Let Singn be the subsemigroup of singular elements of the full transformation semigroup on a totally ordered finite set with n elements. Let be the subsemigroup of all decreasing maps of Singn. In this paper it is shown that is a non-regular abundant semigroup with n − 1 -classes and . Moreover, is idempotent-generated and it is generated by the n(n − 1)/2 idempotents in J*n−1. Let

and

Some recurrence relations satisfied by J*(n, r) and sh (n, r) are obtained. Further, it is shown that sh (n, r) is the complementary signless (or absolute) Stirling number of the first kind.

In this paper we are concerned with qualitative properties of entire solutions to a Schrödinger equation with sublinear nonlinearity and sign-changing potentials. Our analysis considers three distinct cases and we establish sufficient conditions for the existence of infinitely many solutions.

We define a family of Cesàro operators , Reα≧0, and consider the question of their boundedness on Hp spaces. We also consider discrete versions of these operators acting on sequence spaces.

We provide a sufficient and “almost” necessary condition for the existence of a periodic solution of the equation

where F is nondecreasing in u and has a small linear growth as |u|→∞.

We consider a general class of degenerate elliptic-parabolic problems associated with the equation b(υ)t = div a(υ, Dυ) + f. Existence of renormalized solutions is established for general L1 data. Uniqueness of renormalized solutions has already been shown in a previous work.

The paper deals with smooth nonlinear ODE systems in ℝn, ẋ = f(x), such that the derivative f′(x) has a matrix representation of Jacobi type (not necessarily symmetric) with positive off diagonal entries. A discrete functional is introduced and is discovered to be nonincreasing along the solutions of the associated linear variational system ẏ = f′(x(t))y. Two families of transversal cones invariant under the flow of that linear system allow us to prove transversality between the stable and unstable manifolds of any two hyperbolic critical points of the given nonlinear system; it is also proved that the nonwandering points are critical points. A new class of Morse–Smale systems in ℝn is then explicitly constructed.

Associated to every plane curve there is the locus of centres of circles bitangent to that curve, the so-called symmetry set of the curve. We can view this set as the spine of our curve, which can be recovered by taking the envelope of circles of varying radii along this spine. Varying the symmetry set in some isotopy while keeping the radius function fixed may be viewed as crudely modelling motion of the original curve viewed as a biological object. Fixing the symmetry set and varying the radius function can be considered to model growth crudely. In this paper we shall describe the generic changes in the curves which take place in the process of growth and motion, and outline the corresponding results for spheres centred on a space curve. We also use the idea of a stratified Morse function to describe the generic changes which occur in one parameter families of (full) bifurcation sets in the plane. Applying this to the bifurcation set of distance squared functions we find all the transitions of a symmetry set (and evolute) which occur in a generic isotopy of a plane curve.

We study the planar delay differential equation x′(t) = −x(t) + αF(x(t − 1)), for α > 0. An existence theorem for nonconstant periodic solutions is achieved for a certain class of maps F, for α > some α0. Besides a condition of nondegeneracy at x = 0, we assume F is bounded and satisfies a kind of planar negative feedback condition. The nonconstant periodic solutions are associated with nontrivial fixed points of a certain operator defined by the flow in the plase space C([−l, 0], R2). In our approach, the existence of such fixed points depends on the ejectivity of O ϵ C([−1, 0], R2) with respect to that operator. Relaxing the boundedness condition on F, we show the existence of a sequence of values of α, α0 < α1 <…, where a Hopf bifurcation occurs.

In the male chromosome complement of man, Painter (1923) found that one chromosome pair is heteromorphic and forms an asymmetrical or unequal bivalent during meiotic metaphase, and consequently two kinds of gametes are produced. One member of this heteromorphic pair is about three times longer than the other and is designated the X, the other is described as the Y chromosome. Though the sex chromosomes of man were carefully studied and identified with certainty throughout the nuclear cycle of mitosis and meiosis by several investigators (cf. Shiwago and Andres, 1932), no further attempt was made to correlate the peculiar pairing properties of the heteromorphic chromosomes with their qualitative and structural differentiation.

In this paper, a new variational formulation of the Signorini problem with friction is given in terms of the contact stresses. The method corresponds to the direct integral equation approach in classical elastostatic problems. First the displacement and mixed problems are briefly described together with some numerical results. Next the displacements are eliminated by the use of Green's function, and a constrained minimum problem with respect to the normal and tangential tractions on the contact boundary is derived. Then the resulting approximation procedure is studied and certain convergence results are proved. Finally, some remarks on the Signorini problem with Coulomb friction are presented. Numerical results illustrate the theory.

