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Interval Operations Involving NaNs
 Reliable Computing
, 1996
"... Ten years ago IEEE standard [1] for oatingpoint arithmetic became o cial. Each IEEE oatingpoint format supports: its own set of nite real numbers, 1, two distinguished values +0 and;0 and a set of special values called NaNs (NotaNumber). Arithmetic operations include operations on numeric, nonn ..."
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Cited by 10 (3 self)
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Ten years ago IEEE standard [1] for oatingpoint arithmetic became o cial. Each IEEE oatingpoint format supports: its own set of nite real numbers, 1, two distinguished values +0 and;0 and a set of special values called NaNs (NotaNumber). Arithmetic operations include operations on numeric, nonnumeric or mixed operands
Algebraic Solutions to a Class of Interval Equations
 J. UNIVERSAL COMPUTER SCIENCE
, 1998
"... The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplicati ..."
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Cited by 4 (4 self)
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The arithmetic on the extended set of proper and improper intervals is an algebraic completion of the conventional interval arithmetic and thus facilitates the explicit solution of certain interval algebraic problems. Due to the existence of inverse elements with respect to addition and multiplication operations certain interval algebraic equations can be solved by elementary algebraic transformations. The conditionally distributive relations between extended intervals allow that complicated interval algebraic equations, multiincident on the unknown variable, be reduced to simpler ones. In this paper we give the general type of "pseudolinear" interval equations in the extended interval arithmetic. The algebraic solutions to a pseudolinear interval equation in one variable are studied. All numeric and parametric algebraic solutions, as well as the conditions for nonexistence of the algebraic solution to some basic types pseudolinear interval equations in one variable are found. Some examples leading to algebraic solution of the equations under consideration and the extra functionalities for performing true symbolicalgebraic manipulations on interval formulae in a Mathematica package are discussed.
All about Generalized Interval Distributive Relations. I. Complete Proof of the Relations
, 2000
"... The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus e cient solution of interval algebraic problems. This paper generalizes the distributive relations, known by now, on multiplication and addition of p ..."
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Cited by 2 (1 self)
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The arithmetic on an extended set of proper and improper intervals presents algebraic completion of the conventional interval arithmetic allowing thus e cient solution of interval algebraic problems. This paper generalizes the distributive relations, known by now, on multiplication and addition of proper and improper intervals. A complete proof of the main results is presented, demonstrating an original technique based on functional notations and transition formulae between different interval structures. A variety of equivalent forms and different representations are discussed together with some examples. This paper is an extraction from [19] and will be updated permanently to include current improvements, generalizations and applications of the conditionally distributive relations. The second part of the paper is scheduled for the end of 2000 and will include several directions for the application of the generalized distributive relations.
Using Extended Interval Algebra in Discrete Mechanics
"... Abstract: Discrete mechanics deals with discrete mechanical systems, such as cellular automata, in which time proceeds in integer steps and the configuration space is discrete. Directly modeling discrete mechanical systems is a well known alternative to starting from a continuous setting, discretizi ..."
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Abstract: Discrete mechanics deals with discrete mechanical systems, such as cellular automata, in which time proceeds in integer steps and the configuration space is discrete. Directly modeling discrete mechanical systems is a well known alternative to starting from a continuous setting, discretizing the model, and finally force the model to the finite alphabet of a computer. The time evolution of discrete dynamical systems, however, can be calculated exactly. In order to take into account imprecision in the input data and the need to accommodate a finite alphabet, extended interval analysis is introduced in the discrete mechanical systems formulation developed by Baez and Gilliam. It is shown how the EulerLagrange equation must be modified when working with interval input.
Generalized Interval Projection: A New Technique for Consistent Domain Extension
, 2010
"... This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of ..."
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This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of a single punctual solution is of particular interest, because it gives greater freedom for choosing a solution. Our approach is able to build an inner box for the problem starting with a single point solution, by consistently extending the domain of every variable. The key point is a new method called generalized projection. The requirements are that each parameter must occur only once in the system, variable domains must be bounded, and each variable must occur only once in each constraint. Our extension is based on an extended algebraic structure of intervals called generalized intervals, where improper intervals are allowed (e.g. [1,0]). 1
Generalized Interval Projection: A New Technique fo r Consistent Domain Extension, in "IJCAI
, 2007
"... This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of ..."
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This paper deals with systems of parametric equations over the reals, in the framework of interval constraint programming. As parameters vary within intervals, the solution set of a problem may have a non null volume. In these cases, an inner box (i.e., a box included in the solution set) instead of a single punctual solution is of particular interest, because it gives greater freedom for choosing a solution. Our approach is able to build an inner box for the problem starting with a single point solution, by consistently extending the domain of every variable. The key point is a new method called generalized projection. The requirements are that each parameter must occur only once in the system, variable domains must be bounded, and each variable must occur only once in each constraint. Our extension is based on an extended algebraic structure of intervals called generalized intervals, where improper intervals are allowed (e.g. [1,0]). 1
Order Reduction of Linear Interval Systems Using Genetic Algorithm
"... Abstract — This paper presents an algorithm for order reduction of higher order linear interval system into stable lower order linear interval system by means of Genetic algorithm. In this algorithm the numerator and denominator polynomials are determined by minimizing the Integral square error (ISE ..."
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Abstract — This paper presents an algorithm for order reduction of higher order linear interval system into stable lower order linear interval system by means of Genetic algorithm. In this algorithm the numerator and denominator polynomials are determined by minimizing the Integral square error (ISE) using genetic algorithm (GA). The algorithm is simple, rugged and computer oriented. It is shown that the algorithm has several advantages, e.g. the reduced order models retain the steadystate value and stability of the original system. A numerical example illustrates the proposed algorithm. I.