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Topic: Greatest lower bound

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	Infimum -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		Infima of (Any rational or irrational number) real numbers are a common special case that is especially important in (The abstract separation of a whole into its constituent parts in order to study the parts and their relations) analysis.
		Infima are in a precise sense (Click link for more info and facts about dual) dual to the concept of a (Click link for more info and facts about supremum) supremum and thus additional information and examples are found within the corresponding article.
		In this context, especially in (Click link for more info and facts about lattice theory) lattice theory, greatest lower bounds are also called meets.
	www.absoluteastronomy.com /encyclopedia/i/in/infimum.htm (662 words)

	Meet
		The greatest lower bound of a set S is the greatest element b such that for all s in S, b <= s.
		The glb of mutually comparable elements is their minimum but in the presence of incomparable elements, if the glb exists, it will be some other element less than all of them.
		glb is the dual to least upper bound.
	www.linuxguruz.com /foldoc/foldoc.php?Meet (167 words)

	ipedia.com: Supremum Article (Site not responding. Last check: 2007-10-20)
		In analysis the supremum or least upper bound of a set S of real numbers is denoted by sup(S) and is defined to be the smallest real number that is greater than or equal to every number in S.
		The difference between the supremum of a set and the greatest element of a set may not be immediately obvious.
		For an example where there are no greatest but still some maximal elements, consider the set of all subsets of the set of natural numbers (the powerset).
	www.ipedia.com /supremum.html (1393 words)

	Mathwords: Greatest Lower Bound of a Set
		The greatest of all lower bounds of a set of numbers.
		For example, the greatest lower bound of (5, 7) is 5.
		The greatest lower bound of the interval [5, 7] is also 5.
	www.mathwords.com /g/greatest_lower_bound.htm (43 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-20)
		However, it's trivial to show that 1 is its infimum; clearly all elements are greater than or equal to 1, and if we thought that something greater than 1 was a lower bound, it'd be easy to show some member of S which is less than it.
		A couple of last terminology bits: glb is another way of writing inf (sort for "greatest lower bound"), and then there are two other concepts that are similar for upper bounds, sup and lub, which are short for "supremum" and "least upper bound." So what are we talking about here with these approximation algorithms?
		It means we want the greatest lower bound of the set of all numbers r greater than 1 such that RA (I) is less than r for all instances I of the problem.
	www.mathforum.com /library/drmath/view/51435.html (1384 words)

	[No title]
		What about lower bounds and the greatest lower bound (which is the corresponding notion to the least upper bound).
		The element $a$ of $X$ is called the {\bf infimum} or {\bf greatest lower bound} of $S$ if \begin{enumerate} \item[(i)] $a$ is itself a lower bound of $S$, and \item[(ii)] no element $b$ of $X$ which is greater than $a$ can be a lower bound for $S$.
		Since a is an upper bound for the set in the rationals, it is also an upper bound for the set in the reals, and so the square root of 2 is less than or equal to it.
	www.math.unl.edu /~webnotes/src/classes-1997/class06.wfy (1796 words)

	Lattice Model of Information Flow
		There is a highest class H, which is the least upper bound of all classes, and a least class L, which is the greatest lower bound of all classes.
		When external objects, such as files and separately compiled procedures, are bound to a program, the linker must verify that the actual security class of each such object corresponds properly to the security class declared formally for it in the program.
		A universally bounded lattice is a structure consisting of a finite partially ordered set together with least upper and greatest lower bound operators on the set.
	courses.cs.vt.edu /~cs5204/fall99/distributedSys/lattice.html (1457 words)

	Sequences (Site not responding. Last check: 2007-10-20)
		From section 1 we have a least upper bound (greatest lower bound) for any bounded increasing (respectively decreasing) sequence of real numbers.
		Exercise 28 Show that any bounded non-decreasing sequence of real numbers has a least upper bound; a bounded non-increasing sequence has a greatest lower bound.
		} is any bounded sequence of real numbers we have a least upper bound and greatest lower bound for this sequence.
	www.imsc.ernet.in /~kapil/geometry/prereq1/node7.html (273 words)

	# 177 - Meaning of"least upper bound"and"greatest lower bound"
		Items b and c under IFF.2.7 introduce a notion of least upper bound and greatest lower bound for comparison purposes between two attributes.
		The erroneous use of the terms "least upper bound" and "greatest lower bound" conveys the notion that disjoint pairs of categories are omitted - or excluded - from the policy.
		There exist pairwise lower bounds in the set of security attributes, such that, given any two valid security attributes, there is a valid security attribute for which both of the two original valid security attributes are greater than the security attribute.
	niap.nist.gov /cc-scheme/interpretations/draft177.html (434 words)

