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Math and music share one very common property; they are both universal languages of the world. Their scientific relationship represents another powerful tie. The harmonious, unique, and enjoyable tunes we hear are surrounded by scientific and mathematical concepts that make music possible. Exploring this natural connection between math and music allows for a more complete understanding of both these subjects. By analyzing the formulas used to create musical notes, examining how to apply matrices to music, and implementing an inter-disciplinary unit of math and music into the classroom, the reader of this thesis will discover the multitude of similarities that math and music share.

This paper stems from research information accumulated over the summer of 1997 on the topic of ovarian cancer. My role in this was statistical accumulation, computation, and initial analysis. The study appealed to me due to the intertwined usage of math and biology it entailed. Specifically, a background in biology aided in comprehending the affliction of ovarian cancer and the underlying genetics that control it. The original purpose of this study involved compiling statistics on the symptoms, pathology reports, and flow cytometry data accumulated over the past few years from ovarian cancer patients at the Women’s Cancer Center of Northern California. From this, a new method of classifying the extent of the disease in new patients would be developed. The results of analyzing this data could improve upon the current method of diagnosis, which involves stages and grades. This would allow the treating physician a better chance at informing the patient as to the severity of the disease from which she suffers. It would also allow the patient to be more informed as to the chances of survival, the complications, and the realism of reoccurrence. The specific genes that maintain cell cycles and their division, possible mutations, and how all of it is regulated through the expression of certain genetic elements are explained in this paper. The possible benefits of this study include more thorough examinations, better treatments, and informed prognoses. These ideas are experimental, yet their implications could bring about major changes in the treatments of ovarian cancer.

I wanted to examine a specific aspect of the U.S.-Japanese trade relationship that has been cited by some as a source of tension. I chose to examine the unfair trading practice of dumping. Dumping involves selling goods at a price that is below the cost of production. To narrow my topic even further, I decided to examine dumping that was occurring within the flat panel display industry. Flat panel displays are the computer screens that are used in laptop computers, avionics equipment, telecommunications, and military equipment. For a specific case study, I chose to investigate a firm from Beaverton, Oregon that eventually took their dumping claim before the U.S. Department of Commerce and the International Trade Commission.

The purpose of this study was to understand the various mutations of cystic fibrosis on chromosome seven, particularly the most commonly recognized mutation, DeltaF508. Cystic fibrosis (CF) is one of the most common autosomal recessive disorders in Caucasians, affecting about one in 2,500 live births, with a carrier frequency of one in 25. Through this research we demonstrate that DeltaF508 genotype actually can predict disease severity in an ethnically diverse CF population, to an extent. Clinical and genetic data were collected for 214 Cystic Fibrosis patients and entered into a database. The diagnosis of Cystic Fibrosis is based on clinical symptoms, particularly respiratory and digestive complications, and a raise in the electrolytic ion concentration in sweat. Upon complete statistical analysis of the data, including haplotype verification, it can be determined that the DeltaF508 genotype does suggests that patients with two DeltaF508 allele demonstate a greater degree of irregularity with the digestive system.

This paper will present examples of fractal geometry with their respective techniques of analysis in order to illustrate the foundations for a quantitative study of the natural world. The first section studies dimension, both self-similar measure and Hausdorff-Besicovitch measure, emphasizing the idea of dimension as a measure rather than as a number of degrees of freedom. The second section presents random, statistically self-similar fractals. A topologically one-dimensional construction found in fair games of chance leads to the Levy flight, a topologically two-dimensional example that gives a model for the distribution of galaxies as viewed from Earth.

The purpose of this research project was to determine if a particular peer mediation program reduced the incidence of conflict on playgrounds. The difference between observed conflict behaviors at Clancy Elementary School before and after the implementation of the Conflict Management program was examined. The student population (n = 372) of Clancy Elementary consisting of grades K-8, participated in the study. The subjects were observed during recesses before and after the implementation of the Conflict Management program. The data was compared between the two observation periods. Results indicate that the Conflict Management program was effective in reducing conflict behaviors on the playground (p < .05). Both the total number and each type of conflict behavior were less (p < .05) after implementation of Peer Mediation 2 the Conflict Management program.

