**“Mathematics Education and Practice” by Bibhya Sharma**

**1. Teaching Mathematics in an Online Environment**

**Author:** Rajneel Totaram

**Co-authors:** Anjeela Jokhan, Bibhya Sharma and Avinesh Prasad

Recently, an ever swelling number of courses are being taught in an online environment. An online environment is particularly important for distance education. While there are several new technologies available nowadays that foster adequate online teaching and learning, mathematics in an online environment still has its challenges. Mathematics – the language – in itself is its biggest challenge. Unlike most other subjects which can basically be represented using the standard letters and numbers, mathematics has its own unique set of characters, symbols and notations. Hence, authoring mathematical documents requires far more effort. This paper discusses how University of the South Pacific has developed a first year online mathematics course for students from its twelve member countries in the South Pacific region. Activities such as class glossaries, forums and lessons, from the popular learning management system, Moodle, have been successfully used for effective learning and teaching of mathematics. These activities were used to promote student interaction and participation in the course. Pass rates and data from questionnaires and surveys quantify the effectiveness of this mode of delivering mathematics.

**2. Waters Muscle Models for Facial Expression Generation: Some improvements**

**Author: **Dinesh Kumar

**Co-author: **Jito Vanualailai

The animation of facial expressions has been active field of research in computer graphics since the 1970's and there is a wide range of application areas such as character animation in entertainment industry and advertising, low-bandwidth teleconferencing, medicine, face and facial expression recognition to name a few. Different techniques have been developed to control the face deformation to achieve realistic facial expressions of human emotions. Most of these techniques use mathematical equations to deform facial geometry. One such technique is the use of Waters 2D muscle models to deform the facial geometry to achieve the desired results. Waters modeled two types of muscles: vector muscles that forms majority of facial muscles sphincter muscles to model muscles around the mouth. These pseudo-muscles are governed fundamentally by mathematics and mathematical equations. In this paper we study Waters 2D muscle model and suggest some improvements particularly on Water’s Sphincter muscle model and the jaw rotation function.

**3. Enhancing DragMath : An Innovative Tool to Improve Teaching Mathematics Online**

**Author:** Mitesh Sutaria

**Co-authors:** Anjeela Jokhan, Bibhya Sharma, Rajneel Totaram and Avinesh Prasad

There has been a sharp increase in using technologies for teaching mathematical courses online. At The University of the South Pacific (USP), Moodle a Learning Management System (LMS) is used to deliver courses fully/partially online. One of the challenges of using Moodle is the difficulty in expressing the mathematical equations and symbols. An add-on DragMath is a simple equation editor which is integrated with Moodle’s html editor to insert mathematical equations and symbols to the discussion forums. The drawback with this add-on is that the equations and symbols were neither editable nor transferable, hence the users have to retype and re-enter equations and symbols. This paper investigates how we can enhance the DragMath tool to allow mathematical equations to be edited using the DragMath interface and present a user friendly environment for expressing mathematical equations. A further application of this new enhanced feature would be when amalgamated to the Moodle Chat. Users will be able to create a real-time problem solving methodology as well as solve mathematical problems on a step by step basis. The tool has been employed in a fully online first year mathematics course in USP and the activities generated in the Moodle Chat verify its success.

**4. Motion Planning Problem of a Point-Mass Robot using Artificial Neural Network**

**Author:** Avinesh Prasad

**Co-authors:** Bibhya Sharma and Sunil Lal

This paper deals with the motion planning and intelligent control of a mobile point mass robot which moves safely to a designated target in a priori known environment. This environment may involve any number of obstacles of arbitrary shapes and sizes. We describe our approach to solving the motion-planning problem in mobile robot control using an artificial neural networks-based technique. Our method of the construction of a collision-free path for moving robot among obstacles is based on a multi-layer perceptron. A tailored and unique neural network is used to determine the free space and a safe direction for the robot to move in the workspace while avoiding the fixed obstacles. Simulation examples of generated path with the proposed technique will be presented.

**5. Motion Planning and Control of n-link Doubly Non-holonomic Manipulator **

**Author:** Shonal Singh

**Co-authors:** Bibhya Sharma and Jito Vanualailai

The paper essays the design of a new motion planner that will simultaneously manage the multi-task of control and motion planning of n-link doubly non-holonomic manipulators within a constrained environment. This decentralized planner, derived from the Lyapunov-based control scheme provides a solution to the findpath problem by generating collision-free trajectories from an initial to a final configuration in a constrained environment cluttered with fixed obstacles of different shapes and sizes. The results can be viewed as a significant contribution to the intelligent vehicle systems discipline. We demonstrate the efficiency of the new control scheme and its nonlinear algorithm with results through simulations of a couple of interesting situations.

**6. Autonomous Robots in Dynamic Environments **

**Author:** Jai Raj

** Co-authors:** Bibhya Sharma, Shonal Singh and Jito Vanualailai

A new motion planner is designed which provide a feasible solution to the findpath problem and manages simultaneously collision and obstacle avoidances of the multi-vehicle system within a dynamic environment. In this paper the dynamic obstacles include a swarm of point masses, car-like robots and Blindman (Sharma et al (2009). Blindman’s problem involves a one-way avoidance scheme wherein the avoidance is not expected from a blind person crossing the path of an oncoming vehicle. New continuous time-invariant acceleration control laws, formulated via the Lyapunov-based control scheme, guarantee the stability of a system of differential equations governing motion planning and control of multiple nonholonomic car-like robots. The effectiveness of the control scheme and the proposed nonlinear controllers, which take into account kinodynamic constraints, are demonstrated via computer simulations.

