Showing papers on "Symmetry (geometry) published in 2000"

A remarkable periodic solution of the three-body problem in the case of equal masses

Abstract: Using a variational method, we exhibit a surprisingly simple periodic orbit for the newtonian problem of three equal masses in the plane. The orbit has zero angular momentum and a very rich symmetry pattern. Its most surprising feature is that the three bodies chase each other around a flxed eight-shaped curve. Setting aside collinear motions, the only other known motion along a flxed curve in the inertial plane is the \Lagrange relative equilibrium" in which the three bodies form a rigid equilateral triangle which rotates at constant angular velocity within its circumscribing circle. Our orbit visits in turns every \Euler conflguration" in which one of the bodies sits at the midpoint of the segment deflned by the other two (Figure 1). Numerical computations

Model Sets: A Survey

Abstract: Even when reduced to its simplest form, namely that of point sets in euclidean space, the phenomenon of genuine quasi-periodicity appears extraordinary Although it seems unfruitful to try and define the concept precisely, the following properties may be considered as representative: discreteness; extensiveness; finiteness of local complexity; repetitivity; diffractivity; aperiodicity; existence of exotic symmetry (optional)

Bochner-Kahler metrics

Abstract: A Kahler metric is said to be Bochner-Kahler if its Bochner curvature vanishes. This is a nontrivial condition when the complex dimension of the underlying manifold is at least 2. In this article it will be shown that, in a certain well-defined sense, the space of Bochner-Kahler metrics in complex dimension n has real dimension n+1 and a recipe for an explicit formula for any Bochner-Kahler metric will be given. It is shown that any Bochner-Kahler metric in complex dimension n has local (real) cohomogeneity at most n. The Bochner-Kahler metrics that can be `analytically continued' to a complete metric, free of singularities, are identified. In particular, it is shown that the only compact Bochner-Kahler manifolds are the discrete quotients of the known symmetric examples. However, there are compact Bochner-Kahler orbifolds that are not locally symmetric. In fact, every weighted projective space carries a Bochner-Kahler metric. The fundamental technique is to construct a canonical infinitesimal torus action on a Bochner-Kahler metric whose associated momentum mapping has the orbits of its symmetry pseudo-groupoid as fibers.

The “hot spots” conjecture for domains with two axes of symmetry

Abstract: Consider a convex planar domain with two axes of symmetry. We show that the maximum and minimum of a Neumann eigenfunction with lowest nonzero eigenvalue occur at points on the boundary only. We deduce J. Rauch's "hot spots" conjecture in the following form. If the initial temperature distribution is not orthogonal to the first nonzero eigenspace, then the point at which the temperature achieves its maximum tends to the boundary. In fact the maximum point reaches the boundary in finite time if the boundary has positive curvature. Results of this type have already been proved by Bafiuelos and Burdzy [BB] using the heat equation and probabilistic methods to deform initial conditions to eigenfunctions. We introduce here a new technique based on deformation of the domain. An advantage of our method is that it works even in the case of multiple eigenvalues. On the way toward our results, we prove monotonicity properties for Neumann eigenfunctions for symmetric domains that need not be convex and deduce a sharp comparison of eigenvalues with the Dirichlet problem of independent interest.

Symmetries and first integrals of ordinary difference equations

Abstract: This paper describes a new symmetry-based approach to solving a given ordinary difference equation. By studying the local structure of the set of solutions, we derive a systematic method for determining one-parameter Lie groups of symmetries in closed form. Such groups can be used to achieve successive reductions of order. If there are enough symmetries, the difference equation can be completely solved. Several examples are used to illustrate the technique for transitive and intransitive symmetry groups. It is also shown that every linear second-order ordinary difference equation has a Lie algebra of symmetry generators that is isomorphic to sl(3). The paper concludes with a systematic method for constructing first integrals directly, which can be used even if no symmetries are known.

Calabi-Yau manifolds over finite fields. 1.

