Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure

Local and global behaviour of nonlinear equations with natural growth terms

This paper concerns a study of the pointwise behaviour of positive solutions to certain quasi-linear elliptic equations with natural growth terms, under minimal regularity assumptions on the underlying coefficients. Our primary results consist of optimal pointwise estimates for positive solutions of such equations in terms of two local Wolff's potentials.Comment: In memory of Professor Nigel Kalto

Asymptotic behavior of solutions to the σk\sigma_k-Yamabe equation near isolated singularities

$\sigma_k$-Yamabe equations are conformally invariant equations generalizing the classical Yamabe equation. In an earlier work YanYan Li proved that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotically radially symmetric. In this work we prove that an admissible solution with an isolated singularity at $0\in \mathbb R^n$ to the $\sigma_k$-Yamabe equation is asymptotic to a radial solution to the same equation on $\mathbb R^n \setminus \{0\}$. These results generalize earlier pioneering work in this direction on the classical Yamabe equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli et al, we formulate and prove a general asymptotic approximation result for solutions to certain ODEs which include the case for scalar curvature and $\sigma_k$ curvature cases. An alternative proof is also provided using analysis of the linearized operators at the radial solutions, along the lines of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page

Large expert-curated database for benchmarking document similarity detection in biomedical literature search

Document recommendation systems for locating relevant literature have mostly relied on methods developed a decade ago. This is largely due to the lack of a large offline gold-standard benchmark of relevant documents that cover a variety of research fields such that newly developed literature search techniques can be compared, improved and translated into practice. To overcome this bottleneck, we have established the RElevant LIterature SearcH consortium consisting of more than 1500 scientists from 84 countries, who have collectively annotated the relevance of over 180 000 PubMed-listed articles with regard to their respective seed (input) article/s. The majority of annotations were contributed by highly experienced, original authors of the seed articles. The collected data cover 76% of all unique PubMed Medical Subject Headings descriptors. No systematic biases were observed across different experience levels, research fields or time spent on annotations. More importantly, annotations of the same document pairs contributed by different scientists were highly concordant. We further show that the three representative baseline methods used to generate recommended articles for evaluation (Okapi Best Matching 25, Term Frequency-Inverse Document Frequency and PubMed Related Articles) had similar overall performances. Additionally, we found that these methods each tend to produce distinct collections of recommended articles, suggesting that a hybrid method may be required to completely capture all relevant articles. The established database server located at https://relishdb.ict.griffith.edu.au is freely available for the downloading of annotation data and the blind testing of new methods. We expect that this benchmark will be useful for stimulating the development of new powerful techniques for title and title/abstract-based search engines for relevant articles in biomedical research.Peer reviewe

Difficult-To-Treat Periprosthetic Hip Infection: Outcomes of Debridment

Purpose of the study — to compare management efficiency for difficult-to-treat periprosthetic hip joint infection (PJI) during resection arthroplasty with grafting by vastus lateralis pedicle island flap in comparison with insertion of an antimicrobial-loaded cement spacer. Material and Methods. 132 patients were included into the retrospective study who underwent treatment from 2012 until 2018 including removal of orthopaedic implant, radical surgical debridement of infection focus, resection arthroplasty with grafting by vastus lateralis pedicle island muscle flap (PMF group — 57 patients) or insertion of antibacterial-loaded cement spacer (AMS group — 75 patients). The authors examined medical histories, nature of infection process, infection agent type, laboratory data in respect of systemic inflammation, size of bone defects, follow up status and remission of PJI in the late period. Results. 89.4% of patients (n = 51) who underwent grafting by vastus lateralis pedicle island flap had a history of 3 and more prior surgical procedures in the same area. At the same time the share of such patients in the spacer group was only 38.6% (n = 29) (p<0.0001) while the share of patients with two and more recurrences was 78.9% (n = 45) and 25.3% (n = 19), respectively (p<0.0001). No significant variances were observed between the groups in respect of type composition of PJI microbial infection agents. The infection in a vast majority of patients in both groups was caused by microbial association: 77.2% and 72.0% in PMF and AMS groups, respectively. In the early postoperative period secondary revision of surgical site was performed in 35% and 28% of cases in PMF group (n = 20) and AMS group (n = 21), respectively, including due to recurrent infection in 15.8% and 28% of cases, respectively. Stable remission of difficult-to-treat PJI in PMF group was 96.5% and 45.3% in AMS group. Conclusion. Despite some cases that required secondary revisions in early postoperative period the resection arthroplasty in combination with pedicle muscle flap can be considered a surgery of choice for management of recurrent difficultto-treat PJI with feasible re-implantation of prosthesis against the stable remission of infection