Starting from simple generalizations of factorials and binomial coefficients, this book gives a friendly and accessible introduction to q q-analysis, a subject consisting primarily of identities between certain kinds of series and products. Many applications of these identities to combinatorics and number theory are developed in detail. There are numerous exercises to help students appreciate the beauty and power of the ideas, and the history of the subject is kept consistently in view. The book has few prerequisites beyond calculus. It is well suited to a capstone course, or for self-study in combinatorics or classical analysis. Ph.D. students and research mathematicians will also find it useful as a reference.

In May 1984 the Swedish Council for Scientific Research convened a small group of investigators at the scientific research station at Abisko, Sweden, for the purpose of examining various conceptual and mathematical views of the evolution of complex systems. The stated theme of the meeting was deliberately kept vague, with only the purpose of discussing alternative mathematically based approaches to the modeling of evolving processes being given as a guideline to the participants. In order to limit the scope to some degree, it was decided to emphasize living rather than nonliving processes and to invite participants from a range of disciplinary specialities spanning the spectrum from pure and applied mathematics to geography and analytic philosophy. The results of the meeting were quite extraordinary; while there was no intent to focus the papers and discussion into predefined channels, an immediate self-organizing effect took place and the deliberations quickly oriented themselves into three main streams: conceptual and formal structures for characterizing sys tem complexity; evolutionary processes in biology and ecology; the emergence of complexity through evolution in natural lan guages. The chapters presented in this volume are not the proceed ings of the meeting. Following the meeting, the organizers felt that the ideas and spirit of the gathering should be preserved in some written form, so the participants were each requested to produce a chapter, explicating the views they presented at Abisko, written specifically for this volume. The results of this exercise form the volume you hold in your hand.

An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own. An Introduction to Proof through Real Analysis is based on course material developed and refined over thirty years by Professor Daniel J. Madden and was designed to function as a complete text for both first proofs and first analysis courses. Written in an engaging and accessible narrative style, this book systematically covers the basic techniques of proof writing, beginning with real numbers and progressing to logic, set theory, topology, and continuity. The book proceeds from natural numbers to rational numbers in a familiar way, and justifies the need for a rigorous definition of real numbers. The mathematical climax of the story it tells is the Intermediate Value Theorem, which justifies the notion that the real numbers are sufficient for solving all geometric problems. • Concentrates solely on designing proofs by placing instruction on proof writing on top of discussions of specific mathematical subjects • Departs from traditional guides to proofs by incorporating elements of both real analysis and algebraic representation • Written in an engaging narrative style to tell the story of proof and its meaning, function, and construction • Uses a particular mathematical idea as the focus of each type of proof presented • Developed from material that has been class-tested and fine-tuned over thirty years in university introductory courses An Introduction to Proof through Real Analysis is the ideal introductory text to proofs for second and third-year undergraduate mathematics students, especially those who have completed a calculus sequence, students learning real analysis for the first time, and those learning proofs for the first time. Daniel J. Madden, PhD, is an Associate Professor of Mathematics at The University of Arizona, Tucson, Arizona, USA. He has taught a junior level course introducing students to the idea of a rigorous proof based on real analysis almost every semester since 1990. Dr. Madden is the winner of the 2015 Southwest Section of the Mathematical Association of America Distinguished Teacher Award. Jason A. Aubrey, PhD, is Assistant Professor of Mathematics and Director, Mathematics Center of the University of Arizona.

