Remainder terms in the fractional Sobolev inequality

We show that the fractional Sobolev inequality for the embedding $\H \hookrightarrow L^{\frac{2N}{N-s}}(\R^N)$, $s \in (0,N)$ can be sharpened by adding a remainder term proportional to the distance to the set of optimizers. As a corollary, we derive the existence of a remainder term in the weak $L^{\frac{N}{N-s}}$-norm for functions supported in a domain of finite measure. Our results generalize earlier work for the non-fractional case where $s$ is an even integer.Comment: 13 page

CAM use among overweight and obese persons with radiographic knee osteoarthritis

BACKGROUND: Obesity is associated with knee pain and is an independent predictor of incident knee osteoarthritis (OA); increased pain with movement often leads patients to adopt sedentary lifestyles to avoid pain. Detailed descriptions of pain management strategies by body mass index (BMI) level among OA patients are lacking. The objectives were to describe complementary and alternative medicine (CAM) and conventional medication use by BMI level and identify correlates of CAM use by BMI level. METHODS: Using Osteoarthritis Initiative baseline data, 2,675 patients with radiographic tibiofemoral OA in at least one knee were identified. Use of CAM therapies and conventional medications was determined by interviewers. Potential correlates included SF-12, CES-D, Western Ontario and McMaster Universities Osteoarthritis Index, and Knee injury and Osteoarthritis Outcome Score quality of life. Multinomial logistic regression models adjusting for sociodemographic and clinical factors provided estimates of the association between BMI levels and treatment use; binary logistic regression identified correlates of CAM use. RESULTS: BMI was inversely associated with CAM use (45% users had BMI \u3e/=35 kg/m(2); 54% had BMI /m(2)), but positively associated with conventional medication use (54% users had BMI \u3e/=35 kg/m(2); 35.1% had BMI /m(2)). Those with BMI \u3e/=30 kg/m(2) were less likely to use CAM alone or in combination with conventional medications when compared to patients with BMI /m(2). CONCLUSIONS: CAM use is common among people with knee OA but is inversely associated with BMI. Understanding ways to further symptom management in OA among overweight and obese patients is warranted

Racial differences in symptom management approaches among persons with radiographic knee osteoarthritis

Background The extent to which racial differences exist in use of treatments for osteoarthritis (OA) is debatable. The purpose of this study was to describe the differences between African Americans (AA) and Caucasian Americans (CA) in using treatment approaches to manage symptoms among individuals with radiographic-confirmed knee OA. Methods A cross-sectional study was conducted. Using data from the Osteoarthritis Initiative, we identified 508 AA and 2,075 CA with radiographic tibiofemoral OA in at least one knee. Trained interviewers asked questions relating to current OA treatments including seven CAM therapy categories—alternative medical systems, mind-body interventions, manipulation and body-based methods, energy therapies, and three types of biologically based therapies, as well as conventional medications. We categorized participants as: conventional medication only users, CAM only users, users of both and users of neither. Multinomial logistic regression models adjusting for sociodemographics and clinical/functional factors provided estimates of the association between race and treatment use. Results Overall, 16.5% of AA and 24.2% of CA exclusively used CAM to treat OA, 25.0% of AA and 23.8% of CA used CAM in conjunction with conventional medications, and 24.8% of AA and 14.6% of CA exclusively used conventional medications. After control for sociodemographic and clinical factors, AA were less likely than CA to use CAM therapies alone (adjusted odds ratio (OR) of using CAM alone relative to no CAM or conventional treatments: 0.68, 95% confidence interval (CI): 0.48–0.96) or with conventional medications (adjusted OR relative to no CAM or conventional treatments: 0.59, 95%CI: 0.42–0.83). However, no differences in use of conventional medications alone were observed after adjustment of covariates. Conclusion CAM use is common among people with knee OA, but is less likely to be used by AA relative to CA. For effective CAM therapies, targeted outreach to underserved populations including education about benefits of various CAM treatments and providing accessible care may attenuate observed disparities in effective CAM use by race

Families of superhard crystalline carbon allotropes induced via cold-compressed graphite and nanotubes

