An investigation into the use of graphics calculators with pupils in Key Stage 2

This paper outlines an experiment in which pupils in Key Stage 2 were encouraged to use graphics calculators, in particular two simple programs, which helped them develop recall of their tables and allowed them to practise multiplication. The pupils responded very well to the calculators and seemed to have been motivated by them. The pupils did not find them difficult to operate and experienced very few technical problems. The authors concluded that the graphics calculator has considerable potential to enhance the mathematical experience and learning of pupils at this level, and that although the extent of the investigation was fairly limited the results were encouraging enough to justify further work in this area.     

Zwanzig's perturbation method and thermodynamic properties of noble gases at high density

Rayleigh-Schrödinger perturbation theory is applied to compute second and third-order correlation energies using complete active space self-consistent field (CASSCF) zeroth-order wavefunctions. The first-order wavefunction is expanded in a basis of internally contracted configurations. The zeroth-order Hamiltonian is a sum of one-electron effective Fock operators, which are invariant to unitary transformations among the active orbitals. Comparisons with FCI data are made for the singlet-triplet splitting of CH₂ and the barrier height and exothermicity of the F + H₂ reaction. Potential energy functions and spectroscopic data are computed for C₂, CN, CO, CF,N₂, NO, O₂, and F₂ using large basis sets. It is demonstrated that the third-order results (CASPT3) are significantly more accurate than the second-order (CASPT2) ones. The equilibrium distances and harmonic frequencies obtained with CASPT3 are found to be as accurate as MRCI and RCCSD(T) values, while dissociation energies are generally somewhat too small. Modifications of the zeroth-order Hamiltonian as recently proposed by Andersson are found to have a much smaller effect on the third-order energies than on the second-order ones.     

A "partitioned leaping" approach for multiscale modeling of chemical reaction dynamics

We present a novel multiscale simulation approach for modeling stochasticity in chemical reaction networks. The approach seamlessly integrates exact-stochastic and "leaping" methodologies into a single partitioned leaping algorithmic framework. The technique correctly accounts for stochastic noise at significantly reduced computational cost, requires the definition of only three model-independent parameters, and is particularly well suited for simulating systems containing widely disparate species populations. We present the theoretical foundations of partitioned leaping, discuss various options for its practical implementation, and demonstrate the utility of the method via illustrative examples.     

Spin waves and quantum criticality in the frustrated XY pyrochlore antiferromagnet Er2Ti2O7

We report detailed measurements of the low temperature magnetic phase diagram of Er2Ti2O7. Heat capacity and time-of-flight neutron scattering studies of single crystals reveal unconventional low-energy states. Er3+ magnetic ions reside on a pyrochlore lattice in Er2Ti2O7, where local XY anisotropy and antiferromagnetic interactions give rise to a unique frustrated system. In zero field, the ground state exhibits coexisting short and long-range order, accompanied by soft collective spin excitations previously believed to be absent. The application of finite magnetic fields tunes the ground state continuously through a landscape of noncollinear phases, divided by a zero temperature phase transition at micro{0}H{c} approximately 1.5 T. The characteristic energy scale for spin fluctuations is seen to vanish at the critical point, as expected for a second order quantum phase transition driven by quantum fluctuations.     

Investigation of DOTA–Metal Chelation Effects on the Chemical Shift of 129Xe

Recent work has shown that xenon chemical shifts in cryptophane‐cage sensors are affected when tethered chelators bind to metals. Here, we explore the xenon shifts in response to a wide range of metal ions binding to diastereomeric forms of 1,4,7,10‐tetraazacyclododecane‐1,4,7,10‐tetraacetic acid (DOTA) linked to cryptophane‐A. The shifts induced by the binding of Ca²⁺, Cu²⁺, Ce³⁺, Zn²⁺, Cd²⁺, Ni²⁺, Co²⁺, Cr²⁺, Fe³⁺, and Hg²⁺ are distinct. In addition, the different responses of the diastereomers for the same metal ion indicate that shifts are affected by partial folding with a correlation between the expected coordination number of the metal in the DOTA complex and the chemical shift of ¹²⁹Xe. These sensors may be used to detect and quantify many important metal ions, and a better understanding of the basis for the induced shifts could enhance future designs.     

Structural Fluctuations in the spin-liquid state of Tb2Ti2O7

High-resolution x-ray scattering measurements on single crystal Tb2Ti2O7 reveal finite structural correlations at low temperatures. This geometrically frustrated pyrochlore is known to exhibit a spin-liquid or cooperative paramagnetic state at temperatures below approximately 20 K. Parametric studies of structural Bragg peaks appropriate to the Fd3[over ]m space group of Tb2Ti2O7 reveal substantial broadening and peak intensity reduction in the temperature regime 20 K to 300 mK. We also observe a small, anomalous lattice expansion on cooling below a density maximum at approximately 18 K. These measurements are consistent with the development of fluctuations above a cooperative Jahn-Teller, cubic-tetragonal phase transition at very low temperatures.     

Benford’s law and Theil transform of financial data

Among econophysics investigations, studies of religious groups have been of interest. On the one hand, the present paper concerns the Antoinist community financial reports–a community which appeared at the end of the 19-th century in Belgium. Several growth–decay regimes have been previously found over different time spans. However, there is common suspicion about sect finances. In that spirit, the Antoinist community yearly financial reports, income and expenses, are hereby examined through the so-called Benford’s law. The latter is often used as a test about possible accounting wrongdoings. On the other hand, Benford’s law is known to be invariant under scale and base transformation. Therefore, as a further test, of both such data and the use of Benford’s law, the yearly financial reports are nonlinearly remapped through a sort of Theil transformation, i.e. based on a log-transformation. The resulting data is again analyzed along the Benford’s law scheme. Bizarre, puzzling, features are seen. However, it is emphasized that such a non-linear transformation can shift the argument toward a more objective conclusion. In an appendix, some brief discussion is made on why the original Theil mapping should not be used. In a second appendix, an imperfect Benford’s law-like form–better suited for anomalous distributions–is presented.     

Approximating the Quasi-stationary Distribution of the SIS Model for Endemic Infection

Probably the simplest model for endemic infection is the susceptible-infected-susceptible (SIS) logistic model. Long-term behaviour of this model prior to disease extinction is described by the quasi-stationary distribution. This quasi-stationary distribution has been the subject of much previous work, including derivation of a variety of approximations, using both standard distributional forms and specialized approximating formulae. The aim of this paper is to carry out a systematic comparison between approximations. As well as comparing previously available approximations, we derive several new variants. Taking into account both accuracy (measured using total variation distance) and simplicity, and denoting by R ₀ the basic reproduction number, our main findings are: (a) in the subcritical region R ₀ < 1 a geometric distribution approximation is preferred; (b) in the supercritical region R ₀ ≫ 1 a beta-binomial distribution is preferred. Both of these preferred approximations are new.