Uniqueness, mathematical proofs

Mekenyan, O., Bonchev, D. and Balaban, A.T. (1985b). Unique Description of Chemical Structures Based on Hierarchically Ordered Extended Connectivities (HOC Procedures). II. Mathematical Proofs for the HOC Algorithm. J.Comput.Chem., 6,552-561. 

It is certainly true that for any arbitrarily chosen equation, we can calculate what the point described by that equation is, that corresponds to any given data point. Having done that for each of the data points, we can easily calculate the error for each data point, square these errors, and add together all these squares. Clearly, the sum of squares of the errors we obtain by this procedure will depend upon the equation we use, and some equations will provide smaller sums of squares than other equations. It is not necessarily intuitively obvious that there is one and only one equation that will provide the smallest possible sum of squares of these errors under these conditions however, it has been proven mathematically to be so. This proof is very abstruse and difficult. In fact, it is easier to find the equation that provides this least square solution than it is to prove that the solution is unique. A reasonably accessible demonstration, expressed in both algebraic and matrix terms, of how to find the least square solution is available. 

Note, also, that the proof of these theorems, including the 3-D case, can be obtained as a special case of a more general mathematical uniqueness theorem of inverse problems for general partial differential equations. We will outline this more... 

From the mathematical point of view the answer (Toth, Hars 1986b) is a new proof that the Lotka-Volterra model is the only possible two-component bimolecular oscillator — under a different set of assumptions than those made in Subsection 4.5.3. The method of the proof can also be used to prove that the two-dimensional explodator model by Farkas et al. (1985) is almost as unique in its own class. 

A special mathematical problem is that of determining the number of rest points. Rest points can be classified as internal with all variables nonzero or as boundary with at least one variable equal to zero. In a closed chemical system, the internal rest point is unique. Zel dovich (1938) provided the first version of the proof of this uniqueness. Gorban (1980) analyzed the boundary equilibrium points in a closed chemical system. For an open chemical system, isothermal or nonisothermal, multiple internal steady states may exist. 

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