The effect of giving an infinite plate, immersed in a compressible fluid, an impulsive velocity U in its own plane is examined on the assumption that M = U/a0 ≫ 1 where a0 is the velocity of sound in the undisturbed fluid. To begin with the flow field is very complex but very soon after the motion has begun a shock wave moves at right angles to the plate gradually weakening and eventually degenerating into a sound wave. Attention is concentrated on the flow field when there is a distinct strong shock wave, it being assumed, and justified a posteriori, that the flow behind it may be divided into an inviscid region and a boundary layer region. The equation of the shock wave and a complete description of the flow behind it is found when log M, where ν0 is the coefficient of kinematic viscosity of the undisturbed air. An approximate method is also given to enable the solution to be joined up to Van Dyke's (10) which is valid when .

In this paper we shall establish some forms of the closed graph theorem for locally convex spaces, using the approach of Pták(17). Our interest is in classifying pairs of locally convex spaces (E, F) which have the property that every closed graph linear mapping T: E → F is continuous; if (E, F) has this property then we shall say that (E, F) is in the class ℒ if is a particular class of locally convex spaces then ℒ() is the class of all E such that (E, F)∈ℒ for all F ∈ .

1. Introduction. Suppose that we wish to make inferences about an unknown real parameter θ, and that we have a random sample of observations (Xl, X2, …, Xn) from a distribution which depends on θ.

A topological ordered space (X, , <) is a set X endowed with both a topology and a partial order <, and is usually denoted by (X, ), it being understood that the symbol ≤ is used to denote all partial orders.

In the study of integral solutions to Diophantine equations, many problems can be reduced to that of solving the equation

in S-units of the given ring. To accomplish this over number fields, the only known effective method is to use Baker's deep results on linear forms in logarithms, which yield relatively weak upper bounds. For function fields, R. C. Mason [2] has recently given a remarkably strong effective upper bound. In this note we give an independent proof of Mason's bound, relying only on elementary algebraic geometry, principally the Riemann-Hurwitz formula.

In this paper we study Ulrich ideals of and Ulrich modules over Cohen--Macaulay local rings from various points of view. We determine the structure of minimal free resolutions of Ulrich modules and their associated graded modules, and classify Ulrich ideals of numerical semigroup rings and rings of finite CM-representation type.

In the tangle model for DNA site-specific recombination, one is required to solve simultaneous equations for unknown tangles which are summands of observed DNA knots and links. For 0[les ]i[les ]3, given fixed 4-plats Ki where the set {K1, K2, K3} contains at least two distinct 4-plats, let O and R denote unknown 2-string tangles such that {O, R} are the variables in the system of four tangle equations N(O+iR)=Ki, where N is the numerator construction, and nR denotes the tangle sum of n copies of R. Then there is at most one simultaneous solution {O, R} and this solution must be of the form R an integral tangle and O either a rational tangle or the sum of two rational tangles. In addition, if there exists a solution, then at least one of the 4-plats is chiral. We exhibit an algorithm for solving simultaneous tangle equations of this form.

A general theory of piecewise multiharmonic splines is constructed for a class of fractals (post-critically finite) that includes the familiar Sierpinski gasket, based on Kigami's theory of Laplacians on these fractals. The spline spaces are the analogues of the spaces of piecewise Cj polynomials of degree 2j + 1 on an interval, with nodes at dyadic rational points. We give explicit algorithms for effectively computing multiharmonic functions (solutions of Δj+1u = 0) and for constructing bases for the spline spaces (for general fractals we need to assume that j is odd), and also for computing inner products of these functions. This enables us to give a finite element method for the approximate solution of fractal differential equations. We give the analogue of Simpson's method for numerical integration on the Sierpinski gasket. We use splines to approximate functions vanishing on the boundary by functions vanishing in a neighbourhood of the boundary.

Let Q be a semi-regular local ring of dimension d, m be its maximal ideal, and q be an m-primary ideal. Then LQ(Q/qn+1), the length of Q-module Q/qn+1, is equal to the characteristic polynomial PQ(q,n) in n for a sufficiently large value of n:

where ei = ei(q), i = 0,1,2,…, d are integers uniquely determined by q, called normalized Hilbert coefficients of q according to (1). It was shown in (1) that e1(q) is a non-negative integer, and is equal to zero if and only if q is generated by a system of parameters. We shall prove, in this paper, that e2(q) is also a non-negative integer, and that this non-negativity is not necessarily true for other coefficients. We shall give an example with negative e3(q).

Dispersion relations are derived for the triangular, honeycomb and Kagomé lattices using theorems for the eigenvalues of block circulant matrices. Constant frequency contours are plotted for each of the spectral branches of the three lattices and the distribution functions for squared frequencies are sketched.

Existence and regularity results are obtained for the Cauchy problem for the Kadomtsev–Petviashvili–I equation by the inverse spectral method. The initial data are small Schwartz functions which are not assumed to satisfy the zero mass constraint ∫ℝdxq(x, y, 0)=0.

