Loops are nonassociative algebras which can be investigated by using their multiplication groups and inner mapping groups If the inner mapping group of a loop is finite and Abelian, then the multiplication group is a solvable group. It is clear that not all finite Abelian groups can occur as inner mapping groups of loops. In this paper we show that certain finite Abelian groups with a special structure are not isomorphic to inner mapping groups of finite loops. We use our results and show how to construct solvable groups which are not isomorphic to multiplication groups of loops.

The sharp bound for the third Hankel determinant for the coefficients of the inverse function of convex functions is obtained, thus answering a recent conjecture concerning invariance of coefficient functionals for convex functions.

In this paper we give a characterisation of -invariant real hypersurfaces of type A; that is, a tube over a totally geodesic G2(ℂm+1) in complex two-plane Grassmannians G2(ℂm+2) or a ruled real hypersurface foliated by complex hypersurfaces which includes a maximal totally geodesic submanifold G2(ℂm+1) in G2(ℂm+2) in terms of η-parallel shape operator.

A duality theorem, and a converse duality theorem, are proved for a nonlinear fractional program, where the numerator of the objective function involves a concave function, not necessarily differentiable, and also the support function of a convex set, and the denominator involves a convex function, and the support function of a convex set. Various known results are deduced as special cases.

Let G be a graph of order n. Let a, b and k be nonnegative integers such that 1≤a≤b. A graph G is called an (a,b,k)-critical graph if after deleting any k vertices of G the remaining graph of G has an [a,b]-factor. We provide a sufficient condition for a graph to be (a,b,k)-critical that extends a well-known sufficient condition for the existence of a k-factor.

We observe that a solvability criterion for finite groups, conjectured by Miller [The product of two or more groups, Trans. Amer. Math. Soc.12 (1911)] and Hall [A characteristic property of soluble groups, J. London Math. Soc.12 (1937)] and proved by Thompson [Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc.74(3) (1968)], can be sharpened as follows: a finite group is nonsolvable if and only if it has a nontrivial 2-element and an odd p-element, such that the order of their product is not divisible by either 2 or p. We also prove a solvability criterion involving conjugates of odd p-elements. Finally, we define, via a condition on products of p-elements with p′-elements, a formation Pp,p′, for each prime p. We show that P2,2′ (which contains the odd-order groups) is properly contained in the solvable formation.

We prove the following theorem: if G is a locally compact Hausdorff group such that each of its proper closed subgroups has only a finite number of closed subgroups, then G is topologically isomorphic to the circle group.

Let B be a topological algebra with Fréchet space topology, A an algebra with locally convex topology and an algebra of formal power series over A in n commuting indeterminates which carries a Fréchet space topology. In a previous paper the author showed, for the case n = 1, that a homomorphism of B into whose range contains polynomials is necessarily continuous provided the coordinate projections of into A satisfy a certain equicontinuity condition. This result is here extended to the case of general n, and also to weaker topological assumptions.

For a H-type group G, we first give explicit equations for its shortest sub-Riemannian geodesics. We use properties of sub-Riemannian geodesics in G to characterise the isometry group ISO(G) with respect to the Carnot-Carathéodory metric. It turns out that ISO(G) coincides with the isometry group with respect to the standard Riemannian metric of G.

Boyle and Goodearl proved that if R is a left V-ring then R has left Krull dimension if and only if R is left Neotherian. In this paper we extend this result to arbitrary V-modules.

Let $X={\mathcal{C}}_{0}(2{\it\pi})$ or $X=L_{1}[0,2{\it\pi}]$. Denote by ${\rm\Pi}_{n}$ the space of all trigonometric polynomials of degree less than or equal to $n$. The aim of this paper is to prove the minimality of the norm of de la Vallée Poussin’s operator in the set of generalised projections ${\mathcal{P}}_{{\rm\Pi}_{n}}(X,\,{\rm\Pi}_{2n-1})=\{P\in {\mathcal{L}}(X,{\rm\Pi}_{2n-1}):P|_{{\rm\Pi}_{n}}\equiv \text{id}\}$.

In this paper, we give some evaluation formulas for the values of double L-series of Tornheim's type, in terms of the Dirichlet L-values and the Riemann zeta values at positive integers. As special cases, these give the formulas for double L-values given by Terhune.

Every finite group G satisfies a law [x, ry] = [x, sy] for some positive integers r < s. The minimal value of r is called the depth of G. It is well known that groups of depth 1 are abelian. In this paper we prove the following. Let G be a finite group of depth 2. Then G/F(G) is supersoluble, metabelian and has abelian Sylow p-subgroups for all odd primes p. Moreover, lp(G) ≤ 1 for p odd and l2(G2) ≤ 1.

Upper and lower semicontinuity results for the chain recurrent set are shown to remain valid in numerical dynamics with constant stepsizes. It is also pointed out that the chain recurrent set contains numerical ω–limit sets for discretisations with a variable stepsize sequence approaching zero.

We show that every graded ideal of a Leavitt path algebra is graded isomorphic to a Leavitt path algebra. It is known that a graded ideal I of a Leavitt path algebra is isomorphic to the Leavitt path algebra of a graph, known as the generalised hedgehog graph, which is defined based on certain sets of vertices uniquely determined by I. However, this isomorphism may not be graded. We show that replacing the short ‘spines’ of the generalised hedgehog graph with possibly fewer, but then necessarily longer spines, we obtain a graph (which we call the porcupine graph) whose Leavitt path algebra is graded isomorphic to I. Our proof can be adapted to show that, for every closed gauge-invariant ideal J of a graph $C^*$ -algebra, there is a gauge-invariant $*$ -isomorphism mapping the graph $C^*$ -algebra of the porcupine graph of J onto $J.$

This paper presents a necessary condition for a ribbon link to be an homology boundary link and gives a consequent simple counterexample to the conjecture of Smythe that the vanishing of the first Alexander polynomial characterizes homology boundary links among all 2-component links.

In this paper, we prove some new reverse dynamic inequalities of Renaud- and Bennett-type on time scales. The results are established using the time scales Fubini theorem, the reverse Hölder inequality and a time scales chain rule.

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.