Low carb diets

"Proton Soup" wrote in message
...
On Sun, 21 Dec 2003 06:29:45 GMT, "OmegaZero2003"
wrote:

BTW, as a PS to my other answer post, here are some linear systems.

- those characterizable by linear algebra. there are lots of these!
- Hamiltonian oscillators and like systems. (the direction field
specifically)
- continuous-time systems like electrical networks, many mechanical
systems

Only simple RLC electrical networks fall into this category. And even
then, it's just a theoretical assumption over the useful operating
range. Too much current or voltage or flux will flux up your circuit.
Linear electrical networks only exist on paper.

My original point to the OP on the topic was a retort to the statement that
*all* systems are nonlinear.

That is not true.


- any discrete system with a transfer function whose input, response and
output functions depend on one variable
- any systems preserving homogeneity (output proportional to input) and
superposition (a way of combining linear functions such that the result
is a
linear function)

Even non-deterministic systems can be modeled using statistics for linear
dynamics.

But the main point to not be belabored is that there are linear systems
on
nature and manmade (1)

Most systems *are* non-linear but some of those are characterizable using
linear methods to some degree of accuracy; you did make something like
this
point.

(1) Schwarz, Ralph J. and Friedland, Bernard, 1965, Linear Systems, New
York: McGraw-Hill Book Company, 521 pp.

---
Proton Soup

"If I drink water I will have to go to the bathroom and
how can I use the bathroom when my people are in bondage?"
-Saddam Hussein

"Wayne S. Hill" wrote in message
...
Proton Soup wrote:

"OmegaZero2003" wrote:

BTW, as a PS to my other answer post, here are some linear
systems.

- those characterizable by linear algebra. there are lots
of these! - Hamiltonian oscillators and like systems. (the
direction field specifically) - continuous-time systems like
electrical networks, many mechanical systems

Only simple RLC electrical networks

and their analogs in other domains

fall into this category.
And even then, it's just a theoretical assumption over the
useful operating range. Too much current or voltage or flux
will flux up your circuit. Linear electrical networks only
exist on paper.

Exactly. They're (essentially) linear in a linear range.

Welcome to Tautology 101!

Again, I have provided several examples of systems that are chraterizable as
linear. Proton pointedd out one spcific one above.

I provided several categories of linear systems.

My original retort to your original sttement that *all* systems are
non-linear stands: it is not true that all systems are non-linear.

It is very easy to show how a hypothesis that begins: " All..." is false.
I need only provide one counter example.



--
-Wayne

On Sun, 21 Dec 2003 20:26:26 GMT, "OmegaZero2003"
wrote:

"Proton Soup" wrote in message
.. .
On Sun, 21 Dec 2003 06:29:45 GMT, "OmegaZero2003"
wrote:

BTW, as a PS to my other answer post, here are some linear systems.

- those characterizable by linear algebra. there are lots of these!
- Hamiltonian oscillators and like systems. (the direction field
specifically)
- continuous-time systems like electrical networks, many mechanical
systems

Only simple RLC electrical networks fall into this category. And even
then, it's just a theoretical assumption over the useful operating
range. Too much current or voltage or flux will flux up your circuit.
Linear electrical networks only exist on paper.

My original point to the OP on the topic was a retort to the statement that
*all* systems are nonlinear.

That is not true.

Then how about one example of a real physical system that is linear?
I realize that something digital like a NOT gate may be linear, but it
is still an abstraction. It's physical manifestation is something
different. Linearity is just an idealization, a tool that we use.
Even the mechanical systems mentioned are all nonlinear. A system
using masses and dampers and springs can be pieced together to form
mechanical analogues of RLC electrical circuits, but all those systems
are only linear within a threshold.

Sure, there are linear systems, but they're all in our heads.

---
Proton Soup

"If I drink water I will have to go to the bathroom and
how can I use the bathroom when my people are in bondage?"
-Saddam Hussein

OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
OmegaZero2003 wrote:

I would not argue with Steve; he is onto something. He
put it all together and formulated another view of
reality consistent with certain other current views, yet
enabling a look at complexity_from_simplicity that has
heretofore not been appreciated in its scope of
applicability.

That's not clear to me.

Have you read his book?

