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Proceedings of the
Harriett J. Walton Symposium
on Undergraduate Mathematics Research
Volume 13
Editors
Benedict Nmah
Rudy Horne
Farouk Brania
Chaohui Zhang
Department of Mathematics
Morehouse College
Atlanta, GA
2015
ii
Printed in the United States of America
Published by:
Department of Mathematics
Morehouse College
830 Westview Dr., SW
Atlanta, GA 30314 USA
iii
Contents
Preface………………………………………………………………………………...…………vii
Short Biography of Professor Harriett J. Walton……………………………………………...…ix
Arman Green
Proof of the Riemann Mapping Theorem………...……….……………….……..………….……1
Victoria Latimore and Latalya Walden
Stocks as Financial Security, Valuation, Risks, and Retirement Portfolios……………...……...13
Jalen Marshall
Mathematical Modeling of Retardation in an Organic Chain Reaction…………..……………. 18
Trevonta L. Mctyre and Marquis D. Curry
Bond Valuations and Investments Based on Bonds……………………………………………..24
Latalya Walden and Marquis D. Curry
Using Process Capability Estimates and Attribute Data
to Generate Control Charts for Healthcare Delivery and Management………………………….28
Abstracts
Jarret D. Camp
Investigation of Solutions to Differential Equations with Variable Coefficients………………..37
Curtis Clark, Jr.
On 2-2 Graph Achievement Games ……………………………………………………………..37
Adam J. Eiring and Benjamin H. Gaines
Predicting the Steady State Maximum and Minimum Drug Levels in the Blood……………….38
J. R. Gillings
R. Thompson's Group V presented as Permutations of Subintervals of [0, 1]……………...…..38
Coleman Gorham and Robert Weaver
A Look into the RSA Cryptosystem……………………………………………………………..38
Arman Green
A Proof of the Riemann Mapping Theorem……………………………………………………..39
Myles Harper
RSA cryptosystems and public Key data encryption…………………………………….………39
iv
Malik Henry
Random Knot Diagrams…………………………………………………..……………………..39
Jassiem Ifill
Fair division among multiple players or multiple divisible goods ……………………………...40
Talon Johnson
Analytically Understanding Population Dynamics of the
Interaction between T-cells and HIV…………………………………………………………….40
Dorian Kandi
Eigenvectors of Positive Matrices……………………………………………………………….41
Jillian Kuether Computing the minimum norm least squares solution to
a system of linear equations through Gauss-Jordan elimination………………………………...41
Victoria Latimore and Latalya Walden
Stocks as Financial Security, Valuation, Risks and Retirement
Portfolios…………………………………………………………………………………………42
Joshua Manley-Lee
A Model of Multi-store Competition Strategy…………………………………………………...42
Jalen Marshall
Mathematical Modeling of Retardation in Organic Chain Reactions……………………………42
Luis Matos
Fundamental Groups of Coarse Spaces………………………………………………………….43
Christopher McClain
Category Theory and Bridging the Gaps………………………………………………………...43
Trevonta L. Mctyre and Marquis D. Curry
Bond Valuations and Investments Based on Bonds……………………………………………..43
Joseph Park
Bound States in the Radiation Continuum for Periodic Structures………………………………44
Aquia Richburg
Modeling the Brain with Math: Neural Networks and Liquid State Machines………………….44
William Samuels
Understanding L'Hopital's Rule………………………………………………………………….45
v
Crystal Silver, Abebe Mojo and Gabriel Tsegaye
Using a Markov Chain Model to understand the behavior of Student Retention……………..…45
Everett Starling
An Introductory Comparative Analysis of Two Statistical Spectral Estimation Techniques…....46
Latalya Walden and Marquis Curry
Using Process Capability Estimates and Attribute Data to Generate
Control Charts for Healthcare Delivery and Management………………………...…………….46
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Preface
The Department of Mathematics initiated the Harriett J. Walton Symposium on Undergraduate
Mathematics Research to encourage undergraduate students in mathematics research and
practice. We believe that undergraduate research experiences must count among the most
challenging and rewarding experiences for college students. To research and present material
beyond what is traditionally covered in the classroom to explore mathematics independently may
be considered the best career preparation for students regardless of their post-college plans. This
yearly symposium on undergraduate research is dedicated to Professor Harriett J. Walton who,
for forty-two years, served on the faculty of Morehouse College.
The Thirteenth Annual Harriett J. Walton Symposium on Undergraduate Mathematics Research
was held on the campus of Morehouse College in Atlanta, Georgia, USA on March 28, 2015.
Thirteen students from Morehouse College, seven students from Albany State University, four
students from Birmingham-Southern College, three students from Clark Atlanta University, one
student each from University of Florida, Georgia State, University of Georgia, and Kennesaw
State University gave presentations on their research and studies in mathematics and related
fields. The Symposium was sponsored by the Department of Mathematics and the Division of
Science and Mathematics of Morehouse College through the generous support of The
Mathematical Association of America (MAA) Regional Undergraduate Mathematics Conference
Program through National Science Foundation Grant DMS-0846477. This volume contains five
articles and twenty-four abstracts submitted by the Symposium participants and their advisors.
The organizers of the Symposium thank the presenters and their advisors for preparing a
remarkable collection of lectures for the Symposium. We thank the referees for their service to
evaluate and improve the papers before their publication. We thank the administration of
Morehouse College for their generous support, especially Dr. J. K. Haynes, Dean of the Division
of Science and Mathematics, Dr. Garikai Campbell, Provost and Senior Vice President for
Academic Affairs, and Dr. John Wilson Jr., President of Morehouse College. We thank the MAA
for their support, advice, and materials for the Symposium. Special thanks to Morehouse
College Students and Mr. William Barnville for coordinating many aspects of the Symposium.
Finally we thank Professor Walton for attending the Symposium.
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ix
Professor Harriett J. Walton
In September of 1958, Harriett J. Walton joined the faculty of Morehouse College during the
presidency of Benjamin Elijah Mays. She became a member of a team of three persons in the
Department of Mathematics where she worked with the legendary Claude B. Dansby who served
as Department Chair. Dr. Walton and her two colleagues taught all of the mathematics for the
majors as well as the mathematics for non-science students. Dr. Walton relates that two of her
favorite courses that she taught during this period were Abstract Algebra and Number Theory.
The three member mathematics department did an excellent job of preparing their mathematics
majors for graduate school and the other students for success in their respective disciplines. In
fact it was during this period of history that Morehouse gained the reputation of being an
outstanding Institution especially for African American men. As the department grew, Dr.
Walton shifted her attention away from mathematics majors and began to concentrate on
students who needed special attention and care in order to succeed in mathematics. She became
an advisor, mentor, tutor and nurturer to a large number of students matriculating at Morehouse
College. Because of the caring attitude that she had for her students, some of them to this day
refer to her as “Mother Walton”.
Dr. Walton has never been satisfied with mediocrity. Throughout her teaching career she
demonstrated a love for learning. In 1958 when she arrived at Morehouse College she had an
undergraduate degree in mathematics from Clark College in Atlanta, Georgia, a Master of
Science degree in mathematics from Howard University, Washington D.C., and a second
Master's degree in mathematics from Syracuse University. While at Morehouse teaching full
time and raising a family of four children, Dr. Walton earned the Ph.D. degree in Mathematics
Education from Georgia State University. After receiving her doctorate, Dr. Walton realized the
emerging importance of the computer in education so she returned to school and in 1989 earned
a Master’s degree in Computer Science from Atlanta University. She is indeed a remarkable
person.
Dr. Walton’s list of professional activities, awards and accomplishments during her career is
very impressive and too lengthy to be enumerated here. However a few special ones are her
memberships in Alpha Kappa Mu, Beta Kappa Chi, Pi Mu Epsilon, and the prestigious Phi Beta
Kappa Honor Society. Additionally she was selected as a Fulbright Fellow to visit Ghana and
Cameroon in West Africa. Dr. Walton’s professional memberships included the American
Mathematical Society, the Mathematical Association of America, National Council of Teachers
of Mathematics (NCTM) and the National Association of Mathematicians (NAM). She served as
Secretary/Treasurer of NAM for ten years. In May 2000, Dr. Walton retired from Morehouse
College after forty-two (42) years of service.
Proc. H. J. W. S. U. Math. Res., Vol. 13 (2015) 1 - 12
Proof of the Riemann Mapping Theorem
Arman GreenDepartment of Mathematics
Morehouse CollegeAtlanta, GA 30314
Abstract The Riemann Mapping Theorem is a powerful theorem that proves there is aunique analytic function that isomorphically maps a simply connected domain that is notthe entire complex plane to the unit disc with the conditions that the function evaluatedat that specific point is zero and the derivative evaluated at that point is a positive realnumber. In this paper, we will provide some details of key ingredients to help prove theRiemann Mapping Theorem, such as the Open Mapping Theorem, the Local MaximumModulus Principle, and Schwarz Lemma. Afterwards, we will go into depth of the proof ofthe Riemann Mapping Theorem showing how these ingredients are crucial and how theyintertwine with one another.
1. Introduction
The Riemann Mapping Theorem, a powerful and well-known theorem in geometric func-tion theory, was first introduced by the mathematician Bernhard Riemann in 1851. Whenfirst introduced, flaws were found around his original statement; however, Karl Weierstrassand David Hilbert were able to solidify Riemann’s findings and stated the theorem to whatwe know it to be today. The first published proof of the theorem did not come around until1912 by Constantin Carathodory and more proofs using different mathematical areas havebeen published and are available.
In this paper, we will study one version of the proof of the Riemann Mapping Theorem.The proof is very complicated, as it involves many concepts from an undergraduate levelreal analysis and complex analysis course (We used [2] and [3] to help) and it involves newconcepts that one would learn in a graduate level complex analysis course (we used [1] tohelp). We will begin by first introducing some preliminary definitions, with examples, thatone needs to know, followed by some important theorems that are used in the proof of theRiemann Mapping Theorem. Once the preliminaries are out of the way, we will state andprove the theorem. Afterwards, in the conclusion, we will discuss why it is important andhow it is used.
2. Preliminaries
2.1. Conformal Mappings. Before we focus on the Riemann Mapping Theorem and itsproof, we will first go over some of the basic tools and the ingredients needed. The firstconcept we look at is conformal mapping. We give the standard definition of a conformalmapping alongside illustrating some examples (the definition and examples come from [2]).First, we define what an analytic function is.
A function f(z) = u(x, y) + iv(x, y) of the complex variable z = x + iy is analyticat a point z0 if the derivative f ′(z0) exists (i.e. its partial derivatives are continuous and
1
2
it satisfies the Cauchy-Riemann equations: ∂u∂x
= ∂v∂y
and ∂u∂y
= − ∂v∂x
). The function f is
analytic in a domain D (open connected set) if and only if it is analytic at each point ofD.
It is a well-known fact in complex analysis that if f is analytic at z0 then f admits apower series f(z) =
∑∞n=0 an(z − z0)n at z0, where an ∈ C.
A function f(z) is conformal at a point z0 if f is analytic there and f ′(z0) 6= 0. Afunction f(z) defined on a domain D is a conformal mapping, when it is conformal ateach point in D.
To help visualize what a conformal mapping is, we look at the following examples:Consider the function f(z) = ez. Notice that the exponential function is conformal
throughout the entire z plane since (ez)′ = ez 6= 0. Also, we can write the exponentialfunction as
ez = exeiy since z = x+ iy
If we have the lines x = c1 and y = c2 in the z plane, their images under the mapping area positively oriented circle centered at the origin and ray from the origin, respectively (seeFigure 2.1).
Figure 1
The function f(z) = z (the conjugate function) is not conformal. To see this, if a complexvariable z = x + iy, then z = x − iy. In this case, u(x, y) = x andv(x, y) = −y. Howeverlooking at the partial derivatives,
∂u
∂x= 1 6= ∂v
∂y= −1.
