Thesis (Ph. D.)--University of North Texas, May, 2007. / Title from title page display. Includes bibliographical references.

Thesis (Honors)--Liberty University Honors Program, 2007. / Includes bibliographical references. Also available through Liberty University's Digital Commons.

Treatise (D.M.A.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Thesis (M.S.)--Rochester Institute of Technology, 2008. / Typescript. Includes bibliographical references (leaf 17).

Early in the thirteenth century, Leonardo de Pisa, or, Fibonacci, introduced his famous rabbit problem, which may be stated simply as follows: assume that rabbits reproduce at a rate such that one pair is born each month from each pair of adults not less than two months old. If one pair is present initially, and if none die, how many pairs will be present after one year? The solution to the problem gives rise to a sequence {U_n} known as the Classical Fibonacci Sequence. {U_n} is defined by the recurrence relation U_n = U_n-1 + U_n-2, n ≥ 2, U₀ = 0, U₁ = 1 Many properties of this sequence have been derived. A generalized sequence {F_n} can be obtained by retaining the law of recurrence and redefining the first two terms as F₁ = p', F₂ = p' + q' for arbitrary real numbers p' and q'. Moreover, by defining H₁ = p+iq, H₂ = r+is, p,q,r and s real, a complex sequence is determined. Hence, all the properties of the classical sequence can be extended to the complex case. By reducing the classical sequence by a modulus m, many properties of the repeating sequence that results can be derived. The Fibonacci sequence and associated golden ratio occur in communication theory, chemistry, and in nature. / Master of Science

Published Article / In the 13th century an Italian mathematician Fibonacci, also known as Leonardo da Pisa, identified a sequence of numbers that seemed to be repeating and be residing in nature (http://en.wikipedia.org/wiki/Fibonacci) (Kalman, D. et al. 2003: 167). Later a golden ratio was encountered in nature, art and music. This ratio can be seen in the distances in simple geometric figures. It is linked to the Fibonacci numbers by dividing a bigger Fibonacci value by the one just smaller of it. This ratio seems to be settling down to a particular value of 1.618 (http://en.wikipedia.org/wiki/Fibonacci) (He, C. et al. 2002:533) (Cooper, C et al 2002:115) (Kalman, D. et al. 2003: 167) (Sendegeya, A. et al. 2007). Artificial Intelligence or neural networks is the science and engineering of using computers to understand human intelligence (Callan R. 2003:2) but humans and most things in nature abide to Fibonacci numbers and the golden ratio. Since Neural Networks uses the same algorithms as the human brain does, the aim is to experimentally proof that using Fibonacci numbers as weights, and the golden rule as a learning rate, that this might improve learning curve performance. If the performance is improved it should prove that the algorithm for neural network's do represent its nature counterpart. Two identical Neural Networks was coded in LabVIEW with the only difference being that one had random weights and the other (the adapted one) Fibonacci weights. The results were that the Fibonacci neural network had a steeper learning curve. This improved performance with the neural algorithm, under these conditions, suggests that this formula is a true representation of its natural counterpart or visa versa that if the formula is the simulation of its natural counterpart, then the weights in nature is Fibonacci values.

In this thesis we are investigating identities regarding Fibonacci sequences. In particular we are examiningthe so called Pisano period, which is the period for the Fibonacci sequence considered modulo n to repeatitself. The theory shows that it suces to compute Pisano periods for primes. We are also looking atthe same problems for the generalized Pisano period, which can be described as the Pisano period forthe generalized Fibonacci sequence.

Not available / text

Not available / text

Thesis (M.M.)--Bowling Green State University, 2009. / Document formatted into pages; contains 1 score (vi, 29 p.) For clarinet, bassoon, and chamber orchestra (two trumpets, two horns, tenor trombone, bass trombone, percussion, piano, harp, and strings (six first violins, six second violins, four violas, four cellos, and two basses) Includes bibliographical references.