A sequence of lemmas leads to a two-fold characterisation of the syntactic monoid in the title. Some alternatives as well as special cases, in particular when the code consists of a singleton, are considered.

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.

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.

The concept of a Hilbert module (over an H*-algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H*-algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.

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.

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].

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 aim of this paper is to generalise the results of [7] from the prime to the semiprime case. It was shown, for instance, that if M is the annihilator of a simple right module S of projective dimension 1 over a Noetherian prime polynomial identity (PI) ring R then M is either an invertible ideal or an idempotent ideal [7, Proposition 4.2]. One of the main applications of this result was that a prime Noetherian affine PI ring of global dimension less than or equal to 2 is a finite module over its centre. It turns out that this theorem is valid more generally when the ring is semiprime [1, Theorem A]. Clearly this requires [7, Proposition 4.2] also to be strengthened to the semiprime case. We do this by showing that a right invertible maximal ideal in a semiprime Noetherian PI ring is also left invertible (Theorem 3.5).

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.