scholarly journals Three-Complex Numbers and Related Algebraic Structures

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 342
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Algebraic Structure ◽  
Matrix Polynomial ◽  
Three Dimensional ◽  
Basis Vector ◽  
Complex Numbers ◽  
Vector Product ◽  
Exponential Functions ◽  
Spherical Coordinate ◽  
Euclidean Vector Space ◽  
Vector Representations

Three-complex numbers are introduced for using a geometric vector product in the three-dimensional Euclidean vector space R3 and proving its equivalence with a spherical coordinate product. Based upon the definitions of the geometric power and geometric exponential functions, some Euler-type trigonometric representations of three-complex numbers are derived. Further, a general l23−complex algebraic structure together with its matrix, polynomial and variable basis vector representations are considered. Then, the classes of lp3-complex numbers are introduced. As an application, Euler-type formulas are used to construct directional probability laws on the Euclidean unit sphere in R3.

scholarly journals On Complex Numbers in Higher Dimensions

Axioms ◽  
2022 ◽  
Vol 11 (1) ◽  
pp. 22
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Probability Distributions ◽  
Three Dimensional ◽  
Geometric Approach ◽  
Higher Dimensions ◽  
Space Structure ◽  
Complex Numbers ◽  
Vector Product ◽  
Multiplication Rule ◽  
Generalized Complex ◽  
General Vector

The geometric approach to generalized complex and three-dimensional hyper-complex numbers and more general algebraic structures being based upon a general vector space structure and a geometric multiplication rule which was only recently developed is continued here in dimension four and above. To this end, the notions of geometric vector product and geometric exponential function are extended to arbitrary finite dimensions and some usual algebraic rules known from usual complex numbers are replaced with new ones. An application for the construction of directional probability distributions is presented.

Comparison Considerations Toward Investigating the Factors of Load and Age Group on the Maximum Reach Envelope

2020 ◽  
pp. 001872082096501
Author(s):  
Heather Johnston ◽  
Colleen Dewis ◽  
John Kozey
Keyword(s):  
Coordinate System ◽  
Three Dimensional ◽  
Spherical Coordinate System ◽  
Upper Body ◽  
Group Differences ◽  
Anthropometric Measures ◽  
Younger Adults ◽  
Present Measurement ◽  
Spherical Coordinate ◽  
Cylindrical Coordinate

Objective The objectives were to compare cylindrical and spherical coordinate representations of the maximum reach envelope (MRE) and apply these to a comparison of age and load on the MRE. Background The MRE is a useful measurement in the design of workstations and quantifying functional capability of the upper body. As a dynamic measure, there are human factors that impact the size, shape, and boundaries of the MRE. Method Three-dimensional reach measures were recorded using a computerized potentiometric system for anthropometric measures (CPSAM) on two adult groups (aged 18–25 years and 35–70 years). Reach trials were performed holding .0, .5, and 1 kg. Results Three-dimensional Cartesian coordinates were transformed into cylindrical ( r, θ , Z) and spherical ( r, θ, ϕ) coordinates. Median reach distance vectors were calculated for 54 panels within the MRE as created by incremented banding of the respective coordinate systems. Reach distance and reach area were compared between the two groups and the loaded conditions using a spherical coordinate system. Both younger adults and unloaded condition produced greater reach distances and reach areas. Conclusions Where a cylindrical coordinate system may reflect absolute reference for design, a normalized spherical coordinate system may better reflect functional range of motion and better compare individual and group differences. Age and load are both factors that impact the MRE. Application These findings present measurement considerations for use in human reach investigation and design.

scholarly journals Concept to Commercialization of an Artifact for Evaluating Three-Dimensional Imaging Systems per ASTM E3125-17

2018 ◽  
Vol 123 ◽  
Author(s):  
Bala Muralikrishnan ◽  
Prem Rachakonda ◽  
Vincent Lee ◽  
Meghan Shilling ◽  
Daniel Sawyer ◽  
...  
Keyword(s):  
3D Imaging ◽  
Three Dimensional ◽  
National Research Council ◽  
Imaging Systems ◽  
Considerable Time ◽  
Spherical Coordinate ◽  
Laser Scanners ◽  
Time Required ◽  
Test Use ◽  
Range Error

