**Most Current Research Projects**

## Arithmetic - Geometric - Harmonic Mean

There are many different types of mean in mathematics, all having different equations that can be used to find them.

While most people can memorize these equations and the relationships among the types of mean, most never make the connection to geometry.

**Advisor**: Aurel Stan**Reference**: Rudin Walter. Principios de Analisis Matemático. 3a edición 1980. McGraw Hill**Ivan Chavez**

## RSA Encryptions

How does the RSA encryption work? What is the basis behind the use of the encryptions and how mathematicians use primes into this concept? The use of primes, Euclidean algorithm and a computational method or program.

**Advisor**: Warren Sinnott**References**: Davenport, J. H. (2008). Computers And Number Theory. The higher arithmetic: an introduction to the theory of numbers (8th ed., pp. 199-200). Cambridge: Cambridge University Press.**Enid M. Colón**

## Are There Any Perfect Odd Numbers?

In this project I researched about perfect numbers and an unsolved problem: Are there any Perfect Odd numbers? After researching about it, I was able to find a formula that described how a perfect odd number should be composed: N=q^e a_1^{2g_1} a_2^{2g_2} … a_n^{2g_n}, where q and a_i are distinct primes of N, N being a perfect odd number, and e is odd. Then I worked on making a sum with all the possible divisors of N such that: 1+d_1+d_2+ … +d_k=N must be true. The conclusion I arrived was that the sum would be even… But that would be a contradiction. And if my results were true…That would mean that there are no Perfect Odd numbers.

**References**:

- Wolfram MathWorld, the web's most extensive mathemarics resource
- A Study on the Necessary Conditions for Odd Perfect Numbers [pdf], Ben Stevens
- Perfect numbers - a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself
- Sum of Two Squares[pdf], Jahnavi Bhaskar
- Necessary Conditions for the Non-existence of Odd Perfect Numbers [pdf], Jonathan Pearlman; Monday, May 23rd, 2005
- Book IX, David E. Joyce, Clark University
- Sigma Function- Sigma(n) is the sum of the factors of n

**Saed Cruz**

## Earth vs Moon

A mass attached to a spring moves up and down within a unit of time (frequency). Will the frequency be affected by the difference in gravity between the earth and moon? Utilizing Differential Equation and Physics the determination is the frequency remains the same.

**Advisor**: Ulrich Gerlach**References**:

- Elementary Differential Equations and Boundary Value Problems by

William E. Boyce and Richard C. Di Prima - Differential Equations by Kaj L. Nielsen
- Handbook of Differential Equations- Stationary Partial Differential Equations by M. Chipot
- Mechanics by Keith R. Symon.

**Garrett Divens**

## Roulette Game

The purpose of our research is to find a strategy of successfully being profitable in the Vegas game of roulette. Along the way we use the Catalan Number Sequence to find the probability of having a net profit at the end of the game.

**Advisor**: Alexander Diaz**References**: Catlan Numbers [pdf], Tom Davis; November 26, 2006**Jean Carlo Galan and Rafael Gutierrez**

## Borsuk-Ulam Theorem. Topological applications to combinatorics.

Topology has been an abstract area of mathematics. Recently, though, it has found applications in many fields. One of them is combinatorics. This presentation states the theorem and presents topological proofs of the Kneser graph chromatic color and the Necklace splitting problems.

**Advisor**: Mathew Kahle**References**:

- Colouring Kneser Graphs [pdf] - the graph whose vertices correspond to the k-element subsets of a set of n elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint.
- 25 years proof of the Kneser conjecture: The advent of topological combinatorics [pdf], Mark de Longueville (Berlin)
- Lecture 17: The Borsuk-Ulam Theorem [pdf], Anup Rao; November 28, 2011
- Lecture 14: Topological Methods [pdf], Jacob Fox

**Mario Gomez**

## Arzelà Dominated Convergence Theorem for the Riemann Integral

Riemann’s definition of a definite integral gave rise to a number of important developments in analysis. In the course of these developments a remarkable result due to C. Arzelà marked the beginning of a deeper understanding of the continuity properties of the Riemann integral as a function of its integrand. The result of Arzelà we have in mind is the so-called ARZELÀ DOMINATED CONVERGENCE THEOREM for the Riemann integral concerning the passage of the limit under the integral sign.

**Advisor**: Dr. Roberto Quezada Batalla (Universidad Autónoma Metropolitana-I)**References**:

- C. Arzelà, Sulla integrazione per serie, Atti Acc. Lincei Rend., Rome, (4) 1 (1885), 532-537, 596-599.
- S. Banach, The Lebesgue integral in abstract spaces, note in S. Saks, Theory of the Integral, Warsaw 1933 and New York 1937, 320-33
- Arzelà's Dominated Convergence Theorem for the Riemann Integral. W. A. J. Luxemburg, The American Mathematical Monthly, Vol. 78, No. 9 (Nov., 1971), pp. 970-979

**Alejandro Gonzalez-Alba**

## Collatz Conjecture

What it aimed to do, was explain the conjecture in itself, which states that every number will reach 1 if the following conditions are met: if the number is even divided two and if the number is odd multiply it by 3 and add 1 to it making it even and repeat this process as many times as needed.

**Advisors**: Cosmin Roman and Edward Overman**References**: Own research, Wikipedia, Google images**Joel Nunez**

## An inequality between the individual and collective times to complete a job

The inequality between the arithmetic and harmonic means of n positive numbers, can be interpreted as the fact that the time necessary for n workers, each working at a constant speed (productivity), to complete a job working together, is less than or equal to 1/n2 of the sum of the individual times required by the workers laboring alone to complete that job. We show first that if we consider n workers, who are becoming tired in time, and whose productivity decreases exponentially in time at a rate that depends only on the difficulty of the job performed, but not on the workers, then the above inequality still holds. The proof relies on Jensen inequality. Finally, we extend this result to the case in which we have n workers, becoming tired continuously in time, and whose order of productivity remains the same in time, that means, if worker 1 productivity is greater than or equal to worker 2 productivity at time 0, then worker 2 productivity never exceeds worker 1 productivity.

**Advisor**: Aurel Stan**Reference**: materials covered during the SAMMS Real Analysis class period**Dieff Vital and Jose Pastrana**

## Ordinal Arithmetic and Goodstein’s Theorem

A brief introduction to Ordinal Arithmetic was presented. Some important properties of ordinal arithmetic were shown by using examples. Cantor’s Normal Form theorem was presented. Both, Hereditary base n and the method for constructing a Goodstain’s sequence were explained by using an example. The general ideas for proving the Goodstain’s theorem were presented using an example. Ordinals such as α=ω^α were not mentioned. The Kirby-Paris Theorem was presented and explained.

**Advisor**: Ivo Herzog**References**:

- Schloder, J. Ordinal Arithmetic; available at Ordinal Arithmetic [pdf], Julian J Schloder
- Professor Herzog and I worked on the proof of Goodstain’s theorem during our meetings.
- Professor Herzog mentioned the Kirby-Paris theorem during our meetings.

**Rigoberto Zelada**

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