Perfect State Transfer in Laplacian Quantum Walk

For a graph $G$ and a related symmetric matrix $M$, the continuous-time quantum walk on $G$ relative to $M$ is defined as the unitary matrix $U(t) = \exp(-itM)$, where $t$ varies over the reals. Perfect state transfer occurs between vertices $u$ and $v$ at time $\tau$ if the $(u,v)$-entry of $U(\tau)$ has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer: (1) If a $n$-vertex graph has perfect state transfer at time $\tau$ relative to the Laplacian, then so does its complement if $n\tau$ is an integer multiple of $2\pi$. As a corollary, the double cone over any $m$-vertex graph has perfect state transfer relative to the Laplacian if and only if $m \equiv 2 \pmod{4}$. This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009). (2) If a graph $G$ has perfect state transfer at time $\tau$ relative to the normalized Laplacian, then so does the weak product $G \times H$ if for any normalized Laplacian eigenvalues $\lambda$ of $G$ and $\mu$ of $H$, we have $\mu(\lambda-1)\tau$ is an integer multiple of $2\pi$. As a corollary, a weak product of $P_{3}$ with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of $P_{3}$ has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).Comment: 26 pages, 5 figures, 1 tabl

Computer program applying Fourier transforms to the analysis of gamma spectral data

"August 1968.""MIT-3944-2."Includes bibliographical references (pages 86-87)GAMANL, a computer code for automatically identifying the peaks in a complex spectra and determining their centers and areas, is described. The principal feature of the method is a data smoothing technique employing Fourier transforms. The smoothing eliminates most of the random fluctuations without effecting the spectral resolution and makes identification of maxima using a zero slope criterion possible. Using the same Fourier transform with different constants it is possible with a second transformation to improve the spectral resolution. The computer program has been written in FORTRAN IV for the M.IT. IBM 360 model 65 computer and also for the Toshiba Electric Company G.E. 635 computer. The complete analysis of a 4096 channel spectrum containing one hundred twenty peaks requires about 75 seconds of computation time.United States Atomic Energy Commission contract AT(30-1)-394

Thermal neutron capture gamma-ray spectra of the elements

"January 1969."Statement of responsibility on title-page reads: Norman C. Rasmussen, Yoshiyuti Hukai, Tamon Inouye, Victor J. Orphan"Prepared for Air Force Cambridge Research Laboratories, Office of Aerospace Research, United States Air Force, Bedford, Mass.""AFCRL-69-0071."Includes bibliographical referencesAF19(628)5551Project no. 5620, Task no. 562003, Work unit no. 5620030

Cluster Morphologies as a Test of Different Cosmological Models

We investigate how cluster morphology is affected by the cosmological constant in low-density universes. Using high-resolution cosmological N-body/SPH simulations of flat (\Omega_0 = 0.3, \lambda_0 = 0.7, \Lambda CDM) and open (\Omega_0 = 0.3, \lambda_0 = 0, OCDM) cold dark matter universes, we calculate statistical indicators to quantify the irregularity of the cluster morphologies. We study axial ratios, center shifts, cluster clumpiness, and multipole moment power ratios as indicators for the simulated clusters at z=0 and 0.5. Some of these indicators are calculated for both the X-ray surface brightness and projected mass distributions. In \Lambda CDM all these indicators tend to be larger than those in OCDM at z=0. This result is consistent with the analytical prediction of Richstone, Loeb, & Turner, that is, clusters in \Lambda CDM are formed later than in OCDM, and have more substructure at z=0. We make a Kolmogorov-Smirnov test on each indicator for these two models. We then find that the results for the multipole moment power ratios and the center shifts for the X-ray surface brightness are under the significance level (5%). We results also show that these two cosmological models can be distinguished more clearly at z=0 than z = 0.5 by these indicators.Comment: 30pages, 6figures, Accepted for publication in Ap