Modules
for KUCSEC
Below is a list of modules that have been developed for the Keck
Undergraduate Computational Science Educational Consortium. Please
contact the author or Terry Lahm
with any questions.
Computational Science/Mathematics/Scientific
Visualization
Biology
Chemistry
Geology/Environmental Science
Physics
Psychology/Neuroscience
Finance/Economics
Applied Mathematics
System of Linear Equations
LINK TO MODULE
Greg Baker
Department of Mathematics
The Ohio State University
baker@math.mps.ohio-state.edu
To futher our knowledge of science and engineering, we try to formulate
mathematical models that quantify behaviors. We trust these models
only when they predict observed behavior. Normally this requires
the solution of the unknown quantities in our model when written
in a system of equations.
Vector Spaces and
Linear Transformations
LINK TO MODULE
Greg Baker
Department of Mathematics
The Ohio State University
baker@math.mps.ohio-state.edu
What is a vector? The answer may surprize you. But let's start
with the
simplest view of a vector. It is an arrow that records distance
and direction.
By stringing together a sequence of arrows we can provide detailed
directions
for a journey, or outline an object. It is the way we add arrows
to produce
a new arrow that really identi es what a vector is. We can incorporate
this addition property to other quantities such as velocities,
forces, and even
functions. What quickly emerges is that it is the linear combination
of vectors
that allows great diversity in applications and provides deep
understanding
to the nature of solutions to linear problems. This module starts
with the
basic description of vectors and then proceeds to elucidate their
role in the
formation of systems of linear equations.
Fourier Transforms, Fourier Series and
the FFT
LINK TO MODULE
Lisette de Pillis
Harvey Mudd College
depillis@hmc.edu |
Ami Radunskaya
Pomona College
Ami_Radunskaya@pomona.edu |
Introduction: This module on Fourier Transforms
is ideally introduced in a course on modeling or on scientific computing,
or as enrichment for an introductory course in linear algebra, honors
calculus, or engineering. It is self-contained and presented from
an introductory perspective.
Module Objectives: The objectives of this module
include providing the students with an introductory level of familiarity
with the Fourier Transform that will allow them to:
• Recognize mathematical and physical situations in which
the Fourier Transform may be useful for analysis.
• Understand when it is appropriate to use a Continuous Fourier
Transform, and when one should use a Discrete Fourier Transform.
• Understand the connection between the Continuous Fourier
Transform, the Fourier Series, and the Discrete Fourier Transforms.
• Understand how numerically to apply a Fourier Transform
to a set of data, and what the resulting transform represents.
• Be familiar with packaged Fourier Transform routines (we
use MATLAB routines) and be able to use them with understanding.
• Understand computational efficiency issues of the Fourier
Transform.
• Discuss and assess Fourier Transform results.
Evaluating Evidence Using Bayesian
Networks
LINK
TO MODULE INTRODUCTION
LINK
TO ZIP ASSOCIATED FILES
Lisette de Pillis
Harvey Mudd College
depillis@hmc.edu |
Ami Radunskaya
Pomona College
Ami_Radunskaya@pomona.edu |
This module will explore the following problem: Given a DNA mixture
at a crime scene and given the DNA of a suspect, how likely is it
that the suspect was in fact the offender? You will have to consider
possible contamination and whether the fact that the suspect might
be a close relative of the offender might affect the analysis. Bayesian
networks will be used to model the different scenarios, and to produce
likelihoods for the guilt (or innocence) of the suspect. In order
to use Bayesian networks, you will also learn the basics of conditional
probability and Bayes Rule.
Computational
Geology
Well Hydraulics and Capture Zone
Analysis
LINK TO MODULE
Kathryn W. Thorbjarnarson
Dept. of Geological Sciences
San Diego State University
thorbjar@geology.sdsu.edu
To Include:
1. Prediction of drawdowns with Theis equation - EXCEL, program
themselves, I can provide some Visual Basic programs that I have
produced
2. Multiple wells - principle of superposition
3. Boundaries - image wells - EXCEL, program themselves
4. EPA Wellhead Capture Zone software (free and on web) - Simple
(WHPA - DOS based) and more complex (WhAEM 2000 for Windows)
5. Above solutions still assume homogeneous conditions - more complex
simulations with MODFLOW - how wells are represented
A Tale of Two Lakes: Environmental
Mass Balance Modeling
LINK
TO MODULE
Kathryn W. Thorbjarnarson
Dept. of Geological Sciences
San Diego State University
thorbjar@geology.sdsu.edu
This module will explore the use of mass balance modeling to assess
environmental impacts in lakes. Students will explore simple mass-balance
models which result in algebraic solutions. More complex scenarios
will utilize EXCEL for model simulation. The module can be a stand-alone
component for a general education class in environmental science
and/or more complex modeling scenarios can be explored in upper-division
undergraduate hydrology and computer science classes. Knowledge
of basic algebra and the ability to use EXCEL is required. Upon
completion of the module, students will have gained an understanding
of the impacts of water diversions on water bodies through the two
case studies, Mono Lake and Salton Sea.
How Far Will It Go?
Predicting the Extent of Groundwater Plumes
LINK
TO MODULE
LINK
TO ASSOCIATED FILES
Kathryn W. Thorbjarnarson
Dept. of Geological Sciences
San Diego State University
thorbjar@geology.sdsu.edu
This module will explore the use of a solute transport model to
assess the fate of organic contaminants in groundwater. An analytical
solution to the advection-dispersion equation with retardation and
transformation will be implemented in EXCEL. The module can be used
in a lower-division undergraduate computer science class. The module
serves as an introduction to more complex software and modeling
scenarios using the Environmental Protection Agency (EPA)’s
models, BIOSCREEN and BIOCHLOR. Knowledge of basic algebra and the
ability to use EXCEL is required. Upon completion of the module,
students will have gained an understanding possible natural bioremediation
or natural attenuation of groundwater contaminants.
Thermal Conduction - A Tool for Exploring
Geological Processes on the Earth and Other Bodies in our Solar
System
LINK TO MODULE
For answers to this module please contact Eric Grosfils (egrosfils@pomona.edu)
or Terry Lahm (tlahm@capital.edu).
Eric Grosfils
Dept. of Geology
Pomona College
egrosfils@pomona.edu
Thermal conduction is a fundamental physical process, one which
controls many aspects of the volcanic and tectonic evolution of
bodies within our solar system. Using transmission of thermal energy
through the crust of the Earth as an initial, physically intuitive
conceptual model, the module's background material will (a) help
students deduce the thermal conduction equation-a second order differential
which can be constructed from first principles, (b) evaluate volume-adjusted
conduction incorporating internal heat generation and temperature
change, and (c) explore special forms of the equation such as steady
state conduction and thermal diffusion. Analytical solutions of
the problem for a semi-infinite half-space require introduction
of an error function and yield direct insight into useful physical
concepts like the characteristic thermal diffusion distance; solution
illustrations
will include calculation of a thermal boundary layer thickness and
surface heat flow.
Problems to be tackled by the students will likely include: (a)
assessing whether conduction alone is responsible for cooling the
Earth, an approach which yields an age vastly at odds with known
values and which leads students to recognize the importance of radioactive
heating (a problem suitable for an Introductory level geology course);
(b) constraining how long magma flowed through a conduit to produce
an observable, temperature-dependent thermal alteration zone within
the surrounding rock; (c) assessing the isotherm which best defines
the base of the cooling oceanic crust as it moves away from a mid-ocean
ridge spreading center; and, (d) using thermal wave propagation
to constrain the depth to which
subsurface ice on Mars can be melted solely by daily and seasonal
atmospheric temperature variations. As an advanced topic, to be
developed if time permits, students will be introduced to time-dependent
solutions for temperature within host rock adjacent to a magma body
using fast fourier transform methods. In the first iteration of
the module, all problems will be solved using Excel because of its
basic suitability and widespread availability; some components may
later be ported to Matlab to facilitate more advanced exploration.
SPHERICAL MAGMA RESERVOIR FAILURE:
EXPLORING SOME GEOLOGICAL IMPLICATIONS OF STRESS
CONCENTRATION AROUND A PRESSURIZED CAVITY
LINK TO MODULE
For answers to this module please contact Eric Grosfils (egrosfils@pomona.edu)
or Terry Lahm (tlahm@capital.edu).
Eric Grosfils
Dept. of Geology
Pomona College
egrosfils@pomona.edu
This module, which requires only a basic understanding of geology
and calculus, explores a key
process in shallow volcanic systems: shallow magma reservoir failure.
The learning goals are as
follows:
• To introduce you to an important volcanological problem
into which insight can be gained
using both analytical and numerical modeling approaches;
• To provide you with experience assessing the strengths and
limitations involved when using
boundary condition approximations in an analytical model—real
insight can be gained but
real caution must be exercised;
• To introduce you to the process of solving a structural
failure problem under a wide variety
of different conditions using a numerical technique, the finite
element method, and promote
comparing the results from this approach to the analytical results
obtained previously.
The analytical component of the module is implemented in its entirety
using Microsoft Excel.
This component can be used in “stand alone” mode, but
it derives additional value via an extra
numerical modeling section which introduces you to the process
of finite element modeling (employing
the commonly used and inexpensive program FEMLAB, which complements
MATLAB) as a tool for continuing to explore the process of magma
reservoir evolution. Note
that having access to at least one copy of FEMLAB is handy but
FEMLAB need not be available
to use the numerical modeling portion of the module. If FEMLAB
is not available a free viewer
will need to be obtained to allow you to view and explore—fortunately
in considerable depth—
the results from pre-run model simulations.
Stresses and Faulting - Tools for
Exploring Geological Processes on the
Earth
LINK TO MODULE
LINK TO ASSOCIATED FILES
For answers to this module please contact Linda Reinen (lreinen@pomona.edu)
or Terry Lahm (tlahm@capital.edu).
Linda Reinen
Department of Geology
Pomona College
lreinen@pomona.edu
Tectonic forces produce a wide variety of observable geologic phenomenon,
including earthquakes and faulting. In this module, students will
explore the application of classical Newtonian mechanics to the
shallow layers of the Earth in order to understand the formation
and movement of faults. Background information provided with this
model will include information on Newtonian mechanics, rock rheology,
plate tectonics, Mohr diagrams and Coulomb failure. Students will
explore the interrelations between principal, normal and shear stresses;
the angle between the fault plane and the plane normal to the applied
force; and the coefficient of friction of the fault. Students will
use the model they develop to determine why Anderson's hypothesis
for fault formation generally applies to natural systems. Students
will test their models by determining the probable
frictional strength of a natural fault from data collected from
the southern segment of the San Andreas Fault. Possible advanced
topics, if time permits, include: (a) the effects of finite displacement
on the strength and orientation of normal faults, (b) models of
fault strength with a time-dependent term, and (c) slip-rate dependent
models of dynamic friction. All problems will be solved using interactive
graphical techniques in Excel. Some of the advanced topics may require
the use of another package.
Exploring and Visualizing Strain
LINK
TO MODULE
LINK
TO ASSOCIATED FILES
Linda Reinen
Department of Geology
Pomona College
lreinen@pomona.edu
For answers to this module please contact Linda Reinen (lreinen@pomona.edu)
or Terry Lahm (tlahm@capital.edu).
This module is designed for students in an undergraduate course
in Structural Geology. While key concepts are described here, it
is assumed that students will have access to a good structural geology
textbook to augment the information presented here.
• Understand how to quantify strain and apply that knowledge
to real geologic situations.
• Understand the rates at which rocks strain.
• Understand the difference between pure shear, simple shear,
and general shear.
• Explore the differences between incremental and finite strain,
and the clues that rocks can provide to decipher whether the strain
recorded is pure or simple shear, or a combination of both.
• Apply strain path models to decipher the strain history
of natural rocks from different locations, including Norway, Joshua
Tree National Park, California and Ganymede.
This module is implemented using Microsoft Excel and a standard
drawing program, such as Canvas or Adobe Illustrator. These programs
were selected due to their widespread availability and relative
east-of-use.
Landslides and Slope Stability
LINK
TO MODULE
Supporting
Data Files
Zip
File of Supporting Files
John B. Ritter
Wittenberg University
jritter@wittenberg.edu
Landslides are a persistent cause of economic loss in every region
of the United States, amounting to billions of dollars in property
losses annually according to the U.S. Geological Survey (e.g., U.S.
Geological Survey, 1982; Spiker, E.C. and Gori, P.L., 2000; Spiker,
E.C. and Gori, P.L., 2003). One strategy for reducing losses from
landslide hazards is to delineate susceptible areas for planning
and decision-making purposes, the ultimate goal of this module.
The objectives of this module will build toward that goal; they
are to (1) explore different types of mass wasting processes using
on-line resources and recent mass wasting events; (2) evaluate the
sensitivity of slope stability to topographic and earth material
variables according to the infinite slope method; and (3) assess
the spatial variation in slope stability using a Geographic Information
System (GIS). The final objective will focus on the urbanized area
of Cincinnati and Hamilton County in southwestern Ohio. The highest
documented per capita losses due to landslides, amounting to $5.80
per person per year, occur in Hamilton County, in the metropolitan
area of Cincinnati (Fleming and Taylor, 1980). The module uses an
infinite slope stability model to calculate a factor of safety.
