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

Confluence of geodesic paths and separating loops in large planar quadrangulations

We consider planar quadrangulations with three marked vertices and discuss the geometry of triangles made of three geodesic paths joining them. We also study the geometry of minimal separating loops, i.e. paths of minimal length among all closed paths passing by one of the three vertices and separating the two others in the quadrangulation. We concentrate on the universal scaling limit of large quadrangulations, also known as the Brownian map, where pairs of geodesic paths or minimal separating loops have common parts of non-zero macroscopic length. This is the phenomenon of confluence, which distinguishes the geometry of random quadrangulations from that of smooth surfaces. We characterize the universal probability distribution for the lengths of these common parts.Comment: 48 pages, 33 color figures. Final version, with one concluding paragraph and one reference added, and several other small 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