Disjoint

In the unperturbed case, say 7 = 70, let the system have two disjoint invariant sets Bi and B2 associated with two eigenmodes and invariant inea-... 

Lemma 3. Let p M be a probability measure and let X and Y be disjoint sets which are dx- resp. dy-almost invariant with respect to p. Moreover suppose that f (X) n Y = 0 and f Y) n A = 0. Then X UY is 6xuy-almost invariant with respect to p where... 

Each minterm expansion is disjoint from till the others. Thus, in Equation 2.2-4 iiic 4 terms Xi X AU,XyXJ,... 

Events are disjoint if the have no intersection, i.e., they are unrelated. 

If events cannot occur at the same time, they are called disjoint, it is clear from the von... 

Misesian definition of probability that equation 2.4-6 will result for this case as may be demonstrated by adding two ncai-disjoint probabilities (refer to the Veni diagram, Figure 2.4-1). The segments are P(A,) = P(A) P(B), P(Af) = P(A) P B), and... 

The general problem simplifies considerably in the finite field. F[2. Because circuits are always counted at least twice, their number contributes a factor = 0 (mod 2) we see from equation (5.14), therefore, that the only structural information necessary to obtain Pi x) is that of the parity of disjoint edge distributions. Moreover, since there is no way to distribute disjoint edges among an odd number of vertices, equation (5.13) gives... 

In order to construct classes of rules analogous to the two types of value-rules defined above, we partition the local neighborhood into 3 disjoint sets (figure 8.18) 51 i, j) =Vij U Aij U Bij, where... 

Figure 10.4 shows a schematic representation of how Hopfield s net effectively partitions the phase space into disjoint basins of attraction, the attractor states of which represent some desired set of stored patterns. 

CAM-8 also eliminates CAM-6 s need for pipeline buffering (see above) by using partitioning cellular automata (see section 8.1.3) as its basic neighborhood structure. Space is effectively subdivided into disjoint subsets of sites and each site is... 

To make use of the group G we need some way of summarizing those properties of the group that are relevant to the problem. This was provided by Polya in the form of the "cycle index". It is well known that a permutation can be expressed as a product of disjoint... 

The original formulation of de Bruijn s theorem was for a quite general problem of this type, with a broad definition of the "weight" of a mapping. We assume that R is the union of a finite number of pairwise disjoint sets R- (i = 1,. .., k and that // is a direct product of groups //j, where //j acts on / j. For each there is a weight function where n is the number of elements of D that are... 

This is Gershgorin s theorem. Moreover, if any A is separated from the others, and the p s small enough so that the particular disk (2-14) is disjoint from all the others, then that disk contains exactly one root. By applying the theorem to the matrix D(A + B)D l, for a suitable matrix D, it is sometimes possible to obtain even better bounds. 

These subsets are disjoints and cannot be continuously connected by real Lorentz transformations. 

A second approach considers that the regions of equivalent parameter values must enclose parameters for which the loss function is nearly the same or at any rate less different than some threshold. In other words, the equivalence regions should take the form 015(0) < c 5(6) for some appropriate constant of. Note that in this case the shape of the regions would not necessarily be ellipsoidal, or even convex In fact, we might postulate in general the existence of multiple minima surrounded by disjoint equivalence neigh-... 

Many chemical batch production facilities are dedicated to producing a set of products that require for their manufacturing a common set of unit operations. The unit operations are performed in the same sequence for each product. This type of production problem is often solved by configuring the available equipment so that each unit operation is carried out by a fixed set of equipment items that are disjoint from those used in any other operation. If one unit is assigned to each step, this is called a flowshop (Baker, 1974). 

A first distinction which is often made is that between methods focusing on discrimination and those that are directed towards modelling classes. Most methods explicitly or implicitly try to find a boundary between classes. Some methods such as linear discriminant analysis (LDA, Sections 33.2.2 and 33.2.3) are designed to find explicit boundaries between classes while the k-nearest neighbours (A -NN, Section 33.2.4) method does this implicitly. Methods such as SIMCA (Section 33.2.7) put the emphasis more on similarity within a class than on discrimination between classes. Such methods are sometimes called disjoint class modelling methods. While the discrimination oriented methods build models based on all the classes concerned in the discrimination, the disjoint class modelling methods model each class separately. 

S. Wold, Pattern recognition by means of disjoint principal components models. Pattern Recogn., 8 (1976) 127-139. 

M.P. Derde and D.L. Massart, UNEQ a disjoint modelling technique for pattern recognition based on normal distribution. Anal. Chim. Acta, 184 (1986) 33-51. 

Thus, the whole complex of existing experimental data indicates that the major part of polycrystalline contacts in vacuum sintered polycrystalline oxides are provided by bridges of open type. Moreover, the vacuum sintering at moderate temperatures 300 - 350°C leads to formation of a unified pattern (see Fig. 2.4, b) which cannot be disjoint into specific microcrystals and connecting bridges [37, 40]. The structure of adsorbents obtained presents a complex intertwining of branches of various thickness. 

Consider a set X. The topological space (X, T) is called a Hausdorff space if for any two distinct points x, y eX there exist disjoint T-open sets Tx, Ty,... 

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