A note on existentially closed difference fields with algebraically closed fixed field

2001 ◽  
Vol 66 (2) ◽  
pp. 719-721 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Model Companion ◽  
Fixed Field ◽  
Existentially Closed ◽  
Difference Fields

AbstractWe point out that the theory of difference fields with algebraically closed fixed field has no model companion.

Existentially closed models via constructible sets: There are 2ℵ0 existentially closed pairwise non elementarily equivalent existentially closed ordered groups

1996 ◽  
Vol 61 (1) ◽  
pp. 277-284 ◽  
Author(s):  
Anatole Khelif
Keyword(s):  
Open Problem ◽  
General Framework ◽  
Direct Proof ◽  
Model Companion ◽  
Standard Methods ◽  
Division Rings ◽  
Ordered Groups ◽  
Existentially Closed ◽  
Constructible Sets

AbstractWe prove that there are 2χ0 pairwise non elementarily equivalent existentially closed ordered groups, which solve the main open problem in this area (cf. [3, 10]).A simple direct proof is given of the weaker fact that the theory of ordered groups has no model companion; the case of the ordered division rings over a field k is also investigated.Our main result uses constructible sets and can be put in an abstract general framework.Comparison with the standard methods which use forcing (cf. [4]) is sketched.

scholarly journals Existentially closed models of the theory of artinian local rings

1999 ◽  
Vol 64 (2) ◽  
pp. 825-845 ◽  
Author(s):  
Hans Schoutens
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Model Companion ◽  
Closed Model ◽  
Gorenstein Rings ◽  
Existentially Closed ◽  
Finite Set ◽  
Residue Fields

AbstractThe class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms τtl. We show that its existentially closed models are Gorenstein. of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory oτl of all Artinian local Gorenstein rings of length l with algebraically closed residue field is model complete and the theory τtl is companionable, with model-companion oτl.

scholarly journals FIELDS WITH SEVERAL COMMUTING DERIVATIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 1-19 ◽  
Author(s):  
DAVID PIERCE
Keyword(s):  
Natural Number ◽  
Geometric Characterization ◽  
Model Companion ◽  
First Order ◽  
Existentially Closed ◽  
Differential Fields ◽  
Theory Of Fields ◽  
Differential Varieties

AbstractFor every natural numberm, the existentially closed models of the theory of fields withmcommuting derivations can be given a first-order geometric characterization in several ways. In particular, the theory of these differential fields has a model-companion. The axioms are that certain differential varieties determined by certain ordinary varieties are nonempty. There is no restriction on the characteristic of the underlying field.

Existentially closed algebras and boolean products

1988 ◽  
Vol 53 (2) ◽  
pp. 571-596
Author(s):  
Herbert H. J. Riedel
Keyword(s):  
Sufficient Condition ◽  
Subdirectly Irreducible ◽  
Universal Class ◽  
Model Companion ◽  
Universal Horn Class ◽  
Existentially Closed ◽  
Boolean Product ◽  
Boolean Products ◽  
Simple Algebras ◽  
Subdirectly Irreducible Algebras

AbstractA Boolean product construction is used to give examples of existentially closed algebras in the universal Horn class ISP(K) generated by a universal class K of finitely subdirectly irreducible algebras such that Γa(K) has the Fraser-Horn property. If ⟦a ≠ b⟧ ∩ ⟦c ≠ d ⟧ = ∅ is definable in K and K has a model companion of K-simple algebras, then it is shown that ISP(K) has a model companion. Conversely, a sufficient condition is given for ISP(K) to have no model companion.

The structure of algebraically and existentially closed Stone and double Stone algebras

1989 ◽  
Vol 54 (2) ◽  
pp. 363-375 ◽  
Author(s):  
David M. Clark
Keyword(s):  
Finite Algebra ◽  
Amalgamation Property ◽  
Explicit Construction ◽  
Structural Description ◽  
Model Companion ◽  
Existentially Closed ◽  
Boolean Spaces ◽  
Countably Infinite ◽  
Model Completion ◽  
Primitive Recursive

