Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles

Given a simple graph $G=(V,E)$, a subset of $E$ is called a triangle cover if it intersects each triangle of $G$. Let $\nu_t(G)$ and $\tau_t(G)$ denote the maximum number of pairwise edge-disjoint triangles in $G$ and the minimum cardinality of a triangle cover of $G$, respectively. Tuza conjectured in 1981 that $\tau_t(G)/\nu_t(G)\le2$ holds for every graph $G$. In this paper, using a hypergraph approach, we design polynomial-time combinatorial algorithms for finding small triangle covers. These algorithms imply new sufficient conditions for Tuza's conjecture on covering and packing triangles. More precisely, suppose that the set $\mathscr T_G$ of triangles covers all edges in $G$. We show that a triangle cover of $G$ with cardinality at most $2\nu_t(G)$ can be found in polynomial time if one of the following conditions is satisfied: (i) $\nu_t(G)/|\mathscr T_G|\ge\frac13$, (ii) $\nu_t(G)/|E|\ge\frac14$, (iii) $|E|/|\mathscr T_G|\ge2$. Keywords: Triangle cover, Triangle packing, Linear 3-uniform hypergraphs, Combinatorial algorithm

Distance statistics in quadrangulations with a boundary, or with a self-avoiding loop

We consider quadrangulations with a boundary and derive explicit expressions for the generating functions of these maps with either a marked vertex at a prescribed distance from the boundary, or two boundary vertices at a prescribed mutual distance in the map. For large maps, this yields explicit formulas for the bulk-boundary and boundary-boundary correlators in the various encountered scaling regimes: a small boundary, a dense boundary and a critical boundary regime. The critical boundary regime is characterized by a one-parameter family of scaling functions interpolating between the Brownian map and the Brownian Continuum Random Tree. We discuss the cases of both generic and self-avoiding boundaries, which are shown to share the same universal scaling limit. We finally address the question of the bulk-loop distance statistics in the context of planar quadrangulations equipped with a self-avoiding loop. Here again, a new family of scaling functions describing critical loops is discovered.Comment: 55 pages, 14 figures, final version with minor correction

Prevention of falls and fractures in old people by administration of calcium and vitamin d. randomized clinical trial

<p>Abstract</p> <p>Background</p> <p>There are many studies that associate vitamin D serum levels in older persons with muscle strength, physical performance and risk of fractures and falls. However, current evidence is insufficient to make a general recommendation for administrating calcium and vitamin D to older persons. The objective of this study is to determine the effectiveness of calcium and vitamin D supplementation in improving musculoskeletal function and decreasing the number of falls in person aged over 65 years.</p> <p>Methods/Design</p> <p>Phase III, randomized, double blind, placebo-controlled trial to evaluate the efficacy of already marketed drugs in a new indication. It will be performed at Primary Care doctor visits at several Healthcare Centers in different Spanish Health Areas. A total of 704 non-institutionalized subjects aged 65 years or older will be studied (sample size calculated for a statistical power of 80%, alpha error 0.05, annual incidence of falls 30% and expected reduction of 30% to 20% and expected loss to follow up of 20%). The test drug containing 800 IU of vitamin D and 1000 mg of calcium will be administered daily. The control group will receive a placebo. The subjects will be followed up over two years. The primary variable will be the incidence of spontaneous falls. The secondary variables will include: consequences of the falls (fractures, need for hospitalization), change in calcidiol plasma levels and other analytical determinations (transaminases, PTH, calcium/phosphorous, albumin, creatinine, etc.), change in bone mass by densitometry, change in muscle strength in the dominant hand and change in musculoskeletal strength, risk factors for falls, treatment compliance, adverse effects and socio-demographic data.</p> <p>Discussion</p> <p>The following principles have been considered in the development of this Project: the product data are sufficient to ensure that the risks assumed by the study participants are acceptable, the study objectives will probably provide further knowledge on the problem studied and the available information justifies the performance of the study and its possible risk for the participants.</p> <p>If calcium and vitamin D supplementation is effective in the prevention of falls and fractures in the elderly population, a recommendation may be issued with the aim of preventing some of the consequences of falls that affect quality of life and the ensuing personal, health and social costs.</p> <p>Trial Registration</p> <p>ClinicalTrials.gov: <a href="http://www.clinicaltrials.gov/ct2/show/NCT01452243">NCT01452243</a></p> <p>Clinical trial authorized by the Spanish Medicines Agency: EudraCT number 2006-001643-63.</p

