The Funk transform is the integral transform from the space of smooth even functions on the unit sphere S2⊂ℝ3 to itself defined by integration over great circles. One can regard this transform as a limit in a certain sense of the Penrose transform from [Copf ]ℙ2 to [Copf ]ℙ*ast;2. We exploit this viewpoint by developing a new proof of the bijectivity of the Funk transform which proceeds by considering the cohomology of a certain involutive (or formally integrable) structure on an intermediate space. This is the simplest example of what we hope will prove to be a general method of obtaining results in real integral geometry by means of complex holomorphic methods derived from the Penrose transform.

The theory of variational bicomplexes was established at the end of the seventies by several authors [2, 17, 23, 26, 29–32]. The idea is that the operations which take a Lagrangian into its Euler–Lagrange morphism [9, 10, 12, 24] and an Euler–Lagrange morphism into its Helmholtz' conditions of local variationality [1–3, 7, 11, 13, 18, 27] are morphisms of a (long) exact sheaf sequence. This viewpoint overcomes several problems of Lagrangian formulations in mechanics and field theories [21, 28]. To avoid technical difficulties variational bicomplexes were formulated over the space of infinite jets of a fibred manifold. But in this formalism the information relative to the order of the jet where objects are defined is lost.

We refer to the recent formulation of variational bicomplexes on finite order jet spaces [13]. Here, a finite order variational sequence is obtained by quotienting the de Rham sequence on a finite order jet space with an intrinsically defined sub-sequence, whose choice is inspired by the calculus of variations. It is important to find an isomorphism of the quotient sequence with a sequence of sheaves of ‘concrete’ sections of some vector bundle. This task has already been faced locally [22, 25] and intrinsically [33] in the case of one independent variable.

In this paper, we give an intrinsic isomorphism of the variational sequence (in the general case of n independent variables) with a sequence which is made by sheaves of forms on a jet space of minimal order. This yields new natural solutions to problems like the minimal order Lagrangian corresponding to a locally variational Euler–Lagrange morphism and the search of variationally trivial Lagrangians. Moreover, we give a new intrinsic formulation of Helmholtz' local variationality conditions, proving the existence of a new intrinsic geometric object which, for an Euler–Lagrange morphism, plays a role analogous to that of the momentum of a Lagrangian.

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.

Let X be a complex nonsingular projective threefold of general type. Suppose the canonical system of X is composed of a pencil, i.e. dimΦ∼KX∼(X)=1. It is often important to understand birational invariants of X such as pg(X), q(X), h2(OX) and χ(OX) etc. In this paper, we mainly study the irregularity of X.

We may suppose that ∼KX∼ is free of base points. There is a natural fibration f[ratio ]X→C onto a nonsingular curve after the Stein factorization of Φ∼KX∼. Let F be a general fibre of f, then we know that F is a nonsingular projective surface of general type. Set b[ratio ]=g(C) and pg(F), q(F) for the respective invariants of F. The main result is the following theorem.

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.

We shall denote throughout by Ω⊂≠ℝd some conical region with vertex at 0. Let Σ=Σd={x∈ℝd; [mid ]x[mid ]=1} denote the unit sphere and by ΩΣ=Σ∩Ω. Let

formula here

be the expression of the Laplacian in polar coordinates x=(r, σ) (r=[mid ]x[mid ], σ∈Σ) where Δr denote the corresponding r-dimensional spherical Laplacian. For d[ges ]2 we can always solve the first eigenfunction problem for ΩΣ

formula here

with Dirichlet boundary conditions. We shall assume throughout that ∂Ω is sufficiently regular to ensure that u is continuous up to the boundary and vanishes there. With α>0α(α+d−2)=λ the function u(x)=rαu(σ)= [mid ]x[mid ]αu(x/[mid ]x[mid ]) is then harmonic in Ω. It should be observed that α[ges ]1 if Ω is convex. If we further assume, as we shall do in this paper, that ∂Ω is Lipschitz, u(x) (x∈ℝd) is the unique, up tp multiplicative constant positive harmonic function in Ω that vanishes at the boundary ∂Ω This function is homogeneous and is called the réduite of Ω (cf. [1]). We shall denote throughout by α=degΩ the homogenity degree of u. If d=1 the réduite is u(x)≡Cx(x>0).

We shall denote by b(t)∈ℝd the standard Brownian motion in ℝd and by

formula here

the first exit time from Ω. We shall then define the corresponding heat diffusion kernel and the corresponding ‘probability of life’

formula here

We obtain a formula for the essential supremum of the packing dimensions of the sections of sets parallel to a given subspace. This depends on a variant of packing dimension defined in terms of local projections of sets.

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.

Suppose that H=H*[ges ]0 on L2(X, dx) and that e−Ht has an integral kernel K(t, x, y) which is a continuous function of all three variables. It follows from the fact that e−Ht is a non-negative self-adjoint operator that K(t, x, x)[ges ]0 for all t>0 and x∈X. Our main abstract results, Theorems 2 and 3, provide a positive lower bound on K(t, x, x) under suitable general hypotheses. As an application we obtain a explicit positive lower bound on K(t, x, y) when x is close enough to y and H is a higher order uniformly elliptic operator in divergence form acting in L2(RN, dx); see Theorem 6.

We emphasize that our results are not applicable to second order elliptic operators (except in one space dimension). For such operators much stronger lower bounds can be obtained by an application of the Harnack inequality. For higher order operators, however, we believe that our result is the first of its type which does not impose any continuity conditions on the highest order coefficients of the operators.

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.

Existence and regularity results are obtained for the Cauchy problem for the Kadomtsev–Petviashvili–I equation by the inverse spectral method. The initial data are small Schwartz functions which are not assumed to satisfy the zero mass constraint ∫ℝdxq(x, y, 0)=0.

We obtain sharp rates of uniform convergence in the real inversion theorem for Laplace transforms. This entails giving estimates of the Kolmogorov distance between normalized Poisson mixtures and their mixing distribution. Our approach combines probabilistic representations in terms of gamma processes with approximation-theoretic techniques.

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.

We introduce a quasitriangular Hopf algebra or ‘quantum group’ U(B), the double-bosonization, associated to every braided group B in the category of H-modules over a quasitriangular Hopf algebra H, such that B appears as the ‘positive root space’, H as the ‘Cartan subalgebra’ and the dual braided group B* as the ‘negative root space’ of U(B). The choice B=Uq(n+) recovers Lusztig's construction of Uq(g); other choices give more novel quantum groups. As an application, our construction provides a canonical way of building up quantum groups from smaller ones by repeatedly extending their positive and negative root spaces by linear braided groups; we explicitly construct Uq(sl3) from Uq(sl2) by this method, extending it by the quantum-braided plane. We provide a fundamental representation of U(B) in B. A projection from the quantum double, a theory of double biproducts and a Tannaka–Krein reconstruction point of view are also provided.