Differential equations

Bill Harris
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Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Bill Harris » Thu Apr 03, 1997 9:33 am

You raise some very interesting points.

> Differential equations are difficult, confusing, weak, and
> unrealistic. They often mislead students as to the nature of systems.

Thanks for saying this. While Ive studied ODEs and PDEs in school and
used them some at work, I can remember feeling more comfort when working
with analog computers in school to simulate real world processes (now you
can guess my vintage), and I enjoy working (at the relatively low skill
level that I do) with SD simulations to model processes I see today.

> The mathematics alone does not give an adequate feeling for the real world.

I remember as far back as freshman physics learning that good math skills,
without a seat of the pants understanding of the world, wouldnt solve many
problems.

> On teaching dynamics in K-12, we are trying that through the "Road Maps"

> designed for "fun" as Jaideep Mukherjee fears. If they are done as
> intended, with all the book readings and computer exercises, I estimate
> that they will take 150 hours.

This brings me to the point of my question . Some on this list seem to
have argued in favor of SD being practiced only after serious study in a
university program (at the graduate level?) with the opportunity for the
prospective SD person to receive feedback on their modeling efforts. I can
see the benefits of such study. Some in the SD community (vendors,
perhaps?) seem to suggest the "democratic notion" that SD can be done
relatively easily by using todays tools and the admittedly good manuals
they provide. The democratic part of this seems very appealing, but I
recognize the danger of models (any models, not just SD models) not
matching reality.

Your posting seems to suggest a middle ground, that of the Road Maps. Im
part way through them, and I do find they help my thinking and working. I
also recognize that Im not an accomplished modeler (but I may be above the
median of those within half a mile of my desk).

How concerned are you about the work someone does who is working through
the Road Maps and using SD in a practical sense? What risks do you see in
the conclusions they may be drawing which they should take special pains to
avoid? Must those risks be ameliorated through undergraduate or graduate
courses, or is there another way? In particular, some of us have
difficulty in returning to a college campus (financial or personal
responsibility, geographic distance, etc.). Also, I think most (all?) of
us have heard multiple times that (college) educations are about learning
how to learn, so we should be able to do some of this ourselves, right?

I suspect the answer involves a combination of SD theory (e.g., skills in
how to represent reality effectively in a model and how to check the
results of that model for validity) and consulting practice (e.g.,
effective techniques for capturing and representing knowledge which the
"client" has and for working with that client to explore the ramifications
of the model openly and honestly to ensure that the resulting model is
believable).

Thanks for any (more) advice which may help generate more useful models and
more professional modelers.

Bill

PS: Having said all this, I recognize I wouldnt presume to teach myself a
skill such as playing a musical instrument, for example, without a private
teacher unless it were purely for fun. Theres too much about both
technique and about results in music that is really hard to get from
reading and really inefficient and arguably ineffective to get from
observation. These may be primarily characteristics of physical
activities, though.


--
Bill Harris Hewlett-Packard Co.
R&D Engineering Processes Lake Stevens Division
domain: billh@lsid.hp.com M/S 330
phone: (425) 335-2200 8600 Soper Hill Road
fax: (425) 335-2828 Everett, WA 98205-1298


jm62004@Jetson.UH.EDU
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by jm62004@Jetson.UH.EDU » Sun Apr 06, 1997 10:34 am

First of all, it is an honor to get a reply from you, Dr. Forrester.

By raising the "fun" issue, I did not mean that having fun while learning
was bad in some sense, only that sometimes the tendency to make subjects
fun makes explanations too simplistic, without demanding serious effort
from students to really understand what is happening. I am all for fun, as
I had when I took the SD courses at UIUC under Prof. Hannon, while
learning the subject too. In fact, it was the directness and easy-to-use
visual approach of current SD packages that attracted me to SD in the first
place. Only in time with more complex models I learnt that, while "fun", SD
can be as demanding and serious as any other profession. I remember a
remark once by an engineer acquanitance when I mentioned I was working on
World3 extension using Stella: he said - Oh, but Stella is for
hi-schoolers, only for toy models; or by another as: yeah, we do that kind
of work for masters projects but not for serious Ph. D. thesis. I was not
very happy at these remarks, obviously, as I think they did not know what
they were talking about. My question: does presenting something as fun take
away the seriousness of it? I dont think so - else Carl Sagan, Feynmann
etc. wouldnt be so famous/popular.

Thank you for recomending some material on how to teach/learn SD. I will
check it out. There is a lot of food for thought here.

Regards,

Jaideep
jm62004@Jetson.UH.EDU
*******************************************************
Jaideep Mukherjee, Ph. D.
Research Associate
Department of Industrial Engineering
University of Houston
4800 Calhoun Road
Houston, TX 77204-4812, USA

http://www.uh.edu/~jm62004/jm97.html
Office Phone: 713 743 4181; Fax: 713 743 4190
*******************************************************

jforestr@MIT.EDU (Jay W. Forrest
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by jforestr@MIT.EDU (Jay W. Forrest » Sat May 31, 1997 10:00 pm

In (SD0901), Jaideep Mukherjee asked that I comment on experience with
system dynamics in K-12 education, apparently in regard to their not having
had a background in ordinary differential equations.

