Planforms in three dimensions

The spatially periodic, steady-state solutions to systems of partial differential equations (PDE) are calledplanforms. There already exists a partial classification of the planforms for Euclidean equivariant systems of PDE inR ² (see [6, 7]), In this article we attempt to give such a classification for Euclidean equivariant systems of PDE inR ³. Based on the symmetry and spatial periodicity of each planform, 59 different planforms are found.We attempt to find the planforms on all lattices inR ³ that are forced to exist near a steady-state bifurcation from a trivial solution. The proof of our classification uses Liapunov-Schmidt reduction with symmetry (which can be used if we assume spatial periodicity of the solutions) and the Equivariant Branching Lemma. The analytical problem of finding planforms for systems of PDE is reduced to the algebraic problem of computing isotropy subgroups with one dimensional fixed point subspaces.The Navier-Stokes equations and reaction-diffusion equations (with constant diffusion coefficients) are examples of systems of PDE that satisfy the conditions of our classifications. In this article, we show that our classification applies to the Kuramoto-Sivashinsky equation.     

Jordan-Wachspress parameters in three dimensions

This paper describes a new technique which uses the Jordan-Wachspress parameters of the two-dimensional elliptic problem in the three-dimensional one. Thus A.D.I.-like schemes are obtained which, as is shown, converge faster than even the fastest ones known so far.     

Convex source support in three dimensions

This work extends the algorithm for computing the convex source support in the framework of the Poisson equation to a bounded three-dimensional domain. The convex source support is, in essence, the smallest (nonempty) convex set that supports a source that produces the measured (nontrivial) data on the boundary of the object. In particular, it belongs to the convex hull of the support of any source that is compatible with the measurements. The original algorithm for reconstructing the convex source support is inherently two-dimensional as it utilizes M?bius transformations. However, replacing the M?bius transformations by inversions with respect to suitable spheres and introducing the corresponding Kelvin transforms, the basic ideas of the algorithm carry over to three spatial dimensions. The performance of the resulting numerical algorithm is analyzed both for the inverse source problem and for electrical impedance tomography with a single pair of boundary current and potential as the measurement data.     

The ring lemma in three dimensions

Suppose that n cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc. The sharp ring lemma in two dimensions states that no disc has a radius below c _n (R ²) = (F _2n−3−1)⁻¹—where F _k denotes the kth Fibonacci number—and that the lower bound is attained in essentially unique Apollonian configurations. In this article, generalizations of the ring lemma to three dimensions are discussed, a version of the ring lemma in three dimensions is proved, and a natural generalization of the extremal two-dimensional configuration—thought to be extremal in three dimensions—is given. The sharp three-dimensional ring lemma constant of order n is shown to be bounded from below by the two-dimensional constant of order n − 1.     

The two-well problem in three dimensions

We study properties of generalized convex hulls of the set with . If K contains no rank-1 connection we show that the quasiconvex hull of K is trivial if H belongs to a certain (large) neighbourhood of the identity. We also show that the polyconvex hull of K can be nontrivial if H is sufficiently far from the identity, while the (functional) rank-1 convex hull is always trivial. If the second well is replaced by a point then the polyconvex hull is trivial provided that there are no rank-1 connections. Received: March 25, 1999 / Accepted: April 23, 1999     

Planforms in two and three dimensions

Summary When solving systems of PDE with two space dimensions it is often assumed that the solution is spatially doubly periodic. This assumption is usually made in systems such as the Boussinesq equation or reaction-diffusion equations where the equations have Euclidean invariance. In this article we use group theoretic techniques to determine a large class of spatially doubly periodic solutions that are forced to existence near a steady-state bifurcation from a translation-invariant equilibrium.This type of bifurcation problem has been considered by many authors when studying a number of different systems of PDE. Typically, these studies focus at the beginning on equilibria that are spatially periodic with respect to a fixed planar lattice type-such as square or hexagonal. Our focus is different in that we attempt to find all spatially periodic equilibria that bifurcate on all lattices. This point of view leads to some technical simplifications such as being able to restrict to translation free irreducible representations.Of course, many of the types of solutions that we find are well-known-such as hexagon and roll solutions on a hexagonal lattice. This coordinated group theoretic approach does lead, however, to solutions which seem not to have been discussed previously (antisquare solutions on a square lattice) as well as to a more complete classification of the symmetry types of possible solutions. Moreover, our methods extend to triply periodic solutions of PDE with three spatial variables. Some of these results, namely those concerned with primitive cubic lattices, are presented here. The complete results on triply periodic solutions may be found in [6, 7].In honor of Klaus Kirchgässner on the occasion of his sixtieth birthdayResearch supported in part by NSF/DARPA (DMS-8700897) and by the Texas Advanced Research Program (ARP-1100).     