I haven't read it, but have discussed this at length with
someone who has read it, attended Wolfram's lectures, and
discussed it with Wolfram.

Nonlinearity can arise in many different forms. Aside from
quadratic/cubic forms, which you might call "local"
nonlinearities (because the "slope" of the interaction
varies locally),

It does not ahve to; the form and whether it is local or
non-local are orthogonal.

Heh: we're definitely talking past each other.

My point is that there are linear systems.

Yawn.

This makes little sense. It is the complexity of a system
that determines the breadth and depth of a system_state
tree.

Again, we're talking past each other.

A complex system, or a dynamical system need not exhibit
emergent phenomena. The systems that do exhibt emergent
phenomena however, are usually complex dynamical sytems.

I don't know if I've mentioned this in this thread, but the
term "emergent" is not accepted by the bulk of NLD researchers
(mathematicians or physicists).

That something varies smoothly (not descrete steps I presume
you mean), does not mean it does not have distinguisable
states!!!!!! That is what intergation and differentiation
are all about.

Once again, we're talking past one another.

Or VV!! Chaotic systems far from equilibrium. See
Prigogine.

Yeah, yeah.

Can you point me to a ref. where you are reading/getting
this relationship from?

Sorry, I just made it up (but it happens to be true).

Well - I think it is mostly true as I said. But my original
point is that there are linear systems (hell - that is what
LP is for!!).

But - in that case...

I think we should have been discussing the (a)/(b) dichotomy
I mentioned above. Whether nature is discontinuoous or
continuous.

Ack!

True, but I'm referring to much simpler ensembles of
neurons. The computational capability of a NN is directly
traceable to the threshold and saturation characteristics
of the neurons.

Well -err - not necessarily. The computational capability
of brain or any subset depends on how yu think it
characterizes, or *represents* information!!

But a tiny NN is very simple, and it's quite clear how it
stores information. If the neuron activation function were
linear, it would only be able to store y=Ax+b, which contains
very little information.

Right, and this is really important: beyond a narrow
range, the cost of responding linearly to external stimuli
would be too taxing to the organism. Consequently, the
organism lets that mechanism saturate, and turns on a
different one.

Or goes crazy! (Becomes chaotic)

See, I make a living working on systems that are chaotic, so I
don't view them as crazy. For example, the human brain NEEDS
to be chaotic to be functional. Limit cycles are the abnormal
dynamics of brains (epilepsy, etc.).

I don't see as strong a link as you do evidently. There are
far more important aspects that affect robustness and
effectiveness of a system than saturation and threshold.

I wasn't trying to imply that threshold and saturation are
fundamental to robustness of dynamics of nonlinear systems,
but instead that it's fairly easy for self-organizing (e.g.,
biological) systems that have these characteristics to develop
robust dynamics.

Here is another thought. Man-made complex systems are
engineered, usually, to clamp to a safe value(s), all
those parameters that may compromise safety or
efficiency/waste-control.

This is a simple form of saturation.

Yo can clamp well-before a saturation level!

Clamping is functionally a sudden saturation.

Yeesh, enough already!

--
-Wayne

OmegaZero2003 wrote:

My original point to the OP on the topic was a retort to the
statement that *all* systems are nonlinear.

That is not true.

Nonsense. The OP never said any such thing. 8-p

--
-Wayne

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

BTW, as a PS to my other answer post, here are some linear
systems.

- those characterizable by linear algebra. there are lots
of these!

'ang on there, when we refer to linearity in such systems,
we're referring to linearity of the dynamics, i.e.,

x-dot = A * x + b

'ang on there yourself!!! Do you know WHAT characterizes a linear system
mathematically? Do you know the difference between an additive versus a
multiplicative factor? Well there is your answer!

In case you have not understood - will will take it slowly at first. Atend:
here is help for you:

"The response of a [ linear ] system to a sum of inputs is the sum of the
responses to each individual input separately.

