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In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all linear fractional transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.

Definition

The Schwarzian derivative of a function of one complex variable ƒ is defined by

\begin{aligned} (S f) (z) & = {({\frac{f^{″} (z)}{f^{'} (z)}}^{'})}^{'} - \frac{1}{2} {(\frac{f^{″} (z)}{f^{'} (z)})}^{2} \\ = \frac{f^{‴} (z)}{f^{'} (z)} - \frac{3}{2} {(\frac{f^{″} (z)}{f^{'} (z)})}^{2} . \end{aligned}

The alternative notation

{f, z} = (S f) (z)

is frequently used.

Properties

The Schwarzian derivative of any fractional linear transformation

g (z) = \frac{a z + b}{c z + d}

is zero. Conversely, the fractional linear transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which a function fails to be a fractional linear transformation.

If g is a fractional linear transformation, then the composition g o f has the same Schwarzian derivative as f. On the other hand, the Schwarzian derivative of f o g is given by the chain rule

(S (f \circ g)) (z) = (S f) (g (z)) \cdot g^{'} (z)^{2} .

More generally, for any sufficiently differentiable functions f and g

S (f \circ g) = (S (f) \circ g) \cdot (g^{'})^{2} + S (g) .

This makes the Schwarzian derivative an important tool in one-dimensional dynamics ^[1] since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.

Introducing the function of two complex variables^[2]

F (z, w) = \log (\frac{f (z) - f (w)}{z - w}),

its second mixed partial derivative is given by

\frac{\partial^{2} F (z, w)}{\partial z \partial w} = \frac{f^{'} (z) f^{'} (w)}{(f (z) - f (w))^{2}} - \frac{1}{(z - w)^{2}},

and the Schwarzian derivative is given by the formula:

(S f) (z) = {6 \cdot \frac{\partial^{2} F (z, w)}{\partial z \partial w} |}_{z = w} .

The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has

(S w) (v) = - {(\frac{d w}{d v})}^{2} (S v) (w)

which follows from the inverse function theorem, namely that $v^{'} (w) = 1 / w^{'} .$

Differential equation

The Schwarzian derivative has a fundamental relation with a second-order linear ordinary differential equation in the complex plane.^[3] Let $f_{1} (z)$ and $f_{2} (z)$ be two linearly independent holomorphic solutions of

\frac{d^{2} f}{d z^{2}} + Q (z) f (z) = 0 .

Then the ratio $g (z) = f_{1} (z) / f_{2} (z)$ satisfies

(S g) (z) = 2 Q (z)

over the domain on which $f_{1} (z)$ and $f_{2} (z)$ are defined, and $f_{2} (z) \neq 0 .$ The converse is also true: if such a g exists, and it is holomorphic on a simply connected domain, then two solutions $f_{1}$ and $f_{2}$ can be found, and furthermore, these are unique up to a common scale factor.

When a linear second-order ordinary differential equation can be brought into the above form, the resulting Q is sometimes called the Q-value of the equation.

Note that the Gaussian hypergeometric differential equation can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.

Conditions for univalence

If f is a holomorphic function on the unit disc, D, then W. Kraus (1932) and Nehari (1949) proved that a necessary condition for f to be univalent is^[4]

| S (f) | \leq 6 .

Conversely if f(z) is a holomorphic function on D satisfying

| S (f) (z) | \leq 2 (1 - | z |^{2})^{- 2},

then Nehari proved that f is univalent.^[5]

In particular a sufficient condition for univalence is^[6]

| S (f) | \leq 2 .

Conformal mapping of circular arc polygons

The Schwarzian derivative and associated second order ordinary differential equation can be used to determine the Riemann mapping between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines. For polygons with straight edges, this reduces to the Schwarz–Christoffel mapping, which can be derived directly without using the Schwarzian derivative. The accessory parameters that arise as constants of integration are related to the eigenvalues of the second order differential equation. Already in 1890 Felix Klein had studied the case of quadrilaterals in terms of the Lamé differential equation.^[7]^[8]^[9]

Let Δ be a circular arc polygon with angles πα₁, ..., πα_n in clockwise order. Let f : H → Δ be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points a₁, ..., a_n on the real axis. Then p(z) = S(f)(z) is real-valued for x real and not one of the points. By the Schwarz reflection principle p(z) extends to a rational function on the complex plane with a double pole at a_i:

p (z) = \sum_{i = 1}^{n} \frac{(1 - α_{i}^{2})}{2 (z - a_{i})^{2}} + \frac{β_{i}}{z - a_{i}} .

