Semidefinite Approximations of the Matrix Logarithm

© 2018, SFoCM. The matrix logarithm, when applied to Hermitian positive definite matrices, is concave with respect to the positive semidefinite order. This operator concavity property leads to numerous concavity and convexity results for other matrix functions, many of which are of importance in quantum information theory. In this paper we show how to approximate the matrix logarithm with functions that preserve operator concavity and can be described using the feasible regions of semidefinite optimization problems of fairly small size. Such approximations allow us to use off-the-shelf semidefinite optimization solvers for convex optimization problems involving the matrix logarithm and related functions, such as the quantum relative entropy. The basic ingredients of our approach apply, beyond the matrix logarithm, to functions that are operator concave and operator monotone. As such, we introduce strategies for constructing semidefinite approximations that we expect will be useful, more generally, for studying the approximation power of functions with small semidefinite representations

Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix $X$ formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose $X$ into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points $v_1,v_2,...,v_n\in \R^k$ (where $n > k$) determine whether there is a centered ellipsoid passing \emph{exactly} through all of the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a subspace $U$ that ensures any positive semidefinite matrix $L$ with column space $U$ can be recovered from $D+L$ for any diagonal matrix $D$ using a convex optimization-based heuristic known as minimum trace factor analysis. This result leads to a new understanding of the structure of rank-deficient correlation matrices and a simple condition on a set of points that ensures there is a centered ellipsoid passing through them.Comment: 20 page

Superconducting double transition and substantial Knight shift in Sr2RuO₄

Recent nuclear magnetic resonance experiments measuring the Knight shift in Sr2RuO4 have challenged the widely accepted picture of chiral pairing in this superconductor. Here we study the implications of helical pairing on the superconducting state while comparing our results with the available experimental data on the upper critical field and Knight shift. We solve the Bogoliubov–de Gennes equation employing a realistic three-dimensional tight-binding model that captures the experimental Fermi surface very well. In agreement with experiments we find a Pauli limiting to the upper critical field and, at low temperatures and high fields, a second superconducting transition. These transitions, which form a superconducting subphase in the H-T phase diagram are first-order in nature and merge into a single second-order transition at a bicritical point (T∗,H∗),for which we find (0.8 K, 2.4 T) with experiment reporting (0.8 K,∼1.2 T) [Phys. Rev. B93, 184513 (2016)]. Furthermore, we find a substantial drop in the Knight shift in agreement with recent experiments

Impact of infectious diseases consultation on the management of Staphylococcus aureus bacteraemia in children.

OBJECTIVES: Infectious diseases consultation (IDC) in adults with Staphylococcus aureus bacteraemia (SAB) has been shown to improve management and outcome. The aim of this study was to evaluate the impact of IDC on the management of SAB in children. STUDY DESIGN: Observational cohort study of children with SAB. SETTING: Cambridge University Hospitals National Health Service (NHS) Foundation Trust, a large acute NHS Trust in the UK. PARTICIPANTS: All children with SAB admitted to the Cambridge University Hospitals NHS Foundation Trust between 16 July 2006 and 31 December 2012. METHODS: Children with SAB between 2006 and 31 October 2009 were managed by routine clinical care (pre-IDC group) and data were collected retrospectively by case notes review. An IDC service for SAB was introduced in November 2009. All children with SAB were reviewed regularly and data were collected prospectively (IDC group) until 31 December 2012. Baseline characteristics, quality metrics and outcome were compared between the pre-IDC group and IDC group. RESULTS: There were 66 episodes of SAB in 63 children-28 patients (30 episodes) in the pre-IDC group, and 35 patients (36 episodes) in the IDC group. The median age was 3.4 years (IQR 0.2-10.7 years). Patients in the IDC group were more likely to have echocardiography performed, a removable focus of infection identified and to receive a longer course of intravenous antimicrobial therapy. There were no differences in total duration of antibiotic therapy, duration of hospital admission or outcome at 30 or 90 days following onset of SAB. CONCLUSIONS: IDC resulted in improvements in the investigation and management of SAB in children.This work was supported by grants from the UK Clinical Research Collaboration (UKCRC) Translational Infection Research Initiative (TIRI); the Medical Research Council (G1000803), with contributions from the Biotechnology and Biological Sciences Research Council, the National Institute for Health Research (NIHR) on behalf of the UK Department of Health, and the Chief Scientist of the Scottish Government Health Directorate; the Public Health England; and the NIHR Cambridge Biomedical Research Centre

Molecular and Epigenetic Mechanisms Underlying Cognitive and Adaptive Responses to Stress

