Applied mathematical problem solving, modelling, applications, and links to other subjects

The paper will consist of three parts. In part I we shall present some background considerations which are necessary as a basis for what follows. We shall try to clarify some basic concepts and notions, and we shall collect the most important arguments (and related goals) in favour of problem solving, modelling and applications to other subjects in mathematics instruction. In the main part II we shall review the present state, recent trends, and prospective lines of development, both in empirical or theoretical research and in the practice of mathematics instruction and mathematics education, concerning (applied) problem solving, modelling, applications and relations to other subjects. In particular, we shall identify and discuss four major trends: a widened spectrum of arguments, an increased globality, an increased unification, and an extended use of computers. In the final part III we shall comment upon some important issues and problems related to our topic.

Cleanliness versus cow comfort – an insolvable problem?

Cubicle should provide good resting comfort as well as clean udders. Dairy cows in cubicle houses often face a restrictive environment with regard to resting behaviour, whereas cleanliness may still be impaired. This study aimed to determine reliable behavioural measures regarding resting comfort applicable in on-farm welfare assessments. Furthermore, relationships between cubicle design, cow sizes, management factors and udder cleanliness (namely teats and teat tips) were investigated. Altogether 15 resting measures were examined in terms of feasibility, inter-observer reliability (IOR) and consistency of results per farm over time. They were recorded during three farm visits on farms in Germany and Austria with cubicle, deep litter and tie stall systems. Seven measures occurred to infrequently to allow reliable recording within a limited observation time. IOR was generally acceptable to excellent except for 'collisions during lying down', which only showed good IOR after improvement of the definition. Only three measures were acceptably repeatable over time: 'duration of lying down', 'percentage of collisions during lying down' and 'percentage of cows lying partly or completely outside lying area'. These measures were evaluated as suitable animal based welfare measures regarding resting behaviour in the framework of an on-farm welfare assessment protocol. The second part of the thesis comprises a cross-sectional study on resting comfort and cow cleanliness including 23 Holstein Friesian dairy herds with very low within-farm variation in cubicle measures. Height at withers, shoulder width and diagonal body length were measured in 79-100 % of the cows (herd size 30 to115 cows). Based on the 25 % largest animals, compliance with recommendations for cubicle measures was calculated. Cleanliness of different body parts, the udder, teats and teat tips was assessed for each cow in the herd prior to morning milking. No significant correlation was found between udder soiling and teat or teat tip soiling on herd level. The final model of a stepwise regression regarding the percentage of dirty teats per farm explained 58.5 % the variance and contained four factors. Teat dipping after milking which might be associated with an overall clean and accurate management style, deep bedded cubicles, increasing cubicle maintenance times and decreasing compliance concerning total cubicle length predicted lower teat soiling. The final model concerning teat tip soiling explained 46.0 % of the variance and contained three factors. Increasing litter height in the rear part of the cubicle and increased alley soiling which is difficult to explain, predicted for less soiled teat tips, whereas increasing compliance concerning resting length was associated with higher percentages of dirty teat tips. The dependent variable ‘duration of lying down’ was analysed using again stepwise regression. The final model explained 54.8 % of the total variance. Lying down duration was significantly shorter in deep bedded cubicles. Further explanatory though not significant factors in the model were neck-rail height, deep bedding or comfort mattresses versus concrete floor or rubber mats and clearance height of side partitions. In the attempt to create a more comprehensive lying down measure, another analysis was carried out with percentage of ‘impaired lying down’ (i.e. events exceeding 6.3 seconds, with collisions or being interrupted) as dependent variable. The explanatory value of this final model was 41.3 %. An increase in partition length, in compliance concerning cubicle width and the presence of straw within bedding predicted a lower proportion of impaired lying down. The effect of partition length is difficult to interpret, but partition length and height were positively correlated on the study farms, possibly leading to a bigger zone of clear space for pelvis freedom. No associations could be found between impaired lying down and teat or teat tip soiling. Altogether, in agreement with earlier studies it was found that cubicle dimensions in practice are often inadequate with regard to the body dimensions of the cows, leading to high proportions of impaired lying down behaviour, whereas teat cleanliness is still unsatisfactory. Connections between cleanliness and cow comfort are far from simplistic. Especially the relationship between cubicle characteristics and lying down behaviour apparently is very complex, so that it is difficult to identify single influential factors that are valid for all farm situations. However, based on the results of the present study the use of deep bedded cubicles can be recommended as well as improved management with special regard to cubicle and litter maintenance in order to achieve both better resting comfort and teat cleanliness.

