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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 18th, 2018, 4:27 pm

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 metal construction made by mathematicians?
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Is that a physical construction made of atoms or a metal construction made by mathematicians?
Annealing?
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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 18th, 2018, 10:01 pm

Methinks this thread will NOT form a Cauchy sequence!
A convergence sequence ==> it is a Cauchy  sequence.
If not Cauchy then it does not converge.
Some CSs are more equal than others. Just see {1/n} and {(1 + x/n)^n}.
Is there a least convergent CS?
 
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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 19th, 2018, 8:15 am

There's weak convergence, which is not strong / Cauchy. It's an analogue of pointwise convergence in the Hilbert space...
 
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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 19th, 2018, 2:26 pm

Methinks this thread will NOT form a Cauchy sequence!
A convergence sequence ==> it is a Cauchy  sequence.
If not Cauchy then it does not converge.
Some CSs are more equal than others. Just see {1/n} and {(1 + x/n)^n}.
Is there a least convergent CS?
"least convergent CS" seems like a self-inflicted definition. Can you be more precise?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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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 19th, 2018, 2:28 pm

There's weak convergence, which is not strong / Cauchy. It's an analogue of pointwise convergence in the Hilbert space...
You mean, complete spaces?
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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 19th, 2018, 3:04 pm

There's weak convergence, which is not strong / Cauchy. It's an analogue of pointwise convergence in the Hilbert space...
You mean, complete spaces?
Weak topologies. When it converges not in the norm (like Cauchy), but when the sequence of vectors defines the sequence of functionals which converges pointwise everywhere. It can be divergent in the Cauchy sense.
https://en.wikipedia.org/wiki/Weak_conv ... ert_space)
 
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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 19th, 2018, 5:33 pm

There's weak convergence, which is not strong / Cauchy. It's an analogue of pointwise convergence in the Hilbert space...
You mean, complete spaces?
Weak topologies. When it converges not in the norm (like Cauchy), but when the sequence of vectors defines the sequence of functionals which converges pointwise everywhere. It can be divergent in the Cauchy sense.
https://en.wikipedia.org/wiki/Weak_conv ... ert_space)
I don't see how this address T4A's ambiguous question.

I think your post is about convergence of functionals in normed linear spaces != convergence of sequences in  normed linear spaces.
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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 19th, 2018, 6:04 pm

A convergence sequence ==> it is a Cauchy  sequence.
If not Cauchy then it does not converge.
Some CSs are more equal than others. Just see {1/n} and {(1 + x/n)^n}.
Is there a least convergent CS?
"least convergent CS" seems like a self-inflicted definition. Can you be more precise?
Hasn't anyone tried to rank-order CS by some measure of convergence? Here's two possible approaches:

1. How about defining a test that compares two CS to each other. For example, consider a function that is a derived sequence from a CS, epsilon(CS,i), that provides the sequence of epsilon values corresponding to the CS sequence values. This monotonic, non-increasing sequence will be CS, right? Next examine the ratio of two epsilon sequences and the inverse, epsilon(CS_1,i)/epsilon(CS_2,i) and epsilon(CS_2,i)/epsilon(CS_1,i). Is one a CS, is the inverse a CS, or might neither be CSes?

2. Or we could define various families of CS such as: a) CS that take linear time or index count to converge from epsilon to epsilon/2; b) CS that take quadratic time to converge from epsilon to epsilon/2; c) CS that take exponential time to converge from epsilon to epsilon/2, etc. Families might be rank ordered with respect to each other as with the computational complexity classes.
 
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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 19th, 2018, 6:30 pm

You mean, complete spaces?
Weak topologies. When it converges not in the norm (like Cauchy), but when the sequence of vectors defines the sequence of functionals which converges pointwise everywhere. It can be divergent in the Cauchy sense.
https://en.wikipedia.org/wiki/Weak_conv ... ert_space)
I don't see how this address T4A's ambiguous question.

I think your post is about convergence of functionals in normed linear spaces != convergence of sequences in  normed linear spaces.
In a Hilbert space, a sequence of vectors defines a sequence of functionals, and on those functionals one defines weak convergence. My post is a response to both Traden4Alpha's question and your comment "A convergence sequence ==> it is a Cauchy  sequence. If not Cauchy then it does not converge."
 
