Uniqueness and characterization of prime models over sets for totally transcendental first-order theories

1972 ◽  
Vol 37 (1) ◽  
pp. 107-113 ◽  
Author(s):  
Saharon Shelah
Keyword(s):  
Simple Proof ◽  
Finite Sequence ◽  
The Other ◽  
Closed Field ◽  
Prime Model ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Transcendental Theory

If T is a complete first-order totally transcendental theory then over every T-structure A there is a prime model unique up to isomorphism over A. Moreover M is a prime model over A iff: (1) every finite sequence from M realizes an isolated type over A, and (2) there is no uncountable indiscernible set over A in M.The existence of prime models was proved by Morley [3] and their uniqueness for countable A by Vaught [9]. Sacks asked (see Chang and Keisler [1, question 25]) whether the prime model is unique. After proving this I heard Ressayre had proved that every two strictly prime models over any T-structure A are isomorphic, by a strikingly simple proof. From this followsIf T is totally transcendental, M a strictly prime model over A then every elementary permutation of A can be extended to an automorphism of M. (The existence of M follows by [3].)By our results this holds for any prime model. On the other hand Ressayre's result applies to more theories. For more information see [6, §0A]. A conclusion of our theorem is the uniqueness of the prime differentially closed field over a differential field. See Blum [8] for the total transcendency of the theory of differentially closed fields.We can note that the prime model M over A is minimal over A iff in M there is no indiscernible set over A (which is infinite).

Differentially algebraic group chunks

1990 ◽  
Vol 55 (3) ◽  
pp. 1138-1142 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Algebraic Group ◽  
Characteristic Zero ◽  
Quantifier Elimination ◽  
Closed Field ◽  
Zariski Topology ◽  
First Order ◽  
Differential Field ◽  
Closed Fields ◽  
Differential Fields ◽  
Algebraically Closed Fields

We point out that a group first order definable in a differentially closed field K of characteristic 0 can be definably equipped with the structure of a differentially algebraic group over K. This is a translation into the framework of differentially closed fields of what is known for groups definable in algebraically closed fields (Weil's theorem).I restrict myself here to showing (Theorem 20) how one can find a large “differentially algebraic group chunk” inside a group defined in a differentially closed field. The rest of the translation (Theorem 21) follows routinely, as in [B].What is, perhaps, of interest is that the proof proceeds at a completely general (soft) model theoretic level, once Facts 1–4 below are known.Fact 1. The theory of differentially closed fields of characteristic 0 is complete and has quantifier elimination in the language of differential fields (+, ·,0,1, −1,d).Fact 2. Affine n-space over a differentially closed field is a Noetherian space when equipped with the differential Zariski topology.Fact 3. If K is a differentially closed field, k ⊆ K a differential field, and a and are in k, then a is in the definable closure of k ◡ iff a ∈ ‹› (where k ‹› denotes the differential field generated by k and).Fact 4. The theory of differentially closed fields of characteristic zero is totally transcendental (in particular, stable).

Algebraic theories with definable Skolem functions

1984 ◽  
Vol 49 (2) ◽  
pp. 625-629 ◽  
Author(s):  
Lou van den Dries
Keyword(s):  
Nonempty Subset ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
First Order ◽  
Definable Subset ◽  
Skolem Functions ◽  
Algebraic Theories ◽  
Closed Fields ◽  
Definitional Extension

(1.1) A well-known example of a theory with built-in Skolem functions is (first-order) Peano arithmetic (or rather a certain definitional extension of it). See [C-K, pp. 143, 162] for the notion of a theory with built-in Skolem functions, and for a treatment of the example just mentioned. This property of Peano arithmetic obviously comes from the fact that in each nonempty definable subset of a model we can definably choose an element, namely, its least member.(1.2) Consider now a real closed field R and a nonempty subset D of R which is definable (with parameters) in R. Again we can definably choose an element of D, as follows: D is a union of finitely many singletons and intervals (a, b) where – ∞ ≤ a < b ≤ + ∞; if D has a least element we choose that element; if not, D contains an interval (a, b) for which a ∈ R ∪ { − ∞}is minimal; for this a we choose b ∈ R ∪ {∞} maximal such that (a, b) ⊂ D. Four cases have to be distinguished:(i) a = − ∞ and b = + ∞; then we choose 0;(ii) a = − ∞ and b ∈ R; then we choose b − 1;(iii) a ∈ R and b ∈ = + ∞; then we choose a + 1;(iv) a ∈ R and b ∈ R; then we choose the midpoint (a + b)/2.It follows as in the case of Peano arithmetic that the theory RCF of real closed fields has a definitional extension with built-in Skolem functions.

