Four concepts from “geometrical” stability theory in modules

1992 ◽  
Vol 57 (2) ◽  
pp. 724-740 ◽  
Author(s):  
T. G. Kucera ◽  
M. Prest
Keyword(s):  
Model Theory ◽  
Stability Theory ◽  
The Other ◽  
General Stability ◽  
Injective Modules ◽  
Algebraic Problem

In [H1] Hrushovski introduced a number of ideas concerning the relations between types which have proved to be of importance in stability theory. These relations allow the geometries associated to various types to be connected. In this paper we consider the meaning of these concepts in modules (and more generally in abelian structures). In particular, we provide algebraic characterisations of notions such as hereditary orthogonality, “p -internal” and “p-simple”. These characterisations are in the same spirit as the algebraic characterisations of such concepts as orthogonality and regularity, that have already proved so useful. Of the concepts that we consider, p-simplicity is dealt with in [H3] and the other three concepts in [H2].The descriptions arose out of our desire to develop some intuition for these ideas. We think that our characterisations may well be useful in the same way to others, particularly since our examples are algebraically uncomplicated and so understanding them does not require expertise in the model theory of modules. Furthermore, in view of the increasing importance of these notions, the results themselves are likely to be directly useful in the model-theoretic study of modules and, via abelian structures, in more general stability-theoretic contexts. Finally, some of our characterisations suggest that these ideas may be relevant to the algebraic problem of understanding the structure of indecomposable injective modules.

Models and Reality

2019 ◽  
pp. 128-153
Author(s):  
Stephen Yablo
Keyword(s):  
Model Theory ◽  
Expressive Power ◽  
The Other ◽  
Hilary Putnam ◽  
Objective World ◽  
Other Hand ◽  
Cast Doubt

The philosopher Hilary Putnam uses model theory to cast doubt on our ability to engage semantically with an objective world. The role of mathematics for him is to prove this pessimistic conclusion. The present chapter, on the other hand, explores how models can help us to engage semantically with the objective world. Mathematics functions here as an analogy. Among their many other accomplishments, numbers boost the language’s expressive power; they give us access to recondite physical facts. Models, among their many other accomplishments, do the same thing. This is the analogy this chapter attempts to develop.

scholarly journals Pure-injectivity and model theory for G-sets

2009 ◽  
Vol 74 (2) ◽  
pp. 474-488
Author(s):  
Ravi Rajani ◽  
Mike Prest
Keyword(s):  
Model Theory ◽  
Injective Modules ◽  
Ziegler Spectrum

AbstractIn the model theory of modules the Ziegler spectrum, the space of indecomposable pure-injective modules, has played a key role. We investigate the possibility of defining a similar space in the context of G-sets where G is a group.

scholarly journals Epistemicism and modality

2016 ◽  
Vol 46 (4-5) ◽  
pp. 803-835 ◽  
Author(s):  
Juhani Yli-Vakkuri
Keyword(s):  
Model Theory ◽  
Three Dimensional ◽  
The Other ◽  
Object Language ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
Metaphysical Necessity ◽  
Actuality Operator

AbstractWhat kind of semantics should someone who accepts the epistemicist theory of vagueness defended in Timothy Williamson's Vagueness (1994) give a definiteness operator? To impose some interesting constraints on acceptable answers to this question, I will assume that the object language also contains a metaphysical necessity operator and a metaphysical actuality operator. I will suggest that the answer is to be found by working within a three-dimensional model theory. I will provide sketches of two ways of extracting an epistemicist semantics from that model theory, one of which I will find to be more plausible than the other.

scholarly journals Ethical Dilemmas as Seen Through the Major Characters Reflected in "The Danish Girl" Movie Screenplay Written by Lucinda Coxon

2019 ◽  
Vol 8 (2) ◽  
pp. 75-84
Author(s):  
Kemala Putri Anggraeni ◽  
Maria Johana Ari Widayanti
Keyword(s):  
Model Theory ◽  
Sigmund Freud ◽  
Ethical Dilemma ◽  
Qualitative Method ◽  
Ethical Dilemmas ◽  
Psychoanalytic Theory ◽  
The Other ◽  
Tripartite Model ◽  
Daily Lives ◽  
The Impact

