Refimprove date May 2010 No footnotes date May 2010 Artwork image file Convex and Concave.JPG title Convex and Concave artist M. C. Escher year 1955 type Lithography lithograph height metric 27.5 width metric 33.5 metric unit cm Convex and Concave is a Lithography lithograph print by the Netherlands Dutch artist M. C. Escher , first printed in March 1955. It depicts an ornate architectural structure with many stairs, pillars and other shapes. The relative aspects of the objects in the image are distorted in such a way that many of the structure s features can be seen as both convex shapes and concave impressions. This is a very good example of Escher s mastery in creating illusion of Impossible Architectures . The window s, roads, stairs and other shapes can be perceived as opening out in seemingly impossible ways and positions. The trick of using the cubes that appear as the motif in the Flag on right half of this print is easily identified. One can view these features as concave by viewing the image upside down. Note that all additional elements and decoration on the left are consistent with a viewpoint from above, while those on the right with a viewpoint from below hiding half the image makes it very easy to switch between convex and concave. See also Printmaking Sources Locher, J.L. 2000 . The Magic of M. C. Escher . Harry N. Abrams, Inc. ISBN 0 8109 6720 0. M. C. Escher Category Works by M. C. Escher Category 1955 paintings printmaking stub he ... more details
Contents mostly taken from Legendre transformation . In mathematics , convex conjugation is a generalization ... f X to mathbb R cup infty math taking values on the extended real number line , the convex conjugate ... of the convex hull of the function s Epigraph mathematics epigraph in terms of its supporting hyperplane s. http maze5.net ?page id 733 Examples The convex conjugate of an affine function math f x ... cases b, & x a infty, & x ne a. end cases math The convex conjugate of a power function math f ... tfrac 1 p tfrac 1 q 1. math The convex conjugate of the absolute value function math f x left x right ... cases math The convex conjugate of the exponential function math f x , e x math is math f star left x right begin cases x ln x x , & x 0 0 , & x 0 infty , & x 0. end cases math Convex conjugate and Legendre transform of the exponential function agree except that the domain mathematics domain of the convex ... E left min x,X right math has the convex conjugate math begin align f star p int 0 p F 1 q , dq ... function f in particular, math f text inc f math for &fnof nondecreasing. Properties The convex conjugate of a closed convex function is again a closed convex function. The convex conjugate of a polyhedral convex function a convex function with Polyhedron polyhedral epigraph mathematics epigraph is again a polyhedral convex function. Convex conjugation is Order theory order reversing if math ... The convex conjugate of a function is always lower semi continuous . The biconjugate math f math the convex conjugate of the convex conjugate is also the closed convex hull , i.e. the largest lower semi continuous convex function with math f le f math . For Proper convex function proper functions f , f f sup sup if and only if f is convex and lower semi continuous by Fenchel Moreau theorem . Fenchel s inequality For any function f and its convex conjugate f sup sup Fenchel s inequality also known ... . Scaling properties Convex conjugation has the following scaling properties math f x a cdot ... more details
KPR preferences short for King Plosser Rebelo preferences are a certain type of preferences that are used in many macroeconomic models and DSGE models in particular. Having originally been proposed in an article that appeared in the Journal of Monetary Economics in 1988, ref name KPR1998 cite journal last King first Robert G. last2 Plosser first2 Charles I. last3 Rebelo first3 Sergio T. author link year 1988 month May journal Journal of Monetary Economics volume 21 pages 195 232 title Production, Growth and Business Cycles I. The Basic Neoclassical Model doi 10.1016 0304 3932 88 90030 X url http 128.197.153.21 rking EC702 kprjme88a.pdf accessdate ref the corresponding technical appendix detailing their derivation has only been published in 2002. ref name KPR2002 cite journal last King first Robert G. last2 Plosser first2 Charles I. last3 Rebelo first3 Sergio T. author link year 2002 month journal Computational Economics volume 20 issue 1 2 pages 87 116 title Production, Growth and Business Cycles Technical Appendix doi 10.1023 A 1020529028761 url http www.springerlink.com content m28u210825682333 fulltext.pdf accessdate ref Denote consumption with C, leisure with L and the absolute value of the inverse of the intertemporal elasticity of substitution in consumption with math sigma c math . Strict concavity of the utility function implies math sigma c 0 math . For math 0 sigma c 1 ... is increasing and concave if math 0 sigma c 1 math or decreasing and convex if math sigma c 1 math .... In the limit case of math sigma c 1 math the resulting preferences specification is additively separable ... preferences along with Balanced growth equilibrium balanced growth , some studies use the shortcut of introducing ... to other common macroeconomic preference types KPR preferences are one polar case nested in Jaimovich Rebelo preferences . The latter allow to freely scale the wealth effect on the labor supply. The other polar case is the Greenwood Hercowitz Huffman preferences , where the wealth effect ... more details
