Investigating secondary students beliefs about mathematical problem-solving

Many studies over the past 30 years have highlighted the important role of students’ beliefs for successful problem-solving in mathematics. Given the recent emphasis afforded to problem-solving on the reformed Irish secondary school mathematics curriculum, the main aim of this study was to identify Irish students’ (n = 975) beliefs about the field. A quantitative measure of these beliefs was attained through the use of the Indiana Mathematical Belief Scale, an existing 30-item (five-scale) self-report questionnaire. A statistical analysis of the data revealed that students who were further through their secondary education had a stronger belief that not all problems could be solved by applying routine procedures. In contrast, the same students held less positive beliefs than their younger counterparts that they could solve time-consuming problems and that conceptual understanding was important. The analysis also indicated that gender had a significant impact on three of the five belief scales.     

Transition to escape in a system of coupled oscillators

We study a Hamiltonian system of coupled oscillators derived from two forced pendulums, connected with a torsional spring. The uncoupled limit is described by two identical oscillators, each possessing a homoclinic orbit separating bounded from unbounded motion. We focus on intermediate energy levels which lead to detained motions, defined as trajectories that, though unbounded as t → ∞, oscillate within the region defined by the homoclinic orbit of the unperturbed system for a long but finite time. We analyze the existence and behavior of these motions in terms of equipotential surfaces. These curves provide bounds on the motion of the system and are shown to be closed for low energies. However, above some critical energy level the equipotential curves become open. The detained trajectories are shown to arise from the region of phase space that was, for appropriate energies, stochastic. These motions remain within this region for long times before finally “leaking out” of the opening in the equipotential curves and proceeding to infinity.     

On the Dynamics of

We investigate the existence of unbounded solutions and the period-two convergence of solutions of the equation in the title with the parameter γ positive, the remaining parameters nonnegative, and with nonnegative initial conditions.     

Nuclear Uptake of Gold Nanoparticles Deduced Using Dual‐Angle X‐Ray Fluorescence Mapping

Studies into the cell nucleus' incorporation of gold nanoparticles (AuNPs) are often limited by ambiguities arising from conventional imaging techniques. Indeed, it is suggested that to date there is no unambiguous imaging evidence for such uptake in whole cells, particularly at the single nanoparticle level. This shortcoming in understanding exists despite the nucleus being the most important subcellular compartment in eukaryotes and gold being the most commonly used metal nanoparticle in medical applications. Here, dual‐angle X‐ray flouresence is used to show individually resolved nanoparticles within the cell nucleus, finding them to be well separated and 79% of the intranuclear population to be monodispersed. These findings have important implications for nanomedicine, illustrated here through a specific exemplar of the predicted enhancement of radiation effects arising from the observed AuNPs, finding intranuclear dose enhancements spanning nearly five orders of magnitude.