A graph is called radially maximal if it is not complete and the addition of any new edge decreases its radius. Harary and Thomassen [‘Anticritical graphs’, Math. Proc. Cambridge Philos. Soc.79(1) (1976), 11–18] proved that the radius r and diameter d of any radially maximal graph satisfy $r\le d\le 2r-2.$ Dutton et al. [‘Changing and unchanging of the radius of a graph’, Linear Algebra Appl.217 (1995), 67–82] rediscovered this result with a different proof and conjectured that the converse is true, that is, if r and d are positive integers satisfying $r\le d\le 2r-2,$ then there exists a radially maximal graph with radius r and diameter $d.$ We prove this conjecture and a little more.

In this paper, for the convolution and convex combination of harmonic mappings, the radii of univalence, full convexity and starlikeness of order $\unicode[STIX]{x1D6FC}$ are explored. All results are sharp. By way of application, the univalent radius and the Bloch constant of the convolution of two bounded harmonic mappings are obtained.

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.

We report a rare case of a type I Monteggia lesion with an associated fracture of the distal radius and ulna metaphysis in a child. We discuss the mechanism of injury and the importance of thorough clinical examination in childhood injuries.

In children, but not adults with cystic fibrosis (CF), associations between essential fatty acids (FA) and bone mass have been reported. Low bone mineral density (BMD) is common in these patients. Previously we found a normal annual increase of BMD, suggesting a potential for attaining normal bone mass. The aim of the present study was to investigate phospholipid FA pattern in relation to bone in young adult men with CF compared with healthy controls. Fourteen male patients with CF were compared with forty-two healthy controls, using dual-energy X-ray absorptiometry for total bone, lumbar spine and femur and peripheral quantitative computerised tomography for tibia and radius. A questionnaire concerning physical activity and nutrition was used. FA in serum phospholipids were measured using capillary GLC. CF patients did not differ in physical activity and anthropometry from controls. There were no differences in bone parameters between the two groups, but patients chronically colonised with Pseudomonas aeruginosa had lower BMD than non-colonised patients. The trabecular BMD in the tibia differed between patients and controls, but not after adjustment for age and weight. The endosteal circumference of the radius was significantly associated with serum phospholipid concentration of DHA and inversely with the n-6:n-3 FA ratio in CF patients but not in controls. The present study showed that young physically active adult males with classical CF obtained similar bone mass as controls, although influenced by pseudomonas colonisation. The association between DHA and long bone endosteal circumference suggested a later peak bone mass in those with CF compared with controls.

A method to determine the two parameter set of circular cylinders, whosesurfaces contain three given points, is presented in the context of an efficientalgorithm, based on the set of two parameter projections of the points ontoplanar sections, to compute radius and a point where the axes intersect theplane of the given points. The geometry of the surface of points, whose positionvectors represent cylinder radius, r, and axial orientation, isrevealed and described in terms of symmetry and singularity inherent in thetriangle with vertices on the given points. This strongly suggests that, givenone constraint on the axial orientation of the cylinder, there are up to sixcylinders of identical radius on the three given points. A bivariate function,in two of the three line direction Plücker coordinates, is derived to provethis. By specifying r and an axis direction, say, perpendicular to agiven direction, one obtains a sixth order univariate polynomial in one of theline coordinates which yields six axis directions. These ideas are needed in thedesign of parallel manipulators