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.

§§ 32–64. Canal Ship-Waves.

§ 32. To avoid the somewhat cumbrous title “Two-dimensional,” I now use the designation “Canal † Waves” to denote waves in a canal with horizontal bottom and vertical sides, which, if not two-dimensional in their source, become more and more approximately two-dimensional at greater and greater distances from the source. In the present communication the source is such as to render the motion two-dimensional throughout; the two dimensions being respectively perpendicular to the bottom, and parallel to the length of the canal: the canal being straight.

It is shown that the Cesàro averaging operators Cα, Re α > −1, introduced by Stempak, are bounded on the Dirichlet space Da if and only if a > 0, while the associated operators Aα are bounded on Da if and only if −1 < a < 2. This extends results of Galanopoulos, who considered the particular case α = 0 for 0 ≤ a ≤ 1.

In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form

$$Au(x)=\int_{{\open R}^n}\int_{{\open R}^n}e^{{\rm i}(x-y)\cdot\xi}\sigma(x+\tau(y-x),\xi)u(y)\,{\rm d}y\,{\rm d}\xi,$$

$\tau :{\open R}^n\to {\open R}^n$

$\tau (x)=0$

$\tau (x)=x$

$\tau (x)={x}/{2}$

${\open R}^n$

Writers on mammalian anatomy are united in stating that one of the characteristics of the Cetacean family Physeteridæ is the presence of functional teeth in the lower jaw only. Typical statements are those of Flower and Lydekker (1891): “No functional teeth in the upper jaw”; Beddard (1900): “Teeth found in both jaws, but those of lower jaw alone functional; often very reduced in number”; Turner (1912): “Functional teeth in lower, but not in upper jaw.” A possible exception is the Pigmy Whale, Kogia, in which the upper jaw may altogether lack teeth or may possess an anterior rudimentary pair (as in K. simus, Owen) the utility of which is a matter of some doubt.

It is shown that the free product of two residually finite combinatorial strict inverse semigroups in general is not residually finite. In contrast, the free product of a residually finite combinatorial strict inverse semigroup and a semilattice is residually finite.

This paper, extending the work of Stratford [6] considers a boundary layer with uniform pressure when x < x0, and with the pressure in x > x0 so chosen that the layer is just on the point of separation for all x >x0. The required pressure distribution is shown to be

The displacement and momentum thicknesses are also derived as series in powers of ξ (and log ξ), and the shape parameter H then obtained as a similar series. The continuous change in H from the Blasius value (when ξ = 0) towards the Falkner-Skan [3] separation value is convincingly demonstrated, with the aid of the leading terms of an asymptomatic expansion for large ξ.

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.

In 1849 William Swan, subsequently Professor of Natural Philosophy in the University of St Andrews, read a paper on the “Gradual Production of Luminous Impressions on the Eye and other Phenomena of Vision” before the Royal Society of Edinburgh (see Transactions, Vol. XVI.). This paper contains some results of high interest, but I have no recollection of ever having seen it referred to in modern literature on the subject.

In 1907 I had the honour of reading a paper before this Society in which a new genus of iron-bacteria was described. This organism was first discovered in Renfrew, near Glasgow, and subsequently observed in various samples of iron-water from different parts of the country. At Renfrew this organism was at the time of its discovery the sole bacterial occupant of the ironwaters of this neighbourhood. In most samples of these waters—and I have examined hundreds—only the remains of iron-bacteria can be obtained, but occasionally the samples are found to contain iron-bacteria in process of growth and multiplication. After a little experience, it is possible to ascertain which streams contain iron-bacteria in this favourable condition. At Renfrew such streams were then common, and so gave me an opportunity of tracing the life-history of this species. The results of this investigation seemed to me to justify the conclusion that I had obtained a new genus of iron-bacteria.

Any solution to the problem

is shown to decay to zero in the strong topology of the energy space as t→ ∞. The function β is allowed to be nonmonotone and d is a nonnegative function strictly positive on a nonvoid subset of (0, л).

Path formulation can be used to classify and structure efficiently multiparameter bifurcation problems around fundamental singularities: the cores. The non-degenerate umbilic singularities are the generic cores for four situations in corank 2: the general or gradient problems and the Z2-equivariant (general or gradient) problems. Those categories determine an interesting ‘Russian doll’ type of structure in the universal unfoldings of the umbilic singularities.

