Minimal element

In Product Safety Management and Engineering, Willie Hammer points out that to be perfectly safe a product, or in this case, a chemical agent, would have to permit no possibility of causing injury or damage under any circumstances—a situation that is nigh to being impossible. A person therefore must expect some minimal element of risk. How much... 

For example, if we study a simple cycle then formula (13) for steady state is valid with any given accuracy with unite probability for any ordering with the given minimal element k . 

Let us take a weakly ergodic network iV and apply the algorithms of auxiliary systems construction and cycles gluing. As a result we obtain an auxiliary dynamic system with one fixed point (there may be only one minimal sink). In the algorithm of steady-state reconstruction (Section 4.3) we always operate with one cycle (and with small auxiliary cycles inside that one, as in a simple example in Section 2.9). In a cycle with limitation almost all concentration is accumulated at the start of the limiting step (13), (14). Hence, in the whole network almost all concentration will be accumulated in one component. The dominant system for a weekly ergodic network is an acyclic network with minimal element. The minimal element is such a component Amin that there exists an oriented path in the dominant system from any element to Amin- Almost all concentration in the steady state of the network iV will be concentrated in the component Amin-... 

These ideas are evident in an essay of Lavoisier s from 1773, in which he identifies the three different physical states of matter solid, liquid, and gas. Here he makes the crucial distinction between the physical and chemical nature of substances, which confused the ancients and led to their minimal elemental schemes. The same bod/, says Lavoisier, can pass successively through each of these states, and in order to make this phenomenon occur it is necessary only to combine it with a greater or lesser quantity of the matter of fire. ... 

The module M is called artinian if each set of submodules of M has minimal elements. 

First, we again engage in some claim construction, the major observation being that the claim transition is open (comprising) and thus the listed parts are the minimal elements necessary to fulfill the claim language. 

A number of specific approaches to minimizing elemental fractionation have been developed [11,14,54,57,58], It remains impossible to pinpoint the source of fractionation ... 

Table 21.3. The absolute minimal element content of primitive life...

Elements that do not cover any other element are called minimal element, or simply minimals. 

If there is only one minimal element, then this is also called a least element. In Fig. 5, there is no least element, however in Fig. 4 element A is a least element. 

Isolated elements Elements that are both Minimal and Maximal elements. Maximal/Minimal elements which are not isolated, are often called proper maximal/minimal elements. An isolated element in the Hasse diagram, shown in Fig. 12 is f. 

NMIN The number of minimal elements (notation as for the maximal elements). 

With the Hasse diagram of Fig. 19 at hand it is easy to derive the remaining characterizing numbers discussed in former sections. Hence, we find L(E) = 6, H( ) = 7, and NL = H( ) =7. These numbers give an impression in which detail the steps from a minimal element to a maximal element may be disclosed. This informs us here about the maximum possible differentiation in the degree of hazards. 

We further find that NEL = 6, which in the present case coincides with NMAX = 6. The number of minimal elements NMIN = 1. 

Fig. s 10a and 10b shows the Hasse diagrams after single clustering of each attribute (parameter) for AE sediments using a cluster number of FCL=4 and a TMF=0.8. The equivalence classes Kl and K2 in the ecotoxicological evaluation are the only minimal elements, i.e. compared to all samples above their members have the lowest values in all tests. Except K2 all samples are comparable with Kl. The fact that except AE9/94, AE6/94 and AE3/95 all samples above Kl (AE12/92, AE9/93, AE6/95) have been taken at a later date indicates an increasing pollution for AE. Using chemical concentrations for evaluation the temporal trend seems to be contrary. However, both recent samples AE4/01 and AE8/01 are isolated, i.e. not comparable to all other samples. A sensitivity analysis (see e.g. Heininger et al. 2003) shows that the evaluation is most sensitive to the nitrogen content, where omitting this nutrient compensates the isolation of AE4/01 and AE8/01 (see Fig. 11). Moreover, sample AE4/01 is now a minimal element and therefore emphasizes the indication of a decline of the...

Fig. 1 consists of 11 levels, 4 maximal elements, i.e., only those connected to lower-ranked elements (15, 28, 43, 47) and 5 minimal elements, i.e., only those connected to higher-ranked elements (11, 23, 41, 39, 46), respectively. 

D°up and Dlf give a measure of the normalised rank uncertainty, above and below respectively. Note that if u is a priority element (maximal) C(s,u) is not defined, as no element exists above u, thus Dsuup is not defined and only I) "1 can be evaluated. Analogously, if u is a minimal element C(u,t) is not defined, as no element exists below u, thus D f is... 

Diuron (4), experimentally the most toxic chemical has the same value of nCaR and very close value of Mor09m (Mor09m = -0.335). Fenuron (5) and metobromuron (9) are identified as minimal elements, the former is characterised by the lowest nCaR value (nCaR = 1), the latter by the lowest Mor09m value (Mor09m = -1.332). 

First, consider the case where i(r) is a minimal element in ob(/). By Nagata s compactification theorem [34] (see also [27]), there is a factorization... 

Fig. 3 shows a sample program as well as its dependency graphs whose arcs are additionally labeled with the register shared between producer and consumer. The dependency graph of a program P can be viewed as a skeleton of a partial order

execution order of Ps instructions if i < j, then the execution of j cannot be initiated before the execution of i has completed. It also makes explicit the parallelism inherent in P two instructions can (up to structural hazards) be executed concurrently iff they are unordered w.r.t. partial order are the instructions j such that there exists no instruction i < j. 

Next we have to find the minimal element in the set of labeled graphs obtained by inserting a suitable bond according to the definition of P ... 

The minimal element of is 0,1, and we obtain an additional element of the transversal that we want to generate,... 

Definition 10.10. For an arbitrary acyclic category C, let BdC denote the poset whose minimal elements are objects ofC, and whose other elements are all composahle morphism chains consisting of nonidentity morphisms of C. The elementary order relations are given by composing morphisms in the chain, and by removing the first or the last morphism. The poset BdC is called the barycentric subdivision of C. 

Example 10.12. Let P be a totally ordered set with n elements. Clearly, its barycentric subdivision is the Boolean algebra with the minimal element removed BdP = 6. Equation (10.7) implies that... 

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