Independence, dimension and continuity in non-forking frames

2013 ◽  
Vol 78 (2) ◽  
pp. 602-632 ◽  
Author(s):  
Adi Jarden ◽  
Alon Sitton
Keyword(s):  
Continuity Property ◽  
Natural Conditions ◽  
Elementary Class ◽  
Independence Dimension ◽  
The Stability ◽  
Abstract Elementary Class

AbstractThe notion J is independent in (M, M0, N) was established by Shelah, for an AEC (abstract elementary class) which is stable in some cardinal λ and has a non-forking relation, satisfying the good λ-frame axioms and some additional hypotheses. Shelah uses independence to define dimension.Here, we show the connection between the continuity property and dimension: if a non-forking satisfies natural conditions and the continuity property, then the dimension is well-behaved.As a corollary, we weaken the stability hypothesis and two additional hypotheses, that appear in Shelah's theorem.

scholarly journals N⊥ as an abstract elementary class

2007 ◽  
Vol 149 (1-3) ◽  
pp. 25-39 ◽  
Author(s):  
John T. Baldwin ◽  
Paul C. Eklof ◽  
Jan Trlifaj
Keyword(s):  
Elementary Class ◽  
Abstract Elementary Class

AN ABSTRACT ELEMENTARY CLASS NONAXIOMATIZABLE IN

2019 ◽  
Vol 84 (3) ◽  
pp. 1240-1251
Author(s):  
SIMON HENRY
Keyword(s):  
Regular Cardinal ◽  
Vector Spaces ◽  
Left Adjoint ◽  
Scott Topology ◽  
Elementary Class ◽  
Closed Fields ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Bounded Below ◽  
Abstract Elementary Class

AbstractWe show that for any uncountable cardinal λ, the category of sets of cardinality at least λ and monomorphisms between them cannot appear as the category of points of a topos, in particular is not the category of models of a ${L_{\infty ,\omega }}$-theory. More generally we show that for any regular cardinal $\kappa < \lambda$ it is neither the category of κ-points of a κ-topos, in particular, nor the category of models of a ${L_{\infty ,\kappa }}$-theory.The proof relies on the construction of a categorified version of the Scott topology, which constitute a left adjoint to the functor sending any topos to its category of points and the computation of this left adjoint evaluated on the category of sets of cardinality at least λ and monomorphisms between them. The same techniques also apply to a few other categories. At least to the category of vector spaces of with bounded below dimension and the category of algebraic closed fields of fixed characteristic with bounded below transcendence degree.

scholarly journals Shelah's categoricity conjecture from a successor for tame abstract elementary classes

2006 ◽  
Vol 71 (2) ◽  
pp. 553-568 ◽  
Author(s):  
Rami Grossberg ◽  
Monica Vandieren
Keyword(s):  
Abstract Elementary Classes ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Joint Embedding ◽  
Elementary Classes ◽  
Abstract Elementary Class

AbstractWe prove a categoricity transfer theorem for tame abstract elementary classes.Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ, LS(K+}. If K is categorical in λ and λ+, then K is categorical in λ++.Combining this theorem with some results from [37]. we derive a form of Shelah's Categoricity Conjecture for tame abstract elementary classes:Suppose K is χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 ≔ Hanf(K). Ifand K is categorical in somethen K is categorical in μ for all μ .

scholarly journals FORKING AND SUPERSTABILITY IN TAME AECS

2016 ◽  
Vol 81 (1) ◽  
pp. 357-383 ◽  
Author(s):  
SEBASTIEN VASEY
Keyword(s):  
General Construction ◽  
Transfer Theorem ◽  
Elementary Class ◽  
Abstract Elementary Class ◽  
Limit Models

AbstractWe prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.

Soil-Pile Coupling Analysis of Steel Sheet Pile Bulkhead under Horizontal Loads

2015 ◽  
Vol 744-746 ◽  
pp. 1180-1183 ◽  
Author(s):  
Hong Wang ◽  
Peng Liu ◽  
Xiang Liu ◽  
Jin Gang Duan
Keyword(s):  
Large Scale ◽  
Steel Sheet ◽  
Academic Support ◽  
Load Rating ◽  
Sheet Pile ◽  
Natural Conditions ◽  
Coupling Analysis ◽  
Improve Design ◽  
The Stability ◽  
Design And Construction

With the development of steel sheet pile bulkhead gradually toward large-scale and deep water, natural conditions become worse and structural load rating continues to improve, design and construction of steel sheet pile bulkhead needs higher requirements. Then the stability of sheet pile bulkhead under horizontal loads becomes particularly prominent. This paper presents the displacement and stress of steel sheet pile bulkhead structure under different horizontal loads using ANSYS, which provide academic support for the design and construction of steel sheet pile bulkhead.