One advantage of our approach is that we can handle one, two or more parameters using the same framework (even considering some special parameter structure, for instance, some internal hierarchy). We classify the generic bifurcations that occur in those cases with one or two parameters.

When an elastic body is constrained in any manner whatsoever, it is susceptible to vibration, by virtue of its elasticity, when disturbed from its position of equilibrium by an externally applied force. The period and amplitude of such vibration are dependent upon the mass and inertia of the system, the rigidity of the constraints, and upon the nature of thedisturbing force.

When a beam of commercial section is loaded centrally, and subjected to vibrations, the frequency of transverse vibration can be readily determined from a knowledge of the dimensions of the beam, its modulus of elasticity, and the conditions of loading. On the other hand, where the loading is eccentric, the transverse vibration is accompanied by a torsional vibration the frequency of which is very much lower than is indicated by the ordinary elastic theory, due to the inefficiency in torsion of beam sections other than circular. In practice it is not always possible to eliminate the eccentric loading of beams, as for instancewhere power is transmitted through countershafts supported from structural steel-work, and it is with the problem of the torsional vibration of such eccentrically loaded beams that itis proposed here to deal.

In this paper are collected and discussed the observations of the specific gravity of saline solutions of moderate and high dilution made during the course of the last ten years. The instrument employed was the hydrometer, of the type used by me during the cruise of the Challenger in the years 1872 to 1876. The results then obtained were such as to convince me that the instrument, in capable hands, and with strict attention to the obligatory conditions of experiment, gives more exact results than any other instrument in use for such work. In the course of the last forty years the method has been improved in the sense that more and more attention has been paid to the conditions, especially those of temperature, under which the experiment is made, because it was found that the method is so delicate that it responds to every increase of care in experimenting by greater exactness in the results which it furnishes.

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.

We consider the following problem:

where

for x ∈ RN, f(x, t), ft (x, t) ∈ C(RN × R), and f (x, t) ≧ 0 for all x ∈ RN and t ∈ R+, f(x, t) is an odd function of t. We show that if the maximum of Q(x) is achieved at k different points of RN, then for μ large enough the above problem has at least k positive solutions and k nodal solutions.

We prove the persistence of analyticity for classical solutions of the Cauchy problem for quasilinear wave equations with analytic data. Our results show that the analyticity of solutions, stated by the Cauchy–Kowalewski and Ovsiannikov–Nirenberg theorems, lasts until a classical solution exists. Moreover, they show that if the equation and the Cauchy data are analytic only in a part of the space variables, then a classical solution is also analytic in these variables. The approach applies to other quasilinear equations and implies the persistence of the space analyticity (and the partial space analyticity) of their classical solutions.

For a special case of the Sturm-Liouville equation, −(py′)′ + qy = λwy on [0, ∞) with the initial condition y(0) cos α + p(0)y′(0) sin α = 0, α ∈ [0, π), it is shown that, given the spectral derivative for two values of α ∈ [0, π) at a fixed μ = Re{λ} ≥ Λ0, it is possible to uniquely determine . An explicit formula is derived to accomplish this. Further, in a more general case of the Sturm-Liouville problem for μ with both finite and positive, then the following inequality holds

Cases of absence or defect of the corpus callosum are of interest not only because of their great rarity, but because of the light which they throw on the distribution and functions of this commissure, and on the development of the mesial aspects of the cerebral hemispheres.

Whilst examining my “tow-nettings” from the West Coast, I have met with a peculiar Decapod larva, which, so far as I know, is unlike anything previously described. In general appearance it is very like semi-adult forms of Lawifer, having an elongated “neck,” at the tip of which the eyes and antennae are carried. My specimens were obtained from two localities—1st, about half a dozen at about the same stage, from tow-nettings taken in Machrie Bay (west shore of Arran), 17th September 1886; and 2nd, about 20 specimens, some further advanced than those of the previous gathering, which were collected at a depth of 8 to 10 fathoms in the Sound of Mull, 24th August 1887. These were associated with various larval stages of Porcellana, Galathea, and other Decapods.