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In [[geometry]], the '''moment curve''' is an [[algebraic curve]] in ''d''-dimensional [[Euclidean space]] given by the set of points with [[Cartesian coordinate]]s of the form
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The clip point is healthier for a multipurpose work or again woods knife exhibiting how much abuse this knife can take, as well as a distinguished place in Jax Teller's gear on the TELEVISION present Sons of Anarchy.<br><br>The guard is just as essential, if no more so, than the pommel, in protecting the consumer’s hand, The shape of the guard will hold the user’s hand from sliding ahead onto the blade during use. That is especially crucial in moist conditions, such as when fishing or skinning game, when the knife could also be lined in blood. 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wdrivers+a+wood%2Fmetal+file%2C+8%E2%80%B3+ruler%2C+Phillips+screwdriver%2C+and+1%2F4%E2%80%B3+hex+bit+driver.%0D%0A%0D%0AThe+most+exciting+point+about+the+Schrade+SCHF9+Extreme+Survival+Knife+is+the+steel+that+was+made+use+of+to+generate+the+blade.+A+quarter-inch+thick+chunk+of+1095+steel+is+a+steal+for+the+low+price+tag+that+this+knife+is+sold+for.+I+do+not+know+how+they+are+capable+to+present+such+high-high+quality+steel+for+such+a+reasonable+cost%2C+but+I+created+positive+to+load+up+on+a+number+of+of+these+knives.+It+is+not+normally+you+can+come+across+such+fantastic+steel+in+a+workhorse+of+a+blade+like+this+one.+A+Knife+for+Daily&create=Create anodized aluminum] handle. This makes it light-weight, but strong, resisting torque. The Trac-Tec inserts are a patented technique of offering texture in order that the knife doesn’t slip in your fingers, even when they're coated with water or blood.<br><br>Males have been carrying pocket knives for centuries. However with elevated security at the airport and different buildings, knives have been disappearing from men’s pockets. Yet these minor obstacles will not be ample cause to give up carrying a knife completely. The carrying of a pocket knife is a person tradition that should [http://en.wikipedia.org/wiki/Knife machete] be continued. Why a Man Should Carry a Pocket Knife These handsome folding pocket knives make nice items for that special sportsman or an accessory to have readily available always. Distinctive and sensible, these folding pocket knives are a should have for any sportsman. 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At Wild Bill, knives are our specialty and folks trust us to supply wholesale prices on fine quality objects. That's why we have created such a huge stock and packed it with tactical lock backs, straight razors and butterfly knives. Searching for a six-function camper instrument? We have now them, in addition to pocket knives and keychain and novelty knives. Rigorously form the salmon en croute, then flip over and rating the highest, kriss cross. I exploit the back of the knife for this as you don't want to chop by means of to the salmon.
:<math>\left( x, x^2, x^3, \dots, x^d \right).</math><ref>{{harvtxt|Matoušek|2002}}, Definition 5.4.1, p. 97; {{harvtxt|Matoušek|2003}}, Definition 1.6.3, p. 17.</ref>
In the [[Euclidean plane]], the moment curve is a [[parabola]], and in three-dimensional space it is a [[twisted cubic]]. Its closure in projective space is the [[rational normal curve]].
 
Moment curves have been used for several applications in [[discrete geometry]] including [[cyclic polytope]]s, the [[no-three-in-line problem]], and a geometric proof of the [[chromatic number]] of [[Kneser graph]]s.
 
==Properties==
Every [[hyperplane]] intersects the moment curve in a finite set of at most ''d'' points. If a hyperplane intersects the curve in exactly ''d'' points, then the curve crosses the hyperplane at each intersection point. Thus, every finite point set on the moment curve is in [[general position|general linear position]].<ref>{{harvtxt|Edelsbrunner|1987}}, p. 100; {{harvtxt|Matoušek|2002}}, Lemma 5.4.2, p. 97; {{harvtxt|Matoušek|2003}}, Lemma 1.6.4, p. 17.</ref>
 
