Elimination of quantifiers for ordered valuation rings

1987 ◽  
Vol 52 (1) ◽  
pp. 116-128 ◽  
Author(s):  
M. A. Dickmann
Keyword(s):  
Model Theory ◽  
Maximal Ideal ◽  
Valuation Ring ◽  
Quantifier Elimination ◽  
Point Of View ◽  
List Type ◽  
Divisibility Property ◽  
Valuation Rings ◽  
Commutative Domain ◽  
Image Position

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result:Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that:(1) The symbol “∣” is interpreted as the honest divisibility relation: (2) The following divisibility property holds in T:If T admits q.e. in ℒ, then T = RCVR.We do not know at present whether the restriction imposed by condition (2) can be weakened.The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings, for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings:Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent:(a) A is a convexly ordered valuation ring.(b) Every ideal (or, equivalently, principal ideal) is convex in A.(c) A is a valuation ring convex in its field of fractions quot(A).(d) A is a valuation ring and its maximal ideal MA is convex (in A or, equivalently, in quot (A)).(e) A is a valuation ring and its maximal ideal is bounded by ± 1.

A Note on the Quotients of Indecomposable Injective Modules

1969 ◽  
Vol 12 (5) ◽  
pp. 661-665 ◽  
Author(s):  
P. Vámos
Keyword(s):  
Prime Ideal ◽  
Maximal Ideal ◽  
Valuation Ring ◽  
List Type ◽  
Type Number ◽  
Injective Modules ◽  
Factor Module ◽  
Commutative Domain ◽  
Domain I

Let R be a commutative domain, I an ideal of R and write E1 for the infective envelope of R / I. In this note the following theorem will be proved:Theorem. For a prime ideal P of a commutative domain R the following are equivalent:(i)Every factor module of Ep is an indecomposable infective module;(ii)Every non-zero prime ideal P′ ⊆ P is contained in only one maximal ideal M of R, and RM is an almost maximal valuation ring.

Witts Theorem for Quadratic Forms Over Non-Dyadic Discrete Valuation Rings

1977 ◽  
Vol 29 (5) ◽  
pp. 928-936
Author(s):  
David Mordecai Cohen
Keyword(s):  
Bilinear Form ◽  
Maximal Ideal ◽  
Quadratic Forms ◽  
Valuation Ring ◽  
Discrete Valuation Ring ◽  
Symmetric Bilinear Form ◽  
Finitely Generated ◽  
Discrete Valuation Rings ◽  
Valuation Rings ◽  
Group Of Isometries

Let R be a discrete valuation ring, with maximal ideal pR, such that ½ ϵ R. Let L be a finitely generated R-module and B : L × L → R a non-degenerate symmetric bilinear form. The module L is called a quadratic module. For notational convenience we shall write xy = B(x, y). Let O(L) be the group of isometries, i.e. all R-linear isomorphisms φ : L → L such that B((φ(x), (φ(y)) = B(x, y).

Discriminant as a product of local discriminants

2017 ◽  
Vol 16 (10) ◽  
pp. 1750198 ◽  
Author(s):  
Anuj Jakhar ◽  
Bablesh Jhorar ◽  
Sudesh K. Khanduja ◽  
Neeraj Sangwan
Keyword(s):  
Maximal Ideal ◽  
Valuation Ring ◽  
Approximation Theorem ◽  
Discrete Valuation Ring ◽  
Integral Closure ◽  
Weak Approximation ◽  
Separable Extension ◽  
Ideal Formula ◽  
Valuation Rings ◽  
Basic Equality

Let [Formula: see text] be a discrete valuation ring with maximal ideal [Formula: see text] and [Formula: see text] be the integral closure of [Formula: see text] in a finite separable extension [Formula: see text] of [Formula: see text]. For a maximal ideal [Formula: see text] of [Formula: see text], let [Formula: see text] denote respectively the valuation rings of the completions of [Formula: see text] with respect to [Formula: see text]. The discriminant satisfies a basic equality which says that [Formula: see text]. In this paper, we extend the above equality on replacing [Formula: see text] by the valuation ring of a Krull valuation of arbitrary rank and completion by henselization. In the course of proof, we prove a generalization of the well-known weak Approximation Theorem which is of independent interest as well.

