Stahl67 bmp

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WALTER R. STAHL

TABLE I

Mathematical Similaeitt Criteria for Variotjs Types of Models Type of Model    Naturę of Modeling Criteria

I. Analog models

Qualitative physical analogs or simulators

Quantitative physical models

Electrical analogs of physical Systems

Natural physical models

II. Numerical-computational models

Algebraic representations

Ordinary differential equa-tions

Partial differential eąuations

Sets of simultaneous linear differential eąuations

Probabilistic and stochastic eąuations

Biometrical models

General numerical Computer simulation

III. Cybemetic models Analog control

Numerical control

Physical similarity invariants Geometry, materials, relations of parts; some but not all applicable dimensional constants and dimensionless numbers Invariance of all materials and pertinent similarity criteria, usually including geometrie ones; or inyariance of a specific numerical function relating such criteria Inyariance of pertinent dimensional constants and dimensionless numbers, repre-sented by substituted physical constants in the electroanalog model, which are iso-morphic with those of the physical system Inyariance of selected sets of dimensional constants and dimensionless numbers; also of relational and geometrie criteria

Inyariant properties of numerical eąuations

Simple numerical constants, roots of eąuations, meaning of symbols Inyariants under simple transformations of yariables, defining “system constants” Inyariants under complex transformations of seyeral yariables; dimensionless numbers a special case

Inyariants for indiyidual eąuations and of matrices representing similarity transformations for the system of yariables Characteristic inyariants such as Bayes theorem probabilities, correlation function value

Characteristic inyariants such as value of t test or regression coefBcient Pertinent inyariants of any types defined above

Measures of control performance Operational criteria such as gain, integrated error, feedback ratio, or time constants Inyariant properties of eąuations used for control, e.g., ratę, feedback, or time con-stant ratios