Maximisation and minimisation of the Dirichlet integral of a function vanishing on the boundary of a bounded domain are studied, subject to the constraint that the Laplacean be a rearrangement of a given function. When the Laplacean is two-signed, non-existence of minimisers is proved, and some information on the limits of minimising sequences obtained; this contrasts with the known existence of minimisers in the one-signed case. When the domain is a ball and the Laplacean is one-signed, maximisers and minimisers are shown to be radial and monotone. Existence of maximisers is proved subject additionally to a finite number of linear constraints, with particular reference to ideal fluid flows of prescribed angular momentum in a disc.

We consider weakly-coupled elliptic systems of the type

with each fi being either an increasing or a decreasing function of each uj. Assuming the existence of coupled super- and subsolutions, we prove the existence of solutions, and provide a constructive iteration scheme to approximate the solutions. We then apply our results to study the steady-states of two-species interaction in the Volterra–Lotka model with diffusion.

For some singular perturbed Dirichlet and Neumann problems in a bounded smooth domain, we study solutions which have a spike in the interior. We obtain both necessary and sufficient conditions for the existence of interior spike solutions. We use, among others, the methods of projections and viscosity solutions, weak convergence of measures and Liapunov–Schmidt reduction.

We prove a Schauder estimate for solutions of linear second order elliptic equations with linear Venttsel boundary conditions, and establish an existence result for classical solutions for such boundary value problems.

We study the blow-up phenomenon for non-negative solutions to the following parabolic problem:

where 0 < p− = min p ≤ p(x) ≤ max p = p+ is a smooth bounded function. After discussing existence and uniqueness, we characterize the critical exponents for this problem. We prove that there are solutions with blow-up in finite time if and only if p+ > 1.

When Ω = ℝN we show that if p− > 1 + 2/N, then there are global non-trivial solutions, while if 1 < p− ≤ p+ ≤ 1 + 2/N, then all solutions to the problem blow up in finite time. Moreover, in the case when p− < 1 + 2/N < p+, there are functions p(x) such that all solutions blow up in finite time and functions p(x) such that the problem possesses global non-trivial solutions.

When Ω is a bounded domain we prove that there are functions p(x) and domains Ω such that all solutions to the problem blow up in finite time. On the other hand, if Ω is small enough, then the problem possesses global non-trivial solutions regardless of the size of p(x).

The following system of conservation laws is considered:

where σ: ℝ→ℝ is a smooth function monotonically increasing except in an interval. Two criteria for the admissibility of shocks are shown to be independent in the sense that there are shocks satisfying each and violating the other. This contrasts with the corresponding situation for strictly hyperbolic systems (σ'(u)>0 for all u), for which the two criteria are equivalent.

We outline a completely probabilistic study of travelling-wave solutions of the FKPP reaction-diffusion equation that are monotone and connect 0 to 1. The necessary asymptotics of such travelling-waves are proved using martingale and Brownian motion techniques. Recalling the connection between the FKPP equation and branching Brownian motion through the work of McKean and Neveu, we show how the necessary asymptotics and results about branching Brownian motion combine to give the existence and uniqueness of travelling waves of all speeds greater than or equal to the critical speed.

This is the first of a series of three papers devoted to the regularity of solutions of elliptic problems on nonsmooth domains in ℝ3. The present paper introduces various weighted spaces and countably weighted spaces in the neighbourhood of edges and vertices of polyhedral domains, and it concentrates on exploring the structure of these spaces, such as the embeddings of weighted Sobolev spaces, the relation between weighted Sobolev spaces and weighted continuous function spaces, and the relations between the weighted Sobolev spaces and countably weighted Sobolev spaces in Cartesian coordinates and in the spherical and cylindrical coordinates. These well-defined spaces are the foundation for the comprehensive study of the regularity theory of elliptic problems with piecewise analytic data in ℝ3, which are essential for the design of effective computation and the analysis of the h – p version of the finite element method for solving elliptic problems in three-dimensional nonsmooth domains arising from mechanics and engineering.

Using fixed point theorems we discuss continuous solutions of Γ-equivariance for a polynomial-like iterative equation on the real line, where Γ is a closed subgroup of the general linear group GL(R). Our main results guarantee the existence of solutions with certain kinds of symmetry. We show that, under restrictive hypotheses, similar results can be proved in a higher-dimensional case, where the symmetry group is a topologically finitely generated subgroup of the group generated by rotations and dilations in N-dimensional Euclidean space.

In this paper we give the local classification of solution curves of bivalued direction fields determined by the equation

where a and b are smooth functions which we suppose vanish at 0 ∈ ℝ2. Such fields arise on surfaces in Euclidean space, near umbilics, as the principal direction fields, and also in applications of singularity theory to the structure of flow fields and monochromatic-electromagnetic radiation. We give a classification up to homeomorphism (there are three types) but the methods furnish much additional information concerning the fields, via a crucial blowing-up construction.