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Topic: Dense set

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	Density (disambiguation) - Biocrawler (Site not responding. Last check: 2007-10-10)
		A partial order on a set S is said to be dense (or dense-in-itself) if, for all x and y in S for which x < y, there is a z in S such that x < z < y.
		In order theory, a subset B of a partially ordered set A is dense in A if for any x < y in A, there is some z in B such that x < z < y.
		In case the order is a linear order, then B is dense in A in the present sense if and only if B is dense in the order topology on A.
	www.biocrawler.com /encyclopedia/Dense (451 words)

	PlanetMath: dense in-itself
		A simple example of a set which is dense-in-itself but not closed (and hence not a perfect set) is the subset of irrational numbers.
		For similar reasons, the set of rational numbers is also dense-in-itself but not closed.
		This is version 1 of dense in-itself, born on 2004-09-24.
	planetmath.org /encyclopedia/DenseInItself.html (110 words)

	Nowhere dense set - Wikipedia, the free encyclopedia
		For example, the set of rational numbers, as a subset of R has the property that the closure of the interior is empty, but it is not nowhere dense; in fact it is dense in R, which is the opposite notion.
		Every subset of a nowhere dense set is nowhere dense, and the union of finitely many nowhere dense sets is nowhere dense.
		That is, the nowhere dense sets form an ideal of sets, a suitable notion of negligible set.
	en.wikipedia.org /wiki/Nowhere_dense_set (402 words)

	Dense set - Wikipedia, the free encyclopedia
		Equivalently, A is dense in X if the only closed subset of X containing A is X itself.
		An alternative definition in the case of metric spaces is the following: The set A in a metric space X is dense if every x in X is a limit of a sequence of elements in A.
		A metric space M is dense in its completion γM.
	en.wikipedia.org /wiki/Dense_set (206 words)

	[No title]
		The rationals in the set are dense in it, but each rational is an isolated point, as all numbers sufficiently close to a given rational have a large partial quotient either one or two steps after position of the last coefficient of the rational.
		Including a_0 among the partial quotients being restricted to {1,2} gives a set whose smallest element has continued fraction [_1,2_] and largest element [_2,1_] (where the _ are used to mark a periodic block).
		This leads to a Cantor set construction of the set, but one in which the size and location of the segment removed from each interval depends on a parameter representing the history of the construction leading to the interval.
	www.math.niu.edu /~rusin/known-math/98/c1 (1248 words)

	Some nowhere dense sets with positive measure and a strictly monotonic continuous function with a dense set of points ...
		The remaining set of points (if any remain) is nowhere dense, and if the intervals are chosen suitably then the measure of the remaining points will be between 0 and 1.
		Taking k=0 would shrink each interval to a point, so removing a set of measure 0 and leaving a dense set of measure 1, while taking k=2 would remove all the points when the interval around 1/2 is removed leaving the empty set with measure 0.
		The relationship between the measure of the two remainging sets is not quite a coincidence, since the the measure of the interval remaining after given number of steps when k=1 is exactly half that after one fewer step when k=1/2.
	www.btinternet.com /~se16/hgb/nowhere.htm (836 words)

	Structured Gaussian elimination
		Structured Gaussian elimination was very successful on data set K, since it reduced it to set L very quickly (in about 20 minutes for reading the data, roughly the same amount of time for the basic run, and then under an hour to produce the dense set of equations that form set L).
		The performance of the algorithm on set K that is reported in Table 4 is better than that mentioned before (which reduced it to set L, which has 6,006 unknowns).
		For example, the entry for set K in Table 4 was obtained by specifying that Step 3 be applied as soon as the number of heavy columns exceeded 3,200.
	www.farcaster.com /papers/crypto-solve/node5.html (3236 words)

	[No title]
		Moreover, since the set of points at which an arbitrary function is not continuous is an F_sigma set, it follows that the non-continuity points of any derivative must be an F_sigma first category set.
		The set can even have a measure zero complement (which is stronger than having positive measure and being dense), since there exist F_sigma first category sets whose complements have measure zero.
		Indeed, the set can be so large from a measure standpoint that its complement could have Hausdorff dimension zero (even Hausdorff h-measure zero for any given admissible Hausdorff measure function h), for the same reason.
	www.math.niu.edu /~rusin/known-math/00_incoming/deriv_cont (1260 words)

