Parameterizing scalar-tensor theories for cosmological probes

We study the evolution of density perturbations for a class of $f(R)$ models which closely mimic $\Lambda$CDM background cosmology. Using the quasi-static approximation, and the fact that these models are equivalent to scalar-tensor gravity, we write the modified Friedmann and cosmological perturbation equations in terms of the mass $M$ of the scalar field. Using the perturbation equations, we then derive an analytic expression for the growth parameter $\gamma$ in terms of $M$, and use our result to reconstruct the linear matter power spectrum. We find that the power spectrum at $z \sim 0$ is characterized by a tilt relative to its General Relativistic form, with increased power on small scales. We discuss how one has to modify the standard, constant $\gamma$ prescription in order to study structure formation for this class of models. Since $\gamma$ is now scale and time dependent, both the amplitude and transfer function associated with the linear matter power spectrum will be modified. We suggest a simple parameterization for the mass of the scalar field, which allows us to calculate the matter power spectrum for a broad class of $f(R)$ models

SIC~POVMs and Clifford groups in prime dimensions

We show that in prime dimensions not equal to three, each group covariant symmetric informationally complete positive operator valued measure (SIC~POVM) is covariant with respect to a unique Heisenberg--Weyl (HW) group. Moreover, the symmetry group of the SIC~POVM is a subgroup of the Clifford group. Hence, two SIC~POVMs covariant with respect to the HW group are unitarily or antiunitarily equivalent if and only if they are on the same orbit of the extended Clifford group. In dimension three, each group covariant SIC~POVM may be covariant with respect to three or nine HW groups, and the symmetry group of the SIC~POVM is a subgroup of at least one of the Clifford groups of these HW groups respectively. There may exist two or three orbits of equivalent SIC~POVMs for each group covariant SIC~POVM, depending on the order of its symmetry group. We then establish a complete equivalence relation among group covariant SIC~POVMs in dimension three, and classify inequivalent ones according to the geometric phases associated with fiducial vectors. Finally, we uncover additional SIC~POVMs by regrouping of the fiducial vectors from different SIC~POVMs which may or may not be on the same orbit of the extended Clifford group.Comment: 30 pages, 1 figure, section 4 revised and extended, published in J. Phys. A: Math. Theor. 43, 305305 (2010

Retrodictively Optimal Localisations in Phase Space

In a previous paper it was shown that the distribution of measured values for a retrodictively optimal simultaneous measurement of position and momentum is always given by the initial state Husimi function. This result is now generalised to retrodictively optimal simultaneous measurements of an arbitrary pair of rotated quadratures x_theta1 and x_theta2. It is shown, that given any such measurement, it is possible to find another such measurement, informationally equivalent to the first, for which the axes defined by the two quadratures are perpendicular. It is further shown that the distribution of measured values for such a meaurement belongs to the class of generalised Husimi functions most recently discussed by Wuensche and Buzek. The class consists of the subset of Wodkiewicz's operational probability distributions for which the filter reference state is a squeezed vaccuum state.Comment: 11 pages, 2 figures. AMS Latex. Replaced with published versio

Water quality investigation of Loweswater, Cumbria

This is the final report to the Environment Agency: Water Quality Investigation of Loweswater, Cumbria. The project employs palaeolimnological techniques to evaluate the extent of eutrophication at Loweswater

Husimi Transform of an Operator Product

It is shown that the series derived by Mizrahi, giving the Husimi transform (or covariant symbol) of an operator product, is absolutely convergent for a large class of operators. In particular, the generalized Liouville equation, describing the time evolution of the Husimi function, is absolutely convergent for a large class of Hamiltonians. By contrast, the series derived by Groenewold, giving the Weyl transform of an operator product, is often only asymptotic, or even undefined. The result is used to derive an alternative way of expressing expectation values in terms of the Husimi function. The advantage of this formula is that it applies in many of the cases where the anti-Husimi transform (or contravariant symbol) is so highly singular that it fails to exist as a tempered distribution.Comment: AMS-Latex, 13 page

Long quantum channels for high-quality entanglement transfer

High-quality quantum-state and entanglement transfer can be achieved in an unmodulated spin bus operating in the ballistic regime, which occurs when the endpoint qubits A and B are coupled to the chain by an exchange interaction $j_0$ comparable with the intrachain exchange. Indeed, the transition amplitude characterizing the transfer quality exhibits a maximum for a finite optimal value $j_0^{opt}(N)$, where $N$ is the channel length. We show that $j_0^{opt}(N)$ scales as $N^{-1/6}$ for large $N$ and that it ensures a high-quality entanglement transfer even in the limit of arbitrarily long channels, almost independently of the channel initialization. For instance, the average quantum-state transmission fidelity exceeds 90% for any chain length. We emphasize that, taking the reverse point of view, should $j_0$ be experimentally constrained, high-quality transfer can still be obtained by adjusting the channel length to its optimal value.Comment: 12 pages, 9 figure

