scholarly journals Some properties of the polynomially bounded o-minimal expansions of the real field and of some quasianalytic local rings

2020 ◽  
Vol 12 (2) ◽  
pp. 483-491
Author(s):  
M. Berraho
Keyword(s):  
Metric Spaces ◽  
Real Field ◽  
Local Rings ◽  
Topological Properties ◽  
The Real ◽  
Division Theorem

In this paper, we study the Weierstrass division theorem over the rings of smooth germs that are definable in an arbitrary polynomially bounded o-minimal expansion of the real field by giving some criteria for satisfying this theorem. Afterwards, we study some topological properties of some quasianalytic subrings of the ring of smooth germs for the $(x_1)$-adic topology by showing that these rings are separable metric spaces. Also, we cite a criterion for their completeness with respect to the $(x_1)$-adic topology.

scholarly journals Compactness in Metric Spaces

2016 ◽  
Vol 24 (3) ◽  
pp. 167-172
Author(s):  
Kazuhisa Nakasho ◽  
Keiko Narita ◽  
Yasunari Shidama
Keyword(s):  
Real Line ◽  
Metric Spaces ◽  
Topological Properties ◽  
The Real ◽  
The Third ◽  
Norm Space ◽  
Number Sequences ◽  
Sequential Compactness ◽  
Countable Compactness ◽  
Definition Of

Summary In this article, we mainly formalize in Mizar [2] the equivalence among a few compactness definitions of metric spaces, norm spaces, and the real line. In the first section, we formalized general topological properties of metric spaces. We discussed openness and closedness of subsets in metric spaces in terms of convergence of element sequences. In the second section, we firstly formalize the definition of sequentially compact, and then discuss the equivalence of compactness, countable compactness, sequential compactness, and totally boundedness with completeness in metric spaces. In the third section, we discuss compactness in norm spaces. We formalize the equivalence of compactness and sequential compactness in norm space. In the fourth section, we formalize topological properties of the real line in terms of convergence of real number sequences. In the last section, we formalize the equivalence of compactness and sequential compactness in the real line. These formalizations are based on [20], [5], [17], [14], and [4].

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.

Stochastic simulation of the spatio-temporal field of the average daily heat index in Southern Russia

2020 ◽  
Vol 82 ◽  
pp. 149-160
Author(s):  
N Kargapolova
Keyword(s):  
Heat Waves ◽  
Numerical Models ◽  
Heat Index ◽  
Real Field ◽  
Gaussian Field ◽  
The Real ◽  
Southern Russia ◽  
Weather Stations ◽  
Spatio Temporal ◽  
Temporal Field

Numerical models of the heat index time series and spatio-temporal fields can be used for a variety of purposes, from the study of the dynamics of heat waves to projections of the influence of future climate on humans. To conduct these studies one must have efficient numerical models that successfully reproduce key features of the real weather processes. In this study, 2 numerical stochastic models of the spatio-temporal non-Gaussian field of the average daily heat index (ADHI) are considered. The field is simulated on an irregular grid determined by the location of weather stations. The first model is based on the method of the inverse distribution function. The second model is constructed using the normalization method. Real data collected at weather stations located in southern Russia are used to both determine the input parameters and to verify the proposed models. It is shown that the first model reproduces the properties of the real field of the ADHI more precisely compared to the second one, but the numerical implementation of the first model is significantly more time consuming. In the future, it is intended to transform the models presented to a numerical model of the conditional spatio-temporal field of the ADHI defined on a dense spatio-temporal grid and to use the model constructed for the stochastic forecasting of the heat index.

scholarly journals Role Playing In the Field Of Education

2018 ◽  
pp. 1-13
Author(s):  
Yuan Lo
Keyword(s):  
Real Life ◽  
Role Playing ◽  
Real Field ◽  
Real Situation ◽  
The Real ◽  
Artificial Environment ◽  
Life Education

The character and status are presented together. Others have to play the role. The real situation is to be presented in a simple way. It can be understood how to adapt yourself to the real field. The role of the actress is to be revealed. Students get real-life education in the artificial environment. Performances of speech and expression are improved.

