Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
No edit summary
No edit summary
Line 1: Line 1:
[[Image:Maximum modulus principle.png|right|thumb|A plot of the modulus of cos(''z'') (in red) for ''z'' in the [[unit disk]] centered at the origin (shown in blue). As predicted by the theorem, the maximum of the modulus cannot be inside of the disk (so the highest value on the red surface is somewhere along its edge).]]
In [[mathematics]], the '''Cayley transform''', named after [[Arthur Cayley]], has a cluster of related meanings. As originally described by {{Harvtxt|Cayley|1846}}, the Cayley transform is a mapping between [[skew-symmetric matrix|skew-symmetric matrices]] and [[special orthogonal matrix|special orthogonal matrices]]. In [[complex analysis]], the Cayley transform is a [[conformal map]]ping {{Harv|Rudin|1987}} in which the image of the upper complex half-plane is the unit disk {{Harv|Remmert|1991|pp=82ff, 275}}. And in the theory of [[Hilbert space]]s, the Cayley transform is a mapping between [[linear operator]]s {{Harv|Nikol’skii|2001}}.
In [[mathematics]], the '''maximum modulus principle''' in [[complex analysis]] states that if ''f'' is a [[holomorphic function]], then the [[absolute value|modulus]] <math>|f|</math> cannot exhibit a true [[local maximum]] that is properly within the [[domain (mathematics)|domain]] of ''f''.  


In other words, either ''f'' is a [[constant function]], or, for any point ''z''<sub>0</sub> inside the domain of ''f'' there exist other points arbitrarily close to ''z''<sub>0</sub> at which |''f'' | takes larger values.
== Matrix map ==
Among ''n''×''n'' [[square matrix|square matrices]] over the [[real number|reals]], with ''I'' the identity matrix, let ''A'' be any [[skew-symmetric matrix]] (so that ''A''<sup>T</sup>&nbsp;= −''A''). Then ''I''&nbsp;+&nbsp;''A'' is [[invertible matrix|invertible]], and the Cayley transform


==Formal statement==
:<math> Q = (I - A)(I + A)^{-1} \,\!</math>
Let ''f'' be a function holomorphic on some [[connected set|connected]] [[open set|open]] [[subset]] ''D'' of the [[complex plane]] <math>\mathbb{C}</math>  and taking complex values. If ''z''<sub>0</sub> is a point in ''D'' such that
:<math>|f(z_0)|\ge |f(z)|</math>  
for all ''z'' in a [[neighborhood (topology)|neighborhood]] of ''z''<sub>0</sub>, then the function ''f'' is constant on ''D''.


By switching to the [[Multiplicative_inverse|reciprocal]], we can get the '''minimum modulus principle'''. It states that if  ''f'' is holomorphic within a bounded domain ''D'', continuous up to the [[Boundary (topology)|boundary]] of ''D'', and non-zero at all points, then |''f'' (z)| takes its minimum value on the boundary of ''D''.
produces an [[orthogonal matrix]], ''Q'' (so that ''Q''<sup>T</sup>''Q''&nbsp;= ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as <math> Q = (I + A)^{-1}(I - A)</math>. In fact, ''Q'' must have determinant +1, so is special orthogonal. Conversely, let ''Q'' be any orthogonal matrix which does not have −1 as an [[eigenvalue]]; then


Alternatively, the maximum modulus principle can be viewed as a special case of the [[open mapping theorem (complex analysis)|open mapping theorem]], which states that a nonconstant holomorphic function maps open sets to open sets. If |''f''| attains a local maximum at ''z'', then the image of a sufficiently small open neighborhood of ''z'' cannot be open. Therefore, ''f'' is constant.
:<math> A = (I - Q)(I + Q)^{-1} \,\!</math>


==Sketches of proofs==
is a skew-symmetric matrix. The condition on ''Q'' automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing ''Q''&nbsp;= ''A''<sup>c</sup> and ''A''&nbsp;= ''Q''<sup>c</sup>.