	Physics Help and Math Help - Physics Forums - View Single Post - LUB and Nested Interval Equivalancy
		First, the "least upper bound property" is equivalent to the "greatest lower bound property" (If a set of real numbers has a lower bound, then it has a greatest lower bound).
		If X is a set having a lower bound b, -X (multiply each member of X by -1) has upper bound -b and so has a least upper bound, x.
		-x is then the greatest lower bound of X. The left endpoints of the intervals form a set having all the right endpoints as upper bounds therefore, has a least upper bound, a.
	www.physicsforums.com /showpost.php?p=396722&postcount=2 (165 words)

	INFIMUM FACTS AND INFORMATION (Site not responding. Last check: 2007-10-20)
		Consequently the term greatest lower bound (also abbreviated as glb or GLB) is also commonly used.
		Infima are in a precise sense dual to the concept of a supremum and thus additional information and examples are found in that article.
		In analysis the infimum or greatest lower bound of a set ''S'' of real_numbers is denoted by inf(''S'') and is defined to be the biggest real number that is smaller than or equal to every number in ''S''.
	www.whereintheworldisbush.com /infimum (545 words)

	Re: RI # 177 - Meaning of "least upper bound" and "greatest lower bound" (Site not responding. Last check: 2007-10-20)
		However, the definitions for "least upper bound" and "greatest lower bound" are still needed.
		d) Each set of upper bounds has a "least upper bound", which all the elements of that set are greater than or equal to; and e) Each set of lower bounds has a "greatest lower bound", which is greater than or equal to all the elements of that set.
		the defined bounds do not come in pairs, therefore the two instances of "pairwise" should be deleted.
	cio.nist.gov /esd/emaildir/lists/cc-cmt/msg01000.html (272 words)

	Circumnavigating a cube and a tetrahedron
		With some tightening of the outline of why 6 is a lower bound, I believe this would be a proof.
		For some of these, the mid-point will not be the shortest possibility (moving the starting and finishing point towards a corner can in some cases meet the requirements of the question while reducing the length towards sqrt(7) or sqrt(27/4) when it starts from a higher figure).
		There is not a straight line on the unfolded net which is equivalent to the single topological route, but it is possible to draw a succession of four straight lines which have a greatest lower bound of a length of 2+sqrt(28)=7.291...
	www.btinternet.com /~se16/js/circumnavcubetetra.htm (3033 words)

	[No title]
		Bentler, P. A lower bound method for the dimension-free measurement of internal consistency.
		Jackson, P. W., & Agunwamba, C. Lower bounds for the reliability of the total score on a test composed of nonhomogeneous types: I. Algebraic lower bounds.
		Shapiro, A. A note on the asymptotic distribution of the greatest lower bound to reliability.
	www.helsinki.fi /~kvehkala/ref.txt (1844 words)

	Extract 6 (Site not responding. Last check: 2007-10-20)
		So since zero is a lower bound, the greatest of the lower bounds, that is the infimum, must be greater or equal to zero, she concludes.
		we first prove that something has a supremum, so this would have a supremum the set of lower bounds and we have to show that the supremum of this would be equal to the infimum of that.
		K5: This is the greatest lower bound but this is less than...
	www.uea.ac.uk /~m011/thesis/appendices/appendices6/extract6iv.htm (585 words)

	Theoretical part about limits and continuity
		If a set S has an upper bound and a lower bound, we say that the set is bounded.
		Then, the number (l+h/2) is not an upper bound of S and so it is in L. But, the number (l+h/2) is greater than l and so it is in H. This is impossible because L an H have no element in common.
		Hence, f(x) is bounded in a suitable small environment of b.
	www.ping.be /~ping1339/limth.htm (1077 words)

	posetlattice.html
		Define lower bound and greatest lower bound (g.l.b.) is a similar manner.
		Note that being the top element implies uniqueness and that it is the least upper bound.
		has a least upper bound and a greatest lower bound given be set union and intersection,
	www.umsl.edu /~siegel/SetTheoryandTopology/posetlattice.html (108 words)

	Math 554 Sample Homework Assignments
		Prove the equivalence statement for "greatest lower bounds" (we proved in class the similar statement for least upper bounds): If L is the greatest lower bound for a set A, then
		(i) L is a lower bound for A
		the least upper bound property), prove that R satisfies the property that each nonempty set B, which is bounded from below, must have a greatest lower bound.
	www.math.sc.edu /~sharpley/math554/Homework.html (482 words)

	Least Upper Bound (Site not responding. Last check: 2007-10-20)
		We'll prove the existence of the greatest lower bound.
		Now all of s is bounded below by 0, and at least one point in s is as high as 1.
		Therefore t is the greatest lower bound for s.
	www.mathreference.com /top-ms,lub.html (309 words)

	lec11Sept
		the greatest element if it is greater than all other elements in the poset.
		As we saw above, not every subset of a poset has a least upper bound and a greatest lower bound.
		A lattice is a poset such that for every subset consisting of a pair of elements in the poset, the subset has both a least upper bound and a greatest lower bound.
	www.pitt.edu /~vanlehn/cs0441/lec30Oct.html (801 words)

