123 Sentences With "packings" | Random Sentence Generator

Maximally random jammed packings, meanwhile, are a a whole different story.

Higher-dimensional sphere packings are hard to visualize, but they are eminently practical objects: Dense sphere packings are intimately related to the error-correcting codes used by cell phones, space probes and the Internet to send signals through noisy channels.

In most dimensions, the best sphere packings discovered to date didn't even come close to the density limits this method generated.

Given this ridiculously close estimate, it seemed clear that E8 and the Leech lattice must be the best sphere packings in their respective dimensions.

But Cohn and Elkies found that in dimensions eight and 24, the best packings—E8 and the Leech lattice—seemed to practically bump their heads against the ceiling.

The deal, which was made for an undisclosed amount, signals an interest from Samsung in packings its phones with news features to help it stand out above the iPhone.

In 2003, Cohn and Noam Elkies of Harvard University developed a way to estimate just how well E8 and the Leech lattice perform compared to other sphere packings in their respective dimensions.

"In this work, we have shown that twisting a sample of cubic particles is a highly efficient way to achieve ordered packings," the authors write in the paper published recently in Physical Review Letters.

Finding the best packing of equal-sized spheres in a high-dimensional space should be even more complicated than the three-dimensional case Hales solved, since each added dimension means more possible packings to consider.

In every dimension, Cohn and Elkies showed, there is an infinite sequence of "auxiliary" functions that can be used to compute upper limits on how dense sphere packings are allowed to be in that dimension.

Yet mathematicians have long known that two dimensions are special: In dimensions eight and 24, there exist dazzlingly symmetric sphere packings called E8 and the Leech lattice, respectively, that pack spheres better than the best candidates known to mathematicians in other dimensions.

When Cohn and Abhinav Kumar of Stony Brook University carried out extensive numerical calculations on the sequences of auxiliary functions, they found that the best possible sphere packings in dimensions eight and 24 could be at most 83 percent denser than E8 and the Leech lattice.

Original story reprinted with permission from Quanta Magazine, an editorially independent division of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences Mathematicians have been studying sphere packings since at least 1611, when Johannes Kepler conjectured that the densest way to pack together equal-sized spheres in space is the familiar pyramidal piling of oranges seen in grocery stores.

These packings offer high capacity, lower cost, and less sensitivity to solids, while keeping a high performance. Popularity of the packings grew in the 1980s, particularly for revamps in oil and petrochemical plants. These structured packings, made of corrugated metal sheets, had their surfaces treated, chemically or mechanically, in order to enhance their wettability. Consequently, the packings' wetted area increased, improving performance.

In 1987 he proved the Thurston conjecture for circle packings, jointly with Dennis Sullivan.B. Rodin and D. Sullivan, “The convergence of circle packings to the Riemann mapping”, Journal of Differential Geometry, 26 (1987), 349-360.

The second generation appeared at the end of 1950's, with highly efficient wire mesh packings, such as Goodloe ™, Hyperfil ™ and Koch-Sulzer. Until the 1970s, due to their low pressure drop per theoretical stage, those packings were the most widely used in vacuum distillation. However, high cost, low capacity and high sensitivity to solids have prevented wider utilization of wire mesh packings. Corrugated structured packings, introduced by Sulzer by the end of the 1970s, marked the third generation of structured packed columns.

In three dimensions, close-packed structures offer the best lattice packing of spheres, and is believed to be the optimal of all packings. With 'simple' sphere packings in three dimensions ('simple' being carefully defined) there are nine possible definable packings. The 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective real dimensional space.

A similar variety of dense crystalline structures have also been discovered for columnar packings of spheroids through Monte Carlo simulations. Such packings include achiral structures with specific spheroid orientations and chiral helical structures with rotating spheroid orientations.

In 2000, they acquired LC Packings, whose competencies were in LC column packings. LC Packings/Dionex revealed their first monolithic capillary column at the Montreux LC-MS Conference. Earlier that year, another company, Isco, introduced a polystyrene divinylbenzene (PS- DVB) monolith column under the brand SWIFT. In January 2005, Dionex was sold the rights to Teledyne Isco's SWIFT media products, intellectual property, technology, and related assets.

Dense packings of congruent circles in a circle. Discrete Math 1998;181:139–154.

There are 11 circle packings based on the 11 uniform tilings of the plane.[7] In these packings, every circle can be mapped to every other circle by reflections and rotations. The hexagonal gaps can be filled by one circle and the dodecagonal gaps can be filled with 7 circles, creating 3-uniform packings. The truncated trihexagonal tiling with both types of gaps can be filled as a 4-uniform packing.

The fcc and hcp packings are the densest known packings of equal spheres with the highest symmetry (smallest repeat units). Denser sphere packings are known, but they involve unequal sphere packing. A packing density of 1, filling space completely, requires non-spherical shapes, such as honeycombs. Replacing each contact point between two spheres with an edge connecting the centers of the touching spheres produces tetrahedrons and octahedrons of equal edge lengths.

Given two packings for the same graph G, one may apply reflections and Möbius transformations to make the outer circles in these two packings correspond to each other and have the same radii. Then, let v be an interior vertex of G for which the circles in the two packings have sizes that are as far apart as possible: that is, choose v to maximize the ratio r1/r2 of the radii of its circles in the two packings. For each triangular face of G containing v, it follows that the angle at the center of the circle for v in the first packing is less than or equal to the angle in the second packing, with equality possible only when the other two circles forming the triangle have the same ratio r1/r2 of radii in the two packings. But the sum of the angles of all of these triangles surrounding the center of the triangle must be 2π in both packings, so all neighboring vertices to v must have the same ratio as v itself.

