Primes in Short Intervals

ＰｒｉｍｅｓｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓＬｉＨｏｎｇｚｅ（李红泽）（Ｄｅｐｔ，ｏｆＭａｔｈ．，ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎａｎ，Ｓｈａｎｇｄｏｎｇ，２５０１００）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＰａｎＣｈｅｎｇｂｉａｏＲｅｃｅｉｖｅｄ...     

P_2 in Short Intervals

Ｐ＿２ｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓＬｉＨｏｎｇｚｅ（李红泽）（ＳｈａｎｄｏｎｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎａｎ，Ｓｈａｎｇｄｏｎｇ，２５０１００，Ｐ，Ｒ．Ｃ．）Ｐ＿２ｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓ￥ＬｉＨｏｎｇｚｅ（李红泽）（ＳｈａｎｄｏｎｇＵ...     

Bounded Turning in Möbius Structures

Siberian Mathematical Journal - We study a Möbius-invariant generalization, called BTR, of the classical property of bounded turning in a metric space which was introduced by...     

Conjugacy classes in Möbius groups

Let \({\mathbb H^{n+1}}\) denote the n + 1-dimensional (real) hyperbolic space. Let \({\mathbb {S}^{n}}\) denote the conformal boundary of the hyperbolic space. The group of conformal diffeomorphisms of \({\mathbb {S}^{n}}\) is denoted by M(n). Let M _o (n) be its identity component which consists of all orientation-preserving elements in M(n). The conjugacy classification of isometries in M _o (n) depends on the conjugacy of T and T ^?1 in M _o (n). For an element T in M(n), T and T ^?1 are conjugate in M(n), but they may not be conjugate in M _o (n). In the literature, T is called real if T is conjugate in M _o (n) to T ^?1. In this paper we classify real elements in M _o (n). Let T be an element in M _o (n). Corresponding to T there is an associated element T _o in SO(n + 1). If the complex conjugate eigenvalues of T _o are given by \({\{e^{i\theta_j}, e^{-i\theta_j}\}, 0 < \theta_j \leq \pi, j=1,\ldots,k}\) , then {θ₁, . . . , θ _k } are called the rotation angles of T. If the rotation angles of T are distinct from each-other, then T is called a regular element. After classifying the real elements in M _o (n) we have parametrized the conjugacy classes of regular elements in M _o (n). In the parametrization, when T is not conjugate to T ^?1 , we have enlarged the group and have considered the conjugacy class of T in M(n). We prove that each such conjugacy class can be induced with a fibration structure.     

Elliptic elements in Möbius groups

Let G be a nonelementary subgroup of Isom(H ⁿ). In this paper we prove that if G contains elliptic elements and dim(M _G ) is even, then G is discrete if and only if WY (G) is discrete and every nonelementary subgroup of G generated by two elliptic elements is discrete. We also describe an example to show the assumption on dim(M _G ) is necessary. Supported in part by NSFC 10671059 and by Leading academic Discipline program, 211 project for Shanghai University of Finance and Economics (the 3rd phase).     

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.     

Generalized Angles in Ptolemaic Möbius Structures

We show that each Ptolemaic semimetric is Möbius-equivalent to a bounded metric. Introducing generalized angles in Ptolemaic Möbius structures, we study the class of multivalued mappings F: X → 2Y with a lower bound on the distortion of generalized angles. We prove that the inverse mapping to the coordinate function of a quasimeromorphic automorphism of ?? ⁿ lies in this class.     

Primes and Goldbach Numbers in Short Intervals

ＰｒｉｍｅｓａｎｄＧｏｌｄｂａｃｈＮｕｍｂｅｒｓｉｎＳｈｏｒｔＩｎｔｅｒｖａｌｓＬｉＨｏｎｇｚｅ（李红泽）（Ｄｅｐｔ．ｏｆＭａｔｈ．ＳｈａｎｄｏｘｇＵｎｉｖｅｒｓｉｔｙ，Ｊｉｎａｎ，Ｓｈａｎｄｏｎｇ，２５０１００）ＣｏｍｍｕｎｉｃａｔｅｄｂｙＰａｎＣ...     

Möbius inversion formula for monoids with zero

The Möbius inversion formula, introduced during the 19th century in number theory, was generalized to a wide class of monoids called locally finite such as the free partially commutative, plactic and hypoplactic monoids for instance. In this contribution are developed and used some topological and algebraic notions for monoids with zero, similar to ordinary objects such as the (total) algebra of a monoid, the augmentation ideal or the star operation on proper series. The main concern is to extend the study of the Möbius function to some monoids with zero, i.e., with an absorbing element, in particular the so-called Rees quotients of locally finite monoids. Some relations between the Möbius functions of a monoid and its Rees quotient are also provided.     

The Möbius midpoint condition as a test for quasiconformality and the quasimöbius property

The Möbius midpoint condition, introduced by Goldberg in 1974 as a criterion for the quasisymmetry of a mapping of the line onto itself and considered by Aseev and Kuzin in 1998 in the same role for the topological embeddings of the line into ? ⁿ , yields no information on the quasiconformality or quasisymmetry of a topological embedding of ? ^k into ? ⁿ for 1 < k ≤ n. In this article we introduce a Möbius-invariant modification of the midpoint condition, which we call the “Möbius midpoint condition” MMC(f) ≤ H < 1. We prove that if this condition is fulfilled then every homeomorphism of domains in \(\overline {\mathbb{R}^n }\) is K(H)-quasiconformal, while a topological embedding of the sphere \(\overline {\mathbb{R}^k }\) into \(\overline {\mathbb{R}^n }\) (for 1 ≤ k ≤ n) is ω _H-quasimöbius. The quasiconformality coefficient of K(H) and the distortion function ω _H depend only on H and are expressed by explicit formulas showing that K(H) → 1 and ω _H → id as H → 1/2. Since MMC(f) = 1/2 is equivalent to the Möbius property of f, the resulting formulas yield the closeness of the mapping to a Möbius mapping for H near 1/2.     

Riemann Sums and Möbius

Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξ ∈ l^r (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of f_ξR₀, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξ_λ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 ? α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to f_ξ ∈ Λ_α, α < 1/2, ξ ∈ l^r (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies f_ξ ∈ Λ_α, α < 1/4, ξ ∈ l^r (S), 0 < r < 2. The question of whether R₀ ∩ Λ_α ≠ {0} for some positive α > 0 is open.     

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).     

Möbius and triangle semigroups

The purpose of this paper is to describe the structures of the Möbius semigroup induced by the Möbius transformation group (?, SL(2,?)). In particular, we study stabilizer subsemigoups of Möbius semigroup via the triangle semigroup. In this work, we obtained a geometric interpretation of the least contraction coefficient function of the Möbius semigroup via the triangle semigroup and investigated an extension of stabilizer subsemigoups of the Möbius semigroup. Finally, we obtained a factorization of our stabilizer subsemigoups of the Möbius 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.