Operator pencil passing through a given operator

Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight \lo on a manifold $M$. One can consider a pencil of operators \hPi(\Delta)=\{\Delta_\l\} passing through the operator $\Delta$ such that any \Delta_\l is a linear differential operator acting on densities of weight \l. This pencil can be identified with a linear differential operator \hD acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular we analyze the relation between these two concepts, and apply it to the study of \diff(M)-equivariant liftings. Finally we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings.Comment: 32 pages, LaTeX fil

Design of optimal transfers between North and South Pole-sitter orbits

Recent studies have shown the feasibility of an Earth pole-sitter mission, where a spacecraft follows the Earth’s polar axis to have a continuous, hemispherical view of one of the Earth’s Poles. However, due to the tilt of the polar axis, the North and South Poles are alternately situated in darkness for long periods dur-ing the year. This significantly constrains observations and decreases mission scientific return. This paper therefore investigates transfers between north and south pole-sitter orbits before the start of the Arctic and Antarctic winters to maximize scientific return by observing the polar regions only when lit. Clearly, such a transfer can also be employed for the sole purpose of visiting both the North and South Poles with one single spacecraft during one single mission. To enable such a novel transfer, two types of propulsion are proposed, including so-lar electric propulsion (SEP) and a hybridization of SEP with solar sailing. A di-rect optimization method based on pseudospectral transcription is used to find both transfers that minimize the SEP propellant consumption and transfers that trade-off SEP propellant consumption and observation time of the Poles. Also, a feedback control is developed to account for non-ideal properties of the solar sail. It is shown that, for all cases considered, hybrid low-thrust propulsion out-performs the pure SEP case, while enabling a transfer that would not be feasible with current solar sail technology

Orbit period modulation for relative motion using continuous low thrust in the two-body and restricted three-body problems

This paper presents rich new families of relative orbits for spacecraft formation flight generated through the application of continuous thrust with only minimal intervention into the dynamics of the problem. Such simplicity facilitates implementation for small, low-cost spacecraft with only position state feedback, and yet permits interesting and novel relative orbits in both two- and three-body systems with potential future applications in space-based interferometry, hyperspectral sensing, and on-orbit inspection. Position feedback is used to modify the natural frequencies of the linearised relative dynamics through direct manipulation of the system eigenvalues, producing new families of stable relative orbits. Specifically, in the Hill–Clohessy–Wiltshire frame, simple adaptations of the linearised dynamics are used to produce a circular relative orbit, frequency-modulated out-of-plane motion, and a novel doubly periodic cylindrical relative trajectory for the purposes of on-orbit inspection. Within the circular restricted three-body problem, a similar minimal approach with position feedback is used to generate new families of stable, frequency-modulated relative orbits in the vicinity of a Lagrange point, culminating in the derivation of the gain requirements for synchronisation of the in-plane and out-of-plane frequencies to yield a singly periodic tilted elliptical relative orbit with potential use as a Lunar far-side communications relay. The Δv requirements for the cylindrical relative orbit and singly periodic Lagrange point orbit are analysed, and it is shown that these requirements are modest and feasible for existing low-thrust propulsion technology

Supersymmetry on Graphs and Networks

We show that graphs, networks and other related discrete model systems carry a natural supersymmetric structure, which, apart from its conceptual importance as to possible physical applications, allows to derive a series of spectral properties for a class of graph operators which typically encode relevant graph characteristics.Comment: 11 pages, Latex, no figures, remark 4.1 added, slight alterations in lemma 5.3, a more detailed discussion at beginning of sect.6 (zero eigenspace

Education in Blackburn, 1870 - 1914

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How Have Public Sector Pensions Responded to the Financial Crisis?

State and local government pension funds lost nearly $1 trillion in net assets in 2007-08. Average pension funding levels fell from 85 percent in 2007 to 77 percent in 2009. We analyze these patterns by looking at target asset allocations for 29 large public sector plans covering over half of all public pension assets. On average, the standard deviation of target portfolio returns increased slightly and 14 of the 29 plans surveyed increased risk by more than 0.3 percentage points and 8 increased risk by more than one percentage point. Few plans seem to have retreated from risk-taking, while a small number appears to have embraced more risk than before

Automating reuse support in a small company

Software engineering has been facing a crisis for several years now - there is more demand for new software than there is ability to supply. Software reuse is a potential way to tackle the problems caused by the software crisis with its promises of increased productivity and cheaper development costs. Several software reuse successes have been reported, but these have been predominantly in large, well structured companies. However, there are numerous smaller companies that could also benefit from reuse if it were made available to them. This thesis addresses these issues by implementing a reuse programme in a small company. An incremental approach to reuse introduction is adopted, following the Seven Steps to Success, and 'lightweight' processes are recommended to support the reuse programme. A prototype tool set, ReThree- C++, was developed to automate support for the reuse programme. The results of the case study are presented. The reuse programme was successful, with benefits to the company including both increased speed of production and financial gains from selling reusable components. The challenges faced are also identified. Details of the tool set giving automated support for reuse are also presented. The tool set is an approach to reuse repository control which also integrates information abstraction from C++ source code to generate class hierarchy charts and software documentation automatically. It helps developers store, retrieve, understand and use reusable components. The usefulness of the tool set is shown with an experiment and as part of the case study. The purpose of the thesis is to show that small companies can implement reuse, and that the method presented supports the introduction of a reuse programme. It concludes that although challenges were faced, great benefits can be gained by using the method with automated support for reuse in a small company