Interval Of Real Numbers Research Articles

Fuzzy measurement in the Mishnah and the Talmud

I discuss the attitude of Jewish law sources from the 2nd---:5th centuries to the imprecision of measurement. I review a problem that the Talmud refers to, somewhat obscurely, as "impossible reduction". This problem arises when a legal rule specifies an object by referring to a maximized (or minimized) measurement function, e.g., when a rule applies to "the largest part" of a divided whole, or to "the first" incidence that occurs, etc. A problem that is often mentioned is whether there might be hypothetical situations involving more than one maximal (or minimal) value of the relevant measurement and, given such situations, what is the pertinent legal rule. Presumption of simultaneous occurrences or equally measured values are also a source of embarrassment to modern legal systems, in situations exemplified in the paper, where law determines a preference based on measured values. I contend that the Talmudic sources discussing the problem of "impossible reduction" were guided by primitive insights compatible with fuzzy logic presentation of the inevitable uncertainty involved in measurement. I maintain that fuzzy models of data are compatible with a positivistic epistemology, which refuses to assume any precision in the extra-conscious "world" that may not be captured by observation and measurement. I therefore propose this view as the preferred interpretation of the Talmudic notion of "impossible reduction". Attributing a fuzzy world view to the Talmudic authorities is meant not only to increase our understanding of the Talmud but, in so doing, also to demonstrate that fuzzy notions are entrenched in our practical reasoning. If Talmudic sages did indeed conceive the results of measurements in terms of fuzzy numbers, then equality between the results of measurements had to be more complicated than crisp equations. The problem of "impossible reduction" could lie in fuzzy sets with an empty core or whose membership functions were only partly congruent. "Reduction is impossible" may thus be reconstructed as "there is no core to the intersection of two measures". I describe Dirichlet maps for fuzzy measurements of distance as a rough partition of the universe, where for any region A there may be a non-empty set of ăÂ? - _A (upper approximation minus lower approximation), where the problem of "impossible reduction" applies. This model may easily be combined with probabilistic extention. The possibility of adopting practical decision standards based on Â?-cuts (and therefore applying interval analysis to fuzzy equations) is discussed in this context. I propose to characterize the uncertainty that was presumably capped by the old sages as "U-uncertainty", defined, for a non-empty fuzzy set A on the set of real numbers, whose Â?-cuts are intervals of real numbers, as U(A) = 1/h(A) Â? 0 h(A) log [1+μ(Â?A)]dÂ?, where h(A) is the largest membership value obtained by any element of A and μ(Â?A) is the measure of the Â?-cut of A defined by the Lebesge integral of its characteristic function.

Interval measures of power

Standard power indices, introduced by Shapley and Shubik, Banzhaf, Johnston, and others, assign real numbers to the players in a simple game as a quantitative measure of their influence in the voting situation represented by the game. We consider instead two ‘interval notions of power’ that assign intervals of real numbers to the players. The first of these notions originates in the observation that although a particular choice of weights in a weighted game may not provide an accurate measure of power, the interval of all possible weights for a player is, in fact, a reasonable reflection of the player's influence. The resulting weight interval is extended to non-weighted games via the observation that every simple game is an intersection of weighted games. For example, we have shown (Taylor and Zwicker, Games and Economic behavior, 1993, 5, 170–181) that the president's weight interval in the (non-weighted) federal system is ( 15 115 , 17 117 ), suggesting that the president has between 13% and 15% of the power. The development of a combinatorial equivalent, the dispersal interval, yields a uniform method for calculating weight intervals. Our second interval index, the market interval, is based on a modification of Peyton Young's idea that the influence of players can be measured by the sizes of the bribes they can demand when they sell their votes. We show, for example, that the president's market interval in the federal system is (0, 0.2537,…); he can reasonably insist on between 0% and 25% of the total bribe. An iterated version of the market interval, based on rationality assumptions about other players' actions and linked with the notion of a bribery equilibrium, produces potentially smaller intervals that, in some cases, shrink to single points in the limit.

