The class CS of completely simple semigroups forms a variety under the operations of multiplication and inversion (x−1 being the inverse of x in its ℋ-class). We determine a Rees matrix representation of the CS-free product of an arbitrary family of completely simple semigroups and deduce a description of the free completely simple semigroups, whose existence was proved by McAlister in 1968 and whose structure was first given by Clifford in 1979. From this a description of the lattice of varieties of completely simple semigroups is given in terms of certain subgroups of a free group of countable rank. Whilst not providing a “list” of identities on completely simple semigroups it does enable us to deduce, for instance, the description of all varieties of completely simple semigroups with abelian subgroups given by Rasin in 1979. It also enables us to describe the maximal subgroups of the “free” idempotent-generated completely simple semigroups T(α, β) denned by Eberhart et al. in 1973 and to show in general the maximal subgroups of the “V-free” semigroups of this type (which we define) need not be free in any variety of groups.

We study the existence of a positive solution of a p-superlinear equation involving the p-Laplacian operator. The main difficulty here is that the nonlinearity considered does not necessarily verify the well-known Ambrosetti–Rabinowitz condition. As an application, by performing an adequate change of variables we obtain an existence result of a quasilinear equation depending on the gradient.

We consider a singular Sturm—Liouville expression with the indefinite weight sgn x. There is a self-adjoint operator in some Krein space associated naturally with this expression. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgn x)(−d2/dx2 + q) with non-real spectrum accumulating to a real point. The results obtained are applied to several classes of Sturm—Liouville operators.

We present a new method for investigating the Lp-type pullback attractors (2 ≤ p < ∞) of a semilinear heat equation on a time-varying domain under quite general assumptions on the nonlinear and forcing terms. The existing approach does not appear applicable here as it is impossible to show the existence of a pullback absorbing set in Lp space when p is large. A new asymptotic decomposition scheme for a non-autonomous pullback attractor has been introduced. The abstract results and preliminary lemmas are also of independent interest and applicable to other systems.

We consider the homogenisation of transport kinetic equations with a highly periodic oscillating external field. The external field, acting on the particles, consists of a sum of a field deriving from a periodic potential and a bounded periodic perturbation. For the profile function generated by the oscillating solution of the problem, we derive a kinetic model with transmission boundary conditions in the energy variable. In some cases, for example when the field is not perturbed, we deduce a transport kinetic equation with memory effect satisfied by the weak-* limit of the sequence of solutions.

The solution of simultaneous linear algebraic equations, the evaluation of the adjugate or the reciprocal of a given square matrix, and the evaluation of the bilinear or quadratic form reciprocal to a given form, are all special cases of a certain general operation, namely the evaluation of a matrix product H′A-1K, where A is square and non-singular, that is, the determinant | A | is not zero. (Matrix multiplication is like determinant multiplication, but exclusively row-into-column. The matrix H′ is obtained from H by transposition, that is, by changing rows into columns.) The matrices H′ and K may be rectangular. If A is singular, the reciprocal A-1 does not exist; and in such a case the product H′(adj A) K may be required. Arithmetically, the only difference in the computation of H′A-1K and H(adj A) K is that in the latter case a final division of all elements by | A | is not performed.

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equation

Here Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires that

In case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.

We consider a selection–mutation equation for the density of individuals with respect to a continuous phenotypic evolutionary trait. We assume that the competition term for an individual with a given trait depends on the traits of all the other individuals, therefore giving an infinite-dimensional nonlinearity. Mutations are modelled by means of an integral operator. We prove existence of steady states and show that, when the mutation rate goes to zero, the asymptotic profile of the population is a Cauchy distribution.

For a domain Ω in the Euclidean space Rd (d = 2, 3) existence of weak solutions for both interior and exterior Dirichlet boundary value problems of the Ginzburg-Landau equations are established without any restriction on the range of the coupling constant λ, thesize of Ω, or the boundary data. For the critical choice λ = 1, we prove the existence of confined multivortices in a bounded domain by a constructive monotone iteration method.

In this paper we study the existence of least energy solutions to subcritical semilinear elliptic equations of the form

where Ω is an unbounded domain in RN and f is a C1 function, with appropriate superlinear growth. We state general conditions on the domain Ω so that the associated functional has a nontrivial critical point, thus yielding a solution to the equation. Asymptotic results for domains stretched in one direction are also provided.

