Standard geometric realization

Definition 2.27. Given a finite abstract simplicial complex A, we define its standard geometric realization to be the topological space obtained by taking the union of standard a-simplices in for all a A. 

Any topological space that is homeomorphic to the standard geometric realization of A is called the geometric realization of A, and is denoted by Z. ... 

When the abstract simplicial complex A is finite, then we can take its standard geometric realization and take the barycentric subdivisions of the individual n-simplices as just described. This gives the geometric realization of the barycentric subdivision of A. When, on the other hand, the abstract simplicial complex A is infinite, we can take the barycentric subdivisions of its simplices before the gluing and then observe that the gluing process is compatible with the new cell structure hence we will obtain the geometric realization of the barycentric subdivision of A as well. 

Microstructure in Single Phase Polycrystals. We begin with a discussion of the simplest microstructure (at least to characterize geometrically) in which the grain size is the dominant microstructural parameter. We are all used to the idea that the technologies that surround us are constrained by various codes which attempt to standardize products and the processes used to realize them. Such codes exist for everything ranging from skyscrapers to microwave ovens. It is... 

CAD interfaces interconnect internal CAD system components and provide a link to other software and hardware components. They connect internal CAD system components such as geometrical modeling, object representation, and realization of calculation operations to a CAD database tuid standardized CAD files. 

Proof of Theorem 5.3.1 Let a G Af and let V and U be such neighbourhoods of the point x that xGV cV cU, where V is the closure of V and over U (and therefore over V and P) we have a universal covering, and the cotangent bundle is trivial. Let p L Af be a standard projection, p ( ) n (L P) = Introduce the equivalence relation between the near connectedness components Qa if Qa C P and Qp C P, where P C p (U) n (L F) and is linearly connected. FVom each equivalence class choose one element and obtain (from the condition 3) of geometric simplicity) a finite set ( i,..., ). Since the manifold is complete, any element g G Xi M) can be realized by a geodesic loop with the vertex at a point x. If a loop lies in F, then its initial data can be arbitrarily little stirred with the result that the new trajectory lying sufficiently close to the... 

A common means to perform such mapping is to use an approximate method like the Nataf s model (Liu and Der Kiureghian 1986). Once the random variables involved in Equation 1 have been expressed into the standard normal space, it is possible to define the design point (x ) using a geometrical or probabilistic interpretation (see, e.g. (Freudenthal 1956)). In the geometrical interpretation, the design point is defined as the realization in the standard normal space which lies on the limit state function... 

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