===Using the maximum principle for harmonic functions===
This version of the Cayley transform is its own functional inverse, so that ''A''&nbsp;= (''A''<sup>c</sup>)<sup>c</sup> and ''Q''&nbsp;= (''Q''<sup>c</sup>)<sup>c</sup>. A slightly different form is also seen {{Harv|Golub|Van Loan|1996}}, requiring different mappings in each direction (and dropping the superscript notation):
One can use the equality
:log ''f''(''z'') = ln |''f''(''z'')| + i arg ''f''(''z'') 
for complex [[natural logarithm]]s to deduce that ln |''f''(''z'')| is a [[harmonic function]]. Since ''z''<sub>0</sub> is a local maximum for this function also, it follows from the [[maximum principle]] that |''f''(''z'')| is constant. Then, using the [[Cauchy-Riemann equations]] we show that ''f'''(''z'')=0, and thus that ''f''(''z'') is constant as well.


===Using Gauss's mean value theorem===
:<math>\begin{align}
Another proof works by using Gauss's mean value theorem to "force" all points within overlapping open disks to assume the same value. The disks are laid such that their centers form a polygonal path from the value where ''f''(''z'') is maximized to any other point in the domain, while being totally contained within the domain. Thus the existence of a maximum value implies that all the values in the domain are the same, thus ''f''(''z'') is constant.
Q &{}= (I - A)^{-1}(I + A) \\
A &{}= (Q - I)(Q + I)^{-1}
\end{align}</math>


==Physical Interpretation==
The mappings may also be written with the order of the factors reversed {{Harv|Courant|Hilbert|1989|loc=Ch.VII,&nbsp;&sect;7.2}}; however, ''A'' always commutes with (μ''I''&nbsp;±&nbsp;''A'')<sup>−1</sup>, so the reordering does not affect the definition.


A physical interpretation of this principle comes from the [[heat equation]]. That is, since log |''f''(''z'')| is harmonic, it is thus the steady state of a heat flow on the region ''D''. Suppose a strict maximum was attained on the interior of ''D'', the heat at this maximum would be dispersing to the points around it, which would contradict the assumption that this represents the steady state of a system.
=== Examples ===
In the 2×2 case, we have
:<math>
\begin{bmatrix} 0 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 0 \end{bmatrix}
\lrarr
\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} .
</math>
The 180° rotation matrix, −''I'', is excluded, though it is the limit as tan&nbsp;<sup>θ</sup>⁄<sub>2</sub> goes to infinity.


== Applications ==
In the 3×3 case, we have
The maximum modulus principle has many uses in complex analysis, and may be used to prove the following:
:<math>
* The [[fundamental theorem of algebra]].
\begin{bmatrix} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{bmatrix}
* [[Schwarz's lemma]], a result which in turn has many generalisations and applications in complex analysis.
\lrarr
* The [[Phragmén–Lindelöf principle]], an extension to unbounded domains.
\frac{1}{K}
* The [[Borel–Carathéodory theorem]], which bounds an analytic function in terms of its real part.
\begin{bmatrix}
* The [[Hadamard three-lines theorem]], a result about the behaviour of bounded holomorphic functions on a line between two other parallel lines in the complex plane.
  w^2+x^2-y^2-z^2 & 2 (x y-w z) & 2 (w y+x z) \\
  2 (x y+w z) & w^2-x^2+y^2-z^2 & 2 (y z-w x) \\
  2 (x z-w y) & 2 (w x+y z) & w^2-x^2-y^2+z^2
\end{bmatrix} ,
</math>


==References==
where ''K''&nbsp;=&nbsp;''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>, and where ''w''&nbsp;=&nbsp;1. This we recognize as the rotation matrix corresponding to [[quaternion]]
* {{cite book |first=E. C. |last=Titchmarsh |authorlink=E. C. Titchmarsh |title=The Theory of Functions |edition=2nd |year=1939 |publisher=Oxford University Press }} ''(See chapter 5.)''
 