The following graph gives two examples of preferences that are not single peaked. The blue preferences are clearly not single peaked because the preference ranking spikes down for D and then spikes up for E . The green preferences are not single peaked because they have two outcomes that are the most preferred B and C . Such preferences are sometimes called single plateaued. File Singlepeaked2.jpg Interpretations Single peaked preferences have a number of interpretations for different applications. A simple application of ideological preferences is to think of the outcome space ... as little as possible to the stop. Individuals then have single peaked preferences individual math ... that are pro state intervention. Voters have single peaked preferences if they have an ideal ... Banks title Positive Political Theory I Collective Preferences publisher University of Michigan Press ... more details
In economics , Epstein Zin Weil preferences is a specification of recursive utility. A recursive utility function can be constructed from two components a time aggregator that characterizes preferences in the absence of uncertainty and a risk aggregator that defines the certainty equivalent function that characterizes preferences over static gambles and is used to aggregate the risk associated with future utility. With Epstein Zin preferences, the time aggregator is a CES aggregate of current consumption and the certainty equivalent of all future utility, measured in units of current consumption. The certainty equivalent function is a CES aggregate of future utility. Specifically, math U t 1 beta C t frac 1 gamma theta beta E t U t 1 1 gamma frac 1 theta frac theta 1 gamma math See also Elasticity of substitution References The Equity Premium It s Still a Puzzle , Narayana R. Kocherlakota , Journal of Economic Literature , Vol. 34, No. 1. Mar., 1996 , pp. 42 71. JSTOR http links.jstor.org sici?sici 0022 0515 28199603 2934 3A1 3C42 3ATEPISA 3E2.0.CO 3B2 X Substitution, Risk Aversion, and the Temporal Behavior of Consumption Growth and Asset Returns I A Theoretical Framework , Epstein, Larry G. and Stan Zin Zin, Stanley E. , Econometrica , Vol. 57, No. 4. Jul., 1989 , pp. 937 969. Non Expected Utility in Macroeconomics , Weil, Philippe, Quarterly Journal of Economics , Vol. CV, No. 1. Feb., 1990 , pp. 29 42. Category Microeconomics Category Consumer theory microeconomics stub ... more details
Expert subject Economics date November 2008 GHH preferences short for Greenwood Hercowitz Huffman preferences , refer to an economic formula developed by Jeremy Greenwood, Zvi Hercowitz , and Gregory Huffman, in their 1988 paper Investment, Capacity Utilization, and the Real Business Cycle . It describes the macroeconomics macroeconomic impact of technological changes that affect productivity. GHH preferences have Gorman form . Often macroeconomic models assume that agents utility is additively separable in consumption and labor. I.e., frequently the period utility function is something like math u c,l frac c 1 gamma 1 gamma psi frac l 1 theta 1 theta math Where math c math is consumption, math l math is labor e.g., hours worked . Note that this is separable in that the utility loss from working does not directly affect the utility gain or loss from consumption, i.e. the cross derivative of utility with respect to consumption and labor is 0. GHH preferences might instead have a form like math u c,l frac 1 1 gamma left c psi frac l 1 theta 1 theta right 1 gamma math Where now consumption and labor are not additively separable in the same way. With this utility function, the amount you work will actually affect the amount of utility received from consumption, i.e. the cross derivative of utility with respect to consumption and labor is unequal to 0. More generally, the preferences are of the form math u c,l U left c G l right , U 0, U 0, G 0, G 0. math The first order condition of math u c,l math with respect math l math is given by math U left c G l right left frac dc dl G l ... form with math l G 1 w math . As a result, the preferences are exceptionally convenient to work with. Moreover ... Jaimovich Rebelo preferences GHH preferences are not consistent with a Balanced growth equilibrium ... King Plosser Rebelo preferences ref name KPR2002 cite journal last King first Robert G. last2 ... fulltext.pdf accessdate ref and the GHH preferences. References http www.jeremygreenwood.net papers ... more details
