DOCSLIB.ORG

Split Quaternion Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Split Quaternion Matrices View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Repository of the Academy's Library Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 13 (2012), No 2, pp. 223-232 DOI: 10.18514/MMN.2012.364 Split quaternion matrices Yasemin Alagöz, Kür³at Hakan Oral, and Salim Yüce Miskolc Mathematical Notes HU e-ISSN 1787-2413 Vol. 13 (2012), No. 2, pp. 223–232 SPLIT QUATERNION MATRICES YASEMIN ALAGOZ,¨ KURS¸AT¨ HAKAN ORAL, AND SALIM YUCE¨ Received 7 June, 2011 Abstract. In this paper, we consider split quaternions and split quaternion matrices. Firstly, we give some properties of split quaternions. After that we investigate split quaternion matrices using properties of complex matrices. Then we define the complex adjoint matrix of split qua- ternion matrices and we describe some of their properties. Furthermore, we give the definition of q-determinant of split quaternion matrices. 2000 Mathematics Subject Classification: 11R52; 15A33 Keywords: quaternions, split quaternions, matrices of split quaternions 1. INTRODUCTION Sir William Rowan Hamilton introduced the set of quaternions in 1843, which can be represented as H a a0 a1i a2j a3k a0;a1;a2;a3 R Df D C C C W 2 g where i 2 j 2 k2 1 and ijk 1. One can see easily that ij j i k. Thus the setD of quaternionsD D is a skew-field.D Since quaternions are non-commutative,D D they differs from complex numbers .C/ and real numbers .R/. For example, for matrices of quaternions the left eigenvalues and the right eigenvalues must be not the same. Because of this, various authors have been studied quaternion matrices, for example [1], [4], [5] and [6]. After the work of Hamilton, in 1849, James Cockle introduced the set of split quaternions, which can be represented as Hs q q0 q1i q2j q3k q0;q1;q2;q3 R Df D C C C W 2 g where i 2 1; j 2 k2 1 and ijk 1. This set is 4-dimensional like the set D D D D of quaternions. Hs is noncommutative, too. But Hs contains zero-divisors, nilpotent elements and nontrivial idempotents. For more properties of split quatenions the reader is referred to [2] and [3]. In this paper, we will investigate split quaternion matrices. Firstly in Section 2, we will give some properties of split quaternions which will be used in the following sections. After that, in Section 3, we will introduce split quaternion matrices and give some properties of them. Finally in Section 4, we define the complex adjoint matrix c 2012 Miskolc University Press 224 YASEMIN ALAGOZ,¨ KURS¸AT¨ HAKAN ORAL, AND SALIM YUCE¨ of split quaternion matrices and we give some properties of these matrices. Then we will define the determinant of a split quaternion matrices using its adjoint matrix. 2. SPLIT QUATERNIONS A split quaternion q is a vector of the form q a0 a1i a2j a3k D C C C where a0;a1;a2;a3 are real numbers. Split quaternion algebra is an associative, non-commutative, non-division ring with four basis elements 1;i;j;k satisfying the equalities f g i 2 1; j 2 k2 1; D D D ij j i k; jk kj i; ki ik j . D D D D D D Let us denote the algebra of split quaternions by HS . For any q a0 a1i a2j D C C C a3k HS we define Req a0, the real part of q; Imq a1i a2j a3k, the 2 D D C C imaginary part of q. The conjugate of a split quaternion is denoted by q q and it is DN q q a0 a1i a2j a3k. D D The norm of a split quaternion is defined as p qˇ ˇ q qq ˇIqˇ k k D j j D 2 2 2 2 where Iq qq a0 a1 a2 a3.If q 1 then q is called unit split quaternion. D qD C k k D Note that p is a unit split quaternion for q