Dual-complex quaternion representation of gravitoelectromagnetism

2021 ◽  
pp. 2150178
Author(s):  
İsmail Aymaz ◽  
Mustafa Emre Kansu
Keyword(s):  
Differential Equations ◽  
Quaternion Algebra ◽  
Dual Numbers ◽  
Algebraic Structures ◽  
Gravitational Fields ◽  
Complex Quaternion ◽  
Linear Gravity ◽  
Dual Complex ◽  
Quaternion Representation ◽  
Quaternionic Field

In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.

Continuous quaternion fourier and wavelet transforms

2014 ◽  
Vol 12 (04) ◽  
pp. 1460003 ◽  
Author(s):  
Mawardi Bahri ◽  
Ryuichi Ashino ◽  
Rémi Vaillancourt
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Equation ◽  
Wavelet Transforms ◽  
Quaternion Algebra ◽  
Two Dimensional ◽  
Quaternion Fourier Transform ◽  
Fundamental Properties

A two-dimensional (2D) quaternion Fourier transform (QFT) defined with the kernel [Formula: see text] is proposed. Some fundamental properties, such as convolution, Plancherel and vector differential theorems, are established. The heat equation in quaternion algebra is presented as an example of the application of the QFT to partial differential equations. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

scholarly journals Some properties of dark matter field in the complex octonion space

2015 ◽  
Vol 30 (35) ◽  
pp. 1550212 ◽  
Author(s):  
Zi-Hua Weng
Keyword(s):  
Dark Matter ◽  
Electromagnetic Field ◽  
Gravitational Field ◽  
Dark Matter Particle ◽  
Matter Field ◽  
Gravitational Fields ◽  
Existence Region ◽  
Astrophysical Jet ◽  
Complex Quaternion ◽  
Interaction Intensity

The paper aims to consider the electromagnetic adjoint-field in the complex octonion space as the dark matter field, describing some properties of the dark matter, especially the origin, particle category, existence region, force and so forth. Since Maxwell applied the algebra of quaternions to depict the electromagnetic theory, some scholars adopt the complex quaternion and octonion to study the physics property of electromagnetic and gravitational fields. In the paper, by means of the octonion operator, it is found that the gravitational field accompanies with one adjoint-field, whose property is partly similar to that of electromagnetic field. The electromagnetic field accompanies with another adjoint-field, whose feature is partly similar to that of gravitational field. As a result, the electromagnetic adjoint-field can be chosen as one candidate of the dark matter field. According to the electromagnetic adjoint-field, it is able to predict a few properties of the dark matter, for instance, the particle category, interaction intensity, interaction distance, existence region and so forth. The study reveals that the dark matter particle and the gravitational resource will be influenced by the gravitational strength and force. The dark matter field is capable of making a contribution to physics quantities of gravitational field, including the angular momentum, torque, energy, force and so on. Further, there may be comparatively more chances to discover the dark matter in some regions with the ultrastrong field strength, surrounding the neutral star, white dwarf, galactic nucleus, black hole, astrophysical jet and so on.

scholarly journals Classical and quantum aspects of particle propagation in external gravitational fields

2017 ◽  
Vol 26 (11) ◽  
pp. 1750137 ◽  
Author(s):  
Giorgio Papini
Keyword(s):  
Quantum Tunneling ◽  
Vector Potential ◽  
Wave Equations ◽  
Radiation Energy ◽  
Simultaneous Presence ◽  
Gravitational Fields ◽  
Flux Quantization ◽  
Particle Propagation ◽  
Linear Gravity ◽  
Point Vector

In the study of covariant wave equations, linear gravity manifests itself through the metric deviation [Formula: see text] and a two-point vector potential [Formula: see text] itself constructed from [Formula: see text] and its derivatives. The simultaneous presence of the two gravitational potentials is noncontradictory. Particles also assume the character of quasiparticles and [Formula: see text] carries information about the matter with which it interacts. We consider the influence of [Formula: see text] on the dispersion relations of the particles involved, the particles’ motion, quantum tunneling through a horizon, radiation, energy–momentum dissipation and flux quantization.

