Showing papers on "Fibonacci number published in 1975"

Some groups of Fibonacci type

Abstract: The recent resurgence of interest in the groups introduced in Conway (1967), together with their analogues, is recorded in Johnson, Wamsley and Wright (to appear). Let y be the automorphism of the free group Fn = induced by permutation of subscripts in accordance with the cycle y = (l2---n)eSn. Given any word w e F n , w e define Gn(w) to be the group with generators xu---,xn and relators w f , 0 g i ^ n — 1. By taking w = x1x2xf , we obtain the groups studied in Conway (1967), while those of Johnson, Wamsley and Wright (to appear) and Campbell and Robertson (to appear) are given by w = xxx2 ••• xrxr+i and w = X]X2 •••xrx~+k respectively (all subscripts being reduced modulo n to lie in the set {1,2, •••,«}). Dunwoody (to appear) and Mawdesley (1973) are concerned with the cases w = x2xnx~ 1 and w = x^jX^ respectively. In this article, which is largely a paraphrase of Mawdesley (1973), we give a description of these groups for all values of i, j with n S 6.

A simplified recombination scheme for the Fibonacci buddy system

Abstract: A simplified recombination scheme for the Fibonacci buddy system which requires neither tables nor repetitive calculations and uses only two additional bits per buffer is presented.

A note on Fibonacci type groups

Abstract: Let F n be the free group on {ai:i ∊ ℤ n} where the set of congruence classes mod n is used as an index set for the generators. The permutation (1, 2, 3, …, n) of ℤn induces an automorphism θ of F n by permuting the subscripts of the generators. Suppose w is a word in Fn and let N(w) denote the normal closure of {wθi-1:l ≤i≤n}. Define the group Gn(w) by Gn(w)=Fn/N(w) and call wdi-1=l the relation (i) of Gn(w).

Linear recurrences and uniform distribution

Abstract: A necessary and sufficient condition is obtained for the uniform distribution modulo p of a sequence of integers satisfying a linear recurrence relation. Let A = la I' be an infinite sequence of integers. For integers n n =1 m > 2 and r, let A(N, r, m) denote the number of terms a such that n 2. Kuipers, Niederreiter, and Shiue [ 1], [ 21, ['41 have proved that the Fibonacci numbers are uniformly distributed modulo m only for m = 5k, and that the Lucas numbers are not uniformly distributed modulo m for any m > 2. Both the Lucas and Fibonacci numbers satisfy the linear recurrence xn+2 = xn+l + xn. In this note we consider the uniform distribution of an arbitrary linearly recurrent sequence of integers. Theorem 1. Let XI= IxnIn= be a sequence of integers satisfying the linear recurrence xn+2 = ax +1 + bx . Let p be an odd prime. Then the sequence X is uniformly distributed modulo p if and only if p l(a2 + 4b), p{a, and p{ (2x2 ax1). The sequence X is uniformly distributed modulo 2 if and only if 21a,2{b, and 2{(x2 -x1) Proof. The linearly recurrent sequence X is periodic modulo p. If the period of X is not divisible by p, then X is certainly not uniformly distributed modulo p. Zierler [51 showed that if p{ (a2 + 4b), then the period of X is relatively prime to p. If pI(a2 + 4b) and pla, then plb, and so xn 0 (mod p) for all n > 3. If pI(a2+ 4b) and p t a, then Presented to the Society, January 16, 1974 under the title Uniform distribution and linear recurrences; received by the editors February 4, 1974. AMS (MOS) subject classifications (1970). Primary 10A35, 10F99.

Euclid, Fibonacci, and Pascal--Recursed!.

Abstract: Selected algorithms of three famous mathematicians are expressed in recursive computer programmes using a programming language (APL). Specifically: (a) Euclid's greatest common divisior algorithm; (b) Fibonacci's ‘ golden ‘ number series; (c) Pascal's triangle. The programmes are designed to be viewed directly by students in order to elucidate the algorithms. Such programmes can be stimulants for learning mathematics.

On Spanning Trees, Weighted Compositions, Fibonacci Numbers, and Resistor Networks

Abstract: Techniques for enumerating the spanning trees of graphs and multigraphs are presented which involve weighted sums of the spanning forests of certain subgraphs and which both borrow from and contribute to the theories of weighted compositions and Fibonacci numbers, with supplemental reference to electrical resistor network concepts.

Combinatorial relations in the EEG spectrum

Abstract: The mean geometric frequency f′1 of each EEG wave divides its frequency band B=f2−f1 into subbands b and s. For the β wave the values of s, b, and B are elements of a geometric progression with denominator equal to the invariant F of the golden section. It is postulated that all EEG waves are characterized by an SG system of recurrent equations, obtained by combinatorial generalization of a Fibonacci generating function. Theoretical invariants of the SG system coincide with experimental b/s ratios with a standard error of 1%. The SG system predicts the existence of ρ and σ EEG waves (55–118 and 118–225 Hz), which have not yet been found experimentally.