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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-or

Number theory --- Magic squares. --- Number theory. --- Number study --- Numbers, Theory of --- Algebra --- Latin squares and rectangles --- Squares, Magic --- Mathematical recreations

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Humanity's love affair with mathematics and mysticism reached a critical juncture, legend has it, on the back of a turtle in ancient China. As Clifford Pickover briefly recounts in this enthralling book, the most comprehensive in decades on magic squares, Emperor Yu was supposedly strolling along the Yellow River one day around 2200 B.C. when he spotted the creature: its shell had a series of dots within squares. To Yu's amazement, each row of squares contained fifteen dots, as did the columns and diagonals. When he added any two cells opposite along a line through the center square, like 2 and 8, he always arrived at 10. The turtle, unwitting inspirer of the ''Yu'' square, went on to a life of courtly comfort and fame. Pickover explains why Chinese emperors, Babylonian astrologer-priests, prehistoric cave people in France, and ancient Mayans of the Yucatan were convinced that magic squares--arrays filled with numbers or letters in certain arrangements--held the secret of the universe. Since the dawn of civilization, he writes, humans have invoked such patterns to ward off evil and bring good fortune. Yet who would have guessed that in the twenty-first century, mathematicians would be studying magic squares so immense and in so many dimensions that the objects defy ordinary human contemplation and visualization? Readers are treated to a colorful history of magic squares and similar structures, their construction, and classification along with a remarkable variety of newly discovered objects ranging from ornate inlaid magic cubes to hypercubes. Illustrated examples occur throughout, with some patterns from the author's own experiments. The tesseracts, circles, spheres, and stars that he presents perfectly convey the age-old devotion of the math-minded to this Zenlike quest. Number lovers, puzzle aficionados, and math enthusiasts will treasure this rich and lively encyclopedia of one of the few areas of mathematics where the contributions of even nonspecialists count.

Magic squares. --- Mathematical recreations. --- Mathematical puzzles --- Number games --- Recreational mathematics --- Recreations, Mathematical --- Puzzles --- Scientific recreations --- Games in mathematics education --- Magic squares --- Magic tricks in mathematics education --- Latin squares and rectangles --- Squares, Magic --- Number theory --- Mathematical recreations

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The question of composition and spatial qualities arises in every urban design concept or intervention in the spatial structure of urban public squares. How are the essential elements involved: dimension, proportion, alignment, cohesion, accesses, shaping of focus point and of edges like surfaces and materials? How do they contribute to a character of urban space with which residents can identify? Comparing historical examples with current designs aids one in visualizing spatial effect. Similar to a floor planmanualfor buildings,Squares allows the user to evaluate spatial conditions for movement and rest based on comparable existing urban squares. The book offers the planner a comparative example for most conditions (shape, size, location, topography, and so on). Seventy European urban squares are presented and explained with the most important characteristics in a consistent manner in as-built plan, ground plan, section, and axonometric projection.

Places --- Espaces publics --- Parken ; Europa --- Europa --- 711.61 --- 711.4 --- 72.036 --- Pleinen --- Publieke ruimte --- Openbare ruimte --- Stedenbouw --- Twintigste eeuw (architectuur) --- 20ste eeuw (architectuur) --- Public spaces --- Plazas --- Garden squares --- Squares --- Designs and plans --- Dessins et plans --- (Historische) pleinen ; in Europese steden --- Stedenbouw. Ruimtelijke ordening ; pleinen, open ruimten --- City squares --- Piazzas --- Piazze --- Public squares --- Quadrangles (Plazas) --- Quads (Plazas) --- Squares, City --- Squares, Public --- Cities and towns

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Partial least squares structural equation modeling (PLS-SEM) has become a standard approach for analyzing complex inter-relationships between observed and latent variables. Researchers appreciate the many advantages of PLS-SEM such as the possibility to estimate very complex models and the method's flexibility in terms of data requirements and measurement specification. This practical open access guide provides a step-by-step treatment of the major choices in analyzing PLS path models using R, a free software environment for statistical computing, which runs on Windows, macOS, and UNIX computer platforms. Adopting the R software's SEMinR package, which brings a friendly syntax to creating and estimating structural equation models, each chapter offers a concise overview of relevant topics and metrics, followed by an in-depth description of a case study. Simple instructions give readers the "how-tos" of using SEMinR to obtain solutions and document their results. Rules of thumb in every chapter provide guidance on best practices in the application and interpretation of PLS-SEM

Finance --- Least squares. --- R (Computer program language) --- Structural equation modeling. --- Mathematical models. --- Open Access --- PLS-SEM) Using R --- Workbook --- Partial Least Squares Structural Equation Modeling --- R Software Environment

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Life is a chancy proposition: from the movement of molecules to the age at which we die, chance plays a key role in the natural world. Traditionally, biologists have viewed the inevitable "noise" of life as an unfortunate complication. The authors of this book, however, treat random processes as a benefit. In this introduction to chance in biology, Mark Denny and Steven Gaines help readers to apply the probability theory needed to make sense of chance events--using examples from ocean waves to spiderwebs, in fields ranging from molecular mechanics to evolution. Through the application of probability theory, Denny and Gaines make predictions about how plants and animals work in a stochastic universe. Is it possible to pack a variety of ion channels into a cell membrane and have each operate at near-peak flow? Why are our arteries rubbery? The concept of a random walk provides the necessary insight. Is there an absolute upper limit to human life span? Could the sound of a cocktail party burst your eardrums? The statistics of extremes allows us to make the appropriate calculations. How long must you wait to see the detail in a moonlit landscape? Can you hear the noise of individual molecules? The authors provide answers to these and many other questions. After an introduction to the basic statistical methods to be used in this book, the authors emphasize the application of probability theory to biology rather than the details of the theory itself. Readers with an introductory background in calculus will be able to follow the reasoning, and sets of problems, together with their solutions, are offered to reinforce concepts. The use of real-world examples, numerous illustrations, and chapter summaries--all presented with clarity and wit--make for a highly accessible text. By relating the theory of probability to the understanding of form and function in living things, the authors seek to pique the reader's curiosity about statistics and provide a new perspective on the role of chance in biology.

Biomathematics. --- Probabilities. --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Biology