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I attended a very interesting workshop at the research center MFO in Oberwolfach on “Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws”. The title sounds a bit technical, but in plain language we could say: The theme is to survey recent research concerning how mathematics is used to study numerical algorithms involving a special class of equations.
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The topic of this snapshot is interpolation. In the ordinary sense, interpolation means to insert something of a different nature into something else. In mathematics, interpolation means constructing new data points from given data points. The new points usually lie in between the already known points. The purpose of this snapshot is to introduce a particular type of interpolation, namely, polynomial interpolation. This will be explained starting from basic ideas that go back to the ancient Babylonians and Greeks, and will arrive at subjects of current research activity.
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We provide a brief introduction to some basic ideas of Molecular Quantum Dynamics. We discuss the scope, strengths and main applications of this field of science. Finally, we also mention open problems of current interest in this exciting subject.
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Fluid mechanics is the theory of how liquids and gases move around. For the most part, the basic physics are well understood and the mathematical models look relatively simple. Despite this, fluids display a dazzling mystery to their motion. The random-looking, chaotic behavior of fluids is known as turbulence, and it lies far beyond our mathematical understanding, despite a century of intense research.
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Wie würdest du Tennisbälle oder Orangen stapeln? Oder allgemeiner formuliert: Wie dicht lassen sich identische 3-dimensionale Objekte überschneidungsfrei anordnen? Das Problem, welches auch Anwendungen in der digitalen Kommunikation hat, hört sich einfach an, ist jedoch für Kugeln in höheren Dimensionen noch immer ungelöst. Sogar die Berechnung guter Näherungslösungen ist für die meisten Di- mensionen schwierig.
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Anlässlich des internationalen Jahres der Kristallografie wurde ein Artikel aus dem Italienischen ins Deutsche für IMAGINARY übersetzt. Es geht um verschiedene Kristallstrukturen und wie man ursprünglich auf Kristallstrukturen gestoßen ist.
Der orginal Artikel ist in der Zeitschrift XlaTangente am 1. Oktober 2014 verköffentlicht worden. Hier, geht es zum Orginal.
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Formulations of natural phenomena are derived, sometimes, from experimentation and observation. Mathematical methods can be applied to expand on these formulations, and develop them into better models. In the year 1856, the French hydraulic engineer Henry Darcy performed experiments, measuring water flow through a column of sand. He discovered and described a fundamental law: the linear relation between pressure difference and flow rate – known today as Darcy’s law. We describe the law and the evolution of its modern formulation.
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What are the similarities of the mineral pyrite, the dodecahedron as the fifth Platonic solid, and a polynomial of degree 16? This paper explores this connection by using the free software SURFER of the IMAGINARY open mathematics platform, which leads to fascinating pictures displaying transformations from a cube to a dodecahedron, to a rhombic dodecahedron, and to an octahedron, using a single formula. A survey on the ideas and the mathematics behind these visualizations is given.
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Humans have been fascinated by crystals for a long time. Its regular geometry, its special symmetry, but also its diversity in colours surprise and please us. In this article, an overview of the connection between crystals and mathematics is given. It is a contribution to the International Year of Crystallography 2014.
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This snapshot is about zero-dimensional symmetry. Thanks to recent discoveries we now understand such symmetry better than previously imagined possible. While still far from complete, a picture of zero-dimensional symmetry is beginning to emerge.