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By constructing an infinite graph-directed iterated iterated function system associated with a finite iterated function system, we develop a new approach for proving the differentiability of the $L^q$-spectrum and establishing the multifractal formalism of certain self-similar measures with overlaps, especially those defined by similitudes with different contraction ratios. We apply our technique to a well-known class of self-similar measures of generalized finite type.

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Let G=(V,E) be a finite weighted graph, and Ω⊆V be a domain such that Ω^∘≠∅. In this paper, we study the following initial boundary problem for the non-homogenous wave equation ∂^2_t u(t,x) − Δ_Ω u(t,x) = f(t,x), (t,x)∈[0,∞)×Ω^∘ u(0,x)=g(x), x∈Ω^∘, ∂_t u(0,x)=h(x), x∈Ω^∘, u(t,x)=0, (t,x)∈[0,∞)×∂Ω, where Δ_Ω denotes the Dirichlet Laplacian on Ω^∘. Using Rothe's method, we prove that the above wave equation has a unique solution.

Consider a complete abelian category which has an injective cogenerator. If its derived category is left-complete we show that the dual of this derived category satisfies Brown representability. In particular, this is true for the derived category of an abelian AB $4^{*}$ - $n$ category and for the derived category of quasi-coherent sheaves over a nice enough scheme, including the projective finitely dimensional space.