Image Apollonian spheres.jpg thumb right ApollonianspherepackingApollonianspherepacking is the three dimensional equivalent of the Apollonian gasket . The principle of construction is very similar with any four spheres that are cotangent to each other, it is then possible to construct two more spheres that are cotangent to four of them. The fractal dimension is 2.473946. ref Citation first M. last Borkovec first2 W. last2 De Paris first3 R. last3 Peikert author link publication date date year 1994 title The Fractal Dimension of the ApollonianSpherePacking periodical Fractals series publication place place publisher volume 2 issue 4 pages 521 526 url http graphics.ethz.ch peikert papers apollonian.pdf issn doi 10.1142 S0218348X94000739 oclc accessdate 2008 09 15 ref Software for generating and visualization of the apollonianspherepacking ApolFrac. ref http thomasbonner.heliohost.org apolfrac.htm ApolFrac ref References references Category Hyperbolic geometry Category Fractals geometry stub it Impacchettamento di sfere apolloniano ... more details
also Apollonianspherepacking Hermite constant Kissing number problem Spherepacking bound References ...File HCP Oranges.jpg thumb Spherepacking finds practical application in the stacking of orange fruit orange s. In geometry , a spherepacking is an arrangement of non overlapping sphere s within a containing ... dimension al Euclidean space . However, spherepacking problem s can be generalised to consider unequal spheres, n dimensional Euclidean space where the problem becomes circle packing in two dimensions, or hypersphere packing in higher dimensions or to non Euclidean spaces such as hyperbolic space . A typical spherepacking problem is to find an arrangement in which the spheres fill as large a proportion ... rigid or Jamming physics jammed . The locally jammed spherepacking with the lowest density ... packing. Hypersphere packing The spherepacking problem is the three dimensional version of a class ... known regular packing. ref cite journal last Sloane first N. J. A. title The SpherePacking ... packing File Binary spherepacking LS3.png thumb A dense packing of spheres with a radius ratio of 0.648 ... 002 2791 7 ref Other spaces Spherepacking on the corners of a hypercube with the spheres defined by Hamming .... There are other, subtler relationships between Euclidean spherepacking and error correcting codes ... Circle Packing http www.3doro.de e kp.htm Kugelpackungen SpherePacking T.E. Dorozinski http alecjacobson.com graphics hw10b 3D SpherePacking Applet SpherePacking java applet http www.randomwalk.de sphere insphr spheresinsphr.html Densest Packing of spheres into a sphere java applet Category Discrete ... stapeling ja pt Empacotamento de esferas ru simple Spherepacking sv T tpackade ... of the arrangement. As the local density of a packing in an infinite space can vary depending ... degree of symmetry makes it easier to classify them and to measure their densities. Regular packing Image close packing box.svg thumb right 160px HCP lattice left and the FCC lattice right are the two ... more details
. This circle is in the interior of circle with negative curvature. Variations Image Apollonian spheres.jpg thumb left Apollonianspherepacking An Apollonian gasket can also be constructed ... s. The three dimensional equivalent of the Apollonian gasket is the Apollonianspherepacking . Symmetries ... 3 Labels.png Integral Apollonian circle packing defined by circle curvature s of 1, 2, 2, 3 Image ApollonianGasket 3 5 8 8 Labels.png Integral Apollonian circle packing defined by circle ... 6 10 15 19 Labels.png Integral Apollonian circle packing defined by circle curvatures of 6, 10, 15, 19 Image ApollonianGasket 10 18 23 27 Labels.png Integral Apollonian circle packing ... Apollonian circle packing Online experiments with JSXGraph http demonstrations.wolfram.com ApollonianGasket ...Wikibooks Fractals Apollonian fractals In mathematics , an Apollonian gasket or Apollonian net is a fractal ... after Greece Greek mathematician Apollonius of Perga . Construction Image Apollonian gasket.svg thumb An example of an Apollonian gasket An Apollonian gasket can be constructed as follows. Start with three ... these are called Apollonian circles see Descartes theorem . Adding the two Apollonian circles to the original three, we now have five circles. Take one of the two Apollonian circles &ndash say C sub ... 1 sub and C sub 2 sub has its own two Apollonian circles. We already know one of these &ndash it is C ... a total of 3 sup n 1 sup 2 circles after n stages. In the limit, this set of circles is an Apollonian gasket. The Apollonian gasket has a Hausdorff dimension of about 1.3057 http abel.math.harvard.edu ... a radius that is two thirds of this, then the Apollonian gasket has two lines of reflective symmetry ... the Apollonian gasket also has rotational symmetry of degree 2 the symmetry group of this gasket is D sub 2 sub . If all three of the original generating circles have the same radius then the Apollonian ... of the first two Apollonian circles. These lines of symmetry are at angles of 60 degrees to one another ... more details
