JACOBSON RADICAL ALGEBRAS WITH QUADRATIC GROWTH

AbstractWe show that over every countable algebraically closed field $\mathbb{K}$ there exists a finitely generated $\mathbb{K}$-algebra that is Jacobson radical, infinite-dimensional, generated by two elements, graded and has quadratic growth. We also propose a way of constructing examples of algebras with quadratic growth that satisfy special types of relations.</jats:p

On skew braces and their ideals

The first-named author is partially supported by CCP CoDiMa (EP/M022641/1) and the OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541). The second-named author is supported by the ERC Advanced grant 320974. The third-named author is supported by PICT-201-0147, MATH-AmSud 17MATH-01 and ERC Advanced grant 320974.We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series of ideals, prime and semiprime ideals, Baer and Wedderburn radicals and solvability. The paper contains several questions.PostprintPeer reviewe

On a conjecture of Goodearl: Jacobson radical non-nil algebras of Gelfand-Kirillov dimension 2

For an arbitrary countable field, we construct an associative algebra that is graded, generated by finitely many degree-1 elements, is Jacobson radical, is not nil, is prime, is not PI, and has Gelfand-Kirillov dimension two. This refutes a conjecture attributed to Goodearl

A note on the error analysis of classical Gram-Schmidt

An error analysis result is given for classical Gram--Schmidt factorization of a full rank matrix $A$ into $A=QR$ where $Q$ is left orthogonal (has orthonormal columns) and $R$ is upper triangular. The work presented here shows that the computed $R$ satisfies \normal{R}=\normal{A}+E where $E$ is an appropriately small backward error, but only if the diagonals of $R$ are computed in a manner similar to Cholesky factorization of the normal equations matrix. A similar result is stated in [Giraud at al, Numer. Math. 101(1):87--100,2005]. However, for that result to hold, the diagonals of $R$ must be computed in the manner recommended in this work.Comment: 12 pages This v2. v1 (from 2006) has not the biliographical reference set (at all). This is the only modification between v1 and v2. If you want to quote this paper, please quote the version published in Numerische Mathemati

TGFβ upregulates PAR-1 expression and signalling responses in A549 lung adenocarcinoma cells.

The major high-affinity thrombin receptor, proteinase activated receptor-1 (PAR-1) is expressed at low levels by the normal epithelium but is upregulated in many types of cancer, including lung cancer. The thrombin-PAR-1 signalling axis contributes to the activation of latent TGFβ in response to tissue injury via an αvβ6 integrin-mediated mechanism. TGFβ is a pleiotropic cytokine that acts as a tumour suppressor in normal and dysplastic cells but switches into a tumour promoter in advanced tumours. In this study we demonstrate that TGFβ is a positive regulator of PAR-1 expression in A549 lung adenocarcinoma cells, which in turn increases the sensitivity of these cells to thrombin signalling. We further demonstrate that this effect is Smad3-, ERK1/2- and Sp1-dependent. We also show that TGFβ-mediated PAR-1 upregulation is accompanied by increased expression of integrin αv and β6 subunits. Finally, TGFβ pre-stimulation promotes increased migratory potential of A549 to thrombin. These data have important implications for our understanding of the interplay between coagulation and TGFβ signalling responses in lung cancer.Medical Research Council UK (MRC) CASE studentship with Novartis awarded to RCC, MRC Centenary Award awarded to NS and RCC, and MRC Career Development Award G0800340 to CJS

Growth, entropy and commutativity of algebras satisfying prescribed relations

In 1964, Golod and Shafarevich found that, provided that the number of relations of each degree satisfy some bounds, there exist infinitely dimensional algebras satisfying the relations. These algebras are called Golod-Shafarevich algebras. This paper provides bounds for the growth function on images of Golod-Shafarevich algebras based upon the number of defining relations. This extends results from [32], [33]. Lower bounds of growth for constructed algebras are also obtained, permitting the construction of algebras with various growth functions of various entropies. In particular, the paper answers a question by Drensky [7] by constructing algebras with subexponential growth satisfying given relations, under mild assumption on the number of generating relations of each degree. Examples of nil algebras with neither polynomial nor exponential growth over uncountable fields are also constructed, answering a question by Zelmanov [40]. Recently, several open questions concerning the commutativity of algebras satisfying a prescribed number of defining relations have arisen from the study of noncommutative singularities. Additionally, this paper solves one such question, posed by Donovan and Wemyss in [8].Comment: arXiv admin note: text overlap with arXiv:1207.650

A NOTE ON NIL AND JACOBSON RADICALS IN GRADED RINGS

It was shown by Bergman that the Jacobson radical of a Z-graded ring is homogeneous. This paper shows that the analogous result holds for nil rings, namely, that the nil radical of a Z-graded ring is homogeneous. It is obvious that a subring of a nil ring is nil, but generally a subring of a Jacobson radical ring need not be a Jacobson radical ring. In this paper it is shown that every subring which is generated by homogeneous elements in a graded Jacobson radical ring is always a Jacobson radical ring. It is also observed that a ring whose all subrings are Jacobson radical rings is nil. Some new results on graded-nil rings are also obtained