In this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.

Certain classical differential expressions which are singular at a finite end-point (or at an interior point) can be represented as regular, scalar quasi-differential expressions, the best-known examples being the Boyd Equation and Laplace Tidal Wave Equation. We show here that in all such cases the domains of the minimal and maximal operators in the appropriate weighted Hilbert space , for the regularised expression, coincide with the corresponding domains for the expression in its original, singular form.

This is contrasted with a known property of the corresponding expression domains. Whereas for an expression M, the operator domains contain only functions y for which both y and My lie in the appropriate Hilbert space, the expression domain comprises a much larger set of functions with no such restrictions beyond those necessary for My to exist as a function. In the second-order case, the expression domain of the regularisation of a singular expression is known to be a strict subset of the original expression domain, contrasting with the results proved here for the operator domains.

In this paper we consider the problem of accelerating an obstacle in an incompressible viscous fluid from rest to a given speed in a given time with minimum energy expenditure. An existence theorem for the speed trajectory which corresponds to the absolute minimum is provided. The results are valid for arbitrary Reynolds numbers.

An initial boundary value problem of Riemann type is solved for the nonlinear pseudoparabolic equation with two space variables

The complex function H is measurable on ℂ ×I × ℂ5, with I being an interval of the real line ℝ, Lipschitz continuous with respect to the last five variables, with the Lipschitz constant for the last variable being strictly less than one (ellipticity condition). No smallness assumption is needed in the argument.

We study the asymptotic behaviour of Dirichlet problems in domains of R2 bounded by thin layers whose thickness is given by means of an assigned ergodic random function. Using a capacitary method together with ergodic theorems for additive and superadditive processes, we are able to characterise the limit problem precisely.

We study Berman's problem

subjects to the conditions f(0) =f″(0) = f′(1) = f(1)− 1 = 0, which arises from the study of laminar flows in channels with porous walls. Positive (negative) Re denotes the case of suction (injection) flows. Preliminary numerical studies indicated the existence of three different types of solutions and a small portion of mathematical evidence was provided. In this paper, we are able to show that there exist connected sets in the Re-K plane on which different types of solutions occur. In particular, our result verifies that the problem possesses all three types of suction solutions for sufficiently large Re. Moreover, the limiting injection solution is also obtained.

The discrete coagulation-fragmentation equations are a model for the time-evolution of cluster growth. The processes described by the model are the coagulation of clusters via binary interactions and the fragmentation of clusters. The assumptions made on the fragmentation coefficients in this paper have the physical interpretation that surface effects are not important, i.e. it is unlikely that a large cluster will fragment into two large pieces. Since solutions of the initial-value problem are not unique, we have to restrict the class of solutions. With this restriction, we prove that the fragmentation acts as a strong damping mechanism and we obtain results on the asymptotic behaviour of solutions. The main tool used is an estimate on the moments of admissible solutions.

In an operator algebra, the general element of the connected component of the unitary group can beexpressed as a finite product of exponential unitary elements. The recently introduced concept of exponential rank is defined in terms of the number of exponentials required for this purpose. The present paper is concerned with a concept of exponential length, determined not by the number of exponentials but by the sum of the norms of their self-adjoint logarithms. Knowledge of the exponential length of an algebra provides an upper bound for its exponential rank (but not conversely). This is used to estimate the exponential rank of certain algebras of operator-valued continuous functions.

Using some new expressions for the weakly convergent sequences coefficient WCS(X) the lower boundedness

is proved, where δ(-) is the (Clarkson) modulus of convexity. We also define a modulus of noncompact convexity concerning nearly uniformly convex spaces which is used to obtain another lower bound for WCS(X). The computation of this modulus in Ip-spaces shows that our second lower bound is the best possible in these spaces.

In this paper perturbation theory is used to construct systems in four dimensions having two dimensional stable and unstable manifolds which touch along a homoclinic orbit but only with a second order contact.

The probability that a neutron leads to a divergent chain reaction in a nuclear reactor is governed by a nonlinear integro-partial-differential equation [1]. A model case of this equation was completely analysed by Pazy and Rabinowitz [2,3]. The purpose of this paper is to extend their results to the general case and to tackle some related topics.

Let {Er, dr} be a spectral sequence converging to a Hopf algebra H*. We give a method of reconstructing H* from E∞**. By using our method, we determine the mod 2 cohomology of the space of loops on a simply-connected space whose mod 2 cohomology is isomorphic to that of Spin(N) as an algebra over the Steenrod algebra.

We are looking for positive sequences (bn) which have the following property: If (an) is a positive sequence such that , then

We study the equations of viscoelasticity in a multidimensional setting for the ‘no-traction’ boundary data. For the sake of modelling phase transitions we do not assume elliptieity of the stored energy function W. We construct dynamics in W1,2(Ωℝn) globally in time. Next, we study the question of stability for a class of equilibria. Moreover, we show a certain kind of decay in time of solutions for arbitrary initial conditions.

An existence theorem is obtained for a fourth-order semilinear elliptic problem in RN involving the critical Sobolev exponent (N + 4)/(N − 4), N>4. A preliminary result is that the best constant in the Sobolev embedding L2N/(N–4) (RN) is attained by all translations and dilations of (1 + ∣x∣2)(4-N)/2. The best constant is found to be

This paper is the first part in a four-part series which develops the spectral theory for a two-point differential operator L in L2[0, 1] determined by a second order formal differential operator l = −D2 + pD + q and by independent boundary values B1, B2. The differential operator L is classified as belonging to one of five cases, Cases 1–5, according to conditions satisfied by the coefficients of B1, B2. For Cases 1–4 it is shown that if λ = ρ2 is any eigenvalue of L with ∣ρ∣ sufficiently large, then ρ lies in the interior of a horizontal strip (Cases 1–3) or the interior of a logarithmic strip (Case 4), and in each of these cases the generalised eigenfunctions of L are complete in L2[0, 1].

A radiation condition is obtained, and is then used together with weighted Sobolev spaces and the limiting absorption method to establish the unique existence of solutions to the diffraction problem for the wave propagation in the case where the propagation speed is piecewise constant, and the surface separating two media is unbounded.

Uniform asymptotic expansions are derived for solutions of the spheroidal wave equation, in the oblate case where the parameter µ is real and nonnegative, the separation parameter λ is real and positive, and γ is purely imaginary (γ = iu). As u →∞, three types of expansions are derived for oblate spheroidal functions, which involve elementary, Airy and Bessel functions. Let δ be an arbitrary small positive constant. The expansions are uniformly valid for λ/u2 fixed and lying in the interval (0,2), and for λ / u2when 0<λ/u2 < 1, and when 1 = 1≦λ/u2 < 2. The union of the domains of validity of the various expansions cover the half- plane arg (z)≦ = π/2.

A characterisation of the surface energy density for the relaxation in V(Ω; Rp) of the functional

is obtained. A lemma of De Giorgi is used to modify a sequence near the boundary without increasing its total energy.

In this paper we study a class of 2-generator 2-relator groups G(m) and show that they are all finite. Moreover, two infinite subclasses are soluble of derived length 4.