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A radial basis function approximation has the form whereϕ:Rd→R is some given (usually radially symmetric) function, (yj)1n are real coefficients, and the centers (xj)1n are points inRd. For a wide class of functions ϕ, it is known that the interpolation matrixA=(ϕ(xj−xk))j,k=1n is invertible. Further, several recent papers have provided upper bounds on ||A−1||2, where the points (xj)1n satisfy the condition ||xj−xk||2≥δ,j≠k, for some positive constant δ. In this paper we calculate similar upper bounds on ||A−1||2 forp≥1 which apply when ϕ decays sufficiently quickly andA is symmetric and positive definite. We include an application of this analysis to a preconditioning of the interpolation matrixAn = (ϕ(j−k))j,k=1n when ϕ(x)=(x2+c2)1/2, the Hardy multiquadric. In particular, we show that supn ||An−1||∞ is finite. Furthermore, we find that the bi-infinite symmetric Toeplitz matrix enjoys the remarkable property that ||E−1||p = ||E−1||2 for everyp≥1 when ϕ is a Gaussian. Indeed, we also show that this property persists for any function ϕ which is a tensor product of even, absolutely integrable Polya frequency functions.

In [4], deep results were obtained concerning the invertibility of matrices arising from radial basis function interpolation. In particular, the Euclidean distance matrix was shown to be invertible for distinct data. In this paper, we investigate the invertibility of distance matrices generated byp-norms. In particular, we show that, for anyp∈(1, 2), and for distinct pointsx 1,⋯,x n ∈ℛ d , wheren andd may be any positive integers, with the proviso thatn≥2, the matrixA∈ℛ n×n defined by $$A_{ij} = \left\| {x^i - x^j } \right\|_p for1 \leqslant i,j \leqslant n,$$ , satisfies $$( - 1)^{n - 1} \det A > 0.$$ . We also show how to construct, for everyp>2, a configuration of distinct points in some ℛd giving a singularp-norm distance matrix. Thus radial basis function interpolation usingp-norms is uniquely determined by any distinct data forp ∈ (1, 2), but not so forp>2.

This paper discusses the scalability properties of a novel algorithm for the rapid evaluation of radial basis function interpolants. The algorithm is associated with the problem of force calculation in many-body calculations. Contrary to previously developed fast summation schemes including treecodes and fast multipole methods, this algorithm has simple communication patterns which are achieved by exploiting the localisation and smoothness properties of radial basis functions. Thus, the algorithm is scalable even in low bandwidth environments like clusters of workstations and even for relatively small problem sizes.

A radial basis function approximation has the form s ( x ) = is some given function, ( y j ) n 1 are real coefficients, and the centres ( x j ) n 1 are points in ℝ d . For a wide class of functions φ, it is known that the interpolation matrix is invertible (see [5]). Further, several recent papers have provided upper bounds on ‖ A −1 ‖ 2 , where the points ( x j ) n 1 satisfy the condition ‖ x j – x k ‖ 2 ≥ δ, j ≠ k , for some positive constant 6. In this paper, we show that the generalized Fourier transform of φ provides a simple and elegant tool for the derivation of such estimates.

UNLABELLED: The purpose of the study was to compare the ease of insertion of the laryngeal mask airway using the standard uninflated approach or with a fully inflated cuff. Two hundred consecutive patients undergoing anesthesia using a laryngeal mask airway were randomized to have the laryngeal mask inserted using either method. Successful insertion was judged primarily by the clinical function of the airway. The number of insertion attempts to achieve a satisfactory airway and whether an alternative technique was required for success were recorded. On removal of the laryngeal mask, a blind observer noted the presence or absence of blood. Just before leaving the recovery room, each patient was asked whether they had a sore throat. Insertion technique made no difference with regard to first attempt success. However, the presence of blood on the removed masks (P < 0.01) and sore throat (P < 0.01) were less frequent in the inflated cuff group. We conclude that the inflated cuff insertion technique is an acceptable alternative to the standard approach and has the advantage of reducing the incidence of minor pharyngeal mucosal trauma, as evidenced by mucosal bleeding and sore throat. IMPLICATIONS: Insertion of the laryngeal mask airway with the cuff fully inflated is equally successful to the standard uninflated approach in experienced hands. The inflated technique was associated with less minor pharyngeal mucosal trauma and, consequently, a lower incidence of postoperative sore throat. This implies that the inflated technique would be acceptable to the general population of laryngeal mask users.