Therefore, the conjugate function does not satisfy the Cauchy-Riemann equations whichmeans the function is not analytic, hence not conformal.
We introduce the concept of conformal mappings because we will see that the mappingdescribed inside the statement of the Riemann Mapping Theorem is in fact a conformalmapping since the function will be analytic and its derivative is positive at every pointinside of a given domain. We will see why this is true later in the paper.
2.2. Open Mappings and the Open Mapping Theorem. From this point on, every-thing we discuss is in [1]; however, we will present everything together in a way such thatone will be able to understand everything about the Riemann Mapping Theorem. The nextconcept that we will go over is open mapping. This is important to understand because, likea conformal mapping, the function described inside the statement of the Riemann MappingTheorem is an open mapping as well.
If U is an open set and f is a function on U , f is called an open mapping if the imageof every open subset of U is open.
3
Necessary for the proof the Riemann Mapping Theorem, we will look at the Open Map-ping Theorem as it is an application for two key ingredients of proving the Riemann Map-ping Theorem: the Local Maximum Modulus Principle and Schwarz Lemma. However,before we look at and prove the Open Mapping Theorem, we will state the following defi-nition and lemma.
Let f be an analytic function on an open set U , and let f(U) = V . Then f is an analyticisomorphism if V is open and there is an analytic function
g : V → U
such that f and g are inverses to each other. A function f is a local analytic isomorphismat a point z0 if there is an open set U containing z0 such that f is analytically isomorphicon U.
The following lemma gives a criterion for invertibility of a analytic function.
Lemma 2.1.
• Let f(T ) = a1T+ higher terms be a formal power series with a1 6= 0. Then thereexists a unique power series g(T ) such that f(g(T )) = T .• If f is a convergent power series, then g is a convergent power series.• If f is an analytic function on an open set U containing z0 and f ′(z0) 6= 0, then fis an local analytic isomorphism at z0.
We will now turn our attention to stating and proving the Open Mapping Theorem.[Open Mapping Theorem] Let f be analytic on an open set U , and for each point of U ,
f is not constant on a given neighborhood of that point. Then f is an open mapping.
Proof. Let f be an arbitrary analytic function centered at a point z0 and is a convergentnon-constant power series. That is,
f(z) =∑
an(z − z0)n = a1(z − z0) + z2(z − z0)2 + a3(z − z0)3 + ...
Let m be the order of f (note that since f is a non-constant power series, then m = ordf ≥ 1). Thus,
f(z) = am(z − z0)m + am+1(z − z0)m+1 + ... = am(z − z0)m(1 + h(z))
where am 6= 0 and h(z) is also a convergent non-constant power series. Choose a complexnumber a such that am = am. Then
f(z) = am(z − z0)m(1 + h(z)) = [a(z − z0)(1 + h1(z))]m
where1 + h1(z) = (1 + h(z))1/m
and h1(z) is a convergent non-constant power series. Let f1(z) be the function such thatf1(z)m = f(z). That is
f1(z) = a(z − z0)(1 + h1(z)) = a(z − z0) + a(z − z0)h1(z).
By Lemma 2.1, since f ′1(z0) 6= 0, if U is an open set centered at z0 on which f1 converges,then f1(U) contains open set V . The image of V under the map
w → wm
is a open set, which means f(U) contains open set centered at z0. �
This concludes the proof of the Open Mapping Theorem.
4
2.3. Local Maximum Modulus Principle and Schwarz Lemma. A direct applicationof the open mapping theorem will be the the local maximum modulus principle, and asa direct application of the Local Maximum Modulus Principle, one can prove SchwarzLemma. Again, these theorems are two of three major ingredients to proving the RiemannMapping Theorem. We will first look at proving the Local Maximum Modulus Principleand we will continue on to proving Schwarz Lemma.
[Local Maximum Modulus Principle] Let f be analytic on an open set U . Let z0 ∈ U bea maximum for |f |, that is,
|f(z0)| ≥ |f(z)|, for all z ∈ U.Then f is constant in some neighborhood of z0.
Proof. We will assume that the function f is not constant on all neighborhoods of z0. ByTheorem 2.2, f is an open mapping in a arbitrary neighborhood of z0, which means theimage of the neighborhood of z0 is open. Since the image is open, the image contains a disccentered at f(z0) with radius r > 0. Hence the set of numbers |f(z)|, for z in a neighborhoodof z0, contains an open interval around f(z0). However, for some z, |f(z)| > |f(z0)|,contradicting the assumption that |f(z0)| is the maximum for f , and thus, the theorem isproven. �
See Figure 2 that provides visualization of our proof.
Figure 2. Picture Illustrating Local Maximum Modulus Principle
The important matter that we learn from the Local Maximum Modulus Principle is thatthe function defined on an open set cannot have a maximum unless the function is constantthroughout the entire open set. With the Local Maximum Modulus Principle being proven,we will now move to proving Schwarz Lemma.
[Schwarz Lemma] Let f : D(0, 1)→ D(0, 1) be an analytic function of the unit disc intoitself such that f(0) = 0. Then:
• We have |f(z)| ≤ |z| for all z ∈ D.• If for some z0 6= 0 we have |f(z0)| = |z0|, then there is some complex number α of
absolute value 1 such that f(z) = αz.
5
Figure 3. Picture of Schwarz Lemma
Proof. Let f be analytic at 0, then we have a power series representation centered at 0.That is,
f(z) = a1z + a2z2 + ...
The constant term is 0 because we assumed f(0) = 0. Note that
f(z)
z= a1 + a2z + a3z
2 + ...
is an analytic function, and for |z| = r < 1 (the modulus is less than 1 since this is amapping from the unit disc into itself),∣∣∣∣f(z)
z
∣∣∣∣ =|f(z)||z|
=|f(z)|r
<1
r.
This is also true for |z| ≤ r by Theorem 2.3. Therefore, as r approaches 1, the first assertionis proved. Furthermore, if ∣∣∣∣f(z0)
z0
∣∣∣∣ =|f(z0)||z0|
= 1
for some z0 in the unit disc, then by Theorem 2.3 f(z)/z cannot have a maximum unlessit is constant. Thus, there is a constant α such that f(z)/z = α, which would thereforeprove the second assertion and thus proving Schwarz Lemma. �
A picture of what Schwarz Lemma is saying is shown by Figure 3. At this point intime, we are almost able to understanding everything we need to know about the RiemannMapping Theorem. We will need to look at one last ingredient and then we will get intothe main body of this paper.
2.4. Compact Sets in Function Spaces. The final ingredient we need in order to gointo looking at the Riemann Mapping Theorem is understanding what it means for a setto be compact in a specific function space: the space of analytic functions.
If U is an open set and Analytic(U) is the space of analytic functions on U , a subset Φof Analytic(U) is relatively compact or a normal family if every sequence in Φ has auniformly convergent subsequence on every compact subset of U .
Note that these subsequences that converge uniformly do not necessarily need to convergeto an element of Φ. Relatively compact and normal families can be used interchangeably,however, for this paper, we will be consistent with using the phrase “normal family”.
6
A subset Φ of Analytic(U) is uniformly bounded on a compact set K in U if for eachK, there exists a positive number B(K) such that
|f(z)| ≤ B(K) for all f ∈ Φ, z ∈ K.
A subset Φ of Analytic(U) is equicontinuous on a compact set K in U if, given ε, thereexists δ such that if z, z′ ∈ K and |z − z′| < δ, then
|f(z)− f(z′)| < ε for all f ∈ Φ
An equicontinuous family of continuous functions on a compact set is a normal family.This is Ascoli’s theorem from real analysis. In the following theorem, Ascoli’s theorem isused to define a connection between normal families and uniform boundedness.
Let Φ ⊂ Analytic(U), and assume that Φ is uniformly bounded on compact sets in U .Then Φ is a normal family.
This connection between normal families and uniform boundedness will be very impor-tant when we prove the Riemann Mapping Theorem. We will see how later on in thepaper.
3. Riemann Mapping Theorem
3.1. Statement of the Theorem. With all of the preliminary definitions, lemmas, andtheorems out of the way, we are now able to go ahead and look at the Riemann MappingTheorem.
[Riemann Mapping Theorem] Let U be a simply connected open set which is not thewhole complex plane. Then U is analytically isomorphic to the disc of radius 1. Moreprecisely, given z0 ∈ U , there exists a unique analytic isomorphism
f : U → D(0, 1)
of U with the unit disc, such that f(z0) = 0 and f ′(z0) > 0 is a positive real number.
Figure 4. Riemann Mapping Theorem
Figure 4 is a picture of what this powerful theorem is stating. If we have a simplyconnected open set, meaning that there are no holes within the set, and it is not the entirecomplex plane, meaning there is a point not in the set, then I can find a function that isbijective and analytic that maps the open set to the unit disc. To prove this theorem, wewill first prove that this function is unique. Once that is completed, we will look at howwe know this function exists.
7
3.2. Proof of the Theorem. We will first prove that this function stated in the RiemannMapping Theorem is unique. But before we do this, we will provide the following lemma.
Lemma 3.1. Let f : D → D be an analytic automorphism of the unit disc and supposef(α) = 0. Then there exists a real number ρ such that
f(z) = eiρα− z1− αz
To prove uniqueness, let f and g be two analytic isomorphisms of U onto D satisfyingthe conditions f(z0) = g(z0) = 0 with f ′(z0) > 0 and g′(z0) > 0 Then, f ◦g−1 is an analyticautomorphism of the disc with the origin fixed since f ◦ g−1(0) = f(z0) (since g−1(0) = z0)= 0. By Lemma 3.1, we know that α = 0 which means:
f ◦ g−1(z) = eiρα− z1− αz
= −eiρz = βz.
Note that −eiρ = β and the modulus of β is 1 since the modulus of −eiρ = 1 for any numberρ. If we compute the derivative of this function at 0, we find
(f ◦ g−1(0))′ = f ′(g−1(0)) ∗ 1
g′(g−1(0))=f ′(z0)
g′(z0)= β > 0
Since we know that the quotient is positive, and we also know that this quotient equals βsince the derivative of βz is β, which means β is a positive real number. Since β is a positivereal number and |β| = 1, this means that β = 1, thus f ◦g−1 = βz = z = id, meaning f = g.
Now that we proved the uniqueness of this function described in the Riemann MappingTheorem, we will now prove that this type of function exists. However, before we do this,we first make a reduction.
We can find an isomorphism between U and an open subset of the unit disc.
Figure 5. Our Reduction
To see this reduction is true, we’ll assume there exists a point α ∈ C and α 6∈ U . SinceU is simply connected, there exists a single valued branch
g(z) = log(z − α)
8
for z ∈ U which is analytic on U . We know this function exists since z−α 6= 0 since α 6∈ U .This function is also injective since:
g(z1) = g(z2) =⇒ log(z1 − α) = log(z2 − α)
=⇒ elog(z1−α) = elog(z2−α) =⇒ z1 − α = z2 − α=⇒ z1 = z2.
Since g(z) is a single valued branch and if it takes on a value, g(z0), then
g(z) 6= g(z0) + 2πi for all z ∈ U.If it did, meaning,
g(z) = g(z0) + 2πi for some z ∈ U,then
eg(z) = eg(z0)e2πi =⇒ eg(z) = eg(z0) (since e2πi = 1) =⇒ g(z) = g(z0),
which is a contradiction since g(z0) 6= g(z0) + 2πi. Because of this, we know there exists adisc around g(z0) + 2πi such that g takes on no values in this disc. Since g cannot reachpoints inside this disc, then
g(z)− g(z0)− 2πi 6= 0 =⇒ |g(z)− g(z0)− 2πi| > a > 0
=⇒ 1
|g(z)− g(z0)− 2πi|<
1
a.
