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.     

Some new results on transition probability

In this paper, we study the basic properties of stationary transition probability of Markov processes on a general measurable space （E, δ）, such as the continuity, maximum probability, zero point, positive probability set,standardization, and obtain a series of important results such as Continuity Theorem, Representation Theorem, Levy Theorem and so on. These results are very useful for us to study stationary tri-point transition probability on a general measurable space （E, δ）. Our main tools such as Egoroff＇s Theorem, Vitali-Hahn-Saks＇s Theorem and the theory of atomic set and well- posedness of measure are also very interesting and fashionable.     

关于伪压缩映象不动点迭代的一点注记

设$K$是自反的并且具有一致Gateaux可微范数的Banach空间$E$的非空有界闭凸子集.设$T:K\rightarrow K$是一致连续的伪压缩映象.假设$K$的每一非空有界闭凸子集对非扩张映象具有不动点性质.设$\{\lambda_n\}$是$(0,\frac{1}{2}]$中序列满足: (i) $\lim_{n\rightarrow \infty}\lambda_n=0$; (ii) $\sum_{n=0}^{\infty}\lambda_n=\infty$.任给$x_1\in K$,定义迭代序列$\{x_n\}$为:$x_{n+1}=(1-\lambda_n)x_n+\lambda_nTx_n-\lambda_n(x_n-x_1),n\geq 1.$若$\lim_{n\rightarrow \infty}\|x_n-Tx_n\|=0$, 则上述迭代产生的$\{x_n\}$强收敛到$T$的不动点.     

Semigroup wellposedness and asymptotic stability of a compressible Oseen–structure interaction via a pointwise resolvent criterion

In this study, we consider the Oseen structure of the linearization of a compressible fluid–structure interaction (FSI) system for which the interaction interface is under the effect of material derivative term. The flow linearization is taken with respect to an arbitrary, variable ambient vector field. This process produces extra “convective derivative” and “material derivative” terms, which render the coupled system highly nondissipative. We show first a new well-posedness result for the full incorporation of both Oseen terms, which provides a uniformly bounded semigroup via dissipativity and perturbation arguments. In addition, we analyze the long time dynamics in the sense of asymptotic (strong) stability in an invariant subspace (one-dimensional less) of the entire state space, where the continuous semigroup is uniformly bounded. For this, we appeal to the pointwise resolvent condition introduced in Chill and Tomilov [Stability of operator semigroups: ideas and results, perspectives in operator theory Banach center publications, 75 (2007), Institute of Mathematics Polish Academy of Sciences, Warszawa, 71–109], which avoids an immensely technical and challenging spectral analysis and provides a short and relatively easy-to-follow proof.     

The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic solid circular disks or cylinders, rotating at constant angular velocity about a central axis. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic problem for a homogeneous isotropic rotating solid disk or cylinder is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. For the case when the Young"s modulus has a power-law dependence on the radial coordinate, explicit exact solutions are obtained. It is shown that the stress response of the inhomogeneous disk (or cylinder) is significantly different from that of the homogeneous body. For example, the maximum radial and hoop stresses do not, in general, occur at the center as in the case for the homogeneous material. Furthermore, for the case where the Young"s modulus increases with radial distance from the center, it is shown that radially symmetric solutions exist provided the rate of growth of the Young"s modulus is, at most, cubic in the radial variable. It is also shown for the general inhomogeneous isotropic case how the material inhomogeneity may be tailored so that the radial and hoop stress are identical throughout the disk. This revised version was published online in August 2006 with corrections to the Cover Date.     

Strong convergence theorems for nonexpansive semi-groups in Banach spaces

Some strong convergence theorems of explicit composite iteration scheme for nonexpansive semi-groups in the framework of Banach spaces are established.Results presented in the paper not only extend and improve the corresponding results of Shioji- Takahashi,Suzuki,Xu and Aleyner-Reich,but also give a partially affirmative answer to the open questions raised by Suzuki and Xu.     

The problems of nonlinear bending for orthotropic rectangular plate with four clamped edges

（黄家寅）（秦圣立）ＴＨＥＰＲＯＢＬＥＭＳＯＦＮＯＮＬＩＮＥＡＲＢＥＮＤＩＮＧＦＯＲＯＲＴＨＯＴＲＯＰＩＣＲＥＣＴＡＮＧＵＬＡＲＰＬＡＴＥＷＩＴＨＦＯＵＲＣＬＡＭＰＥＤＥＤＧＥＳ￥ＨｕａｎｇＪｉａｙｉｎ;ＱｉｎＳｈｅｎｇｌｉ（ＱｕｆｕＮｏｒｍａｌＵｎ...     

Properties of the random search in global optimization

From theorems which we prove about the behavior of gaps in a set ofN uniformly random points on the interval [0, 1], we determine properties of the random search procedure in one-dimensional global optimization. In particular, we show that the uniform grid search is better than the random search when the optimum is chosen using the deterministic strategy, that a significant proportion of large gaps are contained in the uniformly random search, and that the error in the determination of the point at which the optimum occurs, assuming that it is unique, will on the average be twice as large using the uniformly random search compared with the uniform grid. In addition, some of the properties of the largest gap are verified numerically, and some extensions to higher dimensions are discussed. The latter show that not all of the conclusions derived concerning the inadequacies of the one-dimensional random search extend to higher dimensions, and thaton average the random search is better than the uniform grid for dimensions greater than 6.This paper is based on work started in the Statistics Department of Princeton University when the first author was visiting as a Research Associate. Part of this research was supported by the Office of Naval Research, Contract No. 0014-67-A-0151-0017, and by the US Army Research Office—Durham, Contract No. DA-31-124-ARO-D-215.2 The authors wish to thank B. Omodei for his careful work in preparing the programs for the results of Figs. 1–2 and Table 1. The computations were performed on the IBM 360/50 of the Australian National University's Computer Centre. Thanks are also due to R. Miles for suggestions regarding the extension of the results to multidimensional regions, and to P. A. P. Moran and R. Brent for suggestions regarding the evaluation of the integral ₀ ¹ ... ^0/1 (x ₁ ² + ... +x _n ^/2 )^1/2 dx ₁ ...dx _n.     

On positive invertibility of operators and their decompositions

In Banach spaces ordered by a normal cone that contains interior points the positive invertibility of operators is studied. If there exists a uniformly positive functional then any positively invertible operator A possesses a B -decomposition, i.e., a positive decomposition A = U – V with the properties: U^–1 exists, VU^–1 ≥ 0, Ax ≥ 0, U x ≥ 0 imply x ≥ 0 and r (VU^–1) < 1. Earlier it was shown that the existence of a B -decomposition with r (VU^–1) < 1 is sufficient for the positive invertibility of the operator A. Peris' result is obtained as a special case of the main theorem. The decomposition is demonstrated for a positively invertible operator in a Banach space ordered by an ice cream cone (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Model filled polymers. IX. Synthesis of uniformly crosslinked polystyrene microbeads

Monodisperse sized crosslinked polystyrene (PS) beads prepared by reaction of styrene (S) and divinylbenzene (DVB), in batch emulsion copolymerization in the absence of emulsifiers, are not uniformly crosslinked, because DVB is more reactive than S. For copolymerization of 1 to 10 mol % DVB and S, within each crosslinked PS microbead, the crosslink density varies by a factor exceeding two and decreases with increased conversion. A semicontinuous copolymerization, involving incremental additions of DVB, produces uniformly crosslinked PS beads. For both copolymerization techniques, T_g correlates well with crosslink density and PS beads are spherical and monodisperse in size. © 1992 John Wiley & Sons, Inc.