In this paper we are concerned with the problem of counting rational points of bounded height on rational cubic surfaces. For most such surfaces this question appears much too hard for current methods. We shall therefore examine a particular example, namely the surface

formula here

for which the problem is tractable, though by no means trivial. This surface contains the lines X0=Xi=0, for i=1, 2, 3, which we shall exclude from consideration. We therefore define

formula here

where an integer vector is said to be primitive if its components have no common factor. The function [Nscr ](H) counts points with the most naive height available. This suffices, however, to bring out the key features of the problem. We may observe at once that the condition x0[les ]H is redundant, and that it suffices to assume that (x1, x2, x3) is primitive.

Our principal result is the following theorem.

As usual, let π(x) denote the number of prime numbers [les ]x and ψ(x) the well known Chebyshev's function. Let E(x) denote either (ψ(x)−x)/√x or (π(x)−li x)/(√x/log x), x[ges ]2. The study of E occupies a central place in the theory of primes. A classical result of Littlewood [7] states that E(x)=Ω±(log log log x) as x tends to infinity, showing in particular that E is unbounded. We expect rather erratic behaviour of E, but still one can wonder if it belongs to one of the classic function spaces [Xscr ], necessarily containing some unbounded functions. Let us extend definition of E(x) for x<2 by putting E(x)=0. A natural question is if it belongs to BMO, the space of functions with bounded mean oscillation, see, e.g. [2]. A locally integrable function f on the real line belongs to BMO if there exists a constant C such that for every bounded interval I⊂R we have

formula here

with a suitable constant αI∈R. [mid ]I[mid ] denotes here the length of I. Without any loss in generality one can take

formula here

the average of f over I (cf. [2], chapter VI). BMO is important and intensely studied in the complex analysis. It is obvious that BMO is larger than the space of bounded (measurable) functions and thus it seems a natural candidate for [Xscr ]. E∈BMO would mean that E behaves in a certain predictable way. Otherwise, we obtain another confirmation of the big irregularity in the distribution of primes.

We use certain derivations of a polynomial ring, that do not leave any proper non-zero ideal invariant, to construct simple non-holonomic modules over the nth Weyl algebra. This approach extends to the rings of differential operators of other smooth affine varieties, like smooth quadric surfaces.

Definition [4]. Let A be a noetherian ring, [afr ] an ideal of A and M an A-module. M is said to be [afr ]-cofinite if M has support in V([afr ]) and ExtiA(A/[afr ], M) is a finite A-module for each i.

Remark. (a) If 0→M′→M→M″ →0 is exact and two of the modules in the sequence are [afr ]-cofinite, then so is the third one.

This has the following consequence, which will be used several times.

(b) If f[ratio ]M→N is a homomorphism between two [afr ]-cofinite modules and one of the three modules Ker f, Im f and Coker f is [afr ]-cofinite, then all three of them are [afr ]-cofinite.

Example [5, remark 1·3]. If A is local with maximal ideal [mfr ], then an A-module is [mfr ]-cofinite if and only if it is an artinian A-module.

A link map is a map from a union of spheres into another sphere with pairwise disjoint images. Two link maps are link homotopic if one can be deformed into the other by a homotopy through link maps. This rather crude equivalence relation was introduced in 1954 by Milnor[13] for classical links and an algorithm for the classification problem in this setting was given in 1990 by Habegger and Lin[4]. In the past years, link homotopy in higher dimensions has also attracted much renewed interest. For link maps in a large metastable range, the corresponding classification problem has been reduced to standard questions in homotopy theory by Koschorke[9, 10].

In this paper we consider link maps of 2-spheres into the 4-sphere. This is another interesting case which differs considerably from the two cases above because of the failure of Whitney's trick in dimension four. Here the first contribution was made in 1984 by Fenn and Rolfsen[2] who established the first non-trivial link map of two 2- spheres into the 4-sphere. Their idea has been generalized by Kirk[6] to define his σ-invariant that distinguishes infinitely many different link homotopy classes. For several years, Kirk's invariant remained the strongest invariant for link maps into the 4-sphere. Only recently, it became known that Kirk's invariant suffices to classify two-component link maps in dimension four [12].

