Euclidean distance matrix: Difference between revisions

Revision as of 17:20, 6 August 2013

Name: Jodi Junker
My age: 32
Country: Netherlands
Home town: Oudkarspel
Post code: 1724 Xg
Street: Waterlelie 22

my page - www.hostgator1centcoupon.info

In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice.

Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires.

Riesz spaces have wide ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz Spaces. E.g. the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in Mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.

Definition

A Riesz space E is defined to be a vector space endowed with a partial order " $\leq$ " that, for any $x, y, z \in E$ , satisfies:

$x \leq y$ implies $x + z \leq y + z$ (translation invariance).
For any scalar $α \geq 0$ , $x \leq y$ implies $α x \leq α y$ (positive homogeneity).
For any pair of vectors $x, y \in E$ there exists a supremum (denoted $x \lor y$ ) in E with respect to the partial order " $\leq$ " (lattice structure)

Basic properties

Every Riesz space is a partially ordered vector space, but not every partially ordered vector space is a Riesz space.

Every element f in a Riesz space, E, has unique positive and negative parts, written $f^{+} : = f \lor 0$ and $f^{-} : = (- f) \lor 0$ . Then it can be shown that, $f = f^{+} - f^{-}$ and an absolute value can be defined by $| f | : = f^{+} + f^{-}$ . Every Riesz space is a distributive lattice and has the Riesz decomposition property.

Order convergence

There are a number of meaningful non-equivalent ways to define convergence of sequences or nets with respect to the order structure of a Riesz space. A sequence ${x_{n}}$ in a Riesz space E is said to converge monotonely if it is a monotone decreasing (increasing) sequence and its infimum (supremum) x exists in E and denoted $x_{n} ↓ x$ ( $x_{n} ↑ x$ ).

A sequence ${x_{n}}$ in a Riesz space E is said to converge in order to x if there exists a monotone converging sequence ${p_{n}}$ in E such that $| x_{n} - x | \leq p_{n} ↓ 0$ .

If u is a positive element a Riesz space E then a sequence ${x_{n}}$ in E is said to converge u-uniformly to x for any $ε > 0$ there exists an N such that $| x_{n} - x | < ε u$ for all n>N.

Subspaces

Being vector spaces, it is also interesting to consider subspaces of Riesz spaces. The extra structure provided by these spaces provide for distinct kinds of Riesz subspaces. The collection of each kind structure in a Riesz space (e.g. the collection of all Ideals) forms a distributive lattice.

Ideals

A vector subspace I of a Riesz space E is called an ideal if it is solid, meaning if for any element f in I and any g in E, |g| ≤ |f| implies that g is actually in I. The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset A of E, and is called the ideal generated by A. An Ideal generated by a singleton is called a principal ideal.

Bands and $σ$ -Ideals

A band B in a Riesz space E is defined to be an ideal with the extra property, that for any element f in E for which its absolute value |f| is the supremum of an arbitrary subset of positive elements in B, that f is actually in B. $σ$ -Ideals are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'. Clearly every band is a $σ$ -ideal, but the converse is not true in general.

As with ideals, for every non-empty subset A of E, there exists a smallest band containing that subset, called the band generated by A. A band generated by a singleton is called a principal band.

Disjoint complements

Two elements f,g in a Riesz space E, are said to be disjoint, written $f ⊥ g$ , when $| f | \land | g | = 0$ . For any subset A of E, its disjoint complement $A^{d}$ is defined as the set of all elements in E, that are disjoint to all elements in A. Disjoint complements are always bands, but the converse is not true in general.

Projection bands

A band B in a Riesz space, is called a projection band, if $E = B \oplus B^{d}$ , meaning every element f in E, can be written uniquely as a sum of two elements, $f = u + v$ , with $u \in B$ and $v \in B^{d}$ . There then also exists a positive linear idempotent, or projection, $P_{B} : E \to E$ , such that $P_{B} (f) = u$ .

