Element maximal

Proof. Let m be an element in M 0. Then, by Zorn s Lemma, there exists a maximal element L in the set of all submodules of M not containing m. Since M is assumed to be completely reducible, M possesses a submodule K such that K L = M. 

In the first section of this chapter, we focus on specific characteristics of spherical Coxeter sets such as maximal elements and conjugation. The second section is devoted to an extension theorem for spherical Coxeter sets. Our approach to this theorem (which follows the line of [46]) is partially inspired by a geometrical reasoning provided by Jacques Tits in [37],... 

Proof. The Hilbert basis theorem (A.5) shows that any nonempty collection of ideals in k[X, X ] has a maximal element hence any nonempty... 

We are assuming (L) to have finite valency. As a consequence, (L) is a finite set. In particular, has a maximal element. In the following, we shall... 

Elements, which are not covered by other elements, are called maximal elements, or -as done for example in the chapter by Carl-sen and Walker, p. 153 simply as maximals. In Fig. 5 such elements are f, g, i. 

If there is only one maximal element, then this is also called a greatest element. In Fig. 5, there is no greatest element, however in Fig. 4 element D is a greatest element. 

Draw the maximal elements MAX in the same vertical position. Elements ofM4A-i will located below those of MAX,. 

Fig. 8. The resulting poset and its visualization after subtracting the maximal elements of ( 791, IB) after the start and the first iteration...

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. 

NMAX The number of maximal elements (called number of maximal equivalent classes because this information is related to... 

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

Preferably a maximal element should be chosen as a starting point for the analysis. This choice, however, is not mandatory. Thus, other elements of E or E/y could be chosen too. This selected element is called "key element". We may simultaneously select more than one key element even all elements (no restrictions apply here). For the sake of convenience all key elements are supposed to form a set K (c E). 

The analysis of a key element implies a search of all elements located lower than that of the key element, i.e. all elements that can be reached from the key element by a path, a sequence of connecting edges. (Therefore the selection of maximal elements rather than other elements is more meaningful). These elements together with elements equivalent but not identical to the key element are called successors. The set of all successors of the key element "k" is denoted as G(k,T), A cz IB. We include the information about the actual set of attributes (i.e. the case) by A. Note the similar concept of "down-sets" in Davey Priestley (1990) The order ideal (or down set), generated by the key element will be denoted by 0(k 4). Then it is valid ... 

As the sampling sites with high responses of the test-battery are of most interest, the maximal elements are taken as priority elements, i.e. the equivalence classes 27, 33, 46, 47, 31, 95, 32, 9, 18. From this we conclude that a) the sites 27, 33, 46, 47, 31, 95, 32, 9, 18 are of specific importance, and b) the set of sites 27, 33, 46, 47 has the same profile of scores, thus, they may be remedied by the same methods, whereas the attribute profiles differ among all other priority objects. 

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. 

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. 

The diagram points out chlorbromuron (2) as maximal element, being characterized by the highest nCaR value (nCaR = 3) and the highest Mor09m value (Mor09m = -0.242). 

Important issues for the design of the library synthesis sequence are The molecule should incorporate many different sets of building blocks providing maximal element variability. The... 

Note that this matching can also be found in a functorial way as follows. Let <5 be a chain with [n/2j elements labeled with the numbers 1,..., n/2 in reverse order i.e., 1 labels the maximal element. Define ip P Q hy mapping the cell F e P to the maximal number k such that (2, jg... 

More specifically, let P be a poset, and let P denote the poset obtained from P by augmenting it with a minimal and a maximal element. Let us do as follows ... 

Take the Boolean algebra with the minimal and the maximal elements removed, P = B 0, [n]. A covering relation A y B is a, pair of subsets of [n] such that A B = 1. Let rr = A B, and take x to be the label of A y B. One can check that this is an EL-labeling. Furthermore, there is exactly one weakly decreasing chain, obtained by arranging the elements of the set [n] in decreasing order. 

It is important to note that 3D nonlinear elastic-plastic finite element simulations were performed, while stiffness reduction curves were used for calibration of the material model and for determining (minimal) finite element size. It should also be noted that the largest FEM model had over 0.5 million elements and over 1.6 million DOFs. However, most of simulations were performed with smaller model (with 150 K elements) as it represented mechanics of the problem with appropriate level of accuracy. Working FEM model mesh is shown in Figure 2. The model used, features 484,104 DOFs, 151,264 soil and beam-column elements, and is intended to model appropriately seismic waves of up to 10 Hz, for minimal stiffness degradation of. GjGmax = 0.08, maximum shear strain of y = 1% and with the maximal element size... 

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