A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modelling of biological systems

Biochemical reactions underlying genetic regulation are often modelled as a continuous-time, discrete-state, Markov process, and the evolution of the associated probability density is described by the so-called chemical master equation (CME). However the CME is typically difficult to solve, since the state-space involved can be very large or even countably infinite. Recently a finite state projection method (FSP) that truncates the state-space was suggested and shown to be effective in an example of a model of the Pap-pili epigenetic switch. However in this example, both the model and the final time at which the solution was computed, were relatively small. Presented here is a Krylov FSP algorithm based on a combination of state-space truncation and inexact matrix-vector product routines. This allows larger-scale models to be studied and solutions for larger final times to be computed in a realistic execution time. Additionally the new method computes the solution at intermediate times at virtually no extra cost, since it is derived from Krylov-type methods for computing matrix exponentials. For the purpose of comparison the new algorithm is applied to the model of the Pap-pili epigenetic switch, where the original FSP was first demonstrated. Also the method is applied to a more sophisticated model of regulated transcription. Numerical results indicate that the new approach is significantly faster and extendable to larger biological models.

Year 7 students’ approaches to understanding and solving NAPLAN numeracy problems

This study investigated how the interpretation of mathematical problems by Year 7 students impacted on their ability to demonstrate what they can do in NAPLAN numeracy testing. In the study, mathematics is viewed as a culturally and socially determined system of signs and signifiers that establish the meaning, origins and importance of mathematics. The study hypothesises that students are unable to succeed in NAPLAN numeracy tests because they cannot interpret the questions, even though they may be able to perform the necessary calculations. To investigate this, the study applied contemporary theories of literacy to the context of mathematical problem solving. A case study design with multiple methods was used. The study used a correlation design to explore the connections between NAPLAN literacy and numeracy outcomes of 198 Year 7 students in a Queensland school. Additionally, qualitative methods provided a rich description of the effect of the various forms of NAPLAN numeracy questions on the success of ten Year 7 students in the same school. The study argues that there is a quantitative link between reading and numeracy. It illustrates that interpretation (literacy) errors are the most common error type in the selected NAPLAN questions, made by students of all abilities. In contrast, conceptual (mathematical) errors are less frequent amongst more capable students. This has important implications in preparing students for NAPLAN numeracy tests. The study concluded by recommending that increased focus on the literacies of mathematics would be effective in improving NAPLAN results.

Beyond MOF constraints : multiple constraint set metamodelling for lifecycle management

The management of models over time in many domains requires different constraints to apply to some parts of the model as it evolves. Using EMF and its meta-language Ecore, the development of model management code and tools usually relies on the meta- model having some constraints, such as attribute and reference cardinalities and changeability, set in the least constrained way that any model user will require. Stronger versions of these constraints can then be enforced in code, or by attaching additional constraint expressions, and their evaluations engines, to the generated model code. We propose a mechanism that allows for variations to the constraining meta-attributes of metamodels, to allow enforcement of different constraints at different lifecycle stages of a model. We then discuss the implementation choices within EMF to support the validation of a state-specific metamodel on model graphs when changing states, as well as the enforcement of state-specific constraints when executing model change operations.

LEGO robotics : an authentic problem solving tool?

With the current curriculum focus on correlating classroom problem solving lessons to real-world contexts, are LEGO robotics an effective problem solving tool? This present study was designed to investigate this question and to ascertain what problem solving strategies primary students engaged with when working with LEGO robotics and whether the students were able to effectively relate their problem solving strategies to real-world contexts. The qualitative study involved 23 Grade 6 students participating in robotics activities at a Brisbane primary school. The study included data collected from researcher observations of student problem solving discussions, collected software programs, and data from a student completed questionnaire. Results from the study indicated that the robotic activities assisted students to reflect on the problem solving decisions they made. The study also highlighted that the students were able to relate their problem solving strategies to real-world contexts. The study demonstrated that while LEGO robotics can be considered useful problem solving tools in the classroom, careful teacher scaffolding needs to be implemented in regards to correlating LEGO with authentic problem solving. Further research in regards to how teachers can best embed realworld contexts into effective robotics lessons is recommended.

