Showing papers in "Proceedings of The London Mathematical Society in 1993"

Exponential sums and lattice points III

Abstract: The Gauss circle problem and the Dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve. In the Bombieri?Iwaniec?Mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice. These coincidences are parametrised by integer points close to certain plane curves, the resonance curves. This paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy, the number of integer points minus the area, as in the latest work on single exponential sums. The Bombieri?Iwaniec?Mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range, and it can in principle be used iteratively. We use a bound obtained by Swinnerton-Dyer's approximation determinant method. In the discrepancy estimate $O(R^K (\log R)^{\Lambda })$ in terms of the maximum radius of curvature $R$, we reduce $K$ from 2/3 (classical) and 46/73 (paper II in this series) to 131/208. The corresponding exponent in the Dirichlet divisor problem becomes $K/2 = 131/416$.

The Riemann Zeta-Function and the One-Dimensional Weyl-Berry Conjecture for Fractal Drums

Abstract: Based on his earlier work on the vibrations of 'drums with fractal boundary', the first author has refined M. V. Berry's conjecture that extended from the 'smooth' to the 'fractal' case H. Weyl's conjecture for the asymptotics of the eigenvalues of the Laplacian on a bounded open subset of W (see [16]). We solve here in the one-dimensional case (that is, when n = 1) this 'modified Weyl-Berry conjecture'. We discover, in the process, some unexpected and intriguing connections between spectral geometry, fractal geometry and the Riemann zeta-function. We therefore show that one can 'hear' (that is, recover from the spectrum) not only the Minkowski fractal dimension of the boundary—as was established previously by the first author—but also, under the stronger assumptions of the conjecture, its Minkowski content (a 'fractal' analogue of its 'length'). We also prove (still in dimension one) a related conjecture of the first author, as well as its converse, which characterizes the situation when the error estimates of the aforementioned paper are sharp.

Exponential sums and the Riemann zeta function v

Abstract: A Van der Corput exponential sum is $S = \Sigma \exp (2 \pi i f(m))$, where $m$ has size $M$, the function $f(x)$ has size $T$ and $\alpha = (\log M) / \log T < 1$. There are different bounds for $S$ in different ranges for $\alpha $. In the middle range where $\alpha $ is near ${1\over 2}$, $S = O(\sqrt{M} T^{\theta + \epsilon })$. This $\theta $ bounds the exponent of growth of the Riemann zeta function on its critical line ${\rm Re} s = {1\over 2}$. Van der Corput used an iteration which changed $\alpha$ at each step. The Bombieri?Iwaniec method, whilst still based on mean squares, introduces number-theoretic ideas and problems. The Second Spacing Problem is to count the number of resonances between short intervals of the sum, when two arcs of the graph of $y = f'(x)$ coincide approximately after an automorphism of the integer lattice. In the previous paper in this series [Proc. London Math. Soc. (3) 66 (1993) 1?40] and the monograph Area, lattice points, and exponential sums we saw that coincidence implies that there is an integer point close to some ?resonance curve?, one of a family of curves in some dual space, now calculated accurately in the paper ?Resonance curves in the Bombieri?Iwaniec method?, which is to appear in Funct. Approx. Comment. Math. We turn the whole Bombieri?Iwaniec method into an axiomatised step: an upper bound for the number of integer points close to a plane curve gives a bound in the Second Spacing Problem, and a small improvement in the bound for $S$. Ends and cusps of resonance curves are treated separately. Bounds for sums of type $S$ lead to bounds for integer points close to curves, and another branching iteration. Luckily Swinnerton-Dyer's method is stronger. We improve $\theta $ from 0.156140... in the previous paper and monograph to 0.156098.... In fact $(32/205 + \epsilon , 269/410 + \epsilon)$ is an exponent pair for every $\epsilon > 0$.

The Composition Algebra of a Cyclic Quiver

Abstract: Note that A is also called the quiver of type An_x with cyclic orientation. We denote by Ao the set of vertices of A (and often we will identify Ao with Z/nZ, or also with the set {xx, x2, •-, xn) or just with {1, 2, ..., n}, with arrows xt—>xi+l or i->i + l). There are n one-dimensional representations, corresponding to the vertices of A; these are (up to isomorphism) all the simple objects of T. The simple representation corresponding to the vertex a of A will be denoted by 5(a), and if S' is isomorphic to S(a), we will write [5'] = a. Given a simple representation 5, and I eNu there is (up to isomorphism) a unique indecomposable representation S[l] of length / with top 5, and we obtain in this way all indecomposable representations (again up to isomorphism). It follows that we can index the isomorphism classes in T by the set IT of n-tuples of partitions; the representation of A corresponding to n e U. will be denoted by M{JZ); see § 3.3.

