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

Development of Bis-GMA-free biopolymer to avoid estrogenicity

Objective: Although bisphenol A-glycidyl methacrylate (Bis-GMA)-based dental materials are widely used in dentistry, Estrogenicity from released bisphenol A remains a concern due to possibility of adversely affecting the growth of children and homeostasis of adults. Here, a new family of isosorbide-derived biomonomers were synthesized and experimentally utilized as a matrix of dental sealants to provide physico-mechanical and biological properties comparable to those of a conventional Bis-GMA-based material but without the the potential estrogenicity. / Methods: After synthesis of isosorbide-derived biomonomers (ISDB) by light polymerization, an experimental dental sealant with different silica filler concentrations (0–15 wt%) was characterized and compared to a commercially available Bis-GMA-based sealant. Cytotoxicity and estrogenicity assays were conducted with human oral keratinocytes and estrogen-sensitive MCF-7 cells, respectively. / Results: ISDB-based dental sealants exhibited typical initially smooth surfaces with depth of cure, Vickers hardness, compressive strength/modulus, water resorption/solubility, and flowability comparable to those of the commercial sealant and met the ISO standard for dental sealants and polymer-based restorative materials. Indirect cytotoxicity tests using an extract showed comparable viability among experimental ISDB-based materials and a commercial Bis-GMA-incorporated control. DNA synthesis in MCF-7 cells (a marker of estrogenicity) and the release of bisphenol A under enzymatic incubation were not detected in ISDB-based materials. / Significance: In conclusion, the comparable physico-mechanical properties of ISDB-based materials with their cytocompatibility and lack of estrogenicity suggest the potential usefulness of ISDBs as a newly developed and safe biomaterial