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In the last decade, convolution operators of matrix functions have received unusual attention due to their diverse applications. This monograph presents some new developments in the spectral theory of these operators. The setting is the Lp spaces of matrix-valued functions on locally compact groups. The focus is on the spectra and eigenspaces of convolution operators on these spaces, defined by matrix-valued measures. Among various spectral results, the L2-spectrum of such an operator is completely determined and as an application, the spectrum of a discrete Laplacian on a homogeneous graph is computed using this result. The contractivity properties of matrix convolution semigroups are studied and applications to harmonic functions on Lie groups and Riemannian symmetric spaces are discussed. An interesting feature is the presence of Jordan algebraic structures in matrix-harmonic functions.

Matrix groups. --- Convolutions (Mathematics) --- Convolution transforms --- Transformations, Convolution --- Distribution (Probability theory) --- Functions --- Integrals --- Transformations (Mathematics) --- Group theory --- Matrices --- Functions of complex variables. --- Global differential geometry. --- Functional analysis. --- Operator theory. --- Harmonic analysis. --- Algebra. --- Functions of a Complex Variable. --- Differential Geometry. --- Functional Analysis. --- Operator Theory. --- Abstract Harmonic Analysis. --- Non-associative Rings and Algebras. --- Mathematics --- Mathematical analysis --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Geometry, Differential --- Complex variables --- Elliptic functions --- Functions of real variables --- Differential geometry. --- Nonassociative rings. --- Rings (Algebra). --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Differential geometry

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Algebraic groups are treated in this volume from a group theoretical point of view and the obtained results are compared with the analogous issues in the theory of Lie groups. The main body of the text is devoted to a classification of algebraic groups and Lie groups having only few subgroups or few factor groups of different type. In particular, the diversity of the nature of algebraic groups over fields of positive characteristic and over fields of characteristic zero is emphasized. This is revealed by the plethora of three-dimensional unipotent algebraic groups over a perfect field of positive characteristic, as well as, by many concrete examples which cover an area systematically. In the final section, algebraic groups and Lie groups having many closed normal subgroups are determined.

Linear algebraic groups. --- Lie groups. --- Group theory. --- Groupes linéaires algébriques --- Groupes de Lie --- Groupes, Théorie des --- Linear algebraic groups --- Lie groups --- Group theory --- Geometry --- Algebra --- Mathematical Theory --- Mathematics --- Physical Sciences & Mathematics --- Théorie des groupes --- Groups, Theory of --- Substitutions (Mathematics) --- Groups, Lie --- Algebraic groups, Linear --- Mathematics. --- Algebraic geometry. --- Nonassociative rings. --- Rings (Algebra). --- Topological groups. --- Group Theory and Generalizations. --- Algebraic Geometry. --- Topological Groups, Lie Groups. --- Non-associative Rings and Algebras. --- Lie algebras --- Symmetric spaces --- Topological groups --- Groups, Topological --- Continuous groups --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Algebraic geometry --- Math --- Science --- Geometry, Algebraic --- Algebraic varieties --- Geometry, algebraic. --- Topological Groups. --- Algebra. --- Mathematical analysis

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Basic Algebra and Advanced Algebra systematically develop concepts and tools in algebra that are vital to every mathematician, whether pure or applied, aspiring or established. Together, the two books give the reader a global view of algebra and its role in mathematics as a whole. Key topics and features of Advanced Algebra: *Topics build upon the linear algebra, group theory, factorization of ideals, structure of fields, Galois theory, and elementary theory of modules as developed in Basic Algebra *Chapters treat various topics in commutative and noncommutative algebra, providing introductions to the theory of associative algebras, homological algebra, algebraic number theory, and algebraic geometry *Sections in two chapters relate the theory to the subject of Gröbner bases, the foundation for handling systems of polynomial equations in computer applications *Text emphasizes connections between algebra and other branches of mathematics, particularly topology and complex analysis *Book carries on two prominent themes recurring in Basic Algebra: the analogy between integers and polynomials in one variable over a field, and the relationship between number theory and geometry *Many examples and hundreds of problems are included, along with hints or complete solutions for most of the problems *The exposition proceeds from the particular to the general, often providing examples well before a theory that incorporates them; it includes blocks of problems that illuminate aspects of the text and introduce additional topics Advanced Algebra presents its subject matter in a forward-looking way that takes into account the historical development of the subject. It is suitable as a text for the more advanced parts of a two-semester first-year graduate sequence in algebra. It requires of the reader only a familiarity with the topics developed in Basic Algebra.

Algebra. --- Algebraic number theory. --- Mathematics. --- Algebraic geometry. --- Category theory (Mathematics). --- Homological algebra. --- Field theory (Physics). --- Nonassociative rings. --- Rings (Algebra). --- Number theory. --- Non-associative Rings and Algebras. --- Field Theory and Polynomials. --- Algebraic Geometry. --- Number Theory. --- Category Theory, Homological Algebra. --- Mathematics --- Mathematical analysis --- Number theory --- Geometry, algebraic. --- Number study --- Numbers, Theory of --- Algebra --- Algebraic geometry --- Geometry --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic number theory

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Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.

Banach algebras --- Genetic algebras --- Nonassociative algebras --- Markov processes --- Phytophthora infestans --- Stochastic processes --- Algebra --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Genetics --- Banach algebras. --- Genetic algebras. --- Nonassociative algebras. --- Markov processes. --- Stochastic processes. --- Algebra. --- Genetics. --- Random processes --- Botrytis fallax --- Botrytis infestanus --- Botrytis solani --- Peronospora fintelmannii --- Peronospora infestans --- Peronospora trifurcata --- Phytophthora thalictri --- Potato late blight agent --- Potato late blight fungus --- Analysis, Markov --- Chains, Markov --- Markoff processes --- Markov analysis --- Markov chains --- Markov models --- Models, Markov --- Processes, Markov --- Algebras, Non-associative --- Algebras, Nonassociative --- Non-associative algebras --- Algebras, Genetic --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Mathematics. --- Nonassociative rings. --- Rings (Algebra). --- Probabilities. --- Biomathematics. --- General Algebraic Systems. --- Non-associative Rings and Algebras. --- Probability Theory and Stochastic Processes. --- Mathematical and Computational Biology. --- Mathematical analysis --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Biology --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Math --- Science --- Probabilities --- Phytophthora --- Algebra, Abstract --- Algebras, Linear --- Biomathematics --- Banach spaces --- Topological algebras --- Distribution (Probability theory. --- Distribution functions --- Frequency distribution --- Characteristic functions