Optimal Use of Vitamin D When Treating Osteoporosis

Inadequate serum 25-hydroxyvitamin D (25[OH]D) concentrations are associated with muscle weakness, decreased physical performance, and increased propensity in falls and fractures. This paper discusses several aspects with regard to vitamin D status and supplementation when treating patients with osteoporosis in relation to risks and prevention of falls and fractures. Based on evidence from literature, adequate supplementation with at least 700 IU of vitamin D, preferably cholecalciferol, is required for improving physical function and prevention of falls and fractures. Additional calcium supplementation may be considered when dietary calcium intake is below 700 mg/day. For optimal bone mineral density response in patients treated with antiresorptive or anabolic therapy, adequate vitamin D and calcium supplementation is also necessary. Monitoring of 25(OH)D levels during follow-up and adjustment of vitamin D supplementation should be considered to reach and maintain adequate serum 25(OH)D levels of at least 50 nmol/L, preferably greater than 75 nmol/L in all patients

Efficacy of different doses and time intervals of oral vitamin D supplementation with or without calcium in elderly nursing home residents

Summary: The effect of equivalent oral doses of vitamin D3 600 IU/day, 4200 IU/week and 18,000 IU/month on vitamin D status was compared in a randomized clinical trial in nursing home residents. A daily dose was more effective than a weekly dose, and a monthly dose was the least effective. Introduction: It is assumed that equivalent daily, weekly or monthly doses of vitamin D3 equally influence vitamin D status. This was investigated in a randomized clinical trial in nursing home residents. Methods: The study was performed in ten nursing homes including 338 subjects (76 male and 262 female), with a mean age of 84 (± SD 6.3 years). They received oral vitamin D3 either 600 IU/day, or 4200 IU/week, or 18,000 IU/month or placebo. After 4 months, calcium was added during 2 weeks, 320 mg/day or 640 mg/day or placebo. Outcome: serum levels of 25-hydroxyvitamin D (25(OH)D), parathyroid hormone (PTH) and bone turnover markers. Statistical approach: linear multilevel analysis. Results: At baseline, mean serum 25(OH)D was 25.0 nmol/L (SD 10.9), and in 98%, it was lower than 50 nmol/L. After 4 months, mean serum 25(OH)D levels increased to 62.5 nmol/L (after daily vitamin D3 69.9 nmol/L, weekly 67.2 nmol/L and monthly 53.1 nmol/L, P<0.001 between groups). Median serum PTH levels decreased by 23% (p<0.001). Bone turnover markers did not decrease. Calcium supplementation had no effect on serum PTH and bone turnover. Conclusion: Daily vitamin D was more effective than weekly, and monthly administration was the least effective. © 2007 International Osteoporosis Foundation and National Osteoporosis Foundation

Patient preference and acceptability of calcium plus vitamin D3 supplementation: a randomised, open, cross-over trial

Preference for a drug formulation is important in adherence to long-term medication for chronic illnesses such as osteoporosis. We investigated the preference for and acceptability of chewable tablet containing calcium and vitamin D (Calci Chew D3, Nycomed) compared to that of a sachet containing calcium and vitamin D3 (Cad, Will-Pharma). This open, randomised, cross-over trial was set up to compare the preference and acceptability of two calcium plus vitamin D3 formulations (both with 500 mg calcium and 400/440 IU vitamin D3), given twice a day in patients with osteoporosis. Preference and acceptability were assessed by means of questionnaires. Preference was determined by asking the question, which treatment the patient preferred, and acceptability was measured by scoring five variables, using rating scales. Of the 102 patients indicating a preference for a trial medication, 67% preferred the chewable tablet, 19% the sachet with calcium and vitamin D3, and 15% stated no preference. The significant preference for Calci Chew D3 (p < 0.0001) was associated with higher scores for all five acceptability variables. The two formulations were tolerated equally well. A significant greater number of patients considered the chewable tablet as preferable and acceptable to the sachet, containing calcium and vitamin D3. Trial registration: Current Controlled Trials ISRCTN18822358

Vitamin D and Systemic Lupus Erythematosus: Bones, Muscles, and Joints

Vitamin D3, or cholecalciferol, is the naturally occurring form of vitamin D that is converted in the skin and hydroxylated in the liver and kidney to the active form found in humans. The main role for vitamin D is calcium homeostasis, and low levels of vitamin D result in lower gastrointestinal absorption of calcium. Vitamin D is also critical for mineralization of bone tissue, muscle function, and coordination. Recent studies have found prevention of bone mass loss and reduction in falls and fractures in patients supplemented with vitamin D. A high percentage of systemic lupus erythematosus patients are reported to have insufficient or deficient levels of vitamin D. This paper reviews the biology of vitamin D, its role in calcium homeostasis, and how it contributes to the maintenance of bone, muscle, and joint function in older adults and individuals with systemic lupus erythematosus