A Thompson Group for the Basilica

We describe a Thompson-like group of homeomorphisms of the Basilica Julia set. Each element of this group acts as a piecewise-linear homeomorphism of the unit circle that preserves the invariant lamination for the Basilica. We develop an analogue of tree pair diagrams for this group which we call arc pair diagrams, and we use these diagrams to prove that the group is finitely generated. We also prove that the group is virtually simple.Comment: 23 pages, 31 figure

Rearrangement Groups of Fractals

We construct rearrangement groups for edge replacement systems, an infinite class of groups that generalize Richard Thompson's groups F, T, and V . Rearrangement groups act by piecewise-defined homeomorphisms on many self-similar topological spaces, among them the Vicsek fractal and many Julia sets. We show that every rearrangement group acts properly on a locally finite CAT(0) cubical complex, and we use this action to prove that certain rearrangement groups are of type F infinity.Comment: 48 pages, 37 figure

Quasisymmetries of finitely ramified Julia sets

We develop a theory of quasisymmetries for finitely ramified fractals, with applications to finitely ramified Julia sets. We prove that certain finitely ramified fractals admit a naturally defined class of "undistorted metrics" that are all quasi-equivalent. As a result, piecewise-defined homeomorphisms of such a fractal that locally preserve the cell structure are quasisymmetries. This immediately gives a solution to the quasisymmetric uniformization problem for topologically rigid fractals such as the Sierpi\'nski triangle. We show that our theory applies to many finitely ramified Julia sets, and we prove that any connected Julia set for a hyperbolic unicritical polynomial has infinitely many quasisymmetries, generalizing a result of Lyubich and Merenkov. We also prove that the quasisymmetry group of the Julia set for the rational function $1-z^{-2}$ is infinite, and we show that the quasisymmetry groups for the Julia sets of a broad class of polynomials contain Thompson's group $F$.Comment: 49 pages, 17 figure

Lights Out on a Random Graph

We consider the generalized game Lights Out played on a graph and investigate the following question: for a given positive integer n, what is the probability that a graph chosen uniformly at random from the set of graphs with n vertices yields a universally solvable game of Lights Out? When n ≤ 11, we compute this probability exactly by determining if the game is universally solvable for each graph with n vertices. We approximate this probability for each positive integer n with n ≤ 87 by applying a Monte Carlo method using 1,000,000 trials. We also perform the analogous computations for connected graphs

Mosquito chronological age determination using mid-infrared spectroscopy and chemometrics

Determining a mosquito population’s species composition and age is crucial for estimating the risk of pathogen transmission. At present, age-grading methods are chiefly physiologic and classify the mosquitoes in terms of parity (e.g., nulliparous or parous). Less commonly used chronologic methods (e.g., qPCR or near infrared spectroscopy [NIR]) have limited temporal resolution (NIR) or require consumable reagents and technological expertise with molecular methods. The current lack of robust methods to rapidly evaluate a population’s chronologic age limits our ability to assess pathogen transmission risk in the context of vectorial capacity estimations (i.e., daily survivability). Our current research seeks to develop methods of mosquito age determination utilizing mid-infrared spectroscopy and advanced numerical analysis(chemometrics). Infrared (IR) spectroscopy is a type of vibrational spectroscopy that is both sensitive and information rich. Subtle changes in IR spectra correlate with changes in the biochemistry of mosquitoes as they age. It has been shown that mosquito species can be identified using mid infrared spectroscopy and chemometrics. Using mid-infrared spectroscopy and chemometrics, the chronologic age of Aedes triseriatus mosquitoes were predicted using PLSR and ANN models. Aedes triseriatus were successfully reared into groups of different ages with low uncertainty in the age. Aedes triseriatus spectra were used to create a training dataset and fit models for prediction using PLSR and ANN. PLSR and ANN models were used to predict the age of samples using a validation dataset with SEPsv of 4.3 and 3.3 days respectively. Mean spectra for each age group were used to try and discern a specific chemical underpinning for the performance of these models and to explain why mosquito age could be predicted using PLSR and ANN models. Peaks between 1200 – 1000 cm-1 typically associated with chitin were investigated and the second derivative of mean absorbance by age at 1032 cm-1increased linearly with age

Ultrahigh-temperature osumilite gneisses in southern Madagascar record combined heat advection and high rates of radiogenic heat production in a long-lived high-T orogen

We report the discovery of osumilite in ultrahigh‐temperature (UHT) metapelites of the Anosyen domain, southern Madagascar. The gneisses equilibrated at ~930°C/0.6 GPa. Monazite and zircon U–Pb dates record 80 Ma of metamorphism. Monazite compositional trends reflect the transition from prograde to retrograde metamorphism at 550 Ma. Eu anomalies in monazite reflect changes in fO_2 relative to quartz–fayalite–magnetite related to the growth and breakdown of spinel. The ratio Gd/Yb in monazite records the growth and breakdown of garnet. High rates of radiogenic heat production were the primary control on metamorphic grade at the regional scale. The short duration of prograde metamorphism in the osumilite gneisses (<29 ± 8 Ma) suggests that a thin mantle lithosphere (<80 km) or advective heating may have also been important in the formation of this high‐T, low‐P terrane