Low carb diets

On Sun, 21 Dec 2003 21:35:47 GMT, "OmegaZero2003"
wrote:


"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.

All engineers should know what superposition is. Are you trying to be
condescending?

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.

They only work within a narrowly-bounded input-output range. Too much
signal input and they will break (non-linear).

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.

Two much volume and you will heat the air, changing the transmission
characteristics (speed) of your pressure wave (non-linear).

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

This too will fail linearity at the extremes of yield. A bumper crop
one year could mean an inability to harvest all the fruit before it
goes bad.

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*!!!!

Real systems are always limited on their inputs. Real systems break
down at some point if too much energy enters the system. Linearity is
only a trait of a system over its useful operating range.

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).

In reality, people who make things that work must take nonlinearity
into account. It's the only way to make reliable systems. You may
not be thinking in toto, but I am, because I have to. We know what
linear is. We also know it doesn't really exist. We try to make
things work as linear as possible, because many useful things exhibit
a limited version of it (like amps and filters).

The definition of linearity for that amp doesn't allow for limiting
the input. If it were truly linear, you wouldn't have to place extra
limits on it. So I think you are coming around to accept the
assertion that linear systems don't really exist. I will admit,
though, they are nice on paper.

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)

Good grief. Robot arms are the last thing in the world that would be
linear. There may be some use of linear mathematics in the control
law, but that's it. Linear math in control algorithms is the norm.

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

"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

"Proton Soup" wrote in message
...
On Sun, 21 Dec 2003 21:35:47 GMT, "OmegaZero2003"
wrote:


"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.

All engineers should know what superposition is. Are you trying to be
condescending?

Who? Me?

Sorry. I was probably a little. It seemed elementary considering the
responses seemed to be intelligent but missed major aspects of this.

OmegaZero2003 wrote:

"Proton Soup" wrote...

All engineers should know what superposition is. Are you
trying to be condescending?

Who? Me?

Sorry. I was probably a little. It seemed elementary
considering the responses seemed to be intelligent but
missed major aspects of this.

No they didn't. That was a representation in your own wetware.

--
-Wayne

OmegaZero2003 wrote:

Did you bother to see the papers on computing and
representation I referenced?

Nope! 8-)

See, I have a limited capacity for curiosity, and I suspect
we're arguing about subtleties that verge on the meaningless.
At the very least, a lot is being lost in translation.

I do not care what the neurophsyiological response of a
crayfish CNS is; there ain't any instruments yet that can
tell just what processes and properties in *any* part of a
NN exactly *represent* a "piece" of information - even given
what the definition of "information" is (beyond a
difference).

The problem I have with this is that there's a point, with a
very small number of neurons, where this distinction vanishes.
This is the same problem most researchers have with the idea
of "emergence". Beyond a high level of complexity, the
network representation is probably closest to a high-
dimensional hologram. At a low level of complexity, it's
probably pretty much like an ANN. Is the distinction one of
topological significance, or is it really all the same thing?

In terms of computational neuroscience and ANNs(artificial
neural net), remember that the ANN is a couple orders of
magnitude less sophisticated (at least) (using simple I/O
transforms and connection schemes used to build multi-layer
ANNS) than the real thing in situ.

I never claimed otherwise: I brought ANN's into the
discussion because of the clear understanding of the roles of
threshold and saturation on their behavior.

--
-Wayne

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.


- 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





As several people have pointed out, linear vs. non--linear, per se,
is meaningless. It all depends on the description. Some things are
exactly linear (in the right description.) Sometimes linear is just
local -- but even if local, this can provide quantitative information.
(The example that comes to mind is the tumbling book. Two
directions are stable, one is not. Linear analysis shows this.)


--
Tom Morley | Same roads
| Same rights
| Same rules
AIM: DocTDM

In article ,
Elzinator no one@com wrote:
Lyle McDonald wrote:

Ok, since the book project this was originally written for is unlikely
to ever get done, I figured I'd post it. It's a long (11 pages)
chapter/piece examining the pros/cons of the major dietary camps

Dude, just upload it to your website! That's what it's for.

I had snagged it, fixed typos and formatting, and put in in the archives at:
http://www.trygve.com/mfwalylediet.html

I haven't put in any links to it, but I can if that's okay.

--
soc.singles FAQ [ Nyx Net, free ISP ] Misc.Fitness.Weights page
www.trygve.com/ssfaq.html [ http://www.nyx.net ] www.trygve.com/mfw.html
today's special featu Santa Claus, Fugitive From Justice
on America's Most Wanton: http://www.trygve.com/mostwanton.html

"Tom Morley" wrote in message
link.net...


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.


- 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





As several people have pointed out, linear vs. non--linear, per se,
is meaningless. It all depends on the description. Some things are
exactly linear (in the right description.).

Sometimes linear is just
local -- but even if local, this can provide quantitative information.

Sure - but that is known as a subclassed system. You are only examining it
within a range of I/O/Xfer_function(s). Which is fine as far as it goes,

I was referring to a system that exhibits true linear behavior throughout
all known or extraploated ranges of input.

(The example that comes to mind is the tumbling book. Two
directions are stable, one is not. Linear analysis shows this.)

Yes; although the point that was made , as I now understand, was about
biological system, which certainly have more than the bulk of examples of
non-lineear systems.




--
Tom Morley | Same roads
| Same rights
| Same rules
AIM: DocTDM

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

"Proton Soup" wrote...

All engineers should know what superposition is. Are you
trying to be condescending?

Who? Me?

Sorry. I was probably a little. It seemed elementary
considering the responses seemed to be intelligent but
missed major aspects of this.

No they didn't. That was a representation in your own wetware.

How do you know I am not an ANN!

You still have not shown how robustness is a function of saturation and/or
thresholding.

Although after thinking about it, in an oblique way, one can make up a story
about it - a mind game ala Einstein. I.e., theoretically, I can imagine
that a system can be thought of as robust if it escapes
deterioration/degradation and/or elimination from the context/environment if
it exhibits saturation/thresholding and that prevents state spaces leading
to elimination.

Coming up with a *real* example of such a charaterization in nature is left
to the reader.


--
-Wayne

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

Did you bother to see the papers on computing and
representation I referenced?

Nope! 8-)

See, I have a limited capacity for curiosity, and I suspect
we're arguing about subtleties that verge on the meaningless.
At the very least, a lot is being lost in translation.
OK.


I do not care what the neurophsyiological response of a
crayfish CNS is; there ain't any instruments yet that can
tell just what processes and properties in *any* part of a
NN exactly *represent* a "piece" of information - even given
what the definition of "information" is (beyond a
difference).

The problem I have with this is that there's a point, with a
very small number of neurons, where this distinction vanishes.

First, I suspect that that type of system is uninteresting.

Second, it probably does not exist as a real system in situ' one can take
away only so much of a system auntil it ceases *being* anything like what
you were trying to show in the first place.

Such is the case with a system whose function is representation (and
transformation/translation/signalling).

Third, that distinction is not a quatitative one - it is qualitative. take
away the neurochemical soup for example, and what you show about information
representation is apt to be misleading at best. *Analysis* (in the form of
reductionism)is not always a good approach when dealing with complexity .

This is the same problem most researchers have with the idea
of "emergence". Beyond a high level of complexity, the
network representation is probably closest to a high-

But that is only the network representation; one of several maps, none of
which is the territory.

And most neuroscience researchers or AI researchers for that matter, so not
have a problem with emergence. It is quite well described and accepted.
Again, a really good book is Alwyn Scott's! I recommend it to any scientist
I speak with (just as I recommend Wolfram's work, and Bucky Fuller's
Synergetics).

dimensional hologram. At a low level of complexity, it's
probably pretty much like an ANN. Is the distinction one of
topological significance, or is it really all the same thing?

I don't follow; I don;t think even a highly-connected network like the brain
has properties at the network level (nodes, connections, vertices etc.) that
are appropriate in a discussion about holographical metaphors.

Now, quantum effects, or other field effects -now we're talking.


In terms of computational neuroscience and ANNs(artificial
neural net), remember that the ANN is a couple orders of
magnitude less sophisticated (at least) (using simple I/O
transforms and connection schemes used to build multi-layer
ANNS) than the real thing in situ.

I never claimed otherwise: I brought ANN's into the
discussion because of the clear understanding of the roles of
threshold and saturation on their behavior.

YEs - OK - I understand. But those are man-made systems that, llike I said
above, are so abstracted, or rahter, simplified from the actual NN in the
CNS -in situ with all the attendent functions provided by messenger
molecules, densities , field effects etc., that the analysis has analysed
away any chance of getting a parsimonious, satisfying sanswer to things like
representation in a real brain.



--
-Wayne

In article , Wayne S. Hill
wrote:
Heh: we're definitely talking past each other.


And to about six other groups we don't need traffic with. Can you trim
your headers along with your carbs please?