HS with q 0. The inverse of D q 2 k k ¤ the split quaternionk k q is 1 q q , if Iq 0. D Iq ¤ Theorem 1. Split quaternions satisfy the following properties for the elements p;q and r HS . .i/ q q qq2 , D .ii/ jc cj or jcj c for any complex number c, D 2 D 2 .iii/ q2 Req Imq 2Req Imq, D j j k k C .iv/ .qp/ pq, .v/ .pq/r Dp .qr/, .vi/ pq Dqp in general, .vii/ q ¤ q if and only if q is a real number, D .viii/ q is the inverse of q if a2 a2 a2 a2, Iq 0 C 1 ¤ 2 C 3 .ix/ Every split quaternion q can be uniquely expressed as q c1 c2j; where c1 D C and c2 are complex numbers. Theorem 2. Every split quaternion can be represented by a 2 2 complex matrix. SPLIT QUATERNION MATRICES 225 Proof. Let q HS , then there exist complex numbers a1 and a2 such that q a1 2 D C a2j by Theorem 1.ix/. The linear map fq HS HS is defined by fq .p/ pq W ! D for all p HS . This map is bijective and 2 fq .1/ 1.a1 a2j / a1 a2j D C D C fq .j / j .a1 a2j / a2 a1j . D C D C With this transformation split quaternions are defined as subset of the matrix ring M2 .C/, the set of 2 2 complex matrices:  a a à 0 1 2 HS a1;a2 C . D a2 a1 I 2 0 HS and HS are essentially the same. Note that,  a a à 0 1 2 0 M q a1 a2j HS q HS W D C 2 ! D a2 a1 2 2 is bijective and preserves the operations. Furthermore q detq0 . k k D 3. SPLIT QUATERNION MATRICES The set Mm n .HS / denotes all m n type matrices with split quaternion entries. If m n, then the set of split quaternion matrices is denoted by Mn .HS /. The ordin- D ary matrix addition and multiplication are defined. With these operations Mn .HS / is a ring with a unit. The left (right) scalar multiplication is defined as, for A .ast / D 2 Mm n .HS /, q HS , 2 qA .qast / and Aq .ast q/: D D Here for B Mn o .HS /; p HS we obtain 2 2 q .AB/ .qA/B; D .Aq/B A.qB/; D .pq/A p .qA/: D Mm n .HS / is a left (right) vector space over HS : For A .ast / Mm n .HS /, A .ast / ast Mm n .HS / is the conjugate D 2 D D 2 T T of A; A .ats/ Mn m .HS / is the transpose of A; A A Mn m .HS / is the conjugateD transpose2 of A. D 2 For a square matrix A Mn .HS / if AA AA then A is called a normal matrix; if A A then A is called2 a Hermitian matrix;D if AA I then A is called a unitary D D matrix; for B Mn .HS / if AB BA I then A is an invertible matrix. 2 D D 226 YASEMIN ALAGOZ,¨ KURS¸AT¨ HAKAN ORAL, AND SALIM YUCE¨  i j à  i k à Example 1. Let A , B M2 .HS /. Then, D 0 k D 0 j 2  i 1 à  i j à .i/ A 1 and A D 0 k D 0 k  à  à 1 i 1 i 1 .ii/ A A 1 D 0 k ¤ 0 k D  à  à T 1 i 0 i 0 1T .iii/ A A D 1 k ¤ 1 k D  1 j 1 à  1 j 1 à .iv/ AB C C A B D 0 i ¤ 0 i D  à  à T 1 0 1 0 .v/ .AB/ BT AT D j 1 i ¤ j 1 i D C C With these examples we get the following Corollary for split quaternion matrices. Corollary 1. Let A Mm n .HS / and B Mn o .HS /. Then the followings are satisfied: 2 2 1 .i/ A A 1 in general; ¤ T 1 1T .ii/ A A in general; ¤ .iii/ AB A B in general; ¤ .iv/ .AB/T BT AT in general. ¤ Theorem 3. Let A Mm n .HS / and B Mn o .HS /. Then, T 2 2 .i/ A AT , D .ii/ .AB/ BA, D1 1 1 .iii/ .AB/ B A if A and B are invertible, 1 D 1 .iv/ .A / A if A is invertible. D Proof. .i/,.iii/ and .iv/ can be easily shown. Now we will prove .ii/. .ii/ Let A A1 A2j and B B1 B2j , where A1;A2;B1 and B2 are complex matrices. ThenD C D C .AB/ A1B1 A2B2 A1B2 A2B1 j D C C C T A1B1 A2B2 A1B2 A2B1 j D C C T T A1B1 A2B2 A1B2 A2B1 j D C C T T T T T T T T B1 A1 .B2/ A2 .B2/ .A1/ j B1 .A2/ j . D C On the other hand, for T T T T B B1 .B2/ j and A A1 .A2/ j D D SPLIT QUATERNION MATRICES 227 we can obtain T T Á T T Á B A B1 .B2/ j A1 .A2/ j D T T T T T T T T B1 A1 .B2/ A2 .B2/ .A1/ j B1 .A2/ j .
