Our basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.

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 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 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.

A new method for the mechanical analysis of soils and deposits has been developed. It consists in weighing, from time to time, on a specially constructed balance, the amount of sediment accumulated on a circular disc suspended from the balance near the bottom of a vessel containing an aqueous suspension of the sample. From the “accumulation-curve” thus obtained one finds by a series of mathematical operations a “distribution-curve,” i.e. a curve showing how the amount of particles of a certain size varies with the latter quantity.

This method has been applied to the study of some deep-sea deposits from the Challenger Office. The distribution-curves thus obtained show marked differences for the different samples, and reveal a surprising lack of very fine particles in the deposits from the largest depths, besides suggesting interesting problems for new research.

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.

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 integral maximum principle for the heat equation on a Riemannian manifold is improved and applied to obtain estimates of double integrals of the heat kernel.

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.

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 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.

It is shown that the Cesàro averaging operatorℜα > – 1, satisfies an inequality which immediately implies that it is bounded on certain Hardy spaces including Hp, 0 < p < ∞. This answers an open question of Stempak, who introduced these operators and obtained their boundedness on Hp, 0 < p ≦ 2, for ℜα ≧ 0. The operator which is conjugate to on H2 is also shown to be bounded on Hp for 1 < p < ∞ and ℜα = – 1. This extends a result of Stempak who obtained this boundedness for 2 ≦ p≦ ∞ and ℜα ≧:0.

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 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.

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.

In this note, we study the hyperbolic stochastic damped sine-Gordon equation (SdSG), with a parameter β2 > 0, and its associated Gibbs dynamics on the two-dimensional torus. After introducing a suitable renormalization, we first construct the Gibbs measure in the range 0 < β2 < 4π via the variational approach due to Barashkov-Gubinelli (2018). We then prove almost sure global well-posedness and invariance of the Gibbs measure under the hyperbolic SdSG dynamics in the range 0 < β2 < 2π. Our construction of the Gibbs measure also yields almost sure global well-posedness and invariance of the Gibbs measure for the parabolic sine-Gordon model in the range 0 < β2 < 4π.

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.

We use a simple direct and basic method to prove that there is a unique isosceles trapezoid central configuration of the planar Newtonian four-body problem when two pairs of equal masses are located at adjacent vertices of a trapezoid. Such isosceles trapezoid central configurations are an exactly one-dimensional family. Explicit expressions for masses are given in terms of the size of the quadrilateral.

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.