The classification of small weakly minimal sets. II

1988 ◽  
Vol 53 (2) ◽  
pp. 625-635 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Stable Theory ◽  
Minimal Set ◽  
Minimal Sets ◽  
Finite Set ◽  
Unidimensional Theory ◽  
The Universe ◽  
Countable Models

AbstractThe main result is Vaught's conjecture for weakly minimal, locally modular and non-ω-stable theories. The more general results yielding this are the following.Theorem A. Suppose that T is a small unidimensional theory and D is a weakly minimal set, definable over the finite set B. Then for all finite A ⊂ D there are only finitely many nonalgebraic strong types over B realized in acl(A) ∩ D.Theorem B. Suppose that T is a small, unidimensional, non-ω-stable theory such that the universe is weakly minimal and locally modular. Then for all finite A there is a finite B ⊂ cl(A) such that a ∈ cl(A) iff a ∈ cl(b) for some b ∈ B.Recall the property (S) defined in the abstract of [B1].Theorem C. Let T be as in Theorem B. Then, if T does not satisfy (S), T hasmany countable models.Combining Theorem C and the results in [B1] we obtain Vaught's conjecture for such theories.

On Cantor-Bendixson spectra containing (1,1). II

1985 ◽  
Vol 50 (3) ◽  
pp. 611-618 ◽  
Author(s):  
Annalisa Marcja ◽  
Carlo Toffalori
Keyword(s):  
Boolean Algebras ◽  
Stable Theory ◽  
Definable Subset ◽  
Limit Ordinal ◽  
Image Position ◽  
Countable Models

Let T be a (countable, complete, quantifier eliminable) ω-stable theory; an analysis of T, and consequently a classification of ω-stable theories, can be done by looking at the Boolean algebras B(M) of definable subsets of its countable models M (as usual, we often confuse a definable subset of M with the class of formulas defining it). If M ⊨ T, ∣M∣ = ℵ0, then, for every LM-formula ϕ(v) and for every ordinal α, we define a relation(CB = Cantor-Bendixson, of course) by induction on α:CB-rank ϕ(v) ≥ 0 if ϕ(M) ≠ ∅CB-rank ϕ(v) ≥ λ for λ a limit ordinal, if CB-rank ϕ(v) ≥ for all v < λ;CB-rank ϕ(v)≥ α + 1 if, for all n ∈ ω,(*) there are LM-formulas ϕ0(v), …, ϕn − 1(v) such thatIt is well known that the ω-stability of T implies that, for every consistent LM-formula ϕ(v), there is exactly one ordinal α < ω1 such that CB-rank ϕ(v) ≥ α and CB-rank ϕ(v)≱α + 1. Therefore we define:CB-rank ϕ(v) = αCB-degree ϕ(v) = d if d is the maximal n ∈ ω satisfying (*); andCB-type ϕ(v) = (α, d).

Omitting models

1977 ◽  
Vol 42 (1) ◽  
pp. 29-32
Author(s):  
Ernest Snapper
Keyword(s):  
Homogeneous Model ◽  
Countable Model ◽  
Complete Theory ◽  
Finite Set ◽  
Homogeneous Models ◽  
Definition Of ◽  
General Aspects ◽  
The Universe ◽  
Main Definition ◽  
Countable Models

