Department of Mathematics and Statistics
http://hdl.handle.net/10680/289
2019-09-18T08:03:38ZThe Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem
http://hdl.handle.net/10680/1728
The Brachistochrone Problem: Mathematics for a Broad Audience via a Large Context Problem
Babb, Jeff; Currie, James
Large context problems (LCP) are useful in teaching the history of science. In this article we consider the brachistochrone problem in a context stretching from Euclid through the Bernoullis. We highlight a variety of results understandable by students without a background in analytic geometry. By a judicious choice of methods and themes, large parts of the history of calculus can be made accessible to students in Humanities or Education.
2008-01-01T00:00:00ZThe number of order–preserving maps of fences and crowns (Preprint)
http://hdl.handle.net/10680/1724
The number of order–preserving maps of fences and crowns (Preprint)
Currie, James; Visentin, Terry I.
We perform an exact enumeration of the order-preserving maps of fences (zig-zags) and crowns (cycles). From this we derive asymptotic results.
1991-06-01T00:00:00ZCounting endomorphisms of crown-like orders
http://hdl.handle.net/10680/1723
Counting endomorphisms of crown-like orders
Currie, James D.; Visentin, Terry I.
The authors introduce the notion of crown-like orders and introduce powerful tools for counting the endomorphisms of orders of this type.
2002-12-01T00:00:00ZSuffix conjugates for a class of morphic subshifts
http://hdl.handle.net/10680/1703
Suffix conjugates for a class of morphic subshifts
Currie, James D.; Rampersad, Narad; Saari, Kalle
Let A be a finite alphabet and f: A^* --> A^* be a morphism with an iterative fixed point f^\omega(\alpha), where \alpha{} is in A. Consider the subshift (X, T), where X is the shift orbit closure of f^\omega(\alpha) and T: X --> X is the shift map. Let S be a finite alphabet that is in bijective correspondence via a mapping c with the set of nonempty suffixes of the images f(a) for a in A. Let calS be a subset S^N be the set of infinite words s = (s_n)_{n\geq 0} such that \pi(s):= c(s_0)f(c(s_1)) f^2(c(s_2))... is in X. We show that if f is primitive and f(A) is a suffix code, then there exists a mapping H: calS --> calS such that (calS, H) is a topological dynamical system and \pi: (calS, H) --> (X, T) is a conjugacy; we call (calS, H) the suffix conjugate of (X, T). In the special case when f is the Fibonacci or the Thue-Morse morphism, we show that the subshift (calS, T) is sofic, that is, the language of calS is regular.
2015-09-01T00:00:00ZA Characterization of Fractionally Well-Covered Graphs
http://hdl.handle.net/10680/1701
A Characterization of Fractionally Well-Covered Graphs
Currie, James; Nowakowski, Richard
A graph is called well-covered if every maximal independent set has the same size. One generalization of independent sets in graphs is that of a fractional cover -- attach nonnegative weights to the vertices and require that for every vertex the sum of all the weights in its closed neighbourhood be at least 1. In this paper we consider and characterize fractionally well-covered graphs.
1991-01-01T00:00:00ZAvoiding Patterns in the Abelian Sense
http://hdl.handle.net/10680/1699
Avoiding Patterns in the Abelian Sense
Currie, J.; Linek, V.
We classify all 3 letter patterns that are avoidable in the abelian sense. A short list of four letter patterns for which abelian avoidance is undecided is given. Using a generalization of Zimin words we deduce some properties of ω-words avoiding these patterns.
2001-08-01T00:00:00ZWords without Near-Repetitions
http://hdl.handle.net/10680/1697
Words without Near-Repetitions
Currie, J.; Bendor-Samuel, A.
We find an infinite word w on four symbols with the following property: Two occurrences of any block in w must be separated by more than the length of the block. That is, in any subword of w of the form xyx, the length of y is greater than the length of x. This answers a question of C. Edmunds connected to the Burnside problem for groups.
1992-06-01T00:00:00ZA direct proof of a result of Thue
http://hdl.handle.net/10680/1696
A direct proof of a result of Thue
Currie, James D.
1984-01-01T00:00:00ZThe Complexity of the Simplex Algorithm
http://hdl.handle.net/10680/1695
The Complexity of the Simplex Algorithm
Currie, James
The thesis begins by giving background in linear programming and Simplex methods. Topics covered include the duality theorem, Lemke's algorithm, and the pathological programs of Klee-Minty.
Because of the bad behaviour of Klee-Minty programs, the behaviour of the Simplex algorithm is only good on average. To take such an average, certain assumptions on the distribution of linear programs are introduced and discussed.
A geometrical meaning is given for the number of steps Lemke's algorithm takes to solve a program. This gives rise to a formula bounding the average number of steps taken. This formula is heuristically justified in an original way.
The formula is combinatorially simplified, to get a bound on the complexity of Simplex.
1984-08-01T00:00:00ZClass Numbers and Biquadratic Reciprocity
http://hdl.handle.net/10680/1694
Class Numbers and Biquadratic Reciprocity
Williams, Kenneth S.; Currie, James D.
The research of the first author was supported by Natural Sciences and Engineering Research Council Canada Grant No. A-7233, while that of the second was supported by a Natural Sciences and Engineering Research Council Canada Undergraduate Summer Research Award.
1982-01-01T00:00:00Z