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

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

Tensor products of subspace lattices and rank one density

We show that, if $M$ is a subspace lattice with the property that the rank one subspace of its operator algebra is weak* dense, $L$ is a commutative subspace lattice and $P$ is the lattice of all projections on a separable infinite dimensional Hilbert space, then the lattice $L\otimes M\otimes P$ is reflexive. If $M$ is moreover an atomic Boolean subspace lattice while $L$ is any subspace lattice, we provide a concrete lattice theoretic description of $L\otimes M$ in terms of projection valued functions defined on the set of atoms of $M$. As a consequence, we show that the Lattice Tensor Product Formula holds for \Alg M and any other reflexive operator algebra and give several further corollaries of these results.Comment: 15 page

Validation of 13C NMR measurements of liver glycogen in vivo

The natural abundance 13C NMR intensity of the glycogen C1 resonance was measured in the surgically exposed liver of rabbits in vivo (n = 17) by integration from 98 to 104 ppm and compared double blindedly to the subsequent biochemical measurement. Coil loading was measured each time from a reference sphere at the coil center and the NMR intensity was normalized accordingly. For quantification, the normalized NMR intensity was calibrated using aqueous glycogen solutions ranging from 110 to 1100 μmol glucosyl units/g (n = 14). An in vivo range from 110 to 800 μmol glucosyl units/g wet weight was measured with a highly linear correlation with concentration (r = 0.85, P < 0.001). The in vivo NMR concentration was 0.95 ± 0.05 (mean ± standard error, n = 17) of the concomitant enzymatic measurement of glycogen content. We conclude that the 13C NMR signal of liver glycogen C1 is essentially 100% visible in vivo and that natural abundance 13C NMR spectroscopy can provide reliable noninvasive estimates of in vivo glycogen content over the physiological range of liver glycogen concentrations when using adequate localization and integration procedures

‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal

Heterologous vaccination of BNT162b2 in Ad26. COV2.S-vaccinated healthcare workers elicits long-term humoral immune response

Background. To date, there are no immunological data for the SARS-CoV-2 heterologous vaccination schedule in the South African (SA) population.Objectives. To assess and compare the immunogenicity and reactogenicity of the Jansen Ad26.COV2.S vaccine with the Pfizer/BioNTech- BNT162b2 booster following prime Ad26.COV2.S in 65 SA healthcare workers.Methods. In a prospective, quantitative, cross-sectional trial on individuals &gt;18 years of age vaccinated with a single Ad26.COV2.S dose or single Ad26.COV2.S and a BNT162b2 single-dose/both doses booster, participants filled in a questionnaire on their demographics, type of vaccination, breakthrough infection/s (BTI/s), vaccine reactogenicity, prior SARS-CoV-2 infection and dates of vaccination. Qualitative analysis for presence/absence of anti-S (spike) immunoglobulin G (IgG) was performed using the Euroimmun anti-IgG enzyme-linked immunoassay kit, and anti-S IgG titres were quantitatively assessed using the Abbott IgG Quant II kit.Results. Between 28 October 2021 and 30 November 2021, 65 individuals were enrolled and assigned as either prime Ad26.COV2.S (n=18) or Ad26.COV2.S with a BNT162b2 supplement (n=47) at Charlotte Maxeke Johannesburg Academic Hospital, SA (mean age 45 years (95% confidence interval (CI) 29.5 - 58), 42 women (64.6%) and 23 men (35.4%)). The median IgG titre for the primed Ad26.COV2.S group was 4 272.55 (95% CI 68.40 - 10 351.40) and that for the BNT162b2 supplement group was 7 360.80 (95% CI 4 207.40 - 15 372.60). In the univariate model, the BNT162b2 supplement group showed a significant 1.99 times higher antibody titre factor (95% CI 0.045 - 5.553; p&lt;0.005) than the Ad26.COV2.S group. In both univariate and multivariate models, age, time since prime vaccination, BTI and prior infection failed to show any statistically significant association (p&gt;0.05) with antibody titres in both groups. However, sex (–55.381 (95% CI –76.984 - –13.498; p=0.018) in a multivariate model was found to have a statistically significant association with anti-S IgG titres observed in both groups. Participants who received their first dose of BNT162b2 9 - 10 months after their prime Ad26.COV2.S (n=44) had a higher degree of antibody response than those who received it earlier. Reactogenicity was observed to be manageable, with mild/moderate adverse effects in the study population.Conclusion. A BNT162b2 supplement given in single or two doses as booster in individuals primed with Ad26.COV2.S induced immunological response, with acceptable and manageable reactogenicity. This study provides novel evidence of the highest degree of antibody response in individuals who received a BNT162b2 first dose 9 - 10 months after prime Ad26.COV2.S, implying that a longer time gap between the two vaccines stimulates higher antibody response than a shorter gap, and that this antibody response can persist for as long as 6 months after the last BNT162b2 dose

