Restarting automata with restricted utilization of auxiliary symbols

The restarting automaton is a restricted model of computation that was introduced by Jancar et al. to model the so-called analysis by reduction, which is a technique used in linguistics to analyze sentences of natural languages. The most general models of restarting automata make use of auxiliary symbols in their rewrite operations, although this ability does not directly correspond to any aspect of the analysis by reduction. Here we put restrictions on the way in which restarting automata use auxiliary symbols, and we investigate the influence of these restrictions on their expressive power. In fact, we consider two types of restrictions. First, we consider the number of auxiliary symbols in the tape alphabet of a restarting automaton as a measure of its descriptional complexity. Secondly, we consider the number of occurrences of auxiliary symbols on the tape as a dynamic complexity measure. We establish some lower and upper bounds with respect to these complexity measures concerning the ability of restarting automata to recognize the (deterministic) context-free languages and some of their subclasses.

Anisotropic adaptive resolution of boundary layers for heat conduction problems

We deal with the numerical solution of heat conduction problems featuring steep gradients. In order to solve the associated partial differential equation a finite volume technique is used and unstructured grids are employed. A discrete maximum principle for triangulations of a Delaunay type is developed. To capture thin boundary layers incorporating steep gradients an anisotropic mesh adaptation technique is implemented. Computational tests are performed for an academic problem where the exact solution is known as well as for a real world problem of a computer simulation of the thermoregulation of premature infants.

Shrinking restarting automata

Restarting automata are a restricted model of computation that was introduced by Jancar et.al. to model the so-called analysis by reduction. A computation of a restarting automaton consists of a sequence of cycles such that in each cycle the automaton performs exactly one rewrite step, which replaces a small part of the tape content by another, even shorter word. Thus, each language accepted by a restarting automaton belongs to the complexity class $CSL cap NP$. Here we consider a natural generalization of this model, called shrinking restarting automaton, where we do no longer insist on the requirement that each rewrite step decreases the length of the tape content. Instead we require that there exists a weight function such that each rewrite step decreases the weight of the tape content with respect to that function. The language accepted by such an automaton still belongs to the complexity class $CSL cap NP$. While it is still unknown whether the two most general types of one-way restarting automata, the RWW-automaton and the RRWW-automaton, differ in their expressive power, we will see that the classes of languages accepted by the shrinking RWW-automaton and the shrinking RRWW-automaton coincide. As a consequence of our proof, it turns out that there exists a reduction by morphisms from the language class $cL(RRWW)$ to the class $cL(RWW)$. Further, we will see that the shrinking restarting automaton is a rather robust model of computation. Finally, we will relate shrinking RRWW-automata to finite-change automata. This will lead to some new insights into the relationships between the classes of languages characterized by (shrinking) restarting automata and some well-known time and space complexity classes.

Modellierung und numerische Simulation der Thermoregulation von Früh- und Neugeborenen

Bei frühgeborenen Säuglingen spielt die Thermoregulation zur Aufrechterhaltung einer überlebenswichtigen Körpertemperatur durch Wärmeproduktion, -abgabe bzw. -aufnahme eine entscheidende Rolle. Der Einsatz moderner Inkubatoren soll die körpereigenen Thermoregulatoren unterstützen, und es ist im Hinblick auf verschiedene medizinische Fragestellungen wünschenswert, diesen Prozess modellieren zu können. Wir stellen ein einfaches Modell auf der Basis von partiellen Differentialgleichungen vor und beschreiben detailliert die numerische Simulation mit Hilfe einer Finite-Volumen-Methode. Dazu wird ein zweidimensionales Modell eines Frühgeborenen trianguliert und das Modell diskretisiert. Zahlreiche numerische Resultate zeigen die Qualität unseres Modells.

