scholarly journals Avoiding the projective hierarchy in expansions of the real field by sequences

2005 ◽  
Vol 134 (05) ◽  
pp. 1483-1493 ◽  
Author(s):  
Chris Miller
Keyword(s):  
Real Field ◽  
The Real ◽  
Projective Hierarchy

scholarly journals Expansions of the real field by open sets: definability versus interpretability

2010 ◽  
Vol 75 (4) ◽  
pp. 1311-1325 ◽  
Author(s):  
Harvey Friedman ◽  
Krzysztof Kurdyka ◽  
Chris Miller ◽  
Patrick Speissegger
Keyword(s):  
Measure Theory ◽  
Geometric Measure Theory ◽  
Cantor Set ◽  
Real Field ◽  
The Real ◽  
Geometric Measure ◽  
Projective Hierarchy

AbstractAn open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in (, E) either has interior or is Hausdorff null.

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 Interpreting the projective hierarchy in expansions of the real line

2014 ◽  
Vol 142 (9) ◽  
pp. 3259-3267 ◽  
Author(s):  
Philipp Hieronymi ◽  
Michael Tychonievich
Keyword(s):  
Real Line ◽  
The Real ◽  
Projective Hierarchy

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.

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.

Veblen–Wedderburn hybrid planes

1969 ◽  
Vol 309 (1497) ◽  
pp. 157-170 ◽  
Keyword(s):  
Projective Plane ◽  
Galois Field ◽  
Real Field ◽  
Type I ◽  
Type Ii ◽  
Desarguesian Plane ◽  
The Real ◽  
Desarguesian Planes ◽  
Hughes Plane

The paper describes a method of redistributing the points of the collinear sets in a Desarguesian plane so as to produce a (hybrid) projective plane which is non-Desarguesian. The method is applied to the construction: (i) of a plane over a prescribed subfield of the real field, and (ii) of a plane (over a Galois field) which is proved to be identical with the Hughes plane. On the basis of this construction algebraic relations in the field can be interpreted as incidence relations in the hybrid plane. In order to verify that the planes of type (i) are not isomorphic with Desarguesian planes, some conditions are established which show that all planes of this type (as well as of type (ii)) contain Fano subplanes.

Some advances towards a purely geometrical justification of the use of unreal elements in projective geometry

1931 ◽  
Vol 27 (3) ◽  
pp. 306-325 ◽  
Author(s):  
I. Brahmachari
Keyword(s):  
Projective Geometry ◽  
Ordered Set ◽  
Real Field ◽  
Real Plane ◽  
The Real ◽  
Considerable Advantage ◽  
Geometrical Reasoning ◽  
Successful Attempt

1. Considerable advantage has resulted from the postulation of unreal elements in projective geometry. In the first place these unreal elements were defined in terms of points represented by complex coordinates, and their use in purely geometrical reasoning had become well established before any serious attempt was made to justify this use, independently of algebraic considerations, by providing real representations of the unreal elements. The first successful attempt was that of von Staudt, who represented an unreal element by an elliptic involution associated with an order. In this system an ordered set of four real points is required to specify an unreal point. The system is comparatively simple to deal with in a single real plane, more complicated in a single real [3], and rapidly increases in complexity as the number of dimensions of the real field is increased.

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