These two nice properties allow a whole range of tools to be applicable in
designing linear systems and predicting their behavior. Some more examples
of linear systems in real life:

a) Frequency filters -- circuits which only pass low frequencies and reject
high, or vice-versa.

b) Delays are linear. Echos from faraway canyons are linear. Shout twice as
loud, get an echo twice as loud. Two people shouting at the same time comes
back as two people echoing at the same time.

c) Many different kinds of economic systems -- looking at the apple juice
production (output) vs. the apple crop yield, for example.

d) Limiting cases of non-linear systems for small inputs: Even if the
system's response may not satisfy the equation above exactly, it often will
well enough for small enough inputs. In this case, even if the number of
apples bought by consumers, say, is inversely proportional to the price of
apples, you can still model small changes around a reference price with
linear systems (but beware when the inputs get large!).
(http://van.hep.uiuc.edu/van/qa/secti...20020319233837.
htm)

Note that the last item (d) talks about linearization/signal
decomposition/approximation_techniques of/for non-linear systems, which I am
NOT talking about. We are talking about bona fide LINEAR systems fulfilling
all the mathematical/theoretical properties that decades of science has
determined qualifies a system as "Linear" *by definition*!!!!

I am also not talking about trivial examples of what is a linear system in
toto (including extreme), but is subclassed to non-interesting behavior
(like an amp NOT in saturation mode).



This omits simple algebraic behaviors (because x-dot = 0).

- Hamiltonian oscillators and like systems. (the
direction field specifically)

Not of great interest in biological systems, except as a
backdrop.


We were not restricting the discussion to biological systems.

My original retort to the OP's statement: "All systems are non-linear" was
to say:

That is not true.

It is easy to falsify hypotheses of the form: "All...(x) are (y)" in the
sense that only one counter-example need be provided.

FYI, a special class of linear control systems known as singularly perturbed
control systems, uses the Hamiltonian approach ( recursive) approach based
on t exact pure-slow and pure-fast decoupling of optimal control problems.

Another interesting special class of linear systems::

"Linear systems with non-rational transfer functions (In this project linear
systems described by partial differential equations having non-rational
transfer functions are studied. The aim of this project is to analyse the
dynamic behaviour and properties of linear input-output systems with
non-rational transfer functions (such as flexible robot arms and heat
processes) in the frequensy domain. For some classes of such systems it is
possible to develop the overall transfer function in analytical form by the
transmission matrix method. The interaction between different parts of the
system (including the way they are coupled to each other) can then be
analysed.
The transmission matrix method has been succesfully applied to multi-link
flexible robot arms and to buckling of multi-segment columns. Another
application concerns the stability of heat processes described by parabolic
partial differential equations. Based on the transmission matrix method a
Nyquist stability test was developed for sandwich-layered materials with
linear inner heat source." (M. Vajta DISC Project 1999)



for a linear system characterized using N first-order linear homogeneous
differential equations with constant coefficient can be found at: (with a
little more detailed math):
http://www.mathpages.com/home/kmath440/kmath440.htm

I leave it to the reader to come up with only three biological systems that
are so charaterized (this is an easy quiz to see if you understand what a
linear system is).

You can also look up sparse linear systems for funzies!



- continuous-time systems like electrical networks, many
mechanical systems

Aside from the fact that linearity is an approximation in all
such systems, they all have their nonlinear limits. The
nonlinearity is whatever it is that keeps the systems
operating in their linear range.

- any discrete system with a transfer
function whose input, response and output functions depend
on one variable

Huh?

I think you have to learn the meaning of linear. Note that I am NOT talking
about linearisation of non-linear systems (which seems to be talked about in
this thread instead of the major contentions about what exists.)!!! I am not
talking about signal decomposition to acheive a linear treatment (what you
called piecewise).


Here is some help. Note that he points out some examples in the text.

"Signals, Linear Systems, and Convolution

Professor David Heeger

Characterizing the complete input-output properties of a system by
exhaustive measurement is

usually impossible. Instead, we must find some way of making a finite number
of measurements

that allow us to infer how the system will respond to other inputs that we
have not yet measured.

We can only do this for certain kinds of systems with certain properties. If
we have the right kind

of system, we can save a lot of time and energy by using the appropriate
theory about the system's

responsiveness. Linear systems theory is a good time-saving theory for
linear systems which obey

certain rules. Not all systems are linear, but many important ones are. When
a system qualifies as

a linear system, it is possible to use the responses to a small set of
inputs to predict the response to

any possible input. This can save the scientist enormous amounts of work,
and makes it possible

to characterize the system completely."

