Spectrality of Self-Similar Tiles

We call a set $K \subset {\mathbb R}^s$ with positive Lebesgue measure a {\it spectral set} if $L^2(K)$ admits an exponential orthonormal basis. It was conjectured that $K$ is a spectral set if and only if $K$ is a tile (Fuglede's conjecture). Despite the conjecture was proved to be false on ${\mathbb R}^s$, $s\geq 3$ ([T], [KM2]), it still poses challenging questions with additional assumptions. In this paper, our additional assumption is self-similarity. We study the spectral properties for the class of self-similar tiles $K$ in ${\mathbb R}$ that has a product structure on the associated digit sets. We show that any strict product-form tiles and the associated modulo product-form tiles are spectral sets. As for the converse question, we give a pilot study for the self-similar set $K$ generated by arbitrary digit sets with four elements. We investigate the zeros of its Fourier transform due to the orthogonality, and verify Fuglede's conjecture for this special case.Comment: 22page

Multifractal Structure of Convolution of the Cantor Measure

AbstractThe multifractal structure of measures generated by iterated function systems (IFS) with overlaps is, to a large extend, unknown. In this paper we study the local dimension of the m-time convolution of the standard Cantor measure μ. By using some combinatoric techniques, we show that the set E of attainable local dimensions of μ contains an isolated point. This is rather surprising because when the IFS satisfies the open set condition, the set E is an interval. The result implies that the multifractal formalism fails without the open set condition

Topological Structure of Fractal Squares

Given an integer $n\geq 2$ and a digit set ${\mathcal D}\subsetneq {0,1,...,n-1}^2$, there is a self-similar set $F \subset {\Bbb R}^2$ satisfying the set equation: $F=(F+{\mathcal D})/n$. We call such $F$ a fractal square. By studying a periodic extension $H= F+ {\mathbb Z}^2$, we classify $F$ into three types according to their topological properties. We also provide some simple criteria for such classification.Comment: 17 pages, 12 figure

Apathy and suicide-related ideation 3 months after stroke: a cross-sectional study

Background: Both apathy and suicide are common in poststroke patients. However, the association between poststroke apathy and suicide-related ideation (SI) in Chinese stroke patients is not clear and poorly understood. The aim of this study was to examine the association between apathy and SI in stroke. Methods: A cross-sectional study was conducted to investigate the association in 518 stroke survivors from Acute Stroke Unit of the Prince of Wales Hospital in Hong Kong. Geriatric Mental State Examination-Version A (GMS) and Neuropsychiatric Inventory-apathy subscale (NPI-apathy) were employed to assess poststroke SI and apathy, respectively. Patients’ clinical characteristics were obtained with the following scales: the National Institutes of Health Stroke Scale (NIHSS), the Mini-Mental State Examination (MMSE), and the Geriatric Depression Scale (GDS). Results: Thirty-two (6.2%) stroke survivors reported SI. The SI group had a significantly higher frequency of NPI-apathy than the non-SI group (31.2% vs 5.3%, p \u3c 0.001). The SI group also had higher GDS scores (10.47 ± 3.17 vs 4.24 ± 3.71, p \u3c 0.001). Regression analysis revealed that NPI-apathy (OR 2.955, 95% CI 1.142-7.647, p = 0.025) was a significant predictor of SI. The GDS score also predicted SI (OR 1.436, 95% CI 1.284-1.606, p \u3c 0.001). Conclusions: The current findings show that poststroke apathy is an independent predictor of SI 3 months after stroke. Early screening for and intervention targeting apathy through medication and psychological treatments may be necessary to improve stroke patients’ apathy and reduce SI