On Vessiot's Theory of Partial Differential Equations

The object of research presented here is Vessiot's theory of partial differential equations: for a given differential equation one constructs a distribution both tangential to the differential equation and contained within the contact distribution of the jet bundle. Then within it, one seeks n-dimensional subdistributions which are transversal to the base manifold, the integral distributions. These consist of integral elements, and these again shall be adapted so that they make a subdistribution which closes under the Lie-bracket. This then is called a flat Vessiot connection. Solutions to the differential equation may be regarded as integral manifolds of these distributions. In the first part of the thesis, I give a survey of the present state of the formal theory of partial differential equations: one regards differential equations as fibred submanifolds in a suitable jet bundle and considers formal integrability and the stronger notion of involutivity of differential equations for analyzing their solvability. An arbitrary system may (locally) be represented in reduced Cartan normal form. This leads to a natural description of its geometric symbol. The Vessiot distribution now can be split into the direct sum of the symbol and a horizontal complement (which is not unique). The n-dimensional subdistributions which close under the Lie bracket and are transversal to the base manifold are the sought tangential approximations for the solutions of the differential equation. It is now possible to show their existence by analyzing the structure equations. Vessiot's theory is now based on a rigorous foundation. Furthermore, the relation between Vessiot's approach and the crucial notions of the formal theory (like formal integrability and involutivity of differential equations) is clarified. The possible obstructions to involution of a differential equation are deduced explicitly. In the second part of the thesis it is shown that Vessiot's approach for the construction of the wanted distributions step by step succeeds if, and only if, the given system is involutive. Firstly, an existence theorem for integral distributions is proven. Then an existence theorem for flat Vessiot connections is shown. The differential-geometric structure of the basic systems is analyzed and simplified, as compared to those of other approaches, in particular the structure equations which are considered for the proofs of the existence theorems: here, they are a set of linear equations and an involutive system of differential equations. The definition of integral elements given here links Vessiot theory and the dual Cartan-Kähler theory of exterior systems. The analysis of the structure equations not only yields theoretical insight but also produces an algorithm which can be used to derive the coefficients of the vector fields, which span the integral distributions, explicitly. Therefore implementing the algorithm in the computer algebra system MuPAD now is possible.

Acquiring expert knowledge for the design of conceptual information systems

Conceptual Information Systems unfold the conceptual structure of data stored in relational databases. In the design phase of the system, conceptual hierarchies have to be created which describe different aspects of the data. In this paper, we describe two principal ways of designing such conceptual hierarchies, data driven design and theory driven design and discuss advantages and drawbacks. The central part of the paper shows how Attribute Exploration, a knowledge acquisition tool developped by B. Ganter can be applied for narrowing the gap between both approaches.

Conceptual knowledge discovery and data analysis

In this paper, we discuss Conceptual Knowledge Discovery in Databases (CKDD) in its connection with Data Analysis. Our approach is based on Formal Concept Analysis, a mathematical theory which has been developed and proven useful during the last 20 years. Formal Concept Analysis has led to a theory of conceptual information systems which has been applied by using the management system TOSCANA in a wide range of domains. In this paper, we use such an application in database marketing to demonstrate how methods and procedures of CKDD can be applied in Data Analysis. In particular, we show the interplay and integration of data mining and data analysis techniques based on Formal Concept Analysis. The main concern of this paper is to explain how the transition from data to knowledge can be supported by a TOSCANA system. To clarify the transition steps we discuss their correspondence to the five levels of knowledge representation established by R. Brachman and to the steps of empirically grounded theory building proposed by A. Strauss and J. Corbin.