Wiktionary packing wikibooks Packing & Moving Household Goods TOCright Packing normally refers to the process of Packaging and labeling It can also refer to Packing Packaging and labeling Relocation personal , also known as moving. Moving company , a business that puts goods into boxes to protect them and to make them easier to carry while in transit Meat packing industry , the process of cutting up a slaughtered animal into saleable products. Engineering Packing, also known as an O ring or other type of Seal mechanical , a term for a sealing material Structured packing , i.e. internals of a distillation column Mathematics Spherepacking , a 3 dimensional mathematical problem Packing problem , a recreational mathematics approach Set packing , a classic computing problem. Other uses Backpacking Packing firestopping , the process of installing backer materials, such as mineral wool in service penetrations Packing phallus , the practice of wearing a phallic object inside the clothing to give the appearance of male genitals Data compression , as in bit packing in computers Tourism Concealed carry , or packing, means to carry a concealed weapon An Autism therapies Packing autism therapy involving wrapping children in cold wet sheets. See also lookfrom intitle Pack disambiguation Packer disambiguation disambig ... more details
of Perga , who studied a related circle packing construction. Definition An Apollonian network may ... packing theorem K5 minus edge example.svg thumb left Construction of an Apollonian network from a circle packing Construction from circle packings Apollonian networks are named after Apollonius of Perga ...File Apollonian network.svg thumb 240px An Apollonian network File Goldner Harary graph.svg thumb The Goldner Harary graph , a non Hamiltonian Apollonian network In combinatorics combinatorial mathematics , an Apollonian network is an undirected graph formed by a process of recursively subdividing a triangle into three smaller triangles. Apollonian networks may equivalently be defined as the planar ... , are both Apollonian networks. math K sub 3 sub is formed by starting with a triangle and not performing .... The Goldner Harary graph is an Apollonian network that forms the smallest Hamiltonian cycle ... 357. ref Another more complicated Apollonian network was used by harvtxt Nishizeki 1980 to provide ... Apollonian networks are examples of maximal element maximal planar graph s, graphs to which ... that forms an Apollonian network is an elimination ordering as a chordal graph. This forms an alternative characterization of the Apollonian networks they are exactly the chordal maximal planar graphs ... . ref In an Apollonian network, every maximal clique is a complete graph on four vertices, formed by choosing ... tree math k tree , and Apollonian networks are examples of 3 trees. Not every 3 tree is planar, but the planar 3 trees are exactly the Apollonian networks. Every Apollonian network is also a uniquely ..., but also true, that every uniquely 4 colorable planar graph is an Apollonian network. Therefore, Apollonian networks may also be characterized as the uniquely 4 colorable planar graphs. Apollonian ... k 4 . ref For the characterization of Apollonian networks via unique 4 colorability, see harvtxt Fowler 1998 . The fact that Apollonian networks also minimize the number of colorings with larger numbers ... more details
The Apollonian and Dionysian is a philosophical and literary concept , or dichotomy , based on certain ... the use of the concepts of the Apollonian and Dionysian is famously related to Nietzsche s The Birth ... of Dionysian and Apollonian Kunsttrieben artistic impulses forms dramatic arts, or tragedies ... and reason robs tragedy of its foundation, namely the fragile balance of the Dionysian and Apollonian ... the Apollonian and Dionysian juxtapositions is apparent, Nietzsche claimed in The Birth of Tragedy ... to make order in the Apollonian sense of his unjust and chaotic Dionysian Fate, though he dies unfulfilled ... in Apollonian ideals, which in turn separate him from his essential connection with self. The Dionysian ... connected with the Apollonian. The Dionysian magnifies man, but only so far as he ... experience is of utmost importance in viewing the Dionysian as it is related to the Apollonian ... American humanities scholar Camille Paglia writes about the Apollonian and Dionysian in her