Please note the statement: "Not all systems are linear, but many important
ones are."

And now for a little math:

"Linear Systems

A system or transform maps an input signal x(t) into an output signal y(t):

y(t) = T[x(t)];

where T denotes the transform, a function from input signals to output
signals.

Systems come in a wide variety of types. One important class is known as
linear systems. To

see whether a system is linear, we need to test whether it obeys certain
rules that all linear systems

obey. The two basic tests of linearity are homogeneity and additivity.

4

Homogeneity. As we increase the strength of the input to a linear system,
say we double it,

then we predict that the output function will also be doubled. For example,
if the current injected

to a passive neural membrane is doubled, the resulting membrane potential
fluctuations will double

as well. This is called the scalar rule or sometimes the homogeneity of
linear systems.

Additivity. Suppose we we measure how the membrane potential fluctuates over
time in

response to a complicated time-series of injected current x1(t). Next, we
present a second (different)

complicated time-series x2(t). The second stimulus also generates
fluctuations in the membrane

potential which we measure and write down. Then, we present the sum of the
two currents

x1(t) + x2(t) and see what happens. Since the system is linear, the measured
membrane potential

fluctuations will be just the sum of the fluctuations to each of the two
currents presented separately.

Superposition. Systems that satisfy both homogeneity and additivity are
considered to be

linear systems. These two rules, taken together, are often referred to as
the principle of superposition.

Mathematically, the principle of superposition is expressed as:

T(
x1 + x2) =
T(x1) + T(x2) (2)

Homogeneity is a special case in which one of the signals is absent.
Additivity is a special case in

which
= = 1.

Shift-invariance. Suppose that we inject a pulse of current and measure the
membrane potential

fluctuations. Then we stimulate again with a similar pulse at a different
point in time, and

again we measure the membrane potential fluctuations. If we haven't damaged
the membrane with

the first impulse then we should expect that the response to the second
pulse will be the same as

the response to the first pulse. The only difference between them will be
that the second pulse has

occurred later in time, that is, it is shifted in time. When the responses
to the identical stimulus

presented shifted in time are the same, except for the corresponding shift
in time, then we have

a special kind of linear system called a shift-invariant linear system. Just
as not all systems are

linear, not all linear systems are shift-invariant.

In mathematical language, a system T is shift-invariant if and only if:

y(t) = T[x(t)] implies y(t



"



That is pretty much what I told the OP. Not ALL systems are non-linear.

Period, End of story.

QED and all that.


Gabel, Robert A. and Roberts, Richard A., 1973, Signals and Linear Systems,
New York: John Wiley & Sons, 415 pp.

Gaskill, Jack D., 1978, Linear Systems, Fourier Transforms, and Optics, New
York: John Wiley & Sons, 554 pp.

Lathi, B. P., 1992, Linear Systems and Signals, Carmichael, California:
Berkeley-Cambridge Press, 656 pp.

Lewis, Laurel J., Reynolds, Donald K., Bergseth, F. Robert, Alexandro, Jr.,
Frank J., 1969, Linear Systems Analysis, New York: McGraw-Hill Book Company,
489 pp.

Schwarz, Ralph J. and Friedland, Bernard, 1965, Linear Systems, New York:
McGraw-Hill Book Company, 521 pp








The most common form of discrete systems is the
iteration (strobing based on time or state) of a continuous
system. A single-variable system would be of a form

x-dot = f(x)

Here, f can be (generally is) a nonlinear function of x, so
the system will show nonlinear behavior, both continuously and
discretely.



- any systems preserving homogeneity (output
proportional to input) and superposition (a way of combining
linear functions such that the result is a linear function)

In other words, linear systems.

And there are lots of those!

Do yourself a favor and merely google "linear system", read what you care
to, then come back. I do not believe you do not get this.

I can give you a good start:

Linear System Theory and Design
by Chi-Tsong Chen

Read Chapter 2!


Even non-deterministic systems can be modeled using
statistics for linear dynamics.

Not in their dynamics.

You do not know what you are talking about.

This is the classic engineer's mistake
of characterizing in statistical terms what is not understood.
When you delve into the NLD of such systems, you gain true
insight into what makes them work. I can give examples.

