Hrvoje Šikić

Ruin probabilities and decompositions for general perturbed risk processes

We study a general perturbed risk process with cumulative claims modelled by a subordinator with finite expectation, and the perturba- tion being a spectrally negative Levy process with zero expectation. We derive a Pollaczek-Hinchin type formula for the survival proba- bility of that risk process, and give an interpretation of the formula based on the decomposition of the dual risk process at modified ladder epochs.

Tight Frame Wavelets, their Dimension Functions, MRA Tight Frame Wavelets and Connectivity Properties

This is a continuation of our study of generalized low pass filters and MRA frame wavelets. In this first study we concentrated on the construction of such functions. Here we are particularly interested in the role played by the dimension function. In particular we characterize all semi-orthogonal Tight Frame Wavelets (TFW) by showing that they correspond precisely to those for which the dimension function is non-negative integer-valued. We also show that a TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. We present many examples. In addition we obtain a result concerning the connectivity of TFWs that are MSF tight frame wavelets.

Generalized low pass filters and MRA frame wavelets

A tight frame wavelet ψ is an L2(ℝ) function such that {ψ jk(x)} = {2j/2ψ(2jx −k), j, k ∈ ℤ},is a tight frame for L2 (ℝ).We introduce a class of “generalized low pass filters” that allows us to define (and construct) the subclass of MRA tight frame wavelets. This leads us to an associated class of “generalized scaling functions” that are not necessarily obtained from a multiresolution analysis. We study several properties of these classes of “generalized” wavelets, scaling functions and filters (such as their multipliers and their connectivity). We also compare our approach with those recently obtained by other authors.

The Characterization of Low Pass Filters and Some Basic Properties of Wavelets, Scaling Functions and Related Concepts

AbstractThe “classical” wavelets, those ψ εL2(R) such that {2j/2ψ(2jx−k)}, j,kεZ, is an orthonormal basis for L2 (R), are known to be characterized by two simple equations satisfied by

A stochastic model of eye lens growth

\hat \psi

Further results on the connectivity of Parseval frame wavelets

. The “multiresolution analysis” wavelets (briefly, the MRA wavelets) have a simple characterization and so do the scaling functions that produce these wavelets. Only certain smooth classes of the low pass filters that are determined by these scaling functions, however, appear to be characterized in the literature (see Chapter 7 of [3] for an account of these matters). In this paper we present a complete characterization of all these filters. This somewhat technical result does provide a method for simple constructions of low pass filters whose only smoothness assumption is a Holder condition at the origin. We also obtain a characterization of all scaling sets and, in particular, a description of all bounded scaling sets as well as a detailed description of the class of scaling functions.

The Zak Transform(s)

PURPOSE The mechanisms that regulate the number of cells in the lens and, therefore, its size and shape are unknown. We examined the dynamic relationship between proliferative behavior in the epithelial layer and macroscopic lens growth. METHODS The distribution of S-phase cells across the epithelium was visualized by confocal microscopy and cell populations were determined from orthographic projections of the lens surface. RESULTS The number of S-phase cells in the mouse lens epithelium fell exponentially, to an asymptotic value of approximately 200 cells by 6 months. Mitosis became increasingly restricted to a 300-μm-wide swath of equatorial epithelium, the germinative zone (GZ), within which two peaks in labeling index were detected. Postnatally, the cell population increased to approximately 50,000 cells at 4 weeks of age. Thereafter, the number of cells declined, despite continued growth in lens dimensions. This apparently paradoxical observation was explained by a time-dependent increase in the surface area of cells at all locations. The cell biological measurements were incorporated into a physical model, the Penny Pusher. In this simple model, cells were considered to be of a single type, the proliferative behavior of which depended solely on latitude. Simulations using the Penny Pusher predicted the emergence of cell clones and were in good agreement with data obtained from earlier lineage-tracing studies. CONCLUSIONS The Penny Pusher, a simple stochastic model, offers a useful conceptual framework for the investigation of lens growth mechanisms and provides a plausible alternative to growth models that postulate the existence of lens stem cells.

A full lifespan model of vertebrate lens growth

The size and shape of the ocular lens must be controlled with precision if light is to be focused sharply on the retina. The lifelong growth of the lens depends on the production of cells in the anterior epithelium. At the lens equator, epithelial cells differentiate into fiber cells, which are added to the surface of the existing fiber cell mass, increasing its volume and area. We developed a stochastic model relating the rates of cell proliferation and death in various regions of the lens epithelium to deposition of fiber cells and radial lens growth. Epithelial population dynamics were modeled as a branching process with emigration and immigration between proliferative zones. Numerical simulations were in agreement with empirical measurements and demonstrated that, operating within the strict confines of lens geometry, a stochastic growth engine can produce the smooth and precise growth necessary for lens function.