controversial ... significantly. The Apollonian and Dionysian concepts comprise a dichotomy that serves as the basis of Paglia s theory of art and culture. For Paglia, the Apollonian is light and structured while the Dionysian ..., and unconstrained sex procreation. In contrast, the Apollonian is associated with males, clarity ..., 1990, p. 40 ref She argues that there is a biological basis to the Apollonian Dionysian dichotomy ..., and turning instead to the Apollonian trait of ordered creation. The Dionysian is a force of chaos ... of &ndash or combat with &ndash Chthonianism by socially constructed Apollonian virtues accounts ... Apollo program Apollo . As this anecdote shows, the Apollonian tendency would also seem to include ... According to Peter Sloterdijk , the Dionysian and the Apollonian form a dialectic they are contrasting ... being is determined by the Dionysian Apollonian dialectic. Extending the use of the Apollonian and Dionysian .... Stephen King s usage American novelist Stephen King uses Apollonian and Dionysian analysis in Danse ... more details
Image Apollonian circles.svg thumb right 350px Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points, C and D , and every blue circle separates the two points. This article discusses a family of circles sharing a radical axis , and the corresponding family of orthogonal circles. For other circles associated with Apollonius of Perga , please see the disambiguation page, circles of Apollonius . Apollonian circles are two families of circle s such that every circle in the first family intersects every circle in the second family orthogonal ly, and vice versa. These circles form the basis for bipolar coordinates . They were discovered by Apollonius of Perga , a renowned Ancient Greece Greek geometer . Definition The Apollonian circles are defined in two different ways by a line segment denoted CD . Each circle in the first family the blue circles in the figure is associated with a positive real number r , and is defined as the locus of points X such that the ratio of distances from X to C and to D equals r , math left X mid frac d X,C d X,D r right . math For values of r close to zero, the corresponding circle is close to C , while for values of r close to , the corresponding circle is close to D for the intermediate value r 1, the circle degenerates to a line, the perpendicular bisector of CD ... k pi right math Pencils of circles Both of the families of Apollonian circles are called pencils ... parabolic pencil. It is relatively easy to show using inversion that, in the Apollonian circles ... of the blue Apollonian circles with respect to a circle centered on point C results in a pencil of concentric ... coordinates bipolar coordinate system defined by the Apollonian circles into a polar coordinates ..., so the original Apollonian circles also meet at right angles. Alternatively, ref harvtxt Akopyan ... pencil is hyperbolic, and vice versa in this case the two pencils form a set of Apollonian circles ... more details
Other uses sphere disambiguation Infobox artwork image file Sphere before Sept 11.jpg painting alignment image size 250px title The Sphere alt The Sphere at the Austin J. Tobin Plaza before the September ... and New Jersey PANYNJ The Sphere is a large metal lic sculpture by German people German sculptor Fritz ... dents and holes. Description File Austin Tobin Plaza 1WTC Sphere.jpg thumb 140px The Sphere being cleaned up together with its fountain before the events of 9 11. The Sphere is 25 feet high and cast ... title Koenig s Sphere work publisher Leora Films, Inc. url http www.percyadlon.com film and stage koenigssphere ... , in which The Sphere stood at the place of the Kaaba . ref name Mosque of Commerce cite news last ... nicknamed it The Sphere. ref name Adlon Location history File FEMA 4042 Photograph by Michael Rieger taken on 09 21 2001 in New York.jpg thumb 250px The Sphere after 9 11. After the attacks on September ... to Koenig, made Koenigs Kugel Koenig s Sphere at a time when the sculpture s fate was still uncertain ... retells the story of its creation. At first, Koenig opposed reinstalling The Sphere, considering ... Koenig s Sphere The German Sculptor Fritz Koenig at Ground Zero work publisher Bayerischer Rundfunk ... spokeswoman Vickie Karp, the city is looking to relocate The Sphere by summer 2012 when ... of New York and New Jersey PANYNJ , which owns The Sphere , is considering placing the sculpture ... to place The Sphere until Liberty Park is completed. As of February 2011, PANYNJ had not made an official ... the sculpture to move. Until Liberty Park opens, the Sphere may have to go into storage. ref name ... Center Sphere url http www.dnainfo.com 20110228 downtown 911 family members start petition save world trade center sphere publisher Digital Network Associates dba DNAinfo.com accessdate 28 February 2011 ref ref name Chung cite web last Chung first Jen title World Trade Center Sphere s Uncertain ... demanding the return of The Sphere to the 9 11 Memorial has gained more than 7,123 signatures as of July ... more details
