A model (M, <, …) is κ-like if M has cardinality κ but, for all α ∈ M, the cardinality of {x ∈ M [ratio ] x < a} is strictly less than κ. In this paper we shall give constructions of κ-like models of arithmetic satisfying an arbitrarily large finite part of PA but not PA itself, for various singular cardinals κ. The main results are: (1) for each countable nonstandard M [models ] Π2−Th(PA) with arbitrarily large initial segments satisfying PA and each uncountable κ of cofinality ω there is a cofinal extension K of M which is κ-like; also hierarchical variants of this result for Πn−Th(PA); and (2) for every n [ges ] 1, every singular κ and every M [models ] B[sum ]n+ exp+¬ I[sum ]n there is a κ-like model K elementarily equivalent to M.

In this paper we consider a generalization to algebraic number fields of the classical Hermite constant γn. For this constant we extend the well-known Minkowski bound and study the notion of extreme forms in this setting.

The mod two cohomology of the three connective covering of S3 has the form

formula here

where x2n is in degree 2n and n = 2. If F denotes the homotopy theoretic fibre of the map S3 → B2S1 of degree 2, then the mod2 cohomology of F is also of the same form for n = 1. Notice (cf. Section 7 of the present paper) that the existence of spaces whose cohomology has this form for high values of n would immediately provide Arf invariant elements in the stable stem. Hence, it is worthwhile to determine for what values of n the above algebra can be realized as the mod2 cohomology of some space. The purpose of this paper is to construct a further example of a space with such a cohomology algebra for n = 4 and to show that no other values of n are admissible. More precisely, we prove the following.

Let R be a fully bounded Noetherian ring of finite global dimension. Then we prove that K dim (R) [les ] gldim (R). If, in addition, R is local, in the sense that R/J(R) is simple Artinian, then we prove that R is Auslander-regular and satisfies a version of the Cohen–Macaulay property. As a consequence, we show that a local fully bounded Noetherian ring of finite global dimension is isomorphic to a matrix ring over a local domain, and a maximal order in its simple Artinian quotient ring.

Let [Gscr ] be a Burnside 3-group constructed by Gupta-Sidki. The group algebra ℤ3[[Gscr ]] has a proper ring-quotient which embeds [Gscr ] and is just-infinite and primitive.

Let I be the graded ring of homogeneous rational polynomials in n-variables which are numerical over ℤ. Then I is a subring of Γ, the divided polynomial algebra over ℤ in the n-variables. A consequence of the main theorem is that for any positive integer k, the image of the induced homomorphism I[otimes ](ℤ/kℤ) → Γ [otimes ] (ℤ/kℤ) is a finite graded ring.

Let G be a locally finite group of cardinality ℵn where n is a natural number. Let π(G) be the set of primes p for which G has an element of order p. In [5], Holt conjectures that if k is a finite field with char k ∉ π(G) then

(1) G has cohomological dimension n+1 over k;

(2) Hn+1(G, kG) has cardinality 2ℵn;

(3) Hi(G, kG) = 0 for 0 [les ] i [les ] n.

We show that solutions of analytic elliptic partial differential equations of the form

formula here

in a simply connected domain Ω ⊂ ℝn can be extended holomorphically into [Nscr ] (Ω) ⊂ [Copf ]n, the so-called kernel of the harmonicity hull of Ω. This extends results of Avanissian, Aronszajn etc., on polyharmonic functions and also results of Vekua, Khavinson and Shapiro in ℝ2. We also find the domain of influence for solutions of a certain subclass of these operators, in terms of their Cauchy data on analytic hypersurfaces in [Copf ]n (a complex Huygens' principle). As an application, we investigate reflection properties of these solutions and, in particular, solutions of the Helmholtz equation in ℝ3.

A large number of papers written over the last ten years have concerned the spectral theory of Laplace–Beltrami operators on complete Riemannian manifolds, and of other self-adjoint second order elliptic operators. Much of the interest has centred on the relationship between various types of Sobolev inequality, parabolic Harnack inequalities and the Liouville property on the one hand, and Gaussian heat kernel bounds on the other. For manifolds of bounded geometry there is an important connection between this problem and a corresponding one for discrete Laplacians on graphs. Standard references are [9, 37] and more recent literature can be traced via [5, 16, 32].