The Generalized System of Preferences , or GSP , is a formal system of exemption from the more general rules of the World Trade Organization WTO , formerly, the General Agreement on Tariffs and Trade or GATT . Specifically, it s a system of exemption from the most favored nation principle MFN that obliges WTO member countries to treat the imports of all other WTO member countries no worse than they treat the imports of their most favored trading partner. In essence, MFN requires WTO member countries to treat imports coming from all other WTO member countries equally, that is, by imposing equal tariffs on them, etc. GSP exempts WTO member countries from MFN for the purpose of lowering tariffs for the least developed countries, without also lowering tariffs for rich countries. History The idea of tariff preferences for developing countries was the subject of considerable discussion within the United Nations Conference on Trade and Development UNCTAD in the 1960s. Among other concerns, developing countries claimed that MFN was creating a disincentive for richer countries to reduce and eliminate tariffs and other trade restrictions with enough speed to benefit developing countries. In 1971, the GATT followed the lead of UNCTAD and enacted two waivers to the MFN that permitted tariff preferences to be granted to developing country goods. Both these waivers were limited in time to ten ... to establish systems of trade preferences for other countries, with the caveat that these systems ... done the same with Everything But Arms . See also Global System of Trade Preferences among Developing ...?intItemID 1418&lang 1 title UNCTAD Introduction to Generalized System of Preferences Information ... generalised system of preferences title E.U. Generalised System of Preferences Information from the European ... economy gsp title Japan Generalized System of Preferences Introduction to Japan s GSP program by the Ministry ... System Of Preferences Category United States trade law Category World Trade Organization bg ... more details
In mathematics , a function mathematics function f defined on a convex set convex subset of a real numbers real vector space and taking negative and positive numbers positive values is said to be logarithmically convex or superconvex ref Kingman, J.F.C. 1961. A convexity property of positive matrices. Quart. J. Math. Oxford 2 12,283 284. ref if math log f math is a convex function . A logarithmically convex function f is a convex function since it is the function composition composition of the increasing convex function math exp math and the convex function math log f math , but the converse is not always true. For example math f x x 2 math is a convex function, but math log f x log x 2 2 log x math is not a convex function and thus math f x x 2 math is not logarithmically convex. On the other hand, math f x e x 2 math is logarithmically convex since math log e x 2 x 2 math is convex. An important example of a logarithmically convex function is the gamma function on the positive reals see also the Bohr Mollerup theorem . References Reflist John B. Conway. Functions of One Complex Variable I , second edition. Springer Verlag, 1995. ISBN 0 387 90328 3. PlanetMath attribution id 5664 title logarithmically convex function Category Real analysis es Convexidad logar tmica ... more details
In mathematics, a Schur convex function , also known as S convex , isotonic function and order preserving function is a function mathematics function math f mathbb R d rightarrow mathbb R math , for which if math forall x,y in mathbb R d math where math x math is majorization majorized by math y math , then math f x le f y math . Named after Issai Schur , Schur convex functions are used in the study of majorization . Every function that is Convex function convex and Symmetric function symmetric is also Schur convex. Schur concave function A function math f math is Schur concave if its negative, math f math , is Schur convex. Examples The Shannon entropy function math sum i 1 d P i cdot log 2 frac 1 P i math is Schur concave math sum i 1 d x i k ,k ge 1 math is Schur convex Category Convex analysis Category Inequalities mathanalysis stub ... more details
In mathematics , a convex function is called closed if its epigraph mathematics epigraph is a closed set . Properties A closed convex function f is the pointwise supremum of the collection of all affine function s h such that h f . References R. Tyrrell Rockafellar Rockafellar, R. Tyrell , Convex Analysis , Princeton University Press 1996 . ISBN 0 691 01586 4 Category Convex analysis Category Types of functions mathanalysis stub bs Zatvorena konveksna funkcija ... more details
Merge to Convex function discuss Talk Convex function Proper date March 2011 In mathematical analysis in particular convex analysis and optimization mathematics optimization , a proper convex function is a convex function f taking values in the extended real number line such that math f x infty math for at least one x and math f x infty math for every x . That is, a convex function is proper if its effective domain is nonempty and it never attains math infty math . ref name AB cite book last1 Aliprantis first1 C.D. last2 Border first2 K.C. title Infinite Dimensional Analysis A Hitchhiker s Guide edition 3 publisher Springer year 2007 isbn 978 3 540 32696 0 doi 10.1007 3 540 29587 9 page 254 ref Convex functions that are not proper are called improper convex functions . ref cite book author Rockafellar, R. Tyrrell title Convex Analysis publisher Princeton University Press location Princeton, NJ year 1997 origyear 1970 isbn 978 0 691 01586 6 page 24 ref A proper