Certain representations of the alternating group of degree 6 over the field of 2 elements are shown to be completely reducible.

The behaviour of heat transfer and skin friction is analysed in a compressible laminar boundary layer with external velocity Ue(x)(l + α sin ωt The Mach number M is assumed small but finite so that high frequency flows (s ≫1) in which c =αγM2s/2 = O(1) are considered. Solutions, obtained by matching in the Stokes and Prandtl layers, involve summation of Fourier-like series to give the dominant terms in the heat transfer and skin friction. Results, for c =½, verify that a previous approximate method gives a reasonable description of unsteady heat transfer and skin friction; forc =1 there is a substantial increase in amplitude of heat transfer but little change of phase.

For n ≧ 3 the independence of the associative laws

for n-ary operations (…) is proved.

We discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the cones

We obtain results under various conditions on the Tr, but the following is typical:

THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

In this paper, appropriate definitions are given for the spectra of multiparameter problems in Hilbert space when the operators involved are not necessarily self-adjoint. Notions of adjoint multiparameter problems are introduced, and some properties of the spectrum of a single linear operator are generalised to multiparameter settings.

The results which are achieved are compared with those in the literature which have beendeveloped for multiparameter problems involving self-adjoint operators.

The invariants of pencils of binary cubics were described in an earlier paper by the author. In this note, we give a complete set of covariants for such pencils, and show how each type of pencil may be denned by the identical vanishing of covariants.

In the theory of resolutions of polyhedral homology manifolds to PL manifolds, itis natural to ask whether, when such a resolution exists, it is possible to preserve a given subspace of the original homology manifold under the resolution. The answer is provided in the affirmative so long as the codimension of the subspace is at least 3. In addition, it is shown that if the original dimension was at least 5 then a resolution may be chosen so as to induce an isomorphism of fundamental groups. As a corollary to these results we see that any homology n-manifold homology n-sphere may be PL embedded in S+3.

The periodic quasigeostrophic equations are a coupled system of a second order elliptic equation for a streamfunction and first order hyperbolic equations for the relative potential vorticity and surface potential temperatures, on a three-dimensional domain which is periodic in both horizontal spatial co-ordinates. Such equations are used in both numerical and theoretical studies in meteorology and oceanography. In this paper Schauder estimates and a Schauder fixed point theorem are used to prove the existence and uniqueness of strong, that is classical, solutions of the periodic quasigeostrophic equations for a finite interval of time, which is inversely proportional to the sum of the norms of the initial vorticity and surface temperatures.

Ultradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

An (n, q) graph is a graph on n labelled points and q lines without loops or multiple lines. We write ν(n, q) for the number of smooth (n, q) graphs, i.e. connected graphs without end points, and ν = V(Z, Y) = ∑n,q ν(n,q)ZnYq /n! for the exponential generating function of ν(n,q). We use the Riddell “core and mantle” method to find an explicit form for V (not, as usual with this method, only a functional equation). From this we deduce a partial differential equation satisfied by V. We interpret this equation in purely combinatorial terms. We write Vk = ∑ n ν(n, n + k)Xn/n! and find a recurrence formula for Vk for successive k. We use these and other results to find an asymptotic expansion for ν(n,q) as n→∞ when (q/n) − log n − log log n→ + ∞ and an asymptotic approximation to ν(n,n + k) when 0 < k = o and to log ν(n, n + k) when k < (1−ε).

Some attributes of a new method of defining the convolution and multiplication of generalised functions, too singular to be encompassed by earlier theories, are developed and its extension to the case of several variables is described.

Poincaré's theorem on the sum of the indices of a plane autonomous differential equation at its critical points inside a periodic orbit is here extended to the periodic orbits of higher-dimensional equations under certain conditions. Under the same conditions, higher-dimensional extensions are obtained of some theorems of Bendixson and others which exclude periodic orbits from regions in which the differential equation haspositive divergence. The application of these results to feedback control equations is also considered.

Liouville theorems are obtained for fourth order elliptic systems of the form Δ2U + B Δ U + AU = 0 and for fourth order nonlinear equations of the form Δ2u−q(x) Δu + p(x)f(u) = 0 as a consequence of two related subharmonic functions, the mean value property of subharmonic functions, and a basic Green identity.

A general theory for a class of abundant semigroups is developed. For a semigroup S in this class let E be its set of idempotents and <E> the subsemigroup of S generated by E. When <E> is regular there is a homomorphism with a number of desirable properties from S onto a full subsemigroup of the Hall semigroup T<E>. From this fact, analogues of results in the regular case are obtained for *-simple and ℐ*-simple abundant semigroups.

The investigation of general quasi-adequate semigroups is initiated. These are semigroups which are abundant and in which the idempotents form a subsemigroup. For such a semigroup S we study the minimum good congruence γ such that S/γ is adequate. Results on γ together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi-adequate semigroups.

Let G: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ and T(t) be the Co semigroup generated by G. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {x ε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions for G are satisfied. Proofs are given in terms of the Cayley transformation T = (G + I)(G − I)−1 of G. The results are applied to the damped wave equation utt + γutx + uxxxx + ßuxx = 0, 0 ≦ t < ∞ 0 < x < 1, β, γ ≧ 0, with boundary conditions u(0, t) = ux(0, t) = uxx (1, t) = uxxx(1, t) = 0.

In this paper we study the existence and uniqueness of the two point boundary value problems −(p(x)u′(x))′ = f(x, u(x), u′(x)), xε(0, 1), u′(0)−cu(0) = 0 = u′(1) + du(1), where ∂f/∂u is bounded. above by the least eigenvalue of associated linear problems and ∂f/∂u is bounded. By using monotone techniques to investigate the equivalent problem −(p(x)u′(x))′ + r(x)u(x) = f(x, u(x), u′(x)) + r(x)u(x) where r ε C[0, 1] we show that

gives the optimal bounds for ∂f/∂u and ∂f/u′ when c and d are nonnegative constants.

We consider the continuation of positive solutions of -u" = λs(x)f(u), with appropriate boundary conditions and with positive s and f. We show by an example that bifurcation may occur from the curve of these positive solutions.

An explicit solution is derived for the dual integral equations

when A(ξ) takes the form

It is then shown how the basic technique can be adapted to derive explicit solutions for more general forms of A(ξ) such as

and

and linear combinations of such sums.

We consider ordered regular semigroups in which the order extends the natural order on the idempotents, and which are graced with the presence of a greatest idempotent. This implies that every element has a greatest inverse. An investigation into the properties ofthese special elements allows a description of Green's relations on the subsemigroup generated by the idempotents. This in turn leads to a complete description of the structure of idempotent-generated naturally ordered regular semigroups having a greatest idempotent. The smallest such semigroup that is not orthodox is also described. These results lead us to obtain structure theorems in the general case with the added condition that Green's relations be regular. Finally, necessary and sufficient conditions for such a semigroup to be a Dubreil-Jacotin semigroup are found.