Series: Lecture Notes in Mathematics, Vol. 2136
2015
This monograph presents some cornerstone results in the study of sofic and hyperlinear groups and the closely related Connes' embedding conjecture. These notions, as well as the proofs of many results, are presented in the framework of model theory for metric structures. This point of view, rarely explicitly adopted in the literature, clarifies the ideas therein, and provides additional tools to attack open problems.
Sofic and hyperlinear groups are countable discrete groups that can be suitably approximated by finite symmetric groups and groups of unitary matrices. These deep and fruitful notions, introduced by Gromov and Radulescu, respectively, in the late 1990s, stimulated an impressive amount of research in the last 15 years, touching several seemingly distant areas of mathematics including geometric group theory, operator algebras, dynamical systems, graph theory, and quantum information theory. Several long-standing conjectures, still open for arbitrary groups, are now settled for sofic or hyperlinear groups.
The presentation is self-contained and accessible to anyone with a graduate-level mathematical background. In particular, no specific knowledge of logic or model theory is required. The monograph also contains many exercises, to help familiarize the reader with the topics present.
Series: Lecture Notes in Mathematics, Vol. 2138
2015
The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring.
Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Grobner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented.
Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.
Series: Lecture Notes in Mathematics, Vol. 2144
2015
Focusing on the mathematics that lies at the intersection of probability theory, statistical physics, combinatorics and computer science, this volume collects together lecture notes on recent developments in the area. The common ground of these subjects is perhaps best described by the three terms in the title: Random Walks, Random Fields and Disordered Systems. The specific topics covered include a study of Branching Brownian Motion from the perspective of disordered (spin-glass) systems, a detailed analysis of weakly self-avoiding random walks in four spatial dimensions via methods of field theory and the renormalization group, a study of phase transitions in disordered discrete structures using a rigorous version of the cavity method, a survey of recent work on interacting polymers in the ballisticity regime and, finally, a treatise on two-dimensional loop-soup models and their connection to conformally invariant systems and the Gaussian Free Field. The notes are aimed at early graduate students with a modest background in probability and mathematical physics, although they could also be enjoyed by seasoned researchers interested in learning about recent advances in the above fields.
Series: Lecture Notes in Mathematics, Vol. 2148
Subseries: C.I.M.E. Foundation Subseries
2015
Presenting topics that have not previously been contained in a single volume, this book offers an up-to-date review of computational methods in electromagnetism, with a focus on recent results in the numerical simulation of real-life electromagnetic problems and on theoretical results that are useful in devising and analyzing approximation algorithms. Based on four courses delivered in Cetraro in June 2014, the material covered includes the spatial discretization of Maxwellfs equations in a bounded domain, the numerical approximation of the eddy current model in harmonic regime, the time domain integral equation method (with an emphasis on the electric-field integral equation) and an overview of qualitative methods for inverse electromagnetic scattering problems.
Assuming some knowledge of the variational formulation of PDEs and of finite element/boundary element methods, the book is suitable for PhD students and researchers interested in numerical approximation of partial differential equations and scientific computing.
2016 208 pages
Hardback:
978-1-49-874135-4
25th February 2016
Not yet available
The aim of this book is to describe the essence of fractals and multifractals dynamics in the applications of bioscience and engineering, bioinformatics, and financial time series. To reach these aims, the book will cover the basic idea, history, structure, methodology and analysis of fractals, the relationship between fractals and nonlinear dynamics, the applications of fractal and multifractal analysis in biomedical science, and how fractal signals and images are modelled by Iterated Function system. Illustrations and practical applications are given throughout the text to provide clear interpretations and explanations of various complex fractal objects and fractal modelling.
Fractal ? A Non-Differentiable Geometry. Evolution of fractal ? a basic introduction. Classification of fractal ? a brief review. Deterministic fractal. Random fractal. Note on Connectivity properties. Bibliography. Complex World in View of Nonlinear Dynamics. What is the meaning of complex world?Why nonlinear dynamics is still significant? Describing complex phenomena. Meaning of nonlinear dynamics ? Physical and mathematical perception. Heuristic approach of nonlinear dynamics in site of mathematical knowledge. Connection between fractal and nonlinear dynamics. Bibliography. Measure of Roughness for Deterministic System. Topological dimension. Point wise dimension. Correlation dimension. Generalized correlation dimension ? Multifractal analysis. Bibliography. Measure of Roughness for stochastic system. Stationary time series and Non-stationary time series. Fluctuation analysis. Hurst Exponent. Detrended Fluctuation Analysis. Generalized Hurst Exponent. Multifractal Detrended Fluctuation Analysis. Bibliography. Application in Biomedical Signal. ECG signal and its Fractal analysis. Interpretation of dynamics of EMG signal. Fractals and multifractal analysis of EEG signal. Multifractal analysis of HRV. Bibliography. Application in Bioinformatics and Finance. Method of signal analysis in term of DNA sequence. Discussion on Financial time series in view of stochastic time series. Analysis of fractal and multifractal landscape of DNA sequence. Application of fractal and multifractal analysis in financial time series. Bibliography. Metric Space and Space of Fractals. Basic concept of metric space. Review of complete and compact metric space. Connection between fractals and metric space. Review on contraction mapping and fixed point theorem. Bibliography. Application of Fractal in Image Signal. Introduction: Chaos game. Iterated functions systems. Fractal Interpolation. Modelling signal and image by IFS. Bibliography.