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Topic: Convex combination

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	Convex - Wikipedia, the free encyclopedia
		Convex function, a function with the epigraph (the set of points lying on or above the graph) forming a convex set
		Convex conjugate, is a generalization of the Legendre transformation
		Convex lens, a lens with surfaces that curve outward
	en.wikipedia.org /wiki/Convex (213 words)

	PlanetMath: convex combination
		the convex hull, or convex envelope, or convex closure of
		This is version 8 of convex combination, born on 2001-10-19, modified 2003-09-20.
		A convex combination seems to be a finite dimensional concept.
	planetmath.org /encyclopedia/ConvexCombination.html (222 words)

	Convex set - Wikipedia, the free encyclopedia
		For example, a solid cube is convex, but anything that is hollow or has a dent in it is not convex.
		Closed convex sets can be characterised as the intersections of closed half-spaces (sets of point in space that lie on and to one side of a hyperplane).
		To prove the converse, i.e., every convex set may be represented as such intersection, one needs the supporting hyperplane theorem in the form that for a given closed convex set C and point P outside it, there is a closed half-space H that contains C and not P.
	en.wikipedia.org /wiki/Convex_set (704 words)

	PlanetMath: convex set
		Examples of convex sets on the plane are circles, triangles, and ellipses.
		See Also: convex combination, Carathéodory's theorem, extreme subset of convex set, properties of extreme subsets of a closed convex set
		This is version 13 of convex set, born on 2001-10-15, modified 2006-08-07.
	planetmath.org /encyclopedia/ConvexSet.html (339 words)

	Photolithographic projection apparatus using light in the far ultraviolet - Patent 4302079
		The apex of the convex surface of the beam-splitter plano-convex lens combination is located on an optical path from the object plane to the image plane that is normal to the convex surface at its apex.
		The distance along the above-mentioned optical path from the first surface of the combination to the apex of the convex surface and the distance along the above-mentioned optical path from the apex to the second surface of the combination are mutually approximately equal to the radius of curvature of the convex surface.
		The radius of curvature of the convex surface is -175.1773 mm.
	www.freepatentsonline.com /4302079.html (3593 words)

	s_convex
		Equivalently, a convex combination is a weighted average in which the weights are nonnegative and add to
		A linear combination in which the coefficients have sum 1 is called a barycentric combination.
		Thus a convex combination is a barycentric combination in which the coefficients are also nonnegative.
	www.math.ucla.edu /~baker/149.1.02w/handouts/s_convex/node4.html (96 words)

	PlanetMath: local minimum of convex function is necessarily global
		"local minimum of convex function is necessarily global" is owned by stevecheng.
		Cross-references: continuous, vector, scalar, convex combination, neighborhood, open, extremum, topological vector space, subset, convex, concave function, convex function, local maximum, local minimum
		This is version 10 of local minimum of convex function is necessarily global, born on 2003-04-07, modified 2006-07-05.
	planetmath.org /encyclopedia/ExtremalValueOfConvexconcaveFunctions.html (168 words)

	[No title]
		An equivalent definition of convex hull is the set of points that can be expressed as convex combinations of the points in S.
		The convex hull of a finite set of points is necessarily a bounded, closed, convex polygon.
		Convex hull problem: The (planar) convex hull problem is, given a set of n points P in the plane, output the vertices of the convex hull.
	www.cs.wustl.edu /~pless/506/l2.html (1841 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.
		Let f be a function of many variables with continuous partial derivatives of first and second order on the convex open set S and denote the Hessian of f at the point x by H(x).
	free.prohosting.com /cepr/data/adveco/cvn.html (867 words)

	Brian's Digest: Convexity
		By the way, Rockafellar defines the convex hull of a set S to be the intersection of all convex sets containing S. He defines this for the case S subset of R^n, but the definition may well pertain more generally, since the property that the intersection of convex sets is convex seems to be universal.
		The convex hull of X should be the minimal convex set containing X. Clearly it has to include finite convex combinations of points in X to be convex.
		Consider A(n-1) to be the portion of the convex hull in the n-1 hyperplane.
	www.worms.ms.unimelb.edu.au /digest/convexity.html (1068 words)

	Convexity
		You may already have met a convex function -- one in which the points above the graph of the function (the ``supergraph'') form a convex set.
		An obvious way to generate a convex set is as the intersection of a collection of half planes; hence the set of feasible solutions of a linear programming problem forms a (possibly empty) convex set.
		It is geometrically clear that a (strictly) convex function cannot have two different local minima, while if the same value occurs as a local minimum at different places, the graph must be flat between them, so neither minimum is isolated (and f can't be strictly convex).
	www.maths.abdn.ac.uk /~ran/mx3503/notes/notes/node82.html (461 words)

