Constraint Topology

This section analyzes the topology of timing constraints in a constraint graph G V,E). We describe several concepts that are used in the conflict resolution formulation. Let the target instance operation set 0( ,j)(G) be denoted by G C V, where we dropped the terms t,i) and G for conciseness. The instance operation set O consists of fc = G vertices, denoted by o,-, i = 1. fc. Each vmex Oi O has an associated execution delay 6(o ) that can be fixed or data-dependent In the simplistic case of flat graphs, all elements of G are 

Definition 7.2.1 An operation cluster C of an instance operation set O is the maximal subset of vertices in O that is strongly connected, i.e. there exists a directed path between every pair of vertices in the operation cluster. C denotes the cardinality of C. 

Theorem 7.2.1 A partial order exists among the operations clusters of an operation set. 

Proof Elements of an instance operation set are strongly connected in the constraint graph. Since strong connectivity is an equivalence relation, two operation clusters cannot be connected by a cycle. This is the definition of partial order. 

This partial order over the operation clusters provides the basis for a conflict resolution strategy based on decomposition. Specifically, the problem of finding a valid ordering for an instance operation set is divided into two steps  

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Defect lines in smectics, unlike those in nematics, often do not continuously shrink with time and spontaneously disappear. Instead, there often seems to be a finite energy barrier that must be overcome if a smectic defect is to disappear. This difference between nematics and smectics is a consequence of the layer-spacing constraint that exists in smectics but not nematics. Because of this constraint, topological defects in smectics cannot be removed without ripping layers, and this requires a finite energy. 

The methods described so far for study of polymeric molecules rely on the presence of end sites. For long chain molecules or for cyclic or ring molecules, they are of little use. It turns out that in such cases it is also possible to construct configurational-bias based algorithms in which the source of the bias is not only the potential energy of individual sites in the new, trial position, but also a connectivity constraint. Topological... 

Even when the secondary stmcture of a protein is known, there are a large number of ways that this stmcture can be packed together. Studies dealing with the identification of the topological constraints in the packing of heUces and sheets have revealed certain patterns, but as of this writing accurate prediction is not possible. 

Once the chains become larger and larger, the dynamics of the melt slows down dramatically, due to the topological constraints imposed by the chains on each other. For the chain diffusion one observes a transition... 

A similar anomalous behavior has been detected also in 3d polymer melts but only for rather short chains [41] for longer chains, several regimes occur because of the onset of entanglement (reptation ) effects. In two dimensions, of course, the topological constraints experienced by a chain from... 

There are cases where non-regular lattices may be of advantage [36,37]. The computational effort, however, is substantially larger, which makes the models less flexible concerning changes of boundary conditions or topological constraints. Another direction, which may be promising in the future, is the use of hybrid models, where for example local attachment kinetics are treated on a microscopic atomistic scale, while the transport properties are treated by macroscopic partial differential equations [5,6]. 

A polymer such as polyethylene is a long-chain molecule with repetitions of the same monomer. Due to topological constraints, the crystallization process of polymer chains is expected to be different from that of simple molecules as discussed so far [160]. 

Fig. 8.24 Evolution of a 35 x 35 lattice whose sites are initially randomly seeded with O = 1 with probability p = 1/2. The development proceeds according to T value and OT topology rules defined by code C = (84,36864,2048). The constraints are = 0, A = 10]. The appearance of localized substructures is evidence of a geometrical self-organization.

It is beyond our control how the cross-links are spaced along the polymer chains during the vulcanization process. This extraordinary important fact demands a generalization of the Gibbs formula in statistical mechanics for amorphous materials that have fixed constraints of which the exact topology is unknown. Details of a modified Gibbs formula of polymer networks can be found in the pioneering paper of Deam and Edwards [13]. 

