On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.Comment: 19 page

Thermoelectric Properties of WSi2–SixGe1−X Composites

Thermoelectric properties of the W/Si/Ge alloy system have been investigated with varying concentration levels of germanium and tungsten. The alloys were fabricated by directional solidification with the Bridgman method using boron nitride and fused silica crucibles. The effect of crucible contamination was investigated and found to result in doping the system to suitable levels for thermoelectric applications. The system has been demonstrated as a suitable high temperature p-type thermoelectric material exhibiting high power factors, \u3e3000 μW/m K2. Seebeck coefficients of the system are on the order of +300 μV/K and electrical conductivities of 2.8 × 104 S/m at the optimum operating temperature. The best composition, 0.9 at% W/9.3 at% Ge, achieved a figure of merit comparable to RTG values over the temperature range of interest. The results suggest that W addition can reduce the use of expensive Ge component of the alloy. Reported are the details of processing conditions, microstructure development, and temperature dependent thermoelectric properties. The material system was stable at the temperatures required for NASA’s radioisotope thermoelectric generators

The PAndAS view of the Andromeda satellite system - I. A Bayesian search for dwarf galaxies using spatial and color-magnitude information

We present a generic algorithm to search for dwarf galaxies in photometric catalogs and apply it to the Pan-Andromeda Archaeological Survey (PAndAS). The algorithm is developed in a Bayesian framework and, contrary to most dwarf-galaxy-search codes, makes use of both the spatial and color-magnitude information of sources in a probabilistic approach. Accounting for the significant contamination from the Milky Way foreground and from the structured stellar halo of the Andromeda galaxy, we recover all known dwarf galaxies in the PAndAS footprint with high significance, even for the least luminous ones. Some Andromeda globular clusters are also recovered and, in one case, discovered. We publish a list of the 143 most significant detections yielded by the algorithm. The combined properties of the 39 most significant isolated detections show hints that at least some of these trace genuine dwarf galaxies, too faint to be individually detected. Follow-up observations by the community are mandatory to establish which are real members of the Andromeda satellite system. The search technique presented here will be used in an upcoming contribution to determine the PAndAS completeness limits for dwarf galaxies. Although here tuned to the search of dwarf galaxies in the PAndAS data, the algorithm can easily be adapted to the search for any localised overdensity whose properties can be modeled reliably in the parameter space of any catalog.Comment: 20 pages, 16 figures, 1 table; accepted for publication in ApJ. High res pdf available at https://www.dropbox.com/s/7zk7pme2wunwkjv/PAndAS_dwarf_galaxies.pd

Reciprocal relativity of noninertial frames: quantum mechanics

Noninertial transformations on time-position-momentum-energy space {t,q,p,e} with invariant Born-Green metric ds^2=-dt^2+dq^2/c^2+(1/b^2)(dp^2-de^2/c^2) and the symplectic metric -de/\dt+dp/\dq are studied. This U(1,3) group of transformations contains the Lorentz group as the inertial special case. In the limit of small forces and velocities, it reduces to the expected Hamilton transformations leaving invariant the symplectic metric and the nonrelativistic line element ds^2=dt^2. The U(1,3) transformations bound relative velocities by c and relative forces by b. Spacetime is no longer an invariant subspace but is relative to noninertial observer frames. Born was lead to the metric by a concept of reciprocity between position and momentum degrees of freedom and for this reason we call this reciprocal relativity. For large b, such effects will almost certainly only manifest in a quantum regime. Wigner showed that special relativistic quantum mechanics follows from the projective representations of the inhomogeneous Lorentz group. Projective representations of a Lie group are equivalent to the unitary reprentations of its central extension. The same method of projective representations of the inhomogeneous U(1,3) group is used to define the quantum theory in the noninertial case. The central extension of the inhomogeneous U(1,3) group is the cover of the quaplectic group Q(1,3)=U(1,3)*s H(4). H(4) is the Weyl-Heisenberg group. A set of second order wave equations results from the representations of the Casimir operators

Young accreted globular clusters in the outer halo of M31

We report on Gemini/GMOS observations of two newly discovered globular clusters in the outskirts of M31. These objects, PAndAS-7 and PAndAS-8, lie at a galactocentric radius of ~87 kpc and are projected, with separation ~19 kpc, onto a field halo substructure known as the South-West Cloud. We measure radial velocities for the two clusters which confirm that they are almost certainly physically associated with this feature. Colour-magnitude diagrams reveal strikingly short, exclusively red horizontal branches in both PA-7 and PA-8; both also have photometric [Fe/H] = -1.35 +/- 0.15. At this metallicity, the morphology of the horizontal branch is maximally sensitive to age, and we use the distinctive configurations seen in PA-7 and PA-8 to demonstrate that both objects are very likely to be at least 2 Gyr younger than the oldest Milky Way globular clusters. Our observations provide strong evidence for young globular clusters being accreted into the remote outer regions of M31 in a manner entirely consistent with the established picture for the Milky Way, and add credence to the idea that similar processes play a central role in determining the composition of globular cluster systems in large spiral galaxies in general.Comment: 14 pages, 8 figures, accepted for publication in MNRA