scholarly journals ALMOST GALOIS ω-STABLE CLASSES

2015 ◽  
Vol 80 (3) ◽  
pp. 763-784 ◽  
Author(s):  
JOHN T. BALDWIN ◽  
PAUL B. LARSON ◽  
SAHARON SHELAH
Keyword(s):  
Elementary Class ◽  
Joint Embedding ◽  
Image Position ◽  
Abstract Elementary Class ◽  
Countable Models

AbstractTheorem. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presentable abstract elementary class with Löwenheim–Skolem number ℵ0, satisfying the joint embedding and amalgamation properties in ℵ0. If K has only countably many models in ℵ1, then all are small. If, in addition, k is almost Galois ω-stable then k is Galois ω-stable. Suppose that k = (K, $$\prec_k$$) is an ℵ0-presented almost Galois ω-stable AEC satisfying amalgamation for countable models, and having a model of cardinality ℵ1. The assertion that K is ℵ1-categorical is then absolute.

Upward categoricity from a successor cardinal for tame abstract classes with amalgamation

2005 ◽  
Vol 70 (2) ◽  
pp. 639-660 ◽  
Author(s):  
Olivier Lessmann
Keyword(s):  
Elementary Class ◽  
Uncountable Cardinal ◽  
Abstract Elementary Class

AbstractThis paper is devoted to the proof of the following upward categoricity theorem: Let be a tame abstract elementary class with amalgamation, arbitrarily large models, and countable Löwenheim-Skolem number. If is categorical in ℵ then is categorical in every uncountable cardinal. More generally, wc prove that if is categorical in a successor cardinal λ+ then is categorical everywhere above λ+.

scholarly journals Notes on Quasiminimality and Excellence

2004 ◽  
Vol 10 (3) ◽  
pp. 334-366 ◽  
Author(s):  
John T. Baldwin
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Special Kind ◽  
Order Logic ◽  
First Order Logic ◽  
Structure Theory ◽  
Mathematical Structures ◽  
First Order ◽  
Elementary Class ◽  
Abstract Elementary Class

AbstractThis paper ties together much of the model theory of the last 50 years. Shelah's attempts to generalize the Morley theorem beyond first order logic led to the notion of excellence, which is a key to the structure theory of uncountable models. The notion of Abstract Elementary Class arose naturally in attempting to prove the categoricity theorem for Lω1,ω(Q). More recently, Zilber has attempted to identify canonical mathematical structures as those whose theory (in an appropriate logic) is categorical in all powers. Zilber's trichotomy conjecture for first order categorical structures was refuted by Hrushovski, by the introducion of a special kind of Abstract Elementary Class. Zilber uses a powerful and essentailly infinitary variant on these techniques to investigate complex exponentiation. This not only demonstrates the relevance of Shelah's model theoretic investigations to mainstream mathematics but produces new results and conjectures in algebraic geometry.

scholarly journals Collective behaviour is not robust to disturbance, yet parent and offspring colonies resemble each other in social spiders

2019 ◽  
Author(s):  
David N. Fisher ◽  
James L.L. Lichtenstein ◽  
Raul Costa-Pereira ◽  
Justin Yeager ◽  
Jonathan N. Pruitt
Keyword(s):  
Initial Conditions ◽  
Collective Behaviour ◽  
Group Level ◽  
Natural Conditions ◽  
Social Spider ◽  
The Social ◽  
Parent Colony ◽  
Group Members ◽  
Collective Foraging ◽  
The Stability

AbstractGroups of animals possess phenotypes such as collective behaviour, which may determine the fitness of group members. However, the stability and robustness to perturbations of collective phenotypes in natural conditions is not established. Furthermore, whether group phenotypes are transmitted from parent to offspring groups is required for understanding how selection on group phenotypes contributes to evolution, but parent-offspring resemblance at the group level is rarely estimated. We evaluated robustness to perturbation and parent-offspring resemblance of collective foraging aggressiveness in colonies of the social spider Anelosimus eximius. Among-colony differences in foraging aggressiveness were consistent over time but changed if the colony was perturbed through the removal of individuals, or via their removal and subsequent return. Offspring and parent colony behaviour were correlated, but only once the offspring colony had settled after being translocated. The parent-offspring resemblance was not driven by a shared elevation but could be due to other environmental factors. Laboratory collective behaviour was not correlated with behaviour in the field. Colony aggression seems sensitive to initial conditions and easily perturbed between behavioural states. Despite this sensitivity, offspring colonies have collective behaviour that resembles that of their parent colony, provided they are given enough time to settle into the environment.

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