Geometrization of the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime

The geometrical Kosambi–Cartan–Chern approach has been applied to study the systems of differential equations which arise in quantum-mechanical problems of a particle on the background of non-Euclidean geometry. We calculate the geometrical invariants for the radial system of differential equations arising for electromagnetic and spinor fields on the background of the Schwarzschild spacetime. Because the second invariant is associated with the Jacobi field for geodesics deviation, we analyze its behavior in the vicinity of physically meaningful singular points r = M, ∞. We demonstrate that near the Schwarzschild horizon r = M the Jacobi instability exists and geodesics diverge for both considered problems.

RETROSPECTIVE AND CONTRADICTIONS OF CREATING ARCHITECTURAL PROJECTS IN THE CONTEXT OF DIGITALIZATION

The research article examines the retrospective of the creation of architectural projects, systematizes the experience of architects, reveals contradictions in the development of digital architecture and its relationship with traditional design. At the same time, the problems of the development of digital architecture in the context of the formation of digital civilization are noted. The tendencies of designing objects based on non-Euclidean geometry are revealed, the features of postmodernism and parametrism are systematized, the threats and consequences of the "from figure to form" approach are substantiated.

Data with Non-Euclidean Geometry and its Characterization

Recently, dealing the Non-Euclidean data and its characterization is considered as one of the major issues by researchers. The first problem arises while distinction of among Euclidean and non-Euclidean geometry. The second problem arises with dealing the Non-Euclidean geometry in true, false and uncertain regions. The third problem arises while investigating some pattern in Non-Euclidean data sets. This paper focused on tackling these issues with some real life examples.

Non-Euclidean Geometry Lesson Promotes Mathematical Reasoning

A two-day lesson on taxicab geometry introduces high school students to a unit on proof.

A matemática aplicada à astronomia para o ensino básico: concepções de discentes e relato de experiência de uma oficina

Resumo: Neste artigo relatamos uma experiência de realização de uma oficina para um grupo de 30 estudantes do Ensino Fundamental e Médio que participaram de um projeto de extensão da Universidade Estadual de Maringá (Paraná) denominado TIME – Teoria e Investigação em Matemática Elementar. Na oficina relacionamos matemática e astronomia com uma proposta de construção do instrumento Quadrante. Ademais, com a finalidade de entender as concepções dos discentes sobre geometria nãoeuclidiana e conhecimentos sobre astronomia, elaboramos um questionário respondido previamente. A investigação efetuada se baseou nos pressupostos da pesquisa qualitativa do tipo descritiva. As nossas observações mostram que o trabalho com a astronomia possibilitou aos estudantes desenvolverem noções sobre geometria não euclidiana e euclidiana para o entendimento do funcionamento do instrumento. Além disso, testificamos a fragilidade relacionada ao pouco conhecimento dos estudantes da Educação Básica no que diz respeito aos conceitos básicos de Astronomia.Palavras-chave: Geometria não Euclidiana; Astronomia de Posição; Instrumento Astronômico. Mathematics applied to astronomy in the framework of basic education: conceptions of discents and experience report of a workshopAbstract: In this article we report an experience of conducting a workshop aimed at a group of 30 elementary and high school students who participated in an extension project at the State University of Maringá (Paraná) called TIME – Teoria e Investigação em Matemática Elementar (Theory and Research in Elementary Mathematics). In the workshop we related mathematics and astronomy to a proposal to build the Quadrant instrument. Furthermore,in order to understand students’ conceptions about non-Euclidean geometry and knowledge about astronomy, we prepared a questionnaire previously answered. The investigation carried out is based on the assumptions of qualitative research of the descriptive type. Our observations show that working with astronomy enabled students to develop notions about non-Euclidean and Euclidean geometry to understand how the instrument works. In addition, we testify the fragility related to the lack of knowledge of Basic Education students regarding the basic concepts of Astronomy.Keywords: Non-Euclidean Geometry; Positional Astronomy; Astronomical Instrument.

Knowledge Representation of Mathematics Education Program among Students in Euclidean Parallelism

This research aims to reveal students' perception-based knowledge representation from mathematics programs in Euclidean Parallelism. Students were asked to do parallelism exercises presented in a verbal and picture form. The data were analyzed by knowledge representation theory based on meaning and perception. There were students who have amodal-multimodal-transition hypothesis. Students' assumptions about the verbal and picture information of not-perfectly-drawn parallel lines varied: assuming that angles appear to be the same measure are congruent, the two lines would intersect, considering the two lines parallel but redrawing picture to determine the pair of congruent angles and using other perspectives to interpret the picture. This study recommends action research for geometry learning that provides not-perfectly-drawn parallel lines for students who have amodal and amodalmultimodal- transition hypothesis and observe its effect on their non-Euclidean geometry learning. It may also familiarize students with getting to know parallelism in R3.

On properties of h-differentiable functions

Research in the theory of functions of an h-complex variable is of interest in connection with existing applications in non-Euclidean geometry, theoretical mechanics, etc. This article is devoted to the study of the properties of h-differentiable functions. Criteria for h-differentiability and h-holomorphy are found, formulated and proved a theorem on finite increments for an h-holomorphic function. Sufficient conditions for h-analyticity are given, formulated and proved a uniqueness theorem for h-analytic functions.