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# Non-linear transforms

The monotonic positive mappings, say F„ across repetitively interactive, integrative, and evolutionary (IIE-processes) intra-systems and inter-systems of moral inclusiveness denoted by ‘i’ preserve the self-same kind of IIE-leaming processes along with their circular causation mappings for- evaluating wellbeing. Yet they do not necessarily converge to the identity mapping. That is recursion of knowledge does not form linear mappings. Non-linearity> of the functional relations (mappings) between the variables remain permanent by the induction of knowledge-flows ({0})-values.

Thereby, we can write the following results with all forms of compound mappings:

F, here denotes arbitrary mappings of the f-functions at the ith round of HE within a given learning (recursive) process, a' is scalar to indicate non-linear mappings. It is therefore a function of ({q})-values in the evaluation of wellbeing function.

p' are compounded scalar functional of ({0})-values on p carrying the implication like a' scalar.

y' are compounded scalar functional of ({0})-values on y.

It is likewise the same kinds of recursive mappings for chains of mappings along the IIE-leaming processes. Figures 5.1 and 5.2 can then be extended by indefinitely many integrated mappings explaining as many interrelations as there are emergent variables in the compound maps.

For inter-system learning following evaluation of W(.), the results will be as follows: F,(W) is a monotonic positive transformation of W(.) by the positive mapping F, in the i,h IIE-simulation process. Thus, F^fj) = F^x^O)) represent simulated forms of the estimated functional of X] = fi(x2(0),0). F,(fi) as arbitrary monotonic positive transformation is recursively set by the simulated 0-value in the ith round of IIE-leaming process. F,(.) thus denotes simulacra of non-linear transforms (Fitzpatrick, 2003; Wallerstein, 1998)24 with i = 1,2,.. ,,n. .

F, are non-linear by virtue of the density of {0}-values in its i,h simulated set of variables. That is,

defining evolutionary equilibriums (Grandmont, 1989; Burstein, 1991).25

It can be readily shown by a diagram with 0 on the vertical axis and socio- scientific variable on the horizontal axes that the area under the non-linear surface with {plinr{0}} will include all the sequences of {plirn{0}} under the linear surface; but not vice versa. Consequently, the positive non-linear transformation above the linear domain of learning will yield higher vector- values of {x,(0,x2(0),0}. Consequently, F^Wfx^O, x2(0))))non.lmear karnmg > Fi(W(xi(O,x2(0))))llMar leamng, for 0non.lmear > 0lmear. Linear transformations form deconstruction of non-linear functional.

By virtue of the positive monotonic transformation of the IIE-leaming processes the sets {x(0)} = {(x!(0,x2(0),0)}, {W(0)} = {W(X](0), x2(0))}, {F(x(0),0)} = {F,(W(x1(0),x2(0)))} and their more elementary functionals form topologies. This is proved simply by noting that the continuously differentiable functionals establish neighbourhood evolutionary learning points intra- and inter-systems. This implies that n{x(0)} ф ф. Thereby, by the property of continuous positive monotonic transformation, n{W(0)} ф ф; n{F(x(0),0)} ф ф. Also, in the open learning space of evolutionary {0}-values, any of the mathematical intersections and unions of the given class of sets form similar classes in the learning domain.

Consequently, there are positive monotonic mappings that preserve the namre of the transformations. In the open space of set-theoretic unions and intersections, the universal set of all possible positive monotonic transformations fomi the universal set of all non-linear possibilities. Such a domain of solutions is referred to as the space of socio-scientific ‘everything.’

