Twisted spectral triple for the Standard Model and spontaneous breaking of the Grand Symmetry

Grand symmetry models in noncommutative geometry have been introduced to explain how to generate minimally (i.e. without adding new fermions) an extra scalar field beyond the standard model, which both stabilizes the electroweak vacuum and makes the computation of the mass of the Higgs compatible with its experimental value. In this paper, we use Connes-Moscovici twisted spectral triples to cure a technical problem of the grand symmetry, that is the appearance together with the extra scalar field of unbounded vectorial terms. The twist makes these terms bounded and - thanks to a twisted version of the first-order condition that we introduce here - also permits to understand the breaking to the standard model as a dynamical process induced by the spectral action. This is a spontaneous breaking from a pre-geometric Pati-Salam model to the almost-commutative geometry of the standard model, with two Higgs-like fields: scalar and vector.Comment: References updated, misprint corrected. One paragraph added at the end of the paper to discuss results in the literature since the first version of the paper. 39 pages in Mathematical Physics, Analysis and Geometry (2016

Mathematics of CLIFFORD - A Maple package for Clifford and Grassmann algebras

CLIFFORD performs various computations in Grassmann and Clifford algebras. It can compute with quaternions, octonions, and matrices with entries in Cl(B) - the Clifford algebra of a vector space V endowed with an arbitrary bilinear form B. Two user-selectable algorithms for Clifford product are implemented: 'cmulNUM' - based on Chevalley's recursive formula, and 'cmulRS' - based on non-recursive Rota-Stein sausage. Grassmann and Clifford bases can be used. Properties of reversion in undotted and dotted wedge bases are discussed.Comment: 24 pages, update contains new material included in published versio

Manufacturing a mathematical group: a study in heuristics

I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forwar

On Geometric Applications of Quaternions

Quaternions have become a popular and powerful tool in various engineering fields, such as robotics, image and signal processing, and computer graphics. However, classical quaternions are mostly used as a representation of rotation of a vector in 3-dimensions, and connection between its geometric interpretation and algebraic structures is still not well-developed and needs more improvements. In this study, we develop an approach to understand quaternions multiplication defining subspaces of quaternion H, called as Plane(N) and Line(N), and then, we observe the effects of sandwiching maps on the elements of these subspaces. Finally, we give representations of some transformations in geometry using quaternion

Quaternion Neural Network with Temporal Feedback Calculation: Application to Cardiac Vector Velocity during Myocardial Infarction

Quaternion neural networks have been shownto be useful in image and signal processing applications.Herein, we propose a novel architecture of a neural unit model characterized by its ability of encoding 3-dimensional past information and that facilitates the learning of velocity patterns. We evaluate the implementation of the networkin a study of the cardiac vector velocity and its usefulness in early detection of patients with anterior myocardial infarction. Experimental results show an improvement of the performance in terms of convergence speed and precision when comparing with traditional methods. Furthermore, the network shows successful results in measuring the velocity reduction that is usually observed in vectorcardiogram signals in the presence of myocardial damage. Through a linear discriminant analysis, a pair of 100% / 98% of sensitivity/specificity is met with only two velocity parameters. We conclude that this method is a very promising developmentfor future computational tools devoted to early diagnosis ofheart diseases.Fil: Cruces, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Correa Prado, Raul Oscar. Universidad Nacional de San Juan. Facultad de Ingeniería. Departamento de Electrónica y Automática. Gabinete de Tecnología Médica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Laciar Leber, Eric. Universidad Nacional de San Juan. Facultad de Ingeniería. Departamento de Electrónica y Automática. Gabinete de Tecnología Médica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Arini, Pedro David. Universidad de Buenos Aires. Facultad de Ingeniería. Instituto de Ingeniería Biomédica; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaXXI Congreso Argentino de BioingenieríaCórdobaArgentinaSociedad Argentina de BioingenieríaUniversidad Nacional de Córdob

Memristive crossbars as hardware accelerators: modelling, design and new uses

Digital electronics has given rise to reliable, affordable, and scalable computing devices. However, new computing paradigms present challenges. For example, machine learning requires repeatedly processing large amounts of data; this creates a bottleneck in conventional computers, where computing and memory are separated. To add to that, Moore’s “law” is plateauing and is thus unlikely to address the increasing demand for computational power. In-memory computing, and specifically hardware accelerators for linear algebra, may address both of these issues. Memristive crossbar arrays are a promising candidate for such hardware accelerators. Memristive devices are fast, energy-efficient, and—when arranged in a crossbar structure—can compute vector-matrix products. Unfortunately, they come with their own set of limitations. The analogue nature of these devices makes them stochastic and thus less reliable compared to digital devices. It does not, however, necessarily make them unsuitable for computing. Nevertheless, successful deployment of analogue hardware accelerators requires a proper understanding of their drawbacks, ways of mitigating the effects of undesired physical behaviour, and applications where some degree of stochasticity is tolerable. In this thesis, I investigate the effects of nonidealities in memristive crossbar arrays, introduce techniques of minimising those negative effects, and present novel crossbar circuit designs for new applications. I mostly focus on physical implementations of neural networks and investigate the influence of device nonidealities on classification accuracy. To make memristive neural networks more reliable, I explore committee machines, rearrangement of crossbar lines, nonideality-aware training, and other techniques. I find that they all may contribute to the higher accuracy of physically implemented neural networks, often comparable to the accuracy of their digital counterparts. Finally, I introduce circuits that extend dot product computations to higher-rank arrays, different linear algebra operations, and quaternion vectors and matrices. These present opportunities for using crossbar arrays in new ways, including the processing of coloured images

Singularity-free methods for aircraft flight path optimization using Euler angles and quaternions

The purpose of this work is to rewrite the equations of motion such that even third-class trajectories can be optimized with the current parameter optimization methods. At first the commonly used coordinate systems and Euler angles are presented in Section II. It will be realized that the definition of the Euler angles introduces additional singularities. A short derivation of the commonly-used equations of motion follows for comparison and better understanding of the later derived sets of equations of motion. Section II closes with a reduction of the optimal control problem to a parameter optimization problem. Some characteristic properties and assumptions of the parameter optimization problem are pointed out along with the necessary equations and conditions needed to solve it. Section III introduces several methods that allow integration of second- and third-class trajectories as long as some restrictions are imposed on the allowable trajectories. The first method is the so-called inertial-acceleration method. It is based on the idea that the velocity yaw angle and the velocity pitch angle can be replaced by the velocity components as measured in an inertial reference frame. The so-called two-system method is derived next. It employes the idea of having two sets of equations of motion derived in different reference frames, and thus, having their singularities at different points. In detailed discussions the problems that appear with both methods are explained, and solutions are presented, the emphasis always being on the use of these equations with optimization methods. Section III also includes a method that allows integration of third-class trajectories as long as they can be flown in the vertical plane. This method results directly from the commonly-used equations of motion after removing a restriction on the flight path angle. Because all methods of Section III have still the bank angle as the control, they are referred to here as Euler-angle methods. Section IV presents the quaternion method. Although this method has been investigated first, it is presented last because it yields the best overall solution and because many details and improvements were not found until the other methods were analyzed. Understanding the Euler-angle methods will also help in understanding the properties of the quaternion method. Because the available literature on quaternions is either complex or erroneous, the quaternion is covered in much detail. The concept of the quaternion is explained, and the rules of quaternion algebra are stated in the first two sections. Next, some necessary relationships are developed. It will then be rather straightforward to derive the actual equations of motion. How to use the quaternion method for parameter optimization methods is emphasized in the following sectionsAerospace Engineerin