On the maximum size of minimal definitive quartet sets

In this paper, we investigate a problem concerning quartets; a quartet is a particular kind of tree on four leaves. Loosely speaking, a set of quartets is said to be ‘definitive’ if it completely encapsulates the structure of some larger tree, and ‘minimal’ if it contains no redundant information. Here, we address the question of how large a minimal definitive quartet set on n leaves can be, showing that the maximum size is at least 2n−8 for all n≥4. This is an enjoyable problem to work on, and we present a pretty construction, which employs symmetry.     

Asymptotic properties of random matrices and pseudomatrices

We study the asymptotics of sums of matricially free random variables, called random pseudomatrices, and we compare it with that of random matrices with block-identical variances. For objects of both types we find the limit joint distributions of blocks and give their Hilbert space realizations, using operators called ‘matricially free Gaussian operators’. In particular, if the variance matrices are symmetric, the asymptotics of symmetric blocks of random pseudomatrices agrees with that of symmetric random blocks. We also show that blocks of random pseudomatrices are ‘asymptotically matricially free’ whereas the corresponding symmetric random blocks are ‘asymptotically symmetrically matricially free’, where symmetric matricial freeness is obtained from matricial freeness by an operation of symmetrization. Finally, we show that row blocks of square, block-lower-triangular and block-diagonal pseudomatrices are asymptotically free, monotone independent and boolean independent, respectively.     

Behaviour approximated on subgroups

The recovery of behaviour from its approximation over substructures is fraught with pathology. Here the extent is considered to which the behaviour of a continuous function on a locally compact Abelian group can be approximated by its behaviour on proper closed subgroups. Known results are summarised when the behaviour concerns integrability and the group is the circle; then boundedness and other limiting behaviour ‘at infinity’ are considered for more general groups. It is shown that if a continuous function is bounded on each proper closed subgroup of a connected locally compact Abelian group then it is bounded on the whole group. As befits this Festschrift, the techniques are predominantly topological. In passing we reflect on criteria for the difficult problem of identifying ‘substructures’ in Computer Science.     

Metaphor or Met-Before? The effects of previouos experience on practice and theory of learning mathematics

While the general notion of ‘metaphor’ may offer a thoughtful analysis of the nature of mathematical thinking, this paper suggests that it is even more important to take into account the particular mental structures available to the individual that have been built from experience that the individual has ‘met-before.’ The notion of ‘met-before’ offers not only a principle to analyse the changing meanings in mathematics and the difficulties faced by the learner—which we illustrate by the problematic case of the minus sign—it can also be used to analyse the met-befores of mathematicians, mathematics educators and those who develop theories of learning to reveal implicit assumptions that support our thinking in some ways and act as impediments in others.     

Differential evolution with generalized differentials

In this paper, we study the mutation operation of the differential evolution (DE) algorithm. In particular, we propose the differential of scaled vectors, called the ‘generalized differential’, as opposed to the existing scaled differential vector in the mutation of DE. We derive the probability distribution of points generated by the mutation with ‘generalized differentials’. We incorporate a vector-projection-based exploratory method within the new mutation scheme. The vector projection is not mandatory and it is only invoked if trial points continue to be unsuccessful. An algorithm is then proposed which implements the mutation strategy based on the difference of the scaled vectors as well as the vector projection technique. A numerical study is carried out using a set of 50 test problems, many of which are inspired by practical applications. Numerical results suggest that the new algorithm is superior to DE.     

Transitive digraphs with more than one end

In this paper we investigate infinite, locally finite, connected, transitive digraphs with more than one end. For undirected graphs with these properties it has been shown that they are trees as soon as they are 2-arc transitive. In the case of digraphs the situation is much more involved. We show that these graphs can have both thick and thin ends, even if they are highly arc transitive. Hence they are far away from being ‘tree-like’. On the other hand all known examples of digraphs with more than one end are either highly arc transitive or at most 1-arc transitive. We conjecture that infinite, locally finite, connected, 2-arc transitive digraphs with more than one end are highly arc transitive and prove that this conjecture holds for digraphs with prime in- and out-degree and connected cuts.     

Morita equivalence of semigroups with local units

We prove that two semigroups with local units are Morita equivalent if and only if they have a joint enlargement. This approach to Morita theory provides a natural framework for understanding McAlister’s theory of the local structure of regular semigroups. In particular, we prove that a semigroup with local units is Morita equivalent to an inverse semigroup precisely when it is a regular locally inverse semigroup.     

Diagnostic assessment of children’s proportional reasoning

A proportional reasoning item bank was created from the relevant literature and tested in various forms. Rasch analyses of 303 pupils’ test results were used to calibrate the bank, and data from 84 pupils’ interviews was used to confirm our diagnostic interpretations. A number of sub-tests were scaled, including parallel ‘without models’ and ‘with models’ forms. We provide details of the 13-item ‘without models’ test which was formed from the ‘richest’ diagnostic items and verified on a further test sample (N=212, ages 10-13). Two scales were constructed for this test, one that measures children’s ‘ratio attainment’ and one that measures their ‘tendency for additive strategy.’ Other significant errors — ‘incorrect build-up,’ ‘magical doubling/halving,’ ‘constant sum’ and ‘incomplete reasoning’ — were identified. Finally, an empirical hierarchy of pupils’ attainment of proportional reasoning was formed, incorporating the significant errors and the additive scale.     