Fault tolerance in multisensor networks

Replicating sensors is desirable, not only to tolerate sensor failures, but to increase the average accuracy of the ensemble of replicated sensors beyond that obtainable with a single sensor. Such replication is used in a multi-sensor environment or in a distributed-sensor network. Following Marzullo (1990), the authors have modeled a continuous valued sensor as an interval of real numbers containing the physical value of interest. Given n sensors of which at most f can suffer arbitrary failures, this paper presents an efficient O(n/spl middot/log(n)) fault-tolerant algorithm (J/FTA) whose output is reliable (guaranteed to contain the correct value at all times) and is fairly accurate when f<gilb(/sup 1///sub 2/(n+1)). The output of J/FTA can be either a single-interval or a set-of-intervals, depending on the nature of the multi-sensor environment. J/FTA can be used not only to detect all possibly-faulty sensors but to detect all sets (combinations) of possibly-faulty sensors. This paper proves the following results pertaining to the possibly-faulty sensors identified by J/FTA: the number of sets each containing f possibly-faulty sensors is at most (f+1); the number of sets each containing f or fewer faulty sensors is at most (2f+1); and the number of possibly-faulty sensors identified by J/FTA is at most 2f. These results help to: narrow the search to detect faulty sensors; bound the number of intervals needed to construct an accurate and reliable abstract sensor; and identify at least one correct sensor.

NONSTANDARD APPROACH TO POSSIBILITY MEASURES

Possibility measures have been conceived by Zadeh [1], and developed by, e. g., Dubois, Prade and others, as very simple, if compared with, e.g., probability theory, but still nontrivial and reasonable uncertainty measures. The relatively poor descriptional and operational abilities of possibility measures seem to be closely related to the standard linear ordering relation and the corresponding supremum and infimum operations defined over the unit interval of real numbers. Having discussed, very briefly, the possibilities how to overcome these limitations, we propose and investigate possibility measures taking their values in the well-known Cantor subset of the unit interval but defined with respect to a nonstandard operation of supremum resulting from a nonstandard partial ordering of real numbers from the Cantor set. Such a nonstandard possibility measure is proved to be a sufficient tool to define an infinite sequence of probability measures defined over countable sets, consequently, the distribution of each continuous real-valued random variable defined over a general abstract probability space can be defined by an appropriate nonstandard possibility measure.

The Space of Closed Subsets of a Convergent Sequence

Many topological spaces are simply sets of points (atoms) endowed with a topology. Some spaces, however, have elements that are functions, matrices, or other non-atomic items. Another special type of space has elements that are themselves subsets of another space; these spaces are called hyperspaces. Hyperspaces are metric spaces, and the metric defined on them is called the Hausdorff metric. Hyperspaces wllose points are the closed subsets and hyperspaces whose points are the closed connected subsets (of metric spaces) have been extensively studied. Also, hyperspaces of closed and convex subsets of a bounded convex set in Euclidean space are of great interest in geometry. See Lay [3]. In recent years, geometers and topological dynamicists have explored spaces of closed and bounded subsets of the plane in connection with the study of fractals. One of the major results in the theory is that the hyperspace of closed subsets of a closed interval of real numbers is homeomorphic with the Hilbert cube, and with the space IX, the countable product of unit intervals. For more general information about the Hausdorff metric and spaces of subsets, see Devaney [2] and Sieradski [6]. The main result of this paper is a topological characterization of the space of closed subsets of a convergent sequence of points. The proof given here provides a homneornorphic embedding of the space in the plane, E2. The result was first proved by Pelczynski [5]. He studied arbitrary compact zero-dimensional metric spaces, so his proofs are much more widely applicable, but they are also somewhat technical. Our proof depends only on some well-known results in the theory of metric spaces, and is therefore accessible to advanced undergraduate mathematics majors. We also prove that for no convergent sequence of real numbers is there an isometric embedding of the hyperspace in Euclidean space, E,1, for any n. For other results and discussion, see Nadler [4]. Let (X, d) denote a compact metric space. The hyperspace (2X, D) of X is the metric space whose points are the closed nonempty subsets of X and whose metric is the Hausdorff metric D given by

Generalized set-propagation operations over relations of more than three variables

AbstractThis paper extends previously developed generalized set propagation operations to work over relationships among an arbitrary number of variables, thereby expanding the domain of engineering design problems the theory can address. It then narrows its scope to a class of functions and sets useful to designers solving engineering problems: monotonic algebraic functions and closed intervals of real numbers, proving formulas for computing the operations under these conditions. The work is aimed at the automated optimal design of electro-mechanical systems from catalogs of parts; an electronic example illustrates.

Global optimization by random perturbation of the gradient method with a fixed parameter

The paper deals with the global minimization of a differentiable cost function mapping a ball of a finite dimensional Euclidean space into an interval of real numbers. It is established that a suitable random perturbation of the gradient method with a fixed parameter generates a bounded minimizing sequence and leads to a global minimum: the perturbation avoids convergence to local minima. The stated results suggest an algorithm for the numerical approximation of global minima: experiments are performed for the problem of fitting a sum of exponentials to discrete data and to a nonlinear system involving about 5000 variables. The effect of the random perturbation is examined by comparison with the purely deterministic gradient method.

