Estimating the spectral measure of an extreme value distribution

Let (X₁, Y₁), (X₂, Y₂),…, (X_n, Y_n) be a random sample from a bivariate distribution function F which is in the domain of attraction of a bivariate extreme value distribution function G. This G is characterized by the extreme value indices and its spectral measure or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure. In this paper, we construct an empirical measure, based on the sample, which is a consistent estimator of the spectral measure. We also show for positive extreme value indices the asymptotic normality of the estimator under a suitable 2nd order strengthening of the bivariate domain of attraction condition.     

Experimental Analysis of the Response of a Laser/Phase Doppler Anemometer System to a Partially Atomized Spray

This work describes a systematic approach adopted to establish Laser and Phase Doppler Anemometry, LDA/PDA, experimental techniques that would allow velocity and dropsize measurements to be made over wide velocity and size ranges with confidence in partially atomized sprays. The analysis considers the sprays generated by different gasoline direct injection (GDI) systems injecting into air under atmospheric conditions. The upper limit to the dropsize range in the fuel sprays was confirmed using (a) an Oxford Lasers' VisiSizer and (b) droplets of a known size produced by a mono‐dispersed droplet generator. GDI fuel sprays are highly transient, optically dense and provide a high degree of penetration and atomization. The measurement problem is therefore one of the detection of small, high speed droplets inside a dense cloud of surrounding droplets. Furthermore, under the transients found at the start and end of injection and during high fuel loads, fuel elements in the form of sheets, ligaments and filaments are also injected. These liquid fuel elements subsequently break‐up, downstream from the nozzle, to form droplets of a much larger size class but with a much lower number density [1]. The co‐existence of these liquid fuel elements and the widely different size classes in the spray are considered to pose a problem for dropsize measurements by the PDA technique. In particular: the wide dynamic range of light intensities scattered by the fuel elements and droplets; the trajectory of large drops through the edges of the PDA measurement volume with its Gaussian intensity distribution [2] and the high probability of non spherical droplets. The work concludes that the LDA/PDA measurement technique, as applied here, is robust. It can discriminate between partially and fully atomized sprays, has a high probability of accurately measuring dropsizes larger than the measurement volume and give a realistic indication of ‘sizes’ for non spherical droplets. However, specification of the PDA system parameters must be strictly compatible with the measurement task to yield unambiguous results.     

Singularity and unanimity

Examples are adduced which lead one to ask if the following rule of unanimity makes sense: Given, a classical dynamical problem. Given, that all solutions of the equations of motion (a) run into a singularity [or (b) are free of singularity], except a set of measure zero. Then (rule of unanimity), all solutions of the corresponding quantum mechanical problem are (a) singular [or (b) free of singularity]. If valid, this rule would imply that quantization of Einstein's standard general relativity model for a closed universe gives no escape from the singularity of gravitational collapse.Dedicated to Achille Papapetrou on the occasion of his retirement.     

激光雷达研制及其探测大气气溶胶的实验研究

作为一种激光探测大气气溶胶特征的新技术,采用分子滤波技术的高光变分辨率激光雷达在国际上得到了发展,给出了对这种YAG激光雷达系统的理论分析并建立了实验装置,其接收望远镜孔径为300mm,YAG激光脉冲能量为150mJ。通过数值模拟与初步测量估计了激光雷达性能,测量了大气回向散射信号,测量结果表明实结果与理论分析一致。     

Estimates of Stark broadening of some Si(II) lines

Recent measurements of spectral line widths for the first five multiplets in the visible of singly-ionized silicon, are compared with the corresponding values predicted by the semi-empirical method of Griem, as well as two other Stark broadening theories. Although there is considerable disagreement between different measurements, as well as with and between the theories involved, agreement to better than 30% on average is obtained between these calculations and the computations of Sahal-Bréchot. Some of the experimental and theoretical questions involved are discussed and suggestions are made for future work on the subject.     

