Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations

We show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{ u^{p}\sts{y}}{\pabs{\,x-y\,}{\lambda}} }\diff{y}\, u^{p-1}\sts{x} =0,\quad x\in \R^N \end{equation*} where $u(x)$ is a real-valued function, $N\geq 3$, $0<\lambda<N$, $p=\frac{2N-\lambda}{N-2}$ and {\bm\alpha}\sts{N,\lambda} is a constant. It generalizes the results for the whole range $0<\lambda<N$ in \cite{DY2019dcds, GWY2020na, LTX2021, MWX:Hartree} and confirms an open nondegeneracy problem in \cite{GMYZ2022cvpde}. Firstly, by the stereographic projection and sharp Hardy-Littlewood-Sobolev inequality on the sphere $\S^N$ in \cite{FL2012}, we give an alternative proof of the existence of the extremizer of sharp Hardy-Littlewood-Sobolev inequality in $\R^N$ without use of the rearrangement inequalities in \cite{lieb2001analysis}, which is related to the existence of positive bubble solutions of (NLH). Secondly, by making use of the Green function, we obtain an integral form in $\R^N$ of the corresponding linearized equation around positive bubble solutions under suitable decay condition, and its equivalent integral form on the sphere $\S^N$ via the stereographic projection. Lastly, together with the key spherical harmonic decomposition and the Funk-Hecke formula of the spherical harmonic functions in \cite{AH2012, DX2013book, SteinW:Fourier anal}, we can obtain the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equation (NLH), which is inspired by Frank and Lieb in \cite{FL2012am,FL2012}.Comment: 26 pages. Any comment is welcom

Dynamic analysis and control of strip mill vibration under the coupling effect of roll and rolled piece

According to the “Hill rolling force formula”, taking particular account of the influence from horizontal vibration of rolled piece in roll gap, a dynamic rolling force model is analyzed. Considering the interaction between vibration of strip and roll, the dynamic vibration model of rolling mill is established. On this basis, the time delayed feedback is introduced to control the vibration of the roll system. The amplitude frequency response of the coupled vibration control equation is obtained by using the multiple scales method. Different time delay parameters are selected to test the control effect. Research results show that the unstable vibration of the roll system can be suppressed with appropriate time delay feedback parameters. Because it is simpler and has good control effect in solving nonlinear mechanical vibration, so these results will make a difference for the research of strip mill vibration, and provide theoretical basis for strip steel production

Weight control behaviors in overweight/obese U.S. adults with diagnosed hypertension and diabetes

<p>Abstract</p> <p>Background</p> <p>Obesity is a major risk factor for development and progression of hypertension and diabetes, which often coexist in obese patients. Losing weight by means of energy restriction and physical activity has been effective in preventing and managing these diseases. However, weight control behaviors among overweight/obese adults with these conditions are poorly understood.</p> <p>Methods</p> <p>Using self-reported data from 143,386 overweight/obese participants (aged ≥ 18 years) in the 2003 Behavioral Risk Factor Surveillance System, we examined the proportion of overweight/obese adults who tried to lose weight and their weight control strategies by hypertension and/or diabetes status.</p> <p>Results</p> <p>Among all participants, 58% of those with hypertension, 60% of those with diabetes, and 72% of those with both diseases tried to lose weight, significantly higher than the 50% of those with neither condition (Bonferroni corrected P < 0.017 for all comparisons). The multivariate-adjusted odds ratio (AOR) for trying to lose weight was 1.11 (95% confidence interval [CI]: 1.05–1.17) in participants with hypertension, 1.02 (95% CI: 0.90–1.15) in participants with diabetes, and 1.18 (95% CI: 1.07–1.29) in participants with both diseases (participants with neither condition as the referent). Among 78,446 participants who tried to lose weight, 23% of those with hypertension only and 28% of those with both hypertension and diabetes reported adopting a low fat/low calorie (LF/LC) diet in controlling their weight, significantly higher than 19% of those with neither disease (Bonferroni corrected P < 0.017 for all comparisons). Participants with both diseases had a significantly lower percentage of adopting physical activity in controlling their weight than those with neither condition (6% versus 12%, P < 0.01). After multivariate adjustment, the AOR for adopting a LF/LC diet plus physical activity to lose weight was 1.46 (95% CI: 1.15–1.84) in participants with both diseases. The AOR for adopting a LF/LC diet only to lose weight was 1.72 (95% CI: 1.35–2.20) in participants with both diseases and was 1.21 (95% CI: 1.03–1.40) in participants with hypertension only.</p> <p>Conclusion</p> <p>The proportion of overweight/obese patients with diagnosed hypertension and/or diabetes who attempted to lose weight remains suboptimal and the weight control strategies varied significantly among these patients.</p

