What terminology should I use when refrencing how close a sequence is to a loop for research?
$begingroup$
I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping.
For example, If I was interested in this sequence:
$S_1$ = 10, 21, 32, 23, 14, 25, 36, …
And I also have the sequence:
$S_2$ = 0, 1, 2, 3, 4, 5, 6, 7, ...
I argue that the latter sequence is off by sequence $S_2$. If I respected the order of both of these sequences and subtract the first term of $S_2$
from $S_1$, then I would create a periodic sequence.
($S_1$ - $S_2$) = (10-0), (21-1), (32-2), (23-3), …
($S_1$ - $S_2$) = 10, 20, 30, 20, 10, 20, 30, ...
I am most interested in how this applies to the Collatz Conjecture, because being able to measure a sequence and determine how close its behavior resembles a loop could be used to argue how larger and larger trajectories either point to there being a counter example or suggest that no counter examples can exist.
I am assuming someone else already came up with my idea and I want to find pre-existing work that explores this approach. I believe it is possible I came across a published work already discussing this topic, but I dismissed it because I did not have enough of an understanding of mathematics to understand what they did. I wanted to at least make sure my understanding of the language describing this topic is correct so I can then research the mathematical methods and tools other people are using.
dynamical-systems collatz
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
$endgroup$
add a comment |
$begingroup$
I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping.
For example, If I was interested in this sequence:
$S_1$ = 10, 21, 32, 23, 14, 25, 36, …
And I also have the sequence:
$S_2$ = 0, 1, 2, 3, 4, 5, 6, 7, ...
I argue that the latter sequence is off by sequence $S_2$. If I respected the order of both of these sequences and subtract the first term of $S_2$
from $S_1$, then I would create a periodic sequence.
($S_1$ - $S_2$) = (10-0), (21-1), (32-2), (23-3), …
($S_1$ - $S_2$) = 10, 20, 30, 20, 10, 20, 30, ...
I am most interested in how this applies to the Collatz Conjecture, because being able to measure a sequence and determine how close its behavior resembles a loop could be used to argue how larger and larger trajectories either point to there being a counter example or suggest that no counter examples can exist.
I am assuming someone else already came up with my idea and I want to find pre-existing work that explores this approach. I believe it is possible I came across a published work already discussing this topic, but I dismissed it because I did not have enough of an understanding of mathematics to understand what they did. I wanted to at least make sure my understanding of the language describing this topic is correct so I can then research the mathematical methods and tools other people are using.
dynamical-systems collatz
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
$endgroup$
$begingroup$
I'd try the terms "mixing" or "composing" of sequences, and then the inverse operation as "demixing" or "decomposing". Having a sequence and trying to decompose it into one periodic and one residual part (whether periodic as well or not) reminds me of "Fourieranalysis". And that keyword should provide more ideas for some meaningful terming (and proceeding) in your problem. Perhaps you can find related concepts in the OEIS-community which are involved in analysis, composition and decomposition of sequences (for instance their "JIS"-journal is freely available and possibly has something in it)
$endgroup$
– Gottfried Helms
Jan 31 at 9:55
add a comment |
$begingroup$
I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping.
For example, If I was interested in this sequence:
$S_1$ = 10, 21, 32, 23, 14, 25, 36, …
And I also have the sequence:
$S_2$ = 0, 1, 2, 3, 4, 5, 6, 7, ...
I argue that the latter sequence is off by sequence $S_2$. If I respected the order of both of these sequences and subtract the first term of $S_2$
from $S_1$, then I would create a periodic sequence.
($S_1$ - $S_2$) = (10-0), (21-1), (32-2), (23-3), …
($S_1$ - $S_2$) = 10, 20, 30, 20, 10, 20, 30, ...
I am most interested in how this applies to the Collatz Conjecture, because being able to measure a sequence and determine how close its behavior resembles a loop could be used to argue how larger and larger trajectories either point to there being a counter example or suggest that no counter examples can exist.
I am assuming someone else already came up with my idea and I want to find pre-existing work that explores this approach. I believe it is possible I came across a published work already discussing this topic, but I dismissed it because I did not have enough of an understanding of mathematics to understand what they did. I wanted to at least make sure my understanding of the language describing this topic is correct so I can then research the mathematical methods and tools other people are using.
dynamical-systems collatz
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
$endgroup$
I don't know what language I should use in order to ask what methods already exist that discuss how to take a sequence and assess it's likelihood of looping.