The real numbers β_i are called accessory parameters. They are subject to 3 linear constraints:

\sum β_{i} = 0

\sum 2 a_{i} β_{i} + (1 - α_{i}^{2}) = 0

\sum a_{i}^{2} β_{i} + a_{i} (1 - α_{i}^{2}) = 0

which correspond to the vanishing of the coefficients of $z^{- 1}, z^{- 2}$ and $z^{- 3}$ in the expansion of p(z) around z = ∞. The mapping f(z) can then be written as

f (z) = \frac{u_{1} (z)}{u_{2} (z)},

where $u_{1} (z)$ and $u_{2} (z)$ are linearly independent holomorphic solutions of the linear second order ordinary differential equation

u^{''} (z) + \frac{1}{2} p (z) u (z) = 0 .

There are n−3 linearly independent accessory parameters, which can be difficult to determine in practise.

For a triangle, when n = 3, there are no accessory parameters. The ordinary differential equation is equivalent to the hypergeometric differential equation and f(z) can be written in terms of hypergeometric functions.

For a quadrilateral the accessory parameters depend on one independent variable λ. Writing U(z) = q(z)u(z) for a suitable choice of q(z), the ordinary differential equation takes the form

a (z) U^{''} (z) + b (z) U^{'} (z) + (c (z) + λ) U (z) = 0 .

Thus $q (z) u_{i} (z)$ are eigenfunctions of a Sturm-Liouville equation on the interval $[a_{i}, a_{i + 1}]$ . By the Sturm separation theorem, the non-vanishing of $u_{2} (z)$ forces λ to be the lowest eigenvalue.

Complex structure on Teichmüller space

Universal Teichmüller space is defined to be the space of real analytic quasiconformal mappings of the unit disc D, or equivalently the upper half-plane H, onto itself, with two mappings considered to be equivalent if on the boundary one is obtained from the other by composition with a Möbius transformation. Identifying D with the lower hemisphere of the Riemann sphere, any quasiconformal self-map f of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere $\tilde{f}$ onto itself. In fact $\tilde{f}$ is determined as the restriction to the upper hemisphere of the solution of the Beltrami differential equation

\frac{\partial F}{\partial \overline{z}} = μ (z) \frac{\partial F}{\partial z},

where μ is the bounded measurable function defined by

μ (z) = \frac{\frac{\partial f}{\partial \overline{z}}}{\frac{\partial f}{\partial z}}

on the lower hemisphere, extended to 0 on the upper hemisphere.

Identifying the upper hemisphere with D, Lipman Bers used the Schwarzian derivative to define a mapping

g = S (\tilde{f}),

which embeds universal Teichmüller space into an open subset U of the space of bounded holomorphic functions g on D with the uniform norm. Frederick Gehring showed in 1977 that U is the interior of the closed subset of Schwarzian derivatives of univalent functions.^[10]^[11]^[12]

For a compact Riemann surface S of genus greater than 1, its universal covering space is the unit disc D on which its fundamental group Γ acts by Möbius transformations. The Teichmüller space of S can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions g have the property that

g (z) d z^{2}

is invariant under Γ, so determine quadratic differentials on S. In this way, the Teichmüller space of S is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on S.

Diffeomorphism group of the circle

Let F_λ(S¹) be the space of tensor densities of degree λ on S¹. The group of orientation-preserving diffeomorphisms of S¹, Diff(S¹), acts on F_λ(S¹) via pushforwards. If f is an element of Diff(S¹) then consider the mapping

f \to S (f^{- 1}) .

In the language of group cohomology the chain-like rule above says that this mapping is a 1-cocycle on Diff(S¹)with coefficients in F₂(S¹). In fact

H^{1} (Diff (S^{1}); F_{2}) = R

and the 1-cocycle generating the cohomology is f → S(f⁻¹).

There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra Vect(S¹) of vector fields. This in turn gives the unique non-trivial central extension of Vect(S¹), the Virasoro algebra.