Identification of Problem-based learning components and their effects on graduates' competencies

This study investigated the relationship between higher education and the requirement of the world of work with an emphasis on the effect of problem-based learning (PBL) on graduates' competencies. The implementation of full PBL method is costly (Albanese & Mitchell, 1993; Berkson, 1993; Finucane, Shannon, & McGrath, 2009). However, the implementation of PBL in a less than curriculum-wide mode is more achievable in a broader context (Albanese, 2000). This means higher education institutions implement only a few PBL components in the curriculum. Or a teacher implements a few PBL components at the courses level. For this kind of implementation there is a need to identify PBL components and their effects on particular educational outputs (Hmelo-Silver, 2004; Newman, 2003). So far, however there has been little research about this topic. The main aims of this study were: (1) to identify each of PBL components which were manifested in the development of a valid and reliable PBL implementation questionnaire and (2) to determine the effect of each identified PBL component to specific graduates' competencies. The analysis was based on quantitative data collected in the survey of medicine graduates of Gadjah Mada University, Indonesia. A total of 225 graduates responded to the survey. The result of confirmatory factor analysis (CFA) showed that all individual constructs of PBL and graduates' competencies had acceptable GOFs (Goodness-of-fit). Additionally, the values of the factor loadings (standardize loading estimates), the AVEs (average variance extracted), CRs (construct reliability), and ASVs (average shared squared variance) showed the proof of convergent and discriminant validity. All values indicated valid and reliable measurements. The investigation of the effects of PBL showed that each PBL component had specific effects on graduates' competencies. Interpersonal competencies were affected by Student-centred learning (β = .137; p < .05) and Small group components (β = .078; p < .05). Problem as stimulus affected Leadership (β = .182; p < .01). Real-world problems affected Personal and organisational competencies (β = .140; p < .01) and Interpersonal competencies (β = .114; p < .05). Teacher as facilitator affected Leadership (β = 142; p < .05). Self-directed learning affected Field-related competencies (β = .080; p < .05). These results can help higher education institution and educator to have informed choice about the implementation of PBL components. With this information higher education institutions and educators could fulfil their educational goals and in the same time meet their limited resources. This study seeks to improve prior studies' research method in four major ways: (1) by indentifying PBL components based on theory and empirical data; (2) by using latent variables in the structural equation modelling instead of using a variable as a proxy of a construct; (3) by using CFA to validate the latent structure of the measurement, thus providing better evidence of validity; and (4) by using graduate survey data which is suitable for analysing PBL effects in the frame work of the relationship between higher education and the world of work.

The Computation of Color

This thesis takes an interdisciplinary approach to the study of color vision, focussing on the phenomenon of color constancy formulated as a computational problem. The primary contributions of the thesis are (1) the demonstration of a formal framework for lightness algorithms; (2) the derivation of a new lightness algorithm based on regularization theory; (3) the synthesis of an adaptive lightness algorithm using "learning" techniques; (4) the development of an image segmentation algorithm that uses luminance and color information to mark material boundaries; and (5) an experimental investigation into the cues that human observers use to judge the color of the illuminant. Other computational approaches to color are reviewed and some of their links to psychophysics and physiology are explored.

A Vector Signal Processing Approach to Color

Surface (Lambertain) color is a useful visual cue for analyzing material composition of scenes. This thesis adopts a signal processing approach to color vision. It represents color images as fields of 3D vectors, from which we extract region and boundary information. The first problem we face is one of secondary imaging effects that makes image color different from surface color. We demonstrate a simple but effective polarization based technique that corrects for these effects. We then propose a systematic approach of scalarizing color, that allows us to augment classical image processing tools and concepts for multi-dimensional color signals.

