Product integration with the Clenshaw-Curtis points: Implementation and error estimates

Summary This paper is concerned with the practical implementation of a product-integration rule for approximating , wherek is integrable andf is continuous. The approximation is , where the weightsw _ni are such as to make the rule exact iff is any polynomial of degree n. A variety of numerical examples, fork(x) identically equal to 1 or of the form |–x| with >–1 and ||1, or of the form cosx or sinx, show that satisfactory rates of convergence are obtained for smooth functionsf, even ifk is very singular or highly oscillatory. Two error estimates are developed, and found to be generally safe yet quite accurate. In the special casek(x)1, for which the rule reduces to the Clenshaw-Curtis rule, the error estimates are found to compare very favourably with previous error estimates for the Clenshaw-Curtis rule.     

A Doubly Adaptive Integration Algorithm Using Stratified Rules

A doubly adaptive integration algorithm chooses between a higher order rule applied on the current subinterval or the subdivision of the interval. We describe one such algorithm using a stratified sequence of integration rules. We present a criterion to select the suitable strategy, depending on the type of integrand, using available information.     

A Product Formula Related to Quantum Zeno Dynamics

We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space with the convergence in It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P. The convergence in is demonstrated in the case of a finite-dimensional P. The main result is illustrated in the example where the projection corresponds to a domain in and the unitary group is the free Schrödinger evolution.submitted 21/06/04, accepted 12/10/04     

Categorical Geometry and Integration Without Points

The theory of integration over infinite-dimensional spaces is known to encounter serious difficulties. Categorical ideas seem to arise naturally on the path to a remedy. Such an approach was suggested and initiated by Segal in his pioneering article (Segal, Bull Am Math Soc 71:419–489, 1965). In our paper we follow his ideas from a different perspective, slightly more categorical, and strongly inspired by the point-free topology. First, we develop a general (point-free) concept of measurability (extending the standard Lebesgue integration when applying to the classical σ-algebra). Second (and here we have a major difference from the classical theory), we prove that every finite-additive function μ with values in [0,1] can be extended to a measure on an abstract σ-algebra; this correspondence is functorial and yields uniqueness. As an example we show that the Segal space can be characterized by completely canonical data. Furthermore, from our results it follows that a satisfactory point-free integration arises everywhere where we have a finite-additive probability function on a Boolean algebra.     

Maximal Points of Stable and Related Polynomials

We prove that any polynomial having all its roots in a closed half-plane, whose boundary contains the origin, has either one or two maximal points, and only one if it has at least one root in the open half-plane. This result concerns stable polynomials as well as polynomials having only real roots, including real orthogonal polynomials.     

N个度量空间上的相关不动点的注记

得到了 n个度量空间上的相关不动点的几个结果 .     

A Longitudinal Look at Attitudes and Perceptions Related to the Integration of Mathematics,Science, and Technology Education

The purpose of this study is to provide an in‐depth analysis of attitudes and perceptions related to the integration of mathematics, science, and technology education of preservice teachers preparing to teach STEM disciplines. Longitudinal data by individual cohort and across 7 years of the Integrated Mathematics, Science, and Technology (MSAT) Program are reported, analyzed, and interpreted to help design and improve preservice teacher education programs and improve teaching and learning in STEM classrooms. Results of quantitative analyses indicate that there was generally no change in preservice teacher attitudes and perceptions related to the value of the integration of mathematics, science, and technology education—they clearly valued integration at the onset and at the completion of the program. However, there was a significant change in preservice teacher attitudes and perceptions related to integration feasibility in terms of inefficiency and difficulty. Implications for teacher education programs include: (a) more exposure to concepts, processes, and skills in STEM that are similar, analogous, complementary, or synergistic; (b) familiarity with instructional strategies and access to resources; (c) deeper understanding of content across STEM; and (d) strategies for collaboration and team work to make integrated instruction time more efficient and less difficult to manage.     

Powers of Euler's Product and Related Identities

Ramanujan's partition congruences can be proved by first showing that the coefficients in the expansions of (q; q) ^r satisfy certain divisibility properties when r = 4, 6 and 10. We show that much more is true. For these and other values of r, the coefficients in the expansions of (q; q) ^r satisfy arithmetic relations, and these arithmetic relations imply the divisibility properties referred to above. We also obtain arithmetic relations for the coefficients in the expansions of (q; q) ^r (q ^t; q ^t) ^s , for t = 2, 3, 4 and various values of r and s. Our proofs are explicit and elementary, and make use of the Macdonald identities of ranks 1 and 2 (which include the Jacobi triple product, quintuple product and Winquist's identities). The paper concludes with a list of conjectures.     

Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere S ^r⊂R ^r+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d _n =dim P _n points, where P _n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S ² of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

     

积商求导公式与微分方程求解

对一些特殊类型的一阶或二阶微分方程,从其结构特点出发,联系积商求导公式,可减少求解步骤并降低计算量.     

