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If P ⊂ R d is a rational polytope, then iP (n) := #(nP ∩ Z d) is a quasi-polynomial in n, called the Ehrhart quasi-polynomial of P. The period of iP (n) must divide D(P) = min{n ∈ Z>0 : nP is an integral polytope}. Few examples are known where the period is not exactly D(P). We show that for any D, there is a 2-dimensional triangle P such that D(P) = D but… (More)

We examine two different ways of encoding a counting function, as a rational generating function and explicitly as a function (defined piecewise using the greatest integer function). We prove that, if the degree and number of input variables of the (quasi-polynomial) function are fixed, there is a polynomial time algorithm which converts between the two… (More)

A quasi-polynomial is a function defined of the form q(k) = c d (k) k d + c d−1 (k) k d−1 + · · · + c0(k), where c0, c1,. .. , c d are periodic functions in k ∈ Z. Prominent examples of quasi-polynomials appear in Ehrhart's theory as integer-point counting functions for rational polytopes, and McMullen gives upper bounds for the periods of the cj (k) for… (More)

We describe the first implementation of the Barvinok–Woods (2003) algorithm, which computes a short rational generating function for an integer projection of the set of integer points in a polytope in polynomial time, when the dimension is fixed. The algorithm is based on Kannan's partitioning lemma and the application of set operations to generating… (More)

If P ⊂ R d is a rational polytope, then i P (t) := #(tP ∩ Z d) is a quasi-polynomial in t, called the Ehrhart quasi-polynomial of P. A period of i P (t) is D(P), the smallest D ∈ Z + such that D · P has integral vertices. Often, D(P) is the minimum period of i P (t), but, in several interesting examples, the minimum period is smaller. We prove that, for… (More)

- Kevin M Woods, Alexander Barvinok, Richard Canary, Sergey Fomin, John Stembridge, Satyanarayana Lokam +10 others
- 2004

ACKNOWLEDGEMENTS My thanks to the many people whose thoughts have contributed to this thesis and to my mathematical development, including In particular, my collaborations with Tyrrell McAllister and Herb Scarf have been tremendously invaluable. Many thanks to my doctoral committee, especially John Stembridge for his careful reading of this thesis. I am… (More)

Recombination is an important event in the evolution of HIV. It affects the global spread of the pandemic as well as evolutionary escape from host immune response and from drug therapy within single patients. Comprehensive computational methods are needed for detecting recombinant sequences in large databases, and for inferring the parental sequences. We… (More)

A rational polytope is the convex hull of a finite set of points in R d with rational coordinates. Given a rational polytope P ⊆ R d , Ehrhart proved that, for t ∈ Z ≥0 , the function #(tP ∩ Z d) agrees with a quasi-polynomial L P (t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a… (More)

Given a 1 , a 2 ,. .. , a n ∈ Z d , we examine the set, G, of all non-negative integer combinations of these a i. In particular, we examine the generating function f (z) = b∈G z b. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the… (More)

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