" As for the linkless Lit Shot's ubiquity on Twitter, he said, "We're delighted that people enjoy creating and sharing their own deal 'tombstones' with screenshots, and we're happy to have the focus remain on the author and their deal rather than on us as the reporting site.
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A graph is said to be intrinsically linked if, no matter how it is embedded, the embedding is always linked. Although linkless and flat embeddings are not the same, the graphs that have linkless embeddings are the same as the graphs that have flat embeddings..
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In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three- dimensional analogue of the planar graphs.. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa.
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However, integer-distance straight line embeddings are known to exist for cubic graphs.. raised the question of whether every graph with a linkless embedding in three-dimensional Euclidean space has a linkless embedding in which all edges are represented by straight line segments, analogously to Fáry's theorem for two-dimensional embeddings.
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The number of edges in an n-vertex linkless graph is at most 4n − 10: maximal apex graphs with n > 4 have exactly this many edges, and proved a matching upper bound on the more general class of K6-minor-free graphs. observed that Sachs' question about the chromatic number would be resolved by a proof of Hadwiger's conjecture that any k-chromatic graph has as a minor a k-vertex complete graph. The proof by of the case k = 6 of Hadwiger's conjecture is sufficient to settle Sachs' question: the linkless graphs can be colored with at most five colors, as any 6-chromatic graph contains a K6 minor and is not linkless, and there exist linkless graphs such as K5 that require five colors. The snark theorem implies that every cubic linklessly embeddable graph is 3-edge-colorable.
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Linkless embeddings started being studied within the algorithms research community in the late 1980s through the works of and . Algorithmically, the problem of recognizing linkless and flat embeddable graphs was settled once the forbidden minor characterization was proven: an algorithm of can be used to test in polynomial time whether a given graph contains any of the seven forbidden minors.The application of the Robertson–Seymour algorithm to this problem was noted by . This method does not construct linkless or flat embeddings when they exist, but an algorithm that does construct an embedding was developed by , and a more efficient linear time algorithm was found by .
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A graph has a linkless embedding if and only if it does not have one of the seven graphs of the Petersen family as a minor.
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The planar graphs and the apex graphs are linklessly embeddable, as are the graphs obtained by Y-Δ transforms from these graphs. The YΔY reducible graphs are the graphs that can be reduced to a single vertex by Y-Δ transforms, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices; they are also minor-closed, and include all planar graphs. However, there exist linkless graphs that are not YΔY reducible, such as the apex graph formed by connecting an apex vertex to every degree-three vertex of a rhombic dodecahedron.. There also exist linkless graphs that cannot be transformed into an apex graph by Y-Δ transforms, removal of isolated vertices and degree-one vertices, and compression of degree-two vertices: for instance, the ten-vertex crown graph has a linkless embedding, but cannot be transformed into an apex graph in this way. Related to the concept of linkless embedding is the concept of knotless embedding, an embedding of a graph in such a way that none of its simple cycles form a nontrivial knot.
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If a graph is planar, this is the only way to embed it flatly and linklessly into space: every flat embedding can be continuously deformed to lie on a flat plane. And conversely, every nonplanar linkless graph has multiple linkless embeddings. An apex graph. If the planar part of the graph is embedded on a flat plane in space, and the apex vertex is placed above the plane and connected to it by straight line segments, the resulting embedding is flat.
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In some cases, a graph may be embedded in space in such a way that, for each cycle in the graph, one can find a disk bounded by that cycle that does not cross any other feature of the graph. In this case, the cycle must be unlinked from all the other cycles disjoint from it in the graph. The embedding is said to be flat if every cycle bounds a disk in this way.. A similar definition of a "good embedding" appears in ; see also and . A flat embedding is necessarily linkless, but there may exist linkless embeddings that are not flat: for instance, if G is a graph formed by two disjoint cycles, and it is embedded to form the Whitehead link, then the embedding is linkless but not flat.
