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	PlanetMath: lower bound for sorting
		"lower bound for sorting" is owned by stevecheng.
		Cross-references: properties, total order, necessary, insertion sort, bubble sort, asymptotic bounds for factorial, lower bound, binary tree, generate, order, permutation, depth, height, leaf nodes, represents, internal node, sequence, sorting problem, state, root, tree, class, entire, even, quicksort, heapsort, running, average
		This is version 3 of lower bound for sorting, born on 2002-05-27, modified 2006-05-09.
	planetmath.org /encyclopedia/LowerBoundForSorting.html (479 words)

	Upper bound - Wikipedia, the free encyclopedia
		In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element which is greater than or equal to every element of S.
		On the other hand, a set may have many different upper and lower bounds, and hence one is usually interested in picking out specific elements from the sets of upper or lower bounds.
		This leads to the consideration of least upper bounds (or suprema) and greatest lower bounds (or infima).
	en.wikipedia.org /wiki/Upper_bound (242 words)

	LowerBounds - PineWiki
		Suppose we already have a lower bound S(n) on some problem A, and that there is a reduction from A to some other problem B. Formally, a reduction is a function f such that A(x) = B(f(x)).
		Lower bounds on problems are the really interesting results, since those are the ones that say that the problem is too hard instead of saying that we are too dumb to come up with a better algorithm.
		The difficulty is that it is very hard to write a lower bound proof that covers all possible algorithms, including the ones no one has thought of yet---algorithms are slippery and clever, and there are a lot of them.
	pine.cs.yale.edu /pinewiki/LowerBounds (660 words)

	Infimum - Wikipedia, the free encyclopedia
		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.
		The dual concept of infimum is given by the notion of a supremum or least upper bound.
	en.wikipedia.org /wiki/Infimum (557 words)

	A matrix lower bound
		The matrix lower bound has a natural interpretation in functional analysis, and it satisfies many of the properties that von Neumann stated for it in a restricted case.
		In numerical analysis, the ratio of the matrix norm to the matrix lower bound is a condition number for all consistent systems of linear equations.
		In real analysis, a recursive construction that depends on the matrix lower bound shows that the level sets of continuously differentiable functions lie asymptotically near those of their tangents.
	repositories.cdlib.org /lbnl/LBNL-50635 (194 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-31)
		Date: 06/24/2003 at 14:57:11 From: Doctor Mike Subject: Re: Proof about a bounded set Hi, The infimum is the LARGEST possible lower bound, and the supremum is the SMALLEST possible upper bound, so it is good to use those words only when you mean exactly those specific concepts.
		Assume for notation purposes that the positive "M" is a bound for S. Note, it may NOT be the SMALLEST bound for S. What we are required to prove to show that (b) is true is that S has an upper bound and has a lower bound.
		You correctly observed that M must be an upper bound, and -M must be a lower bound.
	mathforum.org /library/drmath/view/63679.html (437 words)

	Upper bound -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-31)
		The term lower bound is defined (Click link for more info and facts about dually) dually.
		Consider for example the subset of the (The number 1 and any other number obtained by adding 1 to it repeatedly) natural numbers which are greater than a given natural number.
		This leads to the consideration of (Click link for more info and facts about least upper bound) least upper bounds (or suprema) and (Click link for more info and facts about greatest lower bound) greatest lower bounds (or infima).
	www.absoluteastronomy.com /encyclopedia/u/up/upper_bound.htm (274 words)

	LP Addin Documentation
		We use a lower bound of 0 for the constraints.
		In general, the student should be wary of using 0 for lower bounds that are to be redundant as negative constraint coefficients may make the bounds effective rather than redundant.
		The default lower bound is a large negative number, while the default upper bound is a large positive number.
	www.me.utexas.edu /~jensen/or_site/computation/unit/mp_2001/lp_add/two_lim.html (412 words)

	Check Static Lower Bound (Simulink Reference)
		The Check Static Lower Bound block checks that each element of the input signal is greater than (or optionally equal to) a specified lower bound at the current time step.
		The block's parameter dialog box allows you to specify the value of the lower bound and whether the lower bound is inclusive.
		The Check Static Lower Bound block and its companion blocks in the Model Verification library are intended to facilitate creation of self-validating models.
	www.weizmann.ac.il /matlab/toolbox/simulink/slref/checkstaticlowerbound.html (457 words)

	ASAP '96 Explicit Ziv-Zakai Lower Bound for Bearing Estimation Using Planar Arrays - Kristine L. Bell, ... (Site not responding. Last check: 2007-10-31)
		It is a highly nonlinear problem for which calculation of the exact estimation performance is intractable, and lower bounds on the mean square estimation error are used for evaluating performance.
		In this work, the theory of the Ziv-Zakai lower bound is extended to handle vector parameters with arbitrary continuous prior probability density functions, and applied to the bearing estimation problem.
		The resulting bound has a simple closed form expression which is a function of the signal wavelength, the signal-to-noise ratio (SNR), the number of data snapshots, the number of sensors in the array, and the array configuration.
	www.ll.mit.edu /asap/asap_96/abstract/22.html (261 words)

	upper and lower bound for the real roots
		2 15 23 40 because this row is all positive 2 is an upper bound for the roots of the polynomial.
		-7 is lower bound for the real roots.
		2 -3 14 -104 because this row is alternating signs -7 is a lower bound for the roots of the polynomial.
	www.chesapeake.edu /khennayake/MAT113/upperLowerboundforroots.htm (99 words)

