Sir Thomas Muir, 1844–1934
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
36975 1-11 Legalap1 Layout 1
Government Gazette Staatskoerant REPUBLIC OF SOUTH AFRICA REPUBLIEK VAN SUID-AFRIKA Vol. 581 Pretoria, 1 November 2013 No. 36975 PART 1 OF 2 LEGAL NOTICES A WETLIKE SEE PART C SIEN DEEL C KENNISGEWINGS N.B. The Government Printing Works will not be held responsible for the quality of “Hard Copies” or “Electronic Files” submitted for publication purposes AIDS HELPLINE: 0800-0123-22 Prevention is the cure 305305—A 36975—1 2 No. 36975 GOVERNMENT GAZETTE, 1 NOVEMBER 2013 IMPORTANT NOTICE The Government Printing Works will not be held responsible for faxed documents not received due to errors on the fax machine or faxes received which are unclear or incomplete. Please be advised that an “OK” slip, received from a fax machine, will not be accepted as proof that documents were received by the GPW for printing. If documents are faxed to the GPW it will be the sender’s respon- sibility to phone and confirm that the documents were received in good order. Furthermore the Government Printing Works will also not be held responsible for cancellations and amendments which have not been done on original documents received from clients. TABLE OF CONTENTS LEGAL NOTICES Page BUSINESS NOTICES.............................................................................................................................................. 11 Gauteng..................................................................................................................................................... 11 Eastern Cape............................................................................................................................................ -
Chapter 7 Block Designs
Chapter 7 Block Designs One simile that solitary shines In the dry desert of a thousand lines. Epilogue to the Satires, ALEXANDER POPE The collection of all subsets of cardinality k of a set with v elements (k < v) has the ³ ´ v¡t property that any subset of t elements, with 0 · t · k, is contained in precisely k¡t subsets of size k. The subsets of size k provide therefore a nice covering for the subsets of a lesser cardinality. Observe that the number of subsets of size k that contain a subset of size t depends only on v; k; and t and not on the specific subset of size t in question. This is the essential defining feature of the structures that we wish to study. The example we just described inspires general interest in producing similar coverings ³ ´ v without using all the k subsets of size k but rather as small a number of them as possi- 1 2 CHAPTER 7. BLOCK DESIGNS ble. The coverings that result are often elegant geometrical configurations, of which the projective and affine planes are examples. These latter configurations form nice coverings only for the subsets of cardinality 2, that is, any two elements are in the same number of these special subsets of size k which we call blocks (or, in certain instances, lines). A collection of subsets of cardinality k, called blocks, with the property that every subset of size t (t · k) is contained in the same number (say ¸) of blocks is called a t-design. We supply the reader with constructions for t-designs with t as high as 5. -
The Growth of Population in the Province of the Western Cape
Southern Africa Labour and Development Research Unit A Tapestry of People: The Growth of Population in the Province of the Western Cape by Dudley Horner and Francis Wilson WORKING PAPER SERIES Number 21 About the Authors and Acknowledgments Professor Francis Wilson and Dudley Horner are both SALDRU Honorary Research Fellows and were previously respectively director and deputy-director of the research unit. We acknowledge with thanks the Directorate for Social Research & Provincial Population in the Department of Social Development within the Provincial Government of the Western Cape, and particularly Mr Gavin Miller and Dr Ravayi Marindo, who commissioned this study as part of the project on the state of population in the Western Cape Province. We thank, too, Mrs Brenda Adams and Mrs Alison Siljeur for all their assistance with the production of this report. While we have endeavoured to make this historical overview as accurate as possible we would welcome any comments suggesting appropriate amendments or corrections. Recommended citation Horner, D. and Wilson, F. (2008) E A Tapestry of People: The Growth of Population in the Province of the Western Cape. A Southern Africa Labour and Development Research Unit Working Paper Number 21. Cape