Constructive development of the solutions of linear equations in introductory ordinary differential equations

The solution of linear ordinary differential equations (ODEs) is commonly taught in first year undergraduate mathematics classrooms, but the understanding of the concept of a solution is not always grasped by students until much later. Recognising what it is to be a solution of a linear ODE and how to postulate such solutions, without resorting to tables of solutions, is an important skill for students to carry with them to advanced studies in mathematics. In this study we describe a teaching and learning strategy that replaces the traditional algorithmic, transmission presentation style for solving ODEs with a constructive, discovery based approach where students employ their existing skills as a framework for constructing the solutions of first and second order linear ODEs. We elaborate on how the strategy was implemented and discuss the resulting impact on a first year undergraduate class. Finally we propose further improvements to the strategy as well as suggesting other topics which could be taught in a similar manner.

Communicating ecologically positive development

It is difficult to present a paradigm shift from resource efficient to ecologically sustainable design, when many students have not yet thought about what sustainability is, let alone what it implies for the design of the built environment ‘Positive Development’ requires students to think beyond green building to something that does not yet exist. The concept of ecologically positive development suggests a product, building, system or urban area that leaves the ecological base and public estate better off than if no development had occurred. For some years now, I have experimented with communicating this paradigm shift in design to students and professionals ‐ with mixed results. This paper discusses some of the challenges, failures and successes in shifting design studio work from environmentally‐sensitive to eco-positive. The framework underlying this exploration is action research. Conclusions about the success of the strategies used for overcoming perceptual barriers to new typologies of architecture are drawn from recent student feedback. The talk will show examples of student projects that attempt eco-positive development projects.

Harnessing assessment and feedback in the first year to support learning success, engagement and retention

It has been argued that intentional first year curriculum design has a critical role to play in enhancing first year student engagement, success and retention (Kift, 2008). A fundamental first year curriculum objective should be to assist students to make the successful transition to assessment in higher education. Scott (2006) has identified that ‘relevant, consistent and integrated assessment … [with] prompt and constructive feedback’ are particularly relevant to student retention generally; while Nicol (2007) suggests that ‘lack of clarity regarding expectations in the first year, low levels of teacher feedback and poor motivation’ are key issues in the first year. At the very minimum, if we expect first year students to become independent and self-managing learners, they need to be supported in their early development and acquisition of tertiary assessment literacies (Orrell, 2005). Critical to this attainment is the necessity to alleviate early anxieties around assessment information, instructions, guidance, and performance. This includes, for example:  inducting students thoroughly into the academic languages and assessment genres they will encounter as the vehicles for evidencing learning success; and  making expectations about the quality of this evidence clear. Most importantly, students should receive regular formative feedback of their work early in their program of study to aid their learning and to provide information to both students and teachers on progress and achievement. Leveraging research conducted under an ALTC Senior Fellowship that has sought to articulate a research-based 'transition pedagogy' (Kift & Nelson, 2005) – a guiding philosophy for intentional first year curriculum design and support that carefully scaffolds and mediates the first year learning experience for contemporary heterogeneous cohorts – this paper will discuss theoretical and practical strategies and examples that should be of assistance in implementing good assessment and feedback practices across a range of disciplines in the first year.

Mining Negative Relevance Feedback for Information Filtering

It is a big challenge to clearly identify the boundary between positive and negative streams. Several attempts have used negative feedback to solve this challenge; however, there are two issues for using negative relevance feedback to improve the effectiveness of information filtering. The first one is how to select constructive negative samples in order to reduce the space of negative documents. The second issue is how to decide noisy extracted features that should be updated based on the selected negative samples. This paper proposes a pattern mining based approach to select some offenders from the negative documents, where an offender can be used to reduce the side effects of noisy features. It also classifies extracted features (i.e., terms) into three categories: positive specific terms, general terms, and negative specific terms. In this way, multiple revising strategies can be used to update extracted features. An iterative learning algorithm is also proposed to implement this approach on RCV1, and substantial experiments show that the proposed approach achieves encouraging performance.

An effective model of using negative relevance feedback for information filtering

Over the years, people have often held the hypothesis that negative feedback should be very useful for largely improving the performance of information filtering systems; however, we have not obtained very effective models to support this hypothesis. This paper, proposes an effective model that use negative relevance feedback based on a pattern mining approach to improve extracted features. This study focuses on two main issues of using negative relevance feedback: the selection of constructive negative examples to reduce the space of negative examples; and the revision of existing features based on the selected negative examples. The former selects some offender documents, where offender documents are negative documents that are most likely to be classified in the positive group. The later groups the extracted features into three groups: the positive specific category, general category and negative specific category to easily update the weight. An iterative algorithm is also proposed to implement this approach on RCV1 data collections, and substantial experiments show that the proposed approach achieves encouraging performance.

