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Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel’s theorems. The last five chapters present extensions and developments of classical mathematical logic, particularly the concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage. This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines.

Mathematical logic

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Seasonal patterns have been found in a remarkable range of health conditions, including birth defects, respiratory infections and cardiovascular disease. Accurately estimating the size and timing of seasonal peaks in disease incidence is an aid to understanding the causes and possibly to developing interventions. With global warming increasing the intensity of seasonal weather patterns around the world, a review of the methods for estimating seasonal effects on health is timely. This is the first book on statistical methods for seasonal data written for a health audience. It describes methods for a range of outcomes (including continuous, count and binomial data) and demonstrates appropriate techniques for summarising and modelling these data. It has a practical focus and uses interesting examples to motivate and illustrate the methods. The statistical procedures and example data sets are available in an R package called ‘season’. Adrian Barnett is a senior research fellow at Queensland University of Technology, Australia. Annette Dobson is a Professor of Biostatistics at The University of Queensland, Australia. Both are experienced medical statisticians with a commitment to statistical education and have previously collaborated in research in the methodological developments and applications of biostatistics, especially to time series data. Among other projects, they worked together on revising the well-known textbook "An Introduction to Generalized Linear Models," third edition, Chapman Hall/CRC, 2008. In their new book they share their knowledge of statistical methods for examining seasonal patterns in health.

Mathematical statistics

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This book brings together young researchers from a variety of fields within mathematics, philosophy and logic. It discusses questions that arise in their work, as well as themes and reactions that appear to be similar in different contexts. The book shows that a fairly intensive activity in the philosophy of mathematics is underway, due on the one hand to the disillusionment with respect to traditional answers, on the other to exciting new features of present day mathematics. The book explains how the problem of applicability once again plays a central role in the development of mathematics. It examines how new languages different from the logical ones (mostly figural), are recognized as valid and experimented with and how unifying concepts (structure, category, set) are in competition for those who look at this form of unification. It further shows that traditional philosophies, such as constructivism, while still lively, are no longer only philosophies, but guidelines for research. Finally, the book demonstrates that the search for and validation of new axioms is analyzed with a blend of mathematical historical, philosophical, psychological considerations.

Mathematical logic

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The new edition of this influential textbook, geared towards graduate or advanced undergraduate students, teaches the statistics necessary for financial engineering. In doing so, it illustrates concepts using financial markets and economic data, R Labs with real-data exercises, and graphical and analytic methods for modeling and diagnosing modeling errors. Financial engineers now have access to enormous quantities of data. To make use of these data, the powerful methods in this book, particularly about volatility and risks, are essential. Strengths of this fully-revised edition include major additions to the R code and the advanced topics covered. Individual chapters cover, among other topics, multivariate distributions, copulas, Bayesian computations, risk management, multivariate volatility and cointegration. Suggested prerequisites are basic knowledge of statistics and probability, matrices and linear algebra, and calculus. There is an appendix on probability, statistics and linear algebra. Practicing financial engineers will also find this book of interest. David Ruppert is Andrew Schultz, Jr., Professor of Engineering and Professor of Statistical Science at Cornell University, where he teaches statistics and financial engineering and is a member of the Program in Financial Engineering. Professor Ruppert received his PhD in Statistics at Michigan State University. He is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics and won the Wilcoxon prize. He is Editor of the Journal of the American Statistical Association-Theory and Methods and former Editor of the Electronic Journal of Statistics and of the Institute of Mathematical Statistics's Lecture Notes—Monographs. Professor Ruppert has published over 125 scientific papers and four books: Transformation and Weighting in Regression, Measurement Error in Nonlinear Models, Semiparametric Regression, and Statistics and Finance: An Introduction. David S. Matteson is Assistant Professor of Statistical Science at Cornell University, where he is a member of the ILR School, Center for Applied Mathematics, Field of Operations Research, and the Program in Financial Engineering, and teaches statistics and financial engineering. Professor Matteson received his PhD in Statistics at the University of Chicago. He received a CAREER Award from the National Science Foundation and won Best Academic Paper Awards from the annual R/Finance conference. He is an Associate Editor of the Journal of the American Statistical Association-Theory and Methods, Biometrics, and Statistica Sinica. He is also an Officer for the Business and Economic Statistics Section of the American Statistical Association, and a member of the Institute of Mathematical Statistics and the International Biometric Society.

Mathematical statistics

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Mathematical statistics