The group Diff(S¹) and its central extension also appear naturally in the context of Teichmüller theory and string theory.^[13] In fact the homeomorphisms of S¹ induced by quasiconformal self-maps of D are precisely the quasisymmetric homeomorphisms of S¹; these are exactly homeomorphisms which do not send four points with cross ratio 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homoemorphisms QS(S¹) by the subgroup of Möbius transformations Moeb(S¹). (It can also be realized naturally as the space of quasicircles in C.) Since

Moeb (S^{1}) \subset Diff (S^{1}) \subset QS (S^{1})

the homogeneous space Diff(S¹)/Moeb(S¹) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff(S¹) can be identified with the space of Hill's operators on S¹

\frac{d^{2}}{d θ^{2}} + q (θ),

and the coadjoint action of Diff(S¹) invokes the Schwarzian derivative. The inverse of the diffeomorphism f sends the Hill's operator to

\frac{d^{2}}{d θ^{2}} + f^{'} (θ)^{2} q \circ f (θ) + \frac{1}{2} S (f) (θ) .

Notes

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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, Chapter 6, "Teichmüller Spaces"
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, Chapter 10, "The Schwarzian".
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, Section 12, "Mapping of polygons with circular arcs".
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, "On the theory of generalized Lamé functions".
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
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

[1] Weisstein, Eric W. "Schwarzian Derivative." From MathWorld--A Wolfram Web Resource.

