Supersolutions for a class of semilinear heat equations

A semilinear heat equation $u_{t}=\Delta u+f(u)$ with nonnegative initial data in a subset of $L^{1}(\Omega)$ is considered under the assumption that $f$ is nonnegative and nondecreasing and $\Omega\subseteq \R^{n}$. A simple technique for proving existence and regularity based on the existence of supersolutions is presented, then a method of construction of local and global supersolutions is proposed. This approach is applied to the model case $f(s)=s^{p}$, $\phi\in L^{q}(\Omega)$: new sufficient conditions for the existence of local and global classical solutions are derived in the critical and subcritical range of parameters. Some possible generalisations of the method to a broader class of equations are discussed.Comment: Expanded version of the previous submission arXiv:1111.0258v1. 14 page

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.Comment: Minor correction

Effect of hyperfine structure on atomic frequency combs in Pr:YSO

Quantum memory will be a key component in future quantum networks, and atomic frequency combs (AFCs) in rare-earth-doped crystals are one promising platform for realizing this technology. We theoretically and experimentally investigate the formation of AFCs in Pr3+:Y2SiO5, with an overall bandwidth of 120 MHz and tooth spacing ranging from 0.1 MHz to 20 MHz, showing agreement between our calculations and measurements. We observe that the echo efficiency depends crucially on the AFC tooth spacing. Our results suggest approaches to developing a high-efficiency AFC quantum memory.Comment: 20 pages, 7 figure

Association of chronic obstructive pulmonary disease with morbidity and mortality in patients with peripheral artery disease: insights from the EUCLID trial

Background: Patients with chronic obstructive pulmonary disease (COPD) are at increased risk of developing lower extremity peripheral artery disease (PAD) and suffering PAD-related morbidity and mortality. However, the effect and burden of COPD on patients with PAD is less well defined. This post hoc analysis from EUCLID aimed to analyze the risk of major adverse cardiovascular events (MACE) and major adverse limb events (MALE) in patients with PAD and concomitant COPD compared with those without COPD, and to describe the adverse events specific to patients with COPD. Methods: EUCLID randomized 13,885 patients with symptomatic PAD to monotherapy with either ticagrelor or clopidogrel for the prevention of MACE. In this analysis, MACE, MALE, mortality, and adverse events were compared between groups with and without COPD using unadjusted and adjusted Cox proportional hazards model. Results: Of the 13,883 patients with COPD status available at baseline, 11% (n=1538) had COPD. Patients with COPD had a higher risk of MACE (6.02 vs 4.29 events/100 patient-years; p< 0.001) due to a significantly higher risk of myocardial infarction (MI) (3.55 vs 1.85 events/100 patient-years; p< 0.001) when compared with patients without COPD. These risks persisted after adjustment (MACE: adjusted hazard ratio (aHR) 1.30, 95% confidence interval [CI] 1.11– 1.52; p< 0.001; MI: aHR 1.45, 95% CI 1.18– 1.77; p< 0.001). However, patients with COPD did not have an increased risk of MALE or major bleeding. Patients with COPD were more frequently hospitalized for dyspnea and pneumonia (2.66 vs 0.9 events/100 patient-years; aHR 2.77, 95% CI 2.12– 3.63; p< 0.001) and more frequently discontinued study drug prematurely (19.36 vs 12.54 events/100 patient-years; p< 0.001; aHR 1.34, 95% CI 1.22– 1.47; p< 0.001). Conclusion: In patients with comorbid PAD and COPD, the risks of MACE, respiratory-related adverse events, and premature study drug discontinuation were higher when compared with patients without COPD. Registration: ClinicalTrials.gov: NCT01732822

Some results on blow up for semilinear parabolic problems

The authors describe the asymptotic behavior of blow-up for the semilinear heat equation ut=uxx+f(u) in R×(0,T), with initial data u0(x)>0 in R, where f(u)=up, p>1, or f(u)=eu. A complete description of the types of blow-up patterns and of the corresponding blow-up final-time profiles is given. In the rescaled variables, both are governed by the structure of the Hermite polynomials H2m(y). The H2-behavior is shown to be stable and generic. The existence of H4-behavior is proved. A nontrivial blow-up pattern with a blow-up set of nonzero measure is constructed. Similar results for the absorption equation ut=uxx−up, 0<p<1, are discussed

Elderly Patients With Squamous Cell Carcinoma of the Head and Neck and the Benefit of Multimodality Therapy

Limited data are available regarding outcomes in elderly head and neck cancer patients. This retrospective study was designed to characterize head and neck cancer in geriatric patients

Continuity of the blow-up profile with respect to initial data and to the blow-up point for a semilinear heat equation

