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 Expansions of the Real Field by Canonical Products

2019 ◽  
Vol 63 (3) ◽  
pp. 506-521
Author(s):  
Chris Miller ◽  
Patrick Speissegger
Keyword(s):  
Borel Subset ◽  
Connected Components ◽  
Real Field ◽  
Power Functions ◽  
Positive Real ◽  
The Real ◽  
Associated Functions ◽  
Definable Sets ◽  
Canonical Products ◽  
Logarithmic Derivatives

AbstractWe consider expansions of o-minimal structures on the real field by collections of restrictions to the positive real line of the canonical Weierstrass products associated with sequences such as $(-n^{s})_{n>0}$ (for $s>0$) and $(-s^{n})_{n>0}$ (for $s>1$), and also expansions by associated functions such as logarithmic derivatives. There are only three possible outcomes known so far: (i) the expansion is o-minimal (that is, definable sets have only finitely many connected components); (ii) every Borel subset of each $\mathbb{R}^{n}$ is definable; (iii) the expansion is interdefinable with a structure of the form $(\mathfrak{R}^{\prime },\unicode[STIX]{x1D6FC}^{\mathbb{Z}})$ where $\unicode[STIX]{x1D6FC}>1$, $\unicode[STIX]{x1D6FC}^{\mathbb{Z}}$ is the set of all integer powers of $\unicode[STIX]{x1D6FC}$, and $\mathfrak{R}^{\prime }$ is o-minimal and defines no irrational power functions.

Model Completeness for the Real Field with the Weierstrass ℘ Function

2018 ◽  
Vol 61 (3) ◽  
pp. 811-823
Author(s):  
Ricardo Bianconi
Keyword(s):  
Automorphic Forms ◽  
Real Field ◽  
Weierstrass Function ◽  
Model Completeness ◽  
Auxiliary Model ◽  
The Real ◽  
Change Of Variables ◽  
Completeness Result ◽  
Sine Functions ◽  
Derivatives Of

AbstractWe prove model completeness for the expansion of the real field by the Weierstrass ℘ function as a function of the variable z and the parameter (or period) τ. We need to existentially define the partial derivatives of the ℘ function with respect to the variable z and the parameter τ. To obtain this result, it is necessary to include in the structure function symbols for the unrestricted exponential function and restricted sine function, the Weierstrass ζ function and the quasi-modular form E2 (we conjecture that these functions are not existentially definable from the functions ℘ alone or even if we use the exponential and restricted sine functions). We prove some auxiliary model-completeness results with the same functions composed with appropriate change of variables. In the conclusion, we make some remarks about the non-effectiveness of our proof and the difficulties to be overcome to obtain an effective model-completeness result, and how to extend these results to appropriate expansion of the real field by automorphic forms.

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 Model Completeness, Uniform Interpolants and Superposition Calculus

2021 ◽  
Author(s):  
Diego Calvanese ◽  
Silvio Ghilardi ◽  
Alessandro Gianola ◽  
Marco Montali ◽  
Andrey Rivkin
Keyword(s):  
Automated Reasoning ◽  
Quantifier Elimination ◽  
Effective Solution ◽  
Model Completeness ◽  
Present Contribution ◽  
First Order ◽  
Reduction Strategies ◽  
Infinite State Systems ◽  
Infinite State ◽  
Superposition Calculus

AbstractUniform interpolants have been largely studied in non-classical propositional logics since the nineties; a successive research line within the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. This further research line is motivated by the fact that uniform interpolants offer an effective solution to tackle quantifier elimination and symbol elimination problems, which are central in model checking infinite state systems. This was first pointed out in ESOP 2008 by Gulwani and Musuvathi, and then by the authors of the present contribution in the context of recent applications to the verification of data-aware processes. In this paper, we show how covers are strictly related to model completions, a well-known topic in model theory. We also investigate the computation of covers within the Superposition Calculus, by adopting a constrained version of the calculus and by defining appropriate settings and reduction strategies. In addition, we show that computing covers is computationally tractable for the fragment of the language used when tackling the verification of data-aware processes. This observation is confirmed by analyzing the preliminary results obtained using the mcmt tool to verify relevant examples of data-aware processes. These examples can be found in the last version of the tool distribution.

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 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.

Sign in / Sign up

Export Citation Format

Share Document