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.

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.

An Accelerated Cue Combination Principle Accounts for Multi-cue Depth Perception

2020 ◽  
Vol 3 (1) ◽  
pp. 10501-1-10501-9
Author(s):  
Christopher W. Tyler
Keyword(s):  
Belief Propagation ◽  
Probability Distributions ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Fusion Model ◽  
Cue Combination ◽  
Depth Estimate ◽  
Appropriate Behavior ◽  
Depth Cues ◽  
Bayesian Averaging

Abstract For the visual world in which we operate, the core issue is to conceptualize how its three-dimensional structure is encoded through the neural computation of multiple depth cues and their integration to a unitary depth structure. One approach to this issue is the full Bayesian model of scene understanding, but this is shown to require selection from the implausibly large number of possible scenes. An alternative approach is to propagate the implied depth structure solution for the scene through the “belief propagation” algorithm on general probability distributions. However, a more efficient model of local slant propagation is developed as an alternative.The overall depth percept must be derived from the combination of all available depth cues, but a simple linear summation rule across, say, a dozen different depth cues, would massively overestimate the perceived depth in the scene in cases where each cue alone provides a close-to-veridical depth estimate. On the other hand, a Bayesian averaging or “modified weak fusion” model for depth cue combination does not provide for the observed enhancement of perceived depth from weak depth cues. Thus, the current models do not account for the empirical properties of perceived depth from multiple depth cues.The present analysis shows that these problems can be addressed by an asymptotic, or hyperbolic Minkowski, approach to cue combination. With appropriate parameters, this first-order rule gives strong summation for a few depth cues, but the effect of an increasing number of cues beyond that remains too weak to account for the available degree of perceived depth magnitude. Finally, an accelerated asymptotic rule is proposed to match the empirical strength of perceived depth as measured, with appropriate behavior for any number of depth cues.

Stochastic Three-Dimensional Joint Geometry Model and the Properties of REV for a Jointed Rock Mass

2014 ◽  
Vol 1079-1080 ◽  
pp. 266-271
Author(s):  
Wen Hui Tan ◽  
Zhong Hua Sun ◽  
Ning Li ◽  
Xiao Hong Jiang
Keyword(s):  
Rock Mass ◽  
Probability Distributions ◽  
Three Dimensional ◽  
Jointed Rock ◽  
Jointed Rock Mass ◽  
Open Pit ◽  
Mechanical Parameters ◽  
Geometry Model ◽  
Joint Geometry ◽  
3D Network

The lithology of rock mass isnon-homogeneity，anisotropy, andexists size effect. The mechanical parameters of rock mass gotten by engineeringapproaches cannot reflect these properties. Therefore, a newmethod of determining the mechanical parameters of jointed rock mass isproposed: gneiss in Shuichang open-pit mine was selected as a case, thefracture system of the rock mass was measured and analyzed by non-contactmeasuring system of 3GSM and probabilisticmethod,the probability distributions of geometry parameters were analyzed and a 3Djoint geometry model was made by using the program of 3D network modeling.Cubes with different sizes were selected to be tested by tri-axial compressionof numerical simulation with 3DEC based on the 3D network model of joints,thus, the REV and its mechanical parameters were determined, which providedcredible parameters for slope stability analysis.

Spatial Models of Sugars and Their Compounds

2021 ◽  
Vol 11 (2) ◽  
pp. 9-22
Author(s):  
Gennadiy Vladimirovich Zhizhin
Keyword(s):  
Convex Bodies ◽  
Three Dimensional ◽  
Spatial Models ◽  
Higher Dimensions ◽  
Three Dimensional Images

The images of saccharide and polysaccharide molecules in spaces of various dimensions are considered. A method has been developed for obtaining simplified three-dimensional images of sugar molecules and their chains based on their images in spaces of higher dimensions. It was found that three-dimensional images of furanose and pyranose molecules fundamentally differ from each other to form convex and, accordingly, non-convex bodies. This leads to fundamental differences in the structure of polysaccharides from these molecules.

