scholarly journals A FUNDAMENTAL DICHOTOMY FOR DEFINABLY COMPLETE EXPANSIONS OF ORDERED FIELDS

2015 ◽  
Vol 80 (4) ◽  
pp. 1091-1115 ◽  
Author(s):  
ANTONGIULIO FORNASIERO ◽  
PHILIPP HIERONYMI
Keyword(s):  
Ordered Fields ◽  
Complete Field ◽  
Differentiation Theorem

AbstractAn expansion of a definably complete field either defines a discrete subring, or the image of every definable discrete set under every definable map is nowhere dense. As an application we show a definable version of Lebesgue’s differentiation theorem.

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

Comments on the reconnexion rate of magnetic fields

1973 ◽  
Vol 9 (1) ◽  
pp. 49-63 ◽  
Author(s):  
E. N. Parker
Keyword(s):  
Velocity Field ◽  
Magnetic Fields ◽  
Turbulent Flows ◽  
Similarity Solutions ◽  
Neutral Point ◽  
Solar Photosphere ◽  
Complete Field ◽  
Fluid Equations ◽  
The Galaxy

The reconnexion rate of magnetic fields is crucial in understanding the fields found in turbulent flows in the solar photosphere and in the galaxy, and in flare phenomena. This paper examines the behaviour of magnetic fields in the neighbourhood of an X-type neutral point. The treatment is kinematical, specifying the velocity field v and constructing solutions to the hydromagnetic equation for B. The calculations demonstrate that the reconnexion rate is controlled by the diffusion in the near neighbourhood of the neutral point, and is not arbitrarily large, as has been suggested by similarity solutions of the complete field and fluid equations for vanishing diffusion

scholarly journals Ordered fields dense in their real closure and definable convex valuations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lothar Sebastian Krapp ◽  
Salma Kuhlmann ◽  
Gabriel Lehéricy
Keyword(s):  
Ordered Fields ◽  
Real Closure

Abstract In this paper, we undertake a systematic model- and valuation-theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah–Hasson Conjecture (specialized to ordered fields) and provide an example limiting its valuation-theoretic conclusions.

scholarly journals Inaba extension of complete field of characteristic 0

2020 ◽  
Vol 21 (3) ◽  
pp. 59-67
Author(s):  
Vladimirovich Vostokov Sergei ◽  
Borisovich Zhukov Igor ◽  
Yur’evna Ivanova Olga
Keyword(s):  
Complete Field

scholarly journals Ordered fields satisfying Rolle's theorem

1986 ◽  
Vol 30 (1) ◽  
pp. 66-78 ◽  
Author(s):  
Ron Brown ◽  
Thomas C. Craven ◽  
M.J. Pelling
Keyword(s):  
Rolle’S Theorem ◽  
Ordered Fields

scholarly journals Maximal ordered fields of rank $n$

1987 ◽  
Vol 17 (1) ◽  
pp. 157-167 ◽  
Author(s):  
Daiji Kijima ◽  
Mieo Nishi
Keyword(s):  
Ordered Fields

Non-spectral Complete Field Expansion in Two-dimensional Structures

1985 ◽  
Vol 35 (1) ◽  
pp. 1-12 ◽  
Author(s):  
M. P. CARPENTIER ◽  
A. F. DOS SANTOS
Keyword(s):  
Two Dimensional ◽  
Complete Field
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