Perfect colorings of isonemal fabrics using two colors

An isonemal fabric is a weaving in which the symmetry group is transitive on the strands. A perfect coloring with two colors is an assignment of the two colors to the strands so that each symmetry operation preserves or reverses the colors. If a transitive group of symmetries has that property we say the coloring is chromatic. We determine (in terms of thin stripings and thick stripings) the perfect and chromatic colorings for isonemal fabrics using two colors.     

Isonemal prefabrics with no axes of symmetry

This paper refines Richard Roth’s taxonomy of isonemal weaving designs through the final types 33-39 in order to complete the solution of three problems for those designs: which designs exist in various sizes, which prefabrics can be doubled and remain isonemal, and which can be halved and remain isonemal. These types have no symmetry axes but have quarter-turn symmetries. Jean Pedersen’s problem of woven cubes is also discussed.     

Group theory for tetraammineplatinum(II) withC 2v andC 4v point group in the non-rigid system

The non-rigid molecule group theory (NRG) in which the dynamical symmetry operations are defined as physical operations is a new field of chemistry. Smeyers in a series of papers applied this notion to determine the character table of restricted NRG of some molecules. In this work, a simple method is described, by means of which it is possible to calculate character tables for the symmetry group of molecules consisting of a number of NH3 groups attached to a rigid framework. We study the full non-rigid group (f-NRG) of tetraammineplatinum (II) with two separate symmetry groups C2v and C4v. We prove that they are groups of order 216 and 5184 with 27 and 45 conjugacy classes, respectively. Also, we will compute the character tables of these groups.     

Isonemal prefabrics with only parallel axes of symmetry

Isonemal weaving designs, introduced into mathematical literature by Grünbaum and Shephard, were classified into thirty-nine infinite sets, and a small number of exceptions by Richard Roth. This paper refines Roth’s taxonomy for the first ten of these families in order to solve three problems: which designs exist in various sizes, which prefabrics can be doubled and remain isonemal, and which can be halved and remain isonemal.     

Applications of Symmetry and Group Theory for the Investigation of Molecular Vibrations

The application of symmetry and mathematical group theory is a powerful tool for investigating the vibrations of molecules. In this paper, we present an overview of the methods utilized. First we briefly discuss the quantum mechanical nature of vibrations and the experimental methods used. We then present the principal concepts for applying group theory to molecules. The symmetry operations which are used to comprise groups are described and then used to determine the point groups of molecules. The properties of character tables are presented and the method for obtaining a reducible representation for all the motions of a molecule is detailed. This can then be broken down to obtain the irreducible representation which contains the symmetry species of the individual vibrations. The determination of symmetry adapted linear combinations is outlined and the basis for spectroscopic selection rules is presented. The paper concludes by examining how matrix algebra along with symmetry concepts simplifies calculations with molecular force constants.     

Robust heteroclinic cycles

Summary One phenomenon in the dynamics of differential equations which does not typically occur in systems without symmetry is heteroclinic cycles. In symmetric systems, cycles can be robust for symmetry-preserving perturbations and stable. Cycles have been observed in a number of simulations and experiments, for example in rotating convection between two plates and for turbulent flows in a boundary layer. Theoretically the existence of robust cycles has been proved in the unfoldings of some low codimension bifurcations and in the context of forced symmetry breaking from a larger to a smaller symmetry group. In this article we review the theoretical and the applied research on robust cycles.     

On regular polyhedra with hidden symmetries

We consider polyhedral realizations of oriented regular maps with or without self-intersections in E³ whose symmetry group is a subsgroup of small index in their. automorphism group. The four classical kepler-poinsot polyhedra are the only ones of index 1. There are exactly five of Index 2, all with icosahedral symmetry group [W2] as the Kepler-poinsot polyhedra. In this paper we show that there are no such polyhedral realizations with octahedral (tetrahedral) symmetry group and index 2 or 3 (2,3,4,5), which is best possible in the octahedral case.     

Lagrangian reduction and the double spherical pendulum

This paper studies the stability and bifurcations of the relative equilibrium of the double spherical pendulum, which has the circle as its symmetry group. The example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group.Dedicated to Professor Klaus Kirchgässner on the occasion of his 60th birthdayResearch partially supported by a Humboldt award at the Universität Hamburg and by DOE Contract DE-FGO3-88ER25064.     

