Forbidden Families of Minimal Quadratic and Cubic Configurations

A matrix is \emph{simple} if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a \emph{configuration}, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch

Generic method for bijections between blossoming trees and planar maps

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of \alpha-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases all previously known bijections involving blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable maps and simple triangulations and quadrangulations of a k-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and d-angulations of girth d of a k-gon. As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode and generate planar maps. In a recent work, Bernardi and Fusy introduced another unified bijective scheme, we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom

Chiral Models of Weak Scale Supersymmetry

I discuss supersymmetric extensions of the Standard Model containing an extra U(1)' gauge symmetry which provide a solution to the mu-problem and at the same time protect the proton from decaying via dimension 4 operators. Moreover, all fields are protected by chirality and supersymmetry from acquiring high scale masses. The additional requirements of anomaly cancellation and gauge coupling unification imply the existence of extra matter multiplets and that several fields participate in U(1)' symmetry breaking simultaneously. While there are several studies addressing subsets of these requirements, this work uncovers simultaneous solutions to all of them. It is surprising and encouraging that extending the Minimal Supersymmetric Standard Model by a simple U(1) factor solves its major drawbacks with respect to the non-supersymmetric Standard Model, especially as current precision data seem to offer a hint to the existence of its corresponding Z' boson. It is also remarkable that there are many solutions where the U(1)' charges of the known quarks and leptons are predicted to be identical to those E_6 motivated Z' bosons which give the best fit to the data. I discuss the solutions to these constraints including some phenomenological implications.Comment: 19 page

On the sum of the Voronoi polytope of a lattice with a zonotope

A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to be free. Given a set $U$ of free vectors, we give several methods for checking if $P + Z(U)$ is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices. In the case of the root lattice $\mathsf{E}_6$, it is possible to give a more geometric description of the admissible sets of vectors $U$. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of $27$ lines in a cubic. Based on a detailed study of the geometry of $P_V(\mathsf{e}_6)$, we give a simple characterization of the configurations of vectors $U$ such that $P_V(\mathsf{E}_6) + Z(U)$ is a parallelotope. The enumeration yields $10$ maximal families of vectors, which are presented by their description as regular matroids.Comment: 30 pages, 4 figures, 4 table

Selected Topics on Rare Kaon Processes - in the Standard Model and Supergravity -

Rare kaon processes appear to be particularly suitable to study the extensions of the standard model, especially if the possibility for eventual direct evidence becomes unlikely. In this review, we discuss processes that are important as a test of either the standard model or supergravity. Moreover, some of these are important even for both the standard model and for supergravity. Particular attention is paid to the reduction of uncertainties in the calculation, especially the ones coming from the confinement effects. Recent approaches, such as chiral perturbation theory, the large $N_c$-expansion, QCD sum rules and lattice QCD, are discussed. This is found to be the best strategy in view of the fact that supersymmetric effects are rather tiny.Comment: 52 pages (14 figures available upon request), LaTex, LMU-TP 7/89, RBI-TP 4/89, final version July 1992, to be published in Fortschritte der Physi

Two Higgs Pair Heterotic Vacua and Flavor-Changing Neutral Currents

We present a vacuum of heterotic M-theory whose observable sector has the MSSM spectrum with the addition of one extra pair of Higgs-Higgs conjugate superfields. The quarks/leptons have a realistic mass hierarchy with a naturally light first family. The double elliptic structure of the Calabi-Yau compactification threefold leads to two ``stringy'' selection rules. These classically disallow Yukawa couplings to the second Higgs pair and, hence, Higgs mediated flavor-changing neutral currents. Such currents are induced in higher-dimensional interactions, but are naturally suppressed. We show that our results fit comfortably below the observed upper bounds on neutral flavor-changing processes.Comment: 52 pages, 3 figures, 1 table, requires feynm

Geometric Exponents of Dilute Logarithmic Minimal Models

The fractal dimensions of the hull, the external perimeter and of the red bonds are measured through Monte Carlo simulations for dilute minimal models, and compared with predictions from conformal field theory and SLE methods. The dilute models used are those first introduced by Nienhuis. Their loop fugacity is beta = -2cos(pi/barkappa}) where the parameter barkappa is linked to their description through conformal loop ensembles. It is also linked to conformal field theories through their central charges c = 13 - 6(barkappa + barkappa^{-1}) and, for the minimal models of interest here, barkappa = p/p' where p and p' are two coprime integers. The geometric exponents of the hull and external perimeter are studied for the pairs (p,p') = (1,1), (2,3), (3,4), (4,5), (5,6), (5,7), and that of the red bonds for (p,p') = (3,4). Monte Carlo upgrades are proposed for these models as well as several techniques to improve their speeds. The measured fractal dimensions are obtained by extrapolation on the lattice size H,V -> infinity. The extrapolating curves have large slopes; despite these, the measured dimensions coincide with theoretical predictions up to three or four digits. In some cases, the theoretical values lie slightly outside the confidence intervals; explanations of these small discrepancies are proposed.Comment: 41 pages, 32 figures, added reference