Advances in Ultrametric Analysis
This book contains the proceedings of the 14th International Conference on p-adic Functional Analysis, held from June 30-July 4, 2016, at the Universit d'Auvergne, Aurillac, France. Articles included in this book feature recent developments in various areas of non-Archimedean analysis: summatio...
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2018
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| ISBN: | 1470434911, 9781470434915 |
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| Abstract | This book contains the proceedings of the 14th International Conference on p-adic Functional Analysis, held from June 30-July 4, 2016, at the Universit d'Auvergne, Aurillac, France. Articles included in this book feature recent developments in various areas of non-Archimedean analysis: summation of p-adic series, rational maps on the projective line over \mathbb{Q}p, non-Archimedean Hahn-Banach theorems, ultrametric Calkin algebras, G-modules with a convex base, non-compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces, algebras of strictly differentiable functions, inverse function theorem and mean value theorem in Levi-Civita fields, ultrametric spectra of commutative non-unital Banach rings, classes of non-Archimedean K the spaces, p-adic Nevanlinna theory and applications, and sub-coordinate representation of p-adic functions. Moreover, a paper on the history of p-adic analysis with a comparative summary of non-Archimedean fields is presented. Through a combination of new research articles and a survey paper, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area. |
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| AbstractList | This book contains the proceedings of the 14th International Conference on p-adic Functional Analysis, held from June 30-July 4, 2016, at the Universit d'Auvergne, Aurillac, France. Articles included in this book feature recent developments in various areas of non-Archimedean analysis: summation of p-adic series, rational maps on the projective line over \mathbb{Q}p, non-Archimedean Hahn-Banach theorems, ultrametric Calkin algebras, G-modules with a convex base, non-compact Trace class operators and Schatten-class operators in p-adic Hilbert spaces, algebras of strictly differentiable functions, inverse function theorem and mean value theorem in Levi-Civita fields, ultrametric spectra of commutative non-unital Banach rings, classes of non-Archimedean K the spaces, p-adic Nevanlinna theory and applications, and sub-coordinate representation of p-adic functions. Moreover, a paper on the history of p-adic analysis with a comparative summary of non-Archimedean fields is presented. Through a combination of new research articles and a survey paper, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area. |
| Author | Escassut, Alain Shamseddine, Khodr Perez-Garcia, Cristina |
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| SubjectTerms | (Semi-) Fredholm operators Algebraic number theory: local and p-adic fields Arithmetic and non-Archimedean dynamical systems Convergence and divergence of infinite limiting processes Convergence and divergence of series and sequences of functions.-msc Distribution of values, Nevanlinna theory.-msc Dynamical systems and ergodic theory Entire and meromorphic functions, and related topics Field theory and polynomials Functional analysis Functional analysis over fields other than {f R} or {f C} or the quaternions Functional analysis-Congresses Functions of a complex variable General theory of linear operators index theories.-msc Non-Archimedean dynamical systems.-msc non-Archimedean functional analysis.-msc Non-Archimedean valued fields.-msc None of the above, but in this section.-msc Number theory Operator theory Order, lattices, ordered algebraic structures Ordered structures Other (nonclassical) types of functional analysis p-adic analysis-Congresses Polynomial and rational maps.-msc Sequences, series, summability Summability and bases.-msc Topological fields Topological linear spaces and related structures Valued fields.-msc |
