A restricted signature normal form for Hermitian matrices, quasi-spectral decompositions, and applications

In recent years, a number of results on the relationships between the inertias of Hermitian matrices and the inertias of their principal submatrices appeared in the literature. We study restricted congruence transformation of Hermitian matrices M which, at the same time, induce a congruence transformation of a given principal submatrix A of M. Such transformations lead to concept of the restricted signature normal form of M. In particular, by means of this normal form, we obtain short proofs of most of the known inertia theorems and also derive some new results of this type. For some applications, a special class of almost unitary restricted congruence transformations turns out to be useful. We show that, with such transformations, M can be reduced to a quasi-diagonal form which, in particular, displays the eigenvalues of A. Finally, applications of this quasi-spectral decomposition to generalize inverses and Hermitian matrix pencils are discussed

An extension problem for H-unitary matrices with applications to Hermitian Toeplitz matrices

AbstractGiven a Hermitian matrix H, a matrix U is said to be H-unitary if UHHU = H. We consider the following extension problem: If U0 is a rectangular matrix such that UH0HU0 = A, where A is a leading principal submatrix of H, can U0 be extended to an H-unitary matrix? After presenting necessary conditions for a more general situation, we state a necessary and sufficient criterion for this problem and give a description of all its solutions. Finally, these results are used to derive some properties of factorizations of Hermitian Toeplitz matrices

Final technical report : The International Alcohol Control (IAC) Study - Working Meeting, Torino, Italy, 7th and 8th June 2014

This is part of a funding proposal report. Funding received was used to support costs associated with the preparation, facilitation and hosting of the IAC study meeting in Torino, Italy. In addition the financial support enabled three low and middle income (LMIC) researchers and one member of the New Zealand IAC study team to attend the working meeting. IAC survey data was shared from participating countries; cross-country analyses were discussed and planned; and methodological issues encountered by the different research teams were discussed

The Swift Ultra-Violet/Optical Telescope

The UV/Optical Telescope (UVOT) is one of three instruments flying aboard the Swift Gamma-ray Observatory. It is designed to capture the early (approximately 1 minute) UV and optical photons from the afterglow of gamma-ray bursts in the 170-600 nm band as well as long term observations of these afterglows. This is accomplished through the use of UV and optical broadband filters and grisms. The UVOT has a modified Ritchey-Chretien design with micro-channel plate intensified charged-coupled device detectors that record the arrival time of individual photons and provide sub-arcsecond positioning of sources. We discuss some of the science to be pursued by the UVOT and the overall design of the instrument.Comment: 55 Pages, 28 Figures, To be published in Space Science Review

Effect of Hydrocortisone on Mortality and Organ Support in Patients With Severe COVID-19: The REMAP-CAP COVID-19 Corticosteroid Domain Randomized Clinical Trial.

Importance: Evidence regarding corticosteroid use for severe coronavirus disease 2019 (COVID-19) is limited. Objective: To determine whether hydrocortisone improves outcome for patients with severe COVID-19. Design, Setting, and Participants: An ongoing adaptive platform trial testing multiple interventions within multiple therapeutic domains, for example, antiviral agents, corticosteroids, or immunoglobulin. Between March 9 and June 17, 2020, 614 adult patients with suspected or confirmed COVID-19 were enrolled and randomized within at least 1 domain following admission to an intensive care unit (ICU) for respiratory or cardiovascular organ support at 121 sites in 8 countries. Of these, 403 were randomized to open-label interventions within the corticosteroid domain. The domain was halted after results from another trial were released. Follow-up ended August 12, 2020. Interventions: The corticosteroid domain randomized participants to a fixed 7-day course of intravenous hydrocortisone (50 mg or 100 mg every 6 hours) (n = 143), a shock-dependent course (50 mg every 6 hours when shock was clinically evident) (n = 152), or no hydrocortisone (n = 108). Main Outcomes and Measures: The primary end point was organ support-free days (days alive and free of ICU-based respiratory or cardiovascular support) within 21 days, where patients who died were assigned -1 day. The primary analysis was a bayesian cumulative logistic model that included all patients enrolled with severe COVID-19, adjusting for age, sex, site, region, time, assignment to interventions within other domains, and domain and intervention eligibility. Superiority was defined as the posterior probability of an odds ratio greater than 1 (threshold for trial conclusion of superiority >99%). Results: After excluding 19 participants who withdrew consent, there were 384 patients (mean age, 60 years; 29% female) randomized to the fixed-dose (n = 137), shock-dependent (n = 146), and no (n = 101) hydrocortisone groups; 379 (99%) completed the study and were included in the analysis. The mean age for the 3 groups ranged between 59.5 and 60.4 years; most patients were male (range, 70.6%-71.5%); mean body mass index ranged between 29.7 and 30.9; and patients receiving mechanical ventilation ranged between 50.0% and 63.5%. For the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively, the median organ support-free days were 0 (IQR, -1 to 15), 0 (IQR, -1 to 13), and 0 (-1 to 11) days (composed of 30%, 26%, and 33% mortality rates and 11.5, 9.5, and 6 median organ support-free days among survivors). The median adjusted odds ratio and bayesian probability of superiority were 1.43 (95% credible interval, 0.91-2.27) and 93% for fixed-dose hydrocortisone, respectively, and were 1.22 (95% credible interval, 0.76-1.94) and 80% for shock-dependent hydrocortisone compared with no hydrocortisone. Serious adverse events were reported in 4 (3%), 5 (3%), and 1 (1%) patients in the fixed-dose, shock-dependent, and no hydrocortisone groups, respectively. Conclusions and Relevance: Among patients with severe COVID-19, treatment with a 7-day fixed-dose course of hydrocortisone or shock-dependent dosing of hydrocortisone, compared with no hydrocortisone, resulted in 93% and 80% probabilities of superiority with regard to the odds of improvement in organ support-free days within 21 days. However, the trial was stopped early and no treatment strategy met prespecified criteria for statistical superiority, precluding definitive conclusions. Trial Registration: ClinicalTrials.gov Identifier: NCT02735707

