Mathematics

We prove an equidistribution theorem of positive closed currents for a certain class of birational maps f + : P k → P k of algebraic degree d≥2 satisfying ⋃ n≥0 f n − ( I + )∩ ⋃ n≥0 f n + ( I − )=∅ , where f − is the inverse of f + and I ± are the sets of indeterminacy for f ± , respectively.

Consider the set E of endomorphisms of the n-torus endowed with the C 1 topology. A point in E that is persistently singular and robustly transitive is exhibited.

In this paper we present a Hamiltonian perturbation of any completely integrable Hamiltonian system with 2n degrees of freedom ( n≥2 ). The perturbation is C ∞ small but the resulting flow has positive metric entropy and it satisfies KAM non-degeneracy conditions. The key point is that positive entropy can be generated in an arbitrarily small tubular neighborhood of one trajectory.

Here we provide an overview of what is known, and what is not known, about an interesting dynamical system known as the Kepler-Heisenberg problem. The main idea is to pose a version of the classical Kepler problem of planetary motion, but in a sub-Riemannian setting. The result is system which is surprisingly rich and beautiful, mysterious in some ways but tame in others, offering a substantial number of questions which seem non-trivial yet tractable.

We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This extends a previous result of Bergelson, McCutcheon and Zhang. Using this uncountable Roth theorem, we establish the following two additional results. [(i)] We establish a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. [(ii)] We establish a uniform bound on the lower Banach density of the set of double recurrence times along all ? -systems, where ? is any group in a class of uniformly amenable groups. As a special case, we obtain this uniformity over all Z -systems, and our result seems to be novel already in this particular case. Our uncountable Roth theorem is crucial in the proof of both of these results.

We model, analyze, and simulate a novel run-and-tumble model with autochemotaxis, biologically inspired by the phytoplankton Heterosigma akashiwo. Developing a fundamental understanding of planktonic movements and interactions through phototaxis and chemotaxis is vital to comprehending why harmful algal blooms (HABs) start to form and how they can be prevented. We develop a one- and two-dimensional mathematical and computational model reflecting the movement of an ecology of plankton, incorporating both run-and-tumble motion and autochemotaxis. We present a succession of complex and biologically meaningful models combined with a sequence of laboratory and computational experiments that inform the ideas underlying the model. By analyzing the dynamics and pattern formation which are similar to experimental observations, we identify parameters that are significant in plankton's pattern formation in the absence of bulk fluid flow. We find that the precise form of chemical deposition and plankton sensitivity to small chemical gradients are crucial parameters that drive nonlinear pattern formation in the plankton density.

Deep brain stimulation (DBS) is an established method for treating pathological conditions such as Parkinson's disease, dystonia, Tourette syndrome, and essential tremor. While the precise mechanisms which underly the effectiveness of DBS are not fully understood, theoretical studies of populations of neural oscillators stimulated by periodic pulses suggest that this may be related to clustering, in which subpopulations of the neurons are synchronized, but the subpopulations are desynchronized with respect to each other. The details of the clustering behavior depend on the frequency and amplitude of the stimulation in a complicated way. In the present study, we investigate how the number of clusters, their stability properties, and their basins of attraction can be understood in terms of one-dimensional maps defined on the circle. Moreover, we generalize this analysis to stimuli that consist of pulses with alternating properties, which provide additional degrees of freedom in the design of DBS stimuli. Our results illustrate how the complicated properties of clustering behavior for periodically forced neural oscillator populations can be understood in terms of a much simpler dynamical system.

We consider a time-delayed HIV/AIDS epidemic model with education dissemination and study the asymptotic dynamics of solutions as well as the asymptotic behavior of the endemic equilibrium with respect to the amount of information disseminated about the disease. Under appropriate assumptions on the infection rates, we show that if the basic reproduction number is less than or equal to one, then the disease will be eradicated in the long run and any solution to the Cauchy problem converges to the unique disease-free equilibrium of the model. On the other hand, when the basic reproduction number is greater than one, we prove that the disease will be permanent but its impact on the population can be significantly minimized as the amount of education dissemination increases. In particular, under appropriate hypothesis on the model parameters, we establish that the size of the component of the infected population of the endemic equilibrium decreases linearly as a function of the amount of information disseminated. We also fit our model to a set of data on HIV/AIDS from Uganda within the period 1992-2005 in order to estimate the infection, effective response, and information rates of the disease. We then use these estimates to present numerical simulations to illustrate our theoretical findings.

The presence of synchronized clusters in neuron networks is a hallmark of information transmission and processing. The methods commonly used to study cluster synchronization in networks of coupled oscillators ground on simplifying assumptions, which often neglect key biological features of neuron networks. Here we propose a general framework to study presence and stability of synchronous clusters in more realistic models of neuron networks, characterized by the presence of delays, different kinds of neurons and synapses. Application of this framework to the directed network of the macaque cerebral cortex provides an interpretation key to explain known functional mechanisms emerging from the combination of anatomy and neuron dynamics. The cluster synchronization analysis is carried out also by changing parameters and studying bifurcations. Despite some simplifications with respect to the real network, the obtained results are in good agreement with previously reported biological data.

This paper is an announcement of a result followed with explanations of some ideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonian perturbation of any completely integrable Hamiltonian system with 2n degrees of freedom ( n≥2 ). The perturbation is C ∞ small but the resulting flow has positive metric entropy and it satisfies KAM non-degeneracy conditions. The key point is that positive entropy can be generated in an arbitrarily small tubular neighborhood of one trajectory.