Objective bayesian analysis of the Yule–Simon distribution with applications

The Yule–Simon distribution is usually employed in the analysis of frequency data. As the Bayesian literature, so far, has ignored this distribution, here we show the derivation of two objective priors for the parameter of the Yule–Simon distribution. In particular, we discuss the Jeffreys prior and a loss-based prior, which has recently appeared in the literature. We illustrate the performance of the derived priors through a simulation study and the analysis of real datasets.     

Diophantine equations with products of consecutive members of binary recurrences

We prove a finiteness result for the number of solutions of a Diophantine equation of the form \(u_n u_{n+1}\cdots u_{n+k}\pm 1 =\pm u_m^2\), where \(\{ u_n\}_{n\ge 1}\) is a binary recurrent sequence whose characteristic equation has roots which are real quadratic units.     

Existence and multiplicity of positive solutions for a system of difference equations with coupled boundary conditions

We study the existence and multiplicity of positive solutions for a system of nonlinear second-order difference equations subject to coupled multi-point boundary conditions.     

Density lower bound estimate for local minimizer of free interface problem with volume constraint

We prove a density lower bound for some functionals involving bulk and interfacial energies. The bulk energies are convex functions with p-power growth not subjected to any further structure conditions. The interface \(\partial E\) is the boundary of a set \(E\subset \Omega \) such that \(|E|=d\) is prescribed. Then we get \(\mathcal {H}^{n-1}((\partial E{\setminus }\partial E^*)\cup \Omega )=0\).

     

Stability properties of an inverse parabolic problem with unknown boundaries

We treat the stability issue for an inverse problem arising from non-destructive evaluation by thermal imaging. We consider the determination of an unknown portion of the boundary of a thermic conducting body by overdetermined boundary data for a parabolic initial-boundary value problem. We obtain that when the unknown part of the boundary is a priori known to be smooth, the data are as regular as possible and all possible measurements are taken into account, the problem is exponentially ill-posed. Then, we prove that a single measurement with some a priori information on the unknown part of the boundary and minimal assumptions on the data, in particular on the thermal conductivity, is enough to have stable determination of the unknown boundary. Given the exponential ill-posedness, the stability estimate obtained is optimal. AMS 2000 Mathematics Subject Classification. Primary 35R30, Secondary 35B60, 33C90     

Online weighted flow time and deadline scheduling

In this paper we study some aspects of weighted flow time. We first show that the online algorithm Highest Density First is an O(1)-speed O(1)-approximation algorithm for P|r_i,pmtn|∑w_iF_i. We then consider a related Deadline Scheduling Problem that involves minimizing the weight of the jobs unfinished by some unknown deadline D on a uniprocessor. We show that any c-competitive online algorithm for weighted flow time must also be c-competitive for deadline scheduling. We then give an O(1)-competitive algorithm for deadline scheduling.     

Immobilized TiO<Subscript>2</Subscript> nanoparticles produced by flame spray for photocatalytic water remediation

Anatase/rutile mixed-phase titanium dioxide (TiO₂) photocatalysts in the form of nanostructured powders with different primary particle size, specific surface area, and rutile content were produced from the gas-phase by flame spray pyrolysis (FSP) starting from an organic solution containing titanium (IV) isopropoxide as Ti precursor. Flame spray-produced TiO₂ powders were characterized by means of X-ray diffraction, Raman spectroscopy, and BET measurements. As-prepared powders were mainly composed of anatase crystallites with size ranging from 7 to 15 nm according to the synthesis conditions. TiO₂ powders were embedded in a multilayered fluoropolymeric matrix to immobilize the nanoparticles into freestanding photocatalytic membranes. The photocatalytic activity of the TiO₂-embedded membranes toward the abatement of hydrosoluble organic pollutants was evaluated employing the photodegradation of rhodamine B in aqueous solution as test reaction. The photoabatement rate of best performing membranes significantly overcomes that of membranes produced by the same method and incorporating commercial P25-TiO₂.     

On the Dirichlet and Neumann Evolution Operators in ℝ d +

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) associated with a class of nonautonomous elliptic operators (t) with unbounded coefficients defined in I× \(\mathbb{R}_{+}\) (where I is a right-halfline or I=?). We also prove the existence and the uniqueness of a tight evolution system of measures \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) associated with \(G_{\mathcal {N}}(t,s)\) , which turns out to be sub-invariant for \(G_{\mathcal {D}}(t,s)\) , and we study the asymptotic behaviour of the evolution operators \(G_{\mathcal {D}}(t,s)\) and \(G_{\mathcal {N}}(t,s)\) in the L ^p -spaces related to the system \(\left \{\mu _{t}^{\mathcal {N}}\right \}_{t \in I}\) .     

MRICOM–MRI COntrast Modelling using 2D T1–T2 correlation spectra and relaxation signatures

Image contrast is calculated by inputting experimental 2D T₁–T₂ relaxation spectra into the ODIN software interface. The method involves characterising a magnetic resonance imaging pulse sequence with a “relaxation signature” which describes the sensitivity of the sequence to relaxation and is independent of sample parameters. Maximising (or minimising) the overlap between the experimental 2D T₁–T₂ relaxation spectra and the relaxation signature can then be used to maximise image contrast. The concept is illustrated using relaxation signatures for the echo planar imaging and Turbo spin-echo imaging sequences, together with in-vitro 2D T₁–T₂ spectra for liver and cartilage.     

The flipped longitudinal polarization sequence

By flipping the longitudinal magnetization with a chain of 180° pulses it is possible to effectively restore the effects of relaxation so that the same longitudinal magnetization is periodically recovered. The pulse sequence for achieving this, called Flipped LOngitudinal Polarization (FLOP), can be incorporated into any pulse sequence whenever it is desired to stop the attenuation in longitudinal magnetization caused by relaxation. We illustrate its use for fast, single-shot measurements of the longitudinal relaxation time and for three-dimensional T₁ mapping.