This function is bounded, injective since g is injective, and analytic since g is analytic onU . Taking our function
1
|g(z)− g(z0)− 2πi|<
1
aand translating and multiplying it by a small positive real number, we obtain a function fsuch that f(z0) = 0 and |f(z)| < 1 for all z ∈ U , as expressed in Figure 6.
Figure 6. Translating Set to be Subset of Unit Disc
Now that we have shown this reduction is true, WLOG, assume that U is an open subsetof D that contains the origin. Let Φ be the family of all injective analytic maps
f : U → D
such that f(0) = 0 and |f(z)| < 1 for all z ∈ U . We will prove these three steps:
• The family is non-empty and uniformly bounded.• There exists element in family for which |f ′(0)| is maximal.• The element from Step 2 gives desired isomorphism of U onto D.
9
Figure 7. Identity Function in Family
Once we have proven these three steps, we will have completed the proof the RiemannMapping Theorem.
As far as the first step is concerned, this family is not empty, as it contains the identityfunction, and it is uniformly bounded since we’re mapping U ⊂ D into D and |f(z)| < 1for all f ∈ Φ, z ∈ U , as shown in Figure 7. Thus, the first step is completed.
For the second step, we will use the following lemma:
Lemma 3.2. Let U be a connected open set, and let {fn} be a sequence of injective analyticmaps of U into C which converges uniformly on every compact subset of U . Then the limitfunction f is either constant or injective.
Focusing back now on the second step, we know that for f ∈ Φ, |f ′(0)| is bounded usingCauchy’s formula,
fn(z0)
n!=
1
2πi
∫C
f(ζ)
(ζ − z0)n+1dζ.
and the uniform boundedness of the functions f . To see this further, letting n = 1 andz0 = 0 in Cauchy’s formula, we get
2πif ′(0) =
∫γ
f(ζ)
ζ2dζ.
where γ is some small closed circle in the unit disc. Thus,
|2πif ′(0)| = 2π|f ′(0)| =∣∣∣∣ ∫
γ
f(ζ)
ζ2dζ
∣∣∣∣ ≤ ∫γ
|f(ζ)||ζ2|
|dζ|
changing ζ to its polar form, ζ = reiθ, which implies dζ = ireiθdθ. Thus,∫γ
|f(ζ)||ζ2|
|dζ| =∫ 2π
0
|f(reiθ)||ireiθdθ||(reiθ)2|
<
∫ 2π
0
1 ∗ rdθr2
=2π
r.
Therefore,
2π|f ′(0)| < 2π
r=⇒ |f ′(0)| < 1
r.
Since we know that |f ′(0)| is bounded for f ∈ Φ, let λ be the least upper bound of |f ′(0)|for f ∈ Φ and let {fn} be a sequence in Φ such that |f ′n(0)| → λ, Choosing a subsequenceif necessary, by Theorem 2.4, {fn} → f uniformly and
|f ′n(0)| → |f ′(0)| = λ
with |f(z)| ≤ 1 for all z ∈ U . Using Lemma 3.2, f cannot be constant otherwise thederivative is 0 meaning that λ = 0 is a upperbound for all f ∈ Φ, which contradicts that
10
the identity function has a derivative of 1 and 1 > λ. Therefore, f is injective and by thelocal maximum modulus principle,
|f(z)| < 1 for all z ∈ U.
Therefore, f ∈ Φ and |f ′(0)| is maximal in the family Φ.Now all that is left is proving the final step, which is showing that an element f ∈ Φ
such that |f ′(0)| is maximal gives us the desired isomorphism that we are looking for. Wewill show this with the following theorem.
Let f ∈ Φ be such that |f ′(0)| ≥ |g′(0)| for all g ∈ Φ. Then f is an analytic isomorphismof U with the disc.
Proof. We already know that f is injective, so all we need to prove is that f is surjective.
Suppose that f is not surjective. Then there exists a point α ∈ D outisde the image off (as shown in Figure 8).
Figure 8. Our Function f is not Surjective
Let T be an automorphism of the disc such that T (α) = 0. We know there exists anautomorphism by Lemma 3.1. Then T ◦ f is an isomorphism of U onto an open subset ofD that does not contain 0 and is simply connected (see Figure 9). Thus, we can define asquare root function on T (f(U)). Let’s define the square root function as the following:√
T (f(z)) = e12logT (f(z)) for z ∈ U.
Note that this function is injective since T and f are injective and T (f(U)) is contained in
D. Now, let R be an automorphism of D such that R(√T (f(0))) = 0 (see Figure 9). Then
g : z → R(√T (f(z)))
is an injective mapping of U into D and g(0) = 0. Thus g ∈ Φ.If we let S be the square function and we let
ρ(w) = T−1(S(R−1(w))),
thenρ(g(z)) = f(z).
since
ρ(g(z)) = T−1(S(R−1(g(z)))) = T−1(S(√T (f(z))) = T−1(T (f(z)) = f(z)
11
Figure 9. Defining functions f, T, square root, and R
Furthermore ρ : D → D is a map into itself, ρ(0) = 0, and ρ is not injective. Thus, bySchwarz Lemma, we know |ρ′(0)| < 1. But
f ′(0) = ρ′(g(0))g′(0) = ρ′(0)g′(0) =⇒ |f ′(0)| < |g′(0)|contradicting the assumption that |f ′(0)| is maximal. �
Therefore we have found an analytic isomorphism that maps a simply connected openset to the unit disc such that there exists a point z0 ∈ U such that f(z0) = 0 and f ′(z0) > 0is a positive real number. This concludes the proof of the Riemann Mapping Theorem.
4. Conclusion
After looking at the Riemann Mapping Theorem, one can see that it is a conformalmapping since the derivative at any point is a positive real number and it is an openmapping since any open subset of a simply connected open set can be mapped to an opensubset of the unit disc. Now that we have proven the Riemann Mapping Theorem, somemay ask why is this theorem so important. Here is a list of reasons why people care aboutthe Riemann Mapping Theorem.
• It is the easiest way to find and prove a unique analytic isomorphism between twosimply connected sets.• Simply connected open sets can have complicated conditions (such as Dirichlet
boundary conditions), and knowing that a function exists that conformally maps theset to the unit disc in an angle-preserving manner can be useful for mathematicianswho study applied mathematics and work with Partial Differential Equations.• It has applications in other STEM related areas involving Physics and studying the
Steady State Heat Equation to Biology and studying movement of blood throughthe body and how blood clots affect blood travel.
12
Since time was not on our side, getting through the theorem was a feat within itself.However, here are some questions and possible future work ideas that one can look atinvolving the Riemann Mapping Theorem.
• Creating examples of how the Riemann Mapping Theorem is used in geometricfunction theory or any other mathematical course.• What other areas can the Riemann be beneficial to someone besides ones described
previously?• Can we generalize the Riemann Mapping Theorem (i.e. do we necessarily need all
the conditions listed in order for it to be true a Riemann mapping)?
I would be interested in working more on this topic and exploring more areas around howthis Theorem can help someone in different math, or STEM related, areas.
5. Acknowledgments
I would like to thank my advisor Dr. Farouk Brania, for working with me late on mostdays and helping me throughout the semester understanding this project while putting upwith my antics and schedule. I would like to thank other faculty members here in theMorehouse College Department of Mathematics for helping with this project in any meansnecessary and helping me to matriculate to a future mathematician awaiting to head tograduate school.
References
[1] Serge Lang, Graduate Texts in Mathematics: Complex Analysis (3rd Edition). 1993: Springer-Verlag:New York, N.Y..
[2] James Brown and Ruel Churchill, Complex Variables and its Applications (8th Edition). 2008:McGraw-Hill: New York, N.Y.
[3] Robert G. Bartle, The Elements of Real Analysis (2nd Edition). 1976: John Wiley and Sons, Inc.:Champaign, IL.
13
Proc. H. J. W. S. U. Math. Res., Vol. 13 (2015) 13 - 17
Stocks as Financial Security, Valuation, Risks, and RetirementPortfolios
Victoria Latimore and Latalya WaldenDepartment of Mathematics and Computer Science
Albany State UniversityAlbany, GA 31705
Abstract Stocks continue to be popular securities due to their immediate yield of increasein the value of investments. When a company is dully registered to do business thatcompany is allowed to sell stocks to the public or to a restricted population. A holderof the stock of a company is called a shareholder since every stock is equivalent to somepercentage of the company. The company is thus owed by shareholders who possess theshares or equity certificates. Depending on the conditions associated with the stock, ashareholder may sell his stocks at the stock market. The value of the stock at a time t isthe price an individual is willing to pay for the stock. In this paper, we examine commonstock, stock valuation, dividend on stocks, and the role of mixed portfolios in insuringprotection of investments for retirement.
1. Introduction
For many years, trading of financial securities have led to the growth of individual wealthas well as the wealth of nations. In today’s world, there are many and varied financialmarkets in which securities are traded. Such securities include stocks, bonds, options,contracts, currencies, and futures. While an item traded many be bond, stock, or a unit ofcurrency, other types of products traded may be items whose values are derived from othertraded equities or products. So the future values of these latter products depend on thefuture values of the products which they depend upon. For example, the value of a stockoption depends on the future value of the stocks. This item is called a financial derivativeand the equity on which it depends on is called the underlying equity. Financial derivativesinclude options based on stocks, currencies, and bonds.
A stock is a unit share Stock is a security representing ownership in a corporation. Astock certificate is issued as evidence of this ownership. Basically it means that a stockholder has a share in the company it holds stock in. In a sense the stockholders own apiece of the company that it has stock in. Stock shares are traded at stock exchanges suchas the following:
• New York Stock Exchange (NYSE) - Headquartered in New York City, USA. Mar-
ket Capitalization (2011, USD Billions) - 14,242; Trade Value (2011, USD Billions)- 20,161.• NASDAQ OMX - Headquartered in New York City, USA. Market Capitalization
(2011, USD Billions) - 4,687; Trade Value (2011, USD Billions) -13,552.• Tokyo Stock Exchange - Headquartered in Tokyo, Japan. Market Capitalization
(2011, USD Billions) - 3,325; Trade Value (2011, USD Billions) -3,972.• London Stock Exchange - Headquartered in London, Britain. Market Capitalization
(2011, USD Billions) - 3,266; Trade Value (2011, USD Billions) - 2,871.
14
• Shanghai Stock Exchange - Headquartered in Shanghai, China. Market Capitaliza-tion (2011, USD Billions) - 2,357; Trade Value (2011, USD Billions) - 3,658.• Hong Kong Stock Exchange - Headquartered in Hong Kong, China. Market Capi-
talization (2011, USD Billions) - 2,258; Trade Value (2011, USD Billions) - 1,447.• Toronto Stock Exchange - Headquartered in Toronto, Canada. Market Capitaliza-
tion (2011, USD Billions) - 1,912; Trade Value (2011, USD Billions) - 1,542.• BM&F Bovespa - Headquartered in Sao Paulo, Brazil. Market Capitalization (2011,
USD Billions) -1,229; Trade Value (2011, USD Billions) - 931.• Australian Securities Exchange - Headquartered in Sydney, Australia. Market Cap-
italization (2011, USD Billions) - 1,198; Trade Value (2011, USD Billions) - 1,197.• Deutsche Borse - Headquartered in Frankfurt, Germany. Market Capitalization
(2011, USD Billions) - 1,185; Trade Value (2011, USD Billions) - 1,758.
Securities are instruments which give their holders the right to money or property. Secu-rities include stocks, bonds, notes, mortgages. Since stocks are sold on behalf of companiesto raise capital and holders of the stocks become owners of the corporation or company,profits made by the company are shared to the shareholders: so are the headaches accruedby the company. Profit sharing are made in form of dividends, which are shared accordingto how much stocks one holds in the company. Of course one of the disadvantages is thatone can lose money if a stock’s price goes down. The value of a traded stock changes fromday to day. Hence, having insight on the price trend of stock is very essential for a stockowner.
