Dual integral equations of the form

where f(x) and g(x) are given functions, ψ(x) is unknown, k≧0, μ, v and α are real constants, have applications to diffraction theory and also to dynamical problems in elasticity. The special cases v = −μ, α = 0 and v = μ = 0, 0<α2<1 were treated by Ahiezer (1). More recently, equations equivalent to the above were solved by Peters (2) who adapted a method used earlier by Gordon (3) for treating the (extensively studied) case μ = v, k = 0.

The inverse scattering problem for acoustic waves in shallow oceans are different from that in the spaces of R2 and R3 in the way that the “propagating” far-field pattern can only carry the information from the N +1 propagating modes. This loss of information leads to the fact that the far-field pattern operator is not injective. In this paper, we will present some properties of the far-field pattern operator and use this information to construct an injective far-field pattern operator in a suitable subspace of L2(∂Ω). Based on this construction an optimal scheme for solving the inverse scattering problem is presented using the minimizing Tikhonov functional.

At the close of the preceding meeting of the Society a discussion arose concerning the effect of a uniform rise in prices upon the amount of small money necessary for the transaction of business. It was clear that the total amount of money must be increased, but in the case of small money, which is used only when fractional parts of the larger unit are involved, the effect was less obvious. It is the fact, however, that more small money required. The present paper attempts an explanation of that fact.

The basic reciprocity of j-differential and LM-integral

for bounded functions f(x) with simple discontinuities but continuous on the left at each point and for g(x) in the somewhat restricted class B of functions of bounded variation and also left-continuous, was established in (2) and (3); the dot here indicates the lower product of and (jg, g+ (x+)dx), with , and the integral indicated is the RJDS-integral, equivalent to (LM) .

The Kontorovich-Lebedev Transform of a function f(r), 0<r<∞, may be written in a general form as fg(μ) where,

A suitable function f(x) 0 ≤ x < ∞, can be expanded into a Fourier series of Laguerre polynomials , whose interval of orthogonality is 0 ≤ x < ∞. The usual problems as to convergence and, lacking convergence, summability, and also the asymptotic behaviour of Lebesgue constants, arise for such developments. A summary of work on these convergence and summability problems, together with extensive references to the literature, can be found in the standard treatise by G. Szegö (5, especially Chapter IX) to whom many of these results are due.

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

If G is an additive (but not necessarily abelian) group and S is a semigroup of endomorphisms of G, the endomorphism near-ring R of G generated by S consists of all the expressions of the form ɛ1s1+…+ɛnsnwhere ɛi=±1 and si∈S for each i. When functions are written on the right, R forms a distributively generated left near-ring under pointwise addition and composition of functions. A basic reference on near-rings which has a substantial treatment of endomorphism near-rings is [6].

The proof usually given of Von Staudt's Theorem is entirely analytical in character. The following proof is geometrical:—

If A, B, C, D be the vertices of a tetrahedron whose opposite faces are α., β, γ, δ, and if l be any line, to prove that l [ABCD] = l [αβγδ] where lA denotes the plane through the line l and the point A and lα. denotes the intersection of the line l with the plane α.

For N any member of a large class of finite abelian right centralizer near-rings, the subring of the ring End(N) of endomorphisms of (N, +) generated by the set of right multiplication maps on N is explicitly described as a generalized blocked triangular matrix ring, which in some cases turns out to be a structural matrix ring.

We consider the Dirichlet problem for Laplace's equation in a rectangle with a view toward determining the asymptotic behaviour of the solution for the case in which the width of the rectangle is small in comparison with its length. Although the construction of an explicit representation of the solution is an elementary matter, the resulting formula is inconvenient for present purposes, and we accordingly proceed along different lines.

§1. Figure 30. Let AB be any straight line in the plane figure, A′B the position of the same line after a displacement, the point A moving to A′ and B to B′. Since the position of a plane figure in its plane is determined, when the position of a straight line rigidly attached to it is determined, the motion of the plane figure is determined if we determine the motion of the straight line AB.