Town: SALDRU, University of Cape Town ISBN: 978-0-9814123-2-0 © Southern Africa Labour and Development Research Unit, UCT, 2008 Working Papers can be downloaded in Adobe Acrobat format from www.saldru.uct.ac.za. Printed copies of Working Papers are available for R15.00 each plus vat and postage charges. Contact: Francis Wilson - [email protected] Dudley Horner - [email protected] Orders may be directed to: The Administrative Officer, SALDRU, University of Cape Town, Private Bag, Rondebosch, 7701, Tel: (021) 650 5696, Fax: (021) 650 5697, Email: [email protected] A Tapestry of People: The Growth of Population in the Province of the Western Cape by Dudley Horner & Francis Wilson Long Before Van Riebeeck. -
Hadamard and Conference Matrices
Hadamard and conference matrices Peter J. Cameron December 2011 with input from Dennis Lin, Will Orrick and Gordon Royle Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
Some Applications of Hadamard Matrices
ORIGINAL ARTICLE Jennifer Seberry · Beata J Wysocki · Tadeusz A Wysocki On some applications of Hadamard matrices Abstract Modern communications systems are heavily reliant on statistical tech- niques to recover information in the presence of noise and interference. One of the mathematical structures used to achieve this goal is Hadamard matrices. They are used in many different ways and some examples are given. This paper concentrates on code division multiple access systems where Hadamard matrices are used for user separation. Two older techniques from design and analysis of experiments which rely on similar processes are also included. We give a short bibliography (from the thousands produced by a google search) of applications of Hadamard matrices appearing since the paper of Hedayat and Wallis in 1978 and some appli- cations in telecommunications. 1 Introduction Hadamard matrices seem such simple matrix structures: they are square, have entries +1or−1 and have orthogonal row vectors and orthogonal column vectors. Yet they have been actively studied for over 138 years and still have more secrets to be discovered. In this paper we concentrate on engineering and statistical appli- cations especially those in communications systems, digital image processing and orthogonal spreading sequences for direct sequences spread spectrum code division multiple access. 2 Basic definitions and properties A square matrix with elements ±1 and size h, whose distinct row vectors are mutu- ally orthogonal, is referred to as an Hadamard matrix of order h. The smallest examples are: 2 Seberry et al. 111 1 11 − 1 − 1 (1) , , 1 − 11−− − 11− where − denotes −1. Such matrices were first studied by Sylvester (1867) who observed that if H is an Hadamard matrix, then HH H −H is also an Hadamard matrix. -
Stellenbosch University Research Report 2004
STELLENBOSCH UNIVERSITY RESEARCH REPORT 2004 Editor: Senior Director: Research Stellenbosch University Stellenbosch 7602 ISBN 0-7972-1170-5 i FOREWORD The 2004 Research Report provides a comprehensive record of the University's research outputs. In addition to this overall survey of research at the institution, perspectives are provided on the performance of individual faculties and departments. Further statistical perspectives on research output and the nature of research activities are presented in other publications of the Division of Research Development, i.e. Research @ Stellenbosch; and Perspectives of Stellenbosch University. As in the past, a variety of persons and organizations contributed to the University's research programme. Stellenbosch University wishes to express particular recognition to the statutory research councils and commissions, government departments, private enterprises, foundations and individuals for their continued support of our research. As far as research funding is concerned, researchers at South African universities are increasingly dependent upon new sources for research funding. This includes national and international opportunities alike. Stellenbosch University therefore attaches great value to regional cooperation and substantive scientific agreements with other universities local and abroad, and to bilateral research and development agreements between South Africa and other countries. Apart from the active support of research by the University's management, a variety of internal support services provide assistance to researchers. These include the library and information services, capacity building workshops, information technology, infrastructure, the maintenance of laboratories and equipment, central analytical services, and support from the administrative and financial divisions, without which it would be impossible to conduct research of this calibre. We acknowledge with appreciation the dedication and excellent contributions made by Stellenbosch University staff members to the institution's research output. -
