scholarly journals Unexpected imaginaries in valued fields with analytic structure

2013 ◽  
Vol 78 (2) ◽  
pp. 523-542 ◽  
Author(s):  
Deirdre Haskell ◽  
Ehud Hrushovski ◽  
Dugald Macpherson
Keyword(s):  
Analytic Structure ◽  
Valued Fields ◽  
Algebraic Setting

AbstractWe give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries in the corresponding algebraic setting.

scholarly journals Real closed valued fields with analytic structure

2019 ◽  
Vol 63 (1) ◽  
pp. 249-261
Author(s):  
Pablo Cubides Kovacsics ◽  
Deirdre Haskell
Keyword(s):  
Short Proof ◽  
Quantifier Elimination ◽  
Analytic Structure ◽  
Valued Fields

AbstractWe show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are C-minimal.

scholarly journals SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS

2015 ◽  
Vol 16 (3) ◽  
pp. 447-499 ◽  
Author(s):  
Silvain Rideau
Keyword(s):  
Algebraic Closure ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Analytic Structure ◽  
Independence Property ◽  
Valued Fields ◽  
First Order ◽  
Witt Vectors ◽  
First Order Theory

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

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 One-loop string corrections to AdS amplitudes from CFT

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
J.M. Drummond ◽  
H. Paul
Keyword(s):  
Stress Tensor ◽  
Analytic Structure ◽  
Position Space ◽  
Yang Mills ◽  
String Corrections ◽  
The One ◽  
Loop Amplitudes

Abstract We consider α′ corrections to the one-loop four-point correlator of the stress- tensor multiplets in $$ \mathcal{N} $$ N = 4 super Yang-Mills at order 1/N4. Holographically, this is dual to string corrections of the one-loop supergravity amplitude on AdS5 × S5. While this correlator has been considered in Mellin space before, we derive the corresponding position space results, gaining new insights into the analytic structure of AdS loop amplitudes. Most notably, the presence of a transcendental weight three function involving new singularities is required, which has not appeared in the context of AdS amplitudes before. We thereby confirm the structure of string corrected one-loop Mellin amplitudes, and also provide new explicit results at orders in α′ not considered before.

Liquid power: reading the infinity pool as a global semioscape

2021 ◽  
pp. 147035722098482
Author(s):  
Crispin Thurlow
Keyword(s):  
Digital Media ◽  
Cultural Politics ◽  
Primary Objective ◽  
Analytic Structure ◽  
Visual Texts ◽  
Visual Material ◽  
The World

The analytic focus of this article is the highly fashionable ‘infinity pool’, treated here as a visual-material realization of the cultural politics of super-elite mobility. The article is organized around a three-step analytic structure. First, I demonstrate how the infinity pool is mediatized as a status marker, and thus circulated and normalized. Second, I pinpoint the semiotic and ideological ways the infinity pool emerges as a mediated practice. Third, I examines how the infinity pool is also remediated on Instagram and thereby broadcast anew. Throughout, I evidence my analysis with visual texts drawn from a range of commercial, situated and digital media sources. My primary objective is to show how the infinity pool, as a mediatized, mediated and remediated practice, feeds the global semioscape, that more informal, often banal plane of cultural circulation where images, ideas and aesthetic ideals seed themselves all over the place. In this way, and however frivolous or innocuous infinity pools may seem, they also spread a particularly privileged way of looking at, and being in, the world.

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