Numerical techniques for the variable order time fractional diffusion equation

In this paper we consider the variable order time fractional diffusion equation. We adopt the Coimbra variable order (VO) time fractional operator, which defines a consistent method for VO differentiation of physical variables. The Coimbra variable order fractional operator also can be viewed as a Caputo-type definition. Although this definition is the most appropriate definition having fundamental characteristics that are desirable for physical modeling, numerical methods for fractional partial differential equations using this definition have not yet appeared in the literature. Here an approximate scheme is first proposed. The stability, convergence and solvability of this numerical scheme are discussed via the technique of Fourier analysis. Numerical examples are provided to show that the numerical method is computationally efficient. Crown Copyright © 2012 Published by Elsevier Inc. All rights reserved.

Numerical methods for solving a two-dimensional variable-order anomalous subdiffusion equation

Anomalous subdiffusion equations have in recent years received much attention. In this paper, we consider a two-dimensional variable-order anomalous subdiffusion equation. Two numerical methods (the implicit and explicit methods) are developed to solve the equation. Their stability, convergence and solvability are investigated by the Fourier method. Moreover, the effectiveness of our theoretical analysis is demonstrated by some numerical examples. © 2011 American Mathematical Society.

Eliciting metacognitive experiences and reflection in a year 11 chemistry classroom : an activity theory perspective

Concerns regarding students' learning and reasoning in chemistry classrooms are well documented. Students' reasoning in chemistry should be characterized by conscious consideration of chemical phenomenon from laboratory work at macroscopic, molecular/sub-micro and symbolic levels. Further, students should develop metacognition in relation to such ways of reasoning about chemistry phenomena. Classroom change eliciting metacognitive experiences and metacognitive reflection is necessary to shift entrenched views of teaching and learning in students. In this study, Activity Theory is used as the framework for intepreting changes to the rules/customs and tools of the activity systems of two different classes of students taught by the same teacher, Frances, who was teaching chemical equilibrium to those classes in consecutive years. An interpretive methodolgy involving multiple data sources was employed. Frances explicitly changed her pedagogy in the second year to direct students attention to increasingly consider chemical phenomena at the molecular/sub-micro level. Additonally, she asked students not to use the textbook until toward the end of the equilibrium unit and sought to engage them in using their prior knowledge of chemistry to understand their observations from experiments. Frances' changed pedagogy elicited metacognitive experiences and reflection in students and challenged them to reconsider their metacognitive beliefs about learning chemistry and how it might be achieved. While teacher change is essential for science education reform, students are not passive players in the change efforts and they need to be convinced of the viability of teacher pedagogical change in the context of their goals, intentions, and beliefs.

Unsteady natural convection within a differentially heated enclosure of sinusoidal corrugated side walls

Numerically investigation of natural convection within a differentially heated modified square enclosure with sinusoidally corrugated side walls has been performed for different values of Rayleigh number. The fluid inside the enclosure considered is air and is quiescent, initially. The top and bottom surfaces are flat and considered as adiabatic. Results reveal three main stages: an initial stage, a transitory or oscillatory stage and a steady stage for the development of natural convection flow inside the corrugated cavity. The numerical scheme is based on the finite element method adapted to triangular non-uniform mesh element by a non-linear parametric solution algorithm. Investigation has been performed for the Rayleigh number, Ra ranging from 105 to 108 with variation of corrugation amplitude and frequency. Constant physical properties for the fluid medium have been assumed. Results have been presented in terms of the isotherms, streamlines, temperature plots, average Nusselt numbers, traveling waves and thermal boundary layer thickness plots, temperature and velocity profiles. The effects of sudden differential heating and its consequent transient behavior on fluid flow and heat transfer characteristics have been observed for the range of governing parameters. The present results show that the transient phenomena are greatly influenced by the variation of the Rayleigh Number with corrugation amplitude and frequency.

Amphiphilic silicone archtectures via anaerobic thiol - ene chemistry

Despite broad application, few silicone-based surfactants of known structure or, therefore, surfactancy have been prepared because of an absence of selective routes and instability of silicones to acid and base. Herein the synthesis of a library of explicit silicone-poly(ethylene glycol) (PEG) materials is reported. Pure silicone fragments were generated by the B(C(6)F(5))(3)-catalyzed condensation of alkoxysilanes and vinyl-functionalized hydrosilanes. The resulting pure products were coupled to thiol-terminated PEG materials using photogenerated radicals under anaerobic conditions.

Solution methods for advection-diffusion-reaction equations on growing domains and subdomains, with application to modelling skin substitutes

Problems involving the solution of advection-diffusion-reaction equations on domains and subdomains whose growth affects and is affected by these equations, commonly arise in developmental biology. Here, a mathematical framework for these situations, together with methods for obtaining spatio-temporal solutions and steady states of models built from this framework, is presented. The framework and methods are applied to a recently published model of epidermal skin substitutes. Despite the use of Eulerian schemes, excellent agreement is obtained between the numerical spatio-temporal, numerical steady state, and analytical solutions of the model.

Greek or not : the use of symbols and abbreviations in mathematics

The use of symbols and abbreviations adds uniqueness and complexity to the mathematical language register. In this article, the reader’s attention is drawn to the multitude of symbols and abbreviations which are used in mathematics. The conventions which underpin the use of the symbols and abbreviations and the linguistic difficulties which learners of mathematics may encounter due to the inclusion of the symbolic language are discussed. 2010 NAPLAN numeracy tests are used to illustrate examples of the complexities of the symbolic language of mathematics.

A framework for mathematics graphical tasks : The influence of the graphic element on student sense making

Graphical tasks have become a prominent aspect of mathematics assessment. From a conceptual stance, the purpose of this study was to better understand the composition of graphical tasks commonly used to assess students’ mathematics understandings. Through an iterative design, the investigation described the sense making of 11–12-year-olds as they decoded mathematics tasks which contained a graphic. An ongoing analysis of two phases of data collection was undertaken as we analysed the extent to which various elements of text, graphics, and symbols influenced student sense making. Specifically, the study outlined the changed behaviour (and performance) of the participants as they solved graphical tasks that had been modified with respect to these elements. We propose a theoretical framework for understanding the composition of a graphical task and identify three specific elements which are dependently and independently related to each other, namely: the graphic; the text; and the symbols. Results indicated that although changes to the graphical tasks were minimal, a change in student success and understanding was most evident when the graphic element was modified. Implications include the need for test designers to carefully consider the graphics embedded within mathematics tasks since the elements within graphical tasks greatly influence student understanding.

Collective dynamics and control of a 3-D small-world network with time delays

The paper presents a detailed analysis on the collective dynamics and delayed state feedback control of a three-dimensional delayed small-world network. The trivial equilibrium of the model is first investigated, showing that the uncontrolled model exhibits complicated unbounded behavior. Then three control strategies, namely a position feedback control, a velocity feedback control, and a hybrid control combined velocity with acceleration feedback, are then introduced to stabilize this unstable system. It is shown in these three control schemes that only the hybrid control can easily stabilize the 3-D network system. And with properly chosen delay and gain in the delayed feedback path, the hybrid controlled model may have stable equilibrium, or periodic solutions resulting from the Hopf bifurcation, or complex stranger attractor from the period-doubling bifurcation. Moreover, the direction of Hopf bifurcation and stability of the bifurcation periodic solutions are analyzed. The results are further extended to any "d" dimensional network. It shows that to stabilize a "d" dimensional delayed small-world network, at least a "d – 1" order completed differential feedback is needed. This work provides a constructive suggestion for the high dimensional delayed systems.

Teacher aides and their pedagogical contributions in the underachieving Indigenous mathematics classroom

This study examines the pedagogical contributions made by teacher aides in underperforming Indigenous mathematics secondary classrooms. Three teaching teams, each consisting of a teacher and their teacher aide, responded to semi-structured interviews. Their mathematics classrooms were observed for details of pedagogical contributions to the mathematics lessons. It was found that the pedagogical contributions of the teacher aides varied from co-teaching contributions, to the provision of menial support and behaviour management. The techniques used by the teacher aides to provide student feedback, to support behaviour management and to undertake questioning vary greatly, and this variance is also evident in the classroom atmosphere. Teacher aides are providing pedagogical contributions, and are engaged in instructional interactions, and are in a sense “teaching”.

Nagelschmidtite bioceramics with osteostimulation properties : material chemistry activating osteogenic genes and WNT signalling pathway of human bone marrow stromal cells

Bioactive materials with osteostimulation properties are of great importance to promote osteogenic differentiation of human bone marrow stromal cells (hBMSCs) for potential bone regeneration. We have recently synthesized nagelschmidtite (NAGEL, Ca7Si2P2O16) ceramic powders which showed excellent apatite-mineralization ability. The aim of this study was to investigate the interaction of hBMSCs with NAGEL bioceramic bulks and their ionic extracts, and to explore the osteostimulation properties of NAGEL bioceramics and the possible molecular mechanism. The cell attachment, proliferation, bone-related gene expression (ALP, OPN and OCN) and WNT signalling pathways (WNT3a, FZD6, AXIN2 and CTNNB) of hBMSCs cultured on NAGEL bioceramic disks were systematically studied. We further investigated the biological effects of ionic products from NAGEL powders on cell proliferation and osteogenic differentiation of hBMSCs by culturing cells with NAGEL extracts. Furthermore, the effect of NAGEL bioceramics on the osteogenic differentiation in hBMSCs was also investigated with the addition of cardamonin, a WNT inhibitor. The results showed that NAGEL bioceramic disks supported the attachment and proliferation of hBMSCs, and significantly enhanced the bone-related gene expression and WNT signalling pathway of hBMSCs, compared to conventional beta-tricalcium phosphate (β-TCP) bioceramic disks and blank controls. The ionic products from NAGEL powders also significantly promoted the proliferation, bone and WNT-related gene expression of hBMSCs. It was also identified that NAGEL bioceramics could bypass the action of the WNT inhibitor (10 μM) to stimulate the selected osteogenic genes in hBMSCs. Our results suggest that NAGEL bioceramics possess excellent in vitro osteostimulation properties. The possible mechanism for the osteostimulation may be directly related to the released Si, Ca and P-containing ionic products from NAGEL bioceramics which activate bone-related gene expression and WNT signalling pathway of hBMSCs. The present study suggests that NAGEL bioceramics are a potential bone regeneration material with significant osteostimulation capacity.

The Chemistry Discipline Network: One Year On

The Chemistry Discipline Network was funded in mid-2011, with the aim of improving communication between chemistry academics in Australia. In our first year of operation, we have grown to over 100 members, established a web presence, and produced substantial mapping reports on chemistry teaching in Australia. We are now working on the definition of standards for a chemistry degree based on the Threshold Learning Outcomes published by the Learning and Teaching Academic Standards Project.