The bipartite divisor graph B(X), for a set Xof positive integers, and some of its properties have recently been studied. We construct the bipartite divisor graph for the product of subsets of positive integers and investigate some of its properties. We also give some applications in group theory.

For a locally compact group $ \mathcal{G} $, we introduce and study a class of locally convex topologies $\tau $ on the measure algebra $M( \mathcal{G} )$ of $ \mathcal{G} $. In particular, we show that the strong dual of $(M( \mathcal{G} ), \tau )$ can be identified with a closed subspace of the Banach space $M\mathop{( \mathcal{G} )}\nolimits ^{\ast } $; we also investigate some properties of the locally convex space $(M( \mathcal{G} ), \tau )$.

An algebra A is said to be finitely related if the clone Clo(A) of its term operations is determined by a finite set of finitary relations. We prove that each finite idempotent semigroup satisfying the identity xyxzx≈xyzx is finitely related.

We study the existence of solutions for a class of abstract impulsive differential equations. Our technical framework allows us to study partial differential equations with impulsive conditions involving partial derivatives and nonlinear expressions of the solution. Some applications to impulsive partial differential equations are presented.

We characterise solutions f,g:ℝ→ℝ of the functional equation f(x+g(x)y)=f(x)f(y) under the assumption that f is continuous. Our considerations refer mainly to a paper by Chudziak [‘Semigroup-valued solutions of the Goła̧b–Schinzel functional equation’, Abh. Math. Semin. Univ. Hambg. 76, (2006), 91–98], in which the author studied the same equation assuming that g is continuous.

In this paper, we give analogues of the local uncertainty inequality for the Dunkl transform on ℝd, and indicate how the local uncertainty inequality implies a global uncertainty inequality.

In this paper, we give a classification of spacelike submanifolds with parallel normalised mean curvature vector field and linear relation $R= aH+ b$ of the normalised scalar curvature $R$ and the mean curvature $H$ in the de Sitter space ${ S}_{p}^{n+ p} (c)$.

Let u(n) and v(n) be the number of representations of a nonnegative integer n in the forms x2+4y2+4z2 and x2+2y2+2z2, respectively, with x,y,z∈ℤ, and let a4(n) and r3(n) be the number of 4-cores of n and the number of representations of n as a sum of three squares, respectively. By employing simple theta-function identities of Ramanujan, we prove that $u(8n+5)=8a_4(n)=v(8n+5)=\frac {1}{3}r_3(8n+5)$. With the help of this and a classical result of Gauss, we find a simple proof of a result on a4 (n) proved earlier by K. Ono and L. Sze [‘4-core partitions and class numbers’, Acta Arith. 80 (1997), 249–272]. We also find some new infinite families of arithmetic relations involving a4 (n) .

Let G be a finite group. A subset X of G is a set of pairwise noncommuting elements if any two distinct elements of X do not commute. In this paper we determine the maximum size of these subsets in any finite nonabelian metacyclic p-group for an odd prime p.

Let p be an odd prime and let G be a finite p-group such that xZ(G)⊆xG, for all x∈G∖Z(G), where xG denotes the conjugacy class of x in G. Then G has a noninner automorphism of order p leaving the Frattini subgroup Φ(G) elementwise fixed.

Given a morphism T from a Banach algebra ℬ to a commutative Banach algebra 𝒜, a multiplication is defined on the Cartesian product space 𝒜×ℬ perturbing the coordinatewise product resulting in a new Banach algebra 𝒜×Tℬ. The Arens regularity as well as amenability (together with its various avatars) of 𝒜×Tℬ are shown to be stable with respect to T.

We study the question whether a Riemann–Stieltjes integral of a positive continuous function with respect to a nonnegative function of bounded variation is positive.

We study the relationship between generalisations of P-spaces and Volterra (weakly Volterra) spaces, that is, spaces where every two dense Gδ have dense (nonempty) intersection. In particular, we prove that every dense and every open, but not every closed subspace of an almost P-space is Volterra and that there are Tychonoff nonweakly Volterra weak P-spaces. These results should be compared with the fact that every P-space is hereditarily Volterra. As a byproduct we obtain an example of a hereditarily Volterra space and a hereditarily Baire space whose product is not weakly Volterra. We also show an example of a Hausdorff space which contains a nonweakly Volterra subspace and is both a weak P-space and an almost P-space.

Continuous-time discrete-state random Markov chains generated by a random linear differential equation with a random tridiagonal matrix are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses comparison theorems for Carathéodory random differential equations and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself with respect to the Hilbert projective metric. It does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transition probabilities, in which case the attractor is a periodic path.

The unsolved problem of whether there exists a positive constant $c$ such that the number $k(G)$ of conjugacy classes in any finite group $G$ satisfies $k(G) \geq c \log _{2}|G|$ has attracted attention for many years. Deriving bounds on $k(G)$ from (that is, reducing the problem to) lower bounds on $k(N)$ and $k(G/N)$, $N\trianglelefteq G$, plays a critical role. Recently Keller proved the best lower bound known for solvable groups:

\[ k(G)\gt c_{0} \frac {\log _{2}|G|} {\log _{2} \log _{2} |G|}\quad (|G|\geq 4) \]

$k(G/N) \geq \beta [G : N]^{\alpha }$

$k(G/N) \geq \beta (\log [G : N])^{t}$

$G$

$N$

$k(N)$

$k(G)$

Quiver Hopf algebras are classified by means of ramification systems with irreducible representations. This leads to the classification of Nichols algebras over group algebras and pointed Hopf algebras of type one.

We follow the idea of generalising the notion of classical iterated function systems, as presented by Mihail and Miculescu. We give their deliberations a more general setting and, using this general approach, study the generic aspect of the problem of existence of an attractor of a function system.

Let $\{U_n\}$ be given by $U_0=1$ and $U_n=-2\sum _{k=1}^{[n/2]} \binom n{2k}U_{n-2k}\ (n\ge 1)$, where $[\cdot ]$ is the greatest integer function. Then $\{U_n\}$ is analogous to the Euler numbers and $U_{2n}=3^{2n}E_{2n}(\frac 13)$, where $E_m(x)$ is the Euler polynomial. In a previous paper we gave many properties of $\{U_n\}$. In this paper we present a summation formula and several congruences involving $\{U_n\}$.

In this paper, we consider the simultaneous representation of pairs of positive integers. We show that every pair of large positive even integers can be represented in the form of a pair of linear equations in four prime variables and k powers of two. Here, k=63 in general and k=31 under the generalised Riemann hypothesis.

Let $\Gamma $ be a $G$-vertex-transitive graph and let $(u,v)$ be an arc of $\Gamma $. It is known that if the local action $G_v^{\Gamma (v)}$ (the permutation group induced by $G_v$ on $\Gamma (v)$) is permutation isomorphic to the dihedral group of degree four, then either $|G_{uv}|$ is ‘small’ with respect to the order of $\Gamma $ or $\Gamma $is one of a family of well-understood graphs. In this paper, we generalise this result to a wider class of local actions.