can be made to higher dimensions &ndash this is called spherepacking , which usually deals only with identical spheres. The branch of mathematics generally known as circle packing , however, is not overly concerned with dense packing of equal sized circles the densest packing is known but with the geometry ... math 0407145v2 doi 10.1007 s00454 005 1172 4 issue 2 ref clear See also Circle packing in a square Apollonian ...This article describes the packing of circles on surfaces. For the related article on circle packing with a prescribed intersection graph , please see the circle packing theorem . Image Citrus fruits.jpg ... , circle packing is the study of the arrangement of circles of equal or varying sizes on a given surface such that no overlapping occurs and so that all circles touch another. The associated packing ... maximum packing density, it does not have the lowest possible. The worst shape to pack onto a plane is not known, but the smoothed octagon has a packing density of about 0.902414, which is the lowest maximum packing density known of any centrally symmetric convex shape. ref MathWorld urlname SmoothedOctagon title Smoothed Octagon ref Packing densities of concave shapes such as star polygon s can be arbitrarily small. Packings in the plane File Circle packing hexagonal .svg thumb left Identical circles in a hexagonal packing arrangement, the densest packing possible. In two dimensional Euclidean ... density is the hexagon al packing arrangement, ref name ChangWang in which the centres of the circles ... authorlink eprint 1009.4322v1 title A Simple Proof of Thue s Theorem on Circle Packing class math.MG ... packed circles have been identified. Packings on the sphere A related problem is to determine the lowest ... on the surface of a sphere. The Tammes problem is a generalisation of this, dealing with maximising the minimum distance between circles on sphere. This is analogous to distributing non point charges on a sphere. Packings in bounded areas File Circles packed in square 15.svg thumb right Fifteen equal ... more details
Spherical packing may refer to Spherepacking Spherical code disambig Short pages monitor This long comment was added to the page to prevent it being listed on Special Shortpages. It and the accompanying monitoring template were generated via Template Longcomment. Please do not remove the monitor template without removing the comment as well. ... more details
postulated an optimal solution for spherepackingpacking spheres hundreds of years before it was proven ...Expert subject Computer science date November 2008 Puzzles Packing problems are a class of optimization ... issues. Each packing problem has a dual covering problem , which asks how many of the same ... to overlap. In a packing problem, you are given containers usually a single two or three dimensional ... must be packed into this container Usually the packing must be without overlaps between goods and other ... is allowed but should be minimized. Covering Packing Problem Pairs Packing infinite space Many of these problems ... of packing objects as densely as possible in infinite Euclidean space . This problem is relevant ... including tetrahedron packing tetrahedra , ref cite doi 10.1038 nature08641 ref ref cite doi 10.1007 s00454 010 9273 0 ref and unequal sphere dimers. ref cite doi 10.1088 0953 8984 23 19 194103 ref Hexagonal packing of circles File Circle packing hexagonal .svg thumb right The hexagonal packing ... in the circle packing theorem . The related circle packing problem deals with packing circles, possibly of different sizes, on a surface, for instance the plane or a sphere. The N sphere counterparts ... be unused space if you are only packing circles. The most efficient way of packing circles, Circle packing hexagonal packing , produces approximately 91 efficiency. http mathworld.wolfram.com CirclePacking.html Sphere packings in higher dimensions Main Spherepacking In three dimensions, the face centered cubic lattice offers the best lattice packing of spheres, and is believed to be the optimal ... to fill three dimensional space completely, the most natural packing being the cubic honeycomb . No other ... packing Tetrahedra can achieve a packing of at least 85 . One of the best packings of regular dodecahedron ... . class wikitable Solid Maximum known packing density Lowest upper bound for lattice packing density ... Ko rper. Nachr. Akad. Wiss. Go ttingen Math. Phys. KI. II 311 355 1904 . ref Packing in 3 dimensional ... more details