On the use of interval mathematics in fuzzy expert systems

Fuzzy expert systems attempt to model the cognitive processes of human experts. They currently accomplish this by capturing knowledge in the form of linguistic propositions. Real-world problems dictate the need to include mathematical knowledge as well. Pattern matching is a critical part of the inference procedure in expert systems. Matches are made between data clauses, premise clauses, and conclusion clauses, forming an inference chain. Preprocessing the clauses may generate intervals of real numbers which are compared in the fuzzy matching algorithm. These same intervals may be used in arithmetic expressions. the purpose of this article is to devise a method for incorporating arithmetic expressions into inference process of Fuzzy Expert Systems. Interval arithmetic is used to evaluate these expressions. Logical relations between intervals are analyzed using probability theory. © 1994 John Wiley & Sons, Inc.

Interval computation as deduction in chip

Logic programming realizes the ideal of “computation is deduction,” but not when floating-point numbers are involved. In that respect logic programming languages are as careless as conventional computation: they ignore the fact that floating-point operations are only approximate and that it is not easy to tell how good the approximation is. It is our aim to extend the benefits of logic programming to computation involving floating-point arithmetic. Our starting points are the ideas of Cleary and the CHIP programming language. Cleary proposed a relational form of interval arithmetic that was incorporated in BNR Prolog in such a way that variables already bound can be bound again. In this way the usual logical interpretation of computation no longer holds. In this paper we develop a technique for narrowing intervals that we relate both to Cleary's work and to the constraint-satisfaction techniques of artificial intelligence. We then modify CHIP by allowing domains to be intervals of real numbers. To reduce arithmetic primitives with interval domains, we use our interval narrowing technique as an implementation of the looking-ahead inference rule. We show that the result is a system where answers are logical consequences of a declarative logic program, even when floating-point computations have been used. We believe ours is the first system with this property.

EXTENSIONAL PROCESSING OF PROBABILITY MEASURES

The aim of this contribution is to prove that there exists, for each probability measure from a rather wide class of such measures, a uniquely defined and effectively obtainable recursive extensional probability measure, in the sense that probabilities of composed random events are defined as functions of a priori probabilities of composing random events, which can serve as a useful tool to process the original probability measure. The definition and construction are based on the idea of nonstandard probability measures the values of which are infinite binary sequences admitting either the standard or a nonstandard and relative-frequency based interpretation within the unit interval of real numbers.

Weighted Voting, Multicameral Representation, and Power

A simple game ( P, W) can serve as a model of a voting system in which an alternative is pitted against the status quo. In what follows, we investigate the following three aspects of such games as they apply to four real-world examples of voting systems: a characterization of weighted voting systems in terms of the ways in which coalitions can gain or lose by trading players; the application of a graph-theoretic notion of dimension to simple games and voting systems; and the consideration of a way to measure the power of a player as an interval of real numbers. American Mathematical Society Classification Numbers: 90A28, 90D80, 05C65. Journal of Economic Literature Classification Number: C71.

Pure-strategy ε-Nash equilibrium in two-person non-zero-sum games

This paper considers two-person non-zero-sum games where the strategy spaces of players are compact intervals of real numbers and the payoff functions are upper semicontinuous. Under certain quasi-concavity assumptions, it is proved that such games possess ε- Nash equilibria in pure strategies for every ε > 0. AMS 1980 subject classification: 90D05, 90D13.

Lattice of fuzzy subalgebras and closure systems in Ix

Abstract Let X be a non-empty set and I be the unit interval of real numbers. The elements of I x are called fuzzy subsets of X . I x is a lattice with the pointwise ordering of I . Now if A = [ X , F ] is an universal algebra with a specified set F of finitary operations on X , then each operation can be extended to a finitary operation on I x . An element of I x is defined to be a fuzzy subalgebra of A using these extended operations. We also define algebraic closure system in I x . We then show that the lattice of fuzzy subalgebras of A is an algebraic closure system in I x .