By applying the same argument to these other circles in turn, it follows that all circles in both packings have the same ratio. But the outer circles have been transformed to have ratio 1, so r1/r2 = 1 and the two packings have identical radii for all circles.

For further details on these connections, see the book Sphere Packings, Lattices and Groups by Conway and Sloane.

Gaskets are the mechanical seals, or packings, used to prevent the leakage of a gas or fluids from valves.

Discrete geometry includes the study of various sphere packings. Discrete geometry is a subject that has close connections with convex geometry. It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc.

The inversive distance has been used to define the concept of an inversive-distance circle packing: a collection of circles such that a specified subset of pairs of circles (corresponding to the edges of a planar graph ) have a given inversive distance with respect to each other. This concept generalizes the circle packings described by the circle packing theorem, in which specified pairs of circles are tangent to each other.. Although less is known about the existence of inversive distance circle packings than for tangent circle packings, it is known that, when they exist, they can be uniquely specified (up to Möbius transformations) by a given maximal planar graph and set of Euclidean or hyperbolic inversive distances. This rigidity property can be generalized broadly, to Euclidean or hyperbolic metrics on triangulated manifolds with angular defects at their vertices.. However, for manifolds with spherical geometry, these packings are no longer unique.. In turn, inversive-distance circle packings have been used to construct approximations to conformal mappings.

Upper bounds for the density that can be obtained in such binary packings at smaller ratios have also been obtained.

The company and its subsidiaries make a range of sealing products, including gaskets, gasket sheets, high-tech O-rings, molded packings and seals for reciprocating pumps, hydraulics, mechanical seals, braided packings, and custom rubber-molded products. They also offer repairs for mechanical seals. Most of the company's products have short product lifecycles so must be replenished periodically to prevent tools from not working.

The rich variety of such ordered structures can also be obtained by sequential depositioning the spheres into the cylinder. Chan reproduced all dense sphere packings up to D/d < 2.7013 using an algorithm, in which the spheres are placed sequentially dropped inside the cylinder. Mughal et al. also discovered that such structures can be related to disk packings on a surface of a cylinder.

Differently shaped packings have different surface areas and void space between packings. Both of these factors affect packing performance. Another factor in addition to the packing shape and surface area that affects the performance of random or structured packing is liquid and vapor distribution entering the packed bed. The number of theoretical stages required to make a given separation is calculated using a specific vapor to liquid ratio.

Optimal packing fraction for hard spheres of diameter d inside a cylinder of diameter D. Columnar structures arise naturally in the context of dense hard sphere packings inside a cylinder. Mughal et al. studied such packings using simulated annealing up to the diameter ratio of D/d = 2.873 for cylinder diameter D to sphere diameter d. This includes some structures with internal spheres that are not in contact with the cylinder wall.

Despite this difficulty, K. Böröczky gives a universal upper bound for the density of sphere packings of hyperbolic n-space where n ≥ 2. In three dimensions the Böröczky bound is approximately 85.327613%, and is realized by the horosphere packing of the order-6 tetrahedral honeycomb with Schläfli symbol {3,3,6}. In addition to this configuration at least three other horosphere packings are known to exist in hyperbolic 3-space that realize the density upper bound.

Torquato's research work is centered in statistical mechanics and soft condensed matter theory. A common theme of his research is the search for unifying and rigorous principles to elucidate a broad range of physical and biological phenomena. Torquato has made fundamental contributions to our understanding of the randomness of condensed phases of matter through the identification of sensitive order metrics. He is one of the world's experts on packing problems, including pioneering the notion of the "maximally random jammed" state of particle packings, identifying a Kepler-like conjecture for the densest packings of nonspherical particles, and providing strong theoretical evidence that the densest sphere packings in high dimensions (a problem of importance in digital communications) are counterintuitively disordered, not ordered as in our three-dimensional world.

Unlike conventional tray distillation in which every tray represents a separate point of vapor–liquid equilibrium, the vapor–liquid equilibrium curve in a packed column is continuous. However, when modeling packed columns, it is useful to compute a number of "theoretical stages" to denote the separation efficiency of the packed column with respect to more traditional trays. Differently shaped packings have different surface areas and void space between packings. Both of these factors affect packing performance.

When the two parastichy numbers p and q are either consecutive Fibonacci numbers, or Fibonacci numbers that are one step apart from each other in the sequence of Fibonacci numbers, then the third parastichy number will also be a Fibonacci number. For modeling plant growth in this way, spiral packings of tangent circles on surfaces other than the plane, including cylinders and cones, may also be used. Spiral packings of circles have also been studied as a decorative motif in architectural design.

Furthermore, the E8 lattice is the unique lattice (up to isometries and rescalings) with this density. The mathematician Maryna Viazovska proved in 2016 that this density is, in fact, optimal even among irregular packings.

Critical sets for the completion of Latin squaresHorak P.; Dejter I. J. "Completing Latin squares: critical sets, II", Jour. Combin. Des., 15 (2007), 177-83. Almost resolvable maximum packings of complete graphs with 4-cycles.

Construction showing the tight bound for the ring lemma In the geometry of circle packings in the Euclidean plane, the ring lemma gives a lower bound on the sizes of adjacent circles in a circle packing.

Optimal sampling lattices have been studied in higher dimensions. Generally, optimal sphere packing lattices are ideal for sampling smooth stochastic processes while optimal sphere covering latticesJ. H. Conway, N. J. A. Sloane. Sphere packings, lattices and groups.

There are no integer gaskets with D3 symmetry. If the three circles with smallest positive curvature have the same curvature, the gasket will have D3 symmetry, which corresponds to three reflections along diameters of the bounding circle (spaced 120° apart), along with three-fold rotational symmetry of 120°. In this case the ratio of the curvature of the bounding circle to the three inner circles is 2 − 3\. As this ratio is not rational, no integral Apollonian circle packings possess this D3 symmetry, although many packings come close.

