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Everything is a linear model

danielroelfs.com

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Re: Everything is a linear model

#41

Post by jakeinspace »

whatshisface wrote: post #27
sebastianavina wrote: post #21

Earlier quoted context omitted.

"Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas. " and everything turns around the same principles. For example dynamical models and PID controls. yet solving a banana, is the only thing we really know how to do. So we end up fitting everything in our banana models.

I disagree with the implication that linearity is an unnatural concept, it appears whenever the changes being studied are small relative to the key parameters that determine the system. Every system is linear for small perturbations. Even logic gates; in negative feedback they can form passable inverting amplifiers. In a place as big as the universe it is rather common for two things to be very different in scale and…

Isn’t the whole point of chaos sensitivity to initial conditions? Where the output even when minuscule differences varies wildly with input?

Re: Everything is a linear model

#42

Post by whatshisface »

jakeinspace wrote: post #41
whatshisface wrote: post #27

Earlier quoted context omitted.

I disagree with the implication that linearity is an unnatural concept, it appears whenever the changes being studied are small relative to the key parameters that determine the system. Every system is linear for small perturbations. Even logic gates; in negative feedback they can form passable inverting amplifiers. In a place as big as the universe it is rather common for two things to be very different in scale and…

Isn’t the whole point of chaos sensitivity to initial conditions? Where the output even when minuscule differences varies wildly with input?

In chaotic systems, time becomes one of the parameters that has to be small for a linear model to make predictions, but they are still linear for brief times.

Re: Everything is a linear model

#43

Post by sebastianavina »

whatshisface wrote: post #27
sebastianavina wrote: post #21

Earlier quoted context omitted.

"Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas. " and everything turns around the same principles. For example dynamical models and PID controls. yet solving a banana, is the only thing we really know how to do. So we end up fitting everything in our banana models.

I disagree with the implication that linearity is an unnatural concept, it appears whenever the changes being studied are small relative to the key parameters that determine the system. Every system is linear for small perturbations. Even logic gates; in negative feedback they can form passable inverting amplifiers. In a place as big as the universe it is rather common for two things to be very different in scale and…

yeah, we use a lot of small linear systems to model a non linear system.

We just break the problem in small bananas because that's what we can solve, and then solve for those lil'bananas and call it done.

Re: Everything is a linear model

#44

Post by blt »

whatshisface wrote: post #40
waveBidder wrote: post #33

Earlier quoted context omitted.

> Every system is linear for small perturbations. every smooth system, sure, but even continuity is no guarantee of locally linearity.

I've never actually seen a physical example of a system without a continuous first derivative. For example phase transitions, commonly touted as an example of discontinuity, don't actually occur until the matter has gone a bit over the point and a transition nucleates somewhere. The probability of a phase transition is a continuous function of temperature, with continuous derivatives. I'm skeptical that discontinuiti…

Any physical system that makes/breaks contact, such as walking robots. Sure, the foot is not perfectly rigid and technically is a stiff spring. But from a computational perspective, problems still bear all the hallmarks of a discontinuous system such as requiring a very short integration step.

Re: Everything is a linear model

#45

Post by aeonik »

whatshisface wrote: post #40
waveBidder wrote: post #33

Earlier quoted context omitted.

> Every system is linear for small perturbations. every smooth system, sure, but even continuity is no guarantee of locally linearity.

I've never actually seen a physical example of a system without a continuous first derivative. For example phase transitions, commonly touted as an example of discontinuity, don't actually occur until the matter has gone a bit over the point and a transition nucleates somewhere. The probability of a phase transition is a continuous function of temperature, with continuous derivatives. I'm skeptical that discontinuiti…

If I wanted to understand and obtain your intuition about linearity, what would you recommend?

Re: Everything is a linear model

#46

Post by bongodongobob »

SonOfLilit wrote: post #35
sebastianavina wrote: post #21

Earlier quoted context omitted.

"Classification of mathematical problems as linear and nonlinear is like classification of the Universe as bananas and non-bananas. " and everything turns around the same principles. For example dynamical models and PID controls. yet solving a banana, is the only thing we really know how to do. So we end up fitting everything in our banana models.

My favorite moment in university was in the first class of semester 2, where a prof said "lets look at a really small part of our thing, and assume we apply some force to it. This will make it stretch, lets assume the stretching is linear relative to the applied force". I raised my hand and asked "is this assumption supported empirically?" and he said "no, we know it's not always true, but if we don't make it we can'…

Can you give some examples of linear and non-linear materials?

Re: Everything is a linear model

#47

Post by [deleted] »

bongodongobob wrote: post #46
SonOfLilit wrote: post #35

Earlier quoted context omitted.

My favorite moment in university was in the first class of semester 2, where a prof said "lets look at a really small part of our thing, and assume we apply some force to it. This will make it stretch, lets assume the stretching is linear relative to the applied force". I raised my hand and asked "is this assumption supported empirically?" and he said "no, we know it's not always true, but if we don't make it we can'…

Can you give some examples of linear and non-linear materials?

[deleted]

Re: Everything is a linear model

#49

Post by pistachiopro »

bongodongobob wrote: post #46
SonOfLilit wrote: post #35

Earlier quoted context omitted.

My favorite moment in university was in the first class of semester 2, where a prof said "lets look at a really small part of our thing, and assume we apply some force to it. This will make it stretch, lets assume the stretching is linear relative to the applied force". I raised my hand and asked "is this assumption supported empirically?" and he said "no, we know it's not always true, but if we don't make it we can'…

Can you give some examples of linear and non-linear materials?

Practically all materials behave nonlinearly when stretched or compressed a visible amount. For certain structural applications, though, if that happens we've already failed. Linear models work really well for designing big concrete structures and certain metal structures. Sometimes we try to apply linear models to other things, but that's always kind of fishy.
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