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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
:<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]].


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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]]

Revision as of 21:18, 20 January 2014

In geometry, the moment curve is an algebraic curve in d-dimensional Euclidean space given by the set of points with Cartesian coordinates of the form

(x,x2,x3,,xd).[1]

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 polytopes, the no-three-in-line problem, and a geometric proof of the chromatic number of Kneser graphs.

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 linear position.[2]

Applications

The convex hull of any finite set of points on the moment curve is a cyclic polytope.[3] 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 polytopes, 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, Ω(nd/2), among all possible Delaunay triangulations of sets of n points in d dimensions.[4]

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 2d 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.[5]

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 2k + 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 graphs, a problem first solved in a different way by László Lovász.[6]

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, i2 mod p, i3 mod p).[7]

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.[8]

Notes

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References

Template:Refbegin

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    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

Template:Refend

  1. Template:Harvtxt, Definition 5.4.1, p. 97; Template:Harvtxt, Definition 1.6.3, p. 17.
  2. Template:Harvtxt, p. 100; Template:Harvtxt, Lemma 5.4.2, p. 97; Template:Harvtxt, Lemma 1.6.4, p. 17.
  3. Template:Harvtxt; Template:Harvtxt, p. 101; Template:Harvtxt, Lemma 5.4.2, p. 97.
  4. Template:Harvtxt.
  5. Template:Harvtxt, pp. 70–79; Template:Harvtxt, pp. 50–51.
  6. Template:Harvtxt, 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 Template:Harvtxt.
  7. Template:Harvtxt.
  8. Credited by Template:Harvtxt to Paul Erdős.