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When a page, represented by the interval $[0,1],$ is folded right over left $ n$ times, the right-hand fold contains a sequence of points. We specify these points and the order in which they appear in each fold. We also determine exactly where in the folded structure any point in $[0,1]$ appears and, given any point on the bottom line of the structure, which point lies at each level above it.

=b(i+1) c bi c:Thisrepresentsthedif-ference sequence for the integer parts sequence. It was shown that the graph ofM(d)againstdexhibitedaMoir epatternandthatunexpectedlythispatternwasobtainedbysimplyfoldingthefractionalpartsgraphaboutitsmiddle.Thispaperexaminestheself-similaritywithinthepaperfoldingsequenceandreveals yet another interesting pattern within the graph of a paperfolding M(d)againstd:Weshowthatthegraphformshorizontalbands.2. THE PAPERFOLDING OPERATIONTherehavebeenmanystudiesonthepaperfoldingsequence,S=11011001110:::;sincetheseminalpaperbyDavisandKnuth[2]. Itisbasedonthefollowingsimpleoperation: repeatedlyfoldapieceofpaper,rightoverleft,itimes. Whenunfolded,the paper contains v-shaped and inverted v-shaped creases. If we represent a v-shapebya1andaninvertedv-shapebya0;weobtainthefollowingpaperfoldingsubsequenceafterifolds(containing2

Links between the Calkin–Wilf tree and the Stern–Brocot tree are discussed answering the questions: What is the jth vertex in the nth level of the Calkin–Wilf tree? and Where is the vertex rs located in the Calkin–Wilf tree? A simple mechanism is described for converting the jth vertex in the nth level of the Calkin–Wilf tree into the jth entry in the nth level of the Stern–Brocot tree. We also provide a simple method for evaluating terms in the Hyperbinary sequence thus answering a challenge raised in Quantum in September 1997. We also examine successors and predecessors in both trees.

In this paper we discover an efficient method for answering two related questions involving the Stern–Brocot tree: What is the jth term in the nth level of the tree? and What is the exact position of the fraction rs in the tree?

Three equivalent methods of generating the paperfolding sequence are presented as well as a categorization of runs of identical terms. We find all repeated subsequences, the largest repeated subsequences and the spacing of singles, doubles and triples throughout the sequence. The paperfolding sequence is shown to have links to the Binary Reflected Gray Code and the Stern-Brocot tree.

With α irrational the graph of [jα] against integer j, displays interesting patterns of self-matching. This is best seen by comparing the Bernoulli (characteristic Sturmian) or difference sequence 〈βj〉, term by term with the Bernoulli sequence displaced by k terms 〈βj−k〉, where βj=[(j+1)α]−[jα].It is shown that the fraction of such self-matching is the surprisingly simple M(k)=max(|1−2{α}|,|1−2{kα}|).Of particular interest is the graph of M(k) against k as it is seen to exhibit an unexpected Moiré pattern obtained simply by folding the lower half of the graph of {kα} over the upper half.

The binary reflected Gray code function b is defined as follows: If m is a nonnegative integer, then b(m) is the integer obtained when initial zeros are omitted from the binary reflected Gray code of m. This paper examines this Gray code function and its inverse and gives simple algorithms to generate both. It also simplifies Conder's result that the jth letter of the kth word of the binary reflected Gray code of length n is 2^n-2^n^-^j-1@?2^n-2^n^-^j^-^1-k/2@?mod2by replacing the binomial coefficient by k-12^n^-^j^+^1+12.

The graph of [ jτ ] against j displays self matching in that if we displace this graph by a distance of F i , then it is found that the displaced graph matches the original graph except at certain isolated points represented by an interesting Fibonacci function. From this it is shown that the frequency of mismatches is the unexpectedly simple expression 1/( τ i ). The results are proved using lemmas, based on Zeckendorf sums, which have an appeal of their own. These also give simplified solutions to the recurrence of Downey and Griswold. Similar results apply with the Golden Sequence whose jth term is [( j +1) τ ]−[ jτ ].

“If a tree puts forth a new branch after one year, and always rests for a year, producing another new branch only in the year following, and if the same law applies to each branch, then, in the first year we should have only the principal shoot, in the second - two branches, in the third - three, then 5, 8, 13”.

Recently Tognetti and Winley have extended the age dependent integral formulation of demography to account for position dependence. This extended integral model may be used to describe the one dimensional growth of a column of fissioning cells such as in a root. An input to this model is a growth function which characterizes the age, position, and time dependent growth of an average cell. This paper establishes a growth function which is appropriate for a dividing cell in which fission is merely the formation ofa new cell wall. Also this growth function is reconciled with the mean length of dividing mother cells at the time of birth of the average cell.

