Asymptotic behavior of the critical probability for ρ-percolation in high dimensions

We consider oriented bond or site percolation on ℤ ^d ₊. In the case of bond percolation we denote by P _p the probability measure on configurations of open and closed bonds which makes all bonds of ℤ ^d ₊ independent, and for which P _p {e is open} = 1 −P _p e {is closed} = p for each fixed edge e of ℤ ^d ₊. We take X(e) = 1 (0) if e is open (respectively, closed). We say that ρ-percolation occurs for some given 0 < ρ≤ 1, if there exists an oriented infinite path v ₀ = 0, v ₁, v ₂, …, starting at the origin, such that lim inf _{n →∞} (1/n) ∑ _i=1 ⁿ X(e _i ) ≥ρ, where e _i is the edge {v _i−1 , v _i }. [MZ92] showed that there exists a critical probability p _c = p _c (ρ, d) = p _c (ρ, d, bond) such that there is a.s. no ρ-percolation for p < p _c and that P _p {ρ-percolation occurs} > 0 for p > p _c . Here we find lim _{d →∞} d ^1/ρ p _c (ρd, bond) = D ₁ , say. We also find the limit for the analogous quantity for site percolation, that is D ₂ = lim _{d →∞} d ^1/ρ p _c (ρ, d, site). It turns out that for ρ < 1, D ₁ < D ₂ , and neither of these limits equals the analogous limit for the regular d-ary trees. Received: 7 January 1999 / Published online: 14 June 2000     

Is the dying individual the oldest?

Consider a population in which the birth times are a Poisson process with rate γ lifetimes are independent and identically distributed and lifetimes are independent of the birth process. In the paper we provide methods for calculation of several quantities involving the oldest member (senior) of the population. In particular we study the senior's age process and the point process of seniors' deaths obtained by dependent thinning of a Poisson process.     

What's the probability of a rational ratio being irreducible?

We determine the probability that any given rational ratio is irreducible.     

What is the probability an automorphism fixes a group element?

Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group.     

An extension criterion for the local in time solution of the chemotaxis Navier–Stokes equations in the critical Besov spaces

We consider the Keller–Segel model coupled with the incompressible Navier–Stokes equations in the dimension three. Based on the wellposedness result in the critical Besov spaces, we present a result on the extension criterion for the local in time solution in the same functional setting, which is a new result for the model.     

Is the linearized Boltzmann-Enskog operator dissipative?

In this paper, it is shown that the linearized Boltzmann-Enskog collision operator cannot be dissipative in the L₂-space setting contrarily to the linearized Boltzmann operator. Some estimates useful for the spectral theory are given.     

Is the inf-convolution of law-invariant preferences law-invariant?

We analyze the question of whether the inf-convolution of law-invariant risk functionals (preferences) is still law-invariant. In other words, we try to understand whether the representative economic agent (after risk redistribution) only cares about the distribution of the total risk, assuming all individual agents do so. Although the answer to the above question seems to be affirmative for many examples of commonly used risk functionals in the literature, the situation becomes delicate without assuming specific forms and properties of the individual functionals. We illustrate with examples the surprising fact that the answer to the main question is generally negative, even in an atomless probability space. Furthermore, we establish a few very weak conditions under which the answer becomes positive. These conditions do not require any specific forms or convexity of the risk functionals, and they are the richness of the underlying probability space, and monotonicity or continuity of one of the risk functionals. We provide several examples and counter-examples to discuss the subtlety of the question on law-invariance.     

Is the statement of Murphy's law valid?

Murphy's Law is not a law in the formal sense yet popular science often compares it with the Second Law of Thermodynamics as both the statements point toward a more disorganized state with time. In this article, we first construct a mathematically equivalent statement for Murphy's Law and then disprove it using the intuitive idea that energy differences will level off along the paths of steepest descent, or along trajectories of least action. © 2015 Wiley Periodicals, Inc. Complexity 21: 374–380, 2016     

What Is the Complexity of Stieltjes Integration?

We study the complexity of approximating the Stieltjes integral ∫¹₀ f(x) dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show that r(n)≍n^{−min{r, s+1}} and that comp(ε)≍ε^{−1/min{r, s+1}}. We also present an algorithm that computes an ε-approximation at nearly minimal cost.     

What Is the Complexity of Surface Integration?

We study the worst case complexity of computing ε-approximations of surface integrals. This problem has two sources of partial information: the integrand f and the function g defining the surface. The problem is nonlinear in its dependence on g. Here, f is an r times continuously differentiable scalar function of l variables, and g is an s times continuously differentiable injective function of d variables with l components. We must have d⩽l and s⩾1 for surface integration to be well-defined. Surface integration is related to the classical integration problem for functions of d variables that are min{r, s−1} times continuously differentiable. This might suggest that the complexity of surface integration should be Θ((1/ε)^{d/min{r, s−1}}). Indeed, this holds when d<l and s=1, in which case the surface integration problem has infinite complexity. However, if d⩽l and s⩾2, we prove that the complexity of surface integration is O((1/ε)^d/min{r, s}). Furthermore, this bound is sharp whenever d<l.     

What Is the Object of the Encapsulation of a Process?

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this “object” produced by the “encapsulation” of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery?     

Is there chaos in the Spanish labour market?

One could argue that there is a resurgence of the non-linear modelling in economics. Some instruments have been developed to measure the complexity or instability of the analysed systems. At the present work some of these developed techniques are applied to verify the non-linearity present in the time series of Spanish unemployment, as well as to quantify the degree of complexity of the system that has generated the series. Using these techniques we find evidence of chaos in Spanish unemployment time series.     

What Is the Complexity of Solution-Restricted Operator Equations?

We study the worst case complexity of operator equations Lu = f where L: G → X is a bounded linear injection of normed linear spaces. Past work on the complexity of such problems has generally required the class F of problem elements f to be the unit ball of X. However, there are many problems for which this choice of F yields unsatisfactory results. Mixed elliptic—hyperbolic problems are one example. the difficulty being that our technical tools are nor strong enough to give good complexity bounds. Ill-posed problems are another example. because we know that the complexity of computing finite-error approximations is infinite if F is a ball in X. In this paper, we pursue another idea. Rather than directly restrict the class F of problem elements f, we will consider problems that are solution-restricted: i.e., we restrict the class U of solution elements u. In particular, we assume that U is the unit hall of a normed linear space W that is densely, continuously embedded in G. The main idea is that our problem can now be reduced to the standard approximation problem of approximating the embedding of W into G.This allows us to characterize optimal information and algorithms for our problem..We use this idea to study three problems: the Tricomi problem (a mixed hyperbolic— elliptic problem arising in the study of transonic flow), the inverse finite Laplace transform (an ill-posed problem arising. e.g.. in geomathematics), and the backwards heat equation. We determine the problem complexity and derive nearly optimal algorithms for each of these problems.     

On the local convergence of the Douglas–Rachford algorithm

We discuss the Douglas–Rachford algorithm to solve the feasibility problem for two closed sets A,B in \({\mathbb{R}^d}\) . We prove its local convergence to a fixed point when A,B are finite unions of convex sets. We also show that for more general nonconvex sets the scheme may fail to converge and start to cycle, and may then even fail to solve the feasibility problem.