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Mathematical Programming -     

Algorithms for Computing Characters for Symmetric Spaces

Symmetric spaces or more general symmetric k-varieties can be defined as the homogeneous spaces G _k /K _k , where G is a reductive algebraic group defined over a field k of characteristic not 2, K the fixed point group of an involution θ of G and G _k resp. K _k the sets k-rational points of G resp. K. These symmetric spaces have a fine structure of root systems, characters, Weyl groups etc., similar to the underlying algebraic group G. The relationship between the fine structure of the symmetric space and the group plays an important role in the study of these symmetric spaces and their applications. To develop a computer algebra package for symmetric spaces one needs explicit formulas expressing the fine structure of the symmetric space and group in terms of each other. In this paper we consider the case that k is algebraically closed and give explicit algorithmic formulas for expressing the characters of the weight lattice of the symmetric space in terms of the characters of the weight lattice of the group. These algorithms can easily be implemented in a computer algebra package. The root system of the symmetric space can be described as the image of the root system of the group under a projection π derived from an involution θ on . This implies that . Using these formulas for the characters of each of these lattices we show that in fact . A.G. Helminck is partially supported by N.S.F. Grant DMS-0532140.     

Criterion for complete determinacy for concave-convexlike games

Translated from Matematicheskie Zametki, Vol. 49, No. 3, pp. 73–76, March, 1991.     

Continuity Adjustment for Control Charts for Attributes

A unified approach is proposed for making a continuity adjustment on some control charts for attributes, e.g., np-chart and c-chart. through adding a uniform (0,1) random observation to the conventional sample statistic (e.g., npi and ci). The adjusted sample statistic then has a continuous distribution. Consequently, given any Type I risk a (the probability that the sample statistic is on or beyond the control limits), control charts achieving the exact value of a can be readily constructed. Guidelines are given for when to use the continuity adjustment control chart, the conventional Shewhart control chart (with ±3 standard deviations control limits), and the control chart based on the exact distribution of the sample statistic before adjustment.     

多产品公司创建平台邀请竞争对手进行联合销售的策略研究

由于资金分配或生产规模的限制,多产品公司的某类产品与专门销售此类产品的专业产品公司相比,会有一定的不足.以两个产品公司为对象,研究了两个竞争性公司的联合销售模式,即多产品公司投资建设平台,邀请销售单一产品的专业公司在平台上共同销售某类产品.运用主从博弈建立联合销售的基础模型,探讨不同销售模式下的相关投资,并利用数值计算进行决策分析.研究表明,通过创建平台进行联合销售,一方面,消除了消费者的额外购物成本;另一方面,在平台进行联合销售使得两个公司由单纯的竞争关系转化为相互补充,不仅能够吸引更多有不同产品需求的客户,而且能够在一定程度上提高两个产品公司的利润.     

Appropriate problems for learning and for performing—an issue for teacher training

Selecting, modifying or creating appropriate problems for mathematics class has become an activity of increaing importance in the professional development of German mathematics teachers. But rather than asking in general: “What is a good problem?” there should be a stronger emphasis on considering the specific goal of a problem, e.g.: “What are the ingredients that make a problem appropriate for initiating a learning process” or “What are the characteristics that make a problem appropriate for its use in a central test?” We propose a guiding scheme for teachers that turns out to be especially helpful, since the newly introduced orientation on outcome standards a) leads to a critical predominance of test items and b) expects teachers to design adequate problems for specific learning processes (e.g. problem solving, reasoning and modelling activities).     