A diet containing 11.5% (dry matter basis) of a moderately high Cd sewage sludge was fed to six Hereford steers for 106 days to simulate a high sludge intake from sludge-amended soils. At a mean live weight of 460 kg, the steers were slaughtered and liver, kidney, bone, muscle, lung, brain and spleen tissues were collected for heavy metal determinations; tissues were also collected at this time from six control steers fed the same diet with no sludge. The sludge-fed cattle consumed an average of 8.95 kg of diet dry matter/day and gained 1.13 kg/day. This feed intake was slightly higher, but the gain lower than that of control cattle. All gains were equal to those expected from intake of non-sludge diet components. This indicated that the sludge had approximately zero energy value and was acting as a dietary diluent. The sludge metal content (ppm, dry basis) was: Cd, 98; Hg, 18; Pb, 466; Cu, 1,733, and Zn, 1,733. Addition of sludge increased metal contents of the feedlot diet to 30 to 100 times that of the control. Retention of heavy metals in the total animal from sludge ingestion averaged .09%, .06% and .3% for Cd, Hg and Pb; no retention was noted from Cu and Zn. These low fractional retentions increased tissue Cd, Hg and Pb concentrations of liver and kidney tissues by five to 20-fold. Estimates of levels that would enter the human diet from average beef tissue consumption if all feedlot steers were fed sludge are presented for Cd, Hg and Pb. The data indicate that cattle are a moderately effective screen against the entry of these potentially toxic metals into the human diet.

Cattle grazing a sewage sludge disposal site were examined to determine whether, and to what extent, contamination of tissues had occurred as a result of exposure to sewage sludge. The heavy metal and persistent organics content of kidney, liver, muscle, and bone tissues from cattle grazing a sewage sludge disposal site were compared with tissues of cattle from a nearby control ranch. Tissues were analyzed for As, Cd, Cu, Hg, Mo, Ni, Pb, Se, Zn, and 22 persistent organics. The sludge-exposed cattle had an average kidney-Cd content of 16 mg/kg (dry weight), which was significantly higher than the 4 mg/kg found in the control herd, but was not above a range that would be expected from older range cattle. There were no other tissue differences between the two groups with respect to Cd. The average liver-Cu concentration (4.6 mg/kg) of the sludge-exposed cattle was significantly lower than that of the controls (19 mg/kg); however, both groups of animals displayed liver-Cu concentrations that were considerably lower than levels previously considered normal. Mean kidney-Hg and liver-Se concentrations were significantly higher in the control cattle than in the sludge-exposed cows. Kidney-Zn concentrations of the sludge-exposed cattle were significantly higher than those of the controls, but the magnitude of the differences was small. Differences between the two groups of animals in relation to Hg and Se were small, and it is doubtful that any real biological significance can be assigned to these differences. Fat was the only tissue that contained detectable levels of 22 persistent organics. The only significant difference observed was alpha-BHC (hexachlorocylohexane, alpha); the control herd averaged 30 µg/kg (wet weight), whereas the sludge-exposed cattle averaged 10 µg/kg.

As stated in Part I, the snowshed problem on the west slope of the Cascade Range early brought under consideration, by the Chief Engineers of the Railway Company, routes avoiding the snowslide area...

Disturbances of growth and behavior in infants and toddlers of women addicted to heroin during pregnancy have been reported in uncontrolled studies. In this study, 3- to 6-year-old children of heroin-addicted mothers were compared to three other groups matched for age, race, sex, birth weight, and socioeconomic status. Heroin-exposed children weighed less and were shorter than those in the comparison groups; 14% had a head circumference below the third percentile. Heroin-exposed children were rated by parents as less well adjusted than control children and they differed significantly in perceptual measures and on subtests of the Illinois Test of Psycholinguistic Abilities and McCarthy Scales of Children's Abilities relating to the process of organization. These findings suggest that chronic intrauterine exposure to heroin may affect growth and behavior as well as perceptual and learning processes in preschool children.

In this paper we compute the ultraspherical series expansions for the more commonly used radial basis functions. In several special cases we provide asymptotic estimates for the decay rate of the coefficients involved. knowledge of the decay rate of these coefficients is useful because they enable error estimates for spherical interpolation.