Set Theory and its Place in the Foundations of Mathematics:a new look at an old question

This paper reviews the claims of several main-stream candidates to be the foundations of mathematics, including set theory. The review concludes that at this level of mathematical knowledge it would be very unreasonable to settle with any one of these foundations and that the only reasonable choice is a pluralist one

A coalgebraic view of bar recursion and bar induction

We reformulate the bar recursion and induction principles in terms of recursive and wellfounded coalgebras. Bar induction was originally proposed by Brouwer as an axiom to recover certain classically valid theorems in a constructive setting. It is a form of induction on non- wellfounded trees satisfying certain properties. Bar recursion, introduced later by Spector, is the corresponding function defnition principle. We give a generalization of these principles, by introducing the notion of barred coalgebra: a process with a branching behaviour given by a functor, such that all possible computations terminate. Coalgebraic bar recursion is the statement that every barred coalgebra is recursive; a recursive coalgebra is one that allows defnition of functions by a coalgebra-to-algebra morphism. It is a framework to characterize valid forms of recursion for terminating functional programs. One application of the principle is the tabulation of continuous functions: Ghani, Hancock and Pattinson defned a type of wellfounded trees that represent continuous functions on streams. Bar recursion allows us to prove that every stably continuous function can be tabulated to such a tree where by stability we mean that the modulus of continuity is also continuous. Coalgebraic bar induction states that every barred coalgebra is well-founded; a wellfounded coalgebra is one that admits proof by induction

Coalgebra learning via duality

Automata learning is a popular technique for inferring minimal automata through membership and equivalence queries. In this paper, we generalise learning to the theory of coalgebras. The approach relies on the use of logical formulas as tests, based on a dual adjunction between states and logical theories. This allows us to learn, e.g., labelled transition systems, using Hennessy-Milner logic. Our main contribution is an abstract learning algorithm, together with a proof of correctness and termination

Set Theory and Structures

Set-theoretic and category-theoretic foundations represent different perspectives on mathematical subject matter. In particular, category-theoretic language focusses on properties that can be determined up to isomorphism within a category, whereas set theory admits of properties determined by the internal structure of the membership relation. Various objections have been raised against this aspect of set theory in the category-theoretic literature. In this article, we advocate a methodological pluralism concerning the two foundational languages, and provide a theory that fruitfully interrelates a `structural' perspective to a set-theoretic one. We present a set-theoretic system that is able to talk about structures more naturally, and argue that it provides an important perspective on plausibly structural properties such as cardinality. We conclude the language of set theory can provide useful information about the notion of mathematical structure

Skript zur Statistik in den Naturwissenschaften

SIGLEAvailable from TIB Hannover: RN 4868(59) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Asymptotic normality of goodness-of-fit statistics for sparse poisson data

Goodness-of-fit tests for discrete data and models with parameters to be estimated are usually based on Pearson's #chi#"2 or the Likelihood Ratio Statistic, which both are included in the family of Power-Divergence Statistics SD_#lambda#. It is known that SD_#lambda# is asymptotically #chi#"2 distributed for the common sampling schemes, which yield contingency tables being Poisson or conditional Poisson, e.g. product-multinomial, and for an asymptotic approach with the number of cells being fixed. Here a limiting normal distribution of SD_#lambda# for Poisson distributed J x K tables is presented considering an increasing cells approach, i.e. beside the total size the number of covariable groups J increases, whereas the number of categories K and the number of model parameters remains fixed. In contrast to the 'fixed cells' asymptotics an increase of all expected values is not required - the expectations of the cells may be large but need not be, which allows an application of the deduced tests to sparse data. The peculiarity of the here considered approach is that the underlying class of models to test does not specify the marginal distributions of the (covariable) groups and categories - only the associations, i.e. the odds ratios, are modelled with a finite number of parameters. One thus has to deal with an asymptotically infinite number of nuisance parameters. (orig.)SIGLEAvailable from TIB Hannover: RN 4868(51) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman

Estimation of nonparametric risk functions in matched case-control studies

In epidemiological studies one is interested in investigating the probability of disease depending on risk factors and in particular in detecting interactions of risk factors. Within the setting of parametric logistic regression interactions can be modeled only in a clumsy and limited way. Modeling the risk function nonparametrically, estimating it e.g. by a smoothing (thin plate) spline is attractive as a more explorative approach. For prospective studies this amounts to smoothing within the framework and distributional assumptions of generalized regression models (for binary observations). Case-control studies as retrospective studies with exposure to risk factors being observed do not immediately fit into this setting. In the special case of one-to-one matched studies however there is an appropriate likelihood again within the range of generalized models. Inferences will be illustrated using simulated and real data. (orig.)Available from TIB Hannover: RN 4868(49) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekSIGLEDEGerman

A formal derivation of the conditional likelihood for matched case-control studies

SIGLEAvailable from TIB Hannover: RN 4868(55) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDeutsche Forschungsgemeinschaft (DFG), Bonn (Germany)rev. ed.DEGerman