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Department of Mathematics Scranton, Pennsylvania |
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Papers, Presentations, and Research by Undergraduate Students
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Department of Mathematics Scranton, Pennsylvania |
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Papers, Presentations, and Research by Undergraduate Students
Students majoring in mathematics at Marywood University often undertake independent research directed by a faculty member. Usually these projects take the form of papers that are presented at both the Moravian College Student Math Conference (MSMC) held every February and the Marywood Undergraduate Research Forum (MURF) held every April. Additionally, many students have used their paper to satisfy their honors thesis requirement for the Marywood University Honors Program. Such activities significantly augment the transcript and resume of the participating student. Here are abstracts of the student papers directed by Dr. Craig Johnson. Complete copies of any paper can be obtained by phoning (570)348-6291 or e-mailing: johnsonc@marywood.edu
Abstracts of Student Papers
Pitch classes in music theory are congruence classes modulo 12 as defined in number theory. The musical transformation of note inversion Ip across a pivot note p (as used by Bach and other classical composers) can be formulated as a number theoretic function that operates on pitch classes. The effect of each Ip (p = 0, . . ., 11) can be visulized as a reflection across a diagonal through the Circle of Fifths. By studying these reflections, we discovered that sets of these functions formed cosets of a particular subgroup of the dihedral group D12 of order 24. So the study of music theory can be used to introduce groups and subgroups in a very natural way.
Using the number theoretic commands within Maple 9 and its available libraries, procdures were developed for experimentation in the encryption and decryption of short messages. One procedure takes a random prime number of a desired length, computes an exponentiation cipher with a public key, encodes a 10-character message, and then generates the private deciphering key. A second procedure performs decryption given the original prime numbers and the encryption key. A sampling of one of the algorithms is given below (the command lines are given horizontally to save space). For more information contact: Thomas Lodini at: tlodini@echoes.net
encrypt() procedure for Maple 9 with(StringTools): with(numtheory): seed1 := rand(10^10)(); seed2 := rand(10^10)(); prime1 := nextprime(seed1); prime2 := nextprime(seed2); r:=prime1*prime2; r2:=(prime1-1)*(prime2-1); seed3 := rand(10^5)() s:=nextprime(seed3); encrypt:=proc(); global m; local c,t,n,h,j; m:= array(0..10); t:=readstat("Enter a 10 Character String to Encrypt: "); h:=0;c:=0; while c<10 do h:=h+1; j:=substring(t,h..h); n:=Ord(j); m[c]:=(n^s)mod r; print(m[c]); c:=c+1: end do; end proc; d := eval(1/s mod r2);
The position function of a bungee jumper involves modeling acceleration that is affected by gravity, air resistance, and Hooke's Law. These are incorporated into a nonlinear first-order system of differential equations. The problem is to determine whether a jumper will hit the ground given a particular height and stretch constant that is associated with the bungee cord. Acceleration of the jumper is given by a = g - b(x) / m - rv / m, where g = gravity, m = mass, r = air resistance constant, and b(x) = Hooke's force. Let x(t) = position at time t and y(t) = dx/dt = v be velocity. Then the system is:
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dx/dt = y |
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dy/dt = 32 - b(x)/m - r * y/m |
The system was solved using Maple. Because b(x) is a piecewise function, we have two graphs that model the system. The first graph is a phase plane that shows velocity versus position up to the point where the cord is no longer slack. The second graph (shown) is a phase plane that shows velocity vs. position with the initial conditions reset. We changed the initial velocity according to the results of the first graph, and started the jumper off at x=0. We then use this information to predict safe k-values for bungee cords to support people of different masses.
From Galileo to the Spirograph, the cycloid has captured the interest of mathematicians for centuries. This paper examines its innate beauty and its role in both pure and applied mathematics that have made the cycloid one of the most interesting curves in history. We study and solve the brachistochrone, tautochrone, and other problems made famous by Galileo, Roberval, Torricelli, Descartes, Pascal, Huygens, Newton, and the Bernoulli brothers. The associated areas, arclengths, and other characteristics of the cylcoid and the hypocycloid are obtained using caclulus and differential equations and explored for interesting patterns.
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In 1956 Robert Solow created a mathematical model for the economic growth of a country. This model takes into account the country's rate of investment flow per year, the size of its labor force, and its propensity to save. Given certain assumptions, the following differential equation demonstrates the relationship between these factors:
where t represents time, K(t) is the capital accumulation of the composite commodity, s is the constant fraction of output saved, and L(t) = L*eat is the labor force. The function F of capital and labor is assumed to have certain properties that lead to a corresponding differential equation involving the ratio r(t) = K(t)/L(t). This can be solved for r(t) using the software program Maple if we make the reasonable assumption that F = rn. The power n is then determined by selecting the value for which the resulting graph of r(t) best fits a scatterplot of actual U.S. data over a fifty year period. Finally, this function is checked against the analytical solution of the differential equation in r that can be found using the Bernoulli method.
