Dobinski-type relations and the Log-normal distribution

We consider sequences of generalized Bell numbers B(n), n=0,1,... for which there exist Dobinski-type summation formulas; that is, where B(n) is represented as an infinite sum over k of terms P(k)^n/D(k). These include the standard Bell numbers and their generalizations appearing in the normal ordering of powers of boson monomials, as well as variants of the "ordered" Bell numbers. For any such B we demonstrate that every positive integral power of B(m(n)), where m(n) is a quadratic function of n with positive integral coefficients, is the n-th moment of a positive function on the positive real axis, given by a weighted infinite sum of log-normal distributions.Comment: 7 pages, 2 Figure

Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.Comment: 14 pages, 31 reference

Exact and explicit probability densities for one-sided Levy stable distributions

We study functions g_{\alpha}(x) which are one-sided, heavy-tailed Levy stable probability distributions of index \alpha, 0< \alpha <1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_{\alpha}(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_{\alpha}(x) dx = exp(-p^{\alpha}), p>0, for all \alpha = l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of \alpha in order to adapt g_{\alpha}(x) to a given experimental situation.Comment: 4 pages, 3 figures and 1 tabl

Quantum statistical properties of some new classes of intelligent states associated with special quantum systems

Based on the {\it nonlinear coherent states} method, a general and simple algebraic formalism for the construction of \textit{`$f$-deformed intelligent states'} has been introduced. The structure has the potentiality to apply to systems with a known discrete spectrum as well as the generalized coherent states with known nonlinearity function $f (n)$. As some physical appearance of the proposed formalism, a few new classes of intelligent states associated with \textit{`center of-mass motion of a trapped ion'}, \textit{`harmonious states'} and \textit{`hydrogen-like spectrum'} have been realized. Finally, the nonclassicality of the obtained states has been investigated. To achieve this purpose the quantum statistical properties using the Mandel parameter and the squeezing of the quadratures of the radiation field corresponding to the introduced states have been established numerically.Comment: 13page

Random graph states, maximal flow and Fuss-Catalan distributions

For any graph consisting of $k$ vertices and $m$ edges we construct an ensemble of random pure quantum states which describe a system composed of $2m$ subsystems. Each edge of the graph represents a bi-partite, maximally entangled state. Each vertex represents a random unitary matrix generated according to the Haar measure, which describes the coupling between subsystems. Dividing all subsystems into two parts, one may study entanglement with respect to this partition. A general technique to derive an expression for the average entanglement entropy of random pure states associated to a given graph is presented. Our technique relies on Weingarten calculus and flow problems. We analyze statistical properties of spectra of such random density matrices and show for which cases they are described by the free Poissonian (Marchenko-Pastur) distribution. We derive a discrete family of generalized, Fuss-Catalan distributions and explicitly construct graphs which lead to ensembles of random states characterized by these novel distributions of eigenvalues.Comment: 37 pages, 24 figure

Embracing the polypill as a cardiovascular therapeutic: is this the best strategy?

INTRODUCTION: Cardiovascular disease (CVD) is an important cause of mortality and morbidity worldwide. CVD morbidity and mortality are associated with significant financial costs related to hospitalization, medication, and lost productivity. The concept of the 'polypill' for the reduction of cardiovascular risk was proposed in 2000. A polypill is a fixed combination of drugs in a single tablet or capsule. The initial polypill consisted of three different classes of antihypertensive drugs (each at half dose), in addition to aspirin, a statin, and folic acid. The challenge today is to produce polypills containing drugs with established efficacy and complementary actions. Areas covered: The authors provide their expert perspectives on the polypill and consider the randomized clinical trials that have evaluated the safety, efficacy, adherence, and cost-effectiveness of polypills. Expert opinion: The polypill makes prescribing easier by reducing the need for complex treatment algorithms and dose titration. It also appears to be cost-effective. However, there are several issues that need to be addressed before the polypill can be used routinely. A single polypill formulation may not be suitable for all patients. It may be necessary to develop several types of polypill to meet the needs of different patient groups

Low protein wheat for bread making

Nitrogen (N) is the major mineral that determines crop yield, but it is also an important determinant of grain quality, particularly in wheat. It is required for the synthesis of grain proteins, with gluten forming the major protein fraction in wheat grain. Because of the high protein content required for bread making, the requirement for N applied to bread-making wheats may be above the optimum required for yield, by up to 50 kg N/ha. For example, Dampney et al. (1995) reported that to produce grain containing 13% protein, about 60 kgN/ha above the yield optimum was required. N fertiliser is a major cost for farmers, with a high-energy requirement for manufacture and potentially harmful environmental footprint. Therefore, it is important to reduce the requirement for producing breadmaking wheat, either by improving the efficiency of N use within the plant or by developing new types of wheat that allow the use of lower protein contents for bread making. This project focused on the latter strategy. It aimed to identify and characterise types of wheat with good bread-making quality at low grain protein content. Forty wheat genotypes were grown on 6 sites for 2 years, with a sub-set of 30 grown on the same sites for a third year. All were grown in 3 randomised replicate plots and at 2 levels of N fertilisation: 150 kgN/ha (low) and 250 kgN/ha (conventional). This generated over 4000 grain samples that were analysed for protein content. Samples from 4 sites were bulked for detailed analysis, excluding sites associated with technical problems or unusually high or low contents of protein or responses to fertilisation. Whereas all 40 genotypes were studied in the first year, the number was reduced to 30 in year 2 and to 20 in year 3, based on the analysis of the samples from years 1 and 2, respectively. Campden BRI milled the samples and carried out Extensograph and Farinograph analyses of all flours. The mixing and bread-making performances were subsequently determined by 6 commercial partners, who used three different bread-making processes. SE-HPLC analyses of gluten polymer size distribution was determined on all samples from year 1 and the low N samples from years 2 and 3. This comparison showed that five cultivars (called Group 1) performed well at both high and low N and over all three years: Crusoe and Gallant (current UK nabim Group 1), Rumor and Nelson (German varieties bred to show high quality at low grain protein) and Genius (Danish bread-making cultivar). In addition, two cultivars (called Group 2) performed better when grown at low N than at high N: Skyfall (current UK nabim Group 1 cultivar) and Mv Lucilla (Hungarian high protein breadmaking cultivar). A comparison between these two groups of cultivars and the whole set of cultivars was carried out focusing on four parameters: grain N, grain protein deviation (GPD), gluten protein profiles by SE-HPLC and dough rheology (R/E) measured by Extensograph. This showed that: 1. The selected (Groups 1 and 2) wheats had higher %N, GPD, dough elasticity and proportions of glutenin polymers ((%F1+%F2)/(%F3+%F4)) than the non-selected cultivars. 2. In addition, the Group 2 wheats (which performed better at low N) had higher proportions of high molecular weight glutenin polymers (%F1, (%F3+%F4)/%F1). Although these cultivars include two German lines bred to perform well at low N, they also include three highly successful recent UK cultivars: Crusoe, Gallant and Skyfall. Hence, modern cultivars, which have been selected for performance in high-input systems, may also perform well under low N inputs. We conclude that good bread-making performance at low N fertiliser resulted from two factors: efficient translocation of N into the grain and increased proportions of glutenin in gluten, which resulted in greater dough elasticity. Breeding should, therefore, focus on increasing the efficiency of N use combined with high gluten protein elasticity