Components of learning and assessment in linear algebra

Linear algebra provides theory and technology that are the cornerstones of a range of cutting edge mathematical applications, from designing computer games to complex industrial problems, as well as more traditional applications in statistics and mathematical modelling. Once past introductions to matrices and vectors, the challenges of balancing theory, applications and computational work across mathematical and statistical topics and problems are considerable, particularly given the diversity of abilities and interests in typical cohorts. This paper considers two such cohorts in a second level linear algebra course in different years. The course objectives and materials were almost the same, but some changes were made in the assessment package. In addition to considering effects of these changes, the links with achievement in first year courses are analysed, together with achievement in a following computational mathematics course. Some results that may initially appear surprising provide insight into the components of student learning in linear algebra.

Electronic states of trans-polyacetylene, poly(p-phenylene vinylene) and sp-hybridised carbon species in amorphous hydrogenated carbon probed by resonant Raman scattering

Inclusions of sp-hybridised, trans-polyacetylene [trans-(CH)x] and poly(p-phenylene vinylene) (PPV) chains are revealed using resonant Raman scattering (RRS) investigation of amorphous hydrogenated carbon (a-C:H) films in the near IR – UV range. The RRS spectra of trans-(CH)x core Ag modes and the PPV CC-H phenylene mode are found to transform and disperse as the laser excitation energy ћωL is increased from near IR through visible to UV, whereas sp-bonded inclusions only become evident in UV. This is attributed to ћωL probing of trans-(CH)x chain inhomogeneity and the distribution of chains with varying conjugation length; for PPV to the resonant probing of phelynene ring disorder; and for sp segments, to ћωL probing of a local band gap of end-terminated polyynes. The IR spectra analysis confirmed the presence of sp, trans-(CH)x and PPV inclusions. The obtained RRS results for a-C:H denote differentiation between the core Ag trans-(CH)x modes and the PPV phenylene mode. Furthermore, it was found that at various laser excitation energies the changes in Raman spectra features for trans-(CH)x segments included in an amorphous carbon matrix are the same as in bulk trans-polyacetylene. The latter finding can be used to facilitate identification of trans-(CH)x in the spectra of complex carbonaceous materials.

Are combination forecasts of S&P 500 volatility statistically superior?

Forecasting volatility has received a great deal of research attention, with the relative performances of econometric model based and option implied volatility forecasts often being considered. While many studies find that implied volatility is the pre-ferred approach, a number of issues remain unresolved, including the relative merit of combining forecasts and whether the relative performances of various forecasts are statistically different. By utilising recent econometric advances, this paper considers whether combination forecasts of S&P 500 volatility are statistically superior to a wide range of model based forecasts and implied volatility. It is found that a combination of model based forecasts is the dominant approach, indicating that the implied volatility cannot simply be viewed as a combination of various model based forecasts. Therefore, while often viewed as a superior volatility forecast, the implied volatility is in fact an inferior forecast of S&P 500 volatility relative to model-based forecasts.

The jump component of S&P 500 volatility and the VIX index

Much research has investigated the differences between option implied volatilities and econometric model-based forecasts. Implied volatility is a market determined forecast, in contrast to model-based forecasts that employ some degree of smoothing of past volatility to generate forecasts. Implied volatility has the potential to reflect information that a model-based forecast could not. This paper considers two issues relating to the informational content of the S&P 500 VIX implied volatility index. First, whether it subsumes information on how historical jump activity contributed to the price volatility, followed by whether the VIX reflects any incremental information pertaining to future jump activity relative to model-based forecasts. It is found that the VIX index both subsumes information relating to past jump contributions to total volatility and reflects incremental information pertaining to future jump activity. This issue has not been examined previously and expands our understanding of how option markets form their volatility forecasts.