Equivalent operations

Division of matrices is not defined, but the equivalent operation of multiplieation by an inverse matrix (if it exists) is defined. If a matr ix A is multiplied by its own inverse matrix, A , the unit matrix I is obtained. The unit matrix has Is on its prineipal diagonal (the longest diagonal from upper left to lower right) and Os elsewhere for example, a 3 x 3 unit matrix is... 

Column type Exclusion limit (polystyrene equivalent) Operating range (polystyrene equivalent)... 

The main problem here is connected with selecting the regularizator R. Since regularity conditions became operator inequalities, it seems reasonable to choose as R operators of the most simplest structures which are energetically equivalent to the operator A. Let, for instance, A and Aq be energetically equivalent operators with constants 7, and 7j, so that... 

Replacing i by its equivalent operator from equation 3.10 and replacing E by II, the Hamiltonian, equation 3.11, becomes... 

Let us now attempt to review the field experience of seawater RO plants, perhaps the most important aspect of any commercial process, information most critical to the potential plant buyer. My comments here are biased by our experience at the Desalination Technology Transfer Center in St. Croix, where we have been operating seawater RO plants for more than four years. While it is certain that no two facilities have equivalent operational practice, experience, and problems, and I wish to emphasize this, nevertheless, I believe that by discussing the most limited operations information available, in light of our experience, it will make my comments more meaningful. 

There are good reasons for choosing to carry out certain operations in the Laplace domain rather than performing equivalent operations in the time domain. In particular, integration of a function with respect to time is equivalent in the Laplace domain to division by the Laplace variable s. Conversely, differentiation corresponds to multiplication by s. This latter property enables differential equations to be Laplace transformed and then solved by algebraic means. These Laplace domain operations are all more simple than their time domain counterparts. In addition convolution in the time domain is equivcdent to multiplication in the Laplace domain. Formally, this may be represented by eqn. (24), the left-hand side of which is termed the convolution integral. 

The coefficient of the symmetry element, in the top line, tells how many different equivalent operations of this type occur. However, not all symmetry elements of the same type are always equivalent. Thus, in the symmetry group of benzene, the thirteen C2 operations fall into three distinct classes the one dyad perpendicular to the molecular plane, the six that pass through opposite atoms in the ring, and the six that pass in between the atoms, through the centers of the bonds. Each of these operations is equivalent only to the others within the same class. 

The use of Eqs. (27) or (28) to evaluate VMm can become laborious when there is more than one delectron in the atom or ion. Stevens (6) and Bleaney and Stevens (7) have shown that, when all the wave functions belong to the same LS configuration, the matrix elements can all be evaluated by use of the equivalent operator... 

In (a) we divided 50 cents by 30 cents/doz, while in (b) we multiplied 50 cents by 1 doz/30 cents, the reciprocal of 30 cents/doz. These are equivalent operations and you should feel comfortable with either method. In each case, "cents" and doz cancel out, leaving only "apples," as we had hoped. 

The spaces we have defined could (at least piecewise) be mapped onto 72 , and this means precisely the same as saying that their elements x,y,.. e S can be represented by arrays r,s,.. in 72. . This means that any operations to be carried out that involves elements z, y,.. e S, can be replaced by some equivalent operation involving their mappings, which are the arrays r = /(z), a = f(y) etc. However, spaces in general tend to be complicated, and the details of each representation differs. 

Not all aspects of the theory are dealt with in equal depth. Some are just mentioned, some even omitted. For example, the method of effective (equivalent) operators deserves mentioning [321]. It allows one to take into account the main part of relativistic effects but at the same time to preserve the LS coupling used for classification of the energy spectra of the atoms or ions considered. 

A comparison of power requirements of different cycles can only be approximate because of the difficulty in defining equivalent operating conditions for the various cycles. Table 4 and Table 5 present values for different cycles, and for different versions of the mixed refrigerant cycle. 

The wealth of seemingly random flavoproteins can be brought into a certain order by distinguishing them according to the number of redox equivalents operating at once in their input and output reactions (Table... 

The reduction formula can be simplified by grouping the equivalent operations into classes,... 

Introducing two idempotent and mutually exclusive projectors P and Q for the spaces spanned by. and X, eq.(6.1.13) can be converted into an equivalent operator equation. Eq.(6.1.12) implies... 

Fuel research engine evaluation of the performance and emissions of methanol/gasoline blends at equivalent operating conditions to determine any benefits from the slightly altered fuel chemistry. 

Figure 11 displays the energy consumption characteristics of the clear and blended fuels on a thermal efficiency basis. From this information it is apparent that, at equivalent operating conditions, there is essentially no difference in engine efficiency caused by the addition of methanol. The figures also indicate that if the engine can be operated at greater (leaner) equivalence ratios, some increases in thermal efficiency can be expected. 

Nette, P., Alenikov, V., Griffith, R. and da Silva, T., The possible role of the lAEA/WHO SSDL network in implementing the dose equivalent operational quantities into radiation protection practices for individual monitoring. Radiation Prot. Dosim., 28 (1989) 161-165. 

The annual operating time (Table 5) is calculated as full load equivalent operation time per year and indicates the utilisation intensity of the plant. For room heating purposes in Austria it is relatively low (1500 h/a), for big industrial CHP plant it can be up to 7000 h/a (example biomass CHP in pulp mills in Austria). 

It is wise to remember that every operation, mathematical or physical, in real or reciprocal space has an equivalent operation in the other. Often these are not obvious, but they are always there. Molecular replacement, for example, can in theory be carried out in either space, though our computational tools for doing so are much more powerful in reciprocal space. Similarly, structure refinement may be carried out using least squares procedures in reciprocal space or, equivalently, difference Fourier methods in real space. 

Table 1. Stevens multiplicative factors associated with equivalent operators for the ground states of rare earth ions and the calculated Hartree-Fock radial integrals (r > in atomic units of length...

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