	PlanetMath: lowest upper bound
		Greatest lower bound is defined similarly: a greatest lower bound, or infimum, of
		Cross-references: finite, lower bound, property, upper bound, subset, partial ordering
		This is version 8 of lowest upper bound, born on 2001-10-21, modified 2007-02-15.
	planetmath.org /encyclopedia/GreatestLowerBound.html (71 words)

	Lower Bound Techniques in Distributed Computing
		This course studies lower bounds and impossibility results for various problems (and models) in distributed computing.
		An Omega(sqrt(n)) space lower bound for randomized and obstruction-free consensus: the nonanonymous case.
		A lower bound on the number of rounds for randomized consensus in synchronous message-passing systems.
	www.cs.technion.ac.il /~hagit/LowerBounds/index.htm (326 words)

	[No title] (Site not responding. Last check: 2007-10-31)
		And attempt to find a lower bound and an upper bound for the area under this curve and above the interval [1,11] First, let’s determine on which intervals the function is increasing and decreasing.
		[1,5) Lower Bound: As we saw in class, since the function is increasing on [1,5), a lower bound is found by using the left endpoint of each subinterval.
		Lower Bound: 42.0864+10.40832=52.49472 Upper Bound: 43.6224+57.2256=100.848 So the area under the curve and above [1,11] is bigger than 52.49472 but less than 100.848.
	www.unc.edu /~raatzs/IntroductiontoArea.doc (739 words)

	The Random Adversary: A Lower-Bound Technique For Randomized Parallel Algorithms - MacKenzie (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		The Random-Adversary technique is a general method for proving lower bounds on randomized parallel algorithms.
		7.6%: A Lower Bound for the QRQW PRAM - MacKenzie (1997)
		5 A lower bound for the QRQW PRAM - MacKenzie - 1995 ACM
	citeseer.ist.psu.edu /72574.html (852 words)

	Lower Bounds for Dynamic Algebraic Problems (Site not responding. Last check: 2007-10-31)
		We present techniques for showing lower bounds on the worst case time complexity per operation for such problems.
		The first gives lower bounds in a wide range of rather powerful models (for instance history dependent algebraic computation trees over any infinite subset of a field, the integer RAM, and the generalized real RAM model of Ben-Amram and Galil).
		The second technique is the communication complexity technique of Miltersen, Nisan, Safra, and Wigderson which we apply to the setting of dynamic algebraic problems, obtaining similar lower bounds in the word RAM model.
	www.brics.dk /RS/98/11/index.html (230 words)

	Branch and Bound (Site not responding. Last check: 2007-10-31)
		To apply branch and bound, one must have a means of computing a lower bound on an instance of the optimization problem and a means of dividing the feasible region of a problem to create smaller subproblems.
		There must also be a way to compute an upper bound (feasible solution) for at least some instances; for practical purposes, it should be possible to compute upper bounds for some set of nontrivial feasible regions.
		Since it is feasible, it can be used to prune the rest of the tree: if the lower bound for a node exceeds the best known feasible solution, no globally optimal solution can exist in the subspace of the feasible region represented by the node.
	www.cs.sandia.gov /opt/survey/branch-and-bound.html (318 words)

	Lower bound of array was changed to 0 (Visual Basic and Visual C# Concepts)
		Lower bound of array was changed to 0 (Visual Basic and Visual C# Concepts)
		Lower bound of array was changed to 0
		Previous versions of Visual Basic allowed you to set the lower bounds of an array to 1 using the Option Base statement, or to any valid number using the syntax MyArray(lowerbounds to upperbounds) where upperbounds and lowerbounds are numbers.
	msdn.microsoft.com /library/en-us/vbcon/html/vbup1033.asp?frame=true (369 words)

	Greatest lower bound - Computing Reference - eLook.org (Site not responding. Last check: 2007-10-31)
		(glb, meet, infimum) The greatest lower bound of two elements, a and b is an element c such that c <= a and c <= b and if there is any other lower bound c' then c' <= c.
		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.
	www.elook.org /computing/greatest-lower-bound.htm (140 words)

	Lower Bound on Permutation Routing (Site not responding. Last check: 2007-10-31)
		An adaptive routing algorithm is one in which the path a packet takes from its source to its destination may depend on other packets it encounters.
		For a large class of minimal adaptive routing algorithms, we present an Omega(n^2/k^2) bound on the worst case time to route a static permutation of packets on an n x n mesh or torus with nodes that can hold up to k >= 1 packets each.
		This is the first nontrivial lower bound on adaptive routing algorithms.
	cs.washington.edu /research/projects/lis/chaos/www/abstracts/clt.html (241 words)

