It was noted by Klein(4), part I, ch. II, §9, that if a polyhedral group G acts on the 2-sphere identified with P1(ℂ), then the polynomials cΠg∈G(λ − ag μ) form, as a∈ℂ varies, a vector space of dimension 2 over ℂ. G is generated by elements T1, T2 fixing respectively a vertex and a face; T1T2 fixes an edge and these elements have orders p, q, 2 where 1/p + 1/q > ½ Taking a to be a vertex or the mid-point of a face or an edge yields polynomials of the form and which are thus linearly dependent. The ring of all (relatively) invariant polynomials is generated by F1F2 and F3 subject to this single relation.

Let H be a finite dimensional real or complex Hilbert space. We denote by Λ(x, y, z) the area of the triangle with vertices x, y, z ∈ H. A map f: H → H is triangle contractive TC if 0 < α < 1 and for each x, y, z ∈ H either

or

and

and

We prove that if f is TC either there is a fixed point w = f(w) or a fixed line L = ⊃ f(L) We characterize the f which are TC and continuous but have no fixed point.

We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four (Theorem 3.1), and (2) between conformally-flat Riemannian manifolds of dimensions at least three (Corollaries 3.4 and 3.6).

Also, we prove (Proposition 2.5) an integrability result for any real-analytic submersion, from a constant curvature Riemannian manifold of dimension n+2 to a Riemannian manifold of dimension 2, which can be factorised as an n-harmonic morphism with two-dimensional fibres, to a conformally-flat Riemannian manifold, followed by a horizontally conformal submersion, (n≥4).

We present an approach to compute the Hausdorff dimension of a class of sets of real numbers that are defined in terms of nonlinear relations between frequencies of digits in some integer base m. We consider the model case of quadratic perturbations, for which the computations are already rather involved. We show that the Hausdorff dimension is analytic in the parameter determining the perturbation. Our approach also allows to estimate the asymptotic behavior of the Taylor coefficients of the dimension in terms of m.

In this paper, the problem of diffraction of an impulsive P wave by a fluid circular cylinder has been considered. The cylinder is embedded in an unbounded isotropic homogeneous elastic medium and it is filled with inviscid fluid material. The line source, giving rise to the incident front, is situated outside the cylinder parallel to its axis.

The exact solution of the problem is obtained by using the method of dual integral transformations. The solution is evaluated approximately to obtain the motion on the wave front in the shadow zone of the elastic medium. Further, we interpret the approxi mate solutions in terms of Keller's geometrical theory of diffraction. Our result also gives a correction to an earlier investigation of the similar problem by Knopoff and Gilbert(s).

This paper will study certain rational BP operations. We will work in the category of spaces having the homotopy type of CW complexes. Recall that for − 1 ≤ n ≤ ∞ there is defined in this category the homology theory BP(n)*(X). (See (7).) In particular, BP(∞)*(X) = BP*(X) is the usual Brown Peterson homology of X, BP(1)*(X) = l*(X) is the canonical direct summand of connective K-theory with Qp coefficients (Qp are the integers localized at the prime p), BP(0)*(X) = Hℤ*(X) is ordinary homology with Qp coefficients, and BP(−1)*(X) = Hℤ/p*(X) is ordinary homology with ℤp coefficients (ℤp are the integers reduced mod p). For 1 ≤ n ≤ ∞, BP(n)*(X) is a module over BP(n)*(pt) = Qp[v1, …, vn] (deg vs = 2ps − 2). The BP(n) theories are all related via canonical maps p(m, n):BP(m)*(X) → BP(n)*(X) whose performance is described in (7).

In the structure theory of Banach spaces as developed in (1), an important role is played by subspaces which are the ranges of projections having norm properties akin to those of the classical Banach spaces. A linear projection e on a Banach space V is called an M-projection if

and an L-projection if, instead

A closed subspace J of V is called an M-Summand if it is the range of an M-projection and an M-Ideal if J0 is the range of an L-projection in V′. Every M-Summand is an M-Ideal but the reverse is false.

This paper is concerned with what, for lack of a better name, the author proposes to call geometric modules over a field k. These modules are defined as follows. A geometric module M consists of a vector space, also denoted by M, over k, together with a ring of linear transformations A of M into itself subject to the following conditions:

(i) A is a homomorphic image of the ring Sn = k[X1, …, Xn] of polynomials in n indeterminates X1, …, Xn for some value of n;

(ii) M, considered as an A-module, is finitely generated.

It is shown that, at an arbitrary point in a general elastic material and regardless of the prevailing state of deformation, there exists in every plane at least one direction in which a transverse acceleration wave may propagate. The possibility of a transverse wave actually being able to progress in such a specific direction depends upon the associated local wave speed being real and conditions are obtained under which this requirement is satisfied.

