On the wave operators for the critical nonlinear Schr\"odinger equation

8 pages. The published version contains extra remarks.International audienceWe prove that for the mass critical nonlinear Schrodinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one-dimensional case, we show a precise similarity between the mass critical nonlinear Schrodinger equation and a nonlinear Schrodinger equation of derivative type

Efficacy of thalidomide in a girl with inflammatory calcinosis, a severe complication of juvenile dermatomyositis

We report a 14-year-old girl with juvenile dermatomyositis (JDM) complicated by severe inflammatory calcinosis successfully treated with thalidomide. She was diagnosed as JDM when she was 4 years old after a few months of increasing lethargy, muscle pain, muscle weakness, and rash. During three months, clinical manifestations and abnormal laboratory findings were effectively treated with oral prednisolone. However, calcinosis was recognized 18 months after disease onset. Generalized calcinosis rapidly progressed with high fever, multiple skin/subcutaneous inflammatory lesions, and increased level of CRP. Fifty mg/day (1.3 mg/kg day) of oral thalidomide was given for the first four weeks, and then the dose was increased to 75 mg/day. Clinical manifestations subsided, and inflammatory markers had clearly improved. Frequent high fever and local severe pain with calcinosis were suppressed. The levels of FDP-E, IgG, and tryglyceride, which were all elevated before the thalidomide treatment, were gradually returned to the normal range. Over the 18 months of observation up to the present, she has had no inflammatory calcinosis, or needed any hospitalization, although established calcium deposits still remain. Her condition became painless, less extensive and less inflammatory with the CRP level below 3.08 mg/dL. Recent examination by whole-body 18F-FDG-PET-CT over the 15 months of thalidomide treatment demonstrated fewer hot spots around the subcutaneous calcified lesions

On the wave operators for the critical nonlinear Schrödinger equation

We prove that for the L2-critical nonlinear Schrödinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the onedimensional case, we show a precise similarity between the L2-critical nonlinear Schr¨odinger equation and a nonlinear Schr¨odinger equation of derivative type

A Nonlinear Poisson Formula for The Schrödinger Operator

We prove a nonlinear Poisson type formula for the Schrödinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, H1-critical nonlinearities are allowed

Empagliflozin in Patients with Chronic Kidney Disease

Background The effects of empagliflozin in patients with chronic kidney disease who are at risk for disease progression are not well understood. The EMPA-KIDNEY trial was designed to assess the effects of treatment with empagliflozin in a broad range of such patients. Methods We enrolled patients with chronic kidney disease who had an estimated glomerular filtration rate (eGFR) of at least 20 but less than 45 ml per minute per 1.73 m(2) of body-surface area, or who had an eGFR of at least 45 but less than 90 ml per minute per 1.73 m(2) with a urinary albumin-to-creatinine ratio (with albumin measured in milligrams and creatinine measured in grams) of at least 200. Patients were randomly assigned to receive empagliflozin (10 mg once daily) or matching placebo. The primary outcome was a composite of progression of kidney disease (defined as end-stage kidney disease, a sustained decrease in eGFR to < 10 ml per minute per 1.73 m(2), a sustained decrease in eGFR of & GE;40% from baseline, or death from renal causes) or death from cardiovascular causes. Results A total of 6609 patients underwent randomization. During a median of 2.0 years of follow-up, progression of kidney disease or death from cardiovascular causes occurred in 432 of 3304 patients (13.1%) in the empagliflozin group and in 558 of 3305 patients (16.9%) in the placebo group (hazard ratio, 0.72; 95% confidence interval [CI], 0.64 to 0.82; P < 0.001). Results were consistent among patients with or without diabetes and across subgroups defined according to eGFR ranges. The rate of hospitalization from any cause was lower in the empagliflozin group than in the placebo group (hazard ratio, 0.86; 95% CI, 0.78 to 0.95; P=0.003), but there were no significant between-group differences with respect to the composite outcome of hospitalization for heart failure or death from cardiovascular causes (which occurred in 4.0% in the empagliflozin group and 4.6% in the placebo group) or death from any cause (in 4.5% and 5.1%, respectively). The rates of serious adverse events were similar in the two groups. Conclusions Among a wide range of patients with chronic kidney disease who were at risk for disease progression, empagliflozin therapy led to a lower risk of progression of kidney disease or death from cardiovascular causes than placebo