On the representation theory of Möbius groups inR n*

We will solve several fundamental problems of Möbius groupsM(R ⁿ) which have been matters of interest such as the conjugate classification, the establishment of a standard form without finding the fixed points and a simple discrimination method. Let \(g = \left[ {\begin{array}{*{20}c} a &; b \\ c &; d \\ \end{array} } \right]\) be a Clifford matrix of dimensionn, c ≠ 0. We give a complete conjugate classification and prove the following necessary and sufficient conditions:g is f.p.f. (fixed points free) iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|<1 and |E?AE ¹| ≠ 0;g is elliptic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| <1 and |E?AE ¹|=0;g is parabolic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; 0 \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α|=1; andg is loxodromic iff \(g \sim \left[ {\begin{array}{*{20}c} \alpha &; \beta \\ c &; {\alpha '} \\ \end{array} } \right]\) , |α| >1 or rank (E?AE ¹) ≠ rank (E?AE ¹,ac ^?1+c ^?1 d), where α is represented by the solutions of certain linear algebraic equations and satisfies $\left| {c^{ - 1} \alpha '} \right| = \left| {\left( {E - AE^1 } \right)^{ - 1} \left( {\alpha c^{ - 1} + c^{ - 1} \alpha '} \right)} \right|.$      

On Möbius form and Möbius isoparametric hypersurfaces

An umbilic-free hypersurface in the unit sphere is called MSbius isoparametric if it satisfies two conditions, namely, it has vanishing MSbius form and has constant MSbius principal curvatures. In this paper, under the condition of having constant MSbius principal curvatures, we show that the hypersurface is of vanishing MSbius form if and only if its MSbius form is parallel with respect to the Levi-Civita connection of its MSbius metric. Moreover, typical examples are constructed to show that the condition of having constant MSbius principal curvatures and that of having vanishing MSbius form are independent of each other.     

The Möbius invariants for circle packings

Möbius invariants of general circle packings are defined in terms of cross ratios. The necessary and sufficient conditions of existence of circle packings are established by the techniques of Möbius invariants. It is shown that circle packings are uniquely determined, up to Möbius transformations, by their Möbius invariants. The rigidity of infinite circle packings with bounded degree is proved using the approach of Möbius invariants.     

On the Möbius Function of a Pointed Graded Lattice

In this paper, we compute the Möbius function of pointed integer partition and pointed ordered set partition using topological and analytic methods. We show that the associated order complex is a wedge of spheres and compute the associated reduced homology group for each subposet. In addition, we compute the Möbius function of pointed graded lattice and use our method to compute the Möbius function of pointed direct sum decomposition of vector spaces.     

On partial sums of a spectral analogue of the Möbius function

Sankaranarayanan and Sengupta introduced the function μ ^*(n) corresponding to the Möbius function. This is defined by the coefficients of the Dirichlet series 1/L _f (s), where L _f (s) denotes the L-function attached to an even Maaß cusp form f. We will examine partial sums of μ ^*(n). The main result is $\sum_{n\leq x}\mu^{*}(n)=O(x\exp(-A\sqrt{\log x}))$ , where A is a positive constant. It seems to be the corresponding prime number theorem.     

The partition function p(n) in terms of the classical Möbius function

The Ramanujan Journal - In this paper, we investigate decompositions of the partition function p(n) from the additive theory of partitions considering the famous Möbius function $$mu (n)$$...     

On Möbius Duality and Coarse-Graining

We study duality relations for zeta and Möbius matrices and monotone conditions on the kernels. We focus on the cases of families of sets and partitions. The conditions for positivity of the dual kernels are stated in terms of the positive Möbius cone of functions, which is described in terms of Sylvester formulae. We study duality under coarse-graining and show that an \(h\)-transform is needed to preserve stochasticity. We give conditions in order that zeta and Möbius matrices admit coarse-graining, and we prove they are satisfied for sets and partitions. This is a source of relevant examples in genetics on the haploid and multi-allelic Cannings models.     

On the geometry of curves and conformal geodesics in the Möbius space

This article deals with the study of some properties of immersed curves in the conformal sphere \({\mathbb{Q}_n}\), viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein’s Erlangen program. The core of this article is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler–Lagrange equations for any n, we prove an interesting codimension reduction, namely that every conformal geodesic in \({\mathbb{Q}_n}\) lies, in fact, in a totally umbilical 4-sphere \({\mathbb{Q}_4}\). We then extend and complete the work in Musso (Math Nachr 165:107–131, 1994) by solving the Euler–Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.     

