Systems of differential-algebraic equations (DAEs) arise in many areas including chemical engineering, electrical circuit simulation, and robotics. Such systems are routinely generated by simulation and modeling environments, like MapleSim, Matlab/Simulink, and those based on the Modelica language. Before a simulation starts and a numerical solution method is applied, some kind of structural analysis (SA) is performed to determine the structure and the index of a DAE system. Structural analysis methods serve as a necessary preprocessing stage, and among them, Pantelides's graph-theory-based algorithm is widely used in industry. Recently, Pryce's Σ-method is becoming increasingly popular, owing to its straightforward approach and capability of analyzing high-order systems. Both methods are equivalent in the sense that (a) when one succeeds, producing a nonsingular Jacobian, the other also succeeds, and that (b) the two give the same structural index in the case of either success or failure. When SA succeeds, the structural results can be used to perform an index reduction process, or to devise a stage-by-stage solution scheme for computing derivatives or Taylor coefficients up to some order. Although such a success occurs on fairly many problems of interest, SA can fail on some simple, solvable DAEs with an identically singular Jacobian, and give incorrect structural information that usually includes the index. In this thesis, we focus on the Σ-method and investigate its failures. Aiming at making this SA more reliable, we develop two conversion methods for fixing SA failures. These methods reformulate a DAE on which the Σ-method fails into an equivalent problem on which SA is more likely to succeed with a nonsingular Jacobian. The implementation of our methods requires symbolic computations. We also combine our conversion methods with block triangularization of a DAE. Using a block triangular form of a Jacobian sparsity pattern, we identify which diagonal block(s) of the Jacobian is identically singular, and then perform a conversion on each singular block. This approach can reduce the computational cost and improve the efficiency of finding a suitable conversion for fixing SA's failures. / Thesis / Doctor of Philosophy (PhD)

No description available.

The classification of finite semigroups is difficult even for small orders because of their large number. Most finite semigroups are nilpotent of nilpotency rank 3. Formulae for their number up to isomorphism, and up to isomorphism and anti-isomorphism of any order are the main results in the theoretical part of this thesis. Further studies concern the classification of nilpotent semigroups by rank, leading to a full classification for large ranks. In the computational part, a method to find and enumerate multiplication tables of semigroups and subclasses is presented. The approach combines the advantages of computer algebra and constraint satisfaction, to allow for an efficient and fast search. The problem of avoiding isomorphic and anti-isomorphic semigroups is dealt with by supporting standard methods from constraint satisfaction with structural knowledge about the semigroups under consideration. The approach is adapted to various problems, and realised using the computer algebra system GAP and the constraint solver Minion. New results include the numbers of semigroups of order 9, and of monoids and bands of order 10. Up to isomorphism and anti-isomorphism there are 52,989,400,714,478 semigroups with 9 elements, 52,991,253,973,742 monoids with 10 elements, and 7,033,090 bands with 10 elements. That constraint satisfaction can also be utilised for the analysis of algebraic objects is demonstrated by determining the automorphism groups of all semigroups with 9 elements. A classification of the semigroups of orders 1 to 8 is made available as a data library in form of the GAP package Smallsemi. Beyond the semigroups themselves a large amount of precomputed properties is contained in the library. The package as well as the code used to obtain the enumeration results are available on the attached DVD.

<p>SammanfattningSkolverket beslutade 2006 att tillåta symbolhanterande räknare på nationella prov i matematik för alla gymnasieelever från höstterminen 2007. Beslutet debatterades flitigt t.ex. i tidskriften <em>Nämnaren</em>.</p><p>Prototypen till en ny symbolhanterande räknare testades i sju ’pilotklasser’ vid fyra svenska gymnasieskolor från december 2006. Jag har följt två av dessa klasser t.o.m. höstterminen 2007 då de använde den färdiga räknaren TI-Nspire CAS.</p><p>Syftet med denna pilotstudie är att ta reda på hur den nya symbolhanterande räknaren fungerar i matematikundervisningen i de två pilotklasserna på gymnasiets NV-program, främst ur elevernas perspektiv, men också ur lärarnas.</p><p>Tanken är att pröva olika metoder, som kan användas i en större undersökning.</p><p>Bakgrunden ger bl.a. en översikt av de handburna räknehjälpmedlens historia och en kortfattad historik över elektroniska hjälpmedel i den svenska skolan. Tidigare forskning tar upp miniräknaren i skolan och då främst grafräknaren.</p><p>Observationsmetoderna är miniräknardagböcker, diagnostiska test, elev- och lärarenkät. Jag jämför också klassernas kursbetyg och klassificerar ett antal matematikuppgifter m.a.p. möjlig användning av CAS.</p><p>Resultatet gav en mångfacetterad bild med både positiva och negativa sidor. Skolorna behöver ta ställning till vilka klasser som ska använda symbolhanterande räknare och erbjuda lärarna fortbildning. Det behövs metodutveckling, nya läromedel och en reformerad lärarutbildning för att fullt kunna dra nytta av den nya teknikens möjligheter.</p><p>Jag lämnar flera förslag på fortsatt forskning.</p> / kontakt: david_u_strom@hotmail.com