Proximity-to-goal as a constraint on patterns of behaviour in attacker–defender dyads in team games

The aim of this study was to determine whether spatiotemporal interactions between footballers and the ball in 1 vs. 1 sub-phases are influenced by their proximity to the goal area. Twelve participants (age 15.3 ± 0.5 years) performed as attackers and defenders in 1 vs. 1 dyads across three field positions: (a) attacking the goal, (b) in midfield, and (c) advancing away from the goal area. In each position, the dribbler was required to move beyond an immediate defender with the ball towards the opposition goal. Interactions of attacker-defender dyads were filmed with player and ball displacement trajectories digitized using manual tracking software. One-way repeated measures analysis of variance was used to examine differences in mean defender-to-ball distance after this value had stabilized. Maximum attacker-to-ball distance was also compared as a function of proximity-to-goal. Significant differences were observed for defender-to-ball distance between locations (a) and (c) at the moment when the defender-to-ball distance had stabilized (a: 1.69 ± 0.64 m; c: 1.15 ± 0.59 m; P < 0.05). Findings indicate that proximity-to-goal influenced the performance of players, particularly when attacking or advancing away from goal areas, providing implications for training design in football. In this study, the task constraints of football revealed subtly different player interactions than observed in previous studies of dyadic systems in basketball and rugby union.

Commentary on Spence et al. : "Problem solving programme implemented by teachers may prevent depression in the short term, but longer term benefits are unclear"

This is a methodologically exemplary trial of a population based (universal) approach to preventing depression in young people. The programme used teachers in a classroom setting to deliver cognitive behavioural problem solving skills to a cohort of students. We have little knowledge about “best practice” to prevent depression in adolescence. Classroom-based universal approaches appear to offer advantages in recruitment rates and lack of stigmatisation over approaches that target specific groups of at risk students. Earlier research on a universal school-based approach to preventing depression in adolescents showed promise, but employed mental health professionals to teach cognitive behavioural coping skills in small groups.1 Using such an approach routinely would be economically unsustainable. Spence’s trial, with teachers as facilitators, therefore represents a “real world” intervention that could be routinely disseminated.

Problem-solving courts, therapeutic jurisprudence and the constitution : if two is company, is three a crowd?

Court costs, resource-intensive trials, booming prison populations and the obduracy of recidivism rates all present as ugly excesses of the criminal law adversarial paradigm. To combat these excesses, problem-solving courts have evolved with an edict to address the underlying issues that have caused an individual to commit a crime. When a judge seeks to help a problem-solving court participant deal with issues like addiction, mental health or poverty, they are performing a very different role to that of a judicial officer in the traditional court hierarchy. They are no longer the removed, independent arbiter — a problem-solving court judge steps into the ‘arena’ with the participant and makes active use of their judicial authority to assist in rehabilitation and positive behavioural change. Problem-solving court judges employing the principles of therapeutic jurisprudence appreciate that their interaction with participants can have therapeutic and anti-therapeutic consequences. This article will consider how the deployment of therapeutic measures (albeit with good intention) can lead to the behavioural manifestation of partiality and bias on the part of problem-solving court judges. Chapter III of the Commonwealth Constitution will then be analysed to highlight why the operation and functioning of problem solving courts may be deemed unconstitutional. Part IV of this article will explain how a problem-solving court judge who is not acting impartially or independently will potentially contravene the requirements of the Constitution. It will finally be suggested that judges who possess a high level of emotional intelligence will be the most successful in administering an independent and impartial problem solving court.

Computationally efficient methods for solving time-variable-order time-space fractional reaction-diffusion equation

Fractional differential equations are becoming more widely accepted as a powerful tool in modelling anomalous diffusion, which is exhibited by various materials and processes. Recently, researchers have suggested that rather than using constant order fractional operators, some processes are more accurately modelled using fractional orders that vary with time and/or space. In this paper we develop computationally efficient techniques for solving time-variable-order time-space fractional reaction-diffusion equations (tsfrde) using the finite difference scheme. We adopt the Coimbra variable order time fractional operator and variable order fractional Laplacian operator in space where both orders are functions of time. Because the fractional operator is nonlocal, it is challenging to efficiently deal with its long range dependence when using classical numerical techniques to solve such equations. The novelty of our method is that the numerical solution of the time-variable-order tsfrde is written in terms of a matrix function vector product at each time step. This product is approximated efficiently by the Lanczos method, which is a powerful iterative technique for approximating the action of a matrix function by projecting onto a Krylov subspace. Furthermore an adaptive preconditioner is constructed that dramatically reduces the size of the required Krylov subspaces and hence the overall computational cost. Numerical examples, including the variable-order fractional Fisher equation, are presented to demonstrate the accuracy and efficiency of the approach.

