Courcelle's Theorem - A Game-Theoretic Approach

Courcelle's Theorem states that every problem definable in Monadic Second-Order logic can be solved in linear time on structures of bounded treewidth, for example, by constructing a tree automaton that recognizes or rejects a tree decomposition of the structure. Existing, optimized software like the MONA tool can be used to build the corresponding tree automata, which for bounded treewidth are of constant size. Unfortunately, the constants involved can become extremely large - every quantifier alternation requires a power set construction for the automaton. Here, the required space can become a problem in practical applications. In this paper, we present a novel, direct approach based on model checking games, which avoids the expensive power set construction. Experiments with an implementation are promising, and we can solve problems on graphs where the automata-theoretic approach fails in practice.Comment: submitte

Critical examination of cohesive-zone models in the theory of dynamic fracture

We have examined a class of cohesive-zone models of dynamic mode-I fracture, looking both at steady-state crack propagation and its stability against out-of-plane perturbations. Our work is an extension of that of Ching, Langer, and Nakanishi (CLN) (Phys. Rev. E, vol. 53, no. 3, p. 2864 (1996)), who studied a non-dissipative version of this model and reported strong instability at all non-zero crack speeds. We have reformulated the CLN theory and have discovered, surprisingly, that their model is mathematically ill-posed. In an attempt to correct this difficulty and to construct models that might exhibit realistic behavior, we have extended the CLN analysis to include dissipative mechanisms within the cohesive zone. We have succeeded to some extent in finding mathematically well posed systems; and we even have found a class of models for which a transition from stability to instability may occur at a nonzero crack speed via a Hopf bifurcation at a finite wavelength of the applied perturbation. However, our general conclusion is that these cohesive-zone models are inherently unsatisfactory for use in dynamical studies. They are extremely difficult mathematically, and they seem to be highly sensitive to details that ought to be physically unimportant.Comment: 19 pages, REVTeX 3.1, epsf.sty, also available at http://itp.ucsb.edu/~lobkovs

Über kurz oder lang : welche Rolle spielt der Anlagehorizont bei Investitionsentscheidungen?

Die Frage, wie das Risiko einer Investition durch den Anlagehorizont beeinflusst wird, ist insbesondere im Rahmen von Altersvorsorgeentscheidungen von zentraler Bedeutung. In der Anlagepraxis wird häufig auf das Gesetz der großen Zahlen verwiesen, das dafür sorgt, dass eine hinreichend langfristige Investition fast sicher die positive Durchschnittsrendite der Anlage realisiert und damit praktisch risikolos wird. Dem entgegen stehen theoretische Überlegungen, die z.B. aus Sicht der Erwartungsnutzentheorie eine Irrelevanz des Anlagehorizontes propagieren. Ziel der vorliegenden Arbeit ist es, die wichtigsten Argumente in der Diskussion zwischen Praktikern und Theoretikern strukturiert darzustellen, die Ursachen der unterschiedlichen Ergebnisse herauszuarbeiten und damit zu einer Klärung des Sachverhalts beizutragen. Dabei soll auch verdeutlicht werden, warum die Irrelevanz des Anlagehorizontes kein allgemeines Resultat der Erwartungsnutzentheorie ist, sondern auf einer sehr speziellen Annahmenkonstellation berut. Durch eine Erweiterung der Modelle um Aspekte wie Background-Risiko und autokorrelierte Renditen wird die erwartungsnutzenbasierte Analyse nicht nur praxisnäher, sie führt auch zu differenzierteren und keineswegs stets im Widerspruch zur Praktikermeinung stehenden Ergebnissen

Capillary-Wave Model for the Solidification of Dilute Binary Alloys

Starting from a phase-field description of the isothermal solidification of a dilute binary alloy, we establish a model where capillary waves of the solidification front interact with the diffusive concentration field of the solute. The model does not rely on the sharp-interface assumption, and includes non-equilibrium effects, relevant in the rapid-growth regime. In many applications it can be evaluated analytically, culminating in the appearance of an instability which, interfering with the Mullins-Sekerka instability, is similar to that, found by Cahn in grain-boundary motion.Comment: 17 pages, 12 figure

Methyl 5-chloro-2-hydr­oxy-3-(4-methoxy­phen­yl)-4,6-dimethyl­benzoate

In the title compound, C17H17ClO4, the dihedral angle between the mean planes of the two benzene rings is 65.92 (5)°. The methyl ester group lies within the ring plane [deviations of O atoms from the plane = −0.051 (2) and 0.151 (2) Å] due to an intra­molecular O—H⋯O hydrogen bond. In the crystal, mol­ecules are held together by rather weak non-classical inter­molecular C—H⋯O hydrogen bonds, resulting in dimeric units about inversion centers, forming eight- and ten-membered ring systems as R 2 2(8) and R 2 2(10) motifs

Lower Bounds on the Complexity of MSO_1 Model-Checking

One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs. Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width