Unsolved problems

1. Can someone solve this integral equation? Find a non-negative function g defined on [0,1] and a constant a satisfying

for x>0, with further conditions

Motivation for the equation, together with a series-solution and plot, is given in Section 5 of

70. Cowan, R. and Chen, F. K. C. Four interesting problems concerning Markovian shape sequences. Adv. Appl. Prob. 31, 954-968 (1999). Download post-script file.

The equivalent differential equation is also given in this paper. An analytic solution would be desirable.

2. Least dense "full" packings. Consider an infinite ensemble of equal-sized disks with diameter 1, packed together in the plane. A requirement of the packing is that each disk touches at least 4 other disks. In addition, each disk's circumference must be "full" in the sense that the angle at the disk's centre subtended by the centres of any two adjacent neighbours is less than 2p/3. Thus there is no room on the circumference for another disk. In the paper

56. Cowan, R. Constraints on the random packing of disks. J. Appl. Prob. 30, 263-268 (1993), Download post-script file

random ensembles with these properties (and statistical stationarity over the plane) are considered. Regular ensembles with a repeating sub-unit fit into this framework by ensuring that the planar origin is uniformly distributed within the area of the repeating unit.

It is shown that t, defined as the mean number of disk centres in any unit area, must be strictly greater than 0.951327 (see paper for the trigonometric expression). The regular ensemble, whereby every disk has 4 neighbours in the directions of the 4 compass points, has t = 1. Does there exist an ensemble with t < 1?

Readers may prefer to work with non-random ensembles not constrained by the statistical-stationarity condition (which implies that the mean number of disk centres in a domain is invariant under translations of that domain). If working this way, however, one must define t (and the limit of density inside the ball Br as r goes to infinity seems a likely candidate, and so this limit should exist for any example tried).

3. Growth of random nuclei in the planar lattice. Consider the regular rectangular lattice in the plane, with squares of sidelength c and all shaded white. At time zero, each square becomes coloured independently with probability p (or stays white with probability 1 - p). These initial coloured squares are called 'nuclei' and each one has its own distinctive colour. One by one, other squares adjacent to a coloured square become coloured too. In the paper

62. Lee, T. and Cowan, R. A stochastic tessellation of digital space. In Mathematical morphology and its applications to image processing, pp217-224. Eds. J. Serra and P. Soille. Kluwer Publishers (1995).

a class of random models for the choice of which square is next is given. These rules encompass many of the rules used by authors such as Eden, Williams & Bjerknes and Richardson for growth of a single nuclei. Basically each white square with at least one coloured neighbour is given an exponentially-distributed waiting time until it too becomes coloured. The means of these random variates depend on the number of coloured neighbours. Waiting times are resampled if a cell's coloured-neighbour-count rises.

The colour adopted by the "new" square is the same as one of its coloured neighbours. Usually the neighbours are all the same colour, but as clones from different nuclei grow to the point where they are nearly touching, a "new" square may have more than one colour amongst its neighbours. If so, some sensible random rule for determining the new colour is used (details unimportant here).

Eventually the "clonal" growth from a given nucleus ceases, because the clones from other nuclei have taken up all the nearby space and all holes within the clone have been filled. In the paper, beautiful pictures prepared by Thomas Lee show the resulting tessellation of the plane (with clonal cells of very irregular shape).

My questions is: can anyone describe analytically the properties of this tessellation in the limit as c -> 0 and p -> 0 (such that the mean density of nuclei, namely p/c2, remains constant)?

4. Solving an "easy" version of the Gilbert tessellation. Consider a stationary Poisson process of particles in the plane, of intensity 2l. Particles are classified by chance as "horizontal growing" or "vertical growing" (50% chance for each particle, independently). At time zero, growth of a line segment commences at each particle. Growth is bi-directional (ie both ends of the segment are growing) in the chosen horizontal or vertical direction. Each end of each segment grows at the same rate.

Growth of the line stops at one end if that end hits a line which has already grown across its path. The eventual result is a random tessellation comprising rectangular cells. Find the probability distribution of the mature length L of the line segments.

This is a very difficult problem, though not quite as bad as Gilbert's original model (where all directions of growth are allowed). There would be considerable interest in a solution of this "horizontal-vertical version".