Detection and assignment of the glucose signal in 1H NMR difference spectra of the human brain

The difference between 1H NMR spectra obtained during eu- and hyperglycemia exhibited well-resolved glucose peaks between 3 and 4 ppm as demonstrated by comparison with solution spectra. Estimated increases were consistent with recent 13C NMR quantitations of intracerebral glucose. Difference spectra were measured in 36-ml volumes from the human brain every 3 min

Adipocyte JAK2 Regulates Hepatic Insulin Sensitivity Independently of Body Composition, Liver Lipid Content, and Hepatic Insulin Signaling.

Disruption of hepatocyte growth hormone (GH) signaling through disruption of Jak2 (JAK2L) leads to fatty liver. Previously, we demonstrated that development of fatty liver depends on adipocyte GH signaling. We sought to determine the individual roles of hepatocyte and adipocyte Jak2 on whole-body and tissue insulin sensitivity and liver metabolism. On chow, JAK2L mice had hepatic steatosis and severe whole-body and hepatic insulin resistance. However, concomitant deletion of Jak2 in hepatocytes and adipocytes (JAK2LA) completely normalized insulin sensitivity while reducing liver lipid content. On high-fat diet, JAK2L mice had hepatic steatosis and insulin resistance despite protection from diet-induced obesity. JAK2LA mice had higher liver lipid content and no protection from obesity but retained exquisite hepatic insulin sensitivity. AKT activity was selectively attenuated in JAK2L adipose tissue, whereas hepatic insulin signaling remained intact despite profound hepatic insulin resistance. Therefore, JAK2 in adipose tissue is epistatic to liver with regard to insulin sensitivity and responsiveness, despite fatty liver and obesity. However, hepatocyte autonomous JAK2 signaling regulates liver lipid deposition under conditions of excess dietary fat. This work demonstrates how various tissues integrate JAK2 signals to regulate insulin/glucose and lipid metabolism

Localized 13C NMR spectroscopy in the human brain of amino acid labeling from D-[1-13C]glucose

Cerebral metabolism of D[1-13C]glucose was studied with localized 13C NMR spectroscopy during intravenous infusion of enriched [1-13C]glucose in four healthy subjects. The use of three-dimensional localization resulted in the complete elimination of triacylglycerol resonance that originated in scalp and subcutaneous fat. The sensitivity and resolution were sufficient to allow 4 min of time-resolved observation of label incorporation into the C3 and C4 resonances of glutamate and C4 of glutamine, as well as C3 of aspartate with lower time resolution. [4-13C]Glutamate labeled rapidly reaching close to maximum labeling at 60 min. The label flow into [3- 13C]glutamate clearly lagged behind that of [4-13C]glutamate and peaked at t = 110-140 min. Multiplets due to homonuclear 13C-13C coupling between the C3 and C4 peaks of the glutamate molecule were observed in vivo. Isotopomer analysis of spectra acquired between 120 and 180 min yielded a 13C isotopic fraction at C4 glutamate of 27 ± 2% (n = 4), which was slightly less than one-half the enrichment of the C1 position of plasma glucose (63 ± 1%), p < 0.05. By comparison with an external standard the total amount of [4-13C]glutamate was directly quantified to be 2.4 ± 0.1 μmol/ml-brain. Together with the isotopomer data this gave a calculated brain glutamate concentration of 9.1 ± 0.7 μmol/ml, which agrees with previous estimates of total brain glutamate concentrations. The agreement suggests that essentially all of the brain glutamate is derived from glucose in healthy human brain