Application of projection algorithms to differential equations: boundary value problems

The Douglas-Rachford method has been employed successfully to solve many kinds of non-convex feasibility problems. In particular, recent research has shown surprising stability for the method when it is applied to finding the intersections of hypersurfaces. Motivated by these discoveries, we reformulate a second order boundary valued problem (BVP) as a feasibility problem where the sets are hypersurfaces. We show that such a problem may always be reformulated as a feasibility problem on no more than three sets and is well-suited to parallelization. We explore the stability of the method by applying it to several examples of BVPs, including cases where the traditional Newton's method fails

Optimization of resource allocation can explain the temporal dynamics and honesty of sexual signals

In species in which males are free to dynamically alter their allocation to sexual signaling over the breeding season, the optimal investment in signaling should depend on both a male’s state and the level of competition he faces at any given time. We developed a dynamic optimization model within a game‐theoretical framework to explore the resulting signaling dynamics at both individual and population levels and tested two key model predictions with empirical data on three‐spined stickleback (Gasterosteus aculeatus) males subjected to dietary manipulation (carotenoid availability): (1) fish in better nutritional condition should be able to maintain their signal for longer over the breeding season, resulting in an increasingly positive correlation between nutritional status and signal (i.e., increasing signal honesty), and (2) female preference for more ornamented males should thus increase over the breeding season. Both predictions were supported by the experimental data. Our model shows how such patterns can emerge from the optimization of resource allocation to signaling in a competitive situation. The key determinants of the honesty and dynamics of sexual signaling are the condition dependency of male survival, the initial frequency distribution of nutritional condition in the male population, and the cost of signaling

Absence of a self-induced decay effect in 198Au

We report the results of an improved experiment aimed at determining whether the half-life ($T_{1/2}$) of $^{198}$Au depends on the shape of the source. In this experiment, the half-lives of a gold sphere and a thin gold wire were measured after each had been irradiated in the NIST Center for Neutron Research. In comparison to an earlier version of this experiment, both the specific activities of the samples and their relative surface/volume ratios have been increased, leading to an improved test for the hypothesized self-induced decay (SID) effect. We find T_1/2(sphere)/T_1/2(wire) = 0.9993+/-0.0002, which is compatible with no SID effect.Comment: 3 pages, no figure

Multiple yielding processes in a colloidal gel under large amplitude oscillatory stress

Fatigue refers to the changes in material properties caused by repeatedly applied loads. It has been widely studied for, e.g., construction materials, but much less has been done on soft materials. Here, we characterize the fatigue dynamics of a colloidal gel. Fatigue is induced by large amplitude oscillatory stress (LAOStress), and the local displacements of the gel are measured through high-frequency ultrasonic imaging. We show that fatigue eventually leads to rupture and fluidization. We evidence four successive steps associated with these dynamics: (i) the gel first remains solid, (ii) it then slides against the walls, (iii) the bulk of the sample becomes heterogeneous and displays solid-fluid coexistence, and (iv) it is finally fully fluidized. It is possible to homogeneously scale the duration of each step with respect to the stress oscillation amplitude $\sigma_0$. The data are compatible with both exponential and power-law scalings with $\sigma_0$, which hints at two possible interpretations in terms of delayed yielding in terms activated processes or of the Basquin law. Surprisingly, we find that the model parameters behave nonmonotonically as we change the oscillation frequency and/or the gel concentration.Comment: 13 pages, 7 figures, submitted to Soft Matte

Nontangential limits and Fatou-type theorems on post-critically finite self-similar sets

In this paper we study the boundary limit properties of harmonic functions on $\mathbb R_+\times K$, the solutions $u(t,x)$ to the Poisson equation $$\frac{\partial^2 u}{\partial t^2} + \Delta u = 0,$$ where $K$ is a p.c.f. set and $\Delta$ its Laplacian given by a regular harmonic structure. In particular, we prove the existence of nontangential limits of the corresponding Poisson integrals, and the analogous results of the classical Fatou theorems for bounded and nontangentially bounded harmonic functions.Comment: 22 page