We study L^p inequalities that sharpen the triangle inequality for sums of N functions in L^p.

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This paper studies the moment boundedness of solutions of linear stochastic delay differential equations with distributed delay. For a linear stochastic delay differential equation, the rst moment stability is known to be identical to that of the corresponding deterministic delay differential equation. However, boundedness of the second moment is complicated and depends on the stochastic terms. In this paper, the characteristic function of the equation is obtained through techniques of the Laplace transform. From the characteristic equation, suf cient conditions for the second moment to be bounded or unbounded are proposed.

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We study the distribution of harmonic measure on connected Julia sets of unicritical polynomials. Harmonic measure on a full compact set in ℂ is always concentrated on a set which is porous for a positive density of scales. We prove that there is a topologically generic set $\mathcal{A}$ in the boundary of the Mandelbrot set such that for every $c\in \mathcal{A}$ ,$β$>0, and$λ$∈(0,1), the corresponding Julia set is a full compact set with harmonic measure concentrated on a set which is not$β$-porous in scale$λ$^{$n$}for$n$from a set with positive density amongst natural numbers.