Formulae algebraic

Ifb - 4ac iis positive and not a perfect square, the roots are irrational numbers. 

The graph of the parabola y = ax + bx + c opens upward and has a minimum turning point when a is positive, a 0. 

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Algorithms A procedure consisting of a sequence of algebraic formulas and/or logical steps to calculate or determine a given task. [NIH]... 

That property allows us to simplify algebraic formulas. For example, k( + kj could be substituted by max fcj,fc, (with small relative error), or... 

In practice, the solution of polynomial equations is problematic if no simple roots are found by trial and error. In such circumstances the graphical method may be used or, in the cases of a quadratic or cubic equation, there exist algebraic formulae for determining the roots. Alternatively, computer algebra software (such as Maple or Mathematica, for example) can be used to solve such equations... 

As was already mentioned, in theoretical atomic spectroscopy, while considering complex electronic configurations, one has to cope with many sums over quantum numbers of the angular momentum type and their projections (3nj- and ym-coefficients). There are collections of algebraic formulas for particular cases of such sums [9, 11, 88]. However, the most general way to solve problems of this kind is the exploitation of one or another versions of graphical methods [9,11]. They are widely utilized not only in atomic spectroscopy, but also in many other domains of physics (nuclei, elementary particles, etc.) [13],... 

Coefficients of fractional parentage play a fundamental role in the theory of many-electron atoms. There are algebraic formulas for them (see Chapter 16), however, they are not very convenient for practical utilization, and normally tables of their numerical values are used. They can be generated in the recursive way, starting with the formula... 

Formulas (12.21)—(12.26) are fairly simple, the corresponding transformation matrices are expressed only in terms of 6j- and -coefficients, which can be easily calculated with the help of algebraic formulas (see [11]), tables of their numerical values or using standard computer codes. These formulas can be easily generalized to cover the case of a larger number of open shells. 

In Chapter 15, for the CFP with a detached electrons, we obtained a relationship (15.27) whose right side has the form of a vacuum average of a certain product of second-quantized operators q>. To obtain algebraic formulas for CFP, it is necessary to compute this vacuum average by transposing all the annihilation operators to the right side of the creation operators. So, for N = 3, we take into account (for non-repeating terms) the explicit form of operators (15.2) and (15.5), which produce pertinent wave functions out of vacuum, and find (cf. [107])... 

The above recurrence relations allow the numerical values of CFP to be readily found and the availability of the above algebraic formulas for them makes it possible to establish similar algebraic expressions for other quantities in atomic theory, including the CFP with two detached electrons, the submatrix elements of irreducible tensors, and also the matrix elements of the operators of physical quantities [110]. 

The method used to establish the algebraic formulas for CFP is, in broad outline, as follows if at least one wave function is available that is characterized by the quantum numbers (111112) and L, then all the remaining functions with the same quantum numbers can be constructed... 

The expression for quantity g(Rs) in (19.52) follows directly from (5.40). Thus, for p- and d-shells we have simple algebraic formulas for the coefficients of radial integrals (actually, for matrix elements) of all relativistic corrections of the order a2 to the Coulomb energy. For the /-shell such a formula exists only in the case of the orbit-orbit interaction. For the almost filled shell we find... 

The objective of this section is to give algebraic formulae for evaluating the matrix elements of the interaction operators in the basis set of ATs. First, we need to introduce the seniority number and the coefficients of the fractional parentage (CFPs). Then we will use to full advantage the ITO... 

Although the above algebraic formula seems rather complex, in fact it reduces to simple numbers. Therefore, a tabulation of these coefficients is a useful tool (Table 49). To this end one gets the final formulae shown in Table 5. 

In the present section the catafusenes (catacondensed simply connected poly hexes cf. Sect. 5.1) are treated. However, in contrast to the Harary-Read numbers (first column of Table 7) we shall be interested in the numbers of unbranched and branched systems separately. The numbers of unbranched catafusenes (Table 10) are known from algebraic formulas (cf. Sect. 5.2), but now we are interested in the unbranched catacondensed benzenoids and helicenes separately. Likewise we shall treat the numbers of branched catacondensed benzenoids and helicenes separately. 

A computer tape is. .. not unlike an algebraic formula, but this is not an essential limitation, except at the stage when it is being prepared by a human mind. It can be transformed, in the computer, to something much more complex, which is unintelligible to the bare human intellect, and can only be read, so to speak, by another computer. 

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