Efficient preconditioning of the method of lines for solving nonlinear two-sided space-fractional diffusion equations

A standard method for the numerical solution of partial differential equations (PDEs) is the method of lines. In this approach the PDE is discretised in space using �finite di�fferences or similar techniques, and the resulting semidiscrete problem in time is integrated using an initial value problem solver. A significant challenge when applying the method of lines to fractional PDEs is that the non-local nature of the fractional derivatives results in a discretised system where each equation involves contributions from many (possibly every) spatial node(s). This has important consequences for the effi�ciency of the numerical solver. First, since the cost of evaluating the discrete equations is high, it is essential to minimise the number of evaluations required to advance the solution in time. Second, since the Jacobian matrix of the system is dense (partially or fully), methods that avoid the need to form and factorise this matrix are preferred. In this paper, we consider a nonlinear two-sided space-fractional di�ffusion equation in one spatial dimension. A key contribution of this paper is to demonstrate how an eff�ective preconditioner is crucial for improving the effi�ciency of the method of lines for solving this equation. In particular, we show how to construct suitable banded approximations to the system Jacobian for preconditioning purposes that permit high orders and large stepsizes to be used in the temporal integration, without requiring dense matrices to be formed. The results of numerical experiments are presented that demonstrate the effectiveness of this approach.

Numerical methods for solving the multi-term time-fractional wave-diffusion equations

In this paper, the multi-term time-fractional wave diffusion equations are considered. The multiterm time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals [0,1], [1,2), [0,2), [0,3), [2,3) and [2,4), respectively. Some computationally effective numerical methods are proposed for simulating the multi-term time-fractional wave-diffusion equations. The numerical results demonstrate the effectiveness of theoretical analysis. These methods and techniques can also be extended to other kinds of the multi-term fractional time-space models with fractional Laplacian.

A finite volume method for solving the two-sided time-space fractional advection-dispersion equation

The field of fractional differential equations provides a means for modelling transport processes within complex media which are governed by anomalous transport. Indeed, the application to anomalous transport has been a significant driving force behind the rapid growth and expansion of the literature in the field of fractional calculus. In this paper, we present a finite volume method to solve the time-space two-sided fractional advection dispersion equation on a one-dimensional domain. Such an equation allows modelling different flow regime impacts from either side. The finite volume formulation provides a natural way to handle fractional advection-dispersion equations written in conservative form. The novel spatial discretisation employs fractionally-shifted Gr¨unwald formulas to discretise the Riemann-Liouville fractional derivatives at control volume faces in terms of function values at the nodes, while the L1-algorithm is used to discretise the Caputo time fractional derivative. Results of numerical experiments are presented to demonstrate the effectiveness of the approach.

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.

Mixed-dimensional consistent coupling by multi-point constraint equations for efficient multi-scale modeling

Finding an appropriate linking method to connect different dimensional element types in a single finite element model is a key issue in the multi-scale modeling. This paper presents a mixed dimensional coupling method using multi-point constraint equations derived by equating the work done on either side of interface connecting beam elements and shell elements for constructing a finite element multiscale model. A typical steel truss frame structure is selected as case example and the reduced scale specimen of this truss section is then studied in the laboratory to measure its dynamic and static behavior in global truss and local welded details while the different analytical models are developed for numerical simulation. Comparison of dynamic and static response of the calculated results among different numerical models as well as the good agreement with those from experimental results indicates that the proposed multi-scale model is efficient and accurate.

Scheduling techniques to optimise rail operations

In Australia, railway systems play a vital role in transporting the sugarcane crop from farms to mills. The sugarcane transport system is very complex and uses daily schedules, consisting of a set of locomotives runs, to satisfy the requirements of the mill and harvesters. The total cost of sugarcane transport operations is very high; over 35% of the total cost of sugarcane production in Australia is incurred in cane transport. Efficient schedules for sugarcane transport can reduce the cost and limit the negative effects that this system can have on the raw sugar production system. There are several benefits to formulating the train scheduling problem as a blocking parallel-machine job shop scheduling (BPMJSS) problem, namely to prevent two trains passing in one section at the same time; to keep the train activities (operations) in sequence during each run (trip) by applying precedence constraints; to pass the trains on one section in the correct order (priorities of passing trains) by applying disjunctive constraints; and, to ease passing trains by solving rail conflicts by applying blocking constraints and Parallel Machine Scheduling. Therefore, the sugarcane rail operations are formulated as BPMJSS problem. A mixed integer programming and constraint programming approaches are used to describe the BPMJSS problem. The model is solved by the integration of constraint programming, mixed integer programming and search techniques. The optimality performance is tested by Optimization Programming Language (OPL) and CPLEX software on small and large size instances based on specific criteria. A real life problem is used to verify and validate the approach. Constructive heuristics and new metaheuristics including simulated annealing and tabu search are proposed to solve this complex and NP-hard scheduling problem and produce a more efficient scheduling system. Innovative hybrid and hyper metaheuristic techniques are developed and coded using C# language to improve the solutions quality and CPU time. Hybrid techniques depend on integrating heuristic and metaheuristic techniques consecutively, while hyper techniques are the complete integration between different metaheuristic techniques, heuristic techniques, or both.