Elsewhere* I have written:
-------------------------------------------------------
15. Differential Equations vs. Integration

One might ask how it is now possible to teach behavior of complex
dynamic systems in K-12 when the subject has usually been reserved for
college and graduate schools. The answer lies in having realized that the
mathematics of differential equations has been standing in the way.

Differential equations are difficult, confusing, weak, and
unrealistic. They often mislead students as to the nature of systems.
Mathematicians have had difficulty defining a derivative and there is a
reason. Derivatives do not exist except in a mathematicians imagination.
No where in nature does nature take a derivative. Nature only integrates,
that is, accumulates. Casting behavior in terms of differential equations
leaves many students with an ambiguous or even reversed sense of the
direction of causality. I have had MIT students argue that water flows out
of the faucet because the level of water in the glass is rising; that seems
natural to them if the flow has been defined as the derivative of the water
level in the glass.

Any child who can fill a water glass or take toys from a playmate
knows what accumulation means. The levels (stocks) in a system dynamics
model (the rectangles in Figures 4 and 5) are the integrations
(accumulations). By approaching dynamics through the window of
accumulation, students can deal with high-order dynamic systems without
ever discovering that their elders consider such to be very difficult.
--------------------------------------------------------------
* In "System Dynamics and K-12 Teachers", paper D-4665, which will soon
(maybe before end of June) on our web site at:
http://sysdyn.mit.edu/people/jay-forrester.html
-----------------------------------------------------
My objection here is not to having a mathematical dynamics background. I
had such and I believe it has been very useful. That background was in
classical feedback control theory, not optimal control. I do believe that
we need to recast what is now taught as differential equations with the
focus on integration so that the mathematics remains true to the real world
that is being represented.

I find that differential equations have obscured for many students the
direction of causality, even, as above, producing a reversed sense.

The mathematics alone does not give an adequate feeling for the real world.
I had two graduate students in computer science at MIT come over and do a
system dynamics model of what happens to the electrons at the interface of
a transistor. When they were finished, one said, "This is the first time
that I ever understood what was going on."

On teaching dynamics in K-12, we are trying that through the "Road Maps"
series that is available at http://sysdyn.mit.edu. There are now seven
chapters available out of what could be fifty chapters (I do not expect to
get that far). The present seven chapters can not be accused of being
designed for "fun" as Jaideep Mukherjee fears. If they are done as
intended, with all the book readings and computer exercises, I estimate
that they will take 150 hours.

We are receiving some 1000 to 3000 contacts to the web site per week. The
sources of those contacts can not be clearly identified, but we estimate
that two-thirds come from corporations that are using the material for
internal training. One can use exactly the same material at all levels
from middle school to chief executive officers.

Jay W. Forrester
Professor of Management, Emeritus
and Senior Lecturer, Sloan School
Massachusetts Institute of Technology
Room E60-389
Cambridge, MA 02139
tel: 617-253-1571
fax: 617-252-1998

email: jforestr@mit.edu


ijamal@gov.edmonton.ab.ca (Iqbal
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Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by ijamal@gov.edmonton.ab.ca (Iqbal » Wed Jun 04, 1997 10:56 am

Maybe I missed it, but I see the whole issue of the serious study of SD
before modeling is a mute point. It depends on where you are in the cycle
of level of understanding the problem, clear formulation of the problem and
attempted solution, etc. (a typical SD diagram?). If the problem is poorly understood
then experimenting with SD models (or moreso diagramming) helps in better
formulating the problem. The the formal SD training kicks in to translate the
formulation into a working model that can be useful in understanding possible
solutions. The world looks different depending where you are in the loop, and we
seem to have a good idea (from the group discussions) what that looks like.
Thoughts?
............................................................................................................................

Iqbal Jamal
Office of Studies and Budget
City Manager Offices
City of Edmonton
1 Sir Winston Churchill Square
Edmonton, Alberta, Canada
T5J 2R7

(403) 496-8228
ijamal@gov.edmonton.ab.ca

Bill Braun
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Bill Braun » Thu Jun 05, 1997 9:12 am

Hello Iqbal,

I hope this doesnt split the hair too finely...I think of SD as the tool
that leads to an understanding of the problem. Using SD to formulate the
problem statement leads to identifying the leverage points. SD may not
reveal the solutions themselves, but in a later phase, can be used to test
the feasibility of solutions.

Perhaps we are saying the same thing?

Bill
Bill Braun <medprac@hlthsys.com>
---------------------------
Medical Practice Systems Inc. (216) 382-7111 (Voice)
and The Health Systems Group http://www.hlthsys.com
Mergers - Planning - Management Services
Marketing - Managed Care - Education & Training

jforestr@MIT.EDU (Jay W. Forrest
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by jforestr@MIT.EDU (Jay W. Forrest » Thu Jun 05, 1997 7:58 pm

>From "William Steinhurst" <wsteinhu@psd.state.vt.us>
>Difficult and confusing, certainly. Unrealistic, often. But weak?
>Could you clarify?

JWF reply:

They do not deal usefully with nonlinearity or high order systems, and
especially not both together.

