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Approximable Sets
1995
Information and Computation
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Much structural work on NP-complete 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? ...
doi:10.1006/inco.1995.1115
fatcat:3bs654a73rfo5d2yyd74nrseta
Diophantine approximation and badly approximable sets
[article]
2005
arXiv
pre-print
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The classical 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. ...
arXiv:math/0405433v3
fatcat:672ezi3z3rauldm6csop3spyvi
Approximate Set Union Via Approximate Randomization
[article]
2018
arXiv
pre-print
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We develop an randomized 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. ...
arXiv:1802.06204v3
fatcat:ypw2pcndqnauzmvcatmpoeyqvi
Diophantine approximation and badly approximable sets
2006
Advances in Mathematics
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The classical 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. ...
doi:10.1016/j.aim.2005.04.005
fatcat:sqxcszbpprfvdinm77mbxukr5m
Twisted inhomogeneous Diophantine approximation and badly approximable sets
2012
Acta Arithmetica
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For any real pair i, j geq 0 with i+j=1 let Bad(i, j) denote the 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. ...
doi:10.4064/aa151-1-5
fatcat:4j2efmhz3vfhxawsfnsw4se2y4
Probabilistic rough set approximations
2008
International Journal of Approximate Reasoning
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Based on rough membership functions and rough inclusion functions, we revisit probabilistic rough 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. ...
doi:10.1016/j.ijar.2007.05.019
fatcat:kfvuybxvcvej5l2kdrfzsfyguy
Approximation by Enumerable Sets
1955
The American mathematical monthly
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1955] MATHEMATICAL NOTES 573
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.}. ...
doi:10.2307/2307251
fatcat:7b5reljbz5dojgwguzfseazjle
Fuzzy answer sets approximations
2013
Theory and Practice of Logic Programming
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To limit this downside, operators for 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. ...
doi:10.1017/s1471068413000471
fatcat:fpyawy5gqzfg5d4244sqi5claa
Setting equilibrium prices, approximately
2013
ACM SIGecom Exchanges
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We outline recent results, positive and negative, on pricing indivisible goods to 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. ...
doi:10.1145/2509013.2509017
fatcat:mnlqfizrlfgqtkpp6bue4nhoy4
Approximating set multi-covers
2018
European journal of combinatorics (Print)
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2016 pre-print arXiv:1608.01292v1 fatcat:3prpzvljjnfjpdbpi3b7nakz5yOther Versions
We have better 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. ...
doi:10.1016/j.ejc.2017.08.001
fatcat:f2gl4x3rgjcuvgfhspcydqnloy
Approximation on boundary sets
1979
Canadian mathematical bulletin
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We say ^ is singular on bU if \P carries a 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. ...
doi:10.4153/cmb-1979-049-1
fatcat:xgkdgvwm55csxby5fvwgiwhooa
Approximations of fractal sets
1990
Journal of Computational and Applied Mathematics
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Five 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. ...
doi:10.1016/0377-0427(90)90197-8
fatcat:zxeythdw2rhqnfsna4cofkva6q
Approximation of bounded sets
1982
Journal of Approximation Theory
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In this paper, we give a unified treatment of the problem of 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 ...
doi:10.1016/0021-9045(82)90088-0
fatcat:a2mu7lp3mbedll5hrc4lwoqqfy
Effectively approximating measurable sets by open sets
2012
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 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 ...
doi:10.1016/j.tcs.2012.01.011
fatcat:ghy4lhpzhfg3xllcmagl27qody
Approximating residual sets by strongly residual sets
1970
Proceedings of the American Mathematical Society
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(Topological spine = strongly residual 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. ...
doi:10.1090/s0002-9939-1970-0263053-0
fatcat:c37kgdaznzfz5odm7vfpxmr7ke
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