The purpose of this paper is to introduce the notion of “omitting models” and to derive a very natural theorem concerning it (Theorem 1). A corollary of this theorem is the remarkable theorem of Vaught [3] which states that a countable complete theory cannot have precisely two nonisomorphic countable models. In fact, we show that our theorem implies Rosenstein's theorem [2] which, in turn, implies Vaught's theorem.T stands for a countable complete theory whose (countable) language is denoted by L. Following [1], a countably homogeneous model of T is a countable model of T with the property that, for any two n-tuples a1, …, an and b1,…,bn of the universe of whose types are the same, there is an automorphism of which maps ai, on bi, for i = 1, …, n [1, p. 129 and Proposition 3.2.9, p. 131]. “Homogeneous model” always means “countably homogeneous model.” “Type of T” always stands for “n-type of T” where n ≥ s 0, i.e., for the type of some n-tuple of individuals of the universe of some model of T. We often use that two homogeneous models which realize the same types are isomorphic [1, Proposition 3.2.9, p. 131].It is well known that every type of T is realized by at least one countable model of T. The main definition of this paper is:Definition 1. A set of countable models of T is omissible or “may be omitted” if every type of T is realized by at least one countable model of T which is not isomorphic to a model in the set.The main theorem of the paper is:Theorem 1. If a countable complete theory is not ω-categorical, every finite set of its homogeneous models may be omitted.The theorem is proved in §1 and in §2 it is shown how Vaught's and Rosenstein's theorems follow from it. §3 discusses some general aspects of omitting models.

Analysis of statistical coefficients and autoregressive parameters over intrinsic mode functions (IMFs) for epileptic seizure detection

2020 ◽  
Vol 65 (6) ◽  
pp. 693-704
Author(s):  
Rafik Djemili
Keyword(s):  
Involuntary Movement ◽  
Epileptic Seizures ◽  
Intrinsic Mode Functions ◽  
Eeg Signals ◽  
Time Segment ◽  
Feature Extraction Method ◽  
Mode Decomposition ◽  
Finite Set ◽  
Student’S T

AbstractEpilepsy is a persistent neurological disorder impacting over 50 million people around the world. It is characterized by repeated seizures defined as brief episodes of involuntary movement that might entail the human body. Electroencephalography (EEG) signals are usually used for the detection of epileptic seizures. This paper introduces a new feature extraction method for the classification of seizure and seizure-free EEG time segments. The proposed method relies on the empirical mode decomposition (EMD), statistics and autoregressive (AR) parameters. The EMD method decomposes an EEG time segment into a finite set of intrinsic mode functions (IMFs) from which statistical coefficients and autoregressive parameters are computed. Nevertheless, the calculated features could be of high dimension as the number of IMFs increases, the Student’s t-test and the Mann–Whitney U test were thus employed for features ranking in order to withdraw lower significant features. The obtained features have been used for the classification of seizure and seizure-free EEG signals by the application of a feed-forward multilayer perceptron neural network (MLPNN) classifier. Experimental results carried out on the EEG database provided by the University of Bonn, Germany, demonstrated the effectiveness of the proposed method which performance assessed by the classification accuracy (CA) is compared to other existing performances reported in the literature.

Minimal Sets on Graphs and Dendrites

2003 ◽  
Vol 13 (07) ◽  
pp. 1721-1725 ◽  
Author(s):  
Francisco Balibrea ◽  
Roman Hric ◽  
L'ubomír Snoha
Keyword(s):  
Topological Structure ◽  
Partial Characterization ◽  
Minimal Set ◽  
Full Characterization ◽  
Minimal Sets ◽  
Continuous Maps

The topological structure of minimal sets of continuous maps on graphs, dendrites and dendroids is studied. A full characterization of minimal sets on graphs and a partial characterization of minimal sets on dendrites are given. An example of a minimal set containing an interval on a dendroid is given.

Pseudoprojective strongly minimal sets are locally projective

1991 ◽  
Vol 56 (4) ◽  
pp. 1184-1194 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Predicate Symbol ◽  
Infinite Dimension ◽  
Elementary Extension ◽  
Morley Rank ◽  
Minimal Set ◽  
Unary Predicate ◽  
Minimal Sets

AbstractLet D be a strongly minimal set in the language L, and D′ ⊃ D an elementary extension with infinite dimension over D. Add to L a unary predicate symbol D and let T′ be the theory of the structure (D′, D), where D interprets the predicate D. It is known that T′ is ω-stable. We proveTheorem A. If D is not locally modular, then T′ has Morley rank ω.We say that a strongly minimal set D is pseudoprojective if it is nontrivial and there is a k < ω such that, for all a, b ∈ D and closed X ⊂ D, a ∈ cl(Xb) ⇒ there is a Y ⊂ X with a ∈ cl(Yb) and ∣Y∣ ≤ k. Using Theorem A, we proveTheorem B. If a strongly minimal set D is pseudoprojective, then D is locally projective.The following result of Hrushovski's (proved in §4) plays a part in the proof of Theorem B.Theorem C. Suppose that D is strongly minimal, and there is some proper elementary extension D1 of D such that the theory of the pair (D1, D) is ω1-categorical. Then D is locally modular.

