Semi-regular Relative Difference Sets with Large Forbidden Subgroups

Motivated by a connection between semi-regular relative difference sets and mutually unbiased bases, we study relative difference sets with parameters $(m,n,m,m/n)$ in groups of non-prime-power orders. Let $p$ be an odd prime. We prove that there does not exist a $(2p,p,2p,2)$ relative difference set in any group of order $2p^2$, and an abelian $(4p,p,4p,4)$ relative difference set can only exist in the group $\Bbb{Z}_2^2\times \Bbb{Z}_3^2$. On the other hand, we construct a family of non-abelian relative difference sets with parameters $(4q,q,4q,4)$, where $q$ is an odd prime power greater than 9 and $q\equiv 1$ (mod 4). When $q=p$ is a prime, $p>9$, and $p\equiv$ 1 (mod 4), the $(4p,p,4p,4)$ non-abelian relative difference sets constructed here are genuinely non-abelian in the sense that there does not exist an abelian relative difference set with the same parameters

Equiangular lines, mutually unbiased bases, and spin models

We use difference sets to construct interesting sets of lines in complex space. Using (v,k,1)-difference sets, we obtain k^2-k+1 equiangular lines in C^k when k-1 is a prime power. Using semiregular relative difference sets with parameters (k,n,k,l) we construct sets of n+1 mutually unbiased bases in C^k. We show how to construct these difference sets from commutative semifields and that several known maximal sets of mutually unbiased bases can be obtained in this way, resolving a conjecture about the monomiality of maximal sets. We also relate mutually unbiased bases to spin models.Comment: 23 pages; no figures. Minor correction as pointed out in arxiv.org:1104.337

Some Non-Existence Results on Divisible Difference Sets

In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches: (i) character sum arguments similar to the work of Turyn [25] for ordinary difference sets, (ii) involution arguments, and (iii) multipliers in conjunction with results on ordinary difference sets. Among other results, we show that an abelian affine difference set of odd order s (s not a perfect square) in G can exist only if the Sylow 2-subgroup of G is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd order n

A Unifying Construction for Difference Sets

We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, λ,n)=(22d+4(22d+2−1)/3, 22d+1(22d+3+1)/3, 22d+1(22d+1+1)/3, 24d+2) for d⩾0. The construction establishes that a McFarland difference set exists in an abelian group of order 22d+3(22d+1+1)/3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of order pr. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly for r\u3e1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylow p-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups

On the rate-distortion performance and computational efficiency of the Karhunen-Loeve transform for lossy data compression

We examine the rate-distortion performance and computational complexity of linear transforms for lossy data compression. The goal is to better understand the performance/complexity tradeoffs associated with using the Karhunen-Loeve transform (KLT) and its fast approximations. Since the optimal transform for transform coding is unknown in general, we investigate the performance penalties associated with using the KLT by examining cases where the KLT fails, developing a new transform that corrects the KLT's failures in those examples, and then empirically testing the performance difference between this new transform and the KLT. Experiments demonstrate that while the worst KLT can yield transform coding performance at least 3 dB worse than that of alternative block transforms, the performance penalty associated with using the KLT on real data sets seems to be significantly smaller, giving at most 0.5 dB difference in our experiments. The KLT and its fast variations studied here range in complexity requirements from O(n^2) to O(n log n) in coding vectors of dimension n. We empirically investigate the rate-distortion performance tradeoffs associated with traversing this range of options. For example, an algorithm with complexity O(n^3/2) and memory O(n) gives 0.4 dB performance loss relative to the full KLT in our image compression experiment

Astrometry and exoplanets in the Gaia era: a Bayesian approach to detection and parameter recovery

(abridged) We develop Bayesian methods and detection criteria for orbital fitting, and revise the detectability of exoplanets in light of the in-flight properties of Gaia. Limiting ourselves to one-planet systems as a first step of the development, we simulate Gaia data for exoplanet systems over a grid of S/N, orbital period, and eccentricity. The simulations are then fit using Markov chain Monte Carlo methods. We investigate the detection rate according to three information criteria and the delta chi^2. For the delta chi^2, the effective number of degrees of freedom depends on the mission length. We find that the choice of the Markov chain starting point can affect the quality of the results; we therefore consider two limit possibilities: an ideal case, and a very simple method that finds the starting point assuming circular orbits. Using Jeffreys' scale of evidence, the fraction of false positives passing a strong evidence criterion is < ~0.2% (0.6%) when considering a 5 yr (10 yr) mission and using the Akaike information criterion or the Watanabe-Akaike information criterion, and <0.02% (<0.06%) when using the Bayesian information criterion. We find that there is a 50% chance of detecting a planet with a minimum S/N=2.3 (1.7). This sets the maximum distance to which a planet is detectable to ~70 pc and ~3.5 pc for a Jupiter-mass and Neptune-mass planet, respectively, assuming a 10 yr mission, a 4 au semi-major axis, and a 1 M_sun star. The period is the orbital parameter that can be determined with the best accuracy, with a median relative difference between input and output periods of 4.2% (2.9%) assuming a 5 yr (10 yr) mission. The median accuracy of the semi-major axis of the orbit can be recovered with a median relative error of 7% (6%). The eccentricity can also be recovered with a median absolute accuracy of 0.07 (0.06).Comment: 18 pages, 11 figures. New version accepted by A&A for publicatio

