Symbols and Numerics

The user can generate FORTRAN expressions that allow numeric computers to run faster and more efficiently. This saves CPU cycles and makes computing more economical. The user can generate FORTRAN expressions from MACSYMA expressions. The FORTRAN capability is an extremely important feature combining symbolic and numeric capabilities. The trend is clear, and in a few years we will have powerful, inexpensive desktop or notebook computers that merge the symbolic, numeric and graphic capabilities in a scientific workstation. 

The user can explore extremely complex problems that cannot be solved in any other manner. This capability is often thought of as the major use of Computer Algebra systems. However, one should not lose sight of the fact that MACSYMA is often used as an advanced calculator to perform everyday symbolic and numeric problems. It also complements conventional tools such as reference tables or numeric processors. 

An important aspect of our AI application is the attention paid to including well-established Fortran programs and database search methods into the decision structure of an expert system network. Only certain AI software tools (such as TIMM) effectively handle this critical aspect for the analytical instrumentation field at this time (57-60)> The ability to combine symbolic and numeric processing appears to be a major factor in development of multilevel expert systems for practical instrumentation use. Therefore, the expert systems in the EXMAT linked network access factor values and the decisions from EXMATH, an expert system with chemometric/Fortran routines which are appropriate to the nature of the instrumental data and the information needed by the analyst. Pattern recognition and correlation methods are basic capabilities in this field. 

SRL DEXPERT, Systems Research Labs., Inc., Dayton, OH 45440-3639 Integrates LISP algorithms into Fortran or Ada systems. Proc. Workshop on Coupling Symbolic and Numerical Computing in Expert Systems, sponsored by AAAI, Aug 1985, Boeing Computer Services, Bellevue, WA. 

Through the use of chemical symbols and numerical subscripts, the formula of a compound can be written. The simplest formula that may be written is the empirical formula. In this formula, the subscripts are in the form of the simplest whole number ratio of the atoms in a molecule or of the ions in a formula unit. The molecular formula, however, represents the actual number of atoms in a molecule. For example, although CH20 represents the empirical formula of the sugar, glucose, C6H1206, represents the molecular formula. For water, H20, and carbon dioxide, C02, the empirical and the molecular formulas are the same. Ionic compounds are generally written as empirical formulas only for example, common table salt is NaCl. 

Eisenberg, M., Descriptive Simulation Combining Symbolic and Numerical Methods in the Analysis of Chemical Reaction Mechanisms. Technical Report, Laboratory for Computer Science, MIT, Cambridge, Mass., 1989. 

Infroduce fhe engineer briefly to symbolic and numerical soffware for solving mafhe-matical problems of interesf. If should be noted that we have not cited hterature in the text, however, pertinent references are included in the bibhography. 

Mathematica can carry out both symbolic and numerical solutions of equations, including single algebraic equations, simultaneous algebraic equations, and differential equations, which we discuss later. Mathematica contains the rules needed for the symbolic solution of polynomial equations up to the fourth degree, and can... 

Mathematica can solve differential equations both symbolically and numerically. 

In a chemical formula, element symbols and numerical subscripts show the type and number of each atom present in the smallest unit of the substance. There are several types of chemical formulas for a compound ... 

There exist several sources where special formulae for the n/-symbols are presented. One reference compilation is represented by the Landolt-Bornstein series [1]. Of great advantage is the computer system Mathematica which is capable of deriving 3/ - and 6y -symbols in both symbolic and numerical forms [2], Another source using the FORTRAN computer program is available [3]. 

The molecular formula, which shows the element symbols and numerical subscripts, tells you the type and number of each atom in a molecule. As shown in Figure 8.13, there are several different models that can be used to represent a molecule. Note that in the ball-and-stick and space-filling molecular models, atoms of each specific element are represented by spheres of a representative color, as shown in Table R-1 on page 968. These colors are used for identifying the atoms if the chemical symbol of the element is not present. 

Design requirements specify features of our object or system that are required in the case of our specific design. They may be stated in both symbolic and numerical terms. For example, a design requirement may state that our truss should have sufficient stiffness. A symbolic attribute will be used, and this requirement presented as... 

Design constraints identify features of our object or system that must be avoided in our design. They may also be stated using both symbolic and numerical attributes. For example, a constraint may state that concrete may not be used as a material for our bridge. In this case, obviously only a symbolic attribute may be used and the constraint will be presented as... 

Students misunderstandings of chemical equations are also due to their failure to associate the symbols and numerical answers with real objects and events. Thus, students difficulties in understanding chemical reaction equations have been attributed to misunderstanding the symbols themselves, to misunderstanding concepts related to the atomic and electronic structure of matter, and to not understanding the relationship between the equations and the macroscopic level. 

This chapter addresses structural control properties of linear hybrid system models as well as mathematical properties. If the imderlying mathematical model is of the form of a DAE system then its index is an information relevant with regard to its symbolic and numerical processing. 

The metric system, or International System (SI, from Systlme International), is a decimal system of units for measurements of mass, length, time, and other physical quantities. Built around a set of standard units, the metric system uses factors of 10 to express larger or smaller numbers of these units. To express quantities that are larger or smaller than the standard units, prefixes are added to the names of the units. These prefixes represent multiples of 10, making the metric system a decimal system of measurements. Table 2.1 shows the names, symbols, and numerical values of the common prefixes. Some examples of the more commonly used prefixes are... 

Figure 3 Ignition temperature of H2/O2/N2 mixtures over a platinum foil as function of fuel concentration. Comparison of experimental (symbols) and numerical results (line) for a and b see text. 

Detailed description of the entire analytical process including sample preparation, standard preparation, and step-by-step procedure (including any presaturation, pre-equilibration, weights, volumes, extraction times, solvent range adjustments, centrifugation times), method for evaluation and calculation (with a representative calculation with tabulation of all symbols and numerical factors). 

The composition of a compound is given by its chemical formula. A chemical formula indicates the relative numbers of atoms of each kind in a chemical compound by using atomic symbols and numerical subscripts. The chemical formula of a molecular compound is referred to as a molecular formula. 

At a certain stage of Mathcad development it becomes possible to enter some interface operators in tandem, feeding one input/display operator into another. Thus, point 1 of Fig. 6.17 shows the use of the operators and = (symbol and numerical result) in solving the equation. The advantage of a symbolic result is absolute accuracy and numeric shows the distinct locaticm on the number axis (or on a plane in the case of complex number). The tandem use of operators allows us to combine those advantages. 

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