Population and vegetation models often use nonlocal forms of dispersal to describe the spread of individuals and plants. When these long-range effects are modeled by spatially extended convolution kernels, the mathematical analysis of solutions can be simplified by posing the relevant equations on unbounded domains. However, in order to numerically validate these results, these same equations then need to be restricted to bounded sets. Thus, it becomes important to understand what effects, if any, do the different boundary constraints have on the solution. To address this question we present a quadrature method valid for convolution kernels with finite second moments. This scheme is designed to approximate at the same time the convolution operator together with the prescribed nonlocal boundary constraints, which can be Dirichlet, Neumann, or what we refer to as free boundary constraints.  We then apply this scheme to study pulse solutions of an abstract 1-d nonlocal Gray-Scott model as a case study. We consider convolution kernels with exponential and with algebraic decay. We find that, surprisingly, boundary effects can be more prominent when using exponentially decaying kernels.

     For an arbitrary irrational angle \alpha and arbitrary coupling constant \lambda, the Lebesgue measure of the spectrum of the Almost Mathieu operator with these parameters is equal to 4 |1-\lambda|. This was first conjectured in works by S. Aubry and G. Andre (1980), and later established in a series of results by J. Avron, P. H. M. v. Mouche & B. Simon (1990), Jitomirskaya and Last (1998), Jitomirskaya and Krasovsky (2001), Avila and Krikorian (2006).
     The main result of this talk, that is a work in progress with Anton Gorodetski, is devoted to the answer to the following question: what can be said about the measure of the part of the spectrum that is comprised between two given gaps? We show that the dependence on the parameters is piecewise-analytic: it is analytic on any domain when the bounding gaps do not bifurcate. We also show that the moments of the Lebesgue measure, restricted on the spectrum, are polynomials in \lambda and trigonometric polynomials in \alpha.
 

Abstract: In this talk, we present our results for the 1-bit phase retrieval problem: the recovery of a possibly sparse signal $x$ from the observations $y=sign(|Ax|-1)$. This problem is closely related to 1-bit compressed sensing where we observe $y=sign(Ax)$, and phase retrieval where we observe $y=|Ax|$. We show that the major findings in 1-bit compressed sensing theory, including hyperplane tessellation, optimal rate and a very recent efficient and optimal algorithm, can be developed in this phase retrieval setting. This is a joint work with Ming Yuan: https://arxiv.org/abs/2405.04733. 

What sort of algebraic objects do we get if, in place of a standard binary operation on a set, we consider an operation that is multi-valued? A hypergroup is a set with a binary operation into its set of nonempty subsets, satisfying certain axioms (that are similar to standard group axioms/properties). A hyperring is a "ring" but where the underlying group is a hypergroup. One could similarly define a whole world of hyperstructures. We give basic definitions, examples, motivation and talk about some algebraic constructions that we can do with hyperstructures. This is an expository talk.