Results for 'Quaternion'

26 found
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  1.  33
    Quaternion-Loop Quantum Gravity.M. D. Maia, S. S. E. Almeida Silva & F. S. Carvalho - 2009 - Foundations of Physics 39 (11):1273-1279.
    It is shown that the Riemannian curvature of the 3-dimensional hypersurfaces in space-time, described by the Wilson loop integral, can be represented by a quaternion quantum operator induced by the SU(2) gauge potential, thus providing a justification for quaternion quantum gravity at the Tev energy scale.
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  2.  9
    Quaternion-Based Texture Analysis of Multiband Satellite Images: Application to the Estimation of Aboveground Biomass in the East Region of Cameroon.Cedrigue Boris Djiongo Kenfack, Olivier Monga, Serge Moto Mpong & René Ndoundam - 2018 - Acta Biotheoretica 66 (1):17-60.
    Within the last decade, several approaches using quaternion numbers to handle and model multiband images in a holistic manner were introduced. The quaternion Fourier transform can be efficiently used to model texture in multidimensional data such as color images. For practical application, multispectral satellite data appear as a primary source for measuring past trends and monitoring changes in forest carbon stocks. In this work, we propose a texture-color descriptor based on the quaternion Fourier transform to extract relevant (...)
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  3.  23
    Quaternion Physical Quantities.J. Gibson Winans - 1977 - Foundations of Physics 7 (5-6):341-349.
    Quaternions consist of a scalar plus a vector and result from multiplication or division of vectors by vectors. Division of vectors is equivalent to multiplication divided by a scalar. Quaternions as used here consist of the scalar product with positive sign plus the vector product with sign determined by the right-hand rule. Units are specified by the multiplication process. Trigonometric functions are quaternions with units that can satisfy Hamilton's requirements. The square of a trigonometric quaternion is a real number (...)
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  4.  13
    Second Quantized Quaternion Quantum Theory.James D. Edmonds - 1975 - Foundations of Physics 5 (4):643-648.
    The basic structure of a second quantized relativistic quantum theory is outlined. The vector space is over the ring of complex quaternions instead of the usual field of complex numbers. This is motivated by the simple quaternion structure of the Dirac equation.
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  5.  18
    Almost Periodic Synchronization for Quaternion-Valued Neural Networks with Time-Varying Delays.Yongkun Li, Xiaofang Meng & Yuan Ye - 2018 - Complexity 2018:1-13.
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  6.  5
    Existence and Global Exponential Stability of Pseudo Almost Periodic Solutions for Neutral Type Quaternion-Valued Neural Networks with Delays in the Leakage Term on Time Scales.Yongkun Li & Xiaofang Meng - 2017 - Complexity:1-15.
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  7.  6
    Quaternion Algebra on 4D Superfluid Quantum Space-Time: Gravitomagnetism.Valeriy I. Sbitnev - 2019 - Foundations of Physics 49 (2):107-143.
    Gravitomagnetic equations result from applying quaternionic differential operators to the energy–momentum tensor. These equations are similar to the Maxwell’s EM equations. Both sets of the equations are isomorphic after changing orientation of either the gravitomagnetic orbital force or the magnetic induction. The gravitomagnetic equations turn out to be parent equations generating the following set of equations: the vorticity equation giving solutions of vortices with nonzero vortex cores and with infinite lifetime; the Hamilton–Jacobi equation loaded by the quantum potential. This equation (...)
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  8.  42
    Nine-Vectors, Complex Octonion/Quaternion Hypercomplex Numbers, Lie Groups, and the 'Real' World.James D. Edmonds - 1978 - Foundations of Physics 8 (3-4):303-311.
    A “mental” multiplication scheme is given for the super hypercomplex numbers, which extend the 16-element Dirac algebra to 32 elements by appending the complex octonions. This extends the 5-vectors of relativity to 9-vectors. The problems with nonassociativity, for the group structures and wave equation covariance, are explored.
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  9.  28
    Generalized Quaternion Formulation of Relativistic Quantum Theory in Curved Space.James D. Edmonds - 1977 - Foundations of Physics 7 (11-12):835-859.
    A survey is presented of the essential principles for formulating relativistic wave equations in curved spacetime. The approach is relatively simple and avoids much of the philosophical debate about covariance principles, which is also indicated. Hypercomplex numbers provide a natural language for covariance symmetry and the two important kinds of covariant derivative.
