Sum of products

The nth virial coefficient can be written as sums of products of Mayer/-fiinctions integrated over the coordinates and orientations of n particles. The third virial coefficient for spherically syimnetric potentials is... 

Zfv. = sum of product flow rates times respective product values (income)... 

Consider the following example in which the worker risk from a catastrophic accident has been calculated to be 2 X 10 fatalities per year. It is possible to interpret this number in many ways, but one of the most common ways is the following there is one chance in 5000 per year that a worker will be fatally injured at the plant. However, you should be cautious when interpreting single risk estimates that are the sums of products of frequency and consequence of many accidents. The way you believe (and act) may be affected by the frequency/consequence profile that the number represents (see Sections 3.2.4 and 4.2.5.) That is, your reaction to an accident that occurs once every 100 years and kills 1 person (Risk = 10 fatalities per year) and your reaction to an accident that occurs once every 10,000 years and kills 100 people (Risk = 10 fatalities per year) are likely to be very different. 

The sequences involving the i-th component may be separated from the other sequences and this probability factored from the sum of products as shown in equation 2.8-9, where is the cutset with factored out. This property relates the importance measures. Substituting equation 2.8-9 into equation 2.8-2 gives equation... 

The Boolean equation for the probability of a chemical process system failure is R = A (B-i-C (D-rE (B-l-F G+C). Using Table 2.1-1, factor (he equation into a sum of products to get the mincut representation with each of the products representing an accident sequence. 

The simplicity of the final result is the mincut representation (sum of products Section 2.2) depicted as a fault tree in Figure 3.4.4-9. If the single double, and so on to higher redundancy components had been identified, the complex and awkward tree of Figure 3.4.4-S would have been avoided. Some systems are so complex that this cannot be done by observation, but computer analv. is will show simplicities if they exist. 

We now note that the exponential can be written as a sum of products of terms that involve only neighboring spins ... 

Finally, using the physical interpretation of the quantum site-state coefficients ai , we can write down an explicit form for the probability functions p . Since 0 is defined by the list n, we must simply write down the probability that a measurement of the 2r states around a given site F will yield a 2r-tuple which is an element of n. We therefore get sums of products of absolute squares, with individual list elements contributing the terms and list elements... 

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The time-dependent density can be expressed as a sum of products of unnormalized, time-dependent NGSO s analogously with Eq. (3.26)... 

SXY(,) sum of squared residuals per column on diagonal, and sums of products of residuals in cells above diagonal... 

Introducing a complete set of eigenstates of QHmQ, f, we express the resonance-mediated partial-wave amplitude in Eq. (39) as a sum of products of three physically distinct matrix elements,... 

Computation of sums of products of the concentration and population patterns to provide exposure/dosage... 

The theory of the thermodynamics of irreversible systems (Prigogine, 1979 Prigogine and Stengers, 1986) shows that the differential quotient of entropy with time (the change of entropy with time) can be expressed as the sum of products, the terms of which contain a force factor and a flow factor. In chemical systems, the... 

Thus the first row of the result matrix will have the single element resulting from the sum-of-products of the first row of [A] times the column of /i, which will be... 

The coefficients, particularly for large molecules, are often more useful as the sums of products called the density matrix. If Ci p is the coefficient of the pth basis function of symmetry A in the fth molecular orbital, the element (p, q) of the (partial) density matrix for m.o. i is... 

We now look at the concentration profiles for reactions (1) and (2). For simplicity, we shall say that reaction (2) is twice as fast as reaction (1), which is likely - unimolecular reactions are often quite slow. We should note how, at the end of the reaction at the far right-hand side of the profile in Figure 8.19, the total sum of product will be the same as the initial concentration of reactant. Also note how, at all stages during the course of the reaction, the ratio of products will be 2 1, since that was the ratio of the two rate constants. 

For example, at 1200°K — F(2) must be greater than 0.24 eV in order that the two correction terms differ by less than 10%. However, for the small defect concentrations of interest here (e.g. c,w6x 10-5 for Fv — 1.0 eV in copper) the correction due to the vacancy-pair term is unimportant for much smaller values of — Fl2) so that differences are not practically very important. The terms of order (c°)3 are more difficult to compare and the numerical values depend on the relative magnitudes of Fw and F((123 ), as can be seen by noting that the sum of products of /-functions in Bs is equivalent to... 

The trace of the matrix 2", i.e. 2 multiplied by itself n times, gives a sum of products, each composed of n [Pg.52]

In general, the gl contain a sum of products of gk with k < n, which can be found by means of repeated reductions of lower order. That part of —glMz which cannot be reduced to an expression in terms of gk with a lower index is called by Caspers the irreducible principal diagonal part and is written... 

To do this we must study the structure of the operator <0 B( 34> 0) given by expression (25"). This appears as a sum of products of the operators ) preceded from the left by an operator 012. It is necessary to remark that for each of the terms and for every pair of operators S( l), never more than one particle is repeated. These particles common to two S(IT- ) will be articulation points in the graph language defined above and, consequently, all the intermediate states between the operators ) will be k = 0. Furthermore, this remark is general and valid to all orders in the concentration in Cohen s formalism. 

The cost of hydrogen at the pump, i.e., the sum of production, distribution and retail costs, for the same range of options. 

The Gini index is the sum of products of the relative frequencies of one class with the relative frequencies of all other classes. 

The field in each vertical slice m thus can be represented by a weighted sum of products of y-dependent functions, the slab modes for each polarization (2 sketched within slice m- ), and corresponding x-dependent functions. 

Another, often major, advantage of using coded factor levels is that the numerical values involved in matrix manipulations are smaller (especially the products and sums of products), and therefore are simpler to handle and do not suffer as much from round-off errors. 

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