The presented scheme offers several extensions. For example, the model gives a clear route for an additional inclusion of entanglement constraints and packing effects [15]. Again, this can be realized with the successful mean field models based on the conformational tube picture [7,9] where the chains do not have free access to the total space between the cross-links but are trapped in a cage due to the additional topological restrictions, as visualized in the cartoon. 

The model describes the characteristic stress softening via the prestrain-dependent amplification factor X in Equation 22.22. It also considers the hysteresis behavior of reinforced mbbers, since the sum in Equation 22.23 has taken over the stretching directions with ds/dt > 0, only, implying that up and down cycles are described differently. An example showing a fit of various hysteresis cycles of silica-filled ethylene-propylene-diene monomer (EPDM) mbber in the medium-strain regime up to 50% is depicted in Figure 22.12. It must be noted that the topological constraint modulus Gg has... 

Kuhn H., Demidov V.V., Frank-Kame-NETSKii M.D. Rolling-circle amplification under topological constraints. Nucleic Acids Res. 2002 30 574-580... 

To achieve these consistencies, MODEL.LA. provides a series of semantic relationships among its modeling elements, which are defined at different levels of abstraction. For example, the semantic relationship (see 21 1), is-disaggregated-in, triggers the generation of a series of relationships between the abstract entity (e.g., overall plant) and the entities (e.g., process sections) that it was decomposed to. The relationships establish the requisite consistency in the (1) topological structure and (2) the state (variables, terms, constraints) of the systems. For more detailed discussion on how MODEL.LA. maintains consistency among the various hierarchical descriptions of a plant, the reader should consult 21 1. 

The first step in the DG calculations is the generation of the holonomic distance matrix for aU pairwise atom distances of a molecule [121]. Holonomic constraints are expressed in terms of equations which restrict the atom coordinates of a molecule. For example, hydrogen atoms bound to neighboring carbon atoms have a maximum distance of 3.1 A. As a result, parts of the coordinates become interdependent and the degrees of freedom of the molecular system are confined. The acquisition of these distance restraints is based on the topology of a model structure with an arbitrary, but energetically optimized conformation. 

The chemical bonding and the possible existence of non-nuclear maxima (NNM) in the EDDs of simple metals has recently been much debated [13,27-31]. The question of NNM in simple metals is a diverse topic, and the research on the topic has basically addressed three issues. First, what are the topological features of simple metals This question is interesting from a purely mathematical point of view because the number and types of critical points in the EDD have to satisfy the constraints of the crystal symmetry [32], In the case of the hexagonal-close-packed (hep) structure, a critical point network has not yet been theoretically established [28]. The second topic of interest is that if NNM exist in metals what do they mean, and are they important for the physical properties of the material The third and most heavily debated issue is about numerical methods used in the experimental determination of EDDs from Bragg X-ray diffraction data. It is in this respect that the presence of NNM in metals has been intimately tied to the reliability of MEM densities. 

The earliest and simplest approach in this direction starts from Langevin equations with solutions comprising a spectrum of relaxation modes [1-4], Special features are the incorporation of entropic forces (Rouse model, [6]) which relax fluctuations of reduced entropy, and of hydrodynamic interactions (Zimm model, [7]) which couple segmental motions via long-range backflow fields in polymer solutions, and the inclusion of topological constraints or entanglements (reptation or tube model, [8-10]) which are mutually imposed within a dense ensemble of chains. 

With respect to the intensity resolution relationship of NSE, PEB-2 [essentially PE with one ethyl branch every 50 main chain bonds the sample is obtained by saturating 1-4 polybutadiene, the residual 1-2 groups (7%) cause the ethyl branches Mw = 73200 g/mol Mw/Mn = 1.02] has two advantages compared to PEP (1) the Rouse rate W/4 of PEB-2 is more than two times faster than that of PEP at a given temperature [W/pEP (500 K) = 3.3 x 1013 A4s 1 W/pEB (509 K) = 7 x 1013A4s-1] (2) at the same time, the topological constraints are stronger. 

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