Load more

Recommended publications
  • On Regular Matrix Semirings
    Thai Journal of Mathematics Volume 7 (2009) Number 1 : 69{75 www.math.science.cmu.ac.th/thaijournal Online ISSN 1686-0209 On Regular Matrix Semirings S. Chaopraknoi, K. Savettaseranee and P. Lertwichitsilp Abstract : A ring R is called a (von Neumann) regular ring if for every x 2 R; x = xyx for some y 2 R. It is well-known that for any ring R and any positive integer n, the full matrix ring Mn(R) is regular if and only if R is a regular ring. This paper examines this property on any additively commutative semiring S with zero. The regularity of S is defined analogously. We show that for a positive integer n, if Mn(S) is a regular semiring, then S is a regular semiring but the converse need not be true for n = 2. And for n ≥ 3, Mn(S) is a regular semiring if and only if S is a regular ring. Keywords : Matrix ring; Matrix semiring; Regular ring; Regular semiring. 2000 Mathematics Subject Classification : 16Y60, 16S50. 1 Introduction A triple (S; +; ·) is called a semiring if (S; +) and (S; ·) are semigroups and · is distributive over +. An element 0 2 S is called a zero of the semiring (S; +; ·) if x + 0 = 0 + x = x and x · 0 = 0 · x = 0 for all x 2 S. Note that every semiring contains at most one zero. A semiring (S; +; ·) is called additively [multiplicatively] commutative if x + y = y + x [x · y = y · x] for all x; y 2 S: We say that (S; +; ·) is commutative if it is both additively and multiplicatively commutative.
    [Show full text]
  • Quaternions and Cli Ord Geometric Algebras
    Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a .
    [Show full text]
  • 1 Jun 2020 Flat Semimodules & Von Neumann Regular Semirings
    Flat Semimodules & von Neumann Regular Semirings* Jawad Abuhlail† Rangga Ganzar Noegraha‡ [email protected] [email protected] Department of Mathematics and Statistics Universitas Pertamina King Fahd University of Petroleum & Minerals Jl. Teuku Nyak Arief 31261Dhahran,KSA Jakarta12220,Indonesia June 2, 2020 Abstract Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of exact sequences of semimodules and its relationships with other notions of flatness for semimodules over semirings. We also prove that a subtractive semiring over which every right (left) semimodule is e-flat is a von Neumann regular semiring. Introduction Semirings are, roughly, rings not necessarily with subtraction. They generalize both rings and distributive bounded lattices and have, along with their semimodules many applications in Com- arXiv:1907.07047v2 [math.RA] 1 Jun 2020 puter Science and Mathematics (e.g., [1, 2, 3]). Some applications can be found in Golan’s book [4], which is our main reference on this topic. A systematic study of semimodules over semirings was carried out by M. Takahashi in a series of papers 1981-1990. However, he defined two main notions in a way that turned out to be not natural. Takahashi’s tensor products [5] did not satisfy the expected Universal Property. On *MSC2010: Primary 16Y60; Secondary 16D40, 16E50, 16S50 Key Words: Semirings; Semimodules; Flat Semimodules; Exact Sequences; Von Neumann Regular Semirings; Matrix Semirings The authors would like to acknowledge the support provided by the Deanship of Scientific Research (DSR) at King Fahd University of Petroleum & Minerals (KFUPM) for funding this work through projects No.
