Applied and industrial mathematics in Italy: Proc. of the by Primicerio M., et al. (eds.)

By Primicerio M., et al. (eds.)

Commercial arithmetic is evolving into an incredible department of arithmetic. Mathematicians, in Italy specifically, have gotten more and more conscious of this new pattern and are engaged in bridging the space among hugely really good mathematical learn and the rising call for for innovation from undefined. during this appreciate, the contributions during this quantity offer either R&D employees in with a normal view of current talents, and lecturers with state of the art purposes of arithmetic to real-world difficulties, that could even be integrated in complex classes.

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X ∈ C, dimloc μ (x) = p log p + (1 − p) log(1 − p) . 1. Assuming lim 1n log (r1 · · · rn ) exists is a convenience. Similar results can be proved with the local dimension of μ at x replaced by the upper or lower local dimensions and with lim 1n log (r1 · · · rn ) replaced by lim sup 1n log (r1 · · · rn ) (or lim inf). The proof has two parts, a geometric and a probabilistic part. We begin with a geometric lemma which will have other applications. Its significance is to show that under the assumption sup r j < 1/2 we may replace balls by Cantor intervals in the definition of local dimension.

We could also allow the probabilities to vary at different steps. 1 Cantor Sets with Varying Ratios of Dissection Let 0 < r j < 1/2. We denote by C(r j ) 2 the Cantor set with varying ratios of dissection, r j at step j, given by the following iterative Cantor-like construction: Let C0 = [0, 1]. Remove from C0 the open middle interval of length 1 − 2r1 , leaving two closed intervals of lengths r1 . Call these intervals the Cantor intervals of step one and their union C1 . At step j in the construction assume we have inductively constructed C j as a union of 2 j closed intervals of length r1 · · · r j , the Cantor intervals of step j.

Similar results have been obtained for m-fold convolutions of the uniform Cantor measures on the Cantor sets C(1/d) when d ∈ N and, more generally, for self-similar measures generated by an IFS consisting of contractions Fi (x) = x/d + (d − 1)i/d for i = 0, 1, . . , m and probabilities pi > 0, where p0 , pm ≤ pi for all i = 0, m and d ≥ 3 is an integer. The algebraic and combinatorial structure of these self-similar measures can again be used to show that if m ≥ d, then the multifractal spectrum is the union of a closed interval and one (or two) isolated points, the local dimensions at 0, m.

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