Recursion defined

We recursively define a sequence of partially normalized Hamiltonians... 

DEFINITION The operation of primitive recursion defines from functions h and g the function f by the formula ... 

In this section, we introduce situation theoretical notions and objects that are fundamental for fine-grained modelling of information and information components. Situation Theory takes some set-theoretic objects as its basic objects. These basic objects then are used in the recursive construction of more complex situation theoretic objects. Informally, the basic informational pieces, called infons, are composite objects carrying information about relations and objects filling the arguments of the relations, at certain time and space locations. Infons can be basic or complex, by recursively defined system of objects. Infons are the ground, informational content of basic and complex informational objects, the informational content of situated propositions (introduced in Sect. 3), and other objects that carry information about situations. Infons are facts when supported by actual situations, e.g., in real or virtual worlds, theoretical models, or computerized models. 

The many special cases of such recursively defined binary search trees (data structure) each correspond to a search algorithm. Movement through a binary search free is like going to a place in the file. That place has a key, which is to be examined and compared to the one sought. 

On the other hand, the linear combination of atomic orbitals - molecular orbital (LCAO-MO) theory, is actually the same as Hartree-Fock theory. The basic idea of this theory is that a molecular orbital is made of a linear combination of atom-centered basis functions describing the atomic orbitals. The Hartree-Fock procedure simply determines the linear expansion coefficients of the linear combination. The variables in the Hartree-Fock equations are recursively defined, that is, they depend on themselves, so the equations are solved by an iterative procedure. In typical cases, the Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky cases, convergence may be improved by changing the form of the initial guess. Since the equations are solved self-consistently, Hartree-Fock is an example of a self-consistent field (SCF) method. 

Beliefs like (2) and (3) in the above, which can be recursively defined, are called mutual beliefs. Such explanation of cooperation mechanism using one s own cognitive state and a corresponding structure of recursive beliefs can clarify the constitutive meaning of sharing intentions by cooperating team members. 

The n-particle density matrices are not size-extensive, but are instead product separable, just like the wave operator. However, one can factorize any n-body density matrix in spinorbital basis as anti-symmetrized products of 1-particle density matrices and "cumulants which can be recursively defined as the rank n of increases. For now, we denote by u,v., etc., the active spinorbitals. For a CASSCF function Wo, all the core spinorbitals are fully filled in each (f> of Wo, and hence, all the y s corresponding to a 4> will factor out as anti-symmetrized product of y and y with (jia -t- Me) = n, where na and n<, are the number of valence and core occupancies ... 

Note that this fragment does not contain place-holders for modal operators, or something that corresponds to an operator like the fact that. .. (which would make things unnecessary complex) and a considerable number of other operators which might be relevant for scientific discourse. Thus, we should regard the account presented here as a partial account of property structures. We can now recursively define the set of property-structure fem-schemata. I shall talk about terms from now on simpliciter. It should be noted, however, that these terms are schemata for terms. 

Other synthesis systems (such as Cypress [Smith 85]) are recursively defined, and hence infer their own specifications for appearing sub-problems. 

Consider an array of n computation spins, CnCn-i... C2C1, where each computation spin, Q, is attached to a reset spin, (see Fig. 2 for the case of n = 3). To cool c/c, the spin at index fc, to a purification level j e 1... L the procedure Mj k) was recursively defined as follows [2] Mo k) is defined as a single PT step from reset spin rk to computation spin Ck to yield a polarization bias of 0 (the 0 purification level). The procedure Mi (A ) applies Mq to the three... 

The use of the Lanczos recursive method to define the local Hamiltonian and to calculate the density matrix is a common feature shared by several groups. Baroni and Giannozzi suggested a method based on a finite-difference representation of the Hamiltonian, and a recursive Green s function approach to calculate the electron density in real space." For the density at each point, the truncated finite length of the recursion defines the local Hamiltonian for that point. Aoki et al. constructed a bond-order potential method which determines each density matrix element with a recursion in the Green s function. ... 

We recognize that the more general recursion defined in eq. (2.85), facilitates the development of hybrid approaches. 

OPERATION-SET-COMPONENT (a node in the precedence graph) is a member of one of these three subtypes. Members of the SINGLE-ASSEMBLE-OPERATION are atomic objects (nodes that are not recursively defined). Members of RECURSIVE-ASSEMBLE-OPERATION are composite objects (composite nodes in the graph), further defined in terms of their constituents. The type NIL, has exactly one member, nil. Every atomic and composite object ASSEMBLE-OPERATION object points to its next operation(s), through the Next-Operations mapping (attribute). 

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. 

Fibonacci numbers, where the number sought is the nth number in a series defined in terms of relationships to the n-1, n-2, etc. members of the series. All recursive procedures must have a terminating condition, so that they do not call themselves endlessly. 

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 recursion rule itself is straightforward, yet it generates a complex sequence of functions, even though each function in the sequence is defined only by the recursion rule and the value of the preceding function in the sequence ... 

A particular subset of the total recursive functions, the primitive recursive functions, can be computed by a special type of WHILE program, a STEP program or LOOP program. We define a STEP construction ... 

Notice that f can be defined from h and g by primitive recursion as the output of the following program ... 

There are several equivalent ways of defining a recursively enumerable set or predicate. 

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. 

DEFINITION A recursion scheme is a finite set of recursion equations and a designated initial defined function letter F such that ... 

An interpretation of a recursion scheme is defined similarly to an interpretation of a flowchart scheme. The interpretation assigns meanings to constants, predicate letters and basis function letters found in the scheme but does not, of course, assign meanings to defined function letters. A free interpretation is likewise defined as usual, to have as domain the set of all terminal terms over the set of variables, constants and basis function letters found in the scheme. 

The definition of computation in a recursion scheme is a little more complicated than for a flowchart scheme. Computations are defined from the inside out when the equation has nested defined function letters. It has been shown by B. Rosen that evaluating recursion equations from the inside out produces a system with the... 

We can prove the same relationship between computations under arbitrary interpretations and computations under free interpretations that we did for flowchart schemes, defining U(S) for a recursion scheme S in the same way as for a program scheme. We state it without proof. 

Not surprisingly, F defines the factorial under this interpretation. Later in this section we shall see how to verify programs with recursion. This full expansion is somewhat clumsy, and we would probably take the liberty of expressing the computation in this abbreviated form ... 

We can define strong equivalence between recursion or program schemes as before. 

If we try to define a "free" recursion scheme in the same way we defined a free program scheme - every path is an execution sequence - we find that although the intuitive meaning is clear, it is very hard to formalize this concept. Exactly how should one define a "path" in a recursion scheme Or an "execution sequence" It is possible to do so by a moderately complex tree recursion. argument. Instead we will give a "syntactic" definition akin to the one we established as a theorem for program schemes. 

First let us extend the definition of recursion equation as we did the definition of WHILE scheme. Let a Boolean expression be any expression involving predicate terms P(t, ...,tm) where each is a terminal term (not necessarily a variable), and the connectives AND, OR, and NOT. We define a recursion expression inductively, by saying that first any term is a recursion expression, and then that any statement of the form IF Q THEN ELSE E2 is a recursion expression if Q is a Boolean predicate and E and E2 are recursion expressions. 

---