The problem of representing an integer as a sum of squares of integers is one of the oldest and most significant in mathematics. It goes back at least 2000 years to Diophantus, and continues more recently with the works of Fermat, Euler, Lagrange, Jacobi, Glaisher, Ramanujan, Hardy, Mordell, Andrews, and others. Jacobi's elliptic function approach dates from his epic Fundamenta Nova of 1829. Here, the author employs his combinatorial/elliptic function methods to derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi's (1829) 4 and 8 squares identities to 4n2 or 4n(n+1) squares, respectively, without using cusp forms such as those of Glaisher or Ramanujan for 16 and 24 squares. These results depend upon new expansions for powers of various products of classical theta functions. This is the first time that infinite families of non-trivial exact explicit formulas for sums of squares have been found. The author derives his formulas by utilizing combinatorics to combine a variety of methods and observations from the theory of Jacobi elliptic functions, continued fractions, Hankel or Turanian determinants, Lie algebras, Schur functions, and multiple basic hypergeometric series related to the classical groups. His results (in Theorem 5.19) generalize to separate infinite families each of the 21 of Jacobi's explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical theta functions in sections 40-42 of the Fundamental Nova. The author also uses a special case of his methods to give a derivation proof of the two Kac and Wakimoto (1994) conjectured identities concerning representations of a positive integer by sums of 4n2 or 4n(n+1) triangular numbers, respectively. These conjectures arose in the study of Lie algebras and have also recently been proved by Zagier using modular forms. George Andrews says in a preface of this book, `This impressive work will undoubtedly spur others both in elliptic functions and in modular forms to build on these wonderful discoveries.' Audience: This research monograph on sums of squares is distinguished by its diversity of methods and extensive bibliography. It contains both detailed proofs and numerous explicit examples of the theory. This readable work will appeal to both students and researchers in number theory, combinatorics, special functions, classical analysis, approximation theory, and mathematical physics.
Crossing cultures can be a stimulating and rewarding adventure. It can also be a stressful and bewildering experience. This thoroughly revised and updated edition of Furnham and Bochner's classic Culture Shock (1986) examines the psychological and social processes involved in intercultural contact, including learning new culture specific skills, managing stress and coping with an unfamiliar environment, changing cultural identities and enhancing intergroup relations. The book describes the ABCs of intercultural encounters, highlighting Effective, Behavioural and Cognitive components of cross-cultural experience. It incorporates both theoretical and applied perspectives on culture shock and a comprehensive review of empirical research on a variety of cross-cultural travellers, such as tourists, students, business travellers, immigrants and refugees. Minimising the adverse effects of culture shock, facilitating positive msychological outcomes and discussion of selection and training techniques for living and working abroad represent some of the practical issues covered. The Psychology of Culture Shock will provide an essential reference and textbook for courses within psychology, sociology and business training. It will also be a valuable resource for professionals working with culturally diverse populations and acculturating groups such as international students immigrants or refugees.
The British `New Wave' of dramatists, actors and directors in the late 1950s and 1960s created a defining moment in post-war theatre. British Realist Theatre is an accessible introduction to the New Wave, providing the historical and cultural background which is essential for a true understanding of this influential and dynamic era. Drawing upon contemporary sources as well as the plays themselves, Stephen Lacey considers the plays' influences, their impact and their critical receptions. The playwrights discussed include: * Edward Bond * John Osborne * Shelagh Delaney * Harold Pinter
Rutter’s Child and Adolescent Psychiatry has become an established and accepted textbook of child psychiatry. Now completely revised and updated, the fifth edition provides a coherent appraisal of the current state of the field to help trainee and practising clinicians in their daily work. It is distinctive in being both interdisciplinary and international, in its integration of science and clinical practice, and in its practical discussion of how researchers and practitioners need to think about conflicting or uncertain findings. This new edition now offers an entirely new section on conceptual approaches, and several new chapters, including: neurochemistry and basic pharmacology brain imaging health economics psychopathology in refugees and asylum seekers bipolar disorder attachment disorders statistical methods for clinicians This leading textbook provides an accurate and comprehensive account of current knowledge, through the integration of empirical findings with clinical experience and practice, and is essential reading for professionals working in the field of child and adolescent mental health, and clinicians working in general practice and community pediatric settings.
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