known for packings of tetrahedra was less than that of spherepacking spheres , it was suggested that the regular ... density for spherepackingpacking congruent spheres is smaller than that for any other convex ...In geometry , tetrahedron packing is the problem of arranging identical regular tetrahedron tetrahedra ... on the optimal packing fraction of regular tetrahedra is 85.63 . ref name chen2010 Cite arxiv first1 ... Yoav Kallus doi 10.1007 s00454 010 9304 x title Upper bound on the packing density of regular tetrahedra ... Conway and Torquato showed that a packing fraction about 72 can be obtained by constructing a non Bravais lattice packing of tetrahedra with multiple particles with generally different orientations per repeating unit , and thus they showed that the best tetrahedron packing cannot be a lattice packing with one particle per repeating unit such that each particle has a common orientation . ref cite journal last1 Conway first1 J. H. title Packing, tiling, and covering with tetrahedra journal Proceedings ... pnas.0601389103 pmid 16818891 pmc 1502280 bibcode 2006PNAS..10310612C ref These packing constructions almost doubled the optimal Bravais lattice packing fraction 36.73 obtained by Hoylman. ref cite journal last1 Hoylman first1 Douglas J. title The densest lattice packing of tetrahedra journal ... like dice can randomly pack in a finite container up to a packing fraction between 75 and 76 . ref cite ... last4 Chaikin first4 Paul M. title Experiments on the Random Packing of Tetrahedral Dice journal Physical ... with a packing fraction of 77.86 . ref cite journal last1 Chen first1 Elizabeth R. title A Dense Packing ... and Jiao, who compressed Chen s structure using a computer algorithm to a packing fraction of 78.2021 ... a denser random tetrahedron packing with a packing fraction of 82.26 using the same algorithm. ref cite ... MC simulations of initially random systems that at packing densities > 50 an equilibrium fluid ... a packing density as high as 85.03 . ref cite journal last1 Haji Akbari first1 Amir last2 Engel first2 ... more details
For the disambiguation page Packing disambiguation Packing is the process and or the materials used in filling both Penetrant service Penetration firestop penetrations and Joint building building joints with backer materials as approved components within a firestop . gallery Image Ceramic fibre nelson clk installation.jpg Ceramic fibre used as packing to support firestop silicone caulking. Image Pack seal 1.jpg Canned service penetration about to be firestopped. Image Pack seal 2.jpg Packing being installed. Image Pack seal 3.jpg Firestop caulking being applied. Image Pack seal 4.jpg Firestop has been tagged to identify certification information, thus enabling proper Maintenance, repair and operations maintenance . gallery Purpose Packing with inherently fire resistive materials, such as rockwool or ceramic fibre is intended to protect sealant s that would, on their own, be consumed by the fire . Lesser packing, such as foam backer rod or fibreglass are used simply to hold up materials that can survive fire test ing on their own. In both cases, the packing is placed in such a way as to control the exact depth of the materials that top off the seal. Special care Proper packing is particularly important with self leveling sealants. If the packing is not tight enough, the expensive caulking will drip out of the Penetration firestop hole . If the packing surface is not straight enough, the thickness of the caulking will be wrong. If firestopping takes place on plumbing Piping pipe s that have not yet been tested, packing can move the pipes and interfere with the plumbing. It is thus important for the work to be co ordinated between plumbers and firestoppers. See also Firestop ... Annulus firestop External links http www.freepatentsonline.com EP0733384.html Packing as used in a patented ... 3M&operator mfgRefinementSearch&L1 3M Grainger Firestop Packing http www.wardmfg.com Files ULCert W J 1099.pdf Ward Manufacturing UL Listing, Item 4A is Packing Material for Firestop Category Passive ... more details