In this paper we study the varieties of Stone algebras (S, ∧, ∨, *, 0, 1) and double Stone algebras (D, ∧, ∨, *, +, 0, 1). Our primary interest is to give a structural description of the algebraically and existentially closed members of both classes. Our technique is an application of the natural dualities of Davey [6] and Clark and Krauss [5]. This approach gives a description of the desired models as the algebras of all continuous structure-preserving maps from certain structured Boolean spaces into the generating algebra for the variety. In each case the resulting description can be converted in a natural way into a finite ∀∃-axiomatization for these models. For Stone algebras these axioms appeared earlier in Schmid [20], [21] and in Schmitt [22].Since both cases we consider satisfy the amalgamation property, the existentially closed members form a model companion for the variety which is also its model completion. Moreover, it is also ℵ0 categorical and its countably infinite member is the unique countable homogeneous universal model for the variety. In the case of Stone algebras, explicit constructions for this model appear in Schmitt [22] and Weispfenning [23]. We give here an explicit construction for double Stone algebras of S. Hayes.This work was motivated by a problem of Stanley Burris. In [4] Burris and Werner superseded many previous results by showing that for any finite algebra A, the universal Horn class ISP has a model companion. Weispfenning [24], [25] discovered that this model companion is always ℵ0 categorical and has a primitive recursive ∀∃-axiomatization. In spite of these very general theorems, there are few instances in which a structural description of the (any!) existentially closed members of ISP is available. Burris and Werner [4] solve this problem in a special setting.

A Fixed Field/Fixed Length Automated Case Record Using an IBM 1232 Optically Readable Document

1968 ◽  
Vol 07 (03) ◽  
pp. 156-158
Author(s):  
Th. R. Taylor
Keyword(s):  
Clinical Data ◽  
Case Record ◽  
Fixed Field ◽  
Fixed Length

The technique, scope and limitations of a fixed field/fixed length case record utilising the IBM 1232 system is described. The principal problems lie with personnel rather than machinery and with programmes for analysis rather than clinical data.

Helical tomotherapy for advanced esophageal cancer improves target conformity and homogeneity: A comparison with fixed-field intensity-modulated radiotherapy

2015 ◽  
Vol 11 (3) ◽  
pp. 3146-3155
Author(s):  
Luhua Wang
Keyword(s):  
Esophageal Cancer ◽  
Field Intensity ◽  
Helical Tomotherapy ◽  
Intensity Modulated Radiotherapy ◽  
Matched Pair ◽  
Conformity Index ◽  
Target Coverage ◽  
Advanced Esophageal Cancer ◽  
Fixed Field ◽  
Intensity Modulated

Purpose: To evaluate the usefulness of helical tomotherapy (HT) in the treatment of advanced esophageal cancer (EC) and compare target homogeneity, conformity and normal tissue doses between HT and fixed-field intensity-modulated radiotherapy (ff-IMRT).Methods: In all, 23 patients with cT3-4N0-1M0-1a thoracic EC (upper esophagus, 9 patients; middle esophagus, 6; distal esophagus, 6 and esophagogastric junction, 2) who were treated with ff-IMRT (60 Gy in 30 fractions) were re-planned for HT and ff-IMRT with the same clinical require­ments. Comparisons were performed using the Wilcoxon matched-pair signed-rank test.Results: Compared with ff-IMRT, HT significantly reduced the homogeneity index for thoracic, upper, middle and distal ECs by 38%, 31%, 36% and 33%, respectively (P < 0.05). The conformity index was increased by HT for thoracic, upper and middle ECs by 9%, 9% and 18%, respectively (P < 0.05). Target coverage was improved by 1% with HT (P < 0.05). The mean lung dose was significantly reduced by HT for thoracic and upper ECs (P < 0.05). The V20 (volume receiving at least 20 Gy) and higher dose volumes of the lungs were decreased by HT in all cases, but the differences were significant for thoracic, upper and distal ECs (P < 0.05), with reductions of 2.1%, 3.1% and 2.2%, respectively. HT resulted in a larger lung V5 for thoracic, upper, middle and distal ECs, with increases of 3.5%, 1.5%, 7.2% and 3.2%, respectively. Heart sparing was significantly better with HT than with ff-IMRT in terms of the V30 and V40 for thoracic, upper, middle and distal ECs (P < 0.05).Conclusions: Compared to ff-IMRT, HT provides superior target coverage, conformity and homogeneity, with reduced the volume of high doses to the lungs and heart for advanced EC. HT may be a treatment option for advanced EC, especially upper EC.

Existentially closed central extensions of locally finite p-groups

1986 ◽  
Vol 100 (2) ◽  
pp. 281-301 ◽  
Author(s):  
Felix Leinen ◽  
Richard E. Phillips
Keyword(s):  
Central Extensions ◽  
Locally Finite ◽  
Existentially Closed ◽  
Image Position

Throughout, p will be a fixed prime, and will denote the class of all locally finite p-groups. For a fixed Abelian p-group A, we letwhere ζ(P) denotes the centre of P. Notice that A is not a class in the usual group-theoretic sense, since it is not closed under isomorphisms.

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