Splitting groups with cubic Cayley graphs of connectivity two

A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup

On vertex-transitive graphs with a unique hamiltonian cycle

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss some examples with infinitely many ends. In particular, we show each nonabelian free group $F_n$ has a Cayley graph of degree $2n + 2$ that has a unique hamiltonian circle. (A weaker statement had been conjectured by A. Georgakopoulos.) Furthermore, we prove that these Cayley graphs of $F_n$ are outerplanar

Photorefractive Keratectomy for Mild-to-Moderate Myopics with Thin Corneas: A 3-Year Follow-up Results

Background: The purpose of this study was to evaluate the 3-year visual outcomes in mild-to-moderate myopic patients with thin corneas who underwent photorefractive keratectomy (PRK) with or without mitomycin C (MMC).Methods: Thirty myopic eyes with a mean spherical equivalent (SE) of −3.76 ± 1.72 (−6.50 to −1.25) diopter (D) and a mean corneal thickness of 486.03 ± 11.93 (452-499) µm at the thinnest point underwent PRK. MMC was used if they had > 4.0 D myopia. All surgeries were performed with the VISX STAR S4 Excimer Laser (Abbott Medical Optics, Abbott Park, Illinois, USA).Results: Uncorrected visual acuity showed a significant improvement 3 years after surgery when compared to baseline and reached 0.01 ± 0.04 LogMAR (P < 0.001). Best-corrected visual acuity was 0.00 ± 0.01 LogMAR preoperatively and did not change significantly postoperatively. Spherical (P < 0.001) and cylindrical (P < 0.001) error significantly decreased. Manifest refraction SE showed a significant decrease when compared to before the operation and reached −0.08 ± 0.16 D (P < 0.001). At 3 years, mesopic contrast sensitivity was not significantly different from baseline at any spatial frequency. Vertical coma showed a significant decrease and reached −0.10 ± 0.27 µm (P = 0.004). Total coma (P < 0.001), spherical aberration (P < 0.001), and total high order aberrations (P < 0.001) also increased significantly.Conclusions: Based on the 3-year results, PRK (+MMC in patients with SE > 4.0 D) is a safe, effective, and predictable treatment option for mild-to-moderate myopic patients whose minimum corneal thickness is < 500 µm

MEMS 411: Free Kick Robot

The free kick robot was designed to help train the club soccer goalkeeper: Brady Shaeffer, to save shots on goal. After a customer interview with the our client, we aimed our design to shoot accurate and powerful shots in quick succession. In addition the robot was to be easily transportable. Initially, after looking at some patents, the most commercially used soccer training robots involved two parallel disks spinning and shooting out soccer balls. As a result an initial prototype was built with this idea. However, during our concept selection, we realized that the cost of building our machine was out of our 400 dollar budget. Our alternative design focused on a using swinging arm, where energy was powered by a winch and stored through springs