Bregman Voronoi Diagrams

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.     

Psi-floor diagrams and a Caporaso-Harris type recursion

Floor diagrams are combinatorial objects which organize the count of tropical plane curves satisfying point conditions. In this paper we introduce Psi-floor diagrams which count tropical curves satisfying not only point conditions but also conditions given by Psi-classes (together with points). We then generalize our definition to relative Psi-floor diagrams and prove a Caporaso-Harris type formula for the corresponding numbers. This formula is shown to coincide with the classical Caporaso-Harris formula for relative plane descendant Gromov-Witten invariants. As a consequence, we can conclude that in our case relative descendant Gromov-Witten invariants equal their tropical counterparts.     

Classification of Solitary Wave Bifurcations in Generalized Nonlinear Schrödinger Equations

Bifurcations of solitary waves are classified for the generalized nonlinear Schrödinger equations with arbitrary nonlinearities and external potentials in arbitrary spatial dimensions. Analytical conditions are derived for three major types of solitary wave bifurcations, namely, saddle‐node, pitchfork, and transcritical bifurcations. Shapes of power diagrams near these bifurcations are also obtained. It is shown that for pitchfork and transcritical bifurcations, their power diagrams look differently from their familiar solution‐bifurcation diagrams. Numerical examples for these three types of bifurcations are given as well. Of these numerical examples, one shows a transcritical bifurcation, which is the first report of transcritical bifurcations in the generalized nonlinear Schrödinger equations. Another shows a power loop phenomenon which contains several saddle‐node bifurcations, and a third example shows double pitchfork bifurcations. These numerical examples are in good agreement with the analytical results.     

Qualitative possibilistic influence diagrams based on qualitative possibilistic utilities

This paper proposes a new approach for decision making under uncertainty based on influence diagrams and possibility theory. The so-called qualitative possibilistic influence diagrams extend standard influence diagrams in order to avoid difficulties attached to the specification of both probability distributions relative to chance nodes and utilities relative to value nodes. In fact, generally, it is easier for experts to quantify dependencies between chance nodes qualitatively via possibility distributions and to provide a preferential relation between different consequences. In such a case, the possibility theory offers a suitable modeling framework. Different combinations of the quantification between chance and utility nodes offer several kinds of possibilistic influence diagrams. This paper focuses on qualitative ones and proposes an indirect evaluation method based on their transformation into possibilistic networks. The proposed approach is implemented via a possibilistic influence diagram toolbox (PIDT).     

Geometric relations among Voronoi diagrams

Two general classes of Voronoi diagrams are introduced and, along with their modifications to higher order, are shown to be geometrically related. This geometric background, on the one hand, serves to analyse the size and combinatorial structure and, on the other, implies general and efficient methods of construction for various important types of Voronoi diagrams considered in the literature.Research supported by the Austrian Fond zur Foerderung der wissenschaftlichen Forschung.     

A central limit theorem for extremal characters of the infinite symmetric group

The asymptotic behavior of the lengths of the first rows and columns in the random Young diagrams corresponding to extremal characters of the infinite symmetric group is studied. We consider rows and columns with linear growth in n and prove a central limit theorem for their lengths in the case of distinct Thoma parameters. We also prove a more precise statement relating the growth of rows and columns of Young diagrams to a simple independent random sampling model.     

Tableaux and plane partitions of truncated shapes

We consider a new kind of straight and shifted plane partitions/Young tableaux – ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function.     

Logical Analysis of Legislation Using Flow Diagrams

The Department of the Environment/Department of Transport (DOE/DTp) OR group uses flow diagrams to analyse legislation. Taking the bus deregulation sections of the 1985 Transport Act as the main example, this paper shows how a complex course of action prescribed in legal language may be simply represented by a series of yes/no questions. Traffic Area staff who implement the legislation have the diagrams as working guidelines to resolve their own, potentially conflicting, interpretations.     

Students constructing meaning for the number line in small-group discussions: negotiation of essential epistemological issues of visual representations

In mathematics classroom interaction, the multiple meanings of mathematical visual diagrams are often ignored; instead, depending on the given situation, they are read in a well-defined and unitary way. Mathematical visual representations are thus used even less in their epistemological function for learning, but more as pre-given subject matter. The purpose of this paper is to elaborate opportunities for negotiating and clarifying differences, dealing with a great variety of ways of interpreting visual diagrams that are brought into focus in interaction. Theory-based qualitative analyses of two exemplifying video episodes of small-group discussions negotiating their ideas on the topic “number line” show differences of meaning and the importance of conventions followed by mathematical deductions. Two mutually exclusive teacher behaviors within the communicative acts, reconstructed as a dominating way of instruction and a moderating way of focusing, are identified.     