One theorem of Zil′ber's on strongly minimal sets

1985 ◽  
Vol 50 (4) ◽  
pp. 1054-1061 ◽  
Author(s):  
Steven Buechler
Keyword(s):  
Main Part ◽  
Minimal Set ◽  
Minimal Sets ◽  
Image Position

AbstractSuppose D ⊂ M is a strongly minimal set definable in M with parameters from C. We say D is locally modular if for all X, Y ⊂ D, with X = acl(X ∪ C)∩D, Y = acl(Y ∪ C) ∩ D and X ∩ Y ≠ ∅,We prove the following theorems.Theorem 1. Suppose M is stable and D ⊂ M is strongly minimal. If D is not locally modular then inMeqthere is a definable pseudoplane.(For a discussion of Meq see [M, §A].) This is the main part of Theorem 1 of [Z2] and the trichotomy theorem of [Z3].Theorem 2. Suppose M is stable and D, D′ ⊂ M are strongly minimal and nonorthogonal. Then D is locally modular if and only if D′ is locally modular.

THE METAPHOR LONDON-AS-THE-WORLD AS A KEY METAPHOR IN THE STRUCTURE OF THE LONDON TEXT OF ENGLISH LINGUISTIC CULTURE

2020 ◽  
pp. 202-216
Author(s):  
Aleksei V. Sosnin ◽  
◽  
Yuliya V. Balakina ◽  
Keyword(s):  
Semantic Analysis ◽  
Deep Structure ◽  
Literary Canon ◽  
Semiotic Analysis ◽  
Linguistic Expression ◽  
The World ◽  
English Speaking ◽  
The Universe ◽  
Civilized World

The article examines the metaphor London-as-the-World in the structure of the London text of English linguistic culture (i.e., an emic or invariant text for a group of texts related to the British capital). Such an analysis makes it possible to update the most important dimension of the London text: its objects turns out to be a key component of Englishness, being conceptualized as a model of all-English and world processes, as an analogy of the civilized world and the universe. The metaphorical realizations of the London text are seen as the result of conceptual fusion. The research cited in the article is carried out at the junction of the cognitive and semiotic approaches, according to which socially significant mental entities are examined via a semantic analysis of corresponding supertexts. The integration of the cognitive and the semiotic is effected within the framework of unified semantics. Thereby a semiotic analysis of text consists in singling out propositions of diverse degrees of similarity in it, in the selection and classification of predicates with which characters and “things” are endowed in the text, and in the inclusion of individual entities from the text in the general categories, what reveals the picture of the world deep structure from the standpoint of that text. The article draws on the literary canon of New English, and a study into that material educes a continuity in the metaphors and the means of their linguistic expression that were used by the English-speaking community to structure the reality. The article thus postulates the relative stability of London text as a supertextual entity.

Eccentric Periodization: Comparative Perspectives on the Enlightenment and the Baroque

PMLA ◽  
2013 ◽  
Vol 128 (3) ◽  
pp. 690-697 ◽  
Author(s):  
Lois Parkinson Zamora
Keyword(s):  
United States ◽  
Jorge Luis Borges ◽  
The United States ◽  
Critical Discussion ◽  
Organizational Models ◽  
The Enlightenment ◽  
John Wilkins ◽  
Literature Curricula ◽  
The Universe

