Cartesian product operators

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 ... 

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. 

The cartesian product operators do not correspond to a single coherence... 

The Cartesian product operators are the most common operator basis used to understand pulse sequences reduced to one or two phase combinations. This operator formalism is the preferred scheme to describe the effects of hard pulses, the evolution of chemical shift and scalar coupling as well as signal enhancement by polarization transfer. The basic operations can be derived from the expressions in Table 2.4. The evolution due to a rf pulse, chemical shift or scalar coupling can be expressed by equation [2-8]. 

Table 2.4 Shorthand notation and conversion schemes for Cartesian product operators.

A more theoretical description using Cartesian product operators is given in Table 5.23. 

The relaxation interference between the H- N dipolar interaction and the nitrogen-15 CSA is the basis of the transverse relaxation optimized spectroscopy (TROSY), nowadays a standard tool in NMR of larger proteins. Zuiderweg and Rousaki reviewed the field of gradient-enhanced TROSY and described the experiments of this kind in terms of the cartesian product operators. Other midifications of TROSY have also been reported, but are judged to be beyond the scope of this review. 

E. R. P. Zuiderweg and A. Rousaki, Gradient-Enhanced TROSY Described with Cartesian Product Operators, Concepts Magn. Reson., Part A, 2011, 38A, 280. 

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

The product operator formalism is normally based on Cartesian coordinates because that simplifies most of the calculations. However, these operators obscure the coherence order p. For example, we found (Eq. 11.80) that products such as IXSX represent both zero and double quantum coherence. The raising and lowering operators I+ and I are more descriptive in that (as we saw in Eq. 2.8) these operators connect states differing by 1 in quantum number, or coherence order. We can associate I+ and I with p = +1 and — 1, respectively, and (as indicated in Eq. 11.79) the coherences that we normally deal with, Ix and Jy, each include both p = 1. Coherences differing in sign contain partially redundant information, but both are needed to obtain properly phased 2D spectra, in much the same way that both real and imaginary parts of a Fourier transform are needed for phasing. 

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. 

In the vector formalism of Figure A6-2, the effect of chemical shifts (differences in Lar-mor frequencies) is represented by the rotation of a vector in the xy plane from a position on the y axis (not shown) to one ahead of it. The coordinate system is rotating around the z axis at the reference frequency coj., so that the frequency of the magnetization vector moves away from the axis at a rate Aco = (w — tOy), subtending an angle of (Aco)r with the y axis. For simplicity, we will drop the A and refer both to the frequency of the nucleus and its difference from the reference frequency as co, as in eq. A6-2 (corresponding actually to the special case of o)r — 0). In the product operator formalism, the effect of chemical shifts is represented by the operation of the Hamiltonian term HP on magnetization, illustrated for the three Cartesian coordinates by the expressions... 

Taking into account all the relevant criteria spin-1/2 nuclei in the liquid phase can generally be described using CARTESIAN, spherical and shift product operators as shown in Table 2.3. The spherical operators are not shown because they can be easily derived from the shift operators, see Table 2.5. 

Coherence transfer pathways (CT pathway) fall in the domain of spherical product operators instead of CARTESIAN operators. Before proceeding any further it is recommended to a necomer to read section 2.2.2 and for addition information references [2.20 - 2.31]. To illustrate the use of coherence transfer pathways in coherence selection, three pulse sequences will be examined. 

We use the modeling language SMV and the model-checker NuSMV2 3]. SMV enables the declaration of integer variables and constraints on their behavior. NuSMV builds transparently the Cartesian product of the ranges of all variables. When no constraint is declared, all the combinations of variable values (i.e., states) are possible and all transitions between each pair of states are implicitly declared. Constraints are then added to delete undesired states and transitions. As for variables, time is discrete. It is modeled by the operator next (). NuSMV is well-adapted to our variable-oriented modeling approach. Moreover, the implicit transition declaration is convenient for modeling the whole physically possible behavior. 

Suppose we desire a report containing data from the COMPOUND table and corresponding test result data for the two tests. A common way to approach this problem would be to combine the three tables with a natural join operation over registry number. However, since a natural join forms the Cartesian product of the... 

Important algebraic operations of fuzzy sets have been defined Cartesian product, mth power, algebraic sum, bounded sum, bounded difference, and algebraic product. For definitions and elementary examples see Zimmermann. ... 

Given a resource type t T and an allocation a(t), the problem is to compute the number of possible bindings of 0(t) operations to a t) resources. 0 t) is the operation set of t. We assume the condition 1 < a(f) < 0(t) holds because otherwise the allocation is invalid. Without loss of generality, we consider the design space for one resource type only. The extension to support multiple resource types is to form a Cartesian product among the bindings of each resource type. We omit the (t) suffix fitom 0(t) and a i) in the sequel without ambiguity. 

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