The paper is concerned with the problem of a solitary wave moving with constant form and constant velocity c on the surface of an incompressible, inviscid fluid over a horizontal bottom. The motion is assumed to be two-dimensional and irrotational, and if h is the depth of the fluid at infinity and g the acceleration due to gravity, then the Froude number F is defined by

The result that F>1 has recently been proved by Amick and Toland by means of a long and complicated argument. Here we give a short and simple one.

Sufficient conditions are given for the minimal operator T0 generated by the elliptic differential expression τ = ρ−1 (−Σ DjajkDk + q) to be essentially self-adjoint in the weighted Hilbert space , and for all powers of T0 to be essentially self-adjoint.

The components of a two (complex) dimensional Dirac system are studied as trajectories in the complex plane. The system, defined on an interval [a, b) of regular points, is assumed to be of limit circle type at t = b, and to be non-oscillatory there. By introducing moduli ρ1(t) = |y1(t)|, ρ2(t) = |y2(t)| and continuous complex arguments θ1(t) = arg y1(t), θ2(t) = arg y2(t) for the components, the principal result proved is that θ1(t) and θ2(t) are bounded as t → b. Examples show that monotonicity of the argument function θ1(t), which is a feature of the corresponding problem for Sturm–Liouville equations, fails for Dirac systems.

Let P be a set of primes. A nilpotent space X is called “P-universal”, if its P-localization canbe obtained as a direct limit over a sequence of selfmaps of X. A nilpotent fibration is called “fibrewiseP-universal”, if its fibrewise P-localization can be obtained in a similar way as a direct limit of fibrewise maps. In this paper, the following results are proved. Let π:E → B be a nilpotent fibration of connected finite or cofinite spaces. If π is fibrewise P-universal for some proper subset P of the set of primes, then its minimal model (in the sense of D. Sullivan's rational homotopy theory) admits a certaintype of weight decomposition. But the existence of such a weight decomposition implies only that there exists a finite set of primes, such that IT is fibrewise P-universal for all P with . In the absolute case however, i.e.B is a point, the set P can be chosen as the empty set. Thus a result announced by R. Body and D. Sullivan is recovered.

The inequality of Ganelius states that, for suitable functions f and g on an interval [a, b], [inf f + var f]sup ∫ jdg, where the supremum is taken over all sub-intervals J of [a, b]. A more general version of this inequality is derived as well as certain related generalized mean-value theorems.

A detailed description of the totality of distributional solutions of the catenary problem for nonlinear elastic strings is given.

The Schauder continuation method for nonlinear problems is based on appropriate a priori estimates for related linear equations. Recently, in a paper by the present author and G. C. Hsiao, the Hilbert boundary value problem with positive index for nonlinear elliptic systems in the plane was solved by this method but the constructive derivation of the a priori estimate necessarily required a restriction on the ellipticity condition. This is because the norm of the generalized Hilbert transform in the case of positive index is too big. Here, as in a forthcoming paper by G.C. Wen, an indirect and therefore non-constructive proof of the a priori estimate is given which does not require any further restrictions and allows the Hilbert boundary value problem to be solved for nonlinear elliptic systems in general.

It is shown that the index of a congruence subgroup of the modular group cannot be less than the level of the subgroup. This allows a number of existence theorems about non-congruence subgroups.

The level of a subgroup of the modular group can be defined in terms of the action on Q ∪ {∞}. We define a similar action to get information on congruence subgroups. In fact, we get a more powerful result, but this appears to be the most useful version.

Although the regular subsemigroups of a regular semigroup S do not, in general, form a lattice in any naturalway, it is shown that the full regular subsemigroups form a complete sublattice LF of the lattice of all subsemigroups; moreover this lattice has many of the nice features exhibited in (the special case of) the lattice of full inverse subsemigroups of an inverse semigroup, previously studied by one of the authors. In particular, LF is again a subdirect product of the corresponding lattices for each of the principal factors of S.

A description of LF for completely 0-simple semigroups is given. From this, lattice-theoretic properties of LF may be found for completely semisimple semigroups. For instance, for any such combinatorial semigroup, LF is semimodular.

We introduce the notion of a double MS-algebra (L, 0, +) as an MS-algebra (L, 0) whose dual isan MS-algebra (Ld, +), with certain linking conditions concerning the operations x↦x0 and x↦x+. We determine necessary and sufficient conditions whereby an MS-algebra can be made into a double MS-algebra and show that this, when possible, can be done in one and only one way. We also consider the notion of a bistable subvariety of MS-algebras, namely a subvariety R with the property that, for every double MS-algebra (L, 0, +), whenever L, 0 ↦ R, we have (Ld,+)↦ R. Finally, we determine those subvarieties R of MS that are dense (in the sense that every MS-algebra L ↦ R can be made into a double MS-algebra), and those that are sparse (in the sense that if L ↦ R can be made into a double MS-algebra then it belongs to a proper subclass of R).

The paper presents an extension of the well-known Sturm–Picone theorem for self-adjoint equations to the n-dimensional case. The basic domain G ⊆ Rn is possibly unbounded and no regularity hypotheses on the boundary ∂G are required. The coefficients of the elliptic equation may also be unbounded on G.

A group consisting of real continuous functions of one real variable on the interval j = (−∞, ∞) is called planar if through each point of the plane j × j there passes just one element s ∈ .

Every differential oscillatory equation (Q): y″ = Q(t)y (t ∈ j = (−∞, ∞), Q ∈ C(0)) admits functions, called the dispersions of (Q), that transform (Q) into itself. These dispersions are integrals of Kummer's equation (QQ): −{X, t} + Q(X)X′2(t) = Q(t) and form a three-parameter group , known as the dispersion group of (Q). The increasing dispersions of (Q) form a three-parameter group invariant in . The centre of the group is an infinite cyclic group , whose elements, the central dispersions of (Q), describe the position of conjugate points of (Q).

The present paper contains new results concerning the algebraic structure of the group . It provides information on the following: (1) the existence and properties of planar subgroups of a given group and (2) the existence and properties of the groups containing a given planar group . The results obtained are: the planar subgroups of a given group form a system depending on two constants, SQ, such that for all ∈SQ. The equations (Q) whose groups contain the given planar group form a system dependent on one constant, QS, such that for all (Q)∈QS.

By using the Halmos-Wallen description of power partial isometries on Hilbert space, we give a complete description of all monogenic inverse semigroups,ℐ. We also describe the full C*-algebra C*ℐ and the reduced C*-algebra C*(ℐ) with particular emphasis on the case of the free monogenic inverse semigroupℑℐt.

We prove that there exists a complete system of eigenvectors of the eigenvalue problem

for self-adjoint operators Tr and Vrs on separable Hilbert spaces Hr. It is assumed that

(i) the operators Tr have discrete spectrum;

(ii) the operators Vrs are bounded and commute for each r;

(iii) the operators Vrs have the definite sign factor property.

This theorem generalizes and improves a result of Cordes for two-parameter problems. The proof of the theorem depends on an approximation of the given eigenvalue problem by simpler problems, a technique which is related to Atkinson's proof of his expansion theorem.

We prove that there are 22 non-isomorphic subdirectly irreducible double MS-algebras and give a complete description of them.