Holditch Theorem and Steiner Formula for the Planar Hyperbolic Motions

The Steiner formula and the Holditch Theorem for one-parameter closed planar Euclidean motions [1, 5] were expressed by Müller, under the one-parameter closed planar motions in the complex sense. Also, Müller had given Holditch theorem in the complex sense, [6].     

One-Parameter Plane Hyperbolic Motions

Müller [3], in the Euclidean plane , introduced the one parameter planar motions and obtained the relation between absolute, relative, sliding velocities (and accelerations). Also, Müller [11] provided the relation between the velocities (in the sense of Complex) under the one parameter motions in the Complex plane . Ergin [7] considering the Lorentzian plane , instead of the Euclidean plane , and introduced the one-parameter planar motion in the Lorentzian plane and also gave the relations between both the velocities and accelerations. In analogy with the Complex numbers, a system of hyperbolic numbers can be introduced: . Complex numbers are related to the Euclidean geometry, the hyperbolic system of numbers are related to the pseudo-Euclidean plane geometry (space-time geometry), [5,15]. In this paper, in analogy with Complex motions as given by Müller [11], one parameter motions in the hyperbolic plane are defined. Also the relations between absolute, relative, sliding velocities (and accelerations) and pole curves are discussed.      

One-Parameter Homothetic Motions and Euler-Savary Formula in Generalized Complex Number Plane $${\mathbb{C}_{J}}$$

In this paper, we introduce one-parameter homothetic motions in the generalized complex number plane (\({\mathfrak{p}}\)-complex plane)

$$\mathbb{C}_{J}=\left\{x+Jy:\,\,\, x,y \in \mathbb{R},\quad J^2=\mathfrak{p},\quad \mathfrak{p} \in \{-1,0,1\} \right\} \subset \mathbb{C}_\mathfrak{p}$$

where

$$\mathbb{C}_\mathfrak{p}=\{x+Jy:\,\,\, x,y \in \mathbb{R}, \quad J^2=\mathfrak{p}\}$$

such that \({-\infty < \mathfrak{p} < \infty}\). The velocities, accelerations and pole points of the motion are analysed. Moreover, three generalized complex number planes, of which two are moving and the other one is fixed, are considered and a canonical relative system for one-parameter planar homothetic motion in \({\mathbb{C}_{J}}\) is defined. Euler-Savary formula, which gives the relationship between the curvatures of trajectory curves, during the one-parameter homothetic motions, is obtained with the aim of this canonical relative system.

     

Topologisch quasinormale Untergruppen zusammenhängender lokalkompakter Gruppen

A closed subgroupQ of a topological groupG is called topologically quasinormal (tqn) inG if holds for every closed subgroupA ofG. We show that every tqn subgroup of a connected locally compact group is actually a normal subgroup. Besides we prove: a homogeneous spaceG/H of a connected Lie groupG with the property that every non-trivial one-parameter orbit is dense has dimension at most one.     

Areas of two-dimensional moduli spaces

Wolpert's formula expresses the Weil-Petersson -form in terms of the Fenchel-Nielsen coordinates in case of a closed or punctured surface. The area-form in Fenchel-Nielsen coordinates is invariant under the mapping class group on each 2-dimensional Teichmüller space of a surface with singularities, hence areas with respect to it can be calculated for 2-dimensional moduli spaces in cases when the Teichmüller space admits global Fenchel-Nielsen coordinates: The area of the moduli space for the signature is , the definition of signature is generalized to include punctures, cone points and geodesic boundary curves. In case the surface is represented by a Fuchsian group, the area is the classical Weil-Petersson area.     

A New Generalization of the Steiner Formula and the Holditch Theorem

In this study, we first obtained the Steiner area formula in the generalized complex plane. Then, with the aid of this formula, we determined a new approach for the Holditch theorem giving the relationship between the areas formed by points in the generalized complex plane (or p-complex plane). Finally, according to the special values of p = ?1, 0, 1 we examined the cases of the Steiner Formula and Holditch Theorem. In this way, for \({p \in \mathbb{R}}\) we generalized the Steiner Formula and Holditch theorem consisting the Euclidean \({\left({p = -1}\right)}\), Galilean \({\left({p = 0}\right)}\) and Lorentzian \({\left({p = 1}\right)}\) cases.     

