Multiply connected

Retracing the argument used to justify point (2), it is clear that, in a multiply connected space, a given path is only coupled to those paths into which it can be continuously deformed. By definition, these are all the paths that belong to the same homotopy class. Paths belonging to different homotopy classes are thus decoupled from one another [41 5]. For a reactive system with a Cl that has the space of Fig. 1, this means that a path with a given winding number n is coupled to all paths with the same n, but is decoupled from paths with different n. As a result, the Kernel separates into [41-45]... 

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. 

Eq. (55) is the sum of all simple irreducible diagrams that can be formed among the k nodes, every bond representing an ffj function. (Thus every node is multiply connected. If k = 2 then B is exceptional and corresponds to the graph of two nodes and an /l2> bond.) It should be noted that flf is zero when i and j coincide (cf. Eq. (44)). 

The problem at hand is the evaluation of the activity coefficient defined in Eq. (76). It will be assumed that only pairwise interactions between the defects need be considered at the low defect concentrations we have in mind. (The theory can be extended to include non-pairwise forces.23) Then the cluster function R(n) previously defined in Eq. (78) is the sum of all multiply connected diagrams, in which each bond represents an /-function, which can be drawn among the set of n vertices, the /-function being defined by Eqs. (66), (56), and (43). The Helmholtz free energy of interaction of two defects appearing in this definition can be written as... 

Another example of a physical effect of this type is the Aharonov-Bohm effect, which is supported by a multiply connected vacuum configuration such as that described by the 0(3) gauge group [6]. The Aharonov-Bohm effect is a gauge transform of the true vacuum, where there are no potentials. In our notation, therefore the Aharonov-Bohm effect is due to terms such as (1/ )8 , depending on the geometry chosen for the experiment. It is essential for the Aharonov-Bohm effect to exist such that (1/ )8 be physical, and not random. It follows therefore that the vacuum configuration defined by the... 

It is only on this level that the link between helicity and topological quantization [103] can be understood properly. The 0(3) group, like the U(l) group, is multiply connected. The group space of U(l) is a circle [6, p. 105]. As explained earlier in this review, this is not simply connected because a path that goes twice... 

For a multiply connected medium, such as the volume of a torus bounded by the surface 5 is a magnetic surface, the preceding integral becomes... 

In other words, the gauge transform in a multiply connected medium can be represented by a modification of the helicity that denotes how loops C and C2 are linked, since we have... 

Note that Ce cannot be shrunk to zero without passing through VT since ST is a multiply connected surface. From the integral form of the first of (1), we have... 

It is the present writer s opinion that the existence of the obstruction changes the situation entirely. Without the existence of the solenoid in the interferometer, the loop of the two paths can be reduced to a single point and the region occupied by the interferometer is then simply connected. But with the existence of the solenoid, the loop of the two paths cannot be reduced to a single point and the region occupied by this special interferometer is multiply connected. The Aharonov-Bohm effect only exists in the multiply connected scenario. But we should note that the Aharonov-Bohm effect is a physical effect and simple and multiple connectedness are mathematical descriptions of physical situations. 

From a communications viewpoint the CCN is multiply noded (each osteo-cyte is a node) and multiply connected (Figure 4). Each osteocytic process is a connection between two osteocytes, and each osteocyte is multiply connected to a number of osteocytes that are near neighbors. Cell-to-cell communication is considered first below, then some speculative considerations of the ability of the CCN to compute as well as signal are described. It is useful to note the possibility that bone cells, like neurons, may communicate intercellular information by volume transmission, a process that does not require direct cytological contact, but rather utilizes charges in the environment [59, 117, 181]. 

Fig. 5. The Fermi surface of LaFe4Pj2 consists of two hole-like Fermi sheets. The first sheet (a) is nearly spherical with mainly Fe 3d character while the second sheet (b) is multiply connected with mainly P-p character. Both surfaces are centered at the r point. Various high-symmetry directions are noted in (a) (Harima, 1998 Sugawara et al., 2000).

A major part of the complexity arises from the existence of the free surface itself. Not only does the free surface dominate the flow, but it may become multiply connected when sprays, wavebreaking or cavitation occur. There is usually no steady state, and a model for transient flow of the fluid over the obstacle must be used. 

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