Finally, the higher density of the non-linear functionals implies that linear spaces appear as exceptions everywhere and in ‘everything.’ The study of pandemic multidimensional complexity while rearing it towards treatment and cure is just one example. The underlying models of such multidimensional study ensembles can therefore be contained in a set of finite measure, leaving the rest of the complex set of non-linear functionals mathematically ‘measurable almost everywhere’ - AE (Friedman, 1982).26 Thereby, the null-set ф separates the space of non-linear mappings from the space of linear mappings. Then by the positive monotonic transformation, the space of non-linear mappings remains above the space of linear mappings - at a higher level of the dimensions of knowledge, time, and space. In this way, the definition of topology as a mapping that preserves these properties of mappings is established (Maddox, 1970).27

Yet the topology we have described here is on the domain of non-linear mappings {F,} appealing as compounding, shown by ‘o,’ of non-linear functionals (shown earlier). Besides, since {F,} explain simulations out of simulacra of changes in the coefficients of the functionals such as {f’s} and {FJ, therefore, ever}' such functional transformation has probabilistic coefficients that are subject to variations continuously.28 This makes the simulated mappings {F,(f’s)} non-linear ‘foliation’of‘simulation’relations (shown by the simulated version of the estimated coefficients - predictor coefficient values). In mathematical tensor language the evolutionary learning spaces define unions of evolutionary learning spaces. Such set-theoretic unions signify linear aggregation of non-linear relations caused by simulated relations in temis of knowledge-induced dynamic coefficients (continuous variations by simulated predictor values) and their non-linear specifications of the elementary functions of the compound maps.

Now Figure 5.1 assiunes the dynamics shown in Figure 5.2.

Figures 5.1 and 5.2 also explain how the domains of knowledge-induced wellbeing (W(0)) and ‘de-knowledge’-induced mutation functions (W’(0’)) between the treatments towards normalcy and pandemic variables, respectively, remain differentiated in the IIE-leaming perspective. The reversed functions like the f’s and F’s shown in the Figures 5.1 and 5.2 can be expressed in terms the two W-functions as,

W(0)“ • W’(0’)p = 1. (ot,p) are oppositely assigned positive coefficients of the normalcy and mutation states of pandemic, respectively.

The differentiated domains of pandemic normalcy and mutation through the continuous process of learning implies, (d/d0)[a.logW(0) + p.logV’(0’)] ~ 0. This denotes

Figure 5.1 The intra-systemic IIE-leaming process dynamics

Figure 5.2 Evolutionary learning dynamics in non-linear spaces

Expression (5.11) is proved by the fact that a.€W(9)|e > 0; b.€w-(6’)ie) > 0; (d0’/d0) < 0 with monotonicity between the wellbeing (dis-wellbeing by de-knowledge) and the respective knowledge (de-knowledge) parameters. The IIE-leaming and optimal pandemic states between normalcy and mutation depend upon the values of the elasticity coefficients as shown. This also implies the effectiveness of the change in pandemic normalcy caused by the percentage change in the wellbeing function oppositely to the condition of mutation caused by the percentage change in the dis-wellbeing function of pandemic mutation. Expression (5.11) applies to multidisciplinary ensemble but with the normalcy cases of endogenously interrelated treatment and curative variables of wellbeing against the contrary case of exogenous, thereby mutative relationship between the variables of the dis-wellbeing function. Example of the first kind is complementarities between scientific mechanism of pandemic cine and arresting deepening poverty by moral consciousness in policy instrumentation. Example of the second case is inducing racial frontline nursing care to address number of afflictions of coronavirus. This practice reflects an immoral attitude of the use of human service for a wrong purpose:

Doctors, nurses and healthcare workers are literally risking then lives on the frontline with limited resources and yet some people feel the need to impose more hatred and fear.29

In the case of the multidisciplinary idea with differentiated socio-scien- tific fields addressing the pandemic episode there is no episteme of unity of knowledge to address the science-economy-society moral inclusiveness. The utilitarian conception of welfare instead of wellbeing is now used for explaining optimal use of the representation of the welfare function. The result of such optimality yields

wherein, dx, dx, are negative or positive according to whether the relationship between the (x^x^-variables, (i, j) = 1,2,.., n are substitutes or complements, respectively. There is no 0-effect. Marginal substitution according to neoclassical utility theory must necessarily exist, conveying methodological individualism, and thereby an extended meaning of mutation at large.

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