The Fredholm index of locally compact band-dominated operators on Lp(R)L^p (R)

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator $A$ on $L^p(R)$ and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of $A$. The results are applied to operators of convolution type with almost periodic symbol

Geometry of the sample frequency spectrum and the perils of demographic inference

The sample frequency spectrum (SFS), which describes the distribution of mutant alleles in a sample of DNA sequences, is a widely used summary statistic in population genetics. The expected SFS has a strong dependence on the historical population demography and this property is exploited by popular statistical methods to infer complex demographic histories from DNA sequence data. Most, if not all, of these inference methods exhibit pathological behavior, however. Specifically, they often display runaway behavior in optimization, where the inferred population sizes and epoch durations can degenerate to 0 or diverge to infinity, and show undesirable sensitivity of the inferred demography to perturbations in the data. The goal of this paper is to provide theoretical insights into why such problems arise. To this end, we characterize the geometry of the expected SFS for piecewise-constant demographic histories and use our results to show that the aforementioned pathological behavior of popular inference methods is intrinsic to the geometry of the expected SFS. We provide explicit descriptions and visualizations for a toy model with sample size 4, and generalize our intuition to arbitrary sample sizes n using tools from convex and algebraic geometry. We also develop a universal characterization result which shows that the expected SFS of a sample of size n under an arbitrary population history can be recapitulated by a piecewise-constant demography with only k(n) epochs, where k(n) is between n/2 and 2n-1. The set of expected SFS for piecewise-constant demographies with fewer than k(n) epochs is open and non-convex, which causes the above phenomena for inference from data.Comment: 21 pages, 5 figure

A multiprocessor based packet-switch: performance analysis of the communication infrastructure

The intra-chip communication infrastructures are receiving always more attention since they are becoming a crucial part in the development of current SoCs. Due to the high availability of pre-characterized hard-IP, the complexity of the design is moving toward global interconnections which are introducing always more constraints at each technology node. Power consumption, timing closure, bandwidth requirements, time to market, are some of the factors that are leading to the proposal of new solutions for next generation multi-million SoCs. The need of high programmable systems and the high gate-count availability is moving always more attention on multiprocessors systems (MP-SoC) and so an adequate solution must be found for the communication infrastructure. One of the most promising technologies is the Network-On-Chip (NoC) architecture, which seems to better fit with the new demanding complexity of such systems. Before starting to develop new solutions, it is crucial to fully understand if and when current bus architectures introduce strong limitations in the development of high speed systems. This article describes a case study of a multiprocessor based ethernet packet-switch application with a shared-bus communication infrastructure. This system aims to depict all the bottlenecks which a shared-bus introduces under heavy load. What emerges from this analysis is that, as expected, a shared-bus is not scalable and it strongly limits whole system performances. These results strengthen the hypothesis that new communication architectures (like the NoC) must be found