Convergence of weighted averages of relaxed projections

The convergence of the algorithm for solving convex feasibility problem is studied by the method of sequential averaged and relaxed projections. Some results of H. H. Bauschke and J. M. Borwein are generalized by introducing new methods. Examples illustrating these generalizations are given.     

The perimeter of rounded convex planar sets

A convex set is inscribed into a rectangle with sides a and 1/a so that the convex set has points on all four sides of the rectangle. By “rounding” we mean the composition of two orthogonal linear transformations parallel to the edges of the rectangle, which makes a unit square out of the rectangle. The transformation is also applied to the convex set, which now has the same area, and is inscribed into a square. One would expect this transformation to decrease the perimeter of the convex set as well. Interestingly, this is not always the case. For each a we determine the largest and smallest possible increase of the perimeter.      

Set estimation under convexity type assumptions

The problem of estimating a set S from a random sample of points taken within S is considered. It is assumed that S is r-convex, which means that a ball of radius r can go around from outside the set boundary. Under this assumption, the r-convex hull of the sample is a natural estimator of S. We obtain convergence rates for this estimator under both the distance in measure and the Hausdorff metric between sets. It is also proved that the boundary of the estimator consistently estimates the boundary of S, in Hausdorff's sense.     

The numerical range and the spectrum of a product of two orthogonal projections

The aim of this paper is to describe the closure of the numerical range of the product of two orthogonal projections in Hilbert space as a closed convex hull of some explicit ellipses parametrized by points in the spectrum. Several improvements (removing the closure of the numerical range of the operator, using a parametrization after its eigenvalues) are possible under additional assumptions. An estimate of the least angular opening of a sector with vertex 1 containing the numerical range of a product of two orthogonal projections onto two subspaces is given in terms of the cosine of the Friedrichs angle. Applications to the rate of convergence in the method of alternating projections and to the uncertainty principle in harmonic analysis are also discussed.     

Two generalizations of Dykstra’s cyclic projections algorithm

Dykstra’s cyclic projections algorithm allows one to compute best approximations to any pointx in a Hilbert space from the intersectionC = ⋂ _l ^r C _i of a finite number of closed convex setsC _i , by reducing it to a sequence of best approximation problems from theindividual setsC _i . Here we present two generalizations of this algorithm. First we allow the number of setsC _i to beinfinite rather than finite; secondly, we allow arandom, rather than cyclic, ordering of the setsC _i . This author was supported by NSF Grant DMS-9303705.     

Bayesian estimation of the product of two proportions

Suppose that for i = 1,2, a Bernoulli random variable with success probability θ_i is observable from population i. The problem is to estimate θ = θ₁θ₂ using a Bayesian approach with squared error estimation loss in θ. For estimating θ, the best nonrandom sampling scheme, the two-stage sampling scheme, and the optimal sampling scheme are discussed. It is shown that the two-stage sampling scheme is typically asymptotically optimal, and can improve the Bayes risk (over the best nonrandom allocation) up to fifty percent     

A new approach to the reconstruction of images from Radon projections

A new approach is proposed for reconstruction of images from Radon projections. Based on Fourier expansions in orthogonal polynomials of two and three variables, instead of Fourier transforms, the approach provides a new algorithm for the computed tomography. The convergence of the algorithm is established under mild assumptions.     

New bounds on the average distance from the Fermat–Weber center of a planar convex body

The Fermat–Weber center of a planar body Q is a point in the plane from which the average distance to the points in Q is minimal. We first show that for any convex body Q in the plane, the average distance from the Fermat–Weber center of Q to the points in Q is larger than , where Δ(Q) is the diameter of Q. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most . The new bound substantially improves the previous bound of due to Abu-Affash and Katz, and brings us closer to the conjectured value of . We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.     

Pareto分布形状参数的E-Bayes估计和多层Bayes估计及其应用

给出了参数的E-Bayes估计的定义,对Pareto分布在尺度参数已知时,在平方损失下给出了形状参数的E-Bayes估计和多层Bayes估计,并且用Monte Carlo方法给出了模拟算例.最后,结合高尔夫球手收入数据的实际问题进行了计算,结果表明本文提出的方法可行且便于应用.     

DERIVATION AND ANALYSIS OF A NEW RECONSTRUCTION FORMULA FROM CONE-BEAM PROJECTIONS

A new method to compute the first derivative of 3-D Radon transform is given for cone-beam data taken from any orbit. Smith [Ⅰ1] and Grangeat [5] even derived cone-beam inversion formulas which are the basic work in fully 3-D image reconstruction algorithm and are used extensively now. In this paper we will give a new inversion formula and a simple necessary and sufficient condition which guarantees the complete reconstruction algorithm.     

