Ramanujan theta function

In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation invented by William Thurston, who also coined the term "pseudo-Anosov diffeomorphism" when he proved his classification of diffeomorphisms of a surface.

Definition of a measured foliation

A measured foliation F on a closed surface S is a geometric structure on S which consists of a singular foliation and a measure in the transverse direction. In some neighborhood of a regular point of F, there is a "flow box" φ: U → R² which sends the leaves of F to the horizontal lines in R². If two such neighborhoods U_i and U_j overlap then there is a transition function φ_ij defined on φ_j(U_j), with the standard property

${\varphi }_{ij}\circ {\varphi }_j={\varphi }_i,$

which must have the form

$\varphi (x,y)=(f(x,y),c\pm y)$

for some constant c. This assures that along a simple curve, the variation in y-coordinate, measured locally in every chart, is a geometric quantity (i.e. independent of the chart) and permits the definition of a total variation along a simple closed curve on S. A finite number of singularities of F of the type of "p-pronged saddle", p≥3, are allowed. At such a singular point, the differentiable structure of the surface is modified to make the point into a conical point with the total angle πp. The notion of a diffeomorphism of S is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.

Definition of a pseudo-Anosov map

A homeomorphism

$f:S\to S$

of a closed surface S is called pseudo-Anosov if there exists a transverse pair of measured foliations on S, F^s (stable) and F^u (unstable), and a real number λ > 1 such that the foliations are preserved by f and their transverse measures are multiplied by 1/λ and λ. The number λ is called the stretch factor or dilatation of f.

Significance

Thurston constructed a compactification of the Teichmüller space T(S) of a surface S such that the action induced on T(S) by any diffeomorphism f of S extends to a homeomorphism of the Thurston compactification. The dynamics of this homeomorphism is the simplest when f is a pseudo-Anosov map: in this case, there are two fixed points on the Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the Poincaré half-plane. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.

Generalization

Using the theory of train tracks, the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of free groups (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by Bestvina and Handel.

References

• A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988).
• A. Fathi, F. Laudenbach, and V. Poénaru, "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979).
• R.C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197
• 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