The relative-range error test is one of several tests described in the ASTM E3125-2017 standard for performance evaluation of spherical coordinate three-dimensional (3D) imaging systems such as terrestrial laser scanners (TLS). We designed a new artifact, called the plate-sphere target, that allows the realization of the relative-range error tests quickly and efficiently without the need for alignment at each position of the test. Use of a simple planar/plate target requires careful alignment of the target at each position of the relative-range error test, which is labor-intensive and time-consuming. This new artifact significantly reduces the time required to perform the test, from a matter of about 2 h to less than 30 min while resulting in similar test uncertainty values. The plate-sphere target was conceived and initially developed at the National Institute of Standards and Technology (NIST), improved based on feedback from collaborators at the National Research Council (NRC) of Canada and TLS manufacturers, and commercialized by Bal-tec Inc. This new artifact will save users and manufacturers of TLSs considerable time and money.

Three-dimensional quantum mechanics in a curved space based on the q-addition

2019 ◽  
Vol 34 (29) ◽  
pp. 1950177
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Coordinate System ◽  
Three Dimensional ◽  
One Dimension ◽  
Cartesian Coordinate System ◽  
Spherical Coordinate ◽  
Curved Space ◽  
Dimension Formula ◽  
Cartesian Coordinate

In this paper, we extend the theory of the [Formula: see text]-deformed quantum mechanics in one dimension[Formula: see text] into three-dimensional case. We relate the [Formula: see text]-deformed quantum theory to the quantum theory in a curved space. We discuss the diagonal metric based on [Formula: see text]-addition in the Cartesian coordinate system and core radius of neutron star. We also discuss the diagonal metric based on [Formula: see text]-addition in the spherical coordinate system and [Formula: see text]-deformed Heisenberg atom model.

Model Solutions to Quadratic Equations

1986 ◽  
Vol 79 (5) ◽  
pp. 332-336
Author(s):  
Alastair McNaughton
Keyword(s):  
Three Dimensional ◽  
Quadratic Functions ◽  
Complex Numbers ◽  
Quadratic Equations ◽  
Model Solutions

Here is a method of representing quadratic functions by three-dimensional wire models. It enables one to form a simple geometric concept of the location of the imaginary zeros. I have been using this material with my students and have been delighted with the ease with which they respond to it. As a result, their confidence in dealing with complex numbers has increased, their concept of functions has shown much improvement, and they are attacking problems with real insight.

A NOTE ON SLq(2) COVARIANCE AND QUANTUM EUCLIDEAN PLANE

2007 ◽  
Vol 22 (14) ◽  
pp. 1031-1037 ◽  
Author(s):  
Z. BENTALHA ◽  
M. TAHIRI
Keyword(s):  
Euclidean Space ◽  
Quantum Group ◽  
Bilinear Form ◽  
Algebraic Structure ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Three Dimensional ◽  
Central Element ◽  
The Other ◽  
Euclidean Metric

A quantum analogue of SL(2) covariance is given. The method consists of defining a bilinear form on SL q(2) co-module, then we ask the defined form to be invariant under SL q(2) co-actions. We showed that the required invariance leads to the standard algebraic structure of SL q(2) ideal. On the other hand, the geometry of the three-dimensional quantum Euclidean space has been evoked by computing the quantum Euclidean metric, the quantum Euclidean "distance" (the central element) and the relation of orthogonality of SO q(3) quantum group.

scholarly journals Visualising Complex Polynomials: A Parabola Is but a Drop in the Ocean of Quadratics

2018 ◽  
Vol 10 (6) ◽  
pp. 91
Author(s):  
Harry Wiggins ◽  
Ansie Harding ◽  
Johann Engelbrecht
Keyword(s):  
Complex Function ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Complex Polynomials ◽  
Singular Case ◽  
Hyperbolic Paraboloid ◽  
Complex Roots ◽  
Roots Of Polynomials ◽  
Complex Coefficients ◽  
Roots Of A Polynomial

