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Newton's Method Cycles

Based on the paper Sometimes Newton's Method Cycles, we first asked ourselves if there were any Newtonian Method Cycle functions which have non-trivial guesses. We encountered a way to create functions that cycle between a set number of points with any initial, non-trivial guesses when Newton's Method is applied. We exercised these possibilities through the methods of 2-cycles, 3-cycles and 4-cycles. We then generalized these cycles into k-cycles. After generalizing Newton's Method, we found the conditions that skew the cycles into a spiral pattern which will either converge, diverge or become a near-cycle. Once we obtained all this information, we explored additional questions that rose up from our initial exploration of Newton's Method.

Edgara Vanoye & MacKay Martin

The Sum of the Reciprocals of the Squares

A reproduction of a proof by Leonard Euler, which I dug out of the back of a calculus textbook with the help of Brian Burrell. This was in MATH 233 at the University of Massachusetts, Amherst, Fall 2015.

Samuel Castillo

Determining the Speed of Light

When measuring a speed, the most common way to calculate it is by recording
how far something went and the time it took to go that far. In the case of light,
this is very difficult. One could conceivably shine a light over a vast distance
and have someone else record when they see the light, but this would be difficult
even at large distances. The person recording when they see it will need to have
terrific reflexes to accurately measure a correct time as the time will be very
short. A better method involves the use of a quickly rotating mirror and a beam
of light. By aiming a beam of light o the rotating mirror, then reflecting it
o a second stationary mirror back into the rotating mirror, calculations can be
made on the speed of light. After first hitting the rotating mirror, the mirror
will rotate very slightly in the time it takes the beam of light to return and
will reflect back to a different position from where it came from. By measuring
the displacement of the round trip, a measurement of the speed of light can be
made.

sampterson