Suppose S(t) represents the stock price at a time t, a simple stock price model is
dS(t) = µS(t)dt+ σS(t)dB(t)(1.1)
Where µ is the drift parameter which measures the average percent change in stock priceover time, σ is the volatility parameter which measures the randomness of the relativereturn, and B is a Brownian motion process. Equation (1.1) is a stochastic differentialequation.
The common type of stock issued by companies is called common stock. By purchasingthe common stock, the owner has equity of the issuing corporation. Corporations also issuePreferred Stocks. A preferred stock is an equity that pays a fixed dividend, although it isnot guaranteed. As the name indicates, in the case the corporation files for bankruptcy,the bond holders are paid first, followed by holders of preferred stock, and then holders ofcommon stock.
In Section 2, we discuss valuation of stocks and expected return on investment withexamples, and in Section 3, we discuss investment portfolio with various types of stock-backretirement accounts, and in Section 4, we discuss risks associated with stock investments,and in Section 5, we examine history of stocks of stable corporations, and in Section 6, weconclude this paper.
2. Compound Interest and Annuities
Let us recall that compound interest formula is given by
(2.1) Sn = P (1 + i)n
Sn = Future Value, P = Principal, i = Interest = i(m)m
, where i(m) is the normal annualrate. n = total interest period (mt),m = number of conversion periods per year, andt = time.
15
Since P = Sn(1+ i)−n is the present value, v = (1+ i)−nis the present value of $1. Hence
P = Snvn
.Suppose we have a retirement account in which the employee pays in an amount in
regular intervals, such an account is called an annuity. Mortgage payment, rents, and carpayments are other examples of annuities. For the sake of this paper, let us consider asimple annuity such as annuity immediate, the case in which the payment into the accountis made at the ends of interest conversion dates. Suppose $R is the amount paid into theannuity for each period, if such annuity attracts a periodic interest rate of i then the sumof the annuity after n periods is given by
(2.2) S = R+R(1+i)+R(1+i)2 + .........+R(1+i)n−1 =n−1∑j=0
R(1+i)j = R
[(1 + i)n − 1
i
]Many self-employed persons or high earning persons set up Sinking Funds which they
would draw from upon retirement. Sinking fund is a specified future amount (of annuity)which is required at the end of a given period. Suppose Mr. Daniels wants to set up asinking fund work $850,000 at the age of 30 years for retirement at the age of 65. If thisfund yields 6.5% interest rate compounded monthly, what monthly payment is necessaryto have the amount in 35 years?Solution:
$850000 = R
[(1 + .065
12)420 − 1
0.06512
]= (1600.316191)R
Hence
R =$850000
1600.316191= $531.15
So if $531.15 is paid in at the end of every month, the sinking fund amounting to $850,000will be attained at retirement.
Of immense importance is the Present Value of an Annuity, A. The present value of anannuity is given by
(2.3) An = Sn(1 + i)−n = R
[(1 + i)n − 1
i
](1 + i)−n = R
[1− (1 + i)−n
i
]Let us give an example which have relevance to retirements as well as to charities. We wishto determine the present value of an annuity which pays out $1200 quarterly for 35 yearsif the annuity attracts an annual interest rate of 6.25%.Solution:
An = R
[1− (1 + i)−n
i
]= $1200
(1− (1 + .0625
4)−140
.06254
)= $68, 036.30
Here is another example dealing with deferred annuity. Suppose your father establishesa trust fund for your first son, Tony, the day he was born, such that when he gets to theage of 25 and every month after that for the next 30 years, he will receive $500. If the
16
interest rate is 6.5% compounded quarterly, how much is the deposit on the day your sonwas born?Solution:
i =i%(12)
12=
6.5%(12)
12= 0.005416667
The total number of payments n = 360.The interval of deferment is given by q = mt = (12)(25)− 1 = 299. The present value of
this annuity is
x = An = 500a|i.vq = 500 ¯360| 6.5%
12.(1.00541667−299)
= $500(1− 1.00541667−360)
1.00541667(1.00541667−299) = $67, 634.74
The concept Annuity in Perpetuity is very relevant here. Suppose a large number ofpayments n, of an ordinary annuity is made. Then
A∞ = limn→∞
R
[1− (1 + i)−n
i
]=R
i.
Let us present an example. Determine the value of an investment at 8.25 %( 4) whichwill provide $6500 quarterly payment in perpetuity.Solution:
A∞ =R
i=
$65000.0825
4
= $315, 151.52 .
3. Valuation of Stocks
Many preferred stocks behave like perpetuities in that there are no future maturity value.For the sake of our discussion, let us assume that the stocks pay a fixed dividend given by:Υ = D
k, where k is the yield rate. Hence k = D
Υ.
It is well known that some preferred stocks have call dates (that is maturity dates). Thevalue of such stock is therefore the present value of the fixed dividend annuity plus thepresent value of the stock. Preferred stocks are also called zero-growth stocks because oftheir perpetuity nature. The value of such stocks can be written as the sum of presentvalues. That is,
(3.1) Υ = D1(1 + k)−1 +D2(1 + k)−2 + .....+Dm(1 + k)−m =∞∑j=1
Dj(1 + k)−j
17
If the dividend is the same for each period, then Di = D, i = 1, 2, 3, .... and the value ofthe stock becomes the following geometric series
(3.2) Υ =∞∑j=1
Dj(1 + k)−j = D(v + v2 + v3 + ....+ vm + ...) = Dv
1− v=D
k.
Instead of having a constant dividend, most common stocks can experience a growthrate λ for the perpetual dividend. Hence
Υ = Dv +D(1 + λ)v2 +D(1 + λ)2v3 + ....+D(1 + λ)mvm+1 + ....
= Dv(1 + (1 + λ)v + (1 + λ)2v2 + ....+D(1 + λ)mvm + ....
)= Dv
1
1− (1 + λ)v
)=
Dv
k − λ, k > λ.
(3.3)
Let us observe that
(3.4) Υ =Dv
k − λ⇒ k − λ =
D
Υ
Hence
(3.5) k =D
Υ+ λ
k is the expected rate of return.
4. Conclusion
In this paper, we have examined the importance of the stocks in the growth of wealthand some basic mathematical foundations of growth of wealth. More than ever before,mathematicians are finding greater roles in the finance and banking industries. Deeperunderstanding of stock price, price volatility, and stable stock prices is very important inorder to advise clients on how to create optimal financial portfolios to secure their retirementinvestments. For the mathematics major, a course in mathematical finance gives him orher another career pathway.
References
[1] J. Daniel, L. Vaaler, Mathematical Interest Theory( Prentice Hall, New Jersey, 2007).[2] Z. C. Okonkwo, On Global existence of a class of Ito-Volterra equations, , Dynamic Systems and
Applications , Vol. 13(1), 2004, pp.17-24.[3] J. Stampfli, V. Goodman, The Mathematics of Finance: Modeling and Hedging Brookes/Cole, CA,
2001).[4] J. M. Steele, Stochastic Calculus and Financial Applications Springer-Verlag, New York, 2001).[5] R. L. Burden, J.D. Faires , Numerical Analysis Brookes/Cole, CA, 2011).[6] D. G. Zill, A First Course in Differential Equations with Modeling Applications Brookes/Cole, CA,
2009).[7] Okonkwo, Seminar Series, Albany State University, 2011-2012.
18
Proc. H. J. W. S. U. Math. Res., Vol. 13 (2015) 18 - 23
Mathematical Modeling of Retardation in an Organic ChainReaction
Jalen MarshallDepartment of Mathematics
Morehouse CollegeAtlanta, GA 30314
Abstract The three major stages of a chain reaction are initiation, propagation, andtermination. In this article we introduce a less observed stage known as retardation. Re-tardation is observed in more complex systems, when a free radical steals an electron froma non-reactive molecule. The goal of this study is to model retardation in a specific caseas a system of differential equations. Here we use both steady state approximation to helpanalyze these processes and Maple Software to obtain numerical values.
1. Introduction
The field of Chemical Kinetics, covers the analyses of chain reactions using both chemistryand mathematics. Most of these reactions can be described as the initial generation ofreactive molecules–known as free radicals, the forming of molecules via electron pairingfree radicals, and ending with a stable product. We view these reactions as a chain ofindividual chemical reactions. We are able to observe the activity of chemical reactions ona molecular level using assumption presumed in the field of Chemical Kinetics. Consideringdecomposition as a chain reaction, we are concerned with the deterioration of a molecule[M ] ≡ volume of molecule M , and the free radicals Rj, for j = {1, 2, . . .} present in thechain [1]. Over the course of a chain reaction this volume is subject to change. Thus, wemay also consider this variable dependent on time.
We use the Rice-Herzfeld Mechanism and a steady state approximation when approachingthese chain reactions. The Rice-Herzfeld mechanism is commonly used in chemical kineticsto derive rates from complex chain reactions by assuming the chain can be organizedinto three primary stages: initiation, propagation, and termination. We introduce thesekey terms along with others in section 2. In section 3 we model a basic decompositionobserving these primary stages. In Section 4 we consider a basic model for retardation, whichintroduces the addition to our primary stages: retardation. Once we’ve assumed a generalmodel for both decomposition and retardation, in section 5 we approach the new model:potential retardation during acetaldehyde pyrolysis (thermochemical decomposition). Theuse of a steady state approximation helps with simplifying much of the computations. Ineach section we provide examples using either a graph or table of numerical values.
2. Preliminary
In this section we list definitions for terms we encountered.
(1) Chain Reaction - a system of chemical reactions(2) Steady State - we assume molecules are introduced to the system at the same rate
they are removed from the system, so we equate net rates of change to zero.(3) Homogenous Reaction - a reaction in one phase (ie. solid, liquid, or gas)
19
(4) Rate Law - in a homogenous reaction, the rate of change of a substance is directlyproportional to the product of the reactants’ concentrations; given a chemical reac-tion, A + B -¿C[k] C + D
rate = k[A][B], k > 0.
(5) Free Radical - reactive molecule with an odd number of electrons; the opposite is anon-reactive molecule with an even number of electrons.
(6) Initiation - two or more free radicals are generated(7) Propagation - unstable products formed and original free radicals regenerated con-
tinuously(8) Retardation - free radical reacts with a non-reactive molecule, reducing that mole-
cule’s rate of formation(9) Termination - two free radicals combine to form a stable product
3. Modeling Decomposition
In this section we model an elementary decomposition chain reaction [1]. For this purposewe consider the three primary stages of a chain reaction: initiation, propagation, andtermination.
Model 1: Assuming the Rice-Herzfeld mechanism, steady state, and a molecule [M ] in
the initiation phase we can derive expressions for both d[M ]dt
and M(t). During initiation,free radicals are generated from decomposing molecule [M ]
M− > C[k1]R1 +R′1
and this starts the chain reaction. Following this process, the free radicals endure a seriesof reactions during propagation. Unstable products form and free radicals are regenerated.
R1 +M− > C[k2]R2 +R1H,
R2− > C[k3]R1 +M ′.
Eventually two free radicals react to form a stable product during termination
R1 +R1− > C[k4a]P1,
R2 +R2− > C[k4b]P2,
R1 +R2− > C[k4c]P3.
To derive d[M ]dt
we first choose a termination process. In this case we choose the first. Therates of each substance are governed by both their decomposition and formation, so wegovern the rate of [M ] by
M− > C[k1]R1 +R′1,
R1 +M− > C[k2]R2 +R1H.
