Topics in Quaternion Linear Algebra

2014 ◽  
Author(s):  
Leiba Rodman
Keyword(s):  
Linear Algebra ◽  
Complex Analysis ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Number System ◽  
Graduate Course ◽  
Open Problems ◽  
Unpublished Research ◽  
Reference Tool

Quaternions are a number system that has become increasingly useful for representing the rotations of objects in three-dimensional space and has important applications in theoretical and applied mathematics, physics, computer science, and engineering. This is the first book to provide a systematic, accessible, and self-contained exposition of quaternion linear algebra. It features previously unpublished research results with complete proofs and many open problems at various levels, as well as more than 200 exercises to facilitate use by students and instructors. Applications presented in the book include numerical ranges, invariant semidefinite subspaces, differential equations with symmetries, and matrix equations. Designed for researchers and students across a variety of disciplines, the book can be read by anyone with a background in linear algebra, rudimentary complex analysis, and some multivariable calculus. Instructors will find it useful as a complementary text for undergraduate linear algebra courses or as a basis for a graduate course in linear algebra. The open problems can serve as research projects for undergraduates, topics for graduate students, or problems to be tackled by professional research mathematicians. The book is also an invaluable reference tool for researchers in fields where techniques based on quaternion analysis are used.

scholarly journals An overview of augmented visualization: observing the real world as desired

2018 ◽  
Vol 7 ◽  
Author(s):  
Shohei Mori ◽  
Hideo Saito
Keyword(s):  
Real World ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Open Problems ◽  
The Real ◽  
Real Objects ◽  
Point Image ◽  
Set Up ◽  
Background Image

Over 20 years have passed since a free-viewpoint video technology has been proposed with which a user's viewpoint can be freely set up in a reconstructed three-dimensional space of a target scene photographed by multi-view cameras. This technology allows us to capture and reproduce the real world as recorded. Once we capture the world in a digital form, we can modify it as augmented reality (i.e., placing virtual objects in the digitized real world). Unlike this concept, the augmented world allows us to see through real objects by synthesizing the backgrounds that cannot be observed in our raw perspective directly. The key idea is to generate the background image using multi-view cameras, observing the backgrounds at different positions and seamlessly overlaying the recovered image in our digitized perspective. In this paper, we review such desired view-generation techniques from the perspective of free-view point image generation and discuss challenges and open problems through a case study of our implementations.

The Characteristics of Trivariate Vector-Valued Super-Wave Wraps and Applications in Materials Engineering

2014 ◽  
Vol 915-916 ◽  
pp. 1062-1065
Author(s):  
Yong Yi Huang
Keyword(s):  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Riesz Bases ◽  
Time Frequency ◽  
Decomposition Scheme ◽  
Materials Engineering ◽  
Multi Scale ◽  
Dilation Factor ◽  
Vector Valued

The rise of wavelet analysis in applied mathematics is due to its applications and the flex- ibility. We introduce vector-valued wave wraps with multi-scale dilation factor for space , which is the generalization of univariate wavelet packets. A method for constructing a sort of orthogo-nal vector-valued wavelet wraps in three-dimensional space is presented and their orthogonality traits are characterized by virtue of iteration method and time-frequency analysis method. The orthog-onality formulas concerning these wave packets are established. Moreover, it is shown how to obtain new Riesz bases of space from these wave wraps. The notion of multiple pseudo fames for subspaces with integer translation is proposed. The construction of a generalized multiresolution structure of Paley-Wiener subspaces of is investigated. The pyramid decomposition scheme is derived based on a generalized multiresolution structure.

