In this paper expressions are derived for the unknown coefficients in the dual cosine series:

where α is a given constant, and f1(x) and f2(x) are prescribed functions.

1. In obtaining a solution of the differential equations corresponding to the motion of a particle about a position of equilibrium, it is usual to express the displacements in terms of a series of periodic terms, each sine or cosine having for its coefficient a series of powers of small quantities. Korteweg has discussed the general form of such solutions, and, from the developments in series which he has obtained, has deduced certain features of interest. In particular, he has shown that, under certain circumstances, it is possible that certain vibrations of higher order, which are normally of small intensity compared with the principal vibrations, may acquire an abnormally large intensity. Considering the oscillations of a dynamical system having a number of degrees of freedom, and supposing to be the frequencies corresponding to infinitesimal oscillations in the different normal coordinates, Korteweg has shown that these cases of interest arise only when

is zero or very small, where p1, p2,… are small integers, positive or negative; the most important cases occur when

We prove two types of vanishing results for the Euler characteristic.

In this paper we discuss the shape of the quadrature domain of a signed measure for harmonic functions. It is known that the quadrature domain of a positive measure with small support is like a ball if the total measure is large enough. We show that, on the contrary, if the measure is not positive then the quadrature domain can be close to an arbitrary domain. This follows from a lemma concerning linear combinations of harmonic measures.

In spherical, as in plane geometry, transformation by inversion and by similar figures may be used for the deduction of new theorems from known ones, the two methods, however, in the former case becoming identical; and like all other propositions and methods in spherical geometry, these methods of transformation may be dualized. The transformation indicated as the dual of these in the case of the spherical surface is also applicable in piano. The sequel is an account of this method and of some results that may be obtained by means of it.

Concerning your paper written in collaboration with Professor Copson, I found the expansion of

for small values of λ in the following manner: The elementary substitution tan ½φ = sin θ/(et + cos θ)yields

i.e. your equation (5.11).

We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = ∞. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMOx. This turns out to depend strongly on the convexity of the Banach lattice . We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.

Let a meromorphic differential equation

be given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expression

where F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, and

is a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.

We discuss the problem of finding those integers which may be represented by (x + y + z)3/xyz, and also those which may be represented by x/y + y/z + z/x, where x, y, z are integers. For example,

satisfy (x + y + z)3/(xyz) = -47, and

satisfy x/y + y/z + z/x = -86.

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

In (1; p. 38), A. P. Guinand discusses the plane partition function q(n). He observes that q(3), q(6), q(9), q(15), q(18), q(21), and q(24) are respectively 6, 48, 282, 1479, 6879, 29601, 118794, and 451194. As all these are multiples of 3 he suggests the conjecture that q(3n) ≡ 0 (mod 3) for all positive integers n.