Using rate law we express the net rate of M as strictly decomposing
d[M ]
dt= −k1[M ]− k2[R1][M ].(3.1)
20
Next we solve for our free radicals using the same process. We govern the rate of [R1] by
M− > C[k1]R1 +R′1,
R1 +M− > C[k2]R2 +R1H,
R2− > C[k3]R1 +M ′,
R1 +R1− > C[k4a]P1.
Hence, the net rate of change in [R1] is
d[R1]
dt= k1[M ]− k2[R1][M ] + k3[R2].(3.2)
Since we have introduced [R2] to our system of equations, it is necessary that we also definethe rate of [R2]. Using
R2− > C[k3]R1 +M ′,
R1 +M− > C[k2]R2 +R1H,
the net rate of change for [R2] is
d[R2]
dt= k2[R1][M ]− k3[R2].(3.3)
We assume this system is in a steady state, so
d[R1]
dt≈ d[R2]
dt≈ 0
and taking the sum of equations (3.2) and (3.3) we have
d[R1]
dt+
d[R2]
dt= k1[M ]− k4a[R1]2 = 0,
which implies
[R1] ≈√
k1
k4a
[M ].(3.4)
After substituting equation (3.4) into (3.1) we see
d[M ]
dt≈ −k1[M ]− k2
(√k1
k4a
[M ]
)[M ]
and once we simplify this we now have
d[M ]
dt≈ −
(k1[M ] + k2
√k1
k4a
[M ]32
)We are able to derive an expression for [M] as it changes over time, using Bernoulli’sequation and now have
[M ] =
(− K
k1
+ cek12t
)−2
, where K = k2
√k1
k4a
.
In the following example illustrate [M] and choosing specific values for both [M ]0 and ourk constants. For this purpose we use Maple Software
21
4. Modeling Retardation
In this section we will introduce the concept of retardation. Some reactions are morecomplex. Some systems undergo a retardation stage, when a free radical [R] reacts witha non-reactive molecule [P ], which was produced during propagation and reduces thatmolecule’s rate of formation. In this section we consider an elementary model for retardationand we observe a reaction between hydrogen and bromine.
Model 2: Assuming Model 1 and a retardation stage, which includes non-reactive mol-
ecule [P] we can derive an expression for d[P ]dt
. Let [P ] = [HBr] be the product undergoingretardation and let [M ] = [Br2] or [H2]. We are provided the following chain reaction [5].
InitiationBr2 +M− > C[k1]R1 +R1 +M
Propagation
R1 +H2− > C[k2]HBr +R2,
R2 +Br2− > C[k3]HBr +R1,
RetardationR2 +HBr− > C[kr]H2 +R1,
TerminationM +R1 +R1− > C[k5]S +M∗
We derive our system of differential equations to be
d[P ]
dt= k2[R1][H2] + k3[R2][Br2]− kr[R2][P ](4.1)
d[R1]
dt= k1[M ][Br2]− k2[R1][H2] + k3[R2][Br2] + kr[R2][P ]− k5[R1]2[M ](4.2)
d[R2]
dt= k2[R1][H2]− k3[R2][Br2]− kr[R2][P ](4.3)
Using the steady-state approximation we define the free radicals we obtain as follows
[R1] ≈√k1
k5
[Br2](4.4)
and
[R2] =k2[R1][H2]
k3[Br2] + kr[P ]≈k2
(√k1k5
[Br2]
)[H2]
k3[Br2] + kr[P ](4.5)
After substituting for [R1] and [R2] in equation (4.1) using equations (4.4) and (4.5), re-spectively, we see
d[P ]
dt≈ k2
√k1
k5
[Br2][H2] + k3
k2
(√k1k5
[Br2]
)[H2]
k3[Br2] + kr[P ][Br2]− kr
k2
(√k1k5
[Br2]
)[H2]
k3[Br2] + kr[P ][P ]
and once we simplify we see
d[P ]
dt≈ k2
√k1
k5
[Br2]3/2[H2]
(2k3
k3[Br2] + kr[P ]
).
22
In the next example we use Maple Software to approach this rate numerically. If we considerthis rate in the absence of retardation (where kr = 0) we will see that the rate of formationof [HBr] is indeed reduced once [HBr] has undergone retardation (where kr > 0). We showthis controlling for [P ] and choosing values for [Br2] and [H2].
5. Retardation During Acetaldehyde Pyrolysis
Now that we understand a basic model for decomposition, we reference both Model 1 andModel 2 to create a new model. In this section we model retardation during acetaldehydepyrolysis and we reference a provided chain reaction [4]. We are provided the initiation,propagation, and termination processes, which are enough to derive our rates accordingto the Rice-Herzfeld Mechanism. We are not provided a retardation stage, but we proposeone.
Model 3: Assuming both Model 1 and Model 2, we can derive an expression for d[P ]dt
in aspecific case. Let [M ] = [CH3CHO], [R1] = [CH3], [R2] = [CH3CO], [R3] = [CH2CHO],
[R4] = [CHO], [H] = [H], and [P ] = [CH4]. We can derive an expression for d[P ]dt
.Initiation
CH3CHO− > C[k1]CH3 + CHO
Propagation
CHO− > C[k2]CO +H,
H + CH3CHO− > C[k3]H2 + CH3CO,
CH3 + CH3CHO− > C[k4]CH4 + CH3CO,
CH3CO− > C[k5]CH3 + CO,
CH3 + CH3CHO− > C[k6]CH4 + CH2CHO,
CH2CHO− > C[k7]CH2CO +H,
CH2CHO + CH3CHO− > C[k8]CH3CHO + CH3CO,
CH3 + CH3CHO− > C[k9]CH3COCH3 +H
TerminationCH3 + CH3− > C[k10]C2H6.
We propose the following retardation stage
H + CH4− > C[kr]H2 + CH3,
when hydrogen steals an electron from methane. We express the rate of formation in [P ] as
d[P ]
dt= k4[R1][M ] + k6[R1][M ]− kr[H][P ].
Assuming a steady state and approximating for our free radicals we see
(1) [R1] ≈√
k1k10
[M ].
(2) [H] ≈ k7[R3]+k9[R1][M ]+k2[R4]k3[M ]+kr[P ]
.
(3) [R3] ≈k6
√k1k10
[M ]32
k7+k8[M ].
(4) [R4] ≈ k1k2
[M ].
23
which implies
[H] ≈k7
(k6
√k1k10
[M ]32
k7+k8[M ]
)+ k9
(√k1k10
[M ]
)[M ] + k2
k1k2
[M ]
k3[M ] + kr[P ]
Substituting for [R1] and [H] in our expression for d[P ]dt
we now have
d[P ]
dt≈ k4
√k1
k10
[M ][M ]+k6
√k1
k10
[M ][M ]−krk7
(k6
√k1k10
[M ]32
k7+k8[M ]
)+ k9
(√k1k10
[M ]
)[M ] + k2
k1k2
[M ]
k3[M ] + kr[P ][P ],
which implies
d[P ]
dt≈√
k1
k10
[M ]32
(k4 + k6 − kr[P ]
(k7k6
k7+k8[M ]+ k9
k3[M ] + kr[P ]
))− kr[P ]
k1[M ]
k3[M ] + kr[P ].(5.1)
In solving for [M ] we see
d[M ]
dt= −
(k1[M ] +K[M ]
32
)where K =
√k1
k10
(k4 + k6 + k9)
and again by Bernoulli’s Equation we see
[M ] =
(− K
k1
+ cek12t
)−2
, c =1√[M ]0
+K
k1
.(5.2)
We can substitute (5.2) for [M ] in equation (5.1) for better accuracy of d[P ]dt
.If we solve equation (5.1) numerically and control for [P ], we see that the rate of forma-
tion is greater in the absence of retardation. In the next example we use Maple Softwareto know more about equation (5.1).
6. Acknowledgments
I thank the Morehouse College mathematics department for allowing me to cover thistopic. I thank the senior seminar faculty: Dr. Cooper, Dr. Peng, Dr. Lamar, Dr. Wilson,and Dr. Clark. Also, I thank my advisor Dr. Masilamani Sambandham for suggesting thistopic.
References
[1] Anil G. Ladda and G. S. Ladde, An Introduction to Differential Equations. Vol 1, 2000[2] Dennis G. Zill, A First Course in Differential Equations with Modeling Applications. 10th Edition, 2013[3] David W. Oxtoby, H.P. Gillis, and Alan Campion, Principles of Modern Chemistry. 7th Edition, 2012[4] Michael T.H. Liu, and K. J, Laidlder Elementary Processes in Acetaldehyde Pyrolysis. 1967[5] Joseph N. Grima, and Alfred Alquina Chemical Kinetics for Various Processes-Chemical Reactions.
24
Proc. H. J. W. S. U. Math. Res., Vol. 13 (2015) 24 - 27
Bond Valuations and Investments Based on Bonds
Trevonta L. Mctyre and Marquis D. CurryDepartment of Mathematics and Computer Science
Albany State UniversityAlbany, GA 31705
Abstract A bond is an interest yielding security to the holder. Bonds can be issues bycorporations or government agencies, the goal being to raise money for specific or generalpurposes. Common bonds include US Treasury bonds, state government bonds, municipalbonds, mortgage bonds, and debentures. Every bond has an issue date and a maturity datewhich are clearly stated on the promissory note. Certain bonds such as the US Treasurybond have very low risk, and some other bonds have very high risks, for example, Detroitmunicipal bonds. In this paper, we present the nature and properties of bonds, noncallableand callable bonds, bond yields, and reliability on bonds as major parts of retirementportfolios.
1. Introduction
Corporations, governments, and public institutions, including colleges and universities,issue bonds for operations or capital projects. A bond is a negotiable certificate thatacknowledges the indebtedness of the bond issuer to the holder. It is negotiable becausethe ownership of the certificate can be transferred in the secondary market. It is a debtsecurity, in which the authorized issuer owes the holders a debt. The purchaser or holderof the bond is the lender while the party issuing the bond is the borrower. The bondholder therefore expects interest from the borrower. Most of the time, bonds are mademore attractive by the borrower offering interest higher than the nominal rates paid bybanks and other financial institutions. The Face value of the bond is the amount of thebond, usually in denominators of $1000 or multiples of $1000. The redemption value of thebond is the amount paid by party issuing the bond to the holder of the bond on maturitydate or call date. In some cases, the party issuing the bond may redeem the bond earlierthan the call date (sometimes by offering attractive incentives).
The coupon rate is the interest rate stated on the bond. The coupon can be redeemedas cash on the due date automatically. The yield rate is the interest rate earned by thebond holder. If the bond is purchased at face value, then, the yield rate and the bondrate will be the same. However, the purchase value may be different from the face value,in this case, the coupon rate will be different from the yield rate. The interest dates onthe bond are stated on the bond or coupon. Bond interest is usually paid semiannually.The purchase price of the bond may be different from the face value. In the case thebond is purchased after the issue date, the purchase price could be the face value plusthe accumulated interest since the issue date. While government bonds may be purchasedthrough brokerage firms, banks, or the Federal Reserve Bank, Corporate Bonds can onlybe purchased through brokerage firms.
Certain bonds hold certain provisions or characterizations. For example, mortgage bondshave certain asserts as collaterals for the debt- the house or land. Debentures are bondsthat offer no collaterals except the good name of the issuing company. A Convertible bonds
25
are bonds which have all attributes of bonds and in addition the investor has the flexibilityto convert such bonds into specific number of common stocks. Zero coupon bonds are bondswhich pay no interests and are sold like treasury bills. They are purchased on discountbasis and are therefore redeemed at face value. The organization issuing the bond mustprovide the purchaser of the bond all essential information through the brokerage firm orbank. Such information will educate the purchaser on whether the bond is worth buying.Here is a typical bond table entry:
52 weeksHigh Low Name of Cum. Sales High Low Latest Net
Coupon Yield $1000 Change1113
565 UTC 14.25 687 82 76 793
4+33
411.69s15
The purchaser of the bond is usually encouraged to examine the bond rating. Investmentin bonds is risky. The risk is minimized if the investor is able to examine the rating of thebond. Low risk bonds or securities are encouraged. There are two main acclaimed bondrating agencies: Standard and Poor and Moody.
