Статья
Магнитная теория Блоха и некоммутативная геометрия
В данном обзоре описываются конструкции, лежащие в основе применений некоммутативной геометрии к обоснованию дробного квантового эффекта Холла
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) C-algebras. Using a theorem of O. Forsterфn. We also show that the class of HFG algebras is stable under some standard constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum ball. We further concentrate on the algebras of holomorphic functions on the quantum polydisk and on the quantum ball and show that they are isomorphic, in contrast to the classical case. Finally, we interpret our algebras as Fr\'echet algebra deformations of the classical algebras of holomorphic functions on the polydisk and on the ball in C^n.
We introduce and study holomorphically finitely generated (HFG) Fr\'echet algebras, which are analytic counterparts of affine (i.e., finitely generated) complex algebras. Using a theorem of O. Forster, we prove that the category of commutative HFG algebras is anti-equivalent to the category of Stein spaces of finite embedding dimension. We also show that the class of HFG algebras is stable under some natural constructions. This enables us to give a series of concrete examples of HFG algebras, including Arens-Michael envelopes of affine algebras (such as the algebras of holomorphic functions on the quantum affine space and on the quantum torus), the algebras of holomorphic functions on the free polydisk, on the quantum polydisk, and on the quantum polyannulus.
В статье рассматриваются вещественные автономные системы обыкновенных дифференциальных уравнений в окрестности невырожденной особой точки, у которых матрица линейной части имеет два чисто мнимых собственных значения, а остальные собственные значения лежат вне мнимой оси. Исследуется приводимость таких систем к псевдонормальной форме. Уточняется понятие резонанса, вводятся понятия устранимых и неустранимых резонансов. Доказывается, что для таких систем задача о конечно-гладкой эквивалентности решается по конечным отрезкам рядов Тейлора их правых частей.
Foreword by Freeman Dyson All main theoretical aspects and current applications of random matrices are covered Complementing views of leaders in the fields of mathematics and physics Applications in all branches of physics are covered, as well as in mathematics, biology and engineering Includes most important and up to date references Provides a guide for newcomers to the field Introduces those already familiar with random matrix theory with new areas of research With a foreword by Freeman Dyson, the handbook brings together leading mathematicians and physicists to offer a comprehensive overview of random matrix theory, including a guide to new developments and the diverse range of applications of this approach. In part one, all modern and classical techniques of solving random matrix models are explored, including orthogonal polynomials, exact replicas or supersymmetry. Further, all main extensions of the classical Gaussian ensembles of Wigner and Dyson are introduced including sparse, heavy tailed, non-Hermitian or multi-matrix models. In the second and larger part, all major applications are covered, in disciplines ranging from physics and mathematics to biology and engineering. This includes standard fields such as number theory, quantum chaos or quantum chromodynamics, as well as recent developments such as partitions, growth models, knot theory, wireless communication or bio-polymer folding. The handbook is suitable both for introducing novices to this area of research and as a main source of reference for active researchers in mathematics, physics and engineering. Readership: Suitable for mathematicians, physicists, statisticians and engineers. This handbook serves as a reference book for those already familiar with the field, as a guide to the field for newcomers and as an introduction to the wider applications of random matrix theory.
В сборнике представлены полные тексты докладов (статьи) 2-й международной конференции по стохастическим методам и анализу данных (2nd Stochastic Modeling Techniques and Data Analysis International Conference, SMTDA-2012), которая проходила с 5 по 8 июня 2012 года в г. Ханья, Крит, Греция.