If f is a p–th integrable function on the circle group and ω(p; f; δ) is its mean modulus of continuity with exponent p then an extended version of the classical theorem of Jackson states the for each positive integer n, there exists a trigonometric polynomial tn of degree at most n for which

‖f-tn‖p ≤(p; f; 1/n).

In this paper it will be shewn that for G a Hausdorff locally compact abelian group, the algebra L1(G) admits a certain bounded positive approximate unit which, in turn, will be used to prove an analogue of the above result for Lp(G).

For an arbitrary set $X$ (finite or infinite), denote by $\mathcal {I}(X)$ the symmetric inverse semigroup of partial injective transformations on $X$. For $ \alpha \in \mathcal {I}(X)$, let $C(\alpha )=\{ \beta \in \mathcal {I}(X): \alpha \beta = \beta \alpha \}$ be the centraliser of $ \alpha $ in $\mathcal {I}(X)$. For an arbitrary $ \alpha \in \mathcal {I}(X)$, we characterise the transformations $ \beta \in \mathcal {I}(X)$ that belong to $C( \alpha )$, describe the regular elements of $C(\alpha )$, and establish when $C( \alpha )$ is an inverse semigroup and when it is a completely regular semigroup. In the case where $\operatorname {dom}( \alpha )=X$, we determine the structure of $C(\alpha )$in terms of Green’s relations.

A reverse iterated function system is defined as a family of expansive maps {T1,T2,…,Tm} on a uniformly discrete set . An invariant set is defined to be a nonempty set satisfying F=⋃ j=1mTj(F). A computation method for the dimension of the invariant set is given and some questions asked by Strichartz are answered.

We identify the majority of Siegel modular eigenforms in degree four and weights up to 16 as being Duke–Imamoḡlu–Ikeda or Miyawaki–Ikeda lifts. We give two examples of eigenforms that are probably also lifts but of an undiscovered type.

Best possible conditions are given here, under which all solutions of several delay differential equations are oscillatory.

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation

$$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$

$x,y\in {\rm\Omega}$

$H$

$\text{Aut}(G)$

${\rm\Omega}\subset G\times G$

$$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$

$(z,{\it\zeta})\in {\rm\Omega}$

$f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$

${\rm\Omega}\subset \mathbb{C}^{2}$

$0$

In this paper, we count a dual set of Stirling permutations by the number of alternating runs and study properties of the generating functions, including recurrence relations, grammatical interpretations and convolution formulas.

Let Γ be a family of isoclinic finite groups, and let acl(Γ) be the group of autoclinisms of Γ. In this paper we prove the following formula:

where the sum is taken over a complete system of stem groups S in Γ. This result is due to P. Hall, who outlined a proof in his paper, “On groups of automorphisms” (J. reine angew. Math). 182 (1940), 194–204, using presentations, whereas in this paper we consider stem groups in terms of central extensions.

Recently, B.-Y. Chen obtained an inequality for slant surfaces in complex space forms. Further, B.-Y. Chen and one of the present authors proved the non-minimality of proper slant surfaces in non-flat complex space forms. In the present paper, we investigate 3-dimensional proper contact slant submanifolds in Sasakian space forms. A sharp inequality is obtained between the scalar curvature (intrinsic invariant) and the main extrinsic invariant, namely the squared mean curvature.

It is also shown that a 3-dimensional contact slant submanifold M of a Sasakian space form (c), with c ≠ 1, cannot be minimal.

Let $G$ be a finite group and $\mathsf{cd}(G)$ denote the set of complex irreducible character degrees of $G$. We prove that if $G$ is a finite group and $H$ is an almost simple group whose socle is a sporadic simple group $H_{0}$ and such that $\mathsf{cd}(G)=\mathsf{cd}(H)$, then $G^{\prime }\cong H_{0}$ and there exists an abelian subgroup $A$ of $G$ such that $G/A$ is isomorphic to $H$. In view of Huppert’s conjecture, we also provide some examples to show that $G$ is not necessarily a direct product of $A$ and $H$, so that we cannot extend the conjecture to almost simple groups.

