The purpose of this note is to determine some unstable James numbers of Stiefel manifolds. We denote the real numbers by R, the complex numbers by C, and the quaternions by H. Let F be one of these fields with the usual norm, and d = dimRF. Let On, k = On, k(F) be the Stiefel manifold of all orthonormal k–frames in Fn, and q: On, k → Sdn−1 the bundle projection which associates with each frame its last vector. Then the James number O{n, k} = OF{n, k} is defined as the index of q* πdn−1(On, k) in πdn−1(Sdn−1). We already know when O{n, k} is 1 (cf. (1), (2), (3), (13), (33)), and also the value of OK{n, k} (cf. (1), (13), (15), (34)). In this note we shall consider the complex and quaternionic cases. For earlier work see (11), (17), (23), (27), (29), (31) and (32). In (27) we defined the stable James number , which was a divisor of O{n, k}. Following James we shall use the notations X{n, k}, Xs{n, k}, W{n, k} and Ws{n, k} instead of OH{n, k}, , Oc{n, k} and respectively. In (27) we noticed that O{n, k} = Os{n, k} if n ≥ 2k– 1, and determined Xs{n, k} for 1 ≤ k ≤ 4, and also Ws{n, k} for 1 ≤ k ≤ 8. On the other hand Sigrist (31) calculated W{n, k} for 1 ≤ k ≤ 4. He informed the author that W{6,4} was not 4 but 8. Since Ws{6,4} = 4 (cf. § 5 below) this yields that the unstable James number does not equal the stable one in general.

The mod two homology of MSpin, the Spin-cobordism Thom spectrum, has a rich algebraic structure. We will describe it explicitly as a comodule algebra, and give some applications to the ring structure of the Spin-cobordism ring.

An exact sequence resolving the Euler operator of the calculus of variations for partial differential polynomials in several dependent and independent variables is described. This resolution provides a solution to the ‘Inverse problem of the calculus of variations’ for systems of polynomial partial equations.

That problem consists of characterizing those systems of partial differential equations which arise as the Euler-Lagrange equations of some variational principle. It can be embedded in the more general problem of finding a resolution of the Euler operator. In (3), hereafter referred to as I, a solution of this problem was given for the case of one independent and one dependent variable. Here we generalize this resolution to several independent and dependent variables simultaneously. The methods employed are similar in spirit to the algebraic techniques associated with the Gelfand-Dikii transform in I, although are considerably complicated by the appearance of several variables. In particular, a simple algebraic proof of the local exactness of a complex considered by Takens(5), Vinogradov(6), Anderson and Duchamp(1), and others appears as part of the resolution considered here.

Let m ≥ 2 be an integer, and π an irreducible unitary cuspidal representation for GLm(), whose attached automorphic L-function is denoted by L(s, π). Let {λπ(n)}n=1∞ be the sequence of coefficients in the Dirichlet series expression of L(s, π) in the half-plane ℜs > 1. It is proved in this paper that, if π is such that the sequence {λπ(n)}n=1∞ is real, then the first sign change in the sequence {λπ(n)}n=1∞ occurs at some n ≪ Qπ1 + ϵ, where Qπ is the conductor of π, and the implied constant depends only on m and ϵ. This improves the previous bound with the above exponent 1 + ϵ replaced by m/2 + ϵ. A result of the same quality is also established for {Λ(n)aπ(n)}n=1∞, the sequence of coefficients in the Dirichlet series expression of −(L′/L)(s, π) in the half-plane ℜs > 1.

We introduce an associative glueing operation ⊕q on the space of solutions of the Quantum Yang–Baxter Equations of Hecke type. The corresponding glueing operations for the associated quantum groups and quantum vector spaces are also found. The former involves 2×2 quantum matrices whose entries are themselves square or rectangular quantum matrices. The corresponding glueing operation for link-invariants is introduced and involves a state-sum model with Boltzmann weights determined by the link invariants to be glued. The standard su(n) solution, its associated quantum matrix group, quantum space and link-invariant arise at once by repeated glueing of the one-dimensional case.

In this paper we consider the problem of dividing a square of side N into squares of integral side. When we demand that these be unequal, the problem is soluble only for certain N, and has been treated in some detail by various authors (1,3,4,5). If we relax this condition, every side is permissible, for a square of side N may plainly be divided into N2 squares of side 1. It is the object of this paper to reduce this number: in fact we shall show that a square of side N is divisible into some number less than 6 ∛N + 1 squares of integral side.

In papers I(1) and II(2), the author has investigated the Stokes phenomenon associated with the arbitrary constants that occur in the asymptotic expressions of solutions of the nth-order differential equation

and has provided rules for tracing a given linear combination of n such asymptotic expressions across certain defined lines, designated as Stokes lines, anti-Stokes lines and branch-cuts. The object of these investigations is to apply the results to the approximate solution of more complicated linear differential equations of the nth order, and to this end it is an advantage to develop the theory using matrix techniques. In the present paper, the equations are recast into a special matrix form, and it is shown how the results obtained in II may be used in a systematic manner.

We prove the following result: if a Banach space E does not contain l1 and F has the (RDPP), then E ⊗nF has the same property, provided that L(E, F*) = K(E, F*). Hence we prove that if E ⊗n F has the (RDPP) then at least one of the spaces E and F must not contain l1. Some corollaries are then presented as well as results concerning the necessity of the hypothesis L(E, F*) = K(E, F*).

