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

On the spectrum of multiplication operators

We study relations between spectra of two operators that are connected to each other through some intertwining conditions. As application we obtain new results on the spectra of multiplication operators on B(\cl H) relating it to the spectra of the restriction of the operators to the ideal $\mathcal C_2$ of Hilbert-Schmidt operators. We also solve one of the problems, posed in [B.Magajna, Proc. Amer. Math. Soc, 141 2013, 1349-1360] about the positivity of the spectrum of multiplication operators with positive operator coefficients when the coefficients on one side commute. Using the Wiener-Pitt phenomena we show that the spectrum of a multiplication operator with normal coefficients satisfying the Haagerup condition might be strictly larger than the spectrum of its restriction to $\mathcal C_2$.Comment: 12 pages, v2: corrected some typos, to appear in Methods of Funct. Anal. Topo

The effects of hemodynamic lag on functional connectivity and behavior after stroke

Stroke disrupts the brain's vascular supply, not only within but also outside areas of infarction. We investigated temporal delays (lag) in resting state functional magnetic resonance imaging signals in 130 stroke patients scanned two weeks, three months and 12 months post stroke onset. Thirty controls were scanned twice at an interval of three months. Hemodynamic lag was determined using cross-correlation with the global gray matter signal. Behavioral performance in multiple domains was assessed in all patients. Regional cerebral blood flow and carotid patency were assessed in subsets of the cohort using arterial spin labeling and carotid Doppler ultrasonography. Significant hemodynamic lag was observed in 30% of stroke patients sub-acutely. Approximately 10% of patients showed lag at one-year post-stroke. Hemodynamic lag corresponded to gross aberrancy in functional connectivity measures, performance deficits in multiple domains and local and global perfusion deficits. Correcting for lag partially normalized abnormalities in measured functional connectivity. Yet post-stroke FC-behavior relationships in the motor and attention systems persisted even after hemodynamic delays were corrected. Resting state fMRI can reliably identify areas of hemodynamic delay following stroke. Our data reveal that hemodynamic delay is common sub-acutely, alters functional connectivity, and may be of clinical importance

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

Using tasks to explore teacher knowledge in situation-specific contexts

This article was published in the journal, Journal of Mathematics Teacher Education [© Springer] and the original publication is available at www.springerlink.comResearch often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of x+x−1=0 of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore

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

Stronger prediction of motor recovery and outcome post-stroke by cortico-spinal tract integrity than functional connectivity

<div><p>Objectives</p><p>To examine longitudinal changes in structural and functional connectivity post-stroke in patients with motor impairment, and define their importance for recovery and outcome at 12 months.</p><p>Methods</p><p>First-time stroke patients (N = 31) were studied at 1–2 weeks, 3 months, and 12 months post-injury with a validated motor battery and resting-state fMRI to measure inter-hemispheric functional connectivity (FC). Fractional anisotropy (FA) of the cortico-spinal tract (CST) was derived from diffusion tensor imaging as a measure of white matter organization. ANOVAs were used to test for changes in FC, FA, and motor performance scores over time, and regression analysis related motor outcome to clinical and neuroimaging variables.</p><p>Results</p><p>FA of the ipsilesional CST improved significantly from 3 to 12 months and was strongly correlated with motor performance. FA improved even in the absence of direct damage to the CST. Inter-hemispheric FC also improved over time, but did not correlate with motor performance at 12 months. Clinical variables (early motor score, education level, and age) predicted 80.4% of the variation of motor outcome, and FA increased the predictability to 84.6%. FC did not contribute to the prediction of motor outcome.</p><p>Conclusions</p><p>Stroke causes changes to the CST microstructure that can account for behavioral variability even in the absence of demonstrable lesion. Ipsilesional CST undergoes remodeling post-stroke, even past the three-month window when most of the motor recovery happens. FA of the CST, but not inter-hemispheric FC, can improve to the prediction of motor outcome based on early motor scores.</p></div