Computer Aided Tolerance Analysis of Parts and Assemblies

1989 ◽  
Author(s):  
R. T. Scott ◽  
G. A. Gabriele
Keyword(s):  
Real World ◽  
Probability Distributions ◽  
Three Dimensional ◽  
Tolerance Analysis ◽  
The Real ◽  
Assembly Tolerance ◽  
Computer Aided ◽  
Probability Information ◽  
Exact Constraint

Abstract An exact constraint scheme based on the physical contacting constraints of real part mating features is used to represent the process of assembling the parts. To provide useful probability information about how assembly dimensions are distributed when the parts are assembled as intended, the real world constraints that would prevent interference are ignored. This work addresses some limitations in the area of three dimensional assembly tolerance analysis. As a result of this work, the following were demonstrated: 1. Assembly of parts whose assembly mating features are subjected to variation; 2. Assemble parts using a real world set of exact constraints; 3. Provide probability distributions of assembly dimensions.

The Role of the Information Channel in Visual Computing

2020 ◽  
pp. 464-490
Author(s):  
Miquel Feixas ◽  
Mateu Sbert
Keyword(s):  
Information Theory ◽  
Communication Channel ◽  
Probability Distributions ◽  
Global Illumination ◽  
Three Dimensional ◽  
Information Channel ◽  
Visual Computing ◽  
The Social ◽  
Shared Information

Around seventy years ago, Claude Shannon, who was working at Bell Laboratories, introduced information theory with the main purpose of dealing with the communication channel between source and receiver. The communication channel, or information channel as it later became known, establishes the shared information between the source or input and the receiver or output, both of which are represented by random variables, that is, by probability distributions over their possible states. The generality and flexibility of the information channel concept can be robustly applied to numerous, different areas of science and technology, even the social sciences. In this chapter, we will present examples of its application to select the best viewpoints of an object, to segment an image, and to compute the global illumination of a three-dimensional virtual scene. We hope that our examples will illustrate how the practitioners of different disciplines can use it for the purpose of organizing and understanding the interplay of information between the corresponding source and receiver.

scholarly journals Computer-Integrated Manufacturing System for Membrane Structures

2019 ◽  
Vol 258 ◽  
pp. 02025
Author(s):  
Kazuo Yokobori ◽  
Tomo Miura
Keyword(s):  
Membrane Structure ◽  
Three Dimensional ◽  
Numerical Data ◽  
3D Models ◽  
Space Structure ◽  
Computer Integrated Manufacturing ◽  
Support Vector ◽  
Membrane Structures ◽  
Structure Estimation ◽  
Steel Panels

A membrane structure is a space structure composed of a membrane material (fabric or film), cables, and steel frames, among others. It reduces the environmental load for transporting materials and constructions; for instance, compared with conventional roofs that have steel panels or tiles, the membrane structure of a roof is lightweight. Computer analysis and three-dimensional (3D) models are required for determining the stable shape of such tensile structures. It is useful to use computer-integrated systems for the design and manufacturing process because these 3D models consist of numerical data. In this study, we developed a system program based on artificial intelligence methods, with a support vector machine instead of human judgment for the membrane structure estimation and for a probabilistic optimization to predict the differences caused by production loss etc. and compare the results after actual production. And we got close predicted results to the person.

Resonance Analysis of Train–Track–Bridge Interaction Systems with Correlated Uncertainties

2019 ◽  
Vol 20 (01) ◽  
pp. 2050008 ◽  
Author(s):  
Lifeng Xin ◽  
Xiaozhen Li ◽  
Jiaxin Zhang ◽  
Yan Zhu ◽  
Lin Xiao
Keyword(s):  
Probability Distributions ◽  
Three Dimensional ◽  
Estimation Method ◽  
Interaction Model ◽  
Point Estimation ◽  
Statistical Characteristics ◽  
Resonance Analysis ◽  
Track Bridge ◽  
Point Estimation Method ◽  
Nataf Transformation