Local and nonlocal contributions to adhesive energies at metallic interfaces

A self-consistent computation of the adhesive binding energy versus separation is reported for the following contacts: Al(111)-Al(111), Mg(0001)- Mg(0001), Zn(0001)-Zn(0001), and Na(110)-Na(110). Results are obtained for the local density approximation to exchange and correlation as well as for a nonlocal approximation. The resultant shapes of the adhesive binding energy curves varied substantially from metal to metal. The inclusion of nonlocality has little effect on the overall shape of the binding energy curves, but improves agreement of the binding energies as measured from the energy minimum with experimental surface energies.     

On inferring the probability of misclassification by the linear discriminant function

Summary Let the two alternative populationsP ₁ andP ₂ from which the individual with measurements χ may have come beN(μ(1), Σ) andN(μ(2), Σ). Then the classification rule with minimum risk is to assign the individual toP ₁ orP ₂ according as (μ(2)-μ(1))′Σ^-1 x≶(1/2)(μ(2)-μ(1))′Σ^-1(μ(1)+μ(2))+c wherec is a constant depending on the prior probabilities ofP ₁ andP ₂ and the costs of the two kinds of misclassification. The probability of misclassifying an individual fromP ₂ by this rule is π₂₁=Φ(-δ/2+cδ^-1), where Φ(.) is the distribution function of anN(0, 1) and . (Since we are free to choose which population we shall callP ₂, it is not necessary to consider separately the probability of misclassifying an individual fromP ₁.) LetP ₂₁ denote the probability of misclassification of an individual fromP ₂ by the rule derived from the one mentioned by fixing μ(1), μ(2) and Σ at estimates andV and letP ₂₁ ^* be the probability of misclassification of an individual fromP ₂ when the classification rule is the one with minimum risk among those based on . The fiducial distributions of π₂₁,P ₂₁ andP ₂₁ ^* are determined. Point estimates and confidence intervals for π₂₁,P ₂₁ andP ₂₁ ^* are derived. Only easily available tables are needed to make fiducial inferences. An incidental result of some interest elsewhere as well is the distribution of a linear combination of a chi and an independent normal variable.     

A recurrence relation for the square root

The behavior of the sequence x_{n + 1} = x_n(3N − x_n²)/2N is studied for N > 0 and varying real x₀. When 0 < x₀ < (3N)^1/2 the sequence converges quadratically to N^1/2. When x₀ > (5N)^1/2 the sequence oscillates infinitely. There is an increasing sequence β_r, with β₋₁ = (3N)^1/2 which converges to (5N)^1/2 and is such that when β_r < x₀ < β_{r + 1} the sequence {x_n} converges to (−1)^rN^1/2. For x₀ = 0, β₋₁, β₀,… the sequence converges to 0. For x₀ = (5N)^1/2 the sequence oscillates: x_n = (−1)ⁿ(5N)^1/2. The behavior for negative x₀ is obtained by symmetry.     

A nonlinear parabolic free boundary problem

Summary When two immiscible fluids in a porous medium are in contact with one another, an interface is formed and the movement of the fluids results in a free boundary problem for determining the location of the interface along with the pressure distribution throughout the medium. The pressure satisfies a nonlinear parabolic partial differential equation on each side of the interface while the pressure and the volumetric velocity are continuous across the interface. The movement of the interface is related to the pressure through Darcy’s law. Two kinds of boundary conditions are considered. In Part I the pressure is prescribed on the known boundary. A weak formulation of the classical problem is obtained and the existence of a weak solution is demonstrated as a limit of a sequence of classical solutions to certain parabolic boundary value problems. In Part II the same analysis is carried out when the flux is specified on the known boundary, employing special techniques to obtain the uniform parabolicity of the sequence of approximating problems. Entrata in Redazione il 29 novembre 1975. This research was supported in part by the National Science Foundation, the Senior Fellowship Program of the North Atlantic Treaty Organization, the Italian Consiglio Nazionale delle Ricerche, and the Texas Tech. University.