Conditions for isonemal arrays on a Cartesian grid

The concept of an isonemal binary array on a Cartesian grid is examined, and rules for constructing all such arrays are identified. Examples are included for the various possible constructions, and tables of compound twillins of periods six and eight are given.     

The index of symmetry of compact naturally reductive spaces

We introduce a geometric invariant that we call the index of symmetry, which measures how far is a Riemannian manifold from being a symmetric space. We compute, in a geometric way, the index of symmetry of compact naturally reductive spaces. In this case, the so-called leaf of symmetry turns out to be of the group type. We also study several examples where the leaf of symmetry is not of the group type. Interesting examples arise from the unit tangent bundle of the sphere of curvature 2, and two metrics in an Aloff-Wallach 7-manifold and the Wallach 24-manifold.     

On Macbeath-Singerman symmetries of Belyi surfaces with PSL(2,p) as a group of automorphisms

The famous theorem of Belyi states that the compact Riemann surface X can be defined over the number field if and only if X can be uniformized by a finite index subgroup Γ of a Fuchsian triangle group Λ. As a result such surfaces are now called Belyi surfaces. The groups PSL(2,q),q=p ⁿ are known to act as the groups of automorphisms on such surfaces. Certain aspects of such actions have been extensively studied in the literature. In this paper, we deal with symmetries. Singerman showed, using acertain result of Macbeath, that such surfaces admit a symmetry which we shall call in this paper the Macbeath-Singerman symmetry. A classical theorem by Harnack states that the set of fixed points of a symmetry of a Riemann surface X of genus g consists of k disjoint Jordan curves called ovals for some k ranging between 0 and g+1. In this paper we show that given an odd prime p, a Macbetah-Singerman symmetry of Belyi surface with PSL(2,p) as a group of automorphisms has at most     

Symmetry classes of tensors and semi-direct product of finite abelian groups

In the study of symmetry classes of tensors, finding examples of symmetry classes of tensors that possess an o*-basis is of considerable interest. There are only few classes of groups that have been provided a necessary and sufficient condition for having such a basis. There is no general criterion for any finite group yet. In this note, we provide a necessary and sufficient condition for the existence of o*-basis of symmetry classes of tensors associated with semi-direct product of some finite abelian groups and, consequently, their wreath product.     

On the equivalence of Lie symmetries and group representations

We consider families of linear, parabolic PDEs in n dimensions which possess Lie symmetry groups of dimension at least four. We identify the Lie symmetry groups of these equations with the (2n+1)-dimensional Heisenberg group and SL(2,R). We then show that for PDEs of this type, the Lie symmetries may be regarded as global projective representations of the symmetry group. We construct explicit intertwining operators between the symmetries and certain classical projective representations of the symmetry groups. Banach algebras of symmetries are introduced.     

Ovaloids and their symmetry types

A convex body B of dimension n in Euclidean n-space Eⁿ (called hereafter an n-body or simply a body) must have at least n+1 extreme points, this minimum number being attained when B is a simplex. On the other hand, it may happen thatevery point of the frontier B of B in Eⁿ is an extreme point. In this case, we call B anovaloid or n-ovaloid. We study the symmetry classification of n-ovaloids in relation to the symmetry classification of convex n-bodies in general. We show first that the set Oⁿ of n-ovaloids is a dense subset of the space Bⁿ of n-bodies. We show also that for any n-body X there is an ovaloid B with the same symmetry group.While two n-bodies may have conjugate symmetry groups without being symmetry equivalent, we prove that this is not so for n-ovaloids: two n-ovaloids with conjugate symmetry groups are necessarily symmetry equivalent. Consequently, the hierarchy of symmetry types of n-ovaloids conforms exactly to the lattice of conjugacy classes of compact subgroups of O(n) that occur as the symmetry group of some n-body. In particular, the spherical n-ball is the only perfect n-ovaloid.     

