# examples of growth of perturbations in chemical organizations

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Type of Math Object:
Example
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Research

### Typo?

Hi Ray, is there some typo in your reaction formula

A+B –¿ B+B?

For a former chemist, it is incomprehensible.

What are the meanings of the variables x and y?

Thanks,

Jussi

### failure functions and group theory-another example

Let the mother function be the quadratic x*2+1. When x = 4 we get the failure function x = 4 + 17*k ( k belongs to W ). We also have the failure function x = 38 + 289*k . This generates values of x such that f(x)/289 constitute a remainder group isomorphic with Z_17.

### failure functions and indirect primality tests

Let our definition of a failure be a composite number.Let the mother function be x^2+1 ( x belongs to Z). To test whether f(x) is prime or not all we have to do is to test whether x is covered by any of failure functions, x = 1 + 2k, x= 2 + 5k, x= 4 + 17k…….. If the value of x under scutiny is covered by one or more of the failure functions then f(x) is a failure (composite ). Otherwise it is prime. Note we are not directly testing the primality of f(x); we are only testing whether the x under scrutiny is covered by one or more of the failure functions. Here k belongs to Z.

### failure functions and indirect primality tests - II

Let our definition of a failure continue to be a composite number. Let the mother function be the quadratic polynomial x^2 + 7. Then the values of x generated by the failure functions x = 1 +2k, 2 + 7k, 4 +23k etc are such that f(x) is composite. Any value of x not generated by failure functions is such that f(x) is prime and need not be tested for primality. (k belongs to Z ).

### failure functions - refresher

Crudely put failure funcions predict failures. To be more exact failures functions predict the values of the variable when substituted in the original or mother function we get failures in accordance with our definition of a failure. Needless to say the definition of a failure will depend on the problem in hand. The variable itself acts as a function of a specific value of it. Three examples: a) polynomials - let f(x) be a polynomial in x (x belongs to Z). Let our definition of a failure be a composite number. Then x = f(x +kf(x_0)) is a failure function. k belongs to Z.( to be continued)