Let τ be the formally self-adjoint differential operator denned by

where the pr(x) are real-valued, , and p0(x) > 0. Then τ determines a real symmetric linear operator T0, given by T0f = τf, whose domain D(T0) consists of those functions f in the complex space L2(0, ∞) which have compact support and 2n continuous derivatives in (0, ∞) and vanish in some right neighbourhood of x = 0 ((7), p. 27–8). Since D(T0) is dense in L2(0, ∞), T0 has a self-adjoint extension T. We denote by μ the least limit point of the spectrum of T. The operator T may not be unique, but all such T have the same essential spectrum ((7), p. 28) and therefore μ does not depend on the choice of T.

In this paper finite type invariants (also known as Vassiliev invariants) of pure braids are considered from a group-theoretic point of view. New results include a construction of a universal invariant with integer coefficients based on the Magnus expansion of a free group and a calculation of numbers of independent invariants of each type for all pure braid groups.

The conjugate g of a periodic and integrable function f is not necessarily integrable, even when f is monotone inside its fundamental interval. Paley and Wiener, however, proved that g is integrable if f is monotone and odd. A simpler proof was given later by Zygmund‡.

Let D be an integrally closed local Noetherian domain of Krull dimension 2, and let f be a nonzero element of D such that fD has prime radical. We consider when an integrally closed ring H between D and Df is determined locally by finitely many valuation overrings of D. We show such a local determination is equivalent to a statement about the exceptional prime divisors of normalized blow-ups of D and, when D is analytically normal, this property holds for D if and only if it holds for the completion of D. This latter fact, along with MacLane's notion of key polynomials, allows us to prove that in some central cases where D is a regular local ring and f is a regular parameter of D, then H is determined locally by a single valuation. As a consequence, we show that if H is also the integral closure of a finitely generated D-algebra, then the exceptional prime ideals of the extension H/D are comaximal. Geometrically, this translates into a statement about intersections of irreducible components in the closed fiber of the normalization of a proper birational morphism.

1. Introduction. This Department has made extensive observations with a proton magnetometer towed behind a ship (Hill (2)). The magnetometer consists of a bottle of water surrounded by a coil of wire through which a current flows and produces a polarizing field of about 100 oersted. A small proportion of the protons in the water are polarized along the resultant field. When the current is turned off they precess about the Earth's field and produce in the coil an e.m.f. whose frequency is a measure of the field.

An account is given of the seiche motion in a curved lake. It is shown that the periods of the first two modes of oscillation are effectively unchanged in comparison with those for a straight lake having a length equal to that of the median line along the curved lake. Near to the nodal positions, the surface elevation is uniform across the lake. In the neighbourhood of the anti-nodes, which includes the ends of the lake, the wave amplitude is shown to be a maximum at the outside of the bend.

In his well-known treatise, Banach mentions that there exist complete normed bi-orthogonal systems in any finite-dimensional (real) Banach space—a result he attributes to H. Auerbach ((1), p. 238). The purpose of this note is to present a proof of this result (valid for complex as well as real Banach spaces), and to give some applications to the theory of tensor products of Banach spaces.

We describe the formal framework for cellularization functors in triangulated categories and study the preservation of ring and module structures under these functors in stable homotopy categories in the sense of Hovey, Palmieri and Strickland, such as the homotopy category of spectra or the derived category of a commutative ring. We prove that cellularization functors preserve modules over connective rings but they do not preserve rings in general (even if the ring is connective or the cellularization functor is triangulated). As an application of these results, we describe the cellularizations of Eilenberg–Mac Lane spectra and compute all acyclizations in the sense of Bousfield of the integral Eilenberg–Mac Lane spectrum.

We prove that a weak equivalence between two cofibrant (colored) props in chain complexes induces a Dwyer–Kan equivalence between the simplicial localisations of the associated categories of algebras. This homotopy invariance under base change implies that the homotopy category of homotopy algebras over a prop P does not depend on the choice of a cofibrant resolution of P, and gives thus a coherence to the notion of algebra up to homotopy in this setting. The result is established more generally for algebras in combinatorial monoidal dg categories.

Let P(n) = [x1, . . ., xn] = ⊕d≥0Pd(n) be the polynomial algebra viewed as a graded left module over the Steenrod algebra at the prime 2. The grading is by the degree of the homogeneous polynomials Pd(n) of degree d in the n variables x1, . . ., xn. The algebra P(n) realizes the cohomology of the product of n copies of infinite real projective space. We recall that a homogeneous element f of grading d in a graded left -module M is hit if there is a finite sum f = ΣiSqi(hi), called a hit equation, where the pre-images hi ∈ M have grading strictly less than d and the Sqi, called the Steenrod squares, generate . One of the important parts of the hit problem is to check whether a given polynomial in M is hit or not. In this article we study this problem in the 3-variable case.

