The problem of the diffraction of axisymmetric harmonic elastic waves by a rigid circular disc is reduced to finding the solution of a pair of integral equations of the second kind suitable for iteration at low frequencies. Using the principle of contraction mapping, the rate of convergence of the iteration procedure is discussed and the error caused by stopping at any particular stage estimated. Detailed calculations are given for plane compressional waves at normal incidence.

J. Dean introduced the notion of primitive/Seifert-fibered knots and constructions to explain how Dehn surgeries on knots produce small Seifert fibered manifolds. We determine non-hyperbolic, primitive/Seifert-fibered knots, and show that for such knots any integral, small Seifert fibered surgery arises from a primitive/Seifert-fibered construction. We also show that any 2-orbifold $S^2(n_1, n_2, n_3)$ with a pair of indices coprime becomes the base orbifold of a Seifert fibered manifold obtained by Dehn surgery on a hyperbolic, primitive/Seifert-fibered knot in $S^3$.

A contact problem is called tangential when arbitrary tangential displacements are prescribed over a part of the boundary of a transversely isotropic layer, while the tangential stress is zero over the rest of the boundary; the normal stress vanishes all over the boundary. The Generalized Images method is used to give a complete elementary solution to the problem. A new governing integral equation is derived. A particular case of a circular domain of contact is studied in detail. The governing integral equation can be inverted in this case. Approximate formulae are derived for the resultant force and the torque. A direct relationship is established between the integral transform method and the Generalized Images method. A limiting case of general solution gives the solution for an isotropic layer.

We study the pointwise behavior of Birkhoff sums $S_n\phi(x)$ on subshifts of finite type for Hölder continuous functions $\phi$. In particular, we show that for a given equilibrium state $\mu$ associated to a Hölder continuous potential, there are points $x$ such that $S_n\phi(x) - n \mathbb{E}_\mu \phi \sim a n^\beta$ for any $a>0$ and $0< \beta <1$. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of $\mu$ and it is attained by some maximizing equilibrium state $\nu$ such that $\mathbb{E}_\nu \phi = \mathbb{E}_\mu \phi$. On such points the ergodic average $n^{-1}S_n\phi(x)$ converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different choices $(a, \beta)$, are distinct but have the same dimension. This reveals a rich multifractal structure of the symbolic dynamics. As a consequence, we prove that the set of uniform recurrent points, which are close to periodic points, has full dimension. Applications are also given to the study of syndetic numbers, Hardy–Weierstraß functions and lacunary Taylor series.

Let g0, gl, g2,…be a sequence of mutually independent, normally distributed random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of maxima (minima) of the random algebraic curves with the equations

This average is (½(3½ + 1)) log N + O((log N)⅔ (log log N)½), when N is large.

The purpose of this paper and the next is to demonstrate that the ‘perfect Riesz spaces’ of (1) are an effective abstraction of the ‘espaces de Köthe’ of (2). I shall follow the ideas of (1), with certain changes in notation:

If L is a Riesz space and x, y ∈ L, let us denote sup (x, y) by x ∧ y and inf (x, y) by x ∧ y. I shall use the convenient if informal notation xr↓ ((1), section 16·1) and shall in this usage assume that 0 ∈ {r} and that x0 ≥ xτ for all τ. A set A ⊆ L is solid if x ∈ A and |y| ≤ |x| together imply that y ∈ A; A is then balanced. The solid hull of A is the set {y: ∃ x ∈ A, |y| ≤ |x|}; this is the smallest solid set containing A. An ‘ideal’ ((1), section 17) is then a solid subspace.

A mathematical analogue between a special class of problems in steady magnetohydrodynamics and appropriate problems in the theory of the scattering of sound pulses is established and exploited. Rigorous asymptotic expansions for large Hartmann number M are obtained for the exact solution of the prototype problem, the circular cylinder. Asymptotic expansions for large M in the case of a cylinder whose cross-section is an arbitrary closed convex curve are obtained as a straightforward application of the theory of geometrical optics and Keller's geometrical theory of diffraction.

1. Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, and

where γ is Euler's constant. Thus P(x) is the error term in the problem of the lattice points of a circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of a rectangular hyperbola.