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The question of whether K6 has a linkless or flat embedding was posed within the topology research community in the early 1970s by . Linkless embeddings were brought to the attention of the graph theory community by , who posed several related problems including the problem of finding a forbidden graph characterization of the graphs with linkless and flat embeddings; Sachs showed that the seven graphs of the Petersen family (including K6) do not have such embeddings. As observed, linklessly embeddable graphs are closed under graph minors, from which it follows by the Robertson–Seymour theorem that a forbidden graph characterization exists. The proof of the existence of a finite set of obstruction graphs does not lead to an explicit description of this set of forbidden minors, but it follows from Sachs' results that the seven graphs of the Petersen family belong to the set.
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Each of the Petersen family graphs forms a minimal forbidden minor for the family of YΔY-reducible graphs.. However, Neil Robertson provided an example of an apex graph (a linkless embeddable graph formed by adding one vertex to a planar graph) that is not YΔY-reducible, showing that the YΔY- reducible graphs form a proper subclass of the linkless embeddable graphs and have additional forbidden minors. In fact, as Yaming Yu showed, there are at least 68,897,913,652 forbidden minors for the YΔY-reducible graphs beyond the seven of the Petersen family..
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Since every apex graph is linkless embeddable, this shows that there are graphs that are linkless embeddable but not YΔY-reducible and therefore that there are additional forbidden minors for the YΔY-reducible graphs. Robertson's apex graph is shown in the figure. It can be obtained by connecting an apex vertex to each of the degree-three vertices of a rhombic dodecahedron, or by merging two diametrally opposed vertices of a four-dimensional hypercube graph. Because the rhombic dodecahedron's graph is planar, Robertson's graph is an apex graph.
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However, Sachs was unable to prove that these were the only minimal linked graphs, and this was finally accomplished by . The forbidden minor characterization of linkless graphs leads to a polynomial time algorithm for their recognition, but not for actually constructing an embedding. described a linear time algorithm that tests whether a graph is linklessly embeddable and, if so, constructs a flat embedding of the graph. Their algorithm finds large planar subgraphs within the given graph such that, if a linkless embedding exists, it has to respect the planar embedding of the subgraph.
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The suspension systems of stand-up scooters range from simplistic spring based fork systems to the complicated, dampened cam-link and C.I.D.L.ICantilevered Independent Dynamic Linkless Indespension suspension mechanisms or a hybrid combination of wooden deck, coil spring, and dampers.
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Robertson's example of a non-YΔY-reducible apex graph. A connected graph is YΔY-reducible if it can be reduced to a single vertex by a sequence of steps, each of which is a Δ-Y or Y-Δ transform, the removal of a self-loop or multiple adjacency, the removal of a vertex with one neighbor, and the replacement of a vertex of degree two and its two neighboring edges by a single edge.. Like the apex graphs and the linkless embeddable graphs, the YΔY-reducible graphs are closed under graph minors. And, like the linkless embeddable graphs, the YΔY-reducible graphs have the seven graphs in the Petersen family as forbidden minors, prompting the question of whether these are the only forbidden minors and whether the YΔY-reducible graphs are the same as the linkless embeddable graphs. However, Neil Robertson provided an example of an apex graph that is not YΔY-reducible.
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This gradually changed to belt-fed types. Belts were either held in the open by the person, or in a bag or box. Some modern vehicle machine guns used linkless feed systems however. Modern machine guns are commonly mounted in one of four ways.
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Of note, HK recommends that you do not remove the right side slide release for normal maintenance. It is not fitted into a steel insert, only polymer, and repeated removal will loosen the tolerances, resulting in a “rattle” of that part against the weapon. The P2000 is a short recoil-operated locked breech pistol with a modified Browning-type linkless cam action with a vertically tilting barrel also employed in the USP series of self-loading pistols, as well as most modern linkless semi autos. The cold hammer-forged barrel has a polygonal profile while the slide is made from a solid bar of nitro-carburized steel.
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A linkless embedding is an embedding of the graph with the property that any two cycles are unlinked; a knotless embedding is an embedding of the graph with the property that any single cycle is unknotted. The graphs that have linkless embeddings have a forbidden graph characterization involving the Petersen family, a set of seven graphs that are intrinsically linked: no matter how they are embedded, some two cycles will be linked with each other.. A full characterization of the graphs with knotless embeddings is not known, but the complete graph is one of the minimal forbidden graphs for knotless embedding: no matter how is embedded, it will contain a cycle that forms a trefoil knot..
