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Topic: Concavity

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	Concavity - Wikipedia, the free encyclopedia
		In mathematical analysis, concavity is a property of certain geometric figures, and in calculus, a property of certain graphs of functions.
		Equivalently, f(x) is concave on [a, b] iff the function −f(x) is convex on every subinterval of [a, b].
		A concave polygon is often called re-entrant polygon (but in some cases the latter term has a different meaning).
	en.wikipedia.org /wiki/Concavity (391 words)

	Convexity and concavity (Site not responding. Last check: 2007-10-07)
		The twin notions of concavity and convexity are used widely in economic theory, and are also central to optimization theory.
		The fact that it is concave means that the increase in output generated by a one-unit increase in the input is smaller when output is large than when it is small.
		The notions of concavity and convexity are important in optimization theory because, as we shall see, the first-order conditions are sufficient (as well as necessary) for a maximizer of a concave function and for a minimizer of a convex function.
	www.chass.utoronto.ca /~osborne/MathTutorial/CV1.HTM (1352 words)

	Concave GP Faces
		This concavity divides each of the apparent four sides in half, creating a very special and unusual eight-sided pyramid; and it is executed to such an extraordinary degree of precision as to enter the realm of the uncanny.
		Verner agreed: "As in the case of the earlier Red Pyramid, the slightly concave walls were intended to increase the stability of the pyramid's mantle [i.e.
		The evidence indicates that the concavity is a functional feature of the core structure that was hidden from sight when the casing stones were applied.
	www.catchpenny.org /concave.html (1025 words)

	Convexity and concavity for functions of many variables
		Any stationary point of a concave function of a single variable is a global maximizer; any stationary point of a convex function of a single variable is a global minimizer (see an earlier page).
		Convexity and concavity are key also in the case of optima of functions of many variables.
		An definition of a concave function equivalent to the one given previously is sometimes useful.
	free.prohosting.com /cepr/data/adveco/cvn.html (867 words)

	Concavity and the Second Derivative Test - HMC Calculus Tutorial
		Let's determine where the graph of f is concave up and where it is concave down.
		Points on the graph of f where the concavity changes from up-to-down or down-to-up are called inflection points of the graph.
		The Second Derivative Test relates the concepts of critical points, extreme values, and concavity to give a very useful tool for determining whether a critical point on the graph of a function is a relative minimum or maximum.
	www.math.hmc.edu /calculus/tutorials/secondderiv (849 words)

	CONCAVITY (Site not responding. Last check: 2007-10-07)
		This exercise deals with the second derivative and the basic idea of concavity as it relates to economic functions.
		If the second derivative is positive at a given point, the function q = f (p) is concave upward at that point.
		Concavity relates to maximization and convexity to minimization.
	www.mtsu.edu /~dgraddy/micro/pderv3.htm (245 words)

	Concavity
		A function is concave up on [a,b] if it is below its chord line on every subinterval.
		If the function has a second derivative (and hence a first derivative) a simple test for concavity can be based on the sign of the second derivative:
		An inflection point is a point in the domain of f(x) where f(x) changes concavity.
	www.louisville.edu /~grbarn01/205/205LS0404.html (303 words)

	Convexity and concavity
		The importance of concave and convex functions in optimization theory comes from the fact that for a concave function every stationary point is a global maximizer, and for a convex function every stationary point is a global minimizer.
		That is: a nondecreasing concave transformation of an nondecreasing concave function is nondecreasing and concave.
		A point at which a function changes from being convex to concave, or vice versa, is an inflection point.
	free.prohosting.com /cepr/data/adveco/cv1.html (1158 words)

	Visual Calculus / Graphs and Derivatives
		Some examples of finding graphically where a given function is concave upward or concave downward are given.
		f is concave downward on (c, a) and is concave upward on (a, d).
		Quiz on determining which graph is the graph of a function, its derivative and its 2nd derivatives.
	archives.math.utk.edu /visual.calculus/3/graphing.14 (455 words)

	SSRN-Concavity of Utility, Concavity of Welfare, and Redistribution of Income by Louis Kaplow (Site not responding. Last check: 2007-10-07)
		The marginal social value of income redistribution is understood to depend on both the concavity of individuals' utility functions and the concavity of the social welfare function.
		In the pertinent literatures, notably on optimal income taxation and on normative inequality measurement, it seems to be accepted that the role of these two sources of concavity is symmetric with regard to the social concern about inequality in the distribution of income.
		Concavity of utility has a simple, direct effect on the marginal social value of redistribution, as might be expected, whereas concavity of the social welfare function has a more subtle influence, one that in some cases may not be very significant.
	papers.ssrn.com /sol3/papers.cfm?abstract_id=463241 (327 words)

	Lesson #66 (Site not responding. Last check: 2007-10-07)
		Concavity A. concave down — does not hold water concave up — does hold water B. Problem: A coin is in a "cup" formed by 4 matchsticks.
		concave up — they are increasing concave down — they are decreasing The second derivative is a rate of change of the first derivative.
		If the concavity changes from positive to negative or from negative to positive, it is a point of inflection.
	www.pen.k12.va.us /Div/Winchester/jhhs/math/lessons/calculus/day66.html (317 words)