Generators of Orthogonal Groups over Valuation Rings

1981 ◽  
Vol 33 (1) ◽  
pp. 116-128 ◽  
Author(s):  
Hiroyuki Ishibashi
Keyword(s):  
Commutative Ring ◽  
Local Ring ◽  
Maximal Ideal ◽  
Orthogonal Group ◽  
Valuation Ring ◽  
Unit Element ◽  
Orthogonal Groups ◽  
Valuation Rings ◽  
Number Of Factors ◽  
Valuation Domains

Let be a valuation ring with unit element, i.e., is a commutative ring such that for any a and b in , either a divides b or b divides a. We assume 2 is a unit of . V is an n-ary nonsingular quadratic module over , O(V) or On(V) is the orthogonal group on V, and S is the set of symmetries in O(V). We define l(σ) to be the minimal number of factors in the expression of a of O(V) as a product of symmetries on V. For the case where is a field, l(σ) has been determined by P. Scherk [6] and J. Dieudonné [1]. In [3] I have generalized the results of Scherk to orthogonal groups over valuation domains. In the present paper I generalize my results of [3] to orthogonal groups over valuation rings.Since is a valuation ring, it is a local ring with the maximal ideal A which consists of all nonunits of .

The order indiscernibles of divisible ordered abelian groups

1984 ◽  
Vol 49 (1) ◽  
pp. 151-160
Author(s):  
David Rosenthal
Keyword(s):  
Abelian Group ◽  
Model Theory ◽  
Algebraic Structure ◽  
Abelian Groups ◽  
Quantifier Elimination ◽  
Simple Condition ◽  
Algebraic Invariants ◽  
Image Position ◽  
And Algebra ◽  
Simplified Form

There has been much work in developing the interconnections between model theory and algebra. Here we look at a particular example, the divisible ordered abelian groups, and show how the indiscernibles are related to the algebraic structure. Now a divisible ordered abelian group is a model of Th (Q, +, 0, <) and so is linearly ordered by <. Thus the theory is unstable and has a large number of models. It is therefore unrealistic to expect that a simple condition will completely determine a model. Instead we would just like to obtain nice algebraic invariants.Definition. A subset C of a divisible ordered abelian group is a set of (order) indiscernibles iff for every sequence of integers n1,…,nk and for every c1 < … < ck and d1 < … < dk in CNote that this simplified form of indiscernibility is an immediate consequence of quantifier elimination for the theory. The above definition could be formulated in the language of +, 0, < but we have used subtraction as a matter of convenience. Similarly we may also use rational coefficients. Also note that a set of order indiscernibles is usually defined with respect to some external order. But in this case there are only two possibilities: the ordering inherited from or its reverse. So we will always assume that a set of order indiscernibles has the ordering inherited from . We may sometimes refer to a set of indiscernibles as a sequence of indiscernibles if we want to explicitly mention the ordering associated with the set.

scholarly journals On three-variable expanders over finite valuation rings

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Le Quang Ham ◽  
Nguyen Van The ◽  
Phuc D. Tran ◽  
Le Anh Vinh
Keyword(s):  
Valuation Ring ◽  
Product Type ◽  
Quadratic Polynomial ◽  
Valuation Rings

AbstractLet {\mathcal{R}} be a finite valuation ring of order {q^{r}}. In this paper, we prove that for any quadratic polynomial {f(x,y,z)\in\mathcal{R}[x,y,z]} that is of the form {axy+R(x)+S(y)+T(z)} for some one-variable polynomials {R,S,T}, we have|f(A,B,C)|\gg\min\biggl{\{}q^{r},\frac{|A||B||C|}{q^{2r-1}}\bigg{\}}for any {A,B,C\subset\mathcal{R}}. We also study the sum-product type problems over finite valuation ring {\mathcal{R}}. More precisely, we show that for any {A\subset\mathcal{R}} with {|A|\gg q^{r-\frac{1}{3}}} then {\max\{|AA|,|A^{d}+A^{d}|\}}, {\max\{|A+A|,|A^{2}+A^{2}|\}}, {\max\{|A-A|,|AA+AA|\}\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}}, and {|f(A)+A|\gg|A|^{\frac{2}{3}}q^{\frac{r}{3}}} for any one variable quadratic polynomial f.

SURVEY Towards a global view of dynamical systems, for the C1-topology

2011 ◽  
Vol 31 (4) ◽  
pp. 959-993 ◽  
Author(s):  
C. BONATTI
Keyword(s):  
Dynamical Systems ◽  
Compact Manifold ◽  
Global Structure ◽  
Point Of View ◽  
List Type ◽  
Global View ◽  
Local Phenomenon ◽  
Generic Dynamics

AbstractThis paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C1-topology. More precisely, given any compact manifold M, one splits Diff1(M) into disjoint C1-open regions whose union is C1-dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: •either a robust local phenomenon;•or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C1-generic dynamics.

