New Proofs of Davies-Simon's Theorems about Ultracontractivity and Logarithmic Sobolev Inequalities related to Nash Type Inequalities

We present new proofs of two theorems of E.B. Davies and B. Simon about ultracontractivity property for of semigroups of operators and logarithmic Sobolev inequalities with parameter (LSIWP for short) satisfied by the generator of the semigroup. In our proof, we use neither the $L^p$ version of the $LSIWP$ nor Stein's interpolation. Our tool is Nash type inequality. We also present new results. In particular, we show that Nash type inequality and LSIWP are equivalent.Comment: 31 page

Nash type inequalities for fractional powers of non-negative self-adjoint operators

Assuming that a Nash type inequality is satisfied by a non-negative self-adjoint operator $A$, we prove a Nash type inequality for the fractional powers $A^{\alpha}$ of $A$. Under some assumptions, we give ultracontractivity bounds for the semigroup $(T_{t,{\alpha}})$ generated by $-A^{\alpha}$.Comment: January,31 (2002). Submitte

Super-Poincar\'e and Nash-type inequalities for Subordinated Semigroups

We prove that if a super-Poincar\'e inequality is satisfied by an infinitesimal generator $-A$ of a symmetric contracting semigroup then it implies a corresponding super-Poincar\'e inequality for $-g(A)$ with any Bernstein function $g$. We also study the converse statement. We deduce similar results for the Nash-type inequality. Our results applied to fractional powers of $A$ and to $\log(I+A)$ and thus generalize some results of Biroli and Maheux, and Wang 2007. We provide several examples.Comment: submitted. 23p. no figure. Title slightly changed. Results and text improve

Weighted Nash Inequalities

Nash or Sobolev inequalities are known to be equivalent to ultracontractive properties of Markov semigroups, hence to uniform bounds on their kernel densities. In this work we present a simple and extremely general method, based on weighted Nash inequalities, to obtain non-uniform bounds on the kernel densities. Such bounds imply a control on the trace or the Hilbert-Schmidt norm of the heat kernels. We illustrate the method on the heat kernel on \dR naturally associated with the measure with density $C_a\exp(-|x|^a)$, with $

Smoothing Methods for Nonlinear Complementarity Problems

International audienceIn this paper, we present a new smoothing approach to solve general nonlinear complementarity problems. Under the P0 condition on the original problems, we prove some existence and convergence results . We also present an error estimate under a new and general monotonicity condition. The numerical tests confirm the efficiency of our proposed methods

Sexual behaviour of men that consulted in medical outpatient clinics in Western Switzerland from 2005-2006: risk levels unknown to doctors?

<p>Abstract</p> <p>Background</p> <p>To determine male outpatient attenders' sexual behaviours, expectations and experience of talking about their sexuality and sexual health needs with a doctor.</p> <p>Methods</p> <p>A survey was conducted among all male patients aged 18-70, recruited from the two main medical outpatient clinics in Lausanne, Switzerland, in 2005-2006. The anonymous self-administered questionnaire included questions on sexual behaviour, HIV/STI information needs, expectations and experiences regarding discussion of sexual matters with a doctor.</p> <p>Results</p> <p>The response rate was 53.0% (N = 1452). The mean age was 37.7 years. Overall, 13.4% of patients were defined as at STI risk - i.e. having not consistently used condoms with casual partners in the last 6 months, or with a paid partner during the last intercourse - regarding their sexual behaviour in the last year. 90.9% would have liked their physician to ask them questions concerning their sexual life; only 61.4% had ever had such a discussion. The multivariate analysis showed that patients at risk tended to have the following characteristics: recruited from the HIV testing clinic, lived alone, declared no religion, had a low level of education, felt uninformed about HIV/AIDS, were younger, had had concurrent sexual partners in the last 12 months. However they were not more likely to have discussed sexual matters with their doctor than patients not at risk.</p> <p>Conclusion</p> <p>Recording the sexual history and advice on the prevention of the risks of STI should become routine practice for primary health care doctors.</p

