Fixation index

In geometry, Dehn constructed two examples of planes, a semi-Euclidean geometry and a non-Legendrian geometry, that have infinitely many lines parallel to a given one that pass through a given point, but where the sum of the angles of a triangle is at least π. A similar phenomenon occurs in hyperbolic geometry, except that the sum of the angles of a triangle is less than π. Dehn's examples use a non-Archimedean field, so that the Archimedean axiom is violated. They were introduced by Template:Harvs and discussed by Template:Harvtxt.

Dehn's non-archimedean field Ω(t)

To construct his geometries, Dehn used a non-Archimedean ordered Pythagorean field Ω(t), a Pythagorean closure of the field of rational functions R(t), consisting of the smallest field of real-valued rational functions containing the real constants, the identity function t (taking any real number to itself) and closed under the operation ω → √(1+ω²). The field Ω(t) is ordered by putting x>y if the function x is larger than y for sufficiently large reals. An element x of Ω(t) is called finite if m<x<n for some integers m,n, and is called infinite otherwise.

Dehn's semi-Euclidean geometry

The set of all pairs (x, y), where x and y are any (possibly infinite) elements of the field Ω(t), and with the usual metric

$\left|\right|(x,y)\left|\right|=\sqrt{x^2+y^2},$

which takes values in Ω(t), gives a model of Euclidean geometry. The parallel postulate is true in this model, but if the deviation from the perpendicular is infinitesimal (meaning smaller than any positive rational number), the intersecting lines intersect at a point that is not in the finite part of the plane. Hence, if the model is restricted to the finite part of the plane (points (x,y) with x and y finite), a geometry is obtained in which the parallel postulate fails but the sum of the angles of a triangle is π. This is Dehn's semi-Euclidean geometry.

Dehn's non-Legendrian geometry

In the same paper, Dehn also constructed an example of a non-Legendrian geometry where there are infinitely many lines through a point not meeting another line, but the sum of the angles in a triangle exceeds π. Riemann's elliptic geometry over Ω(t) consists of the projective plane over Ω(t), which can be identified with the affine plane of points (x:y:1) together with the "line at infinity", and has the property that the sum of the angles of any triangle is greater than π The non-Legendrian geometry consists of the points (x:y:1) of this affine subspace such that tx and ty are finite (where as above t is the element of Ω(t) represented by the identity function). Legendre's theorem states that the sum of the angles of a triangle is at most π, but assumes Archimedes's axiom, and Dehn's example shows that Legendre's theorem need not hold if Archimedes' axiom is dropped.

References

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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.
  
  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.
  
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