The 20th century was a time of great upheaval and great progress in mathematics. In order to get the overall picture of trends, developments, and results, it is illuminating to examine their manifestations locally, in the personal lives and work of mathematicians who were active during this time. The university archives of Gottingen harbor a wealth of papers, letters, and manuscripts from several generations of mathematicians--documents which tell the story of the historic developments from a local point of view. This book offers a number of essays based on documents from Gottingen and elsewhere--essays which have not yet been included in the author's collected works. These essays, independent from each other, are meant as contributions to the imposing mosaic of the history of number theory. They are written for mathematicians, but there are no special background requirements. The essays discuss the works of Abraham Adrian Albert, Cahit Arf, Emil Artin, Richard Brauer, Otto Grun, Helmut Hasse, Klaus Hoechsmann, Robert Langlands, Heinrich-Wolfgang Leopoldt, Emmy Noether, Abraham Robinson, Ernst Steinitz, Hermann Weyl, and others.

Twentieth-Century Music Theory and Practice introduces a number of tools for analyzing a wide range of twentieth-century musical styles and genres. It includes discussions of harmony, scales, rhythm, contour, post-tonal music, set theory, the twelve-tone method, and modernism. Recent developments involving atonal voice leading, K-nets, nonlinearity, and neo-Reimannian transformations are also engaged. While many of the theoretical tools for analyzing twentieth century music have been devised to analyze atonal music, they may also provide insight into a much broader array of styles. This text capitalizes on this idea by using the theoretical devices associated with atonality to explore music inclusive of a large number of schools and contains examples by such stylistically diverse composers as Paul Hindemith, George Crumb, Ellen Taffe Zwilich, Steve Reich, Michael Torke, Philip Glass, Alexander Scriabin, Ernest Bloch, Igor Stravinsky, Béla Bartók, Sergei Prokofiev, Arnold Schoenberg, Claude Debussy, György Ligeti, and Leonard Bernstein. This textbook also provides a number of analytical, compositional, and written exercises. The aural skills supplement and online aural skills trainer on the companion website allow students to use theoretical concepts as the foundation for analytical listening. Access additional resources and online material here: http: //www.twentiethcenturymusictheoryandpractice.net and https: //www.motivichearing.com/.

The last one hundred years have seen many important achievements in the classical part of number theory. After the proof of the Prime Number Theorem in 1896, a quick development of analytical tools led to the invention of various new methods, like Brun's sieve method and the circle method of Hardy, Littlewood and Ramanujan; developments in topics such as prime and additive number theory, and the solution of Fermat’s problem. Rational Number Theory in the 20th Century: From PNT to FLT offers a short survey of 20th century developments in classical number theory, documenting between the proof of the Prime Number Theorem and the proof of Fermat's Last Theorem. The focus lays upon the part of number theory that deals with properties of integers and rational numbers. Chapters are divided into five time periods, which are then further divided into subject areas. With the introduction of each new topic, developments are followed through to the present day. This book will appeal to graduate researchers and student in number theory, however the presentation of main results without technicalities will make this accessible to anyone with an interest in the area.

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Includes selections from major writers on various approaches to art theory, for example Freud, Jung, Marx, Heidegger.

Schools and circles have been a major force in twentieth-century intellectual movements. They fostered circulation of ideas within and between disciplines, thus altering the shape of intellectual inquiry. This volume offers a new perspective on theoretical schools in the humanities, both as generators of conceptual knowledge and as cultural phenomena. The structuralist, semiotic, phenomenological, and hermeneutical schools and circles have had a deep impact on various disciplines ranging from literary studies to philosophy, historiography, and sociology. The volume focuses on a set of loosely interrelated groups, with a strong literary, linguistic, and semiotic component, but extends to the fields of philosophy and history--the interdisciplinary conjunctions arising from a sense of conceptual kinship. It includes chapters on unstudied or less studied groups, such as Tel Aviv School of poetics and semiotics or the research group Poetics and Hermeneutics. The volume presents a significant supplement to the standard historical accounts of literary, critical, and related theory in the twentieth century. It enhances and complicates our understanding of the twentieth-century intellectual and academic history by showing schools and circles in the state of germination, dialogue, controversy, or decline, in their respective historical and institutional settings, while reaching simultaneously beyond those dense settings to the new cultural and ideological situations of the twenty-first century.

A History of Twentieth-Century Music in a Theoretic-Analytical Context is an integrated account of the genres and concepts of twentieth-century art music, organized topically according to aesthetic, stylistic, technical, and geographic categories, and set within the larger political, social, economic, and cultural framework. While the organization is topical, it is historical within that framework. Musical issues interwoven with political, cultural, and social conditions have had a significant impact on the course of twentieth-century musical tendencies and styles. The goal of this book is to provide a theoretic-analytical basis that will appeal to those instructors who want to incorporate into student learning an analysis of the musical works that have reflected cultural influences on the major musical phenomena of the twentieth century. Focusing on the wide variety of theoretical issues spawned by twentieth-century music, A History of Twentieth-Century Music in a Theoretic-Analytical Context reflects the theoretical/analytical essence of musical structure and design.