SammanfattningSkolverket beslutade 2006 att tillåta symbolhanterande räknare på nationella prov i matematik för alla gymnasieelever från höstterminen 2007. Beslutet debatterades flitigt t.ex. i tidskriften Nämnaren. Prototypen till en ny symbolhanterande räknare testades i sju ’pilotklasser’ vid fyra svenska gymnasieskolor från december 2006. Jag har följt två av dessa klasser t.o.m. höstterminen 2007 då de använde den färdiga räknaren TI-Nspire CAS. Syftet med denna pilotstudie är att ta reda på hur den nya symbolhanterande räknaren fungerar i matematikundervisningen i de två pilotklasserna på gymnasiets NV-program, främst ur elevernas perspektiv, men också ur lärarnas. Tanken är att pröva olika metoder, som kan användas i en större undersökning. Bakgrunden ger bl.a. en översikt av de handburna räknehjälpmedlens historia och en kortfattad historik över elektroniska hjälpmedel i den svenska skolan. Tidigare forskning tar upp miniräknaren i skolan och då främst grafräknaren. Observationsmetoderna är miniräknardagböcker, diagnostiska test, elev- och lärarenkät. Jag jämför också klassernas kursbetyg och klassificerar ett antal matematikuppgifter m.a.p. möjlig användning av CAS. Resultatet gav en mångfacetterad bild med både positiva och negativa sidor. Skolorna behöver ta ställning till vilka klasser som ska använda symbolhanterande räknare och erbjuda lärarna fortbildning. Det behövs metodutveckling, nya läromedel och en reformerad lärarutbildning för att fullt kunna dra nytta av den nya teknikens möjligheter. Jag lämnar flera förslag på fortsatt forskning. / kontakt: david_u_strom@hotmail.com

Der Vortrag zeigt, wie aus einem mathematisch-technischen Sachverhalt ein didaktisch begründetes Unterrichtsbeispiel abgeleitet werden kann und stellt die unterrichtlichen Nutzungsmöglichkeiten vor. Für die digitale Bildbearbeitung spielt die Interpolation eine wichtige Rolle und dient hierbei als Berechnungsverfahren für die Bildgrößenänderung. Interpolation ist ein Approximationsverfahren, bei dem z.B. zu Punkten mit bekannten Koordinaten eine Funktion berechnet wird, die alle diese Punkte erfüllt. Mit dieser Funktion können dann beliebige Zwischenwerte berechnet werden. Dabei bestimmt die Zahl der Datenpunkte die Zahl der notwendigen Polynomterme. Wegen ihrer mathematischen Eigenschaften werden häufig Polynome benutzt. Die Lösung der Aufgabe führt über ein lineares Gleichungssystem zur Bestimmung der Koeffizienten des Polynoms. Der erste Teil des Vortrages befasst sich mit der beispielhaften Darstellung der bikubischen Interpolation und deren Realisierung mittels MathCAD. Es wird gezeigt, wie aus den konkreten Schwärzungswerten eines Digitalbildes für eine Bildvergrößerung ein zu interpolierender Zwischenwert für einen neuen Bildpunkt berechnet wird. Der MathCAD-Wortschatz wird angegeben und notwendige didaktische Vereinfachungen werden beschrieben. Im zweiten Teil werden die Nutzung des Themas als Unterrichtsgegenstand in der Sekundarstufe II in seiner Wechselwirkung zwischen digitaler Bildbearbeitung, Mathematik und Informatik (Nutzung von MathCAD) erläutert, die thematischen Verflechtungsmöglichkeiten aufgezeigt und das didaktische Potential beleuchtet.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

