49 research outputs found
Error estimates for interpolation of rough data using the scattered shifts of a radial basis function
The error between appropriately smooth functions and their radial basis
function interpolants, as the interpolation points fill out a bounded domain in
R^d, is a well studied artifact. In all of these cases, the analysis takes
place in a natural function space dictated by the choice of radial basis
function -- the native space. The native space contains functions possessing a
certain amount of smoothness. This paper establishes error estimates when the
function being interpolated is conspicuously rough.Comment: 12 page
Interpolation by piecewise-linear radial basis functions, I
AbstractIn the two-dimensional plane, a set of points x1, x2, …, xn (called “nodes”) is given. It is desired to interpolate arbitrary data given on the nodes by continuous functions having piecewise-linear (“PL”) structure. For this purpose, one can employ the space of all PL-functions on a rectangular grid generated by the nodes. We study this space first. Next, we investigate the special PL-functions that are linear combinations of functions hi(x) = ∥x − xi∥1, in which the l1-norm on R2 is employed. The “dual” case, involving the two-dimensional l∞-norm, is included in our results, as are certain general interpolating functions of the form (s, t) ↦ F(s − si) + G(t − ti)
Potential medicinal value of some South African seaweeds
Eleven macroalgae were collected from the KwaZulu-Natal coast and nineteen species from the cooler Western Cape coast in March and April 2000. Ethanolic and aqueous extracts were made and tested for biological activity in the Cox-1 anti-inflammatory assay, in a nematode mortality bioassay for anthelminthic activity, an IC50 anticancer assay and a MIC antimicrobial assay. The ethanolic extracts were very active in the Cox-1 anti-inflammatory assay for almost all of the species tested. The aqueous extracts were not active. No anthelminthic mortality was detected in extracts from any of the species tested. Many of the extracts had cytotoxic activity against three cancer cell lines tested, with those from representative species of the Chlorophyta and Rhodophyta being the most effective. The extracts had much lower cytotoxic activity when tested on normal mouse fibroblasts (NIH3T3). Extracts from only a few species had antimicrobial activity with those of the Chlorophyta tested being the most effective against both the Gram-positive and Gram-negative bacteria
Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operators
In this paper we introduce a generalized Sobolev space by defining a
semi-inner product formulated in terms of a vector distributional operator
consisting of finitely or countably many distributional operators
, which are defined on the dual space of the Schwartz space. The types of
operators we consider include not only differential operators, but also more
general distributional operators such as pseudo-differential operators. We
deduce that a certain appropriate full-space Green function with respect to
now becomes a conditionally positive
definite function. In order to support this claim we ensure that the
distributional adjoint operator of is
well-defined in the distributional sense. Under sufficient conditions, the
native space (reproducing-kernel Hilbert space) associated with the Green
function can be isometrically embedded into or even be isometrically
equivalent to a generalized Sobolev space. As an application, we take linear
combinations of translates of the Green function with possibly added polynomial
terms and construct a multivariate minimum-norm interpolant to data
values sampled from an unknown generalized Sobolev function at data sites
located in some set . We provide several examples, such
as Mat\'ern kernels or Gaussian kernels, that illustrate how many
reproducing-kernel Hilbert spaces of well-known reproducing kernels are
isometrically equivalent to a generalized Sobolev space. These examples further
illustrate how we can rescale the Sobolev spaces by the vector distributional
operator . Introducing the notion of scale as part of the
definition of a generalized Sobolev space may help us to choose the "best"
kernel function for kernel-based approximation methods.Comment: Update version of the publish at Num. Math. closed to Qi Ye's Ph.D.
thesis (\url{http://mypages.iit.edu/~qye3/PhdThesis-2012-AMS-QiYe-IIT.pdf}
SARS-CoV-2-specific nasal IgA wanes 9 months after hospitalisation with COVID-19 and is not induced by subsequent vaccination
BACKGROUND: Most studies of immunity to SARS-CoV-2 focus on circulating antibody, giving limited insights into mucosal defences that prevent viral replication and onward transmission. We studied nasal and plasma antibody responses one year after hospitalisation for COVID-19, including a period when SARS-CoV-2 vaccination was introduced. METHODS: In this follow up study, plasma and nasosorption samples were prospectively collected from 446 adults hospitalised for COVID-19 between February 2020 and March 2021 via the ISARIC4C and PHOSP-COVID consortia. IgA and IgG responses to NP and S of ancestral SARS-CoV-2, Delta and Omicron (BA.1) variants were measured by electrochemiluminescence and compared with plasma neutralisation data. FINDINGS: Strong and consistent nasal anti-NP and anti-S IgA responses were demonstrated, which remained elevated for nine months (p < 0.0001). Nasal and plasma anti-S IgG remained elevated for at least 12 months (p < 0.0001) with plasma neutralising titres that were raised against all variants compared to controls (p < 0.0001). Of 323 with complete data, 307 were vaccinated between 6 and 12 months; coinciding with rises in nasal and plasma IgA and IgG anti-S titres for all SARS-CoV-2 variants, although the change in nasal IgA was minimal (1.46-fold change after 10 months, p = 0.011) and the median remained below the positive threshold determined by pre-pandemic controls. Samples 12 months after admission showed no association between nasal IgA and plasma IgG anti-S responses (R = 0.05, p = 0.18), indicating that nasal IgA responses are distinct from those in plasma and minimally boosted by vaccination. INTERPRETATION: The decline in nasal IgA responses 9 months after infection and minimal impact of subsequent vaccination may explain the lack of long-lasting nasal defence against reinfection and the limited effects of vaccination on transmission. These findings highlight the need to develop vaccines that enhance nasal immunity. FUNDING: This study has been supported by ISARIC4C and PHOSP-COVID consortia. ISARIC4C is supported by grants from the National Institute for Health and Care Research and the Medical Research Council. Liverpool Experimental Cancer Medicine Centre provided infrastructure support for this research. The PHOSP-COVD study is jointly funded by UK Research and Innovation and National Institute of Health and Care Research. The funders were not involved in the study design, interpretation of data or the writing of this manuscript
Large-scale phenotyping of patients with long COVID post-hospitalization reveals mechanistic subtypes of disease
One in ten severe acute respiratory syndrome coronavirus 2 infections result in prolonged symptoms termed long coronavirus disease (COVID), yet disease phenotypes and mechanisms are poorly understood1. Here we profiled 368 plasma proteins in 657 participants ≥3 months following hospitalization. Of these, 426 had at least one long COVID symptom and 233 had fully recovered. Elevated markers of myeloid inflammation and complement activation were associated with long COVID. IL-1R2, MATN2 and COLEC12 were associated with cardiorespiratory symptoms, fatigue and anxiety/depression; MATN2, CSF3 and C1QA were elevated in gastrointestinal symptoms and C1QA was elevated in cognitive impairment. Additional markers of alterations in nerve tissue repair (SPON-1 and NFASC) were elevated in those with cognitive impairment and SCG3, suggestive of brain–gut axis disturbance, was elevated in gastrointestinal symptoms. Severe acute respiratory syndrome coronavirus 2-specific immunoglobulin G (IgG) was persistently elevated in some individuals with long COVID, but virus was not detected in sputum. Analysis of inflammatory markers in nasal fluids showed no association with symptoms. Our study aimed to understand inflammatory processes that underlie long COVID and was not designed for biomarker discovery. Our findings suggest that specific inflammatory pathways related to tissue damage are implicated in subtypes of long COVID, which might be targeted in future therapeutic trials
An Introduction to Abstract Analysis
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