On the spectrums of frame multiresolution analyses

We first give conditions for a univariate square integrable function to be a scaling function of a frame multiresolution analysis (FMRA) by generalizing the corresponding conditions for a scaling function of a multiresolution analysis (MRA). We also characterize the spectrum of the ‘central space’ of an FMRA, and then give a new condition for an FMRA to admit a single frame wavelet solely in terms of the spectrum of the central space of an FMRA. This improves the results previously obtained by Benedetto and Treiber and by some of the authors. Our methods and results are applied to the problem of the ‘containments’ of FMRAs in MRAs. We first prove that an FMRA is always contained in an MRA, and then we characterize those MRAs that contain ‘genuine’ FMRAs in terms of the unique low-pass filters of the MRAs and the spectrums of the central spaces of the FMRAs to be contained. This characterization shows, in particular, that if the low-pass filter of an MRA is almost everywhere zero-free, as is the case of the MRAs of Daubechies, then the MRA contains no FMRAs other than itself.     

First vs. best improvement: An empirical study

When applying the 2-opt heuristic to the travelling salesman problem, selecting the best improvement at each iteration gives worse results on average than selecting the first improvement, if the initial solution is chosen at random. However, starting with ‘greedy’ or ‘nearest neighbor’ constructive heuristics, the best improvement is better and faster on average. Reasons for this behavior are investigated. It appears to be better to use exchanges introducing into the solution a very small edge and fairly large one, which can easily be removed later, than two small ones which are much harder to remove.     

Global attractivity and permanence of a delayed SVEIR epidemic model with pulse vaccination and saturation incidence

In this study, we formulate and analyze a new SVEIR epidemic disease model with time delay and saturation incidence, and analyze the dynamic behavior of the model under pulse vaccination. Using the discrete dynamical system determined by the stroboscopic map, we obtain an ‘infection-free’ periodic solution, further, show that the ‘infection-free’ periodic solution is globally attractive for some parameters of the model under appropriate conditions. The permanence of the model is investigated analytically. By computer simulation it is concluded that a large vaccination rate or a short pulse of vaccination or a long latent period are each a sufficient condition for the extinction of the disease.     

Supplier switching decisions

In order to maximize profit, a buying firm should continuously search for and access sources which offer more favorable prices. While the literature is replete with works on the formation and development of buyer–supplier relationships, there is surprisingly only scarce research on the termination of such relationships and supplier switching. Using the concept of switching costs in a principal-agent framework, we at first analyze whether a firm switches the entire or a partial quantity to an alternative supplier when there is either symmetric or asymmetric information about the alternative supplier’s cost structure. Information asymmetry results in inert supplier switching decisions. Subsequently, we extend our model and take competitive reactions of the incumbent supplier and economies of scale effects into consideration. We find conditions under which ‘no’, ‘partial’ and ‘complete’ switching occurs, which depend on the buying firm’s beliefs about the alternative supplier’s unit costs, switching costs, the price offered by the incumbent supplier, and refinements of the price offered by the incumbent supplier due to competitive reactions and economies of scale. Broader implications for supplier relationship management and sourcing strategy decisions are also provided.     

Retrieving seasonally adjusted quarterly growth rates from annual growth rates that are reported quarterly

Most macroeconomic time series are reported on a quarterly basis. However, in many cases, official statistical agencies report annual (or ‘year-on-year’) growth rates rather than the quarterly data themselves. In this paper, we demonstrate how to recover seasonally adjusted quarterly growth rates from annual growth rates that are reported quarterly. As an illustration, we apply the technique to the highly seasonal UK Retail Sales series. Using only the annual growth rates calculated from the underlying data, we find that our procedure generates estimated seasonally adjusted quarterly growth rates that are almost indistinguishable from the ‘true’ seasonally adjusted quarterly growth rates.     

“I’m not very good at solving problems”: An exploration of students’ problem solving behaviours

This paper reports one aspect of a larger study which looked at the strategies used by a selection of grade 6 students to solve six non-routine mathematical problems. The data revealed that the students exhibited many of the behaviours identified in the literature as being associated with novice and expert problem solvers. However, the categories of ‘novice’ and ‘expert’ were not fully adequate to describe the range of behaviours observed and instead three categories that were characteristic of behaviours associated with ‘naïve’, ‘routine’ and ‘sophisticated’ approaches to solving problems were identified. Furthermore, examination of individual cases revealed that each student's problem solving performance was consistent across a range of problems, indicating a particular orientation towards naïve, routine or sophisticated problem solving behaviours. This paper describes common problem solving behaviours and details three individual cases involving naïve, routine and sophisticated problem solvers.     