Phase Transitions for Binomial Sets Under Linear Forms

We revisit the study of the sum and difference sets of a subset of $\mathbb N$ drawn from a binomial model, proceeding under the following more general setting. Given $A \subseteq \{0, 1, \dots, N\}$, an integer $h \geq 2$, and a linear form $L: \mathbb{Z}^h \to \mathbb{Z}$ given by $L(x_1, \dots, x_h) = u_1x_1 + \cdots + u_hx_h$ with nonzero $u_i$, we study the size of $L(A) = \left\{u_1a_1 + \cdots + u_ha_h : a_i \in A \right\}$ and its complement $L(A)^c$ when each element of $\{0, 1, \dots, N\}$ is independently included in $A$ with probability $p(N)$. We identify two phase transition phenomena. The first concerns the relative sizes of $L(A)$ and $L(A)^c$, with $p(N) = N^{-(h-1)/h}$ as the threshold. Asymptotically almost surely, it holds below the threshold that almost all sums generated in $L(A)$ are distinct and almost all possible sums are in $L(A)^c$, and above the threshold that almost all possible sums are in $L(A)$. This generalizes work of Hegarty and Miller and settles their conjecture. The second, which may be expressed in terms of a stochastic process on hypergraphs, concerns the asymptotic behavior of the number of distinct representations in $L(A)$ of a given value, with $p(N) = N^{-\frac{h-2}{h-1}}$ as the threshold.Comment: 34 pages, no figure

Coherence of long-term stratospheric ozone vertical distribution time series used for the study of ozone recovery at a northern mid-latitude station

The coherence of stratospheric ozone time series retrieved from various observational records is investigated at Haute-Provence Observatory (OHP–43.93° N, 5.71° E). The analysis is accomplished through the intercomparison of collocated ozone measurements of Light Detection and Ranging (lidar) with Solar Backscatter UltraViolet(/2) (SBUV(/2)), Stratospheric Aerosol and Gas Experiment II (SAGE~II), Halogen Occultation Experiment (HALOE), Microwave Limb Sounder (MLS) on Upper Atmosphere Research Satellite (UARS) and Aura and Global Ozone Monitoring by Occultation of Stars (GOMOS) satellite observations as well as with in situ ozonesondes and ground-based Umkehr measurements performed at OHP. A detailed statistical study of the relative differences of ozone observations over the whole stratosphere is performed to detect any specific drift in the data. On average, all instruments show their best agreement with lidar at 20–40 km, where deviations are within ±5 %. Discrepancies are somewhat higher below 20 and above 40 km. The agreement with SAGE II data is remarkable since average differences are within ±1 % at 17–41 km. In contrast, Umkehr data underestimate systematically the lidar measurements in the whole stratosphere with a near zero bias at 16–8 hPa (~30 km). Drifts are estimated using simple linear regression for the data sets analysed in this study, from the monthly averaged difference time series. The derived values are less than ±0.5 % yr&lt;sup&gt;&amp;minus;1&lt;/sup&gt; in the 20–40 km altitude range and most drifts are not significant at the 2&lt;i&gt;σ&lt;/i&gt; level. We also discuss the possibilities of extending the SAGE II and HALOE data with the GOMOS and Aura MLS data in consideration with relative offsets and drifts since the combination of such data sets are likely to be used for the study of stratospheric ozone recovery in the future

Switching Pathways for Reversible Ligand Photodissociation in Ru(II) Polypyridyl Complexes with Steric Effects

The effect of a minor difference in ligand structure is shown to have a large effect on the photochemical pathways followed by two ruthenium(II) polypyridyl based complexes [Ru(CH3CN) (LL)](2+), 1 and 2, where LL is MeN4Py (1,1-di(pyridin-2-yl)-N,N-bis (pyridin-2-yl-methyl) ethan-1-amine) or N4Py (1,1-di (pyri din-2-yl)-N,N-bis (pyridin-2-yl-m ethyl) methanamine), respectively. In our earlier report we demonstrated near completely reversible two-way photochromism of 1, in which a pyridyl ring dissociated on irradiation with visible light to form the thermally stable 1P, [Ru(CH3CN)(2)(MeN4Py)](2+). Complex 1 was recovered upon irradiation in the near-UV. Here, we show that the methyl group in the ligand backbone is critical to the reversibility by impeding the dissociation of one of the two sets of pyridyl rings. Irradiation of 2, which does not bear the methyl group, with visible light results in formation of two thermally stable isomers 2a and 2b, which are characterized by UV-vis absorption, FTIR, H-1 NMR spectroscopy, ESI mass spectrometry, and X-ray crystallography. In contrast to 1P, in both 2a and 2b, a different pyridyl moiety is dissociated. Whereas UV irradiation returns 2a to its original state (2), the overall reversibility is limited by the relative stability of 2b. The changes to the structure of 2 made possible by the increased freedom for all four pyridyl moieties to dissociate allows access to coordination modes that are not accessible thermally opening opportunities toward new catalysts for oxidation chemistry, photochromism and photoswitching.</p