We show how some results of the theory of iterated function systems can be derived from the Tarski–Kantorovitch fixed–point principle for maps on partialy ordered sets. In particular, this principle yields, without using the Hausdorff metric, the Hutchinson–Barnsley theorem with the only restriction that a metric space considered has the Heine–Borel property. As a by–product, we also obtain some new characterisations of continuity of maps on countably compact and sequential spaces.

In this article we present a simple method to deduce necessary conditions for weak minimisation of a convex vector program in a Banach space. Our main tool here will be the generalised Jabcobian of Ralph.

Let {E1, … Er} and {F1, … Fs} be partitions of a probability space. We exhibit a natural bijection from the set of efficient ways of verifying the independence of such partitions to the set of spanning trees of the complete bipartite graph Kr, s.

We give direct proofs of meromorphic continuality on the whole complex plane of certain zeta functions ZP, Q (s) and Z (P/Q, s) associated with a pair of polynomials P, Q. We calculate ZP, Q (−q) (q a non-negative integer) and give explicit formulas for the residues of Z (P/Q, s) at poles.

We generalise the Borsuk-Ulam theorem for maps Mn → ℝn.

Everyone knows the Borsuk-Ulam theorem as a simple application of some of the first ideas one encounters in algebraic topology.

We prove that each Polish space is homeomorphic to the ω–limit set of a bounded solution of an autonomous equation x′ = f (x) in l2 × c0, in which f is Lipschitz continuous on bounded sets.

Two 2-cell embeddings i, j of a graph G into surfaces  and ′ are said to be congruent with respect to a subgroup Γ of Aut(G) if there are a homeomorphism h:  → ′ and an automorphism γ ∈ Γ such that h ∘ i = j ∘ γ. In this paper, we compute the total number of congruence classes of 2-cell embeddings of any bouquet of circles into surfaces with respect to a group consisting of graph automorphisms of a bouquet.

The property of σcyclic monotonicity is proposed here to describe subdifferentials of lsc convex functions that are continuous in their domains. It is shown that all monotone operators in R and all densely defined cyclically monotome operators in Rn share this property. Examples of a densely defined maximal cyclically monotone operator in a Hilbert space and of a subdifferential of a convex lsc function in R2 which are not σ-cyclically monotone operators are given.

We consider the stable Steinberg group St (R) over local rings. An element x is called an involution if x2 = 1. We prove that every element δ in St (R) is the product of at most 5 involutions.

In this paper, we count two-block configurations in a general setting. In particular, no restriction is put on the pair repetition factor (that is, the parameter Λ) of the block designs being considered. Besides giving formulae for the counts of the two-block BTD (V, B, R, 3, Λ) configuration classes, a basis for the formulae is given and shown to have size four. Details of the special cases where Λ equals two and three are also presented, along with 2-block BIBD (v, b, r, 3, λ) configuration results.

Let ρ : G →  (H) be an irreducible unitary representation of a compact group G where  (H) is a set of unitary operators of finite dimensional Hilbert space H. For the (p1, …, PL)-Bernoulli shift, the solvability of ρ(φ(x)) g (Tx) = g (x) is investigated, where φ(x) is a step function.

We discuss the problem of finding an analogue of the concept of topological space in supergeometry, motivated by the search for a procedure to compactify supermanifolds along odd coordinates. In particular, we examine the topologies arising naturally on the sets of points of locally ringed superspaces, and show that in the presence of a nontrivial odd sector such topologies are never compact. The main outcome of our discussion is the following new observation: not only the usual framework of supergeometry (the theory of locally ringed spaces), but the more general approach of the functor of points, need to be further enlarged.

Let X be a compact, connected Hausdorff space and f a real valued, symmetric, continuous function on X × X. Then the Gross-Stadje number r (X, f) is the unique real number with the property that for each positive integer n and for all (not necessarily distinct) x1,…,xn in X, there exists some x in X such that . This paper solves the following open question in distance geometry: What is the least upper bound g2(R2) of r (X, d2), where X ranges over all compact, connected subsets of the Euclidean plane with diameter one and where d2 denotes the squared, Euclidean distance. We show: .

By using various kinds of moduli of smoothness, it is established that the multivariate variants of the well-known singular integrals of Picard, Poisson-Cauchy, Gauss-Weierstrass and their Jackson-type generalisations satisfy the “global smoothness preservation” property. The results are extensions of those proved by the authors for the univariate case.

We present several results related to the recently introduced generalised Bernoulli polynomials. Some recurrence relations are given, which permit us to compute efficiently the polynomials in question. The sums , where jk = jk (α) are the zeros of the Bessel function of the first kind of order α, are evaluated in terms of these polynomials. We also study a generalisation of the series appearing in the Euler-MacLaurin summation formula.

In this paper we show that the Thurston distance together with the critical exponent on the Teichmüller space of a convex cocompact hyperbolic 3-manifold distinguishes the different points.

In this paper, we prove the uniqueness of the decaying positive solution on all of Rn for certain second order non linear elliptic equations. This improves earlier work of a number of authors. These problems occur in the theory of peak solutions. In particular, our results apply to a number of non-smooth nonlinearities which occur as limiting equations in population problems.

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

In this paper potential symmetries are sought of the inhomogeneous nonlinear diffusion equations ut = x1−M[xN−1f (u) ux]x. The functional forms of f (u) that admit such symmetries are completely classified. A complete list is presented of the symmetries, which depend on the values of the parameters M and N. We give examples of similarity solutions using potential symmetries. In some cases, the potential symmetries enable us to convert non–invertible mappings of nonlinear partial differential equations to linear ones.

The volume of a general lattice polyhedron P in ℝN can be determined in terms of numbers of lattice points from N − 1 different lattices in P Ehrhart gave a formula for the volume of “polyèdre entier” in even-dimensional spaces involving only N/2 lattices. The aim of this note is to comment on Ehrhart's formula and provide a similar volume formula applicable to lattice polyhedra that are N-dimensional manifolds in ℝN.