Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.Comment: 8 pages, 3 figures, to appear in the proceedings of Integer points in polyhedra, June 11 -- June 15, 2006, Snowbird, U

Motivating Employees in R&D

[Excerpt] A new medicine can take as long as 15 years to develop and may cost a pharmaceutical research company $1.3 billion or more from the laboratory to the pharmacy shelf. The research environment is very different from most other jobs for a host of reasons: the high degree of uncertainty in the research process, the accessibility of individual contributions, and the unpredictable impact of any given final product. As such, the practices employed by pharmaceutical companies to reward and recognize employees in research and development (R&D) functions must reflect these challenges. This report will highlight extrinsic and intrinsic motivators thought to drive innovative behavior. This report will also present additional factors that managers should consider in the design and allocation of rewards and recognition schemes. Lastly, the research offers the best practices of other companies in related industries

On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect

We investigate the problem of computing tensor product multiplicities for complex semisimple Lie algebras. Even though computing these numbers is #P-hard in general, we show that if the rank of the Lie algebra is assumed fixed, then there is a polynomial time algorithm, based on counting the lattice points in polytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based on the ellipsoid algorithm, to decide when the coefficients are nonzero in polynomial time for arbitrary rank. Our experiments show that the lattice point algorithm is superior in practice to the standard techniques for computing multiplicities when the weights have large entries but small rank. Using an implementation of this algorithm, we provide experimental evidence for conjectured generalizations of the saturation property of Littlewood--Richardson coefficients. One of these conjectures seems to be valid for types B_n, C_n, and D_n.Comment: 21 pages, 6 table

Vertices of Gelfand-Tsetlin Polytopes

This paper is a study of the polyhedral geometry of Gelfand-Tsetlin patterns arising in the representation theory \mathfrak{gl}_n \C and algebraic combinatorics. We present a combinatorial characterization of the vertices and a method to calculate the dimension of the lowest-dimensional face containing a given Gelfand-Tsetlin pattern. As an application, we disprove a conjecture of Berenstein and Kirillov about the integrality of all vertices of the Gelfand-Tsetlin polytopes. We can construct for each $n\geq5$ a counterexample, with arbitrarily increasing denominators as $n$ grows, of a non-integral vertex. This is the first infinite family of non-integral polyhedra for which the Ehrhart counting function is still a polynomial. We also derive a bound on the denominators for the non-integral vertices when $n$ is fixed.Comment: 14 pages, 3 figures, fixed attribution

Lattice-point generating functions for free sums of convex sets

Let \J and \K be convex sets in $\R^{n}$ whose affine spans intersect at a single rational point in \J \cap \K, and let \J \oplus \K = \conv(\J \cup \K). We give formulas for the generating function {equation*} \sigma_{\cone(\J \oplus \K)}(z_1,..., z_n, z_{n+1}) = \sum_{(m_1,..., m_n) \in t(\J \oplus \K) \cap \Z^{n}} z_1^{m_1}... z_n^{m_n} z_{n+1}^{t} {equation*} of lattice points in all integer dilates of \J \oplus \K in terms of \sigma_{\cone \J} and \sigma_{\cone \K}, under various conditions on \J and \K. This work is motivated by (and recovers) a product formula of B.\ Braun for the Ehrhart series of \P \oplus \Q in the case where $\P$ and \Q are lattice polytopes containing the origin, one of which is reflexive. In particular, we find necessary and sufficient conditions for Braun's formula and its multivariate analogue.Comment: 17 pages, 2 figures, to appear in Journal of Combinatorial Theory Series

The Fishes of Chadron Creek, Dawes County, Nebraska

This first modern comprehensive survey of fishes collected from Chadron Creek, Dawes County, Nebraska, documents collections made with a small seine and backpack electrofisher during November 2007 and February and March 2008. Chadron Creek’s fish community is of low diversity. The total of 3 collections at each of 9 stations along the length of Chadron Creek resulted in 254 individual fishes, which represented only 7 species within 4 families. Water quality parameters, including dissolved oxygen, pH, conductivity, total dissolved solids, temperature and fecal coliform counts indicate that Chadron Creek is a healthy stream capable of supporting a greater diversity of fishes. Land management practices may be responsible for elevated fecal coliform levels at one locality on the creek. Comparisons of fishes collected herein are made with historical records of fish collected between 1893 and 2000, and show that there are 50% fewer species present than those known from historical accounts

Enumerating Segmented Patterns in Compositions and Encoding by Restricted Permutations

A composition of a nonnegative integer (n) is a sequence of positive integers whose sum is (n). A composition is palindromic if it is unchanged when its terms are read in reverse order. We provide a generating function for the number of occurrences of arbitrary segmented partially ordered patterns among compositions of (n) with a prescribed number of parts. These patterns generalize the notions of rises, drops, and levels studied in the literature. We also obtain results enumerating parts with given sizes and locations among compositions and palindromic compositions with a given number of parts. Our results are motivated by "encoding by restricted permutations," a relatively undeveloped method that provides a language for describing many combinatorial objects. We conclude with some examples demonstrating bijections between restricted permutations and other objects.Comment: 12 pages, 1 figur

Learning from Los Curries

Photography culture is a major part of the tourist experience, whether it is the drunken selfie posted in a chatroom, or the carefully composed street scene published in an upmarket guide book. The pictures shown here, attempt to respond critically to the different dialectics of the camera, bodies, architecture, and light that prevail in the nightime in Magaluf and Palma. These images also aim to transgress the boundaries between these places’ different visual cultures, because they seem outmoded in the way they socially categorise tourists, in what on closer inspection, is evidently becoming part of a more ‘liquid modernity’ than the one still being portrayed (Bauman) (1). This paper expands upon themes raised in a short article published in 2015 (2). 1. Bauman, Z. Culture in a Liquid Modern World. Cambridge: Polity Press; 2011 2. Stringer, B & McAllister J. Guided by the Lights in Magaluf and Palma, Architecture and Culture, 2:3, 417-426; 201