The univalent functions in the diagonal Besov space A_{p}, where 1<p<\infty , are characterized in terms of the distance from the boundary of a point in the image domain. Here A_{2} is the Dirichlet space. A consequence is that there exist functions in A_{p},\ p>2, for which the area of the complement of the image of the unit disc is zero.

1991 Mathematics Subject Classification 30C99, 46E35.

We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relation

with polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

We suggest a way to quantize, using Berezin–Toeplitz quantization, a compact hyperkähler manifold (equipped with a natural 3-plectic form), or a compact integral Kähler manifold of complex dimension n regarded as a (2n−1)-plectic manifold. We show that quantization has reasonable semiclassical properties.

Let

It is familiar that

(1) In the addition theorem [3, p. 363]

where

take θ = π and replace v by v−½. Since

we get

Let M be a compact hypersurface in a Euclidena space ℝn+1. The support function p of M is the component of the position vector field of Min ℝn+1 along the unit normal vector field to M, which is a smooth function defined on M. Let S be the scalar curvature of M. The object of the present paper is to prove the following theorems.

Let V be a countably generated right vector space over a division ring D. If D ≇ ℤ/2ℤ, ℤ/3ℤ, then for any γ ∈ EndD(V), there exists α ∈ AutD(V) such that γ+α, γ−α−1 ∈ AutD(V). This gives a generalization of [D. Zelinsky, Proc. Amer. Math. Soc. 5 (1954), 627–630, Theorem].

Let Pn be the class of normalised polynomials of the form

of degree n which are univalent in U = {|z| < 1}. In this note we discuss the coefficients of polynomials in Pn and in some of its subclasses.

The notation in this paper will be standard and it may be found in [3], for example. In particular, the notation A ⊂′ B stands for the statement “A is an essential submodule of B”. As is customary, we say that a ring R is a Goldie ring when R is both left and right Goldie. Similarly, a ring is noetherian if and only if it is both right and left noetherian, etc.

We study on a contact metric manifold M2n+1(ϕ, ξ, η, g) such that g is a Ricci soliton with potential vector field V collinear with ξ at each point under different curvature conditions: (i) M is of pointwise constant ξ-sectional curvature, (ii) M is conformally flat.

Given an integer g ≥ 2 and a class of finite groups let N(g, ) denote the order of the largest group in that a compact Riemann surface of genus g admits as a group of automorphisms. For the classes of all finite groups, cyclic groups, abelian groups, nilpotent groups, p-groups (given p), soluble groups and finally for metabelian groups, an upper bound for N(g, ) as well as infinite sequences for g for which this bound is attained were found in [5, 6, 7, 8, 13], [4], [10], [15], [16], [1], [2] respectively. This paper deals with that problem for the class of finite supersoluble groups i.e. groups with an invariant series all of whose factors are cyclic. In addition, it goes further by describing exactly those values of g for which the bound is attained. More precisely we prove:

We will denote the dimension of a subspace M of X by dim M and the codimension of M with respect to X by codxM or simply cod M if there is no danger of confusion. The classes of infinite dimensional and closed infinite codimensional subspaces of X will be denoted by and respectively with ℱ(X) and ℱ(X) denoting the classes of finite dimensional and of finite codimensional subspaces of X respectively. For a subspace M of X we denote the injection of M into X by and the quotient map from X onto the quotient space X/M by . Where there is no danger of confusion we will write JM and QM. The injection of X into its completion will be denoted by Jx. Letting X′ denote the continuous dual of X we remark that since X′ is isometric to ()′, these two spaces will be considered identical where convenient. The orthogonal complements of subsets M ⊂ X in X′ and K ⊂ X′ in X will be denoted by M⊥ and ⊥K respectively; M⊥X and X⊥K will be used if there is danger of confusion.

We give a spectral characterization of the norm closure of the class of all weakly hypercyclic operators on a Hilbert space. Analogous results are obtained for weakly supercyclic operators.