Standard and Poor’s Moody’sHigh Grades AAA and AA Aaa and AaMedium Grades A and BBB A and BaaSpeculative Bonds BB and B Ba and BBonds that are in danger CCC, CC, C and D Caa, Ca, and Cof default or are in default
2. Basic Definitions and Preliminary Results
The future value of the amount of compound interest is given by
(2.1) Sn = P (1 + i)n
Sn = Future Value, P = Principal, i = Interest = i(m)m
, where i(m) is the normal annualrate. n = total interest period (mt),m = number of conversion periods per year, andt = time.
Since P = Sn(1+ i)−n is the present value, v = (1+ i)−nis the present value of $1. Hence
P = Snvn
. Of immense importance is the Present Value of an Annuity, A. The present value of anannuity is given by
(2.2) An = Sn(1 + i)−n = R
[(1 + i)n − 1
i
](1 + i)−n = R
[1− (1 + i)−n
i
]Let us give an example which have relevance to retirements as well as to charities. We wishto determine the present value of an annuity which pays out $1500 quarterly for 30 yearsif the annuity attracts an annual interest rate of 7%.
Solution:
An = R
[1− (1 + i)−n
i
]= $1500
(1− (1 + .07
4)−120
.074
)= $75025.63
26
The US Treasury bills are usually issue on discount basis so that the holder receives theface value of the bill. Suppose we have a bid of 95.8 on a T-bill of $1,000,000, then thediscount is
D = $1, 000, 000− .957($1, 000, 000) = $43, 000.
The future value of the bill is the face value. Suppose we wish to determine the Rate ofReturn of a $100,000, 181-day Treasury bill purchased with a bid of 97.25, thenDiscount = Future Value − Present Value.Hence,
D = S − P = $100, 00− .9725($100, 000) = $2, 750.
The discount rate d = DSt
= $2750$100000( 181
360)
= 0.0546961326 = 5.47%
The Present Value of an Annuity, A, is important. The present value of an annuity isgiven by
An = Sn(1 + i)−n = R
[(1 + i)n − 1
i
](1 + i)−n = R
[1− (1 + i)−n
i
]The concept Annuity in Perpetuity is very relevant here. Suppose a large number of
payments n, of an ordinary annuity is made. Then
A∞ = limn→∞
R
[1− (1 + i)−n
i
]=R
i.
Here is an example. Determine the value of an investment at 9.25 %( 4) which will provide$7,500 quarterly payment in perpetuity.
Solution:
A∞ =R
i=
$7, 5000.0925
4
= $324, 424.32 .
3. Bond Valuation
As an investor, the bond holder is interested in having immense knowledge about howhis investment can grow. The bond price (value) computation is determined by two rates:the coupon which is paid semiannually over the life of the bond and the yield rate whichis used to determine the present value of the coupon annuity and the present value of theredemption value. Hence the current bond price is the sum of the two present values. Es-sentially, the price of the bond is given by
Vn = Present Value of the redemption price + Present Value of the Coupon Annuity.
(3.1) Vn = S(1 + i)−n + Fr
(1− (1 + i)−n
i
).
Here F = Face value of the bond.S = The redemption value of the bond of future value of the bond.r = Interest rate paid by the bond each period.n = Number of interest periods to maturity date.i = Current yield.Vn =Value of bond.Fr = Coupon.
27
Let us compute the price of a bond. We want to find the bond price with yield 9.5%(2)for $4000 bond paying 7.5%(2) and maturity date at par in 14 years.Solution:
i =9.5%
2= 4.75% = 0.0475.
Bond rate r = 7.5%2
= 3.75% = .0375.Number of periods to maturity n = 2× 14 = 28The Coupon Fr = .0375($4000) = $150.Future Value S = $4000
V28 = $4000(1.0475−28) + $150(1− (1 + i)−28)
i= $1090.80 + $2296.73 = $3387.53
Let us note the following: this is a discount bond and $612.47 is the discount. Theinvestor buys the bond at $3387.53 and earns $300 a year in interest and $612.47 as capitalgains over 14 years. This gives a total of 9.5% annual return. However, the return per yearis $300+$43.75=$343.75. This gives the adjusted current yield of
$343.75
$3387.53= 10.15%
4. Conclusion
Bonds play a central role in most retirement portfolios. Although bonds in most casesare more predictable than stocks, an investor seeking to purchase bonds must be aware ofthe nature of bonds needed in order to secure his retirement portfolio. Certain bonds suchas the US Treasury bonds have very low risk, but some other bonds have very high risks,for example, Detroit municipal bonds. In this paper, we have examined various types ofbonds and reliability on bonds as major parts of retirement portfolios.
References
[1] J. Daniel, L. Vaaler, Mathematical Interest Theory( Prentice Hall, New Jersey, 2007).[2] Z. C. Okonkwo, On Global existence of a class of Ito-Volterra equations, , Dynamic Systems and
Applications , Vol. 13(1), 2004, pp.17-24.[3] J. Stampfli, V. Goodman, The Mathematics of Finance: Modeling and Hedging Brookes/Cole, CA,
2001).[4] J. M. Steele, Stochastic Calculus and Financial Applications Springer-Verlag, New York, 2001).[5] R. L. Burden, J.D. Faires , Numerical Analysis Brookes/Cole, CA, 2011).[6] D. G. Zill, A First Course in Differential Equations with Modeling Applications Brookes/Cole, CA,
2009).[7] Okonkwo, Seminar Series, Albany State University, 2011-2012.
28
Proc. H. J. W. S. U. Math. Res., Vol. 13 (2015) 28 - 36
Using Process Capability Estimates and Attribute Data toGenerate Control Charts for Healthcare Delivery and
Management
Latalya Walden and Marquis D. CurryDepartment of Mathematics and Computer Science
Albany State UniversityAlbany, GA 31705
Abstract Process capability estimates are made in terms of upper and lower limits ofpopulation distribution such that no more than one in thousand observations lie in eachtail. We shall use process capability procedure to obtain normal estimates of surgery timesfor certain forms of surgeries. Furthermore, we study the use of attribute data encounteredin wide-range of applications to generate control charts, including the count-chart (c-chart)and the u-chart where the u values are obtained by dividing the subgroup count by thesubgroup size. These charts can be applied in the delivery of healthcare for differentsubgroups of the population. Examples are drawn for illustration from data obtained frommonthly counts of patients’ falls and counts on medication errors.
1. Introduction
Processes of measurement of capabilities are associated with systems, be it engineeringsystem or health system. While it is usually not feasible to determine exactly system behav-ior, it is possible to determine an admissible interval of behavior. Hence process capabilitymeasures how well a process produces compared with customer requirements.
Collection and analyses of attribute data are very essential in the practice of healthcare,medicine and pharmacy. Unlike measurement data which uses several statistics such asmeasures of central tendency, variability, and position, attributes data (also known ascount-type data) takes only one statistic, the average, to describe a set of data. This isbecause for attribute data, the standard deviation is a function of the average. Hence,takes only a single chart for an attribute control chart analysis. Attribute charts thereforedo not come in pairs like control charts generated from measurement data. We note thefollowing points for attribute data analyses:
• Quotients involving counted integers, such a monthly rate of patient falls, are at-tribute data.• Variables data can sometimes be converted to attribute data. For example, one can
count those with systolic blood pressure is greater than 140 mm Hg.• Attribute chart centerline is defined as the process capability.• Attribute charts are used with either time-ordered data (for prediction) or with
rational subgroups (for comparison).
In this paper, we discuss the construction of two types of charts: the c-charts and theu-charts. The underlying distribution is the Poisson distribution (named after SimeonPoisson, who developed the distribution). A typical healthcare application is the study ofthe monthly count of patient falls.
29
A count can only exist if there is an area of opportunity in which the count can occur. Thearea of opportunity is called the subgroup size. In the case of a monthly count of patientfalls, it is rational to state the area of opportunity in which the enumerated falls occur (i.e.,the subgroup size) in terms of the monthly number of patient days.C-chart is used in the case whereby the number of patient days each month (i.e. the areaof opportunity or subgroup size) is essentially constant. The u-chart is used when thesubgroups size varies significantly, that is, the number of patients differs significantly frommonth to month.The Poisson distribution is skewed to the right (particularly for small average counts),which is reasonable since counts cannot be negative. Just as with variables data, when thedistribution becomes too skewed, the usual control chart becomes invalid.In Section 2 of this paper, we discuss the Poisson distribution, and in Section 3, examinethe basics of process capabilities. Methods of data collection and generation are discussedin Section 4 while in Section 5, we use the statistical software R to generate the sample ofsurgery times. We conclude this paper in Section 6.
2. The Poisson Distribution
Let us consider the binomial random variable X with parameter b(n, p). The densityfunction is given by
(2.1) f(X) =
(n
X
)PX(1− p)n−X , 0 ≤ X ≤ n.
Now, when n is very large, the computations of the binomial probabilities with the formula
f(X) =
(n
X
)PX(1− p)n−X , 0 ≤ X ≤ n
become prohibitive. For example, to calculate the probability that 24 out of 4000 patientswill not take their medication will involve that we compute
(400024
).
If the chance of success is p = .004, then
(2.2) f(24) =
(4000
24
)(0.004)24(0.996)4000−24.
In such a case, the Poisson distribution can be used to model such problems. In fact, ifn→∞, p→ 0, then np remains a constant. Let
np = λ, p =λ
n,
then
(2.3) b(X,n, p) =
(n
X
)(λ
n
)X (1− λ
n
)n−X(2.4)
b(X,n, p) =
(n
X
)(λ
n
)X (1− λ
n
)n−X=n(n− 1)....(n−X + 1)
X!
(λ
n
)X (1− λ
n
)n−XLet us write (
1− λ
n
)n−X=
((1− λ
n
)−nλ
)−λ(1− λ
n
)−X
30
We can rewrite equation (2.4) in the form
(2.5) b(X,n, p) =1(1− 1
n)(1− 2
n)....(1− X−1
n)
X!λX
((1− λ
n
)−nλ
)−λ(1− λ
n
)−XNote that as n→∞, with X and λ fixed, we find that
1(1− 1
n)(1− 2
n)....(1− X − 1
n)→ 1(
1− λ
n
)−X→ 1 and
(1− λ
n
)− nλ
→ e
and the limiting distribution becomes
(2.6) f(X;λ) =λXe−λ
X!, X = 0, 1, 2, ....
Equation (2.6) is the Poisson Distribution.
3. Basics of Process Capability
The process is capable if is stable and normal. Process capability measures depend ontwo parameters, the mean µ and the standard deviation σ. In most cases of practicalapplications, the upper and lower control limits are given or are known due to historicaldata. If the distribution is a normal distribution, then, most data will lie in the interval[µ − 3σ, µ + 3σ]. So 6σ is an excellent measure of normal process capability. Processcapability ration is given by
(3.1) Cp =Upper Control Limit − Lower Control Limit
6(Standard Deviation)=U − L
6σ.
If Cp = 1, then we have perfect normal control.If Cp < 1, then part of the process output will be out of specification.If Cp > 1, then part of the process output has room.
Suppose a Zenco Nursing home has the following information about their patients missingtheir medication.
Month Number of Subgroup Size u valuesErrors n in Hundreads of dollars
1 20 1.50 13.332 32 2.00 163 35 2.5 144 21 1.25 145 10 1.5 6.676 25 1.5 16.67Total 143 10.5
U = Sum of errorsSum of the doses
= 14310.5
= 13.62 per unit for 100 doses.
