Let $\text{pod}_{-4}(n)$ denote the number of partition quadruples of $n$ where the odd parts in each partition are distinct. We find many arithmetic properties of $\text{pod}_{-4}(n)$ including the following infinite family of congruences: for any integers ${\it\alpha}\geq 1$ and $n\geq 0$,

$$\begin{eqnarray}\text{pod}_{-4}\biggl(3^{{\it\alpha}+1}n+\frac{5\cdot 3^{{\it\alpha}}+1}{2}\biggr)\equiv 0~(\text{mod}~9).\end{eqnarray}$$

$\text{pod}_{-4}(n)$

We consider the number of spanning trees in circulant graphs of ${\it\beta}n$ vertices with generators depending linearly on $n$. The matrix tree theorem gives a closed formula of ${\it\beta}n$ factors, while we derive a formula of ${\it\beta}-1$ factors. We also derive a formula for the number of spanning trees in discrete tori. Finally, we compare the spanning tree entropy of circulant graphs with fixed and nonfixed generators.

Let $g\geq 2$ be a fixed integer. Let $\mathbb{N}$ denote the set of all nonnegative integers and let $A$ be a subset of $\mathbb{N}$. Write $r_{2}(A,n)=\sharp \{(a_{1},a_{2})\in A^{2}:a_{1}+a_{2}=n\}.$ We construct a thin, strongly minimal, asymptotic $g$-adic basis $A$ of order two such that the set of $n$ with $r_{2}(A,n)=2$ has density one.

A subset $X$ of a finite group $G$ is a set of pairwise noncommuting elements if $xy\neq yx$ for all $x\neq y\in X$. If $|X|\geq |Y|$ for any other subset $Y$ of pairwise noncommuting elements, then $X$ is called a maximal subset of pairwise noncommuting elements and the size of such a set is denoted by ${\it\omega}(G)$. In a recent article by Azad et al. [‘Maximal subsets of pairwise noncommuting elements of some finite $p$-groups’, Bull. Iran. Math. Soc.39(1) (2013), 187–192], the value of ${\it\omega}(G)$ is computed for certain $p$-groups $G$. In the present paper, our aim is to generalise these results and find ${\it\omega}(G)$ for some more $p$-groups of interest.

If the centre of a group $G$ is trivial, then so is the centre of its automorphism group. We study the structure of the centre of the automorphism group of a group $G$ when the centre of $G$ is a cyclic group. In particular, it is shown that the exponent of $Z(\text{Aut}(G))$ is less than or equal to the exponent of $Z(G)$ in this case.

The best possible constant in a classical inequality due to Bonsall is established by relating that inequality to Young’s. Further, this extends the range of Bonsall’s inequality and yields a reverse inequality. It also provides a better constant in an inequality of Hardy, Littlewood and Pólya.

This article explores the properties of fractal interpolation functions with variable scaling parameters, in the context of smooth fractal functions. The first part extends the Barnsley–Harrington theorem for differentiability of fractal functions and the fractal analogue of Hermite interpolation to the present setting. The general result is applied on a special class of iterated function systems in order to develop differentiability of the so-called $\boldsymbol{{\it\alpha}}$-fractal functions. This leads to a bounded linear map on the space ${\mathcal{C}}^{k}(I)$ which is exploited to prove the existence of a Schauder basis for ${\mathcal{C}}^{k}(I)$ consisting of smooth fractal functions.

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 $\mathbb{D}$.

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

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

$d$

${\mathcal{M}}$

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

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

$B(x,r)$

$x$

$r$

${\it\delta}$

${\rm\Omega}$

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

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

$p\in [1,\infty )$

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.

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.

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).

We demonstrate that many properties of topological spaces connected with the notion of resolvability are preserved by the relation of similarity between topologies. Moreover, many of them can be characterised by the properties of the algebra of sets with nowhere dense boundary and the ideal of nowhere dense sets. We use these results to investigate whether a given pair of an algebra and an ideal is topological.

We investigate conjectures and questions regarding topological phenomena related to free actions on homotopy spheres and present some affirmative answers.

The main goal is to illustrate that the so-called indirect function of a cooperative game in characteristic function form is applicable to determine the nucleolus for a subclass of coalitional games called compromise stable transferable utility (TU) games. In accordance with the Fenchel–Moreau theory on conjugate functions, the indirect function is known as the dual representation of the characteristic function of the coalitional game. The key feature of a compromise stable TU game is the coincidence of its core with a box prescribed by certain upper and lower core bounds. For the purpose of the determination of the nucleolus, we benefit from the interrelationship between the indirect function and the prekernel of coalitional TU games. The class of compromise stable TU games contains the subclasses of clan games, big boss games and $1$- and $2$-convex $n$-person TU games. As an adjunct, this paper reports the indirect function of clan games for the purpose of determining their nucleolus.

In this paper, we consider signal recovery via $l_{1}$-analysis optimisation. The signals we consider are not sparse in an orthonormal basis or incoherent dictionary, but sparse or nearly sparse in terms of some tight frame $D$. The analysis in this paper is based on the restricted isometry property adapted to a tight frame $D$ (abbreviated as $D$-RIP), which is a natural extension of the standard restricted isometry property. Assuming that the measurement matrix $A\in \mathbb{R}^{m\times n}$ satisfies $D$-RIP with constant ${\it\delta}_{tk}$ for integer $k$ and $t>1$, we show that the condition ${\it\delta}_{tk}<\sqrt{(t-1)/t}$ guarantees stable recovery of signals through $l_{1}$-analysis. This condition is sharp in the sense explained in the paper. The results improve those of Li and Lin [‘Compressed sensing with coherent tight frames via $l_{q}$-minimization for $0<q\leq 1$’, Preprint, 2011, arXiv:1105.3299] and Baker [‘A note on sparsification by frames’, Preprint, 2013, arXiv:1308.5249].