    [Show full text]
  • Subset Semirings
    University of New Mexico UNM Digital Repository Faculty and Staff Publications Mathematics 2013 Subset Semirings Florentin Smarandache University of New Mexico, [email protected] W.B. Vasantha Kandasamy [email protected] Follow this and additional works at: https://digitalrepository.unm.edu/math_fsp Part of the Algebraic Geometry Commons, Analysis Commons, and the Other Mathematics Commons Recommended Citation W.B. Vasantha Kandasamy & F. Smarandache. Subset Semirings. Ohio: Educational Publishing, 2013. This Book is brought to you for free and open access by the Mathematics at UNM Digital Repository. It has been accepted for inclusion in Faculty and Staff Publications by an authorized administrator of UNM Digital Repository. For more information, please contact [email protected], [email protected], [email protected]. Subset Semirings W. B. Vasantha Kandasamy Florentin Smarandache Educational Publisher Inc. Ohio 2013 This book can be ordered from: Education Publisher Inc. 1313 Chesapeake Ave. Columbus, Ohio 43212, USA Toll Free: 1-866-880-5373 Copyright 2013 by Educational Publisher Inc. and the Authors Peer reviewers: Marius Coman, researcher, Bucharest, Romania. Dr. Arsham Borumand Saeid, University of Kerman, Iran. Said Broumi, University of Hassan II Mohammedia, Casablanca, Morocco. Dr. Stefan Vladutescu, University of Craiova, Romania. Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/eBooks-otherformats.htm ISBN-13: 978-1-59973-234-3 EAN: 9781599732343 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One INTRODUCTION 7 Chapter Two SUBSET SEMIRINGS OF TYPE I 9 Chapter Three SUBSET SEMIRINGS OF TYPE II 107 Chapter Four NEW SUBSET SPECIAL TYPE OF TOPOLOGICAL SPACES 189 3 FURTHER READING 255 INDEX 258 ABOUT THE AUTHORS 260 4 PREFACE In this book authors study the new notion of the algebraic structure of the subset semirings using the subsets of rings or semirings.
    [Show full text]
  • Arxiv:1001.0240V1 [Math.RA]
    Fundamental representations and algebraic properties of biquaternions or complexified quaternions Stephen J. Sangwine∗ School of Computer Science and Electronic Engineering, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom. Email: [email protected] Todd A. Ell† 5620 Oak View Court, Savage, MN 55378-4695, USA. Email: [email protected] Nicolas Le Bihan GIPSA-Lab D´epartement Images et Signal 961 Rue de la Houille Blanche, Domaine Universitaire BP 46, 38402 Saint Martin d’H`eres cedex, France. Email: [email protected] October 22, 2018 Abstract The fundamental properties of biquaternions (complexified quaternions) are presented including several different representations, some of them new, and definitions of fundamental operations such as the scalar and vector parts, conjugates, semi-norms, polar forms, and inner and outer products. The notation is consistent throughout, even between representations, providing a clear account of the many ways in which the component parts of a biquaternion may be manipulated algebraically. 1 Introduction It is typical of quaternion formulae that, though they be difficult to find, once found they are immediately verifiable. J. L. Synge (1972) [43, p34] arXiv:1001.0240v1 [math.RA] 1 Jan 2010 The quaternions are relatively well-known but the quaternions with complex components (complexified quaternions, or biquaternions1) are less so. This paper aims to set out the fundamental definitions of biquaternions and some elementary results, which, although elementary, are often not trivial. The emphasis in this paper is on the biquaternions as an applied algebra – that is, a tool for the manipulation ∗This paper was started in 2005 at the Laboratoire des Images et des Signaux (now part of the GIPSA-Lab), Grenoble, France with financial support from the Royal Academy of Engineering of the United Kingdom and the Centre National de la Recherche Scientifique (CNRS).
