ALBERT

Description: Given a Calderón–Zygmund singular integral operator, it is an open question whether the Fefferman inequality holds true with u as an arbitrary weight. This inequality is known as the Muckenhoupt–Wheeden conjecture. In this paper we prove that, for a wider class of operators called Rubio de Francia operators, if || u || ≤1, then where the function is slightly bigger than the identity. Applications of this inequality in the setting of rearrangement invariant spaces are given.

Description: We show that whenever m ≤ n , the space of all continuous rational maps from CP m to CP n has the same homology as the space of all continuous maps between these spaces in dimensions smaller than d (2 n – 2 m + 1) – 1. This improves the result of the paper ‘Spaces of rational maps and the Stone–Weierstrass theorem’ ( Topology 45 (2006), 281–293).

Description: A theorem of Davenport, Mirsky, Newman and Rado shows that there does not exist an exact covering system with distinct moduli. Motivated by this, we raise the question whether or not this is true for covering systems in algebraic number fields. We provide an affirmative answer for certain quadratic fields.

Description: The caustic of a smooth surface in the Euclidean 3-space is the envelope of the normal rays to the surface. It is also the locus of the centres of curvature (the focal points) of the surface. This is why it is also referred to as the focal set of the surface. It has Lagrangian singularities and its generic models are given in [V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of Differentiable Maps , Vol. I, Birkhäuser, Boston, 1986]. The aim of this paper is to define the caustic C ( M ) of a smooth surface M embedded in the Minkowski 3-space and to study its geometry. We denote by the locus of degeneracy (LD) the locus of points on M where the metric is degenerate. If M is a closed surface then its LD is not empty. At a point on the LD the ‘normal’ line to M is lightlike and is tangent to M . Also, the focal set of M is not defined at points on the LD. We define the caustic of M as the bifurcation set of the family of distance-squared functions on M . Then C ( M ) coincides with the focal set of M \ LD and provides an extension of the focal set to the LD. We study the local behaviour of the metric on C ( M ).

Description: Combining probability and first-order logic has been the subject of intensive research during the last 10 years. This artical introduces first-order probabilistic conditional logic (FO-PCL), a first-order extension of a propositional probabilistic logic formalism, which allows for the adequate representation of probabilistic if - then -rules. We demonstrate that our novel formalism allows to represent uncertain knowledge that cannot easily be represented by other formalisms combining first-order logic and probability. Furthermore, as the representation of the models of FO-PCL requires solving a complex entropy-optimization problem, we develop syntactic conditions for its simplification.

Description: This article presents KR eator , a versatile integrated development environment for probabilistic inductive logic programming currently under development. The area of probabilistic inductive logic programming (or statistical relational learning) aims at applying probabilistic methods of inference and learning in relational or first-order representations of knowledge. In the past ten years the community brought forth a lot of proposals to deal with problems in that area, which mostly extend existing propositional probabilistic methods like Bayes Nets and Markov Networks on relational settings. Only few developers provide prototypical implementations of their approaches and the existing applications are often difficult to install and to use. Furthermore, due to different languages and frameworks used for the development of different systems the task of comparing various approaches becomes hard and tedious. KR eator aims at providing a common and simple interface for representing, reasoning and learning with different relational probabilistic approaches. It is a general integrated development environment which enables the integration of various frameworks within the area of probabilistic inductive logic programming and statistical relational learning. Currently, KR eator implements Bayesian logic programs, Markov logic networks and relational maximum entropy under grounding semantics. More approaches will be implemented in the near future or can be implemented by researchers themselves as KR eator is open-source and available under public license. In this article, we provide some background on probabilistic inductive logic programming and statistical relational learning and illustrate the usage of KR eator on several examples using the three approaches currently implemented in KR eator . Furthermore, we give an overview on its system architecture.

Description: A special feature of programs in the action language Golog are non-deterministic constructs such as non-deterministic choice of actions or arguments. It has been shown that in the presence of stochastic actions and rewards reinforcement learning techniques can be applied to obtain optimal choices for those choice-points. In order to avoid an explosion of the state space, an abstraction mechanism is employed that computes first-order state descriptions for the given program. Intuitively, the idea is to generate abstract descriptions that group together states for which the expected reward of executing the program is the same. A current limitation is that a non-deterministic choice of arguments can be handled only if the possible candidates are known in advance. In this article we show how this restriction can be lifted. We also show how a first-order variant of binary decision diagrams can be used to efficiently compute first-order state abstractions. Moreover, we give a completely declarative specification of a learning Golog interpreter that incorporates the presented state-abstraction mechanisms.

Description: Log A B is a family of logics of belief. It holds a middle ground between the expressive, but prone to paradox, syntactical first-order theories and the often inconvenient, but safe, modal approaches. In this report, the syntax and semantics of Log A B are presented. Log A B is algebraic in the sense that it is a language of only terms; there is no notion of a formula, only proposition-denoting terms. The domain of propositions is taken to be a Boolean algebra, which renders classical truth conditions and definitions of consequence and validity theorems about Log A B structures. Log A B is shown to be sufficiently expressive to accommodate complex patterns of reasoning about belief while remaining paradox-free. A number of results are proved regarding paradoxical self-reference. They are shown to strengthen previous results, and to point to possible new approaches to circumventing paradoxes in syntactical theories of belief.

Description: The principle of maximum entropy has proven to be a powerful approach for commonsense reasoning in probabilistic conditional logics on propositional languages. Due to this principle, reasoning is performed based on the unique model of a knowledge base that has maximum entropy. This kind of model-based inference fulfils many desirable properties for inductive inference mechanisms and is usually the best choice for reasoning from an information theoretical point of view. However, the expressive power of propositional formalisms for probabilistic reasoning is limited and in the past few years many proposals have been given for probabilistic reasoning in relational settings. It seems to be a common view that in order to interpret probabilistic first-order sentences, either a statistical approach that counts (tuples of) individuals has to be used, or the knowledge base has to be grounded to make a possible worlds semantics applicable, for a subjective interpretation of probabilities. Most of these proposals of the second type rely on extensions of traditional probabilistic models like Bayes nets or Markov networks whereas there are only few works on first-order extensions of probabilistic conditional logic. Here, we take an approach of lifting maximum entropy methods to the relational case by employing a relational version of probabilistic conditional logic. First, we propose two different semantics and model theories for interpreting first-order probabilistic conditional logic. We address the problems of ambiguity that are raised by the difference between subjective and statistical views, and develop a comprehensive list of desirable properties for inductive model-based probabilistic inference in relational frameworks. Finally, by applying the principle of maximum entropy in the two different semantical frameworks, we obtain inference operators that fulfill these properties and turn out to be reasonable choices for reasoning in first-order probabilistic conditional logic.