Convexity and the Average Curvature of Plane Curves

The average curvature of a rectifiable closed curve in R2 is its total absolute curvature divided by its length. If a rectifiable closed curve in R2 is contained in the interior of a convex set D then its average curvature is at least as large as the average curvature of the simple closed curve D which bounds the convex set.     

Closed Curves in E 2 and S 2 Whose Total Squared Curvatures and Almost Minimal

The total squared curvature of a closed plane curve is minimized by a covering of a circle, if the rotation number of the curve is not zero. We measure the closeness of a closed plane curve to a circle by the difference of its total squared curvature from the minimum. A similar result is obtained for a simple closed curve in the 2-sphere.     

增长曲线模型中最小二乘估计的几种相对效率

对于一般的增长曲线模型Yn×m=Xn×pBp×qZq×m＋ε,ε~（0,δV∑）本文定义了B的最小二乘估计（相对于B的最佳线性无偏估计）的四种相对效率，并得到了它们的下界．     

Wei Hua’s four-parameter potential: Comments and computation of molecular constants αe and εeXe

The value of adjustable parameterC in the four-parameter potentialU(r) =D _e [(1 - exp[-b(r -r _e)])/(1 -C exp[-b(r -r _e)])]² has been expressed in terms of molecular parameters and its significance has been brought out. The potential so constructed, withC derived from the molecular parameters, has been applied to ten electronic states in addition to the states studied by Wei Hua. Average mean deviation for these 25 states has been found to be 3.47 as compared to 6.93, 6.95 and 9.72 obtained from Levine, Varshni and Morse potentials, respectively. Also Dunham’s method has been used to express rotation-vibration interaction constant (α_e) and anharmonicity constant (ω_ex_e) in terms ofC and other molecular constants. These relations have been employed to determine these quantities for 37 electronic states. For α_e, the average mean deviation is 7.2% compared to 19.7% for Lippincott’s potential which is known to be the best to predict these values. Average mean deviation for (ω_ex_e) turns out to be 17.4% which is almost the same as found from Lippincott’s potential function.     

椭圆偏振光直接实验证的数据处理方法

从理论上分析了大学物理吉有关在椭圆偏振光实验里得不到椭圆曲线的原因,提出了得到椭圆曲线的数据处理方法。     

A distributionally robust area under curve maximization model

Area under ROC curve (AUC) is a performance measure for classification models. We propose new distributionally robust AUC models (DR-AUC) that rely on the Kantorovich metric and approximate AUC with the hinge loss function, and derive convex reformulations using duality. The DR-AUC models outperform deterministic AUC and support vector machine models and have superior worst-case out-of-sample performance, thereby showing their robustness. The results are encouraging since the numerical experiments are conducted with small-size training sets conducive to low out-of-sample performance.     

Determination of the in-plane anisotropy field in hexagonal systems via rotational magnetization: Theoretical model and Monte Carlo simulations

The magnetic anisotropy field in thin films with in-plane uniaxial anisotropy can be deduced from the VSM magnetization curves measured in magnetic fields of constant magnitudes. This offers a new possibility of applying rotational magnetization curves to determine the first- and second-order anisotropy constant in these films. In this paper we report a theoretical derivation of rotational magnetization curve in hexagonal crystal system with easy-plane anisotropy based on the principle of the minimum total energy. This model is applied to calculate and analyze the rotational magnetization process for magnetic spherical particles with hexagonal easy-plane anisotropy when rotating the external magnetic field in the basal plane. The theoretical calculations are consistent with Monte Carlo simulation results. It is found that to well reproduce experimental curves, the effect of coercive force on the magnetization reversal process should be fully considered when the intensity of the external field is much weaker than that of the anisotropy field. Our research proves that the rotational magnetization curve from VSM measurement provides an effective access to analyze the in-plane anisotropy constant K ₃ in hexagonal compounds, and the suitable experimental condition to measure K ₃ is met when the ratio of the magnitude of the external field to that of the anisotropy field is around 0.2. Supported by the National Natural Science Foundation of China (Grant Nos. 90505007 and 10774061) Recommended by LI FaShen     

Empirical process approach to some two-sample problems based on ranked set samples

We study the asymptotic properties of both the horizontal and vertical shift functions based on independent ranked set samples drawn from continuous distributions. Several tests derived from these shift processes are developed. We show that by using balanced ranked set samples with bigger set sizes, one can decrease the width of the confidence band and hence increase the power of these tests. These theoretical findings are validated through small-scale simulation studies. An application of the proposed techniques to a cancer mortality data set is also provided. Part of the research was conducted while Kaushik Ghosh was visiting Statistical Research and Applications Branch of the National Cancer Institute on an Intergovernmental Personnel Assignment.     

Integrable semiclassical deformations of general algebraic curves and associated conservation laws

Based on the Lenard relations, we completely classify integrable deformations of general algebraic curves. We construct the general solution of the Lenard relation from the invariance condition with respect to an element of the Galois group of the curve. We give some examples and also some associated conservation laws. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 458–469, June, 2007.     

A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.