Continuants and some decompositions into squares
- Authors: Delorme, Charles , Pineda-Villavicencio, Guillermo
- Date: 2015
- Type: Text , Journal article
- Relation: Integers Vol. 15, no. (2015), p. 1
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- Description: In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number
Boris Mordukhovich, the never tiring traveller, celebrates his sixtieth birthday
- Authors: Henrion, René , Kruger, Alexander , Outrata, Jiri
- Date: 2008
- Type: Text , Journal article
- Relation: Set-Valued Analysis Vol. 16, no. 2-3 (2008), p. 125-127
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Iwasawa's local splitting theorem for pro-Lie groups
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2008
- Type: Text , Journal article
- Relation: Forum Mathematicum Vol. 20, no. 4 (2008), p. 607-629
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- Description: If the nilradical () of the Lie algebra of a pro-Lie group G is finite dimensional modulo the center (), then every identity neighborhood U of G contains a closed normal subgroup N such that G/N is a Lie group and G and N × G/N are locally isomorphic. © Walter de Gruyter 2008.
- Description: C1
Moore bound for mixed networks
- Authors: Nguyen, Minh Hoang , Miller, Mirka
- Date: 2008
- Type: Text , Journal article
- Relation: Discrete Mathematics Vol. 308, no. 23 (Dec 2008), p. 5499-5503
- Full Text: false
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- 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.
Enhancing the image of mathematics by association with simple pleasures from real world contexts
- Authors: Pierce, Robyn , Stacey, Kaye
- Date: 2006
- Type: Text , Journal article
- Relation: ZDM Zentralblatt fur Didaktik der Mathematik Vol. 38, no. 3 (2006), p. 214-225
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- Description: Those who market people or products choose their images very carefully. They create positive associations in the public's mind by photographing their clients with sporting heroes or national icons. In this paper we present a variety of evidence to show that a major and overlooked reason for teachers' use and choice of real world problems is to take advantage of this ‘halo effect’ to improve studients' attitude towards learning mathematics. Analysis of interviews, reports, and results of a brief survey from teachers of middle secondary school classes indicate that they place a very high priority on positive attitudes and hence both choose and enhance real world problems to promote studients' affective engagement through simple pleasures. Pleasant sensory stimuli, generally non-cognitive and peripheral to the situation to be modelled, are used to promote a positive view of mathematics. This is a good strategy for creating enjoyable and memorable lessons, but there is a danger that it may override more substantive learning goals.
- Description: C1
- Description: 2003001563
Sophus Lie's third fundamental theorem and the adjoint functor theorem
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Group Theory Vol. 8, no. 1 (2005), p. 115-133
- Full Text: false
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- Description: The essential attributes of a Lie group G are the associated Lie algebra LðGÞ and the exponential function exp : LðGÞ ! G. The prescription L operates not only on Lie groups but also on morphisms between them: it is a functor. Many features of Lie theory are shared by classes of topological groups which are much larger than that of Lie groups; these classes include the classes of compact groups, locally compact groups, and pro-Lie groups, that is, complete topological groups having arbitrarily small normal subgroups N such that G=N is a (finitedimensional) Lie group. Considering the functor L it is therefore appropriate to contemplate more general classes of topological groups. Certain functorial properties of the assignment of a Lie algebra to a topological group (where possible) will be essential. What is new here is that we will introduce a functorial assignment from Lie algebras to groups and investigate to what extent it is inverse to the Lie algebra functor L. While the Lie algebra functor is well known and is cited regularly, the existence of a Lie group functor available to be cited and applied appears less well known. Sophus Lie’s Third Fundamental Theorem says that for each finite-dimensional real Lie algebra there is a Lie group whose Lie algebra is (isomorphic to) the given one; but even in classical circumstances it is not commonly known that this happens in a functorial fashion and what the precise relationship between the Lie algebra functor and the Lie group functor is.
- Description: C1
- Description: 2003001415
On non-polynomiality of XOR over Zn2
- Authors: Grosek, Otokar , Miller, Mirka , Ryan, Joe
- Date: 2004
- Type: Text , Journal article
- Relation: Tatra Mountains Mathematical Publications Vol. 29, no. (2004), p. 183-191
- Full Text: false
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- Description: C1
- Description: 2003000905
The exponential function of locally connected compact Abelian groups
- Authors: Hofmann, Karl , Morris, Sidney , Poguntke, D.
- Date: 2004
- Type: Text , Journal article
- Relation: Forum Mathematicum Vol. 16, no. 1 (2004), p. 1-16
- Full Text: false
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- Description: It is shown that the following four conditions are equivalent for a compact connected abelian group G :(i)the exponential function of G is open onto its image;(ii)G has arbitrarily small connected direct summands N such that G =N is a .nite dimensional torus;(iii)the arc component G[suba] of the identity is locally arcwise connected;(iv)the character group G G is a torsion free group in which every .nite rank pure subgroup is free and is a direct summand.
- Description: C1
- Description: 2003000909
Projective limits of finite-dimensional Lie groups
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2003
- Type: Text , Journal article
- Relation: Proceedings of the London Mathematical Society Vol. 87, no. 3 (Nov 2003), p. 647-676
- Full Text: false
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- 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: C1
- Description: 2003000376
Undergraduate mathematics curricula - A new approach
- Authors: Giri, Jason , Pierce, Robyn , Turville, Christopher
- Date: 2003
- Type: Text , Journal article
- Relation: New Zealand Journal of Mathematics Vol. 32 , no. Supplementary Issue (2003), p. 155-162
- Full Text: false
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- Description: C1
- Description: 2003000363
Teaching with CAS in a time of transition
- Authors: Kendal, Margaret , Stacey, Kaye , Pierce, Robyn
- Date: 2002
- Type: Text , Journal article
- Relation: International Journal of Computer Algebra in Mathematics Education Vol. 9, no. 2 (2002), p. 113-128
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- Description: Integrating a powerful instrument such as CAS into teaching and learning mathematics requires changes to many aspects of the classroom, which teachers will make from the base of their prior teaching styles and their beliefs about mathematics and how it should be taught. The paper describes the different ways in which two pioneering Australian teachers adapted their teaching to use CAS. One teacher used CAS with the primary goal of increasing understanding but restricted students’ use. The other teacher adopted CAS as an extra technique for solving standard problems, emphasising timesaving routines by hand and with CAS. Through these case studies we comment on the following issues related to teaching with CAS: different ways of organising the classroom, variety in approaches to teaching the use of CAS, the increased range of methods for solving problems and for teaching, the contrast between using of graphics calculators and CAS, the challenge of finding the place of by-hand skills and CAS use, and the curriculum and assessment changes required in schools.
- Description: C1
- Description: 2003000120