Algebraic expressions for

Expand the three detemiinants D, Dt, and for the least squares fit to a linear function not passing through the origin so as to obtain explicit algebraic expressions for b and m, the y-intercept and the slope of the best straight line representing the experimental data. 

Using the expanded determinants from Problem 6, write explicit algebraic expressions for the three minimization parameters a, b, and c for a parabolic curve fit. 

Using the. results (roiri the previous two problems, evaluate the integrals in the answer to Problem 5 and find t as a dosed algebraic expression for the Gaussian trial runetion. 

For cases in which the equiUbtium and operating lines may be assumed linear, having slopes E /and respectively, an algebraic expression for the integral of equation 55 has been developed (41) ... 

It is clear that to develop an explicit algebraic expression for (tj-) or (n) would be exceedingly cumbersome and, as already stated, in this modern day of the digital computer, is unnecessary. A simple program can be written, using equations (6), (7), (8) and (9), that searches for the time (tr) that allows the equivalence defined in... 

Step 8 Solve the Equations. Many material balances can be stated in terms of simple algebraic expressions. For complex processes, matrix-theory techniques and extensive computer calculations will be needed, especially if there are a large number of equations and parameters, and/or chemical reactions and phase changes involved. 

Write the Lineweaver-Burk (double-reciprocal) equivalent of this equation, and from it calculate algebraic expressions for (a) the slope (b) the y-intercepts and (c) the horizontal and vertical coor-... 

Algebraic expressions for run-up distances, Xj, and times to detonation, tj, for the shock initiation of high density PETN pressings, taken... 

Many real reaction systems are not amenable to normal mathematical treatments that give algebraic expressions for concentration versus time, but by no means is the situation hopeless. Such systems need not be avoided. The numerical methods presented... 

The continuous annular chromatograph can be described mathematically by a theoretical plate approach similar to the one developed by Martin and Synge [40] and exemplified by Said [41] for stationary columns [5]. The mathematical description results in algebraic expressions for the elution position of each solute relative to the feed point and for the bandwidth of the eluting zone as a function of the elution position or other system parameters. However, a series of simplifications have to be made in order to describe the CAC with the theoretical plate concept ... 

Apart from the distance variable x that Dunham used in his function V(x) for potential energy, other variables are amenable to production of term coefficients in symbolic form as functions of the corresponding coefficients in a power series of exactly the same form as in formula 16. Through any method to derive algebraic expressions for Dunham coefficients l j, the hamiltonian might have x as its distance variable, but after those expressions are produced they are convertible to contain coefficients of other variables possessing more convenient properties. To replace x, two defined variables are y [38],... 

To convert these data into radial functions, one might apply algebraic expressions for vibrational matrix elements of x to various powers, of form such as... 

The algebraic expressions for the coefficients a, b, and c in terms of the summations calculated from the experimental values are... 

Let us use the matrix least squares method to obtain an algebraic expression for the estimate of Pq in the model y, = Po + r, (see Figure 5.2) with two experiments at two different levels of the factor x,. The initial X, B, R, and F arrays are given in Equation 5.27. Other matrices are... 

Note carefully the logic of this very simple derivation. We want an overall rate r for the single reaction in terms of Ca by eliminating the intermediate Cg in the two-step reaction. We did this by assuming the first reaction in the exact two-step process to be in equilibrium, and we then solved the algebraic expression for Cg in terms of Ca and rate coefficients. We then put this relation into the second reaction and obtained an expression for the overall approximate in terms of the reactant species alone. We eliminated the intermediate from the overall expression by assuming an equilibrium step. 

TABLE 1. Explicit algebraic expressions for the matrix elements elements of (designated by h) and other intermediates (designated by I or i9) used to construct the triply excited moments of the CCSD/EOMCCSD equations, Eq. (60). 

Hint Write the lake cross section in the form (z) = bottom + aiz> anc determine a, from the information given above. Use Fick s first law to express benzene- This yields an algebraic expression for dC / dz. Then, C(z) and SFbenzene= constant can be determined from the boundary conditions. 

To determine how to form a set of trigonally directed hybrid orbitals, we begin in exactly the same way as we did in the MO treatment. We use the three a bonds as a basis for a representation, reduce this representation and obtain the results on page 219. However, we now employ these results differently. We conclude that the s orbital may be combined with two of the p orbitals to form three equivalent lobes projecting from the central atom A toward the B atoms. We find the algebraic expressions for those combinations by the following procedure. 

In the general case, there is no simple algebraic expression for the first term of (5.42), therefore, for its evaluation one must utilize the tables of numerical values of the submatrix elements of operators Uk. The most complete tables, covering also the case of operators Vkl, may be found in [87], For operators, also depending on spin variables, the analogue of formula (5.41) will have the form... 

Clebsch-Gordan coefficients have already occurred several times in our considerations in the Introduction (formula (2)) while generalizing the quasispin concept for complex electronic configurations, while defining a relativistic wave function (formulas (2.15) and (2.16)), in the addition theorem of spherical functions (5.5) and in the definition of tensorial product of two tensors (5.12). Let us discuss briefly their definition and properties. There are a number of algebraic expressions for the Clebsch-Gordan coefficients [9, 11], but here we shall present only one ... 

In angular momentum theory a very important role is played by the invariants obtained while summing the products of the Wigner (or Clebsch-Gordan) coefficients over all projection parameters. Such quantities are called 7-coefficients or 3ny-coefficients. They are invariant under rotations of the coordinate system. A j-coefficient has 3n parameters (n = 1,2,3,...), that is why the notation 3nj-coefficient is widely used. The value n = 1 leads to the trivial case of the triangular condition abc, defined in Chapter 5 after formula (5.25). For n = 2,3,4,... we have 67 -, 9j-, 12j-,. .. coefficients, respectively. 3nj-coefficients (n > 2) may be also defined as sums of 67-coefficients. There are also algebraic expressions for 3nj-coefficients. Thus, 6j-coefficient may be defined by the formula... 

The presence of the repeating terms with the same L, S causes additional difficulties while orthogonalizing the CFP, finding their phase multipliers as well as the relationships between complementary shells. Also the problem of finding the algebraic expressions for the CFP becomes much more complicated (see Chapter 16). 

The most effective way to find the matrix elements of the operators of physical quantities for many-electron configurations is the method of CFP. Their numerical values are generally tabulated. The methods of second-quantization and quasispin yield algebraic expressions for CFP, and hence for the matrix elements of the operators assigned to the physical quantities. These methods make it possible to establish the relationship between CFP and the submatrix elements of irreducible tensorial operators, and also to find new recurrence relations for each of the above-mentioned characteristics with respect to the seniority quantum number. The application of the Wigner-Eckart theorem in quasispin space enables new recurrence relations to be obtained for various quantities of the theory relative to the number of electrons in the configuration. 

In [90] the relationship between eigenvalues of the Casimir operators of higher-rank groups and quantum numbers v, N, L, S is taken into account to work out algebraic expressions for some of the reduced matrix elements of operators (Uk Uk) and (Vkl Vkl). However, the above formulas directly relate the operators concerned, and some of these formulas are not defined by the Casimir operators of respective groups. 

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