Antiinflammatory Therapy with Canakinumab for Atherosclerotic Disease

Background: Experimental and clinical data suggest that reducing inflammation without affecting lipid levels may reduce the risk of cardiovascular disease. Yet, the inflammatory hypothesis of atherothrombosis has remained unproved. Methods: We conducted a randomized, double-blind trial of canakinumab, a therapeutic monoclonal antibody targeting interleukin-1β, involving 10,061 patients with previous myocardial infarction and a high-sensitivity C-reactive protein level of 2 mg or more per liter. The trial compared three doses of canakinumab (50 mg, 150 mg, and 300 mg, administered subcutaneously every 3 months) with placebo. The primary efficacy end point was nonfatal myocardial infarction, nonfatal stroke, or cardiovascular death. RESULTS: At 48 months, the median reduction from baseline in the high-sensitivity C-reactive protein level was 26 percentage points greater in the group that received the 50-mg dose of canakinumab, 37 percentage points greater in the 150-mg group, and 41 percentage points greater in the 300-mg group than in the placebo group. Canakinumab did not reduce lipid levels from baseline. At a median follow-up of 3.7 years, the incidence rate for the primary end point was 4.50 events per 100 person-years in the placebo group, 4.11 events per 100 person-years in the 50-mg group, 3.86 events per 100 person-years in the 150-mg group, and 3.90 events per 100 person-years in the 300-mg group. The hazard ratios as compared with placebo were as follows: in the 50-mg group, 0.93 (95% confidence interval [CI], 0.80 to 1.07; P = 0.30); in the 150-mg group, 0.85 (95% CI, 0.74 to 0.98; P = 0.021); and in the 300-mg group, 0.86 (95% CI, 0.75 to 0.99; P = 0.031). The 150-mg dose, but not the other doses, met the prespecified multiplicity-adjusted threshold for statistical significance for the primary end point and the secondary end point that additionally included hospitalization for unstable angina that led to urgent revascularization (hazard ratio vs. placebo, 0.83; 95% CI, 0.73 to 0.95; P = 0.005). Canakinumab was associated with a higher incidence of fatal infection than was placebo. There was no significant difference in all-cause mortality (hazard ratio for all canakinumab doses vs. placebo, 0.94; 95% CI, 0.83 to 1.06; P = 0.31). Conclusions: Antiinflammatory therapy targeting the interleukin-1β innate immunity pathway with canakinumab at a dose of 150 mg every 3 months led to a significantly lower rate of recurrent cardiovascular events than placebo, independent of lipid-level lowering. (Funded by Novartis; CANTOS ClinicalTrials.gov number, NCT01327846.

UPPER BOUNDS FOR THE BLOW-UP TIME OF THE 2-D PARABOLIC-ELLIPTIC PATLAK-KELLER-SEGEL MODEL OF CHEMOTAXIS

International audienceIn this paper, we obtain upper bounds for the critical time T * of the blowup for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. Such estimates is rarely obtained and discuss in the papers on the subject No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the mass is assumed. The result is expressed not only in terms of the supercritical mass M > 8π, but also in terms of the shape of the initial data

Nash-type inequalities and decay of semigroups of operators

47p.In that paper, we prove an equivalence between Nash-type inequalities and an exponential decay (in the sense of the definition \ref{defent}) for symmetric submarkovian semigroups. This exponential decay generalizes the notion of spectral gap where this number is replaced by a function. We discuss different formulations of the decay associated to the usual Nash inequality in terms of Lyapunov-type functional. We apply this to different classes of ultracontractive semigroups as well as non-ultracontractive semigroups. In particular, we show that any ultracontractive semigroups always satisfy an exponential decay in the sense of \ref{defent}. We treat different classes of examples, one of them containing the Ornstein-Uhlenbeck-type semigroup and ${\Gamma}^*$-semigroup. We apply our results to fractional powers of non-negative self-adjoint semigroup. We derive a simple criterium on the function charaterizing the exponential decay to deduce ultracontractivity property and relations that must satisfy the ultracontractive bounds an heat kernel of the semigroup