Let $T$ be a Banach algebra homomorphism from a Banach algebra ${\mathcal{B}}$ to a Banach algebra ${\mathcal{A}}$ with $\Vert T\Vert \leq 1$. Recently, Bhatt and Dabhi [‘Arens regularity and amenability of Lau product of Banach algebras defined by a Banach algebra morphism’, Bull. Aust. Math. Soc.87 (2013), 195–206] showed that cyclic amenability of ${\mathcal{A}}\times _{T}{\mathcal{B}}$ is stable with respect to $T$, for the case where ${\mathcal{A}}$ is commutative. In this note, we address a gap in the proof of this stability result and extend it to an arbitrary Banach algebra ${\mathcal{A}}$.

Giles and Joseph (Bull. Austral. Math. Soc. 11 (1974), 31–36), proved that the numerical range of an unbounded operator on a Banach space has a certain density property. They showed, in particular, that the numerical range of an unbounded operator on certain Banach spaces is dense in the scalar field. We prove that the numerical range of an unbounded operator on a Banach space is always dense in the scalar field.

We introduce a class of β − v-unfavourable spaces, which contains some known classes of β-unfavourable spaces for topological games of Choquet type. It is proved that every β − v-unfavourable space X is a Namioka space, that is for any compact space Y and any separately continuous function f : x × Y → ℝ there exists a dense in XGδ-set A ⊆ X such that f is jointly continuous at each point of A × Y.

We study a concept of stability under the gap of isomorphic properties of Banach spaces and apply it to obtain some results of stability under compact or small norm perturbation for non-semi-Fredholm operators with closed range.

In this paper we discuss the continuity of the Hausdorff dimension of the invariant set of generalised graph-directed systems given by contractive infinitesimal similitudes on bounded complete metric spaces. We use the theory of positive linear operators to show that the Hausdorff dimension varies continuously with the functions defining the generalised graph-directed system under suitable assumptions.

The necessary and sufficient conditions for atomic orthomodular lattices to have the MacNeille completion modular, or (o)-continuous or order topological, orthomodular lattices are proved. Moreover we show that if in an orthomodular lattice the (o)-convergence of filters is topological then the (o)-convergence of nets need not be topological. Finally we show that even in the case when the MacNeille completion of an orthomodular lattice L is order-topological, then in general the (o)-convergence of nets in does not imply their (o)-convergence in L. (This disproves, also for the orthomodular and order-topological case, one statement in G.Birkhoff's book.)

In a recent paper, Davies and Anderssen (1997) examined the range of relaxation times, on which the linear viscoelasticity relaxation spectrum could be reconstructed, when the oscillatory shear data were only known on a fixed finite interval of frequencies. In particular, they showed that, for such oscillatory shear data, knowledge about the relaxation spectrum could only be recovered on a specific finite interval of relaxation times. They referred to this phenomenon as sampling localisation. The purpose of this note is show how their result can be proved using a duality argument, and, thereby, establish the fundamental nature of sampling localisation in relaxation spectrum recovery.

In his paper “On the structure of ordered real vector spaces” (Publ. Math. Debrecen 4 (1955–56), 334–343), Erdös shows that a totally ordered real vector space of countable dimension is order isomorphic to a lexicographic direct sum of copies of the group of real numbers. Brown, in “Valued vector spaces of countable dimension” (Publ. Math. Debrecen 18 (1971), 149–151), extends the result to a valued vector space of countable dimension and greatly simplifies the proof. In this note it is shown that a finite valued vector lattice of countable dimension is order isomorphic to a direct sum of o–simple totally ordered vector spaces. One obtains as corollaries the result of Erdös and the applications that Brown makes to totally ordered spaces.

Zhou and Feng [‘On symmetric graphs of valency five’, Discrete Math. 310 (2010), 1725–1732] proved that all connected pentavalent 1-transitive Cayley graphs of finite nonabelian simple groups are normal. We construct an example of a nonnormal 2-arc transitive pentavalent symmetric Cayley graph on the alternating group $\text{A}_{39}$. Furthermore, we show that the full automorphism group of this graph is isomorphic to the alternating group $\text{A}_{40}$.

Let C denote the class of all near-rings which have the property that the subnear-ring of constants forms an ideal. Prominent examples are abstract affine near-rings and a generalisation of these by Feigelstock [1]. In this note we show C forms a variety and construct a proper sub-class such that every N ε C can be embedded into some . It turns out that near-rings have an ideal structure which is similar to the ideal structure of abstract affine near-rings, in contrast to the situation for arbitrary elements of C.