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Publications related to Geometric Algebra (2,468)
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The observed anomalous rotation curves and mass distributions of galaxies, conventionally attributed to dark matter, can be explained through the dynamical evolution of ordinary conventional matter. We propose a novel mechanism, termed Velocity-Dependent Black Hole Accretion and Dynamical Ejection, which reshapes galactic mass distributions and rep...
Article
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This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a point-based view to a mirror-based view is a modern movement that allows for a more intuitive representation of geometric and physical entities, with v...
Preprint
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This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a point-based view to a mirror-based view is a modern movement that allows for a more intuitive representation of geometric and physical entities, with v...
Preprint
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Here I present a novel geometric algebra for spacetimes of arbitrary signature, and its representations in terms of nilpotents, idempotents, generalized geometric imaginary numbers, quaternions, and matrices over them. I note in particular noting the algebraic construction naturally implies equal numbers of time and spatial dimensions, or the usual...
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Self-attention mechanisms have revolutionised deep learning architectures, but their mathematical foundations remain incompletely understood. We establish that these mechanisms can be formalised through categorical algebra, presenting a framework that focuses on the linear components of self-attention. We prove that the query, key, and value maps i...
Article
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Graph neural networks (GNNs) have emerged as a prominent approach for capturing graph topology and modeling vertex-to-vertex relationships. They have been widely used in pattern recognition tasks including node and graph label prediction. However, when dealing with graphs from non-Euclidean domains, the relationships, and interdependencies between...
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In this paper we use the power of the outer exponential of a bivector B to see the so-called invariant decomposition from a different perspective. This is deeply connected with the eigenvalues for the adjoint action of B, a fact that allows a version of the Cayley-Hamilton theorem which factorises the classical theorem (both the matrix version and...
Article
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Automatic intelligent skin lesion recognition is crucial for elevating detection accuracy, enhancing diagnostic efficiency, and mitigating the risk of melanoma mortality. Despite advances, current methods often fall short in accuracy and acceptance among clinicians and patients. To improve clinical credibility, we propose an integrated system that...
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The modern algebra concepts are used to construct tables of algebraic spinors related to Clifford algebra multivectors with real and complex coefficients. The following data computed by Mathematica are presented in form of tables for individual Clifford geometric algebras: 1. Initial idempotent; 2. Two-sided ideal; 3. Left ideal basis (otherwise pr...
Article
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The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher dimensions. In this paper, I decompose the Cauchy-Riemann equations for a general Clifford algebra into grades using...
Article
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The concept of slope constitutes a fundamental component of the discourse surrounding linear equations. A subset of students frequently interprets slope merely as an algebraic ratio. This particular context fosters a superficial understanding of slope, as these students typically resort to mechanical memorization of the slope formula. The intent of...
Article
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The geometric errors of industrial robots are key factors affecting positioning accuracy. A new compensation method for industrial robots is proposed based on the kinematics characterizing with conformal geometric algebra (CGA) and measurement strategy with double ball bar (DBB) path point optimization. Firstly, a kinematic error model for the indu...
Preprint
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Explicit formulas to calculate MV functions in a basis-free representation are presented for an arbitrary Clifford geometric algebra Cl(p,q). The formulas are based on analysis of the roots of minimal MV polynomial and covers defective MVs, i.e. the MVs that have non-diagonalizable matrix representations. The method may be generalized straightforwa...
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We introduce the notion of rank of multivector in Clifford geometric algebras of arbitrary dimension without using the corresponding matrix representations and using only geometric algebra operations. We use the concepts of characteristic polynomial in geometric algebras and the method of SVD. The results can be used in various applications of geom...
Article
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The goal of the paper is to introduce a universal approach for calculating integrated information assessment (IIA) in complex systems by utilizing the geometric product from geometric algebra (GA). Traditional models of consciousness try to explain how neural networks and cognitive processes give rise to a unified conscious experience. Quantum mech...
Article
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A novel and fully projective algorithm for a point-in-convex polygon test with computational complexity of O(log N) in 2D is described in this contribution. The polygon vertices and tested points can be given in projective space without conversion to Euclidean space. The proposed algorithm is simple, robust, easy to implement, and invariant to the...
Preprint
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Volume parameterizations abound in recent literature, from the classic voxel grid to the implicit neural representation and everything in between. While implicit representations have shown impressive capacity and better memory efficiency compared to voxel grids, to date they require training via nonconvex optimization. This nonconvex training proce...
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This work examines the problem of extending the one-dimensional analytic signal, which is ubiquitous throughout signal processing, to higher dimensional signals. Bulow et al. and Felsberg et al. have previously used techniques from Clifford algebra and analysis to extend the one-dimensional analytic signal to higher dimensions. However, each author...
