Let φ and ψ be holomorphic self-maps of the unit polydisc Un in the n-dimensional complex space, and denote by Cφ and Cψ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators Cφ−Cψ from Bloch space to bounded holomorphic function space in the unit polydisc. The compactness of the difference is also characterized.

The authors provide precise lower bounds for the completely bounded norm of the operator Ta,b: ß(H) → ß(H) defined by Ta,b (x) = axb + bxa and the injective norm of the corresponding tensor. Further, they compute the norm of the operator x ↦ a*xb + b*xa acting on the space of all conjugate-linear operators on H.

It is a well-known result that if a nonconstant meromorphic function $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f$ on $\mathbb{C}$ and its $l$th derivative $f^{(l)}$ have no zeros for some $l\geq 2$, then $f$ is of the form $f(z)=\exp (Az+B)$ or $f(z)=(Az+B)^{-n}$ for some constants $A$, $B$. We extend this result to meromorphic functions of several variables, by first extending the classic Tumura–Clunie theorem for meromorphic functions of one complex variable to that of meromorphic functions of several complex variables using Nevanlinna theory.

In this paper a new fixed point theorem for upper semicontinuous set-valued mappings with closed acyclic values is established in the setting of an abstract convex structure – called a locally G-convex space, which generalises usual convexity such as locally convex H-spaces, locally convex spaces (locally H-convex spaces), hyperconvex metric spaces and locally convex topological spaces. Our fixed point theorem includes corresponding Fan-Glicksberg type fixed point theorems in locally convex H-spaces, locally convex spaces, hyperconvex metric space and locally convex spaces in the existing literature as special cases.

The existence of travelling wave solutions to equations of a viscous, heat-conducting combustible fluid is proved. The reactions are assumed to be one step exothermic reactions with a natural discontinuous reaction rate function. The problem is studied for a general gas. Instead of assuming the ideal gas conditions we consider a general thermodynamics which is described by a fairly mild set of hypotheses. The existence proof of travelling waves for Chapman-Jouguet detonation reduces to finding specific heteroclinic orbits of a discontinuous system of ordinary differential equations: these heteroclinic orbits connect a rest point corresponding to unburnt state to that of the burnt state. The existence proof for heteroclinic orbits corresponding to Chapman-Jouguet detonation waves is carried out by some general topological arguments in ordinary differential equations theory.

A permutation group is highly transitive if it is n–transitive for every positive integer n. A group G of order-preserving permutations of the rational line Q is highly order-transitive if for every α1 < … < αn and β1 < … < βn in Q there exists g ∈ G such that αig = βi, i = 1, …, n. The free group Fn(2 ≤ η ≤ אo) can be faithfully represented as a highly order-transitive group of order-preserving permutations of Q, and also (reproving a theorem of McDonough) as a highly transitive group on the natural numbers N. If G and H are nontrivial countable groups having faithful representations as groups of order-preserving permutations of Q, then their free product G * H has such a representation which in addition is highly order-transitive. If G and H are nontrivial finite or countable groups and if H has an element of infinite order, then G * H can be faithfully represented as a highly transitive group on N. Some of the representations of Fη on Q can be extended to faithful representations of the free lattice-ordered group Lη.

Slant curves are introduced in three-dimensional warped products with Euclidean factors. These curves are characterised by the scalar product between the normal at the curve and the vertical vector field, and an important feature is that the case of constant Frenet curvatures implies a proper mean curvature vector field. A Lancret invariant is obtained and the Legendre curves are analysed as a particular case. An example of a slant curve is given for the exponential warping function; our example illustrates a proper (that is, not reducible to the two-dimensional) case of the Lancret theorem of three-dimensional hyperbolic geometry. We point out an eventuality relationship with the geometry of relativistic models.

Suppose that $G=(V,E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\unicode[STIX]{x1D6E5}$ be the usual graph Laplacian. Consider the nonlinear Schrödinger equation of the form

$$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u-\unicode[STIX]{x1D6FC}u=f(x,u),\quad u\in W^{1,2}(V),\end{eqnarray}$$

$G$

$f(x,u):V\times \mathbb{R}\rightarrow \mathbb{R}$

$\unicode[STIX]{x1D6FC}$

$G$

$G$

$CD(m,\unicode[STIX]{x1D709})$

$G$

$\unicode[STIX]{x1D6FC}<2\unicode[STIX]{x1D706}_{1}^{2}/m(\unicode[STIX]{x1D706}_{1}-\unicode[STIX]{x1D709})$

$\unicode[STIX]{x1D706}_{1}$

Let R be a ring with an identity and for each x, y in R, (xy)k = xkyk for three consecutive positive integers k. It is shown in this note that R is a commutative ring.

A new result is obtained for the oscillation of second order linear ordinary differential equations with alternating coefficients. This oscillation result extends a recent oscillation criterion due to Kamenev [.Mat. Zametki 23 (1978), 249–251].

The oscillatory character of the solutions of a differential equation with retarded arguments, relevant to an industrial problem, is investigated. It is also proved that one can maintain the oscillatory properties of this equation under proper conditions, if a forcing term is added to it. The results obtained extend already known results on the subject.

We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

The classical formulae for Fredholm integral equations, including expansions in terms of eigenfunctions such as Mercer's Theorem are extended to square-integrable kernels on an arbitrary measure space.

A class of replacement systems is studied which satisfies a “subcommutativity” condition. Examples of systems satisfying this condition are many of the systems of graph-like expressions which have recently been studied in connection with the efficient evaluation of recursive definitions. An optimality theory of subcommutative systems is developed and is used to give conditions which are sufficient to ensure that an evaluation (reduction) procedure is optimal. The optimality theory is also applied to develop conditions under which a given subcommutative system speeds up, in a natural sense, another replacement system.

We determine the possible $\mathbb{Z}_{2}$-cohomology rings of orbit spaces of free actions of $\mathbb{Z}_{2}$ (or fixed point free involutions) on the Dold manifold $P(1,n)$, where $n$ is an odd natural number.

We study best proximity points in the framework of metric spaces with $w$-distances. The results extend, generalise and unify several well-known fixed point results in the literature.

In this paper potential symmetries are sought of the inhomogeneous nonlinear diffusion equations ut = x1−M[xN−1f (u) ux]x. The functional forms of f (u) that admit such symmetries are completely classified. A complete list is presented of the symmetries, which depend on the values of the parameters M and N. We give examples of similarity solutions using potential symmetries. In some cases, the potential symmetries enable us to convert non–invertible mappings of nonlinear partial differential equations to linear ones.

This paper presents three arguments, one involving orientability, and the others Milnor duality and, respectively, the injectivity of cup product into H2 for an abelian group and free finite group actions on homotopy 3-spheres to show that there are high dimensional knot groups which are not the groups of knotted 2-spheres in S4, thus answering a question of Fox (“Some problems in knot theory”, Topology of 3-manifolds and related topics”, 168–176 (Proceedings of the University of Georgia Institute, 1961. Prentice-Hall, Englewood Cliffs, New Jersey, 1962).

In the context of synthetic differential geometry, we present a notion of differential form with values in a group object, typically a Lie group or the group of all diffeomorphisms of a manifold. Natural geometric examples of such forms and the role of their exterior differentiation is given. The main result is a comparison with the classical theory of Lie algebra valued forms.