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The Bushmaster IV requires at 24 volts to operate, fires from a cook off safe open bolt with absolute hangfire protection, and ejects forwards. It fires NATO standard 40×365mm ammunition from an integral linkless feed and has a recoil force of . The ammunition was originally developed for the Bofors 40mm L/70 cannon.
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If a graph G has a linkless or flat embedding, then every minor of G (a graph formed by contraction of edges and deletion of edges and vertices) also has a linkless or flat embedding. Deletions cannot destroy the flatness of an embedding, and a contraction can be performed by leaving one endpoint of the contracted edge in place and rerouting all the edges incident to the other endpoint along the path of the contracted edge. Therefore, by the Robertson–Seymour theorem, the linklessly embeddable graphs have a forbidden graph characterization as the graphs that do not contain any of a finite set of minors. The set of forbidden minors for the linklessly embeddable graphs was identified by : the seven graphs of the Petersen family are all minor- minimal intrinsically linked graphs.
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The KTM 450 EXC is a 4-stroke enduro off-road motorcycle made by KTM. The models are the European road legal 400 EXC and 450 EXC, the US 450 (Close Ratio) XC and 450 (Wide Ratio) XC-W. The European EXC versions have a small headlight, speedometer, tail-light and somewhat softer linkless (PDS) rear suspension. The US EXC is 50-states street legal.
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An 88 by 10 inch (224 x 25 cm) 5.56mm Microgun Pod was also developed for external use on light aircraft and helicopters. The unit had an ammunition capacity of 1,500 to 3,500 rounds and a loaded weight of 186 to 300 pounds. The feeding system was linkless via a helical drum and had a nominal rate of fire of 6,000 rpm. Adjustments to 1,000 or 10,000 rpm were possible if desired.
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The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove. Apex graphs are closed under the operation of taking minors and play a role in several other aspects of graph minor theory: linkless embedding, Hadwiger's conjecture,. YΔY-reducible graphs, and relations between treewidth and graph diameter.
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The seven graphs in the Petersen family. No matter how these graphs are embedded into three-dimensional space, some two cycles will have nonzero linking number. In two dimensions, only the planar graphs may be embedded into the Euclidean plane without crossings, but in three dimensions, any undirected graph may be embedded into space without crossings. However, a spatial analogue of the planar graphs is provided by the graphs with linkless embeddings and knotless embeddings.
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The Tanfoglio Combat/Standard is offered in several variants in full sized, compact and subcompact pistol frame sizes. The steel surface finish can be military blued, polished "Wonder" finish, blue "Wonder" finish or hard chromed finish. The Tanfoglio Combat/Standard is a short recoil operated, locked breech semi-automatic pistol that uses a Browning Hi-Power style linkless system. All T95 Standard and Combat models have the capability of being fired single- and double- action.
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The complete graph K6, the Petersen graph, and the other five graphs in the Petersen family do not have linkless embeddings. Every graph minor of a linklessly embeddable graph is again linklessly embeddable, as is every graph that can be reached from a linklessly embeddable graph by a Y-Δ transform. The linklessly embeddable graphs have the Petersen family graphs as their forbidden minors,. and include the planar graphs and apex graphs.
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These problems were finally settled by ,As previously announced by . who showed that the seven graphs of the Petersen family are the only minimal forbidden minors for these graphs. Therefore, linklessly embeddable graphs and flat embeddable graphs are both the same set of graphs, and are both the same as the graphs that have no Petersen family minor. also asked for bounds on the number of edges and the chromatic number of linkless embeddable graphs.
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The graphs that do not have knotless embeddings (that is, they are intrinsically knotted) include K7 and K3,3,1,1.; . However, there also exist minimal forbidden minors for knotless embedding that are not formed (as these two graphs are) by adding one vertex to an intrinsically linked graph.. One may also define graph families by the presence or absence of more complex knots and links in their embeddings,; . or by linkless embedding in three-dimensional manifolds other than Euclidean space.
