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2016 | Buch

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Mathematical Analysis, Probability and Applications – Plenary Lectures

ISAAC 2015, Macau, China

herausgegeben von: Tao Qian, Luigi G. Rodino

Verlag: Springer International Publishing

Buchreihe : Springer Proceedings in Mathematics & Statistics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book collects lectures given by the plenary speakers at the 10th International ISAAC Congress, held in Macau, China in 2015. The contributions, authored by eminent specialists, present some of the most exciting recent developments in mathematical analysis, probability theory, and related applications. Topics include: partial differential equations in mathematical physics, Fourier analysis, probability and Brownian motion, numerical analysis, and reproducing kernels. The volume also presents a lecture on the visual exploration of complex functions using the domain coloring technique. Thanks to the accessible style used, readers only need a basic command of calculus.

Inhaltsverzeichnis

Frontmatter

A Review of Brownian Motion Based Solely on the Langevin Equation with White Noise

Abstract

We give a historical and mathematical review of Brownian motion based solely on the Langevin equation. We derive the main statistical properties without bringing in external and subsidiary issues, such as temperature, Focker-Planck equations, the Maxwell–Boltzmann distribution, spectral analysis, the fluctuation-dissipation theorem, among many other topics that are typically introduced in discussions of the Langevin equation. The method we use is the formal solution approach, which was the standard method devised by the founders of the field. In addition, we give some relevant historical comments.

L. Cohen

Geometry-Fitted Fourier-Mellin Transform Pairs

Abstract

The construction of novel Fourier/Mellin-type transform pairs that are tailor-made for given planar regions within the special class of circular domains is surveyed. Circular domains are those having boundary components that are either circular arcs or straight lines. The new transform pairs generalize the classical Fourier and Mellin transforms. These geometry-fitted transform pairs can be used to great advantage in solving boundary value problems defined in these domains.

Darren Crowdy

First Order Approach to Estimates for the Stokes Operator on Lipschitz Domains

Abstract

This paper concerns Hodge-Dirac operators \(D_H = d + \delta \) acting in \(L^p(\Omega , \Lambda )\) where \(\Omega \) is a bounded open subset of \(\mathbb {R}^n\) satisfying some kind of Lipschitz condition, \(\Lambda \) is the exterior algebra of \(\mathbb {R}^n, d\) is the exterior derivative acting on the de Rham complex of differential forms on \(\Omega \), and \(\delta \) is the interior derivative with tangential boundary conditions. In \(L^2(\Omega , \Lambda )\), \(d' = \delta \) and \(D_H\) is self-adjoint, thus having bounded resolvent \({\{(I + \mathrm{{it}}{D}_H)\}}_{\{t\in \mathbb {R}\}}\) as well as a bounded functional calculus in \(L^2(\Omega , \Lambda )\). We investigate the range of values \(p_H<p<p^H\) about \(p = 2\) for which \(D_H\) has bounded resolvents and a bounded holomorphic functional calculus in \(L^p(\Omega , \Lambda )\).

Alan McIntosh, Sylvie Monniaux

The Study of Complex Shapes of Fluid Membranes, the Helfrich Functional and New Applications

Abstract

The theoretical study of complex configurations of fluid membranes is reported on the basis of the Helfrich functional. Series of analytical results on the governing equations of closed lipid vesicles and open lipid vesicles with holes are surveyed. The concepts of stress tensor and moment tensor in fluid membranes are investigated from two different viewpoints: the balance of forces (moments) and the generalized variational principle of free energy. Several examples on new applications of the Helfrich functional in understanding the growth mechanism of some mesoscopic structures are illustrated.

Zhong-Can Ou-Yang, Zhan-Chun Tu

Multiplication and Composition in Weighted Modulation Spaces

Abstract

We study the existence of the product of two weighted modulation spaces. For this purpose, we discuss two different strategies. The more simple one allows transparent proofs in various situations. However, our second method allows a closer look onto associated norm inequalities under restrictions in the Fourier image. This will give us the opportunity to treat the boundedness of composition operators.

