Reducing dimensionality

Let us now consider the n-dimensional reducible representation r d which is produced from the function space whose basis functions are Qi> 9i> - Qn> d let us assume that in the reduction of I 1 no irreducible representation of the point group occurs more than once. One way of looking at the reduction is to see it as a change of basis functions from gl9 9i> gn to... 

In Problem 9.14, a two-dimensional reducible representation was found for the group <34. What irreducible representations is this reducible representation the sum of ... 

The condition stated in Definition 2.1 assures that a well-defined n-dimensional reduced model will correspond to each solution (2.12) whenever this condition is violated, the system in Equations (2.7) and (2.8) is said to be in a nonstandard singularly perturbed form. 

With the 12-dimensional reducible representation of the Cartesian displacement vectors of HNNH, the inspection method probably does not work. However, the reduction formula can be used. The reducible representation is ... 

Table 2.1 Division of the permutation matrices of Figure 2.1 into their direct sum components as involving the one-dimensional irreducible set exhibiting the invariance of the nitrogen label [or position on the principal rotational axis] and the three-dimensional reducible set over the hydrogen atom labels [or positions of the hydrogen atoms]. ...

The two other quantities we shall evaluate are the time-evolving (diabatic) electronic populations, Pa t), and two-dimensional reduced densities PaiQi, Qj,t) for the electronic state a. These quantities are defined as follows, using the wavefunc-tion given by (29) ... 

The t2-type orbitals c/2, vr], dx [zx] and dL, [xy] with spin factors give rise to a six-dimensional reducible representation which resolves into the irreducible U and E". It is expedient to choose the set... 

PROTEIN-PROTEIN DOCKING USING THREE-DIMENSIONAL REDUCED REPRESENTATIONS AND BASED ON A GENETIC ALGORITHM... 

The appropriate kind of rotational averaging has been discussed for several years by Bowman and coworkers, "in connection with a hierarchy of dimensionality reducing theories of reactions. Although the BCRLM differs in origin from these dimensionality reducing theories, it resembles them in spirit, and in detail at some levels. Specifically, the application of microcanonical rotational averaging to BCRLM has been presented by Walker and Poliak, and we will review only the final results here. 

Lindstedt RP, Meyer MR A dimensionally reduced reaction mechanism for methanol oxidation. Proc Combust Inst 2002 29 1395-1402. 

In a cyclic model of a crystal with N primitive cells in the main region a band rep is an Np dimensional reducible rep of a space group. An induced rep is a particular case of a band rep as it satisfies both properties 1 and 2 with p = nqU/ n is the dimension of the site-symmetry group irrep for a point q belonging to the set of n, points in the unit cell). 

The atomic displacements of six atoms in the primitive cell generate the 18-dimensional reducible represntation, which contains three acoustic modes and 15 optical modes Aig + A2g + A2U + Big + 2Bi + B2g + Eg + 3i ). Three acoustic modes have zero frequency at the F point and are associated with the translation of the entire crystal along any direction in space. These branches are called acoustic modes as the corresponding vibrations behave as acoustic waves. The translation of... 

It was pointed out earlier that several irreducible representations can be combined into a larger-dimensional reducible representation. Our example was... 

Consider the first-order, two-dimensional reaction system A B C. Let ku k2, and ks be the rate constants for the A-Xo-B, B-to-A, and B-to-C reactions, respectively. In the QSA, 2 h k, the ultra reactive B is maintained at a quasi steady state after the initial transient, so the system becomes one dimensional in that it is essentially governed by the disappearance of A. In the QEA, k k2 k, hence A and B quickly reach equilibrium after startup. The resulting one-dimensional reduced model simply states that the system is dictated by the slow depletion of the equilibrium pool of (A + B) due to a small leak resulting from k, . 

Two-dimensional off-axis reduced stiffness for twisted yarn was developed to three-dimensional reduced stiffness. As described earlier, if the migration structure in the twisted yarn is stationary along the yarn axis, in other words, the yarn axis is the same as the loading axis, the Young s modulus can be calculated through Eq. (10.14) as the orientation angle of each cylindrical element is invariable. 

The first equation of (69) implies that an n-dimensional reducible representation D(R) can be expressed in terms of representations of lower dimensions. Here D(R) is expressed as a direct sum of an ni-dimensional representation Di(R) and an /i2-dimensional representation D R)y where ni + ri2 = n. This suggests the possibility of reducing a reducible representation down to a point where it can be expressed in terms of inequivalent irreducible representations. Furthermore, representations can be constructed from these inequivalent irreducible representations. The second equation of (69) implies that the character of a reducible representation can be written as the sum of the characters of the irreducible representations. From the theory of group representations, one finds that the number of nonequivalent irreducible representations is equal to the number of classes with respect to the relation of conjugation. Also the sum of the squares of the dimensions of the irreducible representations is equal to the order of the group. Then, with the aid of the above theorems one can express this reducibility for a reducible representation D(R) as... 

V. Spiwok and B. Kralova,/. Chem. Phys., 135(22), 224504 (2011). Metadynamics in the Conformational Space Nonhnearly Dimensionally Reduced by Isomap. 

It just remains for us to mention that the investigation of quantum effects in dimensionally reduced electron conductors is an attractive modern field of semiconductor research , characterized by the investigation of the sub-band formation in heterostructures , of the problems of quantum wires and quantum dots, of artificial atoms and artificial molecules . In this context the term nanoelectronics [50,74] is more than just a fashionable slogan, but describes an exciting scientific... 

Marhaba, T.F., Borgaonkar, A.D., and Punburananon, K. (2009). Principal component regression model applied to dimensionally reduced spectral fluorescent signature for the determination of organic character and THM formation potential of source water. 

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