Countable models of nonmultidimensional ℵ0-stable theories

1983 ◽  
Vol 48 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Elisabeth Bouscaren ◽  
Daniel Lascar
Keyword(s):  
Finite Number ◽  
Homogeneous Model ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Stable Theory ◽  
Strong Type ◽  
First Case ◽  
Skolem Functions ◽  
Finite Sequences ◽  
Homogeneous Models

In this paper T will always be a countable ℵ0-stable theory, and in this introduction a model of T will mean a countable model.One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M, if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E, definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L1. In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize.The motivation of this research is to distinguish two classes of ℵ0-Stable theories:(1) theories such that all models are almost homogeneous;(2) theories with 2ℵ0 nonalmost homogeneous models.The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.

scholarly journals On normal subgroups of direct products

1990 ◽  
Vol 33 (2) ◽  
pp. 309-319 ◽  
Author(s):  
F. E. A. Johnson
Keyword(s):  
Automorphism Group ◽  
Representation Theory ◽  
Finite Number ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Direct Products ◽  
Normal Subgroups ◽  
Free Abelian Groups ◽  
Subdirect Products ◽  
Number Of Classes

We investigate the equivalence classes of normal subdirect products of a product of free groups Fn1 × … × Fnk under the simultaneous equivalence relations of commensurability and conjugacy under the full automorphism group. By abelianisation, the problem is reduced to one in the representation theory of quivers of free abelian groups. We show there are infinitely many such classes when k≧3, and list the finite number of classes when k = 2.

A Fast 3-D Visualization Methodology Using Characteristic Views of Objects

1998 ◽  
Vol 08 (01) ◽  
pp. 97-114 ◽  
Author(s):  
Soochan Hwang ◽  
Sang-Young Cho ◽  
Taehyung Wang ◽  
Phillip C.-Y. Sheu
Keyword(s):  
Finite Number ◽  
Real Time ◽  
Time Complexity ◽  
Object Oriented ◽  
Equivalence Classes ◽  
Visualization Method ◽  
Tree Algorithm ◽  
Projected Image ◽  
Characteristic Views ◽  
Run Time

This paper describes a 3-D visualization method based on the concept of characteristic views (CVs). The idea of characteristic views was derived based on the observation that the infinite possible views of a 3-D object can be grouped into a finite number of equivalence classes so that within each class all the views are isomorphic in the sense that they have the same line-junction graphs. To visualize the changes of scenes in real time, the BSP tree algorithm is known to be efficient in a static environment in which the viewpoint can be changed easily. However, if a scene consists of many objects and each object consists of many polygons, the time complexity involved in traversing a BSP tree increases rapidly so that the original BSP tree algorithm may not be efficient. The method proposed in this paper is object-oriented in the sense that, for all viewpoints, at the preprocessing stage the ordering for displaying the objects is determined. At run time, the objects are displayed based on a pre-calculated ordering according to the viewpoint. In addition, a CV is used as a basic 2-D projected image of a 3-D object.

scholarly journals On linear independence for integer translates of a finite number of functions

1993 ◽  
Vol 36 (1) ◽  
pp. 69-85 ◽  
Author(s):  
Rong-Qing Jia ◽  
Charles A. Micchelli
Keyword(s):  
Finite Number ◽  
Linear Combination ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Basis Functions ◽  
Partial Difference ◽  
Integer Translates ◽  
First Case ◽  
Compactly Supported Functions ◽  
Necessary And Sufficient

We investigate linear independence of integer translates of a finite number of compactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some lp space (1 ≦ p ≦ ∞) and we are interested in bounding their lp-norms in terms of the Lp-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

scholarly journals Intima heterogeneity in stress assessment of atherosclerotic plaques

2017 ◽  
Vol 8 (1) ◽  
pp. 20170008 ◽  
Author(s):  
Ali C. Akyildiz ◽  
Lambert Speelman ◽  
Bas van Velzen ◽  
Raoul R. F. Stevens ◽  
Antonius F. W. van der Steen ◽  
...  
Keyword(s):  
Sensitivity Analysis ◽  
Plaque Rupture ◽  
Homogeneous Model ◽  
Sensitivity Analyses ◽  
Homogeneous Material ◽  
Sensitivity Index ◽  
Stress Assessment ◽  
Heterogeneous Models ◽  
Homogeneous Models ◽  
Accuracy Stress