Mobiles Lernen

In ihrer Arbeit "Mobiles Lernen. Analyse des Wissenschaftsprozesses der britischen und deutschsprachigen medienpädagogischen und erziehungswissenschaftlichen Mobile-Learning-Diskussion." zeichnet Judith Seipold die Phasen und Entwicklungslinien der bisherigen und naturwüchsigen Diskussion um das Mobile Lernen nach. Dabei eröffnet sie mit ihrer Analyse des vornehmlich britischen Wissenschaftsprozesses des Mobilen Lernens ab Beginn des 21. Jahrhunderts den Blick auf die Struktur der medienpädagogischen und erziehungswissenschaftlichen Mobile-Learning-Diskussion, auf deren Kontexte, Bezugspunkte, Perspektiven und theoretische Schwerpunkte, aber auch Erfolge und Problembereiche in der praktischen Umsetzung des Lernens mit Mobiltechnologien in formalisierten Lernkontexten wie dem des Schulunterrichts. Auf diese Weise liefert die Autorin eine Systematik, die nicht nur die britische Diskussion für die deutschsprachige Medienpädagogik verfügbar macht, sondern auch eine neue und systematisch begründete Reflexionsebene für eine aktuelle medienpädagogische Entwicklung – das Mobile Lernen – bildet. Zunächst widmet sich die Autorin der Analyse des Wissenschaftsprozesses der bisherigen britischen und teils auch der deutschsprachigen Mobile-Learning-Diskussion. Um diesen Prozess in seiner Struktur greifbar zu machen, skizziert sie die Bezugsdisziplinen, aus denen die medienpädagogische und erziehungswissenschaftliche Mobile-Learning-Forschung schöpft, arbeitet die Legitimationsbasis auf, auf die sich Mobiles Lernen stützt, zeichnet die historische Entwicklung der Diskussion nach, die sowohl Alltagsnutzung von Mobiltechnologien als auch den Fachdiskurs einschließt, und erläutert innerhalb ihres Modells der Phasen und Entwicklungslinien Mobilen Lernens Theorien, Konzepte und Modelle, die in der Mobile-Learning-Community als zentral für Analyse und Planung Mobilen Lernens erachtet werden. Konzepte und Modelle, die in der Mobile-Learning-Diskussion die Rolle der Lernenden in das Zentrum der Betrachtungen rücken und eine explizite Handlungsorientierung unterstützen, sind Kernbereich des theoriebasierten Teils der Arbeit. Überlegungen zu einer „Sozio-kulturellen Ökologie Mobilen Lernens“, zu „user-generated contexts“ und zu einer „kulturökologisch informierten Didaktik des Mobilen Lernens“ sind dabei sowohl reflektierende Einordnung aktueller Konzepte und Modelle als auch theoretische und konzeptionelle Basis für die praktische Umsetzung Mobilen Lernens. Um die Rolle der Handlungskompetenzen, kulturellen Praktiken und Strukturen der Lerner bei der Nutzung von Mobiltechnologien für Lernen analytisch zu fassen, finden im Praxis-Kapitel Planungs- und Analyseschemata unter Rückgriff auf Beispiele aus der Praxisforschung und der Implementierung mobilen Lernens in den Schulunterricht Anwendung. Ihre Erörterungen rahmt die Autorin abschließend kritisch und weist zum einen auf bildungspolitische, strukturelle und handlungspraktische Implikationen hin, die sich aus der Mobile-Learning-Diskussion ergeben; zum anderen lenkt sie das Augenmerk auf Gegensätze und Dialektiken des Mobilen Lernens, die sich im Spannungsfeld zwischen alltäglicher mobiler Mediennutzung und der teils theoretisch informierten schulischen Verwendung von Mobiltechnologien zum Lernen entfalten. Sie gilt es, im weiteren Verlauf der Diskussion zu hinterfragen und aufzulösen.

First-principles electronic theory of non-collinear magnetic order in transition-metal nanowires

The structural, electronic and magnetic properties of one-dimensional 3d transition-metal (TM) monoatomic chains having linear, zigzag and ladder geometries are investigated in the frame-work of first-principles density-functional theory. The stability of long-range magnetic order along the nanowires is determined by computing the corresponding frozen-magnon dispersion relations as a function of the 'spin-wave' vector q. First, we show that the ground-state magnetic orders of V, Mn and Fe linear chains at the equilibrium interatomic distances are non-collinear (NC) spin-density waves (SDWs) with characteristic equilibrium wave vectors q that depend on the composition and interatomic distance. The electronic and magnetic properties of these novel spin-spiral structures are discussed from a local perspective by analyzing the spin-polarized electronic densities of states, the local magnetic moments and the spin-density distributions for representative values q. Second, we investigate the stability of NC spin arrangements in Fe zigzag chains and ladders. We find that the non-collinear SDWs are remarkably stable in the biatomic chains (square ladder), whereas ferromagnetic order (q =0) dominates in zigzag chains (triangular ladders). The different magnetic structures are interpreted in terms of the corresponding effective exchange interactions J(ij) between the local magnetic moments μ(i) and μ(j) at atoms i and j. The effective couplings are derived by fitting a classical Heisenberg model to the ab initio magnon dispersion relations. In addition they are analyzed in the framework of general magnetic phase diagrams having arbitrary first, second, and third nearest-neighbor (NN) interactions J(ij). The effect of external electric fields (EFs) on the stability of NC magnetic order has been quantified for representative monoatomic free-standing and deposited chains. We find that an external EF, which is applied perpendicular to the chains, favors non-collinear order in V chains, whereas it stabilizes the ferromagnetic (FM) order in Fe chains. Moreover, our calculations reveal a change in the magnetic order of V chains deposited on the Cu(110) surface in the presence of external EFs. In this case the NC spiral order, which was unstable in the absence of EF, becomes the most favorable one when perpendicular fields of the order of 0.1 V/Å are applied. As a final application of the theory we study the magnetic interactions within monoatomic TM chains deposited on graphene sheets. One observes that even weak chain substrate hybridizations can modify the magnetic order. Mn and Fe chains show incommensurable NC spin configurations. Remarkably, V chains show a transition from a spiral magnetic order in the freestanding geometry to FM order when they are deposited on a graphene sheet. Some TM-terminated zigzag graphene-nanoribbons, for example V and Fe terminated nanoribbons, also show NC spin configurations. Finally, the magnetic anisotropy energies (MAEs) of TM chains on graphene are investigated. It is shown that Co and Fe chains exhibit significant MAEs and orbital magnetic moments with in-plane easy magnetization axis. The remarkable changes in the magnetic properties of chains on graphene are correlated to charge transfers from the TMs to NN carbon atoms. Goals and limitations of this study and the resulting perspectives of future investigations are discussed.