The spectral duality theory of H.-E. Porst and M. B. Wischnewsky is examined in more generality, and examples based on topological fields are described.

Green's theorem, for line integrals in the plane, is well known, but proofs of it are often complicated. Verblunsky [1] and Potts [2] have given elegant proofs, which depend on a lemma on the decomposition of the interior of a closed rectifiable Jordan curve into a finite collection of subregions of arbitrarily small diameter. The following proof, for the case of Riemann integration, avoids this requirement by making a construction closely analogous to Goursat's proof of Cauchy's theorem. The integrability of Qx—Py is assumed, where P(x, y) and Q(x, y) are the functions involved, but not the integrability of the individual partial derivatives Qx, and py this latter assumption being made by other authors. However, P and Q are assumed differentiable, at points interior to the curve.

Milne-Thomson has used the method of analytic continuation to solve boundary value problems of the annulus in plane elastostatics. However, his use of Cauchy integrals is incorrect, and it is shown in this note that the solution is obtained in terms of Laurent Series expansions. The solution is equivalent to that of Muskhelishvili, but is simpler to use in some applications.

A similar approach is used to solve the boundary value problem of the infinite strip, the solution being given in terms of functions of a complex variable expressed as Fourier integrals.

Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

It is well-known that for any set X, px, the semigroup of all partial transformations on X, can be embedded in Jx∪a for some a ∉ X (see for example Clifford and Preston (1967) and Ljapin (1963)). Recently Magill (1967) has considered a special case of what we call ‘generalised partial transformation semigroups’. We show here that any such semigroup can always be embedded in a full transformation smigroup in which the operation is not in general equal to the usual composition of mappinas. We then examine conditions under which such a semigroup, (J x, θ), is isomorphic to the semigroup, under composition, of all transformations on the same set X.

Let M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

Let S be a Riemann surface of type (p,n) with 3p+n>4 and n≥1. We investigate products of some pseudo-Anosov maps θ and Dehn twists tα on S, and prove that under certain conditions the products tkα∘θ are pseudo-Anosov for all integers k. We also give examples that show that tkα∘θ are not pseudo-Anosov for some integers k.

Gy. Soos [1] and B. Gupta [2] have discussed the properties of Riemannian spaces Vn (n > 2) in which the first covariant derivative of Weyl's projective curvature tensor is everywhere zero; such spaces they call Protective-Symmetric spaces. In this paper we wish to point out that all Riemannian spaces with this property are symmetric in the sense of Cartan [3]; that is the first covariant derivative of the Riemann curvature tensor of the space vanishes. Further sections are devoted to a discussion of projective-symmetric af fine spaces An with symmetric af fine connexion. Throughout, the geometrical quantities discussed will be as defined by Eisenhart [4] and [5].

In a paper [1] of the same title Barnes considered the problem of finding an upper bound for the infimum m+(f) of the non-negative values1 of an indefinite quadratic form f in n variables, of given determinant det(f) ≠ 0 and of signature s. In particular it was announced (and later proved — see [2]) that m+(f) ≦ (16/5)+ for ternary quadratic forms of determinant 1 and signature — 1. A simple consequence of this result is that m+(f) ≦ (256/135)+ for quaternary quadratic forms of determinant — 1 and signature — 2.

For a real vector space V acted on by a group K and fixed x and y in V, we consider the problem of finding the minimum (respectively, maximum) distance, relative to a K-invariant convex function on V, between x and elements of the convex hull of the K-orbit of y. We solve this problem in the case where V is a Euclidean space and K is a finite reflection group acting on V. Then we use this result to obtain an analogous result in the case where K is a maximal compact subgroup of a reductive group G with adjoint action on the vector component ρ of a Cartan decomposition of Lie G. Our results generalize results of Li and Tsing and of Cheng concerning distances to the convex hulls of matrix orbits.

It is the purpose of this note to illustrate by example the results obtained in Finch [1]. All our definitions and notation are as in that paper, and we use them without reference. In his paper, Finch considered orthoposets and, in particular, orthomodular posets that had either the A-property or the B-property.

Let u(x, t) be a smooth function in the domain Q = Ω × (0, L), Ω in n, let Du be the spatial gradient of u(x, t) and let ∇u = (Du, u1). If u(x, t) satisfies the parabolic equation F(u, Du, D2u) = ut, we define w(x, t) by g(w) = │∇u│−1G(∇u) (g is positive and decreasing, G is concave and homogeneous of degree one) and we prove that w(x, t) attains its maximum value on the parabolic boundary of Q. If u(x, t) satisfies the equation Δu + 2h(q2) uiujuij = ut(q2 = │Du│2, 1 + 2q2h(q2) > 0) we prove that qf (u) takes its maximum value on the parabolic boundary of Q provided f satisfies a suitable condition. If u(x, t) satisfies the parabolic equation aij (Du)uij − b(x, t, u, Du) = ut (b is concave with respect to (x, t, u)) we define C(x, y, t, τ) = u(z, θ) − αu(x, t) − βu(y, τ) (0 < α, 0 < β, α + β = 1, z αx +y, θ = αt + βτ) and we prove that if C(x, y, t, r) ≤0 when x, y, z ∈ Ω2 and one of t, τ = 0, and when t, τ ∈ (0, L], and one of x, y, z, ∈ ∂Ω, then it is C(x, y, t, τ) ≤0 everywhere.

