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 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.

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.

Development of mechanical methods for cell-tray propagation and field transplanting of dwarf napiergrass (Pennisetum purpureum Schumach.)

Since dwarf napiergrass (Pennisetum purpureum Schumach.) must be propagated vegetatively due to lack of viable seeds, root splitting and stem cuttings are generally used to obtain true-to-type plant populations. These ordinary methods are laborious and costly, and are the greatest barriers for expanding the cultivation area of this crop. The objectives of this research were to develop nursery production of dwarf napiergrass in cell trays and to compare the efficiency of mechanical versus manual methods for cell-tray propagation and field transplanting. After defoliation of herbage either by a sickle (manually) or hand-mowing machine, every potential aerial tiller bud was cut to a single one for transplanting into cell trays as stem cuttings and placed in a glasshouse over winter. The following June, nursery plants were trimmed to a 25–cm length and transplanted in an experimental field (sandy soil) with 20,000 plants ha^(−1) either by shovel (manually) or Welsh onion planter. Labour time was recorded for each process. The manual defoliation of plants required 44% more labour time for preparing the stem cuttings (0.73 person-min. stemcutting^(−1)) compared to using hand-mowing machinery (0.51 person-min. stem-cutting^(−1)). In contrast, labour time for transplanting required an extra 0.30 person-min. m^(−2) (14%) using the machinery compared to manual transplanting, possibly due to the limited plot size for machinery operation. The transplanting method had no significant effect on plant establishment or plant growth, except for herbage yield 110 days after planting. Defoliation of herbage by machinery, production using a cell-tray nursery and mechanical transplanting reduced the labour intensity of dwarf napiergrass propagation.

Measuring and testing for the systemically important financial institutions

This paper analyzes the measure of systemic importance ∆CoV aR proposed by Adrian and Brunnermeier (2009, 2010) within the context of a similar class of risk measures used in the risk management literature. In addition, we develop a series of testing procedures, based on ∆CoV aR, to identify and rank the systemically important institutions. We stress the importance of statistical testing in interpreting the measure of systemic importance. An empirical application illustrates the testing procedures, using equity data for three European banks.