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The M230 Chain Gun is the "Area Weapon System" on the Boeing AH-64 Apache attack helicopter and is also used on the MH-60L Direct Action Penetrator (DAP). The M230 is mounted on the chin turret. It uses a electric motor to fire linkless ammunition at a rate of 625 (±25) rounds per minute. The practical rate of fire is about 300 rounds per minute with a 10-minute cooling period as the gun is air cooled.
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Improvements include a one-inch longer barrel, integral mount to increase first round hit probability by up to 10 percent, and dual recoil system to enhance accuracy and cope with future hotter propellants. It can have a next-round select, linkless feed system. The 30 mm chain gun can fire Mk310 Programmable Air Burst Munition rounds to attack targets in defilade. The United States Army Research, Development and Engineering Command helped enhance the XM813 mainly for safety and turret integration.
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The international EXC versions have a small headlight, speedometer, tail-light and somewhat softer linkless (PDS) rear suspension. 2007 was the first year the EXC was deemed street legal in the US. As of 2008 the US version of the EXC, with the new XC-4 motor, continued to be sold in street legal trim. In 2012 the US 50-states street legal 530 EXC-R was rebadged the 500 EXC. In mid 2016 the 2017 bikes were introduced adding a 250cc version.
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The Petersen family, the obstruction set for linkless embedding. Some examples of finite obstruction sets were already known for specific classes of graphs before the Robertson–Seymour theorem was proved. For example, the obstruction for the set of all forests is the loop graph (or, if one restricts to simple graphs, the cycle with three vertices). This means that a graph is a forest if and only if none of its minors is the loop (or, the cycle with three vertices, respectively).
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The KTM 300 is a series of 2-stroke enduro off-road motorcycles made by KTM. They are the Penton-inspired 1984 300 GS, the 300 DMX, the dirt only 300 MXC, European road legal 300 EXC and their current US counterpart the 300 XC (close-ratio) and 300 XC-W (wide-ratio gear box) respectively. The KTM 300 is designed for difficult off-road conditions. The 300 EXC and the 300 XC-W have a small headlight, speedometer, tail-light and somewhat softer linkless PDS suspension.
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In 1972, Hughes Helicopters began a company-funded research effort to design a single machine gun to fire the U.S. Army's M50 round.Richardson & Peacock, 1992, pp. 38–40. By April 1973, the program had fired test rounds in more powerful WECOM linked ammunition, from a prototype (A model). In January 1975 a model "C" was added, a linkless version for the proposed Advanced Attack Helicopter YAH-64; the helicopter was eventually adopted as the AH-64 Apache, with the model C as standard armament.
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The USMC and the manufacturer are aware of the problem and in 2011 a linkless feed system developed by Meggit Defense Systems was incorporated into the AH-1W and the Zulus. The system is capable of holding 650 +/-3 rounds in the storage unit with approximately 40 rounds in the feed chute. The M197 is also mounted as a chin turret in Italy's Agusta A129 Mangusta CBT helicopter, with a capacity of 500 rounds. The XM301 was an improved M197 intended to be the gun on the later canceled RAH-66 Comanche.
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Like the original Gatling gun, the entire firing cycle is actuated by cams and powered by the rotation of the barrels. The seven-barrel carriage assembly itself is driven by the aircraft's dual hydraulic system. The GAU-8/A ammunition feed is linkless, reducing weight and avoiding a great deal of potential for jamming. The feed system is double-ended, allowing the spent casings to be recycled back into the ammunition drum, instead of ejected from the aircraft, which would require considerable force to eliminate potential airframe damage.
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The Typhoon also carries a specially developed variant of the Mauser BK-27 27 mm cannon that was developed originally for the Panavia Tornado. This is a single- barrel, electrically fired, gas-operated revolver cannon with a new linkless feed system is located in the starboard wing root, and is capable of firing up to 1700 rounds per minute. There was a proposal on cost grounds in 1999 to limit UK gun-armament fit to the first 53 batch-1 aircraft and not used operationally, but this decision was reversed in 2006.
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The Petersen family. As showed, each of the seven graphs of the Petersen family is intrinsically linked: no matter how each of these graphs is embedded in space, they have two cycles that are linked to each other. These graphs include the complete graph K6, the Petersen graph, the graph formed by removing an edge from the complete bipartite graph K4,4, and the complete tripartite graph K3,3,1. Every planar graph has a flat and linkless embedding: simply embed the graph into a plane and embed the plane into space.
