A generic formula for the values at the boundary points of monic classical orthogonal polynomials

In a previous paper we have determined a generic formula for the polynomial solution families of the well-known differential equation of hypergeometric type σ(x)y"n(x)+τ(x)y'n(x)-λnyn(x)=0. In this paper, we give another such formula which enables us to present a generic formula for the values of monic classical orthogonal polynomials at their boundary points of definition.

Characterization theorem for classical orthogonal polynomials on non-uniform lattices: The functional approach

Using the functional approach, we state and prove a characterization theorem for classical orthogonal polynomials on non-uniform lattices (quadratic lattices of a discrete or a q-discrete variable) including the Askey-Wilson polynomials. This theorem proves the equivalence between seven characterization properties, namely the Pearson equation for the linear functional, the second-order divided-difference equation, the orthogonality of the derivatives, the Rodrigues formula, two types of structure relations,and the Riccati equation for the formal Stieltjes function.

Attitudes and behaviour towards food attributes and organic food – a triangular methodological investigation of occasional organic consumers

Organic food is increasingly available in the conventional food retail, where organic products are offered alongside with various other types of products and compete mainly with conventional and the so-called conventional-plus products. The latter are conventional products displaying particular quality attributes on the product packaging, such as ‘no artificial additives’, or ‘from animal welfare husbandry’. Often, these quality attributes also apply to organic products. Occasional organic consumers might prefer such conventional-plus alternatives that are perceived to be ‘between’ organic and conventional products. The overall objective of this PhD thesis was to provide information about the segment of occasional organic consumers. In particular, the thesis focussed on consumer perceptions and attitudes towards the quality of, and preferences for, organic, conventional and conventional-plus products in two countries: Germany and Switzerland. To achieve these objectives, qualitative and quantitative consumer research was combined in order to explore occasional organic consumers’ perceptions and attitudes as well as to observe their preferences and buying behaviour regarding different types of food products: organic, conventional and conventional-plus products. The qualitative research showed that, depending on single criteria, organic production was both positively as well as negatively assessed by consumers. Consumer perception of organic food was found to be highly selective and primarily focussed on the final stage of the particular production process. A major problem is that consumers are still mostly unfamiliar with factors associated with organic production, have a lack of confidence, and often confuse organic with conventional products. Besides this, consumer expectations of organic products are different from the expectations of conventional products. The quantitative research revealed that attitudes strongly determine consumers’ preferences for organic, conventional and conventional-plus products. Consumer attitudes tended to differ more between organic and conventional choices rather than conventional-plus and conventional choices. Furthermore, occasional organic consumers are heterogeneous in their preferences. They can be grouped into two segments: the consumers in one segment were less price sensitive and preferred organic products. The consumers in the other segment were more price sensitive and rather preferred conventional-plus or conventional products. To conclude, given the selective and subjective nature of consumer perception and the strong focus of consumer perception on the final stage of the food production process, specific additional values of organic farming should be communicated in clear and catchy messages. At the same time, these messages should be particularly focussed on the final stage of organic food production. The communication of specific added values in relation with organic products to improve the perceived price-performance-ratio is important since conventional-plus products represent an interesting alternative particularly for price sensitive occasional organic consumers. Besides this, it is important to strengthen affirmative consumer attitudes towards organic production. Therefore, policy support should emphasise on long-term communication campaigns and education programmes to increase the consumer awareness and knowledge of organic food and farming. Since consumers expect that organic food is regionally or at least domestically produced while they less accept organic imports, policy support of domestic and regional producers is a crucial measure to fill the current gap between the increasing consumer demand of organic food and the stagnation of the domestic and regional organic food supply.

Moments of classical orthogonal polynomials

The aim of this work is to find simple formulas for the moments mu_n for all families of classical orthogonal polynomials listed in the book by Koekoek, Lesky and Swarttouw. The generating functions or exponential generating functions for those moments are given.

On Connection, Linearization and Duplication Coefficients of Classical Orthogonal Polynomials

In this work, we have mainly achieved the following: 1. we provide a review of the main methods used for the computation of the connection and linearization coefficients between orthogonal polynomials of a continuous variable, moreover using a new approach, the duplication problem of these polynomial families is solved; 2. we review the main methods used for the computation of the connection and linearization coefficients of orthogonal polynomials of a discrete variable, we solve the duplication and linearization problem of all orthogonal polynomials of a discrete variable; 3. we propose a method to generate the connection, linearization and duplication coefficients for q-orthogonal polynomials; 4. we propose a unified method to obtain these coefficients in a generic way for orthogonal polynomials on quadratic and q-quadratic lattices. Our algorithmic approach to compute linearization, connection and duplication coefficients is based on the one used by Koepf and Schmersau and on the NaViMa algorithm. Our main technique is to use explicit formulas for structural identities of classical orthogonal polynomial systems. We find our results by an application of computer algebra. The major algorithmic tools for our development are Zeilberger’s algorithm, q-Zeilberger’s algorithm, the Petkovšek-van-Hoeij algorithm, the q-Petkovšek-van-Hoeij algorithm, and Algorithm 2.2, p. 20 of Koepf's book "Hypergeometric Summation" and it q-analogue.

Controlling the transport of an ion: classical and quantum mechanical solutions

The accurate transport of an ion over macroscopic distances represents a challenging control problem due to the different length and time scales that enter and the experimental limitations on the controls that need to be accounted for. Here, we investigate the performance of different control techniques for ion transport in state-of-the-art segmented miniaturized ion traps. We employ numerical optimization of classical trajectories and quantum wavepacket propagation as well as analytical solutions derived from invariant based inverse engineering and geometric optimal control. The applicability of each of the control methods depends on the length and time scales of the transport. Our comprehensive set of tools allows us make a number of observations. We find that accurate shuttling can be performed with operation times below the trap oscillation period. The maximum speed is limited by the maximum acceleration that can be exerted on the ion. When using controls obtained from classical dynamics for wavepacket propagation, wavepacket squeezing is the only quantum effect that comes into play for a large range of trapping parameters. We show that this can be corrected by a compensating force derived from invariant based inverse engineering, without a significant increase in the operation time.