In this paper, our main aim is to discuss the properties of harmonic mappings in the unit ball 𝔹n. First, we characterize the harmonic Bloch spaces and the little harmonic Bloch spaces from 𝔹n to ℂ in terms of weighted Lipschitz functions. Then we prove the existence of a Landau–Bloch constant for a class of vector-valued harmonic Bloch mappings from 𝔹n to ℂn.

Previous work concerning boundary regularity of solutions of the non-parametric least area problem leaves open the question of regularity of solutions at points where the mean curvature of the boundary of the domain vanishes. We here prove that the solutions may be discontinuous at such points, even when the given boundary data is smooth. We also give a sufficient condition which will ensure continuity at such points.

Let X be a separable Banach space which admits a C1-smooth norm, and let G ⊂ X be an open subset. Then any real-valued, bounded and uniformly continuous map on G can be uniformly approximated on G by C1-smooth functions with bounded derivative.

This paper is devoted to a historical study of axioms for group theory. It begins with the emergence of groups in the work of Galois and Cauchy, treats two lines of development discernible in the latter half of the nineteenth century, and concludes with a note about some twentieth century ideas. One of those nineteenth century lines involved Cayley, Dyck and Burnside; the other involved Kronecker, Weber (very strongly), Hölder and Frobenius.

We define a certain class of groups, Ck, which we show to contain the class of all k-free groups. Our main theorem shows that certain amalgamated free products of groups in C3, are again in C3. In the appendix we show that many 3-manifold groups belong to Ck for suitable k.

Dedicated to Filippo Chiarenza

The aim of this note is to prove the unique continuation property for non-negative solutions of the quasilinear elliptic equation

We allow the coefficients to belong to a generalized Kato class.

It is proved that the monoid RN of all partial recursive functions of one variable is finitely generated, and that RN × RNis a cyclic (left and right) RN-act (under the natural diagonal actions s (a, b) = (sa, sb), (a, b) s = (as, bs)). We also construct a finitely presented monoid S such that S × S is a cyclic left and right S-act, and study further interesting properties of diagonal acts and their relationship with power monoids.

We improve the degree of pointwise approximation of continuous functions f(x) by Bernstein operators, when x is close to the endpoints of [0,1]. We apply the new estimate to establish upper and lower pointwise estimates for the test function g(x)=xlog (x)+(1−x)log (1−x). At the end we prove a general statement for pointwise approximation by Bernstein operators.

We study the structure of function fields of plane quartic curves by using projections. Taking a point P ∈ ℙ2, we define the projection from a curve C to a line l with the centre P. This projection induces and extension field k (C)/k (ℙ1). By using this fact, we study the field extension k (C)/k (ℙ1) from a geometrical point of view. In this note, we take up quartic curves with singular points.

We introduce a concept of metric space valued mappings of two variables with finite total variation and define a counterpart of the Hardy space. Then we establish the following Helly type selection principle for mappings of two variables: Let X be a metric space and a commutative additive semigroup whose metric is translation invariant. Then an infinite pointwise precompact family of X-valued mappings on the closed rectangle of the plane, which is of uniformly bounded total variation, contains a pointwise convergent sequence whose limit is a mapping with finite total variation.

The group G is called trifactorisable if G has three subgroups, A, B, and C such that G = AB = BC = CA. Obviously the structure of the group G will be restricted by the structure of these subgroups. In this paper it will be shown that a finite group G is π-separable if and only if it satisfies Dπ and has a trifactorisation with two factors π closed and the third, C say, π-separable. In this case we show that the π- and π′-lengths of G can be at most one more than those of C, and so it is this factor which “controls” the structure of G. Similar results are proved for π-solubility and solubility.

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\mathcal{L}}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K))\in \mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a PST-group if and only if $A^{G}/A_{G}\leq Z_{\infty }(G/A_{G})$ for every subgroup $A\in {\mathcal{L}}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

A strengthened form of the Kantorovich convergence theorem for the modified Newton's method is proved. The result is compared with previously known results.

It is shown that the correspondence which associates with each radical class τ of abelian groups the (radical) class of prime radical rings with additive groups in τ gives a complete classification of those radical classes of rings which are determined (as lower radicals) by zerorings.

Some properties which different definitions or orthogonality in a normed space can possess are considered. It is shown that orthogonality can be defined on any separable space with many of the properties possessed by the usual orthogonality in an inner-product space, but that the possession of a further property forces the space to be isomorphic to a Euclidean space.

In this paper, we prove that there are no warped product proper semi-slant submanifolds such that the spheric submanifold of a warped product is a proper slant. But we show by means of examples the existence of warped product semi-slant submanifolds such that the totally geodesic submanifold of a warped product is a proper slant submanifold in locally Riemannian product manifolds.

We study the closed range property of weighted composition operators on weighted Bergman spaces of infinite order (including the Hardy space of infinite order). We give some necessary and sufficient conditions and find a complete characterisation for weighted composition operators associated with conformal mappings. We also give the corresponding results for composition operators on the Bloch-type spaces. Therefore, the results obtained in this paper also improve and generalise the results of Ghatage, Yan, Zheng and Zorboska.

For a graph G, define the bisection width bw(G) of G as min { eG(A,B): {A,B} partitions V(G) with ‖A| − |B‖ ≤ 1 } where eG(A, B) denotes the number of edges in G with one end in A and one end in B. We show almost every cubic graph G of order n has bw(G) ≥ n/11 while every such graph has bw(G) ≤ (n + 138)/3. We also show that almost every r-regular graph G of order n has bw(G) ≥ crn where cr → r/4 as r → ∞. Our last result is asymtotically correct.

Let Y be a fixed nonempty subset of a set X and let T(X,Y ) denote the semigroup of all total transformations from X into Y. In 1975, Symons described the automorphisms of T(X,Y ). Three decades later, Nenthein, Youngkhong and Kemprasit determined its regular elements, and more recently Sanwong, Singha and Sullivan characterized all maximal and minimal congruences on T(X,Y ). In 2008, Sanwong and Sommanee determined the largest regular subsemigroup of T(X,Y ) when |Y |≠1 and Y ≠ X; and using this, they described the Green’s relations on T(X,Y ) . Here, we use their work to describe the ideal structure of T(X,Y ) . We also correct the proof of the corresponding result for a linear analogue of T(X,Y ) .

The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.