Algorithms for approximation of invariant measures for IFS

We prove convergence of two algorithms approximating invariant measures to iterated function systems numerically. We consider IFSs with finitely many continuous and injective non-overlapping maps on the unit interval. The first algorithm is a version of the Ulam algorithm for IFSs introduced by Strichartz et al. [16]. We obtain convergence in the supremum metric for distribution functions of the approximating eigen-measures to a unique invariant measure for the IFS. We have to make some modifications of the usual way of treating the Ulam algorithm due to a problem concerning approximate eigenvalues, which is part of our more general situation with weights not necessarily being related to the maps of the IFS. The second algorithm is a new recursive algorithm which is an analogue of forward step algorithms in the approximation theory of ODEs. It produces a sequence of approximating measures that converges to a unique invariant measure with geometric rate in the supremum metric. The main advantage of the recursive algorithm is that it runs much faster on a computer (using Maple) than the Ulam algorithm.Mathematics Subject Classification (2000): 37A30, 37C30, 37M25, 47A58Acknowledgement I would like to express my deep gratitude to Andreas Strömbergsson and to the anonymous referee. The referee had several very enlightening comments, which Andreas helped me to deal with. Section 4 is essentially due to Andreas and he also came up with the new Proposition 3 and helped me to improve Lemma 1. Thanks also to Svante Janson, Anders Johansson, Sten Kaijser, Robert Strichartz and Hans Wallin.     

Christoffel-type functions for -orthogonal polynomials for Freud weights

This paper gives upper and lower bounds of the Christoffel-type functions , for the m-orthogonal polynomials for a Freud weight W=e^-Q, which are given as follows. Let a_n=a_n(Q) be the nth Mhaskar–Rahmanov–Saff number, φ_n(x)=max{n^-2/3,1-|x|/a_n}, and d>0. Assume that QC(R) is even, , and for some A,B>1

Then for xR

and for |x|a_n(1+dn^-2/3)

     

The Fermat rule for multifunctions for super efficiency

Using variational analysis, we study the vector optimization problems with objectives being closed multifunctions on Banach spaces or in Asplund spaces. In terms of the coderivatives and normal cones, we present Fermat’s rules as necessary or sufficient conditions for a super efficient solution of the above problems.     

Basin attractors for various methods for multiple roots

There are several methods for approximating the multiple zeros of a nonlinear function when the multiplicity is known. The methods are classified by the order, informational efficiency and efficiency index. Here we consider other criteria, namely the basin of attraction of the method and its dependence on the order. We discuss all known methods of orders two to four and present the basin of attraction for several examples.     

Graphical Diagnostics for Markov Models for Categorical Data

Markov models are widely used as a method for describing categorical data that exhibit stationary and nonstationary autocorrelation. However, diagnostic methods are a largely overlooked topic for Markov models. We introduce two types of residuals for this purpose: one for assessing the length of runs between state changes, and the other for assessing the frequency with which the process moves from any given state to the other states. Methods for calculating the sampling distribution of both types of residuals are presented, enabling objective interpretation through graphical summaries. The graphical summaries are formed using a modification of the probability integral transformation that is applicable for discrete data. Residuals from simulated datasets are presented to demonstrate when the model is, and is not, adequate for the data. The two types of residuals are used to highlight inadequacies of a model posed for real data on seabed fauna from the marine environment.

Supplemental materials, including an R-package RMC with functions to perform the diagnostic measures on the class of models considered in this article, are at the journal’s website. The R-package is also available at CRAN.     

Estimates for the classical parametrix for the Laplacian

We consider the integral operators which were used classically to give a parametrix and remainder for the Laplacian on a Riemannian manifold. Their kernels are defined in terms of the distance function. These operators are shown to be bounded operators on the L₂ Hilbert spaces of differential forms, under the hypotheses that the manifold be complete and of finite volume, and that it satisfy curvature bounds. Furthermore, the remainder is shown to be compact.     

Polynomial cost for solving IVP for high-index DAE

We show that the cost of solving initial value problems for high-index differential algebraic equations is polynomial in the number of digits of accuracy requested. The algorithm analyzed is built on a Taylor series method developed by Pryce for solving a general class of differential algebraic equations. The problem may be fully implicit, of arbitrarily high fixed index and contain derivatives of any order. We give estimates of the residual which are needed to design practical error control algorithms for differential algebraic equations. We show that adaptive meshes are always more efficient than non-adaptive meshes. Finally, we construct sufficiently smooth interpolants of the discrete solution. AMS subject classification (2000) 34A09, 65L80, 68Q25     

A unified theory for homotopy principles for multimaps

In this article, we present a definition of d-essential and d–L-essential maps in completely regular topological spaces and we establish a homotopy property for both d-essential and d–L-essential maps. Also using the notion of extendability, we present new continuation theorems.