The age and position dependent integral formulation of a growing column of cells has been developed to approximate the growth of a root or a thin leaf. This is one of the few reported examples of a one dimensional Lagrangian flow system. Using this as a basis, the Eulerian and Lagrangian continuity equations are compared and contrasted. An attempt is made to rationalize and clarify some of the terminology used in fluid mechanics to make it understandable to a non expert wishing to model in this area. It is found that the continuity equations discard a great deal of information and are unable to account for any age structure in the process.

The two stage model in which an organism may exist for a constant stage as an egg before it becomes an adult is extended to a stochastic model. In the case where the birth and mortality rates are age independent constants, expressions are developed for the time dependent probabilities, variance and extinction probabilities.

Attempts have been made to approximate the solutions of differential-difference equations using a Taylor series expansion of the lag term and ignoring high order derivatives. It is demonstrated that such a technique may lead to serious errors and that it is, perhaps, easier and certainly more valid to apply numerical methods directly.

The long term asymptotic behaviour of a population is evaluated where the age specific fertility behaviour is allowed to change with time. Thus in this article the behaviour of a population is determined with a time dependent net maternity function. It is shown that methods used when the net maternity function was independent of time are still applicable if the change with time is explicit only for the initial population. Further, using the fact that for realistic situations the net maternity function is non-zero over a finite interval α < x < β, it is shown that traditional methods can again be used if the time dependence is associated with ages less than α, the minimum age of childbearing. Recent extensions of Cerone and Keane to include exponential time dependence are utilized and models are presented which are piecewise defined, allowing general and exponential time dependence for the parent and new-born populations respectively. The Sharpe-Lotka single sex determinstic population model is used as the basis for the analysis.

In studying the coupled differential equations for the moments of a stochastic process it is often found that the equation for the j th moment involves higher moments. The usual methods of “decoupling” such a system of equations to obtain estimates of the moments are surveyed and shown generally to result in a system of nonlinear simultaneous differential equations which may be readily solved by numerical methods.Often, estimates of the first and second moments are the main concern. In this case, two further assumptions reported in the literature can be used to simplify the system and avoid the expense of solving the nonlinear equations. These two techniques are evaluated and compared with a new technique. Two processes are analysed, one representing a chemical reaction and the other population growth.

A catastrophe which eliminates a fraction of an age group in a population is investigated, and the asymptotic reduction in the population is obtained. It is shown that the loss of a youthful group is ultimately more damaging than the loss of an older group. An investigation of the transients following a catastrophe shows that they are reduced to less than 1% of the steady state after a small number of generations.

The integral equation used by demographers to describe age dependence in population modeling is extended to account for position dependence. Hence a model is developed which can account for fissioning in a column of cells in which the rates depend on the position of the cells in the column as well as their ages. The integral model gives a basic formulation for describing the one dimensional growth that characterizes a root or thin leaf.

If an organism may exist for a random period as an egg before it becomes an adult, and if the egg and adult mortality, as well as the adult fecundity, are age specific, then the population of eggs and adults as well as their birth rate may be represented in terms of four coupled Volterra integral equations. The one stage integral model is also reviewed and some new results reported.

If an organism may exist for a constant period as an egg before it becomes an adult and if the birth rate per individual and the death rate of an egg and adult are constants, then the number of adults can be represented by the differential delay equation N (t)=γN(t−1)−βN(t) . It is shown that this equation behaves like the Malthusian equation with large t where γ corresponds to the Malthusian birth rate and β corresponds to the Malthusian death rate. With the other parameters held constant the population will inevitably die out if the egg stage is increased beyond a critical value. An expression for the age density function is developed that is shown to asymptotically approach a negative exponential independent of time.

We use child’s addition and cross-differencing to discover significant relationships for diagonals, paths and branches within the Stern–Brocot tree and the Stern–Brocot sequence. This allows us to develop results for continued fraction summation under child’s addition.

A stable population, such that the total birthrateB(t) =Boerot, is abruptly altered by modifying the age-specific birth rate,m(x). The survivor function remains unaltered. The modified population ultimately settles down to a stable behavior, such thatB(t) =B1er1t. It is shown thatB1/B0 = (R0 −R1)/[(r0 −r1)R0Z1], whereR0,R1 are the net reproduction rates before and after the change, and\(\bar Z_1 \) expected age giving birth for the stable population after the change. The age structure and transients resulting from the change are also described. The effect of an abrupt change in the survivor functionl(x) is also investigated for the simple case where the change is caused by alteringl(x) toe−λxl(x). It is shown that the above ratio becomes\(B_1 /B_0 = N_1 /N_0 = [1 - \smallint _0^\infty e^{ - kx} g(x)dx]/\bar Z_1 \lambda \), whereN refers to the numbers in the population,k =r0 + λ, andg(x) =m(x)l(x), the value before the change. A measure for the reproductive worth of the population is also established.