Cryptology is often referred to as the "art and science of secret writing". It is the branch of mathematics that deals with the design and analysis of the techniques used to transmit secret information and therefore it employs a large amount of number theory and statistics. Used since the time of Caesar for military purposes, today the needs fulfilled by cryptology have extended beyond mere privacy to include data integrity, authentication, and non-repudiation. These refer to the accuracy of information in a transmission and the ability to confirm authorship of a received message. This paper reviews some of the classic encryption methods, demonstrates a cryptanalysis of the Vigenere Cipher, and uses the Kasiski and Friedman statistical tests to decrypt an encoded message. Additionally, it examines the invention of Public Key Encryption by Diffie and Hellman in 1976 and mathematically explain the steps in the RSA Algorithm which is one of the most widely used public key cryptosystems in use today.
This paper extends the results of the article "A Home Heating Model for Calculus Students" published in the College Mathematics Journal [Vol. 27, No. 5, pp. 394-397, Nov. 1996]. We examine the problem of heating a three compartment house (two rooms and an attic) using a furnace with constant output that heats either one or both rooms. The following diagram displays the situation for the two heating schemes.
We wish to find the three functions xi(t), i = 1,2,3, each of which gives the temperature of a given compartment of the house as a function of time elapsed after a furnace initiates a heat flow H(t). Initially we assume the temperature of the entire house to be the same as the outside temperature T(t). The mathematical model is based on Newton's Law of Cooling which states that the rate of change of temperature resulting from heat transfer across a single boundary is proportional to the difference in the temperatures on opposite sides of the boundary. For a single boundary this can be written
The ki in the above figures represent the proportionality constants for the indicated walls and ceilings. The values used are based on typical amounts of insulation used in cold weather climates. In the case with just one heated room we get the following 3 x 3 first-order system of differential equations.
dx2/dt = k1[T(t) - x2(t)] + k2[x3(t) - x2(t)] + k3[x1(t) - x2(t)]
dx3/dt = k2[x2(t) - x3(t)] + k2[x2(t) - x3(t)] + k4[T(t) - x3(t)]
After substituting for the k values, T(t), and H(t), this system can be solved analytically for x1(t), x2(t), and x3(t) using linear algebra to find the eigenvalues and corresponding eigenvectors. Upon specifying some initial conditions,the graphs of the component functions of the particular solution were then compared to the numerical solutions obtained from MDEP, a widely used ODE solver created by the Naval Academy at Annapolis. The results matched very closely and so provided a validation of the analytical solutions. Thus this model could prove useful in predicting temperatures of these rooms under various conditions.
Many people in society today do not realize the degree to which mathematics has been integrated into our culture. One field whose influence continues to grow is graph theory. Networks, circuits, and other objects of graph theory provide useful mathematical models for projecting sales within the business world, analyzing communication and transportation problems, explaining psychological relationships, and even for such ordinary purposes as optimizing routes for snowplows and mail trucks. This paper explores the details of these applications.
The accumulation of lead in the body can cause many problems in humans such as anemia, mental retardation, and kidney failure. My research focuses on the use of a mathematical structure to model the distribution of lead in the body. It is a specific case study based on assumed rates of lead intake. The rate at which the lead diffuses throughout the compartments of blood, soft tissue, and bones is derived experimentally. The model was originally proposed in a study by Rabinowitz, Wetherill, and Kopple and can be diagrammed as shown below.
Let the amount of lead in compartment i at any time be denoted by xi(t). The amount of lead transferred from compartment i to compartment j is proportional to the amount xi(t). The transfer coefficient is denoted by aij. The above relationships can be modeled by this first-order system of differential equations.
(Tissue) dx2/dt = a21x1 + a20x2 + a21x1
(Bone) dx3/dt = a13x1 + a31x3
After substituting values for the transfer rates and input rate IL obtained from experimental studies [Rabinowitz et al] the eigenvectors and eigenvalues of the system were found and the method of undetermined coefficients was used to find the particular solution vector. The component functions were then compared to the numerical solutions obtained from MDEP, a widely used ODE solver created by the Naval Academy at Annapolis. The results matched very closely and so provided a validation of the analytical solutions.
The common perception of mathematics in society is that it is primarily concerned with pure computation. However, the development of abstraction in mathematics can be traced over a period of more than six thousand years. The mathematical abstraction of concepts and relationships associated with a natural phenomenon allows us to both make predictions about its behavior as well as provide a logical structure with which to discover new insight. This paper defines and explains the notion of abstraction by providing the history of its development and exploring some examples such as variables, algebra, coordinate geometry, functions, calculus, and Non-Euclidean geometries.
This presentation simulates outcomes of a game of roulette by the use of a computer program written in the C++ programming language. The user selects three inputs: (i) the type of bet chosen from a menu of 14 different options, (ii) the amount of the wager, and (iii) the total amount of money the user is willing to win or lose. Given these inputs the program uses an algorithm employing probability, statistics, and the Monte Carlo method to output the number of spins of the wheel it takes to achieve that betting strategy.
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