* {{springer|author=E.D. Solomentsev|title=Maximum-modulus principle|id=m/m063110}}
:<math> w + \bold{i} x + \bold{j} y + \bold{k} z \,\!</math>
 
(by a formula Cayley had published the year before), except scaled so that ''w''&nbsp;= 1 instead of the usual scaling so that ''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>&nbsp;=&nbsp;1. Thus vector (''x'',''y'',''z'') is the unit axis of rotation scaled by tan&nbsp;<sup>θ</sup>⁄<sub>2</sub>. Again excluded are 180° rotations, which in this case are all ''Q'' which are [[symmetric matrix|symmetric]] (so that ''Q''<sup>T</sup>&nbsp;= ''Q'').
 
=== Other matrices ===
We can extend the mapping to [[complex number|complex]] matrices by substituting "[[unitary matrix|unitary]]" for "orthogonal" and "[[skew-Hermitian matrix|skew-Hermitian]]" for "skew-symmetric", the difference being that the transpose (·<sup>T</sup>) is replaced by the [[conjugate transpose]] (·<sup>H</sup>). This is consistent with replacing the standard real [[inner product]] with the standard complex inner product. In fact, we may extend the definition further with choices of [[adjoint]] other than transpose or conjugate transpose.
 
Formally, the definition only requires some invertibility, so we can substitute for ''Q'' any matrix ''M'' whose eigenvalues do not include −1. For example, we have
:<math>
\begin{bmatrix} 0 & -a & ab - c \\ 0 & 0 & -b \\ 0 & 0 & 0 \end{bmatrix}
\lrarr
\begin{bmatrix} 1 & 2a & 2c \\ 0 & 1 & 2b \\ 0 & 0 & 1 \end{bmatrix} .
</math>
We remark that ''A'' is skew-symmetric (respectively, skew-Hermitian) if and only if ''Q'' is orthogonal (respectively, unitary) with no eigenvalue −1.
 
== Conformal map ==
[[Image:Cayley transform in complex plane.png|thumb|right| 300px|Cayley transform of upper complex half-plane to unit disk]]
In [[complex analysis]], the Cayley transform is a [[mapping (mathematics)|mapping]] of the [[complex plane]] to itself, given by
 
:<math> \operatorname{W} \colon z \mapsto \frac{z-\bold{i}}{z+\bold{i}} . </math>
 
This is a [[linear fractional transformation]], and can be extended to an [[automorphism]] of the [[Riemann sphere]] (the [[complex plane]] augmented with a point at infinity).
 
Of particular note are the following facts:
 
* W maps the upper half plane of '''C''' [[conformal mapping|conformally]] onto the unit disc of '''C'''.
* W maps the real line '''R''' [[injective]]ly into the [[circle group|unit circle]] '''T''' (complex numbers of [[absolute value]] 1).  The image of '''R'''  is '''T''' with 1 removed.
* W maps the upper imaginary axis '''i''' <nowiki>[0, &infin;)</nowiki> [[bijection|bijectively]] onto the half-open interval <nowiki>[−1, +1)</nowiki>.
* W maps 0 to −1.
* W maps the point at infinity to 1.
* W maps −'''i''' to the point at infinity (so W has a [[pole (complex analysis)|pole]] at −'''i''').
* W maps −1 to '''i'''.
* W maps both <sup>1</sup>⁄<sub>2</sub>(−1&nbsp;+&nbsp;√3)(−1&nbsp;+&nbsp;'''i''') and <sup>1</sup>⁄<sub>2</sub>(1&nbsp;+&nbsp;√3)(1&nbsp;−&nbsp;'''i''') to themselves.
 
== Operator map ==
An infinite-dimensional version of an [[inner product space]] is a [[Hilbert space]], and we can no longer speak of [[matrix (mathematics)|matrices]]. However, matrices are merely representations of [[linear operator]]s, and these we still have. So, generalizing both the matrix mapping and the complex plane mapping, we may define a Cayley transform of operators.
:<math>\begin{align}
U &{}= (A - \bold{i}I) (A + \bold{i}I)^{-1} \\
A &{}= \bold{i}(I + U) (I - U)^{-1}
\end{align}</math>
Here the domain of ''U'', dom&nbsp;''U'', is (''A''+'''i'''''I'')&nbsp;dom&nbsp;''A''. See [[self-adjoint operator#Extensions of symmetric operators|self-adjoint operator]] for further details.
 
