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When maths comes out of the shadows


Maths is adopting a higher profile, with an eye-catching poster campaign aimed at people on the move. For four days, a series of posters on the Barcelona, Paris, and Brussels underground systems have posed a number of problems to passengers. Is it possible to make a tour of Königsberg by passing only once over each of its bridges? How can you mix oil and water? Why do tigers have stripes and panthers patches? What do sailors' knots and the actions of viruses have in common? In just a few words, the posters explain the structure of proteins, the secret code of bank cards and the relationship between a mobile phone and a satellite. 'It's a change from advertising and makes you think,' was one student's verdict. Meanwhile, secondary school teachers have been busy writing to the initiative organisers asking for copies of the posters to display in their classrooms so that they can 'try and interest the pupils in mathematics in this way'.'The visual approach to mathematics by poster campaigns is seen as a means of capturing attention, arousing interest, and of stimulating the desire to investigate further,' explains Mireille Chaleyat-Maurel, the coordinator of the Maths in action project and a professor at the Université René Descartes (Paris 5). 'Investigating further can mean, for example, going to a science museum to watch a demonstration, picking up an explanatory leaflet, or visiting the project's Internet site.' On each occasion the objective is the same: 'to render visible the hidden fabric which intertwines mathematics at one level or another with our daily lives'. A mathematics which is as discreet as it is omnipresent. QED.



Turbulent weather!

The growth and track of a storm, the geometrical structure of a cloud and its role in absorbing solar and terrestrial radiation, the optimal integration in a numerical weather forecasting model of measurements made by different instruments operating at different locations (weather stations, satellites, ships and planes) ... There is no doubt about it: mathematical modelling plays a central role in all aspects of modern meteorological science. Essential for describing and understanding the mechanisms of weather and climate, it is used for both analysis and prediction.

Turbulent flows - and atmospheric movements are certainly that - can be modelled by the Navier-Stokes equations. However, since meteorologists cannot always solve these equations, they must use digital simulation using the most powerful computers, and the most sophisticated numerical schemes. Also of use is the mathematical theory of dynamic systems.

It was a meteorologist, E. N. Lorenz, who in 1963 demonstrated that even a simple dynamic system can evolve chaotically, its trajectory highly sensitive to initial conditions. According to his well-known metaphor, the disturbance of the air's flow due to the flight of a butterfly can ultimately lead to a cyclone on the other side of the planet.

Based on an idea of Philippe Courtier (Météo-France) and Claude Basdevant (ENS-Ecole Polytechnique-Paris).



No damage!

The development and testing of new cars is increasingly carried out by virtual experiments using computer simulations. For this purpose, engineers develop virtual vehicle models described through equations, the solution of which requires advanced mathematical methods and powerful computers. Virtual models also allow cars to be tested at a lower cost.

A virtual prototype of a car requires a global mathematical model incorporating the vehicle characteristics as well as its interactions with the road and the air, the description of any obstacles, etc. This results in a system of equations which is solved on a computer using numerical methods.

The complexity of these models generates vast quantities of calculations requiring the use of parallel computers.

Based on an idea of Andreas Frommer - University of Wuppertal, Germany.



Trees and forests

One way to approach the effects of climate change on trees and forests is to try to simulate their development by using models incorporating a large number of parameters while remaining simple enough to be used in practice. This demands close cooperation between mathematicians and forest experts.
The main difficulties stem from the fact that there are different levels of structural hierarchy (forest, tree, leaf, molecule, etc.) all impacting on growth and also different time scales which coexist (forests which may be hundreds of years old, compared to just a few seconds for the metabolism).

Competition for light
Recent years have brought enormous progress in modelling trees and forests. A dynamic model, based on vital and environmental processes, has been successfully applied to the analysis of climate change effects, including forest management in different regions and improving the wood quality.

New models describe tree growth during the year and also make it possible to predict how different trees will compete for light.

Idea and illustrations by Marjo Lipponen. MaDaMe programme - Turku University (Finland).



DNA: beyond the sequence

Over the past 20 years or so, new techniques (DNA sequencing, DNA chips, etc.) have opened up revolutionary perspectives in biology. These techniques generate huge amounts of extremely diverse data (sequences, images, texts, experimental data, bibliographies). Mathematics plays a central role in processing these data, making it possible to extract relevant information. The subject areas most useful to this are algorithms, probability theory, and statistics.

Proteins are long three-dimensionally folded amino acid chains. Their activity essentially depends on the protein form after folding. A knowledge of this three-dimensional structure is necessary for many applications (pharmacology, agriculture, etc.). The most accurate method is to use crystallography techniques, but they are very time consuming. This explains the many attempts to use mathematical or computer models to 'calculate' this form.

Idea and illustrations by Francois Rodolphe, Jean-François Gibrat and Pierre Nicolas - INRA unit - Versailles 'Mathematics, Computing & Genome'.

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