投影矩阵的Kronecker积及其相关的正交表

文[3]证明了当P是素数时投影矩阵τp与P水平对称正交表的矩阵象的Kronecker积所包含的正交表是存在的，本文进一步研究了一个非对称正交表的矩阵象和投影矩阵τp的Kronecker积所包含的正交表的存在性，从而推广了文[3]的结论，并且构造出了一些饱和度很高的混合水平正交表．     

Regularity and Integration of Set-Valued Maps Represented by Generalized Steiner Points

A family of probability measures on the unit ball in generates a family of generalized Steiner (GS-)points for every convex compact set in . Such a “rich” family of probability measures determines a representation of a convex compact set by GS-points. In this way, a representation of a set-valued map with convex compact images is constructed by GS-selections (which are defined by the GS-points of its images). The properties of the GS-points allow to represent Minkowski sum, Demyanov difference and Demyanov distance between sets in terms of their GS-points, as well as the Aumann integral of a set-valued map is represented by the integrals of its GS-selections. Regularity properties of set-valued maps (measurability, Lipschitz continuity, bounded variation) are reduced to the corresponding uniform properties of its GS-selections. This theory is applied to formulate regularity conditions for the first-order of convergence of iterated set-valued quadrature formulae approximating the Aumann integral.      

Blow-Up Estimates at Horizontal Points and Applications

Horizontal points of smooth submanifolds in stratified groups play the role of singular points with respect to the Carnot-Carathéodory distance. When we consider hypersurfaces, they coincide with the well known characteristic points. In two step groups, we obtain pointwise estimates for the Riemannian surface measure at all horizontal points of C ^1,1 smooth submanifolds. As an application, we establish an integral formula to compute the spherical Hausdorff measure of any C ^1,1 submanifold. Our technique also shows that C ² smooth submanifolds everywhere admit an intrinsic blow-up and that the limit set is an intrinsically homogeneous algebraic variety.     

A Geometrical Look at Iterative Methods for Operators with Fixed Points

ABSTRACT

We distinguish classes of operators T with fixed points on a real Hilbert space by comparing the distances of a point x and its image Tx to the (set of) fixed points of T; this leads to a ranking of those classes, based on a nonnegative parameter. That same parameter also lets us conclude about the sign of and an upper bound for a characteristic inner product result that arises in iterative processes to obtain a common fixed point of a set of operators. We use that parameter as the starting point for a geometrically-inclined study of specific iterative algorithms intended to find a common fixed point of operators belonging to such class.     

A Product Integration Approach Based on New Orthogonal Polynomials for Nonlinear Weakly Singular Integral Equations

This paper provides with a generalization of the work by Chelyshkov (Electron. Trans. Numer. Anal. 25(7): 17–26, 2006), who has introduced sequences of orthogonal polynomials over [0,1] which can be expressed in terms of Jacobi polynomials. We develop a new approach of product integration algorithm based on these orthogonal polynomials including the numerical quadratures for solving the nonlinear weakly singular Volterra integral equations. The convergence analysis of the proposed scheme is derived and numerical results are given showing a marked improvement in comparison with recent numerical methods.     

Robust canonical correlations: A comparative study

Summary  Several approaches for robust canonical correlation analysis will be presented and discussed. A first method is based on the definition of canonical correlation analysis as looking for linear combinations of two sets of variables having maximal (robust) correlation. A second method is based on alternating robust regressions. These methods are discussed in detail and compared with the more traditional approach to robust canonical correlation via covariance matrix estimates. A simulation study compares the performance of the different estimators under several kinds of sampling schemes. Robustness is studied as well by breakdown plots.     

Tractability of Integration in Non-periodic and Periodic Weighted Tensor Product Hilbert Spaces

We study strong tractability and tractability of multivariate integration in the worst case setting. This problem is considered in weighted tensor product reproducing kernel Hilbert spaces. We analyze three variants of the classical Sobolev space of non-periodic and periodic functions whose first mixed derivatives are square integrable. We obtain necessary and sufficient conditions on strong tractability and tractability in terms of the weights of the spaces. For the three Sobolev spaces periodicity has no significant effect on strong tractability and tractability. In contrast, for general reproducing kernel Hilbert spaces anything can happen: we may have strong tractability or tractability for the non-periodic case and intractability for the periodic one, or vice versa.     

On Goppa Codes and Weierstrass Gaps at Several Points

We generalize results of Homma and Kim [J. Pure Appl. Algebra Vol. 162, (2001), pp. 273–290] concerning an improvement on the Goppa bound on the minimum distance of certain Goppa codes.This paper was written while the second author was visiting The University of Valladolid (Dpto. Algebra, Geometría y Topología; Fac. de Ciencias) supported by the Grant SB2000-0225 from the Secretaria de Estado de Educacíon y Universidades del Ministerio de Educacíon, Cultura y Deportes de España.     

Peak Points for Pseudoconvex Domains: A Survey

This paper surveys results concerning peak points for pseudoconvex domains. It includes results of Laszlo that have not been published elsewhere.