During this summer project he looked at jammed disordered packings, investigating how M&M;'s arrange in a small volume with Paul Chaikin. Cissé used various techniques to study jammed packings, including magnetic resonance imaging, but in the end used a much simpler approach - painting M&M;'s and analysing how many times they knocked into each other. The result was published in Science. He moved to Urbana for his graduate studies, and earned his PhD under the supervision of single molecule biophysicist Taekjip Ha at the University of Illinois at Urbana–Champaign in 2009.

Old-style packings are now a minor member of this grouping: fine flake graphite in oils or greases for uses requiring heat resistance. A GAN estimate of current US natural graphite consumption in this end use is 7,500 tonnes.

Packings where all spheres are constrained by their neighbours to stay in one location are called rigid or jammed. The strictly jammed sphere packing with the lowest density is a diluted ("tunneled") fcc crystal with a density of only 0.49365.

Similar devices with a non-round cross-sections are called seals, packings or gaskets. See also washers. Automotive cylinder heads are typically sealed by flat gaskets faced with copper. Knife edges pressed into copper gaskets are used for high vacuum.

They also reported a glassy, disordered packing at densities exceeding 78%. For a periodic approximant to a quasicrystal with an 82-tetrahedron unit cell, they obtained a packing density as high as 85.03%. In late 2009, a new, much simpler family of packings with a packing fraction of 85.47% was discovered by Kallus, Elser, and Gravel. These packings were also the basis of a slightly improved packing obtained by Torquato and Jiao at the end of 2009 with a packing fraction of 85.55%, and by Chen, Engel, and Glotzer in early 2010 with a packing fraction of 85.63%.

Tetsuji Shioda, Oberwolfach 2005 Tetsuji Shioda (塩田徹治) is a Japanese mathematician who introduced Shioda modular surfaces and who used Mordell–Weil lattices to give examples of dense sphere packings. He was an invited speaker at the ICM in 1990.

Figure 5: Section of fractionating tower of Figure 4 showing detail of a pair of trays with bubble caps Figure 6: Entire view of a Distillation Column In industrial uses, sometimes a packing material is used in the column instead of trays, especially when low pressure drops across the column are required, as when operating under vacuum. This packing material can either be random dumped packing ( wide) such as Raschig rings or structured sheet metal. Liquids tend to wet the surface of the packing, and the vapors pass across this wetted surface, where mass transfer takes place. Differently shaped packings have different surface areas and void space between packings.

Structured packings have been established for several decades. The first generation of structured packing arose in the early 1940s. In 1953, a patented packing appeared named Panapak™, made of a wavy-form expanded metal sheet. The packing was not successful, due to maldistribution and lack of good marketing.

Only nine particular radius ratios permit compact packing, which is when every pair of circles in contact is in mutual contact with two other circles (when line segments are drawn from contacting circle-center to circle-center, they triangulate the surface). For all these radius ratios a compact packing is known that achieves the maximum possible packing fraction (above that of uniformly-sized discs) for mixtures of discs with that radius ratio. All nine have ratio-specific packings denser than the uniform hexagonal packing, as do some radius ratios without compact packings. It is also known that if the radius ratio is above 0.742, a binary mixture cannot pack better than uniformly-sized discs.

In 1959, Zatkoff formed a company known as Zatkoff Seals & Packings, a supplier to manufacturing companies in the Detroit and Toledo areas. The company grew to eight locations in three states and became "the largest independent distributor of seals in North America." Other companies include Zatkoff Properties, Ltd., and the Roger Zatkoff Company.

In 1994, a new geometry was developed, and called Optiflow. Later, in 1999, an improved structure of corrugated sheet packings, the MellapackPlus, was developed based on CFD simulations and experiment. This new structure, compared with conventional Mellapak, has a lowered pressure drop and maximum useful capacity could be extended up to 50%.

Nastiti (2003), in Ani Triastanti, 2007, p. 34. the Medang kingdom and Airlangga's era Kahuripan kingdom (1000-1049 AD) of Java experienced a long prosperity so that it needed a lot of manpower, especially to bring crops, packings, and send them to ports. Black labor was imported from Jenggi (Zanzibar), Pujut (Australia), and Bondan (Papua).

The project contains small sections of artworks made from similar materials. Such as Firehose Experiment, indicating artworks made from firehoses; similarly, Reformed Packings indicating artworks created by packing materials, etcetera. It is believed that Mallory hoped to express support for nature protection and to draw attention to energy waste through her fame and exhibitions.

An example line-slip structure and its corresponding rolled-out contact network. A line slip is identified by the loss of contacts. For each uniform structure, there also exists a related but different structure, called a line-slip arrangement. The differences between uniform and line-slip structures are marginal and difficult to spot from images of the sphere packings.

The Thurston Conjecture for Circle Packings is his conjecture that the homeomorphism will converge to the Riemann mapping as the radii of the circles tend to zero. The Thurston Conjecture was later proved by Burton Rodin and Dennis Sullivan. This led to a flurry of research on extensions of the circle packing theorem, relations to conformal mappings, and applications.

A partition of points into lines is called a ', and the partition of the spreads of a geometry is called a '. When Hirschfeld considered the problem in his Finite Projective Spaces of Three Dimensions (1985), he noted that some solutions correspond to packings of PG(3,2), essentially as described by Conwell above, and he presented two of them.

The contact network of both packings are identical. For both packing types, it was found that different uniform structures are connected with each other by line slips. Fu et al. extended this work to higher diameter ratios D/d < 4.0 using linear programming and discovered 17 new dense structures with internal spheres that are not in contact with the cylinder wall.