The infinite slope model is based on algebraic manipulation of various
raster data layers to produce a map of factor of safety values and
as such is a deterministic analysis of slope stability. It is particularly
appropriate for slope failures with planar slip surfaces, such as
translational slides, which are common in this area (Baum and Johnson,
1996).
Modeling the Performance
of a Solar Heated Sunroom:
Heat Gain, Storage and Loss
LINK TO MODULE
Supporting Data Files
John E. Petersen
Associate Professor of Environmental Studies and Biology
Oberlin College
john.petersen@oberlin.edu
Alfredo Fernández-González
Associate Professor and Director, N.E.A.T. Laboratory
School of Architect
University of Nevada, Las Vegas
In this module students build, explore and modify dynamic simulation
models of solar gain, heat storage, transfer and loss in a sunroom.
The objective of this module is to provide students with a practical
example of how basic mathematical formulations and a variety of
simplifying assumptions can be combined to develop a model that
can be used to improve system design, analyze system performance,
and explore the efficacy of different management approaches for
optimizing thermal performance. In the process of completing the
exercises associated with this model students will develop an understanding
of the implications of technological and management choices and
an intuition for performance dynamics. At the same time, students
will develop a clearer sense of the practical role that simulation
modeling can play in the design and interpretation of real-world
systems. The exercise will also help students to understand how
interacting physical processes (in this case heat flux via radiation,
conduction and convection) can be “parameterized” by
modelers to balance the goals of mechanistic realism and practical
usability.
Computational
Science
Modeling Malaria
LINK TO MODULE
Angela B. Shiflet and George W. Shiflet
Wofford College
shifletab@wofford.edu
Malaria is caused by protozoan belonging to the genus Plasmodium,
that infects human beings and is transmitted by the Anopheles mosquito.
It is a very old disease (probably prehistoric), originating in
Africa, spreading as humankind migrated to other lands. Some recent
evidence implicates this tiny parasite in the fall of the ancient
Romans, one of the mightiest empires of all time
Mosquitoes are the only vectors for malaria, but only 60 out of
the 380 species of Anopheline mosquitoes, can host malaria-causing
Plasmodium. When a female mosquito bites an infected person, she
sucks up gametocytes along with blood. Once in the mosquito's stomach,
the gametocytes develop into sperm-like male gametes or large, egg-like
female gametes. Fertilization produces an oocyst filled with infectious
sporozoites. When the oocyst matures, it ruptures and the thread-like
sporozoites migrate, by the thousands, to the mosquito's salivary
(saliva-producing) glands. When she again feeds, these sporozoites
will be ready to renew the cycle.
In this module, we develop models of the effects of malaria on
various populations of humans and mosquitoes. After considering
differential equations to model a system, we create a model using
the systems modeling tool STELLA. Projects involve various refinements
of the model.
Modeling Mushroom Fairy Rings
LINK TO
MODULE
LINK
TO MATHEMATICA TUTORIAL
LINK
TO MATHEMATICA TUTORIAL ANSWER
Angela B. Shiflet and George W. Shiflet
Wofford College
shifletab@wofford.edu
Sometimes in a forest or yard, mushrooms seem magically to grow
in circles, which we call "fairy rings". In this module,
we develop simulations for the expansion and interactions of such
mushroom fairy rings. After analyzing the system, formulating the
model, and considering appropriate rules for the spreading of mushrooms,
we create a simulation using the graphical computer algebra system
Mathematica. Projects involve various refinements of the model.
We have designed the module for science and mathematics majors
with the following goals:
1. To study the modeling process
2. To study the simulation process
3. To apply the graphical computer algebra system Mathematica in
simulations
4. To investigate model refinement
Modeling Tumor-Immune Interactions
Mathematics
LINK TO MODULE
Lisette de Pillis
Harvey Mudd College
depillis@hmc.edu |
Ami Radunskaya
Pomona College
Ami_Radunskaya@pomona.edu |
Nonlinear Ordinary Differential Equations, Bifurcation Analysis
Numerics: Various ODE solvers, explicit, implicit, non-stiff, stiff
Time-line: Week 1: Model development and qualitative analysis Week
2: Numerical issues in solving ODE, introduction of explicit, implicit,
stiff solvers Week 3: Bifurcation analysis, numerical continuation
In this module we introduce the Tumor-Immune system interaction
equations developed by Kuznetsov(1994). This model is of particular
interest because the mathematics are straightforward, yet the dynamics
of the system are quite rich. We will begin by developing each component
of the differential equations from biological principals. We will
then take the students through an elementary qualitative analysis
of the system, finding critical points and stabilities. The computational
component will then be introduced, and this will allow us to familiarize
the students with simple explicit, implicit and specialized stiff
numerical ODE solvers. The students will learn about the benefits
and drawbacks of each category of solver, and will be guided through
a basic stability analysis of each solver. Once we start running
numerical experiments, we will have the students experiment with
parameter changes, and discover the impacts on the dynamics of the
system. The system we will be working with has several bifurcation
points, equilibria appear and disappear and stability properties
change. We will examine the numerical problems encountered in the
analysis of these bifurcations by investigating methods of numerical
continuation. Continuation is also important in the numerical detection
of unstable equilibria and separatrices. We will then guide the
students through a discussion of the biological interpretation of
these parameter changes. For example, this model allows for an explanation
of tumor dormancy, a phenomenon which is still not fully understood
biologically.
Computable Performance Metrics
Kris Stewart
Dept. of Computer Science
San Diego State University
stewart@sdsu.edu
Discovering computable metrics for performance for a solution of
a problem from science will depend on the problem, the algorithm
chosen to approximate the problem digitally, the computing platform,
the way this algorithm is implemented in code, and a method for
gathering and analyzing data to validate the theoretical expectations.
We choose to examine a problem with known true solution and a well
understood algorithm to motivate this experiment. Therefore, we
model the heat diffusion on a one dimensional grid u(x) and then
two dimensional grid u(x,y) and Gaussian Elimination with partial
pivoting. This focuses attention on applying the scientific method
to create a computational experiment:
a) predict performance from known mathematical properties for work
and
error
b) discover how to gather data to support the predictions of performance
c) instrument the computer code to gather the data
d) analyze the data to enable forming conclusions
e) write up the results.