Logical aspects of experimental design and analysis

2020 ◽  
pp. 142-151
Author(s):  
Валерий Иванович Хабаров
Keyword(s):  
Model Theory ◽  
Semantic Analysis ◽  
Linear Models ◽  
Intelligent System ◽  
Knowledge Bases ◽  
Object Identification ◽  
Point Of View ◽  
Applied Theory ◽  
Formal Representation ◽  
Active Object

Предложена схема формализации задач активной идентификации объекта с использованием аппарата теории моделей - современного раздела математической логики. Теория моделей позволяет погрузить предмет “планирование и анализ эксперимента” в контекст семантического анализа. Семантический анализ понимается как установление соответствия между миром и его формальным представлением. С этой точки зрения представления об исследуемом объекте выражаются в некоторой прикладной теории. Предложен вывод модели для данной теории как процесс интерпретации, в котором ключевая роль отводится “экспериментатору”. Полученные результаты могут быть использованы при проектировании архитектур интеллектуальных систем для экспериментальных исследований, для построения онтологии эксперимента, создания баз знаний Purpose. The purpose of this work is to formalize the tasks of active object identification based on the apparatus of model theory - a modern section of mathematical logic. Model theory allows putting the subject “planning and analysis of an experiment” in the context of semantic analysis. Semantic analysis is understood as establishing a correspondence between the world and its formal representation. From this point of view, the concept of the object under study is expressed in some applied theory, which allows applying formal methods of model theory to it. Methods. It is assumed that the model is derived for this theory as an interpretation process, in which the key role is assigned to the experimenter. As a research method, it is proposed to use commutative diagrams that reflect the process of interpretation and extension of communication diagrams for the so-called equipped theories of planning and analysis of experiments. Results. The properties of the proposed models are proved and examples for planning a regression experiment are presented as an illustration. It is proved that for linear models it is possible to construct a finitely axiomatization capable theory. Findings, originality. The obtained results can be used in the design of architectures for an intelligent system in experimental research, building an experiment ontology and creation of knowledge bases. These studies will allow using logical programming to implement images of the presented commutative diagrams for equipped theories as applied systems for planning and interpreting the experiment

SHARP WEIGHTED BOUNDS FOR GEOMETRIC MAXIMAL OPERATORS

2016 ◽  
Vol 59 (3) ◽  
pp. 533-547 ◽  
Author(s):  
ADAM OSȨKOWSKI
Keyword(s):  
Maximal Operator ◽  
General Setting ◽  
Maximal Operators ◽  
List Type ◽  
Type Number ◽  
Type Estimate ◽  
Formula Group ◽  
Probability Spaces ◽  
Image Position

AbstractLet $\mathcal{M}$ and G denote, respectively, the maximal operator and the geometric maximal operator associated with the dyadic lattice on $\mathbb{R}^d$. (i)We prove that for any 0 < p < ∞, any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, we have Fefferman–Stein-type estimate $$\begin{equation*} ||G(f)||_{L^p(w)}\leq e^{1/p}||f||_{L^p(\mathcal{M}w)}. \end{equation*} $$ For each p, the constant e1/p is the best possible.(ii)We show that for any weight w on $\mathbb{R}^d$ and any measurable f on $\mathbb{R}^d$, $$\begin{equation*} \int_{\mathbb{R}^d} G(f)^{1/\mathcal{M}w}w\mbox{d}x\leq e\int_{\mathbb{R}^d} |f|^{1/w}w\mbox{d}x \end{equation*} $$ and prove that the constant e is optimal. Actually, we establish the above estimates in a more general setting of maximal operators on probability spaces equipped with a tree-like structure.

Questions on data and the input to GEN

2016 ◽  
Vol 19 (5) ◽  
pp. 889-890 ◽  
Author(s):  
LUIS LÓPEZ
Keyword(s):  
Code Switching ◽  
List Type ◽  
Type Number ◽  
Image Position

The keynote article (Goldrick, Putnam & Schwartz, 2016) discusses doubling phenomena occasionally found in code-switching corpora. Their analysis focuses on an English–Tamil sentence in which an SVO sequence in English is followed by a verb in Tamil, resulting in an apparent VOV structure: (1)

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