Pseudo-Anosov Maps and Invariant Train Tracks in the Disc: A Finite Algorithm

Abstract: Let [f] be an isotopy class of orientation-preserving homeomorphisms of the punctured disc. We construct a finite algorithm which enables one to decide whether the class [f] contains a pseudo-Anosov element. In this case the algorithm defines an invariant train track and the dilatation factor, which gives the minimal topological entropy of the isotropy class

Some Applications of Projective Resolutions of Identity

Abstract: We show that the unit ball K of the bidual of an Asplund space is a Corson compact or contains [0, ω 1 ], and that it has the Namioka property on separate-to-joint continuity. The same results are shown for K a Valdivia compact; a by-product is that all dyadic compacts have the Namioka property. Some connections with weakly compactly generated dual spaces and renormings are given

Topological Properties of Banach Spaces

Abstract: Weakly compact sets in Banach spaces are fragmentable. We introduce the weaker notion of the a-fragmentability of a subset of a Banach space, a notion that has a far wider range of applicability. We prove that all weakly Cech-analytic subsets of a Banach space are a-fragmented. This implies that all Banach spaces having Kadec norms are a-fragmented. However l∞ is not σ-fragmented

Geometric Quotients of Unipotent Group Actions

Abstract: Let G be a unipotent algebraic group over K (a field of characteristic 0) which acts rationally on an affine scheme X = Spec A over K, where A is a commutative K-algebra. The problem of finding sufficient and manageable conditions to guarantee that the geometric quotient X/G exists is of fundamental interest in the theory of moduli spaces for local objects such as isolated singularities or (Cohen-Macaulay) modules over the local ring of a singularity (cf. [L-P], [G-P2], [G-H-P], [H]).

Some congruences for partitions that are p-cores

Abstract: A number of linear congruences modulo r are proved for the number of partitions that are p-cores where p is prime, 5≤p≤23, and r is any prime divisor of 1/2(p-1). Analogous results are derived for the number of irreducible p-modular representations of the symmetric group S n . The congruences are proved using the theory of modular forms

Densities with Gaussian Tails

Abstract: Consider densities f i (t), for i = 1,..., d, on the real line which have thin tails in the sense that, for each i, f i (t)∼γ i (t)e −ψi(t) , where γ i behaves roughly like a constant and ψ i is convex, C 2 , with ψ'→∞ and ψ″>0 and 1/√ψ″ is self-neglecting. (The latter is an asymptotic variation condition.) Then the convolution is of the same form f 1 *...*f d (t)∼γ(t)e −ψ(t) . Formulae for γ, ψ are given in terms of the factor densities and involve the conjugate transform and infimal convolution of convexity theory. The derivations require embedding densities in exponential families and showing that the assumed form of the densities implies asymptotic normality of the exponential families

On the geometry of normal forms in discrete groups

Abstract: We study normal forms in finitely generated groups from the geometric viewpoint of combings. We introduce notions of combability considerably weaker than those commonly in use. We prove that groups which satisfy these conditions are finitely presented and satisfy isoperimetric and isodiametric inequalities of a controlled nature

Positive Braids are Visually Prime

Abstract: It is shown that positive braids represent split or non-prime links only in the obvious ways

Lieb-Thirring Inequalities on the N-Sphere and in the Plane, and Some Applications

Abstract: In this paper we prove the Lieb-Thirring inequalities for a family of scalar functions defined on a sphere S n , which are orthonormal in L 2 (S n ) and have zero mean value, for n ≥ 1. We give explicit values of all the constants involved. In the case of the two-dimensional sphere, we prove the Lieb-Thirring inequalities for an orthonormal family of non-divergent (or irrotational) vector fields with the explicit value of the constant as well. For non-divergent (or irrotational) vector fields defined on the plane R 2 we prove the Lieb-Thirring inequalities with the value of the constant less than was known before. Finally, the rate of growth of the constant is estimated, when a parameter p tends to its limit, and embeddings in the exponential Orlicz spaces are proved. Applications to the dimension of attractors are given

Order and type as measures of growth for convex or entire functions

Abstract: 1. Introduction 2. Order and type in classical complex analysis 3. Relative order and type of convex functions 4. Order and type in duality 5. The infimal convolution 6. The order of an entire function 7. A geometric characterisation of the relative order 8. An extension problem for holomorphic functions 9. Whittaker's decomposition theorem 10. Inequalities for the infimal convolution 11. Inequalities for the infimal convolution, one variable 12. The type of an entire function References

