Topological Groups: Yesterday, Today, Tomorrow

In 1900, David Hilbert asked whether each locally euclidean topological group admits a Lie group structure. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao. It is not possible to describe briefly the richness of the topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl H. Hofmann and Sidney A. Morris, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and ‎Pavel Zalesskii (2012). The 2007 book “The Lie Theory of Connected Pro-Lie Groups” by Karl Hofmann and Sidney A. Morris, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups. The study of free topological groups initiated by A.A. Markov, M.I. Graev and S. Kakutani, has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology. The book “Topological Groups and Related Structures” by Alexander Arkhangelʹskii and Mikhail Tkachenko has a diverse content including much material on free topological groups. Compactness conditions in topological groups, especially pseudocompactness as exemplified in the many papers of W.W. Comfort, has been another direction which has proved very fruitful to the present day

Logics which allow Degrees of Truth and Degrees of Validity

In dieser Dissertation werden Semantiken logischer Systeme, die sowohl Vagheit (im Sinne gradueller Wahrheitsbewertung logischer Formeln) als auch Unsicherheit (im Sinne gradueller Vertrauensbewertung logischer Formeln) auszudrücken erlauben, vom Standpunkt der mathematischen Logik aus betrachtet. Üblicherweise werden zur Repräsentation von Vagheit mehrwertige Logiken verwendet, wobei zur konkreten Wahrheitsbewertung häufig Formeln mit konkreten Wahrheitswerten verknüpft werden. Zur Repräsentation von Unsicherheit (im Sinne unvollständigen Wissens oder Vertrauens) werden Formeln der zweiwertigen Logik mit Vertrauensgraden bewertet. Es wurden eine Vielzahl logischer Systeme mit bewerteten Formeln in der Literatur beschrieben, mit zum Teil stark abweichenden Interpretationen der Struktur und Semantik von Markierungen. Zum Teil wird jedoch die Bedeutung von Markierungen nicht präzise definiert, was die Interpretation von Inferenzergebnissen erschwert bis unmöglich macht. Solange nicht eine präzise quantitative Theorie wie etwa die Wahrscheinlichkeitstheorie zur Erklärung von Markierungswerten verwendet wird, gibt es keine kanonische Erklärung für die Bedeutung einer Markierung. Führt dies dann zu einer Vielzahl möglicher Erklärungen, ohne dass diese anhand präzise dargelegter Kriterien verglichen werden können, so wird der Nutzen solcher gradueller Bewertungen insgesamt fraglich. Ein Weg zur Verbesserung dieser Situation liegt darin, Bewertungssysteme anhand grundlegender Bedeutungsunterschiede der Bewertungen in Klassen einzuteilen. Hier werden besonders Bewertungen des Wahrheitsgehalts sowie des Vertrauens in die Gültigkeit logischer Formeln betrachtet. Es gibt gut ausgearbeitete Theorien bewerteter Logiken, die zu der einen oder der anderen Klasse gehören. In dieser Dissertation wird ein sehr allgemeines System zur Definition von Markierungen zur Bewertung von Wahrheit bzw. Vertrauen beschrieben, zusammen mit den sich daraus ergebenden kanonischen Definitionen des Modellbegriffs sowie der semantischen Folgerung für markierte Formeln. Die resultierenden markierten Logiken sind sehr ausdrucksstark und erlauben sowohl Vagheit als auch Unsicherheit als auch Kombinationen beider gradueller Konzepte zu repräsentieren. Semantiken solcher Logiken werden im Allgemeinen und für interessante Spezialfälle studiert.In this dissertation, the semantics of logical systems which are able to express vagueness and graded truth assessment as well as doubt and graded trust assessment are investigated from the point of view of mathematical logic. Traditionally, logics for modelling graded truth have been many-valued logics which allow truth values between 0 (false) and 1 (true). In applications, sometimes truth values are attached to formulae to assess the truth of the formula. In logics for modelling graded trust, usually trust (or plausibility, or possibility, or belief) degrees are attached to formulae from classical two-valued logic to assess the trust in the knowledge expressed by this formula. Several logical systems using labelled formulae (i. e. formulae to which some label is attached) have been described in the literature, with varying interpretations concerning structure and semantics of labels. In many cases, however, the meaning of a label is not precisely specified, casting doubt on what, from a semantic point of view, is really formalised by labelled formulae or a corresponding inference mechanism. Without a specific background theory for the meaning of labels (as is given, for instance, by probability theory), of course no canonical paradigm for specifying the structure and processing of labels exists. Consequently, several different such paradigms have been developed. Differences between these systems combined with the lack of a precisely defined semantics for labels have led to critique of such logical systems as a whole, because it must seem suspicious if from one and the same knowledge base of labelled formulae, it is possible to infer totally different results, without a clear semantic theory which can explain the differences. There have been attempts to clarify this situation, especially by distinguishing whether a system of labelled logical formulae is used for the representation of graded truth assessment or graded trust (or possibility, necessity, plausibility, uncertainty, belief ) assessment with respect to the states of affairs being modelled. Logical systems which can accomplish one or the other task have been studied and compared. In this dissertation, a very general approach to the definition of labels for expressing graded truth and graded trust is described. This definition gives rise to a canonical definition of the concepts of model and semantic consequence for the resulting logic of labelled formulae. The expressive power of such logics is very high. A label can express uncertainty about truth or trust or any combination of both. A systematic study of the semantics of these logical systems is given here, as well as a discussion and comparison of special cases

Alternatives for jet engine control

Research centered on basic topics in the modeling and feedback control of nonlinear dynamical systems is reported. Of special interest were the following topics: (1) the role of series descriptions, especially insofar as they relate to questions of scheduling, in the control of gas turbine engines; (2) the use of algebraic tensor theory as a technique for parameterizing such descriptions; (3) the relationship between tensor methodology and other parts of the nonlinear literature; (4) the improvement of interactive methods for parameter selection within a tensor viewpoint; and (5) study of feedback gain representation as a counterpart to these modeling and parameterization ideas

Discrete Mathematics and Symmetry

Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group

Estimation and control of non-linear and hybrid systems with applications to air-to-air guidance

Issued as Progress report, and Final report, Project no. E-21-67

Physical modelling of brass instruments using finite-difference time-domain methods

This work considers the synthesis of brass instrument sounds using time-domain numerical methods. The operation of such a brass instrument is as follows. The player's lips are set into motion by forcing air through them, which in turn creates a pressure disturbance in the instrument mouthpiece. These disturbances produce waves that propagate along the air column, here described using one spatial dimension, to set up a series of resonances that interact with the vibrating lips of the player. Accurate description of these resonances requires the inclusion of attenuation of the wave during propagation, due to the boundary layer effects in the tube, along with how sound radiates from the instrument. A musically interesting instrument must also be flexible in the control of the available resonances, achieved, for example, by the manipulation of valves in trumpet-like instruments. These features are incorporated into a synthesis framework that allows the user to design and play a virtual instrument. This is all achieved using the finite-difference time-domain method. Robustness of simulations is vital, so a global energy measure is employed, where possible, to ensure numerical stability of the algorithms. A new passive model of viscothermal losses is proposed using tools from electrical network theory. An embedded system is also presented that couples a one-dimensional tube to the three-dimensional wave equation to model sound radiation. Additional control of the instrument using a simple lip model as well a time varying valve model to modify the instrument resonances is presented and the range of the virtual instrument is explored. Looking towards extensions of this tool, three nonlinear propagation models are compared, and differences related to distortion and response to changing bore profiles are highlighted. A preliminary experimental investigation into the effects of partially open valve configurations is also performed