On subshift presentations

We consider partitioned graphs, by which we mean finite strongly connected directed graphs with a partitioned edge set ${\mathcal E} ={\mathcal E}^- \cup{\mathcal E}^+$. With additionally given a relation $\mathcal R$ between the edges in ${\mathcal E}^-$ and the edges in $\mathcal E^+$, and denoting the vertex set of the graph by ${\frak P}$, we speak of an an ${\mathcal R}$-graph ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+)$. From ${\mathcal R}$-graphs ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^+)$ we construct semigroups (with zero) ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^+)$ that we call ${\mathcal R}$-graph semigroups. We describe a method of presenting subshifts by means of suitably structured labelled directed graphs $({\mathcal V}, \Sigma,\lambda)$ with vertex set ${\mathcal V}$, edge set $\Sigma$, and a label map that asigns to the edges in $\Sigma$ labels in an ${\mathcal R}$-graph semigroup ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$. We call the presented subshift an ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-, {\mathcal E}^-)$-presentation. We introduce a Property $(B)$ and a Property (c), tof subshifts, and we introduce a notion of strong instantaneity. Under an assumption on the structure of the ${\mathcal R}$-graphs ${\mathcal G}_{\mathcal R}({\frak P},{\mathcal E}^-, {\mathcal E}^-)$ we show for strongly instantaneous subshifts with Property $(A)$ and associated semigroup ${\mathcal S}_{\mathcal R}({\frak P},{\mathcal E}^-,{\mathcal E}^-)$, that Properties $(B)$ and (c) are necessary and sufficient for the existence of an ${\mathcal S}_{\mathcal R}({\frak P}, {\mathcal E}^-,{\mathcal E}^-)$-presentation, to which the subshift is topologically conjugate,Comment: 33 page

Primitive prime divisors in the critical orbit of z^d+c

We prove the finiteness of the Zsigmondy set associated to the critical orbit of f(z) = z^d+c for rational values of c by finding an effective bound on the size of the set. For non-recurrent critical orbits, the Zsigmondy set is explicitly computed by utilizing effective dynamical height bounds. In the general case, we use Thue-style Diophantine approximation methods to bound the size of the Zsigmondy set when d >2, and complex-analytic methods when d=2.Comment: This version includes numerous typographical changes and expanded exposition, and a simplified proof of Theorem 6.1. 30 pages, to appear in International Math Research Notice

The critical exponent of the Arshon words

Generalizing the results of Thue (for n = 2) and of Klepinin and Sukhanov (for n = 3), we prove that for all n greater than or equal to 2, the critical exponent of the Arshon word of order $n$ is given by (3n-2)/(2n-2), and this exponent is attained at position 1.Comment: 11 page

On gg-functions for subshifts

A necessary and sufficient condition is given for a subshift presentation to have a continuous $g$-function. An invariant necessary and sufficient condition is formulated for a subshift to posses a presentation that has a continuous $g$-function.Comment: Published at http://dx.doi.org/10.1214/074921706000000329 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

\u3ci\u3eCamellia sinensis\u3c/I\u3e constituents: A Review of Oral Cancer Prevention

Historically, Camellia sinensis (tea) is a plant that has been known to contain antioxidants. Antioxidants such as catechins have been demonstrated to be chemopreventive agents. This review aims to summarize recent findings on the anticancer properties of tea, and its constituents. Since tea is taken orally, and one of the easiest entrances into the human body for microbes is through the oral cavity, this review will focus mainly on oral cancer. Through animal and epidemiological studies, the main active ingredient responsible for the anticancer properties of tea has been determined to be the catechin (-)-epigallocatechin gallate (EGCG). Tea constituents were analyzed through the use of HPLC and confirmed by comparison to authentic standards and mass spectrometry. The results obtained from some studies conflicted with earlier notions that tea catechins act as antioxidants, inhibiting cancer cells. They discovered the catechins to have a pro-oxidant effect, generating reactive oxygen species, such as H2O2. Methods of cancer inhibition were also explored, including cell cycle arrest at certain checkpoints and induction of apoptosis, the active process of cell death. Results from a current study were also examined. Anti-viral and anti-bacterial effects of green and white teas were determined using the plaque method and the Kirby-Bauer, disk-diffusion technique. Results indicated the power of whole tea and tea constituents alongside toothpaste and gum. More than 99% inactivation of viruses was obtained in ten minutes using Tom’s of Maine toothpaste with white tea, whereas infusion of tea into chewing gum yielded over 90% inactivation. Furthermore, distinct zones of inhibition were present for toothpastes and gum treated with tea than for the oral agents by themselves. The future of the research was also briefly discussed. Although many studies have shown beneficial properties of Camellia sinensis, much more epidemiological research remains to be conducted in order to observe the effects on human cancer cells