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.

"I'm still here. Sort of." : Constructed Identities in Paul Auster's City of Glass

Paul Auster’s City of Glass contains a jumble of identities. In fact, the identities are more numerous than the characters, and consequently, characters have several different identities. Some of these identities are obvious constructs, but with others the degree of construction is less evident. Poststructuralist theory, however, puts forward the idea that these seemingly original identities are in fact constructs to the same level as all others. Thus, this essay argues that there are no original identities; identities are constructed by outer factors. This essay discusses three outer factors contributing to the construction of identities, factors commonly discussed in poststructuralist criticism, these three being language, cultural codes and chance.

Sex Trade Involvement in São Paulo, Brazil and Toronto, Canada: Narratives of Social Exclusion and Fragmented Identities

An extensive international literature has been developed regarding the risk trajectories of sex trade-involved children and youth. This literature has not, however, substantially incorporated the narratives of youths regarding their experiences. In this article, the contemporary literature on child and youth sex trade-involvement is reviewed and the findings of a qualitative analysis of the narratives of 14 youth from São Paulo, Brazil and 58 youth from Toronto, Canada are presented. Substantial similarities were found between the groups of narratives with respect to abusive and unstable home experiences, pathways into the sex trade, social exclusion, and the impacts of the sex trade on physical and mental health. Key areas of divergence included the roles of poverty and drug use in entering the sex trade. The implications of shared experiences of social exclusion and fragmented identity across differing sociocultural contexts for policy and intervention are discussed.

Combinatorial identities and quantum state densities of supersymmetric sigma models on N-folds

There is a remarkable connection between the number of quantum states of conformal theories and the sequence of dimensions of Lie algebras. In this paper, we explore this connection by computing the asymptotic expansion of the elliptic genus and the microscopic entropy of black holes associated with (supersymmetric) sigma models. The new features of these results are the appearance of correct prefactors in the state density expansion and in the coefficient of the logarithmic correction to the entropy.

Simple derivation of general Fierz-type identities

General Fierz-type identities are examined and their well-known connection with completeness relations in matrix vector spaces is shown. In particular, I derive the chiral Fierz identities in a simple and systematic way by using a chiral basis for the complex 4 X 4 matrices. Other completeness relations for the fundamental representations of SU(N) algebras can be extracted using the same reasoning. (c) 2005 American Association of Physics Teachers.

Pure spinor superspace identities for massless four-point kinematic factors

Using the pure spinor formalism we prove identities which relate the tree-level, one-loop and two-loop kinematic factors for massless four-point amplitudes. From these identities it follows that the complete supersymmetric one- and two-loop amplitudes are immediately known once the tree-level kinematic factor is evaluated. In particular, the two-loop equivalence with the RNS formalism (up to an overall coefficient) is obtained as a corollary.