Optimal Sobolev type inequalities in Lorentz spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L p,q : the space D 1,p (R n ) , 1\u2009<\u2009p\u2009<\u2009n, embeds into L p 17 ,q (R n ) , p\u2009 64\u2009q\u2009 64\u2009 1e. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case L p 17 ,p (R n ) . Here, we determine optimal constants for the embedding of the space D 1,p (R n ) , 1\u2009<\u2009p\u2009<\u2009n, into the whole Lorentz space scale L p 17 ,q (R n ) , p\u2009 64\u2009q\u2009 64\u2009 1e, including the limiting case q\u2009=\u2009p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems

Discriminants, symmetrized graph monomials, and sums of squares

Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. In the 20-th century this construction was studied by several authors. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent conjecture of F. Sottile and E. Mukhin on discriminant of the derivative of a univariate polynomial, and an interesting example of P. and A. Lax of a graph with 4 edges whose symmetrized graph monomial is non-negative but not a sum of squares. We present detailed information about symmetrized graph monomials for graphs with four and six edges, obtained by computer calculations

Mortality and Cardiovascular Disease among Older Live Kidney Donors

Over the past two decades, live kidney donation by older individuals (≥55 years) has become more common. Given the strong associations of older age with cardiovascular disease (CVD), nephrectomy could make older donors vulnerable to death and cardiovascular events. We performed a cohort study among older live kidney donors who were matched to healthy older individuals in the Health and Retirement Study. The primary outcome was mortality ascertained through national death registries. Secondary outcomes ascertained among pairs with Medicare coverage included death or CVD ascertained through Medicare claims data. During the period from 1996 to 2006, there were 5717 older donors in the United States. We matched 3368 donors 1:1 to older healthy nondonors. Among donors and matched pairs, the mean age was 59 years; 41% were male and 7% were black race. In median follow-up of 7.8 years, mortality was not different between donors and matched pairs (p = 0.21). Among donors with Medicare, the combined outcome of death/CVD (p = 0.70) was also not different between donors and nondonors. In summary, carefully selected older kidney donors do not face a higher risk of death or CVD. These findings should be provided to older individuals considering live kidney donation

Comparative effectiveness and safety of non-vitamin K antagonists for atrial fibrillation in clinical practice: GLORIA-AF Registry

Background and purpose: Prospectively collected data comparing the safety and effectiveness of individual non-vitamin K antagonists (NOACs) are lacking. Our objective was to directly compare the effectiveness and safety of NOACs in patients with newly diagnosed atrial fibrillation (AF). Methods: In GLORIA-AF, a large, prospective, global registry program, consecutive patients with newly diagnosed AF were followed for 3&nbsp;years. The comparative analyses for (1) dabigatran vs rivaroxaban or apixaban and (2) rivaroxaban vs apixaban were performed on propensity score (PS)-matched patient sets. Proportional hazards regression was used to estimate hazard ratios (HRs) for outcomes of interest. Results: The GLORIA-AF Phase III registry enrolled 21,300 patients between January 2014 and December 2016. Of these, 3839 were prescribed dabigatran, 4015 rivaroxaban and 4505 apixaban, with median ages of 71.0, 71.0, and 73.0&nbsp;years, respectively. In the PS-matched set, the adjusted HRs and 95% confidence intervals (CIs) for dabigatran vs rivaroxaban were, for stroke: 1.27 (0.79–2.03), major bleeding 0.59 (0.40–0.88), myocardial infarction 0.68 (0.40–1.16), and all-cause death 0.86 (0.67–1.10). For the comparison of dabigatran vs apixaban, in the PS-matched set, the adjusted HRs were, for stroke 1.16 (0.76–1.78), myocardial infarction 0.84 (0.48–1.46), major bleeding 0.98 (0.63–1.52) and all-cause death 1.01 (0.79–1.29). For the comparison of rivaroxaban vs apixaban, in the PS-matched set, the adjusted HRs were, for stroke 0.78 (0.52–1.19), myocardial infarction 0.96 (0.63–1.45), major bleeding 1.54 (1.14–2.08), and all-cause death 0.97 (0.80–1.19). Conclusions: Patients treated with dabigatran had a 41% lower risk of major bleeding compared with rivaroxaban, but similar risks of stroke, MI, and death. Relative to apixaban, patients treated with dabigatran had similar risks of stroke, major bleeding, MI, and death. Rivaroxaban relative to apixaban had increased risk for major bleeding, but similar risks for stroke, MI, and death. Registration: URL: https://www.clinicaltrials.gov. Unique identifiers: NCT01468701, NCT01671007. Date of registration: September 2013

