There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution. Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions. The main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables. This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.
Mathematical Modelling of Waves in Multi-Scale Structured Media presents novel analytical and numerical models of waves in structured elastic media, with emphasis on the asymptotic analysis of phenomena such as dynamic anisotropy, localisation, filtering and polarisation as well as on the modelling of photonic, phononic, and platonic crystals.
This book outlines a powerful new method in analysis which has already been instrumental in solving complicated partial differential equations arising in various areas of engineering. It is suitable for those working with partial differential equations and their applications, and an undergraduate knowledge of PDE's and functional analysis is assumed.
...text covers static problems of elasticity in dilute composites as well as spectral problems...applications of the mathematical models are in damage mechanics and in problems of design of composite structures used as filters
There are a wide range of applications in physics and structural mechanics involving domains with singular perturbations of the boundary. Examples include perforated domains and bodies with defects of different types. The accurate direct numerical treatment of such problems remains a challenge. Asymptotic approximations offer an alternative, efficient solution. Green’s function is considered here as the main object of study rather than a tool for generating solutions of specific boundary value problems. The uniformity of the asymptotic approximations is the principal point of attention. We also show substantial links between Green’s functions and solutions of boundary value problems for meso-scale structures. Such systems involve a large number of small inclusions, so that a small parameter, the relative size of an inclusion, may compete with a large parameter, represented as an overall number of inclusions. The main focus of the present text is on two topics: (a) asymptotics of Green’s kernels in domains with singularly perturbed boundaries and (b) meso-scale asymptotic approximations of physical fields in non-periodic domains with many inclusions. The novel feature of these asymptotic approximations is their uniformity with respect to the independent variables. This book addresses the needs of mathematicians, physicists and engineers, as well as research students interested in asymptotic analysis and numerical computations for solutions to partial differential equations.
The focus of this book lies at the meeting point of electromagnetic waveguides and photonic crystals. Although these are both widely studied topics, they have been kept apart until recently. The purpose of the first edition of this book was to give state-of-the-art theoretical and numerical viewpoints about exotic fibres which use “photonic crystal effects” and consequently exhibit some remarkable properties.Since that first edition, photonic crystal fibres have become an important and effective optical device. In this second edition, the description of the theoretical and numerical tools used to study these fibres is enhanced, whilst up-to-date information about the properties, applications and fabrication of these fibres is added./a
This book outlines a powerful new method in analysis which has already been instrumental in solving complicated partial differential equations arising in various areas of engineering. It is suitable for those working with partial differential equations and their applications, and an undergraduate knowledge of PDE's and functional analysis is assumed.
Mathematical Modelling of Waves in Multi-Scale Structured Media presents novel analytical and numerical models of waves in structured elastic media, with emphasis on the asymptotic analysis of phenomena such as dynamic anisotropy, localisation, filtering and polarisation as well as on the modelling of photonic, phononic, and platonic crystals.
One of the major questions in the evolution of animals is the transition from unicellular to multicellular organization, which resulted in the emergence of Metazoa through a hypothetical Urmetazoa. The Comparative Embryology of Sponges contains abundant original and literary data on comparative embryology and morphology of the Porifera (Sponges), a group of 'lower Metazoa'. On the basis of this material, original typization of the development of Sponges is given and the problems concerning origin and evolution of Porifera and their ontogenesis are discussed. A morphogenetic interpretation of the body plan development during embryogenesis, metamorphosis and asexual reproduction in Sponges is proposed. Special attention is given to the analysis of characteristic features of the ontogenesis in Porifera. The book pursues three primary goals: 1) generalization of all existing information on individual development of sponges, its classification and a statement according to taxonomical structure of Porifera; 2) revealing of heterogeneity of morphogenesis and peculiarities of ontogeneses in various clades of Porifera, and also their correlations with the organization, both adult sponges, and their larvae; 3) revealing homology of morphogeneses in both Porifera and Eumetazoa, testifying to the general evolutionary roots of multicellular animals, and peculiar features of sponges' morphogeneses and ontogenesis. This book will be of interest to embryologists, zoologists, morphologists and researchers in evolutionary biology.