Independent set Graph coloring Covering number Set packing Ramsey s theorem Set cover problem Spherepacking Steiner system References refbegin 2 Citation last1 Erd s first1 P. last2 Hanani first2 H. title ...In mathematics, a packing in a hypergraph is a Partition of a set partition of the set of the hypergraph s edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k uniform hypergraphs ... packing by R dl nibble is in some sense close to that of the random greedy algorithm. History The problem ... E math consists of exactly math k math vertices. math P math is a hypergraph packing if it is a subset ... of edges that contain both vertices. Theorem There exists an asymptotic packing P of size at least ... hypergraph packing problem, Joel Spencer proposed a random greedy algorithm. In this algorithm ... optimal packing under the above side conditions. Asymptotic packing algorithms There are two famous algorithms for asymptotic packing of k uniform hypegraphs which are random greedy algorithm ... math P math is a packing and it can be shown that its size is math P frac n K 1 math almost ... in form of either packing or covering problem. For math 2 leq l k n math the covering number denoted ... configuration is asymptotically achievable. One may similarly define the packing number math m n,k,l ... the property that every l element set is contained in at most one math A in kappa math . Packing ... hypergraph on math n math vertices, if math k 3 math , there exists a packing math P math covering all vertices but at most math O nD 1 k 1 math . If math k 3 math there exists a packing math P math .... Any Steiner Triple System on math n math vertices contains a packing covering all vertices ... pages 10 13 Citation doi 10.1002 rsa.3240070206 last1 Spencer first1 J. title Asymptotic packing via ... packing and the random greedy algorithm journal Random Structures and Algorithms volume 8 issue 3 ... more details
Orphan date February 2009 unreferenced date August 2008 Hepatic packing is a surgical procedure used in connection with surgery to the liver . The liver is packed to stop the bleeding. med stub Category Traumatology ... more details
Image Pack man silicone packer.png thumb Left The Pack Man Silicone Packer Crossdressing Packing refers to the wearing of padding or a phallic object in the front of a person s pants or underwear, to give the appearance of having male genitals. Packing is commonly practiced by transman trans men i.e. female to male transgender or transsexual people . People who cross dress as male may also pack to do drag king performances. Packers The object used for packing, sometimes called a packer , may take the form of only a penis shape or may also incorporate a false scrotum and testicle s. Packers may be used to make male clothing hang and move correctly. If a packer is not used, the shape of the clothing is sometimes visibly empty . ref http www.ftmguide.org packing.html FTM Guide Packing ref Soft packing The phrase soft packing refers to packing with a device that cannot be used for sexual penetration. Homemade packers may be very low tech and consist of rolled up socks or condoms filled with liquids or gel. There are many commercially available packers, which more realistically emulate the size, shape, color and texture of the penis male genitals . All soft packers are made of soft, flexible material and simulate a flaccid penis. The term packer most frequently describes the soft models not used for sex. A few packers incorporate a receptacle that is inserted under and against the wearer s urethra . The packer has a tube from the receptacle to the tip, which allows the wearer to urinate through the prosthetic. For sex Main Pegging sexual practice Packing hard refers to packing with a device that may be used for sexual penetration . Some such packers are made of firmer material such as silicone , and are flexible enough for packing but firm enough for sexual penetration . Still ... for packers which may be used for sex is packing dildo . A term for the usage of these is pack and play . Slang A slang term related to packing is packing heat . This slang term originally meant carrying ... more details
Image Riempimento strutturato.jpg thumb 250px Structured packing The term structured packing refers to a range of specially designed materials for use in absorption chemistry absorption and distillation columns and chemical reactor s. Structured packings typically consist of thin corrugated metal plates or gauzes arranged in a way that they force fluid s to take complicated paths through the column, thereby creating a large surface area for contact between different phase matter phases . See also packed bed Raschig ring External links http www.rhineruhr.com.au products columnpack.html Rhine Ruhr Structured Packing Category Distillation engineering stub ar de Strukturierte Packung ... more details