This study solves the problem of unsteady free convection from an inclined heated tube both numerically and analytically. The tube is taken to have an elliptic cross-section having a constant heat flux applied to its surface. The surrounding fluid is viscous and incompressible and infinite in extent. The Boussinesq approximation is used to describe the buoyancy force driving the flow. The underlying assumptions made in this work are that the flow remains laminar and two-dimensional for all time. This enables the Navier-Stokes and energy equations to be formulated in terms of the streamfunction, and vorticity. We assume that initially an impulsive heat flux is applied to the surface and that both the tube and surrounding fluid have the same initial temperature. The problem is solved subject to the no-slip and constant heat flux conditions on the surface together with quiescent far-field and initial conditions. An approximate analytical-numerical solution was derived for small times, t and large Grashof numbers, Gr. This was done by expanding the flow variables in a double series in terms of two small parameters and reduces to solving a set of differential equations. The first few terms were solved exactly while the higher-order terms were determined numerically. Flow characteristics presented include average surface temperature plots as well as surface vorticity and surface temperature distributions. The results demonstrate that the approximate analytical-numerical solution is in good agreement with the fully numerical solution for small t and large Gr.

Η κατά συμμετρίες ανάλυση είναι μια σύγχρονή και αποτελεσματική μέθοδος ανάλυσης του μαθηματικού πεδίου των Διαφορικών Εξισώσεων. Στα πλεονεκτήματα της, ο αλγοριθμικός τρόπος με τον οποίο μπορούμε να βρούμε τις συμμετριες ενός συστήματος και η κατακευή λύσεων από αυτές. Όμως, όπως και κάθε άλλη μέθοδος έτσι και αυτή έχει τα μειονεκτήματα της, το μέγεθος και η πολυπλοκότητα των ενδιάμεσων υπολογισμών που απαιτούνται για την εύρεση των συμμετρίων ενός συστήματος αυξάνεται εκθετικά σε σχέση με αυτό. Γεγονός που καθιστά τους υπολογισμούς αυτούς με το χέρι χρονοβόρους και επιρρεπής σε σφάλματα και συνεπώς την ανάγκη για την χρήση αξιόπιστων συμβολικών προγραμμάτων επιτακτική. Για τον σκοπό αυτό αναπτύξαμε το συμβολικό πακέτο Sym για το αλγεβρικό σύστημα Mathematica. Το συμβολικό αυτό πακέτο περιέχει στοιχεία τεχνικής νοημοσύνης και εξιδικευμένες συμβολικές μεθόδους. Στοιχεία που το καθιστούν ένα αποτελεσματικό και ευέλικτο μαθηματικό εργαλείο τόσο στον ερευνητικό τομέα όσο και στην εκπαίδευση. Το παρόν διδακτορικό χωρίζεται σε δύο μέρη, στο πρώτο παρουσιάζουμε τις βασικές έννοιες της κατα συμμετρίες ανάλυσης διαφορικών εξισώσεων και τους λόγους για τους οποίους η χρήση συμβολικών προγραμμάτων βρίσκει πρόσφορο έδαφος. Στο δεύτερο μέρος, παρουσιάζουμε το συμβολικό πακέτο Sym και δύο ερευνητικά αποτελέσματα της χρήσης του. Όσο αναφορά το ίδιο το πακέτο, δίνουμε τα βασικά του χαρακτηριστικά , τον τρόπο λειτουργίας του και τα οφέλη του σε σχέση με τα ήδη υπάρχοντα συμβολικά πακέτα για την εύρεση συμμετριών. Η χρηστικότητα του παρουσιάζεται μέσω δύο ερευνητικών αποτελεσμάτων. Στο πρώτο, εξετάζουμε ενα πρόβλημα από την περιοχή της Γενικής Σχετικότητας, την εύρεση βαρυτικών κυμάτων. Οι συμμετρίες των εξισώσεων πεδίου του Einstein για την μετρική του Bondi καθορίζονται μέσω του Sym και υποβιβάζουμε με αυτές την τάξη του μή γραμμικού συστήματος. Με υποθέσεις εργασίας πάνω στο σύστημα αυτό δίνουμε ειδικές λύσεις οι οποίες είχαν προκύψει παλίοτερα με άλλες μεθόδους. Τέλος, παρουσιάζουμε τις μελλοντικές μας κατευθύνσεις προς την καθορισμό νέων λύσεων με την σωστή φυσική συμπεριφορά που επιβάλει το πρόβλημα. Στο δεύτερο, δίνουμε μια προτότυπη διαδικασία κατηγοριοποίησης διαφορικών εξισώσεων χρησιμοποιώντας τις ένοιες της πλήρους ομάδας συμετρίας και της αξιοσημείωτης κατά Lie διαφορικής εξίσωσης. Με βάση αυτή, επιτυγχάνουμε την συνθέση διαφορικών εξισώσεων κατασκευάζοντας έτσι καινούργιες οικογένεις διαφορικών εξισώσεων περιέχοντες τις αρχικές μας εξισώσεις. / The symmetry analysis is a modern and effective method of mathematical field of differential equations. On its advantages, the algorithmic way for determining the symmetries and constructing solutions. Like any other method it also has its disadvantages; the size and the complexity of the intermediate calculations needed for giving the symmetries is increased exponentially with respect to the equation under investigation. This fact renders the calculations unmanageable by hand and error prone. The need for reliable and fast symbolic tools is apparent. For this reason, we developed a symbolic package called Sym based on the Mathematica program. The package employing artificial intelligent elements and specialized symbolic methods is an effective and versatile mathematical tool ideal for research and education alike. The present thesis consists of two parts; on the first we present the basic notions of the mathematical theory and the reasons that symbolic tools can be utilized. On the second part, we present the symbolic package Sym itself along with two new result employing it. As for the package itself, we give the basic characteristics, its functionality and the benefits using it against the existing programs. Its usefulness is presented through two results. On the first, we study a problem from General Relativity, finding solutions describing gravity waves. The symmetries of the Einstein’s field equations for the radiating Bondi metric are determined from Sym. Using them we reduce the non-linear system. Using specific ansatzes we arrive to specific solutions already found using other methods. Finally, we present our future directions for finding new solutions with the correct physical behavior. On the second, we describe a new procedure for classifying differential equations using the notions of complete symmetry groups and Lie remarkability. Using this procedure we achieved by starting with a set of differential equation to construct a new family that includes the initial set. Future directions include finding a way to link the solutions of the newly constructed family with the solutions of the equations that we use for constructing it.

