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

Research Shows a Majority of People in Hong Kong Support Gay and Lesbian Couples’ Rights, Not Necessarily Marriage

Briefing Paper, Centre for Comparative and Public Law (University of Hong Kong); UNC Legal Studies Research Paper No. 2374875Discussions in Hong Kong about same-sex couples’ rights often focus on the issue of same-sex marriage. However, marriage is not the only way the government could extend rights to same-sex couples. In other parts of the world, governments have conferred rights upon gay and lesbian couples without legalising same-sex marriage. Some governments have developed programs that grant same-sex couples a subset of the rights that married heterosexual couples enjoy, for example the right to hospital visitation, the right to inheritance, and the right to sue in cases of fatal accidents. Other governments have developed programs that give same-sex couples access to all the rights that married heterosexual couples have, without using the word “marriage”. These compromise legal solutions have been given a variety of different names, such as “registered domestic partnerships”, “civil unions”, and “civil partnerships.”1 We conducted a public opinion survey to investigate Hong Kong people’s attitudes towards granting same-sex couples a variety of rights, including but not limited to the right to marry. We found that only 27% of the public completely agreed that same-sex couples should be permitted to marry, and 12% said that they somewhat agreed. However, when we asked about rights more generally, our findings were substantially different: 74% of the public supported granting samesex couples either all or some of the rights that are accorded to heterosexual couples. The remainder of this briefing paper proceeds in two steps. We first provide background on our survey and present our findings. Afterwards, we explore the public policy implications of our research.preprin

The Catch-22 Irony of Child Labor and The Urgency for State Intervention

Child labor (CL) has been found to enhance substantially the poorest households\u27 chances for short-term survival. CL could also - it has been argued - facilitate CWs\u27 socialization and development of moral values and survival skills. This article aims to demonstrate that these short-term benefits are considerably outweighed by the corresponding long-term detrimental repercussions on CWs\u27 health, education, and earning potential in adulthood. While the poorest households might strategically employ CL as a last resort to ensure their short-term survival, it is CL per se that maintains these families in extreme poverty through the impediment to CWs\u27 human capital accumulation, leaving them with few options but continuing to resort to CL to survive-hence, the Catch-Z? irony of CL. It is argued from extant evidence of intergenerational transmission of poverty and CL that without state intervention, these poorest families with CWs would not be likely to escape from this kind of poverty-and-Cl intergenerational cycle, with negative implications for the nation\u27s prospects of achieving long-term socioeconomic development. Thus, this article contends that it would be beneficial for the Brazilian government to treat the elimination of CL as a top national priority

Management of large geospatial datasets

In large simulations, like predicting the movement of ocean particles, it is common that simulation executions are related when they share one or more inputs. When the number of simulations increases, it becomes harder for users who run the simulations to keep track of all the simulations. Also, more storage spaces are wasted if there are multiple copies of the same input files. This thesis describes a system that collects data from previous simulations, allowing users to search for the data they need to run the next simulation. Also, the system identifies the same files that were used in previous simulations, which allows users to re-use these files instead of copying the files to a new simulation folder to use them. Among the simulations that were executed in our current environment, the system identifies around 11\% of input files that are shared by the simulations. Users can refer to the same file to use it instead of copying the file to new simulation folders. The conclusion is that the system helps users who run simulations to reduce their efforts and time to find input files that are used in previous simulations when they set up for a new simulation. Also, it saves storage space on the computing cluster where the simulations run on by identifying the duplicated data