Cyclic variables

Cyclical Variability in Stellar Winds Proceedings, 1997. XXII, 415 pages. 1998. 

Data reduction. We used the log-transformed data in all analyses presented here. The PCA resulted in eight principal components with eigenvalues > 1 and they explained 93.5% of the variation in the original data (Table II). The first three principal components all convey generalized information on chemical structure size (PC 1), degree of branchness (PC 2), and number of cycles (PC 3). PC 1 was positively correlated with all 90 variables (X 32), except for the cyclic variables in which r was as low as. 07 for the 3rd order cyclic variables. PC 2 was positively correlated ( r >. 26) with all cluster variables, but negatively correlated with all path and cyclic variables. PC 3... 

In Section 1.5.1, it was mentioned that the energy of the lowest state of a particle confined in a one-dimensional box is not zero and this residual energy is a consequence of the Uncertainty Principle. Yet the ground state energy of the particle-in-a-ring problem is zero. Does this mean the present result is in violation of the Uncertainty Principle The answer is clearly no, and the reason is as follows. In a one-dimensional box, variable x starts from 0 and ends at a, the length of the box. Hence Ax can at most be a. On the other hand, in a ring, cyclic variable does not lie within a finite domain. In such a situation, the uncertainty in position cannot be estimated. 

In analogy with the one-dimensional analysis, the Jj are defined over complete periods of the orbit in the (qj,Pj) plane, Jj = ptdq j. If one of the separation coordinates is cyclic, its conjugate momentum is constant. The corresponding orbit in the (qj,Pj) plane of phase space is a horizontal straight line, which may be considered as the limiting case of rotational periodicity, for which the cyclic qj always has a natural period of 2-k, and Jj = 2irpj for all cyclic variables. 

Of special importance for the application of the quantum conditions and of the correspondence principle is the case in which the Hamiltonian function is not changed by the rotation as a whole of an atomic system about a fixed direction in space. If we introduce as co-ordinates the azimuth =qf of one of the particles of the system together with the differences of the azimuths of the other particles from , and other magnitudes depending only on the relative position of the particles of the system with respect to the fixed direction in space, will be a cyclic variable and the momentum p conjugated to it is, by 6, the angular momentum of the system... 

We now consider the case where the mechanical system is subjected to a homogeneous external (electric or magnetic) field. The azimuth of a particle of the system about an axis parallel to the field is then a cyclic variable in general H will depend on J<(i, and we have the quantum condition... 

If now a homogeneous magnetic field exists, the potential energy of the system is invariant with respect to a rotation about the direction of the field. The azimuth of an arbitrary point of the system is then a cyclic variable, as proved in 6 and 17, and the corresponding conjugated momentum p is the angular momentum of the system about the direction of the field. 

If the p jlar axis of the co-ordinate system be taken in the direction of the resultant angular momentum P=, /2ir, the angular separation of the line of nodes from a fixed line in the invariable plane is a cyclic variable conjugate to P. For the other co-ordinates let us take the radius vector r of the outer electron and the conjugate momentum p together with the angular separation ifi of the outer electron from the line of nodes and the conjugate momentum... 

D.D. Glover (2003). Expression and cyclic variability of CYP3A4 and CYP3A7 isoforms in human endometrium and cervix during the menstrual cycle. Drug Melab. Dispos. 31, 1-6. 

The balancing of HR systems with internal heating on the rheological principle is virtually impossible to do and can only be approximate. The fundamental problem is the structure of the computer flow model, which allows for the cyclical variability of the thickness of the flowing layer of melt in the hot runners. Approximate methods are therefore used for such systems, chiefly differentiation of the size of the gates. 

Variable-speed vertical Continuousf Cyclic Zero or reduced To 16 ft ... 

Solenoid magnetic separators are designed for batch-type, cyclic, and continuous operation. Devices which can use matrices of expanded metal, grooved plates, steel balls, or filamentaiy metals have been designed. Continuous separators with capacities to 600 t/h for iron ores (similar to the Brazilian hematite) are commercially available (Sala International Inc.). Selection of the method of operation is apphcation-dependent, being based on variables such as temperature, pressure, volume of magnetics in the feed, etc. 

Another way of removing the six translational and rotational degrees of freedom is to use a set of internal coordinates. For a simple acyclic system these may be chosen as 3N — I distances, 3N — 2 angles and 3N -3 torsional angles, as illustrated in the construction of Z-matrices in Appendix E. In internal coordinates the six translational and rotational modes are automatically removed (since only 3N — 6 coordinates are defined), and the NR step can be formed straightforwardly. For cyclic systems a choice of 3A — 6 internal variables which span the whole optimization space may be somewhat more problematic, especially if symmetry is present. 

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