On the growth and range of functions in Möbius invariant spaces

This paper is a continuation of our earlier work and focuses on the structural and geometric properties of functions in analytic Besov spaces, primarily on univalent functions in such spaces and their image domains. We improve several earlier results.     

On prisms,Möbius ladders and the cycle space of dense graphs

For a graph $G$, let $\left|G\right|$ denote its number of vertices, $\delta \left(G\right)$ its minimum degree and $Z_1(G;{\mathbb{F}}_2)$ its cycle space. Call a graph Hamilton-generated if and only if every cycle in $G$ is a symmetric difference of some Hamilton circuits of $G$. The main purpose of this paper is to prove: for every $\gamma >0$ there exists $n_0\in \mathbb{Z}$ such that for every graph $G$ with $\left|G\right|⩾n_0$ vertices,

• (1)if $\delta \left(G\right)⩾(\frac{1}{2}+\gamma )\left|G\right|$ and $\left|G\right|$ is odd, then $G$ is Hamilton-generated,
• (2)if $\delta \left(G\right)⩾(\frac{1}{2}+\gamma )\left|G\right|$ and $\left|G\right|$ is even, then the set of all Hamilton circuits of $G$ generates a codimension-one subspace of $Z_1(G;{\mathbb{F}}_2)$ and the set of all circuits of $G$ having length either $\left|G\right|-1$ or $\left|G\right|$ generates all of $Z_1(G;{\mathbb{F}}_2)$,
• (3)if $\delta \left(G\right)⩾(\frac{1}{4}+\gamma )\left|G\right|$ and $G$ is balanced bipartite, then $G$ is Hamilton-generated.

All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [I.B.-A. Hartman, Long cycles generate the cycle space of a graph, European J. Combin. 4 (3) (1983) 237–246] originates with A. Bondy.     

Some remarks on the theorem of Möbius

H.S.M. Coxeter in his book Introduction to Geometry quotes a theorem of Möbius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.     

Factorizations of Möbius Gyrogroups

In this paper we consider a M?bius gyrogroup on a real Hilbert space (of finite or infinite dimension) and we obtain its factorization by gyrosubgroups and subgroups. It is shown that there is a duality relation between the quotient spaces and the orbits obtained. As an example we will present the factorization of the M?bius gyrogroup of the unit ball in \mathbbRⁿ{\mathbb{R}}^{n} linked to the proper Lorentz group Spin⁺(1, n).     

On the Möbius function of the locally finite poset associated with a numerical semigroup

Let S be a numerical semigroup and let (?,≤ _S ) be the (locally finite) poset induced by S on the set of integers ? defined by x≤ _S y if and only if y?x∈S for all integers x and y. In this paper, we investigate the Möbius function associated to (?,≤ _S ) when S is an arithmetic semigroup.     

Möbius characterization of hemispheres

In this paper we generalize the Möbius characterization of metric spheres as obtained in Foertsch and Schroeder [4] to a corresponding Möbius characterization of metric hemispheres.     

On an invariant Möbius measure and the Gauss-Kuzmin face distribution

We study Möbius measures of the manifold of n-dimensional continued fractions in the sense of Klein. By definition any Möbius measure is invariant under the natural action of the group of projective transformations PGL(n + 1) and is an integral of some form of the maximal dimension. It turns out that all Möbius measures are proportional, and the corresponding forms are written explicitly in some special coordinates. The formulae obtained allow one to compare approximately the relative frequencies of the n-dimensional faces of given integer-affine types for n-dimensional continued fractions. In this paper we make numerical calculations of some relative frequencies in the case of n = 2.     

Willmore hypersurfaces with constant Möbius curvature in R n+1

For an immersed hypersurface ${f : M^n \rightarrow R^{n+1}}$ without umbilical points, one can define the Möbius metric g on f which is invariant under the Möbius transformation group. The volume functional of g is a generalization of the well-known Willmore functional, whose critical points are called Willmore hypersurfaces. In this paper, we prove that if a n-dimensional Willmore hypersurfaces ${(n \geq 3)}$ has constant sectional curvature c with respect to g, then c = 0, n = 3, and this Willmore hypersurface is Möbius equivalent to the cone over the Clifford torus in ${S^{3} \subset R^{4}}$ . Moreover, we extend our previous classification of hypersurfaces with constant Möbius curvature of dimension ${n \ge 4}$ to n = 3, showing that they are cones over the homogeneous torus ${S^1(r) \times S^1(\sqrt{1 - r^2}) \subset S^3}$ , or cylinders, cones, rotational hypersurfaces over certain spirals in the space form R ², S ², H ², respectively.