Diagrammatic Analysis of Interval Linear Equations - Part II: The Two-Dimensional Case and Generalization to n Dimensions

Using the results obtained for the one-dimensional case in Part I (Reliable Computing 9(1) (2003), pp. 1–20) of the paper, an analysis of the two-dimensional relational expression a ₁ x ₁ + a ₂ x ₂ b, where {, , , =}, is conducted with the help of a midpoint-radius diagram and other auxiliary diagrams. The solution sets are obtained with a simple boundary-line selection rule derived using these tools, and are characterized by types of one-dimensional cuts through the solution space. A classification of basic possible solution types is provided in detail. The generalization of the approach for n-dimensional interval systems and avenues for further research are also outlined.     

Solving Overdetermined Systems of Interval Linear Equations

An algorithm is developed to compute interval bounds on the set of all solutions to an overdetermined system of interval linear equations.     

The Hull of Preconditioned Interval Linear Equations

In Ph.D. thesis (Computer Methods for Design Automation, MIT, 1992), C. Bliek gave a method for computing the exact hull of a system of interval linear equations which had been preconditioned using an approximate inverse of the center of the coefficient matrix. In this paper, we simplify both the theoretical procedure and its practical implementation. We give easily verified conditions for regularity of the preconditioned matrix. We describe classes of problems for which preconditioning leaves some or all boundaries of the hull unchanged.     

Inverse Problem of the Interval Linear System of Equations

Given a nonsingular central matrix , a central vector and a prescribed interval solution vector , it is required to find the maximum allowable deviation or so that the solution of the interval linear system is contained in . Special cases for and are considered and bounds on the entries of and are obtained either in a closed form, whenever possible, or via solving a specially designed constrained optimization problem. Received: August 25, 1997; revised February 20, 1999     

A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations

Recently, Ning & Kearfott derived a formula for the interval enclosure of the solution set of linear systems of equations with uncertain data ranging in intervals, in the case when the coefficient matrix is an H-matrix. The enclosure is optimal when the midpoint matrix is diagonal, and when the midpoint is the identity, it reduces to the optimal method for enclosing preconditioned systems found by Hansen and Bliek and simplified by Rohn.An elementary proof of this formula is given using only simple properties of H-matrices and Schur complements. The new proof gives additional insight into why the theorem is true. It is also shown how to preserve rigor in the enclosure when finite precision arithmetic is used.     

Formal Solution to Systems of Interval Linear or Non-Linear Equations

This is the first of two papers which present the Modal Interval Analysis as a framework where the search and interpretation of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this first paper is to show that the modal intervals are a suitable tool to approach problems where logical reference appear. Thus, it is possible to give a logical meaning to general solutions of the system of equations, obtained, in the linear case, by means of an algorithm whose convergence conditions are studied.     

The Optimal Solution of an Interval System of Linear Algebraic Equations

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Algebraic Approach in the "Outer Problem" for Interval Linear Equations

The subject of our work is the classical "outer" problem for the interval linear algebraic System Ax = b with the square interval matrix A: find "outer" coordinate-wise estimates of the united solution set formed by all solutions to the point systems Ax = b with A A and b b. The purpose of this work is to advance a new algebraic approach to the formulated problem, in which it reduces to solving one noninterval (point) equation in the Euclidean space of double dimension. We construct a specialized algorithm (subdifferential Newton method) that implements the new approach, then present results of the numerical tests with it. These results demonstrate that the proposed algebraic approach combines unique computational efficiency with high quality enclosures of the solution set.     

A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters

Consider the systems of linear interval equations whose coefficients are linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in electrical engineering and structure mechanics. A direct method for computing a tight enclosure for the solution set is proposed in this paper. It is proved that for systems with real matrix and interval right-hand vector the method generates the hull of the solution set. For such systems an explicit formula for the hull is also given. Finally some numerical examples are provided to demonstrate the usefulness of the method in structure mechanics.     

