Let G be a sofic group, and let Σ “ pσnqně1 be a sofic approximation to it. For a probability-preserving G-system, a variant of the sofic entropy relative to Σ has recently been defined in terms of sequences of measures on its model spaces that ‘converge’ to the system in a certain sense. Here we prove that, in order to study this notion, one may restrict attention to those sequences that have ...

For a market with an atomless continuum of assets, we formulate the intuitive idea of a ‘‘well-diversified’’ portfolio, and present a notion of ‘‘exact arbitrage’’, strictly weaker than the more conventional notion of ‘‘asymptotic arbitrage’’, and necessary and sufficient for the validity of an APT pricing formula. Our formula involves ‘‘essential’’ risk, one based on a specific index portfolio...

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric gWP on Mγ , the Riemann moduli space of surfaces of genus γ > 1. This space has a singular compactification with respect to gWP, and this metric has crossing cusp-edge singularities along a finite collection of simple normal crossing divisors. We prove first that the scalar...

I provide a background of groups viewed as metric spaces to introduce the notion of asymptotic dimension of a group. I analyze the asymptotic dimension of Z ⊕ Z and the free group on two generators to better understand the concept of asymptotic dimension. The asymptotic dimension of the Heisenberg group, H, asdimH, has been shown to be three using advanced mathematics. I will try to show that a...

It has been observed repeatedly that the standard safety-liveness classi cation for properties of reactive systems does not t for real-time properties. This is because the implicit \liveness" of time shifts the spectrum towards the safety side. While, for example, response | that \something good" will happen eventually | is a classical liveness property, bounded response | that \something good"...

Two-sample tests for multivariate and especially for non-Euclidean data are not fully explored. This paper presents a novel test based on a similarity graph constructed on the pooled observations from the two samples. It can be applied to multivariate data and non-Euclidean data as long as a dissimilarity measure on the sample space can be defined, which can usually be provided by domain expert...

The output distribution, when rate is above capacity, is investigated. It is shown that there is an asymptotic equipartition property (AEP) of the typical output sequences, independently of the specific codebook used, as long as the codebook is typical according to the standard random codebook generation. This equipartition of the typical output sequences is caused by the mixup of input sequenc...

in this paper, we investigate the canonical property of solutions of a system of differentialequations having a singularity and turning point of even order. first, by a replacement, we transform thesystem to the sturm-liouville equation with a turning point. using the asymptotic estimates for a specialfundamental system of solutions of sturm-liouville equation, we study the infinite product rep...

the statistical analysis of spatial data is usually done under gaussian assumption for the underlying random field model. when this assumption is not satisfied, block bootstrap methods can be used to analyze spatial data. one of the crucial problems in this setting is specifying the block sizes. in this paper, we present asymptotic optimal block size for separate block bootstrap to estimate the...