Series: Lecture Notes in Mathematics, Vol. 2017
1st Edition., 2011, X, 502 p. 10 illus., Softcover
ISBN: 978-3-642-18362-1
Due: April 2011
The field of variable exponent function spaces has witnessed an explosive growth in recent years. The standard reference article for basic properties is already 20 years old. Thus this self-contained monograph collecting all the basic properties of variable exponent Lebesgue and Sobolev spaces is timely and provides a much-needed accessible reference work utilizing consistent notation and terminology. Many results are also provided with new and improved proofs. The book also presents a number of applications to PDE and fluid dynamics.
Content Level : Graduate
Keywords : - Electrorheological fluids - Lebesgue spaces - Non-standard growth - Sobolev spaces - Variable exponent
Related subjects : Analysis - Dynamical Systems & Differential Equations
1 Introduction.- 2 A framework for function spaces.- 3 Variable exponent Lebesgue spaces.- 4 The maximal operator.- 5 The generalized Muckenhoupt condition*.- 6 Classical operators.- 7 Transfer techniques.- 8 Introduction to Sobolev spaces.- 9. Density of regular functions.- 10. Capacities.- 11 Fine properties of Sobolev functions.- 12 Other spaces of differentiable functions.- 13 Dirichlet energy integral and Laplace equation.- 14 PDEs and fluid dynamics
Series: Lecture Notes in Mathematics, Vol. 2018
1st Edition., 2011, XII, 124 p. 2 illus., Softcover
ISBN: 978-3-642-18459-8
Due: March 2011
There is an enormous amount of work in the literature about the blow-up behavior of evolution equations. It is our intention to introduce the theory by emphasizing the methods while seeking to avoid massive technical computations. To reach this goal, we use the simplest equation to illustrate the methods; these methods very often apply to more general equations.
Content Level : Graduate
Keywords : - Blow-up - PDE estimates - Semilinear parabolic equation - asymptotic behavior - rate
Related subjects : Analysis - Applications - Dynamical Systems & Differential Equations
1 Introduction.- 2 A review of elliptic theories.- 3 A review of parabolic theories.- 4 A review of fixed point theorems.-5 Finite time Blow-up for evolution equations.- 6 Steady-State solutions.- 7 Blow-up rate.- 8 Asymptotically self-similar blow-up solutions.- 9 One space variable case
Series: Selected Works in Probability and Statistics
1st Edition., 2011, XXXI, 539 p., Hardcover
ISBN: 978-1-4419-8338-1
Due: March 29, 2011
During the second half of the 20th century, Murray Rosenblatt was one of the most celebrated and leading figures in probability and statistics. Among his many contributions, Rosenblatt conducted seminal work on density estimation, central limit theorems under strong mixing conditions, spectral domain methodology, long memory processes and Markov processes. He has published over 130 papers and 5 books, many as relevant today as when they first appeared decades ago. Murray Rosenblatt was one of the founding members of the Department of Mathematics at the University of California at San Diego (UCSD) and served as advisor to over twenty PhD students. He maintains a close association with UCSD in his role as Professor Emeritus.
This volume is a celebration of Murray Rosenblatt's stellar research career that spans over six decades, and includes some of his most interesting and influential papers. Several leading experts provide commentary and reflections on various directions of Murray's research portfolio.
Content Level : Research
Related subjects : Statistical Theory and Methods
Commentary: Discussion of Rosenblatt’s work on Global Measures of Deviations for Density Estimates.- Commentary: Murray Rosenblatt’s contributions to strong mixing.- Commentary: Murray Rosenblatt and cumulant/higher-order/polyspectra.- Commentary: Rosenblatt’s Contribution to Deconvolution.- Commentary: Rosenblatt’s Contributions to Random Walks on Compact Semigroups.- Commentary: The Rosenblatt Process.- On spectral analysis of stationary time series.- Remarks on a multivariate transformation.- Statistical spectral analysis of time series arising from stationary stochastic processes.- Recurrence-time moments in random walks.- A class of stationary processes and a central limit theorem.- A central limit theorem and a strong mixing condition.- Remarks on some nonparametric estimates of a density function.- Some regression problems in time series analysis.- Some purely deterministic processes.- Functions of a Markov process that are Markovian.- Stationary processes as shifts of functions of independent random variables.- Asymptotic distribution of eigenvalues of block Toeplitz matrices.- Limits of convolution sequences of measures on a compact topological semigroup.- Independence and dependence.- Asymptotic behavior of eigenvalues of Toeplitz forms.- Estimation of the bispectrum.- Asymptotic theory of estimates of kthorder spectra.- Remarks on the Burgers equation.- Density estimates and Markov sequences.- Curve estimates.- On some global measures of the deviations of density function estimates.- Asymptotic behavior of a spline estimate of a density function.- Fractional integrals of stationary processes and the central limit theorem.- Limit theorems for Fourier transforms of functionals of Gaussian sequences.- Deconvolution and estimation of transfer function phase and coefficients for non-Gaussian linear processes.- Asymptotic normality, strong mixing and spectral density estimates.- Deconvolution of non-Gaussian linear processes with vanishing spectral values.- Scale renormalization and random solutions of the Burgers equation. On frequency estimation.- Maximum likelihood estimation for noncausal autoregressive processes.- Spectral analysis for harmonizable processes.- Correction: “Spectral analysis for harmonizable processes”.- Estimation for almost periodic processes.- Prolate spheroidal spectral estimates.- Correction: “Estimation for almost periodic processes”.
Series: Encyclopaedia of Mathematical Sciences, Vol. 138
Volume package: Invariant Theory and Algebraic Transformation Groups
1st Edition., 2011, 262 p. 16 illus., Hardcover
ISBN: 978-3-642-18398-0
Due: April 2011
Homogeneous spaces of linear algebraic groups lie at the crossroads of algebraic geometry, theory of algebraic groups, classical projective and enumerative geometry, harmonic analysis, and representation theory. By standard reasons of algebraic geometry, in order to solve various problems on a homogeneous space, it is natural and helpful to compactify it while keeping track of the group action, i.e., to consider equivariant completions or, more generally, open embeddings of a given homogeneous space. Such equivariant embeddings are the subject of this book. We focus on the classification of equivariant embeddings in terms of certain data of "combinatorial" nature (the Luna-Vust theory) and description of various geometric and representation-theoretic properties of these varieties based on these data. The class of spherical varieties, intensively studied during the last three decades, is of special interest in the scope of this book. Spherical varieties include many classical examples, such as Grassmannians, flag varieties, and varieties of quadrics, as well as well-known toric varieties. We have attempted to cover most of the important issues, including the recent substantial progress obtained in and around the theory of spherical varieties.
Content Level : Research
Keywords : algebraic group - equivariant embedding - homogeneous space - reductive group - spherical variety
Related subjects : Algebra
Introduction.- 1 Algebraic Homogeneous Spaces.- 2 Complexity and Rank.- 3 General Theory of Embeddings.- 4 Invariant Valuations.- 5 Spherical Varieties.- Appendices.- Bibliography.- Indices