Recently (3) Mazur proved that if N is a prime number such that some elliptic curve E over Q admits a Q-rational isogeny then N is one of 2, 3, 5, 7, 11, 13, 17, 19, 37, 43, 67 or 163.

As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as a special case.

Let G be a group and A a right G-module. If the additive group A+ of A is a Černikov group, that is, a direct sum of finitely many cyclic and quasi-cyclic groups, we shall call A a Černikov module over G or over the integral group ring . Suppose that A+ is, furthermore, a divisible p-group, where p is a prime. Since the endomorphism ring of a quasi-cyclic p-group is isomorphic to the ring of p-adic integers, we find that is a free -module of finite rank. We can make A* into a right G-module in the usual way, and since A* is actually just the Pontrjagin dual of A, Pontrjagin duality shows that A → A* gives rise to a contravariant equivalence between the categories of divisible Černikov p-torsion modules over and G-modules which are -free of finite rank. Since the latter category is to some extent familiar, at least when G is finite – for its objects determine representations of G over the field of p-adic numbers, a field of characteristic zero – we may hope to exploit this correspondence systematically to study divisible Černikov p-modules. This is our main theme.

In [4], A. Connes has defined the convolution algebra associated to a foliation ℱ of the compact manifold M. Here is the graph or holonomy groupoid of the foliation ℱ (Winkelnkemper [15]). By forming the completion of in its regular representation, he obtains the C*-algebra C*{M, ℱ) associated to the foliation. The completeness of C*(M, ℱ) makes it easier to handle in some analytical contexts, but in others it seems to be too big, and it is necessary to consider instead some carefully selected dense subalgebra (cf. [6]). The purpose of this note is to show that certain spectral functions of leafwise elliptic operators, which might a priori be expected to belong to C*(M, ℱ), in fact belong to the more controllable dense subalgebra . We give a couple of applications, including a proof not passing through C*-algebras of Connes' index theorem for measured foliations [4]. It should be emphasized that the proof of that result offered here is essentially Connes' one, but the presentation may perhaps be more congenial to those who are not C*-algebra specialists.

A function f(x1, x2) of two real variables x1, x2 which are restricted to rational integers will be called discrete harmonic (d.h.) if it satisfies the difference equation

This equation can be considered as the direct analogue either of the differential equation

or of the integral equation

in the notation normally employed to harmonic functions.

This note is concerned with the r-dimensional torus Tr, whose points x are r-tuples (x1, x2, …, xr), where the xi are not numbers but residue-classes modulo 1. The addition of two elements of Tr follows the vector law

The local cohomology theory introduced by Grothendieck(1) is a useful tool for attacking problems in commutative algebra and algebraic geometry. Let A denote a local ring with its unique maximal ideal m. For an ideal I ⊂ A and a finitely generated A-module M we consider the local cohomology modules HiI (M), i є ℤ, of M with respect to I, see Grothendieck(1) for the definition. In particular, the vanishing resp. non-vanishing of the local cohomology modules is of a special interest. For more subtle considerations it is necessary to study the cohomological annihilators, i.e. aiI(M): = AnnΔHiI(M), iєℤ. In the case of the maximal ideal I = m these ideals were used by Roberts (6) to prove the ‘New Intersection Theorem’ for local rings of prime characteristic. Furthermore, we used this notion (7) in order to show the amiability of local rings possessing a dualizing complex. Note that the amiability of a system of parameters is the key step for Hochster's construction of big Cohen-Macaulay modules for local rings of prime characteristic, see Hochster(3) and (4).

The problem of estimating the entry length over which the laminar motion of an electrically conducting, viscous fluid in a pipe under a transverse field adjusts to a steady state from some initial state is linearized by a Rayleigh approximation adapted for cases where there is a pressure gradient. Solutions are found which express the difference between the initial and ultimate velocity profiles as a series of orthogonal, exponentially decaying terms. Approximate entry lengths for the most persistent terms are calculated for flow between two infinite parallel planes and in rectangular tubes when magnetic effects are present. Entry lengths in circular and rectangular pipes and between infinite planes are calculated in the case where magnetic effects are negligible to permit a comparison of the results with those derived by other methods. Pressure losses at entry are considered in the simpler cases.

Isotropic ionospheric propagation of electromagnetic waves yields refractive index profiles via the standard transcendental equations. Some transformations of these equations have long been regarded as standard. Here, every possible transformation is derived by a systematic analysis, yielding all possible refractive index profiles satisfying certain stated criteria regarding the wave frequency.

We use the term ‘measurement’ to refer to the interaction between an object and an apparatus on the basis of which information concerning the initial state of the object may be obtained from information on the resulting state of the apparatus. The quantum theory of measurement is a quantum theoretic investigation of such interactions in order to analyse the correlations between object and apparatus that measurement must establish. Although there is a sizeable literature on quantum measurements there appear to be just two sorts of interactions that have been employed. There are the ‘disturbing’ interactions consistent with the analysis of Landau and Peierls (8) as developed by Pauli (11) and by Landau and Lifshitz (7), and there are the ‘non-disturbing’ interactions explicitly set out by von Neumann ((10), chs. 5, 6), and that dominate the literature. In this paper we shall investigate the most general types of interactions that could possibly constitute measurements and provide a precise mathematical characterization (section 2). We shall then examine an interesting subclass, corresponding to Landau's ideas, that contains both of the above sorts of measurements (section 3). Finally, we shall discuss von Neumann measurements explicitly and explore the purported limitations suggested by Wigner(12) and Araki and Yanase (2). We hope, in this way, to provide a comprehensive basis for discussions of quantum measurements.