Atherosclerotic plaque rupture is recognized as the primary cause of cardiac and cerebral ischaemic events. High structural plaque stresses have been shown to strongly correlate with plaque rupture. Plaque stresses can be computed with finite-element (FE) models. Current FE models employ homogeneous material properties for the heterogeneous atherosclerotic intima. This study aimed to evaluate the influence of intima heterogeneity on plaque stress computations. Two-dimensional FE models with homogeneous and heterogeneous intima were constructed from histological images of atherosclerotic human coronaries ( n = 12). For homogeneous models, a single stiffness value was employed for the entire intima. For heterogeneous models, the intima was subdivided into four clusters based on the histological information and different stiffness values were assigned to the clusters. To cover the reported local intima stiffness range, 100 cluster stiffness combinations were simulated. Peak cap stresses (PCSs) from the homogeneous and heterogeneous models were analysed and compared. By using a global variance-based sensitivity analysis, the influence of the cluster stiffnesses on the PCS variation in the heterogeneous intima models was determined. Per plaque, the median PCS values of the heterogeneous models ranged from 27 to 160 kPa, and the PCS range varied between 43 and 218 kPa. On average, the homogeneous model PCS values differed from the median PCS values of heterogeneous models by 14%. A positive correlation ( R 2 = 0.72) was found between the homogeneous model PCS and the PCS range of the heterogeneous models. Sensitivity analysis showed that the highest main sensitivity index per plaque ranged from 0.26 to 0.83, and the average was 0.47. Intima heterogeneity resulted in substantial changes in PCS, warranting stress analyses with heterogeneous intima properties for plaque-specific, high accuracy stress assessment. Yet, computations with homogeneous intima assumption are still valuable to perform sensitivity analyses or parametric studies for testing the effect of plaque geometry on PCS. Moreover, homogeneous intima models can help identify low PCS, stable type plaques with thick caps. Yet, for thin cap plaques, accurate stiffness measurements of the clusters in the cap and stress analysis with heterogeneous cap properties are required to characterize the plaque stability.

scholarly journals Equivalence Projective Simulation as a Framework for Modeling Formation of Stimulus Equivalence Classes

2020 ◽  
Vol 32 (5) ◽  
pp. 912-968 ◽  
Author(s):  
Asieh Abolpour Mofrad ◽  
Anis Yazidi ◽  
Hugo L. Hammer ◽  
Erik Arntzen
Keyword(s):  
Stimulus Equivalence ◽  
Human Subjects ◽  
Network Models ◽  
Equivalence Classes ◽  
Equivalence Relations ◽  
Learning Context ◽  
Neural Network Models ◽  
Equivalence Theory ◽  
Hypothetical Experiment ◽  
Theory Research

Stimulus equivalence (SE) and projective simulation (PS) study complex behavior, the former in human subjects and the latter in artificial agents. We apply the PS learning framework for modeling the formation of equivalence classes. For this purpose, we first modify the PS model to accommodate imitating the emergence of equivalence relations. Later, we formulate the SE formation through the matching-to-sample (MTS) procedure. The proposed version of PS model, called the equivalence projective simulation (EPS) model, is able to act within a varying action set and derive new relations without receiving feedback from the environment. To the best of our knowledge, it is the first time that the field of equivalence theory in behavior analysis has been linked to an artificial agent in a machine learning context. This model has many advantages over existing neural network models. Briefly, our EPS model is not a black box model, but rather a model with the capability of easy interpretation and flexibility for further modifications. To validate the model, some experimental results performed by prominent behavior analysts are simulated. The results confirm that the EPS model is able to reliably simulate and replicate the same behavior as real experiments in various settings, including formation of equivalence relations in typical participants, nonformation of equivalence relations in language-disabled children, and nodal effect in a linear series with nodal distance five. Moreover, through a hypothetical experiment, we discuss the possibility of applying EPS in further equivalence theory research.

scholarly journals Homogeneous models and almost decidability

1989 ◽  
Vol 46 (3) ◽  
pp. 343-355 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Elementary Theory ◽  
Homogeneous Model ◽  
Point Of View ◽  
Type Spectrum ◽  
Homogeneous Models ◽  
Theoretic Point

AbstractCountable homogeneous models are ‘simple’ objects from a model theoretic point of view. From a recursion theoretic point of view they can be complex. For instance the elementary theory of such a model might be undecidable, or the set of complete types might be recursively complex. Unfortunately even if neither of these conditions holds, such a model still can be undecidable. This paper investigates countable homogeneous models with respect to a weaker notion of decidability called almost decidable. It is shown that for theories that have only countably many type spectra, any countable homogeneous model of such a theory that has a Σ2 type spectrum is almost decidable.