Modelling stereoscopic vision systems for robotic applications

Aquesta tesi s'emmarca dins del projecte CICYT TAP 1999-0443-C05-01. L'objectiu d'aquest projecte és el disseny, implementació i avaluació de robots mòbils, amb un sistema de control distribuït, sistemes de sensorització i xarxa de comunicacions per realitzar tasques de vigilància. Els robots han de poder-se moure per un entorn reconeixent la posició i orientació dels diferents objectes que l'envolten. Aquesta informació ha de permetre al robot localitzar-se dins de l'entorn on es troba per poder-se moure evitant els possibles obstacles i dur a terme la tasca encomanada. El robot ha de generar un mapa dinàmic de l'entorn que serà utilitzat per localitzar la seva posició. L'objectiu principal d'aquest projecte és aconseguir que un robot explori i construeixi un mapa de l'entorn sense la necessitat de modificar el propi entorn. Aquesta tesi està enfocada en l'estudi de la geometria dels sistemes de visió estereoscòpics formats per dues càmeres amb l'objectiu d'obtenir informació geomètrica 3D de l'entorn d'un vehicle. Aquest objectiu tracta de l'estudi del modelatge i la calibració de càmeres i en la comprensió de la geometria epipolar. Aquesta geometria està continguda en el que s'anomena emph{matriu fonamental}. Cal realitzar un estudi del càlcul de la matriu fonamental d'un sistema estereoscòpic amb la finalitat de reduir el problema de la correspondència entre dos plans imatge. Un altre objectiu és estudiar els mètodes d'estimació del moviment basats en la geometria epipolar diferencial per tal de percebre el moviment del robot i obtenir-ne la posició. Els estudis de la geometria que envolta els sistemes de visió estereoscòpics ens permeten presentar un sistema de visió per computador muntat en un robot mòbil que navega en un entorn desconegut. El sistema fa que el robot sigui capaç de generar un mapa dinàmic de l'entorn a mesura que es desplaça i determinar quin ha estat el moviment del robot per tal de emph{localitzar-se} dins del mapa. La tesi presenta un estudi comparatiu dels mètodes de calibració de càmeres més utilitzats en les últimes dècades. Aquestes tècniques cobreixen un gran ventall dels mètodes de calibració clàssics. Aquest mètodes permeten estimar els paràmetres de la càmera a partir d'un conjunt de punts 3D i de les seves corresponents projeccions 2D en una imatge. Per tant, aquest estudi descriu un total de cinc tècniques de calibració diferents que inclouen la calibració implicita respecte l'explicita i calibració lineal respecte no lineal. Cal remarcar que s'ha fet un gran esforç en utilitzar la mateixa nomenclatura i s'ha estandaritzat la notació en totes les tècniques presentades. Aquesta és una de les dificultats principals a l'hora de poder comparar les tècniques de calibració ja què cada autor defineix diferents sistemes de coordenades i diferents conjunts de paràmetres. El lector és introduït a la calibració de càmeres amb la tècnica lineal i implícita proposada per Hall i amb la tècnica lineal i explicita proposada per Faugeras-Toscani. A continuació es passa a descriure el mètode a de Faugeras incloent el modelatge de la distorsió de les lents de forma radial. Seguidament es descriu el conegut mètode proposat per Tsai, i finalment es realitza una descripció detallada del mètode de calibració proposat per Weng. Tots els mètodes són comparats tant des del punt de vista de model de càmera utilitzat com de la precisió de la calibració. S'han implementat tots aquests mètodes i s'ha analitzat la precisió presentant resultats obtinguts tant utilitzant dades sintètiques com càmeres reals. Calibrant cada una de les càmeres del sistema estereoscòpic es poden establir un conjunt de restriccions geomètri ques entre les dues imatges. Aquestes relacions són el que s'anomena geometria epipolar i estan contingudes en la matriu fonamental. Coneixent la geometria epipolar es pot: simplificar el problema de la correspondència reduint l'espai de cerca a llarg d'una línia epipolar; estimar el moviment d'una càmera quan aquesta està muntada sobre un robot mòbil per realitzar tasques de seguiment o de navegació; reconstruir una escena per aplicacions d'inspecció, propotipatge o generació de motlles. La matriu fonamental s'estima a partir d'un conjunt de punts en una imatges i les seves correspondències en una segona imatge. La tesi presenta un estat de l'art de les tècniques d'estimació de la matriu fonamental. Comença pels mètode lineals com el dels set punts o el mètode dels vuit punts, passa pels mètodes iteratius com el mètode basat en el gradient o el CFNS, fins arribar las mètodes robustos com el M-Estimators, el LMedS o el RANSAC. En aquest treball es descriuen fins a 15 mètodes amb 19 implementacions diferents. Aquestes tècniques són comparades tant des del punt de vista algorísmic com des del punt de vista de la precisió que obtenen. Es presenten el resultats obtinguts tant amb imatges reals com amb imatges sintètiques amb diferents nivells de soroll i amb diferent quantitat de falses correspondències. Tradicionalment, l'estimació del moviment d'una càmera està basada en l'aplicació de la geometria epipolar entre cada dues imatges consecutives. No obstant el cas tradicional de la geometria epipolar té algunes limitacions en el cas d'una càmera situada en un robot mòbil. Les diferencies entre dues imatges consecutives són molt petites cosa que provoca inexactituds en el càlcul de matriu fonamental. A més cal resoldre el problema de la correspondència, aquest procés és molt costós en quant a temps de computació i no és gaire efectiu per aplicacions de temps real. En aquestes circumstàncies les tècniques d'estimació del moviment d'una càmera solen basar-se en el flux òptic i en la geometria epipolar diferencial. En la tesi es realitza un recull de totes aquestes tècniques degudament classificades. Aquests mètodes són descrits unificant la notació emprada i es remarquen les semblances i les diferencies entre el cas discret i el cas diferencial de la geometria epipolar. Per tal de poder aplicar aquests mètodes a l'estimació de moviment d'un robot mòbil, aquest mètodes generals que estimen el moviment d'una càmera amb sis graus de llibertat, han estat adaptats al cas d'un robot mòbil que es desplaça en una superfície plana. Es presenten els resultats obtinguts tant amb el mètodes generals de sis graus de llibertat com amb els adaptats a un robot mòbil utilitzant dades sintètiques i seqüències d'imatges reals. Aquest tesi finalitza amb una proposta de sistema de localització i de construcció d'un mapa fent servir un sistema estereoscòpic situat en un robot mòbil. Diverses aplicacions de robòtica mòbil requereixen d'un sistema de localització amb l'objectiu de facilitar la navegació del vehicle i l'execució del les trajectòries planificades. La localització es sempre relativa al mapa de l'entorn on el robot s'està movent. La construcció de mapes en un entorn desconegut és una tasca important a realitzar per les futures generacions de robots mòbils. El sistema que es presenta realitza la localització i construeix el mapa de l'entorn de forma simultània. A la tesi es descriu el robot mòbil GRILL, que ha estat la plataforma de treball emprada per aquesta aplicació, amb el sistema de visió estereoscòpic que s'ha dissenyat i s'ha muntat en el robot. També es descriu tots el processos que intervenen en el sistema de localització i construcció del mapa. La implementació d'aquest processos ha estat possible gràcies als estudis realitzats i presentats prèviament (calibració de càmeres, estimació de la matriu fonamental, i estimació del moviment) sense els quals no s'hauria pogut plantejar aquest sistema. Finalment es presenten els mapes en diverses trajectòries realitzades pel robot GRILL en el laboratori. Les principals contribucions d'aquest treball són: ·Un estat de l'art sobre mètodes de calibració de càmeres. El mètodes són comparats tan des del punt de vista del model de càmera utilitzat com de la precisió dels mètodes. ·Un estudi dels mètodes d'estimació de la matriu fonamental. Totes les tècniques estudiades són classificades i descrites des d'un punt de vista algorísmic. ·Un recull de les tècniques d'estimació del moviment d'una càmera centrat en el mètodes basat en la geometria epipolar diferencial. Aquestes tècniques han estat adaptades per tal d'estimar el moviment d'un robot mòbil. ·Una aplicació de robòtica mòbil per tal de construir un mapa dinàmic de l'entorn i localitzar-se per mitja d'un sistema estereoscòpic. L'aplicació presentada es descriu tant des del punt de vista del maquinari com del programari que s'ha dissenyat i implementat.

