For a well-behaved measure μ, on a locally compact totally ordered set X, with continuous part μc, we make Lp (X, μc) into a commutative Banach bimodule over the totally ordered semigroup algebra Lp (X, μ), in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from Lp (X, μ) into its dual module and in particular shows that if μc is not identically zero then Lp (X, μ) is not weakly amenable. We show that all bounded derivations from L1 (X, μ) into its dual module arise in this way and also describe all bounded derivations from Lp(X, μ) into its dual for 1 < p < ∞ the case that X is compact and μ continuous.

It is shown that every simple complex Lie algebra 𝔤 admits a 1-parameter family 𝔤q of deformations outside the category of Lie algebras. These deformations are derived from a tensor product decomposition for Uq(𝔤)-modules; here Uq(𝔤) is the quantized enveloping algebra of 𝔤. From this it follows that the multiplication on 𝔤q is Uq(𝔤)-invariant. In the special case 𝔤 = (2), the structure constants for the deformation 𝔤 (2)q are obtained from the quantum Clebsch-Gordan formula applied to V(2)q ⊗ V(2)q; here V(2)q is the simple 3-dimensional Uq(𝔤(2))-module of highest weight q2.

For positive integers s and t, let f(s, t) denote the smallest positive integer N such that every 2-colouring of [1, N] = {1, 2,...,N} has a monochromatic homothetic copy of {1, 1 + s, 1 + s + t}.

We show that f (s, t) = 4(s + t) + 1 whenever s/g and t/g are not congruent to 0 (modulo 4), where g = gcd(s, t). This can be viewed as a generalization of part of van der Waerden’s theorem on arithmetic progressions, since the 3-term arithmetic progressions are the homothetic copies of {1, 1 + 1, 1 + 1 + t}. We also show that f (s, t) = 4(s + t) + 1 in many other cases (for example, whenever s > 2t > 2 and t does not divide s), and that f (s, t) ≤ 4 (s + t) + 1 for all s, t.

Thus the set of homothetic copies of {1, 1 + s, 1 + s + t} is a set of triples with a particularly simple Ramsey function (at least for the case of two colours), and one wonders what other “natural” sets of triples, quadruples, etc., have simple (or easily estimated) Ramsey functions.

For positive integers p and q with (p − 2)(q − 2) > 4 there is, in the hyperbolic plane, a group [p, q] generated by reflections in the three sides of a triangle ABC with angles π/p, π/q, π/2. Hyperbolic trigonometry shows that the side AC has length ψ, where cosh ψ = c/s, c = cos π/q, s = sin π/p. For a conformal drawing inside the unit circle with centre A, we may take the sides AB and AC to run straight along radiiwhile BC appears as an arc of a circle orthogonal to the unit circle. The circle containing this arc is found to have radius 1/ sinh ψ = s/z, where z = , while its centre is at distance 1/ tanh ψ = c/z from A. In the hyperbolic triangle ABC, the altitude from AB to the right-angled vertex C is ζ, where sinh ζ = z.

We give a direct proof that wis an A∞+(g) weight if and only if w satisfies a one-sided, weighted reverse Hölder inequality.

If f0: Ω ⊂ ℝm → Sn is a weakly p-harmonic map from a bounded smooth domain Ω in ℝm (with 2 < p < m) into a sphere and if f0 is not stationary p-harmonic, then there exist infinitely many weak solutions of the p-harmonic flow with initial and boundary data f0, i.e., there are infinitely many global weak solutions f :Ω × ℝ → ⊂ Sn of

We also show that there exist non-stationary weakly (m − 1)-harmonic maps f0: Bm → Sm−1.

A characterisation of the range of a homomorphism between two commutative group algebras is presented which implies, among other things, that this range is closed. The work relies mainly on the characterisation of such homomorphisms achieved by P. J. Cohen.

We consider the problem: If K is a compact normal operator on a Hilbert module E, and f ∈ C0(SpK) is a function which is zero in a neighbourhood of the origin, is f(K) of finite rank? We show that this is the case if the underlying C*-algebra is abelian, and that the range of f(K) is contained in a finitely generated projective submodule of E.

An example is constructed to show that the J0-radical of a matrix nearring can be an intermediate ideal. This solves a conjecture put forward in [1].

In this paper, we define the η-invariant for a cusped hyperbolic 3-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

The primary purpose of this paper is to provide necessary and sufficient conditions for certain quadratic polynomials of negative discriminant (which we call Euler-Rabinowitsch type), to produce consecutive prime values for an initial range of input values less than a Minkowski bound. This not only generalizes the classical work of Frobenius, the later developments by Hendy, and the generalizations by others, but also concludes the line of reasoning by providing a complete list of all such primeproducing polynomials, under the assumption of the generalized Riemann hypothesis (GRH).We demonstrate how this prime-production phenomenon is related to the exponent of the class group of the underlying complex quadratic field. Numerous examples, and a remaining conjecture, are also given.

We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question.

Statistics in the 20th century has been enlivened by a passionate, occasionally bitter, and still vibrant debate on the foundations of statistics and in particular on Bayesian vs. frequentist approaches to inference. In 1975 D. V. Lindley predicted a Bayesian 21st century for statistics. This prediction has often been discussed since, but there is still no consensus on the probability of its correctness. Recent developments in the asymptotic theory of statistics are, surprisingly, shedding new light on this debate, and may have the potential to provide a common middle ground.

This paper treats the quasilinear elliptic inequality

where N ≥ 2, m > 1, σ >m− 1, and p:ℝN → (0, ∞) is continuous. Sufficient conditions are given for this inequality to have no positive entire solutions. When p has radial symmetry, the existence of positive entire solutions can be characterized by our results and some known results.

We show that finite dimensional subfactors do not have subdiagonal algebras unless the Jones index is one.