Coalescing systems of non-Brownian particles

A well-known result of Arratia shows that one can make rigorous the notion of starting an independent Brownian motion at every point of an arbitrary closed subset of the real line and then building a set-valued process by requiring particles to coalesce when they collide. Arratia noted that the value of this process will be almost surely a locally finite set at all positive times, and a finite set almost surely if the initial value is compact: the key to both of these facts is the observation that, because of the topology of the real line and the continuity of Brownian sample paths, at the time when two particles collide one or the other of them must have already collided with each particle that was initially between them. We investigate whether such instantaneous coalescence still occurs for coalescing systems of particles where either the state space of the individual particles is not locally homeomorphic to an interval or the sample paths of the individual particles are discontinuous. We give a quite general criterion for a coalescing system of particles on a compact state space to coalesce to a finite set at all positive times almost surely and show that there is almost sure instantaneous coalescence to a locally finite set for systems of Brownian motions on the Sierpinski gasket and stable processes on the real line with stable index greater than one.     

Perturbed Brownian motions

Summary. We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed Brownian motions (Hausdorff dimension of points of monotonicity for example). Received: 17 May 1996 / In revised form: 21 January 1997     

关于从闭区间到完备随机赋范模的抽象值函数的Riemann 可积性的进一步研究

郭铁信和张霞最近引入和研究了从一个闭区间到一个完备随机赋范模的抽象值函数的Riemann积分, 证明了值域几乎处处有界的连续函数是Riemann 可积的. 本文首先给出该结果的一个更简短的证明, 使得我们对于值域的几乎处处有界性有一个更深的认识, 受此启发, 我们进一步构造两个例子, 其一说明值域并非几乎处处有界的连续函数也可以是Riemann 可积的, 另一例子说明连续函数可以非Riemann 可积. 最后, 我们证明从一闭区间到一个满支撑的完备随机赋范模的所有连续函数都Riemann 可积的充要条件是基底概率空间本质上由至多可数原子生成.     

On the pebbling threshold of paths and the pebbling threshold spectrum

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this paper we tighten the gap between the upper and lower bounds for the pebbling threshold for the sequence of paths in the multiset model. We also find the pebbling threshold for the sequence of paths in the binomial model. Finally, we show that the spectrum of pebbling thresholds for graph sequences in the multiset model spans the entire range from n^1/2 to n, answering a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remains unknown.     

随机幂级数的亏函数

研究了十分一般的随机幂级数,并证明了有限级的随机幂级数几乎必然没有亏函数．     

Coexistence for species sharing a predator

A class of equations describing the dynamics of two prey sharing a common predator are considered. Even though the boundary and internal dynamics can exhibit oscillatory behavior, it is shown these equations are permanent if only if they admit a positive equilibrium. Going beyond permanence, a subclass of equations are constructed that are almost surely permanent but not permanent; there exists an attractor in the positive orthant that attracts Lebesgue almost every (but not every) initial condition.     

Lower bounds on two-terminal network reliability

Computing the probability that two nodes in a probabilistic network are connected is a well-known computationally difficult problem. Two strategies are devised for obtaining lower bounds on the connection probability for two terminals. The first improves on the Kruskal-Katona bound by using efficient computations of small pathsets. The second strategy employs efficient algorithms for finding edge-disjoint paths. The resulting bounds are compared; while the edge-disjoint path bounds typically outperform the Kruskal-Katona bounds, they do not always do so. Finally, a method is outlined for developing a uniform bound which combines both strategies.     

No Peace in Our Time

Abstract

Forty young leaders were chosen to be profiled in this issue. In an effort to gain insight from them, they were each asked five questions: What are the major challenges for you as a young leader of the academy? What do you perceive as the major challenges ahead for higher education in the next two decades? Do you have a mentor(s), and if so, how has this helped you in your career? How are you learning to lead? What are your hopes for higher education in the next two decades?

From their responses several items emerged, including a great deal of concern about keeping higher education accessible and rethinking how learning can be prompted more widely. There was strong consensus that mentors had played a vital role in their development, and must continue to do so. Finally, there was strong agreement that preparation for national leadership required the opportunity to practice it, to make mistakes, to take risks, and to learn from their own “real world” choices.

The following profiles surely do not include all the young leaders of the academy, but this group clearly represents many of the strongest and the smartest. Our profound hope is that their numbers grow, as do their opportunities for learning through leading.     

Almost sure estimates for the concentration neighborhood of Sinai’s walk

We consider Sinai’s random walk in random environment. We prove that infinitely often (i.o.) the size of the concentration neighborhood of this random walk is bounded almost surely. We also get that i.o. the maximal distance between two favorite sites is bounded almost surely.     

Two Badly Behaved Percolation Processes on a Nonunimodular Graph

We provide nonunimodular counterexamples to two properties that are well known for automorphism invariant percolation on unimodular transitive graphs. The first property is that the number of encounter points in an infinite cluster is almost surely 0 or ∞. The second property is that speed of random walk on an infinite cluster is almost surely well defined.     

