Margaret Archibald

The largest missing value in a composition of an integer

In this paper we find, asymptotically, the mean and variance for the largest missing value (part size) in a composition of an integer n. We go on to show that the probability that the largest missing value and the largest part of a composition differ by one is relatively high and we find the mean for the average largest value in compositions that have this property. The average largest value of compositions with at least one non-zero missing value is also found, and used to calculate how many distinct values exceed the largest missing value on average.

Blow me down: A new perspective on Aloe dichotoma mortality from windthrow

BackgroundWindthrow, the uprooting of trees during storms associated with strong winds, is a well-established cause of mortality in temperate regions of the world, often with large ecological consequences. However, this phenomenon has received little attention within arid regions and is not well documented in southern Africa. Slow rates of post-disturbance recovery and projected increases in extreme weather events in arid areas mean that windthrow could be more common and have bigger impacts on these ecosystems in the future. This is of concern due to slow rates of post-disturbance recovery in arid systems and projected increases in extreme weather events in these areas. This study investigated the spatial pattern, magnitude and likely causes of windthrown mortality in relation to other forms of mortality in Aloe dichotoma, an iconic arid-adapted arborescent succulent and southern Africa climate change indicator species.ResultsWe found that windthrown mortality was greatest within the equatorward summer rainfall zone (SRZ) of its distribution (mean = 31%, n = 11), and was derived almost exclusively from the larger adult age class. A logistic modelling exercise indicated that windthrown mortality was strongly associated with greater amounts of warm season (summer) rainfall in the SRZ, higher wind speeds, and leptosols. A statistically significant interaction term between higher summer rainfall and wind speeds further increased the odds of being windthrown. While these results would benefit from improvements in the resolution of wind and substrate data, they do support the hypothesised mechanism for windthrow in A. dichotoma. This involves powerful storm gusts associated with either the current or subsequent rainfall event, heavy convective rainfall, and an associated increase in soil malleability. Shallow rooting depths in gravel-rich soils and an inflexible, top-heavy canopy structure make individuals especially prone to windthrown mortality during storms.ConclusionsResults highlight the importance of this previously unrecognised form of mortality in A. dichotoma, especially since it seems to disproportionately affect reproductively mature adult individuals in an infrequently recruiting species. Smaller, more geographically isolated and adult dominated populations in the summer rainfall zone are likely to be more vulnerable to localised extinction due to windthrow events.

The Largest Missing Value in a Sample of Geometric Random Variables

We consider samples of n geometric random variables with parameter 0 p n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

Infinity in Logic and Computation: International Conference, ILC 2007, Cape Town, South Africa, November 3-5, 2007, Revised Selected Papers

Edited in collaboration with FoLLI, the Association of Logic, Language and Information, this volume constitutes a selection of papers presented at the Internatonal Conference on Infinity in Logic and Computation, ILC 2007, held in Cape Town, South Africa, in November 2007. The 7 revised papers presented together with 2 invited talks were selected from 27 initial submissions during two rounds of reviewing and improvement. The papers address all aspects of infinity in automata theory, logic, computability and verification and focus on topics such as automata on infinite objects; combinatorics, cryptography and complexity; computability and complexity on the real numbers; infinite games and their connections to logic; logic, computability, and complexity in finitely presentable infinite structures; randomness and computability; transfinite computation; and verification of infinite state systems.

Variation Statistics on Compositions

In this paper we consider the absolute variation statistics of a composition σ = σ1 ··· σm of n which is a measure of the sum of absolute differences between each consecutive pair of parts in a composition. This and some related statistics which we discuss, can also be interpreted within the context of bargraph polygons and bargraph walks of area n. In this context, absolute variation corresponds to the interior vertical perimeter of a bargraph polygon or to the interior vertical length of a bargraph walk.