3-periodic network topologies

Recently there have been significant advances in mathematical tiling theory which have been applied to more rigorous descriptions of complex 3D (or 3-periodic) network topologies. The reader is referred to the literature for a complete description of these powerful new methods.3 4... 

The first structure of a three-dimensional transition metal network incorporating the oxalate ion was that of [Ni(phen)3][KCo (ox)3] 2H20 (phen= 1,10-phenanthroline) reported by Snow and co-workers in 1971. The true dimensionality of this compound, however, went unrecognized during this period, and the potential of oxalate ions to form three-dimensional networks was not fully realized until 1993, when Decurtins et al. published the crystal structure of the iron(II)-oxalato complex with tris(2,2 -bipyridine)iron(II) cations. This compound has an overall stoichiometry of [Fe (bipy)3] [Fe 2(ox)3] " and forms a three-dimensional anionic polymeric network that is best described with the three-connected decagon network topology. A view of this anionic network is shown in Figure 43. 

In three dimensions a number of large-scale selfassembling periodic structures exist. These include colloidal systems (Tarhan and Watson, 1996) and artificial opals (Vlasov et al, 1997). Unfortunately, these readily available materials do not satisfy the necessary criteria of high index contrast and correct network topology to produce a complete PEG. Theoretical studies, however, indicate the possibility of a complete PEG in closely related structures. Face-centered cubic lattices consisting... 

For a tight binding Hamiltonian we can make a distinction between quantitative (or cellular) and topological disorder. The Hamiltonian is defined by (a) a given set of nearest-neighbor relationships, which may not form a periodic network (this is topological disorder), and (b) matrix elements fluctuating from site to site (this is... 

The active micro-reactors described above cannot be recycled because the SiH moieties cannot be renewed. Recyelable micro-networks may be realized in the form of passive miero-reactors which do not actively take part in the reaction but merely provide the confined reaction space. For this purpose hollow micro-networks are synthesized first, a micro-emulsion of linear poly(dimethyl-siloxane) (PDMS) of low molar mass (M = 2000-3000 g/mol) is prepared and the endgroups are deactivated by reaction with methoxytrimethylsilane. Subsequent addition of trimethoxymethyl-silane leads to core-shell particles with the core formed by linear PDMS surrounded by a crosslinked network shell. Due to the extremely small mesh size of the outer network shell the PDMS ehains become topologically trapped and do not diffuse out of the micro-network over periods of several months (Fig. 3). However, if the mesh size of the outer shell is increased by condensation of trimethoxymethylsilane and dimethoxydimethylsilane the linear PDMS chains readily diffuse out of the network core and are removed by ultrafiltration. The remaining empty or hollow micro-network collapses upon drying (Fig. 4). So far, shape-persistent, hollow particles are prepared of approximately 20 nm radius, which may be viewed as structures similar to crosslinked vesicles. At this stage the reactants cannot be concentrated within the micro-network in respect to the continuous phase. 

The handover scenario is described as follows the PML polls the NML about networks in the local area. The NML sends this information about local networks and their topologies back to the PML. The PML uses this information to decide when, where, and to which network to handover and then gives that information to VHL to execute the handover at the chosen time. The VHL contacts the Reconfiguration layer in the core network to request a channel on the target Base-station. The RAL will acquire a channel on behalf of the mobile terminal. When the TBVH period expires, the VHL then instructs its NAL to handover. It is very important to see that the PML also informs the higher layers of impending handover, giving the new QoS, TBVH, and the IP address on the new network. This is used by the Handover Module in the QoS Framework to initiate QoS and security mechanisms to ensure a smooth vertical handover. 

Over a three-decade period, Feinberg [83] developed the criteria for answering chemical network stability questions, given any reaction with any given number of reactants. The basis for the so-called deficiency theorems is rooted in topology (structure), but in short summary, one takes each reactimi with each set of inputs and outputs and defines new quantities called complexes. After summing over all individual reactions, aU reactants and all complexes, one obtains the criterion. 

The most important benefit of living polymerizations is that they allow preparation of new macromolecules with precisely designed and controlled compositions (homopolymers, random, periodic, block, graft, and gradient copolymers), topologies (linear, star, comb, (hyper )branched, networks, etc), and fimctionalities placed at different parts of macromolecules or various combinations of these. Some of the possibilities are outlined below (244,245). 

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