We report a general scheme to systematically construct two classes of structural families of superhard sp3 carbon allotropes of cold compressed graphite through the topological analysis of odd 5+7 or even 4+8 membered carbon rings stemmed from the stacking of zigzag and armchair chains. Our results show that the previously proposed M, bct-C4, W and Z allotropes belong to our currently proposed families and that depending on the topological arrangement of the native carbon rings numerous other members are found that can help us understand the structural phase transformation of cold-compressed graphite and carbon nanotubes (CNTs). In particular, we predict the existence of two simple allotropes, R- and P-carbon, which match well the experimental X-ray diffraction patterns of cold-compressed graphite and CNTs, respectively, display a transparent wide-gap insulator ground state and possess a large Vickers hardness comparable to diamond.Comment: 5 pages, 4 figures, accepted by Phys. Rev. Let

High pressure evolution of Fe2_{2}O3_{3} electronic structure revealed by X-ray absorption

We report the first high pressure measurement of the Fe K-edge in hematite (Fe$_2$O$_3$) by X-ray absorption spectroscopy in partial fluorescence yield geometry. The pressure-induced evolution of the electronic structure as Fe$_2$O$_3$ transforms from a high-spin insulator to a low-spin metal is reflected in the x-ray absorption pre-edge. The crystal field splitting energy was found to increase monotonically with pressure up to 48 GPa, above which a series of phase transitions occur. Atomic multiplet, cluster diagonalization, and density-functional calculations were performed to simulate the pre-edge absorption spectra, showing good qualitative agreement with the measurements. The mechanism for the pressure-induced phase transitions of Fe$_2$O$_3$ is discussed and it is shown that ligand hybridization significantly reduces the critical high-spin/low-spin pressure.Comment: 5 pages, 4 figures and 1 tabl

Long-term Effects of Use of Prescription Non-steroidal Anti-inflammatory Agents on Symptoms and Disease Progression among Patients with Radiographically Confirmed Osteoarthritis of the Knee

Objective: To estimate the extent to which long-term use of prescription non-steroidal anti-inflammatory agents (NSAIDs) relieve symptoms and delay disease progression among patients with radiographically confirmed osteoarthritis (OA) of the knee. Methods: Using Osteoarthritis Initiative data, we identified participants with confirmed OA at enrollment and evaluated changes in symptoms measured using the Western Ontario and McMaster Universities Arthritis Index, WOMAC (n=1,846) and joint space width measured using serial x-rays and a customized software tool (n=1,116) over 4 years. Covariates included sociodemographics, OA clinical characteristics, indices of general health status, body mass index, and use of other treatments. We adjusted for baseline and time-varying confounders using marginal structural modeling. Results: Six percent initiated NSAID treatment at year one, with half of the initiators being regular users. After adjusting for time-varying confounders with marginal structural models, we found that compared to participants who never reported use of prescription NSAIDs, those reporting use for 3 years had on average 0.88 point decrease (95% Confidence Interval (CI): -0.46 to 2.22) in WOMAC Pain, 0.72 point decrease (95% CI: -0.12 to 1.56) in WOMAC Stiffness, 4.27 points decrease (95% CI: 0.31 to -8.84) in WOMAC Function, and 0.28mm increase (95% CI: -0.06 to 0.62) in joint space width. Conclusions: Long term NSAID use was associated with a priori defined minimally important clinical improvements in stiffness, function and structural degeneration, but not in pain

InGaAs Quantum Well Grown on High-Index Surfaces for Superluminescent Diode Applications

The morphological and optical properties of In0.2Ga0.8As/GaAs quantum wells grown on various substrates are investigated for possible application to superluminescent diodes. The In0.2Ga0.8As/GaAs quantum wells are grown by molecular beam epitaxy on GaAs (100), (210), (311), and (731) substrates. A broad photoluminescence emission peak (~950 nm) with a full width at half maximum (FWHM) of 48 nm is obtained from the sample grown on (210) substrate at room temperature, which is over four times wider than the quantum well simultaneously grown on (100) substrate. On the other hand, a very narrow photoluminescence spectrum is observed from the sample grown on (311) with FWHM = 7.8 nm. The results presented in this article demonstrate the potential of high-index GaAs substrates for superluminescent diode applications