## Advances in Ultrametric Analysis: 12th International Conference on P-adic Functional Analysis, July 2-6, 2012, University of Manitoba, Winnipeg, Manitoba, CanadaThis 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. |

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### Contents

1 | |

Some old and new results on zeros of the derivative of a adic meromorphic function | 23 |

Survey on adic meromorphic functions sharing a small function and additional properties | 31 |

The adic distributions | 45 |

Morphisms between ultrametric Banach algebras and maximal ideals of finite codimension | 63 |

Survey on branched values and exceptional values for adic meromorphic functions | 73 |

GrobmanHartman Theorems for Diffeomorphisms of Banach Spaces over Valued Fields | 79 |

Integral Representations of Continuous Linear Maps On adic Spaces of Continuous Functions | 103 |

Partial Fractional Differentiability | 179 |

Part 1 A review on fractional differentiability in one Variable | 181 |

Part 2 Orthogonal Bases | 185 |

Part 3 Partial fractional differentiability in several variables | 197 |

A Generalized Keller space over a field with a valuation of rank | 205 |

A comprehensive survey of nonarchimedean analysis in Banach spaces over fields with an infinite rank valuation | 215 |

All proper multiplications on the completion of a totally ordered abelian group | 237 |

The Grothendieck approximation theory in nonArchimedean Functional Analysis | 243 |

Subfields of valued complete fields | 123 |

On some classes of nonArchimedean operator algebras | 133 |

Some identities and congruences for Stirling numbers of the second kind | 149 |

Nonmeasurable sets in the LeviCivita field | 163 |

A brief survey of the study of power series and analytic functions on the LeviCivita fields | 269 |

On nonArchimedean generalized power series spaces | 281 |

### Common terms and phrases

Alain Escassut analytic functions approximation property assume Baer ring Banach space bounded branched values Cb(X clopen clopen set compact continuous functions convergence convex subgroup Corollary countable type cs(E deﬁned deﬁnition denote diam{x0 disjoint elements equivalent exists f I g ﬁnd finite ﬁnite subset finite-dimensional ﬁrst ﬁxed FR(E function f functional analysis G-module given Hence Hilbert spaces homeomorphism inﬁnite integer isometry isomorphic Lemma Let f G Levi-Civita field linear map Lipschitz locally convex locally convex spaces Math Mathematics maximal ideal measure meromorphic functions non-Archimedean field normed space operators orthogonal base p-adic analysis polytope power series PROOF proper multiplication Proposition prove pseudoreflexive q G cs(F R-analytic resp satisﬁes seminorm sequence simplex space of countable Stirling numbers subspace Theorem theory topology ultrametric valuation valued field vector spaces zero