For example, If I was interested in this sequence:
$S_1$ = 10, 21, 32, 23, 14, 25, 36, …
And I also have the sequence:
$S_2$ = 0, 1, 2, 3, 4, 5, 6, 7, ...
I argue that the latter sequence is off by sequence $S_2$. If I respected the order of both of these sequences and subtract the first term of $S_2$
from $S_1$, then I would create a periodic sequence.
($S_1$ - $S_2$) = (10-0), (21-1), (32-2), (23-3), …
($S_1$ - $S_2$) = 10, 20, 30, 20, 10, 20, 30, ...
I am most interested in how this applies to the Collatz Conjecture, because being able to measure a sequence and determine how close its behavior resembles a loop could be used to argue how larger and larger trajectories either point to there being a counter example or suggest that no counter examples can exist.
I am assuming someone else already came up with my idea and I want to find pre-existing work that explores this approach. I believe it is possible I came across a published work already discussing this topic, but I dismissed it because I did not have enough of an understanding of mathematics to understand what they did. I wanted to at least make sure my understanding of the language describing this topic is correct so I can then research the mathematical methods and tools other people are using.
dynamical-systems collatz
dynamical-systems collatz
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
asked Dec 12 '18 at 0:45
Griffon Theorist697Griffon Theorist697
30429
30429
$begingroup$
I'd try the terms "mixing" or "composing" of sequences, and then the inverse operation as "demixing" or "decomposing". Having a sequence and trying to decompose it into one periodic and one residual part (whether periodic as well or not) reminds me of "Fourieranalysis". And that keyword should provide more ideas for some meaningful terming (and proceeding) in your problem. Perhaps you can find related concepts in the OEIS-community which are involved in analysis, composition and decomposition of sequences (for instance their "JIS"-journal is freely available and possibly has something in it)
$endgroup$
– Gottfried Helms
Jan 31 at 9:55
add a comment |
$begingroup$
I'd try the terms "mixing" or "composing" of sequences, and then the inverse operation as "demixing" or "decomposing". Having a sequence and trying to decompose it into one periodic and one residual part (whether periodic as well or not) reminds me of "Fourieranalysis". And that keyword should provide more ideas for some meaningful terming (and proceeding) in your problem. Perhaps you can find related concepts in the OEIS-community which are involved in analysis, composition and decomposition of sequences (for instance their "JIS"-journal is freely available and possibly has something in it)
$endgroup$
– Gottfried Helms
Jan 31 at 9:55
$begingroup$
I'd try the terms "mixing" or "composing" of sequences, and then the inverse operation as "demixing" or "decomposing". Having a sequence and trying to decompose it into one periodic and one residual part (whether periodic as well or not) reminds me of "Fourieranalysis". And that keyword should provide more ideas for some meaningful terming (and proceeding) in your problem. Perhaps you can find related concepts in the OEIS-community which are involved in analysis, composition and decomposition of sequences (for instance their "JIS"-journal is freely available and possibly has something in it)
$endgroup$
– Gottfried Helms
Jan 31 at 9:55
$begingroup$
I'd try the terms "mixing" or "composing" of sequences, and then the inverse operation as "demixing" or "decomposing". Having a sequence and trying to decompose it into one periodic and one residual part (whether periodic as well or not) reminds me of "Fourieranalysis". And that keyword should provide more ideas for some meaningful terming (and proceeding) in your problem. Perhaps you can find related concepts in the OEIS-community which are involved in analysis, composition and decomposition of sequences (for instance their "JIS"-journal is freely available and possibly has something in it)
$endgroup$
– Gottfried Helms
Jan 31 at 9:55
add a comment |
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$begingroup$
I'd try the terms "mixing" or "composing" of sequences, and then the inverse operation as "demixing" or "decomposing". Having a sequence and trying to decompose it into one periodic and one residual part (whether periodic as well or not) reminds me of "Fourieranalysis". And that keyword should provide more ideas for some meaningful terming (and proceeding) in your problem. Perhaps you can find related concepts in the OEIS-community which are involved in analysis, composition and decomposition of sequences (for instance their "JIS"-journal is freely available and possibly has something in it)
$endgroup$
– Gottfried Helms
Jan 31 at 9:55