== See also ==
* [[Bilinear transform]]
 
* [[Extensions of symmetric operators]]
 
== References ==
 
* {{Citation
| last=Cayley
| first=Arthur
| author-link=Arthur Cayley
| year=1846
| title=Sur quelques propriétés des déterminants gauches
| journal=[[Journal für die reine und angewandte Mathematik]]
| volume=32
| pages=119–123
| url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D268141
| issn=0075-4102
}}; reprinted as article 52 (pp.&nbsp;332–336) in {{Citation
| last=Cayley
| first=Arthur
| author-link=Arthur Cayley
| year=1889
| title=The collected mathematical papers of Arthur Cayley
| publisher=[[Cambridge University Press]]
| volume=I (1841–1853)
| pages=332–336
| isbn=<!-- none given -->
| url=http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000349
}}
* {{Citation
  | last1=Courant
  | first1=Richard
  | author1-link=Richard Courant
  | last2=Hilbert
  | first2=David
  | author2-link=David Hilbert
  | title=Methods of Mathematical Physics
  | volume=1
  | edition=1st English
  | publisher=Wiley-Interscience
  | year=1989
  | place=New York
  | isbn=978-0-471-50447-4
}}
* {{Citation
  | last1=Golub
  | first1=Gene H.
  | author1-link=Gene H. Golub
  | last2=Van Loan
  | first2=Charles F.
  | author2-link=Charles F. Van Loan
  | title=Matrix Computations
  | edition=3rd
  | publisher=Johns Hopkins University Press
  | year=1996
  | place=Baltimore
  | isbn=978-0-8018-5414-9
}}
* {{Citation
| last=Nikol’skii
| first=N. K.
| contribution=Cayley transform
| contribution-url=http://www.encyclopediaofmath.org/index.php?title=Cayley_transform&oldid=12556
| title=[[Encyclopaedia of Mathematics]]
| year=2001
| publisher=[[Springer-Verlag]]
| isbn=978-1-4020-0609-8<!-- uncertain, web page gives 1402006098, which does not validate -->
}}; translated from the Russian {{Citation
| editor-last=Vinogradov
| editor-first=I. M.
| editor-link=Ivan Matveyevich Vinogradov
| title=Matematicheskaya Entsiklopediya
| place=Moscow
| publisher=Sovetskaya Entsiklopediya
| year=1977
}}
* {{Citation<!-- courtesy of [[User:CSTAR|]] -->
| last=Remmert
| first=Reinhold
| author-link=Reinhold Remmert
| translator=Robert B. Burckel (trans.)<!-- template does not provide for this -->
| title=Theory of Complex Functions
| series=Graduate Texts in Mathematics
| volume='''122''' of ''Graduate Texts in Mathematics'' (''Readings in Mathematics'')<!-- compensate for lack of template "series" support -->
| year=1991
| publisher=[[Springer-Verlag]]
| place=New York
| isbn=978-0-387-97195-7
}}, translated by Robert B. Burckel from {{Citation
| unused_data=Grundwissen Mathematik 5
| last=Remmert
| first=Reinhold
| author-link=Reinhold Remmert
| title=Funktionentheorie I
| edition=2nd
| year=1989
| publisher=[[Springer-Verlag]]
| isbn=978-3-540-51238-7
}}
* {{Citation
| last=Rudin
| first=Walter
| author-link=Walter Rudin
| title=Real and Complex Analysis
| edition=3rd<!-- date was 1966, no edition given -->
| publisher=McGraw-Hill
| year=1987<!-- March 1 -->
| isbn=978-0-07-100276-9
}}


== External links ==
== External links ==
* {{MathWorld | urlname= MaximumModulusPrinciple | title= Maximum Modulus Principle}}
* {{PlanetMath
* [http://math.fullerton.edu/mathews/c2003/LiouvilleMoreraGaussMod.html The Maximum Modulus Principle by John H. Mathews]
| urlname=CayleyTransform
| title=Cayley's parameterization of orthogonal matrices
| id=6535
}}


