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January 14th, 2018, 08:49 PM   #1
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Math Focus: Algebraic Number Theory / Differential Fork Theory
I developed a method of integration and want some thoughts

Over the past week or so I have been writing a somewhat large paper regarding an integration method I thought of. However, in my last non-proof related section I came up with the idea of a differential fork and I am curious if anything regarding that has ever been done before or if there is anything nontrivial that any of you can come up with regarding the periodic differential mentioned in there. I've been trying to come up with an actual expression for it (like with the implied derivative) but I cannot come up with what kind of limit might represent that.

Piecewise Constant Functions in Differential and Functional Equations

I think the title is a bit off kilter now, but oh well. It went a different direction than I originally planned.
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January 14th, 2018, 11:00 PM   #2
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Originally Posted by Typhon View Post
Over the past week or so I have been writing a somewhat large paper regarding an integration method I thought of. However, in my last non-proof related section I came up with the idea of a differential fork and I am curious if anything regarding that has ever been done before or if there is anything nontrivial that any of you can come up with regarding the periodic differential mentioned in there. I've been trying to come up with an actual expression for it (like with the implied derivative) but I cannot come up with what kind of limit might represent that.

Piecewise Constant Functions in Differential and Functional Equations

I think the title is a bit off kilter now, but oh well. It went a different direction than I originally planned.
Hey,
If you want, I can proofread your paper, feel free to contact me if you want this!

Anyway, about differential forks. The first thing I thought of was the Stone-Weierstrass theorem. The Stone-Weierstrass theorem deals with algebra's of continuous functions. Any differential fork is clearly an algebra, so you should probably try to investigate whether the Stone-Weierstrass theorem is something relevant to your theory.

In particular, you should investigate whether any (nontrivial) differential fork separates points. If so, the theorem applies and shows any such differential fork is dense in the continuous functions.
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January 15th, 2018, 08:05 PM   #3
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Math Focus: Algebraic Number Theory / Differential Fork Theory
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Originally Posted by Micrm@ss View Post
Hey,
If you want, I can proofread your paper, feel free to contact me if you want this!

Anyway, about differential forks. The first thing I thought of was the Stone-Weierstrass theorem. The Stone-Weierstrass theorem deals with algebra's of continuous functions. Any differential fork is clearly an algebra, so you should probably try to investigate whether the Stone-Weierstrass theorem is something relevant to your theory.

In particular, you should investigate whether any (nontrivial) differential fork separates points. If so, the theorem applies and shows any such differential fork is dense in the continuous functions.
Thank you for the offer! I will keep it in mind down the road.

I had not considered it as an algebra. I will admit that I know of the major numerical algebrae such as groups, fields, rings, etc. however Ive never seen the term algebra refer to a particular object or structure before.

My thinking was that when you do differential equations and get preliminary constants of integration or antidifferentiation such as C1, C2, etc. you can pretty much always replace them inside a function with themselves as composing constant functions together yields constant functions. Hence, this is why I was thinking a differential fork might generalize and formalize the idea of differentiating a function whilst holding other functions constant in a multivariate sense.

I will look into that. Any thoughts though on the periodic associated derivative? It seems to require macro-knowledge rather than micro-knowledge. I wonder if reordering the domain and the range a la lebesgue integral might be a preliminary step. I'll look into it.

Last edited by Typhon; January 15th, 2018 at 08:12 PM.
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