Some prime factorization results for type II<Subscript>1</Subscript> factors

We prove several unique prime factorization results for tensor products of type II₁ factors coming from groups that can be realized either as subgroups of hyperbolic groups or as discrete subgroups of connected Lie groups of real rank 1. In particular, we show that if is isomorphic to a subfactor in , for some 2r_i,s_j, then mn. Mathematics Subject Classification (2000) Primary 46L10; Secondary 20F67     

The minimal genus problem in rational surfaces CP2 # n[`(CP2 )]CP^2 \# n\overline {CP^2 }

There are three key ingredients in the study of the minimal genus problem for rational surfaces : the generalized adjunction formula, the action of the orthogonal group of the Lorentz space and the geometric construction. In this paper, we prove the uniqueness of the standard form (see Definition 1.1 and Theorem 1.1) of a 2-dimensional homology class under the action of the subgroup of the Lorentz orthogonal group that is realized by the diffeomorphisms of . Using the geometric construction, we determine the minimal genera of some classes (see Theorem 1.2).     

Determining classes of convex bodies by restricted sets of Steiner symmetrizations

In [4] it was shown that a convex body in R ^d (d≥2) is a simplex if and only if each of its Steiner symmetrals has exactly two extreme points outside the corresponding symmetrization space. A natural question arises about restricted sets of symmetrization directions which guarantee this characterization of simplices. Let denote an arbitrary triple of pairwisedistinct great (d-2)-spheres on the unit sphere of R ^d .We shall prove that a convex body K is a simplex if and only if for every direction the corresponding Steiner symmetral of K has the property described above. Weaker conditions characterize additional classes of convex bodies, e.g. (d-2)-fold pyramids over planar, convex 4-gons.     

Distances Between <Emphasis Type="Italic">σ</Emphasis>-Fields on a Probability Space

For a probability space (Ω,ℱ,P) and two sub-σ-fields we consider two natural distances: and . We investigate basic properties of these distances. In particular we show that if a distance (ρ or ) from ℬ to is small then there exists Z∈ℱ with small P(Z), such that for every B∈ℬ there exists such that B∖Z and A∖Z differ by a set of probability zero. This improves results of Neveu (Ann. Math. Stat. 43(4):1369–1371, [1972]), Jajte and Paszkiewicz (Probab. Math. Stat. 19(1):181–201, [1999]).      

Topological dimensions and multifractal Rényi dimensions of Polish spaces: a multifractal Szpilrajn type theorem

In this paper we consider the relationship between the topological dimension and the lower and upper q-Rényi dimensions and of a Polish space X for q ∈ [1, ∞]. Let and denote the Hausdorff dimension and the packing dimension, respectively. We prove that for all analytic metric spaces X (whose upper box dimension is finite) and all q ∈ (1, ∞); of course, trivially, for all q ∈ [1, ∞]. As a corollary to this we obtain the following result relating the topological dimension and the lower and upper q-Rényi dimensions: for all Polish spaces X and all q ∈ [1, ∞]; in (1) and (2) we have used the following notation, namely, for two metric spaces X and Y, we write X ∼ Y if and only if X is homeomorphic to Y. Equality (1) has recently been proved for q = ∞ by Myjak et al. Author’s address: Department of Mathematics, University of St. Andrews, St. Andrews, Fife KY16 9SS, Scotland     

A Topological View of Ramsey Families of Finite Subsets of Positive Integers

If is an initially hereditary family of finite subsets of positive integers (i.e., if and G is initial segment of F then ) and M an infinite subset of positive integers then we define an ordinal index . We prove that if is a family of finite subsets of positive integers such that for every the characteristic function χ_F is isolated point of the subspace

Graph homology of the moduli space of pointed real curves of genus zero

The moduli space parameterizes the isomorphism classes of S-pointed stable real curves of genus zero which are invariant under relabeling by the involution σ. This moduli space is stratified according to the degeneration types of σ-invariant curves. The degeneration types of σ-invariant curves are encoded by their dual trees with additional decorations. We construct a combinatorial graph complex generated by the fundamental classes of strata of . We show that the homology of is isomorphic to the homology of our graph complex. We also give a presentation of the fundamental group of .      