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem
Perron Spectratopes and the Real Nonnegative Inverse Eigenvalue Problem a b, Charles R. Johnson , Pietro Paparella ∗ aDepartment of Mathematics, College of William & Mary, Williamsburg, VA 23187-8795, USA bDivision of Engineering and Mathematics, University of Washington Bothell, Bothell, WA 98011-8246, USA Abstract Call an n-by-n invertible matrix S a Perron similarity if there is a real non- 1 scalar diagonal matrix D such that SDS− is entrywise nonnegative. We give two characterizations of Perron similarities and study the polyhedra (S) := n 1 C x R : SDxS− 0;Dx := diag (x) and (S) := x (S): x1 = 1 , whichf 2 we call the Perron≥ spectracone andgPerronP spectratopef 2, respectively. C Theg set of all normalized real spectra of diagonalizable nonnegative matrices may be covered by Perron spectratopes, so that enumerating them is of interest. The Perron spectracone and spectratope of Hadamard matrices are of par- ticular interest and tend to have large volume. For the canonical Hadamard matrix (as well as other matrices), the Perron spectratope coincides with the convex hull of its rows. In addition, we provide a constructive version of a result due to Fiedler ([9, Theorem 2.4]) for Hadamard orders, and a constructive version of [2, Theorem 5.1] for Sule˘ımanova spectra. Keywords: Perron spectracone, Perron spectratope, real nonnegative inverse eigenvalue problem, Hadamard matrix, association scheme, relative gain array 2010 MSC: 15A18, 15B48, 15A29, 05B20, 05E30 1. Introduction The real nonnegative inverse eigenvalue problem (RNIEP) is to determine arXiv:1508.07400v2 [math.RA] 19 Nov 2015 which sets of n real numbers occur as the spectrum of an n-by-n nonnegative matrix. -
Knowledge and Colonialism: Eighteenth-Century Travellers in South Africa Atlantic World
Knowledge and Colonialism: Eighteenth-century Travellers in South Africa Atlantic World Europe, Africa and the Americas, 1500–1830 Edited by Wim Klooster Clark University and Benjamin Schmidt University of Washington VOLUME 18 Knowledge and Colonialism: Eighteenth-century Travellers in South Africa By Siegfried Huigen LEIDEN • BOSTON 2009 On the cover: “Coba Caffer Captein” (Gordon Atlas, G75). Courtesy of the Rijkspren- tenkabinet, Amsterdam. This book was originally published as Verkenningen van Zuid-Afrika. Achttiende-eeuwse reizigers aan de kaap (2007). This book is printed on acid-free paper. Library of Congress Cataloging-in-Publication Data Huigen, Siegfried. Knowledge and colonialism : eighteenth-century travellers in South Africa / by Siegfried Huigen. p. cm. — (Atlantic world : Europe, Africa, and the Americas, 1500–1830 ; v. 18) Includes bibliographical references and index. ISBN 978-90-04-17743-7 (hbk. : alk. paper) 1. Cape of Good Hope (South Africa)—Description and travel. 2. Cape of Good Hope (South Africa)—Description and travel—Sources. 3. Travelers—South Africa—Cape of Good Hope—History— 18th century. 4. Europeans—South Africa—Cape of Good Hope—History—18th century. 5. Ethnology—South Africa—Cape of Good Hope—History—18th century. 6. Ethnological expeditions—South Africa—Cape of Good Hope—History—18th century. 7. South Africa—History—To 1836. 8. South Africa—Colonial infl uence. 9. South Africa—Description and travel. 10. South Africa—Description and travel— Sources. I. Title. II. Series. DT2020.H85 2009 968.03—dc22 2009017888 ISSN 1570-0542 ISBN 978 90 04 17743 7 Copyright 2009 by Koninklijke Brill NV, Leiden, The Netherlands. Koninklijke Brill NV incorporates the imprints Brill, Hotei Publishers, IDC Publishers, Martinus Nijhoff Publishers and VSP. -
Hadamard Matrices Include
Hadamard and conference matrices Peter J. Cameron University of St Andrews & Queen Mary University of London Mathematics Study Group with input from Rosemary Bailey, Katarzyna Filipiak, Joachim Kunert, Dennis Lin, Augustyn Markiewicz, Will Orrick, Gordon Royle and many happy returns . Happy Birthday, MSG!! Happy Birthday, MSG!! and many happy returns . Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? A matrix attaining the bound is a Hadamard matrix. This is a nice example of a continuous problem whose solution brings us into discrete mathematics. Hadamard's theorem Let H be an n × n matrix, all of whose entries are at most 1 in modulus. How large can det(H) be? Now det(H) is equal to the volume of the n-dimensional parallelepiped spanned by the rows of H. By assumption, each row has Euclidean length at most n1/2, so that det(H) ≤ nn/2; equality holds if and only if I every entry of H is ±1; > I the rows of H are orthogonal, that is, HH = nI. -
