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Dedekind order completion of C(X) by Hausdorff continuous functions
 Quaestiones Mathematicae
"... The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set H ..."
Abstract

Cited by 10 (5 self)
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The concept of Hausdorff continuous interval valued functions, developed within the theory of Hausdorff approximations and originaly defined for interval valued functions of one real variable is extended to interval valued functions defined on a topological space X. The main result is that the set Hft(X) of all finite Hausdorff continuous functions on any topological space X is Dedekind order complete. Hence it contains the Dedekind order completion of the set C(X) of all continuous real functions defined on X as well as the Dedekind order completion of the set Cb(X) of all bounded continuous functions on X. Under some general assumptions about the topological space X the Dedekind order completions of both C(X) and Cb(X) are characterised as subsets of Hft(X). This solves a long outstanding open problem about the Dedekind order completion of C(X). In addition, it has major applications to the regularity of solutions of large classes of nonlinear PDEs.
Rosinger E E : Hausdorff continuous solutions of nonlinear PDEs through the order completion method
 arXiv : math.AP/0406517
"... It was shown in [13] that very large classes of nonlinear PDEs have solutions which can be assimilated with usual measurable functions on the Euclidean domains of definition of the respective equations. In this paper the regularity of these solutions has significantly been improved by showing that t ..."
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Cited by 9 (2 self)
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It was shown in [13] that very large classes of nonlinear PDEs have solutions which can be assimilated with usual measurable functions on the Euclidean domains of definition of the respective equations. In this paper the regularity of these solutions has significantly been improved by showing that they can in fact be assimilated with Hausdorff continuous functions. The method of solution of PDEs is based on the Dedekind order completion of spaces of smooth functions which are defined on the domains of the given equations. 1
Interval Methods That Are Guaranteed to Underestimate (and the resulting new justification of Kaucher arithmetic)
 RELIABLE COMPUTING
, 1995
"... ..."
Extended Interval Arithmetic in IEEE FloatingPoint Environment
 Interval Computations
, 1994
"... This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal interv ..."
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Cited by 2 (1 self)
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This paper describes an implementation of a general interval arithmetic extension, which comprises the following extensions of the conventional interval arithmetic: (1) extension of the set of normal intervals by improper intervals; (2) extension of the set of arithmetic operations for normal intervals by nonstandard operations; (3) extension by infinite intervals. We give a possible realization scheme of such an universal interval arithmetic in any programming environment supporting IEEE floatingpoint arithmetic. A PASCALXSC module is reported which allows easy programming of numerical algorithms formulated in terms of conventional interval arithmetic or of any of the enlisted extended interval spaces, and provides a common base for comparison of such numerical algorithms. 1
Hausdorff Continuous Viscosity Solutions of HamiltonJacobi Equations
, 2005
"... A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the HamiltonJacobi equation is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity solutions can be extended to the wider space of Hausdorff contin ..."
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A new concept of viscosity solutions, namely, the Hausdorff continuous viscosity solution for the HamiltonJacobi equation is defined and investigated. It is shown that the main ideas within the classical theory of continuous viscosity solutions can be extended to the wider space of Hausdorff continuous functions while also generalizing some of the existing concepts of discontinuous solutions.
Solving large classes of nonlinear systems
, 2005
"... It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff continuous functions. The usual NavierStokes equations, as well as ..."
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It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff continuous functions. The usual NavierStokes equations, as well as their various modifications aiming at a realistic modelling, are included as particular cases. The same holds for the critically important constitutive relations in various branches of Continuum Mechanics. The solution method does not involve functional analysis, nor various Sobolev or other spaces of distributions or generalized functions. The general and type independent existence and regularity results regarding solutions presented here have recently been introduced in the literature. ”... provided also if need be that the notion of a solution shall be suitably extended...”
Pretoria
, 2005
"... It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff continuous functions. The usual NavierStokes equations, as well as ..."
Abstract
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It is shown that large classes of nonlinear systems of PDEs, with possibly associated initial and/or boundary value problems, can be solved by the method of order completion. The solutions obtained can be assimilated with Hausdorff continuous functions. The usual NavierStokes equations, as well as their various modifications are included as particular cases. The solution method does not involve functional analysis, nor various Sobolev or other spaces of distributions or generalized functions. The general and type independent existence and regularity results regarding solutions presented here are a first in the literature. ”... provided also if need be that the notion of a solution shall be suitably extended...” cited from Hilbert’s 20th Problem 1. Main ideas of the order completion solution method
B f BIOMATH
"... On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms ..."
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On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms