Transcript Slide 1

Charlotte, North Carolina
Friday, 17 December 2010
An Introduction to the Concepts and
Techniques of Mathematical Modeling
Ben Fusaro
Department of Mathematics
Florida State University
[email protected]
An Introduction to the Concepts and
Techniques of Mathematical Modeling
This workshop has three objectives –
➢ Introduce the concepts and techniques of modeling.
➢ Demonstrate that modeling is a foundation for science,
technology and engineering.
➢ Encourage HS faculty to get their students to take part
in team-oriented applied mathematics contests, such as
COMAP’s Hi-MCM and Moody's Mega Math Challenge.
------------------This workshop is sponsored by the Society for Industrial & Applied
Mathematics. SIAM is funded for this and similar educational
activities by The Moody’s Foundation.
Presentation © 2010 Dr. B.A. Fusaro
Outline
Pure and applied problems – a background.
Definition(s) of model.
Examples.
The modeling process; more examples.
Visual, qualitative, computational.
The Storage Tanks and Flows.
Systems concepts.
A five-stage modeling process.
A Flow problem, Midge Classification, & a Steiner Tree.
A (Very) Brief History of Modeling
Early
Euclid of Alexandria
Aristarchus of Samos
Archimedes of Syracuse
Middle
Kepler (Germany)
Galileo (Italy)
Newton (England)
WWII +
J.G. Kemeny (Hungary), 1926-1992
H.T. Odum (U.S.A.), 1924-2002
B.B. Mandelbrot (Poland), 1924-2010
Have you ever heard of a job where the manager says
something like:
“Here are five (or seven, or ten) problems. Solve them
in 75 minutes and do not -consult with anyone,
refer to your notes,
open a book,
use a grapher/computer,
or
search the Web.”
David Strong, UCLA, ~1998
Genesis of the Math. Contest in Modeling (MCM),
Hi MCM, and the Moody’s Mega Math Challenge.
The Putnam
The Applied Putnam
Clear, crip, precise problems
Real(istic), open-ended;
usually needs clarification.
Unique solutions
Solutions
12 problems, one day
One problem per weekend
Individual
Team
Paper & pencil
Web, books, computers, ...
Brief, symbolic answers
“Executive Summary”
Modeling challenges are often -Ambiguous or incomplete
Require the modeler to make additional assumptions.
Tend to be “open-ended”.
Have multiple, “good enough”, or no solutions.
Require a team to deal with them.
Pure Problems & Modeling Problems
How many three-digit numbers are composed of three
distinct digits such that one digit is the average of the
other two?* Answer: 112.
Chasing the Biofuels Illusion
What is the maximum tonnage that can be expected
from a cord of Southern White Pine...?
______________
* This problem appeared on p.13 of the MAA FOCUS, Oct-Nov
2010. The solution is on p. 22. (It came from the Wells & Faires,
Contest Problem Book IX, MAA 2008.)
Pure Problems and Modeling Problems
A Nutty Problem
(The Brazil Nut Effect)
Photos: Wikipedia
Consumers are often disappointed, as they move down a
can advertised as “No more than 50% peanuts”, to find a
large proportion of peanuts. The FDA has verified that the
labeling is accurate -- the nuts are uniformly distributed
when packed.
Explain this migration of nuts & suggest a solution.
Include an executive summary with your report.
What do most people have in mind when the word
model is used...?
1980 Mr Olympia
Arnold, the Gov
1976 Miss Wales
Sian Adey-Jones
which is not what quite what we have in mind...
A model of an object, process or system is a representation
of it that preserves relevant properties or relations.
➢ A map of Greensboro, NC
➢ A physical replica of the Titanic
➢ Medusa as an octopus (Greek mythology)
Carvaggio, 1597
We would like to tighten our definition of model ...
Animal
“Models”
What is a model of an object, process, or system?
The original definition --
“A model of an object, process, or system is a representation that preserves relevant properties and relations”
is a bit loose for a mathematical model. We want to exclude
physical models, so-called animal models, a photocopy of
a photocopy, etc. Here is our modified definition:
A (mathematical) model of an object, process, or
system is a relatively more abstract representation
of it that preserves relevant properties & relations.
A Visual representation of
Threats to the ozone in Costa Rica
Another Visual Representation of
Threats to the ozone in Costa Rica
Roy Beck, 1996
Numbers USA
Bar Chart of Energy (Kw-Hr) & Line Graph of Temperature (oF)
Energy, Population and the Environment
Digraph from F.S. Roberts, Rutgers University, NJ
Precedence Diagram for a Holiday Dinner
“How to Model It”, Starfield, Smith & Bleloch, 1993, ACM Digital Library.
What the word model might suggest to a mathematician ...
A Matrix
The NYC area has three airports -Kennedy, LaGuardia & Newark.
Avis leases parking space for 500 cars
at each one. A driver may pick up a
car at any airport and return it to any.
The numbers in the table give the
chance – in percents – that a pickup at
an airport will end up at K, L or N.
Return
P
i
c
k
u
p
% K L N










