User:Kieranmrhunt

Mathematical Notes

Gamma Equation

$\prod _{p=2}^{m}(p^{i}-p^{i-1})=\Gamma (m)\Gamma (m+1)^{i-1}$

Note the right hand side can be written as:

$(m-1)!(m!)^{i-1}$

Null Theorem

$\forall \pi (\exists 0\Rightarrow (\exists (k\in \pi ):k\equiv 0))$

Here I define $\pi$ to be the set of all things that exist within the system in question. This is loosely analogous to the wavefunction in quantum mechanics.

Evolutionary Progression Relation

Using a recursive formula to suggest patterns of evolutionary drift can be enhanced by including a term that depends on the history of said progression.

$a_{n}=\underbrace {a_{n-1}+r_{n}-R} _{\text{evolutionary drift}}+\underbrace {M(a_{n-1}-a_{n-m})} _{\text{path history term}}$

Where $r_{n}$ is a random number between 0 and 1, the contribution from fluctuations within the system, $R$ is a number (reducer) of any chosen value for the specific system that controls the positivity of the resultant graph, $M$ is another number (mediator) that controls how much influence the path history has over the progression of the system, and we define $m-1$ to be the order of the equation.

It can be shown that the transition from linear to exponential behaviour of the resulting system occurs at $M=(m-1)^{-1}$ , wherein exponential behaviour quickly dominates as $M$ rises.

Bifurcation Conjecture

Demonstration of bifurcation for $n=2$ .

A graph of the number of ways in which $m$ can be expressed as the sum of $2n+1$ squares plotted against $m$ demonstrates a clear bifurcation (with $n\in \mathbb {N}$ ), particularly as $m$ surpasses 1000. The conjecture is that the definition of the split increases indefinitely with $n$ for suitably large $m$ (i.e. that $n\ll m$ ).

Geographical Notes

A Relationship between Life Expectancy and Poverty

This graph has data from most countries, wherein it plots the poverty fraction (that is, the percentage of population of the country that live in UN defined impoverished standards) against the average life expectancy for a citizen of that country. I thought this graph to be interesting given the loose correlation (which one would expect). The 'best fit' line is $P.F.=0.1{\sqrt {80-L.E.}}$ .

Area and Perimeter

The lower pound of the perimeter of a country, $S$ , given its area $A$ is:

$S=5{\sqrt {A}}$ .