The dimension of planar posets

A partially ordered set (poset) is planar if it has a planar Hasse diagram. The dimension of a bounded planar poset is at most two. We show that the dimension of a planar poset having a greatest lower bound is at most three. We also construct four-dimensional planar posets, but no planar poset with dimension larger than four is known. A poset is called a tree if its Hasse diagram is a tree in the graph-theoretic sense. We show that the dimension of a tree is at most three and give a forbidden subposet characterization of two-dimensional trees.     

On stability of Alfvén discontinuities

We study Alfvén discontinuities for the equations of ideal compressible magnetohydrodynamics (MHD). The Alfvén discontinuity is a characteristic discontinuity for the hyperbolic system of the MHD equations but, as for shock waves, the gas crosses its front. By numerical testing of the Lopatinskii condition, we carry out spectral stability analysis, i.e. we find the parameter domains of stability and violent instability of planar Alfvén discontinuities. We also show that Alfvén discontinuities can be only weakly stable in the sense that the uniform Lopatinskii condition is never satisfied. Copyright © 2008 John Wiley & Sons, Ltd.     

Stability of Front Solutions of the Bidomain Equation

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.© 2016 Wiley Periodicals, Inc.     

Bifurcation of Equilibria in a Class of Planar Piecewise Smooth Systems with 3 Parameters

The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.     

Vesicle computers: Approximating a Voronoi diagram using Voronoi automata

Irregular arrangements of vesicles filled with excitable and precipitating chemical systems are imitated by Voronoi automata - finite-state machines defined on a planar Voronoi diagram. Every Voronoi cell takes four states: resting, excited, refractory and precipitate. A resting cell excites if it has at least one neighbour in an excited state. The cell precipitates if the ratio of excited cells in its neighbourhood versus the number of neighbours exceeds a certain threshold. To approximate a Voronoi diagram on Voronoi automata we project a planar set onto the automaton lattice, thus cells corresponding to data-points are excited. Excitation waves propagate across the Voronoi automaton, interact with each other and form precipitate at the points of interaction. The configuration of the precipitate represents the edges of an approximated Voronoi diagram. We discover the relationship between the quality of the Voronoi diagram approximation and the precipitation threshold, and demonstrate the feasibility of our model in approximating Voronoi diagrams of arbitrary-shaped objects and in constructing a skeleton of a planar shape.     

On region crossing change and incidence matrix

In a recent work of Ayaka Shimizu, she studied an operation named region crossing change on link diagrams, which was proposed by Kishimoto, and showed that a region crossing change is an unknotting operation for knot diagrams. In this paper, we prove that the region crossing change on a 2-component link diagram is an unknotting operation if and only if the linking number of the diagram is even. Besides, we define an incidence matrix of a link diagram via its signed planar graph and its dual graph. By studying the relation between region crossing change and incidence matrix, we prove that a signed planar graph represents an n-component link diagram if and only if the rank of the associated incidence matrix equals c n + 1, where c denotes the size of the graph.     

Voronoi diagram and medial axis algorithm for planar domains with curved boundaries I. Theoretical foundations

In this first installment of a two-part paper, the underlying theory for an algorithm that computes the Voronoi diagram and medial axis of a planar domain bounded by free-form (polynomial or rational) curve segments is presented. An incremental approach to computing the Voronoi diagram is used, wherein a single boundary segment is added to an existing boundary-segment set at each step. The introduction of each new segment entails modifying the Voronoi regions of the existing boundary segments, and constructing the Voronoi region of the new segment. We accomplish this by (i) computing the bisector of the new segment with each of the current boundary segments; (ii) updating the Voronoi regions of the current boundary segments by partitioning them with these bisectors; and (iii) constructing the Voronoi region of the new segment as a union of regions obtained from the partitioning in (ii). When all boundary segments are included, and their Voronoi regions have been constructed, the Voronoi diagram of the boundary is obtained as the union of the Voronoi polygons for each boundary segment. To construct the medial axis of a planar domain, we first compute the Voronoi diagram of its boundary. The medial axis is then obtained from the Voronoi diagram by (i) removing certain edges of the Voronoi diagram that do not belong to the medial axis, and (ii) adding certain edges that do belong to the medial axis but are absent from the Voronoi diagram; unambiguous characterizations for edges in both these categories are given. Details of algorithms based on this theory are deferred to the second installment of this two-part paper.     

Angular Properties of Delaunay Diagrams in Any Dimension

Equiangularity (also called max-min angle criterion) is a well-known property of some planar triangulations that refine the Delaunay diagram. In this paper we generalize the notion of equiangularity to decompositions in inscribable polygons and we show that it characterizes the planar Delaunay diagram, even if more than three sites are cocircular. This result does not extend to higher dimensions. However, we characterize the Delaunay diagram in any dimension by a kind of dual property that we prove both with line angles and with solid angles. We also establish a local equiangularity of Delaunay diagrams in any dimension, and an angular characterization of self-centered diagrams. Finally, we show that these angular properties can, when appropriately defined, be generalized to the farthest point Delaunay diagram. Received April 25, 1996, and in revised form July 31, 1997, and March 18, 1998.     

Nonlinear geometric optics for contact discontinuities and shock waves in one-dimensional Euler system for gas dynamics

The aim of this paper is to study the rigorous theory of nonlinear geometric optics for a contact discontinuity and a shock wave to the Euler system for one-dimensional gas dynamics. For the problem of a contact discontinuity and a shock wave perturbed by a small amplitude, high frequency oscillatory wave train, under suitable stability assumptions, we obtain that the perturbed problem has still a shock wave and a contact discontinuity, and we give their asymptotic expansions.     

