Cognitive trait model for persistent and fine-tuned student modelling in adaptive virtual learning environments : a thesis presented in partial fulfilment of the requirements for the degree of Master of Information Science in Information Systems at Massey University

The increasing need for individualised instructional in both academic and corporate training environment encourages the emergence and popularity of adaptivity in virtual learning environments (VLEs). Adaptivity can be applied in VLEs as adaptivity content presentation, which generates the learning content adaptively to suit the particular learner's aptitude, and as adaptive navigational control, which dynamically modifies the structure of the virtual learning environment presented to the learner in order to prevent overloading the learner's cognitive load. Techniques for both adaptive content presentation and adaptive navigational control need to be integrated in a conceptual framework so their benefits can be synthesised to obtain a synergic result. Exploration space control (ESC) theory attempts to adjust the learning space, called exploration space, to allow the learners to reach an adequate amount of information that their cognitive load is not overloaded. Multiple presentation (MR) approach provides guidelines for the selection of multimedia objects for both the learning content presentation and as navigational links. ESC is further formalised by including the consideration of individual learner's cognitive traits, which are the cognitive characteristics and abilities the learner relevant in the process of learning. Cognitive traits selected in the formalisation include working memory capacity, inductive reasoning skill, associative learning skill, and information processing speed. The formalisation attempts to formulate a guideline on how the learning content and navigational space should be adjusted in order to support a learner with a particular set of cognitive traits. However, in order to support the provision of adaptivity, the learners and their activities in the VLEs need to be profiled; the profiling process is called student modelling. Student models nowadays can be categorised into state models, and process models. State models record learners' progress as states (e.g. learned, not learned), whereas a process model represents the learners in term of both the knowledge they learned in the domain, and the inference procedures they used for completing a process (task). State models and process models are both competence-based, and they do not provide the information of an individual's cognitive abilities required by the formalisation of exploration space control. A new approach of student modelling is required, and this approach is called cognitive trait model (CTM). The basis of CTM lies in the field of cognitive science. The process for the creation of CTM includes the following subtasks. The cognitive trait under inquiry is studied in order to find its indicative signs (e.g. sign A indicates high working memory capacity). The signs are called the manifests of the cognitive trait. Manifests are always in pairs, i.e. if manifest A indicates high working memory capacity, A's inverse, B, would indicates low working memory capacity. The manifests are then translated into implementation patterns which are observable patterns in the records of learner-system interaction. Implementation patterns are regarded as machine-recognisable manifests. The manifests are used to create nodes in a neural network like structure called individualised temperament network (ITN). Every node in the ITN has its weight that conditions and is conditioned by the overall result of the execution of ITN. The output of the ITN's execution is used to update the CTM. A formative evaluation was carried out for a prototype created in this work. The positive results of the evaluation show the educational potential of the CTM approach. The current CTM only cater for the working memory capacity, in the future research more cognitive traits will be studied and included into the CTM

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates

Tame Automorphisms Fixing a Variable of Free Associative Algebras of Rank Three

We study automorphisms of the free associative algebra K over a field K which fix the variable z. We describe the structure of the group of z-tame automorphisms and derive algorithms which recognize z-tame automorphisms and z-tame coordinates

Sealing of micromachined cavities using chemical vapor deposition methods: characterization and optimization

This paper presents results of a systematic investigation to characterize the sealing of micromachined cavities using chemical vapor deposition (CVD) methods. We have designed and fabricated a large number and variety of surface-micromachined test structures with different etch-channel dimensions. Each cavity is then subjected to a number of sequential CVD deposition steps with incremental thickness until the cavity is successfully sealed. At etch deposition interval, the sealing status of every test structure is experimentally obtained and the percentage of structures that are sealed is recorded. Four CVD sealing materials have been incorporated in our studies: LPCVD silicon nitride, LPCVD polycrystalline silicon (polysilicon), LPCVD phosphosilicate glass (PSG), and PECVD silicon nitride. The minimum CVD deposition thickness that is required to successfully seal a microstructure is obtained for the first time. For a typical Type-1 test structure that has eight etch channels-each 10 μm long, 4 μm wide, and 0.42 μm tall-the minimum required thickness (normalized with respect to the height of etch channels) is 0.67 for LPCVD silicon nitride, 0.62 for LPCVD polysilicon, 4.5 for LPCVD PSG, and 5.2 for PECVD nitride. LPCVD silicon nitride and polysilicon are the most efficient sealing materials. Sealing results with respect to etch-channel dimensions (length and width) are evaluated (within the range of current design). When LPCVD silicon nitride is used as the sealing material, test structures with the longest (38 μm) and widest (16 μm) etch channels exhibit the highest probability of sealing. Cavities with a reduced number of etch channels seal more easily. For LPCVD PSG sealing, on the other hand, the sealing performance improves with decreasing width but is not affected by length of etch channels

Polynomial Retracts and the Jacobian Conjecture

Let $K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R$. The presence of other, equivalent, definitions of retracts provides several different methods of studying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $K[x, y]$ up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture. Notably, we prove that if a polynomial mapping $\varphi$ of $K[x,y]$ has invertible Jacobian matrix {\it and } fixes a non-constant polynomial, then $\varphi$ is an automorphism

Affine varieties with equivalent cylinders

A well-known cancellation problem asks when, for two algebraic varieties $V_1, V_2 \subseteq {\bf C}^n$, the isomorphism of the cylinders $V_1 \times {\bf C}$ and $V_2 \times {\bf C}$ implies the isomorphism of $V_1$ and $V_2$. In this paper, we address a related problem: when the equivalence (under an automorphism of ${\bf C}^{n+1}$) of two cylinders $V_1 \times {\bf C}$ and $V_2 \times {\bf C}$ implies the equivalence of their bases $V_1$ and $V_2$ under an automorphism of ${\bf C}^n$? We concentrate here on hypersurfaces and show that this problem establishes a strong connection between the Cancellation conjecture of Zariski and the Embedding conjecture of Abhyankar and Sathaye. We settle the problem for a large class of polynomials. On the other hand, we give examples of equivalent cylinders with inequivalent bases (those cylinders, however, are not hypersurfaces). Another result of interest is that, for an arbitrary field $K$, the equivalence of two polynomials in $m$ variables under an automorphism of $K[x_1,..., x_n], n \ge m,$ implies their equivalence under a tame automorphism of $K[x_1,..., x_{2n}]$.Comment: 12 page

Degree estimate for commutators

Let K be a free associative algebra over a field K of characteristic 0 and let each of the noncommuting polynomials f,g generate its centralizer in K. Assume that the leading homogeneous components of f and g are algebraically dependent with degrees which do not divide each other. We give a counterexample to the recent conjecture of Jie-Tai Yu that deg([f,g])=deg(fg-gf) > min{deg(f),deg(g)}. Our example satisfies deg(g)/2 < deg([f,g]) < deg(g) < deg(f) and deg([f,g]) can be made as close to deg(g)/2 as we want. We obtain also a counterexample to another related conjecture of Makar-Limanov and Jie-Tai Yu stated in terms of Malcev - Neumann formal power series. These counterexamples are found using the description of the free algebra K considered as a bimodule of K[u] where u is a monomial which is not a power of another monomial and then solving the equation [u^m,s]=[u^n,r] with unknowns r,s in K.Comment: 18 page