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Latin Squares and Their Applications, Second edition offers a long-awaited update and reissue of this seminal account of the subject. The revision retains foundational, original material from the frequently-cited 1974 volume but is completely updated throughout. As with the earlier version, the author hopes to take the reader 'from the beginnings of the subject to the frontiers of research'. By omitting a few topics which are no longer of current interest, the book expands upon active and emerging areas. Also, the present state of knowledge regarding the 73 then-unsolved problems given at the end of the first edition is discussed and commented upon. In addition, a number of new unsolved problems are proposed. Using an engaging narrative style, this book provides thorough coverage of most parts of the subject, one of the oldest of all discrete mathematical structures and still one of the most relevant. However, in consequence of the huge expansion of the subject in the past 40 years, some topics have had to be omitted in order to keep the book of a reasonable length. Latin squares, or sets of mutually orthogonal latin squares (MOLS), encode the incidence structure of finite geometries; they prescribe the order in which to apply the different treatments in designing an experiment in order to permit effective statistical analysis of the results; they produce optimal density error-correcting codes; they encapsulate the structure of finite groups and of more general algebraic objects known as quasigroups. As regards more recreational aspects of the subject, latin squares provide the most effective and efficient designs for many kinds of games tournaments and they are the templates for Sudoku puzzles. Also, they provide a number of ways of constructing magic squares, both simple magic squares and also ones with additional properties. - Retains the organization and updated foundational material from the original edition - Explores current and emerging research topics - Includes the original 73 'Unsolved Problems' with the current state of knowledge regarding them, as well as new Unsolved Problems for further study
In 1974 the editors of the present volume published a well-received book entitled ``Latin Squares and their Applications''. It included a list of 73 unsolved problems of which about 20 have been completely solved in the intervening period and about 10 more have been partially solved. The present work comprises six contributed chapters and also six further chapters written by the editors themselves. As well as discussing the advances which have been made in the subject matter of most of the chapters of the earlier book, this new book contains one chapter which deals with a subject (r-orthogonal latin squares) which did not exist when the earlier book was written.The success of the former book is shown by the two or three hundred published papers which deal with questions raised by it.
Design and analysis of experiments/Hinkelmann.-v.1.
Over the past two decades, research in the theory of Latin Squares has been growing at a fast pace, and new significant developments have taken place. This book offers a unique approach to various areas of discrete mathematics through the use of Latin Squares.
This book constitutes the refereed proceedings of the 4th Russian Supercomputing Days, RuSCDays 2018, held in Moscow, Russia, in September 2018. The 59 revised full papers and one revised short paper presented were carefully reviewed and selected from 136 submissions. The papers are organized in topical sections on parallel algorithms; supercomputer simulation; high performance architectures, tools and technologies.
This book provides an introduction to quasigroup theory along with new structural results on some of the quasigroup classes. Many results are presented with some of them from mathematicians of the former USSR. These included results have not been published before in the western mathematical literature. In addition, many of the achievements obtained with regard to applications of quasigroups in coding theory and cryptology are described.
This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Its focus is on orthomorphisms and complete mappings of finite groups, while also offering a complete proof of the Hall–Paige conjecture. The use of latin squares in constructions of nets, affine planes, projective planes, and transversal designs also motivates this inquiry. The text begins by introducing fundamental concepts, like the tests for determining whether a latin square is based on a group, as well as orthomorphisms and complete mappings. From there, it describes the existence problem for complete mappings of groups, building up to the proof of the Hall–Paige conjecture. The third part presents a comprehensive study of orthomorphism graphs of groups, while the last part provides a discussion of Cartesian projective planes, related combinatorial structures, and a list of open problems. Expanding the author’s 1992 monograph, Orthomorphism Graphs of Groups, this book is an essential reference tool for mathematics researchers or graduate students tackling latin square problems in combinatorics. Its presentation draws on a basic understanding of finite group theory, finite field theory, linear algebra, and elementary number theory—more advanced theories are introduced in the text as needed.
It is by no means clear what comprises the "heart" or "core" of algebra, the part of algebra which every algebraist should know. Hence we feel that a book on "our heart" might be useful. We have tried to catch this heart in a collection of about 150 short sections, written by leading algebraists in these areas. These sections are organized in 9 chapters A, B, . . . , I. Of course, the selection is partly based on personal preferences, and we ask you for your understanding if some selections do not meet your taste (for unknown reasons, we only had problems in the chapter "Groups" to get enough articles in time). We hope that this book sets up a standard of what all algebraists are supposed to know in "their" chapters; interested people from other areas should be able to get a quick idea about the area. So the target group consists of anyone interested in algebra, from graduate students to established researchers, including those who want to obtain a quick overview or a better understanding of our selected topics. The prerequisites are something like the contents of standard textbooks on higher algebra. This book should also enable the reader to read the "big" Handbook (Hazewinkel 1999-) and other handbooks. In case of multiple authors, the authors are listed alphabetically; so their order has nothing to do with the amounts of their contributions.
This volume is a record of the papers presented to the fourth British Combinatorial Conference held in Aberystwyth in July 1973. Contributors from all over the world took part and the result is a very useful and up-to-date account of what is happening in the field of combinatorics. A section of problems illustrates some of the topics in need of further investigation.
In 1988, the news of Egmont Köhler's untimely death at the age of 55reached his friends and colleagues. It was widely felt that a lastingmemorial tribute should be organized. The result is the present volume,containing forty-two articles, mostly in combinatorial design theory andgraph theory, and all in memory of Egmont Köhler. Designs and graphswere his areas of particular interest; he will long be remembered for hisresearch on cyclic designs, Skolem sequences, t-designs and theOberwolfach problem. Professors Lenz and Ringel give a detailedappreciation of Köhler's research in the first article of thisvolume. There is, however, one aspect of Egmont Köhler's biographythat merits special attention. Before taking up the study of mathematics atthe age of 31, he had completed training as a musician (studying bothcomposition and violoncello at the Musikhochschule in Berlin), and workedas a cellist in a symphony orchestra for some years. This accounts for hisinterest in the combinatorial aspects of music. His work and lectures inthis direction had begun to attract the interest of many musicians, and hehad commenced work on a book on mathematical aspects of musical theory. Itis tragic indeed that his early death prevented the completion of his work;the surviving paper on the classification and complexity of chordsindicates the loss that his death meant to the area, as he was almostuniquely qualified to bring mathematics and music together, being aprofessional in both fields.