Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
On the binary partition and taking derivatives
#1
Recently we talked about the binary partition function

f(n) - f(n-1) = f(n/2).

And Jay's asymptotic

J ' (x) = J(x/2).

This leads to a general question :
" how to take a derivative of an unsolved equation ? "

I will clarify with the binary partion function as example :

a >= h

where h is the positive infinitesimal.
( I will use 0 for h later , use lim interpretation )


(f_a(x) - f_a(x-a)) / a = f_a(x/2)

The questions are , without solving for the f_a(x) first with respect to x ( asymptoticly ) ,

df/da f_a(x) = ??

f_a(x)/f_(a-1)(x) = ??

df/da f_a(x)/f_(a-1)(x) = ??

and similar ones.

A good techniques for such problems should exist.

Notice Jay conjectured

f_1(x)/f_0(x) ~ C

And with good approximations of both f_0,f_1 that should be easy to prove.

But to show it directly is the goal.


Hence the reason d'être of this thread and its questions.

It is my philosophy of math , that math Always tries to shortcut everything.

Multiply and divide are shortcuts to addition.
Finding shortcuts to addition and multiplication is a motivation for linear algebra and dynamical systems.

Shortcuts to matrix powers led to diagonalization and Jordan forms etc etc.
Asymptotic Shortcuts to counting primes led to PNT.

Series acceleration is another example.

I think you get the idea.
( not having a known " shortcut " ( for computation ) leads to difficult problems in math , for instance collatz conjecture.
I like to count the difficulty of a math problem in terms of unknown shortcuts vs known shortcuts related to the question )

regards

tommy1729
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Isomorphism of newtonian calculus rules for Non-Newtonian (anti)derivatives of hypers Micah 4 869 03/02/2019, 08:23 PM
Last Post: Micah
  1st iterated derivatives and the tetration of 0 Xorter 0 1,057 05/12/2018, 12:34 PM
Last Post: Xorter
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 15,153 08/22/2016, 12:19 AM
Last Post: JmsNxn
  An intresting equation ? Taking squares by equation. tommy1729 0 1,483 05/08/2015, 11:37 PM
Last Post: tommy1729
  Binary partition at oo ? tommy1729 2 3,104 10/07/2014, 07:22 PM
Last Post: tommy1729
  "circular" operators, "circular" derivatives, and "circular" tetration. JmsNxn 2 5,935 06/24/2011, 07:21 PM
Last Post: JmsNxn
  Closed-form derivatives andydude 7 7,419 09/03/2009, 04:02 AM
Last Post: andydude
  q concerning derivatives Gottfried 0 2,154 03/19/2008, 08:28 AM
Last Post: Gottfried



Users browsing this thread: 1 Guest(s)