Processes which propagate faster than exponentially
I.N. Galidakis
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,
--
Ioannis
jbriggs444
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.
Sanny
Throw an Object in a Black hole. It will be faster than a Exponential.
Bye
Sanny
Know the strangest things from computers mouth.
You can just chat with the computer on physics.
Robert
i think some pulse lasers' amplitudes do.
any process that has a combinatorial component might well do. genetics?
Androcles
"I.N. Galidakis" <morp...@olympus.mons> wrote in message
news:1263572066.510809@athprx04...
If you mean faster than y = exp(-t) then yes, y = A.exp(-t) is faster for A
> 1.
Or to put it another way, the isotope carbon 14 has a shorter half life
( 5,730 � 40 years) than the isotope uranium 238, (4.46 billion years).
The human population doubles every 33 years but it once doubled every
100 years. This is due to more people not suffering infant mortality from
disease (countered by modern medicine) and living longer, yet still
reproducing at the same rate (not countered in the third world).
Obviously if you have more births than deaths then you have population
growth that is exponential . If you have less births than deaths then you
have population decline. By upsetting the balance with medicine you
change from exponential growth to super-exponential growth.
The slope of the curve gets steeper for exponential growth (fixed birth
and death rates) and steeper yet for super-exponential growth.
In computer models of locust swarms the exponential growth of
the population always results in them eating all there is and then
starving and dying. Human beings were headed that way naturally
but have hastened their own demise by their use of medicine to
prolong their lives. The land we live on is finite.
Robert
> "I.N. Galidakis" <morph...@olympus.mons> wrote in message
faster than exponential means faster than exponential with *any*
growth constant. but thanks for the malthusian spam
Androcles
"Robert" <robertmar...@gmail.com> wrote in message
news:2493da63-435b-4071...@p8g2000yqb.googlegroups.com...
==========================================
Faster than exponential means faster than exponential.
(period)----->------>------->--------->------->------------^
Thanks for your worthless caveat; now fuck off, wanker.
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Robert
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?
Rod
"I.N. Galidakis" <morp...@olympus.mons> wrote in message
news:1263572066.510809@athprx04...
There seems to be a few examples of double exponential i.e. exp(exp(x))
see
http://en.wikipedia.org/wiki/Double_exponential_function#Physics
factorials or gamma function are also faster then exp
I.N. Galidakis
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
I.N. Galidakis
Thanks. That's one example. So the bound now moves to exp(exp(x)).
--
Ioannis
Robert
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.
Rod
"I.N. Galidakis" <morp...@olympus.mons> wrote in message
a^x = (exp(ln(a)))^x = exp(ln(a) x) = exp(b.x)
so this is not faster than exp
things like exp(a.x^2) are faster and exp(exp(x)) is really fast.
Look up tetration for something super fast but I don't know any physical
process related to it.
I.N. Galidakis
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
Robert
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.
I.N. Galidakis
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
eric gisse
Well power law growth/decay is pretty popular for large scale systems, which
occasionally satisfies your specific criteria. However, a^x is still
exponential.
Robert
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.
Ray Vickson
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
Marvin the Martian
Google "Taylor series". Any real function can be approximated by a series
of exponentials. Thus, your question makes no sense.
Phil Carmody
Anything which goes from 0 to anything strictly positive?
Phil
--
Any true emperor never needs to wear clothes. -- Devany on r.a.s.f1
Dann Corbit
thefatphi...@yahoo.co.uk says...
>
> "I.N. Galidakis" <morp...@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/
I.N. Galidakis
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
I.N. Galidakis
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
Frisbieinstein
An infinite sum of exponentials can increase superexponentially. A
Taylor series is infinite.
Frisbieinstein
Thiotimoline is a fictitious chemical compound conceived by science
fiction author Isaac Asimov.
rabid_fan
> 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?
>
The question concerns natural processes. Thus, we must ask:
How does a natural process produce an exponential rate?
