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In [[projective geometry]], a '''collineation''' is a one-to-one and onto map (a [[bijection]]) from one [[projective space]] to another, or from a projective space to itself, such that the images of [[collinear]] points are themselves collinear. A collineation is thus an ''[[isomorphism]]'' between projective spaces, or an [[automorphism]] from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism.<ref>For instance, {{harvnb|Beutelspacher|Rosenbaum|1998|loc=p.21}}, {{harvnb|Casse|2006|loc=p. 56}} and {{harvnb|Yale|2004|loc=p. 226}}</ref> The set of all collineations of a space to itself form a [[group (mathematics)|group]], called the '''collineation group'''. | |||
==Definition== | |||
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently. | |||
===Linear algebra=== | |||
For a projective space defined in terms of linear algebra (as the projectivization of a vector space), a collineation is a map between the projective spaces that is [[order-preserving]] with respect to inclusion of subspaces. | |||
Formally, let ''V'' be a vector space over a field ''K'' and ''W'' a vector space over a field ''L''. Consider the projective spaces ''PG''(''V'') and ''PG''(''W''), consisting of the [[vector line]]s of ''V'' and ''W''. | |||
Call ''D''(''V'') and ''D''(''W'') the set of subspaces of ''V'' and ''W'' respectively. A collineation from ''PG''(''V'') to ''PG''(''W'') is a map α : ''D''(''V'') → ''D''(''W''), such that: | |||
* α is a bijection. | |||
* ''A'' ⊆ ''B'' ⇔ α(''A'') ⊆ α(''B'') for all ''A'', ''B'' in ''D''(''V'').<ref>Geometers still commonly use an exponential type notation for functions and this condition will often appear as ''A'' ⊆ ''B'' ⇔ ''A''<sup>α</sup> ⊆ ''B''<sup>α</sup> for all ''A'', ''B'' in ''D''(''V'').</ref> | |||
===Axiomatically=== | |||
Given a [[Projective space#Axioms for projective space|projective space defined axiomatically]] in terms of an [[incidence structure]] (a set of points ''P,'' lines ''L,'' and an [[incidence relation]] ''I'' specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function ''f'' between the sets of points and a bijective function ''g'' between the set of lines, preserving the incidence relation.<ref>"Preserving the incidence relation" means that if point ''p'' is on line ''l'' then <math>f(p)</math> is in <math>g(l)</math>; formally, if <math>(p,l) \in I</math> then <math>\big(f(p),g(l)\big) \in I'</math>.</ref> | |||
Every projective space of dimension greater than or equal to three is isomorphic to the [[projectivization]] of a linear space over a [[division ring]], so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the [[non-Desarguesian plane]]s, and this definition permits one to define collineations in such projective planes. | |||
For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set. | |||
===Collineations of the projective line=== | |||
For a projective space of dimension one (a projective line; the projectivization of a vector space of dimension two), all points are collinear, so the collineation group is exactly the [[symmetric group]] of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the [[#Fundamental theorem of projective geometry|fundamental theorem of projective geometry]] holds. | |||
In this definition, when ''V'' has dimension two, a collineation from ''PG''(''V'') to ''PG''(''W'') is a map α : ''D''(''V'') → ''D''(''W''), such that : | |||
* The [[zero object (algebra)|zero subspace]] of ''V'' is mapped to the zero subspace of ''W''. | |||
* ''V'' is mapped to ''W''. | |||
* There is a nonsingular [[semilinear transformation|semilinear map]] β from ''V'' to ''W'' such that, for all ''v'' in ''V'', | |||
:<math>\alpha(\langle v\rangle)=\langle \beta(v)\rangle</math> | |||
This last requirement ensures that collineations are all semilinear maps. | |||
==Types== | |||
The main examples of collineations are projective linear transformations (also known as [[homography|homographies]]) and [[#Automorphic collineations|automorphic collineations]]. For projective spaces coming from a linear space, the [[#Fundamental theorem of projective geometry|fundamental theorem of projective geometry]] states that all collineations are a combination of these, as described below. | |||