For any topological space X and integer n ≥ 1, denote by Cn(X) the configuration space

The symmetric group Sn acts by permuting coordinates on Cn(X) and we are concerned in this note with the induced graded representation of Sn on the cohomology space H*(Cn(X)) = ⊕iHi (Cn(X), ℂ), where Hi denotes (singular or de Rham) cohomology. When X = ℂ, Cn(X) is a K(π, 1) space, where π is the n-string pure braid group (cf. [3]). The corresponding representation of Sn in this case was determined in [5], using the fact that Cn(C) is a hyperplane complement and a presentation of its cohomology ring appears in [1] and in a more general setting, in [8] (see also [2]).

The work of which this paper is an account began as a study of differential equations for functions whose values are random variables of finite variance. It was intended that all questions of convergence should be treated from the standpoint of strong convergence in Hilbert space—familiar to probabilists from the writings of Karhunen(11) and Loève(13) as mean-square convergence. The more general Banach-space approach now adopted was made possible by the discovery of a theorem (Theorem 1 of this paper) which Mr D. G. Kendall, its apparent author, kindly communicated to us.

This paper is a contribution to the study of set functions which, although additive, fail to be measures by not being countably additive. It is motivated by the suggestion (to be found, for instance, in (2), section 2.3) that such functions might be used instead of measures in the theory of probability, and by speculation about the effect which the adoption of such a suggestion would have on the theory of stochastic processes.

Amongst the various modifications of the general theory of relativity (GRT), the scalar-tensor theory of Brans and Dicke (BDT) is treated most seriously ([2], [17], [13]). The BDT is consistent with observations as long as the coupling parameter ω between the scalar and tensor components of gravitation is about equal to or greater than 500 [18]. However, there are no a priori theoretical reasons for excluding other values of ω. In the limit ω → ∞, the BDT reduces to the GRT for a constant BDT-scalar field Φ

A Haken manifold is a compact, orientable, irreducible 3-manifold which contains a properly embedded 2-sided, incompressible surface of positive genus. These manifolds are important in connection with the work of Haken, Waldhausen and the more recent work of Thurston (8). Thus it is interesting to investigate criteria for testing incompressible surfaces on 3-manifolds.

Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.

The type of integral considered, is of frequent occurrence in problems in the kinetic theory of gases and in particular is required in the paper which follows, in calculating viscosity. The result is an extension of one given by Berger, and the analysis follows closely the discussion of Gauss's formula for approximate quadrature in Whittaker and Robinson, The calculus of observations; the novelty lies in the use of Sonine polynomials, which are peculiarly well suited for the discussion of this type of integral.

The ‘Radon measures’ of N. Bourbaki(1) enjoy many striking properties. Among the most important of these is the ‘strong Radon-Nikodým theorem’ that the dual of L1-(μ) can always be identified with L∞(μ) ((1), chap. 5, §5, no. 8, theorem 4). As this is certainly not true of non-σ-finite measures in general, it is natural to ask what are the special properties on which it relies.

The main purposes of this paper are to establish and exploit the result that, over a complete (Noetherian) local ring R of prime characteristic for which the Frobenius homomorphism f is finite, the appropriate restrictions of the Matlis-duality functor provide an equivalence between the category of left modules over the Frobenius skew polynomial ring R[x, f] that are Artinian as R-modules and the category of right R[x, f]-modules that are Noetherian as R-modules.

The general theory of surfaces has been developed without much attention to the simpler aspects of the subject and, in particular, to the problem of determining some elementary criteria of regularity or referability; there is, in fact, only one well-known criterion of this kind, due to Castelnuovo and Enriques. (Theorem 2 of the present work.) It seems desirable, therefore, to establish some tests which require no more knowledge of a surface than its order, sectional genus and normal space. A number of such tests is given in the present paper. Before it was written the author had the advantage of consulting an unpublished manuscript by Professor Comessatti on the classification of surfaces in S4; in this work the regularity of surfaces of order eight is considered and, in part, those of order nine; the four-dimensional case of Theorem 11 and a particular case of Theorem 6 and Theorem 14 are given. Theorems 4 and 5 are also proved by Comessatti. For permission to publish these results and for the help derived from them the author's grateful thanks are due.

It has been found possible to generalize Ising's (4) treatment of the linear chain with nearest-neighbour interaction, and his main result, that such interaction cannot lead to cooperative effects, has been shown to hold for very many, if not all, types of lattice in three dimensions. Reasons are given for presuming a similar result for the second nearest-neighbour model. It is found, however, that even a small interaction of the long-range type can lead to cooperative effects at high temperatures, provided that the nearest-neighbour interaction is sufficiently strong. It is therefore claimed that explicit account must be taken of both types of interaction in any acceptable theory, and a possible method of estimating their relative importance is indicated.

Within the limitations of the small-deflexion plate theory, complex variable methods are used in this paper to obtain an exact solution for the problem of a thin circular plate supported at several interior or boundary points, and subjected to a certain normal loading spread over the area of an eccentric circle, the boundary of the plate being free. The load considered includes as a special case a linearly varying load over the circle and, as the radius of the loaded circle tends to zero, this load can be made to tend to a couple nucleus at its centre. As limiting cases the procedure adopted provides us with solutions appropriate to a circular plate, an infinitely large plate and a half-plane having free boundaries and acted upon by any normal system of concentrated forces and concentrated couples in equilibrium. Formulae for the moments, shears and deflexions relating to special examples are worked out in detail.