The equilibrium positions of an ellipsoid with a cavity containing a liquid on a horizontal plane

The equilibrium positions of an ellipsoid with an ellipsoidal cavity, partially filled with an ideal incompressible liquid, on a horizontal plane in a uniform gravitational field are considered. All trivial and non-trivial equilibrium positions are found and the conditions for their stability are obtained. The results are presented in the form of bifurcation diagrams.     

Zone diagrams in compact subsets of uniformly convex normed spaces

A zone diagram is a relatively new concept which has emerged in computational geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a certain mapping, and neither its uniqueness nor its existence are obvious in advance. It has been studied by several authors, starting with T. Asano, J. Matoušek and T. Tokuyama, who considered the Euclidean plane with singleton sites, and proved the existence and uniqueness of zone diagrams there. In the present paper we prove the existence of zone diagrams with respect to finitely many pairwise disjoint compact sites contained in a compact and convex subset of a uniformly convex normed space, provided that either the sites or the convex subset satisfy a certain mild condition. The proof is based on the Schauder fixed point theorem, the Curtis-Schori theorem regarding the Hilbert cube, and on recent results concerning the characterization of Voronoi cells as a collection of line segments and their geometric stability with respect to small changes of the corresponding sites. Along the way we obtain the continuity of the Dom mapping as well as interesting and apparently new properties of Voronoi cells.     

Symmetry factors of Feynman diagrams for scalar fields

We calculate the symmetry factors of diagrams for real and complex scalar fields in general form using an analysis of the Wick expansion for Green’s functions. We separate two classes of symmetry factors: factors corresponding to connected diagrams and factors corresponding to vacuum diagrams. The symmetry factors of vacuum diagrams play an important role in constructing the effective action and phase transitions in cosmology. In the complex scalar field theory, diagrams with different topologies can contribute the same, and the inverse symmetry factor for the total contribution is therefore the sum of the inverse symmetry factors.     

Markov processes on partitions

We introduce and study a family of Markov processes on partitions. The processes preserve the so-called z-measures on partitions previously studied in connection with harmonic analysis on the infinite symmetric group. We show that the dynamical correlation functions of these processes have determinantal structure and we explicitly compute their correlation kernels. We also compute the scaling limits of the kernels in two different regimes. The limit kernels describe the asymptotic behavior of large rows and columns of the corresponding random Young diagrams, and the behavior of the Young diagrams near the diagonal. Our results show that recently discovered analogy between random partitions arising in representation theory and spectra of random matrices extends to the associated time– dependent models.     

Metalogical Decorations of Logical Diagrams

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of (sub)contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams (viz. Aristotelian and duality diagrams) and four kinds of metalogical decorations (viz. those based on the opposition, implication, Aristotelian and duality relations).     

Quantum field theory meets Hopf algebra

This paper provides a primer in quantum field theory (QFT) based on Hopf algebra and describes new Hopf algebraic constructions inspired by QFT concepts. The following QFT concepts are introduced: chronological products, S ‐matrix, Feynman diagrams, connected diagrams, Green functions, renormalization. The use of Hopf algebra for their definition allows for simple recursive derivations and leads to a correspondence between Feynman diagrams and semi‐standard Young tableaux. Reciprocally, these concepts are used as models to derive Hopf algebraic constructions such as a connected coregular action or a group structure on the linear maps from S (V) to V. In many cases, noncommutative analogues are derived (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The KOH terms and classes of unimodal N-modular diagrams

We show how certain suitably modified N-modular diagrams of integer partitions provide a nice combinatorial interpretation for the general term of Zeilberger?s KOH identity. This identity is the reformulation of O?Hara?s famous proof of the unimodality of the Gaussian polynomial as a combinatorial identity. In particular, we determine, using different bijections, two main natural classes of modular diagrams of partitions with bounded parts and length, having the KOH terms as their generating functions. One of our results greatly extends recent theorems of J. Quinn et al., which presented striking applications to quantum physics.