Article
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Multibody systems are characterized by two distinguishing features: system components undergo finite motions and these components are connected by mechanical joints that impose restrictions on their relative motion. Clearly, kinematics plays a fundamental role in the analysis of these systems: it is required to describe the arbitrary motion of syst...
Article
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In a companion paper, the fundamentals of vector space geometric algebra and plane-based geometric algebra were developed. Plane-based geometric algebra was found to be well suited for describing the kinematics of multibody dynamics: all required operations are performed using algebraic manipulations only and singularities are avoided altogether. T...
Article
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We compute and explore the full geometric product of two oriented points in conformal geometric algebra Cl(4, 1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.
Article
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This paper presents an approach for extracting points from conic intersections by using the concept of pencils. This method is based on QC2GA—the two-dimensional version of QCGA (Quadric Conformal Geometric Algebra)—that is demonstrated to be equivalent to GAC (Geometric Algebra for Conics). A new interpretation of QC2GA and its objects based on pe...
Article
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Stereotactic systems have traditionally used Cartesian coordinate combined with linear algebraic mathematical models to navigate the brain. Previously, the development of a novel stereotactic system allowed for improved patient comfort, reduced size, and carried through a simplified interface for surgeons. The system was designed with a work envelo...
Article
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This article seeks to introduce new readers interested in robotics to some of the tools available in the field. It specifically focuses on the study and comparison of kinematic analysis in robotic systems, with an emphasis on forward kinematics using a range of techniques. These techniques range from traditional methods to advanced ones. In particu...
Preprint
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We show that the Lorentz-Equivariant Geometric Algebra Transformer (L-GATr) yields state-of-the-art performance for a wide range of machine learning tasks at the Large Hadron Collider. L-GATr represents data in a geometric algebra over space-time and is equivariant under Lorentz transformations. The underlying architecture is a versatile and scalab...
Article
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Este proyecto de investigación se realizó en el Centro de Educación Básica General José Santos Puga de la provincia de Veraguas, República de Panamá. Una meta para los estudiantes de noveno grado era reforzar el proceso de factorización de trinomios usando álgebra geométrica. La investigación se inició con la aplicación de pre-tests en los que los...
Article
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The rotating equilibrium solutions of N identical point vortices are the stationary energy states of a higher-dimensional object projected to the x−y plane, in this case, an (N−1)-dimensional regular simplex. Parameterizing the point vortex Hamiltonian with this fundamental geometrical object leads to a simple bivector condition, which contains the...
Article
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We reorganize, simplify and expand the theory of contractions or interior products of multivectors, and related topics like Hodge star duality. Many results are generalized and new ones are given, like: geometric characterizations of blade contractions and regressive products, higher-order graded Leibniz rules, determinant formulas, improved comple...
Article
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The theory of contractions of multivectors, and star duality, was reorganized in a previous article, and here we present some applications. First, we study inner and outer spaces associated to a general multivector M via the equations v∧M=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amss...
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The article analyzes geometric structures on smooth inner product vector bundles determined by geometric algebras, a class that includes matrix, group with two-cocycle, and Euclidean Clifford algebras. The objects of interest are the derivations and linear connections on such bundles compatible with the underlying structures. Part one of the artic...
Article
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The field of multi-source remote sensing observation is becoming increasingly dynamic through the integration of various remote sensing data sources. However, existing deep learning methods face challenges in differentiating between internal and external relationships and capturing fine spatial features. These models often struggle to effectively c...
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We apply a well known technique of theoretical physics, known as geometric algebra or Clifford algebra, to linear electrical circuits with nonsinusoidal voltages and currents. We rederive from the first principles the geometric algebra approach to the apparent power decomposition. The important new point consists of endowing the space of Fourier ha...
Preprint
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This paper proposes a novel approach to word embeddings in Transformer models by utilizing spinors from geometric algebra. Spinors offer a rich mathematical framework capable of capturing complex relationships and transformations in high-dimensional spaces. By encoding words as spinors, we aim to enhance the expressiveness and robustness of languag...
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This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for...
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In this paper we discuss the dynamic effects of the varying frames. The differential of frame or basis vectors is always equivalent to a linear transformation of the frame, and the linear transformation is not the same in different contexts. In differential geometry, the linear transformation is the connection operator. While in quantum mechanics,...
Article
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Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC...
Preprint
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We introduce STAResNet, a ResNet architecture in Spacetime Algebra (STA) to solve Maxwell's partial differential equations (PDEs). Recently, networks in Geometric Algebra (GA) have been demonstrated to be an asset for truly geometric machine learning. In [1], GA networks have been employed for the first time to solve partial differential equations...
Preprint
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The last two decades, since the seminal work of Selig, has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its in...
Article
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Synopsis In this article I argue that i is a quantity associated with the two-dimensional real number plane, whether as a vector, a bi-vector, a point or a transformation (rotation). This position provides a foundation for the complex numbers and accounts for complex numbers in some equations of applied mathematics and physics. I also argue that co...
