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January 14th, 2018, 08:49 PM  #1 
Newbie Joined: Jan 2018 From: somewhere Posts: 14 Thanks: 2 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 nonproof 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. 
January 14th, 2018, 11:00 PM  #2  
Senior Member Joined: Oct 2009 Posts: 436 Thanks: 147  Quote:
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 StoneWeierstrass theorem. The StoneWeierstrass theorem deals with algebra's of continuous functions. Any differential fork is clearly an algebra, so you should probably try to investigate whether the StoneWeierstrass 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.  
January 15th, 2018, 08:05 PM  #3  
Newbie Joined: Jan 2018 From: somewhere Posts: 14 Thanks: 2 Math Focus: Algebraic Number Theory / Differential Fork Theory  Quote:
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 macroknowledge rather than microknowledge. 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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