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In mathematics, a '''Meyer set''' or '''almost lattice''' is a set [[Delone set|relatively dense]] ''X'' of points in the [[Euclidean plane]] or a higher-dimensional [[Euclidean space]] such that its [[Minkowski sum|Minkowski difference]] with itself is [[Delone set|uniformly discrete]]. Meyer sets have several equivalent characterizations; they are named after [[Yves Meyer]], who studied them as a mathematical model for [[quasicrystal]]s.<ref name="moody">{{citation | |||
| last = Moody | first = Robert V. | authorlink = Robert Moody | |||
| contribution = Meyer sets and their duals | |||
| location = Dordrecht | |||
| mr = 1460032 | |||
| pages = 403–441 | |||
| publisher = Kluwer Academic Publishers | |||
| series = NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences | |||
| title = The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995) | |||
| url = http://www.math.ualberta.ca/~rvmoody/psFiles/moody.ps | |||
| volume = 489 | |||
| year = 1997}}.</ref><ref>{{citation | |||
| last = Lagarias | first = J. C. | authorlink = Jeffrey Lagarias | |||
| issue = 2 | |||
| journal = Communications in Mathematical Physics | |||
| mr = 1400744 | |||
| pages = 365–376 | |||
| title = Meyer's concept of quasicrystal and quasiregular sets | |||
| url = http://projecteuclid.org/getRecord?id=euclid.cmp/1104286996 | |||
| volume = 179 | |||
| year = 1996}}.</ref> | |||
==Definition and characterizations== | |||
A subset ''X'' of a [[metric space]] is relatively dense if there exists a number ''r'' such that all points of ''X'' are within distance ''r'' of ''X'', and it is uniformly discrete if there exists a number ''ε'' such that no two points of ''X'' are within distance ''ε'' of each other. A set that is both relatively dense and uniformly discrete is called a [[Delone set]]. When ''X'' is a subset of a [[vector space]], its [[Minkowski sum|Minkowski difference]] ''X'' − ''X'' is the set {''x'' − ''y'' | ''x'', ''y'' in ''X''} of differences of pairs of elements of ''X''.<ref>Moody gives different definitions for relative density and uniform discreteness, specialized to locally compact groups, but remarks that these definitions coincide with the usual ones for real vector spaces.</ref> | |||
With these definitions, a Meyer set may be defined as a relatively dense set ''X'' for which ''X'' − ''X'' is uniformly discrete. Equivalently, it is a Delone set for which ''X'' − ''X'' is Delone,<ref name="moody"/> or a Delone set ''X'' for which there exists a finite set ''F'' with ''X'' − ''X'' ⊂ 'X'' + ''F''<ref name="m7">{{harvtxt|Moody|1997}}, Section 7.</ref> | |||
Some additional equivalent characterizations involve the set | |||
:<math>X^\epsilon = \{y\mid |x\cdot y - 1| \le\varepsilon \text{ for all } x\in X\}.</math> | |||
defined for a given ''X'' and ''ε'', and approximating (as ''ε'' approaches zero) the definition of the [[reciprocal lattice]] of a [[Lattice (group)|lattice]]. A relatively dense set ''X'' is a Meyer set if and only if | |||
* For all ''ε'' > 0, ''X''<sup>''ε''</sup> is relatively dense, or equivalently | |||
* There exists an ''ε'' with 0 < ''ε'' < 1/2 for which ''X''<sup>''ε''</sup> is relatively dense.<ref name="moody"/> | |||
A [[Character (mathematics)|character]] of an additively closed subset of a vector space is a function that maps the set to the unit circle in the plane of [[complex number]]s, such that the sum of any two elements is mapped to the product of their images. A set ''X'' is a [[harmonious set]] if, for every character ''χ'' on the additive closure of ''X'' and every ''ε'' > 0, there exists a continuous character on the whole space that ''ε''-approximates ''χ''. Then a relatively dense set ''X'' is a Meyer set if and only if it is harmonious.<ref name="moody"/> | |||
==Examples== | |||
Meyer sets include | |||
*The points of any [[lattice (group)|lattice]] | |||
*The vertices of any rhombic [[Penrose tiling]]<ref>{{harvtxt|Moody|1997}}, Section 3.2.</ref> | |||
*The [[Minkowski sum]] of another Meyer set with any nonempty [[finite set]]<ref name="m7"/> | |||
*Any relatively dense subset of another Meyer set<ref>{{harvtxt|Moody|1997}}, Corollary 6.7.</ref> | |||
==References== | |||
{{reflist}} | |||
[[Category:Metric geometry]] | |||
[[Category:Crystallography]] | |||
[[Category:Lattice points]] | |||
Revision as of 16:38, 15 January 2014
In mathematics, a Meyer set or almost lattice is a set relatively dense X of points in the Euclidean plane or a higher-dimensional Euclidean space such that its Minkowski difference with itself is uniformly discrete. Meyer sets have several equivalent characterizations; they are named after Yves Meyer, who studied them as a mathematical model for quasicrystals.[1][2]
Definition and characterizations
A subset X of a metric space is relatively dense if there exists a number r such that all points of X are within distance r of X, and it is uniformly discrete if there exists a number ε such that no two points of X are within distance ε of each other. A set that is both relatively dense and uniformly discrete is called a Delone set. When X is a subset of a vector space, its Minkowski difference X − X is the set {x − y | x, y in X} of differences of pairs of elements of X.[3]
With these definitions, a Meyer set may be defined as a relatively dense set X for which X − X is uniformly discrete. Equivalently, it is a Delone set for which X − X is Delone,[1] or a Delone set X for which there exists a finite set F with X − X ⊂ 'X + F[4]
Some additional equivalent characterizations involve the set
defined for a given X and ε, and approximating (as ε approaches zero) the definition of the reciprocal lattice of a lattice. A relatively dense set X is a Meyer set if and only if
- For all ε > 0, Xε is relatively dense, or equivalently
- There exists an ε with 0 < ε < 1/2 for which Xε is relatively dense.[1]
A character of an additively closed subset of a vector space is a function that maps the set to the unit circle in the plane of complex numbers, such that the sum of any two elements is mapped to the product of their images. A set X is a harmonious set if, for every character χ on the additive closure of X and every ε > 0, there exists a continuous character on the whole space that ε-approximates χ. Then a relatively dense set X is a Meyer set if and only if it is harmonious.[1]
Examples
Meyer sets include
- The points of any lattice
- The vertices of any rhombic Penrose tiling[5]
- The Minkowski sum of another Meyer set with any nonempty finite set[4]
- Any relatively dense subset of another Meyer set[6]
References
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- ↑ 1.0 1.1 1.2 1.3 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.
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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. - ↑ Moody gives different definitions for relative density and uniform discreteness, specialized to locally compact groups, but remarks that these definitions coincide with the usual ones for real vector spaces.
- ↑ 4.0 4.1 Template:Harvtxt, Section 7.
- ↑ Template:Harvtxt, Section 3.2.
- ↑ Template:Harvtxt, Corollary 6.7.