In this paper, we discuss a necessary and sufficient condition for common fixed points of two nonexpansive mappings. We then prove a convergence theorem to a common fixed point. Finally, we discuss the existence of a nonexpansive retraction onto the set of common fixed points of nonexpansive mappings. In these theorems, we do not assume the strict (uniform) convexity of the norm of the Banach space.

We prove a general theorem on dissections of quotients of theta-functions. As corollaries, we establish six q-series identities that were conjectured by M.D. Hirschhorn:

Sufficient conditions are provided in order that some classes of finite soluble groups, defined by properties of the Sylow normalisers, are saturated formations.

We consider some generalisations of the Cahn—Hilliard equation based on constitutive equations derived by M. Gurtin in (1996) with a logarithmic free energy. Compared to the classical Cahn—Hilliard equation (see [4, 5]), these models take into account the work of internal microforces and the anisotropy of the material. We obtain the existence and uniqueness of solutions results and then prove the existence of finite dimensional attractors.

We study a quantitative approach to weak noncompactness of operators under the Cobos-Peetre polygon interpolation method for Banach N-tuples. In the case of operators acting between two J-spaces or two K-spaces obtained by this method we prove logarithmically convex-type inequalities for certain operator seminorm vanishing on the subspace of weakly compact operators. Geometrically speaking, in these estimates only some triangles inscribed in the polygon are involved. For operators acting from a J-space to a K-space we prove logarithmically convex-type estimates where all polygon vertices are included. In particular, the estimates obtained here give the new proofs of the results showing the relation between distribution of weakly compact operators among polygon vertices and weak compactness of operators under interpolation.

Let T be a tree, f: T → T be a continuous map. We show that if f is pointwise chain recurrent (that is, every point of T is chain recurrent under f), then either fan is identity or fan is turbulent if Fix(f) ∩ End(T) = ∅ or else fan−1 is identity or fan−1 is turbulent if Fix(f) ∩ End(T) ≠  . Here n denotes the number of endpoints of T and, an denotes the minimal common multiple of 2,3,…,n.

We consider certain subsets of a semigroup S, defined mainly by conditions involving regularity preservation. In particular, the regular base B(S) of S may be regarded as a generalisation of the zero ideal in a semigroup with zero; if it non-empty then S is E-inversive. The other subsets considered are related in a natural way either to B(S) or to the set RP(S) of regularity-preserving elements in S. In a regular semigroup (equipped with the Hartwig-Nambooripad order) each of these subsets contains either minimal elements only or maximal elements only. The relationships between the subsets are discussed, and some characterisations of completely simple semigroups are obtained.

Let X be an infinite set and suppose א0 ≤ q ≤ |X|. The Baer-Levi semigroup on X is the set of all injective ‘total’ transformations α: X → X such that |X\Xα| = q. It is known to be a right simple, right cancellative semigroup without idempotents, its automorphisms are “inner”, and some of its congruences are restrictions of Malcev congruences on I(X), the symmetric inverse semigroup on X. Here we consider algebraic properties of the semigroup consisting of all injective ‘partial’ transformations α of X such that |X\Xα| = q: in particular, we descried the ideals and Green's relations of it and some of its subsemigroups.

We give a characterisation of Banach spaces X containing a subspace with a shrinking Markushevich basis {xγ, fγ}γ∈Γ. This gives a sufficient condition for X to have a renorming with Mazur's intersection property.

We have constructed all simple Lie superalgebras from some triple systems. In particular, we shall discuss simple Lie superalgebras of P(n), Q(n) and Cartan types W(n), S(n), H(n) in this article.

Let X be a separable Banach space which admits a C1-smooth norm, and let G ⊂ X be an open subset. Then any real-valued, bounded and uniformly continuous map on G can be uniformly approximated on G by C1-smooth functions with bounded derivative.

In this note, a formula for the characteristic polynomial of any (regular or irregular) graph covering is described.

We give some characterisations of rings R whose modules with composition series are all cyclic. In particular, we prove that all left R-modules of finite length are cyclic if and only if R has no nonzero Artinian factor rings.

We consider the class N of groups in which the normaliser of every subgroup is normal, and the class C of groups in which the commutator subgroup normalises every subgroup. It is clear that C ⊆ N, and it is known that groups in the class N are nilpotent of class at most 3. We show that every finite p-group in N is also in C, provided that p ≥ 5, and we give an example showing that this is not true for p = 2.

We consider the space Sn of all nonempty bounded closed normal subsets of the cone where is the set of all vectors x ∈ Rn with nonnegative coordinates. We equip the space Sn with the Hausdorff metric and show that most elements of Sn are, in fact, strictly normal. More precisely, we show that the complement of the collection of all stricly normal elements of Sn is a σ-porous subset of Sn.

We prove that, given a quasi-primitive complex representation D for a finite group G, the possible ways of decomposing D as an inner tensor product of two projective representations of G are parametrised in terms of the group structure of G. More explicitly, we construct a bijection between the set of such decompositions and a particular interval in the lattice of normal subgroups of G.