We show that the left joint spectrum of an arbitrary n-tuple of hyponormal Hilbert space operators can be obtained from the spectral set γ introduced by McIntosh and Pryde. A dual statement for cohyponormal operators is also true. The result is a generalisation of a theorem proved by Pryde and the author for normal operators.

We prove that the Fridman invariant defined using the Carathéodory pseudodistance does not always go to 1 near strongly Levi pseudoconvex boundary points and it always goes to 0 near nonpseudoconvex boundary points. We also discuss whether Fridman invariants can be extended continuously to some boundary points of domains constructed by deleting compact subsets from other domains.

For a graph G, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of G. Given a positive integer m and a fixed graph H, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$, as G ranges over all graphs on m edges that contain no copy of H. We prove bounds on $f(m,H)$ for some bipartite graphs H and give a bound for a conjecture of Alon et al. [‘MaxCut in H-free graphs’, Combin. Probab. Comput. 14 (2005), 629–647] concerning $f(m,K_{4,s})$.

We give sufficient conditions for a homogeneous superposition operator to be a continuous mapping between Orlicz-Sobolev spaces. This extends a result of Marcus and Mizel concerning mappings between Sobolev spaces.

A graph $\Gamma $ is called $(G, s)$ -arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $ , where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$ . A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$ -arc-transitive but not $(\text{Aut} (\Gamma ), 3)$ -arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$ , ${\rm M} _{11}$ , $\text{M} _{23}$ and $A_{11}$ . However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.

Fibonacci varieties were introduced by one of us in 1978 and a natural generalisation was studied shortly afterwards. We carry this investigation one stage further by giving a description of the free objects in these varieties. This is done in terms of the n-abelian groups of Levi.

Let Ω⊂ℝN be a smooth bounded domain and let f⁄≡0 be a possibly discontinuous and unbounded function. We give a necessary and sufficient condition on f for the existence of positive solutions for all λ>0 of Dirichlet periodic parabolic problems of the form Lu=h(x,t,u)+λf(x,t), where h is a nonnegative Carathéodory function that is sublinear at infinity. When this condition is not fulfilled, under some additional assumptions on h we characterize the set of λs for which the aforementioned problem possesses some positive solution. All results remain true for the corresponding elliptic problems.

The conditions in the strong law of large numbers given by Li et al. [‘A strong law for B-valued arrays’, Proc. Amer. Math. Soc.123 (1995), 3205–3212] for B-valued arrays are relaxed. Further, the compact logarithm rate law and the bounded logarithm rate law are discussed for the moving average process based on an array of random elements.

We study non-commutative convolution measure algebras satisfying the condition in the title and having an involution with a non-degenerate finite dimensional *-representation. We show first that the group algebra L1(G) of a locally compact group G satisfies these conditions. Then we show that to a given algebra A with the above conditions there corresponds a locally compact group G such that A is a * and L-subalgebra of M(G) and such that the enveloping C*-algebra of A is *isomorphic to C*(G). Finally we show for certain groups that L1(G) is the only example of such algebras, thus giving a characterisation of L1(G).

We characterize the boundedness and compactness of the extended Cesàro operator Tg from H∞ to the mixed norm space and Bloch-type space (or little Bloch-type space), where g is a given holomorphic function in the unit ball of Cn and Tg is defined by .

The asymptotic behavior of nonoscillatory solutions of nth order nonlinear functional differential equations

is investigated. Sufficient conditions are provided which ensure that all nonoscillatory solutions approach zero as t → ∞.

Let p be a prime number and let k be a field of characteristic not equal to p. Assuming k contains the appropriate roots of unity, we characterise the non-cyclic Galois extensions of k of degree p3. Concrete examples of such extensions are given for each possible case which can occur, up to isomorphism.

The equality between the closures of the sets of periodic and of recurrent points (called the periodic-recurrent property) is extended from mappings of a tree to mappings defined on a λ-dendroid obtained as a compactification of the complement of a finite subset of a tree provided that the components of the remainder have the same finite depth and each has the periodic-recurrent property.

In this paper we establish operator quasilinearity properties of some functionals associated with Davis–Choi–Jensen’s inequality for positive maps and operator convex or concave functions. Applications for the power function and the logarithm are provided.

Let $A$ be a subset of $\mathbb{N}$ , the set of all nonnegative integers. For an integer $h\geq 2$ , let $hA$ be the set of all sums of $h$ elements of $A$ . The set $A$ is called an asymptotic basis of order $h$ if $hA$ contains all sufficiently large integers. Otherwise, $A$ is called an asymptotic nonbasis of order $h$ . An asymptotic nonbasis $A$ of order $h$ is called a maximal asymptotic nonbasis of order $h$ if $A\cup \{a\}$ is an asymptotic basis of order $h$ for every $a\notin A$ . In this paper, we construct a sequence of asymptotic nonbases of order $h$ for each $h\geq 2$ , each of which is not a subset of a maximal asymptotic nonbasis of order $h$ .