From SICs and MUBs to Eddington

This is a survey of some very old knowledge about Mutually Unbiased Bases (MUB) and Symmetric Informationally Complete POVMs (SIC). In prime dimensions the former are closely tied to an elliptic normal curve symmetric under the Heisenberg group, while the latter are believed to be orbits under the Heisenberg group in all dimensions. In dimensions 3 and 4 the SICs are understandable in terms of elliptic curves, but a general statement escapes us. The geometry of the SICs in 3 and 4 dimensions is discussed in some detail.Comment: 12 pages; from the Festschrift for Tony Sudber

Environmental change and atmospheric contamination across China as indicated by lake sediments (Joint Project Q741)

In China, anthropogenic impact from changes in water quality, land-use and atmospheric deposition vary both spatially and temporally. There is a gradient across the country from the populous lowlands in the east, where considerable long-term impact on water bodies has resulted from centuries of agricultural and aquacultural practices superimposed by more recent, rapid industrial growth; to the mountainous west where many areas remain minimally impacted and any anthropogenic impact is restricted to long-range transport of atmospheric pollutants and change in climate. Currently, however, there is little information on temporal trends in atmospheric deposition. The importance of water resources in China cannot be overestimated and therefore determining the extent, rate and direction of change in water quality is a national priority. In the absence of long-term monitoring programmes, lake sediments remain the only way whereby this information can be retrospectively determined at the necessary temporal scale to ascertain whether the causes of any detriment in quality are as a result of natural changes, or due to agricultural or industrial impacts. Therefore, the aim of this research programme is to use lake sediments from three regions of China to detennine the extent to which impacts to lakes have changed through time and the causes of these changes. The project focuses on an east - west transect along the Yangtse River. The Middle and Lower Reaches of the Yangtse are undergoing exceptionally rapid economic and industrial development and this region is receiving a great deal of attention as the Three Gorges Dam undergoes construction. The three areas in which lakes are to be studied in this proposal are: l. Jiangsu Province in east China north of Shanghai. Lowland sites. Lakes in this area are likely to have been impacted by long-term agriculture and may therefore may have become eutrophic. Atmospheric deposition may also be significant from local and regional industrial sources. 2. The upper reaches of the Yangtse River, in Sichuan and Yunnan Provinces. Lakes at l-2000m. Lakes in this area may have been impacted by agriculture, but where possible sites will be selected where direct impact is minimal. Lakes may therefore be mesotrophic but sources of atmospherically derived industrial pollutants are likely to be remote. 3. The Tibetan Plateau. Lakes at 4 - 5000m. 'Control' sites in a pristine area with minimal human influence. Sediment cores from these sites will be used to detem1ine background levels of atmospherically deposited contaminants. Oligotrophic (low nutrient) sites on sensitive geology (low acid neutralising capacity e.g. granites) will be selected where possible. By including earlier collaborative work between the participating institutes (funded by the Royal Society and the Chinese Academy of Sciences) on the lake sediment records on the Jianghan Plain and research currently being undertaken on lakes in the Shennonggjia region in western Hubei (NNFSC funded), this study will produce a transect of lakes from five regions providing a unique database on the historical impact of human activity on the freshwater environment

A Re-Examination of the Taxonomic Boundaries of \u3cem\u3eSymphysia\u3c/em\u3e (Ericaceae)

DNA sequence data were generated for the nuclear ITS region for Symphysia racemosa and for 26 additional Vaccinieae representing 12 sections in the genus Vaccinium plus one species from each of five additional segregate genera. Our focus is on the placement of S. racemosa relative to Vaccinium sensu scricto and Vaccinium sect. Oreades (represented by V. poasanum). Maximum parsimony analysis of 608 bp of nrITS region suggests that S. racemosa and V. poasanum form a well-supported clade in spite of substantial morphological divergence. Futhermore, this clade is a sister group to a clade consisting of all segregate genera examined. These molecular results led us to undertake a morphological cladistic analysis of all of the other Central American green-flowered taxa. We suggest that the genus Symphysia should be expanded to encompass these 15 taxa, despite the lack of phylogenetic resolution within this group. This will necessitate eight new combinations, via. Symphysia almedae (= V. almedae), Symphysia costaricensis (= V. costaricense), Symphysia jefensis (= V. jefense), Symphysia orosiensis (= V. orosiense), Symphysia ovata (= Lateropora ovata), Symphysia perardua (= V. santafeënsis), Symphysia poasana (= Vaccinium poasanum), Symphysia santafeënsis (= L. santafeënsis), and Symphysia tubulifera (= L. tubulifera)

The Lie Algebraic Significance of Symmetric Informationally Complete Measurements

Examples of symmetric informationally complete positive operator valued measures (SIC-POVMs) have been constructed in every dimension less than or equal to 67. However, it remains an open question whether they exist in all finite dimensions. A SIC-POVM is usually thought of as a highly symmetric structure in quantum state space. However, its elements can equally well be regarded as a basis for the Lie algebra gl(d,C). In this paper we examine the resulting structure constants, which are calculated from the traces of the triple products of the SIC-POVM elements and which, it turns out, characterize the SIC-POVM up to unitary equivalence. We show that the structure constants have numerous remarkable properties. In particular we show that the existence of a SIC-POVM in dimension d is equivalent to the existence of a certain structure in the adjoint representation of gl(d,C). We hope that transforming the problem in this way, from a question about quantum state space to a question about Lie algebras, may help to make the existence problem tractable.Comment: 56 page