Transfer principles for pseudo real closed e-fold ordered fields

1986 ◽  
Vol 51 (4) ◽  
pp. 981-991 ◽  
Author(s):  
Şerban A. Basarab
Keyword(s):  
Elementary Theory ◽  
Field Extension ◽  
Equivalent Model ◽  
Ordered Field ◽  
Ordered Fields ◽  
Order Language ◽  
Irreducible Variety ◽  
Famous Paper ◽  
Definition Of ◽  
Geometric Definition

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.

The elementary theory of free pseudo p-adically closed fields of finite corank

1991 ◽  
Vol 56 (2) ◽  
pp. 484-496 ◽  
Author(s):  
Ido Efrat
Keyword(s):  
Rational Point ◽  
Elementary Theory ◽  
Valuation Rings ◽  
Order Language ◽  
Irreducible Variety ◽  
Closed Fields ◽  
Free Profinite Group ◽  
Unary Relation ◽  
Adic Valuation ◽  
Almost All

Let be p-adic closures of a countable Hilbertian field K. The main result of [EJ] asserts that the field has the following properties for almost all σ1,…,σe + m ϵ G(K) (in the sense of the unique Haar measure on G(K)e+m):(a) Kσ is pseudo p-adically closed (abbreviation: PpC), i.e., each nonempty absolutely irreducible variety defined over Kσ has a Kσ-rational point, provided that it has a simple rational point in each p-adic closure of Kσ.(b) G(Kσ) ≅ De,m, where De,m is the free profinite product of e copies Γ1,…, Γe of G(ℚp) and a free profinite group of rank m.(c) Kσ has exactly e nonequivalent p-adic valuation rings. They are the restrictions Oσ1,…, Oσe of the unique p-adic valuation rings on , respectively.In this paper we show that this result is in a certain sense the best possible. More precisely, we first show that the class of fields which satisfy (a)–(c) above is elementary in the appropriate language e(K), which is the ordinary first-order language of rings augmented by constant symbols for the elements of K and by e new unary relation symbols (interpreted as e p-adic valuation rings).

Groundwork for weak analysis

2002 ◽  
Vol 67 (2) ◽  
pp. 557-578 ◽  
Author(s):  
António M. Fernandes ◽  
Fernando Ferreira
Keyword(s):  
Elementary Theory ◽  
Real Variable ◽  
Number System ◽  
Order Theory ◽  
Continuous Real Function ◽  
The Real ◽  
Ordered Field ◽  
Ordered Fields ◽  
Intermediate Value ◽  
Real Number System

AbstractThis paper develops the very basic notions of analysis in a weak second-order theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we show how to interpret the theory BTFA in Robinson's theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q.

scholarly journals A geometric definition of Gabrielov numbers

2013 ◽  
Vol 27 (2) ◽  
pp. 447-460 ◽  
Author(s):  
Wolfgang Ebeling ◽  
Atsushi Takahashi
Keyword(s):  
Definition Of ◽  
Gabrielov Numbers ◽  
Geometric Definition

scholarly journals Conformal mapping of the Misner–Sharp mass from gravitational collapse

2016 ◽  
Vol 25 (07) ◽  
pp. 1650081 ◽  
Author(s):  
Fayçal Hammad
Keyword(s):  
Conformal Mapping ◽  
Gravitational Collapse ◽  
Conformal Transformation ◽  
Conformal Mappings ◽  
Conformal Transformations ◽  
Theories Of Gravity ◽  
Definition Of ◽  
Geometric Definition

The conformal transformation of the Misner–Sharp mass is reexamined. It has recently been found that this mass does not transform like usual masses do under conformal mappings of spacetime. We show that when it comes to conformal transformations, the widely used geometric definition of the Misner–Sharp mass is fundamentally different from the original conception of the latter. Indeed, when working within the full hydrodynamic setup that gave rise to that mass, i.e. the physics of gravitational collapse, the familiar conformal transformation of a usual mass is recovered. The case of scalar–tensor theories of gravity is also examined.

scholarly journals Optimal Decision Trees on Simplicial Complexes

10.37236/1900 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Jakob Jonsson
Keyword(s):  
Decision Tree ◽  
Decision Trees ◽  
Simplicial Complex ◽  
Elementary Theory ◽  
Simplicial Complexes ◽  
Optimal Decision ◽  
Property A ◽  
Recursive Definition ◽  
Topological Combinatorics ◽  
Definition Of

We consider topological aspects of decision trees on simplicial complexes, concentrating on how to use decision trees as a tool in topological combinatorics. By Robin Forman's discrete Morse theory, the number of evasive faces of a given dimension $i$ with respect to a decision tree on a simplicial complex is greater than or equal to the $i$th reduced Betti number (over any field) of the complex. Under certain favorable circumstances, a simplicial complex admits an "optimal" decision tree such that equality holds for each $i$; we may hence read off the homology directly from the tree. We provide a recursive definition of the class of semi-nonevasive simplicial complexes with this property. A certain generalization turns out to yield the class of semi-collapsible simplicial complexes that admit an optimal discrete Morse function in the analogous sense. In addition, we develop some elementary theory about semi-nonevasive and semi-collapsible complexes. Finally, we provide explicit optimal decision trees for several well-known simplicial complexes.

