LST

HyperChein has two synch ron ons transit meth ods im piemen ted. The linear synchronous transit method (LST) searches for a maximum along a linear path between reactants and products. It may happen that this method will end up with a structure having two or more negative eigenvalues. The quadratic synchronous transit method (QSTlisan improvement of LST approach and searches for a maximum along a parabola connecting reactants and products, and for a minimum in all directions perpendicular to the parabola. 

The work was supported by INTAS grant Open Call 03-51-6278, NATO Linkage grant LST.NUKR.CLG 980621 and the grants of the Program Sensor Systems and Technologies funded by NAS of Ukraine. 

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In a recent version, the LST or QST algorithm is used to find an estimate of the maximum, and a Newton method is then used to complete the optimization (Peng and Schlegel, 1993). 

The Linear Synchronous Transit (LST) method forms the geometry difference vector between the reactant and product, and locates the highest energy structure along this line. The assumption is that all variables change at the same rate along tire reaction path. 

While there is, at present, no known CA analogue of a Froebenius-Perron construction, a systematic n -order approximation to the invariant probability distributions for CA systems is readily obtainable from the local structure theory (LST), developed by Gutowitz, et.al. [guto87a] LST is discussed in some detail in section 5.3. 

The LST is a finitely parameterized model of the action of a given CA rule, >, on probability measures on the space of configurations on an arbitrary lattice. In a very simple manner - which may be thought of as a generalization of the simple mean field theory (MFT) introduced in section 3.1.3. - the LST provides a sequence of approximations of the statistical features of evolving CA patterns. 

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. 

Behavioral Classification We will shortly see that many rules may in fact give rise to the same set of recursive equations of a given order. This suggests that the LST provides an alternative behavioral classification scheme to the four (thus far... 

Complexity Engineering Specific applications of CA to physical problems typically involve enormous efforts spent on obtaining just the right set of rules appropriate for a given problem. By succinctly suinniarizing the statistical behavior of a well-defined class of rules for a variety of lattice structures, the LST equations may be used to effectively guide searches for particular rules displaying the set of desired statistical behaviors. 

We will have more to say about both of these two possibilities later. In the remaining paragraphs of this section, we introduce (and remind ourselves of) some basic terminology, outline the LST for one dimensional systems and provide a few simple examples of its use. The section concludes with a brief discussion of some subtleties needed to define a LST for systems with dimension d > 1. 

The LST alleviates this problem by systematically approximating the probabilities of Bm, with M > N, from the set of probabilities of smaller blocks, Bi, B2,. .., Bn- In this way, order correlation information is used to predict the statistical properties of evolving patterns for arbitrarily large times. The outline of the approach begins with a formal definition of block probability functions. 

Having thus outlined tiie fcnmal theoretical structure of one-dimensional LST, we now turn to a more pragmatic discussion of its use and applications. 

The A -order LST is defined by the same infinite system of equations appearing in equation 5.66, but with = rn -order Bayesian extended probability... 

The general LST aJgorithni for approximating the temporal evolution of A-block measures using An ) is therefore as follows ... 

Repeat steps 2 and 3 to generate the LST approximation of the temporal evolution. 

As a simple example, consider the A -order LST equations for range k = 2, r = 1 CA. Since the inverse image of an A-block is a block of size N + 2), we need to apply the operator 7TAf >Af+i to twice in succession to define the N + 2)-block probabilities ... 

Probabilities for blocks Bj with length j < N may be calculated by appealing to the Kolmogorov consistency conditions (equation 5.68). We now examine some low order LST approximations in more detail. 

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