Abstract. In this paper we obtain theorems which give the Hausdorff dimension of the invariant set for a finite family of contraction mappings which are "infinitesimal similitudes" on a complete, perfect metric space. Our work generalizes the graph-directed construction of Mauldin and Williams (1988) and is related in its general setting to results of Schief (1996), but differs crucially in that the mappings need not be similitudes. We use the theory of positive linear operators and generalizations of the Krein-Rutman theorem to characterize the Hausdorff dimension as the unique value of σ > 0 for which r(L σ ) = 1, where L σ , σ > 0, is a naturally associated family of positive linear operators and r(L σ ) denotes the spectral radius of L σ . We also indicate how these results can be generalized to countable families of infinitesimal similitudes. The intent here is foundational: to derive a basic formula in its proper generality and to emphasize the utility of the theory of positive linear operators in this setting. Later work will explore the usefulness of the basic theorem and its functional analytic setting in studying questions about Hausdorff dimension.

This paper develops the theory of vibrational control Of "lhJear time Systems with arbitrarily large but bounded delay. theory for fast Oscillating, differential delay equations is presented and then applied to vibrational control. conditions B1p given which ensure the e*ne ot parametric vibrations that stabilize nonlinear time lag systems. Transient behavior is also discussed. Illustrative examples are given which Show 1) the f-bfiv Ofthe theory to hP0-t aPPfiatiOm and 2, the differences in the presented and the eKisw known theory for vibrational control of ordinary differential equations. the vibrations depended only on time (and not on the value of the state), there no longer was a need to take measurements of concentration, thus reducing the cost of the reaction even more. For similar reasons, the vibrational control described by [61 has many benefits. A number of practical, important systems, however, are best described by including time delays in their states. In particular, if the exothermic reaction vibrationally controlled in [5] includes a recycle stream, as in [7], the model must

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The effect of low-dose ionizing radiation exposure on leukemia incidence remains poorly understood. Possible dose-response curves for various forms of leukemia are largely based on cohorts of atomic bomb survivors. Animal studies can contribute to an improved understanding of radiation-induced acute myeloid leukemia (rAML) in humans. In male CBA/H mice, incidence of rAML can be described by a two-hit model involving a radiation-induced deletion with Sfpi1 gene copy loss and a point mutation in the remaining Sfpi1 allele. In the present study (historical) mouse data were used and these processes were translated into a mathematical model to study photon-induced low-dose AML incidence in male CBA/H mice following acute exposure. Numerical model solutions for low-dose rAML incidence and diagnosis times could respectively be approximated with a model linear-quadratic in radiation dose and a normal cumulative distribution function. Interestingly, the low-dose incidence was found to be proportional to the modeled number of cells carrying the Sfpi1 deletion present per mouse following exposure. After making only model-derived high-dose rAML estimates available to extrapolate from, the linear-quadratic model could be used to approximate low-dose rAML incidence calculated with our mouse model. The accuracy in estimating low-dose rAML incidence when extrapolating from a linear model using a low-dose effectiveness factor was found to depend on whether a data transformation was used in the curve fitting procedure.

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Abstract. It is shown that for an SDE in a Hilbert space, eventual compactness of the driving semigroup together with compact perturbations can be used to establish the existence of an invariant measure. The result is applied to stochastic functional differential equations and the heat equation perturbed by delay and noise, which are both shown to be driven by an eventually compact semigroup.