459 research outputs found
Connectivity in bridge-addable graph classes: the McDiarmid-Steger-Welsh conjecture
A class of graphs is bridge-addable if given a graph G in the class, any graph obtained by adding an edge between two connected components of G is also in the class. We prove a conjecture of McDiarmid, Steger, and Welsh, that says that if is any bridge-addable class of graphs on n vertices, and is taken uniformly at random from , then is connected with probability at least , when n tends to infinity. This lower bound is asymptotically best possible since it is reached for forests. Our proof uses a “local double counting” strategy that may be of independent interest, and that enables us to compare the size of two sets of combinatorial objects by solving a related multivariate optimization problem. In our case, the optimization problem deals with partition functions of trees relative to a supermultiplicative functional.Postprint (author's final draft
Sufficient Conditions for Tuza's Conjecture on Packing and Covering Triangles
Given a simple graph , a subset of is called a triangle cover if
it intersects each triangle of . Let and denote the
maximum number of pairwise edge-disjoint triangles in and the minimum
cardinality of a triangle cover of , respectively. Tuza conjectured in 1981
that holds for every graph . 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 of triangles covers all edges in . We
show that a triangle cover of with cardinality at most can be
found in polynomial time if one of the following conditions is satisfied: (i)
, (ii) , (iii)
.
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
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