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 B_{r} 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. InMathematical 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/c^{2}, 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 h_{0}, h_{1}, h_{2},
... . Firstly h_{0} = 1, then

Can anyone solve for h_{n} ? 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 Biology122 ,205-224 (1985).

Here, cessation of division is modelled by a sequence of numbers
m_{0}, m_{1}, m_{2}, ... , where m_{n}
:= the expected number of dividing daughters for a mother belonging
to generation n. The generation time distributions are a sequences
of functions F_{0}, F_{1}, F_{2}, ...
, where F_{n} 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 Proliferation28,373-391 (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,
G^{n} 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 F_{0},
F_{1}, F_{2}, ... . So there is not enough information
in the data to estimate these distributions. We can, however,
estimate the m_{n} sequence and the means of the F_{n}
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 Kinetics20153-159 (1987).

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