A representation theory for the Hankel transformation Hv, defined for suitable f and for v > −1, by

on the spaces ℒμ, p defined by the norm

is developed. Using this, representation theories are developed for some related transformations.

In this paper, we use the Lyapunov–Schmidt reduction and the S1 × S1-index which is due to Chenkui Zhong to prove that any exact symplectic diffeomorphisms on T2k × CPn × CPm have at least 1 + min {m, n} fixed points.

Let 1 < p, q < ∞. It is shown for complex scalars that there are no nontrivial M-ideals in ℒ(Lp[0, 1]) if p ≠ 2, and is the only nontrivial M-ideal in .

In this paper we show that each quasiperiodic standing wave solution of the real Ginzburg–Landau equation which is on the global branch emanating from the Eckhaus unstable periodic orbit is itself unstable. A rigorous proof of the instability is given by showing that the linearised operator about such a solution has spectrum which contains an interval along the unstable axis of the spectral plane. The proof employs some geometric and topological methods arising from a dynamical systems approach to the analysis of the eigenvalue problem for the linearised operator.

We show that for every finite set A and for every natural number n, there exists a natural number N such that every word of length N over the alphabet A has, for every permutation π of the numbers 1,…,n, a representation of the form Xw1 … wnzwπ(1) … wπ(n) Y, where X, Y are words and w1,…,wn, z are nonempty words over A.

In this paper, we consider an n-dimensional semilinear equation of parabolic type with a discontinuous source term arising from combustion theory. We prove local existence for a classical solution having a ‘regular’ free boundary. In this regard, the free boundary is a surface through which the discontinuous source term exhibits a switch-like behaviour. We specify conditions under which this solution and its free boundary are global in time; moreover, we exhibit a special domain for which, for t tending to infinity, such a global-in-time solution converges, together with its free boundary, to the solution of the stationary problem and to its regular free boundary (which is proved to exist), respectively. We also prove uniqueness and continuous dependence theorems.

This paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form, whose cost functional is of Lagrangian type. Our aim is to prove the existence of solutions and derive some optimality conditions. To attain this second objective, we accomplish the sensitivity analysis of the state equation with respect to the control, proving that, under some assumptions, this relation is Gâteaux differentiable. Finally, a regularising procedure along with Ekeland's variational principle allow us to treat some other problems for which this differentiability property cannot be stated.

We classify completely integrable holonomic systems of first-order differential equations for one real-valued function by equivalence under the group of point transformations in the sense of Sophus Lie. In order to pursue the classification, we use the notion of one parameter Legendrian unfoldings which induces a special class of divergent diagrams of map germs which are called integral diagrams. Our normal forms are represented by integral diagrams.

It is shown that under the action of a geometric subgroup of and , for a germ f satisfying a certain finiteness condition, given a germ p, if the tangent spaces of f and f + p are equal for all t ∈ [0, 1], then f and f + p are -equivalent.

Most of the development of shape theory was in the so-called outer shape theory, where the shape of spaces is described with the help of some outside objects.

This paper belongs to the so-called inner shape theory, in which the shape of spaces is described intrinsically without the use of any outside gadgets. We give a description of shape theory that does not need absolute neighbourhood retracts. We prove that the category ℋN whose objects are topological spaces and whose morphisms are proximate homotopy classes of proximate nets is naturally equivalent to the shape category h. The description of the category ℋN for compact metric spaces was given earlier by José M. R. Sanjurjo. We also give three applications of this new approach to shape theory.

We study a class of AC-Stark Hamiltonians H1(t) = H0(t) + V, where H0(t)= −Δ + E · x cos ωt. For a class of repulsive potentials, we show that the wave operators exist and are unitary, provided that |E|/ω2 is small. If |E|/ω2 is sufficiently large, the result remains true as long as V is sufficiently small.

A correspondence of a semigroup S is any subsemigroup of S × S, and the set of all correspondences of S, with the operations of composition and involution and the relation of set-theoretic inclusion, forms the bundle of correspondences of S, denoted by (S). For semigroups S and T, any isomorphism of (S) onto (T) is called a -isomorphism of S upon T. Similar notion can be introduced for other types of algebras and in the general frame of category theory. The principal goal of this paper is to study -isomorphisms of completely regular semigroups (that is, unions of groups) and of one other interesting class of semigroups.

Local models are given for the singularities which can appear on the trajectories of general one-dimensional motions of the plane or space. Versal unfoldings of these model singularities give simple pictures describing the family of trajectories arising from small deformations of the tracing point.