Kaplan classes of a certain family of functions

We give the complete characterization of members of Kaplan classes of products of power functions with all zeros symmetrically distributed in \(\mathbb{T} := \{z \in\mathbb{C} : |z| = 1\}\) and weakly monotonic sequence of powers. In this way we extend Sheil-Small’s theorem. We apply the obtained result to study univalence of antiderivative of these products of power functions.

The Monotonic Sequence Theorem and Measurement of Lengths and Areas in Axiomatic Non-Standard Hyperrational Analysis

This paper lies in the framework of axiomatic non-standard analysis based on the non-standard arithmetic axiomatic theory. This arithmetic includes actual infinite numbers. Unlike the non-standard model of arithmetic, this approach does not take models into account but uses an axiomatic research method. In the axiomatic theory of non-standard arithmetic, hyperrational numbers are defined as triplets of hypernatural numbers. Since the theory of hyperrational numbers and axiomatic non-standard analysis is mainly published in Russian, in this article we give a brief review of its basic concepts and required results. Elementary hyperrational analysis includes defining and evaluating such notions as continuity, differentiability and integral calculus. We prove that a bounded monotonic sequence is a Cauchy sequence. Also, we solve the task of line segment measurement using hyperrational numbers. In fact, this allows us to approximate real numbers using hyperrational numbers, and shows a way to model real numbers and real functions using hyperrational numbers and functions.

Invariant syllable skeleton, complex segments and word edges

San Duanmu'sSyllable Structure: The Limits of Variationraises a number of questions that are of general interest for phonological theory. Of special interest here are: the genesis and management of linearity in complex segments, the place of analogy (or paradigm uniformity) in grammar, the role of morphology in accounting for phonological patterns, the balance of static (distributional patterns) and dynamic (phonological processes) evidence for syllable structure, the role of stress in syllabification, and the import of corpus-based data for phonological analysis. In each case, Duanmu's proposals are evaluated according to their intrinsic consistency, the empirical record and the relevant body of literature. Alternative ways of handling the phenomena are offered, and these are fairly traditional in most cases. Duanmu's book is particularly relevant in the current constitution of the field where the see-saw movement between computation and representations seems to swing back in direction of the latter after having long been immobilised on the computational end. Standing clearly on the representational side, the theory exposed in the book aims to show that all surface strings may be reduced to a fixed and invariant syllable template, C(onsonant)V(owel)X. This enterprise is interesting especially in presence of another representationally-oriented theory, CVCV (Lowenstamm 1996, Scheer 2004), which also aims at reducing surface variation to an invariant syllabic skeleton, made of a monotonic sequence of CV units. However, the CVX and the CVCV templates are quite distinct, and the strategies that are used in order to accommodate the surface string are opposite (shrinking in the former case, expanding in the latter).

Axiomatisations of the average and a further generalisation of monotonic sequences

A bounded monotonic sequence is convergent. This paper shows that a bounded sequence which is g-monotonic (to be defined) also converges. The proof generalises one attributed to Professor R. A. Rankin by Copson [1]. The theorem requires two definitions: the first axiomatises the notion of “average“ and the second generalises the concept of monotonicity.

On a generalisation of monotonic sequences

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequalitythen it is convergent. This I was able to prove by considering the corresponding difference equation