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Approximable Sets

R. Beigel, M. Kummer, F. Stephan
1995 Information and Computation  
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]

Simon Kristensen, Rebecca Thorn, Sanju Velani
2005 arXiv   pre-print
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]

Bin Fu, Pengfei Gu, Yuming Zhao
2018 arXiv   pre-print
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

Simon Kristensen, Rebecca Thorn, Sanju Velani
2006 Advances in Mathematics  
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

Stephen Harrap
2012 Acta Arithmetica  
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

Yiyu Yao
2008 International Journal of Approximate Reasoning  
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

R. M. Redheffer
1955 The American mathematical monthly  
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  
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

Brendan Lucier
2013 ACM SIGecom Exchanges  
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

Márton Naszódi, Alexandr Polyanskii
2018 European journal of combinatorics (Print)  
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

James Li Ming Wang
1979 Canadian mathematical bulletin  
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

Serge Dubuc, Abdelkader Elqortobi
1990 Journal of Computational and Applied Mathematics  
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

J.H. Freilich, H.W. McLaughlin
1982 Journal of Approximation Theory  
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

Chris J. Conidis
2012 Theoretical Computer Science  
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

D. A. Moran
1970 Proceedings of the American Mathematical Society  
(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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