Kurt Gödel was an intellectual giant. His Incompleteness Theorem turned not only mathematics but also the whole world of science and philosophy on its head. Shattering hopes that logic would, in the end, allow us a complete understanding of the universe, Gödel's theorem also raised many provocative questions: What are the limits of rational thought? Can we ever fully understand the machines we build? Or the inner workings of our own minds? How should mathematicians proceed in the absence of complete certainty about their results? Equally legendary were Gödel's eccentricities, his close friendship with Albert Einstein, and his paranoid fear of germs that eventually led to his death from self-starvation. Now, in the first book for a general audience on this strange and brilliant thinker, John Casti and Werner DePauli bring the legend to life.
“If I find a Ukrainian who is worthy of sitting at the same table with me, I must have him shot,” declared Nazi commissar Erich Koch. To the Nazi leaders, the Ukrainians were Untermenschen—subhumans. But the rich land was deemed prime territory for Lebensraum expansion. Once the Germans rid the country of Jews, Roma, and Bolsheviks, the Ukrainians would be used to harvest the land for the master race. Karel Berkhoff provides a searing portrait of life in the Third Reich’s largest colony. Under the Nazis, a blend of German nationalism, anti-Semitism, and racist notions about the Slavs produced a reign of terror and genocide. But it is impossible to understand fully Ukraine’s response to this assault without addressing the impact of decades of repressive Soviet rule. Berkhoff shows how a pervasive Soviet mentality worked against solidarity, which helps explain why the vast majority of the population did not resist the Germans. He also challenges standard views of wartime eastern Europe by treating in a more nuanced way issues of collaboration and local anti-Semitism. Berkhoff offers a multifaceted discussion that includes the brutal nature of the Nazi administration; the genocide of the Jews and Roma; the deliberate starving of Kiev; mass deportations within and beyond Ukraine; the role of ethnic Germans; religion and national culture; partisans and the German response; and the desperate struggle to stay alive. Harvest of Despair is a gripping depiction of ordinary people trying to survive extraordinary events.
The material presented in this book corresponds to a semester-long course, ``Linear Algebra and Differential Equations'', taught to sophomore students at UC Berkeley. In contrast with typical undergraduate texts, the book offers a unifying point of view on the subject, namely that linear algebra solves several clearly-posed classification problems about such geometric objects as quadratic forms and linear transformations. This attractive viewpoint on the classical theory agrees well with modern tendencies in advanced mathematics and is shared by many research mathematicians. However, the idea of classification seldom finds its way to basic programs in mathematics, and is usually unfamiliar to undergraduates. To meet the challenge, the book first guides the reader through the entire agenda of linear algebra in the elementary environment of two-dimensional geometry, and prior to spelling out the general idea and employing it in higher dimensions, shows how it works in applications such as linear ODE systems or stability of equilibria. Appropriate as a text for regular junior and honors sophomore level college classes, the book is accessible to high school students familiar with basic calculus, and can also be useful to engineering graduate students.
This authoritative biography of Kurt Goedel relates the life of this most important logician of our time to the development of the field. Goedel's seminal achievements that changed the perception and foundations of mathematics are explained in the context of his life from the turn of the century Austria to the Institute for Advanced Study in Princeton.
A landmark book that changed the story of Poland’s role in the Holocaust On July 10, 1941, in Nazi-occupied Poland, half of the town of Jedwabne brutally murdered the other half: 1,600 men, women, and children—all but seven of the town’s Jews. In this shocking and compelling classic of Holocaust history, Jan Gross reveals how Jedwabne’s Jews were murdered not by faceless Nazis but by people who knew them well—their non-Jewish Polish neighbors. A previously untold story of the complicity of non-Germans in the extermination of the Jews, Neighbors shows how people victimized by the Nazis could at the same time victimize their Jewish fellow citizens. In a new preface, Gross reflects on the book’s explosive international impact and the backlash it continues to provoke from right-wing Polish nationalists who still deny their ancestors’ role in the destruction of the Jews.