CODING PARTITIONS OF REGULAR SETS

2009 ◽  
Vol 19 (08) ◽  
pp. 1011-1023 ◽  
Author(s):  
MARIE-PIERRE BÉAL ◽  
FABIO BURDERI ◽  
ANTONIO RESTIVO
Keyword(s):  
Finite Number ◽  
Free Product ◽  
Canonical Decomposition ◽  
Minimal Length ◽  
Regular Monoid ◽  
Finite Sequences ◽  
Regular Sets ◽  
Regular Classes

A coding partition of a set of words partitions this set into classes such that whenever a sequence, of minimal length, has two distinct factorizations, the words of these factorizations belong to the same class. The canonical coding partition is the finest coding partition that partitions the set of words in at most one unambiguous class and other classes that localize the ambiguities in the factorizations of finite sequences. We prove that the canonical coding partition of a regular set contains a finite number of regular classes and we give an algorithm for computing this partition. From this we derive a canonical decomposition of a regular monoid into a free product of finitely many regular monoids.

scholarly journals Effects of Collagen Heterogeneity on Myocardial Infarct Mechanics in a Multiscale Fiber Network Model

2019 ◽  
Vol 141 (9) ◽  
Author(s):  
Christopher E. Korenczuk ◽  
Victor H. Barocas ◽  
William J. Richardson
Keyword(s):  
Myocardial Infarction ◽  
Finite Element ◽  
Homogeneous Model ◽  
Mechanical Anisotropy ◽  
Local Alignment ◽  
Fiber Density ◽  
Heterogeneous Model ◽  
Fiber Alignment ◽  
Sample Shape ◽  
Homogeneous Models

The scar that forms after a myocardial infarction is often characterized by a highly disordered architecture but generally exhibits some degree of collagen fiber orientation, with a resulting mechanical anisotropy. When viewed in finer detail, however, the heterogeneity of the sample is clear, with different subregions exhibiting different fiber orientations. In this work, we used a multiscale finite element model to explore the consequences of the heterogeneity in terms of mechanical behavior. To do so, we used previously obtained fiber alignment maps of rat myocardial scar slices (n = 15) to generate scar-specific finite element meshes that were populated with fiber models based on the local alignment state. These models were then compared to isotropic models with the same sample shape and fiber density, and to homogeneous models with the same sample shape, fiber density, and average fiber alignment as the scar-specific models. All simulations involved equibiaxial extension of the sample with free motion in the third dimension. We found that heterogeneity led to a lower degree of mechanical anisotropy and a higher level of local stress concentration than the corresponding homogeneous model, and also that fibers failed in the heterogeneous model at much lower macroscopic strains than in the isotropic and homogeneous models. Taken together, these results suggest that scar heterogeneity may impair myocardial mechanical function both in terms of anisotropy and strength, and that individual variations in scar heterogeneity could be an important consideration for understanding scar remodeling and designing therapeutic interventions for patients after myocardial infarction.

Strongly 2-dimensional theories

1988 ◽  
Vol 53 (3) ◽  
pp. 931-936
Author(s):  
Akito Tsuboi
Keyword(s):  
Simple Structure ◽  
Stable Theory ◽  
Morley Rank ◽  
Dimensional Theory ◽  
Finite Sequences ◽  
Strongly Regular ◽  
Definition Of ◽  
Condition B

In [8], we have shown the equivalence of almost strong minimality and strong unidimensionality. More precisely, we proved:Theorem [8]. Let T be a countable stable theory. Then the following two conditions are equivalent:(i) T is almost strongly minimal;(ii) T can be extended to a theory such that any two nonalgebraic types are not almost orthogonal.In the present paper, we define the notion of strong 2-dimensionality (of T). We show that if T is strongly 2-dimensional then T is ω-stable and its model has a simple structure. Roughly speaking, in a model of a strongly 2-dimensional theory, one of the following holds: (a) every element is in acl (δi is strongly minimal), or (b) every element is in acl (δ is strongly regular). Shelah's definition of 2-dimensionality does not imply even superstability. (See Exercise 5.5 in [6, Chapter V, §5].) We show also that condition (a) above implies strong 2-dimensionality of T. However condition (b) does not imply strong 2-dimensionality in general.Our notations and conventions are standard. T is always countable and stable. We work in . A,B,… are used to denote small subsets of . , … are used to denote finite sequences of elements in . δ, φ,… are used to denote formulas (with parameters), p, q, … are used to denote types (with parameters). The fact that p is a nonforking (forking) extension of q is denoted by p ⊃nfq(p ⊃fq). If p is stationary, p∣A denotes the type in S(A) which is parallel to p. (or ) denotes the set of realizations of p (or δ). The Morley rank of p is denoted by RM(p).

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