K-Theory of Azumaya Algebras over Schemes

Let X be a connected, noetherian scheme and A{script} be a sheaf of Azumaya algebras on X, which is a locally free O{script}-module of rank a. We show that the kernel and cokernel of K(X) ? K(A{script}) are torsion groups with exponent a for some m and any i = 0, when X is regular or X is of dimension d with an ample sheaf (in this case m = d + 1). As a consequence, K(X, Z/m) ? K(A{script}, Z/m), for any m relatively prime to a. © 2013 Copyright Taylor and Francis Group, LLC.

Sets of multiplicity and closable multipliers on group algebras

We undertake a detailed study of the sets of multiplicity in a second countable locally compact group G and their operator versions. We establish a symbolic calculus for normal completely bounded maps from the space B(L-2(G)) of bounded linear operators on L-2 (G) into the von Neumann algebra VN(G) of G and use it to show that a closed subset E subset of G is a set of multiplicity if and only if the set E* = {(s,t) is an element of G x G : ts(-1) is an element of E} is a set of operator multiplicity. Analogous results are established for M-1-sets and M-0-sets. We show that the property of being a set of multiplicity is preserved under various operations, including taking direct products, and establish an Inverse Image Theorem for such sets. We characterise the sets of finite width that are also sets of operator multiplicity, and show that every compact operator supported on a set of finite width can be approximated by sums of rank one operators supported on the same set. We show that, if G satisfies a mild approximation condition, pointwise multiplication by a given measurable function psi : G -> C defines a closable multiplier on the reduced C*-algebra G(r)*(G) of G if and only if Schur multiplication by the function N(psi): G x G -> C, given by N(psi)(s, t) = psi(ts(-1)), is a closable operator when viewed as a densely defined linear map on the space of compact operators on L-2(G). Similar results are obtained for multipliers on VN(C).

Ideals of A(G) and bimodules over maximal abelian selfadjoint algebras

This paper is concerned with weak⁎ closed masa-bimodules generated by A(G)-invariant subspaces of VN(G). An annihilator formula is established, which is used to characterise the weak⁎ closed subspaces of B(L2(G)) which are invariant under both Schur multipliers and a canonical action of M(G) on B(L2(G)) via completely bounded maps. We study the special cases of extremal ideals with a given null set and, for a large class of groups, we establish a link between relative spectral synthesis and relative operator synthesis.

A class of almost C_0(K)-C*-algebras

We consider in this paper the family of exponential Lie groups Gn,µ, whose Lie algebra is an extension of the Heisenberg Lie algebra by the reals and whose quotient group by the centre of the Heisenberg group is an ax + b-like group. The C*-algebras of the groups Gn,µ give new examples of almost C0(K)-C*-algebras.

Development of a doped titania immobilised thin film multi tubular photoreactor

This paper describes a novel doped titania immobilised thin film multi tubular photoreactor which has been developed for use with liquid, vapour or gas phase media. In designing photocatalytic reactors measuring active surface area of photocatalyst within the unit is one of the critical design parameters. This dictate greatly limits the applicability of any semi-conductor photocatalyst in industrial applications, as a large surface area equates to a powder catalyst. This demonstration of a thin film coating, doped with a rare earth element, novel photoreactor design produces a photocatalytic degradation of a model pollutant (methyl orange) which displayed a comparable degradation achieved with P25 TiO2. The use of lanthanide doping is reported here in the titania sol gel as it is thought to increase the electron hole separation therefore widening the potential useful wavelengths within the electromagnetic spectrum. Increasing doping from 0.5% to 1.0% increased photocatalytic degradation by ∼17% under visible irradiation. A linear relationship has been seen between increasing reactor volume and degradation which would not normally be observed in a typical suspended reactor system. © 2012 Elsevier B.V.

