Group one-dimensional

Fig. 8 CPK models of (A) p-structural Ala and (B) a-helical Leu derived from PEPCON. The black atoms present carbonyl carbons. A linear and planar solute such as pentacene provides more effective interaction area with the carbonyl groups one-dimensionally-aligned on the rigid main chain than a disk-like solute such as coronene (A). On the other hand, the carbonyl groups of Leu are covered with their bulky residual groups (B).

H. P. Flatt, (AlM-6) A Multi-Group, One Dimensional Diffusion Theory Code, Unpublished Atomics Ihtema-tional Report (1960). 

For the analysis of these assemblies, the BPG code was used extensively along with a four-group, one-dimensional neutron diffusion code. The thermal flux depression factors were obtained from a one-velocity Monte Carlo program. A resonance integral program was used to calculate epithermal absorption parameters for U-238 and U-235,... 

H. P. FLATTand D. C. BALLER, AIM-S-A Mutti-Group, One Dimensional Diffusion Equation Code, NAA-SR-4694, North American Aviation (March 1 50). 

H. P. FLATT and D. C BALLER, AIM-5-A Multi-group One Dimensional Diffusion Theory Code, NAA-SR-4694. 

In each class, the rare earth minerals are basically arranged according to the sizes of the anionic groups i.e., in the order of isolated anionic groups, one-dimensional chains, columns, two-dimensional sheets, plates, and three-dimensional frameworks. 

This means that these matriees are really a eombination of two separate group representations (mathematieally, it is ealled a direet sum representation). We say that Dl ) is redueible into a one-dimensional representation DlO and a three-dimensional representation formed by the 3x3 submatriees whieh we will eall Dl ). 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

Another one-dimensional representation of the group ean be obtained by taking rotation about the Z-axis (the C3 axis) as the objeet on whieh the symmetry operations aet ... 

These one-dimensional matriees ean be shown to multiply together just like the symmetry operations of the C3V group. They form an irredueible representation of the group (beeause it is one-dimensional, it ean not be further redueed). Note that this one-dimensional representation is not identieal to that found above for the Is N-atom orbital, or the Ti funetion. 

We have found three distinet irredueible representations for the C3V symmetry group two different one-dimensional and one two dimensional representations. Are there any more An important theorem of group theory shows that the number of irredueible representations of a group is equal to the number of elasses. Sinee there are three elasses of operation, we have found all the irredueible representations of the C3V point group. There are no more. 

The additional coupling (9.8 Hz) of the cyclopropane proton A at <5// = 1.60 is the result of a vicinal H atom in the side-chain. This contains a methyl group B, a vinyl group C and an additional substituted ethenyl group D, as may be seen from the one dimensional H and C NMR spectra and from the CH COSY diagram. 

The typical chemical shift values and CH coupling constants in the one-dimensional NMR spectra reveal three functional groups ... 

Any extended part of a linear polymer molecule exhibits a strong anisotropy of many properties since its atoms and atomic groups are oriented and the macromolecule is actually a one-dimensional crystal . The parallel packing of these parts during the formation of a uniaxially oriented polymer substance imparts the anisotropie properties of a single molecule to the whole polymeric material. 

Case (b). This case includes no single-group representations, and all real one-dimensional, double-group representations. This includes only the following. 

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