Decision Trees with Continuous Distribution

Decision trees are often used as a convenient way to visualize, and then solve, a decision problem. A standard problem approached in this way is a decision as to whether or not to sample, followed by a decision whether or not to engage in an activity. This approach has been limited to a small number of sample outcomes to be practical. This paper develops an approach which permits a continuous distribution of sample outcomes to be used. A new distribution, the "skewed parabolic distribution" is introduced as a flexible way of representing the judgemental (probabilistic) beliefs of the assessor. This new distribution is compared with the beta distribution. An algorithm, which is easily programmed and is practical for hand computation with a calculator and logarithm tables, is developed for the complete solution of a decision tree problem using this skewed parabolic distribution. An example is given. It is concluded that the skewed parabolic distribution makes use of continuous distributions practical for decision trees, and has advantages over the beta for this purpose.     

An Ingham type proof for the boundary observability of a N−d wave equation

The boundary observability of the wave equation has been studied by many authors. A method of choice is to use the multiplier method (cf. Komornik (1994)). Recently, in Ramdani et al. (2005), a first Fourier based proof is given in the case where the domain is a square, thanks to a new Hautus type test. We give here a new self-contained proof with an Ingham type approach in the more general case where the domain is a product of intervals; this leads, in contrary to the proof in Ramdani et al., to explicit time and constants. However, we do not reach the optimal time which can be obtained for this problem by the multiplier method. To cite this article: M. Mehrenberger, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

奇数阶亚循环p-群分类的一个证明

一个有限p群p称为亚循环的,如果P有一个循环的正规子群A,使得PA是循环的.1973年King在文[2]中对亚循环p群进行了分类;在文[4,5]中M.F.NewmanandMingYaoXu发现了这些群的新的表示,给出了新的分类方法,这种方法是由p群生成算法(见文[3])得到的.本文的目的是给这些结果的另一种证明,与p群生成算法是不相关的.     

On Euler numbers, polynomials and related p-adic integrals

In this note we give a new proof of Witt's formula for Euler numbers, which are related to some known or new identities involving the Euler numbers. We also obtain a brief proof of a classical result on Euler numbers modulo of two due to M.A. Stern using the approach of p-adic integration, which was recently proved by G. Liu, and Z.-W. Sun. Finally some explicit formulas for Genocchi numbers are proved and applications are given.     

A proof of strongly uniform termination for Gödel's T by methods from local predicativity

We estimate the derivation lengths of functionals in G?del's system of primitive recursive functionals of finite type by a purely recursion-theoretic analysis of Schütte's 1977 exposition of Howard's weak normalization proof for . By using collapsing techniques from Pohlers' local predicativity approach to proof theory and based on the Buchholz-Cichon and Weiermann 1994 approach to subrecursive hierarchies we define a collapsing f unction so that for (closed) terms of G?del's we have: If reduces to then By one uniform proof we obtain as corollaries: A derivation lengths classification for functionals in , hence new proof of strongly uniform termination of . A new proof of the Kreisel's classific ation of the number-theoretic functions which can be defined in , hence a classification of the provably total functions of Peano Arithmetic. A new proof of Tait's results on weak normalization for . A new proof of Troelstra's result on strong normalization for . Additionally, a slow growing analysis of G?del's is obtained via Girard's hierarchy comparison theorem. This analyis yields a contribution to two open pro blems posed by Girard in part two of his book on proof theory. For the sake of completeness we also mention the Howard Schütte bound on derivation lengths for the simple typed -calculus. Received August 4, 1995     

Macdonald identities and multidimensional theta-functions

A new proof of the combinatorial Macdonald identities is presented. It is shown that one may regard these identities as a decomposition of certain multidimensional theta-functions into infinite products. The proof is based on some analytical properties of theta-functions. It is briefly discussed how one can modify the proof in order to replace analytical arguments by formal ones involving only operations with formal series.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 240, 1997, pp. 67–77.The author is grateful to A. M. Vershik and Yu. M. Bazlov for their interest and helpful comments.This research was supported by ISSEP (grant No. A96-1965).     

Stein's method and Plancherel measure of the symmetric group

We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.

     

混合体积、对数凹序列和B样条函数

本文主要通过样条函数方法研究与之相关的离散几何学和组合学问题.在离散几何学方面主要考虑超立方体切面(cube slicing)体积和混合体(mixed volume)的样条表示,利用B样条函数的几何解释,将超立方体切面问题转化为与之等价的样条函数问题,分别给出Laplace和P′olya关于超立方体切面定理的样条证明,将样条函数与混合体积联系起来,给出一类混合体积的样条解释.利用这种解释可以得到一类具有对数凹性质的组合序列,从而部分地回答了Schmidt和Simion所提出的关于混合体积的公开问题.在组合数学方面主要考虑多种组合多项式与样条函数的关联以及组合序列对数凹性质的样条方法研究.本文借助丰富的样条函数理论,不但验证了离散几何学和组合数学中很多现有的结果,而且得到了一系列离散数学对象的新性质,建立了离散数学问题与具有连续性特质的样条函数之间的内在联系.     

Vertices of Specht modules and blocks of the symmetric group

This paper studies the vertices, in the sense defined by J.A. Green, of Specht modules for symmetric groups. The main theorem gives, for each indecomposable non-projective Specht module, a large subgroup contained in one of its vertices. A corollary of this theorem is a new way to determine the defect groups of symmetric groups. The main theorem is also used to find the Green correspondents of a particular family of simple Specht modules; as a corollary, this gives a new proof of the Brauer correspondence for blocks of the symmetric group. The proof of the main theorem uses the Brauer homomorphism on modules, as developed by M. Broué, together with combinatorial arguments using Young tableaux.     