The collection of all projection bands in a Riesz space forms a Boolean algebra. Some spaces do not have non-trivial projection bands (e.g. $C ([0, 1])$ ), so this Boolean algebra may be trivial.

Projection properties

There are numerous projection properties that Riesz spaces may have. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band.

The so-called main inclusion theorem relates these properties. Super Dedekind completeness implies Dedekind completeness; Dedekind completeness implies both Dedekind $σ$ -completeness and the projection property; Both Dedekind $σ$ -completeness and the projection property separately imply the principal projection property; and the principal projection property implies the Archimedean property.

None of the reverse implications hold, but Dedekind $σ$ -completeness and the projection property together imply Dedekind completeness.

Examples

The space of continuous real valued functions with compact support on a topological space X with the pointwise partial order defined by f ≤ g when f(x) ≤ g(x) for all x in X, is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless X satisfies further conditions (e.g. being extremally disconnected).

Any Lp space with the (almost everywhere) pointwise partial order is a Dedekind complete Riesz space.

The space $ℝ^{2}$ with the lexicographical order is a non-Archimedean Riesz space.

Properties

Riesz spaces are lattice ordered groups
Every Riesz space is a distributive lattice

References

Bourbaki, Nicolas; Elements of Mathematics: Integration. Chapters 1–6; ISBN 3-540-41129-1
Riesz, Frigyes; Sur la décomposition des opérations fonctionelles linéaires, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930) pp. 143–148
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:Functional Analysis

