Topology trivial

Now we should decompose C and next S into pieces that can be put together to form slices that are topologically trivial and thus subject to the standard non-Abelian Stokes theorem. Such decomposition is shown in Fig. 9. Explicitly, this reads as... 

In Figure 19 is given the ORTEP representation of the unknotted, topologically trivial complex Cu2(m-86) + made of two disconnected 43 rings. 

Similar considerations hold for the topologically trivial rotaxanes. 

Figure 5.13. (a) Entanglement of disclinations in a medium with a trihedron of vectors as the order parameter (b) topologically trivial and (c) nontrivial. 

The concept of OP space helps us to analyze complicated configurations of the cholesteric order parameter even when these configurations are topologically trivial, i.e., equivalent to an undistorted cholesteric or nematic. 

The problem of perception complete structures is related to the problem of their representation, for which the basic requirements are to represent as much as possible the functionality of the structure, to be unique, and to allow the restoration of the structure. Various approaches have been devised to this end. They comprise the use of molecular formulas, molecular weights, trade and/or trivial names, various line notations, registry numbers, constitutional diagrams 2D representations), atom coordinates (2D or 3D representations), topological indices, hash codes, and others (see Chapter 2). 

The need for simple descriptions of complicated organic ligands has led to the evolution of some trivial nomenclature systems, such as those for crown ethers (e.g. 76) 72AG(E)16) and cryptands 73MI10200), which have become quite elaborate 8OMII0200). These systems are intended primarily to indicate topology, and the positions of potential donor atoms, and are not particularly appropriate for general use. 

What is the complete behavioral specification of a given transition rule on arbitrary lattices Which topologies yield trivial or complex dynamics What fraction of topologies of given size induce a particular variety of dynamical behavior ... 

The simplest, from the viewpoint of topological structure, are the linear polymers. Depending on the number m of the types of monomeric units they differentiate homopolymers (m=1) and copolymers (m>2). In the most trivial case molecules in a homopolymer are merely identified by the number l of monomeric units involved, whereas the composition of a copolymer macromolecule is defined by vector 1 with components equal to the numbers of mono-... 

The short calculations we presented here are all 3-dimensional cavities with trivial topology ( For example of not like the region between co-axial cylinders or cones, co-centric spheres or tori (Ahmedov and Duru, 2003) ). The known results ( including the present ones ) for three dimensional cavities are... 

The second chapter is devoted to computing the Betti numbers of Hilbert schemes of points. The main tool we want to use are the Weil conjectures. In section 2.1 we will study the structure of the closed subscheme of X which parametrizes subschemes of length nonl concentrated in a variable point of X. We will show that (X )rei is a locally trivial fibre bundle over X in the Zariski topology with fibre Hilbn( [[xi,... arj]]). We will then also globalize the stratification of Hilbn( [[xi,..., x ]]) from section 1.3 to a stratification of Some of the strata parametrize higher order data of smooth m-dimensional subvarieties Y C X for m < d. In chapter 3 we will study natural smooth compactifications of these strata. 

Lemma 2.1.4. n (X " j)red — X is a locally trivial fibre bundle in the Zariski topology with fibre Hilbn(J )red. 

A feature of theories for tree-like polymers is the disentanglement transition , which occurs when the tube dilation becomes faster than the arm-retraction within it. In fact this will happen even for simple star polymers, but very close to the terminal time itself when very little orientation remains in the polymers. In tree-like polymers, it is possible that several levels of molecule near the core are not effectively entangled, and instead relax via renormalised Rouse dynamics (in other words the criterion for dynamic dilution of Sect. 3.2.5 occurs before the topology of the tree becomes trivial). In extreme cases the cores may relax by Zimm dynamics, when the surroundings fail to screen even the hydro-dynamic interactions between the slowest sections of the molecules. 

There is no systematic nomenclature developed for molecular sieve materials. The discoverer of a synthehc species based on a characteristic X-ray powder diffraction pattern and chemical composihon typicaUy assigns trivial symbols. The early syn-thehc materials discovered by Milton, Breck and coworkers at Uruon Carbide used the modem Lahn alphabet, for example, zeoHtes A, B, X, Y, L. The use of the Greek alphabet was inihated by Mobil and Union Carbide with the zeoHtes alpha, beta, omega. Many of the synthetic zeoHtes which have the structural topology of mineral zeoHte species were assigned the name of the mineral, for example, syn-thehc mordenite, chabazite, erionite and offretite.The molecular sieve Hterature is replete with acronyms ZSM-5, -11, ZK-4 (Mobil), EU-1, FU-1, NU-1 (ICI), LZ-210, AlPO, SAPO, MeAPO, etc. (Union Carbide, UOP) and ECR-1 (Exxon). The one pubHcaHon on nomenclature by lUPAC in 1979 is Hmited to the then-known zeoHte-type materials [3]. 

Locally in the etale topology on S we can choose a rigidified line bundle (i.e. a line bundle trivialized along the origin) L on X such that A(L) = A (see [GIT] Definition 6.2). This... 

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