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.

Existence results are established for second-order boundary value problems for ordinary differential equations on non-compact intervals.

In 1988, Howie and Marques-Smith studied Pm, a Rees quotient semigroup of transformations associated with a regular cardinal m, and described the elements which can be written as a product of nilpotents in Pm. In 1981, Marques proved that if Δm denotes the Malcev congruence on Pm, then Pm/Δm is congruence-free for any infinite m. In this paper, we describe the products of nilpotents in Pm when m is nonregular, and determine all the congruences on Pm when m is an arbitrary infinite cardinal. We also investigate when a nilpotent is a product of idempotents.

In this paper we analyse the change of stability of Schrödinger semigroups with indefinite potentials when a coupling parameter varies. Generically, the change of stability takes place at a principal eigenvalue associated with the problem. The uniqueness of the principal eigenvalue is shown for several classes of potentials.

We prove the existence of a global attractor for the problem

where f is ‘coercive at infinity’ and satisfies the growth condition

while g ϵL2(R3).

This paper considers the existence of axisymmetric solitary waves in an inviscid and incompressible rotating fluid bounded by a rigid cylinder. It has been obtained by many experiments and formal derivations that this flow has internal solitary waves in the fluid when equilibrium state at infinity satisfies certain conditions. This paper gives a rigorous proof of the existence of solitary wave solutions for the exact equations governing the flow under such conditions at infinity, and shows that the first-order approximations of the solitary wave solutions for the exact equations are solitary wave solutions derived formally using long-wave approximation. The ideas in the proof of the existence of solitary waves in two-dimensional stratified fluids are used and a main difficulty from the singularity at axis of rotation is overcome.

Complex and chaotic structures in certain dynamical systems in biology arise as a consequence of noncomplete integrability of two-degree-of-freedom Hamiltonian systems. A study of this problem is made using Ziglin theory and implemented with the aid of the Kovacic algorithm.

We establish the existence of positive solutions with two peaks being located on the boundary of the domain for the problem −Δu + λu = up in antipodal invariant domains including ball domains with Neumann boundary conditions. Here p is the critical Sobolev exponent (N + 2)/(N − 2). The shape of the solutions and the location of the peaks are also studied.

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.

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.

The real difference equation an+2 − (λ|an+1| + μan+1) + an = 0 may be interpreted as a dynamical system Φ:(an, an+1) ↦ (an+1, an+2) acting in the plane. The set ΛP of points (λ, μ) for which the mapping Φ is periodic has a rich structure. In this paper, we derive some geometric properties of ΛP (for example, we show that it is unbounded and uncountable), and we derive criteria for Φ to be periodic. We also investigate when Φ is conjugate to a rotation of the plane, and we describe how the rotation numbers of the corresponding circle maps Φ/|Φ| are related to the structure of ΛP.

We study the problem when a ring which is an extension of a commutative idempotent ring by a commutative idempotent ring is commutative. In particular, we answer Sands' question showing that the class of commutative idempotent rings whose every homomorphic image has zero annihilator is a maximal but not the largest radical class consisting of commutative idempotent rings.

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.

We give a concise approach to generalising the inequalities of Wirtinger, Hardy, Weyl and Opial by using the well-known inequality: if X and Y are non-negative, then

for p > 1 (0 < p < 1), respectively.

In the first part of this paper, Yau's estimates for positive L-harmonic functions and Li and Yau's gradient estimates for the positive solutions of a general parabolic heat equation on a complete Riemannian manifold are obtained by the use of Bakry and Emery's theory. In the second part we establish a heat kernel bound for a second-order differential operator which has a bounded and measurable drift, using Girsanov's formula.

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.

For a smooth, simple closed curve α in the plane, the perpendicular bisector map P associates to each pair of distinct points (p, q) on α the perpendicular bisector of the chord joining p and q. To a pair (p, p), the map P associates the normal to α at p. The set of critical values of this map is the union of the dual of the symmetry set of α and the dual of the evolute. (The symmetry set is the locus of the centres of circles bitangent to α.) We study the mapP and use it to give a complete list of the transitions which take place on the dual of the symmetry set and the dual of the evolute, as α varies in a generic one-parameter family of plane curves.