Disjoint sets

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... 

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... 

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... 

The variables are divided into three not necessarily disjoint sets,... 

Two finite disjoint sets of states, and K2, a designated initial state q0 in U Kj, an accepting state qa and a rejecting state qr with neither qa nor qr in U 1C,, ... 

Those reflections which are phased comprise the basis set R the disjoint set, the non-basis set, of unphased amplitudes is EJ the phase... 

To illustrate this point, consider a composite system composed of two noninteracting subsystems, one with p electrons (subsystem A) and the other with q = N — p electrons (subsystem B). This would be the case, for example, in the limit that a diatomic molecule A—B is stretched to infinite bond distance. Because subsystems A and B are noninteracting, there must exist disjoint sets Ba and Bb of orthonormal spin orbitals, one set associated with each subsystem, such that the composite system s Hamiltonian matrix can be written as a direct sum. 

Every permutation can be written as a product, in the group sense, of cycles, which are represented by disjoint sets of integers. The S5mibol (12) represents the interchange of objects 1 and 2 in the set. This is independent of the number of objects. 

Illustration 2.1.5 Figure 2.4(a) illustrates two sets which are separable, but which are neither disjoint or convex. It should be noted that separability does not imply that the sets are disjoint. Also, two disjoint sets are not in general separable as shown in Figure 2.4(b). 

Multiplication of the Dirac characters produces a linear combination of Dirac characters (see eq. (4.2.8)), as do the operations of addition and scalar multiplication. The Dirac characters therefore satisfy the requirements of a linear associative algebra in which the elements are linear combinations of Dirac characters. Since the classes are disjoint sets, the Nc Dirac characters in a group G are linearly independent, but any set of N< I 1 vectors made up of sums of group elements is necessarily linearly dependent. We need, therefore, only a satisfactory definition of the inner product for the class algebra to form a vector space. The inner product of two Dirac characters i lj is defined as the coefficient of the identity C in the expansion of the product il[ ilj in eq. (A2.2.8),... 

Unlike the disjoint sets of approaches to taxonomy and nomenclature for "organic chemistry" vs. "inorganic chemistry" vs. "polymer chemistry", etc., which form the cornerstone of all of the various nomenclature systems in common usage today, a common graph theory based, bi-parametric, alternating code of atoms and bonds that is equally applicable to each of these individual domains is proposed. In this system the detailed formula will be all of the name that is needed. Advantages to such an approach include ... 

Extrinsically non-planar graphs are non-planar by virtue of their embedding in the 3D-space. They may be intrinsically planar at the same time (Fig. 6) this is the case of links (45) which are homeomorphic to disjoint sets of circuit graphs [71,72] (46) and knots (47a and b) which are homeomorphic to a circuit graph (48). 

But there is more than one of these minimal systems of generat ing graphs A- In the case of Hve-membered rings, one needs at least three graphs A, having no disjoint set of edges. 

Contractions between the creation operators or the annihilation operators vanish identically because of the Fermi-Dirac statistics obeyed by electrons [cf. Eq. (43)] and as in the single-excitation operator case, contractions between creation and annihilation operators are zero, because the indices belong to disjoint sets [cf. Eq. (44)]. Hence, Eq. (62) becomes... 

Since both the particle and hole labels are in general permuted, and it is often the case that the permutations are independent, it is efficacious to (always) write the permutation operator as the binary product of operators acting on the disjoint sets,... 

Transactions at disjoint sets of access points can be carried out in parallel. With global synchronism, where each message sent arrives in the next round, the simplification is not too unrealistic, because the entities participating in a transaction are kept quite busy and transactions terminate quickly. With more realistic models of time, one would have to permit overlapping transactions at a single access point. However, this would introduce many standard tasks of multi-processing into signature schemes, which is of no use in a classification. 

Classifications that enable us to partition chemical entities into mutually disjoint sets, that is to divide them up in such a way that each entity belongs to one and only one subset, are clearly of fundamental importance to chemists. In mathematical terminology this situation is described by saying that an equival-... 

Because of the rejection of the discontinuous reflections in the spacetime, the topological space of this group must be connected-, that is, it cannot be decomposed into two or more disjoint sets. 

Another important classification of conjugated diradicals depends on whether their Hiickel NBMOs can or cannot be confined to a disjoint sets of atoms.446 As discussed in... 

Now, suppose we apply the transformations to the distinct one-electron integrals and add up the results to generate a one-electron matrix. The result is a one-electron matrix with the correct symmetry but the elements are all doubled-, a matrix of 2hij. It is easy to see why this doubling has occurred the result of applying the symmetry operations to the distinct elements does not result in a disjoint set of new elements-, repetitions occur as they do in the list of transformed orbitals. Of course, these repetitions must occur mth the molecular symmetry-, if hn occurs twice, so must 22 and In our case both hn and hi2 occur twice but this need not be so in general there will be a characteristic number of repetitions of each of the symmetry-distinct matrix elements. Thus the method is clear, we must... 

This simple fact ought to make us alert to the possibility that the optimum wavefunction of our chosen form might not be localised but might simply be a Cl-type wavefunction using disjoint sets of delocalised MOs. Just as the SCF MOs of a single-determinant wavefunction may be transformed to a more chemically appealing localised set, it may be necessary to transform the expansion of the pair functions, which were introduced on the basis of an expected localised description, into a localised form. 

A 2D correlation map may represent the interactions within a set of nuclei, leading to an auto-oorrelatlon map, or between two disjoint sets in the form of a cross-correlation map. He would like to mention four important applications of 2H correlated speotroscoDV. 

Not only does each element eventually get sent back to itself, but by applying / three tintes, the set A = 0,1 2,3 has been divided into two smaller disjoint sets. 

In a segmented contraction each primitive as a rule is used only in one contracted function, i.e. the primitive set of functions is partitioned into disjoint sets. In some cases it may be necessary to duplicate one or two PGTOs in two adjacent CGTOs. The contraction coefficients can be determined by a variational optimization of the atomic HF energy, where both the exponents and contraction coefficients are optimized simultaneously. It should be noted that this optimization often produces multiple minima, and selecting a suitable optimum solution may be non-trivial. ... 

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