The ingredients to a Bayesian analysis comprise the sampling model {Pθ : θ ∈ Ω} for the response X, the prior Π for the parameter θ, and the observed value X = x. This is equivalent to specifying the joint model Pθ ×Π for (X, θ) and the observed value X = x. In many situations we may also have a loss function but we ignore this here as it is not material to our discussion. We refer to Pθ ×Π as ...

We provide a corrected proof of [4, Theorem 2.2], which preserves the validity of the theorem exactly under those assumptions as stated in the original paper. 1. Corrigendum. This Corrigendum concerns the proof of [4, Theorem 2.2]. In the original proof we used [1, Theorem 2.4] and [3, Proposition 3.2], which require more restrictive conditions than necessary. We provide here an elementary maxi...

The existence of mixed curvature measures of two sets in R with positive reach introduced in [6] is discussed. An example shows that the non-osculating condition from [6] does not ensure the locally bounded variation of the mixed curvature measures. Further, some sufficient conditions for the local boundedness of mixed curvature measures involving absolute curvature measures are presented. For ...

We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R , into a transform domain with continuous scale a > 0, location b ∈ R, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) = 〈f, γabθ〉 where the inner products project f onto analyzing elements called curvelets γabθ which are smooth and of rapid decay away from an a by √ ...

New upper bounds for the number of independent sets in graphs are obtained. c © 2006 Published by Elsevier Ltd 1. Definitions and the statement of results All graphs under consideration are finite, undirected, and simple. The vertices are considered as numbered. Denote the degree of a vertex v by σ(v). G is called an (n, k, θ)-graph if it has n vertices and k ≤ σ(v) ≤ k + θ for any vertex v. Ev...

We consider analytically the domain of the three mixing angles Θ ij and the CP phase δ for three flavour neutrino oscillations both in vacuum and matter. Similarly to the quark sector, it is necessary and sufficient to let all the mixing angles Θ 12 , Θ 13 , Θ 23 and δ be in the range 0, π 2 and 0 ≤ δ < 2π, respectively. To exploit the full range of δ will be important in future when more preci...

ion of states. We adopt intervals of integers, I = {[m,n] | m ∈ N,n ∈ N∪{∞}∧m ≤ n}. Over intervals we consider the standard order ⊑I , such that I ⊑I J iff min(I),max(I)∈ J. Moreover, we use ⊔I for the corresponding l.u.b.. The abstract states are defined by replacing multiplicities with intervals of multiplicities. Therefore, an abstract state is a function M◦ : X → I . We also use M ◦ for the...

It is a truism that all of mathematics can be expressed in the language of set theory, i.e. the predicate calculus (including equality) with the single two-place relation ∈. As with any other countable language, the language of set theory is subject to the Löwenheim-Skolem Theorem, which asserts that if we have some axioms – e.g. the usual axioms of set theory (ZFC) – formulated in this languag...

Detecting termination of a distributed computation is a fundamental problem in distributed systems. We present two optimal algorithms for detecting termination of a non-diffusing distributed computation for an arbitrary topology. Both algorithms are optimal in terms of message complexity and detection latency. The first termination detection algorithm has to be initiated along with the underlyi...

Hampel (1974) introduced a very general procedure to derive optimal robust M-estimates for oneparameter families of distributions. The optimal estimate is obtained by minimizing the asymptotic variance among M-estimates which are Fisher-consistent and have gross error sensitivity (GES) bounded by a given constant. Stahel (1981) generalized the optimal M-estimates for the case of families which ...