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m Various citation cleanup (identifiers mostly), replaced: | url=http://dx.doi.org/10.1006/aama.2000.0700 → | doi=10.1006/aama.2000.0700, | id={{MR|1783553}} → | mr=1783553, typos fixed: , → ,, added Empty section (1) t using AWB
 
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In [[mathematics]], the '''measurable Riemann mapping theorem''' is a theorem proved in 1960 by [[Lars Ahlfors]] and [[Lipman Bers]] in [[complex analysis]] and [[geometric function theory]]. Contrary to its name, it is not a direct generalization of the [[Riemann mapping theorem]], but instead a result concerning [[quasiconformal mapping]]s and solutions of the [[Beltrami equation]]. The result was prefigured by earlier results of [[Charles Morrey]] from 1938 on quasi-linear [[elliptic partial differential equation]]s.  
 
The theorem of Alhfors and Bers states that if μ is a bounded measurable function on '''C''' with <math>\|\mu\|_\infty < 1</math>, then there is a
unique solution ''f'' of the Beltrami equation
 
:<math> \partial_{\overline{z}} f(z) = \mu(z) \partial_z f(z)</math>
 
for which ''f'' is a quasiconformal homeomorphism of '''C''' fixing the points 0, 1 and ∞. A similar result is true with '''C''' replaced by the [[unit disk]] '''D'''. Their proof used the [[Beurling transform]], a [[singular integral operator]].
==References==
*{{citation|first=Lars|last=Ahfors|first2=Lipman|last2=Bers|title=Riemann mapping's theorem for variable metrics|journal= Annals of Math.|volume= 72|year=1960|pages= 385–404}}
*{{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966}}
*{{citation|title=Elliptic partial differential equations and quasiconformal mappings in the plane|volume= 48|series= Princeton mathematical series|
first=Kari|last= Astala|first2= Tadeusz |last2=Iwaniec|first3= Gaven|last3= Martin|publisher=Princeton University Press|year= 2009|
isbn=0-691-13777-3|pages=161–172}}
*{{citation|last=Carleson|first=L.|last2= Gamelin|first2= T. D. W.|title=Complex dynamics|series=
Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year=1993|isbn=0-387-97942-5}}
*{{citation
  | title = On the solutions of quasi-linear elliptic partial differential equations
  | first = Charles B. Jr.
  | last = Morrey
  | authorlink = Charles B. Morrey, Jr.
  | journal = [[Transactions of the American Mathematical Society]]
  | volume = 43
  | year = 1938
  | pages = 126–166
  | doi = 10.2307/1989904
  | issue = 1
  | jstor = 1989904
  | jfm = 62.0565.02
  | mr = 1501936
  | zbl = 0018.40501
}}
*{{citation|title=When ellipses look like circles: the measurable Riemann mapping theorem|first=Saeed|last=Zakeri|first2=Mahmood|last2=Zeinalian|url=http://www.math.qc.edu/~zakeri/papers/ahl-bers.pdf|journal=Nashr-e-Riazi|volume= 8 |year=1996|pages=5–14}}
 
[[Category:Theorems in complex analysis]]
 
{{mathanalysis-stub}}

Revision as of 04:38, 29 September 2012

In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and solutions of the Beltrami equation. The result was prefigured by earlier results of Charles Morrey from 1938 on quasi-linear elliptic partial differential equations.

The theorem of Alhfors and Bers states that if μ is a bounded measurable function on C with μ<1, then there is a unique solution f of the Beltrami equation

zf(z)=μ(z)zf(z)

for which f is a quasiconformal homeomorphism of C fixing the points 0, 1 and ∞. A similar result is true with C replaced by the unit disk D. Their proof used the Beurling transform, a singular integral operator.

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.

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    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
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    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

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  • 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.

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    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.

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  • 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.

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    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.

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  • 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.

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    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.

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    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.

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