We define performance to be measured by work, in terms of amount
of CPU time used, and by error, in terms of deviation from the known
true solution.
Students develop an understanding of rectangular discretization
to transform the continuous statement of the heat diffusion equation
into a discrete approximation using finite differences. Students
come to understand the relationship between the known number of
algebraic operations of Gaussian Elimination and the measurable
metric of work, the CPU timer. Students also come to understand
the behavior of error, as predicted by the truncation error of the
discrete approximation, and the
computable difference from the known solution.
Pharmacokinetics: Mathematical
Analysis of Drug Distribution in Living Organisms
LINK
TO MODULE
Ignatios Vakalis
Department of Math, Computer Science, and Physics
Capital University
ivakalis@capital.edu
Pharmacokinetics is the study of time course of a drug or a metabolite
in different fluids, tissues and of the mathematical relationships
required to develop models to interpret such data.
The mathematical theory of drug phenomena is a branch of the mathematical
theory of metobolism. Even though drugs are not normal metablites
they do affect different metabolic processes. While the drug is
acting, it does take part in some phases of metabolism.
The theory of drug phenomena can be subdivided into two categories:
· The modeling of distribution of drugs in the organism;
· The biochemical kinetics of the interaction of the drug
with different components of the organism.
The first category will be used as the background info, while the
module focus on the mathematical treatment of the second category.
Image
Reconstruction in Emission Tomography I. Non-iterative Methods
LINK
TO MODULE
ZIP
of Supporting Data Files
Edward J. Soares
Department of Mathematics & Computer Science
College of the Holy Cross
esoares@holycross.edu
Single-photon emission computed tomography (SPECT) is a non-invasive
imaging modality that yields information regarding the function
of organs within the body. This information is in the form of pixelated
images, which have been reconstructed from projection measurements
at various angular positions around the patient. The image, which
represents the bio-distribution of radio-pharmaceuticals that have
been administered to the patient, can be used as a diagnostic tool.
Since cancerous or damaged tissue can have a different metabolic
rate compared with healthy tissue, anomalies exhibit themselves
as exceptionally dark or bright regions within the image. An important
problem in SPECT is the reconstruction problem: How do we obtain
the image from the projection data measurements?
We model the data acquisition process as a matrix transformation
Ax=b, where the ij th element of the matrix A describes
how object pixel j contributes to detector measurement i .
The reconstruction problem simply reduces to solving a linear system
of equations. The inversion methods that this module will focus
on are so-called "Non-iterative Methods", meaning 1)
Gaussian Elimination/Back-substitution, 2) Matrix Inversion, and
3) Singular Value Decomposition/Pseudo-inversion. Each is a standard
technique used to solve linear systems of equations from matrix
theory, and have both theoretical and computational advantages
and disadvantages. For example, Matrix Inversion is not possible
if the number of object and data pixels is not equal or if the
matrix is singular (theoretical constraints). However, even if
they are equal and the matrix is non-singular, Matrix Inversion
may not be practical from a numerical standpoint, as round-off
errors in determining the matrix inverse would effectively make
the matrix singular.
The overall goal is to supply the students with data measurements
and have them determine the object that produced the data. The
obvious challenge here is that they don't know the correct answer
(and there may be several "correct" answers). After being
given the data measurements and the parameters of the imaging protocol
(the number of object and data pixels, which may be unequal, and
the number and orientation of the detector acquisition angles),
the students can then:
1) Determine the elements of the projector matrix A
2) Implement the "non-iterative" solution methodologies using the
appropriate software
3) Compute exact or approximate solutions
4) Interpret and explain the results
Some of the questions students can answer include:
1) Is the system over-, under-, or even-determined? What solution
methods may apply?
2) How many solutions exist? None? One? An infinite number?
3) Can the data be fit exactly?
4) If SVD is used, does the system have any zero singular values? How will
this impact the measurement and inversion process?
Image
Reconstruction in Emission Tomography
II. Iterative Methods
LINK
TO MODULE
ZIP
of Supporting Data Files
Edward J. Soares
Department of Mathematics & Computer Science
College of the Holy Cross
esoares@holycross.edu
Single-photon emission computed tomography (SPECT) is a non-invasive
imaging modality that yields information regarding the function
of organs within the body. This information is in the form of pixelated
images, which have been reconstructed from projection measurements
at various angular positions around the patient. The image, which
represents the bio-distribution of radio-pharmaceuticals that have
been administered to the patient, can be used as a diagnostic tool.
Since cancerous or damaged tissue can have a different metabolic
rate compared with healthy tissue, anomalies exhibit themselves
as exceptionally dark or bright regions within the image. An important
problem in SPECT is the reconstruction problem: How do we obtain
the image from the projection data measurements?
We model the data acquisition process as a matrix transformation
Ax=b, where the ijth element of the matrix A describes how object
pixel j contributes to detector measurement i. The reconstruction
problem simply reduces to solving a linear system of equations.
The inversion methods that this module will focus on are so-called "Iterative
Methods", meaning the 1) Jacobi method, 2) Gauss-Seidel method,
3) Landweber method and 4) Simultaneous Iterative Reconstruction
Technique. The first two are standard linear techniques used to
solve linear systems of equations from matrix theory, and all have
both theoretical and computational advantages and disadvantages.
For example, the Jacobi and Gauss-Seidel methods cannot be used
if the number of object and data pixels is not equal or if one
of the eigenvalues of a related matrix is greater than one (theoretical
constraints). However, even if they are equal and the eigenvalue
requirement is satisfied, these mthods may not be practical from
a numerical standpoint, as round-off errors might not yield convergence
of the iteration scheme
The overall goal is to supply the students with data measurements
and have them determine the object that produced the data. The
obvious challenge here is that they don't know the correct answer
(and there may be several "correct" answers). After being
given the data measurements and the parameters of the imaging protocol
(the number of object and data pixels, which may be unequal, and
the number and orientation of the detector acquisition angles),
the students can then:
1) Determine the elements of the projector matrix A
2) Implement
the "iterative" solution methodologies
using the appropriate software
3) Compute exact or approximate solutions
4) Interpret and explain
the results
Some of the questions students can answer include:
1) Is the system over-, under-, or even-determined? What solution
methods may apply?
2) How many solutions exist? None? One? An infinite
number?
3) Can the data be fit exactly?
4) How should I choose the values
for the acceleration parameter or the maximum iteration number?
