Abstract: The Michaelis-Menten
Equation is useful in biological kinetics. This talk shall develop
the basic logic and the algebraic steps in a modern derivation
of this equation and consider its application to the processing
of drugs in the body.
Abstract: Gene duplications
are the basis for genetic evolution as evidenced by a number
of genome wide duplication events in increasingly evolved lineages.
Mathematical models, such as the Wright-Fisher and Moran, illustrate
the possible change in allele frequencies from generation to
generation. These results can be extended to the retention of
gene duplicates within an individual and throughout a population.
The resolution of gene redundancy and the mechanisms reviewed
in this research are defined fully by mathematics and are based
on the intricacies of biological interactions. A new user-defined
C++ program proposed herein predicts the fate of duplicated genes
with varying initial conditions.
Abstract: In calculus textbooks,
formulas are developed for the length of a curve in the plane
and for the area of a surface in three-space. Many textbooks,
including Stewart’s calculus book that we use at Miami University,
take different approaches to these two very similar mathematical
situations. One approach connects the dots along a curve to get
a polygonal approximation, while the other approximates via tangential
considerations. This raises the question of why don’t we take
the same approach in both situations? We shall look at these
differing approaches, compare them, and prove that each leads
to the expected mathematical conclusions. Upon doing so, we remain
with the curiosity of this common inconsistency.
Abstract: Despite being better
known for his literary works, Charles Ludwig Dodgson (a.k.a.
Lewis Carroll) produced many mathematical innovations. His work
mainly focused on the fields of Euclidean Geometry, Determinants,
Trigonometry, Symbolic Logic and Riddles. Additionally, doing
pioneering work with symbolic logic and teaching it as a mathematical
subject. In this talk we will explore some of the symbolic logic
problems found in his best known work Alice in Wonderland and
it’s sequel Through the Looking Glass, and look at an early approach
to solving symbolic logic problems.
Abstract: The ability to
characterize the interior of an object without damaging the object
is an invaluable tool in industry. One useful technique of recent
interest is impedance imaging, or equivalently, steady-state
thermal imaging. The idea, in thermal terms, is to use temperature
measurements on the boundary of an object---specifically, imposed
thermal energy fluxes and measured boundary temperatures---to
determine interior structure, for example, to find internal cracks
or voids.
Abstract: The union-closed
sets conjecture, also known as Frankl's conjecture, states that
any family of sets F, which is closed under unions, has an element
in at least half its sets, with equality holding only if F is
a Boolean lattice. The conjecture has been proved for families
with at most 40 sets, and families where the largest set in F
has size at most 7. A number of other partial results and equivalent
statements are known. We will present an interesting characterization
of union-closed families, and give a number of statements which
imply Frankl's conjecture. Time allowing, we will survey past
results regarding the conjecture. This is joint work with Noah
Prince and Doug West.
Abstract: This presentation will
demonstrate an interactive model of Widmark's linear equation
for calculating blood alcohol content using a participant's total
body water, hypothetical rate of alcohol consumption, and average
rate of elimination. A regression equation using gender, weight,
height, and age will be used to calculate total body water. The
model will illustrate the value of spreadsheet software, such
as Microsoft Excel, in modeling and graphing discrete dynamical
systems.
Abstract: A minor of a graph G is
a subgraph which can be obtained through vertex and edge deletions
and edge contractions. One important open problem in graph theory
is the Hadwiger Conjecture, which states that every k-chromatic
graph has a complete minor of order k. Kostochka and Thomason
found the average vertex degree which forces a complete order k minor.
In this talk, I will survey old results and present a new theorem
of Kostochka and Prince. No previous familiarity with graph theory
is necessary for attendees.
Abstract: I will first discuss basic
facts regarding edge detection in digital images. Following that,
I will illustrate a heuristics based algorithm using edge detection
to focus a digital camera to varying scene conditions. This discussion
will be followed by a demonstrastion of the algorithm. Time permitting
I may also quickly sketch out an algorithm for adapting the camera
to changing lighting conditions.
Abstract: In this talk we discuss
some applications of Maple in teaching Algebra and Number Theory.
