This is the second of the two volumes (the first was also published by Springer in 2014) and it covers problems in five core topics of mathematical analysis: Function Spaces, Nonlinear and Multivalued Maps, Smooth and Nonsmooth Calculus, Degree Theory and Fixed Point Theory, and Variational and Topological Methods. Each of five topics corresponds to a different chapter with inclusion of the basic theory and accompanying main definitions and results, followed by suitable comments and remarks for better understanding of the material. Problems are presented for each topic, with solutions available at the end of each chapter. The entire collection of problems offers a balanced and useful picture for the application surrounding each topic. This encyclopedic coverage of exercises in nonlinear mathematical analysis is the first of its kind and is accessible to a wide readership. Graduate students will find the collection of problems valuable in preparation for their preliminary or qualifying exams as well as for testing their deeper understanding of the material. Problems are denoted by degree of difficulty. Lecturers teaching courses that include one or all of the above mentioned topics will find the exercises of great help in course preparation. Researchers in nonlinear analysis may find this work useful as a summary of analytic theories published in one accessible volume.
In the reporting period we published 239 papers (of which 68 journals in the top quarter of the SCI list). Below we list a selection of papers on topology and geometry (1-5) and nonlinear analysis (6-10), with their COBISS ID’s, where their abstracts are posted: 1.SMREKAR, Jaka. CW towers and mapping spaces. Topology and its Applications, 2015, vol. 194, str. 93-117. [COBISS.SI-ID 17413721]. 2.GOVC, Dejan. On the definition of the homological critical value. Journal of homotopy and related structures, ISSN 2193-8407, 2016, vol. 11, iss. 1, str. 143-151. [COBISS.SI-ID 17228633] 3.LAMPRET, Leon, VAVPETIČ, Aleš. (Co)homology of Lie algebras via algebraic Morse theory. Journal of algebra, 2016, vol. 463, str. 254-277. [COBISS.SI-ID 17743193], 4.PAVEŠIĆ, Petar. Extension of functors to fibrewise pointed spaces. Topological Methods in Nonlinear Analysis, 2015, vol. 45, no. 1, str. 91-102. [COBISS.SI-ID 17282649]. 5.KING, Henry C., KNUDSON, Kevin, MRAMOR KOSTA, Neža. Birth and death in discrete Morse theory. Journal of symbolic computation, 2017, vol. 78, str. 41-60. [COBISS.SI-ID 17737817]. 6. REPOVŠ, Dušan. Stationary waves of Schrödinger-type equations with variable exponent. Analysis and applications, 2015, vol. 13, iss. 6, str. 645-661. [COBISS.SI-ID 17144409] 7.BOUREANU, Maria-Magdalena, RǍDULESCU, Vicenţiu, REPOVŠ, Dušan. On a p(.)-biharmonic problem with no-flux boundary condition. Computers & Mathematics with Applications, 2016, vol. 72, iss. 9, str. 2505-2515. [COBISS.SI-ID 17789785]. 8.PAPAGEORGIOU, Nikolaos Socrates, RǍDULESCU, Vicenţiu, REPOVŠ, Dušan. On a class of parametric (p, 2)-equations. Applied mathematics and optimization, 2016, 36 str. [COBISS.SI-ID 17592153]. 9.MOLICA BISCI, Giovanni, REPOVŠ, Dušan, SERVADEI, Raffaella. Nontrivial solutions of superlinear nonlocal problems. Forum mathematicum, 2016, vol. 28, iss. 6, str. 1095-1110. [COBISS.SI-ID 17671001]. 10.CENCELJ, Matija, REPOVŠ, Dušan, VIRK, Žiga. Multiple perturbations of a singular eigenvalue problem. Nonlinear Analysis, Theory, Methods and Applications, 2015, vol. 119, str. 37-45. [COBISS.SI-ID 17086297].