Rigorous steps towards holography in asymptotically flat spacetimes

Scalar QFT on the boundary $\Im^+$ at null infinity of a general asymptotically flat 4D spacetime is constructed using the algebraic approach based on Weyl algebra associated to a BMS-invariant symplectic form. The constructed theory is invariant under a suitable unitary representation of the BMS group with manifest meaning when the fields are interpreted as suitable extensions to $\Im^+$ of massless minimally coupled fields propagating in the bulk. The analysis of the found unitary BMS representation proves that such a field on $\Im^+$ coincides with the natural wave function constructed out of the unitary BMS irreducible representation induced from the little group $\Delta$, the semidirect product between SO(2) and the two dimensional translational group. The result proposes a natural criterion to solve the long standing problem of the topology of BMS group. Indeed the found natural correspondence of quantum field theories holds only if the BMS group is equipped with the nuclear topology rejecting instead the Hilbert one. Eventually some theorems towards a holographic description on $\Im^+$ of QFT in the bulk are established at level of $C^*$ algebras of fields for strongly asymptotically predictable spacetimes. It is proved that preservation of a certain symplectic form implies the existence of an injective $*$-homomorphism from the Weyl algebra of fields of the bulk into that associated with the boundary $\Im^+$. Those results are, in particular, applied to 4D Minkowski spacetime where a nice interplay between Poincar\'e invariance in the bulk and BMS invariance on the boundary at $\Im^+$ is established at level of QFT. It arises that the $*$-homomorphism admits unitary implementation and Minkowski vacuum is mapped into the BMS invariant vacuum on $\Im^+$.Comment: 62 pages, amslatex, xy package; revised section 2 and the conclusions; corrected some typos; added some references; accepted for pubblication on Rev. Math. Phy

Major Substructure in the M31 Outer Halo: the South-West Cloud

We undertake the first detailed analysis of the stellar population and spatial properties of a diffuse substructure in the outer halo of M31. The South-West Cloud lies at a projected distance of ~100 kpc from the centre of M31, and extends for at least ~50 kpc in projection. We use Pan-Andromeda Archaeological Survey photometry of red giant branch stars to determine a distance to the South-West Cloud of 793 +/- 45 kpc. The metallicity of the cloud is found to be [Fe/H] = -1.3 +/- 0.1. This is consistent with the coincident globular clusters PAndAS-7 and PAndAS-8, which have metallicities determined using an independent technique of [Fe/H] = -1.35 +/- 0.15. We measure a brightness for the Cloud of M_V = -12.1 mag; this is ~75 per cent of the luminosity implied by the luminosity-metallicity relation. Under the assumption that the South-West Cloud is the visible remnant of an accreted dwarf satellite, this suggests that the progenitor object was amongst M31's brightest dwarf galaxies prior to disruption.Comment: 13 pages, 9 figures, accepted for publication in MNRA

Projective Fourier Duality and Weyl Quantization

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 102Kb, 44 pages, no figures. requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with text improvements and crucial typos correction

Fourier Duality as a Quantization Principle

The Weyl-Wigner prescription for quantization on Euclidean phase spaces makes essential use of Fourier duality. The extension of this property to more general phase spaces requires the use of Kac algebras, which provide the necessary background for the implementation of Fourier duality on general locally compact groups. Kac algebras -- and the duality they incorporate -- are consequently examined as candidates for a general quantization framework extending the usual formalism. Using as a test case the simplest non-trivial phase space, the half-plane, it is shown how the structures present in the complete-plane case must be modified. Traces, for example, must be replaced by their noncommutative generalizations - weights - and the correspondence embodied in the Weyl-Wigner formalism is no more complete. Provided the underlying algebraic structure is suitably adapted to each case, Fourier duality is shown to be indeed a very powerful guide to the quantization of general physical systems.Comment: LaTeX 2.09 with NFSS or AMSLaTeX 1.1. 97Kb, 43 pages, no figures. requires subeqnarray.sty, amssymb.sty, amsfonts.sty. Final version with (few) text and (crucial) typos correction

PAndAS in the mist: The stellar and gaseous mass within the halos of M31 and M33

Large scale surveys of the prominent members of the Local Group have provided compelling evidence for the hierarchical formation of massive galaxies, revealing a wealth of substructure that is thought to be the debris from ancient and on-going accretion events. In this paper, we compare two extant surveys of the M31-M33 subgroup of galaxies; the Pan-Andromeda Archaeological Survey (PAndAS) of the stellar structure, and a combination of observations of the HI gaseous content, detected at 21cm. Our key finding is a marked lack of spatial correlation between these two components on all scales, with only a few potential overlaps between stars and gas.The paucity of spatial correlation significantly restricts the analysis of kinematic correlations, although there does appear to the HI kinematically associated with the Giant Stellar Stream where it passes the disk of M31. These results demonstrate that that different processes must significantly influence the dynamical evolution of the stellar and HI components of substructures, such as ram pressure driving gas away from a purely gravitational path. Detailed modelling of the offset between the stellar and gaseous substructure will provide a determination of the properties of the gaseous halo of M31 and M33.Comment: 11 pages, 6 figures. Accepted for publication in the Astrophysical Journal. Figure quality reduced. High quality version available at http://www.physics.usyd.edu.au/~gfl/Arxiv_Papers/PAndAS_Mist