We discuss number theoretic and algebraic algorithms that are
used in Maple procedures and demonstrate how some Maple procedures
work step by step. This allows students to understand the applications
of some well known theoretical topics. At the same time, Maple
allows students to quickly solve some problems using Maple for
algebraic transformations that would otherwise be impossible
to do by hand during the class time for these topics. We will
demonstrate the possibilities within Maple that help students
to solve problems concerning the number of roots of polynomials
that depend on parameters using continuous fractions.
S.O.S – Help I’m lost!of
Essentially Equivalent Lattice Paths Mike Husband and Mike Yates, Siena Heights University
Abstract: The presentation
will give a brief history of the Coast Guard Auxiliary Search
and Rescue Procedure and the rationale behind a project that
was offered by a local Coast Guard instructor. The presenters
will explain and demonstrate algorithms developed for both a
graphing calculator and computer (Maple) to assist in the search
and rescue process.
Abstract: We used properties of the
maximal covering group of a cwatset to investigate the construction
of perfect cwatsets. This leads to a simple proof of a generalization
of a Lagrange-like result of Biss and suggests a potential technique
of constructing cwatsets from combining different groups from
binary space. We also proved the conjecture that there exists
a cwatset with no covering group of the same cardinality as the
cwatset.
Abstract: One of the major concerns
in cryptology today is data authentication. Authentication confirms
to the receiver that the message has come from the alleged source
and has not been altered in storage or transit. One way to do
this is through the use of smart cards. The smart card authenticates
itself to the system through the use of an algorithm only after
the user has authenticated himself to the smart card. I will
explain how this can be done by the use of a zero knowledge proof.
In this method, the user can authenticate himself by proving
that he does in fact know the key without actually revealing
the key or any new information that is not already known. I will
also talk about several examples, including the Fiat-Shamir Protocol,
which is the best known protocol of this kind.
Abstract: Studying how a disease spreads
through a population is important for determining countermeasures
to an epidemic. As an introduction, we will look briefly at the
basic SIR and SIRS epidemic models, after which we will analyze
a model taken from Murray’s Mathematical Biology for the
spread of the AIDS virus in a male population. In our analysis,
we will look at conditions for an epidemic, determining and classifying
equilibrium states, and implications of the model.
Abstract: One of the constant debates
in baseball statistics has been “Who is the greatest hitter,
pitcher, player, etc.?” No matter what the debate is over, whether
it is offense or defense, everyone wants to know who the best
is, and every player always wants to strive to be the best at
what they do. This talk will address the argument of “Who is
the greatest hitter of all time?” from a statistical basis. By
looking at previous formulas, a new model will be developed to
finally solve the debate.
Abstract: In this talk, we discuss
one application of Maple in teaching Algebra. We concentrate
on systems of linear equations. It is well known that such systems
of linear equations either have unique solutions, no solutions,
or infinitely many solutions. If some parameters are placed within
the system, then these parameters can be changed to cause one
of those situations. The purpose of our talk is to explore what
happens to the solutions set of a system as the values of parameters
approach certain values that would yield infinitely many solutions.
Using Maple, we will clearly demonstrate how to understand where
the infinite solution set exists in circumstances when a person
is limited to approximate values of parameters that are close
to those required for the infinite solution set.
Knot Invariants Ben Lundell, University of Illinois at Urbana-Champaign
Abstract: A fundamental problem in
knot theory is proving whether two distinct knot projections
represent distinct knots. Knot theorists strive to discover properties
that are invariant under the projection chosen for the knot.
Unfortunately, calculating and invariant can be quite a cumbersome
task. This talk will cover several types of invariants and relationships
among them. Specifically, we will present results from a conjecture
relating the crossing number, determinant, and embeddability
number of a knot.
Abstract: The Transparent Grid Termination
(TGT) enables computationally efficient Finite Difference (FD)
numeric solutions of Poisson’s equation for open boundary problems.
