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Use Of Calculators In High School Mathematics, Alex Mcdaniel
Use Of Calculators In High School Mathematics, Alex Mcdaniel
Williams Honors College, Honors Research Projects
For many collegiate level mathematics courses at universities across the country, students are not allowed to use calculators. This is often a huge adjustment for students who have spent their entire high school career using calculators. Whether it is graphing or simple arithmetic, students relied on calculators in high school, but for college, the tool of a calculator may not be available for them. This leads to a multitude of questions: Should calculators be used in high school? Do calculators help students learn or are they simply a tool to get answers? Are calculators beneficial in some courses, but not …
Modelling Potential Fluctuations In Double Layer Graphene Systems As A Periodic Oscillation In Electron Density & Its Effect On Coulomb Drag, Ryan Bogucki
Williams Honors College, Honors Research Projects
An expression for the drag transresistivity in a graphene double layer system exhibiting potential fluctuations modelled as a periodic oscillation in electron density is derived. Our model starts from the Coulombic interaction and we derive the correlation between a sinusoidal fluctuation in electron density in the first layer and the induced electron density in the second layer. Previous models in the literature have employed an arbitrary correlation between each layer’s electron density, and the model presented is the first attempt in the literature to explicitly derive this correlation. Recent experiments have found that the drag transresistivity in graphene double layers …
The Rsa Cryptosystem, Rodrigo Iglesias
The Rsa Cryptosystem, Rodrigo Iglesias
Williams Honors College, Honors Research Projects
This paper intends to present an overview of the RSA cryptosystem. Cryptosystems are mathematical algorithms that disguise information so that only the people for whom the information is intended can read it. The invention of the RSA cryptosystem in 1977 was a significant event in the history of cryptosystems. We will describe in detail how the RSA cryptosystem works and then illustrate the process with a realistic example using fictional characters. In addition, we will discuss how cryptosystems worked prior to the invention of RSA and the advantage of using RSA over any of the previous cryptosystems. This will help …
An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime Divisors, Jacob Liddy
An Algorithm To Determine All Odd Primitive Abundant Numbers With D Prime Divisors, Jacob Liddy
Williams Honors College, Honors Research Projects
An abundant number is said to be primitive if none of its proper divisors are abundant. Dickson proved that for an arbitrary positive integer d there exists only finitely many odd primitive abundant numbers having exactly d prime divisors. In this paper we describe a fast algorithm that finds all primitive odd numbers with d unique prime divisors. We use this algorithm to find all the number of odd primitive abundant numbers with 6 unique Divisors. We use this algorithm to prove that an odd weird number must have at least 6 prime divisors.
Using Mathematical Research Methods To Solve A Problem In Music Theory Instruction, Specifically, The Teaching Of Secondary Dominant Chords, Angela Ulrich
Williams Honors College, Honors Research Projects
The mathematical method for research is used to find a solution to a problem in music theory: understanding and identifying secondary dominant chords. By reviewing and assessing the teaching methods of university professors and theory textbooks, and comparing those findings with student reviews, a new method for teaching the concept is developed. The proposed system incorporates aural, visual, and kinetic exercises to serve every learner. The literature review and sample unit plan are followed by a possible procedure for testing the effectiveness of the new method.
Subgroups Of Finite Wreath Product Groups For P=3, Jessica L. Gonda
Subgroups Of Finite Wreath Product Groups For P=3, Jessica L. Gonda
Williams Honors College, Honors Research Projects
Let M be the additive abelian group of 3-by-3 matrices whose entries are from the ring of integers modulo 9. The problem of determining all the normal subgroups of the regular wreath product group P=Z9≀(Z3 × Z3) that are contained in its base subgroup is equivalent to the problem of determining the subgroups of M that are invariant under two particular endomorphisms of M. In this thesis we give a partial solution to the latter problem by implementing a systematic approach using concepts from group theory and linear algebra.
New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe
New Facets Of The Balanced Minimal Evolution Polytope, Logan Keefe
Williams Honors College, Honors Research Projects
The balanced minimal evolution (BME) polytope arises from the study of phylogenetic trees in biology. It is a geometric structure which has a variant for each natural number n. The main application of this polytope is that we can use linear programming with it in order to determine the most likely phylogenetic tree for a given genetic data set. In this paper, we explore the geometric and combinatorial structure of the BME polytope. Background information will be covered, highlighting some points from previous research, and a new result on the structure of the BME polytope will be given.
Philosophy Of Mathematics: Theories And Defense, Amy E. Maffit
Philosophy Of Mathematics: Theories And Defense, Amy E. Maffit
Williams Honors College, Honors Research Projects
In this paper I discuss six philosophical theories of mathematics including logicism, intuitionism, formalism, platonism, structuralism, and moderate realism. I also discuss problems that arise within these theories and attempts to solve them. Finally, I attempt to harmonize the best features of moderate realism and structuralism, presenting a theory that I take to best describe current mathematical practice.