Isaac Ma and I have some results for an even simpler model, whereby the growth of Eastward-growing lines is halted only by Southward-growing lines (and vice versa). Westward and Northward have the same reciprocity. Our result is an explicit answer if we could find a simple solution for the following recurrence relationship for the real number sequence h0, h1, h2, ... . Firstly h0 = 1, then

Can anyone solve for hn ? The sequence starts 1, 1/2, 1/3, 29/120, 11/60, ... .

5. Fat and thin rectangles. Consider the Markov process of rectangles constructed as follows. Start with a rectangle of any non-square shape. On a long side of the rectangle place a uniformly-distributed point P. Divide the rectangle into two rectangles with a line through P orthogonal to the long side. Select the thinner of the two rectangles formed as the next in sequence, "thinner" being defined as the one with the smaller shape X (see below). Iterate the process with this new rectangle.

X, the shape of a rectangle, is defined as the ratio of shorter to longer side. What is the equilibrium distribution of X on [0,1] for the shape of this "thinner-choice" sequence of rectangles.

If instead, one selects the fatter of the two rectangles after the cut, that is the rectangle with the higher shape value, the limiting shape has pdf given by
f(x) = (2 - 1/x)/(1 - log 2)        if 1/2 < x < 1,
= 0                                 otherwise.
See the paper

68. Chen, F. K. C. and Cowan, R. Invariant distributions for shapes in sequences of randomly-divided rectangles. Adv. Appl. Prob. 31, 1-14 (1999). Download post-script file

These fatter/thinner problems were both unsolved using another natural probability measure for the cutting line, namely where the cutting line is an equally-likely choice from the space of all lines which divide the rectangle into two rectangles.

6. A problem for statisticians. During development of a biological tissue, cells divide into two for a while but gradually there is cell specialisation and cessation of cell division. The generation time of cells is highly variable during this process. Also, while this growth process is happening, it is common for the probability distribution of generation times to shift; mean generation times gradually get longer. The slowing in growth of cell numbers is compounded by these two phenomena -- cessation of division and lengthening generation times.

The problem I pose is "how to sort out the roles of these compounding factors from data which can be collected on (a) the cell population size and (b) the number of non-dividing cells?".

One model is the age-dependent branching process discussed in this context in

35. Cowan, R. and Morris, V. B. Cell population dynamics during the differentiative phase of tissue development. J. Theoretical Biology 122 , 205-224 (1985).

Here, cessation of division is modelled by a sequence of numbers m0, m1, m2, ... , where mn := the expected number of dividing daughters for a mother belonging to generation n. The generation time distributions are a sequences of functions F0, F1, F2, ... , where Fn is defined as the distribution function of generation times for dividing cells of generation n. Non-dividing cells live for a variable time having d.f. G (with large mean).

Data can be collected on cell numbers at time t and also on the proportion of cells at time t which are non-dividing cells. For chick retinal tissues, such data are published in

63. Morris, V. B. and Cowan, R. An analysis of the growth of the retinal cell population in embryonic chicks yielding proliferative ratios and cell-cycle times for successive generations of cell cycles. Cell Proliferation 28, 373-391 (1995).

where there is also a fit to the branching-process model. The fitting technique involves an expressions for the mean population size at t and the expected number of non-dividing cells -- N cells -- at t (derived in paper #35). Starting with K cells in generation 0, these are:

where

and

Since N cells usually outlive the duration of the experiment, Gn can be replaced by 1 for simplicity. Now these theoretical curves, after initial wiggly behaviour, settle down to a shape which is not highly dependent on the detailed shape of F0, F1, F2, ... . So there is not enough information in the data to estimate these distributions. We can, however, estimate the mn sequence and the means of the Fn distributions. Paper #63 does this in a trial-and-error way. Can the fitting procedure be done automatically?

Footnote: The same problem is approached in another way, where the rates of cells producing "dividing" and "non-dividing" daughters are modelled as functions of "time" (not functions of "generation"), in

38. Cowan, R. and Morris, V. B. Determination of proliferative parameters from growth curves. Cell &Tissue Kinetics 20 153-159 (1987).

7. A problem concerning generalisations of the Borel probability distribution is given on another page (link).

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