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Infinite dimensional holomorphy and applications

Holomorphic functions --- Domains of holomorphy

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Problems arising from the study of holomorphic continuation and holomorphic approximation have been central in the development of complex analysis in finitely many variables, and constitute one of the most promising lines of research in infinite dimensional complex analysis. This book presents a unified view of these topics in both finite and infinite dimensions.

Banach spaces. --- Domains of holomorphy. --- Holomorphic functions. --- Banach spaces --- Domains of holomorphy --- Holomorphic functions

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This textbook presents a collection of interesting and sometimes original exercises for motivated students in mathematics. Written in the same spirit as Volume 1, this second volume of Mathematical Tapas includes carefully selected problems at the intersection between undergraduate and graduate level. Hints, answers and (sometimes) comments are presented alongside the 222 “tapas” as well as 8 conjectures or open problems. Topics covered include metric, normed, Banach, inner-product and Hilbert spaces; differential calculus; integration; matrices; convexity; and optimization or variational problems. Suitable for advanced undergraduate and graduate students in mathematics, this book aims to sharpen the reader’s mathematical problem solving abilities.

Mathematics. --- Mathematics, general. --- Math --- Science --- Convex domains. --- Combinatorial optimization. --- Calculus.

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Cossos convexos --- Dominis convexos --- Geometria diferencial --- Convex bodies. --- Convex domains

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This volume is a sequel to “Manis Valuation and Prüfer Extensions I,” LNM1791. The Prüfer extensions of a commutative ring A are roughly those commutative ring extensions R / A,where commutative algebra is governed by Manis valuations on R with integral values on A. These valuations then turn out to belong to the particularly amenable subclass of PM (=Prüfer-Manis) valuations. While in Volume I Prüfer extensions in general and individual PM valuations were studied, now the focus is on families of PM valuations. One highlight is the presentation of a very general and deep approximation theorem for PM valuations, going back to Joachim Gräter’s work in 1980, a far-reaching extension of the classical weak approximation theorem in arithmetic. Another highlight is a theory of so called “Kronecker extensions,” where PM valuations are put to use in  arbitrary commutative  ring extensions in a way that ultimately goes back to the work of Leopold Kronecker.

Commutative algebra. --- Commutative rings. --- Prüfer rings. --- Commutative semihereditary domains --- Commutative semihereditary entire rings --- Domains, Commutative semihereditary --- Domains, Prüfer --- Entire rings, Commutative semihereditary --- Prüfer domains --- Prüfer's domains --- Prüfer's rings --- Semihereditary domains, Commutative --- Semihereditary entire rings, Commutative --- Rings (Algebra) --- Algebra --- Algebra. --- Commutative Rings and Algebras. --- Mathematics --- Mathematical analysis