An Analysis of Shopping Choice behavior Using Fuzzy Multiattribute Attitude Model

There have been a number of investigations of the impact of the location of a town, city, or trading area as a determinant of store patronage. For example, it was found that the large majority living in small towns shopped beyond the town boundalies (Engel et al., 1986; Nakanishi, 1983). However, it is suggested that the consumer's attitude (or mental image) related to store or shopping area is more important in explaining shopping behavior and preferences than is actual location.Attitude is a central concept in consumer psychology, and defined as sum total of evaluation of the perceived attributes of alternatives. Traditional approaches to the measurement of attitude have involved method such as the semantic differential, the Likert scale, or the Thurstone scale. Althoguh insights into ambiguous nature of attitudes were identified early in the development of attitude measurement, the subsequent methods used failed to capture this ambiguity, no doubt because traditional mathematics was not well developed for dealing with ambiguity of judgment.We propose a fuzzy multiattribute attitude model (FMA model) for explaining shopping choice behavior. The FMA model is an extension of traditional multiattribute attitude model such as Fishbein model, and the application of the fuzzy set theory to the multiattribute attitude model.A fuzzy set A is defined as follows. Let X denote a universal set. Then, the membership function, μA by which a fuzzy set A is defined has the formWhere [0, 1] denotes the interval of real numbers from 0 to 1, inclusive. A convex and normalized fuzzy set whose membership function is piecewise continuous is called a fuzzy number. Fuzzy rating data we postulated in the FMA model can be approximated to fuzzy numbers.Symbolically, the FMA model can be expressed aswhere: A basic principle that allow the generalization of crisp mathematical concepts to the fuzzy framework is known as the extension principle proposed by Zadeh (1975). It provides the means for any function f that maps points x1, x2, /…, xn in the crisp set X to the crisp set Y to be generalized such that it maps fuzzy subsets of X to Y. According to the extension principle, the membership function for the FMA model can be expressed as follows.Where: In order to test the FMA model, we use the fuzzy rating method (Hesketh et al., 1988; Takemura, 1990, 1991). The fuzzy rating scale presents respondents with the option of indicating a preferred point and then asks them to extend the rating to the left or light if they wish. By making certain simplifying assumption, the fuzzy rating data can be viewed as fuzzy numbers, hence making possible the use of fuzzy set theoretic operations. For the purpose of explaining shopping choice behavior, a survey study has been undertaken in Kyoto city, Japan.

Fuzzy prime ideals of rings

The notions of fuzzy ideal and fuzzy prime ideal of a ring with truth values in a complete lattice satisfying the infinite meet distributive law are introduced which generalize the existing notions with truth values in the unit interval of real numbers. A procedure to construct any fuzzy ideal from a family of ideals with certain conditions is established. All fuzzy prime (maximal) ideals of a given ring are determined by establishing a one-to-one correspondence between fuzzy prime (maximal) ideals and the pairs ( P, α), where P is a prime (maximal) ideal of the ring and α is a prime element (dual atom) in the complete lattice.

Standard threads and distributivity

On a given standard thread (J, ∘), all operations * over which ∘ distributes are determined and among such operations those which are continuous are identified. A standard thread is a topological semigroup on a closed real number interval whose largest element is an identity and smallest element is a zero for the semigroup. A quotient operation can be defined forx ⩾ y on a standard thread by $$\frac{y}{x}: = \min \{ w \in J|y = x \circ w\} .$$ The operations * in question are shown to be generated by pairs of functionsp, q:J → J such that $$x * y = \left\{ {\begin{array}{*{20}c} {x \circ q\left( {\frac{y}{x}} \right) if x \geqslant y} \\ {p\left( {\frac{x}{y}} \right) \circ y if x \leqslant y.} \\ \end{array} } \right.$$ Those functionsp andq which generate operations * over which ∘ distributes are completely identified.

Optimal algorithms for seeking the interval of greatest values for a class of functions

Optimal algorithms are constructed for finding the interval of greatest values for a class of functions, defined in an interval of real numbers with ordered values.

On the Computation of the Confluent Hypergeometric Function at Densely Spaced Points

An algorithm is described for computing the values of the confluent hypergeometric function at several points spaced densely along an interval of real numbers. Numerical data obtained by using this algorithm and pertaining to a commonly made approximation for the first-passage problem of the response amplitude of a lightly damped and randomly excited linear oscillator are presented.

Multipliers of certain convolution algebras over locally compact semigroups

In this paper we extend a result of Johnson and Lahr(3), which characterizes the multiplier algebra of L1(a, b) (the algebra of Lebesgue integrable functions on the interval of real numbers from a to b, under order convolution) to the L1 algebra of a general totally ordered semigroup. Similar work has been done in (l), but under more restrictive conditions.

Optimal strategies for seeking the domain of greatest values for a class of functions

OPTIMAL strategies for seeking the domain of greatest values are constructed for a class of functions, defined in an interval of real numbers with ordered values.