It is true in the Solovay model that every countable ordinal-definable set of sets of reals contains only ordinal-definable elements.

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the secondorder theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordin...

We present ordinal measures of association for establishing visual correspondence in images. Linear correspondence measures like correlation and the sum of squared differences are known to be fragile. Ordinal measures, which are based on relative ordering of intensity values in windows, have demonstrable robustness to depth discontinuities, occlusion, and noise. The relative ordering of intensi...

An α-automaton (for α some ordinal) is an automaton similar to a Muller automaton that processes words of length α. A structure is called α-automatic if it can be presented by α-automata (completely analogous to the notion of automatic structures which can be presented by the well-known finite automata). We call a structure ordinal-automatic if it is α-automatic for some ordinal α. We continue ...

In this paper we address the problem of aggregating outranking statements from multiple preference criteria of ordinal significance. The concept of ordinal concordance of a global outranking situation is defined and an operational test for its presence is developed. Finally, we propose a new kind of robustness analysis for global outranking statements integrating classical dominance, ordinal an...

We study the problem of ordinal embedding: given a set of ordinal constraints of the form distance(i, j) < distance(k, l) for some quadruples (i, j, k, l) of indices, the goal is to construct a point configuration x̂1, ..., x̂n in R that preserves these constraints as well as possible. Our first contribution is to suggest a simple new algorithm for this problem, Soft Ordinal Embedding. The key fe...

A didactic on latent growth curve modeling for ordinal outcomes is presented. The conceptual aspects of modeling growth with ordinal variables and the notion of threshold invariance are illustrated graphically using a hypothetical example. The ordinal growth model is described in terms of 3 nested models: (a) multivariate normality of the underlying continuous latent variables (yt) and its rela...

We present a novel ordinal co-occurrence matrix framework for the purpose of content-based texture retrieval. Several particularizations of the framework will be derived and tested for retrieval purposes. Features obtained using the framework represent the occurrence frequency of certain ordinal relationships at different distances and orientations. In the ordinal co-occurrence matrix framework...

Let α be an ordinal. Then α­1, the set of all ordinals equal to or preceding α, is a compact Hausdorff topological space. The space of all real-valued continuous functions on α­1 is commonly denoted by C(α). In this paper, we study the symmetric sequence spaces which are isomorphic to subspaces of C(α) for some countable ordinal α. In the next section, we prove a general result which shows that...

Most discussions of ordinal variables in the sociological literature debate the suitability of linear regression and structural equation methods when some variables are ordinal. Largely ignored in these discussions are methods for ordinal variables that are natural extensions of probit and logit models for dichotomous variables. If ordinal variables are discrete realizations of unmeasured conti...