Hee Oh (, born 1969) is a South Korean mathematician who works in dynamical systems. She has made contributions to dynamics and its connections to number theory. She is a student of homogeneous dynamics and has worked extensively on counting and equidistribution for Apollonian circle packings, Sierpinski carpets and Schottky dances. She is currently the Abraham Robinson Professor of Mathematics at Yale University.

A version of the ring lemma with a weaker bound was first proven by Burton Rodin and Dennis Sullivan as part of their proof of William Thurston's conjecture that circle packings can be used to approximate conformal maps. Lowell Hansen gave a recurrence relation for the tightest possible lower bound, and Dov Aharonov found a closed-form expression for the same bound.

Because the earliest lower bound known for packings of tetrahedra was less than that of spheres, it was suggested that the regular tetrahedra might be a counterexample to Ulam's conjecture that the optimal density for packing congruent spheres is smaller than that for any other convex body. However, the more recent results have shown that this is not the case.

Fractionation Research Inc. (FRI) is an industry cooperative organization that researches the performance of industrial-scale mass transfer devices such as trays, packings and other column internals. Its objective is to facilitate the design of more economical distillation, absorption and stripping systems. Before the formation of FRI, such research was performed on a small scale by universities or private companies.

The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane. In one dimension it is packing line segments into a linear universe. In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions.

A packing of PG(3,2) is a partition of the 35 lines into 7 disjoint spreads of 5 lines each, and corresponds to a solution for all seven days. There are 240 packings of PG(3,2), that fall into two conjugacy classes of 120 under the action of PGL(4,2) (the collineation group of the space); a correlation interchanges these two classes.

He has co-authored The Physics of Foams, Oxford University Press (2000) with Stefan Hutzler, and The Pursuit of Perfect Packing, IoP Press (2000) with Tomaso Aste. In this context he published several scientific articles on cylinder sphere packings. In 2005, he was awarded the premier award of the Royal Irish Academy, the Cunningham Medal. Previous winners include William Rowan Hamilton.

Polyphenylene sulfide Polyphenylene sulfide (PPS) is an organic polymer consisting of aromatic rings linked by sulfides. Synthetic fiber and textiles derived from this polymer resist chemical and thermal attack. PPS is used in filter fabric for coal boilers, papermaking felts, electrical insulation, film capacitors, specialty membranes, gaskets, and packings. PPS is the precursor to a conductive polymer of the semi-flexible rod polymer family.

In practice, suboptimal rectangular packings are often used to simplify decoding. Circle packing has become an essential tool in origami design, as each appendage on an origami figure requires a circle of paper.TED.com lecture on modern origami "Robert Lang on TED." Robert J. Lang has used the mathematics of circle packing to develop computer programs that aid in the design of complex origami figures.

A smoothed octagon. The family of maximally dense packings of the smoothed octagon. The smoothed octagon is a region in the plane conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes.K. Reinhardt, Über die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurven, Abh. Math. Sem. Hamburg 10, 216-230 (1934).

Monoliths exhibit no shear forces or eddying effects. High interconnectivity of the mesopores allows for multiple avenues of convective flow through the column. Mass transport of solutes through the column is relatively unaffected by flow rate. This is completely at odds to traditional particulate packings, whereby eddy effects and shear forces contribute greatly to the loss of resolution and capacity, as seen in the vanDeemter curve.

Structure of the packing: In the general strip packing problem, the structure of the packing is irrelevant. However, there are applications that have explicit requirements on the structure of the packing. One of these requirements is to be able to cut the items from the strip by horizontal or vertical edge to edge cuts. Packings that allow this kind of cutting are called guillotine packing.

Silica fibers are fibers made of sodium silcate (water glass). They are used in heat protection (including asbestos substitution) and in packings and compensators. They can be made such that they are substantially free from non- alkali metal compounds. Sodium silicate fibres may be used for subsequent production of silica fibres, which is better than producing the latter from a melt containing SiO2 or by acid-leaching of glass fibres.

Greenwood, 658 Two forms (α-, β-) of cyclo-S9 are known, one of which has been characterized.Steudel, 8 Two forms of cyclo-S18 are known where the conformation of the ring is different. To differentiate these structures, rather than using the normal crystallographic convention of α-, β-, etc., which in other cyclo-Sn compounds refer to different packings of essentially the same conformer, these two conformers have been termed endo- and exo-.

Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.

Although polyhedra and tessellations had been studied for many years by people such as Kepler and Cauchy, modern discrete geometry has its origins in the late 19th century. Early topics studied were: the density of circle packings by Thue, projective configurations by Reye and Steinitz, the geometry of numbers by Minkowski, and map colourings by Tait, Heawood, and Hadwiger. László Fejes Tóth, H.S.M. Coxeter and Paul Erdős, laid the foundations of discrete geometry.

She is responsible for several patents in the area, including one to improve lactation and another to improve feed utilisation. Phillips joined the Waters Corporation in 1984, where she worked in research and development until 1996. At Waters Corporation Phillips was a member of the Chemical Research and Development department, developing chromatography packing materials and bioseparations. Under her leadership the team developed the AccellPlus exchange packings, which could be used to separate proteins.

Sphere packing on the corners of a hypercube (with the spheres defined by Hamming distance) corresponds to designing error-correcting codes: if the spheres have radius t, then their centers are codewords of a (2t + 1)-error-correcting code. Lattice packings correspond to linear codes. There are other, subtler relationships between Euclidean sphere packing and error- correcting codes. For example, the binary Golay code is closely related to the 24-dimensional Leech lattice.