It is well known that a uniform spherical shell produces the same gravitational field outside itself as that of a concentric uniform solid sphere with the same total mass. A given external field may therefore be produced by a variety of mass systems even under the restriction that the bodies of the system should have a prescribed uniform density. It can be asked what further conditions on the system are sufficient to ensure its uniqueness. The problem is of interest in connexion with the geological interpretation of local anomalies in the earth's gravitational field and work has already been done on it by several authors, (3), (6), (7), (8), (9). The results proved so far have all been restricted to systems consisting of a single body and the purpose of the present paper is to extend one of these results to include more general mass systems. The example of the spherical shell, mentioned above, might suggest that the absence of cavities in the bodies is a sufficient additional condition to ensure uniqueness. This is not so however. Celmiņš (3) gives an example of two different mass systems which have the same external field, each system consisting of three bodies of the same uniform density without any cavities in them.

In a previous paper (afterwards referred to as Paper I) tests have been given for the significance of some quantities found statistically. The results are given in the form P(q | θh)/P (˜ q|θh); here h denotes the previous knowledge and θ the experimental evidence used, while q is the hypothesis that all the variations outstanding can be attributed to accidental error or random variation, and ˜q the hypothesis that at least part of them is systematic. It has been supposed in the analysis that q and ˜q are equally probable on the information h; but if they are not, the only alteration is that the ratios evaluated now represent

If successive batches of relevant information are available the total effect on the probability of q can therefore be got by multiplying the values of

given by the investigations separately. In each case the assumption that q has prior probability ½ is really a practical working rule rather than a statement of fact.

In this paper we advertise the category of Γ-spaces as a convenient framework for doing ‘algebra’ over ‘rings’ in stable homotopy theory. Γ-spaces were introduced by Segal [Se] who showed that they give rise to a homotopy category equivalent to the usual homotopy category of connective (i.e. (−1)-connected) spectra. Bousfield and Friedlander [BF] later provided model category structures for Γ-spaces. The study of ‘rings, modules and algebras’ based on Γ-spaces became possible when Lydakis [Ly] introduced a symmetric monoidal smash product with good homotopical properties. Here we develop model category structures for modules and algebras, set up (derived) smash products and associated spectral sequences and compare simplicial modules and algebras to their Eilenberg–MacLane spectra counterparts.

During the past several years, new types of geometric measure and dimension have been introduced; the packing measure and dimension, see [Su], [Tr] and [TT1]. These notions are playing an increasingly prevalent role in various aspects of dynamics and measure theory. Packing measure is a sort of dual of Hausdorff measure in that it is defined in terms of packings rather than coverings. However, in contrast to Hausdorff measure, the usual definition of packing measure requires two limiting procedures, first the construction of a premeasure and then a second standard limiting process to obtain the measure. This makes packing measure somewhat delicate to deal with. The question arises as to whether there is some simpler method for defining packing measure and dimension. In this paper, we find a basic limitation on this possibility. We do this by determining the descriptive set-theoretic complexity of the packing functions. Whereas the Hausdorff dimension function on the space of compact sets is Borel measurable, the packing dimension function is not. On the other hand, we show that the packing dimension functions are measurable with respect to the σ-algebra generated by the analytic sets. Thus, the usual sorts of measurability properties used in connection with Hausdorff measure, for example measures of sections and projections, remain true for packing measure.

In 1988, E. Witten proposed an array of invariants (quantum G-invariants) for a 3-manifold associated with a compact simple Lie group G based on the quantum field theory [22]. N. Reshetikhin and V. Turaev [20] gave a mathematical approach to these invariants for G = SU(2) by using representations of the quantum group Uq(sl(2, ℂ)). See also [2, 11, 12, 14, 15, 16] for other approaches to these invariants: [16] may be the simplest way (for knot theorists) to understand the existence of these invariants.

We establish necessary and sufficient conditions for the separability of a system of second-order differential equations into independent one-dimensional second-order equations. The characterization of this property is given in terms of geometrical objects which are directly related to the system and relatively easy to compute. The proof of the main theorem is constructive and thus yields a practical procedure for constructing coordinates in which the system decouples.

Let x = ℘′(u) and y = ½℘(u), where the Weierstrass ℘-function has invariants g2 = 4A, g3 = 4B and differentiation with respect to the parameter u is denoted by a dash, so that

1. The problem of determining the distribution of stress in the neighbourhood of a penny-shaped crack defined in terms of cylindrical coordinates (ρ, φ, z) by 0 ≤ ρ ≤ α, z = 0, has been considered by Sneddon ((2)) and Sack ((1)). In the latter paper the solution is derived only in the case in which the stress field is due to the application of constant pressure to the faces of the crack. In the former paper the analysis given applies to an axisymmetric distribution of pressure p(ρ) applied to both the upper and lower face of the penny-shaped cavity, but the calculation of the stress intensity factor

and of the energy W required to open up the crack is a complicated matter even in the case in which p(ρ) is a constant.