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+In [[mathematics]], the '''Schwarzian derivative''', named after the German mathematician [[Hermann Schwarz]], is a certain operator that is invariant under all [[linear fractional transformation]]s.  Thus, it occurs in the theory of the [[complex projective line]], and in particular, in the theory of [[modular forms]] and [[hypergeometric functions]]. It plays an important role in the theory of [[univalent function]]s, [[conformal mapping]] and [[Teichmüller space]]s.
+==Definition==
+The Schwarzian derivative of a function of one [[complex variable]] ''&fnof;'' is defined by
+:<math>\begin{align}
+(Sf)(z) & = \left( \frac{f''(z)}{f'(z)}\right)'  - \frac{1}{2}\left({f''(z)\over f'(z)}\right)^2 \\
+& = \frac{f'''(z)}{f'(z)}-\frac{3}{2}\left({f''(z)\over f'(z)}\right)^2.
+\end{align}</math>
+The alternative notation
+:<math>\{f,z\} = (Sf)(z)</math>
+is frequently used.
+==Properties==
+The Schwarzian derivative of any [[fractional linear transformation]]
+: <math>g(z) = \frac{az + b}{cz + d}</math>
+is zero. Conversely, the fractional linear transformations are the only functions with this property. Thus, the Schwarzian derivative precisely measures the degree to which  a function fails to be a fractional linear transformation.
+If ''g'' is a fractional linear transformation, then the composition ''g'' <small>o</small> ''f'' has the same Schwarzian derivative as ''f''. On the other hand, the Schwarzian derivative of ''f'' <small>o</small> ''g'' is given by the [[chain rule]]
+: <math>(S(f \circ g))(z) = (Sf)(g(z)) \cdot g'(z)^2.</math>
+<!--:{{bigmath|(S(<VAR >f</VAR > &#8728; <VAR >g</VAR >))(<VAR >z</VAR >) {{=}} (S<VAR >f</VAR >)(<VAR >g</VAR >(z)) &sdot; <VAR >g</VAR >&prime;(<VAR >z</VAR >)&sup2;}}-->
+More generally, for any sufficiently differentiable functions ''f'' and ''g''
+: <math>S(f \circ g) = \left( S(f)\circ g\right ) \cdot(g')^2+S(g).</math>
+<!--:{{bigmath|(S(<VAR >f</VAR > &#8728; <VAR >g</VAR >))(<VAR >z</VAR >) {{=}} (S<VAR >f</VAR >)(<VAR >g</VAR >(z)) &sdot; <VAR >g</VAR >&prime;(<VAR >z</VAR >)&sup2; + S(<VAR >g</VAR >)}}-->
+This makes the Schwarzian derivative an important tool in one-dimensional [[Dynamical system|dynamics]] <ref>[http://mathworld.wolfram.com/SchwarzianDerivative.html  Weisstein, Eric W. "Schwarzian Derivative." From MathWorld--A Wolfram Web Resource.]</ref> since it implies that all iterates of a function with negative Schwarzian will also have negative Schwarzian.
+Introducing the function of two complex variables<ref>{{harvnb|Schiffer|1966}}</ref>
+:<math>F(z,w)= \log \left ( \frac{f(z)-f(w)}{z-w} \right ),</math>
+its second mixed partial derivative is given by
+:<math> \frac{\partial^2 F(z,w)}{\partial z \partial w} = {f^\prime(z)f^\prime(w)\over(f(z)-f(w))^2}-{1\over(z-w)^2},</math>
+and the Schwarzian derivative is given by the formula:
+:<math> (Sf)(z)= \left. 6 \cdot {\partial^2 F(z,w)\over \partial z \partial w}\right\vert_{z=w}.</math>
+The Schwarzian derivative has a simple inversion formula, exchanging the dependent and the independent variables. One has
+:<math>(Sw)(v) = -\left(\frac{dw}{dv}\right)^2 (Sv)(w)</math>
+which follows from the [[inverse function theorem]], namely that <math>v'(w)=1/w'.</math>
+==Differential equation==
+The Schwarzian derivative has a fundamental relation with a second-order linear [[Complex differential equation|ordinary differential equation in the complex plane]].<ref>{{harvnb|Hille|1976}}</ref> Let <math>f_1(z)</math> and <math>f_2(z)</math> be two [[Wronskian|linearly independent]] [[holomorphic]] solutions of
+:<math>\frac{d^2f}{dz^2}+ Q(z) f(z)=0.</math>
+Then the ratio <math>g(z)=f_1(z)/f_2(z)</math> satisfies
+:<math>(Sg)(z) = 2Q(z)</math>
+over the domain on which <math>f_1(z)</math> and <math>f_2(z)</math>  are defined, and <math>f_2(z) \ne 0.</math> The converse is also true: if such a ''g'' exists, and it is holomorphic on a [[simply connected]] domain, then two solutions <math>f_1</math> and <math>f_2</math> can be found, and furthermore, these are unique [[up to]] a common scale factor.
+When a linear second-order ordinary differential equation can be brought into the above form, the resulting ''Q'' is sometimes called the '''Q-value''' of the equation.