Consolidation of contextual memories after a stressful encounter is essential for the survival of an organism and in allowing a more appropriate response to be elicited should the perceived threat reoccur. Recent evidence has explored the complex role that epigenetic mechanisms play in the formation of such memories, and the underlying signaling pathways are becoming more apparent. The glucocorticoid receptor (GR) has been shown to play a key role in these events having both genomic and non-genomic actions in the brain. GR has been shown to interact with the extracellular signal-regulated kinase mitogen-activated protein kinase (ERK MAPK) signaling pathway which, in concert, drives epigenetic modifications and chromatin remodeling, resulting in gene induction and memory consolidation. Evidence indicates that stressful events can have an effect on the offspring in utero, and that epigenetic marks altered early in life may persist into adulthood. A new and controversial area of research, however, suggests that epigenetic modifications could be inherited through the germline, a concept known as transgenerational epigenetics. This review explores the role that epigenetic processes play in the central nervous system, specifically in the consolidation of stress-induced memories, the concept of transgenerational epigenetic inheritance, and the potential role of epigenetics in revolutionizing the treatment of stress-related disorders through the emerging field of pharmacoepigenetics and personalized medical treatment

Rifampicin and clarithromycin (extended release) versus rifampicin and streptomycin for limited Buruli ulcer lesions: a randomised, open-label, non-inferiority phase 3 trial.

BACKGROUND: Buruli ulcer is a neglected tropical disease caused by Mycobacterium ulcerans infection that damages the skin and subcutis. It is most prevalent in western and central Africa and Australia. Standard antimicrobial treatment with oral rifampicin 10 mg/kg plus intramuscular streptomycin 15 mg/kg once daily for 8 weeks (RS8) is highly effective, but streptomycin injections are painful and potentially harmful. We aimed to compare the efficacy and tolerability of fully oral rifampicin 10 mg/kg plus clarithromycin 15 mg/kg extended release once daily for 8 weeks (RC8) with that of RS8 for treatment of early Buruli ulcer lesions. METHODS: We did an open-label, non-inferiority, randomised (1:1 with blocks of six), multicentre, phase 3 clinical trial comparing fully oral RC8 with RS8 in patients with early, limited Buruli ulcer lesions. There were four trial sites in hospitals in Ghana (Agogo, Tepa, Nkawie, Dunkwa) and one in Benin (Pobè). Participants were included if they were aged 5 years or older and had typical Buruli ulcer with no more than one lesion (caterories I and II) no larger than 10 cm in diameter. The trial was open label, and neither the investigators who took measurements of the lesions nor the attending doctors were masked to treatment assignment. The primary clinical endpoint was lesion healing (ie, full epithelialisation or stable scar) without recurrence at 52 weeks after start of antimicrobial therapy. The primary endpoint and safety were assessed in the intention-to-treat population. A sample size of 332 participants was calculated to detect inferiority of RC8 by a margin of 12%. This study was registered with ClinicalTrials.gov, NCT01659437. FINDINGS: Between Jan 1, 2013, and Dec 31, 2017, participants were recruited to the trial. We stopped recruitment after 310 participants. Median age of participants was 14 years (IQR 10-29) and 153 (52%) were female. 297 patients had PCR-confirmed Buruli ulcer; 151 (51%) were assigned to RS8 treatment, and 146 (49%) received oral RC8 treatment. In the RS8 group, lesions healed in 144 (95%, 95% CI 91 to 98) of 151 patients, whereas lesions healed in 140 (96%, 91 to 99) of 146 patients in the RC8 group. The difference in proportion, -0·5% (-5·2 to 4·2), was not significantly greater than zero (p=0·59), showing that RC8 treatment is non-inferior to RS8 treatment for lesion healing at 52 weeks. Treatment-related adverse events were recorded in 20 (13%) patients receiving RS8 and in nine (7%) patients receiving RC8. Most adverse events were grade 1-2, but one (1%) patient receiving RS8 developed serious ototoxicity and ended treatment after 6 weeks. No patients needed surgical resection. Four patients (two in each study group) had skin grafts. INTERPRETATION: Fully oral RC8 regimen was non-inferior to RS8 for treatment of early, limited Buruli ulcer and was associated with fewer adverse events. Therefore, we propose that fully oral RC8 should be the preferred therapy for early, limited lesions of Buruli ulcer. FUNDING: WHO with additional support from MAP International, American Leprosy Missions, Fondation Raoul Follereau France, Buruli ulcer Groningen Foundation, Sanofi-Pasteur, and BuruliVac

Polynomial-sized Semidefinite Representations of Derivative Relaxations of Spectrahedral Cones

We give explicit polynomial-sized (in $n$ and $k$) semidefinite representations of the hyperbolicity cones associated with the elementary symmetric polynomials of degree $k$ in $n$ variables. These convex cones form a family of non-polyhedral outer approximations of the non-negative orthant that preserve low-dimensional faces while successively discarding high-dimensional faces. More generally we construct explicit semidefinite representations (polynomial-sized in $k,m$, and $n$) of the hyperbolicity cones associated with $k$th directional derivatives of polynomials of the form $p(x) = \det(\sum_{i=1}^{n}A_i x_i)$ where the $A_i$ are $m\times m$ symmetric matrices. These convex cones form an analogous family of outer approximations to any spectrahedral cone. Our representations allow us to use semidefinite programming to solve the linear cone programs associated with these convex cones as well as their (less well understood) dual cones.Comment: 20 pages, 1 figure. Minor changes, expanded proof of Lemma

The Role of Innate APOBEC3G and Adaptive AID Immune Responses in HLA-HIV/SIV Immunized SHIV Infected Macaques

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