In this paper we propose a mathematical model to describe the evolution of glioma cells taking into account the viscoelastic properties of brain tissue. The mathematical model is established considering that the glioma cells are of two phenotypes: migratory and proliferative. The evolution of the migratory cells is described by a diffusion-reaction equation of non Fickian type deduced considering a mass conservation law with a non Fickian migratory mass flux. The evolution of the proliferative cells is described by a reaction equation. A stability analysis that leads to the design of efficient protocols is presented. Numerical simulations that illustrate the behavior of the mathematical model are included.

The muscle contraction, operation of ATP synthase, maintaining the shape of a cell are believed to be secured by motor proteins, which can be modelled using the Brownian ratchet mechanism. We consider the randomly flashing ratchet model of a Brownian motor, where the particles can be in two states, only one of which is sensitive the applied spatially periodic potential (the mathematical setting is a pair of weakly coupled reaction-diffusion and Fokker-Planck equations). We prove that this mechanism indeed generates unidirectional transport by showing that the amount of mass in the wells of the potential decreases/increases from left to right. The direction of transport is unambiguously determined by the location of each minimum of the potential with respect to the so-called diffusive mean of its adjacent maxima. The transport can be generated not only by an asymmetric potential, but also by a symmetric potential and asymmetric transition rates, and as a consequence of the general result we derive explicit conditions when the latter happens. When the transitions are localized on narrow active sites in the protein conformation space, we find a more explicit characterization of the bulk transport direction, and infer that some common preconditions of the motor effect are redundant.

The objective of the article was to answer the following question: ‘What are students’ difficulties in Basic Education evidenced in teaching proposals that used problem-solving to apply mathematical content and, from these difficulties, what understandings are revealed about the teaching process?’ In the bibliographical research modality, we selected six master’s degree dissertations. After describing the teaching proposals, we analyzed the students’ difficulties in the problem-solving process. The results showed that most of the difficulties are in understanding problems, precisely because of the poor formation of mathematical concepts and the lack of knowledge about the meaning of words. We also verified difficulties in using formulas and operating with mathematical algorithms. To overcome such difficulties, it is understood that the process of teaching with problem-solving should avoid following a simple content resumption/revision. We conclude that it is necessary to incorporate an approach of formation of mathematical concepts and procedures.