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1.
The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivique I, Comptes rendus de l'Académie des Sciences 339 (2004) 411-416].  相似文献   

2.
We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [a, b] ? ?, f and its analytic continuation f? satisfy the same properties that can be expressed in the language of real closed ordered fields. If f is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.  相似文献   

3.
In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isoparametric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition, when the function is defined on a compact analytic manifold M.  相似文献   

4.
Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.  相似文献   

5.
In this paper, by making use of the familiar concept of neighborhoods of p-valently analytic functions, we prove coefficient bounds, distortion inequalities and associated inclusion relations for the (nδ)-neighborhoods of a family of p-valently analytic functions and their derivatives, which is defined by means of a certain general family of non-homogenous Cauchy-Euler differential equations.  相似文献   

6.
By making use of the principle of subordination between analytic functions and the generalized fractional differintegral operator, we introduce and investigate some new subclasses of p-valently analytic functions in the open unit disk. Such results as inclusion relationships, integral-preserving properties, convolution properties, subordination and superordination properties, and sandwich theorems for these classes are derived.  相似文献   

7.
In this paper existence of analytic solutions of a nonlinear iterative equations is studied when given functions are all analytic and when given functions have poles. As well as in many previous works, we reduce this problem to finding analytic solutions of a functional equation without iteration of the unknown function f. For technical reasons, in previous works an indeterminate constant related to the eigenvalue of the linearized f at its fixed point O is required to fulfill the Diophantine condition that O is an irrationally neutral fixed point of f. In this paper the case of rationally neutral fixed points is also discussed, where the Diophantine condition is not required.  相似文献   

8.
Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on Z, a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.  相似文献   

9.
In the present paper we will introduce a new approach to multivariate interpolation by employing polyharmonic functions as interpolants, i.e. by solutions of higher order elliptic equations. We assume that the data arise from C or analytic functions in the ball BR. We prove two main results on the interpolation of C or analytic functions f in the ball BR by polyharmonic functions h of a given order of polyharmonicity p.  相似文献   

10.
In this paper, we introduced two new subclasses of the function class Σ of bi-univalent functions analytic in the open unit disc defined by convolution. Furthermore, we find estimates on the coefficients ∣a2∣ and ∣a3∣ for functions in these new subclasses.  相似文献   

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