HEAT FLOW ON THE JOVIAN SATELLITE
EUROPA
LINK
TO MODULE
MARC GOULET, ALEX SMITH, ANDREW T. PHILLIPS, PAUL J. THOMAS
Departments of Math, Computer Science, and Physics
University of Wisconsin-Eau Claire
gouletmr@uwec.edu
The presence of a liquid ocean on Europa, one of the moons of Jupiter
discovered by Galileo, is a topic of interest to many planetary
scientists including physicist, chemists, geologists and biologists
[Lucchita and Soderblom, 1982; Cassen et al., 1979]. This ocean
is expected to lie beneath an icy crust of unknown thickness. D.
Stevenson, a prominent planetary scientist, says [Stevenson, 2000],
"The possibility of water beneath this ice, perhaps as little
as 10 km below the surface, has excited those interested in extraterrestrial
environments for life and established a major role for Europa in
NASA's plans for outer solar system missions."
ABLATION, AEROBRAKING AND AIRBURSTING
OF A HYPERSONIC PROJECTILE IN EARTH'S ATMOSPHERE
LINK
TO MODULE
PAUL J. THOMAS, MARC GOULET, ANDREW T. PHILLIPS, ALEX SMITH
Departments of Math, Computer Science, and Physics
University of Wisconsin-Eau Claire
thomaspj@uwec.edu
All major bodies in the solar system have been shaped by a continuous
rain of impacting objects from space. These impactors are composed
of material left over from the formation of the planets, and the
rate of impact was extremely high during the early history of the
solar system, a period of time 4.5-3.8 Gya (billion years ago) known
as the Heavy Bombardment era. The craters formed by impacts during
the Heavy Bombardment era are common on all planetary objects where
erosion and geological activity has been sufficiently small for
these ancient fea-
tures to survive: Mercury, the highlands of the Earth's Moon, the
southern hemisphere of Mars and many of the satellites of the outer
solar system.
COMPUTATIONAL ANALYSIS
OF ORBITAL MOTION
IN GENERAL RELATIVITY AND NEWTONIAN
PHYSICS
LINK
TO MODULE
MARC GOULET, ALEX SMITH, PAUL J. THOMAS, BRANDON BARRETTE
Departments of Math, Computer Science, and Physics
University of Wisconsin-Eau Claire
gouletmr@uwec.edu
This module is intended as a stand-alone component of a second,
project-based course in computational science. The students should
have had a course in di®erential equations, and an interest
in physics,
astronomy or mathematics. It assumes some proficiency with the symbolic,
visualization and programming capabilities of Maple, as might
be taught in a first course in computational science. The module
is
implemented in its entirety using Maple.
The learning goals are as follows:
• To review the classical Newtonian theory of orbits.
• To solve, visualize and analyze the Newtonian di®erential
equation
whose solutions are Keplerian orbits.
• To modify the Newtonian di®erential equation to model
General
Relativity (GR) e®ects (post-Newtonian correction).
• To introduce the general formalism for GR.
• To see how the modified Newtonian di®erential equation
is consistent
with this formalism when we use the exact solution to
Einstein’s field equations called the Schwarzschild solution.
• To apply the formalism to visualize and analyze orbits around
a Kerr (rotating) black hole.
Spread of SARS
LINK
TO MODULE
LINK TO STELLA MODEL 1
LINK TO STELLA
MODEL 2
Angela B. Shiflet and George W. Shiflet
Wofford College
shifletab@wofford.edu
Severe Acute Respiratory Syndrome (SARS), which emerged in 2003,
is a highly infectious disease caused by a coronavirus. The high
infection and mortality rates and the lack of treatment alarmed
health officials throughout the world. Quick action by organizations,
like WHO and CDC, to contain the outbreak averted a public health
catastrophe.
In this module, we develop a simplified model (SIR) of the spread
of an infectious disease before considering a more involved model
of SARS. For the former, after analyzing the system and formulating
the model with appropriate differential equations, we create a model
using the systems modeling tool STELLA. For the latter, we build
on the earlier model to perform the analysis and much of the model
formulation, but leave the completion of the model to the student.
Projects involve various refinements of the models along with additional
problems.
Computable Performance
Metrics
Module 0b: Floating Point Precision
MACHAR – Test for IEEE 754
Kris Stewart
Computer Science Department
San Diego State University
stewart@sdsu.edu
LINK
TO MODULE
LINK TO ASSOCIATED
FILES
Module 0: Floating Point Precision covered the task to find “computable”
ways to detect the actual manner in which a computer performed arithmetic,
developed by W.J. Cody [1] in 1988. In that module, a minimal test
was examined to compute a machine’s unit round-off, the value
eps so that 1.0 + eps = 1.0. Considering the extent to which the
FORTRAN subroutine MACHAR can compute the properties of a processor,
some may wish to examine the full range of values computed. This
can be used to test if the user’s computer platform actually
follows the IEEE 754 Floating Point Standard [2].
Computational
Finance
The Anatomy of a Large Company acquiring
a Small Privately-owned Company Description
LINK TO
MODULE
Hal Green
School of Management
Capital University
www.hjgconsult.com
Over the past twenty years, many major companies have been on an
acquisition binge. These companies have, many times, changed themselves
and in the process changed their respective industries. This module
will examine a process by which these organizations go about acquiring
companies.
Prerequisites: Since this module will cover acquisition techniques,
the student should have a cursory knowledge of financial analysis
(including the mathematical derivation of NPV and IRR), familiarity
with cash flow, and ability to use some sort of spreadsheet package
(Lotus, Excel, etc.) NOTE: CHECK PREREQUISITES FOR FINANCIAL MANAGEMENT
COURSE
Problem Statement: A company has a strategy. As part of that strategy,
this company wants to enter new markets or expand current markets.
While this may done by internal growth, the company feels that internal
growth may be risky and expensive. The company, therefore, decides
to expand through acquisitions. However, what is the process, how
does the company determine what to pay for an acquisition, and what
does the company need to know? This module will try to answer some
of those questions.
Contents
· The strategy; the process begins
· Development of Cashflow, Exercise I
· Developing a capitalization rate (discount rate), Exercise
II
· The Offering Memorandum, what is in it.
· Developing company value (IRR/NPV). Describe similarities
of NPV/IRR.
Exercise III & IV. Development of synergies.