Genomic architecture of adaptive color pattern divergence and convergence in Heliconius butterflies

Identifying the genetic changes driving adaptive variation in natural populations is key to understanding the origins of biodiversity. The mosaic of mimetic wing patterns in Heliconius butterflies makes an excellent system for exploring adaptive variation using next-generation sequencing. In this study, we use a combination of techniques to annotate the genomic interval modulating red color pattern variation, identify a narrow region responsible for adaptive divergence and convergence in Heliconius wing color patterns, and explore the evolutionary history of these adaptive alleles. We use whole genome resequencing from four hybrid zones between divergent color pattern races of Heliconius erato and two hybrid zones of the co-mimic Heliconius melpomene to examine genetic variation across 2.2 Mb of a partial reference sequence. In the intergenic region near optix, the gene previously shown to be responsible for the complex red pattern variation in Heliconius, population genetic analyses identify a shared 65-kb region of divergence that includes several sites perfectly associated with phenotype within each species. This region likely contains multiple cis-regulatory elements that control discrete expression domains of optix. The parallel signatures of genetic differentiation in H. erato and H. melpomene support a shared genetic architecture between the two distantly related co-mimics; however, phylogenetic analysis suggests mimetic patterns in each species evolved independently. Using a combination of next-generation sequencing analyses, we have refined our understanding of the genetic architecture of wing pattern variation in Heliconius and gained important insights into the evolution of novel adaptive phenotypes in natural populations.

Are distortion levels a problem with digital signal processing hearing aids and frequency modulation (FM) systems?

The purpose of this study was to examine objective and subjective distortion present when frequency modulation (FM) systems were coupled with four digital signal processing (DSP) hearing aids. Electroacoustic analysis and subjective listening tests by experienced audiologists revealed that distortion levels varied across hearing aids and channels.

Weak constraints in four-dimensional variational data assimilation

The formulation of four-dimensional variational data assimilation allows the incorporation of constraints into the cost function which need only be weakly satisfied. In this paper we investigate the value of imposing conservation properties as weak constraints. Using the example of the two-body problem of celestial mechanics we compare weak constraints based on conservation laws with a constraint on the background state.We show how the imposition of conservation-based weak constraints changes the nature of the gradient equation. Assimilation experiments demonstrate how this can add extra information to the assimilation process, even when the underlying numerical model is conserving.

The role of schizotypy and creativity in a group problem-solving task

The well-studied link between psychotic traits and creativity is a subject of much debate. The present study investigated the extent to which schizotypic personality traits - as measured by O-LIFE (Oxford-Liverpool Inventory of Feelings and Experiences) - equip healthy individuals to engage as groups in everyday tasks. From a sample of 69 students, eight groups of four participants - comprised of high, medium, or low-schizotypy individuals - were assembled to work as a team to complete a creative problem-solving task. Predictably, high scorers on the O-LIFE formulated a greater number of strategies to solve the task, indicative of creative divergent thinking. However, for task success (as measured by time taken to complete the problem) an inverted U shaped pattern emerged, whereby high and low-schizotypy groups were consistently faster than medium schizotypy groups. Intriguing data emerged concerning leadership within the groups, and other tangential findings relating to anxiety, competition and motivation were explored. These findings challenge the traditional cliche that psychotic personality traits are linearly related to creative performance, and suggest that the nature of the problem determines which thinking styles are optimally equipped to solve it. (C) 2009 Elsevier Ltd. All rights reserved.

Four-dimensional data assimilation

Numerical weather prediction can be regarded as an initial value problem whereby the governing atmospheric equations are integrated forward from fully determined initial values of the meteorological parameters. However, in spite of the considerable improvements of the observing systems in recent years, the initial values are known only incompletely and inaccurately and one of the major tasks of any forecasting centre is to determine the best possible initial state from available observations.

