Processes which propagate faster than exponentially
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From: eric gisse on 15 Jan 2010 15:29 I.N. Galidakis wrote: > Robert wrote: > >> Well, for anyone that hasn't kill filed me, the "worthless caveat" is >> essential to the OP's question. >> >> Would >> >> as x->oo, for all t, f(x)/e^xt -> oo >> >> be an adequate definition for a function with faster than exponential >> growth? > > Yes. Sorry for the (created) confusion. > > As far as I am concerned, I mean anything like: f(x) ~ a^x, with a > e. > > Thanks again to all the responders. Well power law growth/decay is pretty popular for large scale systems, which occasionally satisfies your specific criteria. However, a^x is still exponential.
From: Robert on 15 Jan 2010 15:32 On 15 Jan, 19:48, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > Robert wrote: > > On 15 Jan, 19:27, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > >> Robert wrote: > >>> On 15 Jan, 19:09, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > >>>> Robert wrote: > >>>>> Well, for anyone that hasn't kill filed me, the "worthless caveat" is > >>>>> essential to the OP's question. > > >>>>> Would > > >>>>> as x->oo, for all t, f(x)/e^xt -> oo > > >>>>> be an adequate definition for a function with faster than exponential > >>>>> growth? > > >>>> Yes. Sorry for the (created) confusion. > > >>>> As far as I am concerned, I mean anything like: f(x) ~ a^x, with a > e. > > >>>> Thanks again to all the responders. > >>>> -- > >>>> Ioannis > > >>> in that case androcles is right and there are millions of examples. so > >>> the question is pretty trivial. > > >>> a^x, even with a > e, *is* exponential growth. > > >>> a^x = e^(x ln(a)). in other words all the a not being e does is change > >>> the growth constant, but it's still exponential. > > >> Sorry, typo on my part. Sentence should read: > > >>> As far as I am concerned, I mean anything greater than: f(x) ~ a^x, with a > > > >> e. > > >> But I am interested in these cases as well. Any examples with a > e? > >> -- > >> Ioannis > > > well. your question then doesn't have much physical meaning. as any > > real world exponential growth has a growth factor k (and an initial > > value A) > > > A e^(k t) > > > this scales the process to whatever units you are using. because of > > the scaling anything with a > e could, with a different scale, be > > written as a = e or even a < e. > > > they are essentially the same. > > Excellent. So is there then general agreement that there are no *naturally > occuring* (excluding Rod's examples on Wiki which I am not sure they qualify as > *natural*) processes in nature which propagate faster than exponential? > -- > Ioannis not agreement from me. i'd say everything that actually happens is *natural*. so if a pulsed lasers amplitude can be modelled with a faster than exponential function then that is a good candidate, so long as the model is sound. but i do wonder if there are any combinatorial processes in nature that are faster. seeing as, as someone above points out, the factorial is faster than exponential.
From: Ray Vickson on 15 Jan 2010 15:40 On Jan 15, 11:48 am, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > Robert wrote: > > On 15 Jan, 19:27, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > >> Robert wrote: > >>> On 15 Jan, 19:09, "I.N. Galidakis" <morph...(a)olympus.mons> wrote: > >>>> Robert wrote: > >>>>> Well, for anyone that hasn't kill filed me, the "worthless caveat" is > >>>>> essential to the OP's question. > > >>>>> Would > > >>>>> as x->oo, for all t, f(x)/e^xt -> oo > > >>>>> be an adequate definition for a function with faster than exponential > >>>>> growth? > > >>>> Yes. Sorry for the (created) confusion. > > >>>> As far as I am concerned, I mean anything like: f(x) ~ a^x, with a > e. > > >>>> Thanks again to all the responders. > >>>> -- > >>>> Ioannis > > >>> in that case androcles is right and there are millions of examples. so > >>> the question is pretty trivial. > > >>> a^x, even with a > e, *is* exponential growth. > > >>> a^x = e^(x ln(a)). in other words all the a not being e does is change > >>> the growth constant, but it's still exponential. > > >> Sorry, typo on my part. Sentence should read: > > >>> As far as I am concerned, I mean anything greater than: f(x) ~ a^x, with a > > > >> e. > > >> But I am interested in these cases as well. Any examples with a > e? > >> -- > >> Ioannis > > > well. your question then doesn't have much physical meaning. as any > > real world exponential growth has a growth factor k (and an initial > > value A) > > > A e^(k t) > > > this scales the process to whatever units you are using. because of > > the scaling anything with a > e could, with a different scale, be > > written as a = e or even a < e. > > > they are essentially the same. > > Excellent. So is there then general agreement that there are no *naturally > occuring* (excluding Rod's examples on Wiki which I am not sure they qualify as > *natural*) processes in nature which propagate faster than exponential? > -- > Ioannis Just one more point of clarification: what about a function like f(t) = t^n * exp(t) for n > 1? This does not have exponential growth, but is bounded above by a function with exponential growth for large enough t. We have exp(k*t) > t^n * exp(t) if t/ln(t) > n/(k-1). In _some_ subjects (such as computational complexity theory) a function like f(t) would be considered as having "exponential growth", although it might more accurately be described as having "at most exponential growth". So, you want to exclude such functions as well when you want faster-than-exponential. R.G. Vickson
From: Marvin the Martian on 15 Jan 2010 15:42 On Fri, 15 Jan 2010 18:14:25 +0200, I.N. Galidakis wrote: > Apologies for the crosspost, but this is related to many areas. Is > anyone aware of any physical/chemical/nuclear processes which propagate > at rates faster than exponential? > > From my search so far, it appears that the fastest processes available, > like cancer and viruses in biology, and nuclear explosions and supernova > explosions in physics all propagate at most exponentially. > > Many thanks, Google "Taylor series". Any real function can be approximated by a series of exponentials. Thus, your question makes no sense.
From: Phil Carmody on 15 Jan 2010 16:06
"I.N. Galidakis" <morpheus(a)olympus.mons> writes: > Apologies for the crosspost, but this is related to many areas. Is anyone aware > of any physical/chemical/nuclear processes which propagate at rates faster than > exponential? > > From my search so far, it appears that the fastest processes available, like > cancer and viruses in biology, and nuclear explosions and supernova explosions > in physics all propagate at most exponentially. Anything which goes from 0 to anything strictly positive? Phil -- Any true emperor never needs to wear clothes. -- Devany on r.a.s.f1 |