Robert Leslie Ellisʼs work on philosophy of science and the foundations of probability theory

The goal of this paper is to provide an extensive account of Robert Leslie Ellis?s largely forgotten work on philosophy of science and probability theory. On the one hand, it is suggested that both his ‘idealist’ renovation of the Baconian theory of induction and a ‘realism’ vis-à-vis natural kinds were the result of a complex dialogue with the work of William Whewell. On the other hand, it is shown to what extent the combining of these two positions contributed to Ellis?s reformulation of the metaphysical foundations of traditional probability theory. This parallel is assessed with reference to the disagreement between Ellis and Whewell on the nature of (pure) mathematics and its relation to scientific knowledge.     

Roll cutting in the curtain industry,or: A well-solvable allocation problem

We study the problem of cutting a number of pieces of the same length from n rolls of different lengths so that the remaining part of each utilized roll is either sufficiently short or sufficiently long. A piece is ‘sufficiently short’, if it is shorter than a pre-specified threshold value δ_min, so that it can be thrown away as it cannot be used again for cutting future orders. And a piece is ‘sufficiently long’, if it is longer than a pre-specified threshold value δ_max (with δ_max > δ_min), so that it can reasonably be expected to be usable for cutting future orders of almost any length. We show that this problem, faced by a curtaining wholesaler, is solvable in O(nlogn) time by analyzing a non-trivial class of allocation problems.     

Characterizing the Single-Valuedness of Multifunctions

We characterize the local single-valuedness and continuity of multifunctions (set-valued mappings) in terms of their premonotonicity and lower semicontinuity. This result completes the well-known fact that lower semicontinuous, monotone multifunctions are single-valued and continuous. We also show that a multifunction is actually a Lipschitz single-valued mapping if and only if it is premonotone and has a generalized Lipschitz property called Aubin continuity. The possible single-valuedness and continuity of multifunctions is at the heart of some of the most fundamental issues in variational analysis and its application to optimization. We investigate the impact of our characterizations on several of these issues; discovering exactly when certain generalized subderivatives can be identified with classical derivatives, and determining precisely when solutions to generalized variational inequalities are locally unique and Lipschitz continuous. As an application of our results involving generalized variational inequalities, we characterize when the Karush–Kuhn–Tucker pairs associated with a parameterized optimization problem are locally unique and Lipschitz continuous.     

Predicting DAX trends from Dow Jones data by methods of the mathematical theory of democracy

The mathematical theory of democracy operates on the indices of popularity and universality which are used to find socially optimal representatives and representative bodies. Regarded mathematically, neither the ‘society’, nor its ‘representatives’ are necessarily human, so that some objects can represent the behavior of other objects. This idea is applied to predicting the DAX-trends (German stock index) from the actual Dow-Jones data; the current fluctuations of stock prices in New York are regarded as representative indicators of future stock price fluctuations in Frankfurt. In particular, it is found that American Express anticipates on the average the price ±fluctuations of 2/3 of the DAX stocks. The statistical significance of the null hypothesis that such a bias from the uncertainty 50% can occur by chance is only 3.5%.     

Coupling functions for domino tilings of Aztec diamonds

The inverse Kasteleyn matrix of a bipartite graph holds much information about the perfect matchings of the system such as local statistics which can be used to compute local and global asymptotics. In this paper, we consider three different weightings of domino tilings of the Aztec diamond and show using recurrence relations, that we can compute the inverse Kasteleyn matrix. These weights are the one-periodic weighting where the horizontal edges have one weight and the vertical edges have another weight, the q^vol$q^{\mathrm{vol}}$ weighting which corresponds to multiplying the product of tile weights by q if we add a ‘box’ to the height function and the two-periodic weighting which exhibits a flat region with defects in the center.     

A group based solution strategy for multi-physics simulations in parallel

Multi-physics simulation often requires the solution of a suite of interacting physical phenomena, the nature of which may vary both spatially and in time. For example, in a casting simulation there is thermo-mechanical behaviour in the structural mould, whilst in the cast, as the metal cools and solidifies, the buoyancy induced flow ceases and stresses begin to develop. When using a single code to simulate such problems it is conventional to solve each ‘physics’ component over the whole single mesh, using definitions of material properties or source terms to ensure that a solved variable remains zero in the region in which the associated physical phenomenon is not active. Although this method is secure, in that it enables any and all the ‘active’ physics to be captured across the whole domain, it is computationally inefficient in both scalar and parallel. An alternative, known as the ‘group’ solver approach, involves more formal domain decomposition whereby specific combinations of physics are solved for on prescribed sub-domains. The ‘group’ solution method has been implemented in a three-dimensional finite volume, unstructured mesh multi-physics code, which is parallelised, employing a multi-phase mesh partitioning capability which attempts to optimise the load balance across the target parallel HPC system. The potential benefits of the ‘group’ solution strategy are evaluated on a class of multi-physics problems involving thermo-fluid–structural interaction on both a single and multi-processor systems. In summary, the ‘group’ solver is a third faster on a single processor than the single domain strategy and preserves its scalability on a parallel cluster system.