Quaternion-Based Algorithm for Direct Kinematic Model of a Kawasaki FS10E Articulated Arm Robot

2015 ◽  
Vol 762 ◽  
pp. 249-254 ◽  
Author(s):  
Stelian Popa ◽  
Alexandru Dorin ◽  
Florin Adrian Nicolescu ◽  
Andrei Mario Ivan
Keyword(s):  
Degrees Of Freedom ◽  
Dimensional Space ◽  
Kinematic Model ◽  
Three Dimensional ◽  
Number System ◽  
Simulation Software ◽  
Six Degrees Of Freedom ◽  
Mathematical Algorithm ◽  
Development And Validation ◽  
Three Dimensional Space

This article follows a detailed description of development and validation for the direct kinematic model of six degrees of freedom articulated arm robot - Kawasaki FS10E model. The development of the kinematic model is based on widely used Denavit-Hartenberg notation, but, after the initial parameter identification, the mathematical algorithm itself follows an approach that uses the quaternion number system, taking advantage of their efficiency in describing spatial rotation - providing a convenient mathematical notation for expressing rotations and orientations of objects in three-dimensional space. The proposed algorithm concludes with two quaternion-based relations that express both the position of robot tool center point (TCP) position and end-effector orientation with respect to robot base coordinate system using Denavit-Hartenberg parameters and joint values as input data. Furthermore, the developed direct kinematic model was validated using the programming and offline simulation software Kawasaki PC Roset.

scholarly journals Algorithm for Calculating the Global Minimum of a Smooth Function of Several Variables

2021 ◽  
Vol 8 (4) ◽  
pp. 591-596
Author(s):  
Zhetkerbay Kaidassov ◽  
Zhailan S. Tutkusheva
Keyword(s):  
Global Minimum ◽  
Dimensional Space ◽  
Applied Mathematics ◽  
Three Dimensional ◽  
Computational Experiments ◽  
Quadrature Formulas ◽  
Cubature Formulas ◽  
Integration Formulas ◽  
C Programming ◽  
Approximate Integration

Every year the interest of theorists and practitioners in optimisation problems is growing, and extreme problems are found in all branches of science. Local optimisation problems are well studied and there are constructive methods for their solution. However, global optimisation problems do not meet the requirements in practice; therefore, the search for the global minimum remains one of the major challenges for computational and applied mathematics. This study discusses the search for the global minimum of multidimensional and multiextremal problems with high precision. Mechanical quadrature formulas, that is, the formulas for approximate integration were applied to calculate the integrals. Of all the approximate integration formulas, the Sobolev lattice cubature formulas with a regular boundary layer were chosen. In multidimensional examples, the Sobolev formulas are optimal. Computational experiments were carried out in the most popular C++ programming language. Based on the computational experiments, a new algorithm was proposed. In three-dimensional space, the calculations of the global minimum have been described using specific examples. Computational experiments show that the proposed algorithm works for multiextremal problems with the same amount of time as for convex ones.

Three Dimensional structure and organization of diploid chromosomes by optical section microscopy

1987 ◽  
Vol 45 ◽  
pp. 633-635
Author(s):  
David A. Agard ◽  
Yasushi Hiraoka ◽  
John W. Sedat
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Developmental State ◽  
Mitotic Cell ◽  
Biological Properties ◽  
Dimensional Structure ◽  
Specific Gene ◽  
Three Dimensional Structure ◽  
Complementary Techniques ◽  
Dynamic Structures

In an effort to understand the complex relationship between structure and biological function within the nucleus, we have embarked on a program to examine the three-dimensional structure and organization of Drosophila melanogaster embryonic chromosomes. Our overall goal is to determine how DNA and proteins are organized into complex and highly dynamic structures (chromosomes) and how these chromosomes are arranged in three dimensional space within the cell nucleus. Futher, we hope to be able to correlate structual data with such fundamental biological properties as stage in the mitotic cell cycle, developmental state and transcription at specific gene loci.Towards this end, we have been developing methodologies for the three-dimensional analysis of non-crystalline biological specimens using optical and electron microscopy. We feel that the combination of these two complementary techniques allows an unprecedented look at the structural organization of cellular components ranging in size from 100A to 100 microns.

