Elections with Three Candidates Four Candidates and Beyond: Counting Ties in the Borda Count with Permutahedra and Ehrhart Quasi-Polynomials

In voting theory, the Borda count’s tendency to produce a tie in an election varies as a function of n, the number of voters, and m, the number of candidates. To better understand this tendency, we embed all possible rankings of candidates in a hyperplane sitting in m-dimensional space, to form an (m - 1)-dimensional polytope: the m-permutahedron. The number of possible ties may then be determined computationally using a special class of polynomials with modular coefficients. However, due to the growing complexity of the system, this method has not yet been extended past the case of m = 3. We examine the properties of certain voting situations for m ≥ 4 to better understand an election’s tendency to produce a Borda tie between all candidates

トーリック曲面上のある特異曲線に対応するトロピカル曲線について

Tohoku University石川昌治課

Convex Polytopes and Enumeration

AbstractThis is an expository paper on connections between enumerative combinatorics and convex polytopes. It aims to give an essentially self-contained overview of five specific instances when enumerative combinatorics and convex polytopes arise jointly in problems whose initial formulation lies in only one of these two subjects. These examples constitute only a sample of such instances occurring in the work of several authors. On the enumerative side, they involved ordered graphical sequences, combinatorial statistics on the symmetric and hyperoctahedral groups, lattice paths, Baxter, André, and simsun permutations,q-Catalan andq-Schröder numbers. From the subject of polytopes, the examples involve the Ehrhart polynomial, the permutohedron, the associahedron, polytopes arising as intersections of cubes and simplices with half-spaces, and thecd-index of a polytope

Controlling wildlife damage by diffusing beaver population : a bioeconomic application of the distributed parameter control model

The beaver population in the Southeastern United States has caused severe damage to valuable timber land through dam-building and flooding of bottom-land forest. Traditionally, beavers have been trapped by small group of people as a source of their livelihood. The low pelt price in the recent years has failed to stimulate adequate trapping pressure, and thus, resulted in increased beaver population and damage losses. The low trapping pressure has left the burden of nuisance control on property owners. Since the beaver population is mobile, extermination of beavers from affected parcels results in migration of beavers from neighboring less controlled parcels to less populated controlled parcels. This backward migration of beavers from uncontrolled habitat to controlled habitat imposes a negative diffusion externality on the owners of controlled parcels because they have to incur the future cost of trapping immigrating beavers. Unless all the land owners agree to control the beaver population simultaneously, the diffusion externality could result in a low incentive for control of beaver population on the part of individual land owners, causing a wedge between social and private needs for controlling beaver population. This study attempts to develop a bioeconomic model that incorporates dispersive population dynamics of beavers into the design of a cost-minimizing trapping strategy. While recognizing the need for several management options, depending on the land owners attitude about beavers, this study focuses its attention on the situation where all the land owners in a given habitat share common interest of controlling beaver nuisance, and collectively agree to place the area-wide control decision in the hands of a public agency, on a cost sharing basis. The model is based on the notion that the public manager attempts to minimize the present value combined costs of beaver damage and trapping over a finite period of time subject to spatiotemporal dynamics of beaver population. The time and spatial dynamics of beaver population is summarized by the parabolic diffusive Volterra-Lotka partial differential equation. Thus, the current problem is a typical distributed parameter control problem. The cost-minimizing area-wide trapping model is capable of characterizing the beaver control strategy that leaves enough beavers after taking into account the net migration at each location and time, so as to strike the optimal balance between timber damage and trapping cost. The marginality condition governing this tradeoff requires that the marginal damage savings from the beavers trapped at each location equal the marginal costs of trapping. The marginal savings from trapping activity, in turn, is measured as the imputed nuisance value (shadow price) of the beaver stock in a unit area. The optimality system for this problem that characterizes the optimal control is solved numerically. The validity of the theoretical model is empirically examined using the bioeconomic data collected for the Wildlife Management Regions of the New York State Department of Environmental Conservation. The empirical simulation generated discrete values for the optimal beaver densities and trapping rates across all the individual operational units over time. The entire distribution of optimal beaver densities does gradually and smoothly decline over the period of time. The unevenness of the initial population distribution smoothes out eventually across the beaver habitat. At each geographical location, towards the end of the planning period optimal trapping rate will become zero, whereas the population density asymptotically approaches zero. The sensitivity analysis where the cost and damage parameters of the model are alternated between high and low values indicates that an increase in the damage potential of beavers could substantially increase the net present value total cost. On the other hand, an increase in the cost of beaver trapping adds only marginally to the total cost, conserving more number of beavers. The geographical variation in the beaver damage potential has a noticeable reflection on the spatial distribution of trapping rates, with little impact on the optimal densities. The areas with higher beaver damage potentials require more intensive trapping operation

The diffusion of polymers in porous materials as studied by dynamic light scattering/

The diffusion of polymers in porous materials as studied by dynamic light scattering

