Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets

In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set ${\mathcal{D}}=\{0,1,b\}v$, where $b>1$ and $v\in {\mathbb{R}}^2$ such that $\{v, Av\}$ is linearly independent. We discuss the domain of the digit $b$ to determine the connectedness of $T(A,{\mathcal{D}})$. Especially, a complete characterization is obtained when we restrict $b$ to be an integer. Some results on the general case of $|\det (A)|> 3$ are obtained as well.Comment: 15 pages, 10 figure

Boundaries of Disk-like Self-affine Tiles

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega$, we find the generalized Hausdorff dimension $\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q|$ where $\rho(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log \rho/\log |q|$ where $\rho$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.Comment: 26 pages, 11 figure

Meta-analysis Followed by Replication Identifies Loci in or near CDKN1B, TET3, CD80, DRAM1, and ARID5B as Associated with Systemic Lupus Erythematosus in Asians

Systemic lupus erythematosus (SLE) is a prototype autoimmune disease with a strong genetic involvement and ethnic differences. Susceptibility genes identified so far only explain a small portion of the genetic heritability of SLE, suggesting that many more loci are yet to be uncovered for this disease. In this study, we performed a meta-analysis of genome-wide association studies on SLE in Chinese Han populations and followed up the findings by replication in four additional Asian cohorts with a total of 5,365 cases and 10,054 corresponding controls. We identified genetic variants in or near CDKN1B, TET3, CD80, DRAM1, and ARID5B as associated with the disease. These findings point to potential roles of cell-cycle regulation, autophagy, and DNA demethylation in SLE pathogenesis. For the region involving TET3 and that involving CDKN1B, multiple independent SNPs were identified, highlighting a phenomenon that might partially explain the missing heritability of complex diseases

Pre-service teachers' attitudes towards mathematics and mathematics education

published_or_final_versionabstracttocEducationMasterMaster of Educatio

Retinal Nerve Fiber Layer Measurements in Myopia: An Optical Coherence Tomography Study

PURPOSE. To evaluate the relationship between retinal nerve fiber layer (RNFL) thickness measured by optical coherence tomography (OCT) and the axial length/refractive error of the eye. METHODS. A total of 115 eyes of 115 healthy subjects, comprising 75 eyes with high myopia (spherical equivalent [SE] Ͻ Ϫ6.0 D) and 40 eyes with low to moderate myopia (SE between Ϫ6.0 D and Ϫ0.5D), were analyzed in this cross-sectional study. Total average and mean clock hour RNFL thicknesses were measured by OCT and compared between the two myopia groups. Associations between RNFL measurements and axial length and spherical equivalent were evaluated by linear regression analysis. RESULTS. The RNFL measurements were significantly lower in the high myopia group compared with those of the low-tomoderate myopia group at 12, 1, and 7 o'clock (right eye orientation). Apart from the temporal clock hours, significant correlations were evident between RNFL measurements and the axial length and spherical equivalent. The average RNFL thickness decreased with increasing axial length (r ϭ Ϫ0.314, P ϭ 0.001) and negative refractive power (r ϭ 0.291, P ϭ 0.002). A significant proportion of myopic eyes were classified as outside normal limits, with reference to the normative database. The most frequently abnormal sector was at 2 o'clock, where 16.5% of myopic eyes were outside normal limits. CONCLUSIONS. RNFL measurements vary with the axial length/ refractive error of the eye. Analysis of RNFL thickness in the evaluation of glaucoma should always be interpreted with reference to the refractive status. Although the normative database provided by OCT has been helpful in identifying ocular diseases involving the RNFL, it may not be reliable in the analysis of myopic eyes. (Invest Ophthalmol Vis Sci. 2006;47: 5171-5176