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Approximable Sets

1995
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Information and Computation
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Much structural work on NP-complete

doi:10.1006/inco.1995.1115
fatcat:3bs654a73rfo5d2yyd74nrseta
*sets*has exploited SAT's d-self-reducibility. ... In particular no p-selective*set*is NP-hard under polynomialtime n o(1) -tt reductions unless P = NP. In addition, no easily countable*set*is NP-hard under Turing reductions unless P = NP. ... Both are proper subclasses of*approximable**sets*. Are there natural*approximable**sets*, for instance NP-complete*sets*? ...##
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Diophantine approximation and badly approximable sets
[article]

2005
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arXiv
*
pre-print

The classical

arXiv:math/0405433v3
fatcat:672ezi3z3rauldm6csop3spyvi
*set*\Bad of 'badly*approximable*' numbers in the theory of Diophantine*approximation*falls within our framework as do the*sets*\Bad(i,j) of simultaneously badly*approximable*numbers. ... We consider 'natural' classes of badly*approximable*subsets of \Omega. Loosely speaking, these consist of points in \Omega which 'stay clear' of some given*set*of points in X. ... In the case i = j = 1/2, the*set*under consideration is simply the standard*set*of badly*approximable*pairs. ...##
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Approximate Set Union Via Approximate Randomization
[article]

2018
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arXiv
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pre-print

We develop an randomized

arXiv:1802.06204v3
fatcat:ypw2pcndqnauzmvcatmpoeyqvi
*approximation*algorithm for the size of*set*union problem A_1∪ A_2∪...∪ A_m, which given a list of*sets*A_1,... ... Our algorithm gives an*approximation*scheme with O(·()^O(1)) running time and O( m) rounds, where m is the number of*sets*. ... Our*approximation*ratio depends on the*approximation*ratio for the input*set*sizes and bias of random generator of each input*set*. ...##
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Diophantine approximation and badly approximable sets

2006
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Advances in Mathematics
*

The classical

doi:10.1016/j.aim.2005.04.005
fatcat:sqxcszbpprfvdinm77mbxukr5m
*set*Bad of 'badly*approximable*' numbers in the theory of Diophantine*approximation*falls within our framework as do the*sets*Bad(i, j ) of simultaneously badly*approximable*numbers. ... We consider 'natural' classes of badly*approximable*subsets of . Loosely speaking, these consist of points in which 'stay clear' of some given*set*of points in X. ... The*set*Bad * (R, , ) is easily seen to be a generalization of the classical*set*Bad of badly*approximable*numbers. ...##
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Twisted inhomogeneous Diophantine approximation and badly approximable sets

2012
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Acta Arithmetica
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For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the

doi:10.4064/aa151-1-5
fatcat:4j2efmhz3vfhxawsfnsw4se2y4
*set*of (i, j)-badly*approximable*pairs. ... A new characterization of the*set*Bad(i, j) in terms of 'well-*approximable*' vectors in the area of 'twisted' inhomogeneous Diophantine*approximation*is established. ... We now introduce the general badly*approximable**set*to which the results of [15] relate. ...##
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Probabilistic rough set approximations

2008
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International Journal of Approximate Reasoning
*

Based on rough membership functions and rough inclusion functions, we revisit probabilistic rough

doi:10.1016/j.ijar.2007.05.019
fatcat:kfvuybxvcvej5l2kdrfzsfyguy
*set**approximation*operators and present a critical review of existing studies. ... Results from existing studies are reviewed, synthesized and critically analyzed, and new results on the decision-theoretic rough*set*model are reported. ... Yao for their kind collaborations on the decision-theoretic rough*set*models and the anonymous reviewers for their constructive comments. ...##
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Approximation by Enumerable Sets

1955
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The American mathematical monthly
*

1955] MATHEMATICAL NOTES 573

doi:10.2307/2307251
fatcat:7b5reljbz5dojgwguzfseazjle
*APPROXIMATION*BY ENUMERABLE*SETS*R. M. ... If }-d, = ©, then there is an enumerable*set*{r,} (depending on 12" but not on £) such that every*set*E is*approximated*by {rn} within d.}. ...##
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Fuzzy answer sets approximations

2013
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Theory and Practice of Logic Programming
*

To limit this downside, operators for

doi:10.1017/s1471068413000471
fatcat:fpyawy5gqzfg5d4244sqi5claa
*approximating*fuzzy answer*sets*can be introduced: Given a FASP program, these operators compute lower and upper bounds for all atoms in the program such that all ... answer*sets*are between these bounds. ... The focus of this paper is on operators for*approximating*fuzzy answer*sets*. ...##
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Setting equilibrium prices, approximately

