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46
Definable sets in ordered structures
 Bull. Amer. Math. Soc. (N.S
, 1984
"... Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of m ..."
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Cited by 96 (7 self)
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Abstract. This paper introduces and begins the study of a wellbehaved class of linearly ordered structures, the ^minimal structures. The definition of this class and the corresponding class of theories, the strongly ©minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of Cminimal ordered groups and rings. Several other simple results are collected in §3. The primary tool in the analysis of ¿¡minimal structures is a strong analogue of "forking symmetry, " given by Theorem 4.2. This result states that any (parametrically) definable unary function in an (5minimal structure is piecewise either constant or an orderpreserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0categorical ¿¡¡minimal structures (Theorem 6.1). 1. Introduction. The
On Logics with Two Variables
 Theoretical Computer Science
, 1999
"... This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable ..."
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Cited by 41 (8 self)
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This paper is a survey and systematic presentation of decidability and complexity issues for modal and nonmodal twovariable logics. A classical result due to Mortimer says that the twovariable fragment of firstorder logic, denoted FO 2 , has the finite model property and is therefore decidable for satisfiability. One of the reasons for the significance of this result is that many propositional modal logics can be embedded into FO 2 . Logics that are of interest for knowledge representation, for the specification and verification of concurrent systems and for other areas of computer science are often defined (or can be viewed) as extensions of modal logics by features like counting constructs, path quantifiers, transitive closure operators, least and greatest fixed points etc. Examples of such logics are computation tree logic CTL, the modal ¯calculus L¯ , or popular description logics used in artificial intelligence. Although the additional features are usually not firstorder...
Walrasian Equilibrium without Survival: Equilibrium, Efficiency, and Remedial Policy
 welfare and development: A Festschrift in honour of Amartya K. Sen
, 1995
"... Standard general equilibrium theory excludes starvation by assuming that everybody can survive without trade. Because trade cannot harm consumers, they can therefore also survive with trade. Here this assumption is abandoned, and equilibria in which not everybody survives are investigated. A simple ..."
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Cited by 19 (15 self)
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Standard general equilibrium theory excludes starvation by assuming that everybody can survive without trade. Because trade cannot harm consumers, they can therefore also survive with trade. Here this assumption is abandoned, and equilibria in which not everybody survives are investigated. A simple example is discussed, along with possible policies which might reduce starvation. Thereafter, for economies with a continuum of agents, the usual results are established — existence of equilibrium, the two fundamental efficiency theorems of welfare economics, and core equivalence. Their validity depends on some special but not very stringent assumptions needed to deal with natural nonconvexities in each consumer’s feasible set.
Using The `RCC' Formalism To Describe The Topology Of Spherical Regions
, 1996
"... This research report concerns the topology of 2dimensional regions embedded in spherical surfaces, such as that of the Earth (`spherical regions'). It shows that the RCC (RegionConnection Calculus) firstorder logic formalism for qualitative spatial representation and reasoning is sufficiently exp ..."
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Cited by 13 (2 self)
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This research report concerns the topology of 2dimensional regions embedded in spherical surfaces, such as that of the Earth (`spherical regions'). It shows that the RCC (RegionConnection Calculus) firstorder logic formalism for qualitative spatial representation and reasoning is sufficiently expressive to support a rich topological taxonomy of uniformly twodimensional regions forming parts of a spherical surface such as the Earth's. However, there are potentially useful constraints on the topology of such regions which the language of RCC cannot capture. Furthermore, the spherical model of the RCC axiom set developed here permits the construction of a proof that the theory axiomatised by this axiom set is undecidable (a result also derivable from (Grzegorczyk 1951).
Places of Algebraic Function Fields in Arbitrary Characteristic
, 2003
"... We consider the Zariski space of all places of an algebraic function field F of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discret ..."
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Cited by 11 (6 self)
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We consider the Zariski space of all places of an algebraic function field F of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper Embedding problems over large fields. We also study the question whether a field K is existentially closed in an extension field L if L admits a Krational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasicompact and that it is a spectral space.
Definable sets in algebraically closed valued fields. Part I: elimination of imaginaries
, 2002
"... It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in K of certain definable Rsubmodules of K (for all n 1). The proof involves the development of a theory of independenc ..."
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Cited by 10 (4 self)
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It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in K of certain definable Rsubmodules of K (for all n 1). The proof involves the development of a theory of independence for unary types, which play the role of 1types, followed by an analysis of germs of definable functions from unary sets to the sorts.
Differential arcs and regular types in differential fields
 J. REINE ANGEW. MATH
, 2007
"... We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety in characteristic zero is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ1,..., δn) is a differentially closed field of characteristic zero with n ..."
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Cited by 8 (3 self)
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We introduce differential arc spaces in analogy to the algebraic arc spaces and show that a differential variety in characteristic zero is determined by its arcs at a point. Using differential arcs, we show that if (K, +, ×, δ1,..., δn) is a differentially closed field of characteristic zero with n commuting derivations and p ∈ S(K) is a regular type over K, then either p is locally modular or there is a definable subgroup G ≤ (K, +) of the additive group having a regular generic type that is nonorthogonal to p.