Jay W. Forrester
Professor of Management, Emeritus
and Senior Lecturer, Sloan School
Massachusetts Institute of Technology
Room E60-389
Cambridge, MA 02139
tel: 617-253-1571
fax: 617-252-1998

email: jforestr@mit.edu

ijamal@gov.edmonton.ab.ca (Iqbal
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by ijamal@gov.edmonton.ab.ca (Iqbal » Fri Jun 06, 1997 10:53 am

>I hope this doesnt split the hair too finely...I think of SD as the tool
>that leads to an understanding of the problem. Using SD to formulate the
>problem statement leads to identifying the leverage points. SD may not
>reveal the solutions themselves, but in a later phase, can be used to test
>the feasibility of solutions.

I believe so. "Learning from experiments" and designing "experiments from learning"
would seem to be parts of the overall cyclical relationships (loops) between learning,
problem definition and solution. Perspectives and use of tools would depend on where
you are in the cycle?

Cheers

............................................................................................................................

Iqbal Jamal
Office of Studies and Budget
City Manager Offices
City of Edmonton
1 Sir Winston Churchill Square
Edmonton, Alberta, Canada
T5J 2R7

(403) 496-8228
ijamal@gov.edmonton.ab.ca

Bruce Skarin
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Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Bruce Skarin » Sun Apr 07, 2002 4:27 pm

I know this topic has been pretty much been beaten to death, but I thought
I might share some final thoughts for those still confused about the
implications of this thread.

I liked Finns statement:

"Our models are often useful and interesting maps of the way the real
world works.

But the map is not the territory."

Like a few others I have spent probably far too much time contemplating
the relationship between rates of change and stocks. In reading the
discussions regarding temperature, photons, inductors, time, and the like
I couldnt help but think about entropy and the second law of
thermodynamics. Of course I think I slept through most of my chemistry
courses, so I visited: http://www.2ndlaw.com/ which offers a fairly
comical review of the subject. I was disturbed to discover I had been
mislead!

The reason I thought of entropy was because of that common misnomer that
everything in the universe is moving to chaos or disorder. If this were
true then regardless of having a "true" derivative, we might at least know
the direction of flow. I was somewhat surprised to discover that even this
principle is conditional. Moreover it is really more related to the
condition of two or more state variables! Yes heat or energy does tend to
spread out, but only if there is enough "space" for that to happen.

What I finally pulled out of this thread is that Jay was indeed
reinforcing an important point about dynamics. - In the case of any
differential equation there are always at least TWO integrations involved.
There is one stock that is decreasing and one that is increasing. As a
result, rates of change are only possible because of the conditions of
stocks.

Where I believe myself and others often get confused is that many of our
models are indeed only concerned with how things change. As a result we
often neglect that stocks or accumulations that provide the potential for
change.

Another confusing point for me is relativity. While examples like the
Doppler shift seem to provide "instantaneous" means for measuring a rate
of change, it is really only measuring the shift of a frequency relative
to a previous measurement.

There is certainly more to it than this, but I do believe that it is
incredibly important to scrutinize differential equations in respect the
related integrations (even if they are not to be included in the model
(clouds)).

-Bruce
From: Bruce Skarin <bskarin@WPI.EDU>

Mabel Fong
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

differential equations

Post by Mabel Fong » Sat Apr 13, 2002 3:32 pm

Hi Guys!

I have been following your postings with hopes of
discovering just how much I know about System
Dynamics. I have taken a course in Numerical Methods
and Control Theory from my local EE department in
hopes of becoming a better economic theorist; and have
taken the on-line course posted by SUNY Albany. But,
since formal SD offerings are not available on my
campus, I am wondering if my ‘little knowledge’ is not
perhaps a dangerous thing.

May I presume to test myself by entering the
discussion on whether or not derivatives exist? It
seems to me that they do not; and that the better
argument for this position would be rooted in Zeno’s
paradoxes. Instantaneous rates of change are
categorically beyond detection in the wild because
there are no such things as ‘points’ in time or space.
There are only volumes of space and intervals of
time. So long as space and time are no more than
abstractions we impose upon the universe (as opposed
to the quantum phenomena speculated upon by some
physicists) all volumes and intervals can be divided
without limit. The derivative would not be
calculable, even as an abstraction, unless we can
presume this property on behalf of time and space.

If this contribution has not revealed me to be a
complete idiot, I would very much like to test myself
further by inviting your consideration of certain
conundra arising from my attempts to get economic
principles operating within a dynamic perspective. Is
this the place for me?

Vty,

Mabel Fong
may_belle_66@yahoo.com

Niall Palfreyman
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Niall Palfreyman » Wed Apr 24, 2002 12:45 pm

Jay Forrester wrote:

> Mathematicians have had some difficulty defining a derivative, and
> there is a reason. There is no such thing. Nowhere in nature or
> human affairs is a true derivative to be found. Nature only
> integrates or accumulates. No instrument can instantaneously
> measure a rate; all such instruments contain some form of
> integration.

I really like this statement - its one of those upside-down-turning
statements which refreshes and sets me mulling and musing in the
shower and (unfortunately) at 3 oclock in the morning. I also _think_
I agree with it wholeheartedly, however I find I have a number of
issues with it which I cant quite clear in my mind. A list follows ...