The focus of this paper is a presentation of patterns that occur in the world in which we live and how they are explained in mathematics. The major pattern is that of the Fibonacci sequence, which was first discovered by Leonardo Pisano. A brief history of Leonardo (Fibonacci) will precede topics such as the equiangular spiral, naturally occurring patterns, the golden proportion and the golden ratio. Next is a short introduction to Blaise Pascal and how the Fibonacci sequence fits into his famous Pascal’s Triangle. Finally, fractals will be discussed along with background knowledge of Benoit Mandelbrot, the newest mathematician of his time.

This thesis reviews past research on gender differences in mathematics achievement and attitudes toward mathematics. The review is divided into three categories: gender differences that have been found in mathematics, possible causes of those gender differences, and ways to eliminate the gender differences. A small community in Montana was tested for gender differences in mathematics achievement and confidence in the learning of mathematics. The subjects consisted of 55 females and 56 males in grades eight and nine. The subjects’ scores on the Fennema-Sherman Confidence in the Learning of Mathematics Scale and the CTBS Mathematics Achievement Test were compared to each other and to other research results. No significant (a =.05) gender differences in mathematics achievement or in confidence in learning mathematics were found in the sample, and a positive correlation between the two variables existed. The results from this particular small town contradict past studies that show a gender difference in favor of males in the field of mathematics.

“How can students compete in a mathematical society when they leave school knowing so little mathematics?” This quote is taken from Everybody Counts: A Report to the Nation on the Future of Mathematics Education, and sums up the current concerns of many mathematics educators. The past decade has been filled with disturbing reports, including An Agenda for Action. A Nation at Risk, and Everybody Counts, that deal with the status of the American educational system. In addition to learning that Johnny cannot read, we learned that Johnny also cannot add or subtract. Throughout American high schools, student achievement in mathematics has fallen dramatically. Nowhere is this more clear, perhaps, than in the high school geometry classroom. Many people would argue that geometry, as it is taught has become the downfall of a wide majority of young students. Typically consisting of a year filled with ‘proofs’, geometry sometimes succeeds not in teaching students to think logically, but in teaching students to dislike mathematics. Rather than learning to create and construct solutions, students often resort to simply memorizing solutions, in order to successfully complete the course. If we as educators ever hope to improve the status of mathematics education in American high schools, we must eradicate the habit of memorizing information just to “get through the 1 course.” Memorization is not learning; until students start learning (and start wanting to learn), achievement in high school geometry will never improve. I support the theory that the method of teaching high school geometry, currently a teacher-oriented, proof-based method, be altered to include student-oriented, hands-on learning activities. In this thesis, I will explore the research supporting hands-on learning and present ways to implement hands-on activities into the curriculum.

here exists a group of practical problems in computer science today for which no one has yet been able to find reasonable algorithmic solutions (those solvable in a reasonable amount of time), yet no one has been able to prove that no such solutions exist, either. These problems readily admit naive, or obvious solutions, but these solutions would take entirely unreasonable amounts of time, say thousands of years, to terminate. Scientists have simply not been able to find other algorithms that run faster and more efficiently, but, as noted, have also been unable to prove their nonexistence. This puzzle has teased computer scientists for decades now, and it continues to be of great interest. Why are these problems so important? There are approximately 1000 problems known to fall into this uncertainty category1, and most of them have many practical uses. For example, if we could solve the Traveling Salesman problem, we could immediately map out the most efficient route for delivery companies such as UPS or Federal Express. If we could figure out the Bin Packing problem, lumber yards would be able to more efficiently fill orders from large pieces of wood. Being able to solve the Graph Coloring problem would make scheduling final exams for a university much quicker and easier. These problems are discussed in length in the pages that follow. One of the most interesting aspects of these problems is that if scientists could find a fast running algorithm to solve any one of this special category of problems, they would know that a fast algorithm necessarily exists for all of them! Conversely, if they could prove that no such algorithm exists for one of them, they would know that there will never be a fast algorithm to solve any of them, and they could concentrate their efforts on finding algorithms to approximate the solutions. This class of problems, commonly called NPcomplete problems, will be a major topic of this paper. Turing machines, which will be the second focus of

Probably the greatest alarm to the American Public concerning their education system came about with declining SAT and ACT test scores. The rash declines sparked the realization for the need to examine the school system. The SAT, Scholastic Aptitude Test, and ACT, American College Test, are given to high school college-bound seniors, with one or both required for admission. The SAT is scored on a scale of 200 to 800 in the areas of verbal and mathematic skills. The ACT grading scale ranges from 1 to 36, with an average composite score taken from the areas of English, Math, Social Studies, and the Natural Sciences.''' With average annual scores stable through the 1940's and 1950's, the 1960's and 1970's marked drastic declines.

