Processes which propagate faster than exponentially
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From: Dann Corbit on 15 Jan 2010 19:43 In article <87ockv5b8a.fsf(a)kilospaz.fatphil.org>, thefatphil_demunged(a)yahoo.co.uk says... > > "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? E.g., I had zero apples now I have one apple. The rate of increase was infinite. The Dirac delta function was invented for a reason (it pops up in all sorts of interesting places), and it has infinite slope. I have a jar with ten marbles of one color in it (say 'red'). If I pull out all the marbles how many different color sequences can I get: 1. Now, I add ten new marbles of a different color (for example, 'blue'). If I pull out all the marbles one at a time after shaking how many different color sequences can I get: Essentially, we have 2^20 in binary different color combinations. Now, I add ten new marbles of a different color (for example, 'yellow'). If I pull out all the marbles how many different color sequences can I get: Essentially, we have 3^30 in ternary different color combinations. If we continue adding new marbles of different colors in this manner, the number of possible color combinations we get grows faster than exponential, because both the base and the exponent are increasing. One could argue that almost every important thing (compute power, total world information volume, etc.) grows at a superexponential rate {faster than exponential}. See Ray Kurzweil's site: http://singularity.com/
From: I.N. Galidakis on 15 Jan 2010 20:34 Dann Corbit wrote: > In article <87ockv5b8a.fsf(a)kilospaz.fatphil.org>, > thefatphil_demunged(a)yahoo.co.uk says... >> >> "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? > > E.g., I had zero apples now I have one apple. The rate of increase was > infinite. > > The Dirac delta function was invented for a reason (it pops up in all > sorts of interesting places), and it has infinite slope. > > I have a jar with ten marbles of one color in it (say 'red'). > If I pull out all the marbles how many different color sequences can I > get: > 1. > > Now, I add ten new marbles of a different color (for example, 'blue'). > If I pull out all the marbles one at a time after shaking how many > different color sequences can I get: > Essentially, we have 2^20 in binary different color combinations. > > Now, I add ten new marbles of a different color (for example, 'yellow'). > If I pull out all the marbles how many different color sequences can I > get: > Essentially, we have 3^30 in ternary different color combinations. > > If we continue adding new marbles of different colors in this manner, > the number of possible color combinations we get grows faster than > exponential, because both the base and the exponent are increasing. Isn't that the sequence a(n)=n^{n+10}? Maple reports: >f:=x->x^(x+10); >g:=x->exp(exp(x)); then > limit(f(x)/g(x),x=infinity); 0 so it looks to me like SUB-double-exponential. > One could argue that almost every important thing (compute power, total > world information volume, etc.) grows at a superexponential rate {faster > than exponential}. See Ray Kurzweil's site: > http://singularity.com/ I like his theory :-) -- Ioannis
From: I.N. Galidakis on 15 Jan 2010 21:21 I.N. Galidakis wrote: > Dann Corbit wrote: >> In article <87ockv5b8a.fsf(a)kilospaz.fatphil.org>, >> thefatphil_demunged(a)yahoo.co.uk says... >>> >>> "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? >> >> E.g., I had zero apples now I have one apple. The rate of increase was >> infinite. >> >> The Dirac delta function was invented for a reason (it pops up in all >> sorts of interesting places), and it has infinite slope. >> >> I have a jar with ten marbles of one color in it (say 'red'). >> If I pull out all the marbles how many different color sequences can I >> get: >> 1. >> >> Now, I add ten new marbles of a different color (for example, 'blue'). >> If I pull out all the marbles one at a time after shaking how many >> different color sequences can I get: >> Essentially, we have 2^20 in binary different color combinations. >> >> Now, I add ten new marbles of a different color (for example, 'yellow'). >> If I pull out all the marbles how many different color sequences can I >> get: >> Essentially, we have 3^30 in ternary different color combinations. >> >> If we continue adding new marbles of different colors in this manner, >> the number of possible color combinations we get grows faster than >> exponential, because both the base and the exponent are increasing. > > Isn't that the sequence a(n)=n^{n+10}? > > Maple reports: >> f:=x->x^(x+10); >> g:=x->exp(exp(x)); > > then >> limit(f(x)/g(x),x=infinity); > > 0 > > so it looks to me like SUB-double-exponential. Sorry, I just had a bout with severe stupidity. The sequence looks like: a(n)=n^{10*n}, (NOT n^{n+10}) for which Maple gives: > f:=x->x^(10*x); > g:=x->exp(exp(x)); limit(g(x)/f(x),x=infinity); 0 so looks like it's actually SUPER-double-exponential. -- Ioannis
From: Frisbieinstein on 15 Jan 2010 22:49 On Jan 16, 4:42 am, Marvin the Martian <mar...(a)ontomars.org> wrote: > 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. An infinite sum of exponentials can increase superexponentially. A Taylor series is infinite.
From: Frisbieinstein on 15 Jan 2010 22:51
On Jan 16, 12:54 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > On Jan 15, 11:14 am, "I.N. Galidakis" <morph...(a)olympus.mons> 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. > > Some processes are too fast to even have a decent way to categorize > the rate. > > Take, for instance, the chemical core of a nuclear device. The pieces > are set off simultaneously so that the reaction need not progress from > a single point of ignition. The limit on the reaction rate is the > number of detonators used and the precision with which they can be set > off. Rather than being a log, a cube root, a square root or linear in > the reactant size, the reaction time can be held constant. > > That's without considering Thiotimoline, a substance which, when > purified by repeated resublimation has a solubility reaction rate that > goes endochronic. Thiotimoline is a fictitious chemical compound conceived by science fiction author Isaac Asimov. |