Reunitising hundredths: prototypic and nonprototypic representations

This paper reports on a study in which 29 Year 6 students (selected from the top 30% of 176 Year 6 students) were individually interviewed to explore their ability to reunitise hundredths as tenths (Behr, Harel, Post & Lesh, 1992) when represented by prototypic (PRO) and nonprototypic (NPRO) models. The results showed that 55.2% of the students were able to unitise both models and that reunitising was more successful with the PRO model. The interviews revealed that many of these students had incomplete, fragmented or non-existent structural knowledge of the reunitising process and often relied on syntactic clues to complete the tasks. The implication for teaching is that instruction should not be limited to PRO representations of the part/whole notion of fraction and that the basic structures (equal parts, link between name and number of equal parts) of the part/whole notion needs to be revisited often.

Fractions, reunitisation and the number-line representation

Centre for Mathematics and Science Education, QUT, Brisbane, Australia This paper reports on a study in which Years 6 and 10 students were individually interviewed to determine their ability to unitise and reunitise number lines used to represent mixed numbers and improper fractions. Only 16.7% of the students (all Year 6) were successful on all three tasks and, in general, Year 6 students outperformed Year 8 students. The interviews revealed that the remaining students had incomplete, fragmented or non-existent structural knowledge of mixed numbers and improper fractions, and were unable to unitise or reunitise number lines. The implication for teaching is that instruction should focus on providing students with a variety of fraction representations in order to develop rich and flexible schema for all fraction types (mixed numbers, and proper and improper fractions).

Lie group symmetry analysis for granular media stress equations

The Airy stress function, although frequently employed in classical linear elasticity, does not receive similar usage for granular media problems. For plane strain quasi-static deformations of a cohesionless Coulomb–Mohr granular solid, a single nonlinear partial differential equation is formulated for the Airy stress function by combining the equilibrium equations with the yield condition. This has certain advantages from the usual approach, in which two stress invariants and a stress angle are introduced, and a system of two partial differential equations is needed to describe the flow. In the present study, the symmetry analysis of differential equations is utilised for our single partial differential equation, and by computing an optimal system of one-dimensional Lie algebras, a complete set of group-invariant solutions is derived. By this it is meant that any group-invariant solution of the governing partial differential equation (provided it can be derived via the classical symmetries method) may be obtained as a member of this set by a suitable group transformation. For general values of the parameters (angle of internal friction and gravity g) it is found there are three distinct classes of solutions which correspond to granular flows considered previously in the literature. For the two limiting cases of high angle of internal friction and zero gravity, the governing partial differential equation admit larger families of Lie point symmetries, and from these symmetries, further solutions are derived, many of which are new. Furthermore, the majority of these solutions are exact, which is rare for granular flow, especially in the case of gravity driven flows.

Automated proof for formal indistinguishability and its applications

Bana et al. proposed the relation formal indistinguishability (FIR), i.e. an equivalence between two terms built from an abstract algebra. Later Ene et al. extended it to cover active adversaries and random oracles. This notion enables a framework to verify computational indistinguishability while still offering the simplicity and formality of symbolic methods. We are in the process of making an automated tool for checking FIR between two terms. First, we extend the work by Ene et al. further, by covering ordered sorts and simplifying the way to cope with random oracles. Second, we investigate the possibility of combining algebras together, since it makes the tool scalable and able to cover a wide class of cryptographic schemes. Specially, we show that the combined algebra is still computationally sound, as long as each algebra is sound. Third, we design some proving strategies and implement the tool. Basically, the strategies allow us to find a sequence of intermediate terms, which are formally indistinguishable, between two given terms. FIR between the two given terms is then guaranteed by the transitivity of FIR. Finally, we show applications of the work, e.g. on key exchanges and encryption schemes. In the future, the tool should be extended easily to cover many schemes. This work continues previous research of ours on use of compilers to aid in automated proofs for key exchange.

Automatic region-of-interest detection and prioritisation for visually optimised coding of low bit rate videos

The increasing popularity of video consumption from mobile devices requires an effective video coding strategy. To overcome diverse communication networks, video services often need to maintain sustainable quality when the available bandwidth is limited. One of the strategy for a visually-optimised video adaptation is by implementing a region-of-interest (ROI) based scalability, whereby important regions can be encoded at a higher quality while maintaining sufficient quality for the rest of the frame. The result is an improved perceived quality at the same bit rate as normal encoding, which is particularly obvious at the range of lower bit rate. However, because of the difficulties of predicting region-of-interest (ROI) accurately, there is a limited research and development of ROI-based video coding for general videos. In this paper, the phase spectrum quaternion of Fourier Transform (PQFT) method is adopted to determine the ROI. To improve the results of ROI detection, the saliency map from the PQFT is augmented with maps created from high level knowledge of factors that are known to attract human attention. Hence, maps that locate faces and emphasise the centre of the screen are used in combination with the saliency map to determine the ROI. The contribution of this paper lies on the automatic ROI detection technique for coding a low bit rate videos which include the ROI prioritisation technique to give different level of encoding qualities for multiple ROIs, and the evaluation of the proposed automatic ROI detection that is shown to have a close performance to human ROI, based on the eye fixation data.

The sphynx's new riddle : how to relate the canonical formula of myth to quantum interaction

We introduce Claude Lévi Strauss' canonical formula (CF), an attempt to rigorously formalise the general narrative structure of myth. This formula utilises the Klein group as its basis, but a recent work draws attention to its natural quaternion form, which opens up the possibility that it may require a quantum inspired interpretation. We present the CF in a form that can be understood by a non-anthropological audience, using the formalisation of a key myth (that of Adonis) to draw attention to its mathematical structure. The future potential formalisation of mythological structure within a quantum inspired framework is proposed and discussed, with a probabilistic interpretation further generalising the formula

Concurrent validity of accelerations measured using a tri-axial inertial measurement unit while walking on firm, compliant and uneven surfaces

Although accelerometers are extensively used for assessing gait, limited research has evaluated the concurrent validity of these devices on less predictable walking surfaces or the comparability of different methods used for gravitational acceleration compensation. This study evaluated the concurrent validity of trunk accelerations derived from a tri-axial inertial measurement unit while walking on firm, compliant and uneven surfaces and contrasted two methods used to remove gravitational accelerations: i) subtraction of the best linear fit from the data (detrending), and; ii) use of orientation information (quaternions) from the inertial measurement unit. Twelve older and twelve younger adults walked at their preferred speed along firm, compliant and uneven walkways. Accelerations were evaluated for the thoracic spine (T12) using a tri-axial inertial measurement unit and an eleven-camera Vicon system. The findings demonstrated excellent agreement between accelerations derived from the inertial measurement unit and motion analysis system, including while walking on uneven surfaces that better approximate a real-world setting (all differences <0.16 m.s−2). Detrending produced slightly better agreement between the inertial measurement unit and Vicon system on firm surfaces (delta range: −0.05 to 0.06 vs. 0.00 to 0.14 m.s−2), whereas the quaternion method performed better when walking on compliant and uneven walkways (delta range: −0.16 to −0.02 vs. −0.07 to 0.07 m.s−2). The technique used to compensate for gravitational accelerations requires consideration in future research, particularly when walking on compliant and uneven surfaces. These findings demonstrate trunk accelerations can be accurately measured using a wireless inertial measurement unit and are appropriate for research that evaluates healthy populations in complex environments.