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An apex graph, formed by adding a single vertex to a planar graph, also has a flat and linkless embedding: embed the planar part of the graph on a plane, place the apex above the plane, and draw the edges from the apex to its neighbors as line segments. Any closed curve within the plane bounds a disk below the plane that does not pass through any other graph feature, and any closed curve through the apex bounds a disk above the plane that does not pass through any other graph feature. If a graph has a linkless or flat embedding, then modifying the graph by subdividing or unsubdividing its edges, adding or removing multiple edges between the same pair of points, and performing Y-Δ transforms that replace a degree-three vertex by a triangle connecting its three neighbors or the reverse all preserve flatness and linklessness. In particular, in a cubic planar graph (one in which all vertices have exactly three neighbors, such as the cube) it is possible to make duplicates of any independent set of vertices by performing Y-Δ transforms, adding multiple copies of the resulting triangle edges, and then performing the reverse Δ-Y transforms.
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The PL-15 is a 9mm short recoil-operated, locked-breech handgun that uses a modified Browning cam-lock system adapted from the Hi-Power pistol. The firearm's locking mechanism uses a linkless, vertically tilting barrel with a rectangular breech that locks into the ejection port cut-out in the slide. It features a double-action only (DAO) trigger, requiring a long and heavy trigger pull that puts the hammer into a cocked state and then releases it with every shot. The trigger pull weight is 4 kg, and the full length of the trigger pull is 7 mm.
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The CZ 75 is a short recoil operated, locked breech pistol. It uses the Browning linkless cam locking system similar to that used in the Browning Hi-Power pistol, where the barrel and slide are locked together on firing, using locking lugs milled into the barrel mating with recesses in the roof of the slide. An enclosed cam track integral with the barrel is actuated by the slide release lever's transverse pin. After the first few millimetres of the recoil stroke, the barrel is cammed downwards at the rear, enabling the slide to continue the recoil stroke and eject the spent cartridge.
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By repeatedly simplifying the graph whenever such a subgraph is found, they reduce the problem to one in which the remaining graph has bounded treewidth, at which point it can be solved by dynamic programming. The problem of efficiently testing whether a given embedding is flat or linkless was posed by . It remains unsolved, and is equivalent in complexity to unknotting problem, the problem of testing whether a single curve in space is unknotted. Testing unknottedness (and therefore, also, testing linklessness of an embedding) is known to be in NP but is not known to be NP-complete..
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The Colin de Verdière graph invariant is an integer defined for any graph using algebraic graph theory. The graphs with Colin de Verdière graph invariant at most μ, for any fixed constant μ, form a minor-closed family, and the first few of these are well-known: the graphs with μ ≤ 1 are the linear forests (disjoint unions of paths), the graphs with μ ≤ 2 are the outerplanar graphs, and the graphs with μ ≤ 3 are the planar graphs. As conjectured and proved, the graphs with μ ≤ 4 are exactly the linklessly embeddable graphs. A linkless apex graph that is not YΔY reducible.
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It was observed that in fact several edges may be drawn in the same "page"; the book thickness of the graph is the minimum number of halfplanes needed for such a drawing. Alternatively, any finite graph can be drawn with straight-line edges in three dimensions without crossings by placing its vertices in general position so that no four are coplanar. For instance, this may be achieved by placing the ith vertex at the point (i,i2,i3) of the moment curve. An embedding of a graph into three- dimensional space in which no two of the cycles are topologically linked is called a linkless embedding.
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A very rare "first-generation" Glock 17 "Cutaway" pistol used for demonstrations and training with law enforcement agencies in the 1980s. The Glock 17 is a 9 mm short recoil–operated, locked-breech semi-automatic pistol that uses a modified Browning cam-lock system adapted from the Hi-Power pistol. The firearm's locking mechanism uses a linkless, vertically tilting barrel with a rectangular breech that locks into the ejection port cut-out in the slide. During the recoil stroke, the barrel moves rearward initially locked together with the slide about until the bullet leaves the barrel and chamber pressure drops to a safe level.
