Resonances Induced by Broken Symmetry in a System with a Singular Potential

We study a three-dimensional nonrelativistic quantum system with a delta-type potential. The support of this potential is determined by a circle and straight line in ${\mathbb{R}^3}$ . We show that a special symmetry of our system induces embedded eigenvalues, and breaking this symmetry leads to resonances.     

Covariant Isotropy of Grothendieck Toposes and Extensive Categories

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. In order to do so, we first extend previous techniques for computing covariant isotropy from locally finitely presentable categories to locally presentable categories. As a consequence, we also obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor. We conclude by providing a more categorical approach to show that these characterizations also extend to any extensive category.

     

Coloring Quasicrystals with Prescribed Symmetries and Frequencies

We show how to color the tiles in a heirarchical tiling system so that the resulting system is not only repetitive (i.e., has the local isomorphism property) but has prescribed color symmetries as well. Received March 9, 1998, and in revised form August 3, 1998.     

Symmetry and Localization in Periodic Crystals: Triviality of Bloch Bundles with a Fermionic Time-Reversal Symmetry

Acta Applicandae Mathematicae - We describe some applications of group- and bundle-theoretic methods in solid state physics, showing how symmetries lead to a proof of the localization of electrons...     

偏序集中的弱理想和弱滤子

在偏序集上利用上集算子和下集算子引入了弱理想,弱滤子,弱素理想,弱素滤子,弱极大理想和弱极大滤子等概念,研究了它们的若干性质,同时给出了偏序集上的(DPI),(BPI),(DMI),(BUF)公理并建立了它们与Zorn引理之问的相互关系.     

The Riemannian Manifolds with Boundary and Large Symmetry

Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most 1/2 dim M（dim M - 1）. Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.     

Generalized Quadrangles with a Spread of Symmetry

We present a common construction for some known infinite classes of generalized quadrangles. Whether this construction yields other (unknown) generalized quadrangles is an open problem. The class of generalized quadrangles obtained this way is characterized in two different ways. On the one hand, they are exactly the generalized quadrangles having a spread of symmetry. On the other hand, they can be characterized in terms of the group of projectivities with respect to a spread. We explore some properties of these generalized quadrangles. All these results can be applied to the theory of the glued near hexagons, a class of near hexagons introduced by the author in De Bruyn (1998) On near hexagons and spreads of generalized quadrangles, preprint.     

Symmetry analysis of differential equations with Mathematica

We will discuss three different methods for finding symmetry solutions based on the Fréchet derivative common to each procedure. The methods discussed are Lie's standard procedure of symmetry analysis, the nonclassical method, and the derivation of potential symmetries. A ferromagnet in a strong external field represented by a nonlinear telegraph equation serves as an example describing the application of all three methods. The symmetry methods discussed are realized in a Mathematica package called MathLie performing all of the required calculations.     

Drawing Trees with Perfect Angular Resolution and Polynomial Area

We study methods for drawing trees with perfect angular resolution, i.e., with angles at each node $v$ equal to $2\pi /d(v)$ . We show:

1. Any unordered tree has a crossing-free straight-line drawing with perfect angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each edge is represented by a circular arc) with perfect angular resolution and polynomial area.

Thus, our results explore what is achievable with straight-line drawings and what more is achievable with Lombardi-style drawings, with respect to drawings of trees with perfect angular resolution.     

Isotropy Representations, Eigenvalues of a Casimir Element, and Commutative Lie Subalgebras

Let h be a reductive subalgebra of a semisimple Lie algebrag and C_h U(h) be the Casimir element determined by the restrictionof the Killing form on g to h. The paper studies eigenvaluesof C_h on the isotropy representation mg/h. Some general estimatesconnecting the eigenvalues and the Dynkin indices of m are given.If h is a symmetric subalgebra, it is shown that describingthe maximal eigenvalue of C_h on exterior powers of m is connectedwith possible dimensions of commutative Lie subalgebras in m,thereby extending a result of Kostant. In this situation, aformula is also given for the maximal eigenvalue of C_h on m.More generally, a similar picture arises if h = g, where isan automorphism of finite order m and m is replaced by the eigenspaceof corresponding to a primitive mth root of unity.     

Reduction and Reconstruction Aspects of Second-Order Dynamical Systems with Symmetry

We examine the reduction process of a system of second-order ordinary differential equations which is invariant under a Lie group action. With the aid of connection theory, we explain why the associated vector field decomposes in three parts and we show how the integral curves of the original system can be reconstructed from the reduced dynamics. An illustrative example confirms the results.      

Symmetry classes of tensors associated with certain groups

We discuss the existence of an orthogonal basis consisting of decomposable vectors for some symmetry classes of tensors associated with certain subgroups of the full symmetric group The dimensions of these symmetry classes of tensors are also given     

弱Orlicz鞅空间的弱原子分解及其应用

对3类由凹函数生成的弱Orlicz鞅空间建立了相应的弱原子分解.作为应用,首先给出了这些弱Orlicz鞅空间上次线性算子有界的一个充分条件,并在此基础上证明了一些弱型鞅不等式,然后证明了关于这些弱Orlicz鞅空间的Marcinkiewicz型插值定理.     

Fundamental Groups of Manifolds with Positive Sectional Curvature and Torus Symmetry

In 1965, Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories.