The paper computes the Hausdorff dimension of sets of real numbers which are close to infinitely many real algebraic integers of bounded degree. It also investigates the distribution of real algebraic integers of bounded degree, which are proved to be evenly spaced.

Two results, the Auslander–Buchsbaum and Bass Theorems from the homological theory of commutative noetherian rings, are generalized to important classes of non-commutative ℕ-graded algebras over fields. As a corollary to the generalized Auslander–Buchsbaum Theorem, it is found that under weak (so-called χ−) conditions, a non-commutative ℕ-graded connected noetherian algebra of finite global dimension is in fact Artin–Schelter regular.

The paper presents diverse methods for estimating the covering number of a precompact subset of a Banach space when the entropy of the set of its extremal points is already known. In the case of a Hilbert space, the Gelfand diameters of the subset are also estimated.

As is well known, the classical Titchmarsh–Weyl m-function for second order differential operators admits a generalisation to considerably larger classes of differential equations. In the applications, however, the use of the general M-matrix often turns out to be rather cumbersome. The paper interprets the m-function in terms of Hilbert space notions, and shows that, in a sense that is made precise, the classical m-function can be recovered as a part of the more complicated one. Applications of this trick lead to results on spectral multiplicity and on stability properties of the spectrum.

A group G is conjugacy separable if whenever x and y are non-conjugate elements of G, there exists some finite quotient of G in which the images of x and y are non- conjugate. It is known that free products of conjugacy separable groups are again conjugacy separable [19, 12]. The property is not preserved in general by the formation of free products with amalgamation; but in [15] a method was introduced for showing that under certain circumstances, the free product of two conjugacy separable groups G1 and G2 amalgamating a cyclic subgroup is again conjugacy separable. The main result of [15] states that this is the case if G1 and G2 are free-by-finite or finitely generated and nilpotent-by-finite. We show here that the same conclusion holds for groups G1 and G2 in a considerably wider class, including, in particular, all polycyclic-by-finite groups. (This answers a question posed by C. Y. Tang, Problem 8.70 of the Kourovka Notebook [7], as well as two questions recently asked by Kim, MacCarron and Tang in G. Kim, J. MacCarron and C. Y. Tang, ‘On generalised free products of conjugacy separable groups’, J. Algebra 180 (1996) 121–135.)

Homogeneous fragmentations describe the evolution of a unit mass that breaks down randomly into pieces as time passes. They can be thought of as continuous time analogues of a certain type of branching random walk, which suggests the use of time-discretization to shift known results from the theory of branching random walks to the fragmentation setting. In particular, this yields interesting information about the asymptotic behaviour of fragmentations.

On the other hand, homogeneous fragmentations can also be investigated using a powerful technique of discretization of space due to Kingman, namely, the theory of exchangeable partitions of $\mathbb{N}$. Spatial discretization is especially well suited to the direct development for continuous times of the conceptual method of probability tilting of Lyons, Pemantle and Peres.

It is considered whether $ L\,{=}\,\limsup_{n\to\infty} n \snormo{T^{n+1}-T^n}\,\lt\,\infty$ implies that the operator $T$ is power-bounded. It is shown that this is so if $L\lt1/e$, but it does not necessarily hold if $L\,{=}\,1/e$. As part of the methods, a result of Esterle is improved, showing that if $\sigma(T)\,{=}\,\{1\}$ and $T\,{\ne}\,I$, then $\liminf_{n\to\infty} n \snormo{T^{n+1}-T^n} \ge 1/e$. The constant $1/e$ is sharp. Finally, a way to create many generalizations of Esterle's result is described, and also many conditions are given on an operator which imply that its norm is equal to its spectral radius.

In this paper we extend two theorems from [2] on p-adic subanalytic sets, where p is a fixed prime number, Qp is the field of p-adic numbers and Zp is the ring of p- adic integers. One of these theorems [2, 3.32] says that each subanalytic subset of Zp is semialgebraic. This is extended here as follows.

Let T be a d×d matrix with integral coefficients. Then T determines a self-map of the d-dimensional torus X = ℝd/ℤd. Choose for each natural number n a ball B(n) in X and suppose that B(n+1) has smaller radius than B(n) for all n. Now let W be the set of points x ∈ X such that Tn(x) ∈ B(n) for infinitely many n ∈ ℕ. The Hausdorff dimension of W is studied by analogy with the Jarník–Besicovitch theorem on the dimension of the set of well-approximable real numbers. The dimension depends on the quantity

formula here

A complete description is given only when the matrix is diagonalizable over ℚ. In other cases a result is obtained for sufficiently large τ. The results, in as far as they go, show that the Hausdorff dimension of W is a strictly decreasing, continuous function of τ which is piecewise of the form (Aτ+B)/(Cτ+D). The numbers A, B, C and D which arise in this way are typically sums of logarithms of the absolute values of eigenvalues of T.

It is shown that every special Lagrangian cone in $\C^3$ determines, and is determined by, a primitive harmonic surface in the 6-symmetric space ${\rm SU}_3/{\rm SO}_2$. For cones over tori, this allows the classification theory of harmonic tori to be used to describe the construction of all the corresponding special Lagrangian cones. A parameter count is given for the space of these, and some of the examples found recently by Joyce are put into this context.

At the regional conference held at the University of California, Irvine, in 1985 [24], Harald Upmeier posed three basic questions regarding derivations on JB*-triples:

(1) Are derivations automatically bounded?

(2) When are all bounded derivations inner?

(3) Can bounded derivations be approximated by inner derivations?