Parameterizing higher-order processes on names and processes

2019 ◽  
Vol 53 (3-4) ◽  
pp. 153-206
Author(s):  
Xian Xu
Keyword(s):  
Lambda Calculus ◽  
Higher Order ◽  
The Other ◽  
First Order ◽  
Full Abstraction

Parameterization extends higher-order processes with the capability of abstraction and application (like those in lambda-calculus). As is well-known, this extension is strict, meaning that higher-order processes equipped with parameterization are strictly more expressive than those without parameterization. This paper studies strictly higher-order processes (i.e., no name-passing) with two kinds of parameterization: one on names and the other on processes themselves. We present two main results. One is that in presence of parameterization, higher-order processes can interpret first-order (name-passing) processes in a quite elegant fashion, in contrast to the fact that higher-order processes without parameterization cannot encode first-order processes at all. We present two such encodings and analyze their properties in depth, particularly full abstraction. In the other result, we provide a simpler characterization of the standard context bisimilarity for higher-order processes with parameterization, in terms of the normal bisimilarity that stems from the well-known normal characterization for higher-order calculus. As a spinoff, we show that the bisimulation up-to context technique is sound in the higher-order setting with parameterization.

Prime models and almost decidability

1986 ◽  
Vol 51 (2) ◽  
pp. 412-420 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
Order Theory ◽  
The Other ◽  
Prime Model ◽  
First Order ◽  
Decidable Theory ◽  
Complete Extension ◽  
First Order Theory ◽  
Additional Constant ◽  
Open Question ◽  
Countable Models

This paper introduces and investigates a notion that approximates decidability with respect to countable structures. The paper demonstrates that there exists a decidable first order theory with a prime model that is not almost decidable. On the other hand it is proved that if a decidable complete first order theory has only countably many complete types, then it has a prime model that is almost decidable. It is not true that every decidable complete theory with only countably many complete types has a decidable prime model. It is not known whether a complete decidable theory with only countably many countable models up to isomorphism must have a decidable prime model. In [1] a weaker result was proven—if every complete extension, in finitely many additional constant symbols, of a theory T fails to have a decidable prime model, then T has 2ω nonisomorphic countable models. The corresponding statement for saturated models is false, even if all the complete types are recursive, as was shown in [2]. This paper investigates a variation of the open question via a different notion of effectiveness—almost decidable.A tree Tr will be a subset of ω<ω that is closed under predecessor. For elements f, g in ω<ω ∪ ωω, ƒ ⊲ g iffdf ∀i < lh(ƒ)[ƒ(i) = g(i)].

scholarly journals A VALUATION THEORETIC CHARACTERIZATION OF RECURSIVELY SATURATED REAL CLOSED FIELDS

2015 ◽  
Vol 80 (1) ◽  
pp. 194-206 ◽  
Author(s):  
PAOLA D’AQUINO ◽  
SALMA KUHLMANN ◽  
KAREN LANGE
Keyword(s):  
Abelian Group ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Theoretic Characterization ◽  
Image Position ◽  
Recursively Saturated

AbstractWe give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization for κ-saturation, for a cardinal $\kappa \ge \aleph _0 $. Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.

On definable subsets of p-adic fields

1976 ◽  
Vol 41 (3) ◽  
pp. 605-610 ◽  
Author(s):  
Angus MacIntyre
Keyword(s):  
Cross Section ◽  
Finite Union ◽  
Point Of View ◽  
Closed Field ◽  
Theoretical Point ◽  
First Order ◽  
Real Closed Fields ◽  
Closed Fields ◽  
Definable Sets ◽  
The Difference

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p-adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy.In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p-adic fields. We shall outline a new treatment of the p-adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p-adically closed fields.We want to describe the definable subsets of p-adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K). In particular, if X is infinite, X has nonempty interior.Now, there is an analogous question for p-adically closed fields. If K is p-adically closed, what are the definable subsets of K? To the best of our knowledge, this question has not been answered until now.What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p-adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

scholarly journals Grouplike minimal sets in ACFA AND in TA

2010 ◽  
Vol 75 (4) ◽  
pp. 1462-1488
Author(s):  
Alice Medvedev
Keyword(s):  
Algebraic Geometry ◽  
Algebraic Curve ◽  
Model Companion ◽  
Full Characterization ◽  
Closed Fields ◽  
Transcendental Theory ◽  
Phd Thesis ◽  
Algebraically Closed Fields ◽  
Injective Endomorphism

AbstractThis paper began as a generalization of a part of the author's PhD thesis about ACFA and ended up with a characterization of groups definable in TA. The thesis concerns minimal formulae of the form x Є A ∧ σ(x) = f(x) for an algebraic curve A and a dominant rational function f: A → σ(A). These are shown to be uniform in the Zilber trichotomy, and the pairs (A, f) that fall into each of the three cases are characterized. These characterizations are definable in families. This paper covers approximately half of the thesis, namely those parts of it which can be made purely model-theoretic by moving from ACFA, the model companion of the class of algebraically closed fields with an endomorphism, to TA, the model companion of the class of models of an arbitrary totally-transcendental theory T with an injective endomorphism, if this model-companion exists. A TA analog of the characterization of groups definable in ACFA is obtained in the process. The full characterization of the cases of the Zilber trichotomy in the thesis is obtained from these intermediate results with heavy use of algebraic geometry.

