Let K be a closed, normal cone with nonempty interior int(K) in a Banach space X. Let Σ = {x ∈ int(K) : q(x) = 1} where q: int(K) → (0,∞) is continuous and homogeneous of degree 1 and it is usually assumed that Σ is bounded in norm. In this framework there is a complete metric d, Hilbert’s projective metric, defined on Σ and a complete metric d, Thompson’s metric, defined on int(K). We study pr...

We study phase transitions of the ferromagnetic q-state Potts chain with random nearest-neighbor couplings having a variance Δ^{2} and with homogeneous long-range interactions, which decay with distance as a power r^{-(1+σ)}, σ>0. In the large-q limit the free-energy of random samples of length L≤2048 is calculated exactly by a combinatorial optimization algorithm. The phase transition stays fi...

Let (Ω,B, P ) be a probability space, A ⊂ B a sub-σ-field, and μ a regular conditional distribution for P given A. Necessary and sufficient conditions for μ(ω)(A) to be 0–1, for all A ∈A and ω ∈A0, where A0 ∈ A and P (A0) = 1, are given. Such conditions apply, in particular, when A is a tail sub-σ-field. Let H(ω) denote the Aatom including the point ω ∈ Ω. Necessary and sufficient conditions fo...

This paper investigates aspects of measure and randomness in the context of locale theory (point-free topology). We prove that every measure (σ-continuous valuation) μ, on the σ-frame of opens of a fitted σ-locale X, extends to a measure on the lattice of all σ-sublocales of X (Theorem 1). Furthermore, when μ is a finite measure with μ(X) = M , the σ-locale X has a smallest σ-sublocale of measu...

1. Recall the spectrum of a bounded linear operator T ∈ B(X) on a Banach space X, more generally a Banach algebra element a ∈ A: 1.1 σ(a) = σA(a) = {λ ∈ C : a− λ 6∈ A−1}, where of course 1.2 a ∈ A−1 ⇐⇒ ∃a′, a′′ ∈ A, a′a = 1 = aa′′ : for example for square matrices A = Cn×n we have the well worn cliche 1.3 σ(a) = σA(a) = {λ ∈ C : det(a− λ) = 0}; for continuous functions A = C(Ω) on compact Hausd...

Fairly rapid environmental changes call for continuous surveillance and decision making, areas where IT technologies can be valuable. In the aforementioned context this work describes the application of a novel classifier, namely σFLNMAP, for estimating the ozone concentration level in the atmosphere. In a series of experiments on meteorological and air pollutants data, the σ–FLNMAP classifier ...

For a Hecke pair (G, H) and a character σ of H we consider a generalised Hecke algebra Hσ(G, H), which we study by embedding the given Hecke pair in a Schlichting completion (Gσ, Hσ) that comes equipped with a continuous extension σ on Hσ. The image of σ in Cc(Gσ) is a non-full projection pσ, and Hσ(G, H) is isomorphic to pσCc(Gσ)pσ. We study the structure and properties of C *-completions of a...

We prove that, for each countable ordinal ξ ≥ 1, there exist some Σ0ξ-complete ω-powers, and some Π0ξ-complete ω-powers, extending previous works on the topological complexity of ω-powers [Fin01, Fin03, Fin04, Lec01, Lec05, DF06]. We prove effective versions of these results; in particular, for each recursive ordinal ξ < ω 1 there exist some recursive sets A ⊆ 2 such that A∞ ∈ Π 0ξ \Σ 0 ξ (resp...

The paper aims at finding widely and smoothly defined nonparametric location and scatter functionals. As a convenient vehicle, maximum likelihood estimation of the location vector μ and scatter matrix Σ of an elliptically symmetric t distribution on R with degrees of freedom ν > 1 extends to an M-functional defined on all probability distributions P in a weakly open, weakly dense domain U . Her...