<!-- This illustrates the pitfall of not restricting the definition of collineation. Correlations are not automorphisms of the projective space. | |||
A [[duality (projective geometry)|duality]] is a collineation from a projective space onto its dual space, taking points to hyperplanes (and vice versa) and preserving incidence. A [[correlation (projective geometry)|correlation]] is a duality from a projective space onto itself (this implies that the space is self-dual). A [[pole and polar|polarity]] is an [[involution (mathematics)|involutory]] correlation. | |||
--> | |||
===Projective linear transformations=== | |||
Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are projective linear transformations. PGL is in general a proper subgroup of the collineation group. | |||
===Automorphic collineations=== | |||
An '''{{Visible anchor|automorphic collineation}}''' is a map that, in coordinates, is a [[field automorphism]] applied to the coordinates. | |||
==Fundamental theorem of projective geometry== | |||
{{see also|Homography#Fundamental theorem of projective geometry}} | |||
If the geometric dimension of a [[Pappus's hexagon theorem|pappian]] projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the [[projective semilinear group]], which is the [[semidirect product]] of homographies by automorphic collineations. | |||
In particular, the collineations of ''PG''(2, '''R''') are exactly the homographies, as '''R''' has no nontrivial automorphisms (that is, Gal('''R'''/'''Q''') is trivial). | |||
Suppose φ is a semilinear nonsingular map from ''V'' to ''W'', with the dimension of ''V'' at least three. Define α : ''D''(''V'') → ''D''(''W'') by saying that {{Nowrap begin}}''Z''<sup>α</sup> = { φ(''z'') | ''z'' ∈ ''Z'' }{{Nowrap end}} for all ''Z'' in ''D''(''V''). As φ is semilinear, one easily checks that this map is properly defined, and further more, as φ is not singular, it is bijective. It is obvious now that α is a collineation. We say α is induced by φ. | |||
The fundamental theorem of projective geometry states the converse: | |||
Suppose ''V'' is a vector space over a field ''K'' with dimension at least three, ''W'' is a vector space over a field ''L'', and α is a collineation from ''PG''(''V'') to ''PG''(''W''). This implies ''K'' and ''L'' are isomorphic fields, ''V'' and ''W'' have the same dimension, and there is a semilinear map φ such that φ induces α. | |||
For <math>n\geq 3,</math> the collineation group is the [[projective semilinear group]], <math>P\Gamma L</math> – this is PGL, twisted by [[field automorphism]]s; formally, the [[semidirect product]] <math>P\Gamma L \cong PGL \rtimes \operatorname{Gal}(K/k),</math> where ''k'' is the [[prime field]] for ''K.'' | |||
===Linear structure=== | |||
Thus for ''K'' a prime field (<math>\mathbb{F}_p</math> or <math>\mathbb{Q}</math>), we have <math>PGL = P\Gamma L,</math> but for ''K'' not a prime field (such as <math>\mathbb{F}_{p^n}</math> for <math>n\geq 2</math> or <math>\mathbb{C}</math>), the projective linear group is in general a proper subgroup of the collineation group, which can be thought of as "transformations preserving a projective ''semi''-linear structure". Correspondingly, the quotient group <math>P\Gamma L/PGL \cong \operatorname{Gal}(K/k)</math> corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure. Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and PΓL, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup <math>PGL < P\Gamma L,</math> these choices forming a [[torsor]] over <math>\operatorname{Gal}(K/k).</math> | |||
==History== | |||
The idea of a [[line (geometry)|line]] was abstracted to a [[ternary relation]] determined by [[collinearity|collinear points]]. According to [[Wilhelm Blaschke]]<ref>[[Felix Klein]] (1926, 1949) ''Vorlesungen über Höhere Geometrie'', edited by Blaschke, Seite 138</ref> it was [[August Möbius]] that first abstracted this essence of geometrical transformation: | |||
:What do our geometric transformations mean now? Möbius threw out and fielded this question already in his ''Barycentric Calculus'' (1827). There he spoke not of ''transformations'' but of ''permutations'' [Verwandtschaften], when he said two elements drawn from a domain were ''permuted'' when they were interchanged by an arbitrary equation. In our particular case, linear equations between homogeneous point coordinates, Möbius called a permutation [Verwandtschaft] of both point spaces in particular a ''collineation''. This signification would be changed later by [[Michel Chasles|Chasles]] to ''homography''. Möbius’ expression is immediately comprehended when we follow Möbius in calling points [[collinear]] when they lie on the same line. Mobius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight. | |||