@@ Line 1: / Line 1: @@
-Nice to satisfy you, my name is Ling and I completely dig that title. Her family members life in [http://www.ag.state.mn.us/consumer/cars/mncarlaws/mncarlaws_2.asp Delaware] but  [http://indianapolisfaith.org/UserProfile/tabid/320/userId/242895/Default.aspx extended auto warranty] car warranty she needs to transfer because of her family members.  extended car warranty After being  [http://www.battdouglas.com/ActivityFeed/MyProfile/tabid/61/userId/6189/Default.aspx auto warranty]  extended car [http://www.mpp.com/warranty-plans/used-car-warranties.htm warranty] out of my job for years I grew to become a production and distribution officer but I strategy on altering it. The preferred pastime for him and his kids is to [http://www.consumerreports.org/cro/magazine/2014/04/extended-warranties-for-cars-are-an-expensive-game/ generate] and now he is attempting to make cash with it.<br><br>Feel free to surf to my web page; [http://royalcajun.com/UserProfile/tabid/57/userId/17325/Default.aspx extended auto warranty]
+{{redirect|Vector lattice|the concept of <u>lattice vector</u>|Bravais lattice}}
+In [[mathematics]], a '''Riesz space''', '''lattice-ordered vector space''' or '''vector lattice''' is a [[ordered vector space|partially ordered vector space]] where the [[order structure]] is a [[lattice (order)|lattice]].
+Riesz spaces are named after [[Frigyes Riesz]] who first defined them in his 1928 paper ''Sur la décomposition des opérations fonctionelles linéaires''.
+Riesz spaces have wide ranging applications. They are important in [[measure theory]], in that important results are special cases of results for Riesz Spaces. E.g. the [[Radon–Nikodym theorem]] follows as a special case of the [[Freudenthal spectral theorem]]. Riesz spaces have also seen application in [[Mathematical economics]] through the work of Greek-American economist and mathematician [[Charalambos D. Aliprantis]].
+== Definition ==
+A Riesz space ''E'' is defined to be a vector space endowed with a [[partial order]] "<math>\leq</math>" that, for any <math>x,y,z\in E</math>, satisfies:
+# <math>x\leq y</math> implies <math>x + z\leq y + z</math> ([[translation invariance]]).
+# For any scalar <math>\alpha \geq 0</math>, <math>x\leq y</math> implies <math>\alpha x \leq \alpha y </math>   ([[Homogeneous_function#Positive_homogeneity|positive homogeneity]]).
+# For any pair of vectors <math>x,y\in E</math>  there exists a [[supremum]] (denoted <math>x\vee y</math>) in ''E'' with respect to the partial order "<math>\leq</math>" ([[Lattice (order)|lattice structure]])
+== Basic properties ==
+Every Riesz space is a [[ordered vector space|partially ordered vector space]], but not every partially ordered vector space is a Riesz space.
+Every element ''f'' in a Riesz space, ''E'', has unique positive and negative parts, written <math>f^+:=f\vee 0</math> and <math>f^-:=(-f)\vee 0</math>. Then it can be shown that, <math>f=f^+-f^-</math> and an absolute value can be defined by <math>|f|:=f^++f^-</math>. Every Riesz space is a [[distributive lattice]] and has the [[Approximately finite dimensional C*-algebra#The Effros-Handelman-Shen theorem|Riesz decomposition property]].
+== Order convergence ==
+There are a number of meaningful non-equivalent ways to define convergence of sequences or nets with respect to the order structure of a Riesz space. A sequence <math>\{x_n\}</math> in a Riesz space ''E'' is said to '''converge monotonely''' if it is a [[monotone sequence|monotone]] decreasing (increasing) sequence and its [[infimum]] (supremum)  ''x'' exists in ''E'' and denoted <math>x_n\downarrow x</math>   (<math>x_n\uparrow x</math>).
+A sequence <math>\{x_n\}</math> in a Riesz space ''E'' is said to '''converge in order''' to ''x'' if there exists a monotone converging sequence <math>\{p_n\}</math> in ''E'' such that <math>|x_n-x|\leq p_n\downarrow 0</math>.
+If ''u'' is a positive element a Riesz space ''E'' then a sequence <math>\{x_n\}</math> in ''E'' is said to '''converge u-uniformly''' to ''x'' for any <math>\varepsilon >0</math> there exists an ''N'' such that <math>|x_n-x|<\varepsilon u</math> for all ''n>N''.
+== Subspaces ==
+Being vector spaces, it is also interesting to consider subspaces of Riesz spaces. The extra structure provided by these spaces provide for distinct kinds of Riesz subspaces. The collection of each kind structure in a Riesz space (e.g. the collection of all Ideals) forms a [[distributive lattice]].
+=== Ideals ===
+A vector subspace ''I'' of a Riesz space ''E'' is called an ''ideal'' if it is ''solid'', meaning if for any element ''f'' in ''I'' and any ''g'' in ''E'', ''|g| ≤ |f|'' implies that ''g'' is actually in ''I''. The intersection of an arbitrary collection of ideals is again an ideal, which allows for the definition of a smallest ideal containing some non-empty subset ''A'' of ''E'', and is called the ideal ''generated'' by ''A''. An Ideal generated by a singleton is called a '''principal ideal'''.
