Band 119, Jahrgang 2017
Heft 2
Guido Schneider
Vorwort, Seite 69
Michael Struwe
Variational Methods in Geometry, ab Seite 71
Michael Struwe, Diese E-Mail-Adresse ist vor Spambots geschützt! Zur Anzeige muss JavaScript eingeschaltet sein!, Mathematik ETH-Zürich, Zürich, Switzerland
- Mathematics Subject Classification: 49-02, 53-02, 58-02, 58E05, 58E10, 58E12, 49Q05, 58E20
Abstract: Variational principles are ubiquitous in nature. Many geometric objects such as geodesics or minimal surfaces allow variational characterizations. We recall some basic ideas in the calculus of variations, also relevant for some of the most advanced research in the field today, and show how a subtle variation of standard methods can lead to surprising improvements, with numerous applications.
The original article is available on SpringerLink.
Hiroshi Maehara, Horst Martini
Geometric Probability on the Sphere, ab Seite 93
Hiroshi Maehara, Ryukyu University, Nishihara, Japan
Horst Martini, Diese E-Mail-Adresse ist vor Spambots geschützt! Zur Anzeige muss JavaScript eingeschaltet sein!, Faculty of Mathematics, Chemnitz, University of Technology, Chemnitz, Germany
- Keywords: Coverage problem, Crofton’s formula, Random great circle, Random spherical polygon, Santaló’s chord theorem, Sylvester’s problem
- Mathematics Subject Classification: 60D05, 52A22, 53C65
Abstract: This is an exposition of results and methods from geometric probability on the surface of a ball (i.e., on a sphere) in three-dimensional space. We tried to make our arguments simple and intuitive. We present many concrete results together with their (mainly) elementary proofs, and also several new results are derived. In addition, the reader will also find various interesting unsolved problems.
The original article is available on SpringerLink.
Heft 1
Guido Schneider
Vorwort, Seite 1
Roland Speicher
Free Probability Theory
And Its Avatars in Representation Theory, Random Matrices, and Operator Algebras; also Featuring: Non-commutative Distributions, ab Seite 3
Roland Speicher, Diese E-Mail-Adresse ist vor Spambots geschützt! Zur Anzeige muss JavaScript eingeschaltet sein!, Fachrichtung Mathematik, Universität des Saarlandes, Saarbrücken, Germany
- Keywords: Free probability theory,, Random matrix, Asymptotic representation theory, Non-commutative distribution
- Mathematics Subject Classification: 46L54
Abstract: This article is an invitation to the world of free probability theory. This theory was introduced by Dan Voiculescu at the beginning of the 1980?s and has developed since then into a vibrant and very active theory which connects with many different branches of mathematics. We will motivate Voiculescu?s basic notion of ?freeness?, and relate it with problems in representation theory, random matrices, and operator algebras. The notion of ?non-commutative distributions? is one of the central objects of the theory and awaits a deeper understanding.
The original article is available on SpringerLink.
Ben Schweizer
Resonance Meets Homogenization
Construction of Meta-Materials with Astonishing Properties, ab Seite 31
Ben Schweizer, Fakultät für Mathematik, TU Dortmund, Dortmund, Germany, Diese E-Mail-Adresse ist vor Spambots geschützt! Zur Anzeige muss JavaScript eingeschaltet sein!
- Keywords: Meta-materials, Resonance, Homogenization, Helmholtz equation, Maxwell’s equations, Sound absorbers, Negative index materials
- Mathematics Subject Classification: 78M40, 35B27, 35B34
Abstract: Meta-materials are assemblies of small components. Even though the single component consists of ordinary materials, the meta-material may behave effectively in a way that is not known from ordinary materials. In this text, we discuss some meta-materials that exhibit unusual properties in the propagation of sound or light. The phenomena are based on resonance effects in the small components. The small (sub-wavelength) components can be resonant to the wave-length of an external field if they incorporate singular features such as a high contrast or a singular geometry. Homogenization theory allows to derive effective equations for the macroscopic description of the meta-material and to verify its unusual properties. We discuss three examples: Sound-absorbing materials, optical materials with a negative index of refraction, perfect transmission through grated metals.
The original article is available on SpringerLink.