On Dedekind complete o-minimal structures

1987 ◽  
Vol 52 (1) ◽  
pp. 156-164
Author(s):  
Anand Pillay ◽  
Charles Steinhorn
Keyword(s):  
Complete Model ◽  
Boolean Combination ◽  
Minimal Theory

AbstractFor a countable complete o-minimal theory T, we introduce the notion of a sequentially complete model of T. We show that a model of T is sequentially complete if and only if ≺ for some Dedekind complete model . We also prove that if T has a Dedekind complete model of power greater than , then T has Dedekind complete models of arbitrarily large powers. Lastly, we show that a dyadic theory—namely, a theory relative to which every formula is equivalent to a Boolean combination of formulas in two variables—that has some Dedekind complete model has Dedekind complete models in arbitrarily large powers.

Definability of types, and pairs of O-minimal structures

1994 ◽  
Vol 59 (4) ◽  
pp. 1400-1409 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Elementary Proof ◽  
Elementary Extension ◽  
Complete Model ◽  
Unary Predicate ◽  
Minimal Theory ◽  
Simple Set

AbstractLet T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L* be L together with a unary predicate P. Let T* be the L*-theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an ⅼMⅼ+-saturated elementary extension of N (and M is the interpretation of P). Using the definability of types result, we show that T* is complete and we give a simple set of axioms for T*. We also show that for every L*-formula ϕ(x) there is an L-formula ψ(x) such that T* ⊢ (∀x)(P(x) → (ϕ(x) ↔ ψ(x)). This yields the following result:Let M be a Dedekind complete model of T. Let ϕ(x, y) be an L-formula where l(y) – k. Let X = {X ⊂ Mk: for some a in an elementary extension N of M, X = ϕ(a, y)N ∩ Mk}. Then there is a formula ψ(y, z) of L such that X = {ψ(y, b)M: b in M}.

Dedekind Completeness and the Algebraic Complexity of o-Minimal Structures

1992 ◽  
Vol 44 (4) ◽  
pp. 843-855 ◽  
Author(s):  
Alan Mekler ◽  
Matatyahu Rubin ◽  
Charles Steinhorn
Keyword(s):  
Algebraic Complexity ◽  
Elementary Extension ◽  
Ordered Structure ◽  
Complete Model ◽  
Boolean Combination ◽  
Free Variables ◽  
Definable Subset ◽  
Dedekind Completeness

AbstractAn ordered structure is o-minimal if every definable subset is the union of finitely many points and open intervals. A theory is o-minimal if all its models are ominimal. All theories considered will be o-minimal. A theory is said to be n-ary if every formula is equivalent to a Boolean combination of formulas in n free variables. (A 2-ary theory is called binary.) We prove that if a theory is not binary then it is not rc-ary for any n. We also characterize the binary theories which have a Dedekind complete model and those whose underlying set order is dense. In [5], it is shown that if T is a binary theory, is a Dedekind complete model of T, and I is an interval in , then for all cardinals K there is a Dedekind complete elementary extension of , so that . In contrast, we show that if T is not binary and is a Dedekind complete model of T, then there is an interval I in so that if is a Dedekind complete elementary extension of .

scholarly journals Generic derivations on o-minimal structures

2020 ◽  
pp. 2150007
Author(s):  
Antongiulio Fornasiero ◽  
Elliot Kaplan
Keyword(s):  
Complete Model ◽  
Minimal Theory ◽  
Elementary Formula ◽  
Ordered Fields ◽  
Differential Fields ◽  
Model Completion ◽  
Open Core ◽  
Theory Formula

Let [Formula: see text] be a complete, model complete o-minimal theory extending the theory [Formula: see text] of real closed ordered fields in some appropriate language [Formula: see text]. We study derivations [Formula: see text] on models [Formula: see text]. We introduce the notion of a [Formula: see text]-derivation: a derivation which is compatible with the [Formula: see text]-definable [Formula: see text]-functions on [Formula: see text]. We show that the theory of [Formula: see text]-models with a [Formula: see text]-derivation has a model completion [Formula: see text]. The derivation in models [Formula: see text] behaves “generically”, it is wildly discontinuous and its kernel is a dense elementary [Formula: see text]-substructure of [Formula: see text]. If [Formula: see text], then [Formula: see text] is the theory of closed ordered differential fields (CODFs) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that [Formula: see text] has [Formula: see text] as its open core, that [Formula: see text] is distal, and that [Formula: see text] eliminates imaginaries. We also show that the theory of [Formula: see text]-models with finitely many commuting [Formula: see text]-derivations has a model completion.

scholarly journals Verification and Strategy Synthesis for Coalition Announcement Logic

2021 ◽  
Author(s):  
Natasha Alechina ◽  
Hans van Ditmarsch ◽  
Rustam Galimullin ◽  
Tuo Wang
Keyword(s):  
Model Checking ◽  
Proof Of Concept ◽  
Model Checker ◽  
Complete Model ◽  
Concept Model ◽  
The Family ◽  
Winning Strategies ◽  
Epistemic Goals

AbstractCoalition announcement logic (CAL) is one of the family of the logics of quantified announcements. It allows us to reason about what a coalition of agents can achieve by making announcements in the setting where the anti-coalition may have an announcement of their own to preclude the former from reaching its epistemic goals. In this paper, we describe a PSPACE-complete model checking algorithm for CAL that produces winning strategies for coalitions. The algorithm is implemented in a proof-of-concept model checker.

On flexible link manipulators: modeling and analysis using the algebra of rotations

Robotica ◽  
1994 ◽  
Vol 12 (4) ◽  
pp. 371-382 ◽  
Author(s):  
F. Xi ◽  
R.G. Fenton
Keyword(s):  
Nonlinear Coupling ◽  
Flexible Link ◽  
Complete Model ◽  
Modeling And Analysis ◽  
Displacement Equation

SUMMARYIn this paper, a complete model of the elasto-kinematics is formulated in terms of a new kinematic notation, called the algebra of rotations. Based on this formulation, the elegant and concise expressions are derived for the displacement equation and especially the Jacobians governing the motion mapping between the manipulator tip and joint variables as well as link deflections. Introduction of the elasto-kinematics into the elasto-dynamics can directly take into consideration the nonlinear coupling between joint variables and link deflections, and thus improve the result of the elasto-dynamics.

Analysis of SiC CVD Growth in a Horizontal Hot-Wall Reactor by Experiment and 3D Modelling

2007 ◽  
Vol 556-557 ◽  
pp. 61-64
Author(s):  
Y. Shishkin ◽  
Rachael L. Myers-Ward ◽  
Stephen E. Saddow ◽  
Alexander Galyukov ◽  
A.N. Vorob'ev ◽  
...  
Keyword(s):  
Growth Rate ◽  
Growth Process ◽  
Three Dimensional ◽  
Chemical Species ◽  
3D Modelling ◽  
Complete Model ◽  
Cvd Growth ◽  
Dimensional Simulation ◽  
Insight Into ◽  
Hot Zone

A fully-comprehensive three-dimensional simulation of a CVD epitaxial growth process has been undertaken and is reported here. Based on a previously developed simulation platform, which connects fluid dynamics and thermal temperature profiling with chemical species kinetics, a complete model of the reaction process in a low pressure hot-wall CVD reactor has been developed. Close agreement between the growth rate observed experimentally and simulated theoretically has been achieved. Such an approach should provide the researcher with sufficient insight into the expected growth rate in the reactor as well as any variations in growth across the hot zone.

Sign in / Sign up

Export Citation Format

Share Document