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This book presents a wide panorama of methods to investigate the spectral properties of block operator matrices. Particular emphasis is placed on classes of block operator matrices to which standard operator theoretical methods do not readily apply: non-self-adjoint block operator matrices, block operator matrices with unbounded entries, non-semibounded block operator matrices, and classes of block operator matrices arising in mathematical physics.

In this paper, the problem of unipotency for the matrix group generated by two matrices is examined. By employing matrix logarithms as a tool, various combinatorial formulas for matrices were derived by selecting different primitive elements. Key conclusions were then reached through the organization and simplification of these formulas. It was ultimately demonstrated, based on these conclusions, that a matrix group G generated by two matrices, where the Jordan blocks do not exceed third order, must be unipotent if each primitive element of G is unipotent and has an order of six or less.

In many physical problems several scales are present in space or time, caused by inhomogeneity of the medium or complexity of the mechanical process. A fundamental approach is to first construct micro-scale models, and then deduce the macro-scale laws and the constitutive relations by properly averaging over the micro-scale. The perturbation method of multiple scales can be used to derive averaged equations for a much larger scale from considerations of the small scales. In the mechanics of multiscale media, the analytical scheme of upscaling is known as the Theory of Homogenization.

This book gives an up-to-date exposition on the theory of oblique derivative problems for elliptic equations. The modern analysis of shock reflection was made possible by the theory of oblique derivative problems developed by the author. Such problems also arise in many other physical situations such as the shape of a capillary surface and problems of optimal transportation. The author begins the book with basic results for linear oblique derivative problems and work through the theory for quasilinear and nonlinear problems. The final chapter discusses some of the applications. In addition, notes to each chapter give a history of the topics in that chapter and suggestions for further reading.

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic.

Differential-algebraic equations (DAEs) provide an essential tool for system modeling and analysis within different fields of applied sciences and engineering. This book addresses modeling issues and analytical properties of DAEs, together with some applications in electrical circuit theory.

For a long time, Soviet students learned ‘pure’ mathematics in their mathematics classrooms, while applications of mathematics were introduced in their science (mainly physics) classrooms. This approach was a part of a uniform and rigid national curriculum. Even when in the 1990s the world was moving towards including applications in school mathematics, Russian students continued to engage in pure mathematics learning in their mathematics classrooms. It was physics teachers’ responsibility to teach applications of mathematics; therefore, physics courses were highly mathematics-intensive, making extensive use of mathematics from algebra to calculus in the formulation of scientific laws and the investigation of their consequences. The collapse of the Soviet Union and some liberalization in educational policy led to changes in graduation requirements in mathematics and physics as well as diversity in mathematics and physics curricula in schools. Based on a review of textbooks, standards, curriculum documents, and other resources, in this paper we analyze changes that affected the teaching and learning of mathematics in physics classrooms in Russia from the Soviet period to the present.

This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein–Gordon equations, KdV equations as well as Navier–Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods.

This book introduces the method of automorphic descent, providing an explicit inverse map to the (weak) Langlands functorial lift from generic, cuspidal representations on classical groups to general linear groups. The essence of this method is the study of certain Fourier coefficients of Gelfand–Graev type, or of Fourier–Jacobi type when applied to certain residual Eisenstein series. This book contains a complete account of this automorphic descent, with complete, detailed proofs. The book will be of interest to graduate students and mathematicians, who specialize in automorphic forms and in representation theory of reductive groups over local fields. Relatively self-contained, the content of some of the chapters can serve as topics for graduate students seminars.

It would have been easy for a less imaginative historian of mathematics than Herbert Mehrtens to have portrayed the work of Hilbert, Hausdorff, and other modernists as pioneers, and those who did not subscribe to their program as people who failed, were not good enough to make the turn, and were eventually and convincingly left behind. That he did not do so is not only because this would have been a shallow, selective view of the facts: it is incompatible with his Foucauldian approach to the relations between knowledge and power. Instead, he defined what I see as the most intriguing category of actor in his Moderne—Sprache—Mathematik (1990), the Gegenmoderner, or counter-moderns. The three men who characterize this position are Felix Klein, Henri Poincaré, and Luitzen Brouwer, and each merits a section in the book. Of the three, Poincaré is the hardest to contain within that category. The range of his work, the nature of his influence, and the shifting standards by which mathematical significance has been evaluated by mathematicians, historians of mathematics, and society at large, all contribute to the problem. After thirty years, the methodological presumptions and aspirations of historians of mathematics have also changed, and I shall suggest that one way to appreciate the richness of Mehrtens’ book, to gain insight into what is meant by mathematical modernism, and to acknowledge a generation of work by other historians since 1990, is to re-examine aspects of Poincaré’s life and work and scholarship about him. Prodded by remarks by Leo Corry, Moritz Epple, and David Rowe, I shall suggest that the simple but useful dichotomy modern/counter-modern must be seen as a way into a more complicated situation, one in which different aspects of mathematics, specifically applied mathematics and the relationship of mathematics to contemporary physics, require fresh accounts of the role of modern mathematics in society.