Some Estimates for 2-Dimensional Infinite and Bounded Dilute Random Lorentz Gases

We present a (mostly) rigorous approach to unbounded and bounded (open) dilute random Lorentz gases. Relying on previous rigorous results on the dilute (Boltzmann–Grad) limit we compute the asymptotics of the Lyapunov exponent in the unbounded case. For the bounded open case in a circular region we give here an incomplete rigorous analysis which gives the asymptotics for large radius of the escape rate and of the rescaled quasi-invariant (q.i., or quasi-stationary) measure. We finally give a complete proof on existence and asymptotic properties of the q.i. measure in a one-dimensional caricature.     

Positive Toeplitz Operators Between the Harmonic Bergman Spaces

On the setting of the upper half space we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact positive Toeplitz operators taking a harmonic Bergman space b ^p into another b ^q for 1<p<, 1<q<. The case p=1 or q=1 seems more intriguing and is left open for further investigation. Also, we give criteria for positive Toeplitz operators acting on b ² to be in the Schatten classes. Some applications are also included.     

On the Existence of Invariant Measures for Networks with String Transitions

Recently a new class of Markov network processes was introduced, characterized by so-called string transitions. These are continuous-time Markov processes on a discrete state space. It is known that they possess an invariant measure of a special form, called a product-form, provided that a certain system of so-called traffic equations possesses a solution. Little is known about the existence of solutions of the traffic equations. The present paper deals with this question, focussing on the most important special case of unit vector string transitions. It is shown for open networks with unit vector string transitions of bounded lengths that the traffic equations possess a solution. Furthermore, it is shown for a prominent example of a network featuring signals and batch services that the traffic equations possess a solution.     

Large Scale and Heavy Traffic Asymptotics for Systems with Unreliable Servers

The asymptotic behaviour of the M/M/n queue, with servers subject to independent breakdowns and repairs, is examined in the limit where the number of servers tends to infinity and the repair rate tends to 0, such that their product remains finite. It is shown that the limiting two-dimensional Markov process corresponds to a queue where the number of servers has the same stationary distribution as the number of jobs in an M/M/ queue. Hence, the limiting model is referred to as the M/M/[M/M/] queue. Its numerical solution is discussed.Next, the behaviour of the M/M/[M/M/] queue is analysed in heavy traffic when the traffic intensity approaches 1. The convergence of the (suitably normalized) process of the number of jobs to a diffusion is proved.     

Compact range property and operators on -algebras

We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.

     

Singular measures with absolutely continuous convolution squares on locally compact groups

Saeki's result states that on any locally compact nondiscrete group there exist continuous singular measures, with respect to the left Haar measure, with in for all . This paper gives a new and short proof of this using Rademacher-Riesz products.

     

Solution of a functional equation arising from utility that is both separable and additive

The problem of determining all utility measures over binary gambles that are both separable and additive leads to the functional equation

The following conditions are more or less natural to the problem: strictly increasing, strictly decreasing; both map their domains onto intervals ( onto a , onto ); thus both are continuous, , , , , . We determine, however, the general solution without any of these conditions (except , , both into). If we exclude two trivial solutions, then we get as general solution (, ; for ), which satisfies all the above conditions. The paper concludes with a remark on the case where the equation is satisfied only almost everywhere.