1. "Mathematicians have had some difficulty defining a derivative."
Really? I find that surprising. Id have thought the definition of an
integral posed more problems than the definition of a derivative. Can
you say exactly what problems? And can you give me a general reference?

2. "Nowhere in nature or human affairs is a true derivative to be
found." A (possible) counterexample: The motion of a ball on a hillside
takes no notice of how high the hill is, but only of how steep the slope
is. I think my point here is: Could it be that your statement applies
only to rates, and not to spatial gradients? Im concerned with
modelling reaction-diffusion systems, and I find gradients fundamental
to what Im doing, since there it is the spatial concentration gradient
which drives the diffusive flows. However Im _very_ open to having that
particular mindset changed. Is it possible that the integrals of grad,
div and curl are more basic and practically useful than their
differential counterparts? Im all ears if you have any thoughts on this
issue.

3. "No instrument can instantaneously measure a rate." Im not sure its
possible to measure _anything_ instantaneously, is it? Any measurement
demands that the measuring apparatus relaxes to an equilibrium state
which represents the measured value, and the detection of this
equilibrium state demands two measurements which should be identical to
each other. I hope Im not just being picky here - its just that I have
a feeling that the instaneousness is not exactly the crux of the point
youre making. On the other hand, although I have an awareness that
there _is_ a crux, I cant quite get at what it is. Force is a rate, but
I can measure it in one by balancing it with a spring of known
elasticity.

Thank you for the food for thought. Id be interested in any answers to
these points.

Cheers,
Niall Palfreyman.
From: Niall Palfreyman <
niall.palfreyman@fh-weihenstephan.de>




carolus
Junior Member
Posts: 18
Joined: Wed Mar 31, 2004 5:14 pm

Differential equations

Post by carolus » Fri Apr 26, 2002 6:09 pm

SDrs,

Ed Gallaher posed a very interesting question:
* Can anyone provide examples that contradict Jays assertion?

in which Jays assertion is:

>>> Mathematicians have had some difficulty defining a derivative, and
>>> there is a reason. There is no such thing. Nowhere in nature or
>>> human affairs is a true derivative to be found. Nature only
>>> integrates or accumulates. No instrument can instantaneously
>>> measure a rate; all such instruments contain some form of
>>> integration.
>>
How about time?
A derivative is defined as an instantaneously measured rate,
indicating a change of something over time.
Velocity, for example, is the derivative of position: change of position
over time.
If we really want to measure _instantaneously_ velocity, we have a problem.
We need at least a fragment of time in order to know the change of position.
And if we make this fragment of time zero, there is no movement: the world
freezes.

But, what if we want to know the change of time over time?
Maybe this sounds absurd. Who would like to know?
I suppose that this derivative in fact tells us what time it IS:
in hours, minutes and seconds since midnight.
Isnt this the meaning of the statement: it is 5 oclock (monday-morning)?

Probably the english language strengthens the problem here at hand.
After all, time is used in two different meanings: time as changing unit
and time as a value

To contradict this, one could pose that time is only a human invention:
it doesnt really exist. We live in a permanent present and we cant get
hold
of time in nature: time is not really accumulating; the invention of
time is
just passing by..
The fact, however, that we are able to remember, links memories to
(a.o.) a certain moment in time, thus providing us on time=t with feedback
on things that happened at time=t-x.
That implies that the past exists, or did exist, and we refer to it using
(dates and) clock references.

Am I totally wrong in suggesting - then - that the statement that
something happened at the moment that my calender/clock showed
13.46 hrs (GMT), september 11th in the year 2001
is a true instantaneoulsy measured derivative?


--
m.vr.gr.,

mr Carolus Grütters
Law & IT
http://www.jur.kun.nl
it/

Centre for Migration Law
http://www.jur.kun.nl/cmr/

University of Nijmegen
Nijmegen, The Netherlands
email: c.grutters@jur.kun.nl

"Jay W. Forrester"
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by "Jay W. Forrester" » Wed Jul 17, 2002 9:14 pm

Several writers have recently implied that differential equations are
a more fundamentally correct way of representing dynamic systems than
the integrations in a system dynamics formulation. As examples,

Christopher Barker wrote:
> Standard SD approaches use stock formulae that attempt to
>approximate the differential equations underlying the
>system......... example is well outside any attempt to approximate
>the systems differential equations,

Neil Douglas wrote:
>The algorithms used by the software (most commonly the Euler
>algorithm) seek to solve this system in a numerical fashion (most
>differential equations of interest to a system dynamicist are
>insoluble analytically).

I believe the view that differential equations are more theoretically
founded than the integral formulations should be reversed.
Differential equations is a difficult subject. Mathematicians have
had some difficulty defining a derivative, and there is a reason.
There is no such thing. No where in nature or human affairs is a
true derivative to be found. Nature only integrates or accumulates.
No instrument can instantaneously measure a rate; all such
instruments contain some form of integration.

The adoption of differential equations is perhaps the greatest
disservice ever to the understanding of behavior. As one evidence of
the unreality of differential equations, observe that, when one wants
to solve such an equation on a computer, it must be converted to a
set of integrations.