Group Classification and Exact Solutions of Nonlinear Wave Equations

We perform complete group classification of the general class of quasi linear wave equations in two variables. This class may be seen as a broad generalization of the nonlinear d'Alembert, Liouville, sin/sinh-Gordon and Tzitzeica equations. In this way we derived a number of new genuinely nonlinear invariant models with high symmetry properties. In particular, we obtain four classes of nonlinear wave equations admitting five-dimensional invariance groups. Applying the symmetry reduction technique we construct multi-parameter families of exact solutions of these equations.     

On the Elliptic Genera of Manifolds of Spin(7) Holonomy

Superstring compactification on a manifold of Spin(7) holonomy gives rise to a 2d worldsheet conformal field theory with an extended supersymmetry algebra. The \({\mathcal{N} = 1}\) superconformal algebra is extended by additional generators of spins 2 and 5/2, and instead of just superconformal symmetry one has a c = 12 realization of the symmetry group \({\mathcal{S}W(3/2,2)}\). In this paper, we compute the characters of this supergroup and decompose the elliptic genus of a general Spin(7) compactification in terms of these characters. We find suggestive relations to various sporadic groups, which are made more precise in a companion paper.     

The symmetry approach on economic systems

This paper studies the symmetry on the dynamic equations of economic systems. Symmetry group transforms a solution of the equations to other solutions. Thus we can understand the transformational relations among these solutions. Symmetry describes the invariance of the dynamic equations. Then the symmetry provides more information for the inner structure of the economic systems. The complexity of the socioeconomic phenomena is discussed by using dynamic systems with symmetry. For the calculation of the symmetry, the prolongation technique is adopted. This method reformulates the basic problem of finding solutions of differential equations in a more geometrical form which is suited to the investigation of symmetry groups. Some examples of dynamic equations by using symmetry analysis are also illustrated.     

Maximal Strings in the Crystal Graph of Spin Representations of the Symmetric and Alternating Groups

We define a block-reduced version of the crystal graph of spin representations of the symmetric and alternating groups, and separate it into layers, each obtained by translating the previous layer and, possibly, adding new defect zero blocks. We demonstrate that each layer has weight-preserving central symmetry, and study the sequence of weights occurring in the maximal strings.

The Broué conjecture, that a block with abelian defect group is derived equivalent to its Brauer correspondent, has been proven for blocks of cyclic defect group and verified for many other blocks. This article is part of a study of the spin block case.     

Generating simple convex Venn diagrams

In this paper we are concerned with producing exhaustive lists of simple monotone Venn diagrams that have some symmetry (non-trivial isometry) when drawn on the sphere. A diagram is simple if at most two curves intersect at any point, and it is monotone if it has some embedding on the plane in which all curves are convex. We show that there are 23 such 7-Venn diagrams with a 7-fold rotational symmetry about the polar axis, and that 6 of these have an additional 2-fold rotational symmetry about an equatorial axis. In the case of simple monotone 6-Venn diagrams, we show that there are 39 020 non-isomorphic planar diagrams in total, and that 375 of them have a 2-fold symmetry by rotation about an equatorial axis, and amongst these we determine all those that have a richer isometry group on the sphere. Additionally, 270 of the 6-Venn diagrams also have the 2-fold symmetry induced by reflection about the center of the sphere.Since such exhaustive searches are prone to error, we have implemented the search in a couple of ways, and with independent programs. These distinct algorithms are described. We also prove that the Grünbaum encoding can be used to efficiently identify any monotone Venn diagram.     

Forced symmetry breaking of homoclinic cycles in a PDE with O(2) symmetry

We perform a numerical study of solutions near homoclinic orbits for forced symmetry breaking of a PDE with O(2) symmetry to one with SO(2) symmetry. Taking particular care of the consequences of the continuous group action, we concentrate on the Kuramoto-Sivashinsky equation with spatially periodic boundary conditions. The breakup of structurally stable homoclinic cycles is investigated via the introduction of flux term that breaks the reflectional symmetry while retaining the translational symmetry. In particular, we note that although Chossat (1993) has proved that generic perturbations cause the appearance of quasiperiodic orbits, for the simplest possible flux terms this is not the case. We compare these results with numerical simulations of a Galerkin approximation of the equations.