Summary.Continuously indexed Gaussian fields (GFs) are the most important ingredient in spatial statistical modelling and geostatistics. The specification through the covariance function gives an intuitive interpretation of the field properties. On the computational side, GFs are hampered with the big n problem, since the cost of factorizing dense matrices is cubic in the dimension. Although computational power today is at an all time high, this fact seems still to be a computational bottleneck in many applications. Along with GFs, there is the class of Gaussian Markov random fields (GMRFs) which are discretely indexed. The Markov property makes the precision matrix involved sparse, which enables the use of numerical algorithms for sparse matrices, that for fields in R 2 only use the square root of the time required by general algorithms. The specification of a GMRF is through its full conditional distributions but its marginal properties are not transparent in such a parameterization. We show that, using an approximate stochastic weak solution to (linear) stochastic partial differential equations, we can, for some GFs in the Matérn class, provide an explicit link , for any triangulation of R d , between GFs and GMRFs, formulated as a basis function representation. The consequence is that we can take the best from the two worlds and do the modelling by using GFs but do the computations by using GMRFs. Perhaps more importantly, our approach generalizes to other covariance functions generated by SPDEs, including oscillating and non-stationary GFs, as well as GFs on manifolds. We illustrate our approach by analysing global temperature data with a non-stationary model defined on a sphere.

A total of 2895 threaded, cylindrical titanium implants have been inserted into the mandible or the maxilla and 124 similar implants have been installed in the tibial, temporal or iliac bones in man for various bone restorative procedures. The titanium screws were implanted without the use of cement, using a meticulous technique aiming at osseointegration--a direct contact between living bone and implant. Thirty-eight stable and integrated screws were removed for various reasons from 18 patients. The interface zone between bone and implant was investigated using X-rays, SEM, TEM and histology. The SEM study showed a very close spatial relationship between titanium and bone. The pattern of the anchorage of collagen filaments to titanium appeared to be similar to that of Sharpey's fibres to bone. No wear products were seen in the bone or soft tissues in spite of implant loading times up to 90 months. The soft tissues were also closely adhered to the titanium implant, thereby forming a biological seal, preventing microorganism infiltration along the implant. The implants in many cases had been allowed to permanently penetrate the gingiva and skin. This caused no adverse tissue effects. An intact bone-implant interface was analyzed by TEM, revealing a direct bone-to-implant interface contact also at the electron microscopic level, thereby suggesting the possibility of a direct chemical bonding between bone and titanium. It is concluded that the technique of osseointegration is a reliable type of cement-free bone anchorage for permanent prosthetic tissue substitutes. At present, this technique is being tried in clinical joint reconstruction. In order to achieve and to maintain such a direct contact between living bone and implant, threaded, unalloyed titanium screws of defined finish and geometry were inserted using a delicate surgical technique and were allowed to heal in situ, without loading, for a period of at least 3--4 months.

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