A general-method for tracking analysis and its application to meteorological data

Recent interest in the validation of general circulation models (GCMs) has been devoted to objective methods. A small number of authors have used the direct synoptic identification of phenomena together with a statistical analysis to perform the objective comparison between various datasets. This paper describes a general method for performing the synoptic identification of phenomena that can be used for an objective analysis of atmospheric, or oceanographic, datasets obtained from numerical models and remote sensing. Methods usually associated with image processing have been used to segment the scene and to identify suitable feature points to represent the phenomena of interest. This is performed for each time level. A technique from dynamic scene analysis is then used to link the feature points to form trajectories. The method is fully automatic and should be applicable to a wide range of geophysical fields. An example will be shown of results obtained from this method using data obtained from a run of the Universities Global Atmospheric Modelling Project GCM.

Developing the next-generation climate system models: challenges and achievements

Although climate models have been improving in accuracy and efficiency over the past few decades, it now seems that these incremental improvements may be slowing. As tera/petascale computing becomes massively parallel, our legacy codes are less suitable, and even with the increased resolution that we are now beginning to use, these models cannot represent the multiscale nature of the climate system. This paper argues that it may be time to reconsider the use of adaptive mesh refinement for weather and climate forecasting in order to achieve good scaling and representation of the wide range of spatial scales in the atmosphere and ocean. Furthermore, the challenge of introducing living organisms and human responses into climate system models is only just beginning to be tackled. We do not yet have a clear framework in which to approach the problem, but it is likely to cover such a huge number of different scales and processes that radically different methods may have to be considered. The challenges of multiscale modelling and petascale computing provide an opportunity to consider a fresh approach to numerical modelling of the climate (or Earth) system, which takes advantage of the computational fluid dynamics developments in other fields and brings new perspectives on how to incorporate Earth system processes. This paper reviews some of the current issues in climate (and, by implication, Earth) system modelling, and asks the question whether a new generation of models is needed to tackle these problems.