Generic bifurcation of refracted systems

In this article we discuss some qualitative and geometric aspects of non-smooth dynamical systems theory. Our goal is to study the diagram bifurcation of typical singularities that occur generically in one parameter families of certain piecewise smooth vector fields named Refracted Systems. Such systems has a codimension-one submanifold as its discontinuity set.     

Stability of discontinuity structures described by a generalized KdV–Burgers equation

The stability of discontinuities representing solutions of a model generalized KdV–Burgers equation with a nonmonotone potential of the form φ(u) = u⁴–u² is analyzed. Among these solutions, there are ones corresponding to special discontinuities. A discontinuity is called special if its structure represents a heteroclinic phase curve joining two saddle-type special points (of which one is the state ahead of the discontinuity and the other is the state behind the discontinuity).The spectral (linear) stability of the structure of special discontinuities was previously studied. It was shown that only a special discontinuity with a monotone structure is stable, whereas special discontinuities with a nonmonotone structure are unstable. In this paper, the spectral stability of nonspecial discontinuities is investigated. The structure of a nonspecial discontinuity represents a phase curve joining two special points: a saddle (the state ahead of the discontinuity) and a focus or node (the state behind the discontinuity). The set of nonspecial discontinuities is examined depending on the dispersion and dissipation parameters. A set of stable nonspecial discontinuities is found.     

A Möbius-Invariant Power Diagram and Its Applications to Soap Bubbles and Planar Lombardi Drawing

We use three-dimensional hyperbolic geometry to define a form of power diagram for systems of circles in the plane that is invariant under Möbius transformations. By applying this construction to circle packings derived from the Koebe–Andreev–Thurston circle packing theorem, we show that every planar graph of maximum degree three has a planar Lombardi drawing (a drawing in which the edges are drawn as circular arcs, meeting at equal angles at each vertex). We use circle packing to construct planar Lombardi drawings of a special class of 4-regular planar graphs, the medial graphs of polyhedral graphs, and we show that not every 4-regular planar graph has a planar Lombardi drawing. We also use these power diagrams to characterize the graphs formed by two-dimensional soap bubble clusters (in equilibrium configurations) as being exactly the 3-regular bridgeless planar multigraphs, and we show that soap bubble clusters in stable equilibria must in addition be 3-connected.     

Generalized Hopf Bifurcation for Planar Filippov Systems Continuous at the Origin

In this paper, we study the existence of periodic orbits bifurcating from stationary solutions of a planar dynamical system of Filippov type. This phenomenon is interpreted as a generalized Hopf bifurcation. In the case of smoothness, Hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis. This method does not carry over to nonsmooth systems, due to the lack of linearization at the origin which is located on the line of discontinuity. In fact, generalized Hopf bifurcation is determined by interactions between the discontinuity of the system and the eigen-structures of all subsystems. With the help of geometrical observations for a corresponding piecewise linear system, we derive an analytical method to investigate the existence of periodic orbits that are obtained by searching for the fixed points of return maps.     

一类非线性波方程的光滑与非光滑行波解

讨论了一类偏微分方程的行波解。该方程的行波方程对应于一个平面三次多项式系统,因而可将行波解的研究化为对平面系统所定义的相轨线的拓扑分类研究。应用平面动力系统理论在三参数空间内作定性分析,首先获得三次多项式系统的完整拓扑分类,再将相平面分析的结果返回到非线性波解u(ξ) 。考虑到解关于变量ξ=x-ct在“奇线”近旁的不连续性,可得到各种光滑与非光滑行波的存在条件。     

Planar lattices and planar graphs

It is shown that a finite lattice is planar if and only if the (undirected) graph obtained from its (Hasse) diagram by adding an edge between its least and greatest elements is a planar graph.     

Binary polyhedral groups and Euclidean diagrams

Recently, J. McKay [7] has observed that the irreducible complex representations of the binary polyhedral groups can be arranged in order to form the vertices of a Euclidean diagram in such a way that the tensor product of any irreducible representation M with the standard two-dimensional representation is the direct sum of the irreducible representations which are the neighbors of M in the diagram, and he asked for an explanation. In this note, we will show that any self-dual two-dimensional representation gives rise to a generalized Euclidean diagram, and that this in fact can be used to give a proof of the classification theorem of the binary polyhedral groups which at the same time furnishes a list of the irreducible representations and also gives the minimal splitting field.     

Another explanation for the bias observed in the filter rule test

Substantial bias in profits is observed when we apply Alexander's filter rule to the piecewise linear function formed by the linear interpolation of a past daily (weekly or monthly) stock price sequence. The only explanation for this phenomenon reported up to now is the possible discontinuity of the original price path. This paper demonstrates that the autocorrelation generated by the linear interpolation procedure causes this phenomenon even if the original path is a realization of the Brownian motion. It is also shown that the bias for the TOPIX index in the Tokyo Stock Exchange is substantially explained in our theoretical framework.     

The greedy triangulation can be computed from the Delaunay triangulation in linear time

The greedy triangulation of a finite planar point set is obtained by repeatedly inserting a shortest diagonal that does not cross those already in the plane. The Delaunay triangulation, which is the straight-line dual of the Voronoi diagram, can be produced in O(nlogn) worst-case time, and often even faster, by several practical algorithms. In this paper we show that for any planar point set S, if the Delaunay triangulation of S is given, then the greedy triangulation of S can be computed in linear worst-case time (and linear space).     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

SOME APPLICATIONS OF PLANAR GRAPH IN KNOT THEORY

The relationship between a link diagram and its corresponding planar graph is briefly reviewed.A necessary and sufficient condition is given to detect when a planar graph corresponds to a knot.The rela...