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

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

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

Behavioural clusters and predictors of performance during recovery from stroke

We examined the patterns and variability of recovery post-stroke in multiple behavioral domains. A large cohort of first time stroke patients with heterogeneous lesions was studied prospectively and longitudinally at 1-2 weeks, 3 months and one year post-injury with structural MRI to measure lesion anatomy and in-depth neuropsychological assessment. Impairment was described at all timepoints by a few clusters of correlated deficits. The time course and magnitude of recovery was similar across domains, with change scores largely proportional to the initial deficit and most recovery occurring within the first three months. Damage to specific white matter tracts produced poorer recovery over several domains: attention and superior longitudinal fasciculus II/III, language and posterior arcuate fasciculus, motor and corticospinal tract. Finally, after accounting for the severity of the initial deficit, language and visual memory recovery/outcome was worse with lower education, while the occurrence of multiple deficits negatively impacted attention recovery

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

Reduced synthesis in harmonic analysis and compact synthesis in operator theory

The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f −1(0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given

A Non-cognitive Behavioral Model for Interpreting Functional Neuroimaging Studies

The dominant model for interpreting brain imaging experiments, which we refer to as the Standard Cognitive Model (SCM), assumes that the brain is organized in support of mental processes that control behavior. However, functional neuroimaging experiments of cognitive tasks have not shown clear anatomic segregation between mental processes originally proposed by this model. This failing has been blamed on limitations in imaging technology and non-linearity in the brain’s implementation of these processes. However, the validity of the underlying cognitive models used to describe the brain has rarely been questioned or directly tested against imaging results. We propose an alternative model of brain function, that we term the Non-cognitive Behavioral Model (NBM), which correlates observed human behavior directly with measured brain activity without making assumptions about intervening cognitive processes. Our model derives from behavioral psychology but is extended to include brain activity, in addition to behavior, as observables. A further extension is the role of neuroplasticity, as opposed to innate cognitive processes, in developing the brain’s support of cognitive behavior. We present the theoretical basis with which the SCM maps cognitive processes onto functional magnetic resonance and positron emission tomography images and compare and contrast with the NBM. We also describe how the NBM can be used experimentally to study how the brain supports behavior. Two applications are presented that support the usefulness of the NBM. In one, the NBM use of the total functional imaging signal (not just the differences between states) provides a stronger correlation of neural activity with the behavioral state of consciousness than the SCM approach in both anesthesia and coma. The second example reviews studies of facial and object recognition that provide evidence for the NBM proposal that neuroplasticity and experience play key roles in the brain’s support of recognition and other behaviors. The conclusions regarding neuroplasticity are then generalized to explain the incomplete functional segregation observed in the application of the SCM to neuroimaging

Structural Disconnections Explain Brain Network Dysfunction after Stroke

Stroke causes focal brain lesions that disrupt functional connectivity (FC), a measure of activity synchronization, throughout distributed brain networks. It is often assumed that FC disruptions reflect damage to specific cortical regions. However, an alternative explanation is that they reflect the structural disconnection (SDC) of white matter pathways. Here, we compare these explanations using data from 114 stroke patients. Across multiple analyses, we find that SDC measures outperform focal damage measures, including damage to putative critical cortical regions, for explaining FC disruptions associated with stroke. We also identify a core mode of structure-function covariation that links the severity of interhemispheric SDCs to widespread FC disruptions across patients and that correlates with deficits in multiple behavioral domains. We conclude that a lesion\u27s impact on the structural connectome is what determines its impact on FC and that interhemispheric SDCs may play a particularly important role in mediating FC disruptions after stroke