An analogue of the Magnus problem for associative algebras

We prove an analogue of Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by $n+k$ generators and $k$ relations and has an $n$-element system of generators, then this algebra is a free algebra of rank $n$.

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.

On the weak amenability of β(X)

We investigate the weak amenability of the Banach algebra ß(X) of all bounded linear operators on a Banach space X. Sufficient conditions are given for weak amenability of this and other Banach operator algebras with bounded one-sided approximate identities.

Semiring induced valuation algebras: Exact and approximate local computation algorithms

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra.There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.

Two problems from the Polishchuk and Positselski book on Quadratic algebras

In the book ’Quadratic algebras’ by Polishchuk and Positselski [23] algebras with a small number of generators (n = 2, 3) are considered. For some number r of relations possible Hilbert series are listed, and those appearing as series of Koszul algebras are specified. The first case, where it was not possible to do, namely the case of three generators n = 3 and six relations r = 6 is formulated as an open problem. We give here a complete answer to this question, namely for quadratic algebras with dimA_1 = dimA_2 = 3, we list all possible Hilbert series, and find out which of them can come from Koszul algebras, and which can not. As a consequence of this classification, we found an algebra, which serves as a counterexample to another problem from the same book [23] (Chapter 7, Sec. 1, Conjecture 2), saying that Koszul algebra of finite global homological dimension d has dimA_1 > d. Namely, the 3-generated algebra A given by relations xx + yx = xz = zy = 0 is Koszul and its Koszul dual algebra A^! has Hilbert series of degree 4: HA! (t) = 1 + 3t + 3t^2 + 2t^3 + t^4, hence A has global homological dimension 4.

Turning the categories inside-out: Complex identifications and multiple interactions in religious ethnography

This paper challenges the fixed boundaries that ethnographers have often constructed between religious insiders and outsiders. Drawing on Neitz's observations, it argues that the main task of reflexive fieldwork is locating the self in relation to ambiguous and shifting boundaries. We offer a comparative analysis of the experiences of two differently socially located researchers to illustrate how religious identity emerges as a continuum, on which one's place is negotiated with one's research participants. We also examine the importance of intersecting multiple identities. Finally, the paper questions whether social identity categories are the primary way that we relate with our respondents. It explores the spiritual and emotional dimensions of research relationships and argues that these may transform, reinforce and generally interact with social identities. Comparing our experiences, we outline the consequences of these reflections for data gathering and analysis.

Graphs of relations and Hilbert series

We are discussing certain combinatorial and counting problems related to quadratic algebras. First we give examples which confirm the Anick conjecture on the minimal Hilbert series for algebras given by $n$ generators and $\frac {n(n-1)}{2}$ relations for $n \leq 7$. Then we investigate combinatorial structure of colored graph associated to relations of RIT algebra. Precise descriptions of graphs (maps) corresponding to algebras with maximal Hilbert series are given in certain cases. As a consequence it turns out, for example, that RIT algebra may have a maximal Hilbert series only if components of the graph associated to each color are pairwise 2-isomorphic.

Structural information content of networks: Graph entropy based on local vertex functionals

In this paper we define the structural information content of graphs as their corresponding graph entropy. This definition is based on local vertex functionals obtained by calculating-spheres via the algorithm of Dijkstra. We prove that the graph entropy and, hence, the local vertex functionals can be computed with polynomial time complexity enabling the application of our measure for large graphs. In this paper we present numerical results for the graph entropy of chemical graphs and discuss resulting properties. (C) 2007 Elsevier Ltd. All rights reserved.