Over the last decades, the resonance-related dynamics for bridge systems subjected to a moving train has been researched and discussed from mechanics, physics and mathematics. In the current work, new perspectives of train-induced resonance analysis are investigated through introducing random propagation process into the train–bridge dynamic interactions. Besides, the Nataf-transformation-based point estimation method is applied to generate pseudorandom variables following arbitrarily correlated probability distributions. A three-dimensional (3D) nonlinear train-ballasted track–bridge interaction model founded on fundamental physical and mechanical principles is employed to convey and depict train–bridge interactions with random properties considered. After that, extensive applications are illustrated in detail for revealing the statistical characteristics of the so-called “random resonance”. Numerical results show that the critical train speeds associated with resonance and cancelation are random in essence owing to the variability of system parameters; the correlation between parameters exerts obvious influences on system dynamic behaviors; the last vehicle of a train will be in more violent vibrations compared to the front vehicles; the influences of track irregularities on the wheel–rail interactions are significantly greater than those of resonance.

scholarly journals Structural transformations in quasicrystals induced by higher dimensional lattice transitions

2012 ◽  
Vol 468 (2141) ◽  
pp. 1452-1471 ◽  
Author(s):  
Giuliana Indelicato ◽  
Tom Keef ◽  
Paolo Cermelli ◽  
David G. Salthouse ◽  
Reidun Twarock ◽  
...  
Keyword(s):  
Three Dimensional ◽  
Higher Dimensions ◽  
Structural Transformations ◽  
Icosahedral Symmetry ◽  
Local Transformation ◽  
Project Method ◽  
Higher Dimensional ◽  
Transformation Mechanisms ◽  
Transition Paths ◽  
Penrose Tilings

We study the structural transformations induced, via the cut-and-project method, in quasicrystals and tilings by lattice transitions in higher dimensions, with a focus on transition paths preserving at least some symmetry in intermediate lattices. We discuss the effect of such transformations on planar aperiodic Penrose tilings, and on three-dimensional aperiodic Ammann tilings with icosahedral symmetry. We find that locally the transformations in the aperiodic structures occur through the mechanisms of tile splitting, tile flipping and tile merger, and we investigate the origin of these local transformation mechanisms within the projection framework.

Design of a Massively Reconfigurable Origami Space Structure Incorporating Shape Memory Alloys

2012 ◽  
Author(s):  
Darren Hartl ◽  
Kathryn Lane ◽  
Richard Malak
Keyword(s):  
Shape Memory ◽  
Three Dimensional ◽  
Passive Layer ◽  
Space Structure ◽  
Integration Scheme ◽  
Cylindrical Shape ◽  
Material System ◽  
Analysis Model ◽  
Element Analysis ◽  
Design Assessment

The subject of origami design has recently garnered increasing attention from the science, mathematics, and engineering communities. Mathematically rigorous frameworks have been developed that allow the identification of folding patterns needed to obtain a final three-dimensional goal shape. However, relatively little research exists on the problem of understanding the behavioral aspects of the material system undergoing the folding operations. This work considers the design and analysis of a novel concept for a self-folding material system. The system consists of an active, self-morphing laminate structure that includes thermally actuated shape memory alloy (SMA) layers and a compliant passive layer. Multiple layers allow folds in both directions (e.g., cross-folds). The layers are configured to allow continuously variable folding operations based only on which regions are heated. For the purposes of demonstration, an example problem is considered whereby an autonomous planetary landing craft is designed that can be stored in a flat sheet configuration, morph using a set of folds into a stable shape for safe descent through a gaseous atmosphere, and then, once landed, morph again toward a cylindrical shape for the purpose of rolling locomotion. We examine the effects of fold width, layer thicknesses, and activation parameters on the geometric configurations that can be obtained. The design efforts are supported by realistic morphing structural analysis tools. These include a comprehensive and accurate three-dimensional constitutive model for SMAs implemented into a finite element analysis (FEA) framework (the Abaqus Unified FEA suite) using a robust and efficient numerical integration scheme. Shell elements and laminate theory are used to increase the computational efficiency of the analysis. Model pre-processing, submission, and post-processing scripting methods are used to automate the design assessment tasks.

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