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The present text book contains a collection of six high-quality articles. In particular, this book is devoted to Linear Mathematics by presenting problems in Applied Linear Algebra of general or special interest.

Linear algebraic groups. --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Algebra

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In recent years, there has been an increased interest in exploring the connections between various disciplines of mathematics and theoretical physics such as representation theory, algebraic geometry, quantum field theory, and string theory. One of the challenges of modern mathematical physics is to understand rigorously the idea of quantization. The program of quantization by branes, which comes from string theory, is explored in the book. This open access book provides a detailed description of the geometric approach to the representation theory of the double affine Hecke algebra (DAHA) of rank one. Spherical DAHA is known to arise from the deformation quantization of the moduli space of SL(2,C) flat connections on the punctured torus. The authors demonstrate the study of the topological A-model on this moduli space and establish a correspondence between Lagrangian branes of the A-model and DAHA modules. The finite-dimensional DAHA representations are shown to be in one-to-one correspondence with the compact Lagrangian branes. Along the way, the authors discover new finite-dimensional indecomposable representations. They proceed to embed the A-model story in an M-theory brane construction, closely related to the one used in the 3d/3d correspondence; as a result, modular tensor categories behind particular finite-dimensional representations with PSL(2,Z) action are identified. The relationship of Coulomb branch geometry and algebras of line operators in 4d N = 2* theories to the double affine Hecke algebra is studied further by using a further connection to the fivebrane system for the class S construction. The book is targeted at experts in mathematical physics, representation theory, algebraic geometry, and string theory. This is an open access book.

Mathematical physics. --- Algebraic geometry. --- Quantum physics. --- Mathematical Physics. --- Algebraic Geometry. --- Quantum Physics. --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Algebraic geometry --- Geometry --- Physical mathematics --- Mathematics

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"This book presents copious and sometimes unexpected examples of sheaf theory, a mathematical tool with promising applications in data science and engineering and in efforts to apply category theory more widely"--

Sheaf theory --- Cohomology, Sheaf --- Sheaf cohomology --- Sheaves, Theory of --- Sheaves (Algebraic topology) --- Algebraic topology --- Sheaf theory sheaves --- category theory categories --- applied category theory --- topology --- geometry --- logic --- mathematics --- Sheaf theory. --- MATHEMATICS / Logic --- MATHEMATICS / Topology --- MATHEMATICS / Algebra / General

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The question of reconstructing a geometric shape from spectra of operators (such as the Laplace operator) is decades old and an active area of research in mathematics and mathematical physics. This book focusses on the case of compact Riemannian manifolds, and, in particular, the question whether one can find finitely many natural operators that determine whether two such manifolds are isometric (coverings). The methods outlined in the book fit into the tradition of the famous work of Sunada on the construction of isospectral, non-isometric manifolds, and thus do not focus on analytic techniques, but rather on algebraic methods: in particular, the analogy with constructions in number theory, methods from representation theory, and from algebraic topology. The main goal of the book is to present the construction of finitely many “twisted” Laplace operators whose spectrum determines covering equivalence of two Riemannian manifolds. The book has a leisure pace and presents details and examples that are hard to find in the literature, concerning: fiber products of manifolds and orbifolds, the distinction between the spectrum and the spectral zeta function for general operators, strong isospectrality, twisted Laplacians, the action of isometry groups on homology groups, monomial structures on group representations, geometric and group-theoretical realisation of coverings with wreath products as covering groups, and “class field theory” for manifolds. The book contains a wealth of worked examples and open problems. After perusing the book, the reader will have a comfortable working knowledge of the algebraic approach to isospectrality. This is an open access book.

Isomorphisms (Mathematics) --- Categories (Mathematics) --- Group theory --- Morphisms (Mathematics) --- Set theory --- Global analysis (Mathematics). --- Manifolds (Mathematics). --- Number theory. --- Group theory. --- Algebraic topology. --- Geometry, Differential. --- Global Analysis and Analysis on Manifolds. --- Manifolds and Cell Complexes. --- Number Theory. --- Group Theory and Generalizations. --- Algebraic Topology. --- Differential Geometry. --- Differential geometry --- Topology --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Number study --- Numbers, Theory of --- Geometry, Differential --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Varietats de Riemann --- Isometria (Matemàtica)

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Our world today is becoming increasingly complex, and technical devices are getting ever smaller and more powerful. The high density of electronic components together with high clock frequencies leads to unwanted side-effects like crosstalk, delayed signals and substrate noise, which are no longer negligible in chip design and can only insufficiently be represented by simple lumped circuit models. As a result, different physical phenomena have to be taken into consideration since they have an increasing influence on the signal propagation in integrated circuits. Computer-based simulation methods play thereby a key role. The modelling and analysis of complex multi-physics problems typically leads to coupled systems of partial differential equations and differential-algebraic equations (DAEs). Dynamic iteration and model order reduction are two numerical tools for efficient and fast simulation of coupled systems. Formodelling of low frequency electromagnetic field, we use magneto-quasistatic (MQS) systems which can be considered as an approximation to Maxwells equations. A spatial discretization by using the finite element method leads to a DAE system. We analyze the structural and physical properties of this system and develop passivity-preserving model reduction methods. A special block structure of the MQS model is exploited to to improve the performance of the model reduction algorithms.

Technology. --- Applied science --- Arts, Useful --- Science, Applied --- Useful arts --- Science --- Industrial arts --- Material culture --- model order reduction --- dynamic iteration --- magneto-quasistatic systems --- differential algebraic equations --- finite element method