concave function is any function g such that math f g math is a proper convex function. Properties For every proper convex function f on R sup n sup there exist some b in R sup n sup and in R such that math f x ge x cdot b beta math for every x . Citation needed date October 2011 The sum of two proper convex functions is not necessarily proper or convex. For instance if the sets math A subset X math and math B subset X math are convex set s in the vector space X , then the Characteristic function convex analysis indicator function s math I A math and math I B math are proper convex functions, but math I A I B math is not convex unless math A cup B math is convex , and is possibly identically equal to math infty math if math A cup B X math i.e. are complimentary halfspace s . The infimal convolute infimal convolution of two proper convex functions is convex but not necessarily proper convex. Citation needed date October 2011 References Reflist Category Convex analysis Category Types of functions fr Fonction propre ... more details
File Pentagon.svg right thumb 150px An example of a convex polygon a regular polygon regular pentagon In geometry , a polygon can be either convex or concave non convex . Convex polygons A convex polygon is a simple polygon whose Interior topology interior is a convex set . ref http www.mathopenref.com polygonconvex.html Definition and properties of convex polygons with interactive animation. ref The following properties of a simple polygon are all equivalent to convexity Every internal angle is less than or equal to 180 Degree angle degrees . Every line segment between two vertex geometry vertices remains inside or on the boundary of the polygon. A simple polygon is strictly convex if every internal angle is strictly less than 180 degrees. Equivalently, a polygon is strictly convex if every line segment between two nonadjacent vertices of the polygon is strictly interior to the polygon except at its endpoints. Every nondegenerate triangle is strictly convex. Concave or non convex polygons File Simple polygon.svg thumb 150px An example of a concave polygon. A simple polygon that is not convex is called concave , ref citation first Jeffrey J. last McConnell year 2006 title Computer Graphics Theory Into Practice isbn 0763722502 page 130 . ref non convex ref Citation last Leff first .... ref It is always possible to cut a concave polygon into a set of convex polygons. clarify date September 2011 A polynomial time algorithm for finding a decomposition into as few convex polygons as possible ... Optimal convex decompositions title Computational Geometry year 1985 editor first G.T. editor last Toussaint ... . ref See also Convex hull Cyclic polygon Tangential polygon References references External links mathworld urlname ConvexPolygon title Convex polygon mathworld urlname ConcavePolygon title Concave polygon http www.rustycode.com tutorials convex.html Category Convex geometry Category Polygons ar bs Konveksni poligon ca Pol gon convex et Kumer hulknurk es Pol gono convexo eo Konveksa ... more details
Algorithms that construct convex hull s of various objects have a broad range of applications in mathematics and computer science , see Convex hull Applications Convex hull applications . In computational geometry , numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexity computational complexities . Computing the convex hull means that a non ambiguous and efficient data structure representation of the required convex shape is constructed ... of input points, and h , the number of points on the convex hull. Planar case Consider the general ..., then their convex hull is a convex polygon whose vertices are some of the points in the input set. Its .... In some applications it is convenient to represent a convex polygon as an intersection ... in the plane the lower bound on the computational complexity of finding the convex hull represented as a convex polygon is easily shown to be the same as for sorting using the following reduction ... is a monotone function monotone curve it is easy to see that the vertices of the convex hull ... their sorted order. Therefore in the general case the convex hull of n points cannot be computed ... can be performed however, in this model, convex hulls cannot be computed at all. Sorting also requires ... suitable for convex hulls, and in this model convex hulls also require &Omega n log n time. ref name ps Preparata, Shamos, Computational Geometry , Chapter Convex Hulls Basic Algorithms ref However, in models ... by using integer sorting algorithms, planar convex hulls can also be computed more quickly the Graham scan algorithm for convex hulls consists of a single sorting step followed by a linear ... a convex hull as a function the input size n is lower bounded by &Omega n log n . However, the complexity of some convex hull algorithms can be characterized in terms of both input size n and the output ... n . The lower bound on worst case running time of output sensitive convex hull algorithms was established ... more details