	PlanetMath: dense set
		Cross-references: reals, irrationals, rationals, metric, metric space, open set, intersects, closure, topological space, subset
		This is version 5 of dense set, born on 2002-01-03, modified 2006-02-14.
		Object id is 1192, canonical name is Dense.
	planetmath.org /encyclopedia/Dense.html (44 words)

	Topological Preliminaries
		A set S is called thin if none of its points satisfies that condition.
		Thus, boundary (the set of all boundary points) is shared by a set and its complement.
		The set Q of all rational numbers is dense in R, thick, neither open nor closed, and without internal points.
	www.cut-the-knot.org /do_you_know/topology.shtml (759 words)

	ASSIGNMENT 5 SOLUTIONS
		For each of the following sets, decide whether or not the set is dense in [0,1].
		S1 is the set of all real numbers in [0,1] except those of the form 1/2^n for n=1,2,3...
		S_2 is the set of all rationals in [0,1] of the form p/(2^n) where p and n are natural numbers.
	math.bu.edu /people/bob/MA471/solutions/Hwk5-solns.html (1292 words)

	Density (disambiguation) - Wikipedia, the free encyclopedia
		Density and dense usually refer to a measure of how much of some entity is "in" a fixed amount of other space.
		In case the order is a linear order, then B is dense in A in the present sense if and only if B is a dense set with respect to the order topology on A.
		In forcing (mathematics), a subset D of a forcing notion (P, ≤) is called dense in P if for any p in P there is d in D with d≤p.
	en.wikipedia.org /wiki/Dense (374 words)

	First and Second Category
		The set u is first category, or meager, if it is a countable union of nowhere dense sets.
		Since the countable union of countable sets is countable, the countable union of first category sets is first category.
		If every point acts as a nowhere dense set then s becomes first category, which is a contradiction.
	www.mathreference.com /top-ms,cat12.html (651 words)

	Dense
		By the definition you gave, (1, 2) is not dense: for example, the interval (7, 96) does not contain a point of (1, 2).
		If you have two sets which are dense, then their intersection would also be dense if some part of one set is a subset of the other.
		Given any set not dense in the reals, I can find two sets that are dense in the reals, and have that set as their intersection.
	www.physicsforums.com /showthread.php?t=60708 (1481 words)

	Baire Category
		is a countable union of nowhere dense sets.
		Synonyms: A meager set is a set of first category.
		be a countable collection of nowhere dense sets.
	www.math.unl.edu /~s-bbockel1/922-notes/node4.html (307 words)

	Dense, Separable (Site not responding. Last check: 2007-10-10)
		A set s is dense in t if it intersects every nonempty open set in t.
		A space is separable if it contains a countable dense set.
		The reals are separable, since the rationals are dense in the reals, and the rationals are countable.
	www.mathreference.com /top,dense.html (44 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		Take a set K such that K+K =R, and do the usual construction, but pick elements for B from K (let r be outside the span of the part already built, write r = k1+k2, adjoin one (if enough) or both of them as new elements, repeat).
		In fact, Erdos proves this set is a Lusin set (has countable intersection with every nowhere dense set).
		CH implies there exists a Hamel basis for which the set of all its nonnegative rational linear combinations is first category in the reals.
	www.math.niu.edu.cob-web.org:8888 /~rusin/known-math/99/hamel (3445 words)

	what is a cardinality of the sets?
		What cardinality has a set of all monotone maps R to R? here R is
		No, the set of mappings from Q to Q is uncountable, and the same goes for
		The cardinality of the set C_n of continuous functions with exacty N
	www.groupsrv.com /science/about153744.html (1376 words)

	Statistical Methods for Mapping Quantitative Trait Loci From a Dense Set of Markers -- Dupuis and Siegmund 151 (1): 373 ...
		crosses using a dense set of markers were introduced by
		The set of all values q that are not rejected by this test is
		sets of 350 tomatoes and calculated the average size and the
	www.genetics.org /cgi/content/full/151/1/373 (7129 words)