Set packing is a classical NP complete problem in computational complexity theory and combinatorics , and was one ... of S . Then, the set packing problem asks if some k subsets in the list are pairwise disjoint sets ... math and a family math mathcal S math of subsets of math mathcal U math , a packing is a subfamily math ... disjoint. In the set packing decision problem , the input is a pair math mathcal U , mathcal S math and an integer math k math the question is whether there is a set packing of size math k math or more. In the set packing optimization problem , the input is a pair math mathcal U , mathcal S math , and the task is to find a set packing that uses the most sets. The problem is clearly in NP complexity ... problem optimization version of the problem, maximum set packing , asks for the maximum number of pairwise ... linear program , belongs to the class of packing problem s, and its dual linear program is the set cover problem . ref harvtxt Vazirani 2001 ref Covering Packing Problem Pairs Integer linear program formulation The maximum set packing problem can be formulated as the following integer linear ... packing or not Example As a simple example, suppose you re at a convention of foreign ambassadors .... A maximum set packing will choose the largest possible number of ambassadors under the desired constraint .... Heuristics and related problems Set packing is one among a family of problems related to covering ... the minimum number of such sets. The maximum set packing need not cover every possible element. One advantage of the set packing problem is that even if it s hard for some k , it s not hard to find ... the sets it intersects. We continually do this until no sets are left, and we have a set packing of some size, although it may not be the maximum set packing. Although no algorithm can always produce ... problem is about as easy as set packing. Karp originally showed set packing NP complete ... The set packing problem is not only NP complete, but its optimization version general maximum ... more details
In mathematics , the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space . Packing dimension is in some sense duality mathematics dual to Hausdorff dimension , since packing dimension is constructed by packing small open ball s inside the given subset, whereas Hausdorff dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982. Definitions Let X , d be a metric space with a subset S &sube X and let s &ge 0. The s dimensional packing pre measure of S is defined to be math P 0 s S lim delta downarrow 0 sup left left ... measure leads to a bona fide measure the s dimensional packing measure of S is defined to be math P ... i.e., the packing measure of S is the infimum of the packing pre measures of countable covers of S . Having done this, the packing dimension dim sub P sub S of S is defined analogously to the Hausdorff ... align math An example The following example is the simplest situation where Hausdorff and packing ... that the Hausdorff and the packing dimensions of the set math K math are given respectively by math ... Cantor like math K math set has Hausdorff dimension math d 1 math and packing dimension math d 2 math ... function h 0, 0, , let the packing pre measure of S with dimension ... balls with text diameters leq delta text and centres in S end matrix right math and define the packing ... bigcup j in J S j, J text countable right . math The function h is said to be an exact packing ... of n dimensional Euclidean space R sup n sup with its usual metric, then the packing dimension ... MB S . math This result is interesting because it shows how a dimension derived from a measure packing ..., that the packing dimension is not equal to the box dimension. For example, the set of rational number rationals Q has box dimension one and packing dimension zero. See also Hausdorff dimension ... more details
Onesource date June 2008 Image Apple and Pear packing plant Pateros WA.jpg thumb 475px An apple and pear packing house in Pateros, Washington . A packing house is a facility where fruit is received and processed prior to distribution to market. Bulk fruit such as apples, oranges, pears, and the like is delivered to the plant via trucks or wagons, where it is dumped into receiving bins and sorted for quality and size. In the case of citrus , ripe fruit with a greenish tint is placed in special storage rooms where ethylene gas is used to bring out the color. Obvious culls fruit that is not suitable to sell for eating due to cosmetic defects is removed and sold for juice or other uses. Fruit that is ready to be packed into crates or flats is run through a washer and then air dried. A light coating of natural wax is applied to help the fruit retain moisture and enhance its appeal. The fruit is transported via conveyor belts to the grading tables where it is visually sorted into three grades top quality , average , and orchard run , and is then carried via belts to the packing tables. During the late 19th century top grade fruit would be wrapped in printed tissue paper a technique developed in the town of Orange, California in the 1880s and placed in the boxes so that the printed names showed between the slats. In the 1920s tissue wrappers were replaced with printed logos, and ultimately paper stickers. Packed fruit is designated by size, based on the average number of pieces it takes to fill a box. In the days of wooden crates, sizes generally ranged from the 100s to the 390s, demonstrating ... shipped. Today, packing is often still performed by hand, even though Sunkist Growers, Incorporated Sunkist developed robotic packing machines in the 1980s. Packed boxes are stored in a pre cooler to prepare ... . See also Packing house is also used to refer to a slaughterhouse . References cite web title Citrus ... index.htm accessdate August 7, 2006 DEFAULTSORT Packing House Category Agricultural buildings Category ... more details
