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Ordered algebraic structures --- 51 <082.1> --- Mathematics--Series --- Algebra, Homological --- Characteristic functions. --- Fonctions caractéristiques --- Algèbre homologique --- Characteristic functions --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Probabilities --- Homological algebra --- Algebra, Abstract --- Homology theory --- Fonctions caractéristiques. --- Algèbre homologique.

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This book tells the story of the Riemann hypothesis for function fields (or curves) starting with Artin's 1921 thesis, covering Hasse's work in the 1930s on elliptic fields and more, and concluding with Weil's final proof in 1948. The main sources are letters which were exchanged among the protagonists during that time, found in various archives, mostly the University Library in Göttingen. The aim is to show how the ideas formed, and how the proper notions and proofs were found, providing a particularly well-documented illustration of how mathematics develops in general. The book is written for mathematicians, but it does not require any special knowledge of particular mathematical fields.

Number theory. --- History of Mathematical Sciences. --- Number Theory. --- Number study --- Numbers, Theory of --- Algebra --- Riemann hypothesis. --- Characteristic functions. --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Probabilities --- Riemann's hypothesis --- Numbers, Prime --- Mathematics. --- History. --- Annals --- Auxiliary sciences of history --- Math --- Science

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The notions of transfer function and characteristic functions proved to be fundamental in the last fifty years in operator theory and in system theory. Moshe Livsic played a central role in developing these notions, and the book contains a selection of carefully chosen refereed papers dedicated to his memory. Topics include classical operator theory, ergodic theory and stochastic processes, geometry of smooth mappings, mathematical physics, Schur analysis and system theory. The variety of topics attests well to the breadth of Moshe Livsic's mathematical vision and the deep impact of his work.

Characteristic functions. --- Light -- Scattering. --- Scattering (Physics). --- Transfer functions. --- Mathematics --- Calculus --- Mathematical Statistics --- Physical Sciences & Mathematics --- Characteristic functions --- Scattering (Mathematics) --- Transfer functions --- Operator theory --- Ergodic theory --- Functions, Transfer --- Scattering theory (Mathematics) --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Mathematics. --- Operator theory. --- Operator Theory. --- Probabilities --- Automatic control --- Control theory --- Boundary value problems --- Differential equations, Partial --- Scattering operator --- Functional analysis

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In a certain sense characteristic functions and correlation functions are the same, the common underlying concept is positive definiteness. Many results in probability theory, mathematical statistics and stochastic processes can be derived by using these functions. While there are books on characteristic functions of one variable, books devoting some sections to the multivariate case, and books treating the general case of locally compact groups, interestingly there is no book devoted entirely to the multidimensional case which is extremely important for applications. This book is intended to fill this gap at least partially. It makes the basic concepts and results on multivariate characteristic and correlation functions easily accessible to both students and researchers in a comprehensive manner. The first chapter presents basic results and should be read carefully since it is essential for the understanding of the subsequent chapters. The second chapter is devoted to correlation functions, their applications to stationary processes and some connections to harmonic analysis. In Chapter 3 we deal with several special properties, Chapter 4 is devoted to the extension problem while Chapter 5 contains a few applications. A relatively large appendix comprises topics like infinite products, functional equations, special functions or compact operators.

Characteristic functions. --- Correlation (Statistics) --- Variables (Mathematics) --- Multivariate analysis. --- Multivariate distributions --- Multivariate statistical analysis --- Statistical analysis, Multivariate --- Analysis of variance --- Mathematical statistics --- Matrices --- Mathematical constants --- Mathematics --- Least squares --- Probabilities --- Regression analysis --- Statistics --- Instrumental variables (Statistics) --- Characteristic formula of an ideal --- Characteristic Hilbert functions --- Functions, Characteristic --- Functions, Hilbert --- Hilbert characteristic functions --- Hilbert functions --- Hilbert's characteristic functions --- Hilbert's functions --- Postulation formula --- Graphic methods --- Characteristic Functions. --- Fourier Transform. --- Moment Problem. --- Probability Distribution.