Contemporary mathematicians view geometry as an [[incidence structure]] with an [[automorphism group]] consisting of mappings of the underlying space that preserve [[incidence (geometry)|incidence]]. Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists. | |||
As mentioned by Blaschke and Klein, [[Michel Chasles]] preferred the term ''homography'' to ''collineation''. A distinction between the terms arose when the distinction was clarified between the [[real projective plane]] and the [[complex projective line]]. Since there are no non-trivial field automorphisms of the [[real number]] field, all the collineations are homographies in the real projective plane.,<ref>{{harvnb|Casse|2006|loc=p. 64, Corollary 4.29}}</ref> however due to the field automorphism [[complex conjugation]], not all collineations of the complex projective line are homographies. In applications such as [[computer vision]] where the underlying field is the real number field, ''homography'' and ''collineation'' can be used interchangeably. | |||
==Anti-homography== | |||
The operation of taking the [[complex conjugate]] in the [[complex plane]] amounts to a [[reflection (mathematics)|reflection]] in the [[real line]]. With the notation ''z''* for the conjugate of ''z'', an '''anti-homography''' is given by | |||
:<math>f(z) = \frac {a z^* + b}{c z^* + d}.</math> | |||
Thus an anti-homography is the composition of conjugation with an [[homography]], and so is an example of a collineation which is not an homography. For example, geometrically, the mapping <math>f(z) = 1/z^*</math> amounts to [[inversive geometry#Circle inversion|circle inversion]].<ref>{{harvnb|Morley|Morley|1933|loc=p. 38}}</ref> The transformations of [[inversive geometry]] of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.<ref>{{harvnb|Blair|2000}}, [http://books.google.com/books?id=sxHKpaAuclgC&pg=PA43 p. 43]; {{harvnb|Schwerdtfeger|2012}}, [http://books.google.com/books?id=4XE2_AqxYVkC&pg=PA42 p. 42].</ref> | |||
==Notes== | |||
{{Reflist}} | |||
==References== | |||
* {{citation|first1=Albrecht|last1=Beutelspacher|first2=Ute|last2=Rosenbaum|authorlink1=Albrecht Beutelspacher|title=Projective Geometry / From Foundations to Applications|publisher=Cambridge University Press|year=1998|isbn=0-521-48364-6}} | |||
* {{citation|title=Inversion Theory and Conformal Mapping|volume=9|series=Student mathematical library|first=David E.|last=Blair|publisher=American Mathematical Society|year=2000|isbn=9780821826362}} | |||
* {{citation|first=Wilhelm|last=Blaschke|authorlink=Wilhelm Blaschke|title=Projective Geometrie|year=1948|publisher=Wolfenbütteler Verlagsanstalt}} | |||
* {{citation|first=Rey|last=Casse|title=Projective Geometry / An Introduction|publisher=Oxford University Press|year=2006|isbn=9780199298860}} | |||
* {{citation|first1=Frank|last1=Morley|authorlink1=Frank Morley|first2=F.V.|last2=Morley|year=1933|title=Inversive Geometry|place=London|publisher=G. Bell and Sons}} | |||
* {{citation|title=Geometry of Complex Numbers|first=Hans|last=Schwerdtfeger|publisher=Courier Dover Publications|year=2012|isbn=9780486135861}} | |||
* {{citation|first=Paul B.|last=Yale|title=Geometry and Symmetry|publisher=Dover|year=2004|origyear=first published 1968|isbn=0-486-43835-X}} | |||
==External links== | |||
* {{PlanetMath|urlname=Collineation|title=projectivity}} | |||
{{Fundamental theorems}} | |||
[[Category:Projective geometry]] | |||
[[Category:Fundamental theorems|*Collineation]] | |||
Revision as of 08:44, 9 January 2014
In projective geometry, a collineation is a one-to-one and onto map (a bijection) from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. A collineation is thus an isomorphism between projective spaces, or an automorphism from a projective space to itself. Some authors restrict the definition of collineation to the case where it is an automorphism.[1] The set of all collineations of a space to itself form a group, called the collineation group.