Image Orthogonal convex hull.svg thumb The orthogonal convex hull of a point set In Euclidean geometry , a set math K subset R n math is defined to be orthogonally convex if, for every line L that is parallel ..., a point, or a single interval. Unlike ordinary convex set s, an orthogonally convex set is not necessarily connectedness connected . The orthogonal convex hull of a set math S subset R n math is the intersection of all connected orthogonally convex supersets of S . These definitions are made by analogy with the classical theory of convexity, in which K is convex set convex if, for every line L ... the lines for which this property is required to hold, so every convex set is orthogonally convex but not vice versa. For the same reason, the orthogonal convex hull itself is a subset of the convex hull of the same point set. A point p belongs to the orthogonal convex hull of S if and only if each ... convex hull is also known as the rectilinear convex hull , or the x y convex hull . Example The figure shows a set of 16 points in the plane and the orthogonal convex hull of these points. As can be seen in the figure, the orthogonal convex hull is a polygon with some degenerate edges connecting ... convex hull edges are horizontal or vertical. In this example, the orthogonal convex hull is connected. Algorithms Several authors have studied algorithms for constructing orthogonal convex ... Wood 1984 harvtxt Karlsson Overmars 1988 . By the results of these authors, the orthogonal convex ... orthogonal convexity to restricted orientation convexity , in which a set K is defined to be convex ... is closely related to the orthogonal convex hull. If a finite point set in the plane has a connected orthogonal convex hull, that hull is the tight span for the Manhattan distance on the point set. However ... y convex hull of a set of x y polygons year 1982 . citation last1 Nicholl first1 T. M. last2 Lee first2 ... journal BIT pages 456 471 title On the X Y convex hull of a set of X Y polygons volume 23 year ... more details
A Set mathematics set C in a real number real or complex number complex vector space is said to be absolutely convex if it is convex set convex and balanced set balanced . Properties A set math C math is absolutely convex if and only if for any points math x 1, , x 2 math in math C math and any numbers math lambda 1, , lambda 2 math satisfying math lambda 1 lambda 2 leq 1 math the sum math lambda 1 x 1 lambda 2 x 2 math belongs to math C math . Since the intersection of any collection of absolutely convex sets is absolutely convex then for any subset A of a vector space one can define its absolutely convex hull to be the intersection of all absolutely convex sets containing A . Absolutely convex hull The absolutely convex hull of the set A assumes the following representation math mbox absconv A left sum i 1 n lambda i x i n in N, , x i in A, , sum i 1 n lambda i leq 1 right math . References cite book last Robertson first A.P. coauthors W.J. Robertson title Topological vector spaces series Cambridge Tracts in Mathematics volume 53 year 1964 publisher Cambridge University Press pages 4 6 See also Wikibooks Algebra Vector spaces vector geometric , for vectors in physics Vector field Category Abstract algebra Category Linear algebra Category Group theory Category Convex geometry de Absolutkonvexe Menge nl Absoluut convexe verzameling pt Conjunto absolutamente convexo ... more details
The dynamic convex hull problem is a class of dynamic problem algorithms dynamic problem s in computational geometry . The problem consists in the maintenance, i.e., keeping track, of the convex hull for the dynamically changing input data, i.e., when input data elements may be inserted, deleted, or modified. Problems of this class may be distinguished by the types of the input data and the allowed types of modification of the input data. Planar point set It is easy to construct an example for which the convex hull contains all input points, but after the insertion of a single point the convex hull becomes a triangle. And conversely, the deletion of a single point may produce the opposite drastic change of the size of the output. Therefore if the convex hull is required to be reported in traditional way as a polygon, the lower bound for the worst case computational complexity of the recomputation of the convex hull is math Omega N math , since this time is required for a mere reporting of the output. This lower bound is attainable, because several general purpose convex hull algorithms run in linear time when input points are sorting ordered in some way and logarithmic time methods ... of the convex hull in an amount of time per update that is much smaller than linear. For many ... of applications finding the convex hull is a step in an algorithm for the solution of the overall problem. The selected representation of the convex hull may influence on the computational complexity of further operations of the overall algorithm. For example, the point in polygon query for a convex ... would be impossible for convex hulls reported by the set of it vertices without any additional information. Therefore some research of dynamic convex hull algorithms involves the computational complexity of various geometric search problems with convex hulls stored in specific kinds of data structures ... Convex Hull 2002 , a http www.brics.dk BRICS dissertation Category Convex hull algorithms uk ... more details