Anticoagulant selection in relation to the SAMe-TT2R2 score in patients with atrial fibrillation. the GLORIA-AF registry

Aim: The SAMe-TT2R2 score helps identify patients with atrial fibrillation (AF) likely to have poor anticoagulation control during anticoagulation with vitamin K antagonists (VKA) and those with scores &gt;2 might be better managed with a target-specific oral anticoagulant (NOAC). We hypothesized that in clinical practice, VKAs may be prescribed less frequently to patients with AF and SAMe-TT2R2 scores &gt;2 than to patients with lower scores. Methods and results: We analyzed the Phase III dataset of the Global Registry on Long-Term Oral Antithrombotic Treatment in Patients with Atrial Fibrillation (GLORIA-AF), a large, global, prospective global registry of patients with newly diagnosed AF and ≥1 stroke risk factor. We compared baseline clinical characteristics and antithrombotic prescriptions to determine the probability of the VKA prescription among anticoagulated patients with the baseline SAMe-TT2R2 score &gt;2 and ≤ 2. Among 17,465 anticoagulated patients with AF, 4,828 (27.6%) patients were prescribed VKA and 12,637 (72.4%) patients an NOAC: 11,884 (68.0%) patients had SAMe-TT2R2 scores 0-2 and 5,581 (32.0%) patients had scores &gt;2. The proportion of patients prescribed VKA was 28.0% among patients with SAMe-TT2R2 scores &gt;2 and 27.5% in those with scores ≤2. Conclusions: The lack of a clear association between the SAMe-TT2R2 score and anticoagulant selection may be attributed to the relative efficacy and safety profiles between NOACs and VKAs as well as to the absence of trial evidence that an SAMe-TT2R2-guided strategy for the selection of the type of anticoagulation in NVAF patients has an impact on clinical outcomes of efficacy and safety. The latter hypothesis is currently being tested in a randomized controlled trial. Clinical trial registration: URL: https://www.clinicaltrials.gov//Unique identifier: NCT01937377, NCT01468701, and NCT01671007

A Moser-type inequality in Lorentz-Sobolev spaces for unbounded domains in R^N

We derive a Pohoaev-Trudinger type embedding for the Lorentz-Sobolev space W1"0LN,q(\u3a9), for general domains \u3a9 86RN and in particular for \u3a9=RN. Precisely, we first prove that the corresponding inequality is domain independent and then, by constructing explicit concentrating sequences \ue0 la Moser, we establish that the embedding inequality is sharp and we exhibit the best constant

Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth

We consider a Kirchhoff type elliptic problem; {-(1 + alpha integral(Omega) vertical bar del u vertical bar(2)dx) Delta u = f(x, u), u >= 0 in Omega, u = 0 on partial derivative Omega, where Omega subset of R-2 is a bounded domain with a smooth boundary partial derivative Omega, alpha > 0 and f is a continuous function in (Omega) over bar x R. Moreover, we assume f has the Trudinger-Moser growth. We prove the existence of solutions of (P), so extending a former result by de Figueiredo-Miyagaki-Ruf [11] for the case alpha = 0 to the case alpha > 0. We emphasize that we also show a new multiplicity result induced by the nonlocal dependence. In order to prove this, we carefully discuss the geometry of the associated energy functional and the concentration compactness analysis for the critical case

Schr\uf6dinger\u2013Newton equations in dimension two via a Pohozaev\u2013Trudinger log-weighted inequality

We study the following Choquard type equation in the whole plane (C)-\u394u+V(x)u=(I2 17F(x,u))f(x,u),x 08R2where I2 is the Newton logarithmic kernel, V is a bounded Schr\uf6dinger potential and the nonlinearity f(x,&nbsp;u), whose primitive in u vanishing at zero is F(x,&nbsp;u), exhibits the highest possible growth which is of exponential type. The competition between the logarithmic kernel and the exponential nonlinearity demands for new tools. A proper function space setting is provided by a new weighted version of the Pohozaev\u2013Trudinger inequality which enables us to prove the existence of variational, in particular finite energy solutions to (C)