+=== Bands and <math>\sigma</math>-Ideals ===
+A ''band'' ''B'' in a Riesz space ''E'' is defined to be an ideal with the extra property, that for any element ''f'' in ''E'' for which its absolute value ''|f|'' is the supremum of an arbitrary subset of positive elements in ''B'', that ''f'' is actually in ''B''. ''<math>\sigma</math>-Ideals'' are defined similarly, with the words 'arbitrary subset' replaced with 'countable subset'. Clearly every band is a <math>\sigma</math>-ideal, but the converse is not true in general.
+As with ideals, for every non-empty subset ''A'' of ''E'', there exists a smallest band containing that subset, called ''the band generated by A''. A band generated by a singleton is called a '''principal band'''.
+=== Disjoint complements ===
+Two elements ''f,g'' in a Riesz space ''E'', are said to be '''disjoint''', written <math>f\bot g</math>, when <math>|f|\wedge |g|=0</math>. For any subset ''A'' of ''E'', its disjoint complement <math>A^d</math> is defined as the set of all elements in ''E'', that are disjoint to all elements in ''A''. Disjoint complements are always bands, but the converse is not true in general.
+=== Projection bands ===
+A band ''B'' in a Riesz space, is called a ''projection band'', if <math>E=B\oplus B^d</math>, meaning every element ''f'' in ''E'', can be written uniquely as a sum of two elements, <math>f=u+v</math>, with <math>u\in B</math> and <math>v\in B^d</math>. There then also exists a positive linear idempotent, or ''projection'', <math>P_B:E\to E</math>, such that <math>P_B(f)=u</math>.
+The collection of all projection bands in a Riesz space forms a [[Boolean algebra]]. Some spaces do not have non-trivial projection bands (e.g. <math>C([0,1])</math>), so this Boolean algebra may be trivial.
+== Projection properties ==
+There are numerous projection properties that Riesz spaces may have. A Riesz space is said to have the (principal) projection property if every (principal) band is a projection band.
+The so-called '''main inclusion theorem''' relates these properties. Super [[Dedekind complete]]ness implies Dedekind completeness; Dedekind completeness implies both Dedekind <math>\sigma</math>-completeness and the projection property; Both Dedekind <math>\sigma</math>-completeness and the projection property separately imply the principal projection property; and the principal projection property implies the [[Archimedean property]].
+None of the reverse implications hold, but Dedekind <math>\sigma</math>-completeness and the projection property together imply Dedekind completeness.
+== Examples ==
+* The space of continuous real valued functions with [[compact support]] on a topological space ''X'' with the [[pointwise]] [[partial order]] defined by ''f'' ≤ ''g'' when ''f(x)'' ≤ ''g(x)'' for all ''x'' in ''X'', is a Riesz space. It is Archimedean, but usually does not have the principal projection property unless ''X'' satisfies further conditions (e.g. being [[extremally disconnected]]).
+* Any [[Lp space]] with the ([[almost everywhere]]) pointwise partial order is a Dedekind complete Riesz space.
+* The space <math>\mathbb{R}^2</math> with the [[lexicographical order]] is a non-Archimedean Riesz space.
+== Properties ==
+* Riesz spaces are [[lattice ordered group]]s
+* Every Riesz space is a [[distributive lattice]]
+== See also ==
+* [[Partially ordered space]]
+* [[Ordered vector space]]
+* [[Convex cone]]
+== References ==
+* [[Bourbaki, Nicolas]]; <cite>Elements of Mathematics: Integration. Chapters 1–6</cite>; ISBN 3-540-41129-1
+* Riesz, Frigyes; <cite>Sur la décomposition des opérations fonctionelles linéaires</cite>, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli  (1930)  pp.&nbsp;143–148
+* {{citation
+  | last=Sobolev
+  | first=V. I.
+  | author-link=
+  | contribution=Riesz space
+  | title=[[Encyclopaedia of Mathematics|Encyclopædia of Mathematics]]
+  | publisher=[[Springer Verlag|Springer]]
+  | year=2001
+  | isbn=978-1-4020-0609-8
+  | url=http://eom.springer.de/R/r082290.htm
+}}
+* {{Citation
+  | last = Zaanen  | first = Adriaan C.
+  | year = 1996
+  | title = Introduction to Operator Theory in Riesz spaces
+  | publisher = [[Springer Verlag|Springer]]
+  | isbn = 3-540-61989-5
+}}
+{{Functional Analysis}}
+[[Category:Functional analysis]]
+[[Category:Ordered groups]]

Euclidean distance matrix: Difference between revisions

Revision as of 17:20, 6 August 2013

Contents

Definition

Basic properties

Order convergence

Subspaces

Ideals

Bands and $σ$ -Ideals

Disjoint complements

Projection bands

Projection properties

Examples

Properties

See also

References

Navigation menu

Euclidean distance matrix: Difference between revisions

Revision as of 17:20, 6 August 2013

Definition

Basic properties

Order convergence

Subspaces

Ideals

Bands and σ-Ideals

Disjoint complements

Projection bands

Projection properties

Examples

Properties

See also

References

Navigation menu

Search

Bands and $σ$ -Ideals