Refined Asymptotics for the subcritical Keller-Segel system and Related Functional Inequalities

We analyze the rate of convergence towards self-similarity for the subcritical Keller-Segel system in the radially symmetric two-dimensional case and in the corresponding one-dimensional case for logarithmic interaction. We measure convergence in Wasserstein distance. The rate of convergence towards self-similarity does not degenerate as we approach the critical case. As a byproduct, we obtain a proof of the logarithmic Hardy-Littlewood-Sobolev inequality in the one dimensional and radially symmetric two dimensional case based on optimal transport arguments. In addition we prove that the one-dimensional equation is a contraction with respect to Fourier distance in the subcritical case

Cartuja 98, A Technological Park located at the site of Sevilla's World's Fair

The Cartuja 93 Technological Park was created in 1993 and was located at the site of the World’s Fair that took place in Sevilla in 1992. The Park’s emergence and formation, between 1993 and 1999, was a slow and dificult process and during this period it was very common among social scientists to refer to it as as a failure, as the deserted Cartuja. At present, however, it seems that Cartuja 93 is consolidated and has become an urban Tecnological Park. Within Sevilla’s technopolis, research and development centers, technology transfer units and high technology and advanced service firms are located. The paper describes the formation and consolidation process of Cartuja 93, analyses the changes and transformation of the technopolis and asses the results of the Technological Park.

Functional inequalities, thick tails and asymptotics for the critical mass Patlak-Keller-Segel model

We investigate the long time behavior of the critical mass Patlak-Keller-Segel equation. This equation has a one parameter family of steady-state solutions $\rho_\lambda$, $\lambda>0$, with thick tails whose second moment is not bounded. We show that these steady state solutions are stable, and find basins of attraction for them using an entropy functional ${\mathcal H}_\lambda$ coming from the critical fast diffusion equation in $\R^2$. We construct solutions of Patlak-Keller-Segel equation satisfying an entropy-entropy dissipation inequality for ${\mathcal H}_\lambda$. While the entropy dissipation for ${\mathcal H}_\lambda$ is strictly positive, it turns out to be a difference of two terms, neither of which need to be small when the dissipation is small. We introduce a strategy of "controlled concentration" to deal with this issue, and then use the regularity obtained from the entropy-entropy dissipation inequality to prove the existence of basins of attraction for each stationary state composed by certain initial data converging towards $\rho_\lambda$. In the present paper, we do not provide any estimate of the rate of convergence, but we discuss how this would result from a stability result for a certain sharp Gagliardo-Nirenberg-Sobolev inequality.Comment: This version of the paper improves on the previous version by removing the small size condition on the value of the second Lyapunov functional of the initial data. The improved methodology makes greater use of techniques from optimal mass transportation, and so the second and third sections have changed places, and the current third section completely rewritte

Goal Driven Interaction (GDI) vs. Direct Manipulation (MD), an empirical comparison

Interacción'15, September 07-09, 2015, Vilanova i la Geltrú, Spain ACM 978-1-4503-3463-1/15/09. http://dx.doi.org/10.1145/2829875.2829892This paper presents a work in process about Goal Driven Interaction (GDI), a style of interaction intended for inexperienced, infrequent and occasional users, whose main priorities are to use a system and achieve their goals without cost in terms of time or effort. GDI basic philosophy is to guide the user about the "what" to do and the "how" to do it in each moment of the interaction process, without requiring from the user a previous knowledge to use the interface. This interaction style was introduced in previous work, where a description of its characteristics and the most appropriate user interface for it, were described. Those works included a methodology for the analysis and synthesis of the whole interactive process through a language of specification. This paper presents partial results we are collecting in real user testing, with the main aim of comparing GDI with direct manipulation interfaces (MD), nevertheless the most extended and commonly regarded as the most suitable for novice and experienced users.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

On global minimizers of repulsive-attractive power-law interaction energies

We consider the minimisation of power-law repulsive-attractive interaction energies which occur in many biological and physical situations. We show existence of global minimizers in the discrete setting and get bounds for their supports independently of the number of Dirac Deltas in certain range of exponents. These global discrete minimizers correspond to the stable spatial profiles of flock patterns in swarming models. Global minimizers of the continuum problem are obtained by compactness. We also illustrate our results through numerical simulations.Comment: 14 pages, 2 figure

One dimensional Fokker-Planck reduced dynamics of decision making models in Computational Neuroscience

We study a Fokker-Planck equation modelling the firing rates of two interacting populations of neurons. This model arises in computational neuroscience when considering, for example, bistable visual perception problems and is based on a stochastic Wilson-Cowan system of differential equations. In a previous work, the slow-fast behavior of the solution of the Fokker-Planck equation has been highlighted. Our aim is to demonstrate that the complexity of the model can be drastically reduced using this slow-fast structure. In fact, we can derive a one-dimensional Fokker-Planck equation that describes the evolution of the solution along the so-called slow manifold. This permits to have a direct efficient determination of the equilibrium state and its effective potential, and thus to investigate its dependencies with respect to various parameters of the model. It also allows to obtain information about the time escaping behavior. The results obtained for the reduced 1D equation are validated with those of the original 2D equation both for equilibrium and transient behavior