· Sensitivity analysis
· Other issues
· Acquisition due diligence
· Appendix I & II: Legal Issues, Due Diligence List
Cash Flow Analysis and Capital Asset Pricing Model
LINK
TO MODULE
LINK TO ZIP ASSOCIATED
FILES
Rick Percy
rpercy@insight.rr.com
The first goal for the module is for the student to be able to
make financial decisions similar to what a financial manager or
an investment manager of a firm would make. To do this the student
must learn to analyze cash flows of varied projects, investments,
and capital budgets. Among the methodologies employed will be the
criteria of Net Present Value (NPV) and Internal Rate of Return
(IRR). The concepts of Present Value and Future Value computation
will be acquired prior to the Cash Flow analysis. The theory of
interest and how it works in the market is explored to better understand
its use in Present Value computations. The second portion of this
module will emphasize evaluation, generation, and interpretation
of the Capital Asset Pricing Model (CAPM). To facilitate financial
analysis and valuation of risky assets based on the model it will
be necessary to introduce the concept of return- risk analysis and
linear regression analysis prior to its application. Students will
learn about risk-free assets, forming optimal portfolios given a
set of investments. Emphasis is given to the use of models and real
world examples with careful attention given to assumptions in models
and likely violations of these assumptions in the real world.
Option Pricing
LINK
TO MODULE
LINK
TO ZIP ASSOCIATED FILE
Rick Percy
rpercy@insight.rr.com
This module analyzes the topic of option pricing. Binomial tree
models and the Black-Scholes formulae are used to price call options
on an underlying asset. . The put-call parity theorem is used to
value put options. Background information on options is provided
as well as the theoretical implications of the models. Assessment
of the models is evaluated through on an investigation of the strengths
and shortcomings of the models compared with their empirical performance.
The module evaluates the differences between European and American
options using mathematical models. In addition, the module examines
expansions of the model including adaptation to stocks that pay
dividends, the market for foreign exchange, and options on portfolios
such as market indices. In assessing the strengths and shortcomings
of the models investigated, the existence of alternative models
is introduced to allow the interested student pathways to learning
after the module.
Computational
Chemistry
Kinetics Module
LINK TO MODULE
LINK
TO WEBSITE
Anna Dara Bowen
San Diego Supercomputer Center
adb@sdsc.edu
Goal: Aid students taking General Chemistry in understanding chemical
kinetics through a series of interactive 'experiments'.
This module consists of a series of Interactive Graphing Activities
that allow students to explore the equations used in chemical kinetics.
The final module will allow students to complete a numerical analysis
of a simple first order reactiontroduce the steady state approximation.entrations
of end with a rigorous computational analysis ummer. The interactive
graphing 'activities' are designed to be a tool for completing the
questions about determining rate constant, concentration and related
quantities.
Activity 1: Simple Determining Reaction Rate Experiment
This interactive activity will allow students to 'measure' concentrations
of reactants and products at different times.
They will be able to modify the initial amounts of reactants. They
will plot the graph of concentration as a function of time ( on
the computer ), and determine the slope. This module will also introduce
the steady state approximation.
Activity 2: Determining Rate Constants and concentrations of different
types of reactions
Interactive graphing activity that allows students to analyze the
graphs of more complex reactions and compare and contrast the meaning
of the resulting graph. (covers simple first- and second-order reactions,
consecutive irreversible first-order reactions, reversible first-order
reactions, competitive first-order reactions).
Activity 3: Reaction Rates and Temp: The Arrhenius Equation
Part 1: Demonstaration of Activation Energy:
Tutorial style demonstration of collision theory including an Interactive
Graph of Fraction of Collisions vsare and contrast the resulting
graph.anhey interactive potential energy profile as well as a graph
of fraction of collisions with a particular energy at various temperatures.
Part 2: Using Arrhenius Equation
Interactive activities that explore using rate constants to find
the Activation Energy (using a plot of log k vs 1/T )
Activity 4: Use the fourth order Runge Kutta method used (a) to
solve a simple first order reaction; demonstrating the comparison
between the analytic solution and the numerical solution, and (b)
to give concentration-time data for a nested set of elementary reactions
which form a complex mechanism. This latter feature is actually
useful for analysis of research data.
Quantum Chemistry in the Environment
LINK TO MODULE
LINK
TO WEBSITE
Anna Dara Bowen
San Diego Supercomputer Center
adb@sdsc.edu
This unit will introduce how quantum mechanical calculations can
be used to investigate chemical problems. An online computational
chemistry portal to GAMESS will be used to run calculations and
explore the major common computational methods and calculations
used in computational chemistry. These methods will be used to investigate
several molecules that are important in the environment.
The module is designed to be accessible to undergraduates taking
General Chemistry when they encounter quantum theory for the first
time. The goal of this module is to familiarize students with computational
chemistry so that they will be able to run calculations and determine
if the results are reasonable. To accomplish this, the module starts
out with a background section (entitled 'learn' in the on-line module)
to get students acquainted with the basic background of quantum
mechanics used in computational chemistry software packages. There
is a tutorial section (entitled 'apply' in the on-line module) that
is meant to be a guide to how to complete a computational analysis
using the GAMESS portal. Finally, the students will apply there
knowledge (entitled 'solve' in the on-line module) to a computational
chemistry project that will use an on-line web portal to GAMESS
and allow students to run calculations on there own and interpret
the results.
Scientific Visualization
Marching Cubes
LINK TO MODULE
David Reed
Math, Computer Science and Physics
Capital University
dreed@capital.edu
This module is longer and more complex than many of
the other modules. Most of the modules involve using VTK to visualize
data. VTK is a general purpose toolkit that supports many different
data types and many different algorithms. Its generic
visualization pipeline structure allows different algorithms to
be pieced together easily. The downside of this is that it is not
optimized for any specific task and is thus slower and takes more
memory to do the same task than an algorithm optimized for a specific
visualization algorithm. This module examines many of the memory
and low-level programming details that VTK hides by having the student
develop a complete program for visualizing isosurfaces.
Scientific Visualization
LINK
TO MODULE OVERVIEW
LINK
TO VOLUME RENDERING MODULE
LINK
TO VECTOR FIELDS (Part A)
LINK
TO VECTOR FIELDS (Part B)
Raghu Machiraju
Computer and Information Science
The Ohio State University
raghu@cis.ohio-state.edu
In this module the aspiring student will explore the use of volume
rendering as a tool. Volume rendering allows one to gain insights
into the data generated from a simulation or obtained from a medical
or seismic scanner. Both vector and scalar filed visualization techniques
will be considered for study. The emphasis in the module shifts
to discovery or mining of features present in the data. The programming
examples will use vtk and adequate material will be included.
Computational
Neuroscience and Psychology
Artificial Neural Networks
LINK
TO MODULE
LINK
TO MATHEMATICA FILES
Flip Philips
Department of Psychology
Skidmore College
flip@skidmore.edu
This module introduces the student to the mathematical foundation,
biological foundation, the structure and the function of artificial
neural networks (ANNs). There are a wide variety of approaches to
the construction of artificial neural networks including the multi-layer
perceptron, learning vector quantization, and the Hopfield network.