Development and illustrative outputs of the Community Integrated Assessment System (CIAS), a multi-institutional modular integrated assessment approach for modelling climate change.

This paper describes the development and first results of the “Community Integrated Assessment System” (CIAS), a unique multi-institutional modular and flexible integrated assessment system for modelling climate change. Key to this development is the supporting software infrastructure, SoftIAM. Through it, CIAS is distributed between the community of institutions which has each contributed modules to the CIAS system. At the heart of SoftIAM is the Bespoke Framework Generator (BFG) which enables flexibility in the assembly and composition of individual modules from a pool to form coupled models within CIAS, and flexibility in their deployment onto the available software and hardware resources. Such flexibility greatly enhances modellers’ ability to re-configure the CIAS coupled models to answer different questions, thus tracking evolving policy needs. It also allows rigorous testing of the robustness of IA modelling results to the use of different component modules representing the same processes (for example, the economy). Such processes are often modelled in very different ways, using different paradigms, at the participating institutions. An illustrative application to the study of the relationship between the economy and the earth’s climate system is provided.

Spatial econometric issues for bio-economic and land-use modelling

We survey the literature on spatial bio-economic and land-use modelling and assess its thematic development. Unobserved site-specific heterogeneity is a feature of almost all the surveyed works, and this feature, it seems, has stimulated significant methodological innovation. In an attempt to improve the suitability with which the prototype incorporates heterogeneity, we consider modelling alternatives and extensions. We discuss solutions and conjecture others.

Spatial econometric issues for bio-economic and land-use modelling

We survey the literature on spatial bio-economic and land-use modelling and assess its thematic development. Unobserved site-specific heterogeneity is a feature of almost all the surveyed works, and this feature, it seems, has stimulated significant methodological innovation. In an attempt to improve the suitability with which the prototype incorporates heterogeneity, we consider modelling alternatives and extensions. We discuss solutions and conjecture others.

Crystal and molecular structure of the thioether-analogue of PEEK from X-ray powder data and diffraction-modelling

Analysis of X-ray powder data for the melt-crystallisable aromatic poly(thioether thioether ketone) [-S-Ar-S-Ar-CO-Ar](n), ('PTTK', Ar= 1,4-phenylene), reveals that it adopts a crystal structure very different from that established for its ether-analogue PEEK. Molecular modelling and diffraction-simulation studies of PTTK show that the structure of this polymer is analogous to that of melt-crystallised poly(thioetherketone) [-SAr-CO-Ar](n) in which the carbonyl linkages in symmetry-related chains are aligned anti-parallel to one another. and that these bridging units are crystallographically interchangeable. The final model for the crystal structure of PTTK is thus disordered, in the monoclinic space group 121a (two chains per unit cell), with cell dimensions a = 7.83, b = 6.06, c = 10.35 angstrom, beta = 93.47 degrees. (c) 2005 Elsevier Ltd. All rights reserved.

Mechanical effects of plant cell wall enzymes on xyloglucan/cellulose composites

Xyloglucan-acting enzymes are believed to have effects on type I primary plant cell wall mechanical properties. In order to get a better understanding of these effects, a range of enzymes with different in vitro modes of action were tested against cell wall analogues (bio-composite materials based on Acetobacter xylinus cellulose and xyloglucan). Tomato pericarp xyloglucan endo transglycosylase (tXET) and nasturtium seed xyloglucanase (nXGase) were produced heterologously in Pichia pastoris. Their action against the cell wall analogues was compared with that of a commercial preparation of Trichoderma endo-glucanase (EndoGase). Both 'hydrolytic' enzymes (nXGase and EndoGase) were able to depolymerise not only the cross-link xyloglucan fraction but also the surface-bound fraction. Consequent major changes in cellulose fibril architecture were observed. In mechanical terms, removal of xyloglucan cross-links from composites resulted in increased stiffness (at high strain) and decreased visco-elasticity with similar extensibility. On the other hand, true transglycosylase activity (tXET) did not affect the cellulose/xyloglucan ratio. No change in composite stiffness or extensibility resulted, but a significant increase in creep behaviour was observed in the presence of active tXET. These results provide direct in vitro evidence for the involvement of cell wall xyloglucan-specific enzymes in mechanical changes underlying plant cell wall re-modelling and growth processes. Mechanical consequences of tXET action are shown to be complimentary to those of cucumber expansin.