TO VEGETATION ANALYSIS Principles, practice and interpretation D.R.CAUSTON Department of Botany and Microbiology, University College of Wales, Aberystwyth London UNWIN HYMAN Boston Sydney Wellington © D. R. Causton, 1988 This book is copyright under the Berne Convention. No reproduction without permission. All rights reserved. Published by the Academic Division of Unwin Hyman Ltd 15/17 Broadwick Street, London W1V 1FP, UK Allen & Unwin Inc., 8 Winchester Place, Winchester, Mass. 01890, USA Allen & Unwin (Australia) Ltd, 8 Napier Street, North Sydney, NSW 2060, Australia Allen & Unwin (New Zealand) Ltd in association with the Port Nicholson Press Ltd, 60 Cambridge Terrace, Wellington, New Zealand First published in 1988 British Library Cataloguing in Publication Data Causton, David R. An introduction to vegetation analysis: principles, practice and intepretation. 1. Botany-Ecology-Mathematics I. Title 581.5'247 QK901 ISBN-13: 978-0-04-581025-3 e-ISBN-13: 978-94-011-7981-2 DOl: 10.1007/978-94-011-7981-2 Library of Congress Cataloging-in-Publication Data Causton, David R. An introduction to vegetation analysis. Bibliography: p. Includes index. 1. Botany-Ecology-Methodology. 2. Plant communities-Research-Methodology. 3. Vegetation surveys. 4. Vegetation classification. I. Title. QK90I.C33 1987 581.5 87-19327 ISBN-13: 978-0-04-581025-3 Typeset in 10 on 12 point Times by Mathematical Composition Setters Ltd, Salisbury and Biddies of Guildford Preface This book has been written to help students and their teachers, at various levels, to understand the principles, some of the methods, and ways of interpreting vegetational and environmental data acquired in the field.

This book is a simple yet thorough introduction to Q methodology, a research technique designed to capture the subjective or first-person viewpoints of its participants. Watts and Stenner outline the key theoretical concepts developed by William Stephenson, the founder of Q methodology, including subjectivity, concourse theory and abduction. They then turn to the practicalities of delivering high quality Q methodological research. Using worked examples throughout, the reader is guided through: • important design issues • the conduct of fieldwork • all the analytic processes of Q methodology, including factor extraction, factor rotation and factor interpretation. Drawing on helpful conceptual introductions to potentially difficult statistical concepts and a step-by-step guide to running Q methodological analyses using dedicated software, this book enables interested readers to design, manage, analyse, interpret and publish their own Q methodological research.

p-adic numbers are of great theoretical importance in number theory, since they allow the use of the language of analysis to study problems relating toprime numbers and diophantine equations. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own (a large appendix with hints on the problem is included). Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.

Problems after each chapter

Originally published in 1997, An Introduction to Mathematical Analysis provides a rigorous approach to real analysis and the basic ideas of complex analysis. Although the approach is axiomatic, the language is evocative rather than formal, and the proofs are clear and well motivated. The author writes with the reader always in mind. The text includes a novel and simplified approach to the Lebesgue integral, a topic not usually found in books at this level. The problems are scattered throughout the text, and are designed to get the student actively involved in the development at every stage. "This Introduction to Mathematical Analysis is a very carefully written and well organized presentation of the major theorems in classical real and complex analysis. I can find no fault whatever pertaining to the level of rigor or mathematical precision of the manuscript. All in all I think this is a fine text." Reviewer from Portland State "To summarize I think this text is very good. Its strengths are many. The choices of the problems and examples are well made. The proofs are very to the point and the style makes the text very readable." Reviewer from Michigan State "H. S. Bear seems to be one of the best kept secrets around. His writing in general is superb. This book is a well organized first course in analysis broken into digestible chunks and surprisingly thorough. It covers the basic topics and then introduces the reader to complex analysis and later to Lebesgue integration." James M. Cargal Professor Bear obtained his degree at the University of California, Berkeley with a thesis in functional analysis. He has held permanent positions at several major western universities, as well as visiting appointments at Princeton, the University of California, San Diego, and Erlangen-Nurnberg, Germany. All of these venues involved a ridiculous amount of bad weather, so he went to the University of Hawaii as department chairman in 1969. He served as department chairman for five years, and later served a term as graduate chairman. He has numerous research and expository publications in the areas of functional analysis, real and complex analysis, and measure theory.