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The Dodge Rampage was a subcompact, unibody coupe utility based on Chrysler's L platform and manufactured from 1982 to 1984. First released as a 1982 model, the Rampage was later joined for 1983 by its rebadged variant, the Plymouth Scamp. The Rampage borrowed the car's unibody construction and the front fascia from the sporty 024/Charger variant, and used the suspension from the Omni/Horizon with coil struts and a linkless sway bar at the front, and leaf springs with shock absorbers at the rear. It was available with a Chrysler built and designed 2.2 L carbureted inline-four engine with and a curb weight of around .
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The Bren Ten is a short recoil operated, locked breech semi-automatic pistol that uses a Browning Hi-Power style linkless system. The pistol has the capability of being fired single- or double action and features a reversible frame-mounted combat thumb manual safety that locks the sear so the trigger cannot be moved rearward as well as an internal firing pin block safety which stops the firing pin from traveling forward. The manual safety allows the pistol to be carried with the hammer back, ready for use just by switching the safety off, a configuration known as condition one. The Bren Ten has adjustable iron sights with three dots for increased visibility.
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The XM813 Bushmaster is based on the Mk44 and is offered as an upgrade for M1126 Stryker and M2 Bradley vehicles, as well as having been a contender to be the primary armament of the GCV Infantry Fighting Vehicle. Improvements include a 2.54 cm (1 inch) longer barrel, integral mount to increase first round hit probability by up to 10%, a dual recoil system to enhance accuracy and cope with future hotter propellants, and a Meggitt linkless dual feed ammunition system. The 30 mm chain gun can fire Mk310 Programmable Air Burst Munition rounds to attack targets in defilade. The United States Army Research, Development and Engineering Command helped enhance the XM813 mainly for safety and turret integration.
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Most common are the John Browning tilting barrel designs based on either the swinging link and locking lugs as used in the M1911 pistol or the linkless cam design used in the Hi Power and CZ 75. Other designs are the locking block design found in the Walther P38 and Beretta 92, rollers in the MG42, or a rotating barrel used in the Beretta 8000 and others. An unusual variant is the toggle bolt design of the Borchardt C-93 and its descendant, the Luger pistol. While the short recoil design is most common in pistols, the very first short-recoil–operated firearm was also the first machine gun, the Maxim gun.
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Some weapons such as the M134 Minigun and related designs use a hybrid mechanism to strip rounds from disintegrating belts into a linkless feed system or a specialized delinker to allow for more reliable feeding at extreme rates of fire. Modern infantry machine guns often have feed systems allowing the use of linked ammunition as well as other forms of feed like from magazines or drums. In some instances—like the FN Minimi/M249 SAW and the IMI Negev—the feed system requires no modification to fire with either mechanism. Other designs—like the Heckler and Koch HK21-based designs or MG34—require the exchange of modular parts to allow belt or alternate feeding.
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The M197 went into service on later marks of the AH-1 Cobra, using either the M97 or A/A49E-7 armament subsystems, and was also fitted in a ventral turret on the U.S. Marine Corps YOV-10D Bronco NOGS. It is also the basis of the GPU-2/A gun pod, which incorporates the cannon, a battery and electric drive motor, and 300 rounds of linkless ammunition. In the Cobra, the weapon is supplied with a magazine of 700 linked rounds, with a total capacity including feeder system of 750 rounds. It has a cyclic rate of fire of 730+-50 rounds per minute. Standard practice is to fire the cannon in 30- to 50-round bursts.
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The Tanfoglio Force is offered in several variants in full sized, compact and subcompact pistol frame sizes. The Tanfoglio Force is a short recoil operated, locked breech semi-automatic pistol that uses a Browning Hi- Power style linkless system. Most models have the capability of being fired single- and double-action and feature an ambidextrous frame-mounted combat "switch" style manual safety that locks the sear so the trigger cannot be moved rearward as well as an internal firing pin block safety which stops the firing pin from travelling forward. The manual safety allows the pistol to be carried with the hammer back, ready for use just by switching the safety off, a configuration known as condition one.