We construct a one-parameter family of sets in ℝ2 generated by a disjoint union of open squares. We study the spectral counting function associated to a variational Dirichlet eigenvalue problem on a set from this family and show that the spectral asymptotics depend not only on the Minkowski dimension of the boundary, but also on whether the values of a specific function of the parameter are rational or irrational. Furthermore, we significantly sharpen the results in the rational case.

We give a necessary and sufficient condition on an elliptic symbol σ of order m to ensure that the unique closed extension in Lp(ℝn) for 1 < p < ∞, of the pseudo-differential operator Tσ, initially defined on the Schwartz space, is a Fredholm operator from Lp(ℝn) into Lp(ℝn) with domain Hm, p, where Hm, p is the Lp Sobolev space of order m.

We study the boundedness of the Hardy–Littlewood maximal function in weighted Lorentz spaces, using some new techniques on distribution and rearrangement functions. Our goal is to give a unified version of some well-known weighted inequalities in ℝn and ℝ+, showing the relation between the theories.

In order to obtain the linear relations among the 24-solutions for the p-differential equation of Riemann, Barnes [12] discovered an elegant method which uses the following identity for the classical Γ function.

For complex numbers α, β, γ, δ the equality

formula here

holds and the path of integration from −i∞ to +i∞ is curved so that the poles of Γ(γ−s) Γ(δ−s) lie on the right and the poles of Γ(α+s) Γ(β+s) lie on the left.

Suppose that A is a C*-algebra and C is a unital abelian C*-subalgebra which is isomorphic to a unital subalgebra of the centre of M(A), the multiplier algebra of A. Letting Ω = &Ccirc;, so that we may write C = C(Ω), we call A a C(Ω)-algebra (following Blanchard [7]). Suppose that B is another C(Ω)-algebra, then we form A[otimes ]CB, the algebraic tensor product of A with B over C as follows: A [otimes ] B is the algebraic tensor product over [Copf ], IC = {[sum ]ni=1 (fi [otimes ] 1−1[otimes ] fi)x| fi∈C, x∈A[otimes ]B} is the ideal in A[otimes ]B generated by 〈f[otimes ]1−1[otimes ] f|f∈C〉, and A [otimes ]CB = A[otimes ]B/IC. Then A[otimes ]CB is an involutive algebra over [Copf ], and we shall be interested in deciding when A[otimes ]CB is a pre-C*-algebra; that is, when is there a C*-norm on A[otimes ]CB? There is a C*-semi-norm, which we denote by ‖·‖C-min, which is minimal in the sense that it is dominated by any semi-norm whose kernel contains the kernel of ‖·‖C-min. Moreover, if A [otimes ]CB has a C*-norm, then ‖·‖C-min is a C*-norm on A[otimes ]CB. The problem is to decide when ‖·‖C-min is a norm. It was shown by Blanchard [7, Proposition 3.1] that when A and B are continuous fields and C is separable, then ‖·‖C-min is a norm. In this paper we show that ‖·‖C-min is a norm when C is a von Neumann algebra, and then we examine some consequences.

There is extensive literature concerning approximately finite-dimensional (AF) C*-algebras. For example see [1, 5]. In recent years study has focused on non-self-adjoint subalgebras of AF C*-algebras ([4, 8, 9], for example) and this paper continues that theme. We define T(m, n) to be the block upper triangular subalgebra of Mm+n with self-adjoint part equal to Mm [oplus ] Mn. The class of algebras that are considered here are norm closures of ascending chains of algebras of the form T(m, n) and are referred to as uniformly T2 algebras. This class of algebras properly contains the matroid C*-algebras of Dixmier.

The close relationship between the notions of positive forms and representations for a C*-algebra A is one of the most basic facts in the subject. In particular the weak containment of representations is well understood in terms of positive forms: given a representation π of A in a Hilbert space H and a positive form φ on A, its associated representation π φ is weakly contained in π (that is, ker π φ ⊃ ker π) if and only if φ belongs to the weak* closure of the cone of all finite sums of coefficients of π. Among the results on the subject, let us recall the following ones. Suppose that A is concretely represented in H. Then every positive form φ on A is the weak* limit of forms of the type x [map ] [sum ]ki=1 〈ξi, xξi〉 with the ξi in H; moreover if A is a von Neumann subalgebra of ℒ(H) and φ is normal, there exists a sequence (ξi)i [ges ] 1 in H such that φ (x) = [sum ]i [ges ] 1 〈ξi, xξi〉 for all x.