Processes which propagate faster than exponentially

Robert wrote:
On 15 Jan, 19:09, "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.
--
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

On 15 Jan, 19:27, "I.N. Galidakis" wrote:
Robert wrote:
On 15 Jan, 19:09, "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.
--
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.

Robert wrote:
On 15 Jan, 19:27, "I.N. Galidakis" wrote:
Robert wrote:
On 15 Jan, 19:09, "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.
--
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

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.

On 15 Jan, 19:48, "I.N. Galidakis" wrote:
Robert wrote:
On 15 Jan, 19:27, "I.N. Galidakis" wrote:
Robert wrote:
On 15 Jan, 19:09, "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.
--
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.

On Jan 15, 11:48*am, "I.N. Galidakis" wrote:
Robert wrote:
On 15 Jan, 19:27, "I.N. Galidakis" wrote:
Robert wrote:
On 15 Jan, 19:09, "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.
--
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

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.

"I.N. Galidakis" 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

In article ,
says...

"I.N. Galidakis" 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/

Dann Corbit wrote:
In article ,
says...

"I.N. Galidakis" 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