Ackermann functions and transfinite ordinals

A set of binary operators are defined and shown to be equivalent to Ackermann functions. The same set of operators are used to develop a notation for writing the sequence of transfinite ordinals.     

Primitive iteration and unary functions

Programming practice suggests a general notion of primitive iteration which subsumes the for-until-do construct as well as all known primative iteration operators. This leads to new iterative characterizations of primitive computable functions usable in computer science.     

Bounded iteration and unary functions

The set of unary functions of complexity classes defined by using bounded primitive recursion is inductively characterized by means of bounded iteration. Elementary unary functions, linear space computable unary functions and polynomial space computable unary functions are then inductively characterized using only composition and bounded iteration. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Regular splittings and monotone iteration functions

Summary In this paper we introduce the set of so-called monotone iteration functions (MI-functions) belonging to a given function. We prove necessary and sufficient conditions in order that a given MI-function is (in a precisely defined sense) at least as fast as a second one.Regular splittings of a function which were initially introduced for linear functions by R.S. Varga in 1960 are generating MI-functions in a natural manner.For linear functions every MI-function is generated by a regular splitting. For nonlinear functions, however, this is generally not the case.     

On topologization of transfinite convergence of sequences of real functions

Summary Let κ be an infinite regular cardinal. We are concerned with the question when the κ-convergence is topologyzable. In particular, we show that if<span style='font-size:10.0pt;font-family:"Lucida Sans Unicode"'>cis regular then<span style='font-size:10.0pt; font-family:"Lucida Sans Unicode"'>c-convergence is topologyzable by the <<span style='font-size:10.0pt;font-family: "Lucida Sans Unicode"'>c-box topology on R^R. Thus under CH the transfinite convergence is topologyzable. On the other hand, under MA+?CH the transfinite convergence is not topologyzable.     

Additive functions on arithmetic progressions with large moduli

In this paper we study additive functions on arithmetic progressions with large moduli. We are able to improve some former results given by Elliott.     

General divisor functions in arithmetic progressions to large moduli

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.     

Adequateness and interpretability of objective functions in ordinal data analysis

Objective functions that are applied in ordinal data analysis must be adequate, i.e. carefully adapted to the structure of the observed data. In addition, any analysis of data that is based upon objective functions must lead to interpretable results. After a general characterization of adequate objective functions in ordinal data analysis, therefore, the particular problems of constructing adequate and interpretable dissimilarity coefficients and correlation coefficients in ordinal data analysis, stress measures (stress functions) in non-metric scaling and generalized stress measures or correlation coefficients in any theory of rank estimation will be discussed.