Let B be a Banach space and X ⊂ B a normal cone such that the norm is monotone on X for the order determinated by X.

We study the sup, denoted by i(X), of the q ≥ 1 such that, for each E> 0 and each n, there are x1, …, xn in X such that:

for all a1, …, an ≥ 0, where ‖ ‖q is the norm in lq.

We prove that i(X) is the inf of the p for which we have:

The proof use a similar theorem of Kirvine, concerning Banach Riesz spaces. Here conical measures are a useful tool. We establish a link with a preceding work in which we adapt the Maurey theory factorisation of operators with values in a LP space, to the case of normal cones, contained in a Banach space.

Let D be a (2 × 2) matrix with distinct eigenvalues λ1 and λ2. There is a basic and well known functional equation which provides a formula for constructing the matrix g (D), for any ℂ-valued function g defined on a subset of ℂ containing {λ1,λ2}, namely .

This equation is used to give a direct and transparent proof of the following fact due to Anderson: A pair of (2 × 2) selfadjoint matrices A1 and A2 commute if and only if the Weyl functional calculus of the pair (A1,A2), which is a matrix-valued distribution, has order zero (that is, is a measure).

We give a characterization of monoidal closed structures, “determined by systems of filters” on the category of topological spaces and continuous maps. The method we use to introduce suitable topologies on the product set X × Y of spaces X and Y, and on the set of all continuous maps from X to Y, is essentially that of Wilker.

Following Dwork's indications, in this work we give a further elaboration and a list of corrections for the article “G-fonctions et cohomologie des hypersurfaces singulières”. Moreover we extend the contents of that work to quasi-homogeneous polynomials.

As a special case of more general results, it is proved in this note that, if α is any real number and δ any positive number, then there exists a positive integer X such that the inequality

has infinitely many solutions in positive integers h and Yh.

The method depends on the study of infinite sequences of real linear forms in a fixed number of variables. It has relations to that used by Kronecker in the proof of his classical theorem and can be generalised.

If f(x) is defined on [−1, 1], let H¯1 n(f, x) denote the Lagrange interpolation polynomial of degree n (or less) for f which agrees with f at the n+1 equally spaced points xk, n = −1 + (2k)/n (0 ≤ k ≤ n). A famous example due to S. Bernstein shows that even for the simple function h(x) = │x│, the sequence H¯1 n (h, x) diverges as n → ∞ for each x in 0 < │x│ < 1. A generalisation of Lagrange interpolation is the Hermite-Fejér interpolation polynomial H¯mn (f, x), which is the unique polynomial of degree no greater than m(n + 1) – 1 which satisfies (f, Xk, n) = δo, pf(xk, n) (0 ≤ p ≤ m − 1, 0 ≤ k ≤ n). In general terms, if m is an even number, the polynomials H¯mn(f, x) seem to possess better convergence properties than the H¯1 n (f, x). Nevertheless, D.L. Berman was able to show that for g(x) ≡ x, the sequence H¯2n(g, x) diverges as n → ∞ for each x in 0 < │x│. In this paper we extend Berman's result by showing that for any even m, H¯mn(g, x) diverges as n → ∞ for each x in 0 < │x│ < 1. Further, we are able to obtain an estimate for the error │H¯mn(g, x) – g(x)│.

We construct frame starters in Z2n − {0, n}, for n ≡ 0, 1 mod 4, where Z2n denotes the cyclic group of order 2n. We also construct left frame starters in Q2n − {e, αn}, where Q2n is the dicyclic group of order 4n and αn is the unique element of order 2 in Q2n.

We consider a semigroup S acting as affine continuous maps on a compact convex set X. F denotes the corresponding set of fixed points. Let exX and exF denote the corresponding sets of extreme points. If X is a simplex, conditions are given which ensure that when x ε F, the maximal measure representing x invariant under S. We also prove exF = F ∩ exX under conditions involving extreme amenability of S. Topological properties of exF are also studied.

We prove that the restriction of a given orthogonal-complete metric space to the closure of the orbit induced by the origin point with respect to an orthogonal-preserving and orthogonal-continuous map is a complete metric space. Then we show that many existence results on fixed points in orthogonal-complete metric spaces can be proved by using the corresponding existence results in complete metric spaces.

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\alpha \in I(V)$ , we denote the domain and the range of $\alpha $ by ${\mathop {\textrm {dom}}}\,\alpha $ and ${\mathop {\textrm {im}}}\,\alpha $ , and we call the cardinals $g(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {dom}}}\,\alpha $ and $d(\alpha )={\mathop {\textrm {codim}}}\,{\mathop {\textrm {im}}}\,\alpha $ the ‘gap’ and the ‘defect’ of $\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

Examples of normed barrelled spaces E or quasicomplete barrelled spaces E are given such that there is a non-continuous linear map from the space E into itself with closed graph.

Andersson obtained weighted Lp estimates for on analytic polyhedra. We provide an example to show why they cannot be improved.

Let Mn be the “polygon space” introduced by Kirwan and Klyachko. In this paper, we give new results on the topology of Mn for odd n. We determine πq(Mn) (q ≤ n − 3). Then we describe Mn in the oriented cobordism ring . We also give new and elementary proofs of the result on the ring structure of H*(Mn/Sn; Q), where Sn denotes the symmetric group acting naturally on Mn.

A characterisation of a smoothing spline is sought in a convex closed set C of Hilbert space: , T and A are linear operators. A representation of the solution is obtained in the terms of the kernels of the above operators, of the dual operators T*, A* and of the dual cone C0. A particular case is considered when T is the differential operator and A is the operator-trace of a function.

In J. Math. Anal. Appl. 61 (1977), 466–474, Leader has given a fixed point principle for an operator f:X → X, where X is a metric-space, based on conditional uniform equivalence of orbits. We generalize this principle for two mappings f1 and f2 to give common fixed point results in two different ways. Further we derive an f-generalized fixed point theorem for two commuting mappings.