	[No title]
		The distribution of the convex combination is translation invariant and does not depend on the heights.
		Special cases of RAP are a type of smoothing process (when the convex combination is deterministic) and the voter model (when the convex combination concentrates on one of the neighbors chosen at random).
		We show that when the convex combination is neither deterministic nor concentrating on one of the neighbors the variance of the height at the origin at time $t$ is proportional to the number of returns to the origin of a symmetric random walk of dimension $d$.
	www.ma.utexas.edu /mp_arc/papers/97-198 (5531 words)

	Computing Maximum-Area Sections of Convex Polyhedra - Background
		In this section, we will let K be a convex polyhedron and A(a) the sectional area function of K. That is, A(a) is the area of the section obtained by intersecting K with the plane x = a.
		Theorem (unimodality of sectional area of convex polyhedra): The sectional area function A(a) of a convex polyhedron K is a strictly unimodal function.
		In this section, we will show that every convex near-drum K with n vertices, whose inner vertices have total degree k, can, in O(n) time, be decomposed into a set of drums (possibly nonconvex) with O(n) vertices in total, leaving behind a single convex near-drum with O(k) vertices.
	cgm.cs.mcgill.ca /~perouz/cs507/Projects/SamuliHeilala/background.html (3200 words)

	Convex Combination Kohenen Map
		It only adjusts the winning neuron (which due to the addition of random noise may be one of the neurons that would normally never win).
		Both the random noise method and the convex combination method start with all neurons initialized to 1/sqrt(n) where n is the number of inputs and both methods adjust only the winning neuron.
		The convex combination method changes the input so that it becomes: alpha * x + [1/sqrt(n)](1 - alpha), where x is the original input, alpha is a small value that gradually increases to 1.
	www2.ics.hawaii.edu /~chin/491-1/kohenen.html (333 words)

	MINGLE Homepage
		Michael Floater has developed a linear method by convex combinations for the parameterization of polyhedral surfaces.
		We have then considered the extension of the result in the three-dimensional case: we have defined convex combination maps over tetrahedralizations and shown with a counterexample that these are not necessarily one-to-one.
		From this counterexample appears a necessary (maybe sufficient) condition for the map to be one-to-one and the current task is to establish sufficient conditions.
	www.cs.technion.ac.il /~vitus/mingle/pham-trong.html (161 words)

	Computing Maximum-Area Sections of Convex Polyhedra - Definitions
		, the convex hull of a set of points is the convex polygon obtained by stretching a rubber band around the points and letting it contract until it comes into contact with the "exterior" points.
		A polygon P is convex if it is simple (noncrossing) and if for all points x, y in P, the closed line segment xy is also in P. In more graphic terms, a polygon is convex if its boundary has no dents or depressions, when viewed from the outside.
		A polyhedron K is convex if for all points x, y in K, the closed line segment xy is also in K. In more graphic terms, a polyhedron is convex if its boundary has no dents or depressions, when viewed from the outside.
	cgm.cs.mcgill.ca /~perouz/cs507/Projects/SamuliHeilala/definitions.html (1863 words)

	EJP Vol 3 (1998) Paper 6 (Abstract) (Site not responding. Last check: 2007-10-25)
		At rate $1$ the $x$-th height is updated to a random convex combination of the heights of the `neighbors' of $x$.
		We show that, when the convex combination is neither deterministic nor concentrating on one site, the variance of the height at the origin at time $t$ is proportional to the number of returns to the origin of a symmetric random walk of dimension $d$.
		Under mild conditions on the distribution of the random convex combination, this gives variance of the order of $t^{1/2}$ in dimension $d=1$, $\log t$ in dimension $d=2$ and bounded in $t$ in dimensions $d\ge 3$.
	www.math.utah.edu:8080 /pub/ejpecp/EjpVol3/paper6.abs.html (326 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		We introduced the idea of convex combinations (of two points, or of multiple points).
		The set of all rays forms a cone, where each ray in the cone can be written as a nonnegative linear combination of extreme rays.
		Hence, no extreme ray can be written as a nonnegative linear combination of other rays (except for simply taking scalar multiples of the same ray).
	www.ise.ufl.edu /esi6417/Summaries2005/090605_LPNO.doc (212 words)

	Convex Combination (Site not responding. Last check: 2007-10-25)
		Convex combination,7 inch,acrylic convex,concave convex,convex function,convex group,convex hull,convex mirror,convex mirror concave,convex...
		Convex combination,combined insurance life,jason white and nfl combine,punching combination,combinations,non convex polygon,insurance combined,fused...
		Weights in convex combination methods are important parameters in the estimation of partial derivative values in three dimensional scattered...
	www.convexcombination.info (273 words)

