Adic spaces

2020 ◽  
pp. 7-16
Author(s):  
Peter Scholze ◽  
Jared Weinstein
Keyword(s):  
Formal Schemes ◽  
Geometric Objects ◽  
Analytic Varieties

This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.

Meet fractal curves with 1C:MathKit

2020 ◽  
pp. 38-48
Author(s):  
O. M. Korchazhkina
Keyword(s):  
Methodological Approach ◽  
Fractal Curve ◽  
Iterative Processes ◽  
Geometric Figures ◽  
Modern Natural ◽  
Geometric Objects ◽  
Ict Tools ◽  
Subject Areas ◽  
The Subject ◽  
And Mathematics

The article presents a methodological approach to studying iterative processes in the school course of geometry, by the example of constructing a Koch snowflake fractal curve and calculating a few characteristics of it. The interactive creative environment 1C:MathKit is chosen to visualize the method discussed. By performing repetitive constructions and algebraic calculations using ICT tools, students acquire a steady skill of work with geometric objects of various levels of complexity, comprehend the possibilities of mathematical interpretation of iterative processes in practice, and learn how to understand the dialectical unity between finite and infinite parameters of flat geometric figures. When students are getting familiar with such contradictory concepts and categories, that replenishes their experience of worldview comprehension of the subject areas they study through the concept of “big ideas”. The latter allows them to take a fresh look at the processes in the world around. The article is a matter of interest to schoolteachers of computer science and mathematics, as well as university scholars who teach the course “Concepts of modern natural sciences”.

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals Edge Models with the CAD Software: Creating a New Context for Mathematics in Elementary School

2021 ◽  
Author(s):  
Felicitas Pielsticker ◽  
Ingo Witzke ◽  
Amelie Vogler
Keyword(s):  
Elementary Schools ◽  
Digital Media ◽  
Computer Aided Design ◽  
Subjective Experience ◽  
Fourth Graders ◽  
Point Of View ◽  
Toulmin Model ◽  
Geometric Objects ◽  
Design Software ◽  
Aided Design

AbstractDigital media have become increasingly important in recent years and can offer new possibilities for mathematics education in elementary schools. From our point of view, geometry and geometric objects seem to be suitable for the use of computer-aided design software in mathematics classes. Based on the example of Tinkercad, the use of CAD software — a new and challenging context in elementary schools — is discussed within the approach of domains of subjective experience and the Toulmin model. An empirical study examined the influence of Tinkercad on fourth-graders’ development of a model of a geometric solid and related reasoning processes in mathematics classes.

The geometry of H4 polytopes

2020 ◽  
Vol 20 (3) ◽  
pp. 433-444
Author(s):  
Tomme Denney ◽  
Da’Shay Hooker ◽  
De’Janeke Johnson ◽  
Tianna Robinson ◽  
Majid Butler ◽  
...  
Keyword(s):  
Unimodular Lattice ◽  
Geometric Objects ◽  
Even Unimodular Lattice

AbstractWe describe the geometry of an arrangement of 24-cells inscribed in the 600-cell. In Section 7 we apply our results to the even unimodular lattice E8 and show how the 600-cell transforms E8/2E8, an 8-space over the field F2, into a 4-space over F4 whose points, lines and planes are labeled by the geometric objects of the 600-cell.

scholarly journals The geometry of null-like disformal transformations

2019 ◽  
Vol 16 (11) ◽  
pp. 1950180 ◽  
Author(s):  
I. P. Lobo ◽  
G. G. Carvalho
Keyword(s):  
Vector Field ◽  
Conformal Transformation ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Fields ◽  
Spinor Structure ◽  
Schwarzschild Metric ◽  
Killing Vectors ◽  
Symmetry Properties ◽  
Geometric Objects

Motivated by the hindrance of defining metric tensors compatible with the underlying spinor structure, other than the ones obtained via a conformal transformation, we study how some geometric objects are affected by the action of a disformal transformation in the closest scenario possible: the disformal transformation in the direction of a null-like vector field. Subsequently, we analyze symmetry properties such as mutual geodesics and mutual Killing vectors, generalized Weyl transformations that leave the disformal relation invariant, and introduce the concept of disformal Killing vector fields. In most cases, we use the Schwarzschild metric, in the Kerr–Schild formulation, to verify our calculations and results. We also revisit the disformal operator using a Newman–Penrose basis to show that, in the null-like case, this operator is not diagonalizable.

scholarly journals EXTENDED ABSOLUTE PARALLELISM GEOMETRY

2008 ◽  
Vol 05 (07) ◽  
pp. 1109-1135 ◽  
Author(s):  
NABIL. L. YOUSSEF ◽  
A. M. SID-AHMED
Keyword(s):  
Tangent Bundle ◽  
Special Form ◽  
Vector Fields ◽  
A Priori ◽  
Building Blocks ◽  
Absolute Parallelism ◽  
Linearly Independent ◽  
Geometric Objects ◽  
Directional Argument ◽  
Curvature Tensors

In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.

ProgramCRYC3Dfor geometric computing in crystallography

2008 ◽  
Vol 41 (2) ◽  
pp. 479-480 ◽  
Author(s):  
Ludmila Urzhumtseva ◽  
Alexandre Urzhumtsev
Keyword(s):  
Computer Program ◽  
Three Dimensional ◽  
Teaching Tool ◽  
Online Help ◽  
Principal Mode ◽  
Geometric Characteristics ◽  
Simple Program ◽  
Or Groups ◽  
Geometric Objects ◽  
Everyday Work

The computer programCRYC3Dworks with three-dimensional crystallographic geometric objects or groups of them and calculates their basic geometric characteristics by a simple click in the menu. In particular, this includes vector operations in both direct and reciprocal spaces and cell transformations. Collecting basic crystallographic operations in a single and simple program helps crystallographers to avoid looking for `fast-and-dirty' scripts or using large and unwieldy packages and may be useful in everyday work. When running the program in its principal mode, macro-operations are accompanied by a list of elementary geometric operations. This feature, together with the presence of a single-command mode and online help, may be useful also as a teaching tool.

scholarly journals Entropy and Geometric Objects

Proceedings ◽  
2017 ◽  
Vol 2 (4) ◽  
pp. 153
Author(s):  
Georg J. Schmitz
Keyword(s):  
Geometric Objects

scholarly journals A motivic version of the theorem of Fontaine and Wintenberger

2018 ◽  
Vol 155 (1) ◽  
pp. 38-88 ◽  
Author(s):  
Alberto Vezzani
Keyword(s):  
Galois Groups ◽  
Rational Homotopy ◽  
Mixed Characteristic ◽  
Homotopy Invariant ◽  
Analytic Varieties

We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field $K$ of mixed characteristic and over the associated (tilted) perfectoid field $K^{\flat }$ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of $K$ and $K^{\flat }$ are isomorphic.

Sign in / Sign up

Export Citation Format

Share Document