Recall that the first homology group H1(G) of a group G is the derived quotient G/[G, G]. The first homology groups of the mapping class groups of closed orientable surfaces are well known. Let F be a closed orientable surface of genus g. Recall that the extended mapping class group [Mscr ]*F of the surface F is the group of the isotopy classes of self-homeomorphisms of F. The mapping class group [Mscr ]F of F is the subgroup of [Mscr ]*F consisting of the isotopy classes of orientation-preserving self-homeomorphisms of F. It is well known that [Mscr ]F is trivial if F is a sphere. Hence the first homology group of the mapping class group of a sphere is trivial. If the genus of F is at least three, then H1([Mscr ]F) is again trivial. This result is due to Powell [P]. The group H1([Mscr ]F) is Z10 if the genus of F is two, proved by Mumford [Mu], and Z12 if F is a torus. When a problem about orientable surfaces is solved, it is natural to ask the corresponding problem for nonorientable surfaces. This is our motivation for the present paper.

This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter −A.

There is an application to Penrose's binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane and the matrices are determined by a type of spinor transport which generalizes to links in any 3-manifold.

A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold.

Let I, I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp be monomial ideals of a polynomial ring R = K[X1,. . ., Xr] and Ln = I+∩jIn1j + ⋅ ⋅ ⋅ + ∩jIpjn. It is shown that the ai-invariant ai(R/Ln) is asymptotically a quasi-linear function of n for all n ≫ 0, and the limit limn→∞ad(R/Ln)/n exists, where d = dim(R/L1). A similar result holds if I11,. . ., I1q1,. . ., Ip1,. . ., Ipqp are replaced by their integral closures. Moreover all limits also exist.

As a consequence, it is shown that there are integers p > 0 and 0 ≤ e ≤ d = dim R/I such that reg(In) = pn + e for all n ≫ 0 and pn ≤ reg(In) ≤ pn + d for all n > 0 and that the asymptotic behavior of the Castelnuovo–Mumford regularity of ordinary symbolic powers of a square-free monomial ideal is very close to a linear function.

It is generally conjectured that if α1, α2 …, αk are algebraic numbers for which no equation of the form

is satisfied with rational ri not all zero, and if K > 1 + l/k, then there are only finitely many sets of integers p1, p2, …, pkq, q > 0, such that

This result would be best possible, for it is well known that (1) has infinitely many solutions when K = 1 + 1/k. † If α1, α2, …, αk are elements of an algebraic number field of degree k + 1 the result can be deduced easily (see Perron (11)). The famous theorem of Roth (13) asserts the truth of the conjecture in the case k = 1 and this implies that for any positive integer k, (1) certainly has only finitely many solutions if K > 2. Nothing further in this direction however has hitherto been proved.‡

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

Throughout this paper p denotes an odd prime. The groups we shall consider are central extensions of a cyclic subgroup by Cp ⊕ Cp, and may be presented as

The group P(n) is defined for each n ≥ 3, and has order pn. The group P(3) is the non-abelian group of order p3 and exponent p. The mod-p cohomology rings of the other groups of order p3 have been known for some time (see [14] or [5] for that of the non-abelian metacyclic group), and so this paper completes the calculation of the mod-p cohomology rings of the groups of order p3.

Wishart and Hirschfeld have considered the following problem (1). Given n points in order on a line, suppose that they may be ‘black’ or ‘white’ independently with probability p and q = 1 − p. What is the probability distribution of the number of black-white joins? They give an exact solution to this problem and prove that it tends to normality as n increases. This result is of interest in several branches of science (Mood (2)). In the present note we consider the analogous problem in more than one dimension. This has applications in physics and agriculture (compare, Ising (3), Cochran (4) and Todd (5)). Another problem of similar type occurs when we consider the number of black points to be fixed and their arrangement to be random, but we do not consider this here. A problem of arrangements of similar type has been considered by Onsager.

It is shown how to convert the integral equation for the pressure distribution on a circular disc in an axisymmetric sound wave into a singular integral equation of the first kind. This singular integral equation has a simple kernel and can be transformed easily into an integral equation of the second kind which is especially useful at high frequencies. Although a direct iteration on this last integral equation fails, an indirect method is devised which ensures that each iterate is of lower order than the preceding at high frequencies. The form of the general term in the iteration is given together with the first terms of its asymptotic behaviour. It is relatively simple to estimate the error caused by stopping at any particular iterate.

Detailed calculations are made for a plane wave striking the disc at normal incidence.

A very general exact solution of Einstein's equation Rik = σξiξk, is given in terms of the Kerr-Schild metric, gik = ηij + Hξiξk, where ξi is a shearfree geodetic null congruence. The Kerr-Schild solution for Rik = 0 is derived as a particular case.