Contributions to the theory of Ehrhart polynomials

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 89-91).In this thesis, we study the Ehrhart polynomials of different polytopes. In the 1960's Eugene Ehrhart discovered that for any rational d-polytope P, the number of lattice points, i(P,m), in the mth dilated polytope mP is always a quasi-polynomial of degree d in m, whose period divides the least common multiple of the denominators of the coordinates of the vertices of P. In particular, if P is an integral polytope, i(P, m) is a polynomial. Thus, we call i(P, m) the Ehrhart (quasi-)polynomial of P. In the first part of my thesis, motivated by a conjecture given by De Loera, which gives a simple formula of the Ehrhart polynomial of an integral cyclic polytope, we define a more general family of polytopes, lattice-face polytopes, and show that these polytopes have the same simple form of Ehrhart polynomials. we also give a conjecture which connects our theorem to a well-known fact that the constant term of the Ehrhart polynomial of an integral polytope is 1. In the second part (joint work with Brian Osserman), we use Mochizuki's work in algebraic geometry to obtain identities for the number of lattice points in different polytopes. We also prove that Mochizuki's objects are counted by polynomials in the characteristic of the base field.by Fu Liu.Ph.D

Lattice Polytopes and Triangulations

Das Existenzproblem unimodularer Triangulierungen von Gitterpolytopen wird untersucht. Diese Triangulierungen entsprechen nicht-diskrepanten Auflösungen torischer Singularitäten - Auflösungen, die den kanonischen Divisor erhalten. Nach der Einführung der grundlegenden Konzepte und Methoden gliedert sich die Arbeit in drei Kapitel. Kapitel 2: Die sogenannten leeren Gittersimplizes sind die Hindernisse für unimodulare Triangulierungen. Es ist bekannt, daß ihre Gitterweite durch eine Konstante w(d) beschränkt ist, die nur von der Dimension abhängt. Eine andere Konstante, W(d) - die maximale Weite fast leerer Simplizes, wird eingeführt. Die Konstruktion einer unendlichen Familie d-dimensionaler leerer Simplizes aus einem fast leeren (d-1)-dimensionalen Simplex zeigt die Monotonie beider Konstanten und widerlegt eine Vermutung von Bárány. Eine Computersuche in Dimension 4 liefert genau ein leeres Simplex der Weite 4 und suggeriert, daß die Determinante leerer Weite-3-Simplizes nie größer als 179 ist. Zusammen mit dem Beweis von W(3)=2 ist dies ein Indiz für eine modifizierte Vermutung. Kapitel 3: Die Polytope zu torischen lokal vollständigen Durchschnitten werden unimodular trianguliert. Dies verallgemeinert ein Resultat von Dais, Henk und Ziegler. Darüberhinaus sind diese Polytope Koszulsch. Kapitel 4: Die stringtheoretischen Hodge Zahlen von Batyrev und Dais werden für zwei Serien von (Hyperflächen in den projektiven torischen Varietäten zu) reflexiven Polytopen berechnet. Die erste Serie bilden die pseudosymmetrischen Fano Polytope. Ihre polar-dualen erzeugen glatte Hyperflächen, so daß nach Spiegelsymmetrie Formeln von Danilov und Khovanskii benutzt werden können. Diese dualen Polytope erlauben unimodulare Triangulierungen und sind Koszulsch. Die zweite Serie besteht aus Pyramiden über reflexiven Polytopen.The existence problem for unimodular triangulations of lattice polytopes is investigated. These triangulations correspond to the crepant resolutions of toric singularities - resolutions that preserve the canonical divisor. After the introduction of the basic concepts and tools, the thesis is divided into three chapters. Chapter 2. The so called empty lattice simplices are the obstacles to a unimodular triangulation. It is known that their lattice width is bounded by a constant w(d) that only depends on the dimension. Another constant, W(d) - the maximal width of almost empty simplices, is introduced. The construction of an infinite family of d-dimensional empty simplices out of an almost empty (d-1)-dimensional simplex shows the monotonicity of both constants and disproves a conjecture of Bárány. A computer search in dimension 4 yields exactly one empty simplex of width 4 and suggests that the determinant of empty width 3 simplices does not exceed 179. Together with a proof of W(3)=2 this supports a modified conjecture. Chapter 3. A unimodular triangulation is constructed for the polytopes that are associated with toric local complete intersections, thus generalizing a result of Dais, Henk and Ziegler. Furthermore, these polytopes are shown to have the Koszul property. Chapter 4. The string theoretic Hodge numbers of Batyrev and Dais are computed for two series of (hypersurfaces in the projective toric varieties corresponding to) reflexive polytopes. The first series is given by the pseudo symmetric Fano polytopes. Their polar duals give rise to smooth hypersurfaces, so by mirror-symmetry, formulae of Danilov and Khovanskii can be used. These dual polytopes admit unimodular triangulations and they have the Koszul property. The second series consists of pyramids over reflexive polytopes. I