WEAK ISOMETRIES OF HAMMING SPACES

In this thesis we study weak isometries of Hamming spaces. These are permutations of a Hamming space that preserve some but not necessarily all distances. We wish to find conditions under which a weak isometry is in fact an isometry. This type of problem was first posed by Beckman and Quarles for Rn. In chapter 2 we give definitions pertinent to our research. The 3rd chapter focuses on some known results in this area with special emphasis on papers by V. Krasin as well as S. De Winter and M. Korb who solved this problem for the Boolean cube, that is, the binary Hamming space. We attempted to generalize some of their methods to the non-boolean case. The 4th chapter has our new results and is split into two major contributions. Our first contribution shows if n=p or p < n2, then every weak isometry of Hnq that preserves distance p is an isometry. Our second contribution gives a possible method to check if a weak isometry is an isometry using linear algebra and graph theory.

Measuring energy spectra of TeV gamma-ray emission from the Cygnus region of our galaxy with Milagro

High energy gamma rays can provide fundamental clues to the origins of cosmic rays. In this thesis, TeV gamma-ray emission from the Cygnus region is studied. Previously the Milagro experiment detected five TeV gamma-ray sources in this region and a significant excess of TeV gamma rays whose origin is still unclear. To better understand the diffuse excess the separation of sources and diffuse emission is studied using the latest and most sensitive data set of the Milagro experiment. In addition, a newly developed technique is applied that allows the energy spectrum of the TeV gamma rays to be reconstructed using Milagro data. No conclusive statement can be made about the spectrum of the diffuse emission from the Cygnus region because of its low significance of 2.2 σ above the background in the studied data sample. The entire Cygnus region emission is best fit with a power law with a spectral index of α=2.40 (68% confidence interval: 1.35-2.92) and a exponential cutoff energy of 31.6 TeV (10.0-251.2 TeV). In the case of a simple power law assumption without a cutoff energy the best fit yields a spectral index of α=2.97 (68% confidence interval: 2.83-3.10). Neither of these best fits are in good agreement with the data. The best spectral fit to the TeV emission from MGRO J2019+37, the brightest source in the Cygnus region, yields a spectral index of α=2.30 (68% confidence interval: 1.40-2.70) with a cutoff energy of 50.1 TeV (68% confidence interval: 17.8-251.2 TeV) and a spectral index of α=2.75 (68% confidence interval: 2.65-2.85) when no exponential cutoff energy is assumed. According to the present analysis, MGRO J2019+37 contributes 25% to the differential flux from the entire Cygnus at 15 TeV.