On the structure of spaces of vector-valued Lipschitz functions

We analyse the strong connections between spaces of vector-valued Lipschitz functions and spaces of linear continuous operators. We apply these links to study duality, Schur properties and norm attainment in the former class of spaces as well as in their canonical predualsComment: 22 page

A note on the spectrum of Lipschitz operators and composition operators on Lipschitz spaces

Fix a metric space $M$ and let $\mathrm{Lip}_0(M)$ be the Banach space of complex-valued Lipschitz functions defined on $M$. A weighted composition operator on $\mathrm{Lip}_0(M)$ is an operator of the kind $wC_f : g \mapsto w \cdot g \circ f$, where $w : M \to \mathbb C$ and $f: M \to M$ are any map. When such an operator is bounded, it is actually the adjoint operator of a so-called weighted Lipschitz operator $w\widehat{f}$ acting on the Lipschitz-free space $\mathcal F(M)$. In this note, we study the spectrum of such operators, with a special emphasize when they are compact. Notably, we obtain a precise description in the non-weighted $w \equiv 1$ case: the spectrum is finite and made of roots of unity

Injectivity of Lipschitz operators

Any Lipschitz map $f\colon M \to N$ between metric spaces can be "linearised" in such a way that it becomes a bounded linear operator $\widehat{f}\colon \mathcal F(M) \to \mathcal F(N)$ between the Lipschitz-free spaces over $M$ and $N$. The purpose of this note is to explore the connections between the injectivity of $f$ and the injectivity of $\widehat{f}$. While it is obvious that if $\widehat{f}$ is injective then so is $f$, the converse is less clear. Indeed, we pin down some cases where this implication does not hold but we also prove that, for some classes of metric spaces $M$, any injective Lipschitz map $f\colon M \to N$ (for any $N$) admits an injective linearisation. Along our way, we study how Lipschitz maps carry the support of elements in free spaces and also we provide stronger conditions on $f$ which ensure that $\widehat{f}$ is injective

Hubble Space Telescope Observations of the Associated Absorption Systems in Q0122+0338

We have studied a spectrum of Q0122+0338 (z = 1.202) obtained by the FOS on board HST. We present the analysis of three associated absorption systems at z = 1.207, 1.199 and 1.166. The most complex of these at z = 1.207 shows strong absorption from the highly ionized transitions of Lyman alpha, Lyman beta, N V, O VI, Si III, Si IV, and possibly P V. We derive minimal ionic column densities and compare them with those predicted from numerical photoionization models. We find that conditions in the absorbing gas are consistent with an absorber with metallicity twice solar and a total absorbing column density of N(H) = 2*10^19 cm^2. The kinematics of the absorption lines in the z = 1.207 system suggest that a correlation exists between the relative velocity and the creation ionization potential energy for each transition. This is evidence that a complex, multi-component absorber exists. Althought the location of the absorber is uncertain, we consider the origin of this absorption system using the available data and discuss how the high-ionization, and high-metallicity indicate that the absorber may be intrinsic to Q0122+0338.Comment: 16 pages, 5 figures, uses emulateapj.sty, accepted for publication in Ap

The role of unintegrated DNA in HIV infection

Integration of the reverse transcribed viral genome into host chromatin is the hallmark of retroviral replication. Yet, during natural HIV infection, various unintegrated viral DNA forms exist in abundance. Though linear viral cDNA is the precursor to an integrated provirus, increasing evidence suggests that transcription and translation of unintegrated DNAs prior to integration may aid productive infection through the expression of early viral genes. Additionally, unintegrated DNA has the capacity to result in preintegration latency, or to be rescued and yield productive infection and so unintegrated DNA, in some circumstances, may be considered to be a viral reservoir. Recently, there has been interest in further defining the role and function of unintegrated viral DNAs, in part because the use of anti-HIV integrase inhibitors leads to an abundance of unintegrated DNA, but also because of the potential use of non-integrating lentiviral vectors in gene therapy and vaccines. There is now increased understanding that unintegrated viral DNA can either arise from, or be degraded through, interactions with host DNA repair enzymes that may represent a form of host antiviral defence. This review focuses on the role of unintegrated DNA in HIV infection and additionally considers the potential implications for antiviral therapy

COVID-19 trajectories among 57 million adults in England: a cohort study using electronic health records

BACKGROUND: Updatable estimates of COVID-19 onset, progression, and trajectories underpin pandemic mitigation efforts. To identify and characterise disease trajectories, we aimed to define and validate ten COVID-19 phenotypes from nationwide linked electronic health records (EHR) using an extensible framework. METHODS: In this cohort study, we used eight linked National Health Service (NHS) datasets for people in England alive on Jan 23, 2020. Data on COVID-19 testing, vaccination, primary and secondary care records, and death registrations were collected until Nov 30, 2021. We defined ten COVID-19 phenotypes reflecting clinically relevant stages of disease severity and encompassing five categories: positive SARS-CoV-2 test, primary care diagnosis, hospital admission, ventilation modality (four phenotypes), and death (three phenotypes). We constructed patient trajectories illustrating transition frequency and duration between phenotypes. Analyses were stratified by pandemic waves and vaccination status. FINDINGS: Among 57 032 174 individuals included in the cohort, 13 990 423 COVID-19 events were identified in 7 244 925 individuals, equating to an infection rate of 12·7% during the study period. Of 7 244 925 individuals, 460 737 (6·4%) were admitted to hospital and 158 020 (2·2%) died. Of 460 737 individuals who were admitted to hospital, 48 847 (10·6%) were admitted to the intensive care unit (ICU), 69 090 (15·0%) received non-invasive ventilation, and 25 928 (5·6%) received invasive ventilation. Among 384 135 patients who were admitted to hospital but did not require ventilation, mortality was higher in wave 1 (23 485 [30·4%] of 77 202 patients) than wave 2 (44 220 [23·1%] of 191 528 patients), but remained unchanged for patients admitted to the ICU. Mortality was highest among patients who received ventilatory support outside of the ICU in wave 1 (2569 [50·7%] of 5063 patients). 15 486 (9·8%) of 158 020 COVID-19-related deaths occurred within 28 days of the first COVID-19 event without a COVID-19 diagnoses on the death certificate. 10 884 (6·9%) of 158 020 deaths were identified exclusively from mortality data with no previous COVID-19 phenotype recorded. We observed longer patient trajectories in wave 2 than wave 1. INTERPRETATION: Our analyses illustrate the wide spectrum of disease trajectories as shown by differences in incidence, survival, and clinical pathways. We have provided a modular analytical framework that can be used to monitor the impact of the pandemic and generate evidence of clinical and policy relevance using multiple EHR sources. FUNDING: British Heart Foundation Data Science Centre, led by Health Data Research UK