A hyperbolicity criterion for subgroups of $\mathcal{RF}(G)$

This paper continues the investigation of the groups RF(G)\mathcal{RF}(G) first introduced in the forthcoming book of Chiswell and Müller “A Class of Groups Universal for Free ℝ-Tree Actions” and in the article by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009). We establish a criterion for a family {H_s}\{\mathcal{H}_{\sigma}\} of hyperbolic subgroups H_s ￡ RF(G)\mathcal{H}_{\sigma}\leq\mathcal{RF}(G) to generate a hyperbolic subgroup isomorphic to the free product of the H_s\mathcal{H}_{\sigma} (Theorem 1.2), as well as a local-global principle for local incompatibility (Theorem 4.1). In conjunction with the theory of test functions as developed by Müller and Schlage-Puchta (Abh. Math. Semin. Univ. Hambg. 79:193–227, 2009), these results allow us to obtain a necessary and sufficient condition for a free product of real groups to embed as a hyperbolic subgroup in RF(G)\mathcal{RF}(G) for a given group G (Corollary 5.4). As a further application, we show that the centralizers associated with a family of pairwise locally incompatible cyclically reduced functions in RF(G)\mathcal{RF}(G) generate a hyperbolic subgroup isomorphic to the free product of these centralizers (Corollary 5.2).     

Subwords in Reverse-Complement Order

We examine finite words over an alphabet of pairs of letters, where each word w₁w₂ ... w_t is identified with its reverse complement where ( ). We seek the smallest k such that every word of length n, composed from Γ, is uniquely determined by the set of its subwords of length up to k. Our almost sharp result (k~ 2n = 3) is an analogue of a classical result for “normal” words. This problem has its roots in bioinformatics. Received October 19, 2005     

The Generalized Holditch Theorem for the Homothetic Motions on the Planar Kinematics

W. Blaschke and H. R. Müller [4, p. 142] have given the following theorem as a generalization of the classic Holditch Theorem: Let E/E be a 1-parameter closed planar Euclidean motion with the rotation number and the period T. Under the motion E/E, let two points A = (0, 0), B = (a + b, 0) E trace the curves k _A, k _B E and let F _A, F _B be their orbit areas, respectively. If F _X is the orbit area of the orbit curve k of the point X = (a, 0) which is collinear with points A and B then

Velocity Averaging Lemmas in Hyperbolic Sobolev Spaces for the Kinetic Transport Equation with Velocity Field on the Sphere

We show how the methods of [6–8] can be used to prove velocity averaging lemmas in hyperbolic Sobolev spaces for the kinetic transport equation . Here v is allowed to vary in the whole space and the velocity field lies on the unit sphere. We work in dimensions and, in contrast with [6, 8], we allow right-hand sides with velocity derivatives in any direction and not necessarily tangential to the sphere.      

On Limit Sets and Discreteness Criteria for Nonelementary Subgroups of M（R^n）

In this paper, we will study the nonelementary groups of MSbius transformations in R^n and some properties are obtained. Also in this paper we will prove several theorems about discreteness criteria and group convergence of nonelementary groups of M（R^n）.     

On quasiconformal deformations of the universal hyperbolic solenoid

We investigate the Teichmüller metric and the complex structure on the Teichmüller space (H _∞) of the universal hyperbolic solenoid H _∞. In particular, a version of the Reich-Strebel inequality for H _∞ is obtained. As a consequence, we show that the Teichmüller type Beltrami coefficients determine unique geodesics in (H _∞), and we compute the infinitesimal form of the Teichmüller metric. In addition, we show that a Beltrami coefficient is Teichmüller extremal if and only if it is infinitesimally extremal. Finally, we show that the Kobayashi metric on (H _∞) equals the Teichmüller metric.     

Augmented Teichmüller spaces and orbifolds

We study complex analytic properties of the augmented Teichmüller spaces [`(T)]<sub>g,n</sub>{\overline{\mathcal{T}}_{g,n}} obtained by adding to the classical Teichmüller spaces T<sub>g,n</sub>{\mathcal{T}_{g,n}} points corresponding to Riemann surfaces with nodal singularities. Unlike T<sub>g,n</sub>{\mathcal{T}_{g,n}}, the space [`(T)]_g,n{\overline{\mathcal{T}}_{g,n}} is not a complex manifold (it is not even locally compact). We prove, however, that the quotient of the augmented Teichmüller space by any finite index subgroup of the Teichmüller modular group has a canonical structure of a complex orbifold. Using this structure, we construct natural maps from [`(T)]{\overline{\mathcal{T}}} to stacks of admissible coverings of stable Riemann surfaces. This result is important for understanding the cup-product in stringy orbifold cohomology. We also establish some new technical results from the general theory of orbifolds which may be of independent interest.