• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 exponential distributivity quickfur Junior Fellow Posts: 22 Threads: 1 Joined: Feb 2008 02/22/2008, 06:51 PM bo198214 Wrote:[...] Another idea would then be to investigate the distribution law for exponentiation, i.e. whether there is an operation # such that (a^b)#c=(a#c)^(b#c). As a side node I give a proof that there are only trivial operations # satisfying this property. [...] Now to the proof, if there would be such an operation then we can consider the function f(x)=x#c (fixed c) which satisfies f(a^b)=f(a)^f(b). [...] Summarizing Proposition: Any operation # defined on the positive real numbers such that (a^b)#c=(a#c)^(b#c) (for each a,b,c) and such that f(x)=x#c is continuous (for each c), is either given by x#c=x or by x#c=1 for all x,c.Hmm, interesting. I was going to ask what happens if we consider f(x)=c#x instead, but it doesn't change the conclusion because we assume the same operator #. So we established that a higher operation cannot distribute over exponentiation without becoming trivial. I wonder what is the root cause of this... maybe it has something to do with the non-associativity of exponentiation? « Next Oldest | Next Newest »

 Messages In This Thread exponential distributivity - by bo198214 - 02/22/2008, 12:36 PM RE: exponential distributivity - by quickfur - 02/22/2008, 06:51 PM RE: exponential distributivity - by bo198214 - 02/22/2008, 07:17 PM RE: exponential distributivity - by quickfur - 02/23/2008, 12:29 AM RE: exponential distributivity - by JmsNxn - 09/22/2011, 03:27 PM

 Possibly Related Threads... Thread Author Replies Views Last Post Math overflow question on fractional exponential iterations sheldonison 4 4,095 04/01/2018, 03:09 AM Last Post: JmsNxn An exponential "times" table MikeSmith 0 1,904 01/31/2014, 08:05 PM Last Post: MikeSmith exponential baby Mandelbrots? sheldonison 0 2,109 05/08/2012, 06:59 PM Last Post: sheldonison help with a distributivity law JmsNxn 3 3,907 09/22/2011, 03:32 PM Last Post: JmsNxn Base 'Enigma' iterative exponential, tetrational and pentational Cherrina_Pixie 4 10,007 07/02/2011, 07:13 AM Last Post: bo198214 Two exponential integrals Augustrush 2 4,534 11/10/2010, 06:44 PM Last Post: Augustrush HELP NEEDED: Exponential Factorial and Tetrations rsgerard 5 8,578 11/13/2009, 02:27 AM Last Post: rsgerard Exponential factorial mike3 3 6,125 10/07/2009, 02:04 AM Last Post: andydude Additional super exponential condition bo198214 4 5,556 10/21/2008, 03:40 PM Last Post: martin exponential polynomial interpolation Gottfried 3 7,193 07/16/2008, 10:32 PM Last Post: andydude

Users browsing this thread: 1 Guest(s)