Eigenvalues of matrices related to the octonions

A pseudo real matrix representation of an octonion, which is based on two real matrix represen- tations of a quaternion, is considered. We study how some operations defined on the octonions change the set of eigenvalues of the matrix obtained if these operations are performed after or before the matrix representation. The established results could be of particular interest to researchers working on estimation algorithms involving such operations.info:eu-repo/semantics/publishedVersio

Bounds for the zeros of unilateral octonionic polynomials

In the present work it is proved that the zeros of a unilateral octo- nionic polynomial belong to the conjugacy classes of the latent roots of an appropriate lambda-matrix. This allows the use of matricial norms, and matrix norms in particular, to obtain upper and lower bounds for the zeros of unilateral octonionic polynomials. Some results valid for complex and/or matrix polynomials are extended to octonionic polyno- mials.R. Serˆodio and P. D. Beites were supported by FCT (Funda¸c˜ao para a Ciˆencia e a Tecnologia, Portugal), project UIDB/00212/2020 of CMA-UBI (Centro de Matem´atica e Aplica¸c˜oes da Universidade da Beira Interior). P. D. Beites was also supported by MINECO (Ministerio de Asuntos Econ´omicos y Transformaci´on Digital, Spain), project MTM2017-83506-C2-2-P. The authors would like to thank Maria Helena Serˆodio and Mary Garfield for clarifying some language queries.info:eu-repo/semantics/publishedVersio

Intersection of a Double Cone and a Line in the Split-Quaternions Context

This is a work on an application of the real split-quaternions to Spatial Analytic Geometry. Concretely, the intersection of a double cone and a line, which can be the empty set, a point, two points or a line, is studied in the real split-quaternions setting.MTM2013-45588-C3-1-P (Spain)info:eu-repo/semantics/publishedVersio

Uniqueness of the Fock quantization of a free scalar field on S1S^1 with time dependent mass

We analyze the quantum description of a free scalar field on the circle in the presence of an explicitly time dependent potential, also interpretable as a time dependent mass. Classically, the field satisfies a linear wave equation of the form $\ddot{\xi}-\xi"+f(t)\xi=0$. We prove that the representation of the canonical commutation relations corresponding to the particular case of a massless free field ($f=0$) provides a unitary implementation of the dynamics for sufficiently general mass terms, $f(t)$. Furthermore, this representation is uniquely specified, among the class of representations determined by $S^1$-invariant complex structures, as the only one allowing a unitary dynamics. These conclusions can be extended in fact to fields on the two-sphere possessing axial symmetry. This generalizes a uniqueness result previously obtained in the context of the quantum field description of the Gowdy cosmologies, in the case of linear polarization and for any of the possible topologies of the spatial sections.Comment: 13 pages, typos corrected, version accepted for publication in Physical Review

Convertible subspaces that arise from different numberings of the vertices of a graph

In this paper, we describe subspaces of generalized Hessenberg matrices where the determinant is convertible into the permanent by affixing ± signs. These subspaces can arise from different numberings of the vertices of a graph. With this numbering process, we obtain some well-known sequences of integers. For instance, in the case of a path of length n, we prove that the number of these subspaces is the (n + 1)th Fibonacci number.info:eu-repo/semantics/publishedVersio

Higher education methodologies: from urban planning to mathematical issues

This article will be focused on higher education learning and teaching methodologies, based on the experience of the Master Degree in Civil Engineering at the University of Beira Interior, in Covilhã (Portugal). It aims to present the results of the practises used by scholar of urban planning and mathematical issues, both regarding the civil engineering research field. Actually, there are some similarities in between the research process features of urban planning domain and mathematics field in order to promote the students’ success. In fact, these both scientific subjects follow analogous tasks in their research processes, not only regarding the same starting point which is the definition of the research problem, but also observing the final phase, which is based on the findings of results, or the proposed solution. It joins scholars from the department of civil engineering and architecture, experts in spatial analysis and scholars from the department of mathematics of the University of Beira Interior (UBI). One case study will be presented as an example and pioneer research of the application of these methodological approaches. It will be focused on the urban planning experiences, associated with postgraduate teachings. It is related to a PhD thesis in Civil Engineering focused on urban planning issues.info:eu-repo/semantics/publishedVersio

Research methodologies focused on urban planning and mathematical issues

This article aims to present an interdisciplinary approach about the research methodologies used at the civil engineering research field, in the domains of urban planning and mathematics. Actually, there are some similarities in between the research process features of urban planning and mathematics. In fact, these both scientific subjects follow analogous tasks in their research process, which have the same starting point with the definition of the research problem and the final phase, based on the proposed solution. It joins scholars from the department of civil engineering and architecture, experts in spatial analysis and scholars form the department of mathematics of the University of Beira Interior. Two case studies will be presented as examples of the application of these methodological approaches, both of them focused on the urban planning researches, associated with postgraduate teachings, one is related to a PhD thesis and the other one relates to a master degree dissertation.info:eu-repo/semantics/publishedVersio

Data’s Hidden Data: Qualitative Revelations of Sports Efficiency Analysis brought by Neural Network Performance Metrics

In the study of effectiveness and efficiency of an athlete’s performance, intelligent systems can be applied on qualitative approaches and their performance metrics provide useful information on not just the quality of the data, but also reveal issues about the observational criteria and data collection context itself. 2000 executions of two similar exercises, with different levels of complexity, were collected through a single inertial sensor applied on the fencer’s weapon hand. After the signals were split into their key segments through Dynamic Time Warping, the extracted features and respective qualitative evaluations were fed into a Neural Network to learn the patterns that distinguish a good from a bad execution. The performance analysis of the resulting models returned a prediction accuracy of 76.6% and 72.7% for each exercise, but other metrics pointed to the data suffering from high bias. This points towards an imbalance in the qualitative criteria representation of the bad executions, which can be explained by: i) reduced number of samples; ii) ambiguity in the definition of the observation criteria; iii) a single sensor being unable to fully capture the context without taking the actions of the other key body segments into account

Conjugacy and geometry I : foot of the perpendicular, distance and gram determinant

In this note on space geometry, the Gram determinant is used for expressing distances, vectors whose magnitude equals those distances and best approximation points. Three cases are considered: distances from a point to a line and to a plane and distances between two skew lines. (Symbolic) determinants occur in the expressions of the feet of perpendiculars and in the representation of the vectors materializing the distances. Because best approximation problems often require the use of subspaces, in order to solve the general cases of the proposed problems, we make extensive use of the conjugacy principle much present in Mathematics. The main purpose of this paper, focused on the resolution of distance problems in tridimensional geometry, is to provide the acquisition of spatial abilities through the proposed constructive approach. The obtained results, which could be a starting point and give clues for solving more advanced geometry problems, are applicable in several fields of practical sciences, such as the Coordinate Metrology, for instance. Moreover, this paper may be a window for coming across with a diversity of scalar products.info:eu-repo/semantics/publishedVersio