**7. Formation Control in Constrained Environments**

**Author:** Bibhya Sharma

**Co-authors:** Jito Vanualailai and Shonal Singh

Motion planning and control of autonomous robots have been widely researched in the last two decades. Injecting constraints which govern formations of multi-agents and translating these formations within constrained environments provide newer challenges to researchers. This paper will design new decentralized control laws that ensure collision free trajectories of a team of nonholonomic car-like robots fixed in a prescribed formation in a constrained environment such as a tunnel. Derived from the Lyapunov-based control scheme, the controllers work within an overarching leader-follower framework to generate either split/rejoin or expansion/contraction of the formation, as feasible solutions of the tunnel passing problem. In either scenario, the prescribed formation is re-established after the tunnel has been passed. Avoidance of the tunnel walls, either straight or curved, will be accomplished via the minimum distance technique. The new controllers can be viewed as a significant contribution to the intelligent vehicle systems discipline.

**8. Innovative Online Assessment Methods for Mathematics**

**Author:** Deepak Bhartu

**Co-authors:** Anjeela Jokhan, Bibhya Sharma, Rajneel Totaram and Avinesh Prasad

E-Learning technology is having a major impact on the teaching-learning systems. E-Learning which refers to learning (online) via systems built on computer and electronic technologies has offered the potential for students to do self-study and self-testing which can assist them in their learning process. Mathematics has particularly been restricted to paper based learning , however, E-Learning in mathematics can help students learn concepts better, reduce attrition, find the subject enjoyable and real, and maintain interest throughout. In addition to the current state of knowledge about the learning of mathematics, the main purpose of the research is to discuss ways of improving the process of learning through a modified assessment portfolio. Thus, this paper focuses on the use of an online computer aided assessment system known as STACK to develop assessments in mathematics and provide feedback to the students at the same-time.

**9. An Optimum Multivariate Stratified Sampling Design in Presence of Non-response Using Dynamic Programming**

**Author:** M. G. M. Khan

**Co-authors:** Shazia Ghufran, Saman Khowaja and M.J. Ahsan

Hansen and Hurwitz (1946) suggested a technique for eliciting responses from a sub-sample of non-respondents. Khare (1987) applied this procedure in stratified sampling and addresses the problem of optimum allocation in the presences on non-response. In multivariate stratified sample survey, where more than one characteristic are defined on each unit of the population, the individual optimum allocations cannot be applied for one reason or the other. In such situations the surveyor needs an allocation that is optimal, in some sense, for all characteristics. Such an allocation is based on a compromise criterion and hence is called a compromise allocation. Many compromise allocations are available in the sampling literature. Khan et al (2008) formulated a problem of determining an optimum compromise allocation in the presence of non-response as a Nonlinear Programming Problem (NLPP). Ignoring the non-negativity restrictions, they worked out a solution using the “Lagrange Multipliers Technique”. In the present paper the NLPP of Khan et al (2008) is reformulated with the added constraints that take care of the problems of over-sampling and the problems of estimating the stratum variances. Due to the added constraints including the non-negativity restrictions, which are more practical, the Lagrange Multipliers Techniques cannot always provide the solution to the problem under consideration. In the present paper we developed a procedure to solve this problem using a “Dynamic Programming Technique”. A numerical example is also worked out to illustrate the computational details of the problem.

**11. Enhancing eLearning: Progressive Feedback via ANNs**

**Author:** Shaveen Singh

**Co-authors:** Sunil Lal, Anjeela Jokhan, Bibhya Sharma and Avinesh Prasad

Universities all over the world are increasingly motivated towards using learning management systems (LMS) to support and improve learning within their institutions. Different technologies are integrated within modern LMS to support the divergent learning styles of auditory learners, visual learners, and kinesthetic learners. However, what truly makes eLearning pedagogy different from traditional teaching methodology is that it is self-paced and suited to different types of learners. In courses which lack regular formal assessments, Machine Learning (ML) algorithms can be used to intelligently examine the user interaction in the LMS to diagnose each student's academic progression. In this study, we model the behavior of students of a first year mathematics course, using data obtained from the logs of Moodle, a widely used LMS, using an artificial neural network (ANN). The data is combined with information on the final evaluation of the students to obtain a prediction map that permits rating a students' ability to surpass a course. Such swift feedback can be immensely helpful to identify under-performing students during the semester and take corrective actions before it becomes too late.

**12. Effectiveness of Group-based Learning in Pacific Tertiary Students – Case ****Study of a USP Course**

**Author:** Shymal Chandra

Facilitating learning at the tertiary level can be a varied and challenging task. While it may encompass more of independent learning, many different learning styles exist that can work well in one environment or more. This paper attempts to experiment how group-based learning can be effective among tertiary students in the South Pacific using a mathematics course at University of the South Pacific as a case study. It will also aim to determine the reasons behind it and present findings as ideas or alternate means to teaching in this multi-cultural community.

**13. A Flocking Algorithm via the Lyapunov method**

**Author:** Salsabil Nusair

In nature and artificial science, aggregate motion “Swarm Intelligence” is composed of many individuals that coordinate using decentralization controls and self organizations. Such behavior emerges from a flock of birds, ants foraging, herd of animals, and schools of fish.

In our presentation, we consider a Lagrange swarm model “Individual- Based” where the motion of each individual is determined by attraction on long range distance and repulsion on short range distance moving with the velocity of the swarm’s centroid. Based on this heuristic rules we construct a simple Lyapunov function governed by ODEs, for a 3D Flocking Algorithm which exhibits self organized behavior. To measure the strength of the cohesion of the swarm we utilized three parameters, Cohesion parameter: measures the strength of the swarm cohesion, Coupling parameter: interaction between any two individuals, and Convergence parameter: instantaneous velocity of an individual with respect to the swarm centroid. We vary these parameters using computer simulations to show self- organized 3D motions.

*** The list will be updated periodically.**