scholarly journals LOCALLY CONSTANT FUNCTIONS IN C-MINIMAL STRUCTURES

2015 ◽  
Vol 80 (1) ◽  
pp. 207-220
Author(s):  
PABLO CUBIDES KOVACSICS
Keyword(s):  
Ultrametric Space ◽  
Valued Fields ◽  
Minimal Structure ◽  
Canonical Tree

AbstractLet M be a C-minimal structure and T its canonical tree (which corresponds in an ultrametric space to the set of closed balls with radius different than ∞ ordered by inclusion). We present a description of definable locally constant functions f : M → T in C-minimal structures having a canonical tree with infinitely many branches at each node and densely ordered branches. This provides both a description of definable subsets of T in one variable and analogues of known results in algebraically closed valued fields.

Preliminaries

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Valued Fields ◽  
Definable Set ◽  
Valued Field ◽  
Galois Coverings ◽  
Definable Type ◽  
Definable Sets ◽  
Background Material

This chapter provides some background material on definable sets, definable types, orthogonality to a definable set, and stable domination, especially in the valued field context. It considers more specifically these concepts in the framework of the theory ACVF of algebraically closed valued fields and describes the definable types concentrating on a stable definable V as an ind-definable set. It also proves a key result that demonstrates definable types as integrals of stably dominated types along some definable type on the value group sort. Finally, it discusses the notion of pseudo-Galois coverings. Every nonempty definable set over an algebraically closed substructure of a model of ACVF extends to a definable type.

Existence

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Division Algebra ◽  
Quadratic Forms ◽  
Structure Theorem ◽  
Quadratic Extension ◽  
Quadratic Space ◽  
Division Algebras ◽  
Separable Extension ◽  
Valued Fields ◽  
Quaternion Division Algebra

This chapter proves that Bruhat-Tits buildings exist. It begins with a few definitions and simple observations about quadratic forms, including a 1-fold Pfister form, followed by a discussion of the existence part of the Structure Theorem for complete discretely valued fields due to H. Hasse and F. K. Schmidt. It then considers the generic unramified cases; the generic semi-ramified cases, the generic ramified cases, the wild unramified cases, the wild semi-ramified cases, and the wild ramified cases. These cases range from a unique unramified quadratic space to an unramified separable quadratic extension, a tamely ramified division algebra, a ramified separable quadratic extension, and a unique unramified quaternion division algebra. The chapter also describes ramified quaternion division algebras D₁, D₂, and D₃ over K containing a common subfield E such that E/K is a ramified separable extension.

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.

Fuzzy linear spaces over valued fields

1991 ◽  
Vol 42 (3) ◽  
pp. 351-354 ◽  
Author(s):  
Sudarsan Nanda
Keyword(s):  
Linear Spaces ◽  
Valued Fields
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