Abstract: We study zeta-functions for a one parameter family of quintic threefolds defined over finite fields and for their mirror manifolds and comment on their structure. The zeta-function for the quintic family involves factors that correspond to a certain pair of genus 4 Riemann curves. The appearance of these factors is intriguing since we have been unable to `see' these curves in the geometry of the quintic. Having these zeta-functions to hand we are led to comment on their form in the light of mirror symmetry. That some residue of mirror symmetry survives into the zeta-functions is suggested by an application of the Weil conjectures to Calabi-Yau threefolds: the zeta-functions are rational functions and the degrees of the numerators and denominators are exchanged between the zeta-functions for the manifold and its mirror. It is clear nevertheless that the zeta-function, as classically defined, makes an essential distinction between Kahler parameters and the coefficients of the defining polynomial. It is an interesting question whether there is a `quantum modification' of the zeta-function that restores the symmetry between the Kahler and complex structure parameters. We note that the zeta-function seems to manifest an arithmetic analogue of the large complex structure limit which involves 5-adic expansion.

Host−Guest Symmetry and Charge Matching in Two Germanates with Intersecting Three-Dimensional Channels

Abstract: Two germanates reported here illustrate important effects of host−guest symmetry matching (or mismatching) on the host framework symmetry, the ordering of extraframework species, and the coordination number of germanium. For a strong structure-directing agent, the symmetry of the host framework is dictated by the symmetry of the ordered guest molecules, whereas for a relatively weaker structure-directing agent, the siting of guest molecules must conform to the symmetry of the host framework through specific molecular conformation or multiple orientations.

Intriguing Gold TrifluorideMolecular Structure of Monomers and Dimers: An Electron Diffraction and Quantum Chemical Study

Abstract: The molecular geometry of monomeric and dimeric gold trifluoride, AuF3 and Au2F6, has been determined by gas-phase electron diffraction and high-level quantum chemical calculations. Both experiment and computation indicate that the ground-state structure of AuF3 has C2v symmetry, rather than 3-fold symmetry, with one shorter and two longer Au−F bonds and an almost T-shaped form, due to a first-order Jahn−Teller effect. CASSCF calculations show the triplet D3h symmetry structure, 3A‘, to lie about 42 kcal/mol above the 1A1 symmetry ground state and the D3h symmetry singlet, 1A‘, even higher than the triplet state, by about a further 13 kcal/mol. The molecule has a typical “Mexican-hat”-type potential energy surface with three equal minimum-energy structures around the brim of the hat, separated by equal-height transition structures, about 3.6 kcal/mol above the minimum energy. The geometry of the transition structure has also been calculated. The dimer has a D2h symmetry planar, halogen-bridged geometry, w...

Monotonicity and symmetry results for $p$-Laplace equations and applications

Abstract: In this paper we prove monotonicity and symmetry properties of positive solutions of the equation $ - div (|Du|^{p-2}Du)=f(u)$, $1 0 $, $ c \geq 0 $, $ q \geq p-1 $. As a consequence we get an extension to the $p$--Laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.

A study on the fourth q-Painlev\'e equation

Abstract: A q-difference analogue of the fourth Painleve equation is proposed. Its symmetry structure and some particular solutions are investigated.

Solutions of equivariance for a polynomial-like iterative equation

Abstract: Using fixed point theorems we discuss continuous solutions of Γ-equivariance for a polynomial-like iterative equation on the real line, where Γ is a closed subgroup of the general linear group GL(R). Our main results guarantee the existence of solutions with certain kinds of symmetry. We show that, under restrictive hypotheses, similar results can be proved in a higher-dimensional case, where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in N-dimensional Euclidean space.

Virtual symmetry reduction

Abstract: We provide a general method for ameliorating state explosion via symmetry reduction in certain asymmetric systems, such as systems with many similar, but not identical, processes. The method applies to systems whose structures (i.e., state transition graphs) have more state symmetries than arc symmetries. We introduce a new notion of "virtual symmetry" that strictly subsumes earlier notions of "rough symmetry" and "near symmetry" (Emerson and Trefler, 1999). Virtual symmetry is the most general condition under which the structure of a system is naturally bisimilar to its quotient by a group of state symmetries. We give several example systems exhibiting virtual symmetry that are not amenable to symmetry reduction by earlier techniques: a one-lane bridge system, where the direction with priority for crossing changes dynamically; an abstract system with asymmetric communication network; and a system with asymmetric resource sharing motivated from the drinking philosophers problem. These examples show that virtual symmetry reduction applies to a significantly broader class of asymmetric systems than could be handled before.

