**Contemporary Mathematics**

Volume: 596;
2013;
291 pp;
Softcover

MSC: Primary 46; 30; 12; 32; 26; 11; 47; 06;

Print ISBN: 978-0-8218-9142-1

Product Code: CONM/596

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

**Electronic ISBN: 978-1-4704-1024-7
Product Code: CONM/596.E**

List Price: $100.00

AMS Member Price: $80.00

MAA Member Price: $90.00

# Advances in Ultrametric Analysis

Share this page *Edited by *
*Khodr Shamseddine*

This volume contains papers based on lectures
given at the 12th International Conference on \(p\)-adic Functional
Analysis, which was held at the University of Manitoba on July 2–6,
2012.

The articles included in this book feature recent developments in
various areas of non-archimedean analysis: branched values and zeros
of the derivative of a \(p\)-adic meromorphic function,
\(p\)-adic meromorphic functions
\(f^{\prime}P^{\prime}(f), g^{\prime}P^{\prime}(g)\) sharing a
small function, properties of
composition of analytic functions, partial fractional
differentiability, morphisms between ultrametric Banach algebras of
continuous functions and maximal ideals of finite dimension, the
\(p\)-adic \(q\)-distributions, Banach spaces over
fields with an infinite rank valuation, Grobman-Hartman theorems for
diffeomorphisms of Banach spaces over valued fields, integral
representations of continuous linear maps on \(p\)-adic spaces
of continuous functions, non-Archimedean operator algebras,
generalized Keller spaces over valued fields, proper multiplications
on the completion of a totally ordered abelian group, the Grothendieck
approximation theory in non-Archimedean functional analysis,
generalized power series spaces, measure theory and the study of power
series and analytic functions on the Levi-Civita fields.

Through a combination of new research articles and survey papers,
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.

#### Readership

Graduate students and researchers interested in non-archimedean analysis and related topics.

# Table of Contents

## Advances in Ultrametric Analysis

- Preface vii8 free
- Affine Invariant Measures in Levi-Civita Vector Spaces and the Erdös Obtuse Angle Theorem 110 free
- Some old and new results on zeros of the derivative of a 𝑝-adic meromorphic function 2332
- Survey on 𝑝-adic meromorphic functions 𝑓’𝑃’(𝑓),𝑔’𝑃’(𝑔) sharing a small function and additional properties 3140
- The 𝑝-adic 𝑞-distributions 4554
- Morphisms between ultrametric Banach algebras and maximal ideals of finite codimension 6372
- Survey on branched values and exceptional values for 𝑝-adic meromorphic functions 7382
- Grobman-Hartman Theorems for Diffeomorphisms of Banach Spaces over Valued Fields 7988
- 1. Introduction and statement of main results 7988
- 2. Preliminaries and notation 8291
- 3. Passage from one perturbation to another 8493
- 4. Hölder property of the conjugacies 8695
- 5. Proof of Theorem A 8998
- 6. Proof of Theorem B 8998
- 7. Parameter dependence of the conjugacy 91100
- Appendix A. Existence of ultrametric adapted norms 96105
- Appendix B. Proof of Lemma 7.1 97106
- Appendix C. Proof of Lemma 7.4 97106
- Appendix D. Proof of Lemma 7.10 99108
- References 100109

- Integral Representations of Continuous Linear Maps On 𝑝-adic Spaces of Continuous Functions 103112
- Subfields of valued, complete fields 123132
- On some classes of non-Archimedean operator algebras 133142
- Some identities and congruences for Stirling numbers of the second kind 149158
- Non-measurable sets in the Levi-Civita field 163172
- Partial Fractional Differentiability 179188
- Part 1. A review on fractional differentiability in one Variable 181190
- Part 2. Orthogonal Bases 185194
- Part 3. Partial fractional differentiability in several variables 197206
- A Generalized Keller space over a field with a valuation of rank 𝛼>𝜔 205214
- A comprehensive survey of non-archimedean analysis in Banach spaces over fields with an infinite rank valuation 215224
- All proper multiplications on the completion of a totally ordered abelian group 237246
- The Grothendieck approximation theory in non-Archimedean Functional Analysis 243252
- Introduction 243252
- 1. Preliminaries 244253
- 2. Basic facts and (counter)examples 246255
- 3. Bounded approximation properties and pseudoreflexivity 249258
- 4. Characterizations 250259
- 5. Hereditary properties 253262
- 6. The 1-BAP for spaces of countable type and the FDDP 255264
- 7. The 1-BAP for equivalent (polar) norms 257266
- 8. Problems 259268
- 9. Comparison with the classical case 262271
- References 266275

- A brief survey of the study of power series and analytic functions on the Levi-Civita fields 269278
- On non-Archimedean generalized power series spaces 281290