	Concavity and the Second Derivative Test: Sec. 3.4 (Site not responding. Last check: 2007-10-07)
		The graph of f is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.
		If the concavity changes, meaning the sign changes, this is a possible point of inflection.
		If the interval is positive, the graph is concave up; if it is negative, the graph is concave down.
	www.kent.k12.wa.us /staff/DavidWright/calculus/book/34 (268 words)

	Concavity and the Second Derivative
		Conversely, if the graph is concave up or down, then the derivative is monotonic.
		Suppose that f is twice differentiable on the open interval (a,b).
		What is being said about the concavity of that function.
	oregonstate.edu /instruct/mth251/cq/Stage7/Lesson/concavity.html (255 words)

	Concavity and Points of Inflection
		We will say that the graph of f(x) is concave down on I iff f '(x) is decreasing on I.
		Usually graphs have regions which are concave up and others which are concave down.
		Thus there are often points at which the graph changes from being concave up to concave down, or vice versa.
	www.sosmath.com /calculus/diff/der15/der15.html (356 words)

	Logarithmic Concavity And (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		We observe that for any logarithmically concave finite sequence a 0, a 1, : : :, an of positive integers there is a representation of the Lie algebra sl 2 (C) from which this logarithmic concavity follows.
		Thus, in applying this strategy to prove logarithmic concavity, the only issue is to construct such a representation naturally from given combinatorial data.
		As an example, we do this when a j is the number of j-element stable sets in a claw-free graph, reproving a theorem of...
	citeseer.ist.psu.edu /302959.html (250 words)

	Saint Thomas Aquinas
		And these differ because snub is bound up with matter (for what is snub is a concave nose), while concavity is independent of perceptible matter." (1025a28-32) The objects of natural philosophy are defined like ‘snub’ and the objects of mathematics like ‘concave’.
		This makes it clear that the way in which natural things are separated from sensible matter is the way in which the definition common to many things abstracts from the singular characteristics of each.
		Mathematical things, on the analogy of ‘concave’, do not have sensible matter in their definitions.
	plato.stanford.edu /entries/aquinas (11428 words)

	Concavity Cuts for Disjoint Bilinear Programming (Site not responding. Last check: 2007-10-07)
		We pursue the study of concavity cuts for the disjoint bilinear programming problem.
		This optimization problem has two equivalent symmetric linear maxmin reformulations, leading to two sets of concavity cuts.
		We next propose a branch and bound algorithm which make use of concavity cuts.
	www.caam.rice.edu /~charlesa/TR99788.html (146 words)

	Exambot - Concavity and Inflections
		Mathematics > Differential Calculus > Function Graphs > Concavity and Inflections
		The function f (x) is given by f (x)= x 5 - 10 kx 4 + 25 k 2 x 3, where k is a positive constant....
		Using the second derivative test, determine the extrema of the equation and the points of inflection, then draw the graph.
	www.exambot.com /cgi/topic/show.cgi/math/difc/graf/coin (575 words)

	cooltech.iafrica.com \| new ideas \| bizarre patents Mattress with a concavity for the breasts
		A mattress having a wedge-shaped mattress body with an inclined upper surface, the upper surface divided by a transverse breast concavity into a head-supporting portion and a body-supporting portion.
		The mattress has a flat position in which the concavity is open to receive breasts as well as an upright position in which the concavity functions as a hinging mechanism and the mattress is held in the upright position by adjustable mechanisms.
		In the flat position tha mattress may be used in the prone position, laying on the back, or in a yoga position.
	cooltech.iafrica.com /inventions/bizarrepatents/216432.htm (233 words)

	EconPapers: Concavity of Utility, Concavity of Welfare, and Redistribution of Income
		EconPapers: Concavity of Utility, Concavity of Welfare, and Redistribution of Income
		Concavity of Utility, Concavity of Welfare, and Redistribution of Income
		Abstract: The marginal social value of income redistribution is understood to depend on both the concavity of individuals' utility functions and the concavity of the social welfare function.
	econpapers.repec.org /paper/nbrnberwo/10005.htm (334 words)

	Glossary of research economics
		CDFC: Stands for Concavity of distribution function condition.
		concavity of distribution function condition: A property of a distribution function-utility function pair.
		The grid approach is necessary because the problem is not concave.
	econterms.com /econtent.html (14743 words)

	Keith Price Bibliography Concavity Detection (Site not responding. Last check: 2007-10-07)
		Sklansky, J. Measuring Concavity on a Rectangular Mosaic,
		Sklansky, J. Cordella, L.P., and Levialdi, S. Parallel Detection of Concavities in Cellular Blobs,
		Fill the concavities to get the convex hull.
	iris.usc.edu /Vision-Notes/bibliography/twod283.html (145 words)

	EconPapers: Quadratic Concavity and Determinacy of Equilibrium
		Working Paper: Quadratic Concavity and Determinacy of Equilibrium (1999)
		Journal Article: Quadratic Concavity and Determinacy of Equilibrium (2002)
		This item may be available elsewhere in EconPapers: Search for items with the same title.
	econpapers.repec.org /paper/wpawuwpge/9912001.htm (111 words)

	Joseph L. Brandell: Personal Website (Site not responding. Last check: 2007-10-07)
		Navigate to the concepts of maxima, minima, extreme points and other chapter 4 concepts!
		The three bullets on the first page, “Critical Points,” “Global Extrema” and “Concavity and Points of Inflection” are presented very well.
		If you need a refresher, this is the place to go!
	www.brandellcentral.org /links (549 words)

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