Diffraction contrast of dislocations in icosahedral quasicrystals

1990 ◽  
Vol 48 (4) ◽  
pp. 454-455
Author(s):  
K. Urban ◽  
Z. Zhang ◽  
M. Wollgarten ◽  
D. Gratias
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Characterization ◽  
Extinction Condition ◽  
Burgers Vector ◽  
Diffraction Contrast ◽  
Lattice Geometry ◽  
Reference Space ◽  
Icosahedral Quasicrystals ◽  
The One

Recently dislocations have been observed by electron microscopy in the icosahedral quasicrystalline (IQ) phase of Al65Cu20Fe15. These dislocations exhibit diffraction contrast similar to that known for dislocations in conventional crystals. The contrast becomes extinct for certain diffraction vectors g. In the following the basis of electron diffraction contrast of dislocations in the IQ phase is described. Taking account of the six-dimensional nature of the Burgers vector a “strong” and a “weak” extinction condition are found.Dislocations in quasicrystals canot be described on the basis of simple shear or insertion of a lattice plane only. In order to achieve a complete characterization of these dislocations it is advantageous to make use of the one to one correspondence of the lattice geometry in our three-dimensional space (R3) and that in the six-dimensional reference space (R6) where full periodicity is recovered . Therefore the contrast extinction condition has to be written as gpbp + gobo = 0 (1). The diffraction vector g and the Burgers vector b decompose into two vectors gp, bp and go, bo in, respectively, the physical and the orthogonal three-dimensional sub-spaces of R6.

Flavin radicals, conformational sampling and robust design principles in interprotein electron transfer: the trimethylamine dehydrogenase-electron-transferring flavoprotein complex

2004 ◽  
Vol 71 ◽  
pp. 1-14
Author(s):  
David Leys ◽  
Jaswir Basran ◽  
François Talfournier ◽  
Kamaldeep K. Chohan ◽  
Andrew W. Munro ◽  
...  
Keyword(s):  
Electron Transfer ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Kinetic Studies ◽  
Molecular Assembly ◽  
Conformational Sampling ◽  
Trimethylamine Dehydrogenase ◽  
Key Residues ◽  
Electron Transferring Flavoprotein ◽  
Electron Transfer Complex

TMADH (trimethylamine dehydrogenase) is a complex iron-sulphur flavoprotein that forms a soluble electron-transfer complex with ETF (electron-transferring flavoprotein). The mechanism of electron transfer between TMADH and ETF has been studied using stopped-flow kinetic and mutagenesis methods, and more recently by X-ray crystallography. Potentiometric methods have also been used to identify key residues involved in the stabilization of the flavin radical semiquinone species in ETF. These studies have demonstrated a key role for 'conformational sampling' in the electron-transfer complex, facilitated by two-site contact of ETF with TMADH. Exploration of three-dimensional space in the complex allows the FAD of ETF to find conformations compatible with enhanced electronic coupling with the 4Fe-4S centre of TMADH. This mechanism of electron transfer provides for a more robust and accessible design principle for interprotein electron transfer compared with simpler models that invoke the collision of redox partners followed by electron transfer. The structure of the TMADH-ETF complex confirms the role of key residues in electron transfer and molecular assembly, originally suggested from detailed kinetic studies in wild-type and mutant complexes, and from molecular modelling.

Matrix Completions, Moments, and Sums of Hermitian Squares

2011 ◽  
Author(s):  
Mihály Bakonyi ◽  
Hugo J. Woerdeman
Keyword(s):  
Linear Algebra ◽  
Undergraduate Students ◽  
Complex Analysis ◽  
Measure Theory ◽  
Complex Function ◽  
Complex Function Theory ◽  
Extensive Discussion ◽  
The Subject ◽  
Processing Control ◽  
Developing Area

Intensive research in matrix completions, moments, and sums of Hermitian squares has yielded a multitude of results in recent decades. This book provides a comprehensive account of this quickly developing area of mathematics and applications and gives complete proofs of many recently solved problems. With MATLAB codes and more than two hundred exercises, the book is ideal for a special topics course for graduate or advanced undergraduate students in mathematics or engineering, and will also be a valuable resource for researchers. Often driven by questions from signal processing, control theory, and quantum information, the subject of this book has inspired mathematicians from many subdisciplines, including linear algebra, operator theory, measure theory, and complex function theory. In turn, the applications are being pursued by researchers in areas such as electrical engineering, computer science, and physics. The book is self-contained, has many examples, and for the most part requires only a basic background in undergraduate mathematics, primarily linear algebra and some complex analysis. The book also includes an extensive discussion of the literature, with close to six hundred references from books and journals from a wide variety of disciplines.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

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