These three questions had all been answered in the binary cases. Question 1 was answered affirmatively by Sakai [17] for C*-algebras and by Upmeier [23] for JB-algebras. Question 2 was answered by Sakai [18] and Kadison [12] for von Neumann algebras and by Upmeier [23] for JW-algebras. Question 3 was answered by Upmeier [23] for JB-algebras, and it follows trivially from the Kadison–Sakai answer to question 2 in the case of C*-algebras.

In the ternary case, both question 1 and question 3 were answered by Barton and Friedman in [3] for complex JB*-triples. In this paper, we consider question 2 for real and complex JBW*-triples and question 1 and question 3 for real JB*-triples. A real or complex JB*-triple is said to have the inner derivation property if every derivation on it is inner. By pure algebra, every finite-dimensional JB*-triple has the inner derivation property. Our main results, Theorems 2, 3 and 4 and Corollaries 2 and 3 determine which of the infinite-dimensional real or complex Cartan factors have the inner derivation property.

A theorem due to Hardy states that, if f is a function on R such that |f(x)| [les ] C e−α|x|2 for all x in R and |fˆ(ξ)| [les ] C e−β|ξ|2 for all ξ in R, where α > 0, β > 0, and αβ > 1/4, then f = 0. A version of this celebrated theorem is proved for two classes of Lie groups: two-step nilpotent Lie groups and harmonic NA groups, the latter being a generalisation of noncompact rank-1 symmetric spaces. In the first case the group Fourier transformation is considered; in the second case an analogue of the Helgason–Fourier transformation for symmetric spaces is considered.

The object of the paper is a thorough analysis of the WP-Bailey tree, a recent extension of classical Bailey chains. The paper begins by observing how the WP-Bailey tree naturally requires a finite number of classical $q$ -hypergeometric transformation formulas. It then shows how to move beyond this closed set of results, and in the process, heretofore mysterious identities of Bressoud are explicated. Next, WP-Bailey pairs are used to provide a new proof of a recent formula of Kirillov. Finally, the relation between the approach in the paper and that of Burge is discussed.

The Skolem–Mahler–Lech theorem states that if $f(n)$ is a sequence given by a linear recurrence over a field of characteristic 0, then the set of m such that $f(m)$ is equal to 0 is the union of a finite number of arithmetic progressions in $m\ge 0$ and a finite set. We prove that if X is a subvariety of an affine variety Y over a field of characteristic 0 and q is a point in Y, and $\sigma$ is an automorphism of Y, then the set of m such that $\sigma^m({\bf q})$ lies in X is a union of a finite number of complete doubly-infinite arithmetic progressions and a finite set. We show that this is a generalisation of the Skolem–Mahler–Lech theorem.

Let G be a simple algebraic group over an algebraically closed field K of characteristic p with p[ges ]0. Then G contains finitely many conjugacy classes of unipotent elements. There is a unique class of unipotent elements of largest dimension – the class of regular unipotent elements; the centralizer of any such element is a unipotent group of dimension equal to the rank of G. For example, if G is the linear group SL (V) on the finite dimensional vector space V over K, then the regular unipotent elements are precisely the unipotent matrices consisting of a single Jordan block.

In 1920 Chowla made the following conjecture. Let pn denote the nth prime; if q [ges ] 3, (q, a) = 1 then there are infinitely many pairs of consecutive primes pn and pn+1 such that

formula here

By considering the sum

formula here

where χ is the non-principal character modulo 4 or 6, it is possible to prove the conjecture for q = 4 and q = 6 (a = ±1). In this paper we prove Chowla's conjecture for all q and a with (q, a) = 1. Moreover, we shall show that for any k there exist ‘strings’ of congruent primes such that

formula here

For each modulus q the method used applies best to the following two sets of residue classes:

formula here

Larger values of k in terms of pn+1 can be found for residue classes belonging to these sets.

A linear autonomous system of differential equations $\dot{x}=Ax$ can be transformed to its Jordan normal form, that is, the transformed system is in block diagonal form and the blocks correspond to different eigenvalues. This result is generalized to arbitrary nonautonomous linear systems $\dot{x}=A(t)x$ with a locally integrable matrix function $A:{\bb R}\longrightarrow {\bb R}^{N\times N}$ .

Let F be a non-Archimedean local field and G be the group of F-points of a connected reductive group defined over F. Let M be an F-Levi subgroup of G and P = MN be a parabolic subgroup with Levi decomposition P = MN. Jacquet, or truncated, restriction gives a functor from the category of smooth representations of G to that of M. The main result describes this functor in terms of homomorphisms and localizations of Hecke algebras attached to certain compact open subgroups of G and M. This leads to new and straightforward proofs of some fundamental results. The first computes the smooth dual of a Jacquet module of a smooth representation of G, generalizing the corresponding result for admissible representations due to Harish-Chandra and Casselman. The second identifies the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.

A formula for the interior $\varepsilon$-neighborhood of the classical von Koch snowflake curve is computed in detail. This function of $\varepsilon$ is shown to match quite closely with earlier predictions of what it should be, but is also much more precise. The resulting ‘tube formula’ is expressed in terms of the Fourier coefficients of a suitable nonlinear and periodic analog of the standard Cantor staircase function and reflects the self-similarity of the Koch curve. As a consequence, the possible complex dimensions of the Koch snowflake are computed explicitly.

Several authors have proved Lefschetz type formulas for the local Euler obstruction. In particular, a result of this type has been proved that turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of the paper is to determine what prevents the local Euler obstruction from satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ‘defect’) of such functions. An interpretation of this defect is given in terms of vanishing cycles, which allows it to be calculated algebraically. When the function has an isolated singularity, the invariant can be defined geometrically, via obstruction theory. This invariant unifies the usual concepts of the Milnor number of a function and the local Euler obstruction of an analytic set.