Topologically Torsion Elements of the Circle Group

Let (m _n ) be a faithfully enumerated sequence of integers with m _n  | m _n+1 for every n ∈ ?. We describe the topologically (m _n )-torsion elements of the circle group 𝕋 = ?/? (written additively), namely, those elements x ∈ 𝕋 such that m _n x coverges to 0.     

The topology of a semisimple Lie group is essentially unique

We study locally compact group topologies on simple and semisimple Lie groups. We show that the Lie group topology on such a group S is very rigid: every “abstract” isomorphism between S and a locally compact and σ-compact group Γ is automatically a homeomorphism, provided that S is absolutely simple. If S is complex, then noncontinuous field automorphisms of the complex numbers have to be considered, but that is all. We obtain similar results for semisimple groups.     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group

Let $G$ be a second countable locally compact abelian group. The aim of this paper is to characterize the left translates on the Heisenberg group $\mathbb{H}\left(G\right)$ to be frames and Riesz bases in terms of the group Fourier transform.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

The structure of shift invariant spaces on a locally compact abelian group

We investigate shift invariant subspaces of L²(G), where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.     

A Note on the Uniqueness of Koebe–Andreev–Thurston Theorem

In this paper, we present a new proof of the uniqueness of Koebe-Andreev-Thurston theorem. Our method is based on the argument principle in complex analysis and reviews the connection between the circle packing theorem and complex analysis.     

A geometric characterization of Vassiliev invariants

It is a well-known paradigm to consider Vassiliev invariants as polynomials on the set of knots. We prove the following characterization: a rational knot invariant is a Vassiliev invariant of degree if and only if it is a polynomial of degree on every geometric sequence of knots. Here a sequence with is called geometric if the knots coincide outside a ball , inside of which they satisfy for all and some pure braid . As an application we show that the torsion in the braid group over the sphere induces torsion at the level of Vassiliev invariants: there exist knots in that can be distinguished by -invariants of finite type but not by rational invariants of finite type. In order to obtain such torsion invariants we construct over a universal Vassiliev invariant of degree for knots in .

     

A characterization of the simple group PSL<Subscript>5</Subscript>(5) by the set of its element orders

Let G be a finite group and let ω(G) denote the set of the element orders of G. For the simple group PSL₅(5) we prove that if G is a finite group with ω(G) = ω(PSL₅(5)), then either G ? PSL₅(5) or G ? PSL₅(5): 〈θ〉 where θ is a graph automorphism of PSL₅(5) of order 2.     

Products of roots of the identity

It is proved that every invertible bounded linear operator on a complex infinite-dimensional Hilbert space is a product of five -th roots of the identity for every 2$">. For invertible normal operators four factors suffice in general.

     

A new characterization of Suzuki-Ree group

It is proved that a Suzuki-Ree group can he characterized by the bet of its order components Project partially supported by the National Natural Science Foundation of China and Ph. D. Foundation of Southwest China Normal University.     

A heuristic for the circle packing problem with a variety of containers

In this paper we present a heuristic algorithm based on the formulation space search method to solve the circle packing problem. The circle packing problem is the problem of finding the maximum radius of a specified number of identical circles that can be fitted, without overlaps, into a two-dimensional container of fixed size. In this paper we consider a variety of containers: the unit circle, unit square, rectangle, isosceles right-angled triangle and semicircle. The problem is formulated as a nonlinear optimization problem involving both Cartesian and polar coordinate systems.Formulation space search consists of switching between different formulations of the same problem, each formulation potentially having different properties in terms of nonlinear optimization. As a component of our heuristic we solve a nonlinear optimization problem using the solver SNOPT.Our heuristic improves on previous results based on formulation space search presented in the literature. For a number of the containers we improve on the best result previously known. Our heuristic is also a computationally effective approach (when balancing quality of result obtained against computation time required) when compared with other work presented in the literature.     

A metric characterization of the irrationals via a group operation

Let S be a separable metric space with a compatible metric d that satisfies: For each point x ? S and each nonnegative real number r there exists a unique point y ? S such that d(x,y) = r.In this paper spaces that meet the above criterion are investigated. It is shown that, under the assumption of completeness, this metric property characterizes the space of irrationals.     

The amenability constant of the Fourier algebra

For a locally compact group , let denote its Fourier algebra and its dual object, i.e., the collection of equivalence classes of unitary representations of . We show that the amenability constant of is less than or equal to and that it is equal to one if and only if is abelian.

     

Capelli elements of the group algebra

Inspired by the Capelli-type identities for group determinants researched by Tôru Umeda, we give Capelli identities for irreducible representations of any finite group, and Capelli elements of the group algebra associated with these identities. These elements construct a basis of the centre of the group algebra.     

A formula for the Whitehead group of a three-dimensional crystallographic group

We give an explicit formula for the Whitehead group of a three-dimensional crystallographic group Γ in terms of the Whitehead groups of the virtually infinite cyclic subgroups of Γ.     

On the Square Integrability of Quasi Regular Representation on Semidirect Product Groups

Let H be a locally compact group and K be a locally compact abelian group. Also let G=H× _τ K denote the semidirect product group of H and K, respectively. Then the unitary representation (U,L ²(K)) on G defined by is called the quasi regular representation. The properties of this representation in the case K=(ℝ ⁿ ,+), have been studied by many authors under some specific assumptions. In this paper we aim to consider a general case and extend some of these properties when K is an arbitrary locally compact abelian group. In particular we wish to show that the two conditions (i) , and (ii) the stabilizers H ^ω are compact for a.e. ; both are necessary for square integrability of U. Furthermore, we shall consider some sufficient conditions for the square integrability of U. Also, for the square integrability of subrepresentations of U, we will introduce a concrete form of the Duflo-Moore operator.