Illustration of a columnar structure assembled by golf balls. A columnar structure or crystal is a cylindrical arrangement that forms, in the context of cylinder sphere packings, inside or on the surface of a columnar confinement. Spheres of identical size d assemble on the surface of a cylinder to an ordered columnar structure, if the cylinder diameter is of similar order of magnitude. A typical ordered columnar structure assembles by sequentially dropping golf balls inside a tube.

Much of his work concerns the Geometry of Numbers, Hausdorff Measures, Analytic Sets, Geometry and Topology of Banach Spaces, Selection Theorems and Finite Dimensionsl Convex Geometry. In the theory of Banach spaces and summability, he proved the Dvoretzky-Rogers lemma and the Dvoretzky-Rogers theorem, both with Aryeh Dvoretzky. He constructed a counterexample to a conjecture related to the Busemann–Petty problem. In the geometry of numbers, the Rogers bound is a bound for dense packings of spheres.

Vibration of a random loose packing can result in the arrangement of spherical particles into regular packings, a process known as granular crystallisation. Such processes depend on the geometry of the container holding the spherical grains. When spheres are randomly added to a container and then compressed, they will generally form what is known as an "irregular" or "jammed" packing configuration when they can be compressed no more. This irregular packing will generally have a density of about 64%.

Structures are known which exceed the close packing density for radius ratios up to 0.659786. Upper bounds for the density that can be obtained in such binary packings have also been obtained. In many chemical situations such as ionic crystals, the stoichiometry is constrained by the charges of the constituent ions. This additional constraint on the packing, together with the need to minimize the Coulomb energy of interacting charges leads to a diversity of optimal packing arrangements.

The company A. Fischer was founded in 1923 in Dietikon by Alois Fischer for the production of sheet metal packings. In 1952, production began to produce carton and paper containers of up to 25 liters capacity. In 1955, the company, with 15 employees, moved into a new building in Mutschellen. In 1959, the company was transformed into the "Fischer Söhne AG" company. In 1960, the first extrusion blow-molding plant for 10-liter containers was put into operation.

Reference to Australia and native Australian people has been recorded in 10th century AD Java. According to Waharu IV inscription (931 AD) and Garaman insription (1053 AD),Nastiti (2003), in Ani Triastanti, 2007, p. 39.Nastiti (2003), in Ani Triastanti, 2007, p. 34. the Medang kingdom and Airlangga's era Kahuripan kingdom (1000-1049 AD) of Java experienced a long prosperity so that it needed a lot of manpower, especially to bring crops, packings, and send them to ports.

After a PhD at ETH Zurich, he held faculty positions at Technical University Munich, University of Cambridge and at University of Milan. In 2015 he was elected a Fellow of Queens' College, Cambridge. Zaccone contributed to various areas of condensed matter physics. He is known for his work on the atomic theory of elasticity and viscoelasticity of amorphous solids, in particular for having developed the microscopic theory of elasticity of random sphere packings and elastic random networks.

Nastiti (2003), in Ani Triastanti, 2007, p. 34. the Medang kingdom and Airlangga's era Kahuripan kingdom (1000-1049 AD) of Java experienced a long prosperity so that it needed a lot of manpower, especially to bring crops, packings, and send them to ports. Black labor was imported from Jenggi (Zanzibar), Pujut (Australia), and Bondan (Papua). According to Naerssen, they arrived in Java by trading (bought by merchants) or being taken prisoner during a war and then made slaves.

Collamer's best known landmark was the Collamer Dam. The first dam was built in 1845 out of "wooden timbers with the chinks closed with packings of small sticks and straw," originally constructed to provide power for a gristmill."Eel Important to Eel," Bulletin of the Whitley County Historical Society, (February 1968) 8. In 1904 the dam that stood until January 2020 was built by O. Grant Gandy and Company, and was used to power Collamer and Sidney for a time.

Quinn went to Williams College as an undergraduate, graduating in 1985. She earned a master's degree from the University of Illinois at Chicago in 1987, and completed her doctorate at the University of Wisconsin–Madison in 1993. Her dissertation, Colorings and Cycle Packings in Graphs and Digraphs, was supervised by Richard A. Brualdi. She taught at Occidental College until 2005, when she gave up her position as full professor and department chair to move with her husband, biologist Mark Martin, to Washington.

From its inception until 1978, Virginia Slims saw a steady increase in market share to 1.75% (3.9% of all female smokers). With the introduction of Lights in 1978, the market share increased to 2.5%. Other packings, including 120's, Ultra Lights, and Superslims helped push the market share to a peak of 3.1% (nearly 7% of female smokers) in 1989. With increased competition from other brands, notably Capri and Misty, the brand lost ground but stabilized at around 2.4% though 2003.

When the smaller particle sizes are used to separate biomolecules, backpressures increase further because of the large molecule size. In monoliths, where backpressures are low and channel sizes are large, small molecule separations are less efficient. This is demonstrated by the dynamic binding capacities, a measure of how much sample can bind to the surface of the stationary phase. Dynamic binding capacities of monoliths for large molecules can be an order of ten times greater than that for particulate packings.

In tune with the varied demands of the market, the above range of products are available in different weights and quantity in poly packings and cans. Meat Products of India Ltd. is a Kerala Government owned company engaged in production and marketing of various meat and meat products derived from pork, beef, chicken, muton, rabbit and quails. It holds MFPO Licence No.1 under A. The products are manufactured under strict veterinary supervision from selected animals free from zoonotic disease.

Robert W. Brooks (1986) Robert Wolfe Brooks (Washington, D.C., September 16, 1952 - Montreal, September 5, 2002) was a mathematician known for his work in spectral geometry, Riemann surfaces, circle packings, and differential geometry. He received his Ph.D. from Harvard University in 1977; his thesis, The smooth cohomology of groups of diffeomorphisms, was written under the supervision of Raoul Bott. He worked at the University of Maryland (1979-1984), then at the University of Southern California, and then, from 1995, at the Technion in Haifa.