| TableOfContents | 4. Some open problems and some partial answers -- 5. Comparison with the classical theory -- References -- -modules with a convex base -- 1. Preliminaries -- 2. Basic concepts -- 3. Continuity of -modules with convex base -- 4. Maximal tight -submodules -- References -- A journey throughout the history of -adic numbers -- 1. Introduction -- 2. How Hensel was led to create the -adic numbers -- 3. Definition and properties of -adic numbers -- 4. -adic absolute values -- 5. The -adic numbers at the foundation of quantum mechanics -- 6. Conclusion -- References -- On some classes of non-Archimedean Köthe spaces -- 1. Introduction -- 2. Preliminaries -- 3. Results -- References -- On the sub-coordinate representation of -adic functions -- 1. Introduction -- 2. Sub-coordinate representation -- 3. Examples of the sub-coordinate representation -- References -- Back Cover adic meromorphic functions ' '( ), ' '( ) sharing a small function, ignoring multiplicity -- 1. Introduction and Main Results -- 2. Basic Results -- 3. Specific Lemmas -- 4. Proof of Theorems -- Acknowledgement -- References -- Spectra of commutative non-unital Banach rings -- 1. Introduction, notation and preliminary -- 2. Double centalizer algebras and unitalizations -- 3. Semi-simplicity -- 4. Comparison with the prime spectra -- 5. Representations on continuous sections of bundles of Banach algebras -- Acknowledgement -- References -- Ultrametric Calkin algebras -- 1. The completely continuous linear operators -- 2. Ultrametric Calkin algebras -- 3. Ultrametric ultraproducts of Banach spaces -- 4. A linear representation of the Calkin algebra when the Banach space is infinite dimensional of countable type -- 5. Some additional remarks -- References -- On summation of -adic series -- 1. Introduction -- 2. Series and -adic invariant summation in integer points -- 3. Functional summation formula and polynomials _{ }^{ }( -- ) -- Concluding remarks -- Acknowledgements -- References -- Spectrum of ultrametric Banach algebras of strictly differentiable functions -- Introduction and preliminaries -- Acknowledgements -- References -- Classical -adic Nevanlinna theory and Nevalinna Theory out of a hole -- I. Classical theory -- II. Nevanlinna Theory out of a hole -- Acknowledgement -- References -- The distance preserving mappings and isometrics defined on non-Archimedean Banach spaces -- 1. Introduction -- 2. Results -- References -- Rational map +1/ on the projective line over ℚ_{ } -- 1. Introduction -- 2. Preliminaries -- 3. Dynamical structures -- References -- Non-Archimedean Hahn-Banach theorems and injective Banach spaces -- Introduction -- 1. Preliminaries -- 2. Non-Archimedean injective Banach spaces -- 3. Hereditary properties Cover -- Title page -- Contents -- Preface -- Summary on non-Archimedean valued fields -- Introduction. -- 1. Ultrametric Spaces. -- - 1.1 The Strong Triangle Inequality and its Consequences. -- - 1.2 Compactness and Separability. -- - 1.3 Ultrametrizability. -- - 1.4 Spherical Completeness. -- 2. Valued fields. -- - 2.1 Archimedean and non-Archimedean valuations. -- - 2.2 Completion Theorem. -- - 2.3 Completion of Archimedean valued fields. -- - 2.4 Completion of non-Archimedean valued fields. -- - 2.5 Incompleteness of \QQ and ( ). -- - 2.6 Compact and Locally compact valued fields. -- 3. Ordered fields. -- - 3.1 Formally real fields. -- - 3.2 General Hahn fields and the Embedding theorem. -- - 3.3 Hahn Fields and Levi-Civita fields. -- - 3.4 Real-closed field extensions of \RR. -- 4. Algebraic closure of valued fields and their completions. -- - 4.1 General Results. -- - 4.2 Examples. -- 5. General valuations. -- 6. Maximal, Pseudo and Spherical completeness. -- - 6.1 Pseudo-completeness. -- - 6.2 Spherical completeness. -- - 6.3 Maximal completeness. -- 7. Structures of Maximally Complete fields. -- - 7.1 Equal characteristic case. -- - 7.2 Mixed characteristic case. -- 8. Catalog of fields. -- 9. Classification of fields. -- References -- Non-compact Trace class operators and Schatten-class operators in -adic Hilbert spaces -- 1. Introduction -- 2. Background -- 3. Schatten-class Operators in Classical Hilbert spaces -- 4. Schatten Class operators in -adic Hilbert spaces -- 5. Completely Continuous operators and Schatten-class operators -- 6. Trace and Schatten-class operators -- Acknowledgments -- References -- Calculus on a non-Archimedean field extension of the real numbers: inverse function theorem, intermediate value theorem and mean value theorem -- 1. Introduction -- 2. WLUD Functions -- 3. Calculus Theorems -- References |
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