In many students, differential equations inculcate a reversed sense
of causality. I have had MIT students argue that water flows from
the faucet because the water in the glass is rising, instead of the
water flowing because the level of water in the glass is not yet to
the target value. After all, the flow is the derivative of the water
level, so the flow is because the water is rising. They are often
unable to see the distinction.

Students are able to work with higher order nonlinear feedback
systems at the 5th grade level because of the integration approach.
Any student who can fill a water glass or take toys away from a
playmate knows what accumulation or integration is.

Integrations and the feedback around them are the fundamental
structure of the real world. Differential equations are a perverse,
unrealistic creature of the mathematicians imagination.

This viewpoint was first called to my attention by Gordon S. Brown
who was my mentor at MIT, director of the Servomechanisms Laboratory,
and later head of the Electrical Engineering Department, and Dean of
Engineering.
--
---------------------------------------------------------
From: "Jay W. Forrester" <
jforestr@MIT.EDU>
Jay W. Forrester
Professor of Management
Sloan School
Massachusetts Institute of Technology
Room E60-156
Cambridge, MA 02139

"Raymond T. Joseph"
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by "Raymond T. Joseph" » Sun Jul 21, 2002 5:23 pm

After much work in systems theory, I read JWFs books on industrial dynamics
and gained insight into viewing my environment. This has been a basis of
much of the work I have done since. As such, I would like to contribute
something. I hope the following helps.

It doesnt matter whether nature builds differential equations (DEs) or
mathematicians can appropriately define derivatives. The fact that systems
can be well described by DEs is sufficient to use them as a tool not only to
predict the behavior of a system but to learn about the relationships
between subsystems.

Just the concept that nature integrates (well) suggests that DEs are
appropriate for dynamic system modeling. Integration is how DEs are solved
/ simulated. An appropriate analogy is that if nature integrates, the
system must be a DE - only DEs are solved by integration.

The statement "Differential equations is a difficult subject" may be
controversial, but it is irrelevant as to the applicability in modeling
systems. Rather than contrive a relationship of system components, it would
be more productive to teach modelers simplified methods of identifying,
exposing, and representing the DEs. Actually, there is a large set of
definitions of derivatives - not just the simple functional differences
learned in the first year of calculus but pseudo derivatives for
discontinuous functions and derivatives in non-Euclidean spaces (quite
suitable for systems with specific, pervasive non-linearities).

Derivatives are found (in abundance) in nature. And thus provide
derivative meters. For example, an (electrical) inductor has the
relationship that a change in current flow through it creates a voltage
across it. Change the speed of an electron in motion and it radiates
electromagnetic energy in proportion to the change (an example is the free
electron laser). A persons interest (attention) in a particular media
presentation is proportional to information rate. In general, purchases are
typically discounted in proportion to the purchasing rate.

Are these pure derivatives? . . . Does it matter? Almost all derivatives
in nature are bandlimited. That is, a derivative is taken with respect to
some variable up to a certain rate of change, then some other mechanism
takes over. But again this is fine, we usually know at what point this
occurs and can account for it in the model, or the model is never operated
in that region.

* As one evidence of the unreality of differential equations, observe that,
* when one wants to solve such an equation on a computer, it must be
* converted to a set of integrations.

As I stated above, a DE is solved by integration. Another analogy is that
between the calculus and algebra. We know that the area of a rectangle is
the product of the length of two adjacent sides. If we want to find the
length of one of the sides when we know the area and the length of the other
side, we divide the known area by the length of the known side. The
solution is achieved through division. This does not mean that the
underlying system does not follow a multiplicative process.

* . . . students argue that water flows from the faucet because the water in
the glass is rising,
* instead of the water flowing because the level of water in the glass is
not yet to the
* target value. After all, the flow is the derivative of the water level,
so the flow is because the
* water is rising. They are often unable to see the distinction.

Water flows because the pressure upstream of the faucet (valve) is higher
than the pressure downstream of it and the flow conductance of the valve is
non zero. So the problem here is not one of derivatives and integration but
incomplete problem definition. The causality between the flow and the level
in the glass is the feedback controller who monitors the water level and
actuates the faucet valve.

* Integrations and the feedback around them are the fundamental structure of
the real world.
* Differential equations are a perverse, unrealistic creature of the
mathematicians imagination.

Yes, DEs are a perverse recreation of mathemagicians;~) As in playing a
violin, one only gets better with practice. Disciplines that live by DEs
get more practice and thus are more productive. We can see a great example
of this in the electronics industry where a large portion of each
advancement is based on DEs. And what has been the result? Moores Law, a
doubling of performance every 18 months. How does this compare with
disciplines that force ever increasing complex structures such as neural
nets to describe systems in stead of using differential relationships?

Actually, all of this is irrelevant. Differential relationships are
modeled by derivative and integral operations only due to our bias in
choosing systems of measurement. For example, when looking at the lines on
a sheet of ruled paper, we typically state that the lines are spaced 1/4
inch apart. We could say that there are four lines per inch. This is the
difference between looking at how a variable changes in the time domain
versus the frequency domain. As in Fourier and La Place transforms, the
differential operators change from derivatives and integrals to
multiplication and division. It all the same.

It is only perception - - All is light, Everything is decaying exponentials.

