Volume 9 Issue 1 June 2010

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https://doi.org/10.33697/ajur.2010.006

Editorial: Complexity in Our Everyday World

https://doi.org/10.33697/ajur.2010.007

Author(s):

C. C. Chancey

Affiliation:

American Journal of Undergraduate Research, University of Northern Iowa, Cedar Falls, Iowa 50614-0150 USA


A Lesson Learned from Outer Billiard Derivatives

https://doi.org/10.33697/ajur.2010.008

Author(s):

Samuel Otten and Filiz Doğru

Affiliation:

Department of Mathematics Grand Valley State University, 1 Campus Drive Allendale, Michigan 49401-6495, USA

ABSTRACT:

As students move onward and upward through collegiate mathematics they are often impressed by the power of advanced techniques (aren’t they?), techniques allowing problems that were previously difficult or near-impossible to be solved with relative ease. For example, once students have learned the residue theorem in complex analysis they are able to elegantly evaluate integrals that were unwieldy in calculus. Among practicing mathematicians there is also a tendency to look to new or powerful results when trying to unlock a problem within one’s own research. This article presents a new theorem concerning derivatives within the outer billiard dynamical system, and in so doing serves as a reminder that higher-powered mathematics are not always needed and do not necessarily produce more satisfying proofs. Sometimes it is beneficial to keep elementary approaches in mind.


The Applicability of the River Continuum Concept to the Upper Reaches of a Neotropical Lower Montane Stream

https://doi.org/10.33697/ajur.2010.009

Author(s):

Nicholas Skaff

Affiliation:

Department of Biology, Tufts University, Medford, Massachusetts 02155 USA

ABSTRACT:

The applicability of the River Continuum Concept (RCC) to the upper reaches of Quebrada Máquina, a lower montane stream in Monteverde, Costa Rica, was examined. Macroinvertebrate samples were taken from ten points along the stream from first through fourth order segments. The families of the collected individuals were then categorized based on functional feeding group. The similarity between the families found at each collection point was calculated, along with correlations between the functional groups and various stream characteristics. In most cases, RCC predictions did not apply to Quebrada Máquina. The first (first order) and last (fourth order) sample points were 92% similar despite RCC predictions of substantial divergence in relative functional group abundance. This digression from the RCC predictions may be caused by the relatively few differences in stream characteristics between first and fourth order sections. Specifically, the observed similarities and correlations may have been determined by local scale heterogeneity of the stream characteristics.


On the Number of Rational Iterated Pre-Images of -1 under Quadratic Dynamical Systems

https://doi.org/10.33697/ajur.2010.010

Author(s):

Trevor Hyde

Affiliation:

Department of Mathematics, Amherst College, Amherst, Massachusetts 01002 USA

ABSTRACT:

For the class of functions fc(x)=x^2+c, we prove a conditional bound on the number of rational solutions to fc^N(x) =−1 and make computational conjectures for a bound on the number of rational solutions to  fc^N(x)=a fora in a specific subset of the rationals.