Image Convex metric illustration2.png right thumb An illustration of a convex metric space. In mathematics , convex metric spaces are, intuitively, metric space s with the property any segment joining ... becomes an equality. A convex metric space is a metric space X , d such that, for any two distinct ... and its analogues for other dimensions, are convex metric spaces. Given any two distinct points math .... Image Circle as convex metric space.png right thumb A circle as a convex metric space. Any convex set in a Euclidean space is a convex metric space with the induced Euclidean norm. For closed set s the Contraposition ... distance is a convex metric space, then it is a convex set this is a particular case of a more general statement to be discussed below . A circle is a convex metric space, if the distance between ... Let math X, d math be a metric space which is not necessarily convex . A subset math S math of math ... metric segments between any two distinct points in the space, then it is a convex metric space. The Contraposition converse is not true, in general. The rational number s form a convex metric space ... up of rational numbers only. If however, math X, d math is a convex metric space, and, in addition ... X math there exists a metric segment connecting them which is not necessarily unique . Convex metric spaces and convex sets As mentioned in the examples section, closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex sets. It is then natural to think of convex ... this way does not have one of the most important properties of Euclidean convex sets, that being that the intersection of two convex sets is convex. Indeed, as mentioned in the examples section, a circle ... metric space complete convex metric space. Yet, if math x math and math y math are two points ... arcs into which these points split the circle , and those two arcs are metrically convex, but their intersection is the set math x, y math which is not metrically convex. See also Intrinsic metric ... more details
A convex lattice polytope also called Z polyhedron or Z polytope is a geometry geometric object playing an important role in discrete geometry and combinatorial commutative algebra . It is a polytope in a Euclidean space R sup n sup which is a convex hull of finitely many points in the integer lattice Z sup n sup &sub R sup n sup . Such objects are prominently featured in the theory of toric variety toric varieties , where they correspond to polarized projective toric varieties. Examples An n dimensional simplex &Delta in R sup n sup is the convex hull of n 1 points that do not lie on a single affine hyperplane. The simplex is a convex lattice polytope if and only if the vertices have integral coordinates. The corresponding toric variety is the n dimensional projective space P sup n sup . The unit cube in R sup n sup , whose vertices are the 2 sup n sup points all of whose coordinates are 0 or 1 , is a convex lattice polytope. The corresponding toric variety is the Segre embedding of the n fold product of the projective line P sup 1 sup . In the special case of two dimensional convex lattice polytopes in R sup 2 sup , they are also known as convex lattice polygons . In algebraic geometry, an important instance of lattice polytopes called the Newton polytopes are the convex hulls of the set math A math which consists of all the exponent vectors appearing in a collection of monomials. For example, consider the polynomial of the form math axy bx 2 cy 5 d math with math a,b,c,d neq 0 math has a lattice equal to the triangle math rm conv 1,1 , 2,0 , 0,5 , 0,0 . math See also Normal polytope Pick s theorem Ehrhart polynomial Integer points in convex polyhedra References Ezra Miller, Bernd Sturmfels , Combinatorial commutative algebra . Graduate Texts in Mathematics, 227. Springer Verlag, New York, 2005. xiv 417 pp. ISBN 0 387 22356 8 geometry stub Category Polytopes Category Lattice points ... more details
In mathematics , more precisely in complex analysis , the holomorphically convex hull of a given compact set in the n dimension al complex number complex space C sup n sup is defined as follows. Let math G subset mathbb C n math be a domain an open set open and connected set connected Set mathematics set , or alternatively for a more general definition, let math G math be an math n math dimensional complex analytic manifold . Further let math mathcal O G math stand for the set of holomorphic function s on math G. math For a compact set math K subset G math , the holomorphically convex hull of math K math is math hat K G z in G big left f z right leq sup w in K left f w right mbox for all f in mathcal O G . math One obtains a narrower concept of polynomially convex hull by requiring in the above definition that f be a polynomial . The domain math G math is called holomorphically convex if for every math K subset G math compact in math G math , math hat K G math is also compact in math G math . Sometimes this is just abbreviated as holomorph convex . When math n 1 math , any domain math G math is holomorphically convex since then math hat K G math is the union of math K math with the relatively compact components of math G setminus K subset G math . Also note that being holomorphically convex is the same as being a domain of holomorphy The Cartan Thullen theorem . These concepts are more important in the case n 1 of several complex variables . See also Stein manifold Pseudoconvexity References Lars H rmander . An Introduction to Complex Analysis in Several Variables , North Holland Publishing Company, New York, New York, 1973. Steven G. Krantz. Function Theory of Several Complex Variables , AMS Chelsea Publishing, Providence, Rhode Island, 1992. PlanetMath attribution id 6798 title Holomorphically convex Category Several complex variables ... more details
