As we cannot have knots in two dimensions, and as Prof. Klein has proved that they cannot exist in space of four dimensions, it would appear that the investigation of their properties belongs to that class of problems for which the methods of quaternions were specially devised. The equation

where ϕ is a periodic function, of course represents any endless curve whatever. Now the only condition to which variations of this function (looked on as corresponding to deformations of the knot) is subject, is that no two values of ρ shall ever be equal even at a stage of the deformation. Subject to this proviso, ϕ may suffer any changes whatever—retaining of course its periodicity. Some of the simpler results of a study of this novel problem in the theory of equations were given,—among others the complete representation of any knot whatever by three closed plane curves, non-autotomic and (if required) non-intersecting.

§ 1. Let X and Y be two metals, of which X is electrically positive to Y in the Volta-contact series for dry metals, and make an incomplete circuit of them as indicated in the diagram, with X and Y metallically connected at an interface J, and free surfaces II, KCC exposed, with ether or air, or any gas or insulating fluid, between them. Let CC be a movable slab of the Y-metal, resting frictionlessly on a fixed surface KK of the game metal. If left to itself the movable slab would, in virtue of electric attraction, as we shall see, oscillate on the two sides of the middle position in which the whole of its upper surface is opposite to II. We shall suppose it held by applied force F in any position, or moved, or allowed to move, from any one position to any other, at our pleasure.

§ 2. Suppose now our apparatus to be given with no excess of either electricity above the other, and to be insulated in air or ether at a distance from all other bodies great in comparison with its own dimensions, and with no electrified body near enough to it to produce sensible electrification through influence. Every part of the X-surface will be found positively and every part of the Y-surface negatively electrified, provided each surface is of perfectly uniform Volta-quality throughout its extent; but the electric surface-densities of the opposite electrifications will be everywhere exceedingly small, except in and near the portions of II and CC closely opposed to one another.

The fishes now exhibited were brought by the Rev. Alexander Robb; D.D., from Old Calabar; West Africa. They were taken in the fresh water of the great Old Calabar River, near Ikorofiong, about a hundred miles or so, by the windings of the river, from the bar near its mouth. The Rev. Dr Robb resides at Ikorofiong, which is one of the stations of the Calabar mission of the United Presbyterian Church.

We prove the existence of multi-soliton and kink-multi-soliton solutions of the Euler–Korteweg system in dimension one. Such solutions behave asymptotically in time like several travelling waves far away from each other. A kink is a travelling wave with different limits at ±∞. The main assumption is the linear stability of the solitons, and we prove that this assumption is satisfied at least in the transonic limit. The proof relies on a classical approach based on energy estimates and a compactness argument.

This paper proves some extensions of Brenier's theorem that an integrable vector-valued function u, satisfying a nondegeneracy condition, admits a unique polar factorisation u = u# ° s. Here u# is the monotone rearrangement of u, equal to the gradient of a convex function almost everywhere on a bounded connected open set Y with smooth boundary, and s is a measure-preserving mapping. We show that two weaker alternative hypotheses are sufficient for the existence of the factorisation; that u# be almost injective (in which case s is unique), or that u be countably degenerate (which allows u to have level sets of positive measure). We allow Y to be any set of finite positive Lebesgue measure. Our construction of the measure-preserving map s is especially simple.

The aim of this paper is to link the analytic results of Brezis et al., Demengel and Ignat relative to $W^{1,1}$-mappings from $B^n$ into $S^1$ to the measure-theoretical geometric results in our previous work. The paper also contains a few remarks about mappings in $W^{1,p}$, $p\geq2$, with values in $S^2$.

This paper deals with initial value problems in ℝ2 which are governed by a hyperbolic differential equation consisting of a nonlinear first order part and a linear second order part. The second order part of the differential operator contains a small factor ε and can therefore be considered as a perturbation of the nonlinear first order part of the operator.

The existence of a solution u together with pointwise a priori estimates for this solution are established by applying a fixed point theorem for nonlinear operators in a Banach space.

It is shown that the difference between the solution u and the solution w of the unperturbed nonlinear initial value problem (which follows from the original problem by putting ε = 0) is of order ε, uniformly in compact subsets of ℝ2 where w is sufficiently smooth.

Let y(n) + a1y(n−1 +…+ an−1y(1) + an = 0 (*) be a linear ordinary differential equation of order n. A (relative) differential invariant of (*) is a differential polynomial function π(xi) defined on the solution space of (*) satisfying: there is an integer g such that for all invertible linear transformations α of V into itself, π(αxi) = (det α)βπ(xi). We prove in a purely algebraic manner the following two theorems: A. The differential invariants of (*) are generated algebraically by the Wronskian W and the coefficients ala2, …, an of (*). B. Every generic differential relation (i.e. differential relation which holds for every linear ordinary differential equation of order n) among W, a1 …, an can be deduced algebraically from Abel's identity, W′ = −a1W. The second theorem may be considered as an algebraic version of the existence theorem for linear ordinary differential equations.

