The Cereal Box Problem Revisited

The Cereal Box problem is fascinating for students of ail levels. Rich in mathematical content, this problem offers students the opportunity to collect data, make conjectures, and derive mathematical models. Using Monte Carlo methods, the Cereal Box problem is investigated in this paper, using both an experimental and theoretical framework. This investigation extends previous considerations of the Cereal Box problem. Using empirical data, students can discover patterns and relationships that help them understand the origin of the theoretical solution to the problem. Building on experimental findings, a theoretical model is derived, showing that the expected solution of the Cereal Box problem is formed from the sum of successive geometric series.     

Cubature on Wiener Space: Pathwise Convergence

Cubature on Wiener space (Lyons and Victoir in Proc. R. Soc. Lond. A 460(2041):169–198, 2004) provides a powerful alternative to Monte Carlo simulation for the integration of certain functionals on Wiener space. More specifically, and in the language of mathematical finance, cubature allows for fast computation of European option prices in generic diffusion models. We give a random walk interpretation of cubature and similar (e.g. the Ninomiya–Victoir) weak approximation schemes. By using rough path analysis, we are able to establish weak convergence for general path-dependent option prices.     

The Wiener Sequential Testing Problem with Finite Horizon

We present a solution of the Bayesian problem of sequential testing of two simple hypotheses about the mean value of an observed Wiener process on the time interval with finite horizon. The method of proof is based on reducing the initial optimal stopping problem to a parabolic free-boundary problem where the continuation region is determined by two continuous curved boundaries. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundaries can be characterized as a unique solution of the coupled system of two nonlinear integral equations.     

Tangent Processes on Wiener Space

This paper deals with the study of the Malliavin calculus of Euclidean motions on Wiener space (i.e. transformations induced by general measure-preserving transformations, called “rotations”, and H-valued shifts) and the associated flows on abstract Wiener spaces.     

A Nonlinear Deformation of Wiener Space

Based on the complex hyperbolic geometry associated with discrete series of SU(1, 1), we construct a quasi-invariant and ergodic measure on infinite product of Poincaré disc and a hyperbolic analogue of numerical Wiener space which turns out to be a nonlinear deformation of the Wiener space. An integration by parts formula is established. We also investigate the orthogonal decomposition of the L ²-holomorphic functions which is an analogue of the Wiener–Itô–Segal decomposition. In the zero-curvature and large spin limit, we recover the linear Wiener space.     

The Multi-Story Space Assignment Problem

The Multi-Story Space Assignment Problem (MSAP) is an innovative formulation of the multi-story facility assignment problem that allows one to model the location of departments of unequal size within multi-story facilities as a Generalized Quadratic 3-dimensional Assignment Problem (GQ3AP). Not only can the MSAP generate the design of the location of the departments in the facility, the MSAP also includes the evacuation planning for the facility. The formulation, background mathematical development, and computational experience with a branch and bound algorithm for the MSAP are also presented.     

Hankel Operators over the Complex Wiener Space

This work introduces and investigates (small) Hankel operators H _b on the Hilbert space of holomorphic, square integrable Wiener functionals. A regularity condition on the symbol b, which guarantees the boundedness of H _b , is provided. The symbols b for which H _b is of Hilbert–Schmidt type are characterized, and a representation of H _b by an integral operator is given. The proofs employ the hypercontractivity of the Ornstein–Uhlenbeck semigroup, together with approximations by finitely many variables. These results extend known results from a finite-dimensional context.     

Relatively Compact Sets on Abstract Wiener Space

In this note, we obtain a sufficient and necessary condition for a set in an abstract Winner space （X, H, μ） to be relatively compact in L^2（X, μ）. Meanwhile, we give a sufficient condition for relative compactness in L^P（X, μ） for p〉1. We also provide an example of Da Prato-Malliavin Nualart to show the result.     

Time-Localization of Random Distributions on Wiener Space

A nuclear space of distributions on Wiener space was constructed by Gorostiza and Nualart [10] as a framework for studying weak convergence of trajectorial fluctuations of particle systems. A basic problem in recovering the usual time-evolution results from the trajectorial ones consists in associating in a unique way an -valued process to a random distribution on by localizing it at each time t [0,1]. In this paper we solve this problem for a large class of random distributions which includes trajectorial fluctuation limits of some systems of diffusions.     

About Widths of Wiener Space in theLq-Norm

We calculate the average Kolmogorov and linearn-widths of the Wiener space in theL_q-norm. For the case 1 ≤q< ∞, then-widthsd_ndecrease asymptotically asn^-1/2.     

Evaluation Formulas for a Conditional Feynman Integral Over Wiener Paths in Abstract Wiener Space

In this paper, we introduce a simple formula for conditional Wiener integrals over , the space of abstract Wiener space valued continuous functions. Using this formula, we establish various formulas for a conditional Wiener integral and a conditional Feynman integral of functionals on in certain classes which correspond to the classes of functionals on the classical Wiener space introduced by Cameron and Storvick. We also evaluate the conditional Wiener integral and conditional Feynman integral for functionals of the form which are of interest in Feynman integration theories and quantum mechanics.     

Shape from Shadows: A Hilbert Space Setting

We study the problem of recovering a surface from the shadows it casts on itself when lighted by the sun at various times of the day. Shadows can create both linear and nonlinear information. We will show how to incorporate both types of information in the solution. The problem is formulated and solved in a Hilbert space setting and the spline algorithm interpolating the data that result from the shadows is constructed. This algorithm is optimal in terms of the approximation error and has low cost. We furthermore derive optimal information for this problem.     

空间曲线的双切圆问题

Peter和Duan研究了平面曲线双切圆问题,Duau还研究了Rn中的(n-1)维闭子流形的双切球问题.本文则研究了空间曲线的双切圆问题,并利用几何分析的方法,获得了如果空间曲线是优良曲线则必定存在双切圆的结果,推广了平面曲线必定存在双切圆的结果.     

拟Hermite插值在一重积分Wiener空间下的平均误差

本文在L_p范数逼近意义下确定了一种拟Hermite插值多项式列在一重积分Wiener空间下平均误差的弱渐近阶.结果显示在信息基复杂性的意义下,若可允许信息泛函为Hermite数据,则这种插值多项式列的平均误差弱等价于相应的最小非自适应信息半径.     

Grnwald插值算子在Wiener空间下的平均误差

得到了以第二类Chebyshev多项式的零点为插值结点组的Grnwald插值多项式在Wiener空间下平均误差的一个估计.     

Finite Rank Hankel Operators over the Complex Wiener Space

This work studies finite rank Hankel operators H _b on a Hilbert space of holomorphic, square integrable Wiener functionals. The main tool to investigate these operators is their unitary equivalent representation on the Hilbert space of skeletons. The finite rank property is characterized in terms of a functional equation for the symbol b, which generalizes the well known equation b(z+w)=b(z)b(w). Also finite rank symbols of polynomial type are characterized in terms of their chaos expansions.     

The Average Errors for Hermite Interpolation on the 1-Fold Integrated Wiener Space

For the weighted approximation in Lp-norm,the authors determine the weakly asymptotic order for the p-average errors of the sequence of Hermite interpolation based on the Chebyshev nodes on the 1-fold integrated Wiener space.By this result,it is known that in the sense of information-based complexity,if permissible information functionals are Hermite data,then the p-average errors of this sequence are weakly equivalent to those of the corresponding sequence of the minimal p-average radius of nonadaptive information.