Cartesian product

Each point in a Dugundji space represents a chemical constitution for which there can exist a multiplicity of stereoisomers, corresponding to a set of CC-matrices associated with the BE-matrix of the constitution a). Owing to their interdependence, the constitutional and stereochemical features of chemical systems do not form a cartesian product, whereas the configurations and conformations can be represented as cartesian products. 

In other words, the regulation problem is equivalent to the problem of finding a subset Z of the Cartesian product R x R on which the output tracking error e t) is zeroed, and an input signal which makes attractive and invariant this subset (Figure 3). The trajectories described by the state and the input on the invariant subset Z, are thereafter referred as the steady state... 

Remark 3. Equation (32) is known as the Francis-Isidori-Byrnes equation (FIB) [8] and is the nonlinear version of equation (10) used to find the subset Z on the Cartesian product R x called, so far, the zero tracking error submanifold. 

Applying this relation to the Cartesian product of all frames of discernment of each input i.e., Q = i i x 172 x. .. x if there are n sensors), it is possible to refine all available belief functions into a common reference set and then to combine them to produce a unique belief structure that, in addition, is able to indicate the conflict between information sources. 

We are now ready to combine the 2 p-boxes. To do so, we construct the Cartesian product of the 2 collections of interval-mass pairs in the matrix shown in Table 6.2. 

Cartesian product of 2 collections of interval-mass pairs... 

Fig. 2. Trajectories of some Cartesian product operators of a two-spin system under a 270°...

Figure 1 shows the pulse sequence of the C HSQC experiment supplemented by a spin-lock pulse to suppress the signals from C-bound protons. The experiment is readily described in terms of Cartesian product operators [9]. For a two spin system consisting of a proton spin H coupled to a C spin C, the relevant coherence transfer pathway is... 

The use of spin-lock pulses for water suppression is illustrated with the NOESY and ROESY pulse sequences (fig. 5). Using the Cartesian product operator description [9], the effect of the NOESY pulse sequence of fig. 5(A) is readily illustrated ... 

One way to combine vector spaces is to take a Cartesian sum. (Mathematicians sometimes call this a Cartesian product. Another common term is direct sum.)... 

X7ne can, however, speak of the Cartesian product of sets, without vector space operations. 

Exercise 2.4 Show that for any natural number n, the Cartesian product C is a complex vector space of dimension n. Then show that C with the usual addition but with scalar multiplication by real numbers only is a real vector space of dimension 2n. 

Show that this multiplication makes Gi x G2 into a group. The group G y G is called the Cartesian product group. 

Exercise 4.2 Show that the Cartesian product of groups defined in Exercise 4.1 is associative, i.e., for any groups G, G2 and Gj, the group (Gi x G2) X G3 is isomorphic to the group Gi x (G2 x G3 ). Conclude that for any natural number n, n-fold products of groups are well defined. 

Exercise 4.3 Show that the set ofly. diagonal special unitary matrices is a group and that it is isomorphic to the group T x T. (See Exercise 4.1 for the definition of the Cartesian product of groups.)... 

Below the pulse sequence we can show the desired coherence level, p, at each stage of the pulse sequence. This diagram defines the coherence pathway that is desired for a particular NMR experiment. Coherence order is mixed for Cartesian product operators... 

The following pages show the 15 Cartesian product operators for a spin system consisting of two /-coupled protons I (Ha) and S (Hb) (Fig. A.l). Each operator is represented in six ways the product operator symbol, an energy diagram with transitions, a vector diagram, a spectrum, a density matrix, and the coherence order. 

Product Operator. For multiple-quantum coherences, the pure zero-quantum and double-quantum states are shown with their Cartesian product operator equivalents. 

With a I-frame, one introduces a time axis appearing as a parameter in Eq. (1), the space component provides a mean to define a multidimensional Euclidean configuration space, x = x1,..., x ), that is, sets of real numbers. The space dimension is determined by the number of degrees of freedom related to constitutive elements of the material system these coordinates belong to an abstract cartesian product space, whereas origin and relative orientations of I-frames belong to laboratory space. Spin degrees of freedom are separately handled. 

When fuzzy sets are defined on universal sets that are Cartesian products of two or more sets, they are called fuzzy relations. For any Cartesian product of n sets, the relations are called n-dimensional. From the standpoint of fuzzy relations, ordinary fuzzy sets may be viewed as degenerate, one-dimensional relations. All concepts and operations applicable to fuzzy sets are applicable to fuzzy relations as well. However, fuzzy relations involve additional concepts and operations that emerge from their multidimensionality. 

One may think that this generalization is an unnecessary complication, because in all the following examples, 4 is in fact a pair (x, y). However, when group structures on these sets are introduced, the group structure on the Cartesian product in the factoring case is not a direct xxxluct... 

The Cartesian product of the middle-thirds Cantor set with itself. 

This pattern of dots and gaps is a topological Cantor set. Since each dot corresponds to one layer of the complex, our model of the Rdssler attractor is a Cantor set of suifaces. More precisely, the attractor is locally topologically equivalent to the Cartesian product of a ribbon and a Cantor set. This is precisely the structure we would expect, based on our earlier work with... 

The MCSCF wavefunctions discussed in this chapter consists of linear expansions in terms of JV-electron basis functions. These N-electron functions depend on 4N variables and are spanned by the space of the Cartesian product of the spatial and spin coordinates for each electron. The total wavefunction must be antisymmetric with respect to interchange of the coordinates of any pair of electrons. This is because electrons are half-integer spin particles called fermions and all fermion wavefunctions must satisfy this property. This is treated as a constraint on the class of admissible basis functions in the MCSCF... 

Klavzar, S. (2007) On ihe PI index Pl-partitions and Cartesian product graphs. MATCH Commun. Math. Comput. Chem., 57, 573-586. 

The idea is to limit the evaluation of the cartesian product to at most a few initial steps in the match workflow, e.g., one matcher, and to eliminate all element pairs with very low similarity from further processing since they are very unlikely to match. This idea is especially suitable for workflows with sequential matchers where the first matcher can evaluate the cartesian product, but all highly dissimilar element pairs are excluded in the evaluation of subsequent matchers and the combination of match results. 

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