K 80 10 10
L 30 20 50
N 20 60 20










[ 278.5, 114.5, 107.0 ]
H. Anton, Applied Finite Math., 1988
What the word model might suggest to a mathematician ...
Ordinary Differential Equations
Mechanical System -mass on a spring
Electrical System -L-R-C Circuit
mD2 y  bDy  ky  F (t)
LD2Q  RDQ  1 Q  E(t)
C
What word “model” might suggest to a mathematician ...
Maxwell’s Equations for the ElectroMagnetic Field
E  (1/ c) B/ t
B  (1/ c) E/ t  (4 / c) J
E  4
B  0
E is the electric field, B is the magnetic field
J is the current density, ρ is the charge density
The Modeling Process
The Modeling Trade-off
Simplicity
Structure
Complexity
Information
Generality
Specificity
The Modeling Process – another view
Computer Predictions, Oden, & Moser & Ghattas, SIAM News, November 2010
Some Hydrocarbon Fuels
Molecular or algebraic formulas for alkanes -Methane
C1 H 4
Ethane
C2 H 6
Propane
C3 H 8
Butane
C4 H10
(Natural or Marsh gas)
(“Bottled” gas)
Structural or graphical formulation
Methane
Ethane
Propane
Butane
=C,
=H
How to go from Butane
C4H10
to the “next” hydrocarbon, Pentane ... ?
Pentane
C5H12
We all use plenty of one alkane in our cars -Octane.
( 87 Octane gasoline means that 87% of the
fuel is octane.)
Write down two representations for Octane:
Molecular (Algebraic)
Structural (Graphical) .
How to go from a typical structure with n Carbon atoms
to a general Molecular formula . . . ?
Drop the H’s at the ends, leaving two H’s for each C. This
gives 2n Hydrogen atoms, & the formula becomes CnH2n .
Add back in the two H’s at the end to get CnH2n+2 .
From Alkanes to Alcohol
Replace the ball-and-stick model of methane C1 H4
by a simpler, more explicit model -H
Then replace the right hydrogen
H
C
by an OH group (hydroxyl) to get
methanol, C1 H3 OH.
H
OH
H
Replaces the corresponding hydrogen atom in ethane C2 H6
by an OH group yields ethanol, C2 H5 OH.
Ethanol is also known as grain alcohol.
Some species do not interact, such as water
buffaloes and cheetahs, while other species do,
such as lions and hyenas. How many two-way
interactions are possible among n species . . . ?
n=2
n=3
1=1
2+ 1 = 3
How about
n = 4 species?
3+2 +1=6
. . . and
n = 5 species?
4 + 3 + 2 + 1 = 10
The number of possible two-way
interactions among n species
Species
Interactions
2
1 = 1
3
2+1 = 3
4
5
3 + (2 + 1) = 3 + 3 = 6
4 + (3 + (2 + 1)) = 4 + 6 = 10
6
n
Closed form: C(n,2) 
n!  n (n -1)
2
2!(n -2)!
Three Deadly Sins of Modeling
Confusing curve-fitting with modeling.
Identifying curve-fitting is modeling.
Passing off curve-fitting as modeling.
Fitting a curve to data before “looking out
the window” is a branch of pure analysis.
After the context, meaning or qualitative aspects of
data are examined, curve-fitting could be a next step.
Date 19
oF
109
20 21
22 23 24 25 26 27 28 29
113 114 113 113 113 120 122 118 118 108
Phoenix, AZ
Some method -- cubic splines, Fourier series, a 6th
degree poynomial -- fits an attractive curve . . .
However...
Phoenix, AZ
. . . diurnal temperature tends to be oscillatory . . .
Visual
A blueprint, graph, map, pattern, picture, sketch, ...
Qualitative
A “wise-guy” definition –
Know the answer to a problem before you solve it.
Computational
Calculations, operations done on a calculator or computer.
The Storage Tank
A tank represents a storage device. The letter Q will
stand for the quantity of content or stock in this device.
Loss
Gasoline evaporates
Money is taxed
Soil erodes
Output (& Loss)
Gasoline is pumped
Money is withdrawn
Soil grows crops
Tanks and Flow
The tank symbol is used to indicate any storage
where outflows are proportional to its contents - Radioactive material decays in proportion to the
material present.
 Leaf litter is decomposed by fungi and bacteria
in proportion to the quantity of leaves.
 Many of us spend discretionary savings in
proportion to how much money is in our account.
The Storage Tank
Outflow is
proportional
to the contents Q
Qualitative Graphing
The Storage Tank
Outflow is
proportional
to the contents Q
Qualitative Graphing
The Loser -- a Sitting Car Battery
We start with a fully-charged battery,
Q0 = 120 amp-hr. It is known that a
sitting battery loses ~5% of its energy,
or charge Q, each month.
How much of the original charge , in
amp-hr, will be left at the end of -a) 1 month; b) 3 months; c) 1 year ...?
At the end of the 1st month, 95% = .95 of 120 amp-hr
is left, or 114 a-h. At the end of the 2 nd month,
95% of 95% = .95 ×.95 = .95 2 of 120 a-h is left.
Real-world Tanks Leak
