In this paper, we investigate the value distribution of difference polynomials and prove some difference analogues of results of Hayman and the Brück conjecture.

Let R be a ring, S a strictly ordered monoid, and ω:S→End(R) a monoid homomorphism. The skew generalized power series ring R[[S,ω]] is a common generalization of (skew) polynomial rings, (skew) power series rings, (skew) Laurent polynomial rings, (skew) group rings, and Mal’cev–Neumann Laurent series rings. We study the (S,ω)-Armendariz condition on R, a generalization of the standard Armendariz condition from polynomials to skew generalized power series. We resolve the structure of (S,ω)-Armendariz rings and obtain various necessary or sufficient conditions for a ring to be (S,ω)-Armendariz, unifying and generalizing a number of known Armendariz-like conditions in the aforementioned special cases. As particular cases of our general results we obtain several new theorems on the Armendariz condition; for example, left uniserial rings are Armendariz. We also characterize when a skew generalized power series ring is reduced or semicommutative, and we obtain partial characterizations for it to be reversible or 2-primal.

We give derivative-free characterizations for bounded and compact generalized composition operators between (little) Zygmund type spaces. To obtain these results, we extend Pavlović’s corresponding result for bounded composition operators between analytic Lipschitz spaces.

Answering a question of W. Arendt and M. Kunze in the negative, we construct a Banach space X and a norm closed weak* dense subspace Y of the dual X′ of X such that X, endowed with the Mackey topology μ(X,Y ) of the dual pair 〈X,Y 〉, is not complete.

In this paper, we present a constructive proof of the amenability of C(X), where X is a compact space.

In this paper, we explore the monotone Lindelöf property of two kinds of linearly ordered extensions of monotonically Lindelöf generalized ordered spaces. In addition, we construct nonseparable monotonically Lindelöf spaces using the Bernstein set, which generalizes Corollary 4 of Levy and Matveev [‘Some more examples of monotonically Lindelöf and not monotonically Lindelöf spaces’, Topology Appl.154 (2007), 2333–2343].

We discuss the problem of constructing elements of multiplicative high order in finite fields of large degree over their prime field. We obtain such elements by evaluating rational functions on elliptic curves, at points whose order is small with respect to their degree. We discuss several special cases, including an old construction of Wiedemann, giving the first nontrivial estimate for the order of the elements in this construction.

It is well known that an exponential real function, which is Lebesgue measurable (Baire measurable, respectively) or bounded on a set of positive Lebesgue measure (of the second category with the Baire property, respectively), is continuous. Here we consider bounded on ‘big’ set solutions of an equation generalizing the exponential equation as well as the Goła̧b–Schinzel equation. Moreover, we unify results into a more general and abstract case.

A two-level defect–correction method for the steady-state Navier–Stokes equations with a high Reynolds number is considered in this paper. The defect step is accomplished in a coarse-level subspace Hm by solving the standard Galerkin equation with an artificial viscosity parameter σ as a stability factor, and the correction step is performed in a fine-level subspace HM by solving a linear equation. H1 error estimates are derived for this two-level defect–correction method. Moreover, some numerical examples are presented to show that the two-level defect–correction method can reach the same accuracy as the standard Galerkin method in fine-level subspace HM. However, the two-level method will involve much less work than the one-level method.

Using potential theory, we prove the existence of filling disks of an algebroid function of finite order defined in the unit disk.

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

Let G be a graph, and k a positive integer. Let h:E(G)→[0,1] be a function. If ∑ e∋xh(e)=k holds for each x∈V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh={e∈E(G)∣h(e)>0}. In this paper we use neighbourhoods to obtain a new sufficient condition for a graph to have a fractional k-factor. Furthermore, this result is shown to be best possible in some sense.

In this paper, four new discreteness criteria for isometric groups on complex hyperbolic spaces are proved, one of which shows that the Condition C hypothesis in Cao [‘Discrete and dense subgroups acting on complex hyperbolic space’, Bull. Aust. Math. Soc.78 (2008), 211–224, Theorem 1.4] is removable; another shows that the parabolic condition hypothesis in Li and Wang [‘Discreteness criteria for Möbius groups acting on II’, Bull. Aust. Math. Soc.80 (2009), 275–290, Theorem 3.1] is not necessary.

We give a characterization of a minimal real hypersurface with respect to the condition for the sectional curvature.

In this note we give an example of a Hausdorff, star compact space with point-countable base which is not metrizable.

The classical variational analysis of curvature energy functionals, acting on spaces of curves of a Riemannian manifold, is extremely complicated, and the procedure usually can not be completely developed under such a degree of generality. Sometimes this difficulty may be overcome by focusing on specific actions in real space forms. In this note, we restrict ourselves to quadratic Lagrangian energies acting on the space of closed curves of the 2-sphere. We solve the Euler–Lagrange equation and show that there exists a two-parameter family of closed critical curves. We also discuss the stability of the circular critical points. Since, even for this class of energies, the complete variational analysis is quite involved, we use instead a numerical approach to provide a useful method of visualization of relevant aspects concerning uniqueness, stability and explicit representation of the closed critical curves.

The group of any nontrivial torus knot, hyperbolic 2-bridge knot, or hyperbolic knot with unknotting number one contains infinitely many elements, none of which is the automorphic image of another, such that each normally generates the group.

We introduce a differential operator on quasimodular polynomials that corresponds to the derivative operator on quasimodular forms. We then prove that such a differential operator is compatible with a heat operator on Jacobi-like forms in certain cases. These results show in those cases that the derivative operator on quasimodular forms corresponds to a heat operator on Jacobi-like forms.