The class of Schur functions consists of analytic functions on the unit disk that are bounded by $1$. The Schur algorithm associates to any such function a sequence of complex constants, which is much more useful than the Taylor coefficients. There is a generalization to matrix-valued functions and a corresponding algorithm. These generalized Schur functions have important applications to the theory of linear operators, to signal processing and control theory, and to other areas of engineering. In this book, Alpay looks at matrix-valued Schur functions and their applications from the unifying point of view of spaces with reproducing kernels. This approach is used here to study the relationship between the modeling of time-invariant dissipative linear systems and the theory of linear operators. The inverse scattering problem plays a key role in the exposition. The point of view also allows for a natural way to tackle more general cases, such as nonstationary systems, non-positive metrics, and pairs of commuting nonself-adjoint operators. This is the English translation of a volume originally published in French by the Societe Mathematique de France. Translated by Stephen S. Wilson.
There are incredibly rich connections between classical analysis and number theory. For instance, analytic number theory contains many examples of asymptotic expressions derived from estimates for analytic functions, such as in the proof of the Prime Number Theorem. In combinatorial number theory, exact formulas for number-theoretic quantities are derived from relations between analytic functions. Elliptic functions, especially theta functions, are an important class of such functions in this context, which had been made clear already in Jacobi's Fundamenta nova. Theta functions are also classically connected with Riemann surfaces and with the modular group $\Gamma = \mathrm{PSL (2,\mathbb{Z )$, which provide another path for insights into number theory. Farkas and Kra, well-known masters of the theory of Riemann surfaces and the analysis of theta functions, uncover here interesting combinatorial identities by means of the function theory on Riemann surfaces related to the principal congruence subgroups $\Gamma(k)$. For instance, the authors use this approach to derive congruences discovered by Ramanujan for the partition function, with the main ingredient being the construction of the same function in more than one way. The authors also obtain a variant on Jacobi's famous result on the number of ways that an integer can be represented as a sum of four squares, replacing the squares by triangular numbers and, in the process, obtaining a cleaner result. The recent trend of applying the ideas and methods of algebraic geometry to the study of theta functions and number theory has resulted in great advances in the area. However, the authors choose to stay with the classical point of view. As a result, their statements and proofs are very concrete. In this book the mathematician familiar with the algebraic geometry approach to theta functions and number theory will find many interesting ideas as well as detailed explanations and derivations of new and old results. Highlights of the book include systematic studies of theta constant identities, uniformizations of surfaces represented by subgroups of the modular group, partition identities, and Fourier coefficients of automorphic functions. Prerequisites are a solid understanding of complex analysis, some familiarity with Riemann surfaces, Fuchsian groups, and elliptic functions, and an interest in number theory. The book contains summaries of some of the required material, particularly for theta functions and theta constants. Readers will find here a careful exposition of a classical point of view of analysis and number theory. Presented are numerous examples plus suggestions for research-level problems. The text is suitable for a graduate course or for independent reading.
Today’s rapidly flowing global economy, hit by recession following the financial crisis of 2008/9, means the geographical economic perspective has never been more important. An Introduction to Economic Geography comprehensively guides you through the core issues and debates of this vibrant and exciting area, whilst also exploring the range of approaches and paradigms currently invigorating the wider discipline. Rigorous and accessible, the authors demystify and enliven a crucial subject for geographical study. Underpinned by the themes of globalisation, uneven development and place, the text explores the diversity and vitality of contemporary economic geography. It balances coverage of 'traditional' areas such as regional development and labour markets with insight into new and evolving topics like neoliberalism, consumption, creativity and alternative economic practices. An Introduction to Economic Geography is an essential textbook for undergraduate students taking courses in Economic Geography, Globalisation Studies and more broadly in Human Geography. It will also be of key interest to anyone in Planning, Business and Management Studies and Economics.
