1 84 MoNES Herman and Robert L. Schoenfeld 



which is of the fonn 



mMI = MIH (11) 



where 



|a I = |ax^| 



|A'| = \UDA-^ (12 



Equation (12) represents a mapping of the matrix |A| in terms of the variations 

 in the unknown a^ only. 



By applying the restriction that every fractional turnover rate must be 



positive, 



r,, ^ 



A',,^iA',i (13) 



i = \ 



equations (9) and (12) limit the range of values of the variables in the matrices 

 \P\ and \D\. Since these variables are all independent, they represent a co-ordinate 

 space of dimension equal to their number. Every point in this space specifies a 

 set of values for the variables in the matrices |P| and \D\ and, thus, defines a 

 model through equations (9) and (12). The restrictions on the range of values 

 of the variables as expressed by equation (13) correspond to a region in the 

 co-ordinate space in which all physically meaningful models must lie. 



The choice of the starting point for the transformations indicated above is 

 completely arbitrary and does not affect the final result. Any mathematically 

 consistent model leads to a region in the mapping space corresponding to 

 proper physical models. 



III. UNCERTAINTY MAPPINGS IN GENERALIZED SPACE 



We now wish to examine the problem from a somewhat different point of 

 view. The system is represented by n^ X^^, generally independent of each other. 

 We can, therefore, consider the X^^ to represent an n^ dimensional space, and any 

 point in that space as a specific model of the system. It was also indicated 

 earlier that the data could be represented by a set of invariants composed of 

 n oij and {n^ — n) A^j or a total of n^ invariants. Hence, the transformation 

 from the data space to the X^^ space is dimensionally consistent and unique. 



This means that a complete set of A^j and a^ corresponds to a point in the 

 \X\ space, and vice versa. By definition, however, the values of the A,,- must all 

 be positive. Consequently all the models must lie in a restricted region of the 

 \X\ hyperspace. This restriction carries over to the data space, limiting the region 

 in which the Aj^j and a^ may lie. 



Any specified A^j or a_, implies a one dimensional constraint in the data 

 space. This carries over as a one dimensional constraint in the \X\ space, and 

 restricts all models to a surface in the hyperspace. If, however, the value of 

 A^j or oij is known only within a certain range, the surface has correspondingly a 

 certain thickness. 



When several A^^ or a^ are known, the dimensions of the space in which all 

 models must he is reduced by a corresponding number. Statistical uncertainties 



