A unified framework for generalized multicategories

Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the "lax algebras" or "Kleisli monoids" relative to a "monad" on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous examples, while at the same time simplifying and clarifying much of the theory.Comment: 76 pages; final version, to appear in TA

Reduced spectral synthesis and compact operator synthesis

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it with other exceptional sets in operator algebra theory, studied previously. We show that a closed subset $E$ of a second countable locally compact group $G$ satisfies reduced local spectral synthesis if and only if the subset $E^* = \{(s,t) : ts^{-1}\in E\}$ of $G\times G$ satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten $p$-classes.Comment: 43 page

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 $\mathcal{B}(L^2(G))$ 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 $E\subseteq G$ is a set of multiplicity if and only if the set $E^* = \{(s,t)\in G\times G : ts^{-1}\in E\}$ 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 $\psi : G\to \mathbb{C}$ defines a closable multiplier on the reduced C*-algebra $C_r^*(G)$ of $G$ if and only if Schur multiplication by the function $N(\psi) : G\times G\to \mathbb{C}$, 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 $VN(G)$.Comment: 51 page

Closable Multipliers

Let (X,m) and (Y,n) be standard measure spaces. A function f in $L^\infty(X\times Y,m\times n)$ is called a (measurable) Schur multiplier if the map $S_f$, defined on the space of Hilbert-Schmidt operators from $L_2(X,m)$ to $L_2(Y,n)$ by multiplying their integral kernels by f, is bounded in the operator norm. The paper studies measurable functions f for which $S_f$ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if f is of Toeplitz type, that is, if f(x,y)=h(x-y), x,y in G, where G is a locally compact abelian group, then the closability of f is related to the local inclusion of h in the Fourier algebra A(G) of G. If f is a divided difference, that is, a function of the form (h(x)-h(y))/(x-y), then its closability is related to the "operator smoothness" of the function h. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.Comment: 35 page

The role of the clinical chemistry service

Larger numbers of increasingly diversified laboratory investigations have demanded an improved performance in laboratory services. The implementation of automation has alleviated the analytical problems. The formidable clerical task involving professional staff can be handled conveniently, more accurately and more quickly than by manual methods by electronic data processing systems, which are briefly reviewed. Organisation of the clinical chemistry service in the UK is returning to a centralised system, which has  become necessary in the face of escalating expense in buying equipment and the running costs of the Health Service. Serious shortage of specialised personnel is alleviated by the adoption of a rationalised system with the formation of group laboratory services. Application of some of these ideas may help in planning improvement of the clinical chemistry services in South Africa.S. Afr. Med. J., 48, 1712 (1974)

A Philosophical Analysis of Neuroenergetics

Neuroimaging studies measure cerebral rates of glucose and oxygen consumption yet they are often interpreted to support philosophical positions about mental processes presumed to represent behavior. Cognitive Neuroscience has claimed that Neuroimaging resolves Descartes’ dualism of Mind and Matter by mapping such presumed mental processes onto the brain. Bennett & Hacker point out that this is nonsense because a person, not the brain, performs observable activities such as remembering, planning or voting. This analysis is supported by the continuing inability of neuroimaging to locate the modular brain activities postulated to perform mental activity. Pragmatist philosophy emphasizes that such postulated representations are contingent hypothesis invented to deal with the world in everyday life and are not definable as proposed by Cognitive Neuroscience. A recent philosophy of Mechanisms, reflecting what biologists actually do when explaining observable phenomena, looks for explanations not from postulated representations but from multi-level, multi-disciplinary mechanisms of metabolism and energetics which are actually measured in neuroimaging. The value of this philosophy is illustrated by the experimental relations between cerebral energetics in disorders of consciousness and the response to stimuli

Noise diffraction patterns eliminated in coherent optical systems

Lens rotation technique of noise diffraction pattern elimination spreads diffracted energy, normally concentrated over small area of image, over much larger annular area. Technique advantages include simplified lens selecting process, reduced clean room requirements, and low cost equipment requirements

The molecular structure of isocyanic acid from microwave and infra-red absorption spectra

Experimental investigations of the infra-red and microwave spectra of the slightly asymmetric rotor, HNCO, have been made, and the structure of the molecule has been determined

Elimination of coherent noise in a coherent light imaging system

Optical imaging systems using coherent light introduce objectionable noise into the output image plane. Dust and bubbles on and in lenses cause most of the noise in the output image. This noise usually appears as bull's-eye diffraction patterns in the image. By rotating the lens about the optical axis these diffraction patterns can be essentially eliminated. The technique does not destroy the spatial coherence of the light and permits spatial filtering of the input plane