The aim of this note is to give a sharp upper bound on the ratio

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where ϕ is a nonconstant eigenfunction for the Laplace–Beltrami operator on a connected compact Riemannian manifold without boundary. This ratio is always positive, since maxϕ>0 and minϕ<0 for every nonconstant eigenfunction. We assume that maxϕ[ges ]−minϕ, in order to simplify the notation. For the case of a two-dimensional manifold with nonnegative Ricci curvature, our theorem implies that the above ratio is less than the ratio of the maximum divided by the absolute value of the minimum of the Bessel function of order zero.

The proof is based on a gradient estimate from a previous paper of the author (see [5]), which in turn was proved using the maximum principle technique. In contrast to the standard applications of gradient estimates, which are based on integration along geodesics, we arrive at a contradiction by integrating the gradient estimate over small spheres centred at a point where the absolute value of the eigenfunction attains its maximum.

The main motivation for our work is that the ratio of the maximum and the minimum of an eigenfunction plays a role in estimates of the corresponding eigenvalues (see [5] and [7]). More precisely, our theorem implies that there are minimizing sequences of compact manifolds such that the first eigenvalues of the manifolds approach the corresponding lower bound for the first eigenvalue obtained in [5, Theorem 2] for every possible ratio of the maximum and the minimum of the corresponding eigenfunction.

Akemann and Anderson made a conjecture about ‘paving’ projections in finite-dimensional matrix algebras which, if true, would settle the well-known Kadison–Singer problem. Their conjecture is falsified in this paper by an explicit sequence of counterexamples.

In [2], H. Furstenberg studied a distal action of a locally compact group G on a compact metric space X, and established a structure theorem. As a consequence, he showed that if G is abelian, then a simply connected space X does not admit a minimal distal G-action.

In this paper we concern ourselves with a nonsingular flow ϕ={ϕt} on a closed 3-manifold M. Recall that ϕ is called distal if for any distinct two points x, y ∈ M, the distance d(ϕtx, ϕty) is bounded away from 0. The distality depends strongly upon the time parametrization. For example, there exists a time parametrization of a linear irrational flow on T2 which yields a nondistal flow [4, 6].

The presentation of the quantum cohomology of the moduli space of stable vector bundles of rank two and odd degree with fixed determinant over a Riemann surface of genus $g>2$ is obtained. The argument avoids the use of gauge theory, providing an alternative proof to that given by the author in Duke Math. J. 98 (1999) 525–540.

An example is constructed of a manifold for which two quadratic forms corresponding to the fourth-order invariant operators are not equivalent.

By using a more efficient way of ‘almost diagonalizing’, known results on small fractional parts of general real cubic forms in a small number of variables, are improved. In particular, for ternary forms the method gives a bound superior to that arising from simply reducing to a cubic form in one variable.

Frank Smithies was born in Edinburgh, on 10 March 1912. His father, also Frank Smithies, and his mother, Mary (née Blakemore) had met through their involvement in the socialist movement. They had two children, Frank and his sister Violet, some four years younger.

Let q and N be integers, let a be an integer coprime to q, and let zN be defined implicitly by q = (log N)log22−ZN √(log2N). We show that for large N, an integer n has at least one divisor d with q [les ] d [les ] N and d ≡ a(mod q) with probability approximately Φ(zN), where Φ denotes the distribution function of the Gaussian Law. This solves a conjecture of Hall.

In this paper, a proof is given that certain boundedness properties of operators yield distributional estimates that have exponential decay at infinity. Such distributional estimates imply local exponential integrability, and apply to many operators such as $m$-linear Calderón–Zygmund operators and their maximal counterparts.

A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

A deterministic self-similar Cantor set F defines a natural probability space, the elements of which are neighbourhoods of various sizes of all points in F. Considering random variables on this probability space, some porosity and density parameters are explicitly calculated.

Tao has shown that hard summation (summing Fourier-type series by using terms in order of decreasing size of the Fourier coefficients) works for wavelets, but the present author has shown that it fails for classical Fourier series. This paper, which is intended for a general audience, exhibits the underlying ideas in the context of Haar and Walsh series, where many of the proofs simplify.