The penetrating component of cosmic radiation is now believed to consist of mesons—particles with a mass about one-tenth of that of a proton—while the soft component is attributed to electrons. By collisions with atomic electrons a fast meson occasionally produces fast secondary electrons, “knock-on” electrons, thus giving rise to a certain amount of the soft component. It was first suggested by Bhabha(1) that fast electrons produced in this way may be responsible for the electron showers known to be associated with the penetrating particles. In the present note we calculate the ratio, which we denote by R, of the average number of electrons which accompany a meson due to the knock-on process, making use of the simplification which results from the fact that the rate of loss of energy by fast electrons in ionization and excitation and also the emission of soft quanta are nearly independent of their energy. In virtue of this, the total length of electron track, and therefore the magnitude of R, is only slightly affected by the cascade process of shower production, a process which causes the energy of a fast electron to be shared among a number of electrons. The difficulties associated with a treatment of the cascade process may therefore be avoided in calculating R.

Configurations of points in higher space have been known and studied for some time. In what follows we shall classify those configurations of points lying on the hyper-sphere in [4] which may be said to possess the group property. In this classification we shall seek to enumerate only those configurations which are essentially different from this point of view, in order to bring out any geometrical differences which might throw light on the fundamental problems of the theory of linear groups.

We show that under certain general conditions, short sums of ℓ-adic trace functions over finite fields follow a normal distribution asymptotically when the origin varies, generalising results of Erdős–Davenport, Mak–Zaharescu and Lamzouri. In particular, this applies to exponential sums arising from Fourier transforms such as Kloosterman sums or Birch sums, as we can deduce from the works of Katz. By approximating the moments of traces of random matrices in monodromy groups, a quantitative version can be given as in Lamzouri's article, exhibiting a different phenomenon than the averaging from the central limit theorem.

This paper continues the investigation of the hit problem, started in [5], for the algebra of symmetric polynomials B(n) viewed as a left ${\cal A}$-module graded by degree, where ${\cal A}$ denotes the Steenrod algebra over the field of two elements ${\bb F}_2$. We recall that a homogeneous element f of grading d in a graded left ${\cal A}$-module M is hit if there is a hit equation in the form of a finite sum $f=\sum_{k>0}Sq^{k}(h_k)$, where the homogeneous elements hk in M have grading less than d and the Sqk are the Steenrod squares, which generate ${\cal A}$. We denote by Q = Q(M) = ${\bb F}_2\otimes_{\cal A}$M the quotient of the module M by the hit elements, where ${\bb F}_2$ is here viewed as a right ${\cal A}$-module concentrated in grading 0. Then Q is a graded vector space over ${\bb F}_2$ and a basis for Q lifts to a minimal generating set for M as a module over ${\cal A}$. The hit problem is to find minimal generating sets for M and criteria for elements to be hit. We recall that B(n) = ${\bb F}_{2}\[\sigma_1,\ldots,\sigma_n\]$ is the polynomial subalgebra of P(n) = ${\bb F}_{2}\[x_1,\ldots,x_n\]$ generated by the elementary symmetric functions $\sigma_i$ in the variables xj . In particular, $\sigma_n=x_1\cdots x_n$. The algebras P(n) and B(n) realize respectively the cohomology of the product of n copies of infinite real projective space and the cohomology of the classifying space BO(n) of the orthogonal group O(n) over ${\bb F}_2$, where the usual grading in cohomology corresponds to degree in the polynomial algebra. The ideal M(n) in B(n), generated by $\sigma_n$, can be identified with the cohomology H*(MO(n), ${\bb F}_2$) of the Thom space MO(n) in positive dimensions. It is also convenient to introduce the notation L(n) for the polynomials in P(n) divisible by $\sigma_n$. Topologically, L(n) corresponds in positive degrees to the cohomology of the n-fold smash product of infinite real projective space. From the topological point of view, ${\cal A}$ is the algebra of universal stable operations in ordinary cohomology with ${\bb F}_2$ coefficients and this explains the action of ${\cal A}$ on P(n), L(n), B(n) and M(n). However, the whole subject may be treated in a purely algebraic fashion [5, 11].

The Fourier binest algebra is defined as the intersection of the Volterra nest algebra on L2(ℝ) with its conjugate by the Fourier transform. Despite the absence of nonzero finite rank operators this algebra is equal to the closure in the weak operator topology of the Hilbert–Schmidt bianalytic pseudo-differential operators. The (non-distributive) invariant subspace lattice is determined as an augmentation of the Volterra and analytic nests (the Fourier binest) by a continuum of nests associated with the unimodular functions exp(−isx2/2) for s>0. This multinest is the reflexive closure of the Fourier binest and, as a topological space with the weak operator topology, it is shown to be homeomorphic to the unit disc. Using this identification the unitary automorphism group of the algebra is determined as the semi-direct product ℝ2×κℝ for the action κt(λ, μ)=(etλ, e−t μ).

