The Variety of Leibniz Algebras Defined by the Identity x(y(zt)) ≡ 0

2000 Mathematics Subject Classification: Primary: 17A32; Secondary: 16R10, 16P99, 17B01, 17B30, 20C30

Asymptotic Behaviour of Colength of Varieties of Lie Algebras

This project was partially supported by RFBR, grants 99-01-00233, 98-01-01020 and 00-15-96128.

Exponents of Subvarieties of Upper Triangular Matrices over Arbitrary Fields are Integral

Partially supported by grant RFFI 98-01-01020.

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.

q-Leibniz Algebras

2000 Mathematics Subject Classification: Primary 17A32, Secondary 17D25.

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.

constructing finite algebras with falcon

The generation of models and counterexamples is an important form of reasoning. In this paper, we give a formal account of a system, called FALCON, for constructing finite algebras from given equational axioms. The abstract algorithms, as well as some implementation details and sample applications, are presented. The generation of finite models is viewed as a constraint satisfaction problem, with ground instances of the axioms as constraints. One feature of the system is that it employs a very simple technique, called the least number heuristic, to eliminate isomorphic (partial) models, thus reducing the size of the search space. The correctness of the heuristic is proved. Some experimental data are given to show the performance and applications of the system.

Two-dimensional Banach spaces with polynomial numerical index zero

We study two-dimensional Banach spaces with polynomial numerical indices equal to zero.

Molecular analysis of endothelial progenitor cell (EPC) subtypes reveals two distinct cell populations with different identities

BACKGROUND: The term endothelial progenitor cells (EPCs) is currently used to refer to cell populations which are quite dissimilar in terms of biological properties. This study provides a detailed molecular fingerprint for two EPC subtypes: early EPCs (eEPCs) and outgrowth endothelial cells (OECs). METHODS: Human blood-derived eEPCs and OECs were characterised by using genome-wide transcriptional profiling, 2D protein electrophoresis, and electron microscopy. Comparative analysis at the transcript and protein level included monocytes and mature endothelial cells as reference cell types. RESULTS: Our data show that eEPCs and OECs have strikingly different gene expression signatures. Many highly expressed transcripts in eEPCs are haematopoietic specific (RUNX1, WAS, LYN) with links to immunity and inflammation (TLRs, CD14, HLAs), whereas many transcripts involved in vascular development and angiogenesis-related signalling pathways (Tie2, eNOS, Ephrins) are highly expressed in OECs. Comparative analysis with monocytes and mature endothelial cells clusters eEPCs with monocytes, while OECs segment with endothelial cells. Similarly, proteomic analysis revealed that 90% of spots identified by 2-D gel analysis are common between OECs and endothelial cells while eEPCs share 77% with monocytes. In line with the expression pattern of caveolins and cadherins identified by microarray analysis, ultrastructural evaluation highlighted the presence of caveolae and adherens junctions only in OECs. CONCLUSIONS: This study provides evidence that eEPCs are haematopoietic cells with a molecular phenotype linked to monocytes; whereas OECs exhibit commitment to the endothelial lineage. These findings indicate that OECs might be an attractive cell candidate for inducing therapeutic angiogenesis, while eEPC should be used with caution because of their monocytic nature.

C∗-Segal algebras with order unit

We introduce the notion of a (noncommutative) C *-Segal algebra as a Banach algebra (A, {norm of matrix}{dot operator}{norm of matrix} A) which is a dense ideal in a C *-algebra (C, {norm of matrix}{dot operator}{norm of matrix} C), where {norm of matrix}{dot operator}{norm of matrix} A is strictly stronger than {norm of matrix}{dot operator}{norm of matrix} C onA. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C *-Segal algebras with order unit is determined.

Finite dimensional semigroup quadratic algebras with the minimal number of relations

A quadratic semigroup algebra is an algebra over a field given by the generators x_1, . . . , x_n and a finite set of quadratic relations each of which either has the shape x_j x_k = 0 or the shape x_j x_k = x_l x_m . We prove that a quadratic semigroup algebra given by n generators and d=(n^2+n)/4 relations is always infinite dimensional. This strengthens the Golod–Shafarevich estimate for the above class of algebras. Our main result however is that for every n, there is a finite dimensional quadratic semigroup algebra with n generators and d_n relations, where d_n is the first integer greater than (n^2+n)/4 . That is, the above Golod–Shafarevich-type estimate for semigroup algebras is sharp.

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.