For n = 2.5, we have 3σ = 3√
13.622.5
= 7.0 . Hence
UCL = U + 3σ = 13.62 + 7 = 20.62
LCL = U − 3σ = 13.62− 7 = 6.62
The other limits can be computed analogously.
31
4. Data Collection and Generation
Simulation Using the Graphing Calculator. In order to obtain the range of surgerytime, we examined several websites including the following:
http://www.ncbi.nlm.nih.gov/pubmed/10374091
http://www.spine-health.com/forum/treatment/back-surgery-and-neck-surgery/how-long-will-it-take
By examining the minimum and maximum amount of time it takes, we computed themean of the minimum times obtained and the mean of the maximum times obtained. Usethese two, we were able to us the random number generator to obtain a distribution ofpatient surgery times. Essentially, we the surgery times ranged from 25 minutes as theminimum and 185 minutes as the maximum.
Below is data for the c-chart and for the u-chart obtained from Statistical Process Con-trol for Healthcare by Hart, M., K., Hart, R. F. (2002).
Data for U- Chart
Month Numbers Number of Errors1 200 132 200 203 200 214 200 105 200 86 200 177 200 98 200 169 200 1110 200 1311 200 13
32
Data for U- ChartMonth Data for U - chart Subgroup Size Base Number
Number of Errors n in Hundreads of dollars1 13 1.252 20 1.503 21 1.254 10 1.255 8 1.56 17 1.57 9 1.258 16 1.259 11 1.2510 7 1.25Total 132 13.25 total
33
5. Computer Simulation of Surgery Times and Their C-Charts
(Computer Simulation Using R-Software)
Using the upper and lower bounds of surgery times for various types of surgeries, com-puter simulation was performed for each type of surgery data. Large amount of sample datawas obtained for each case. Various statistics were computed for each data and interpretedaccordingly. Controls Charts are drawn using standard procedures.Surgery Type: C-SectionSimulated Sample Space
48 39 44 56 26 37 56 40 29 45 55 42 45 40 48 42 42 56 26 52 25 25 43 50 41 32 43 60 3827 31 58 55 29 51 35 37 48 59 32 41 42 42 42 45 57 37 28 58 26 45 28 36 40 59 42 46 33 2642 57 52 59 54 35 29 49 41 25 34 56 41 54 46 50 44 50 31 44 60 52 46 30 28 60 42 37 51 6042 45 32 9 42 42 55 39 41 48 58 48 45 53 28 32 42 58 30 50 28 34 49 25 48 42 60 32 39 5655 40 46 30 52 41 51 47 26 42 43 32 33 37 47 36 52 46 32 46 49 32 34 36 57 39 37 25 30 3630 45 38 31 40 49 32 27 36 41 26 36 46 43 39 57 59 59 51 54 54 57 37 34 42 58 56 43 49 4125 55 60 57 57 27 60 59 45 56 30 29 50 31 53 53 54 60 36 41 27 33 40 51 49 55 52 59 54 5030 43 44 28 28 32 54 47 42 57 44 55 56 57 26 33 57 26 35 32 55 55 34 51 31 53 38 42 54 5430 60 45 48 49 28 48 34 52 46 30 51 46 60 51 53 59 51 29 49 43 28 58 48 51 48 37 46 3060 60 28 31 44 53 38 39 35 55 25 49 31 26 30 59 28 49 26 59 45 44 30 50 46 43 55 55 33 39 27 28
Mean=43.23
Mean+3*sd=74.92737
Max=60
Mean-3*sd=11.53263
34
Min=25
35
Surgery Type: Heart SurgerySimulated Sample Space
230 260 246 261 247 261 234 246 184 250 232 216 186 226 194 255 233 246 261 274 194195 285 249 232 214 266 266 286 279 224 235 214 202 296 266 214 210 260 268 231 204 288266 293 286 247 217 260 234 247 269 260 299 258 197 206 288 208 181 238 198 184 237 280231 298 200 271 287 181 299 261 273 220 280 280 221 282 272 229 261 284 289 199 261 196255 197 189 238 280 243 184 294 286 286 185 186 234 182 250 300 261 293 197 285 201 203290 180 277 290 208 285 264 265 198 240 236 264 211 265 192 242 276 223 281 250 291 295289 274 205 219 287 211 191 277 182 224 262 227 212 215 244 247 291 286 245 235 281 187226 248 282 288 180 227 200 231 194 259 230 228 184 201 267 190 194 264 191 286 215 217209 249 215 186 281 293 230 297 225 211 298 230 199 234 285 229 193 229 272 235 284 193256 205 294 240 229 239 188 296 187 221 259 252 225 254 290 274 194 197 182 278 229 229297 269 252 239 227 200 210 253 196 242 201 248 230 254 279 202 190 220 205 262 190 223246 181 279 225 261 284 244 200 221 225 194 249 258 202 256 257 267 278 257 286 274 244263 219 237 278 185 237 277 272 191 294 208 292 245 243 250 263 258 293 201 245 198 183199 287 282 191 198 185 253 278 189 207 201 233 279 297 240
Mean=240.3867Mean+3*sd=346.0488Max=300Mean-3*sd=134.7245Min=180
36
6. Conclusion
Many health institutions, including nursing homes and hospitals, have challenges in usingtheir empirical data to manage their service delivery processes. Although some use standardbenchmarks to manage their healthcare delivery, many are unable to collect this data andsome of their management teams are unaware of the importance of data collection in theimprovement of their services. U-charts and c-charts generated from empirical data arevery essential. Those processes which are not in control would need to be examined. Forexample, a hospital in which patients miss medications could have unintended consequencesincluding high number of loss of lives. This could attract closure of the hospital and jailtime for the management. Finding adequate process capability measures is very important.
References
[1] M. K. Hart, R. F. Hart, Statistical Process Control for Healthcare, Statistical Process Control forHealthcare, (2002).
[2] Z. Okonkwo, (2014). MATH 3314-Mathematical Statistics Class notes.[3] Z. Okonkwo, (2015). Project Seminar Series on Statistical Process Control .
37
Proc. H. J. W. S. U. Math. Res., Vol. 13 (2015) 37 - 46
ABSTRACTS
Jarret D. CampDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Tawaner LamarTitle: Investigation of Solutions to Differential Equations with Variable Coefficients
An equation containing the derivatives of one or more dependent variables with respectto one or more independent variables is said to be a partial differential equation. Most or-dinary differential equations with variable coefficients are not possible to solve analytically.However, some special cases do exist such as the Cauchy-Euler equation, Bessels equationand the Legendre equation. In this investigation, we examine the Simply-Supported Beamequation with variable coefficients.
Curtis Clark,Jr.Department of MathematicsMorehouse CollegeAdvisor: Dr. Curtis ClarkTitle: On 2-2 Graph Achievement Games
Let F be a graph with no isolated vertices. The 2-2 F-achievement game on the com-plete graph Kn is described as follows. Player A first colors at most two edges of Kn green.Then Player B colors at most two different edges of Kn red. They continue alternatelycoloring the edges with Player A coloring at most two edges green and Player B coloring atmost two different edges red. The graph F is achievable on Kn if Player A can make a copyof F in his color. The minimum n such that F is achievable on Kn is the 2-2 achievementnumber of F denoted a(F ). The 2-2 move number of F , m(F ), is the least number of edgesthat must be colored by Player A to make F on the complete graph with a(F ) vertices.The numbers a(F ) and m(F ) are determined for some small graphs and paths.
38
Adam J. Eiring and Benjamin H. GainesDepartment of MathematicsSouthern CollegeAdvisor: Dr. Jeff BartonTitle: Predicting the Steady State Maximum and Minimum Drug Levels in the Blood
We did our senior research in the field of pharmacokinetics which is the study of howdrugs move through the body including dissolution, absorption and elimination. In thistalk, we examine a discrete, two-compartment model for an orally administered drug. Weassume that the absorption and elimination are both first order processes. Our goal was todevelop a formula to predict the maximum and minimum steady state drug levels in theblood based on a given drug dosage and frequency. We were able to successfully derive aformula for the minimum drug level in the blood. Our derivation makes extensive use ofthe geometric series formula.
J. R. GillingsDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Chuang PengTitle: R. Thompson’s Group V presented as Permutations of Subintervals of [0, 1]
We introduce R. Thompsons group V and express it as a collection of bijections on theinterval [0, 1] that have specific restrictions. The dyadic rationals are also introduced inorder to offer a detailed explaination about elements of R. Thompsons group V. We showhow elements of V interact, what their structure is and how they fit the provided definitions.We also offer a visual presentation of V and its elements to aid our explaination. We provethat elements of V permute partitions of [0, 1] formed with endpoints that are elements ofthe dyadic rationals and show that as a result, the definition of V can be loosened from“a collection of bijections on the interval [0, 1] to “a collection of bijections on the dyadicrationals in [0, 1].
Coleman Gorham and Robert WeaverDepartment of MathematicsBirmingham-Southern CollegeAdvisior: Dr. Jeff BartonTitle: A Look into the RSA Cryptosystem
Cryptology, the study of communicating with secret codes, influences many aspects ofour daily lives, including ATM transactions and online credit card purchases. A problemin many cryptosystems is that of key transmission. If the encryption and decryption keysare the same, then before two individuals can exchange secret messages, one must sendthe other the key and this introduces a security risk. If the key is intercepted, then allencrypted messages may be read. Our project examines the Rivest, Shamir, and Adleman(RSA) cryptosystem, which is a public-key encryption system. In a public-key system, theencryption and decryption keys are different. Anyone may encrypt a message to be sentto anyone else because each individuals encryption key is made public. However, only theintended recipient can decrypt a message because the decryption key is kept secret. Our
39
research involves the history, evolution and process of the RSA as well as a real worldexample to show our understanding of how the system works.
Arman GreenDepartment of MathematicsMorehouse CollegeAdvisors: Dr. Farouk BraniaTitle: A Proof of the Riemann Mapping Theorem
The Riemann Mapping Theorem is a powerful theorem that proves there is a unique ana-lytic function that isomorphically maps a point in a simply connected domain that is notthe entire complex plane to the unit disk with the conditions that the function evaluated atthat specific point is zero and the derivative at that point is greater than zero. In this talk,I will provide some details of the proofs of the Open Mapping Theorem, The MaximumModulus Principle and Schwarz Lemma, which are important results on analytic functions,and which provide the building blocks of the the Riemann Mapping Theorem.
Myles HarperDepartment of MathematicsMorehouse CollegeAdvisors: Dr. George YuhaszTitle: RSA cryptosystems and public Key data encryption
In this talk, we will discuss the RSA scheme and public key cryptosystems. First welook into a background of crypto-analysis and then examine the mathematical backbone ofthe RSA scheme. From there, we will begin to understand how public key cryptosystemswork and why they are an effective way to protect information. Lastly, we will then lookat various applications of public key cryptosystems such as passwords and data transfer.
Malik HenryDepartment of MathematicsUniversity of GeorgiaAdvisor: Dr. Jason CantarellaTitle: Random Knot Diagrams
In this paper, we will take a look at knots as topological figures. We will show thatrandom knot diagrams can be constructed using the star diagram model and we will provemany properties of random knot diagrams beginning with stick crossings and ending withEulers characteristic equation.
40
Jassiem IfillDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Duane CooperTitle: Fair division among multiple players or multiple divisible goods
Fair division is the partioning of a divisible good among two or more people, or “play-ers. Within Fair Division, there are various types of division including simple fair divisionand envy-free division. Moreover, depending on the number of players, different algorithmsexist to guarantee various types of divisions such as the Cut and Choose algorithm, theTrimming algorithm, the three player envy-free algorithm and More. However, these al-gorithms are for the common problem of only dividing one divisible good among a fewplayers. As such, this begs the question of how one or a group would go about dividingup multiple divisible goods among two players, or how to evenly divide a good such thatmultiple people have a consensus about its portions. Through the introduction and usageof simplices, polytopes, triangulations, and various other terms, theorems, and lemmas.We will attempt to delve deeper into these applications of Fair Divisions.