Taxobox name Convex horseshoe bat image status CR status system IUCN2.3 regnum Animalia phylum Chordata classis Mammalia ordo Chiroptera familia Rhinolophidae genus Rhinolophus species R. convexus binomial Rhinolophus convexus binomial authority Csorba, 1997 synonyms range map Convex Horseshoe Bat area.png range map caption Convex Horseshoe Bat range The convex horseshoe bat Rhinolophus convexus is a species of bat in the Rhinolophidae family. It is Endemism endemic to Malaysia . References Reflist Chiroptera Specialist Group 2000. http www.iucnredlist.org search details.php 40037 all Rhinolophus convexus . http www.iucnredlist.org 2006 IUCN Red List of Threatened Species. Downloaded on 30 July 2007. Rhinolophidae Category Endemic fauna of Malaysia Category Bats of Malaysia Category Rhinolophus Bat stub es Rhinolophus convexus eu Rhinolophus convexus nl Rhinolophus convexus ... more details
Image Vector norms.svg frame right The unit ball in the middle figure is strictly convex, while the other two balls are not they contain a line segment as part of their boundary . In mathematics , a strictly convex space is a normed vector space normed topological vector space V , for which the unit ball is a strictly convex set . Put another way, a strictly convex space is one for which, given any two points x and y in the boundary topology boundary B of the unit ball B of V , the affine line L x , y passing through x and y meets B only at x and y . Strict convexity is somewhere between an inner product space all inner product spaces are strictly convex and a general normed space all strictly convex normed spaces are normed spaces in terms of structure. It also guarantees the uniqueness of a best approximation to an element in X strictly convex out of Y a subspace of X if indeed such an approximation exists. Properties A Banach space V , is strictly convex if and only if the modulus of convexity for V , satisfies 2 1. A Banach space V , is strictly convex if and only if x y and x y 1 together imply that x y 2. A Banach space V , is strictly convex if and only if x y and x y 1 together imply that x 1 &minus y < 1 for all 0 < < 1. A Banach space V , is strictly convex if and only if x 0 and y 0 and x y x y together imply that x cy for some constant c 0 . References cite journal last Goebel first Kazimierz title Convexity of balls and fixed point theorems ... 1970 pages 269&ndash 274 Category Convex analysis Category Normed spaces de Strikt konvexer Raum ru ... more details
In mathematics , uniformly convex spaces are common examples of reflexive space reflexive Banach space s. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a normed vector space so that, for every math epsilon 0 math there is some math delta 0 math so that for any two vectors with math x 1 math and math y 1, math math x y 2 delta math implies math x y epsilon. math Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. Properties The Milman Pettis theorem states that every uniformly convex Banach space is reflexive space reflexive , while the converse is not true. If math f n n 1 infty math is a sequence in a uniformly convex Banach space which converges weakly to math f math and satisfies math f n to f math , then math f n math converges strongly to math f math , that is, math f n f to 0 math . A Banach space math X math is uniformly convex if and only if its dual math X math is uniformly smooth . Examples Every Hilbert space is uniformly convex. Hanner s inequalities imply that lp space L sup p sup spaces math 1 p infty math are uniformly convex. Conversely, math L infty math is not uniformly convex. For example, in math mathbb R 2 math consider math x 1,1 math and math y 0,1 math . Then math x infty y infty 1 math and math x y infty 1,2 infty 2 math , but math x y infty 1,0 infty 1 math . See also Modulus and characteristic of convexity References Cite journal first J. A. last Clarkson title Uniformly convex spaces journal Trans. Amer. Math. Soc. volume 40 year 1936 pages 396 414 doi 10.2307 1989630 jstor 1989630 issue 3 publisher American Mathematical Society postscript None . Cite journal first O. last Hanner title On the uniform ... an equivalent uniformly convex norm journal Israel Journal of Mathematics volume 13 issue 3 4 year ... functional analysis Colloquium publications, 48. American Mathematical Society. Category Convex ... more details