2013
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ACM SIGecom Exchanges
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We outline recent results, positive and negative, on pricing indivisible goods to

doi:10.1145/2509013.2509017
fatcat:mnlqfizrlfgqtkpp6bue4nhoy4
*approximately*maximize social welfare. ... We will review some well-known impossibility results, then turn to*approximation*methods as a way to circumvent them. First, we must establish what it means to*set*"appropriate" prices. ... We therefore turn to the paradigm of*approximation*and ask whether there exist prices that result in*approximately*optimal social welfare. ...##
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Approximating set multi-covers

2018
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European journal of combinatorics (Print)
*

We have better

doi:10.1016/j.ejc.2017.08.001
fatcat:f2gl4x3rgjcuvgfhspcydqnloy
*approximation*for larger f . Corollary 1.3. ... Let K, L and T be bounded Borel measurable*sets*in R d and let Λ ⊂ R d be a finite*set*with K ⊆ Λ + T . Then , where the case f = 1 is considered. ...##
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Approximation on boundary sets

1979
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Canadian mathematical bulletin
*

We say ^ is singular on bU if \P carries a

doi:10.4153/cmb-1979-049-1
fatcat:xgkdgvwm55csxby5fvwgiwhooa
*set*of full harmonic measure on b 17, to a*set*of zero harmonic measure on bV i for each /. ... Note that there are Dirichlet algebras A(U) such that A( V) is not Dirichlet for a homeomorphism ^, e.g., the "string of beads" ( [4] , p. 145)*sets*. ...##
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Approximations of fractal sets

1990
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Journal of Computational and Applied Mathematics
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Five

doi:10.1016/0377-0427(90)90197-8
fatcat:zxeythdw2rhqnfsna4cofkva6q
*approximations*of fractal*sets*A are compared. The first three were described by Hutchinson, Barnsley-Den&o and Williams, respectively. ... Two new other*approximations*are still efficient in all circumstances. For any algorithm, error estimates in the*approximations*are found. ... Then the*set*B = {Fix(g) : g E S(C)}, the*set*of fixed points of functions g of S(C), is an*approximation*of the attractor A. Lemma 5. ...##
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Approximation of bounded sets

1982
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Journal of Approximation Theory
*

In this paper, we give a unified treatment of the problem of

doi:10.1016/0021-9045(82)90088-0
fatcat:a2mu7lp3mbedll5hrc4lwoqqfy
*approximating*a family of elements belonging to a space of real-valued functions simultaneously by a single element from a specified*approximating*... Specifically, if F denotes a uniformly bounded subset of a linear vector space X with norm ]I I(, and V denotes a nonempty convex subset of X, we seek an element uO E V, designated a best simultaneous*approximation*... We also note that since u is continuous on K, both u and --c are U.S.C. on That is to say, the original problem of*approximating*the*set*F is the same as that of*approximating*the U.S.C. function U:(L ...##
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Effectively approximating measurable sets by open sets

2012
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Theoretical Computer Science
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We examine an effective version of the standard fact from analysis which says that, for any ε > 0 and any Lebesgue-measurable subset of Cantor space, X ⊆ 2 ω , there is an open

doi:10.1016/j.tcs.2012.01.011
fatcat:ghy4lhpzhfg3xllcmagl27qody
*set*U ε ⊆ 2 ω , U ε ⊇ X ... Some of the most recent results in algorithmic randomness relate the algorithmic randomness properties of a*set*A ⊆ ω to its ability to effectively (i.e., computably)*approximate*Borel*sets*with respect ... Questions regarding*approximating*Borel*sets*(with respect to Lebesgue measure) via effectively open and closed*sets*have been considered by various mathematicians in recent years, including [1, 6] and ...##
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Approximating residual sets by strongly residual sets

1970
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Proceedings of the American Mathematical Society
*

(Topological spine = strongly residual

doi:10.1090/s0002-9939-1970-0263053-0
fatcat:c37kgdaznzfz5odm7vfpxmr7ke
*set*.) This local separation property is satisfied whenever R is an ANR, or when dim 7?£dim M -2. Received by the editors April 7, 1969. ... The main result contained herein represents some progress toward determining which residual*sets*are strongly residual: it shows that a residual*set*possessing a certain property akin to semi-localconnectedness ... can be enlarged by an arbitrarily small amount to form a*set*which is strongly residual. ...
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