Showing papers on "Fibonacci number published in 1977"

Phyllotaxis and the Fibonacci Series

Abstract: The principal conclusion is that Fibonacci phyllotaxis follows as a mathematical necessity from the combination of an expanding apex and a suitable spacing mechanism for positioning new leaves. I have considered an inhibitory spacing mechanism at some length, as it is a plausible candidate. However, the same treatment would apply equally well to depletion of, or competition for, a compound by developing leaves, and could no doubt accommodate other ingredients. The mathematical principles involved are clear when it is assumed that only two leaves (the contacts) position a new leaf. There is some experimental evidence for this assumption. Nonetheless, it is not a precondition for Fibonacci phyllotaxis, since a computer model shows that this pattern is generated even when many leaves contribute to inhibition at a given point. Indeed, the Fibonacci pattern seems to be a robust and stable mathematical phenomenon, a finding which goes some way to explaining its widespread occurrence throughout the plant kingdom.

The Fibonacci groups. II

Abstract: Since the appearance of the article [15] to which this is a sequel, considerable progress has been made in the study of the groups F(r, n) of the title. It is therefore our intention to give a brief account of these developments before proceeding to our main theme, which is to apply the elegant and powerful methods of small cancellation theory to these groups. This has a variety of consequences, perhaps the most important of which is the generalisation to arbitrary r of Lyndon's proof that F(2, n) is infinite for n ≧ 11.

Method and device for reduction of fibonacci p-codes to minimal form

Abstract: An arrangement for reduction of Fibonacci p-codes to minimal form, which performs, in succession, all convolutions and all devolutions of bits of the original Fibonacci p-code of a number, whereto the original combination of binary signals corresponds. The original combination of binary signals is handled during the convolution operation so that a binary signal corresponding to a 0 value of the lth digit of the original Fibonacci p-code of the number, as well as binary signals corresponding to l values of the (l - i)th and the (l - p- 1)th digit of the original Fibonacci p-code of the number, are substituted by their inverse signals. The original combination of binary signals is handled during the devolution operation so that a binary signal corresponding to a 1 value of the lth digit of the original Fibonacci p-code of the number, as well as binary signals corresponding to 0 values of the digits (l - p) through (l - 2p), inclusive, of the original Fibonacci p-code of the number, are substituted by their inverse signals.

Analog-Fibonacci p-code converter

Abstract: An analog-Fibonacci p-code converter comprises a code readout device having a control input receiving a code readout pulse, as well as a device for realization of a Fibonacci p-code table having an input receiving an analog value being converted, the device being coupled to message inputs of the code readout device. The converter also comprises a device for conversion of the readout combination of binary signals which corresponds to a quantitized analog value into a minimal form of the Fibonacci p-code of the serial number of the quantitized analog value which is connected with the inputs thereof to the outputs of the code readout device corresponding to places of the Fibonacci p-code, beginning with the pth place, and the outputs of this device are message outputs of the analog-Fibonacci p-code converter which correspond to respective places of the Fibonacci p-code, beginning with the pth place. The device for conversion of the readout combination of binary signals comprises a plurality of similar functional stages for identification of a pair of neighboring unity places in the selected combination of binary signals. In addition, the analog-Fibonacci p-code converter comprises a device for checking the converted code combination of binary signals corresponding to a minimal form of the Fibonacci p-code. The checking device has its inputs connected to the outputs of the code readout device and of the device for conversion of the readout combination of binary signals, and has its output which is the check output of the analog-Fibonacci p-code converter.

An optimal strategy for searching the best fault-plane solution using wave-amplitude data

Abstract: This paper presents an optimum search procedure known as the Fibonacci Technique for abstracting the earthquake-source parameters from the amplitude data of seismic waves. The power of the method has been demonstrated by determining the fault-plane solution of a deep-focus earthquake using the P -wave spectral amplitude data.

Fibonacci code adder

Abstract: A Fibonacci code adder comprising an n-digit half-adder which includes at least two inputs through which addends of numbers represented in the Fibonacci code minimal form are introduced and also includes intermediate sum and carry outputs coupled to analogous inputs of a rewriting device having their intermediate sum and carry outputs coupled to analogous inputs of a Fibonacci code converter whose outputs produce the codeword of the final sum.

Cathode-ray tube display

Abstract: A cathode-ray tube display comprising a control unit whose intensity control output, horizontal sweep control output and vertical sweep control output are coupled, respectively, to the data input of an intensity control unit, to the counting input of a first Fibonacci code pulse counter, and to the counting input of a second Fibonacci code pulse counter, the output of the intensity control unit being coupled to the data input of a cathode-ray tube, data outputs of the first and second Fibonacci code pulse counters being coupled, respectively, to the inputs of first and second pulse amplifiers via first and second Fibonacci code digital-analog converters, fault acknowledgement outputs of the first and second Fibonacci code pulse counters being coupled via a checking OR gate to the fault acknowledgement input of the control unit, and transient outputs of the first and second Fibonacci code pulse counters being coupled via a blanking OR gate to the blanking input of the intensity control unit.

******* composites and primes among powers of fibonacci numbers, increased or decreased by one

Abstract: Here, we raise the question of when F™ ± 1 is composite. First, if k£0 (mod 3), then F k is odd, F™ is odd, and F™± 7 is even and hence composite. Now, suppose we deal with F™ k ± I Since A n-B n always has (A-B) as a factor, we see that F™ k-1 m is composite except when (A-B)= 1; that is, for k= 1. Thus, Theorem 1. F™-1 is composite, k ± 3. We return to F™ k + I For/77 odd, then A m + B m is known to have the factor (A + B), so that F™ k + 1 m has the factor (F^k + 11 and hence is composite. If m is even, every even m except powers of 2 can be written in the form (2j + 1)2' = m, so that F3k + 1 m = (F 2 3k) 2i+1 + (1 2) 2i+1. which, from the known factors of A m + B m , m odd, must have (Fj k + 1) as a factor, and hence, F™ k + /is composite.. This leaves only the case F™ k + /, where m = 2'. When k= 1, we have the Fermat primes,? 2 + /, prime for / = 0, 1, 2, 3, 4 but composite for / = 5, 6. It is an unsolved problem whether or not 2 2 ' + 1 has other prime values. We note in passing that, when k = 2,F 6 =8 = 2 3 , and 8 m ± 1 = (2 3) ^ ± 1 = (2 m) 3 ± 7 is always composite , since A 3 ± B is always factorable. It is th ought that F g + 1 is a prime. Since F 3k =Q (mod 10),Ar=0(mod 5) , / ^ , + / = 102'-t+1. Since F?/, = 6 (mod 10), i> 2, k^ 0 (mod 5), F 2 ' + 1 has the form 10t + 7, k4ft (mod 5). We can summarize these remarks as Theorem 2. F™ + 1 is composite, k ?3s, F™ k + 1 is composite, m =f 2'. It is worthwhile to note the actual factors in at least one case. Since Fk+2F k-2-F 2 k = (~V k+1 F k+lFk-l~ F k = (-V moving F k to the right-hand side and then …

On the Matrix Approach to Fibonacci Numbers and the Fibonacci Pseudoprimes.

Abstract: : A study of those integers n such that L sub n = 1 (mod n), where I sub n are the Lucas numbers. (Author)