Let ∆ be the finite difference Laplacian associated to the lattice Z d . For dimension d ≥ 3, a ≥ 0 and L a sufficiently large positive dyadic integer, we prove that the integral kernel of the resolvent G a := (a − ∆) −1 can be decomposed as an infinite sum of positive semi-definite functions V n of finite range, V n (x−y) = 0 for |x−y| ≥ O(L) n . Equivalently, the Gaussian process on the lattice with covariance G a admits a decomposition into independent Gaussian processes with finite range covariances. For a = 0, V n has a limiting scaling formx−y L n as n → ∞. As a corollary, such decompositions also exist for fractional powers (The results of this paper give an alternative to the block spin renormalization group on the lattice.
We consider a point process i + ξ i , where i Î \mathbbZiZ and the ξ i ’s are i.i.d. random variables with compact support and variance σ 2. This process, with a suitable rescaling of the distribution of ξ i ’s, is well known to converge weakly, for large σ, to the Poisson process. We then study a simple queueing system with this process as arrival process. If the variance σ 2 of the random translations ξ i is large but finite, the resulting queue is very different from the Poisson case. We provide the complete description of the system for traffic intensity ϱ = 1, where the average length of the queue is proved to be finite, and for ϱ < 1 we propose a very effective approximated description of the system as a superposition of a fast process and a slow, birth and death, one. We found interesting connections of this model with the statistical mechanics of Fermi particles. This model is motivated by air traffic systems
received her Ph.D. in applied mathematics from the University of Alabama at Birmingham in 2012. Her current research focuses on the Errors-In-Variables (EIV) model and fitting geometric curves and surfaces to observed data points. Before joining the University of Virginia (UVA), she worked as an assistant professor at Black Hills State University for two years. In her current role as an APMA faculty member at UVA, she teaches applied math courses to engineering students. Her goals in teaching are to help students develop the confidence in their own ability to do mathematics and to make mathematics a joyful and successful experience.
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