Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice

Let Π be a homogenous Markov specification associated with a countable state space S and countably infinite parameter space A possessing a neighbor relation ~ such that (A,~) is the regular tree with d +1 edges meeting at each vertex. Let $\text{g}$(π)be the simplex of corresponding Markov random fields. We show that if Π satisfies a ‘boundedness’ condition then $\text{g}$(π).We further study the structure of $\text{g}$(π) when Π is either attractive or repulsive with respect to a linear ordering on S. When d = 1, so that (A, ~) is the one-dimensional lattice, we relax the requirement of homogeneity to that of stationarity; here we give sufficient conditions for $\text{g}$(π) and for $\text{g}$(π)to have precisely one member.     

Modeling of atmospheric radiative transfer with polarization and its application to the remote sensing of tropospheric ozone

Light reflected or transmitted by a planetary atmosphere contains information about particles and molecules in the atmosphere. Therefore, accurate modeling of the radiation field may be used to retrieve information on atmospheric composition. In this paper, a multi-layer model for a vertically inhomogeneous atmosphere is implemented by using the doubling-adding method for a plane-parallel atmosphere. By studying the degree of linear polarization of the transmitted and reflected solar light in the Huggins bands, we find significant differences between tropospheric ozone and stratospheric ozone. The effects of tropospheric ozone change on the linear polarization are 10 times more than that of the same amount of stratospheric ozone change. We also show the aerosol effect on the linear polarization, but this effect is wavelength independent as compared to that caused by the tropospheric ozone change. The results provide a theoretical basis for the retrieval of tropospheric ozone from measurement of linear polarization of the scattered sunlight both from the ground and from a satellite.     

An Axiomatisation of the Conditionals of Post's Many Valued Logics

The paper provides a method for a uniform complete Hilbert-style axiomatisation of Post's (m, u)-conditionals and Post's negation, where m is the number of truth values and u is the number of designated truth values (cf. [5]). The main feature of the technique which we employ in this proof generalises the well-known Kalmár Lemma which was used by its author in his completeness argument for the ordinary, two-valued logic (cf. [2]).     

The Influence of the Annealing Temperature on the Properties of Sn<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>Fe<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>O<Subscript>2</Subscript> Powders Evidenced by EMR Spectroscopy

Electron magnetic resonance (EMR) investigations of Sn_{1− x} Fe _x O₂ (x = 0.07) powders annealed at different temperatures are reported. EMR spectra show the presence of both the isolated paramagnetic Fe³⁺ ions incorporated into the SnO₂ lattice and a ferromagnetically ordered component. By increasing the annealing temperature, the EMR spectrum attributed to the ferromagnetically coupled Fe³⁺ ions diminishes probably due to the migration of Fe ions and the formation of the second phase. The presence of the second phase was evidenced by X-ray diffraction measurements.     

CVaR norm and applications in optimization

This paper introduces the family of CVaR norms in \({\mathbb {R}}^{n}\) , based on the CVaR concept. The CVaR norm is defined in two variations: scaled and non-scaled. The well-known \(L_{1}\) and \(L_{\infty }\) norms are limiting cases of the new family of norms. The D-norm, used in robust optimization, is equivalent to the non-scaled CVaR norm. We present two relatively simple definitions of the CVaR norm: (i) as the average or the sum of some percentage of largest absolute values of components of vector; (ii) as an optimal solution of a CVaR minimization problem suggested by Rockafellar and Uryasev. CVaR norms are piece-wise linear functions on \({\mathbb {R}}^{n}\) and can be used in various applications where the Euclidean norm is typically used. To illustrate, in the computational experiments we consider the problem of projecting a point onto a polyhedral set. The CVaR norm allows formulating this problem as a convex or linear program for any level of conservativeness.     

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t ? au(t)/t ² = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (?∞,?1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×?. It is shown that S _c = {u ∈ S: u′(T) = ?c} is nonempty and compact for each c ≥ 0 and S = ∪ _c≥0 S _c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S _c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.     

Two-point boundary value problems for the generalized Bagley-Torvik fractional differential equation

We investigate the fractional differential equation u″ + A ^c D ^α u = f(t, u, ^c D ^μ u, u′) subject to the boundary conditions u′(0) = 0, u(T)+au′(T) = 0. Here α ∈ (1, 2), µ ∈ (0, 1), f is a Carathéodory function and ^c D is the Caputo fractional derivative. Existence and uniqueness results for the problem are given. The existence results are proved by the nonlinear Leray-Schauder alternative. We discuss the existence of positive and negative solutions to the problem and properties of their derivatives.