When almost all sets are difference dominated

We investigate the relationship between the sizes of the sum and difference sets attached to a subset of {0,1,…,N}, chosen randomly according to a binomial model with parameter p(N), with N^?1 = o(p(N)). We show that the random subset is almost surely difference dominated, as N → ∞, for any choice of p(N) tending to zero, thus confirming a conjecture of Martin and O'Bryant. The proofs use recent strong concentration results. Furthermore, we exhibit a threshold phenomenon regarding the ratio of the size of the difference to the sumset. If p(N) = o(N^?1/2) then almost all sums and differences in the random subset are almost surely distinct and, in particular, the difference set is almost surely about twice as large as the sumset. If N^?1/2 = o(p(N)) then both the sum and difference sets almost surely have size (2N + 1) ? O(p(N)^?2), and so the ratio in question is almost surely very close to one. If p(N) = c · N^?1/2 then as c increases from zero to infinity (i.e., as the threshold is crossed), the same ratio almost surely decreases continuously from two to one according to an explicitly given function of c. We also extend our results to the comparison of the generalized difference sets attached to an arbitrary pair of binary linear forms. For certain pairs of forms f and g, we show that there in fact exists a sharp threshold at c_f,g · N^?1/2, for some computable constant c_f,g, such that one form almost surely dominates below the threshold and the other almost surely above it. The heart of our approach involves using different tools to obtain strong concentration of the sizes of the sum and difference sets about their mean values, for various ranges of the parameter p. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Online paths in mathematics teacher training: new resources and new skills for teacher educators

This paper considers the work carried out by online teacher educators and their professional development. We use the theoretical perspective of the documentational approach that focuses, in this case, on the interaction between teacher educators and the resources they use for their online training work. We thus study the following issues: (1) What kinds of resources do online teacher educators need, and how are such resources modified according to the educators?? specific skills and needs? (2) What specific skills are needed for setting up online training for mathematics teachers and how do these skills evolve as teacher education resources are used? We consider both questions simultaneously, while presenting results from a study within a specific teacher training programme in France that proposes ??training paths?? on a national platform. These ??paths?? are resources designed for teacher educators. We follow the appropriation of two training paths by two educator teams. The ways in which these educator teams were able to appropriate the paths give insights into the teacher educators?? skills and, as well, into the resources they need. By looking at their use of resources (as online mathematics teacher educators), we observe and analyse professional geneses, leading to the development of new skills.     

Towards Nikishin’s theorem on the almost sure convergence of rearrangements of functional series

Necessary and sufficient conditions are found for the almost sure convergence of almost all simple rearrangements of a series of Banach space valued random variables. The results go back to Nikishin’s well-known theorem on the existence of an almost surely convergent rearrangement of a numerical random series. An example is also given of a numerical random series with general term tending to zero almost surely such that this series converges in probability and any its rearrangement diverges almost surely.     

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Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is ??Let be a correlated random walk in random environment. For the sub-linear regime, that is, almost surely but , we show that there is $0s. This result characterizes the slowdown property of the walk.     

Monotonicity and phase diagram for multirange percolation on oriented trees

We consider Bernoulli bond percolation on oriented regular trees, where besides the usual short bonds, all bonds of a certain length are added. Independently, short bonds are open with probability p and long bonds are open with probability q. We study properties of the critical curve which delimits the set of pairs (p,q) for which there are almost surely no infinite paths. We also show that this curve decreases with respect to the length of the long bonds.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.     

Randomly generated distributions

A new scheme for randomly generating probability distributions on the interval [0, 1] is introduced. The scheme can also be viewed as a way to generate homeomorphisms at random. Conditions are given so that a continuous measure with full support is generated almost surely. Geometric properties of the generated probability measures are examined, including the dimension and derivative structure of the measures and their respective distribution functions. For example, we give conditions so that almost all the distribution functions of the measures generated are strictly singular. Applications include determining average case errors for numerical methods of equation solving and Bayesian statistics. Research supported by NSF Grant DMS-9303888.     

La singularité mutuelle presque sûre du spectre des transformations d’Ornstein

It is shown that, almost surely with respect to the product measure, the maximal spectral type of Ornstein’s transformations are mutually singular so that these transformations are mutually disjoint in Furstenberg’s sense.      

On the convergence of uncertain sequences

Uncertain variables are measurable functions from uncertainty spaces to the set of real numbers. In this paper, a new kind of convergence, convergence uniformly almost surely (convergence uniformly a.s.), is presented. Then, relations between convergence uniformly almost surely and convergence almost surely (convergence a.s.), convergence in measure, convergence in mean, and convergence in distribution are discussed.     

The lyapunov spectrum and stable manifolds for stochastic linear delay equations

A class of linear stochastic retarded functional differential equations is considered. These equations have diffusion coefficients that do not look into the past. It is shown that the trajectories of such equations form a continuous linear cocycle on the underlying state space. At times greater than the delay the cocycle is almost surely compact. Consequently, using an infinite-dimensional Oseledec multiplicative ergodic theorem of Ruelle, the existence of a countable non-random Lyapunov spectrum is proved. In the hyperbolic case it is shown that the state space admits an almost sure saddle-point splitting which is cocycle-invariant and corresponds to an exponential dichotomy for the stochastic flow