Due to the

noncommutativity of quaternions, there are two types of eigenvalues and linear maps: left and right eigenvalues of quaternion matrices and left and right quaternion linear maps.

But in [26], even Shor's ordering finding processes is assumed available as an oracle, one cannot solve the F[P.sup.G.sub.g, f] problem since the order-lifting sub-process is blocked by the

noncommutativity of G .

One can see that the

noncommutativity of the space does not affect the commutator between current density variables.

For the former, [mathematical expression not reproducible] holds, while for the latter ones, since they are just dynamical counterparts of the energy and time, there is no reason to demand

noncommutativity apriori; that is, their commutator depends on the particular choice of these functions of coordinate and momentum.

Through the

noncommutativity of the quaternion algebra when difficulties arise already in determining the quaternion determinant, the problem of the determinantal representation of generalized inverses only now can be solved due to the theory of column-row determinants introduced in [28, 29].

Rodriguez-Lopez, "

Noncommutativity of mappings in hybrid fixed point results," Boundary Value Problems, vol.

Note that, in general, due to rotation

noncommutativity, the rotation (1) and angular velocity vectors (4) cannot be considered as real vectors.

Note that the

noncommutativity is not intrinsic to the overall scheme, but results from the special way the permutations are generated, namely, by deploying the two-dimensional discrete Cat Map.

The so-called "quantum question" equality presented by Wang et al shows how the very same quantum probability theory that explain otherwise mysterious

noncommutativity of measurements in physics, can also provide excellent measurement predictions for question order effects in social and behavioral science experiments.

Note that, due to the

noncommutativity of matrix multiplication and the inclusion of [ARFact-Matrix] algorithm, the code of this spectral algorithm is much more elaborate than the code of [ASpecPaired-Scalar] algorithm.

Moreover, they gave the first arguments on resisting Shor's quantum algorithm attacks based on

noncommutativity (see Remark 11).