Conference Paper
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This work introduces a novel approach in power quality analysis by fitting three-phase voltage curves with ellipses using Geometric Algebra for Conics (GAC). Recognizing the limitations of current linear algebra-based methods, this paper highlights the need for further exploration into more effective techniques. Geometric Algebra offers a potent, e...
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The Geometric Algebra Fulcrum Library (GA-FuL) version 1.0 is introduced in this paper as a comprehensive computational library for geometric algebra (GA) and Clifford algebra (CA), in addition to other classical algebras. As a sophisticated software system, GA-FuL is useful for practical applications requiring numerical or symbolic prototyping, op...
Research
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Despite all the flack that string theory folks are getting for the failure to connect theory and experiment, it is possible that they've got the math right, just in the wrong representation, and is just what they claim - the unification. This abstract to the October Boston APS meeting outlines what their model might look like when no longer lost in...
Article
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Adaptive filtering algorithms are currently successfully employed in a number of fields.But, a disadvantage of traditional real-valued fixed step-size adaptive filtering algorithm is that it is unable to meet both requirement of convergence rate and the steady-state error.In this article, we presented cosine function and geometric algebra(GA) based...
Article
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Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majorit...
Article
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The Hansen problem is a classic and well-known geometric challenge in geodesy and surveying involving the determination of two unknown points relative to two known reference locations using angular measurements. Traditional analytical solutions rely on cumbersome trigonometric calculations and are prone to propagation errors. This paper presents a...
Preprint
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Modelling the propagation of electromagnetic signals is critical for designing modern communication systems. While there are precise simulators based on ray tracing, they do not lend themselves to solving inverse problems or the integration in an automated design loop. We propose to address these challenges through differentiable neural surrogates...
Presentation
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Null vectors are metric-free and define oriented rays on the Minkowski-Einstein light-cone in special relativity, and in higher dimensional null cones. The concepts of local and global duality on a set of nullvectors makes possible a new classification scheme of real and complex Clifford geometric algebras. It is conjectured that Finsler Geometry i...
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From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time.
Preprint
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We present, for the first time, a novel theoretical approach to address the problem of correspondence free multivector cloud registration in conformal geometric algebra. Such formalism achieves several favorable properties. Primarily, it forms an orthogonal automorphism that extends beyond the typical vector space to the entire conformal geometric...
Article
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Alternative mathematical explorations in quantum computing can be of great scientific interest, especially if they come with penetrating physical insights. In this paper, we present a critical revisitation of our application of geometric (Clifford) algebras (GAs) in quantum computing as originally presented in [C. Cafaro and S. Mancini, Adv. Appl....
Book
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The book has been written for two reasons: as a graduate textbook on the most important geometric, algebraic and group structures, Chapters 1–3, and as a research monograph on Cayley-Klein groups and categories, Chapters 3–7. The text in Chapters 3–7 is a compilation and extension of works that the author has done in Vienna for the eleven years [12...
Article
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In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We...
Article
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This paper introduces a novel method for solving the resection problem in two and three dimensions based on conformal geometric algebra (CGA). Advantage is taken because of the characteristics of CGA, which enables the representation of points, lines, planes, and volumes in a unified mathematical framework and offers a more intuitive and geometric...
Article
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Multi-loop coupling mechanisms (MCMs) have been widely used in spacedeployable antennas. However, the mobility of MCMs is difficult to analyze due to their complicated structure and coupled limbs. This paper proposes a general method for calculating the mobility of MCMs using geometric algebra (GA). For the independent limbs in the MCM, the twist s...
Article
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Hand dysfunction is a common symptom in stroke patients. This paper presents a robotic device which assists the rehabilitation process in order to reduce the need of physical therapy, i.e., a 3UPS/S parallel robotic device is employed for repetitive robot-assisted rehabilitation. Euler angle representation was used to solve the robot’s inverse kine...
Preprint
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Extracting scientific understanding from particle-physics experiments requires solving diverse learning problems with high precision and good data efficiency. We propose the Lorentz Geometric Algebra Transformer (L-GATr), a new multi-purpose architecture for high-energy physics. L-GATr represents high-energy data in a geometric algebra over four-di...
Article
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Using the powerful language of geometric algebra, we present an observationally symmetric derivation of the strong correlations predicted by the entangled singlet state in a deterministic and locally causal model, usually also referred to as a local-realistic model, in which the physical space is assumed to be a quaternionic 3-sphere, or S3\documen...
Article
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To achieve 3D skeleton-based human motion prediction, attention-based methods have encouraged performance due to the observation that attention parts in the past state influence future actions. The model can identify the most relevant information for motion prediction by introducing an attention mechanism. However, existing methods tend to address...