Ritonavir was originally dispensed as an ordinary capsule that did not require refrigeration. This contained a crystal form of ritonavir that is now called form I. However, like many drugs, crystalline ritonavir can exhibit polymorphism, i.e., the same molecule can crystallize into more than one crystal type, or polymorph, each of which contains the same repeating molecule but in different crystal packings/arrangements. The solubility and hence the bioavailability can vary in the different arrangements, and this was observed for forms I and II of ritonavir.

Also, the new design allowed for easier repairs of the pump or the packings which were being "eaten away" by new materials and chemistry affecting the industry. (Schneller) This divorced air motor pump design eventually lead to the invention of the PileDriver pump series. The same air motors used with larger capacity pump tubes enabled these new pumps to output extremely heavy materials such as mastics, adhesives and sealant. Today, PileDriver pumps are used to pump materials used in the construction of everything from appliances to automobiles.

Therefore, it was concluded that clusters of these specific numbers of rare gas atoms dominate due to their exceptional stability. The concept was also successfully applied to explain the monodispersed occurrence of thiolate-protected gold clusters; here the outstanding stability of specific cluster sizes is connected with their respective electronic configuration. The term magic numbers is also used in the field of nuclear physics. In this context, magic numbers often represent three-dimensional figurate numbers such as the octahedral numbers: they count the numbers of spheres in sphere packings of Platonic solids and related polyhedra...

The eight helices of the globin fold core share significant nonlocal structure, unlike other structural motifs in which amino acids close to each other in primary sequence are also close in space. The helices pack together at an average angle of about 50 degrees, significantly steeper than other helical packings such as the helix bundle. The exact angle of helix packing depends on the sequence of the protein, because packing is mediated by the sterics and hydrophobic interactions of the amino acid side chains near the helix interfaces.

FRI has tested various types of fractionation trays and packings (both generic and proprietary designs), with the objective of developing correlations for predicting tray efficiencies and pressure drops. This information is needed to design fractionators, absorbers and strippers. It has also tested performance of other column internals such as liquid distributors. It periodically reports its findings to the member companies. The OSU library maintains an unrestricted collection of FRI progress reports, plant tests, topical reports, consultants’ reports and annual reports that were issued during the period 1954 – 1970.

Two piece ball valves are generally slightly reduced (or standard) bore, they can be either throw-away or repairable. The 3 piece design allows for the center part of the valve containing the ball, stem & seats to be easily removed from the pipeline. This facilitates efficient cleaning of deposited sediments, replacement of seats and gland packings, polishing out of small scratches on the ball, all this without removing the pipes from the valve body. The design concept of a three piece valve is for it to be repairable.

Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing. In 2016, Maryna Viazovska announced a proof that the E8 lattice provides the optimal packing (regardless of regularity) in eight-dimensional space, and soon afterwards she and a group of collaborators announced a similar proof that the Leech lattice is optimal in 24 dimensions.

Although the concept of circles and spheres can be extended to hyperbolic space, finding the densest packing becomes much more difficult. In a hyperbolic space there is no limit to the number of spheres that can surround another sphere (for example, Ford circles can be thought of as an arrangement of identical hyperbolic circles in which each circle is surrounded by an infinite number of other circles). The concept of average density also becomes much more difficult to define accurately. The densest packings in any hyperbolic space are almost always irregular.

This means that parties to the Convention are required to prohibit the export of hazardous wastes to parties which have prohibited the import of such wastes via the notification procedure in Article 13 of the Convention. In places such as India, however, there continues to be a high use of friable or dust-based asbestos in compressed asbestos fiber (CAF) gaskets, ropes, cloth, gland packings, millboards, insulation, brake liners, and other products which are being exported without adequate knowledge and information to the other countries. Asbestos use is prevalent in India because there is no effective enforcement of the rules.

It is sent to another column rectifying the argon to the desired purity from which liquid is returned to the same location in the LP column. Use of modern structured packings which have very low pressure drops enable argon with less than 1 ppm impurities. Though argon is present in less to 1% of the incoming, the air argon column requires a significant amount of energy due to the high reflux ratio required (about 30) in the argon column. Cooling of the argon column can be supplied from cold expanded rich liquid or by liquid nitrogen.

Castle Forbes Bay's landscape is still shaped by apple orchards, though many of them are now disused or empty of trees. A large number of pickers huts, garages and apple packings dot the scenery, considered the possibly best preserved orcharding landscape in Australia. The township was named for the ship the Castle Forbes, which in 1836 mistakenly entered the Huon River instead of the Derwent River, and was forced to unload passengers at the bay after being unable to continue onwards. A small cemetery, now unused and maintained by Castle Forbes Bay Landcare, is located in the locality.

Thomas Callister Hales (born June 4, 1958) is an American mathematician working in the areas of representation theory, discrete geometry, and formal verification. In representation theory he is known for his work on the Langlands program and the proof of the fundamental lemma over the group Sp(4) (many of his ideas were incorporated into the final proof, due to Ngô Bảo Châu). In discrete geometry, he settled the Kepler conjecture on the density of sphere packings and the honeycomb conjecture. In 2014, he announced the completion of the Flyspeck Project, which formally verified the correctness of his proof of the Kepler conjecture.

The term structured packing refers to a range of specially designed materials for use in absorption and distillation columns and chemical reactors. Structured packings typically consist of thin corrugated metal plates or gauzes arranged in a way that force fluids to take complicated paths through the column, thereby creating a large surface area for contact between different phases. Structured packing is formed from corrugated sheets of perforated embossed metal, plastic (including PTFE) or wire gauze. The result is a very open honeycomb structure with inclined flow channels giving a relatively high surface area but with very low resistance to gas flow.