There is no classification of the universe that is not arbitrary and conjectural. … But the impossibility of penetrating the divine scheme of the universe cannot dissuade us from outlining human schemes, even though we are aware that they are provisional.—Jorge Luis Borges, “The Analytical Language of John Wilkins” (104)Our professional practice of grouping cultural products into historical categories has recently been the subject of lively critical discussion, as well as some consternation. Here I want to consider how, and how well, periodization organizes knowledge in the field of comparative literature. Organizing knowledge is what scholars do in all disciplines, of course, but the organizational models differ according to our objects of study. Historians may be the most dependent on schemata of periodization, but literary scholars aren't far behind. Literature curricula in United States universities are largely organized according to diachronic historical categories, whether they are labeled by centuries or by rubrics tied to a historical period's style or ideology or political circumstances. This is not surprising since European periodic categories long precede the establishment of curricula in the United States. Periods are powerful because they carry with them their own historical accumulations and applications, and they become dialectical as we engage their diverse cultural and historical meanings. For this reason, they can be particularly useful to comparatists. Indeed, to speak of any period at all is to make a comparative statement. One period necessarily implies others, each period a part that exists in relation to other parts and to an implied whole—a provisional “classification of the universe,” to quote from the passage by Jorge Luis Borges that I take as my epigraph.

scholarly journals COUNTABLE MODELS OF THE THEORIES OF BALDWIN–SHI HYPERGRAPHS AND THEIR REGULAR TYPES

2019 ◽  
Vol 84 (3) ◽  
pp. 1007-1019
Author(s):  
DANUL K. GUNATILLEKA
Keyword(s):  
Large Body ◽  
Countable Model ◽  
Stable Theory ◽  
Generic Structure ◽  
Elementary Chain ◽  
Regular Type ◽  
Image Position ◽  
Original Class ◽  
Countable Models

AbstractWe continue the study of the theories of Baldwin–Shi hypergraphs from [5]. Restricting our attention to when the rank δ is rational valued, we show that each countable model of the theory of a given Baldwin–Shi hypergraph is isomorphic to a generic structure built from some suitable subclass of the original class used in the construction. We introduce a notion of dimension for a model and show that there is a an elementary chain $\left\{ {\mathfrak{M}_\beta :\beta \leqslant \omega } \right\}$ of countable models of the theory of a fixed Baldwin–Shi hypergraph with $\mathfrak{M}_\beta \preccurlyeq \mathfrak{M}_\gamma $ if and only if the dimension of $\mathfrak{M}_\beta $ is at most the dimension of $\mathfrak{M}_\gamma $ and that each countable model is isomorphic to some $\mathfrak{M}_\beta $. We also study the regular types that appear in these theories and show that the dimension of a model is determined by a particular regular type. Further, drawing on a large body of work, we use these structures to give an example of a pseudofinite, ω-stable theory with a nonlocally modular regular type, answering a question of Pillay in [11].

scholarly journals On Primitive Extensions of Rank 3 of Symmetric Groups

1966 ◽  
Vol 27 (1) ◽  
pp. 171-177 ◽  
Author(s):  
Tosiro Tsuzuku
Keyword(s):  
Permutation Group ◽  
Symmetric Groups ◽  
Transitive Group ◽  
Permutation Groups ◽  
Minimal Set ◽  
Finite Set ◽  
Rank 2 ◽  
One To One

1. Let Ω be a finite set of arbitrary elements and let (G, Ω) be a permutation group on Ω. (This is also simply denoted by G). Two permutation groups (G, Ω) and (G, Γ) are called isomorphic if there exist an isomorphism σ of G onto H and a one to one mapping τ of Ω onto Γ such that (g(i))τ=gσ(iτ) for g ∊ G and i∊Ω. For a subset Δ of Ω, those elements of G which leave each point of Δ individually fixed form a subgroup GΔ of G which is called a stabilizer of Δ. A subset Γ of Ω is called an orbit of GΔ if Γ is a minimal set on which each element of G induces a permutation. A permutation group (G, Ω) is called a group of rank n if G is transitive on Ω and the number of orbits of a stabilizer Ga of a ∊ Ω, is n. A group of rank 2 is nothing but a doubly transitive group and there exist a few results on structure of groups of rank 3 (cf. H. Wielandt [6], D. G. Higman M).

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