Paul Cull

The Mobius cubes

The Mobius cubes are hypercube variants that give better performance with the same number of links and processors. We show that the diameter of the Mobius cubes is about one half the diameter of the equivalent hypercube, and that the average number of steps between processors for a Mobius cube is about two-thirds of the average for a hypercube. We give an efficient routing algorithm for the Mobius cubes. This routing algorithm finds a shortest path and operates in time proportional to the dimension of the cube. We also give efficient broadcast algorithms for the Mobius cubes. We show that the Mobius cubes contain ring networks and other networks. We report results of simulation studies on the dynamic message-passing performance of the hypercube, the Twisted Cube of P.A.J. Hilbers et al. (1987), and the Mobius cubes. Our results are in agreement with S. Abraham (1990), showing that the Twisted Cube has worse dynamic performance than the hypercube, but our results show that the 1-Mobius cube has dynamic performance superior to that of the hypercube. This contradicts current literature, which implies that twisted cube variants will have worse dynamic performance. >

Global stability of population models

Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.

Local and global stability for population models

In general, local stability does not imply global stability. We show that this is true even if one only considers population models.We show that a population model is globally stable if and only if it has no cycle of period 2. We also derive easy to test sufficient conditions for global stability. We demonstrate that these sufficient conditions are useful by showing that for a number of population models from the literature, local and global stability coincide.We suggest that the models from the literature are in some sense “simple”, and that this simplicity causes local and global stability to coincide.

Mathematical analysis of the asymptotic behavior of the leslie population matrix model

Let L be a Leslie population matrix. Leslie (1945) and others have shown that the matrix L has a leading positive eigenvalue λ0 and that in general: lim ⁡ t → ∞ L t X λ 0 t = γ X λ 0 , ( 1 ) where X λ 0 is an eigenvector corresponding to λ0, X is any initial population vector, and γ is a scalar quantity determined by X. In this article we generalize (1) exhaustively by removing the mild restrictions on the fertility rates which most writers impose. The result is an oscillatory limit of a kind first noted by Bernardelli (1941) and Lewis (1942) and described by Bernardelli as “population waves”. We calculate in terms of λ0 and the entries of the matrix L the values of this oscillatory limit as well as its time-independent average over one period. This calculation includes as its leading special case the result of (1), confirming incidentally that γ is non-zero. To stabilize a population, the matrix L must be adjusted so that λ0 = 1. The limits calculated for the oscillatory and non-oscillatory cases then have maximum significance since they represent the limiting population vectors. We discuss a simple scheme for accomplishing stabilization which yields as a byproduct an easily accessible scalar measure of Ls tendency to promote population growth. The reciprocal of this measure is the familiar net reproduction rate.

Error-correcting codes on the towers of Hanoi graphs

Abstract A perfect one-error-correcting code on a graph is a subset of the vertices so that no two vertices in the subset are adjacent and each vertex not in the subset is adjacent to exactly one vertex in the subset. We show that the Towers of Hanoi puzzle defines an infinite family of graphs, and that each such graph supports a perfect one-error-correcting code. We show that these codes are essentially unique. Our characterization of the codewords as those ternary strings with an even number of 1s and an even number of 2s, makes generation and decoding computationally easy. In particular, decoding can be carried out by a two-pass finite state machine. We also show that determining if a graph can support a perfect one-error-correcting code is an NP-complete problem.

Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.

The periodic limit for the Leslie model

Abstract If adjacent fertility rates in the Leslie population model are not assumed to be strictly positive, population distributions need not converge to the so-called stable age distribution. Instead the asymptotic behavior of a distribution may be periodic, taking a form which Bernardelli [1] called “population waves.” Here we state and interpret the main theorems describing this phenomenon and discuss objections which stand in the way of applying it to population studies.

The mathematical biophysics of Nicolas Rashevsky.

N. Rashevsky (1899-1972) was one of the pioneers in the application of mathematics to biology. With the slogan: mathematical biophysics : biology :: mathematical physics ; physics, he proposed the creation of a quantitative theoretical biology. Here, we will give a brief biography, and consider Rashevskys contributions to mathematical biology including neural nets and relational biology. We conclude that Rashevsky was an important figure in the introduction of quantitative models and methods into biology.

On generalized twisted cubes

Abstract We briefly survey some of the results on twisted cubes which have been published in the last decade. We point out that a number of n-dimensional twisted cubes with diameter about n 2 are known. These twisted cubes are better than the recent ⌈ 2n 3 ⌉ diameter twisted cube given by F. Chedid and R. Chedid (1993). We discuss embedding other networks within twisted cubes and point out an error in one supposed embedding given by Chedid and Chedid.

Exact learning of unordered tree patterns from queries

We consider learning tree patterns from queries extending our preceding work [Amoth, Cull, & Tadepalli, 1998]. The instances in this paper are unordered trees with nodes labeled by constant identifiers. The concepts are tree patterns and unions of tree patterns (unordered forests) with leaves labeled with constants or variables. A tree pattern matches any tree with its variables replaced with constant subtrees. A negative result for learning with equivalence and membership/subset queries is shown for unordered trees where a successful match requires the number of children in the pattern and instance to be the same. Unordered trees and forests are shown to be learnable with an alternative matching semantics that allows an instance to have extra children at each node.