==Applications==
The [[convex hull]] of any finite set of points on the moment curve is a [[cyclic polytope]].<ref>{{harvtxt|Gale|1963}}; {{harvtxt|Edelsbrunner|1987}}, p. 101; {{harvtxt|Matoušek|2002}}, Lemma 5.4.2, p. 97.</ref> Cyclic polytopes have the largest possible number of faces for a given number of vertices, and in dimensions four or more have the property that their edges form a [[complete graph]]. More strongly, they are [[neighborly polytope]]s, meaning that each set of at most ''d''/2 vertices of the polytope forms one of its faces. Sets of points on the moment curve also realize the maximum possible number of simplices, <math>\Omega(n^{\lceil d/2\rceil})</math>, among all possible [[Delaunay triangulation]]s of sets of ''n'' points in ''d'' dimensions.<ref>{{harvtxt|Amenta|Attali|Devillers|2007}}.</ref>
 
In the [[Euclidean plane]], it is possible to divide any area or measure into four equal subsets, using the [[ham sandwich theorem]]. Similarly but more complicatedly, any volume or measure in three dimensions may be partitioned into eight equal subsets by three planes. However, this result does not generalize to five or more dimensions, as the moment curve provides examples of sets that cannot be partitioned into 2<sup>''d''</sup> subsets by ''d'' hyperplanes. In particular, in five dimensions, sets of five hyperplanes can partition segments of the moment curve into at most 26 pieces. It is not known whether four-dimensional partitions into 16 equal subsets by four hyperplanes are always possible, but it is possible to partition 16 points on the four-dimensional moment curve into the 16 orthants of a set of four hyperplanes.<ref>{{harvtxt|Edelsbrunner|1987}}, pp. 70–79; {{harvtxt|Matoušek|2003}}, pp. 50–51.</ref>
 
A construction based on the moment curve can be used to prove a lemma of Gale, according to which, for any positive integers ''k'' and ''d'', it is possible to place 2''k''&nbsp;+&nbsp;''d'' points on a ''d''-dimensional sphere in such a way that every open hemisphere contains at least ''k'' points. This lemma, in turn, can be used to calculate the [[chromatic number]] of the [[Kneser graph]]s, a problem first solved in a different way by [[László Lovász]].<ref>{{harvtxt|Matoušek|2003}}, Section 3.5, Gale's Lemma and Schrijver's Theorem, pp. 64–67. The use of Gale's lemma for the coloring problem is due to {{harvtxt|Bárány|1978}}.</ref>
 
The moment curve has also been used in [[graph drawing]], to show that all ''n''-vertex graphs may be drawn with their vertices in a three-dimensional integer grid of side length O(''n'') and with no two edges crossing. The main idea is to choose a prime number ''p'' larger than ''n'' and to place vertex ''i'' of the graph at coordinates
:(''i'', ''i''<sup>2</sup>&nbsp;mod&nbsp;''p'', ''i''<sup>3</sup>&nbsp;mod&nbsp;''p'').<ref>{{harvtxt|Cohen|Eades|Lin|Ruskey|1997}}.</ref>
A similar construction using the moment curve modulo a prime number, but in two dimensions rather than three, provides a linear bound for the [[no-three-in-line problem]].<ref>Credited by {{harvtxt|Roth|1951}} to [[Paul Erdős]].</ref>
 