An ethical dilemma is an interesting topic to discuss since it always occurs to us in our daily lives whether we realise it or not. The study is aimed to explain the ethical dilemmas of the major characters in The Danish Girl movie screenplay by Lucinda Coxon. This study used psychoanalytic theory of the tripartite model theory by Sigmund Freud and employed qualitative method. The result was the ethical dilemmas of the major characters occurred before they made a decision in their actions. This study revealed that the ethical dilemma happened to them because they needed to think about the impact of their actions on other people or themselves. It could prevent them from doing or saying inappropriate things which could hurt someone's feelings or harm themselves. Another result in this study was that the ethical dilemmas that happened to the major characters mostly represent ego more than the other two parts of psyche. They could control what they wanted to say or do since they thought about the possible things that could happen to each choice before they made the decision. Keywords: Characters, Ethical Dilemma, Movie Screenplay, Psychoanalytic Theory, The Danish Girl

scholarly journals Effect of vehicle motion stability after impact/crash on traffic safety

2018 ◽  
Vol 231 ◽  
pp. 05005
Author(s):  
Eligiusz Mieloszyk ◽  
Anita Milewska ◽  
Sławomir Grulkowski
Keyword(s):  
Dynamical Systems ◽  
Traffic Safety ◽  
Stability Theory ◽  
Road Traffic ◽  
Motion Stability ◽  
General Stability ◽  
Vehicle Motion

The article presents the application of general stability theory to the study of road traffic stability immediately after an impact (crash, collision). It turns out that when modelling a collision, vehicles can be treated as colliding masses and dynamical systems can be assigned to this phenomenon.

Non-Archimedean Tame Topology and Stably Dominated Types (AM-192)

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Algebraic Geometry ◽  
Model Theory ◽  
Stability Theory ◽  
Analytic Geometry ◽  
Previous Knowledge ◽  
Real Numbers ◽  
Algebraic Dynamics ◽  
Theoretic Foundations ◽  
Rational Polytope ◽  
Deformation Retraction

Over the field of real numbers, analytic geometry has long been in deep interaction with algebraic geometry, bringing the latter subject many of its topological insights. In recent decades, model theory has joined this work through the theory of o-minimality, providing finiteness and uniformity statements and new structural tools. For non-archimedean fields, such as the p-adics, the Berkovich analytification provides a connected topology with many thoroughgoing analogies to the real topology on the set of complex points, and it has become an important tool in algebraic dynamics and many other areas of geometry. This book lays down model-theoretic foundations for non-archimedean geometry. The methods combine o-minimality and stability theory. Definable types play a central role, serving first to define the notion of a point and then properties such as definable compactness. Beyond the foundations, the main theorem constructs a deformation retraction from the full non-archimedean space of an algebraic variety to a rational polytope. This generalizes previous results of V. Berkovich, who used resolution of singularities methods. No previous knowledge of non-archimedean geometry is assumed and model-theoretic prerequisites are reviewed in the first sections.

Diophantine Geometry from Model Theory

2001 ◽  
Vol 7 (1) ◽  
pp. 37-57 ◽  
Author(s):  
Thomas Scanlon
Keyword(s):  
Model Theory ◽  
Stability Theory ◽  
Function Field ◽  
Algebraic Groups ◽  
Diophantine Geometry ◽  
Diophantine Problems ◽  
Theory Of Fields ◽  
Model Theory Of Fields

Abstract§1. Introduction. With Hrushovski's proof of the function field Mordell-Lang conjecture [16] the relevance of geometric stability theory to diophantine geometry first came to light. A gulf between logicians and number theorists allowed for contradictory reactions. It has been asserted that Hrushovski's proof was simply an algebraic argument masked in the language of model theory. Another camp held that this theorem was merely a clever one-off. Still others regarded the argument as magical and asked whether such sorcery could unlock the secrets of a wide coterie of number theoretic problems.In the intervening years each of these prejudices has been revealed as false though such attitudes are still common. The methods pioneered in [16] have been extended and applied to a number of other problems. At their best, these methods have been integrated into the general methods for solving diophantine problems. Moreover, the newer work suggests limits to the application of model theory to diophantine geometry. For example, all such known applications are connected with commutative algebraic groups. This need not be an intrinsic restriction, but its removal requires serious advances in the model theory of fields.