Problems in four-dimensional data assimilation

With the introduction of new observing systems based on asynoptic observations, the analysis problem has changed in character. In the near future we may expect that a considerable part of meteorological observations will be unevenly distributed in four dimensions, i.e. three dimensions in space and one in time. The term analysis, or objective analysis in meteorology, means the process of interpolating observed meteorological observations from unevenly distributed locations to a network of regularly spaced grid points. Necessitated by the requirement of numerical weather prediction models to solve the governing finite difference equations on such a grid lattice, the objective analysis is a three-dimensional (or mostly two-dimensional) interpolation technique. As a consequence of the structure of the conventional synoptic network with separated data-sparse and data-dense areas, four-dimensional analysis has in fact been intensively used for many years. Weather services have thus based their analysis not only on synoptic data at the time of the analysis and climatology, but also on the fields predicted from the previous observation hour and valid at the time of the analysis. The inclusion of the time dimension in objective analysis will be called four-dimensional data assimilation. From one point of view it seems possible to apply the conventional technique on the new data sources by simply reducing the time interval in the analysis-forecasting cycle. This could in fact be justified also for the conventional observations. We have a fairly good coverage of surface observations 8 times a day and several upper air stations are making radiosonde and radiowind observations 4 times a day. If we have a 3-hour step in the analysis-forecasting cycle instead of 12 hours, which is applied most often, we may without any difficulties treat all observations as synoptic. No observation would thus be more than 90 minutes off time and the observations even during strong transient motion would fall within a horizontal mesh of 500 km * 500 km.

Four-dimensional data assimilation and balanced dynamics

The problem of spurious excitation of gravity waves in the context of four-dimensional data assimilation is investigated using a simple model of balanced dynamics. The model admits a chaotic vortical mode coupled to a comparatively fast gravity wave mode, and can be initialized such that the model evolves on a so-called slow manifold, where the fast motion is suppressed. Identical twin assimilation experiments are performed, comparing the extended and ensemble Kalman filters (EKF and EnKF, respectively). The EKF uses a tangent linear model (TLM) to estimate the evolution of forecast error statistics in time, whereas the EnKF uses the statistics of an ensemble of nonlinear model integrations. Specifically, the case is examined where the true state is balanced, but observation errors project onto all degrees of freedom, including the fast modes. It is shown that the EKF and EnKF will assimilate observations in a balanced way only if certain assumptions hold, and that, outside of ideal cases (i.e., with very frequent observations), dynamical balance can easily be lost in the assimilation. For the EKF, the repeated adjustment of the covariances by the assimilation of observations can easily unbalance the TLM, and destroy the assumptions on which balanced assimilation rests. It is shown that an important factor is the choice of initial forecast error covariance matrix. A balance-constrained EKF is described and compared to the standard EKF, and shown to offer significant improvement for observation frequencies where balance in the standard EKF is lost. The EnKF is advantageous in that balance in the error covariances relies only on a balanced forecast ensemble, and that the analysis step is an ensemble-mean operation. Numerical experiments show that the EnKF may be preferable to the EKF in terms of balance, though its validity is limited by ensemble size. It is also found that overobserving can lead to a more unbalanced forecast ensemble and thus to an unbalanced analysis.

Behavioural responses of the seven-spot ladybird Coccinella septempunctata to plant headspace chemicals collected from four crop Brassicas and Arabidopsis thaliana, infested with Myzus persicae

