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Mahler measure, a height function for polynomials, is the central theme of this book. It has many interesting properties, obtained by algebraic, analytic and combinatorial methods. It is the subject of several longstanding unsolved questions, such as Lehmer’s Problem (1933) and Boyd’s Conjecture (1981). This book contains a wide range of results on Mahler measure. Some of the results are very recent, such as Dimitrov’s proof of the Schinzel–Zassenhaus Conjecture. Other known results are included with new, streamlined proofs. Robinson’s Conjectures (1965) for cyclotomic integers, and their associated Cassels height function, are also discussed, for the first time in a book. One way to study algebraic integers is to associate them with combinatorial objects, such as integer matrices. In some of these combinatorial settings the analogues of several notorious open problems have been solved, and the book sets out this recent work. Many Mahler measure results are proved for restricted sets of polynomials, such as for totally real polynomials, and reciprocal polynomials of integer symmetric as well as symmetrizable matrices. For reference, the book includes appendices providing necessary background from algebraic number theory, graph theory, and other prerequisites, along with tables of one- and two-variable integer polynomials with small Mahler measure. All theorems are well motivated and presented in an accessible way. Numerous exercises at various levels are given, including some for computer programming. A wide range of stimulating open problems is also included. At the end of each chapter there is a glossary of newly introduced concepts and definitions. Around the Unit Circle is written in a friendly, lucid, enjoyable style, without sacrificing mathematical rigour. It is intended for lecture courses at the graduate level, and will also be a valuable reference for researchers interested in Mahler measure. Essentially self-contained, this textbook should also be accessible to well-prepared upper-level undergraduates.
What role do metaphors play in philosophical language? Are they impediments to clear thinking and clear expression, rhetorical flourishes that may well help to make philosophy more accessible to a lay audience, but that ought ideally to be eradicated in the interests of terminological exactness? Or can the images used by philosophers tell us more about the hopes and cares, attitudes and indifferences that regulate an epoch than their carefully elaborated systems of thought? In Paradigms for a Metaphorology, originally published in 1960 and here made available for the first time in English translation, Hans Blumenberg (1920-1996) approaches these questions by examining the relationship between metaphors and concepts. Blumenberg argues for the existence of "absolute metaphors" that cannot be translated back into conceptual language. "Absolute metaphors" answer the supposedly naïve, theoretically unanswerable questions whose relevance lies quite simply in the fact that they cannot be brushed aside, since we do not pose them ourselves but find them already posed in the ground of our existence. They leap into a void that concepts are unable to fill. An afterword by the translator, Robert Savage, positions the book in the intellectual context of its time and explains its continuing importance for work in the history of ideas.