Simple closed subset

In the second section, we show that, if L is a finite set and does not contain thin elements, the closed subset generated by L is a direct product of simple closed subsets each of which is generated by the elements of L which it contains. 

Theorem 11.2.4 tells us that, in order to investigate Coxeter sets without thin elements, it is enough to look at Coxeter sets which generate a simple closed subset. 

The following lemma is a result about simple closed subsets generated by a Coxeter set. It will be needed in the proof of Lemma 12.3.1. 

A closed subset T of S is called simple if T has exactly two normal closed subsets, namely 1 and T. 

Recall that a closed subset T of S different from 1 is called simple if 1 and T are the only normal closed subsets of T. 

Proof. Assume that (L) is not simple. Then, by definition, (L) possesses a normal closed subset T with 1 T L). 

Multiple chain robotic systems can take many forms, some of them quite complex. Simple closed-chain mechanisms are a subset of multiple chain systems with specific structural characteristics. In this section, a model for simple closed-chain mechanisms is described, and the nature of the simulation problem for these mechanisms is discussed. 

It is of vital interest for a wider applicability of CTCB to examine how these two mechanisms can be accommodated in OCT. In Section 3, we shall argue that the mutual decoupling status of several subsets of basis functions, manifesting itself by the absence of any external communications (bond orders) in the whole system, calls for the separate unit normalization of its input probabilities since such fragments constitute the mutually nonbonded (closed) building blocks of the molecular electronic structure. It will be demonstrated, using simple hydrides as an illustrative example, that the fulfillment of this requirement dramatically improves the agreement with the accepted chemical intuition and the alternative bond multiplicity concepts formulated in the MO theory. 

If A is k[S] for some Sslt", the definition makes S homeomorphic to its image as a subset of Spec A. Furthermore, the image is dense for if a closed set Z(I) contains S, each fin I vanishes at all points of S, so I = 0. As in (5.1), it follows that Spec A is irreducible iff S is irreducible. Simple topology also shows that Spec A is connected if S is. The converse of this is not true, and the last section shows that we don t want it to be true. 

Hydrogen-bonded clusters are an important subset of the large number of neutral and ionic binary complexes (often called van der Waals molecules ) that can now be characterized by modern gas-phase spectroscopic and molecular beam techniques. These complexes often exhibit puzzling structural properties that seem to defy simple rationalization or prediction. As an illustrative example, we consider here the simple n-n binary complexes formed from isoelectronic closed-shell diatomic species CO (carbon monoxide) and NO (nitrosyl cation). 

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