Signals from the future

Advanced interactions do not lead to dynamical inconsistencies in closed systems. They are, however, incompatible with the existence of truly random processes.Work supported in part by U.S. Army Research Office (Durham).NATO Fellow. On leave from: Physics Department, Indiana University, Bloomington, Indiana 47401.     

New Coins From Old: Computing With Unknown Bias

Suppose that we are given a function f : (0, 1)→(0,1) and, for some unknown p∈(0, 1), a sequence of independent tosses of a p-coin (i.e., a coin with probability p of “heads”). For which functions f is it possible to simulate an f(p)-coin? This question was raised by S. Asmussen and J. Propp. A simple simulation scheme for the constant function f(p)≡1/2 was described by von Neumann (1951); this scheme can be easily implemented using a finite automaton. We prove that in general, an f(p)-coin can be simulated by a finite automaton for all p ∈ (0, 1), if and only if f is a rational function over ℚ. We also show that if an f(p)-coin can be simulated by a pushdown automaton, then f is an algebraic function over ℚ; however, pushdown automata can simulate f(p)-coins for certain nonrational functions such as . These results complement the work of Keane and O’Brien (1994), who determined the functions f for which an f(p)-coin can be simulated when there are no computational restrictions on the simulation scheme. * Supported by a Miller Fellowship. † Supported in part by NSF Grant DMS-0104073 and by a Miller Professorship. ‡ This work is supported under a National Science Foundation Graduate Research Fellowship.     

Biased tug-of-war, the biased infinity Laplacian, and comparison with exponential cones

We prove that if ${U\subset \mathbb {R}^n}$ is an open domain whose closure ${\overline U}$ is compact in the path metric, and F is a Lipschitz function on ?U, then for each ${\beta \in \mathbb {R}}$ there exists a unique viscosity solution to the β-biased infinity Laplacian equation $$\beta |\nabla u| + \Delta_\infty u=0$$ on U that extends F, where ${\Delta_\infty u= |\nabla u|^{-2} \sum_{i,j} u_{x_i}u_{x_ix_j} u_{x_j}}$ . In the proof, we extend the tug-of-war ideas of Peres, Schramm, Sheffield and Wilson, and define the β-biased ${\epsilon}$ -game as follows. The starting position is ${x_0 \in U}$ . At the kth step the two players toss a suitably biased coin (in our key example, player I wins with odds of ${\exp(\beta\epsilon)}$ to 1), and the winner chooses x _k with ${d(x_k,x_{k-1}) < \epsilon}$ . The game ends when ${x_k \in \partial U}$ , and player II pays the amount F(x _k ) to player I. We prove that the value ${u^{\epsilon}(x_0)}$ of this game exists, and that ${\|u^\epsilon - u\|_\infty \to 0}$ as ${\epsilon \to 0}$ , where u is the unique extension of F to ${\overline{U}}$ that satisfies comparison with β-exponential cones. Comparison with exponential cones is a notion that we introduce here, and generalizing a theorem of Crandall, Evans and Gariepy regarding comparison with linear cones, we show that a continuous function satisfies comparison with β-exponential cones if and only if it is a viscosity solution to the β-biased infinity Laplacian equation.     

Mobile geometric graphs: detection, coverage and percolation

We consider the following dynamic Boolean model introduced by van den Berg et al. (Stoch. Process. Appl. 69:247–257, 1997). At time 0, let the nodes of the graph be a Poisson point process in ${\mathbb{R}^d}$ with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point—fixed or moving—is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.     

Structural and optical spectroscopy of LiNbO3:Tm nanocrystals embedded in a SiO2 glass matrix

Transparent SiO₂:Li₂O:Nb₂O₅ glass doped with Tm³⁺ has been prepared by the sol–gel method, and heat-treated in air (HT) at temperatures between 500 and 800 °C. X-ray diffraction (XRD) patterns and Raman spectroscopy show SiO₂ and LiNbO₃ phases in samples HT above 650 °C, and a NbTmO₄ phase for T > 750 °C. The XRD SEM analysis show increasing particle size and number with the increase of HT temperature. Intra-4f¹² transitions due to Tm³⁺ ion dispersed in the matrix are observed in samples with T > 650 °C. The luminescence is dominated by the ¹G₄ → ³F₄ (～650 nm), ¹D₂ → ³F₃ (～780 nm), ³H₄ → ³H₆ (～800 nm), ³H₅ → ³H₆ (～1200 nm) and ³H₄ → ³F₄ (～1500 nm) transitions under resonant excitation to the ion levels.     

Low-density ferromagnetism in biased bilayer graphene

We compute the phase diagram of a biased graphene bilayer. The existence of a ferromagnetic phase is discussed with respect to both carrier density and temperature. We find that the ferromagnetic transition is first-order, lowering the value of U relatively to the usual Stoner criterion. We show that in the ferromagnetic phase the two planes have unequal magnetization and that the electronic density is holelike in one plane and electronlike in the other.     

hcp‐Co Nanowires Grown on Metallic Foams as Catalysts for Fischer–Tropsch Synthesis

The Fischer–Tropsch synthesis (FTS) is a structure‐sensitive exothermic reaction that enables catalytic transformation of syngas to high quality liquid fuels. Now, monolithic cobalt‐based heterogeneous catalysts were elaborated through a wet chemistry approach that allows control over nanocrystal shape and crystallographic phase, while at the same time enables heat management. Copper and nickel foams have been employed as supports for the epitaxial growth of hcp‐Co nanowires directly from a solution containing a coordination compound of cobalt and stabilizing ligands. The Co/Cu_foam catalyst was tested for Fischer–Tropsch synthesis in a fixed‐bed reactor, showing stability and significantly superior activity and selectivity towards C5+ compared to a Co/SiO₂‐Al₂O₃ reference catalyst under the same conditions.     

Well-posedness for the nonlocal nonlinear Schrödinger equation

We establish local well-posedness for small initial data in the usual Sobolev spaces Hs(R), s?1, and global well-posedness in H¹(R), for the Cauchy problem associated to the nonlocal nonlinear Schrödinger equation