This Reference eBook deals with an existing classification of a nanosized structure and an analysis of its properties. It summarizes an information about how a grain size affects physical, mechanical, thermal, and other properties of a nanostructured material. A basic method, which is employed for a fabrication of an isolated nanoparticle, an ultradisperse powder, a compact nanocystalline, nanoporous, and amorphous material, a fullerene, a nanotube, and a nanostuctured coating, is considered. Investigation methods, which are applied to study the nanostructured material, are briefly described. A modern understanding of a formation of the nanostructured and nanocomposite coating, which are fabricated using a ion-plasma deposition method, is reposted. A potential application of the nanostructured material and coating in a field of engineering is demonstrated. Readership: Graduate, Postgraduate, Ph.D. Students, Researchers and Industry Professionals. The eBook contains 9 Chapters, 87 Figures, 14 Tables, 411 References-totally, 155 Pages. It was approved by two Scientific Boards from National Kharkov University and Sumy State University. A main content of this eBook is a basis for lectures presented for students at Sumy State University (the Physical-Technical Faculty), Kharkov National University (the Physical Faculty), Omsk State University (the Physical Faculty, the Department of Material Science), East-Kazakhstan State Technical University (Ust-Kamenogorsk, Kazakhstan), and Moscow State University (the Physical Faculty, Moscow, Russia). A source of this eBook is original papers of leading world-known scientists, who ware involved in a field of new nano composite material fabrication, nanotechnologies, and researches. This version had not been published elsewhere. It is interesting for a wide circle of specialists, Masters, aspirants, scientific researchers, and a technical staff of Higher Education System, Research Institutes and Laboratories. It covers recent data since 2008 to 2010 year.
Incomplete second order linear differential equations in Banach spaces as well as first order equations have become a classical part of functional analysis. This monograph is an attempt to present a unified systematic theory of second order equations y" (t) + Ay' (t) + By (t) = 0 including well-posedness of the Cauchy problem as well as the Dirichlet and Neumann problems. Exhaustive yet clear answers to all posed questions are given. Special emphasis is placed on new surprising effects arising for complete second order equations which do not take place for first order and incomplete second order equations. For this purpose, some new results in the spectral theory of pairs of operators and the boundary behavior of integral transforms have been developed. The book serves as a self-contained introductory course and a reference book on this subject for undergraduate and post- graduate students and research mathematicians in analysis. Moreover, users will welcome having a comprehensive study of the equations at hand, and it gives insight into the theory of complete second order linear differential equations in a general context - a theory which is far from being fully understood.
The essential aim of this book is to consider a wide set of problems arising in the mathematical modeling of mechanical systems under unilateral constraints. In these investigations elastic and non-elastic deformations, friction and adhesion phenomena are taken into account. All the necessary mathematical tools are given: local boundary value problem formulations, construction of variational equations and inequalities and their transition to minimization problems, existence and uniqueness theorems, and variational transformations (Friedrichs and Young-Fenchel-Moreau) to dual and saddle-point search problems.
This research monograph provides a brief overview of the authors' research in the area of ordered granular media over the last decade. The exposition covers one-dimensional homogeneous and dimer chains in great detail incorporating novel analytical tools and experimental results supporting the analytical and numerical studies. The proposed analytical tools have since been successfully implemented in studying two-dimensional dimers, granular dimers on on-site perturbations, solitary waves in Toda lattices to name a few. The second part of the monograph dwells on weakly coupled homogeneous granular chains from analytical, numerical and experimental perspective exploring the interesting phenomenon of Landau-Zener tunneling in granular media. The final part of the monograph provides a brief introduction to locally resonant acoustic metamaterials incorporating internal rotators and the resulting energy channeling mechanism in unit-cells and in one- and two-dimensional lattices. The monograph provides a comprehensive overview of the research in this interesting domain. However, this exposition is not all exhaustive with regard to equally exciting research by other researchers across the globe, but we provide an exhaustive list of references for the interested readers to further explore in this direction.
This text illustrates how different methods of finite group theory including representation theory, cohomology theory, combinatorial group theory and local analysis are combined to construct one of the last of the sporadic finite simple groups - the fourth Janko group J_4. Aimed at graduates and researchers in group theory, geometry and algebra, Ivanov's approach is based on analysis of group amalgams and the geometry of the complexes of these amalgams with emphasis on the underlying theory. An indispensible resource, this book will be a unique and essential reference for researchers in the area. The author is a leading researcher in the field.
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