An exponential rate arises according to the model where
the rate is proportional to the amount of substance:
dx/dt = k * x
If we assume that x = 1 at t = 0, the solution becomes:
x = exp(k*t)
So to find processes that would be faster than exponential
(if they exist) we can create models where the rate is
proportional to quantities greater than the linear amount,
i.e.:
dx/dt = k * x^2, with x(0)=1
The solution is x = 1/(1-k*t) which increases faster than
exponential.
dx/dt = k * x^3, with x(0)=1
The solution is x = 1/sqrt(1-2k*t) which increases faster than
exponential.
dx/dt = k * exp(x), with x(0)=1
The solution is x = ln(1/e-k*t) which increases faster than
exponential.
We can easily create these models that all lead to a faster
rate than the exponential. Whether or not they actually exist
in the natural world is another story.
John Park
Missing minus sign, I think.
>
> We can easily create these models that all lead to a faster
> rate than the exponential. Whether or not they actually exist
> in the natural world is another story.
>
Second-order rate equations are commmon enough in chemistry and
third-order rates are not unknown. If only one reactant is involved or the
initial concentrations of all reactants are equal, the simple expressions
you give for these cases follow automatically (to appropriate levels of
approximation as always).
--John Park
Tom Potter
"Rod" <rodrod...@hotmail.com> wrote in message
news:FX24n.4612$fc3....@newsfe26.ams2...
This is a very interesting thread
as it deals with "becoming", "being", and "going".
Although it is easy to imagine how populations can increase and decrease,
and even become extinct, it is more difficult to conceptualize how a
population
goes from zero to something.
( The shattering of something bigger that came before?)
I suggest that the Weibull Distribution Function
models reality much better than the exponential,
the double exponential,
or any other function.
Many Power Point files that explain the Weibull Distribution Function
and it's application to real world situations
such as life, death, infant mortality, wear out, reliability, health care,
etc.
can be downloaded from the following web site.
http://juhuj.com/open-file-ppt-convert-ppt-download-weibull.htm
The following web page discusses how the exponential function
is associated with entropy and the growth and decay of populations.
http://www.tompotter.us/entropy.html
Excerpt
=====
"time, in the sense of aging, is related to entropy, and entropy, in turn,
is related to populations and the number of ways a population can combine.
The direction of time is determined by comparing the exponential decays of
populations to one another, and the events used to define periods and
intervals are ordered along populations decay and growth curves. The
fundamental population units of nature seems to be standing waves, and
macroscopic populations such as atomic particles, atoms, molecules, body
cells, rats, people and stars seem to be aggregates of standing waves. A
population may consist of particles, atoms, molecules, rats, etc. When
factors exist that affect a population, the population changes. Factors can
make populations grow or diminish in number to zero, and beyond!"
--
Tom Potter
--
http://tdp1001.wiki.zoho.com/
http://tdp1001.wordpress.com/
http://tdp1001.spaces.live.com
http://www.tompotter.us/misc.html
http://webspace.webring.com/people/st/tdp1001
http://notsocrazyideas.blogspot.com
-----------------------------------------------
Andrew Usher
> 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?
First, this is not really a mathematical question. Of course equations
may be defined that grow arbitrarily rapidly.
Second, any exponential growth process in the real world can only
maintain such growth for a short time, and this would apply even more
to super-exponential processes.
Third, if one requires only super-exponential growth _in time_
(there's really no such thing as even exponential growth in space),
there's an obvious example: any exothermic chemical chain reaction.
Since the growth would be exponential if temperature were constant,
but temperature is also increasing rapidly, the progress of the whole
process is faster than exponential (until the concentration of
reactive particles has reached its peak).
Andrew Usher
Tom Potter
observation that the Weibull function
is far superior to the exponential function in modeling reality,
I am starting a new thread in order to make my point.
An important parameter of the Weibull distributions is the SHAPE PARAMETER.
Weibull distributions with a SHAPE PARAMETER < 1
model INFANT MORTALITIES.
Weibull distributions with SHAPE PARAMETER = 1
model RANDOM FAILURES.
Weibull distributions with SHAPE PARAMETER > 1
model the WEAROUT of a system.
These comprise the three sections of the so-called "bathtub curve."
