We construct travelling waves in the Burgers equation with the fractional Laplacian $(D^{2})^{\unicode[STIX]{x1D6FC}}$ , $\unicode[STIX]{x1D6FC}\in (1/2,1)$ . This is done by first constructing odd solutions $u_{\unicode[STIX]{x1D700}}$ of $uu^{\prime }=K_{\unicode[STIX]{x1D700}_{1}}\ast u-k_{\unicode[STIX]{x1D700}_{1}}u+\unicode[STIX]{x1D700}_{2}u^{\prime \prime }$ , $u(-\infty )=u_{c}>0$ , with $K_{\unicode[STIX]{x1D700}_{1}}\ast u-k_{\unicode[STIX]{x1D700}_{1}}u$ nonsingular, and then passing to the limit $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2}\rightarrow 0$ , to give $K_{\unicode[STIX]{x1D700}_{1}}\ast u_{\unicode[STIX]{x1D700}}-k_{\unicode[STIX]{x1D700}_{1}}u_{\unicode[STIX]{x1D700}}\rightarrow (D^{2})^{\unicode[STIX]{x1D6FC}}u_{0}$ pointwise. The proof relies on operator splitting.

We investigate surjective solutions of the functional equation

$$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$

$f:X\rightarrow Y$

${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$

${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$

Extending recent results by Cascales et al. [‘Plasticity of the unit ball of a strictly convex Banach space’, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat.110(2) (2016), 723–727], we demonstrate that for every Banach space $X$ and every collection $Z_{i},i\in I$ , of strictly convex Banach spaces, every nonexpansive bijection from the unit ball of $X$ to the unit ball of the sum of $Z_{i}$ by $\ell _{1}$ is an isometry.

Motivated by the notion of Ulam stability, we investigate some inequalities connected with the functional equation

$$\begin{eqnarray}f(xy)+f(x\unicode[STIX]{x1D70E}(y))=2f(x)+h(y),\quad x,y\in G,\end{eqnarray}$$

$f$

$h$

$(G,\cdot )$

$(E,+)$

$\unicode[STIX]{x1D70E}:G\rightarrow G$

$G$

$\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}(x))=x$

$x\in G$

$\unicode[STIX]{x1D70E}$

$\unicode[STIX]{x1D70E}$

For a Tychonoff space $X$ , let $\mathbb{V}(X)$ be the free topological vector space over $X$ , $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$ , then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

New inequalities relating the norm $n(X)$ and the numerical radius $w(X)$ of invertible bounded linear Hilbert space operators were announced by Hosseini and Omidvar [‘Some inequalities for the numerical radius for Hilbert space operators’, Bull. Aust. Math. Soc.94 (2016), 489–496]. For example, they asserted that $w(AB)\leq$ $2w(A)w(B)$ for invertible bounded linear Hilbert space operators $A$ and $B$ . We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form $n(X^{-1})=n(X)^{-1}$ . We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example, $w(AB)=w(A)w(B)$ .

We show that if $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ has the weak Haagerup property, then both $M$ and $\unicode[STIX]{x1D6E4}$ have the weak Haagerup property, and if $\unicode[STIX]{x1D6E4}$ is an amenable group, then the weak Haagerup property of $M$ implies that of $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ . We also give a condition under which the weak Haagerup property for $M$ and $\unicode[STIX]{x1D6E4}$ implies that of $M\,\bar{\rtimes }_{\unicode[STIX]{x1D6FC}}\,\unicode[STIX]{x1D6E4}$ .

Let $N$ be a fixed positive integer and $f:\mathbb{R}\rightarrow \mathbb{C}$ . As a generalisation of the superstability of the exponential functional equation we consider the functional inequalities

$$\begin{eqnarray}\displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D719}(x), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \big|f\big(\!\sqrt[N]{x^{N}+y^{N}}\big)-f(x)f(y)\big|\leq \unicode[STIX]{x1D713}(x,y) & \displaystyle \nonumber\end{eqnarray}$$

$x,y\in \mathbb{R}$

$\unicode[STIX]{x1D719}:\mathbb{R}\rightarrow \mathbb{R}^{+}$

$\unicode[STIX]{x1D713}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{+}$

We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.

Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}>0$ . We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$ , if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$ , then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

We show that every topological grading of a $C^{\ast }$ -algebra by a discrete abelian group is implemented by an action of the compact dual group.

To explore the difficulties of classifying actions with the tracial Rokhlin property using K-theoretic data, we construct two $\mathbb{Z}_{2}$ actions $\unicode[STIX]{x1D6FC}_{1},\unicode[STIX]{x1D6FC}_{2}$ on a simple unital AF algebra $A$ such that $\unicode[STIX]{x1D6FC}_{1}$ has the tracial Rokhlin property and $\unicode[STIX]{x1D6FC}_{2}$ does not, while $(\unicode[STIX]{x1D6FC}_{1})_{\ast }=(\unicode[STIX]{x1D6FC}_{2})_{\ast }$ , where $(\unicode[STIX]{x1D6FC}_{i})_{\ast }$ is the induced map by $\unicode[STIX]{x1D6FC}_{i}$ acting on $K_{0}(A)$ for $i=1,2$ .

For every rotation $\unicode[STIX]{x1D70C}$ of the Euclidean space $\mathbb{R}^{n}$ ( $n\geq 3$ ), we find an upper bound for the number $r$ such that $\unicode[STIX]{x1D70C}$ is a product of $r$ rotations by an angle $\unicode[STIX]{x1D6FC}$ ( $0<\unicode[STIX]{x1D6FC}\leq \unicode[STIX]{x1D70B}$ ). We also find an upper bound for the number $r$ such that $\unicode[STIX]{x1D70C}$ can be written as a product of $r$ full rotations by an angle  $\unicode[STIX]{x1D6FC}$ .

The theory of almost invariant half-spaces for operators on Banach spaces was begun recently and is now under active development. Much less attention has been given to almost invariant half-spaces for operators on Hilbert space, where some techniques and results are available that are not present in the more general context of Banach spaces. In this note, we begin such a study. Our much simpler and shorter proofs of the main theorems have important consequences for the matricial structure of arbitrary operators on Hilbert space.

For a convex domain, we use Klain’s cyclic rearrangement to obtain a sequence of convex domains with increasing area and the same perimeter which converges to a disk. As a byproduct, we give a proof of the classical isoperimetric inequality in the plane.

We use a unified approach to study the boundedness of fractional integral operators on $\unicode[STIX]{x1D6FC}$ -modulation spaces and find sharp conditions for boundedness in the full range.

We consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under Möbius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. Finally, we establish the quasi-invariance properties of both metrics under quasiconformal maps.

We use the pointwise Lipschitz constant to define an upper Lyapunov exponent for maps on metric spaces different to that given by Kifer [‘Characteristic exponents of dynamical systems in metric spaces’, Ergodic Theory Dynam. Systems3(1) (1983), 119–127]. We prove that this exponent reduces to that of Bessa and Silva on Riemannian manifolds and is not larger than that of Kifer at stable points. We also prove that it is invariant along orbits in the case of (topological) diffeomorphisms and under topological conjugacy. Moreover, the periodic orbits where this exponent is negative are asymptotically stable. Finally, we estimate this exponent for certain hyperbolic homeomorphisms.

A generalised topology is a collection of subsets of a given nonempty set containing the empty set and arbitrary unions of the elements in the collection. By using the concept of hereditary classes, a generalised topology can be extended to a new one, called a generalised topology via a hereditary class. We study continuity on generalised topological spaces via hereditary classes in various situations.

We give a simple proof of the strong law of large numbers with rates, assuming only finite variance. This note also serves as an elementary introduction to the theory of large deviations, assuming only finite variance, even when the random variables are not necessarily independent.