Topological deformations

What is the dynamical response of a system to a sequence of systematically applied minimal topological deformations of a l-dirn lattice ... 

Fig. 3.44 Example of the range deperident behavior of rule T20 the system evolves as class-2 for r=l, class-4 for r = 2 and class-3 for r 3. The various possible intermediate (or transitional) behaviors can be studied by successively applying we.ll-defined minimal topological deformations to the initial lattice.

Individual cluster coefficients, as well as integrals, cannot in general be freely reordered in expressions for matrix elements. In particular, the rearrangement of coefficients can be seen to correspond to a topological deformation of a diagram that alters the sequence of topologically distinct... 

Figure S. Two stereoisomeric (but isomorphic) toroidal fullerenes formed b> gluing pairs of sides in different order in (a) the rectangle is folded down into the plane of the paper and glued top to bottom in (b) the rectangle is rotated 90 anticlockwise first. Both cylinders are then bent outwards from the paper before joining to form a torus. The two stereoisomers can be Interchanged by a formal topological deformation process, but not by any process that is physically plausible. [This illustration is an adaptation of one used by Klein (Ref. 20a). Copyright 1994 American Chemical Society.)...

Let two densities pi(r) and p2(r) from be given. Both of them are represented by the corresponding bundles of curves. Let us choose the unit vector 6r and in these bundles, the projections of pi(r) and p2(f) onto Cri the curves pj (r) and P2W which are, according to (Diii), continuously differentiable functions of r = Irl. Hence, they are homotopicaUy equivalent, or equivalently, there does exist such topological deformation that maps or deforms pj (r) into P2(r). Formally,... 

The geometry of open SWNT s is qualitatively different than that of the single-cage and multi-cage fullerenes. In the latter cases, each of the fullerenes may be topologically deformed onto the surface of a sphere. But, an open cylinder nanotube may not be so deformed. The Descartes-Euler formula does not hold for an open-ended nanotube. What then becomes of the dual concept, based upon the Descartes-Euler formula, when we come to open nanotube geometry Evidently a generalization of the Descartes-Euler formula is required since it occurs that... 

In the deformed state, the variables in the Hamiltonian change from ( R , r ) to ( R , Ar ). However, the distribution p( r ) of finding the topology r depends solely on how the material is made instantaneously at thermal equilibrium (i.e., at constant temperature T, pressure p, etc.) i.e., p( r ) does not depend on the external deformation tensor A. Then, the final answer for the free energy of the deformed network is... 

Most of the relevant features of the charge density distribution can be elegantly elucidated by means of the topological analysis of the total electron density [43] nevertheless, electron density deformation maps are still a very effective tool in charge density studies. This is especially true for all densities that are not specified via a multipole model and whose topological analysis has to be performed from numerical values on a grid. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

In system 1, the 3-D dynamic bubbling phenomena in a gas liquid bubble column and a gas liquid solid fluidized bed are simulated using the level-set method coupled with an SGS model for liquid turbulence. The computational scheme in this study captures the complex topological changes related to the bubble deformation, coalescence, and breakup in bubbling flows. In system 2, the hydrodynamics and heat-transfer phenomena of liquid droplets impacting upon a hot flat surface and particle are analyzed based on 3-D level-set method and IBM with consideration of the film-boiling behavior. The heat transfers in... 

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. 

Soft and topologically adaptable supramolecular modules that allow host lattices to achieve dense packing through low-energy deformations, while retaining their inherent dimensionality and supramolecular connectivity, can facilitate the systematic design of molecular inclusion compounds. This is demonstrated by the ability to predict and maintain architectural control in a series of lamellar host frameworks. 

Topological analysis of the total density has a considerable advantage over the use of the deformation densities in that it is reference-density independent. There is no need to define hybridized atoms to analyze the nature of covalent bonding, and the ambiguity when using the standard deformation density, noted above in the discussion on propellanes, does not occur. 

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