Let X be a compact, connected, oriented topological G-manifold, where G is a compact connected Lie group. Assume that the fixed point set is finite but nonempty, the action is otherwise free, and the orbit space is a manifold. It follows that either G = U(1) = S1 and dimX =4 or G = Sp(1) = S3 and dimX = 8, and the number of fixed points is even. The authors prove that these ∪(1)-manifolds (respectively, Sp

(1)-manifolds) are classified up to orientation-preserving equivariant homeomorphism by (1) the orientation-preserving homeomorphism type of their orbit 3-manifolds (respectively, 5-manifolds), and

(2) the (even) number of fixed points.

Both the homeomorphism type in (1) and the even number in (2) are arbitrary, and all the examples are constructed. The smooth analog for U(1) is also proved.

Popov has recently introduced a class of subspaces of Lp(μ) (μ nonatomic) which generalise the finite codimensional ones, and proved that for p ≠ 2 any projection onto such a subspace has a norm strictly greater than one. In this paper we give the quantitative version of Popov's result computing the best possible lower bound for the norms of the considered projections.

A short elementary proof is given of the theorem of Marggraff which states that a primitive permutation group which contains a cycle fixing k points is (k+1)-fold transitive. It is then shown that the method of proof actually yields a generalization of Marggraff's theorem.

We study the existence and uniqueness of ${\mathcal{S}}$-asymptotically periodic solutions for a general class of abstract differential equations with state-dependent delay. Some examples related to problems arising in population dynamics are presented.

New oscillation criteria for nonlinear differential equations with deviating arguments of the form

n even, are established.

The Hall projective planes of odd order are characterized in terms of their translations, collineations which fix all the points of a Baer subplane, and involutory homologies.

By applying a minimax inequality on H-spaces from our earlier work, new generalisations of well-known intersection theorems concerning sets with convex sections and minimax inequalities of von Neumann type are obtained. Our results generalise the corresponding results of Ben-El-Mechaiekh, Deguire and Granas, Fan, Liu, Shih-Tan, Sion and Tarafdar.

Let r be a given prime. Then a monoid M is the endomorphism monoid of a field of characteristic r if and only if either M is a finite cyclic group or M is a right cancellative monoid and M has an element of infinite order in its centre. The main lemma is the technical base of the present and other papers.

We study a Dirichlet problem involving the weak Laplacian on the Sierpiński gasket, and we prove the existence of at least two distinct nontrivial weak solutions using Ekeland’s Variational Principle and standard tools in critical point theory combined with corresponding variational techniques.

The main results proved in this note are the following:

(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and co-Hopfian.

(ii) There is no functorial imbedding of groups (respectively finitely generated groups) into Hopfian groups.

(iii) We prove a result which implies in particular that if the double orientable cover N of a closed non-orientable aspherical manifold M has a co-Hopfian fundamental group then π1(M) itself is co-Hopfian.

Little seems to be known about the digits or sequences of digits in the decimal representation of a given irrational number like √2 or π. There is no difficulty in constructing an irrational number such that in its decimal representation certain digits or sequences of digits do not occur. On the other hand, well known theorems by Tchebychef, Kronecker, and Weyl imply that some integral multiple of the given irrational number always has any given finite sequence of digits occuring at least once in its decimal representation: for the fractional parts of the multiplies of the number lie dense in the interval (0, 1).

Let E be a Banach space. The averaging interval AI(E) is defined as the set of positive real numbers α, with the following property: For each n ∈ ℕ and for all (not necessarily distinct) x1, x2, … xn ∈ E with ∥x1∥ = ∥x2∥ = … = ∥xn∥ = 1, there is an x ∈ E, ∥x∥ = 1, such that

It follows immediately, that AI(E) is a (perhaps empty) interval included in the closed interval [1,2]. For example in this paper it is shown that AI(E) = {α} for some 1 < α < 2, if E has finite dimension. Furthermore a complete discussion of AI(C(X)) is given, where C(X) denotes the Banach space of real valued continuous functions on a compact Hausdorff space X. Also a Banach space E is found, such that AI(E) = [1,2].

Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

In this paper, new minimax theorems for mixed lower-upper semicontinuous functions in probabilistic metric spaces are given. As applications, we utilise these results to show the existence of solutions of abstract variational inequalities, implicit variational inequalities and saddle point problems, and the existence of coincidence points in probabilistic metric spaces.

We apply recent results of Vaaler that simultaneously bound linear forms so as to obtain a ‘Siegel lemma’, sharper than those that have appeared in the literature, and in a shape convenient for application in transcendence theory.

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

In this paper we investigate the analogue of the classical badly approximable setup in which the distance to the nearest integer ‖⋅‖ is replaced by the sup norm |⋅|. In the case of one linear form we prove that the hybrid badly approximable set is of full Hausdorff dimension.

For a modulus function φ, we define the Hardy-Orlicz space H (φ). Two main questions are discussed in this paper. First, when is a linear map mg: H (φ) → H (φ), mg (f) = g.f an isometry? Second, when is H (φ) = H1?

Some inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P)∑ ni=1pixi)≤P∑ ni=1(f(xi)/pi) is given in which p1,…,pn are positive numbers, P=∑ ni=1pi and f is a Q-class function. The notion of the jointly Q-class function is introduced and some Jensen type inequalities for these functions are proved. Some Ostrowski and Hermite–Hadamard type inequalities related to Q-class functions are presented as well.