Imaginary in all directions: an elegant formulation of special relativity and classical electrodynamics

A suitable parameterization of space-time in terms of one complex and three quaternionic imaginary units allows Lorentz transformations to be implemented as multiplication by complex-quaternionic numbers rather than matrices. Maxwell's equations reduce to a single equation.Comment: 8 page

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

The beliefs, practices and developments of three teachers of science in the primary school

This thesis explores the beliefs of three teachers about effective primary science practice, and the ways these develop in the context of a climate where there are few school resources for primary science training. The research proposes a new theoretical model combining Lave and Wenger’s communities of practice (1991) with Bourdieu’s science capital social theory (2004) to explore the impact of science knowledge, culture and science social contacts on the development of beliefs about science, science teaching and the teachers’ position and agency in the school to enable learning from others. A case study methodology was used, with three local primary teachers, employing participative observation and semi-structured interviews to gather data on beliefs on primary science and professional development. The data was evaluated using thematic analysis.The three teachers identified a range of beliefs about effective teaching strategies in primary science, including enquiry, linking ideas to observables in activities and dialogic learning approaches which appeared to be influenced by each teacher’s type and amount of science capital.There appeared to be little science expertise or CPD in schools, to support science teaching especially during the first years when teaching strategies are established. Some teachers were not part of any external ‘community of practice’ and found online resources unhelpful for developing pedagogy.The combined theoretical model was found to be effective in recognising the teachers’ previous experience in science and its impact on their present beliefs

Fast Decoders for Topological Quantum Codes

We present a family of algorithms, combining real-space renormalization methods and belief propagation, to estimate the free energy of a topologically ordered system in the presence of defects. Such an algorithm is needed to preserve the quantum information stored in the ground space of a topologically ordered system and to decode topological error-correcting codes. For a system of linear size L, our algorithm runs in time log L compared to L^6 needed for the minimum-weight perfect matching algorithm previously used in this context and achieves a higher depolarizing error threshold.Comment: 4 pages, 4 figure

Solving a "Hard" Problem to Approximate an "Easy" One: Heuristics for Maximum Matchings and Maximum Traveling Salesman Problems

We consider geometric instances of the Maximum Weighted Matching Problem (MWMP) and the Maximum Traveling Salesman Problem (MTSP) with up to 3,000,000 vertices. Making use of a geometric duality relationship between MWMP, MTSP, and the Fermat-Weber-Problem (FWP), we develop a heuristic approach that yields in near-linear time solutions as well as upper bounds. Using various computational tools, we get solutions within considerably less than 1% of the optimum. An interesting feature of our approach is that, even though an FWP is hard to compute in theory and Edmonds' algorithm for maximum weighted matching yields a polynomial solution for the MWMP, the practical behavior is just the opposite, and we can solve the FWP with high accuracy in order to find a good heuristic solution for the MWMP.Comment: 20 pages, 14 figures, Latex, to appear in Journal of Experimental Algorithms, 200

Quantum computing with nearest neighbor interactions and error rates over 1%

Large-scale quantum computation will only be achieved if experimentally implementable quantum error correction procedures are devised that can tolerate experimentally achievable error rates. We describe a quantum error correction procedure that requires only a 2-D square lattice of qubits that can interact with their nearest neighbors, yet can tolerate quantum gate error rates over 1%. The precise maximum tolerable error rate depends on the error model, and we calculate values in the range 1.1--1.4% for various physically reasonable models. Even the lowest value represents the highest threshold error rate calculated to date in a geometrically constrained setting, and a 50% improvement over the previous record.Comment: 4 pages, 8 figure

Deuteron Dipole Polarizabilities and Sum Rules

The scalar, vector, and tensor components of the (generalized) deuteron electric polarizability are calculated, as well as their logarithmic modifications. Several of these quantities arise in the treatment of the nuclear corrections to the deuterium Lamb shift and the deuterium hyperfine structure. A variety of second-generation potential models are used and a (subjective) error is assigned to the calculations. The zero-range approximation is used to analyze a subset of the results, and a simple relativistic version of this approximation is developed.Comment: 14 pages, LaTex - submitted to Physical Review

Contemplations on Dirac's equation in quaternionic coordinates

A formulation of Dirac's equation using complex-quaternionic coordinates appears to yield an enormous gain in formal elegance, as there is no longer any need to invoke Dirac matrices. This formulation, however, entails several peculiarities, which we investigate and attempt to interpret

Three Additional Quiescent Low-Mass X-ray Binary Candidates in 47 Tucanae

We identify through their X-ray spectra one certain (W37) and two probable (W17 and X4) quiescent low-mass X-ray binaries (qLMXBs) containing neutron stars in a long Chandra X-ray exposure of the globular cluster 47 Tucanae, in addition to the two previously known qLMXBs. W37's spectrum is dominated by a blackbody-like component consistent with radiation from the hydrogen atmosphere of a 10 km neutron star. W37's lightcurve shows strong X-ray variability which we attribute to variations in its absorbing column depth, and eclipses with a probable 3.087 hour period. For most of our exposures, W37's blackbody-like emission (assumed to be from the neutron star surface) is almost completely obscured, yet some soft X-rays (of uncertain origin) remain. Two additional candidates, W17 and X4, present X-ray spectra dominated by a harder component, fit by a power-law of photon index ~1.6-3. An additional soft component is required for both W17 and X4, which can be fit with a 10 km hydrogen-atmosphere neutron star model. X4 shows significant variability, which may arise from either its power-law or hydrogen-atmosphere spectral component. Both W17 and X4 show rather low X-ray luminosities, Lx(0.5-10 keV)~5*10^{31} ergs/s. All three candidate qLMXBs would be difficult to identify in other globular clusters, suggesting an additional reservoir of fainter qLMXBs in globular clusters that may be of similar numbers as the group of previously identified objects. The number of millisecond pulsars inferred to exist in 47 Tuc is less than 10 times larger than the number of qLMXBs in 47 Tuc, indicating that for typical inferred lifetimes of 10 and 1 Gyr respectively, their birthrates are comparable.Comment: Accepted for publication in ApJ. 13 pages, 7 figures (2 color