Let M be a compact, orientable 3-manifold with ∂M a torus. If r is a slope on ∂M (the isotopy class of an unoriented essential simple loop), then we can form the closed 3-manifold M(r) by gluing a solid torus Vr to M along their boundaries in such a way that r bounds a disc in Vr. We say that M(r) is obtained from M by r-Dehn filling.

Assume now that M contains no essential sphere, disc, torus or annulus. Then, by Thurston's Geometrization Theorem for Haken manifolds [T1, T2], M is hyperbolic, in the sense that int M has a complete hyperbolic structure of finite volume. Furthermore, M(r) is hyperbolic for all but finitely many r [T1, T2] and the precise nature of the set of exceptional slopes E(M)={r: M(r) is not hyperbolic} has been the subject of a considerable amount of investigation. The maximal observed value of e(M)=[mid ]E(M)[mid ] (the cardinality of E(M)) is 10, realized, apparently uniquely, by the exterior of the figure eight knot [T1].

Let Δ(r1, r2) denote as usual the minimal geometric intersection number of two slopes r1 and r2. If [Sscr ] is any set of slopes, then clearly any upper bound for Δ([Sscr ])=max{Δ(r1, r2): r1, r2∈[Sscr ]} gives one for [mid ][Sscr ][mid ]. For example, one can check (using [GLi, lemma 2·1]) that for 1[les ]Δ([Sscr ])[les ]10, the maximum values of [mid ][Sscr ][mid ] are as given in Table 1.

In particular, any upper bound for Δ(M)=Δ(E(M)) gives a corresponding bound for e(M). (The maximal observed value of Δ(M) is 8, realized by the figure eight knot exterior and the figure eight sister manifold [T1, HW].)

If M(r) is not hyperbolic, then it is either reducible (contains an essential sphere), toroidal (contains an essential torus), a small Seifert fibre space (one with base S2 and at most three singular fibres), or a counterexample to the Geometrization Conjecture [T1, T2]. A survey of the presently known upper bounds on the distances Δ(r1, r2) between various classes of exceptional slopes r1 and r2, and the maximal values realized by known examples, is given in [Go2]. (See also [Wu2] for a discussion of the additional cases that arise when M has more than one boundary component.) In the present note we prove the following theorem, which deals with one further pair of possibilities.

We present a new class of curves which are self-approaching in the following sense. For any three consecutive points a, b, c on the curve the point b is closer to c than a to c. This is a generalisation of curves with increasing chords which are self- approaching in both directions. We show a tight upper bound of 5·3331… for the length of a self-approaching curve over the distance between its endpoints.

The sheaves [Cscr ]*M and [Cscr ]d,*M are constructed for the microlocal analysis of singularities of ultradistributions and ultradifferentiable functions. The suppleness of these sheaves is proved.

In previous papers [4, 6], B.-Y. Chen introduced a Riemannian invariant δM for a Riemannian n-manifold Mn, namely take the scalar curvature and subtract at each point the smallest sectional curvature. He proved that every submanifold Mn in a Riemannian space form Rm(ε) satisfies: δM[les ][n2(n−2)]/ 2(n−1)H2+[half](n+1)(n−2)ε. In this paper, first we classify constant mean curvature hypersurfaces in a Riemannian space form which satisfy the equality case of the inequality. Next, by utilizing Jacobi's elliptic functions and theta function we obtain the complete classification of conformally flat hypersurfaces in Riemannian space forms which satisfy the equality.

A new Möbius-invariant metric δG defined on an open set G⊂[Rmacron]n with at least two boundary points is introduced. This metric coincides with the hyperbolic metric if G is the unit ball. Some inequalities between this and other metrics are proved.

The behaviour of δG under K-quasiconformal maps is also studied. In particular, a generalization of the Schwarz lemma is obtained for K-quasiconformal maps defined on quasiballs.

