Analysis

mathemathical subject image: 

Mathematische Modellierung von Krebswachstum

Submitted by IMAGINARY on

Krebs ist eine der größten Herausforderungen der modernen Medizin. Der WHO zufolge starben 2012 weltweit 8,2 Millionen Menschen an Krebs. Bis heute sind dessen molekulare Mechanismen nur in Teilen verstanden, was eine erfolgreiche Behandlung erschwert. Mathematische Modellierung und Computersimulationen können helfen, die Mechanismen des Tumorwachstums besser zu verstehen. Sie eröffnen somit neue Chancen für zukünftige Behandlungsmethoden.

Aperiodic Order and Spectral Properties

Submitted by IMAGINARY on

Periodic structures like a typical tiled kitchen floor or the arrangement of carbon atoms in a diamond crystal certainly possess a high degree of order. But what is order without periodicity? In this snapshot, we are going to explore highly ordered structures that are substantially nonperiodic, or aperiodic. As we construct such structures, we will discover surprising connections to various branches of mathematics, ma- terials science, and physics. Let us catch a glimpse into the inherent beauty of aperiodic order! 

Winkeltreue zahlt sich aus

Submitted by IMAGINARY on

Nicht nur Seefahrerinnen, auch Computergrafikerinnen und Physikerinnen wissen Winkeltreue zu schätzen. Doch beschränkte Rechenkapazitäten und Vereinfachungen in theoretischen Modellen erfordern es, winkeltreue Abbildungen nur mit einer überschaubaren Datenmenge zu beschreiben. Entsprechende Theorien werden in der diskreten Mathematik untersucht. Im Folgenden lade ich Sie auf eine Reise in die faszinierende Welt der winkeltreuen Abbildungen ein.

News on quadratic polynomials

Submitted by IMAGINARY on

Many problems in mathematics have remained unsolved because of missing links between mathematical disciplines, such as algebra, geometry, analysis, or number theory. Here we introduce a recently discovered result concerning quadratic polynomials, which uses a bridge between algebra and analysis. We study the iterations of quadratic polynomials, obtained by computing the value of a polynomial for a given number and feeding the outcome into the exact same polynomial again. These iterations of polynomials have interesting applications, such as in fractal theory. 

Towards a Mathematical Theory of Turbulence in Fluids

Submitted by IMAGINARY on

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. 

Swarming robots

Submitted by IMAGINARY on

When lots of robots come together to form shapes, spread in an area, or move in one direction, their motion has to be planned carefully. We discuss how mathematicians devise strategies to help swarms of robots behave like an experienced, coordinated team. 

Quantum diffusion

Submitted by IMAGINARY on

If you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see something that is never observed in the real world. Such diffusive and irreversible behaviour is ubiquitous in nature.

Curriculum development in university mathematics: where mathematicians and education collide

Submitted by IMAGINARY on

This snapshot looks at educational aspects of the design of curricula in mathematics. In particular, we examine choices textbook authors have made when introducing the concept of the completness of the real numbers. Can significant choices really be made? Do these choices have an effect on how people learn, and, if so, can we understand what they are?

Darcy’s law and groundwater flow modelling

Submitted by IMAGINARY on

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.

Operator theory and the singular value decomposition

Submitted by IMAGINARY on

This is a snapshot about operator theory and one of its fundamental tools: the singular value decomposition (SVD). The SVD breaks up linear transformations into simpler mappings, thus unveiling their  geometric properties. This tool has become important in many areas of applied mathematics for its ability to organize information. We discuss the SVD in the concrete situation of linear transformations of the plane (such as rotations, reflections, etc.).

Seiten