    [Show full text]
  • Derivations of a Matrix Ring Containing a Subring of Triangular Matrices S
    ISSN 1066-369X, Russian Mathematics (Iz. VUZ), 2011, Vol. 55, No. 11, pp. 18–26. c Allerton Press, Inc., 2011. Original Russian Text c S.G. Kolesnikov and N.V. Mal’tsev, 2011, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2011, No. 11, pp. 23–33. Derivations of a Matrix Ring Containing a Subring of Triangular Matrices S. G. Kolesnikov1* andN.V.Mal’tsev2** 1Institute of Mathematics, Siberian Federal University, pr. Svobodnyi 79, Krasnoyarsk, 660041 Russia 2Institute of Core Undergraduate Programs, Siberian Federal University, 26 ul. Akad. Kirenskogo 26, Krasnoyarsk, 660041 Russia Received October 22, 2010 Abstract—We describe the derivations (including Jordan and Lie ones) of finitary matrix rings, containing a subring of triangular matrices, over an arbitrary associative ring with identity element. DOI: 10.3103/S1066369X1111003X Keywords and phrases: finitary matrix, derivations of rings, Jordan and Lie derivations of rings. INTRODUCTION An endomorphism ϕ of an additive group of a ring K is called a derivation if ϕ(ab)=ϕ(a)b + aϕ(b) for any a, b ∈ K. Derivations of the Lie and Jordan rings associated with K are called, respectively, the Lie and Jordan derivations of the ring K; they are denoted, respectively, by Λ(K) and J(K). (We obtain them by replacing the multiplication in K with the Lie multiplication a ∗ b = ab − ba and the Jordan one a ◦ b = ab + ba, respectively.) In 1957 I. N. Herstein proved [1] that every Jordan derivation of a primary ring of characteristic =2 is a derivation. His theorem was generalized by several authors ([2–9] et al.) for other classes of rings and algebras.
    [Show full text]
  • Arithmetic of Hyperbolic 3-Manifolds
    Arithmetic of Hyperbolic 3-Manifolds Alan Reid Rice University Hyperbolic 3-Manifolds and Discrete Groups Hyperbolic 3-space can be defined as 3 H = f(z; t) 2 C × R : t > 0g dsE and equipped with the metric ds = t . Geodesics are vertical lines perpendicular to C or semi-circles perpendicular to C. Codimension 1 geodesic submanifolds are Euclidean planes in 3 H orthogonal to C or hemispheres centered on C. 3 The full group of orientation-preserving isometries of H can be identified with PSL(2; C). Abuse of notation We will often view Kleinian groups as groups of matrices! A discrete subgroup of PSL(2; C) is called a Kleinian group. 3 Such a group acts properly discontinuously on H . If Γ is torsion-free (does not contain elements of finite order) then Γ acts freely. 3 In this latter case we get a quotient manifold H =Γ and a 3 3 covering map H ! H =Γ which is a local isometry. 3 We will be interested in the case when H =Γ has finite volume. Note: If Γ is a finitely generated subgroup of PSL(2; C) there is always a finite index subgroup that is torsion-free. Examples 1. Let d be a square-freep positive integer, and Od the ring of algebraic integers in Q( −d). Then PSL(2; Od) is a Kleinian group. 3 Indeed H =PSL(2; Od) has finite volume but is non-compact. These are known as the Bianchi groups. d = 1 (picture by Jos Leys) 2. Reflection groups. Tessellation by all right dodecahedra (from the Not Knot video) 3.Knot Complements The Figure-Eight Knot Thurston's hyperbolization theorem shows that many knots have complements that are hyperbolic 3-manifolds of finite volume.
