Quotients, exactness, and nuclearity in the operator system category

We continue our study of tensor products in the operator system category. We define operator system quotients and exactness in this setting and refine the notion of nuclearity by studying operator systems that preserve various pairs of tensor products. One of our main goals is to relate these refinements of nuclearity to the Kirchberg conjecture. In particular, we prove that the Kirchberg conjecture is equivalent to the statement that every operator system that is (min,er)-nuclear is also (el,c)-nuclear. We show that operator system quotients are not always equal to the corresponding operator space quotients and then study exactness of various operator system tensor products for the operator system quotient. We prove that an operator system is exact for the min tensor product if and only if it is (min,el)-nuclear. We give many characterizations of operator systems that are (min,er)-nuclear, (el,c)-nuclear, (min,el)-nuclear and (el,max)-nuclear. These characterizations involve operator system analogues of various properties from the theory of C*-algebras and operator spaces, including the WEP and LLP.

Hypercyclic and mixing operator semigroups

We describe a class of topological vector spaces admitting a mixing uniformly continuous operator group $\{T_t\}_{t\in\C^n}$ with holomorphic dependence on the parameter $t$. This result covers those existing in the literature. We also
describe a class of topological vector spaces admitting no supercyclic strongly continuous operator semigroups $\{T_t\}_{t\geq 0}$.

BIT-LEVEL SYSTOLIC ARRAY CIRCUIT FOR MATRIX VECTOR MULTIPLICATION.

A bit-level systolic array for computing matrix x vector products is described. The operation is carried out on bit parallel input data words and the basic circuit takes the form of a 1-bit slice. Several bit-slice components must be connected together to form the final result, and authors outline two different ways in which this can be done. The basic array also has considerable potential as a stand-alone device, and its use in computing the Walsh-Hadamard transform and discrete Fourier transform operations is briefly discussed.

A VLSI architecture for multiplication, division and square root

A high-performance VLSI architecture to perform combined multiply-accumulate, divide, and square root operations is proposed. The circuit is highly regular, requires only minimal control, and can be reconfigured for every cycle. The execution time for each operation is the same. The combination of redundancy and pipelining results in a throughput independent of the wordsize of the array. With current CMOS technology, throughput rates in excess of 80 million operations per second are achievable.

Multiplication of microbes below 0.690 water activity: implications for terrestrial and extraterrestrial life

Since a key requirement of known life forms is available water (water activity; aw), recent searches for signatures of past life in terrestrial and extraterrestrial environments have targeted places known to have contained significant quantities of biologically available water. However, early life on Earth inhabited high-salt environments, suggesting an ability to withstand low water-activity. The lower limit of water activity that enables cell division appears to be ∼ 0.605 which, until now, was only known to be exhibited by a single eukaryote, the sugar-tolerant, fungal xerophile Xeromyces bisporus. The first forms of life on Earth were, though, prokaryotic. Recent evidence now indicates that some halophilic Archaea and Bacteria have water-activity limits more or less equal to those of X. bisporus. We discuss water activity in relation to the limits of Earth's present-day biosphere; the possibility of microbial multiplication by utilizing water from thin, aqueous films or non-liquid sources; whether prokaryotes were the first organisms able to multiply close to the 0.605-aw limit; and whether extraterrestrial aqueous milieux of ≥ 0.605 aw can resemble fertile microbial habitats found on Earth.

Manual mapping of drumlins in synthetic landscapes to assess operator effectiveness

Mapped topographic features are important for understanding processes that sculpt the Earth’s surface. This paper presents maps that are the primary product of an exercise that brought together 27 researchers with an interest in landform mapping wherein the efficacy and causes of variation in mapping were tested using novel synthetic DEMs containing drumlins. The variation between interpreters (e.g. mapping philosophy, experience) and across the study region (e.g. woodland prevalence) opens these factors up to assessment. A priori known answers in the synthetics increase the number and strength of conclusions that may be drawn with respect to a traditional comparative study. Initial results suggest that overall detection rates are relatively low (34–40%), but reliability of mapping is higher (72–86%). The maps form a reference dataset.

High-radix systolic modular multiplication on reconfigurable hardware

The overall aim of the work presented in this paper has been to develop Montgomery modular multiplication architectures suitable for implementation on modern reconfigurable hardware. Accordingly, novel high-radix systolic array Montgomery multiplier designs are presented, as we believe that the inherent regular structure and absence of global interconnect associated with these, make them well-suited for implementation on modern FPGAs. Unlike previous approaches, each processing element (PE) comprises both an adder and a multiplier. The inclusion of a multiplier in the PE means that the need to pre-compute or store any multiples of the operands is avoided. This also allows very high-radix implementations to be realised, further reducing the amount of clock cycles per modular multiplication, while still maintaining a competitive critical delay. For demonstrative purposes, 512-bit and 1024-bit FPGA implementations using radices of 2(8) and 2(16) are presented. The subsequent throughput rates are the fastest reported to date.

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).