Maximilian Reich, Winfried Sickel

A Reproducing Kernel Theory with Some General Applications

Abstract

In this paper, some essences of the general theory of reproducing kernels from the viewpoint of general applications and general interest will be introduced by our recent results, that are presented in the plenary talk.

Saburou Saitoh

Sparse Approximation by Greedy Algorithms

Abstract

It is a survey on recent results in constructive sparse approximation. Three directions are discussed here: (1) Lebesgue-type inequalities for greedy algorithms with respect to a special class of dictionaries, (2) constructive sparse approximation with respect to the trigonometric system, (3) sparse approximation with respect to dictionaries with tensor product structure. In all three cases constructive ways are provided for sparse approximation. The technique used is based on fundamental results from the theory of greedy approximation. In particular, results in the direction (1) are based on deep methods developed recently in compressed sensing. We present some of these results with detailed proofs.

V. Temlyakov

The Bi-free Extension of Free Probability

Abstract

Free probability is a noncommutative probability theory adapted to variables with the highest degree of noncommutativity. The theory has connections with random matrices, combinatorics, and operator algebras. Recently, we realized that the theory has an extension to systems with left and right variables, based on a notion of bi-freeness. We provide a look at the development of this new direction. The paper is an expanded version of the plenary lecture at the 10th ISAAC Congress in Macau.

Dan-Virgil Voiculescu

Stability of the Prandtl Boundary Layers

Abstract

This note is to survey our recent study on the stability and instability of the Prandtl boundary layers in incompressible viscous flows near a physical boundary. Both of the two-dimensional and three-dimensional problems are considered. First, we present an energy method for studying the well-posedness of the two-dimensional Prandtl boundary layer equations in Sobolev spaces under the Oleinik monotonicity condition on the tangential velocity field. Then, we give an instability result for the Prandtl equations in three space variables, which shows that the monotonicity condition of tangential velocity fields is not sufficient for the well-posedness of the three-dimensional Prandtl equations. Later, we present a well-posedness result of the three-dimensional Prandtl equations for a special structured flow. These results show that a shear flow is linearly and nonlinearly stable for the three-dimensional Prandtl equations, if and only if, the tangential velocity field direction is invariant with respect to the normal variable.

Y.-G. Wang

Visual Exploration of Complex Functions

Abstract

The technique of domain coloring allows one to represent complex functions as images on their domain. It endows functions with an individual face and may serve as simple and efficient tool for their visual exploration. The emphasis of this paper is on phase plots, a special variant of domain coloring. Though these images utilize only the argument (phase) of a function and neglect its modulus, analytic (and meromorphic) functions are uniquely determined by their phase plot up to a positive constant factor. Following (Wegert in Not AMS 58:78–780, 2011 [49], Wegert in Visual Complex Functions. An Introduction with Phase Portraits, Springer Basel, 2012 [53]), we introduce phase plots and several of their modifications and explain how properties of functions can be reconstructed from these images. After a survey of related results, the main part is devoted to a number of applications which illustrate the usefulness of phase plots in teaching and research.

Elias Wegert

Integral Transform Approach to Time-Dependent Partial Differential Equations

Abstract

In this review, we present an integral transform that maps solutions of some class of the partial differential equations with time independent coefficients to solutions of more complicated equations, which have time-dependent coefficients. We illustrate this transform by applications to several model equations. In particular, we give applications to the generalized Tricomi equation, the Klein–Gordon and wave equations in the curved spacetimes such as Einstein-de Sitter, de Sitter, anti-de Sitter, and the spacetimes of the black hole embedded into de Sitter universe.

Karen Yagdjian

Titel: Mathematical Analysis, Probability and Applications – Plenary Lectures
herausgegeben von: Tao Qian
Luigi G. Rodino
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-41945-9
Print ISBN: 978-3-319-41943-5
DOI: https://doi.org/10.1007/978-3-319-41945-9