Some ANNs are classified as feedforward while others are recurrent
(i.e., feedback) depending on how data are processed through the
network. ANN types are classified by their method of learning (or
training), and some ANNs employ supervised training while others
are referred to as unsupervised or self-organizing. The advantage
of ANNs lies in their resistance to input data distortions and their
capability of learning. Neural networks are unlike artificial intelligence
software in that they are trained to learn relationships in the
data they have been given. Just like a child learns the difference
between a chair and a table by being shown examples, a neural net
learns by being given a training set. Due to its complex, non-linear
structure, the ANN can find relationships in data where humans can't.
ANNs are collections of mathematical models that emulate some of
the observed physical properties of the nervous system and draw
on the analogies of adaptive learning in the nervous system. An
ANN is composed of a large number of interconnected processing elements
that are analogous to neurons that are tied together with weighted
connections that are analogous to synapses. Learning in the nervous
system involves adjustments to synaptic connections between neurons.
This is true of ANNs as well. Learning typically occurs by example
through training, or exposure to a truth set of input/output data
where the training algorithm iteratively adjusts the connection
weights (synapses). These connection weights "store the knowledge"
necessary to solve specific problems.
Interactive Schedule
LINK
TO MODULE
Andrea M. Karkowski
Behavioral Sciences Department
Capital University
akarkows@capital.edu
Most behavioral scientists treat typical schedules of reinforcement,
such as Variable Interval and Variable Ratio schedules, as separate
and discrete entities. While this approach may facilitate the development
of an understanding of such schedules, both in the classroom and
in the laboratory, it belies the true character of scheduled reinforcement
as it exists in the natural environment. That is, few behaviors
are mediated by a pure interval or ratio schedule. To alleviate
this discrepancy, Berger (1988) established a continuum of behavioral-temporal
reinforcer contingencies, the Interactive Schedule: fo = Fbx/C where
fo is the instantaneous frequency of reinforcement, Fb is the mean
response rate since the last reinforcer, x identifies the point
on the continuum that the organism is experiencing, and C is the
set of contingencies established for the value of x. Using the Interactive
Schedule, students will explore the resulting relationships among
responses, time, and reinforcement along the continuum. Students
will learn about operant conditioning and simple and complex schedules
of reinforcement. They will also explore the matching law and behavioral
economics, which highlight the
complexity of behavior-reward contingencies. Due to these complex
relationships, the simple schedules of reinforcement are frequently
rendered insufficient, and thus, the interactive schedule is needed
to more fully accommodate the complex relationships. The problem
that students will explore is how the basic schedules of reinforcement
can be incorporated into a model that more closely depicts the complexity
of response-reinforcer contingencies.
Sex, Sex Role, and Relationships
LINK
TO MODULE
LINK TO
STELLA FILE
Andrea M. Karkowski, Sarah Stith, Renée Walling, Megan Anders
Behavioral Sciences Department
Capital University
akarkows@capital.edu
Developing and maintaining satisfying intimate and romantic relationships
during adolescence and throughout adulthood is important for both
physical and psychological well-being. For example, Popovic (2005)
reports that “a close, satisfying relationship is often considered
as the essential factor in adults’ health, ability to adapt,
happiness, and sense of meaning in life” (p. 35). McAdams
(1988) asserts that the relationship characteristics of self-disclosure,
caring, trust, and commitment directly affect psychological health.
While the benefits of healthy romantic relationships are numerous,
the costs of such relationships, when things go wrong, can be devastating,
physically, psychologically, economically, and professionally. For
example, Flannagan et al. (2005) report that dissatisfaction with
a romantic relationship can diminish the positive psychological
benefits of the relationship. According to Savard et al. (2006),
“Love relations are central to human development and a sudden
worsening of their status (e.g., threats of separation, financial
difficulties, infidelity) may create a life context that promotes
personal characteristics typifying psychopathy: hatred, arrogance,
envy, mistrust, and destructiveness” (p. 939). Popovic also
indicates that a lack of close relationships results in an increased
susceptibility to stress and stressors, feelings of powerlessness,
loneliness, and substance abuse. In this module, students examine
a variety of factors associated with heterosexual romantic relationships.
Students explore a STELLA® model that was created from the empirical
results of a factor analysis of college students’ attitudes
about romantic relationships. The model also includes components
for sex and sex role, among other variables. Students then refine
the model to account for other variables that contribute to attitudes
toward romantic relationships.
Discounting of Delayed
Reinforcers under Asymmetrical Choice Conditions
LINK TO MODULE
Andrew J. Velkey, II
avelkey@cnu.edu
757-594-7927
&
Leonardis L. Bruce
Many researchers treat the study of behavior as if a “black
box” were the entity underlying these behaviors (e.g. “free
will”, etc.). This approach may be useful in a more qualitative
setting, but a quantitative approach to understanding behavior must
be predicated upon the establishment that the determinants of behavior
are found in the interaction of an organism and its environment.
The study of behavior ultimately is the study of choice, for even
in the highly-controlled setting of the Skinner box, the rat must
still choose whether or not to press the lever at any particular
instance (i.e. asymmetrical choice or “Hobson’s Choice”).
The rate at which the rat presses the lever is affected not only
by the delivery of reinforcement for lever pressing but also by
the value placed upon the reinforcing event. It follows that the
quantitative study of behavior must include a framework for systematically
exploring this valuation of reinforcers in order to better understand
the manner in which organisms apportion their behavior in different
settings.
Computational
Biology
Modeling the Cardiovascular System using
Stella®
LINK TO MODULE
Karl Romstedt
Department of Biological Sciences
Captial University
kromsted@capital.edu
This module teaches concepts regarding cardiovascular function
and modeling. It is intended for a college audience but requires
only basic skills in biology, mathematics and programming. The module
will include lecture/discussion periods, computer laboratories and
hands-on experimentation. It is designed to be a stand-alone unit
that can be integrated anywhere into the course on Computational
Biology. If it is used after the section on statistics and curve-fitting,
however, these concepts can be incorporated into group projects
involving physiological experimentation and data collection. Little
in the way of explicit prerequisites are required since the intention
is to cover the required biology and computational science during
the module. Some questions may require using the web or other references
to investigate
Gene Identification
LINK TO MODULE
LINK TO APPENDIX
Chuck Daniels
Department of Microbiology
The Ohio State University
Daniels.7@osu.edu
The advent of rapid DNA sequencing methods has revolutionized molecular
life sciences. Within the last decade the genomes, or genetic blueprints,
of nearly 100 organisms have been completed and the stream of information
continues to grow. For the first time, modern biologists have the
opportunity to predict all of the capabilities of an organism based
on its gene content. Despite its promise, deciphering these genomes
is a problem not yet within the reach of current experimental techniques
and the biologists have turned to computer scientists and mathematicians
to help in developing new methods for analysis and storage of genome
data. This has led to formation of a new discipline, Bioinformatics.
