Preservice teachers’ creation of dynamic geometry sketches to understand trigonometric relationships

Dynamic geometry software can help teachers highlight mathematical relationships in ways not possible with static diagrams. However, these opportunities are mediated by teachers' abilities to construct sketches that focus users' attention on the desired variant or invariant relationships. This paper looks at two cohorts of preservice secondary mathematics teachers and their attempts to build dynamic geometry sketches that highlighted the trigonometric relationship between the angle and slope of a line on the coordinate plane. We identify common challenges in the construction of these sketches and present examples for readers to interact with that highlight these issues. Lastly, we discuss ways that mathematics teacher educators can help beginning teachers understand common pitfalls in the building of dynamic geometry sketches, which can cause sketches not to operate as intended

Clifford-based spectral action and renormalization group analysis of the gauge couplings

The Spectral Action Principle in noncommutative geometry derives the actions of the Standard Model and General Relativity (along with several other gravitational terms) by reconciling them in a geometric setting, and hence offers an explanation for their common origin. However, one of the requirements in the minimal formalism, unification of the gauge coupling constants, is not satisfied, since the basic construction does not introduce anything new that can change the renormalization group (RG) running of the Standard Model. On the other hand, it has been recently argued that incorporating structure of the Clifford algebra into the finite part of the spectral triple, the main object that encodes the complete information of a noncommutative space, gives rise to five additional scalar fields in the basic framework. We investigate whether these scalars can help to achieve unification. We perform a RG analysis at the one-loop level, allowing possible mass values of these scalars to float from the electroweak scale to the putative unification scale. We show that out of twenty configurations of mass hierarchy in total, there does not exist even a single case that can lead to unification. In consequence, we confirm that the spectral action formalism requires a model-construction scheme beyond the (modified) minimal framework.Comment: 20 pages, 1 figure, 1 table of results; matches the published versio

The effects of spatially distributed ionisation sources on the temperature structure of HII region

Spatially resolved studies of star forming regions show that the assumption of spherical geometry is not realistic in most cases, with a major complication posed by the gas being ionised by multiple non-centrally located stars or star clusters. We try to isolate the effects of multiple non-centrally located stars on the temperature and ionisation structure of HII regions, via the construction of 3D photoionisation models using the 3D Monte Carlo photoionisation code MOCASSIN. We find that the true temperature fluctuations due to the stellar distribution (as opposed to the large-scale temperature gradients due to other gas properties) are small in all cases and not a significant cause of error in metallicity studies. Strong emission lines from HII regions are often used to study the metallicity of star-forming regions. We compare integrated emission line spectra from our models and quantify any systematic errors caused by the simplifying assumption of a single, central location for all ionising sources. We find that the dependence of the metallicity indicators on the ionisation parameter causes a clear bias, due to the fact that models with a fully distributed configuration of stars always display lower ionisation parameters than their fully concentrated counterparts. The errors found imply that the geometrical distribution of ionisation sources may partly account for the large scatter in metallicities derived using model-calibrated empirical methods.Comment: 13 pages, 6 figures, Accepted by MNRA

NASTRAN data generation of helicopter fuselages using interactive graphics

The development and implementation of a preprocessor system for the finite element analysis of helicopter fuselages is described. The system utilizes interactive graphics for the generation, display, and editing of NASTRAN data for fuselage models. It is operated from an IBM 2250 cathode ray tube (CRT) console driven by an IBM 370/145 computer. Real time interaction plus automatic data generation reduces the nominal 6 to 10 week time for manual generation and checking of data to a few days. The interactive graphics system consists of a series of satellite programs operated from a central NASTRAN Systems Monitor. Fuselage structural models including the outer shell and internal structure may be rapidly generated. All numbering systems are automatically assigned. Hard copy plots of the model labeled with GRID or elements ID's are also available. General purpose programs for displaying and editing NASTRAN data are included in the system. Utilization of the NASTRAN interactive graphics system has made possible the multiple finite element analysis of complex helicopter fuselage structures within design schedules