A very rapid in situ Kikuchi technique to determine relative crystal misorientation

1988 ◽  
Vol 46 ◽  
pp. 830-831
Author(s):  
J. I. Bennetch
Keyword(s):  
Electron Beam ◽  
Analytical Techniques ◽  
Superplastic Forming ◽  
The Other ◽  
Kikuchi Pattern ◽  
Kikuchi Line ◽  
Line Analysis

In a recent study of the superplastic forming (SPF) behavior of certain Al-Li-X alloys, the relative misorientation between adjacent (sub)grains proved to be an important parameter. It is well established that the most accurate way to determine misorientation across boundaries is by Kikuchi line analysis. However, the SPF study required the characterization of a large number of (sub)grains in each sample to be statistically meaningful, a very time-consuming task even for comparatively rapid Kikuchi analytical techniques.In order to circumvent this problem, an alternate, even more rapid in-situ Kikuchi technique was devised, eliminating the need for the developing of negatives and any subsequent measurements on photographic plates. All that is required is a double tilt low backlash goniometer capable of tilting ± 45° in one axis and ± 30° in the other axis. The procedure is as follows. While viewing the microscope screen, one merely tilts the specimen until a standard recognizable reference Kikuchi pattern is centered, making sure, at the same time, that the focused electron beam remains on the (sub)grain in question.

Rapid Isolation of High Molecular Weight Urokinase from Native Human Urine

1982 ◽  
Vol 47 (03) ◽  
pp. 197-202 ◽  
Author(s):  
Kurt Huber ◽  
Johannes Kirchheimer ◽  
Bernd R Binder
Keyword(s):  
Molecular Weight ◽  
High Molecular Weight ◽  
Human Urine ◽  
Gel Filtration ◽  
Chain Structure ◽  
The Other ◽  
Single Chain ◽  
Apparent Homogeneity ◽  
Sequential Adsorption

SummaryUrokinase (UK) could be purified to apparent homogeneity starting from crude urine by sequential adsorption and elution of the enzyme to gelatine-Sepharose and agmatine-Sepharose followed by gel filtration on Sephadex G-150. The purified product exhibited characteristics of the high molecular weight urokinase (HMW-UK) but did contain two distinct entities, one of which exhibited a two chain structure as reported for the HMW-UK while the other one exhibited an apparent single chain structure. The purification described is rapid and simple and results in an enzyme with probably no major alterations. Yields are high enough to obtain purified enzymes for characterization of UK from individual donors.

scholarly journals Thematic Maps for the Variation of Bearing Capacity of Soil Using SPTs and MATLAB

Geosciences ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 329
Author(s):  
Mahdi O. Karkush ◽  
Mahmood D. Ahmed ◽  
Ammar Abdul-Hassan Sheikha ◽  
Ayad Al-Rumaithi
Keyword(s):  
Bearing Capacity ◽  
Mean Squared Error ◽  
Ground Level ◽  
The Other ◽  
Thematic Maps ◽  
First Order ◽  
Squared Error ◽  
Order Polynomial ◽  
Interpolation Polynomials ◽  
Penetration Tests

The current study involves placing 135 boreholes drilled to a depth of 10 m below the existing ground level. Three standard penetration tests (SPT) are performed at depths of 1.5, 6, and 9.5 m for each borehole. To produce thematic maps with coordinates and depths for the bearing capacity variation of the soil, a numerical analysis was conducted using MATLAB software. Despite several-order interpolation polynomials being used to estimate the bearing capacity of soil, the first-order polynomial was the best among the other trials due to its simplicity and fast calculations. Additionally, the root mean squared error (RMSE) was almost the same for the all of the tried models. The results of the study can be summarized by the production of thematic maps showing the variation of the bearing capacity of the soil over the whole area of Al-Basrah city correlated with several depths. The bearing capacity of soil obtained from the suggested first-order polynomial matches well with those calculated from the results of SPTs with a deviation of ±30% at a 95% confidence interval.

Sign in / Sign up

Export Citation Format

Share Document