E-Bayesian estimation of the reliability derived from Binomial distribution

This paper introduces a new parameter estimation method, named E-Bayesian estimation method, to estimate reliability derived from Binomial distribution. The definition of E-Bayesian estimation of the reliability is proposed, the formulas of E-Bayesian estimation and hierarchical Bayesian estimation of the reliability are also provided. Finally, it is shown, through a numerical example, that the new method is much simpler than hierarchical Bayesian estimation in practice.     

The rate of convergence for the cyclic projections algorithm III: Regularity of convex sets

The cyclic projections algorithm is an important method for determining a point in the intersection of a finite number of closed convex sets in a Hilbert space. That is, for determining a solution to the “convex feasibility” problem. This is the third paper in a series on a study of the rate of convergence for the cyclic projections algorithm. In the first of these papers, we showed that the rate could be described in terms of the “angles” between the convex sets involved. In the second, we showed that these angles often had a more tractable formulation in terms of the “norm” of the product of the (nonlinear) metric projections onto related convex sets.In this paper, we show that the rate of convergence of the cyclic projections algorithm is also intimately related to the “linear regularity property” of Bauschke and Borwein, the “normal property” of Jameson (as well as Bakan, Deutsch, and Li’s generalization of Jameson’s normal property), the “strong conical hull intersection property” of Deutsch, Li, and Ward, and the rate of convergence of iterated parallel projections. Such properties have already been shown to be important in various other contexts as well.     

对二阶抽样中两个定理的扩充及其应用

本文对德斯·拉奇(DesRaj)就不放回二阶抽样的估计量、估计量的方差及其估计提出的两个定理做了一些扩充,并对拉奥(Rao)-哈特利(Hartley)-科克伦(Cochran)提出的二阶RHC估计量运用扩充后的定理得到其方差估计量。     

Least-squares linear estimation of signals from observations with Markovian delays

The least-squares linear estimation of signals from randomly delayed measurements is addressed when the delay is modeled by a homogeneous Markov chain. To estimate the signal, recursive filtering and fixed-point smoothing algorithms are derived, using an innovation approach, assuming that the covariance functions of the processes involved in the observation equation are known. Recursive formulas for filtering and fixed-point smoothing error covariance matrices are obtained to measure the goodness of the proposed estimators.     

Measure of segments which intersect a convex body from rotational formulae

Classical problems in integral geometry and geometric probability involve the kinematic measure of congruent segments of fixed length within a convex body in R³. We give this measure from rotational formulae; that is, from isotropic plane sections through a fixed point. From this result we also obtain a new rotational formula for the volume of a convex body; which is proved to be equivalent to the wedge formula for the volume.     

Numerical simulation two phase flows of casting filling process using SOLA particle level set method

Mould filling process is a typical gas–liquid metal two phase flow phenomenon. Numerical simulation of the two phase flows of mould filling process can be used to properly predicate the back pressure effect, the gas entrapment defects, and better understand the complex motions of the gas phase and the liquid phase. In this paper, a novel sharp interface incompressible two phase numerical model for mould filling process is presented. A simple ghost fluid method like discretization method and a density evaluation method at face centers of finite difference staggered grid are proposed to overcome the difficulties when solving two phase Navier–Stokes equations with large-density ratio and large-viscosity ratio. A new mass conservation particle level set method is developed to capture the gas–liquid metal phase interface. The classical pressure-correction based SOLA algorithm is modified to solve the two phase Navier–Stokes equations. Two numerical tests including the Zalesak disk problem and the broken dam problem are used to demonstrate the accuracy of the present method. The numerical method is then adopted to simulate three mould filling examples including two high speed CCD camera imaging water filling experiments and an in situ X-ray imaging experiment of pure aluminum filling. The simulation results are in good agreement with the experiments.     

Asymptotic distribution of residual autocorrelations from estimation of ARMA processes by Gram-Schmidt orthogonalization

One computationally efficient procedure for obtaining maximum likelihood parameter estimates for an ARMA process is based on the Gram-Schmidt orthogonalization of the space generated by the finite series of observations. This paper shows that the asymptotic distribution of the autocorrelations of the resulting residuals coincides with that for least-square residuals.     

股票收益率波动性的非参数核回归估计及对中国股市的实证分析

非参数回归估计是研究非线性时间序列的一种有用工具．在混合相依样本的条件下，基于非参数核回归估计方法来研究收益率的波动性，利用改良的交叉核实函数选取光滑参数，并给出上海和深圳股市实证分析的一些有趣结果．     

Reconstruction of a graph from 2-vicinities of its vertices

We prove that a connected graph of diameter at least 4 and of girth 7 or more (in particular, a tree) can be exactly reconstructed from metric balls of radius 2 of all its vertices. On the other hand, there exist graphs of diameter 3 and of girth 6 which are not reconstructible. This new graph theory problem is motivated by reconstruction of chemical compounds.