Symbolic manipulation

Since biological systems can reasonably cope with some of these problems, the intuition behind neural nets is that computing systems based on the architecture of the brain can better emulate human cognitive behavior than systems based on symbol manipulation. Unfortunately, the processing characteristics of the brain are as yet incompletely understood. Consequendy, computational systems based on brain architecture are highly simplified models of thek biological analogues. To make this distinction clear, neural nets are often referred to as artificial neural networks. 

SETS is unique in that it performs symbolic manipulation of Boolean equations usi... 

This integral can be solved analytically. Its solution is a good test for symbolic manipulators such as Mathematica or Maple. We illustrate its solution using classical methods. Differentiating Equation (8.1) gives... 

Furthermore, a variable classification strategy based on an output set assignment algorithm and the symbolic manipulation of process constraints is discussed. It manages any set of unmeasured variables and measurements, such as flowrates, compositions, temperatures, pure energy flows, specific enthalpies, and extents of reaction. Although it behaves successfully for any relationship between variables, it is well suited to nonlinear systems, which are the most common in process industries. 

From the classification it was found that, for this specific problem, there are 10 redundant and 6 nonredundant measured variables, and all the unmeasured process variables are determinable. Symbolic manipulation of the equations allowed us to obtain the three redundant equations used in the reconciliation problem ... 

KARMA is an interactive computer assisted drug design tool that incorporates quantitative structure-activity relationships (QSAR), conformational analysis, and three-dimensional graphics. It represents a novel approach to receptor mapping analysis when the x-ray structure of the receptor site is not known, karma utilizes real time interactive three-dimensional color computer graphics combined with numerical computations and symbolic manipulation techniques from the field of artificial intelligence. 

However, the typical scientist s appreciation of the computer may be too narrow. Computers are much more than fast adders and multipliers they are symbol manipulators of a very general kind. 

Repeat Example 8.1 and obtain an analytical solution for the case of first-order reaction and pressure-driven flow between flat plates. Feel free to use software for the symbolic manipulations, but do substantiate your results. 

Hints First convince yourself that there is an optimal solution by considering the limiting cases of rj near zero, where a large hole can almost double the catalyst activity, and of rj near 1, where any hole hurts because it removes catalyst mass. Then convert Equation (10.33) to the form appropriate to an infinitely long cylinder. Brush up on your Bessel functions or trust your symbolic manipulator if you go for an analytical solution. Figuring out how to best display the results is part of the problem. 

MAC S Symbolic Manipulation System (Macsyma), 1985, Symbolics Inc., Cambridge Massachusetts. 

There are approximately 40 three loop diagrams to be computed for this reason the choice of the appropriate strategy is of significant importance. Any approach has to rely heavily on the use of symbolic manipulation computer programs. Our principal tool in this calculation is the so called integration-by-parts technique [9], which is common in perturbative calculations in high energy physics. 

All the above steps have been automated (in simple terms this means, that getting either the three loop g — 2 or the slope of the Dirac form factor is just a matter of changing a single line in the computer code). The program is written in Form, a symbolic manipulation language created by J.A.M. Vermaseren. Organization of the program is as follows ... 

J. A. M. Vermaseren Symbolic Manipulation with FORM, Version 2, CAN, Amsterdam, 1991... 

A growing number of works in computer science address the descriptive simulation of physicochemical systems through novel integration of (1) numerical simulations, (2) symbolic manipulations, and (3) analytic knowledge from mathematics. Typical examples of this attitude are the following ... 

A number of standard computer programs easily handle problems of this type such as spreadsheet packages, Matlab, Mathcad, Polymath, and so on as well as symbolic manipulators such as Mathematica, Maple, Derive, etc. Most statistic packages and equation solvers will also solve linear equations and have a simple user interface. 

Transform the Hamiltonian to the normal form described above up to the desired degree of accuracy using a symbolic manipulator. The Hamiltonian is now in a new coordinate system that we will call the normal form coordinates. ... 

Symbolic-to-symbolic transformations are used in various symbolic manipulations, including natural language processing and rule-based system implementation. See Refs. 54 and 140. 

Warning This calculation involves massive amounts of algebra, but if you do it correctly, you ll be rewarded by seeing many wonderful cancellations. Teach yourself Mathematica, Maple, or some other symbolic manipulation language, and do the problem on the computer.)... 

The SELECTOR module is responsible for transforming the internal representation of the reaction system into a form which can readily be solved by the SOLVER module. The equations that are represented by the original FLUX matrix, generated in the INPUT module, may be in an unsolvable form. For example, unknown constant parameters may appear in the same equation with as yet unsolved variable parameters. Also, if there are equilibrium assumptions made about certain reactions, the associated rate constants must be eliminated. Finally, if there are unsolvable parameters, they must be identified, and the associated equations must be eliminated. This process involves a rearrangement of the equations that represent the reaction system, using the FLUX matrix. Other rearrangements may be possible by examining the rate expressions, but the symbol manipulative capability that is needed to accomplish tliis is not yet available in CRAMS. 

The NONLIN module is responsible for intializing the concentration vector, C(t), for l i NRCT. Here NRCT is the number of reactants. If there are no equilibrium reactions, then C i) is set to IC i), the initial concentration vector, for 1 < f < NRCT. If equilibrium reactions do exist, then the type (2) equations (with derivatives set to zero) and the Type (1) and Type (3) equations are all solved simultaneously for the equilibrium concentrations of all reactants. Because the equilibrium equations are generally nonlinear, the Newton-Raphson iteration method is used to solve these equations. Also, since there is no symbol manipulation capability in the current version of CRAMS, numerical differentiation is used to calculate the required partial derivatives. That is, the rate expressions cannot at this time be automatically differentiated by analytical methods. A three point differentiation formula is used 27) ... 

Each Type (5) equation is solved by one separate use of the CURFIT module. In the present version of CRAMS, the maximum tolerance for each dependent variable is arbitrarily set to one percent of its value. This is done because some of the variable parameters in the rate expressions, which are the dependent variables, may have been generated by one of the other modules and at this time such cases cannot be identified. However, it is planned to implement a symbol manipulating capability that can identify such cases. When this capability is implemented all the options allowed for specifying maximum tolerances will be allowed. 

Implementing the algorithm sketched above in the computer symbolic manipulation program FORM, as exemplified in Appendix A, and applying the method to the second-harmonic-generation (SHG) process, which is described by the interaction Hamiltonian Hi given by (55), one can easily calculate subsequent terms of the series (92). Restricting the calculations to the fourth-order terms, we get... 

Let us start with the short-time approximation in which we can use the symbolic manipulation computer program described in Appendix A to find the corrections coming from the quantum fluctuations of the fields. The operator formulas (94) and (95) are valid also for the degenerate downconversion because the two processes are governed by the same Hamiltonian, but now initially the second-harmonic mode is populated while the fundamental mode is initially in the vacuum state. Assuming that the pump mode at the frequency 2oo is in a coherent state fi0) (p0 = /Ni,exp(k )h)), we have... 

The sign of the linear terms in (153) and (154) depends on the sign of Imp0, and this sign decides whether the quadrature is squeezed. These examples illustrate the effectiveness of the symbolic manipulation programs in obtaining such expansions. Previously such calculations have been performed by hand. This approach belongs to the standard methods of quantum optics, and many results based on the power series expansion have been discussed in the book [62], so we restrict ourselves to these few examples only. 

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