Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup _n→∞(S_n/n^κ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.This work is partially supported by ARC Grant DP0210572.     

The hitting distributions of a half real line for two-dimensional random walks

For every two-dimensional random walk on the square lattice Z ² having zero mean and finite variance we obtain fine asymptotic estimates of the probability that the walk hits the negative real line for the first time at a site (s,0), when it is started at a site far from both (0,s) and the origin.     

An analog of Wald’s identity for random walks with infinite mean

We deduce an analog of the classical Wald’s identity ES _τ = EτEξ in the case of the infinite mean of summands. We find the conditions on τ under which Emin(S _τ , x) ～ EτE min(ξ, x) as x→∞.     

Analysis of wavy flow and blood oxygenation in two-dimensional channels with constriction

A steady wavy incompressible Newtonian fluid flow in a channel with irregular surfaces is studied to understand the abnormal flow conditions caused by the boundary irregularities in diseased vessels. Analytical solutions are obtained under the assumption that the spread of the surface roughness to be large compared to the mean width of the channel. Expressions for the stream function, vorticity, the wall shear stress distribution and viscous energy loss per unit cycle are derived and the effects of various pertinent parameters upon them have been investigated for symmetric and non-symmetric channels with graphical representations. In order to determine the effects of the wall roughness upon the blood oxygenation in a membrane oxygenator, the haemodynamical solution is used. It is found that oxygen concentration increases with increase of channel constriction due to increase of cell-plasma and cell-cell interaction as well as waviness of flow field and this is predicted graphically.     

Throughput limits from the asymptotic profile of cyclic networks with state-dependent service rates

We consider networks where at each node there is a single exponential server with a service rate which is a non-decreasing function of the queue length. The asymptotic profile of a sequence of networks consists of the set of persistent service rates, the limiting customer-to-node ratio, and the limiting service-rate measure. For a sequence of cyclic networks whose asymptotic profile exists, we compute upper and lower bounds for the limit points of the sequence of throughputs as functions of the limiting customer-to-node ratio. We then find conditions under which the limiting throughput exists and is expressible in terms of the asymptotic profile. Under these conditions, we determine the limiting queue-length distributions for persistent service rates. In the absence of these conditions, the limiting throughput need not exist, even for increasing sequences of cyclic networks.      

On the existence of ordered couplings of random sets — with applications

Letψ andϕ be two given random closed sets in a locally compact second countable topological spaceS. (They need not be based on the same probability space.) The main result gives necessary and sufficient conditions on the distributions ofψ andϕ, for the existence of two random closed sets and , based on the same probability space and such that their distributions coincide with those ofψ andϕ, resp., and a.s. This coupling result tells us in particular when a probability distribution onS is selectionable w.r.t. (the distribution of) a random closed set. An existence result for realizable thinnings of a simple point process is obtained by specializing it to supports of random measures. The coupling result is extended to random variables in a countably based continuous poset. As examples we mention various kinds of random capacities — in particular random measures — and random compact (saturated) sets. Moreover, the extended result tells us when a probability distribution onS is selectionable w.r.t. the distribution of a random compact (saturated) set.     

Study on multi-objective optimization of flow allocation in a multi-commodity stochastic-flow network with unreliable nodes

A long distance transportation problem was abstracted to a resource flow allocation problem upon a stochastic-flow network with unreliable nodes. The objectives were the probability that transmission was successful and transportation cost. In order to solve constructed model, a multi-objective genetic algorithm was propounded. Tested by examples, the algorithm well solved the flow allocation problem in a stochastic-flow network.     

Front tracking for a model of immiscible gas flow with large data

In this paper we study front tracking for a model of one dimensional, immiscible flow of several isentropic gases, each governed by a gamma-law. The model consists of the p-system with variable gamma representing the different gases. The main result is the convergence of a front tracking algorithm to a weak solution, thereby giving existence as well. This convergence holds for general initial data with a total variation satisfying a specific bound. The result is illustrated by numerical examples.     