Two of the seven non- isomorphic solutions to this problem can be stated in terms of structures in the Fano 3-space, PG(3,2), known as packings. A spread of a projective space is a partition of its points into disjoint lines, and a packing is a partition of the lines into disjoint spreads. In PG(3,2), a spread would be a partition of the 15 points into 5 disjoint lines (with 3 points on each line), thus corresponding to the arrangement of schoolgirls on a particular day. A packing of PG(3,2) consists of seven disjoint spreads and so corresponds to a full week of arrangements.

In analytical chromatography, the goal is to separate and uniquely identify each of the compounds in a substance. Alternatively, preparative scale chromatography is a method of purification of large batches of material in a production environment. The basic methods of separation in HPLC rely on a mobile phase (water, organic solvents, etc.) being passed through a stationary phase (particulate silica packings, monoliths, etc.) in a closed environment (column); the differences in reactivity among the solvent of interest and the mobile and stationary phases distinguish compounds from one another in a series of adsorption and desorption phenomena. The results are then visually displayed in a resulting chromatogram.

Experiment shows that dropping the spheres in randomly will achieve a density of around 65% . However, a higher density can be achieved by carefully arranging the spheres as follows. Start with a layer of spheres in a hexagonal lattice, then put the next layer of spheres in the lowest points you can find above the first layer, and so on. At each step there are two choices of where to put the next layer, so this natural method of stacking the spheres creates an uncountably infinite number of equally dense packings, the best known of which are called cubic close packing and hexagonal close packing.

She chose to run away from home to her grandfather and enlisted the aid of Poggwydd to hide some of the packings she would need. Unfortunately, it also resulted in her becoming stuck with the Gnome as a travelling companion. In addition, she was joined by the mysterious cat Edgewood Dirk who seemed to be able to come and go as it pleases, and refused to talk or appear other than an ordinary cat except when alone with her. When Misty arrived at her grandfather's domain of the lake country, he allowed her to stay but refused to take her side against her father.

Many of the cross-sections of the Leech lattice, including the Coxeter-Todd lattice and Barnes-Wall lattice, in 12 and 16 dimensions, were found much earlier than the Leech lattice. discovered a related odd unimodular lattice in 24 dimensions, now called the odd Leech lattice, one of whose two even neighbors is the Leech lattice. The Leech lattice was discovered in 1965 by , by improving some earlier sphere packings he found . calculated the order of the automorphism group of the Leech lattice, and, working with John G. Thompson, discovered three new sporadic groups as a by-product: the Conway groups, Co1, Co2, Co3.

While the circle has a relatively low maximum packing density of 0.9069 on the Euclidean plane, it does not have the lowest possible, even among centrally-symmetric convex shapes. The "worst" such shape to pack onto a plane has not been determined, 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. (Packing densities of concave shapes such as star polygons can be arbitrarily small.) The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.

The red points form part of an ε-net for the Euclidean plane, where ε is the radius of the large yellow disks. The blue disks of half the radius are disjoint, and the yellow disks together cover the whole plane, satisfying the two definitional requirements on an ε-net. In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals.

The reason of the attack is because that place had goods suitable for their country and for China, such as ivory, tortoise shells, panther skins, and ambergris, and also because they wanted black slaves from Bantu people (called Zeng or Zenj by Arabs, Jenggi by Javanese) who were strong and make good slaves. According to Waharu IV inscription (931 AD) and Garaman insription (1053 AD),Nastiti (2003), in Ani Triastanti, 2007, p. 39.Nastiti (2003), in Ani Triastanti, 2007, p. 34. the Medang kingdom and Airlangga's era Kahuripan kingdom (1000-1049 AD) of Java experienced a long prosperity so that it needed a lot of manpower, especially to bring crops, packings, and send them to ports.

The reason of the attack is because that place had goods suitable for their country and for China, such as ivory, tortoise shells, panther skins, and ambergris, and also because they wanted black slaves from Bantu people (called Zeng or Zenj by Malay, Jenggi by Javanese) who were strong and make good slaves. According to Waharu IV inscription (931 AD) and Garaman insription (1053 AD),Nastiti (2003), in Ani Triastanti, 2007, p. 39.Nastiti (2003), in Ani Triastanti, 2007, p. 34. the Medang kingdom and Airlangga's era Kahuripan kingdom (1000-1049 AD) of Java experienced a long prosperity so that it needed a lot of manpower, especially to bring crops, packings, and send them to ports.

Circle packings, as studied in this book, are systems of circles that touch at tangent points but do not overlap, according to a combinatorial pattern of adjacencies specifying which pairs of circles should touch. The circle packing theorem states that a circle packing exists if and only if the pattern of adjacencies forms a planar graph; it was originally proved by Paul Koebe in the 1930s, and popularized by William Thurston, who rediscovered it in the 1970s and connected it with the theory of conformal maps and conformal geometry. As a topic, this should be distinguished from sphere packing, which considers higher dimensions (here, everything is two dimensional) and is more focused on packing density than on combinatorial patterns of tangency. The book is divided into four parts, in progressive levels of difficulty.

In this case, different extensions of this pattern to larger maximal planar graphs will lead to different packings, which can be mapped to each other by corresponding circles. The book explores the connection between these mappings, which it calls discrete analytic functions, and the analytic functions of classical mathematical analysis. The final part of the book concerns a conjecture of William Thurston, proved by Burton Rodin and Dennis Sullivan, that makes this analogy concrete: conformal mappings from any topological disk to a circle can be approximated by filling the disk by a hexagonal packing of unit circles, finding a circle packing that adds to that pattern of adjacencies a single outer circle, and constructing the resulting discrete analytic function. This part also includes applications to number theory and the visualization of brain structure.