Raymond T. Joseph, PE
rtjoseph@ev1.net
Aarden Control Engineering and Science

Tom Fiddaman
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Tom Fiddaman » Wed Jul 24, 2002 1:25 pm

>Derivatives are found (in abundance) in nature. And thus provide
>derivative meters. For example, an (electrical) inductor has the
>relationship that a change in current flow through it creates a voltage
>across it. Change the speed of an electron in motion and it radiates
>electromagnetic energy in proportion to the change (an example is the free
>electron laser). A persons interest (attention) in a particular media
>presentation is proportional to information rate. In general, purchases are
>typically discounted in proportion to the purchasing rate.

Ive long suspected that Jay is right and that inductors actually are
fundamentally integrators, but with a short time constant (speed of light /
size of device?). Perhaps someone on the list can shed a little light (or
synchrotron radiation) on this.

However, leaving the domain of physics, I think the last two examples at
best have no bearing on Jays assertion:
- A persons interest in a particular media is proportional to their
perception of the information content, which they can determine only by
observing the stream over some period of time.
- The fact that bulk purchases are discounted has little to do with
dynamics; its just a nonlinearity. Purchases are discounted not with
respect to some instantaneous purchase rate, but according to the size of a
particular transaction or expected size of a group of future transactions -
which sounds like a stock to me either way.

I have yet to see a counter-example to Jays proposition in the sphere of
human affairs, though there are of course times when its a modeling
convenience to assume instantaneous measurement or perception.

Tom
From: Tom Fiddaman <tom@vensim.com>

Ed Gallaher
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Ed Gallaher » Thu Jul 25, 2002 1:52 pm

>Jay Forrester wrote:
>> Mathematicians have had some difficulty defining a derivative, and
>> there is a reason. There is no such thing. Nowhere in nature or
>> human affairs is a true derivative to be found. Nature only
>> integrates or accumulates. No instrument can instantaneously
>> measure a rate; all such instruments contain some form of
>> integration.

I heard these statements from Jay some years ago, and they were
startling! In medicine, we often hear that cells respond to the rate
of change of some sensory input, which would appear to contradict
Jays assertion.

Also, I contemplated my speedometer, and the police radar gun. Of
course we can "instantaneously" measure a rate!

Or can we . . . ?

Lets see; some mechanical speedometesr include a rotating cable
acting on a centrifigal weight to move a needle. So we are really
measuring the accumulation of centrifugal (centripital?) force.

An electronic speedo acts as an electrical generator to produce a
current, which in turn accumulates in a capacitor. The resulting
voltage across the capacitor is then presented on an analog
voltmeter. Again, the message that reaches our eyes has at its
foundation an accumulation.

The radar gun provides an "instantaneous signal" to the policeman,
indicating the cars speed as it goes by. But how does radar work?
The signal leaves the source, reflects off the car, and returns to
the source indicating "distance A from the source"; the next signal
indicates "distance B from the source". Positions A and B are the
result of integrating velocity over time. The difference between A
and B, divided by a very short, very accurate time interval, provides
the measurement of velocity . . . and leads to the traffic ticket
|:-(

Can anyone provide examples that contradict Jays assertion?

Ed Gallaher
From: Bill Harris <bill_harris@facilitatedsystems.com>

Jay Forrest
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by Jay Forrest » Fri Jul 26, 2002 10:17 am

In his response, Niall compares induced voltage
otation of an electric
field to velocity. As a ChE I am out of my element with voltage. I am
confident I understand the concept of velocity better than voltage, so I
began parsing Nialls comments with Velocity.

Velocity seems more like an arbitrary property, like temperature, than a
tangible entity. For a specific mass I could have a velocity stock
(effectively inertia). While velocity is a /dt function, it would seem to
operate by integrating "nudges" of positive and negative force (propulsion
and friction for example.

Personally I think of voltage as a tangible entity (well at least static
voltage) based on a population of electrons. I dont know if induced
voltage is "tangible" or not. In any event I am not sure tangibility makes
much difference. If induced voltage exists only as a direct function of
d(LinkedFlux)/dt that seems to suggest that induced voltage is
proportionate to d(flux). A changing flux would seem to imply something
causing it (the flux) to change so that the induced voltage is an
integrator of the changing forces of the flux field (with appropriate
conversion factors/etc.) As a result d(flux)/dt (and thereby induced
voltage) seems to be an integrator of external electromagnetic forces.

Seems like the fact a /dt is involved is relatively immaterial and
misleading as to the "derivative /integrative nature of induced voltage.

Looking forward to EEs helping me understand what I am missing!
Jay Forrest

From: Jay Forrest <jay@jayforrest.com>

hazhir@MIT.EDU
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by hazhir@MIT.EDU » Fri Jul 26, 2002 10:56 am

Niall Palfreyman asks:

>What exactly is it that gives me the gut feeling that (for example)
>distance is "more fundamental" that velocity? It has something to do
>with ease of measurement, but I dont quite know what.
>

This question sounds interesting to me: We say that nature does not
differentiate, it seems to me that the nature we are talking about is not
"The Nature" itself, but how we perceive it based on our senses, (socially
constructed) concepts and words. We dont have direct access to "Nature" to
know what it does (in fact differentiation and integration are also labels
and categories we are using to cut the world name the "Nature" (I am an
engineer by training, so I dont go too far to say: Create the "Reality"
:-) )). In the evolution of our culture, language and categories we use to
perceive the "Nature", we are probably biased towards making these
categories and concepts based on what we can perceive: I cant perceive the
instantanous rate of flow in the river (because of how biologically I sense
things), so I cant discribe it to my fellow caveman 20000 years ago, so I
will come up with the term water
iver etc much before I come up with
anything to name its rate of flow... as a result level variables become
more fundamental in how we see the world than rates. Then it takes a long
time for the evolution of the culture and language to feel the need for
naming rates of change, thats how the derivatives become secondary, dont
seem "Natural" and we say "Nature" does not differentiate.