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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 19th, 2018, 7:08 pm

Is there a least convergent CS?
"least convergent CS" seems like a self-inflicted definition. Can you be more precise?
Hasn't anyone tried to rank-order CS by some measure of convergence?  Here's two possible approaches:

1. How about defining a test that compares two CS to each other.  For example, consider a function that is a derived sequence from a CS, epsilon(CS,i), that provides the sequence of epsilon values corresponding to the CS sequence values.  This monotonic, non-increasing sequence will be CS, right?  Next examine the ratio of two epsilon sequences and the inverse, epsilon(CS_1,i)/epsilon(CS_2,i) and epsilon(CS_2,i)/epsilon(CS_1,i).  Is one a CS, is the inverse a CS, or might neither be CSes?

2. Or we could define various families of CS such as: a) CS that take linear time or index count to converge from epsilon to epsilon/2; b) CS that take quadratic time to converge from epsilon to epsilon/2; c) CS that take exponential time to converge from epsilon to epsilon/2, etc.  Families might be rank ordered with respect to each other as with the computational complexity classes.
These are good questions and getting to heart of computational schemes.
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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 19th, 2018, 9:08 pm

"least convergent CS" seems like a self-inflicted definition. Can you be more precise?
Hasn't anyone tried to rank-order CS by some measure of convergence?  Here's two possible approaches:

1. How about defining a test that compares two CS to each other.  For example, consider a function that is a derived sequence from a CS, epsilon(CS,i), that provides the sequence of epsilon values corresponding to the CS sequence values.  This monotonic, non-increasing sequence will be CS, right?  Next examine the ratio of two epsilon sequences and the inverse, epsilon(CS_1,i)/epsilon(CS_2,i) and epsilon(CS_2,i)/epsilon(CS_1,i).  Is one a CS, is the inverse a CS, or might neither be CSes?

2. Or we could define various families of CS such as: a) CS that take linear time or index count to converge from epsilon to epsilon/2; b) CS that take quadratic time to converge from epsilon to epsilon/2; c) CS that take exponential time to converge from epsilon to epsilon/2, etc.  Families might be rank ordered with respect to each other as with the computational complexity classes.
These are good questions and getting to heart of computational schemes.
Yes, they are preliminary questions that may lead to a more formal framework for analyzing rate of convergence.

They may also lead to empirical tools for analyzing the convergence properties of physical systems which is part of the rationale for this thread.
 
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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 20th, 2018, 7:59 am

Yes. Let me think about it for  bit to get a good example. We are really entering a discussion of Constructive Analysis that states that all theorems must be computable/construct the solution.

In fact, the result in my very first post (General Convergence Principle) is constructive (in this case we use a dt instead of n = 1,2,3,...)
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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 20th, 2018, 1:45 pm

The ratio and inverse, epsilon(CS_1,i)/epsilon(CS_2,i) and epsilon(CS_2,i)/epsilon(CS_1,i), would likely have three possible states:

1. epsilon(CS_1,i)/epsilon(CS_2,i) converges to 0 and epsilon(CS_2,i)/epsilon(CS_1,i) diverges to ∞

2. epsilon(CS_1,i)/epsilon(CS_2,i) diverges to ∞ and epsilon(CS_2,i)/epsilon(CS_1,i) converges to 0

3. both epsilon(CS_1,i)/epsilon(CS_2,i) and epsilon(CS_2,i)/epsilon(CS_1,i) converge to some non-zero value.

The only computational issue is what to do if both epsilon(CS_1,i)=0 and epsilon(CS_2,i)=0.
 
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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 25th, 2018, 7:37 am

P.S.  Steady-state solutions prove bumble bees cannot fly.  But then bumble bees, like most living things including people and their economies, are NOT in a steady state.
It's off-topic here, but the bumble-bee model has a single point of failure.
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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 25th, 2018, 7:57 am

I'm producing an infinite sequence of posts on the unreality of infinite sequences, what?
Hold on, have you forgotten your Vico cycles (time-series for medievaluists)? Do the converge of just repeat themselves?
Ah, so Vico disproved the existence of Cauchy sequences?????????
Hold your horses! Not everything is a Cauchy sequence. Those who have read their Vico (and FW for that matter!) will know it is a recurrent sequence, similar to

[$]x_{n+1} \: = \: rx_{n} (1 - x_{n})[$]

A Vico sequence is bounded but not monotonic.
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

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