A new characterization of Möbius transformations by use of Apollonius hexagons

The purpose of this paper is to give a new characterization of Möbius transformations from the standpoint of conformal mappings. To this end a new concept of Apollonius hexagons on the complex plane is used.

     

A new characteristic of Möbius transformations by use of polygons having type A

In the hyperbolic plane Möbius transformations can be characterized by Lambert quadrilaterals, i.e., a continuous bijection which maps Lambert quadrilaterals to Lambert quadrilaterals must be Möbius. In this paper we generalize this result to the case of polygons with n sides having type A, that is, having exactly two non-right interior angle.     

A new characteristic of Möbius transformations in hyperbolic geometry

We present a new characterization of Möbius transformations by using two classes of hyperbolic geometric objects: Lambert quadrilaterals and Saccheri quadrilaterals. The proof is based on a geometric approach.     

A characterization of Möbius transformations

Let be an integer and let be a domain of . Let be an injective mapping which takes hyperspheres whose interior is contained in to hyperspheres in . Then is the restriction of a Möbius transformation.

     

A characterization of Möbius transformations by use of hyperbolic regular polygons

In this paper we present a new characterization of Möbius transformations by use of hyperbolic regular polygons.     

Banach algebras with involution and Möbius transformations

This paper introduces the use of Potapov's Möbius transformations in the study of Banach algebras with involution.     

The Möbius Inverse Monoid

We show that the Möbius transformations generate anF-inverse monoid whose maximum group image is the Möbius group. We describe the monoid in terms of McAlister triples.     

Examples of Möbius-like groups which are not Möbius groups

In this paper we give two basic constructions of groups with the following properties:

(a)

, i.e., the group is acting by orientation preserving homeomorphisms on ;

(b)

every element of is Möbius-like;

(c)

, where denotes the limit set of ;

(d)

is discrete;

(e)

is not a conjugate of a Möbius group.

Both constructions have the same basic idea (inspired by Denjoy): we start with a Möbius group (of a certain type) and then we change the underlying circle upon which acts by inserting some closed intervals and then extending the group action over the new circle. We denote this new action by . Now we form a new group which is generated by all of and an additional element whose existence is enabled by the inserted intervals. This group has all the properties (a) through (e).

     

Möbius functions and semigroup representation theory

This paper explores several applications of Möbius functions to the representation theory of finite semigroups. We extend Solomon's approach to the semigroup algebra of a finite semilattice via Möbius functions to arbitrary finite inverse semigroups. This allows us to explicitly calculate the orthogonal central idempotents decomposing an inverse semigroup algebra into a direct product of matrix algebras over group rings. We also extend work of Bidigare, Hanlon, Rockmore and Brown on calculating eigenvalues of random walks associated to certain classes of finite semigroups; again Möbius functions play an important role.     

On the Möbius Algebra of Geometric Lattices

The Möbius algebra of a poset was introduced by Solomon and also studied by Greene in the special case of lattices. Let denote a geometric lattice. Our key idea is to consider all the characteristic polynomials of upper intervals in as components of one object, which is multiplicative. Now, by simple algebraic means we obtain new identities involving the characteristic polynomial and the Tutte polynomial of a geometric lattice.     

Bounded Functions in Möbius Invariant Dirichlet Spaces

Forp∈(0, 1), letQ_p(Q_{p, 0}) be the space of analytic functionsfon the unit diskΔwith sup_w∈Δ ‖f°?_w‖_p<∞ (lim_|w|→1 ‖f°?_w‖_p=0), where ‖·‖_pmeans the weighted Dirichlet norm and?_wis the Möbius map ofΔonto itself with?_w(0)=w. In this paper, we prove the Corona theorem for the algebraQ_p∩H^∞(Q_{p, 0}∩H^∞); then we provide a Fefferman–Stein type decomposition forQ_p(Q_{p, 0}), and finally we describe the interpolating sequences forQ_p∩H^∞(Q_{p, 0}∩H^∞)).     

The Möbius function and the residue theorem

A classical conjecture of Bouniakowsky says that a non-constant irreducible polynomial in Z[T] has infinitely many prime values unless there is a local obstruction. Replacing Z[T] with κ[u][T], where κ is a finite field, the obvious analogue of Bouniakowsky's conjecture is false. All known counterexamples can be explained by a new obstruction, and this obstruction can be used to fix the conjecture. The situation is more subtle in characteristic 2 than in odd characteristic. Here, we illustrate the general theory for characteristic 2 in some examples.     

Möbius inversion formulas for flows of arithmetic semigroups

We define a convolution-like operator which transforms functions on a space X via functions on an arithmetical semigroup S, when there is an action or flow of S on X. This operator includes the well-known classical Möbius transforms and associated inversion formulas as special cases. It is defined in a sufficiently general context so as to emphasize the universal and functorial aspects of arithmetical Möbius inversion. We give general analytic conditions guaranteeing the existence of the transform and the validity of the corresponding inversion formulas, in terms of operators on certain function spaces. A number of examples are studied that illustrate the advantages of the convolutional point of view for obtaining new inversion formulas.     

符号空间的拟移位和Moebius带上的奇怪吸引子

给出了双边符号空间上的一种新的拟移位映射，得到了它与传统的移位映射拓扑共轭．类似Smale马蹄，对这种拟移位映射给出了一个模型，即在Moebius带上给出一类映射．同时刻划了这类映射的吸引子的结构及动力学行为．     

Deformation of hypersurfaces preserving the Möbius metric and a reduction theorem

A hypersurface without umbilics in the (n+1)$(n+1)$-dimensional Euclidean space f:Mⁿ→Rⁿ⁺¹$f:M^n\to R^{n+1}$ is known to be determined by the Möbius metric g and the Möbius second fundamental form B   up to a Möbius transformation when n?3$n?3$. In this paper we consider Möbius rigidity for hypersurfaces and deformations of a hypersurface preserving the Möbius metric in the high dimensional case n?4$n?4$. When the highest multiplicity of principal curvatures is less than n−2$n-2$, the hypersurface is Möbius rigid. When the multiplicities of all principal curvatures are constant, deformable hypersurfaces and the possible deformations are also classified completely. In addition, we establish a reduction theorem characterizing the classical construction of cylinders, cones, and rotational hypersurfaces, which helps to find all the non-trivial deformable examples in our classification with wider application in the future.     

广义的Möbius反演公式及其在一个物理逆问题中的应用

用偏序集上广义的Möbius反演公式去求解一类物理逆问题(晶体对势反演).这种方法是解决此类问题的一般性数学方法.文章中给出的两个应用实例说明了这种方法的有效性.     

On iterates of Möbius transformations on fields

Let be a quadratic polynomial over a splitting field , and be the set of zeros of . We define an associative and commutative binary relation on so that every Möbius transformation with fixed point set is of the form ``plus' for some . This permits an easy proof of Aitken acceleration as well as generalizations of known results concerning Newton's method, the secant method, Halley's method, and higher order methods. If is equipped with a norm, then we give necessary and sufficient conditions for the iterates of a Möbius transformation to converge (necessarily to one of its fixed points) in the norm topology. Finally, we show that if the fixed points of are distinct and the iterates of converge, then Newton's method converges with order 2, and higher order generalizations converge accordingly.