Values Ofk Research Articles

A proposed variation of the matching technique

A matching procedure is proposed by which a judge may use the same items in different matches. TheK items in one group most likely to include the correct match with each item in the other are selected. Inclusion of the correct match among theK items chosen is defined as a success. The distribution of the number of successes is discussed. Tables are presented showing the number of successes needed for significance for various values ofK and ofN, the number of items in each group.

The shortest path through many points

ABSTRACTWe prove that the length of the shortest closed path throughnpoints in a bounded plane region of areavis ‘almost always’ asymptotically proportional to √(nv) for largen; and we extend this result to bounded Lebesgue sets ink–dimensional Euclidean space. The constants of proportionality depend only upon the dimensionality of the space, and are independent of the shape of the region. We give numerical bounds for these constants for various values ofk; and we estimate the constant in the particular casek= 2. The results are relevant to the travelling-salesman problem, Steiner's street network problem, and the Loberman—Weinberger wiring problem. They have possible generalizations in the direction of Plateau's problem and Douglas' problem.

Stability of a Numerical Solution of Differential Equations

Stability of a Numerical Solution of Differential Equations

The magnetic susceptibility of oxygen in a clathrate compound

Cooke, Meyer, Wolf, Evans andRichards (Proc. Roy. Soc.,225, 112 (1954)) have measured at low temperatures the susceptibility of oxygen dissolved in aβ-quinol compound, wherein the oxygen molecules are 8 A apart, and the deviation of the susceptibility from that in the gaseous state may hence be small. They find that16O18O and16O16O have the same susceptibility, although at low temperatures the magnetic behavior in the gaseous state should be different, inasmuch as only odd values ofK are allowed for16O16O. The present paper shows that the magnetic behavior can be explained on the assumption that at helium temperatures the O2 molecules do not rotate freely, but instead are aligned along a particular spatial direction. Calculations by MissO’Brien show that the susceptibility curves can be fitted by assuming the axis of the O2 molecule is «frozen» in space and that the spin-spin parameter is about 25 percent lower than in the gas. This apparent reduction can be explained on the basis that the molecule is not rigidly frozen, and instead has a potential barrier of about 50 cm−1.

Some remarks on integral equations with kernels: L (ξ 1 - x 1 ,..., ξ n - x n ; α)

The paper discusses integral equations of the type k (ξ 1 ... ξ n ) = ∫ -∞ ∞ ... ∫ -∞ ∞ ψ ( x 1 ... x n ) L (ξ 1 -x 1 ... ξ n -x n ;α) d x 1 ... d x n , where L is L -integrable, and ψ and k are bounded. Since rapidly oscillating ψ have a small k , and since measurements of k are necessarily uncertain within non-zero limits of experimental error, very different ψ are consistent with any given set of measurements of k . Thus ψ is not determined by measurements of k . Instead of ψ, partial information about ψ that is not sensitive to rapid oscillations of ψ, can be obtained from k . In the present paper we consider smoothed versions of ψ, and their applications to gravity survey and the theory of surface waves. (1) For given L we construct normalized smoothing functions μ( x 1 ... x n ), so that ψ( x 1 ... x n ) = ∫ -∞ ∞ ... ∫ -∞ ∞ ψ( u 1 ... u n )μ (x 1 - u 1 ... x n - u n ) d u 1 .... d u n can be calculated from measurements of k( (ξ 1 ... ξ n ). The method is applied to gravity survey, where the distribution ψ of masses on a plane ∑' is to be calculated from the normal force k on another (parallel) plane∑. (2) By studying suitable smoothing functions we get a lower bound for the maximum modulus of the functions ψ which are consistent with given experimental values of k . The bound is large if k is known to vary rapidly. The bound is also large if α is large and if the Fourier transform of L tends to 0 when α→∞. The results are applied to gravity survey where now we consider the normal force on ∑ due to masses of bounded density distributed in the space below ∑', where ∑' is itself below ∑. If the normal force is not uniform the distance between ∑ and ∑' must not be too large, the estimate depending on the bound for the density. Also fairly general conditions are imposed on ψ so that ψ that is approximately determined by measurements of k , and an example from the theory of propagation in dispersive media is given where such conditions may be justified. The gist of the paper is contained in theorems A and B.