I have more examples in my head than you can provide. I worked on this for
20 years.


But the main point to not be belabored is that there are
linear systems on nature and manmade (1)

Most systems *are* non-linear but some of those are
characterizable using linear methods to some degree of
accuracy; you did make something like this point.

That is an approximation that people sometimes find useful to
make. That doesn't make it so, especially when you take the
system outside of the limited context in which you placed it
for convenience.

There are systems that are approximated as linear; that is called
*linearization* and there is a gamut of math to deal with how to do that
properly!

But there are many systems that are inherently and demonstrably linear.



--
-Wayne

"Proton Soup" wrote in message
...
On Sun, 21 Dec 2003 20:26:26 GMT, "OmegaZero2003"
wrote:

"Proton Soup" wrote in message
.. .
On Sun, 21 Dec 2003 06:29:45 GMT, "OmegaZero2003"
wrote:

BTW, as a PS to my other answer post, here are some linear systems.

- those characterizable by linear algebra. there are lots of these!
- Hamiltonian oscillators and like systems. (the direction field
specifically)
- continuous-time systems like electrical networks, many mechanical
systems

Only simple RLC electrical networks fall into this category. And even
then, it's just a theoretical assumption over the useful operating
range. Too much current or voltage or flux will flux up your circuit.
Linear electrical networks only exist on paper.

My original point to the OP on the topic was a retort to the statement
that
*all* systems are nonlinear.

That is not true.

Then how about one example of a real physical system that is linear?
I realize that something digital like a NOT gate may be linear, but it
is still an abstraction. It's physical manifestation is something
different. Linearity is just an idealization, a tool that we use.
Even the mechanical systems mentioned are all nonlinear. A system
using masses and dampers and springs can be pieced together to form
mechanical analogues of RLC electrical circuits, but all those systems
are only linear within a threshold.

See my recent post. There are examples there.

I think you have to understand the *mathematical definition* of a linear
system to understand anything further. I provided that edication in that
recent post.

There are many systems that neet that definition. I am not talking about
approximation techniques, subclassing a system into non-extreme behavior
ranges, or linearization techniques.


Sure, there are linear systems, but they're all in our heads.

And sometimes not even there

---
Proton Soup

"If I drink water I will have to go to the bathroom and
how can I use the bathroom when my people are in bondage?"
-Saddam Hussein

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

My original point to the OP on the topic was a retort to the
statement that *all* systems are nonlinear.

That is not true.

Nonsense. The OP never said any such thing. 8-p

--
-Wayne


Here is the context; I see you were pointing to biological systems.



"Elzinator wrote:

"OmegaZero2003" wrote...

This is very similar to the issues facing cancer
researchers. Three very different mechanisms/theories using
separate processes all interacting to produce the endpoint.

Biological systems are more complex than most realize:
feedback loops, negative and positive regulators, redundant
and overlapping pathways, etc.

And, they're all nonlinear. That is, they are rife with
thresholds and saturation effects. This makes them very, very
(very) complicated, but has a lot to do with their effectiveness
and robustness."

See where you said they (biological) are all non-linear?

There are biological systems that are linear.

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
OmegaZero2003 wrote:

I would not argue with Steve; he is onto something. He
put it all together and formulated another view of
reality consistent with certain other current views, yet
enabling a look at complexity_from_simplicity that has
heretofore not been appreciated in its scope of
applicability.

That's not clear to me.

Have you read his book?

I haven't read it, but have discussed this at length with
someone who has read it, attended Wolfram's lectures, and
discussed it with Wolfram.

Nonlinearity can arise in many different forms. Aside from
quadratic/cubic forms, which you might call "local"
nonlinearities (because the "slope" of the interaction
varies locally),

It does not ahve to; the form and whether it is local or
non-local are orthogonal.

Heh: we're definitely talking past each other.

My point is that there are linear systems.

Yawn.

This makes little sense. It is the complexity of a system
that determines the breadth and depth of a system_state
tree.

Again, we're talking past each other.

A complex system, or a dynamical system need not exhibit
emergent phenomena. The systems that do exhibt emergent
phenomena however, are usually complex dynamical sytems.