Definition
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated differently.
Linear algebra
For a projective space defined in terms of linear algebra (as the projectivization of a vector space), a collineation is a map between the projective spaces that is order-preserving with respect to inclusion of subspaces.
Formally, let V be a vector space over a field K and W a vector space over a field L. Consider the projective spaces PG(V) and PG(W), consisting of the vector lines of V and W. Call D(V) and D(W) the set of subspaces of V and W respectively. A collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that:
- α is a bijection.
- A ⊆ B ⇔ α(A) ⊆ α(B) for all A, B in D(V).[2]
Axiomatically
Given a projective space defined axiomatically in terms of an incidence structure (a set of points P, lines L, and an incidence relation I specifying which points lie on which lines, satisfying certain axioms), a collineation between projective spaces thus defined then being a bijective function f between the sets of points and a bijective function g between the set of lines, preserving the incidence relation.[3]
Every projective space of dimension greater than or equal to three is isomorphic to the projectivization of a linear space over a division ring, so in these dimensions this definition is no more general than the linear-algebraic one above, but in dimension two there are other projective planes, namely the non-Desarguesian planes, and this definition permits one to define collineations in such projective planes.
For dimension one, the set of points lying on a single projective line defines a projective space, and the resulting notion of collineation is just any bijection of the set.
Collineations of the projective line
For a projective space of dimension one (a projective line; the projectivization of a vector space of dimension two), all points are collinear, so the collineation group is exactly the symmetric group of the points of the projective line. This is different from the behavior in higher dimensions, and thus one gives a more restrictive definition, specified so that the fundamental theorem of projective geometry holds.
In this definition, when V has dimension two, a collineation from PG(V) to PG(W) is a map α : D(V) → D(W), such that :
- The zero subspace of V is mapped to the zero subspace of W.
- V is mapped to W.
- There is a nonsingular semilinear map β from V to W such that, for all v in V,
This last requirement ensures that collineations are all semilinear maps.
Types
The main examples of collineations are projective linear transformations (also known as homographies) and automorphic collineations. For projective spaces coming from a linear space, the fundamental theorem of projective geometry states that all collineations are a combination of these, as described below.
Projective linear transformations
Projective linear transformations (homographies) are collineations (planes in a vector space correspond to lines in the associated projective space, and linear transformations map planes to planes, so projective linear transformations map lines to lines), but in general not all collineations are projective linear transformations. PGL is in general a proper subgroup of the collineation group.
Automorphic collineations
An Template:Visible anchor is a map that, in coordinates, is a field automorphism applied to the coordinates.
Fundamental theorem of projective geometry
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If the geometric dimension of a pappian projective space is at least 2, then every collineation is the product of a homography (a projective linear transformation) and an automorphic collineation. More precisely, the collineation group is the projective semilinear group, which is the semidirect product of homographies by automorphic collineations.
In particular, the collineations of PG(2, R) are exactly the homographies, as R has no nontrivial automorphisms (that is, Gal(R/Q) is trivial).
