Hypothetical networks

To help answer this question, there has been a long history of geometric and mathematical analysis of observed and hypothetical periodic networks. As a 

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Early theories of Guth, Kuhn, Wall and others proceeded on the assumption that the microscopic distribution of end-to-end vectors of the chains should reflect the macroscopic dimensions of the specimen, i.e., that the chain vectors should be affine in the strain. The pivotal theory of James and Guth (1947), put forward subsequently, addressed a network of Gaussian chains free of all interactions with one another, the integrity of the chains which precludes one from the space occupied by another being deliberately left out of account. Hypothetical networks of this kind came to be known later as phantom networks (Flory, 1964,... 

Example 2.1. Compilation of rate equations for a hypothetical network. Assume the network is... 

FBA is attractive as a predictive tool. That is, once the model is constructed, one can predict the metabolic behavior under a number of different conditions. The basic principle underlying FBA is the steady-state conservation of mass, energy, and redox potential. A dynamic mass balance can be written arormd each metabolite (A,) within a metabolic network. Fig. 4 shows a hypothetical network with the fluxes (V) affecting a metabolite (A,). The dynamic mass balance for A,- is ... 

Oscillations may involve periods with more than a single wave, and may be chaotic (with no recurring periods at all). Aperiodic behavior is called chaos, but is not random The seeming randomness results from the fact that minutes difference in starting conditions can lead to drastically different behavior (butterfly effect of meteorology). Even relatively simple hypothetical networks can produce chaotic behavior. However, while periodic oscillation are possible in networks with only two independent mathematical variables, chaos requires at least three. 

Figure 1. In this hypothetical network of chemical reactions, called a reaction graph, smaller molecules (A and B) are combined to form larger molecules (AA, AB, etc.) which in turn are combined to form still larger molecules (BAB, BBA, BABB, etc.). Simultaneously, these longer molecules are broken down into simple substrates again. For each reaction, a line leads from the two substrates to a square denoting the reaction, an arrow leads from the reaction square to the product. (Since reactions are reversible, the use of arrows is meant to distinguish substrates from products only in one direction of the chemical flow.) Since the products of some reactions are substrates of further reactions, the result is a web of interlinked reactions. (Reproduced with permission from reference 9. Copyright 2000 Oxford University Press.)...

Suppose that C is, in fact, obtained from a mixture of other points contained in X. This simation is shown in Figure 6.4 for a set containing six distinct concentrations Cj, C2,. .., Cg obtained from the hypothetical network. If C is a mixture of the set, then it must be expressible by ... 

Figure 11 (a) Model of the hypothetical network expected to form with NTCDI and melamine, (b), (c), and (d) Models of three of... 

The present project was designed to provide a better understanding of the mechanism of entrapment and hysteresis. To this end, mathematical models of various hypothetical networks of different connectivity have been developed. 

Figure 5.4. Example of a small region of a hypothetical protein interaction network. Each letter represents a different yeast protein. The white boxes and gray boxes represent genes that are involved in the same function while the hatched boxes indicate proteins of unknown function. The A protein is likely to be involved in the same process as the white box protein and the J protein is likely to be involved in the same process as the gray box proteins because of the multiple interactions within the network. The connection between the E and I proteins indicates communication between the cellular processes. Figure adapted from Hazbun and Fields (2001).

Fig. 10.11 The modified tree of life still has the usual tree-like structure and also confirms that the eukaryotes originally took over mitochondria and chloroplasts from bacteria. It does, however, also show a network of links between the branches. The many interconnections indicate a frequent transfer of genes between unicellular organisms. The modified tree of life is not derived, as had previously been assumed, from a single cell (the hypothetical primeval cell ). Instead, the three main kingdoms are more likely to have developed from a community of primitive cells with different genomes (Doolittle, 2000)...

Figure 4.9 Forces acting in the x, y and z directions on a hypothetical diamond-shaped portion of a fibre network.

Even though k2 is a hypothetical rate constant for many reaction chain systems within the overall network of reactions in the reacting media and hence... 

Interpenetrating polymer networks of PMAA/PNIPAAm were prepared. The material exhibited both pH and temperature sensitivities. The temperature sensitivity was verified by DSC studies. The permeation studies showed a significant size exclusion phenomenon of the IPNs. The influence of pH and temperature on the IPN permeability was investigated. A hypothetical mechanism was presented to explain the high permeability at high temperature and high pH. 

From these equations one can work out what sets of atyx numbers are possible for known or hypothetical boranes B H n+m and so rationalize known atomic networks or predict new ones. For example, the atyx numbers for the compounds shown in Fig. 4 are B4Hxq, 4012 BsHu, 3203 andBgHio, 4220. 

Hypothetical behaviour in which chains can move freely through one another when a network is deformed. 

Figure 13. a-Cyclodextrin-based bimotecular ion channels, (a) Hexatcar-boxymethyh-substituted a-cydodextrin combined with dioctadecyldimethylam-monium cation to afford hexa ion pairs 9, hexa(stearoyl)substituted a-cyclodextrin tO, and hexa trimethylcholyl)-substituted a-cydodextrin 11. (b) A hypothetical side view for the passage of metal ion through a hydrophilic channel wall comprised of ether network convergently extended from steroidal backbone. 

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