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Up-to-date Interval Arithmetic: From Closed Intervals to Connected Sets of Real Numbers
[chapter]

2016
*
Lecture Notes in Computer Science
*

To a great deal this can be seen as an extension

doi:10.1007/978-3-319-32152-3_39
fatcat:7efbekt3h5dkhiwoomd2phn62m
*of*arithmetic for closed*real**intervals*to*open*and halfopen*real**intervals*. ... Here*interval*arithmetic just deals with connected sets*of**real*numbers. These can be closed,*open*, half-*open*, bounded or unbounded. ... He gratefully acknowledges frequent e-mail exchange with John Gustafson on the contents*of*the paper. ...##
###
Real number computability and domain theory
[chapter]

1993
*
Lecture Notes in Computer Science
*

Domain theory is a useful tool to study higher

doi:10.1007/3-540-57182-5_33
fatcat:zvjqtdqh5vgpld2rrdpedx7wie
*order**computability*on*real*numbers. ... An important result*in*this paper is the proof that every*computable*functional on*real*numbers is continuous w.r.t. the compact*open*topology on the function space. ...*In*this representation a*real*number is defined as the limit*of*a*computable*sequence*of*rational*intervals*. ...##
###
Real Number Computability and Domain Theory

1996
*
Information and Computation
*

Domain theory is a useful tool to study higher

doi:10.1006/inco.1996.0046
fatcat:cb73dq2qfnacrmmlhetmw5rmou
*order**computability*on*real*numbers. ... An important result*in*this paper is the proof that every*computable*functional on*real*numbers is continuous w.r.t. the compact*open*topology on the function space. ] ...*In*this representation a*real*number is defined as the limit*of*a*computable*sequence*of*rational*intervals*. ...##
###
Interval Representations

2004
*
TEMA
*

The concept is often applied

doi:10.5540/tema.2004.05.02.0317
fatcat:nqk5w3nirjhjbekue5od5c63wa
*in**interval*mathematics and captures the essence*of*that theory; namely:*Interval*analysis is a language that designates*computations*with*real*numbers. ... The idea*of**interval*objects as representation*of**real*objects is defined and its relation with some aspects*of**interval*analysis is showed. ... Introduction One*of*the ideas behind*interval*analysis is that it is a kind*of*language which designates*real**computations*,*in*the sense that an*interval*[a, b] represents any*real*number r ∈ [a, b] 3 ...##
###
An induction principle over real numbers

2016
*
Archive for Mathematical Logic
*

We give a constructive proof

doi:10.1007/s00153-016-0513-8
fatcat:7n33mbyqqbflpa2vappslmulka
*of*the*open*induction principle on*real*numbers, using bar induction and enumerative*open*sets. We comment the algorithmic content*of*this result. ...*Real*numbers and enumerative*open*sets. The set R*of**real*numbers is axiomatically defined as a Cauchy complete Archimedean*ordered*field. ... We call*open*induction over*real*numbers an elementary lemma*of**real*analysis which states that an*open*set*of*the closed*interval*[0, 1] satisfying an inductive property*in*fact covers entirely the space ...##
###
Computability and the morphological complexity of some dynamics on continuous domains

2008
*
Theoretical Computer Science
*

The partially

doi:10.1016/j.tcs.2008.01.048
fatcat:pbesijwghngjzioblranl242ty
*ordered*set*of*compact*intervals*provides a convenient embedding space for the analysis*of*some Dynamical Systems. ... Crucial dynamical properties are transferred to it, while allowing an investigation*of*stability and chaoticity,*in*terms*of**computability*,*in*particular*in*the presence*of*singularities. ... (Preliminary or revised versions*of*Longo's papers are downloadable from: http://www.di.ens.fr/users/longo or Search: Google: Giuseppe Longo). ...##
###
A computable approach to measure and integration theory

2007
*
22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
*

This leads us to the

doi:10.1109/lics.2007.5
dblp:conf/lics/Edalat07
fatcat:s6w4uc3hhvde7mnx3aihs2wxfq
*interval*versions*of*the main results*in*classical measure theory. ... We introduce a*computable*framework for Lebesgue's measure and integration theory*in*the spirit*of*domain theory. ... μ on the space if the μ-measure*of*the inverse image*of*each*open*dyadic rational*interval*is a*computable**real*number. ...##
###
A computable approach to measure and integration theory

2009
*
Information and Computation
*

This leads us to the

doi:10.1016/j.ic.2008.05.003
fatcat:poxmj243evb2xmndlzuog3o53i
*interval*versions*of*the main results*in*classical measure theory. ... We introduce a*computable*framework for Lebesgue's measure and integration theory*in*the spirit*of*domain theory. ... μ on the space if the μ-measure*of*the inverse image*of*each*open*dyadic rational*interval*is a*computable**real*number. ...##
###
Linear Models of Computation and Program Learning
[article]

