The estimation or the computation of the Hausdorff dimension of self-affine fractals is of considerable interest. An almost-sure formula for that has been given by K.J. Falconer. However, the precise ...Hausdorff dimension formulas have been given only in special cases. This problem is even unsolved for a general integral self-affine set F, which is generated by an n×n integer expanding matrix T (not necessarily a similitude) and a finite set A⊂Rn of integer vectors so that F=T−1(F+A). In this paper, we focus on the pivotal case F⊂R2 and show that the Hausdorff dimension of F is the limit of a monotonic sequence of McMullen-type dimensions by introducing a process, which we call the fractal perturbation (or deflection) method. In fact, the perturbation method is developed to deal with the pathological case where the characteristic polynomial of T is irreducible over Z. We also consider certain examples of exceptional self-affine fractals for which the Hausdorff dimension is less than the upper bound given by Falconer's formula. These examples show that our approach leads to highly non-trivial computation.
Let F be an integral self-affine set (not necessarily a self-similar set) satisfying F=T(F+A), where T−1 is an integer expanding matrix and A is a finite set of integer vectors. For “totally ...disconnected F”, in 1992, Falconer obtained formulas for lower and upper bounds for the Hausdorff dimension of F. In order to have such bounds for arbitrary F, we consider an extension of Falconer’s formulas to certain graph directed sets and define new bounds. For a very few classes of self-affine sets, the Hausdorff dimension and Falconer’s upper bound are known to be different. In this paper, we present a new such class by using the new upper bound, and show that our upper bound is the box dimension for that class. We also study the computation of those bounds.
We consider certain discrete dynamical systems, conventional number systems and redundant number systems for Z2, which are associated with certain 2×2 integral expansive matrices and collinear digit ...sets for them. Using integral self-affine sets in Rn we also give a geometric characterization to attractors of certain discrete dynamical systems and associated redundant number systems for Zn.
This paper explores the workforce development issues that arose in the course of an Australian repeat pilot study. The aim of the pilot study was to introduce, within a different setting, a planned ...approach to the assessment of, and interventions in, emotional states of service users that may lead to episodes of behavioural disturbance within psychiatric units. The pilot study necessitated training of staff in the use of an assessment tool. During the course of the study, a novel element was encountered with regard to staff understanding of service user involvement in treatment. This element, presented here as 'integral self-intervention', emerged in conjunction with the development of two wall charts: an acute arousal management process chart for staff, and a patient safety chart for service users. The paper will outline the collaborative process towards the partial realisation of this element of integral self-intervention, and associated workforce development issues.
A measurable set Q ⊂Rnis a wavelet set for an expansive matrix A if F−1(ΧQ) is an A-dilation wavelet. Dai, Larson, and Speegle 7 discovered the existence of wavelet sets inRnassociated with any real ...n ×n expansive matrix. In this work, we construct a class of compact wavelet sets which do not contain the origin and which are, up to a certain linear transformation, finite unions of integer translates of an integral selfaffine tile associated with the matrix B = At. Some of these wavelet sets may have good potential for applications because of their tractable geometric shapes.
Let 0<
α⩽2 and let
T⊆
R
. Let {
X(
t),
t∈
T} be a linear fractional
α-stable (0<
α⩽2) motion with scaling index
H (0<
H<1) and with symmetric
α-stable random measure. Suppose that
ψ is a bounded real ...function with compact support
a,
b and at least one null moment. Let the sequence of the discrete wavelet coefficients of the process
X be
D
j,k=
∫
R
X(t)ψ
j,k(t)
dt,
j,k∈
Z
.
We use a stochastic integral representation of the process
X to describe the wavelet coefficients as
α-stable integrals when
H−1/
α>−1. This stochastic representation is used to prove that the stochastic process of wavelet coefficients
{D
j,k,
k∈
Z}
, with fixed scale index
j∈
Z
, is strictly stationary. Furthermore, a property of self-similarity of the wavelet coefficients of
X is proved. This property has been the motivation of several wavelet-based estimators for the scaling index
H.
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