Just a few thoughts...

Best,
Hazhir




Hazhir Rahmandad
Ph.D. Student
System Dynamics Group
Sloan School of Management
M.I.T
E53-364A
hazhir@mit.edu

"Raymond T. Joseph"
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by "Raymond T. Joseph" » Sat Jul 27, 2002 1:49 am

> Ive long suspected that Jay is right and that inductors actually are
fundamentally integrators

I have never heard Jay say that but then I live 2000 miles away. As stated
earlier, the voltage drop across an inductor is proportional to the time
rate of change of the current through it - the derivative. I cant image
any one mistaking the derivative for an integral. Rather than blaming Jay
for the statement, please explain your understanding.

As previously stated, it is all totally irrelevant. Nature doesnt
differentiate or integrate; nature doesnt even sum. We are the only
mathematicians. We see patterns in our environment (correlations) and
propose mathematical models that seem to make sense of it all. This is all
totally irrelevant to nature; it is our construct. If it works, that is
sufficient.

But all of this is off the point of Jays original post:
>. . . that differential equations are more theoretically founded than the
integral formulations should be reversed

Again, this is just a loss of direction. These arent two opposing
methodologies; they are tightly coordinated processes. The differential
equation (DE) is a statement of the problem; the integral formulation is
the solution method to the problem. This is the fundamental relationship
here. Can anyone suggest how this is not the case?

Well, one could jump in and write down a solution (integral formulation) to
simulate the operation of a system. But then one may ask what system this
is a simulation of. Another person may state the problem in terms of DEs
and another person may state it as a relationship between input and output
behavior. Either way, a system has been described. Neither one provides a
solution. These are only problem statements. By writing "down a solution
(integral formulation)" one is only conveying a solution method to
something that has an integral formulation solution that looks like this.
This is not a very good communication method.

Further more, by writing down the DEs, many characteristics of a system can
be deduced with out doing a simulation. Rejecting all this information
because the subject seems difficult is very short sighted. The idea is to
help people understand systems. If the only method of understanding a
system is through simulation, we are in deep sh . . . trouble. Exhaustive
searches would have to be done to find solutions to real problems. Due to
problem size of real (interesting?) systems, exhaustive searches would take
too long to solve if they could ever be solved.

A system could be simulated by integral formulations. Or the problem could
have been stated in another measurement system (say inverse time, i.e. 1/t)
and the solution could be carried out algebraically. We could come up with
numerous other methods but two should be sufficient to do a comparison. But
these are only mathematical constructs to describe a system. Nature doesnt
know how to read any of these methods. Nature doesnt do any math. It is
useless to say one method is better than another because this is the way
nature does it. Nature doesnt do math!

Having fifth grade students simulate systems with integral formulations is a
great way to give them a feel for dynamics, it doesnt mean that they have
all the tools necessary to solve the problems of their lives.

Raymond T. Joseph, PE
rtjoseph@ev1.net
Aarden Control Engineering and Science


"Raymond T. Joseph"
Junior Member
Posts: 14
Joined: Fri Mar 29, 2002 3:39 am

Differential equations

Post by "Raymond T. Joseph" » Sat Aug 03, 2002 4:51 pm

<snip . . snip>
> Am I totally wrong in suggesting - then - that the statement that
> something happened at the moment that my calender/clock showed
> 13.46 hrs (GMT), september 11th in the year 2001
> is a true instantaneoulsy measured derivative?
mr Carolus Grütters

Lets see if I can differentiate the meanings of the above:
1) The statement is an instantaneously measured derivative
2) Or, the happening at the specified moment is an instantaneously measured
derivative

1) The statement marked a concept in time and space. It didnt show me a
value of a rate of change.
2) The happening was driven by change of time, where as you stated, would
not have happened if time was not changing. Is being driven by a derivative
the same thing as measuring the derivative? Measuring something implies
that a comparison can be made between different instances and being able to
show some ordered relationship. An example might be that water flowing out
the end of a pipe projects farther distances when the flow rate is high as
compared to when the flow rate is low. As to the occurrence of an event
allows us to detect that time, in fact, has changed, it doesnt tell us how
much it changed. And, for everyday experiences, it is unlikely that we
would ever detect time occurring at different rates. Where as we see
occurrences all day long, I dont see them providing a means to detect dt/dt
having different values.


<snip>
* However with respect to spatial gradients Im still flummoxed,
* and Id _love_ to see some examples and thoughts from people regarding these.
* Niall Palfreyman.