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  10.  3
    Antiperiodic Solutions for Quaternion-Valued Shunting Inhibitory Cellular Neural Networks with Distributed Delays and Impulses.Nina Huo & Yongkun Li - 2018 - Complexity 2018:1-12.
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  11. Flat-Space Metric in the Quaternion Formulation of General Relativity.C. Marcio do Amaral - 1969 - Rio De Janeiro, Centro Brasileiro De Pesquisas Físicas.
     
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  12.  47
    Time as a Geometric Concept Involving Angular Relations in Classical Mechanics and Quantum Mechanics.Juan Eduardo Reluz Machicote - 2010 - Foundations of Physics 40 (11):1744-1778.
    The goal of this paper is to introduce the notion of a four-dimensional time in classical mechanics and in quantum mechanics as a natural concept related with the angular momentum. The four-dimensional time is a consequence of the geometrical relation in the particle in a given plane defined by the angular momentum. A quaternion is the mathematical entity that gives the correct direction to the four-dimensional time.Taking into account the four-dimensional time as a vectorial quaternionic idea, we develop a (...)
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  13.  62
    Division Algebras and Quantum Theory.John C. Baez - 2012 - Foundations of Physics 42 (7):819-855.
    Quantum theory may be formulated using Hilbert spaces over any of the three associative normed division algebras: the real numbers, the complex numbers and the quaternions. Indeed, these three choices appear naturally in a number of axiomatic approaches. However, there are internal problems with real or quaternionic quantum theory. Here we argue that these problems can be resolved if we treat real, complex and quaternionic quantum theory as part of a unified structure. Dyson called this structure the ‘three-fold way’. It (...)
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  14.  11
    Wave-Mechanical Model for Chemistry.Jan C. A. Boeyens - 2015 - Foundations of Chemistry 17 (3):247-262.
    The strength and defects of wave mechanics as a theory of chemistry are critically examined. Without the secondary assumption of wave–particle duality, the seminal equation describes matter waves and leaves the concept of point particles undefined. To bring the formalism into line with the theory of special relativity, it is shown to require reformulation in hypercomplex algebra that imparts a new meaning to electron spin as a holistic spinor, eliminating serious current misconceptions in the process. Reformulation in the curved space–time (...)
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  15.  31
    Formalism, Hamilton and Complex Numbers.John O'Neill - 1986 - Studies in History and Philosophy of Science Part A 17 (3):351.
    The development and applicability of complex numbers is often cited in defence of the formalist philosophy of mathematics. This view is rejected through an examination of hamilton's development of the notion of complex numbers as ordered pairs of reals, And his later development of the quaternion theory, Which subsequently formed the basis of vector analysis. Formalism, By protecting informal assumptions from critical scrutiny, Constrained rather than encouraged the development of mathematics.
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  16. Cosmological Choices.David Finkelstein - 1982 - Synthese 50 (3):399 - 420.
    Present physics is a mix of theories of time, logic, and matter. These may have a common origin in a unitary quantum cosmology founded on process alone. A quantum theory of sets, or something like it, is helpful for such a cosmology, and one is constructed by adding superposition to a slightly reformulated classical set theory. There is an elementary or atomic process in such theories. The size of its characteristic time is estimated from the mass spectrum, although this gives (...)
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  17.  67
    Nearly Orthosymmetric Ortholattices and Hilbert Spaces.R. Mayet & S. Pulmannová - 1994 - Foundations of Physics 24 (10):1425-1437.
    The theory of nearly orthosymmetric ortholattices generalizes the theory of orthosymmetric ortholattices defined by one of the authors in a previous paper. In this theory, some equations allow one to distinguish complex Hilbertian lattices from real and quaternion ones.
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  18.  18
    From the Geometry of Pure Spinors with Their Division Algebras to Fermion Physics.Paolo Budinich - 2002 - Foundations of Physics 32 (9):1347-1398.
    The Cartan equations defining simple spinors (renamed “pure” by C. Chevalley) are interpreted as equations of motion in compact momentum spaces, in a constructive approach in which at each step the dimensions of spinor space are doubled while those of momentum space increased by two. The construction is possible only in the frame of the geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and then momentum spaces result compact, isomorphic to invariant-mass-spheres (...)
