Podoski and Szegedy [‘On finite groups whose derived subgroup has bounded rank’, Israel J. Math.178 (2010), 51–60] proved that for a finite group $G$ with rank $r$, the inequality $[G:Z_{2}(G)]\leq |G^{\prime }|^{2r}$ holds. In this paper we omit the finiteness condition on $G$ and show that groups with finite derived subgroup satisfy the same inequality. We also construct an $n$-capable group which is not $(n+1)$-capable for every $n\in \mathbf{N}$.

In the present paper, a coupled algorithm refining recursively the Hermite–Hadamard inequality on a simplex is investigated. Our approach allows us to express the integral mean value $M_{f}$ of a convex function $f$ on a simplex as both the limit of sequences and sum of series involving iterative lower and upper bounds of $M_{f}$. Two examples of interest are discussed.

The main aim of this paper is to establish the Lipschitz continuity of the $(K,K^{\prime })$-quasiconformal solutions of the Poisson equation ${\rm\Delta}w=g$ in the unit disk $\mathbb{D}$.

By applying the theory of exponential dichotomies and contraction mapping, we establish some existence and uniqueness results for weighted pseudo almost periodic solutions of some differential equations with piecewise constant arguments. For this purpose, we also describe some basic properties of weighted pseudo almost periodic sequences.

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound

$$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$

$d$

${\mathcal{M}}$

$V_{{\rm\Omega}}(x,r)$

$B(x,r)\cap {\rm\Omega}$

$B(x,r)$

$x$

$r$

${\it\delta}$

${\rm\Omega}$

$e^{-t{\mathcal{A}}}$

$L^{p}({\rm\Omega})$

$p\in [1,\infty )$

This paper is devoted to the study of the local existence, uniqueness, regularity, and continuous dependence of solutions to a logistic equation with memory in the Bessel potential spaces.

Let $f$ be a transcendental meromorphic function with at least one direct tract. In this note, we investigate the structure of the escaping set which is in the same direct tract. We also give a theorem about the slow escaping set.

Let $G$ be a commutative group and $\mathbb{C}$ the field of complex numbers, $\mathbb{R}^{+}$ the set of positive real numbers and $f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$. In this paper, we first consider the Levi-Civitá functional inequality

$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$

${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$

${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$

$0<t_{1}\leq t_{2}$

$0<s_{1}\leq s_{2}$

$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$

$u,v,w,k$

$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$

$\circ$

$\otimes$

${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$

${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$

We obtain some results on approximate solutions of the generalised linear functional equation $\sum _{i=1}^{m}L_{i}f(\sum _{j=1}^{n}a_{ij}x_{j})=0$ for functions mapping a normed space into a normed space. We show that, under suitable assumptions, the approximate solutions are in fact exact solutions. The theorems correspond to and complement recent results on the hyperstability of generalised linear functional equations.

We prove that if ${\it\rho}$ is an irreducible positive definite function in the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\Vert {\it\rho}\Vert _{B(G)}=1$, then the iterated powers $({\it\rho}^{n})$ as a sequence of unital completely positive maps on the group $C^{\ast }$-algebra converge to zero in the strong operator topology.

We use the best constants in the Khintchine inequality to generalise a theorem of Kato [‘Similarity for sequences of projections’, Bull. Amer. Math. Soc.73(6) (1967), 904–905] on similarity for sequences of projections in Hilbert spaces to the case of unconditional Schauder decompositions in $\ell _{p}$ spaces. We also sharpen a stability theorem of Vizitei [‘On the stability of bases of subspaces in a Banach space’, in: Studies on Algebra and Mathematical Analysis, Moldova Academy of Sciences (Kartja Moldovenjaska, Chişinău, 1965), 32–44; (in Russian)] in the case of unconditional Schauder decompositions in any Banach space.

Grafakos and Sansing [‘Gabor frames and directional time–frequency analysis’, Appl. Comput. Harmon. Anal.25 (2008), 47–67] have shown how to obtain directionally sensitive time–frequency decompositions in $L^{2}(\mathbb{R}^{n})$ based on Gabor systems in $L^{2}(\mathbb{R})$. The key tool is the ‘ridge idea’, which lifts a function of one variable to a function of several variables. We generalise their result in two steps: first by showing that similar results hold starting with general frames for $L^{2}(\mathbb{R}),$ in the settings of both discrete frames and continuous frames, and second by extending the representations to Sobolev spaces. The first step allows us to apply the theory to several other classes of frames, for example wavelet frames and shift-invariant systems, and the second one significantly extends the class of examples and applications. We consider applications to the Meyer wavelet and complex B-splines. In the special case of wavelet systems we show how to discretise the representations using ${\it\epsilon}$-nets.

In this paper we improve the inequalities obtained by Chen in 2009 for the Euler–Mascheroni constant.

A nonreflexive Banach space may have a weakly uniformly rotund dual. The aim of this paper is to determine alternative characterisations and study further implications of this property in higher duals.

The Dunkl transform ${\mathcal{F}}_{k}$ is a generalisation of the usual Fourier transform to an integral transform invariant under a finite reflection group. The goal of this paper is to prove a strong uncertainty principle for orthonormal bases in the Dunkl setting which states that the product of generalised dispersions cannot be bounded for an orthonormal basis. Moreover, we obtain a quantitative version of Shapiro’s uncertainty principle on the time–frequency concentration of orthonormal sequences and show, in particular, that if the elements of an orthonormal sequence and their Dunkl transforms have uniformly bounded dispersions then the sequence is finite.

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 ${\mathcal{A}}$.

We give a positive answer to the question of Bouras [‘Almost Dunford–Pettis sets in Banach lattices’, Rend. Circ. Mat. Palermo (2) 62 (2013), 227–236] concerning weak compactness of almost Dunford–Pettis sets in Banach lattices. That is, every almost Dunford–Pettis set in a Banach lattice $E$ is relatively weakly compact if and only if $E$ is a $\mathit{KB}$-space.

It is an open question whether every derivation of a Fréchet GB$^{\ast }$-algebra $A[{\it\tau}]$ is continuous. We give an affirmative answer for the case where $A[{\it\tau}]$ is a smooth Fréchet nuclear GB$^{\ast }$-algebra. Motivated by this result, we give examples of smooth Fréchet nuclear GB$^{\ast }$-algebras which are not pro-C$^{\ast }$-algebras.

We prove that if a uniformly bounded (or equidistantly uniformly bounded) Nemytskij operator maps the space of functions of bounded ${\it\varphi}$-variation with weight function in the sense of Riesz into another space of that type (with the same weight function) and its generator is continuous with respect to the second variable, then this generator is affine in the function variable (traditionally, in the second variable).

In this paper, for an arbitrary $\ell ^{1}$-Munn algebra $\mathfrak{A}$ over a Banach algebra $A$ with a sandwich matrix $P$, we characterise all homomorphisms from $\mathfrak{A}$ to a commutative Banach algebra $B$. Especially, we study the character space of this algebra. Then, as an application, its character amenability is investigated. Finally, we apply these results to certain semigroups, which are called Rees matrix semigroups.