Talon JohnsonDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Shelby WilsonTitle: Analytically Understanding Population Dynamics of the Interaction between T-cellsand HIV
HIV is a sexually transmitted disease that weakens ones immune system allowing otherpathogens to affect ones body, ultimately resulting in the development of AIDS. A nonlin-ear mathematical model of differential equations with piecewise constants will show us therate in population. The solutions will be analytically solved through ordinary differentialequation techniques. We will analyze the solution of a standard differential equation modelof T-cell population. Furthermore, we will analyze multiple model of increasing complexityin order to study the dynamics of HIV.
41
Dorian KandiDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Ulrica WilsonTitle: Eigenvectors of Positive Matrices
In linear algebra, an eigenvector is a vector whose product, when multiplied by a squarematrix, is a scalar multiple of the vector itself. We call this scalar an eigenvalue. Modernmatrix theory only restricts this vector to being nonzero. However, eigenvalues and eigen-vectors have special properties when the parent matrix is strictly positive. This paper willexamine the impact of positivity on eigenpairs of a matrix and highlight the differencesthat result from nonnegative matrices versus positive matrices.
Jillian KuetherDepartment of MathematicsKennesaw State UniversityAdvisor: Dr. Jun JiTitle: Computing the minimum norm least squares solution to a system of linear equationsthrough Gauss-Jordan elimination
One of the simplest and most common ways to compute the solution x = A−1b to anon-singular system of linear equations of the form Ax = b where x is unknown is theGauss-Jordan elimination. For a system of linear equations with a singular square or rect-angular matrix, the system may not have any vectors satisfying Ax = b or may havemultiple solutions. Thus, the situation becomes more complicated, as the traditional in-verse matrix A−1 does not exist. A vector that minimizes both || b − Ax || and || x ||always exists and is unique and is called the minimum norm, least squares solution to thesystem of linear equations. It has been shown that the minimum norm least squares solu-tion is indeed A†b, the product of the Moore-Penrose inverse of A and the right-hand sidevector b. The minimum norm, least squares solution is used in curve fitting and numerousaspects of statistical analysis. In particular, it is useful in regression analysis and linearapproximation. This solution can be calculated through the simple use of Gauss-JordanElimination and the construction of bordered matrices as outlined in this paper. Comparedto other widely used methods for calculating the minimum norm, least squares solution forlinear systems, this proposed algorithm is especially easy to calculate by hand and mostclosely resembles the procedure used for finding the solution to a square, non-singular sys-tem of linear equations. While the method based on the QR decomposition is accurateand stable, it is very difficult to execute all the steps by hand and is almost always doneusing software. Other methods that can be computed by hand often take more work toconclude that the solution found is in fact the minimum norm, least squares solution. Thisprocedure for computing A†b will always return the incredibly useful, unique minimumnorm, least squares solution.
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Victoria Latimore and Latalya WaldenDepartment of Mathematics and Computer ScienceAlbany State UniversityAdvisor: Dr. Zephyrinus C. OkonkwoTitle: Stocks as Financial Securtiy, Valuation, Risks and Retirement Porfolios
Stocks continue to be popular securities due to their immediate yield of increase in thevalue of investments. When a company is dully registered to do business, that company isallowed to sell stocks to the public or to a restricted population. A holder of the stock ofa company is called a shareholder since every stock is equivalent to some percentage of thecompany. The company is thus owned by shareholders who possess the shares or equitycertificates. Depending on the conditions associated with the stock, a shareholder may sellhis stocks at the stock market. The value of the stock at a time t is the price an individualis willing to pay for the stock. In this paper, we examine common stock, stock valuation,dividend on stocks and the role of mixed portfolios in insuring protection of investmentsfor retirement.
Joshua Manley − LeeDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Johnson KakeuTitle: A Model of Multi-store Competition Strategy
Harold Hotellings Linear City model of minimum product differentiation assumes thereare two distinct Firms (Firm A and Firm B), located on a linear city, competing in priceand location. In this model, we investigate a multi-firm situation in which the executivemanagement team at Firm A decides to open another franchise, on the same linear city.Will it be more profitable for Firm A to have two franchises when competing against oneFranchise (Firm B)? If so, how will Firm Bs price and location respond to the new change?
Jalen MarshallDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Masilamani SambandhamTitle: Mathematical Modeling of Retardation in Organic Chain Reactions
Chemical reactions play an active role as dynamic systems in everyday life. Most Ofthese reactions can be described as the initial formation of free radicals, the forming ofmolecules via electron pairing free radicals and ending with a yielded product. For thisstudy, we consider a fourth conditional process: retardation. Retardation is observed ina more complex system when a free radical takes an electron from a formed molecule, es-sentially reversing the chain reaction. The goal of this study is to model retardation asa system of differential equations. We utilize undergraduate level chemistry, including asteady state approximation to help us with this task.
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Luis MatosDepartment of MathematicsGeorgia State UniversityAdvisor: Dr. Jeremy BrazasTitle: Fundamental Groups of Coarse Spaces
Finite spaces are topological spaces with only finitely many points and are closely re-lated to order theory. It is quite surprising that the homotopy theory of finite space ishighly non-trivial. In fact, the fundamental group of a finite TO space can be any finitelygenerated group. In this talk, I will discuss the finite analogue of the unit circle and usethis to construct a space that acts as a coarse version of the Hawaiian earring. This pro-vides a new surprising example: a space with only countably many points but which hasan uncountable fundamental group
Christopher McClainDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Keith PenrodTitle: Category Theory and Bridging the Gaps
In this presentation, we will present an overview of category theory. We will provide afew basic theorem proofs that set the basis for category theory. We will also explain whatit takes for something to be a category and the 4 tests which it has to be considered acategory. We will explore objects and explain the different types of objects. I will pro-vide different examples using different groups to prove that they are categories. I will alsoshow right inverses and left inverses that exist categories and I will also show homomorphicgroups. My research is still ongoing, so there is still more information to be discovered.
Trevonta L. Mctyre and Marquis D. CurryDepartment of Mathematics and Computer ScienceAlbany State UniversityAdvisor: Dr. Zephyrinus C. OkonkwoTitle: Bond Valuations and Investments Based on Bonds
A bond is an interest yielding security to the holder. Bonds can be issued by corporationsor government agencies, the goal being to raise money for specific or general purposes.Common bonds include US Treasury bonds, state government bonds, municipal bonds,mortgage bonds and debentures. Every bond has an issue date and a maturity date whichare clearly stated on the promissory note. Certain bonds such as the US Treasury bondhave very low risk and some bonds have very high risk, for example, Detroit municipalbonds. In this paper, we present the nature and properties of bonds, noncallable andcallable bonds, bond yields and reliability on bonds as major parts of retirement porfolios.
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Joseph ParkDepartments of Mathematics, Physics and PhilosophyUniversity of FloridaAdvisor: Dr. Sergei ShabanovTitle: Bound States in the Radiation Continuum for Periodic Structures
All optical data-processing could diminish the limitations of computational power, a per-vasive problem in computational research. The biggest obstacle is developing an opticalanalog of a transistor. My research advisor, mathematical physicist professor Sergei Sha-banov, has made significant progress toward this end investigating bound states of electro-magnetic waves in the radiation continuum. It was proved that the interaction betweentrapped electromagnetic modes can lead to scattering resonances of negligible width, whichare the bound states in the radiation continuum first discovered in quantum systems byvon Neumann and Wigner. It was then shown in a double array of subwavelength dielectriccylinders that by varying the spatial parameters toward the critical value, the near field canbe amplified in certain regions. The present study is the generalized system of an arbitrarynumber of arrays, two parallel 2D lattices of spherical scatters and analogous systems forelastodynamic and/or acoustic waves. The main fields of study involved are mathematicalphysics, scattering theory, functional analysis, operator theory, electromagnetism, acousticsand elastodynamics. Other potential applications include large amplification of electromag-netic fields within photonic structures and, hence, enhancement of nonlinear phenomena,impurity detection, biosensing, as well as perfect filters and waveguides for a particularfrequency.
Aquia RichburgDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Shelby WilsonTitle: Modeling the Brain with Math: Neural Networks and Liquid State Machines
Neural networks are useful models for programming computers on how to learn tasks.A perceptron is a basic machine that has linear input and output. The perceptron trainingalgorithm is an artificial learning algorithm that separates data into two predefined classes.If a set of data is linearly separable, there exists a line (hyperplane in higher dimensions)where the data is partitioned on either side. In this talk, we will show that, given a set oflinearly separable data, the perceptron training algorithm will converge to a line (hyper-plane) that correctly separates the data into their respective classes regardless of the initialweight vector.
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William SamuelsDepartment of MathematicsMorehouse CollegeAdvisor: Dr. Steven PedersonTitle: Understanding L’Hopital’s Rule
The purpose of this investigation is to understand LHopitals Rule from a theory perspec-tive using theorems and corollaries. In Calculus I, LHopitals Rule is stated and appliedto certain limit problems but does not have to be proven. LHopitals Rule is applied ifsubstitution into the limiting of the function leads to an indeterminate form. By usingthe rule, it makes the limit calculations less difficult by differentiating the numerator anddenominator. My intention is to understand why LHopitals Rule works using theoremsthat are learned in Real Analysis (Advanced Calculus).
Crystal Silver, Abebe Mojo and Gabriel TsegayeDepartment of MathematicsClark Atlanta UniversityAdvisor: Dr. Charles PierreTitle: Using a Markov Chain Model to understand the behavior of Student Retention
Dr. Charles Pierre and his graduate operations research class, consisting of Mr. AbebeMojo, Ms. Crystal Silver and Mr. Gabriel Tsegaye, were able to determine predictors ofthe length of time it took a student to graduate from Clark Atlanta University (CAU)circa 2006 by using data from the Universitys Trend book, a fact book created, under theapproval of the universitys president, Dr. Carlton E. Brown, under the watchful eye ofthe provost and vice president for academic affairs, Dr. James A. Hefner and under thedirection of the vice president of the Office of Planning, Assessment and Research (OPAR),Mr. Narendra H. Patel. They used Markovian tools to interpret the probabilities that weregleamed from the Trend book.
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Everett StarlingDepartment of Mathematics and Computer ScienceAlbany State UniversityAdvisor: Dr. Robert Steven OworTitle: An Introductory Comparative Analysis of Two Statistical Spectral Estimation Tech-niques
Fast, secure and accurate spectral estimation techniques are vital for the management ofsignal processing in small portable devices and embedded communicating microprocessors.As more and more devices become part of “The Internet of Things, the need for speed,security and accuracy increases. For this reason, several techniques are being developed forfast, secure and accurate estimation of spectral waves. This paper compares and analyzestwo promising techniques, namely the Burg Estimators and Yule- Walker Equations.
Latalya Walden and Marquis CurryDepartment of Mathematics and Computer ScienceAlbany State UniversityAdvisor: Dr. Zephyrinus C. OkonkwoTitle: Using Process Capability Estimates and Attribute Data to Generate Control Chartsfor Healthcare Delivery and Management
Process capability estimates are made in terms of upper and lower limits of populationdistribution such that no more than one in a thousand observations lie in each tail. Weshall use the process capability procedure to obtain normal estimates of surgery times forcertain forms of surgeries. Furthermore, we study the use of attribute data encountered ina wide-range of applications to generate control charts, including the count-chart (c-chart)and the u-chart where the u-values are obtained by dividing the subgroup count by thesubgroup size. These charts can be applied in the delivery of healthcare for different sub-groups of the population. Examples are drawn for illustration from data obtained frommonthly counts of patients falls and counts on medication errors.
