The classical problem of parameter estimation

The most popular and most important inverse problem is the estimation of reaction rate constants, see, for example, Deuflhard et al. (1981) Hosten (1979), or Vajda et al. 1987). Using the terminology introduced above is the function that gives the solution of the kinetic differential equation as a function of the reaction, while F o provides the values of the solution at discrete time points together with a certain error. In this case a subset of V with the same mechanism is delineated and the aim is to select a reaction from this set in such a way that the solution of the kinetic differential equation be as close to the measurements as possible by a prescribed, usually quadratic, norm. As the solution is a nonlinear function of the parameters, therefore a final solution to the general problem seems to be unobtainable both because a global optimum usually cannot be determined and because the estimates cannot be well-characterised from the statistical point of view. In addition to these problems, reaction rate consants only have a physicochemical meaning if they are universal, i.e. the reaction rate constant of a concrete elementary reaction must be the same whenever it is estimated from any complex chemical reaction. 

Verify that a polynomial differential equation without negative crosseffect can be induced by a reaction (called canonic reaction) that is constructed in the following way  

Show that the generalised Volterra equation (4.26) in the case M = 3 is conservative if and only if the matrix v is of the form 

The property that the vector on the right-hand side of a differential equation points into the interior of the first orthant is obviously a necessary condition for negative cross-effects to be absent. Give a polynomial example showing that it is not sufficient. 

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