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.

The Hartman–Wintner problem on asymptotic equivalence of fundamental systems of solutions (FSSs) for two Sturm–Liouville equations is studied. The following results are obtained: a criterion of asymptotic equivalence of FSSs, and sufficient conditions of asymptotic equivalence of FSSs which are expressed in terms of the coefficients of the considered equations only.

A differential operator (1) Ly = y(n) + an-1(t)y(n-1) +…+ao(t)y is said to be disconjugate on an interval I if every non-trivial solution of Ly = 0 has less than n zeros on I, multiple zeros being counted according to their multiplicity. An algorithm for the construction of disconjugate operators of type (1) is given here. All such operators are obtained at least in the case when the interval I is open or compact.

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.

For Laplacians defined by measures on a bounded domain in ℝn, we prove analogues of the classical eigenvalue estimates for the standard Laplacian: lower bound of sums of eigenvalues by Li and Yau, and gaps of consecutive eigenvalues by Payne, Pólya and Weinberger. This work is motivated by the study of spectral gaps for Laplacians on fractals.

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.

§§ 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.

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.

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.

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

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

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

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

$\tau (x)=0$

$\tau (x)=x$

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

${\open R}^n$

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.

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.