The Quotient of a Free Product of Groups by a Single High-Powered Relator

Abstract: Let A, B be groups, and r eA *B a cyclically reduced word of the form xUyU l for some word U and letters x, y, where Jt = _y = l. We call this form exceptional. If m = 4 or m = 5, and G is the one-relator product group (A*B)/N(r) (where N(-) denotes normal closure), then it was stated in [2,3] (Theorem E) that the cohomology of G is linked to that of A * B and of the finite triangle group G(2, 3, m) = {a, b\\ a = b = (ab) = 1) by a certain MayerVietoris sequence. The proof of this theorem involves so-called spherical pictures over G that are induced from the finite subgroup G(2, 3, m). This proof fails if r is multiply exceptional, in the sense that both r and some cyclic permutation of r have exceptional forms, because then more spherical pictures arise than are dealt with in [2,3]. An example of such a word r is ababab~ where A = {a\\ a= 1) and B = (b\\ 6 = 1), since both r and its cyclic permutation ab~abab are exceptional. Theorem E of [2,3] is correct only under the stronger hypothesis that r is uniquely exceptional (i.e. not multiply exceptional). This problem does not affect the other results of [2,3]. A more detailed discussion is given in [1].

Affine Distance-Transitive Groups

Abstract: A fruitful way of studying groups is by means of their actions on graphs. A very special and interesting class of groups are the ones acting distance-transitively on some graph. A group G acting on a connected graph F = (VT, ET) is said to act distance-transitively if its action on each of the sets {(x, y)\ x, y e VT, d(x, y) = i} is transitive. Here VT and ET stand for the vertices and edges of T respectively, d stands for the usual distance in T, and i runs through {0, 1, 2, ..., diam(F)}. A graph is called a distance-transitive graph if it admits a group acting distancetransitively on it. Moreover, in this case the group is called a distance-transitive group. In general there is no hope of classifying all distance-transitive graphs, as observed by Cameron (cf. §7.8 of Brouwer, Cohen, and Neumaier [8]). However, if there are only finitely many vertices, then a complete classification is in sight. First, if the group G acts imprimitively on a distance-transitive graph T, then by a result of Smith [31], there is a natural way to construct a new graph from T that admits a distance-transitive group acting primitively on it. In the classification project we usually restrict ourselves to the primitive groups. For many of the known primitive distance-transitive graphs it is known whether they have an imprimitive parent or not (see Hemmeter [18] and van Bon and Brouwer [6]). Next notice that when the diameter is equal to 1, the graph is just a clique, and G acts distance-transitively on it if and only if G is a 2-transitive group. These groups are known (cf. Cameron [10], Cohen and Zantema [11], Hering [19], Kan tor [21], Liebeck [24], Curtis, Kantor and Seitz [13] and many others). Their classification depends on the classification of finite simple groups. When the valency of the graph equals 2, that is, each vertex is adjacent to precisely two others, the graph is a polygon and G must be a dihedral group. In the case where G is a rank-3 permutation group, there are two distance-transitive graphs associated with it, one being the complement of the other. This case has been completely settled by Foulser [15], Foulser and Kallaher [16], Kantor and Liebler [22], Liebeck [25] and Liebeck and Saxl [28]. For the general primitive case, the first determination of the structure of G was made by Praeger, Saxl and Yokoyama [29] in 1987. Their theorem, based on the O'Nan Scott Theorem [9, 26, 30], states that if G acts primitively and distance-transitively on a finite graph F of diameter at least 2 and valency at least 3, then either F is a Hamming graph and G is a wreath product, or G is almost simple or affine. By the classification of finite simple groups and the knowledge of their automorphism groups and maximal subgroups, a full classification of groups and graphs in the

Cleft Extensions of Hopf Algebras, II

Abstract: Let K be a p-adic field, let G and U be groups of order p, and let D (respectively M) be a Hopf order in the group algebra KG (respectively in the algebra of maps U→K). We use the algebraic machinery of [2] to determine the cleft Hopf algebra extensions of M by D, and investigate which of these cleft extensions are Hopf orders in a group algebra

C*-Algèbres De Kac Et Algèbres De Kac

Abstract: S. Baaj and G. Skandalis have proved that, to every Kac algebra, as studied by J.-M. Schwartz and the first author, corresponds a canonical C * -Kac algebra, as studied by the second author. This article proves the converse result. So, we have then a complete proof that the C * version and the von Neumann version of this theory are exactly equivalent, as are, in the commutative case, thanks to A. Weil's theorem, locally compact groups and measured groups with a left-invariant measure

The Index Change and Global Bifurcation for Flows with a First Integral

Abstract: The existence, continuation and bifurcation of solutions of fixed period r of one-parameter families of ordinary differential equations with a first integral may be studied using the topological degrees defined in [6] and [7] and the change of index formula given in [8]. The purpose here is threefold. First, suppose that (f(x), ⊇V(x)) = 0, so that V is a first integral for x = f(x). We examine how the behaviour of f restricts that of V and vice versa, and how the periodic spectra of linear ordinary differential equations are affected by the presence of a linear or non-linear (which occurs surprisingly naturally) first integral. Second, we give classes of one-parameter families of ordinary differential equations where the change of index in [8] is non-zero