scholarly journals Geometric Invariants Under the Möbius Action of the Group SL(2;R)

2021 ◽  
Vol 45 (6) ◽  
pp. 925-941
Author(s):  
DEBAPRIYA BISWAS ◽  
◽  
SANDIPAN DUTTA
Keyword(s):  
Homogeneous Spaces ◽  
Geometric Invariants ◽  
Double Numbers ◽  
Erlangen Program ◽  
Geometric Objects

In this paper we have introduced new invariant geometric objects in the homogeneous spaces of complex, dual and double numbers for the principal group SL(2; ℝ), in the Klein’s Erlangen Program. We have considered the action as the Möbius action and have taken the spaces as the spaces of complex, dual and double numbers. Some new decompositions of SL(2; ℝ) have been used.

A hierarchy of classification methods for patterns*

1981 ◽  
Vol 154 (3-4) ◽  
Author(s):  
Branko Grünbaum ◽  
G. C Shephard
Keyword(s):  
Symmetry Groups ◽  
Classification Methods ◽  
Methodical Approach ◽  
Erlangen Program ◽  
Geometric Objects ◽  
Crystallographic Symmetry ◽  
Circle Packings ◽  
Systematic Framework

AbstractIn the spirit of Felix Klein's “Erlangen Program” a methodical approach is developed to three levels of classification of geometric objects. These three levels are successive refinements of the classification by the crystallographic (symmetry) groups. The approach provides a systematic framework for most kinds of previously classified objects, such as dot patterns (= lattice complexes), circle or ellipse patterns (= circle packings, ellipse packings), tilings (tessellations) with equivalent tiles (or edges, or vertices), and many others.

Computational Micrograph Registration with Sieve Processes

1996 ◽  
Vol 54 ◽  
pp. 440-441
Author(s):  
P.J. Phillips ◽  
J. Huang ◽  
S. M. Dunn
Keyword(s):  
Planar Graph ◽  
Efficient Algorithm ◽  
Spatial Organization ◽  
Spatial Arrangement ◽  
Stereo Image ◽  
Image Model ◽  
Geometric Invariants ◽  
Point Set

In this paper we present an efficient algorithm for automatically finding the correspondence between pairs of stereo micrographs, the key step in forming a stereo image. The computation burden in this problem is solving for the optimal mapping and transformation between the two micrographs. In this paper, we present a sieve algorithm for efficiently estimating the transformation and correspondence.In a sieve algorithm, a sequence of stages gradually reduce the number of transformations and correspondences that need to be examined, i.e., the analogy of sieving through the set of mappings with gradually finer meshes until the answer is found. The set of sieves is derived from an image model, here a planar graph that encodes the spatial organization of the features. In the sieve algorithm, the graph represents the spatial arrangement of objects in the image. The algorithm for finding the correspondence restricts its attention to the graph, with the correspondence being found by a combination of graph matchings, point set matching and geometric invariants.

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.

scholarly journals Gauge × gauge = gravity on homogeneous spaces using tensor convolutions

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
L. Borsten ◽  
I. Jubb ◽  
V. Makwana ◽  
S. Nagy
Keyword(s):  
Homogeneous Spaces ◽  
Gauge Fields ◽  
Convolution Product ◽  
Tensor Fields ◽  
Einstein Universe ◽  
Gauge Transformations ◽  
Yang Mills ◽  
Static Einstein Universe ◽  
Definition Of ◽  
Gauge Gravity

Abstract A definition of a convolution of tensor fields on group manifolds is given, which is then generalised to generic homogeneous spaces. This is applied to the product of gauge fields in the context of ‘gravity = gauge × gauge’. In particular, it is shown that the linear Becchi-Rouet-Stora-Tyutin (BRST) gauge transformations of two Yang-Mills gauge fields generate the linear BRST diffeomorphism transformations of the graviton. This facilitates the definition of the ‘gauge × gauge’ convolution product on, for example, the static Einstein universe, and more generally for ultrastatic spacetimes with compact spatial slices.

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