Learning analysis by reduction from positive data

Analysis by reduction is a linguistically motivated method for checking correctness of a sentence. It can be modelled by restarting automata. In this paper we propose a method for learning restarting automata which are strictly locally testable (SLT-R-automata). The method is based on the concept of identification in the limit from positive examples only. Also we characterize the class of languages accepted by SLT-R-automata with respect to the Chomsky hierarchy.

Artificial boundary conditions for the Stokes and Navier-Stokes equations in domains that are layer-like at infinity

Artificial boundary conditions are presented to approximate solutions to Stokes- and Navier-Stokes problems in domains that are layer-like at infinity. Based on results about existence and asymptotics of the solutions v^infinity, p^infinity to the problems in the unbounded domain Omega the error v^infinity - v^R, p^infinity - p^R is estimated in H^1(Omega_R) and L^2(Omega_R), respectively. Here v^R, p^R are the approximating solutions on the truncated domain Omega_R, the parameter R controls the exhausting of Omega. The artificial boundary conditions involve the Steklov-Poincare operator on a circle together with its inverse and thus turn out to be a combination of local and nonlocal boundary operators. Depending on the asymptotic decay of the data of the problems, in the linear case the error vanishes of order O(R^{-N}), where N can be arbitrarily large.

A crack on the interface of piezo-electric bodies

Singularities of elastic and electric fields are investigated at the tip of a crack on the interface of two anisotropic piezo-electric media under various boundary conditions on the crack surfaces. The Griffith formulae are obtained for increments of energy functionals due to growth of the crack and the notion of the energy release matrix is introduced. Normalization conditions for bases of singular solution are proposed to adapt them to the energy, stress, and deformation fracture criteria. Connections between these bases are determined and additional properties of the deformation basis related to the notion of electric surface enthalpy are established.

An approximation method using approximate approximations

The aim of this paper is to extend the method of approximate approximations to boundary value problems. This method was introduced by V. Maz'ya in 1991 and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the Laplace equation in two dimensions using approximate approximations. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström's method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.

Asymptotics of number fields and the Cohen-Lenstra heuristics

We study the asymptotics conjecture of Malle for dihedral groups Dl of order 2l, where l is an odd prime. We prove the expected lower bound for those groups. For the upper bounds we show that there is a connection to class groups of quadratic number fields. The asymptotic behavior of those class groups is predicted by the Cohen-Lenstra heuristics. Under the assumption of this heuristic we are able to prove the expected upper bounds.

Orthogonal polynomials and recurrence equations, operator equations and factorization

This article surveys the classical orthogonal polynomial systems of the Hahn class, which are solutions of second-order differential, difference or q-difference equations. Orthogonal families satisfy three-term recurrence equations. Example applications of an algorithm to determine whether a three-term recurrence equation has solutions in the Hahn class - implemented in the computer algebra system Maple - are given. Modifications of these families, in particular associated orthogonal systems, satisfy fourth-order operator equations. A factorization of these equations leads to a solution basis.

Subpixel-Filterung für eine autostereoskopische Multiperspektiven-3-D-Darstellung hoher Qualität