This module will examine the "language" of genes and illustrate
how basic statistical methods can be applied to the problem of gene
prediction. The merger of computational sciences with biology, and
the challenges facing Bioinformatics, will also be explored through
the use of analysis tools available at the National Center for Biotechnology
Information (NCBI).
Protein Identification
LINK TO MODULE Chuck Daniels
Department of Microbiology
The Ohio State University
Daniels.7@osu.edu
How is the biological identity of a potential gene product predicted?
This is one of the primary problems faced by bioinformaticians.
The genetic code provides rules for the prediction of open reading
frames; however, these data do not allow assignment of a function
to the gene product. Current predictive methods depend on the identification
of homologs or related sequences that have already been identified
and are present in the NCBI databases. The problem is reduced to
the questions: Are there any related sequences present in the databases?
And if so, is this relationship sufficiently significant that an
assignment of function can be made? Methods for the identification
of gene products are based on a series of tools that measure the
relatedness of nucleic acid or protein sequences. This is achieved
by finding the best alignment between two macromolecules. The alignment
problem will be present to the students as a progression of methods
starting with simple matrix comparisons visualized as dot plots,
then extended to introduce global alignments, which utilized dynamic
programming methods. The dynamic programming approach is similar
to the familiar “traveling salesman” problem. The final step is
to apply these methods to query a large database, such as NCBI,
searching for related molecules. In this later step we will also
address the problem of determining the significance of the match.
For this we will introduce concepts of probability. Biological
science students will see the difficulties and the limitations
in assigning gene identities based solely on sequence information.
CS/Math students will see the applications of statistical tools
to the problem of pattern matching. The importance of data management
and the visualization of sequence alignment will also be emphasized.
The problems will be designed to illustrate the basic concepts
of alignment and then extended to searching larger databases. The
NCBI site provides a search tool, Basic Local Alignment Search
Tool or BLAST, which allows users to query the NCBI databases.
This is the commonly used method. There are numerous options for
homework and project assignments, and many of these can be built
to be extensions of the assignments from the preceding module.
Classification of Hybrids
Using Genetic Markers and Maximum-Likelihood Estimates
LINK
TO MODULE
Andrew T. Phillips, C. Alex Buerkle, Marc R. Goulet, Alexander
J. Smith, Paul J. Thomas
Departments of Math, Computer Science, Biology, and Physics
University of Wisconsin-Eau Claire
phillipa@uwec.edu
Organisms exist as distinct species, or groups of individuals that
are genetically, and often reproductively, isolated from other groups.
Such species harbor distinct features and retain these features
over long periods of time. However, many species are also capable
of reproducing with members of other species, resulting in hybrid
individuals. As an example, when horses and donkeys reproduce, a
hybrid offspring commonly known as the mule results. Generally though,
mules are unable to reproduce, and hence they do not lead to mixing
of the genes of the parental species.
Endangered Species and Island
Biogeography: An Introductory Computational Approach to Modeling
Populations and Habitats for Kirtland’s Warbler
LINK
TO MODULE
Census
Excel File
Zip
of Supporting Shape Files
Timothy L. Lewis, Ph.D.
Department of Biology
Wittenberg University
tlewis@wittenberg.edu
Conserving endangered species often requires a balancing of the
biological needs of endangered populations against the human desires
for economic and recreational opportunities. At least some biological
aspects of every species are poorly understood, some do have large
data sets that can be difficult to interpret. All are confounded
by human interactions. Computational science allows visualization,
analysis, and interpretation of large data sets in ways that can
inform these complex biological and environmental problems. This
module will allow students to explore one of the fundamental paradigms
of conservation biology, island biogeography, and apply that theoretical
ecological concept to a real-world problem by creating models for
habitat management. Specifically this module reviews island biogeography
as it applies to forest fragmentation in northern Michigan and uses
the related concepts to explore applications to preservation of
the endangered Kirtland’s warbler (Dendroica kirtlandii).
Students will learn fundamental ecological concepts, visualize and
analyze large spatial data sets of an endangered species using a
free but sophisticated geographic information system (GIS), and
develop an environmental impact statement formatted output to explain
recommendations based on their analysis of the data and developed
models.
Computational
Physics
Modeling the Hydrogen Atom
LINK TO MODULE
LINK TO MATLAB EXAMPLES
David Joiner and Robert Panoff
The National Computational Science Institute
Shodor Educational Foundation
djoiner@scan.shodor.org or rpanoff@shodor.org
This module presents the problem of determining the most likely
position, or orbital, of an electron around a hydrogen atom. The
mechanics of quantum particles are determined by a partial differential
equation known as Schrödinger's wave equation. This sets up
an eigenvalue problem that limits the possible energies of the electron.
The computational solution of Schrödinger's equation for hydrogen
is typically done via separation of variables, in which the radial
solution is separated from the angular solution, in effect reducing
the problem to 1 dimension and allowing for standard numerical integration
techniques to be applied.
Diffusion Limited Aggregation
LINK
TO MODULE
LINK TO DATA
ZIP FILE
This module presents the problem of growing aggregate structures
one particle at a time through random processes. Such structures
are seen throughout nature, through examples such as electrodeposition,
dielectric breakdown, and snowflake formation. The main algorithm
for modeling these aggregate structures is diffusion-limited aggregation
(DLA). DLA models cover a wide range of phenomena and size sales,
and variations range from lattice based models to models that allow
free movement, models in multiple dimensions, and models that change
how particles stick to the growing aggregate.
Spontaneous Magnetization
- Using the Ising Model and Monte Carlo simulations to study the
magnetization phase transition
LINK
TO MODULE
LINK TO DATA
ZIP FILE
John Phillips
Capital University
jphillip@capital.edu
This module provides an introduction to spontaneous magnetization
in ferromagnetic materials. It studies this phenomena by using the
Ising model and Monte Carlo modeling techniques. Although the topic
matter is probably not familiar to most students, the techniques
used are approachable by anyone who has completed a typical first
year introductory physics course.
For more information, please contact
Andrea Karkowski (614-236-6449)
Terry Lahm (614-236-6800)
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