+Note that the Gaussian [[hypergeometric differential equation]] can be brought into the above form, and thus pairs of solutions to the hypergeometric equation are related in this way.
+==Conditions for univalence==
+If ''f'' is a [[holomorphic function]] on the unit disc, '''D''', then W. Kraus (1932) and [[Zeev Nehari|Nehari]] (1949) proved that a ''necessary condition'' for ''f'' to be [[univalent function|univalent]] is<ref>{{harvnb|Lehto|1987|p=60}}</ref>
+:<math>|S(f)| \le 6.</math>
+Conversely if ''f''(''z'') is a holomorphic function on '''D''' satisfying
+:<math> |S(f)(z)| \le 2(1-|z|^2)^{-2},</math>
+then Nehari proved that ''f'' is univalent.<ref>{{harvnb|Duren|1983}}</ref>
+In particular a ''sufficient condition'' for univalence is<ref>{{harvnb|Lehto|1987|p=90}}</ref>
+:<math> |S(f)|\le 2.</math>
+==Conformal mapping of circular arc polygons==
+The Schwarzian derivative and associated second order ordinary differential equation can be used to determine the [[Riemann mapping]] between the upper half-plane or unit circle and any bounded polygon in the complex plane, the edges of which are circular arcs or straight lines.  For polygons with straight edges, this reduces to the [[Schwarz–Christoffel mapping]], which can be derived directly without using the Schwarzian derivative. The ''accessory parameters'' that arise as constants of integration are related to the [[Spectral theory of ordinary differential equations|eigenvalues]] of the second order differential equation. Already in 1890 [[Felix Klein]] had studied the case of quadrilaterals in terms of the [[Lamé function|Lamé differential equation]].<ref>{{harvnb|Nehari|1953}}</ref><ref>{{harvnb|von Koppenfels|Stallmann|1959}}</ref><ref>{{harvnb|Klein|1922}}</ref>
+Let Δ be a circular arc polygon with angles πα<sub>1</sub>, ..., πα<sub>''n''</sub> in clockwise order. Let ''f'' : '''H''' → Δ be a holomorphic map extending continuously to a map between the boundaries. Let the vertices correspond to points ''a''<sub>1</sub>, ..., ''a<sub>n</sub>'' on the real axis. Then ''p''(''z'') = ''S''(''f'')(''z'') is real-valued for ''x'' real and not one of the points. By the [[Schwarz reflection principle]] ''p''(''z'') extends to a rational function on the complex plane with a double pole at ''a<sub>i</sub>'':
+:<math> p(z)=\sum_{i=1}^n \frac{(1-\alpha_i^2)}{2(z-a_i)^2} + \frac{\beta_i}{z-a_i}.</math>
+The real numbers β<sub>''i''</sub> are called ''accessory parameters''. They are subject to ''3'' linear constraints:
+:<math>\sum \beta_i=0</math>
+:<math> \sum 2a_i \beta_i + \left ( 1-\alpha_i^2 \right ) =0</math>
+:<math> \sum a_i^2 \beta_i + a_i \left ( 1-\alpha_i^2 \right ) =0</math>
+which correspond to the vanishing of the coefficients of <math> z^{-1}, z^{-2}</math> and  <math>z^{-3}</math> in the expansion of ''p''(''z'') around ''z'' = ∞. The mapping ''f''(''z'') can then be written as
+:<math> f(z) = {u_1(z)\over u_2(z)},</math>
+where <math>u_1(z)</math> and <math>u_2(z)</math> are linearly independent holomorphic solutions of the linear second order ordinary differential equation
+:<math> u^{\prime\prime}(z) + \tfrac{1}{2} p(z)u(z)=0.</math>
+There are ''n''−3 linearly independent accessory parameters, which can be difficult to determine in practise.
+For a triangle, when ''n'' = 3, there are no accessory parameters. The ordinary differential equation is equivalent to the [[hypergeometric differential equation]] and ''f''(''z'') can be written in terms of [[hypergeometric function]]s.
+For a quadrilateral the accessory parameters depend on one independent variable λ. Writing ''U''(''z'') = ''q''(''z'')''u''(''z'') for a suitable choice of ''q''(''z''), the ordinary differential equation takes the form
+:<math> a(z) U^{\prime\prime}(z) + b(z) U^\prime(z) +(c(z)+\lambda)U(z)=0.</math>
+Thus <math>q(z) u_i(z)</math> are eigenfunctions of a [[Sturm-Liouville equation]] on the interval <math>[a_i,a_{i+1}]</math>. By the [[Sturm separation theorem]], the non-vanishing of <math>u_2(z)</math> forces λ to be the lowest eigenvalue.
+==Complex structure on Teichmüller space==