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In 1972, Hughes Helicopters began a company-funded research effort to design a single machine gun to fire the U.S. Army's M50 round. By April 1973, the program had fired test rounds in more powerful WECOM linked ammunition, from a prototype (A model). In January 1975 a model "C" was added, a linkless version for the proposed Advanced Attack Helicopter YAH-64; the helicopter was later adopted as the Hughes Model 77/AH-64A Apache, with the model C (receiving the designation M230 chain gun) as standard armament. In 1976, Hughes Helicopters successfully patented the chain gun and it has since been further developed into several other system of different caliber sizes.
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First steps toward determining the computational complexity were undertaken in proving that the problem is in larger complexity classes, which contain the class P. By using normal surfaces to describe the Seifert surfaces of a given knot, showed that the unknotting problem is in the complexity class NP. claimed the weaker result that unknotting is in AM ∩ co- AM; however, later they retracted this claim.Mentioned as a "personal communication" in reference [15] of . In 2011, Greg Kuperberg proved that (assuming the generalized Riemann hypothesis) the unknotting problem is in co- NP, and in 2016, Marc Lackenby provided an unconditional proof of co-NP membership. The unknotting problem has the same computational complexity as testing whether an embedding of an undirected graph in Euclidean space is linkless.
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The basic armament of the F-104 was the M61 Vulcan autocannon. As the first aircraft to carry the weapon, testing of the Starfighter revealed issues with the initial version of the M61: the Gatling-mechanism cannon suffered problems with its linked ammunition, being prone to misfeed and presenting a foreign object damage (FOD) hazard as discarded links were occasionally sucked into the engine. A linkless ammunition feed system was developed for the upgraded M61A1 installed in the F-104C; the M61A1 has subsequently been used by a wide variety of American combat aircraft. Open weapons bay of a left The cannon, mounted in the lower part of the port fuselage, was fed by a 725-round drum behind the pilot's seat.
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The original Steyr M9 'Second generation' Steyr M9-A1 The Steyr M series of pistols employs the mechanically locked Browning short recoil method of operation with a linkless, vertically dropping barrel. The cold-hammer-forged conventional rifled barrel is locked to the slide by means of a single rectangular lug around the barrel chamber that recesses into the ejection port in the slide. When fired, the recoil impulse from the ignited cartridge drives the barrel and slide back, locked together until the bullet leaves the barrel and pressures drop to a safe level. A locking block integrated into the frame then engages a lug at the base of the chamber and drives the barrel downward, separating it from the slide and terminating any further rearward movement while the slide continues back in a straight line.
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The BK 27 (also BK27 or BK-27) (German acronym for Bordkanone, "on-board cannon") is a caliber revolver cannon manufactured by Mauser (now part of Rheinmetall) of Germany. It was developed in the late 1960s for the MRCA (Multi Role Combat Aircraft) program that ultimately became the Panavia Tornado. The BK 27 is a gas-operated cannon firing a new series of 27×145 mm cartridges with a typical projectile weight of 260 g (9.2 oz), and a total weight for the complete round of 516 g (1.14 lb). Most models use a linked feed system for the ammunition; however, the Eurofighter Typhoon makes use of a specially developed variant of the BK 27 that uses a linkless feed system instead, which is intended to improve reliability.
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The A-100 is a recoil-operated, locked breech design with improved Browning-style linkless locking. The lock up is similar to the Browning High Power or CZ-75 with locking lugs on the top of the barrel which mate with lugs in the slide (not with a single lug in the ejection port like a SIG P22X.) The pistol's safety features include a manual decocking lever, firing pin block, and hammer safety. The A-100 lacks a manual safety, but depends entirely on a deliberate pull of its trigger to fire it; in this way it is similar to the SIG P22X series of pistols. The double-action trigger pull is relatively long and heavy, thereby precluding the need for a safety as the probability of an inadvertent discharge is low.
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However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither nor the complete bipartite graph as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions. As part of the Petersen family, plays a similar role as one of the forbidden minors for linkless embedding.. In other words, and as Conway and Gordon proved, every embedding of into three-dimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any three- dimensional embedding of contains a Hamiltonian cycle that is embedded in space as a nontrivial knot.