Existence of solutions to the biaxial nematic liquid crystals with two order parameter tensors

The recent paper considers some mathematical problems on a tensor model with two order parameters to the biaxial nematic liquid crystals. The energy in this model contains a cubic term and is unbounded from below. We study the gradient flow generated by this energy in two dimensions. We focus on establishing global existence of classical and weak solutions.     

奇数维可压缩液晶流方程解的逐点估计

首先, 本文利用标准的能量估计方法得到高维(3 维及以上) 的液晶流方程组小初值经典解的整体存在性. 然后, 本文运用Green 函数方法, 得到奇数维情形(3 维及以上) 该解的逐点估计. 该结果表明, 密度ρ和动量m同Navier-Stokes 方程组一样满足一般Huygens 原理, 而单位向量场d则没有这种现象, 其有着与热方程的解类似的时空估计.     

Chern-Connes character for the invariant Dirac operator in odd dimensions

In this paper we give a proof of the Lefschetz fixed point formula of Freed for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced by Lafferty et al. and then we generalize this formula under the noncommutative geometry framework.     

Symmetric versus asymmetric equilibria in symmetric supermodular games

This paper investigates the general properties of symmetric n-player supermodular games with complete-lattice action spaces. In particular, we examine the extent to which all pure strategy Nash equilibria tend to be symmetric for the general case of multi-dimensional strategy spaces. As asymmetric equilibria are possible even for strictly supermodular games, we investigate whether some symmetric equilibrium would always Pareto dominate all asymmetric equilibria. While this need not hold in general, we identify different sufficient conditions, each of which guarantees that such dominance holds: 2-player games with scalar action sets, uni-signed externalities, identical interests, and superjoin payoffs. Various illustrative examples are provided. Finally, some economic applications are discussed. The first version of this paper was completed while M. Jakubczyk and M. Knauff were visiting junior scholars at CORE, Louvain-la-Neuve, Belgium, financed through the Marie-Curie Early Stage Training program of the European Union (under contract no HPMT-CT-2001-00327), which is hereby gratefully acknowledged. The presentation of the revised version of this paper has benefitted from detailed and careful suggestions by two anonymous referees and William Thomson (as editor) of this Journal.     

Some third-order rotatable designs in three dimensions

Summary Some new third-order rotatable designs in three dimensions are derived from some of the available third-order rotatable designs in two dimensions. When these designs are used the results of the experiments performed according to the two-dimentional designs need not be discarded. Some of these designs may be performed sequentially in all three factors, starting with a one-dimensional design. Further, these third-order rotatable designs require a smaller number of points than most of the available three-dimensional third-order rotatable designs.     

A combinatorial Yamabe flow in three dimensions

A combinatorial version of Yamabe flow is presented based on Euclidean triangulations coming from sphere packings. The evolution of curvature is then derived and shown to satisfy a heat equation. The Laplacian in the heat equation is shown to be a geometric analogue of the Laplacian of Riemannian geometry, although the maximum principle need not hold. It is then shown that if the flow is nonsingular, the flow converges to a constant curvature metric.     

The 2-center problem in three dimensions

Let P be a set of n points in ${\mathbb{R}}^3$. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present a randomized algorithm for computing a 2-center of P that runs in $O(\beta (r^{?})n^2{\log }^{4}n\log \log n)$ expected time; here $\beta (r)=1/{(1?r/r_0)}^3$, $r^{?}$ is the radius of the 2-center balls of P, and $r_0$ is the radius of the smallest enclosing ball of P. The algorithm is near quadratic as long as $r^{?}$ is not too close to $r_0$, which is equivalent to the condition that the centers of the two covering balls be not too close to each other. This improves an earlier slightly super-cubic algorithm of Agarwal, Efrat, and Sharir (2000) [2] (at the cost of making the algorithm performance depend on the center separation of the covering balls).     

Fast evaluation of polyharmonic splines in three dimensions

M. J. D. Powell ^{This paper concerns the fast evaluation of radial basis functions.It describes the mathematics of hierarchical and fast multipolemethods for fast evaluation of splines of the form where is a positive integer andp is a low-degree polynomial. Splines s of this form are polyharmonicsplines in 3 and have been found to be very useful for providingsolutions to scattered data interpolation problems in 3. Asit is now well known, hierarchical methods reduce the incrementalcost of a single extra evaluation from O(N) to O(log N) operationsand reduce the cost of a matrix–vector product (evaluationof s at all the centres) from O(N2) to O(N log N) operations.We give appropriate far- and near-field expansions, togetherwith error estimates, uniqueness theorems and translation formulae.A hierarchical code based on these formulae is detailed andsome numerical results are given.}     

The Angel and the Devil in three dimensions

Our main aim in this paper is to show that, in Conway's Angel and Devil game, an Angel of sufficient speed can always escape in three dimensions. We also prove some related results and make some conjectures.