OK, lets look at the pressure in a vertical pipe filled with water. Say the
pipe is 10 tall. At the base of the pipe, I would measure 10 ft of
hydraulic head. One foot above that, I would measure 9 ft of head. Of
course, these can each be made an instantaneous measurement with respect to
time. But by definition, they are not local measurements - measurements at
a true point. Pressure has to be measured over an area. In the domain we
are working in (height of the pipe) the measurement takes place over a
finite extent of that domain. As such, it is not instantaneous, not local,
it is an extended measurement.

Now that we see this, what is the relevance of a measurement being extended?


<snip>
> single cell "differentiation". What is the accumulator or flow in this case?
Warren Tignor

Cells express themselves through their active DNA. This expression is what
makes one cell different from another - differentiation. Not all of a cells
DNA is immediately available for use (expression). Most of it is tied up so
it cant be transcribed to RNA. A large variety of chemicals directly
attach to the DNA to inhibit expression. Thus it is the cells chemical
environmental history that determines the current expression -
differentiation.

For example, in embryonic development, when there are only a few cells,
their environments look pretty much the same. As soon as there are enough
that some are relatively inside the group and others are on the outside
edges, their environments are different and they begin to differentiate.
The stocks are the chemicals. More specifically, the chemical
concentrations. Being living entities, the cells produce chemicals from
substrates. They produce these at certain rates dependant upon, wouldnt
you know, the chemical environment and recent history of nuclear
expressions. Note that temperature, pressure, light and other things can
effect the chemical environment.

<snip>
* If induced voltage exists only as a direct function of d(LinkedFlux)/dt
* that seems to suggest that induced voltage is proportionate to d(flux).
* A changing flux would seem to imply something causing it (the flux)
* to change so that the induced voltage is an integrator of the changing
* forces of the flux field (with appropriate conversion factors/etc.)
* As a result d(flux)/dt (and thereby induced voltage) seems to be an
* integrator of external electromagnetic forces.
<snip>
Jay Forrest

OK, lets start with some EE IDs:
Inductor -
1) volts = d flux /dt
2) flux = inductance * current

>From 2) (assuming constant inductance)
d flux / dt = inductance * d current / dt
substituted into 1)

3) volts = d flux /dt = inductance * d current / dt

Capacitor -
4) current = d charge / dt
5) charge = capacitance * volts

>From 5) (assuming constant capacitance)
d charge / dt = capacitance * d volts / dt
substitute into 4)
6) current = d charge / dt = capacitance * d volts / dt

These are the equations that model the behavior of these two devices. In
order to calculate the specific behavior of such a device under a given
condition, a modeler might choose to integrate equation 3) to find the
current in the inductor due to a given voltage input. Or integrate equation
6) to obtain the capacitor voltage due to a given current input.

But lets say we have a capacitor and we want to know the current that will
be produced by subjecting the device to a given voltage input. What do we
do? Well, we know
current = capacitance * d volts / dt
To simulate the behavior, we can differentiate the voltage input, multiple
by the capacitance and we would find the current. What is the difficulty
here. Well, if the voltage has sudden jumps, the derivative is not defined.
What does this mean? It means that the voltage across a capacitor can not
change instantly (jump), it must change smoothly.

>From the capacitor example, if we are having a problem because we are
required to differentiate a discontinuous signal, maybe the problem isnt
that math doesnt define this derivative but that we are asking the physical
system to do something it can not do - it is not physically realizable.


I am afraid that Finn Jackson has removed all the fun by recognizing that

* [Jay is correct in that] the response is "of course not, because in order to
* measure a rate per time you have to have allow at least a little it of time to pass"

So lets put some fun back in. Can an integral be measured instantaneously?
Oh course not. It would be meaningless to as "what is the integral over the
last zero instant". The answer is always zero. So if an instantaneous
measurement is a requirement for a derivative to be real, is it not a
sufficient question for the reality of the integral?

Now that we have played this game for a while, does anyone find any
relevance in the question of the ability to take an instantaneous
measurement?

> . . . it seems to me that the nature we are talking about is not
> "The Nature" itself, but how we perceive it . . .

It looks like Hazhir has captured the essence here. Since each of us has a
different perception of the world around us, we must have a different
perception of Nature. If Bobby says nature doesnt differentiate, he is
right - his view of nature does not differentiate. If Sue says that nature
does sum, she is right - her view of nature doesnt sum. To make a blanket
statement that nature is . . . (fill in the blank) takes us way back to the
caveman where the world revolves around me - I am the most important thing
in the universe.

All of this is just perception and thus subjective. We can make it anything
we wish. Does it make sense to try to arrive at a consensus of our mutual
views of nature? I think we have already identified that the views are
significantly different.

If we wish to convey understanding to others in the simplest terms possible,
relying on subjective reasoning would require that we make everyone the
same - the same experiences, training, . . . whatever. Then we would all
have the same subjective view of nature and we could all agree on what it
looks like. Further, we could build models that would be intuitively
understood by everyone.

It is unlikely that we could (even if we wanted to) give everyone the same
experiences. Thus it would be beneficial to base our conveyance of modeling
methodology on something a little less subjective. Dont you know, someone
has already started such a method. It called math.


Raymond T. Joseph, PE
rtjoseph@ev1.net
Aarden Control Engineering and Science

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