Let Ω be the set {1, 2, [ctdot ], n}, and let [emptyv ] be the empty set. Let [Gscr ] be the family of all non-empty sets of subsets of Ω. For [Ascr ]∈[Gscr ] and X⊆Ω, put

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An important discovery is the following.

Let G be either a free product with amalgamation A[midast ]CB or an HNN group A[midast ]C, where all normal subgroups of C are finitely generated. Suppose that both A and B have no non-trivial finitely generated normal subgroups of infinite indices. We show that if G contains a finitely generated normal subgroup N which intersects A or B non-trivially but is not contained in C, then the index of N in G is finite.

Cet article présente des minorations universelles de la première valeur propre non nulle du laplacien des formes différentielles sur les domaines euclidiens ou hémisphériques convexes ou les hypersurfaces euclidiennes convexes.

The semilinear elliptic eigenvalue problem with superlinear pure power nonlinearity is considered. This problem is treated from the standpoint of $L^2$-theory and the precise asymptotic formula for the eigenvalue parameter $\lambda \,{=}\, \lambda(\alpha)$ as $\alpha \,{\to}\, \infty$ is established, where $\alpha$ is the $L^2$-norm of the solution $u$ associated with $\lambda$.

Let $\sigma$ be a non-degenerate positive $M_n$-valued measure on a locally compact group $G$ with $\|\sigma\|\,{=}\,1$. An $M_n$-valued Borel function $f$ on $G$ is called $\sigma$-harmonic if $f(x) = \int_G f(xy^{-1})\,d\sigma(y)$ for all $x\,{\in}\,G$. Given such a function $f$ which is bounded and left uniformly continuous on $G$, it is shown that every central element in $G$ is a period of $f$. Further, it is shown that $f$ is constant if $G$ is nilpotent or central.

In the early 1930s, Wiener proved that if f(x) is a strictly positive periodic function whose Fourier series is absolutely convergent, then the Fourier series of g(x)=1/f(x) is also absolutely convergent [8, pp. 10–14]. This phenomenon can be easily understood nowadays using Banach algebra techniques (see, for example, [4, pp. 202–203]). In fact, these techniques allow us to study the absolute convergence of g(x)=F(f(x)), where F is holomorphic in an open subset of [Copf ] that contains the range of f(x) (for x∈ℝ). In this context, Wiener's original problem corresponds to the choice F(z)=1/z.

In this work we want to analyse the constraints on the simultaneous rate of vanishing of the Fourier coefficients fˆ(n) and ĝ(n) as n→∞. We shall focus on g=1/f, but we shall also study the general case g=F(f). In either case, there are obviously no constraints when f is a constant function.

Although this problem does not seem to be directly related to uncertainty inequalities for the Fourier Transform, we observe that there are some analogies, both in the nature of the results and in the proof techniques. The general fact with which we are dealing is that fˆ(n) and ĝ(n) cannot vanish too quickly at the same time as n→∞, unless f(x) is constant. The general fact that underlies uncertainty inequalities is that a non-periodic function ϕ(x) and its Fourier Transform ϕcirc;(u) cannot vanish too quickly at the same time as x→∞ and u→∞, unless ϕ(x) is zero (almost everywhere). For a simple introduction to some aspects of uncertainty inequalities, see [5]; for a thorough and recent introduction to this vast subject, see [3].

It is known that, if f is a hyperbolic rational function, then the Hausdorff, packing and box dimensions of the Julia set, J(f), are equal. In this paper it is shown that, for a hyperbolic transcendental meromorphic function f, the packing and upper box dimensions of J(f) are equal, but can be strictly greater than the Hausdorff dimension of J(f).

Let $D\subset H^n(-k^2)$ be a convex compact subset of the hyperbolic space $H^n(-k^2)$ with non-empty interior and smooth boundary. It is shown that the volume of D can be estimated by the total curvature of $\partial D$. More precisely, $(n-1)k^n{\rm Vol}(D)+ {\rm Vol}(S^{n-1})\leq \int_{\partial D}K$, where K denotes the Gauss–Kronecker curvature of $\partial D$ and Vol$(S^{n-1})$?> denotes the Euclidean volume of the sphere.