Unreferenced date April 2010 Trousseau packing is the specialised packaging for wedding gifts given to the bride and her to be family including her husband. These gifts are ceremoniously given away a few days before marriage. The tradition in India is to give away articles that the new couple would need as part of their life together. The items may include a car, jewelry, bed, dressing table, clothes, Saree etc. Dowry is an illegal practice, but India marriages being deep rooted in tradition, follow the practice of giving dowry to the groom s family. Specialty about trousseau packing Unlike other gift packaging, items are wrapped in traditional materials like Gold, Silver paper. They are decorated with ornaments. Traditionally, articles for the bridegroom s family are in 11,21,51,101 or 201 quantities. These numbers are thought to be lucky the extra 1 signifies Sagan or luck. Category Wedding gifts ... more details
burn also ruins the flavor and texture of foods. Vacuum packing reduces freezer burn by preventing ... organism s can proliferate, potentially causing food safety problems. Vacuum packing is often ... more details
Expert subject Optics date July 2011 Image Poincare Sphere with Polarizations.svg thumb 256px Poincare sphere with 6 common types of polarizations labeled. See the Jones calculus Jones vectors Jones Vector for a detailed list of the 6 common polarization types. Poincar sphere , often spelled Poincare sphere without the accent may refer to Poincar sphere , in optics, is a graphical tool for visualizing different types of Polarization waves polarized light Poincar homology sphere , in mathematics, an example of a Homology sphere Poincar homology sphere homology sphere In condensed matter physics, the Poincar sphere is also known as the Bloch sphere . Disambiguation ru ... more details
Inner Sphere can refer to Inner sphere complex , a type of ion surface binding Inner sphere electron transfer , a chemical reaction involving closely associated atoms A region of space in the BattleTech universe disambig ... more details
line O P sup 1 sup . 23 sphere A highly dense spherepacking is possible in 24 dimensional space ...DISPLAYTITLE n sphere Image Sphere wireframe.svg thumb 2 sphere wireframe as an orthogonal projection Image Hypersphere coord.PNG right thumb Just as a stereographic projection can project a sphere s surface to a plane, it can also project the surface of a 3 sphere into 3 space. This image shows three ... , an n sphere is a generalization of the surface of an ordinary sphere to arbitrary dimension . For any natural number n , an n sphere of radius r is defined as the set of points in n ... space. In particular a 0 sphere is a pair of points that are the ends of a line segment , a 1 sphere is a circle in the plane, and a 2 sphere is an ordinary sphere in three dimensional space. Spheres of dimension n 2 are sometimes called hyperspheres , with 3 sphere s sometimes known as glomes . The n sphere of unit radius centered at the origin is called the unit n sphere , denoted S sup n sup . The unit n sphere is often referred to as the n sphere. An n sphere is the surface or boundary of an n ... 1 sphere. Description For any natural number n , an n sphere of radius r is defined as the set ... 1 dimensional space. In particular a 0 sphere is a pair of points c &minus r , c r , and is the boundary of a line segment 1 ball . a 1 sphere is a circle of radius r centered at c , and is the boundary of a disk 2 ball . a 2 sphere is an ordinary 2 dimensional sphere in 3 dimensional Euclidean space, and is the boundary of an ordinary ball 3 ball . a 3 sphere is a sphere in 4 dimensional ... , x sub 1 sub , x sub 2 sub , , x sub n 1 sub that define an n sphere, S sup n sub is represented by the equation .... The above n sphere exists in n 1 dimensional Euclidean space and is an example of an n manifold . The volume form of an n sphere of radius r is given by math omega 1 over r sum j 1 n 1 ... enclosed by an n sphere is called an n 1 Ball mathematics ball . An n 1 ball ... more details
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