Recursion examples

Task P Split E into two complementary (but possibly empty) sub-sets E and E, such that every example of E is admissible wrt a=0 and b =1. The set E is relative the non-recursive examples of r n where Y is effectively constmcted from all the constituents of the decomposition of X, while the set E is relative the non-recursive examples of r n where Y is not constructed from all the constituents of ... 

Bl) The metrics effect is very significant in special theoretical examples, like a freely joined chain. In simulations of polymer solutions of alkanes, however, it only slightly affects the static ensemble properties even at high temperatures [21]. Its possible role in common biological applications of MD has not yet been studied. With the recently developed fast recursive algorithms for computing the metric tensor [22], such corrections became affordable, and comparative calculations will probably appear in the near future. 

If the sampling time is 0.1 seconds, the values of the discrete-time state transition and control matrices AfT) and BfT) calculated in Example 9.8 may be used in the recursive solution. 

For each failure time, calculate the corresponding cumulative hazard value, which is the sum of its hazard value and the hazard values of all preceding failure times. This calculation is done recursively by simple addition. For example, for the generator fan failure at... 

Thus, whenever the set A has a manifest self-similarity, so that, like the Cantor set, it can be defined by a recursive geometric construction, Dfractal oan be easily calculated from this relation. The Koch Curve, for example, the first three steps in the construction of which are shown in figure 2.2, has a length L which scales as... 

The LST, on the other hand, explicitly takes into account all correlations (up to an arbitrary order) that arise between different cells on a given lattice, by considering the probabilities of local blocks of N sites. For one dimensional lattices, for example, it is simply formulated as a set of recursive equations expressing the time evolution of the probabilities of blocks of length N (to be defined below). As the order of the LST increases, so does the accuracy with which the LST is able to predict the statistical behavior of a given rule. 

From this first example, it is obvious that, apart from a CA-like pai allel updating of an ever-increasing number of sites and the set of recursive symbol-strings to which such a dynamics inevitably leads, there is no real internal geometry, as such. Each string at time t remains essentially a static collection of abstract symbols. 

These statements are a consequence of the recursion relations obtained by identifying the coefficients of the power series expansion on the right- and left-hand side of the equation. For example, in (4.6), the coefficient of x" is (n > 1) on the left-hand side, and on the right-hand side a polynomial in R, . [cf. (2.56)], which implies the uniqueness. The coefficients of the polynomial mentioned are non-negative the term occurs, coming from x/, thus Rj > n-i statements that the coefficients are... 

Rather similar was the paper [PolG36a] which also derives asymptotic formulae for the number of several kinds of chemical compounds, for example the alcohols and benzene and naphthalene derivatives. Unlike the paper previously mentioned, this one gives proofs of the recursion formulae from which the asymptotic results are derived. A third paper on this topic [PolG36] covers the same sort of ground but ranges more broadly over the chemical compounds. Derivatives of anthracene, pyrene, phenanthrene, and thiophene are considered as well as primary, secondary, and tertiary alcohols, esters, and ketones. In this paper Polya addresses the question of enumerating stereoisomers -- a topic to which we shall return later. 

Software to predict the properties of formulated products is made more powerful by a recursive procedure which can use formulas stored in files as raw materials. Particular care must be taken with program flow control and data structures for the recursion to be effective. This paper illustrates these issues using an example derived from a working formulation system for coatings development. 

A recursive modelling procedure will let the chemist use formulated products as a raw materials. In this paper, we illustrate this use of recursion, and provide a working example that highlights some of the complexities encountered when using this technique. 

Recursion, when used in the context of computer programs, refers to a procedure that calls itself as a subprocedure. The classic examples cited in programming texts Q, ate computation of the factorial function and... 

The table below illustrates these issues by comparing how a recursive subroutine must handle data which is available from a database, such as the cost of a raw material, data that is calculated for the formulated product, such as PBR, and data for intermediate products. (The variable names shown in the table are part of the example procedure given in the appendix.) Compare with the previous table for a non-recursive modelling procedure s data structure. 

Source and use of data within recursive modelling procedure Examples of variable names shown in ITALICS ... 

The appendix shows an example of a recursive procedure to calculate the pigment to binder ratio for a non-reactive coating formulation. We assume that a raw material data base contains raw material code, density, cost per unit weight, solids volume fraction, and pigment solids volume fraction. Formulas are stored in files that contain the identification and amount of... 

Equation (4) is a three-term recursion for propagating a wave packet, and, assuming one starts out with some 4>(0) and (r) consistent with Eq. (1), then the iterations of Eq. (4) will generate the correct wave packet. The difficulty, of course, is that the action of the cosine operator in Eq. (4) is of the same difficulty as evaluating the action of the exponential operator in Eq. (1), requiring many evaluations of H on the current wave packet. Gray [8], for example, employed a short iterative Lanczos method [9] to evaluate the cosine operator. However, there is a numerical simplification if the representation of H is real. In this case, if we decompose the wave packet into real and imaginary parts. 

In this chapter we discuss the principles of the Kalman filter with reference to a few examples from analytical chemistry. The discussion is divided into three parts. First, recursive regression is applied to estimate the parameters of a measurement equation without considering a systems equation. In the second part a systems equation is introduced making it necessary to extend the recursive regression to a Kalman filter, and finally the adaptive Kalman filter is discussed. In the concluding section, the features of the Kalman filter are demonstrated on a few applications. 

Unlike the previous two examples, this is a one-term recursion formula. Hence, the series that is constructed from the value of no is a particular solution of Eq. (135). Once again, however, because of the problem of convergence, the series must be terminated after a finite number of terms. The condition for it to break off after the term in pv is given by... 

The third column of Table 1 is calculated by applying the recursion relation to the values shown in the second column, etc. It corresponds to the method of Milne It is apparent that the convergence becomes much more rapid with each successive column. For this particular example foe same Limiting values is obtained as either n or m becomes very large. 

This evolution of a complex set of numbers from something very simple is rather like a recursion rule. For example, the wave function for a harmonic oscillator contains the Hermite polynomial, Hb(t/), which satisfies the recursion relation ... 

These equivalence relations are not the sane. Obviously strong equivalence implies recursive equivalence and recursive equivalence implies finite equivalence. None of the reverse inclusions hold. We are already in a position to show that finite equivalence does not imply recursive equivalence. Consider the scheme of P of Example II-3 which halts for all finite interpretations but diverges for some infinite hit recursive interpretations (for the interpretation of P we saw that caused divergence is obviously recursive). We can diagram P as ... 

We shall conclude this chapter with further examples and a slightly different approach to verification. At the end of Chapter VII (on recursion augmented schemes) we discuss one method of extending verification to programs with procedures and calls. 

EXAMPLE VII-3 CONSTRUCTION OF FLOWCHART P EQUIVALENT TO A LINEAR RECURSION SCHEME S. ... 

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