The Pisier algebra [Pscr ] consists of those continuous functions f on the unit circle [ ] for which

formula here

is continuous on [ ] almost surely, that is, for almost every choice of a sequence of signs (±1, ±1, …). In this paper, we prove that spectral synthesis holds in [Pscr ]. Moreover, we show that certain closed ideals in [Pscr ] with infinite hull have bounded approximate identities and give an example of a closed ideal in [Pscr ] which does not have a bounded approximate identity.

The problem of scalar Dirichlet diffraction of a plane wave by an elliptic disc is discussed. A scheme is given whereby the low frequency expansion of the scattered field may be readily obtained. Series expansions are obtained for the far-field amplitude up to and including the second order in the wave number. The first two terms of the scattering cross-section are also derived.

In this paper, we give an alternative approach to Royden–Earle–Kra–Markovic's characterization of biholomorphic automorphisms of Teichmüller space of Riemann surface of analytically finite type.

1. Let ℳ be an additive category. (We refer to ((1), Ch. IX) for the definition of an additive category and associated terms. In particular a sequence

is exact if i = kerp and p = coker i.) We write End M for Hom(M, M). A trace on ℳ with values in an abelian group G is a collection of (abelian group) homomorphisms

one for each M ∈ ℳ, satisfying the following two conditions:

(i) Exactness. Given a commutative diagram with exact rows,

then tA(f) + tC(h) = tB(g).

We exploit a recent approach to Brascamp–Lieb inequalities, due to Caffarelli [5], and reconsider earlier approaches to establish stochastic domination inequalities between Gaussian variables and random variables with density of the form $g\cdot h, g$ a Gaussian density and $h$ a log-concave or log-convex function. These extend to inequalities on random vectors via a classical result by Prékopa and Leindler and they complement the Brascamp–Lieb moment inequalities. Some applications to a class of Gibbs measures, the anharmonic crystals, are developed.

The requirement that the graph of a function be invariant under a linear map is equivalent to a functional equation of f. For area preserving maps M(det (M) = 1), the functional equation is equivalent to an (easily solved) linear one, or to a quadratic one of the form

for all Here 2C = Trace (M). It is shown that (Q) admits continuous solutions ⇔ M has real eigenvalues ⇔ (Q) has linear solutions f(x) = λx ⇔ |C| ≥ 1. For |c| = 1 or C < – 1, (Q) only admits a few simple solutions. For C > 1, (Q) admits a rich supply of continuous solutions. These are parametrised by an arbitrary function, and described in the sense that a construction is given for the graphs of the functions which solve (Q).

Let ϵ > 0 and be a family of finite subsets of the Cantor set . Following D.H. Fremlin, we say that is ϵ-filling over if is hereditary and for every F ⊆ finite there exists G ⊆ F such that G ϵ and . We show that if is ϵ-filling over and C-measurable in , then for every P ⊆ perfect there exists Q ⊆ P perfect with . A similar result for weaker versions of density is also obtained.

One can study the local topology of images of complex analytic maps and the restriction of such a map to a real subspace. Conditions are given for the topology of the image multiple point spaces of the complex and real maps to have the same homology or homotopy type.

In a recent paper I discussed plane congruences of order two in [4] and obtained congruences of types (2, 6)1, (2, 6)2, (2, 5), (2, 4) and (2, 3). The method employed was due to Segre, who showed that a plane congruence of order two in [4] has in general a curve locus of singular points which is met by each plane in five points. Then, if we can find a curve in [4], composite or not, with an ∞2 system of quadrisecant planes of which two pass through an arbitrary point, the planes must all meet a residual curve, and we shall have obtained a congruence of the second order and a fifth incidence theorem.

Our principal result deals with a function φ defined on the ‘non-negative half’ X+ of a discrete ordered group X, the dual of a compact Abelian connected group G. Suppose φ vanishes outside a countable set S = {χ1, χ2 …} and has the following property: for every t in a set 0 ⊂ (0, 1) of positive Lebesgue measure, the function on X+ defined to be 0 off S and to be rn(t)φ(χn) for χ = χn, where rn is the nth Rademacher function, is the restriction to X+ of a Fourier–Stieltjes transform. We conclude that φ ∈ ℓ2(X+).

The aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ, gΣ) embedded into a complete five dimensional manifold (M5, g) with positive scalar curvature R and nonnegative Ricci curvature. Under a suitable choice, we have $area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ, gΣ) is isometric to a standard sphere ($\mathbb{S}$4, gcan) and in a neighbourhood of Σ, (M5, g) splits as ((-ϵ, ϵ) × $\mathbb{S}$4, dt2 + gcan) and the Riemannian covering of (M5, g) is isometric to $\Bbb{R}$ × $\mathbb{S}$4.