scholarly journals Forces in the complex octonion curved space

2016 ◽  
Vol 13 (06) ◽  
pp. 1650076 ◽  
Author(s):  
Zi-Hua Weng
Keyword(s):  
Covariant Derivative ◽  
Flat Space ◽  
Gravitational Theory ◽  
Electromagnetic Theory ◽  
Field Source ◽  
Curved Space ◽  
Gravitational Fields ◽  
Gravitational Theories ◽  
Complex Quaternion ◽  
Connection Coefficient

The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.

scholarly journals Simulation of an Exoskeleton with a Hybrid Linear Gravity Compensator

2020 ◽  
Vol 24 (3) ◽  
pp. 66-78
Author(s):  
A. E. Karlov ◽  
A. A. Postolny ◽  
A. V. Fedorov ◽  
S. F. Jatsun
Keyword(s):  
Numerical Simulation ◽  
Mathematical Model ◽  
Differential Equations ◽  
Kinematic Model ◽  
Hybrid Approach ◽  
Second Order ◽  
System Parameters ◽  
Second Order Differential Equations ◽  
Gravity Compensator ◽  
Linear Gravity

Purpose of research. Development of a mathematical model of an exoskeleton equipped with a hybrid linear gravity compensator (HLGC), dynamic analysis on the example of a typical exoskeleton application scenario (in the process of lifting a load), obtaining time patterns of changes in system parameters, including electric drive torques allowing assessment of power plan power consumption and energy efficiency. The article deals with the challenging issue of improving the efficiency of the exoskeletal suit by means of HLGC. The use of a hybrid approach makes it possible to increase the efficiency of assisting the exoskeletal suit when performing various technological operations, for example, when lifting a load, when tilting and holding. Methods. When developing a mathematical model, an original approach was used to form the motion trajectory of the exoskeleton sectors during operation, based on the use of seventh-order polynomials. The paper uses a mathematical model represented by a system of second-order differential equations that connects the moments acting on the operator and the exoskeleton, the angular accelerations of the operator's back and the exoskeleton. Results. During numerical simulation, time diagrams of changes in system parameters, angles of rotation of exoskeleton hinges, moments that occur in a hybrid LGC, as well as graphs of current consumption of engines when performing lift and tilt with a load are obtained. Conclusion. In the course of the research, a kinematic model of an exoskeleton suit equipped with a GLGC was developed, second-order differential equations describing the dynamic behavior of the electromechanical system were written, and numerical simulation was performed to estimate the forces and energy consumption in the exoskeleton hinges and the drive of the hybrid linear gravity compensator.

scholarly journals Differential equations and expansions for quaternionic modular forms in the discriminant 6 case

2012 ◽  
Vol 15 ◽  
pp. 385-399 ◽  
Author(s):  
Srinath Baba ◽  
Håkan Granath
Keyword(s):  
Differential Equations ◽  
Explicit Formula ◽  
Modular Forms ◽  
Quaternion Algebra ◽  
Recursive Formula ◽  
Unit Group ◽  
Rational Structure ◽  
Taylor Coefficients ◽  
Taylor Expansions ◽  
Modular Polynomials

AbstractWe study the differential structure of the ring of modular forms for the unit group of the quaternion algebra over ℚ of discriminant 6. Using these results we give an explicit formula for Taylor expansions of the modular forms at the elliptic points. Using appropriate normalizations we show that the Taylor coefficients at the elliptic points of the generators of the ring of modular forms are all rational and 6-integral. This gives a rational structure on the ring of modular forms. We give a recursive formula for computing the Taylor coefficients of modular forms at elliptic points and, as an application, give an algorithm for computing modular polynomials.

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