	Boundedness (Site not responding. Last check: 2007-10-20)
		Least Upper Bound Axiom for the Real Numbers: If S is nonempty and has an upper bound, then it has a least upper bound.
		If S is a nonempty subset of the real numbers and S has a lower bound, then it has a greatest lower bound.
		If S is a nonempty subset of the rational numbers that has an upper bound, then it has a rational least upper bound.
	www.ms.uky.edu /~lee/ma502/notes7/node2.html (233 words)

	Väitöskirjani lähteillä / Kimmo Vehkalahti
		Jackson, P. W., and Agunwamba, C. Lower bounds for the reliability of the total score on a test composed of nonhomogeneous types: I. Algebraic lower bounds.
		Woodhouse, B., and Jackson, P. Lower bounds for the reliability of the total score on a test composed of nonhomogeneous types: II.
		Woodward, J. A., and Bentler, P. A statistical lower bound to population reliability.
	www.helsinki.fi /~kvehkala/ref.html (2116 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Upper semilattice of binary strings with the relation ``x is simple conditional to y'' Andrei Muchnik, Andrei Romashchenko, Alexander Shen, Nikolai Vereshagin We study the properties of the set of binary strings with the relation ``the Kolmogorov complexity of x conditional to y is small''.
		We prove that there are pairs of strings which have no greatest common lower bound with respect to this preorder.
		We present several examples when the greatest common lower bound exists but its complexity is much less than mutual information (extending G'acs and K¨orner result).
	www.illc.uva.nl /Publications/ResearchReports/CT-1997-04.abstract.txt (90 words)

	The Lipschitz Constant (Site not responding. Last check: 2007-10-20)
		Technically, the lipschitz constant for f is the greatest lower bound of all k satisfying the above criteria.
		Since the values of k are bounded below, let's see why the greatest lower bound is also a lipschitz constant.
		Thus the greatest lower bound is a valid lipschitz constant.
	www.mathreference.com /top-ms,lip.html (443 words)

	Physics Help and Math Help - Physics Forums - LUB and Nested Interval Equivalancy
		12-08-2004 02:35 PM First, the "least upper bound property" is equivalent to the "greatest lower bound property" (If a set of real numbers has a lower bound, then it has a greatest lower bound).
		Because A is bounded, it is contained in some interval [a, b] which, by lemma 2 is compact.
		If there were no x in A larger than p (so that p is an upper bound on A), then we have x in (p- delta,delta) which say that p is the least upper bound of A- which contradicts the hypothesis that A has no LUB.
	www.physicsforums.com /printthread.php?t=55944 (1046 words)

	L.U.B. implies Connectedness (Site not responding. Last check: 2007-10-20)
		We show that, if our ordered field satisfies the Least Upper Bound property, then, considering it with the order topology (ultimately equivalent to the metric topology), it cannot be expressed as the disjoint union of two nonempty open subsets.
		Therefore U has a greatest lower bound, call it g.
		If g were in B, then since B is open, there is a number g-epsilon in B which is also greater than a.
	www.math.uaa.alaska.edu /~mathclub/mc3.html (122 words)

	Efficient Implementation of Lattice Operations - Ait-Kaci, Boyer, Lincoln, Nasr (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Abstract: Lattice operations such as greatest lower bound (GLB), least upper bound (LUB), and relative complementation (BUTNOT) are becoming more and more important in programming languages supporting object inheritance.
		Although those encoding schemes are successful in their particular fields, research is ongoing in the quest for general purpose,...
		...an algorithm to 3 compute the greatest lower bound and least upper bound of any two class schemas in the inheritance lattice (c.f.
	citeseer.ist.psu.edu /ait-kaci89efficient.html (586 words)

	Upper Semi-Lattice of Binary Strings With the Relation "x is Simple Conditional to Y" (ResearchIndex) (Site not responding. Last check: 2007-10-20)
		Abstract: We study the properties of the set of binary strings with the relation "the Kolmogorov complexity of x conditional to y is small".
		We prove that there are pairs of strings which have no greatest common lower bound with respect to this pre-order.
		We present several examples when the greatest common lower bound exists but its complexity is much less than mutual information (extending G'acs and Korner result [2]).
	citeseer.ist.psu.edu /76880.html (370 words)

	BBC - KS3 Bitesize - SOS Teacher Maths shape,space numbers (Site not responding. Last check: 2007-10-20)
		Calculate the least upper bound and greatest lower bound of the expansion.
		So the expansion could be any length between these extremes.
		The upper bound = 45.75 - 44.75 = 1.0cm
	www.bbc.co.uk /schools/ks3bitesize/sosteacher/maths/46900.shtml (135 words)

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