	Orthonormal Sets and Bases
		is an orthonormal set, then there is a basis containing
		is a directed set with this partial order, that is, if
		Recall We say a metric space is separable if it has a countable dense subset.
	www.math.unl.edu /~s-bbockel1/928/node5.html (220 words)

	Maxim R. Burke - Continuous functions which take a somewhere dense set of values on every open set
		Continuous functions which take a somewhere dense set of values on every open set
		We study the class of Tychonoff spaces that can be mapped continuously into the real line in such a way that the preimage of every nowhere dense set is nowhere dense.
		We also give examples of spaces which have nowhere constant continuous maps into the real line and are not in this class.
	www.cms.math.ca /Events/winter98/w98-abs/node164.html (124 words)

	Perlegen Sciences: Perlegen's Approach (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-10)
		Statistical data analysis of results using proprietary software to identify the genetic loci responsible for the disease or drug response.
		In order to successfully identify the genetic causes of common traits, such association studies rely on the analysis of a dense set of common SNPs.
		This density is required because in many regions of the genome, linkage disequilibrium does not extend far.
	www.perlegen.com.cob-web.org:8888 /science/perlegen.html (292 words)

	strictly decreasing function with zero derivative on dense subset
		For 0dense set with positive measure.
		Let f(k) be the measure of that set - this has the desired properties:
		unequal at a countable set of points where the end point of a removed
	sci4um.com /about1719.html (1030 words)

	CJM - Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
		CJM - Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
		Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set
		We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A cap C is nonmeager in C.
	journals.cms.math.ca /cgi-bin/vault/view/burke4426 (99 words)

	MCS Software (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-10)
		City is a set of software, tools, and procedures that we have developed to help run the clusters in the MCS Division.
		The dgsol code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds.
		The use of projected searches, in particular, allows TRON to examine faces of the feasible set by generating a small number of minor iterates, even for problems with a large number of variables.
	www-fp.mcs.anl.gov.cob-web.org:8888 /division/software (1981 words)

	Solution
		of real numbers is dense if for any open interval
		We've seen in class that between any two (different) numbers (i.e., within any open interval) there is a rational number and there is an irrational number.
		is not dense because, for example, it's intersection with the interval
	www.math.toronto.edu /drorbn/classes/0304/157AnalysisI/Final/Solution.html (542 words)

	Loop-Free Routing Using a Dense Label Set in Wireless Networks (Site not responding. Last check: 2007-10-10)
		The protocols guarantee loop-freedom at every instant by ensuring that node labels are always in topological order, and thus induce a directed acyclic graph (DAG).
		The novel feature of SLR is that it uses a dense ordinal set with a strict partial order to label nodes.
		We present a practical, finitely dense implementation that uses a destination-controlled sequence number.
	csdl.computer.org /comp/proceedings/icdcs/2004/2086/00/20860380abs.htm (236 words)

	The Wavelet Digest :: View topic - Dense Set
		The Wavelet Digest :: View topic - Dense Set
		How to prove that set of scaling functions form a dense set
		This page was created in 0.200143 seconds : 18 queries executed : GZIP compression disabled
	www.wavelet.org /phpBB2/viewtopic.php?t=5843 (65 words)

	justtatas: Day Ten (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-10)
		In fact, Canyonlands, which is far larger, was quite different at least at this northern end (the southern part is accessed closer to the Grand Canyon than here.)
		Where Arches is a fairly tightly packed and dense set of towers and balanced rocks and pinnacles, Canyonlands was more prairie than desert as we approached.
		The desert ecosystem of the Canyonlands area we were in (Island in the Sky — Mike did not take to the name) is actually in places 1000 feet below where the tourists (“users”) are.
	www.justtatas.com.cob-web.org:8888 /archives/000049.html (1407 words)

	Equidistributed Sequence -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.isi.jhu.edu) (Site not responding. Last check: 2007-10-10)
		However, dense sets need not necessarily be equidistributed.
		The top set of above plots show the values of
		Similarly, the bottom set of above plots show a histogram of the distribution of
	mathworld.wolfram.com.cob-web.org:8888 /EquidistributedSequence.html (353 words)

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