Eigenvalue Estimates of Laplacians Defined by Fractal Measures

We study various lower and upper estimates for the first eigenvalue of Dirichlet Laplacians defined by positive Borel measures on bounded open subsets of Euclidean spaces. These Laplacians and the corresponding eigenvalue estimates differ from classical ones in that the defining measures can be singular. By using properties of self-similar measures, such as Strichartz\u27s second-order self-similar identities, we improve some of the eigenvalue estimates

Spectral Asymptotics of Laplacians Associated to One-Dimensional Iterated Function Systems with Overlaps

We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition

Spectral Asymptotics of Laplacians Defined by Fractal Measures and Some Applications

We report some results concerning spectral asymptotics of fractal Laplacians defined one-dimensional self-similar measures with overlaps. We also discuss some applications of the theory, including heat kernel estimates and wave propagation speed. Part of this work is joint with Qingsong Gu, Jiaxin Hu, Wei Tang and Yuanyuan Xie

Separation conditions for iterated function systems with overlaps on Riemannian manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau \textit{et al.} in [Monatsch. Math. 156 (2009), 325-355]. We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin-Yau [Comm. Anal. Geom. 13 (2005), 821--843] and Lau-Ngai [Adv. Math. 208 (2007), 647-671] on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai \textit{et al.} in [Nonlinearity 23 (2010), 2333--2350]

Self-Similar Measures Associated to {IFS} with Non-Uniform Contraction Ratios

In this paper we study the absolute continuity of self-similar measures defined by iterated function systems (IFS) whose contraction ratios are not uniform. We introduce a transversality condition for a multi-parameter family of IFS and study the absolute continuity of the corresponding self-similar measures. Our study is a natural extension of the study of Bernoulli convolutions by Solomyak, Peres, et al

One-dimensional wave equations defined by fractal Laplacians

We study one-dimensional wave equations defined by a class of fractal Laplacians. These Laplacians are defined by fractal measures generated by iterated function systems with overlaps, such as the well-known infinite Bernoulli convolution associated with the golden ratio and the 3-fold convolution of the Cantor measure. The iterated function systems defining these measures do not satisfy the post-critically finite condition or the open set condition. By using second-order self-similar identities introduced by Strichartz et al., we discretize the equations and use the finite element and central difference methods to obtain numerical approximations to the weak solutions. We prove that the numerical solutions converge to the weak solution, and obtain estimates for the rate of convergence

Vertices of Self-Similar Tiles

The set Vn of n-vertices of a tile T in \Rd is the common intersection of T with at least n of its neighbors in a tiling determined by T. Motivated by the recent interest in the topological structure as well as the associated canonical number systems of self-similar tiles, we study the structure of Vn for general and strictly self-similar tiles. We show that if T is a general self-similar tile in \R2 whose interior consists of finitely many components, then any tile in any self-similar tiling generated by T has a finite number of vertices. This work is also motivated by the efforts to understand the structure of the well-known L\\u27evy dragon. In the case T is a strictly self-similar tile or multitile in \Rd, we describe a method to compute the Hausdorff and box dimensions of Vn. By applying this method, we obtain the dimensions of the set of n-vertices of the L\\u27evy dragon for all n≥1

A class of self-affine tiles in Rd\mathbb{R}^d that are dd-dimensional tame balls

We study a family of self-affine tiles in $\mathbb{R}^d$ ($d\ge2$) with noncollinear digit sets, which naturally generalizes a class studied originally by Deng and Lau in $\mathbb{R}^2$ and its extension to \mathbb{R}^3} by the authors. By using Brouwer's invariance of domain theorem, along with a tool which we call horizontal distance, we obtain necessary and sufficient conditions for the tiles to be $d$-dimensional tame balls. This answers positively the conjecture in an earlier paper by the authors stating that a member in a certain class of self-affine tiles is homeomorphic to a $d$-dimensional ball if and only if its interior is connected.Comment: 56 pages, 17 figure

On 2-Reptiles in the Plane

We classify all rational 2-reptiles in the plane. We also establish properties concerning rational reptiles in the plane in general

Herpes zoster related hospitalization after inactivated (CoronaVac) and mRNA (BNT162b2) SARS-CoV-2 vaccination: A self-controlled case series and nested case-control study

BACKGROUND: Stimulation of immunity by vaccination may elicit adverse events. There is currently inconclusive evidence on the relationship between herpes zoster related hospitalization and COVID-19 vaccination. This study aimed to evaluate the effect of inactivated virus (CoronaVac, Sinovac) and mRNA (BNT162b2, BioNTech/Fosun Pharma) COVID-19 vaccine on the risk of herpes zoster related hospitalization. METHODS: Self-controlled case series (SCCS) analysis was conducted using the data from the electronic health records in Hospital Authority and COVID-19 vaccination records in the Department of Health in Hong Kong. We conducted the SCCS analysis including patients with a first primary diagnosis of herpes zoster in the hospital inpatient setting between February 23 and July 31, 2021. A confirmatory analysis by nested case-control method was also conducted. Each herpes zoster case was randomly matched with ten controls according to sex, age, Charlson comorbidity index, and date of hospital admission. Conditional Poisson regression and logistic regression models were used to assess the potential excess rates of herpes zoster after vaccination. FINDINGS: From February 23 to July 31, 2021, a total of 16 and 27 patients were identified with a first primary hospital diagnosis of herpes zoster within 28 days after CoronaVac and BNT162b2 vaccinations. The incidence of herpes zoster was 7.9 (95% Confidence interval [CI]: 5.2–11.5) for CoronaVac and 7.1 (95% CI: 4.1–11.5) for BNT162b2 per 1,000,000 doses administered. In SCCS analysis, CoronaVac vaccination was associated with significantly higher risk of herpes zoster within 14 days after first dose (adjusted incidence rate ratio [aIRR]=2.67, 95% CI: 1.08–6.59) but not in other periods afterwards compared to the baseline period. Regarding BNT162b2 vaccination, a significantly increased risk of herpes zoster was observed after first dose up to 14 days after second dose (0-13 days after first dose: aIRR=5.23, 95% CI: 1.61–17.03; 14–27 days after first dose: aIRR=5.82, 95% CI: 1.62–20.91; 0-13 days after second dose: aIRR=5.14, 95% CI: 1.29–20.47). Using these relative rates, we estimated that there has been an excess of approximately 5 and 7 cases of hospitalization as a result of herpes zoster after every 1,000,000 doses of CoronaVac and BNT162b2 vaccination, respectively. The findings in the nested case control analysis showed similar results. INTERPRETATION: We identified an increased risk of herpes zoster related hospitalization after CoronaVac and BNT162b2 vaccinations. However, the absolute risks of such adverse event after CoronaVac and BNT162b2 vaccinations were very low. In locations where COVID-19 is prevalent, the protective effects on COVID-19 from vaccinations will greatly outweigh the potential side effects of vaccination. FUNDING: The project was funded by Research Grant from the Food and Health Bureau, The Government of the Hong Kong Special Administrative Region (Ref. No.COVID19F01). FTTL (Francisco Tsz Tsun Lai) and ICKW (Ian Chi Kei Wong)’s posts were partly funded by D(2)4H; hence this work was partly supported by AIR@InnoHK administered by Innovation and Technology Commission