The corners of the smoothed octagon can be found by rotating three regular octagons whose centres form a triangle with constant area. By considering the family of maximally dense packings of the smoothed octagon, the requirement that the packing density remain the same as the point of contact between neighbouring octagons changes can be used to determine the shape of the corners. In the figure, three octagons rotate while the area of the triangle formed by their centres remains constant, keeping them packed together as closely as possible. For regular octagons, the red and blue shapes would overlap, so to enable the rotation to proceed the corners are clipped by a point that lies halfway between their centres, generating the required curve, which turns out to be a hyperbola.

The first part introduces the subject visually, encouraging the reader to think about packings not just as static objects but as dynamic systems of circles that change in predictable ways when the conditions under which they are formed (their patterns of adjacency) change. The second part concerns the proof of the circle packing theorem itself, and of the associated rigidity theorem: every maximal planar graph can be associated with a circle packing that is unique up to Möbius transformations of the plane. More generally the same result holds for any triangulated manifold, with a circle packing on a topologically equivalent Riemann surface that is unique up to conformal equivalence. The third part of the book concerns the degrees of freedom that arise when the pattern of adjacencies is not fully triangulated (it is a planar graph, but not a maximal planar graph).

Each of these steps may be performed with simple trigonometric calculations, and as Collins and Stephenson argue, the system of radii converges rapidly to a unique fixed point for which all covering angles are exactly 2π. Once the system has converged, the circles may be placed one at a time, at each step using the positions and radii of two neighboring circles to determine the center of each successive circle. describes a similar iterative technique for finding simultaneous packings of a polyhedral graph and its dual, in which the dual circles are at right angles to the primal circles. He proves that the method takes time polynomial in the number of circles and in log 1/ε, where ε is a bound on the distance of the centers and radii of the computed packing from those in an optimal packing.

However, the space of realizations of locally-square spiral packings is infinite-dimensional, unlike the Doyle spirals which can be determined only by a constant number of parameters. It is also possible to describe spiraling systems of overlapping circles that cover the plane, rather than non-crossing circles that pack the plane, with each point of the plane covered by at most two circles except for points where three circles meet at 60^\circ angles, and with each circle surrounded by six others. These have many properties in common with the Doyle spirals. The Doyle spiral, in which the circle centers lie on logarithmic spirals and their radii increase geometrically in proportion to their distance from the central limit point, should be distinguished from a different spiral pattern of disjoint but non-tangent unit circles, also resembling certain forms of plant growth such as the seed heads of sunflowers.

Expanded graphite is made by immersing natural flake graphite in a bath of chromic acid, then concentrated sulfuric acid, which forces the crystal lattice planes apart, thus expanding the graphite. The expanded graphite can be used to make graphite foil or used directly as "hot top" compound to insulate molten metal in a ladle or red-hot steel ingots and decrease heat loss, or as firestops fitted around a fire door or in sheet metal collars surrounding plastic pipe (during a fire, the graphite expands and chars to resist fire penetration and spread), or to make high-performance gasket material for high-temperature use. After being made into graphite foil, the foil is machined and assembled into the bipolar plates in fuel cells. The foil is made into heat sinks for laptop computers which keeps them cool while saving weight, and is made into a foil laminate that can be used in valve packings or made into gaskets.

The circle packing theorem implies that every polyhedral graph can be represented as the graph of a polyhedron that has a midsphere. A stronger form of the circle packing theorem asserts that any polyhedral graph and its dual graph can be represented by two circle packings, such that the two tangent circles representing a primal graph edge and the two tangent circles representing the dual of the same edge always have their tangencies at right angles to each other at the same point of the plane. A packing of this type can be used to construct a convex polyhedron that represents the given graph and that has a midsphere, a sphere tangent to all of the edges of the polyhedron. Conversely, if a polyhedron has a midsphere, then the circles formed by the intersections of the sphere with the polyhedron faces and the circles formed by the horizons on the sphere as viewed from each polyhedron vertex form a dual packing of this type.

Modern combinatorics reveals that the number of ways to place the pieces of the Suter board to reform their square, allowing them to be turned over, is 17,152; the number is considerably smaller – 64 – if pieces are not allowed to be turned over. The sharpness of some angles in the Suter board makes fabrication difficult, while play could be awkward if pieces with sharp points are turned over. For the Codex board (again as with Tangram) there are three ways to pack the pieces: as two unit squares side by side; as two unit squares one on top of the other; and as a single square of side the square root of two. But the key to these packings is forming isosceles right triangles, just as Socrates gets the slave boy to consider in Plato's Meno – Socrates was arguing for knowledge by recollection, and here pattern recognition and memory seem more pertinent than a count of solutions.

This shape was called a Siamese dodecahedron in the paper by Hans Freudenthal and B. L. van der Waerden (1947) which first described the set of eight convex deltahedra.. The dodecadeltahedron name was given to the same shape by , referring to the fact that it is a 12-sided deltahedron. There are other simplicial dodecahedra, such as the hexagonal bipyramid, but this is the only one that can be realized with equilateral faces. Bernal was interested in the shapes of holes left in irregular close-packed arrangements of spheres, so he used a restrictive definition of deltahedra, in which a deltahedron is a convex polyhedron with triangular faces that can be formed by the centers of a collection of congruent spheres, whose tangencies represent polyhedron edges, and such that there is no room to pack another sphere inside the cage created by this system of spheres. This restrictive definition disallows the triangular bipyramid (as forming two tetrahedral holes rather than a single hole), pentagonal bipyramid (because the spheres for its apexes interpenetrate, so it cannot occur in sphere packings), and icosahedron (because it has interior room for another sphere).