Root lattices and quasicrystals

Abstract: It is shown how root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models. All non-periodic symmetries observed so far are covered in minimal embedding with maximal symmetry.

Point symmetries in the Hartree-Fock approach. I. Densities, shapes, and currents

Abstract: Three mutually perpendicular symmetry axes of the second order, inversion, and time reversal can be used to construct a double point group denoted by ${D}_{2h}^{\mathrm{TD}}.$ Properties of this group are analyzed in relation to the symmetry and symmetry-breaking effects within the mean-field (Hartree-Fock) theories, both in even and odd fermion systems. We enumerate space symmetries of local one-body densities, and symmetries of electromagnetic moments, that appear when some or all of the ${D}_{2h}^{\mathrm{TD}}$ elements represent self-consistent mean-field symmetries.

On the Zagreb Indices as Complexity Indices

Abstract: Two original Zagreb indices, denoted M(1) and M(2), and introduced in 1972, were symmetry-modified by summing up only degrees (SMM(1)) or edge-weights (SMM(2)) of symmetry nonequivalent vertices or edges of graphs. Their dependence on the structural features and symmetry of molecular graphs is studied. They were also compared to eight other complexity indices (RCI, TC, TC1, BT, BI, twc, wcx) on nine graphs (A, B, C, D, E, F, G, H, I) with five vertices that were earlier studied by several research groups. The TC, TC1 and Nt produce exactly the same complexity ordering of nine graphs (I > H > G > F > E > D > C > B > A). The ordering produced by M(1) is different from this ordering in that it cannot discriminate E and F, and C and D. Likewise, M(2) and twc produce exactly the same ordering and the latter ordering differs from the former only in the reverse order of E and F. Orderings produced by SMM(1) and SMM(2) differ considerably from orderings given by TC, TC1 and N(t) or M(2) and twc.

RSPS version 4.0: a semi-interactive vector-search program for solving heavy-atom derivatives

Abstract: A program for inspection and interpretation of the Patterson function is described. The program is mainly intended for finding heavy-atom positions from difference Patterson maps, but may also be used to locate molecules with non-crystallographic symmetry when the local axis is nearly parallel to a crystallographic symmetry axis. Options are available for vector-based methods to locate heavy-atom sites, for finding sets from a list of possible heavy-atom positions and for checking of potential solutions. Both crystallographic and non-crystallographic symmetry may be used, either independently or in conjunction.

Symmetry Breaking in Software Patterns

Abstract: Patterns have a longstanding identity in the scientific community as results of a phenomenon called symmetry breaking. This article proposes a formalism for software patterns through connections from software patterns to symmetry and symmetry breaking. Specifically, we show (1) the ties from Alexander's work to symmetry and symmetry-breaking foundations; (2) many programming languages provide constructs that support symmetry; (3) software patterns are the results of symmetry breaking, compensating for design shortfalls in programming languages. The proposed pattern formalism may be useful as a foundation for pattern taxonomies, and to differentiate patterns as a design discipline from heuristics, rules, and arbitrary micro-architectures.

Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles

Abstract: Wigner's classical theorem on symmetry transformations plays a fundamental role in quantum mechanics. It can be formulated, for example, in the following way: Every bijective transformation on the set L of all 1-dimensional subspaces of a Hilbert space H which preserves the angle between the elements of L is induced by either a unitary or an antiunitary operator on H. The aim of this paper is to extend Wigner's result from the 1-dimensional case to the case of n-dimensional subspaces of H with n fixed.

In Our Own Image: Personal Symmetry in Discovery

Abstract: Preface. 1. Introduction. 2. Perception and Symmetry. 3. Johannes Kepler. 4. Buckminster Fuller. 5. Linus Pauling. 6. Aleksandr Kitaiigorodskii. 7. Desmond Bernal. 8. Pierre Curie. Epilogue. Index.

Ground states of semilinear elliptic equations: a geometric approach

Abstract: We consider the problem Δu+u p +u q =0, in R N 0 as |x|→+∞, where 1 N/(N−2) and then let q approach (N+2)/(N−2) . If we fix q and then let p be close enough to N/(N−2) then no solutions exist.