We often think of a leak in terms of a liquid or a gas, as in the
BP Gulf oil leak, or air leaking from a tire. Anything that is
stored (banked, cached, saved), can suffer a loss from a leak -❖ energy, as in a drafty house or un-insulated water heater;
❖ soil, due to erosion, over-plowing, or toxic chemicals;
❖ money, as in theft, or in taxation that returns no services;
❖ information, due to noise, interference, other scrambling;
❖ structure, as in an ill or aging animal, a termite-infested
house, or a social group with a troublesome member.
A system is a set of interactive parts
Pair of pliers
Needle and thread
Pond
Social club
Drawing compass
Power grid
Tic Tac Toe
Pig Latin
Candle
Chess
Latin
Cat
Systems of interest are usually those
that are complex enough to give rise
to emergent properties.
Some System Properties
Closed and Open Systems
Taxonomy
Local, Global
Stability
Feedback
Self-organization
Unintended Consequences
Scale
Invariants
Closed and Open Systems
Stock, Storage, or Tank
No energy exchange
Energy exchange
Closed and Open Systems
An unplugged refrigerator and the solar system approximate
closed systems (over relatively short periods of time).
A candle imports oxygen and exports CO2, C and heat.
A pond takes in air, water and sediment, while yielding
fish, insects and water plants.
Pyramid of Trophic Levels
J = joules (1 Cal = 4186 joules)
How much solar energy does it take a coal-fired
power plant to produce one kw-hr of electricity ...?
The vertical path to the “ground”, or sink, is lost energy -thermal, chemical or biological pollution.
The take-home message: It requires 120,000 kw-hr of solar
energy to produce one kw-hr.of electricity –
Input :Output = Solar:Elec. = 120000 :1
Classification, Taxonomy
... provides a framework for thought & perceptions.
Chemistry
The Periodic Table of Elements
Acid, Neutral, Alkali
Endothermic, Exothermic reactions
Physics
Kinetic or Active Energy -Waterfall, arrow in flight, lava.
Potential or Passive (stored) energy -Lake, stretched bow, magma.
Taxonomy
Two Kingdoms of Life
(Carl Linnaeus, 1707-1778)
Five Kingdoms of Life
(R.H. Whittaker, 1920–1980)
Taxonomy
The Periodic Table
Two rows (*Lanthanides, **Actinides) are not shown.
Local, Global
Business
short range
long range
Economics
micro
macro
Military
tactics
strategy
Geography
local
regional
global
Stability – How to roll with the punch
Stable or robust system
Small changes in the context or initial
conditions cause small changes in its behavior.
Unstable or sensitive system
Small changes in the context or initial
conditions cause large changes in its behavior.
Stability (cont.)
J. Gleick wrote in his book, Chaos, that a Chinese
butterfly's flapping wings could trigger a tornado in
the Midwest.
His book title comes from a term for certain complex,
unstable systems -- chaotic systems.
For the want of a nail the kingdom was lost.
Nail got loose ... horse lost shoe ... horse got lame ...
message was late ... defeated in battle ... kingdom lost!
(Capewell nail)
3’’
(In)Stability Demonstrated by an Ordinary Box
Spin the box around its shortest major axis and toss
the box in the air. Try to gain height & avoid wobbling.
Repeat the action for the longest major axis.
Now, do the same for the intermediate major axis.
Feedback
A Feedback Loop or cycle is a sequence of responses
and co-responses that modify behavior.
In a Laurel & Hardy comedy film, Tit for Tat -Hardy taps Laurel, who responds with a some-what
stronger tap.
Hardy returns with a still stronger tap, and soon they
are doing serious escalation.
The responses keep getting stronger, and the violence
spirals up.
Feedback
A spotted hyena breaks a tooth. This will make it harder to
crush bones, an essential source of calcium. The tooth structure
will tend to weaken. This will make it still harder to crush
bones, so the tooth structure will weaken still more. The hyena's
bone-crunching performance (and vitality) spirals down.
A Question of Scale
How volume and surface scale with length (radius)
How area
scales
with length
Invariants
Stream meander
Run / Width = 10
Two common invariants -C
C = πD
D
C/D=π
= 3.14159…
The speed c of light in a
vacuum or “empy” space is
c = 300,000 km per sec.
Invariants
How many heartbeats in a lifetime?
Felis cattus (180)
Loxodonta africana (30) Wikipedia
Homo sap. (60)
A Five-Stage Modeling Process
1) Draw a diagram
2) Plot a qualitative graph
3) Write a flow equation
4) Compute a solution of this equation
5) Plot a quantitative graph
Flow from a Storage Tank

The flow of Q will be denoted by Q. (“Q–dot”)
So we can write the

Flow of Q is proportional to Q as Q

Q=-cQ

Q = - bQ - cQ

Q.
Q = - kQ
The Prudent Spender
Maria starts with $3000 in her bank account. Her rich
uncle promises to deposit $1200 at the end of each month
providing she withdraws only 20% of her $3000 balance
per month. Naturally, Maria agrees.
Assume that taxes & fees are offset by interest.
During the first few months, do you expect her balance
to increase, decrease or stay about the same?
What happens to Q down the road . . .?
A Five-Stage Modeling Process
1) Draw a diagram
Verbal to Visual
 A bank account with steady deposits and
withdrawals proportional to the balance.
 Steadily falling leaves accumulate on a forest
floor. Gardeners collect them, bacteria & mold
attack them.
Verbal to Visual
(cont.)
Target
Bank
Leaves
J0
Q
b-path
c-path
Steady deposits
Balance
Withdrawals
Fees, taxes
Steadily falling
Pile
Collected
Bacteria, mold
A Five-Stage Modeling Process
1) Draw a diagram
2) Plot a qualitative graph
Visual to Qualitative
9000
Q
6000
3000
T
1
5
10
15
A Five-Stage Modeling Process
1) Draw a diagram
2) Plot a qualitative graph
3) Write a flow equation
The Flow Equation
1) Net Flow = Inflow – Outflow
Our mantra

2)
3)
Q = J0 - bQ - cQ
Outflow

Q = J0 - (b + c)Q
Algebra

4)
Q = J0 - kQ
( k = b+c )
Flow Equation --

Notation
Q = J0 - kQ
Start Value -- Q(0) = Q0
Q
A Five-Stage Modeling Process
1) Draw a diagram
2) Plot a qualitative graph
3) Write a flow equation
4) Compute a solution of this equation
Model #1 Tank fed by Constant Flow Source
Q = Jo - kQ  1200  0.20 Q BTU / min
Q(0) = Qo  3000 BTU
T
ΔQ
Q
ΔQ = 1200 - Q/ 5
0
...
3000
1200 - 3000/ 5 = 1200 - 600 = 600
1
600
3600
1200 - 3600/ 5 = 1200 - 720 = 480
2
480
4080
1200 - 4080/5 = 1200 - 816 = 384
3
384
4
307
5
4464
1200 - 4464/5 = 1200 - 893 = 307
4771
k=b+c
A Five-Stage Modeling Process
1) Draw a diagram
2) Plot a qualitative graph
3) Write a flow equation
4) Compute a solution of this equation
5) Plot a quantitative graph
5) Plotting Energy vs Time
Carry out the calculations in the table to t = 7.
Graph the points for t = 0, 1, . . . , 7.
Use this method to complete the table (& graph).
Use this method to examine the behavior of Q
for Start Values Q(0) = 5000, 7000, 6000.
Two Scenarios
 A fuel dump delivers a proportion of its store to a field
station weekly. The field station supplies vehicles with a
proportion of its fuel each week.