International audienceWe consider blow-up solutions for semilinear heat equations with Sobolev subcritical power nonlinearity. Given a blow-up point $\hat{a}$, we have from earlier literature, the asymptotic behavior in similarity variables. Our aim is to discuss the stability of that behavior, with respect to perturbations in the blow-up point and in initial data. Introducing the notion of ``profile order", we show that it is upper semicontinuous, and continuous only at points where it is a local minimum

Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables

[EN] Let H-infinity be the set of all ordinary Dirichlet series D = Sigma(n) a(n)(n-1) ann-s representing bounded holomorphic functions on the right half plane. A completely multiplicative sequence (b(n)) of complex numbers is said to be an l(1)-multiplier for H-infinity whenever Sigma(n vertical bar)a(n)b(n vertical bar) < infinity for every D is an element of H-infinity. We study the problem of describing such sequences (b(n)) in terms of the asymptotic decay of the subsequence (b(pj)), where p(j) denotes the j th prime number. Given a completely multiplicative sequence b = (b(n)) we prove (among other results): b is an l(1)-multiplier for H-infinity provided vertical bar b(pj)vertical bar < 1 for all j and (lim(n)) over bar 1/log(n) Sigma(n)(j=1) b(p j)*(2) < 1, and conversely, if b is an l(1)-multiplier for H-infinity, then vertical bar b(pj)vertical bar < 1 for all j and (lim(n)) over bar 1/log(n) Sigma(n)(j=1) b(p j)*(2) <= 1 (here b* stands for the decreasing rearrangement of b). Following an ingenious idea of Harald Bohr it turns out that this problem is intimately related with the question of characterizing those sequences z in the infinite dimensional polydisk D-infinity (the open unit ball of l(infinity)) for which every bounded and holomorphic function f on D-infinity has an absolutely convergent monomial series expansion Sigma(alpha) partial derivative alpha f (0)/alpha! z alpha. Moreover, we study analogous problems in Hardy spaces of Dirichlet series and Hardy spaces of functions on the infinite dimensional polytorus T-infinity.The second, fourth and fifth authors were supported by MINECO and FEDER Project MTM2014-57838-C2-2-P. The fourth author was also supported by PrometeoII/2013/013. The fifth author was also supported by project SP-UPV20120700.Bayart, F.; Defant, A.; Frerick, L.; Maestre, M.; Sevilla Peris, P. (2017). Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables. Mathematische Annalen. 368(1-2):837-876. https://doi.org/10.1007/s00208-016-1511-1S8378763681-2Aleman, A., Olsen, J.-F., Saksman, E.: Fatou and brother Riesz theorems in the infinite-dimensional polydisc. arXiv:1512.01509Balasubramanian, R., Calado, B., Queffélec, H.: The Bohr inequality for ordinary Dirichlet series. 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Understanding Study Drug Discontinuation Through EUCLID

Introduction: Disparities in the care and outcomes of peripheral artery disease (PAD) have been well-established. In part this is due to disparities in enrollment of PAD trial cohorts. However, less attention has been paid to non-random protocol non-adherence after enrollment, which may lead to inaccurate estimates of treatment effects and reduce generalizability of study results. We aimed to ascertain characteristics associated with premature study drug discontinuation in a PAD cohort.Methods: Using data from EUCLID (Examining Use of Ticagrelor in Peripheral Artery Disease), factors associated with study drug discontinuation were assessed using univariable and multivariable Cox proportional hazards models with time to study drug discontinuation as the outcome of interest. Relationships between study drug discontinuation and major adverse cardiovascular events (MACE; cardiovascular death, myocardial infarction, ischemic stroke), major adverse limb events (MALE; acute limb ischemia, major amputation, and lower extremity revascularization), and all-cause hospitalization were assessed.Results: Of 13,842 eligible EUCLID participants, 3,886 (28.1%) prematurely and permanently discontinued study drug over a maximum follow-up of 42 months (annualized rate of 13.2 discontinuations per 100 patient-years). In a multivariable model, premature study drug discontinuation was associated with older age (aHR 1.16, 95%CI 1.14-1.19), eligibility based on prior lower extremity revascularization rather than ABI/TBI criteria (aHR 1.14, 95%CI 1.06-1.23), CLI status (aHR 1.23, 95%CI 1.06-1.42), COPD (aHR 1.36, 95%CI 1.24-1.49), and geographic region. In a multivariable analysis, study drug discontinuation was significantly associated with MACE (aHR 3.27, 95%CI 2.90-3.67, p p Conclusions: This analysis of EUCLID demonstrates that premature, permanent discontinuation of study drug is relatively common in more than a quarter of PAD patients, is unevenly distributed based on geography and other baseline characteristics, and is associated with worse outcomes in a clinical trial context. Study teams leading future PAD trials may want to address the possibility of study drug discontinuation prospectively, as a proactive approach may help investigators to maintain study cohort diversity and representativeness without sacrificing power and precision.</p