I don't know if I've mentioned this in this thread, but the
term "emergent" is not accepted by the bulk of NLD researchers
(mathematicians or physicists).


I would not agreee with that. it may (certainly is I will say) be the case
that what is characterized *as* emergent* is not agreed to.

There is a difference.

That there is a definition/theory of emergence is indisputable. That such
has been defined with mathematical characterizations of systems is also
true.

What usually is confused or, better, *conflated* by those whom you may be
speaking, is "emergence" and "synergistic phenomena".




That something varies smoothly (not descrete steps I presume
you mean), does not mean it does not have distinguisable
states!!!!!! That is what intergation and differentiation
are all about.

Once again, we're talking past one another.

Or VV!! Chaotic systems far from equilibrium. See
Prigogine.

Yeah, yeah.

Can you point me to a ref. where you are reading/getting
this relationship from?

Sorry, I just made it up (but it happens to be true).

Well - I think it is mostly true as I said. But my original
point is that there are linear systems (hell - that is what
LP is for!!).

But - in that case...

I think we should have been discussing the (a)/(b) dichotomy
I mentioned above. Whether nature is discontinuoous or
continuous.

Ack!

True, but I'm referring to much simpler ensembles of
neurons. The computational capability of a NN is directly
traceable to the threshold and saturation characteristics
of the neurons.

Well -err - not necessarily. The computational capability
of brain or any subset depends on how yu think it
characterizes, or *represents* information!!

But a tiny NN is very simple, and it's quite clear how it
stores information.

No is isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

It isn't clear at all how brain *represents* information. This is one of
the biggest challenges facing neuroscience; how to span the gap between the
psychological phenomena/overt behavior (verbal behavior etc.) and the NCCs
(the neural correlates of consciousness)!!!!!!!!!!!!

And all those silly extrapolations of how much the brain can compute if the
brain can compute all day (based on such simplistic notions as you seem to
think are veridical), have been shown to be silly!

See Chalmer's page on such at:
http://www.u.arizona.edu/~chalmers/biblio.html

BTW, I have read ALL of those papers. This is my life's work.


If the neuron activation function were
linear, it would only be able to store y=Ax+b, which contains
very little information.

Right, and this is really important: beyond a narrow
range, the cost of responding linearly to external stimuli
would be too taxing to the organism. Consequently, the
organism lets that mechanism saturate, and turns on a
different one.

Or goes crazy! (Becomes chaotic)

See, I make a living working on systems that are chaotic, so I
don't view them as crazy.

I was using the term loosely.

For example, the human brain NEEDS
to be chaotic to be functional. Limit cycles are the abnormal
dynamics of brains (epilepsy, etc.).

I agree.




I don't see as strong a link as you do evidently. There are
far more important aspects that affect robustness and
effectiveness of a system than saturation and threshold.

I wasn't trying to imply that threshold and saturation are
fundamental to robustness of dynamics of nonlinear systems,
but instead that it's fairly easy for self-organizing (e.g.,
biological) systems that have these characteristics to develop
robust dynamics.


Hmmm - OK - let me think about that.


Here is another thought. Man-made complex systems are
engineered, usually, to clamp to a safe value(s), all
those parameters that may compromise safety or
efficiency/waste-control.

This is a simple form of saturation.

Yo can clamp well-before a saturation level!

Clamping is functionally a sudden saturation.

No! Not theoretically or practically.

Control systems are clamped at a setpoint that assures continued, NORMAL
operation, in the vast majority of cases.


Yeesh, enough already!

--
-Wayne

"OmegaZero2003" wrote in message
s.com...

"Wayne S. Hill" wrote in message
...
OmegaZero2003 wrote:

"Wayne S. Hill" wrote...
OmegaZero2003 wrote:

I would not argue with Steve; he is onto something. He
put it all together and formulated another view of
reality consistent with certain other current views, yet
enabling a look at complexity_from_simplicity that has
heretofore not been appreciated in its scope of
applicability.

That's not clear to me.

Have you read his book?

I haven't read it, but have discussed this at length with
someone who has read it, attended Wolfram's lectures, and
discussed it with Wolfram.

Nonlinearity can arise in many different forms. Aside from
quadratic/cubic forms, which you might call "local"
nonlinearities (because the "slope" of the interaction
varies locally),

It does not ahve to; the form and whether it is local or
non-local are orthogonal.

