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Latin Squares

Author: József Dénes
Publisher: Elsevier
ISBN: 9780080867861
Size: 11.38 MB
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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.

Latin Squares And Their Applications

Author: A. Donald Keedwell
Publisher: Elsevier
ISBN: 0444635580
Size: 33.76 MB
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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

Discrete Mathematics Using Latin Squares

Author: Charles F. Laywine
Publisher: John Wiley & Sons
ISBN: 9780471240648
Size: 18.79 MB
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An intuitive and accessible approach to discrete mathematics using Latin squares In the past two decades, researchers have discovered a range of uses for Latin squares that go beyond standard mathematics. People working in the fields of science, engineering, statistics, and even computer science all stand to benefit from a working knowledge of Latin squares. Discrete Mathematics Using Latin Squares is the only upper-level college textbook/professional reference that fully engages the subject and its many important applications. Mixing theoretical basics, such as the construction of orthogonal Latin squares, with numerous practical examples, proofs, and exercises, this text/reference offers an extensive and well-rounded treatment of the topic. Its flexible design encourages readers to group chapters according to their interests, whether they be purely mathematical or mostly applied. Other features include: An entirely new approach to discrete mathematics, from basic properties and generalizations to unusual applications 16 self-contained chapters that can be grouped for custom use Coverage of various uses of Latin squares, from computer systems to tennis and golf tournament design An extensive range of exercises, from routine problems to proofs of theorems Extended coverage of basic algebra in an appendix filled with corresponding material for further investigation. Written by two leading authorities who have published extensively in the field, Discrete Mathematics Using Latin Squares is an easy-to-use academic and professional reference.

Latin Squares

Author: József Dénes
Publisher: North Holland
ISBN: 9780444888990
Size: 16.35 MB
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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.

Automated Reasoning And Its Applications

Author: Robert Veroff
Publisher: MIT Press
ISBN: 9780262220552
Size: 47.79 MB
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The contributors are among the world's leading researchers inautomated reasoning. Their essays cover the theory, software system design, and use of these systems to solve real problems. The primary objective of automated reasoning (which includes automated deduction and automated theorem proving) is to develop computer programs that use logical reasoning for the solution of a wide variety of problems, including open questions. The essays in Automated Reasoning and Its Applications were written in honor of Larry Wos, one of the founders of the field. Wos played a central role in forming the "culture" of automated reasoning at Argonne National Laboratory. He and his colleagues consistently seek to build systems that search huge spaces for solutions to difficult problems and proofs of significant theorems. They have had numerous notable successes. The contributors are among the world's leading researchers in automated reasoning. Their essays cover the theory, software system design, and use of these systems to solve real problems. Contributors Robert S. Boyer, Shang-Ching Chou, Xiao-Shan Gao, Lawrence Henschen, Deepak Kapur, Kenneth Kunen, Ewing Lusk, William McCune, J Strother Moore, Ross Overbeek, Lawrence C. Paulson, Hantao Zhang, Jing-Zhong Zhang

Orthogonal Latin Squares Based On Groups

Author: Anthony B. Evans
Publisher: Springer
ISBN: 3319944304
Size: 57.50 MB
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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.

Encyclopaedia Of Mathematics

Author: Michiel Hazewinkel
Publisher: Springer Science & Business Media
ISBN: 940151237X
Size: 77.84 MB
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This ENCYCLOPAEDIA OF MA THEMA TICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.