    [Show full text]
  • Right Ideals of a Ring and Sublanguages of Science
    RIGHT IDEALS OF A RING AND SUBLANGUAGES OF SCIENCE Javier Arias Navarro Ph.D. In General Linguistics and Spanish Language http://www.javierarias.info/ Abstract Among Zellig Harris’s numerous contributions to linguistics his theory of the sublanguages of science probably ranks among the most underrated. However, not only has this theory led to some exhaustive and meaningful applications in the study of the grammar of immunology language and its changes over time, but it also illustrates the nature of mathematical relations between chunks or subsets of a grammar and the language as a whole. This becomes most clear when dealing with the connection between metalanguage and language, as well as when reflecting on operators. This paper tries to justify the claim that the sublanguages of science stand in a particular algebraic relation to the rest of the language they are embedded in, namely, that of right ideals in a ring. Keywords: Zellig Sabbetai Harris, Information Structure of Language, Sublanguages of Science, Ideal Numbers, Ernst Kummer, Ideals, Richard Dedekind, Ring Theory, Right Ideals, Emmy Noether, Order Theory, Marshall Harvey Stone. §1. Preliminary Word In recent work (Arias 2015)1 a line of research has been outlined in which the basic tenets underpinning the algebraic treatment of language are explored. The claim was there made that the concept of ideal in a ring could account for the structure of so- called sublanguages of science in a very precise way. The present text is based on that work, by exploring in some detail the consequences of such statement. §2. Introduction Zellig Harris (1909-1992) contributions to the field of linguistics were manifold and in many respects of utmost significance.
    [Show full text]
  • Hypercomplex Numbers
    Hypercomplex numbers Johanna R¨am¨o Queen Mary, University of London [email protected] We have gradually expanded the set of numbers we use: first from finger counting to the whole set of positive integers, then to positive rationals, ir- rational reals, negatives and finally to complex numbers. It has not always been easy to accept new numbers. Negative numbers were rejected for cen- turies, and complex numbers, the square roots of negative numbers, were considered impossible. Complex numbers behave like ordinary numbers. You can add, subtract, multiply and divide them, and on top of that, do some nice things which you cannot do with real numbers. Complex numbers are now accepted, and have many important applications in mathematics and physics. Scientists could not live without complex numbers. What if we take the next step? What comes after the complex numbers? Is there a bigger set of numbers that has the same nice properties as the real numbers and the complex numbers? The answer is yes. In fact, there are two (and only two) bigger number systems that resemble real and complex numbers, and their discovery has been almost as dramatic as the discovery of complex numbers was. 1 Complex numbers Complex numbers where discovered in the 15th century when Italian math- ematicians tried to find a general solution to the cubic equation x3 + ax2 + bx + c = 0: At that time, mathematicians did not publish their results but kept them secret. They made their living by challenging each other to public contests of 1 problem solving in which the winner got money and fame.