Norms of basic elementary operators on algebras of regular operators

We show that if E is an atomic Banach lattice with an ordercontinuous norm, A, B ∈ L^r(E) and M_A,_B is the operator on L^r(E) defined by M_A,_B(T) = AT B then ||M_A,_B||r = ||A||_r||B||_r but that there is no real α > 0 such that ||MA,B || ≥ α ||A||r||B ||r.

Two dimensional unital Riesz algebras, their representations and norms

We describe all two dimensional unital Riesz algebras and study representations of them in Riesz algebras of regular operators. Although our results are not complete, we do demonstrate that very varied behaviour can occur even though all these algebras can be given a Banach lattice algebra norm.

Spectral isometries into commutative Banach algebras

We determine the structure of spectral isometries between unital Banach algebras under the hypothesis that the codomain is commutative.

Fabrication and evaluation of NiO/Y2O3-stabilized-ZrO2 hollow fibers for anode-supported micro-tubular solid oxide fuel cells

In this work NiO/3mol% Y₂O₃-ZrO₂ (3YSZ) and NiO/8mol% Y₂O₃-ZrO₂ (8YSZ) hollow fibers were prepared by phase-inversion. The effect of different kinds of YSZ (3YSZ and 8YSZ) on the porosity, electrical conductivity, shrinkage and flexural strength of the hollow fibers were systematically evaluated. When compared with Ni-8YSZ the porosity and shrinkage of Ni-3YSZ hollow fibers increases while the electrical conductivity decreases, while at the same time also exhibiting enhanced flexural strength. Single cells with Ni-3YSZ and Ni-8YSZ hollow fibers as the supported anode were successfully fabricated showing maximum power densities of 0.53 and 0.67Wcm^-2 at 800°C, respectively. Furthermore, in order to improve the cell performance, a Ni-8YSZ anode functional layer was added between the electrolyte and Ni-YSZ hollow fiber. Here enhanced peak power densities of 0.79 and 0.73Wcm^-2 were achieved at 800°C for single cells with Ni-3YSZ and Ni-8YSZ hollow fibers, respectively.

Optimal 5-step nilpotent quadratic algebras

By the Golod–Shafarevich theorem, an associative algebra $R$ given by $n$ generators and $<n^2/3$ homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer $n$, there is an algebra $R$ given by $n$ generators and $\lceil n^2/3\rceil$ homogeneous quadratic relations such that $R$ is 5-step nilpotent.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).

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.

Linnaeus was right all along: Ulva and Enteromorpha are not distinct genera.

Ulva, one of the first Linnaean genera, was later circumscribed to consist of green seaweeds with distromatic blades, and Enteromorpha Link was established for tubular forms. Although several lines of evidence suggest that these generic constructs are artificial, Ulva and Enteromorpha have been maintained as separate genera. Our aims were to determine phylogenetic relationships among taxa currently attributed to Ulva, Enteromorpha, Umbraulva Bae et I.K. Lee and the monotypic genus Chloropelta C.E. Tanner, and to make any nomenclatural changes justified by our findings. Analyses of nuclear ribosomal internal transcribed spacer DNA (ITS nrDNA) (29 ingroup taxa including the type species of Ulva and Enteromorpha), the chloroplast-encoded rbcL gene (for a subset of taxa) and a combined data set were carried out. All trees had a strongly supported clade consisting of all Ulva, Enteromorpha and Chloropelta species, but Ulva and Enteromorpha were not monophyletic. The recent removal of Umbraulva olivascens (P.J.L. Dangeard) Bae et I.K. Lee from Ulva is supported, although the relationship of the segregate genus Umbraulva to Ulvaria requires further investigation. These results, combined with earlier molecular and culture data, provide strong evidence that Ulva, Enteromorpha and Chloropelta are not distinct evolutionary entities and should not be recognized as separate genera. A comparison of traits for surveyed species revealed few synapomorphies. Because Ulva is the oldest name, Enteromorpha and Chloropelta are here reduced to synonymy with Ulva, and new combinations are made where necessary.