Constructing quasisymmetric functions via graph-directed iterated function systems

Using Tukia’s method for representing a quasisymmetric function as a quasisymmetric sieve, we generalize his modification to the Salem scheme and find a sufficient condition for the collection of functions that realize a structure parametrization of a graph-directed function system of a particular form (a one-dimensional multizipper) to consist of quasisymmetric functions. We give an asymptotically sharp estimate for the quasisymmetry coefficient of these functions in terms of the dilation coefficients of the mappings constituting a given multizipper, which yields a substantially more general method for constructing quasisymmetric functions than Tukia’s construction.     

Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covitz-Nadler sense. In this situation, such operators T possess fixed points satisfying the relation x∈Tx. We introduce an iterative method involving projections that guarantees convergence from any starting point x₀∈X to a point x∈XT, the set of all fixed points of a multifunction operator T. We also prove a continuity result for fixed point sets XT as well as a “generalized collage theorem” for contractive multifunctions. These results can then be used to solve inverse problems involving contractive multifunctions. Two applications of contractive multifunctions are introduced: (i) integral inclusions and (ii) iterated multifunction systems.     

On generalized cluster sets of functions and multifunctions

This paper introduces and studies generalized cluster sets (g-cluster sets) of functions and multifunctions on GTS, which unifies the existing notions of cluster sets, θ-cluster sets, δ-cluster sets, S-cluster sets, s-cluster sets, p-cluster sets and many more. Several properties of the functions and multifunctions as well as their range and domain spaces are observed via degeneracies of their g-cluster sets. Characterizations of g-cluster sets through filterbases and grills on a typical class of GTS’s are also obtained. Moreover, μ-compactness of a GTS is characterized through g-cluster sets of multifunctions.     

On indexed families of multifunctions generated by families of functions

We present some necessary and some sufficient conditions for a family of multifunctions generated by two families of real functions to be a collapsing, an expanding family or an iteration semigroup. Some properties of set-valued iteration groups generated by commuting homeomorphisms not embeddable in an iteration group are given.     

Composition of entire functions with finite iterated order

In this paper, we investigate the growth of two composite entire functions of finite iterated order and a series of comparative growths of logp+qT(r,f(g)) (p,q∈N) with logpT(r,f) and logqT(r,g). At the end of this paper, we apply some growth results into the factorization of the solutions of linear differential equation. We achieve some results which are the improvements and extensions of the previous results.     

Affine and eclipsing multifunctions

In this paper, we want to compare two classes of multifunctions which can be used as approximating multifunctions in differentiability theory: affine and eclipsing multifunctions. We show how the notion of eclipsing multifunctions is an extension of affine multifunctions, and what kinds of difficulties arise in this extension.     

Counting solutions of polynomial systems via iterated fibrations

The paper introduces a new polynomial to count the solutions of a system of polynomial equations and inequations over an algebraically closed field of characteristic zero based on the triangular decomposition algorithm by J. M. Thomas of the nineteen-thirties. In the special case of projective varieties examples indicate that it is a finer invariant than the Hilbert polynomial. Received: 8 March 2008; Revised: 12 August 2008     

The limits of sequences of iterated overshoot distribution functions

In this paper, we deal with sequences of iterated overshoot distribution functions. Under certain norming conditions and under the assumption that these sequences are convergent, the limits are completely characterized. The paper can be considered as a continuation of the work by Harkness and Shantaram [4].As has been indicated by the referee, Shantaram and Harkness recently published a continuation of their work (see [5]). The present paper, however, is more general than [5]. The attention of the authors was directed to this problem while studying the behaviour of market demand transmitted through a chain of stock points.     

Invex multifunctions and duality

For optimization problems with multifunction objective and constraints, duality theorems are proved for analogs of the Wolfe and Mond–Weir dual problems, assuming that the multifunctions satisfy a generalization of the invex property for functions. Several characterizations of generalized invexity are obtained.     

Systems of two iterated functions over skew field of quaternions

Systems of linear iterated functions f ₀(z) = qz + a, f ₁(z) = qz + b over the field of complex numbers have been investigated since 1985 (Barnsley and Harrington). Much attention is paid to the question of the connection of their attractors. We consider systems of iterated functions f ₀(z) = qzp + a, f ₁(z) = qzp + b over the skew field of quaternions. We simplify the form of such systems and study the structure of their attractors.     

Extending paired multifunctions

As a rule, the classical Michael-type selection theorems for the existence of single-valued selections are analogues and, in certain respects, generalisations of ordinary extension theorems. In contrast to this, the theorems for the existence of multi-selections deal with natural generalisations of cover properties of topological spaces. This paper continues the study of the latter problem, and its main purpose is to furnish a mapping characterisation of a cover-extension property—the so-called Katětov spaces.     

Frame properties of systems arising via iterated actions of operators

Motivated by recent progress in dynamical sampling we prove that every frame which is norm-bounded below can be represented as a finite union of sequences ${\{{(T_j)}^n{\phi }_j\}}_{n=0}^{\infty },j=1,\dots ,J$ for some bounded operators $T_j$ and elements ${\phi }_j$ in the underlying Hilbert space. The result is optimal, in the sense that it turns out to be problematic to replace the collection of generators ${\phi }_1,\dots ,{\phi }_J$ by a singleton: indeed, for linearly independent frames we prove that we can represent the frame in terms of just one system ${\{T^n\phi \}}_{n=0}^{\infty }$, but unfortunately this representation often forces the operator T to be unbounded. Several examples illustrate the connection of the results to typical frames like Gabor frames and wavelet frames, as well as generic constructions in arbitrary separable Hilbert spaces.