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 $\mathbf{P}$ consisting of finitely or countably many distributional operators $P_n$, 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 $G$ with respect to $L:=\mathbf{P}^{\ast T}\mathbf{P}$ now becomes a conditionally positive definite function. In order to support this claim we ensure that the distributional adjoint operator $\mathbf{P}^{\ast}$ of $\mathbf{P}$ is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function $G$ 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 $s_{f,X}$ to data values sampled from an unknown generalized Sobolev function $f$ at data sites located in some set $X \subset \mathbb{R}^d$. 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 $\mathbf{P}$. 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}

Can sand dunes be used to study historic storm events?

Knowing the long-term frequency of high magnitude storm events that cause coastal inundation is critical for present coastal management, especially in the context of rising sea levels and potentially increasing frequency and severity of storm events. Coastal sand dunes may provide a sedimentary archive of past storm events from which long-term frequencies of large storms can be reconstructed. This study uses novel portable optically stimulated luminescence (POSL) profiles from coastal dunes to reconstruct the sedimentary archive of storm and surge activity for Norfolk, UK. Application of POSL profiling with supporting luminescence ages and particle size analysis to coastal dunes provides not only information of dunefield evolution but also on past coastal storms. In this study, seven storm events, two major, were identified from the dune archive spanning the last 140 years. These appear to correspond to historical reports of major storm surges. Dunes appear to be only recording (at least at the sampling resolution used here) the highest storm levels that were associated with significant flooding. As such the approach seems to hold promise to obtain a better understanding of the frequency of large storms by extending the dune archive records further back to times when documentation of storm surges was sparse

Clinical Science Unique risks for mortality in patients with end-stage renal disease undergoing nonemergent colorectal surgery

Abstract BACKGROUND: The aim of this study was to identify unique risk factors for mortality in patients with end-stage renal disease undergoing nonemergent colorectal surgery. METHODS: A multivariate logistic regression model predicting 30-day mortality was constructed for patients with end-stage renal disease undergoing nonemergent colorectal procedures. Data were obtained from the National Surgical Quality Improvement Program (2005Program ( -2010. RESULTS: Among the 394 patients analyzed, those with serum creatinine levels .7.5 mg/dL had .07 times the adjusted mortality risk of those with levels ,3.5 mg/dL. For colorectal surgery patients, the average serum creatinine level was 5.52 6 2.6 mg/dL, and mortality was 13% (n 5 50). CONCLUSIONS: High serum creatinine was associated with a lower risk for mortality in patients with end-stage renal disease, even though creatinine is often considered a risk factor for surgery. These results show how variables from a patient-centered subpopulation can differ in meaning from the general population. 4 This database is a tool that records perioperative risk factors for addressing surgical morbidity and mortality. Our objective was to use the NSQIP data to create a multivariate model to identify unique risk factors predicting mortality in patients with ESRD undergoing nonemergent colorectal procedures

Instant luminescence chronologies? High resolution luminescence profiles using a portable luminescence reader

Establishing a robust chronology is fundamental to most palaeoenvironmental studies. However, the number and positioning of dated points is critical. Using a portable luminescence reader, it is possible to rapidly generate high resolution down core relative age profiles. Profiles of portable luminescence data from two coastal dunes were evaluated and compared with the results of particle size analysis, stratigraphy, and an independent historical chronology. Results show that, even in young samples, portable luminescence data is dominated by an age related signal which in homogeneous sediment need not be corrected for moisture, feldspar content changes or grain size. Profiles therefore provide relative chronologies from which accumulation phases can be established, and from which better targeted sampling and comparison to other sites could be undertaken. Even though they do not provide instant absolute chronologies, field-based portable luminescence profiling of Late Quaternary sites hold much potential to improve the resultant chronologies

Metastable States in Spin Glasses and Disordered Ferromagnets

We study analytically M-spin-flip stable states in disordered short-ranged Ising models (spin glasses and ferromagnets) in all dimensions and for all M. Our approach is primarily dynamical and is based on the convergence of a zero-temperature dynamical process with flips of lattice animals up to size M and starting from a deep quench, to a metastable limit. The results (rigorous and nonrigorous, in infinite and finite volumes) concern many aspects of metastable states: their numbers, basins of attraction, energy densities, overlaps, remanent magnetizations and relations to thermodynamic states. For example, we show that their overlap distribution is a delta-function at zero. We also define a dynamics for M=infinity, which provides a potential tool for investigating ground state structure.Comment: 34 pages (LaTeX); to appear in Physical Review

Equivalence of 2 effective graft-versus-host disease prophylaxis regimens: Results of a prospective double-blind randomized trial

AbstractWe have previously demonstrated a decrease in the incidence of acute graft-versus-host disease (GVHD) with the addition of methotrexate (MTX) to cyclosporine (CSP) and prednisone (PSE) chemotherapy in patients with leukemia. We have now completed a prospective randomized trial comparing the 3-drug regimen (CSP/MTX/PSE, including 3 doses of MTX) to the standard 2-drug regimen (CSP/MTX, including 4 doses of MTX) to investigate the benefit of PSE used up front for the prevention of acute and chronic GVHD. In the trial, 193 patients were randomized and 186 were included in the final analysis. All patients received a bone marrow graft from a fully histocompatible sibling donor. The preparatory regimen consisted of fractionated total-body irradiation (fTBI) and etoposide in all but 13 patients, who received fTBI and cyclophosphamide. The patients were randomized to receive either CSP/MTX/PSE or CSP/MTX. The 2 groups were well balanced with respect to diagnosis, disease stage, age, donor-recipient sex, and parity. In an intent-to-treat analysis, the incidence of acute GVHD was 18% (95% confidence interval [CI] 12-28) for the CSP/MTX/PSE group compared with 20% (CI 10-26) for the CSP/,MTX group (P = .60), with a median follow up of 2.2 years. Overall survival was 65% for those receiving CSP/MTX/PSE and 72% for those receiving CSP/MTX (P = .10); the relapse rate was 15% for the CSP/MTX/PSE group and 12% for the CSP/MTX group (P = .83). The incidence of chronic GVHD was similar (46% versus 52%; P = .38), with a follow-up of 0.7 to 6.0 years. Of interest, 21 patients went off study due to GVHD (5 in the CSP/MTX/PSE group and 16 in the CSP/MITX group [P = .02]), and 11 patients went off study because of alveolar hemorrhage (3 in the CSP/MTX/PSE group and 8 in the CSP/MTX group [P = .22]). The addition of PSE did not result in a higher incidence of infectious complications, bacterial (66% versus 58%), viral (77% versus 66%), or fungal (20% versus 20%), in those receiving CSP/MTX/PSE versus CSP/MTX, respectively. These data suggest that the addition of PSE was associated with a somewhat lower incidence of early posttransplantation complications but did not have a positive impact on the incidence of acute or chronic GVHD or event-free or overall survival.Biol Blood Marrow Transplant 2000;6(3):254-61