Let I be a countably infinite set of points in ℝ which we can write as I={ui: i∈ℤ}, with ui<ui+1 for every i and where ui→±∞ if i→±∞. Consider a continuous-time Markov chain Y={Y(t): t[ges ]0} with state space I such that:

Y is driftless; and

Y jumps only between nearest neighbours.

We remember that the simple symmetric random-walk, when repeatedly rescaled suitably in space and time, looks more and more like a Brownian motion. In this paper we explore the convergence properties of the Markov chain Y on the set I under suitable space-time scalings. Later, we consider some cases when the set I consists of the points of a renewal process and the jump rates assigned to each state in I are perhaps also randomly chosen.

This work sprang from a question asked by one of us (Sibson) about ‘driftless nearest-neighbour’ Markov chains on countable subsets I of ℝd, work of Sibson [7] and of Christ, Friedberg and Lee [2] having identified examples of such chains in terms of the Dirichlet tessellation associated with I. Amongst methods which can be brought to bear on this d-dimensional problem is the theory of Dirichlet forms. There are potential problems in doing this because we wish I to be random (for example, a realization of a Poisson point process), we do not wish to impose artificial boundedness conditions which would clearly make things work for certain deterministic sets I. In the 1-dimensional case discussed here and in the following paper by Harris, much simpler techniques (where we embed the Markov chain in a Brownian motion using local time) work very effectively; and it is these, rather than the theory of Dirichlet forms, that we use.

The purpose of this paper is to determine the optimal order of decrease of the L2 norm for the following degenerate and singular oscillatory integral operators defined by

formula here

for f∈C∞0(ℝ), where ψ(x, y) is a smooth compactly supported function on ℝ2, S(x, y) is a homogeneous polynomial of degree n

formula here

and K(x, y) is a distribution kernel which satisfies the following hypotheses:

K is a C2 function away from the diagonal and the estimates

formula here

hold for 0<μ<1 and i=1, or 2.

This paper basically uses the methods of [6], which are very general and can be widely used. The main idea of these methods is that the sizes of operator norms after the decomposition are estimated in different ways and the decomposition is then summed back by balancing the estimates of two types. The first type estimate takes into account the support sizes of the integrand after the decomposition. The second type estimate is delicate, depending on how to keep track of the distance to the singular varieties of phase function. In our case, we have to include the singular variety arising from distribution kernel as well. The analytic tool of the second type estimate is the operator version of the Van der Corput lemma [7]. We notice that the crux of matters is to locate the singular varieties of phase function and kernel. For higher dimensional situations, the problem is very difficult. One expects that the powerful theory of algebraic geometry may play a significant role. Here we would like to remark that Mather's results [2] may be useful although the geometrical shapes of singular varieties are basically undecidable.

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges. In this theory, solutions of a partial differential equation are sections of a fibre bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection on Y, a covariant Hamiltonian density [Hscr ] is then intrinsically defined on the primary constraint manifold P[Lscr ], the image of the multisymplectic version of the Legendre transformation. One views P[Lscr ] as a subbundle of J1(Y)*, the affine dual of J1(Y)*, the first jet bundle of Y. A canonical multisymplectic (n+2)-form Ω[Hscr ] is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original partial differential equation as well as the Euler–Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ω(μ) defined by Bridges are a particular coordinate representation for a single multisymplectic (n+2)-form and, in the presence of symmetries, can be assembled into Ω[Hscr ]. A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to P[Lscr ] with the conservation laws of the system. These conservation laws are defined by our generalized Noether's theorem which recovers the vanishing of the divergence of the vector of n+1 distinct momentum mappings defined by Bridges and, when applied to water waves, recovers Whitham's conservation of wave action. In our view, the multisymplectic structure provides the natural setting for studying dispersive wave propagation problems, particularly the instability of water waves, as discovered by Bridges. After developing the theory, we show its utility in the study of periodic pattern formation and wave instability.