We prove that if the order-one differential operator $S=\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i + \gamma$, with $\beta_i,\gamma \in K[x_1,\ldots,x_n]$, generates a maximal left ideal of the Weyl algebra $A_n(K)$, then $S$ does not admit any Darboux differential operator in $K[x_1,\ldots,x_n]\langle \partial_2,\ldots,\partial_n\rangle $; hence in particular, the derivation $\partial_1 + \sum_{i=2}^{n} \beta_i\partial_i$ does not admit any Darboux polynomial in $K[x_1,\ldots,x_n]$. We show that the converse is true when $\beta_i \in K[x_1,x_i]$, for every $i=2,\ldots,n$. Then, we generalize to $K[x_1,\ldots,x_n]$ the classical result of Shamsuddin that characterizes the simple linear derivations of $K[x_1,x_2]$. Finally, we establish a criterion for the left ideal generated by $S$ in $A_n(K)$ to be maximal in terms of the existence of polynomial solutions of a finite system of differential polynomial equations.

All groups G considered in this paper are finite and all representations of G are defined over the field of complex numbers. The reader unfamiliar with projective representations is referred to [9] for basic definitions and elementary results.

Let Proj(G, α) denote the set of irreducible projective characters of a group G with cocycle α. In a previous paper [3] the author showed that if G is a (p, α)-group, that is the degrees of the elements of Proj(G, α) are all powers of a prime number p, then G is solvable. However Isaacs and Passman in [8] were able to give structural information about a group G for which ξ(1) divides pe for all ξ ∈ Proj(G, 1), where 1 denotes the trivial cocycle of G, and indeed classified all such groups in the case e = l. Their results rely on the fact that G has a normal abelian p-complement, which is false in general if G is a (p, α)-group; the alternating group A4 providing an easy counter-example for p = 2.

Using the notion of discrete Morse function introduced by R. Forman for finite cw-complexes, we generalize it to the infinite 2-dimensional case in order to get the corresponding version of the well-known discrete Morse inequalities on a non-compact triangulated 2-manifold without boundary and with finite homology. We also extend them for the more general case of a non-compact triangulated 2-pseudo-manifold with a finite number of critical simplices and finite homology.

A compact Klein surface X is a compact surface with a dianalytic structure. Such a surface is said to be q-hyperelliptic if it admits an involution \phi , that is an order two automorphism, such that X/ < \phi > has algebraic genus q. A Klein surface of genus 1 and k boundary components is a k-bordered torus.

By means of NEC groups, q-hyperelliptic k-bordered tori are studied and a geometrical description of their associated Teichmüller spaces is given.

The Wielandt subgroup w(G) of a group G is defined to be the intersection of the normalizers of all the subnormal subgroups of G. If G is a group satisfying the minimal condition on subnormal subgroups then Wielandt [10] showed that w(G) contains every minimal normal subgroup of G, and so contains the socle of G, and, later, Robinson [6] and Roseblade [9] proved that w(G) has finite index in G.

Let C be a non-empty closed convex subset of a reflexive and strictly convex Banach space E which also has a weakly continuous duality map Jφ(x) with the gauge φ. Let S and T be non-expansive mappings from C into itself such that F = F(S) ∩ F(T) ≠ ∅. Let {αn} and {βn} be sequences in (0, 1). Let {xn} be a sequence defined by where u ∈ C is a given point. Assume that the following restrictions imposed on the control sequences are satisfied:

We study finiteness conditions on essential extensions of simple modules over the quantum plane, the quantised Weyl algebra and Noetherian down-up algebras. The results achieved improve the ones obtained by Carvalho et al. (Carvalho et al., Injective modules over down-up algebras, Glasgow Math. J. 52A (2010), 53–59) for down-up algebras.

The Minkowski question mark function ?(x) arises as a real distribution of rationals in the Farey tree. We examine the generating function of moments of ?(x). It appears that the generating function is a direct dyadic analogue of period functions for Maass wave forms and it is defined in the cut plane \ (1, ∞). The exponential generating function satisfies an integral equation with kernel being the Bessel function. The solution of this integral equation leads to the definition of dyadic eigenfunctions, arising from a certain Hilbert–Schmidt operator. Finally, we describe p-adic distribution of rationals in the Stern–Brocot tree. Surprisingly, the Eisenstein series G2(z) does manifest in both real and p-adic cases.