Planning in the face of ‘deep divisions’: a view from Beirut, Lebanon

This article discusses planning in the global South-East while focusing on the specific context of social divides, political turmoil and conflict situations. The article proposes a five-way framework based on political science and planning to theory to analyse such contexts. The article explores the case of Beirut, Lebanon that has undergone several episodes of internal and external conflicts resulting in a society splintered along sectarianism. Three Two case studies of open urban spaces and their public activities are analysed using the five-way framework The discussion indicates how economic liberalism that is prevalent in countries of the South-East, along with place-based identities, interest-based identities, consensus orientated processes and institutionalism might facilitate a cultivation of deep values away from a narrowly constructed identity. The article argues that planners should understand the options for positive action that aim to bridge deep divisions and suggests that the five-way framework provides a reference for contextualising in different ways to suit particular contexts. Therefore, the framework is not necessarily restricted to the South-East but could be applicable to any context which manifests deep divisions.

Skew-symmetric identities of octonions

Quadratic alternative superalgebras are introduced and their super-identities and central functions on one odd generator are described. As a corollary, all multilinear skew-symmetric identities and central polynomials of octonions are classified. (C) 2008 Elsevier B.V. All rights reserved.

An application of lattice basis reduction to polynomial identities for algebraic structures

The authors` recent classification of trilinear operations includes, among other cases, a fourth family of operations with parameter q epsilon Q boolean OR {infinity}, and weakly commutative and weakly anticommutative operations. These operations satisfy polynomial identities in degree 3 and further identities in degree 5. For each operation, using the row canonical form of the expansion matrix E to find the identities in degree 5 gives extremely complicated results. We use lattice basis reduction to simplify these identities: we compute the Hermite normal form H of E(t), obtain a basis of the nullspace lattice from the last rows of a matrix U for which UE(t) = H, and then use the LLL algorithm to reduce the basis. (C) 2008 Elsevier Inc. All rights reserved.

Polynomial identities for the ternary cyclic sum

We simplify the results of Bremner and Hentzel [J. Algebra 231 (2000) 387-405] on polynomial identities of degree 9 in two variables satisfied by the ternary cyclic sum [a, b, c] abc + bca + cab in every totally associative ternary algebra. We also obtain new identities of degree 9 in three variables which do not follow from the identities in two variables. Our results depend on (i) the LLL algorithm for lattice basis reduction, and (ii) linearization operators in the group algebra of the symmetric group which permit efficient computation of the representation matrices for a non-linear identity. Our computational methods can be applied to polynomial identities for other algebraic structures.

POLYNOMIAL AND GROUP IDENTITIES IN ALTERNATIVE LOOP ALGEBRAS

We investigate polynomial identities on an alternative loop algebra and group identities on its (Moufang) unit loop. An alternative loop ring always satisfies a polynomial identity, whereas whether or not a unit loop satisfies a group identity depends on factors such as characteristic and centrality of certain kinds of idempotents.

POLYNOMIAL IDENTITIES OF FINITE DIMENSIONAL SIMPLE ALGEBRAS

Let F be an algebraically closed field and let A and B be arbitrary finite dimensional simple algebras over F. We prove that A and B are isomorphic if and only if they satisfy the same identities.

ASSOCIATIVE IDENTITIES OF OCTONIONS

We work to find a basis of identities for an octonion algebra modulo an associator ideal of a free alternative algebra, or, in other words, a basis for an associative replica of an ideal of identities of an octonion algebra.

Star-group identities and groups of units

Analogous to *-identities in rings with involution we define *-identities in groups. Suppose that G is a torsion group with involution * and that F is an infinite field with char F not equal 2. Extend * linearly to FG. We prove that the unit group U of FG satisfies a *-identity if and only if the symmetric elements U(+) satisfy a group identity.

Group identities on symmetric units

Let F be an infinite field of characteristic different from 2, G a group and * an involution of G extended by linearity to an involution of the group algebra FG. Here we completely characterize the torsion groups G for which the *-symmetric units of FG satisfy a group identity. When * is the classical involution induced from g -> g(-1), g is an element of G, this result was obtained in [ A. Giambruno, S. K. Sehgal, A. Valenti, Symmetric units and group identities, Manuscripta Math. 96 (1998) 443-461]. (C) 2009 Elsevier Inc. All rights reserved.

Alternative Identities and Foreign Language Learning

My D-essay has the working title “Alternative Identities and Foreign Language Learning”. I have chosen this area because I have noticed a certain reluctance among Swedish students to use the foreign language English in English classes. They often seem embarrassed to express themselves in a language which is not their mother tongue, but they seem less embarrassed when they are allowed to act somebody else. These two observations converge into a focus of discussion on the matter, which will be supported by a minor study of my own, by extracts from other people’s essays on the matter, and by an overview of current litterature on language, identity and drama.The aim of my essay is to compare Swedish students’ willingness to use the foreign language English when acting minor plays in school, as themselves and as a chosen character, and to investigate the possibility of improving students’ willingness to use a foreign language, when given the opportunity to do so through acting somebody else.