A regular semigroup S is called V-regular if for any elements a, b ∈ S and any inverse (ab)′ of ab, there exists an inverse a′ of a and an inverse b′ of b such that (ab)′ = b′a′. A characterization of a V-regular semigroup is given in terms of its partial band of idempotents. The strongly V-regular semigroups form a subclass of the class of V-regular semigroups which may be characterized in terms of their biordered set of idempotents. It is shown that the class of strongly V-regular semigroups comprises the elementary rectangular bands of inverse semigroups (including the completely simple semigroups), a special class of orthodox semigroups (including the inverse semigroups), the strongly regular Baer semigroups (including the semigroups that are the multiplicative semigroup of a von Neumann regular ring), the full transformation semigroup on a set, and the semigroup of all partial transformations on a set.

We study the self-adjoint eigenvalue problem W(λ)x = 0, (*), in Hilbert space for one equation in two parameters. Here

is bounded below with compact resolvent for each λ = (λ1, λ2). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W(λ)= ≧0 and we investigate the geometry of the set Z0 of such λ. We also discuss higher order solution sets Zi where the ith eigenvalue of W(λ) vanishes, deriving various asymptotic results in a unified fashion.

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.

The proof of Lemma 6 (and thus of Theorem 9) has a gap in it. While m(B) → 0 as r → ∞ for each fixed h in Ni, it is not clear (and probably false) that this holds uniformly for h in . However Lemma 6 (and thus Theorem 9) holds with only trivial modifications of the given proof if one of the following holds: (i) ygi(y) → ∞ as |y| → ∞; (ii) m{x∈Ω: h(x) = 0} = 0 for every h in (iii) Niis one dimensional; (iv) there is a subset A of Ω such that h(x) = 0 if x ∈ A and h ∈ and m{x∈Ω\A: h(x) = 0} = 0 for every h ∈ . Assumption (ii) holds under very weak conditions. For example, the methods in [1] and regularity theorems imply that (ii) holds if there is a closed subset T of Ω of measure zero such that either (a) Ω\T is connected and aij (i, j = 1, …, n) are locally Lipschitz continuous on Ω\T or (b) for each component A of Ω\T, the aij have Lipschitz extensions to Ā and T is a “nice” set. (For example, it suffices to assume that T is a smooth submanifold of Ω though much weaker conditions would suffice.) Remember that we are assuming Ω is connected.

We study, in unbounded domains Ω⊂Rn, an elliptic semilinear problem with homogeneous boundary conditions. We assume that the nonlinear term f(x, u, Du) satisfies some condition of quadratic growth with respect to Du. We prove, in the framework of weighted Sobolev spaces, that, if and are respectively a subsolution and a supersolution of our problem, then there exists a least solution ū and a greatest solution û in the ordered interval and we obtain some multiplicity results.

The paper deals with explicit estimates concerning certain circles in the complex plane which were associated with Sturm–Liouville problems by H. Weyl. By the use of Riccati equations instead of linear integral equations, improvements are obtained for results of Everitt and Halvorsen concerning the behaviour of the Titchmarsh–Weyl m-coefficient.

Let y(n) + a1y(n−1 +…+ an−1y(1) + an = 0 (*) be a linear ordinary differential equation of order n. A (relative) differential invariant of (*) is a differential polynomial function π(xi) defined on the solution space of (*) satisfying: there is an integer g such that for all invertible linear transformations α of V into itself, π(αxi) = (det α)βπ(xi). We prove in a purely algebraic manner the following two theorems: A. The differential invariants of (*) are generated algebraically by the Wronskian W and the coefficients ala2, …, an of (*). B. Every generic differential relation (i.e. differential relation which holds for every linear ordinary differential equation of order n) among W, a1 …, an can be deduced algebraically from Abel's identity, W′ = −a1W. The second theorem may be considered as an algebraic version of the existence theorem for linear ordinary differential equations.

The H-spaces considered have no homology p-torsion and are rationally equivalent as H-spaces to products of even dimensional Eilenberg–Maclane spaces. We obtain conditions which ensure that if the cohomology with coefficients in the ring of integers localized at the prime p is a polynomial algebra, then the Pontrjagin ring with these same coefficients is polynomial. A topological consequence is that BSUP has just one homotopy associative, homotopy commutative H-structure.

We study the quasistatic behaviour of an elastoviscoplastic material submitted to a cyclic load. We prove the weak convergence of the solution of the stress evolution problem towards a periodical one when t → + ∞.

Several definiteness conditions in the multiparameter spectral literature are discussed. It is shown that some of these conditions permit simplifying transformations of the eigenvalues, leading to further definiteness properties. Geometrical equivalents for the algebraic conditions are established in terms of separation of convex cones. As a result, the relationship between the standard left and right definiteness conditions is clarified.

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.

A variational formulation of the Hele—Shaw flow model of the point injection of fluid into a laminar cell is introduced. The analysis concerning the existence, uniqueness and regularity of a solution to the variational problem is presented.

Let Mn be the semigroup, under composition, of endomorphisms of an n-dimensional vector space. Let E be the set of idempotents of Mn. It has been shown that each singular element of Mn() may be expressed as a composition of elements of E. In this paper the minimum number of idempotents needed in this composition is determined. This is given in terms of n and one parameter dependent on the element. Further, it is shown that En–1⊂<E>, while En=<E>.

Consider the differential expression

where p and w > 0 are real-valued and q is complex-valued on I. A number of criteria are established for certain extensions of the minimal operator generated by τ in the weighted Hilbert space to be maximal dissipative.

In this paper we determine the asymptotic behaviour of entropy numbers of embedding maps between Besov sequence spaces and Besov function spaces. The results extend those of M. Š. Birman, M. Z. Solomjak and H. Triebel originally formulated in the language of ε-entropy. It turns out that the characterization of embedding maps between Besov spaces by entropy numbers can be reduced to the characterization of certain diagonal operators by their entropy numbers.

Finally, the entropy numbers are applied to the study of eigenvalues of operators acting on a Banach space which admit a factorization through embedding maps between Besov spaces.

The statements of this paper are obtained by results recently proved elsewhere by the author.

Absolutely square integrable solutions are determined for the equation = λ y where the ζn−r(x) are holomorphic in a sector of the complex plane and have asymptotic expansions as x approaches infinity. It is shown that the number of such solutions depends upon the roots of the characteristic equation and their multiplicity, and upon the sign of the derivative of the characteristic polynomial. Application is made to formally symmetric ordinary differential operators.

Dirichlet, Neumann and mixed boundary value problems for the equation uxx −uyy = 0 are considered for a variety of rectangular domains. Uniqueness of solutions to these non-well-posed problems is considered by separation of variables methods. The question of uniqueness is also discussed for domains other than rectangles.