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In How Economics Became a Mathematical Science E. Roy Weintraub traces the history of economics through the prism of the history of mathematics in the twentieth century. As mathematics has evolved, so has the image of mathematics, explains Weintraub, such as ideas about the standards for accepting proof, the meaning of rigor, and the nature of the mathematical enterprise itself. He also shows how economics itself has been shaped by economists’ changing images of mathematics. Whereas others have viewed economics as autonomous, Weintraub presents a different picture, one in which changes in mathematics—both within the body of knowledge that constitutes mathematics and in how it is thought of as a discipline and as a type of knowledge—have been intertwined with the evolution of economic thought. Weintraub begins his account with Cambridge University, the intellectual birthplace of modern economics, and examines specifically Alfred Marshall and the Mathematical Tripos examinations—tests in mathematics that were required of all who wished to study economics at Cambridge. He proceeds to interrogate the idea of a rigorous mathematical economics through the connections between particular mathematical economists and mathematicians in each of the decades of the first half of the twentieth century, and thus describes how the mathematical issues of formalism and axiomatization have shaped economics. Finally, How Economics Became a Mathematical Science reconstructs the career of the economist Sidney Weintraub, whose relationship to mathematics is viewed through his relationships with his mathematician brother, Hal, and his mathematician-economist son, the book’s author.
This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.
Updated throughout, this revised edition contains 25% new material covering progress made in the field over the past decade.
The American Journal of Mathematics publishes research papers and articles of broad appeal covering the major areas of contemporary mathematics.
Systems Factorial Technology: A Theory Driven Methodology for the Identification of Perceptual and Cognitive Mechanisms explores the theoretical and methodological tools used to investigate fundamental questions central to basic psychological and perceptual processes. Such processes include detection, identification, classification, recognition, and decision-making. This book collects the tools that allow researchers to deal with the pervasive model mimicry problems which exist in standard experimental and theoretical paradigms and includes novel applications to not only basic psychological questions, but also clinical diagnosis and links to neuroscience. Researchers can use this book to begin using the methodology behind SFT and to get an overview of current uses and future directions. The collected developments and applications of SFT allow us to peer inside the human mind and provide strong constraints on psychological theory. - Provides a thorough introduction to the diagnostic tools offered by SFT - Includes a tutorial on applying the method to reaction time data from a variety of different situations - Introduces novel advances for testing the significance of SFT results - Incorporates new measures that allow for the relaxation of the high accuracy criterion - Examines tools to expand the scope of SFT analyses - Applies SFT to a spectrum of different cognitive domains across different sensory modalities