In an earlier paper (3) the author developed a theory of the neighbourhoods of a local domain, in which the concept of the first neighbourhood ring played a central role. This notion has since proved useful in connexion with certain one-dimensional problems, but it has emerged, in the process, that a considerable advantage would be gained if the theory could be freed from the assumption that the basic ring was to be without zero-divisors. This parallels the situation in the geometry of plane curves, where it is desirable, so far as is possible, that results and methods should apply with equal facility to reducible as well as to irreducible curves. Accordingly Part I of the present paper is devoted to a fresh account of the first neighbourhood ring from a more general standpoint than that used previously. Besides the greater generality thus obtained, the theory is extended by the addition of some new results. Furthermore, the proof of one of the main results of the original paper ((3), Theorem 10) has been considerably simplified. Of course, the ideas of (3) reappear here in a modified form, but, to spare the reader the irritating fragmentation of the subject which would otherwise be necessary, the revised account has been made independent of the earlier one. To this extent, Part I is self-contained.

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatM

and more generally,

R. M. W. Wood has proved a conjecture we made about dimensions of A generators of H*(RP∞ × … × RP∞). In this note we give some of the consequences of this conjecture.

Let Kτ be a framed knot in S3 representing the 4-manifold which is obtained by attaching a 2-handle onto B4 along K with the framing τ (see (1)). Tristram (2) proved the following non-imbedding theorem for the generator of ).

Let ${\rm q}$ be a Lie algebra and ξ a linear form on ${\rm q}$. Denote by ${\rm q}_{\xi}$ the set of all $x \in {\rm q}$ such that $\xi([{\rm q},x]) = 0$. In other words, ${\rm q}_{\xi}=\{x\in {\rm q} \mid({\rm ad}^*x)\cdot = 0\}$, where ${\rm ad}^*: {\rm q}\rightarrow {\rm g I}({\rm q}^*)$ is the coadjoint representation of ${\rm q}$. The index of ${\rm q}$, ind ${\rm q}$, is defined by \[ {\rm ind\, q} = \min_{\xi\in {\rm q}^*}\dim {\rm q}\xi. \]

In 1955 Noll (7) gave a definition of an anisotropic fluid. Noll's idea is developed in more detail and the results of applying the theory to some simple flows are compared with those obtained by Ericksen (1) from his theory of transversely isotropic fluids.

1. The formulae usually given for the scattering of electrons by atoms are calculated without taking into account the radiative forces. It has been suggested by the author in a former paper that in the case of the large-angle scattering of electrons with velocity comparable with that of light (for which case the scattering is entirely nuclear), these forces might influence the scattering by an appreciable amount, which would explain the divergence between the theoretical formula and the results of Chadwick and Mercier, which are some 40 per cent. greater. In this paper we shall show that for electrons–of any velocity the influence of these forces cannot be greater than 2 or 3 per cent., and so cannot be invoked to explain discrepancies between theory and experiment.

In this paper, as a suitable application of the well-known generalized maximum principle of Omori–Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson–Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

Bethe's method, used in the problem of order-disorder transitions in alloys, is applied formally to the problem of molecular rotations in solids. To apply this method, we assume that the solid is entirely homogeneous, and that the state of rotation is the same throughout the solid. Without this assumption, the application of this method is impossible.

A particular form of the mutual potential energy between two neighbouring molecules has been chosen, and classical statistics is employed throughout. The calculations are made entirely after the manner of Bethe, and the similarities of the two problems are pointed out. The result is that there is a critical temperature, and also a discontinuity in the specific heat of the magnitude of some ten times the Boltzmann constant per molecule, arising from the sudden setting in of the rotations as the temperature is increased beyond the critical point. Agreement with the experiments is bad, indicating that a more profound theory is necessary.

In many applied probability models, one is concerned with a sequence {Xn: n > 1} of independent random variables (r.v.'s) with a common distribution function (d.f.), F say. When making statistical inferences within such a model, one frequently must do so on the basis of observations X1, X2,…, XN where the sample size N is a r.v. For example, N might be the number of observations that it was possible to take within a given period of time or within a fixed cost of experimentation. In cases such as these it is not uncommon for statisticians to use fixed-sample-size techniques, even though the random sample size, N, is not independent of the sample. It is therefore important to investigate the operating characteristics of these techniques under random sample sizes. Much work has been done since 1952 on this problem for techniques based on the sum, X1 + … + XN (see, for example, the references in (3)). Also, for techniques based on max(X1, X2, …, XN), results have been obtained independently by Barndorff-Nielsen(2) and Lamperti(9).

We obtain weak and strong Gaussian approximations for logarithmic averages of indicators of normalized partial sums. The proofs are based on invariance principles for integrals of an Ornstein–Uhlenbeck process and on strong approximations of normalized partial sums by Orstein–Uhlenbeck processes.