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  19.  53
    Hypercomplex Quantum Mechanics.L. P. Horwitz - 1996 - Foundations of Physics 26 (6):851-862.
    The fundamental axioms of the quantum theory do not explicitly identify the algebraic structure of the linear space for which orthogonal subspaces correspond to the propositions (equivalence classes of physical questions). The projective geometry of the weakly modular orthocomplemented lattice of propositions may be imbedded in a complex Hilbert space; this is the structure which has traditionally been used. This paper reviews some work which has been devoted to generalizing the target space of this imbedding to Hilbert modules of a (...)
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  20.  70
    Clifford Algebras in Symplectic Geometry and Quantum Mechanics.Ernst Binz, Maurice A. de Gosson & Basil J. Hiley - 2013 - Foundations of Physics 43 (4):424-439.
    The necessary appearance of Clifford algebras in the quantum description of fermions has prompted us to re-examine the fundamental role played by the quaternion Clifford algebra, C 0,2 . This algebra is essentially the geometric algebra describing the rotational properties of space. Hidden within this algebra are symplectic structures with Heisenberg algebras at their core. This algebra also enables us to define a Poisson algebra of all homogeneous quadratic polynomials on a two-dimensional sub-space, $\mathbb{F}^{a}$ of the Euclidean three-space. This (...)
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  21.  4
    Heuristics and Inferential Microstructures: The Path to Quaternions.Emiliano Ippoliti - 2019 - Foundations of Science 24 (3):411-425.
    I investigate the construction of the mathematical concept of quaternion from a methodological and heuristic viewpoint to examine what we can learn from it for the study of the advancement of mathematical knowledge. I will look, in particular, at the inferential microstructures that shape this construction, that is, the study of both the very first, ampliative inferential steps, and their tentative outcomes—i.e. small ‘structures’ such as provisional entities and relations. I discuss how this paradigmatic case study supports the recent (...)
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  22.  14
    The Precessional Frequency of a Gyroscope in the Quaternionic Formulation of General Relativity.Mendel Sachs - 1989 - Foundations of Physics 19 (1):105-108.
    The precessional frequency of a gyroscope in a reference frame that orbits about a gravitational body is compared between Einstein's tensor formulation of general relativity and the author's quaternion generalization—obtained from a factorization of the tensor form. The difference in predictions then suggests an experiment that could choose which of these formulations of general relativity is more valid in the analysis of gyroscopic motion.
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  23.  13
    The Inexactness of Time.G. Miller - 1973 - Foundations of Physics 3 (3):389-398.
    The differential aging effect is shown to be valid in any physically reasonable extension of the special theory of relativity which includes a description of accelerating observers. Einstein's controversial assumption—the clock hypothesis—is avoided. Instead, it is sufficient to assume accessibility—that it is possible to travel from one inertial observer to another and then return to the first in a reasonable manner. Since Minkowski space-time displays this accessibility property, there must be an error in Sachs's quaternion development of general relativity. (...)
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  24.  7
    Symbolic Expressions of the Human Cognitive Architecture.Enidio Ilario, Alfredo Pereira Jr & Valdir Gonzalez Paixão Jr - 2016 - Dialogue and Universalism 26 (1):107-120.
    We briefly review and discuss symbolic expressions of the cognitive architecture of the human mind/brain, focusing on the Quaternion, the Axis Mundi and the Tree of Life, and elaborate on a quaternary diagram that expresses a contemporary worldview. While traditional symbols contain vertical and horizontal dimensions related to transcendence and immanence, respectively, in the contemporary interpretation the vertical axis refers to diachronic processes as biological evolution and cultural history, while the horizontal axis refers to synchronic relations as the interactions (...)
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  25.  8
    Quaternionic Particle in a Relativistic Box.Sergio Giardino - 2016 - Foundations of Physics 46 (4):473-483.
    This study examines quaternion Dirac solutions for an infinite square well. The quaternion result does not recover the complex result within a particular limit. This raises the possibility that quaternionic quantum mechanics may not be understood as a correction to complex quantum mechanics, but it may also be a structure that can be used to study phenomena that cannot be described through the framework of complex quantum mechanics.
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  26.  25
    Linear and Geometric Algebra.Alan Macdonald - 2012 - North Charleston, SC: CreateSpace.
    This textbook for the second year undergraduate linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics. -/- Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much (...)
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