In this paper, the duality theory for vector lattices developed by the author in (2) is applied to the study of vector lattices freely generated by distributive lattices.

If L is a finite distributive lattice defined by a set of generators X, subject to a finite set of relations R, then the vector lattice FVL(L) freely generated by the elements of L subject to the relations governing intersection and union in L is a finitely generated projective vector lattice, being a quotient of the vector lattice freely generated by elements of X by the principal ideal generated by the relations R. Thus, there is a polyhedron Q, such that FVL(L) is isomorphic with the vector lattice of real valued polyhedral functions on Q, under pointwise operations, and the isomorphism class of FVL(L) determines and is determined by the polyhedral type of Q. The principal result of this paper shows that the polyhedral class to which Q belongs can be simply characterised in terms of the nerve of the poset P of join-irreducible elements of L. Explicitly, up to polyhedral equivalence, Q is a suspension of the nerve of P.

We introduce and compare four procedures for defining the order-continuity of a function from one topological ordered space into another, where each reduces to the usual conception when the orderings of the two spaces are trivial. Chiefly for the purposes of this comparison, we use the idea of an ‘order-connected’ space, and in the course of investigating under which types of order-continuous functions this property is preserved, we are helped in assessing their relative importance.

In this paper a fundamental problem of thermal transport in a channel-gas flow has been studied. The most general solution in terms of integral formulae has been obtained when the duct is bounded by an arbitrary closed contour, the prescribed normal temperature gradient at the wall varies arbitrarily along the periphery, and the specified additional heat generation varies in an arbitrary manner from point to point in a cross-section. The power series solution is also discussed, and is given in a separate section. One of the objects of the present study is to look forward to the qualitative as well as quantitative effects of viscous dissipation and work of compression. Several results and all the graphs of Tao ((13)) have also been corrected.

1. Introduction. We introduce the notion of a weakly prime compact convex set, and we develop a reduction theory for spaces A(K). The notion is less restrictive in general than the prime compact convex sets of Chu, but gives a finer reduction than the Bishop and Silov decompositions forA(K) (12). The natural analogue for uniform algebras is related to the concept of weakly analytic sets due to Arenson, but unlike maximal weakly analytic sets the maximal weakly prime sets are always generalized peak sets; the uniform algebra can always be retrieved from the restrictions of the algebra to the maximal weakly prime sets.

Let K be a field and let R be a regular domain containing K. Let G be a finite subgroup of the group of automorphisms of R. We assume that |G| is invertible in K. Let RG be the ring of invariants of G. Let I be an ideal in RG. Fix i ⩾ 0. If RG is Gorenstein then:

1. (i) injdimRGHiI(RG) ⩽ dim Supp HiI(RG);

2. (ii) $H^j_{\mathfrak{m}}$(HiI(RG)) is injective, where $\mathfrak{m}$ is any maximal ideal of RG;

3. (iii) μj(P, HiI(RG)) = μj(P′, HiIR(R)) where P′ is any prime in R lying above P.

Using the concept of $s$-formality we are able to extend the bounds of a Theorem of Miller and show that a compact $k$-connected $(4k+3)$- or $(4k+4)$-manifold with $b_{k+1}=1$ is formal. We study $k$-connected $n$-manifolds, $n=4k+3, 4k+4$, with a hard Lefschetz-like property and prove that in this case if $b_{k+1}=2$, then the manifold is formal, while, in $4k+3$-dimensions, if $b_{k+1}=3$ all Massey products vanish. We finish with examples inspired by symplectic geometry and manifolds with special holonomy.

In this paper, we give lower bounds for the fundamental tone of open sets in minimal submanifolds immersed into warped product spaces of type Nn ×f Qq, where f ∈ C∞(N). This setting allows us to deal, among other things, with minimal submanifolds bounded by cylinders, cones, spheres and pseudo-hyperbolic spaces where most of these examples are not covered in the literature. Applications also include the study of the essential spectrum of hyperbolic graphs over compact regions of the boundary at infinity.

C. A. Rogers and J. E. Jayne have asked whether, given a Polish space and an analytic subset A of which is not a Borel set, there is always a compact subset K of such that, A ∩ K is not Borel. In this paper we give both a proof, using Martin's axiom and the negation of the continuum hypothesis, of and a counter-example, using the axiom of constructibility, to the conjecture of Rogers and Jayne, which set theory with the axiom of choice is thus powerless to decide.

Let the generalized hypergeometric function of one variable be denoted by

where (a)m is the Pochhammer symbol ((1, 3)).

