On Series of Ordinals and Combinatorics

This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑_bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑_bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element subsets for each natural number n ≥ 2. Moreover, series ∑_β<αf(β) are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic.     

Importance of the choice of assumptions and models in the estimation of analytical quality specifications

The validity of any model depends on its ability to imagine the situation or problem to which it is applied. Further, the assumptions made in relation to the model are determining for the actual outcome. Within the field of clinical biochemistry a lot of models for analytical quality specifications, based on a variety of concepts and ’clinical settings’, have been proposed. A hierarchical structure for application of these approaches and models has been agreed on at several occasions in 1999. In this hierarchy, the highest rank is given to evaluation of analytical quality specifications based on ’clinical settings’/’clinical outcome’ models, followed by specifications based on biological variation and on ’clinicians opinions’. This contribution, deals with the problems of combining random and systematic errors and the implications of application of different models to a variety of clinical settings. Received: 1 June, 2002 Accepted: 17 July 2002 Presented at the European Conference on Quality in the Spotlight in Medical Laboratories, 7–9 October 2001, Antwerp, Belgium     

Graphical Models for Ordinal Data

This article considers a graphical model for ordinal variables, where it is assumed that the data are generated by discretizing the marginal distributions of a latent multivariate Gaussian distribution. The relationships between these ordinal variables are then described by the underlying Gaussian graphical model and can be inferred by estimating the corresponding concentration matrix. Direct estimation of the model is computationally expensive, but an approximate EM-like algorithm is developed to provide an accurate estimate of the parameters at a fraction of the computational cost. Numerical evidence based on simulation studies shows the strong performance of the algorithm, which is also illustrated on datasets on movie ratings and an educational survey.     

Continuous Utility Representation Theorems in Arbitrary Concrete Categories

In this paper the continuous utility representation problem will be discussed in arbitrary concrete categories. In particular, generalizations of the utility representation theorems of Eilenberg, Debreu and Estévez and Hervés will be presented that also hold if the codomain of a utility function is an arbitrary totally ordered set and not just the real line. In addition, we shall prove and apply a general result on the characterization of structures that have the property that every continuous total preorder has a continuous utility representation. Finally, generalizations of the utility representation theorems of Debreu and Eilenberg will be discussed that are valid if we consider arbitrary binary relations and allow a utility function to have values in an arbitrary totally ordered set.      

A potential approach for ordinal games

We introduce the class of ordinal games with a potential, which are characterized by the absence of weak improvement cycles, the same condition used by Voorneveld and Norde (1997) for ordinal potential games.     

A synergy of multicriteria techniques to assess additive value models

The assessment of additive value functions in Multicriteria Decision Aid (MCDA) has to face issues of legitimacy and technical difficulties when real decision makers are involved. This paper presents a synergy of three complementary techniques to assess additive models on the whole criteria space. The synergy includes a revised MACBETH technique, the standard MAUT trade-off analysis and UTA-based methods for the assessment of both the marginal value functions and the weighting factors. The paper uses a set of original robustness measures and rules associated with revised MACBETH and UTA in order to manage multiple linear programming solutions and to extract robust conclusions from them. Finally, to illustrate the methods’ synergy, an application example is presented, dealing with the planning of metro extension lines.     

ORDINAL NUMBERS IN ARITHMETIC PROGRESSION

The class of all ordinal numbers can be partitioned into two subclasses in such a way that neither subclass contains an arithmetic progression of order type ω, where an arithmetic progression of order type τ means an increasing sequence of ordinal numbers (ß + δγ)γ<γ<>r, δ ≠ 0.     

Stochastic DEA with ordinal data applied to a multi-attribute pricing problem

Multiple attribute pricing problems are highly challenging due to the dynamic and uncertain features in the associated market. In this paper, we address the condominium multiple attribute pricing problem using data envelopment analysis (DEA). In this study, we simultaneously consider stochastic variables, non-discretionary variables, and ordinal data, and present a new type of DEA model. Based on our proposed DEA, an effective performance measurement tool is developed to provide a basis for understanding the condominium pricing problem, to direct and monitor the implementation of pricing strategy, and to provide information regarding the results of pricing efforts for units sold as well as insights for future building design. A case study is executed on a leading Canadian condominium developer.     

Injections into function spaces over ordinals

We investigate when and how function spaces over subspaces of ordinals admit continuous injections into each other. To formulate our results let τ be an uncountable regular cardinal. We prove, in particular, that: (1) If A and B are disjoint stationary subsets of τ then Cp(A) does not admit a continuous injection into Cp(B); (2) For A⊂ω₁, admits a continuous injection into iff A is countable or ω₁ embeds into A (which, in its turn, is equivalent to the statement “ embeds into ”).