Question about mathematical "proof"

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Tomaso
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Question about mathematical "proof"

Postby Tomaso » Mon Nov 13, 2017 12:50 pm UTC

Hi

I have been working on a certain proof for a while now, hobby-wise. If my proof is correct, then I have found a shorter and more elegant way to solve an already-solved theorem.

- I have an equation consisting of functions, with variables and parameters. There is a claim for the behaviour of this equation.

- Now I modify that equation in a certain way and get a new equation. One variable (a positive real number) disappears from the system, but I add a new parameter "a" to the system (also a positive real number). So, I remove a degree of freedom from the system and add a degree of freedom to the system.

- There is a certain critical threshold for "a". That threshold depends on the other parameters (not variables!) of the system and on some mathematical constants.

- If "a" > that threshold, then the new equation shows a behavior that makes it possible to determine whether the claim is true or not. (And my result is in accordance with literature: The claim is true.)

- If "a" < that threshold, then the new equation shows a behavior where it is on principle impossible to determine whether the claim is true or not.



My question is:
When I introduce the new parameter "a", can I just arbitrarily define it as being above that threshold?
e.g. a -> inf ?

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gmalivuk
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Re: Question about mathematical "proof"

Postby gmalivuk » Mon Nov 13, 2017 1:16 pm UTC

It depends on what 'a' represents and on what your equations are. If you won't tell us those things, we can't help you.
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Tomaso
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Re: Question about mathematical "proof"

Postby Tomaso » Mon Nov 13, 2017 3:22 pm UTC

gmalivuk wrote:It depends on what 'a' represents and on what your equations are. If you won't tell us those things, we can't help you.


"a" is the broadness of a gaussian curve. When I introduce a gaussian into the system, the proportion between the other parameters of the other functions and the broadness of the gaussian determine the further behaviour of the system.
With a fixed broadness I can prove individual cases up to a certain limit. But for a general proof I would need the broadness so large that I can cover proofs for any generic case.

My question is simple: When I introduce the gaussian curve, am I allowed to define the broadness as "big enough for all intents and purposes?"



I would rather not go into more detail, because if this works I would like to publish it. And if it doesn't work... I have a whole pile of failed ideas. :D

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Re: Question about mathematical "proof"

Postby doogly » Mon Nov 13, 2017 3:30 pm UTC

Sigma is the standard deviation of a Gaussian - don't make a new definition for a "broadness" if an existing concept can work for you.

Can the a>threshold condition be related to a condition which was known to be obeyed in the original equation? If not, you are introducing things with this step and weakening the scope.
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Tomaso
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Re: Question about mathematical "proof"

Postby Tomaso » Mon Nov 13, 2017 5:01 pm UTC

doogly wrote:Sigma is the standard deviation of a Gaussian - don't make a new definition for a "broadness" if an existing concept can work for you.

Can the a>threshold condition be related to a condition which was known to be obeyed in the original equation? If not, you are introducing things with this step and weakening the scope.


No, the standard deviation is pretty much "brand-new" and unrelated to the earlier problem. That was my worry: That making the sigma arbitrarily big would be "handwaving".

And I had hoped that this blasted monster would be dead and defeated! Turns out I will have to draw my mighty pen once more!

Thanks anyway, doogly.

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gmalivuk
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Re: Question about mathematical "proof"

Postby gmalivuk » Mon Nov 13, 2017 9:01 pm UTC

Pro-Tip: If the existing theorem took a long time for trained mathematicians to prove, and you're tinkering with it as a hobby and asking for help on a web forum, then you almost certainly did not find a more elegant proof. Either it's not a proof at all (more likely), or whatever you did find boils down to the same proof that's already out there (less likely but I suppose possible).

You may as well just tell us what it is (or at least what you're trying to prove) and learn something from any mistakes you've made. On the off chance that it is something new, you can use your forum posts to demonstrate that it's your proof so no one else beats you to publishing.
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