This paper is an account of the methods that have been used with the EDSAC for the solution of algebraic equations. Three repetitive or iterative methods are examined: Bernoulli's method, the root-squaring method, and the Newton-Raphson method. Experience with the EDSAC has shown that, as in hand computing, quadratically convergent methods are to be preferred to those less rapidly convergent. In particular, the Newton-Raphson method has proved the most useful. Several examples are given in the appendix.

In this note it is shown that any positive root of the transcendental equation

is definable as a continuous increasing function of the real variable ν, provided ν is positive. Here Jν and Yν denote respectively the Bessel functions of the first and second kind of order ν, and k is a positive constant. Watson ((4)) has established the corresponding result for the simpler equations Jν(z) = 0 and Jν(z) cosα − Yν(z) sinα = 0, where α is a constant. The extension of the result to the positive roots of equation (1) is important because this equation occurs quite frequently in physical problems.

We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.

It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

The construction of geometric surfaces via labelled complexes was introduced by Thurston[16, chapter 13], and subsequent applications and developments have appeared in [1, 3, 4, 5, 14, 15]. The basic idea of using labelled complexes to produce geometric structures is that the vertices of a simplicial triangulation of a surface can be labelled with positive real numbers that collectively determine a metric of constant curvature ±1 or 0, with possible singularities at vertices, by using the label values to identify 2-simplices of the triangulation with geometric triangles. Beardon and Stephenson[1] developed a particularly simple method for producing non-singular surfaces via labelled complexes that is modelled after the classical Perron method for producing harmonic functions, and they applied their method in [2] to construct a fairly comprehensive theory of circle packings in general Riemann surfaces. This Perron method was developed more fully by Stephenson and the author in [3, 4] and applied to the study of circle packing points in moduli space. At about the same time and independently of Beardon, Stephenson, and Bowers, Carter and Rodin [5] and Doyle [8] developed the method for flat surfaces and Minda and Rodin [14] developed the method for finite type surfaces. Minda and Rodin [14] applied their development to give partial solutions to the labelled complex version of the classical Schwarz-Picard problem that concerns the construction of singular hyperbolic metrics on surfaces with prescribed singularities. In this paper, we modify the aforementioned approaches and examine the upper Perron method for producing non-singular geometric surfaces. This upper method has several advantages over the Perron method as developed previously and provides a complete solution to the labelled complex version of the Schwarz-Picard problem.

All curves of the title are calculated up to an equivalence relation which is coarser than the relation of isogeny between the associated Jacobian varieties.

In (3) the second author has shown that all sub-Hopf-algebras of the mod 2 Steenrod algebra have a certain form; this resulthas been used in (1) and (2). The analogous result for the mod p Steenrod algebra, where p is an odd prime, is contained in (6). The object of this paper is to give a shorter proof of a slightly sharper result; we construct all the subalgebras in question. In the mod 2 case this sharper result has also been obtained by Anderson and Davis (2).

Throughout this paper, let M be a finitely generated module over a Noether local ring (A, [mfr ]) with dim M = d. Let x = (x1, …, xd) be a system of parameters of M and n = (n1, …, nd) ∈ ℕd a d-tuple of positive integers. This paper is concerned with the following two points of view.

First, it is well-known that, the difference between lengths and multiplicities

formula here

considered as a function in n, gives a lot of information on the structure of M. This function in general is not a polynomial in n for all n1, …, nd large enough (n [Gt ] 0 for short). But, it was shown in [C3] that the least degree of all polynomials in n bounding above IM (n, x) is independent of the choice of x. This numerical invariant is denoted by p(M) and called the polynomial type of the module M. By [C2] and [C3] this polynomial type does not change by the [mfr ]-adic completion Mˆ of M and p(M) is just equal to the dimension of the non-Cohen–Macaulay locus of Mˆ (see [C2, C3, C4, CM] for more details). Therefore a module M is Cohen–Macaulay or generalized Cohen–Macaulay if and only if p(M) = − ∞ or p(M) [les ] 0, respectively, where we set by − ∞ the degree of the zero-polynomial. However, one knows little about the structure of M when p(M) > 0.

Second, following Sharp and Hamieh ([SH]), we consider the difference

formula here

as a function in n, where

formula here

is the cyclic submodule of the module of generalized fractions U(M)−d−1d+1M defined in [SZ1].