Recursive properties

In our case, we define a formula as consisting of raw materials and/or other formulas. We develop a modelling procedure we call to determine formula characteristics from raw material properties, and give it the recursive property of being able to call itself when it encounters a formula used as a raw material. The procedures terminate when all formulas are resolved into basic raw materials. This terminating condition cannot be met if any formula contains a reference to itself, either directly or indirectly, through another formula. 

The recursive property of the Kalman filter allows the detection of such model deviations, and offers the possibility of disregarding the measurements in the region where the model is invalid. This filter is the so-called adaptive Kalman filter. 

In this chapter we start the discussion of an alternative model for programs, designed to reflect recursive properties of programming languages. We shall see that this model does indeed represent an augmentation of the flowchart model we have been studying up to now. One topic of concern will be when recursion equations can be translated into flowchart form - when recursion schemes are flowchartable. 

For a non-recursive property Pj, a normalized successful derivation is an instance of the following derivation schema ... 

If clause Q is not recursive, then everything related to r(s) may simply be ignored. A slight variation of this schema can be established for recursive properties. 

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. 

Recursive Techniques in Property Information Retrieval and Calculation for Computer-Aided... 

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. 

Figure 1 shows a flow chart for part of a recursive modelling procedure, illustrated in this paper, which accepts as input a formula consisting of constituent raw material codes or formula names, and quantities. The procedure retrieves property data for each raw material in order to perform the required calculations. When the procedure encounters a constituent that is a formulated product, it calls itself using that product as input. The output of the procedure consists of the calculated properties of the formula, including those properties of the formula that would be retrieved from data files for non-formulated or purchased raw materials. By returning this latter set of properties, the procedure can treat formulas as raw materials. 

Stockfisch TP. Partially unified multiple property recursive partitioning (PUMP-RP) a new method for predicting and understanding drug selectivity. J Chem Inf Comput Sci 2003 43 1608-13. 

The way we have stated the domain theory for the state-space representation has enabled us to avoid making explicit reference to the alphabet symbol properties. However, if in other formulations we need to refer to these properties, we would again use a recursive parsing of the list of symbols to enable generalization over the size of the alphabet. 

More specifically, the basic notions of a Turing Machine, of computable functions and of undecidable properties are needed for Chapter VI (Decision Problems) the definitions of recursive, primitive recursive and partial recursive functions are helpful for Section F of Chapter IV and two of the proofs in Chapter VI. The basic facts regarding regular sets, context-free languages and pushdown store automata are helpful in Chapter VIII (Monadic Recursion Schemes) and in the proof of Theorem 3.14. For Chapter V (Correctness and Program Verification) it is useful to know the basic notation and ideas of the first order predicate calculus a highly abbreviated version of this material appears as Appendix A. 

Suppose we have a class C of objects for which a property P may or may not be true and recursive set A of names of C such that C = ca a e A. Each member of C has at least one name some members may have several or even infinitely many names. However, each name in A refers to exactly one member of C. Then we say that P is decidable for C - really for the set of names A fear C - if the set... 

A property P is partially decidable if the set a e A P holds for cal is recursively enumerable. That is, P is partially decidable if there is a Turing machine which halts and gives answer YES on input a if c has property P and... 

Statements (1), (2) and (3) declare that partial recursive function val(P,I,n) is nowhere defined it is known that it is not partially decidable whether a partial recursive function is everywhere undefined. Statement (4) says that partial recursive function val(P,I,n) is not total recursive while statements (5) and (6) say that it is total recursive neither property is partially decidable for partial recursive functions as defined by, e.g., Turing machines. 

What does all this mean in a practical way Certainly we have no intention of writing programs for all partially computable functions indeed time and space considerations do not allow execution of all partially computable functions. The functions actually computed form a very small subset of the primitive recursive functions. We do not know, however, whether they fall into a class for which partial correctness is partially decidable one suspects not. In any case, since we obtain our undecidability results for programs with very simple structure, there can be nothing in the structure of "real" programs which will allow us by and of itself to conclude that the properties of interest are at least partially decidable. 

Bessel functions have many interesting properties that will be presented here without proof, e.g. the recursion formula... 

A straightforward generalization uses a recursive definition of port and connector, allowing us to make connections either at the level of individual events and properties or at higher-level bundles of them. 

Additional properties of this function are the recursion formula... 

As indicated by the Kronecker deltas in the above equation, the resulting Hamiltonian matrix is extremely sparse and its action onto a vector can be readily computed one term at a time.12,13 This property becomes very important for recursive diagonalization methods, which rely on matrix-vector multiplication ... 

---