76,085 research outputs found
Strong splitting in stable homogeneous models
We study elementary submodels of a stable homogeneous structure. We improve
the independence relation defined in [T. Hyttinen, On nonstructure of
elementary submodels of a stable homogeneous structure, Fundamenta
Mathematicae, 156(1998): 167-182]. We apply this to prove a structure theorem.
We also show that dop and sdop are essentially equivalent, where the negation
of dop is the property we use in our structure theorem and sdop implies
nonstructure
Shelah's Categoricity Conjecture from a successor for Tame Abstract Elementary Classes
Let K be an Abstract Elemenetary Class satisfying the amalgamation and the
joint embedding property, let \mu be the Hanf number of K. Suppose K is tame.
MAIN COROLLARY: (ZFC) If K is categorical in a successor cardinal bigger than
\beth_{(2^\mu)^+} then K is categorical in all cardinals greater than
\beth_{(2^\mu)^+}.
This is an improvment of a Theorem of Makkai and Shelah ([Sh285] who used a
strongly compact cardinal for the same conclusion) and Shelah's downward
categoricity theorem for AECs with amalgamation (from [Sh394]).Comment: 19 page
Limit Models in Strictly Stable Abstract Elementary Classes
In this paper, we examine the locality condition for non-splitting and
determine the level of uniqueness of limit models that can be recovered in some
stable, but not superstable, abstract elementary classes. In particular we
prove:
Suppose that is an abstract elementary class satisfying
1. the joint embedding and amalgamation properties with no maximal model of
cardinality .
2. stabilty in .
3. .
4. continuity for non--splitting (i.e. if and is a
limit model witnessed by for some limit
ordinal and there exists so that does
not -split over for all , then does not -split over
).
For and limit ordinals both with cofinality , if satisfies symmetry for non--splitting (or just
-symmetry), then, for any and that are
and -limit models over , respectively, we have that
and are isomorphic over .Comment: This article generalizes some results from arXiv:1507.0199
Discrete models of force chain networks
A fundamental property of any material is its response to a localized stress
applied at a boundary. For granular materials consisting of hard, cohesionless
particles, not even the general form of the stress response is known. Directed
force chain networks (DFCNs) provide a theoretical framework for addressing
this issue, and analysis of simplified DFCN models reveal both rich
mathematical structure and surprising properties. We review some basic elements
of DFCN models and present a class of homogeneous solutions for cases in which
force chains are restricted to lie on a discrete set of directions.Comment: 17 pages, 6 figures, dcds-B.cls; Minor corrections to version 2, but
including an important factor of 2; Submitted to Discrete and Continuous
Dynamical Systems B for special issue honoring David Schaeffe
Reaction-diffusion models for biological pattern formation
We consider the use of reaction-diffusion equations to model biological pattern formation and describe the derivation of the reaction-terms for several illustrative examples. After a brief discussion of the Turing instability in such systems we extend the model formulation to incorporate domain growth. Comparisons are drawn between solution behaviour on growing domains and recent results on self-replicating patterns on domains of fixed size
- …