==Notes==
{{reflist|colwidth=30em}}
 
==References==
{{refbegin|colwidth=30em}}
*{{citation
| last1 = Amenta | first1 = Nina
| last2 = Attali | first2 = Dominique
| last3 = Devillers | first3 = Olivier
| contribution = Complexity of Delaunay triangulation for points on lower-dimensional polyhedra
| location = New York
| mr = 2485262
| pages = 1106–1113
| publisher = ACM
| title = Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms
| year = 2007}}.
*{{citation
| last = Bárány | first = I. | authorlink = Imre Bárány
| doi = 10.1016/0097-3165(78)90023-7
| issue = 3
| journal = [[Journal of Combinatorial Theory]] | series = Series A
| mr = 514626
| pages = 325–326
| title = A short proof of Kneser's conjecture
| volume = 25
| year = 1978}}.
*{{citation
| last1 = Cohen | first1 = R. F.
| last2 = Eades | first2 = P. | author2-link = Peter Eades
| last3 = Lin | first3 = Tao
| last4 = Ruskey | first4 = F. | author4-link = Frank Ruskey
| doi = 10.1007/BF02522826
| issue = 2
| journal = [[Algorithmica]]
| mr = 1425733
| pages = 199–208
| title = Three-dimensional graph drawing
| volume = 17
| year = 1997}}.
*{{citation
| last = Edelsbrunner | first = Herbert | authorlink = Herbert Edelsbrunner
| isbn = 3-540-13722-X
| location = Berlin
| mr = 904271
| publisher = Springer-Verlag
| series = EATCS Monographs on Theoretical Computer Science
| title = Algorithms in Combinatorial Geometry
| volume = 10
| year = 1987}}.
*{{citation
| last = Gale | first = David | author-link = David Gale
| contribution = Neighborly and cyclic polytopes
| editor-last = Klee | editor-first = Victor | editor-link = Victor Klee
| location = Providence, R.I.
| mr = 0152944
| pages = 225–232
| publisher = American Mathematical Society
| series = Symposia in Pure Mathematics
| title = Convexity, Seattle, 1961
| volume = 7
| year = 1963}}.
*{{citation
| last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)
| isbn = 978-0-387-95373-1
| publisher = Springer-Verlag
| series = [[Graduate Texts in Mathematics]]
| title = Lectures on Discrete Geometry
| volume = 212
| year = 2002}}.
*{{citation
| last = Matoušek | first = Jiří | author-link = Jiří Matoušek (mathematician)
| isbn = 978-3-540-00362-5
| publisher = Springer
| series = Universitext
| title = Using the Borsuk-Ulam Theorem: Lectures on Topological Methods in Combinatorics and Geometry
| year = 2003}}.
*{{citation
| last = Roth | first = K. F. | authorlink = Klaus Roth
| doi = 10.1112/jlms/s1-26.3.198
| issue = 3
| journal = [[Journal of the London Mathematical Society]]
| pages = 198–204
| title = On a problem of Heilbronn
| volume = 26
| year = 1951}}.
{{refend}}
 
[[Category:Algebraic curves]]

Latest revision as of 22:40, 9 December 2014

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Ken Onion is without doubt one of the greatest knife makers in the USA. His knife could be present in private collections in addition to in pocket of wanted clients. Ken is concerned within the phases from design to manufacturing so that each knife meets his expectations. Being a marine corp and data about knife & it’s manufacturing keeps him ahead of all other corporations. Kershaw leek series of knives are very talked-about as of late and Kershaw Ken Onion Leek Folding Knife is certainly one of them. It has speedSafe ambidextrous assisted opening system. There are lots of deal with styles which is on the market in leek series.

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The guard is just as essential, if no more so, than the pommel, in protecting the consumer’s hand, The shape of the guard will hold the user’s hand from sliding ahead onto the blade during use. That is especially crucial in moist conditions, such as when fishing or skinning game, when the knife could also be lined in blood. Blade The Kershaw Blur Knife is a folding knife that has an anodized aluminum handle. This makes it light-weight, but strong, resisting torque. The Trac-Tec inserts are a patented technique of offering texture in order that the knife doesn’t slip in your fingers, even when they're coated with water or blood.

Males have been carrying pocket knives for centuries. However with elevated security at the airport and different buildings, knives have been disappearing from men’s pockets. Yet these minor obstacles will not be ample cause to give up carrying a knife completely. The carrying of a pocket knife is a person tradition that should machete be continued. Why a Man Should Carry a Pocket Knife These handsome folding pocket knives make nice items for that special sportsman or an accessory to have readily available always. Distinctive and sensible, these folding pocket knives are a should have for any sportsman. From collectible heirlooms to sensible basics, we've got the perfect folding knife.

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As knife connoisseurs, we know that not all tactical folding knives are created equal. At Wild Bill, knives are our specialty and folks trust us to supply wholesale prices on fine quality objects. That's why we have created such a huge stock and packed it with tactical lock backs, straight razors and butterfly knives. Searching for a six-function camper instrument? We have now them, in addition to pocket knives and keychain and novelty knives. Rigorously form the salmon en croute, then flip over and rating the highest, kriss cross. I exploit the back of the knife for this as you don't want to chop by means of to the salmon.