MA-351 Homework 1
Due at 4:59pm in my mailbox in HA 245, Monday, September 11, 2006
Calculations necessary for these problems may be done either by
hand or with Maple.
Solutions may be submitted in person in class,
or you may email an ASCII text,
Maple Worksheet (.mws),
html, or postscipt/pdf-formatted document to
me (kaltofen@math.ncsu.edu).
Note my office hours on my schedule.
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Consider the Fibonacci numbers
f0=1, f1=1 such that
fi+2 = fi+1 + fi is satisfied
for any integer i. For i ≥ 0 we have the Fibonacci
numbers from class, but it is possible from the recursion to determine uniquely
the numbers with negative subscripts.
- Please write down the numbers f-1,
f-2, f-3, f-4,
f-5, f-6, f-7.
- If by gk we denote f-k
for k ≥ 0, please write a linear recursion
for gk+2.
- Following the procedure from class, from the recursion
given in b. and the initial values in a.,
please compute a closed form formula
for gk.
- Prove that the number of edges in the n-dimensional
hypercube is n 2n-1.
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Consider the 5-dimensional hypercube as presented in class.
In class the vertices were the integers from 1 to 32.
The task is to relabel the vertices in the drawing as 5-digit
binary numbers (sequences of 5 integers from 0 and 1)
in such a way that the Hamming distance between
adjacent vertices is exactly one.