Diagram of topological spaces

Definition 15.1. The following data constitute what is called a diagram of topological spaces T> over a trisp A ... 

Definition 15.3. Let T> be a diagram of topological spaces over a trisp A. A colimit ofV is the quotient space colim D = where the... 

Again, the topological spaces and continuous maps can be replaced with any other category. In fact, when C is a category, a diagram of topological spaces over C is nothing but a functor T> C Top. 

Proof of Theorem 19.16. For convenience of notation we set d = h(X). Let ip X be a cellular Z2-map, and consider the commuting diagram of topological spaces and continuous maps shown in Figure 19.3, where the vertical arrows correspond to quotient maps. 

The subunits are arranged in the crystals as homotetramers with D2 symmetry. The structure of a subunit is shown schematically in Fig. 1 (87). Each subunit of 552 amino acid residues has a globular shape with dimensions of 49 x 53 x 65 A and is built up of three domains arranged sequentially on the polypeptide chain, tightly associated in space. The folding of all three domains is of a similar jS-barrel type. It is distantly related to the small blue copper proteins, for example, plastocyanin or azurin. Domain 1 is made up of two four-stranded jS-sheets (Fig. lb), which form a jS-sandwich structure. Domain 2 consists of a six-stranded and a five-stranded jS-sheet. Finally, domain 3 is built up of two five-stranded jS-sheets that form the jS-barrel structure and a four-stranded j8-sheet that is an extension at the N-terminal part of this domain. A topology diagram of ascorbate oxidase for all three domains and of the related structures of plastocyanin and azurin is shown in Fig. 2. Ascorbate oxidase contains seven helices. Domain 2 has a short a-helix (aj) between strands A2 and B2. Domain 3 exhibits five short a-helices that are located between strands D3 and E3 (a ), 13 and J3 (a ), and M3 and N3 (a ) as well as at the C terminus (ag and a ). Helix 2 connects domain 2 and domain 3. 

Figure 21. Mutant space high-value contour near local optimum. Diagram is multiply branched tree with different macromolecular sequences at vertices. Each line joins neighboring sequences whose values are within 0.5 of locally optimum sequence at lower center for linearized fitness function of type 2 [Eqn. (IV.7)] and reference fold that is cruciform, like tRNA, for sequence of length 72. Over 1300 branches shown extending up to 10 mutant shells away from central optimum. Better sequence (labeled optimum) was found in tenth mutant shell. Non-random sampling of mutant sequences demonstrated typical of population sampling in quasispecies model. Note small number of ridges that penetrate deeply into surrounding mutant space. (Additional connected paths due to hypercube topology of mutant space not shown.)...

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

Assume that X is a topological space with base point x, and assume that A and B are path-connected open subspaces of X such that AC B is also path connected, x AC B, and X = AU B. Then ni(X,x) is the colimit of the following diagram ... 

We note that when considering a poset P, in order to specify a diagram of spaces over A P), by the commutativity condition we need only give a topological space T> x) for every x G P and to describe a continuous map V x —> y) V x) —> T> y) for each pair x,y P such that x covers y. 

It is now time to formalize an important property of homotopy colimits their flexibility with respect to homotopy type. It turns out that if the topological spaces in a diagram are replaced by homotopy equivalent ones in a coherent way (as a diagram map), then the homotopy type of the homotopy colimit... 

The second result concerns the fact that any interfacial curvature can not be associated with any interfacial distance when an ideal structure is built in a curved space, they are indeed related by a specific relation in each topological class. This set of relations is represented in Fig. 11, it is such that, when the frustration varies monotonically with the parameters of the phase diagram, the topologies must appear along a sequence similar to that observed in phase diagrams. For instance, when the water content is increased in a system with an amphiphile with one chain, a... 

SmA phases, and SmA and SmC phases, meet tlie line of discontinuous transitions between tire N and SmC phase. The latter transition is first order due to fluctuations of SmC order, which are continuously degenerate, being concentrated on two rings in reciprocal space ratlier tlian two points in tire case of tire N-SmA transition [18,19 and 20], Because tire NAC point corresponds to the meeting of lines of continuous and discontinuous transitions it is an example of a Lifshitz point (a precise definition of tliis critical point is provided in [18,19 and 20]). The NAC point and associated transitions between tire tliree phases are described by tire Chen-Lubensky model [97], which is able to account for tire topology of tire experimental phase diagram. In tire vicinity of tire NAC point, universal behaviour is predicted and observed experimentally [20]. 

Figure 1. (a) Diagram illustrating the topology of the iV-dimensional nuclear coordinate space... 

A continuous connected group may be simply connected or multiply connected, depending on the topology of the parameter space. A subset of the euclidean space Sn is said to be k-fold connected if there are precisely k distinct paths connecting any two points of the subset which cannot be brought into each other by continuous deformation without going outside the subset. A schematic of four-fold connected space is shown in the lower diagram. 

A radius-diameter diagram is defined as a bivariate distribution of the - data set compounds in the space defined by the molecular radius and diameter it provides a summary of the similarities among the molecule chemical shapes in the topological or geometrical space. 

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