scholarly journals Плюригармонические определимые функции в некоторых $o$-минимальных расширениях вещественного поля

2021 ◽  
pp. 35-40
Author(s):  
M. Berraho
Keyword(s):  
Harmonic Functions ◽  
Positive Answer ◽  
Real Field ◽  
Real Numbers ◽  
Pluriharmonic Function ◽  
The Real ◽  
Division Theorem ◽  
Minimal Structure ◽  
Real Analytic Functions ◽  
Real Analytic

In this paper, we first try to solve the following problem: If a pluriharmonic function $f$ is definable in an arbitrary o-minimal expansion of the structure of the real field $\overline{\mathbb{R}}:=(\mathbb{R},+,-,.,0,1,<)$, does this function be locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of $\mathbb{R}^n$. We obtain the same answer for an o-minimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at $0\in\mathbb{R}^n$ which are definable in the o-minimal structure $(\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})$, here $\alpha_1,\ldots,\alpha_p$ are irrational real numbers.

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

scholarly journals On the Construction of Large Algebras Not Contained in the Image of the Borel Map

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Céline Esser ◽  
Gerhard Schindl
Keyword(s):  
Sequence Space ◽  
Analytic Functions ◽  
General Setting ◽  
Smooth Functions ◽  
The Real ◽  
Cauchy Product ◽  
Real Analytic Functions ◽  
Borel Map ◽  
Real Analytic ◽  
The Stability

AbstractThe Borel map $$j^{\infty }$$j∞ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $$j^{\infty }$$j∞ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $$j^{\infty }$$j∞ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $$j^{\infty }$$j∞ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

scholarly journals Extension sets for real analytic functions and applications to Radon transforms

1996 ◽  
Vol 19 (4) ◽  
pp. 625-632
Author(s):  
Vernor Arguedas ◽  
Ricardo Estrada
Keyword(s):  
Entire Function ◽  
Analytic Functions ◽  
Radon Transforms ◽  
Support Theorem ◽  
The Real ◽  
Real Analytic Functions ◽  
Small Set ◽  
Real Analytic

The real analytic character of a functionf(x,y)is determined from its behavior along radial directionsfθ(s)=f(scosθ,ssinθ)forθ∈E, whereEis a “small” set. A support theorem for Radon transforms in the plane is proved. In particular iffθextends to an entire function forθ∈Eandf(x,y)is real analytic inℝ2then it also extends to an entire function inℂ2.

Homomorphisms on Function Algebras

1994 ◽  
Vol 46 (4) ◽  
pp. 734-745 ◽  
Author(s):  
M. I. Garrido ◽  
J. Gómez Gil ◽  
J. A. Jaramillo
Keyword(s):  
Banach Space ◽  
Topological Space ◽  
Wide Class ◽  
Rational Functions ◽  
Positive Answer ◽  
Algebra Homomorphism ◽  
Function Algebras ◽  
Real Analytic Functions ◽  
Real Analytic ◽  
Separable Spaces

AbstractLet A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism φ: A → R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or Cm-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal).

scholarly journals Composite quasianalytic functions

2018 ◽  
Vol 154 (9) ◽  
pp. 1960-1973 ◽  
Author(s):  
André Belotto da Silva ◽  
Edward Bierstone ◽  
Michael Chow
Keyword(s):  
Real Field ◽  
Composite Function ◽  
The Real ◽  
Quasianalytic Class ◽  
Minimal Structure ◽  
Carleman Class ◽  
Composition Condition ◽  
Loss Of Regularity ◽  
Quasianalytic Functions

We prove two main results on Denjoy–Carleman classes: (1) a composite function theorem which asserts that a function $f(x)$ in a quasianalytic Denjoy–Carleman class ${\mathcal{Q}}_{M}$, which is formally composite with a generically submersive mapping $y=\unicode[STIX]{x1D711}(x)$ of class ${\mathcal{Q}}_{M}$, at a single given point in the source (or in the target) of $\unicode[STIX]{x1D711}$ can be written locally as $f=g\circ \unicode[STIX]{x1D711}$, where $g(y)$ belongs to a shifted Denjoy–Carleman class ${\mathcal{Q}}_{M^{(p)}}$; (2) a statement on a similar loss of regularity for functions definable in the $o$-minimal structure given by expansion of the real field by restricted functions of quasianalytic class ${\mathcal{Q}}_{M}$. Both results depend on an estimate for the regularity of a ${\mathcal{C}}^{\infty }$ solution $g$ of the equation $f=g\circ \unicode[STIX]{x1D711}$, with $f$ and $\unicode[STIX]{x1D711}$ as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

Multiplicative norms on Banach algebras

1951 ◽  
Vol 47 (3) ◽  
pp. 473-474 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Banach Algebras ◽  
Real Field ◽  
Complex Numbers ◽  
Normed Algebra ◽  
Real Numbers ◽  
The Real ◽  
Image Position ◽  
Real Quaternions

1. Mazur(1) has shown that any normed algebra A over the real field in which the norm is multiplicative in the sense thatis equivalent (i.e. algebraically isomorphic and isometric under one and the same mapping) to one of the following algebras: (i) the real numbers, (ii) the complex numbers, (iii) the real quaternions, each of these sets being regarded as normed algebras over the real field. Completeness of A is not assumed by Mazur. A relevant discussion is given also in Lorch (2).

scholarly journals The Pfaffian closure of an o-minimal structure

1999 ◽  
Vol 1999 (508) ◽  
pp. 189-211 ◽  
Author(s):  
Patrick Speissegger
Keyword(s):  
Real Field ◽  
The Real ◽  
Minimal Structure ◽  
Pfaffian Equations

Abstract Every o-minimal expansion of the real field has an o-minimal expansion in which the solutions to Pfaffian equations with definable C1 coefficients are definable.

scholarly journals Counting algebraic points in expansions of o-minimal structures by a dense set

2020 ◽  
Author(s):  
Pantelis E Eleftheriou
Keyword(s):  
Rational Points ◽  
Real Field ◽  
Elementary Substructure ◽  
The Real ◽  
Semialgebraic Set ◽  
Minimal Structure ◽  
Infinite Set ◽  
Dense Set ◽  
The Way

Abstract The Pila–Wilkie theorem states that if a set $X\subseteq \mathbb{R}^n$ is definable in an o-minimal structure $\mathcal{R}$ and contains ‘many’ rational points, then it contains an infinite semialgebraic set. In this paper, we extend this theorem to an expansion $\widetilde{\mathcal{R}}=\langle {\mathcal{R}}, P\rangle$ of ${\mathcal{R}}$ by a dense set P, which is either an elementary substructure of ${\mathcal{R}}$, or it is $\mathrm{dcl}$-independent, as follows. If X is definable in $\widetilde{\mathcal{R}}$ and contains many rational points, then it is dense in an infinite semialgebraic set. Moreover, it contains an infinite set which is ${\emptyset}$-definable in $\langle \overline{\mathbb{R}}, P\rangle$, where $\overline{\mathbb{R}}$ is the real field. Along the way we introduce the notion of the ‘algebraic trace part’ $X^{{\, alg}}_t$ of any set $X\subseteq \mathbb{R}^n$, and we show that if X is definable in an o-minimal structure, then $X^{{\, alg}}_t$ coincides with the usual algebraic part of X.

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