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.)

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.

The fact that smooth Schubert varieties in partial flag manifolds are iterated fiber bundles over Grassmannians is used to give a simple presentation for their integral cohomology ring, generalizing Borel's presentation for the cohomology of the partial flag manifold itself. More generally, such a presentation is shown to hold for a larger class of subvarieties of the partial flag manifolds (which are called subvarieties defined by inclusions). The Schubert varieties which lie within this larger class are characterized combinatorially by a pattern avoidance condition.

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.

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.

The paper explores dualizing differential graded (DG) modules over DG algebras. The focus is on DG algebras that are commutative local, and finite. One of the main results established is that, for this class of DG algebras, a finite DG module is dualizing precisely when its Bass number is 1. As a corollary, one obtains that the Avramov–Foxby notion of Gorenstein DG algebras coincides with that due to Frankild and Jørgensen. One other key result is that, under suitable hypotheses, any two dualizing DG modules are quasiisomorphic up to a suspension. In addition, it is established that a number of naturally occurring DG algebras possess dualizing DG modules.

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 moduli space of parabolic bundles with fixed determinant over a smooth curve of genus greater than one is proved to be rational whenever one of the multiplicities of the quasi-parabolic structure equals one. This gives a new proof that the moduli space of vector bundles of coprime rank and degree is stably rational, a result originally due to Ballico, and the bound on the level is strong enough to deduce rationality in many cases, extending results of Newstead.

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.

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 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.

Poincaré's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms homotopic to the identity, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempted classification, and in fact we define a system as $\rho$-bounded if such deviations are bounded and as $\rho$-unbounded otherwise. For the $\rho$-bounded case we prove a close analogue of Poincaré's result: if the rotation number is rationally related to the rotation rate on the base then there exists an invariant strip (the appropriate analogue for fixed or periodic points in this context), otherwise the system is semi-conjugate to an irrational translation of the torus. In the $\rho$-unbounded case, where neither of these two alternatives can occur, we show that the dynamics are always topologically transitive.

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.

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.

The paper considers n-dimensional hypersurfaces with constant scalar curvature of a unit sphere Sn−1(1). The hypersurface Sk(c1)×Sn−k(c2) in a unit sphere Sn+1(1) is characterized, and it is shown that there exist many compact hypersurfaces with constant scalar curvature in a unit sphere Sn+1(1) which are not congruent to each other in it. In particular, it is proved that if M is an n-dimensional (n > 3) complete locally conformally flat hypersurface with constant scalar curvature n(n−1)r in a unit sphere Sn+1(1), then r > 1−2/n, and

(1) when r ≠ (n−2)/(n−1), if

then M is isometric to S1(√1−c2)×Sn−1(c), where S is the squared norm of the second fundamental form of M;

(2) there are no complete hypersurfaces in Sn+1(1) with constant scalar curvature n(n−1)r and with two distinct principal curvatures, one of which is simple, such that r = (n−2)/(n−1) and

Four methods for constructing anti-Pasch Steiner triple systems are developed. The first generalises a construction of Stinson and Wei to obtain a general singular direct product construction. The second generalises the Bose construction. The third employs a construction due to Lu. The fourth employs Wilson-type inflation techniques using Latin squares having no subsquares of order 2. As a consequence of these constructions we are able to produce anti-Pasch systems of order v for v 31 or 7 (mod 18), for v ≡ 49 (mod 72), and for many other values of v.

We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation

formula here

subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh–Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.

The weak type 1 for the Mehler maximal function is studied via a precise estimate for the ‘maximal kernel’. This, in turn, allows the geometry involved in this setting to be described.

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.

For each $p$ in $[1, \infty)$ let ${\bf E}_p$ denote the closure of the region of holomorphy of the Ornstein–Uhlenbeck semigroup $\{{\cal H}_t : t >0\}$ on $L^p$ with respect to the Gaussian measure. Sharp weak type and strong type estimates are proved for the maximal operator $f \mapsto {{\cal H}^*}_pf=\sup\{\vert {\cal H}_zf\vert :z\in {\bf E}_p\}$ and for a class of related operators. As a consequence, a new and simpler proof of the weak type 1 estimate is given for the maximal operator associated to the Mehler kernel.