On the geometric definition of quasiconformality

2021 ◽  
Vol 62 (5) ◽  
pp. 965-982
Author(s):  
V. V. Aseev
Keyword(s):  
Definition Of ◽  
Geometric Definition

Recursive categoricity and persistence

1986 ◽  
Vol 51 (2) ◽  
pp. 430-434 ◽  
Author(s):  
Terrence Millar
Keyword(s):  
First Order ◽  
Existential Sentence ◽  
Order Language ◽  
Recursive Structures ◽  
Definition Of

This paper is concerned with recursive structures and the persistance of an effective notion of categoricity. The terminology and notational conventions are standard. We will devote the rest of this paragraph to a cursory review of some of the assumed conventions. If θ is a formula, then θk denotes θ if k is zero, and ¬θ if k is one. If A is a sequence with domain a subset of ω, then A∣n denotes the sequence obtained by restricting the domain of A to n. For an effective first order language L, let {ci∣i<ω} be distinct new constants, and let {θi∣i<ω} be an effective enumeration of all sentences in the language L ∪ {ci∣j<ω}. An infinite L-structure is recursive iff it has universe ω and the set is recursive, where cn is interpreted by n. In general we say that a set of formulas is recursive if the set of its indices with respect to an enumeration such as above is recursive. The ∃-diagram of a structure is recursive if the structure is recursive and the set and θi is an existential sentence} is also recursive. The definition of “the ∀∃-diagram of is recursive” is similar.

scholarly journals Crossed Products and Ramification

1966 ◽  
Vol 28 ◽  
pp. 85-111 ◽  
Author(s):  
Susan Williamson
Keyword(s):  
Galois Group ◽  
Galois Extension ◽  
Valuation Ring ◽  
Field Extension ◽  
Integral Closure ◽  
Crossed Product ◽  
Crossed Products ◽  
Quotient Field ◽  
Rank One ◽  
Definition Of

Introduction. Let S be the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, and let G denote the Galois group of the quotient field extension. Auslander and Rim have shown in [3] that the trivial crossed product Δ (1, S, G) is an hereditary order if and only if 5 is a tamely ramified extension of R. And the author has proved in [7] that if the extension S of R is tamely ramified then the crossed product Δ(f, 5, G) is a Π-principal hereditary order for each 2-cocycle f in Z2(G, U(S)). (See Section 1 for the definition of Π-principal hereditary order.) However, the author has exhibited in [8] an example of a crossed product Δ(f, S, G) which is a Π-principal hereditary order in the case when S is a wildly ramified extension of R.

Finite Spaces of Signatures

1989 ◽  
Vol 41 (5) ◽  
pp. 808-829 ◽  
Author(s):  
Victoria Powers
Keyword(s):  
Quadratic Forms ◽  
Local Rings ◽  
Character Group ◽  
Witt Rings ◽  
Ordered Field ◽  
Spaces Of Orderings ◽  
Ordered Fields ◽  
Skew Fields ◽  
Finite Spaces ◽  
Field Setting

Marshall's Spaces of Orderings are an abstract setting for the reduced theory of quadratic forms and Witt rings. A Space of Orderings consists of an abelian group of exponent 2 and a subset of the character group which satisfies certain axioms. The axioms are modeled on the case where the group is an ordered field modulo the sums of squares of the field and the subset of the character group is the set of orders on the field. There are other examples, arising from ordered semi-local rings [4, p. 321], ordered skew fields [2, p. 92], and planar ternary rings [3]. In [4], Marshall showed that a Space of Orderings in which the group is finite arises from an ordered field. In further papers Marshall used these abstract techniques to provide new, more elegant proofs of results known for ordered fields, and to prove theorems previously unknown in the field setting.

On the notion of algebraic closedness for noncommutative groups and fields

1971 ◽  
Vol 36 (3) ◽  
pp. 441-444 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Wide Class ◽  
First Order ◽  
Division Rings ◽  
Ordered Field ◽  
Ordered Fields ◽  
Skew Fields

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

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