Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space

We consider groups Γ generated by inversions in a pair of asymptotic complex hyperplanes in complex hyperbolic spaceH _? ⁿ . We show that there exists a Γ-invariant real hypersurfaceF ?H _? ⁿ such that the Dirichlet fundamental polyhedron for Γ centered at z₀ has two sides (resp. infinitely many sides) if and only ifz ₀ ∈F (resp.z ₀ ?F). The Dirichlet regions are determined explicitly in terms of coordinates on Γ-invariant horospheres and the geometry ofH _? ⁿ is developed in terms of these horospherical coordinates.     

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

On the Davenport constant and on the structure of extremal zero-sum free sequences

Let $G = C_{n_1 } \oplus \cdots \oplus C_{n_r }$ with 1 < n ₁ | ?? | n _r be a finite abelian group, d*(G) = n ₁ +??+n _r ?r, and let d(G) denote the maximal length of a zerosum free sequence over G. Then d(G) ?? d*(G), and the standing conjecture is that equality holds for G = C _n ^r . We show that equality does not hold for C ₂ ?? C _2n ^r , where n ?? 3 is odd and r ?? 4. This gives new information on the structure of extremal zero-sum free sequences over C _2n ^r .     

Derived space of L p (G, H)

Let G denote a locally compact abelian group and H a separable Hilbert space. Let L _p (G, H), 1 ≤ p < ∞, be the space of H-valued measurable functions which are in the usual L _p space. Motivated by the work of Helgason [1], Figa-Talamanca [11] and Bachelis [2, 3], we have defined the derived space of the Banach space L _p (G, H) and have studied its properties. Similar to the scalar case, we prove that if G is a noncompact, locally compact abelian group, then L _p ⁰ (G, H) = {0} holds for 1 ≤ p < 2. Let G be a compact abelian group and Γ be its dual group. Let S _p (G, H) be the L ₁(G) Banach module of functions in L _p (G, H) having unconditionally convergent Fourier series in L _p -norm. We show that S _p (G, H) coincides with the derived space L _p ⁰ (G, H), as in the scalar valued case. We also show that if G is compact and abelian, then L _p ⁰ (G, H) = L ₂(G, H) holds for 1 ≤ p ≤ 2. Thus, if F ∈ L _p (G, H), 1 ≤ p < 2 and F has an unconditionally convergent Fourier series in L _p -norm, then F ∈ L ₂(G, H). Let Ω be the set of all functions on Γ taking only the values 1, ?1 and Ω* be the set of all complex-valued functions on Γ having absolute value 1. As an application of the derived space L _p ⁰ (G, H), we prove the following main result of this paper. Let G be a compact abelian group and F be an H-valued function on the dual group Γ such that $$ \sum \omega (\gamma )F(\gamma )\gamma $$ is a Fourier-Stieltjes series of some measure µ ∈ M(G, H) for every scalar function ω such that |ω(γ)| = 1. Then F ∈ l ²(Γ, H).     

Generic representations of abelian groups and extreme amenability

If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group $\overline {\pi (\Gamma )} $ for the generic π ∈ Hom(Γ, G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $\overline {\pi (\Gamma )} $ ; in the other two, we show that the generic $\overline {\pi (\Gamma )} $ is extremely amenable. We also show that if Γ is torsionfree, the centralizer of the generic π is as small as possible, extending a result of Chacon and Schwartzbauer from ergodic theory.     

The index of singular integral operators in reflexive Orlicz spaces

We consider the Banach algebra $\mathfrak{A}$ of singular integral operators with matrix piecewise continuous coefficients in the reflexive Orlicz spaceL _M ⁿ (Γ). We assume that Γ belongs to a certain wide subclass of the class of Carleson curves; this subclass includes curves with cusps, as well as curves of the logarithmic spiral type. We obtain an index formula for an arbitrary operator from the algebra $L_M^n (\Gamma )$ in terms of the symbol of this operator.     

A constructive description of Hölder classes on closed Jordan curves

Let Γ be a closed, Jordan, rectifiable curve, whose are length is commensurable with its subtending chord, leta ε int Γ, and let R_n(a) be the set of rational functions of degree ≤n, having a pole perhaps only at the pointa. Let Λ^α(Γ), 0 < α < 1, be the Hölder class on Γ. One constructs a system of weights γ_n(z) > 0 on Γ such that f∈Λ^α(Γ) if and only if for any nonnegative integer n there exists a function R_n, R_n ε R_n(a) such that ¦f(z) ? R_n(z)¦ ≤ c_f·γ_n(z), z ε Γ. It is proved that the weights γ_n cannot be expressed simply in terms of ρ ₁ ⁺ /n^(z) and ρ ₁ ^- /n^(z), the distances to the level lines of the moduli of the conformal mappings of ext Γ and int Γ on \(\mathbb{C}\backslash \mathbb{D}\) .     

Determinants of parabolic bundles on Riemann surfaces

LetX be a compact Riemann surface andM _s ^p (X) the moduli space of stable parabolic vector bundles with fixed rank, degree, rational weights and multiplicities. There is a natural Kähler metric onM _s ^p (X). We obtain a natural metrized holomorphic line bundle onM _s ^p (X) whose Chern form equalsmr times the Kähler form, wherem is the common denominator of the weights andr the rank.     