This talk describes recent theoretical and computational enhancements
that extend and enrich the use of TGT methods. Non-Dirichlet
boundary conditions, such as uniform fields, can be accepted
to simulate conditions other than open boundaries. The TGT method
enables solving a smaller solution region while simulating a
surrounding buffer region (where there are no sources.) The potentials
on the boundary between the solution region and buffer region
are computed using TGT coefficients. The derivation of these
coefficients and their use in solving Ax=b is discussed. Additionally,
the topic of non-zero boundary conditions and how this impacts
the current TGT algorithm is discussed.
Abstract: Group extensions play a fundamental
role in much of group theory, even though many people don't realize
that they are using the concept. Indeed part 2 of the Hölder
Program says: "Given groups H and K, find all ways
of constructing a group G such that H is a normal
subgroup of G and G/H is isomorphic to K." This
talk will begin with a look at group extensions, followed by
a look at a generalization called a group enlargement, consider
each in the realm of Galois Theory, and finally consider applications
to quasi p-groups.
Abstract: Sub-exponential factorization
algorithms have gained prominence in recent years, with the most
recent accomplishment being the factorization of RSA-576. Many
times, these sub-exponential algorithms rely on exponential algorithms
to factor auxiliary numbers. Among these, Shanks' Square Form
Factorization Algorithm (SQUFOF) is one of the most attractive.
Introduce about 20 years ago by Daniel Shanks, SQUFOF is based
on the theory of binary quadratic forms to find a factorization
involving computations with relatively small numbers. Thus the
advantage is a computational one. Shanks stated without proof
the average running time of SQUFOF. This running time has been
computationally consistent, yet a proof has not surfaced. In
this talk, I give a brief overview of the SQUFOF algorithm and
study the numerical evidence pointing to the correctness of Shanks'
expression. Using the theory of binary quadratic forms, and results
form algebraic and analytic number theory, I derive my own expression
on the running time. I also show how my expression can be transformed
into Shank's expression, assuming the existence of certain quantities.
Abstract: There is a way of assigning
to a ring R a (spectral) space such that ring homomorphisms get
taken to continuous maps. This assignment turns out to be a functor
called Spec. In this talk we will investigate how the action
of the Galois group Gal(F/E) on the polynomial ring F[x] transforms
to an action on the space Spec(F[x]). In particular, we will
show that the assignment takes the fixed points of the action
on F[x] to the orbit space of the action on Spec(F[x]) and similarly
the orbit space is taken to the fixed points. This talk will
introduce many to category theoretic ideas such as pushouts,
pullbacks and contravariance. An undergraduate course in abstract
algebra and a basic knowledge of topological spaces that one
would get in an analysis course should provide enough background
for this talk.
Abstract: Nonlinear partial differential
equations are used to model a wide variety of phyiscal phenomena.
Of these equations, the Korteweg-de Vries (KdV) equation arises
in the study of shallow water wave propagation. Symmetry analysis
is one of the most powerful techniques used to generate exact
solutions of these equations. There are two approaches to symmetry
analysis: the classical and nonclassical methods. Typically,
the nonclassical is more difficult to perform than the classical
method, but it may give rise to exact solutions that the classical
method does not. The goal of this project was to determine whether
or not the nonclassical method will give rise to solutions unobtainable
by the classical method.
Abstract: This presentation will discuss
the double bubble problem with an emphasis on the use of computers
to assist in problem solving techniques. Then, several conjectures
about possible solutions for the triple bubble problem will be
discussed, along with how computer models led to better conjectures
for future work.
Abstract: Ciphers are methods used
to turn plain text messages into secret messages. One way of
doing this is to rearrange the order of the letters in the message
to create the secret message. Such ciphers are called transposition
ciphers. We will explain these with examples, including the grille,
the rail fence, and the scytale, and discuss their cryptoanalysis.
Abstract: In this talk, we consider
tilings of the upper half plane by hyperbolic polygons. We combine
the polygons of our tiling to form Klein surfaces, and show how
these surfaces possess automorphisms. It is known that if Y is
a bordered Klein surface of algebraic genus p, the order of its
automorphism group must be less than or equal to 12(p-1). We
conclude the talk by examining ways to construct Klein surfaces
that possess this maximal number of automorphisms.
