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Hello,
My name is Joseph Galante and I would like to formally request that my paper, intitled "Generalized Cantor Expansions" be published in the RHIT Mathematics Journal. The article is attached to this email (zipped) in MS Word format. The work for the paper was completed during sophomore year at the University of Rochester, between the dates 11/02 and 5/03 and was put into its final form this summer (2003). The mathematician who is sponsoring me is Ken McMurdy at the University of Rochester.
Abstract:
There are many ways to represent a number, commonly known as base expansions. The most frequently used base is ten, which is the basis for our decimal number system. However a more uncommon way to represent a number is the so called cantor expansion of the number. This system uses factorials rather than numbers to powers as the basis for the system, and it can be shown that this produces a unique expansion for every natural number. However, if you view factorials as products, then it becomes natural to ask what happens if you use other types of products as bases.
This paper explores that question and shows there are an uncountably infinite number of bases which can be used to represent the natural, and real numbers uniquely. By using these new and interesting types of bases, it becomes possible to formulate bases in which all rational numbers have a terminating expansion.
About the Author:
My name is Joseph Galante and am junior mathematics major at the University of Rochester. I initially learned of the Cantor Base Expansion from Kenneth Rosen's Discrete Mathematics textbook. During sophomore year, I mentioned to Prof. Ken McMurdy that there are probably more types of strange expansions and he encouraged me to explore this further. It ultimately resulted in the paper, completed during my free time sophomore year. At college, I am the VP of the Society for Undergraduate Math Majors (SUMS) and I am the webmaster for my Tae Kwon Do club. In the future I plan to pursue mathematics with hopes of graduate school.
Thank you for your time in reading this email and processing my request.
Sincerely,
Joseph Galante
Dear Editors,
I would like to recommend a paper by Joe Galante entitled "Generalized Cantor Expansions". Joe was a student of mine in honors calculus as a freshman in the Fall of 2001. The work was done in the Fall of 2002 and Spring of 2003, though, while Joe was a sophomore. It's actually a very nice story how the work came about. Joe had a class right after my Discrete Math class and in the same room. So we would sometimes chat for a few minutes about what I had been covering. One day I explained what the Cantor Expansion of a natural number was and casually wondered out loud if there were actually many such expansions with their own individual merits. Joe picked up on the suggestion and worked on the problem unprompted by me. A couple of weeks later he showed me his initial result, an uncountable collection expansions for which the Cantor Expansion and the usual base expansions were examples. I was very excited and immediately suggested that he should write a paper and submit it to an undergraduate journal for publication.
A semester has passed since then, and Joe has now proven some results which further put his generalized Cantor expansions in perspective. For example, he has shown that fractions and even real numbers can also be represented in this way. I must warn that I have not carefully checked every detail of the paper and do recommend that it be carefully refereed (as I'm sure it will). I have a great deal of confidence in Joe from his course work, though. I will be surprised if there are more than a few minor mistakes.
Overall I think the paper is very appropriate for the RHIT journal. The subject matter (digit-related number theory) should be accessible to many undergraduates, and yet the results are laid out and proven in a very sophisticated way, with carefully stated (and well motivated) definitions and theorems. I will look forward to hearing how it stands up to the refereeing process, but I honestly feel very good about the paper.
Sincerely,
Ken McMurdy
Assistant Professor
University of Rochester
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