§§ 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.

Dactylopterus volitans, the so-called “flying gurnard,” is not included by Günther in the genus Triglidæ, as its name might imply, but is assigned to the small allied family of Cataphracti.

Günther diagnoses it as follows :—“Dactylopterus, no lateral line, pectoral fins very large, an organ of flight, with the upper portion detached and shorter, granular teeth in the jaws, none on the palate, air bladder divided into two lateral halves, each with a large muscle.”

Before proceeding to describe in detail the swimming bladder and other anatomical peculiarities, it may be well to state, that the skull is provided with a superficial bony covering which projects backwards over the region of the “shoulder” in two flattened plates, each terminating in a spine (seen in fig. 2). Also that the first four vertebræ of the column have coalesced so as to form a rigid tube, the neural spines being united as a vertical plate, which for convenience I have termed the neural plate.

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 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.

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

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

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

$\tau (x)=0$

$\tau (x)=x$

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

${\open R}^n$

In the beginning of January 1865, the author received from the Rev. Alexander Robb, Old Calabar, a package of specimens of natural history preserved in spirits. Among these were two small ganoid fish. They were, however, imperfect, having been torn across near the anal region, and their caudal extremities were wanting. The characters of the fish could not, therefore, be completely determined. The author, however, exhibited them at a meeting of the Royal Physical Society, on the 22d March 1865, and stated that they were allied to the genus Polypterus; but from various differences in character, to be afterwards detailed, and especially the great relative length of their bodies, and the apparently total absence of ventral fins, he would place them provisionally in a new genus, which, from their general aspect and form, he designated Erpetoichthys, the reptile or serpent fish; and the species, from the locality where it was found, he named E. Calabaricus.

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.

Conditions are obtained under which a graph G', obtained from a graph G by the attachment of a pendant edge, may be regarded as a perturbation of G. In this situation the index of G' is expressed as the sum of a convergent series whose terms are determined by the spectrum and certain angles of G.

In the paper referred to in the title [2] an open question was raised concerning the equality of the largest left hereditary radical and the largest right hereditary radical contained in each of certain radicals. In this addendum an affirmative answer is provided to this question.

We introduce the concept of flat ridges for submanifolds of codimension 2 from the viewpoint of their contact with hyperplanes. We characterize them geometrically, studying some of their properties. In particular, we see that the highest-order flat ridges coincide with the flattenings of the asymptotic lines and from this we obtain some lower bounds for their numbers under appropriate conditions.

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

We investigate how the fixed point algebra of a C*-dynamical system can differ from the underlying C*-algebra. For any exact group Γ and any infinite group Λ, we construct an outer action of Λ on the Cuntz algebra 𝒪2 whose fixed point algebra is almost equal to the reduced group C*-algebra ${\rm C}_{\rm r}^* (\Gamma)$. Moreover, we show that every infinite group admits outer actions on all Kirchberg algebras whose fixed point algebras fail the completely bounded approximation property.

We construct new optimal $L^{p}$ Hardy-type inequalities for elliptic Schrödinger-type operators with a potential term.

In the third Book of The Wars of the Jews, Chap. VIII. § 7, we are told that Josephus managed to save himself and a companion out of a total of 41 men, the majority of whom had resolved on self-extermination (to avoid falling into the hands of Vespasian) provided their leader died with them. The passage is very obscure, and in a sense self-contradictory, but it obviously suggests deliberate fraud of some kind on Josephus' part.

“And now,” said he, “since it is resolved among you that you will die, come on, let us commit our mutual deaths to determination by lot. He whom the lot falls to first, let him be killed by him that hath the second lot, and thus fortune shall make its progress through us all; nor shall any of us perish by his own right hand, for it would be unfair if, when the rest are gone, some-body should repent and save himself.” Whiston, Works of Flavius Josephus, IV. 39.

Bachet, in No. XXIII. of his Problèmes plaisants et délectables, makes a definite hypothesis as to the possible nature of the lot here spoken of; so that the problem, as we have it, is really his.

“Supposons qu'il ordonna que comptant de 3 en 3 on tuerait toujours le troisième, …. il faut que Josèphe se mît le trenteunième après celui par lequel on commençait a compter, au cas qu'il visât à demeurer en vie lui tout seul.

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.

We obtain a measure representation for a functional arising in the context of optimal design problems under linear growth conditions. The functional in question corresponds to the relaxation with respect to a pair $(\chi,u)$, where $\chi$ is the characteristic function of a set of finite perimeter and $u$ is a function of bounded deformation, of an energy with a bulk term depending on the symmetrized gradient as well as a perimeter term.

The present paper is concerned with the infimum of the norm of potentials for Sturm–Liouville eigenvalue problems with Dirichlet boundary condition such that the first two eigenvalues are known. The explicit quantity of the infimum is given by the two eigenvalues.

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.

Unilateral ligation of the vas deferens in the mammal is a method now commonly employed in the surgical treatment of senility. The demonstrably beneficial results that follow the operation are explained on the assumptions that the testis is an endocrine gland; that if the gametogenic function is embarrassed, the interstitial tissue, regarded as the source of the sex-hormone, undergoes hyperplasia; and that associated with this increase in amount of tissue there is an increased elaboration of the internal secretion.