Almost Sure Frequency Independence of the Dimension of the Spectrum of Sturmian Hamiltonians

We consider the spectrum of discrete Schr\"odinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.Comment: 12 pages, to appear in Commun. Math. Phy

Spectral analysis of a family of binary inflation rules

The family of primitive binary substitutions defined by 1 ↦ 0 ↦ 01m with integer m ∈ ℕ is investigated. The spectral type of the corresponding diffraction measure is analysed for its geometric realisation with prototiles (intervals) of natural length. Apart from the well-known Fibonacci inflation (m=1), the inflation rules either have integer inflation factors, but non-constant length, or are of non-Pisot type. We show that all of them have singular diffraction, either of pure point type or essentially singular continuous

The Impact of Prolonged Inflammation on Wound Healing

The treatment of chronic wounds still challenges modern medicine because of these wounds’ heterogenic pathophysiology. Processes such as inflammation, ischemia and bacterial infection play major roles in the progression of a chronic wound. In recent years, preclinical wound models have been used to understand the underlying processes of chronic wound formation. However, the wound models used to investigate chronic wounds often lack translatability from preclinical models to patients, and often do not take exaggerated inflammation into consideration. Therefore, we aimed to investigate prolonged inflammation in a porcine wound model by using resiquimod, a TLR7 and TLR8 agonist. Pigs received full thickness excisional wounds, where resiquimod was applied daily for 6 days, and untreated wounds served as controls. Dressing change, visual documentation and wound scoring were performed daily. Biopsies were collected for histological as well as gene expression analysis. Resiquimod application on full thickness wounds induced a visible inflammation of wounds, resulting in delayed wound healing compared to non-treated control wounds. Gene expression analysis revealed high levels of IL6, MMP1 and CD68 expression after resiquimod application, and histological analysis showed increased immune cell infiltration. By using resiquimod, we were able to show that prolonged inflammation delayed wound healing, which is often observed in chronic wounds in patients. The model we used shows the importance of inflammation in wound healing and gives an insight into the progression of chronic wounds

Renormalisation of Pair Correlations and Their Fourier Transforms for Primitive Block Substitutions

For point sets and tilings that can be constructed with the projection method, one has a good understanding of the correlation structure, and also of the corresponding spectra, both in the dynamical and in the diffraction sense. For systems defined by substitution or inflation rules, the situation is less favourable, in particular beyond the much-studied class of Pisot substitutions. In this contribution, the geometric inflation rule is employed to access the pair correlation measures of self-similar and self-affine inflation tilings and their Fourier transforms by means of exact renormalisation relations. In particular, we look into sufficient criteria for the absence of absolutely continuous spectral contributions, and illustrate this with examples from the class of block substitutions. We also discuss the Frank-Robinson tiling, as a planar example with infinite local complexity and singular continuous spectrum