In a recent paper In a recent paper the authors proved a multiplier theorem for Hardy spaces Hp (G), 0 < p ≤ 1, defined on a locally compact Vilenkin group G. The assumptions on the multiplier were expressed in terms of the “norms” of certain Herz spaces K (1/p − 1/?r, r, p) with r restricted to 1 ≤ r < ∞ and p < r. In the present paper we show how this restriction on r may be weakened to p ≤ r ∞. Furthermore, we present two modifications of our main theorem and compare these with certain results for multipliers on LP (Rn)-spaces, 1 < p < ∞, due to Seeger and to Cowling, Fendler and Foumier. We also discuss the sharpness of some of our results.

There are 11 closed 4-manifolds which admit geometries of compact type (S4, CP2 or S2 × S2) and two other closely related bundle spaces (S2 × S2 and the total space of the nontrivial RP2-bundle over S2). We show that the homotopy type of such a manifold is determined up to an ambiguity of order at most 4 by its quadratic 2-type, and this in turn is (in most cases) determined by the Euler characteristic, fundamental group and Stiefel-Whitney classes. In (at least) seven of the 13 cases, a PL 4-manifold with the same invariants as a geometric manifold or bundle space must be homeomorphic to it.

Existence and properties of incompressible surfaces in 3-dimensional manifolds are surveyed. Some conjectures of Waldhausen and Thurston concerning such surfaces are stated. An outline is given of the proof that such surfaces can be pulled back by non-zero degree maps between 3-manifolds. The effect of surgery on immersed, incompressible surfaces and on hierarchies is discussed. A characterisation is given of the immersed, incompressible surfaces previously studied by Hass and Scott, which arise naturally with cubings of non-positive curvature.

We present structural properties of the complex associative algebra generated by the canonical commutation relations in exponential form. In particular, we show it to be a central simple algebra that lacks zero divisors and is not Noetherian on either side; in addition, we determine explicitly its units and its automorphisms.

We provide a number of explicit examples of small volume hyperbolic 3-manifolds and 3-orbifolds with various geometric properties. These include a sequence of orbifolds with torsion of order q interpolating between the smallest volume cusped orbifold (q = 6) and the smallest volume limit orbifold (q → ∞), hyperbolic 3-manifolds with automorphism groups with large orders in relation to volume and in arithmetic progression, and the smallest volume hyperbolic manifolds with totally geodesic surfaces. In each case we provide a presentation for the associated Kleinian group and exhibit a fundamental domain and an integral formula for the co-volume. We discuss other interesting properties of these groups.

We give an account of the minimal volume of the plane, as defined by Gromov, and first computed by Bavard and Pansu. We also describe some related geometric inequalities.

We prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.

Let 1 ≤ M ≤ N − 1 be integers and K be a convex, symmetric set in Euclidean N-space. Associated with K and M, Mahler identified the Mth compound body of K, (K)m, in Euclidean (MN)-space. The compound body (K)M is describable as the convex hull of a certain subset of the Grassmann manifold in Euclidean (MN)-space determined by K and M. The sets K and (K)M are related by a number of well-known inequalities due to Mahler.

Here we generalize this theory to the geometry of numbers over the adèle ring of a number field and prove theorems which compare an adelic set with its adelic compound body. In addition, we include a comparison of the adelic compound body with the adelic polar body and prove an adelic general transfer principle which has implications to Diophantine approximation over number fields.

We consider a variety of algebras with two binary commutative and associative operations. For each integer n ≥ 0, we represent the partitions on an n-element set as n-ary terms in the variety. We determine necessary and sufficient conditions on the variety ensuring that, for each n, these representing terms be all the essentially n-ary terms and moreover that distinct partitions yield distinct terms.

A ring R is called a (proper) quotient no-zero-divisor ring if every (proper) nonzero factor ring of R has no zero-divisors. A characterization of a quotient no-zero-divisor ring is given. Using it, the additive groups of quotient no-zero-divisor rings are determined. In addition, for an arbitrary positive integer n, a quotient no-zero-divisor ring with exactly n proper ideals is constructed. Finally, proper quotient no-zero-divisor rings and their additive groups are classified.

A closed irreducible 3-manifold M is topologically rigid if any homotopy equivalent irreducible 3-manifold is homeomorphic to M. A construction is given which produces infinitely many non-Haken topologically rigid 3-manifolds

A conjecture on the Hausdorff dimension for Markov attractors of disjoint hyperbolic iterated function systems was given by Ellis and Branton. This paper proves the conjecture and generalizes the result to more general cases.

Some new results concerning tubular sets are presented, with applications to the convergence of the Polya algorithm in the contexts of simultaneous approximation and approximation of multivariate functions by univariate functions. (The Polya algorithm constructs a best uniform approximation from the limit, as p → ∞, of best Lp approximations.)

Let denote the kth successive inhomogeneous minima for positive values of real indefinite ternary quadratic forms of type (2, 1). Here it is proved that for the class of zero forms, All the critical forms have also been obtained. is already known. For non-zero forms it is proved that .

A 2-handle addition on the boundary of a hyperbolic 3-manifold M is called degenerating if the resulting manifold is not hyperbolic. There are examples that some manifolds admit infinitely many degenerating handle additions. But most of them are not ‘basic’. (See Section 1 for definitions). Our first main theorem shows that there are only finitely many basic degenerating handle additions. We also study the case that one of the handle additions produces a reducible manifold, and another produces a ∂-reducible manifold, showing that in this case either the two attaching curves are disjoint, or they can be isotoped into a once-punctured torus. A byproduct is a combinatorial proof of a similar known result about degenerating hyperbolic structures by Dehn filling.

Let A be a subring of a commutative ring B. If the natural mapping from the prime spectrum of B to the prime spectrum of A is injective (respectively bijective) then the pair (A, B) is said to have the injective (respectively bijective) Spec-map. We give necessary and sufficient conditions for a pair of rings A and B graded by a free abelian group to have the injective (respectively bijective) Spec-map. For this we first deal with the polynomial case. Let l be a field and k a subfield. Then the pair of polynomial rings (k[X], l[X]) has the injective Spec-map if and only if l is a purely inseparable extension of k.

Let k ≥ 3 and n > 6k be positive integers. The equations, with integer coefficients, have nontrivial p-adic solutions for all p > Ck8, where C is some positive constant. Further, for values k≥ K we can take C = 1 + O(K-½).

Using elementary methods we give a new proof of a result concerning the special form of the character of the bounded peripheral element which arises at an end of a curve component of the character variety of a knot complement.

We construct a functional calculus, g → g(A), for functions, g, that are the sum of a Stieltjes function and a nonnegative operator monotone function, and unbounded linear operators, A, whose resolvent set contains (−∞, 0), with {‖r(r + A)−1‖ ¦ r > 0} bounded. For such functions g, we show that –g(A) generates a bounded holomorphic strongly continuous semigroup of angle θ, whenever –A does.

We show that, for any Bernstein function f, − f(A) generates a bounded holomorphic strongly continuous semigroup of angle π/2, whenever − A does.

We also prove some new results about the Bochner-Phillips functional calculus. We discuss the relationship between fractional powers and our construction.

In this paper, we show the existence of solutions of functional equations fix ∈ Sx ∪ Tx and x = fix ∈ Sx ∪ Tx under certain nonlinear hybrid contraction and asymptotic regularity conditions, generalize and improve a recent result due to Kaneko concerning common fixed points of multivalued mappings weakly commuting with a single-valued mapping and satisfying a generalized contraction type. Some related results are also obtained.