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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 3:02 am

katastrofa wrote:
Measurement amounts to constructing a density matrix of the statistical ensemble (trivial - diagonal in classical case), which you investigate. Actually, the description of the quantum measurement *process* (involving the quantum evolution) also generally requires the Cauchy sequence. Any measurement or optimisation, I would add - if you think about it long enough, they mean the same :-)
Is that a physical construction made of atoms or a mental construction made by mathematicians?
 
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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 3:47 am

Do you want me to answer with a front or rear part of my brain? :-)

Anyway, I'm afraid we have triggered Cuchulainn's e^5 obsession...
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 8:50 am

katastrofa wrote:
Do you want me to answer with a front or rear part of my brain? :-)

Anyway, I'm afraid we have triggered Cuchulainn's e^5 obsession...

Image
This thread is the big general theory stuff :D
 
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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 10:13 am

Big general theories come from physicists (mathematicians only prove them).
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 10:22 am

katastrofa wrote:
Big general theories come from physicists (mathematicians only prove them).

And later generations of physicists disprove them.
What caused String Theory to implode?

String theory is a powerful, well-motivated idea and deserves much of the work that has been devoted to it. If it has so far failed, the principal reason is that its intrinsic flaws are closely tied to its strengths—and, of course, the story is unfinished, since string theory may well turn out to be part of the truth. The real question is not why we have expended so much energy on string theory but why we haven't expended nearly enough on alternative approaches.

more hype 

http://www.math.columbia.edu/~woit/wordpress/
 
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 11:09 am

So, people use Cauchy sequences; they just don't know it yet?

Cauchy sequences have amazing properties that can be used to understand the behavior of a system as time progresses. They are heavily used in fields like satellite design, manufacturing, construction, treatment plants, and so on.

Let’s say we want to analyze how a structure is going to respond to weather conditions in the next couple of decades. Based on historical data, if we know that this can be modeled as a Cauchy sequence, we can extract a lot of insights. The goal is to model the system and make predictions about how this structure is going to react under various conditions. If we know that a given physical process can be described using a Cauchy sequence, we know that the process will converge to a single point.
 
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 11:57 am

katastrofa wrote:
Big general theories come from physicists (mathematicians only prove them).
Exactly!

But the mathematicians can only "prove" that the theory has logical internal consistency and produce derived implications. It's not a proof that the theory is true for the physical world. The whole thing then goes back the physicists for testing. And even the physicist can't "prove" (in the black-and-white logical sense of math) that the theory is correct. At best, they can show the theory is not inconsistent with the available evidence although that evidence may be prone to any of a range of sampling, methodological, and measurement errors.

That is, they can only fail to disprove the theory.
Last edited by Traden4Alpha on May 17th, 2018, 12:54 pm
 
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outrun
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 12:52 pm

Traden4Alpha wrote:
katastrofa wrote:
Big general theories come from physicists (mathematicians only prove them).
Exactly!

But the mathematicians can only "prove" that the theory has logical internal consistency and produce derived implications. It's not a proof that the theory is true for the physical world. The whole thing then goes back the physicists for testing. And even the physicist can't "prove" (in the black-and-white logical sense of math) the theory is correct. At best, they can show the theory is not inconsistent with the available evidence which may be prone to any fo a range of sampling, methodological, and measurement errors.

That is, they can fail to disprove the theory.

Exactly! You can't use it to say something about "approximation to an unknown function".
Unknown means that you assume no prior and also that you need to sample it, and samples don't give any guarantee about future samples. In the real world you are in the realms of sampling and statistics. You don't know how the function behaves between samples (interpolation), you don't know if repeated samples will give the same outcome (stationarity: the function you approximate in RL are eg non-stationary) and there is the finite resolution of samples: measurement and representation error.

However the main point is that you can't generalize and give guarantees on a set of samples.
 
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 1:51 pm

However the main point is that you can't generalize and give guarantees on a set of samples.

This is the world of Statistics. This is something completely different. AFAIK Statistics doesn't operate on sequences..

You can if it is natural or artificial evolution. Populations are Cauchy sequences. They converge to a single best individual (Cauchy) in the population, thus making it a complete metric space using the fitness function as a metric. aka Cauchy annealing schedule.

But some people don't believe in evolution. 
 
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katastrofa
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 2:18 pm

Cuchulainn wrote:
So, people use Cauchy sequences; they just don't know it yet?

Cauchy sequences have amazing properties that can be used to understand the behavior of a system as time progresses. They are heavily used in fields like satellite design, manufacturing, construction, treatment plants, and so on.

Let’s say we want to analyze how a structure is going to respond to weather conditions in the next couple of decades. Based on historical data, if we know that this can be modeled as a Cauchy sequence, we can extract a lot of insights. The goal is to model the system and make predictions about how this structure is going to react under various conditions. If we know that a given physical process can be described using a Cauchy sequence, we know that the process will converge to a single point.

My answer to your question would be yes, but I don't understand the two paragraphs you have quoted. The first would be clearer to me if it was about Cauchy boundary conditions (as used in hyperbolic PDEs in physics of systems the quoted author listed, I suppose), the second I don't understand - I would need to ask the person what they mean.
PS, there are fuzzy Cauchy sequences in statistics, afair.
 