Informacinių ir komunikacinių technologijų (IKT) įtakota ugdymo technologijų raida pakeitė visų sričių, tarp jų ir matematikos ugdymo turinį. Konstruktyvistinę mokymosi paradigmą, pagrindinius šiuolaikinius didaktinius matematikos mokymo principus – suprantamumą, sąmoningą ir aktyvų žinių perėmimą, vaizdumą, mokymo diferencijavimą, individualizavimą ir kt. – gali įgyvendinti tik atitinkamos kompetencijos matematikos mokytojai. Šalies pedagogų kompiuterinio raštingumo standarto, numatančio mokytojų gebėjimus naudoti IKT ugdymo procese, vienas iš reikalavimų yra „Žinoti pagrindinių ugdymui naudojamų kompiuterinių programų tipus, gebėti analizuoti jų privalumus ir trūkumus. Mokėti pritaikyti svarbiausias bendrosios paskirties ir mokomąsias kompiuterines programas ugdymo procese“. Šio bendro reikalavimo taikymas matematikos mokymui – tai ne tik sudėtingų matematikai taikytinų technologijų žinojimo ir įvaldymo, bet ir gebėjimų tinkamai taikyti jas matematikos mokyme reikalavimas. Informacinių technologijų (IT) ištakos yra pirmosiose skaičiavimo mašinose, kurios buvo kurtos spręsti taikomuosius matematikos uždavinius. Ir asmeninių kompiuterių (AK) pirmoji paskirtis buvo matematinių uždavinių sprendimas. Todėl matematikai skirtų šiuolaikinių technologijų yra žymiai daugiau, negu taikytinų kituose moksluose. Dar daugiau, yra programų sistemų, kuriose, atrodo, realizuota visa matematika: jos ne tik skaičiuoja, pertvarko, prastina, vaizduoja, bet ir sprendžia specifinius bene visų... [toliau žr. visą tekstą] / The survey summarizes investigations of the author on the technological tools and their educational features applicable to mathematics. The purpose of the investigations is enhancement of the technological and educational competence of mathematics teachers. The main problems of the investigation are: characterization of the main technological tools applicable in mathematics at the university level, description which abilities, knowledge, and skills of application in teaching comprise the technological and educational ICT competence of a contemporary mathematician. Development of educational technologies, influenced by information and communication technologies (ICT), has changed the content of all spheres of education, and the content of mathematical education among them. A constructivist paradigm of learning, the main didactic principles of mathematics teaching: understanding, conscious and active acquiring of knowledge, visualization, differentiation and individualization of learning, etc., can realize only mathematics teachers of the adequate competence. The Standard of Computer Literacy of educators in Lithuania provides abilities of teachers to apply ICT in educational process. One of the requirements is “To know the main types of computer programmes useful for education, to have the ability to analyze their merits and demerits, to apply them in the educational process”. The application of the general requirement in teaching mathematics means not only the knowledge and... [to full text]