Geometric approach to stable homotopy groups of spheres. The Adams–Hopf invariants

In this paper, a geometric approach to stable homotopy groups of spheres based on the Pontryagin–Thom construction is proposed. From this approach, a new proof of the Hopf-invariant-one theorem of J. F. Adams for all dimensions except 15, 31, 63, and 127 is obtained. It is proved that for n > 127, in the stable homotopy group of spheres Π _n , there is no element with Hopf invariant one. The new proof is based on geometric topology methods. The Pontryagin–Thom theorem (in the form proposed by R. Wells) about the representation of stable homotopy groups of the real, projective, infinite-dimensional space (these groups are mapped onto 2-components of stable homotopy groups of spheres by the Kahn–Priddy theorem) by cobordism classes of immersions of codimension 1 of closed manifolds (generally speaking, nonoriented) is considered. The Hopf invariant is expressed as a characteristic class of the dihedral group for the self-intersection manifold of an immersed codimension-1 manifold that represents the given element in the stable homotopy group. In the new proof, the geometric control principle (by M. Gromov) for immersions in the given regular homotopy classes based on the Smale–Hirsch immersion theorem is required. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 3–15, 2007.     

多项式带余除法定理的一种新证明

利用线性空间的理论,在高观点下给出了多项式带余除法定理的一个新证明.它避免了使用构造法这种技巧性较高的数学方法,使学生比较容易轻松掌握.     

有偏总体的极差控制图

根据加权标准差方法建立有偏总体的极差控制图,它基于有偏总体来计算对应于正态分布的控制图常数,根据样本数据的偏度来计算上下控制限,对于总体是对称分布,该控制图退化为标准的休哈特控制图.最后,用蒙特卡洛方法给出了改进的控制图常数.     

Generalized solutions of the Vlasov-Poisson system with singular data

We study spherically symmetric solutions of the Vlasov-Poisson system in the context of algebras of generalized functions. This allows to model highly concentrated initial configurations and provides a consistent setting for studying singular limits of the system. The proof of unique solvability in our approach depends on new stability properties of the system with respect to perturbations.     

Robustness of Tube Formula Based Confidence Bands

Abstract

Simultaneous confidence bands of a regression curve may be used to quantify the uncertainty of an estimate of the curve. The tube formula for volumes of tubular neighborhoods of a manifold provides a very powerful method for obtaining such bands at a prescribed level, when errors are Gaussian. This article studies robustness of the tube formula for non-Gaussian errors. The formula holds without modification for an error vector with a spherically symmetric distribution. Simulations are used for a variety of independent non-Gaussian error distributions. The results are acceptable for contaminated and heavy tailed error distributions. The formula can break down in some extreme cases for discrete and highly skewed errors. Computational issues involved in applying the tube formula are also discussed.     

A companion matrix approach to the study of zeros and critical points of a polynomial

In this paper, we introduce a new type of companion matrices, namely, D-companion matrices. By using these D-companion matrices, we are able to apply matrix theory directly to study the geometrical relation between the zeros and critical points of a polynomial. In fact, this new approach will allow us to prove quite a number of new as well as known results on this topic. For example, we prove some results on the majorization of the critical points of a polynomial by its zeros. In particular, we give a different proof of a recent result of Gerhard Schmeisser on this topic. The same method allows us to prove a higher order Schoenberg-type conjecture proposed by M.G. de Bruin and A. Sharma.     

基于尺度赋权方差的有偏总体EWMA均值控制图

基于尺度赋权方差法给出总体分布有偏并假定总体分布未知情况下休哈特均值控制图的控制限.基于此研究结果,将其推广到EWMA均值控制图,给出总体分布有偏并假定总体分布未知情况下EWMA均值控制图的控制界限,并以质量特性值服从对数正态分布为例,给出EWMA均值控制图的控制效果的模拟分析.     

Sylvester方程在矩阵扰动分析中的应用

§1.引言 矩阵扰动分析的研究对于矩阵论的发展及数值分析问题计算结果的分析和处理都有重要意义.有关特征值、广义特征值及最小二乘问题的主要研究结果均含于[1]中,[5]运用二次方程根的判别法通过对代数Ricatti方程的解的估计给出了QR分解因子及Cholesky因子的扰动分析,但论证方法及所得结果都比较复杂且所求条件很强.[3]和     

An iterative method for the finite-time bilinear-quadratic control problem

For bilinear control systems with quadratic cost, the so-called bilinear-quadratic problems, a feedback controller for the finite-time case is designed. An iteration procedure in close proximity to the Riccati approach is presented, and the proof of convergence is outlined. The potential of the new method is discussed, and the design procedure is illustrated for two examples.     

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The log-normal distribution is a common choice for modeling positively skewed data arising from many practical applications.This article introduces a new method of constructing confidence interval for a common mean shared by several log-normal populations through confidence distributions, which combines all information from independent sources. We develop a non-trivial weighting approach by taking account of the sample variances of related quantities to enhance efficiency. Combined confidence distributions are used to construct confidence intervals for the common mean and a simplified version of one existing method is also proposed. We conduct simulation studies to evaluate the performance of the proposed methods in comparison with existing methods. Our simulation results show that the weighting approach yields shorter interval length than the non-weighting approach. The newly proposed confidence intervals perform very well in terms of empirical coverage probability and average interval length. Finally, applications of the proposed methodology is illustrated through three real data examples.