Cycles in American History - 9
Proof of the General
The Proof
It is now time to get to the proof that Fourier Transforms differ from Laplace Transforms. More work will be needed to examine on republican and democratic shifts or the American context.
The first step is to make rigorous the Fourier Transform.
1. A Fourier Transform is a Reimann Integral. It also must have a result ℝ which in this case is the communication between two entities. For this proof, an FT is understood to be an interaction between 2 entities.
2. We can represent over the interval [-L,L] using the Fourier Series where n is a constant, [-L,L] is the interval over a period of sin and cos:
3. When changing the sum to an interval:
4. In biology this form represents the generational model by Haldane:
Since Haldane notes that in humanity the sex ratio is equal, and we can set 2 𝜋ik to the curve of breeding, this then becomes the Population Pyramid.
Thus:
If:
Since from a biological standpoint, the two must be the same.
5. With the addition of an “outside” which contributes members a rate that is analogous to 2 𝜋ik inside, we can then modify the vector chain to a modified Markov Chain with the difference that new entities enter the Markov Chain, while old entities exit by entering into a Markov Chain that self-recurs.
This then is an FT of a Modified Markov Chain with a Markov Matrix as the result because if an arbitrary f is a function that acts as a process transforming X, then G(f) is a block diagonal matrix is equivalent to obtaining the FT of f.
6. So, a new Markov Chain is started by two already established Markov Chains sending a Diracian delta signal. This means that the Haldane function is satisfied in equaling 1.
7. This means that:
And that:
8. Note this means F(𝜔) is not square-integrable because it is at a point, but the delta function is not (L^2)(ℝ)
9. The event that causes reproduction is, however, like the Dirac delta function not well-behaved. (No comment)
10. The difference between the Haldane and the Population Pyramid is that there are constraints on the Population Pyramid for example, money is not a parameter of the Haldane but is in the PP.
11. If this were a Real number then the Population Pyramid would still be a Fourier Transform. If not, then an LT is required.
12. Money is an ℝ. The expectation of Money is an ℝ. But a private expectation not communicated to the other partner is not in ℝ. Thus, the Population Pyramid must have an LT.
13. Markov Chain, in particular context cannot be simple. That is be position at T does not contain all of the information that can be ascribed to any particular entity. This can be proven by myriad examples such as Pompei, where people did not know that they were going to die from some external event. This means that classical birth-death Markov chains also do not work because they assume that any new entity is exactly the same as any old entity.
14. This means that the reproduction function is a Poisson Process with the 𝜔 being the time of pregnancy but the longer term is an agreement (or lack thereof) to reproduce with one entity.
15. But this means again that an LT must happen.
16. This means that a generalized FT minus the actual Population Pyramid produces ℝ of the result of the LT. For example, the difference between the Soviet Union’s predicted FT and the actual.
17. Thus:
Where the L value represents ℝ of the LT.
18. Since this is a Poisson Process in the Markov Space, therefore:
The conjugate of L(|x|) when added to or multiplied by L(|x|) will result in an ℝ.
This means that any FT as a bottom-up number can be checked by calculating the
and adding its complement.
21. While a bottom-up change in the birth-rate maybe stimulated by an LT on the PP, such as a Depression, it still must go through the Fourier Transform. Thus, the cause may be different from the FT which must be present.
22. The first attempt is the Birth-Death Markov Chain. While, for an FT this might be sufficient it fails on the LT because the individual birth must include all events. Then Birth-Death Markov Chain is insufficient for a PP because the Total Number does not take into account previous experience which may, or may not, be an ℝ.
23. Thus, we must replace the Simplified BD with one that has each individual member as a separate. Of course, a normal human life span is the Fourier Series which can be shown to be periodicities over the entire “BD span” (though we note that, in our case, it is conception to death) span. We will call this a Laplace Normal Form.
24. The LPF has the LT in the PP.
25. LNF has experiences for each entity which the CBD model does not have. This leads us to the conclusion that the is more than bottom-up FT changes. This we will call, in the American context, the difference between bottom-up “democratic” and top-down “republican” shifts while turning to middle out elsewhere.
26. This leads up to the idea that LT comes from other than bottom-up FT.
27. This means the LPN only increments rather than increments and decrements CBD. Thus, we may use the same FT transforms with a small addition that the number of new entities is drawn from the Dirac delta of existing entities, and for every given country a number of new entries from and to other countries will have the Markov by equal to 0. Tacking λ as the number of new entities and μ as the number of old entities dying:
𝑁(𝑡+Δ𝑡)=𝑁(𝑡)+𝑁(𝑡)λΔ𝑡−𝑁(𝑡)μΔ𝑡
28. ∴ We see from Strauss & Howe that they do not have this form of the PP nor of the difference between FT and LT. We will later show that they do not have a specific LT which we shall term “power,” for want of a better one.
The next step is to prove that Laplace Transforms can be different from Fourier Transforms in the in the PP paradigm, and that that a more specific PP for the US is required.