Article
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Quantum simulation qubit models of electronic Hamiltonians rely on specific transformations in order to take into account the fermionic permutation properties of electrons. These transformations (principally the Jordan–Wigner transformation (JWT) and the Bravyi–Kitaev transformation) correspond in a quantum circuit to the introduction of a suppleme...
Article
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With regards to the problem of multidimensional signal processing in the field of adaptive filtering, geometric algebra based higher-order statistics algorithms were proposed. For instance, to express a multidimensional signal as a multi-vector, the adaptive filtering algorithms described in this work leverage all of the benefits of GA theory in mu...
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Typhoons, as highly destructive natural disasters, significantly impact society and economy, making accurate prediction of their intensity crucial. Artificial intelligence technologies, such as machine learning, have been extensively applied in the meteorological domain due to their advantages in processing large-scale datasets and learning complex...
Article
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The geometric algebra lift of conventional quantum mechanics qubits is the gamechanging quantum leap forward potentially kicking from the quantum computing market big fishes (IBM, Microsoft, Google, dozens of smaller ones) investing billions in elaborating quantum computing devices. It brings into reality a kind of physical field spreading through...
Conference Paper
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We propose DPGNN, a differentiable physics-and geometry-aided pipeline for the solution of 2D partial differential equations describing the turbulent flow of a fluid around obstacles. We build upon the work of [1], that proposes a hybrid framework through a dif-ferentiable solver-in-the-loop strategy called differentiable physics-assisted neural ne...
Article
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The study of signal processing has recently devoted significantly more attention to adaptive filtering techniques. By addressing the shortcoming of the conventional geometric algebra-based fixed step-size least mean square algorithm that is unable to satisfy in terms of both reducing steady-state error and a faster convergence speed, simultaneously...
Article
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This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first...
Article
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This work is the result of a master’s investigation in Brazil, which discusses the teaching of the parabola in the initial training of mathematics teachers. Our theoretical framework addresses the relationship between intuition and the dialectics of the theory of didactic situations, which supports the analysis of the results of this study Its obje...
Article
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The aim of this work is to define quaternion curves and surfaces and their conjugates via operators in Euclidean projective geometric algebra (EPGA). In this space, quaternions were obtained by the geometric product of vector fields. New vector fields, which we call trajectory curves and surfaces, were obtained by using this new quaternion operator...
Article
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We extend Bloch sphere formalism to pure two-qubit systems. Combining insights from Geometric Algebra and the analysis of entanglement in different conjugate bases we identify two Bloch sphere geometry that is suitable for representing maximally entangled states. It turns out that the relative direction of the coordinate axes of the two Bloch spher...
Article
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The orthogonal operators defined as similarity transformations on Euclidean space E can also be considered as group actions on the Clifford Algebra. In this paper, we investigate the finite subgroup of Euclidian space E of Geometric Algebra over a finite dimension vector space E. The hierarchy of the finite subgroups of Clifford Algebra C(E) is dep...
Preprint
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From the table of the standard genetic code at 64 codons, 61 amino acids and 3 Stop signals, it is invested a reduced triangular matrix of the genetic code comprising just twice fifty-five entities. In relationship with number theory, many symmetrical and asymmetrical geometric sub-configurations of this triangular matrix show that the amino acids...
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Navier-Stokes equations are based on Newton’s second law and the Stokes hypothesis. In this paper, we have derived the fluid dynamic equations by applying the powerful tool of the Euler-Lagrangian approach, based on the principle of least action. The new equation highlights the incompleteness of the Navier-Stokes equations. The main reason is that...
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Power flow study is critical to electrical power system analysis. This study solves the power flow problem using a mathematical framework based on geometric algebra (GA). Geometric algebra expands on the concepts of linear algebra and geometry to offer a comprehensive and more flexible mathematical framework. In contrast, complex numbers are a cont...
Conference Paper
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Many differential equations describing physical phenomena are intrinsically geometric in nature. It has been demonstrated how this geometric structure of data can be captured effectively through networks sitting in Geometric Algebra (GA) that work with multivectors, making them suitable candidates to solve differential equations. GA networks howeve...
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We present a geometric theory of everything, using the generalization of Pythagoras' theorem. We look at the theory in the geometric algebra framework and consider it outside of that framework. The theory of everything action Using the generalization of Pythagoras' theorem [1], we propose the following action X 2 ≡ X ∅ 2 + X I 2 + X II 2 + X III 2...
Preprint
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This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first...
Article
Full-text available
This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intui...
Article
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Multi-loop coupling mechanisms (MCMs) have been widely used in architectural design antennas and space-deployable. This paper proposes a general method for actuation selection of MCMs using geometric algebra. The constraint space is obtained by collineation and dual operators of twist space. This method determines additional constraints by analyzin...