Nets of Schrödinger C0-semigroups (Sε)ε with the polynomial growth with respect to ε are used for solving the Cauchy problem (∂t − Δ)U + VU = f(t, U), U(0, x) = U0(x) in a suitable generalized function algebra (or space), where V and U0 are singular generalized functions while f satisfies a Lipschitz-type condition. The existence of distribution solutions is proved in appropriate cases by the means of white noise calculus as well as classical energy estimates.

A reduction method is used to prove the existence and uniqueness of strong solutions to stochastic Kolmogorov–Petrovskii–Piskunov (KPP) equations, where the initial condition may be anticipating. The asymptotic behaviour of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions.

Functional and functional differential equations in a Banach space X are related to systems of operators A(t) in C = C(−r, 0; X), given by

Conditions are sought on F such that A(t) generates an evolution system U(t, s)φThis system gives the segments of solution for φ in a certain domain which is determined.

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

Ophelia cluthensis is a Polychæte worm which has been known for some years from specimens taken from the sands around Millport, and particularly from Kames Bay. It was described for the first time by McGuire in 1935, and its general appearance may be seen in the Plate, figs. A, B.

We consider a developable surface normal to a surface along a curve on the surface. We call it a normal developable surface along the curve on the surface. We investigate the uniqueness and the singularities of such developable surfaces. We discover two new invariants of curves on a surface that characterize these singularities.

We consider the diffusive logistic equation

supplemented by the nonlinear boundary condition

where α is a non-negative, non-decreasing function with α([0, 1]) ⊆ [0, 1]. When regarded as an ecological model for an organism inhabiting a focal patch of its habitat, the assumptions on α are intended to capture a tendency on the part of the organism to remain in the habitat patch when it encounters the patch boundary that increases with species density. Such a mechanism has been suggested in the ecological literature as a means by which the dynamics of the organism at the scale of the patch might differ from its local dynamics within the patch. Building upon earlier examinations of the boundary-value problem by Cantrell and Cosner, we detail in this paper the global disposition of biologically relevant equilibria when both 0 and 1 (the local carrying capacity within the patch) are equilibria. Our analysis relies on global bifurcation theory and estimates for elliptic and parabolic partial differential equations.

A new method for studying inequalities of the type ‖y(r)‖2<ε‖Sk−ry(k)‖2 + K(ε)‖S−ry‖2 and ‖y′‖2≦ Kp(S)‖Sy″‖ ‖S−1y‖ is presented here. With this new approach we obtain new and far reaching extensions of previously known inequalities of this sort as well as simpler proofs of the known cases. In addition we obtain an inequality of type ‖Sy′‖<ε‖(Sy′)′‖ + K(ε)‖y‖ for a general class of functions S. Also we give an elementary operator-theoretic proof of Everitt's characterization of the best constant as well as all cases of equality for

We consider the mass-critical non-linear Schrödinger equation on non-compact metric graphs. A quite complete description of the structure of the ground states, which correspond to global minimizers of the energy functional under a mass constraint, is provided by Adami, Serra and Tilli in [R. Adami, E. Serra and P. Tilli. Negative energy ground states for the L2-critical NLSE on metric graphs. Comm. Math. Phys. 352 (2017), 387–406.] , where it is proved that existence and properties of ground states depend in a crucial way on both the value of the mass, and the topological properties of the underlying graph. In this paper we address cases when ground states do not exist and show that, under suitable assumptions, constrained local minimizers of the energy do exist. This result paves the way to the existence of stable solutions in the time-dependent equation in cases where the ground state energy level is not achieved.

(1) Ligia, like other iposods, moults in two stages. First the coverin of the abdomen with that of the posterior three thoracic segments is es uviated. About four days later the anterior covering is thrown off. Th external changes that occur in the cuticle as it ages, and the behaviou of the animal during moult, are described.

(2) At the moult the cuticle splits in two main directions: (i) transvers between the fourth and fifth (free) thoracic segments; (ii) longitudina at the coxotergal junctions of (free) thoracic segments 2 to 7, there being no coxotergal split in segment 1. Coxotergal splitting is an arthrostracai if not a peracaridan, feature.

(3) Ligiæ that have fasted for many weeks in sea-water may still moult normally. It follows that onset of the moult is determined by an underlying cyclic change, not as Réaumur suggested by simple growth of the animal. A second moult during the period of fast was not observed.