+[[Universal Teichmüller space]] is defined to be the space of [[real analytic]] [[quasiconformal mapping]]s of the unit disc '''D''', or equivalently the [[upper half-plane]] '''H''', onto itself, with two mappings considered to be equivalent if on the boundary one is obtained from the other by composition with a [[Möbius transformation]]. Identifying '''D''' with the lower hemisphere of the [[Riemann sphere]], any quasiconformal self-map ''f'' of the lower hemisphere corresponds naturally to a conformal mapping of the upper hemisphere <math>\tilde{f}</math> onto itself. In fact <math>\tilde{f}</math> is determined as the restriction to the upper hemisphere of the solution of the [[Beltrami differential equation]]
+:<math> \frac{\partial F}{\partial \overline{z}} = \mu(z) \frac{\partial F}{\partial z},</math>
+where μ is the bounded measurable function defined by
+:<math>\mu(z) = {{\partial f\over \partial \overline{z}}\over{\partial f\over \partial z}}</math>
+on the lower hemisphere, extended to 0 on the upper hemisphere.
+Identifying the upper hemisphere with '''D''', [[Lipman Bers]] used the Schwarzian derivative to define a [[Bers embedding|mapping]]
+:<math> g= S(\tilde{f}),</math>
+which embeds universal Teichmüller space into an open subset ''U'' of the space of bounded holomorphic functions ''g'' on '''D''' with the [[uniform norm]].  [[Frederick Gehring]] showed in 1977 that ''U'' is the interior of the closed subset of Schwarzian derivatives of univalent functions.<ref>{{harvnb|Ahlfors|1966}}</ref><ref>{{harvnb|Lehto|1987}}</ref><ref>{{harvnb|Imayoshi|Taniguchi|1992}}</ref>
+For a [[compact Riemann surface]] ''S'' of genus greater than 1, its [[universal covering space]] is the unit disc '''D''' on which its fundamental group Γ acts by Möbius transformations. The [[Teichmüller space]] of ''S'' can be identified with the subspace of the universal Teichmüller space invariant under Γ. The holomorphic functions ''g'' have the property that
+:<math>g(z) dz^{2}</math>
+is invariant under Γ, so determine [[quadratic differential]]s on ''S''. In this way, the Teichmüller space of ''S'' is realized as an open subspace of the finite-dimensional complex vector space of quadratic differentials on ''S''.
+==Diffeomorphism group of the circle==
+Let ''F''<sub>λ</sub>('''S'''<sup>1</sup>) be the space of [[tensor density|tensor densities]] of degree λ on '''S'''<sup>1</sup>. The group of orientation-preserving diffeomorphisms of '''S'''<sup>1</sup>, Diff('''S'''<sup>1</sup>), acts on ''F''<sub>λ</sub>('''S'''<sup>1</sup>) via [[pushforward (differential)|pushforwards]]. If ''f'' is an element of Diff('''S'''<sup>1</sup>) then consider the mapping
+:<math>f \to S(f^{-1}).</math>
+In the language of [[group cohomology]] the chain-like rule above says that this mapping is a 1-cocycle on Diff('''S'''<sup>1</sup>)with coefficients in ''F''<sub>2</sub>('''S'''<sup>1</sup>). In fact
+:<math>H^1(\text{Diff}(\mathbf{S}^1);F_2) = \mathbf{R}</math>
+and the 1-cocycle generating the cohomology is ''f'' → ''S''(''f''<sup>−1</sup>).
+There is an infinitesimal version of this result giving a 1-cocycle for the Lie algebra Vect('''S'''<sup>1</sup>) of [[vector field]]s. This in turn gives the unique non-trivial central extension of Vect('''S'''<sup>1</sup>), the [[Virasoro algebra]].
+The group Diff('''S'''<sup>1</sup>) and its central extension also appear naturally in the context of Teichmüller theory and [[string theory]].<ref>{{harvnb|Pekonen|1995}}</ref> In fact the homeomorphisms of '''S'''<sup>1</sup> induced by quasiconformal self-maps of '''D''' are precisely the [[quasisymmetric map|quasisymmetric homeomorphisms]] of '''S'''<sup>1</sup>; these are exactly homeomorphisms which do not send four points with [[cross ratio]] 1/2 to points with cross ratio near 1 or 0. Taking boundary values, universal Teichmüller can be identified with the quotient of the group of quasisymmetric homoemorphisms QS('''S'''<sup>1</sup>) by the subgroup of Möbius transformations Moeb('''S'''<sup>1</sup>). (It can also be realized naturally as the space of [[quasicircle]]s in '''C'''.) Since
+:<math>\text{Moeb}(\mathbf{S}^1)\subset \text{Diff}(\mathbf{S}^1) \subset \text{QS}(\mathbf{S}^1)</math>