Die stereoskopische 3-D-Darstellung beruht auf der naturgetreuen Präsentation verschiedener Perspektiven für das rechte und linke Auge. Sie erlangt in der Medizin, der Architektur, im Design sowie bei Computerspielen und im Kino, zukünftig möglicherweise auch im Fernsehen, eine immer größere Bedeutung. 3-D-Displays dienen der zusätzlichen Wiedergabe der räumlichen Tiefe und lassen sich grob in die vier Gruppen Stereoskope und Head-mounted-Displays, Brillensysteme, autostereoskopische Displays sowie echte 3-D-Displays einteilen. Darunter besitzt der autostereoskopische Ansatz ohne Brillen, bei dem N≥2 Perspektiven genutzt werden, ein hohes Potenzial. Die beste Qualität in dieser Gruppe kann mit der Methode der Integral Photography, die sowohl horizontale als auch vertikale Parallaxe kodiert, erreicht werden. Allerdings ist das Verfahren sehr aufwendig und wird deshalb wenig genutzt. Den besten Kompromiss zwischen Leistung und Preis bieten präzise gefertigte Linsenrasterscheiben (LRS), die hinsichtlich Lichtausbeute und optischen Eigenschaften den bereits früher bekannten Barrieremasken überlegen sind. Insbesondere für die ergonomisch günstige Multiperspektiven-3-D-Darstellung wird eine hohe physikalische Monitorauflösung benötigt. Diese ist bei modernen TFT-Displays schon recht hoch. Eine weitere Verbesserung mit dem theoretischen Faktor drei erreicht man durch gezielte Ansteuerung der einzelnen, nebeneinander angeordneten Subpixel in den Farben Rot, Grün und Blau. Ermöglicht wird dies durch die um etwa eine Größenordnung geringere Farbauflösung des menschlichen visuellen Systems im Vergleich zur Helligkeitsauflösung. Somit gelingt die Implementierung einer Subpixel-Filterung, welche entsprechend den physiologischen Gegebenheiten mit dem in Luminanz und Chrominanz trennenden YUV-Farbmodell arbeitet. Weiterhin erweist sich eine Schrägstellung der Linsen im Verhältnis von 1:6 als günstig. Farbstörungen werden minimiert, und die Schärfe der Bilder wird durch eine weniger systematische Vergrößerung der technologisch unvermeidbaren Trennelemente zwischen den Subpixeln erhöht. Der Grad der Schrägstellung ist frei wählbar. In diesem Sinne ist die Filterung als adaptiv an den Neigungswinkel zu verstehen, obwohl dieser Wert für einen konkreten 3-D-Monitor eine Invariante darstellt. Die zu maximierende Zielgröße ist der Parameter Perspektiven-Pixel als Produkt aus Anzahl der Perspektiven N und der effektiven Auflösung pro Perspektive. Der Idealfall einer Verdreifachung wird praktisch nicht erreicht. Messungen mit Hilfe von Testbildern sowie Schrifterkennungstests lieferten einen Wert von knapp über 2. Dies ist trotzdem als eine signifikante Verbesserung der Qualität der 3-D-Darstellung anzusehen. In der Zukunft sind weitere Verbesserungen hinsichtlich der Zielgröße durch Nutzung neuer, feiner als TFT auflösender Technologien wie LCoS oder OLED zu erwarten. Eine Kombination mit der vorgeschlagenen Filtermethode wird natürlich weiterhin möglich und ggf. auch sinnvoll sein.

On the equivariant Tamagawa number conjecture for abelian extensions of a quadratic imaginary field

Let k be a quadratic imaginary field, p a prime which splits in k/Q and does not divide the class number hk of k. Let L denote a finite abelian extention of k and let K be a subextention of L/k. In this article we prove the p-part of the Equivariant Tamagawa Number Conjecture for the pair (h0(Spec(L)),Z[Gal(L/K)]).

Stability of preconditioned finite volume schemes at low Mach numbers

In [4], Guillard and Viozat propose a finite volume method for the simulation of inviscid steady as well as unsteady flows at low Mach numbers, based on a preconditioning technique. The scheme satisfies the results of a single scale asymptotic analysis in a discrete sense and comprises the advantage that this can be derived by a slight modification of the dissipation term within the numerical flux function. Unfortunately, it can be observed by numerical experiments that the preconditioned approach combined with an explicit time integration scheme turns out to be unstable if the time step Dt does not satisfy the requirement to be O(M2) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Dt=O(M), M to 0, which results from the well-known CFL-condition. We present a comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M-2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. Thereby, we present statements for both the standard preconditioner used by Guillard and Viozat [4] and the more general one due to Turkel [21]. The theoretical results are after wards confirmed by numerical experiments.

An algebraic proof of Iitaka's conjecture C2,1

We give a proof of Iitaka's conjecture C2,1 using only elementary methods from algebraic geometry.