Heh: we're definitely talking past each other.

My point is that there are linear systems.

Yawn.

This makes little sense. It is the complexity of a system
that determines the breadth and depth of a system_state
tree.

Again, we're talking past each other.

A complex system, or a dynamical system need not exhibit
emergent phenomena. The systems that do exhibt emergent
phenomena however, are usually complex dynamical sytems.

I don't know if I've mentioned this in this thread, but the
term "emergent" is not accepted by the bulk of NLD researchers
(mathematicians or physicists).


I would not agreee with that. it may (certainly is I will say) be the case
that what is characterized *as* emergent* is not agreed to.

There is a difference.

That there is a definition/theory of emergence is indisputable. That such
has been defined with mathematical characterizations of systems is also
true.

What usually is confused or, better, *conflated* by those whom you may be
speaking, is "emergence" and "synergistic phenomena".




That something varies smoothly (not descrete steps I presume
you mean), does not mean it does not have distinguisable
states!!!!!! That is what intergation and differentiation
are all about.

Once again, we're talking past one another.

Or VV!! Chaotic systems far from equilibrium. See
Prigogine.

Yeah, yeah.

Can you point me to a ref. where you are reading/getting
this relationship from?

Sorry, I just made it up (but it happens to be true).

Well - I think it is mostly true as I said. But my original
point is that there are linear systems (hell - that is what
LP is for!!).

But - in that case...

I think we should have been discussing the (a)/(b) dichotomy
I mentioned above. Whether nature is discontinuoous or
continuous.

Ack!

True, but I'm referring to much simpler ensembles of
neurons. The computational capability of a NN is directly
traceable to the threshold and saturation characteristics
of the neurons.

Well -err - not necessarily. The computational capability
of brain or any subset depends on how yu think it
characterizes, or *represents* information!!

But a tiny NN is very simple, and it's quite clear how it
stores information.

No is isn't!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

It isn't clear at all how brain *represents* information. This is one of
the biggest challenges facing neuroscience; how to span the gap between
the
psychological phenomena/overt behavior (verbal behavior etc.) and the NCCs
(the neural correlates of consciousness)!!!!!!!!!!!!

And all those silly extrapolations of how much the brain can compute if
the
brain can compute all day (based on such simplistic notions as you seem to
think are veridical), have been shown to be silly!

See Chalmer's page on such at:
http://www.u.arizona.edu/~chalmers/biblio.html

BTW, I have read ALL of those papers. This is my life's work.

Since I know you are not gonna wade through 2000 papers, here is the part
that pertains:

http://www.u.arizona.edu/~chalmers/biblio/4.html#4.2

What is key is *how* the brain represents information and that is a subject
of *intense* debate and research!!!





If the neuron activation function were
linear, it would only be able to store y=Ax+b, which contains
very little information.

Right, and this is really important: beyond a narrow
range, the cost of responding linearly to external stimuli
would be too taxing to the organism. Consequently, the
organism lets that mechanism saturate, and turns on a
different one.

Or goes crazy! (Becomes chaotic)

See, I make a living working on systems that are chaotic, so I
don't view them as crazy.

I was using the term loosely.

For example, the human brain NEEDS
to be chaotic to be functional. Limit cycles are the abnormal
dynamics of brains (epilepsy, etc.).

I agree.




I don't see as strong a link as you do evidently. There are
far more important aspects that affect robustness and
effectiveness of a system than saturation and threshold.

I wasn't trying to imply that threshold and saturation are
fundamental to robustness of dynamics of nonlinear systems,
but instead that it's fairly easy for self-organizing (e.g.,
biological) systems that have these characteristics to develop
robust dynamics.


Hmmm - OK - let me think about that.


Here is another thought. Man-made complex systems are
engineered, usually, to clamp to a safe value(s), all
those parameters that may compromise safety or
efficiency/waste-control.

This is a simple form of saturation.

Yo can clamp well-before a saturation level!

Clamping is functionally a sudden saturation.

No! Not theoretically or practically.

Control systems are clamped at a setpoint that assures continued, NORMAL
operation, in the vast majority of cases.


Yeesh, enough already!

--
-Wayne