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.

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.

scholarly journals INTEGRATION AND CELL DECOMPOSITION IN P-MINIMAL STRUCTURES

2016 ◽  
Vol 81 (3) ◽  
pp. 1124-1141 ◽  
Author(s):  
PABLO CUBIDES KOVACSICS ◽  
EVA LEENKNEGT
Keyword(s):  
Cell Decomposition ◽  
Poincaré Series ◽  
Weak Version ◽  
Poincare Series ◽  
Skolem Functions ◽  
Direct Corollary ◽  
Image Position ◽  
And Function ◽  
Constructible Functions

AbstractWe show that the class of ${\cal L}$-constructible functions is closed under integration for any P-minimal expansion of a p-adic field $\left( {K,{\cal L}} \right)$. This generalizes results previously known for semi-algebraic and subanalytic structures. As part of the proof, we obtain a weak version of cell decomposition and function preparation for P-minimal structures, a result which is independent of the existence of Skolem functions. A direct corollary is that Denef’s results on the rationality of Poincaré series hold in any P-minimal expansion of a p-adic field $\left( {K,{\cal L}} \right)$.

scholarly journals CELL DECOMPOSITION AND CLASSIFICATION OF DEFINABLE SETS INp-OPTIMAL FIELDS

2017 ◽  
Vol 82 (1) ◽  
pp. 120-136 ◽  
Author(s):  
LUCK DARNIÈRE ◽  
IMMANUEL HALPUCZOK
Keyword(s):  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Extreme Value ◽  
Property A ◽  
Definable Subset ◽  
Skolem Functions ◽  
Definable Sets ◽  
A Cell ◽  
Preparation Theorem ◽  
Definable Functions

AbstractWe prove that forp-optimal fields (a very large subclass ofp-minimal fields containing all the known examples) a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining the two we get a preparation theorem for definable functions onp-optimal fields satisfying the Extreme Value Property, from which it follows that infinite sets definable over such fields are in definable bijection iff they have the same dimension.

scholarly journals Effective Cylindrical Cell Decompositions for Restricted Sub-Pfaffian Sets

2020 ◽  
Author(s):  
Gal Binyamini ◽  
Nicolai Vorobjov
Keyword(s):  
Betti Numbers ◽  
Cell Decomposition ◽  
Upper Bounds ◽  
Geometric Complexity ◽  
Cylindrical Cell ◽  
Cell Decompositions ◽  
Minimal Structure ◽  
Natural Measure ◽  
Definition Of

Abstract The o-minimal structure generated by the restricted Pfaffian functions, known as restricted sub-Pfaffian sets, admits a natural measure of complexity in terms of a format  ${{\mathcal{F}}}$, recording information like the number of variables and quantifiers involved in the definition of the set, and a degree  $D$, recording the degrees of the equations involved. Khovanskii and later Gabrielov and Vorobjov have established many effective estimates for the geometric complexity of sub-Pfaffian sets in terms of these parameters. It is often important in applications that these estimates are polynomial in $D$. Despite much research done in this area, it is still not known whether cell decomposition, the foundational operation of o-minimal geometry, preserves polynomial dependence on $D$. We slightly modify the usual notions of format and degree and prove that with these revised notions, this does in fact hold. As one consequence, we also obtain the first polynomial (in $D$) upper bounds for the sum of Betti numbers of sets defined using quantified formulas in the restricted sub-Pfaffian structure.

Sheaves of continuous definable functions

1988 ◽  
Vol 53 (4) ◽  
pp. 1165-1169 ◽  
Author(s):  
Anand Pillay
Keyword(s):  
Real Closed Field ◽  
Spectral Space ◽  
Real Spectrum ◽  
Closed Field ◽  
Elementary Extension ◽  
Definable Set ◽  
Definable Subset ◽  
Skolem Functions ◽  
Minimal Structure ◽  
Definable Functions

Let M be an o-minimal structure or a p-adically closed field. Let be the space of complete n-types over M equipped with the following topology: The basic open sets of are of the form Ũ = {p ∈ Sn (M): U ∈ p} for U an open definable subset of Mn. is a spectral space. (For M = K a real closed field, is precisely the real spectrum of K[X1, …, Xn]; see [CR].) We will equip with a sheaf of LM-structures (where LM is a suitable language). Again for M a real closed field this corresponds to the structure sheaf on (see [S]). Our main point is that when Th(M) has definable Skolem functions, then if p ∈ , it follows that M(p), the definable ultrapower of M at p, can be factored through Mp, the stalk at p with respect to the above sheaf. This depends on the observation that if M ≺ N, a ∈ Nn and f is an M-definable (partial) function defined at a, then there is an open M-definable set U ⊂ Nn with a ∈ U, and a continuous M-definable function g:U → N such that g(a) = f(a).In the case that M is an o-minimal expansion of a real closed field (or M is a p-adically closed field), it turns out that M(p) can be recovered as the unique quotient of Mp which is an elementary extension of M.

scholarly journals Exploring the “minimal” structure of a functional ADAMTS13 by mutagenesis and small-angle X-ray scattering

Blood ◽  
2019 ◽  
Vol 133 (17) ◽  
pp. 1909-1918 ◽  
Author(s):  
Jian Zhu ◽  
Joshua Muia ◽  
Garima Gupta ◽  
Lisa A. Westfield ◽  
Karen Vanhoorelbeke ◽  
...  
Keyword(s):  
Small Angle ◽  
Columba Livia ◽  
Allosteric Activation ◽  
Complement Components ◽  
X Ray ◽  
X Ray Scattering ◽  
Morphogenetic Protein ◽  
Minimal Structure ◽  
Von Willebrand ◽  
Ray Scattering

Abstract Human ADAMTS13 is a multidomain protein with metalloprotease (M), disintegrin-like (D), thrombospondin-1 (T), Cys-rich (C), and spacer (S) domains, followed by 7 additional T domains and 2 CUB (complement components C1r and C1s, sea urchin protein Uegf, and bone morphogenetic protein-1) domains. ADAMTS13 inhibits the growth of von Willebrand factor (VWF)–platelet aggregates by cleaving the cryptic Tyr1605-Met1606 bond in the VWF A2 domain. ADAMTS13 is regulated by substrate-induced allosteric activation; without shear stress, the distal T8-CUB domains markedly inhibit VWF cleavage, and binding of VWF domain D4 or selected monoclonal antibodies (MAbs) to distal ADAMTS13 domains relieves this autoinhibition. By small angle X-ray scattering (SAXS), ADAMTS13 adopts a hairpin-like conformation with distal T7-CUB domains close to the proximal MDTCS domains and a hinge point between T4 and T5. The hairpin projects like a handle away from the core MDTCS and T7-CUB complex and contains distal T domains that are dispensable for allosteric regulation. Truncated constructs that lack the T8-CUB domains are not autoinhibited and cannot be activated by VWF D4 but retain the hairpin fold. Allosteric activation by VWF D4 requires T7, T8, and the 58–amino acid residue linker between T8 and CUB1. Deletion of T3 to T6 produced the smallest construct (delT3-6) examined that could be activated by MAbs and VWF D4. Columba livia (pigeon) ADAMTS13 (pADAMTS13) resembles human delT3-6, retains normal activation by VWF D4, and has a SAXS envelope consistent with amputation of the hairpin containing the dispensable T domains of human ADAMTS13. Our findings suggest that human delT3-6 and pADAMTS13 approach a “minimal” structure for allosterically regulated ADAMTS13.

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.

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