Crisis stability or general stability? Assessing Northeast Asia’s absence of war and prospects for liberal transition

2015 ◽  
pp. 1-23 ◽  
Author(s):  
Jong Kun Choi
Keyword(s):  
Northeast Asia ◽  
Security Studies ◽  
The Other ◽  
Military Power ◽  
Military Capability ◽  
Delicate Balance ◽  
General Stability ◽  
Other Hand ◽  
Logic Of Inquiry ◽  
Stability Concepts

AbstractIs the relative long peace of Northeast Asia a result of crisis stability or general stability? The article introduces two stability concepts – crisis and general stability. Crisis stability occurs when both sides in military crisis are secure due to military capability and are able to wait out a surprise attack fully confident of the ability to respond with a punishing counter attack. On the other hand, general stability prevails when two powers greatly prefer peace even to a victorious war whether crisis stability exist or not, simply because war has become inconceivable as a means of solving any political disagreements and conflicts. While crisis stability entails delicate balance of military power from the deterrence literature of security studies, general stability bases its logic of inquiry on constructivism where the idea of war aversion – categorically rejecting war as a means to end conflicts – becomes the prevailing norm. Therefore, this article empirically examines how Northeast Asia has sustained its peace through crisis stability and presents a new trend toward general stability

Injective modules over Mori domains

2003 ◽  
Vol 40 (1-2) ◽  
pp. 33-40
Author(s):  
L. Fuchs
Keyword(s):  
Injective Module ◽  
The Other ◽  
Injective Hull ◽  
Maximum Condition ◽  
Direct Sum ◽  
Injective Modules ◽  
Special Case ◽  
False Claim

Injective modules are considered over commutative domains. It is shown that every injective module admits a decomposition into two summands, where one of the summands contains an essential submodule whose elements have divisorial annihilator ideals, while the other summand contains no element with divisorial annihilator. In the special case of Mori domains (i.e., the divisorial ideals satisfy the maximum condition), the first summand is a direct sum of a S-injective module and a module that has no such summand. The former is a direct sum of indecomposable injectives, while the latter is the injective hull of such a direct sum. Those Mori domains R are characterized for which the injective hull of Q/R is S-injective (Q denotes the field of quotients of R) as strong Mori domains, correcting a false claim in the literature.

The relative expressive power of some logics extending first-order logic

1979 ◽  
Vol 44 (2) ◽  
pp. 129-146 ◽  
Author(s):  
John Cowles
Keyword(s):  
Model Theory ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
The Other ◽  
First Order Logic ◽  
Finite Sets ◽  
Limited Ability ◽  
First Order ◽  
Second Order Logic

In recent years there has been a proliferation of logics which extend first-order logic, e.g., logics with infinite sentences, logics with cardinal quantifiers such as “there exist infinitely many…” and “there exist uncountably many…”, and a weak second-order logic with variables and quantifiers for finite sets of individuals. It is well known that first-order logic has a limited ability to express many of the concepts studied by mathematicians, e.g., the concept of a wellordering. However, first-order logic, being among the simplest logics with applications to mathematics, does have an extensively developed and well understood model theory. On the other hand, full second-order logic has all the expressive power needed to do mathematics, but has an unworkable model theory. Indeed, the search for a logic with a semantics complex enough to say something, yet at the same time simple enough to say something about, accounts for the proliferation of logics mentioned above. In this paper, a number of proposed strengthenings of first-order logic are examined with respect to their relative expressive power, i.e., given two logics, what concepts can be expressed in one but not the other?For the most part, the notation is standard. Most of the notation is either explained in the text or can be found in the book [2] of Chang and Keisler. Some notational conventions used throughout the text are listed below: the empty set is denoted by ∅.

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