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An embedding of a graph into three-dimensional space consists of a mapping from the vertices of the graph to points in space, and from the edges of the graph to curves in space, such that each endpoint of each edge is mapped to an endpoint of the corresponding curve, and such that the curves for two different edges do not intersect except at a common endpoint of the edges. Any finite graph has a finite (though perhaps exponential) number of distinct simple cycles, and if the graph is embedded into three-dimensional space then each of these cycles forms a simple closed curve. One may compute the linking number of each disjoint pair of curves formed in this way; if all pairs of cycles have zero linking number, the embedding is said to be linkless.; ; .
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Two capabilities not demonstrated were its linkless ammunition with airburst capabilities; airburst rounds increase lethality by enabling engagement of targets in defilade when they would otherwise only be suppressed by fire.30-mm weapon system on display - Army.mil, 16 September 2014 In early 2015, the U.S. Army approved an upgrade for 81 Stryker vehicles of a Stryker Brigade Combat Team deployed in Europe to be upgunned with the Mk44 30 mm Bushmaster cannon to increase their lethality against other light armored vehicles used by Russia in the theater.US Army: Strykers Need Bigger Gun to Fight Russia - Defensenews.com, 24 July 2015 The cannons will be installed by 2018, and may be the first step in adding the Bushmaster to the entire active fleet of about 1,000 Strykers;Firepower Upgrade Planned for GDLS Strykers - Defensenews.com, 14 October 2015 the XM813 variant will be used with the Strykers.
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The TA-50 version has a three- barrel cannon version of the M61 Vulcan mounted internally behind the cockpit, which fires linkless 20 mm ammunition. Wingtip rails can accommodate the AIM-9 Sidewinder missile, and a variety of additional weapons can be mounted to underwing hardpoints. Compatible air-to-surface weapons include the AGM-65 Maverick missile, Hydra 70 and LOGIR rocket launchers, CBU-58 and Mk-20 cluster bombs, and Mk-82, −83, and −84 general-purpose bombs. The FA-50 can be externally fitted with Rafael's Sky Shield or LIG Nex1's ALQ-200K ECM pods, Sniper or LITENING targeting pods, and Condor 2 reconnaissance pods to further improve the fighter's electronic warfare, reconnaissance, and targeting capabilities. Other improved weapon systems include SPICE multifunctional guidance kits, Textron CBU-97/105 Sensor Fuzed Weapon with WCMD tail kits, JDAM, JDAM-ER for more comprehensive air-to-ground operations, and AIM-120 missiles for BVR air-to-air operations.
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Robertson's irreducible apex graph, showing that the YΔY- reducible graphs have additional forbidden minors beyond those in the Petersen family. A minor of a graph G is another graph formed from G by contracting and removing edges. As the Robertson–Seymour theorem shows, many important families of graphs can be characterized by a finite set of forbidden minors: for instance, according to Wagner's theorem, the planar graphs are exactly the graphs that have neither the complete graph K5 nor the complete bipartite graph K3,3 as minors. Neil Robertson, Paul Seymour, and Robin Thomas used the Petersen family as part of a similar characterization of linkless embeddings of graphs, embeddings of a given graph into Euclidean space in such a way that every cycle in the graph is the boundary of a disk that is not crossed by any other part of the graph.. Horst Sachs had previously studied such embeddings, shown that the seven graphs of the Petersen family do not have such embeddings, and posed the question of characterizing the linklessly embeddable graphs by forbidden subgraphs.. Robertson et al.
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The linking number of two closed curves in three-dimensional space is a topological invariant of the curves: it is a number, defined from the curves in any of several equivalent ways, that does not change if the curves are moved continuously without passing through each other. The version of the linking number used for defining linkless embeddings of graphs is found by projecting the embedding onto the plane and counting the number of crossings of the projected embedding in which the first curve passes over the second one, modulo 2.. The projection must be "regular", meaning that no two vertices project to the same point, no vertex projects to the interior of an edge, and at every point of the projection where the projections of two edges intersect, they cross transversally; with this restriction, any two projections lead to the same linking number. The linking number of the unlink is zero, and therefore, if a pair of curves has nonzero linking number, the two curves must be linked. However, there are examples of curves that are linked but that have zero linking number, such as the Whitehead link.
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