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Calvin Tromp and Rob Davis lead us through a mapping task that engages students in significant learning in the final weeks of the year: this paper reports on a classroom investigation of a sequence of cross-disciplinary mapping lessons undertaken by Grade Five students at Black Hill Primary, a Victorian State Primary School in Ballarat
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Algebraic-geometric (AG) codes over finite fields with respect to the Lee metric have been studied. A lower bound on the minimum Lee distance is derived, which is a Lee-metric version of the well-known Goppa bound on the minimum Hamming distance of AG codes. The bound generalises a lower bound on the minimum Lee distance of Lee-metric BCH and Reed-Solomon codes, which have been successfully used for protecting against bitshift and synchronisation errors in constrained channels and for error control in partial-response channels.
Description:
Education policies and practices developed for urban populations are not always effective when implemented in regional and remote locations. Despite government policy initiatives that may provide for collaboration across communities, a singular issue is that a diversity of solutions may be required rather than a 'one size fits all' approach. This article presents a potential solution to this problem through engaging collaboration and co-creation to optimise educational opportunities in initial teacher education in Australia. Qualitative analysis of a collaborative and co-created process of enhancement, lesson development and reflection brings together the every-day problem-solving processes used by pre-service teachers and classroom students with those used by research scientists and community experts. A consequence of such a process that benefits regional and remote communities is the development of collaborative networks founded in co-creation of educational opportunities and based on daily life in local communities.
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Delamination is a typical failure mode of composite materials caused by weak bonding. It arises when a crack initiates and propagates under a destructive loading. Given the physical law characterizing the properties of the interlayer adhesive between the bonded bodies, we consider the problem of computing the propagation of the crack front and the stress field along the contact boundary. This leads to a hemivariational inequality, which after discretization by finite elements we solve by a nonconvex bundle method, where upper- C 1 criteria have to be minimized. As this is in contrast with other classes of mechanical problems with non-monotone friction laws and in other applied fields, where criteria are typically lower- C 1 , we propose a bundle method suited for both types of nonsmoothness. We prove its global convergence in the sense of subsequences and test it on a typical delamination problem of material sciences.
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Abstract As with many mathematics teacher educators, I purposefully set about challenging the curriculum and pedagogical status-quo; I wanted my Bachelor of Education pre-service teachers (PSTs) to experience learning and teaching mathematics in a connected and powerful way. I initiated was Roundtable Reflective Inquiry (RRI) sessions as one key practice that enabled PSTs to reflect on their mathematics teaching experience in an ongoing, structured manner. PSTs generated discussions based on critical events and interactions, and my aim was for PSTs to learn through their own experience rather than mine. My role was to facilitate, synthesise and challenge (often) firmly held assumptions. However, as this chapter will reveal, an often unexpected outcome of pre-service teacher education is teacher educator enrichment, knowledge and transformation. While my intention, and indeed the initial focus of my self-study was to enhance PST learning and teaching in mathematics, it soon became evident that I was learning as a teacher educator and my new approach was contributing to my own pedagogical development as a teacher educator. Learning in mathematics teacher education is about coming to know and self-study enabled me as teacher educator researcher to notice, to challenge my assumptions and to name new knowledge
Description:
For a topological group G we define N to be the set of all normal subgroups modulo which G is a finite-dimensional Lie group. Call G a pro-Lie group if, firstly, G is complete, secondly, N is a filter basis, and thirdly, every identity neighborhood of G contains some member of N. It is easy to see that every pro-Lie group G is a projective limit of the projective system of all quotients of G modulo subgroups from N. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup N of a pro-Lie group G, for any one parameter subgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.
Description:
In the 1990s, two major issues emerged globally for tertiary mathematics educators: declining standards in the mathematical proficiency of students at tertiary-entry level and the advocates for the incorporation of IT into the processes of teaching and learning. Addressing these issues required significant reform of both curriculum content and classroom practice to ensure that the technology was used appropriately and effectively. These reforms were largely implemented by staff from tertiary Schools of Mathematics. Rather than leading the reform, the Education community lagged behind and in some instances has taken research into the use of IT in Mathematics Education in a questionable direction. In this paper, we outline what we contend has been a failure on the behalf of the Education community to tackle and address problems experienced by mathematics educators (particularly at the tertiary level) and make some suggestions for the directions of future research in Mathematics Education.
Description:
In the 1990s, two major issues emerged globally for tertiary mathematics educators: declining standards in the mathematical proficiency of students at tertiary-entry level and the advocates for the incorporation of IT into the processes of teaching and learning. Addressing these issues required significant reform of both curriculum content and classroom practice to ensure that the technology was used appropriately and effectively. These reforms were largely implemented by staff from tertiary Schools of Mathematics. Rather than leading the reform, the Education community lagged behind and in some instances has taken research into the use of IT in Mathematics Education in a questionable direction. In this paper, we outline what we contend has been a failure on the behalf of the Education community to tackle and address problems experienced by mathematics educators (particularly at the tertiary level) and make some suggestions for the directions of future research in Mathematics Education.
Description:
A large pile of Lego sits on the table, and you watch as young children are actively engaged in creating vehicles. Moments of quick chatter are followed by a quick run around the room to test their latest vehicle, then intense concentration as they try to find the next perfect piece.
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This paper addresses some fundamental issues in nonconvex analysis. By using pure complementary energy principle proposed by the author, a class of fully nonlinear partial differential equations in nonlinear elasticity is able to convert a unified algebraic equation, a complete set of analytical solutions are obtained in dual space for 3-D finite deformation problems governed by generalized neo-Hookean model . Both global and local extremal solutions to the nonconvex variational problem are identified by a triality theory. Connection between challenges in nonlinear analysis and NP-hard problems in computational science is revealed. Results show that Legendre–Hadamard condition can only guarantee ellipticity for generalized convex problems. For nonconvex systems, the ellipticity depends not only on the stored energy, but also on the external force field. Uniqueness is proved based on a generalized quasiconvexity and a generalized ellipticity condition. Application is illustrated for nonconvex logarithm stored energy.
Description:
Selection is an important and critical aspect in evolutionary computation. This paper presents a novel parental selection technique that includes the advantages of both the deterministic and the stochastic selection techniques and helps to reduce the loss of diversity by distributing the reproduction opportunity among all the members of the population. Moreover, the proposed selection strategy promotes the concept of non-random mating by clustering the population into groups according to the fitness values and then by persuading the mating between individuals from different groups based on performance determined dynamically over the evolution. Computational results using widely used benchmark functions show significant improvements in the convergence characteristics of the proposed selection method over two well-known selection techniques.
Description:
Mixed graphs contain both undirected as well as directed links between vertices and therefore are an interesting model for interconnection communication networks. In this paper, we establish the Moore bound for mixed graphs, which generalizes both the directed and the undirected Moore bound. Crown Copyright (C) 2007 Published by Elsevier B.V. All rights reserved.
Description:
We propose a flexible approach for computing the resolvent of the sum of weakly monotone operators in real Hilbert spaces. This relies on splitting methods where strong convergence is guaranteed. We also prove linear convergence under Lipschitz continuity assumption. The approach is then applied to computing the proximity operator of the sum of weakly convex functions, and particularly to finding the best approximation to the intersection of convex sets.