	1.3 Polyhedral Geometry
		Finally, if the combination is both conical and affine, then it is a convex combination.
		Definition 1.8 [special sets] S is a (linear) subspace /cone /affine subspace /convex set if every finite linear/conical/affine/convex combination of elements of S is in S. Proposition 1.9 The intersection of convex sets is a convex set.
		Definition 1.10 [spans `n' hulls] The (linear) span /conical hull /affine span /convex hull is the set of all (finite) linear /conical /affine /convex combinations of elements of S. Let A be an m×n matrix and b
	www.ms.uky.edu /~sills/webprelim/sec013.html (1309 words)

	[No title]
		The convex hull is the smallest polygon or polyhedra (filled and convex), enclosing a set of points P[i].
		This special combination is called the barycentre and is in fact the "centre of mass" of the n equal mass points P[i].
		All these special curves and surfaces use barycentric combinations to make sure that their result is independent of the origin, just like a line or plane is. I can't go further on that, but if you read a tutorial on one of these, keep that in mind.
	www.flipcode.com /articles/gprimer1_issue09-pf.shtml (2814 words)

	cc_convex
		Prove that the image of a convex set under an affine transformation is convex.
		to show that the image of a convex set under an orthogonal or oblique projection is still convex.
		An interesting convex polyhedron is the rhombic dodecahedron, with twelve sides, each a rhombus, and fourteen vertices.
	www.math.ucla.edu /~baker/149.1.03s/handouts/cc_convex/node7.html (1275 words)

	Michael Floater: Surface parameterization
		A closely related `convex combination' method was developed jointly with Craig Gotsman (JCAM 1999) for morphing pairs of compatible planar triangular meshes, i.e., continuously deforming one mesh into the other without foldover.
		2003) that convex combination maps over triangular meshes are one-to-one when the domain boundary is mapped to a convex polygon.
		Valerie Pham-Trong and I extended the simple proof of Tutte's theorem to general planar graphs ("tilings") and also showed that convex combination maps over tetrahedral meshes (in 3D) are NOT necessarily one-to-one (AiCM 2004).
	heim.ifi.uio.no /~michaelf/focus/param.html (909 words)

	On-Line Geometric Modeling Notes
		These notes discuss affine combinations of points, barycentric coordinates of points and vectors, convex combinations, convex sets, and the convex hull of a set of points.
		Given any set of points, we say that the set is a convex set, if given any two points of the set, any convex combination of these two points is also in the set.
		If one looked closely at the coordinates of the point, one would find that this point could be written as a convex combination of the other five.
	www.css.tayloru.edu /~btoll/s99/424/res/ucdavis/CAGDNotes/Affine-Barycentric-and-Convex/Affine-Barycentric-and-Convex.html (673 words)

	[No title]
		Find convex combination of X1 and X2 when (=1/2 and call it X(1/2).
		Find convex combination of X1 and X2 when (=1/4 and call it X(1/4).
		Using convex combination defined in ‘a’, define a convex set S. c.
	www.sp.uconn.edu /~arh98001/Final214.doc (594 words)

	INTRODUCTION
		By considering the convex combination of all optimal CPF's, this integer solution can be obtained.
		Recording this combination of basic and nonbasic variables means recording the variables that are basic and those that are not.
		This combination of basic and nonbasic variables has been considered before so there's no reason to try it again.
	www.cs.virginia.edu /~bec4f/research/oldresearch/cpf/cpf.htm (3367 words)

	The Convex Hull and the Gale-Transform
		is a face of the convex hull of the points
		the origin can be represented as a linear combination of the points
		But by theorem 1 the kernel is spanned by the columns of
	www.inf.ethz.ch /personal/fischerk/code/gt/tut/node3.html (179 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		It means something like: a point in a convex set is a convex combination of the extreme points of the set, where the convex combination is not a finite sum but a kind of integral.
		> It means something like: a point in a convex set is a convex combination > of the extreme points of the set, where the convex combination is not a > finite > sum but a kind of integral.
		It is long out of print, but university libraries probably have copies.
	www.math.niu.edu /~rusin/known-math/00_incoming/choquet (298 words)

	Convex Hulls (Site not responding. Last check: 2007-10-25)
		This definition generalizes to convex combination of n points.
		, the convex hull is the convex combination of all the points of
		Extreme Points: These are the minimal or maximal points along some direction and are used to represent the output of convex hulls.
	www.cs.unc.edu /~smp/COMP205/LECTURES/GEOM/lec11/node2.html (173 words)

	Review for Test
		There will be one proof (choose one out of two).
		If S and T are convex sets then S intersect T is convex.
		Find the convex hull of the extreme points.
	www.saintmarys.edu /~psmith/338act9.html (683 words)

	Convex Combination Maps - Floater (ResearchIndex)
		Abstract: Piecewise linear maps over triangulations are used extensively in geometric modelling and computer graphics.
		This short note surveys recent progress on the important question of when such maps are one-to-one, central to which are convex combination maps.
		Floater M.S. "Convex combination maps." Algorithms for Approximation IV.
	citeseer.ist.psu.edu /485845.html (317 words)

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