Representations basis for

Any set of algebraic functions or vectors may serve as the basis for a representation of a group. In order to use them for a basis, we consider them to be the components of a vector and then determine the matrices which show how that vector is transformed by each symmetry operation. The resulting matrices, naturally, constitute a representation of the group. We have previously used the coordinates jc, y, and z as a basis for representations of groups C2r (page 78) and T (page 74). In the present case it will be easily seen that the matrices for one operation in each of the three classes are as follows ... 

As we can see by comparing the HH-Clar structures and k-Clar structures they offer a distinctive basis for representation of benzenoid hydrocarbons. Each of the two approaches have their own merits, and as we will see in the next section, although the structures arising in the two models are quite different, the two approaches can be related in some respect. Both approaches use the same basis Kekule structures and differ... 

Figure 5-5. Cartesian displacement vectors as basis for representation of the water molecule.

A DNA strand could act as the input tape, containing a sequence of the bases A, T, G, and C. This sequence could be artificially created as a code for any kind of data. Modern digital computers use two states, 0 and 1, as the basis for representation of input data. A string of O s and I s, properly weighted, can represent any number or letter of the alphabet, for instance. With four different bases, a DNA computer could be quaternary rather than binary. Of course, the input DNA strand could be restricted to use only two of the four bases, which would make it easier to translate program instructions already developed for a binary computer into the program for DNA computation, but the use of aU four bases makes data representation much more compact. For instance, the numbers 0,1, 2, and 3 can be represented in binary by 00,01,10, and 11, or in the DNA s four bases as A, T, G, and C. This saves one space, in the operation of the computer. 

E, identity transformation, 3, 18, 51 Eigenfunctions, as a basis for representation of point groups, 112 classification of, 3, 14ff degeneracy of, 9... 

A number of reactions (usually, but not exclusively anodic ones) show self-passivation effects so that linear relations of Figures 4a or 4b are no longer an adequate basis for representation of the relative yields. This type of case is illustrated in Figure 4d. 

The BE-matrix also provides the basis for the matrix representation of chemical... 

To this pom t, th e basic approxmi alien is th at th e total wave I lnic-tion IS a single Slater determinant and the resultant expression of the molecular orbitals is a linear combination of atomic orbital basis functions (MO-LCAO). In other words, an ah miiio calculation can be initiated once a basis for the LCAO is chosen. Mathematically, any set of functions can be a basis for an ah mitio calculation. However, there are two main things to be considered m the choice of the basis. First one desires to use the most efficient and accurate functions possible, so that the expansion (equation (49) on page 222). will require the few esl possible term s for an accurate representation of a molecular orbital. The second one is the speed of tW O-electron integral calculation. 

The matrix Rij,kl = Rik Rjl represents the effeet of R on the orbital produets in the same way Rik represents the effeet of R on the orbitals. One says that the orbital produets also form a basis for a representation of the point group. The eharaeter (i.e., the traee) of the representation matrix Rij,id appropriate to the orbital produet basis is seen to equal the produet of the eharaeters of the matrix Rik appropriate to the orbital basis Xe (R) = Xe(R)Xe(R) whieh is, of eourse, why the term "direet produet" is used to deseribe this relationship. 

These veetors form the basis for a redueible representation. Evaluate the eharaeters for this redueible representation under the symmetry operations of the D h group. 

We could take any set of functions as a basis for a group representation. Commonly used sets include coordinates (x,y,z) located on the atoms of a polyatomic molecule (their symmetry treatment is equivalent to that involved in treating a set of p... 

Before considering other concepts and group-theoretical machinery, it should once again be stressed that these same tools can be used in symmetry analysis of the translational, vibrational and rotational motions of a molecule. The twelve motions of NH3 (three translations, three rotations, six vibrations) can be described in terms of combinations of displacements of each of the four atoms in each of three (x,y,z) directions. Hence, unit vectors placed on each atom directed in the x, y, and z directions form a basis for action by the operations S of the point group. In the case of NH3, the characters of the resultant 12x12 representation matrices form a reducible representation... 

Figure 2.1 served as the basis for our initial analysis of viscosity, and we return to this representation now with the stipulation that the volume of fluid sandwiched between the two plates is a unit of volume. This unit is defined by a unit of contact area with the walls and a unit of separation between the two walls. Next we consider a shearing force acting on this cube of fluid to induce a unit velocity gradient. According to Eq. (2.6), the rate of energy dissipation per unit volume from viscous forces dW/dt is proportional to the square of the velocity gradient, with t]q (pure liquid, subscript 0) the factor of proportionality ... 

Fig. 1. Schematic representation showing the basis for classification of toxic effects into local and systemic by single or repeated exposures.

Fig. 3. Schematic representation showing the anatomical basis for differences in the quantitative supply of absorbed material to the Hver. By swallowing (oral route), the main fraction of the absorbed dose is transported direcdy to the Hver. FoUowing inhalation or dermal exposure, the material passes to the pulmonary circulation and thence to the systemic circulation, from which only a portion passes to the Hver. This discrepancy in the amount of absorbed material passing to the Hver may account for differences in toxicity of a material by inhalation and skin contact, compared with its toxicity by swallowing, if metaboHsm of the material in the Hver is significant in its detoxification or metaboHc activation.

The Chemical Abstracts Service has institutionalized the use of graphic representations for identification of and retrieval of information about chemical compounds through their Graphical Data Stmcture (GDS) and connection table (CT). Two information packages. Messenger and STN Express, are the basis for this on-line retrieval system (42). 

Variations in measurable properties existing in the bulk material being sampled are the underlying basis for samphng theory. For samples that correctly lead to valid analysis results (of chemical composition, ash, or moisture as examples), a fundamental theoiy of sampling is applied. The fundamental theoiy as developed by Gy (see references) employs descriptive terms reflecting material properties to calculate a minimum quantity to achieve specified sampling error. Estimates of minimum quantity assumes completely mixed material. Each quantity of equal mass withdrawn provides equivalent representation of the bulk. 

The decomposition approach is used, it is necessary to represent the way in which the various task elements and other possible failures are combined to give the failure probability of the task as a whole. Generally, the most common form of representation is the event tree (see Section 5.7). This is the basis for THERP, which will be described in the next section. Fault trees are only used when discrete human error probabilities are combined with hardware failure probabiliHes in applications such as CPQRA (see Figure 5.2). 

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