A reservoir gets a fixed percentage of water daily from a
large, isolated aquifer. Every day the reservoir dispenses a
certain percentage of its water to an urban area.
Verbal to Visual
A fuel dump delivers a proportion of its store to a field
station weekly. The field station supplies vehicles with
a proportion of its fuel each week.
A reservoir gets a percentage of water daily from a large
isolated aquifer. The reservoir sends a percentage of its
water to an urban area.
Verbal to Visual
(cont.)
Target
E
Q
a-path
b-path
c-path
Dump
Fuel
---
Deliveries
---
---
Station
---
Fuel
---
Deliveries
Leakage
Aquifer
Water
---
Deliveries
---
---
---
Water
---
Deliveries
Leakage
Reservoir
Visual to Qualitative
Flow Equation
Net Flow = Inflow – Outflow
Our mantra

E = -aE

Q = aE - (b + c)Q

Q = aE - kQ
Flow Equations -- E = -aE ,
E
Outflow
Q
Notation
(k = b+c)

Outflow

Q = aE - kQ
Start Values -- E(0) = E0 , Q(0) = Q0
Computation
E = -aE = -.30 E gal/day, E(0) = 800 gal
Q = aE – kQ = .30E - .25Q gal/day, Q(0) = 250 gal
T
0
1
ΔE
---240
2
-168
3
-117.6
4
5
E
800
560
392
ΔQ
---
Q
250
177.5 427.5
ΔE = -.30E
ΔQ = .30E - .25Q
-.3∙800 = -240
240 - 250/4 = 177.5
-.3∙560 = -168
168 - 427.5/4 = 61.1
61.1 488.6
-.3∙392 = -117.6 117.6 - 488.6/4 = - 4.6
274.4
-4.6
484.0
COMAP H.S. products http://www.comap.com/search.cgi?words=himap
HiMCM Project Director
William P. Fox
Naval Postgraduate School
Monterrey, CA 93943
http://www.comap.com/
highschool/contests/
himcm/ index.html
(2011 announcement
not yet available)
Moodys Mega Math Challenge
Project Director
Michelle Montgomery
Society for Industrial and
Applied Mathematics
Philadelphia, PA 19104
http://M3Challenge.siam.org
The Midge Classification Challenge
Biologists W.L. Grogan of Salisbury Univ., and W.W. Wirth
of the Smithsonian Institute, do research on biting midges.
Note for Lee and Michelle –
The following four slides (93, 94, 95, 96) were put on separate
8.5”x11” handouts.
An appendix of various and sundry items is not included – this
monster is already big enough.
The Midge Classification Challenge
Grogan and Wirth were doing field work & captured
18 biting midges. They agreed that nine of the midges
belonged to an antenna-dominated species and six
belonged to a wing-dominated species.
The were sure that each of the left-overs belonged to
one of the two species but which one...?
The challenge -- Take a look at their antenna-wing
data and see if you can help them out.
The Midge Classification Challenge
w
w
w
w
w
w
?
?
?
a
a
a
a
a
a
a
a
a
Ant
1.14
1.18
1.20
1.26
1.28
1.30
1.24
1.28
1.40
1.24
1.36
1.38
1.38
1.38
1.40
1.48
1.54
1.56
Wng
1.78
1.96
1.86
2.00
2.00
1.96
1.80
1.84
2.04
1.72
1.74
1.64
1.82
1.90
1.70
1.82
1.82
2.08
Minimal Spanning Trees for a Communications Network
The cost for a communication line between two stations is proportional
to the length of the line.
The cost for conventional minimal spanning trees of a set of stations can
often be cut by introducing “phantom” stations and then constructing a
Steiner tree. A network with n stations never requires more than n – 2
phantoms to construct the cheapest Steiner tree. Two simple cases are
shown in Figure 1.
For local networks, it is often necessary to use rectilinear or “checkerboard” distances. Distances in this metric are computed as shown in
Figure 2.
Suppose you wish to design a minimum-cost spanning tree for a local
network with 9 stations with coordinates -- (0, 15), (5, 20), (16, 24),
(20, 20), (33, 25), (23, 11), (35, 7), (25, 0), (10, 3).
Moreover, the coordinates of all phantom stations must be integers. Find
a minimal-cost tree for the network.
Minimal Spanning Trees for a Communications Network
Concepts & Techniques of Math. Modeling
December 2010 Workshop
Happy Modeling (Teams)
[email protected]