Uniqueness for weak flow of H—systems with dirichlet boundary conditons

It is proved that the weak flow of H—system with a Dirichlet boundary condition is unique and regular up to the time of energy concentration. Moreover, if the global weak solution satisfies certain energy inequality, then it is unique for all the time and is regular with the exception of finitely many points     

Generating uniform random vectors over a simplex with implications to the volume of a certain polytope and to multivariate extremes

A uniform random vector over a simplex is generated. An explicit expression for the first moment of its largest spacing is derived. The result is used in a proposed diagnostic tool which examines the validity of random number generators. It is then shown that the first moment of the largest uniform spacing is related to the dependence measure of random vectors following any extreme value distribution. The main result is proved by a geometric proof as well as by a probabilistic one.     

Global optimization of truss topology with discrete bar areas—Part I: theory of relaxed problems

This two part paper considers the classical problem of finding a truss design with minimal compliance subject to a given external force and a volume bound. The design variables describe the cross-section areas of the bars. While this problem is well-studied for continuous bar areas, we treat here the case of discrete areas. This problem is of major practical relevance if the truss must be built from pre-produced bars with given areas. As a special case, we consider the design problem for a single bar area, i.e., a 0/1-problem. In contrast to heuristic methods considered in other approaches, Part I of the paper together with Part II present an algorithmic framework for the calculation of a global optimizer of the underlying large-scaled mixed integer design problem. This framework is given by a convergent branch-and-bound algorithm which is based on solving a sequence of nonconvex continuous relaxations. The main issue of the paper and of the approach lies in the fact that the relaxed nonlinear optimization problem can be formulated as a quadratic program (QP). Here the paper generalizes and extends the available theory from the literature. Although the Hessian of this QP is indefinite, it is possible to circumvent the non-convexity and to calculate global optimizers. Moreover, the QPs to be treated in the branch-and-bound search tree differ from each other just in the objective function. In Part I we give an introduction to the problem and collect all theory and related proofs for the treatment of the original problem formulation and the continuous relaxed problems. The implementation details and convergence proof of the branch-and-bound methodology and the large-scale numerical examples are presented in Part II.     

Global optimization of truss topology with discrete bar areas—Part II: Implementation and numerical results

A classical problem within the field of structural optimization is to find the stiffest truss design subject to a given external static load and a bound on the total volume. The design variables describe the cross sectional areas of the bars. This class of problems is well-studied for continuous bar areas. We consider here the difficult situation that the truss must be built from pre-produced bars with given areas. This paper together with Part I proposes an algorithmic framework for the calculation of a global optimizer of the underlying non-convex mixed integer design problem. In this paper we use the theory developed in Part I to design a convergent nonlinear branch-and-bound method tailored to solve large-scale instances of the original discrete problem. The problem formulation and the needed theoretical results from Part I are repeated such that this paper is self-contained. We focus on the implementation details but also establish finite convergence of the branch-and-bound method. The algorithm is based on solving a sequence of continuous non-convex relaxations which can be formulated as quadratic programs according to the theory in Part I. The quadratic programs to be treated within the branch-and-bound search all have the same feasible set and differ from each other only in the objective function. This is one reason for making the resulting branch-and-bound method very efficient. The paper closes with several large-scale numerical examples. These examples are, to the knowledge of the authors, by far the largest discrete topology design problems solved by means of global optimization.     

Homogenization of the diffusion equation with nonlinear flux condition on the interior boundary of a perforated domain — the influence of the scaling on the nonlinearity in the effective sink-source term

In this paper, we study the asymptotic behavior of solutions u _ε of the initial boundary value problem for parabolic equations in domains W_e ì \mathbbRⁿ {\Omega_\varepsilon } \subset {\mathbb{R}^n} , n ≥ 3, perforated periodically by balls with radius of critical size ε ^α , α = n/(n − 2), and distributed with period ε. On the boundary of the balls a nonlinear third boundary condition is imposed. The weak convergence of the solutions u _ε to the solution of an effective equation is given. Furthermore, an improved approximation for the gradient of the microscopic solutions is constructed, and a corrector result with respect to the energy norm is proved.