The Dynamics and Geometry of Semi-Hyperbolic Rational Semigroups

We deal with various classes of finitely generated semi-hyperbolic rational semigroups $G$ acting on the Riemann sphere. Our primary tool is the associated skew product map $\tilde f$. For every real number $t\ge 0$ we define the topological pressure $P(t)$ ascribed to the (possibly) unbounded potential $-t\log|\tilde f'|$. We show that for some non-degenerate open (in $[0,+\infty)$) interval containing $0$ and the Hausdorff dimension of the Julia set $J(G)$, the function $t\mapsto P(t)$ is real-analytic. We further show that for all $t$ in such an interval there exist a unique $t$-conformal measure $m_t$ and a unique Borel probability $\tilde f$-invariant measure $\mu_t$ absolutely continuous with respect to $m_t$. We then show that $\mu_t$ and $m_t$ are equivalent measures and that the dynamical system $(\tilde f,\mu_t)$ is metrically exact (thus ergodic), and that for H\"older continuous observables $(\tilde f,\mu_t)$ satisfies the CLT, LIL, and the Exponential Decay of Correlations. We prove a Variational Principle for the potentials $-t\log|\tilde f'|$, and we characterize the measures $\mu_t$ as their unique equilibrium states. Concerning geometry, we first introduce the Nice Open Set Condition, and we prove that each *semi-hyperbolic rational semigroup satisfying this condition is of finite type. We then introduce the class of non-exceptional semigroups and perform a full multifractal analysis of the equilibrium states $\mu_t$. In particular, we show that the corresponding multifractal spectrum is non-trivial for every non-exceptional totally non-recurrent rational semigroup satisfying the Nice Open Set Condition. Finally, we settle a long standing problem in the theory of rational semigroups by proving that for our class of semigroups the Hausdorff dimension of each fiber Julia set is strictly smaller than the Hausdorff dimension of the global Julia set of the semigroup.Comment: 151 pages, 2 figure

Health Promotion and Illness Prevention

Abstract Those with unhealthy lifestyle behaviors are likely to have a shorter life expectancy and experience an onset of disability sooner. An unhealthy lifestyle also increases the chance of heart disease, diabetes, and cancer. We can decrease our chances of all of those by taking care of our physical, mental, social, emotional, spiritual, and environmental health. Health promotion and illness prevention gives us the choice to better our own lives and the quality of our lives. This can be done by the actions we take daily that influence our health and prevent illness. Some of these actions include activities such as exercising 30 minutes a day, 3-5 days a week. For others it may be as simple as changing their consumption by making healthier food choices. There are other habits that if stopped will promote health and prevent illness such as not using tobacco

Assessing the Relationship Between Student and Faculty Perceptions of Student Engagement at Central Mountain College

This study compared the perceptions of students versus faculty at Central Mountain College with regard to the issue of student engagement. Central Mountain College participated in the Community College Survey of Student Engagement and the Community College Faculty Survey of Student Engagement during the spring semesters of 2009, 2011, and 2013. The institution was provided with aggregate results from these survey administrations by the Center for Community College Student Engagement. Prior to this study, the survey results had not been accumulatively evaluated by the institution. The study aimed to determine areas where there was congruence and incongruence between the students and the faculty so as to be able to target problem areas for improvement and to reinforce successful practices. A descriptive analysis of the survey results was conducted utilizing a framework known as the Crosswalk Tool which was produced by the Center for Community College Student Engagement. A report of institutional activities that coincided with the timing of the survey administration, and which could have had impacts on student and faculty perceptions was also prepared. The study found little variation within student and faculty perceptions of student engagement during the three survey administrations. Despite significant physical changes to campus, and organizational changes to the institution, there appeared to be minimal impact to the two groups’ perceptions of what was taking place in the classroom. Areas of disagreement centered on student effort and involvement in their learning. These differences in perspectives highlight the need for more open communication between faculty and students, and expectations that are made clearer and more attainable to students. The study also suggested that more collaboration and congruence between the expectations of the K-12 system and higher education institutions would ease the transition to college and perhaps improve student engagement. Faculty may need to complete additional training in the area of classroom management and student success initiatives to enhance the level of engagement in their classrooms. Advisor: Jody Isernhage

Opioid growth factor modulates angiogenesis

AbstractObjective: Induced angiogenesis has recently been attempted as a therapeutic modality in patients with occlusive arterial atherosclerotic disease. We investigated the possible role of endogenous opioids in the modulation of angiogenesis. Methods: Chick chorioallantoic membrane was used as an in vivo model to study angiogenesis. Fertilized chick eggs were incubated for 3 days, explanted, and incubated for an additional 2 days. Three-millimeter methylcellulose disks were placed on the surface of the chorioallantoic membrane; each disk contained opioid growth factor ([Met5]-enkephalin; 5 μg), the short-acting opioid receptor antagonist naloxone (5 μg), opioid growth factor and naloxone together (5 μg of each), the long-acting opioid antagonist naltrexone (5 μg), or distilled water (control). A second series of experiments was performed with distilled water, the angiogenic inhibitor retinoic acid (1 μg), and vascular endothelial growth factor (1 μg) to further evaluate our model. The developing vasculature was imaged 2 days later with a digital camera and exported to a computer for image analysis. Total number of blood vessels, total vessel length, and mean vessel length were measured within a 100-mm2 region surrounding each applied disk. Immunocytochemical analysis was performed with antibodies directed against opioid growth factor and its receptor (OGFr). Results: Opioid growth factor had a significant inhibitory effect on angiogenesis, both the number of blood vessels and the total vessel length being decreased (by 35% and 20%, respectively) in comparison with control levels (P <.005). The simultaneous addition of naloxone and opioid growth factor had no effect on blood vessel growth, nor did naloxone alone. Chorioallantoic membranes exposed to naltrexone displayed increases of 51% and 24% in blood vessel number and length, respectively, in comparison with control specimens (P <.005). These results indicate that the opioid growth factor effects are receptor mediated and tonically active. Immunocytochemistry demonstrated the presence of both opioid growth factor and OGFr within the endothelial cells and mesenchymal cells of the developing chorioallantoic membrane vessel wall. Retinoic acid significantly reduced the number and the total length of blood vessels, whereas vascular endothelial growth factor increased both the number and the length of blood vessels in comparison with the controls (P <.0001). The magnitude of opioid growth factor's effects were comparable to those seen with retinoic acid, whereas inhibition of opioid growth factor with naltrexone induced an increase in total vessel length comparable to that for vascular endothelial growth factor. Conclusions: These results demonstrate for the first time that endogenous opioids modulate in vivo angiogenesis. Opioid growth factor is a tonically active peptide that has a receptor-mediated action in regulating angiogenesis in developing endothelial and mesenchymal vascular cells. (J Vasc Surg 2000;32:364-73.

Extreme Value Theory with Spectral Techniques: application to a simple attractor

We give a brief account of application of extreme value theory in dynamical systems by using perturbation techniques associated to the transfer operator. We will apply it to the baker's map and we will get a precise formula for the extremal index. We will also show that the statistics of the number of visits in small sets is compound Poisson distributed.Comment: 15