Recursion higher order

The higher order n-body functions V(3>, V(4),... can be obtained recursively considering sets of three, four,... molecules and applying equation (48) successively. Thus, one can write... 

In this Section we discuss the ROPM integral equations for the recursive scheme introduced in Sec. 2.2, restricting the analysis to the order s. An extension to higher order can proceed along the same lines. Expanding the energy and the potential into a power series with respect to A-,... 

The same definition applies for Lgh, where the function g plays the same role as / in (5.8). Of course, higher-order Lie derivatives can be defined in a recursive way, e.g.,... 

Similarly, higher-order Lie derivatives can be defined recursively as... 

A higher-order derivative of a function is defined recursively in terms of the lower-order derivative. 

Accordingly difference and recursion equation of higher order can be transferred into the Af-system also. Multiplying eq. (5.17) by one finds... 

The higher-order En can be calculated from a recursion relation involving the moments of the coordinates. This relation is similar in form to that found for the groxmd state [13], although some significant modifications are necessary to treat excited states. In the case of the ground state we only needed moments of the form... 

The same problem is compounded in architectural descriptions, where DDD uses recursive networks of streams to model synchronous hardware. Animation of streams in Scheme is done using function closures and continuations to implement call-by-need behavior. Neither of these higher-order objects can be expressed directly in Nqthm, nor can nonterminating function systems. 

As was the case with Nqthm, any embedding of DDD expressions is hindered by the fact that neither simultaneous recursion nor nontermination is allowed in function definitions. Higher order types open additional avenues for attacking this problem, but any such attack imposes a layer of interpretation to be maintained either within the proof management system, or in the user s mind. 

This new notation is interesting as it emphasizes that the explicit form of the individual operators is of little importance. Moreover, we will understand that higher-order terms can be constructed in a recursive manner so that only the explicit expressions in Eqs. (11.37)-(11.39) need to be known (see also chapter 12). 

A completely different approach is advocated by [Hagiya 90]. He re-formulates Summers recurrence relation detection mechanism in a logic framework, using higher-order unification in a type theory with a recursion operator. The method is extended to synthesizing deductive proofs-by-induction from concrete sample proofs. 

We define divided differences of higher orders recursively... 

Together with (9.5.32), this expression allows us to generate recursively the higher-order multipole moments from those of lower orders, noting that all Hermite integrals with f > e are zero ... 

Thus Eq. (8.454) is used as a recursion formula to generate the W s of higher index m. Eventually an interface between two regions is reached, and it is necessary to invoke the conditions of continuity of flux and current in order to cross the boundary and proceed with the recursion formula (8.454) as applied to the second region. Let the interface be at radial position R = Ar then, using the definitions above, we have... 

Method B. For higher model orders, a recursive approach proposed in Atkinson (1989) can be used. 

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