For a typical convex body in Ed a typical shadow boundary under parallel illumination has infinite (d - 2)-dimensional Hausdorff measurewhile having Hausdorff dimension d 2.

In this note, we investigate those Hausdorff measures which obey a simple scaling law. Consider a continuous increasing function θ defined on with θ(0)= 0 and let be the corresponding Hausdorff measure. We say that obeys an order α scaling law provided whenever K⊂ and c> 0, then

or, equivalently, if T is a similarity map of with similarity ratio c:

A high Reynolds number theory is developed for a viscous fluid flowing through an elastic channel. Unlike the flow through rigid symmetric channels, the viscous flow through a symmetric elastic channel is found to admit free-interaction solutions, due solely to the interaction of the boundary layer with the elastic channel wall. The assumption of symmetry is found to be general providing that the streamwise extent of the channel collapse dilation is larger than O(K17) and the channel is allowed to deviate only slightly from a straight channel. These free-interactions are believed to be the viscous initiation of a sudden collapse or dilation of the channel, commonly observed in experiment. The collapse of the channel is found to occur over a wide range of possible streamwise length scales from O(l) to O(K). For a rigid channel which is coated with a thin elastic solid, the equations are found to reduce to the hypersonic strong interaction problem of triple-deck theory. The hypersonic triple-deck is known to admit both compressive and expansive free-interactions. The expansive free-interaction is found to correspond to a sudden collapse of the channel and an acceleration of the flow within the core of the channel. A cha nnel that is backed by a stagnant constant pressure fluid is also examined. For this problem, the pressure is proportional to the negativeof the fourth derivative of the channel wall displacement. This structure is also found to admit compressive and or expansive free-interactions, depending on whether the internal pressure within the channel is less than or greater than the constant pressure external to the channel. Terminal forms are developed for the expansive free-interaction and compared with numerical calculations.

Introduction. This paper describes a natural way to associate fractal setsto a certain class of absolutely convergent series in In Theorem 1 we give sufficient conditions for such series. Theorem 2 shows that each analytic function gives a different fractal series for each number in a certain open set. Theorem 3 gives the Hausdorff dimension of the associated sets to fractal series, under suitable conditions on the series. This theorem can be applied to some standard series in analysis, such as the binomial, exponential and trigonometrical complex series. The associated sets to geometrical complex series are selfsimilar sets previously studied by M. F. Barnsley from a different (dynamical) point of view (see refs. [5], [6]).

We consider a metric space (X, ρ) of a certain class studied by H. Federer in “Geometric Measure Theory”. Let Ф be any derivation basis on X, which is formed by open balls and is ρ-fine. We show that Ф allows mutual derivation of two arbitrary Borel regular measures on X, which are σ-finite and finite-valued on bounded sets. The proof is based on the so-called De Giorgi property studied in a previous paper.