One of the problems encountered when teaching complex numbers arises from an inability to visualise the complex roots, the so-called "imaginary" roots of a polynomial. Being four dimensional, it is problematic to visualize graphs and roots of polynomials with complex coefficients in spite of many attempts through centuries. An innovative way is described to visualize the graphs and roots of functions, by restricting the domain of the complex function to those complex numbers that map onto real values, leading to the concept of three dimensional sibling curves. Using this approach we see that a parabola is but a singular case of a complex quadratic.  We see that sibling curves of a complex quadratic lie on a three-dimensional hyperbolic paraboloid. Finally, we show that the restriction to a real range causes no loss of generality.

scholarly journals Sensorimotor Self-organization via Circular-Reactions

2021 ◽  
Vol 15 ◽  
Author(s):  
Dongcheng He ◽  
Haluk Ogmen
Keyword(s):  
Visual Processing ◽  
Reference Frames ◽  
Three Dimensional ◽  
Self Organization ◽  
Motor Systems ◽  
Sensory Inputs ◽  
Spatial Environment ◽  
Motor Activities ◽  
Motor Actions ◽  
Vector Representations

Newborns demonstrate innate abilities in coordinating their sensory and motor systems through reflexes. One notable characteristic is circular reactions consisting of self-generated motor actions that lead to correlated sensory and motor activities. This paper describes a model for goal-directed reaching based on circular reactions and exocentric reference-frames. The model is built using physiologically plausible visual processing modules and arm-control neural networks. The model incorporates map representations with ego- and exo-centric reference frames for sensory inputs, vector representations for motor systems, as well as local associative learning that result from arm explorations. The integration of these modules is simulated and tested in a three-dimensional spatial environment using Unity3D. The results show that, through self-generated activities, the model self-organizes to generate accurate arm movements that are tolerant with respect to various sources of noise.

scholarly journals Translation, modulation and dilation systems in set-valued signal processing

2018 ◽  
Vol 10 (1) ◽  
pp. 143-164 ◽  
Author(s):  
H. Levent ◽  
Y. Yilmaz
Keyword(s):  
Signal Processing ◽  
Linear Space ◽  
Algebraic Structure ◽  
Compact Convex ◽  
Inner Product ◽  
Complex Numbers ◽  
Real Numbers ◽  
Aumann Integral ◽  
Definition Of ◽  
Interval Valued

In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$.

scholarly journals Three-Dimensional Modeling and Control of a Tethered UAV-Buoy System

2021 ◽  
Author(s):  
Ahmad Kourani ◽  
Naseem Daher
Keyword(s):  
Control System ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Robotic System ◽  
Three Dimensions ◽  
Spherical Coordinate ◽  
Nonlinear Dynamical ◽  
Modeling And Control ◽  
Three Dimensional Modeling ◽  
And Control

Abstract This work presents the nonlinear dynamical model and motion controller of a system consisting of an unmanned aerial vehicle (UAV) that is tethered to a floating buoy in the three-dimensional (3D) space. Detailed models of the UAV, buoy, and the coupled tethered system dynamics are presented in a marine environment that includes surface-water currents and oscillating gravity waves, in addition to wind gusts. This work extends the previously modeled planar (vertical) motion of this novel robotic system to allow its free motion in all three dimensions. Furthermore, a Directional Surge Velocity Control System (DSVCS) is hereby proposed to allow both the free movement of the UAV around the buoy when the cable is slack, and the manipulation of the buoy’s surge velocity when the cable is taut. Using a spherical coordinate system centered at the buoy, the control system commands the UAV to apply forces on the buoy at specific azimuth and elevation angles via the tether, which yields a more appropriate realization of the control problem as compared to the Cartesian coordinates where the traditional x- , y- , and z -coordinates do not intuitively describe the tether’s tension and orientation. The proposed robotic system and controller offer a new method of interaction and collaboration between UAVs and marine systems from a locomotion perspective. The system is validated in a virtual high-fidelity simulation environment, which was specifically developed for this purpose, while considering various settings and wave scenarios.

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