Factors Associated with Vitamin D Deficiency and Inadequacy among Women of Childbearing Age in the United States

Objective. To examine the prevalence and correlates of vitamin D deficiency and inadequacy among US women of childbearing age. Methods. Data from 1,814 female participants (20–44 y) in the 2003–2006 NHANES were analyzed to estimate the age-adjusted prevalence and prevalence ratios with 95% confidence intervals (CIs) for vitamin D deficiency (defined as serum 25-hydroxyvitamin D [25(OH)D] <12.0 ng/mL) and inadequacy (defined as 25(OH)D: 12.0–<20.0 ng/mL). Results. The age-adjusted prevalence was 11.1% (95% CI: 8.8–14.0%) for vitamin D deficiency and 25.7% (95% CI: 22.3–29.5%) for vitamin D inadequacy. Race/ethnicity other than non-Hispanic white and obesity were associated with increased risks, whereas dietary supplement use, milk consumption of ≥1 time/day, and potential sunlight exposure during May-October were associated with decreased risks for both vitamin D deficiency and inadequacy (P < 0.05). Current smoking and having histories of diabetes and cardiovascular disease were also associated with an increased risk for vitamin D deficiency (P < 0.05). Conclusions. Among women of childbearing age, periconceptional intervention programs may focus on multiple risk factors for vitamin D deficiency and inadequacy to ultimately improve their vitamin D nutrition

Investigation of Tribological Performances for Porous Structure of Diatom Frustule with FSI Method

Tribological performances of the diatom frustule are investigated with the liquid-solid interaction (FSI) method. Take, for example, the representative Coscinodiscus sp. shell; the diatom frustule with the porous structure is achieved by the scanning electron microscope (SEM). Based on the frustule, the representative diatom frustule is modeled. Further, tribological performances of the diatom at its different geometry sizes and velocities are solved with FSI method and compared with corresponding values for the nonporous structure. The numerical result shows that the existence of the porous structure of the diatom helps to reduce friction between it and ambient water and to increase its load-carrying capacity

Serum Non-high-density lipoprotein cholesterol concentration and risk of death from cardiovascular diseases among U.S. adults with diagnosed diabetes: the Third National Health and Nutrition Examination Survey linked mortality study