+the [[homogeneous space]] Diff('''S'''<sup>1</sup>)/Moeb('''S'''<sup>1</sup>) is naturally a subspace of universal Teichmüller space. It is also naturally a complex manifold and this and other natural geometric structures are compatible with those on Teichmüller space. The dual of the Lie algebra of Diff('''S'''<sup>1</sup>) can be identified with the space of [[Hill's equation|Hill's operators]] on '''S'''<sup>1</sup>
+:<math>{d^2\over d\theta^2} + q(\theta),</math>
+and the [[coadjoint action]] of Diff('''S'''<sup>1</sup>) invokes the Schwarzian derivative. The inverse of the diffeomorphism ''f'' sends the Hill's operator to
+:<math>{d^2\over d\theta^2} + f^\prime(\theta)^2 \,q\circ f(\theta) + \tfrac{1}{2} S(f)(\theta).</math>
+==Notes==
+{{reflist|2}}
+==References==
+*{{citation|first=Lars|last=Ahlfors|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966|pages=117–146}}, Chapter 6, "Teichmüller Spaces"
+*{{citation|last=Duren|first=Peter L.|title=Univalent functions|publisher=Springer-Verlag|series=Grundlehren der Mathematischen Wissenschaften |volume=259|year= 1983|isbn= 0-387-90795-5|pages=258–265}}
+*{{citation|first=Einar|last=Hille|authorlink=Einar Hille|title=Ordinary differential equations in the complex domain|publisher=Dover|year=1976|isbn=0-486-69620-0|pages=374–401}}, Chapter 10,  "The Schwarzian".
+*{{citation|first=Y.|last=Imayoshi|first2=M.|last2=Taniguchi|title=An introduction to Teichmüller spaces|publisher=Springer-Verlag|year=1992|isbn=4-431-70088-9}}
+*{{citation|first=W.|last=von Koppenfels|first2=F.|last2=Stallmann|title=Praxis der konformen Abbildung|year=1959|publisher=Springer-Verlag|series=Die Grundlehren der mathematischen Wissenschaften|volume=100|pages=114–141}}, Section 12, "Mapping of polygons with circular arcs".
+*{{citation|first=Felix|last=Klein|authorlink=Felix Klein|title=Collected works|volume=2|pages=540–549|url=http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN237843552|publisher=Springer-Verlag
+|year=1922}}, "On the theory of generalized Lamé functions".
+*{{citation|first=Otto|last=Lehto|title=Univalent functions and  Teichmüller spaces|publisher=Springer-Verlag|year=1987|isbn=0-387-96310-3|pages=50–59, 111–118, 196–205}}
+*{{citation | last1=Nehari | first1=Zeev |authorlink=Zeev Nehari| title=The Schwarzian derivative and schlicht functions | doi=10.1090/S0002-9904-1949-09241-8  | id={{MathSciNet | id = 0029999}} | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=545–551}}
+*{{citation|first=Zeev|last=Nehari|authorlink=Zeev Nehari|title=Conformal mapping|publisher=Dover|year=1952|isbn=0-486-61137-X|pages=189–226}}
+*{{citation|first=V. |last=Ovsienko|first2=S.|last2= Tabachnikov |title=Projective Differential Geometry Old and New|publisher= Cambridge University Press|year=2005|isbn=0-521-83186-5}}
+* {{citation | first1 = Valentin | last1 = Ovsienko | first2 = Sergei | last2 = Tabachnikov | title = What Is . . . the Schwarzian Derivative? | journal = AMS Notices | volume = 56 | issue = 01 | pages = 34–36 | year = 2009
+    | url = http://www.ams.org/notices/200901/tx090100034p.pdf }}
+*{{citation|last=Pekonen|first=Osmo|title=Universal Teichmüller space in geometry and physics|journal=J. Geom. Phys.|volume=15|year= 1995|pages= 227–251|
+url=http://www.sciencedirect.com/science/article/pii/039304409400007Q|doi=10.1016/0393-0440(94)00007-Q}}
+* {{citation|title=Half-Order Differentials on Riemann Surfaces|first=
+Menahem|last= Schiffer|journal=SIAM Journal on Applied Mathematics|volume=14|pages=922–934|year=1966|jstor=2946143}}
+[[Category:Projective geometry]]
+[[Category:Modular forms]]
+[[Category:Ordinary differential equations]]
+[[Category:Complex analysis]]
+[[Category:Conformal mapping]]

Mutation (genetic algorithm): Difference between revisions

Revision as of 06:43, 3 March 2013

Contents

Definition

Properties

Differential equation

Conditions for univalence

Conformal mapping of circular arc polygons

Complex structure on Teichmüller space

Diffeomorphism group of the circle

Notes

References

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Mutation (genetic algorithm): Difference between revisions

Revision as of 06:43, 3 March 2013

Definition

Properties

Differential equation

Conditions for univalence

Conformal mapping of circular arc polygons

Complex structure on Teichmüller space

Diffeomorphism group of the circle

Notes

References

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