For the poetry collection Self portrait in a Convex Mirror book Infobox Painting image file Parmigianino Selfportrait.jpg title Self portrait in a Convex Mirror artist Parmigianino year c. 1524 type Oil on convex panel diameter 24.4 museum Kunsthistorisches Museum Self portrait in a Convex Mirror c. 1524 is a painting by the Italian late Renaissance artist Parmigianino . It is housed in the Kunsthistorisches Museum , Vienna , Austria . History The work is mentioned by Late Renaissance art biographer Giorgio Vasari , who lists it as one of three small size paintings that the artist brought to Rome with him in 1525. Vasari relays that the self portrait was created by Parmigianino as an example to showcase his talent to potential customers. ref cite book authorlink Giorgio Vasari first Giorgio last Vasari title Lives of the Most Excellent Painters, Sculptors, and Architects chapter Francesco Mazzuoli year 1568 ref The portrait was donated to pope Clement VII , and later to writer Pietro Aretino , in whose house Vasari himself, then still a child, saw it. It was later acquired by Vicentine sculptor Valerio Belli and, after his death in 1546, by his son Elio. Through the intercession of Andrea Palladio , in 1560 the work went to Venetian sculptor Alessandro Vittoria , who assigned it in heritage to emperor Rudolf II, Holy Roman Emperor Rudolf II . It arrived in Prague in 1608, and later it become part of the Habsburg imperial collections in Vienna 1777 , although attributed to Correggio . Description The painting depicts the young artist then sixteen in the middle of a room, distorted by the use of a convex mirror . The hand in the foreground is greatly elongated and distorted by the mirror. The work was painted on a specially prepared convex panel in order to mimic the curve of the mirror used. commons category Self portrait in a Convex Mirror See also Self portrait Self portrait in a Convex Mirror book Self portrait in a Convex Mirror by John Ashbery the portrait is the subject ... more details
In the mathematical field of graph theory , a convex bipartite graph is a bipartite graph with specific properties. A bipartite graph, U V , E , is said to be convex over the vertex set U if U can be enumerate d such that for all v V the vertices adjacent to v are consecutive. Convexity over V is defined analogously. A bipartite graph U V , E that is convex over both U and V is said to be biconvex or doubly convex . Formal definition Let G U V , E be a bipartite graph, i.e, the vertex set is U V where U V . Let N sub G sub v denote the neighborhood of a vertex v V . The graph G is convex over U if and only if there exists a bijective mapping, f U 1, 2, ..., U &minus 1, U , such that for all v V , for any two vertices x , y N sub G sub v U there does not exist a z N sub G sub v such that f x f z f y . See also Convex plane graph References cite journal author W. Lipski Jr. coauthors Franco P. Preparata year 1981 month August title Efficient algorithms for finding maximum matchings in convex bipartite graphs and related problems journal Acta Informatica volume 15 issue 4 pages 329 346 doi 10.1007 BF00264533 url http www.springerlink.com content u18656lrg6424n3u accessdate 2009 07 20 cite journal author Ten hwang Lai coauthors Shu shang Wei year 1997 month April title Bipartite permutation graphs with application to the minimum buffer size problem journal Discrete Applied Mathematics volume 74 issue 1 pages 33 55 doi 10.1016 S0166 218X 96 00014 5 url http citeseer.ist.psu.edu old lai94bipartite.html accessdate 2009 07 20 cite book title Efficient graph representations author Jeremy P. Spinrad year 2003 publisher American Mathematical ... PA128&lpg PA128&dq 22a bipartite graph is a convex graph 22 accessdate 2009 07 20 cite book title ... more details
no footnotes date October 2011 In the field of mathematics known as convex analysis , the characteristic function of a set is a convex function that indicates the membership or non membership of a given element in that set. It is similar to the usual indicator function , and one can freely convert between the two, but the characteristic function as defined below is better suited to the methods of convex analysis. Definition Let math X math be a set mathematics set , and let math A math be a subset of math X math . The characteristic function of math A math is the function math chi A X to mathbb R cup infty math taking values in the extended real number line defined by math chi A x begin cases 0, & x in A infty, & x not in A. end cases math Relationship with the indicator function Let math mathbf 1 A X to mathbb R math denote the usual indicator function math mathbf 1 A x begin cases 1, & x in A 0, & x not in A. end cases math If one adopts the conventions that for any math a in mathbb R cup infty math , math a infty infty math and math a infty infty math math frac 1 0 infty math and math frac 1 infty 0 math then the indicator and characteristic functions are related by the equations math mathbf 1 A x frac 1 1 chi A x math and math chi A x infty left 1 mathbf 1 A x right . math Bibliography cite book last Rockafellar first R. T. authorlink R. Tyrrell Rockafellar title Convex Analysis publisher Princeton University Press location Princeton, NJ year 1997 origyear 1970 isbn 9780691015866 Category Convex analysis ... more details
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