Genomic reconstruction of the SARS-CoV-2 epidemic in England.

The evolution of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) virus leads to new variants that warrant timely epidemiological characterization. Here we use the dense genomic surveillance data generated by the COVID-19 Genomics UK Consortium to reconstruct the dynamics of 71 different lineages in each of 315 English local authorities between September 2020 and June 2021. This analysis reveals a series of subepidemics that peaked in early autumn 2020, followed by a jump in transmissibility of the B.1.1.7/Alpha lineage. The Alpha variant grew when other lineages declined during the second national lockdown and regionally tiered restrictions between November and December 2020. A third more stringent national lockdown suppressed the Alpha variant and eliminated nearly all other lineages in early 2021. Yet a series of variants (most of which contained the spike E484K mutation) defied these trends and persisted at moderately increasing proportions. However, by accounting for sustained introductions, we found that the transmissibility of these variants is unlikely to have exceeded the transmissibility of the Alpha variant. Finally, B.1.617.2/Delta was repeatedly introduced in England and grew rapidly in early summer 2021, constituting approximately 98% of sampled SARS-CoV-2 genomes on 26 June 2021

Iterative methods for ill-conditioned Toeplitz Matrices

. In this paper we study the use of the Sine Transform for preconditioning linear Toeplitz systems. We consider Toeplitz matrices with a real generating function that is nonnegative with only a small number of zeros. Then we can define a preconditioner of the form S n S n where S n is the matrix describing the discrete Sine transform and is a diagonal matrix. If we have full knowledge about f then we can show that the preconditioned system is of bounded condition number independly of n. We can obtain the same result for the case that we know only the position and order of the zeros of f . If we only know the matrix and its coefficients t j , we present Sine transform preconditioners that show in many examples the same numerical behaviour. Key Words. Toeplitz matrix, Sine Transform, preconditioned conjugate gradient method AMS(MOS) Subject Classifications. 65F10,65N06 Iterative methods for ill-conditioned Toeplitz matrices 1. Introduction We are concerned with the numerical solutio..

Iterative Methods for Toeplitz-like Matrices

. In this paper we will give a survey on iterative methods for solving linear equations with Toeplitz matrices. We introduce a new class of Toeplitz matrices for which clustering of eigenvalues and singular values can be proved. We consider optimal (!)- circulant preconditioners as a generalization of the circulant preconditioner. For positive definite Toeplitz matrices, especially in the real case, there is a hard competition between the fast, superfast, and iterative solvers. Therefore, it is necessary to get optimal implementations of the iterative solver. We will show different ways to get improved preconditioned conjugate gradient algorithms, and compare the number of flops for the three concurrent methods. Furthermore, we show different ways to deal with nearsingular Toeplitz matrices. Key Words. Toeplitz matrix, Fourier Transform, preconditioned conjugate gradient method AMS(MOS) Subject Classifications. 65F10,65N06 0. Introduction We consider linear equations of the form Tn..