[[Category:Mathematical principles]]
[[Category:Conformal mapping]]
[[Category:Theorems in complex analysis]]
[[Category:Transforms]]


[[de:Maximumprinzip (Mathematik)]]
[[fr:Transformation de Cayley]]
[[it:Trasformata di Cayley]]
[[ru:Преобразование Мёбиуса#Примеры]]
[[uk:Перетворення Келі]]

Revision as of 23:22, 12 August 2014

In mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by Template:Harvtxt, the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping Template:Harv in which the image of the upper complex half-plane is the unit disk Template:Harv. And in the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators Template:Harv.

Matrix map

Among n×n square matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix (so that AT = −A). Then I + A is invertible, and the Cayley transform

produces an orthogonal matrix, Q (so that QTQ = I). The matrix multiplication in the definition of Q above is commutative, so Q can be alternatively defined as . In fact, Q must have determinant +1, so is special orthogonal. Conversely, let Q be any orthogonal matrix which does not have −1 as an eigenvalue; then

is a skew-symmetric matrix. The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing Q = Ac and A = Qc.

This version of the Cayley transform is its own functional inverse, so that A = (Ac)c and Q = (Qc)c. A slightly different form is also seen Template:Harv, requiring different mappings in each direction (and dropping the superscript notation):

The mappings may also be written with the order of the factors reversed Template:Harv; however, A always commutes with (μI ± A)−1, so the reordering does not affect the definition.

Examples

In the 2×2 case, we have

The 180° rotation matrix, −I, is excluded, though it is the limit as tan θ⁄2 goes to infinity.

In the 3×3 case, we have

where K = w2 + x2 + y2 + z2, and where w = 1. This we recognize as the rotation matrix corresponding to quaternion

(by a formula Cayley had published the year before), except scaled so that w = 1 instead of the usual scaling so that w2 + x2 + y2 + z2 = 1. Thus vector (x,y,z) is the unit axis of rotation scaled by tan θ⁄2. Again excluded are 180° rotations, which in this case are all Q which are symmetric (so that QT = Q).

Other matrices

We can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose (·T) is replaced by the conjugate transpose (·H). This is consistent with replacing the standard real inner product with the standard complex inner product. In fact, we may extend the definition further with choices of adjoint other than transpose or conjugate transpose.

Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include −1. For example, we have

We remark that A is skew-symmetric (respectively, skew-Hermitian) if and only if Q is orthogonal (respectively, unitary) with no eigenvalue −1.

Conformal map

Cayley transform of upper complex half-plane to unit disk

In complex analysis, the Cayley transform is a mapping of the complex plane to itself, given by

This is a linear fractional transformation, and can be extended to an automorphism of the Riemann sphere (the complex plane augmented with a point at infinity).

Of particular note are the following facts:

  • W maps the upper half plane of C conformally onto the unit disc of C.
  • W maps the real line R injectively into the unit circle T (complex numbers of absolute value 1). The image of R is T with 1 removed.
  • W maps the upper imaginary axis i [0, ∞) bijectively onto the half-open interval [−1, +1).
  • W maps 0 to −1.
  • W maps the point at infinity to 1.
  • W maps −i to the point at infinity (so W has a pole at −i).
  • W maps −1 to i.
  • W maps both 1⁄2(−1 + √3)(−1 + i) and 1⁄2(1 + √3)(1 − i) to themselves.

Operator map

An infinite-dimensional version of an inner product space is a Hilbert space, and we can no longer speak of matrices. However, matrices are merely representations of linear operators, and these we still have. So, generalizing both the matrix mapping and the complex plane mapping, we may define a Cayley transform of operators.

Here the domain of U, dom U, is (A+iI) dom A. See self-adjoint operator for further details.

See also

References

  • 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; reprinted as article 52 (pp. 332–336) in 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; translated from the Russian 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, translated by Robert B. Burckel from 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

fr:Transformation de Cayley it:Trasformata di Cayley ru:Преобразование Мёбиуса#Примеры uk:Перетворення Келі

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=47606"