On the sum of powers of terms of a linear recurrence sequence

Let (F _n ) _n??0 be the Fibonacci sequence given by F _n+2 = F _n+1 + F _n , for n ?? 0, where F ₀ = 0 and F ₁ = 1. There are several interesting identities involving this sequence such as F _n ² + F _n+1 ² = F _2n+1, for all n ?? 0. In a very recent paper, Marques and Togbé proved that if F _n ^s + F _n+1 ^s is a Fibonacci number for all sufficiently large n, then s = 1 or 2. In this paper, we will prove, in particular, that if (G _m ) _m is a linear recurrence sequence (under weak assumptions) and G _n ^s + ... + G _n+k ^s ?? (G _m ) _m , for infinitely many positive integers n, then s is bounded by an effectively computable constant depending only on k and the parameters of G _m .     

On the relation of generalized Valiron summability to Cesàro summability

A family (V _a ^k ) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (B_α,γ), (E _ρ, (T_α), (S _β) and (V _a) are members of this family. In §3 some properties of the (B _α,γ) and (V _a ^k ) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V _a ^k ) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If s _n (n ≥ 0) isa real sequence satisfying $$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$ , and if s_n →s (V _a ^k ) thens _n → s (C, 2ρ).     

Estimates for wave and Klein-Gordon equations on modulation spaces

We prove that the fundamental semi-group eit(m 2I+|Δ|)1/2(m = 0) of the Klein-Gordon equation is bounded on the modulation space M ps,q(Rn) for all 0 < p,q ∞ and s ∈ R.Similarly,we prove that the wave semi-group eit|Δ|1/2 is bounded on the Hardy type modulation spaces μsp,q(Rn) for all 0 < p,q ∞,and s ∈ R.All the bounds have an asymptotic factor tn|1/p 1/2| as t goes to the infinity.These results extend some known results for the case of p 1.Also,some applications for the Cauchy problems related to the semi-group eit(m2I+|Δ|)1/2 are obtained.Finally we discuss the optimum of the factor tn|1/p 1/2| and raise some unsolved problems.     

On σ-algebras related to the measurability of compositions

Given a measurable space (T, F), a set X, and a map ?: T → X, the σ-algebras N _Ф = ?_?∈Φ N _?, and M _Φ = ?_?∈Φ N _?, where G _?(t) = (t, ?(t)) and Φ ? X ^T , are considered. These σ-algebras are used to characterize the (F, B, ?)-measurability of the compositions g ○ ? and f о G _?, where g: X → Y, f: T × X → Y, and (Y, ?) is a measurable space. Their elements are described without using the operations ? ^?1 and G _? ^?1 .     

On inner automorphisms and certain central automorphisms of groups

Let G be a group, let M and N be two normal subgroups of G. We denote by Aut _N ^M (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = Aut _N ^M (G), Aut _N ^M (G) ≤ Inn(G) or Inn(G) ≤ Aut _N ^M (G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”.     

Regular Submanifolds in Conformal Space ${\mathbb Q}^n_p$

The authors study the regular submanifolds in the conformal space Q_p~n and introduce the submanifold theory in the conformal space Q_p~n.The first variation formula of the Willmore volume functional of pseudo-Riemannian submanifolds in the conformal spaceQ_p~n is given.Finally,the conformal isotropic submanifolds in the conformal space Q_p~n are classified.     

Techniques of computations of Dolbeault cohomology of solvmanifolds

We consider semi-direct products ${\mathbb{C}^{n}\ltimes_{\phi}N}$ of Lie groups with lattices Γ such that N are nilpotent Lie groups with left-invariant complex structures. We compute the Dolbeault cohomology of direct sums of holomorphic line bundles over G/Γ by using the Dolbeaut cohomology of the Lie algebras of the direct product ${\mathbb{C}^{n}\times N}$ . As a corollary of this computation, we can compute the Dolbeault cohomology H ^p,q (G/Γ) of G/Γ by using a finite dimensional cochain complexes. Computing some examples, we observe that the Dolbeault cohomology varies for choices of lattices Γ.     

Signed k-independence in graphs

Let k ≥ 2 be an integer. A function f: V(G) → {?1, 1} defined on the vertex set V(G) of a graph G is a signed k-independence function if the sum of its function values over any closed neighborhood is at most k ? 1. That is, Σ _x∈N[v] f(x) ≤ k ? 1 for every v ∈ V(G), where N[v] consists of v and every vertex adjacent to v. The weight of a signed k-independence function f is w(f) = Σ _v∈V(G) f(v). The maximum weight w(f), taken over all signed k-independence functions f on G, is the signed k-independence number α _s ^k (G) of G. In this work, we mainly present upper bounds on α _s ^k (G), as for example α _s ^k (G) ≤ n ? 2?(Δ(G) + 2 ? k)/2?, and we prove the Nordhaus-Gaddum type inequality $\alpha _S^k \left( G \right) + \alpha _S^k \left( {\bar G} \right) \leqslant n + 2k - 3$ , where n is the order, Δ(G) the maximum degree and $\bar G$ the complement of the graph G. Some of our results imply well-known bounds on the signed 2-independence number.     

The reconstruction problem for certain infinite graphs

We are concerned with the notion of the degree-type (D _G ⁱ )_i∈ω of a graphG, whereD _G ⁱ is defined to be the number of vertices inG with degreei. In the first section the following results are proven:

1. IfG is a connected, locally finite, countably infinite graph such that there exists ani so thatD _G ⁱ andD _G ⁱ⁺¹ are both finite and different from 0, thenG is reconstructible.
2. Locally finite, countably infinite graphsG, for which infinitely manyD _G ⁱ are different from 0 but only finitely manyD _G ⁱ are infinite, are reconstructible.

In the second section we give some results about the reconstructibility of certain locally finite countably infinite interval graphs and show that a reconstruction of a planar, infinite graph has to be planar too.     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.