Stiefel Whitney classes for real representations of GL2(𝔽q)

We compute the total Stiefel Whitney class for a real representation [Formula: see text] of [Formula: see text], where [Formula: see text] is odd. The obstruction class of [Formula: see text] is defined to be the Stiefel Whitney class of lowest positive degree that does not vanish. We provide an expression for the obstruction class of [Formula: see text] in terms of its character values if [Formula: see text].

On Hermitian Solutions of the Generalized Quaternion Matrix Equation A X B + C X D = E

The paper deals with the matrix equation A X B + C X D = E over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and sufficient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a unificated algebraic technique for finding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.

Analysis of the Possibilities of Tire-Defect Inspection Based on Unsupervised Learning and Deep Learning

At present, inspection systems process visual data captured by cameras, with deep learning approaches applied to detect defects. Defect detection results usually have an accuracy higher than 94%. Real-life applications, however, are not very common. In this paper, we describe the development of a tire inspection system for the tire industry. We provide methods for processing tire sidewall data obtained from a camera and a laser sensor. The captured data comprise visual and geometric data characterizing the tire surface, providing a real representation of the captured tire sidewall. We use an unfolding process, that is, a polar transform, to further process the camera-obtained data. The principles and automation of the designed polar transform, based on polynomial regression (i.e., supervised learning), are presented. Based on the data from the laser sensor, the detection of abnormalities is performed using an unsupervised clustering method, followed by the classification of defects using the VGG-16 neural network. The inspection system aims to detect trained and untrained abnormalities, namely defects, as opposed to using only supervised learning methods.

Towards Unification of General Relativity and Quantum Theory: Dendrogram Representation of the Event-Universe

Following Smolin, we proceed to unification of general relativity and quantum theory by operating solely with events, i.e., without appealing to physical systems and space-time. The universe is modelled as a dendrogram (finite tree) expressing the hierarchic relations between events. This is the observational (epistemic) model; the ontic model is based on p-adic numbers (infinite trees). Hence, we use novel mathematics—not only space-time but even real numbers are not in use. Here, the p-adic space (which is zero dimensional) serves as the base for the holographic image of the universe. In this way our theory relates to p-adic physics; in particular, p-adic string theory and complex disordered systems (p-adic representation of Parisi matrix for spin glasses). Our Dendrogramic-Holographic (DH) theory matches perfectly with the Mach’s principle and Brans-Dicke theory. We found surprising informational interrelation between the fundamental constants, h, c, G, and their DH-analogues, h(D), c(D), G(D). DH-theory is part of Wheeler’s project on the information restructuring of physics. It is also a step towards the Unified Field theory. The universal potential V is nonlocal, but this is relational DH-nonlocality. V can be coupled to the Bohm quantum potential by moving to the real representation. This coupling enhanced the role of the Bohm potential.

Three special kinds of least squares solutions for the quaternion generalized Sylvester matrix equation

<abstract><p>In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation $ AXB+CYD = E $. By integrating real representation of a quaternion matrix with $ \mathcal{H} $-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply $ \mathcal{H} $-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of $ \mathcal{H} $-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.</p></abstract>

Principles for certain geomechanics problems solution during overworking of mine workings

This paper deals with two relevant and interrelated directions of coal mining development. Firstly, the resource-saving complete extraction of reserves, and secondly, the intensification of mining activity through mining operations conducted on several seams. The objective of overworking of mine workings with a real representation of mining-and-geological and mining-engineering conditions has been formulated. Five peculiarities of the geomechanical models construction for overworking of mine works have been revealed and substantiated. The technology for performing a series of computational experiments is proposed. The task of the protecting pillar influence, which is located in overlying coal seam, on the state of mine working placed in the underlying coal seam has been solved. An optimal scheme for its maintenance is proposed.

The least squares Bisymmetric solution of quaternion matrix equation $ AXB = C $

<abstract><p>In this paper, the idea of partitioning is used to solve quaternion least squares problem, we divide the quaternion Bisymmetric matrix into four blocks and study the relationship between the block matrices. Applying this relation, the real representation of quaternion, and M-P inverse, we obtain the least squares Bisymmetric solution of quaternion matrix equation $ AXB = C $ and its compatable conditions. Finally, we verify the effectiveness of the method through numerical examples.</p></abstract>

Educating creative activity as a factor in the civilization of an individual

The article reveals the problem of educating creative activity as a factor in the civil socialization of the individual comprehends the phenomenon of creative activity and civil socialization, interprets their real representation and sociocultural potential for the development of civil society in modern Russia. The authors analyze the positions of scientists on the essence of the key terms of this problem and the results of a survey of students on the individual activity, factors that contribute to and inhibit the successful socialization of youth, as well as effective forms, methods, means of educating creative activity, on the importance of project, research, volunteer, analytical, evaluative and reflective activities in the educational process of the university. The main problems and prospects for the creative activity in the sociocultural space are summarized.

Algebraic Techniques for Least Squares Problems in Elliptic Complex Matrix Theory and Their Applications

In this study, we introduce concepts of norms of elliptic complex matrices and derive the least squares solution, the pure imaginary least squares solution, and the pure real least squares solution with the least norm for the elliptic complex matrix equation AX=B by using the real representation of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic numbers are generalized form of complex and so real numbers. Thus, the obtained results extend, generalize and complement some known least squares solutions results from the literature.