<p>Abstract</p> <p>Background</p> <p>Non-high-density lipoprotein cholesterol (non-HDL-C) measures all atherogenic apolipoprotein B-containing lipoproteins and predicts risk of cardiovascular diseases (CVD). The association of non-HDL-C with risk of death from CVD in diabetes is not well understood. This study assessed the hypothesis that, among adults with diabetes, non-HDL-C may be related to the risk of death from CVD.</p> <p>Methods</p> <p>We analyzed data from 1,122 adults aged 20 years and older with diagnosed diabetes who participated in the Third National Health and Nutrition Examination Survey linked mortality study (299 deaths from CVD according to underlying cause of death; median follow-up length, 12.4 years).</p> <p>Results</p> <p>Compared to participants with serum non-HDL-C concentrations of 35 to 129 mg/dL, those with higher serum levels had a higher risk of death from total CVD: the RRs were 1.34 (95% CI: 0.75-2.39) and 2.25 (95% CI: 1.30-3.91) for non-HDL-C concentrations of 130-189 mg/dL and 190-403 mg/dL, respectively (<it>P </it>= 0.003 for linear trend) after adjustment for demographic characteristics and selected risk factors. In subgroup analyses, significant linear trends were identified for the risk of death from ischemic heart disease: the RRs were 1.59 (95% CI: 0.76-3.32) and 2.50 (95% CI: 1.28-4.89) (<it>P </it>= 0.006 for linear trend), and stroke: the RRs were 3.37 (95% CI: 0.95-11.90) and 5.81 (95% CI: 1.96-17.25) (<it>P </it>= 0.001 for linear trend).</p> <p>Conclusions</p> <p>In diabetics, higher serum non-HDL-C concentrations were significantly associated with increased risk of death from CVD. Our prospective data support the notion that reducing serum non-HDL-C concentrations may be beneficial in the prevention of excess death from CVD among affected adults.</p

Use of folic acid and vitamin supplementation among adults with depression and anxiety: a cross-sectional, population-based survey

<p>Abstract</p> <p>Background</p> <p>Evidence suggests that folate deficiency may be causatively linked to depressive symptoms. However, little is known on the status of use of folic acid and vitamin supplements among people with mental disorders. This study examined the prevalence and the likelihood of use of folic acid or vitamin supplements among adults with depression and anxiety in comparison to those without these conditions.</p> <p>Methods</p> <p>Using data from 46, 119 participants (aged ≥ 18 years) in the 2006 Behavioral Risk Factor Surveillance System survey, we estimated the adjusted prevalence and odds ratios with 95% confidence intervals for taking folic acid and vitamin supplements among those with ever diagnosed depression (n = 8, 019), ever diagnosed anxiety (n = 5, 546) or elevated depressive symptoms (n = 3, 978, defined as having a depression severity score of ≥ 10 on the Patient Health Questionnaire-8 diagnostic algorithm).</p> <p>Results</p> <p>Overall, women were more likely than men to take folic acid supplements 1-4 times/day (50.2% versus 38.7%, P < 0.001) and vitamin supplements (62.5% versus 49.8%, P < 0.001). After multivariate adjustment, men with ever diagnosed depression or anxiety were 42% and 83%, respectively, more likely to take folic acid supplements < 1 time/day; 44% and 39%, respectively, more likely to take folic acid supplements 1-4 times/day; and 40% and 46%, respectively, more likely to take vitamin supplements compared to men without these conditions (P < 0.05 for all comparisons). Women with ever diagnosed depression were 13% more likely to take folic acid supplements 1-4 times/day and 15% more likely to take vitamin supplements than women without this condition (P < 0.05 for both comparisons). Use of folic acid and vitamin supplements did not differ significantly by elevated depressive symptoms in either sex.</p> <p>Conclusion</p> <p>The prevalence and the likelihood of taking folic acid and vitamin supplements varied substantially by a history of diagnosed depression among both men and women and by a history of diagnosed anxiety among men, but not by presence of elevated depressive symptoms in either sex.</p

Global well-posedness for Schr\"odinger equation with derivative in H1/2(R)H^{{1/2}}(\R)

In this paper, we consider the Cauchy problem of the cubic nonlinear Schr\"{o}dinger equation with derivative in $H^s(\R)$. This equation was known to be the local well-posedness for $s\geq \frac12$ (Takaoka,1999), ill-posedness for $s<\frac12$ (Biagioni and Linares, 2001, etc.) and global well-posedness for $s>\frac12$ (I-team, 2002). In this paper, we show that it is global well-posedness in H^{1/2(\R). The main approach is the third generation I-method combined with some additional resonant decomposition technique. The resonant decomposition is applied to control the singularity coming from the resonant interaction.Comment: 31pages; In this version, we change some expressions in Englis