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Consider a Conservation Law and a Hamilton-Jacobi equation with a flux/Hamiltonian depending also on the space variable. We characterize first the attainable set of the two equations and, second, the set of initial data evolving at a prescribed time into a prescribed profile. An explicit example then shows the deep differences between the cases of x-independent and x-dependent fluxes/Hamiltonians. [ABSTRACT FROM AUTHOR]

Health-related quality of life (HRQoL) represents one of the most concerning aspects for cancer patients. The Healthy Eating Index (HEI) is an a priori diet quality index directly associated with health outcomes and HRQoL in cancer survivors in North American populations. We evaluated, in a Mediterranean population, the baseline associations between HEI-2015 and HRQoL in 492 women with breast cancer recruited in a DEDiCa lifestyle trial. Dietary data were obtained from 7-day food records; HRQoL was assessed through the European Organisation for Research and Treatment of Cancer Quality of Life Questionnaire Core 30 (EORTC QLQ C30) and the C30 Summary Score (SumSc). Analysis of variance and multivariable linear and log-gamma regression models were performed. Mean and standard deviation for HEI-2015 score was 68.8 ± 11.2; SumSc was 81.5 ± 12.9. Women with lower HEI-2015 score had higher BMI, were more frequently exposed to tobacco smoke and had fewer years of education. Patients with a HEI-2015 score greater than 68.7 (median value) showed a significant increase in SumSc of 4% ( p = 0.02). HEI-2015 components also associated with SumSc were beans and greens (β = 1.04; p = 0.02). Weak associations were found for total vegetables and saturated fats. Higher diet quality in breast cancer survivors was associated with higher overall HRQoL in this cross-sectional analysis.

Objective: To report cross-sectionally serum levels of 25-hydroxyvitamin D [25(OH)D] in women living in Italy within 12 months from breast cancer (BC) diagnosis., Methods: Baseline data were obtained from 394 women diagnosed with primary BC, enrolled from 2016 to 2019 in a lifestyle trial conducted in Italy. Subjects' characteristics were compared between two 25(OH)D concentrations (hypovitaminosis D<20 and ≥20 ng/mL) with the Chi-squared test or Fisher's exact test for small-expected counts. Using multiple logistic regression-adjusted models, we estimated odds ratios (ORs) of hypovitaminosis D with 95% confidence intervals (CIs) in the total sample and in the unsupplemented subgroup., Results: Hypovitaminosis D was found in 39% of all subjects, 60% in unsupplemented subjects, and 10% in supplemented subjects. Increasing ORs of hypovitaminosis D were found with increasing body mass index, 25-30, >30, and ≥35 versus <25 kg/m2 (ORs: 2.50, 4.64, and 5.81, respectively, in the total cohort and ORs: 2.68, 5.38, and 7.08 in the unsupplemented); living in the most southern Italian region (OR 2.50, 95%CI 1.22-5.13); and with hypertriglyceridemia (OR 2.46; 95%CI 1.16-5.22), chemotherapy history (OR 1.86, 95%CI 1.03-3.38), and inversely with anti-estrogenic therapy (OR 0.43, 95%CI 0.24-0.75) in the total sample., Conclusions: Hypovitaminosis D in women recently diagnosed with BC and participating in a lifestyle trial in Italy was widespread and highest with obesity, hypertriglyceridemia, and chemotherapy use. Considering that hypovitaminosis D is a risk factor for lower efficacy of bone density treatments and possibly BC mortality, our results suggest the need to promptly address and treat vitamin D deficiency.