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During the last two decades several remarkable new results were discovered about harmonic measure in the complex plane. This book provides a careful survey of these results and an introduction to the branch of analysis which contains them. Many of these results, due to Bishop, Carleson, Jones, Makarov, Wolff and others, appear here in paperback for the first time. The book is accessible to students who have completed standard graduate courses in real and complex analysis. The first four chapters provide the needed background material on univalent functions, potential theory, and extremal length, and each chapter has many exercises to further inform and teach the readers.

Functions of complex variables. --- Potential theory (Mathematics)

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Themainobjectiveofthisbookistogiveacollectionofcriteriaavailablein the spectral theory of selfadjoint operators, and to identify the spectrum and its components in the Lebesgue decomposition. Many of these criteria were published in several articles in di?erent journals. We collected them, added some and gave some overview that can serve as a platform for further research activities. Spectral theory of Schr¨ odinger type operators has a long history; however the most widely used methods were limited in number. For any selfadjoint operatorA on a separable Hilbert space the spectrum is identi?ed by looking atthetotalspectralmeasureassociatedwithit;oftenstudyingsuchameasure meant looking at some transform of the measure. The transforms were of the form f,?(A)f which is expressible, by the spectral theorem, as ?(x)dµ (x) for some ?nite measureµ . The two most widely used functions? were the sx ?1 exponential function?(x)=e and the inverse function?(x)=(x?z) . These functions are “usable” in the sense that they can be manipulated with respect to addition of operators, which is what one considers most often in the spectral theory of Schr¨ odinger type operators. Starting with this basic structure we look at the transforms of measures from which we can recover the measures and their components in Chapter 1. In Chapter 2 we repeat the standard spectral theory of selfadjoint op- ators. The spectral theorem is given also in the Hahn–Hellinger form. Both Chapter 1 and Chapter 2 also serve to introduce a series of de?nitions and notations, as they prepare the background which is necessary for the criteria in Chapter 3.

Operator theory --- Functional analysis --- Partial differential equations --- Differential equations --- Mathematical physics --- Quantum mechanics. Quantumfield theory --- differentiaalvergelijkingen --- quantumfysica --- analyse (wiskunde) --- Laplacetransformatie --- functies (wiskunde) --- wiskunde --- fysica --- Potential theory (Mathematics). --- Mathematical physics. --- Quantum theory. --- Differential equations, partial. --- Operator theory. --- Functional analysis. --- Potential Theory. --- Mathematical Methods in Physics. --- Quantum Physics. --- Partial Differential Equations. --- Operator Theory. --- Functional Analysis. --- Physics. --- Quantum physics. --- Partial differential equations. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Thermodynamics --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Potential theory (Mathematics) --- Scattering (Mathematics) --- Spectral theory (Mathematics)

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Quadrature domains were singled out about 30 years ago by D. Aharonov and H.S. Shapiro in connection with an extremal problem in function theory. Since then, a series of coincidental discoveries put this class of planar domains at the center of crossroads of several quite independent mathematical theories, e.g., potential theory, Riemann surfaces, inverse problems, holomorphic partial differential equations, fluid mechanics, operator theory. The volume is devoted to recent advances in the theory of quadrature domains, illustrating well the multi-facet aspects of their nature. The book contains a large collection of open problems pertaining to the general theme of quadrature domains.

Functions of complex variables. --- Potential theory (Mathematics) --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Complex variables --- Elliptic functions --- Functions of real variables --- Global analysis (Mathematics). --- Operator theory. --- Numerical analysis. --- Mathematical physics. --- Analysis. --- Operator Theory. --- Numerical Analysis. --- Mathematical Methods in Physics. --- Theoretical, Mathematical and Computational Physics. --- Physical mathematics --- Physics --- Functional analysis --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematics --- Mathematical analysis. --- Analysis (Mathematics). --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- 517.1 Mathematical analysis

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This volume contains two of the three lectures that were given at the 33rd Probability Summer School in Saint-Flour (July 6-23, 2003). Amir Dembo’s course is devoted to recent studies of the fractal nature of random sets, focusing on some fine properties of the sample path of random walk and Brownian motion. In particular, the cover time for Markov chains, the dimension of discrete limsup random fractals, the multi-scale truncated second moment and the Ciesielski-Taylor identities are explored. Tadahisa Funaki’s course reviews recent developments of the mathematical theory on stochastic interface models, mostly on the so-called abla varphi interface model. The results are formulated as classical limit theorems in probability theory, and the text serves with good applications of basic probability techniques.

Probabilities. --- Mathematical statistics. --- Probabilités --- Statistique mathématique --- Mathematics. --- Differential equations, partial. --- Potential theory (Mathematics). --- Distribution (Probability theory). --- Statistics. --- Probability Theory and Stochastic Processes. --- Measure and Integration. --- Potential Theory. --- Statistics for Engineering, Physics, Computer Science, Chemistry & Geosciences. --- Partial Differential Equations. --- Mathematical Statistics --- Mathematics --- Physical Sciences & Mathematics --- Statistical analysis --- Statistical data --- Statistical methods --- Statistical science --- Distribution functions --- Frequency distribution --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Partial differential equations --- Math --- Measure theory. --- Partial differential equations. --- Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences. --- Econometrics --- Mathematical analysis --- Mechanics --- Science --- Probability --- Statistical inference --- Combinations --- Chance --- Least squares --- Mathematical statistics --- Risk --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Distribution (Probability theory. --- Characteristic functions --- Probabilities --- Statistics . --- Distribution (Probability theory)