One version of the theorem allows one to conclude that two chain complex functors are naturally homotopic and another that two such functors are object-wise homologous. Neither is entirely satisfactory. The purpose of this paper is to provide a uniform account of these two, fixing what is unsatisfactory and also finding intermediate forms of the theorem. If the lag term smaller than the stepsize when solving these problems, the standard explicit equations associated with the underlying CRK formula become implicitly defined . To solve these implicit equations, we develop an iteration scheme based on the use of special 2-step interpolant to generate the initial guess.
Numerical tests show favorable aspects compared with other approaches.
We approach these problems via the basic concept of surprise. As a special case the theory leads to a combination of the posterior probability and Bayes factor approaches to Bayesian hypothesis testing. This is achieved within the context of proper Bayesian inference which is not the case for other attempts at resolving these problems. Several historical examples are analyzed in order to show that mistakes are genuine in most pioneering works and they become a source of creativity. Then it is shown that mistakes play a fundamental role in both creative teaching and creative learning.