Suppose φ is a semilinear nonsingular map from V to W, with the dimension of V at least three. Define α : D(V) → D(W) by saying that Template:Nowrap beginZα = { φ(z) | z ∈ Z }Template:Nowrap end for all Z in D(V). As φ is semilinear, one easily checks that this map is properly defined, and further more, as φ is not singular, it is bijective. It is obvious now that α is a collineation. We say α is induced by φ.
The fundamental theorem of projective geometry states the converse:
Suppose V is a vector space over a field K with dimension at least three, W is a vector space over a field L, and α is a collineation from PG(V) to PG(W). This implies K and L are isomorphic fields, V and W have the same dimension, and there is a semilinear map φ such that φ induces α.
For the collineation group is the projective semilinear group, – this is PGL, twisted by field automorphisms; formally, the semidirect product where k is the prime field for K.
Linear structure
Thus for K a prime field ( or ), we have but for K not a prime field (such as for or ), the projective linear group is in general a proper subgroup of the collineation group, which can be thought of as "transformations preserving a projective semi-linear structure". Correspondingly, the quotient group corresponds to "choices of linear structure", with the identity (base point) being the existing linear structure. Given a projective space without an identification as the projectivization of a linear space, there is no natural isomorphism between the collineation group and PΓL, and the choice of a linear structure (realization as projectivization of a linear space) corresponds to a choice of subgroup these choices forming a torsor over
History
The idea of a line was abstracted to a ternary relation determined by collinear points. According to Wilhelm Blaschke[4] it was August Möbius that first abstracted this essence of geometrical transformation:
- What do our geometric transformations mean now? Möbius threw out and fielded this question already in his Barycentric Calculus (1827). There he spoke not of transformations but of permutations [Verwandtschaften], when he said two elements drawn from a domain were permuted when they were interchanged by an arbitrary equation. In our particular case, linear equations between homogeneous point coordinates, Möbius called a permutation [Verwandtschaft] of both point spaces in particular a collineation. This signification would be changed later by Chasles to homography. Möbius’ expression is immediately comprehended when we follow Möbius in calling points collinear when they lie on the same line. Mobius' designation can be expressed by saying, collinear points are mapped by a permutation to collinear points, or in plain speech, straight lines stay straight.
Contemporary mathematicians view geometry as an incidence structure with an automorphism group consisting of mappings of the underlying space that preserve incidence. Such a mapping permutes the lines of the incidence structure, and the notion of collineation persists.
As mentioned by Blaschke and Klein, Michel Chasles preferred the term homography to collineation. A distinction between the terms arose when the distinction was clarified between the real projective plane and the complex projective line. Since there are no non-trivial field automorphisms of the real number field, all the collineations are homographies in the real projective plane.,[5] however due to the field automorphism complex conjugation, not all collineations of the complex projective line are homographies. In applications such as computer vision where the underlying field is the real number field, homography and collineation can be used interchangeably.
Anti-homography
The operation of taking the complex conjugate in the complex plane amounts to a reflection in the real line. With the notation z* for the conjugate of z, an anti-homography is given by
Thus an anti-homography is the composition of conjugation with an homography, and so is an example of a collineation which is not an homography. For example, geometrically, the mapping amounts to circle inversion.[6] The transformations of inversive geometry of the plane are frequently described as the collection of all homographies and anti-homographies of the complex plane.[7]
Notes
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References
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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 - 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
External links
- ↑ For instance, Template:Harvnb, Template:Harvnb and Template:Harvnb
- ↑ Geometers still commonly use an exponential type notation for functions and this condition will often appear as A ⊆ B ⇔ Aα ⊆ Bα for all A, B in D(V).
- ↑ "Preserving the incidence relation" means that if point p is on line l then is in ; formally, if then .
- ↑ Felix Klein (1926, 1949) Vorlesungen über Höhere Geometrie, edited by Blaschke, Seite 138
- ↑ Template:Harvnb
- ↑ Template:Harvnb
- ↑ Template:Harvnb, p. 43; Template:Harvnb, p. 42.