1 Insects using olfactory stimuli to forage for prey/hosts are proposed to encounter a ‘reliability–detectability problem’, where the usability of a stimulus depends on its reliability as an indicator of herbivore presence and its detectability. 2 We investigated this theory using the responses of female seven-spot ladybirds Coccinella septempunctata (Coleoptera: Coccinellidae) to plant headspace chemicals collected from the peach-potato aphid Myzus persicae and four commercially available Brassica cultivars; Brassica rapa L. cultivar ‘turnip purple top’, Brassica juncea L. cultivar ‘red giant mustard’, Brassica napus L. cultivar ‘Apex’, Brassica napus L. cultivar ‘Courage’ and Arabidopsis thaliana. For each cultivar/species, responses to plants that were undamaged, previously infested by M. persicae and infested with M. persicae, were investigated using dual-choice Petri dish bioassays and circular arenas. 3 There was no evidence that ladybirds responded to headspace chemicals from aphids alone. Ladybirds significantly preferred headspace chemicals from B. napus cv. Apex that were undamaged compared with those from plants infested with aphids. For the other four species/cultivars, there was a consistent trend of the predators being recorded more often in the half of the Petri dish containing plant headspace chemicals from previously damaged and infested plants compared with those from undamaged ones. Furthermore, the mean distance ladybirds walked to reach aphid-infested A. thaliana was significantly shorter than to reach undamaged plants. These results suggest that aphid-induced plant chemicals could act as an arrestment or possibly an attractant stimulus to C. septempunctata. However, it is also possible that C. septempunctata could have been responding to aphid products, such as honeydew, transferred to the previously damaged and infested plants. 4 The results provide evidence to support the ‘reliability–detectability’ theory and suggest that the effectiveness of C. septempunctata as a natural enemy of aphids may be strongly affected by which species and cultivar of Brassica are being grown.

Application of complex extreme learning machine to multiclass classification problems with high dimensionality: a THz spectra classification problem

We extend extreme learning machine (ELM) classifiers to complex Reproducing Kernel Hilbert Spaces (RKHS) where the input/output variables as well as the optimization variables are complex-valued. A new family of classifiers, called complex-valued ELM (CELM) suitable for complex-valued multiple-input–multiple-output processing is introduced. In the proposed method, the associated Lagrangian is computed using induced RKHS kernels, adopting a Wirtinger calculus approach formulated as a constrained optimization problem similarly to the conventional ELM classifier formulation. When training the CELM, the Karush–Khun–Tuker (KKT) theorem is used to solve the dual optimization problem that consists of satisfying simultaneously smallest training error as well as smallest norm of output weights criteria. The proposed formulation also addresses aspects of quaternary classification within a Clifford algebra context. For 2D complex-valued inputs, user-defined complex-coupled hyper-planes divide the classifier input space into four partitions. For 3D complex-valued inputs, the formulation generates three pairs of complex-coupled hyper-planes through orthogonal projections. The six hyper-planes then divide the 3D space into eight partitions. It is shown that the CELM problem formulation is equivalent to solving six real-valued ELM tasks, which are induced by projecting the chosen complex kernel across the different user-defined coordinate planes. A classification example of powdered samples on the basis of their terahertz spectral signatures is used to demonstrate the advantages of the CELM classifiers compared to their SVM counterparts. The proposed classifiers retain the advantages of their ELM counterparts, in that they can perform multiclass classification with lower computational complexity than SVM classifiers. Furthermore, because of their ability to perform classification tasks fast, the proposed formulations are of interest to real-time applications.

Intersections of Jane Jacobs’ conditions for diversity and low-carbon urban systems: a look at four global cities

Countless cities are rapidly developing across the globe, pressing the need for clear urban planning and design recommendations geared towards sustainability. This article examines the intersections of Jane Jacobs’ four conditions for diversity with low-carbon and low-energy use urban systems in four cities around the world: Lyon (France), Chicago (United-States), Kolkata (India), and Singapore (Singapore). After reviewing Jacobs’ four conditions for diversity, we introduce the four cities and describe their historical development context. We then present a framework to study the cities along three dimensions: population and density, infrastructure development/use, and climate and landscape. These cities differ in many respects and their analysis is instructive for many other cities around the globe. Jacobs’ conditions are present in all of them, manifested in different ways and to varying degrees. Overall we find that the adoption of Jacobs' conditions seems to align well with concepts of low-carbon urban systems, with their focus on walkability, transit-oriented design, and more efficient land use (i.e., smaller unit sizes). Transportation sector emissions seems to demonstrate a stronger influence from the presence of Jacobs' conditions, while the link was less pronounced in the building sector. Kolkata, a low-income, developing world city, seems to possess many of Jacobs' conditions, while exhibiting low per capita emissions - maintaining both of these during its economic expansion will take careful consideration. Greenhouse gas mitigation, however, is inherently an in situ problem and the first task must therefore be to gain local knowledge of an area before developing strategies to lower its carbon footprint.