Weierstrass Division in Quasianalytic Local Rings

1976 ◽  
Vol 28 (5) ◽  
pp. 938-953 ◽  
Author(s):  
C. L. Childress
Keyword(s):  
Local Ring ◽  
Analytic Functions ◽  
Local Rings ◽  
Division Theorem ◽  
Background Material

In this paper we consider the problem of extending the Weierstrass division theorem to quasianalytic local rings of germs of functions of k real variables which properly contain the local ring of germs of analytic functions. After some background material (§ 2) and some technical preliminaries (§ 3), we show by examples that the so-called generic division theorem fails in such rings if k ≧ 1 and that the Weierstrass division theorem fails in such rings if k ≧ 2 (§ 4).

Uniform model-completeness for the real field expanded by power functions

2010 ◽  
Vol 75 (4) ◽  
pp. 1441-1461
Author(s):  
Tom Foster
Keyword(s):  
Real Field ◽  
Power Functions ◽  
Model Completeness ◽  
The Real ◽  
Unary Function ◽  
First Order ◽  
Existential Formula ◽  
Uniform Model

AbstractWe prove that given any first order formula ϕ in the language L′ = {+, ·, <,(fi)iЄI, (ci)iЄI}, where the fi are unary function symbols and the ci are constants, one can find an existential formula Ψ such that φ and Ψ are equivalent in any L′-structure

Pfaffian differential equations over exponential o-minimal structures

2002 ◽  
Vol 67 (1) ◽  
pp. 438-448 ◽  
Author(s):  
Chris Miller ◽  
Patrick Speissegger
Keyword(s):  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Integration ◽  
Real Field ◽  
Asymptotic Behavior Of Solutions ◽  
The Real ◽  
Unary Function ◽  
Ordered Fields ◽  
Component Function

In this paper, we continue investigations into the asymptotic behavior of solutions of differential equations over o-minimal structures.Let ℜ be an expansion of the real field (ℝ, +, ·).A differentiable map F = (F1,…, F1): (a, b) → ℝi is ℜ-Pfaffian if there exists G: ℝ1+l → ℝl definable in ℜ such that F′(t) = G(t, F(t)) for all t ∈ (a, b) and each component function Gi: ℝ1+l → ℝ is independent of the last l − i variables (i = 1, …, l). If ℜ is o-minimal and F: (a, b) → ℝl is ℜ-Pfaffian, then (ℜ, F) is o-minimal (Proposition 7). We say that F: ℝ → ℝl is ultimately ℜ-Pfaffian if there exists r ∈ ℝ such that the restriction F ↾(r, ∞) is ℜ-Pfaffian. (In general, ultimately abbreviates “for all sufficiently large positive arguments”.)The structure ℜ is closed under asymptotic integration if for each ultimately non-zero unary (that is, ℝ → ℝ) function f definable in ℜ there is an ultimately differentiable unary function g definable in ℜ such that limt→+∞[g′(t)/f(t)] = 1- If ℜ is closed under asymptotic integration, then ℜ is o-minimal and defines ex: ℝ → ℝ (Proposition 2).Note that the above definitions make sense for expansions of arbitrary ordered fields.

scholarly journals Real Tropicalization and Analytification of Semialgebraic Sets

2020 ◽  
Author(s):  
Philipp Jell ◽  
Claus Scheiderer ◽  
Josephine Yu
Keyword(s):  
Inverse Limit ◽  
Point Of View ◽  
General Point ◽  
Real Spectrum ◽  
Closed Field ◽  
Topological Properties ◽  
Absolute Value ◽  
The Real ◽  
Semialgebraic Set ◽  
Refined Version

Abstract Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $S \subseteq K^n$ we define a space $S_r^{{\operatorname{an}}}$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $S$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $X$ is an algebraic variety, we show that $X_r^{{\operatorname{an}}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.

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