2015
*
arXiv
*
pre-print

We look at the recent advances

arXiv:1512.04639v1
fatcat:b4oap7ko2bhvzmvr6b6lgamqbm
*in*the "sampling the samplers" paradigm*in*higher-*order*probabilistic programming. ... We consider two classes*of**computations*which admit taking linear combinations*of*execution runs: probabilistic sampling and generalized animation. ... The*open*sets*of*the upper topology on the*reals*is a particular representation*of**real*numbers extended with ±∞, the same is true about the*open*sets*of*the lower topology on the*reals*. ...##
###
Characterizing Computable Analysis with Differential Equations

2008
*
Electronical Notes in Theoretical Computer Science
*

The functions

doi:10.1016/j.entcs.2008.12.004
fatcat:bvmaynycpnetbi45zvb6fsapjq
*of**Computable*Analysis are defined by enhancing the capacities*of*normal Turing Machines to deal with*real*number inputs. ...*In*a similar model, Shannon's General Purpose Analog*Computer*, Bournez et. al. 2007 [3] also characterize the functions*of**Computable*Analysis. ... For an*open**interval*G*of*G, we consider its closureḠ, which is*in*the domain*of*g. The minimum, m, and the maximum, M ,*of*g onḠ are*computable*. ...##
###
Interval Domains and Computable Sequences: A Case Study of Domain Reductions

2012
*
Computer journal
*

The

doi:10.1093/comjnl/bxs121
fatcat:hx2g6bxfjrevvgpvrufrszikcq
*interval*domain as a model*of*approximations*of**real*numbers is not unique,*in*fact, there are many variations*of*the*interval*domain. We study these variations with respect to domain reductions. ... A (directed) complete partial*order*, abbreviated CPO, is a partial*order*, D = (D; , ⊥), such that ⊥ is the least element*in*D and any directed set A ⊆ D has a supremum, Downloaded from Definition 2.5. ... It has been used for semantics*of**computable**reals*,*interval*arithmetic and constraint satisfaction over the*reals*. ...##
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Using Constraint Programming for Reconfiguration of Electrical Power Distribution Networks
[chapter]

2005
*
Lecture Notes in Computer Science
*

We analyze the performance

doi:10.1007/978-3-540-31845-3_22
fatcat:awrwh2bjkregfejwkktm6mli2a
*of*our system*in*canonical distribution networks*of*up to 60 nodes. ... A variety*of*approximation*computational*models have recently been proposed. We describe a constraint programming model for this problem, using the MOzArt system. ... Future Work We implemented*in*CRE2 a network*of*around 600 nodes corresponding to six electric circuits*of*the power system for the city*of*Buenaventura*in*Colombia. ...##
###
Interval Interpolation By Newtons Divided Differences

2014
*
Journal of Mathematics and Computer Science
*

*In*this paper, we present applied

*of*

*interval*algebra operation

*in*interpolation, when the support points are

*intervals*. We

*compute*interpolation polynomial that coefficients are

*interval*. ... We

*compute*interpolation polynomial by Newton's divided difference formula. ... Conclusion We present applied

*of*

*interval*algebra operation

*in*interpolation, when the support points are

*intervals*. We

*compute*interpolation polynomial with Newton`s divided differences. ...

##
###
Foundation of a computable solid modelling

2002
*
Theoretical Computer Science
*

*In*practice, correctness

*of*algorithms

*in*

*computational*geometry is usually proved using the unrealistic

*Real*RAM machine model

*of*

*computation*, which allows comparison

*of*

*real*numbers, with the undesirable ... This framework is equipped with a well deÿned and realistic notion

*of*

*computability*which re ects the observable properties

*of*

*real*solids. ... Acknowledgements The ÿrst author has been supported by EPSRC and would like to thank Reinhold Heckmann for constructing a regular

*open*set, on the

*real*line, with a boundary

*of*non-zero Lebesgue measure ...

##
###
GiNaCRA: A C++ Library for Real Algebraic Computations
[chapter]

2011
*
Lecture Notes in Computer Science
*

*In*contrast to other similar tools, our

*open*source library aids exact,

*real*algebraic

*computations*based on an appropriate data type representing

*real*zeros

*of*polynomials. ... It is based on the C++ library GiNaC, supporting the symbolic representation and manipulation

*of*polynomials. ... It supports different representations

*of*

*real*algebraic numbers (

*order*, sign, and

*interval*representation [14, p. 327] ). ...

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