Low carb diets

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

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.

- 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? 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.

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

Not in their dynamics. 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.

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.

--
-Wayne

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

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

AND I LOVE to argue or debate or discuss Mathematics.
Why, me and my trusty Mathematica app have been through
many wars together. Akk Steven Wolfram about what that
might mean.

Go argue it with him. Some people think he's really onto
something, but I have my doubts.

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?


No, you're misreading me. I said the same thing both
times.

I copied and pasted your original statement.

I must have been unclear the first time, because I intended
the same meaning both times.

That non-linearity itself has a lot to do with thier
effectiveness and robustness.

It does, but the nature of the nonlinearity has a lot to do
with it.

What does that mean?

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.

the global behaviors of threshold and saturation
phenomena are common themes in biological systems.

Sure.

My point is that there are linear systems.


Perhaps you can elaborate. I would like to know what you
thin thresholds and saturation effects have to do with
linearity such that they help constitute a property or
process of robustness and effectiveness.

I really don't want to get into this too deeply (not why I
come here), but threshold and saturation phenomena remap an
infinite range of possibilities into a modest finite range.
Since a biological system can only act within such a range,
this permits the system to respond to very broad ranges of
environments.

The possible system states have little to do with whether a
system is linear or non-linear.

Au contraire. If a system is linear, it must accommodate an
infinite range of input variables linearly. Thus, the output
range has to be of infinite extent, and cannot exhibit
different types of states.

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


However, complextity is all about such.

You've got to be careful here. I take it you're referring to
complex dynamical systems that exhibit self-organizing so-
called emergent behaviors.

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.


A mass of nitrogen molecules is a
counter example: it never does anything "emergent",

That is what I said just above.

and so
doesn't (normally) have distinguishably different states.
That is, given N molecules in a box of size V and temperature
T, it exerts a pressure P. This varies in a simple and smooth
manner from above the boiling point to the neighborhood of
dissociation.

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.

Not only that but ther are clever theories purporting to show:

a) everything is quatal/descrete to the finest level of description
b) everything is analog/no_quantal_states to the finest level of
description.

Both positions are far from established given our level of instrumentality.


The difference between a "simple" complex
system and one capable of self-organization is the way it
approaches equilibrium in the face of large disequilibrium.

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


The system does this by employing different
mechanisms or strategies in different ranges of external
influence (with each mechanism triggered by its own
threshold,

I agree with this. But how does that (threshold and
saturation) affect robustness and saturation directly. They
are parameters constraining response yes and I get your
point here, but a response to a perturbation using, say,
Green's Theroem to determine such (where the result of
solved SPDEs will eventually converge to zero - meaning the
system will reach a minima on a mapping -
energy/complexity/activity/etc), in terms of its robustness
to that perturbation (ability to so converge/relax), will
not have threshold and saturation terms in those equations.
Similarly for the effectiveness parameter(s) (again, in tems
of? meeting a goal (if an intensional system), surviving an
environment?) . If what you mean *is* a system's
effectiveness in surviving perturbations of an environment
without becoming unstable, there are aharmonic
mutivibrator-characterized systems that can tend to chaos or
to stable systems with zippo to do with. There are many
other complex systems that do not reach such extrema
(saturation) in their response, nor are they especially
threshold-based system. For example, the brain can detect
one photon of light (via the VC) when such impinges upon a
photoreceptor. That is the smallest threshold one can
imagine - a pseudo-infinitely-small threshold in the *sense*
that it is representative of the quanta of em energy. No
telling if any brain has actually detected *only* one photon
at a "time" of course, but the point is one of
threshold-based systems. You have to make a quantum leap to
get to that threshold arr, arr!

There are also discontinuous processes that "jump" right
over "thresholds".

True.

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.

Instead!

;^)


Remember, I see dead dimensions.

and limited by saturation). For example, for a room-
temperature environment, the body maintains core
temperature using different strategies than in very cold or
very hot conditions.

Note that this is what makes neural networks into
computational engines.

That is one level of description - or -one view of what
brain does among several. I have a bit of experience
constructing ANNs for process control and there are levels
of description of brain that are not also characterizable as
a TME (Turing Machine Equivalent).

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!!

IFF it is based on the go/no-go neuronal firing model, and IFF one
establishes as a premise that such is representative of 1 "bit" of
information (which is problematic itself, since the binary system is one of
*represenation* of higher-order/more-complex information represenation
schemes - that is, it is a mapping itself!!!), then one can estimate the
computational extent of that group of neurons (assuming further that one can
charaterize the NN architecture (its connection scheme) in sufficient
detail.


A little bit goes a long way - haha!



Without threshold and saturation phenomena, a NN would be
useless.

Threshold is apparent in the neuronal characterization of
all-or-nothing firings (which itself is a function of
humongous complexity); however, that one aspect of the
messenger processes (first or second) of the brain.

I cannot see where it has the import ascribed WRT
robustness or effectiveness (towards a goal for example).

Then open your eyes.

I think we're arguing past each other, something that wouldn't
happen if we actually discussed this in person.

More than likely! I like to waves hands and draw on boards!!



Saturation is an example of an extrema - a perturbation
causing a behavior point, and subsequent behavior points
that are the same or similar magnatude until the system
relaxes. The system simply has no differential response to
continuing stimula.

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) There are examples of systems that
persist in their output without switching. This kind of system exists on an
organismic level and in vary large systems - like socail systems and
economics.

It all depends on the feedback mechanisms (FF/FB) and *where* in the input
stream the FB occurs, and *whether* that point in the input stream has the
capability to clamp or switch its input-processing algorithm.



Again, this is orthogonal to robustness and effectiveness of
a system (in terms of - we have not defined except as my
intial take on what each means earlier.

No, it's key.

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 guess it depends on how you have learned to view system sciences, control
systems, and biological systems etc.


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!


That, and the other
characteristics I mentioned (no single point of failure,
graceful degradation etc.) make a system robust (in the face
of error or failure). Threshold and saturation are not part
of that consideration except as knowledge that can be
employed to determine startpopints (states), endpoints (end
states), and the PID coefficients affecting operation. When
a P/I/D/PI/PD/PID process goes awry, the PID and any
cascaded processes/points to which it is related/connected
get reset to some clamp value(s) and a good system will
transfer control to simple LL-based controllers and/or
simple interlocks completely divorced from the other control
system (isolatability is another aspect of robustness).

You don't see the analogy?

Huh? Between what and what?


A good ref on all of this is the classic N. Weiner's
Cybernetics Second Edition: or the Control and Communication
in the Animal and the Machine

Any good book on control systems theory incorporating the
good ole PID controller strategy should give more insight
into the parameters affecting system control, especially
systems with feedback.

Well, yeah, but they provide little insight into profound
nonlinearity.

Weiner's book does. ANd if you learn about how control systems are
engineered to deal with "profound non-linearities", then you will see what
is important and what is not (from a control standpoint that is.)

The books/people that do provide the "insight" however include another I
mentioned - Alwyn Scott's work.


--
-Wayne

"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.

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Berkeley-Cambridge Press, 656 pp.

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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.