3 ms·

What I find cool is that it seems that all sorts of growth rates in nature (and even computer science) can be expressed in terms of elementary mathematical func

What I find cool is that it seems that all sorts of growth rates in nature (and even computer science) can be expressed in terms of elementary mathematical functions. Things can grow at exponential rates, logarithmic rates, hyperbolic rates, etc., and we actually have the terminology to express all these types of growth curves, whether they apply to bacterial colonies, or a nautilus shell, or a tree-trunk's diameter. Growth is described by log x, or a^x, or x^n, or sin x, or other functions we've already catalogued, and there's nothing "in-between" these curves that isn't already covered. Are there any natural generative processes that produce/grow at a rate that isn't modellable by a linear combination of basic functions? Have we essentially discovered all the possible functions that describe how things grow or accumulate? Do more complicated functions sometimes describe changes in nature? I'm no mathematician, but I'd enjoy learning about this, because it puzzled me back in school.

textminer 14y ago
I've wondered this, too! It helps that elementary functions are in some sense dense in the set of all other functions. Results like Stone-Weierstrass show that on a compact interval, any function can be arbitrarily-well approximated by a polynomial. Other things that might interest you-- how one might know if a function even has an antiderivative that can be expressed as an elementary function: http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differen...
setrofim_ 14y ago
> Are there any natural generative processes that produce/grow at a rate that isn't modellable by a linear combination of basic functions? Have we essentially discovered all the possible functions that describe how things grow or accumulate? Do more complicated functions sometimes describe changes in nature? Global weather patterns come to mind. E.g. how hurricanes, tornadoes, etc. form and change. As I understand it (and I'm no expert) there is some funky math involved in modelling those, and we're not really there yet. I'm not sure if those fall under "generative processes" though.
zby 14y ago
This is because we try to fit these patterns into elementary functions, and we do that because it is most useful.
anywhichway 14y ago
Natural generative processes come down to growth force. When you write them down as the results of your most basic observations of them, like "Growth is exactly proportional to the population size" you end up putting everything in terms of these "forces" which yields a differential equation p'(t) = k p(t) which has a solution of p(t) = c e^(kt). There are a number of basic tweaks that reflect real life situations that can quickly make these functions unsolvable by elementary functions. One example are time-varying carrying capacity. If you're on an island that generates a limited amount of resources each year you get the basic carrying capicity logistic function p'(t)=p(t)(1-p(t)) which is solved by p(t) = 1/(1+c e^t). But if that your population has a delayed affect on the available resources you get p'(t)=p(t)(1-p(t-t1)) which can't be expressed as elemantary equations. http://en.wikipedia.org/wiki/Logistic_function#Time-varying_carrying_capacity http://en.wikipedia.org/wiki/Logistic_function#Time-varying_... Also, preditor-prey models which occur naturally where you have two populations that depend on each other which can't be reduced to elementary functions either: http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation

textminer 14y ago

I've wondered this, too! It helps that elementary functions are in some sense dense in the set of all other functions. Results like Stone-Weierstrass show that on a compact interval, any function can be arbitrarily-well approximated by a polynomial. Other things that might interest you-- how one might know if a function even has an antiderivative that can be expressed as an elementary function: http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra) http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differen...

setrofim_ 14y ago

> Are there any natural generative processes that produce/grow at a rate that isn't modellable by a linear combination of basic functions? Have we essentially discovered all the possible functions that describe how things grow or accumulate? Do more complicated functions sometimes describe changes in nature? Global weather patterns come to mind. E.g. how hurricanes, tornadoes, etc. form and change. As I understand it (and I'm no expert) there is some funky math involved in modelling those, and we're not really there yet. I'm not sure if those fall under "generative processes" though.

zby 14y ago

This is because we try to fit these patterns into elementary functions, and we do that because it is most useful.

anywhichway 14y ago

Natural generative processes come down to growth force. When you write them down as the results of your most basic observations of them, like "Growth is exactly proportional to the population size" you end up putting everything in terms of these "forces" which yields a differential equation p'(t) = k p(t) which has a solution of p(t) = c e^(kt). There are a number of basic tweaks that reflect real life situations that can quickly make these functions unsolvable by elementary functions. One example are time-varying carrying capacity. If you're on an island that generates a limited amount of resources each year you get the basic carrying capicity logistic function p'(t)=p(t)(1-p(t)) which is solved by p(t) = 1/(1+c e^t). But if that your population has a delayed affect on the available resources you get p'(t)=p(t)(1-p(t-t1)) which can't be expressed as elemantary equations. http://en.wikipedia.org/wiki/Logistic_function#Time-varying_carrying_capacity http://en.wikipedia.org/wiki/Logistic_function#Time-varying_... Also, preditor-prey models which occur naturally where you have two populations that depend on each other which can't be reduced to elementary functions either: http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation