Stevens postulated that we can use the responses of a participant in a ratio scaling experiment directly to construct a psychophysical function representing the participant’s sensations. Although Stevens’ methods of constructing measurement scales are widely used in the behavioral sciences, the problem of which scale type is appropriate to describe ratio scaling data is still unresolved. To deal with this problem, we develop a theoretical framework to specify the scale type attained by Stevens’ direct scaling methods. It is shown, under fairly mild background assumptions, that the behavioral axioms presented in this paper are necessary and sufficient for the psychophysical functions to be ordinal-, interval-, log-interval-, or ratio-scales. Furthermore, suggestions on how to test these behavioral axioms are provided.

We prove that immersions of planar domains are uniquely specified by their Jacobian determinant, curl function and boundary values. This settles the two-dimensional version of an outstanding conjecture related to a particular grid generation method in computer graphics.

In the context of self-stabilizing processes, that is processes attracted by their ownlaw, living in a potential landscape, we investigate different properties of the invariantmeasures. The interaction between the process and its law leads to nonlinear stochasticdifferential equations. In [S. Herrmann and J. Tugaut. Electron. J. Probab.15 (2010) 2087–2116], the authors proved that, for linearinteraction and under suitable conditions, there exists a unique symmetric limit measureassociated to the set of invariant measures in the small-noise limit. The aim of thisstudy is essentially to point out that this statement leads to the existence, as the noiseintensity is small, of one unique symmetric invariant measure for the self-stabilizingprocess. Informations about the asymmetric measures shall be presented too. The main keyconsists in estimating the convergence rate for sequences of stationary measures usinggeneralized Laplace’s method approximations.

Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón–Lozanovskiĭ construction. Factorization theorems for operators in spaces more general than the Lebesgue ${{L}^{p}}$ spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt ${{A}_{p}}$ weights to reflexive Orlicz spaces. However, it turns out that for the scales far from ${{L}^{p}}$-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón–Lozanovskiĭ construction are involved in the proofs.