    [Show full text]
  • Problems in Abstract Algebra
    STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth 10.1090/stml/082 STUDENT MATHEMATICAL LIBRARY Volume 82 Problems in Abstract Algebra A. R. Wadsworth American Mathematical Society Providence, Rhode Island Editorial Board Satyan L. Devadoss John Stillwell (Chair) Erica Flapan Serge Tabachnikov 2010 Mathematics Subject Classification. Primary 00A07, 12-01, 13-01, 15-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/stml-82 Library of Congress Cataloging-in-Publication Data Names: Wadsworth, Adrian R., 1947– Title: Problems in abstract algebra / A. R. Wadsworth. Description: Providence, Rhode Island: American Mathematical Society, [2017] | Series: Student mathematical library; volume 82 | Includes bibliographical references and index. Identifiers: LCCN 2016057500 | ISBN 9781470435837 (alk. paper) Subjects: LCSH: Algebra, Abstract – Textbooks. | AMS: General – General and miscellaneous specific topics – Problem books. msc | Field theory and polyno- mials – Instructional exposition (textbooks, tutorial papers, etc.). msc | Com- mutative algebra – Instructional exposition (textbooks, tutorial papers, etc.). msc | Linear and multilinear algebra; matrix theory – Instructional exposition (textbooks, tutorial papers, etc.). msc | Group theory and generalizations – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA162 .W33 2017 | DDC 512/.02–dc23 LC record available at https://lccn.loc.gov/2016057500 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space
    Split Quaternions and Spacelike Constant Slope Surfaces in Minkowski 3- Space Murat Babaarslan and Yusuf Yayli Abstract. A spacelike surface in the Minkowski 3-space is called a constant slope surface if its position vector makes a constant angle with the normal at each point on the surface. These surfaces completely classified in [J. Math. Anal. Appl. 385 (1) (2012) 208-220]. In this study, we give some relations between split quaternions and spacelike constant slope surfaces in Minkowski 3-space. We show that spacelike constant slope surfaces can be reparametrized by using rotation matrices corresponding to unit timelike quaternions with the spacelike vector parts and homothetic motions. Subsequently we give some examples to illustrate our main results. Mathematics Subject Classification (2010). Primary 53A05; Secondary 53A17, 53A35. Key words: Spacelike constant slope surface, split quaternion, homothetic motion. 1. Introduction Quaternions were discovered by Sir William Rowan Hamilton as an extension to the complex number in 1843. The most important property of quaternions is that every unit quaternion represents a rotation and this plays a special role in the study of rotations in three- dimensional spaces. Also quaternions are an efficient way understanding many aspects of physics and kinematics. Many physical laws in classical, relativistic and quantum mechanics can be written nicely using them. Today they are used especially in the area of computer vision, computer graphics, animations, aerospace applications, flight simulators, navigation systems and to solve optimization problems involving the estimation of rigid body transformations. Ozdemir and Ergin [9] showed that a unit timelike quaternion represents a rotation in Minkowski 3-space.
    [Show full text]
  • The Geometry and Topology of Arithmetic Hyperbolic 3-Manifolds
    The geometry and topology of arithmetic hyperbolic 3-manifolds Alan W. Reid∗ May 12, 2007 1 Introduction This paper is based on three lectures given by the author at the RIMS Symposium, “Topology, Com- plex Analysis and Arithmetic of Hyperbolic Spaces” held at the Research Institute for Mathematical Sciences, Kyoto University, in December 2006. The goal of the lectures was to describe recent work on understanding topological and geometric properties of arithmetic hyperbolic 3-manifolds, and the connections with number theory. This is the theme of the paper. Our discussion of topological aspects of arithmetic hyperbolic 3-orbifolds is motivated by the following central conjectures from 3-manifold topology: Conjecture 1.1. Let M be a closed hyperbolic 3-manifold, then M is virtually Haken; ie M has a finite sheeted cover N which contains embedded incompressible surface (necessarily of genus at least 2). Conjecture 1.2. Let M be a closed hyperbolic 3-manifold, then M has a finite sheeted cover N for which b1(N) > 0 (where b1(N) is the first Betti number of N). We can also rephrase Conjecture 1.2 to say that vb1(M) > 0 where vb1(M) is defined by vb1(M) = sup{b1(N) : N is a finite cover of M}, and is called the virtual first Betti number of M. Conjecture 1.3. Let M be a closed hyperbolic 3-manifold, then vb1(M) = ∞. Conjecture 1.4. Let M be a closed hyperbolic 3-manifold, then π1(M) is large; that is to say, some finite index subgroup of π1(M) admits a surjective homomorphism onto a non-abelian free group.
    [Show full text]