If you plot a set of points for any failure,
or for that matter any binary condition,
you can use the Weibull distribution to
determine if the data represents a set of things
( Objects, effects, or binary conditions)
that are new, mature, or dying.
As contiguous things form an environment,
and will increase in a population if the environment is nurturing,
and will decrease in a population if the environment is non-nurturing,
the question becomes, how do things come into being.
( Go from a zero number of things to N number of things.)
I suggest that things come into being as a result of the random failures,
as a random failure is a non-linear event unlike a thing's
normal interaction with its' normal environment.
Note that if A and B combine in a non-linear system,
a new term A * B arises.
Example:
(A + B )^2 = A^2 + 2AB + B^2
1. RANDOM FAILURES create new life forms.
2. The new life forms become part of the existing environment.
3. Life leads to death.
4. Death provides substance for preceding life.
5. Reality is a parallel-series combination of BATHTUB CURVES.
Note that the Weibull Distribution
applies to the birth and death of stars,
just as it does to the birth and death of living things,
and the failure of non-living things,
and that by sampling a FEW events,
the Weibull Distribution allows comprehensive, ACCURATE
projections to be made about sets of living and non-living things,
and it provides the tool needed to isolate, compare, and control
BECOMING, BEING and GOING.
Uncle Al
>
> Considering that no one commented on my
> observation that the Weibull function
> is far superior to the exponential function in modeling reality,
Dog turd, cat turd, Potty turd - all turds.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/qz4.htm
Tom Potter
"Uncle Al" <Uncl...@hate.spam.net> wrote in message
news:4B55D154...@hate.spam.net...
> Tom Potter wrote:
>>
>> Considering that no one commented on my
>> observation that the Weibull function
>> is far superior to the exponential function in modeling reality,
> [snip crap]
>
> Dog turd, cat turd, Potty turd - all turds.
>
> --
> Uncle Al
To a hungry dog
everything looks like a turd.
I trust that in order to prove his point,
that my pal Uncle Al will explain what function better describes,
birth, infant mortality, random failures, and wear out
( The natural life curve of things, animal, vegetable, machines, stars,
etc.)
better than the Weibull function.
jbriggs444
Indeed. I had hoped that the references to repeated resublimation and
endochronicity would make the tongue-in-cheek nature of the remark
clear, even to those fortunate few who can still look forward to
reading the article below for the first time.
A. Asimov: "The Endochronic Properties of Resublimated Thiotimoline",
The Journal of Astounding Science Fiction, March, 1948.
jbriggs444
A Taylor series is not a sum of exponentials. It is a sum of
polynomials.
As I recall, a not-identically zero, real-valued function may have a
derivitive of zero to all degrees. Its Taylor series then has not one
damned thing to do with its value except at zero.
jbriggs444
Suppose you have 10 miles of oil in the Alaskan pipeline bearing down
on a closed valve at a rate of one meter per second buffered by a 2
meter air bubble at 1 atmosphere directly adjacent to the valve.
(Some idiot closed the valve abruptly)
Assume that all relevant safety devices have been disabled and no
other air bubbles exist.
I think that the rate of increase of air pressure with respect to time
and with respect to distance are both super-exponential (or are best
modelled by a curve that is super-exponential) right up to the point
where the pipe breaks.
Andrew Usher
> Suppose you have 10 miles of oil in the Alaskan pipeline bearing down
> on a closed valve at a rate of one meter per second buffered by a 2
> meter air bubble at 1 atmosphere directly adjacent to the valve.
> (Some idiot closed the valve abruptly)
>
> Assume that all relevant safety devices have been disabled and no
> other air bubbles exist.
>
> I think that the rate of increase of air pressure with respect to time
> and with respect to distance are both super-exponential (or are best
> modelled by a curve that is super-exponential) right up to the point
> where the pipe breaks.
Yes, this is another example. One needs only to formulate the diff.eq.
to see that this is super-exponential in the ideal case.
If the oil has infinite motivating force and the pipe infinite
strength, we have P = 1/(t'-t) where t' is when the bubble is crushed.
This is equivalent to
P' = P^2
which clearly grows faster than the exponential P' = P .
Andrew Usher