We completely classify all the twistor harmonic (non-minimal) Lagrangian immersions of compact surfaces in the complex projective plane [Copf]ℙ2, i.e. those Lagrangian immersions such that their twistor lifts to the twistor space over [Copf]ℙ2 are harmonic maps.

Write F for the finite field, , having 2m elements. Let W2(F) denote the Witt vectors of length two over F (for a definition, see [4] or [10], §10). Write F(q) for the truncated polynomial ring, F[t]/(tq).

In this paper, we prove that the full automorphism group of the ${\mathbb Z}_2$-orbifold of the Barnes–Wall lattice vertex operator algebra of central charge 32 has the shape 227.E6(2). In order to identify the group structure, we introduce a graph structure on the Griess algebra and show that it is a rank 3 graph associated to E6(2).

Let ${\mathcal I}$ be an arbitrary ideal in ${\mathbb C}$[[x, y]]. We use the Newton algorithm to compute by induction the motivic zeta function of the ideal, yielding only few poles, associated to the faces of the successive Newton polygons. We associate a minimal Newton tree to ${\mathcal I}$, related to using good coordinates in the Newton algorithm, and show that it has a conceptual geometric interpretation in terms of the log canonical model of ${\mathcal I}$. We also compute the log canonical threshold from a Newton polygon and strengthen Corti's inequalities.

The main result of the paper is that the real characters of a group G of type FPm (Fm respectively) that do not vanish on a normal locally polycyclic-by-finite subgroup represent elements of the geometric invariant [sum ]m(G, ℤ) ([sum ]m(G) respectively). In the case m = 2 a stronger result is proved. Some consequences of the main result are considered.

Consider a product system over the positive cone of a quasi-lattice ordered group. We construct a Fell bundle over an associated groupoid so that the cross-sectional algebra of the bundle is isomorphic to the Nica–Toeplitz algebra of the product system. Under the additional hypothesis that the left actions in the product system are implemented by injective homomorphisms, we show that the cross-sectional algebra of the restriction of the bundle to a natural boundary subgroupoid coincides with the Cuntz–Nica–Pimsner algebra of the product system. We apply these results to improve on existing sufficient conditions for nuclearity of the Nica–Toeplitz algebra and the Cuntz–Nica–Pimsner algebra, and for the Cuntz–Nica–Pimsner algebra to coincide with its co-universal quotient.

Since the pioneer work of Prof. R. H. Fowler(1) on the statistical foundations of the Debye-Hückel theory of strong electrolytes (2), several attempts have been made to obtain as much information as possible on the question of the equation of state of electrolytes from merely thermodynamical and dimensional considerations.

The technique of sunny collapsing was introduced by Zeeman in (1) in order to prove the unknotting of PL ball pairs with co-dimension ≥ 3. Since then the technique has been used a number of times. In each case the method was used to prove special cases of the Theorem in this paper. Apart from providing a more unified treatment of various applications of this one technique, the theorem has important applications to the theory of relative regular neighbourhoods.

The aim of this paper is to complement results by Wolfart [14] about algebraic values of the classical hypergeometric series

for rational parameters a, b, c and algebraic arguments z. Wolfart essentially determines the set of a, b, c ∈ ℚ,z ∈ ℚ for which F(a, b, c; z) ∈ ℚ and indicates, in a joint paper with F. Beukers[1], that some of these values can be expressed in terms of special values of modular forms. This method yields a few strikingly explicit identities like

but it does not give general statements about the nature of the algebraic values in question. In this paper we identify F(a, b, c; z) as a generator of a Kummer extension of a certain number field depending on z, which in particular bounds its degree as an algebraic number in terms of the degree of z. Our theorem in §2 seems to be the most precise statement one can make in general but sometimes improvements are possible as we point out at the end of §2.

Let π denote a finite group. It is well known that every element of the projective class group K0 ℤπ may be realized as Wall obstruction of a finitely dominated complex with fundamental group π (cf. (13)). We will study two subgroups N0ℤπ and Nℤπ of K0ℤπ, which are closely related to the Wall obstruction of nilpotent spaces. If the group π is nilpotent and if S denotes the set of elements x ∈ K0ℤπ which occur as Wall obstructions of nilpotent spaces, then

It turns out that in many instances one has N0,ℤπ = Nℤπ (cf. Section 3) and one obtains hence new information on S. The main theorem (2·4) provides a systematic way of constructing finitely dominated nilpotent (or even simple) spaces with non-vanishing Wall obstructions.