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The study of arithmetic differential operators is a novel and promising area of mathematics. This complete introduction to the subject starts with the basics: a discussion of p-adic numbers and some of the classical differential analysis on the field of p-adic numbers leading to the definition of arithmetic differential operators on this field. Buium's theory of arithmetic jet spaces is then developed succinctly in order to define arithmetic operators in general. Features of the book include a comparison of the behaviour of these operators over the p-adic integers and their behaviour over the unramified completion, and a discussion of the relationship between characteristic functions of p-adic discs and arithmetic differential operators that disappears as soon as a single root of unity is adjoined to the p-adic integers. This book is essential reading for researchers and graduate students who want a first introduction to arithmetic differential operators over the p-adic integers.

Differential operators. --- Arithmetic functions. --- p-adic numbers. --- Numbers, p-adic --- Number theory --- p-adic analysis --- Functions, Arithmetic --- Functions of complex variables --- Operators, Differential --- Differential equations --- Operator theory

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p-adic numbers --- Mathematical Sciences --- General and Others --- Numbers, p-adic --- Algebraic number theory --- Algebraic number theory. --- p-adic numbers. --- Number theory --- p-adic analysis --- Nombres p-àdics --- Teoria algebraica de nombres

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This 2010 book was the first devoted to the theory of p-adic wavelets and pseudo-differential equations in the framework of distribution theory. This relatively recent theory has become increasingly important in the last decade with exciting applications in a variety of fields, including biology, image analysis, psychology, and information science. p-Adic mathematical physics also plays an important role in quantum mechanics and quantum field theory, the theory of strings, quantum gravity and cosmology, and solid state physics. The authors include many new results, some of which constitute new areas in p-adic analysis related to the theory of distributions, such as wavelet theory, the theory of pseudo-differential operators and equations, asymptotic methods, and harmonic analysis. Any researcher working with applications of p-adic analysis will find much of interest in this book. Its extended introduction and self-contained presentation also make it accessible to graduate students approaching the theory for the first time.

p-adic numbers. --- p-adic analysis. --- Distribution (Probability theory) --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Numbers, p-adic --- Number theory --- p-adic analysis

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Functional analysis --- Number theory --- p-adic analysis --- p-adic numbers --- Functions, Zeta --- #WWIS:d.d. Prof. L. Bouckaert/ALTO --- 511 --- Zeta functions --- Numbers, p-adic --- Analysis, p-adic --- Algebra --- Calculus --- Geometry, Algebraic --- Functions, Zeta. --- p-adic analysis. --- p-adic numbers. --- 511 Number theory --- P-adic analysis. --- P-adic numbers.

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Ordinary differential equations --- Differential equations --- p-adic numbers --- 517.9 --- Equations, Differential --- Bessel functions --- Calculus --- Numbers, p-adic --- Number theory --- p-adic analysis --- Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.91 Differential equations --- 517.9 Differential equations. Integral equations. Other functional equations. Finite differences. Calculus of variations. Functional analysis --- 517.91. --- Numerical solutions --- 517.91