The Elements of the Theory of Geometric Constructions

2001 ◽  
pp. 11-16
Keyword(s):  
Geometric Constructions

scholarly journals A note on moduli spaces of conformal classes for flat tori of higher dimension and on their conformal multiplication

2021 ◽  
Author(s):  
Anna Gori ◽  
Alberto Verjovsky ◽  
Fabio Vlacci
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Higher Dimension ◽  
Geometric Constructions ◽  
Flat Torus ◽  
Flat Tori

AbstractMotivated by the theory of complex multiplication of abelian varieties, in this paper we study the conformality classes of flat tori in $${\mathbb {R}}^{n}$$ R n and investigate criteria to determine whether a n-dimensional flat torus has non trivial (i.e. bigger than $${\mathbb {Z}}^{*}={\mathbb {Z}}{\setminus }\{0\}$$ Z ∗ = Z \ { 0 } ) semigroup of conformal endomorphisms (the analogs of isogenies for abelian varieties). We then exhibit several geometric constructions of tori with this property and study the class of conformally equivalent lattices in order to describe the moduli space of the corresponding tori.

Geometric Constructions of Gallager Codes

2003 ◽  
pp. 147-162
Author(s):  
Shu Lin ◽  
Jun Xu ◽  
Heng Tang ◽  
Yu Kou
Keyword(s):  
Geometric Constructions ◽  
Gallager Codes

scholarly journals THE LOCALLY COVARIANT DIRAC FIELD

2010 ◽  
Vol 22 (04) ◽  
pp. 381-430 ◽  
Author(s):  
KO SANDERS
Keyword(s):  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Dirac Field ◽  
Quantum Field ◽  
Geometric Constructions ◽  
Covariant Quantum ◽  
Dimensional Spacetime ◽  
Stress Energy ◽  
Hadamard States ◽  
Stress Energy Momentum Tensor

We describe the free Dirac field in a four-dimensional spacetime as a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, using a representation independent construction. The freedom in the geometric constructions involved can be encoded in terms of the cohomology of the category of spin spacetimes. If we restrict ourselves to the observable algebra, the cohomological obstructions vanish and the theory is unique. We establish some basic properties of the theory and discuss the class of Hadamard states, filling some technical gaps in the literature. Finally, we show that the relative Cauchy evolution yields commutators with the stress-energy-momentum tensor, as in the scalar field case.

scholarly journals Novel pedagogical tool for simultaneous learning of plane geometry and R programming

2018 ◽  
Vol 4 ◽  
pp. e25485 ◽  
Author(s):  
Álvaro Briz-Redón ◽  
Ángel Serrano-Aroca
Keyword(s):  
Programming Language ◽  
Undergraduate Students ◽  
R Package ◽  
Plane Geometry ◽  
Cost Ratio ◽  
Geometric Constructions ◽  
Pedagogical Tool ◽  
Benefit Cost Ratio ◽  
Wide Range ◽  
R Programming

Programming a computer is an activity that can be very beneficial to undergraduate students in terms of improving their mental capabilities, collaborative attitudes and levels of engagement in learning. Despite the initial difficulties that typically arise when learning to program, there are several well-known strategies to overcome them, providing a very high benefit-cost ratio to most of the students. Moreover, the use of a programming language usually raises the interest of students to learn any specific concept, which has caused that many teachers around the world employ a programming language as a learning environment to treat almost every possible topic. Particularly, mathematics can be taught and learnt while using a suitable programming language. The R programming language is endowed with a wide range of capabilities that allow its use to learn different kind of concepts while programming. Therefore, complex subjects such as mathematics could be learnt with the help of this powerful programming language. In addition, since the R language provides numerous graphical functions, it could be very useful to acquire simultaneously basic plane geometry and programming knowledge at the undergraduate level. This paper describes the LearnGeom R package, a novel pedagogical tool, which contains multiple functions to learn geometry in R at different levels of difficulty, from the most basic geometric objects to high-complexity geometric constructions, while developing numerous programming skills.

scholarly journals Some geometric constructions on Frobenius Weil bundles

2014 ◽  
Vol 35 ◽  
pp. 143-149
Author(s):  
Miroslav Doupovec ◽  
Miroslav Kureš
Keyword(s):  
Geometric Constructions

Features of Distance Learning in Geometric and Graphic Disciplines Using Methods of Constructive Geometric Modeling

2021 ◽  
pp. 46-56
Author(s):  
E. Boyashova
Keyword(s):  
Distance Learning ◽  
Geometric Modeling ◽  
Information Technologies ◽  
Graphical Representation ◽  
Teaching Experience ◽  
Educational Process ◽  
Descriptive Geometry ◽  
Geometric Constructions ◽  
Professional Activities

The article is devoted to the peculiarities of teaching the discipline "Descriptive geometry" in the conditions of distance learning, it examines the application of information technologies in the educational process in geometric and graphic disciplines. Increasing the speed of information processes, reducing the number of hours for mastering the discipline. the conditions of distance learning set new tasks for teachers and dictate their requirements for teaching graphic disciplines and the use of teaching experience in a new reality; there is a need to introduce and develop new forms of education without losing the quality of education. Geometric-graphic disciplines occupy one of the important places in technical education, the complexity of the study of which lies in the development of a graphical representation of phenomena, objects and processes by methods of constructive geometric modeling. The knowledge and skills acquired by students contribute to the development of spatial, imaginative and rational thinking, which is necessary for future professional activities. Descriptive geometry is a discipline that is not easy to master on your own without a conscious understanding of the logic and sequence of geometric constructions, without deep knowledge of theoretical foundations and constant, repeated implementation of practical tasks. The acquisition of practical skills in mastering the methods of discipline has become more difficult in the current epidemiological situation. In modern conditions of distance learning, the use of the Simplex geometric modeling system made it possible to develop and propose a new concept of geometric-graphic interaction, which significantly reduced the time for completing and checking educational tasks in real time. The proposed technology reveals the deep informational essence of the studied discipline "Descriptive Geometry" and becomes a powerful research tool for students. The integration of traditional teaching methods in the graphic preparation of students with computer and communication facilities increases the possibilities of communication and improves the quality of teaching.

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