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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)

The classic algorithms of Needleman-Wunsch and Smith-Waterman find a maximum a posteriori probability alignment for a pair hidden Markov model (PHMM). To process large genomes that have undergone complex genome rearrangements, almost all existing whole genome alignment methods apply fast heuristics to divide genomes into small pieces that are suitable for… (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)

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)

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)

Directed and undirected graphical models, also called Bayesian networks and Markov random fields, respectively, are important statistical tools in a wide variety of fields, ranging from computational biology to probabilistic artificial intelligence. We give an upper bound on the number of inference functions of any graphical model. This bound is polynomial… (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)

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)

We generalize a theorem of Nymann that the density of points in Z d that are visible from the origin is 1/ζ(d), where ζ(a) is the Riemann zeta function P ∞ i=1 1/i a. A subset S ⊂ Z d is called primitive if it is a Z-basis for the lattice Z d ∩ span R (S), or, equivalently, if S can be completed to a Z-basis of Z d. We prove that if m points in Z d are… (More)

The purpose of this study was to determine whether antiorthostatic suspension of C3HeB/FeJ mice for a period of 11 days affected macrophage and spleen cell function. We found that antiorthostatic suspension did not alter macrophage secretion of prostaglandin E2, tumor necrosis factor alpha, and interleukin-1. Antiorthostatic suspension also did not affect… (More)