Crumpet structures: experimental and modelling studies

Crumpets are made by heating fermented batter on a hot plate at around 230°C. The characteristic structure dominated by vertical pores develops rapidly: structure has developed throughout around 75% of the product height within 30s, which is far faster than might be expected from transient heat conduction through the batter. Cooking is complete within around 3 min. Image analysis based on results from X-ray tomography shows that the voidage fraction is approximately constant and that there is continual coalescence between the larger pores throughout the product although there is also a steady level of small bubbles trapped within the solidified batter. We report here experimental studies which shed light on some of the mechanisms responsible for this structure, together with some models of key phenomena.Three aspects are discussed here: the role of gas (carbon dioxide and nitrogen) nuclei in initiating structure development; convective heat transfer inside the developing pores; and the kinetics of setting the batter into an elastic solid structure. It is shown conclusively that the small bubbles of carbon dioxide resulting from the fermentation stage play a crucial role as nuclei for pore development: without these nuclei, the result is not a porous structure, but rather a solid, elastic, inedible, gelatinized product. These nuclei are also responsible for the tiny bubbles which are set in the final product. The nuclei form the source of the dominant pore structure which is largely driven by the, initially explosive, release of water vapour from the batter together with the desorption of dissolved carbon dioxide. It is argued that the rapid evaporation, transport and condensation of steam within the growing pores provides an important mechanism, as in a heat pipe, for rapid heat transfer, and models for this process are developed and tested. The setting of the continuous batter phase is essential for final product quality: studies using differential scanning calorimetry and on the kinetics of change in the visco-elastic properties of the batter suggest that this process is driven by the kinetics of gelatinization. Unlike many thermally driven food processes the rates of heating are such that gelatinization kinetics cannot be neglected. The implications of these results for modelling and for the development of novel structures are discussed.

Adaptive modelling, estimation and fusion from data: a neurofuzzy approach

In a world of almost permanent and rapidly increasing electronic data availability, techniques of filtering, compressing, and interpreting this data to transform it into valuable and easily comprehensible information is of utmost importance. One key topic in this area is the capability to deduce future system behavior from a given data input. This book brings together for the first time the complete theory of data-based neurofuzzy modelling and the linguistic attributes of fuzzy logic in a single cohesive mathematical framework. After introducing the basic theory of data-based modelling, new concepts including extended additive and multiplicative submodels are developed and their extensions to state estimation and data fusion are derived. All these algorithms are illustrated with benchmark and real-life examples to demonstrate their efficiency. Chris Harris and his group have carried out pioneering work which has tied together the fields of neural networks and linguistic rule-based algortihms. This book is aimed at researchers and scientists in time series modeling, empirical data modeling, knowledge discovery, data mining, and data fusion.

Generic unified modelling process for developing semantically rich, dynamic and temporal models

Models play a vital role in supporting a range of activities in numerous domains. We rely on models to support the design, visualisation, analysis and representation of parts of the world around us, and as such significant research effort has been invested into numerous areas of modelling; including support for model semantics, dynamic states and behaviour, temporal data storage and visualisation. Whilst these efforts have increased our capabilities and allowed us to create increasingly powerful software-based models, the process of developing models, supporting tools and /or data structures remains difficult, expensive and error-prone. In this paper we define from literature the key factors in assessing a model’s quality and usefulness: semantic richness, support for dynamic states and object behaviour, temporal data storage and visualisation. We also identify a number of shortcomings in both existing modelling standards and model development processes and propose a unified generic process to guide users through the development of semantically rich, dynamic and temporal models.