A notion of a normal autometrized algebra is introduced which generalises the concepts of Boolean geometry. Brouwerian geometry, autometrized lattice ordered groups, semi-Brouwerian geometry, etc. The notions of ideals and congruence relations are introduced in normal autometrized algebras and a one to one correspondence between ideals and congruence relations is established. Some other common properties of the above geometries are also obtained for normal autometrized algebras.

Let S be a Riemann surface of finite type. Let ω be a pseudo-Anosov map of S that is obtained from Dehn twists along two families {A,B} of simple closed geodesics that fill S. Then ω can be realized as an extremal Teichmüller mapping on a surface of the same type (also denoted by S). Let ϕ be the corresponding holomorphic quadratic differential on S. We show that under certain conditions all possible nonpuncture zeros of ϕ stay away from all closures of once punctured disk components of S∖{A,B}, and the closure of each disk component of S∖{A,B} contains at most one zero of ϕ. As a consequence, we show that the number of distinct zeros and poles of ϕ is less than or equal to the number of components of S∖{A,B}.

A left ideal on any C*-algebra is an example of an operator algebra with a right contractive approximate indentiy (r.c.a.i.). Indeed, left ideal in C*-algebras may be charcterized as the class of such operator algebras, which happen also to be triple systems. Conversely, we show here and in a sequel to this paper, that operator algebras with r.c.a.i. shoulod be studied in terms of a certain let ideal of a C*-algebra. We study left ideals from the perspective of ‘Hamana theory’ and using the multiplier algebras of an operator space studied elsewhere by the author. More generally, we develop some general theory for operator algebras which have a 1-sided identity or approzimate indentity, including a Banach-Stone theorem for these algebras, and an analysis of the ‘multiplier operator algebra’.

A crack is assumed to be the union of two smooth plane surfaces of which various parts may be in contact, while the remainder will not. Such a crack in an isotropic elastic solid is an obstacle to the propagation of plane pulses of the scalar and vector velocity potential so that both reflected and diffracted fields will be set up. In spite of the non-linearity which is present because the state of the crack, and hence the conditions to be applied at the surfaces, is a function of the dependent variables, it is possible to separate incident step-function pulses into either those of a tensile or a compressive nature and the associated scattered field may then be calculated. One new feature which arises is that following the arrival of a tensile field which tends to open up the crack there is necessarily a scattered field which causes the crack to close itself with the velocity of free surface waves.

The automorphisms of the canonical core UHF subalgebra ℱn of the Cuntz algebra 𝒪n do not necessarily extend to automorphisms of 𝒪n. Simple examples are discussed within the family of infinite tensor products of (inner) automorphisms of the matrix algebras Mn. In that case, necessary and sufficient conditions for the extension property are presented. Also addressed is the problem of extending to 𝒪n the automorphisms of the diagonal 𝒟n, which is a regular maximal abelian subalgebra with Cantor spectrum. In particular, it is shown that there exist product-type automorphisms of 𝒟n that do not extend to (possibly proper) endomorphisms of 𝒪n.

Here we prove that, for a $2$-primal ring $R$, the Laurent series ring $R((x))$ is a clean ring if and only if $R$ is a semiregular ring with $J(R)$ nil. This disproves the claim in K. I. Sonin [‘Semiprime and semiperfect rings of Laurent series’, Math. Notes 60 (1996), 222–226] that the Laurent series ring over a clean ring is again clean. As an application of the result, it is shown that, for a $2$-primal ring $R$, $R((x))$ is semiperfect if and only if $R((x))$ is semiregular if and only if $R$ is semiperfect with $J(R)$ nil.

We consider a regression relation of the from wherein y(t) and x(t) are real (column) vectors of q and p components and e(t) is real and is generated by a stationary generalised vector process of q components with zero mean and covariance function (a q rowed matrix) Γ(t–s) = E{x(s)x(t)′}. (See Hannan (1970; pages 23–26, 91–94) and references therein for definitions of terms used.) We assume e(t) to be independent of x(s) for all s, t. Thus we may regard x(t) as a fixed time function and not stochastic and we shall henceforth do that. We take Γ(t) to be continuous and to correspond to an absolutely continuous spectral function with spectral density which is uniformly bounded and continuous. Then we have We do not exclude the possibility that for theyth diagonal element, fjj, of fwe have

Let G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of finite variation on G. For a function f ∈ C(G, X) and μ ∈ Mc(G, X′), we define the notion of convolution f * μ. A function f ∈ C(G, X) is called mean-periodic if there exists a non-trivial measure μ ∈ Mc(G, X′) such that f * μ = 0. For μ ∈ Mc(G, X′), let M P(μ) = {f ∈ C(G, X): f * μ = 0} and let M P(G, X) = ∪μ M P(μ). In this paper we analyse the following questions: Is M P(G, X) ≠ 0? Is M P(G, X) ≠ C(G, X)? Is M P(G, X) dense in C(G, X)? Is M P(μ) generated by ‘exponential monomials’ in it? We answer these questions for the groups G = ℝ, the real line, and G = T, the circle group. Problems of spectral analysis and spectral synthesis for C(ℝ, X) and C(T, X) are also analysed.