Not all children can learn at the same rate or through the same methods. Everyone, whether adult or child, male or female, has areas in which they are gifted, average, and deficient. In one way or another, individuals learn to compensate for those areas in which they are impaired. Truly, not everyone can be a Babe Ruth, Albert Einstein, or George Washington; however, when a child's handicap affects his ability to learn, and he knows of no methods to help him compensate for this academic loss, something must be done to help the child. All children are at different levels maturationally and intellectually, even though they may be similar chronologically. Gaps between some children's learning abilities are so wide that learning problems exist. When students' performances do not match their potentials, this is often an indication of the presence of learning disabilities. Many techniques are available that attempt to remediate children's learning problems, but not all work for all students. The purpose of this paper is to identify various learning disabilities and to explore some possible uses of the computer in assisting handicapped children to reach their potentials.

This thesis deals with an introduction to the functionals in the calculus of variations. Since calculus of variations is a relatively advanced topic in mathematics, this thesis is written with the attempt to introduce some of the oasis notions with regard to what the calculus of variations is. To do this, an attempt is made to try to draw an analogy between the calculus of variations and ordinary maximum and minimum theory of functions which a reader may be very familiar with, and see how some of the ideas and methods used in ordinary maximum and minimum problems are carried over to the calculus of variations. The thesis begins with the consideration of maximum and minimum theory of functions. This is done by two examples. This is followed by a consideration of a problem that is caused by the point of inflection in ordinary maximum and minimum theory. With all these considerations, one hopes to shed some light as to how some of these ideas are used in the calculus of variations. Next one shall see what the calculus of variations is and see how a problem in calculus of variations is dealt with. To do this, an example concerning the finding of a shortest distance curve Joining two given fixed points in a two dimensional Euclidean space is considered.In this example, one shall see in fact how one can change a problem in the calculus of variations into an ordinary maximum and minimum problem, and try to draw some analogy between the two. Following this, one shall see how the Euler’s differential equation is derived in general for a fixed end points problem in two dimensional Euclidean space. Here because of the limitations of this thesis, only some specific functions, namely, functions that are continuous and twice differentiable are dealt with. Some remarks about Euler’s differential equation will be given. The form of Euler’s equation for n dependent variables and one independent variable will be stated without proof here. A simple application of Euler’s equation to p

The non-Riemannian geometry of Weyl is an outgrowth of Levi- Civita’s concept of parallelism. It is based on the concept of linear displacement. In Weyl’s geometry length is non-transferable and, in light of this indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separated by infintesmal intervals. We must therefore fix at every point of space certain measuring rods which are to serve as a unit of length when we measure lengths situated by their side. The totality of these unit rods constitute what is known as a gauge system. These gauges may be selected arbitrarily. We will give a detailed discussion of the gauge system later in the thesis. Moreover, since Weyl’s geometry is a generalization of Riemann’s, we might first consider some of the foundations of that geometry which are applicable to the former. This we shall do by a process of abstraction. We shall proceed from a Euclidean space defined on the more familiar Cartesian co-ordinates to one defined on the general or curvilinear co-ordinates of three dimensions. We then proceed to a space of n dimensions and to the Riemann space. We shall discuss the theory of tensors from an intuitive standpoint using these geometrical foundations. The body of the thesis will involve a detailed discussion of the geometrical foundations of Weylian geometry and some of its important geometrical properties. The conclusion will be synthesis of the most important physical implications of Weylian geometry to the general theory of relativity.The non-Riemannian geometry of Weyl is an outgrowth of Levi- Civita’s concept of parallelism. It is based on the concept of linear displacement. In Weyl’s geometry length is non-transferable and, in light of this indeterminateness that surrounds the comparison of lengths in different places, we must confine ourselves to the comparison of lengths at any one place or at points separate

Nearly everything a person does, every action taken, is the result of making a decision. With the exception of a few physiological actions, such as breathing, every action involves a decision, even though some are very minor decisions because the consequences are not too important. For these minor decisions, intuition instead of conscious thought is used. Because of the importance of decision-making, both in everyday situations and in complex business matters, the theory of decision-making has been developed extensively in recent years. It has often been said that the key to understanding any decision-making process is to discover the ways in which the decision-maker simplifies the complex ideas into workable conceptions of the decision problem. And this is just what Decision Theory does.