O-MINIMALISM

2014 ◽  
Vol 79 (2) ◽  
pp. 355-409 ◽  
Author(s):  
HANS SCHOUTENS
Keyword(s):  
Cell Decomposition ◽  
Dimension Theory ◽  
Pigeonhole Principle ◽  
Euler Characteristics ◽  
Grothendieck Ring ◽  
First Order ◽  
Minimal Structure ◽  
Related Notion ◽  
Depth Analysis

AbstractThis paper is devoted to o-minimalism, the study of the first-order properties of o-minimal structures. The main protagonists are the pseudo-o-minimal structures, that is to say, the models of the theory of all o-minimal L-structures, but we start with a more in-depth analysis of the well-known fragment DCTC (Definable Completeness/Type Completeness), and show how it already admits many of the properties of o-minimal structures: dimension theory, monotonicity, Hardy structures, and quasi-cell decomposition, provided one replaces finiteness by discreteness in all of these. Failure of cell decomposition leads to the related notion of a eukaryote structure, and we give a criterium for a pseudo-o-minimal structure to be eukaryote.To any pseudo-o-minimal structure, we can associate its Grothendieck ring, which in the non-o-minimal case is a nontrivial invariant. To study this invariant, we identify a third o-minimalistic property, the Discrete Pigeonhole Principle, which in turn allows us to define discretely valued Euler characteristics. As an application, we study certain analytic subsets, called Taylor sets.

b-MINIMALITY

2007 ◽  
Vol 07 (02) ◽  
pp. 195-227 ◽  
Author(s):  
RAF CLUCKERS ◽  
FRANÇOIS LOESER
Keyword(s):  
Characteristic Zero ◽  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Dimension Theory ◽  
Minimal Cell ◽  
Euler Characteristics ◽  
Valued Fields ◽  
Henselian Valued Fields ◽  
A Cell ◽  
Abstract Notion

We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are also b-minimal, but b-minimality leaves more room for nontrivial expansions. The b-minimal setting is intended to be a natural framework for the construction of Euler characteristics and motivic or p-adic integrals. The b-minimal cell decomposition is a generalization of concepts of Cohen [11], Denef [15], and the link between cell decomposition and integration was first made by Denef [13].

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

scholarly journals TOPOLOGICAL CELL DECOMPOSITION AND DIMENSION THEORY IN P-MINIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 347-358 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
LUCK DARNIÈRE ◽  
EVA LEENKNEGT
Keyword(s):  
Algebraic Geometry ◽  
Open Subset ◽  
Cell Decomposition ◽  
Real Algebraic Geometry ◽  
Dimension Theory ◽  
Definable Set ◽  
Image Position ◽  
Definable Function

AbstractThis paper addresses some questions about dimension theory for P-minimal structures. We show that, for any definable set A, the dimension of $\bar A\backslash A$ is strictly smaller than the dimension of A itself, and that A has a decomposition into definable, pure-dimensional components. This is then used to show that the intersection of finitely many definable dense subsets of A is still dense in A. As an application, we obtain that any definable function $f:D \subseteq {K^m} \to {K^n}$ is continuous on a dense, relatively open subset of its domain D, thereby answering a question that was originally posed by Haskell and Macpherson.In order to obtain these results, we show that P-minimal structures admit a type of cell decomposition, using a topological notion of cells inspired by real algebraic geometry.

Combinatorics with Definable Sets: Euler Characteristics and Grothendieck Rings

2000 ◽  
Vol 6 (3) ◽  
pp. 311-330 ◽  
Author(s):  
Jan Krajíček ◽  
Thomas Scanlon
Keyword(s):  
Euler Characteristic ◽  
Order Structure ◽  
Euler Characteristics ◽  
Grothendieck Ring ◽  
Open Problems ◽  
Partially Ordered ◽  
Counting Functions ◽  
Finite Structures ◽  
Definable Sets ◽  
Grothendieck Rings

AbstractWe recall the notions of weak and strong Euler characteristics on a first order structure and make explicit the notion of a Grothendieck ring of a structure. We define partially ordered Euler characteristic and Grothendieck ring and give a characterization of structures that have non-trivial partially ordered Grothendieck ring. We give a generalization of counting functions to locally finite structures, and use the construction to show that the Grothendieck ring of the complex numbers contains as a subring the ring of integer polynomials in continuum many variables. We prove the existence of a universal strong Euler characteristic on a structure. We investigate the dependence of the Grothendieck ring on the theory of the structure and give a few counter-examples. Finally, we relate some open problems and independence results in bounded arithmetic to properties of particular Grothendieck rings.

Experimental Study on Tire Non-Steady State Slip Properties

2013 ◽  
Vol 753-755 ◽  
pp. 1736-1744
Author(s):  
Jie Liu ◽  
Xiao Ling Jia
Keyword(s):  
Linear System ◽  
Dynamic Performance ◽  
Lateral Force ◽  
Slip Angle ◽  
Test Results ◽  
Relaxation Length ◽  
Side Slip Angle ◽  
First Order ◽  
Depth Analysis ◽  
Yaw Angle

As for the two typical inputs of pure side slip angle and pure yaw angle, this paper presents the in-depth analysis of lateral force, aligning torque and relaxation length respectively within the domains of distance and spacial frequency, and also explains the test results by theoretical model. Within the small side slip angle, tire is a first-order linear system. Relaxation length is equivalent to the time constant of linear system, which decreases as slip angle increases. It indicates the dynamic performance of tire system.

scholarly journals CHARACTERIZING O-MINIMAL GROUPS IN TAME EXPANSIONS OF O-MINIMAL STRUCTURES

2019 ◽  
pp. 1-26
Author(s):  
Pantelis E. Eleftheriou
Keyword(s):  
Elliptic Curve ◽  
Independent Set ◽  
Dimension Theory ◽  
Maximal Dimension ◽  
Minimal Group ◽  
Full Characterization ◽  
Minimal Structure ◽  
Definable Groups ◽  
Topological Closure

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.

Approximate Euler characteristic, dimension, and weak pigeonhole principles

2004 ◽  
Vol 69 (1) ◽  
pp. 201-214
Author(s):  
Jan Krajíček
Keyword(s):  
Euler Characteristic ◽  
Characteristic Dimension ◽  
Order Structure ◽  
Dimension Function ◽  
Pigeonhole Principle ◽  
First Order ◽  
Definable Set ◽  
Definable Sets

AbstractWe define the notion of approximate Euler characteristic of definable sets of a first order structure. We show that a structure admits a non-trivial approximate Euler characteristic if it satisfies weak pigeonhole principle : two disjoint copies of a non-empty definable set A cannot be definably embedded into A, and principle CC of comparing cardinalities: for any two definable sets A, B either A definably embeds in B or vice versa. Also, a structure admitting a non-trivial approximate Euler characteristic must satisfy .Further we show that a structure admits a non-trivial dimension function on definable sets if and only if it satisfies weak pigeonhole principle : for no definable set A with more than one element can A2 definably embed into A.

scholarly journals A P-MINIMAL STRUCTURE WITHOUT DEFINABLE SKOLEM FUNCTIONS

2017 ◽  
Vol 82 (2) ◽  
pp. 778-786 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
KIEN HUU NGUYEN
Keyword(s):  
Cell Decomposition ◽  
Skolem Functions ◽  
Minimal Structure

AbstractWe show there are intermediate P-minimal structures between the semialgebraic and subanalytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are P-minimal structures which do not admit classical cell decomposition.

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