Matrix Theory
Matrix Theory Xingzhi Zhan +VEHYEXI7XYHMIW MR1EXLIQEXMGW :SPYQI %QIVMGER1EXLIQEXMGEP7SGMIX] Matrix Theory https://doi.org/10.1090//gsm/147 Matrix Theory Xingzhi Zhan Graduate Studies in Mathematics Volume 147 American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 15-01, 15A18, 15A21, 15A60, 15A83, 15A99, 15B35, 05B20, 47A63. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-147 Library of Congress Cataloging-in-Publication Data Zhan, Xingzhi, 1965– Matrix theory / Xingzhi Zhan. pages cm — (Graduate studies in mathematics ; volume 147) Includes bibliographical references and index. ISBN 978-0-8218-9491-0 (alk. paper) 1. Matrices. 2. Algebras, Linear. I. Title. QA188.Z43 2013 512.9434—dc23 2013001353 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 a chapter 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. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. -
On the Classification of Hadamard Matrices of Order 32 1 Introduction
On the classi¯cation of Hadamard matrices of order 32 H. Kharaghania;1 B. Tayfeh-Rezaieb aDepartment of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K3M4, Canada bSchool of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran December 5, 2009 Abstract All equivalence classes of Hadamard matrices of order at most 28 have been found by 1994. Order 32 is where a combinatorial explosion occurs on the number of Hadamard matrices. We ¯nd all equivalence classes of Hadamard matrices of order 32 which are of certain types. It turns out that there are exactly 13,680,757 Hadamard matrices of one type and 26,369 such matrices of another type. Based on experience with the classi¯cation of Hadamard matrices of smaller order, it is expected that the number of the remaining two types of these matrices, relative to the total number of Hadamard matrices of order 32, to be insigni¯cant. AMS Subject Classi¯cation: 05B20, 05B05, 05B30. Keywords: Hadamard matrices, classi¯cation of combinatorial objects, isomorph-free gen- eration, orderly algorithm. 1 Introduction A Hadamard matrix of order n is a (¡1; 1) square matrix H of order n such that HHt = nI, where Ht is the transpose of H and I is the identity matrix. It is well known that the order of a Hadamard matrix is 1,2 or a multiple of 4. The old Hadamard conjecture states that the converse also holds, i.e. there is a Hadamard matrix for any order which is divisible by 4. -
The Swiss in Southern Africa 1652-1970
ADOLPHE LINDER THE SWISS IN SOUTHERN AFRICA 1652-1970 PART I ARRIVALS AT THE CAPE 1652-1819 IN CHRONOLOGICAL SEQUENCE Originally published 1997 by Baselr Afrika Bibliographien, Basel Revised for Website 2011 © Adolphe Linder 146 Woodside Village 21 Norton Way Rondebosch 7700 South Africa Paper size 215x298 mm Face 125x238 Font Times New Roman, 10 Margins Left and right 45 mm, top and bottom 30 mm Face tailored to show full page width at 150% enlargement 1 CONTENTS 1. Prologue ………………………………………………………………………………2 2. Chronology 1652-1819 ……………………………………………………………….6 3. Introduction 3.1 The spelling of Swiss names ………………………………………………….6 3.2 Swiss origine of arrivals………………………………………………………..7 3.3 Location of Swiss at the Cape…………………………………………………..8 3.4 Local currency …………………………………………………………………8 3.5 Glossary ………………………………………………………………………..8 4. Short history of arrivals during Company rule 1652-1795 4.1 Establishment of the settlement at the Cape…………………………………….10 4.2 The voyage to the Cape …………………………………………………………10 4.3 Company servants……………………………………………………………….12 4.4 Swiss labour migration to the Netherlands ……………………………………..12 4.5 Recruitment for the Company …………………………………………………..17 4.6 In Company service …………………………………………………………….17 4.7 Freemen………………………………………………………………………….22 4.8 Crime and punishment ………………………………………….…………….. 25 4.9 The Swiss Regiment Meuron at the Cape 1783-1795 …………………………..26 4.10 The end of the Dutch East India Company ……………………………………..29 4.11 Their names live on …………………………………………………………….29 5. Summary of Swiss arrivals during First British Occupation 1695-1803…………….29 6. Summary of Swiss arrivals during Batavian rule 1803-1806 ………………………30 7. Summary of Swiss arrivals during first fourteen years of British colonial rule, 1806-1819……………………………………………………………………………30 8. Personalia 1652-1819 ……………………………………………………………….31 9.