State-of-the-Art Review on Opinion Mining from Online Customers’ Feedback

Dealing with the ever-growing information overload in the Internet, Recommender Systems are widely used online to suggest potential customers item they may like or find useful. Collaborative Filtering is the most popular techniques for Recommender Systems which collects opinions from customers in the form of ratings on items, services or service providers. In addition to the customer rating about a service provider, there is also a good number of online customer feedback information available over the Internet as customer reviews, comments, newsgroups post, discussion forums or blogs which is collectively called user generated contents. This information can be used to generate the public reputation of the service providers’. To do this, data mining techniques, specially recently emerged opinion mining could be a useful tool. In this paper we present a state of the art review of Opinion Mining from online customer feedback. We critically evaluate the existing work and expose cutting edge area of interest in opinion mining. We also classify the approaches taken by different researchers into several categories and sub-categories. Each of those steps is analyzed with their strength and limitations in this paper.

A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

Krylov subspace methods for approximating functions of symmetric positive definite matrices with applications to applied statistics and anomalous diffusion

Matrix function approximation is a current focus of worldwide interest and finds application in a variety of areas of applied mathematics and statistics. In this thesis we focus on the approximation of A^(-α/2)b, where A ∈ ℝ^(n×n) is a large, sparse symmetric positive definite matrix and b ∈ ℝ^n is a vector. In particular, we will focus on matrix function techniques for sampling from Gaussian Markov random fields in applied statistics and the solution of fractional-in-space partial differential equations. Gaussian Markov random fields (GMRFs) are multivariate normal random variables characterised by a sparse precision (inverse covariance) matrix. GMRFs are popular models in computational spatial statistics as the sparse structure can be exploited, typically through the use of the sparse Cholesky decomposition, to construct fast sampling methods. It is well known, however, that for sufficiently large problems, iterative methods for solving linear systems outperform direct methods. Fractional-in-space partial differential equations arise in models of processes undergoing anomalous diffusion. Unfortunately, as the fractional Laplacian is a non-local operator, numerical methods based on the direct discretisation of these equations typically requires the solution of dense linear systems, which is impractical for fine discretisations. In this thesis, novel applications of Krylov subspace approximations to matrix functions for both of these problems are investigated. Matrix functions arise when sampling from a GMRF by noting that the Cholesky decomposition A = LL^T is, essentially, a `square root' of the precision matrix A. Therefore, we can replace the usual sampling method, which forms x = L^(-T)z, with x = A^(-1/2)z, where z is a vector of independent and identically distributed standard normal random variables. Similarly, the matrix transfer technique can be used to build solutions to the fractional Poisson equation of the form ϕn = A^(-α/2)b, where A is the finite difference approximation to the Laplacian. Hence both applications require the approximation of f(A)b, where f(t) = t^(-α/2) and A is sparse. In this thesis we will compare the Lanczos approximation, the shift-and-invert Lanczos approximation, the extended Krylov subspace method, rational approximations and the restarted Lanczos approximation for approximating matrix functions of this form. A number of new and novel results are presented in this thesis. Firstly, we prove the convergence of the matrix transfer technique for the solution of the fractional Poisson equation and we give conditions by which the finite difference discretisation can be replaced by other methods for discretising the Laplacian. We then investigate a number of methods for approximating matrix functions of the form A^(-α/2)b and investigate stopping criteria for these methods. In particular, we derive a new method for restarting the Lanczos approximation to f(A)b. We then apply these techniques to the problem of sampling from a GMRF and construct a full suite of methods for sampling conditioned on linear constraints and approximating the likelihood. Finally, we consider the problem of sampling from a generalised Matern random field, which combines our techniques for solving fractional-in-space partial differential equations with our method for sampling from GMRFs.

Stability and convergence of an implicit numerical method for the non-linear fractional reaction-subdiffusion process

In this paper, we consider the following non-linear fractional reaction–subdiffusion process (NFR-SubDP): Formula where f(u, x, t) is a linear function of u, the function g(u, x, t) satisfies the Lipschitz condition and 0Dt1–{gamma} is the Riemann–Liouville time fractional partial derivative of order 1 – {gamma}. We propose a new computationally efficient numerical technique to simulate the process. Firstly, the NFR-SubDP is decoupled, which is equivalent to solving a non-linear fractional reaction–subdiffusion equation (NFR-SubDE). Secondly, we propose an implicit numerical method to approximate the NFR-SubDE. Thirdly, the stability and convergence of the method are discussed using a new energy method. Finally, some numerical examples are presented to show the application of the present technique. This method and supporting theoretical results can also be applied to fractional integrodifferential equations.

New Approach on Robust Delay-Dependent H∞ Control for Uncertain T-S Fuzzy Systems With Interval Time-Varying Delay

This paper investigates the robust H∞ control for Takagi-Sugeno (T-S) fuzzy systems with interval time-varying delay. By employing a new and tighter integral inequality and constructing an appropriate type of Lyapunov functional, delay-dependent stability criteria are derived for the control problem. Because neither any model transformation nor free weighting matrices are employed in our theoretical derivation, the developed stability criteria significantly improve and simplify the existing stability conditions. Also, the maximum allowable upper delay bound and controller feedback gains can be obtained simultaneously from the developed approach by solving a constrained convex optimization problem. Numerical examples are given to demonstrate the effectiveness of the proposed methods.

Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.