The main aim of the present note is to evaluate certain definite integrals involving Meijer's G-function. The results proved are of general character and include as particular cases some of the results given earlier by MacRobert ((3), p. 81). For the definitions and properties of the G-functions see (1), §§5·3 and 5·3·1.

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined by

where ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.

The non-linear graviton construction of Penrose (9) and the ℋ-space construction of Newman (6) are two complementary techniques for constructing complex four dimensional space-times with quadratic metric and anti-self-dual curvature tensor.

In the former, the space-time is the space of holomorphic sections of a complex fibre space obtained by deforming part of flat twistor space. In the latter the space-time is the space of regular solutions of a differential equation, the good cut equation.

Pathologies arise in the non-linear graviton construction when the normal bundle of a holomorphic section changes. This is reflected in the ℋ-space construction by a change in the character of the solutions of the linearized good cut equation, the Newman equation.

We prove a new structure theorem which we call the Countable Layer Theorem. It says that for any compact group G we can construct a countable descending sequence G = Ω0(G) ⊇ … ⊇ Ωn(G) … of closed characteristic subgroups of G with two important properties, namely, that their intersection ∩∞n=1 Ωn(G) is Z0(G0), the identity component of the center of the identity component G0 of G, and that each quotient group Ωn−1(G)/Ωn(G), is a cartesian product of compact simple groups (that is, compact groups having no normal subgroups other than the singleton and the whole group).

In the special case that G is totally disconnected (that is, profinite) the intersection of the sequence is trivial. Thus, even in the case that G is profinite, our theorem sharpens a theorem of Varopoulos [8], who showed in 1964 that each profinite group contains a descending sequence of closed subgroups, each normal in the preceding one, such that each quotient group is a product of finite simple groups. Our construction is functorial in a sense we will make clear in Section 1.

Let ∑un be a convergent infinite series which is not summable in finite form. In principle its sum can be found, to within any preassigned error ε, by adding numerically a sufficient number of terms; but if the series is slowly convergent, the ‘sufficient number’ of terms may be prohibitively large. A plan to deal with this case is to separate the series into a ‘main part’ u0+u1+ … +un−1 and a ‘remainder’ Rn = un+un+1+…; the main part is evaluated by direct summation, while the remainder is transformed analytically into a series which is more rapidly ‘convergent’, in the practical sense, and so evaluated. For example, the Euler-Maclaurin sum-formula gives such a transformation. It commonly happens that the new form of the remainder Rn is a divergent series, but that it represents Rn asymptotically as n ˜ ∞. It is for this reason that the transformation is applied to Rn instead of to the whole series; for practical use we have to choose n sufficiently large for the error inherent in the use of the asymptotic series to be below the preassigned bound ε.

The present paper is a companion to one which appeared in these Proceedings some months ago. Since that time the characteristics of the Hyper-Octohedral group of order 2n.n! have been calculated by Dr Alfred Young, who has kindly allowed me to make use of his results.

Throughout this paper k will be an algebraically closed field, R the ring k[[X1, …, XN]] and (kN, 0) the k-scheme Spec(R).

The expansion of a general function in a series of squares of Bessel functions has been considered by Neumann. As, however, there appear to be only a fairly small number of known expansions of this type, it is perhaps worth while adding to their number. I give here some new expansions of this type, and deduce some simple inequalities satisfied by Bessel coefficients of low order.

Several recent papers have been concerned with a group G, of order 28.36.5.7, which can be represented as a collineation group in five dimensions. This collineation group is generated by harmonic inversions (projections) which leave fixed a point (the vertex) and a prime (said to be conjugate to the vertex). There are 126 projections in G and the set of 126 vertices form a configuration which is described in a paper (Hamill (3)) in which the operations of G are expressed as products of at most six projections. The group leaves invariant six algebraically independent primals; these have been determined (Todd (6)), and the equations of the simplest are given. The simplest invariant is a sextic primal, and some properties of this have been recorded in an earlier paper (Hartley (4)).

Theorem. if a ≥ 2, b ≥ 1, and a + b ≥ ℵ0, then every collection of more than ab sets each of cardinal not exceeding b contains a subcollection of more than ab sets every two of which have the same intersection.