We consider an elliptic boundary problem in a bounded region Ω ⊂ ℝn wherein the spectral parameter is multiplied by a real-valued weight function with the property that it, together with its reciprocal, is essentially bounded in Ω. The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Then, fixing our attention upon the operator induced on L2(Ω) by the boundary problem under null boundary conditions, we establish results pertaining to the asymptotic behaviour of the eigenvalues of this operator under weaker smoothness assumptions than have hitherto been supposed.

We give an alternative self-contained proof of the homogenization theorem for periodic multi-parameter integrals that was established by the authors. The proof in that paper relies on the so-called compactness method for Γ-convergence, while the one presented here is by direct verification: the candidate to be the limit homogenized functional is first exhibited and the definition of Γ-convergence is then verified. This is done by an extension of bounded gradient sequences using the Acerbi et al. extension theorem from connected sets, and by the adaptation of some localization and blow-up techniques developed by Fonseca and Müller, together with De Giorgi's slicing method.

This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

It is shown that there exists an uncountable number of rings R with the properties (1) if r∈R then 2r = 0 and r2 = 0, (2) if r ∈ R then there exist s, t in R with r = st.

In the coordinatization of lattices by Baer semigroups, two notable gaps that remain to be filled concern the coordinatization of modular and distributive lattices. In this paper we present coordinatizations of modular lattices. In a subsequent paper we shall deal with the distributive case. Here we show that a bounded lattice L is modular if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) then there exist idempotents ē, ∊ S such that ēS = eS, S = fS and e¯, commute; equivalently, if and only if L can be coordinatized by a Baer semigroup S such that if eS, fS ∊ R(S) with e idempotent then there is an idempotent such that S = fS and e = ee.

The suggestion is made that in certain cases the light particles emitted in ternary fission may be released otherwise than in the ground state—and, in particular, that low-lying particle-unstable states may be involved. This suggestion is examined quantitatively in respect of the release of 8Be in the first excited short-lived state of 2·9 MeV excitation, and is shown to provide a satisfactory interpretation of the observation of quaternary fission in the α,α mode recently reported by Kataria et al. (1973). It appears probable that the other modes of quaternary fission observed by these authors should be similarly interpreted.

This paper is concerned with quasilinear hyperbolic systems of conservation type and establishes two main results. The first is that when a general system is considered in one space dimension and time, then the exceptional nature of a characteristic field implies the coincidence of a shock with one of the characteristics of that field. The second result involves the demonstration by example that quasilinear hyperbolic systems of conservation type may possess solutions that become unbounded after only a finite time, even though they are exceptional.

Lie algebras endowed with an action by automorphisms of any of the symmetric groups S3 or S4 are considered, and their decomposition into a direct sum of irreducible modules for the given action is studied. In the case of S3-symmetry, the Lie algebras are coordinatized by some non-associative systems, which are termed generalized Malcev algebras, as they extend the classical Malcev algebras. These systems are endowed with a binary and a ternary product, and include both the Malcev algebras and the Jordan triple systems.

We prove the convergence in certain weighted spaces in momentum space of eigenfunctions of $H=T-\lambda V$ as the energy goes to an energy threshold. We do this for three choices of kinetic energy $T$, namely the non-relativistic Schrödinger operator, the pseudorelativistc operator $\sqrt{-\Delta+m^2}-m$, and the Dirac operator.

In this paper, we consider the question of existence of solutions and their regularity properties for a large class of stochastic evolution equations governed by B-evolutions involving two different Hilbert spaces. This allows dynamic boundary conditions together with noisy boundary data. They cover also stochastic boundary value problems. Our results are illustrated by two practical examples.

We analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.

In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.

We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.

The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

In this paper we study constrained variational problems that are principally motivated by nonlinear elasticity theory. We examine, in particular, the relationship between the positivity of the Jacobian det ∇u and the uniqueness and regularity of energy minimizers u that are either twist maps or shear maps. We exhibit explicit twist maps, defined on two-dimensional annuli, that are stationary points of an appropriate energy functional and whose Jacobian vanishes on a set of positive measure in the annulus. Within the class of shear maps we precisely characterize the unique global energy minimizer $u_{\sigma }: \Omega \to {\open R}^2$ in a model, two-dimensional case. We exploit the Jacobian constraint $\det \nabla u_{\sigma} \gt 0$ a.e. to obtain regularity results that apply ‘up to the boundary’ of domains with corners. It is shown that the unique shear map minimizer has the properties that (i) $\det \nabla u_{\sigma }$ is strictly positive on one part of the domain Ω, (ii) $\det \nabla u_{\sigma } = 0$ necessarily holds on the rest of Ω, and (iii) properties (i) and (ii) combine to ensure that $\nabla u_{\sigma }$ is not continuous on the whole domain.