We show that for arbitrary metric spaces X and Y the following dimension inequalities hold:

where ‘dim’ denotes Hausdorff dimension and ‘Dim’ denotes packing dimension. The main idea of the proof is to use modified constructions of the Hausdorff and packing measure to deduce appropriate inequalities for the measure of X × Y.

The effect of non-linearity on standing edge waves is studied on the basis of shallow water theory. Four problems are considered: the decay of free edge waves and the forcing of edge waves by an incident wave of double the frequency, a synchronous incident wave and by a side-wall wavemaker. Hysteresis effects are predicted for all types of forcing.

1. Introduction. In (1) we proved that the direct sum of a finite number of unique factorization rings is a unique factorization ring (UFR), and in particular that the direct sum of a finite number of unique factorization domains (UFD's) is a UFR. The converse, however, does not hold i.e. not every UFR can be expressed as a direct sum of UFD's. Here we investigate the structure of UFR's and show that every UFR is a finite direct sum of UFD's and of special UFR's. There is thus a relationship with the structure theorem for principal ideal rings ((2), p. 245).

In this paper we will study the homological properties of the enveloping algebra U = U (Sl2(ℂ)), with particular reference to the homological dimension of simple U-modules and the global dimension of the primitive factor rings of U.

The breakdown of stability in a porous homogeneous medium of a fluid subject to a vertical temperature gradient was examined by Lapwood (4). However, in natural conditions the fluids move generally in inhomogeneous media and then it is important to deal with the problem of stability in these media; the purpose of this paper is to consider just such a problem.

The velocity distribution in steady rectilinear plastic flow of a Bingham solid between coaxial circular cylinders in relative motion can be obtained exactly by simple arguments. On the basis of the general equations of plastic flow, an approximate velocity distribution is derived below for the case of eccentric circular cylinders as rigid boundaries and zero pressure gradient by making use of a conformal transformation. This approximation is valid only if a dimensionless parameter S (which is proportional to the yield value) is sufficiently small, and there is flow in the whole region between the cylinders. The differences in velocity distribution in the cases of Bingham solid (S finite) and Newtonian liquid (S zero) are illustrated by a particular example. It is shown that the effect of a finite yield value is to cause the contours of equal velocity in a normal section to become more nearly concentric with the inner boundary as the yield value increases from zero.

A typical Riesz product on the circle is the weak* limit

where – 1 ≤ rk ≤ 1, øk ∈ R, λT is Haar measure, and the positive integers nk satisfy nk+1/nk ≥ 3. A classical result of Zygmund (11) implies that either µ is absolutely continuous with respect to λT (when ) or µ is purely singular (when ).

Let I be an ideal of a commutative ring R and M an R-module. It is well known that the I-adic completion functor ΛI defined by ΛI(M) = lim←tM/ItM is an additive exact covariant functor on the category of finitely generated R-modules, provided R is Noetherian. Unfortunately, even if R is Noetherian, ΛI is neither left nor right exact on the category of all R-modules. Nevertheless, we can consider the sequence of left derived functors {LIi} of ΛI, in which LI0 is right exact, but in general LI0 ≠ ΛI. Therefore the computation of these functors is in general very difficult. For the case that R is a local Noetherian ring with the maximal ideal [mfr ] and I is generated by a R-regular sequence, Matlis proved in [9, 10] that

where D(−) = HomR(−; E(R/[mfr ])) is the Matlis dual functor, and that

In [18, 5] A.-M. Simon shows that LIo(M) = M and LIi(M) = 0 for i > 0, provided that M is complete with respect to the I-adic topology.

Later, Greenlees and May [3] using the homotopy colimit, or telescope, of the cochain of Koszul complexes to define so-called local homology groups of a module M (over a commutative ring R) by

where x is a finitely generated system of I. Then they showed, under some conditions on x which are satisfied when R is Noetherian, that HI[bull ](M) ≅ LI[bull ](M). Recently, Tarrío, López and Lipman [1] have presented a sheafified derived-category generalization of Greenlees–May results for a quasi-compact separated scheme. The purpose of this paper is to study, with elementary methods of homological and commutative algebra, local homology modules for the category of Artinian modules over Noetherian rings.

Suleῐmanova (3) describes a geometrical technique* for discussing the conditions under which a given set of n + 1 numbers can be a set of characteristic roots of a positive stochastic matrix of order n + 1. Suleῐmanova states that, using this technique, it is possible to prove that a set of n + 1 real numbers 1, λ1, λ2, …, λn, where | λi | < 1 for i = 1, 2, …, n, is a set of characteristic roots of a positive stochastic matrix provided that the sum of the moduli of the negative numbers of the set is less than unity. No indication of the method to be employed is given in (3), and it would seem that the problem is a difficult one. In the present paper I give a full discussion of how Suleῐmanova's technique may be applied to solve the problem for third-order matrices, and simply an indication of the solution for fourth-order matrices; and in, the latter case I point out that Suleῐmanova's result may be replaced by a stronger one. For second-order matrices the problem is almost trivial. As a preliminary I prove two general theorems.