	Citations: Lower bound gradient estimates for rst-order Hamilton-Jacobi equations and applications to the regularity of ... (Site not responding. Last check: 2007-10-31)
		This lower bound estimate implies that fattening cannot occur, i.
		O.Ley, Lower bound gradient estimates for rst-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts, Adv.
		Lower bound gradient estimates for rst-order Hamilton-Jacobi equations and applications to the regularity of propagating fronts, preprint.
	citeseer.ist.psu.edu /context/1228395/0 (699 words)

	A Fundamental Lower Bound to the Performance of Phase Estimators Over Rician-Fading Channels (Site not responding. Last check: 2007-10-31)
		We present a fundamental lower bound to the mean-square error of estimators of the phase of a sine wave passed through a Rician-fading channel.
		The bound is derived by analyzing the performance of the Bayes minimum mean-square error phase estimator, and therefore represents the best performance attainable by physically realizable phase estimators over a Rician channel.
		Moeneclaey, "A fundamental lower bound to the performance of practical joint carrier and bit synchronizers," IEEE Trans.
	www.comsoc.org /comm/private/1997/jul/775_45comm07-reggiannini.html (276 words)

	The Random Adversary: A Lower-Bound Technique For Randomized Parallel Algorithms - MacKenzie (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		The Random-Adversary technique is a general method for proving lower bounds on randomized parallel algorithms.
		7.6%: A Lower Bound for the QRQW PRAM - MacKenzie (1997)
		5 A lower bound for the QRQW PRAM - MacKenzie - 1995
	citeseer.lcs.mit.edu /72574.html (793 words)

	An Attainable Lower Bound for the Best Normal Approximation
		Lower bounds for the distance of a complex $n \times n$ matrix $A$ from the variety of normal matrices are established.
		The weaker version gives a lower bound of the form ${\rm dep}(A)/\sqrt n$, where ${\rm dep}(A)$ is Henrici's "departure from normality.
		Recall that ${\rm dep}(A)$ itself is an upper bound for the distance at issue.
	epubs.siam.org /sam-bin/dbq/article/23230 (172 words)

	Lower-bound and upper-bound markers
		It is often convenient to define explicitly the lower bound and upper bound of a type map, and override the definition given on page Lower-bound and upper-bound markers.
		To achieve this, we add two additional ``pseudo-datatypes,'' MPI_LB and MPI_UB, that can be used, respectively, to mark the lower bound or the upper bound of a datatype.
		Similarly, the upper bound of Typemap is defined to be
	www.mpi-forum.org /docs/mpi-11-html/node57.html (468 words)

	# 177 - Meaning of"least upper bound"and"greatest lower bound"
		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 upper bounds in the set of security attributes, such that, given any two valid security attributes, there is a valid security attribute that is greater than or equal to the two valid security attributes; and
		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)

	A Lower Bound for Randomized Algebraic Decision Trees - Grigoriev, Karpinksi, auf, Smolensky (ResearchIndex) (Site not responding. Last check: 2007-10-31)
		Abstract: We extend the lower bounds on the depth of algebraic decision trees to the case of randomized algebraic decision trees (with two-sided error) for languages being finite unions of hyperplanes and the intersections of halfspaces, solving a long standing open problem.
		As an application, among other things, we derive, for the first time, an \Omega\Gamma n 2) randomized lower bound for the Knapsack Problem which was previously only known for deterministic algebraic decision trees.
		3 A Lower Bound (context) - Dobkin, Lipton
	citeseer.lcs.mit.edu /80559.html (457 words)

	Lower-bound and Upper-bound Markers
		extentderived datatype, extent lower boundderived datatype, lower bound upper boundderived datatype, upper bound
		MPI_TYPE_LB returns the lower bound of a datatype, in bytes, relative to the datatype origin.
		MPI_TYPE_UB returns the upper bound of a datatype, in bytes, relative to the datatype origin.
	zikova.cvut.cz /parallel/mpi-book/node87.html (425 words)

	asymptotic lower bound (Site not responding. Last check: 2007-10-31)
		Definition: An asymptotic bound, as function of the size of the input, on the best (fastest, least amount of space used, etc.) an algorithm can possibly achieve to solve a problem.
		That is, no algorithm can use fewer resources than the bound.
		Sandeep Kumar Shukla, "asymptotic lower bound", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/asymptoticLowerBound.html (110 words)

	How long before superintelligence? (Site not responding. Last check: 2007-10-31)
		The Bekenstein bound gives an upper limit on the amount of information that can be contained within any given volume using a given amount of energy.
		Since space colonization would allow at most a polynomial (~t^3) expansion rate (assuming the maximal speed is bounded by the speed of light), the exponential increase of available computational power cannot be continued indefinitely, unless new physics is forthcoming.
		If Moore's law continues to hold then the lower bound will be reached sometime between 2004 and 2008, and the upper bound between 2015 and 2024.
	mikeai.nm.ru /english/superint.html (5458 words)

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