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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.
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The study of mathematical biology attempts to use mathematical models to draw useful conclusions about biological systems. Here, we consider the modelling of brain tumour spread with the ultimate goal of improving treatment outcomes.
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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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In this snapshot, we will consider the problem of finding the number of solutions to a given system of polynomial equations. This question leads to the theory of Newton polytopes and Newton-Okounkov bodies of which we will give a basic notion.
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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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Cet instantané traite de la symétrie zéro-dimensionnelle. Grâce à des découvertes récentes, nous comprenons mieux ces symétries que ce qui était auparavant cru possible. Bien qu’encore très incomplète, une vision de la symétrie zéro-dimensionnelle commence à émerger.
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Cet instantané concerne la théorie des opérateurs et un de ses outils fondamentaux : la décomposition en valeur singulières (ou SVD, de l’anglais « Singular Value Decomposition »). La SVD factorise les applications linéaires en applications plus simples et dévoile leurs propriétés géométriques. De par sa capacité à organiser l’information, cet outil est devenu important dans de nombreux domaines des mathématiques appliquées.
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En mécanique quantique, contrairement à la mécanique classique, il n’est pas possible de faire des prédictions précises quant au comportement d’un système. On est obligé de travailler avec les probabilités. En conséquence, c’est une tâche importante que de déterminer les probabilités de base d’un système donné. Dans cet instantané, nous présenterons un résultat d’unicité récent concernant ces probabilités.
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A friend of mine, an expert in statistical genomics, told me the following story: At a dinner party, an attractive lady asked him, “What do you do for a living?” He replied, “I model.” As my friend is a handsome man, the lady did not question his statement and continued, “What do you model?” “Genes.” She then looked at him up and down and said, “Mh, you must be very much in demand.” “Yes, very much so, especially after I helped discover a new culprit gene for a common childhood disease.” The lady looked puzzled.