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 2:35 pm

outrun wrote:
Traden4Alpha wrote:
katastrofa wrote:
Big general theories come from physicists (mathematicians only prove them).
Exactly!

But the mathematicians can only "prove" that the theory has logical internal consistency and produce derived implications. It's not a proof that the theory is true for the physical world. The whole thing then goes back the physicists for testing. And even the physicist can't "prove" (in the black-and-white logical sense of math) the theory is correct. At best, they can show the theory is not inconsistent with the available evidence which may be prone to any fo a range of sampling, methodological, and measurement errors.

That is, they can fail to disprove the theory.

Exactly! You can't use it to say something about "approximation to an unknown function".
Unknown means that you assume no prior and also that you need to sample it, and samples don't give any guarantee about future samples. In the real world you are in the realms of sampling and statistics. You don't know how the function behaves between samples (interpolation), you don't know if repeated samples will give the same outcome (stationarity: the function you approximate in RL are eg non-stationary) and there is the finite resolution of samples: measurement and representation error.

However the main point is that you can't generalize and give guarantees on a set of samples.
Exactly!

I choked on my coffee when I first read the hypothetical example of modeling "how a structure is going to respond to weather conditions in the next couple of decades." There's nothing about the weather that justifies pretending that historical data can be used for prediction. Weather is extremely non-stationary on all time-scales. Moreover, the weather phenomena that affect buildings (e.g., hurricanes) seem to follow a power-law distribution -- the tails are extremely fat. The Cauchy distribution seems more applicable that a Cauchy sequence to this example.

The entire example could be summarized as "assume a falsehood that lets us use our pretty math."
 
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outrun
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 2:56 pm

Cuchulainn wrote:
However the main point is that you can't generalize and give guarantees on a set of samples.

This is the world of Statistics. This is something completely different. AFAIK Statistics doesn't operate on sequences..

You can if it is natural or artificial evolution. Populations are Cauchy sequences. They converge to a single best individual (Cauchy) in the population, thus making it a complete metric space using the fitness function as a metric. aka Cauchy annealing schedule.

But some people don't believe in evolution. 

I don't think a population evolve towards a single best individual in the real world. The measure of 'best" is non-stationary (a comet hit the Earth and it went dark and everything changed), pluriform and conflicting: an investment banker might best at robbing money from clients, but is he also the best in weight lifting? Depending on the measure you get different answers about who is best (but you can pick one). Evolution means that the population changes is size, that properties constantly change (evolve=non-stationary), and that it might even die out like T-Rex did.

All you can do is observe what happens during evolution in the real world, you can't know what will be best (fish that eat plastic will do good these days, maybe not if people start to clean things up). You will need to do observations and measure in the real works (the distinctive  topic of this thread) and then it's statistics.

This is different from doing computation with mathematic abstractions of the real work and solving those mathematical models.
 
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 3:53 pm

Cuchulainn wrote:
However the main point is that you can't generalize and give guarantees on a set of samples.

This is the world of Statistics. This is something completely different. AFAIK Statistics doesn't operate on sequences..

You can if it is natural or artificial evolution. Populations are Cauchy sequences. They converge to a single best individual (Cauchy) in the population, thus making it a complete metric space using the fitness function as a metric. aka Cauchy annealing schedule.

But some people don't believe in evolution. 
Evolution is another example of a non-linear system that generates divergence, not convergence. It may be the best evidence of how dangerously wrong Cauchy sequences are for many categories of systems because on some timescales, the population of genotypes do seem to be converging toward "best" condition. But then something happens (a new competitor evolves, climate changes, species become toxic, disease sweeps the population, etc.) and whole chunks of the genome change quite quickly.

At the gene level, the "convergence model" is provably false. For example, the gene for sickle cell anemia causes debilitating problems if the person inherits two copies of it. And yet evolution will never drive this "bad mutation" out of the population because people who have no copies of this gene are more susceptible to dying of malaria. At best, the "equilibrium" condition is that all three genotypes persisting in the population. But even this equilibrium is highly unstable under the forces of evolution of both humans (who might evolve another defense against malaria or an adjunct that fixes the double-copy sickle cell problem) and the malaria parasite (which might evolve a counter measure to the sickle gene defense).

I could go on because there are quite a large number of phenomena that prevent convergence in evolution.
 
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Cuchulainn
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 7:56 pm

I choked on my coffee when I first read the hypothetical example of modeling "how a structure is going to respond to weather conditions in the next couple of decades." There's nothing about the weather that justifies pretending that historical data can be used for prediction.


 
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Traden4Alpha
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Re: Does Cauchy sequence ever occur in the real world or is it all the minds of mathematicians?

May 17th, 2018, 8:42 pm

Cuchulainn wrote:
I choked on my coffee when I first read the hypothetical example of modeling "how a structure is going to respond to weather conditions in the next couple of decades." There's nothing about the weather that justifies pretending that historical data can be used for prediction.



LOL!
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