Regularity, Asymptotic Solutions and Travelling Waves analysis in a porous medium system to model the interaction between invasive and invaded species

This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use of the selfsimilar structure that permits to show the existence of a diffusive front. The solutions are then studied within the Travelling Waves (TW) domain showing the existence of potential and exponential profiles in the stable connection that converges to the stationary solutions in which the invasive specie predominates. The TW profiles are shown to exist based on the geometry perturbation theory together with an analytical-topological argument in the phase plane. The finding of an exponential decaying rate (related with the advection and diffusion parameters) in the invaded specie TW is not trivial in the non-linear diffusion case and reflects the existence of a TW trajectory governed by the invaded specie runaway (in the direction of the advection) and the diffusion (acting along a finite speed front or support).

DUALIZING INVOLUTIONS ON THE METAPLECTIC GL(2) à la TUPAN

Let F be a non-Archimedean local field of characteristic zero. Let G = GL(2, F) and $3\widetildeG = \widetilde{GL}(2,F)$ be the metaplectic group. Let τ be the standard involution on G. A well-known theorem of Gelfand and Kazhdan says that the standard involution takes any irreducible admissible representation of G to its contragredient. In such a case, we say that τ is a dualizing involution. In this paper, we make some modifications and adapt a topological argument of Tupan to the metaplectic group $\widetildeG$ and give an elementary proof that any lift of the standard involution to $\widetildeG$ ; is also a dualizing involution.

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

In this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

Profile of a touch-down solution to a nonlocal MEMS model

In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on [Formula: see text] : [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] and [Formula: see text]. In this work, we have succeeded to construct a solution which quenches in finite time [Formula: see text] only at one interior point [Formula: see text]. In particular, we give a description of the quenching behavior according to the following final profile [Formula: see text] The construction relies on some connections between the quenching phenomenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The method is inspired by the work of Merle and Zaag [Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] with a suitable modification. In addition to that, the proof relies on two main steps: A reduction to a finite-dimensional problem and a topological argument based on index theory. The main difficulty and novelty of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite-dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data.

Topology for gaze analyses - Raw data segmentation

Recent years have witnessed a remarkable growth in the way mathematics, informatics, and computer science can process data. In disciplines such as machine learning, pattern recognition, computer vision, computational neurology, molecular biology, information retrieval, etc., many new methods have been developed to cope with the ever increasing amount and complexity of the data. These new methods offer interesting possibilities for processing, classifying and interpreting eye-tracking data. The present paper exemplifies the application of topological arguments to improve the evaluation of eye-tracking data. The task of classifying raw eye-tracking data into saccades and fixations, with a single, simple as well as intuitive argument, described as coherence of spacetime, is discussed, and the hierarchical ordering of the fixations into dwells is shown. The method, namely identification by topological characteristics (ITop), is parameter-free and needs no pre-processing and post-processing of the raw data. The general and robust topological argument is easy to expand into complexsettings of higher visual tasks, making it possible to identify visual strategies. As supplementary file an interactive demonstration of the method can be downloaded,

(In-)stability of singular equivariant solutions to the Landau–Lifshitz–Gilbert equation

In this paper, we use formal asymptotic arguments to understand the stability properties of equivariant solutions to the Landau–Lifshitz–Gilbert model for ferromagnets. We also analyse both the harmonic map heatflow and Schrödinger map flow limit cases. All asymptotic results are verified by detailed numerical experiments, as well as a robust topological argument. The key result of this paper is that blowup solutions to these problems are co-dimension one and hence both unstable and non-generic.

HEAVY TRAFFIC LIMITS VIA BROWNIAN EMBEDDINGS

For theGI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

Sudden cardiac arrest and a problem in topology

In the May 1983 issue of Scientific American, A.T. Winfree published an article which gave an explanation of sudden cardiac arrest in terms of topology. His topological argument was modified in his 1987 book called “When time breaks down”. In this paper we fill what appears to be a gap in Winfree's topological arguments.

"MUTUAL FRACTIONAL STATISTICS" AND "RELATIVE ANYONS"

A topological argument similar to that of Leinaas and Myrheim implies that a non-trivial statistical phase factor can also arise from exchanging twice a pair of distinguishable particles in two dimensions. Some general properties of this phase factor are deduced. Wilczek's model for anyons and the Laughlin theory for the quasiparticles in the fractional quantum Hall ground states are examined in light of these properties, and the former is generalized for systems containing many species of anyons. The statistical properties of holons and spinons relative to each other are briefly discussed as an example.