Interval Estimations of Solution Sets to Real-Valued Systems of Linear or Non-Linear Equations

This is a second paper devoted to present the Modal Interval Analysis as a framework where the search of formal solutions for a set of simultaneous interval linear or non-linear equations is started on, together with the interval estimations for sets of solutions of real-valued systems in which coefficients and right-hand sides belong to certain intervals. The main purpose of this second paper is to show that the modal intervals are a suitable tool to approach problems where logical references appear, for example, to find interval estimates of a special class of generalized sets of solutions of real-valued linear and non-linear systems, the UE-solution sets.     

ON THE MEASURE OF FUZZINESS AND NEGATION Part I: Membership in the Unit Interval

We investigate the meaning of the concept of fuzziness. It is suggested that fuzziness can be related to the lack of distinction between a set and its negation. Using fuzzy subsets with grades of membership in the unit interval, measures of fuzziness are suggested. First, a class of measures based on metric distances is used to derive numeric measures of fuzziness of a fuzzy set. Finally, the concept of compatibility is used to develop linguistic measures of fuzziness.     

A Comment on the Shape of the Solution Set for Systems of Interval Linear Equations with Dependent Coefficients

This article is a short supplement to our previously published paper, in which we proved that each semialgebraic set can be represented as a projection of a solution set of some system of interval linear equations with dependent coefficients. The new result says that interval occurring can be chosen as narrow as wanted. The new result is proved by a simple linear transformation.     

Interval Branch and Bound Algorithm for Finding the First-Zero-Crossing-Point in One-Dimensional Functions

In this paper, new ideas have been incorporated to a basic interval branch-and-bound algorithm which solves the problem of finding zeros in one-dimensional functions. These new ideas are based on the combination of a new rejection criterion, a selection strategy and an easy-to-obtain precondition of the problem at hand. The methodology described here focuses on finding the first zero crossing point, allowing the search of other zero crossing points to be avoided. In addition, a heuristic subdivision criterion has been proposed that, compared to bisection rule, provides improvements in most of the forty problems that have been tested.     

线性方程组迭代法在流处理器上的映射与分析

斯坦福大学的Imagine流处理器具有很强的计算能力,如何将该体系结构应用在科学计算领域是当前研究的热点。解线性方程组的迭代法在工程和科学计算的各个领域中有着十分广泛的应用,该算法具有较好的计算密集性和并行性,十分适合流处理器的计算模型。本文分别针对系数矩阵的规模大小和稠密程度,介绍了Jacobi和Seidel迭代在流处
理器上的映射。实验结果表明,迭代算法能高效地开发Imagine的计算能力,取得较高的性能加速。     

From Home to Home Page: New Challenges to Basic Notions of Privacy and Property

The notion of "home" is a critical organizing principle for how we understand the relationship between individuals and society at large. A wide range of cultural and legal practices enshrine the home with various protections that ensure us a degree of privacy, security, solitude, and control over our lives there. As our private papers migrate to the Web, activities that we could previously conduct in "private" have seemingly been thrown open to unfettered public view. Additionally, as increasingly powerful communication technologies reach into our homes, places that were once private become more connected to public life     

Smart Rehabilitation Devices: Part I - Force Tracking Control

Resistance exercise has been widely reported to have positive rehabilitation effects for patients with neuromuscular and orthopaedic conditions. This article presents prototypes of smart variable resistance exercise devices using magneto-rheological fluid dampers. An intelligent supervisory control for regulating the resistive force or torque of the device is developed, and is validated both numerically and experimentally. The device provides both isometric and isokinetic strength training for the human joints including knee, elbow, hip, and ankle.     

一种不确定时段的扩展时段对序逻辑：时间Petri网模型表示和线性推理

针对点－时段时序逻辑的不足,提出了一种新的时段时序逻辑－－扩展时段时序逻辑,对不确定时间段发生的事件具有较好的描述能力。时间Petri网模型表示的引入,增强了扩展时段时序逻辑的描述直观性及分析能力,为进行线性推理提供了有利的工具。同时还提出了几种变迁间的实施推理规则。运用这些规则可以简化复杂时序关系的Petri网模型,并在线性时间复杂度内定量地得到各变迁间的时序逻辑关系,因而是一种行这有效的方法。