Sums of two squares and a power

We extend results of Jagy and Kaplansky and the present authors and show that for all $k\geq 3$ there are infinitely many positive integers $n$, which cannot be written as $x^2+y^2+z^k=n$ for positive integers $x,y,z$, where for $k\not\equiv 0 \bmod 4$ a congruence condition is imposed on $z$. These examples are of interest as there is no congruence obstruction itself for the representation of these $n$. This way we provide a new family of counterexamples to the Hasse principle or strong approximation.Comment: 6 pages, to appear in the memorial volume "From Arithmetic to Zeta-Functions - Number Theory in Memory of Wolfgang Schwarz

Modelling cross-reactivity and memory in the cellular adaptive immune response to influenza infection in the host

The cellular adaptive immune response plays a key role in resolving influenza infection. Experiments where individuals are successively infected with different strains within a short timeframe provide insight into the underlying viral dynamics and the role of a cross-reactive immune response in resolving an acute infection. We construct a mathematical model of within-host influenza viral dynamics including three possible factors which determine the strength of the cross-reactive cellular adaptive immune response: the initial naive T cell number, the avidity of the interaction between T cells and the epitopes presented by infected cells, and the epitope abundance per infected cell. Our model explains the experimentally observed shortening of a second infection when cross-reactivity is present, and shows that memory in the cellular adaptive immune response is necessary to protect against a second infection.Comment: 35 pages, 12 figure

Dialysis initiation, modality choice, access, and prescription: conclusions from a Kidney Disease: Improving Global Outcomes (KDIGO) Controversies Conference

Globally, the number of patients undergoing maintenance dialysis is increasing, yet throughout the world there is significant variability in the practice of initiating dialysis. Factors such as availability of resources, reasons for starting dialysis, timing of dialysis initiation, patient education and preparedness, dialysis modality and access, as well as varied \u201ccountry-specific\u201d factors significantly affect patient experiences and outcomes. As the burden of end-stage kidney disease (ESKD) has increased globally, there has also been a growing recognition of the importance of patient involvement in determining the goals of care and decisions regarding treatment. In January 2018, KDIGO (Kidney Disease: Improving Global Outcomes) convened a Controversies Conference focused on dialysis initiation, including modality choice, access, and prescription. Here we present a summary of the conference discussions, including identified knowledge gaps, areas of controversy, and priorities for research. A major novel theme represented during the conference was the need to move away from a \u201cone-size-fits-all\u201d approach to dialysis and provide more individualized care that incorporates patient goals and preferences while still maintaining best practices for quality and safety. Identifying and including patient-centered goals that can be validated as quality indicators in the context of diverse health care systems to achieve equity of outcomes will require alignment of goals and incentives between patients, providers, regulators, and payers that will vary across health care jurisdictions

Implementing core outcomes in kidney disease: report of the Standardized Outcomes in Nephrology (SONG) implementation workshop

There are an estimated 14,000 randomized trials published in chronic kidney disease. The most frequently reported outcomes are biochemical endpoints, rather than clinical and patient-reported outcomes including cardiovascular disease, mortality, and quality of life. While many trials have focused on optimizing kidney health, the heterogeneity and uncertain relevance of outcomes reported across trials may limit their policy and practice impact. The international Standardized Outcomes in Nephrology (SONG) Initiative was formed to identify core outcomes that are critically important to patients and health professionals, to be reported consistently across trials. We convened a SONG Implementation Workshop to discuss the implementation of core outcomes. Eighty-two patients/caregivers and health professionals participated in plenary and breakout discussions. In this report, we summarize the findings of the workshop in two main themes: socializing the concept of core outcomes, and demonstrating feasibility and usability. We outline implementation strategies and pathways to be established through partnership with stakeholders, which may bolster acceptance and reporting of core outcomes in trials, and encourage their use by end-users such as guideline producers and policymakers to help improve patient-important outcomes